Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 461907 13 pageshttpdxdoiorg1011552013461907

Research ArticleDesigning of Adaptive Model-Free Controller Based onOutput Error and Feedback Linearization

S Pezeshki1 M A Badamchizadeh1 S Ghaemi1 and M A Poor2

1 Control Engineering Department Faculty of Electrical and Computer Engineering University of Tabriz Tabriz 5166614776 Iran2Department of Mechanical and Industrial Engineering Concordia University Montreal QC Canada

Correspondence should be addressed to M A Badamchizadeh mbadamchitabrizuacir

Received 20 September 2012 Accepted 30 December 2012

Academic Editor Shukai Duan

Copyright copy 2013 S Pezeshki et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper introduces a new approach to design Model-Free Adaptive Controller (MFAC) using adaptive fuzzy procedure as afeedback linearization based on output error The basic idea is to transfer the control signal to an appropriate surface and thendepending on the output error of system the control signal changes around this surface Some examples are provided as well toillustrate the efficiency of the proposed approach The obtained simulation results have shown good performances of the proposedcontroller

1 Introduction

Classical control method is based onmathematical equationsof the system however this method suffers from some draw-backs For example in thismethod the performance of systemcan be affected by unmodeled dynamics of system andor bylarge delays Model-Free Adaptive Control (MFAC) as a partof modern control theory shows superiorities compared tomodel-based methods MFAC is an adaptive control methodand needs no information about system model It only uses119868119874 data for controller design and therefore it is a nonlinearcontroller designed without the need to mathematical modelof controlled system [1ndash3]

In 1994 Han and Wang introduced model-free topic andproved the stability of MFAC controllers In 1993-1994 HouZhongsheng represented applications of these controllersWith using pseudo-Jacobian matrix nonlinear systems arereplaced to near the rail line of controlled system and thenthe 119868119874 data of controlled system are used to design MFACcontroller [4]

In 1995 Jagannathan suggested a fuzzy stable controllerfor a limited class of nonlinear system in the form of 119909(119896 +1) = 119891(119909(119896)) + 119906(119896) where 119891(119909) is the unknown nonlinearfunction In 1996 this controller was used for general classof nonlinear system in the form of 119909(119896 + 1) = 119891(119909(119896)) +

119892(119909(119896))119906(119896) [5]

Another idea is introduced in 2000 [6] which uses aneural network as an adaptive controller for stabilization ofsystemTheir proposed algorithmdoes not need any complextuning and can be applied to any controllable multiinputsystems

Compared with other adaptive control schemes theMFAC approach has several advantages which make thismethod suitable for control applications First MFAC justdepends on the real-timemeasurement data of the controlledplant Second MFAC does not require any other testingsignals and any training process Third MFAC is simple andeasily implementable with small computational burden andhas strong robustness FourthMFACapproach does not needa specific controller for each specific process Finally theMFAC has been successfully implemented in many practicalapplications for example chemical industry linear motorcontrol injectionmodeling process PH value control and soon [4]

The main contribution of this work is to introduce anew MFAC approach based on output error and feedbacklinearization The proposed model has rapid monotonictracking error convergence robust stability and good distur-bance rejection

The rest of the paper is organized as follows in Section 2adaptive fuzzy control based on feedback linearization is

2 Abstract and Applied Analysis

described The proposed model-free adaptive controller andits Lyapunov stability are represented in Sections 3 and4 respectively in Section 5 simulation results are given tohighlight advantages of MFAC method Finally conclusion isstated in Section 6

2 Feedback Linearization by Means ofAdaptive Fuzzy Controller

In feedback linearization method control law is determinedin such a way that nonlinear terms eliminate the systemdynamic and replace it with appropriate reference input asseen below (for simplifying 119909(119905) = 119909)

(119905) = 119891 (119909) + 119892 (119909) 119906 (119905) 997888rarr 119906 (119905)

= 119892minus1

(119909) (minus119891 (119909) + 119907 (119905))

(1)

Conceptually Adaptive Feedback Linearization (AFL)method is the same as nonadaptive counterpart but AFLuses adaptation law for updating controller parameter toenhance system performance Notice that both methodsshould have complete description of system dynamic butour proposed method does not need any information ofsystem dynamic Due to the fact that it is difficult to obtaina mathematical model of the system and typically there existsome numerical approximations the object of this paper isto control systems just with 119868119874 data in order to improvesystem performance In order to reach this goal it is requiredto estimate unknown nonlinear function of system dynamic119891(119909) and 119892(119909) with 119891(119909) and 119892(119909) or estimate controllaw 119906(119905) with (119909) Estimation algorithm in this approachutilizes basic fuzzy membership function As an instance forestimating 119891(119909) we have

119891 (119909) 997888rarr 119891 (119909) =sum119901

119894=0

119888119894

120583119894

sum119901

119894=0

120583119894

(2)

where 119901 is the number of fuzzy roles 120583119894

(119909) is membershipfunction for 119894th fuzzy role also 119888

119894

is and the result of 119894thfuzzy role for system that is generally considered as linearcombination of some series of continuous functions such as119911(119909)

119911119896

(119909) isin 119877 119896 = 1 2 119898 minus 1

119888119894

= 1198861198940

+ 119886119894119894

1199111

(119909) + sdot sdot sdot + 119886119894119898minus1

119911119898minus1

(119909)

(3)

119911 = [1 1199111

(119909) 1199112

(119909) sdot sdot sdot 119911119898minus1

(119909)]119879

119860119879

=[[

[

11988610

11988611

sdot sdot sdot 1198861119898minus1

1198861199010

1198861199011

sdot sdot sdot 119886119901119898minus1

]]

]

119888 = 119911119879

119860

(4)

Using (3) (4) 119891(119909) can be rewritten as

120585 =

[1205831

1205832

sdot sdot sdot 120583119901

]

sum119901

119894=0

120583119894

119891 (119909) = 119911119879

sdot 119860 sdot 120577 + 120576 (119909) 997904rArr 119891 (119909) = 119911119879

sdot 119860 sdot 120577

(5)

where 120576(119909) is the approximation error of fuzzy approach119860(119905)is updated by the adaptation law below

(119905) = minus119876minus1

sdot 119911 sdot 120585119879

sdot es (6)

119876 is a constant coefficient that usually is selected small es isthe difference between output 119909 and reference output 119909ref

es = 119909 minus 119909ref (7)

It should be pointed that AFC guarantees that 119860(119905)

approaches to its desired value 119860lowast(119905)Notice that the initial values of 119860(119905) can be chosen as

119860(0) = 0 or any selected value by information that isacquired from system dynamic or even can be determinedfrom another available control approach for instance thecolumns of 119860(0) can be taken as 119896

119891

(119896119891

is obtained fromstate feedback method 119906 = minus119896

119891

119909) or for example if we wantto estimate 119891(119909) function the columns of 119860(0) are better tocontain coefficients of system states of 119891(119909) [7] Fuzzy rulesare expressed as follows

If 119883 is 119865119894

then 119888119894

= 119891119894

(119911) (8)

119891119894

(119911) is related to 119894th row of 119911119879119860 matrix So the unknownnonlinear function 119891(119909) is estimated as 119891(119909) by fuzzyapproach

Adaptive-fuzzy control method has been divided to twoparts direct and indirect approachs this paper uses indirectmethod

21 Indirect Adaptive Fuzzy Controller (IAFC) First considerthe affine system equations given below

(119905) = 119891 (119909) + 119892 (119909) 119906 (119905)

(119905) = (120572 (119905) + 120572 (119909)) + (120573 (119905) + 120573 (119909)) lowast 119906 (119905)

(9)

120572(119909) 120573(119909) are known parts of system dynamic (which areobtained from experimental and numerical methods) and120572(119905) 120573(119905) are system equations that must be identifiedConsidering the above discussions 120572(119909) and 120573(119909) can beobtained as

120572 (119909) = 119911119879

120572

sdot 119860lowast

120572

sdot 120585120572

+ 119889120572

(119909)

120573 (119909) = 119911119879

120573

sdot 119860lowast

120573

sdot 120585120573

+ 119889120573

(119909)

(10)

119889120572

(119909) 119889120573

(119909) are approximated errors between real systemand fuzzy system with upper bound values |119889

120572

(119909)| le 119863120572

(119909)|119889120573

(119909)| le 119863120573

(119909) (119863120572

(119909) 119863120573

(119909) are known values) Soestimation of system parameters can be rewritten as follows

(119909) = 119911119879

120572

sdot 119860120572

sdot 120585120572

120573 (119909) = 119911119879

120573

sdot 119860120573

sdot 120585120573

(11)

Abstract and Applied Analysis 3

119860120572

119860120573

are matrices that will be updated adaptively with thefollowing updating laws

120572

(119905) = minus119876minus1

sdot 119911 sdot 120585119879

120572

sdot es

120573

(119905) = minus119876minus1

sdot 119911 sdot 120585119879

120573

sdot es sdot 119906 (119905)(12)

and finally the control law is described as follows

119906 (119905) =1

120573 (119905) + 120573 (119909)

(minus [120572 (119905) + (119909)] + 119907 (119905)) (13)

22 Direct Adaptive Fuzzy Controller (DAFC) Control law119906(119905) in feedback linearization method can be obtained asfollows

119906 (119905) =1

120573 (119909)(minus120572 (119909) + 119907 (119905)) (14)

The goal is to estimate 119906(119905) with consideration of basic fuzzyfunction as seen below

(119905) = 119911119879

119906

sdot 119860lowast

119906

sdot 120585119906

+ 119889119906

(119909) (15)

where (119905) is the estimated signal and 119889119906

(119909) is the errorbetween the described 119906 by fuzzymethod and desired (appro-priate) 119906lowast this error is restricted by upper known bound|119889119906

(119909)| le 119863119906

(119909) since practically it is difficult to define119863119906

(119909) so its value is specified and adjusted in designingprocess repetition this adjustment procedure continues untilthe performance of controller indicates that 119863

119906

(119909) comesclose to proper value Finally updating law 119860

119906

(119905) is obtainedas follows

119906

(119905) = minus119876minus1

119906

sdot 119911119879

119906

sdot 120585119906

sdot es (16)

The initial value of 119860119906

(119905)may be considered as 119860119906

(0) = 0

Remark 1 Thismethod is suitable for minimum phase plantsand also has the following limitation for 120573(119909)

minusinfin lt 1205731

lt 120573 (119909) lt 1205732

lt infin (17)

3 Proposed Approach

In the previous section an adaptive fuzzy controller usingfuzzy basis functions is described for stabilizing the con-trolled system In our introducedmethod the same approachhas been employed but our method does not need to utilizefuzzy basis functions for estimating system parametersInstead three proposed rules based on output errors areconsidered to control the plant

Consider a system with general form

(119905) = 119891 (119909 (119905)) + 119892 (119909 (119905)) 119906 (119905) (18)

At first nonlinear functions 119891 and 119892 must be estimated insuch a way that controller does not rely on system dynamicIt should be noted that in this paper the system is consideredcontinues time but the output is sampled and these sampled

points are used in proposed controller The estimations ofsystem dynamic can be defined as

119891 (119896) = 119879

119891

(119896)

119892 (119896) = 119879

119892

(119896)

(19)

And the updating values of 119892

119891

are given by

119891

(119896 + 1) = 119891

(119896) + 120572119903119879

(119896 + 1)

119892

(119896 + 1) = 119892

(119896) + 120573119906119888

(119896) 119903119879

(119896 + 1) 119868 = 1

119892

(119896) 119868 = 0

(20)

where 120572119891

120572119892

are constant coefficients with respect to 119891

119892

It should also be noted that usually these parameters donot need to be changed for different system dynamics Thevalue of 119868 will be defined in (27) 119903(119905) is the so-called ldquofilteredtracking errorrdquo which is the sum of errors as defined below

119903 (119896) = 119890119899

(119896) + 1205821

119890119899minus1

(119896) + sdot sdot sdot + 120582119899minus1

1198902

(119896) + 120582119899

1198901

(119896)

1198901

(119896) = 119910 (119896) minus 119910119889

(119896)

1198902

(119896) = 119910 (119896) minus 119910119889

(119896 + 1)

119890119899

(119896) = 119910 (119896) minus 119910119889

(119896 + 119899 minus 1)

(21)

where 120582119894

119894 = 1 119899 are coefficients of the predictive errorAnd119910

119889

is the desired output For eliminating nonlinear termsof system dynamic feedback linearization method has beenexploited for this purpose 119906(119896) is chosen as follows

119906 (119896) = 119892minus1

(119896) (minus119891 (119896) + 119907 (119896)) (22)

where 119907(119896) is defined by

119907 (119896) = 119896119907

119903 (119896) + 119910119889

(119896 + 119899) minus

119901minus2

sum

119894=0

120582119894+1

119890119899minus119894

(119896) (23)

In (23) 119896119907

is the tracking error coefficient By choosingappropriate 119896

119907

119903(119896) is converged into zero and at last itcauses 119907(119896) to be restricted In other words if the bigger 119896

119907

is opted better performance of output and quick tendencyto the desired value are achieved while increasing the controlsignal 119906 it should be noted that if 119896

119907

is chosen very large itcan make the system unstable

As it is seen from (22) the problem of singularity occurswhen 119892 is near zero and the control signal can becomeunbounded To solve this problem separate controller com-prising two parts is proposed in the following form

119906 (119896)

=

119906119888

(119896) + 5 (119906119903

(119896) minus 119906119888

(119896)) exp (120574 (1003817100381710038171003817119906119888 (119896)1003817100381710038171003817 minus 119904))

119868 = 1

119906119903

(119896) minus 5 (119906119903

(119896) minus 119906119888

(119896)) exp (minus120574 (1003817100381710038171003817119906119888 (119896)1003817100381710038171003817 minus 119904))

119868 = 0

(24)

4 Abstract and Applied Analysis

In the above equation 119904 is a constant value that its amountimpacts on control signal 119906 119906

119888

119906119903

are defined as

119906119888

(119896) = 119892minus1

(119896) (minus119891 (119896) + 119907 (119896)) (25)

119906119903

(119896) = minus1205831003817100381710038171003817119906119888 (119896)

1003817100381710038171003817 sign(119903 (119896))

119892 (26)

where 120583 is robust control signal coefficient that the decreasingor increasing of this coefficient affects quantity of final controlsignal 119906 and also 119892 is a constant value that its small amountdecreases the effect of 119906

119903

on final control signal 119906 that is 119906119903

will become more effective factor for determining 119906 and 119868 isgiven by the equation below

119868 = 11003817100381710038171003817119892 (119896)

1003817100381710038171003817 gt 1198921003817100381710038171003817119906119888 (119896)

1003817100381710038171003817 lt 119904

0 otherwise(27)

Here 119892 is lower bound of 119892 Proper selection of 119892 prevents 119892to be singular

Remark 2 120574 120583 and 119904 are designing parameters with thefollowing conditions [5] 120574 lt ln(2119904) 120583 gt 0 119904 gt 0

Remark 3 If 119906119888

is greater than 1120574 it is assumed to be equalto 1120574

The major principles of proposed MFA controller areconstructed based on the three following experimental ruleswhich make controller produce appropriate control signal

119906 (119896) =

119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896)1003816100381610038161003816 le

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

minus119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896)1003816100381610038161003816 gt

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

minus120578119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 gt

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

(28)

where 120576 is threshold value for changing final control signal119906 in the above rules These laws utilize appropriate last levelcontrol signal (if error becomes less than specified value thatis called 120576) for using at next level Indeed if error at time 119896 and119896 minus 1 becomes less than 120576 and also error at time 119896 is greaterthan the previous time the controller applies negative valueof previous control signal for current time

Regarding adaptive fuzzy controllers [5 7] that esti-mate unknown nonlinear functions of the system by fuzzyalgorithm and when new reference input is applied to thesystem this estimation becomes better But the sensitivity ofproposed controller on estimation of nonlinear functions isless than that of AFCmethodThis is because output error hassignificant role in control effort rather than significant role ofestimation in AFC

4 Lyapunov Stability for the Proposed Model-Free Adaptive Controller

Let the Lyapunov function candidate 119881 be given by [8]

119881 = 119903119879

(119896) 119903 (119896) +1

120572tr (119879119891

(119896) 119891

(119896))

+1

120573tr (119879119892

(119896) 119892

(119896))

(29)

(a) For the first region 119868 = 1At first we should obtain the difference of Lyapunov

function in 119896 and 119896 + 1 level So general Lyapunov functionΔ119881 = Δ119881

1

+ Δ1198812

+ Δ1198813

can be acquired as follows

Δ1198811

= 119903119879

(119896 + 1) 119903 (119896 + 1) minus 119903119879

(119896) 119903 (119896)

Δ1198812

=1

120572tr (119879119891

(119896 + 1) 119891

(119896 + 1) minus 119879

119891

(119896) 119891

(119896))

Δ1198813

=1

120573tr (119879119892

(119896 + 1) 119892

(119896 + 1) minus 119879

119892

(119896) 119892

(119896))

(30)

If we prove thatΔ119881 le 0 it will demonstrate that 119892

119891

and 119903are bounded and they are able to stabilize system Δ119881

1

Δ1198812

and Δ119881

3

are obtained as follows

Δ1198811

= minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896))119879

(119879

119891

+ 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ (119879

119891

)119879

119879

119891

+ (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ (119892119906119889

)119879

119892119906119889

+ 1205761015840119879

1205761015840

+ 2119891

(119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ 2(119879

119892

119906119888

)119879

(119892119906119889

+ 1205761015840

) + 2(119892119906119889

)119879

1205761015840

Δ1198812

= minus (2 minus 120572) (119879

119891

)119879

119879

119891

+ 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120572) 119891

(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

Δ1198813

= minus (2 minus 120573) (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ 120573(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120573) (119879

119892

119906119888

)119879

(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

(31)

With combination of the above three Lyapunov functions(32) is established

Δ119881 = minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896) minus (1 minus 120572) (119879

119891

)119879

119879

119891

+ 2(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

Abstract and Applied Analysis 5

+ 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ 2120572119891

(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

minus (1 minus 120573) (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ 120573(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

+ 2120573(119879

119892

119906119888

)119879

(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

(32)

By simplifying (32) and using the below definition

120578 = 120572 + 120573

10038171003817100381710038171199061198881003817100381710038171003817

2

119868 = 1

120572 119868 = 0(33)

Inequality (34) is acquired as follows

Δ119881 le minus119903119879

(119896) (119868 minus (1 + 120578) 119896119879

119907

119896119907

) 119903 (119896)

+ 2120578(119896119907

119903 (119896))119879

(119892119906119889

+ 1205761015840

)

+ (1 + 120578) (119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

+120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

(34)

with assuming that 119896119907max is the maximum value of 119896

119907

and byresimplifying (34) inequality (35) is stated as given below

Δ119881 le minus (119868 minus 1198861

1198962

119907max) 119903 (119896)2 + 21198862119896119907max 119903 (119896) (119892119906119889 + 1205761015840)+ 1198861

(119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(35)

where 1198861

and 1198862

are described as

1198861

= 1 + 120578 +120578

1 minus 120578 119886

2

= 120578 +120578

1 minus 120578 (36)

By considering Lemma A4 in Appendix A

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817=10038171003817100381710038171003817119892119906119889

+ 1205761015840

10038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896)

(37)

And with substituting inequality (37) in (35)

Δ119881 le minus (119868 minus 1198861

1198962

119907max) 119903 (119896)2+ 21198862

119896119907max 119903 (119896) (1198880 + 1198881 119903 (119896))

+ 1198861

(1198880

+ 1198881

119903 (119896))119879

(1198880

+ 1198881

119903 (119896))

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(38)

With simplifying the above inequality the below equation isobtained

Δ119881 le minus (119868 minus 1198863

) 119903 (119896)2

+ 21198864

119903 (119896) + 1198865

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(39)

where 1198863

1198864

and 1198865

are defined as

1198863

= 1198861

1198962

119907max + 211988621198881119896119907max + 119886111988821

1198864

= 1198862

1198880

119896119907max + 119886111988801198881

1198865

= 1198861

1198882

0

(40)

By assuming that 120578 lt 1 the below statement becomesnegative

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

lt 0

(41)

As regards 120572 gt 0 120573 gt 0 120578 becomes positive too andfinally 119886

1

1198862

1198863

1198864

1198865

gt 0 totally the first term in (39) minus(119868 minus1198863

)119903(119896)2

+ 21198864

119903(119896) + 1198865

becomes nonpositive if it obeysthe below condition

119903 (119896) gt 1205751199031

(42)

in which 1205751199031

is acquired from the below inequality

100381710038171003817100381712057511990311003817100381710038171003817 gt

1198865

+ radic1198862

5

+ 1198866

(1 minus 1198864

)

(1 minus 1198864

) (43)

Thus both terms of inequality (39) have nonpositive valuesand finally Δ119881 le 0

(b) For the second region 119868 = 0

6 Abstract and Applied Analysis

In this region the proposed Lyapunov functions Δ1198811

Δ1198812

and Δ1198813

are obtained like (30) and they are stated asfollows

Δ1198811

= minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896))119879

(119879

119891

+ 119892119906119889

+ 1205761015840

) + (119879

119891

)119879

119879

119891

+ (119892119906119889

)119879

119892119906119889

+ 1205761015840119879

1205761015840

+ 2119891

(119892119906119889

+ 1205761015840

) + 2(119892119906119889

)119879

1205761015840

Δ1198812

= minus (2 minus 120572) (119879

119891

)119879

119879

119891

+ 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120572) 119891

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

Δ1198813

= 0

(44)

By combination of the above three equations (and definitionof 120578) the statement is demonstrated below

Δ119881 = minus 119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

+1

1 minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(45)

with the consideration of Lemma A41003817100381710038171003817119892119906 minus 119892119906119862

1003817100381710038171003817 le 1198890 + 1198891 119903 (119896) (46)

By substituting inequality (46) in (45)

Δ119881 le minus (119868 minus 1198870

) 119903 (119896)2

+ 21198871

119903 (119896) + 1198872

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2 (47)

in which 1198870

1198871

and 1198872

are as follows

1198870

= 1198962

119907max + 21198891 (1198891 + 119896119907max) + (1198891 + 119896119907max)21 minus 120572

1198871

= 2 (1198890

+ 1205761015840

) 1198891

+4 minus 2120572

1 minus 120572(1198890

+ 1205761015840

) (1198891

+ 119896119907max)

1198872

=3 minus 2120572

1 minus 120572(1198890

+ 1205761015840

)2

(48)

By the assumption that 120578 lt 1 the statement below becomesnegative

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

lt 0 (49)

As regards120572 gt 0 120573 gt 0 120578 becomes positive too andfinally 1198870

1198871

1198872

gt 0The first term in (39) minus(119868minus1198870

)119903(119896)2

+21198871

119903(119896)+

1198872

becomes nonpositive if it obeys the below condition

119903 (119896) gt 1205751199032

(50)

in which 1205751199032

is obtained from the relation below

100381710038171003817100381712057511990321003817100381710038171003817 gt

1198871

+ radic1198872

1

+ 1198872

(1 minus 1198870

)

(1 minus 1198870

) (51)

As a result both terms of inequality (47) have nonpositivevalues which means Δ119881 le 0 And finally if ||119903(119896)|| gt

max(1205751199031

1205751199032

) Lyapunov function becomes negative

5 Implementation of MFA Controller

Example 4 Consider MIMO system with 2 input-2 outputsystem state equations which are given below [8]

1

(119905) = 1199092

(119905) + (1 + 1199092

(119905)2

) 1199061

(119905) + 1198891

(119905)

2

(119905) = minus 1199091

(119905) minus (01 minus exp (minus1199091

(119905)2

minus 1199092

(119905)2

)) 1199092

(119905)

+ (1 + 1199091

(119905)2

) 1199062

(119905) + 1198892

(119905)

1199101

(119905) = 1199091

(119905) + 1198991

(119905) 1199102

(119905) = 1199092

(119905) + 1198992

(119905)

119899 (119905) = measurement noise

1198891

(119905) 1198892

(119905) = disturbance(52)

Select control parameters as follows

120583 = 10 120572 = 1 120573 = 1 119904 = 2 120574 = 05

119892 = 8 119896119907

= [10 0 0 10] 1205821

= 001 120576 = 01

120578 = 0 119891

= [1 1] 119892

= [1 0 0 1]

(53)

Simulation results of MIMO system without disturbances1198891

(119905) 1198892

(119905) = 0 are demonstrated in Figure 1 In Figure 2external disturbances are applied to the system with power1198891

(119905) = 04 1198892

(119905) = 04 and finally the system with existenceof white Gaussian measurement noise (power 00001) andexternal disturbance is shown in Figure 3 also desired inputsare assumed 119910

1119889

= 11199102119889

= cos(2120587119905)

The aim of using this MIMO system which is broughtby [8] (along with two control signals) is surveying couplingeffect in multiinput multioutput systems As it is understoodfrom the result of Figure 1 caused by MFA controller and theresult of AFC in [8] it can be said that system outputs ofMFA controller do not have any overshoot in its responsebut in AFC this problem is observed As it is seen fromFigure 2 MFA controller rejects disturbance without anynoticeable effort in control signals and in the outputs Alsoit can be deduced from Figure 3 that the measurement noisehas intense effect on system outputs and control signals

Abstract and Applied Analysis 7

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

Time (s)

System output 1Desired output 1

Out

put1

(a)

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

Time (s)

System output 2Desired output 2

Out

put2

(b)

0 05 1 15 2 25 3 35 4 45 5minus10010

20

30

40

50

60

70

Time (s)

Cont

rol p

ower

1

(c)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20020

40

60

80

(d)

Figure 1 Simulation results for MIMO system without disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs1 and 2

(approximately became twice) but it should be noted that theproposed controller is able to track desired output and yetremains stable As it is demonstrated in Figures 1 2 and 3the disadvantage of this controller is oscillating control signalProfits of MFAC in addition of good (also rapid) trackingand system stabilizing are decreasing settling time of outputresponse

Example 5 The second system under study is Maglev trainwhich is shown in Figure 4 Using the notations given inFigure 4 the vertical dynamics is described by [9]

System state equations are described as follows

1198981198892

119911 (119905)

1198891199052= minus119865 (119894 119911 119905) + 119891

119889

+ 119898119892

= minus1205830

1198732

119886119898

4[119894 (119905)

119911 (119905)]

2

+ 119891119889

+ 119898119892

119889119894 (119905)

119889119905=119894 (119905)

119911 (119905)

119889119911 (119905)

119889119905

minus2

1205830

1198732119886119898

119911 (119905) (119877119898

119894 (119905) minus 119906 (119905))

(54)

Here 119911 is the distance between rail and train 119894 is the current ofwindings 119898 is the train mass 119891

119889

is the force of disturbance1205830

is the permeability coefficient which equals 4120587 lowast 10minus7 119873the number of turns in winding 119886

119898

is the section surface ofwindings and 119877

119898

is the wire resistance in windings

Consider the state vector as below thus the aboveequations convert to new state equations

1

(119905) = 1199092

(119905)

2

(119905) = minus1205830

1198732

119886119898

4119898[1199093

(119905)

1199091

(119905)]

2

+1

119898119908 + 119892

3

(119905) = minus2119877119898

1205830

1198732119886119898

1199093

(119905) 1199091

(119905) +1199092

(119905) 1199093

(119905)

1199091

(119905)

+21199091

(119905) 119906 (119905)

1205830

1198732119886119898

119910 (119905) = 1199092

(119905) + 119899 (119905)

119899 (119905) = measurement noise119908 (119905) = external disturbance

(55)

8 Abstract and Applied Analysis

Time (s)

Out

put 1

System output 1Desired output 1

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

(c)

0

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20

(d)

Figure 2 Simulation results for MIMO system with disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs 1and 2

Maglev trains work in one of two ways both methods arebased on the same concept but involve different approaches

(1) Electromagnetic Suspension is based on magneticattraction it is very complex and somewhat unstable

(2) Electrodynamic Suspension is based on the repulsionofmagnetsThemagnetic levitation force balances theweight of the car at a stable position Controlling EMSis more difficult than EDS train because normality ofthis dynamic state is unstable

Magnetic trains have two important issues levitationand propulsion the target of controller is first goes uptrain to desired level along with guarantee stabilizing againstsome uncertainty such as wind and changing train mass inboltroads And second adjust train speed at the workingfrequency of magnets In this paper just levitation part(important section of train in controlling) is discussed Thetrain in this example goes from 10mm to 16mm level By

considering these assumption values for train and controlleras follows

119898 = 15 kg 119873 = 280 turns 119886119898

= 1024m2

119877119898

= 11Ω 120583 = 32 120572 = 1 120573 = 1

119904 = 140 120574 = 01 119892 = 9 119896119907

= 1 1205821

= 0

120576 = 0001 120578 = 1 119891

= minus 4 119892

= 1

(56)

The result of simulating without considering disturbancehas been displayed in Figure 5 in Figure 6 input step asexternal disturbance with amplitude 119889(119905) = 1 is appliedto the system for Figure 7 sinusoidal measurement noisewith amplitude 1mm in presence of disturbance is con-sidered Also white Gaussian noise in existence of externaldisturbance is applied to the system with power 8 times 10minus9(it is considered small because 119868119874 amplitude is small)which is shown in Figure 8 The purpose of presentingthis system is studying on sort of complicated practicalsystem dynamics such as Maglev in which they have some

Abstract and Applied Analysis 9

Time (s)

Out

put 1

System output 1Desired output 1

0 051 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

0102030405060

Time (s)

Cont

rol p

ower

1

0 05 1 15 2 25 3 35 4 45 5minus30minus20minus10

(c)

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5

minus20minus40minus60minus80(d)

Figure 3 Simulation results for MIMO system with the measurement noise and disturbance (a) (b) system outputs 1 and 2 (c) (d) controlsignals 119906 relative to inputs 1 and 2

Figure 4 Vertical slice of magnetic levitation and rail in a magnetic train [9]

nonlinear terms like existence of first state in denominator(that affect directly the destabilizing system) and so forthAs it is demonstrated in Figure 5(a) the system outputthat is distance between rail and train has no overshootafter about 25 second it goes from 10 to 16mm smoothlySuch as the last example it is acquired from Figure 6 thatdisturbance did not have any effect on the output control

effort and settling time In Figure 7 it is illustrated thatsinusoidal measurement noise with no changes appears intooutput but no noticeable alternation in control signal hasbeen seen And finally in Figure 8 an impalpable effectof white Gaussian measurement noise on output and con-trol signal can be shown in this example some repeatedbenefits which are seen in Maglev system response are

10 Abstract and Applied AnalysisD

istan

ce (m

)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 5 Simulation result for Maglev without disturbance (inwhich it goes from 10 to 16mm) (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 6 Simulation result for Maglev with disturbance (a)Distance between train and rail (b) control signal

expressed such as appropriate system stabilizing good andrapid tracking disturbance rejection and small settlingtime

6 Conclusion

Model-free adaptive controller is an adaptive controller thatjust uses system outputs and does not require another systemstates (so does not need observer) and with this as model-free controller it performs good and also has some meritssuch as good stability appropriate tracking robust againstuncertainty disturbance rejection and good decoupling andthe biggest advantage of MFA controller is no requiringto system dynamic and even does not need to any priorexperiment about system

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 7 Simulation result for Maglev with sinuous measurementnoise at 15Hz and disturbance (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 8 Simulation result for Maglev train with white Gaussianmeasurement noise along with disturbance (a) Distance betweentrain and rail (b) control signal

Appendices

A Some RequirementsLemma A1 For each time 119896 119909(119896) is bounded by

119909 (119896) le 1198890

+ 1198891

119903 (119896) (A1)

which is 119903(119896) called ldquotracking errorrdquo and 1198890

1198891

are computablepositive constants

Abstract and Applied Analysis 11

Proof [see [8 Lemma 32]] Let 119909(119896) isin 119880 in which 119880 is acompact subset of 119877119899 Assume that ℎ(119909(119896)) isin 119862infin119880 that isℎ(119909(119896)) is a smooth function 119880 rarr 119877 so that the Taylorseries expansion of ℎ(119909(119896)) exists Then using the bound on119909(119896) express the function ℎ(119909(119896)) on a compact set as linearform of parameters Thus the upper bound for ℎ(119909(119896)) canbe obtained as

ℎ (119909 (119896)) le 1198880

+ 1198881

119903 (119896) (A2)

where 1198880

and 1198881

are constant matrices [8]

Remark A2 120576119892

120576119891

are estimation errors of 119892 119891 respectivelyand 119889 is external disturbance These errors are bounded andhave these definite bounds

10038171003817100381710038171003817120576119891

10038171003817100381710038171003817le 120576119873119891

10038171003817100381710038171003817120576119892

10038171003817100381710038171003817le 120576119873119892

119889 le 119889119872

(A3)

Remark A3 According to (27) if 119868 = 1 then 119906119888

le 119904 so 119906119888

is bounded and if 119868 = 0 then 119906119888

gt 119904 but by consideringthe mentioned condition at Remark 3 in Section 5 in whichif 119906119888

is greater than 1120574 it is assumed to be equal 1120574 Thus itcan be deduced that in both regions and in all of times 119906

119888

isbounded and its bounds are shown as follows

10038171003817100381710038171199061198881003817100381710038171003817 le

119904 119868 = 1

1

120574119868 = 0

(A4)Lemma A4 With regards to that the proposed controlalgorithm is divided into two regions 119868 = 0 and 119868 = 1 Wewill prove the following relationship which is established in eachregion

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896) 119868 = 1

1003817100381710038171003817119892119906 minus 1198921199061198621003817100381710038171003817 le 1198890 + 1198891 119903 (119896) 119868 = 0

(A5)

Proof (a) First region 119868 = 1 or 119906119888

le 119904 and 119892(119909) gt 119892In consideration of Lemma A1 it is demonstrated that

119892(119909) le 11988801

+ 11988811

119903(119896) where 11988801

11988811

are constant matrices

10038171003817100381710038171199061198881003817100381710038171003817 le 119904 997888rarr

10038171003817100381710038171199061199031003817100381710038171003817 le

120583

119892

10038171003817100381710038171199061198881003817100381710038171003817 997888rarr bounded 1003817100381710038171003817119906119903

1003817100381710038171003817 (A6)

By using the result of relation (A6) the following inequalityis obtained

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A7)

As regards 119906119888

le 119904 rarr 120574(119906119888

minus 119904) le 0

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(10038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198882

(A8)

By considering (A7)1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817

1003817100381710038171003817(119906 minus 119906119862)1003817100381710038171003817

le 1198882

(11988801

+ 11988811

119903 (119896)) = 11988802

+ 11988812

119903 (119896)

(A9)

Also by utilizing Remark 3 (A10) is stated as follows10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le10038171003817100381710038171003817120576119891

10038171003817100381710038171003817+10038171003817100381710038171003817120576119892

10038171003817100381710038171003817

10038171003817100381710038171199061198881003817100381710038171003817 + 119889

le 120576119873119891

+ 120576119873119892

+ 119889119872

= 120576119873

(A10)

By using (A9) and (A10) the following inequality is acquired10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 +10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le 11988802

+ 11988812

119903 (119896) + 120576119873

le 1198880

+ 1198881

119903 (119896)

(A11)

Thus the assumption is proved in the first region(b) Second region 119868 = 0 or 119906

119888

ge 119904 and 119892(119909) lt 119892 andthe mentioned condition for the proposed MFA controller inRemark 3 in Section 5 is stated as follows

if 119906119888

gt1

120574997904rArr 119906119888

=1

120574 (A12)

In consideration of Lemma A1 it is demonstrated that119892(119909) le 119888

01

+ 11988811

119903(119896) where 11988801

11988811

are constant matricesBy using the result of relation (A6) and the condition that isgiven above the following inequality is obtained

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890minus120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A13)

As regards 119906119888

gt 119904 rarr minus120574(119906119888

minus 119904) le 0

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(310038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198892

(A14)

By considering (B5)1003817100381710038171003817119892119906

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817 119906 le 1198892 (11988801 + 11988811 119903 (119896))

= 11988902

+ 11988912

119903 (119896)

10038171003817100381710038171198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921003817100381710038171003817

10038171003817100381710038171003817100381710038171003817

1

120574

10038171003817100381710038171003817100381710038171003817

le 1198893

(A15)

By using (A10) and (A15) the following inequality isobtained

1003817100381710038171003817119892119906 minus 1198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921199061003817100381710038171003817 +

10038171003817100381710038171198921199061198881003817100381710038171003817

le 11988901

+ 11988911

119903 (119896) + 1198893

le 1198890

+ 1198891

119903 (119896)

(A16)

Finally the assumption is proved in the second region Thusinequality (A5) is correct in both regions

B Obtain 119903(119896)

In consideration of error definition 119903(119896) in (21)

119903 (119896 + 1) = 119890119899

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

I+ 1205821

119890119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

II+ sdot sdot sdot + 120582

119899minus1

1198901

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

III

(B1)

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

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Volume 2013

2 Abstract and Applied Analysis

described The proposed model-free adaptive controller andits Lyapunov stability are represented in Sections 3 and4 respectively in Section 5 simulation results are given tohighlight advantages of MFAC method Finally conclusion isstated in Section 6

2 Feedback Linearization by Means ofAdaptive Fuzzy Controller

In feedback linearization method control law is determinedin such a way that nonlinear terms eliminate the systemdynamic and replace it with appropriate reference input asseen below (for simplifying 119909(119905) = 119909)

(119905) = 119891 (119909) + 119892 (119909) 119906 (119905) 997888rarr 119906 (119905)

= 119892minus1

(119909) (minus119891 (119909) + 119907 (119905))

(1)

Conceptually Adaptive Feedback Linearization (AFL)method is the same as nonadaptive counterpart but AFLuses adaptation law for updating controller parameter toenhance system performance Notice that both methodsshould have complete description of system dynamic butour proposed method does not need any information ofsystem dynamic Due to the fact that it is difficult to obtaina mathematical model of the system and typically there existsome numerical approximations the object of this paper isto control systems just with 119868119874 data in order to improvesystem performance In order to reach this goal it is requiredto estimate unknown nonlinear function of system dynamic119891(119909) and 119892(119909) with 119891(119909) and 119892(119909) or estimate controllaw 119906(119905) with (119909) Estimation algorithm in this approachutilizes basic fuzzy membership function As an instance forestimating 119891(119909) we have

119891 (119909) 997888rarr 119891 (119909) =sum119901

119894=0

119888119894

120583119894

sum119901

119894=0

120583119894

(2)

where 119901 is the number of fuzzy roles 120583119894

(119909) is membershipfunction for 119894th fuzzy role also 119888

119894

is and the result of 119894thfuzzy role for system that is generally considered as linearcombination of some series of continuous functions such as119911(119909)

119911119896

(119909) isin 119877 119896 = 1 2 119898 minus 1

119888119894

= 1198861198940

+ 119886119894119894

1199111

(119909) + sdot sdot sdot + 119886119894119898minus1

119911119898minus1

(119909)

(3)

119911 = [1 1199111

(119909) 1199112

(119909) sdot sdot sdot 119911119898minus1

(119909)]119879

119860119879

=[[

[

11988610

11988611

sdot sdot sdot 1198861119898minus1

1198861199010

1198861199011

sdot sdot sdot 119886119901119898minus1

]]

]

119888 = 119911119879

119860

(4)

Using (3) (4) 119891(119909) can be rewritten as

120585 =

[1205831

1205832

sdot sdot sdot 120583119901

]

sum119901

119894=0

120583119894

119891 (119909) = 119911119879

sdot 119860 sdot 120577 + 120576 (119909) 997904rArr 119891 (119909) = 119911119879

sdot 119860 sdot 120577

(5)

where 120576(119909) is the approximation error of fuzzy approach119860(119905)is updated by the adaptation law below

(119905) = minus119876minus1

sdot 119911 sdot 120585119879

sdot es (6)

119876 is a constant coefficient that usually is selected small es isthe difference between output 119909 and reference output 119909ref

es = 119909 minus 119909ref (7)

It should be pointed that AFC guarantees that 119860(119905)

approaches to its desired value 119860lowast(119905)Notice that the initial values of 119860(119905) can be chosen as

119860(0) = 0 or any selected value by information that isacquired from system dynamic or even can be determinedfrom another available control approach for instance thecolumns of 119860(0) can be taken as 119896

119891

(119896119891

is obtained fromstate feedback method 119906 = minus119896

119891

119909) or for example if we wantto estimate 119891(119909) function the columns of 119860(0) are better tocontain coefficients of system states of 119891(119909) [7] Fuzzy rulesare expressed as follows

If 119883 is 119865119894

then 119888119894

= 119891119894

(119911) (8)

119891119894

(119911) is related to 119894th row of 119911119879119860 matrix So the unknownnonlinear function 119891(119909) is estimated as 119891(119909) by fuzzyapproach

Adaptive-fuzzy control method has been divided to twoparts direct and indirect approachs this paper uses indirectmethod

21 Indirect Adaptive Fuzzy Controller (IAFC) First considerthe affine system equations given below

(119905) = 119891 (119909) + 119892 (119909) 119906 (119905)

(119905) = (120572 (119905) + 120572 (119909)) + (120573 (119905) + 120573 (119909)) lowast 119906 (119905)

(9)

120572(119909) 120573(119909) are known parts of system dynamic (which areobtained from experimental and numerical methods) and120572(119905) 120573(119905) are system equations that must be identifiedConsidering the above discussions 120572(119909) and 120573(119909) can beobtained as

120572 (119909) = 119911119879

120572

sdot 119860lowast

120572

sdot 120585120572

+ 119889120572

(119909)

120573 (119909) = 119911119879

120573

sdot 119860lowast

120573

sdot 120585120573

+ 119889120573

(119909)

(10)

119889120572

(119909) 119889120573

(119909) are approximated errors between real systemand fuzzy system with upper bound values |119889

120572

(119909)| le 119863120572

(119909)|119889120573

(119909)| le 119863120573

(119909) (119863120572

(119909) 119863120573

(119909) are known values) Soestimation of system parameters can be rewritten as follows

(119909) = 119911119879

120572

sdot 119860120572

sdot 120585120572

120573 (119909) = 119911119879

120573

sdot 119860120573

sdot 120585120573

(11)

Abstract and Applied Analysis 3

119860120572

119860120573

are matrices that will be updated adaptively with thefollowing updating laws

120572

(119905) = minus119876minus1

sdot 119911 sdot 120585119879

120572

sdot es

120573

(119905) = minus119876minus1

sdot 119911 sdot 120585119879

120573

sdot es sdot 119906 (119905)(12)

and finally the control law is described as follows

119906 (119905) =1

120573 (119905) + 120573 (119909)

(minus [120572 (119905) + (119909)] + 119907 (119905)) (13)

22 Direct Adaptive Fuzzy Controller (DAFC) Control law119906(119905) in feedback linearization method can be obtained asfollows

119906 (119905) =1

120573 (119909)(minus120572 (119909) + 119907 (119905)) (14)

The goal is to estimate 119906(119905) with consideration of basic fuzzyfunction as seen below

(119905) = 119911119879

119906

sdot 119860lowast

119906

sdot 120585119906

+ 119889119906

(119909) (15)

where (119905) is the estimated signal and 119889119906

(119909) is the errorbetween the described 119906 by fuzzymethod and desired (appro-priate) 119906lowast this error is restricted by upper known bound|119889119906

(119909)| le 119863119906

(119909) since practically it is difficult to define119863119906

(119909) so its value is specified and adjusted in designingprocess repetition this adjustment procedure continues untilthe performance of controller indicates that 119863

119906

(119909) comesclose to proper value Finally updating law 119860

119906

(119905) is obtainedas follows

119906

(119905) = minus119876minus1

119906

sdot 119911119879

119906

sdot 120585119906

sdot es (16)

The initial value of 119860119906

(119905)may be considered as 119860119906

(0) = 0

Remark 1 Thismethod is suitable for minimum phase plantsand also has the following limitation for 120573(119909)

minusinfin lt 1205731

lt 120573 (119909) lt 1205732

lt infin (17)

3 Proposed Approach

In the previous section an adaptive fuzzy controller usingfuzzy basis functions is described for stabilizing the con-trolled system In our introducedmethod the same approachhas been employed but our method does not need to utilizefuzzy basis functions for estimating system parametersInstead three proposed rules based on output errors areconsidered to control the plant

Consider a system with general form

(119905) = 119891 (119909 (119905)) + 119892 (119909 (119905)) 119906 (119905) (18)

At first nonlinear functions 119891 and 119892 must be estimated insuch a way that controller does not rely on system dynamicIt should be noted that in this paper the system is consideredcontinues time but the output is sampled and these sampled

points are used in proposed controller The estimations ofsystem dynamic can be defined as

119891 (119896) = 119879

119891

(119896)

119892 (119896) = 119879

119892

(119896)

(19)

And the updating values of 119892

119891

are given by

119891

(119896 + 1) = 119891

(119896) + 120572119903119879

(119896 + 1)

119892

(119896 + 1) = 119892

(119896) + 120573119906119888

(119896) 119903119879

(119896 + 1) 119868 = 1

119892

(119896) 119868 = 0

(20)

where 120572119891

120572119892

are constant coefficients with respect to 119891

119892

It should also be noted that usually these parameters donot need to be changed for different system dynamics Thevalue of 119868 will be defined in (27) 119903(119905) is the so-called ldquofilteredtracking errorrdquo which is the sum of errors as defined below

119903 (119896) = 119890119899

(119896) + 1205821

119890119899minus1

(119896) + sdot sdot sdot + 120582119899minus1

1198902

(119896) + 120582119899

1198901

(119896)

1198901

(119896) = 119910 (119896) minus 119910119889

(119896)

1198902

(119896) = 119910 (119896) minus 119910119889

(119896 + 1)

119890119899

(119896) = 119910 (119896) minus 119910119889

(119896 + 119899 minus 1)

(21)

where 120582119894

119894 = 1 119899 are coefficients of the predictive errorAnd119910

119889

is the desired output For eliminating nonlinear termsof system dynamic feedback linearization method has beenexploited for this purpose 119906(119896) is chosen as follows

119906 (119896) = 119892minus1

(119896) (minus119891 (119896) + 119907 (119896)) (22)

where 119907(119896) is defined by

119907 (119896) = 119896119907

119903 (119896) + 119910119889

(119896 + 119899) minus

119901minus2

sum

119894=0

120582119894+1

119890119899minus119894

(119896) (23)

In (23) 119896119907

is the tracking error coefficient By choosingappropriate 119896

119907

119903(119896) is converged into zero and at last itcauses 119907(119896) to be restricted In other words if the bigger 119896

119907

is opted better performance of output and quick tendencyto the desired value are achieved while increasing the controlsignal 119906 it should be noted that if 119896

119907

is chosen very large itcan make the system unstable

As it is seen from (22) the problem of singularity occurswhen 119892 is near zero and the control signal can becomeunbounded To solve this problem separate controller com-prising two parts is proposed in the following form

119906 (119896)

=

119906119888

(119896) + 5 (119906119903

(119896) minus 119906119888

(119896)) exp (120574 (1003817100381710038171003817119906119888 (119896)1003817100381710038171003817 minus 119904))

119868 = 1

119906119903

(119896) minus 5 (119906119903

(119896) minus 119906119888

(119896)) exp (minus120574 (1003817100381710038171003817119906119888 (119896)1003817100381710038171003817 minus 119904))

119868 = 0

(24)

4 Abstract and Applied Analysis

In the above equation 119904 is a constant value that its amountimpacts on control signal 119906 119906

119888

119906119903

are defined as

119906119888

(119896) = 119892minus1

(119896) (minus119891 (119896) + 119907 (119896)) (25)

119906119903

(119896) = minus1205831003817100381710038171003817119906119888 (119896)

1003817100381710038171003817 sign(119903 (119896))

119892 (26)

where 120583 is robust control signal coefficient that the decreasingor increasing of this coefficient affects quantity of final controlsignal 119906 and also 119892 is a constant value that its small amountdecreases the effect of 119906

119903

on final control signal 119906 that is 119906119903

will become more effective factor for determining 119906 and 119868 isgiven by the equation below

119868 = 11003817100381710038171003817119892 (119896)

1003817100381710038171003817 gt 1198921003817100381710038171003817119906119888 (119896)

1003817100381710038171003817 lt 119904

0 otherwise(27)

Here 119892 is lower bound of 119892 Proper selection of 119892 prevents 119892to be singular

Remark 2 120574 120583 and 119904 are designing parameters with thefollowing conditions [5] 120574 lt ln(2119904) 120583 gt 0 119904 gt 0

Remark 3 If 119906119888

is greater than 1120574 it is assumed to be equalto 1120574

The major principles of proposed MFA controller areconstructed based on the three following experimental ruleswhich make controller produce appropriate control signal

119906 (119896) =

119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896)1003816100381610038161003816 le

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

minus119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896)1003816100381610038161003816 gt

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

minus120578119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 gt

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

(28)

where 120576 is threshold value for changing final control signal119906 in the above rules These laws utilize appropriate last levelcontrol signal (if error becomes less than specified value thatis called 120576) for using at next level Indeed if error at time 119896 and119896 minus 1 becomes less than 120576 and also error at time 119896 is greaterthan the previous time the controller applies negative valueof previous control signal for current time

Regarding adaptive fuzzy controllers [5 7] that esti-mate unknown nonlinear functions of the system by fuzzyalgorithm and when new reference input is applied to thesystem this estimation becomes better But the sensitivity ofproposed controller on estimation of nonlinear functions isless than that of AFCmethodThis is because output error hassignificant role in control effort rather than significant role ofestimation in AFC

4 Lyapunov Stability for the Proposed Model-Free Adaptive Controller

Let the Lyapunov function candidate 119881 be given by [8]

119881 = 119903119879

(119896) 119903 (119896) +1

120572tr (119879119891

(119896) 119891

(119896))

+1

120573tr (119879119892

(119896) 119892

(119896))

(29)

(a) For the first region 119868 = 1At first we should obtain the difference of Lyapunov

function in 119896 and 119896 + 1 level So general Lyapunov functionΔ119881 = Δ119881

1

+ Δ1198812

+ Δ1198813

can be acquired as follows

Δ1198811

= 119903119879

(119896 + 1) 119903 (119896 + 1) minus 119903119879

(119896) 119903 (119896)

Δ1198812

=1

120572tr (119879119891

(119896 + 1) 119891

(119896 + 1) minus 119879

119891

(119896) 119891

(119896))

Δ1198813

=1

120573tr (119879119892

(119896 + 1) 119892

(119896 + 1) minus 119879

119892

(119896) 119892

(119896))

(30)

If we prove thatΔ119881 le 0 it will demonstrate that 119892

119891

and 119903are bounded and they are able to stabilize system Δ119881

1

Δ1198812

and Δ119881

3

are obtained as follows

Δ1198811

= minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896))119879

(119879

119891

+ 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ (119879

119891

)119879

119879

119891

+ (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ (119892119906119889

)119879

119892119906119889

+ 1205761015840119879

1205761015840

+ 2119891

(119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ 2(119879

119892

119906119888

)119879

(119892119906119889

+ 1205761015840

) + 2(119892119906119889

)119879

1205761015840

Δ1198812

= minus (2 minus 120572) (119879

119891

)119879

119879

119891

+ 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120572) 119891

(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

Δ1198813

= minus (2 minus 120573) (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ 120573(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120573) (119879

119892

119906119888

)119879

(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

(31)

With combination of the above three Lyapunov functions(32) is established

Δ119881 = minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896) minus (1 minus 120572) (119879

119891

)119879

119879

119891

+ 2(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

Abstract and Applied Analysis 5

+ 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ 2120572119891

(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

minus (1 minus 120573) (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ 120573(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

+ 2120573(119879

119892

119906119888

)119879

(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

(32)

By simplifying (32) and using the below definition

120578 = 120572 + 120573

10038171003817100381710038171199061198881003817100381710038171003817

2

119868 = 1

120572 119868 = 0(33)

Inequality (34) is acquired as follows

Δ119881 le minus119903119879

(119896) (119868 minus (1 + 120578) 119896119879

119907

119896119907

) 119903 (119896)

+ 2120578(119896119907

119903 (119896))119879

(119892119906119889

+ 1205761015840

)

+ (1 + 120578) (119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

+120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

(34)

with assuming that 119896119907max is the maximum value of 119896

119907

and byresimplifying (34) inequality (35) is stated as given below

Δ119881 le minus (119868 minus 1198861

1198962

119907max) 119903 (119896)2 + 21198862119896119907max 119903 (119896) (119892119906119889 + 1205761015840)+ 1198861

(119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(35)

where 1198861

and 1198862

are described as

1198861

= 1 + 120578 +120578

1 minus 120578 119886

2

= 120578 +120578

1 minus 120578 (36)

By considering Lemma A4 in Appendix A

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817=10038171003817100381710038171003817119892119906119889

+ 1205761015840

10038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896)

(37)

And with substituting inequality (37) in (35)

Δ119881 le minus (119868 minus 1198861

1198962

119907max) 119903 (119896)2+ 21198862

119896119907max 119903 (119896) (1198880 + 1198881 119903 (119896))

+ 1198861

(1198880

+ 1198881

119903 (119896))119879

(1198880

+ 1198881

119903 (119896))

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(38)

With simplifying the above inequality the below equation isobtained

Δ119881 le minus (119868 minus 1198863

) 119903 (119896)2

+ 21198864

119903 (119896) + 1198865

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(39)

where 1198863

1198864

and 1198865

are defined as

1198863

= 1198861

1198962

119907max + 211988621198881119896119907max + 119886111988821

1198864

= 1198862

1198880

119896119907max + 119886111988801198881

1198865

= 1198861

1198882

0

(40)

By assuming that 120578 lt 1 the below statement becomesnegative

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

lt 0

(41)

As regards 120572 gt 0 120573 gt 0 120578 becomes positive too andfinally 119886

1

1198862

1198863

1198864

1198865

gt 0 totally the first term in (39) minus(119868 minus1198863

)119903(119896)2

+ 21198864

119903(119896) + 1198865

becomes nonpositive if it obeysthe below condition

119903 (119896) gt 1205751199031

(42)

in which 1205751199031

is acquired from the below inequality

100381710038171003817100381712057511990311003817100381710038171003817 gt

1198865

+ radic1198862

5

+ 1198866

(1 minus 1198864

)

(1 minus 1198864

) (43)

Thus both terms of inequality (39) have nonpositive valuesand finally Δ119881 le 0

(b) For the second region 119868 = 0

6 Abstract and Applied Analysis

In this region the proposed Lyapunov functions Δ1198811

Δ1198812

and Δ1198813

are obtained like (30) and they are stated asfollows

Δ1198811

= minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896))119879

(119879

119891

+ 119892119906119889

+ 1205761015840

) + (119879

119891

)119879

119879

119891

+ (119892119906119889

)119879

119892119906119889

+ 1205761015840119879

1205761015840

+ 2119891

(119892119906119889

+ 1205761015840

) + 2(119892119906119889

)119879

1205761015840

Δ1198812

= minus (2 minus 120572) (119879

119891

)119879

119879

119891

+ 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120572) 119891

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

Δ1198813

= 0

(44)

By combination of the above three equations (and definitionof 120578) the statement is demonstrated below

Δ119881 = minus 119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

+1

1 minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(45)

with the consideration of Lemma A41003817100381710038171003817119892119906 minus 119892119906119862

1003817100381710038171003817 le 1198890 + 1198891 119903 (119896) (46)

By substituting inequality (46) in (45)

Δ119881 le minus (119868 minus 1198870

) 119903 (119896)2

+ 21198871

119903 (119896) + 1198872

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2 (47)

in which 1198870

1198871

and 1198872

are as follows

1198870

= 1198962

119907max + 21198891 (1198891 + 119896119907max) + (1198891 + 119896119907max)21 minus 120572

1198871

= 2 (1198890

+ 1205761015840

) 1198891

+4 minus 2120572

1 minus 120572(1198890

+ 1205761015840

) (1198891

+ 119896119907max)

1198872

=3 minus 2120572

1 minus 120572(1198890

+ 1205761015840

)2

(48)

By the assumption that 120578 lt 1 the statement below becomesnegative

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

lt 0 (49)

As regards120572 gt 0 120573 gt 0 120578 becomes positive too andfinally 1198870

1198871

1198872

gt 0The first term in (39) minus(119868minus1198870

)119903(119896)2

+21198871

119903(119896)+

1198872

becomes nonpositive if it obeys the below condition

119903 (119896) gt 1205751199032

(50)

in which 1205751199032

is obtained from the relation below

100381710038171003817100381712057511990321003817100381710038171003817 gt

1198871

+ radic1198872

1

+ 1198872

(1 minus 1198870

)

(1 minus 1198870

) (51)

As a result both terms of inequality (47) have nonpositivevalues which means Δ119881 le 0 And finally if ||119903(119896)|| gt

max(1205751199031

1205751199032

) Lyapunov function becomes negative

5 Implementation of MFA Controller

Example 4 Consider MIMO system with 2 input-2 outputsystem state equations which are given below [8]

1

(119905) = 1199092

(119905) + (1 + 1199092

(119905)2

) 1199061

(119905) + 1198891

(119905)

2

(119905) = minus 1199091

(119905) minus (01 minus exp (minus1199091

(119905)2

minus 1199092

(119905)2

)) 1199092

(119905)

+ (1 + 1199091

(119905)2

) 1199062

(119905) + 1198892

(119905)

1199101

(119905) = 1199091

(119905) + 1198991

(119905) 1199102

(119905) = 1199092

(119905) + 1198992

(119905)

119899 (119905) = measurement noise

1198891

(119905) 1198892

(119905) = disturbance(52)

Select control parameters as follows

120583 = 10 120572 = 1 120573 = 1 119904 = 2 120574 = 05

119892 = 8 119896119907

= [10 0 0 10] 1205821

= 001 120576 = 01

120578 = 0 119891

= [1 1] 119892

= [1 0 0 1]

(53)

Simulation results of MIMO system without disturbances1198891

(119905) 1198892

(119905) = 0 are demonstrated in Figure 1 In Figure 2external disturbances are applied to the system with power1198891

(119905) = 04 1198892

(119905) = 04 and finally the system with existenceof white Gaussian measurement noise (power 00001) andexternal disturbance is shown in Figure 3 also desired inputsare assumed 119910

1119889

= 11199102119889

= cos(2120587119905)

The aim of using this MIMO system which is broughtby [8] (along with two control signals) is surveying couplingeffect in multiinput multioutput systems As it is understoodfrom the result of Figure 1 caused by MFA controller and theresult of AFC in [8] it can be said that system outputs ofMFA controller do not have any overshoot in its responsebut in AFC this problem is observed As it is seen fromFigure 2 MFA controller rejects disturbance without anynoticeable effort in control signals and in the outputs Alsoit can be deduced from Figure 3 that the measurement noisehas intense effect on system outputs and control signals

Abstract and Applied Analysis 7

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

Time (s)

System output 1Desired output 1

Out

put1

(a)

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

Time (s)

System output 2Desired output 2

Out

put2

(b)

0 05 1 15 2 25 3 35 4 45 5minus10010

20

30

40

50

60

70

Time (s)

Cont

rol p

ower

1

(c)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20020

40

60

80

(d)

Figure 1 Simulation results for MIMO system without disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs1 and 2

(approximately became twice) but it should be noted that theproposed controller is able to track desired output and yetremains stable As it is demonstrated in Figures 1 2 and 3the disadvantage of this controller is oscillating control signalProfits of MFAC in addition of good (also rapid) trackingand system stabilizing are decreasing settling time of outputresponse

Example 5 The second system under study is Maglev trainwhich is shown in Figure 4 Using the notations given inFigure 4 the vertical dynamics is described by [9]

System state equations are described as follows

1198981198892

119911 (119905)

1198891199052= minus119865 (119894 119911 119905) + 119891

119889

+ 119898119892

= minus1205830

1198732

119886119898

4[119894 (119905)

119911 (119905)]

2

+ 119891119889

+ 119898119892

119889119894 (119905)

119889119905=119894 (119905)

119911 (119905)

119889119911 (119905)

119889119905

minus2

1205830

1198732119886119898

119911 (119905) (119877119898

119894 (119905) minus 119906 (119905))

(54)

Here 119911 is the distance between rail and train 119894 is the current ofwindings 119898 is the train mass 119891

119889

is the force of disturbance1205830

is the permeability coefficient which equals 4120587 lowast 10minus7 119873the number of turns in winding 119886

119898

is the section surface ofwindings and 119877

119898

is the wire resistance in windings

Consider the state vector as below thus the aboveequations convert to new state equations

1

(119905) = 1199092

(119905)

2

(119905) = minus1205830

1198732

119886119898

4119898[1199093

(119905)

1199091

(119905)]

2

+1

119898119908 + 119892

3

(119905) = minus2119877119898

1205830

1198732119886119898

1199093

(119905) 1199091

(119905) +1199092

(119905) 1199093

(119905)

1199091

(119905)

+21199091

(119905) 119906 (119905)

1205830

1198732119886119898

119910 (119905) = 1199092

(119905) + 119899 (119905)

119899 (119905) = measurement noise119908 (119905) = external disturbance

(55)

8 Abstract and Applied Analysis

Time (s)

Out

put 1

System output 1Desired output 1

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

(c)

0

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20

(d)

Figure 2 Simulation results for MIMO system with disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs 1and 2

Maglev trains work in one of two ways both methods arebased on the same concept but involve different approaches

(1) Electromagnetic Suspension is based on magneticattraction it is very complex and somewhat unstable

(2) Electrodynamic Suspension is based on the repulsionofmagnetsThemagnetic levitation force balances theweight of the car at a stable position Controlling EMSis more difficult than EDS train because normality ofthis dynamic state is unstable

Magnetic trains have two important issues levitationand propulsion the target of controller is first goes uptrain to desired level along with guarantee stabilizing againstsome uncertainty such as wind and changing train mass inboltroads And second adjust train speed at the workingfrequency of magnets In this paper just levitation part(important section of train in controlling) is discussed Thetrain in this example goes from 10mm to 16mm level By

considering these assumption values for train and controlleras follows

119898 = 15 kg 119873 = 280 turns 119886119898

= 1024m2

119877119898

= 11Ω 120583 = 32 120572 = 1 120573 = 1

119904 = 140 120574 = 01 119892 = 9 119896119907

= 1 1205821

= 0

120576 = 0001 120578 = 1 119891

= minus 4 119892

= 1

(56)

The result of simulating without considering disturbancehas been displayed in Figure 5 in Figure 6 input step asexternal disturbance with amplitude 119889(119905) = 1 is appliedto the system for Figure 7 sinusoidal measurement noisewith amplitude 1mm in presence of disturbance is con-sidered Also white Gaussian noise in existence of externaldisturbance is applied to the system with power 8 times 10minus9(it is considered small because 119868119874 amplitude is small)which is shown in Figure 8 The purpose of presentingthis system is studying on sort of complicated practicalsystem dynamics such as Maglev in which they have some

Abstract and Applied Analysis 9

Time (s)

Out

put 1

System output 1Desired output 1

0 051 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

0102030405060

Time (s)

Cont

rol p

ower

1

0 05 1 15 2 25 3 35 4 45 5minus30minus20minus10

(c)

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5

minus20minus40minus60minus80(d)

Figure 3 Simulation results for MIMO system with the measurement noise and disturbance (a) (b) system outputs 1 and 2 (c) (d) controlsignals 119906 relative to inputs 1 and 2

Figure 4 Vertical slice of magnetic levitation and rail in a magnetic train [9]

nonlinear terms like existence of first state in denominator(that affect directly the destabilizing system) and so forthAs it is demonstrated in Figure 5(a) the system outputthat is distance between rail and train has no overshootafter about 25 second it goes from 10 to 16mm smoothlySuch as the last example it is acquired from Figure 6 thatdisturbance did not have any effect on the output control

effort and settling time In Figure 7 it is illustrated thatsinusoidal measurement noise with no changes appears intooutput but no noticeable alternation in control signal hasbeen seen And finally in Figure 8 an impalpable effectof white Gaussian measurement noise on output and con-trol signal can be shown in this example some repeatedbenefits which are seen in Maglev system response are

10 Abstract and Applied AnalysisD

istan

ce (m

)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 5 Simulation result for Maglev without disturbance (inwhich it goes from 10 to 16mm) (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 6 Simulation result for Maglev with disturbance (a)Distance between train and rail (b) control signal

expressed such as appropriate system stabilizing good andrapid tracking disturbance rejection and small settlingtime

6 Conclusion

Model-free adaptive controller is an adaptive controller thatjust uses system outputs and does not require another systemstates (so does not need observer) and with this as model-free controller it performs good and also has some meritssuch as good stability appropriate tracking robust againstuncertainty disturbance rejection and good decoupling andthe biggest advantage of MFA controller is no requiringto system dynamic and even does not need to any priorexperiment about system

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 7 Simulation result for Maglev with sinuous measurementnoise at 15Hz and disturbance (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 8 Simulation result for Maglev train with white Gaussianmeasurement noise along with disturbance (a) Distance betweentrain and rail (b) control signal

Appendices

A Some RequirementsLemma A1 For each time 119896 119909(119896) is bounded by

119909 (119896) le 1198890

+ 1198891

119903 (119896) (A1)

which is 119903(119896) called ldquotracking errorrdquo and 1198890

1198891

are computablepositive constants

Abstract and Applied Analysis 11

Proof [see [8 Lemma 32]] Let 119909(119896) isin 119880 in which 119880 is acompact subset of 119877119899 Assume that ℎ(119909(119896)) isin 119862infin119880 that isℎ(119909(119896)) is a smooth function 119880 rarr 119877 so that the Taylorseries expansion of ℎ(119909(119896)) exists Then using the bound on119909(119896) express the function ℎ(119909(119896)) on a compact set as linearform of parameters Thus the upper bound for ℎ(119909(119896)) canbe obtained as

ℎ (119909 (119896)) le 1198880

+ 1198881

119903 (119896) (A2)

where 1198880

and 1198881

are constant matrices [8]

Remark A2 120576119892

120576119891

are estimation errors of 119892 119891 respectivelyand 119889 is external disturbance These errors are bounded andhave these definite bounds

10038171003817100381710038171003817120576119891

10038171003817100381710038171003817le 120576119873119891

10038171003817100381710038171003817120576119892

10038171003817100381710038171003817le 120576119873119892

119889 le 119889119872

(A3)

Remark A3 According to (27) if 119868 = 1 then 119906119888

le 119904 so 119906119888

is bounded and if 119868 = 0 then 119906119888

gt 119904 but by consideringthe mentioned condition at Remark 3 in Section 5 in whichif 119906119888

is greater than 1120574 it is assumed to be equal 1120574 Thus itcan be deduced that in both regions and in all of times 119906

119888

isbounded and its bounds are shown as follows

10038171003817100381710038171199061198881003817100381710038171003817 le

119904 119868 = 1

1

120574119868 = 0

(A4)Lemma A4 With regards to that the proposed controlalgorithm is divided into two regions 119868 = 0 and 119868 = 1 Wewill prove the following relationship which is established in eachregion

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896) 119868 = 1

1003817100381710038171003817119892119906 minus 1198921199061198621003817100381710038171003817 le 1198890 + 1198891 119903 (119896) 119868 = 0

(A5)

Proof (a) First region 119868 = 1 or 119906119888

le 119904 and 119892(119909) gt 119892In consideration of Lemma A1 it is demonstrated that

119892(119909) le 11988801

+ 11988811

119903(119896) where 11988801

11988811

are constant matrices

10038171003817100381710038171199061198881003817100381710038171003817 le 119904 997888rarr

10038171003817100381710038171199061199031003817100381710038171003817 le

120583

119892

10038171003817100381710038171199061198881003817100381710038171003817 997888rarr bounded 1003817100381710038171003817119906119903

1003817100381710038171003817 (A6)

By using the result of relation (A6) the following inequalityis obtained

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A7)

As regards 119906119888

le 119904 rarr 120574(119906119888

minus 119904) le 0

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(10038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198882

(A8)

By considering (A7)1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817

1003817100381710038171003817(119906 minus 119906119862)1003817100381710038171003817

le 1198882

(11988801

+ 11988811

119903 (119896)) = 11988802

+ 11988812

119903 (119896)

(A9)

Also by utilizing Remark 3 (A10) is stated as follows10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le10038171003817100381710038171003817120576119891

10038171003817100381710038171003817+10038171003817100381710038171003817120576119892

10038171003817100381710038171003817

10038171003817100381710038171199061198881003817100381710038171003817 + 119889

le 120576119873119891

+ 120576119873119892

+ 119889119872

= 120576119873

(A10)

By using (A9) and (A10) the following inequality is acquired10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 +10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le 11988802

+ 11988812

119903 (119896) + 120576119873

le 1198880

+ 1198881

119903 (119896)

(A11)

Thus the assumption is proved in the first region(b) Second region 119868 = 0 or 119906

119888

ge 119904 and 119892(119909) lt 119892 andthe mentioned condition for the proposed MFA controller inRemark 3 in Section 5 is stated as follows

if 119906119888

gt1

120574997904rArr 119906119888

=1

120574 (A12)

In consideration of Lemma A1 it is demonstrated that119892(119909) le 119888

01

+ 11988811

119903(119896) where 11988801

11988811

are constant matricesBy using the result of relation (A6) and the condition that isgiven above the following inequality is obtained

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890minus120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A13)

As regards 119906119888

gt 119904 rarr minus120574(119906119888

minus 119904) le 0

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(310038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198892

(A14)

By considering (B5)1003817100381710038171003817119892119906

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817 119906 le 1198892 (11988801 + 11988811 119903 (119896))

= 11988902

+ 11988912

119903 (119896)

10038171003817100381710038171198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921003817100381710038171003817

10038171003817100381710038171003817100381710038171003817

1

120574

10038171003817100381710038171003817100381710038171003817

le 1198893

(A15)

By using (A10) and (A15) the following inequality isobtained

1003817100381710038171003817119892119906 minus 1198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921199061003817100381710038171003817 +

10038171003817100381710038171198921199061198881003817100381710038171003817

le 11988901

+ 11988911

119903 (119896) + 1198893

le 1198890

+ 1198891

119903 (119896)

(A16)

Finally the assumption is proved in the second region Thusinequality (A5) is correct in both regions

B Obtain 119903(119896)

In consideration of error definition 119903(119896) in (21)

119903 (119896 + 1) = 119890119899

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

I+ 1205821

119890119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

II+ sdot sdot sdot + 120582

119899minus1

1198901

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

III

(B1)

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

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Abstract and Applied Analysis 3

119860120572

119860120573

are matrices that will be updated adaptively with thefollowing updating laws

120572

(119905) = minus119876minus1

sdot 119911 sdot 120585119879

120572

sdot es

120573

(119905) = minus119876minus1

sdot 119911 sdot 120585119879

120573

sdot es sdot 119906 (119905)(12)

and finally the control law is described as follows

119906 (119905) =1

120573 (119905) + 120573 (119909)

(minus [120572 (119905) + (119909)] + 119907 (119905)) (13)

22 Direct Adaptive Fuzzy Controller (DAFC) Control law119906(119905) in feedback linearization method can be obtained asfollows

119906 (119905) =1

120573 (119909)(minus120572 (119909) + 119907 (119905)) (14)

The goal is to estimate 119906(119905) with consideration of basic fuzzyfunction as seen below

(119905) = 119911119879

119906

sdot 119860lowast

119906

sdot 120585119906

+ 119889119906

(119909) (15)

where (119905) is the estimated signal and 119889119906

(119909) is the errorbetween the described 119906 by fuzzymethod and desired (appro-priate) 119906lowast this error is restricted by upper known bound|119889119906

(119909)| le 119863119906

(119909) since practically it is difficult to define119863119906

(119909) so its value is specified and adjusted in designingprocess repetition this adjustment procedure continues untilthe performance of controller indicates that 119863

119906

(119909) comesclose to proper value Finally updating law 119860

119906

(119905) is obtainedas follows

119906

(119905) = minus119876minus1

119906

sdot 119911119879

119906

sdot 120585119906

sdot es (16)

The initial value of 119860119906

(119905)may be considered as 119860119906

(0) = 0

Remark 1 Thismethod is suitable for minimum phase plantsand also has the following limitation for 120573(119909)

minusinfin lt 1205731

lt 120573 (119909) lt 1205732

lt infin (17)

3 Proposed Approach

In the previous section an adaptive fuzzy controller usingfuzzy basis functions is described for stabilizing the con-trolled system In our introducedmethod the same approachhas been employed but our method does not need to utilizefuzzy basis functions for estimating system parametersInstead three proposed rules based on output errors areconsidered to control the plant

Consider a system with general form

(119905) = 119891 (119909 (119905)) + 119892 (119909 (119905)) 119906 (119905) (18)

At first nonlinear functions 119891 and 119892 must be estimated insuch a way that controller does not rely on system dynamicIt should be noted that in this paper the system is consideredcontinues time but the output is sampled and these sampled

points are used in proposed controller The estimations ofsystem dynamic can be defined as

119891 (119896) = 119879

119891

(119896)

119892 (119896) = 119879

119892

(119896)

(19)

And the updating values of 119892

119891

are given by

119891

(119896 + 1) = 119891

(119896) + 120572119903119879

(119896 + 1)

119892

(119896 + 1) = 119892

(119896) + 120573119906119888

(119896) 119903119879

(119896 + 1) 119868 = 1

119892

(119896) 119868 = 0

(20)

where 120572119891

120572119892

are constant coefficients with respect to 119891

119892

It should also be noted that usually these parameters donot need to be changed for different system dynamics Thevalue of 119868 will be defined in (27) 119903(119905) is the so-called ldquofilteredtracking errorrdquo which is the sum of errors as defined below

119903 (119896) = 119890119899

(119896) + 1205821

119890119899minus1

(119896) + sdot sdot sdot + 120582119899minus1

1198902

(119896) + 120582119899

1198901

(119896)

1198901

(119896) = 119910 (119896) minus 119910119889

(119896)

1198902

(119896) = 119910 (119896) minus 119910119889

(119896 + 1)

119890119899

(119896) = 119910 (119896) minus 119910119889

(119896 + 119899 minus 1)

(21)

where 120582119894

119894 = 1 119899 are coefficients of the predictive errorAnd119910

119889

is the desired output For eliminating nonlinear termsof system dynamic feedback linearization method has beenexploited for this purpose 119906(119896) is chosen as follows

119906 (119896) = 119892minus1

(119896) (minus119891 (119896) + 119907 (119896)) (22)

where 119907(119896) is defined by

119907 (119896) = 119896119907

119903 (119896) + 119910119889

(119896 + 119899) minus

119901minus2

sum

119894=0

120582119894+1

119890119899minus119894

(119896) (23)

In (23) 119896119907

is the tracking error coefficient By choosingappropriate 119896

119907

119903(119896) is converged into zero and at last itcauses 119907(119896) to be restricted In other words if the bigger 119896

119907

is opted better performance of output and quick tendencyto the desired value are achieved while increasing the controlsignal 119906 it should be noted that if 119896

119907

is chosen very large itcan make the system unstable

As it is seen from (22) the problem of singularity occurswhen 119892 is near zero and the control signal can becomeunbounded To solve this problem separate controller com-prising two parts is proposed in the following form

119906 (119896)

=

119906119888

(119896) + 5 (119906119903

(119896) minus 119906119888

(119896)) exp (120574 (1003817100381710038171003817119906119888 (119896)1003817100381710038171003817 minus 119904))

119868 = 1

119906119903

(119896) minus 5 (119906119903

(119896) minus 119906119888

(119896)) exp (minus120574 (1003817100381710038171003817119906119888 (119896)1003817100381710038171003817 minus 119904))

119868 = 0

(24)

4 Abstract and Applied Analysis

In the above equation 119904 is a constant value that its amountimpacts on control signal 119906 119906

119888

119906119903

are defined as

119906119888

(119896) = 119892minus1

(119896) (minus119891 (119896) + 119907 (119896)) (25)

119906119903

(119896) = minus1205831003817100381710038171003817119906119888 (119896)

1003817100381710038171003817 sign(119903 (119896))

119892 (26)

where 120583 is robust control signal coefficient that the decreasingor increasing of this coefficient affects quantity of final controlsignal 119906 and also 119892 is a constant value that its small amountdecreases the effect of 119906

119903

on final control signal 119906 that is 119906119903

will become more effective factor for determining 119906 and 119868 isgiven by the equation below

119868 = 11003817100381710038171003817119892 (119896)

1003817100381710038171003817 gt 1198921003817100381710038171003817119906119888 (119896)

1003817100381710038171003817 lt 119904

0 otherwise(27)

Here 119892 is lower bound of 119892 Proper selection of 119892 prevents 119892to be singular

Remark 2 120574 120583 and 119904 are designing parameters with thefollowing conditions [5] 120574 lt ln(2119904) 120583 gt 0 119904 gt 0

Remark 3 If 119906119888

is greater than 1120574 it is assumed to be equalto 1120574

The major principles of proposed MFA controller areconstructed based on the three following experimental ruleswhich make controller produce appropriate control signal

119906 (119896) =

119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896)1003816100381610038161003816 le

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

minus119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896)1003816100381610038161003816 gt

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

minus120578119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 gt

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

(28)

where 120576 is threshold value for changing final control signal119906 in the above rules These laws utilize appropriate last levelcontrol signal (if error becomes less than specified value thatis called 120576) for using at next level Indeed if error at time 119896 and119896 minus 1 becomes less than 120576 and also error at time 119896 is greaterthan the previous time the controller applies negative valueof previous control signal for current time

Regarding adaptive fuzzy controllers [5 7] that esti-mate unknown nonlinear functions of the system by fuzzyalgorithm and when new reference input is applied to thesystem this estimation becomes better But the sensitivity ofproposed controller on estimation of nonlinear functions isless than that of AFCmethodThis is because output error hassignificant role in control effort rather than significant role ofestimation in AFC

4 Lyapunov Stability for the Proposed Model-Free Adaptive Controller

Let the Lyapunov function candidate 119881 be given by [8]

119881 = 119903119879

(119896) 119903 (119896) +1

120572tr (119879119891

(119896) 119891

(119896))

+1

120573tr (119879119892

(119896) 119892

(119896))

(29)

(a) For the first region 119868 = 1At first we should obtain the difference of Lyapunov

function in 119896 and 119896 + 1 level So general Lyapunov functionΔ119881 = Δ119881

1

+ Δ1198812

+ Δ1198813

can be acquired as follows

Δ1198811

= 119903119879

(119896 + 1) 119903 (119896 + 1) minus 119903119879

(119896) 119903 (119896)

Δ1198812

=1

120572tr (119879119891

(119896 + 1) 119891

(119896 + 1) minus 119879

119891

(119896) 119891

(119896))

Δ1198813

=1

120573tr (119879119892

(119896 + 1) 119892

(119896 + 1) minus 119879

119892

(119896) 119892

(119896))

(30)

If we prove thatΔ119881 le 0 it will demonstrate that 119892

119891

and 119903are bounded and they are able to stabilize system Δ119881

1

Δ1198812

and Δ119881

3

are obtained as follows

Δ1198811

= minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896))119879

(119879

119891

+ 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ (119879

119891

)119879

119879

119891

+ (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ (119892119906119889

)119879

119892119906119889

+ 1205761015840119879

1205761015840

+ 2119891

(119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ 2(119879

119892

119906119888

)119879

(119892119906119889

+ 1205761015840

) + 2(119892119906119889

)119879

1205761015840

Δ1198812

= minus (2 minus 120572) (119879

119891

)119879

119879

119891

+ 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120572) 119891

(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

Δ1198813

= minus (2 minus 120573) (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ 120573(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120573) (119879

119892

119906119888

)119879

(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

(31)

With combination of the above three Lyapunov functions(32) is established

Δ119881 = minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896) minus (1 minus 120572) (119879

119891

)119879

119879

119891

+ 2(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

Abstract and Applied Analysis 5

+ 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ 2120572119891

(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

minus (1 minus 120573) (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ 120573(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

+ 2120573(119879

119892

119906119888

)119879

(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

(32)

By simplifying (32) and using the below definition

120578 = 120572 + 120573

10038171003817100381710038171199061198881003817100381710038171003817

2

119868 = 1

120572 119868 = 0(33)

Inequality (34) is acquired as follows

Δ119881 le minus119903119879

(119896) (119868 minus (1 + 120578) 119896119879

119907

119896119907

) 119903 (119896)

+ 2120578(119896119907

119903 (119896))119879

(119892119906119889

+ 1205761015840

)

+ (1 + 120578) (119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

+120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

(34)

with assuming that 119896119907max is the maximum value of 119896

119907

and byresimplifying (34) inequality (35) is stated as given below

Δ119881 le minus (119868 minus 1198861

1198962

119907max) 119903 (119896)2 + 21198862119896119907max 119903 (119896) (119892119906119889 + 1205761015840)+ 1198861

(119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(35)

where 1198861

and 1198862

are described as

1198861

= 1 + 120578 +120578

1 minus 120578 119886

2

= 120578 +120578

1 minus 120578 (36)

By considering Lemma A4 in Appendix A

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817=10038171003817100381710038171003817119892119906119889

+ 1205761015840

10038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896)

(37)

And with substituting inequality (37) in (35)

Δ119881 le minus (119868 minus 1198861

1198962

119907max) 119903 (119896)2+ 21198862

119896119907max 119903 (119896) (1198880 + 1198881 119903 (119896))

+ 1198861

(1198880

+ 1198881

119903 (119896))119879

(1198880

+ 1198881

119903 (119896))

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(38)

With simplifying the above inequality the below equation isobtained

Δ119881 le minus (119868 minus 1198863

) 119903 (119896)2

+ 21198864

119903 (119896) + 1198865

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(39)

where 1198863

1198864

and 1198865

are defined as

1198863

= 1198861

1198962

119907max + 211988621198881119896119907max + 119886111988821

1198864

= 1198862

1198880

119896119907max + 119886111988801198881

1198865

= 1198861

1198882

0

(40)

By assuming that 120578 lt 1 the below statement becomesnegative

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

lt 0

(41)

As regards 120572 gt 0 120573 gt 0 120578 becomes positive too andfinally 119886

1

1198862

1198863

1198864

1198865

gt 0 totally the first term in (39) minus(119868 minus1198863

)119903(119896)2

+ 21198864

119903(119896) + 1198865

becomes nonpositive if it obeysthe below condition

119903 (119896) gt 1205751199031

(42)

in which 1205751199031

is acquired from the below inequality

100381710038171003817100381712057511990311003817100381710038171003817 gt

1198865

+ radic1198862

5

+ 1198866

(1 minus 1198864

)

(1 minus 1198864

) (43)

Thus both terms of inequality (39) have nonpositive valuesand finally Δ119881 le 0

(b) For the second region 119868 = 0

6 Abstract and Applied Analysis

In this region the proposed Lyapunov functions Δ1198811

Δ1198812

and Δ1198813

are obtained like (30) and they are stated asfollows

Δ1198811

= minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896))119879

(119879

119891

+ 119892119906119889

+ 1205761015840

) + (119879

119891

)119879

119879

119891

+ (119892119906119889

)119879

119892119906119889

+ 1205761015840119879

1205761015840

+ 2119891

(119892119906119889

+ 1205761015840

) + 2(119892119906119889

)119879

1205761015840

Δ1198812

= minus (2 minus 120572) (119879

119891

)119879

119879

119891

+ 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120572) 119891

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

Δ1198813

= 0

(44)

By combination of the above three equations (and definitionof 120578) the statement is demonstrated below

Δ119881 = minus 119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

+1

1 minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(45)

with the consideration of Lemma A41003817100381710038171003817119892119906 minus 119892119906119862

1003817100381710038171003817 le 1198890 + 1198891 119903 (119896) (46)

By substituting inequality (46) in (45)

Δ119881 le minus (119868 minus 1198870

) 119903 (119896)2

+ 21198871

119903 (119896) + 1198872

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2 (47)

in which 1198870

1198871

and 1198872

are as follows

1198870

= 1198962

119907max + 21198891 (1198891 + 119896119907max) + (1198891 + 119896119907max)21 minus 120572

1198871

= 2 (1198890

+ 1205761015840

) 1198891

+4 minus 2120572

1 minus 120572(1198890

+ 1205761015840

) (1198891

+ 119896119907max)

1198872

=3 minus 2120572

1 minus 120572(1198890

+ 1205761015840

)2

(48)

By the assumption that 120578 lt 1 the statement below becomesnegative

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

lt 0 (49)

As regards120572 gt 0 120573 gt 0 120578 becomes positive too andfinally 1198870

1198871

1198872

gt 0The first term in (39) minus(119868minus1198870

)119903(119896)2

+21198871

119903(119896)+

1198872

becomes nonpositive if it obeys the below condition

119903 (119896) gt 1205751199032

(50)

in which 1205751199032

is obtained from the relation below

100381710038171003817100381712057511990321003817100381710038171003817 gt

1198871

+ radic1198872

1

+ 1198872

(1 minus 1198870

)

(1 minus 1198870

) (51)

As a result both terms of inequality (47) have nonpositivevalues which means Δ119881 le 0 And finally if ||119903(119896)|| gt

max(1205751199031

1205751199032

) Lyapunov function becomes negative

5 Implementation of MFA Controller

Example 4 Consider MIMO system with 2 input-2 outputsystem state equations which are given below [8]

1

(119905) = 1199092

(119905) + (1 + 1199092

(119905)2

) 1199061

(119905) + 1198891

(119905)

2

(119905) = minus 1199091

(119905) minus (01 minus exp (minus1199091

(119905)2

minus 1199092

(119905)2

)) 1199092

(119905)

+ (1 + 1199091

(119905)2

) 1199062

(119905) + 1198892

(119905)

1199101

(119905) = 1199091

(119905) + 1198991

(119905) 1199102

(119905) = 1199092

(119905) + 1198992

(119905)

119899 (119905) = measurement noise

1198891

(119905) 1198892

(119905) = disturbance(52)

Select control parameters as follows

120583 = 10 120572 = 1 120573 = 1 119904 = 2 120574 = 05

119892 = 8 119896119907

= [10 0 0 10] 1205821

= 001 120576 = 01

120578 = 0 119891

= [1 1] 119892

= [1 0 0 1]

(53)

Simulation results of MIMO system without disturbances1198891

(119905) 1198892

(119905) = 0 are demonstrated in Figure 1 In Figure 2external disturbances are applied to the system with power1198891

(119905) = 04 1198892

(119905) = 04 and finally the system with existenceof white Gaussian measurement noise (power 00001) andexternal disturbance is shown in Figure 3 also desired inputsare assumed 119910

1119889

= 11199102119889

= cos(2120587119905)

The aim of using this MIMO system which is broughtby [8] (along with two control signals) is surveying couplingeffect in multiinput multioutput systems As it is understoodfrom the result of Figure 1 caused by MFA controller and theresult of AFC in [8] it can be said that system outputs ofMFA controller do not have any overshoot in its responsebut in AFC this problem is observed As it is seen fromFigure 2 MFA controller rejects disturbance without anynoticeable effort in control signals and in the outputs Alsoit can be deduced from Figure 3 that the measurement noisehas intense effect on system outputs and control signals

Abstract and Applied Analysis 7

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

Time (s)

System output 1Desired output 1

Out

put1

(a)

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

Time (s)

System output 2Desired output 2

Out

put2

(b)

0 05 1 15 2 25 3 35 4 45 5minus10010

20

30

40

50

60

70

Time (s)

Cont

rol p

ower

1

(c)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20020

40

60

80

(d)

Figure 1 Simulation results for MIMO system without disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs1 and 2

(approximately became twice) but it should be noted that theproposed controller is able to track desired output and yetremains stable As it is demonstrated in Figures 1 2 and 3the disadvantage of this controller is oscillating control signalProfits of MFAC in addition of good (also rapid) trackingand system stabilizing are decreasing settling time of outputresponse

Example 5 The second system under study is Maglev trainwhich is shown in Figure 4 Using the notations given inFigure 4 the vertical dynamics is described by [9]

System state equations are described as follows

1198981198892

119911 (119905)

1198891199052= minus119865 (119894 119911 119905) + 119891

119889

+ 119898119892

= minus1205830

1198732

119886119898

4[119894 (119905)

119911 (119905)]

2

+ 119891119889

+ 119898119892

119889119894 (119905)

119889119905=119894 (119905)

119911 (119905)

119889119911 (119905)

119889119905

minus2

1205830

1198732119886119898

119911 (119905) (119877119898

119894 (119905) minus 119906 (119905))

(54)

Here 119911 is the distance between rail and train 119894 is the current ofwindings 119898 is the train mass 119891

119889

is the force of disturbance1205830

is the permeability coefficient which equals 4120587 lowast 10minus7 119873the number of turns in winding 119886

119898

is the section surface ofwindings and 119877

119898

is the wire resistance in windings

Consider the state vector as below thus the aboveequations convert to new state equations

1

(119905) = 1199092

(119905)

2

(119905) = minus1205830

1198732

119886119898

4119898[1199093

(119905)

1199091

(119905)]

2

+1

119898119908 + 119892

3

(119905) = minus2119877119898

1205830

1198732119886119898

1199093

(119905) 1199091

(119905) +1199092

(119905) 1199093

(119905)

1199091

(119905)

+21199091

(119905) 119906 (119905)

1205830

1198732119886119898

119910 (119905) = 1199092

(119905) + 119899 (119905)

119899 (119905) = measurement noise119908 (119905) = external disturbance

(55)

8 Abstract and Applied Analysis

Time (s)

Out

put 1

System output 1Desired output 1

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

(c)

0

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20

(d)

Figure 2 Simulation results for MIMO system with disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs 1and 2

Maglev trains work in one of two ways both methods arebased on the same concept but involve different approaches

(1) Electromagnetic Suspension is based on magneticattraction it is very complex and somewhat unstable

(2) Electrodynamic Suspension is based on the repulsionofmagnetsThemagnetic levitation force balances theweight of the car at a stable position Controlling EMSis more difficult than EDS train because normality ofthis dynamic state is unstable

Magnetic trains have two important issues levitationand propulsion the target of controller is first goes uptrain to desired level along with guarantee stabilizing againstsome uncertainty such as wind and changing train mass inboltroads And second adjust train speed at the workingfrequency of magnets In this paper just levitation part(important section of train in controlling) is discussed Thetrain in this example goes from 10mm to 16mm level By

considering these assumption values for train and controlleras follows

119898 = 15 kg 119873 = 280 turns 119886119898

= 1024m2

119877119898

= 11Ω 120583 = 32 120572 = 1 120573 = 1

119904 = 140 120574 = 01 119892 = 9 119896119907

= 1 1205821

= 0

120576 = 0001 120578 = 1 119891

= minus 4 119892

= 1

(56)

The result of simulating without considering disturbancehas been displayed in Figure 5 in Figure 6 input step asexternal disturbance with amplitude 119889(119905) = 1 is appliedto the system for Figure 7 sinusoidal measurement noisewith amplitude 1mm in presence of disturbance is con-sidered Also white Gaussian noise in existence of externaldisturbance is applied to the system with power 8 times 10minus9(it is considered small because 119868119874 amplitude is small)which is shown in Figure 8 The purpose of presentingthis system is studying on sort of complicated practicalsystem dynamics such as Maglev in which they have some

Abstract and Applied Analysis 9

Time (s)

Out

put 1

System output 1Desired output 1

0 051 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

0102030405060

Time (s)

Cont

rol p

ower

1

0 05 1 15 2 25 3 35 4 45 5minus30minus20minus10

(c)

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5

minus20minus40minus60minus80(d)

Figure 3 Simulation results for MIMO system with the measurement noise and disturbance (a) (b) system outputs 1 and 2 (c) (d) controlsignals 119906 relative to inputs 1 and 2

Figure 4 Vertical slice of magnetic levitation and rail in a magnetic train [9]

nonlinear terms like existence of first state in denominator(that affect directly the destabilizing system) and so forthAs it is demonstrated in Figure 5(a) the system outputthat is distance between rail and train has no overshootafter about 25 second it goes from 10 to 16mm smoothlySuch as the last example it is acquired from Figure 6 thatdisturbance did not have any effect on the output control

effort and settling time In Figure 7 it is illustrated thatsinusoidal measurement noise with no changes appears intooutput but no noticeable alternation in control signal hasbeen seen And finally in Figure 8 an impalpable effectof white Gaussian measurement noise on output and con-trol signal can be shown in this example some repeatedbenefits which are seen in Maglev system response are

10 Abstract and Applied AnalysisD

istan

ce (m

)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 5 Simulation result for Maglev without disturbance (inwhich it goes from 10 to 16mm) (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 6 Simulation result for Maglev with disturbance (a)Distance between train and rail (b) control signal

expressed such as appropriate system stabilizing good andrapid tracking disturbance rejection and small settlingtime

6 Conclusion

Model-free adaptive controller is an adaptive controller thatjust uses system outputs and does not require another systemstates (so does not need observer) and with this as model-free controller it performs good and also has some meritssuch as good stability appropriate tracking robust againstuncertainty disturbance rejection and good decoupling andthe biggest advantage of MFA controller is no requiringto system dynamic and even does not need to any priorexperiment about system

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 7 Simulation result for Maglev with sinuous measurementnoise at 15Hz and disturbance (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 8 Simulation result for Maglev train with white Gaussianmeasurement noise along with disturbance (a) Distance betweentrain and rail (b) control signal

Appendices

A Some RequirementsLemma A1 For each time 119896 119909(119896) is bounded by

119909 (119896) le 1198890

+ 1198891

119903 (119896) (A1)

which is 119903(119896) called ldquotracking errorrdquo and 1198890

1198891

are computablepositive constants

Abstract and Applied Analysis 11

Proof [see [8 Lemma 32]] Let 119909(119896) isin 119880 in which 119880 is acompact subset of 119877119899 Assume that ℎ(119909(119896)) isin 119862infin119880 that isℎ(119909(119896)) is a smooth function 119880 rarr 119877 so that the Taylorseries expansion of ℎ(119909(119896)) exists Then using the bound on119909(119896) express the function ℎ(119909(119896)) on a compact set as linearform of parameters Thus the upper bound for ℎ(119909(119896)) canbe obtained as

ℎ (119909 (119896)) le 1198880

+ 1198881

119903 (119896) (A2)

where 1198880

and 1198881

are constant matrices [8]

Remark A2 120576119892

120576119891

are estimation errors of 119892 119891 respectivelyand 119889 is external disturbance These errors are bounded andhave these definite bounds

10038171003817100381710038171003817120576119891

10038171003817100381710038171003817le 120576119873119891

10038171003817100381710038171003817120576119892

10038171003817100381710038171003817le 120576119873119892

119889 le 119889119872

(A3)

Remark A3 According to (27) if 119868 = 1 then 119906119888

le 119904 so 119906119888

is bounded and if 119868 = 0 then 119906119888

gt 119904 but by consideringthe mentioned condition at Remark 3 in Section 5 in whichif 119906119888

is greater than 1120574 it is assumed to be equal 1120574 Thus itcan be deduced that in both regions and in all of times 119906

119888

isbounded and its bounds are shown as follows

10038171003817100381710038171199061198881003817100381710038171003817 le

119904 119868 = 1

1

120574119868 = 0

(A4)Lemma A4 With regards to that the proposed controlalgorithm is divided into two regions 119868 = 0 and 119868 = 1 Wewill prove the following relationship which is established in eachregion

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896) 119868 = 1

1003817100381710038171003817119892119906 minus 1198921199061198621003817100381710038171003817 le 1198890 + 1198891 119903 (119896) 119868 = 0

(A5)

Proof (a) First region 119868 = 1 or 119906119888

le 119904 and 119892(119909) gt 119892In consideration of Lemma A1 it is demonstrated that

119892(119909) le 11988801

+ 11988811

119903(119896) where 11988801

11988811

are constant matrices

10038171003817100381710038171199061198881003817100381710038171003817 le 119904 997888rarr

10038171003817100381710038171199061199031003817100381710038171003817 le

120583

119892

10038171003817100381710038171199061198881003817100381710038171003817 997888rarr bounded 1003817100381710038171003817119906119903

1003817100381710038171003817 (A6)

By using the result of relation (A6) the following inequalityis obtained

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A7)

As regards 119906119888

le 119904 rarr 120574(119906119888

minus 119904) le 0

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(10038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198882

(A8)

By considering (A7)1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817

1003817100381710038171003817(119906 minus 119906119862)1003817100381710038171003817

le 1198882

(11988801

+ 11988811

119903 (119896)) = 11988802

+ 11988812

119903 (119896)

(A9)

Also by utilizing Remark 3 (A10) is stated as follows10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le10038171003817100381710038171003817120576119891

10038171003817100381710038171003817+10038171003817100381710038171003817120576119892

10038171003817100381710038171003817

10038171003817100381710038171199061198881003817100381710038171003817 + 119889

le 120576119873119891

+ 120576119873119892

+ 119889119872

= 120576119873

(A10)

By using (A9) and (A10) the following inequality is acquired10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 +10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le 11988802

+ 11988812

119903 (119896) + 120576119873

le 1198880

+ 1198881

119903 (119896)

(A11)

Thus the assumption is proved in the first region(b) Second region 119868 = 0 or 119906

119888

ge 119904 and 119892(119909) lt 119892 andthe mentioned condition for the proposed MFA controller inRemark 3 in Section 5 is stated as follows

if 119906119888

gt1

120574997904rArr 119906119888

=1

120574 (A12)

In consideration of Lemma A1 it is demonstrated that119892(119909) le 119888

01

+ 11988811

119903(119896) where 11988801

11988811

are constant matricesBy using the result of relation (A6) and the condition that isgiven above the following inequality is obtained

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890minus120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A13)

As regards 119906119888

gt 119904 rarr minus120574(119906119888

minus 119904) le 0

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(310038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198892

(A14)

By considering (B5)1003817100381710038171003817119892119906

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817 119906 le 1198892 (11988801 + 11988811 119903 (119896))

= 11988902

+ 11988912

119903 (119896)

10038171003817100381710038171198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921003817100381710038171003817

10038171003817100381710038171003817100381710038171003817

1

120574

10038171003817100381710038171003817100381710038171003817

le 1198893

(A15)

By using (A10) and (A15) the following inequality isobtained

1003817100381710038171003817119892119906 minus 1198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921199061003817100381710038171003817 +

10038171003817100381710038171198921199061198881003817100381710038171003817

le 11988901

+ 11988911

119903 (119896) + 1198893

le 1198890

+ 1198891

119903 (119896)

(A16)

Finally the assumption is proved in the second region Thusinequality (A5) is correct in both regions

B Obtain 119903(119896)

In consideration of error definition 119903(119896) in (21)

119903 (119896 + 1) = 119890119899

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

I+ 1205821

119890119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

II+ sdot sdot sdot + 120582

119899minus1

1198901

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

III

(B1)

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

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4 Abstract and Applied Analysis

In the above equation 119904 is a constant value that its amountimpacts on control signal 119906 119906

119888

119906119903

are defined as

119906119888

(119896) = 119892minus1

(119896) (minus119891 (119896) + 119907 (119896)) (25)

119906119903

(119896) = minus1205831003817100381710038171003817119906119888 (119896)

1003817100381710038171003817 sign(119903 (119896))

119892 (26)

where 120583 is robust control signal coefficient that the decreasingor increasing of this coefficient affects quantity of final controlsignal 119906 and also 119892 is a constant value that its small amountdecreases the effect of 119906

119903

on final control signal 119906 that is 119906119903

will become more effective factor for determining 119906 and 119868 isgiven by the equation below

119868 = 11003817100381710038171003817119892 (119896)

1003817100381710038171003817 gt 1198921003817100381710038171003817119906119888 (119896)

1003817100381710038171003817 lt 119904

0 otherwise(27)

Here 119892 is lower bound of 119892 Proper selection of 119892 prevents 119892to be singular

Remark 2 120574 120583 and 119904 are designing parameters with thefollowing conditions [5] 120574 lt ln(2119904) 120583 gt 0 119904 gt 0

Remark 3 If 119906119888

is greater than 1120574 it is assumed to be equalto 1120574

The major principles of proposed MFA controller areconstructed based on the three following experimental ruleswhich make controller produce appropriate control signal

119906 (119896) =

119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896)1003816100381610038161003816 le

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

minus119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816 lt 120576

10038161003816100381610038161198901 (119896)1003816100381610038161003816 gt

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

minus120578119906 (119896 minus 1) if 10038161003816100381610038161198901 (119896)1003816100381610038161003816 gt

10038161003816100381610038161198901 (119896 minus 1)1003816100381610038161003816

(28)

where 120576 is threshold value for changing final control signal119906 in the above rules These laws utilize appropriate last levelcontrol signal (if error becomes less than specified value thatis called 120576) for using at next level Indeed if error at time 119896 and119896 minus 1 becomes less than 120576 and also error at time 119896 is greaterthan the previous time the controller applies negative valueof previous control signal for current time

Regarding adaptive fuzzy controllers [5 7] that esti-mate unknown nonlinear functions of the system by fuzzyalgorithm and when new reference input is applied to thesystem this estimation becomes better But the sensitivity ofproposed controller on estimation of nonlinear functions isless than that of AFCmethodThis is because output error hassignificant role in control effort rather than significant role ofestimation in AFC

4 Lyapunov Stability for the Proposed Model-Free Adaptive Controller

Let the Lyapunov function candidate 119881 be given by [8]

119881 = 119903119879

(119896) 119903 (119896) +1

120572tr (119879119891

(119896) 119891

(119896))

+1

120573tr (119879119892

(119896) 119892

(119896))

(29)

(a) For the first region 119868 = 1At first we should obtain the difference of Lyapunov

function in 119896 and 119896 + 1 level So general Lyapunov functionΔ119881 = Δ119881

1

+ Δ1198812

+ Δ1198813

can be acquired as follows

Δ1198811

= 119903119879

(119896 + 1) 119903 (119896 + 1) minus 119903119879

(119896) 119903 (119896)

Δ1198812

=1

120572tr (119879119891

(119896 + 1) 119891

(119896 + 1) minus 119879

119891

(119896) 119891

(119896))

Δ1198813

=1

120573tr (119879119892

(119896 + 1) 119892

(119896 + 1) minus 119879

119892

(119896) 119892

(119896))

(30)

If we prove thatΔ119881 le 0 it will demonstrate that 119892

119891

and 119903are bounded and they are able to stabilize system Δ119881

1

Δ1198812

and Δ119881

3

are obtained as follows

Δ1198811

= minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896))119879

(119879

119891

+ 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ (119879

119891

)119879

119879

119891

+ (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ (119892119906119889

)119879

119892119906119889

+ 1205761015840119879

1205761015840

+ 2119891

(119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ 2(119879

119892

119906119888

)119879

(119892119906119889

+ 1205761015840

) + 2(119892119906119889

)119879

1205761015840

Δ1198812

= minus (2 minus 120572) (119879

119891

)119879

119879

119891

+ 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120572) 119891

(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

Δ1198813

= minus (2 minus 120573) (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ 120573(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120573) (119879

119892

119906119888

)119879

(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

(31)

With combination of the above three Lyapunov functions(32) is established

Δ119881 = minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896) minus (1 minus 120572) (119879

119891

)119879

119879

119891

+ 2(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

Abstract and Applied Analysis 5

+ 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ 2120572119891

(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

minus (1 minus 120573) (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ 120573(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

+ 2120573(119879

119892

119906119888

)119879

(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

(32)

By simplifying (32) and using the below definition

120578 = 120572 + 120573

10038171003817100381710038171199061198881003817100381710038171003817

2

119868 = 1

120572 119868 = 0(33)

Inequality (34) is acquired as follows

Δ119881 le minus119903119879

(119896) (119868 minus (1 + 120578) 119896119879

119907

119896119907

) 119903 (119896)

+ 2120578(119896119907

119903 (119896))119879

(119892119906119889

+ 1205761015840

)

+ (1 + 120578) (119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

+120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

(34)

with assuming that 119896119907max is the maximum value of 119896

119907

and byresimplifying (34) inequality (35) is stated as given below

Δ119881 le minus (119868 minus 1198861

1198962

119907max) 119903 (119896)2 + 21198862119896119907max 119903 (119896) (119892119906119889 + 1205761015840)+ 1198861

(119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(35)

where 1198861

and 1198862

are described as

1198861

= 1 + 120578 +120578

1 minus 120578 119886

2

= 120578 +120578

1 minus 120578 (36)

By considering Lemma A4 in Appendix A

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817=10038171003817100381710038171003817119892119906119889

+ 1205761015840

10038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896)

(37)

And with substituting inequality (37) in (35)

Δ119881 le minus (119868 minus 1198861

1198962

119907max) 119903 (119896)2+ 21198862

119896119907max 119903 (119896) (1198880 + 1198881 119903 (119896))

+ 1198861

(1198880

+ 1198881

119903 (119896))119879

(1198880

+ 1198881

119903 (119896))

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(38)

With simplifying the above inequality the below equation isobtained

Δ119881 le minus (119868 minus 1198863

) 119903 (119896)2

+ 21198864

119903 (119896) + 1198865

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(39)

where 1198863

1198864

and 1198865

are defined as

1198863

= 1198861

1198962

119907max + 211988621198881119896119907max + 119886111988821

1198864

= 1198862

1198880

119896119907max + 119886111988801198881

1198865

= 1198861

1198882

0

(40)

By assuming that 120578 lt 1 the below statement becomesnegative

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

lt 0

(41)

As regards 120572 gt 0 120573 gt 0 120578 becomes positive too andfinally 119886

1

1198862

1198863

1198864

1198865

gt 0 totally the first term in (39) minus(119868 minus1198863

)119903(119896)2

+ 21198864

119903(119896) + 1198865

becomes nonpositive if it obeysthe below condition

119903 (119896) gt 1205751199031

(42)

in which 1205751199031

is acquired from the below inequality

100381710038171003817100381712057511990311003817100381710038171003817 gt

1198865

+ radic1198862

5

+ 1198866

(1 minus 1198864

)

(1 minus 1198864

) (43)

Thus both terms of inequality (39) have nonpositive valuesand finally Δ119881 le 0

(b) For the second region 119868 = 0

6 Abstract and Applied Analysis

In this region the proposed Lyapunov functions Δ1198811

Δ1198812

and Δ1198813

are obtained like (30) and they are stated asfollows

Δ1198811

= minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896))119879

(119879

119891

+ 119892119906119889

+ 1205761015840

) + (119879

119891

)119879

119879

119891

+ (119892119906119889

)119879

119892119906119889

+ 1205761015840119879

1205761015840

+ 2119891

(119892119906119889

+ 1205761015840

) + 2(119892119906119889

)119879

1205761015840

Δ1198812

= minus (2 minus 120572) (119879

119891

)119879

119879

119891

+ 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120572) 119891

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

Δ1198813

= 0

(44)

By combination of the above three equations (and definitionof 120578) the statement is demonstrated below

Δ119881 = minus 119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

+1

1 minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(45)

with the consideration of Lemma A41003817100381710038171003817119892119906 minus 119892119906119862

1003817100381710038171003817 le 1198890 + 1198891 119903 (119896) (46)

By substituting inequality (46) in (45)

Δ119881 le minus (119868 minus 1198870

) 119903 (119896)2

+ 21198871

119903 (119896) + 1198872

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2 (47)

in which 1198870

1198871

and 1198872

are as follows

1198870

= 1198962

119907max + 21198891 (1198891 + 119896119907max) + (1198891 + 119896119907max)21 minus 120572

1198871

= 2 (1198890

+ 1205761015840

) 1198891

+4 minus 2120572

1 minus 120572(1198890

+ 1205761015840

) (1198891

+ 119896119907max)

1198872

=3 minus 2120572

1 minus 120572(1198890

+ 1205761015840

)2

(48)

By the assumption that 120578 lt 1 the statement below becomesnegative

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

lt 0 (49)

As regards120572 gt 0 120573 gt 0 120578 becomes positive too andfinally 1198870

1198871

1198872

gt 0The first term in (39) minus(119868minus1198870

)119903(119896)2

+21198871

119903(119896)+

1198872

becomes nonpositive if it obeys the below condition

119903 (119896) gt 1205751199032

(50)

in which 1205751199032

is obtained from the relation below

100381710038171003817100381712057511990321003817100381710038171003817 gt

1198871

+ radic1198872

1

+ 1198872

(1 minus 1198870

)

(1 minus 1198870

) (51)

As a result both terms of inequality (47) have nonpositivevalues which means Δ119881 le 0 And finally if ||119903(119896)|| gt

max(1205751199031

1205751199032

) Lyapunov function becomes negative

5 Implementation of MFA Controller

Example 4 Consider MIMO system with 2 input-2 outputsystem state equations which are given below [8]

1

(119905) = 1199092

(119905) + (1 + 1199092

(119905)2

) 1199061

(119905) + 1198891

(119905)

2

(119905) = minus 1199091

(119905) minus (01 minus exp (minus1199091

(119905)2

minus 1199092

(119905)2

)) 1199092

(119905)

+ (1 + 1199091

(119905)2

) 1199062

(119905) + 1198892

(119905)

1199101

(119905) = 1199091

(119905) + 1198991

(119905) 1199102

(119905) = 1199092

(119905) + 1198992

(119905)

119899 (119905) = measurement noise

1198891

(119905) 1198892

(119905) = disturbance(52)

Select control parameters as follows

120583 = 10 120572 = 1 120573 = 1 119904 = 2 120574 = 05

119892 = 8 119896119907

= [10 0 0 10] 1205821

= 001 120576 = 01

120578 = 0 119891

= [1 1] 119892

= [1 0 0 1]

(53)

Simulation results of MIMO system without disturbances1198891

(119905) 1198892

(119905) = 0 are demonstrated in Figure 1 In Figure 2external disturbances are applied to the system with power1198891

(119905) = 04 1198892

(119905) = 04 and finally the system with existenceof white Gaussian measurement noise (power 00001) andexternal disturbance is shown in Figure 3 also desired inputsare assumed 119910

1119889

= 11199102119889

= cos(2120587119905)

The aim of using this MIMO system which is broughtby [8] (along with two control signals) is surveying couplingeffect in multiinput multioutput systems As it is understoodfrom the result of Figure 1 caused by MFA controller and theresult of AFC in [8] it can be said that system outputs ofMFA controller do not have any overshoot in its responsebut in AFC this problem is observed As it is seen fromFigure 2 MFA controller rejects disturbance without anynoticeable effort in control signals and in the outputs Alsoit can be deduced from Figure 3 that the measurement noisehas intense effect on system outputs and control signals

Abstract and Applied Analysis 7

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

Time (s)

System output 1Desired output 1

Out

put1

(a)

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

Time (s)

System output 2Desired output 2

Out

put2

(b)

0 05 1 15 2 25 3 35 4 45 5minus10010

20

30

40

50

60

70

Time (s)

Cont

rol p

ower

1

(c)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20020

40

60

80

(d)

Figure 1 Simulation results for MIMO system without disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs1 and 2

(approximately became twice) but it should be noted that theproposed controller is able to track desired output and yetremains stable As it is demonstrated in Figures 1 2 and 3the disadvantage of this controller is oscillating control signalProfits of MFAC in addition of good (also rapid) trackingand system stabilizing are decreasing settling time of outputresponse

Example 5 The second system under study is Maglev trainwhich is shown in Figure 4 Using the notations given inFigure 4 the vertical dynamics is described by [9]

System state equations are described as follows

1198981198892

119911 (119905)

1198891199052= minus119865 (119894 119911 119905) + 119891

119889

+ 119898119892

= minus1205830

1198732

119886119898

4[119894 (119905)

119911 (119905)]

2

+ 119891119889

+ 119898119892

119889119894 (119905)

119889119905=119894 (119905)

119911 (119905)

119889119911 (119905)

119889119905

minus2

1205830

1198732119886119898

119911 (119905) (119877119898

119894 (119905) minus 119906 (119905))

(54)

Here 119911 is the distance between rail and train 119894 is the current ofwindings 119898 is the train mass 119891

119889

is the force of disturbance1205830

is the permeability coefficient which equals 4120587 lowast 10minus7 119873the number of turns in winding 119886

119898

is the section surface ofwindings and 119877

119898

is the wire resistance in windings

Consider the state vector as below thus the aboveequations convert to new state equations

1

(119905) = 1199092

(119905)

2

(119905) = minus1205830

1198732

119886119898

4119898[1199093

(119905)

1199091

(119905)]

2

+1

119898119908 + 119892

3

(119905) = minus2119877119898

1205830

1198732119886119898

1199093

(119905) 1199091

(119905) +1199092

(119905) 1199093

(119905)

1199091

(119905)

+21199091

(119905) 119906 (119905)

1205830

1198732119886119898

119910 (119905) = 1199092

(119905) + 119899 (119905)

119899 (119905) = measurement noise119908 (119905) = external disturbance

(55)

8 Abstract and Applied Analysis

Time (s)

Out

put 1

System output 1Desired output 1

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

(c)

0

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20

(d)

Figure 2 Simulation results for MIMO system with disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs 1and 2

Maglev trains work in one of two ways both methods arebased on the same concept but involve different approaches

(1) Electromagnetic Suspension is based on magneticattraction it is very complex and somewhat unstable

(2) Electrodynamic Suspension is based on the repulsionofmagnetsThemagnetic levitation force balances theweight of the car at a stable position Controlling EMSis more difficult than EDS train because normality ofthis dynamic state is unstable

Magnetic trains have two important issues levitationand propulsion the target of controller is first goes uptrain to desired level along with guarantee stabilizing againstsome uncertainty such as wind and changing train mass inboltroads And second adjust train speed at the workingfrequency of magnets In this paper just levitation part(important section of train in controlling) is discussed Thetrain in this example goes from 10mm to 16mm level By

considering these assumption values for train and controlleras follows

119898 = 15 kg 119873 = 280 turns 119886119898

= 1024m2

119877119898

= 11Ω 120583 = 32 120572 = 1 120573 = 1

119904 = 140 120574 = 01 119892 = 9 119896119907

= 1 1205821

= 0

120576 = 0001 120578 = 1 119891

= minus 4 119892

= 1

(56)

The result of simulating without considering disturbancehas been displayed in Figure 5 in Figure 6 input step asexternal disturbance with amplitude 119889(119905) = 1 is appliedto the system for Figure 7 sinusoidal measurement noisewith amplitude 1mm in presence of disturbance is con-sidered Also white Gaussian noise in existence of externaldisturbance is applied to the system with power 8 times 10minus9(it is considered small because 119868119874 amplitude is small)which is shown in Figure 8 The purpose of presentingthis system is studying on sort of complicated practicalsystem dynamics such as Maglev in which they have some

Abstract and Applied Analysis 9

Time (s)

Out

put 1

System output 1Desired output 1

0 051 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

0102030405060

Time (s)

Cont

rol p

ower

1

0 05 1 15 2 25 3 35 4 45 5minus30minus20minus10

(c)

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5

minus20minus40minus60minus80(d)

Figure 3 Simulation results for MIMO system with the measurement noise and disturbance (a) (b) system outputs 1 and 2 (c) (d) controlsignals 119906 relative to inputs 1 and 2

Figure 4 Vertical slice of magnetic levitation and rail in a magnetic train [9]

nonlinear terms like existence of first state in denominator(that affect directly the destabilizing system) and so forthAs it is demonstrated in Figure 5(a) the system outputthat is distance between rail and train has no overshootafter about 25 second it goes from 10 to 16mm smoothlySuch as the last example it is acquired from Figure 6 thatdisturbance did not have any effect on the output control

effort and settling time In Figure 7 it is illustrated thatsinusoidal measurement noise with no changes appears intooutput but no noticeable alternation in control signal hasbeen seen And finally in Figure 8 an impalpable effectof white Gaussian measurement noise on output and con-trol signal can be shown in this example some repeatedbenefits which are seen in Maglev system response are

10 Abstract and Applied AnalysisD

istan

ce (m

)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 5 Simulation result for Maglev without disturbance (inwhich it goes from 10 to 16mm) (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 6 Simulation result for Maglev with disturbance (a)Distance between train and rail (b) control signal

expressed such as appropriate system stabilizing good andrapid tracking disturbance rejection and small settlingtime

6 Conclusion

Model-free adaptive controller is an adaptive controller thatjust uses system outputs and does not require another systemstates (so does not need observer) and with this as model-free controller it performs good and also has some meritssuch as good stability appropriate tracking robust againstuncertainty disturbance rejection and good decoupling andthe biggest advantage of MFA controller is no requiringto system dynamic and even does not need to any priorexperiment about system

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 7 Simulation result for Maglev with sinuous measurementnoise at 15Hz and disturbance (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 8 Simulation result for Maglev train with white Gaussianmeasurement noise along with disturbance (a) Distance betweentrain and rail (b) control signal

Appendices

A Some RequirementsLemma A1 For each time 119896 119909(119896) is bounded by

119909 (119896) le 1198890

+ 1198891

119903 (119896) (A1)

which is 119903(119896) called ldquotracking errorrdquo and 1198890

1198891

are computablepositive constants

Abstract and Applied Analysis 11

Proof [see [8 Lemma 32]] Let 119909(119896) isin 119880 in which 119880 is acompact subset of 119877119899 Assume that ℎ(119909(119896)) isin 119862infin119880 that isℎ(119909(119896)) is a smooth function 119880 rarr 119877 so that the Taylorseries expansion of ℎ(119909(119896)) exists Then using the bound on119909(119896) express the function ℎ(119909(119896)) on a compact set as linearform of parameters Thus the upper bound for ℎ(119909(119896)) canbe obtained as

ℎ (119909 (119896)) le 1198880

+ 1198881

119903 (119896) (A2)

where 1198880

and 1198881

are constant matrices [8]

Remark A2 120576119892

120576119891

are estimation errors of 119892 119891 respectivelyand 119889 is external disturbance These errors are bounded andhave these definite bounds

10038171003817100381710038171003817120576119891

10038171003817100381710038171003817le 120576119873119891

10038171003817100381710038171003817120576119892

10038171003817100381710038171003817le 120576119873119892

119889 le 119889119872

(A3)

Remark A3 According to (27) if 119868 = 1 then 119906119888

le 119904 so 119906119888

is bounded and if 119868 = 0 then 119906119888

gt 119904 but by consideringthe mentioned condition at Remark 3 in Section 5 in whichif 119906119888

is greater than 1120574 it is assumed to be equal 1120574 Thus itcan be deduced that in both regions and in all of times 119906

119888

isbounded and its bounds are shown as follows

10038171003817100381710038171199061198881003817100381710038171003817 le

119904 119868 = 1

1

120574119868 = 0

(A4)Lemma A4 With regards to that the proposed controlalgorithm is divided into two regions 119868 = 0 and 119868 = 1 Wewill prove the following relationship which is established in eachregion

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896) 119868 = 1

1003817100381710038171003817119892119906 minus 1198921199061198621003817100381710038171003817 le 1198890 + 1198891 119903 (119896) 119868 = 0

(A5)

Proof (a) First region 119868 = 1 or 119906119888

le 119904 and 119892(119909) gt 119892In consideration of Lemma A1 it is demonstrated that

119892(119909) le 11988801

+ 11988811

119903(119896) where 11988801

11988811

are constant matrices

10038171003817100381710038171199061198881003817100381710038171003817 le 119904 997888rarr

10038171003817100381710038171199061199031003817100381710038171003817 le

120583

119892

10038171003817100381710038171199061198881003817100381710038171003817 997888rarr bounded 1003817100381710038171003817119906119903

1003817100381710038171003817 (A6)

By using the result of relation (A6) the following inequalityis obtained

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A7)

As regards 119906119888

le 119904 rarr 120574(119906119888

minus 119904) le 0

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(10038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198882

(A8)

By considering (A7)1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817

1003817100381710038171003817(119906 minus 119906119862)1003817100381710038171003817

le 1198882

(11988801

+ 11988811

119903 (119896)) = 11988802

+ 11988812

119903 (119896)

(A9)

Also by utilizing Remark 3 (A10) is stated as follows10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le10038171003817100381710038171003817120576119891

10038171003817100381710038171003817+10038171003817100381710038171003817120576119892

10038171003817100381710038171003817

10038171003817100381710038171199061198881003817100381710038171003817 + 119889

le 120576119873119891

+ 120576119873119892

+ 119889119872

= 120576119873

(A10)

By using (A9) and (A10) the following inequality is acquired10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 +10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le 11988802

+ 11988812

119903 (119896) + 120576119873

le 1198880

+ 1198881

119903 (119896)

(A11)

Thus the assumption is proved in the first region(b) Second region 119868 = 0 or 119906

119888

ge 119904 and 119892(119909) lt 119892 andthe mentioned condition for the proposed MFA controller inRemark 3 in Section 5 is stated as follows

if 119906119888

gt1

120574997904rArr 119906119888

=1

120574 (A12)

In consideration of Lemma A1 it is demonstrated that119892(119909) le 119888

01

+ 11988811

119903(119896) where 11988801

11988811

are constant matricesBy using the result of relation (A6) and the condition that isgiven above the following inequality is obtained

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890minus120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A13)

As regards 119906119888

gt 119904 rarr minus120574(119906119888

minus 119904) le 0

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(310038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198892

(A14)

By considering (B5)1003817100381710038171003817119892119906

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817 119906 le 1198892 (11988801 + 11988811 119903 (119896))

= 11988902

+ 11988912

119903 (119896)

10038171003817100381710038171198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921003817100381710038171003817

10038171003817100381710038171003817100381710038171003817

1

120574

10038171003817100381710038171003817100381710038171003817

le 1198893

(A15)

By using (A10) and (A15) the following inequality isobtained

1003817100381710038171003817119892119906 minus 1198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921199061003817100381710038171003817 +

10038171003817100381710038171198921199061198881003817100381710038171003817

le 11988901

+ 11988911

119903 (119896) + 1198893

le 1198890

+ 1198891

119903 (119896)

(A16)

Finally the assumption is proved in the second region Thusinequality (A5) is correct in both regions

B Obtain 119903(119896)

In consideration of error definition 119903(119896) in (21)

119903 (119896 + 1) = 119890119899

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

I+ 1205821

119890119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

II+ sdot sdot sdot + 120582

119899minus1

1198901

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

III

(B1)

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

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Abstract and Applied Analysis 5

+ 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

+ 2120572119891

(119896119907

119903 (119896) + 119879

119892

119906119888

+ 119892119906119889

+ 1205761015840

)

minus (1 minus 120573) (119879

119892

119906119888

)119879

(119879

119892

119906119888

)

+ 120573(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)119879

times (119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

+ 2120573(119879

119892

119906119888

)119879

(119896119907

119903 (119896) + 119879

119891

+ 119892119906119889

+ 1205761015840

)

(32)

By simplifying (32) and using the below definition

120578 = 120572 + 120573

10038171003817100381710038171199061198881003817100381710038171003817

2

119868 = 1

120572 119868 = 0(33)

Inequality (34) is acquired as follows

Δ119881 le minus119903119879

(119896) (119868 minus (1 + 120578) 119896119879

119907

119896119907

) 119903 (119896)

+ 2120578(119896119907

119903 (119896))119879

(119892119906119889

+ 1205761015840

)

+ (1 + 120578) (119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

+120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

(34)

with assuming that 119896119907max is the maximum value of 119896

119907

and byresimplifying (34) inequality (35) is stated as given below

Δ119881 le minus (119868 minus 1198861

1198962

119907max) 119903 (119896)2 + 21198862119896119907max 119903 (119896) (119892119906119889 + 1205761015840)+ 1198861

(119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(35)

where 1198861

and 1198862

are described as

1198861

= 1 + 120578 +120578

1 minus 120578 119886

2

= 120578 +120578

1 minus 120578 (36)

By considering Lemma A4 in Appendix A

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817=10038171003817100381710038171003817119892119906119889

+ 1205761015840

10038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896)

(37)

And with substituting inequality (37) in (35)

Δ119881 le minus (119868 minus 1198861

1198962

119907max) 119903 (119896)2+ 21198862

119896119907max 119903 (119896) (1198880 + 1198881 119903 (119896))

+ 1198861

(1198880

+ 1198881

119903 (119896))119879

(1198880

+ 1198881

119903 (119896))

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(38)

With simplifying the above inequality the below equation isobtained

Δ119881 le minus (119868 minus 1198863

) 119903 (119896)2

+ 21198864

119903 (119896) + 1198865

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(39)

where 1198863

1198864

and 1198865

are defined as

1198863

= 1198861

1198962

119907max + 211988621198881119896119907max + 119886111988821

1198864

= 1198862

1198880

119896119907max + 119886111988801198881

1198865

= 1198861

1198882

0

(40)

By assuming that 120578 lt 1 the below statement becomesnegative

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

+ 119879

119892

119906119888

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

lt 0

(41)

As regards 120572 gt 0 120573 gt 0 120578 becomes positive too andfinally 119886

1

1198862

1198863

1198864

1198865

gt 0 totally the first term in (39) minus(119868 minus1198863

)119903(119896)2

+ 21198864

119903(119896) + 1198865

becomes nonpositive if it obeysthe below condition

119903 (119896) gt 1205751199031

(42)

in which 1205751199031

is acquired from the below inequality

100381710038171003817100381712057511990311003817100381710038171003817 gt

1198865

+ radic1198862

5

+ 1198866

(1 minus 1198864

)

(1 minus 1198864

) (43)

Thus both terms of inequality (39) have nonpositive valuesand finally Δ119881 le 0

(b) For the second region 119868 = 0

6 Abstract and Applied Analysis

In this region the proposed Lyapunov functions Δ1198811

Δ1198812

and Δ1198813

are obtained like (30) and they are stated asfollows

Δ1198811

= minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896))119879

(119879

119891

+ 119892119906119889

+ 1205761015840

) + (119879

119891

)119879

119879

119891

+ (119892119906119889

)119879

119892119906119889

+ 1205761015840119879

1205761015840

+ 2119891

(119892119906119889

+ 1205761015840

) + 2(119892119906119889

)119879

1205761015840

Δ1198812

= minus (2 minus 120572) (119879

119891

)119879

119879

119891

+ 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120572) 119891

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

Δ1198813

= 0

(44)

By combination of the above three equations (and definitionof 120578) the statement is demonstrated below

Δ119881 = minus 119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

+1

1 minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(45)

with the consideration of Lemma A41003817100381710038171003817119892119906 minus 119892119906119862

1003817100381710038171003817 le 1198890 + 1198891 119903 (119896) (46)

By substituting inequality (46) in (45)

Δ119881 le minus (119868 minus 1198870

) 119903 (119896)2

+ 21198871

119903 (119896) + 1198872

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2 (47)

in which 1198870

1198871

and 1198872

are as follows

1198870

= 1198962

119907max + 21198891 (1198891 + 119896119907max) + (1198891 + 119896119907max)21 minus 120572

1198871

= 2 (1198890

+ 1205761015840

) 1198891

+4 minus 2120572

1 minus 120572(1198890

+ 1205761015840

) (1198891

+ 119896119907max)

1198872

=3 minus 2120572

1 minus 120572(1198890

+ 1205761015840

)2

(48)

By the assumption that 120578 lt 1 the statement below becomesnegative

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

lt 0 (49)

As regards120572 gt 0 120573 gt 0 120578 becomes positive too andfinally 1198870

1198871

1198872

gt 0The first term in (39) minus(119868minus1198870

)119903(119896)2

+21198871

119903(119896)+

1198872

becomes nonpositive if it obeys the below condition

119903 (119896) gt 1205751199032

(50)

in which 1205751199032

is obtained from the relation below

100381710038171003817100381712057511990321003817100381710038171003817 gt

1198871

+ radic1198872

1

+ 1198872

(1 minus 1198870

)

(1 minus 1198870

) (51)

As a result both terms of inequality (47) have nonpositivevalues which means Δ119881 le 0 And finally if ||119903(119896)|| gt

max(1205751199031

1205751199032

) Lyapunov function becomes negative

5 Implementation of MFA Controller

Example 4 Consider MIMO system with 2 input-2 outputsystem state equations which are given below [8]

1

(119905) = 1199092

(119905) + (1 + 1199092

(119905)2

) 1199061

(119905) + 1198891

(119905)

2

(119905) = minus 1199091

(119905) minus (01 minus exp (minus1199091

(119905)2

minus 1199092

(119905)2

)) 1199092

(119905)

+ (1 + 1199091

(119905)2

) 1199062

(119905) + 1198892

(119905)

1199101

(119905) = 1199091

(119905) + 1198991

(119905) 1199102

(119905) = 1199092

(119905) + 1198992

(119905)

119899 (119905) = measurement noise

1198891

(119905) 1198892

(119905) = disturbance(52)

Select control parameters as follows

120583 = 10 120572 = 1 120573 = 1 119904 = 2 120574 = 05

119892 = 8 119896119907

= [10 0 0 10] 1205821

= 001 120576 = 01

120578 = 0 119891

= [1 1] 119892

= [1 0 0 1]

(53)

Simulation results of MIMO system without disturbances1198891

(119905) 1198892

(119905) = 0 are demonstrated in Figure 1 In Figure 2external disturbances are applied to the system with power1198891

(119905) = 04 1198892

(119905) = 04 and finally the system with existenceof white Gaussian measurement noise (power 00001) andexternal disturbance is shown in Figure 3 also desired inputsare assumed 119910

1119889

= 11199102119889

= cos(2120587119905)

The aim of using this MIMO system which is broughtby [8] (along with two control signals) is surveying couplingeffect in multiinput multioutput systems As it is understoodfrom the result of Figure 1 caused by MFA controller and theresult of AFC in [8] it can be said that system outputs ofMFA controller do not have any overshoot in its responsebut in AFC this problem is observed As it is seen fromFigure 2 MFA controller rejects disturbance without anynoticeable effort in control signals and in the outputs Alsoit can be deduced from Figure 3 that the measurement noisehas intense effect on system outputs and control signals

Abstract and Applied Analysis 7

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

Time (s)

System output 1Desired output 1

Out

put1

(a)

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

Time (s)

System output 2Desired output 2

Out

put2

(b)

0 05 1 15 2 25 3 35 4 45 5minus10010

20

30

40

50

60

70

Time (s)

Cont

rol p

ower

1

(c)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20020

40

60

80

(d)

Figure 1 Simulation results for MIMO system without disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs1 and 2

(approximately became twice) but it should be noted that theproposed controller is able to track desired output and yetremains stable As it is demonstrated in Figures 1 2 and 3the disadvantage of this controller is oscillating control signalProfits of MFAC in addition of good (also rapid) trackingand system stabilizing are decreasing settling time of outputresponse

Example 5 The second system under study is Maglev trainwhich is shown in Figure 4 Using the notations given inFigure 4 the vertical dynamics is described by [9]

System state equations are described as follows

1198981198892

119911 (119905)

1198891199052= minus119865 (119894 119911 119905) + 119891

119889

+ 119898119892

= minus1205830

1198732

119886119898

4[119894 (119905)

119911 (119905)]

2

+ 119891119889

+ 119898119892

119889119894 (119905)

119889119905=119894 (119905)

119911 (119905)

119889119911 (119905)

119889119905

minus2

1205830

1198732119886119898

119911 (119905) (119877119898

119894 (119905) minus 119906 (119905))

(54)

Here 119911 is the distance between rail and train 119894 is the current ofwindings 119898 is the train mass 119891

119889

is the force of disturbance1205830

is the permeability coefficient which equals 4120587 lowast 10minus7 119873the number of turns in winding 119886

119898

is the section surface ofwindings and 119877

119898

is the wire resistance in windings

Consider the state vector as below thus the aboveequations convert to new state equations

1

(119905) = 1199092

(119905)

2

(119905) = minus1205830

1198732

119886119898

4119898[1199093

(119905)

1199091

(119905)]

2

+1

119898119908 + 119892

3

(119905) = minus2119877119898

1205830

1198732119886119898

1199093

(119905) 1199091

(119905) +1199092

(119905) 1199093

(119905)

1199091

(119905)

+21199091

(119905) 119906 (119905)

1205830

1198732119886119898

119910 (119905) = 1199092

(119905) + 119899 (119905)

119899 (119905) = measurement noise119908 (119905) = external disturbance

(55)

8 Abstract and Applied Analysis

Time (s)

Out

put 1

System output 1Desired output 1

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

(c)

0

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20

(d)

Figure 2 Simulation results for MIMO system with disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs 1and 2

Maglev trains work in one of two ways both methods arebased on the same concept but involve different approaches

(1) Electromagnetic Suspension is based on magneticattraction it is very complex and somewhat unstable

(2) Electrodynamic Suspension is based on the repulsionofmagnetsThemagnetic levitation force balances theweight of the car at a stable position Controlling EMSis more difficult than EDS train because normality ofthis dynamic state is unstable

Magnetic trains have two important issues levitationand propulsion the target of controller is first goes uptrain to desired level along with guarantee stabilizing againstsome uncertainty such as wind and changing train mass inboltroads And second adjust train speed at the workingfrequency of magnets In this paper just levitation part(important section of train in controlling) is discussed Thetrain in this example goes from 10mm to 16mm level By

considering these assumption values for train and controlleras follows

119898 = 15 kg 119873 = 280 turns 119886119898

= 1024m2

119877119898

= 11Ω 120583 = 32 120572 = 1 120573 = 1

119904 = 140 120574 = 01 119892 = 9 119896119907

= 1 1205821

= 0

120576 = 0001 120578 = 1 119891

= minus 4 119892

= 1

(56)

The result of simulating without considering disturbancehas been displayed in Figure 5 in Figure 6 input step asexternal disturbance with amplitude 119889(119905) = 1 is appliedto the system for Figure 7 sinusoidal measurement noisewith amplitude 1mm in presence of disturbance is con-sidered Also white Gaussian noise in existence of externaldisturbance is applied to the system with power 8 times 10minus9(it is considered small because 119868119874 amplitude is small)which is shown in Figure 8 The purpose of presentingthis system is studying on sort of complicated practicalsystem dynamics such as Maglev in which they have some

Abstract and Applied Analysis 9

Time (s)

Out

put 1

System output 1Desired output 1

0 051 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

0102030405060

Time (s)

Cont

rol p

ower

1

0 05 1 15 2 25 3 35 4 45 5minus30minus20minus10

(c)

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5

minus20minus40minus60minus80(d)

Figure 3 Simulation results for MIMO system with the measurement noise and disturbance (a) (b) system outputs 1 and 2 (c) (d) controlsignals 119906 relative to inputs 1 and 2

Figure 4 Vertical slice of magnetic levitation and rail in a magnetic train [9]

nonlinear terms like existence of first state in denominator(that affect directly the destabilizing system) and so forthAs it is demonstrated in Figure 5(a) the system outputthat is distance between rail and train has no overshootafter about 25 second it goes from 10 to 16mm smoothlySuch as the last example it is acquired from Figure 6 thatdisturbance did not have any effect on the output control

effort and settling time In Figure 7 it is illustrated thatsinusoidal measurement noise with no changes appears intooutput but no noticeable alternation in control signal hasbeen seen And finally in Figure 8 an impalpable effectof white Gaussian measurement noise on output and con-trol signal can be shown in this example some repeatedbenefits which are seen in Maglev system response are

10 Abstract and Applied AnalysisD

istan

ce (m

)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 5 Simulation result for Maglev without disturbance (inwhich it goes from 10 to 16mm) (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 6 Simulation result for Maglev with disturbance (a)Distance between train and rail (b) control signal

expressed such as appropriate system stabilizing good andrapid tracking disturbance rejection and small settlingtime

6 Conclusion

Model-free adaptive controller is an adaptive controller thatjust uses system outputs and does not require another systemstates (so does not need observer) and with this as model-free controller it performs good and also has some meritssuch as good stability appropriate tracking robust againstuncertainty disturbance rejection and good decoupling andthe biggest advantage of MFA controller is no requiringto system dynamic and even does not need to any priorexperiment about system

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 7 Simulation result for Maglev with sinuous measurementnoise at 15Hz and disturbance (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 8 Simulation result for Maglev train with white Gaussianmeasurement noise along with disturbance (a) Distance betweentrain and rail (b) control signal

Appendices

A Some RequirementsLemma A1 For each time 119896 119909(119896) is bounded by

119909 (119896) le 1198890

+ 1198891

119903 (119896) (A1)

which is 119903(119896) called ldquotracking errorrdquo and 1198890

1198891

are computablepositive constants

Abstract and Applied Analysis 11

Proof [see [8 Lemma 32]] Let 119909(119896) isin 119880 in which 119880 is acompact subset of 119877119899 Assume that ℎ(119909(119896)) isin 119862infin119880 that isℎ(119909(119896)) is a smooth function 119880 rarr 119877 so that the Taylorseries expansion of ℎ(119909(119896)) exists Then using the bound on119909(119896) express the function ℎ(119909(119896)) on a compact set as linearform of parameters Thus the upper bound for ℎ(119909(119896)) canbe obtained as

ℎ (119909 (119896)) le 1198880

+ 1198881

119903 (119896) (A2)

where 1198880

and 1198881

are constant matrices [8]

Remark A2 120576119892

120576119891

are estimation errors of 119892 119891 respectivelyand 119889 is external disturbance These errors are bounded andhave these definite bounds

10038171003817100381710038171003817120576119891

10038171003817100381710038171003817le 120576119873119891

10038171003817100381710038171003817120576119892

10038171003817100381710038171003817le 120576119873119892

119889 le 119889119872

(A3)

Remark A3 According to (27) if 119868 = 1 then 119906119888

le 119904 so 119906119888

is bounded and if 119868 = 0 then 119906119888

gt 119904 but by consideringthe mentioned condition at Remark 3 in Section 5 in whichif 119906119888

is greater than 1120574 it is assumed to be equal 1120574 Thus itcan be deduced that in both regions and in all of times 119906

119888

isbounded and its bounds are shown as follows

10038171003817100381710038171199061198881003817100381710038171003817 le

119904 119868 = 1

1

120574119868 = 0

(A4)Lemma A4 With regards to that the proposed controlalgorithm is divided into two regions 119868 = 0 and 119868 = 1 Wewill prove the following relationship which is established in eachregion

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896) 119868 = 1

1003817100381710038171003817119892119906 minus 1198921199061198621003817100381710038171003817 le 1198890 + 1198891 119903 (119896) 119868 = 0

(A5)

Proof (a) First region 119868 = 1 or 119906119888

le 119904 and 119892(119909) gt 119892In consideration of Lemma A1 it is demonstrated that

119892(119909) le 11988801

+ 11988811

119903(119896) where 11988801

11988811

are constant matrices

10038171003817100381710038171199061198881003817100381710038171003817 le 119904 997888rarr

10038171003817100381710038171199061199031003817100381710038171003817 le

120583

119892

10038171003817100381710038171199061198881003817100381710038171003817 997888rarr bounded 1003817100381710038171003817119906119903

1003817100381710038171003817 (A6)

By using the result of relation (A6) the following inequalityis obtained

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A7)

As regards 119906119888

le 119904 rarr 120574(119906119888

minus 119904) le 0

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(10038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198882

(A8)

By considering (A7)1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817

1003817100381710038171003817(119906 minus 119906119862)1003817100381710038171003817

le 1198882

(11988801

+ 11988811

119903 (119896)) = 11988802

+ 11988812

119903 (119896)

(A9)

Also by utilizing Remark 3 (A10) is stated as follows10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le10038171003817100381710038171003817120576119891

10038171003817100381710038171003817+10038171003817100381710038171003817120576119892

10038171003817100381710038171003817

10038171003817100381710038171199061198881003817100381710038171003817 + 119889

le 120576119873119891

+ 120576119873119892

+ 119889119872

= 120576119873

(A10)

By using (A9) and (A10) the following inequality is acquired10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 +10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le 11988802

+ 11988812

119903 (119896) + 120576119873

le 1198880

+ 1198881

119903 (119896)

(A11)

Thus the assumption is proved in the first region(b) Second region 119868 = 0 or 119906

119888

ge 119904 and 119892(119909) lt 119892 andthe mentioned condition for the proposed MFA controller inRemark 3 in Section 5 is stated as follows

if 119906119888

gt1

120574997904rArr 119906119888

=1

120574 (A12)

In consideration of Lemma A1 it is demonstrated that119892(119909) le 119888

01

+ 11988811

119903(119896) where 11988801

11988811

are constant matricesBy using the result of relation (A6) and the condition that isgiven above the following inequality is obtained

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890minus120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A13)

As regards 119906119888

gt 119904 rarr minus120574(119906119888

minus 119904) le 0

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(310038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198892

(A14)

By considering (B5)1003817100381710038171003817119892119906

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817 119906 le 1198892 (11988801 + 11988811 119903 (119896))

= 11988902

+ 11988912

119903 (119896)

10038171003817100381710038171198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921003817100381710038171003817

10038171003817100381710038171003817100381710038171003817

1

120574

10038171003817100381710038171003817100381710038171003817

le 1198893

(A15)

By using (A10) and (A15) the following inequality isobtained

1003817100381710038171003817119892119906 minus 1198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921199061003817100381710038171003817 +

10038171003817100381710038171198921199061198881003817100381710038171003817

le 11988901

+ 11988911

119903 (119896) + 1198893

le 1198890

+ 1198891

119903 (119896)

(A16)

Finally the assumption is proved in the second region Thusinequality (A5) is correct in both regions

B Obtain 119903(119896)

In consideration of error definition 119903(119896) in (21)

119903 (119896 + 1) = 119890119899

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

I+ 1205821

119890119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

II+ sdot sdot sdot + 120582

119899minus1

1198901

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

III

(B1)

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

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Volume 2013

6 Abstract and Applied Analysis

In this region the proposed Lyapunov functions Δ1198811

Δ1198812

and Δ1198813

are obtained like (30) and they are stated asfollows

Δ1198811

= minus119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896))119879

(119879

119891

+ 119892119906119889

+ 1205761015840

) + (119879

119891

)119879

119879

119891

+ (119892119906119889

)119879

119892119906119889

+ 1205761015840119879

1205761015840

+ 2119891

(119892119906119889

+ 1205761015840

) + 2(119892119906119889

)119879

1205761015840

Δ1198812

= minus (2 minus 120572) (119879

119891

)119879

119879

119891

+ 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus 2 (1 minus 120572) 119891

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

Δ1198813

= 0

(44)

By combination of the above three equations (and definitionof 120578) the statement is demonstrated below

Δ119881 = minus 119903119879

(119896) (119868 minus 119896119879

119907

119896119907

) 119903 (119896)

+ 2(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119892119906119889

+ 1205761015840

)

+1

1 minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

(45)

with the consideration of Lemma A41003817100381710038171003817119892119906 minus 119892119906119862

1003817100381710038171003817 le 1198890 + 1198891 119903 (119896) (46)

By substituting inequality (46) in (45)

Δ119881 le minus (119868 minus 1198870

) 119903 (119896)2

+ 21198871

119903 (119896) + 1198872

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2 (47)

in which 1198870

1198871

and 1198872

are as follows

1198870

= 1198962

119907max + 21198891 (1198891 + 119896119907max) + (1198891 + 119896119907max)21 minus 120572

1198871

= 2 (1198890

+ 1205761015840

) 1198891

+4 minus 2120572

1 minus 120572(1198890

+ 1205761015840

) (1198891

+ 119896119907max)

1198872

=3 minus 2120572

1 minus 120572(1198890

+ 1205761015840

)2

(48)

By the assumption that 120578 lt 1 the statement below becomesnegative

minus (1 minus 120578)

10038171003817100381710038171003817100381710038171003817

119879

119891

minus120578

1 minus 120578(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)

10038171003817100381710038171003817100381710038171003817

2

lt 0 (49)

As regards120572 gt 0 120573 gt 0 120578 becomes positive too andfinally 1198870

1198871

1198872

gt 0The first term in (39) minus(119868minus1198870

)119903(119896)2

+21198871

119903(119896)+

1198872

becomes nonpositive if it obeys the below condition

119903 (119896) gt 1205751199032

(50)

in which 1205751199032

is obtained from the relation below

100381710038171003817100381712057511990321003817100381710038171003817 gt

1198871

+ radic1198872

1

+ 1198872

(1 minus 1198870

)

(1 minus 1198870

) (51)

As a result both terms of inequality (47) have nonpositivevalues which means Δ119881 le 0 And finally if ||119903(119896)|| gt

max(1205751199031

1205751199032

) Lyapunov function becomes negative

5 Implementation of MFA Controller

Example 4 Consider MIMO system with 2 input-2 outputsystem state equations which are given below [8]

1

(119905) = 1199092

(119905) + (1 + 1199092

(119905)2

) 1199061

(119905) + 1198891

(119905)

2

(119905) = minus 1199091

(119905) minus (01 minus exp (minus1199091

(119905)2

minus 1199092

(119905)2

)) 1199092

(119905)

+ (1 + 1199091

(119905)2

) 1199062

(119905) + 1198892

(119905)

1199101

(119905) = 1199091

(119905) + 1198991

(119905) 1199102

(119905) = 1199092

(119905) + 1198992

(119905)

119899 (119905) = measurement noise

1198891

(119905) 1198892

(119905) = disturbance(52)

Select control parameters as follows

120583 = 10 120572 = 1 120573 = 1 119904 = 2 120574 = 05

119892 = 8 119896119907

= [10 0 0 10] 1205821

= 001 120576 = 01

120578 = 0 119891

= [1 1] 119892

= [1 0 0 1]

(53)

Simulation results of MIMO system without disturbances1198891

(119905) 1198892

(119905) = 0 are demonstrated in Figure 1 In Figure 2external disturbances are applied to the system with power1198891

(119905) = 04 1198892

(119905) = 04 and finally the system with existenceof white Gaussian measurement noise (power 00001) andexternal disturbance is shown in Figure 3 also desired inputsare assumed 119910

1119889

= 11199102119889

= cos(2120587119905)

The aim of using this MIMO system which is broughtby [8] (along with two control signals) is surveying couplingeffect in multiinput multioutput systems As it is understoodfrom the result of Figure 1 caused by MFA controller and theresult of AFC in [8] it can be said that system outputs ofMFA controller do not have any overshoot in its responsebut in AFC this problem is observed As it is seen fromFigure 2 MFA controller rejects disturbance without anynoticeable effort in control signals and in the outputs Alsoit can be deduced from Figure 3 that the measurement noisehas intense effect on system outputs and control signals

Abstract and Applied Analysis 7

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

Time (s)

System output 1Desired output 1

Out

put1

(a)

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

Time (s)

System output 2Desired output 2

Out

put2

(b)

0 05 1 15 2 25 3 35 4 45 5minus10010

20

30

40

50

60

70

Time (s)

Cont

rol p

ower

1

(c)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20020

40

60

80

(d)

Figure 1 Simulation results for MIMO system without disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs1 and 2

(approximately became twice) but it should be noted that theproposed controller is able to track desired output and yetremains stable As it is demonstrated in Figures 1 2 and 3the disadvantage of this controller is oscillating control signalProfits of MFAC in addition of good (also rapid) trackingand system stabilizing are decreasing settling time of outputresponse

Example 5 The second system under study is Maglev trainwhich is shown in Figure 4 Using the notations given inFigure 4 the vertical dynamics is described by [9]

System state equations are described as follows

1198981198892

119911 (119905)

1198891199052= minus119865 (119894 119911 119905) + 119891

119889

+ 119898119892

= minus1205830

1198732

119886119898

4[119894 (119905)

119911 (119905)]

2

+ 119891119889

+ 119898119892

119889119894 (119905)

119889119905=119894 (119905)

119911 (119905)

119889119911 (119905)

119889119905

minus2

1205830

1198732119886119898

119911 (119905) (119877119898

119894 (119905) minus 119906 (119905))

(54)

Here 119911 is the distance between rail and train 119894 is the current ofwindings 119898 is the train mass 119891

119889

is the force of disturbance1205830

is the permeability coefficient which equals 4120587 lowast 10minus7 119873the number of turns in winding 119886

119898

is the section surface ofwindings and 119877

119898

is the wire resistance in windings

Consider the state vector as below thus the aboveequations convert to new state equations

1

(119905) = 1199092

(119905)

2

(119905) = minus1205830

1198732

119886119898

4119898[1199093

(119905)

1199091

(119905)]

2

+1

119898119908 + 119892

3

(119905) = minus2119877119898

1205830

1198732119886119898

1199093

(119905) 1199091

(119905) +1199092

(119905) 1199093

(119905)

1199091

(119905)

+21199091

(119905) 119906 (119905)

1205830

1198732119886119898

119910 (119905) = 1199092

(119905) + 119899 (119905)

119899 (119905) = measurement noise119908 (119905) = external disturbance

(55)

8 Abstract and Applied Analysis

Time (s)

Out

put 1

System output 1Desired output 1

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

(c)

0

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20

(d)

Figure 2 Simulation results for MIMO system with disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs 1and 2

Maglev trains work in one of two ways both methods arebased on the same concept but involve different approaches

(1) Electromagnetic Suspension is based on magneticattraction it is very complex and somewhat unstable

(2) Electrodynamic Suspension is based on the repulsionofmagnetsThemagnetic levitation force balances theweight of the car at a stable position Controlling EMSis more difficult than EDS train because normality ofthis dynamic state is unstable

Magnetic trains have two important issues levitationand propulsion the target of controller is first goes uptrain to desired level along with guarantee stabilizing againstsome uncertainty such as wind and changing train mass inboltroads And second adjust train speed at the workingfrequency of magnets In this paper just levitation part(important section of train in controlling) is discussed Thetrain in this example goes from 10mm to 16mm level By

considering these assumption values for train and controlleras follows

119898 = 15 kg 119873 = 280 turns 119886119898

= 1024m2

119877119898

= 11Ω 120583 = 32 120572 = 1 120573 = 1

119904 = 140 120574 = 01 119892 = 9 119896119907

= 1 1205821

= 0

120576 = 0001 120578 = 1 119891

= minus 4 119892

= 1

(56)

The result of simulating without considering disturbancehas been displayed in Figure 5 in Figure 6 input step asexternal disturbance with amplitude 119889(119905) = 1 is appliedto the system for Figure 7 sinusoidal measurement noisewith amplitude 1mm in presence of disturbance is con-sidered Also white Gaussian noise in existence of externaldisturbance is applied to the system with power 8 times 10minus9(it is considered small because 119868119874 amplitude is small)which is shown in Figure 8 The purpose of presentingthis system is studying on sort of complicated practicalsystem dynamics such as Maglev in which they have some

Abstract and Applied Analysis 9

Time (s)

Out

put 1

System output 1Desired output 1

0 051 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

0102030405060

Time (s)

Cont

rol p

ower

1

0 05 1 15 2 25 3 35 4 45 5minus30minus20minus10

(c)

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5

minus20minus40minus60minus80(d)

Figure 3 Simulation results for MIMO system with the measurement noise and disturbance (a) (b) system outputs 1 and 2 (c) (d) controlsignals 119906 relative to inputs 1 and 2

Figure 4 Vertical slice of magnetic levitation and rail in a magnetic train [9]

nonlinear terms like existence of first state in denominator(that affect directly the destabilizing system) and so forthAs it is demonstrated in Figure 5(a) the system outputthat is distance between rail and train has no overshootafter about 25 second it goes from 10 to 16mm smoothlySuch as the last example it is acquired from Figure 6 thatdisturbance did not have any effect on the output control

effort and settling time In Figure 7 it is illustrated thatsinusoidal measurement noise with no changes appears intooutput but no noticeable alternation in control signal hasbeen seen And finally in Figure 8 an impalpable effectof white Gaussian measurement noise on output and con-trol signal can be shown in this example some repeatedbenefits which are seen in Maglev system response are

10 Abstract and Applied AnalysisD

istan

ce (m

)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 5 Simulation result for Maglev without disturbance (inwhich it goes from 10 to 16mm) (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 6 Simulation result for Maglev with disturbance (a)Distance between train and rail (b) control signal

expressed such as appropriate system stabilizing good andrapid tracking disturbance rejection and small settlingtime

6 Conclusion

Model-free adaptive controller is an adaptive controller thatjust uses system outputs and does not require another systemstates (so does not need observer) and with this as model-free controller it performs good and also has some meritssuch as good stability appropriate tracking robust againstuncertainty disturbance rejection and good decoupling andthe biggest advantage of MFA controller is no requiringto system dynamic and even does not need to any priorexperiment about system

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 7 Simulation result for Maglev with sinuous measurementnoise at 15Hz and disturbance (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 8 Simulation result for Maglev train with white Gaussianmeasurement noise along with disturbance (a) Distance betweentrain and rail (b) control signal

Appendices

A Some RequirementsLemma A1 For each time 119896 119909(119896) is bounded by

119909 (119896) le 1198890

+ 1198891

119903 (119896) (A1)

which is 119903(119896) called ldquotracking errorrdquo and 1198890

1198891

are computablepositive constants

Abstract and Applied Analysis 11

Proof [see [8 Lemma 32]] Let 119909(119896) isin 119880 in which 119880 is acompact subset of 119877119899 Assume that ℎ(119909(119896)) isin 119862infin119880 that isℎ(119909(119896)) is a smooth function 119880 rarr 119877 so that the Taylorseries expansion of ℎ(119909(119896)) exists Then using the bound on119909(119896) express the function ℎ(119909(119896)) on a compact set as linearform of parameters Thus the upper bound for ℎ(119909(119896)) canbe obtained as

ℎ (119909 (119896)) le 1198880

+ 1198881

119903 (119896) (A2)

where 1198880

and 1198881

are constant matrices [8]

Remark A2 120576119892

120576119891

are estimation errors of 119892 119891 respectivelyand 119889 is external disturbance These errors are bounded andhave these definite bounds

10038171003817100381710038171003817120576119891

10038171003817100381710038171003817le 120576119873119891

10038171003817100381710038171003817120576119892

10038171003817100381710038171003817le 120576119873119892

119889 le 119889119872

(A3)

Remark A3 According to (27) if 119868 = 1 then 119906119888

le 119904 so 119906119888

is bounded and if 119868 = 0 then 119906119888

gt 119904 but by consideringthe mentioned condition at Remark 3 in Section 5 in whichif 119906119888

is greater than 1120574 it is assumed to be equal 1120574 Thus itcan be deduced that in both regions and in all of times 119906

119888

isbounded and its bounds are shown as follows

10038171003817100381710038171199061198881003817100381710038171003817 le

119904 119868 = 1

1

120574119868 = 0

(A4)Lemma A4 With regards to that the proposed controlalgorithm is divided into two regions 119868 = 0 and 119868 = 1 Wewill prove the following relationship which is established in eachregion

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896) 119868 = 1

1003817100381710038171003817119892119906 minus 1198921199061198621003817100381710038171003817 le 1198890 + 1198891 119903 (119896) 119868 = 0

(A5)

Proof (a) First region 119868 = 1 or 119906119888

le 119904 and 119892(119909) gt 119892In consideration of Lemma A1 it is demonstrated that

119892(119909) le 11988801

+ 11988811

119903(119896) where 11988801

11988811

are constant matrices

10038171003817100381710038171199061198881003817100381710038171003817 le 119904 997888rarr

10038171003817100381710038171199061199031003817100381710038171003817 le

120583

119892

10038171003817100381710038171199061198881003817100381710038171003817 997888rarr bounded 1003817100381710038171003817119906119903

1003817100381710038171003817 (A6)

By using the result of relation (A6) the following inequalityis obtained

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A7)

As regards 119906119888

le 119904 rarr 120574(119906119888

minus 119904) le 0

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(10038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198882

(A8)

By considering (A7)1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817

1003817100381710038171003817(119906 minus 119906119862)1003817100381710038171003817

le 1198882

(11988801

+ 11988811

119903 (119896)) = 11988802

+ 11988812

119903 (119896)

(A9)

Also by utilizing Remark 3 (A10) is stated as follows10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le10038171003817100381710038171003817120576119891

10038171003817100381710038171003817+10038171003817100381710038171003817120576119892

10038171003817100381710038171003817

10038171003817100381710038171199061198881003817100381710038171003817 + 119889

le 120576119873119891

+ 120576119873119892

+ 119889119872

= 120576119873

(A10)

By using (A9) and (A10) the following inequality is acquired10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 +10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le 11988802

+ 11988812

119903 (119896) + 120576119873

le 1198880

+ 1198881

119903 (119896)

(A11)

Thus the assumption is proved in the first region(b) Second region 119868 = 0 or 119906

119888

ge 119904 and 119892(119909) lt 119892 andthe mentioned condition for the proposed MFA controller inRemark 3 in Section 5 is stated as follows

if 119906119888

gt1

120574997904rArr 119906119888

=1

120574 (A12)

In consideration of Lemma A1 it is demonstrated that119892(119909) le 119888

01

+ 11988811

119903(119896) where 11988801

11988811

are constant matricesBy using the result of relation (A6) and the condition that isgiven above the following inequality is obtained

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890minus120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A13)

As regards 119906119888

gt 119904 rarr minus120574(119906119888

minus 119904) le 0

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(310038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198892

(A14)

By considering (B5)1003817100381710038171003817119892119906

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817 119906 le 1198892 (11988801 + 11988811 119903 (119896))

= 11988902

+ 11988912

119903 (119896)

10038171003817100381710038171198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921003817100381710038171003817

10038171003817100381710038171003817100381710038171003817

1

120574

10038171003817100381710038171003817100381710038171003817

le 1198893

(A15)

By using (A10) and (A15) the following inequality isobtained

1003817100381710038171003817119892119906 minus 1198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921199061003817100381710038171003817 +

10038171003817100381710038171198921199061198881003817100381710038171003817

le 11988901

+ 11988911

119903 (119896) + 1198893

le 1198890

+ 1198891

119903 (119896)

(A16)

Finally the assumption is proved in the second region Thusinequality (A5) is correct in both regions

B Obtain 119903(119896)

In consideration of error definition 119903(119896) in (21)

119903 (119896 + 1) = 119890119899

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

I+ 1205821

119890119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

II+ sdot sdot sdot + 120582

119899minus1

1198901

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

III

(B1)

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

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Abstract and Applied Analysis 7

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

Time (s)

System output 1Desired output 1

Out

put1

(a)

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

Time (s)

System output 2Desired output 2

Out

put2

(b)

0 05 1 15 2 25 3 35 4 45 5minus10010

20

30

40

50

60

70

Time (s)

Cont

rol p

ower

1

(c)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20020

40

60

80

(d)

Figure 1 Simulation results for MIMO system without disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs1 and 2

(approximately became twice) but it should be noted that theproposed controller is able to track desired output and yetremains stable As it is demonstrated in Figures 1 2 and 3the disadvantage of this controller is oscillating control signalProfits of MFAC in addition of good (also rapid) trackingand system stabilizing are decreasing settling time of outputresponse

Example 5 The second system under study is Maglev trainwhich is shown in Figure 4 Using the notations given inFigure 4 the vertical dynamics is described by [9]

System state equations are described as follows

1198981198892

119911 (119905)

1198891199052= minus119865 (119894 119911 119905) + 119891

119889

+ 119898119892

= minus1205830

1198732

119886119898

4[119894 (119905)

119911 (119905)]

2

+ 119891119889

+ 119898119892

119889119894 (119905)

119889119905=119894 (119905)

119911 (119905)

119889119911 (119905)

119889119905

minus2

1205830

1198732119886119898

119911 (119905) (119877119898

119894 (119905) minus 119906 (119905))

(54)

Here 119911 is the distance between rail and train 119894 is the current ofwindings 119898 is the train mass 119891

119889

is the force of disturbance1205830

is the permeability coefficient which equals 4120587 lowast 10minus7 119873the number of turns in winding 119886

119898

is the section surface ofwindings and 119877

119898

is the wire resistance in windings

Consider the state vector as below thus the aboveequations convert to new state equations

1

(119905) = 1199092

(119905)

2

(119905) = minus1205830

1198732

119886119898

4119898[1199093

(119905)

1199091

(119905)]

2

+1

119898119908 + 119892

3

(119905) = minus2119877119898

1205830

1198732119886119898

1199093

(119905) 1199091

(119905) +1199092

(119905) 1199093

(119905)

1199091

(119905)

+21199091

(119905) 119906 (119905)

1205830

1198732119886119898

119910 (119905) = 1199092

(119905) + 119899 (119905)

119899 (119905) = measurement noise119908 (119905) = external disturbance

(55)

8 Abstract and Applied Analysis

Time (s)

Out

put 1

System output 1Desired output 1

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

(c)

0

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20

(d)

Figure 2 Simulation results for MIMO system with disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs 1and 2

Maglev trains work in one of two ways both methods arebased on the same concept but involve different approaches

(1) Electromagnetic Suspension is based on magneticattraction it is very complex and somewhat unstable

(2) Electrodynamic Suspension is based on the repulsionofmagnetsThemagnetic levitation force balances theweight of the car at a stable position Controlling EMSis more difficult than EDS train because normality ofthis dynamic state is unstable

Magnetic trains have two important issues levitationand propulsion the target of controller is first goes uptrain to desired level along with guarantee stabilizing againstsome uncertainty such as wind and changing train mass inboltroads And second adjust train speed at the workingfrequency of magnets In this paper just levitation part(important section of train in controlling) is discussed Thetrain in this example goes from 10mm to 16mm level By

considering these assumption values for train and controlleras follows

119898 = 15 kg 119873 = 280 turns 119886119898

= 1024m2

119877119898

= 11Ω 120583 = 32 120572 = 1 120573 = 1

119904 = 140 120574 = 01 119892 = 9 119896119907

= 1 1205821

= 0

120576 = 0001 120578 = 1 119891

= minus 4 119892

= 1

(56)

The result of simulating without considering disturbancehas been displayed in Figure 5 in Figure 6 input step asexternal disturbance with amplitude 119889(119905) = 1 is appliedto the system for Figure 7 sinusoidal measurement noisewith amplitude 1mm in presence of disturbance is con-sidered Also white Gaussian noise in existence of externaldisturbance is applied to the system with power 8 times 10minus9(it is considered small because 119868119874 amplitude is small)which is shown in Figure 8 The purpose of presentingthis system is studying on sort of complicated practicalsystem dynamics such as Maglev in which they have some

Abstract and Applied Analysis 9

Time (s)

Out

put 1

System output 1Desired output 1

0 051 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

0102030405060

Time (s)

Cont

rol p

ower

1

0 05 1 15 2 25 3 35 4 45 5minus30minus20minus10

(c)

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5

minus20minus40minus60minus80(d)

Figure 3 Simulation results for MIMO system with the measurement noise and disturbance (a) (b) system outputs 1 and 2 (c) (d) controlsignals 119906 relative to inputs 1 and 2

Figure 4 Vertical slice of magnetic levitation and rail in a magnetic train [9]

nonlinear terms like existence of first state in denominator(that affect directly the destabilizing system) and so forthAs it is demonstrated in Figure 5(a) the system outputthat is distance between rail and train has no overshootafter about 25 second it goes from 10 to 16mm smoothlySuch as the last example it is acquired from Figure 6 thatdisturbance did not have any effect on the output control

effort and settling time In Figure 7 it is illustrated thatsinusoidal measurement noise with no changes appears intooutput but no noticeable alternation in control signal hasbeen seen And finally in Figure 8 an impalpable effectof white Gaussian measurement noise on output and con-trol signal can be shown in this example some repeatedbenefits which are seen in Maglev system response are

10 Abstract and Applied AnalysisD

istan

ce (m

)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 5 Simulation result for Maglev without disturbance (inwhich it goes from 10 to 16mm) (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 6 Simulation result for Maglev with disturbance (a)Distance between train and rail (b) control signal

expressed such as appropriate system stabilizing good andrapid tracking disturbance rejection and small settlingtime

6 Conclusion

Model-free adaptive controller is an adaptive controller thatjust uses system outputs and does not require another systemstates (so does not need observer) and with this as model-free controller it performs good and also has some meritssuch as good stability appropriate tracking robust againstuncertainty disturbance rejection and good decoupling andthe biggest advantage of MFA controller is no requiringto system dynamic and even does not need to any priorexperiment about system

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 7 Simulation result for Maglev with sinuous measurementnoise at 15Hz and disturbance (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 8 Simulation result for Maglev train with white Gaussianmeasurement noise along with disturbance (a) Distance betweentrain and rail (b) control signal

Appendices

A Some RequirementsLemma A1 For each time 119896 119909(119896) is bounded by

119909 (119896) le 1198890

+ 1198891

119903 (119896) (A1)

which is 119903(119896) called ldquotracking errorrdquo and 1198890

1198891

are computablepositive constants

Abstract and Applied Analysis 11

Proof [see [8 Lemma 32]] Let 119909(119896) isin 119880 in which 119880 is acompact subset of 119877119899 Assume that ℎ(119909(119896)) isin 119862infin119880 that isℎ(119909(119896)) is a smooth function 119880 rarr 119877 so that the Taylorseries expansion of ℎ(119909(119896)) exists Then using the bound on119909(119896) express the function ℎ(119909(119896)) on a compact set as linearform of parameters Thus the upper bound for ℎ(119909(119896)) canbe obtained as

ℎ (119909 (119896)) le 1198880

+ 1198881

119903 (119896) (A2)

where 1198880

and 1198881

are constant matrices [8]

Remark A2 120576119892

120576119891

are estimation errors of 119892 119891 respectivelyand 119889 is external disturbance These errors are bounded andhave these definite bounds

10038171003817100381710038171003817120576119891

10038171003817100381710038171003817le 120576119873119891

10038171003817100381710038171003817120576119892

10038171003817100381710038171003817le 120576119873119892

119889 le 119889119872

(A3)

Remark A3 According to (27) if 119868 = 1 then 119906119888

le 119904 so 119906119888

is bounded and if 119868 = 0 then 119906119888

gt 119904 but by consideringthe mentioned condition at Remark 3 in Section 5 in whichif 119906119888

is greater than 1120574 it is assumed to be equal 1120574 Thus itcan be deduced that in both regions and in all of times 119906

119888

isbounded and its bounds are shown as follows

10038171003817100381710038171199061198881003817100381710038171003817 le

119904 119868 = 1

1

120574119868 = 0

(A4)Lemma A4 With regards to that the proposed controlalgorithm is divided into two regions 119868 = 0 and 119868 = 1 Wewill prove the following relationship which is established in eachregion

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896) 119868 = 1

1003817100381710038171003817119892119906 minus 1198921199061198621003817100381710038171003817 le 1198890 + 1198891 119903 (119896) 119868 = 0

(A5)

Proof (a) First region 119868 = 1 or 119906119888

le 119904 and 119892(119909) gt 119892In consideration of Lemma A1 it is demonstrated that

119892(119909) le 11988801

+ 11988811

119903(119896) where 11988801

11988811

are constant matrices

10038171003817100381710038171199061198881003817100381710038171003817 le 119904 997888rarr

10038171003817100381710038171199061199031003817100381710038171003817 le

120583

119892

10038171003817100381710038171199061198881003817100381710038171003817 997888rarr bounded 1003817100381710038171003817119906119903

1003817100381710038171003817 (A6)

By using the result of relation (A6) the following inequalityis obtained

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A7)

As regards 119906119888

le 119904 rarr 120574(119906119888

minus 119904) le 0

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(10038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198882

(A8)

By considering (A7)1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817

1003817100381710038171003817(119906 minus 119906119862)1003817100381710038171003817

le 1198882

(11988801

+ 11988811

119903 (119896)) = 11988802

+ 11988812

119903 (119896)

(A9)

Also by utilizing Remark 3 (A10) is stated as follows10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le10038171003817100381710038171003817120576119891

10038171003817100381710038171003817+10038171003817100381710038171003817120576119892

10038171003817100381710038171003817

10038171003817100381710038171199061198881003817100381710038171003817 + 119889

le 120576119873119891

+ 120576119873119892

+ 119889119872

= 120576119873

(A10)

By using (A9) and (A10) the following inequality is acquired10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 +10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le 11988802

+ 11988812

119903 (119896) + 120576119873

le 1198880

+ 1198881

119903 (119896)

(A11)

Thus the assumption is proved in the first region(b) Second region 119868 = 0 or 119906

119888

ge 119904 and 119892(119909) lt 119892 andthe mentioned condition for the proposed MFA controller inRemark 3 in Section 5 is stated as follows

if 119906119888

gt1

120574997904rArr 119906119888

=1

120574 (A12)

In consideration of Lemma A1 it is demonstrated that119892(119909) le 119888

01

+ 11988811

119903(119896) where 11988801

11988811

are constant matricesBy using the result of relation (A6) and the condition that isgiven above the following inequality is obtained

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890minus120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A13)

As regards 119906119888

gt 119904 rarr minus120574(119906119888

minus 119904) le 0

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(310038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198892

(A14)

By considering (B5)1003817100381710038171003817119892119906

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817 119906 le 1198892 (11988801 + 11988811 119903 (119896))

= 11988902

+ 11988912

119903 (119896)

10038171003817100381710038171198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921003817100381710038171003817

10038171003817100381710038171003817100381710038171003817

1

120574

10038171003817100381710038171003817100381710038171003817

le 1198893

(A15)

By using (A10) and (A15) the following inequality isobtained

1003817100381710038171003817119892119906 minus 1198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921199061003817100381710038171003817 +

10038171003817100381710038171198921199061198881003817100381710038171003817

le 11988901

+ 11988911

119903 (119896) + 1198893

le 1198890

+ 1198891

119903 (119896)

(A16)

Finally the assumption is proved in the second region Thusinequality (A5) is correct in both regions

B Obtain 119903(119896)

In consideration of error definition 119903(119896) in (21)

119903 (119896 + 1) = 119890119899

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

I+ 1205821

119890119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

II+ sdot sdot sdot + 120582

119899minus1

1198901

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

III

(B1)

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

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Volume 2013

8 Abstract and Applied Analysis

Time (s)

Out

put 1

System output 1Desired output 1

0 05 1 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

(c)

0

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

05 1 15 2 25 3 35 4 45 5minus80minus60minus40minus20

(d)

Figure 2 Simulation results for MIMO system with disturbance (a) (b) system outputs 1 and 2 (c) (d) control signals 119906 relative to inputs 1and 2

Maglev trains work in one of two ways both methods arebased on the same concept but involve different approaches

(1) Electromagnetic Suspension is based on magneticattraction it is very complex and somewhat unstable

(2) Electrodynamic Suspension is based on the repulsionofmagnetsThemagnetic levitation force balances theweight of the car at a stable position Controlling EMSis more difficult than EDS train because normality ofthis dynamic state is unstable

Magnetic trains have two important issues levitationand propulsion the target of controller is first goes uptrain to desired level along with guarantee stabilizing againstsome uncertainty such as wind and changing train mass inboltroads And second adjust train speed at the workingfrequency of magnets In this paper just levitation part(important section of train in controlling) is discussed Thetrain in this example goes from 10mm to 16mm level By

considering these assumption values for train and controlleras follows

119898 = 15 kg 119873 = 280 turns 119886119898

= 1024m2

119877119898

= 11Ω 120583 = 32 120572 = 1 120573 = 1

119904 = 140 120574 = 01 119892 = 9 119896119907

= 1 1205821

= 0

120576 = 0001 120578 = 1 119891

= minus 4 119892

= 1

(56)

The result of simulating without considering disturbancehas been displayed in Figure 5 in Figure 6 input step asexternal disturbance with amplitude 119889(119905) = 1 is appliedto the system for Figure 7 sinusoidal measurement noisewith amplitude 1mm in presence of disturbance is con-sidered Also white Gaussian noise in existence of externaldisturbance is applied to the system with power 8 times 10minus9(it is considered small because 119868119874 amplitude is small)which is shown in Figure 8 The purpose of presentingthis system is studying on sort of complicated practicalsystem dynamics such as Maglev in which they have some

Abstract and Applied Analysis 9

Time (s)

Out

put 1

System output 1Desired output 1

0 051 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

0102030405060

Time (s)

Cont

rol p

ower

1

0 05 1 15 2 25 3 35 4 45 5minus30minus20minus10

(c)

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5

minus20minus40minus60minus80(d)

Figure 3 Simulation results for MIMO system with the measurement noise and disturbance (a) (b) system outputs 1 and 2 (c) (d) controlsignals 119906 relative to inputs 1 and 2

Figure 4 Vertical slice of magnetic levitation and rail in a magnetic train [9]

nonlinear terms like existence of first state in denominator(that affect directly the destabilizing system) and so forthAs it is demonstrated in Figure 5(a) the system outputthat is distance between rail and train has no overshootafter about 25 second it goes from 10 to 16mm smoothlySuch as the last example it is acquired from Figure 6 thatdisturbance did not have any effect on the output control

effort and settling time In Figure 7 it is illustrated thatsinusoidal measurement noise with no changes appears intooutput but no noticeable alternation in control signal hasbeen seen And finally in Figure 8 an impalpable effectof white Gaussian measurement noise on output and con-trol signal can be shown in this example some repeatedbenefits which are seen in Maglev system response are

10 Abstract and Applied AnalysisD

istan

ce (m

)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 5 Simulation result for Maglev without disturbance (inwhich it goes from 10 to 16mm) (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 6 Simulation result for Maglev with disturbance (a)Distance between train and rail (b) control signal

expressed such as appropriate system stabilizing good andrapid tracking disturbance rejection and small settlingtime

6 Conclusion

Model-free adaptive controller is an adaptive controller thatjust uses system outputs and does not require another systemstates (so does not need observer) and with this as model-free controller it performs good and also has some meritssuch as good stability appropriate tracking robust againstuncertainty disturbance rejection and good decoupling andthe biggest advantage of MFA controller is no requiringto system dynamic and even does not need to any priorexperiment about system

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 7 Simulation result for Maglev with sinuous measurementnoise at 15Hz and disturbance (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 8 Simulation result for Maglev train with white Gaussianmeasurement noise along with disturbance (a) Distance betweentrain and rail (b) control signal

Appendices

A Some RequirementsLemma A1 For each time 119896 119909(119896) is bounded by

119909 (119896) le 1198890

+ 1198891

119903 (119896) (A1)

which is 119903(119896) called ldquotracking errorrdquo and 1198890

1198891

are computablepositive constants

Abstract and Applied Analysis 11

Proof [see [8 Lemma 32]] Let 119909(119896) isin 119880 in which 119880 is acompact subset of 119877119899 Assume that ℎ(119909(119896)) isin 119862infin119880 that isℎ(119909(119896)) is a smooth function 119880 rarr 119877 so that the Taylorseries expansion of ℎ(119909(119896)) exists Then using the bound on119909(119896) express the function ℎ(119909(119896)) on a compact set as linearform of parameters Thus the upper bound for ℎ(119909(119896)) canbe obtained as

ℎ (119909 (119896)) le 1198880

+ 1198881

119903 (119896) (A2)

where 1198880

and 1198881

are constant matrices [8]

Remark A2 120576119892

120576119891

are estimation errors of 119892 119891 respectivelyand 119889 is external disturbance These errors are bounded andhave these definite bounds

10038171003817100381710038171003817120576119891

10038171003817100381710038171003817le 120576119873119891

10038171003817100381710038171003817120576119892

10038171003817100381710038171003817le 120576119873119892

119889 le 119889119872

(A3)

Remark A3 According to (27) if 119868 = 1 then 119906119888

le 119904 so 119906119888

is bounded and if 119868 = 0 then 119906119888

gt 119904 but by consideringthe mentioned condition at Remark 3 in Section 5 in whichif 119906119888

is greater than 1120574 it is assumed to be equal 1120574 Thus itcan be deduced that in both regions and in all of times 119906

119888

isbounded and its bounds are shown as follows

10038171003817100381710038171199061198881003817100381710038171003817 le

119904 119868 = 1

1

120574119868 = 0

(A4)Lemma A4 With regards to that the proposed controlalgorithm is divided into two regions 119868 = 0 and 119868 = 1 Wewill prove the following relationship which is established in eachregion

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896) 119868 = 1

1003817100381710038171003817119892119906 minus 1198921199061198621003817100381710038171003817 le 1198890 + 1198891 119903 (119896) 119868 = 0

(A5)

Proof (a) First region 119868 = 1 or 119906119888

le 119904 and 119892(119909) gt 119892In consideration of Lemma A1 it is demonstrated that

119892(119909) le 11988801

+ 11988811

119903(119896) where 11988801

11988811

are constant matrices

10038171003817100381710038171199061198881003817100381710038171003817 le 119904 997888rarr

10038171003817100381710038171199061199031003817100381710038171003817 le

120583

119892

10038171003817100381710038171199061198881003817100381710038171003817 997888rarr bounded 1003817100381710038171003817119906119903

1003817100381710038171003817 (A6)

By using the result of relation (A6) the following inequalityis obtained

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A7)

As regards 119906119888

le 119904 rarr 120574(119906119888

minus 119904) le 0

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(10038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198882

(A8)

By considering (A7)1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817

1003817100381710038171003817(119906 minus 119906119862)1003817100381710038171003817

le 1198882

(11988801

+ 11988811

119903 (119896)) = 11988802

+ 11988812

119903 (119896)

(A9)

Also by utilizing Remark 3 (A10) is stated as follows10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le10038171003817100381710038171003817120576119891

10038171003817100381710038171003817+10038171003817100381710038171003817120576119892

10038171003817100381710038171003817

10038171003817100381710038171199061198881003817100381710038171003817 + 119889

le 120576119873119891

+ 120576119873119892

+ 119889119872

= 120576119873

(A10)

By using (A9) and (A10) the following inequality is acquired10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 +10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le 11988802

+ 11988812

119903 (119896) + 120576119873

le 1198880

+ 1198881

119903 (119896)

(A11)

Thus the assumption is proved in the first region(b) Second region 119868 = 0 or 119906

119888

ge 119904 and 119892(119909) lt 119892 andthe mentioned condition for the proposed MFA controller inRemark 3 in Section 5 is stated as follows

if 119906119888

gt1

120574997904rArr 119906119888

=1

120574 (A12)

In consideration of Lemma A1 it is demonstrated that119892(119909) le 119888

01

+ 11988811

119903(119896) where 11988801

11988811

are constant matricesBy using the result of relation (A6) and the condition that isgiven above the following inequality is obtained

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890minus120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A13)

As regards 119906119888

gt 119904 rarr minus120574(119906119888

minus 119904) le 0

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(310038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198892

(A14)

By considering (B5)1003817100381710038171003817119892119906

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817 119906 le 1198892 (11988801 + 11988811 119903 (119896))

= 11988902

+ 11988912

119903 (119896)

10038171003817100381710038171198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921003817100381710038171003817

10038171003817100381710038171003817100381710038171003817

1

120574

10038171003817100381710038171003817100381710038171003817

le 1198893

(A15)

By using (A10) and (A15) the following inequality isobtained

1003817100381710038171003817119892119906 minus 1198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921199061003817100381710038171003817 +

10038171003817100381710038171198921199061198881003817100381710038171003817

le 11988901

+ 11988911

119903 (119896) + 1198893

le 1198890

+ 1198891

119903 (119896)

(A16)

Finally the assumption is proved in the second region Thusinequality (A5) is correct in both regions

B Obtain 119903(119896)

In consideration of error definition 119903(119896) in (21)

119903 (119896 + 1) = 119890119899

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

I+ 1205821

119890119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

II+ sdot sdot sdot + 120582

119899minus1

1198901

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

III

(B1)

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

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Abstract and Applied Analysis 9

Time (s)

Out

put 1

System output 1Desired output 1

0 051 15 2 25 3 35 4 45 50

02

04

06

08

1

12

(a)

Time (s)

Out

put 2

System output 2Desired output 2

0 05 1 15 2 25 3 35 4 45 5minus15minus1minus05005

1

15

(b)

0102030405060

Time (s)

Cont

rol p

ower

1

0 05 1 15 2 25 3 35 4 45 5minus30minus20minus10

(c)

0

20

40

60

80

Time (s)

Cont

rol p

ower

2

0 05 1 15 2 25 3 35 4 45 5

minus20minus40minus60minus80(d)

Figure 3 Simulation results for MIMO system with the measurement noise and disturbance (a) (b) system outputs 1 and 2 (c) (d) controlsignals 119906 relative to inputs 1 and 2

Figure 4 Vertical slice of magnetic levitation and rail in a magnetic train [9]

nonlinear terms like existence of first state in denominator(that affect directly the destabilizing system) and so forthAs it is demonstrated in Figure 5(a) the system outputthat is distance between rail and train has no overshootafter about 25 second it goes from 10 to 16mm smoothlySuch as the last example it is acquired from Figure 6 thatdisturbance did not have any effect on the output control

effort and settling time In Figure 7 it is illustrated thatsinusoidal measurement noise with no changes appears intooutput but no noticeable alternation in control signal hasbeen seen And finally in Figure 8 an impalpable effectof white Gaussian measurement noise on output and con-trol signal can be shown in this example some repeatedbenefits which are seen in Maglev system response are

10 Abstract and Applied AnalysisD

istan

ce (m

)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 5 Simulation result for Maglev without disturbance (inwhich it goes from 10 to 16mm) (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 6 Simulation result for Maglev with disturbance (a)Distance between train and rail (b) control signal

expressed such as appropriate system stabilizing good andrapid tracking disturbance rejection and small settlingtime

6 Conclusion

Model-free adaptive controller is an adaptive controller thatjust uses system outputs and does not require another systemstates (so does not need observer) and with this as model-free controller it performs good and also has some meritssuch as good stability appropriate tracking robust againstuncertainty disturbance rejection and good decoupling andthe biggest advantage of MFA controller is no requiringto system dynamic and even does not need to any priorexperiment about system

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 7 Simulation result for Maglev with sinuous measurementnoise at 15Hz and disturbance (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 8 Simulation result for Maglev train with white Gaussianmeasurement noise along with disturbance (a) Distance betweentrain and rail (b) control signal

Appendices

A Some RequirementsLemma A1 For each time 119896 119909(119896) is bounded by

119909 (119896) le 1198890

+ 1198891

119903 (119896) (A1)

which is 119903(119896) called ldquotracking errorrdquo and 1198890

1198891

are computablepositive constants

Abstract and Applied Analysis 11

Proof [see [8 Lemma 32]] Let 119909(119896) isin 119880 in which 119880 is acompact subset of 119877119899 Assume that ℎ(119909(119896)) isin 119862infin119880 that isℎ(119909(119896)) is a smooth function 119880 rarr 119877 so that the Taylorseries expansion of ℎ(119909(119896)) exists Then using the bound on119909(119896) express the function ℎ(119909(119896)) on a compact set as linearform of parameters Thus the upper bound for ℎ(119909(119896)) canbe obtained as

ℎ (119909 (119896)) le 1198880

+ 1198881

119903 (119896) (A2)

where 1198880

and 1198881

are constant matrices [8]

Remark A2 120576119892

120576119891

are estimation errors of 119892 119891 respectivelyand 119889 is external disturbance These errors are bounded andhave these definite bounds

10038171003817100381710038171003817120576119891

10038171003817100381710038171003817le 120576119873119891

10038171003817100381710038171003817120576119892

10038171003817100381710038171003817le 120576119873119892

119889 le 119889119872

(A3)

Remark A3 According to (27) if 119868 = 1 then 119906119888

le 119904 so 119906119888

is bounded and if 119868 = 0 then 119906119888

gt 119904 but by consideringthe mentioned condition at Remark 3 in Section 5 in whichif 119906119888

is greater than 1120574 it is assumed to be equal 1120574 Thus itcan be deduced that in both regions and in all of times 119906

119888

isbounded and its bounds are shown as follows

10038171003817100381710038171199061198881003817100381710038171003817 le

119904 119868 = 1

1

120574119868 = 0

(A4)Lemma A4 With regards to that the proposed controlalgorithm is divided into two regions 119868 = 0 and 119868 = 1 Wewill prove the following relationship which is established in eachregion

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896) 119868 = 1

1003817100381710038171003817119892119906 minus 1198921199061198621003817100381710038171003817 le 1198890 + 1198891 119903 (119896) 119868 = 0

(A5)

Proof (a) First region 119868 = 1 or 119906119888

le 119904 and 119892(119909) gt 119892In consideration of Lemma A1 it is demonstrated that

119892(119909) le 11988801

+ 11988811

119903(119896) where 11988801

11988811

are constant matrices

10038171003817100381710038171199061198881003817100381710038171003817 le 119904 997888rarr

10038171003817100381710038171199061199031003817100381710038171003817 le

120583

119892

10038171003817100381710038171199061198881003817100381710038171003817 997888rarr bounded 1003817100381710038171003817119906119903

1003817100381710038171003817 (A6)

By using the result of relation (A6) the following inequalityis obtained

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A7)

As regards 119906119888

le 119904 rarr 120574(119906119888

minus 119904) le 0

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(10038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198882

(A8)

By considering (A7)1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817

1003817100381710038171003817(119906 minus 119906119862)1003817100381710038171003817

le 1198882

(11988801

+ 11988811

119903 (119896)) = 11988802

+ 11988812

119903 (119896)

(A9)

Also by utilizing Remark 3 (A10) is stated as follows10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le10038171003817100381710038171003817120576119891

10038171003817100381710038171003817+10038171003817100381710038171003817120576119892

10038171003817100381710038171003817

10038171003817100381710038171199061198881003817100381710038171003817 + 119889

le 120576119873119891

+ 120576119873119892

+ 119889119872

= 120576119873

(A10)

By using (A9) and (A10) the following inequality is acquired10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 +10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le 11988802

+ 11988812

119903 (119896) + 120576119873

le 1198880

+ 1198881

119903 (119896)

(A11)

Thus the assumption is proved in the first region(b) Second region 119868 = 0 or 119906

119888

ge 119904 and 119892(119909) lt 119892 andthe mentioned condition for the proposed MFA controller inRemark 3 in Section 5 is stated as follows

if 119906119888

gt1

120574997904rArr 119906119888

=1

120574 (A12)

In consideration of Lemma A1 it is demonstrated that119892(119909) le 119888

01

+ 11988811

119903(119896) where 11988801

11988811

are constant matricesBy using the result of relation (A6) and the condition that isgiven above the following inequality is obtained

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890minus120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A13)

As regards 119906119888

gt 119904 rarr minus120574(119906119888

minus 119904) le 0

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(310038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198892

(A14)

By considering (B5)1003817100381710038171003817119892119906

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817 119906 le 1198892 (11988801 + 11988811 119903 (119896))

= 11988902

+ 11988912

119903 (119896)

10038171003817100381710038171198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921003817100381710038171003817

10038171003817100381710038171003817100381710038171003817

1

120574

10038171003817100381710038171003817100381710038171003817

le 1198893

(A15)

By using (A10) and (A15) the following inequality isobtained

1003817100381710038171003817119892119906 minus 1198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921199061003817100381710038171003817 +

10038171003817100381710038171198921199061198881003817100381710038171003817

le 11988901

+ 11988911

119903 (119896) + 1198893

le 1198890

+ 1198891

119903 (119896)

(A16)

Finally the assumption is proved in the second region Thusinequality (A5) is correct in both regions

B Obtain 119903(119896)

In consideration of error definition 119903(119896) in (21)

119903 (119896 + 1) = 119890119899

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

I+ 1205821

119890119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

II+ sdot sdot sdot + 120582

119899minus1

1198901

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

III

(B1)

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Game Theory

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Complex Systems

Journal of

ISRN Operations Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Abstract and Applied Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Industrial MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

OptimizationJournal of

ISRN Computational Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Complex AnalysisJournal of

ISRN Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Geometry

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

thinspAdvancesthinspin

DecisionSciences

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Algebra

ISRN Mathematical Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

10 Abstract and Applied AnalysisD

istan

ce (m

)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 5 Simulation result for Maglev without disturbance (inwhich it goes from 10 to 16mm) (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 6 Simulation result for Maglev with disturbance (a)Distance between train and rail (b) control signal

expressed such as appropriate system stabilizing good andrapid tracking disturbance rejection and small settlingtime

6 Conclusion

Model-free adaptive controller is an adaptive controller thatjust uses system outputs and does not require another systemstates (so does not need observer) and with this as model-free controller it performs good and also has some meritssuch as good stability appropriate tracking robust againstuncertainty disturbance rejection and good decoupling andthe biggest advantage of MFA controller is no requiringto system dynamic and even does not need to any priorexperiment about system

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 7 Simulation result for Maglev with sinuous measurementnoise at 15Hz and disturbance (a) Distance between train and rail(b) control signal

Dist

ance

(m)

Time (s)0 05 1 15 2 25 3

001001200140016

(a)

5

15

Time (s)

Cont

rol p

ower

0 05 1 15 2 25 3minus15minus5(v

olt)

(b)

Figure 8 Simulation result for Maglev train with white Gaussianmeasurement noise along with disturbance (a) Distance betweentrain and rail (b) control signal

Appendices

A Some RequirementsLemma A1 For each time 119896 119909(119896) is bounded by

119909 (119896) le 1198890

+ 1198891

119903 (119896) (A1)

which is 119903(119896) called ldquotracking errorrdquo and 1198890

1198891

are computablepositive constants

Abstract and Applied Analysis 11

Proof [see [8 Lemma 32]] Let 119909(119896) isin 119880 in which 119880 is acompact subset of 119877119899 Assume that ℎ(119909(119896)) isin 119862infin119880 that isℎ(119909(119896)) is a smooth function 119880 rarr 119877 so that the Taylorseries expansion of ℎ(119909(119896)) exists Then using the bound on119909(119896) express the function ℎ(119909(119896)) on a compact set as linearform of parameters Thus the upper bound for ℎ(119909(119896)) canbe obtained as

ℎ (119909 (119896)) le 1198880

+ 1198881

119903 (119896) (A2)

where 1198880

and 1198881

are constant matrices [8]

Remark A2 120576119892

120576119891

are estimation errors of 119892 119891 respectivelyand 119889 is external disturbance These errors are bounded andhave these definite bounds

10038171003817100381710038171003817120576119891

10038171003817100381710038171003817le 120576119873119891

10038171003817100381710038171003817120576119892

10038171003817100381710038171003817le 120576119873119892

119889 le 119889119872

(A3)

Remark A3 According to (27) if 119868 = 1 then 119906119888

le 119904 so 119906119888

is bounded and if 119868 = 0 then 119906119888

gt 119904 but by consideringthe mentioned condition at Remark 3 in Section 5 in whichif 119906119888

is greater than 1120574 it is assumed to be equal 1120574 Thus itcan be deduced that in both regions and in all of times 119906

119888

isbounded and its bounds are shown as follows

10038171003817100381710038171199061198881003817100381710038171003817 le

119904 119868 = 1

1

120574119868 = 0

(A4)Lemma A4 With regards to that the proposed controlalgorithm is divided into two regions 119868 = 0 and 119868 = 1 Wewill prove the following relationship which is established in eachregion

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896) 119868 = 1

1003817100381710038171003817119892119906 minus 1198921199061198621003817100381710038171003817 le 1198890 + 1198891 119903 (119896) 119868 = 0

(A5)

Proof (a) First region 119868 = 1 or 119906119888

le 119904 and 119892(119909) gt 119892In consideration of Lemma A1 it is demonstrated that

119892(119909) le 11988801

+ 11988811

119903(119896) where 11988801

11988811

are constant matrices

10038171003817100381710038171199061198881003817100381710038171003817 le 119904 997888rarr

10038171003817100381710038171199061199031003817100381710038171003817 le

120583

119892

10038171003817100381710038171199061198881003817100381710038171003817 997888rarr bounded 1003817100381710038171003817119906119903

1003817100381710038171003817 (A6)

By using the result of relation (A6) the following inequalityis obtained

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A7)

As regards 119906119888

le 119904 rarr 120574(119906119888

minus 119904) le 0

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(10038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198882

(A8)

By considering (A7)1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817

1003817100381710038171003817(119906 minus 119906119862)1003817100381710038171003817

le 1198882

(11988801

+ 11988811

119903 (119896)) = 11988802

+ 11988812

119903 (119896)

(A9)

Also by utilizing Remark 3 (A10) is stated as follows10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le10038171003817100381710038171003817120576119891

10038171003817100381710038171003817+10038171003817100381710038171003817120576119892

10038171003817100381710038171003817

10038171003817100381710038171199061198881003817100381710038171003817 + 119889

le 120576119873119891

+ 120576119873119892

+ 119889119872

= 120576119873

(A10)

By using (A9) and (A10) the following inequality is acquired10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 +10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le 11988802

+ 11988812

119903 (119896) + 120576119873

le 1198880

+ 1198881

119903 (119896)

(A11)

Thus the assumption is proved in the first region(b) Second region 119868 = 0 or 119906

119888

ge 119904 and 119892(119909) lt 119892 andthe mentioned condition for the proposed MFA controller inRemark 3 in Section 5 is stated as follows

if 119906119888

gt1

120574997904rArr 119906119888

=1

120574 (A12)

In consideration of Lemma A1 it is demonstrated that119892(119909) le 119888

01

+ 11988811

119903(119896) where 11988801

11988811

are constant matricesBy using the result of relation (A6) and the condition that isgiven above the following inequality is obtained

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890minus120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A13)

As regards 119906119888

gt 119904 rarr minus120574(119906119888

minus 119904) le 0

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(310038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198892

(A14)

By considering (B5)1003817100381710038171003817119892119906

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817 119906 le 1198892 (11988801 + 11988811 119903 (119896))

= 11988902

+ 11988912

119903 (119896)

10038171003817100381710038171198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921003817100381710038171003817

10038171003817100381710038171003817100381710038171003817

1

120574

10038171003817100381710038171003817100381710038171003817

le 1198893

(A15)

By using (A10) and (A15) the following inequality isobtained

1003817100381710038171003817119892119906 minus 1198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921199061003817100381710038171003817 +

10038171003817100381710038171198921199061198881003817100381710038171003817

le 11988901

+ 11988911

119903 (119896) + 1198893

le 1198890

+ 1198891

119903 (119896)

(A16)

Finally the assumption is proved in the second region Thusinequality (A5) is correct in both regions

B Obtain 119903(119896)

In consideration of error definition 119903(119896) in (21)

119903 (119896 + 1) = 119890119899

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

I+ 1205821

119890119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

II+ sdot sdot sdot + 120582

119899minus1

1198901

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

III

(B1)

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Game Theory

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Complex Systems

Journal of

ISRN Operations Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Abstract and Applied Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Industrial MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

OptimizationJournal of

ISRN Computational Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Complex AnalysisJournal of

ISRN Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Geometry

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

thinspAdvancesthinspin

DecisionSciences

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Algebra

ISRN Mathematical Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Abstract and Applied Analysis 11

Proof [see [8 Lemma 32]] Let 119909(119896) isin 119880 in which 119880 is acompact subset of 119877119899 Assume that ℎ(119909(119896)) isin 119862infin119880 that isℎ(119909(119896)) is a smooth function 119880 rarr 119877 so that the Taylorseries expansion of ℎ(119909(119896)) exists Then using the bound on119909(119896) express the function ℎ(119909(119896)) on a compact set as linearform of parameters Thus the upper bound for ℎ(119909(119896)) canbe obtained as

ℎ (119909 (119896)) le 1198880

+ 1198881

119903 (119896) (A2)

where 1198880

and 1198881

are constant matrices [8]

Remark A2 120576119892

120576119891

are estimation errors of 119892 119891 respectivelyand 119889 is external disturbance These errors are bounded andhave these definite bounds

10038171003817100381710038171003817120576119891

10038171003817100381710038171003817le 120576119873119891

10038171003817100381710038171003817120576119892

10038171003817100381710038171003817le 120576119873119892

119889 le 119889119872

(A3)

Remark A3 According to (27) if 119868 = 1 then 119906119888

le 119904 so 119906119888

is bounded and if 119868 = 0 then 119906119888

gt 119904 but by consideringthe mentioned condition at Remark 3 in Section 5 in whichif 119906119888

is greater than 1120574 it is assumed to be equal 1120574 Thus itcan be deduced that in both regions and in all of times 119906

119888

isbounded and its bounds are shown as follows

10038171003817100381710038171199061198881003817100381710038171003817 le

119904 119868 = 1

1

120574119868 = 0

(A4)Lemma A4 With regards to that the proposed controlalgorithm is divided into two regions 119868 = 0 and 119868 = 1 Wewill prove the following relationship which is established in eachregion

10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le 1198880

+ 1198881

119903 (119896) 119868 = 1

1003817100381710038171003817119892119906 minus 1198921199061198621003817100381710038171003817 le 1198890 + 1198891 119903 (119896) 119868 = 0

(A5)

Proof (a) First region 119868 = 1 or 119906119888

le 119904 and 119892(119909) gt 119892In consideration of Lemma A1 it is demonstrated that

119892(119909) le 11988801

+ 11988811

119903(119896) where 11988801

11988811

are constant matrices

10038171003817100381710038171199061198881003817100381710038171003817 le 119904 997888rarr

10038171003817100381710038171199061199031003817100381710038171003817 le

120583

119892

10038171003817100381710038171199061198881003817100381710038171003817 997888rarr bounded 1003817100381710038171003817119906119903

1003817100381710038171003817 (A6)

By using the result of relation (A6) the following inequalityis obtained

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A7)

As regards 119906119888

le 119904 rarr 120574(119906119888

minus 119904) le 0

1003817100381710038171003817119906 minus 1199061198881003817100381710038171003817 le

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(10038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198882

(A8)

By considering (A7)1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817

1003817100381710038171003817(119906 minus 119906119862)1003817100381710038171003817

le 1198882

(11988801

+ 11988811

119903 (119896)) = 11988802

+ 11988812

119903 (119896)

(A9)

Also by utilizing Remark 3 (A10) is stated as follows10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817le10038171003817100381710038171003817120576119891

10038171003817100381710038171003817+10038171003817100381710038171003817120576119892

10038171003817100381710038171003817

10038171003817100381710038171199061198881003817100381710038171003817 + 119889

le 120576119873119891

+ 120576119873119892

+ 119889119872

= 120576119873

(A10)

By using (A9) and (A10) the following inequality is acquired10038171003817100381710038171003817119892 (119906 minus 119906

119862

) + 120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le1003817100381710038171003817119892 (119906 minus 119906119862)

1003817100381710038171003817 +10038171003817100381710038171003817120576119891

+ 120576119892

119906119888

+ 11988910038171003817100381710038171003817

le 11988802

+ 11988812

119903 (119896) + 120576119873

le 1198880

+ 1198881

119903 (119896)

(A11)

Thus the assumption is proved in the first region(b) Second region 119868 = 0 or 119906

119888

ge 119904 and 119892(119909) lt 119892 andthe mentioned condition for the proposed MFA controller inRemark 3 in Section 5 is stated as follows

if 119906119888

gt1

120574997904rArr 119906119888

=1

120574 (A12)

In consideration of Lemma A1 it is demonstrated that119892(119909) le 119888

01

+ 11988811

119903(119896) where 11988801

11988811

are constant matricesBy using the result of relation (A6) and the condition that isgiven above the following inequality is obtained

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2119890minus120574(119906119888minus119904)1003817100381710038171003817100381710038171003817 (A13)

As regards 119906119888

gt 119904 rarr minus120574(119906119888

minus 119904) le 0

119906 le10038171003817100381710038171199061199031003817100381710038171003817 +

1003817100381710038171003817100381710038171003817

119906119903

minus 119906119888

2

1003817100381710038171003817100381710038171003817le1

2(310038171003817100381710038171199061199031003817100381710038171003817 +

10038171003817100381710038171199061198881003817100381710038171003817) le 1198892

(A14)

By considering (B5)1003817100381710038171003817119892119906

1003817100381710038171003817 le10038171003817100381710038171198921003817100381710038171003817 119906 le 1198892 (11988801 + 11988811 119903 (119896))

= 11988902

+ 11988912

119903 (119896)

10038171003817100381710038171198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921003817100381710038171003817

10038171003817100381710038171003817100381710038171003817

1

120574

10038171003817100381710038171003817100381710038171003817

le 1198893

(A15)

By using (A10) and (A15) the following inequality isobtained

1003817100381710038171003817119892119906 minus 1198921199061198881003817100381710038171003817 le

10038171003817100381710038171198921199061003817100381710038171003817 +

10038171003817100381710038171198921199061198881003817100381710038171003817

le 11988901

+ 11988911

119903 (119896) + 1198893

le 1198890

+ 1198891

119903 (119896)

(A16)

Finally the assumption is proved in the second region Thusinequality (A5) is correct in both regions

B Obtain 119903(119896)

In consideration of error definition 119903(119896) in (21)

119903 (119896 + 1) = 119890119899

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

I+ 1205821

119890119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

II+ sdot sdot sdot + 120582

119899minus1

1198901

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

III

(B1)

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

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Abstract and Applied Analysis

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Industrial MathematicsJournal of

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Mathematical Problems in Engineering

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Complex AnalysisJournal of

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Algebra

ISRN Mathematical Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete MathematicsJournal of

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Volume 2013

12 Abstract and Applied Analysis

(I) 119890119899

(119896 + 1) = 119909119899

(119896 + 1) minus 119910119889

(119896 + 119899) = 119891(119909) + 119892(119909)119906(119896) +

119889(119896) minus 119910119889

(119896 + 119899)

(II) 1205821

119890119899minus1

(119896 + 1) = 1205821

[119909119899minus1

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119909119899(119896) minus 119910119889

(119896 + 119899 minus 1)] =

1205821

119890119899

(119896)

(III) 120582119901minus1

1198901

(119896 + 1) = 120582119901minus1

[1199091

(119896 + 1)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

1199092(119896) minus 119910119889

(119896 minus 1)] =

120582119901minus1

1198902

(119896)

Thus by substituting (I) (II) and (III) in the above equation

119903 (119896 + 1) = 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896) minus 119910119889

(119896 + 119875)

+ 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

(B2)

Equation (25) can be rewritten as follows

119907 (119896) = 119892 (119896) 119906119888

(119896) + 119891 (119896) (B3)

By using (23) (25) and (B2)

119903 (119896 + 1) = 119907 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) 119906 (119896) + 119889 (119896)

minus 119910119889

(119896 + 119875) + 1205821

119890119875

(119896) + sdot sdot sdot + 120582119875minus1

1198902

(119896)

= 119896119907

119903 (119896) minus 119907 (119896) + 119891 (119909) + 119892 (119909) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

(B4)

Also by considering the above equation the tracking errors119891

and 119892

are acquired as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + (119892119906 minus 119892119906119888

)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906119888

minus 119892119906119888

)

+ 119889 (119896) + 119892119906119889

(B5)

where 119906119889

= 119906minus119906119888

By the following definitions about the119892(119909)119891 119891(119909) and 119892 (B5) is rewritten as follows

119891 (119909) = 119906119879

119891

+ 120576119891

(119909) 997904rArr 119891 = 119879

119891

119892 (119909) = 119906119879

119892

+ 120576119892

(119909) 997904rArr 119892 = 119879

119892

119891

= 119906119891

minus 119891

119892

= 119906119892

minus 119892

119903 (119896 + 1) = 119896119907

119903 (119896) + (119906119879

119891

minus 119879

119891

) + (119906119879

119892

119906119888

minus 119879

119892

119906119888

)

+ 120576119891

+ 120576119892

119906 + 119889 (119896) + 119892119906119889

= 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 120576119891

+ 120576119892

119906 + 119892119906119889

(B6)

with substitution of 1205761015840 = 120576119891

+ 120576119892

119906119888

+ 119889 in (B6) final relationof tracking error is demonstrated as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + 119879

119891

+ 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

(B7)

Remark B1 In consideration of (20) in this region thefollowing equalities are given

119891

= 119906119891

minus 119891

997904rArr 119891

(119896 + 1) = (1 minus 120572) 119891

(119896)

minus 120572(119896119907

119903 (119896) + 119879

119892

119906119888

+ 1205761015840

+ 119892119906119889

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = (1 minus 120573) 119879

119892

(119896) 119906119888

minus 120573(119896119907

119903 (119896) + 119879

119891

+ 1205761015840

+ 119892119906119889

)119879

(B8)

By using equation (B4) in the second region tracking errors119891

119892

are obtained as follows

119903 (119896 + 1) = 119896119907

119903 (119896) + (119891 (119909) minus 119891) + (119892 (119909) 119906 minus 119892119906119888

) + 119889 (119896)

= 119896119907

119903 (119896) + (119891 (119909) minus 119891) + 119892119906119889

+ 119889 (119896)

(B9)

Remark B2 By utilizing (20) in this region 119891

and 119892

areacquired as follows

119891

= 119906119891

minus 119891

119891

(119896 + 1) = (1 minus 120572) 119891

(119896) minus 120572(119896119907

119903 (119896) + 119892119906119889

+ 1205761015840

)119879

119892

= 119906119892

minus 119892

997904rArr 119892

(119896 + 1) = 119892

(119896)

(B10)

References[1] M Yan C Xue and X Xie ldquoA novel model free adaptive

controller with tracking differentiatorrdquo in Proceedings of theIEEE International Conference onMechatronics and Automation(ICMA rsquo09) pp 4191ndash4196 Changchun China August 2009

[2] A Xu Y Zheng Y Song and M Liu ldquoAn improved modelfree adaptive control algorithmrdquo in Proceedings of the 5th Inter-national Conference on Natural Computation (ICNC rsquo09) pp70ndash74 Tianjian China August 2009

[3] Q Gao D Ren C Dong and Z Jia ldquoThe Study of Model-FreeAdaptive Controller based on dSPACErdquo in Proceedings of the2nd International Symposiumon Intelligent InformationTechnol-ogy Application pp 608ndash611 Shanghai China December 2008

[4] H Zhongsheng and J Shangtai ldquoA novel data-driven controlapproach for a class of discrete-time nonlinear systemsrdquo IEEETransactions on Control Systems Technology vol 19 no 6 pp1549ndash1558 2011

[5] S Jagannathan ldquoAdaptive fuzzy logic control of feedback lin-earizable discrete-time nonlinear systemsrdquo in Proceedings of theIEEE International Symposium on Intelligent Control DearbornSeptember 1996

[6] S G Cheng ldquoModel-free adaptive process controlrdquo UnitedStates Patent 6055524 Washington United States Patent andTrademark Office of theUnited States Department of Com-merce 2000

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Game Theory

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Complex Systems

Journal of

ISRN Operations Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Abstract and Applied Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Industrial MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

OptimizationJournal of

ISRN Computational Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Complex AnalysisJournal of

ISRN Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Geometry

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

thinspAdvancesthinspin

DecisionSciences

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Algebra

ISRN Mathematical Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Abstract and Applied Analysis 13

[7] ROrdonez J Zumberge J T Spooner andKM Passino ldquoAda-ptive fuzzy control experiments and comparative analysesrdquoIEEE Transactions on Fuzzy Systems vol 5 no 2 pp 167ndash1881997

[8] S Jagannathan ldquoAdaptive fuzzy logic control of feedbacklinearizable discrete-time dynamical systems under persistenceof excitationrdquo Automatica vol 34 no 11 pp 1295ndash1310 1998

[9] P K Sinha and A N Pechev ldquoNonlinear Hinfin controllersfor electromagnetic suspension systemsrdquo IEEE Transactions onAutomatic Control vol 49 no 4 pp 563ndash568 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Game Theory

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Complex Systems

Journal of

ISRN Operations Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Abstract and Applied Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Industrial MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

OptimizationJournal of

ISRN Computational Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Complex AnalysisJournal of

ISRN Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Geometry

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

thinspAdvancesthinspin

DecisionSciences

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Algebra

ISRN Mathematical Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Game Theory

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Complex Systems

Journal of

ISRN Operations Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Abstract and Applied Analysis

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Industrial MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

OptimizationJournal of

ISRN Computational Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Complex AnalysisJournal of

ISRN Combinatorics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Geometry

ISRN Applied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

thinspAdvancesthinspin

DecisionSciences

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2013

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Algebra

ISRN Mathematical Physics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2013

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2013