research article finite frequency vibration control for...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 598489 12 pageshttpdxdoiorg1011552013598489

Research ArticleFinite Frequency Vibration Control for Polytopic ActiveSuspensions via Dynamic Output Feedback

Yifu Zhang1 Gongdai Liu2 Pinchao Wang2 and Hamid Reza Karimi3

1 Research Institute of Intelligent Control and Systems Harbin Institute of Technology Harbin 150001 China2 School of Astronautics Harbin Institute of Technology Harbin 150001 China3Department of Engineering Faculty of Engineering and Science University of Agder N-4898 Grimstad Norway

Correspondence should be addressed to Yifu Zhang yifuzhang19gmailcom

Received 18 June 2013 Accepted 25 July 2013

Academic Editor Hui Zhang

Copyright copy 2013 Yifu Zhang 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 presents a disturbance attenuation strategy for active suspension systems with frequency band constraints wheredynamic output feedback control is employed in consideration that not all the state variables can be measured on-line In viewof the fact that human are sensitive to the virbation between 4ndash8Hz in vertical direction the 119867

infincontrol based on generalized

Kalman-Yakubovich-Popov (KYP) lemma is developed in this specific frequency in order to achieve the targeted disturbanceattenuation Moreover practical constraints required in active suspension design are guaranteed in the whole time domain At theend of the paper the outstanding performance of the system using finite frequency approach is confirmed by simulation

1 Introduction

By reason of rough road conditions passengers in thecar are often in a vibration environment which negativelyimpacts the comfort mental and physical health of themand suspensions are crucial to attenuate the disturbancetransferred to passengers [1ndash3] Hence various approachesvarious approaches have been developed that aim to enhancesuspensionsrsquo performance such as adaptive control [4] robustcontrol [5 6] and fuzzy control [7] Generally speakingthere are three types of suspensions passive semiactive andactive suspensions Compared with the other two kinds ofsuspensions active suspensions have a greater potential toimprove ride comfort and to guarantee the ride safety due tothe existence of active actuator

There are three performance requirements for activesuspension systems One is the ride comfort which requiresisolation of vibration from road another one is handling per-formance mainly described by road holding which restrictsthe hop of wheel in order to ensure continuous contact ofwheels to road the last one is the sprung displacement whichlimits the suspension stroke within an allowable band

However these requirements are usually conflicting Forinstance a large suspension displacementmay exist if a better

rider comfort performance is required A variety of controlstrategies have been applied to cope with this conflict [8ndash12]In particular because the 119867

infinnorm index can measure the

vibration attenuation performance of system appropriately[13] many suspension problems are considered by 119867

infincon-

trol theory [14ndash22] In this paper the handling performanceand the suspension stroke are regarded as constraints and theride comfort is deemed as the main index to optimize

Although various control strategies have been applied topromote ride comfort performance of suspension systemsfew of them notice the fact that due to the human bodystructure and other factors people are more sensitive todisturbances in 4ndash8Hz than other frequency in verticaldirection (ISO2361) Therefore it is considerable to developa finite frequency strategy to reduce the negative effectcaused by disturbances in 4ndash8Hz The generalized Kalman-Yakubovich-Popov (KYP) lemma which has been used tosolve practical problems [23 24] is applied to achieve thefinite frequency control to active suspensions

It should be mentioned that the parameters of passengersincluding model for suspension system could vary due to themass change of passengers so how to guarantee the perfor-mance of suspension with varying parameters is worth dis-cussing Meanwhile considering that the mass of passengers

2 Mathematical Problems in Engineering

can be accessed on line in this paper the suspension modelis described as a polytope and then a parameter-dependentcontrol law is proposed to assure the above performancerequirements which can also reach a lower conservativenessthan control law based on quadratic stability and constantparameter feedback at the same time In addition thoughstate feedback may attain a superior performance comparedwith static output feedback measuring some statesmay bringburden to the systems Thusly constructing dynamic outputfeedback is desirable which can achieve a relative enhancedperformance andmeanwhile reduce state-measuring sensors

The paper is organized as follows In Section 2 the state-space model for quarter car suspension is presented InSection 3 the theorems which can be used to design thedynamic feedback controller are illustrated The simulationis presented in Section 4 and the conclusion is in Section 5

Notation For a matrix 119875 119875119879 119875lowast 119875minus1 119875minus119879 and 119875perp standfor its transpose conjugate transpose inverse transposedinverse and orthogonal complement respectively sym(119875)denotes 119875+119875119879 119875 gt 0 (119875 lt 0)means that matrix 119875 is positive(negative) definite For a matrix sdot

119894stands for the 119894th line

of the matrix 119866(119895120596)infin

stands for the119867infin

norm of transferfunction matrix 119866 For matrices 119875 and 119876 119875 otimes 119876 means theKronecker product In symmetric block matrices or complexmatrix expressions we use an asterisk (lowast) to represent a termthat is induced by symmetry

2 Quarter Car Suspension Model

The model of a quarter car suspension is shown in Figure 1119898119904and119898

119906stand for sprung and unsprungmass respectively

119911119904 119911119906 and 119911

119903denote the sprung unsprung displacement and

disturbance displacement from the road respectively 119896119904 119896119905

119888119904 and 119888

119905are the stiffnesses and dampings of the suspension

system respectivelyThe input of the controller is denoted by119906

Based on the law of Newton the motion equation ofsuspension can be denoted as

119898119904119904(119905) + 119888

119904[119904(119905) minus

119906(119905)] + 119896

119904[119911119904(119905) minus 119911

119906(119905)] = 119906 (119905)

119898119906119906(119905) minus 119888

119904[119904(119905) minus

119906(119905)] minus 119896

119904[119911119904(119905) minus 119911

119906(119905)]

+ 119896119905[119911119906(119905) minus 119911

119903(119905)] minus 119888

119905[119903(119905) minus

119906(119905)] = minus119906 (119905)

(1)

Define the following state variables and the disturbanceinput

1205851(119905) = 119911

119904(119905) minus 119911

119906(119905) 120585

2(119905) = 119911

119906(119905) minus 119911

119903(119905)

1205853(119905) =

119904(119905) 120585

4(119905) =

119906(119905) 119908 (119905) =

119903(119905)

(2)

Then (1) is equivalent to

120585 (119905) = 119860 (119898119904) 120585 (119905) + 119861 (119898

119904) 119906 (119905) + 119861

1(119898119904) 119908 (119905) (3)

Vehicle ms

ks

kt

mu

ct

ucs

zu

zs

zr

ControllerSuspension

Wheel

Tyre

Figure 1 The quarter car model

where

120585 (119905) = [1205851(119905) 1205852(119905) 1205853(119905) 1205854(119905)]119879

119860 (119898119904) =

[[[[[[[

[

0 0 1 minus1

0 0 0 1

minus119896119904

119898119904

0 minus119888119904

119898119904

119888119904

119898119904

119896119904

119898119906

minus119896119905

119898119906

119888119904

119898119906

minus119888119904+ 119888119905

119898119906

]]]]]]]

]

119861 (119898119904) = [0 0

1

119898119904

minus1

119898119906

]

119879

1198611(119898119904) = [0 minus1 0

119888119905

119898119906

]

119879

(4)

Define

1199111199001(119905) =

119904(119905) (5)

which reflects the acceleration of 119898119904that contains the body

mass of passengers and seat In the design of control law forsuspension system body acceleration is the main index thatneeds to be optimized

Thehandling performance requires continuous contact ofwheel to road whichmeans that the suspension system needsto guarantee that dynamic tire load is less than static loadnamely

119896119905(119911119906(119905) minus 119911

119903(119905)) lt (119898

119904+ 119898119906) 119892 (6)

The stroke of the suspension could not be so large that itmay exceed the maximum which can be formulated as

1003816100381610038161003816119911119904 (119905) minus 119911119906 (119905)1003816100381610038161003816 lt 119911max (7)

The state space expression is described integrally as120585 (119905) = 119860 (119898

119904) 120585 (119905) + 119861 (119898

119904) 119906 (119905) + 119861

1119908 (119905)

1199111199001(119905) = 119862

1(119898119904) 120585 (119905) + 119863

1(119898119904) 119906 (119905)

1199111199002(119905) = 119862

2(119898119904) 120585 (119905)

119910 (119905) = 119862120585 (119905)

(8)

Mathematical Problems in Engineering 3

where 119860 119861 and 1198611are same with the definition in (4) and

1198621(119898119904) = [minus

119896119904

119898119904

0 minus119888119904

119898119904

119888119904

119898119904

]

1198622(119898119904) =

[[[

[

1

119911max0 0 0

0119896119905

(119898119904119892 + 119898

119906119892)

0 0

]]]

]

1198631(119898119904) =

1

119898119904

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

(9)

1199111199001(119905) reflects the acceleration output of 119898

119904 1199111199002(119905) rep-

resents for the relative (normalized) constraints output 119910(119905)stands for the output of measurable states

The parameter-varyingmodel is depicted as the followingpolytopic form

(119860 (119898119904) 119861 (119898

119904) 1198611(119898119904) 1198621(119898119904) 1198622(119898119904) 1198631(119898119904))

=

2

sum

119894=1

120582119894(119860119894 119861119894 1198611119894 1198621119894 1198622119894 1198631119894)

(10)

where(1198601 1198611 11986111 11986211 11986221 11986311)

= (119860 (119898119904max) 119861 (119898119904max) 1198611 (119898119904max)

1198621(119898119904max) 1198622 (119898119904max) 1198631 (119898119904max) )

(1198602 1198612 11986112 11986212 11986222 11986312)

= (119860 (119898119904min) 119861 (119898119904min) 1198611 (119898119904min)

1198621(119898119904min) 1198622 (119898119904min) 1198631 (119898119904min) )

1205821=

1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1205822=

1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

(11)

and 119898119904max 119898119904min stand for the maximum and minimum of

119898119904 respectivelyThe form of dynamic output feedback controller is

described as120578 (119905) = 119860

119870(119898119904) 120578 (119905) + 119861

119870(119898119904) 119910 (119905)

119906 (119905) = 119862119870(119898119904) 120578 (119905) + 119863

119870(119898119904) 119910 (119905)

(12)

Substituting (12) into (8) we get

(119905) = 119860 (119898119904) 119909 (119905) + 119861 (119898

119904) 119908 (119905)

1199111199001(119905) = 119862

1(119898119904) 119909 (119905)

1199111199002(119905) = 119862

2(119898119904) 119909 (119905)

(13)

where

119909 (119905) = [120585 (119905)

120578 (119905)]

119860 (119898119904)

= [119860 (119898119904) + 119861 (119898

119904)119863119870(119898119904) 119862 119861 (119898

119904) 119862119870(119898119904)

119861119870(119898119904) 119862 119860

119870(119898119904)

]

119861 (119898119904) = [

1198611(119898119904)

0]

1198621(119898119904)

= [1198621(119898119904) + 1198631(119898119904)119863119870(119898119904) 119862 119863

1(119898119904) 119862119870(119898119904)]

1198622(119898119904) = [119862

2(119898119904) 0]

(14)Denote

119866 (119895120596) = 1198621(119898119904) (119895120596119868 minus 119860 (119898

119904))minus1

119861 (119898119904) (15)

as the transfer function from 119908(119905) to 1199111199001(119905)

The119867infin

norm of transfer function matrix 119866 is applied todepict the ride comfort performance of suspension systemwhich is defined as

1003817100381710038171003817119866 (119895120596)1003817100381710038171003817infin

= sup0notin119908isin119871

2[0+infin)

1003817100381710038171003817119911119900210038171003817100381710038171198712

1199081198712

(16)

where

1003817100381710038171003817119911119900210038171003817100381710038171198712

= (int

+infin

0

10038171003817100381710038171199111199002 (119905)1003817100381710038171003817

2

119889119905)

12

1199081198712

= (int

+infin

0

119908 (119905)2

119889119905)

12

(17)

The control target is summarized as followsFor a certain 120574 design a dynamic output feedback in the

form of (12) which satisfies(I) The closed loop system is asymptotically stable(II) The finite frequency (from 120596

1to 1205962) 119867infin

norm fromroad disturbance to vehicle acceleration is less than 120574namely

1003817100381710038171003817119866 (119895120596)1003817100381710038171003817infin

1205961lt120596lt120596

2

lt 120574 (18)

(III) The relative constraint responses shown in the follow-ing can be satisfied as long as the disturbance energy isless than the maximum of the 2-norm of disturbanceinput denoted as 119908max that is

10038161003816100381610038161199111199002 (119905)119896

1003816100381610038161003816 lt 1 119896 = 1 2 (19)

3 Dynamic Output Feedback ControllerDesign for Polytopic Suspension System

In this section we will derive three theorems for the design ofoutput feedback controller that satisfies (19) and one theoremof full frequency controller used for comparison


31 Finite Frequency Design

Theorem 1 For the given parameters 120574 120578 120588 gt 0 if thereexist symmetric matrices 119875(119898

119904) positive definite symmetric

matrices 119875119878(119898119904) 119876(119898

119904) and general matrix 119882(119898

119904) satisfy-

ing

[[[[[[

[

minus sym (119882 (119898119904)) minus119882

119879

(119898119904) 119860 (119898

119904) + 119875119878(119898119904) minus119882

119879

(119898119904) minus119882

119879

(119898119904) 119861 (119898

119904)

lowast minus119875119878(119898119904) 0 0

lowast lowast minus119875119878(119898119904) 0

lowast lowast lowast minus120578119868

]]]]]]

]

lt 0 (20)

[

Ω1Ω2

lowast Ω3

] lt 0 (21)

[

minus119868 radic120588 1198622 (119898119904)119896

lowast minus119875119878(119898119904)

] lt 0 119896 = 1 2 (22)

where

Ω1= [

[

minus12059611205962119876 (119898119904) minus sym (119860

119879

(119898119904)119882 (119898

119904)) 119875 (119898

119904) minus 119895120596

119888119876 (119898119904) + 119882

119879

(119898119904)

119875 (119898119904) + 119895120596

119888119876 (119898119904) + 119882(119898

119904) minus119876 (119898

119904)

]

]

Ω2= [

[

minus119882119879

(119898119904) 119861 (119898

119904) 119862119879

1(119898119904)

0 0

]

]

Ω3= [

minus1205742

119868 0

0 minus119868

]

120596119888=(1205961+ 1205962)

2

(23)

then a dynamic output feedback controller exists which sat-isfies the requirements of (I) (II) and (III) with 119908max =

120588120578

Proof By using Schur complement inequality (20) is equiv-alent to

[Γ minus119882

119879

(119898119904) 119860 (119898

119904) + 119875119878(119898119904)

lowast minus119875119878(119898119904)

] lt 0 (24)

where

Γ =1

120578119882119879

(119898119904) 119861 (119898

119904) 119861119879

(119898119904)119882 (119898

119904)

+ 119882119879

(119898119904) 119875minus1

119878(119898119904)119882 (119898

119904) + sym (119882 (119898

119904))

(25)

Multiplying (24) by diagminus119882minus119879(119898119904) 119875minus119879

119878(119898119904) from the

left side and by diagminus119882minus1(119898119904) 119875minus1

119878(119898119904) from the right side

respectively we obtain

[

[

1

120578119861 (119898119904) 119861119879

(119898119904) + 119875minus1

119878(119898119904) minus sym (119871 (119898

119904)) 119860 (119898

119904) 119875minus1

119878(119898119904) + 119871119879

(119898119904)

lowast minus119875minus1

119878(119898119904)

]

]

lt 0 (26)

where 119871 = minus119882minus1


Applying reciprocal projection theorem (see Appendix A)and choosing 119878

119879

(119898119904) = 119860(119898

119904)119875minus1

119878(119898119904) Ψ(119898

119904) =

(1120578)119861(119898119904)119861119879

(119898119904) inequality (26) is equivalent to

119860 (119898119904) 119875minus1

119878(119898119904) + 119875minus1

119878(119898119904) 119860119879

(119898119904) +

1

120578119861 (119898119904) 119861119879

(119898119904) lt 0

(27)

Multiplying the above inequality from both the left andthe right sides by 119875

119878(119898119904) we get

119875119878(119898119904) 119860 (119898

119904) + 119860119879

(119898119904) 119875119878(119898119904)

+1

120578119875119878(119898119904) 119861 (119898

119904) 119861119879

(119898119904) 119875119878(119898119904) lt 0

(28)

which guarantees

119875119878(119898119904) 119860 (119898

119904) + 119860119879

(119898119904) 119875119878(119898119904) lt 0 (29)

obviously From the standard Lyapunov theory forcontinuous-time linear system the closed-loop system(13) is asymptotically stable with 119908(119905) = 0

By substitution inequality (21) is equivalent to

[119868 119865119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

+ sym (119865119860(119898119904)119882 (119898

119904) 119877) lt 0

(30)

where

119865119860(119898119904) =

[[

[

minus119860119879

(119898119904)

119868

minus119861119879

(119898119904)

]]

]

119865119861(119898119904) = [

[

119862119879

1(119898119904)

0

0

]

]

119877 = [119868 0 0]

Ω (119898119904) = 119879 [

Φ otimes 119875119904(119898119904) + Ψ otimes 119876 (119898

119904) 0

0 Π]119879119879

Π = [119868 0

0 minus1205742

119868] Φ = [

0 1

1 0]

Ψ = [minus1 119895120596

119888

minus119895120596119888minus12059611205962

] 119879 =

[[[

[

0 119868 0 0

119868 0 0 0

0 0 0 119868

0 0 119868 0

]]]

]

(31)

Based on projection lemma (see Appendix A) inequality(30) is equivalent to

119865perp

119860(119898119904) [119868 119865

119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

(119865perp

119860(119898119904))119879

lt 0

(32)

(119877119879

)perp

[119868 119865119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

(119877119879perp

)119879

lt 0 (33)

Noting that inequality (33) is eternal establishment wejust need to consider inequality (32) which is equivalent to

119865119879

(119898119904)Ω (119898

119904) 119865 (119898

119904) lt 0 (34)

where

119865 (119898119904) = [

119868 119860119879

(119898119904) 0 119862

119879

1(119898119904)

0 119861119879

(119898119904) 119868 0

]

119879

(35)

Rewrite inequality (34) as

[[[

[

119860 (119898119904) 119861 (119898

119904)

119868 0

1198621(119898119904) 0

0 119868

]]]

]

119879

[Φ otimes 119875 (119898

119904) + Ψ otimes 119876 (119898

119904) 0

0 Π]

times

[[[

[

119860 (119898119904) 119861 (119898

119904)

119868 0

1198621(119898119904) 0

0 119868

]]]

]

lt 0

(36)

and then we obtain

[119860 (119898119904) 119861 (119898

119904)

119868 0]

119879

(Φ otimes 119875 (119898119904) + Ψ otimes 119876 (119898

119904))

times [119860 (119898119904) 119861 (119898

119904)

119868 0] + [

1198621(119898119904) 0

0 119868]

119879

times Π[1198621(119898119904) 0

0 119868] lt 0

(37)

Applying generalized KYP lemma (see Appendix A) weget

Ξlowast

ΠΞ lt 0 1205961lt 120596 lt 120596

2 (38)

where

Ξ = [1198621(119898119904) 0

0 119868] [(119895120596119868 minus (119860 (119898

119904))minus1

119861 (119898119904)

119868] (39)

namely

sup1205961lt120596lt120596

2

1003817100381710038171003817119866 (119895120596)1003817100381710038171003817infin

lt 120574 (40)

Select

119881 (119905) = 119909119879

(119905) 119875119878(119898119904) 119909 (119905) (41)

as the Lyapunov function we obtain

(119905) = 2119909119879

(119905) 119875119878(119898119904) 119860 (119898

119904) 119909 (119905)

+ 2119909119879

(119905) 119875119878(119898119904) 119861 (119898

119904) 119908 (119905)

(42)

Applying the following inequality

2119909119879

(119905) 119875119878(119898119904) 119861 (119898

119904) 119908 (119905)

le1

120578119909119879

(119905) 119875119904(119898119904) 119861 (119898

119904) 119861119879

(119898119904) 119875119878(119898119904) (119909) 119905

+ 120578119908119879

(119905) 119908 (119905) forall120578 gt 0

(43)


and (28) to (42) we get

(119905) le 120578119908119879

(119905) 119908 (119905) (44)

Integrate (44) from 0 to 119905

119881 (119905) minus 119881 (0) le 120578int

119905

0

119908119879

(119905) 119908 (119905) 119889119905 = 120578119908 (119905)2le 120578119908max

(45)

Substituting (41) into (45) with 119881(0) = 0 we get

119909119879

(119905) 119875119878(119898119904) 119909 (119905) le 119881 (0) + 120578119908max = 120588 (46)

which is equivalent to

119909119879

(119905) 119909 (119905) le 120588 (47)

where 119909(119905) = 11987512119878(119898119904)119909(119905)

Noting that

max 10038161003816100381610038161199111199002 (119905)1198961003816100381610038161003816

2

= max100381710038171003817100381710038171003817119909119879

(119905) 119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

times 1198622(119898119904)119896

119875minus12

119878(119898119904) 119909(119905)

100381710038171003817100381710038172

le 120588120579max (119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

1198622(119898119904)119896

119875minus12

119878(119898119904))

(48)

where 120579max stands for the maximum eigenvalue we cantherefore guarantee constraints mentioned in (19) as long as

120588119875minus12

119878(119898119904) 1198622(119898119904)119879

1198961198622(119898119904)119896119875minus12

119878(119898119904) lt 119868 (49)

which is equivalent to (22) by applying Schur complement

Before giving a convex expression which can be solved byLMI Toolbox we firstly perform transformation to inequali-ties (20) (21) and (22)

Decompose matrix119882(119898119904) for convenience in the follow-

ing form

119882(119898119904) = [

119883 (119898119904) 119884 (119898

119904)

119880 (119898119904) 119881 (119898

119904)]

119882minus1

(119898119904) = [

119872(119898119904) 119866 (119898

119904)

119867 (119898119904) 119871 (119898

119904)]

(50)

According to the literature [25] we assume that both119880(119898119904) and119867(119898

119904) are invertible without loss of generality

Denote

Δ (119898119904) = [

119868 119872 (119898119904)

0 119867 (119898119904)] (51)

and perform the congruence transformation to inequalities(20) (21) and (22) by

1198691(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) Δ119879

(119898119904) 119868

1198692(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) 119868 119868

1198693(119898119904) = diag 119868 Δ119879 (119898

119904)

(52)

respectivelyDenote

119876 (119898119904) = Δ119879

(119898119904) 119876 (119898

119904) Δ (119898

119904)

(119898119904) = Δ119879

(119898119904) 119875 (119898

119904) Δ (119898

119904)

119878(119898119904) = Δ119879

(119898119904) 119875119878(119898119904) Δ (119898

119904)

119860 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119860 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119860 (119898

119904) + 119861119870(119898119904) 119862 119860

119870(119898119904)

119860 (119898119904) + 119861 (119898

119904)119863119870(119898119904) 119862 119860 (119898

119904)119872 (119898

119904) + 119861 (119898

119904) 119862119870(119898119904)]

119861 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119861 (119898

119904)

= [119883119879

(119898119904) 1198611(119898119904)

1198611(119898119904)

]

1198621(119898119904) = 1198621(119898119904) Δ (119898

119904)

= [1198621(119898119904) + 1198631(119898119904)119863119870(119898119904) 119862 119862

1(119898119904)119872 (119898

119904) + 1198631(119898119904) 119862119870(119898119904)]

1198622(119898119904) = 1198622(119898119904) Δ (119898

119904) = [119862

2(119898119904) 1198622(119898119904)119872 (119898

119904)]


(119898119904) = Δ119879

(119898119904)119882 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119885 (119898

119904)

119868 119872 (119898119904)]

(53)

then we get the following theorem

Theorem 2 For the given parameters 120574 120578 120588 gt 0 if there existsymmetric matrix (119898

119904) positive definite symmetric matrices

119878(119898119904) 119876(119898

119904) and general matrices 119860

119870(119898119904) 119861119870(119898119904)

119862119870(119898119904)119863119870(119898119904) (119898

119904)119872(119898

119904)119883(119898

119904) 119885(119898

119904) satisfying

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

lt 0 (54)

[[[[

[

minus12059611205962119876 (119898119904) minus sym (119860 (119898

119904)) (119898

119904) minus 119895120596

119888119876 (119898119904) +

119879

(119898119904) minus119861 (119898

119904) 119862119879

1(119898119904)

(119898119904) + 119895120596

119888119876 (119898119904) + (119898

119904) minus119876 (119898

119904) 0 0

minus119861119879

(119898119904) 0 minus120574

2

119868 0

1198621(119898119904) 0 0 minus119868

]]]]

]

lt 0 (55)

[minus119868 radic1205881198622 (119898119904)

119896

lowast minus119878(119898119904)

] lt 0 119896 = 1 2 (56)

then a dynamic output feedback controller exists which satisfiesthe requirements of (I) (II) and (III) with 119908max = 120588120578

The corresponding controller in the form of (12) can begiven by

119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904)

= 119880minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)

(57)Though a parameter-dependent controller can be

designed via Theorem 2 it is difficult to obtain targetedmatrices in real time as 119898

119904varies Therefore we give a

tractable LMI-based theorem as follows

Theorem 3 For the given parameters 120574 120578 and 120588 gt 0 if thereexist symmetric matrix

119894 positive definite symmetric matrices

119878119894 119876119894 and general matrices119860

119870119894 119861119870119894 119862119870119894 119863119870119894 119894119872119894 119883119894 119885119894

(119894 = 1 2) satisfying

[11986911198692

lowast 1198693

] lt 0 1 le 119894 le 119895 le 2 (58)

[1198701+ 11989511987021198703

lowast 1198704

] lt 0 1 le 119894 le 119895 le 2 (59)

[minus119868 radic1205881198622119894119895 + 1198622119895119894

119896

lowast minus119878119894minus 119878119895

] lt 0 119896 = 1 2 1 le 119894 le 119895 le 2

(60)

where

1198691= [

sym (119894+ 119895) minus119860

119894119895minus 119860119895119894+ 119878119894+ 119878119895


]

1198692= [

minus119879

119894minus 119879

119895minus119861119894119895minus 119861119895119894

0 0]

1198693= [

minus119878119894minus 119878119895

0

lowast minus2120578119868]

1198701= [

minus12059611205962(119876119894+ 119876119895) minus sym (119860

119894119895+ 119860119895119894) 119894+ 119895+ 119879

119894+ 119879

119895

119894+ 119895+ 119894+ 119895

minus119876119894minus 119876119895

]

1198702= [

0 minus120596119888(119876119894+ 119876119895)

120596119888(119876119894+ 119876119895) 0

]


1198703= [

minus119861119894119895minus 119861119895119894119862119879

1119894119895+ 119862119879

1119895119894

0 0]

1198704= [

minus21205742

119868 0

0 minus2119868]

119860119894119895= [

119883119879

119894119860119895+ 119861119870119894119862 119860

119870119894

119860119894+ 119861119894119863119870119895119862 119860119894119872119895+ 119861119894119862119870119895

]

119861119894119895= [

119883119879

1198941198611119895

1198611119894

]

1198621119894119895= [1198621119894+ 1198631119894119863119870119895119862 1198621119894119872119895+ 1198631119894119862119870119895]

1198622119894119895= [11986221198941198622119894119872119895]

119894= [

119883119879

119894119885119894

119868 119872119894

]

(61)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904) = 119880

minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)

(62)

where

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904) )

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(63)

and 120582119894(119894 = 1 2) can be calculated by (11)

Proof we just prove that inequality (58) is sufficient toinequality (54) for simplicity

Denote

Ψ (119898119904) =

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0

lowast lowast minus (119898119904) 0


]]]

]

(64)

which stands for the left of inequality (54)Inequality (58) is equivalent to

Ψ119894119895+ Ψ119895119894lt 0 1 le 119894 lt 119895 le 2

Ψ119894119894lt 0 119894 = 1 2

(65)

where

Ψ119894119895=

[[[

[

sym (119894) minus119860

119894119895+ 119878119894minus119879

119894minus119861119894119895

lowast minus119878119894

0 0

lowast lowast minus119878119894

0


]]]

]

1 le 119894 le 119895 le 2

(66)

Assume that

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904))

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(67)

and then we get

Ψ (119898119904) =

2

sum

119894=1

2

sum

119895=1

120582119894120582119895Ψ119894119895=

2

sum

119894=1

1205822

119894Ψ119894119894+ 12058211205822(Ψ12+ Ψ21) (68)

which is negative definite by inequality (65) that is

Ψ (119898119904) lt 0 (69)

Remark 4 119880(119898119904) 119867(119898

119904) should be chosen to meet the

definition that is

119880119879

(119898119904)119867 (119898

119904) = 119885 (119898

119904) minus 119883119879

(119898119904)119872 (119898

119904) (70)

However the value of 119880(119898119904) and 119867(119898

119904) can be chosen

variously for the given 119885(119898119904) 119867(119898

119904) and 119872(119898

119904) In this

paper we use the singular value decomposition approach

Remark 5 Based on [26] for real matrices 1198781and 119878

2 1198781+

1198951198782lt 0 is equivalent to [ 1198781 1198782

minus11987821198781

] lt 0 Thusly (59) can beconverted into real matrix inequality by defining

1198781= [

11987011198703

lowast 1198704

] 1198782= [

11987020

0 0] (71)


Remark 6 The matrices of dynamic output feedback con-troller 119860

119870(119898119904) 119861119870(119898119904) 119862119870(119898119904) 119863119870(119898119904) and Lyapunov

matrix 119875119878(119898119904) are dependent nonlinearly on parameter 119898

119904

For instance as can be seen in the above deduction matrix119863119870(119898119904) in Theorem 3 can be formulated as

119863119870(119898119904) = 119863

119870(119898119904) =

2

sum

119894=1

120582119894119863119870119894= 12058211198631198701+ 12058221198631198702

=1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1198631198701

+1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

1198631198702

(72)

where 1198631198701

and 1198631198702

are the corresponding matrix solutionsof LMIs in Theorem 3 and the value of matrix 119863

119870(119898119904) may

vary with the change of119898119904

Remark 7 Inequalities from (58) to (60) can be simplifiedfrom 12 LMIs to 4 LMIs if the following matrix variables arechosen as

1= 2 1198781

= 1198782 1198761= 1198762 1198601198701

= 1198601198702

1198611198701= 1198611198702 1198621198701= 1198621198702 1198631198701= 1198631198702 1= 21198721= 1198722

1198831= 1198832 and119885

1= 1198852 However this simplification will keep

the matrices for the designed controller constant for all 119898119904

and an invariant Lyapunov function rather than a parameter-dependent one has to be used for the whole domain whichwill bring a larger conservativeness thanTheorem 3

32 Full Frequency Design Based on the theorem of [27] weformulate Theorem 8 without proof for simplicity

Theorem 8 For the given parameter 120574 gt 0 if there existpositive definite symmetric matrices 119884

119862119894 119883119862119894

and generalmatrices 119860

119862119894 119861119862119894 119862119862119894 119863119862119894 (119894 = 1 2) satisfying

[[

[

sym (119860119890119894119895+ 119860119890119895119894) 119861119890119894119895

119862119879

1198901119894119895+ 119862119879

1198901119895119894


119868 0


]]

]

lt 0

1 le 119894 le 119895 le 2

(73)

[minus2119868 radic1205881198621198902119894119895 + 1198621198902119895119894

119896


] lt 0

119896 = 1 2 1 le 119894 le 119895 le 2

(74)

where

119860119890119894119895= [

119860119894119883119862119895+ 119861119894119862119862119895

119860119894+ 119861119894119863119862119895119862

119860119862119894

119884119862119894119860119895+ 119861119862119894119862]

119861119890119894119895= [

1198611119894

1198841198621198941198611119895

]

1198621198901119894119895

= [1198621119894119883119862119895+ 11986311198941198621198621198951198621119894+ 1198631119894119863119862119895119862]

1198621198902119894119895

= [11986221198941198831198621198951198622119894]

119875119862119894

= [119883119862119894

119868

119868 119884119862119894

]

(75)

then a dynamic output feedback controller exists which satisfies(I) and (III) and 119866(119895120596)

infinlt 120574 with 119908max = 120588120578


119863119870(119898119904) = 119863

119862(119898119904) (76)

119862119870(119898119904) = (119862

119862(119898119904) minus 119863119862(119898119904) 119862119883119862(119898119904))119872minus119879

119862(119898119904)

(77)

119861119870(119898119904) = 119873

minus1

119862(119898119904) (119861119862(119898119904) minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904))

(78)

119860119870(119898119904)

= 119873minus1

119862(119898119904) [119860119862(119898119904) minus 119884119862(119898119904) 119860 (119898

119904)119883119862(119898119904)

minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904) 119862119883119862(119898119904)

minus 119873119862(119898119904) 119861119870(119898119904) 119862119883119862(119898119904)

minus119884119862(119898119904) 119861 (119898

119904) 119862119870(119898119904)119872119879

119862]

times119872minus119879

119862(119898119904)

(79)

where

(119860119862(119898119904) 119861119862(119898119904) 119862119862(119898119904) 119863119862(119898119904) 119883119862(119898119904) 119884119862(119898119904))

=

2

sum

119894=1

120582119894(119860119862119894 119861119862119894 119862119862119894 119863119862119894 119883119862119894 119884119862119894)

(80)


The same approach mentioned in Remark 4 is used toobtain the value of119873

119862(119898119904) and119872

119862(119898119904) which should meet

the following equation

119873119862(119898119904)119872119879

119862(119898119904) = 119868 minus 119884

119862(119898119904)119883119862(119898119904) (81)

4 A Design Example

In this section we will show how to apply the above theoremsto design finite frequency controller and full frequencycontroller for a specific suspension systemThe parameter forthe suspension is shown in Table 1

We first choose the following parameters 120588 = 1 120578 =

10000 119911max = 01m1205961= 8120587rads and120596

2= 16120587radsThen

the matrices in (12) can be calculated by applying Theorem 3


Table 1 The model parameters of active suspensions

119896119904

119896119905

119888119904

119888119905

119898119904max 119898

119904min 119898119904

119898119906

18 kNm 200 kNm 1 kNm 10Nm 384 kg 254 kg 320 kg 40 kg

Frequency (Hz)

Max

imum

sing

ular

val

ues (

abs)

10minus1

100

101

102

Finite frequencyFull frequencyOpen loop

4ndash8 Hz

25

20

15

10

5

0

Figure 2 The curves of maximum singular values

and Remark 7 for finite frequency controller and Theorem 8for full frequency controller (shown in Appendix B) with theleast guaranteed 119867

infinperformances 120574 = 254 for Theorem 3

120574 = 273 for Remark 6 and 120574 = 1098 for Theorem 8Then the curves of maximum singular values of systems

using open-loop finite frequency (parameter-dependent)and full frequency controllers are shown in Figure 2 Com-pared with the other two curves (the open-loop system andthe system with full frequency controller) the system withfinite frequency controller has the least119867

infinnorm of the three

systems in 4ndash8Hz which indicates that the finite frequencycontroller has a better effect on the attenuation of targetedfrequency disturbance

In order to examine the performance of finite frequencycontroller we assume that the disturbance is in the form of

119908 (119905) = 119860 sin (2120587119891119905) 119905 isin [0 119879]

0 119905 notin [0 119879] (82)

where 119860 119891 stand for the amplitude and the frequency ofdisturbance respectively and 119879 = 1119891

Suppose that 119860 is 04m and 119891 is 5Hz and then thebody acceleration and the relative constraints responses tothis disturbance are shown in Figures 3 4 and 5 respectively

It is obvious that the body acceleration response for finitefrequency controller decreases faster with respect to timethan the other two controllers and at the same time both therelative dynamic tire load response and relative suspensionstroke response are within the allowable range which satisfyrequirement (III) namely |119911

1199002(119905)119896| lt 1 119896 = 1 2

5 Conclusion

In this paper we manage to design a dynamic outputfeedback controller for active suspensions with practical

0 02 04 06 08 1 12 14 16 18 2

005

115

225

Time (seconds)

minus25

minus1

minus2

minus15

minus05


Body

accle

ratio

n (m

s2)

Figure 3 The time response of body acceleration

0 02 04 06 08 1 12 14 16 18 2

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

0

Relat

ive d

ynam

ic ti

re lo

ad

Figure 4 The time response of relative dynamic tire load

0 02 04 06 08 1 12 14 16 18 2

0

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

Rela

tive s

uspe

nsio

n str

oke

Figure 5 The time response of relative suspension stroke


constraints included This controller particularly diminishesdisturbance at 4ndash8Hz which is the frequency sensitive bandfor human Besides for the reason that the controller is aparameter-dependent one it has a smaller conservativenessthan controller designed on the basis of quadratic stabilityand constant parameter feedbackThe excellent performanceof the closed-loop system with finite frequency controller hasbeen demonstrated by simulation

Appendices

A Related Lemmas

Lemma A1 (generalized KYP lemma [28]) Let matrices Θ119865 Φ and Ψ be given Denote by 119873

120596the null space of 119879

120596119865

where 119879120596= [119868 minus119895120596119868] The inequality

119873lowast

120596Θ119873120596lt 0 120596 isin [120596

11205962] (A1)

holds if and only if there exist 119875 119876 gt 0 such that

119865lowast

(Φ otimes 119875 + Ψ otimes 119876) 119865 lt 0 (A2)

where

Φ = [0 1

1 0] Ψ = [

minus1 119895120596119888

minus119895120596119888minus12059611205962

] (A3)

with 120596119888= (1205961+ 1205962)2

Lemma A2 (projection lemma [25]) Let Γ Λ and Θ begiven there exists a matrix satisfying

sym (ΓΛΘ) + Θ lt 0 (A4)

if and only if the following two conditions hold

Γperp

Θ(Γperp

)119879

lt 0 (Λ119879

)perp

Θ((Λ119879

)perp

)

119879

lt 0 (A5)

Lemma A3 (reciprocal projection lemma [25]) Let 119875 beany given positive definite matrixThe following statements areequivalent

(1) Ψ + 119878 + 119878119879 lt 0

(2) The LMI problem

[Ψ + 119875 minus [119882

119904] 119878119879

+119882119879

lowast minus119875] lt 0 (A6)

is feasible with respect to119883

B Controller Matrices

The parameter matrices of the dynamic output feedbackcontroller for Theorem 3

119860119870=

[[[

[

minus14821 minus012731 minus11424 000255

minus22844 minus13692 13701 002030

20381 15159 minus21603 minus047598

24991 times 109

minus74319 times 107

42012 times 106


]]]

]

119861119870=

[[[

[

99943 10908 44969

minus31706 minus29568 minus89161

minus63200 11564 minus11258




]]]

]

119862119870= [38144 minus68619 17589 35961]

119863119870= [minus18676 times 10

5


minus86675]

(B1)


119860119870=

[[[

[

minus12734 26225 21367 minus031502

minus36286 14954 36984 069500

minus11101 minus84507 minus65487 minus00771


]]]

]

119861119870=

[[[

[


17709 12215 11303

minus90855 34487 minus80239

64417 times 105

71058 times 106


]]]

]

119862119870= [51819 minus60403 minus59041 10015]

119863119870= [25422 17208 26098]

(B2)

References

[1] D Hrovat ldquoSurvey of advanced suspension developments andrelated optimal control applicationsrdquoAutomatica vol 33 no 10pp 1781ndash1817 1997

[2] W Sun H Gao andO Kaynak ldquoAdaptive backstepping controlfor active suspension systems with hard constraintsrdquo IEEETransactions on Mechatronics vol 18 pp 1072ndash1079 2013


[3] W Sun H Gao and B Yao ldquoAdaptive robust vibration controlof full-car active suspensions with electrohydraulic actuatorsrdquoIEEE Transactions on Control Systems Technology 2013

[4] I Fialho and G J Balas ldquoRoad adaptive active suspensiondesign using linear parameter-varying gain-schedulingrdquo IEEETransactions on Control Systems Technology vol 10 no 1 pp43ndash54 2002

[5] H Zhang Y Shi and A Saadat Mehr ldquoRobust static outputfeedback control and remote PID design for networked motorsystemsrdquo IEEETransactions on Industrial Electronics vol 58 no12 pp 5396ndash5405 2011

[6] H Zhang Y Shi and A Saadat Mehr ldquoRobust non-fragiledynamic vibration absorbers with uncertain factorsrdquo Journal ofSound and Vibration vol 330 no 4 pp 559ndash566 2011

[7] M Yu S B Choi X M Dong and C R Liao ldquoFuzzy neuralnetwork control for vehicle stability utilizing magnetorheolog-ical suspension systemrdquo Journal of Intelligent Material Systemsand Structures vol 20 no 4 pp 457ndash466 2009

[8] D E Williams and W M Haddad ldquoActive suspension controlto improve vehicle ride and handlingrdquoVehicle SystemDynamicsvol 28 no 1 pp 1ndash24 1997

[9] A G Ulsoy D Hrovat and T Tseng ldquoStability robustness ofLQ and LQG active suspensionsrdquo Journal of Dynamic SystemsMeasurement and Control Transactions of the ASME vol 116no 1 pp 123ndash131 1994

[10] W Sun H Gao Sr and O Kaynak ldquoFinite frequency119867infincon-

trol for vehicle active suspension systemsrdquo IEEETransactions onControl Systems Technology vol 19 no 2 pp 416ndash422 2011

[11] W Sun J Li Y Zhao and H Gao ldquoVibration control foractive seat suspension systems via dynamic output feedbackwith limited frequency characteristicrdquoMechatronics vol 21 no1 pp 250ndash260 2011

[12] H Chen and K-H Guo ldquoConstrained 119867infin

control of activesuspensions an LMI approachrdquo IEEE Transactions on ControlSystems Technology vol 13 no 3 pp 412ndash421 2005

[13] H R Karimi N Luo M Zapateiro and L Zhang ldquo119867infincontrol

design for building structures under seismic motion withwireless communicationrdquo International Journal of InnovativeComputing Information and Control vol 7 no 9 pp 5269ndash5284 2011

[14] H Chen Z-Y Liu and P-Y Sun ldquoApplication of constrained119867infin

control to active suspension systems on half-car modelsrdquoJournal of Dynamic Systems Measurement and Control Trans-actions of the ASME vol 127 no 3 pp 345ndash354 2005

[15] H Du J Lam and K Y Sze ldquoNon-fragile output feedback119867infin

vehicle suspension control using genetic algorithmrdquo Engineer-ing Applications of Artificial Intelligence vol 16 no 7-8 pp 667ndash680 2003

[16] M Yamashita K Fujimori K Hayakawa and H KimuraldquoApplication of 119867

infincontrol to active suspension systemsrdquo

Automatica vol 30 no 11 pp 1717ndash1729 1994[17] H Zhang Y Shi and M Liu ldquo119867

infinstep tracking control for

networked discrete-time nonlinear systems with integral andpredictive actionsrdquo IEEE Transactions on Industrial Informaticsvol 9 pp 337ndash345 2013

[18] M C Smith and F-C Wang ldquoController parameterization fordisturbance response decoupling application to vehicle activesuspension controlrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 3 pp 393ndash407 2002

[19] W Sun Y Zhao J Li L Zhang and H Gao ldquoActive suspensioncontrol with frequency band constraints and actuator input

delayrdquo IEEE Transactions on Industrial Electronics vol 59 no1 pp 530ndash537 2012

[20] W Sun Z Zhao andH Gao ldquoSaturated adaptive robust controlof active suspension systemsrdquo IEEE Transactions on IndustrialElectronics vol 60 pp 3889ndash3896 2013

[21] H Gao J Lam andCWang ldquoMulti-objective control of vehicleactive suspension systems via load-dependent controllersrdquoJournal of Sound and Vibration vol 290 no 3-5 pp 654ndash6752006

[22] D Sammier O Sename and L Dugard ldquoSkyhook and 119867infin

control of semi-active suspensions some practical aspectsrdquoVehicle System Dynamics vol 39 no 4 pp 279ndash308 2003

[23] T Iwasaki S Hara and H Yamauchi ldquoDynamical systemdesign from a control perspective finite frequency positive-realness approachrdquo IEEE Transactions on Automatic Controlvol 48 no 8 pp 1337ndash1354 2003

[24] B Hencey and A G Alleyne ldquoA KYP lemma for LMI regionsrdquoIEEE Transactions on Automatic Control vol 52 no 10 pp1926ndash1930 2007

[25] P Apkarian H D Tuan and J Bernussou ldquoContinuous-timeanalysis eigenstructure assignment and H

2synthesis with

enhanced linear matrix inequalities (LMI) characterizationsrdquoIEEE Transactions on Automatic Control vol 46 no 12 pp1941ndash1946 2001

[26] P Gahinet A Nemirovskii A Laub and M Chilali LMIControl Toolbox Users Guide The Math Works Natick MassUSA 1995

[27] C Scherer P Gahinet and M Chilali ldquoMultiobjective output-feedback control via LMI optimizationrdquo IEEE Transactions onAutomatic Control vol 42 no 7 pp 896ndash911 1997

[28] J Collado R Lozano and R Johansson ldquoOn Kalman-Yakubovich-Popov Lemma for stabilizable systemsrdquo IEEETransactions on Automatic Control vol 46 no 7 pp 1089ndash10932001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


can be accessed on line in this paper the suspension modelis described as a polytope and then a parameter-dependentcontrol law is proposed to assure the above performancerequirements which can also reach a lower conservativenessthan control law based on quadratic stability and constantparameter feedback at the same time In addition thoughstate feedback may attain a superior performance comparedwith static output feedback measuring some statesmay bringburden to the systems Thusly constructing dynamic outputfeedback is desirable which can achieve a relative enhancedperformance andmeanwhile reduce state-measuring sensors

The paper is organized as follows In Section 2 the state-space model for quarter car suspension is presented InSection 3 the theorems which can be used to design thedynamic feedback controller are illustrated The simulationis presented in Section 4 and the conclusion is in Section 5

Notation For a matrix 119875 119875119879 119875lowast 119875minus1 119875minus119879 and 119875perp standfor its transpose conjugate transpose inverse transposedinverse and orthogonal complement respectively sym(119875)denotes 119875+119875119879 119875 gt 0 (119875 lt 0)means that matrix 119875 is positive(negative) definite For a matrix sdot

119894stands for the 119894th line

of the matrix 119866(119895120596)infin

stands for the119867infin

norm of transferfunction matrix 119866 For matrices 119875 and 119876 119875 otimes 119876 means theKronecker product In symmetric block matrices or complexmatrix expressions we use an asterisk (lowast) to represent a termthat is induced by symmetry

2 Quarter Car Suspension Model

The model of a quarter car suspension is shown in Figure 1119898119904and119898

119906stand for sprung and unsprungmass respectively

119911119904 119911119906 and 119911

119903denote the sprung unsprung displacement and

disturbance displacement from the road respectively 119896119904 119896119905

119888119904 and 119888

119905are the stiffnesses and dampings of the suspension

system respectivelyThe input of the controller is denoted by119906

Based on the law of Newton the motion equation ofsuspension can be denoted as

119898119904119904(119905) + 119888

119904[119904(119905) minus

119906(119905)] + 119896

119904[119911119904(119905) minus 119911

119906(119905)] = 119906 (119905)

119898119906119906(119905) minus 119888

119904[119904(119905) minus

119906(119905)] minus 119896

119904[119911119904(119905) minus 119911

119906(119905)]

+ 119896119905[119911119906(119905) minus 119911

119903(119905)] minus 119888

119905[119903(119905) minus

119906(119905)] = minus119906 (119905)

(1)

Define the following state variables and the disturbanceinput

1205851(119905) = 119911

119904(119905) minus 119911

119906(119905) 120585

2(119905) = 119911

119906(119905) minus 119911

119903(119905)

1205853(119905) =

119904(119905) 120585

4(119905) =

119906(119905) 119908 (119905) =

119903(119905)

(2)

Then (1) is equivalent to

120585 (119905) = 119860 (119898119904) 120585 (119905) + 119861 (119898

119904) 119906 (119905) + 119861

1(119898119904) 119908 (119905) (3)

Vehicle ms

ks

kt

mu

ct

ucs

zu

zs

zr

ControllerSuspension

Wheel

Tyre

Figure 1 The quarter car model

where

120585 (119905) = [1205851(119905) 1205852(119905) 1205853(119905) 1205854(119905)]119879

119860 (119898119904) =

[[[[[[[

[

0 0 1 minus1

0 0 0 1

minus119896119904

119898119904

0 minus119888119904

119898119904

119888119904

119898119904

119896119904

119898119906

minus119896119905

119898119906

119888119904

119898119906

minus119888119904+ 119888119905

119898119906

]]]]]]]

]

119861 (119898119904) = [0 0

1

119898119904

minus1

119898119906

]

119879

1198611(119898119904) = [0 minus1 0

119888119905

119898119906

]

119879

(4)

Define

1199111199001(119905) =

119904(119905) (5)

which reflects the acceleration of 119898119904that contains the body

mass of passengers and seat In the design of control law forsuspension system body acceleration is the main index thatneeds to be optimized

Thehandling performance requires continuous contact ofwheel to road whichmeans that the suspension system needsto guarantee that dynamic tire load is less than static loadnamely

119896119905(119911119906(119905) minus 119911

119903(119905)) lt (119898

119904+ 119898119906) 119892 (6)

The stroke of the suspension could not be so large that itmay exceed the maximum which can be formulated as

1003816100381610038161003816119911119904 (119905) minus 119911119906 (119905)1003816100381610038161003816 lt 119911max (7)

The state space expression is described integrally as120585 (119905) = 119860 (119898

119904) 120585 (119905) + 119861 (119898

119904) 119906 (119905) + 119861

1119908 (119905)

1199111199001(119905) = 119862

1(119898119904) 120585 (119905) + 119863

1(119898119904) 119906 (119905)

1199111199002(119905) = 119862

2(119898119904) 120585 (119905)

119910 (119905) = 119862120585 (119905)

(8)



1198621(119898119904) = [minus

119896119904

119898119904

0 minus119888119904

119898119904

119888119904

119898119904

]

1198622(119898119904) =

[[[

[

1

119911max0 0 0

0119896119905

(119898119904119892 + 119898

119906119892)

0 0

]]]

]

1198631(119898119904) =

1

119898119904

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

(9)


119904 1199111199002(119905) rep-



(119860 (119898119904) 119861 (119898

119904) 1198611(119898119904) 1198621(119898119904) 1198622(119898119904) 1198631(119898119904))

=

2

sum

119894=1

120582119894(119860119894 119861119894 1198611119894 1198621119894 1198622119894 1198631119894)

(10)

where(1198601 1198611 11986111 11986211 11986221 11986311)

= (119860 (119898119904max) 119861 (119898119904max) 1198611 (119898119904max)

1198621(119898119904max) 1198622 (119898119904max) 1198631 (119898119904max) )

(1198602 1198612 11986112 11986212 11986222 11986312)

= (119860 (119898119904min) 119861 (119898119904min) 1198611 (119898119904min)

1198621(119898119904min) 1198622 (119898119904min) 1198631 (119898119904min) )

1205821=

1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1205822=

1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

(11)



described as120578 (119905) = 119860

119870(119898119904) 120578 (119905) + 119861

119870(119898119904) 119910 (119905)

119906 (119905) = 119862119870(119898119904) 120578 (119905) + 119863

119870(119898119904) 119910 (119905)

(12)


(119905) = 119860 (119898119904) 119909 (119905) + 119861 (119898

119904) 119908 (119905)

1199111199001(119905) = 119862

1(119898119904) 119909 (119905)

1199111199002(119905) = 119862

2(119898119904) 119909 (119905)

(13)

where

119909 (119905) = [120585 (119905)

120578 (119905)]

119860 (119898119904)

= [119860 (119898119904) + 119861 (119898

119904)119863119870(119898119904) 119862 119861 (119898

119904) 119862119870(119898119904)

119861119870(119898119904) 119862 119860

119870(119898119904)

]

119861 (119898119904) = [

1198611(119898119904)

0]

1198621(119898119904)

= [1198621(119898119904) + 1198631(119898119904)119863119870(119898119904) 119862 119863

1(119898119904) 119862119870(119898119904)]

1198622(119898119904) = [119862

2(119898119904) 0]

(14)Denote

119866 (119895120596) = 1198621(119898119904) (119895120596119868 minus 119860 (119898

119904))minus1

119861 (119898119904) (15)


The119867infin


1003817100381710038171003817119866 (119895120596)1003817100381710038171003817infin


2[0+infin)

1003817100381710038171003817119911119900210038171003817100381710038171198712

1199081198712

(16)

where

1003817100381710038171003817119911119900210038171003817100381710038171198712

= (int

+infin

0

10038171003817100381710038171199111199002 (119905)1003817100381710038171003817

2

119889119905)

12

1199081198712

= (int

+infin

0

119908 (119905)2

119889119905)

12

(17)



1to 1205962) 119867infin


1003817100381710038171003817119866 (119895120596)1003817100381710038171003817infin

1205961lt120596lt120596

2

lt 120574 (18)


10038161003816100381610038161199111199002 (119905)119896

1003816100381610038161003816 lt 1 119896 = 1 2 (19)







matrices 119875119878(119898119904) 119876(119898


119904) satisfy-

ing

[[[[[[

[

minus sym (119882 (119898119904)) minus119882

119879

(119898119904) 119860 (119898

119904) + 119875119878(119898119904) minus119882

119879

(119898119904) minus119882

119879

(119898119904) 119861 (119898

119904)

lowast minus119875119878(119898119904) 0 0



]]]]]]

]

lt 0 (20)

[

Ω1Ω2

lowast Ω3

] lt 0 (21)

[

minus119868 radic120588 1198622 (119898119904)119896

lowast minus119875119878(119898119904)

] lt 0 119896 = 1 2 (22)

where

Ω1= [

[

minus12059611205962119876 (119898119904) minus sym (119860

119879

(119898119904)119882 (119898

119904)) 119875 (119898

119904) minus 119895120596

119888119876 (119898119904) + 119882

119879

(119898119904)

119875 (119898119904) + 119895120596

119888119876 (119898119904) + 119882(119898

119904) minus119876 (119898

119904)

]

]

Ω2= [

[

minus119882119879

(119898119904) 119861 (119898

119904) 119862119879

1(119898119904)

0 0

]

]

Ω3= [

minus1205742

119868 0

0 minus119868

]

120596119888=(1205961+ 1205962)

2

(23)


120588120578


[Γ minus119882

119879

(119898119904) 119860 (119898

119904) + 119875119878(119898119904)

lowast minus119875119878(119898119904)

] lt 0 (24)

where

Γ =1

120578119882119879

(119898119904) 119861 (119898

119904) 119861119879

(119898119904)119882 (119898

119904)

+ 119882119879

(119898119904) 119875minus1

119878(119898119904)119882 (119898

119904) + sym (119882 (119898

119904))

(25)


119878(119898119904) from the




[

[

1

120578119861 (119898119904) 119861119879

(119898119904) + 119875minus1

119878(119898119904) minus sym (119871 (119898

119904)) 119860 (119898

119904) 119875minus1

119878(119898119904) + 119871119879

(119898119904)


119878(119898119904)

]

]

lt 0 (26)




119879

(119898119904) = 119860(119898

119904)119875minus1

119878(119898119904) Ψ(119898

119904) =

(1120578)119861(119898119904)119861119879


119860 (119898119904) 119875minus1

119878(119898119904) + 119875minus1

119878(119898119904) 119860119879

(119898119904) +

1

120578119861 (119898119904) 119861119879

(119898119904) lt 0

(27)


119878(119898119904) we get

119875119878(119898119904) 119860 (119898

119904) + 119860119879

(119898119904) 119875119878(119898119904)

+1

120578119875119878(119898119904) 119861 (119898

119904) 119861119879

(119898119904) 119875119878(119898119904) lt 0

(28)

which guarantees

119875119878(119898119904) 119860 (119898

119904) + 119860119879

(119898119904) 119875119878(119898119904) lt 0 (29)



[119868 119865119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

+ sym (119865119860(119898119904)119882 (119898

119904) 119877) lt 0

(30)

where

119865119860(119898119904) =

[[

[

minus119860119879

(119898119904)

119868

minus119861119879

(119898119904)

]]

]

119865119861(119898119904) = [

[

119862119879

1(119898119904)

0

0

]

]

119877 = [119868 0 0]

Ω (119898119904) = 119879 [

Φ otimes 119875119904(119898119904) + Ψ otimes 119876 (119898

119904) 0

0 Π]119879119879

Π = [119868 0

0 minus1205742

119868] Φ = [

0 1

1 0]

Ψ = [minus1 119895120596

119888

minus119895120596119888minus12059611205962

] 119879 =

[[[

[

0 119868 0 0

119868 0 0 0

0 0 0 119868

0 0 119868 0

]]]

]

(31)


119865perp

119860(119898119904) [119868 119865

119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

(119865perp

119860(119898119904))119879

lt 0

(32)

(119877119879

)perp

[119868 119865119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

(119877119879perp

)119879

lt 0 (33)


119865119879

(119898119904)Ω (119898

119904) 119865 (119898

119904) lt 0 (34)

where

119865 (119898119904) = [

119868 119860119879

(119898119904) 0 119862

119879

1(119898119904)

0 119861119879

(119898119904) 119868 0

]

119879

(35)


[[[

[

119860 (119898119904) 119861 (119898

119904)

119868 0

1198621(119898119904) 0

0 119868

]]]

]

119879

[Φ otimes 119875 (119898

119904) + Ψ otimes 119876 (119898

119904) 0

0 Π]

times

[[[

[

119860 (119898119904) 119861 (119898

119904)

119868 0

1198621(119898119904) 0

0 119868

]]]

]

lt 0

(36)

and then we obtain

[119860 (119898119904) 119861 (119898

119904)

119868 0]

119879

(Φ otimes 119875 (119898119904) + Ψ otimes 119876 (119898

119904))

times [119860 (119898119904) 119861 (119898

119904)

119868 0] + [

1198621(119898119904) 0

0 119868]

119879

times Π[1198621(119898119904) 0

0 119868] lt 0

(37)


Ξlowast

ΠΞ lt 0 1205961lt 120596 lt 120596

2 (38)

where

Ξ = [1198621(119898119904) 0

0 119868] [(119895120596119868 minus (119860 (119898

119904))minus1

119861 (119898119904)

119868] (39)

namely

sup1205961lt120596lt120596

2

1003817100381710038171003817119866 (119895120596)1003817100381710038171003817infin

lt 120574 (40)

Select

119881 (119905) = 119909119879

(119905) 119875119878(119898119904) 119909 (119905) (41)


(119905) = 2119909119879

(119905) 119875119878(119898119904) 119860 (119898

119904) 119909 (119905)

+ 2119909119879

(119905) 119875119878(119898119904) 119861 (119898

119904) 119908 (119905)

(42)


2119909119879

(119905) 119875119878(119898119904) 119861 (119898

119904) 119908 (119905)

le1

120578119909119879

(119905) 119875119904(119898119904) 119861 (119898

119904) 119861119879

(119898119904) 119875119878(119898119904) (119909) 119905

+ 120578119908119879

(119905) 119908 (119905) forall120578 gt 0

(43)



(119905) le 120578119908119879

(119905) 119908 (119905) (44)


119881 (119905) minus 119881 (0) le 120578int

119905

0

119908119879

(119905) 119908 (119905) 119889119905 = 120578119908 (119905)2le 120578119908max

(45)


119909119879

(119905) 119875119878(119898119904) 119909 (119905) le 119881 (0) + 120578119908max = 120588 (46)


119909119879

(119905) 119909 (119905) le 120588 (47)

where 119909(119905) = 11987512119878(119898119904)119909(119905)

Noting that

max 10038161003816100381610038161199111199002 (119905)1198961003816100381610038161003816

2

= max100381710038171003817100381710038171003817119909119879

(119905) 119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

times 1198622(119898119904)119896

119875minus12

119878(119898119904) 119909(119905)

100381710038171003817100381710038172

le 120588120579max (119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

1198622(119898119904)119896

119875minus12

119878(119898119904))

(48)


120588119875minus12

119878(119898119904) 1198622(119898119904)119879

1198961198622(119898119904)119896119875minus12

119878(119898119904) lt 119868 (49)




ing form

119882(119898119904) = [

119883 (119898119904) 119884 (119898

119904)

119880 (119898119904) 119881 (119898

119904)]

119882minus1

(119898119904) = [

119872(119898119904) 119866 (119898

119904)

119867 (119898119904) 119871 (119898

119904)]

(50)



Denote

Δ (119898119904) = [

119868 119872 (119898119904)

0 119867 (119898119904)] (51)


1198691(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) Δ119879

(119898119904) 119868

1198692(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) 119868 119868

1198693(119898119904) = diag 119868 Δ119879 (119898

119904)

(52)

respectivelyDenote

119876 (119898119904) = Δ119879

(119898119904) 119876 (119898

119904) Δ (119898

119904)

(119898119904) = Δ119879

(119898119904) 119875 (119898

119904) Δ (119898

119904)

119878(119898119904) = Δ119879

(119898119904) 119875119878(119898119904) Δ (119898

119904)

119860 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119860 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119860 (119898

119904) + 119861119870(119898119904) 119862 119860

119870(119898119904)

119860 (119898119904) + 119861 (119898

119904)119863119870(119898119904) 119862 119860 (119898

119904)119872 (119898

119904) + 119861 (119898

119904) 119862119870(119898119904)]

119861 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119861 (119898

119904)

= [119883119879

(119898119904) 1198611(119898119904)

1198611(119898119904)

]

1198621(119898119904) = 1198621(119898119904) Δ (119898

119904)

= [1198621(119898119904) + 1198631(119898119904)119863119870(119898119904) 119862 119862

1(119898119904)119872 (119898

119904) + 1198631(119898119904) 119862119870(119898119904)]

1198622(119898119904) = 1198622(119898119904) Δ (119898

119904) = [119862

2(119898119904) 1198622(119898119904)119872 (119898

119904)]


(119898119904) = Δ119879

(119898119904)119882 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119885 (119898

119904)

119868 119872 (119898119904)]

(53)




119878(119898119904) 119876(119898


119870(119898119904) 119861119870(119898119904)

119862119870(119898119904)119863119870(119898119904) (119898

119904)119872(119898

119904)119883(119898

119904) 119885(119898

119904) satisfying

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

lt 0 (54)

[[[[

[

minus12059611205962119876 (119898119904) minus sym (119860 (119898

119904)) (119898

119904) minus 119895120596

119888119876 (119898119904) +

119879

(119898119904) minus119861 (119898

119904) 119862119879

1(119898119904)

(119898119904) + 119895120596

119888119876 (119898119904) + (119898

119904) minus119876 (119898

119904) 0 0

minus119861119879

(119898119904) 0 minus120574

2

119868 0

1198621(119898119904) 0 0 minus119868

]]]]

]

lt 0 (55)

[minus119868 radic1205881198622 (119898119904)

119896

lowast minus119878(119898119904)

] lt 0 119896 = 1 2 (56)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904)

= 119880minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)








119870119894 119861119870119894 119862119870119894 119863119870119894 119894119872119894 119883119894 119885119894


[11986911198692

lowast 1198693

] lt 0 1 le 119894 le 119895 le 2 (58)

[1198701+ 11989511987021198703

lowast 1198704

] lt 0 1 le 119894 le 119895 le 2 (59)

[minus119868 radic1205881198622119894119895 + 1198622119895119894

119896


] lt 0 119896 = 1 2 1 le 119894 le 119895 le 2

(60)

where

1198691= [

sym (119894+ 119895) minus119860

119894119895minus 119860119895119894+ 119878119894+ 119878119895


]

1198692= [

minus119879

119894minus 119879

119895minus119861119894119895minus 119861119895119894

0 0]

1198693= [

minus119878119894minus 119878119895

0


1198701= [

minus12059611205962(119876119894+ 119876119895) minus sym (119860

119894119895+ 119860119895119894) 119894+ 119895+ 119879

119894+ 119879

119895

119894+ 119895+ 119894+ 119895

minus119876119894minus 119876119895

]

1198702= [

0 minus120596119888(119876119894+ 119876119895)

120596119888(119876119894+ 119876119895) 0

]


1198703= [

minus119861119894119895minus 119861119895119894119862119879

1119894119895+ 119862119879

1119895119894

0 0]

1198704= [

minus21205742

119868 0

0 minus2119868]

119860119894119895= [

119883119879

119894119860119895+ 119861119870119894119862 119860

119870119894

119860119894+ 119861119894119863119870119895119862 119860119894119872119895+ 119861119894119862119870119895

]

119861119894119895= [

119883119879

1198941198611119895

1198611119894

]

1198621119894119895= [1198621119894+ 1198631119894119863119870119895119862 1198621119894119872119895+ 1198631119894119862119870119895]

1198622119894119895= [11986221198941198622119894119872119895]

119894= [

119883119879

119894119885119894

119868 119872119894

]

(61)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904) = 119880

minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)

(62)

where

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904) )

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(63)



Denote

Ψ (119898119904) =

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

(64)


Ψ119894119895+ Ψ119895119894lt 0 1 le 119894 lt 119895 le 2

Ψ119894119894lt 0 119894 = 1 2

(65)

where

Ψ119894119895=

[[[

[

sym (119894) minus119860

119894119895+ 119878119894minus119879

119894minus119861119894119895


0 0


0


]]]

]

1 le 119894 le 119895 le 2

(66)

Assume that

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904))

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(67)

and then we get

Ψ (119898119904) =

2

sum

119894=1

2

sum

119895=1

120582119894120582119895Ψ119894119895=

2

sum

119894=1

1205822

119894Ψ119894119894+ 12058211205822(Ψ12+ Ψ21) (68)


Ψ (119898119904) lt 0 (69)

Remark 4 119880(119898119904) 119867(119898


definition that is

119880119879

(119898119904)119867 (119898

119904) = 119885 (119898

119904) minus 119883119879

(119898119904)119872 (119898

119904) (70)




119904) and 119872(119898

119904) In this



2 1198781+


minus11987821198781


1198781= [

11987011198703

lowast 1198704

] 1198782= [

11987020

0 0] (71)



119870(119898119904) 119861119870(119898119904) 119862119870(119898119904) 119863119870(119898119904) and Lyapunov


119904


119863119870(119898119904) = 119863

119870(119898119904) =

2

sum

119894=1

120582119894119863119870119894= 12058211198631198701+ 12058221198631198702

=1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1198631198701

+1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

1198631198702

(72)

where 1198631198701

and 1198631198702


119870(119898119904) may



1= 2 1198781

= 1198782 1198761= 1198762 1198601198701

= 1198601198702

1198611198701= 1198611198702 1198621198701= 1198621198702 1198631198701= 1198631198702 1= 21198721= 1198722

1198831= 1198832 and119885






119862119894 119883119862119894


119862119894 119861119862119894 119862119862119894 119863119862119894 (119894 = 1 2) satisfying

[[

[

sym (119860119890119894119895+ 119860119890119895119894) 119861119890119894119895

119862119879

1198901119894119895+ 119862119879

1198901119895119894


119868 0


]]

]

lt 0

1 le 119894 le 119895 le 2

(73)

[minus2119868 radic1205881198621198902119894119895 + 1198621198902119895119894

119896


] lt 0

119896 = 1 2 1 le 119894 le 119895 le 2

(74)

where

119860119890119894119895= [

119860119894119883119862119895+ 119861119894119862119862119895

119860119894+ 119861119894119863119862119895119862

119860119862119894

119884119862119894119860119895+ 119861119862119894119862]

119861119890119894119895= [

1198611119894

1198841198621198941198611119895

]

1198621198901119894119895

= [1198621119894119883119862119895+ 11986311198941198621198621198951198621119894+ 1198631119894119863119862119895119862]

1198621198902119894119895

= [11986221198941198831198621198951198622119894]

119875119862119894

= [119883119862119894

119868

119868 119884119862119894

]

(75)


infinlt 120574 with 119908max = 120588120578


119863119870(119898119904) = 119863

119862(119898119904) (76)

119862119870(119898119904) = (119862

119862(119898119904) minus 119863119862(119898119904) 119862119883119862(119898119904))119872minus119879

119862(119898119904)

(77)

119861119870(119898119904) = 119873

minus1

119862(119898119904) (119861119862(119898119904) minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904))

(78)

119860119870(119898119904)

= 119873minus1

119862(119898119904) [119860119862(119898119904) minus 119884119862(119898119904) 119860 (119898

119904)119883119862(119898119904)

minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904) 119862119883119862(119898119904)

minus 119873119862(119898119904) 119861119870(119898119904) 119862119883119862(119898119904)

minus119884119862(119898119904) 119861 (119898

119904) 119862119870(119898119904)119872119879

119862]


119862(119898119904)

(79)

where

(119860119862(119898119904) 119861119862(119898119904) 119862119862(119898119904) 119863119862(119898119904) 119883119862(119898119904) 119884119862(119898119904))

=

2

sum

119894=1

120582119894(119860119862119894 119861119862119894 119862119862119894 119863119862119894 119883119862119894 119884119862119894)

(80)



119862(119898119904) and119872



119873119862(119898119904)119872119879

119862(119898119904) = 119868 minus 119884

119862(119898119904)119883119862(119898119904) (81)

4 A Design Example



10000 119911max = 01m1205961= 8120587rads and120596

2= 16120587radsThen




119896119904

119896119905

119888119904

119888119905

119898119904max 119898

119904min 119898119904

119898119906


Frequency (Hz)

Max

imum

sing

ular

val

ues (

abs)

10minus1

100

101

102


4ndash8 Hz

25

20

15

10

5

0









119908 (119905) = 119860 sin (2120587119891119905) 119905 isin [0 119879]

0 119905 notin [0 119879] (82)




1199002(119905)119896| lt 1 119896 = 1 2

5 Conclusion


0 02 04 06 08 1 12 14 16 18 2

005

115

225

Time (seconds)

minus25

minus1

minus2

minus15

minus05


Body

accle

ratio

n (m

s2)


0 02 04 06 08 1 12 14 16 18 2

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

0

Relat

ive d

ynam

ic ti

re lo

ad


0 02 04 06 08 1 12 14 16 18 2

0

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

Rela

tive s

uspe

nsio

n str

oke




Appendices

A Related Lemmas



120596119865


119873lowast

120596Θ119873120596lt 0 120596 isin [120596

11205962] (A1)


119865lowast


where

Φ = [0 1

1 0] Ψ = [

minus1 119895120596119888

minus119895120596119888minus12059611205962

] (A3)

with 120596119888= (1205961+ 1205962)2




Γperp

Θ(Γperp

)119879

lt 0 (Λ119879

)perp

Θ((Λ119879

)perp

)

119879

lt 0 (A5)


(1) Ψ + 119878 + 119878119879 lt 0

(2) The LMI problem

[Ψ + 119875 minus [119882

119904] 119878119879

+119882119879





119860119870=

[[[

[


minus22844 minus13692 13701 002030

20381 15159 minus21603 minus047598

24991 times 109


42012 times 106


]]]

]

119861119870=

[[[

[

99943 10908 44969


minus63200 11564 minus11258




]]]

]

119862119870= [38144 minus68619 17589 35961]

119863119870= [minus18676 times 10

5


minus86675]

(B1)


119860119870=

[[[

[

minus12734 26225 21367 minus031502

minus36286 14954 36984 069500



]]]

]

119861119870=

[[[

[


17709 12215 11303

minus90855 34487 minus80239

64417 times 105

71058 times 106


]]]

]

119862119870= [51819 minus60403 minus59041 10015]

119863119870= [25422 17208 26098]

(B2)

References





































2synthesis with












Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198621(119898119904) = [minus

119896119904

119898119904

0 minus119888119904

119898119904

119888119904

119898119904

]

1198622(119898119904) =

[[[

[

1

119911max0 0 0

0119896119905

(119898119904119892 + 119898

119906119892)

0 0

]]]

]

1198631(119898119904) =

1

119898119904

119862 = [

[

1 0 0 0

0 1 0 0

0 0 1 0

]

]

(9)


119904 1199111199002(119905) rep-



(119860 (119898119904) 119861 (119898

119904) 1198611(119898119904) 1198621(119898119904) 1198622(119898119904) 1198631(119898119904))

=

2

sum

119894=1

120582119894(119860119894 119861119894 1198611119894 1198621119894 1198622119894 1198631119894)

(10)

where(1198601 1198611 11986111 11986211 11986221 11986311)

= (119860 (119898119904max) 119861 (119898119904max) 1198611 (119898119904max)

1198621(119898119904max) 1198622 (119898119904max) 1198631 (119898119904max) )

(1198602 1198612 11986112 11986212 11986222 11986312)

= (119860 (119898119904min) 119861 (119898119904min) 1198611 (119898119904min)

1198621(119898119904min) 1198622 (119898119904min) 1198631 (119898119904min) )

1205821=

1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1205822=

1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

(11)



described as120578 (119905) = 119860

119870(119898119904) 120578 (119905) + 119861

119870(119898119904) 119910 (119905)

119906 (119905) = 119862119870(119898119904) 120578 (119905) + 119863

119870(119898119904) 119910 (119905)

(12)


(119905) = 119860 (119898119904) 119909 (119905) + 119861 (119898

119904) 119908 (119905)

1199111199001(119905) = 119862

1(119898119904) 119909 (119905)

1199111199002(119905) = 119862

2(119898119904) 119909 (119905)

(13)

where

119909 (119905) = [120585 (119905)

120578 (119905)]

119860 (119898119904)

= [119860 (119898119904) + 119861 (119898

119904)119863119870(119898119904) 119862 119861 (119898

119904) 119862119870(119898119904)

119861119870(119898119904) 119862 119860

119870(119898119904)

]

119861 (119898119904) = [

1198611(119898119904)

0]

1198621(119898119904)

= [1198621(119898119904) + 1198631(119898119904)119863119870(119898119904) 119862 119863

1(119898119904) 119862119870(119898119904)]

1198622(119898119904) = [119862

2(119898119904) 0]

(14)Denote

119866 (119895120596) = 1198621(119898119904) (119895120596119868 minus 119860 (119898

119904))minus1

119861 (119898119904) (15)


The119867infin


1003817100381710038171003817119866 (119895120596)1003817100381710038171003817infin


2[0+infin)

1003817100381710038171003817119911119900210038171003817100381710038171198712

1199081198712

(16)

where

1003817100381710038171003817119911119900210038171003817100381710038171198712

= (int

+infin

0

10038171003817100381710038171199111199002 (119905)1003817100381710038171003817

2

119889119905)

12

1199081198712

= (int

+infin

0

119908 (119905)2

119889119905)

12

(17)



1to 1205962) 119867infin


1003817100381710038171003817119866 (119895120596)1003817100381710038171003817infin

1205961lt120596lt120596

2

lt 120574 (18)


10038161003816100381610038161199111199002 (119905)119896

1003816100381610038161003816 lt 1 119896 = 1 2 (19)







matrices 119875119878(119898119904) 119876(119898


119904) satisfy-

ing

[[[[[[

[

minus sym (119882 (119898119904)) minus119882

119879

(119898119904) 119860 (119898

119904) + 119875119878(119898119904) minus119882

119879

(119898119904) minus119882

119879

(119898119904) 119861 (119898

119904)

lowast minus119875119878(119898119904) 0 0



]]]]]]

]

lt 0 (20)

[

Ω1Ω2

lowast Ω3

] lt 0 (21)

[

minus119868 radic120588 1198622 (119898119904)119896

lowast minus119875119878(119898119904)

] lt 0 119896 = 1 2 (22)

where

Ω1= [

[

minus12059611205962119876 (119898119904) minus sym (119860

119879

(119898119904)119882 (119898

119904)) 119875 (119898

119904) minus 119895120596

119888119876 (119898119904) + 119882

119879

(119898119904)

119875 (119898119904) + 119895120596

119888119876 (119898119904) + 119882(119898

119904) minus119876 (119898

119904)

]

]

Ω2= [

[

minus119882119879

(119898119904) 119861 (119898

119904) 119862119879

1(119898119904)

0 0

]

]

Ω3= [

minus1205742

119868 0

0 minus119868

]

120596119888=(1205961+ 1205962)

2

(23)


120588120578


[Γ minus119882

119879

(119898119904) 119860 (119898

119904) + 119875119878(119898119904)

lowast minus119875119878(119898119904)

] lt 0 (24)

where

Γ =1

120578119882119879

(119898119904) 119861 (119898

119904) 119861119879

(119898119904)119882 (119898

119904)

+ 119882119879

(119898119904) 119875minus1

119878(119898119904)119882 (119898

119904) + sym (119882 (119898

119904))

(25)


119878(119898119904) from the




[

[

1

120578119861 (119898119904) 119861119879

(119898119904) + 119875minus1

119878(119898119904) minus sym (119871 (119898

119904)) 119860 (119898

119904) 119875minus1

119878(119898119904) + 119871119879

(119898119904)


119878(119898119904)

]

]

lt 0 (26)




119879

(119898119904) = 119860(119898

119904)119875minus1

119878(119898119904) Ψ(119898

119904) =

(1120578)119861(119898119904)119861119879


119860 (119898119904) 119875minus1

119878(119898119904) + 119875minus1

119878(119898119904) 119860119879

(119898119904) +

1

120578119861 (119898119904) 119861119879

(119898119904) lt 0

(27)


119878(119898119904) we get

119875119878(119898119904) 119860 (119898

119904) + 119860119879

(119898119904) 119875119878(119898119904)

+1

120578119875119878(119898119904) 119861 (119898

119904) 119861119879

(119898119904) 119875119878(119898119904) lt 0

(28)

which guarantees

119875119878(119898119904) 119860 (119898

119904) + 119860119879

(119898119904) 119875119878(119898119904) lt 0 (29)



[119868 119865119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

+ sym (119865119860(119898119904)119882 (119898

119904) 119877) lt 0

(30)

where

119865119860(119898119904) =

[[

[

minus119860119879

(119898119904)

119868

minus119861119879

(119898119904)

]]

]

119865119861(119898119904) = [

[

119862119879

1(119898119904)

0

0

]

]

119877 = [119868 0 0]

Ω (119898119904) = 119879 [

Φ otimes 119875119904(119898119904) + Ψ otimes 119876 (119898

119904) 0

0 Π]119879119879

Π = [119868 0

0 minus1205742

119868] Φ = [

0 1

1 0]

Ψ = [minus1 119895120596

119888

minus119895120596119888minus12059611205962

] 119879 =

[[[

[

0 119868 0 0

119868 0 0 0

0 0 0 119868

0 0 119868 0

]]]

]

(31)


119865perp

119860(119898119904) [119868 119865

119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

(119865perp

119860(119898119904))119879

lt 0

(32)

(119877119879

)perp

[119868 119865119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

(119877119879perp

)119879

lt 0 (33)


119865119879

(119898119904)Ω (119898

119904) 119865 (119898

119904) lt 0 (34)

where

119865 (119898119904) = [

119868 119860119879

(119898119904) 0 119862

119879

1(119898119904)

0 119861119879

(119898119904) 119868 0

]

119879

(35)


[[[

[

119860 (119898119904) 119861 (119898

119904)

119868 0

1198621(119898119904) 0

0 119868

]]]

]

119879

[Φ otimes 119875 (119898

119904) + Ψ otimes 119876 (119898

119904) 0

0 Π]

times

[[[

[

119860 (119898119904) 119861 (119898

119904)

119868 0

1198621(119898119904) 0

0 119868

]]]

]

lt 0

(36)

and then we obtain

[119860 (119898119904) 119861 (119898

119904)

119868 0]

119879

(Φ otimes 119875 (119898119904) + Ψ otimes 119876 (119898

119904))

times [119860 (119898119904) 119861 (119898

119904)

119868 0] + [

1198621(119898119904) 0

0 119868]

119879

times Π[1198621(119898119904) 0

0 119868] lt 0

(37)


Ξlowast

ΠΞ lt 0 1205961lt 120596 lt 120596

2 (38)

where

Ξ = [1198621(119898119904) 0

0 119868] [(119895120596119868 minus (119860 (119898

119904))minus1

119861 (119898119904)

119868] (39)

namely

sup1205961lt120596lt120596

2

1003817100381710038171003817119866 (119895120596)1003817100381710038171003817infin

lt 120574 (40)

Select

119881 (119905) = 119909119879

(119905) 119875119878(119898119904) 119909 (119905) (41)


(119905) = 2119909119879

(119905) 119875119878(119898119904) 119860 (119898

119904) 119909 (119905)

+ 2119909119879

(119905) 119875119878(119898119904) 119861 (119898

119904) 119908 (119905)

(42)


2119909119879

(119905) 119875119878(119898119904) 119861 (119898

119904) 119908 (119905)

le1

120578119909119879

(119905) 119875119904(119898119904) 119861 (119898

119904) 119861119879

(119898119904) 119875119878(119898119904) (119909) 119905

+ 120578119908119879

(119905) 119908 (119905) forall120578 gt 0

(43)



(119905) le 120578119908119879

(119905) 119908 (119905) (44)


119881 (119905) minus 119881 (0) le 120578int

119905

0

119908119879

(119905) 119908 (119905) 119889119905 = 120578119908 (119905)2le 120578119908max

(45)


119909119879

(119905) 119875119878(119898119904) 119909 (119905) le 119881 (0) + 120578119908max = 120588 (46)


119909119879

(119905) 119909 (119905) le 120588 (47)

where 119909(119905) = 11987512119878(119898119904)119909(119905)

Noting that

max 10038161003816100381610038161199111199002 (119905)1198961003816100381610038161003816

2

= max100381710038171003817100381710038171003817119909119879

(119905) 119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

times 1198622(119898119904)119896

119875minus12

119878(119898119904) 119909(119905)

100381710038171003817100381710038172

le 120588120579max (119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

1198622(119898119904)119896

119875minus12

119878(119898119904))

(48)


120588119875minus12

119878(119898119904) 1198622(119898119904)119879

1198961198622(119898119904)119896119875minus12

119878(119898119904) lt 119868 (49)




ing form

119882(119898119904) = [

119883 (119898119904) 119884 (119898

119904)

119880 (119898119904) 119881 (119898

119904)]

119882minus1

(119898119904) = [

119872(119898119904) 119866 (119898

119904)

119867 (119898119904) 119871 (119898

119904)]

(50)



Denote

Δ (119898119904) = [

119868 119872 (119898119904)

0 119867 (119898119904)] (51)


1198691(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) Δ119879

(119898119904) 119868

1198692(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) 119868 119868

1198693(119898119904) = diag 119868 Δ119879 (119898

119904)

(52)

respectivelyDenote

119876 (119898119904) = Δ119879

(119898119904) 119876 (119898

119904) Δ (119898

119904)

(119898119904) = Δ119879

(119898119904) 119875 (119898

119904) Δ (119898

119904)

119878(119898119904) = Δ119879

(119898119904) 119875119878(119898119904) Δ (119898

119904)

119860 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119860 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119860 (119898

119904) + 119861119870(119898119904) 119862 119860

119870(119898119904)

119860 (119898119904) + 119861 (119898

119904)119863119870(119898119904) 119862 119860 (119898

119904)119872 (119898

119904) + 119861 (119898

119904) 119862119870(119898119904)]

119861 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119861 (119898

119904)

= [119883119879

(119898119904) 1198611(119898119904)

1198611(119898119904)

]

1198621(119898119904) = 1198621(119898119904) Δ (119898

119904)

= [1198621(119898119904) + 1198631(119898119904)119863119870(119898119904) 119862 119862

1(119898119904)119872 (119898

119904) + 1198631(119898119904) 119862119870(119898119904)]

1198622(119898119904) = 1198622(119898119904) Δ (119898

119904) = [119862

2(119898119904) 1198622(119898119904)119872 (119898

119904)]


(119898119904) = Δ119879

(119898119904)119882 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119885 (119898

119904)

119868 119872 (119898119904)]

(53)




119878(119898119904) 119876(119898


119870(119898119904) 119861119870(119898119904)

119862119870(119898119904)119863119870(119898119904) (119898

119904)119872(119898

119904)119883(119898

119904) 119885(119898

119904) satisfying

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

lt 0 (54)

[[[[

[

minus12059611205962119876 (119898119904) minus sym (119860 (119898

119904)) (119898

119904) minus 119895120596

119888119876 (119898119904) +

119879

(119898119904) minus119861 (119898

119904) 119862119879

1(119898119904)

(119898119904) + 119895120596

119888119876 (119898119904) + (119898

119904) minus119876 (119898

119904) 0 0

minus119861119879

(119898119904) 0 minus120574

2

119868 0

1198621(119898119904) 0 0 minus119868

]]]]

]

lt 0 (55)

[minus119868 radic1205881198622 (119898119904)

119896

lowast minus119878(119898119904)

] lt 0 119896 = 1 2 (56)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904)

= 119880minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)








119870119894 119861119870119894 119862119870119894 119863119870119894 119894119872119894 119883119894 119885119894


[11986911198692

lowast 1198693

] lt 0 1 le 119894 le 119895 le 2 (58)

[1198701+ 11989511987021198703

lowast 1198704

] lt 0 1 le 119894 le 119895 le 2 (59)

[minus119868 radic1205881198622119894119895 + 1198622119895119894

119896


] lt 0 119896 = 1 2 1 le 119894 le 119895 le 2

(60)

where

1198691= [

sym (119894+ 119895) minus119860

119894119895minus 119860119895119894+ 119878119894+ 119878119895


]

1198692= [

minus119879

119894minus 119879

119895minus119861119894119895minus 119861119895119894

0 0]

1198693= [

minus119878119894minus 119878119895

0


1198701= [

minus12059611205962(119876119894+ 119876119895) minus sym (119860

119894119895+ 119860119895119894) 119894+ 119895+ 119879

119894+ 119879

119895

119894+ 119895+ 119894+ 119895

minus119876119894minus 119876119895

]

1198702= [

0 minus120596119888(119876119894+ 119876119895)

120596119888(119876119894+ 119876119895) 0

]


1198703= [

minus119861119894119895minus 119861119895119894119862119879

1119894119895+ 119862119879

1119895119894

0 0]

1198704= [

minus21205742

119868 0

0 minus2119868]

119860119894119895= [

119883119879

119894119860119895+ 119861119870119894119862 119860

119870119894

119860119894+ 119861119894119863119870119895119862 119860119894119872119895+ 119861119894119862119870119895

]

119861119894119895= [

119883119879

1198941198611119895

1198611119894

]

1198621119894119895= [1198621119894+ 1198631119894119863119870119895119862 1198621119894119872119895+ 1198631119894119862119870119895]

1198622119894119895= [11986221198941198622119894119872119895]

119894= [

119883119879

119894119885119894

119868 119872119894

]

(61)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904) = 119880

minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)

(62)

where

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904) )

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(63)



Denote

Ψ (119898119904) =

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

(64)


Ψ119894119895+ Ψ119895119894lt 0 1 le 119894 lt 119895 le 2

Ψ119894119894lt 0 119894 = 1 2

(65)

where

Ψ119894119895=

[[[

[

sym (119894) minus119860

119894119895+ 119878119894minus119879

119894minus119861119894119895


0 0


0


]]]

]

1 le 119894 le 119895 le 2

(66)

Assume that

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904))

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(67)

and then we get

Ψ (119898119904) =

2

sum

119894=1

2

sum

119895=1

120582119894120582119895Ψ119894119895=

2

sum

119894=1

1205822

119894Ψ119894119894+ 12058211205822(Ψ12+ Ψ21) (68)


Ψ (119898119904) lt 0 (69)

Remark 4 119880(119898119904) 119867(119898


definition that is

119880119879

(119898119904)119867 (119898

119904) = 119885 (119898

119904) minus 119883119879

(119898119904)119872 (119898

119904) (70)




119904) and 119872(119898

119904) In this



2 1198781+


minus11987821198781


1198781= [

11987011198703

lowast 1198704

] 1198782= [

11987020

0 0] (71)



119870(119898119904) 119861119870(119898119904) 119862119870(119898119904) 119863119870(119898119904) and Lyapunov


119904


119863119870(119898119904) = 119863

119870(119898119904) =

2

sum

119894=1

120582119894119863119870119894= 12058211198631198701+ 12058221198631198702

=1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1198631198701

+1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

1198631198702

(72)

where 1198631198701

and 1198631198702


119870(119898119904) may



1= 2 1198781

= 1198782 1198761= 1198762 1198601198701

= 1198601198702

1198611198701= 1198611198702 1198621198701= 1198621198702 1198631198701= 1198631198702 1= 21198721= 1198722

1198831= 1198832 and119885






119862119894 119883119862119894


119862119894 119861119862119894 119862119862119894 119863119862119894 (119894 = 1 2) satisfying

[[

[

sym (119860119890119894119895+ 119860119890119895119894) 119861119890119894119895

119862119879

1198901119894119895+ 119862119879

1198901119895119894


119868 0


]]

]

lt 0

1 le 119894 le 119895 le 2

(73)

[minus2119868 radic1205881198621198902119894119895 + 1198621198902119895119894

119896


] lt 0

119896 = 1 2 1 le 119894 le 119895 le 2

(74)

where

119860119890119894119895= [

119860119894119883119862119895+ 119861119894119862119862119895

119860119894+ 119861119894119863119862119895119862

119860119862119894

119884119862119894119860119895+ 119861119862119894119862]

119861119890119894119895= [

1198611119894

1198841198621198941198611119895

]

1198621198901119894119895

= [1198621119894119883119862119895+ 11986311198941198621198621198951198621119894+ 1198631119894119863119862119895119862]

1198621198902119894119895

= [11986221198941198831198621198951198622119894]

119875119862119894

= [119883119862119894

119868

119868 119884119862119894

]

(75)


infinlt 120574 with 119908max = 120588120578


119863119870(119898119904) = 119863

119862(119898119904) (76)

119862119870(119898119904) = (119862

119862(119898119904) minus 119863119862(119898119904) 119862119883119862(119898119904))119872minus119879

119862(119898119904)

(77)

119861119870(119898119904) = 119873

minus1

119862(119898119904) (119861119862(119898119904) minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904))

(78)

119860119870(119898119904)

= 119873minus1

119862(119898119904) [119860119862(119898119904) minus 119884119862(119898119904) 119860 (119898

119904)119883119862(119898119904)

minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904) 119862119883119862(119898119904)

minus 119873119862(119898119904) 119861119870(119898119904) 119862119883119862(119898119904)

minus119884119862(119898119904) 119861 (119898

119904) 119862119870(119898119904)119872119879

119862]


119862(119898119904)

(79)

where

(119860119862(119898119904) 119861119862(119898119904) 119862119862(119898119904) 119863119862(119898119904) 119883119862(119898119904) 119884119862(119898119904))

=

2

sum

119894=1

120582119894(119860119862119894 119861119862119894 119862119862119894 119863119862119894 119883119862119894 119884119862119894)

(80)



119862(119898119904) and119872



119873119862(119898119904)119872119879

119862(119898119904) = 119868 minus 119884

119862(119898119904)119883119862(119898119904) (81)

4 A Design Example



10000 119911max = 01m1205961= 8120587rads and120596

2= 16120587radsThen




119896119904

119896119905

119888119904

119888119905

119898119904max 119898

119904min 119898119904

119898119906


Frequency (Hz)

Max

imum

sing

ular

val

ues (

abs)

10minus1

100

101

102


4ndash8 Hz

25

20

15

10

5

0









119908 (119905) = 119860 sin (2120587119891119905) 119905 isin [0 119879]

0 119905 notin [0 119879] (82)




1199002(119905)119896| lt 1 119896 = 1 2

5 Conclusion


0 02 04 06 08 1 12 14 16 18 2

005

115

225

Time (seconds)

minus25

minus1

minus2

minus15

minus05


Body

accle

ratio

n (m

s2)


0 02 04 06 08 1 12 14 16 18 2

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

0

Relat

ive d

ynam

ic ti

re lo

ad


0 02 04 06 08 1 12 14 16 18 2

0

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

Rela

tive s

uspe

nsio

n str

oke




Appendices

A Related Lemmas



120596119865


119873lowast

120596Θ119873120596lt 0 120596 isin [120596

11205962] (A1)


119865lowast


where

Φ = [0 1

1 0] Ψ = [

minus1 119895120596119888

minus119895120596119888minus12059611205962

] (A3)

with 120596119888= (1205961+ 1205962)2




Γperp

Θ(Γperp

)119879

lt 0 (Λ119879

)perp

Θ((Λ119879

)perp

)

119879

lt 0 (A5)


(1) Ψ + 119878 + 119878119879 lt 0

(2) The LMI problem

[Ψ + 119875 minus [119882

119904] 119878119879

+119882119879





119860119870=

[[[

[


minus22844 minus13692 13701 002030

20381 15159 minus21603 minus047598

24991 times 109


42012 times 106


]]]

]

119861119870=

[[[

[

99943 10908 44969


minus63200 11564 minus11258




]]]

]

119862119870= [38144 minus68619 17589 35961]

119863119870= [minus18676 times 10

5


minus86675]

(B1)


119860119870=

[[[

[

minus12734 26225 21367 minus031502

minus36286 14954 36984 069500



]]]

]

119861119870=

[[[

[


17709 12215 11303

minus90855 34487 minus80239

64417 times 105

71058 times 106


]]]

]

119862119870= [51819 minus60403 minus59041 10015]

119863119870= [25422 17208 26098]

(B2)

References





































2synthesis with












Volume 2014




Journal of











Journal of


Function Spaces






Algebra













matrices 119875119878(119898119904) 119876(119898


119904) satisfy-

ing

[[[[[[

[

minus sym (119882 (119898119904)) minus119882

119879

(119898119904) 119860 (119898

119904) + 119875119878(119898119904) minus119882

119879

(119898119904) minus119882

119879

(119898119904) 119861 (119898

119904)

lowast minus119875119878(119898119904) 0 0



]]]]]]

]

lt 0 (20)

[

Ω1Ω2

lowast Ω3

] lt 0 (21)

[

minus119868 radic120588 1198622 (119898119904)119896

lowast minus119875119878(119898119904)

] lt 0 119896 = 1 2 (22)

where

Ω1= [

[

minus12059611205962119876 (119898119904) minus sym (119860

119879

(119898119904)119882 (119898

119904)) 119875 (119898

119904) minus 119895120596

119888119876 (119898119904) + 119882

119879

(119898119904)

119875 (119898119904) + 119895120596

119888119876 (119898119904) + 119882(119898

119904) minus119876 (119898

119904)

]

]

Ω2= [

[

minus119882119879

(119898119904) 119861 (119898

119904) 119862119879

1(119898119904)

0 0

]

]

Ω3= [

minus1205742

119868 0

0 minus119868

]

120596119888=(1205961+ 1205962)

2

(23)


120588120578


[Γ minus119882

119879

(119898119904) 119860 (119898

119904) + 119875119878(119898119904)

lowast minus119875119878(119898119904)

] lt 0 (24)

where

Γ =1

120578119882119879

(119898119904) 119861 (119898

119904) 119861119879

(119898119904)119882 (119898

119904)

+ 119882119879

(119898119904) 119875minus1

119878(119898119904)119882 (119898

119904) + sym (119882 (119898

119904))

(25)


119878(119898119904) from the




[

[

1

120578119861 (119898119904) 119861119879

(119898119904) + 119875minus1

119878(119898119904) minus sym (119871 (119898

119904)) 119860 (119898

119904) 119875minus1

119878(119898119904) + 119871119879

(119898119904)


119878(119898119904)

]

]

lt 0 (26)




119879

(119898119904) = 119860(119898

119904)119875minus1

119878(119898119904) Ψ(119898

119904) =

(1120578)119861(119898119904)119861119879


119860 (119898119904) 119875minus1

119878(119898119904) + 119875minus1

119878(119898119904) 119860119879

(119898119904) +

1

120578119861 (119898119904) 119861119879

(119898119904) lt 0

(27)


119878(119898119904) we get

119875119878(119898119904) 119860 (119898

119904) + 119860119879

(119898119904) 119875119878(119898119904)

+1

120578119875119878(119898119904) 119861 (119898

119904) 119861119879

(119898119904) 119875119878(119898119904) lt 0

(28)

which guarantees

119875119878(119898119904) 119860 (119898

119904) + 119860119879

(119898119904) 119875119878(119898119904) lt 0 (29)



[119868 119865119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

+ sym (119865119860(119898119904)119882 (119898

119904) 119877) lt 0

(30)

where

119865119860(119898119904) =

[[

[

minus119860119879

(119898119904)

119868

minus119861119879

(119898119904)

]]

]

119865119861(119898119904) = [

[

119862119879

1(119898119904)

0

0

]

]

119877 = [119868 0 0]

Ω (119898119904) = 119879 [

Φ otimes 119875119904(119898119904) + Ψ otimes 119876 (119898

119904) 0

0 Π]119879119879

Π = [119868 0

0 minus1205742

119868] Φ = [

0 1

1 0]

Ψ = [minus1 119895120596

119888

minus119895120596119888minus12059611205962

] 119879 =

[[[

[

0 119868 0 0

119868 0 0 0

0 0 0 119868

0 0 119868 0

]]]

]

(31)


119865perp

119860(119898119904) [119868 119865

119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

(119865perp

119860(119898119904))119879

lt 0

(32)

(119877119879

)perp

[119868 119865119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

(119877119879perp

)119879

lt 0 (33)


119865119879

(119898119904)Ω (119898

119904) 119865 (119898

119904) lt 0 (34)

where

119865 (119898119904) = [

119868 119860119879

(119898119904) 0 119862

119879

1(119898119904)

0 119861119879

(119898119904) 119868 0

]

119879

(35)


[[[

[

119860 (119898119904) 119861 (119898

119904)

119868 0

1198621(119898119904) 0

0 119868

]]]

]

119879

[Φ otimes 119875 (119898

119904) + Ψ otimes 119876 (119898

119904) 0

0 Π]

times

[[[

[

119860 (119898119904) 119861 (119898

119904)

119868 0

1198621(119898119904) 0

0 119868

]]]

]

lt 0

(36)

and then we obtain

[119860 (119898119904) 119861 (119898

119904)

119868 0]

119879

(Φ otimes 119875 (119898119904) + Ψ otimes 119876 (119898

119904))

times [119860 (119898119904) 119861 (119898

119904)

119868 0] + [

1198621(119898119904) 0

0 119868]

119879

times Π[1198621(119898119904) 0

0 119868] lt 0

(37)


Ξlowast

ΠΞ lt 0 1205961lt 120596 lt 120596

2 (38)

where

Ξ = [1198621(119898119904) 0

0 119868] [(119895120596119868 minus (119860 (119898

119904))minus1

119861 (119898119904)

119868] (39)

namely

sup1205961lt120596lt120596

2

1003817100381710038171003817119866 (119895120596)1003817100381710038171003817infin

lt 120574 (40)

Select

119881 (119905) = 119909119879

(119905) 119875119878(119898119904) 119909 (119905) (41)


(119905) = 2119909119879

(119905) 119875119878(119898119904) 119860 (119898

119904) 119909 (119905)

+ 2119909119879

(119905) 119875119878(119898119904) 119861 (119898

119904) 119908 (119905)

(42)


2119909119879

(119905) 119875119878(119898119904) 119861 (119898

119904) 119908 (119905)

le1

120578119909119879

(119905) 119875119904(119898119904) 119861 (119898

119904) 119861119879

(119898119904) 119875119878(119898119904) (119909) 119905

+ 120578119908119879

(119905) 119908 (119905) forall120578 gt 0

(43)



(119905) le 120578119908119879

(119905) 119908 (119905) (44)


119881 (119905) minus 119881 (0) le 120578int

119905

0

119908119879

(119905) 119908 (119905) 119889119905 = 120578119908 (119905)2le 120578119908max

(45)


119909119879

(119905) 119875119878(119898119904) 119909 (119905) le 119881 (0) + 120578119908max = 120588 (46)


119909119879

(119905) 119909 (119905) le 120588 (47)

where 119909(119905) = 11987512119878(119898119904)119909(119905)

Noting that

max 10038161003816100381610038161199111199002 (119905)1198961003816100381610038161003816

2

= max100381710038171003817100381710038171003817119909119879

(119905) 119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

times 1198622(119898119904)119896

119875minus12

119878(119898119904) 119909(119905)

100381710038171003817100381710038172

le 120588120579max (119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

1198622(119898119904)119896

119875minus12

119878(119898119904))

(48)


120588119875minus12

119878(119898119904) 1198622(119898119904)119879

1198961198622(119898119904)119896119875minus12

119878(119898119904) lt 119868 (49)




ing form

119882(119898119904) = [

119883 (119898119904) 119884 (119898

119904)

119880 (119898119904) 119881 (119898

119904)]

119882minus1

(119898119904) = [

119872(119898119904) 119866 (119898

119904)

119867 (119898119904) 119871 (119898

119904)]

(50)



Denote

Δ (119898119904) = [

119868 119872 (119898119904)

0 119867 (119898119904)] (51)


1198691(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) Δ119879

(119898119904) 119868

1198692(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) 119868 119868

1198693(119898119904) = diag 119868 Δ119879 (119898

119904)

(52)

respectivelyDenote

119876 (119898119904) = Δ119879

(119898119904) 119876 (119898

119904) Δ (119898

119904)

(119898119904) = Δ119879

(119898119904) 119875 (119898

119904) Δ (119898

119904)

119878(119898119904) = Δ119879

(119898119904) 119875119878(119898119904) Δ (119898

119904)

119860 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119860 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119860 (119898

119904) + 119861119870(119898119904) 119862 119860

119870(119898119904)

119860 (119898119904) + 119861 (119898

119904)119863119870(119898119904) 119862 119860 (119898

119904)119872 (119898

119904) + 119861 (119898

119904) 119862119870(119898119904)]

119861 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119861 (119898

119904)

= [119883119879

(119898119904) 1198611(119898119904)

1198611(119898119904)

]

1198621(119898119904) = 1198621(119898119904) Δ (119898

119904)

= [1198621(119898119904) + 1198631(119898119904)119863119870(119898119904) 119862 119862

1(119898119904)119872 (119898

119904) + 1198631(119898119904) 119862119870(119898119904)]

1198622(119898119904) = 1198622(119898119904) Δ (119898

119904) = [119862

2(119898119904) 1198622(119898119904)119872 (119898

119904)]


(119898119904) = Δ119879

(119898119904)119882 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119885 (119898

119904)

119868 119872 (119898119904)]

(53)




119878(119898119904) 119876(119898


119870(119898119904) 119861119870(119898119904)

119862119870(119898119904)119863119870(119898119904) (119898

119904)119872(119898

119904)119883(119898

119904) 119885(119898

119904) satisfying

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

lt 0 (54)

[[[[

[

minus12059611205962119876 (119898119904) minus sym (119860 (119898

119904)) (119898

119904) minus 119895120596

119888119876 (119898119904) +

119879

(119898119904) minus119861 (119898

119904) 119862119879

1(119898119904)

(119898119904) + 119895120596

119888119876 (119898119904) + (119898

119904) minus119876 (119898

119904) 0 0

minus119861119879

(119898119904) 0 minus120574

2

119868 0

1198621(119898119904) 0 0 minus119868

]]]]

]

lt 0 (55)

[minus119868 radic1205881198622 (119898119904)

119896

lowast minus119878(119898119904)

] lt 0 119896 = 1 2 (56)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904)

= 119880minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)








119870119894 119861119870119894 119862119870119894 119863119870119894 119894119872119894 119883119894 119885119894


[11986911198692

lowast 1198693

] lt 0 1 le 119894 le 119895 le 2 (58)

[1198701+ 11989511987021198703

lowast 1198704

] lt 0 1 le 119894 le 119895 le 2 (59)

[minus119868 radic1205881198622119894119895 + 1198622119895119894

119896


] lt 0 119896 = 1 2 1 le 119894 le 119895 le 2

(60)

where

1198691= [

sym (119894+ 119895) minus119860

119894119895minus 119860119895119894+ 119878119894+ 119878119895


]

1198692= [

minus119879

119894minus 119879

119895minus119861119894119895minus 119861119895119894

0 0]

1198693= [

minus119878119894minus 119878119895

0


1198701= [

minus12059611205962(119876119894+ 119876119895) minus sym (119860

119894119895+ 119860119895119894) 119894+ 119895+ 119879

119894+ 119879

119895

119894+ 119895+ 119894+ 119895

minus119876119894minus 119876119895

]

1198702= [

0 minus120596119888(119876119894+ 119876119895)

120596119888(119876119894+ 119876119895) 0

]


1198703= [

minus119861119894119895minus 119861119895119894119862119879

1119894119895+ 119862119879

1119895119894

0 0]

1198704= [

minus21205742

119868 0

0 minus2119868]

119860119894119895= [

119883119879

119894119860119895+ 119861119870119894119862 119860

119870119894

119860119894+ 119861119894119863119870119895119862 119860119894119872119895+ 119861119894119862119870119895

]

119861119894119895= [

119883119879

1198941198611119895

1198611119894

]

1198621119894119895= [1198621119894+ 1198631119894119863119870119895119862 1198621119894119872119895+ 1198631119894119862119870119895]

1198622119894119895= [11986221198941198622119894119872119895]

119894= [

119883119879

119894119885119894

119868 119872119894

]

(61)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904) = 119880

minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)

(62)

where

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904) )

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(63)



Denote

Ψ (119898119904) =

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

(64)


Ψ119894119895+ Ψ119895119894lt 0 1 le 119894 lt 119895 le 2

Ψ119894119894lt 0 119894 = 1 2

(65)

where

Ψ119894119895=

[[[

[

sym (119894) minus119860

119894119895+ 119878119894minus119879

119894minus119861119894119895


0 0


0


]]]

]

1 le 119894 le 119895 le 2

(66)

Assume that

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904))

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(67)

and then we get

Ψ (119898119904) =

2

sum

119894=1

2

sum

119895=1

120582119894120582119895Ψ119894119895=

2

sum

119894=1

1205822

119894Ψ119894119894+ 12058211205822(Ψ12+ Ψ21) (68)


Ψ (119898119904) lt 0 (69)

Remark 4 119880(119898119904) 119867(119898


definition that is

119880119879

(119898119904)119867 (119898

119904) = 119885 (119898

119904) minus 119883119879

(119898119904)119872 (119898

119904) (70)




119904) and 119872(119898

119904) In this



2 1198781+


minus11987821198781


1198781= [

11987011198703

lowast 1198704

] 1198782= [

11987020

0 0] (71)



119870(119898119904) 119861119870(119898119904) 119862119870(119898119904) 119863119870(119898119904) and Lyapunov


119904


119863119870(119898119904) = 119863

119870(119898119904) =

2

sum

119894=1

120582119894119863119870119894= 12058211198631198701+ 12058221198631198702

=1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1198631198701

+1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

1198631198702

(72)

where 1198631198701

and 1198631198702


119870(119898119904) may



1= 2 1198781

= 1198782 1198761= 1198762 1198601198701

= 1198601198702

1198611198701= 1198611198702 1198621198701= 1198621198702 1198631198701= 1198631198702 1= 21198721= 1198722

1198831= 1198832 and119885






119862119894 119883119862119894


119862119894 119861119862119894 119862119862119894 119863119862119894 (119894 = 1 2) satisfying

[[

[

sym (119860119890119894119895+ 119860119890119895119894) 119861119890119894119895

119862119879

1198901119894119895+ 119862119879

1198901119895119894


119868 0


]]

]

lt 0

1 le 119894 le 119895 le 2

(73)

[minus2119868 radic1205881198621198902119894119895 + 1198621198902119895119894

119896


] lt 0

119896 = 1 2 1 le 119894 le 119895 le 2

(74)

where

119860119890119894119895= [

119860119894119883119862119895+ 119861119894119862119862119895

119860119894+ 119861119894119863119862119895119862

119860119862119894

119884119862119894119860119895+ 119861119862119894119862]

119861119890119894119895= [

1198611119894

1198841198621198941198611119895

]

1198621198901119894119895

= [1198621119894119883119862119895+ 11986311198941198621198621198951198621119894+ 1198631119894119863119862119895119862]

1198621198902119894119895

= [11986221198941198831198621198951198622119894]

119875119862119894

= [119883119862119894

119868

119868 119884119862119894

]

(75)


infinlt 120574 with 119908max = 120588120578


119863119870(119898119904) = 119863

119862(119898119904) (76)

119862119870(119898119904) = (119862

119862(119898119904) minus 119863119862(119898119904) 119862119883119862(119898119904))119872minus119879

119862(119898119904)

(77)

119861119870(119898119904) = 119873

minus1

119862(119898119904) (119861119862(119898119904) minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904))

(78)

119860119870(119898119904)

= 119873minus1

119862(119898119904) [119860119862(119898119904) minus 119884119862(119898119904) 119860 (119898

119904)119883119862(119898119904)

minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904) 119862119883119862(119898119904)

minus 119873119862(119898119904) 119861119870(119898119904) 119862119883119862(119898119904)

minus119884119862(119898119904) 119861 (119898

119904) 119862119870(119898119904)119872119879

119862]


119862(119898119904)

(79)

where

(119860119862(119898119904) 119861119862(119898119904) 119862119862(119898119904) 119863119862(119898119904) 119883119862(119898119904) 119884119862(119898119904))

=

2

sum

119894=1

120582119894(119860119862119894 119861119862119894 119862119862119894 119863119862119894 119883119862119894 119884119862119894)

(80)



119862(119898119904) and119872



119873119862(119898119904)119872119879

119862(119898119904) = 119868 minus 119884

119862(119898119904)119883119862(119898119904) (81)

4 A Design Example



10000 119911max = 01m1205961= 8120587rads and120596

2= 16120587radsThen




119896119904

119896119905

119888119904

119888119905

119898119904max 119898

119904min 119898119904

119898119906


Frequency (Hz)

Max

imum

sing

ular

val

ues (

abs)

10minus1

100

101

102


4ndash8 Hz

25

20

15

10

5

0









119908 (119905) = 119860 sin (2120587119891119905) 119905 isin [0 119879]

0 119905 notin [0 119879] (82)




1199002(119905)119896| lt 1 119896 = 1 2

5 Conclusion


0 02 04 06 08 1 12 14 16 18 2

005

115

225

Time (seconds)

minus25

minus1

minus2

minus15

minus05


Body

accle

ratio

n (m

s2)


0 02 04 06 08 1 12 14 16 18 2

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

0

Relat

ive d

ynam

ic ti

re lo

ad


0 02 04 06 08 1 12 14 16 18 2

0

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

Rela

tive s

uspe

nsio

n str

oke




Appendices

A Related Lemmas



120596119865


119873lowast

120596Θ119873120596lt 0 120596 isin [120596

11205962] (A1)


119865lowast


where

Φ = [0 1

1 0] Ψ = [

minus1 119895120596119888

minus119895120596119888minus12059611205962

] (A3)

with 120596119888= (1205961+ 1205962)2




Γperp

Θ(Γperp

)119879

lt 0 (Λ119879

)perp

Θ((Λ119879

)perp

)

119879

lt 0 (A5)


(1) Ψ + 119878 + 119878119879 lt 0

(2) The LMI problem

[Ψ + 119875 minus [119882

119904] 119878119879

+119882119879





119860119870=

[[[

[


minus22844 minus13692 13701 002030

20381 15159 minus21603 minus047598

24991 times 109


42012 times 106


]]]

]

119861119870=

[[[

[

99943 10908 44969


minus63200 11564 minus11258




]]]

]

119862119870= [38144 minus68619 17589 35961]

119863119870= [minus18676 times 10

5


minus86675]

(B1)


119860119870=

[[[

[

minus12734 26225 21367 minus031502

minus36286 14954 36984 069500



]]]

]

119861119870=

[[[

[


17709 12215 11303

minus90855 34487 minus80239

64417 times 105

71058 times 106


]]]

]

119862119870= [51819 minus60403 minus59041 10015]

119863119870= [25422 17208 26098]

(B2)

References





































2synthesis with












Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119879

(119898119904) = 119860(119898

119904)119875minus1

119878(119898119904) Ψ(119898

119904) =

(1120578)119861(119898119904)119861119879


119860 (119898119904) 119875minus1

119878(119898119904) + 119875minus1

119878(119898119904) 119860119879

(119898119904) +

1

120578119861 (119898119904) 119861119879

(119898119904) lt 0

(27)


119878(119898119904) we get

119875119878(119898119904) 119860 (119898

119904) + 119860119879

(119898119904) 119875119878(119898119904)

+1

120578119875119878(119898119904) 119861 (119898

119904) 119861119879

(119898119904) 119875119878(119898119904) lt 0

(28)

which guarantees

119875119878(119898119904) 119860 (119898

119904) + 119860119879

(119898119904) 119875119878(119898119904) lt 0 (29)



[119868 119865119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

+ sym (119865119860(119898119904)119882 (119898

119904) 119877) lt 0

(30)

where

119865119860(119898119904) =

[[

[

minus119860119879

(119898119904)

119868

minus119861119879

(119898119904)

]]

]

119865119861(119898119904) = [

[

119862119879

1(119898119904)

0

0

]

]

119877 = [119868 0 0]

Ω (119898119904) = 119879 [

Φ otimes 119875119904(119898119904) + Ψ otimes 119876 (119898

119904) 0

0 Π]119879119879

Π = [119868 0

0 minus1205742

119868] Φ = [

0 1

1 0]

Ψ = [minus1 119895120596

119888

minus119895120596119888minus12059611205962

] 119879 =

[[[

[

0 119868 0 0

119868 0 0 0

0 0 0 119868

0 0 119868 0

]]]

]

(31)


119865perp

119860(119898119904) [119868 119865

119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

(119865perp

119860(119898119904))119879

lt 0

(32)

(119877119879

)perp

[119868 119865119861(119898119904)]Ω (119898

119904) [119868 119865

119861(119898119904)]119879

(119877119879perp

)119879

lt 0 (33)


119865119879

(119898119904)Ω (119898

119904) 119865 (119898

119904) lt 0 (34)

where

119865 (119898119904) = [

119868 119860119879

(119898119904) 0 119862

119879

1(119898119904)

0 119861119879

(119898119904) 119868 0

]

119879

(35)


[[[

[

119860 (119898119904) 119861 (119898

119904)

119868 0

1198621(119898119904) 0

0 119868

]]]

]

119879

[Φ otimes 119875 (119898

119904) + Ψ otimes 119876 (119898

119904) 0

0 Π]

times

[[[

[

119860 (119898119904) 119861 (119898

119904)

119868 0

1198621(119898119904) 0

0 119868

]]]

]

lt 0

(36)

and then we obtain

[119860 (119898119904) 119861 (119898

119904)

119868 0]

119879

(Φ otimes 119875 (119898119904) + Ψ otimes 119876 (119898

119904))

times [119860 (119898119904) 119861 (119898

119904)

119868 0] + [

1198621(119898119904) 0

0 119868]

119879

times Π[1198621(119898119904) 0

0 119868] lt 0

(37)


Ξlowast

ΠΞ lt 0 1205961lt 120596 lt 120596

2 (38)

where

Ξ = [1198621(119898119904) 0

0 119868] [(119895120596119868 minus (119860 (119898

119904))minus1

119861 (119898119904)

119868] (39)

namely

sup1205961lt120596lt120596

2

1003817100381710038171003817119866 (119895120596)1003817100381710038171003817infin

lt 120574 (40)

Select

119881 (119905) = 119909119879

(119905) 119875119878(119898119904) 119909 (119905) (41)


(119905) = 2119909119879

(119905) 119875119878(119898119904) 119860 (119898

119904) 119909 (119905)

+ 2119909119879

(119905) 119875119878(119898119904) 119861 (119898

119904) 119908 (119905)

(42)


2119909119879

(119905) 119875119878(119898119904) 119861 (119898

119904) 119908 (119905)

le1

120578119909119879

(119905) 119875119904(119898119904) 119861 (119898

119904) 119861119879

(119898119904) 119875119878(119898119904) (119909) 119905

+ 120578119908119879

(119905) 119908 (119905) forall120578 gt 0

(43)



(119905) le 120578119908119879

(119905) 119908 (119905) (44)


119881 (119905) minus 119881 (0) le 120578int

119905

0

119908119879

(119905) 119908 (119905) 119889119905 = 120578119908 (119905)2le 120578119908max

(45)


119909119879

(119905) 119875119878(119898119904) 119909 (119905) le 119881 (0) + 120578119908max = 120588 (46)


119909119879

(119905) 119909 (119905) le 120588 (47)

where 119909(119905) = 11987512119878(119898119904)119909(119905)

Noting that

max 10038161003816100381610038161199111199002 (119905)1198961003816100381610038161003816

2

= max100381710038171003817100381710038171003817119909119879

(119905) 119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

times 1198622(119898119904)119896

119875minus12

119878(119898119904) 119909(119905)

100381710038171003817100381710038172

le 120588120579max (119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

1198622(119898119904)119896

119875minus12

119878(119898119904))

(48)


120588119875minus12

119878(119898119904) 1198622(119898119904)119879

1198961198622(119898119904)119896119875minus12

119878(119898119904) lt 119868 (49)




ing form

119882(119898119904) = [

119883 (119898119904) 119884 (119898

119904)

119880 (119898119904) 119881 (119898

119904)]

119882minus1

(119898119904) = [

119872(119898119904) 119866 (119898

119904)

119867 (119898119904) 119871 (119898

119904)]

(50)



Denote

Δ (119898119904) = [

119868 119872 (119898119904)

0 119867 (119898119904)] (51)


1198691(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) Δ119879

(119898119904) 119868

1198692(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) 119868 119868

1198693(119898119904) = diag 119868 Δ119879 (119898

119904)

(52)

respectivelyDenote

119876 (119898119904) = Δ119879

(119898119904) 119876 (119898

119904) Δ (119898

119904)

(119898119904) = Δ119879

(119898119904) 119875 (119898

119904) Δ (119898

119904)

119878(119898119904) = Δ119879

(119898119904) 119875119878(119898119904) Δ (119898

119904)

119860 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119860 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119860 (119898

119904) + 119861119870(119898119904) 119862 119860

119870(119898119904)

119860 (119898119904) + 119861 (119898

119904)119863119870(119898119904) 119862 119860 (119898

119904)119872 (119898

119904) + 119861 (119898

119904) 119862119870(119898119904)]

119861 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119861 (119898

119904)

= [119883119879

(119898119904) 1198611(119898119904)

1198611(119898119904)

]

1198621(119898119904) = 1198621(119898119904) Δ (119898

119904)

= [1198621(119898119904) + 1198631(119898119904)119863119870(119898119904) 119862 119862

1(119898119904)119872 (119898

119904) + 1198631(119898119904) 119862119870(119898119904)]

1198622(119898119904) = 1198622(119898119904) Δ (119898

119904) = [119862

2(119898119904) 1198622(119898119904)119872 (119898

119904)]


(119898119904) = Δ119879

(119898119904)119882 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119885 (119898

119904)

119868 119872 (119898119904)]

(53)




119878(119898119904) 119876(119898


119870(119898119904) 119861119870(119898119904)

119862119870(119898119904)119863119870(119898119904) (119898

119904)119872(119898

119904)119883(119898

119904) 119885(119898

119904) satisfying

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

lt 0 (54)

[[[[

[

minus12059611205962119876 (119898119904) minus sym (119860 (119898

119904)) (119898

119904) minus 119895120596

119888119876 (119898119904) +

119879

(119898119904) minus119861 (119898

119904) 119862119879

1(119898119904)

(119898119904) + 119895120596

119888119876 (119898119904) + (119898

119904) minus119876 (119898

119904) 0 0

minus119861119879

(119898119904) 0 minus120574

2

119868 0

1198621(119898119904) 0 0 minus119868

]]]]

]

lt 0 (55)

[minus119868 radic1205881198622 (119898119904)

119896

lowast minus119878(119898119904)

] lt 0 119896 = 1 2 (56)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904)

= 119880minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)








119870119894 119861119870119894 119862119870119894 119863119870119894 119894119872119894 119883119894 119885119894


[11986911198692

lowast 1198693

] lt 0 1 le 119894 le 119895 le 2 (58)

[1198701+ 11989511987021198703

lowast 1198704

] lt 0 1 le 119894 le 119895 le 2 (59)

[minus119868 radic1205881198622119894119895 + 1198622119895119894

119896


] lt 0 119896 = 1 2 1 le 119894 le 119895 le 2

(60)

where

1198691= [

sym (119894+ 119895) minus119860

119894119895minus 119860119895119894+ 119878119894+ 119878119895


]

1198692= [

minus119879

119894minus 119879

119895minus119861119894119895minus 119861119895119894

0 0]

1198693= [

minus119878119894minus 119878119895

0


1198701= [

minus12059611205962(119876119894+ 119876119895) minus sym (119860

119894119895+ 119860119895119894) 119894+ 119895+ 119879

119894+ 119879

119895

119894+ 119895+ 119894+ 119895

minus119876119894minus 119876119895

]

1198702= [

0 minus120596119888(119876119894+ 119876119895)

120596119888(119876119894+ 119876119895) 0

]


1198703= [

minus119861119894119895minus 119861119895119894119862119879

1119894119895+ 119862119879

1119895119894

0 0]

1198704= [

minus21205742

119868 0

0 minus2119868]

119860119894119895= [

119883119879

119894119860119895+ 119861119870119894119862 119860

119870119894

119860119894+ 119861119894119863119870119895119862 119860119894119872119895+ 119861119894119862119870119895

]

119861119894119895= [

119883119879

1198941198611119895

1198611119894

]

1198621119894119895= [1198621119894+ 1198631119894119863119870119895119862 1198621119894119872119895+ 1198631119894119862119870119895]

1198622119894119895= [11986221198941198622119894119872119895]

119894= [

119883119879

119894119885119894

119868 119872119894

]

(61)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904) = 119880

minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)

(62)

where

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904) )

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(63)



Denote

Ψ (119898119904) =

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

(64)


Ψ119894119895+ Ψ119895119894lt 0 1 le 119894 lt 119895 le 2

Ψ119894119894lt 0 119894 = 1 2

(65)

where

Ψ119894119895=

[[[

[

sym (119894) minus119860

119894119895+ 119878119894minus119879

119894minus119861119894119895


0 0


0


]]]

]

1 le 119894 le 119895 le 2

(66)

Assume that

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904))

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(67)

and then we get

Ψ (119898119904) =

2

sum

119894=1

2

sum

119895=1

120582119894120582119895Ψ119894119895=

2

sum

119894=1

1205822

119894Ψ119894119894+ 12058211205822(Ψ12+ Ψ21) (68)


Ψ (119898119904) lt 0 (69)

Remark 4 119880(119898119904) 119867(119898


definition that is

119880119879

(119898119904)119867 (119898

119904) = 119885 (119898

119904) minus 119883119879

(119898119904)119872 (119898

119904) (70)




119904) and 119872(119898

119904) In this



2 1198781+


minus11987821198781


1198781= [

11987011198703

lowast 1198704

] 1198782= [

11987020

0 0] (71)



119870(119898119904) 119861119870(119898119904) 119862119870(119898119904) 119863119870(119898119904) and Lyapunov


119904


119863119870(119898119904) = 119863

119870(119898119904) =

2

sum

119894=1

120582119894119863119870119894= 12058211198631198701+ 12058221198631198702

=1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1198631198701

+1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

1198631198702

(72)

where 1198631198701

and 1198631198702


119870(119898119904) may



1= 2 1198781

= 1198782 1198761= 1198762 1198601198701

= 1198601198702

1198611198701= 1198611198702 1198621198701= 1198621198702 1198631198701= 1198631198702 1= 21198721= 1198722

1198831= 1198832 and119885






119862119894 119883119862119894


119862119894 119861119862119894 119862119862119894 119863119862119894 (119894 = 1 2) satisfying

[[

[

sym (119860119890119894119895+ 119860119890119895119894) 119861119890119894119895

119862119879

1198901119894119895+ 119862119879

1198901119895119894


119868 0


]]

]

lt 0

1 le 119894 le 119895 le 2

(73)

[minus2119868 radic1205881198621198902119894119895 + 1198621198902119895119894

119896


] lt 0

119896 = 1 2 1 le 119894 le 119895 le 2

(74)

where

119860119890119894119895= [

119860119894119883119862119895+ 119861119894119862119862119895

119860119894+ 119861119894119863119862119895119862

119860119862119894

119884119862119894119860119895+ 119861119862119894119862]

119861119890119894119895= [

1198611119894

1198841198621198941198611119895

]

1198621198901119894119895

= [1198621119894119883119862119895+ 11986311198941198621198621198951198621119894+ 1198631119894119863119862119895119862]

1198621198902119894119895

= [11986221198941198831198621198951198622119894]

119875119862119894

= [119883119862119894

119868

119868 119884119862119894

]

(75)


infinlt 120574 with 119908max = 120588120578


119863119870(119898119904) = 119863

119862(119898119904) (76)

119862119870(119898119904) = (119862

119862(119898119904) minus 119863119862(119898119904) 119862119883119862(119898119904))119872minus119879

119862(119898119904)

(77)

119861119870(119898119904) = 119873

minus1

119862(119898119904) (119861119862(119898119904) minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904))

(78)

119860119870(119898119904)

= 119873minus1

119862(119898119904) [119860119862(119898119904) minus 119884119862(119898119904) 119860 (119898

119904)119883119862(119898119904)

minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904) 119862119883119862(119898119904)

minus 119873119862(119898119904) 119861119870(119898119904) 119862119883119862(119898119904)

minus119884119862(119898119904) 119861 (119898

119904) 119862119870(119898119904)119872119879

119862]


119862(119898119904)

(79)

where

(119860119862(119898119904) 119861119862(119898119904) 119862119862(119898119904) 119863119862(119898119904) 119883119862(119898119904) 119884119862(119898119904))

=

2

sum

119894=1

120582119894(119860119862119894 119861119862119894 119862119862119894 119863119862119894 119883119862119894 119884119862119894)

(80)



119862(119898119904) and119872



119873119862(119898119904)119872119879

119862(119898119904) = 119868 minus 119884

119862(119898119904)119883119862(119898119904) (81)

4 A Design Example



10000 119911max = 01m1205961= 8120587rads and120596

2= 16120587radsThen




119896119904

119896119905

119888119904

119888119905

119898119904max 119898

119904min 119898119904

119898119906


Frequency (Hz)

Max

imum

sing

ular

val

ues (

abs)

10minus1

100

101

102


4ndash8 Hz

25

20

15

10

5

0









119908 (119905) = 119860 sin (2120587119891119905) 119905 isin [0 119879]

0 119905 notin [0 119879] (82)




1199002(119905)119896| lt 1 119896 = 1 2

5 Conclusion


0 02 04 06 08 1 12 14 16 18 2

005

115

225

Time (seconds)

minus25

minus1

minus2

minus15

minus05


Body

accle

ratio

n (m

s2)


0 02 04 06 08 1 12 14 16 18 2

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

0

Relat

ive d

ynam

ic ti

re lo

ad


0 02 04 06 08 1 12 14 16 18 2

0

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

Rela

tive s

uspe

nsio

n str

oke




Appendices

A Related Lemmas



120596119865


119873lowast

120596Θ119873120596lt 0 120596 isin [120596

11205962] (A1)


119865lowast


where

Φ = [0 1

1 0] Ψ = [

minus1 119895120596119888

minus119895120596119888minus12059611205962

] (A3)

with 120596119888= (1205961+ 1205962)2




Γperp

Θ(Γperp

)119879

lt 0 (Λ119879

)perp

Θ((Λ119879

)perp

)

119879

lt 0 (A5)


(1) Ψ + 119878 + 119878119879 lt 0

(2) The LMI problem

[Ψ + 119875 minus [119882

119904] 119878119879

+119882119879





119860119870=

[[[

[


minus22844 minus13692 13701 002030

20381 15159 minus21603 minus047598

24991 times 109


42012 times 106


]]]

]

119861119870=

[[[

[

99943 10908 44969


minus63200 11564 minus11258




]]]

]

119862119870= [38144 minus68619 17589 35961]

119863119870= [minus18676 times 10

5


minus86675]

(B1)


119860119870=

[[[

[

minus12734 26225 21367 minus031502

minus36286 14954 36984 069500



]]]

]

119861119870=

[[[

[


17709 12215 11303

minus90855 34487 minus80239

64417 times 105

71058 times 106


]]]

]

119862119870= [51819 minus60403 minus59041 10015]

119863119870= [25422 17208 26098]

(B2)

References





































2synthesis with












Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(119905) le 120578119908119879

(119905) 119908 (119905) (44)


119881 (119905) minus 119881 (0) le 120578int

119905

0

119908119879

(119905) 119908 (119905) 119889119905 = 120578119908 (119905)2le 120578119908max

(45)


119909119879

(119905) 119875119878(119898119904) 119909 (119905) le 119881 (0) + 120578119908max = 120588 (46)


119909119879

(119905) 119909 (119905) le 120588 (47)

where 119909(119905) = 11987512119878(119898119904)119909(119905)

Noting that

max 10038161003816100381610038161199111199002 (119905)1198961003816100381610038161003816

2

= max100381710038171003817100381710038171003817119909119879

(119905) 119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

times 1198622(119898119904)119896

119875minus12

119878(119898119904) 119909(119905)

100381710038171003817100381710038172

le 120588120579max (119875minus12

119878(119898119904) 1198622(119898119904)119879

119896

1198622(119898119904)119896

119875minus12

119878(119898119904))

(48)


120588119875minus12

119878(119898119904) 1198622(119898119904)119879

1198961198622(119898119904)119896119875minus12

119878(119898119904) lt 119868 (49)




ing form

119882(119898119904) = [

119883 (119898119904) 119884 (119898

119904)

119880 (119898119904) 119881 (119898

119904)]

119882minus1

(119898119904) = [

119872(119898119904) 119866 (119898

119904)

119867 (119898119904) 119871 (119898

119904)]

(50)



Denote

Δ (119898119904) = [

119868 119872 (119898119904)

0 119867 (119898119904)] (51)


1198691(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) Δ119879

(119898119904) 119868

1198692(119898119904) = diag Δ119879 (119898

119904) Δ119879

(119898119904) 119868 119868

1198693(119898119904) = diag 119868 Δ119879 (119898

119904)

(52)

respectivelyDenote

119876 (119898119904) = Δ119879

(119898119904) 119876 (119898

119904) Δ (119898

119904)

(119898119904) = Δ119879

(119898119904) 119875 (119898

119904) Δ (119898

119904)

119878(119898119904) = Δ119879

(119898119904) 119875119878(119898119904) Δ (119898

119904)

119860 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119860 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119860 (119898

119904) + 119861119870(119898119904) 119862 119860

119870(119898119904)

119860 (119898119904) + 119861 (119898

119904)119863119870(119898119904) 119862 119860 (119898

119904)119872 (119898

119904) + 119861 (119898

119904) 119862119870(119898119904)]

119861 (119898119904) = Δ119879

(119898119904)119882119879

(119898119904) 119861 (119898

119904)

= [119883119879

(119898119904) 1198611(119898119904)

1198611(119898119904)

]

1198621(119898119904) = 1198621(119898119904) Δ (119898

119904)

= [1198621(119898119904) + 1198631(119898119904)119863119870(119898119904) 119862 119862

1(119898119904)119872 (119898

119904) + 1198631(119898119904) 119862119870(119898119904)]

1198622(119898119904) = 1198622(119898119904) Δ (119898

119904) = [119862

2(119898119904) 1198622(119898119904)119872 (119898

119904)]


(119898119904) = Δ119879

(119898119904)119882 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119885 (119898

119904)

119868 119872 (119898119904)]

(53)




119878(119898119904) 119876(119898


119870(119898119904) 119861119870(119898119904)

119862119870(119898119904)119863119870(119898119904) (119898

119904)119872(119898

119904)119883(119898

119904) 119885(119898

119904) satisfying

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

lt 0 (54)

[[[[

[

minus12059611205962119876 (119898119904) minus sym (119860 (119898

119904)) (119898

119904) minus 119895120596

119888119876 (119898119904) +

119879

(119898119904) minus119861 (119898

119904) 119862119879

1(119898119904)

(119898119904) + 119895120596

119888119876 (119898119904) + (119898

119904) minus119876 (119898

119904) 0 0

minus119861119879

(119898119904) 0 minus120574

2

119868 0

1198621(119898119904) 0 0 minus119868

]]]]

]

lt 0 (55)

[minus119868 radic1205881198622 (119898119904)

119896

lowast minus119878(119898119904)

] lt 0 119896 = 1 2 (56)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904)

= 119880minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)








119870119894 119861119870119894 119862119870119894 119863119870119894 119894119872119894 119883119894 119885119894


[11986911198692

lowast 1198693

] lt 0 1 le 119894 le 119895 le 2 (58)

[1198701+ 11989511987021198703

lowast 1198704

] lt 0 1 le 119894 le 119895 le 2 (59)

[minus119868 radic1205881198622119894119895 + 1198622119895119894

119896


] lt 0 119896 = 1 2 1 le 119894 le 119895 le 2

(60)

where

1198691= [

sym (119894+ 119895) minus119860

119894119895minus 119860119895119894+ 119878119894+ 119878119895


]

1198692= [

minus119879

119894minus 119879

119895minus119861119894119895minus 119861119895119894

0 0]

1198693= [

minus119878119894minus 119878119895

0


1198701= [

minus12059611205962(119876119894+ 119876119895) minus sym (119860

119894119895+ 119860119895119894) 119894+ 119895+ 119879

119894+ 119879

119895

119894+ 119895+ 119894+ 119895

minus119876119894minus 119876119895

]

1198702= [

0 minus120596119888(119876119894+ 119876119895)

120596119888(119876119894+ 119876119895) 0

]


1198703= [

minus119861119894119895minus 119861119895119894119862119879

1119894119895+ 119862119879

1119895119894

0 0]

1198704= [

minus21205742

119868 0

0 minus2119868]

119860119894119895= [

119883119879

119894119860119895+ 119861119870119894119862 119860

119870119894

119860119894+ 119861119894119863119870119895119862 119860119894119872119895+ 119861119894119862119870119895

]

119861119894119895= [

119883119879

1198941198611119895

1198611119894

]

1198621119894119895= [1198621119894+ 1198631119894119863119870119895119862 1198621119894119872119895+ 1198631119894119862119870119895]

1198622119894119895= [11986221198941198622119894119872119895]

119894= [

119883119879

119894119885119894

119868 119872119894

]

(61)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904) = 119880

minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)

(62)

where

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904) )

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(63)



Denote

Ψ (119898119904) =

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

(64)


Ψ119894119895+ Ψ119895119894lt 0 1 le 119894 lt 119895 le 2

Ψ119894119894lt 0 119894 = 1 2

(65)

where

Ψ119894119895=

[[[

[

sym (119894) minus119860

119894119895+ 119878119894minus119879

119894minus119861119894119895


0 0


0


]]]

]

1 le 119894 le 119895 le 2

(66)

Assume that

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904))

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(67)

and then we get

Ψ (119898119904) =

2

sum

119894=1

2

sum

119895=1

120582119894120582119895Ψ119894119895=

2

sum

119894=1

1205822

119894Ψ119894119894+ 12058211205822(Ψ12+ Ψ21) (68)


Ψ (119898119904) lt 0 (69)

Remark 4 119880(119898119904) 119867(119898


definition that is

119880119879

(119898119904)119867 (119898

119904) = 119885 (119898

119904) minus 119883119879

(119898119904)119872 (119898

119904) (70)




119904) and 119872(119898

119904) In this



2 1198781+


minus11987821198781


1198781= [

11987011198703

lowast 1198704

] 1198782= [

11987020

0 0] (71)



119870(119898119904) 119861119870(119898119904) 119862119870(119898119904) 119863119870(119898119904) and Lyapunov


119904


119863119870(119898119904) = 119863

119870(119898119904) =

2

sum

119894=1

120582119894119863119870119894= 12058211198631198701+ 12058221198631198702

=1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1198631198701

+1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

1198631198702

(72)

where 1198631198701

and 1198631198702


119870(119898119904) may



1= 2 1198781

= 1198782 1198761= 1198762 1198601198701

= 1198601198702

1198611198701= 1198611198702 1198621198701= 1198621198702 1198631198701= 1198631198702 1= 21198721= 1198722

1198831= 1198832 and119885






119862119894 119883119862119894


119862119894 119861119862119894 119862119862119894 119863119862119894 (119894 = 1 2) satisfying

[[

[

sym (119860119890119894119895+ 119860119890119895119894) 119861119890119894119895

119862119879

1198901119894119895+ 119862119879

1198901119895119894


119868 0


]]

]

lt 0

1 le 119894 le 119895 le 2

(73)

[minus2119868 radic1205881198621198902119894119895 + 1198621198902119895119894

119896


] lt 0

119896 = 1 2 1 le 119894 le 119895 le 2

(74)

where

119860119890119894119895= [

119860119894119883119862119895+ 119861119894119862119862119895

119860119894+ 119861119894119863119862119895119862

119860119862119894

119884119862119894119860119895+ 119861119862119894119862]

119861119890119894119895= [

1198611119894

1198841198621198941198611119895

]

1198621198901119894119895

= [1198621119894119883119862119895+ 11986311198941198621198621198951198621119894+ 1198631119894119863119862119895119862]

1198621198902119894119895

= [11986221198941198831198621198951198622119894]

119875119862119894

= [119883119862119894

119868

119868 119884119862119894

]

(75)


infinlt 120574 with 119908max = 120588120578


119863119870(119898119904) = 119863

119862(119898119904) (76)

119862119870(119898119904) = (119862

119862(119898119904) minus 119863119862(119898119904) 119862119883119862(119898119904))119872minus119879

119862(119898119904)

(77)

119861119870(119898119904) = 119873

minus1

119862(119898119904) (119861119862(119898119904) minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904))

(78)

119860119870(119898119904)

= 119873minus1

119862(119898119904) [119860119862(119898119904) minus 119884119862(119898119904) 119860 (119898

119904)119883119862(119898119904)

minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904) 119862119883119862(119898119904)

minus 119873119862(119898119904) 119861119870(119898119904) 119862119883119862(119898119904)

minus119884119862(119898119904) 119861 (119898

119904) 119862119870(119898119904)119872119879

119862]


119862(119898119904)

(79)

where

(119860119862(119898119904) 119861119862(119898119904) 119862119862(119898119904) 119863119862(119898119904) 119883119862(119898119904) 119884119862(119898119904))

=

2

sum

119894=1

120582119894(119860119862119894 119861119862119894 119862119862119894 119863119862119894 119883119862119894 119884119862119894)

(80)



119862(119898119904) and119872



119873119862(119898119904)119872119879

119862(119898119904) = 119868 minus 119884

119862(119898119904)119883119862(119898119904) (81)

4 A Design Example



10000 119911max = 01m1205961= 8120587rads and120596

2= 16120587radsThen




119896119904

119896119905

119888119904

119888119905

119898119904max 119898

119904min 119898119904

119898119906


Frequency (Hz)

Max

imum

sing

ular

val

ues (

abs)

10minus1

100

101

102


4ndash8 Hz

25

20

15

10

5

0









119908 (119905) = 119860 sin (2120587119891119905) 119905 isin [0 119879]

0 119905 notin [0 119879] (82)




1199002(119905)119896| lt 1 119896 = 1 2

5 Conclusion


0 02 04 06 08 1 12 14 16 18 2

005

115

225

Time (seconds)

minus25

minus1

minus2

minus15

minus05


Body

accle

ratio

n (m

s2)


0 02 04 06 08 1 12 14 16 18 2

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

0

Relat

ive d

ynam

ic ti

re lo

ad


0 02 04 06 08 1 12 14 16 18 2

0

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

Rela

tive s

uspe

nsio

n str

oke




Appendices

A Related Lemmas



120596119865


119873lowast

120596Θ119873120596lt 0 120596 isin [120596

11205962] (A1)


119865lowast


where

Φ = [0 1

1 0] Ψ = [

minus1 119895120596119888

minus119895120596119888minus12059611205962

] (A3)

with 120596119888= (1205961+ 1205962)2




Γperp

Θ(Γperp

)119879

lt 0 (Λ119879

)perp

Θ((Λ119879

)perp

)

119879

lt 0 (A5)


(1) Ψ + 119878 + 119878119879 lt 0

(2) The LMI problem

[Ψ + 119875 minus [119882

119904] 119878119879

+119882119879





119860119870=

[[[

[


minus22844 minus13692 13701 002030

20381 15159 minus21603 minus047598

24991 times 109


42012 times 106


]]]

]

119861119870=

[[[

[

99943 10908 44969


minus63200 11564 minus11258




]]]

]

119862119870= [38144 minus68619 17589 35961]

119863119870= [minus18676 times 10

5


minus86675]

(B1)


119860119870=

[[[

[

minus12734 26225 21367 minus031502

minus36286 14954 36984 069500



]]]

]

119861119870=

[[[

[


17709 12215 11303

minus90855 34487 minus80239

64417 times 105

71058 times 106


]]]

]

119862119870= [51819 minus60403 minus59041 10015]

119863119870= [25422 17208 26098]

(B2)

References





































2synthesis with












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(119898119904) = Δ119879

(119898119904)119882 (119898

119904) Δ (119898

119904)

= [119883119879

(119898119904) 119885 (119898

119904)

119868 119872 (119898119904)]

(53)




119878(119898119904) 119876(119898


119870(119898119904) 119861119870(119898119904)

119862119870(119898119904)119863119870(119898119904) (119898

119904)119872(119898

119904)119883(119898

119904) 119885(119898

119904) satisfying

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

lt 0 (54)

[[[[

[

minus12059611205962119876 (119898119904) minus sym (119860 (119898

119904)) (119898

119904) minus 119895120596

119888119876 (119898119904) +

119879

(119898119904) minus119861 (119898

119904) 119862119879

1(119898119904)

(119898119904) + 119895120596

119888119876 (119898119904) + (119898

119904) minus119876 (119898

119904) 0 0

minus119861119879

(119898119904) 0 minus120574

2

119868 0

1198621(119898119904) 0 0 minus119868

]]]]

]

lt 0 (55)

[minus119868 radic1205881198622 (119898119904)

119896

lowast minus119878(119898119904)

] lt 0 119896 = 1 2 (56)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904)

= 119880minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)








119870119894 119861119870119894 119862119870119894 119863119870119894 119894119872119894 119883119894 119885119894


[11986911198692

lowast 1198693

] lt 0 1 le 119894 le 119895 le 2 (58)

[1198701+ 11989511987021198703

lowast 1198704

] lt 0 1 le 119894 le 119895 le 2 (59)

[minus119868 radic1205881198622119894119895 + 1198622119895119894

119896


] lt 0 119896 = 1 2 1 le 119894 le 119895 le 2

(60)

where

1198691= [

sym (119894+ 119895) minus119860

119894119895minus 119860119895119894+ 119878119894+ 119878119895


]

1198692= [

minus119879

119894minus 119879

119895minus119861119894119895minus 119861119895119894

0 0]

1198693= [

minus119878119894minus 119878119895

0


1198701= [

minus12059611205962(119876119894+ 119876119895) minus sym (119860

119894119895+ 119860119895119894) 119894+ 119895+ 119879

119894+ 119879

119895

119894+ 119895+ 119894+ 119895

minus119876119894minus 119876119895

]

1198702= [

0 minus120596119888(119876119894+ 119876119895)

120596119888(119876119894+ 119876119895) 0

]


1198703= [

minus119861119894119895minus 119861119895119894119862119879

1119894119895+ 119862119879

1119895119894

0 0]

1198704= [

minus21205742

119868 0

0 minus2119868]

119860119894119895= [

119883119879

119894119860119895+ 119861119870119894119862 119860

119870119894

119860119894+ 119861119894119863119870119895119862 119860119894119872119895+ 119861119894119862119870119895

]

119861119894119895= [

119883119879

1198941198611119895

1198611119894

]

1198621119894119895= [1198621119894+ 1198631119894119863119870119895119862 1198621119894119872119895+ 1198631119894119862119870119895]

1198622119894119895= [11986221198941198622119894119872119895]

119894= [

119883119879

119894119885119894

119868 119872119894

]

(61)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904) = 119880

minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)

(62)

where

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904) )

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(63)



Denote

Ψ (119898119904) =

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

(64)


Ψ119894119895+ Ψ119895119894lt 0 1 le 119894 lt 119895 le 2

Ψ119894119894lt 0 119894 = 1 2

(65)

where

Ψ119894119895=

[[[

[

sym (119894) minus119860

119894119895+ 119878119894minus119879

119894minus119861119894119895


0 0


0


]]]

]

1 le 119894 le 119895 le 2

(66)

Assume that

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904))

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(67)

and then we get

Ψ (119898119904) =

2

sum

119894=1

2

sum

119895=1

120582119894120582119895Ψ119894119895=

2

sum

119894=1

1205822

119894Ψ119894119894+ 12058211205822(Ψ12+ Ψ21) (68)


Ψ (119898119904) lt 0 (69)

Remark 4 119880(119898119904) 119867(119898


definition that is

119880119879

(119898119904)119867 (119898

119904) = 119885 (119898

119904) minus 119883119879

(119898119904)119872 (119898

119904) (70)




119904) and 119872(119898

119904) In this



2 1198781+


minus11987821198781


1198781= [

11987011198703

lowast 1198704

] 1198782= [

11987020

0 0] (71)



119870(119898119904) 119861119870(119898119904) 119862119870(119898119904) 119863119870(119898119904) and Lyapunov


119904


119863119870(119898119904) = 119863

119870(119898119904) =

2

sum

119894=1

120582119894119863119870119894= 12058211198631198701+ 12058221198631198702

=1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1198631198701

+1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

1198631198702

(72)

where 1198631198701

and 1198631198702


119870(119898119904) may



1= 2 1198781

= 1198782 1198761= 1198762 1198601198701

= 1198601198702

1198611198701= 1198611198702 1198621198701= 1198621198702 1198631198701= 1198631198702 1= 21198721= 1198722

1198831= 1198832 and119885






119862119894 119883119862119894


119862119894 119861119862119894 119862119862119894 119863119862119894 (119894 = 1 2) satisfying

[[

[

sym (119860119890119894119895+ 119860119890119895119894) 119861119890119894119895

119862119879

1198901119894119895+ 119862119879

1198901119895119894


119868 0


]]

]

lt 0

1 le 119894 le 119895 le 2

(73)

[minus2119868 radic1205881198621198902119894119895 + 1198621198902119895119894

119896


] lt 0

119896 = 1 2 1 le 119894 le 119895 le 2

(74)

where

119860119890119894119895= [

119860119894119883119862119895+ 119861119894119862119862119895

119860119894+ 119861119894119863119862119895119862

119860119862119894

119884119862119894119860119895+ 119861119862119894119862]

119861119890119894119895= [

1198611119894

1198841198621198941198611119895

]

1198621198901119894119895

= [1198621119894119883119862119895+ 11986311198941198621198621198951198621119894+ 1198631119894119863119862119895119862]

1198621198902119894119895

= [11986221198941198831198621198951198622119894]

119875119862119894

= [119883119862119894

119868

119868 119884119862119894

]

(75)


infinlt 120574 with 119908max = 120588120578


119863119870(119898119904) = 119863

119862(119898119904) (76)

119862119870(119898119904) = (119862

119862(119898119904) minus 119863119862(119898119904) 119862119883119862(119898119904))119872minus119879

119862(119898119904)

(77)

119861119870(119898119904) = 119873

minus1

119862(119898119904) (119861119862(119898119904) minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904))

(78)

119860119870(119898119904)

= 119873minus1

119862(119898119904) [119860119862(119898119904) minus 119884119862(119898119904) 119860 (119898

119904)119883119862(119898119904)

minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904) 119862119883119862(119898119904)

minus 119873119862(119898119904) 119861119870(119898119904) 119862119883119862(119898119904)

minus119884119862(119898119904) 119861 (119898

119904) 119862119870(119898119904)119872119879

119862]


119862(119898119904)

(79)

where

(119860119862(119898119904) 119861119862(119898119904) 119862119862(119898119904) 119863119862(119898119904) 119883119862(119898119904) 119884119862(119898119904))

=

2

sum

119894=1

120582119894(119860119862119894 119861119862119894 119862119862119894 119863119862119894 119883119862119894 119884119862119894)

(80)



119862(119898119904) and119872



119873119862(119898119904)119872119879

119862(119898119904) = 119868 minus 119884

119862(119898119904)119883119862(119898119904) (81)

4 A Design Example



10000 119911max = 01m1205961= 8120587rads and120596

2= 16120587radsThen




119896119904

119896119905

119888119904

119888119905

119898119904max 119898

119904min 119898119904

119898119906


Frequency (Hz)

Max

imum

sing

ular

val

ues (

abs)

10minus1

100

101

102


4ndash8 Hz

25

20

15

10

5

0









119908 (119905) = 119860 sin (2120587119891119905) 119905 isin [0 119879]

0 119905 notin [0 119879] (82)




1199002(119905)119896| lt 1 119896 = 1 2

5 Conclusion


0 02 04 06 08 1 12 14 16 18 2

005

115

225

Time (seconds)

minus25

minus1

minus2

minus15

minus05


Body

accle

ratio

n (m

s2)


0 02 04 06 08 1 12 14 16 18 2

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

0

Relat

ive d

ynam

ic ti

re lo

ad


0 02 04 06 08 1 12 14 16 18 2

0

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

Rela

tive s

uspe

nsio

n str

oke




Appendices

A Related Lemmas



120596119865


119873lowast

120596Θ119873120596lt 0 120596 isin [120596

11205962] (A1)


119865lowast


where

Φ = [0 1

1 0] Ψ = [

minus1 119895120596119888

minus119895120596119888minus12059611205962

] (A3)

with 120596119888= (1205961+ 1205962)2




Γperp

Θ(Γperp

)119879

lt 0 (Λ119879

)perp

Θ((Λ119879

)perp

)

119879

lt 0 (A5)


(1) Ψ + 119878 + 119878119879 lt 0

(2) The LMI problem

[Ψ + 119875 minus [119882

119904] 119878119879

+119882119879





119860119870=

[[[

[


minus22844 minus13692 13701 002030

20381 15159 minus21603 minus047598

24991 times 109


42012 times 106


]]]

]

119861119870=

[[[

[

99943 10908 44969


minus63200 11564 minus11258




]]]

]

119862119870= [38144 minus68619 17589 35961]

119863119870= [minus18676 times 10

5


minus86675]

(B1)


119860119870=

[[[

[

minus12734 26225 21367 minus031502

minus36286 14954 36984 069500



]]]

]

119861119870=

[[[

[


17709 12215 11303

minus90855 34487 minus80239

64417 times 105

71058 times 106


]]]

]

119862119870= [51819 minus60403 minus59041 10015]

119863119870= [25422 17208 26098]

(B2)

References





































2synthesis with












Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198703= [

minus119861119894119895minus 119861119895119894119862119879

1119894119895+ 119862119879

1119895119894

0 0]

1198704= [

minus21205742

119868 0

0 minus2119868]

119860119894119895= [

119883119879

119894119860119895+ 119861119870119894119862 119860

119870119894

119860119894+ 119861119894119863119870119895119862 119860119894119872119895+ 119861119894119862119870119895

]

119861119894119895= [

119883119879

1198941198611119895

1198611119894

]

1198621119894119895= [1198621119894+ 1198631119894119863119870119895119862 1198621119894119872119895+ 1198631119894119862119870119895]

1198622119894119895= [11986221198941198622119894119872119895]

119894= [

119883119879

119894119885119894

119868 119872119894

]

(61)



119863119870(119898119904) = 119863

119870(119898119904)

119862119870(119898119904) = (119862

119870(119898119904) minus 119863119870(119898119904) 119862119872(119898

119904))119867minus1

(119898119904)

119861119870(119898119904) = 119880

minus119879

(119898119904) (119861119870(119898119904) minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904))

119860119870(119898119904) = 119880

minus119879

(119898119904) [119860119870(119898119904) minus 119883119879

(119898119904) 119860 (119898

119904)119872 (119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904)119863119870(119898119904) 119862119872(119898

119904)

minus 119880119879

(119898119904) 119861119870(119898119904) 119862119872(119898

119904)

minus 119883119879

(119898119904) 119861 (119898

119904) 119862119870(119898119904)119867 (119898

119904)]

times 119867minus1

(119898119904)

(62)

where

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904) )

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(63)



Denote

Ψ (119898119904) =

[[[

[

sym ( (119898119904)) minus119860 (119898

119904) + 119878(119898119904) minus

119879

(119898119904) minus119861 (119898

119904)

lowast minus119878(119898119904) 0 0



]]]

]

(64)


Ψ119894119895+ Ψ119895119894lt 0 1 le 119894 lt 119895 le 2

Ψ119894119894lt 0 119894 = 1 2

(65)

where

Ψ119894119895=

[[[

[

sym (119894) minus119860

119894119895+ 119878119894minus119879

119894minus119861119894119895


0 0


0


]]]

]

1 le 119894 le 119895 le 2

(66)

Assume that

(119860119870(119898119904) 119861119870(119898119904) 119862119870(119898119904)

119863119870(119898119904) 119872 (119898

119904) 119883 (119898

119904) 119885 (119898

119904))

=

2

sum

119894=1

120582119894(119860119870119894 119861119870119894 119862119870119894 119863119870119894119872119894 119883119894 119885119894)

(67)

and then we get

Ψ (119898119904) =

2

sum

119894=1

2

sum

119895=1

120582119894120582119895Ψ119894119895=

2

sum

119894=1

1205822

119894Ψ119894119894+ 12058211205822(Ψ12+ Ψ21) (68)


Ψ (119898119904) lt 0 (69)

Remark 4 119880(119898119904) 119867(119898


definition that is

119880119879

(119898119904)119867 (119898

119904) = 119885 (119898

119904) minus 119883119879

(119898119904)119872 (119898

119904) (70)




119904) and 119872(119898

119904) In this



2 1198781+


minus11987821198781


1198781= [

11987011198703

lowast 1198704

] 1198782= [

11987020

0 0] (71)



119870(119898119904) 119861119870(119898119904) 119862119870(119898119904) 119863119870(119898119904) and Lyapunov


119904


119863119870(119898119904) = 119863

119870(119898119904) =

2

sum

119894=1

120582119894119863119870119894= 12058211198631198701+ 12058221198631198702

=1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1198631198701

+1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

1198631198702

(72)

where 1198631198701

and 1198631198702


119870(119898119904) may



1= 2 1198781

= 1198782 1198761= 1198762 1198601198701

= 1198601198702

1198611198701= 1198611198702 1198621198701= 1198621198702 1198631198701= 1198631198702 1= 21198721= 1198722

1198831= 1198832 and119885






119862119894 119883119862119894


119862119894 119861119862119894 119862119862119894 119863119862119894 (119894 = 1 2) satisfying

[[

[

sym (119860119890119894119895+ 119860119890119895119894) 119861119890119894119895

119862119879

1198901119894119895+ 119862119879

1198901119895119894


119868 0


]]

]

lt 0

1 le 119894 le 119895 le 2

(73)

[minus2119868 radic1205881198621198902119894119895 + 1198621198902119895119894

119896


] lt 0

119896 = 1 2 1 le 119894 le 119895 le 2

(74)

where

119860119890119894119895= [

119860119894119883119862119895+ 119861119894119862119862119895

119860119894+ 119861119894119863119862119895119862

119860119862119894

119884119862119894119860119895+ 119861119862119894119862]

119861119890119894119895= [

1198611119894

1198841198621198941198611119895

]

1198621198901119894119895

= [1198621119894119883119862119895+ 11986311198941198621198621198951198621119894+ 1198631119894119863119862119895119862]

1198621198902119894119895

= [11986221198941198831198621198951198622119894]

119875119862119894

= [119883119862119894

119868

119868 119884119862119894

]

(75)


infinlt 120574 with 119908max = 120588120578


119863119870(119898119904) = 119863

119862(119898119904) (76)

119862119870(119898119904) = (119862

119862(119898119904) minus 119863119862(119898119904) 119862119883119862(119898119904))119872minus119879

119862(119898119904)

(77)

119861119870(119898119904) = 119873

minus1

119862(119898119904) (119861119862(119898119904) minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904))

(78)

119860119870(119898119904)

= 119873minus1

119862(119898119904) [119860119862(119898119904) minus 119884119862(119898119904) 119860 (119898

119904)119883119862(119898119904)

minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904) 119862119883119862(119898119904)

minus 119873119862(119898119904) 119861119870(119898119904) 119862119883119862(119898119904)

minus119884119862(119898119904) 119861 (119898

119904) 119862119870(119898119904)119872119879

119862]


119862(119898119904)

(79)

where

(119860119862(119898119904) 119861119862(119898119904) 119862119862(119898119904) 119863119862(119898119904) 119883119862(119898119904) 119884119862(119898119904))

=

2

sum

119894=1

120582119894(119860119862119894 119861119862119894 119862119862119894 119863119862119894 119883119862119894 119884119862119894)

(80)



119862(119898119904) and119872



119873119862(119898119904)119872119879

119862(119898119904) = 119868 minus 119884

119862(119898119904)119883119862(119898119904) (81)

4 A Design Example



10000 119911max = 01m1205961= 8120587rads and120596

2= 16120587radsThen




119896119904

119896119905

119888119904

119888119905

119898119904max 119898

119904min 119898119904

119898119906


Frequency (Hz)

Max

imum

sing

ular

val

ues (

abs)

10minus1

100

101

102


4ndash8 Hz

25

20

15

10

5

0









119908 (119905) = 119860 sin (2120587119891119905) 119905 isin [0 119879]

0 119905 notin [0 119879] (82)




1199002(119905)119896| lt 1 119896 = 1 2

5 Conclusion


0 02 04 06 08 1 12 14 16 18 2

005

115

225

Time (seconds)

minus25

minus1

minus2

minus15

minus05


Body

accle

ratio

n (m

s2)


0 02 04 06 08 1 12 14 16 18 2

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

0

Relat

ive d

ynam

ic ti

re lo

ad


0 02 04 06 08 1 12 14 16 18 2

0

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

Rela

tive s

uspe

nsio

n str

oke




Appendices

A Related Lemmas



120596119865


119873lowast

120596Θ119873120596lt 0 120596 isin [120596

11205962] (A1)


119865lowast


where

Φ = [0 1

1 0] Ψ = [

minus1 119895120596119888

minus119895120596119888minus12059611205962

] (A3)

with 120596119888= (1205961+ 1205962)2




Γperp

Θ(Γperp

)119879

lt 0 (Λ119879

)perp

Θ((Λ119879

)perp

)

119879

lt 0 (A5)


(1) Ψ + 119878 + 119878119879 lt 0

(2) The LMI problem

[Ψ + 119875 minus [119882

119904] 119878119879

+119882119879





119860119870=

[[[

[


minus22844 minus13692 13701 002030

20381 15159 minus21603 minus047598

24991 times 109


42012 times 106


]]]

]

119861119870=

[[[

[

99943 10908 44969


minus63200 11564 minus11258




]]]

]

119862119870= [38144 minus68619 17589 35961]

119863119870= [minus18676 times 10

5


minus86675]

(B1)


119860119870=

[[[

[

minus12734 26225 21367 minus031502

minus36286 14954 36984 069500



]]]

]

119861119870=

[[[

[


17709 12215 11303

minus90855 34487 minus80239

64417 times 105

71058 times 106


]]]

]

119862119870= [51819 minus60403 minus59041 10015]

119863119870= [25422 17208 26098]

(B2)

References





































2synthesis with












Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119870(119898119904) 119861119870(119898119904) 119862119870(119898119904) 119863119870(119898119904) and Lyapunov


119904


119863119870(119898119904) = 119863

119870(119898119904) =

2

sum

119894=1

120582119894119863119870119894= 12058211198631198701+ 12058221198631198702

=1119898119904minus 1119898

119904min1119898119904max minus 1119898119904min

1198631198701

+1119898119904max minus 1119898119904

1119898119904max minus 1119898119904min

1198631198702

(72)

where 1198631198701

and 1198631198702


119870(119898119904) may



1= 2 1198781

= 1198782 1198761= 1198762 1198601198701

= 1198601198702

1198611198701= 1198611198702 1198621198701= 1198621198702 1198631198701= 1198631198702 1= 21198721= 1198722

1198831= 1198832 and119885






119862119894 119883119862119894


119862119894 119861119862119894 119862119862119894 119863119862119894 (119894 = 1 2) satisfying

[[

[

sym (119860119890119894119895+ 119860119890119895119894) 119861119890119894119895

119862119879

1198901119894119895+ 119862119879

1198901119895119894


119868 0


]]

]

lt 0

1 le 119894 le 119895 le 2

(73)

[minus2119868 radic1205881198621198902119894119895 + 1198621198902119895119894

119896


] lt 0

119896 = 1 2 1 le 119894 le 119895 le 2

(74)

where

119860119890119894119895= [

119860119894119883119862119895+ 119861119894119862119862119895

119860119894+ 119861119894119863119862119895119862

119860119862119894

119884119862119894119860119895+ 119861119862119894119862]

119861119890119894119895= [

1198611119894

1198841198621198941198611119895

]

1198621198901119894119895

= [1198621119894119883119862119895+ 11986311198941198621198621198951198621119894+ 1198631119894119863119862119895119862]

1198621198902119894119895

= [11986221198941198831198621198951198622119894]

119875119862119894

= [119883119862119894

119868

119868 119884119862119894

]

(75)


infinlt 120574 with 119908max = 120588120578


119863119870(119898119904) = 119863

119862(119898119904) (76)

119862119870(119898119904) = (119862

119862(119898119904) minus 119863119862(119898119904) 119862119883119862(119898119904))119872minus119879

119862(119898119904)

(77)

119861119870(119898119904) = 119873

minus1

119862(119898119904) (119861119862(119898119904) minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904))

(78)

119860119870(119898119904)

= 119873minus1

119862(119898119904) [119860119862(119898119904) minus 119884119862(119898119904) 119860 (119898

119904)119883119862(119898119904)

minus 119884119862(119898119904) 119861 (119898

119904)119863119862(119898119904) 119862119883119862(119898119904)

minus 119873119862(119898119904) 119861119870(119898119904) 119862119883119862(119898119904)

minus119884119862(119898119904) 119861 (119898

119904) 119862119870(119898119904)119872119879

119862]


119862(119898119904)

(79)

where

(119860119862(119898119904) 119861119862(119898119904) 119862119862(119898119904) 119863119862(119898119904) 119883119862(119898119904) 119884119862(119898119904))

=

2

sum

119894=1

120582119894(119860119862119894 119861119862119894 119862119862119894 119863119862119894 119883119862119894 119884119862119894)

(80)



119862(119898119904) and119872



119873119862(119898119904)119872119879

119862(119898119904) = 119868 minus 119884

119862(119898119904)119883119862(119898119904) (81)

4 A Design Example



10000 119911max = 01m1205961= 8120587rads and120596

2= 16120587radsThen




119896119904

119896119905

119888119904

119888119905

119898119904max 119898

119904min 119898119904

119898119906


Frequency (Hz)

Max

imum

sing

ular

val

ues (

abs)

10minus1

100

101

102


4ndash8 Hz

25

20

15

10

5

0









119908 (119905) = 119860 sin (2120587119891119905) 119905 isin [0 119879]

0 119905 notin [0 119879] (82)




1199002(119905)119896| lt 1 119896 = 1 2

5 Conclusion


0 02 04 06 08 1 12 14 16 18 2

005

115

225

Time (seconds)

minus25

minus1

minus2

minus15

minus05


Body

accle

ratio

n (m

s2)


0 02 04 06 08 1 12 14 16 18 2

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

0

Relat

ive d

ynam

ic ti

re lo

ad


0 02 04 06 08 1 12 14 16 18 2

0

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

Rela

tive s

uspe

nsio

n str

oke




Appendices

A Related Lemmas



120596119865


119873lowast

120596Θ119873120596lt 0 120596 isin [120596

11205962] (A1)


119865lowast


where

Φ = [0 1

1 0] Ψ = [

minus1 119895120596119888

minus119895120596119888minus12059611205962

] (A3)

with 120596119888= (1205961+ 1205962)2




Γperp

Θ(Γperp

)119879

lt 0 (Λ119879

)perp

Θ((Λ119879

)perp

)

119879

lt 0 (A5)


(1) Ψ + 119878 + 119878119879 lt 0

(2) The LMI problem

[Ψ + 119875 minus [119882

119904] 119878119879

+119882119879





119860119870=

[[[

[


minus22844 minus13692 13701 002030

20381 15159 minus21603 minus047598

24991 times 109


42012 times 106


]]]

]

119861119870=

[[[

[

99943 10908 44969


minus63200 11564 minus11258




]]]

]

119862119870= [38144 minus68619 17589 35961]

119863119870= [minus18676 times 10

5


minus86675]

(B1)


119860119870=

[[[

[

minus12734 26225 21367 minus031502

minus36286 14954 36984 069500



]]]

]

119861119870=

[[[

[


17709 12215 11303

minus90855 34487 minus80239

64417 times 105

71058 times 106


]]]

]

119862119870= [51819 minus60403 minus59041 10015]

119863119870= [25422 17208 26098]

(B2)

References





































2synthesis with












Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896119904

119896119905

119888119904

119888119905

119898119904max 119898

119904min 119898119904

119898119906


Frequency (Hz)

Max

imum

sing

ular

val

ues (

abs)

10minus1

100

101

102


4ndash8 Hz

25

20

15

10

5

0









119908 (119905) = 119860 sin (2120587119891119905) 119905 isin [0 119879]

0 119905 notin [0 119879] (82)




1199002(119905)119896| lt 1 119896 = 1 2

5 Conclusion


0 02 04 06 08 1 12 14 16 18 2

005

115

225

Time (seconds)

minus25

minus1

minus2

minus15

minus05


Body

accle

ratio

n (m

s2)


0 02 04 06 08 1 12 14 16 18 2

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

0

Relat

ive d

ynam

ic ti

re lo

ad


0 02 04 06 08 1 12 14 16 18 2

0

01

02

03

04

Time (seconds)

minus04

minus03

minus02

minus01

Rela

tive s

uspe

nsio

n str

oke




Appendices

A Related Lemmas



120596119865


119873lowast

120596Θ119873120596lt 0 120596 isin [120596

11205962] (A1)


119865lowast


where

Φ = [0 1

1 0] Ψ = [

minus1 119895120596119888

minus119895120596119888minus12059611205962

] (A3)

with 120596119888= (1205961+ 1205962)2




Γperp

Θ(Γperp

)119879

lt 0 (Λ119879

)perp

Θ((Λ119879

)perp

)

119879

lt 0 (A5)


(1) Ψ + 119878 + 119878119879 lt 0

(2) The LMI problem

[Ψ + 119875 minus [119882

119904] 119878119879

+119882119879





119860119870=

[[[

[


minus22844 minus13692 13701 002030

20381 15159 minus21603 minus047598

24991 times 109


42012 times 106


]]]

]

119861119870=

[[[

[

99943 10908 44969


minus63200 11564 minus11258




]]]

]

119862119870= [38144 minus68619 17589 35961]

119863119870= [minus18676 times 10

5


minus86675]

(B1)


119860119870=

[[[

[

minus12734 26225 21367 minus031502

minus36286 14954 36984 069500



]]]

]

119861119870=

[[[

[


17709 12215 11303

minus90855 34487 minus80239

64417 times 105

71058 times 106


]]]

]

119862119870= [51819 minus60403 minus59041 10015]

119863119870= [25422 17208 26098]

(B2)

References





































2synthesis with












Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Appendices

A Related Lemmas



120596119865


119873lowast

120596Θ119873120596lt 0 120596 isin [120596

11205962] (A1)


119865lowast


where

Φ = [0 1

1 0] Ψ = [

minus1 119895120596119888

minus119895120596119888minus12059611205962

] (A3)

with 120596119888= (1205961+ 1205962)2




Γperp

Θ(Γperp

)119879

lt 0 (Λ119879

)perp

Θ((Λ119879

)perp

)

119879

lt 0 (A5)


(1) Ψ + 119878 + 119878119879 lt 0

(2) The LMI problem

[Ψ + 119875 minus [119882

119904] 119878119879

+119882119879





119860119870=

[[[

[


minus22844 minus13692 13701 002030

20381 15159 minus21603 minus047598

24991 times 109


42012 times 106


]]]

]

119861119870=

[[[

[

99943 10908 44969


minus63200 11564 minus11258




]]]

]

119862119870= [38144 minus68619 17589 35961]

119863119870= [minus18676 times 10

5


minus86675]

(B1)


119860119870=

[[[

[

minus12734 26225 21367 minus031502

minus36286 14954 36984 069500



]]]

]

119861119870=

[[[

[


17709 12215 11303

minus90855 34487 minus80239

64417 times 105

71058 times 106


]]]

]

119862119870= [51819 minus60403 minus59041 10015]

119863119870= [25422 17208 26098]

(B2)

References





































2synthesis with












Volume 2014




Journal of











Journal of


Function Spaces






Algebra











































2synthesis with












Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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