research article fuzzy modeling and control for a class of ...which is very popular. based on t-s...
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Research ArticleFuzzy Modeling and Control for a Class ofInverted Pendulum System
Liang Zhang,1 Xiaoheng Chang,1 and Hamid Reza Karimi2
1 College of Engineering, Bohai University, Jinzhou 121013, China2Department of Engineering, Faculty of Engineering and Science, The University of Agder, 4898 Grimstad, Norway
Correspondence should be addressed to Liang Zhang; [email protected]
Received 14 December 2013; Accepted 24 December 2013; Published 29 January 2014
Academic Editor: Ming Liu
Copyright ยฉ 2014 Liang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Focusing on the issue of nonlinear stability control system about the single-stage inverted pendulum, the T-S fuzzy model isemployed. Firstly, linear approximation method would be applied into fuzzy model for the single-stage inverted pendulum. Atthe same time, for some nonlinear terms which could not be dealt with via linear approximation method, this paper will adopt fanrange method into fuzzy model. After the T-S fuzzy model, the PDC technology is utilized to design the fuzzy controller secondly.Numerical simulation results, obtained byMatlab, demonstrate the well-controlled effectiveness based on the proposedmethod forthe model of T-S fuzzy system and fuzzy controller.
1. Introduction
Traditional control theory has perfect control ability forexplicitly controlled system, however, which is a little weak todescribe too complex or difficult systemaccurately.Therefore,many researchers seek ways to resolve this problem; thoseresearchers have also focused on fuzzy mathematics andapplied it to control problems. Zadeh [1] created fuzzy math-ematics on an uncertainty system of control which is greatcontribution. Since the 70s, some practical controllers appearin succession, so that we have a big step forward in the controlfield. A number of control design approaches using adaptivecontrol [2โ4], sliding mode control [5, 6],๐ป
โ[7โ9], optimal
control [10โ12], control based data driven [10, 13โ15], andfuzzy control [16, 17]. The inverted pendulum system iscontrolled by the method of fuzzy control and realizes steadycontrol.The inverted pendulum is a typical automatic controlin the field of controlled object [18], which is multivariableand nonlinear and strong coupling characteristics, and soon. The inverted pendulum system reveals a natural unstableobject, which can accomplish the stability and good perfor-mance by the control methods.
For the stability control of inverted pendulum system,the establishment of the model takes an important role. T-S fuzzy control [19] is the most popular one of the most
promising methods based on modeling of fuzzy controlresearch platform. At present, the T-S fuzzy control is oneof the methods for nonlinear system control research [20],which is very popular. Based on T-S fuzzy model of invertedpendulum system modeling and control have a certainresearch. For inverted pendulum system based on T-S fuzzymode, there are two methods [21]: the first one is the fan ofnonlinear method. Although this method has high precisionin describing the nonlinear system, it obtains many fuzzyrules.Thus it brings to the controller design difficulty, especi-ally for the nonlinear term system. The second one is linearapproximation modeling method, the method at the expenseof the modeling accuracy and less number of rules of T-Sfuzzy model. Since the secondmethod can obtain a simple T-S fuzzy model, so in the inverted pendulum systemmodelingit is widely applied, but there is a very important problem,which is that if for one type of inverted pendulum system itcontains the approximate method to deal with the nonlinearterm, then the fuzzy modeling becomes the key to study.
Based on the above analysis anddiscussion, this thesiswillcarry the fuzzy modeling and control on inverted pendulumsystem of complex nonlinear term. For this point, sectornonlinear and linear approximation method will be adoptedin the T-S fuzzy modeling of some inverted pendulums andthe design of fuzzy controller.The fuzzymodeling and control
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014, Article ID 936868, 6 pageshttp://dx.doi.org/10.1155/2014/936868
2 Abstract and Applied Analysis
method can achieve the stability control of the single invertedpendulum system through the simulation.
2. Fuzzy Modeling for the InvertedPendulum System
Assume that the carโs quality is ๐, the pendulumโs quality is๐, the pendulumโs length is ๐, the pendulumโs angle is ๐ at an
instant (the angle between the pendulum rod and the verticaldirection), the initial displacement is ๐ฅ, ๐ = 9.8 m/s2 is thegravity constant, the level for control is forced acting on thecar is ๐น, ๐ = 1/(๐ + ๐), and the inverted pendulumโs statespace is as follows:
(๏ฟฝ๏ฟฝ1(๐ก)
๏ฟฝ๏ฟฝ2(๐ก)
) = (
๐ฅ2(๐ก)
1
(4๐/3) โ ๐๐๐cos2 (๐ฅ1(๐ก))
[๐ sin (๐ฅ1(๐ก)) โ
๐๐๐๐ฅ2
2(๐ก) sin2 (๐ฅ
1(๐ก))
2โ ๐ฅ2(๐ก) โ ๐๐ข (๐ก)]
) , (1)
where ๐ is ๐ฅ1(๐ก), ๐ is ๐ฅ
2(๐ก), ๐น = ๐ข(๐ก), and ๐ฅ
1(๐ก) โ (0, ยฑ๐/2),
๐ฅ2(๐ก) โ [โ๐ผ, ๐ผ].When ๐ฅ
1(๐ก) = ยฑ๐/2, the system is uncontrollable, so
we take ๐ฅ1(๐ก) โ [โ88
โ
, 88โ
] as the range. For this invertedpendulum system, T-S fuzzy model can be considered asfollows:
๐ ๐
: if ๐ฅ1(๐ก) is ๐๐
1, . . . ๐ฅ๐(๐ก) is ๐๐
๐
then ๏ฟฝ๏ฟฝ (๐ก) = ๐ด๐๐ฅ (๐ก) + ๐ต
๐๐ข (๐ก) ,
(2)
where ๐ฅ(๐ก) = [๐ฅ1(๐ก) ๐ฅ2(๐ก) โ โ โ ๐ฅ
๐(๐ก)]๐
โ ๐ ๐ is the state vari-
ables for the fuzzy system,๐๐๐is the fuzzy sets and where ๐ =
1, 2, . . . , ๐, ๐ = 1, 2, . . . , ๐, the input vector is ๐ข(๐ก) โ ๐ ๐, ๐ด๐โ
๐ ๐ร๐, ๐ต
๐โ ๐ ๐ร๐ are coefficient matrix for the system. The
number of fuzzy rules for the system is ๐.The total fuzzy control system is as follows:
๏ฟฝ๏ฟฝ (๐ก) =โ๐
๐=1๐๐(๐ฅ (๐ก)) (๐ด
๐๐ฅ (๐ก) + ๐ต
๐๐ข (๐ก))
โ๐
๐=1๐๐(๐ฅ (๐ก))
, (3)
where ๐๐(๐ฅ(๐ก)) = โ
๐
๐=1๐๐
๐(๐ฅ๐(๐ก)), and ๐
๐
๐(๐ฅ๐(๐ก)) is denotes
the membership degree, and where ๐ฅ๐(๐ก) for ๐
๐
๐. The
โ๐(๐ฅ(๐ก)) = ๐
๐(๐ฅ(๐ก))/โ
๐
๐=1๐๐(๐ฅ(๐ก)) and (2) will be as follows:
๏ฟฝ๏ฟฝ (๐ก) =
๐
โ
๐=1
โ๐(๐ฅ (๐ก)) (๐ด
๐๐ฅ (๐ก) + ๐ต
๐๐ข (๐ก)) , (4)
where โ๐(๐ฅ(๐ก)) โฅ 0 and โ
๐
๐=1โ๐(๐ฅ(๐ก)) = 1.
There is an important nonlinear term in this invertedpendulum system, in other words ๐ฅ
2
2(๐ก)sin2(๐ฅ
1(๐ก)), which
should be paid more attention. The nonlinear term cannotbe conducted through the linear approximation method onthis inverted pendulum system.Thus, the thesis will combinethe linear approximation method with the fan of nonlinearmethod to establish the fuzzymodel.Theprocess is as follows.
(1) If ๐ฅ1(๐ก) is about 0, through approximate treatment
the systemwith the linear approximationmethod, thefuzzy model of system can be obtained as follows:
then
(๏ฟฝ๏ฟฝ1(๐ก)
๏ฟฝ๏ฟฝ2(๐ก)
) = (
0 1
๐
(4๐/3) โ ๐๐๐
โ1
(4๐/3) โ ๐๐๐
)(๐ฅ1(๐ก)
๐ฅ2(๐ก)
)
+ (
0
โ๐
(4๐/3) โ ๐๐๐
)๐ข (๐ก)
(๏ฟฝ๏ฟฝ1(๐ก)
๏ฟฝ๏ฟฝ2(๐ก)
) = (
0 1
3๐
4๐ โ 3๐๐๐
โ3
4๐ โ 3๐๐๐
)(๐ฅ1(๐ก)
๐ฅ2(๐ก)
)
+ (
0
โ3๐
4๐ โ 3๐๐๐
)๐ข (๐ก) .
(5)
(2) If ๐ฅ1(๐ก) is about ยฑ๐/2, and consider the fan of nonli-
near method, and ๐ง(๐ก) = ๐ฅ2(๐ก)sin2(๐ฅ
1(๐ก)),
then
max๐ฅ1(๐ก), ๐ฅ2(๐ก)
๐ง (๐ก) = ๐ฅ2(๐ก) sin2 (๐ฅ
1(๐ก)) โก ๐
1= 0.2595. (6)
Then
min๐ฅ1(๐ก), ๐ฅ2(๐ก)
๐ง (๐ก) = ๐ฅ2(๐ก) sin2 (๐ฅ
1(๐ก)) โก ๐
2= โ0.2595. (7)
(1) If ๐ง(๐ก) is ๐1, through the linear approximation
method and the fan of nonlinear method toapproximate treatment, the fuzzy model of sys-tem can be obtained as follows:
(๏ฟฝ๏ฟฝ1(๐ก)
๏ฟฝ๏ฟฝ2(๐ก)
) = (
0 1
๐ (2/๐)
4๐/3โ(
1
4๐/3
๐๐๐๐1
2+
1
4๐/3))(
๐ฅ1(๐ก)
๐ฅ2(๐ก)
)
+ (
0
โ๐
4๐/3
)๐ข (๐ก) ,
(๏ฟฝ๏ฟฝ1(๐ก)
๏ฟฝ๏ฟฝ2(๐ก)
)=(
0 1
3๐
2๐๐โ(
3๐๐๐1
8+3
4๐))(
๐ฅ1(๐ก)
๐ฅ2(๐ก)
)+(
0
โ3๐
4๐
) ๐ข (๐ก) .
(8)
Abstract and Applied Analysis 3
Rule 2
00
Rule 1
1
โ๐
2
๐
2
Figure 1: Membership functions of two-rule model.
(2) If ๐ง(๐ก) is ๐2, through the linear approximation
method and the fan of nonlinear method toapproximate treatment, the fuzzy model of sys-tem can be obtained as follows:
(๏ฟฝ๏ฟฝ1(๐ก)
๏ฟฝ๏ฟฝ2(๐ก)
)=(
0 1
๐ (2/๐)
4๐/3โ(
1
4๐/3
๐๐๐๐2
2+
1
4๐/3))(
๐ฅ1(๐ก)
๐ฅ2(๐ก)
)
+ (
0
โ๐
4๐/3
)๐ข (๐ก) ,
(๏ฟฝ๏ฟฝ1(๐ก)
๏ฟฝ๏ฟฝ2(๐ก)
)=(
0 1
3๐
2๐๐โ(
3๐๐๐2
8+3
4๐))(
๐ฅ1(๐ก)
๐ฅ2(๐ก)
)+(
0
โ3๐
4๐
)๐ข (๐ก) .
(9)
Here define membership functions. For the part of lin-ear approximation, the membership function is shown asFigure 1.
Rule 1: Consider
๐ป1(๐ก) =
{{
{{
{
2
๐๐ฅ1(๐ก) + 1, (โ
๐
2โค ๐ฅ1(๐ก) โค 0)
โ2
๐๐ฅ1(๐ก) + 1, (0 < ๐ฅ
1(๐ก) โค
๐
2)
. (10)
Rule 2: Consider
๐ป2(๐ก) = 1 โ ๐ป
1(๐ก) . (11)
Figure 2 is the membership function for the fan of nonlinearmethod. ๐ง(๐ก) can be rewritten as ๐ง(๐ก) = โ
2
๐=1๐ธ๐(๐ง(๐ก))๐
๐, where
๐ธ1(๐ง(๐ก)) = (๐ง(๐ก)โ๐
2)/(๐1โ๐2) and๐ธ
2(๐ง(๐ก)) = (๐
1โ๐ง(๐ก))/(๐
1โ๐2).
The membership functions ๐ธ1(๐ง(๐ก)) and ๐ธ
2(๐ง(๐ก)) will meet
the equation ๐ธ1(๐ง(๐ก)) + ๐ธ
2(๐ง(๐ก)) = 1.
In conclusion, the finally fuzzy model for the system willbe shown as follows.
Rule 1: if ๐ฅ1(๐ก) tends to 0, then ๏ฟฝ๏ฟฝ(๐ก) = ๐ด
1๐ฅ(๐ก)+๐ต
1๐ข(๐ก).
Rule 2: if ๐ฅ1(๐ก) tends to ยฑ(๐/2)(|๐ฅ
1(๐ก)| < ๐/2) and
๐ง(๐ก) takes the maximum value, then ๏ฟฝ๏ฟฝ(๐ก) = ๐ด2๐ฅ(๐ก) +
๐ต2๐ข(๐ก).
E2(z(t)) E1(z(t))
Negative Positive
โa a0
1
z(t)0
Figure 2: Membership function for the fan of nonlinear method.
Table 1: Function parameters.
Parameter Function Value๐ Themass of the cart 1.096 kg๐ Themass of the pendulum 0.109 kg๐ The length of the pendulum 0.25m
๐The angle of the pendulum fromthe vertical
๐น The force applied to the cart
Rule 3: if ๐ฅ1(๐ก) tends to ยฑ(๐/2)(|๐ฅ
1(๐ก)| < ๐/2) and
๐ง(๐ก) takes the minimum value, then ๏ฟฝ๏ฟฝ(๐ก) = ๐ด3๐ฅ(๐ก) +
๐ต3๐ข(๐ก).
The function and value of every parameter are shown inTable 1.
All the parameters defined in Table 1 are taken to account,then the system coefficient matrix can be obtained as follow
๐ด1= (
0 1
3๐
4๐ โ 3๐๐๐
โ3
4๐ โ 3๐๐๐
) = (0 1
31.5397 โ3.2183) ,
๐ต1= (
0
โ3๐
4๐ โ 3๐๐๐
) = (0
โ2.6708) ,
๐ด2= (
0 1
3๐
2๐๐โ (
3๐๐๐1
8+
3
4๐)) = (
0 1
18.7166 โ3.0088) ,
๐ต2= (
0
โ3๐
4๐
) = (0
โ2.4896) ,
๐ด3= (
0 1
๐ (2/๐)
4๐/3โ(
1
4๐/3
๐๐๐๐2
2+
1
4๐/3))
= (0 1
18.7166 โ2.9912) ,
๐ต3= (
0
โ๐
4๐/3
) = (0
โ2.4896) .
(12)
4 Abstract and Applied Analysis
0 5 10 15 20โ8
โ6
โ4
โ2
0
2
4
Time (s)
x(d
eg)
x1(t)
x2(t)
Figure 3: Simulation result without the controller.
The model rules can be represented as follows:
โ1(๐ก) = ๐ป
1(๐ก) ,
โ2(๐ก) = ๐ป
2(๐ก) ร ๐ธ
1(๐ง (๐ก)) = ๐ป
2(๐ก) ร [
๐ง (๐ก) โ ๐2
๐1โ ๐2
] ,
โ3(๐ก) = ๐ป
2(๐ก) ร ๐ธ
2(๐ง (๐ก)) = ๐ป
2(๐ก) ร [
๐1โ ๐ง (๐ก)
๐1โ ๐2
] .
(13)
For the T-S model of the control object, a parallel distributedcompensation control scheme (PDC) is employed. And theregulations are described as follow:
๐ ๐
: if ๐ฅ1(๐ก) is ๐๐
1, . . . ๐ฅ๐(๐ก) is ๐๐
๐
then ๐ข (๐ก) = ๐พ๐๐ฅ (๐ก) .
(14)
Here the fuzzy controller and the fuzzy system adopt the samefuzzy rule. The overall model for the fuzzy controller is asfollows:
๐ข (๐ก) =
๐
โ
๐=1
โ๐(๐ฅ (๐ก)) ๐พ
๐๐ฅ (๐ก) . (15)
The closed control system can be obtained by combining (2)and (4):
๏ฟฝ๏ฟฝ =
๐
โ
๐=1
๐
โ
๐=1
โ๐โ๐(๐ด๐+ ๐ต๐๐พ๐) ๐ฅ. (16)
3. Based on Linear Matrix Inequalities (LMI)and the Matlab Simulation
Without the controller, the simulation output curves ofthe angular velocity and angular acceleration are shown inFigure 3.
Figure 3 shows that the inverted pendulum system isunstable without the controller.
In the following, by using the linear matrix inequalitiestechnique [23], the fuzzy controller is designed.
Let us define the Lyapunov function as (๐ฅ(๐ก)) =
๐ฅ๐
(๐ก)๐๐ฅ(๐ก),๐ > 0, then the stable criterion of the system for(16) is as follows:๐
โ
๐=1
๐
โ
๐=1
โ๐โ๐(๐ด๐+ ๐ต๐๐พ๐)๐
๐ + ๐
๐
โ
๐=1
๐
โ
๐=1
โ๐โ๐(๐ด๐+ ๐ต๐๐พ๐) < 0.
(17)
Define๐ = ๐โ1; we can obtain (18) by multiplying๐ on both
sides contemporary:
๐
๐
โ
๐=1
๐
โ
๐=1
โ๐โ๐(๐ด๐+ ๐ต๐๐พ๐)๐
+
๐
โ
๐=1
๐
โ
๐=1
โ๐โ๐(๐ด๐+ ๐ต๐๐พ๐)๐ < 0.
(18)
After defining ๐พ๐๐ = ๐
๐, the system stability discriminant
conditions can be obtained. And this will guarantee a positivedefinite matrix ๐ and matrix ๐
๐can be searched, then the
following matrix inequality can be established:
๐๐ด๐
๐+ ๐๐
๐๐ต๐
๐+ ๐ด๐๐ + ๐ต
๐๐๐< 0,
๐๐ด๐
๐+๐๐
๐๐ต๐
๐+๐ด๐๐+๐ต๐๐๐+๐๐ด๐
๐+๐๐
๐๐ต๐
๐+๐ด๐๐+๐ต๐๐๐< 0,
๐ = 1, 2, . . . , ๐, ๐ < ๐,
(19)
where the stable controller will be obtained from (20):
๐พ๐= ๐๐๐โ1
. (20)
Through solving (19) by linear matrix inequality with LMI ofMatlab [24], we can obtain
๐ = [1.1127 โ0.4874
โ0.4874 1.9975] ,
๐1= [14.6725 โ 7.9846] ,
๐2= [9.7868 โ 5.9077] ,
๐3= [9.7836 โ 5.8945] .
(21)
Moreover, taking advantage of (20), the fuzzy controller gainwill be obtained and shown as follows:
๐พ1= [12.8038 โ 0.8733] ,
๐พ2= [8.3975 โ 0.9086] ,
๐พ3= [8.3975 โ 0.9020] .
(22)
Put the controller gain into (16); design the simulationprogram in the simulink environment. Here the initial value๐ฅ(0) = [โ0.01 โ0.1] is selected. The results of the fuzzy con-trol simulation of the level single inverted pendulum systemare shown in Figures 4, 5, and 6.
Simulation results show that the system responses con-verge to the equilibriumpoint, which indicates that the designof the controller is stable.
Abstract and Applied Analysis 5
0 5 10 15 20โ0.06
โ0.04
โ0.02
0
0.02
0.04
Time (s)
x1(t)
x1(t)
Figure 4: Simulation result of the angle.
0 5 10 15 20โ0.1
โ0.05
0
0.05
0.1
Time (s)
x2(t)
x2(t)
Figure 5: Simulation result of the angular velocity.
0 5 10 15 20โ0.8
โ0.6
โ0.4
โ0.2
0
0.2
0.4
0.6
Time (s)
u(t)
u(t)
Figure 6: Simulation result of the controller.
4. Conclusion
This thesis takes a class of an inverted pendulum systemas the research object. The system fuzzy model was estab-lished by the methods that combining with the linearizationapproximation processing and fan-shaped interval, and thenthe fuzzy controller was designed. Matlab-Simulink soft-ware toolbox was employed to be on computer simulation.The results show that it achieved a stable control of thesingle-stage inverted pendulum system through fuzzy controlmethod on the basis of this fuzzy model. This model hasthe advantages of less fuzzy rules, high precision, and simplestructure. The research results can provide an effective wayfor the subsequent instability in other nonlinear systemmodeling and fuzzy control.
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
This work is supported by the National Nature ScienceFoundation of China under Grant (Project no. 61104071),by the Program for Liaoning Excellent Talents in University,China, under Grant (Project no. LJQ2012095), by the OpenProgram of the Key Laboratory of Manufacturing IndustrialIntegrated Automation, Shenyang University, China, underGrant (Project no. 1120211415), and Natural Science Foun-dation of Liaoning, China (Project no. 2013020044). Theauthors highly appreciate the above financial supports.
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