Open access peer-reviewed chapter

Fuzzy Control of Nonlinear Systems with General Performance Criteria

By Xin Wang, Edwin E. Yaz, James Long and Tim Miller

Submitted: November 18th 2011Reviewed: May 14th 2012Published: September 27th 2012

DOI: 10.5772/48298

Downloaded: 2171

1. Introduction

Research on control of non-linear systems over the years has produced many results: control based on linearization, global feedback linearization, non-linear Hcontrol, sliding mode control, variable structure control, state dependent Riccati equation control, etc [Khalil, 2002]. This chapter will focus on fuzzy control techniques. Fuzzy control systems have recently shown growing popularity in non-linear system control applications. A fuzzy control system is essentially an effective way to decompose the task of non-linear system control into a group of local linear controls based on a set of design-specific model rules. Fuzzy control also provides a mechanism to blend these local linear control problems all together to achieve overall control of the original non-linear system. In this regard, fuzzy control technique has its unique advantage over other kinds of non-linear control techniques. Latest research on fuzzy control systems design is aimed to improve the optimality and robustness of the controller performance by combining the advantage of modern control theory with the Takagi-Sugeno fuzzy model [7, 8, 9, 10, 13, 14].

In this chapter, we address the non-linear state feedback control design of both continuous-time and discrete-time non-linear fuzzy control systems using the Linear Matrix Inequality (LMI) approach. We characterize the solution of the non-linear control problem with the LMI, which provides a sufficient condition for satisfying various performance criteria. A preliminary investigation into the LMI approach to non-linear fuzzy control systems can be found in [7, 8, 13]. The purpose behind this novel approach is to convert a non-linear system control problem into a convex optimization problem which is solved by a LMI at each time. The recent development in convex optimization provides efficient algorithms for solving LMIs. If a solution can be expressed in a LMI form, then there exist optimization algorithms providing efficient global numerical solutions [3]. Therefore if the LMI is feasible, then LMI control technique provides globally stable solutions satisfying the corresponding mixed performance criteria [4, 6, 15, 16, 17, 18, 19, 20]. We further propose to employ mixed performance criteria to design the controller guaranteeing the quadratic sub-optimality with inherent stability property in combination with dissipative type of disturbance attenuation.

In the following sections, we first introduce the Takagi-Sugeno fuzzy modelling for non-linear systems in both continuous time and discrete time. We then propose the general performance criteria in section 3. Then, the LMI control solutions are derived to characterize the optimal and robust fuzzy control of continuous time and discrete time non-linear systems, respectively. The inverted pendulum system control is used as an illustrative example to demonstrate the effectiveness and robustness of our proposed approaches.

The following notation is used in this work: xRndenotes n-dimensional real vector with norm x=(xTx)1/2where (.)Tindicates transpose. A0for a symmetric matrix denotes a positive semi-definite matrix. L2and l2denotes the space of infinite sequences of finite dimensional random vectors with finite energy, i.e. 0xt2<in continuous-time, and Σk=0xk2<in discrete-time, respectively.

2. Takagi-Sugeno system model

The importance of the Takagi-Sugeno fuzzy system model is that it provides an effective way to decompose a complicated non-linear system into local dynamical relations and express those local dynamics of each fuzzy implication rule by a linear system model. The overall fuzzy non-linear system model is achieved by fuzzy “blending” of the linear system models, so that the overall non-linear control performance is achieved. Both of the continuous-time and the discrete-time system models are summarized below.

2.1. Continuous-time Takagi-Sugeno system model

The ithrule of the Takagi-Sugeno fuzzy model can be expressed by the following forms:

Model Rulei:

If φ1(t)

isMi1,φ2(t)

isMi2,...and φp(t)isMip,

Then the input-affine continuous-time fuzzy system equation is:

x˙(t)=Aix(t)+Biu(t)+Fiw(t)y(t)=Cix(t)+Diu(t)+Ziw(t)i=1,2,3,...,rE1

where x(t)Rnis the state vector, u(t)Rmis the control input vector, y(t)Rqis the performance output vector, w(t)Rsis L2type of disturbance, ris the total number of model rules, Mijis the fuzzy set. The coefficient matrices areAiRn×n,BiRn×m,FiRn×s,CiRq×n,DiRq×m,ZiRq×s. And φ1,...,φpare known premise variables, which can be functions of state variables, external disturbance and time.

It is assumed that the premises are not the function of the input vectoru(t), which is needed to avoid the defuzzification process of fuzzy controller. If we use φ(t)to denote the vector containing all the individual elementsφ1(t),φ2(t),...,φp(t), then the overall fuzzy system is

x˙t=i=1rgiφtAixt+Biut+Fiwti=1rgiφt=i=1rhiφtAixt+Biut+FiwtE2
 y(t)=i=1rgi(φ(t))[Cix(t)+Diu(t)+Ziw(t)]i=1rgi(φ(t))=i=1rhi(φ(t))[Cix(t)+Diu(t)+Ziw(t)]E3

where

φ(t)=[φ1(t),φ2(t),...,φp(t)]E4
gi(φ(t))=j=1pMij(φj(t))E5
hi(φ(t))=gi(φ(t))i=1rgi(φ(t))E6

for all timet. The term Mij(φj(t))is the grade membership function of ϕj(t)inMij.

Since, the following properties hold

i=1rgi(φ(t))>0gi(φ(t))0,i=1,2,3,...,rE7

We have

i=1rhi(φ(t))=1hi(φ(t))0,i=1,2,3,...,rE8

for all timet.

It is assumed that the state feedback is available and the non-linear state feedback control input is given by

u(t)=-i=1rhi(φ(t))Kix(t)E9

Substituting this into the system and performance output equation, we have

x˙(t)=i=1rj=1rhi(φ(t))hj(φ(t))(Ai-BiKj)x(t)+i=1rhi(φ(t))Fiw(t)y(t)=i=1rj=1rhi(φ(t))hj(φ(t))(Ci-DiKj)x(k)+i=1rhi(φ(t))Ziw(t)E10

Using the notation

Gij=Ai-BiKjHij=Ci-DiKjE11

then the system equation becomes

x˙(t)=i=1rj=1rhi(φ(t))hj(φ(t))Gijx(t)+i=1rhi(φ(t))Fiw(t)y(t)=i=1rj=1rhi(φ(t))hj(φ(t))Hijx(t)+i=1rhi(φ(t))Ziw(t)E12
(11)

2.2. Discrete-time Takagi-Sugeno system model

At time stepk, the ithrule of the Takagi-Sugeno fuzzy model can be expressed by the following forms:

Model Rulei:

If φ1(k)isMi1,φ2(k)isMi2,...and φp(k)isMip,

Then the input-affine discrete-time fuzzy system equation is:

x(k+1)=Aix(k)+Biu(k)+Fiw(k)y(k)=Cix(k)+Diu(k)+Ziw(k)i=1,2,3,...,rE13

where x(k)Rnis the state vector, u(k)Rmis the control input vector, y(k)Rqis the performance output vector, w(k)Rsis l2type of disturbance, ris the total number of model rules, Mijis the fuzzy set. The coefficient matrices areAiRn×n,BiRn×m,FiRn×s,CiRq×n,DiRq×m,ZiRq×s. And φ1,...,φpare known premise variables which can be functions of state variables, external disturbance and time.

It is assumed that the premises are not the function of the input vectoru(k), which is needed to avoid the defuzzification process of fuzzy controller. If we use φ(k)to denote the vector containing all the individual elementsφ1(k),φ2(k),...,φp(k), then the overall fuzzy system is

x(k+1)=i=1rgi(φ(k))Aix(k)+Biu(k)+Fiw(k)i=1rgi(φ(k))=i=1rhi(φ(k))Aix(k)+Biu(k)+Fiw(k)y(k)=i=1rgi(φ(k))Cix(k)+Diu(k)+Ziw(k)i=1rgi(φ(k))=i=1rhi(φ(k))Cix(k)+Diu(k)+Ziw(k)E14

where

φ(k)=[φ1(k),φ2(k),...,φp(k)]E15
gi(φ(k))=j=1pMij(φj(k))E16
hi(φ(k))=gi(φ(k))i=1rgi(φ(k))E17

for allk. The term Mij(φj(k))is the grade membership function of ϕj(k)inMij.

Since, the following properties hold

i=1rgi(φ(k))>0gi(φ(k))0,i=1,2,3,...,rE18

We have

i=1rhi(φ(k))=1hi(φ(k))0,i=1,2,3,...,rE19

for all

kE20
.

It is assumed that the state feedback is available and the non-linear state feedback control input is given by

u(k)=-i=1rhi(φ(k))Kix(k)E21

Substituting this into the system and performance output equation, we have

x(k+1)=i=1rj=1rhi(φ(k))hj(φ(k))(Ai-BiKj)x(k)+i=1rhi(φ(k))Fiw(k)y(k)=i=1rj=1rhi(φ(k))hj(φ(k))(Ci-DiKj)x(k)+i=1rhi(φ(k))Ziw(k)E22

Using the notation

Gij=Ai-BiKjHij=Ci-DiKjE23

then the system equation becomes

x(k+1)=i=1rj=1rhi(φ(k))hj(φ(k))Gijx(k)+i=1rhi(φ(k))Fiw(k)y(k)=i=1rj=1rhi(φ(k))hj(φ(k))Hijx(k)+i=1rhi(φ(k))Ziw(k)E24
(22)

3. General performance criteria

In this section, we propose the general performance criteria for non-linear control design, which yields a mixed Non-Linear Quadratic Regular (NLQR) in combination with Hor dissipative performance index. The commonly used system performance criteria, including bounded-realness, positive-realness, sector boundedness and quadratic cost criterion, become special cases of the general performance criteria. Both the continuous-time and discrete-time general performance criteria are given below:

3.1. Continuous-time general performance criteria

Consider the quadratic Lyapunov function

V(t)=xT(t)Px(t)>0E25

for the following difference inequality

V˙(t)+xT(t)Qx(t)+uT(t)Ru(t)+αyT(t)y(t)-βyT(t)w(t)+γwT(t)w(t)0E26

with Q>0,R>0functions ofx(t).

Note that upon integration over time from 0 toTf, (24) yields

V(Tf)+0Tf[(xT(t)Qx(t)+uT(t)Ru(t)]dt+0Tf[αyT(t)y(t)-βyT(t)w(t)+γwT(t)w(t)]dtV(0)E27

By properly specifying the value of weighing matrices Q,R,Ci,Di,Ziandα,β,γ, mixed performance criteria can be used in non-linear control design, which yields a mixed Non-linear Quadratic Regulator (NLQR) in combination with dissipative type performance index with disturbance reduction capability. For example, if we takeα=1,β=0,γ<0, (25) yields

V(Tf)+0Tf[(xT(t)Qx(t)+uT(t)Ru(t)+yT(t)y(t)]dt+V(0)-γ0Tf[wT(t)w(t)]dtE28

which is a mixed NLQR-HDesign [16, 17, 18].

Other possible performance criteria which can be used in this framework with various design parameters α,β,γare given in Table.1. Design coefficients αand γcan be maximized or minimized to optimize the controller behavior. It should also be noted that the satisfaction of any of the criteria in Table 1 will also guarantee asymptotic stability of the controlled system.

3.2. Discrete-time general performance criteria

Consider the quadratic Lyapunov function

V(k)=xT(k)Px(k)E29

for the following difference inequality

V(k+1)-V(k)+xT(k)Qx(k)+uT(k)Ru(k)+αyT(k)y(k)-βyT(k)w(k)+γwT(k)w(k)0E30

with Q>0,R>0functions ofx(k).

Note that upon summation overk, (28) yields

V(N)+k=0N-1(xT(k)Qx(k)+uT(k)Ru(k)+αyT(k)y(k)-βyT(k)w(k)+γwT(k)w(k))V(0)E31

By properly specifying the value of weighing matrices Q,R,Ci,Di,Ziandα,β,γ, mixed performance criteria can be used in non-linear control design, which yields a mixed Non-linear Quadratic Regulator (NLQR) in combination with dissipative type performance index with disturbance reduction capability. For example, if we takeα=1,β=0,γ<0, (29) yields

V(N)+k=0N-1(xT(k)Qx(k)+uT(k)Ru(k)+αyT(k)y(k))V(0)-γk=0N-1wT(k)w(k)E32

which is a mixed NLQR-HDesign [16, 17, 18]. In (19), γcan be minimized to achieve a smaller l2-l2or Hgain for the closed loop system.

Other possible performance criteria which can be used in this framework with various design parameters α,β,γare given in Table.1. Design coefficients αand γcan be maximized or minimized to optimize the controller behavior. It should also be noted that the satisfaction of any of the criteria in Table 1 will also guarantee asymptotic stability of the controlled system.

αβγPerformance Criteria
10<0NLQR-H  Design
10NLQR-Passivity  Design
1"/>0NLQR-Input  Strict  Passivity  Design
"/>010NLQR-Output  Strict  Passivity  Design
"/>01"/>0NLQR-Very  Strict  Passivity

Table 1.

Various performance criteria in a general framework

4. Fuzzy LMI control of continuous time non-linear systems with general performance criteria

The main results of this chapter are summarized in section 4 and section 5. The following theorem provides the fuzzy LMI control to the continuous time non-linear systems with general performance criteria.

Theorem 1 Given the system model and performance output (2) and control input (8), if there exist matrices S=P-1>0for allt0, such that the following LMI holds:

Λ11Λ12Λ13Λ14Λ15Λ22Λ2300*I00**R-10***I0E33

where

Λ11=-12[SAiT-MjBiT+SAjT-MiTBjT+AiS-BiMj+AjS-BjMi]E34
Λ12=-12(Fi+Fj)+β4[SCiT-MjDiT+SCjT-MiTDjT]E35
Λ13=12α1/2[SCiT-MjDiT+SCjT-MiTDjT]E36
Λ14=12(MiT+MjT)E37
Λ15=SQT/2E38
Λ22=-γI+12β(Zi+Zj)TE39
Λ23=12α1/2[Zi+Zj]TE40

using the notation

Mi=KiP-1=KiSE41

then inequality (24) is satisfied.

Proof

By applying system model and performance output (2)(11), and state feedback input (8), the performance index inequality (24) becomes

[i=1rj=1rhi(φ(t))hj(φ(t))Gijx(t)+i=1rhi(φ(t))Fiw(t)]TPx(t)+E42
xT(t)P[i=1rj=1rhi(φ(t))hj(φ(t))Gijx(t)+i=1rhi(φ(t))Fiw(t)]+E43
xT(t)Qx(t)+[-i=1rhiφ(t)Kix(t)]TR[-i=1rhiφ(t)Kix(t)]E44
α[i=1rj=1rhi(φ(t))hj(φ(t))Hijx(t)+i=1rhi(φ(t))Ziw(t)]TE45
×[i=1rj=1rhi(φ(t))hj(φ(t))Hijx(t)+i=1rhi(φ(t))Ziw(t)]E46
-β[i=1rj=1rhi(φ(t))hj(φ(t))Hijx(t)+i=1rhi(φ(t))Ziw(t)]T×w(t)E47
+γwT(t)w(t)0E48

Inequality (34) is equivalent to

xT(t)wT(t)×Δ11Δ12Δ22×x(t)w(t)0E49

where

Δ11=(ijhihjGij)TP+P(ijhihjGij)+Q+[ihiKi]TR[ihiKi]+E50
α[ijhihjHij]T[ijhihjHij]E51
Δ12=P(ihiFi)+α[ijhihjHij]T[ihiZi]-β2[ijhihjHij]TE52
Δ22=γI+α[ihiZi]T[ihiZi]-β[ihiZi]TE53

Inequality (35) can be rewritten as

Θ11Θ12Θ22-α[ijhihjHij]TihiZiT×[[ijhihjHij][[ihiZi]]0E54

where

Θ11=-(ijhihjGij)TP-P(ijhihjGij)-Q-[ihiKi]TR[ihiKi]E55
Θ12=-P(ihiFi)+β2[ijhihjHij]TE56
Θ22=-γI+β[ihiZi]TE57

By applying Schur complement to inequality (37), we have

Θ11Θ12α1/2[ijhihjHij]TΘ22α1/2[ihiZi]T*I0E58

Similarly, inequality (39) can also be written as

Φ11Φ12α1/2[ijhihjHij]TΦ22α1/2[ihiZi]T*I-[ihiKi]T00R[ihiKi]000E59

where

Φ11=-(ijhihjGij)TP-P(ijhihjGij)-QE60
Φ12=-P(ihiFi)+β2[ijhihjHij]TE61
Φ22=-γI+β[ihiZi]TE62

By applying Schur complement again to (40), we have

Φ11Φ12α1/2[ijhihjHij]T[ihiKi]T*Φ22α1/2[ihiZi]T0**I0***R-10E63

Equivalently, we have

ijhihj×Γ11Γ12Γ13Γ14*Γ22Γ230**I0***R-10E64

where

Γ11=-12[(Ai-BiKj)+(Aj-BjKi)]TP-12P[(Ai-BiKj)+(Aj-BjKi)]-QE65
Γ12=-12P(Fi+Fj)+β4[(Ci-DiKj)+(Cj-DjKi)]TE66
Γ13=-12α1/2[(Ci-DiKj)+(Cj-DjKi)]TE67
Γ14=-12(Ki+Kj)TE68
Γ22=-γI+12β(Zi+Zj)TE69
Γ23=12α1/2(Zi+Zj)TE70

Therefore, we have the following LMI

Γ11Γ12Γ13Γ14Γ22Γ230*I0**R-10E71

By multiplying both sides of the LMI above by the block diagonal matrixdiag{S,I,I,I}, whereS=P-1, and using the notation

Mi=KiP-1=KiSE72

we obtain

X11X12X13X14X22X230*I0**R-10E73

where

X11=-12[SAiT-MjBiT+SAjT-MiTBjT+AiS-BiMj+AjS-BjMi]-SQSE74
X12=-12(Fi+Fj)+β4[SCiT-MjTDiT+SCjT-MiTDjT]E75
X13=12α1/2[SCiT-MjTDiT+SCjT-MiTDjT]E76
X14=12(MiT+MjT)E77
X22=-γI+12β(Zi+Zj)TE78
X23=12α1/2(Zi+Zj)TE79

By applying Schur complement again, the final LMI is derived

Λ11Λ12Λ13Λ14Λ15Λ22Λ2300*I00**R-10***I0E80

where

Λ11=-12[SAiT-MjBiT+SAjT-MiTBjT+AiS-BiMj+AjS-BjMi]E81
Λ12=-12(Fi+Fj)+β4[SCiT-MjDiT+SCjT-MiTDjT]E82
Λ13=12α1/2[SCiT-MjDiT+SCjT-MiTDjT]E83
Λ14=12(MiT+MjT)E84
Λ15=SQT/2E85
Λ22=-γI+12β(Zi+Zj)TE86
Λ23=12α1/2[Zi+Zj]TE87

Hence, if the LMI (49) holds, inequality (24) is satisfied. This concludes the proof of the theorem.

Remark 1: For the chosen performance criterion, the LMI (49) need to be solved at each time to find matricesS,M, by using relation (33), we can find the feedback control gain, therefore, the feedback control can be found to satisfy the chosen criterion.

5. Fuzzy LMI control of discrete time non-linear systems with general performance criteria

This section summarizes the main results for fuzzy LMI control of discrete time non-linear systems with general performance criteria:

Theorem 2: Given the closed loop system and performance output (13), and control input (19), if there exist matrices S=P-1>0for allk0, such that the following LMI holds:

Ξ11Ξ12Ξ13Ξ14Ξ15Ξ16Ξ22Ξ23Ξ2400*S000**I00***R-10****I0E88

where

Ξ11=SE89
Ξ12=β4(CiS-DiYj+CjS-DjYi)TE90
Ξ13=12(AiS-BiYj+AjS-BjYi)TE91
Ξ14=12α1/2(CiS-DiYj+CjS-DjYi)TE92
Ξ15=12(Yi+Yj)TE93
Ξ16=SQT/2E94
Ξ22=-γI+β2(Zi+Zj)TE95
Ξ23=12α1/2(Fi+Fj)TE96
Ξ24=12α1/2(Zi+Zj)TE97

and

S(k+1)>S(k)E98

whereS(k)=P-1(k), then (28) is satisfied with the feedback control gain being found by

K(k)=Y(k)P(k)E99

Proof

The performance index inequality (28) can be explicitly written as

[i=1rj=1rhi(φ(k))hj(φ(k))Gijx(k)+i=1rhi(φ(k))Fiw(k)]TE100
×P×[i=1rj=1rhi(φ(k))hj(φ(k))Gijx(k)+i=1rhi(φ(k))Fiw(k)]E101
-xT(k)Px(k)+xT(k)Qx(k)+[-i=1rhi(φ(k))Kix(k)]TR[-i=1rhi(φ(k))Kix(k)]+E102
α[i=1rj=1rhi(φ(k))hj(φ(k))Hijx(k)+i=1rhi(φ(k))Ziw(k)]TE103
×[i=1rj=1rhi(φ(k))hj(φ(k))Hijx(k)+i=1rhi(φ(k))Ziw(k)]E104
-β[i=1rj=1rhi(φ(k))hj(φ(k))Hijx(k)+i=1rhi(φ(k))Ziw(k)]T×w(k)E105
+γwT(k)w(k)0E106

Equivalently,

xT(k)wT(k)-P+Q00γIx(k)w(k)+E107
xT(k)wT(k)(ijhihjGij)    (ihiFi)T×P×(ijhihjGij)    (ihiFi)x(k)w(k)+E108
+xT(k)[ihiKi]TR[ihiKi]x(k)+E109
αxT(k)wT(k)(ijhihjHij)    (ihiZi)T×(ijhihjHij)    (ihiZi)x(k)w(k)+E110
-βxTkwTkijhihjHij   ihiZiTwk0E111

which can be written, after collecting terms, as

xT(k)wT(k)Υ11Υ12Υ22x(k)w(k)+E112
xT(k)wT(k)(ijhihjGij)    (ihiFi)T×P×(ijhihjGij)    (ihiFi)x(k)w(k)+E113
αxT(k)wT(k)(ijhihjHij)    (ihiZi)T×(ijhihjHij)    (ihiZi)x(k)w(k)0E114

where

Υ11=P-Q-[ihiKi]TR[ihiKi]E115
Υ12=β2[ijhihjHij]TE116
Υ22=-γI+β[ihiZi]TE117

Equivalently, we have

Υ11Υ12Υ22-(ijhihjGij)    (ihiFi)T×P×(ijhihjGij)    (ihiFi)-E118
α(ijhihjHij)    (ihiZi)T×(ijhihjHij)    (ihiZi)0E119

By applying Schur complement, we obtain

Υ11Υ12(ijhihjGij))TΥ22(ihiFi)T*P-1-α(ijhihjHij)    (ihiZi)T×(ijhihjHij)    (ihiZi)0E120

By applying Schur complement again, we obtain

Υ11Υ12(ijhihjGij))Tα1/2(ijhihjHij)T*Υ22(ihiFi)Tα1/2(ihiZi)T**P-10***I0E121

Equivalently, the following inequality holds

Ψ11Ψ12(ijhihjGij))Tα1/2(ijhihjHij)TΨ22(ihiFi)Tα1/2(ihiZi)T*P-10**I-E122
(ihiKi)T000×R×(ihiKi)0000E123

where

Ψ11=P-QE124
Ψ12=β2[ijhihjHij]TE125
Ψ22=-γI+β[ihiZi]TE126

By applying Schur complement one more time, we have

Ψ11Ψ12(ijhihjGij))Tα1/2(ijhihjHij)T(ihiKi)T*Ψ22(ihiFi)Tα1/2(ihiZi)T0**P-100***I0****R-10E127

By factoring out the ijhi(φk)hj(φk)term, we have

Ω11Ω12Ω13Ω14Ω15Ω22Ω23Ω240*P-100**I0***R-10E128

where

Ω11=P-QE129
Ω12=β4[Hji+Hij]TE130
Ω13=12(Gji+Gij))TE131
Ω14=12α1/2(Hij+Hji)TE132
Ω15=12(Ki+Kj)TE133
Ω22=-γI+β2(Zi+Zj)TE134
Ω23=12(Fi+Fj)TE135
Ω24=12α1/2(Zi+Zj)TE136

By pre-multiplying and post-multiplying the matrix with the block diagonal matrixdiag(S,I,I,I,I), whereS=P-1, and applying Schur complement again, the following LMI result is obtained

Ξ11Ξ12Ξ13Ξ14Ξ15Ξ16Ξ22Ξ23Ξ2400*S000**I00***R-10****I0E137

where

Ξ11=SE138
Ξ12=β4(CiS-DiYj+CjS-DjYi)TE139
Ξ13=12(AiS-BiYj+AjS-BjYi)TE140
Ξ14=12α1/2(CiS-DiYj+CjS-DjYi)TE141
Ξ15=12(Yi+Yj)TE142
Ξ16=SQT/2E143
Ξ22=-γI+β2(Zi+Zj)TE144
Ξ23=12α1/2(Fi+Fj)TE145
Ξ24=12α1/2(Zi+Zj)TE146

whereS(k)=P-1(k), then (28) is satisfied with the feedback control gain being found by

K(k)=Y(k)P(k)E147
(69)

6. Application to the inverted pendulum system

The inverted pendulum on a cart problem is a benchmark control problem used widely to test control algorithms. A pendulum beam attached at one end can rotate freely in the vertical 2-dimensional plane. The angle of the beam with respect to the vertical direction is denoted at angleθ. The external force uis desired to set angle of the beam θ(x1) and angular velocity θ˙(x2) to zero while satisfying the mixed performance criteria. A model of the inverted pendulum on a cart problem is given by [1, 9]:

x˙1=x2+ε1wE148
x˙2=gsin(x1)-amLx22sin(2x1)/2-acos(x1)u4L/3-amLcos2(x1)+ε2wE149

where x1is the angle of the pendulum from vertical direction, x2is the angular velocity of the pendulum, gis the gravity constant, mis the mass of the pendulum, Mis the mass of the cart, Lis the length of the center of mass (the entire length of the pendulum beam equals2L), uis the external force, control input to the system, wis the L2type of disturbance, a=1m+Mis a constant, and ε1.ε2is the weighing coefficients of disturbance.

Due to the system non-linearity, we approximate the system using the following two-rule fuzzy model:

continuous-time fuzzy model

Rule 1: If |x1(t)|is close to zero,

Then

x˙(t)=A1x(t)+B1u(t)+F1w(t)E150

Rule 2: If |x1(t)|is close toπ/2,

Then

x˙(t)=A2x(t)+B2u(t)+F2w(t)E151

where

A1=01g4L/3-amL0B1=0-a4L/3-amLF1=ε1ε2E152
A2=012gπ(4L/3-amLδ2)0B1=0-aδ4L/3-amLδ2F1=ε1ε2with  δ=cos(80o)E153

discrete-time fuzzy model

Rule 1: If |x1(k)|is close to zero,

Then

x(k+1)=A1x(k)+B1u(k)+F1w(k)E154

Rule 2: If |x1(k)|is close toπ/2,

Then

x(k+1)=A2x(k)+B2u(k)+F2w(k)E155

where

A1=1TgT4L/3-amL1B1=0-aT4L/3-amLF1=ε1Tε2TE156
A2=1T2gTπ(4L/3-amLδ2)1B2=0-aδT4L/3-amLδ2F2=ε1Tε2TE157
with  δ=cos(80o),Sampling  time  T=0.001E158

The following values are used in our simulation:

M=8kg,m=2kg,L=0.5m,g=9.8m/s2,ε1=1,ε2=0E159

and the initial condition ofx1(0)=π/6,x2(0)=-π/6. The membership function of Rule 1 and Rule 2 is shown below in Fig.1.

Figure 1.

Membership functions of Rule 1 and Rule 2.

Figure 2.

Angle trajectory of the inverted pendulum.

Figure 3.

Angular velocity trajectory of the inverted pendulum.

Figure 4.

Control input applied to the inverted pendulum.

The feedback control gain can be found from (31)(51) by solving the LMI at each time. The following design parameters are chosen to satisfy:

Mixed NLQR-Hcriteria:

C=[1      1],D=[1],Q=diag[1001],R=1,α=1,β=0,γ=-5E160

Mixed NLQR-passivitycriteria:

C=[1      1],D=[1],Q=diag[1001],R=1,α=1,β=5,γ=0E161

The mixed criteria control performance results are shown in the Figs.2-4. From these figures, we find that the novel fuzzy LMI control has satisfactory performance. The mixed NLQR-Hcriteria control has a smaller overshoot and a faster response than the one with passivity property. The new technique controls the inverted pendulum very well under the effect of finite energy disturbance. It should also be noted that the LMI fuzzy control with mixed performance criteria satisfies global asymptotic stability.

7. Summary

This chapter presents a novel fuzzy control approach for both of continuous time and discrete time non-linear systems based on the LMI solutions. The Takagi-Sugeno fuzzy model is applied to decompose the non-linear system. Multiple performance criteria are used to design the controller and the relative weighting matrices of these criteria can be achieved by choosing different coefficient matrices. The optimal control can be obtained by solving LMI at each time. The inverted pendulum is used as an example to demonstrate its effectiveness. The simulation studies show that the proposed method provides a satisfactory alternative to the existing non-linear control approaches.

© 2012 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Xin Wang, Edwin E. Yaz, James Long and Tim Miller (September 27th 2012). Fuzzy Control of Nonlinear Systems with General Performance Criteria, Fuzzy Controllers - Recent Advances in Theory and Applications, Sohail Iqbal, Nora Boumella and Juan Carlos Figueroa Garcia, IntechOpen, DOI: 10.5772/48298. Available from:

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