中 華 大 學

中 華 大 學 碩 士 論 文

題目： 具 隨機不確定性的離散時間隨機T-S模糊 模型之最佳控制

Optimal Control for Discrete-time Stochastic T-S fuzzy Model with Stochastic Uncertainty

系 所 別：電機工程學系碩士班 學號姓名：M09501030 林炫亨 指導教授：李 柏 坤 博士

中華民國 九十七 年 八 月

具隨機不確定性的離散時間隨機T-S模糊模型之 最佳控制

Optimal Control for Discrete-time Stochastic T-S fuzzy Model with Stochastic Uncertainty

研 究 生：林炫亨 Student：Hsuan-Heng Lin 指導教授：李柏坤 博士 Advisor：Dr. Bore-Kuen Lee

中華大學

電機工程學系碩士班

碩士論文

A Thesis

Submitted to Institute of Electrical Engineering Chung Hua University

In Partial Fulfillment of the Requirements For the Degree of

Master of Science In

Electrical Engineering August 2008

Hsin-Chu, Taiwan, Republic of China

中 華 民 國 九 十 七 年 八 月

具隨機不確定性的離散時間隨機T-S模糊模型 之最佳控制

研 究 生：林炫亨 指導教授：李柏坤 博士

中華大學

電機工程學系碩士班

中文摘要

在本論文中,具狀態相依雜訊的離散時間非線性隨機T-S模糊模型之H^∞動態

輸出回授控制被著手討論。我們所討論之模糊T-S模型具有隨機不確定項,即在系 統矩陣,輸入矩陣,與輸出矩陣中的狀態相依雜訊。首先,當模糊模型中的假定變

數可取得時, 我們使用與T-S模糊系統中相同假定變數之動態模糊H^∞輸出回授控

制,已達到規定控制系統能符合H^∞所要求之具體提出的性能指標。其次, 當模糊

模型中的假定變數無法取得時,基於估測器之模糊H^∞狀態回授控制被提出。由上

面兩種情況,我們推導出充分的條件去描述線性矩陣不等式(LMI)以保證閉迴路 系統之穩定度。上述提出之模糊控制將由模擬證明之。

Optimal Control for Discrete-time Stochastic T-S fuzzy Model with Stochastic Uncertainty

Student : Hsuan-Heng Lin Advisor : Dr. Bore-Kuen Lee

Institute of Electrical Engineering Chung Hua University

Abstract

In this thesis, H^∞ dynamic output feedback control for discrete-time nonlinear stochastic T-S fuzzy model with state-dependent noise is attacked. We consider the fuzzy T-S models has stochastic uncertainties, i.e., state-dependent noise, in the system matrix, input matrix, and output matrix. First, when the premise variables in the fuzzy plant model are available, an H^∞ fuzzy dynamic output feedback controller, which uses the same premise variables as the T-S fuzzy model, is proposed for regulation of the controlled system to meet the H^∞ control performance specification.

Next, when the premise variables for building the fuzzy plant model are not available, a fuzzy H^∞ observer-based state feedback controller, in which the premise variables are the estimated version of the premise variables in the T-S fuzzy model, is proposed.

For the two cases, we conduct sufficient conditions described by linear matrix inequalities (LMI) to ensure stability of the closed-loop system. Performance of the proposed fuzzy controller is verified by simulation study.

Acknowledgement

I would like to express my sincere gratitude for my advisor , Dr. Bore-Kuen Lee for his helpful advice , patient guidance, encouragement, and valuable support during the course of the research. I am obliged to my classmates for their helpful discussions and all my friends for their listening to my mood. Finally I want to express my sincere gratitude for my parents for their encouragement and suggestions.

1 Introduction 1 1.1 Literature survey . . . 1 1.2 Motivation and Objective . . . 3 1.3 Organization of the Thesis . . . 4

2 Optimal H₁ Output Feedback Control of Stochastic Fuzzy Systems 5 2.1 The Stochastic T-S Fuzzy Model . . . 5 2.2 Optimal H₁Output Feedback Control with measurable premise variables 8 2.3 Optimal H₁ Output Feedback Control without measurable premise

variables . . . 17 2.4 Simulation Example . . . 23

3 Conclusion and Discussion 29

4 Appendix 31

4.1 Constructing a Fuzzy Design Model Using a Nonlinear Model . . . . 31

1

List of Figures

2.1 Membership function of the premise variable in Example 1. . . 25

2.2 Response of the state xk in Example 1. . . 27

2.3 Response of the state ^x_k in Example 1. . . 27

2.4 Response of the state uk in Example 1. . . 28

i

In this thesis, H₁ dynamic output feedback control for discrete-time nonlinear sto- chastic T-S fuzzy model with state-dependent noise is attacked. We consider the fuzzy T-S models has stochastic uncertainties, i.e., state-dependent noise, in the system matrix, input matrix, and output matrix. First, when the premise variables in the fuzzy plant model are available, an H₁ fuzzy dynamic output feedback controller, which uses the same premise variables as the T-S fuzzy model, is proposed for regulation of the con- trolled system to meet the H₁ control performance speci…cation. Next, when the premise variables for building the fuzzy plant model are not available, a fuzzy H₁ observer-based state feedback controller, in which the premise variables are the estimated version of the premise variables in the T-S fuzzy model, is proposed. For the two cases, we conduct suf-

…cient conditions described by linear matrix inequalities (LMI) to ensure stability of the closed-loop system. Performance of the proposed fuzzy controller is veri…ed by simulation study.

Keyword: LMI, Optimal Control, State Dependent Noise

Chapter 1

Introduction

1.1 Literature survey

Usually, random signals such as process noise and measurement noise are involved in real control systems. When we try to estimate a suitable model for the control system under the stochastic environment, stochastic modeling errors are inevitable. It is also possible that a system has inherent random parameters such as a biological system.

Recently, some researchers have turned their attention to the topic of robust control design to reduce e¤ects of the model uncertainties and stochastic signals especially for nonlinear systems. To deal with model uncertainty, H₁control for nonlinear systems, originally assuming state vector is available, is a very popular topic recently. Robust H₁control design for deterministic nonlinear systems based on state feedback to deal with uncertainty can be referred to [1] and [2]. For a stochastic system with state- dependent noise, a robust control scheme aiming at mixed H2 and H₁ performance is proposed in [34][35] assuming that all states are available.

1

As not all the system states are measurable in practice, this arises the estimator design problem [3]-[4] to estimate unknow system state. However, for state estimation of nonlinear stochastic systems, it is a di¢ cult and complex problem, due to stochastic modeling uncertainty and the nonlinearity of system dynamics. Survey of literature concerning state estimation of nonlinear stochastic systems can be described by two parts as in the following.

(i) State Estimation for nonlinear stochastic systems without state- dependent noise: Early pioneer work for this problem can be traced back to the celebrated extended Kalman …lter [7] where the continuous-time nonlinear dynamics is linearized at the current estimated state such that the conventional Kalman …lter [8] can be applied. Recently, as the T-S fuzzy system [13] can approximate various nonlinear systems with arbitrary accuracy (the universal approximation property) [14]

[15], state estimation based on the T-S fuzzy model for nonlinear systems have also been applied widely. In [16], a fuzzy …lter theory based on the T-S fuzzy model was developed to minimize the worst-case state estimation error with respect to bounded disturbances and noises with unknown statistic properties. A fuzzy Kalman …lter is developed in [17] for state estimation of nonlinear discrete-time stochastic systems.

Similarly, in [18], a fuzzy Kalman …ler is developed for linear uncertain systems where the uncertainty is considered as a fuzzy set with a suitable membership function. The optimal fuzzy Kalman …lter for a standard stochastic T-S fuzzy model is established in [19].

(ii) State Estimation for nonlinear stochastic systems with state-dependent noise: Observer design for nonlinear stochastic systems with state-dependent noise

3 can be traced back to [36] where mean-square stability of estimation error is analyzed based on Lyapunov method. Minimax …ltering for nonlinear stochastic systems with state-dependent noise is studied in [37] and robust H₁ …ltering problem is attacked in [38]. Filtering for stochastic T-S fuzzy model with state-dependent noise in the system matrix and output matrix has been studied in [20].

For observer-based control problem, analysis of robustness of the closed-loop con- trol system using state feedback control scheme based on the extended Kalman …lter has been studied in [31]. An observer-based H₁control design for the continuous-time stochastic nonlinear system is proposed in [5] by solvinga certain Hamilton-Jacobi inequality (HJI). Furthermore, based on the observer design, a suboptimal output feedback control is presented in [39]. Meanwhile, an observer-based H2/H₁ output feedback control is considered in [40].

1.2 Motivation and Objective

Although T-S fuzzy model has been used for output feedback control for nonlinear stochastic systems [21][22], it seems that little attention has been paid to fuzzy dy- namic output feedback control for uncertain stochastic T-S fuzzy systems with state- dependent noise. In this study, for nonlinear stochastic systems with state-dependent noises in the system matrix, input matrix, and output matrix, we shall discuss two cases of H₁ dynamic output feedback control. First, when the premise variables in the fuzzy plant model are available, an H₁fuzzy dynamic output feedback controller, which uses the same premise variables as the T-S fuzzy model, is proposed for reg-

ulation of the controlled system. Next, when the premise variables for building the fuzzy plant model are not available, we shall propose a fuzzy H₁observer-based state feedback controller, in which the premise variables are the estimated version of the premise variables in the T-S fuzzy model. We also expect to solve the H₁ dynamic output feedback control via the LMI technique to improve design e¢ ciency.

1.3 Organization of the Thesis

The remainder of this thesis is organized as follows. In Section 2.1, the considered stochastic T-S fuzzy model with state-dependent noise is de…ned. Some assumptions with respect to the T-S fuzzy model are also discussed. In Section 2.2, with the measurable premise variables, we consider H₁ feedback control problem by using a fuzzy dynamic controller. On the other hand, when the premise variables in the fuzzy model are unavailable, an H₁ observer-based output feedback control scheme is proposed in Section 2.3. Simulation study is discussed in Section 2.4. Conclusions and discussions are given in Chapter 3. Finally, the technique of fuzzy modeling which is introuduced in [41] is placed in the Appendix.

Chapter 2

Optimal H ₁ Output Feedback Control of Stochastic Fuzzy Systems

2.1 The Stochastic T-S Fuzzy Model

Consider a class of discrete-time nonlinear stochastic system as following form:

x_k+1 = f₁(x_k) + g₁(x_k) u_k+ w_k y_k = f₂(x_k) + v_k

(2.1)

where xk = (x_1k; :::; x_mk)^T 2 R^{m 1} denotes variables of system state, yk denotes the measurement output of system, uk = (u_1k; :::; u_nk)^T 2 R^{n 1} denotes control input signal, the f1(x_k) , f2(x_k) and g1(x_k) are nonlinear function vectors of appropriate dimensions, wk and vk denotes external disturbance and measurement noise, respec-

5

tively. The wk and vk are assumed to be m 1vector-valued zero-mean white-noise processes, with

E 8>

>>

>>

>>

>>

>>

<

>>

>>

>>

>>

>>

>: 2 66 66 66 4

x₀ w_k v_k

3 77 77 77 5

2 66 66 66 66 66 4

x₀ w_j v_j 1

3 77 77 77 77 77 5

T9

>>

>>

>>

>>

>>

>=

>>

>>

>>

>>

>>

>;

= 2 66 66 66 4

0 0 0 0

0 R_{w kj} 0 0

0 0 R_{v kj} 0

3 77 77 77 5

(2.2)

We shall approximate the above system by a stochastic fuzzy dynamic model proposed by Takagi and Sugeno [43], and the T-S fuzzy model will be employed to deal with various control design problems for the nonlinear stochastic system (2.1).

The system can be represented by a stochastic T-S fuzzy model composed of the following rules:

System Rule i; 1 i L:

IF z1;k is Fi1 and z2;k is Fi2 and ... and zg;k is Fi;g, THEN xk+1 = (Ai+ Ai) xk+ (Bi+ Bi) uk+ wk

y_k = (C_i+ C_i) x_k + v_k

(2.3)

where L is the number of IF-THEN rules, g denotes the number of premise variables, F_ij denotes the fuzzy set for 1 i L and 1 j g, and zk = [z_1;k; : : : ; z_g;k]^T is the vector of premise variables; Ai, Bi C_i are system matrices of appropriate dimen- sions, and A_i(k) = _Aiq_i(k); B_i(k) = _Biq_i(k); C_i(k) = _ciq_i(k). The driving noises qi(k) is an i:i:d: (independent and identically distributed) processes of normal

7 distribution N (0; 1). Then the overall stochastic T-S fuzzy system is equivalent to

x_k+1 = XL

i=1

h_i(z_k)f(Aⁱ+ A_i) x_k+ (B_i+ B_i) u_k+ w_kg

y_k = XL

i=1

h_i(z_k)f(Cⁱ+ C_i) x_k + v_kg (2.4)

where

hi(zk) = ⁱ(z_k) PL

i=1 i(z_k) ⁱ(zk) = Qg j=1

Fij(zj;k) ; (2.5) and Fij(z_j;k)is the grade of membership of zj;k in Fij. Undel the assumption

PL

i=1 i(z_k) > 0 (2.6)

for any zk; then we get

h_i(z_k) 0; for i = 1; 2; ; L

and

PL

i=1h_i(z_k) = 1 (2.7)

Some assumptions with respect to the T-S fuzzy model in (2.4) are made in the following.

(A1) Efqⁱ(k)q_j(m)g = ² (i j) (k m); so that Ef A^Ti (k) A_j(m)g =

2 T

Ai Aj (i j) (k m), and Ef Ci^T(k) C_j(k)g = ^{2 T}Ci Cj (i j) :

(A2) Efw^kw^T_jg = R^w (k j), Efv^kv^T_jg = R^v (k j), and E fq²(k)jF^{k 1}g =

2, E fq(k)jF^{k 1}g = 0:

(A3) The premise variables z1;k ... zg;k are measurable with respect to the algebra F^k generated by Yk , fy⁰; :::; y_kg. The input u^k is also F^k measurable:

In this part, we summarize some basic materials concerning conditional proba- bility. The materials presented in this part are extracted from [44]. Let z be the -algebra generated by the random variable and let E fxg exist. Let fzⁱg¹i=1 be a nondecreasing -algebras. Some properties of conditional expectation are summarized in the following:

1. E fE fxjz¹gg = E fxg :

2. If x is z¹ measurable and jxj < 1 a.s., E fjxj jz¹g < 1 a.s., then E x^Tyjz¹ = x^TEfyjz¹g :

3. If z¹ z² z; then E fE fxjz²g jz¹g = E fxjz¹g :

Lemma 1 If random variable x is independent of random variables y and z, then

Efxyjzg = E fxg E fyjzg

2.2 Optimal H

Output Feedback Control with measurable premise variables

For the approximated stochastic T-S model in (2.4), we consider the following fuzzy output feedback controller:

Controller Rule i: 1 i L

9

IF z1;k is Fi1 and z2;k is Fi2 and ... and zg;k is Fi;g, THEN ^x_k+1 = ^A_ix^_k+ ^B_iy_k

u_k = ^C_ix^_k

(2.8)

where ^x_k 2 R^{m 1} is the controller’s state vector, ^A_i, ^B_i and ^C_i are parameters of the controller which are to be determined. Hence, the overall fuzzy output feedback controller is represented as follows:

^ x_k+1 =

PL j=1

h_j(z_k) A^_jx^_k+ bB_jy_k u_k =

PL j=1

h_j(z_k) ^C_jx^_k

(2.9)

Combining the fuzzy model in (2.4) and the controller in (2.9), the close-loop system is

x_k+1 = PL i=1

PL j=1

h_i(z_k) h_j(z_k)h

(A_i+ A_i) x_k+ (B_i+ B_i) ^C_jx^_k+ w_ki

^ x_k+1 =

PL i=1

PL i=1

h_i(z_k) h_j(z_k)h

A^_jx^_k+ ^B_j(C_i+ C_i) x_k+ ^B_jv_ki (2.10) After some algebraic manipulations, the augmented system can be expressed as the following form:

x_k+1 = XL

i=1

XL j=1

h_i(z_k) h_j(z_k)h

F~_ij(k)x_k+ ~E_j(k) ~w_ki

(2.11) where

x_k+1 = 2 66 4

x_k+1

^ x_k+1

3 77 5

F~_ij(k) = 2 66 4

A_i+ A_i (B_i+ B_i) bC_j b

B_j(C_i+ C_i) A^_j

3 77 5 ;

E~_j(k) = 2 66 4

I 0 0 B^_j

3 77 5 ; ~w_k =

2 66 4

w_k v_k

3 77 5

(2.12)

The randomly time-varying matrix ~F_ij(k) can be further expressed as

F~_ij(k) = F_ij + _ijq_i(k)

where

F_ij = 2 66 4

A_i B_iCb_j Bb_jC_i A^_j

3 77 5

ij = 2 66 4

Ai BiC^_j B^_{j Ci} 0

3 77 5

Before presenting our results, some useful lemmas are reviewed as follows.

Lemma 2 [45] For any matrices (or vectors) X and Y with appropriate dimensions, we have

X^TY + Y^TX X^TW ¹X + Y^TW Y (2.13)

where W is any symmetric positive de…nite matrix.

Lemma 3 For any positive matrix P and any matrix Xij; we have

" _L X

i=1

XL j=1

h_i(z_k) h_j(z_k) X_ij;k

#^T P

" _L X

m=1

XL n=1

h_m(z_k) h_n(z_k) X_ij;k

#

XL i=1

XL j=1

h_i(z_k) h_j(z_k) (X_ij;k)^T P (X_ij;k) (2.14)

11 Proof. :

XL i=1

XL j=1

h_i(z_k) h_j(z_k) X_ij;k

!^T P

XL m=1

XL n=1

h_m(z_k) h_n(z_k) X_mn;k

!

= XL

i=1

XL j=1

XL m=1

XL n=1

h_i(z_k) h_j(z_k) h_m(z_k) h_n(z_k) X_ij;k^T P X_mn;k

= XL

i=1

XL j=1

XL m=1

XL n=1

h_i(z_k) h_j(z_k) h_m(z_k) h_n(z_k) 1

2(X_ij;k^T P¹²P¹²X_mn;k + X_mn;kP¹²P¹²X_ij;k) XL

i=1

XL j=1

XL m=1

XL n=1

h_i(z_k) h_j(z_k) h_m(z_k) h_n(z_k) 1

2(X_ij;k^T P X_ij;k+ X_mn;k^T P X_mn;k)

= 1 2

XL i=1

XL j=1

h_i(z_k) h_j(z_k) X_ij;k^T P X_ij;k+ 1 2

XL m=1

XL n=1

h_m(z_k) h_n(z_k) X_mn;k^T P X_mn;k

= XL

i=1

XL j=1

h_i(z_k) h_j(z_k) X_ij;k^T P X_ij;k

The proof is complete.

Lemma 4 [42] The linear matrix inequality (LMI) 2

66 4

M (~x) S (~x) S^T(~x) N (~x)

3 77

5 0 (2.15)

where M (~x) = M^T (~x), N (~x) = N^T (~x), and S (~x) depends on ~x, is equivalent to N (~x) 0; M (~x) S (~x) N^y(~x) S^T (~x) 0

and

S (~x) I N (~x) N^y(~x) = 0 (2.16)

where N^y(~x) denotes the Moore-Penrose inverse of N (~x).

In the following, we shall discuss the optimal H₁ control design problem for the stochastic fuzzy system in (2.11) . From (2.11), the augmented system including the system can be expressed as follows

x_k+1 = XL

i=1

XL j=1

h_i(z_k) h_j(z_k)h

F~_ij(k)x_k+ ~E_j(k) ~w_ki

(2.17)

Let us consider the following suboptimal H₁ control performance for the augmented system

E ( _N

X

k=0

x^T_k+1Qx_k+1 )

E (

x^T₀P x₀+ ² XN

k=0

~ w^T_kw~_k

)

(2.18)

where is a prescribed noise attenuation level, and P is a positive de…nite weighting

matrix. Here we set Q = 2 66 4

Q₁ 0 0 Q₂

3 77 5 :

Theorem 1 If there exists a positive real number and a positive de…nite matrices R with

R = 2 66 4

R₁ R₂ R^T₂ R₃

3 77

5 (2.19)

13 such that the following linear matrix inequalities hold

2 66 66 66 66 66 66 66 66 66 66 66 66 66 4

2I R₁ Q₁ R₂ 0 0 A^T_i C_i^TBb_j^{T T}_A

i

T Ci

B^^T_j R^T₂ 2I R₃ Q₂ 0 0 Cb_j^TB_i^T A^^T_j C^_j^{T T}_B

i 0

0 0 ²I 0 I 0 0 0

0 0 0 ²I 0 Bb_j^T 0 0

A_i B_iCb_j I 0 R₁ R₂ 0 0

Bb_jC_i A^_j 0 Bb_j R^T₂ R₃ 0 0

Ai BiC^_j 0 0 0 0 R₁= ² R₂= ²

B^_{j Ci} 0 0 0 0 0 R^T₂= ² R₃= ²

3 77 77 77 77 77 77 77 77 77 77 77 77 77 5

> 0

(2.20)

for 1 i L and 1 j L, then the suboptimal H₁ control performance in (2.18) is attained.

Proof. : Let us choose a Lyapunov function for the system (2.18) as

V (x_k) = E x^T_kP x_k (2.21)

where P is a positive de…nite matrix. By the Lyapunov function, we obtain

V (x_k+1) V (x_k)

= E 8<

:

"

PL i=1

PL j=1

h_i(z_k) h_j(z_k) F~_ij(k)x_k+ ~E_jw~_k

#T

P PL

m=1

PL p=1

hm(zk) hp(zk) F~mp(k)xk+ ~Epw~k x^T_kP xk

(2.22)

Using Lemma 3, we can reduce(2.22) as V (x_k+1) V (x_k)

E 8>

><

>>

: PL i=1

PL j=1

hi(zk) hj(zk) 2 66 4

x_k

~ w_k

3 77 5

T 8

>>

<

>>

: 2 66 4

F_ij^T(k) + ^T_ijq_i(k) E~_j^T

3 77 5 P

F_ij(k) + _ijq_i(k) E~_j

2 66 4

P 0 0 0

3 77 5

9>

>=

>>

; 2 66 4

x_k

~ w_k

3 77 5

9>

>=

>>

;

(2.23)

Let Fk⁰ be the -algebra spanned by fy^s; x_s; w_s; v_sgs k: Applying the conditional mean operator E jFk⁰ to (2.23), we can get

= E

8>

><

>>

: E

8>

><

>>

: PL i=1

PL j=1

h_i(z_k) h_j(z_k) 2 66 4

x_k

~ wk

3 77 5

T 8

>>

<

>>

: 0 BB

@ 2 66 4

F_ij^T E~_j^T

3 77 5

+ 2 66 4

T ij

0 3 77 5 qⁱ(k)

1 CC

A P Fij E~j + _ij 0 q_i(k) 2

66 4

P 0 0 0

3 77 5

9>

>=

>>

; 2 66 4

xk

~ w_k

3 77 5 F^k0

9>

>=

>>

; 9>

>=

>>

;

(2.24)

= E

8>

><

>>

: PL i=1

PL j=1

h_i(z_k) h_j(z_k) 2 66 4

x_k

~ w_k

3 77 5

T

E 8>

><

>>

: 2 66 4

F_ij^T E~_j^T

3 77

5 P F_ij E~_j + 2

66 4

T ij

0 3 77

5 P ^ij 0 q_i²(k) 2 66 4

P 0 0 0

3 77 5

9>

>=

>>

; Fk⁰

9>

>=

>>

; 2 66 4

x_k

~ w_k

3 77 5

9>

>=

>>

;

(2.25)

= E 8>

><

>>

: PL i=1

PL j=1

h_i(z_k) h_j(z_k) 2 66 4

x_k

~ w_k

3 77 5

T 8

>>

<

>>

: 2 66 4

F_ij^T E~_j^T

3 77

5 P F_ij E~_j

+ ² 2 66 4

T ij

0 3 77

5 P ^ij 0

2 66 4

P 0 0 0

3 77 5

9>

>=

>>

; 2 66 4

x_k

~ w_k

3 77 5

9>

>=

>>

;

(2.26)

15 Then, we have the following form

V (x_k+1) V (x_k)

E 8>

><

>>

: PL i=1

PL j=1

h_i(z_k) h_j(z_k) 2 66 4

x_k

~ wk

3 77 5

T8

>>

<

>>

: 2 66 4

F_ij^T E~_j^T

3 77

5 P F_ij E~_j

+ ² 2 66 4

T ij

0 3 77

5 P ^ij 0 2 66 4

P 0 0 0

3 77 5

9>

>=

>>

; 2 66 4

x_k

~ w_k

3 77 5

9>

>=

>>

;

(2.27)

If there exist a positive matrix Q and a scalar such that 2

66 4

F_ij^T E~_j^T

3 77

5 P F_ij E~_j + ² 2 66 4

T ij

0 3 77

5 P ^ij 0 2

66 4

P 0 0 0

3 77 5 <

2 66 4

Q 0

0 ²I 3 77 5

(2.28)

then

V (x_k+1) V (x_k)

E 8>

><

>>

: 2 66 4

xk

~ w_k

3 77 5

T2 66 4

Q 0

0 ²I 3 77 5

2 66 4

xk

~ w_k

3 77 5

9>

>=

>>

;

= E x^T_kQx_k+ ²w~_k^Tw~_k

(2.29)

Summing (2.29) from k = 0 to k = N , we have

V (x_{N +1}) V (x₀) E

( _N

X

k=0

x^T_kQx_k+ ² XN k=0

~ w_k^Tw~_k

)

and by the de…nition of the Lyapunov function V (xk)in (2.21), we get

E ( _N

X

k=0

x^T_kQx_k )

E (

x^T₀P x₀ + ² XN

k=0

~ w^T_kw~_k

)

(2.30)

Therefore, the H₁ control performance is achieved with a prescribed ² provided that the inequalities (2.28) hold. By the Schur complement; (2.28) is equivalent to

2 66 66 66 66 66 4

P Q 0 F_ij^{T T}_ij 0 ²I E~_j^T 0 F_ij E~_j P ¹ 0

ij 0 0 ^P₂¹

3 77 77 77 77 77 5

> 0 (2.31)

Now let R = P ¹ so that (2.31) is equivalent to 2

66 66 66 66 66 4

R ¹ Q 0 F_ij^{T T}_ij 0 ²I E~_j^T 0 F_ij E~_j R 0

ij 0 0 ^R2

3 77 77 77 77 77 5

> 0 (2.32)

With the following inequality

R ¹² R¹²

T

R ¹² R¹² 0for R > 0 (2.33)

which implies

R ¹ 2I R; (2.34)

we can rewrite (2.32) as follows 2 66 66 66 66 66 4

2I R Q 0 F_ij^{T T}_ij 0 ²I E~_j^T 0 F_ij E~_j R 0

ij 0 0 ^R2

3 77 77 77 77 77 5

> 0 (2.35)

With the structure of the matrix R as de…ned in (2.19), the inequality (2.20) can be obtained. This completes the proof.

17

2.3 Optimal H

Output Feedback Control without measurable premise variables

In the previous section, the output feedback fuzzy controller is assumed to have the same premise variables zk as those of the fuzzy system model (2.8). This means that the premise variables of fuzzy system model in (2.8) are assumed to be measurable.

However, in general, it is di¢ cult to derive an accurate fuzzy system model by im- posing that all premise variables are measurable.Thus, we select our stochastic T-S fuzzy model controller rules as follows:

Controller Rule j: 1 i L

IF ^z_1;k is Fi1 and ^z_2;k is Fi2 and ... and ^z_g;k is Fj;g, THEN ^x_k+1 = A_jx^_k+ B_ju_k+ L_j(y_k y^_k)

^ y_k =

PL s=1

h_s(^z_k) C_sx^_k

(2.36)

where [^z_1;k; :::; ^z_g;k] is an estimate of [z1;k; :::; z_g;k], ^x_kis an estimate of xk, and Lj is the observer gain for the j-th observer rule. Then, we can obtain the following form:

^ xk+1 =

PL j=1

hj(^zk)fA^jx^k+ Bjuk+ Lj(yk y^k)g u_k=

PL s=1

h_s(^z_k) K_sx^_k

(2.37)

where Ks is the control gain for the s-th controller rule. Let us denote ~x_k = x_k x^_k

and then we get

^ x_k+1 =

PL j=1

h_j(^z_k) A_jx^_k+ B_ju_k+ L_j PL i=1

h_i(z_k)f(Cⁱ+ C_i) x_k + v_kg PL

s=1

h_s(^z_k) C_sx^_k

= PL i=1

PL j=1

PL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k)fA^jx^_k+ L_jC_ix_k+ B_ju_k+ L_j C_ix_k L_jC_sx^_k+ L_jv_kg

(2.38)

Using the above equation 2.38, we can obtain

~ x_k+1 =

PL i=1

PL j=1

PL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k)f(Aⁱ+ A_i L_jC_i) x_k A_jx^_k (B_i+ B_i B_j) u_k+ w_k L_j C_ix_k+ L_jC_sx^_k L_jv_kg

(2.39)

The last equation can be rearranged as

~ x_k+1

= PL i=1

PL j=1

PL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k)f(Aⁱ A_j L_jC_i+ L_jC_s) x_k+ (A_j L_jC_s) ~x_k

+ ( Ai Lj Ci) xk+ Biuk+ (Bi Bj) uk+ (wk Ljvk)

= PL i=1

PL j=1

PL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k)f[(Aⁱ A_j) + (B_i B_j) K_S+ L_j(C_s C_i)] x_k

+ [(A_j L_jC_s) (B_i B_j) K_S] ~x_k + ( A_i L_j C_i+ B_iK_S) x_k B_iK_Sx^_k+ (w_k L_jv_k)g (2.40)

and xk+1 can be expressed as

x_k+1 = PL i=1

PL j=1

PL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k)f(Aⁱ+ A_i) x_k+

(B_i+ B_i) K_sx^_k+ w_kg

= PL i=1

PL j=1

PL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k)f(Aⁱ+ B_iK_s) x_k B_iK_sx~_k + ( A_i+ B_iK_s) x_k+ B_iK_sx~_k+ w_kg

(2.41)

After some algebraic manipulations, the augmented system can be expressed as the

19 following form:

x_k+1 = XL

i=1

XL j=1

XL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k) ~F_ijs(k)x_k+ ~E_j(k) ~w_k

where

x_k+1= 2 66 4

xk+1

~ x_k+1

3 77 5

F~_ijs(k) = 2 66 66 66 4

A_i+ B_iK_s B_iK_s

(Ai Aj) + (Bi Bj) KS

+L_j(C_s C_i)

(A_j L_jC_s) (B_i B_j) K_S 3 77 77 77 5

+ 2 66 4

A_i+ B_iK_s B_iK_s Ai Lj Ci+ BiKs BiKs

3 77 5

E~_j(k) = 2 66 4

I 0

I L_j 3 77 5 ; ~w_k =

2 66 4

w_k v_k

3 77 5

(2.42)

The randomly time-varying matrix ~F_ij(k) can be further expressed as F~_ijs(k) = F_ijs+ _ijsq_i(k)

where

F_ij(k) = 2 66 4

A_i+ B_iK_s B_iK_s

M₁ M₂

3 77

5 (2.43)

M₁ = (A_i A_j) + (B_i B_j) K_s+ L_j(C_s C_i) (2.44) M₂ = (A_j L_jC_s) (B_i B_j) K_s (2.45)

ij = 2 66 4

Ai + _BiK_{s Bi}K_s

Ai L_{j Ci} + _BiK_{s Bi}K_s 3 77 5

With the closed-loop equation being obtained, we now present the H₁control design.

Theorem 2 If there exists a positive real number and a positive de…nite matrices R with

R = 2 66 4

R1 R2

R^T₂ R₃ 3 77

5 (2.46)

such that the following linear matrix inequalities hold 2

66 66 66 66 66 4

2I R Q 0 F_ijs^{T T}_ijs 0 ²I E~_j^T 0 F_ijs E~_j R 0

ijs 0 0 ^R₂

3 77 77 77 77 77 5

> 0 (2.47)

for 1 i L and 1 j L, then the suboptimal H₁ control performance in (2.18) is attained.

Proof. : Let us choose a Lyapunov function for the system (2.18) as

V (x_k) = E x^T_kP x_k (2.48)

where P is a positive de…nite matrix. By the Lyapunov function, we obtain

V (x_k+1) V (x_k)

= E 8<

:

"

PL i=1

PL j=1

PL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k) F~_ijs(k)x_k+ ~E_jw~_k

#T

P

"

PL

^{=1

PL

^

|=1

PL

^ s=1

h^{(zk) h^|(^zk) h^s(^zk) F~^{^|^s(k)xk+ ~E^|w~k

#

x^T_kP xk

)

(2.49)

21 Using Lemma 3, we can reduce (2.49) as

V (x_k+1) V (x_k)

E 8>

><

>>

: PL i=1

PL j=1

PL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k) 2 66 4

xk

~ w_k

3 77 5

T 8

>>

<

>>

: 2 66 4

F_ijs^T + ^T_ijsqj(k) E~_j^T

3 77 5 P

F_ijs+ _ijsq_j(k) E~_j

2 66 4

P 0 0 0

3 77 5

9>

>=

>>

; 2 66 4

x_k

~ wk

3 77 5

9>

>=

>>

;

(2.50)

Let Fk⁰ be the -algebra spanned by fy^s; x_s; w_s; v_sgs k: Applying the conditional mean operator E jFk⁰ to (2.50), we can get

E 8>

><

>>

: E

8>

><

>>

: PL i=1

PL j=1

PL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k) 2 66 4

xk

~ w_k

3 77 5

T 8

>>

<

>>

: 0 BB

@ 2 66 4

F_ijs^T E~_j^T

3 77 5

+ 2 66 4

T ijs

0 3 77 5 q^j(k)

1 CC

A P F_ijs E~_j + _ijs 0 q_j(k) 2

66 4

P 0 0 0

3 77 5

9>

>=

>>

; 2 66 4

x_k

~ w_k

3 77 5 F^k0

9>

>=

>>

; 9>

>=

>>

;

(2.51)

= E

8>

><

>>

: PL i=1

PL j=1

PL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k) 2 66 4

x_k

~ w_k

3 77 5

T

E 8>

><

>>

: 2 66 4

F_ijs^T E~_j^T

3 77

5 P F_ijs E~_j + 2

66 4

T ijs

0 3 77

5 P ^ijs 0 q²_j(k) 2 66 4

P 0 0 0

3 77 5

9>

>=

>>

; Fk⁰

9>

>=

>>

; 2 66 4

x_k

~ w_k

3 77 5

9>

>=

>>

;

(2.52)

Then, we have the following form

V (x_k+1) V (x_k)

E 8>

><

>>

: PL i=1

PL j=1

PL s=1

h_i(z_k) h_j(^z_k) h_s(^z_k) 2 66 4

x_k

~ wk

3 77 5

T 8

>>

<

>>

: 2 66 4

F_ijs^T E~_j^T

3 77

5 P F_ijs E~_j

+ ² 2 66 4

T ijs

0 3 77

5 P ^ijs 0 2 66 4

P 0 0 0

3 77 5

9>

>=

>>

; 2 66 4

x_k

~ w_k

3 77 5

9>

>=

>>

;

(2.53)

If there exist a positive matrix Q and a scalar such that 2

66 4

F_ijs^T E~_j^T

3 77

5 P F_ijs E~_j + ² 2 66 4

T ijs

0 3 77

5 P ^ijs 0 2

66 4

P 0 0 0

3 77 5 <

2 66 4

Q 0

0 ²I 3 77 5

(2.54)

then

V (x_k+1) V (x_k)

E 8>

><

>>

: 2 66 4

xk

~ w_k

3 77 5

T2 66 4

Q 0

0 ²I 3 77 5

2 66 4

xk

~ w_k

3 77 5

9>

>=

>>

;

= E x^T_kQx_k+ ²w~_k^Tw~_k

(2.55)

Summing (2.55) from k = 0 to k = N , we have

V (x_{N +1}) V (x₀) E

( _N

X

k=0

x^T_kQx_k+ ² XN k=0

~ w_k^Tw~_k

)

and by the de…nition of the Lyapunov function V (xk)in (2.21), we get

E ( _N

X

k=0

x^T_kQx_k )

E (

x^T₀P x₀ + ² XN

k=0

~ w^T_kw~_k

)

(2.56)

23 Therefore, the H₁control performance is achieved with a prescribed ² provided that

the inequalities (2.56) hold. By the Schur complement; (2.54) is equivalent to 2

66 66 66 66 66 4

P Q 0 F_ijs^{T T}_ijs 0 ²I E~_j^T 0 F_ij E~_j P ¹ 0

ijs 0 0 ^P₂¹

3 77 77 77 77 77 5

> 0 (2.57)

We can use (2.33) to rewrite (2.57) as follows 2

66 66 66 66 66 4

2I R Q 0 F_ijs^{T T}_ijs 0 ²I E~_j^T 0 F_ijs E~_j R 0

ijs 0 0 ^R2

3 77 77 77 77 77 5

> 0 (2.58)

With the structure of the matrix R as de…ned in (2.19), the inequality (2.47) can be obtained. This completes the proof.

2.4 Simulation Example

In this section, two simulation examples are given to con…rm the performance of the proposed fuzzy H₁ control for the stochastic fuzzy system.

Example 3 1

In this example,we assume that the premise variable zk is the plant output yk so that it is available in the controller design. Therefore we can apply the results in

Section 2.2 to design the fuzzy H₁ controller. Consider the following stochastic T-S fuzzy system:

Rule 1:

IF yk is F11;

THEN xk+1 = (A₁+ A₁(k)) x_k+ (B₁+ B₁)u_k+ w_k, y_k = (C₁+ C₁(k))x_k +v_k.

Rule 2:

IF yk is F12;

THEN xk+1 = (A₂+ A₂(k)) x_k+ (B₂+ B₂)u_k+ w_k, y_k = (C₂+ C₂(k))x_k +v_k.

Rule 3:

IF yk is F13;

THEN xk+1 = (A₃+ A₃(k)) x_k+ (B₃+ B₃)u_k+ w_k, y_k = (C₃+ C₃(k))x_k +v_k.

The related matrices in the above fuzzy system are de…ned as follows.

A₁ = 2 66 4

0:8 0:3 0:1 0:6

3 77

5 A² = 2 66 4

0:8 0:034 0:02 0:5

3 77

5 A³ = 2 66 4

0:3 0:04 0:03 0:5

3 77 5

B₁ = 1 2

T

; B₂ = 1 3

T

; B₃ = 2 1

T

;

C₁ = 1 3 ; C₂ = 2 1 ; C₃ = 1 2 ;

A1 = 2 66 4

0:2828 0:1414 0:1414 0:4243

3 77

5 ; ^A2 = 2 66 4

0:1414 0:2828 0:2828 0:24243

3 77

5 ; ^A3 = 2 66 4

0:2828 0:28228 0:1414 0:1414

3 77 5

25

-6 -4 -2 0 2 4 6

0 0.2 0.4 0.6 0.8 1

zk

Membership functions used in Example 1

F1 F

2 F

3

Figure 2.1: Membership function of the premise variable in Example 1.

B1 = 2 66 4

0:7071 0:4243

3 77

5 ; ^B2 = 2 66 4

0:4243 0:2828

3 77

5 ; ^B3 = 2 66 4

0:5657 0:2828

3 77 5

C1 = 0:4243 0:2828 ; _C2 = 0:1414 0:2828 ; _C3 = 0:2828 0:5657 The membership function for the premise variable is shown in Fig. 2.1.

The noises wk and vk are zero-mean Gaussian white noise with covariance matrix Rw =

2 66 4

1 0 0 1

3 77

5 and variance R^v = 1, respectively. The white process qi(k) is zero- mean with variance ² = 0:02: The initial condition of the sate x(k) is given by

2 66 4

x₁(0) x₂(0)

3 77 5 =

2 66 4

1 1

3 77 5