中 華 大 學

中 華 大 學 碩 士 論 文

題目：隨機模糊 T-S 模式的適應最小變異控制 Adaptive Minimum Variance Control of Stochastic

Fuzzy T-S Modes

系 所 別：電機工程學系碩士班 學號姓名：M09401031 邱政宏 指導教授：李 柏 坤 博士

中華民國 九十六 年 八 月

Adaptive Minimum Variance Control of Stochastic Fuzzy T-S Modes

研 究 生：邱政宏 Student：Chung-Hung Chiu 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 2007

Hsin-Chu, Taiwan, Republic of China

中 華 民 國 九 十 六 年 八 月

隨機模糊 T-S 模式的適應最小變異控制

研 究 生：邱政宏 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

在這一次研究中是引用隨機 T-S 模糊 ARMAX 模式的適應最小變異控 制技巧。在模糊 ARMAX 模式中，模糊的一步估測模式是最先成熟的。

隨機梯度演算法被提議用來判別關於一步估測的參數。針對適應控制 方法，最小變異控制是用來找一個控制法則使得輸出追蹤到參考訊 號。適應隨機模糊控制系統的穩定度和效能分析有嚴格推導。模擬討 論也證實這些結果。

Stochastic Fuzzy T-S Modes

Student：Chung-Hung Chiu Advisor：Dr. Bore-Kuen Lee

Institute of Electrical Engineering Chung Hua University

Abstract

Adaptive minimum variance control for stochastic T-S fuzzy ARMAX model is addressed in this study. From the fuzzy ARMAX model, a fuzzy one-step ahead prediction model is first developed. A stochastic gradient algorithm is then proposed to identify the parameters of the related one-step-ahead predictor. Under the direct adaptive control scheme, minimum variance control is applied to find the control law to make the output track a desired reference signal. Stability and performance of the adaptive stochastic fuzzy control system are rigorously derived. Simulation study is also made to verify the developed results.

Acknowledgement

I would like to express my sincere gratitude to 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 Dr.

Chi-Kuang Hwang, and my classmates for their helpful discussions and comments.

Finally, I want to express my sincere gratitude for my parents for their encouragement and suggestions.

1 Introduction 4

1.1 Survey of Related Literature . . . 4

1.2 Motivation and Objective . . . 5

1.3 Organization of the Thesis . . . 6

1.4 Notations and De…nitions . . . 6

2 System modeling and problem formulation 8 3 Adaptive Minimum Variance Control of T-S Fuzzy Model 12 3.1 Stability of Stochastic T-S Fuzzy Systems . . . 12

3.2 Optimal predictor of stochastic fuzzy systems . . . 17

3.3 Stochastic Gradient Algorithm . . . 19

3.4 Adaptive Minimum Variance Control . . . 23

3.5 Analysis of Stability And Tracking Performance . . . 25

3.6 Simulation Study . . . 27

4 Conclusions 30

1

2

A Appendix 31

A.1 Proof of Theorem 1 . . . 31

A.2 Proof of Theorem 2 . . . 34

A.3 Proof of Lemma 1 . . . 35

A.4 Proof of Lemma 2 . . . 38

A.5 Proof of Theorem 3 . . . 39

A.6 Proof of Lemma 3 . . . 42

A.7 Proof of Lemma 4 . . . 44

A.8 Proof of Lemma 5 . . . 48

3.1 Membership functioms of Example 1 . . . 28 3.2 Output y(t) and its prediction y(t) of Example 1. . . 29 3.3 The reference signal y (t) and the output y(t) of Example 1 is shown

in the upper trace. The tracking error is shown in the lower trace. . . 29

3

Chapter 1

Introduction

1.1 Survey of Related Literature

Recently, based on the Takagi-Sugeno model, fuzzy modeling for nonlinear dynamic systems and identi…cation problem are discussed in [1]-[3]. Meanwhile, fuzzy control scheme has been employed for tracking control of nonlinear systems based on the adaptive feedback linearization techniques [4]-[11]. In the previously mentioned liter- ature, the external disturbances or noises are considered to be deterministic for the convenience of control design. However, in many practical applications [12][13], exter- nal noises are inevitable and are more adequately described by random processes. In this situation, the systems to be controlled are always modeled by stochastic systems.

A nonlinear stochastic system can be approximated by a fuzzy stochastic system [14]-[19]. However, it is more di¢ cult to design a control law to achieve the optimal tracking of fuzzy stochastic systems because the membership functions of the fuzzy stochastic system are also functions of the random premise variables. This will make

4

the identi…cation problem and the control design of the stochastic fuzzy systems more di¢ cult and complicated.

Up to date, the stochastic fuzzy modeling and control issues are rarely addressed in the literature. A stochastic adaptive control scheme for the state-space T-S fuzzy model based on the LQG control theory is proposed in [20]. Non-adaptive LQG fuzzy controllers are also considered in [14] and [15]. On the other hand, the NARMAX (nonlinear ARMAX) model has been presented for modeling nonlinear processes.

The NARMAX model can be reduced to a quasi-ARMAX system by linearization or approximation. Fuzzy system identi…cation and nonlinear model predictive control based on the quasi-ARMAX model are discussed in [16][17][18]. Besides the quasi- ARMAX model, the fuzzy ARMAX model has been used to forecast the short-term load of a power system in [19]. However, these results proposed by the mentioned literature are given without vigorous proofs. Solid proof of the stability and tracking performance of the fuzzy ARX model for deterministic systems can be referred to [21].

1.2 Motivation and Objective

Adaptive minimum variance control for stochastic T-S fuzzy ARMAX model is ad- dressed in this study. From the fuzzy ARMAX model, a fuzzy one-step ahead pre- diction model is …rst developed. A stochastic gradient algorithm is then proposed to identify the parameters of the related one-step-ahead predictor. Under the direct adaptive control scheme, minimum variance control is applied to …nd the control law

6 to make the output track a desired reference signal. Stability and performance of the adaptive stochastic fuzzy control system are rigorously derived. Simulation study is also made to verify the developed results.

1.3 Organization of the Thesis

The remainder of the paper is organized as follows. System description and problem formulation for the identi…cation of fuzzy ARMAX systems is described in Chapter 2. General stochastic stability results of the T-S fuzzy model is attacked in Section 3.1. Then the one-step ahead predictor for the fuzzy ARMAX model is introduced in Section 3.2. Based on the developed predictor, a stochastic gradient algorithm, together with the parameter convergence properties, for identifying the parameters in the optimal one-step ahead predictor are given in Section 3.3. Adaptive minimum variance control design is discussed in Section 3.4. Stability and tracking performance of the adaptive minimum variance fuzzy control system are proved in Section 3.5.

Simulation study is discussed in Section 3.6. Conclusions and discussions are given in Chapter 4. Finally, several technical proofs of the lemmas and theorems are placed in the Appendix.

1.4 Notations and De…nitions

Let kxk be the Euclidean norm of a vector x and kAk is the associated matrix induced norm for a matrix A: The minimal and maximal eigenvalues of a matrix A with

real eigenvalue are denoted as min(A) and max(A), respectively. For a random vector x and a deterministic or random matrix A, the norm kAkms is de…ned as kAk²ms , sup_Efkxk²_g=1EfkAxk²g: A stochastic process x(k) is said to be mean square bounded if sup_kE kx(k)k² < 1. The abbreviation a: s: means almost surely.

Let A(q ¹) be a polynomial with A(q ¹) = Xⁿ

i=0a_iq ⁱ: The companion matrix A

associated with the polynomial A(q ¹) is de…ned as

A= 2 66 64

0_{(n 1) 1} I_{n 1} a_n a_{n 1} a₁

3 77 75

Chapter 2

System modeling and problem formulation

A nonlinear stochastic system can be divided into several local linear stochastic sys- tems according to their operation regions. Local linear systems using ARX, ARMAX, and NARMAX models to approximate nonlinear stochastic systems can be referred to [21][18][19][15]. A fuzzy stochastic model can be employed to interpolate local linear stochastic systems for a nonlinear stochastic system via the smoothing of fuzzy basis functions. This fuzzy stochastic model is described by fuzzy if-then rules and will be used here to deal with the stochastic tracking problem of nonlinear stochas- tic systems. The i-th rule of this fuzzy stochastic model for nonlinear discrete-time

8

stochastic systems is proposed as the following fuzzy ARMAX form:

Plant Rule i:

If z1(k) is Fi1 and z2(k) is Fi2 and and zg0(k) is Fig0

Then A_i(q ¹)y(k + 1) = B_i(q ¹)u(k) + C_i(q ¹)w(k + 1)

(2.1)

for i = 1; 2; :::; L; where Fij is the fuzzy set, z1(k); z₂(k); ; z_g0(k) are the premise variables, and L is the number of if-then rules. Polynomials Ai(q ¹); B_i(q ¹); and C_i(q ¹) are de…ned, respectively, as follows

A_i(q ¹) = a_i0+ a_i1q ¹+ :::::: + a_inq ⁿ; a_i0 = 1 B_i(q ¹) = b_i0+ b_i1q ¹+ :::::: + b_imq ^m;

C_i(q ¹) = c_i0+ c_i1q ¹+ c_i2q ²+ :::::: + c_ilq ^l; c_i0= 1

(2.2)

for i = 1; 2; :::; L where q ¹ denotes the delay operator, i.e., q ¹y(k) = y(k 1).

Without loss of generality, Ci(q ¹) can be taken to have roots inside the unit circle [12][13]. y(k) is the output measurement, u(k) the control input, and the noise process w(k) will be taken to satisfy the following assumptions [12][13]:

E[w(k + 1)j z^k] = 0; a: s: (2.3)

E[w²(k + 1)j z^k] = ²_w; a: s: (2.4) lim sup

N !1

1 N

XN k=1

w²(k) K_w <1; a: s: (2.5)

where E denotes the expectation, z^k denotes the sub- algebra generated from the data set fy(s)g^{s k}. Note that z^k is increasing, i.e., z^k z^k+1:We also assume that u(k) is z^k measurable. For the premise variables zi(k); 1 i g₀; we assume that they are z^k measurable, i.e., zi(k) depends on the data set fy(s); u(s)g^{s k}. Using

10 the smoothing property of the conditional mean [22], conditions (2.3) and (2.4) imply that w(k) is also a white process with zero mean and variance ²_w:Note that condition (2.5) implies

1 N

XN k=1

w²(k) K_w; a:s:, for N N_w (2.6)

where Nw is a su¢ ciently large integer.

Given the input/output sequences fu(k)g and fy(k)g, the stochastic fuzzy system (2.1) is equivalent to

y(k + 1) = XL

i=1

hi(z(k))f(1 Aⁱ(q ¹))y(k + 1) + Bi(q ¹)u(k) + Ci(q ¹)w(k + 1)g (2.7)

where z(k) = [z1(k) z₂(k) ::: z_g0(k)] and, for 1 i L;

i(z(k)) =

g0

Y

j=1

F_ij(z_j(k)) (2.8)

h_i(z(k)) = ⁱ(z(k)) PL

i=1 i(z(k)) (2.9)

Where the function Fij(z_j(k)) is the grade of membership of zj(k) in Fij: For (2.8) and (2.9), we assume that

h_i(z(k)) 0 , XL

i=1

h_i(z(k)) = 1 (2.10)

The physical meaning of (2.7) is that the L local linear stochastic subsystems are interpolated by the fuzzy basis functions hi(z(k)); for i = 1; 2; : : : ; L:

In the sequel, we shall …rst attack the identi…cation problem for estimating the parameters related to the fuzzy ARMAX model (2.1). To attack this problem, we shall need some mathematical tools concerning the stochastic stability of the T-S stochastic fuzzy system. After obtaining the estimates of the parameters, the design objective

for the stochastic fuzzy system in (2.7) is to determine the adaptive control input u(k) so as to bring the output y(k + 1) to optimally track a desired bounded output command y (k+1), which is speci…ed beforehand. Based on the identi…ed parameters, the control objective is to choose the input u(k), as a function of fy(s); u(s 1)gs k, to minimize the mean square error

J1(k + 1) = Ef[y(k + 1) y (k + 1)]²jz^kg (2.11)

between the the output y(k + 1) of stochastic fuzzy system and the desired output y (k + 1).

Chapter 3

Adaptive Minimum Variance Control of T-S Fuzzy Model

3.1 Stability of Stochastic T-S Fuzzy Systems

In order to deal with the identi…cation problem of the T-S stochastic fuzzy system, the stability issue of the stochastic fuzzy system must be addressed …rst. The results will be used in the sequel for analysis of the optimal predictor and the identi…cation algorithm for the stochastic fuzzy ARMAX model. Since the fuzzy ARMAX model, such as in (2.7), can be transformed into a state-space stochastic fuzzy model and stability is easier to discuss from the state-space perspective, we consider a forced T-S fuzzy system in the state-space form as follows

x(k + 1) = XL

i=1

h_i(z(k))A_ix(k) + v(k + 1) (3.1)

12

where fv(k)g is the stochastic forced term. It is assumed that fv(k)g is z^k measurable.

Recall that z(k) is z^k measurable, i.e., E fhⁱ(z(k))jz^kg = hⁱ(z(k)) a.s. Then x(k) is also z^{k 1} measurable. Also we make the assumption that v(k) is a process with uniformly bounded average power

sup

k

E kv(k)k² = ²_v <1 (3.2)

Before deriving the stability result for the forced stochastic fuzzy model (3.1), we need to …rst consider the following unforced stochastic system

x(k + 1) = A(k)x(k) (3.3)

where A(k) is z^k measurable and the sequence kA(k)k² is uniformly bounded. A su¢ cient conditions concerning the stability of the unforced system in (3.3) is given below.

Theorem 1 If there exists a sequence of symmetric positive de…nite matrices fP(k)g with 0 < ^min_P I P(k) ^max_P I < 1 and P(k) being z^k measurable such that the matrix inequality

P(k) A^T(k)EfP(k + 1)jz^kg A(k) > 0; 8k (3.4)

holds for some with 0 < < 1, then the stochastic fuzzy system (3.3) is mean square exponentially stable with

E kx(k)k²

max P min P

k k0E kx(k⁰)k² ; 8k k0 (3.5)

where k0 is an arbitrary initial time, x(k0) is an arbitrary initial condition. Further- more, it is also exponentially stable in the sense that

kx(k)k c₁(p

)^{k k0}kx(k⁰)k ; k K₁; a:s: (3.6)

14 for some positive almost surely bounded random variable c1 > 0 and a su¢ ciently large integer K1:

Proof. The proof is given in Appendix A.1.

Note that the following unforced system

x(k + 1) = XL

i=1

h_i(z(k))A_ix(k); (3.7)

which is related to (3.1), is a special case of (3.3) by identifying

A(k) = XL

i=1

h_i(z(k))A_i

Hence we have the following corollary.

Corollary 1 If there exist symmetric positive de…nite matrices Pi; 1 i L; such that the linear matrix inequalities

i;j , 2 66 4

P_i A^T_i P_j P_jA_i P_j

3 77

5 > 0; 1 i; j L (3.8)

hold where is a positive real number with 0 < < 1, then condition (3.4) is satis…ed.

Therefore, for the system in (3.7), the stability properties in (3.5) and (3.6) hold.

Proof. Since Pj is positive de…nite, by Schur complement, condition (3.8) is equiv- alent to

P_i A^T_i P_jA_i > 0 (3.9)

De…ne P(k) = XL

i=1

h_i(z(k))P_i. It follows P(k) is z^k measurable and ^min_P I P(k)

max

P Iwhere ^min_P = min

1 i L( _min(P_i))and ^max_P = max

1 i L( _max(P_i)):Due to the properties

of hi( ) in (2.10), we also have

Efh^j(z(k + 1))jz^kg 0 , XL

j=1

Efh^j(z(k + 1))jz^kg = 1

Now applying the operation XL

i=1

XL j=1

h_i(z(k))Efh^j(z(k + 1))jz^kg to both sides of (3.9), we have

P(k) A^T(k)EfP(k + 1)jz^kg A(k) > 0; 8k:

By using Theorem 1, the proof is completed.

In the following, we shall see that, provided the matrix inequalities in (3.4) hold, the stochastic system in (3.3) behaves like a linear time-varying system. With the system A(k), the system response of the system in (3.3) can be described by

x(k + 1) = (k + 1; k₀)x(k₀); k k₀ 0

where (k + 1; k₀) can be regarded as the transition matrix [24] and is de…ned as

(k + 1; k₀), A(k)A(k 1) A(k₀) (3.10)

and (k; k) , I: The following corollary directly follows from the de…nition of the norm k kms and inequality (3.5).

Corollary 2 If there exists a sequence of symmetric positive de…nite matrices fP(k)g with 0 < ^min_P I P(k) ^max_P I < 1 and P(k) being z^k measurable such that the matrix inequality (3.4) hold for some with 0 < < 1, then the upper bounds of the induced norm of (k; k₀) in the mean square and almost sure senses are given by

k (k; k⁰)kms

s max P min P

(p

)^{k k0}; 8k k₀ (3.11) k (k; k⁰)k c₂(p

)^{k k0}; k K₁; a:s: (3.12)

16 for some positive almost surly bounded random variable c2 and a su¢ ciently large integer K1.

Now consider the following stochastic system

x(k + 1) = [A(k)x(k) + B(k)u_s(k)]

y_s(k) = [C(k)x(k) + D(k)u_s(k)]

(3.13)

where the sequences kA(k)k² ; kB(k)k² ; kC(k)k² , and kD(k)k² are uni- formly bounded.

Theorem 2 For the stochastic system in (3.13), there exists a sequence of symmetric positive de…nite matrices fP(k)g with 0 < ^minP I P(k) ^max_P I <1 and P(k) being z^k measurable such that the matrix inequality (3.4) hold for some with 0 < < 1, then we have

1 N

XN k=1

ky^s(k)k² K₂ N

XN k=1

ku^s(k)k²+K₃

N ; a: s:; for N K₁ (3.14) where K1 is a su¢ ciently large number, 0 < K2 <1, and 0 K₃ <1:

Proof. The proof is given in Appendix A.2.

Note that the following general T-S fuzzy state-space system x(k + 1) =

XL i=1

h_i(z(k)) [A_ix(k) + B_iu_s(k)]

y_s(k) = XL

i=1

h_i(z(k)) [C_ix(k) + D_iu_s(k)]

(3.15)

is a special case of the systems in (3.13) by identifying A(k) =

XL i=1

h_i(z(k))A_i; B(k) = XL

i=1

h_i(z(k))B_i

C(k) = XL

i=1

h_i(z(k))C_i; D(k) = XL

i=1

h_i(z(k))D_i

Therefore, by combining the results in Corollary 1, Corollary 2, and Theorem 2, the following corollary can be easily obtained.

Corollary 3 If there exist symmetric positive de…nite matrices Pi; 1 i L; such that the matrix inequality (3.8) holds for some with 0 < < 1, then, for the stochastic fuzzy system (3.15), the inequality (3.14) holds.

Stability analysis of the state-space stochastic fuzzy system (3.1) is very useful for the optimal tracking design of the fuzzy ARMAX model in (2.7). For system identi…cation based on the prediction error method [26] for the fuzzy ARMAX model, the optimal fuzzy prediction must be …rst established as in the next section.

3.2 Optimal predictor of stochastic fuzzy systems

In this section, the prediction problem of the fuzzy ARMAX model in (2.7) will be addressed. This will result in a fuzzy predictor model which will be suitable for parameter estimation and optimal tracking design of fuzzy ARMAX systems. The optimal fuzzy predictor for the fuzzy ARMAX model has been studied in [27]. The results in that reference are brie‡y summarized in the following.

Assumption 1: Let C;ibe the companion matrix associated with the polynomial C_i(q ¹). Assume that there exist symmetric positive matrices PC;i; 1 i L, such that the set of matrix inequalities

2 66 4

CPC;i T C;iPC;j

P_{C;j C;i} P_C;j 3 77

5 > 0; 1 i; j L (3.16)

is solvable for some C with 0 < C < 1:

18 De…nition 1 A fuzzy polynomialPL

i=1h_i(z(k))C_i(q ¹) with C_i(q ¹) being manic and h_i( ) satisfying (2.10) is stable if the LMI condition (3.16) holds.

Under the above de…nition, Assumption 1 means that PL

i=1h_i(z(k))C_i(q ¹) is a stable fuzzy polynomial. Let y⁰(k + 1jk) denote the conditional mean of y(k + 1) given the data set fu(s); y(s)gs k, i.e., y⁰(k + 1j k) , E fy(k + 1)j z^kg : De…ne the polynomial i(q ¹); 1 i L; as

C_i(q ¹) A_i(q ¹) = q ¹ _i(q ¹) (3.17)

where

i(q ¹) = _i0+ _i1q ¹+ + q ^{(n 1)}; n = max(n; l)

Under Assumption 1 on the fuzzy ARMAX model (2.7), the optimal one-step ahead optimal predictor of y(k + 1) given the data set fu(s); y(s)gs k is y⁰(k + 1j k) which satis…es the following equation

y⁰(k + 1jk) = XL

i=1

h_i(z(k))f 1 C_i(q ¹) y⁰(k + 1jk) + ⁱ(q ¹)y(k) + B_i(q ¹)u(k)g (3.18) with the prediction error

y(k + 1) y⁰(k + 1jk) = w(k + 1) (3.19)

Equation (3.18) de…nes a unique fuzzy prediction model corresponding to the fuzzy ARMAX model (2.7). From human-operation point of view, the fuzzy prediction model is more feasible than the fuzzy ARMAX model since we can use the current and past measurement data fu(s); y(s)gs k to predict the future response y(k + 1) of

the stochastic fuzzy system; while using the fuzzy ARMAX model (2.7), the statistical properties of the noise process w(k) should be speci…ed in advance.

3.3 Stochastic Gradient Algorithm

Following from the fuzzy prediction model represented by (3.18), the stochastic gra- dient algorithm in [13] will be used to identify the parameters. First, rearrange the prediction model (3.18) as follows

y⁰(k + 1jk) = XL

i=1

h_i(z(k)) ^T₀(k) _i0 = ^T₀(k) ₀ (3.20)

where

0(k) = y⁰(kjk 1) y⁰(k l + 1jk l) y(k) y(k n + 1) u(k) u(k m) ^T

i0 = c_i1 c_{il i0 i(n 1)} b_i0 b_im ^T ; for 1 i L

0(k) = h₁(z(k)) ^T₀(k) h₂(z(k)) ^T₀(k) h_L(z(k)) ^T₀(k) ^T

0 = ^T₁₀ ^T₂₀ ^T_L0 ^T

Note that (3.20) represents a pseudo linear regression form for the fuzzy ARMAX prediction model (3.18) because the component y⁰(k i + 1jk i) in ₀(k) depends on the true parameter vector 0. According to the pseudo linear regression form (3.20), the proposed stochastic gradient algorithm to identify the true parameter vector 0 is given by, for k 1;

b(k) = b(k 1) + (k 1)

r(k 2) + ^T(k 1) (k 1) h

y(k) ^T(k 1)b(k 1) i

(3.21)

20 where the regression vector (k) and r (k 1) are de…ned as

(k) = h₁(z(k)) ^T(k) h₂(z(k)) ^T(k) h_L(z(k)) ^T(k) ^T (3.22) (k) = [ y(k) y(k l + 1) y(k) y(k n + 1) u(k) u(k m)](3.23)^T

y(k) = ^T(k 1)b(k) (3.24)

r(k 1) = r(k 2) + ^T(k 1) (k 1) (3.25)

For the initial conditions, b(0) can be arbitrarily chosen and r( 1) must be a positive scalar. By its de…nition, the variable y(k) can be regarded as a posterior estimate of y(k):

Before proceeding to analyze the stochastic gradient algorithm, some useful de…- nitions are made as follows

b

y(k) = ^T(k 1)b(k 1) (3.26)

e(k) = y(k) by(k) (3.27)

(k) = y(k) y(k) (3.28)

&(k) = (k) w(k) (3.29)

e(k) = b(k) ₀ (3.30)

(k) = ^T(k 1)e(k) (3.31)

The variables by(k) and y(k) are the a prior and the a posteriori predictions of y(k), respectively. Accordingly, e(k) and (k) are termed as the a prior and the a posteriori prediction errors, respectively. Using (3.19), the quantity &(k) can be rewritten as

&(k) = y⁰(kjk 1) y(k)and thus it accounts for the deviation between the optimal prediction y⁰(kjk 1) and the a posteriori prediction y(k): In the extreme case, if

&(k) = 0, then (k) = ₀(k) and the pseudo linear regression form (3.20) becomes y(k + 1) = ^T(k) ₀ which is a linear regression form.

Lemma 1 For the stochastic gradient algorithm in (3.21)-(3.23), we have

(i) lim

N !1

XN k=1

T(k 1) (k 1)

r(k 1)r(k 2) <1 (3.32)

(ii) (k) = r(k 2)

r(k 1)e(k) (3.33)

(iii) Ef (k)w(k)j z^{k 1}g = ^T(k 1) (k 1) r(k 1)

2

w; a: s: (3.34) (iv)

XL i=1

h_i(z(k 1))C_i(q ¹)&(k) = (k) (3.35)

Proof. The proof is given in Appendix A.3.

In addition to the results in Lemma 1, we shall need the following assumptions in order to obtain the properties of the parameter estimate b(k):

Assumption 2 : For each i, 1 i L; system Ci(q ¹) is input strictly passive (ISP) [13].

In (3.35), the signals &(k) and (k) are related by the fuzzy polynomialPL

i=1h_i(z(k 1))C_i(q ¹). As shall be shown in the next lemma, Assumption 2 implies a passivity condition for that fuzzy polynomial.

Lemma 2 Consider the fuzzy system in (3.35). With Assumption 2 that Ci(q ¹) is input strictly passive (ISP), we have

Xk j=1

(j)&(j) &²(j) 0; f or k 1 (3.36)

for some > 0:

Proof. The proof is given in Appendix A.4.

22 Theorem 3 Under Assumption 2, for the stochastic gradient algorithm in (3.21)- (3.25), we have the parameter di¤erence convergence

lim

N !1

XN k=1

b(k) b(k 1) ² <1; a: s: (3.37)

and the normalized prediction error convergence

N !1lim XN k=1

[e(k) w(k)]²

r(k 1) <1; a: s: (3.38)

Proof. The proof is given in Appendix A.5.

With the property in (3.38), it is possible to attain further results of the stochastic gradient algorithm by imposing an additional key condition. The following stochastic key technical lemma is quoted from [13].

Lemma 3 With the property in (3.38), if there exist positive constants Ka1, Ka2; and N such that

1

Nr(N 1) K_a1+ K_a2 N

XN k=1

[e(k) w(k)]²; a: s:; for N N (3.39)

then

(i) lim

N !1

1 N

XN k=1

[e(k) w(k)]² = 0; a: s: (3.40)

(ii) lim sup

k!1

r(N 1)

N <1; a: s: (3.41)

(iii) lim

N !1

1 N

XN k=1

E [y(k) y(k)]b ²j z^{k 1} = ²_w; a: s: (3.42)

If, in addition, condition (2.5) is strengthened to the following ergodic condition

E w⁴(k)j z^{k 1} <1; a: s: (3.43)

then

(iv) lim

N !1

1 N

XN k=1

[y(k) by(k)]² = ²_w; a: s: (3.44)

Proof. The proof is given in Appendix A.6.

3.4 Adaptive Minimum Variance Control

With the stochastic gradient algorithm for identifying parameters in the stochastic fuzzy predictor model, we are ready to propose adaptive fuzzy controller. The ob- jective of the adaptive control system is to design u (k) to minimize the mean-square error between the output y (k) and the desired output command y (k) at any time instance k, i.e., the cost function in (2.11) is minimized. We shall construct a di- rect adaptive control. Therefore, we shall …rst discuss the structure of the minimum variance controller by assuming that the system parameters are given. For the fuzzy stochastic system (2.7) having the optimal one-step ahead prediction form in (3.18), the minimum variance tracking control minimizing the cost function J1(k + 1) in (2.11) is given by [27]

u(k) = 1

b₀(k)

( _L

X

i=1

h_i(z(k)) B_i(q ¹) b_i0 u(k) + y (k + 1) XL

i=1

h_i(z(k)) _i(q ¹)y(k) + (1 C_i(q ¹))y⁰(k + 1jk) )

(3.45)

where b0(k) =PL

i=1h_i(z(k))b_i0. The e¤ect of the control law in (3.45) is to give

y⁰(k + 1jk) = y (k + 1) = ^T(k) ₀ (3.46)

24 i.e., the predicted output is forced to be equal to the desired output. Moreover, with (3.46), the control law (3.45) can be rewritten as

u(k) = 1

b₀(k)

( _L

X

i=1

h_i(z(k)) B_i(q ¹) b_i0 u(k) (3.47)

+ XL

i=1

h_i(z(k)) C_i(q ¹)y (k + 1) _i(q ¹)y(k) )

Now suppose that the estimated parameters,b^ij(k); ^b_ij(k), and ^c_ij(k)are obtained by using the stochastic gradient algorithm at time k. Accordingly, de…ne the following polynomials

bⁱ(k; q ¹) = bⁱ⁰(k) +bⁱ¹(k)q ¹+ :::::: +b^{i(n 1)}(k)q ^{(n 1)}; B^_i(k; q ¹) = ^b_i0(K) + ^b_i1(k)q ¹+ :::::: + ^b_im(k)q ^m;

C^_i(k; q ¹) = 1 + ^c_i1(k)q ¹+ ^c_i2(k)q ²+ :::::: + ^c_il(k)q ^l; (c_i0 = 1)

Let bb₀(k) = PL

i=1h_i(z(k))^b_i0(K): Based on the above estimated polynomials, the adaptive minimum variance control law under the certainty equivalent principle is given by

u(k) = 1

bb₀(k)

( _L

X

i=1

h_i(z(k))h b

B_i(k; q ¹) bb_i0(k)i u(k)

+ XL

i=1

h_i(z(k))h

Cb_i(k; q ¹)y (k + 1) bⁱ(k; q ¹)y(k)i)

(3.48)

in which the control law is derived from the following equation

y (k + 1) = ^T(k)b(k) (3.49)

3.5 Analysis of Stability And Tracking Performance

In this section, stability and tracking performance of the proposed adaptive stochastic fuzzy control system will be discussed. As the output y(k) is demanded to track arbitrary bounded reference signal y (k), some minimum-phase-like property of the stochastic fuzzy system in (2.7) is required in order to ensure internal stability of the adaptive control system. Therefore, we make the following assumption.

Assumption 3: (i) Assume that these exists a positive number b0;min such that 0 < b_0;min b_0;i and 0 < b0;min jb⁰(k)j : (2) Let B;ie be the companion matrix associated with the polynomial eB_i(q ¹)which is de…ned as eB_i(q ¹) = ^B_b^{i(q 1)}

0;min :Assume that there exist m m symmetric positive de…nite matrices P_B;ie ; 1 i L; of the form

P_B;ie = 2 66 4

P¹¹_e

B;i 0_{(m 1) 1} 0_{1 (m 1)} P²²_e

B;i

3 77

5 (3.50)

such that the matrix inequalities 2

66 4

BeP_B;ie ^T_e

B;iP_B;je P_{B;je B;ie} P_B;je

3 77

5 > 0; 1 i; j L (3.51)

hold for some _Be with 0 < _Be < 1.

Based on Assumption 3, we have the following results which will be used to prove stability of the adaptive control system.

Lemma 4 Under Assumption 3, for the stochastic fuzzy system in (2.7), we have, for N N ;

1 N

NX1 k=1

ku(k)k² K₆ N

N +1X

k=1

ky(k)k²+ K₇; a.s. (3.52)

26 where 0 < K6 <1; 0 < K⁷ <1 and N is a su¢ cient large number:

Proof. The proof is given in Appendix A.7.

Lemma 5 Under Assumption 1-Assumption 3, there exist …nite positive con- stants K8 to K13 such that

(i) 1 N

XN k=1

y²(k) K₈ N

XN k=1

[e(k) w(k)]²+ K₉; a:s: (3.53)

(ii) 1 N

XN k=1

y²(k) K₁₀ N

XN k=1

[e(k) w(k)]² + K₁₁; a:s: (3.54)

(iii) r(N 1) N

K_a2 N

XN k=1

[e(k) w(k)]²+ K_a1; a:s: (3.55)

for N N :

Proof. The proof is given in Appendix (A.8).

Finally, with the above lemma, we have the following tracking performance and global convergence result.

Theorem 4 For the stochastic fuzzy system in (2.7) with Assumption 1-Assumption 3, the adaptive minimum variance control algorithm is globally convergent with

(i) lim sup

k!1

1 N

XN k=1

y²(k) <1; a: s: (3.56)

(ii) lim sup

k!1

1 N

XN k=1

u²(k) <1; a: s: (3.57)

(iii) lim

N !1

1 N

XN k=1

E [y(k) y (k)]²j z^{k 1} = ²_w a: s: (3.58)

Furthermore, if (3.43) holds, then the result (3.58) is strengthened to

(iv) lim

N !1

1 N

XN k=1

[y(k) y (k)]² = ²_w a: s: (3.59)

Proof. Since, with (3.55), the stochastic key technical lemma (Lemma 3) holds, we have

N !1lim 1 N

XN k=1

[e(k) w(k)]² = 0; a: s:

Therefore, (3.56) and (3.57) follow from (3.52) and (3.53). Finally, as the objective of the adaptive minimum variance control is to set ^y(k) = y (k), (3.58) and (3.59) also follow from (3.42) and (3.44), respectively. This completes the proof.

3.6 Simulation Study

In this section, three simulation examples are given to verify the proposed adaptive minimum variance control algorithm.

Example 1 Adaptive control for T-S fuzzy systems Consider the following stochastic fuzzy system:

If z(k) is F_i

then A_i(q ¹)y(k + 1) = B_i(q ¹)u(k) + C_i(q ¹)w(k + 1); f or i = 1; 2; 5

where

A₁(q ¹) = 1 0:27q ¹+ 0:011q ²; B₁(q ¹) = 1 0:2q ¹; C₁(q ¹) = 1 0:135q ¹ A2(q ¹) = 1 0:33q ¹+ 0:023q ²; B2(q ¹) = 1 0:3q ¹; C2(q ¹) = 1 0:165q ¹ A₃(q ¹) = 1 0:362q ¹+ 0:0288q ²; B₃(q ¹) = 1 0:4q ¹; C₃(q ¹) = 1 0:18q ¹ A₄(q ¹) = 1 0:39q ¹+ 0:035q ²; B₄(q ¹) = 1 0:5q ¹; C₄(q ¹) = 1 0:195q ¹ A₅(q ¹) = 1 0:44q ¹+ 0:0468q ²; B₅(q ¹) = 1 0:6q ¹; C₅(q ¹) = 1 0:22q ¹

28

-100 -8 -6 -4 -2 0 2 4 6 8 10

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 3.1: Membership functioms of Example 1

and w(k) is a zero-mean Gaussian white noise with w = 0:01. The membership function for the fuzzy logic set Fi is given in Fig 3.1 and the premise variable z(k) is chosen as z(k) = y(k): We choose y (k + 1) = sin(₁₀₀² ) + 3 sin ₁₀₀⁶ as the reference signal. to estimation parameter and …nd the u (k) to yield y (k + 1) = y (k + 1) : The simulation results are shown in Fig 3.2 and Fig 3.3. Fig 3.2 shows the output y(k) and the prediction y(k) together with the estimation error. Fig 3.3 shows the output y(k) and the desired output command y (k).

100 200 300 400 500 600 700 800 900 1000 -4

-2 0 2 4

y and ybar

time(k)

output predictor

100 200 300 400 500 600 700 800 900 1000

-0.4 -0.2 0 0.2 0.4

e

Figure 3.2: Output y(t) and its prediction y(t) of Example 1.

100 200 300 400 500 600 700 800 900 1000

-4 -2 0 2 4

ystar and y

time(k)

object output

100 200 300 400 500 600 700 800 900 1000

-0.4 -0.2 0 0.2 0.4

y-ystar

Figure 3.3: The reference signal y (t) and the output y(t) of Example 1 is shown in the upper trace. The tracking error is shown in the lower trace.

Chapter 4

Conclusions

Adaptive minimum variance control for stochastic T-S fuzzy ARMAX model is ad- dressed in this study. From the fuzzy ARMAX model, a fuzzy one-step ahead pre- diction model is …rst developed. A stochastic gradient algorithm is then proposed to identify the parameters of the related one-step-ahead predictor. Under the direct adaptive control scheme, minimum variance control is applied to …nd the control law to make the output track a desired reference signal. Stability and performance of the adaptive stochastic fuzzy control system are rigorously derived. Simulation study is also made to verify the developed results.

30

Appendix

A.1 Proof of Theorem 1

Proof. First de…ne a Lyapunov function as

V (x(k)) = x^T(k)P(k)x(k) (A.1)

which is uniformly positive de…nite and

min

P kx(k)k² V (x(k)) ^max_P kx(k)k² (A.2)

With the de…nition of V (x(k)); it follows that

V (x(k + 1)) = x^T(k) A^T(k)P(k + 1)A(k) x(k) (A.3)

Note that the terms x(k), P(k), and A(k) are all z^k measurable. Now applying the conditional mean operator E f j z^kg to the both sides of (A.3) and using (??), we

31

32 have, almost surely,

EfV (x(k + 1)) j z^kg

= x^T(k) A^T(k)EfP(k + 1)jz^kg A(k) x(k) x^T(k)P(k)x(k)

= V (x(k))

(A.4)

Note that as E kA(k)k² and E kP(k)k² are uniformly bounded, E fV (x(k + 1)) j z^kg and E fP(k + 1)jz^kg are well de…ned. Apply the conditional expectation operator Ef j z^{k 1}g again to the both sides of (A.4) and recall that the sequence of the algebra z^k is increasing. With the smoothing properties [13] of conditional mean and inequality (A.4), it follows that almost surely

EfV (x(k + 1)) j z^{k 1}g ²V (x(k 1))

Continuing this procedure by sequentially applying E f j z^{k 2}g, E f j z^{k 3}g, , Ef j z^k0g ; one can obtain almost surely

EfV (x(k + 1)) j z^k0g ^{k+1 k0}V (x(k₀)) (A.5) Now taking expectation of the last inequality to yield

EfV (x(k))g ^{k k0}EfV (x(k⁰))g

Finally, using the fact of (A.2), inequality (3.5) is obtained.

Now we turn to prove the almost sure exponential stability (3.6). Clearly, it is trivial if x(k0) = 0. Now assume that the initial condition x(k0) is nonzero. By Chebyshev’s inequality [22], for any k> 0, we have

Probn

kx(k)k

kx(k0)k > _ko

En

kx(k)k² kx(k0)k²

o

= ²_k

= En

1

kx(k0)k²E kx(k)k² j z^k0

o

= ²_k

(A.6)

where ProbfAg is the probability measure of the event A: With (A.2) and (A.5), one can get

E kx(k)k² j z^k0

max P

min P

k k0

kx(k⁰)k²; a:s:

With the last inequality, (A.6) can be reduced to

Prob kx(k)k

kx(k⁰)k > _k 1

2 k

max P min P

k k0 (A.7)

Now choose the sequence k as k = ₀ ^{(k k}₁ ⁰⁾⁼² for any 0 > 0 and 1 > : Then inequality (A.7) implies that

X1 k=k0

Probn

kx(k)k > ^{0 (k k}1 ⁰⁾⁼²kx(k⁰)ko 1

2 0

max P min P

X1 k=k0

(

1

)^{k k0}

As 1 < 1, it follows that X1 k=k0

Probn

kx(k)k > ^{0 (k k}1 ⁰⁾⁼²kx(k⁰)ko

<1

and consequentially, by the Borel-Cantelli Lemma [22], we obtain that

Probn

[^{k K1}n

kx(k)k > ^{0 (k k}1 ⁰⁾⁼²kx(k⁰)koo

= 0

for some su¢ ciently large K1, any 0 > 0; and any 1 > :This means that for any sample path with bounded initial state x(k0), we have

kx(k)k c₁(p

)^{k k0}kx(k⁰)k ; k K₁; a:s:

for any initial condition x(k0); some positive bounded random variable c1; and a su¢ ciently large integer K1: This completes the proof.

34

A.2 Proof of Theorem 2

Proof. Suppose that kA(k)k A_L;kB(k)k B_L;kC(k)k C_L;and kD(k)k D_L for all k. Using the de…nition of the transition matrix de…ned in (3.10), the response of the output ys(k) of the fuzzy system in (3.15) can be represented by

y_s(k) = C(k) (k; 0)x(0) + D(k)u_s(k) + C(k) Xk 1

j=0

(k; j + 1)B(j)u_s(j)

Applying the results in Corollary 2, for k K₁, we have

ky^s(k)k C_Lc₂p k

kx(0)k + D^Lku^s(k)k + C^LB_L Xk 1

j=0

k (k; j + 1)k ku^s(j)k ; a: s:

By the Cauchy-Schwartz inequality, the last inequality leads to

ky^s(k)k² 3 8<

:C_L²c²₂ ^kkx(0)k²+ D_L² ku^s(k)k²+ C_L²B_L²

"_{k 1} X

j=0

k (k; j + 1)k ku^s(j)k

#²9

=

;; a: s:

c₃ ^k+ 3D²_Lku^s(k)k²+ 3C_L²B_L² Xk 1

j=0

k (k; j + 1)k Xk 1

j=0

k (k; j + 1)k ku^s(j)k²; a: s:

(A.8)

where c3 is de…ned as c3 = 3C_L²c²₂kx(0)k²:Considering the change of index i = k j;

the …rst summation term in the last inequality can be rearranged as Xk 1

j=0

k (k; j + 1)k = Xk

i=1

k (k; k i + 1)k

=

K1

X

i=1

k (k; k i + 1)k + Xk i=K1+1

k (k; k i + 1)k (A.9)

With the transition matrix de…ned in (3.10), it follows that k (k; k i + 1)k A^{i 1}_L for i K₁: On the other hand, for i > K1; inequality (3.12) ensures that k (k; k i + 1)k c₂p i 1

; a:s: and thus

lim

k!1

Xk 1 j=0

k (k; j + 1)k

K1

X

i=1

A^{i 1}_L + c₂ X1 i=K1+1

p ^{i 1}

= c₄ <1; a: s: (A.10)

where

c₄ = 1 A^K_L¹ 1 AL

+ c₂ p ^K1

1 p

Taking the summation operation _N¹ PN

k=1 on both sides of (A.8) and using (A.10), one can get

1 N

XN k=1

ky^s(k)k² 1 N

c₃

1 +3D²_L N

XN k=1

ku^s(k)k²+3C_L²B_L²c₄ N

XN k=1

Xk 1 j=0

k (k; j + 1)k ku^s(j)k²; a: s:

(A.11) in which the double summation term can be rearranged as follows

XN k=1

Xk 1 j=0

k (k; j + 1)k ku^s(j)k² =

NX1 j=0

XN k=j+1

k (k; j + 1)k ku^s(j)k²

With the same argument made from (A.9) to (A.10), it is easy to see that XN

k=j+1

k (k; j + 1)k

X1 k=j+1

k (k; j + 1)k c₄ <1; a: s: (A.12)

Therefore, following from (A.11) and (A.12), inequality (3.14) can be attained with K₃ = ₁^c3 + 3C_L²B_L²c²₄ku^s(0)k² = 3C_L² ₁^c22 kx(0)k²+ B_L²c²₄ku^s(0)k² K₂ = max (3D_L²; 3C_L²B²_Lc²₄)

A.3 Proof of Lemma 1

Proof. (i) Using (3.25), we can get X1

k=1

T (k 1) (k 1) r (k 1) r (k 2) =

X1 k=1

r (k 1) r (k 2) r (k 1) r (k 2)

= X1 k=1

1 r (k 2)

1 r (k 1) 1

r ( 1) <1 for r ( 1) = r^o

36 Therefore, condition (3.32) is valid.

(ii) From equation (3.21) and using (3.26), we can obtain

b(k) = b(k 1) + (k 1)

r(k 2) + ^T(k 1) (k 1)[y(k) by(k)]

Then multiply ^T (k 1)to both sides of the above equation and use (3.27) to get

T(k 1) b(k) = ^T(k 1) b(k 1) +

T (k 1) (k 1) r(k 1) e (k) Subtracting y (k) from both sides of the above equation, we have

y(k) y(k) = y(k) by(k)

T(k 1) (k 1) r(k 1) e (k) Using (3.25) and (3.27), we can get the following equation

(k) = 1

T (k 1) (k 1)

r(k 1) e (k)

= r(k 2) r(k 1)e (k) This completes the proof.

(iii) From (3.21), (3.25), and (3.26), we can get b(k) = b(k 1) + (k 1)

r(k 1)e (k) (A.13)

Subtracting 0 from (A.13) and multiplying by w (k) ^T (k 1) give

w (k) ^T (k 1) e (k) = w (k) ^T (k 1) e (k 1) +

T (k 1) (k 1)

r(k 1) [e (k) w (k) + w (k)] w (k)(A.14) Now taking the conditional mean Ef j z^{k 1}g on (A.14), we have

Ef (k) w (k)j z^{k 1}g = En

w (k) ^T (k 1) e (k 1)j z^{k 1}o +E

T (k 1) (k 1)

r(k 1) [e (k) w (k) + w (k)] w (k)j z^{k 1} (A.15)

Particularly, by (3.27), the term e (k) w (k) can be represented as

e (k) w (k) = y (k) y (k)b w (k)

= XL

i=1

h_i(z (k 1)) (1 A_i(q ¹))y(k) + B_i(q ¹)u(k 1) +C_i(q ¹)w(k) y (k)b w (k)

= XL

i=1

h_i(z (k 1)) (1 A_i(q ¹))y(k) + B_i(q ¹)u(k 1)

+ Ci(q ¹) 1 w(k) y (k)b (A.16)

From (A.16), the term e (k) w (k) is z^{k 1} measurable. Therefore, by properties of the noise w(k) in (2.3) and (2.4), it follows that (3.34) is concluded.

(iv) Rewrite (2.7) to get

XL i=1

h_i(z(k 1))A_i(q ¹)y(k) = XL

i=1

h_i(z(k 1)) B_i(q ¹)u(k 1) + C_i(q ¹)w(k) (A.17) Substituting (3.17) into (A.17), we have

XL i=1

h_i(z(k 1))[C_i(q ¹) q ¹ _i(q ¹)]y(k) = XL

i=1

h_i(z(k 1)) B_i(q ¹)u(k 1) +C_i(q ¹)w(k)

which leads to

XL i=1

h_i(z(k 1))C_i(q ¹)[y(k) w(k)] = XL

i=1

h_i(z(k 1)) q ¹ _i(q ¹)y(k) +B_i(q ¹)u(k 1)

From (3.28) and (3.29), we subtract PL

i=1h_i(z(k 1))C_i(q ¹)y(k)from both sides

38 of the above equation to get

XL i=1

h_i(z(k 1))C_i(q ¹)[y(k) y(k) w(k)] = XL

i=1

h_i(z(k 1)) q ¹ _i(q ¹)y(k) C_i(q ¹)y(k) + B_i(q ¹)u(k 1)

and thus XL

i=1

h_i(z(k 1))C_i(q ¹)&(k) = XL

i=1

h_i(z(k 1)) (C_i(q ¹) 1)y(k) +q ¹ _i(q ¹)y(k) + B_i(q ¹)u(k 1) y(k)

Using (3.23), we can get the following equation XL

i=1

h_i(z(k 1))C_i(q ¹)&(k) = XL

i=1

h_i(z(k 1)) ^T(k 1) _i0 y(k)

= ^T(k 1) ₀ ^T(k 1)b(k) = ^T(k 1)e(k)

= (k)

This completes the proof.

A.4 Proof of Lemma 2

Proof. First de…ne _i(k) = C_i(q ¹)&(k) for 1 i L: With the fuzzy system (3.35), we have

(k) = XL

i=1

h_i(z(k 1)) _i(k) (A.18)

As Ci(q ¹) is ISP [13], for any i, there is a positive number i such that Xk

j=1

&(j) _i(j) _i&²(j) 0 (A.19)