## 中 華 大 學 碩 士 論 文

### 題目： 具 隨機不確定性的連續時間隨機T-S模糊模型 的最佳控制

### Optimal Control for Continuous-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：M09501008 賀世儒 指導教授：李 柏 坤 博士

### 中華民國九十七年八月

### 具隨機不確定性的連續時間隨機 T-S 模糊模型 的最佳控制

### Optimal Control for Continuous-time Stochastic T-S fuzzy Model with Stochastic

### Uncertainty

研 究 生：賀世儒 Student：Shih-Ju Ho

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

### 具隨機不確定性的連續時間隨機 T-S 模糊模型 的最佳控制

研 究 生：賀世儒 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

本論文裡，我們對於一種具有狀態相依雜訊的連續時間非線性隨機系統提出 模糊輸出回授強健控制設計的方法。基於 T-S 模糊動態模型，我們利用模糊

估測器與模糊控制器去達到我們所要求的 控制系統的效能，並將問題歸納為

HJI(Hamilton-Jacobi inequality)條件式。為了簡化計算上的問題與複雜度，

我們使用 LMI(linear matrix inequality)條件式去取代 HJI 條件式來求得我們

控制器與估測器中的問題參數。在廣義的動態輸出回授方法下，對於 控制器

的設計問題我們也做出了討論與研究。

*H*_{∞}

*H*_{∞}

*H*_{∞}

**Optimal Control for Continuous-time ** **Stochastic T-S fuzzy Model with **

**Stochastic Uncertainty **

Student : Shih-Ju Ho Advisor : Dr. Bore-Kuen Lee

Institute of Electrical Engineering Chung Hua University

### Abstract

In this study, we propose a robust fuzzy output feedback control design method for a class of continuous-time nonlinear stochastic systems with state-dependent noise. Based on Takagi and Sugeno (T-S) fuzzy dynamic model, fuzzy estimator and fuzzy controller are developed to achieve the control system performance by meeting the Hamilton-Jacobi inequality (HJI). However, for reducing the complicated computation, the controller gain matrices and estimator gain matrices can be obtained via solving some related linear matrix inequalities (LMI) instead of the Hamilton-Jacobi inequality (HJI). The control design under generalized dynamic output feedback scheme is also studied.

*H*_{∞}

*H*_{∞}

*H*_{∞}

**Acknowledgement **

I would like to express my sincere gratitude for my advisor , Dr. Chi-Kuang Hawang and 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.

## Contents

1 Introduction 1

1.1 Literature survey . . . 1 1.2 Motivation and objective . . . 4 1.3 Organization of the thesis . . . 5

2 Some Basic Stochastic System Theories 7

2.1 Wiener-Lévy Process . . . 7 2.2 Itô-type Stochastic Di¤erential Equation . . . 9

3 Settlement of the H_{1} Output Feedback Control Design Problems 13
3.1 H_{1} Observer-based Control Design with Available Premise Variables 13
3.1.1 Problem Formulation . . . 13
3.1.2 H_{1} Observer-based Control Design . . . 17
3.2 H_{1} Control Design under Generalized Output Feedback Scheme with

Available Premise Variables . . . 24
3.2.1 System Formulation . . . 24
3.2.2 H_{1} Generalized Output Feedback Control Design . . . 26

1

3.3 Simulation Study . . . 32

3.3.1 Example of H_{1} Generalized Output Feedback Control . . . . 32

3.3.2 Example of H_{1} Observer-based Control . . . 34

4 Conclusion 43 5 Appendix A 44 5.1 A.1 Proof of Lemma 3 . . . 44

5.2 A.2 Proof of Theorem 2 . . . 45

5.3 A.3 Proof of Theorem 3 . . . 46

5.4 A.4 Proof of Theorem 4 . . . 47

## List of Figures

3.1 The trajectory of system state x1(t): . . . 35

3.2 The trajectory of system state x2(t): . . . 35

3.3 The derivative of measurement output signal dy(t): . . . 36

3.4 The trajectory of controller state ^x_{1}(t): . . . 36

3.5 The trajectory of controller state ^x_{2}(t): . . . 37

3.6 The control signal u(t): . . . 37

3.7 The white noise signal n(t)dt = dW (t): . . . 38

3.8 The external disturbance signal v(t): . . . 38

3.9 The trajectory of system state x1(t): . . . 41

3.10 The trajectory of system state x2(t): . . . 42

i

In this study, we propose a robust H_{1} fuzzy output feedback control design method
for a class of continuous-time nonlinear stochastic systems with state-dependent noise.

Based on Takagi and Sugeno (T-S) fuzzy dynamic model, fuzzy estimator and fuzzy
controller are developed to achieve the H_{1} control system performance by meeting the
Hamilton-Jacobi inequality (HJI). However, for reducing the complicated computation,
the controller gain matrices and estimator gain matrices can be obtained via solving some
related linear matrix inequalities (LMI) instead of the Hamilton-Jacobi inequality (HJI).

The H_{1}control design under generalized dynamic output feedback scheme is also studied.

Simulation studies are provided to illustrate my main results.

Keywords : optimal control, robust control, LMI

## Chapter 1

## Introduction

### 1.1 Literature survey

State feedback control design have been proposed for many years. Robust control
design problem originally based on state feedback control for linear systems to deal
with systen uncertainties is also well studied since 1980’s. By extending the method-
ology of the robust control for linear systems, robust H_{1} control design problem for
nonlinear systems [01][02] has been a very popular topic in the last two decades.

As not all the system states are measurable in practice, this leads to the estimator design problem [03]-[04] to estimate unknow system state. For modeling a real sys- tem, a nonlinear stochastic system is comparatively more and more interested than a deterministic system since stochastic signals and random parameters may exist in real systems. Therefore, for state estimation of nonlinear stochastic systems, it is a di¢ cult and complex problem due to the nonlinearity of system dynamics and the inherent stochastic properties in the system.

1

To deal the e¤ect of romdom signals upon state estimation, early pioneer work for this problem can be traced back to the celebrated extended Kalman …lter [07]

where the continuous-time nonlinear dynamics is linearized at the current estimated state such that the conventional Kalman …lter [08] can be applied. When the nonlin- ear system has a nonlinearity satisfying a cone-bounded condition, the upper bound and lower bound of the mean-square state estimation error are discussed in [09] and [10], respectively. In [11], a Lyapunov-like method is proposed to construct the ob- servers of both the continuous-time and discrete-time nonlinear stochastic systems.

Furthermore, as the fuzzy system [13] has can be used to approximate various nonlin- ear systems with arbitrary accuracy (the universal approximation property) [14][15], state estimation based on the T-S fuzzy model for nonlinear systems has also been widely applied in various …elds. 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. Also based on the stochastic T-S fuzzy model, 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.

In the presence of process noise and measurement noise, analysis of robustness
of the closed-loop control system using state feedback control scheme based on the
extended Kalman …lter has been studied in [29]. With comparison study of various
methods of state estimation, an H_{1} controller design problem for continuous-time
stochastic nonlinear systems, which is solved by using the connection between L2-gain

3 property and the solution in a certain Hamilton-Jacobi inequality (HJI), is attacked in [05].

However, in addition to random signals, model imprecision comes from actual un- certainty or from the purposeful choice of a simpli…ed representation of the dynamics system. Recently, some researches have turned their attention to the topic of robust control design to reduce e¤ect of the system uncertainties. A class of linear stochas- tic problems with state and control dependent noise was …rst studied by [30] and robust control problem was discussed in the presence of stochastic uncertainty [31].

In addition, if there are random parameters in a system, the system is said to have
state-dependent noise or multiplicative noise. For such a system, a robust control
scheme aiming at mixed H2 and H_{1} performance is proposed in [32] assuming that
all states are available and the mixed control for stochastic linear systems with state
dependent noises has been proposed in [33]. When the states are not all measured,
observer design for nonlinear stochastic systems with state-dependent noise can be
traced back to [34] 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 [35] and robust H_{1}…ltering problem is attacked in [36].

Furthermore, based on the observer design, a suboptimal output feedback control
is presented in [37]. Meanwhile, an observer-based H2/H_{1} output feedback control
is considered in [38]. For the output feedback control design problem, the solution
to the H_{1} …ltering problem or the H_{1} control problem is characterized in terms of
Hamilton-Jacobi inequality (HJI) such as in [36] and [38]. However, until now, it is
di¢ cult to solve HJI problem directly. This problem can be mitigated if the control

design is made with respect to a Takagi and Sugeno (T-S) fuzzy system which is an
approximation of the considered nonlinear stochastic system. By fuzzy approach,
a robust H_{1} …ltering design is proposed in [39] and robust stabilization using state
feedback is analyzed in [40] for nonlinear stochastic systems with state-dependent
noise.

Besides the state feedback control scheme, dynamic output feedback control, which is to directly design a dynamic controller based on measurement output, has been concerned by more and more researchers lately. The dynamic output feedback con- trol problem for nonlinear systems is more di¢ cult than the state feedback control problem.

### 1.2 Motivation and objective

Compared with the state feedback scheme in [40], H_{1} output feedback control based
on the fuzzy approach for a class of continuous-time nonlinear stochastic systems with
state-dependent noise will be proposed in this study.

First, we shall consider the fuzzy observer-based state feedback scheme. It will
be shown that the H_{1} control system design problem can be attacked by solving
bilinear matrix inequalities (BMI) which correspond to the Hamilton-Jacobi inequal-
ity. Moreover, for the BMI, we adopt a two-stage method from [41] to obtain the
estimator gain matrices and the controller gain matrices separately by solving two
sets of linear matrix inequalities (LMI). Simulation study will be made to guarantee
the H_{1} robust control performance for the considered nonlinear stochastic systems

5 with state-dependent noise.

Next, we shall present a generalized dynamic output feedback scheme. We con-
struct a fuzzy dynamic output feedback controller to achieve the H_{1} control system
performance by solving linear matrix inequalities (LMI) directly. The H_{1} control
system performance will also be veri…ed by simulation example.

### 1.3 Organization of the thesis

The remainder of this thesis is organized as follows. Some basic stochastic process and
system theories are described in Chapter 2. In Section 2.1, the Wiener-Lévy process
is introduced and in Section 2.2, we shall discuss the Itô-type stochastic di¤erential
equation for representinging a sotchastic dynamic system. The main results are in
Chapter 3. In Section 3.1, we shall present the H_{1} observer-based control scheme
under the assumptions that all the premise variables are available. The considered
nonlinear stochastic system togrther with the H_{1} control problem will be speci…ed in
Subsection 3.1.1. In Subsection 3.1.2, we shall present the H_{1}observer-based control
design method. Furthermore, we propose an H_{1}control design under the generalized
output feedback scheme with available premise variables to attain the H_{1} control
system performance in Section 3.2. Then, formulation of the new problem is made in
Subsection 3.2.1. In Section 3.2.2, we shall present the H_{1} dynamic output feedback
control design. In Section 3.3, simulation studies are made. The H_{1}dynamic output
feedback control performance is exhibited in Subsection 3.3.1. And the H_{1} observer-
based control design performance is also concluded in Subsection 3.3.2. Moreover,

conclusions are made in Section 4. Finally, several technical proofs of the lemmas and theorems are placed in the Appendix.

## Chapter 2

## Some Basic Stochastic System Theories

### 2.1 Wiener-Lévy Process

Before introducing Wiener-Lévy process, we represent some probability notations.

Sample space is denoted by ; and F is called a -algebra de…ned on which is a collection have the following properties:

(a): 2 F:

(b): Let A be a subset of and A^{c} be its complement. Then A 2 F ) A^{c} 2 F:

(c): If a countable number of subsets belong to F, then so is their union.

i.e. Ai 2 F )S_{1}

i=1A_{i} 2 F:

7

The map P : F ! [0; 1] is de…ned a mathematical measure and is called a probability measure de…ned on ( ; F) with the properties as follows:

(a) P (A) = 0; for any event A;

(b) P ( ) = 1 where is null set,
(c) P (S_{1}

i=1A_{i}) = P_{1}

i=1P (A_{i}), if all Ai 2 F with A^{i}T

A_{j} = , 8i 6= j:

Consequently, we call ( , F, P ) a probability space. Also, let I be a nonnegative internal of the real number R: We often consider any stochastic process X = X(t; w) as a mapping from I to R where w 2 . Thus, we say that X(t) is de…ned on I for brevity. After previous introductions, we next discuss a special type of second order process: the Wiener-Lévy process.

The Wiener-Lévy process is a mathematical description of the so-called Brownian motion, which is denoted by W (t).

De…nition 1 : Let W (t), t 2 I, be a random process de…ned on I . W (t) is called a standard Wiener-Lévy (W-L) process, if

(a): W (0) = 0:

(b): E fW (t)g = 0:

(c): The trajectory of W (t) is continuous almost sure with t 2 I.

(d): W (t) has independent increments fW (t) W ( )g for all t, 2 I

with t; and this set of increments is stationary with the correlation function
E jW (t) W ( )j^{2} =jt j for all t, 2 I:

Horeover, W (t) is not di¤erentiable almost everywhere. Fortunately, the derivative of a general W-L vector can be de…ned by the generalized function theory [42]. The

9 formal derivative is de…ned as dW (t) = n(t) dt where n(t) is called a white noise process.

Corollary 1 : A white noise process n(t), t 2 [0, T ) with T 1, is the generalized derivative of a general Wiener-Lévy process, W (t), in the sense of distributions:

dW (t) = n(t) dt or W (t) = Z t

0

n( )d ,

which has a covariance function given by the delta function.

R_{n}(s; t) = (t s), s; t2 [0, T ) :

Consequently, a white noise process has the property that the values n(s) and n(t) are independent for all s 6= t in (0; T ):

White noise processes play a very important role in stochastic control theory, because they o¤er an excellent description of the random noise, which is a certain kind of rapidly ‡uctuating random phenomenon, and in which the correlation of the two neighboring states of the process changes very rapidly with respect to the time variable.

### 2.2 Itô-type Stochastic Di¤erential Equation

Because of random process is nowhere di¤erentiable by using ordinary calculus or
Riemann-Stieltjes integration, Itô integral is a reasonable form for the stochastic
integral. In this section, we discuss the formulation of stochastic di¤erential equations
in the sense of Itô. We …rst give some notations. Let W (t); t 2 [t^{0}, T ], be an

n-dimensional standard W-L vector and G(t) be an m n matrix-valued random process whose elements are all of second order. Besides, we will use the inner product and norm de…ned as

hX; Y i = E XY^{T} and kXk =p

tracehX; Xi

Furthermore, the concept of convergence in mean square (m. s.) is introduced in the following de…nition.

De…nition 2 : A sequence fX^{n}j n = 1, 2, g of second-order random variables is
said to be converge in mean square to a second order random variable X, if

n!1lim kX^{n} Xk = 0:

Here, X is called the mean square (m.s.) limit of the sequence fX^{n}g.

Finally, we need a de…nition to complete the preparations of introducing Itô inte- gral.

De…nition 3 : Let W (t), t 0, be an n-dimensional standard W-L process de…ned
on a probability space ( , F, P ). Also, let F^{t}be the -algebra of subsets of generated
by W (s) for all s : 0 s t. Then, F^{t} is the smallest -algebra containing all the
events

n

wj ^W_{i}(s; w) < a, i = 1, , n, a 2 R, 0 s to

where ^W_{i}(s; w) is the i th component of the vector W (s; w): An random process X(t)
is said to be adapted to F^{t}if X(t) is F^{t} measurable for each t 0. G(t) is nonantici-
pating if G(t) is independent of the -algebra generated by F (W (t + s) W (t) j s > 0)
for all t 2 [0, 1] :

11
Partition the time interval [t0, T ] as t0 < t_{1} < t_{2} < < t_{p} = T, and denote

= max

1 i pjt^{i} t_{i 1}j : To simplify the notation in applying this argument, we will use

! 0 to represent the limiting process of partitions and the number of partitioned time intervals p ! 1: After the above notations, the following de…nition can be exhibited.

De…nition 4 [43]: Let G(t) be a nonanticipating second-order random process. The Itô integral of G(t) with respect to W (t) , denoted

Z T t0

G(t)dW (t), is de…ned to be the m. s. limit of the following …nite sums:

Z T t0

G(t)dW (t), lim

!0

Pp

i=1[G(t_{i 1})(W (t_{i}) W (t_{i 1}))] (2.1)

It can be shown by certain martingale arguments [44] that for any second-order nonan- ticipating random process G(t), t 0, the above limit uniquely exists, independently of the partition of the interval. Hence, the Iˆto integral (2.1) is well de…ned.

In order to realize the di¤erence between Riemann-Stieltjes integral and Itô inte- gral, we consider a stochastic di¤erential equation as of the following form:

dX(t) = X(t)dW (t)
X(0) = X_{0}; t 2 [0, 1]

(2.2)

and we expect to have a solution in the form

X(t) = X_{0}+
Z t

0

X( )dW ( ); t2 [0, 1]

Hence, (2.2) can be rewritten as:

d Z t

0

X( )dW ( ) = X(t)dW (t) (2.3)

For the case of Riemann-Stieltjes integral, we can get the following solution of (2.3):

dX(t) = X(t)dW (t)

! Z t

0

dX( ) X( ) =

Z t 0

dW (t)

! X(t) = X^{0}eW (t) W (0)

(2.4)

where X(t) and W (t) are functions depending on time t. On the other hand, the answer of Itô integral can be written as

X(t) = X_{0}eW (t) W (0) (t=2) (2.5)

where X(t) is a second-order random process and W (t) is a standard W-L process.

We …nd the correction is the term t

2 between the ordinary calculus and stochastic calculus.

## Chapter 3

## Settlement of the H _{1} Output

## Feedback Control Design Problems

### 3.1 H

_{1}

### Observer-based Control Design with Avail- able Premise Variables

### 3.1.1 Problem Formulation

Consider a continuous-time nonlinear stochastic system speci…ed by Iˆto-type stochas- tic di¤erential equation

8>

><

>>

:

dx (t) = (f1(x (t)) + g1(x(t))v(t) + p(x(t))u(t))dt + h1(x(t))dW (t)
dy(t) = (f_{2}(x (t)) + g_{2}(x(t))v(t))dt + h_{2}(x(t))dW (t)

(3.1)

where x(t) is the system state, y(t) is the measurement output, u(t) is the control
input signal, W (t) is the standard Wiener-Lévy process, and v(t) 2 L^{2}(R_{+})is external

13

disturbance signal. The Hilbert space L^{2}(R_{+}) contain any stochastic process f (t) as
an element under the requirement that

kf(t)k^{2}L2 , E Z _{1}

0

f^{T}(t)f (t)dt <1

Remark 1 : Physically, we only can obtain measurement output signal denoted by Y (t) with mathematical illustration

y(t) = Z

0

Y ( )d :

Hence, we use (3.37) to be our continuous-time nonlinear stochastic system.

We assume that f1(x); g_{1}(x); h_{1}(x), p(x); f2(x); g_{2}(x); and h2(x) are smooth
functions with f1(0) = g_{1}(0) = h_{1}(0) = f_{2}(0) = g_{2}(0) = h_{2}(0) = 0. We can use Takagi
and Sugeno fuzzy model to approximate the original nonlinear stochastic system (3.1).

The plant rule of the T-S fuzzy model can be represented as the following form:

Plant Rule j: for j = 1; 2; : : : L

If z1(t) is Fj1, , and zg(t) is Fjg then

dx(t) = (A_{j}x(t) + B_{1j}v(t) + B_{j}u(t)) dt + D_{1j}x(t)dW (t)
dy(t) = (C_{j}x(t) + B_{2j}v(t)) dt + D_{2j}x(t)dW (t)

(3.2)

where Aj, B1j, Bj, D1j, Cj, B2j, as well as D2j are known constant matrices of
appropriate dimensions, Fji is the fuzzy set, and L is the number of Fuzzy-If-Then
rules. The terms z1(t); z_{2}(t) : : : ; and zg(t) are the premise variables. Then by using
singleton fuzzi…er, product inference, and center average defuzzi…er, the fuzzy system
can be represented as following form:

15

dx(t) = PL j=1

h_{j}(z(t)) [(A_{j}x(t) + B_{1j}v(t) + B_{j}u(t)) dt + D_{1j}x(t)dW (t)] (3.3)

and

dy(t) = PL j=1

h_{j}(z(t)) [(C_{j}x(t) + B_{2j}v(t)) dt + D_{2j}x(t)dW (t)] (3.4)

where

j(z(t)) = Qg i=1

F_{ji}(z_{i}(t))
h_{j}(z(t)) = ^{j}(z(t))

PL

i=1 i(z(t))

z(t) = z_{1}(t); z_{2}(t); z_{g}(t)

T

(3.5)

and Fji(z_{i}(t))is the degree of membership of premise variable zi(t)in Fji. We assume
that

j(z(t)) 0; for j = 1; 2; : : : L and

PL

i=1 i(z(t)) > 0; for all t.

Therefore, we get

h_{j}(z(t)) 0; for i = 1; 2; : : : L (3.6)

and

PL j=1

h_{j}(z(t)) = 1: (3.7)

Based on the Takagi and Sugeno fuzzy model [13], we adopt the fuzzy estimator to deal with state estimation problem of the nonlinear stochastic system (3.3):

Estimator Rule i: for i = 1; 2; : : : L If z1(t) is Fi1, ; and zg(t) is Fig; then

d^x(t) = (A_{i}x(t) + B^ _{i}u(t))dt + L_{i}(dy(t) d^y(t))

(3.8)

where Li is the estimator gain for the i-th estimator rule and

d^y(t) = PL j=1

h_{j}(z(t))C_{j}x(t)dt:^ (3.9)

The initial state of ^x(t)is given as ^x(0) = 0. Then we can obtain the overall estimator as the following form:

d^x(t) = PL i=1

h_{i}(z(t)) [(A_{i}x(t) + B^ _{i}u(t))dt + L_{i}(dy(t) d^y(t))] (3.10)

In order to deal with control system design problem, we employ the following fuzzy controller:

Controller Rule j: for j = 1; 2; : : : L
If z1(t) is Fj1 and zg(t)is Fjg then
u(t) = K_{j}x(t)^

(3.11)

where Kj is the controller gain for the j-th control rule. Hence, we obtain the fuzzy controller represented by the following form:

17

u(t) = PL j=1

h_{j}(z(t))K_{j}x(t)^ (3.12)

Then we de…ne the state estimate error as

e(t) = x(t) x(t)^ (3.13)

and the augmented system state vector ~x(t) as ~x(t) = x^{T}(t) e^{T}(t) : The vector
of variables to be controlled is denoted by (t) and is de…ned as

(~x(t)) = G~x(t)

where

G = 2 66 4

G_{2} 0

0 G

3 77

5 : (3.14)

Given the stochastic T-S fuzzy model in (3.2) with zero initial state, the H_{1} control
design problem is to …nd the controller gains Ki and the estimator gains Li, for
1 i L, such that

k (~x(t))k^{2}L2

2kv(t)k^{2}L2

where ^{2} is a prescribed energy attenuation ratio of k (~x(t))k^{2}L2 over kv(t)k^{2}L2.

### 3.1.2 H

_{1}

### Observer-based Control Design

To attack the H_{1} control system design problem, we …rst combine (3.3) and (3.13)
to yield the augmented system which can be written as:

d~x(t) =PL i=1

PL

j=1h_{i}(z(t))h_{j}(z(t)) A_{ij}x(t) + B~ _{ij}v(t) dt + D_{ij}x(t)dW (t)~
(3.15)

where

A_{ij} =
2
66
4

A_{i}+ B_{i}K_{j} B_{i}K_{j}

0 A_{i} L_{i}C_{j}

3 77

5 ; D^{ij} =
2
66
4

D_{1i} 0

D_{1i} L_{i}D_{2j} 0
3
77
5 ;

B_{ij} =
2
66
4

B_{1i}

B_{1i} L_{i}B_{2j}
3
77
5 :

(3.16)

The closed-loop system in (3.15) can be expressed in a more compact form as in the following:

d~x (t) = (f (~x (t)) + g(~x(t))v(t))dt + h(~x(t))dW (t) (3.17)

where

f (~x (t)) = PL i=1

PL j=1

h_{i}(z(t))h_{j}(z(t))A_{ij}x(t)~

g(~x(t)) = PL i=1

PL j=1

hi(z(t))hj(z(t))Bij

h(~x(t)) = PL i=1

PL j=1

h_{i}(z(t))h_{j}(z(t))D_{ij}x(t)~ (3.18)

To …nd the estimator gain Li and the controller gain Kj, we shall borrow an important lemma which has been derived in [36].

Lemma 1 For the system (3.17) and the control variables (~x(t)), if there exists a
positive function V (~x)2 C^{2}(R^{n}) and V (0) = 0 solving the following HJI

(@V

@ ~x)^{T}f + 1
2

2(@V

@ ~x)^{T}gg^{T}(@V

@ ~x) + 1

2k (~x)k^{2} +1

2h^{T}(@^{2}V

@ ~x^{2})h < 0 (3.19)
for some > 0, then :

(i) : for the case of v(t) = 0, the equilibrium point x 0 of the system (3.1) is

19 globally asymptotically stable in probability.

(ii) : 8v(t) 2 L^{2}(R_{+}) with v(t) 6= 0, the following inequality (3.20) holds for some

> 0 when the initial state ~x(0) 6= 0.

k (~x(t))k^{2}L2 2E [V (~x(0))] + ^{2}kv(t)k^{2}L2 (3.20)

Note that in (3.20), if x(0) = 0 and thus ~x(0) = 0, then we have

k (~x(t))k^{2}L2

2kv(t)k^{2}L2 (3.21)

which is the desired H_{1}control system performance as de…ned in the previous section.

To solve the HJI in (3.19), we shall …nd some su¢ cient condition such that the gain matrices Li and Kj can be obtained by solving two sets of linear matrix inequalities (LMI). Then the design procedure can be easily implemented by using the Matlab LMI toolbox. To this end, we need the following lemma:

Lemma 2 : For any vectors a and b, the following inequality holds for any real number .

a^{T}b + b^{T}a ^{2}a^{T}a + ^{2}b^{T}b

Lemma 3 : Let Xij be any matrices and P = P^{T} > 0. Then we have

"

PL i=1

PL j=1

h_{i}( (t))h_{j}( (t))X_{ij}

#T

P PL m=1

PL n=1

h_{m}( (t))h_{n}( (t))X_{mn}
PL

i=1

PL j=1

h_{i}( (t))h_{j}( (t))X_{ij}^{T}P X_{ij}

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

Now, we are ready to present our main result.

Theorem 1 Consider the augmented system (3.15). If P = P^{T} > 0 is the common
solution of the following matrix inequalities:

2 66 66 66 66 66 4

P A_{ij} + A^{T}_{ij}P P B_{ij} D_{ij}^{T}P G^{T}

B_{ij}^{T}P ^{2}I 0 0

P D_{ij} 0 P 0

G 0 0 I

3 77 77 77 77 77 5

< 0 (3.22)

for all i and j (i ; j = 1; : : : ; L); then:

(i) : For the case of v(t) = 0, the equilibrium point ~x 0 of the augmented system (3.15) is globally asymptotically stable in probability.

(ii) : 8v(t) 2 L^{2}(R_{+}) with v(t)6= 0; the following inequality (3.23) holds

k (~x(t))k^{2}L2 2E ~x^{T}(0)P ~x(0) + ^{2}kv(t)k^{2}L2 (3.23)

for some > 0.

Proof. : Consider a Lyapunov candidate function

V (~x) = 1

2x~^{T}P ~x (3.24)

which belongs to C^{2}(R^{2 n}) and V (0) = 0: By Lemma 3 and (3.17), if (3.19) holds,
i.e.,

1 2

PL i=1

PL

j=1h_{i}(z(t))h_{j}(z(t))~x^{T} P A_{ij}+ A^{T}_{ij}P ~x + 1
2

2PL i=1

PL

j=1h_{i}(z(t))h_{j}(z(t))

~

x^{T}P B_{ij} PL
m=1

PL

n=1h_{m}(z(t))h_{n}(z(t)) B_{mn}^{T} P ~x + 1

2 G~x ^{2}+1
2
PL

i=1

PL

j=1h_{i}(z(t))h_{j}(z(t))h

~
x^{T}D^{T}_{ij}i

P PL m=1

PL

n=1h_{m}(z(t))h_{n}(z(t)) D_{mn}x < 0~
(3.25)

21 then (i) and (ii) can be concluded. By Lemma 3, a su¢ cient condition for the inequality (3.25) is given by:

1 2

PL i=1

PL j=1

h_{i}(z(t))h_{j}(z(t))n

~
x^{T} h

P A_{ij} + A^{T}_{ij}P + ^{2}P B_{ij}B_{ij}^{T}P + G^{T}G + D^{T}_{ij}P D_{ij}i

~ xo

< 0 (3.26) Therefore if the following matrix inequalities

P A_{ij} + A^{T}_{ij}P + ^{2}P B_{ij}B^{T}_{ij}P + G^{T}G + D_{ij}^{T}P D_{ij} < 0 (3.27)

hold for 1 i L and 1 j L, then we can obtain (i) and (ii). By Schur complement [45], inequality (3.27) can be rewritten as

2 66 66 66 66 66 4

P A_{ij} + A^{T}_{ij}P P B_{ij} D_{ij}^{T}P G^{T}

B_{ij}^{T}P ^{2}I 0 0

P D_{ij} 0 P 0

G 0 0 I

3 77 77 77 77 77 5

< 0 (3.28)

Therefore, the proof is completed.

From the above analysis, the important work we should do is solving the common
solution P = P^{T} > 0 from (3.28) to attain the H_{1} fuzzy control performance (3.23)
for a given attenuation ratio ^{2}. For the convenience of design, we would like to solve
the estimator gain Li and the controller gain Kj separately. Hence, we let

P = 2 66 4

P_{1} 0
0 P2

3 77

5 : (3.29)

where P1 = P_{1}^{T} > 0 and P2 = P_{2}^{T} > 0: Then (3.28) can be rewritten as follows:

M_{ij} =
2
66
66
66
66
66
66
66
66
66
66
66
4

G1ij P1BiKj P1B1i D_{1i}^{T}P1 Hij G^{T}_{2} 0
K_{j}^{T}B_{i}^{T}P_{1} G_{2ij} W_{ij} 0 0 0 G^{T}

B_{1i}^{T}P_{1} W_{ij}^{T} ^{2}I 0 0 0 0

P1D1i 0 0 P1 0 0 0

H_{ij}^{T} 0 0 0 P_{2} 0 0

G_{2} 0 0 0 0 I 0

0 G 0 0 0 0 I

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

< 0 (3.30)

where

G_{1ij} = P_{1}A_{i}+ A^{T}_{i} P_{1}+ P_{1}B_{i}K_{j}+ K_{j}^{T}B_{i}^{T}P_{1}
G_{2ij} = P_{2}A_{i}+ A^{T}_{i} P_{2} P_{2}L_{i}C_{j} C_{j}^{T}L^{T}_{i} P_{2}
W_{ij} = P_{2}B_{1i} P_{2}L_{i}B_{2j}

H_{ij} = D^{T}_{1i}P_{2} D_{2j}^{T}L^{T}_{i} P_{2}

It should be noted that (3.30) is not a linear matrix inequality (LMI) which can’t
be successfully solved using the LMI technique for obtaining (P1; K_{j}; P_{2}; L_{i}). We
shall adopt a two-stage method as suggested in [41] to separately obtain (P1; K_{j})and
(P2; Li). For the two-stage method, a lemma quoted from [46] is …rst introduced.

Lemma 4 The inequalities (3.30) is solvable if the following LMI holds :

M_{ii} < 0; (3.31)

1

L 1M_{ii}+ 1

2(M_{ij} + M_{ji}) < 0; (3.32)
for 1 i6= j L:

23

Lemma 4 implies the submatrix G1ij in Mij satisfying 8>

><

>>

:

G_{1ii} < 0 , i = 1; 2; : : : L
1

L 1G_{1ii}+1

2(G_{1ij}+ G_{1ji}) < 0 , 1 i6= j L

(3.33)

which is equivalent to

A_{i}W_{1}+ W_{1}A^{T}_{i} + B_{i}Y_{i}+ Y_{i}^{T}B_{i}^{T} < 0 (3.34)

1

L 1 A_{i}W_{1}+ W_{1}A^{T}_{i} + B_{i}Y_{i}+ Y_{i}^{T}B_{i}^{T} +1
2

h

A_{i}W_{1}+ W_{1}A^{T}_{i} + B_{i}Y_{j} + Y_{j}^{T}B_{i}^{T}
+AjW1+ W1A^{T}_{j} + BjYi+ Y_{i}^{T}B_{j}^{T}

i

< 0 , 1 i6= j L

(3.35)

where W1 = P_{1} ^{1} and Yj = K_{j}P_{1} ^{1}. After solving (P1; K_{j})from (3.34) and (3.35), we
substitute (P1; K_{j})into (3.31) and (3.32) to get a set of LMI. Similarly, we can solve
(P_{2}; Z_{i} = P_{2}L_{i}) easily from the following problems:

Solve LMI’s (3.31) and (3.32)
subject to P2 = P_{2}^{T} > 0

(3.36)

Based on the above discussion, a design procedure for controlling a given nonlinear stochastic system in (3.1) is given below.

step 1): Design a T-S fuzzy system and choose an appropriate initial value of ^{2}.
step 2): Use (3.34) and (3.35) to obtain (P1; K_{j}).

step 3): Substitute (P1; K_{j}) into (3.31) and (3.32) to solve (P2; L_{i}).

step 4): Try to decrease ^{2} and repeat step 3) until (P2; L_{i}) can not be found.

step 5): Construct the fuzzy controller and the fuzzy estimator.

### 3.2 H

_{1}

### Control Design under Generalized Out- put Feedback Scheme with Available Premise Variables

### 3.2.1 System Formulation

Consider a class of continuous-time nonlinear stochastic systems represented by Itô- type stochastic di¤erential equation

dx(t) = (f1(x(t)) + g1(x(t))v(t) + p(x(t))u(t))dt + (h(x(t)) + l(x(t))u(t))dW (t)
dy(t) = (f_{2}(x (t)) + g_{2}(x(t))v(t))dt

(3.37)
where x(t) is the system state, dy(t) is the derivative of measurement output signal,
u(t)is the control input signal, W (t) is the standard Wiener-Lévy process. We assume
that f1(x(t)), g1(x(t)), f2(x(t)), g2(x(t)), p(x(t)), h(x(t)) as well as l(x(t)) are smooth
functions with f1(0) = g_{1}(0) = f_{2}(0) = g_{2}(0) = p(0) = h(0) = l(0) = 0. The process
v(t)2 L^{2}(H)is the external disturbance signal and L^{2}(H)is the Hilbert space which
contains any stochastic process f (t) satis…ed that

kf(t)k^{2}L2 , E Z _{1}

0

f^{T}(t)f (t)dt <1:

In order to approximate the original nonlinear stochastic system (3.37) for further control system design, a T-S stochastic fuzzy dynamic model, which is proposed by

25 Takagi and Sugeno [13], will be used. By the T-S fuzzy model [13], we get the plant rule of the following form:

Plant Rule i for i = 1; 2; : : : ; L :

If 1(t) is Fi1, , and g(t) is Fig; then

dx(t) = (A_{i}x(t) + B_{1i}v(t) + B_{2i}u(t)) dt + (4A^{i}x(t) +4B^{2i}u(t))dW (t)
dy(t) = (C1ix(t) + D1iv(t))dt

(3.38)

where 1(t), 2(t), . . . , and g(t) are premise variables, Ai, B1i, B2i, 4A^{i}, 4B^{2i}, C1i

and D1i are known constant matrices of appropriate dimensions, Fij is the fuzzy set, and L is the number of Fuzzy-If-Then rules. Then the fuzzy systems with singleton fuzzi…er, product inference, and the center average defuzzi…er are inferred as follows:

dx(t) = PL i=1

h_{i}( (t))f[A^{i}x(t) + B_{1i}v(t) + B_{2i}u(t)]dt + [4A^{i}x(t) +4B^{2i}u(t)]dW (t)g
dy(t) =

PL i=1

h_{i}( (t))fC^{1i}x(t) + D_{1i}v(t)gdt

(3.39) where

i( (t)) = Qg j=1

F_{ij}( _{j}(t))
h_{i}( (t)) = ^{i}( (t))

PL

i=1 i( (t))
(t) = ^{T}

1(t); ^{T}_{2}(t); ^{T}_{g}(t)

T

(3.40)

and Fij( _{j}(t)) is the grade of membership of j(t) in Fij. We presume that
PL

i=1 i( (t)) > 0, for any (t) Therefore, we get

h_{i}( (t)) 0, for i = 1; 2; : : : L (3.41)

and

PL i=1

hi( (t)) = 1. (3.42)

### 3.2.2 H

_{1}

### Generalized Output Feedback Control Design

Based on the fuzzy model (3.39), we construct a fuzzy output feedback controller to deal with the control system design problem as described by:

Controller Rule i for i = 1; 2; : : : ; L:

If 1(t)is Fi1, , and g(t) is Fig then
d^x(t) = ^A_{ij}x(t)dt + ^^ B_{i}dy(t)

u(t) = ^C_{i}x(t)^

(3.43)

where ^A_{ij}, ^B_{i};and ^C_{i} are the controller matrices for the i-th controller rule and ^x(t)
is the state of the controller. By the fuzzy systems mentioned above, we can obtain
the overall controller as the following form:

d^x(t) = PL i=1

h_{i}( (t))f ^A_{ij}x(t)dt + ^^ B_{i}dy(t)g (3.44)

and

u(t) = PL i=1

h_{i}( (t)) ^C_{i}x(t)^ (3.45)

Substituting (3.45) into the system (3.39), the Itô-type stochastic di¤erential equa- tion of the system state can be expressed as:

dx(t) = PL i=1

PL j=1

h_{i}( (t))h_{j}( (t))n

[A_{i}x(t) + B_{1i}v(t) + B_{2i}C^_{j}x(t)]dt^

+[4A^{i}x(t) +4B^{2i}C^_{j}x(t)]dW (t)^ o (3.46)
Similarly, (3.44) can be rearranged into an appropriate form for the design purpose:

27

d^x(t) = PL i=1

PL j=1

h_{i}( (t))h_{j}( (t))n

[ ^A_{ij}x(t) + ^^ B_{i}C_{1j}x(t) + ^B_{i}D_{1j}v(t)]dto

(3.47) With (3.46) and (3.47), we establish the augmented closed-loop system as follows:

d~x(t)

= PL i=1

PL j=1

h_{i}( (t))h_{j}( (t))
8>

><

>>

: 2 66 4

Ai B2iC^j

B^_{i}C_{1j} A^_{ij}
3
77
5

2 66 4

x(t)

^ x(t)

3 77 5 dt

+ 2 66 4

B_{1i}
B^iD1j

3 77

5 v(t)dt + 2 66 4

4A^{i} 4B^{2i}C^_{j}

0 0

3 77 5

2 66 4

x(t)

^ x(t)

3 77

5 dW (t) 9>

>=

>>

;

= PL i=1

PL j=1

h_{i}( (t))h_{j}( (t))n

[ ~A_{ij}x(t) + ~~ B_{ij}v(t)]dt + ~D_{ij}x(t)dW (t)~ o

(3.48)

where

~ x(t) =

2 66 4

x(t)

^ x(t)

3 77

5 ; A~ij = 2 66 4

A_{i} B_{2i}C^_{j}
B^_{i}C_{1j} A^_{ij}

3 77

5 ; B~ij = 2 66 4

B_{1i}
B^_{i}D_{1j}

3 77 5 ;

D~_{ij} =
2
66
4

4A^{i} 4B^{2i}C^_{j}

0 0

3 77 5 :

(3.49)

The H_{1} output feedback control design problem based on the fuzzy output feed-
back control is to …nd the appropriate controller matrices ^A_{ij}, ^B_{i}; and ^C_{i} for i,
j = 1; 2; : : : L, and attenuate the e¤ect of the external disturbance v(t) on the control
variable z(t) = m(~x(t)) in the sense of energy under a prescribed attenuation level

2, such that

km(~x(t))k^{2}L2

2kv(t)k^{2}L2

where m(~x(t)) = k~x(t)k^{2}L2.In order to establish the H_{1}output feedback control design
for the stochastic system (3.48), we need an important theorem as below.

Theorem 2 : Consider the nonlinear stochastic system (3.48) with the controlled
variable z(t) = m(~x(t)): If there exist a positive function V (~x(t)) 2 C^{2}(R^{n}) and
V (0) = 0 to the following Hamilton-Jacobi inequality (HJI)

(@V

@ ~x)^{T}f + ^{2}(@V

@ ~x)^{T}gg^{T}(@V

@ ~x) + 1

2h^{T}(@^{2}V

@ ~x^{2})h +km(~x(t))k^{2} < 0 (3.50)
then the H_{1} control system performance

km(~x(t))k^{2}L2 EV (~x(0)) + ^{2}kv(t)k^{2}L2 (3.51)

holds for some > 0 when the initial augmented system state ~x(0) 6= 0 with v(t) 6= 0:

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

Remark 2 : Suppose that ~x(0) = 0. Then the H_{1}control system performance (3.51)
becomes

km(~x(t))k^{2}L2

2kv(t)k^{2}L2 (3.52)

.

Note that the closed-loop system in (3.48) can be expressed in a compact form as follows:

d~x(t) = [f (~x(t)) + g(~x(t))v(t)]dt + h(~x(t))dW (t) (3.53) where

f (~x(t)) = PL i=1

PL j=1

h_{i}( (t))h_{j}( (t))[ ~A_{ij}x(t)]~
g(~x(t)) =

PL i=1

PL j=1

h_{i}( (t))h_{j}( (t))[ ~B_{ij}]
h(~x(t)) =

PL i=1

PL j=1

h_{i}( (t))h_{j}( (t))[ ~D_{ij}x(t)]~

(3.54)

29
Then, by using Theorem 2, the desired H_{1} control system performance of the sto-
chastic system (3.48) can be guaranteed. However, the controller gains ^A_{ij}, ^B_{i}; and
C^_{i} for i, j = 1; 2; : : : L, are not easy to obtain by solving the HJI in (3.50) directly. To
avoid complicated computations from solving the HJI, we need to …nd some e¤ective
ways to solve the matrices ^A_{ij}, ^B_{i}and ^C_{i}. To this end, a condition which can simplify
the H_{1} control system design is o¤ered in the following.

Theorem 3 : For the augmented system (3.48), if there exists a matrix P = P^{T} > 0;

which is the common solution of the following matrix inequalities:

2 66 66 66 66 66 4

ij P ~B_{ij} D~_{ij}^{T} I
B~^{T}_{ij}P 1

4

2I 0 0

D~ij 0 P ^{1} 0

I 0 0 I

3 77 77 77 77 77 5

< 0 (3.55)

where _{ij} = (P ~A_{ij} + ~A^{T}_{ij}P ), for i, j = 1; 2; : : : L, then the H_{1} control system perfor-
mance

km(~x(t))k^{2}L2 Ef~x^{T}(0)P ~x(0)g + ^{2}kv(t)k^{2}L2 (3.56)
is guaranteed for some > 0.

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

However, the H_{1} control design problem, which is to …nd the common solution
P = P^{T} > 0, controller matrices ^A_{ij}, ^B_{i}; and ^C_{i} for i, j = 1; 2; : : : L from the matrix
inequalities (MI) (3.55), can not be obtained by using the linear matrix inequality
(LMI) technique. Therefore, as quoted from [48], we specify the matrix P as the
following form:

P = 2 66 4

X Y ^{1} X

Y ^{1} X X Y ^{1}

3 77

5 > 0 (3.57)

where X = X^{T} > 0 and Y = Y^{T} > 0. Then we propose a theorem to transfer the
matrix inequalities (MI) (3.55) into a su¢ cient condition under which can be solved
by the LMI technique so that the system (3.48) can possesses the H_{1} control system
performance (3.56).

Theorem 4 : Consider the augmented system (3.48). If there exist matrices X =
X^{T} > 0, Y = Y^{T} > 0, B_{i}; and Ci, i = 1; 2; : : : ; L such that the following linear
matrix inequalities are satis…ed, then the H_{1} control system performance (3.56) can
be guaranteed.

2 66 4

X I

I Y

3 77

5 > 0 (3.58)

ii < 0 i = 1; 2; : : : L

ij + _{ji}< 0 i < j L

(3.59)

31 where

ij = 2 66 66 66 66 66 66 66 66 66 66 66 4

A_{11} 0 B_{1i} D_{11} 0 Y Y

0 A_{22} XB_{1i}+ B_{i}D_{1j} 4A^{T}i 0 I 0

B_{1i}^{T} (XB_{1i}+ B_{i}D_{1j})^{T} 1
4

2I 0 0 0 0

D_{11}^{T} 4A^{i} 0 X 2I I Y 0 0

0 0 0 I Y Y 0 0

Y I 0 0 0 I 0

Y 0 0 0 0 0 I

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

A_{11} = A_{i}Y + Y A^{T}_{i} + B_{2i}C_{j} + C_{j}^{T}B_{2i}^{T}
A_{22} = XA_{i}+ A^{T}_{i} X + B_{i}C_{1j} + C_{1j}^{T}B_{i}^{T}

D_{11} = Y4A^{T}i + C_{j}^{T}4B2i^{T} (3.60)

The matrices in the output feedback controller can be obtained via

B^i = (Y ^{1} X) ^{1}Bi (3.61)

C^_{j} = C_{j}Y ^{1} (3.62)

A^_{ij} = (Y ^{1} X) ^{1}M_{ij}Y ^{1} (3.63)

where

Mij = A^{T}_{i} XAiY XB2iCj BiC1jY (3.64)

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

Apparently, (3.59) are linear matrix inequalities (LMI). Thus variables X, Y , Bi

and Cj;for i, j = 1; 2; : : : ; L, can be easily obtained, for example, by using the Matlab LMI toolbox. Hence, substituting X, Y , Bi; as well as Cj into (3.63), we can get the

variable ^A_{ij}: Moreover, using the fact ^B_{i} = (Y ^{1} X) ^{1}B_{i} and ^C_{j} = C_{j}Y ^{1}; we
can determine controller gains ^A_{ij}, ^B_{i} and ^C_{i} to complete the fuzzy controller for
achieving H_{1} control system performance (3.56).

According to previous discussion, a design procedure for a class of nonlinear sto-
chastic systems in (3.37) to attain H_{1} control system performance is given below.

step 1): Design a T-S fuzzy system for the nonlinear stochastic system and choose
an appropriate initial value of ^{2}.

step 2): Use (3.58)-(3.64) to obtain ^Bi and ^Cj.
step 3): Substitute ^B_{i} and ^C_{j} into (3.64) to get ^A_{ij}.

step 4): If ^B_{i}, ^C_{j}; and ^A_{ij} are solvable, then decrease ^{2} and repeat step 2)-step
3) until ^B_{i}, ^C_{j} and ^A_{ij} can not be found.

step 5): Construct the fuzzy controller.

### 3.3 Simulation Study

In this section, two simulation examples are given to verify the proposed design methods for the considered stochastic system.

### 3.3.1 Example of H

_{1}

### Generalized Output Feedback Control

Example 1 : Now we consider a fuzzy model de…ned as follows:

Rule (1): If x1 is about 10, then

dx(t) = (A_{1}x(t) + B_{11}v(t) + B_{21}u(t)) dt + (4A^{1}x(t) +4B^{21}u(t))dW (t)
dy(t) = (C_{11}x(t) + D_{11}v(t))dt

(3.65)

33

Rule (2): If x1 is about 0, then

dx(t) = (A_{2}x(t) + B_{12}v(t) + B_{22}u(t)) dt + (4A^{2}x(t) +4B^{22}u(t))dW (t)
dy(t) = (C_{12}x(t) + D_{12}v(t))dt

(3.66)

Rule (3): If x1 is about 10, then

dx(t) = (A_{3}x(t) + B_{13}v(t) + B_{23}u(t)) dt + (4A^{3}x(t) +4B^{23}u(t))dW (t)
dy(t) = (C_{13}x(t) + D_{13}v(t))dt

(3.67)

where

A_{1} =
2
66
4

3 2

2 5

3 77

5 ; A^{1} =
2
66
4

5 1

1 5

3 77

5 ; A^{1} =
2
66
4

3 2

2 5

3 77 5 ;

B_{11} =
2
66
4

0:6 0:6

3 77

5, B^{12} = B_{11}, B13 = B_{11}, B21 =
2
66
4

1 0

3 77

5, B^{22}= B_{21}, B23= B_{21},

C_{11} = 1 5 , C12= 1 3 , C13= 1 5 , D11 = 1, D_{12} = 1, D_{13} = 1;

4A^{1} =
2
66
4

1 0:2 0:2 1

3 77

5, 4A^{2} =
2
66
4

0:5 0

0 0:5

3 77

5, 4A^{3} =
2
66
4

1 0:2

0:2 1 3 77 5,

4B^{21} =
2
66
4

0:1 0

3 77

5, 4B^{22} =
2
66
4

0 0:1

3 77

5, 4B^{23}=
2
66
4

0:1 0

3 77 5,

As aforementioned, W (t) is the standard Wiener-Lévy process with zero mean and unit variance, v(t) is an external normal distribution disturbance with zero mean as well as unit variance, x(t) is the system state with initial values x(0) = 1 1

T

and u(t) is the control input signal. We select triangular membership functions for the fuzzy system. Then given an initial = 50; by solving (3.58) and (3.59), one can

try to decrease until (3.59) is infeasible using the Matlab LMI toolbox. With this procedure, we get

A^_{11} =
2
66
4

71:2879 83:3525 48:0539 264:8019

3 77

5, A^_{21}=
2
66
4

87:5629 72:0551 66:9169 214:9316

3 77 5,

A^_{31} =
2
66
4

60:2617 22:2286 7:0874 77:4969

3 77

5, A^_{12}=
2
66
4

72:9620 85:3559 48:0199 264:7612

3 77 5,

A^_{22} =
2
66
4

89:2369 74:0585 66:8828 214:8908

3 77

5, A^_{32}=
2
66
4

61:9357 24:2320 7:0533 77:4561

3 77 5,

A^_{13} =
2
66
4

71:2879 83:3525 48:0539 264:8019

3 77

5, A^_{23}=
2
66
4

87:5629 72:0551 66:9169 214:9316

3 77 5,

A^_{33} =
2
66
4

60:2617 22:2286 7:0874 77:4969

3 77

5, B^_{1} =
2
66
4

16:7809 47:7780

3 77

5, B^_{2} =
2
66
4

22:0439 63:3803

3 77 5,

B^_{3} =
2
66
4

3:8015 10:6997

3 77

5, C^_{1} = 13:3032 0:4984 , ^C_{2} = 13:8553 1:1592 ,

C^_{3} = 13:3032 0:4984 , X
2
66
4

1:1018 0:1202 0:1202 1:1165

3 77 5, Y =

2 66 4

1:3724 0:1625 0:1625 1:3627

3 77 5 :

with the …nal = 2:5. The simulation results are shown in Figure 3:1 to Figure 3:8.

### 3.3.2 Example of H

_{1}

### Observer-based Control

Example 2 : Consider the nonlinear stochastic system as follows:

35

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6 1 .8 2

0 0 .2 0 .4 0 .6 0 .8 1

b lu e == x 1

d ata1

Figure 3.1: The trajectory of system state x1(t):

0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6 1 .8 2

- 0 .9 - 0 .8 - 0 .7 - 0 .6 - 0 .5 - 0 .4 - 0 .3 - 0 .2 - 0 .1 0

b lu e == x 2

d ata1

Figure 3.2: The trajectory of system state x2(t):

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6 1 .8 2 - 3

- 2 - 1 0 1 2

b lu e == de r iv a tiv e of me a s u r e men t o utpu t s ig na l d y ( t)

d ata1

Figure 3.3: The derivative of measurement output signal dy(t):

0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6 1 .8 2

- 0 .6 - 0 .4 - 0 .2 0 0 .2 0 .4

g r ee n == c o n tr o lle r s ta te of x 1 h at

d ata1

Figure 3.4: The trajectory of controller state ^x_{1}(t):