Robust Control of Time-Delay Systems: A Linear ...

國立成功大學電機工程學系博士論文使用線性矩陣不等式法探討時延系統的強健控制

Robust Control of Time-Delay Systems: A Linear Matrix Inequality Approach

研究生: 盧建余指導教授: 蔡聖鴻

Student: Chien-Yu Lu Advisor: Jason Sheng-Hong Tsai

Department of Electrical Engineering National Cheng Kung University Tainan, Taiwan, R.O.C. Dissertation for Doctor of Philosophy June, 2004

中華民國九十三年六月

使用線性矩陣不等式法探討強健時延系統控制盧建余* 蔡聖鴻** 國立成功大學電機工程學系摘要本論文旨在使用線性矩陣不等式法探討強健時延系統控制。其研究主題包括針對一系列不確定性時延系統、二維離散狀態延遲蘿莎模式、隨機時延系統及中性型隨機時延系統探討其穩定性、控制器設計、濾波器設計及追蹤控制設計。在穩定性分析方面，藉由狀態轉移矩陣法，我們提出創新的穩定法則。根據 Lyapunov-Krasovskii 法及結合線性矩陣不等式法，可推得簡單且改善時延相關的穩定法則。濾波器設計方面，提出一種新的濾波器設計方法，使得時延系統達到穩定且同時能滿足狀 H ∞ 的強健性能指標。追蹤控制方面，針對時延系統，藉由線性矩陣不等式法設計一種強健追蹤器以保証閉迴路的追蹤能滿足 H ∞ 的性能指標。至於二維離散狀態延遲系統方面，則提出一種強健的二維動態控制器，使得閉迴路穩定同時滿足 H ∞ 的性能指標。在隨機時延系統控制方面，以線性矩陣不等式推衍而得到的時延相關控制器亦被設計出，最後本論文亦提出中性型隨機線性與非線性系統的強健穩定化控制器。 * 研究生 ** 指導教授 i

Robust Control of Time-Delay Systems: A Linear Matrix Inequality Approach Chien-Yu Lu* and Jason Sheng-Hong Tsai** Department of Electrical Engineering National Cheng Kung University, Tainan, Taiwan, R.O.C.

Abstract A complete study of the robust control of time-delay systems via the linear matrix inequality approach is presented in this dissertation. This includes the developments of the stability analysis/controller design/filtering problem/tracking control for a series of uncertain systems with time delay, two-dimensional (2-D) discrete state-delayed systems described by Roesser model, a class of stochastic systems with time delay and a class of neutral stochastic systems. A novel state transformation matrix has been proposed for the stability analysis. Based on the Lyapunov-Krasovskii functionals combining with linear matrix inequality (LMI) techniques, simple and improved delay-dependent robust stability criteria are derived. For the filtering design, asymptotically stabilizing Kalman filtering and H ∞ filtering approaches are presented for the system with time delay. For the robust H ∞ tracking control, the proposed tracker obtained in terms of LMIs guarantees the stability of closed-loop systems and makes the output to approach the command reference input with a specified H ∞ performance. For the two-dimensional discrete state-delayed system, a 2-D dynamic output feedback controller is designed to achieve the closed-loop system asymptotic stability and a specified H ∞ performance using the LMI approach. Also, we develop a 2-D filtering approach with a specified H ∞ performance measure. For the stochastic system with time delay, a state feedback controller is designed to guarantee the closed-loop system remaining stable for all admissible uncertainties. Finally, we propose a state feedback controller to stabilize the class of neutral stochastic nonlinear systems with a specified H ∞ performance.

* the student ** the advisor ii

Acknowledgment

I would like to thank my advisor Professor Jason Sheng Hong Tsai for this ardent and invaluable guidance throughout the course of this research work. The information and suggestion given by him make my advanced and fruitful study. I am deeply indebted to Professor Ten-Jen Su for his consistency support and kind assistance. I also wish to express the most sincere appreciation to my parents, my wife and my children (Liang-Yu and Liang-Tzu) for their supports and encouragement during the entire course of this dissertation. Without their continual encouragement and help, I would not be able to complete this dissertation successfully. This dissertation is dedicated to them, in love and gratitude.

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Contents Page Abstract --------------------------------------------------------------------------------------------- ii Acknowledgement ------------------------------------------------------------------------------ iii Contents ---------------------------------------------------------------------------------------------iv List of Figures ------------------------------------------------------------------------------------ vii Chapter 1 Introduction ------------------------------------------------------------------------ 1 1.1 Time-Delay Systems -------------------------------------------------------------------------- 1 1.2 Delay-Independent/Delay-Dependent Conditions ---------------------------------------- 4 1.3 Mathematical Modelling --------------------------------------------------------------------- 5 1.4 Linear Matrix Inequality (LMI) ------------------------------------------------------------ 11 1.5 Contribution of the Dissertation ------------------------------------------------------------ 15 1.6 Brief Sketch of the Contents ---------------------------------------------------------------- 16

Chapter 2 Robust Stability and Robust Stabilization for Systems with Time Delay ----------------------------------------------------------------------- 17 2.1 Introduction ----------------------------------------------------------------------------------- 17 2.2 Delay-Dependent Stability ------------------------------------------------------------------ 18 2.2.1 System Description-------------------------------------------------------------------- 19 2.2.2 Main Results --------------------------------------------------------------------------- 20 2.2.3 Examples ------------------------------------------------------------------------------- 26 2.3 Delay-Dependent Stabilization ------------------------------------------------------------- 29 2.3.1 Preliminaries --------------------------------------------------------------------------- 29 2.3.2 Main Results --------------------------------------------------------------------------- 30 2.3.3 Examples ------------------------------------------------------------------------------- 33 2.4 Stability of Cellular Neural Networks with Time-Varying Delay---------------------- 35 2.4.1 Model of Cellular Neural Networks with Delay (DCNNs) ---------------------- 35 2.4.2 Existence of the Equilibrium and Stability of DCNNs --------------------------- 36 2.4.3 Examples ------------------------------------------------------------------------------- 38 2.5 Summary -------------------------------------------------------------------------------------- 39

Chapter 3 Robust Filtering for a Class of Uncertain Interval Systems with Time Delay ---------------------------------------------------------------- 40 3.1 Introduction ----------------------------------------------------------------------------------- 40 3.2 Robust Filtering for Delay-Dependent Interval Systems-------------------------------- 41 3.2.1 Problem Formulation and Assumptions -------------------------------------------- 42

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3.2.2 Main Results --------------------------------------------------------------------------- 45 3.2.3 An Illustrative Example--------------------------------------------------------------- 51 3.3 Delay-Dependent Robust H ∞ Filtering for Interval Systems with State Delays -- 54 3.3.1 Problem Formulation and Assumptions -------------------------------------------- 54 3.3.2 Main Results --------------------------------------------------------------------------- 56 3.3.3 An Illustrative Example--------------------------------------------------------------- 63 3.4 Summary -------------------------------------------------------------------------------------- 64

Chapter 4 Robust H ∞ Tracking for Systems with Time Delay ------------ 65 4.1 Introduction ----------------------------------------------------------------------------------- 65 4.2 Delay-Dependent Robust H ∞ Tracking for Uncertain Continuous Time-Delay Systems -------------------------------------------------------------------------------------- 66 4.2.1 System Description-------------------------------------------------------------------- 66 4.2.2 Main Results --------------------------------------------------------------------------- 69 4.2.3 An Illustrative Example--------------------------------------------------------------- 76 4.3 Robust H ∞ Tracking for Discrete Time-Delay Systems ------------------------------ 78 4.3.1 System Description-------------------------------------------------------------------- 78 4.3.2 Main Results --------------------------------------------------------------------------- 81 4.3.3 An Illustrative Example--------------------------------------------------------------- 87 4.4 Summary -------------------------------------------------------------------------------------- 89

Chapter 5 Two-Dimensional Robust Stabilization for State-Delayed Systems ------------------------------------------------------------------------------ 90 5.1 Introduction ----------------------------------------------------------------------------------- 90 5.2 Preliminaries ---------------------------------------------------------------------------------- 91 5.3 Main Results ---------------------------------------------------------------------------------- 92 5.4 An Illustrative Example ------------------------------------------------------------------- 104 5.5 Summary ------------------------------------------------------------------------------------ 106

Chapter 6 Two-Dimensional Robust H ∞ Filtering for State-Delayed Systems ------------------------------------------------------------------------ 107 6.1 Introduction --------------------------------------------------------------------------------- 107 6.2 Preliminaries -------------------------------------------------------------------------------- 108 6.3 Main Results -------------------------------------------------------------------------------- 109 6.4 An Illustrative Example --------------------------------------------------------------------117 6.5 Summary -------------------------------------------------------------------------------------118

Chapter 7 Robust Stabilization of Uncertain Stochastic Systems ----------119 7.1 Introduction----------------------------------------------------------------------------------119

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7.2 An LMI-Based Approach for Robust Stabilization of Uncertain Stochastic Systems with Time-Varying Delays ---------------------------------------------------------------- 120 7.2.1 Preliminaries ------------------------------------------------------------------------- 120 7.2.2 Main Results ------------------------------------------------------------------------- 122 7.2.3 An Illustrative Example------------------------------------------------------------- 126 7.3 Robust Stabilization of Uncertain Stochastic Neutral Interval Systems with Multiple Delays---------------------------------------------------------------------------------------- 127 7.3.1 Preliminaries-------------------------------------------------------------------------- 127 7.3.2 Main Results ------------------------------------------------------------------------- 129 7.3.3 Examples ----------------------------------------------------------------------------- 136 7.4 Delay-Dependent Robust H ∞ Control for Nonlinear Stochastic Neutral Systems with State Delays --------------------------------------------------------------------------7.4.1 Preliminaries ------------------------------------------------------------------------7.4.2 Main Results ------------------------------------------------------------------------7.4.3 An Illustrative Example------------------------------------------------------------7.5 Summary ------------------------------------------------------------------------------------

138 138 139 146 148

Chapter 8 Conclusions---------------------------------------------------------------------- 149 References --------------------------------------------------------------------------------------- 152 Biography ---------------------------------------------------------------------------------------- 160

vi

List of Figures

Page Figure 3.1 Responses of error dynamics---------------------------------------------------------- 52 Figure 3.2 Real state variable x1 and its estimate xˆ1 ----------------------------------------- 53 Figure 3.3 Real state variable x 2 and its estimate xˆ 2 ---------------------------------------- 53 Figure 4.1 Real state x1 (t ) , reference state xr1 (t ) --------------------------------------------- 77 Figure 4.2 Real state x 2 (t ) , reference state xr 2 (t ) --------------------------------------------- 78 Figure 4.3 Real state x1 (t ) , reference state xr1 (t ) --------------------------------------------- 88 Figure 4.4 Real state x 2 (t ) , reference state xr 2 (t ) --------------------------------------------- 89

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Chapter 1 Introduction

The purpose of this chapter is to introduce the overview of this dissertation and the basic concept of the linear matrix inequality.

1.1 Time-Delay Systems Time delays occur in different industrial systems such as mechanical systems, electrical systems, metallurgical systems, chemical systems, economic systems and biological system [17, 31, 47, 51]. A lot of attention has been paid to the class of dynamical systems with time delay. Among the problems that have been tackled are described as follows: ‧Stability problem, ‧Stabilization with different types of controllers like memoryless state feedback controller, memory state feedback controllers, output feedback controllers, etc., ‧ H ∞ control problem, ‧Filtering problem, ‧Tracking control problem, and ‧ Stochastic control problem. The robustness of these problems has also been addressed. Interesting results have been reported in the literature. The problem of stability has been extensively studied and delay-independent and delay-dependent conditions have been developed [16, 18, 19, 28, 44, 57]. In the case of uncertain linear time delay, robust stability has been also considered by many researchers in [24, 29, 39, 42, 43, 76, 94]. For the stabilization problem, readers can refer the literature [21, 35, 48, 91]. Robust stabilization has also attracted many researchers from the control community. Some

1

interesting results in this direction can be found [23, 46, 60, 96]. Notice that different types of controllers have been considered. For systems with time delay and exogenous disturbances, H ∞ control is one solution for eliminating the effect of the external disturbance. This type of problem has been extensively studied [15, 25, 49, 58, 61, 68]. For the robust case, readers can consult [36, 69, 94, 99]. Filtering is an important control problem that has attracted many researchers also. The references [3, 76, 77, 97] summarize the mainstream in this direction of research. For the robust filtering readers can consult [12, 22, 26, 80, 88, 93]. Moreover, tracking control is also an important research topic that researchers have concentrated their efforts on this problem [11, 9, 66, 70, 71, 78, 81]. As for stochastic systems, readers can refer literature [4, 31, 45, 52, 53, 54, 55] for the stochastic stability, stabilization and their robustness. To focus our idea on the topic we are dealing with, let us consider the following example from classic control theory. We consider a model that can be used to describe a general class of single input/ single output systems that arises in process control. The transfer function of this model is

G ( s) = K

−τs

e , Ts + 1

(1.1)

where K is the system gain, τ is the time delay in the system, and T is the system time constant. Let y (t ) and u (t ) denote respectively the system output and input at time t respectively. Based on the classical control theory, we have T y& (t ) + y (t ) = Ku (t − τ ) .

(1.2)

If we assume that the initial condition is zero and letting x(t ) = y (t ) , the corresponding state space representation is given by x& (t ) = Ax (t ) + Bu (t − τ ) ,

(1.3)

y (t ) = Cx (t ) ,

(1.4)

x(σ ) = φ (σ ) = 0 , σ ∈ [ −τ , 0]

(1.5)

2

where φ (⋅) is the initial function, A =

−1 K , B= and C = 1 . T T

Remark 1.1: The time delay is only on the control variable u (t ) . Such a situation arises in many systems like fluid systems where, for instance, long pipes are used to transport fluids. As a second example let us consider a dynamical system with the following state space representation x& (t ) = Ax (t ) + Bu (t ) ,

(1.6)

where x(t ) ∈ Rn is the state vector at time t , u (t ) ∈ Rm is the input vector at time t , and A and B are constant matrices with appropriate dimensions. Let us assume that the system is stabilizable and we have complete access to the state vector at time t , or the system allows us to get an estimate of the state vector at time t . Then, a state feedback controller can be used to control our system and get the desired performances. However, since this requires the measurement or the estimation of the state vector the controller has to the following form u (t ) = Kx (t − τ ) ,

(1.7)

where K is the controller gain to be designed. Plugging this controller into the previous state space representation we get x& (t ) = Ax (t ) + Ad x (t − τ ) ,

(1.8)

with Ad = BK . Remark 1.2: We have considered only constant time delay in the two examples. In general, time delay can also be time-varying. In some parts of the monograph, sometimes time delay will be considered time-varying. These two examples show that dynamical systems with time delay can be easily described by a state space representation, and this will be our choice for the rest of the dissertation.

3

1.2 Delay-Independent/Delay-Dependent Conditions In the stability analysis of time-delay systems, stability criteria reported so far can be classified into categories according to their dependence upon the size of delays. The criteria which do not carry information on the delays are called delay-independent criteria, and those carrying information on the delays are referred to delay-dependent criteria. Based on all results given above, we have the following definitions in detail: Definitions 1.1 (Delay-independent set) [59]: The set S ∞ defined by S ∞ = { ( A, Ad ) : ∑

asymptotic ally stabl e ∀τ ≥ 0 }

(1.9)

is called the delay-independent stability set in the parameter space ( A, Ad ) in (1.8).

S∞

If a triplet ∑ satisfies the condition ( A, Ad ) ∈ S ∞ , we shall say that the triplet is stable.

Remark 1.3: Delay-independent stability has been largely treated in the literature, see, for instance, the guided tour in [59]. Indeed, it represents the easiest robustness analysis if the delay value is not well known (uncertain models). The next step is to define the delay-dependent stability set. Definition 1.2 (Delay-dependent set) [59]: Assume that there exists a delay value τ 0 ≥ 0 such that ( A, Ad ) ∈ S (τ 0) . Then the set S τ defined by asymptotic ally satble ∀τ ∈ (τ 0 − δ 1 , τ 0 + δ 2) S τ = { ( A, Ad ) : ∑ and unstable at least in one point τ ∈ {τ 0 − δ 1 , τ 0 + δ 2} ,

(1.10) τ 0 ≥ δ 1 ≥ 0 , δ 2 > 0 , with either δ 2 infinite, or δ 1 > 0 } is called the delay-dependent stability set in the parameter space ( A, Ad ) in (1.8). If a triplet ∑ satisfies the condition ( A, Ad ) ∈ S τ , we shall say that the triplet is S τ stable.

4

1.3 Mathematical Modelling In this section we present models of some typical systems featuring time-delay behavior. These systems have the common property that the growth of some parts (future states) of the underlying model depends not only on the present state, but also on the delayed (past history)/delayed input. Therefore, we provide in the sequel some representative system models. 1.3.1

Stream Water Quality

In practice, it is important to keep water quality in streams standard. This can be measured by the concentrations of some water biochemical constituents [47]. Let z (t ) and q (t ) be the concentrations per unit volume of biological oxygen demand (BOD) and dissolved oxygen (DO), respectively, at time t . For simplicity, we consider that the stream has a constant flow rate and the water is well mixed. We further assume that there exists τ > 0 such that the (BOD, DO) concentrations entering at time t are equal to the corresponding concentrations τ time units ago. Using mass balance concentration, the growth of (BOD, DO) can be expressed as z& (t ) = − K c (t ) z (t ) + υ −1[Q e ( m + u1 (t )) + Q s z (t − τ ) − (Q s + Q e) z (t )] + ξ 1 (t ) ,

(1.11)

q& (t ) = − K d (t ) z (t ) + K r (t )[q d − q (t )] + υ −1[Q s q (t − τ ) − (Q s + Q e) q (t )] + u 2 (t ) + ξ 2 (t ) , (1.12)

where K c (t ) is the BOD decay rate, K r (t ) is the BOD re-aeration rate, K d (t ) is the BOD deoxygenation rate, q d is the DO saturation concentration, Qs is the stream flow rate, Qe is the effluent flow rate, υ is the constant volume of water in stream, m is constant, u1 (t ) and u 2 (t ) are the controls and ξ 1 (t ) and ξ 2 (t ) are random disturbances affecting the growth of BOD and DO. Using state-space format, model (1.11) and (1.12) can be cast into x& (t ) = f [ x (t ), x (t − τ ), u (t )] ,

(1.13)

which represents a nonlinear system with time-varying state delay.

5

1.3.2

Vehicle Following Systems

A simple version of vehicle following models for throttle control purpose can be described by [47] x& (t ) = v(t ) ,

(1.14)

v&(t ) = m−1[T n (t ) − T l ] ,

(1.15)

where x(t ) is the position of vehicle, v (t ) is the speed of vehicle, T n (t ) is the force produced by the vehicle engine, m is the mass of the vehicle and T l is the total load torque on the engine. For simplicity, consider that T l is constant. In terms of the throttle input u (t ) , the engine dynamics can be expressed as dynamics −1 T& n (t ) = − τ [T n (t ) + u (t )] .

(1.16)

Here, τ represents the vehicle's engine time constant when the vehicle is traveling with a speed v (t ) . Combining (1.15) and (1.16), we obtain

−1 T& n (t ) = − τ [ mv&(t ) + T l + u (t )] .

(1.17)

To proceed further, we differentiate (1.17) and set T& l = 0 to get −1 a& (t ) = − τ −1 a (t ) + ( mτ ) [u (t − h) − T l ] ,

(1.18)

where a (t ) = T& n (t ) . By incorporating the effect of actuator delay, due to fueling delay and transport factor, we express (1.14), (1.15) and (1.18) into the form ⎡ x& (t ) ⎤ ⎡0 ⎢v&(t ) ⎥ = ⎢0 ⎢ ⎥ ⎢ ⎢⎣a& (t )⎥⎦ ⎢⎣0

⎡ 0 ⎤ ⎡ x(t ) ⎤ ⎡ 0 ⎤ 0 ⎤ ⎢ ⎥ ⎢ ⎥ ⎥ 1 ⎥ ⎢⎢v(t ) ⎥⎥ + ⎢ 0 ⎥u (t − h) + ⎢ 0 ⎥Tl , ⎢− (mτ )−1⎥ 0 − τ −1⎥⎦ ⎢⎣a (t )⎥⎦ ⎢⎣(mτ )−1⎥⎦ ⎣ ⎦

1 0

(1.19)

where h is the total throttle delay. Model (1.19) represents a linear system with constant input delay.

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1.3.3

Heat Exchanger Dynamics

A different case study (always coming from chemistry) is the heat exchanger consisting of a tube carrying water placed in a current of hot air. If we assume that the velocity of the flow water, the air temperature and the inlet water temperature are constant ( w, T a and T in , respectively) , then the outlet water temperature T w can be controlled by the velocity of the hot air over the tube. The equations describe this phenomenon under the assumption that there is no mixing and axial conduction [57] ⎞ ⎛ t T w = (T in − T a ) exp⎜⎜ − ∫ h(θ )dθ ⎟⎟ + T a , ⎠ ⎝ t −L w

(1.20)

where L is the length of the tube, and h describes some heat exchange rates, assumed to be independent of the difference T in − T a . If we assume that w and h are functions of T w : w = φ (T w) , h = ϕ (T w) , by differentiating both sides in (1.20) , we obtain L ⎞ dTw ⎛ = − T w (t )⎜ ϕ (T w) − ϕ (T w (t − )) ⎟ . w ⎠ dt ⎝

(1.21)

If we assume ϕ is defined as

ϕ ( x ) = B − Ax , A, B ∈ R ,

(1.22)

equation (1.21) becomes L ⎞ dTw ⎛ = A T w ⎜ T w (t ) − T w (t − ) ⎟ . w ⎠ dt ⎝

(1.23)

A first step to analyze (1.23) is to consider the delay equation x& (t ) = αx(t )( x(t ) − x (t − τ )) ,

α ,τ ∈ R , τ > 0 .

Model (1.24) represents a nonlinear system with constant state delay.

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(1.24)

1.3.4

Electrical-Circuit Models The transients of a (lossless) LC transmission line are described by the equations

(normalized length line) ∂v ∂i + L = 0, ∂λ ∂t

(1.25a)

∂i ∂v +C = 0, ∂λ ∂t v (0, t ) = − E (t ) − R 0 i (0, t ) ,

(1.25b)

v(1, t ) = v1 (t ) , i (1, t ) = ϕ (v1 (t ), v&1 (t )) , i ( λ , 0) = i 0 ( λ ) ,

v ( λ , 0) = v 0 ( λ ) , 0 ≤ λ ≤ 1 ,

(1.25c) (1.25d) (1.25e) (1.25f)

with a given source E (t ) and a given (linear) map ϕ (acting on v1 and , eventually, on its derivatives). Thus, for example, ϕ can be described using capacitors (C1) and /or some (one-port) resistors ( R1)

ϕ (v1 (t ), v&1 (t )) = C1 v&1 (t ) + R1 v1 (t ) ,

(1.26)

(only first derivative information, the nonlinear characteristic of the tunnel diode). Such system corresponds to a tunnel diode circuit containing lossless transmission lines and it was already considered in [57]. Via the change v (t , λ ) = u1 (λ , t ) + u 2 (λ , t ) ,

i (λ , t ) =

(1.27)

C [u1 (λ , t ) − u 2 (λ , t )] , L

(1.28)

the system (1.25) can be written as ∂ u1 ∂u + ξ (λ ) 1 = 0 , ∂t ∂λ

(1.29a)

∂u ∂u 2 − ξ (λ ) 2 = 0 , ∂t ∂λ u1 (0, t ) + α 1 u 2 (0, t ) = φ1 (t , x(t )) ,

(1.29b)

u 2 (1, t ) + α 2 u1 (1, t ) = φ 2 (t , x(t )) , x& = f (t , m(t ), x (t ), u 2 (0, t ), u1 (1, t )) ,

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(1.29c) (1.29d) (1.29e)

x(0) = x0 ∈ Rn , ui (λ , 0) = γ i (λ ) , 0 ≤ λ ≤ 1 , t > 0 ,

(1.29f)

where ξ (λ ) > 0 , 0 ≤ λ ≤ 1 (normalized) and m(t ) is a control signal or a forcing term(defined by the application). By integration along the characteristics with u1 (0, t ) = η1 (t ) , u 2 (1, t ) = η 2 (t ) , the partial differential equations described above lead to the following model

⎛1 C⎞ C ⎟ v1 (t ) − 2 η1 (t − LC ) , C1 v&1 (t ) = −⎜⎜ + ⎟ L L ⎝ R1 ⎠ η1 (t ) = −

1 − R0 1 + R0

C L (t − LC ) , η2 C L

(1.30)

(1.31)

η 2 (t ) = v1 (t ) − η1 (t − LC ) .

(1.32)

We may eliminate the variable η1 from the equations, and, thus, we shall obtain ( x1 and x 2 are defined by v1 and η 2 , respectively)

x&1 (t ) = −

1 − R0

C⎞ 1⎛1 1 C ⎜ + ⎟ x1 (t ) − 2 ⎜ ⎟ L⎠ C1 ⎝ R1 C1 L 1 + R0

x2 (t ) = x1 (t ) +

1 − R0 1 + R0

C L (t − τ ) , x2 C L

C L (t − τ ) , x2 C L

(1.33)

(1.34)

where the new variable x1 and x 2 are appropriately defined, and τ = 2 LC . Using state-space format, model (1.33) and (1.34) can be cast into x&1 (t ) = A x1 (t ) + B x 2 (t − τ ) ,

(1.35)

x 2 (t ) = C x1 (t ) + D x 2 (t − τ ) .

(1.36)

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We replace (1.36) by the following d [ x 2 (t ) − D x 2 (t − τ ) − C x1 (t )] = 0 , dt

(1.37)

since x1 (t ) is differential, we may write (1.37) as follows d [ x 2 (t ) − D x 2 (t − τ )] = CA x1 (t ) + CB x 2 (t − τ ) . dt

(1.38)

Combining (1.36) and (1.38), we obtain the neutral system

d ⎛ ⎡ x1 (t ) ⎤ ⎡0 ⎜ − dt ⎜⎝ ⎢⎣ x2 (t )⎥⎦ ⎢⎣0

0 ⎤ ⎡ x1 (t − τ ) ⎤ ⎞ ⎡ A ⎟= D ⎥⎦ ⎢⎣ x2 (t − τ )⎥⎦ ⎟⎠ ⎢⎣CA

0 ⎤ ⎡ x1 (t ) ⎤ ⎡0 + 0⎥⎦ ⎢⎣ x2 (t )⎥⎦ ⎢⎣0

B ⎤ ⎡ x1 (t − τ ) ⎤ . CB ⎥⎦ ⎢⎣ x2 (t − τ )⎥⎦

(1.39)

1.3.5 Decision of Manufacturing Systems

Planning constitutes a crucial part in the decision-making of manufacturing systems. It requires careful modeling of the underlying processes of sales, inventory and production. Due to the nature of manufacturing systems, there are inherent time delays between production on one hand and sales plus inventory on the other hand. In addition, there are uncertainties in the identification of the various economic ratios and coefficients. Following [47], we consider a factory that produces two kinds of products ( j = 1, 2) sharing common resources and raw materials like color TV and black/white TV, PC and laptop computer. During the kth period (quarter or season), we let s jk : Amount of sales of product j . a jk : Advertisement cost spent for product j . c jk : Amount of inventory of product j .

p jk : Production of product j . Let x(k ) = [ s1T j sT2 j c1T j pT2 j ]T ,

(1.40)

and

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u (k ) = [ p1T, k +1 pT2, k +1 a1Tk pTa 2 k ]T .

(1.41)

The effect of advertisements on sales in the marketing process and the interlink between inventory and production in the production process (assuming one-period gestation delay) can then be expressed dynamically by a linear model of the form x( k + 1) = A0 x (k ) + Am x ( k − m + 1) + ∆ A0 x( k ) + ∆ Am x (k − m + 1) + Bu ( k ) + ∆Bu ( k − m + 1) ,

(1.42)

where ∆ A0 x ( k ) , ∆ Am x ( k − m + 1) and ∆Bu ( k − m + 1) denote, respectively, the uncertain amount of sales, inventory of product and change in advertisements cost and m ≥ 2 stands for the amount of delay between making a decision and realizing its effect on production. It is readily seen that the above model fits nicely into the discrete-time delay format.

1.4 Linear Matrix Inequality (LMI) The basic idea of the LMI method is to formulate a given problem with linear objective and linear matrix inequality (LMI) constraints [5]. Through the idea, one can specify and solve several engineering problems that can be formulated as one of the following generic LMI problems [4] 1. The feasibility problem: A linear matrix inequality has the form m

F ( z) = F 0 + ∑ zi F i < 0 ,

(1.43)

i =1

where z = ( z1 ,L, z m) ∈ Rm is the variable vector to be determined and the symmetric matrices F i ∈ Rn×n , 0 ≤ i ≤ m are given. The inequality symbol in (1.43) means that F (z ) is negative-definite, i.e., vT F ( z )v < 0 for all nonzero v ∈ R n . For example, a linear system with the following dynamic x& (t ) = Ax (t ) ,

(1.44)

where x(t ) ∈ R2 and A ∈ R 2×2 , is stable if and only if there exists a symmetric and positive-definite matrix P > 0 such that

11

T (1.45) A P + PA < 0 . This problem, in fact, can be solved using the LMI toolbox [14]. Let us now see how we

⎡a1 a2 ⎤ can put this problem in the form of (1.43) For this purpose, let A = ⎢ ⎥ and a 3⎦ ⎣a 2 ⎡ z1 z2 ⎤ suppose that P = ⎢ ⎥ , where z1 , z 2 and z3 are design parameters. Then, z 3⎦ ⎣z2 ⎡2 a1 T A P + PA = z1 ⎢ ⎣a 2

⎡ 2 a2 a2⎤ + z2 ⎢ ⎥ 0⎦ ⎣a1 + a3

⎡0 a1 + a3⎤ + z3 ⎢ ⎥ 2 a2 ⎦ ⎣a 2

a2⎤ . 2 a3 ⎥⎦

(1.46)

Therefore, (1.46) is a standard LMI feasibility problem that can be solved using the function “feasp＂of the LMI toolbox. A typical situation for the feasibility problem (FEASP) is the stability test for dynamical systems. In fact, based on control theory a system with the (1.44) is stable if and only if there exists a symmetric and positive-definite matrix P > 0 such that (1.45) is satisfied. The goal in this case is to find a matrix P > 0 such that the inequality (1.45) is satisfied. The “feasp＂can be used to solve this problem. 2. The linear optimization problem: Minimization of a linear objective under LMI constraints (MINCX) is another interesting problem that we have used extensively. The MINCX problem is stated as follows:

minimize

T c x

subject to F ( x) < 0 ,

(1.47)

where c ∈ R m is a given vector. This problem can be solved using “mincx＂ of LMI toolbox. To provide an example using “mincx＂, let us consider an H 2 control problem. For this purpose let us assume the system dynamics are given by x& (t ) = Ax (t ) + Bw(t ) ,

(1.48)

y (t ) = Cx (t ) ,

(1.49)

where w(t ) is a white noise disturbance with unit covariance. Suppose that the H 2 performance is defined by 2 ⎛1 t ⎞ H 2 = lim E ⎜ ∫ yT ( s ) y ( s )ds ⎟ . t →∞ ⎝t 0 ⎠

(1.50)

12

Then, it can be shown that the solution to this problem is given by [5] H 2 = min{tr (CP C T : AP + P AT + B BT < 0}. 2

(1.51)

Obviously, this optimization problem is equivalent to minimizing tr (Q ) subject to AP + P AT + B BT < 0 ,

(1.52)

CP C T ≤ Q .

(1.53)

Lemma 1.1: (Schur Complement) Given constant matrices Ω1 , Ω2 and Ω3 , then T −1 Ω1 + Ω3 Ω2 Ω3 < 0 if and only if

⎡Ω1 ⎢ ⎣ Ω3

T ⎡− Ω2 Ω3 ⎤ ⎥ < 0 or ⎢ T − Ω2 ⎦ ⎣ Ω3

Ω3 ⎤ ⎥ < 0, Ω1 ⎦

(1.54)

where Ω1 = Ω1T > 0 and Ω2 = ΩT2 > 0 . Using Lemma 1.1, (1.53) is equivalent to

⎡− Q ⎢ T ⎣P C

CP ⎤ ⎥ < 0. −P ⎦

(1.55)

3. The generalized eigenvalue minimization problem: The generalized eigenvalue minimization problem (GEVP) is the third interesting problem used extensively. This problem is stated as follows: minimize λ

subject to F 1 ( x ) < λ F 2 ( x ) ,

(1.56)

where F 1 ( x) and F 2 ( x ) are two matrices of form (1.44). The GEVP is quasiconvex with respect to the design parameters x and λ , which can be solved by using “gevp＂ of the LMI toolbox. To provide an example using “gevp＂, the decay rate of (1.45) is defined as the largest γ such that lim eγt x(t ) = 0 . Let us consider a Lyapunov function t →∞

candidate V ( x(t )) = xT (t ) Px(t ) . If we can establish that

13

dV ( x (t )) ≤ −2γV ( x (t )) dt

(1.57)

holds for all trajectories, then the decay rate of system (1.45) is at least γ . Note that Eq. (1.57) holds if and only if T A P + PA + 2γP ≤ 0 ,

(1.58)

we conclude that the largest lower bound on the decay rate can be found by solving the GEVP in P and γ maximize γ subject to P > 0 and AT P + PA + 2γP ≤ 0 .

(1.59)

To solve this optimization problem, let us rewrite (1.58) as 2P ≤

1

γ

(− PA − AT P ) .

(1.60)

Note that most of stable problems using LMIs proposed in this dissertation are FEASP. The above framework is particularly attractive for the following reasons [57]. 1. Efficient numerical solution: Optimization problems can be solved very efficiently using recent interior-point methods for convex optimization (the global optimum is found in polynomial time). This brings a numerical solution to problems when no analytical or closed-form solution is known. 2. Robustness against uncertainty: The approach is very well suited for problems with uncertainty in the data. Based on a deterministic description of uncertainty (with detailed structure and hard bounds), a systematic procedure enables us to formulate an optimization problem that yields a robust solution. This statement has implications for a wide scope of engineering problems, where measurements, modeling errors, etc., are often present. 3. Multicriteria problems: The approach enables us to impose many different (possibly conflicting) specifications in the design process, allowing us to explore trade-offs and analyze limits of performance and feasibility. This offers a drastic advantage over design methods that rely on a single criterion deemed to reflect all design constraints, the choice of a relevant criterion is sometimes a nontrivial task.

14

4. Wide applicability: The techniques used in the approach are relevant far beyond control and estimation. This opens exciting avenues of research where seemingly very different problems are analyzed and solved in a unified framework. For example, the method known in LMI-based control can be successfully applied in combinatorial optimization, leading to efficient relaxations of hard problems.

1.5 Contributions of the Dissertation The contributions of this dissertation are summarized as follows. 1. The new stability criteria for a class of uncertain linear time-delay systems with time-invariant/ time-varying delays are introduced. Via the state transformation matrix, simple and improved delay-dependent robust stability criteria, which are given in terms of quadratic forms of state and LMIs, are derived. 2. We have proposed the stability criterion for cellular neural networks with time-varying delay by linear matrix inequalities (LMIs). A sufficient condition related to the exponentially asymptotic delay-dependent stability for cellular neural networks with delay (DCNNs) is given in a new direction. 3. For the filtering design, asymptotically stabilizing Kalman filtering and H ∞ filtering approaches are presented for the uncertain interval system with dependence of delay in terms of LMIs. 4. For the robust H ∞ tracking control, the proposed tracker obtained in terms of LMIs guarantees the stability of closed-loop system and makes the output to approach the command reference input with a specified H ∞ performance. 5. The stabilization for a class of two-dimensional (2-D) discrete state-delayed systems described by the Roesser model is presented. Our attention is focused on the design of 2-D dynamic output feedback controller such that the closed-loop system is designed to approach asymptotic stability and a specified H ∞ performance using a linear matrix

15

inequality approach.

6. The robust H ∞ filtering for a class of two-dimensional (2-D) discrete state-delayed systems described by the Roesser model is proposed. Our goal is to design a 2-D linear filter that ensures both the asymptotic stability and a specified H ∞ performance of the filtering error dynamics using a linear matrix inequality approach. 7. The stability problem and the stabilization problem are solved by means of LMIs approaches for a class of uncertain stochastic systems with time delay in the state. Moreover, we propose the robust stabilization for a class of uncertain neutral stochastic interval systems. Finally, we present the robust H ∞ control for a class of neutral stochastic nonlinear systems.

1.6 Brief Sketch of the Contents This dissertation is organized as follows. In Chapter 2, we have established delay-dependent sufficient conditions for a class uncertain time-delay systems. In Chapter 3, delay-dependent sufficient conditions are established for Kalman filtering and H ∞ filtering approaches for a class of uncertain interval systems with time delay via LMIs. In Chapter 4, we established delay-dependent sufficient conditions to design a robust controller which guarantees the stability of closed-loop system and makes the output to approach the command reference input with a specified H ∞ performance. We study the 2-D dynamic output feedback H ∞ control problem in Chapter 5. Based on the 2-D bounded realness property, solvability conditions for the H ∞ control of 2-D discrete systems described by the Roesser model are given in terms of LMIs. In Chapter 6, The design of a linear 2-D filter that guarantees a prescribed level of a noise attenuation for any energy bounded noise input are introduced. We discussed the robust delay-dependent stability for a class of uncertain stochastic systems with time delays and for a class of uncertain neutral stochastic systems with time delay in Chapter 7. Finally, conclusions are included in Chapter 8.

16

Chapter 2 Robust Stability and Robust Stabilization for Systems with Time Delay

This chapter first deals with the robust stability for a class of uncertain time-delay systems with time-invariant and time-varying multiple state delays. Moreover, we extend the proposed theory to discuss the robust stabilization for a class of uncertain discrete time systems with time delay. Finally, the stability for cellular neural networks with time-varying delay is introduced. The results are given in terms of linear matrix inequalities. Eight examples are provided to demonstrate the effectiveness of the proposed approach [44, 74, 84, 98].

2.1 Introduction Dynamic systems with time-delay are common in chemical processes and long transmission lines in pneumatic, hydraulic, and rolling mill systems. A major subject in the analysis of linear dynamical systems with time-delay is related to the stability. The criteria for asymptotic stability of such systems can be classified as delay-independent, which do not include any information on the size of delay, for example [73, 75], and delay-dependent, obtained via Lyapunov equation, which include such information on the size of delay, for example [58, 73]. Lyapunov-Razumikhin function approach is employed to obtain the results [7, 62, 73] which are not analytical criteria owing to no relation to the system parameters and are somewhat conservative when delays are small. The stability criteria have been proposed [7, 31, 39] via LMI approach. Niculescu [58] proposed a new H ∞ memoryless control and α stability constrained for time delay systems via Lyapunov-Krasovskii functional and LMI approach. However, if α = 0 , the stability criterion becomes delay independent. The delay-dependent robust stability criteria, which are included some supplementary conditions, are developed in [39] by Lyapunov function

17

method, Leibniz-Newton formula and matrix norm. The delay-dependent criteria have also been addressed in the time-varying delay case in [74]. For discrete time-delay systems, we refer the literature [16, 35, 92]. In [16], necessary and sufficient conditions with delay-free have been obtained through different LMI techniques and procedures for the construction of stabilizing controllers proposed. Xu et al. [92] presented necessary and sufficient conditions with state delay which have been obtained through different LMI approaches for the construction of stabilizing controllers proposed, but it has not been presented delay-dependent conditions yet. The delay-dependent stability for discrete-time systems with time delay is considered in [35]. However, the robust stability and stabilization problems are not investigated in [35]. With regard to cellular neural networks with delay (DCNNs) proposed in [65] have found applications in many areas including classification of patterns and processing of moving images. When used as a pattern classifier, the DCNN is required to possess a unique and globally asymptotically stable equilibrium point [64]. Liao and Wang [37] proposed a sufficient condition for global stability of cellular neural networks with delay but the output of the cell is a piecewise linear function and the time delay is delay-independent. Zhang et al. [100] introduced a wider adaptive range which the output of the cell was not piecewise linear function, and the results do not depend on the size of the time delay, and the time-delay terms of cellular neural networks are constant. This chapter deals with robust stability criteria for a class of uncertain time-delay systems. Based on Lyapunov –Krasovskii functionals combined with LMI techniques, we obtain simple and improved delay-dependent stability criteria. Our results, which are given in terms of quadratic forms of state and LMI, are more informative. The new criteria do not include any supplementary conditions on the system matrices and are less conservative than existing stability criteria. Moreover, we extend the proposed theory to study the robust stabilization for a class of discrete time-delay systems and the exponentially asymptotic delay-dependent stability for cellular neural networks with time-varying delays.

2.2 Delay-Dependent Stability In this section, we provide new stability criteria for a class of uncertain linear time-delay systems with time-invariant delays. Based on Lyapunov-Krasovskii functionals combining with LMI techniques, simple and improved delay-dependent robust stability criteria, which are given in terms of quadratic forms of state and LMI, are derived. Our results shown by two examples are less conservative than the existing stability criteria.

18

2.2.1 System Description Consider the following uncertain time-delay systems described by k

x& (t ) = ( A0 + ∆ A0 (t )) x(t ) + ∑ ( Ai + ∆ Ai (t )) x(t − hi ) ,

(2.1)

x (t ) = θ (t ) , ∀t ∈ [ − h,0] ,

(2.2)

i =1

where x(t ) ∈ Rn is the state vector and A j , j = 0, 1,..., k are known constant matrices with appropriate dimensions, ∆ A j (t ) , j = 0, 1,..., k are matrix functions representing the uncertainties in the matrices A j , j = 0, 1,..., k

∆ A j (t ) = D j F j (t ) E j , j = 0, 1,..., k ,

(2.3)

where F j (t ) ∈ R k j× g j are unknown real time-varying matrices with Lebesgue measure elements bounded by T F j (t ) F j (t ) ≤ I , ∀t , j = 0, 1,..., k ,

(2.4)

D j and E j are known real constant matrices, hi , i = 1,..., k , are the unknown constant delay terms, but bounded 0 ≤ hi ≤ h , θ (t ) is a smooth vector-valued initial function in

− h ≤ t ≤ 0 . The main aim of this section is to develop delay-dependent conditions for robust stability of the uncertain time-delay system (2.1). More specifically, our objective is to determine bounds for the time-delay by using different arrangements of Lyapunov-Krasovskii functionals and LMI methods. The following matrix inequality will be essential for the proofs. Lemma 2.1: Let D , E and F be real matrices of appropriate dimensions with F ≤ 1 , then we have the following:

DFE + E T F T DT ≤ ε −1 D DT + ε E T E , for any scalar ε > 0 .

19

(2.5)

2.2.2 Main Results In this subsection, we describe our method for determining the robust stability criteria of uncertain time-delay system (2.1)-(2.2). The main results are given in the following Theorems. Theorem 2.1: Consider the uncertain delay system (2.1) with ∆ A j (t ) = D j F j (t ) E j , j = 0 ,1,...,k and F j ≤ 1 . For all delays hi ∈ [0, h ] are given constants, this system is

robustly stable if there exist symmetric and positive-definite matrices P > 0 , Ri > 0 , , i = 1,...,k and scalars ε j > 0 , j = 0,1,..., k such that the following LMI ⎡X1 ⎢ T ⎢M 1 ⎢ M T2 ⎢ T ⎣⎢ M 3

M1

M2 0

− N1 0

− N2

0

0

M3 ⎤ ⎥ 0 ⎥ 0 , where k

X 1 = A0 P + P A0 + 2αP + ∑ Ri + ε 0 E 0 E 0 , T

T

i =1

αh αh M 1 = [P D1 . . . P Dk ] , N 1 = diag [e−2αh ε 1 ,..., e−2αh ε k ] , M 2 = [e P A1 . . . e P Ak ],

T T N 2 = diag [R1 − ε 1 E1 E1 , L , R k − ε k E k E k ] , M 3 = P D0 and N 3 = ε 0 .

Proof: Via the state transformation matrix z (t ) = eα t x(t ), t > 0 ,

(2.7)

where α > 0 is stability degree, to transform (2.1) into z& (t ) = α eα t x(t ) + eα t x& (t ) k

= α z(t) + eα t [( A0 + ∆ A0) x(t ) + ∑ ( Ai + ∆ Ai ) x(t − hi )] i =1

20

k

= ( A0 + α I + ∆ A0) z (t ) + ∑ eα hi ( Ai + ∆ Ai ) z (t − hi ) . i =1

(2.8)

First let us consider the time-delay system of (2.8), using the Lyapunov-Krasovskii functional candidate in the following form [31], we can write t

k

T ∫ z (θ ) Ri z (θ )dθ .

V ( z (t ), z (t − hi )) = zT (t ) Pz (t ) + ∑

i =1 t − hi

(2.9)

The time derivative of (2.9) along the trajectory of (2.8) is given by k

k

i =1

i =1

V& = z&T (t ) Pz (t ) + zT (t ) Pz& (t ) + ∑ zT (t ) Ri z (t ) − ∑ zT (t − hi ) Ri z (t − hi ).

(2.10)

Using Lemma 2.1, we have

V& ≤ X T SX < 0,

(2.11)

where X = [ zT (t ) zT (t − h1) ... zT (t − h k ) ]T ,

(2.12)

⎡M S=⎢ T ⎣G

(2.13)

G⎤ ⎥, L⎦

where T

k

M = ( A0 +α I) P + P( A0 + α I ) + ∑ Ri + ε 0−1 P D0 DT0 P + ε 0 E T0 E 0 i =1

k

+ ∑ ε i−1 e2α hiP Di DTi P, i =1

G = [eα h1 P A1 L eα h k P Ak ] and L = − diag [R1 − ε 1 E1T E1 , L , R k − ε k E Tk E k ] . Finally, using Schur complements, with some efforts, we can show that (2.6) guarantees the negativeness of V& whenever (2.12) is not zero, which immediately implies the asymptotic stability of the system (2.1). Remark 2.1: If we let k = 1 , then the system (2.1) is a single state delay, and (2.6) can be transformed into (2.14) as LMI problem on a single state delay,

21

⎡X ⎢ T ⎢M 1 ⎢ T ⎢M 2 T ⎣⎢ M 3

M1 − N1

M3 ⎤ ⎥ 0 ⎥ 0 , R i > 0 , Qi 0 > 0 , i = 1,...,k and scalars ε j > 0 , j = 0 ,...,k such that the following LMI ⎡ Xˆ ⎢ T ⎢ Mˆ 1 ⎢ T ⎢ Mˆ 2 ⎢ ˆT ⎣M 3

Mˆ 1 − Nˆ 1

Mˆ 2 0

0

− Nˆ 2

0

0

Mˆ 3 ⎤ ⎥ 0 ⎥ ⎥ 0 , where k

k

i =1

i =1

Xˆ = AT0 P + P A0 + 2αP + ∑ Ri + ε 0 E T0 E 0 + ∑ h Qio, αh αh −2αh −2αh Mˆ 1 = [P D1L P D k ] , Nˆ 1 = diag [e ε 1 , L , e ε k ] , Mˆ 2 = [e P A1L e P Ak ] , T T Nˆ 2 = diag [R1 − ε 1 E1 E1 , L , R k − ε k E k E k ] , Mˆ 3 = P D0 and Nˆ 3 = ε 0 .

Proof: Consider the time-delay system of (2.1), using the improved Lyapunov-Krasovskii functional candidate in the following form, we can write

22

k

V ( z (t ), z (t − hi )) = zT (t ) Pz (t ) + ∑

t

T ∫ z (θ ) Ri z (θ )dθ

i =1 t − h i

k

0

t

+ ∑ ∫ ∫ zT ( ρ ) Qi0 z ( ρ )dρ dβ . i =1 − h i t + β

(2.16)

The time derivative of (2.16) along the trajectory of (2.8) is given by k

k

i =1

i =1

V& = z&T (t ) Pz (t ) + zT (t ) Pz& (t ) + ∑ zT (t ) Ri z (t ) − ∑ zT (t − hi ) Ri z (t − hi ) k

k

t

+ ∑ zT (t ) hi Qi 0z (t ) − ∑ ∫ zT ( β ) Qi 0 z ( β )dβ . i =1

i =1 t − h i

(2.17)

Using Lemma 2.1, we obtain t

k

V& ≤ X T S X − ∑ ∫ zT ( β ) Qi 0 z ( β )dβ .

(2.18)

i =1 t − h i

k

t

Since the last term ∑ ∫ zT ( β ) Q i 0 z ( β )dβ in (2.18) is positive-definite [33], one has i =1 t − hi

V& ≤ X T S X < 0,

(2.19)

where X = [ zT (t ) zT (t − h1) ... zT (t − h k ) ]T ,

(2.20)

⎡M S =⎢ T ⎣G

(2.21)

G⎤ ⎥, L⎦

with k

T

M = ( A0 +α I) P + P( A0 + α I ) + ∑ Ri + ε 0−1 P D0 DT0 P + ε 0 E T0 E 0 i =1

k

k

i =1

i =1

+ ∑ ε i−1 e2α hi P Di DTi P + ∑ hi Qi 0 , G = [eα h1 P A1L eα h k P Ak ] and L = − diag [R1 − ε 1 E1T E1 , L , R k − ε k ETk E k ] .

23

Finally, using Schur complements, with some efforts, we can show that (2.15) guarantees the negativeness of V& whenever (2.20) is not zero, which immediately implies the asymptotic stability of the system (2.1). Remark 2.2: If we let k = 1 , then the system (2.1) is a single state delay, and (2.15) can be transformed into (2.22) as LMI problem on a single state delay, ⎡Ω ⎢ T ⎢Ω1 ⎢ΩT2 ⎢ T ⎢⎣Ω3

Ω1 ~ −N 1 0 0

Ω2 0 ~ −N 2 0

Ω3 ⎤ ⎥ 0 ⎥ 0 , Ri > 0 , i = 1,L ,k and scalars ε j > 0 , j = 0 ,1,L ,k such that the following LMI ⎡ X 11 ⎢ T ⎢ M 11 ⎢ M T22 ⎢ T ⎢⎣ M 33

M 11

M 22 0

− N 11 0

− N 22

0

0

M 33 ⎤ ⎥ 0 ⎥ 0 and d j < 1 , j = 0 ,1,L , k , where k

X 11 = A0 P + P A0 + 2αP + ∑ Ri + ε 0 E 0 E 0 , T

T

i =1

αh αh M 11 = [P D1L P Dk ] , N 11 = diag [e2 αh ε1 ,L , e2αh ε k ] , M 22 = [e P A1L e P Ak ], T T N 22 = diag [(1− d 1) R1 − ε1 E1 E1 L (1 − d k ) R k − ε k E k E k ] , M 33 = P D 0 and N 33 = ε 0 .

The proof of Theorem 2.3 is similar to Theorem 2.1. Theorem 2.4: Consider the uncertain delay system (2.23). For all delays hi (t ) ∈ [0, h] given constants, this system is robustly stable if there exist symmetric and positive-definite matrices P > 0 , R i > 0 , Qi 0 > 0 , i = 1,..., k and scalars ε j > 0 , j = 0 ,..., k such that the following LMI

25

⎡ X~11 ⎢ ~T ⎢ M 11 ⎢ ~T ⎢ M 22 ~T ⎢M ⎣ 33

~ M 11 ~ −N 11 0 0

~ M 22 0 ~ −N 22 0

~ M 33 ⎤ ⎥ 0 ⎥ 0 and d j < 1 , j = 0 ,1,L , k , where k k T T ~ X 11 = A0 P + P A0 + 2αP + ∑ Ri + ε 0 E 0 E 0 + ∑ h Qio, i =1

i =1

~ ~ 2 αh 2 αh M 11 = [P D1L P D k ] , N 11 = diag [e ε1L e ε k ], ~ αh αh T T ~ M 22 = [e P A1L e P Ak ], N 22 = diag [(1− d 1) R1 − ε1 E1 E1 ,L , (1 − d k ) R k − ε k E k E k ] ,

~ ~ M 33 = P D0 and N 33 = ε 0 , The proof of Theorem 2.4 is similar to Theorem 2.2. 2.2.3 Examples

Example 2.1: Consider the uncertain time-delay system 0 0 ⎡− 1 + 0.5 cos t ⎡− 2 + 0.5 cos t ⎤ ⎤ x(t − h) . x& (t ) = ⎢ x +⎢ ⎥ −1 − 1 + 0.6sint ⎥⎦ 0 − 2.5 + 0.6sint ⎦ ⎣ ⎣

(2.29)

The uncertainty can be described by ∆ A0 = D 0 F (t ) E 0 and ∆ A1 = D1 F (t ) E1 with ⎡ 0.5 D0 = E 0 = ⎢ 0 ⎢⎣

⎡ 0.5 0 ⎤ ⎥ , D1 = E1 = ⎢ 0.6 ⎥⎦ 0 ⎢⎣

0 ⎤ ⎡cos t ⎥ , F (t ) = ⎢ 0.6 ⎥⎦ ⎣ 0

0 ⎤ . sint ⎥⎦

Applying Theorem 2.1 to this uncertain time-delay system, it is found, using the software package LMI Lab, that this system is robustly stable for any constant time-delay satisfying 0 ≤ h ≤ 10.6724 . The delay bounds for guaranteeing asymptotic stability of the system given in [7] is 0 ≤ h ≤ 0.8570 and in [73] is 0 ≤ h ≤ 0.1381 . The result of our approach is less conservative than the results in [7] and [73]. Finally, by the trial and error,

26

it is found that this system is stable for the time-delay satisfying 0 ≤ h ≤ 11.3762 . So, the admissible maximal bounded error is 0.7038 . Example 2.2: Consider the following uncertain time delay system: x& (t ) = [ A0 + ∆ A0 (t )] x (t ) + [ A1 + ∆ A1 (t )] x (t − h) ,

(2.30)

where 0⎤ ⎡− 2 , A0 = ⎢ − 3⎥⎦ ⎣ 1 matrices satisfying

⎡− 1.4 A1 = ⎢ ⎣− 0.8

0 ⎤ − 1.5⎥⎦

and ∆ A0 (t ) and

∆ A1 (t )

are uncertain

∆ A0 (t ) ≤ 0.2 , ∆ A1 (t ) ≤ 0.2 , ∀t ,

(2.31)

0 ⎤ ⎡1 , E 0 = E1 = ⎢ ⎥ 0.2⎦ ⎣0

(2.32)

⎡0.2 D0 = D1 = ⎢ ⎣0

0⎤ . 1⎥⎦

Applying Theorem 2.2 to this uncertain time-delay system, it is found, using the software package LMI Lab., that this system is robustly stable for any constant time-delay satisfying 0 ≤ h ≤ 1.3686 . We note that the result of [41] guarantees the robust stability of (2.30) when 0 ≤ h ≤ 0.3142 , whereas by the method of [72] the time-delay is 0 ≤ h ≤ 0.2117 and [43] is 0 ≤ h ≤ 0.3025 . This example shows again that our method is an improvement of the previous results of [43], [41] and [72]. Finally, it is found, by the trial and error, that this system is stable for the constant time-delay satisfying 0 ≤ h ≤ 1.9000 . So, the admissible maximal bounded error is 0.5314 .

Example 2. 3: Consider the uncertain time-delay system ⎡− 2 + 0.5 cos t x& (t ) = ⎢ 0 ⎣ (2.33)

0 ⎤ ⎡− 1 + 0.5 cos t x+⎢ ⎥ − 2.5 + 0.6 sin t ⎦ ⎣ −1

0 ⎤ x(t − h(t )) − 1 + 0.6 sin t ⎥⎦

The uncertainty can be described by ∆ A0 = D 0 F (t ) E 0 and ∆ A1 = D1 F (t ) E1 with ⎡ sqrt (0.5) 0 ⎤ ⎡cos t 0 ⎤ ⎡ sqrt (0.5) 0 ⎤ , D1 = E1 = ⎢ , F (t ) = ⎢ D0 = E 0 = ⎢ ⎥ ⎥ . (2.34) ⎥ 0 sqrt (0.6)⎦ 0 sqrt (0.6)⎦ ⎣ ⎣ 0 sin t ⎦ ⎣

27

.

Applying Theorem 2.3 to this uncertain time-delay system, it is found, using the software package LMI Lab, that this system is robustly stable for time-varying time-delay satisfying 0 ≤ h(t ) ≤ 11.0136 . The delay bound for guaranteeing asymptotic stability of the system given in [7] is 0 ≤ h(t ) ≤ 0.8570 and [41] is 0 ≤ h(t ) ≤ 0.1035 and [73] is 0 ≤ h(t ) ≤ 0.1381 . Therefore, our result is less conservative than the results in the above mentioned literature [7], [41], [73].

Example 2. 4: Consider the following uncertain time delay system provided in [72]: x& (t ) = [ A0 + ∆ A0 (t )] x (t ) + [ A1 + ∆ A1 (t )] x (t − h(t )) ,

(2.35)

where ⎡− 2 A0 = ⎢ ⎣ 1

0⎤ 0⎤ ⎡− 1 , A1 = ⎢ ⎥ ⎥ −3⎦ ⎣− 0.8 − 1⎦

and ∆ A0 (t ) and ∆ A1 (t ) are uncertain matrices satisfying

∆ A0 (t ) ≤ 0.2 , ∆ A1 (t ) ≤ 0.2 , ∀t . ⎡0.2 D0 = D1 = ⎢ ⎣0

0⎤ ⎡1 , E 0 = E1 = ⎢ ⎥ 0.2⎦ ⎣0

(2.36)

0⎤ . 1⎥⎦

(2.37)

Applying Theorem 2.4 to this uncertain time-delay system, it is found, using the software package LMI Lab, that this system is robustly stable for time-varying time-delay satisfying 0 ≤ h(t ) ≤ 3.0369 . We note that the result of [41] guarantees the robust stability of (2.35) when 0 ≤ h(t ) ≤ 0.0433 , whereas by the method of [72] the time-delay is 0 ≤ h(t ) ≤ 0.3188 . This example shows again that our method is an improvement of the previous results [41], [72].

28

2.3 Delay-Dependent Stabilization This section develops delay-dependent stability and stabilization conditions for a class of uncertain discrete time systems with time delay. In the stabilization case, we are seeking a state feedback control law u (k ) that asymptotically stabilizes the closed-loop system. 2.3.1 Preliminaries

Consider the uncertain linear discrete time system x( k + 1) = ( A + ∆A) x (k ) + ( Ad + ∆ Ad ) x ( k − h) + ( B + ∆B )u ( k ) ,

(2.38)

x(k ) = θ (k ) ,

(2.39)

∀k ∈ [−h , 0] ,

where x(k ) ∈ Rn is the state vector, u (k ) ∈ Rm is the control input, A , Ad and B are known constant matrices with appropriate dimensions, ∆A , ∆ Ad and ∆B are unknown matrices with appropriate dimensions which represent the system uncertainties. h is bounded delay time satisfying 0 ≤ h ≤ h . θ ( k ) is any given initial data of x(k ) = θ (k ) ∀k ∈ [−h , 0] . Here, we assume that the uncertainties can be described as follows

[∆A

∆ Ad

∆B ] = MF (k )[N 1

N2

N 3] ,

(2.40)

where M , N 1 , N 2 and N 3 are known real constant matrices and F ( k ) is an unknown real-valued matrix satisfying T F (k ) F (k ) ≤ I ,

∀k > 0 .

It is assumed that all the elements of

(2.41)

F ( k ) are Lebesgue measurable.

∆A , ∆ Ad and

∆B are said to be admissible if both (2.40) and (2.41) hold.

Lemma 2.2: Let Φ be a symmetric matrix and matrices M , E be given with appropriate dimensions, and F satisfies F T F ≤ I . Then, we have Φ + MFE + E T F T M T < 0 holds if and only if there exists a scalar ε > 0 such that

29

Φ + εM M T + ε −1 ET E < 0 . 2.3.2 Main Results

(2.42)

In this subsection, we first state robust stability criteria for the uncertain discrete time system with u ( k ) ≡ 0 . 2.3.2.1 Robust Stability

We propose the robust stability for uncertain discrete time system as follows. Theorem 2.5: Given a time-delay h satisfying 0 ≤ h ≤ h , system (2.38) with u ( k ) ≡ 0 ~ is robustly stable, if there exist symmetric positive-definite matrices X > 0 , S > 0 and scalar ε > 0 such that the following LMI ~ ⎡−X +S ⎢ 0 ⎢ ⎢ α ⎢ e AX ⎢⎣ eα ( 2 h +1) N 1 X

0 ~ −S α ( h +1)

Ad X α ( 3 h +1) e N2 X e

X N 1T ⎤ ⎥ α ( h +1) α ( 3 h +1) X ATd X N T2 ⎥ e e ⎥ 0 . Proof: Let u ( k ) ≡ 0 , using the state transformation matrix

z (k ) = eα k x(k ), k > 0 ,

(2.44)

where α > 0 is stability degree, to transform (2.38) into z (k + 1) = eα ( k +1) x(k + 1) = eα ( k +1) [( A + ∆A) x(k ) + ( Ad + ∆ Ad ) x(k − h)] = eα ( A + ∆A) z (k ) + eα ( h +1) ( Ad + ∆ Ad ) z (k − h) = eα A z (k ) + eα ( h +1) A d z (k − h) ,

(2.45)

where A = A + ∆A and A d = Ad + ∆ Ad . First let us consider the time-delay system of (2.38), using the Lyapunov-Krasovskii functional candidate in the following form, we can write h

V = zT (k ) Pz (k ) + ∑ xT (k − i ) Sx(k − i ) , i =1

30

(2.46)

along trajectories of (2.45) and making the difference of (2.47), the generator ∆V for the evolution of V is therefore given by ∆V = V ( k + 1) − V ( k )

= zT (k + 1) P z (k + 1) − zT (k ) P z (k ) + zT (k ) Sz (k ) − zT (k − h) Sz (k − h) = zT (k )(− P + S + e2α A T PA ) z (k ) + eα eα ( h +1) zT (k ) A T P A d z (k − h) + eα eα ( h +1) zT (k − h) A Td PA z (k ) + e2α ( h +1) zT (k − h)( A Td P A d − S ) z (k − h) .

(2.47)

From (2.47), one has

∆V = π T Σπ ,

(2.48)

where

π ( k ) = [ zT ( k ) zT ( k − h) ]T , T A P Ad ⎤ ⎥, T 2α ( h +1) P S − ⎥⎦ e Ad Ad

⎡ − P + S + e 2α A T P A Σ=⎢ T α α ( h +1) ⎢⎣ e e A d PA

α

α ( h +1)

e e

(2.49)

The requirement ∆V < 0 , ∀ π ≠ 0 implies Σ < 0 . Using Lemma 1.1, it is expressed as

⎡− P + S ⎢ 0 ⎢ ⎢ αA ⎣ e

e

T

α ( h +1)

−Q α ( h +1)

⎤ ⎥ e A ⎥ < 0. − P −1 ⎥⎦ α

e A

0 Ad

T d

(2.50)

We then extend equation (2.50) as α T ⎡− P + S 0 e A ⎤ ⎡0 ⎤ ⎢ ⎢ ⎥ α ( h+1) α ( h +1) T ⎥ −Q 0 N2 e Ad ⎥ + ⎢0 ⎥ F (k )[eα N 1 e ⎢ ⎢ eα A eα ( h +1) A − P −1 ⎥⎦ ⎢⎣ M ⎥⎦ d ⎣

⎤ ⎡eα N 1T ⎥ ⎢ + ⎢eα ( h +1) N T2 ⎥ F T (k )[0 ⎥ ⎢ ⎥⎦ ⎢⎣ 0

0

M

T

]< 0 .

0]

(2.51)

31

Based on Lemma 2.2, equation (2.51) holds if and only if ⎡− P + S ⎢ 0 ⎢ ⎢ eα A ⎢ α ( 2 h +1) N1 ⎣⎢ e

α

−S e

α ( h +1)

Ad

α ( 3 h +1)

N2

e

α ( 2 h +1)

N1 ⎤ α ( h +1) T α ( 3 h +1) T⎥ e e N2⎥ Ad < 0. ⎥ 0 − P −1 + ε e− 2αh M M T ⎥ 0 −ε ⎦⎥ T

e A

0

e

T

(2.52)

. Pre- and Post-multiplying both sides of (2.52) by diag ( X , X , I , I ) and denoting X = P −1 ~ and S = XSX yield (2.43). With some efforts, we are able to show that (2.43) guarantees the negativeness, which immediately implies the robust stability of the discrete time systems. 2.3.2.2 Robust Stabilization

Given the system (2.38) with the control u ( k ) = Fx (k ) , where the matrix F ∈ R m×n , using the state transformation matrix (2.44), system (2.38) becomes

z (k + 1) = eα A f z (k ) + eα ( h+1) A d z (k − h) ,

(2.53)

where A f = ( A + ∆A) + ( B + ∆B) F .

Applying Theorem 2.5 to the above matrices, therefore, we are able to find the quadratic term (2.54) ∆V = π T Σ1 π , where ⎡− P + S + e2α A Tf P A f Σ1 = ⎢ T α α ( h +1) ⎢⎣ e e Ad P A f

T A f P Ad ⎤ ⎥, T 2α ( h +1) e A d P A d − S ⎥⎦

α α ( h +1)

e e

Similarly, we can obtain

32

(2.55)

⎡ −P+S ⎢ 0 ⎢ ⎢ α ( A + BF ) ⎢ e ⎢⎣ eα (2 h +1) ( N 1 + N 3 F )

α e ( A + BF ) α ( h +1)

−S α ( h +1)

Ad

α ( 3 h +1)

N2

e e

( N 1 + N 3 F )T ⎤ ⎥ α ( 3 h +1) T e N2 ⎥ < 0 , (2.56) ⎥ 0 ⎥ ⎥⎦ −ε

α (2 h +1)

T

0

e

T

e Ad − 2αh −1 − P +ε e M MT 0

Pre- and Post-multiplying both sides of (2.56) by diag ( X , X , I , I ) and denoting X = P −1 , ~ S = XSX and Y = FX yield ~ ⎡ −X +S ⎢ 0 ⎢ ⎢ α ( AX + BY ) ⎢e ⎢⎣ eα ( 2 h +1) ( N 1 X + N 3 Y )

0 ~ −S α ( h +1)

Ad X α ( 3 h +1) e N2 X e

α T T T e (X A + Y B )

α ( 2 h +1)

e

α ( h +1) X ATd e − ( X − ε e− 2αh M M T ) 0

( X N 1T + Y T N T3 )⎤ ⎥ α (3 h +1) X N T2 ⎥ e ⎥ 0 , S > 0 and scalar ε > 0 such that the LMI (2.57) holds for a given α > 0 . Then, a suitable stabilizing control law is

given by u ( k ) = Fx (k ) ,

where F = Y X −1 and Y ∈ R m×n . 2.3.3 Examples

Example 2.5: Consider an uncertain discrete time-delay system x( k + 1) = ( A + ∆A) x ( k ) + ( Ad + ∆ Ad ) x (k − h) ,

(2.58)

where ⎡− 0.5 − 0.4⎤ ⎡0.2 ⎡ 0.3 0.1⎤ , M =⎢ A=⎢ , Ad = ⎢ ⎥ ⎥ ⎣ 0.2 − 0.6 ⎦ ⎣0.1 ⎣− 0.1 0.1 ⎦ ⎡0.1 N2 = ⎢ ⎣ 0

0.3 ⎤ . 0.2⎥⎦

33

0.1⎤ ⎡0.1 , N1 = ⎢ ⎥ 0.2⎦ ⎣0.2

0.3 ⎤ , 0.2⎥⎦

Applying Theorem 2.5, by the software package LMI Lab., let h = 0.1 and α = 1 , it can then be verified that ⎡0.0805 X =⎢ ⎣0.0326

− 0.0055⎤ , ε = 1.1136 . 0.5501⎥⎦

0.0326⎤ ~ ⎡ 0.1049 , S =⎢ 0.3059⎥⎦ ⎣− 0.0055

Therefore, the system under study (2.58) is robust stable with normal-bounded uncertainties. Example 2.6: Consider an uncertain discrete time system with state delay x( k + 1) = ( A + ∆A) x (k ) + ( Ad + ∆ Ad ) x ( k − h) + ( B + ∆B )u ( k ) ,

where ⎡ 1 − 0.6⎤ ⎡0.5 A=⎢ , Ad = ⎢ ⎥ 0.5 ⎦ ⎣0.4 ⎣0.6 ⎡0.1 N1 = ⎢ ⎣0.2

0.3 ⎤ ⎡0.1 , N2 = ⎢ ⎥ 0.2⎦ ⎣ 0

(2.59)

0.2 ⎤ ⎡0.1 , B=⎢ ⎥ 0.4⎦ ⎣0

0.2⎤ ⎡0.2 , M =⎢ ⎥ 0.1 ⎦ ⎣0.1

0.3 ⎤ ⎡0.1 , N3 = ⎢ ⎥ 0.2⎦ ⎣0.2

0.3 ⎤ , h = 1, α = 1. 0.1⎥⎦

0.1⎤ , 0.2⎥⎦

According to Theorem 2.6, by the software package LMI Lab., we can obtain ⎡ 0.069 X =⎢ ⎣− 0.0177

− 0.0177⎤ ⎡− 0.0241 , Y =⎢ ⎥ 0.0592⎦ ⎣− 0.0257

− 0.0157 ⎤ ~ ⎡3.9469 2.0768⎤ , S =⎢ ⎥, − 0.0128⎥⎦ ⎣2.0768 1.6011 ⎦

ε = 0.4538 . The system (2.59) under study is robustly stabilizable, the correspondent robustly stabilizing control law is ⎡− 0.4519 u (k ) = ⎢ ⎣− 0.4642

− 0.3997 − 0.3556

34

⎤ ⎥ x(k ) . ⎦

2.4 Stability of Cellular Neural Networks with Time-Varying Delay In this section, the stability for cellular neural networks with time-varying delay is introduced by linear matrix inequality (LMI). A sufficient condition related to the exponentially asymptotic stability for DCNNs is proposed. This condition is less restrictive than that given in the literature. 2.4.1 Model of Cellular Neural Networks with Delay (DCNNs)

The dynamic behavior of a continuous time DCNN can be described by the following retarded functional differential state equations

x& (t ) = − x (t ) + Ay ( x (t )) + By ( x (t − τ (t ))) + u ,

where

T x(t ) = [ x1 (t ), L , x n (t )] ∈ R n ,

(2.60)

y ( x(t )) = [ y1 ( x1 (t )),L, y n ( xn (t )) ]T ∈ Rn

is

the

output vector, A = [aij ] and B = [bij ] are known constant matrices with appropriate dimensions, A is the feedback matrix, B is the delayed feedback matrix, u is the constant

control

input

vector

with

appropriate

dimensions

and

τ (t ) = [τ 1 (t ), L , τ n (t ) ]T ∈ R n is time-varying bounded delay time satisfying 0 ≤ τ i (t ) ≤ hi and τ&i (t ) ≤ d i < 1 ,

i = 1, L , n , where

hi

and

di

are constant scalars. Let

h = max{hi , i = 1,L , n} and d = max{d i , i = 1,L, n}. yi ( xi (t )) is defined as follows (a) For ∀ x1 , x 2 ∈ R , yi ( xi (t )) satisfies Lipschitz condition:

∃ M i > 0, yi ( x1) − yi ( x2) ≤ M i x1 − x2 . (b) yi ( xi (t )) is a bounded monotonic nondecreasing function.

Lemma 2.3 [10]: There exists an equilibrium for the DCNN (2.60).

Remark 1: The Lemma 2.3 only shows the existence of an equilibrium. The uniqueness of the equilibrium will follow from the exponential asymptotic stability to be established below. Based on the Lemma 2.3, we assume that the system (2.60) has an equilibrium point x∗ for a given u . Let z (t ) = x(t ) − x∗ , then the system (2.60) can be transformed into the

35

form of z& (t ) = − z (t ) + AΨ ( z (t )) + BΨ ( z (t − τ (t ))) ,

(2.61)

where z (t ) = [ z1 (t ), L , z n (t ) ]T , Ψ ( z (t )) = [ Ψ1 ( z1 (t )), L , Ψ n ( z n (t )) ]T , ∗ Ψ i ( z i (t )) = y i ( z i (t ) + x ) − y i ( z i (t ))

and

Ψ i ( z i (t )) ≤ M i z i (t ) .

(2.62)

Definition 2.1: If there exist constants α > 0 and β > 0 such that z (t ) ≤ β e −αt sup z(θ ) , ∀t > 0 , -h ≤θ ≤ 0

(2.63)

then system (2.61) is said to be exponentially stable, where α is called the degree of exponential stability. Lemma 2.4: For any vectors or matrices z and y with appropriate dimensions and any

positive constant ε , the following inequality is satisfied: 2 z T y ≤ ε z T z + ε −1 y y . T

(2.64)

2.4.2 Existence of the Equilibrium and Stability of DCNNs

In this subsection, we present a sufficient condition for the existence of unique equilibrium point and for the exponential stability of the system (2.60).

Theorem 2.7: For a DCNN defined by (2.61), delay τ i (t ) satisfying 0 ≤ τ i (t ) ≤ hi ≤ h and τ&i (t ) ≤ d i ≤ d < 1 , i = 1, L , n , the origin is the unique equilibrium point and it is

exponentially asymptotically stable such that the LMI

36

⎡ AT A + (2 + h) ΣT Σ + 2(α − 1) I ⎢ αh T e B ⎣

αh e B⎤ ⎥ < 0, −µ ⎦

(2.65)

holds for given scalars α > 0 and M i > 0 , i = 1, L , n , where Σ = diag ( M i ) , i = 1, L , n , µ = (1 − d ) , and I is the identity matrix with compatible dimension.

Proof: In order to show that the origin of (2.61) is exponentially asymptotically stable, the Lyapunov functional is given by

V ( z (t )) = e

2αt

t

2αθ T T z (t ) z (t ) + ∫ e Ψ ( z (θ ))Ψ ( z (θ ))dθ t −τ ( t )

t

+ ∫ ( s − t + τ (t )) e2αs ΨT ( z ( s))Ψ ( z ( s))ds . t −τ ( t )

(2.66)

The derivative of this functional along the solution of system (2.61) is V& = e2αt {(2α zT (t ) z (t ) + z&T (t ) z (t ) + zT (t ) z& (t ) + ΨT ( z (t ))Ψ ( z (t )) − e−2ατ ( t ) (1 − τ&(t )) ΨT ( z (t − τ (t )))Ψ ( z (t − τ (t ))) t

+ τ (t ) ΨT ( z (t ))Ψ ( z (t ))} − (1 − τ&(t )) ∫ e2αs ΨT ( z ( s))Ψ ( z ( s))ds . t −τ ( t )

(2.67)

From (2.61), (2.62) and Lemma 2.4, combining τ&i (t ) ≤ d i ≤ d < 1 and 0 ≤ τ i (t ) ≤ hi ≤ h , one has V& ≤ e2αt {zT (t )[2(α − 1) I + AT A + µ −1 e2αh B BT + ( 2 + h) ΣT Σ] z (t )} t

− (1 − d ) ∫ e2αs ΨT ( z ( s ))Ψ ( z ( s ))ds .

(2.68)

t −τ ( t )

t

Since the last term (1 − d ) ∫ e2αs ΨT ( z ( s ))Ψ ( z ( s))ds in (2.68) is positive definite, we have t −τ ( t )

V& ≤ e2αt {zT (t )[2(α − 1) I + AT A + µ −1 e2αh B BT + (2 + h) ΣT Σ] z (t )} .

37

(2.69)

From (2.69), we can get V& ≤ e2αt zT (t )Ωz (t ) ,

(2.70)

with Ω = 2(α − 1) I + AT A + µ −1 e2αh B BT + ( 2 + h) ΣT Σ ,

(2.71)

The requirement V& < 0 , ∀ z (t ) ≠ 0 implies Ω < 0 . Using Schur complement, it is expressed as

⎡ AT A + (2 + h) ΣT Σ + 2(α − 1) I Ω=⎢ αh T e B ⎣

αh e B⎤ ⎥ < 0. −µ ⎦

(2.72)

From [5], we are able to show that (2.65) guarantees the negativeness, which immediately implies the exponentially asymptotic stability of (2.61). 2.4.3 Examples

Example2. 7: Consider the following matrices

⎡− 0.1 0.3⎤ ⎡0.5 0.4 ⎤ ⎡0.1 , B=⎢ A=⎢ , Σ=⎢ ⎥ ⎥ ⎣ 0 − 0.1⎦ ⎣0.15 0.5 ⎦ ⎣0

0 ⎤ . 0.1⎥⎦

Applying Theorem 2.7 to this (2.65), by the LMI Toolbox, we find delay satisfying 0 ≤ τ 1 (t ) ≤ 0.3623 and 0 ≤ τ 2 (t ) ≤ 0.3623 , then the (2.61) has a unique and exponentially asymptotically stable equilibrium point.

Example2. 8: We consider the following matrices

0.3 ⎤ ⎡− 0.25 ⎡0.25 A=⎢ , B=⎢ ⎥ ⎣ 0.15 − 0.25⎦ ⎣0.15

0.3 ⎤ ⎡0.1 , Σ=⎢ ⎥ 0.25 ⎦ ⎣0

0 ⎤ . 0.1⎥⎦

Based on Theorem 2.7, we find delay satisfying 0 ≤ τ 1 (t ) ≤ 1.5947 and 0 ≤ τ 2 (t ) ≤ 1.5947 , then the (2.61) has a unique and exponentially asymptotically stable equilibrium point.

38

2.5 Summary This chapter first deals with the problem of robust stability criteria for a class of uncertain linear time-delay systems. Based on Lyapunov-Krasovskii functionals combining with LMI techniques, simple and improved delay-dependent stability are derived. Two examples are illustrated to show that the criteria perform much better than existing stability criteria. Also, we propose that controllers are capable of guaranteeing the closed-loop system stabilization by the formulation of a linear matrix inequality for a class of discrete time-delay systems. Further, a sufficient condition related to the existence of unique equilibrium point and its exponentially asymptotic stability for cellular network with delay (DCNNs) has been derived. The output of the cell is not a piecewise linear function. Therefore, the sufficient condition established in Theorem 2.7 has a wider adaptive range and is delay-dependent.

39

Chapter 3 Robust Filtering for a Class of Uncertain Interval Systems with Time Delay

In this chapter we first study the problem of robust filtering for a class of linear continuous-time interval systems with delay dependent conditions. Moreover, we extend the proposed theory to discuss the robust H ∞ filtering for a class of linear continuous-time interval systems with delay dependent conditions [26, 80].

3.1 Introduction Kalman filtering is one of the most popular estimation approaches. In the past three decades, considerable effort has been devoted to its theory and applications. This filtering method assures that both the state equation and output measurement are subjected to stationary Gaussian noises. The applications of the Kalman filtering theory may be found in a large spectrum of different fields ranging from various engineering problems to biology, geoscience, economics, and management, etc. Recently, the robust state estimation arises from the desire to estimate unmeasurable state variables when the plant model has uncertain parameters. A robust Kalman filtering estimation has also been introduced by [90] for linear systems with norm-bounded parameter uncertainty in both the state and measurement matrices. The delayed state is very often the cause of instability and poor performance of system, increasing interests have recently been devoted to the robust and H ∞ controller design problems of the linear uncertain state delayed systems for linear discrete-time [22] and continuous-time cases [50], respectively. The results of [22] and [50] are derived by a guaranteed finite upper bound on the error covariance. Wang and Burnham [88] address the robust filter design problem for a class of nonlinear time-delay stochastic systems. Sufficient conditions are proposed to guarantee the existence of desired robust exponential

40

filters, which are derived in terms of the solutions to algebraic Riccati inequalities. On the other hand, systems with interval parameters have been extensively discussed and many achievements have been made in the control design and stability analysis. For some representative prior work on this topic, we refer the reader to [38, 43, 95] and the references therein. To the best of the author’s knowledge, however, so far there are few papers that not only deal with interval systems with delay dependence but also take robust filtering into account. As to H ∞ filtering problem for time-delay systems, De Souza et al. [12] introduced to design a stable linear filtering, assuring asymptotic stability and a prescribed H ∞ performance level for the filtering error system, irrespective of the uncertainties and the time delays. Wang et al. [87] addressed the robust H ∞ filtering problems for continuous time-delay systems with structured uncertainty via an algebraic Riccati matrix inequality approach. However, to the best of authors’ knowledge, the problem of robust H ∞ filtering in the context interval systems with state delays has not been fully investigated in the literature. So, the goal of this chapter is to finish these gaps. The purpose of this chapter is to consider the state-estimation problem for a class of linear continuous time-delay interval systems. For robust filtering problem, our attention is focused on the design of a linear state estimator; the dynamics of the estimation error is stochastically exponentially stable in the mean square, independent of the time delay. Sufficient conditions are proposed to guarantee the existence of desired robust exponential filtering. For robust H ∞ filtering problems, we derive a sufficient condition for the existence of an asymptotically stable linear filter that ensures asymptotic stability and a prescribed H ∞ performance for the filtering error system. Finally, a solution based on linear matrix inequalities (LMIs) results is then developed to obtain G and K matrices of the filter. Two numerical examples demonstrate the validity of two theoretical results.

3.2 Robust Filtering for Delay-Dependent Interval Systems This section studies the problem of robust filtering for a class of linear continuous-time interval systems with delay dependent conditions. By employing a Lyapunov-Krasovskii functional approach, it is proved that the dynamics of the estimation error is stochastically exponentially stable in the mean square. Sufficient conditions are proposed to guarantee the existence of the desired robust filters by solving linear matrix inequality which is delay-dependent. A numerical example is worked out to illustrate the validness of the theoretical results.

41

3.2.1 Problem Formulation and Assumptions We consider a class of interval time-delay systems represented by x& (t ) = AI x (t ) + AdI x (t − h(t )) + E1 w(t )

(3.1)

y (t ) = C I x (t ) + D I x (t − h(t )) + E 2 v (t ) ,

(3.2)

on t ≥ 0 , where AI ∈ [ A, A] , AdI ∈ [ Ad , Ad ] , C I ∈ [C , C ] , D I ∈ [ D, D ] .

Let's introduce A ≡

1 1 ( A + A ) and Am ≡ ( A − A ) . 2 2

Clearly, all the elements of Am are nonnegative. Moreover, AI can be written as

AI ≡ A + ∆ A with ∆A ∈ [− Am , Am ] . Similarly, we introduce Ad , Adm , ∆ Ad , C , C m , ∆C , D , Dm and ∆D . Then (3.1) and (3.2) can be written as x& (t ) = [ A + ∆A] x(t ) + [ Ad + ∆ Ad ] x (t − h(t )) + E1 w(t )

(3.3)

y (t ) = [C + ∆C ] x (t ) + [ D + ∆D ]x (t − h(t )) + E 2 v (t ) ,

(3.4)

where x(t ) ∈ Rn is the state, y(t ) ∈ Rm is the measured output, w(t ) ∈ Rn and v(t ) ∈ Rm are the process and measurement noises, respectively. A, Ad , C , D , E1 and E 2 are known constant matrices with appropriate dimensions. ∆A, ∆ Ad , ∆C and ∆D are the uncertain parts of AI , AdI , C I and D I , respectively. h(t ) denotes the delay in the state and it is assumed that there exist positive numbers h such that 0 ≤ h(t ) ≤ h , h&(t ) ≤ d < 1 . In fact, ∆A will take a particular deterministic matrix within [− Am , Am ] but we just do not know which particular one, and similarly for ∆ Ad , ∆C , ∆D . When we talk about

42

the solution of (3.3) and (3.4), we mean the solution when ∆A , ∆ Ad , ∆C and ∆D take particular matrices within their matrix intervals [38]. Remark 3.1: If ∆A ∈ [− Am , Am ] , then ∆A ≤ Am , where the operator norm for any matrix Z , i.e.

Z , denotes

Z = sup{Zx : x = 1} = λ max ( Z T Z ) , where λ max ( Z T Z )

means the largest eigenvalue of Z T Z . We make the following assumptions on the process and measurement noises: Assumption 3.1. ∀t , s ≥ 0 ( a ) E[ w(t )] = 0 , E[ w(t ) wT ( s )] = Wδ (t − s ), W > 0 ,

(3.5)

(b) E[v (t )] = 0 , E[v(t ) vT ( s )] = Vδ (t − s ), V > 0 ,

(3.6)

(c) E[ x(0) wT (t )] = 0 , E[ x(0) vT (t )] = 0 ,

(3.7)

(d ) E[ w(t ) vT ( s )] = 0 ,

(3.8)

where E (⋅) represents the expectation and δ (⋅) is the Dirac function. We consider the full-order linear filter of the form x&ˆ (t ) = Gxˆ (t ) + Ky (t ) ,

(3.9)

where G and K are constant matrices to be designed with appropriate dimensions. Let the error state be e(t ) = x(t ) − xˆ (t ) .

(3.10)

After substituting (3.3)-(3.4) and (3.9) into (3.10), one has e&(t ) = Ge(t ) + [( A + ∆A) − K (C + ∆C ) − G ] x (t )

+ [( Ad + ∆ Ad ) − K ( D + ∆D )] x (t − h(t )) + [ E1 w(t ) − K E 2 v (t )] .

43

(3.11)

We introduce the extended state vector ⎡ x(t )⎤ ⎥. ⎣e(t ) ⎦

φ (t ) = ⎢

(3.12)

It follows from (3.3)-(3.4) and (3.11) that

φ&(t ) = ( Aφ + ∆ Aφ )φ (t ) + ( Adφ + ∆ Adφ )φ (t − h(t )) + Bφ π (t ) ,

(3.13)

where A ⎡ Aφ = ⎢ ⎣ A − G − KC

0⎤ ⎡ ∆A , ∆ Aφ = ⎢ ⎥ G⎦ ⎣∆A − K∆C

0⎤ 0⎤ ⎡ Ad , Adφ = ⎢ ⎥ ⎥, 0⎦ ⎣ Ad − KD 0⎦

0⎤ 0 ⎤ ⎡ ⎡ E1 ⎡ w(t )⎤ ∆Ad , Bφ = ⎢ , π (t ) = ⎢ ∆Adφ = ⎢ ⎥ ⎥ ⎥, ⎣∆ Ad − K∆D 0 ⎦ ⎣ E1 − K E 2 ⎦ ⎣v(t ) ⎦ and π (t ) is a stationary zero-mean white noise vector. The initial state is specified as < φ (0), φ ( s) >=< φ 0 ,ψ (s) > , where ψ (⋅) ∈ L 2 [− h, 0] which is assumed to be a zero-mean Gaussian random vector. Definition 3.1 [54]: For the system (3.13) and ψ (⋅) ∈ L 2 [− h, 0] , the trivial solution is asymptotically stable in the mean square if

lim E φ (t ;ψ ) = 0 2

(3.14)

t →∞

and is exponentially stable in the mean square if there exists constants γ > 0 and β > 0 such that E φ (t ;ψ ) ≤ γ e− β t sup E ψ (θ ) . 2

2

(3.15)

− h ≤θ ≤ 0

Definition 3.2 [88]: We say that the filter (3.9) is exponentially stable if for every ψ (⋅) ∈ L 2 [ − h, 0] , the corresponding extended system (3.13) is exponentially stable in mean square.

44

The purpose of this section is to consider an exponential robust filtering for the linear time-delay interval system (3.3)-(3.4). Specifically, we propose the state-estimator G and K using linear matrix inequality approach, such that the extended system (3.13) is

exponentially stable in the mean square with delay dependent conditions. 3.2.2 Main Results In this subsection we state the analysis and design of the filter. 3.2.2.1 Filter Analysis Here we will introduce the conditions under which the estimation error is stochastically exponentially stable in the mean square. Theorem 3.1: Let the filter parameters G and K be given, then there exist positive scalars d , ε i > 0 , i = 1, L ,5 and symmetric positive definite matrix P > 0 , Q > 0 , R > 0 such that the following matrix inequality holds for a given α > 0 Aφ P + P Aφ + 2αP + (ε 1 + ε 2) e T

2αh

I + ε 1−1 e−2αh P M Aφ P + ε −21 e−2αh P M Cφ P

+ ε 3−1 e−2αh P M Adφ P + ε −41 e−2αh P M Dφ P + ε 5−1 e−2αh P Adφ ATdφ P + hR + Q < 0 ,

(3.16)

4αh

e (ε 3 + ε 4 + ε 5) ⋅ I , I is an identity matrix. Here Aφ , Adφ are defined in 1− d the above and

where Q =

M Aφ

⎡ Am ATm =⎢ T ⎣ Am Am

⎡0 M Dφ = ⎢ ⎣0

T ⎡0 Am Am ⎤ = , M C φ ⎢ T⎥ Am Am ⎦ ⎣0

⎡ Adm ATdm ⎤ ⎥ , M Adφ = ⎢ T K C m C Tm K T ⎦ ⎣ Adm Adm 0

Adm Adm ⎤ , T ⎥ Adm Adm ⎦ T

⎤ ⎥. K Dm DTm K T ⎦ 0

Then the extended system (3.13) is exponentially stable in the mean square for all admissible parameter uncertainties ∆A , ∆ Ad , ∆C and ∆D .

45

Proof: We consider the Lyapunov-Krasovskii functional 0

t

t

V (φ (t ), t ) = e2αt φ T (t ) Pφ (t ) + ∫ e2αθ φ T (θ )Qφ (θ )dθ + ∫ ∫ e2αuφ (u ) Rφ (u )dudβ T

,

− h( t ) t + β

t −h (t )

(3.17) Along the trajectories of (3.13) and making use of the derivative of (3.17), we have T V& = 2α e2αt φ T (t ) Pφ (t ) + 2 e2αt φ& (t ) Pφ (t ) + e2αt φ T (t )Qφ (t ) + h(t ) e2αt φ T (t ) Rφ (t ) t

T − (1 − h&(t )) e2α ( t − h ( t )) φ T (t − h(t ))Qφ (t − h(t )) − (1 − h&(t )) ∫ e2αsφ ( s )Rφ ( s )ds .

(3.18)

t − h (t )

Using Lemma 2.4 and substituting (3.13) into (3.18), one has T V& ≤ e 2 α t {φ ( t ) AφT P + P Aφ + 2α P + (ε 1 + ε 2 ) e 2 α h + ε 1− 1 e − 2 α h P M

Aφ

P

+ ε 2−1 e−2αh P M Cφ P + ε 3−1 e−2αh P M Adφ P + ε 4−1 e−2αh P M Dφ P + ε 5−1 e−2αh P Adφ ATdφ P + hR + Q ]φ (t ) + φ T (t − h(t ))[(ε 3 + ε 4 + ε 5) e2αh − (1 − d ) e−2αh Q ]φ (t − h(t )) t

+ 2 φ T (t ) P Bφ π (t )} − (1 − d ) ∫ e2αsφ ( s ) Rφ ( s )ds . T

(3.19)

t −h (t )

Since Q =

4αh

t T e (ε 3 + ε 4 + ε 5) ⋅ I and the last term (1 − d ) ∫ e2αsφ ( s ) Rφ ( s )ds in (3.19) is 1− d t − h (t )

positive-definite, then we have

V& ≤ e2αt {φ T (t )[ AφT P + P Aφ + 2αP + (ε 1 + ε 2) e2αh + ε 1−1 e−2αh P M Aφ P + ε −2 1 e−2αh P M Cφ P + ε 3−1 e−2αh P M Adφ P + ε 4−1 e−2αh P M Dφ P

+ ε 5−1 e−2αh P Adφ ATdφ P + hR + Q]φ (t ) + 2 φ T (t ) P Bφ π (t )} ,

46

(3.20)

let Γ = AφT P + P Aφ + 2αP + (ε 1 + ε 2) e2αh + ε 1−1 e−2αh P M Aφ P + ε −21 e−2αh P M Cφ P + ε 3−1 e−2αh P M Adφ P + ε −4 1 e−2αh P M Dφ P + ε 5−1 e−2αh P Adφ ATdφ P + hR + Q < 0 .

(3.21)

We obtain

V& ≤ e2αt {− λ min (−Γ) φ T (t )φ (t ) + 2 φ T (t ) P Bφ π (t )},

(3.22)

which implies that the extended system (3.13) is asymptotically stable in mean square, provided (3.16) is satisfied for a given α > 0 . 3.2.2.2 Filter Design In this subsection we design the filter parameters G and K by using LMI techniques. For the sake of simplicity, we define the following parameters: ~ A = A + ε 1−1 e−2αh Am ATm P1 + ε 3−1 e−2αh Adm ATdm P1 + ε 5−1 e−2αh Ad ATd P1 , ~ C = C + ε 5−1 e2αh D ATd P1 , ~ L = ε −21 C m C Tm + ε −41 D m DTm + ε 5−1 D DT , η = C + ε 5−1 e−2αh D ATd P 2 ,

⎡ P1 P=⎢ ⎣0

0⎤ ⎡ R1 ⎥ , R = ⎢0 P2⎦ ⎣

0⎤ ⎥, R2⎦

(3.23)

(3.24)

where P1 , P2 , R1 , R2 are symmetric positive-definite matrices, and U ∈ R p× p is any arbitrarily orthogonal matrix (i.e., U U T = I ). Theorem 3.2: Given a time-delay h . Consider the robust filtering controlled system (3.13), if there exist symmetric positive-definite matrices P1 , P2 , R1 , R2 , S ∈ R n× p and scalars ε i > 0, i = 1, L ,5 such that the following LMI holds for a given α > 0 and

d >0,

47

(i) The symmetric positive-definite matrix P1 satisfies

⎡M M1 M 2⎤ ⎥ ⎢ T 0 ⎥ < 0, ⎢M 1 − N 1 ⎢⎣ M T2 0 − N 2 ⎥⎦

(3.25)

and (ii) Symmetric positive-definite matrix P2 satisfies ⎡Ω ⎢ T ⎢Ω1 ⎢ΩT2 ⎢ T ⎣⎢Ω3

Ω1 − G1

Ω2 0

0

− G2

0

0

Ω3 ⎤ ⎥ 0 ⎥ < 0, 0 ⎥ ⎥ − G 3⎦⎥

(3.26)

with

M = AT P1 + P1 A + 2α P1 + (ε 1 + ε 2) e2αh + M 1 = [e

−α h

P1 Am , e

− αh

P1 Adm , e

−α h

4αh

e (ε 3 + ε 4 + ε 5) I , 1− d

P1 Ad ] , M 2 = h R1 , N 1 = diag [ε 1 , ε 3 , ε 5] ,

4αh ~ e ~T 2αh (ε 3 + ε 4 + ε 5) I , N 2 = R1 , Ω = A P2 + P 2 A + 2α P 2 + (ε 1 + ε 2) e + 1− d −α h −α h − αh Ω1 = [e P 2 Am , e P 2 Adm , e P 2 Ad ] , G1 = diag [ε 1 , ε 3 , ε 5] , Ω2 = S , G 2 = I ,

Ω3 = h R 2 , G 3 = R 2 .

Then the robust filtering with parameters

K = P −2 1 [η L −1 + SU L − 2 ]

(3.27)

~ ~ G = A − KC

(3.28)

T

1

and

will yield the augmented system (3.13) to be asymptotically stable in the mean square.

48

Proof: Using (3.16), we obtain

⎡Γ11 Γ12 ⎤ Γ=⎢ T ⎥ < 0, ⎣Γ12 Γ22⎦

(3.29)

where −1 −2αh

Γ11 = A P1 + P1 A + 2α P1 + (ε 1+ ε 2) e2αh + ε 1 e T

T

P1 Am Am P1

+ ε 3−1 e−2αh P1 Adm ATdm P1 + ε 5−1 e−2αh P1 Ad ATd P1 + Q + h R1 , −1 −2αh

Γ12 = ( A − G − KC ) P 2 + ε 1 e−2αh P1 Am Am P 2 + ε 3 e T

−1

T

(3.30)

T

P1 Adm Adm P 2

+ ε 5−1 e−2αh P1 Ad ATd P 2 − ε 5−1 e−2αh P1 Ad DT K T P 2 , Γ 22 = G P 2 + P 2 G + 2αP + (ε 1 + ε 2) e T

2αh

(3.31)

+ Q + ε 1−1 e−2αh P 2 Am AmT P 2

− ε 2−1 e−2αh P 2 K C m C Tm K T P 2 + ε 3−1 e−2αh P 2 Adm ATdm P 2 + ε 4−1 e−2αh P 2 K D m DTm K T P 2

+ ε 5−1 e−2αh P2 Ad Ad T P2 − ε 5−1 e−2αh P2 Ad DT K T P2 − ε 5−1 e−2αh P2 ( Ad DT K T ) P2 T

+ ε 5−1 e−2αh P 2 KD DT K T P 2 + h R 2 .

(3.32)

A sufficient condition to satisfy (3.29) is that Γ11 < 0 ,

(3.33a)

Γ 22 < 0 , and

(3.33b)

Γ12 = 0 .

(3.33c)

One has −1 −2αh

Γ11 = A P1 + P1 A + 2α P1 + (ε 1+ ε 2) e2αh + ε 1 e T

T

P1 Am Am P1

+ ε 3−1 e−2αh P1 Adm ATdm P1 + ε 5−1 e−2αh P1 Ad ATd P1 + Q + h R1 < 0 .

(3.34)

Using the Schur complement, inequality (3.34) can be transformed into the form (3.25).

49

Next, we consider the following inequality Γ 22 = G P 2 + P 2 G + 2αP + (ε 1 + ε 2) e T

2αh

+ Q + ε 1−1 e−2αh P 2 Am AmT P 2

− ε 2−1 e−2αh P 2 K C m C Tm K T P 2 + ε 3−1 e−2αh P 2 Adm ATdm P 2 + ε 4−1 e−2αh P 2 K D m DTm K T P 2

+ ε 5−1 e−2αh P2 Ad Ad T P2 − ε 5−1 e−2αh P2 Ad DT K T P2 − ε 5−1 e−2αh P2 ( Ad DT K T ) P2 T

+ ε 5−1 e−2αh P 2 KD DT K T P 2 + h R 2 < 0 .

(3.35)

Substituting (3.28) into (3.35), we find that ~ −2αh T T ~T −1 −1 −1 T T T A P 2 + P 2 A + e P 2 (ε 1 Am Am + ε 3 Adm Adm + ε 5 Ad Ad ) P 2 − ( P 2 K )η − η ( P 2 K ) + e−2αh ( P 2 K ) L ( P 2 K ) + (ε 1 + ε 2) e2αh + Q + 2α P 2 + h R 2 < 0 . T

(3.36)

From the (3.27), we can obtain T

⎡( K ) 12 − ηT − 12 ⎤ ⋅ ⎡( K ) 12 −ηT − 12 ⎤ = ( SU ) ( SU )T = S T . S L ⎥ ⎢ P2 L L ⎥ ⎢⎣ P2 L ⎦ ⎣ ⎦ Therefore (3.36) can be rewritten as ~ −2αh T −1 ~T T −1 −1 −1 T T T A P 2 + P 2 A + e P 2 (ε 1 Am Am + ε 3 Adm Adm + ε 5 Ad Ad ) P 2 − η L η + S S

+ (ε 1 + ε 2) e2αh + Q + 2α P 2 + h R 2 < 0 .

(3.37)

Since the term η T L −1η in (3.37) is positive-definite, we have

~ −2αh ~T −1 −1 −1 T T T T A P 2 + P 2 A + e P 2 (ε 1 Am Am + ε 3 Adm Adm + ε 5 Ad Ad ) P 2 + S S

+ (ε 1 + ε 2) e2αh + Q + 2α P 2 + h R 2 < 0 .

(3.38)

Using the Schur complement formula, (3.38) can be considered in the form (3.26). Finally, we substitute (3.28) into (3.31) to satisfy Γ12 = 0 , therefore, Γ < 0 . Moreover, it follows from Theorem 3.1 that the augmented system (3.13) is exponentially stable in the mean square with delay-dependent conditions. This completes the proof of this Theorem 3.2.

50

□ Remark 3.2: The proposed design procedures are then summarized as follows:

(1) Read given data h , A , Ad , C , D , Am , Adm , C m , Dm . (2) Solve (3.26) for P1 , P2 , S and ε i > 0, i = 1, L ,5 . Then, we are able to get ~ ~ A, C, L and η from (3.23).

(3) Compute K . (4) Compute G . 3.2.3 An Illustrative Example

Example 3.1: Consider the second-order system such that x& (t ) = [ A + ∆A] x(t ) + [ Ad + ∆ Ad ]x (t − h(t )) + w(t ),

(3.39)

y (t ) = [C + ∆C ] x(t ) + [ D + ∆D ]x (t − h(t )) + v(t ),

(3.40)

0⎤ ⎡1 0⎤ ⎡0.1 0 ⎤ ⎡− 1 ⎡− 2 1 ⎤ , D=⎢ , C=⎢ A=⎢ , Ad = ⎢ ⎥, ⎥ ⎥ ⎥ ⎣0 1⎦ ⎣ 0 0.1⎦ ⎣− 0.9 − 1⎦ ⎣ 0 − 2⎦ ⎡0.1 0 ⎤ ⎡0.1 0⎤ ⎡0.1 , Adm = ⎢ , Cm = ⎢ Am = ⎢ ⎥ ⎥ ⎣ 0 0.1⎦ ⎣0 0.1⎦ ⎣ 0

0 ⎤ ⎡0.1 , Dm = ⎢ ⎥ 0.1⎦ ⎣0

0 ⎤ , h = 2. 0.1⎥⎦

Solving matrix inequality (3.25) and (3.26) by using the LMI-control Toolbox [14], we obtain ⎡ 11.8024 − 5.7641⎤ ⎡ 1.7994 − 0.8128 ⎤ ⎡2.2059 − 0.1192⎤ , P2 = ⎢ , S=⎢ P1 = ⎢ ⎥ ⎥, ⎥ 3.9345⎦ ⎣− 5.7641 ⎣− 0.8216 0.7866⎦ ⎣− 0.1192 1.6998 ⎦

ε 1 = 1.8793 , ε 2 = 6.3712 , ε 3 = 1.9605 , ε 4 = 4.2089 and ε 5 = 4.0891 . The orthogonal matrix U is defined as U = I 2 . Therefore, we can obtain the expected filter parameters form (3.27) and (3.28) as ⎡0.6966 K=⎢ ⎣0.4629

0.5721⎤ ⎡− 1.2499 0.8218 ⎤ , G=⎢ ⎥. ⎥ 1.5552⎦ ⎣ 0.4218 − 0.4499⎦

51

Based on the above numerical solution, the responses of errors ei , ei = xi − xˆ i , i = 1, 2 are shown in Fig. 1, respectively. The real state variables xi , i = 1, 2 and its estimation state variables xˆ i , i = 1, 2 are depicted in Fig. 2 and Fig. 3, respectively. The simulation results imply that the desired purpose is well achieved.

Fig. 3.1 Responses of error dynamics ( e1 : solid, e2 : dashed)

52

Fig. 3.2 Real state variable x1 (solid) and its estimate xˆ1 (dashed)

Fig. 3.3 Real state variable x 2 (solid) and its estimate xˆ 2 (dashed)

53

3.3 Delay-Dependent Robust

H ∞ Filtering for Interval Systems with

State Delays This section deals with the problem of robust H ∞ filtering for a class of linear continuous-time interval systems with delay dependence. The problem aims at designing a stable linear filtering assuring asymptotic stability and a prescribed H ∞ performance level for the filtering error system. A sufficient condition for the existence of such a filter is developed in terms of linear matrix inequalities. A numerical example demonstrates the validity of the theoretical results. 3.3.1 Problem Formulation and Assumptions

We consider a class of interval time-delay systems represented by x& (t ) = AI x (t ) + AdI x(t − h) + E1 w(t ),

(3.40)

y (t ) = C I x (t ) + C dI x(t − h) + E 2 w(t ),

(3.41)

on t ≥ 0 , where AI ∈ [ A, A] , AdI ∈ [ AdI , AdI ] , C I ∈ [C , C] , D I ∈ [D , D] .

Let's introduce A ≡

1 1 ( A + A ) and Am ≡ ( A − A ) . 2 2

Clearly, all the elements of Am are nonnegative. Moreover, AI can be written as

AI ≡ A + ∆ A with ∆A ∈ [− Am , Am ] . Similar representations are applicable to Ad , Adm , ∆ Ad , C , C m , ∆C and C d , C dm , ∆ C d . Then (3.40) and (3.41) can be written as

54

x& (t ) = [ A + ∆A] x(t ) + [ Ad + ∆ Ad ]x (t − h) + E1 w(t ),

(3.43)

y (t ) = [C + ∆C ] x (t ) + [C d + ∆ C d ] x (t − h) + E 2 w(t ),

(3.44)

where x(t ) ∈ Rn is the state, y(t ) ∈ Rm is the measured output, w(t ) ∈ Rn is the measurement noises, respectively. A, Ad , C , C d , E1 and E 2 are known constant matrices with appropriate dimensions. ∆A, ∆ Ad , ∆C and ∆ C d are the uncertain parts of AI , AdI , C I and C dI , respectively. h denotes the delay in the state and it is assumed that there exists a positive number h such that 0 ≤ h ≤ h . In fact, ∆A will take a particular deterministic matrix within [− Am , Am ] but we just do not know which particular one, and similar situations exist for ∆ Ad , ∆C , ∆ C d . When we talk about the solution of (3.43) and (3.44), we mean the solution are evaluated when ∆A, ∆ Ad , ∆C and ∆ C d take particular matrices within their matrix intervals.

We now consider the full-order linear filter of the form x&ˆ (t ) = Gxˆ (t ) + Ky (t ) ,

(3.45)

where xˆ (t ) is the estimated state, G and K are constant matrices to be designed with appropriate dimensions. Let the error state be e(t ) = x(t ) − xˆ (t ) .

(3.46)

After substituting (3.43)-(3.44) and (3.45) into (3.46), one has e&(t ) = Ge(t ) + [( A + ∆A) − K (C + ∆C ) − G ] x (t )

+ [( Ad + ∆ Ad ) − K (C d + ∆ C d )] x (t − h) + ( E1 − K E 2) w(t ) .

(3.47)

Let z (t ) = Le (t ) represent the output error state, where L is a pre-specified constant matrix. We introduce the extended state vector ⎡ x(t )⎤ ⎥. ⎣e(t ) ⎦

φ (t ) = ⎢

(3.48)


55

φ&(t ) = ( Aφ + ∆ Aφ )φ (t ) + ( Adφ + ∆ Adφ )φ (t − h) + Bφ w(t ) ,

(3.49)

z (t ) = Cφ φ (t ) ,

(3.50)

where A ⎡ Aφ = ⎢ ⎣ A − G − KC

0⎤ ⎡ ∆A , ∆ Aφ = ⎢ ⎥ G⎦ ⎣∆A − K∆C

0 ⎤ 0⎤ ⎡ Ad , Adφ = ⎢ ⎥ ⎥, 0⎦ ⎣ Ad − K Cd 0⎦

0⎤ ⎡ E1 ⎤ ⎡ ∆Ad , Bφ = ⎢ ∆Adφ = ⎢ ⎥ , C φ = [0 L ] . ⎥ ⎣ E1 − K E 2 ⎦ ⎣∆ Ad − K∆ C d 0 ⎦ The purpose of this section is to construct the filter parameters G and K which satisfies all admissible interval parameters ∆A , ∆ Ad , ∆C , ∆D , the augmented system (3.49) and (3.50) is asymptotically stable and guarantees the H ∞ norm upper bound constraint

⋅ denotes H zw (s) ∞ ≤ γ simultaneously, where H zw ( s) ∞ := supw∈R σ max [H zw ( jw)] , σ max [] ⋅ , and γ < 1 is a given positive constant. the largest singular value of [] 3.3.2 Main Results

In this subsection we describe the analysis and design of the filter 3.3.2.1 Filter Analysis

In this subsection we will derive the conditions under which the estimation error is stable and guarantees the upper bound of H ∞ norm. We first give some results which are essential for the development of our main results. Definition 3.3 If there exist α > 0 and β > 0 such that x (t ) ≤ β e−αt sup x (θ ) , ∀t > 0 , 2

2

(3.51)

− h ≤θ ≤ 0

then system (3.49) is said to be exponentially stable, where α > 0 is called the degree of exponential stability.

56

Lemma 3.1 The state delayed interval system (3.49) is stable for all h ≥ 0 , if there exist positive definite matrices P , Q and R which satisfy the following inequality

( Aφ + ∆ Aφ )T P + P( Aφ + ∆ Aφ ) + Q + 2αP + h R + e2αh P( Adφ + ∆ Adφ ) Q−1 ( Adφ + ∆ Adφ )T P < 0 .

(3.52)

Proof: We consider the following Lyapunov-Krasovskii functional for the system (3.49), t

0 t

t −h

− ht + β

V (φ (t ), t ) = e2αt φ T (t ) Pφ (t ) + ∫ e2αθ φ T (θ )Qφ (θ )dθ + ∫ ∫ e2αu φ (u ) Rφ (u )dudβ . T

(3.53)

Based on (3.49) and making use of the derivative of (3.53), we have T V& = 2α e2αt φ T (t ) Pφ (t ) + 2 e2αt φ& (t ) Pφ (t ) + e2αt φ T (t )Qφ (t ) + h e2αt φ T (t ) Rφ (t )

t

− e2α (t −h ) φ T (t − h)Qφ (t − h) − ∫ e2αsφ ( s )Rφ ( s )ds . T

(3.54)

t −h

t

Since the last term ∫ e2αsφ ( s ) Rφ ( s)ds in (3.54) is positive-definite, then we have T

t −h

V& ≤ e2αt φ T (t )[( Aφ + ∆ Aφ )T P + P( Aφ + ∆ Aφ ) + 2αP + Q + h R]φ (t ) + 2 e2αt φ T (t ) P( Adφ + ∆ Adφ )φ (t − h) + 2 e2αt φ T (t ) P Bφ w(t ) − e2αt φ T (t − h) e−2αh Qφ (t − h) .

(3.55)

Hence, the augmented system (3.49) with w(t ) = 0 is stable if the following inequality is satisfied: T M P( Adφ + ∆ Adφ )⎤ ⎡φ (t ) ⎤ φ (t ) ⎤ ⎡ 2αt ⎡ & V ≤e ⎢ ⎢ ⎥⎢ ⎥ ⎥ < 0, T − e− 2αh Q ⎦⎥ ⎣φ (t − h)⎦ ⎣φ (t − h)⎦ ⎢⎣( Adφ + ∆ Adφ ) P

57

(3.56)

where M = ( Aφ + ∆ Aφ ) P + P( Aφ + ∆ Aφ ) + 2αP + Q + h R . T

The intermediate matrix in (3.56) is negative definite if the following inequality is satisfied

M + e2αh P( Adφ + ∆ Adφ ) Q−1 ( Adφ + ∆ Adφ )T P < 0 .

(3.57)

Indeed, equation (3.57) is equivalent to (3.52). Lemma 3.1 provides a delay-dependent stability criterion since the inequality condition (3.52) does include the upper bound of delays h . To establish the H ∞ performance for the augmented system, we now present the next Lemma which guarantees the H ∞ norm bound of the transfer function H (s) zw ,

i.e., H zw (s) ∞ ≤ γ . Lemma 3.2 For a given positive constant γ , if there exist positive definite matrices P , Q and R satisfying the following inequality

( Aφ + ∆ Aφ )T P + P( Aφ + ∆ Aφ ) + Q + h R + e2αh P( Adφ + ∆ Adφ ) Q−1 ( Adφ + ∆ Adφ )T P +

1

γ

Cφ Cφ + T

1

γ

P Bφ BφT P < 0 ,

(3.58)

for all admissible parameter interval matrices ∆ Aφ and ∆ Adφ , then the system (3.49)(3.50) is robustly asymptotically stable and meets H zw (s) ∞ ≤ γ . Proof: Lemma 3.2 can be readily proved along the same line of the proof for Theorem 1 in [36]. Lemma 3.3 Assume inequality ∆A∆ AT ≤ Am ATm is satisfied, where Am is an n × n real matrix. Then, for any positive scalar ε > 0 and a positive definite matrix Q implies that

Q − ε −1 I > 0 holds. Thus, one has −1

−1

( A + ∆A) Q ( A + ∆A ) ≤ A(Q − ε −1 I ) AT + ε Am ATm . T

58

(3.59)

Lemma 3.4: Let x ∈ R n , y ∈ Rm , S ∈ R n×m and σ > 0 , then one has 2 xT Sy ≤ σ xT x + σ −1 y S T Sy . T

(3.60)

Now we present our main results as follows. 3.3.2.1 Filter Design

In this subsection we design the filter parameters G and K by using LMI techniques. For the sake of simplicity, we define the following parameters T ~ A = A + σ 1−1 e− 2αh Am ATm P1 + Ad [(Q1−ε −1 I )−1] ATd P1 + 2ε Adm ATdm P1

+ γ −1 E1 E1T P1 ,

(3.61)

~ C = C + γ −1 E 2 E1 P1 + C d [(Q1 − ε −1 I )−1 ]T ATd P1 ,

(3.62)

Σ = σ 2−1 e−2αh C m CTm + C d (Q1 − ε −1 I )−1 CTd + 2ε C dm C Tdm + γ −1 E 2 ET2 ,

(3.63)

~

η = C + γ −1 E 2 E1T P2 + C d (Q1 − ε −1 I )−1 ATd ,

⎡ P1 P=⎢ ⎣0

0⎤ ⎡Q1 0 ⎤ ⎡ R1 Q = , ⎢ ⎥,R = ⎢ ⎥ P2⎦ ⎣0 ⎣0 Q 2 ⎦

(3.64)

0⎤ ⎥, R2⎦

(3.65)

where P1 , P2 , Q1 , Q2 , R1 , R2 are symmetric positive-definite matrices. U ∈ R p× p is any arbitrary orthogonal matrix (i.e., U U T = I ). Theorem 3.3 Considering the system (3.43)-(3.44), if there exist symmetric positive-definite matrices P1 , P2 , Q1 , Q2 , R1 , R2 , S ∈ R n× p and positive scalars γ , µ , σ i , i = 1, 2 for a given time-delay h such that the following two LMIs hold for a

specified α > 0

59

(i) The symmetric positive-definite matrix P1 satisfies ⎡M ⎢ T ⎢M 1 ⎢⎣ M T2

M1 − N1 0

M 2⎤ ⎥ 0 ⎥ < 0, − N 2 ⎥⎦

(3.66)

where

M = AT P1 + P1 A + +2α P1 + σ 1 e2αh I + σ 2 e2αh I + Q1 , 2α h M 1 = [ P1 Am , 2 P1 Adm , P1 E1 , h R1] , N 1 = diag [σ 1 e , µ , γ , R1] ,

M 2 = P1 Ad , N 2 = (Q1 − µI ) , (ii) Symmetric positive-definite matrix P2 satisfies ⎡Ω ⎢ T ⎢Ω1 ⎢⎣ΩT2

Ω2 ⎤ ⎥ − G1 0 ⎥ < 0 , 0 − G 2 ⎥⎦ Ω1

(3.67)

where

~ ~T + A 2αh 2αh Ω= A P2 P 2 + 2α P 2 + σ 1 e I + σ 2 e I + Q2 , 2αh Ω1 = [ P 2 Am , P 2 Ad , 2 P 2 Adm] , G1 = diag[σ 1 e , (Q1 − µI ), µ ] ,

T Ω2 = [ L , P 2 E1 , S , h R 2] , G 2 = diag [γ , γ , I , R 2] ,

then filter (3.45) follows with parameters K = P −21[η Σ −1 + SU Σ − 2 ] , T

1

(3.68)

~ ~ G = A − KC , where η is defined in (3.64). As a result, filter (3.45) yields

(3.69)

(a) The augmented system (3.49) and (3.50) is asymptotically stable and (b)

H zw (s) ∞ ≤ γ .

60

Proof: From Lemma 3.2 and Lemma 3.3, combining (3.61)-(3.65), one has

⎡Γ11 Γ12 ⎤ Γ=⎢ T ⎥ < 0, ⎣Γ12 Γ22⎦

(3.70)

where −1 −2αh

Γ11 = A P1 + P1 A + +2α P1 + σ 1e2αh I +σ 2 e2αh I +σ 1 e T

+ P1 Ad (Q1 − ε −1 )−1 ATd P + 2ε P1 Adm ATdm P1 + Q1 + −1 −2αh

Γ12 = ( A − G − KC ) P 2 + σ 1 e T

+ 2ε P1 Adm ATdm P 2 + Γ 22 = G P 2 + P 2 G + σ 1 e T

1

γ

1

γ

T

P1 Am Am P1

P1 E1 E1 P1 + h R1 , T

−1 T T −1 P1 Am Am P 2 + P1 Ad (Q1 − ε I ) ( Ad − K C d ) P 2

P1 E1 ( E1 − K E 2 ) P 2 ,

2α h

(3.71)

T

(3.72)

I + σ 2 e2αh I + σ 1−1 e−2αh P 2 Am AmT P 2

+ σ 2−1 e−2αh P2 K C m CTm K T P2 + P2 ( Ad − K C d )(Q1 − ε −1 I )−1 ( Ad − K C d )T P2 + 2ε P 2 K C dm C Tdm K T P 2 + 2ε P 2 Adm ATdm P 2 + Q 2 + 1 + [ P 2 ( E1 − K E 2)( E1 − K E 2 )T P 2] + h R 2 .

γ

1

γ

T L L

(3.73)

A sufficient condition to satisfy (3.70) is that Γ11 < 0 ,

(3.74a)

Γ 22 < 0 , and

(3.74b)

Γ12 = 0 ,

(3.74c)

which implies

61

−1 −2αh

Γ11 = A P1 + P1 A + +2α P1 + σ 1e2αh I +σ 2 e2αh I +σ 1 e T

+ P1 Ad (Q1 − ε −1 )−1 ATd P + 2ε P1 Adm ATdm P1 + Q1 +

1

γ

T

P1 Am Am P1

P1 E1 E1 P1 + h R1 < 0 . T

(3.75)

Using the Schur complement and denoting µ = ε −1 , inequality (3.75) can be transformed into the form (3.66). Next, we consider the inequality as follows

Γ 22 = G P 2 + P 2 G + σ 1 e T

2α h

I + σ 2 e2αh I + σ 1−1 e−2αh P 2 Am AmT P 2

+ σ 2−1 e−2αh P2 K C m CTm K T P2 + P2 ( Ad − K C d )(Q1 − ε −1 I )−1 ( Ad − K C d )T P2 + 2ε P 2 K C dm C Tdm K T P 2 + 2ε P 2 Adm ATdm P 2 + Q 2 +

1

γ

T L L

1 + [ P 2 ( E1 − K E 2)( E1 − K E 2 )T P 2] + h R 2 < 0 .

(3.76)

γ

Substituting (3.69) into (3.76), one has 1 T ~ ~T 2αh 2αh −1 − 2αh T Am Am A P 2 + P 2 A + 2α P 2 + σ 1 e I + σ 2 e I + L L + Q 2 + h R 2 + P 2 [σ 1 e

γ

+ Ad (Q1 − ε −1 I )−1 ATd + 2ε Adm ATdm +

1

γ

T T T E1 E1 ] P 2 − ( P 2 K )η − η ( P 2 K )

+ ( P 2 K )Σ ( P 2 K ) < 0 . T

(3.77)

From (3.68), we obtain T

⎡( K ) 12 − ηT − 12 ⎤ ⋅ ⎡( K ) 12 −ηT − 12 ⎤ = ( SU ) ( SU )T = S T . S Σ ⎥ Σ ⎥ ⎢ P2 Σ ⎢⎣ P 2 Σ ⎦ ⎦ ⎣ Then, equation (3.77) becomes 1 T ~ ~T 2αh 2αh A P 2 + P 2 A + 2α P 2 + σ 1 e I + σ 2 e I + L L + Q2 + h R 2

γ

+ P 2 [σ 1−1 e− 2αh Am ATm + Ad (Q1 − ε −1 I )−1 ATd + 2ε Adm ATdm + − η T Σ −1η + S S T < 0 .

1

γ

T

E1 E1 ] P 2 (3.78)

62

Since the term η T Σ −1η in (3.78) is positive-definite, we have

1 T ~ ~T 2αh 2αh A P 2 + P 2 A + 2α P 2 + σ 1 e I + σ 2 e I + L L + Q2 + h R 2

γ

+ P 2 [σ 1−1 e− 2αh Am ATm + Ad (Q1 − ε −1 I )−1 ATd + 2ε Adm ATdm +

1

γ

T T E1 E1 ] P 2 + S S .

(3.79)

Using the Schur complement formula, equation (3.79) can be easily considered in the form (3.67). Moreover, we also can have Γ12 = 0 by substituting (3.69) into (3.72). Finally, we immediately get Γ < 0 . The system (3.49) and (3.50) is robustly asymptotically stable and

H zw (s) ∞ ≤ γ . So, we complete the proof of Theorem 3.3.

□

Remark 3.3 The proposed design procedures are then summarized as follows:

(i) Read given data h , γ , A , Ad , C , Am , Adm , C m , C dm , L . ~ ~ (ii) Solve inequality (3.16) for P1 , σ i > 0 , i = 1, 2 , µ , and to obtain A, C, Σ .

(iii) Solve inequality (3.17) for P2 to find η . (iv) Compute K and G . 3.3.3 An Illustrative Example

Example 3.2: Consider the second-order system x& (t ) = [ A + ∆A] x(t ) + [ Ad + ∆ Ad ] x (t − h) + w(t ),

(3.80)

y (t ) = [C + ∆C ] x (t ) + [C d + ∆ C d ] x (t − h) + w(t ),

(3.81)

⎡− 1 ⎡− 2 1⎤ A=⎢ , Ad = ⎢ ⎥ ⎣− 1 ⎣ 0 − 1⎦

0⎤ ⎡1 0⎤ ⎡0.1 0⎤ , Cd = ⎢ , C=⎢ ⎥ , L = [0.5 0.4] , ⎥ ⎥ − 1⎦ ⎣0 1⎦ ⎣ 0 0.1⎦

⎡0.1 0⎤ ⎡0.1 ⎡0.1 0 ⎤ , Adm = ⎢ , Cm = ⎢ Am = ⎢ ⎥ ⎥ ⎣0 0.1⎦ ⎣ 0 ⎣ 0 0.1⎦

0 ⎤ ⎡0.1 , C dm = ⎢ ⎥ 0.1⎦ ⎣0

0 ⎤ , h = 2. 0.1⎥⎦

The purposes here are to design the robust H ∞ filtering in the structure (3.45), the

63

filtering process is asymptotically stable and the transfer function from exogenous disturbances to error state outputs meets the pre-specified upper bound constrain

H zw ( s) ∞ ≤ γ = 0.8 . Solving matrix inequality (3.66) by using the LMI-control toolbox, we obtain ⎡ 0.7361 P1 = ⎢ ⎣− 0.0002

− 0.0002 ⎤ ⎡2.6696 , Q1 = ⎢ ⎥ 0.6919⎦ ⎣0.4024

0.4024⎤ , σ 1 = 1.044 , σ 2 = 8.9694 , 2.2484⎥⎦

µ = 0.9576 . Next, solving matrix inequality (3.67) by using the LMI-control Toolbox gives ⎡ 0.8359 − 0.0555⎤ ⎡ 0.6335 − 0.0215⎤ ⎡ 0.5919 − 0.0252⎤ , Q2 ⎢ , S=⎢ P2 = ⎢ ⎥ ⎥. ⎥ 0.7303⎦ ⎣− 0.0555 ⎣− 0.0189 0.5882⎦ ⎣− 0.0252 0.3857⎦ The orthogonal matrix U is defined as U = I 2 . Therefore, we obtain the expected filter parameters form (3.68) and (3.69) as follows ⎡36.9278 K =⎢ ⎣2.3091

2.3249 ⎤ ⎡− 21.7050 , G=⎢ ⎥ 56.4329⎦ ⎣− 14.7635

9.4072 ⎤ . − 22.5153⎥⎦

3.4 Summary The robust Kalman filtering for a class of uncertain time-delay interval systems has been proposed in this chapter with delay-dependent conditions. It has been presented that the Kalman filtering achieves the robust exponential stability. The analysis and design issues of the Kalman filtering are determined by solving LMIs. Moreover, the robust H ∞ filtering for a class of time-delay interval systems with delay dependence has also been proposed in this chapter. It has been presented that the filtering achieves the robust stability to satisfy the prescribed H ∞ performance level for the filtering error system. The analysis and design issues of the filtering are determined by solving two LMIs. Finally, two numerical examples are employed to illustrate the theoretical validness.

64

Chapter 4 Robust H ∞ Tracking for Systems with Time Delay

This chapter is to design a tracking control so that an H ∞ tracking performance, which is related to the tracking state error for some bounded reference inputs, is formulated for a class of uncertain time-delay continuous systems and for a class of uncertain discrete time-delay systems [81].

4.1 Introduction It is well known that the problem of robust stabilization of state delayed systems with uncertain parameters have received considerable attention of many researchers [16, 35, 39, 44, 73, 92]. It is also well-known that time delay arises quite naturally in various engineering systems, such as the turbojet engine, microwave oscillator, nuclear reactor, rolling mill, chemical process, long transmission lines in pneumatic, hydraulic systems [31]. A number of estimations and control problems relating to time-delay systems have been addressed by many researchers [12, 47, 80]. In general, the tracking control design is more difficult than the stabilization control design. Furthermore, by using the above-mentioned approaches of estimation, recently, attention has been focused on the problem of asymptotic tracking [66, 70] for non-time-delay systems. Further, in [11], nominal systems with a single time delay in the state has been developed, and the problem is to find a controller which tracks a given reference input with zero steady state error. Chen and Yu [9] proposed a time-delay feedback control of the non-delayed chaotic systems. To understand to what extent the time-delay feedback control method is useful for chaotic systems, some easily applied analytic (sufficient) conditions have been derived in that paper for both stabilization and tracking problems. However, control designs based on a nominal system may not work well in practice since the system parameters may deviate from their nominal values. This has led to an interest in robust controller design, i.e. the controller must be designed to achieve desired performance when there are perturbations in

65

the system. The robust asymptotic tracking problem for time-delay nominal systems was addressed in [78], where two different ways to design a tracking controller were adopted. With regard to the robust H ∞ tracking performance in terms of linear matrix inequalities and the dependence on the size of the delays have not been discussed. Recently, the interests are focused on using the linear matrix inequalities (LMIs) [5] approach to deal with the above problems [78]. However, the robust H ∞ delay-dependent continuous systems of asymptotic tracking in terms of LMI and the robust H ∞ discrete systems of asymptotic tracking in terms of LMI have not been fully investigated, which are still open and remain challenging. Motivated by the aforementioned concerns, the main aim of this chapter is to design tracking controls for a class of uncertain delay-dependent continuous systems with a time-varying time delay in the state and for a class of discrete systems with time delay in the state. The system parameter uncertainties are unknown but bounded. An H ∞ tracking performance, which is related to the tracking state error for some bounded reference inputs, is formulated. Sufficient conditions for the solvability of two problems are obtained in terms of LMIs. The desired controller can be constructed through a convex optimization problem, that it can be efficiently implemented using standard numerical algorithms.

4.2 Delay-Dependent Robust H ∞ Tracking for Uncertain Continuous Time-Delay Systems This section considers a tracking problem for a class of uncertain time-delay continuous systems. 4.2.1 System Description Consider the following uncertain time-delay systems described by x& (t ) = ( A + ∆A(t )) x (t ) + ( Ad + ∆ Ad (t )) x (t − h(t )) + ( B + ∆B (t ))u (t ) + w(t ) ,

(4.1)

where x(t ) ∈ Rn is the state vector, u (t ) ∈ Rm is the control input; A , B , Ad , and C are known constant matrices with appropriate dimensions, ∆A(t ) , ∆ Ad (t ) and ∆B (t ) are unknown matrices representing time-varying parameter uncertainties and are assumed

66

to be of the form

[∆A

∆B] = MF (t )[N 1

∆ Ad

N2

N3 ],

(4.2)

where M , N 1 , N 2 , and N 3 are known real constant matrices and F (t ) is an unknown real-valued time-varying matrix satisfying T F (t ) F (t ) ≤ I , ∀t ,

(4.3)

h(t ) is the unknown time-varying delay term but bounded, 0 ≤ h(t ) ≤ h , h&(t ) ≤ d < 1 ,

and w(t ) denotes the unknown but bounded disturbance. Consider a desired reference model as follows [34, 86] x& r (t ) = Ar x r (t ) + Adr x r (t − h(t )) + r (t ) , ∀ t ≥ 0 ,

(4.4)

where x r (t ) is the desired reference state, Ar is a specific asymptotically stable matrix, Adr is a known constant matrix with an appropriate dimension, and r (t ) is the reference input. Let the state error be e (t ) = x (t ) − x r (t ) .

(4.5)

In this case, the state feedback controller is designed as u (t ) = Ke(t ) .

(4.6)

After substituting (4.5) and (4.6) into (4.1), one has x& (t ) = ( A + ∆A(t )) x(t ) + ( Ad + ∆ Ad (t )) x (t − h(t )) + ( B + ∆B (t )) Ke (t ) + w(t ) .

(4.7)

We introduce the extended state vector and the external disturbance vector as follows ⎡ w(t )⎤ ⎡ x(t ) ⎤ , ξ (t ) = ⎢ ⎥. ⎥ ⎣r (t ) ⎦ ⎣ x r (t )⎦

η (t ) = ⎢

(4.8)

It follows from (4.4) and (4.7)-(4.8) that

η& (t ) = A c (t )η (t ) + A cd (t )η (t − h(t )) + ξ (t ) , where

67

(4.9)

A c (t ) = A + MF (t ) N 1 ,

(4.9a)

A cd (t ) = A d + MF (t ) N 2

(4.9b)

and with ⎡ A + BK − BK ⎤ ⎡ Ad , Ad = ⎢ A=⎢ ⎥ 0 Ar ⎦ ⎣ ⎣0

N 2 = [N 2

0 ⎤ ⎡M ⎤ ⎥ , M = ⎢ 0 ⎥ , N 1 = [N 1 + N 3 K Adr ⎦ ⎣ ⎦

− N 3 K ],

0] .

(4.10)

The purpose of this section is to construct the state feedback controller which satisfies all admissible uncertainties, the asymptotically stable augmented system (4.9) and guarantees the H ∞ tracking performance related to the tracking state error e(t ) , simultaneously, ∞

∞

0

0

T 2 T ∫ e (t )e(t )dt ≤ γ ∫ ξ (t )ξ (t )dt ,

(4.11)

where γ is a prescribed attenuation level. The physical meaning of (4.11) is that the effect of any ξ (t ) on the tracking state error e(t ) must be attenuated below a desired level γ from the viewpoint of energy, no matter what ξ (t ) is, i.e., the L 2 gain from ξ (t ) to e(t ) must be equal to or less than a prescribed value γ 2 .

Definition 4.1: If there exist α > 0 and β > 0 such that 2

2

η (t ) ≤ β e−αt sup η (θ ) , ∀t > 0 , − h ≤θ ≤ 0

(4.12)

then the extended system (4.9) is said to be exponentially stable, where η (θ ) is a continuous vector valued initial function, θ ∈ [ − h, 0] , and α > 0 is called the degree of exponential stability. The following matrix inequality will be essential for the proof. Lemma 4.1: Let Y be a symmetric matrix, and D, E be given matrices of appropriate dimensions, and F be a real matrix of appropriate dimension with F ≤ 1 . Then we have

68

(i)

For

ε >0

any

,

DFE + ET F T DT ≤ ε −1 D DT + ε ET E

.

(4.13)

(ii) For any matrix P > 0 and scalar ε > 0 satisfying P − εD DT > 0 , −1

( A + DFE ) P −1 ( A + DFE ) ≤ AT ( P − εD DT ) A + ε −1 E T E . T

(4.14)

4.2.2 Main Results In this section, we describe our method for determining the tracking control design with the extended uncertain time-delay system (4.9). The main results are given in the following theorems. Theorem 4.1: For a given h , the extended system (4.9) is said to be quadratically robustly exponential stable if there exist symmetric and positive-definite matrices P > 0 , Q > 0 and R > 0 and scalars α > 0 and 0 < d < 1 such that ⎡ Ω ⎢ T ⎣ A cd (t ) P

P A cd (t ) ⎤ ⎥ 0.

Proof: We consider the following Lyapunov-Krasovskii functional candidate as 0

t

t

V (η (t ), t ) = e2αt η T (t ) Pη (t ) + ∫ e2αθ η T (θ )Qη (θ )dθ + ∫ ∫ e2αuη (u ) Rη (u )dudβ . T

− h( t ) t + β

t − h (t )

(4.16) From (4.9) with ξ (t ) = 0 , equation (4.9) can be rewritten as

η& (t ) = A c (t )η (t ) + A cd (t )η (t − h(t )) . Based on (4.17) and making using of the derivative of V (η (t ), t ) , one has

69

(4.17)

V& = 2α e2αt η T (t ) Pη (t ) + 2 e2αt η&T (t ) Pη (t ) + e2αt η T (t )Qη (t ) + h(t ) e2αt η T (t ) Rη (t ) t

− (1 − h&(t )) e2α ( t − h ( t ))η T (t − h(t ))Qη (t − h(t )) − (1 − h&(t )) ∫ e2αsη ( s )Rη ( s )ds . T

(4.18)

t −h(t )

Thus

{

}

t

T V& = e2αt η T Πη − ∫ e2αsη ( s ) Rη ( s )ds ,

(4.19)

t − h (t )

where

η = [η T (t )

η T (t − h(t )) ]T ,

⎡ Ω Π=⎢ T ⎣ A cd (t ) P

⎤ ⎥, − (1 − d ) e− 2αh Q ⎦ P A cd (t )

with Ω = A Tc (t ) P + P Ac (t ) + 2αP + hR + Q . t

Since the last term ∫ e2αsη ( s ) Rη ( s )ds in (4.19) is positive definite, we obtain T

t − h (t )

V& ≤ e2αt {η T Π η } ,

(4.20)

hence, the augmented system (4.9) with ξ (t ) = 0 is stable. If the following inequality is satisfied ⎡ Ω ⎢ T ⎣ A cd (t ) P

P A cd (t ) ⎤ ⎥ < 0. − (1 − d ) e− 2αh Q ⎦

This concludes the proof of Theorem 4.1.

(4.21)

□

Theorem 4.1 provides a delay-dependent exponential stability criterion since the inequality condition (4.15) does include the delay h . To establish the H ∞ tracking performance for the augmented system, first, we define

70

⎡ S S =⎢ ⎣− S

− S⎤ , S ⎥⎦

(4.22)

where S > 0 . Next, we introduce the H ∞ tracking performance related to tracking error e(t ) as follows ∞

∞

∫ e e (t ) Se(t )dt = ∫ e2αt ( x(t ) − x r (t )) S ( x(t ) − x r (t ))dt 2αt T

0

T

0

∞

∞

0

0

T T = ∫ e2αtη (t ) S η (t )dt = ∫ (e2αtη (t ) S η (t ) + V& )dt .

Based on (4.9) and making using of the derivative of V (η (t ), t ) in (4.16), we obtain ∞

∞

0

0

{

}

T 2 T T 2αt ∫ e2αt e (t ) Se(t )dt ≤ ∫ e η Ση + γ ξ (t )ξ (t ) dt ,

(4.23)

with ⎡ Ω Σ=⎢ T ⎣ A cd (t ) P

⎤ ⎥, − (1 − d ) e− 2αh Q ⎦ P A cd (t )

(4.24)

where Ω = A Tc (t ) P + P Ac (t ) + 2αP + hR + Q + γ −2 PP + S . By Σ < 0 , which implies ∞

∞

0

0

T 2 2αt T ∫ e2αt e (t ) Se(t )dt ≤ γ ∫ e ξ (t )ξ (t )dt .

(4.25)

Also, equation (4.25) implies (4.12) is satisfied. Therefore, the robust H ∞ tracking control performance is approached with a prescribed γ 2 . Now we are in a position to provide a solution to the robust H ∞ tracking control problem for the augmented system (4.9).

71

Theorem 4.2: Consider the uncertain system (4.1). If there exist scalars ε 1 > 0 and

~ ~ ~ > 0 , ~ > 0 and β > 0 and matrices X 1 > 0 , X 2 > 0 , Q1 > 0 , Q 2 > 0 , R~1 > 0 , R S1 2

~ S 2 > 0 and Y such that the following LMIs hold for given constants α > 0 , d > 0 and h > 0, (i)

The symmetric positive-definite matrix X 1 satisfies (4.26) ⎡ Φ Φ1 ⎢ T ⎢ Φ1 − Ζ1 ⎢ X 1 ΦT 0 2 ⎢ 0 ⎢⎣ 0

0 ⎤ Φ2 X 1 ⎥ 0 0 ⎥ < 0, T ~ −Ζ X1M ⎥ 1 ⎥ M X 1 − β ⎥⎦

(4.26)

where ~ ~ + ~ + γ −2 I Φ = X 1 AT + A X 1 + Y T BT + BY + ε 1 e2αh M M T + 2α X 1 + Q1 + h R S1 1 −1

+ β (1 − d ) e2αh N 2 N T2 ,

~ T T αh 2αh ~ T Φ1 = X 1 N 1 + Y N 3 , Φ 2 = e Ad , Ζ1 = ε 1 e , Ζ1 = (1 − d ) Q1 , and (ii)

Symmetric positive-definite matrix X 2 satisfies ⎡ Φ ⎢ T ⎢ Φ1 T ⎢ ⎢ X 2 Φ2 ⎢⎣ 0

Φ1 − Ζ1 0 0

Φ2 X 2 0 ~ −Ζ 2 M X2

⎤ ⎥ 0 ⎥ 0 , X 2 > 0 , Q1 > 0 , Q 2 > 0 , and Y such that the following LMIs hold for a given constant γ > 0 , (i) The symmetric positive-definite matrix X 1 > 0 satisfies ⎡ X1 X1 ⎢ ~ − Q1 ⎢ X1 ⎢ LX 0 1 ⎢ 0 ⎢N1 X 1 + N 2 Y ⎢ 0 ⎣⎢ A X 1 + BY

T T T X1 A + Y B ⎤ ⎥ 0 ⎥ 0 ⎥ 0 satisfies ⎡ − X2 X2 ⎢ ~ − Q2 ⎢ X2 ⎢ LX 0 2 ⎢ ⎢N 2 K s X 2 0 ⎢B 0 ⎢ Ks X 2 ⎢⎣ Ar X 2 0

T

X2L 0

T

T

X 2 Ks N2 0

T X 2 Ks B 0

−I

0

0

0 0

−ε 0

0

0

0

81

− Ω11 0

T X 2 Ar ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ < 0, 0 ⎥ 0 ⎥⎥ − Ω22 ⎥⎦

(4.58)

~ ~ where Ω11 = X 1 − εM M T − Ad Q1 ATd − γ −2 I , Ω22 = X 2 − Adr Q 2 ATdr − γ −2 I , then the robust H ∞ tracking control problem is solvable. Furthermore, when (4.57) and (4.58) are satisfied, the controller is K s = Y X 1−1 and a suitable reference model is given as (4.45).

In order to show that (4.50) is robustly stable with a H ∞ tracking control performance, it is required that the associated Hamiltonian

Proof:

H (η , ξ , k ) = ∆V (η ( k )) + zT ( k ) z ( k ) − γ 2 ξ T ( k )ξ ( k ) < 0 , ∀ k > 0 ,

(4.59)

where V (η ( k )) is given as follows k −1

V (η (k )) = η T (k ) Pη (k ) + ∑η T (i )Qη (i ) , i=k −h

with

P = PT > 0

and

Q=Q >0 T

(4.60)

are weighting matrices. By evaluating the

first-forward difference ∆V (η (k )) = V (η ( k + 1)) − V (η ( k )) along the solution of (4.50) and arranging terms, we get ∆V (η (k )) = ΠT (k ) Σ Π(k ) ,

(4.61)

where

Π ( k ) = [η T ( k ) ξ T ( k ) η T ( k − h) ]T ,

⎡− P + Q + A Tc (k ) P A c (k ) ⎢ Σ=⎢ P A c (k ) T ⎢ A cd P A c (k ) ⎣

A c (k ) P P T

A cd (k ) P T

⎤ ⎥ ⎥. T ⎥ A cd (k ) P A cd (k ) − Q ⎦ A c (k ) P A cd (k ) P A cd (k ) T

(4.62)

Performing the standard matrix manipulations yields

H (η , ξ , k ) = ΠT (k ) Σ1 Π (k ) , with

82

(4.63)

⎡− P + Q + A Tc (k ) P A c (k ) + C T C ⎢ P A c (k ) Σ1 = ⎢ ⎢ T A cd P A c (k ) ⎣

A c (k ) P −γ2I + P T

A (k ) P T cd

⎤ ⎥ ⎥. ⎥ T A cd (k ) P A cd (k ) − Q ⎦ A c (k ) P A cd (k ) P A cd (k ) T

(4.64)

The requirement H (η , ξ , k ) < 0 , ∀ Π ( k ) ≠ 0 implies Σ1 < 0 . Using Lemma 2.3, it is expressed as ⎡− P + Q + C T C 0 ⎢ −γ2I 0 ⎢ ⎢ 0 0 ⎢ ⎢⎣ I A C (k )

A c (k )⎤ ⎥ I ⎥ ⎥ < 0. T A cd ⎥ − P −1 ⎥⎦

0

T

0 −Q A cd

(4.65)

We then extend equation (4.65) as ⎡− P + Q + C T C 0 ⎢ −γ2I 0 ⎢ ⎢ 0 0 ⎢ ⎢⎣ A I

⎡ N 1T ⎤ ⎢ ⎥ 0 + ⎢ ⎥ F T (k ) 0 ⎢0 ⎥ ⎢ ⎥ ⎢⎣0 ⎥⎦

[

0

T A ⎤ ⎡0 ⎤ ⎥ ⎢ ⎥ I ⎥ ⎢0 ⎥ + F (k )[N 1 0 T ⎥ ⎢ ⎥ 0 A cd ⎥ ⎢ ⎥ − P −1⎥⎦ ⎣ M ⎦

0 0 −Q A cd

0

M

T

0

0]

]< 0.

(4.66)

Based on Lemma 2.4, equation (4.66) holds if and only if ⎡− P + Q + C T C + ε −1 N 1T N 1 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢⎣ A

0

0

−γ2I 0

0 −Q

I

A cd

⎤ ⎥ ⎥ ⎥ 0 and Q = diag{Qh , Qv} > 0 such that the LMI ⎡− S ⎢ T ⎢A S ⎢ TS ⎢ Ad ⎢B T S ⎢ ⎣⎢0

SA

S Ad

SB

−S +Q

0

0

−Q

0

0 0

0

−γ I

C

0

D

0 ⎤ T ⎥ C ⎥ 0 ⎥ 0 , Y = diag{Y h , Y v} > 0 , R = diag {R h , R v} > 0 , G = diag{G h , G v} > 0 satisfying (5.37), (5.38) and

⎡X h ⎢I ⎣

I⎤ ⎡X v ⎥ ≥ 0, ⎢ I Y h⎦ ⎣

I⎤ ⎡ Rh ⎥ ≥ 0, ⎢ I Y v⎦ ⎣

I ⎤ ⎡ Rv ⎥ ≥ 0, ⎢ I Gh⎦ ⎣

I ⎤ ⎥ ≥ 0, Gv⎦

(5.40)

then the 2-D H ∞ control problem of the system (Σ ∆ ) is solvable; that is, there exists ⎡ Dc C c ⎤ output feedback controller gain K = ⎢ ⎥ such that controller (Σc ) stabilizes ⎣ Bc Ac ⎦ system (Σ ∆ ) and the closed-loop system satisfies z 2 < γ 2 w 2 . 2

2

Proof: There exists matrices P h > 0 , P v > 0 , Z h > 0 and Z v > 0 satisfying (5.34) if and only if X h − Y −h1 > 0 , X v − Y v−1 > 0 , R h − G −h1 > 0 , and Rv − G v−1 > 0 hold, which is equal to (5.40). The rest of the proof follows from above discussion.

Theorem 5.2 provides a sufficient condition for the existence of the stabilizing controller, rather than develops a design method. However, a stabilizing controller can be designed using the following algorithm. 1. Compute a feasible solution X , R , Y , and G satisfying (5.37), (5.38), and (5.40). 2. Using the singular value decomposition, compute real matrices X h12 and X h22 such that X h12 X h−122 X Th12 = X h − Y h−1 .

100

⎡X h 3. Construct a positive-definite matrix P h > 0 of the form P h = ⎢ T ⎣ X h12

X h12 ⎤ ⎥ . Similarly, X h 22 ⎦

get matrices P v > 0 , Z h > 0 , and Z v > 0 . Thus, we have P = diag {P h , P v} > 0 and Z = diag {Z h , Z v} > 0 . 4. With P and Z given, solve the feasible problem (5.24) to obtain K since it becomes an LMI structure. Now we apply the above-mentioned problem to the problem of dynamic output feedback stabilization of 2-D systems (Σ) with norm bounded uncertainties (5.1). Combining the system (Σ) with the dynamic output feedback controller (5.20)-(5.21), one has ⎡η h (i + 1, j )⎤ ~ ⎡η h (i, j )⎤ ~ ⎡η hd (i − d h , j )⎤ ~ ⎥ + B w(i, j ) , ⎥ + Ad ⎢ v ⎥ = A⎢ v ⎢ v ⎢⎣η (i, j ) ⎥⎦ ⎢⎣η (i, j + 1) ⎥⎦ ⎢⎣η d (i, j − d v ) ⎥⎦

h ~ ⎡η (i, j )⎤ ~ z (i, j ) = C ⎢ v ⎥ + Dw(i, j ) , ⎢⎣η (i, j ) ⎥⎦

(5.41)

where

~ ~ ~ ~ T ~ = Ω ( A ) T + ΩM A = Ω( A0 + B00 K C 00) ΩT + ΩMF N~1 ΩT , A F N2Ω , d 1I0 Ω

~ ~ ~ ~ B = Ω( B 0 + B 00 K D 2) + ΩMF N 3 , C = (C 0 + D1 K C 00) ΩT , D = D11 + D1 K D 2 ,

in which ~ ⎡M ⎤ ~ M = ⎢ ⎥, N 1 = [N 1 ⎣0 ⎦

0] , N~ 2 = [N 2

0] .

Applying Theorem 5.1 and Lemma 2.4 to above matrices, results in a nonlinear matrix inequality in terms of K and P ,

101

⎡− P PA P Ad PB ⎢ T ~T ~ 0 ⎢ A P − P + Z + ε N1 N1 0 ⎢ T ~T ~ −Z +ε N 0 0 2 N2 ⎢ Ad P ⎢ TP 0 0 − γ 2 I + ε N T3 N 3 ⎢B ⎢0 C D 0 ⎢ ~T P 0 0 0 ⎢⎣ 3 M

~ 3PM ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ < 0. 0 ⎥ ⎥ 0 ⎥ ⎥ − ε ⎥⎦

0 C

T

0 D −I

T

0

(5.42)

We therefore consider another version of LMI condition, which is derived from Lemma 5.1 and Lemma 4.2 as follows:

~T ~ ~ Π⊥ Φ Π⊥ < 0 ,

(5.43)

~T ~ ~ Θ⊥ Φ Θ⊥ < 0 ,

(5.44)

and

where ⎡0 ⎢0 ⎢ ⎢W 1 ⎢ ~ =⎢0 Π⊥ ⎢ 0 ⎢ ⎢W 2 ⎢0 ⎢ ⎢⎣0

I 0 0 0 0 0 0 0

0 I 0 0 0 0 0 0

0 0 0 0 I 0 0 0

0 0 0 0 0 0 I 0

0⎤ ⎡W 3 ⎢0 ⎥ 0⎥ ⎢ ⎢0 0⎥ ⎢ ⎥ 0 ⎥ ~ T ⎢0 , Θ⊥ = ⎢0 0⎥ ⎢ ⎥ 0⎥ ⎢0 ⎢W 4 ⎥ 0 ⎢ ⎥ I ⎥⎦ ⎢⎣0

⎡ ⎡ − X #⎤ ⎡ A⎤ P⎢ ⎥ ⎢⎢ ⎥ ⎣ 0⎦ ⎢⎣ # # ⎦ ⎢[AT 0 ]P − X + R + ε N 1T N1 ⎢ 0 # ⎢ ~ ⎢ T Φ = ⎢[Ad 0]P 0 ⎢[B1T 0]P 0 ⎢ 0 C1 ⎢ ⎢ T 0 ⎢ 3 [M 0]P ⎢⎣

0 0 I 0 0 0 0 0

0 0 0 I 0 0 0 0

#

⎡ Ad ⎤ P⎢ ⎥ ⎣0 ⎦ 0

#

0

0

0 − R + ε N T2 N 2

0 0 0 0 I 0 0 0

0 0 0 0 0 I 0 0

⎡ B1⎤ P⎢ ⎥ ⎣ 0⎦ 0 0

0

0

T

0

0

0

102

T

⎡M ⎤ 3P ⎢ ⎥ ⎣0⎦ 0

0

0

As a result, one has Theorem 5.3 as follows:

0 C1 0

0 − γ 2 I + ε N T3 N3 0 D11

0

0⎤ 0 ⎥⎥ 0⎥ ⎥ 0⎥ , 0⎥ ⎥ 0⎥ 0⎥ ⎥ I ⎥⎦

0

D11 −I 0

0

0 −ε

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎡ Dc C c ⎤ gain K = ⎢ ⎥ , X = diag{ X h , X v} > 0 , ⎣ Bc Ac ⎦ Y = diag{Y h , Y v} > 0 , R = diag {R h , R v} > 0 , and G = diag{G h , G v} > 0 satisfying (5.42), I⎤ I⎤ I ⎤ ⎡X h ⎡X v ⎡ Rh ⎡ Rv I ⎤ and ⎢ ≥ 0, ⎢ ≥ 0, ⎢ ≥ 0, ⎢ ⎥ ⎥ ⎥ ⎥ ≥ 0 , which implies that G h⎦ Gv⎦ Y h⎦ Y v⎦ ⎣I ⎣I ⎣I ⎣I controller (5.20)-(5.21) stabilizes system (Σ) for a given scalar ε > 0 and the closed-loop Theorem

5.3:

If

there

exist

system satisfies z 2 < γ 2 w 2 , then LMIs 2

2

T T ⎡ H 0 C1 D11 A X ⎢ T T 0 ⎢ D11 C1 − H 1 B1 X ~T ⎢ XA X B1 −X Ad E0 ⎢ T Τ ⎢ 0 0 Αd − (− R + ε N 2 N 2) ⎢ ⎢⎣ 0 0 0 3 MT X

0 ⎤ ⎥ 0 ⎥ 0 ⎥ E~ 0 < 0 , ⎥ 3 XM ⎥ ⎥ − ε ⎥⎦

(5.45)

and ⎡ L ⎢ 3 MT ⎢ ⎢ T T ⎢( Ad G N 2 ) ~T ⎢ T Γ1 E1 ⎢ T ⎢ B1 ⎢ T T ⎢ Y A ⎢ 0 ⎣

3M

T

Ad G N 2

Γ1

B1

−ε

0

0

D11

0

− L0

0

0

0

− L1

0

0 D 0

0

0

− L2

0

0

Y C1

0

0

0

0

T 11

T

⎤ ⎥ 0 0 ⎥ ⎥ 0 0 ⎥ 0 0 ⎥ E~1 < 0 , ⎥ 0 ⎥ C1 Y ⎥ − L3 G N 1T ⎥ ( G N 1T )T − L4 ⎥ ⎦ AY

T

0

where T T H = − X + R + ε N 1T N 1 + C1T C1 , H 1 = γ 2 I − D11 D11 − ε N 3 N 3 ,

L = −Y + AY AT + Ad G ATd , L0 = (ε −1 − N 2 G N T2 ) , T T Γ1 = AY C1 , L1 = I − C1 Y C1 ,

2 −1 T T L 2 = γ I − ε N 3 N 3 , L3 = G − Y , L4 = (ε − N 1 G N 1 ) ,

103

(5.46)

⎡W 1 ⎢ ⎢W 2 ~ = ⎢0 E0 ⎢ ⎢0 ⎢⎣0

0

0

0

0

I

0

0

I

0

0

⎡W 3 ⎢ 0⎤ ⎢W 4 ⎥ ⎢0 0⎥ ⎢ ~ 0 ⎥ , E 1 = ⎢0 ⎥ ⎢0 0⎥ ⎢ I ⎥⎦ ⎢0 ⎢0 ⎣

have a set of feasible solutions

0

0

0

0

0 I

0 0

0 0

0 0

0

I

0

0

0 0

0 0

I 0

0 I

0

0

0

0

0⎤ 0 ⎥⎥ 0⎥ ⎥ 0⎥ , 0⎥ ⎥ 0⎥ I ⎥⎦

X = diag{ X h , X v} > 0 ,

Y = diag{Y h , Y v} > 0 ,

⎡ Dc C c ⎤ R = diag {R h , R v} > 0 , G = diag{G h , G v} > 0 , and K = ⎢ ⎥ . On the other hand, if ⎣ Bc Ac ⎦ there exist matrices X = diag{ X h , X v} > 0 , Y = diag{Y h , Y v} > 0 , R = diag {R h , R v} > 0 , ⎡ Dc C c ⎤ G = diag{G h , G v} > 0 , and K = ⎢ ⎥ satisfying (5.45) and (5.46), then there exist ⎣ Bc Ac ⎦ ⎡ Dc C c ⎤ matrices P > 0 ( P = ΩT SΩ) and K = ⎢ ⎥ satisfying (5.42). ⎣ Bc Ac ⎦

5.4 An Illustrative Example Let us consider a system described by (5.1)-(5.3) and suppose that the system data are given as follows: 1⎤ ⎡0 ⎤ ⎡ 1 ⎡0.1⎤ ⎡0.2 0.1⎤ ⎡0 , B1 = ⎢ ⎥ , B 2 = ⎢ ⎥ , C1 = ⎢ A=⎢ , Ad = ⎢ ⎥ ⎥ ⎣1 ⎦ ⎣− 1 ⎣0.8⎦ ⎣0.3 − 0.1⎦ ⎣0.5 0.4⎦

C 2 = [− 1

⎡0 ⎤ ⎡0 ⎤ ⎡1 0] , D11 = ⎢ ⎥ , D12 = ⎢ ⎥ , D 21 = 1 , M = ⎢ ⎣1 ⎦ ⎣1 ⎦ ⎣0

⎡0 N1 = ⎢ ⎣1

1⎤ ⎡0.1 , N2 = ⎢ ⎥ 0⎦ ⎣0.1

0⎤ , 1⎥⎦

0⎤ , 0⎥⎦

0⎤ ⎡0.1⎤ , N 3 = ⎢ ⎥ , ε = 0.1 , d h = 0.5 , d v = 1 . ⎥ 0⎦ ⎣ 1⎦

By employing the Matlab LMI Toolbox, we solve LMIs (5.45, (5.46), and (5.40) to yield 0 ⎤ 0 ⎤ 0 ⎤ ⎡4.9345 ⎡72.9140 ⎡62.3862 , , R=⎢ X =⎢ , Y =⎢ ⎥ ⎥ 6.2052⎥⎦ 33.9757⎦ 34.8965⎦ ⎣ 0 ⎣ 0 ⎣ 0

104

0⎤ ⎡15.3269 G=⎢ . 5.2566⎥⎦ ⎣ 0 Using the singular value decomposition, we obtain the following positive-definite matrices 0 ⎤ ⎡4.9435 ⎡62.3862 10.8563 0 ⎢3.1312 ⎢ 10.8563 ⎥ 2 0 0 ⎥ ⎢ P= , Z =⎢ ⎢ 0 ⎢ 0 0 34.8965 5.6245⎥ ⎢ ⎢ ⎥ 0 5.6245 1 ⎦ ⎣ 0 ⎣ 0

3.1312 0 0 ⎤ 2 0 0 ⎥⎥ . 0 6.2052 2.4368⎥ ⎥ 0 2.4368 1 ⎦

With the P and Z , solve (5.42) to yield the solution ⎡0.6952 K = ⎢⎢ 0.0185 ⎢⎣ 0.0046

0.5129 − 0.1064 0.0839

− 0.3095 ⎤ − 0.0175 ⎥⎥ . − 0.0178⎥⎦

Thus, we get a set of stabilizing gains as follows ⎡− 0.1064 Ac = ⎢ ⎣ 0.0839

− 0.0175⎤ ⎡0.0185⎤ , Bc = ⎢ ⎥ ⎥ , C c = [0.6952 0.5129] , − 0.0178 ⎦ ⎣0.0046⎦

Dc = [− 0.3095] . The closed-loop 2-D discrete state-delayed system under controller (5.20) and (5.21) with this set of gains has a noise attenuation level γ = 0.07 .

105

5.5 Summary Solutions for the H ∞ control and robust stabilization problem for 2-D discrete state-delayed systems described by the Roesser model with norm-bounded uncertainties and delays have been newly presented in this chapter. Sufficient conditions in term of LMIs are derived to ensure that control of 2-D systems using the H ∞ performance specification has a systematic solution. An example is worked out to illustrate the validness of the theoretical results.

106

Chapter 6 Robust H ∞ Filtering for Two-Dimensional Discrete State-Delayed Systems

In this chapter we deal with the problem of robust H ∞ filtering for a class of two-dimensional (2-D) discrete state-delayed systems described by the Roesser model. The aim of this chapter is to design a 2-D linear filter that ensures both the asymptotic stability and a specified H ∞ performance of the filtering error dynamics using a linear matrix inequality approach. A numerical example is given to demonstrate the theoretical results.

6.1 Introduction Two-dimensional (2-D) filter’s state estimator design has been an important research topic and addressed in numerous references [27, 89]. The 2-D Kalman filtering approach is one of the most popular ways on this topic. A common characteristic of the standard Kalman filtering algorithm is that an exact internal model of the system is available and the exogenous input signals are assumed to be Gaussian noises with known statistics. In some applications, however, the noise sources may not be exactly known which limit the scope of the Kalman filtering approach. A Complement to the Kalman filtering is the H ∞ filtering which does not require a priori knowledge of statistical properties, except the variance is assumed to be bounded. It has been shown that the H ∞ filtering technique provides a guaranteed noise attenuation level [56], that is, the study of the H ∞ filtering problem is concerned with the design of estimators ensuring that the filtering error system is asymptotically stable and the L 2 -induced gain from the noise signal to the estimation error is less than a prescribed level. On the other hand, it is well-known that delays arise quite naturally in propagation phenomena, population dynamics or engineering systems such as chemical processes, long transmission lines in pneumatic systems et al. [32]. Many control problems relating to

107

state-delayed 2-D systems have been addressed by a number of authors [8, 10, 79]. Recently, a great deal of attention has been devoted to the study of 2-D H ∞ filtering systems. Tuan et al. [85] have presented the robust H ∞ filtering problem for 2-D discrete systems. However, for state-delayed 2-D discrete systems and the problem of robust H ∞ filtering have not been fully investigated, which are still open and remain challenging. In this chapter, we develop a 2-D filtering approach with an H ∞ performance measure for 2-D discrete systems described by the 2-D Roesser models [63]. The class of 2-D discrete systems under consideration is described by a delayed linear state space model. The delays appear in both state and measured output equations. We present the design of a linear filter such that the filtering error dynamics are stable and the L 2 -induced gain from the noise signal to the estimation error is below a prescribed level. Sufficient conditions are proposed to guarantee the existence of desired filters which are derived in terms of the solutions to LMI [5].

6.2 Preliminaries Consider a 2-D discrete delayed systems described by Rosser model [79]:

⎡ xh (i − d h , j )⎤ ⎡ xh (i, j )⎤ ⎡ xh (i + 1, j )⎤ ( ) ( ) A A + + ∆ = + ∆ (Σ) : ⎢ v Ad ⎢ v ⎥ + B1 w(i, j ) , ⎢ v ⎥ Ad ⎥ ( , 1 ) ( , ) ( i , j ) i j i j − + x d x x v ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (6.1)

⎡ xh (i − d h , j )⎤ ⎡ x h (i, j )⎤ y (i , j ) = (C + ∆C ) ⎢ v ⎥ + B 2 w(i, j ) , ⎥ + (C d + ∆ C d ) ⎢ v ⎣ x (i, j − d v) ⎦ ⎣ x (i, j ) ⎦ ⎡ x h (i − d h , j )⎤ ⎡ xh (i, j )⎤ z (i , j ) = ( E + ∆E ) ⎢ v + + ∆ ( ) Ed ⎢ v ⎥, ⎥ Ed − i j ( , ) ( i , j ) x x d v ⎣ ⎦ ⎣ ⎦

(6.2)

(6.3)

where xh ∈ Rn1 , xv ∈ Rn 2 are the horizontal and vertical state vectors, respectively, w(i, j ) ∈ Rq is the disturbance input, z (i, j ) ∈ R p is the controlled output, and y (i, j ) ∈ Rl is the measurement output, A , Ad , Bk , C , C d , E and E d are known real constant matrices, ∆A , ∆ Ad , ∆C , ∆ C d , ∆E and ∆ E d are unknown matrices representing parameter uncertainties and are of the form

108

⎡∆A ⎢∆C ⎢ ⎢⎣∆E

∆ Ad ⎤ ⎡ M 1 ⎤ ∆ C d ⎥⎥ = ⎢⎢ M 2 ⎥⎥ F [N 1 ∆ E d ⎥⎦ ⎢⎣ M 3 ⎥⎦

N 2] ,

(6.4)

where M 1 , M 2 , M 3 , N 1 and N 2 are known real matrices and F is an unknown matrix satisfying F T F ≤ I . d h and d v are positive integers representing horizontal and vertical delays of the systems, respectively.

6.3 Main Results In this section we propose two kinds of systems for nominal system and uncertain system. 6.3.1 Systems with Exactly Known Matrices

The main purpose of this subsection is to find the H ∞ filtering problem formulated. We first present the performance analysis result of the system (6.1) and (6.3) with exactly known matrices. The nominal 2-D discrete state-delayed system of (Σ) can be written as

⎡ x h (i + 1, j )⎤ (Σ1) : ⎢ v ⎥= i j ( , 1 ) + x ⎣ ⎦

⎡ xh (i − d h , j )⎤ ⎡ x h (i, j )⎤ A⎢ v + ⎥ + B1 w(i, j ) , ⎥ Ad ⎢ v i j ( , ) ( i , j ) − x x d v ⎣ ⎦ ⎣ ⎦

(6.5)

⎡ x h (i − d h , j )⎤ ⎡ xh (i, j )⎤ y (i , j ) = C ⎢ v + ⎥ + B2 w(i, j ) , ⎥ Cd ⎢ v i j ( , ) ( i , j ) − x x d v ⎣ ⎦ ⎣ ⎦

(6.6)

⎡ x h (i − d h , j )⎤ ⎡ xh (i, j )⎤ z (i , j ) = E ⎢ v ⎥. ⎥ + Ed ⎢ v ⎣ x (i, j − d v) ⎦ ⎣ x (i, j ) ⎦

(6.7)

109

Definition 6.1: Given a positive scalar γ , the 2-D system (6.5) and (6.7) with zero boundary condition is said to have an H ∞ performance γ if it is asymptotically stable [27] and its l 2 induced norm is bounded by γ , i.e.

sup 0 ≠ w∈l 2

z

2

w2

0 and Q > 0 such that SA ⎡− S ⎢ T ⎢A S − S + Q ⎢ ATd S 0 ⎢ T 0 ⎢ B1 S ⎢ 0 E ⎣

S Ad 0

S B1

−Q

0

0 E ed

0 −γ I 0 2

0 ⎤ T ⎥ E ⎥ T Ed ⎥ < 0 . ⎥ 0 ⎥ − I ⎥⎦

(6.9)

Proof: We first establish the asymptotic stability of the system (Σ1) with w(i, j ) ≡ 0 .

Let’s define a 2-D Lyapunov function candidate as

⎡ x h (i, j )⎤ ⎡ x h (i, j )⎤ ⎡ x h (i, j )⎤ i −1 h T h V (⎢ v ⎥) = ⎢ v ⎥ + ∑ x (l , j ) Q h x (l , j ) ⎥ S⎢ v l = i − d ( i , j ) h ⎦ ⎣ x (i, j ) ⎦ ⎣ x (i, j ) ⎦ ⎣ x T

j −1

+ ∑ xv (i, k )T Q v xv (i, k ) .

(6.10)

k = j −d v

⎡ xh (i, j )⎤ The difference of V ( ⎢ v ⎥ ) along the solution of (6.5) with w(i, j ) ≡ 0 is then given ⎣ x (i, j ) ⎦ by

⎡ x h (i, j )⎤ ⎡ xh (i + 1, j )⎤ ⎡ x h (i, j )⎤ ∆V ( ⎢ v ⎥) . ⎥) − V (⎢ v ⎥) = V (⎢ v ⎣ x (i, j ) ⎦ ⎣ x (i, j + 1) ⎦ ⎣ x (i, j ) ⎦

110

(6.11)

⎡ xh (i, j )⎤ ⎡ xh (i − d h , j )⎤ = We can define x = ⎢ v and xd ⎢ v ⎥ ⎥ , then, by using (6.5)-(6.7) and ( i , j ) ( i , j − ) x x d v ⎣ ⎦ ⎣ ⎦ completing the squares, one has ⎡x ⎤ ∆V ( x) = [xT xTd ]W ⎢ ⎥ , ⎣ xd ⎦

(6.12)

where

⎡ AT SA − S + Q W =⎢ T ⎣ Ad SA

T A S Ad T Ad S Ad − Q

⎤ ⎥. ⎦

(6.13)

From (6.9), it is easy to show that SA ⎡− S ⎢ T ⎢A S − S + Q ⎢⎣ ATd S 0

S Ad ⎤ ⎥ 0 ⎥ < 0. − Q ⎥⎦

(6.14)

By Schur complement, equation (6.14) implies that W < 0 and hence ∆V ( x ) < 0 , which guarantees the asymptotic stability of the system Σ1 with w(i, j ) ≡ 0 . Next, we shall show z < γ 2 w under zero boundary condition. To establish the 2

2

H ∞ performance, we introduce J (i, j ) = zT (i, j ) z (i, j ) − γ 2 wT (i, j ) w(i, j ) , = ∆V ( x ) + zT (i, j ) z (i, j ) − γ 2 wT (i, j ) w(i, j ) ,

(6.15)

where V ( x (0)) = 0 and V ( x(∞)) → 0 . Substituting (6.5)-(6.7) and (6.12) into (6.14), we obtain J (i, j ) = [x x T

T d

⎡x ⎤ w] Γ ⎢⎢ xd ⎥⎥ , ⎢⎣ w ⎥⎦

(6.16)

where

111

⎡ AT SA + C1T C1 − S + Q ⎢ T Γ=⎢ Ad SA T ⎢ SA + D11 C1 ⎣

T A S Ad T Ad S Ad − Q T

B1 S Ad

⎤ ⎥ ⎥. 2 T T − γ I + D11 D11 + B1 S B1⎥⎦ T T A S B1 + C1 D11 T Ad S B1

By Schur complement, we can verify that the LMI (6.8) guarantees Γ < 0 , and hence J (i, j ) < 0 for any non-zero w(i, j ) , which implies that z 2 < γ 2 w 2 . So, we complete 2

2

□

above the proof. Next, we consider the linear filter of the form

⎡ xˆ h (i − d h , j )⎤ ⎡ xˆ h (i, j )⎤ ⎡ xˆ h (i + 1, j )⎤ ⎥ + L{y (i, j ) ⎥ + Ad ⎢ v ⎢ v ⎥ = A⎢ v ⎣ xˆ (i, j − d v) ⎦ ⎣ xˆ (i, j ) ⎦ ⎣ xˆ (i, j + 1) ⎦ ⎡ xˆ h (i, j )⎤ ⎡ xˆ h (i − d h , j )⎤ ⎫⎪ − C⎢ v − ⎥ Cd ⎢ v ⎥⎬ , i j i j ( , ) ( , ) − ˆ ˆ x x d v ⎣ ⎦ ⎣ ⎦ ⎪⎭

⎡ xˆ h (i − d h , j )⎤ ⎡ xˆ h (i, j )⎤ zˆ(i, j ) = E ⎢ v ⎥, ⎥ + Ed ⎢ v ⎣ xˆ (i, j − d v) ⎦ ⎣ xˆ (i, j ) ⎦

(6.17)

(6.18)

where L is constant matrix to be designed with an appropriate dimension. Let ~ x (i, j ) = x (i, j ) − xˆ (i, j ) .

(6.19)

From (6.5)-(6.7), (6.17) and (6.18), we have h h h ⎡~ ⎡~ ⎡~ x (i + 1, j )⎤ x (i, j )⎤ x (i − d h , j )⎤ ⎥ + Bw(i, j ) , ⎥ + ( Ad − L C d ) ⎢ ~v ⎢ ~v ⎥ = ( A − LC ) ⎢ ~v ⎣ x (i, j − d v) ⎦ ⎣ x (i, j ) ⎦ ⎣ x (i, j + 1) ⎦

(6.20)

where B = B1 − L B 2 .

(6.21)

We introduce the extended state vector

⎡Ωh (i, j )⎤ ~ Ω=⎢ v ⎥ , z (i, j ) = z (i, j ) − zˆ (i, j ) , ⎣Ω (i, j ) ⎦

112

(6.22)

where

⎡ xh (i, j ) ⎤ Ω (i, j ) = ⎢ ~ h ⎥, ⎣ x (i, j )⎦

(6.23a)

⎡ xv (i, j ) ⎤ Ω (i, j ) = ⎢ ~v ⎥. ⎣ x (i, j )⎦

(6.23b)

h

and v


⎡Ωh (i − d h , j )⎤ ⎡Ωh (i + 1, j )⎤ ⎡Ωh (i, j )⎤ = + ⎥ + Be w(i, j ) , ⎢ v ⎥ Ae ⎢ v ⎥ Aed ⎢ v i j i j i j ( , 1 ) ( , ) ( , ) − + d Ω Ω Ω v ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(6.24)

⎡Ωh (i − d h , j )⎤ ⎡Ωh (i, j )⎤ ~ z (i, j ) = E e ⎢ v + ⎥ ⎥ E ed ⎢ v ⎣Ω (i, j − d v) ⎦ ⎣Ω (i, j ) ⎦

(6.25)

where 0 ⎤ ⎡A ⎡ Ad , Aed = ⎢ Ae = ⎢ ⎥ ⎣0 A − LC ⎦ ⎣0

E e = [0

E ] , E ed = [0

0 ⎤ ⎡ B1⎤ , Be = ⎢ ⎥ , ⎥ A − L Cd ⎦ ⎣B ⎦

E d ].

(6.26)

Via Theorem 6.1, we have that if there exist symmetric and positive-definite matrices S e = diag {S h , S v} > 0 and Qe = diag{Qh , Qv} > 0 such that the LMI ⎡− S e ⎢ T ⎢ Ae S e ⎢ ATed S e ⎢ ⎢ BTe S e ⎢ ⎣ 0

S e Ae − S e + Qe 0

S e Aed 0 − Qe

⎤ ⎥ E ⎥ T E ed ⎥ < 0 ⎥ 0 ⎥ ⎥ −I ⎦

0

S e Be 0

T e

0

0

0

−γ I

Ee

E ed

0

2

holds, then the extended system is asymptotically stable and satisfies

(6.27)

z 2 0 and Qe = diag{Qh , Qv} > 0 satisfying (6.28) and L = S v−1 M , then the 2-D H ∞ filtering problem of the system (Σ)

is solvable; that is, when (6.28) is satisfied, a feasible linear filter is presented as (6.17) and (6.18). Proof: The proof of the Theorem 6.2 follows from above discussion.

Theorem 6.2 provides a sufficient condition for the design of the H ∞ filtering problem. 6.3.2 Systems with Uncertain Matrices

This subsection is devoted to further extend the results found in the above section to systems with uncertain matrices in (6.1)-(6.3).

114

⎡ xh (i − d h , j )⎤ ⎡ xh (i, j )⎤ ⎡ xh (i + 1, j )⎤ A A ( ) ( ) + + ∆ = + ∆ (Σ ) : ⎢ v Ad ⎢ v ⎥ + B1 w(i, j ) , ⎢ v ⎥ Ad ⎥ i j i j ( , 1 ) ( , ) ( i , j ) − + x x x d v ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎡ xh (i − d h , j )⎤ ⎡ x h (i, j )⎤ y (i , j ) = (C + ∆C ) ⎢ v ⎥ + B 2 w(i, j ) , ⎥ + (C d + ∆ C d ) ⎢ v ⎣ x (i, j − d v) ⎦ ⎣ x (i, j ) ⎦ ⎡ x h (i − d h , j )⎤ ⎡ xh (i, j )⎤ z (i , j ) = ( E + ∆E ) ⎢ v + + ∆ ( ) Ed ⎢ v ⎥. ⎥ Ed − i j ( , ) ( i , j ) x x d v ⎣ ⎦ ⎣ ⎦

(6.30)

(6.31)

(6.32)

x (i, j ) = x(i, j ) − xˆ (i, j ) . From (6.30)-(6.32), (6.17) and (6.18), equation (6.20) can Set ~ be extended to h h h ⎡~ ⎡~ ⎡~ x (i + 1, j )⎤ x (i, j )⎤ x (i − d h , j )⎤ ⎥ ⎥ + ( Ad − L C d ) ⎢ ~v ⎢ ~v ⎥ = ( A − LC ) ⎢ ~v ⎣ x (i, j − d v) ⎦ ⎣ x (i, j ) ⎦ ⎣ x (i, j + 1) ⎦

⎡ xh (i − d h , j )⎤ ⎡ xh (i, j )⎤ ( L ) + ∆ − ∆ + (∆A − L∆C ) ⎢ v Cd ⎢ v Ad ⎥ + Bw(i, j ) . ⎥ ( i , j ) ( i , j ) − x x d v ⎣ ⎦ ⎣ ⎦

(6.33)

From the extended state vector, one has

⎡Ωh (i − d h , j )⎤ ⎡Ωh (i, j )⎤ ⎡Ωh (i + 1, j )⎤ ⎥ + Be w(i, j ) , ⎥ + ( Aed + ∆ Aed ) ⎢ v ⎢ v ⎥ = ( Ae + ∆ Ae) ⎢ v ⎣Ω (i, j − d v) ⎦ ⎣Ω (i, j ) ⎦ ⎣Ω (i, j + 1) ⎦

(6.34)

⎡Ωh (i − d h , j )⎤ ⎡Ωh (i, j )⎤ ~ + + ∆ ( ) z (i, j ) = ( E e + ∆ E e) ⎢ v E ed ⎢ v ⎥, ⎥ E ed − ( , ) ( i , j ) i j d Ω Ω v ⎣ ⎦ ⎣ ⎦

(6.35)

where 0⎤ 0⎤ ⎡ ∆A ⎡ ∆ Ad , ∆ Aed = ⎢ ∆ Ae = ⎢ ⎥ ⎥ , ∆ E e = [∆E ⎣∆A − L∆C 0⎦ ⎣∆ Ad − L∆ C d 0⎦

∆ E ed = [∆ E d

0] .

115

0] ,

Define ⎡∆ Ae ⎢∆ ⎣ Ee

∆ Aed ⎤ ⎡ M 1e ⎤ F [N 1e = ∆ E ed ⎥⎦ ⎢⎣ M 3 ⎥⎦

N 2e]

where ⎤ ⎡ M1 M 1e = ⎢ ⎥ , N 1e = [N 1 ⎣M 1 − L M 2⎦

0 ] , N 2 e = [N 2

0] and F T F ) ≤ I .

Then, one has the following result. Applying Theorem 6.1 and Lemma 2.4 to above matrices yields a nonlinear matrix inequality in terms of L and S e as ⎡− S e ⎢ T ⎢ Ae S e ⎢ ATed S e ⎢ ⎢ BTe S e ⎢ ⎢ 0 ⎢⎣ M 1Te S e

S e Ae Γ1

S e Aed

ε N N 2e

T 2e

ε N N 1e

Γ2

0

0

Ee 0

E ed 0

0

S e Be 0

T 1e

E

T e T

0

E ed

−γ2I

0

0 −I

0

M3

T

S e M 1e ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 and Qe = diag{Qh , Qv} > 0 , L , and scalar ε > 0 satisfying (6.36), which implies that

the extended system (6.34) and (6.35) under filter (6.17)and (6.18) is asymptotically stable and the estimation error satisfies Definition 6.1, then LMI (6.37) has a set of feasible solutions S h > 0 , S v > 0 , Q h > 0 , Qv > 0 , and M . Proof: The proof follows directly from the above derivation.

6.4 An Illustrative Example Example 6.1: Let us consider a system described by (6.1)-(6.3) and suppose that the system data are given as follows: ⎡0.3 A=⎢ ⎣ 1

0⎤ ⎡0.5⎤ ⎡1 ⎤ ⎡ 1 ⎡0.2 0.1⎤ , B1 = ⎢ ⎥ , B 2 = ⎢ ⎥ , C = ⎢ , Ad = ⎢ ⎥ ⎥ 0.2⎦ ⎣0.3⎦ ⎣0.2⎦ ⎣0.8 ⎣0.3 − 0.1⎦

⎡0 Cd = ⎢ ⎣1

− 1⎤ ⎡0.1 , E=⎢ ⎥ 1⎦ ⎣0.5

⎡0.2 M2 = ⎢ ⎣0.1

0.3⎤ ⎡0 − 0.1⎤ ⎡0.2 , Ed = ⎢ , M1 = ⎢ ⎥ ⎥ 1⎦ 0.1⎦ ⎣0.6 ⎣0.1

0.3⎤ ⎡0.3 , M3 = ⎢ ⎥ 0.2 ⎦ ⎣0.2

0.1⎤ ⎡0.1 , N1 = ⎢ ⎥ 0.1⎦ ⎣0.2

117

0.3 ⎤ , 0.2⎥⎦

0.5⎤ , 0.2 ⎥⎦

0.1⎤ , 0.2⎥⎦

⎡0.1 N2 = ⎢ ⎣ 0

0.3 ⎤ , d h = 0 .5 , d v = 1 . 0.2⎥⎦

By employing the Matlab LMI Tool, we solve (6.37) to yield ⎡ 0.5214 − 0.1229⎤ ⎡ 0.2693 − 0.3021 ⎤ , Sv = ⎢ , Sh = ⎢ ⎥ 0.6674⎥⎦ 0.5700⎦ ⎣− 0.1229 ⎣ − 0.3021 ⎡ 0.6188 Qh = ⎢ ⎣− 0.0677

− 0.0677 ⎤ ⎡ 0.6787 , Qv = ⎢ ⎥ − 0.7099⎦ ⎣− 0.0295

− 0.0295⎤ ⎡ 0.1943 , M =⎢ ⎥ − 0.6864⎦ ⎣ 0.1186

0.1186⎤ , 0.0801⎥⎦

ε = 2.6697 . Thus, we get L = S v−1 M as ⎡0.4335 L=⎢ ⎣0.2576

0.2674⎤ . 0.1693⎥⎦

The closed-loop 2-D discrete state-delayed systems under controller L = S v−1 M have noise attenuation level γ = 2.7851 .

6.5 Summary Solutions for the H ∞ filtering problem for 2-D discrete delayed systems described by the Roesser model are presented. A sufficient condition in term of LMI is derived to ensure that the desired filter of 2-D systems is not only stable but also satisfying a prescribed H ∞ performance. An example is worked out to illustrate the validness of the theoretical results.

118

Chapter 7 Robust Stabilization of Uncertain Stochastic Systems

In this chapter we deal with the robust exponential delay-dependent stabilization for a class of uncertain stochastic time-delay systems. The parameter uncertainty is norm-bounded and the delays are time varying. We then extend the proposed theory to discuss the robust stabilization of neutral stochastic interval systems with dependence of delay. Finally, attention is focused on the design of a state feedback controller that ensures not only the robust stability but also a prescribed H ∞ performance level for a class of neutral stochastic nonlinear time-delay system for all admissible uncertainties [45, 82, 83].

7.1 Introduction The problem of delayed systems has been investigated over the years. The phenomena of time-delay are very often encountered in different technical systems, such as electric, pneumatic, and hydraulic networks, chemical processes, long transmission lines, etc. A delay in the control or state-evolution laws may cause an undesirable system transient response or even instability. Consequently, the analysis of stability for this class of system has been of interest to many researchers. Numerous reports have been published on this matter [7, 41, 73]. [7] and [41] present delay-dependent criteria in terms of LMIs which were deterministic systems. In the aspect of robustness of stochastic stability, Haussman [20] studied the robust stability for stochastic systems. Whereas the delay case was not presented. The stability of linear stochastic differential equation with time-delay systems via Riccati equations was introduced in [54]. However, to the best of the authors’ knowledge, the problem of robust stabilization for uncertain stochastic time-delay systems combined with linear matrix inequalities remains open. With regard to the stability of time-delay systems of neutral type has also received some attention during the past years due to their theoretical importance as well as the extensive applications of these systems in many areas such as population dynamic model,

119

microwave oscillator, nuclear reactor, and so on [31, 40, 47, 49, 57]. Moreover, neutral stochastic uncertain time-delay systems were introduced and the problems of the stability have been also studied by [52, 53]. However, if the parameters of stochastic systems are estimated using confidence intervals, the systems will get stochastic interval equations and the study of such systems are much more complicated [38, 55], so far, the problems of robust stochastic neutral interval systems with delay dependence have not yet been fully investigated, which are more involved and still open. Similar system which incorporates H ∞ for a class of neutral stochastic nonlinear time-delay systems with dependence of delay has not been fully investigated also and this problem is still open and remains challenging. This chapter first deals with robust stabilization for a class of stochastic uncertain time-delay systems with dependence of delay via LMIs. Moreover, we study the robust stabilization for a class of neutral stochastic interval systems with dependence of delay in terms of LMIs. Finally, we investigate the robust H ∞ control for a class of neutral stochastic nonlinear systems with dependence of delay in terms of LMIs.

7.2 An LMI-based Approach for Robust Stabilization of Uncertain Stochastic Systems with Time-varying Delays In this section we introduce the robust stability of uncertain linear stochastic differential delay systems with delay dependence. The parameter uncertainty is norm-bounded and the delays are time varying. We then extend the proposed theory to discuss the robust stabilization of uncertain stochastic differential delay systems. 7.2.1 Preliminaries Consider the following uncertain stochastic system with time-varying time delays described by dx (t ) = [( A + ∆A(t )) x(t ) + ( Ad + ∆ Ad (t )) x(t − h(t )) + ( B + ∆B (t )) u (t ) ]dt + [(C + ∆C (t )) x(t ) + ( D + ∆D (t )) x(t − h(t ))] dw(t ) x(t ) = φ (t ),

∀t ∈ [ − h,0 ] ,

(7.1)

where x(t ) ∈ Rn is the state vector, A , Ad and B are known constant matrices with appropriate dimensions, ∆A(t ) , ∆ Ad (t ) and ∆B (t ) are unknown matrices with

120

appropriate dimensions which represent the system uncertainties, ∆C (t ) and ∆D (t ) are also unknown matrices with appropriate dimensions which represent the uncertainties of stochastic perturbations. w(t ) is a scalar Brownian motion defined on the probability space (Ω , F , {F t}t≥0 , P ) . h(t ) is time-varying bounded delay time satisfying 0 ≤ h(t ) ≤ h , h&(t ) ≤ d < 1 . φ (t ) is any given initial data x(t ) = φ (t ) ∀t ∈ [ − h,0 ] in L2F 0 ([−h,0]; Rn) . It

is well known that (7.1) has a uniqe solution, denoted by x (t ; φ ) , that is square integrable. In this chapter, we assume that the uncertainties can be described as follows:

[∆A

∆ Ad ∆B ∆C ∆D] = MF (t )[N 1 N 2

N3

N4

N 5],

(7.2)

where M and N i , i = 1, L ,5 are known constant real matrices of appropriate dimensions, and F (t ) ∈ Rk×l is an unknown matrix function with Lebesgue measurable elements and satisfies F T (t ) F (t ) ≤ I , in which I is the identity matrix with compatible dimension. The following is necessary in the theory of stochastic differential equations. Definition 7.1: If there exist α > 0 and γ > 0 such that 2

2

E x(t ) ≤ γ e −αt sup E φ (t ) , t > 0 .

(7.3)

− h ≤θ ≤ 0

Then, the system (7.1) is said to be exponentially mean square stable The main results hinge on the following fact. Fact 7.1 [30], [53]: The trivial solution of a stochastic differential equation dx (t ) = F ( x (t ), x (t − τ ), t ) dt + G ( x (t ), x (t − τ ), t ) dw(t ) x(t ) = φ (t ),

∀t ∈ [ −τ , 0 ] ,

(7.4)

on t ∈ [t 0 , T ] with initial data f (φ , t ) = F (φ (0), φ ( −τ ), t ) and g (φ , t ) = G (φ (0), φ ( −τ ), t ) ,

(7.5)

F : Rn × Rn × [t 0 , T ] → Rn G : Rn × Rn × [t 0 , T ] → Rn×m and where for (φ , t ) ∈ C([−τ , 0]; Rn) × [t 0 , T ] , is globally asymptotically stable in probability if there exists a function V (t , x ) ∈ C 2 ( R+ × Rn ) which is positive definite in the Lyapunov sense, and satisfies the generator LV as follows

121

LV =

∂V 1 + gradV F + (tr G GT ) HessV < 0 , ∂t 2

(7.6)

for x ≠ 0 and V (t , x) → +∞ as x → ∞ . The matrix HessV is the Hessian matrix of the second-order partial derivatives. This fact is analogous to the well-known theorem of Lyapunov for deterministic systems. 7.2.2 Main Results In this section, we first state robust stability criteria for the uncertain stochastic time-delay system (7.1) with u (t ) ≡ 0 . Finally, we propose the robust stabilization. 7.2.2.1 Robust Stability The purpose here is to introduce delay-dependent conditions for robust exponential stability of system (7.1) via the Lyapunov-Krasovskii functional approach. Theorem 7.1: Given a h , system (7.1) with u (t ) ≡ 0 is robustly exponentially stable, if there exist symmetric positive-definite matrices P , R and Q and scalars ε i > 0 , i = 1, 2, 3 , such that the following LMI ⎡Ω Π 0 ⎤ MP MP C T P ⎢ T ⎥ T − Ω1 0 0 0 ⎥ D P ⎢Π ⎢M T P 0 − ε 1 e− 2αh 0 0 0 ⎥ ⎢ ⎥ < 0, 0 0 − ε 2 e− 2αh 0 0 ⎥ ⎢M T P ⎢P C −P 0 0 PD PM ⎥⎥ ⎢ T ⎢⎣ 0 0 0 0 M P − ε 3 ⎥⎦ holds for a given α > 0 , where Ω = AT P + PA + 2αP + hQ + R + ε 1 e−2αh N 1T N 1 + ε 3 N T4 N 4 , Π = P Ad + ε 3 N T4 N 5 , Ω1 = (1 − d ) e

−2αh

R − ε 2 e−2αh N T2 N 2 − ε 3 N T5 N 5 .

122

(7.7)

Proof:

Considering the Lyapunov-Krasovskii functional [31] t

0

V ( x) = e2αt xT (t ) Px(t ) + ∫ e2αθ xT (θ ) Rx(θ )dθ + ∫

t

T ∫ e2αs x ( s )Qx( s )dsdβ .

(7.8)

−h (t ) t + β

t − h (t )

Along trajectories of (7.1) with u (t ) ≡ 0 and making use of the It oˆ -differential rule [30], the generator LV for the evolution of V is therefore given by LV = e2αt xT (t ) R x (t ) − e2α ( t − h (t )) xT (t − h(t ))(1 − h&(t )) Rx (t − h(t )) + h(t ) e2αt xT (t )Q x (t ) + 2 e2αt xT (t ) P[( A + ∆A(t )) x (t ) + ( Ad + ∆ Ad (t )) x (t − h(t ))]

{

T

+ e2αt [(C + ∆C (t )) x(t ) + ( D + ∆D(t )) x(t − h(t ))]

t

⋅ P[(C + ∆C (t )) x(t ) + ( D + ∆D(t )) x(t − h(t ))] } − (1 − h&(t )) ∫ e2αβ xT ( β )Q x( β )dβ . t − h (t )

(7.9) Using Lemma 4.1, one has 2 eαh xT (t ) P∆A(t ) x(t ) e−αh ≤ ε 1−1 e2αh xT (t ) PM M T Px(t ) + ε 1 e−2αh xT (t ) N 1T N 1 x(t ) , 2 eαh xT (t ) P∆ Ad (t ) x(t − h(t )) e−αh ≤ ε 2−1 e2αh xT (t ) PM M T Px(t ) + ε 2 e−2αh xT (t − h(t )) N T2 N 2 x(t − h(t )) ,

(7.10) (7.11)

and T

[(C + ∆C (t )) x(t ) + ( D + ∆D(t )) x(t − h(t ))] ⋅ P[(C + ∆C (t )) x(t ) + ( D + ∆D (t )) x(t − h(t ))]

[

]

~ T ~ = Ψ T (W + MFN ) P (W + MFN ) Ψ ,

(7.12)

where ~ Ψ = [ xT (t ) xT (t − h(t )) ]T , W = [C D ] , N = [N 4 N 5] .

From Lemma 4.1, we obtain ~ T ~ ~ −1 ~T N . (W + MFN ) P (W + MFN ) ≤ W T ( P −1 − ε 3−1 M M T ) W + ε 3 N

(7.13)

We define

⎡ S Γ=⎢ T ⎣ Ad P

P Ad ⎤ ⎥, − S0 ⎦

(7.14)

123

where S = AT P + PA + 2αP + ε 1−1 e−2αh PM M T P + ε 2−1 e−2αh PM M T P + R + hQ + ε 1 e−2αh N 1T N 1 , −2αh −2αh T S 0 = (1 − d ) e R − ε 2 e N 2 N 2 . Therefore ~ −1 LV ≤ e2αt Ψ T [Γ + W T ( P −1 − ε 3−1 M M T ) W + ε 3 N~ T N ]Ψ t

− (1 − d ) ∫ e2αβ xT ( β )Q x( β )dβ .

(7.15)

t −h(t )

t

Since the last term − (1 − d ) ∫ e2αβ xT ( β )Q x( β )dβ in (7.15) is positive-definite, one has t −h(t )

~ −1 LV ≤ e2αt Ψ T [Γ + W T ( P −1 − ε 3−1 M M T ) W + ε 3 N~ T N ]Ψ .

(7.16)

The requirement LV < 0 , ∀ Ψ ≠ 0 implies ~ −1 ~T N Γ + W T ( P −1 − ε 3−1 M M T ) W + ε 3 N < 0.

(7.17)

According to the Schur complement, equation (7.17) can be rewritten as ⎡Ω Π MP MP ⎢ T − Ω1 0 0 ⎢Π − h α 2 T ⎢M P 0 − ε1e 0 ⎢ T 0 0 − ε 2 e− 2αh ⎢M P ⎢ 0 0 D ⎢C ⎢⎣ 0 0 0 0

C

T T

D 0

0 − P −1 M

T

0 ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 , ⎣ Ad P − Z 0 ⎦ with

Z = AT P + PA + PBK + ( PBK )T + 2αP + ε 1−1 e2αh PM M T P + ε −21 e2αh PM M T P + ε 3−1 e2αh PM M T P + ε 1 e− 2αh N 1T N 1 + ε 2 e− 2αh K T N T3 N 3 K + hQ + R, Z 0 = (1 − d ) e

−2αh

~ R − ε 3 e−2αh N T2 N 2 , W = [C D ] , N = [N 4 N 5] .

It follows from the Schur complement that (7.20) is equivalent to ~ ⎡ Z P Ad ⎢ T ⎢ Ad P − Z 0 ⎢ 0 ⎢ N1 ⎢N 3 K 0 ⎢ D ⎢ C ⎢⎣ N 4 N5

T

N1 0

( N 3 K )T 0

C

T T

− ε 1−1 e2αh

0

D 0

0 0 0

− ε 2−1 e2αh

0

0 0

− G2 0

T N4 ⎤ ⎥ T N5⎥ 0⎥ ⎥ < 0, 0⎥ ⎥ 0⎥ − ε ⎥⎦

where ~ T Z = AT P + PA + PBK + ( PBK ) + 2αP + ε 1−1 e2αh PM M T P + ε −21 e2αh PM M T P

+ ε 3−1 e2αh PM M T P + hQ + R , G 2 = P−1 − εM M T .

125

(7.21)

Pre- and post-multiplying the (7.21) by diag ( P−1 , P−1 , I , I , I , I ) and denoting X = P −1 , ~ ~ Q = XQX , R = XRX , Y = KX , β 1 = ε 1−1 , β 2 = ε −2 1 and β 3 = ε 3−1 , result in

⎡ G ⎢ T ⎢ X Ad ⎢ 0 ⎢ ⎢N1 X ⎢ ⎢N 3Y ⎢CX ⎢ ⎢⎣ N 4 X

Ad X − G1

X N T2

N2 X

− β 3 e2 αh

0

X N 1T 0

Y N3 0

0

0

− β 1 e2 αh

T

T

X CT 0 X DT 0

0

0

0

0

0

0

0

DX

0

0

− G2

N5

0

0

0

0

− β 2 e2 αh

0

X N T4 ⎤ ⎥ 0 ⎥ T N5 ⎥ ⎥ 0 ⎥ 0 , i = 1,L ,3 ,

and ε such that the following LMIs (7.22) holds for a given α > 0 . Then, a suitable stabilizing control law is given by u (t ) = Kx(t ) = Y X −1 x(t ) , Y ∈ R m×n , where K = Y X −1 .

7.2.3 An Illustrative Example Example 7.1: Consider a two–dimensional uncertain stochastic differential equation dx (t ) = [ Ax (t ) + Ad x (t − h(t )) + ( B + ∆B (t )) u (t ) ]dt + [ ∆C (t ) x (t ) + ∆D (t ) x (t − h(t ))] dw(t )

(7.23)

126

0⎤ ⎡ − 2 0⎤ ⎡ 1 1⎤ ⎡− 1.4 , B=⎢ A=⎢ , Ad = ⎢ ⎥ ⎥, ⎥ ⎣1 − 3 ⎦ ⎣− 1 1⎦ ⎣− 0.8 − 1.5⎦ ⎡1 N3 = N4 = N5 = ⎢ ⎣0

⎡0.2 M =⎢ ⎣0

0⎤ , 0.2⎥⎦

0⎤ , α = 0.1 , h = 2 . 1⎥⎦

Applying the software package LMI to (7.23), it is found that, ⎡ 0.2154 − 0.0269⎤ ⎡4.7085 0.5200⎤ ~ ⎡ 0.8631 − 0.0293⎤ P=⎢ , X =⎢ ⎥ ⎥ , R = ⎢− 0.0293 0.8721⎥ , 0.2437⎦ ⎣− 0.0269 ⎣0.5200 4.1612⎦ ⎦ ⎣ ⎡− 0.5408 0.4703 ⎤ ~ ⎡ 0.3298 − 0.0240 ⎤ Q=⎢ , Y =⎢ ⎥ ⎥ , ε = 1.1330 , ε 1 = 0.8931 , 0.3577 ⎦ ⎣− 0.0240 ⎣− 0.6950 − 0.6019⎦

ε 2 = 1.2926 , ε 3 = 1.1274 . Therefore, this system (7.23) is robustly exponentially stabilizable. Moreover, the correspondent robustly stabilizing control law is ⎡− 2.3019 1.6756 ⎤ u (t ) = ⎢ ⎥ x(t ) . ⎣− 3.5852 − 2.8659⎦

7.3 Robust Stabilization of Uncertain Stochastic Neutral Interval Systems with Multiple Delays In this section, we first deal with the robust stability of uncertain linear stochastic neutral interval systems. The parameter uncertainties are unknown and the delays are time-invariant. Then, we extend the proposed theory to discuss the robust stabilization of uncertain stochastic neutral interval systems. The proposed results are given in terms of linear matrix inequalities. Two examples are worked out to illustrate the validness of the theoretical results. 7.3.1 Preliminaries The interval systems are introduced as follows [43, 95], if A = [ a ij ]u×v and B = [b ij ]u × v are two matrices with property that aij ≤ bij for all 1 ≤ i ≤ u , 1 ≤ j ≤ v , we

127

define the u × v interval matrix [ A, B ] by [ A, B ] = {C = [cij ]u×v : aij ≤ cij ≤ bij for all i, j}.

Let [ A, A ],

[ B, B ],

(7.24)

[C , C ] be given n × n interval matrices. Consider a linear

stochastic neutral interval delay system l

l

i =1

i =1

d [ x(t ) + ∑ C iI x(t − hi )] = [ A0 I x(t ) + ∑ AiI x(t − hi )]dt + [ D I x(t ) + E I x (t − τ )]dw(t ) ,

(7.25)

in which real t ≥ 0 and integer l ≥ 1 , hi is time-delay term, where A0 I ∈ [ A0, A 0], AiI ∈ [ Ai, A i ], D I ∈ [ D, D ] , E I ∈ [ E , E ] and C iI ∈ [C i, C i ] .

Thus we introduce 1 1 ( A0 + A 0) and A0 m ≡ ( A 0 − A0) . 2 2 Clearly, all the elements of A0 m are nonnegative. Moreover, A0 I can be written as A0 ≡

A0 I ≡ A0 + ∆ A0 with ∆A0 ∈ [ − A0 m , A0 m ] .

Similarly, we introduce Ai , Aim , ∆ Ai , C i , C im , ∆ C i and D, D m , ∆D , E , E m , ∆ E , i = 1, L , l . Then equation (7.25) can be written as l

l

i =1

i =1

d [ x(t ) + ∑ (C i + ∆ C i ) x(t − hi ] = [( A0 + ∆ A0) x(t ) + ∑ ( Ai + ∆ Ai ) x(t − hi )]dt + [( D + ∆D ) x (t ) + ( E + ∆E ) x (t − τ )]dw(t ) ,

(7.26)

where A0 , Ai , D, E , and C i are all deterministic matrices while ∆ A0 , ∆ Ai , ∆D, ∆E and ∆ C i represent the uncertain parts of A0 I , AiI , D I , E I and C iI , respectively. In fact, ∆ A0 will take a particular deterministic matrix within [− A0 m , A0 m ] but we just do not know which particular one, and similarly for ∆ Ai , ∆D, ∆E and ∆ C i . When we talk about the solution of (7.26), we mean the solution when ∆ A0 , ∆ Ai , ∆D, ∆E and ∆ C i take particular matrices within their matrix intervals.

128

We consider the uncertain linear stochastic neutral interval differential delay equation l

l

i =1

i =1

d [ x(t ) + ∑ (C i + ∆ C i )x(t − hi )] = [ A0 x(t ) + ∑ Aix(t − hi ) + Bu (t ) ]dt + [∆Dx(t ) + ∆Ex(t − τ )] dw(t ) x(t ) = φ (t ),

∀t ∈ [ − h,0 ] ,

(7.27)

where x(t ) ∈ Rn is the state vector, Ai , i = 0, 1,L , l , B , C i , C im , i = 1,L , l , Dm and E m are known constant matrices with appropriate dimensions, ∆ C i ≡ [ − C im , C im ] , i = 1, L , l , ∆ D ≡ [− Dm , Dm] and ∆E ≡ [ − E m , E m ] , w(t ) is a scalar Brownian motion defined on the probability space (Ω , F , {F t}t ≥0 , P ) [32], u (t ) is a control input with appropriate dimensions. hi and τ are independently time-invariant bounded delay times satisfying 0 ≤ hi ≤ h and 0 ≤ τ ≤ h . φ (t ) is a continuous vector valued initial data x (t ) = φ (t ) ∀t ∈ [ − h,0] . Fact 7.2 [53]: The trivial solution of a neutral stochastic differential equation d [ x (t ) − G ( x (t − h)] = f (t , x (t ), x (t − h)) dt + g (t , x(t ), x (t − h)) dw(t ),

(7.28)

f : R+ × Rn × Rn → Rn , g : R+ × Rn × Rn → Rn×m and G : R n → R n sufficiently differentiable maps, is globally asymptotically stable in probability if there exists a function V (t , x ) ∈ C 2 ( R+ × Rn ) which is positive definite in the Lyapunov sense, and

with

satisfies LV (t , x, y ) =

∂V (t , x − G ( y )) + gradV (t , x − G ( y )) f (t , x, y ) ∂t

+

1 tr g (t , x, y ) g T (t , x, y ) HessV (t , x − G ( y )) ≤ 0 , 2

for x ≠ 0 . The matrix HessV

(7.29)

is the Hessian matrix of the second-order partial

derivatives. This fact is analogous to the well-known theorem of Lyapunov for deterministic systems. 7.3.2 Main Results In this subsection, we first state robust stability criteria for the stochastic interval time-delay system with u (t ) ≡ 0 . Equation (7.27) can be written in the following form

129

l

t

i =1

t − hi

d [ x(t ) + ∑ ((C i + ∆ C i ) x(t − hi ) + Ai ∫ x( s )ds )] l

= [( A0 + ∑ Ai ) x(t )]dt + [∆Dx(t ) + ∆Ex(t − τ )] dw(t ) . i =1

(7.30)

Define the operator Z : C ([−h = max{hi},0], Rn) → Rn as l

t

i =1

t −hi

Z ( x(t )) = x(t ) + ∑ [(C i + ∆ C i ) x(t − hi ) + Ai ∫ x( s )ds ] .

(7.31)

We have the following results. Remark 7.1: The operator Z is stable if the difference-integral system l

t

i =1

t − hi

x(t ) + ∑ [(C i + ∆ C i ) x(t − hi ) + Ai ∫ x( s)ds] = 0

(7.32)

is asymptotically stable. The stability of (7.31) is equivalent to the fact that there exists a δ > 0 such that all solutions λ of the associated characteristic equation 0 l ⎛ ⎡ ⎞⎤ det ⎢ I + ∑ ⎜⎜ (C i + ∆ C i ) ⋅ e− s hi + Ai ∫ esθ dθ ⎟⎟⎥ = 0, −hi ⎠⎦ ⎣ i =1 ⎝

s∈C ,

(7.33)

satisfy Re( λ ) ≤ −δ < 0 (δ > 0) . A sufficient condition is that the inequality (7.34) holds 0 l ⎛ ⎞ l ∑ ⎜⎜ (C i + ∆ C i ) ⋅ e− s hi + Ai ∫ esθ dθ ⎟⎟ ≤ ∑ ( C i + C im + hi Ai ) < 1 . i =1 ⎝ −hi ⎠ i =1

(7.34)

7.3.2.1 Robust Stability In this subsection we propose robust stability for neutral stochastic interval system as follows. Theorem 7.3: Given scalars hi > 0 , i = 0, 1,L , l , system (7.27) with u (t ) ≡ 0 is robustly stable in probability, if the operator Z is stable and there exist symmetric positive-definite matrices P and R i , i = 1,L , l , scalar constants α > 0 , δ > 0 , ε > 0 , such that the following LMI

130

⎡M ⎢ T ⎢M 1 ⎢ M T2 ⎢ T ⎢⎣ M 3

M1 − N1

M2 0

0

− N2

0

M 3⎤ ⎥ 0 ⎥ 0 , where T

l

l

l

l

i =0

i =1

i =0

T

l

l

M = ( ∑ Ai ) P + P( ∑ Ai ) + ∑ δ ( ∑ Ai ) ( ∑ Ai ) + ρ (1+ ε ) E m E m + ρ DTm D m + ∑ Ri , i =0

T

i =0

i =1

T T T M 1 = [ 2 h1 A1 P,L , 2 hl Al P, ρ D m ] , N 1 = diag [ h1 δ ,L , hl δ , ρε ] ,

T

l

T

l

M 2 = [( ∑ Ai ) P,L, ( ∑ Ai ) P] , N 2 = diag [α , L , α ] , i =0

i =0

T T M 3 = [( ∑ Ai ) P C 1 ,L , ( ∑ Ai ) P C l ] , N 3 = diag [R1 − α C1m C1m , L , Rl − α C lm C lm ].

T

l

T

l

i =0

i =0

Proof: Let u (t ) = 0 , equation (7.27) can be written in the following form: l

t

i =1

t − hi

d [ x(t ) + ∑ ((C i + ∆ C i ) x(t − hi ) + Ai ∫ x( s )ds )] l

= [( A0 + ∑ Ai ) x(t )]dt + [∆Dx(t ) + ∆Ex(t − τ )] dw(t ) . i =1

Let

l

t

i =1

t −hi

Z ( x(t )) = x(t ) + ∑ [(C i + ∆ C i ) x(t − hi ) + Ai ∫ x( s )ds ]

(7.36)

and consider the proposed

Lyapunov-Krasovskii type functional [31] l

t

l

t

V = Z T ( x(t )) PZ ( x(t )) + ∑ ∫ xT (θ ) Ri x(θ )dθ + 2 ∑ ∫ [( s − t + hi ) xT ( s )γ ATi PP Ai x( s )]ds i =1 t − h i

i =1 t − h i

t

+ { ∫ [ xT ( β )( ρ (1 + ε ) E Tm E m) x( β )]dβ } .

(7.37)

t −τ

Along trajectories of (7.36) and making use of the It oˆ -differential rule [30], then the generator LV for the evolution of V is therefore given by

131

l

l

l

i =1

i =1

i =1

LV = ∑ xT (t ) Rix(t ) − ∑ xT (t − hi (t )) Rix(t − hi (t )) + 2 ∑ xT (t ) hi γ ATi PP Ai x(t ) l

t

− 2 ∑ ∫ [ xT ( s )γAi PP Ai x( s )]ds + ρ (1 + ε ) xT (t ) E Tm E m x(t ) T

i =1 t − h i

− xT (t − τ (t )) ρ (1 + ε )( E Tm E m) x(t − τ (t ))] T

l

t

i =1

t − hi

l

+ 2 {x(t ) + ∑ [(C i + ∆ C i ) x(t − hi (t )) + Ai ∫ x( s )ds ]} P ( ∑ Ai ) x(t ) i =0

+ [∆Dx (t ) + ∆Ex (t −τ (t ))] P [ ∆Dx (t ) + ∆Ex (t − τ (t ))] . T

(7.38)

Set P ≤ ρI , one has l

l

l

i =1

i =0

i =0

T

LV ≤ xT (t ){2∑ γ hi ATi PT P Ai + ρ (1 + ε ) E Tm E m + P ( ∑ Ai ) + ( ∑ Ai ) P

l

l

i =1

i =0

T

T

l

l

l

i =0

i =1

i =0

l

l

i =0

i =1

+ ∑ α ( ∑ Ai ) PP( ∑ Ai ) + ∑ γ −1 hi ( ∑ Ai ) ( ∑ Ai ) + ∑ Ri T

l

l

i =1

i =0

T

+ ρ (1+ ε −1) D m D m}x(t ) + ∑ 2 xT (t ) ( ∑ Ai ) P C i x(t − hi ) l

l

t

− ∑ xT (t- hi )[ Ri − α C Tim C im]x(t − hi ) − ∑ ∫ [ xT ( s )γ ATi PP Ai x( s )]ds . i =1

i =1 t − h i

l

(7.39)

t

Since the last term ∑ ∫ [ xT ( s )γ ATi PP Ai x( s )]ds is positive definite in (7.39), we rewrite i =1 t − h i

(7.39) in the following form

⎡ X 11 LV ≤ X T ⎢ T ⎣ X 12

X 12 ⎤ ⎥X , − X 22 ⎦

(7.40)

T

where X = [ xT (t ), xT (t − h1), L, xT (t − hl )] , with

⎡ X 11 ⎢ T ⎣ X 12

X 12 ⎤ ⎥ < 0, − X 22⎦

(7.41)

where

132

l

T

l

l

−1 T T T T X 11 = P( ∑ Ai ) + ( ∑ Ai ) P + ∑ 2 hi γ Ai P P Ai + ρ (1 + ε ) E m E m + ρ (1 + ε ) D m D m i =0 l

i =0

i =1

T

l

l

l

i =0

i =1

T

l

−1

l

l

i =0

i =1

+ ∑ α ( ∑ Ai ) PP( ∑ Ai ) + ∑ γ ( ∑ Ai ) ( ∑ Ai ) + ∑ Ri , −1

i =1

i =0

i =0

T T X 12 = [( ∑ Ai ) P C 1 , L, ( ∑ Ai ) P C l ] , X 22 = diag [R1 − α C1m C1m , L, Rl − α C lm C lm ].

l

T

l

i =0

T

i =0

From (7.41), denoting δ = γ −1 and according to the Schur complement, with some efforts, we are able to show that (7.41) guarantees the negativeness, which immediately implies the robust stability of the stochastic neutral interval system. Because of that the operator Z is stable, therefore, equation (7.27) is asymptotically stable. Now we apply the problem of the previous section to the state feedback design of stochastic neutral interval time delay systems. 7.3.2.2 Robust Feedback Stabilization

Given the system (7.27) with the control u ( t ) = Kx (t ) , where the matrix K ∈ R m×n , system (7.27) becomes l

t

i =1

t − hi

d [ x(t ) + ∑ ((C i + ∆ C i ) x(t − hi ) + Ai ∫ x( s)ds)] l

= [( A f + ∑ Ai ) x(t )]dt + [∆Dx(t ) + ∆Ex(t − τ )] dw(t ) , i =1

(7.42)

where

A f = A0 + BK . Applying Theorem 7.3 to the above matrices, results in a nonlinear matrix inequality because of the term PBK . We therefore consider another version of LMI condition which is derived from (7.35). In order to obtain an LMI, we have to correct ourselves to the case of T T Ri = ρ (ε 1 C i C i + ε 2 C im C im) , i = 1, L , l , where ε 1 and ε 2 are scalar parameters. Therefore, we are able to find the quadratic term

133

l

l

i =1

i =1

W = ∑ ρ (ε 1C Ti C i + ε 2 C Tim C im) + 2∑ α hi ATi PT P Ai l

l

i =0

i =0

T

+ ρ (1 + ε 0) E Tm E m + ρ (1 + ε 0−1) DTm D m + P ( BK + ∑ Ai ) + ( BK + ∑ Ai ) P T

l

l

i =1

i =0

l

l

l

i =0

i =1

i =0

T

l

+ ∑ ρε 1−1( BK + ∑ Ai ) ( BK + ∑ Ai ) + ∑ ρε −21 ( BK + ∑ Ai ) ( BK + ∑ Ai ) l

l

i =1

i =0

T

i =0

l

+ ∑ α −1 hi ( BK + ∑ Ai ) ( BK + ∑ Ai ) < 0 .

(7.43)

i =0

It follows from the Schur complement that (7.43) is equivalent to ~ ⎡M ⎢ ~T ⎢M 1 ⎢ ~T ⎢M 2 ⎢ ~T ⎢M 3 ~T ⎢M 4 ⎢ T ~ ⎢M 5 ⎢ ~T ⎢M 6 ⎢ ~T ⎣M 7

~ M1 ~ −N 1 0 0

~ M2 0 ~ −N 2 0

~ M3 0

~ M4 0

~ M5 0

~ M6 0

0 − N~ 3 0

0

0

0

0

0

0

0

0

0 ~ −N 4 0

0

0

0 − N~ 5 0

0

0

0

0

0

0

0

0

0

0

0 ~ −N 6 0

~ M7 ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ < 0, 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ~ ⎥ −N 7 ⎦

where T l l ~ ~ = [ , ], M = P( BK + ∑ Ai ) + ( BK + ∑ Ai ) P , M E m Dm 1 i =0

i =0

−1 ~ −1 −1 ~ N 1 = diag[( ρ (1 + ε 0) ) , ( ρ (1 + ε 0 ) ] , M 2 = [ 2 h1αP A1 ,L , 2 hl αP Al ] , −1 −1 ~ ~ N 2 = diag[αh1 ,L, αhl ] , M 3 = [C1 ,L, C l ] , N 3 = diag[( ρ ε 1) ,L , ( ρ ε 1) ] , −1 −1 ~ ~ M 4 = [C1m ,L, C lm] , N 4 = diag[( ρ ε 2) ,L , ( ρ ε 2) ] , T T ~ = ⎡( BK + ∑l ) ,L , ( BK + ∑l ) ⎤ , ~ = diag[( ρ −1)−1 ,L , ( ρ −1)−1] , A A i i ⎥ ε1 ε1 N5 M5 ⎢ i =0 i =0 ⎣ ⎦ T T ~ = ⎡( BK + ∑l ) , L, ( BK + ∑l ) ⎤ , ~ = diag[( ρ −1)−1 ,L , ( ρ −1)−1] , A A i i ⎥ ε2 ε2 N6 M6 ⎢ i =0 i =0 ⎣ ⎦ T T ~ = ⎡ ( BK + ∑l ) ,L, ( BK + ∑l ) ⎤ , ~ = diag [ ,L , α ] . h h αh1 hl A A i i ⎥ l N7 M7 ⎢ 1 i =0 i =0 ⎣ ⎦

134

(7.44)

Pre- and post-multiplying both sides of (7.44) by diag ( P−1 , I , I , I , I , I , I ) and denoting X = P −1 , Y = KX yield (7.45),

⎡Ω Ω1 ⎢ T ⎢ Ω1 − Σ1 ⎢ ΩT2 0 ⎢ T 0 ⎢ Ω3 ⎢ T 0 ⎢ Ω4 T ⎢ Ω5 0 ⎢ T 0 ⎢ Ω6 ⎢ ΩT 0 ⎣ 7

Ω2 0

Ω3 0

Ω4 0

Ω5 0

Ω6 0

− Σ2

0

0

0

0

0

− Σ3

0

0

0

0

0

− Σ4

0

0

0

0

0

0

0

0

0

− Σ5 0

0

0

0

0

Ω7 ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ < 0. 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ − Σ7 ⎥⎦

− Σ6 0

(7.45)

where l

l

i =0

i =0

Ω = BY + Y T BT + ∑ X ATi + ∑ Ai X , −1 Ω1 = [ X E m , X Dm] , Σ1 = diag[( ρ (1+ ε 0)) , ( ρ (1+ ε 0−1)) ] , −1

Ω 2 = [ 2α h1 A1 ,L , 2α hl Al ] , Σ2 = diag[αh1 ,L, α hl ] , −1

−1

Ω3 = [ X C1 , L , X C l ] , Σ3 = diag [( ρ ε 1) ,L , ( ρ ε 1) ] , −1

−1

Ω 4 = [ X C1m , L , X C lm ] , Σ 4 = diag [( ρ ε 2) ,L , ( ρ ε 2) ] , l

l

i =0

i =0

l

l

i =0

i =0

−1

−1

T T T T T T Ω5 = [Y B + ∑ XAi ,L , Y B + ∑ X Ai ] , Σ5 = diag[( ρ ε 1−1) ,L , ( ρ ε 1−1) ] ,

−1

−1

T T T T T T Ω6 = [Y B + ∑ XAi , L, Y B + ∑ XAi ] , Σ6 = diag[( ρ ε 2−1) ,L , ( ρ ε 2−1) ] , T T T T T T Ω7 = [h1 (Y B + ∑ XAi ),L , hl (Y B + ∑ XAi )] , Σ7 = diag[α h1 ,L , α hl ]. l

l

i =0

i =0

We show that (7.45) guarantees the negativeness, which immediately implies that the closed-loop stochastic neutral interval time delay system (7.42) is robust stability and the stochastic neutral interval time delay system (7.27) is robustly stabilizable. We obtain the following.

135

Theorem 7.4: Given scalars hi > 0 , i = 0 , 1,L , l , system (7.27) is robustly stable in probability, if the operator Z is stable and there exist symmetric positive-definite matrix X and scalar α > 0 such that the LMI (7.45) holds for given scalars ρ > 0 , and ε i > 0 , i = 0, 1, 2 . Then, a suitable stabilizing control law is given by u (t ) = Kx(t ) = Y X −1 x(t ) , Y ∈ R m×n , where K = Y X −1 .

7.3.3 Examples

Example 7.2: Consider a linear stochastic neutral interval delay system 2

d [ x(t ) + ∑ (C i + ∆ C i )x(t − hi )] i =1

2

= [ A0 x(t ) + ∑ Aix(t − hi ) ]dt + [∆Dx(t ) + ∆Ex(t − τ )] dw(t ) , i =1

(7.46)

where ⎡ 1 0.05⎤ ⎡0.1 ⎡− 2 1 ⎤ , A1 = ⎢ , A2 = ⎢ A0 = ⎢ ⎥ ⎥ ⎣0.05 1 ⎦ ⎣0 ⎣ 0 − 2⎦

0 ⎤ ⎡ 0.1 0.05⎤ , C1 = ⎢ ⎥ ⎥, 0.1⎦ ⎣0.05 0.1 ⎦

⎡0.05 0 ⎤ C2 = ⎢ ⎥ , ∆ C i = [ − C im , C im ] , i = 1,2 , ∆ D = [− Dm , Dm] , ⎣ 0 0.05⎦ ⎡0.1 0 ⎤ ⎡0.1 0 ⎤ ⎡0.1 0 ⎤ , C 2m = ⎢ , Dm = ⎢ ∆E = [ − E m , E m ] , C1m = ⎢ ⎥ ⎥ ⎥, ⎣ 0 0.1⎦ ⎣ 0 0.1⎦ ⎣ 0 0.1⎦ ⎡0.1 0 ⎤ Em = ⎢ ⎥ , h1 = 0.33 , h2 = 1.0 and ρ = 1 . Applying Theorem 7.3, by the software ⎣ 0 0.1⎦ package LMI Lab., we find the solutions of the LMI as ⎡0.3052 0.2422⎤ ⎡0.1268 0.0170⎤ ⎡ 0.1087 − 0.0037 ⎤ , R2 = ⎢ P=⎢ , R1 = ⎢ ⎥ ⎥ ⎥, ⎣0.2422 0.7342⎦ ⎣0.0170 0.1516⎦ ⎣− 0.0037 0.1336 ⎦

ε = 0.1167 , δ = 18.8752 , α = 4.1492 . Therefore, this implies (7.35) is asymptotically stable in probability for any h1 ∈ [0, 0.33] and h 2 ∈ [0, 1.0] .

136

Example 7.3: Consider a two–dimensional stochastic neutral interval differential equation 2

d [ x(t ) + ∑ (C i + ∆ C i )x(t − hi )] i =1

2

= [ A0 x(t ) + ∑ Aix(t − hi ) + Bu (t ) ]dt + [∆Dx(t ) + ∆Ex(t − τ )] dw(t ) ,

(7.47)

i =1

where ⎡0 ⎡− 2 1⎤ , A1 = ⎢ A0 = ⎢ ⎥ ⎣1 ⎣ 0 − 2⎦

1⎤ ⎡ 0.1 − 0.05⎤ ⎡1 , A2 = ⎢ , B=⎢ ⎥ ⎥ 0⎦ ⎣ 0.05 0.1 ⎦ ⎣1

⎡0.1 0 ⎤ ⎡0.1 ⎡ 0 0.2⎤ , C2 = ⎢ , C1m = ⎢ C1 = ⎢ ⎥ ⎥ ⎣ 0 0.1⎦ ⎣ 0 ⎣0.2 0 ⎦ ⎡0.1 Dm = ⎢ ⎣ 0

0⎤ ⎡0.1 , Em = ⎢ ⎥ 0.1⎦ ⎣ 0

0⎤ ⎡0.1 , C 2m = ⎢ ⎥ 0.1⎦ ⎣ 0

0⎤ , 1⎥⎦ 0⎤ , 0.1⎥⎦

0⎤ , h1 = 0.4 , h 2 = 0.89 , ρ = 4 , ε 0 = 0.9878 , 0.1⎥⎦

ε 1 = 1.0556 , ε 2 = 1.2009 . Using Theorem 7.4 to this uncertain stochastic neutral interval delay equations, we find the solutions of the (7.45) as ⎡ 0.7710 − 0.2215⎤ ⎡ 0.1138 X =⎢ , Y =⎢ ⎥ 0.3779⎦ ⎣− 0.2215 ⎣− 0.4029

− 0.1443 ⎤ , α = 4.0559 . − 0.1187 ⎥⎦

We found that this system is robustly stabilizable for any time-delay h1 ∈ [0, 0.4] and h 2 ∈ [0, 0.89] and the correspondent robustly stabilizing control law is ⎡ 0.0445 − 0.3552⎤ u (t ) = ⎢ ⎥ x(t ) . ⎣− 0.7945 − 0.9464⎦

137

7.4 Delay-Dependent Robust H ∞ Control for Nonlinear Stochastic Neutral Systems with State Delays This section deals with the problem of robust stability and robust H ∞ control for a class of uncertain delay-dependent neutral stochastic nonlinear systems. The nonlinearities are assumed to satisfy the global Lipschitz conditions with both states and perturbations. Attention first is focused on investigating a sufficient condition for designing a state feedback controller which stabilizes the uncertain neutral stochastic nonlinear system. Then, we guarantee an robust H ∞ -norm bound constraint on the disturbance attenuation. The proposed results are given in terms of linear matrix inequalities. An example is worked out to illustrate the validness of the theoretical results. 7.4.1 Preliminaries

Consider the uncertain nonlinear stochastic neutral differential delay equation d [ x (t ) + Cx (t − h)] = [( A + ∆A(t )) x (t ) + ( Ah + ∆ Ah (t )) x (t − h) + F ( x (t ), x (t − h)) + Bu (t ) + E1 v (t )]dt + [G ( x (t ), x (t − h)) + E 2 v (t )] dw(t ), z (t ) = Lx (t ) + L h x (t − h) ,

(7.48)

where x(t ) ∈ Rn is the state vector, u (t ) is a control input with appropriate dimension, v (t ) is the noise signal which belongs to L 2 [0, ∞ ) with appropriate dimension, w(t ) is a scalar Brownian motion defined on the probability space (Ω , F , {F t}t≥0 , P ) , z (t ) is the controlled

output

with

appropriate

dimension,

F (⋅, ⋅) : Rn × Rn → Rn F

and

G(⋅, ⋅) : R × R → RnG are known nonlinear functions, A , B , C , Ah , E1 and E2 are known constant matrices with appropriate dimensions, h is time-invariant bounded delay times satisfying 0 ≤ h ≤ h . ∆A(t ) and ∆ Ah (t ) are unknown matrices representing n

n

time-varying parameter uncertainty , and are assumed to be of the form

[∆A(t )

∆ Ah (t )] = MF (t )[N 1

N 2] ,

(7.49)

where F T (t ) F (t ) ≤ I .

138

Assumption 7.1 (Lipschitz condition)

(1) F (0, 0) = 0 , G (0, 0) = 0 , (2) F ( x1 , x2) − F ( y1 , y 2) ≤ g1 ( x1 − y1) + g 2 ( x2 − y 2) , (3) G ( x1 , x2) − G ( y1 , y 2) ≤ g 3 ( x1 − y1) + g 4 ( x2 − y 2) .

for all x1 , x 2 , y1 , y 2 ∈ R n , where g1 , g 2 , g 3 and g 4 are known real constant matrices. 7.4.2 Main Results

In this subsection, we first state robust stabilization problem for the stochastic time-delay system with v(t ) ≡ 0 and u (t ) = Kx (t ) . Equation (7.48) can be written in the following form t

d [ x(t ) + Cx(t − h) + Ah ∫ x( s)ds)] t −h

= [( A f + Ah + ∆A(t )) x(t ) + ∆ Ah x(t − h) + F ( x(t ), x(t − h)]dt + [G ( x(t ), x (t − h))] dw(t ) ,

(7.50)

where

A f = A + BK . Define the operator Z : C ([−h ,0], R n) → R n as t

Z ( x(t )) = x(t ) + Cx(t − h) + Ah ∫ x( s)ds .

(7.51)

t −h

7.4.2.1 Robust Stabilization

In this subsection we describe robust stabilization for neutral stochastic nonlinear system as follows.

139

Theorem 7.5: Given scalar h > 0 , system (7.50) is robustly stable in probability, if the operator Z is stable and there exist symmetric positive-definite matrix X , scalars α > 0 and β i > 0 , i = 1, L ,3 such that the following LMI

⎡Ω I ⎢ − ε1 ⎢ I ⎢ T 0 ⎢ Ω1 ⎢X 0 ⎢ 0 ⎢ hX ⎢ T 0 ⎢ Ω2 ⎢ 0 0 ⎢ T 0 ⎢ Ω3 T ⎢α T Γ1 ⎢ M N2 ⎣⎢ N 1 X

α M

0

Ω3 0

0

0

0

0

0

0

0

Γ2 0

0

0

0

Σ3 − Σ4

0

0

0

0

0

− Σ5

0

0 0

0 0

Ω1 0

X

hX

0

− Σ1

0

0

0

Ω2 0

0

0

−R

0

0

− (Q )−1

0

0

0

0

− Σ2

−1

0

0

0

0

0

0

Σ 0

T

0 0

0 0

0 0

Γ2 0

T 3

Γ1

−α 0

X N 1T ⎤ ⎥ T N2 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 , Q > 0 , and scalar ρ > 0 , where T Ω = BY + Y T BT + ( A + Ah ) X + X ( A + Ah ) ,

T Ω1 = [ 2 X g1

2 X g1T

2h X g1T

2 ρX g T3 ] , Σ1 = diag[ β 1 , β 2 , β 3 , ρ ] ,

T T T T Ω 2 = ρ [ X ( A + Ah ) + Y B ]C , Ω3 = ρh[ X ( A + Ah ) + Y B ] Ah , T

T

T T 2 T Σ 2 = R − ρ β 2 C C − 2 ρ g 4 g 4 , Σ3 = [ 2 g 2

2 g T2

2h g T2 ] ,

2 T T T Σ4 = diag[ β 1 , β 2 , β 3 ] , Σ5 = h(Q − ρ β 3 Ah Ah) , Γ1 = ρ αC M , Γ2 = ραh Ah .

Then, a suitable stabilizing control law is given by u (t ) = Kx(t ) = Y X −1 x(t ) , Y ∈ R m×n , where (7.53) K = Y X −1 .

Proof:

Let

t

Z ( x(t )) = x(t ) + Cx(t − h) + Ah ∫ x( s)ds) and t −h

Lyapunov-Krasovskii type functional

140

consider

the

proposed

t

0 t

t −h

− ht + β

V = Z T ( x(t )) PZ ( x(t )) + ∫ xT (θ ) Rx(θ )dθ + ∫ ∫ xT ( ρ ) Rx( ρ )dρdβ .

(7.54)

Along trajectories of (7.50) and making use of the It oˆ -differential rule, then one has the generator LV for the evolution of V as t

LV = xT (t ) Rx(t ) − xT (t − h) Rx(t − h) + h xT (t )Qx(t ) − ∫ xT ( β )Qx( β )dβ t −h

T

t

+ 2 {x(t ) + Cx (t − h) + Ah ∫ x(θ )dθ ] } P{( A + BK + Ah + ∆A(t )) x(t ) t −h

T

+ ∆ Ah (t ) x (t − h) + F ( x (t ), x (t − h)} + [G ( x(t ), x (t − h)] P[G ( x (t ), x(t − h)] .

(7.55)

From equation (7.55), one has 2 xT (t ) P∆A(t ) x(t ) + 2 xT (t ) P∆ Ah (t ) x(t − h) + 2 xT (t − h) C T P∆A(t ) x(t ) t

t

t −h

t −h

+ 2 xT (t − h) C T P∆ Ah x(t − h) + 2 ∫ xT (θ ) ATh P∆A(t ) x(t )dθ + 2 ∫ xT (θ ) ATh P∆ Ah (t ) x(t )dθ ⎡ x(t ) ⎤ 1 t T T T = ∫ [x (t ) x (t − h) x (θ )] Γ ⎢⎢ x(t − h)⎥⎥ dθ , h t −h ⎢⎣ x(θ ) ⎥⎦

(7.56)

with ⎡ N 1T ⎤ ⎡ PM ⎤ ⎢ ⎥ ⎢ ⎥ Γ = ⎢C T PM ⎥ F (t )[N 1 N 2 0] + ⎢ N T2 ⎥ F (t )[M T P M T PC h M T P Ah ]. ⎢ 0⎥ ⎢⎣h ATh PM ⎥⎦ ⎣ ⎦

Using Assumption 7.1, we have

F ( x(t ), x(t − h)) ≤ g1 x(t ) + g 2 x(t − h) ,

(7.57)

and

G ( x(t ), x(t − h)) ≤ g 3 x(t ) + g 4 x(t − h) .

(7.58)

Hence 2

2

2

F ( x(t ), x (t − h)) ≤ 2 g 1 x (t ) + 2 g 2 x (t − h) ,

and

141

(7.59)

2

2

2

G ( x (t ), x (t − h)) ≤ 2 g 3 x (t ) + 2 g 4 x(t − h) .

(7.60)

By Lemma 2.4 in [41], Lemmas 2.1 and 2.6 in Chapter 2 and P ≤ ρI , we can find that ⎡ x(t ) ⎤ 1 t T ~ ⎢ T T LV ≤ ∫ [x (t ) x (t − h) x (θ )] (U + Γ) ⎢ x(t − h)⎥⎥ dθ , h t −h ⎢⎣ x(θ ) ⎥⎦

(7.61)

where ⎡φ 11 ⎢ U = ⎢φ 21 ⎢⎣φ 31

φ 12 φ 22 φ 32

φ 13 ⎤ ⎥ φ 23⎥ , φ 33 ⎥⎦

with T T φ 11 = P ( A + Ah) + ( A + Ah ) P + PBK + ( PBK ) + ε 1−1 PP

+ 2(ε 1 + ε 2 + h ε 3) g1T g1 + 2 ρ g T3 g 3 + R + hQ,

φ12 = φ T21 = ρ ( A + Ah + BK )T C , φ13 = φ T31 = ρh( A + Ah + BK )T Ah , φ 22 = − R + ρ 2 ε −21 CT C + 2(ε 1 + ε 2 + h ε 3) g T2 g 2 + 2 ρ g T4 g 4 , φ 23 = φ T32 = 0 , φ 33 = −hQ + ρ 2 ε 3−1 h ATh Ah , and ⎡ PM ⎤ ⎡ N 1T ⎤ ⎥ ⎢ ⎢ ⎥ ~ Γ = α ⎢ ρC T M ⎥[M T P ρ M T C ρh M T Ah ] + α −1 ⎢ N T2 ⎥[N 1 N 2 0]. ⎢ ρh ATh M ⎥ ⎢ 0⎥ ⎣ ⎦ ⎦ ⎣ From equation (7.61), we obtain

~ U +Γ 0 , i = 1, L ,3 yield (7.52). We show that (7.52) guarantees the negativeness, which immediately implies that the closed-loop stochastic neutral nonlinear time delay system (7.50) is robustly stabilizable. 7.4.2.2 Robust H ∞ Performance

In this subsection, we extend the robust stabilization results developed in the previous section to the case of robust H ∞ performance as follows. Theorem 7.6: Given scalar h > 0 , system (7.48) is robustly stable in probability, if the operator Z is stable and there exist symmetric positive-definite matrix X , scalars α > 0 and β i > 0 , i = 1,L ,4 such that the following LMI

143

~ ~ ⎡Ω I h X X LT Ω1 X ⎢ 0 0 0 0 ⎢I − ε 1 ⎢~T ~ 0 −Σ 0 0 0 1 ⎢Ω1 ⎢X 0 0 − R −1 0 0 ⎢ ⎢ hX 0 0 0 − (Q )−1 0 ⎢ 0 0 0 0 −I ⎢ LX T ⎢~ 0 0 0 0 0 ⎢Ω 2 ⎢0 0 0 0 0 0 ⎢ ~T 0 0 0 0 0 ⎢Ω3 ⎢ T T T 0 0 0 Ψ1 Ψ 2 ⎢ E1 ⎢αM T Π1T ΠT2 0 0 0 ⎢ 0 0 0 ⎢⎣ N 1 X N 2 0

~ Ω2 0

0

~ Ω3 0

0

0

0

0

0

0

0

0

0 0 ~ ~ − Σ 2 Σ3 ~T ~ Σ3 − Σ 4 0 0

αM

E1 Ψ1

Π1

0

Ψ2 0

Π2 0

0

0

0

0

0

0

0

0

0

0

0 0

0

0

0 ~ −Σ 5 0

0

0

0

0 ~ −Σ 6 0

0

0

0

0

0 −α 0

X N 1T ⎤ ⎥ T N2 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ 0 , Q > 0 and scalar ρ > 0 , where ~ T Ω = BY + Y T BT + ( A + Ah ) X + X ( A + Ah ) , T ~ Ω1 = [ 2 X g1

2 X g1T

2h X g1T

2 ρ X g T3

2 ρ X g T3 ] ,

~ Σ1 = diag [β 1 , β 2 , β 3 , β 4 , I ] , T T ~ ~ T T T T T Ω2 = ρX ( A + Ah ) C + X L Ld + ρ Y B C , Ω3 = ρh[ X ( A + Ah ) + Y B ] Ah , T T 2 T T ~ ~ Σ 2 = R − ρ β 2 C C − L d L d − 2 ρ g 4 g 4 , Σ3 = [ 2 g 2

2 g T2

2h g T2

2 ρ g T4 ] ,

2 T ~ T ~ ~ Σ4 = diag [β 1 , β 2 , β 3 , β 4] , Σ5 = h(Q − ρ β 3 Ah Ah) , Σ6 = γ I − ρ (1 + β 4) E 2 E 2 , T T T Π1 = ρα C M , Π 2 = ρhα Ah , Ψ1 = ρ C E1 , Then the state feedback controller

u (t ) = Kx(t ) = Y X −1 x(t )

Ψ 2 = ρh Ah E1 . T

(7.65)

stabilizes system (7.48) and guarantees that the closed-loop system has a prescribed level γ > 0. Proof: Applying the controller (7.65) to (7.48), we obtain the resulting closed-loop system in the following

144

d [ x (t ) + Cx (t − h)] = [( A f + ∆A(t )) x(t ) + ( Ah + ∆ Ah (t )) x(t − h) + F ( x(t ), x(t − h)) + E1 v(t )]dt + [G ( x(t ), x(t − h)) + E 2 v (t )] dw(t ), z (t ) = Lx (t ) + L h x (t − h) ,

(7.66)

where

A f = A + BK . For showing that (7.48) is robustly stable with a disturbance attenuation γ > 0 , we evaluate the associated Hamiltonian H ( x, v, t ) to satisfy H ( x, v, t ) = LV ( x (t )) + zT (t ) z (t ) − γ 2 vT (t )v (t ) < 0 .

(7.67)

Along trajectories of (7.48) and making use of the It oˆ -differential rule, one has ⎡ x(t ) ⎤ ⎢ x(t − h)⎥ 1 t T T T T ⎥ dθ , H ( x, v, t ) ≤ ∫ [x (t ) x (t − h) x (θ ) v (t )](W + Γ1) ⎢ ⎢ x(θ ) ⎥ h t −h ⎥ ⎢ ⎦ ⎣v(t ) where ~ ⎡φ 11 ⎢~ ⎢φ 21 W = ⎢~ ⎢φ 31 ⎢φ~ ⎣ 41

~

~

~

φ 14 ⎤

φ 12

φ 13

φ 22

φ 23 φ 24⎥

φ 32

φ 33 φ 34 ⎥ ~ ~ φ 43 φ 44⎥⎦

~

~

~

φ 42

~

~

~ ⎥

~ ⎥,

with T T ~ φ 11 = P( A + Ah) + ( A + Ah ) P + PBK + ( PBK ) + ε 1−1 PP

+ 2(ε 1 + ε 2 + h ε 3) g1T g1 + 2 ρ (1 + ε 4) g T3 g 3 + LT L + R + hQ, T T ~ ~T ~ ~T φ 12 = φ 21 = ρ ( A + Ah + BK ) C + LT Lh , φ 13 = φ 31 = ρh( A + Ah + BK ) Ah ,

~ ~T φ 14 = φ 41 = P E1 ,

~

φ 22 = − R + ρ ε −21 C T C + 2(ε 1 + ε 2 + h ε 3) g 2 g 2 + 2 ρ (1 + ε 4) g 4 g 4 + LTh Lh , 2

T

T

145

(7.68)

~ φ 24 = ρ CT E1 ,

~ ~T φ 23 = φ 32 = 0 ,

~ 2 ~ ~ φ 33 = −hQ + ρ ε 3−1 h ATh Ah , φ 34 = φ 43 = ρh ATh E1 , T

~ 2 φ 44 = − γ I + ρ (1 + ε −41) E T2 E 2 , and ⎡ PM ⎤ ⎡ N 1T ⎤ ⎢ T ⎥ ⎢ T⎥ ρC M ⎥ T T −1 ⎢ N 2 ⎥ T ⎢ [N N 0 0] . Γ1 = α ⎢ M P ρM C ρh M Ah 0 + α ⎢ 0⎥ 1 2 ρh ATh M ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ 0 ⎥⎦ ⎣⎢ 0 ⎦⎥

[

]

From equation (7.68), one has W + Γ1 < 0 .

(7.69)

Applying Schur complement to (7.69), taking the congruence transformation with

diag ( P−1 , I , I , I , I , I , I , I , I , I , I ) and letting X = P −1 , Y = KX , and β i = ε i−1 , i = 1,L ,4 easily verify the (7.64). This completes the proof of the Theorem 7.6.

□

7.4.3 An Illustrative Example

Example 7.4: Consider a two–dimensional stochastic neutral interval differential equation d [ x(t ) + Cx (t − h)] = [( A + ∆A(t )) x(t ) + ( Ah + ∆ Ah (t )) x(t − h) + F ( x(t ), x(t − h)) + Bu (t ) + E1 v(t )]dt + [G ( x (t ), x (t − h)) + E 2 v (t )] dw(t ), z (t ) = Lx (t ) + L h x (t − h) ,

(7.70)

⎡0 ⎡− 2 1⎤ A=⎢ , A1 = ⎢ ⎥ ⎣1 ⎣ 0 − 2⎦

1⎤ ⎡1 , B=⎢ ⎥ 0⎦ ⎣1

⎡0.5 E2 = ⎢ ⎣0.5

− 0.1⎤ 0.3⎤ ⎡0 ⎡0.2 , Lh = ⎢ , M =⎢ ⎥ ⎥ 1⎦ 0.1 ⎦ ⎣0.6 ⎣0.1

0.2⎤ ⎡0.1 L=⎢ ⎥ 0.3 ⎦ ⎣0.5

0⎤ ⎡0.5 ⎡ 0 0.2⎤ , E1 = ⎢ , C=⎢ ⎥ ⎥ 1⎦ ⎣0.5 ⎣0.2 0 ⎦

146

0.1⎤ , 0.2 ⎥⎦

0.3⎤ , 0.2 ⎥⎦

⎡0.1 N1 = ⎢ ⎣0.2

0.3 ⎤ ⎡0.1 0.3 ⎤ , N2 = ⎢ . ⎥ 0.2⎥⎦ 0.2⎦ ⎣0

The nonlinearities F ( x (t ), x (t − h)) and G ( x (t ), x (t − h)) are assumed to satisfy Assumption 7.1 with ⎡0.3 g1 = ⎢ ⎣0.1 It

is

0⎤ ⎡0.2 , g2 = ⎢ ⎥ 0.2⎦ ⎣0.3 assumed

that

0.2⎤ ⎡0.6 , g3 = ⎢ ⎥ 0.1 ⎦ ⎣0.1 h ∈ [0, 0.75]

.

0.1 ⎤ ⎡0.1 , g4 = ⎢ ⎥ 0.2⎦ ⎣0.2 Denoting

0.2⎤ . 0.1 ⎥⎦

⎡2.7873 1.4247 ⎤ R=⎢ ⎥ ⎣1.4247 2.7702⎦

and

⎡4.6302 1.6658 ⎤ Q=⎢ ⎥ , using the Matlab LMI Control Toolbox to solve the Theorem 7.4, ⎣1.6658 3.9188⎦ we find the solution as ⎡− 3.0422 − 0.6183⎤ ⎡ 0.0241 − 0.0006 ⎤ X =⎢ , Y =⎢ ⎥ , β 1 = 2.9290 , ⎥ 0.0239⎦ ⎣ 0.1024 − 2.2455⎦ ⎣− 0.0006

β 2 = 1.1373 , β 3 = 1.1976 , β 4 = 1.1107 , α = 2.4202 . Therefore, the correspondent robustly H ∞ stabilizing control law is ⎡− 127.2214 − 29.1051⎤ u (t ) = ⎢ x(t ) . − 93.9717⎥⎦ ⎣ 1.8792 We guarantee that the closed-loop system has a prescribed level γ = 2.45 .

147

7.5 Summary Applying a linear matrix inequality formulation, this chapter has created delay-dependent results and presented insights into the problems of robust exponential stability analysis and robust feedback synthesis for a class of uncertain stochastic systems with time-varying delay. It has been further established that controllers are capable of guaranteeing the closed-loop system stabilization. With regard to neutral stochastic interval systems with time delays in the state, it has been established that controllers are capable of guaranteeing the closed-loop system stabilization by linear matrix inequalities formulation. In addition, we also have investigated the problem of robust H ∞ control design for a class of neutral stochastic nonlinear systems with time delay. A delay-dependent sufficient condition for the solvability of this problem has been presented. The desired controller, constructed by solving a certain LMI, ensures not only the robust stabilization but also a H ∞ -norm bound constraint on the disturbance attenuation for all admissible uncertainties. Four numerical examples have shown the validness of the proposed approach.

148

Chapter 8 Conclusions and Future Research

Five topics of research for uncertain systems with time delays have been investigated in this dissertation. First, we have provided novel stability criteria for a class of uncertain linear time-delay systems with time-invariant delays and time varying delays. Also, we extended the results to cellular neural networks with time-varying time delays. Next, we proposed Kalman filtering to get a good estimate for the filtering error systems, and we employed the H ∞ theory to solve this filtering problem. Besides, we designed a tracking controller to satisfy an H ∞ tracking performance. The next in order, we designed a dynamic output feedback controller for a 2-D discrete state-delayed system, such that the closed-loop system is asymptotically stable and satisfies a specified H ∞ performance. Also, the 2-D H ∞ filtering problem is investigated for 2-D discrete state-delayed system. Finally, we dealt with the class of stochastic system with time delay and provide some results on stability and stabilization via LMIs. In addition, the stabilization problem for a class of neutral stochastic interval system with time delay was proposed and the H ∞ control problem for a class of neutral stochastic nonlinear system with time delay was investigated also. In Chapter 2, we have proposed novel stability criteria for a class of uncertain linear time-delay systems. Based on Lyapunov-Krasovskii functionals combining with LMI techniques, simple and improved delay-dependent robust stability criteria, which are given in terms of quadratic forms of state and LMI, are derived. Our results shown by some examples are less conservative than the existing stability criteria. Using the aforementioned results, we develop stabilization conditions for a class of uncertain discrete time systems with time delay. In the stabilization case, we are seeking a state feedback control law u (k ) that asymptotically stabilizes the closed-loop system. Further, the stability for cellular neural networks with time-varying delay (DCNNs) which is practical for circuitry implementation is introduced by linear matrix inequality (LMI). A sufficient condition related to the exponentially asymptotic stability for DCNNs is proposed. This condition is less restrictive than that given in the literature. In Chapter 3, state estimation is a subject of

149

great practical and theoretical importance which has received much attention in recent years. The Kalman filtering approach is one of the most popular ways to deal with this topic. This approach provides an optimal estimation of some desired variables of a dynamic system from available measurement in the sense that the covariance of the estimation errors is minimized. It should be pointed out that the Kalman filtering approach is based on the assumptions that the system under consideration is exactly known and its disturbances are stationary Gaussian noises with known statistics. Also, to deal with the estimation problem for systems without exact knowledge of the statistics of the noise signals, H ∞ filtering has been introduced as an alternative. We considered the problem of robust Kalman filtering/ H ∞ filtering for a class of state-delayed interval systems with delay dependence. The parameter uncertainties appear in all the matrices of the system model. The main aim was to design a filter such that filtering error dynamics are robustly stable/the L 2 -induced gain from the noise signals to the estimation error is less that a prescribed level for all admissible uncertainties. A robust H ∞ tracking controller has newly developed in Chapter 4 to deal with the problem of state-feedback controller design for a class of delay-dependent state-delayed systems with time-varying norm-bounded uncertainty/a class of uncertain discrete time-delay systems. A robust tracking controller is proposed. A sufficient condition for the solvability of this problem has obtained in terms of LMIs. The desired controller can be constructed through a convex optimization problem that can be efficiently implemented using standard numerical algorithms. Feedback control of two-dimensional (2-D) state delayed systems has been a problem of considerable importance in both theory and practical applications. In Chapter 5, we have presented a state-space solution to the problem of H ∞ control of 2-D state-delayed system. For a linear discrete time 2-D state-delayed system described by a 2-D state-space Roesser model, a 2-D dynamic output feedback controller is designed to achieve the close-loop system asymptotic stability and a specified H ∞ performance using the linear matrix inequalities (LMIs) approach. We further have given a solution for robust stabilization of 2-D state-delayed systems subject to a class of norm bounded uncertainties. Also, in Chapter 6, we focused on the H ∞ filtering problem for 2-D discrete state-delayed systems described by the Roesser model. We have concerned with the design of a 2-D filter that guarantees a prescribed level of H ∞ noise attenuation for any energy bounded noise input. We first concerned with delay-dependent stability for a class of stochastic systems with time delay in Chapter 7. We then extend the proposed theory to discuss the robust stabilization of uncertain stochastic differential delay systems. First, a stability criterion, expressed in terms of LMI, is derived based on the Lyapunov-Krasovskii functional method. Also, we exploited stabilization for a class of uncertain neutral stochastic linear

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interval systems. Further, we dealt with the problem of robust H ∞ control for neutral stochastic nonlinear system subject to parameter uncertainties and time delay in the state. Attention has focused on the design of a controller which ensures not only the robust stochastic but also a prescribed H ∞ performance level for the neutral stochastic nonlinear system for all admissible uncertainties and time delays. A sufficient condition for the solvability of this problem is derived and an LMI approach is developed. The desired robust H ∞ controller can be constructed through a convex optimization problem that can be efficiently solved by using standard numerical algorithms.

Further Research Directions The followings are some research directions that are worthy to be studied in the future. i.

Adaptive control problems for a class of uncertain systems with time delay against controller gain variations,

ii.

Stochastic cellular neural networks with time delay,

iii.

Digital redesign for a class of uncertain systems with time delay,

iv.

Tracking control for a class of neutral systems,

v.

Control laws for a continuous 2-D uncertain Roesser model with state delay,

vi.

Filtering problems for a continuous 2-D Roesser model with state delay,

vii.

Filtering problems for a class of stochastic neutral systems.

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Biography

Chien-Yu Lu was born in Taiwan, R.O.C., on May 28, 1964. He received B.S. degree in Industrial Education & Technology from the Changhua University of Education and M.S. degree in Biomedical Engineering from the Cheng-Kung University in 1991 and 1994, respectively. Since 2000, he started his graduate study for Ph.D. degree at the Department of Electrical Engineering, National Cheng-Kung University, Tainan, Taiwan, R.O.C..

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