An LMI approach to non-fragile robust optimal guaranteed cost control of 2D discrete uncertain systems

Abstract

This paper addresses the problem of non-fragile robust optimal guaranteed cost control for a class of two-dimensional discrete systems described by the general model with norm-bounded uncertainties. Based on Lyapunov method, a new linear matrix inequality (LMI)-based criterion for the existence of non-fragile state feedback controller is established. Furthermore, a convex optimization problem with LMI constraints is formulated to select a non-fragile robust optimal guaranteed cost controller, which minimizes the upper bound of the closed-loop cost function. The merit of the proposed criterion in aspect of conservativeness over a recently reported criterion is demonstrated with the help of illustrative examples.

Keywords

2D discrete system general model guaranteed cost control linear matrix inequality uncertain systems

Introduction

In recent years, due to wide variety of applications of two-dimensional (2D) discrete systems in many areas such as filtering, image processing, seismographic data processing, thermal processes, gas absorption, water stream heating, river pollution modeling, etc. (Bose, 1982; Bracewell, 1995; Fornasini, 1991; Kaczorek, 1985; Lu and Antoniou, 1992), the research on 2D discrete systems have received considerable attention. Many authors have proposed and analysed the linear state space models for 2D discrete systems. The popular models are the Attasi model (Attasi, 1973), Roesser model (Rosser, 1975), Fornasini–Marchesini first model (Fornasini and Marchesini, 1976), Fornasini–Marchesini second model (Fornasini and Marchesini, 1978) and Kurek model or the general model (GM) (Kurek, 1985). The stability properties of 2D discrete systems have been investigated extensively (Anderson et al., 1986; Bauer and Jury, 1990; Bhaya et al. 2001; Bose, 1995; Du and Xie, 1999; Dumistrescu, 2008; Hinamoto, 1993, 1997; Hu and Liu, 2005; Huang and Xiang, 2013a, 2013b; Kar, 2008; Kar and Singh, 1997, 2001, 2003; Liu, 2008; Liu and Michel, 1994; Lu, 1994; Ooba, 2000; Singh, 2007; Thamvichai and Bose, 2002; Xiao and Hill, 1996; Xiang and Huang, 2013; Zhou, 2006).

On the other hand, there has been a growing interest in the study of guaranteed cost control problem for 2D discrete uncertain systems. The guaranteed cost control technique aims to design a robust controller such that the closed-loop system is asymptotically stable and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. A lot of effort has been devoted to this research area and many significant results have been obtained (Dhawan and Kar, 2007a, 2007b, 2007c, 2010, 2011; Guan et al., 2001; Tiwari and Dhawan, 2012a, 2012b; Ye et al., 2009). In recent years, the problem of non-fragile control has been an attractive topic in theory analysis and practical implement. The aim of non-fragile control problem is to design a controller for a given system such that the controller is insensitive to some amount of error with respect to its gain. Since controller fragility is basically the performance deterioration of a feedback control system due to inaccuracies in controller implementation, the non-fragile control problem for one-dimensional (1D) and 2D systems has been investigated in Dhawan (2012); Haddad and Corrado (2000); Lien (2007a, 2007b); Park (2004); Sharma and Dhawan (2012); Wu et al. (2012); Xu et al. (2009); and Yang and Wang (2001). Recently, the non-fragile robust guaranteed cost control problem for a class of 2D discrete systems described by the GM (Kurek, 1985) has been studied in Ye et al. (2011) and a linear matrix inequality (LMI)-based sufficient condition for the existence of a non-fragile robust guaranteed cost controller has been presented. However, the approach of Ye et al. (2011) is not suitable for the design of non-fragile robust optimal guaranteed cost controller that renders the corresponding guaranteed cost as small as possible. Non-fragile robust optimal guaranteed cost control for 2D discrete uncertain systems described by the GM is an important problem. To the best of the author’s knowledge, such problem has not yet been addressed in the literature.

This paper, therefore, addresses the non-fragile robust optimal guaranteed cost control problem for a class of 2D discrete uncertain systems described by the GM with norm-bounded uncertainties. The approach adopted here is as follows: we first develop an LMI-based criterion for the existence of non-fragile robust guaranteed cost controller in terms of feasible solution to a certain LMI. Furthermore, a convex optimization problem is introduced to find the non-fragile robust optimal guaranteed cost controller, which minimizes the upper bound of the closed-loop cost function. The paper is organized as follows. In the next section, we formulate the problem of non-fragile robust guaranteed cost control for a class of 2D discrete uncertain systems described by the GM and recall some useful results. An LMI-based criterion for the design of non-fragile robust optimal guaranteed cost controller is presented, and it is shown with the help of illustrative examples that the present approach is less conservative than Ye et al. (2011).

Notation

Throughout the paper, the following notations are used: $R^{n}$ denotes real vector space of dimension n, $R^{n \times m}$ is the set of n×m real matrices, the superscript T stands for matrix transposition, 0 denotes null matrix or null vector of appropriate dimension, I is the identity matrix of appropriate dimension, G < 0 stands for the matrix G is symmetric and negative definite, diag{….} stands for a block diagonal matrix and $λ_{max}$ ( G ) stands for maximum eigenvalue of matrix G .

Problem formulation and preliminaries

This paper deals with the problem of non-fragile robust optimal guaranteed cost control for a class of 2D discrete uncertain systems described by the GM (Kurek, 1985). Specifically, the system under consideration is given by

\begin{matrix} x (i + 1, j + 1) = (A_{1} + Δ A_{1}) x (i + 1, j) \\ + (A_{2} + Δ A_{2}) x (i, j + 1) + (A_{0} + Δ A_{0}) x (i, j) \\ + B_{1} u (i + 1, j) + B_{2} u (i, j + 1) + B_{0} u (i, j) \end{matrix}

(1a)

A = [\begin{matrix} A_{1} & A_{2} & A_{0} \end{matrix}]

(1b)

where $x (i, j) \in R^{n}$ and $u (i, j) \in R^{m}$ are the state and the control input, respectively. The matrices $A_{1}, A_{2}, A_{0} \in R^{n \times n}$ and $B_{1}, B_{2}, B_{0} \in R^{n \times m}$ are known constant matrices representing the nominal plant. The matrices $Δ A_{1}$ , $Δ A_{2}$ and $Δ A_{0}$ represent parameter uncertainties in the system matrices, which are assumed to be of the form

[\begin{matrix} Δ A_{1} & Δ A_{2} & Δ A_{0} \end{matrix}] = H F (i, j) [\begin{matrix} E_{1} & E_{2} & E_{0} \end{matrix}]

(1c)

where $H \in R^{n \times k}$ , $E_{1}, E_{2}, E_{0} \in R^{l \times n}$ are known structural matrices of uncertainty and $F (i, j) \in R^{k \times l}$ is an unknown matrix representing parameter uncertainty, which satisfies

F^{T} (i, j) F (i, j) \leq I (or equivalently, ‖ F (i, j) ‖ \leq 1)

(1d)

The matrices $H$ , $E_{1}$ , $E_{2}$ and $E_{0}$ characterize how the uncertain parameter in $F (i, j)$ enter the matrices $A_{1}$ , $A_{2}$ and $A_{0}$ . Observe that $F (i, j)$ can always be restricted as (1d) by appropriately choosing $H$ , $E_{1}$ , $E_{2}$ and $E_{0}$ . Therefore, without loss of generality, one can always choose $F (i, j)$ as in (1d).

It is assumed that the system (1a) has a finite set of initial conditions (Ye et al., 2011), i.e. there exist two positive integers $r_{1}$ and $r_{2}$ such that

x (i, 0) = 0, i > r_{2}, x (0, j) = 0, j > r_{1} .

(1e)

Associated with the uncertain system (1a) is the cost function (Ye et al., 2011)

J = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [η^{T} R η] + \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [ξ^{T} W ξ]

(2a)

where

\begin{array}{l} η = [\begin{matrix} x (i + 1, j) \\ x (i, j + 1) \\ x (i, j) \end{matrix}], ξ = [\begin{matrix} u (i + 1, j) \\ u (i, j + 1) \\ u (i, j) \end{matrix}], \\ R = [\begin{matrix} R_{1} & 0 & 0 \\ 0 & R_{2} & 0 \\ 0 & 0 & R_{0} \end{matrix}] > 0, W = [\begin{matrix} W_{1} & 0 & 0 \\ 0 & W_{2} & 0 \\ 0 & 0 & W_{0} \end{matrix}] > 0, \end{array}

and

R_{1}, R_{2}, R_{0} \in R^{n \times n}, W_{1}, W_{2}, W_{0} \in R^{m \times m}

(2b)

Suppose the system state is available for feedback, the aim of this paper is to develop a procedure to design a non-fragile state feedback control law

u (i, j) = (K + Δ K) x (i, j)

(3a)

where $K \in R^{m \times n}$ is the nominal controller gain and $Δ K$ represents the gain perturbation, which is assumed to be of the form

Δ K = H_{k} F_{k} (i, j) E_{k}

(3b)

where $H_{k} \in R^{m \times g}$ , $E_{k} \in R^{h \times n}$ are known constant matrices and $F_{k} (i, j) \in R^{g \times h}$ is an unknown matrix satisfying

F_{k}^{T} (i, j) F_{k} (i, j) \leq I (or equivalently, ‖ F_{k} (i, j) ‖ \leq 1)

(3c)

for system (1) with the cost function (2) such that the closed-loop system

\begin{array}{l} x (i + 1, j + 1) = (A_{1} + B_{1} K + Δ A_{1} + B_{1} Δ K) x (i + 1, j) \\ + (A_{2} + B_{2} K + Δ A_{2} + B_{2} Δ K) x (i, j + 1) \\ + (A_{0} + B_{0} K + Δ A_{0} + B_{0} Δ K) x (i, j) \end{array}

(4)

is asymptotically stable and the closed-loop cost function

J = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} η^{T} \tilde{R} η

(5a)

where

\tilde{R} = [\begin{matrix} R_{1} + {(K + Δ K)}^{T} W_{1} (K + Δ K) & 0 & 0 \\ 0 & R_{2} + {(K + Δ K)}^{T} W_{2} (K + Δ K) & 0 \\ 0 & 0 & R_{0} + {(K + Δ K)}^{T} W_{0} (K + Δ K) \end{matrix}]

(5b)

satisfies $J \leq J^{*}$ , where $J^{*}$ is some specified constant.

Definition 1. Consider the system (1) and cost function (2), if there exist a control law $u^{*} (i, j)$ and a positive scalar $J^{*}$ such that for all admissible uncertainties, the resulting closed-loop system (4) is asymptotically stable and its cost function (5) satisfies $J \leq J^{*}$ , then $J^{*}$ is said to be a guaranteed cost and $u^{*} (i, j)$ is said to be a guaranteed cost control law for the uncertain system (1).

Now, we recall some useful related results on the stability of 2D discrete uncertain systems described by the GM.

Lemma 1 (Bose, 1982; Fornasini and Marchesini, 1976; Kar and Singh, 2003). The 2D discrete uncertain system (1) is globally asymptotically stable if and only if

\det {I - z_{1} (A_{2} + H F (i, j) E_{2}) - z_{2} (A_{1} + H F (i, j) E_{1}) - z_{1} z_{2} (A_{0} + H F (i, j) E_{0})} \neq 0

for all (z_{1}, z_{2}, F (i, j)) \in \bar{U^{2}}

(6)

where $\bar{U^{2}} = {(z_{1}, z_{2}, F (i, j)) : | z_{1} | \leq 1, | z_{2} | \leq 1, ‖ F (i, j) ‖ \leq 1}$ .

As an extension of the result for the global asymptotic stability condition of the 2D discrete systems given in Kar and Singh (2003), one can easily arrive at the following lemma.

Lemma 2 (Kar and Singh, 2003). The system (4) is asymptotically stable if there exist $n \times n$ positive definite symmetric matrices $P$ , $P_{1}$ and $P_{0}$ satisfying

\begin{matrix} Γ = {[\begin{matrix} ({\tilde{A}}_{1} + Δ {\tilde{A}}_{1}) & ({\tilde{A}}_{2} + Δ {\tilde{A}}_{2}) & ({\tilde{A}}_{0} + Δ {\tilde{A}}_{0}) \end{matrix}]}^{T} \\ X P [\begin{matrix} ({\tilde{A}}_{1} + Δ {\tilde{A}}_{1}) & ({\tilde{A}}_{2} + Δ {\tilde{A}}_{2}) & ({\tilde{A}}_{0} + Δ {\tilde{A}}_{0}) \end{matrix}] \end{matrix}

- [\begin{matrix} P_{1} & 0 & 0 \\ 0 & P - P_{1} - P_{0} & 0 \\ 0 & 0 & P_{0} \end{matrix}] < 0

(7)

for all admissible uncertainties (1c) and (3b) satisfying (1d) and (3c), respectively, where ${\tilde{A}}_{1} = A_{1} + B_{1} K$ , ${\tilde{A}}_{2} = A_{2} + B_{2} K$ , ${\tilde{A}}_{0} = A_{0} + B_{0} K$ , $Δ {\tilde{A}}_{1} = Δ A_{1} + B_{1} Δ K$ , $Δ {\tilde{A}}_{2} = Δ A_{2} + B_{2} Δ K$ , $Δ {\tilde{A}}_{0} = Δ A_{0} + B_{0} Δ K$ .

On the basis of above lemma, we have the following definition.

Definition 2. A non-fragile state feedback control law (3a) is said to define a quadratic guaranteed cost control associated with cost matrix $0 < P = P^{T} \in R^{n \times n}$ for the system (4) and cost function (5), if there exist a $3 n \times 3 n$ positive definite symmetric matrix $\tilde{R}$ given by (5b) and $n \times n$ positive definite symmetric matrices $P_{1}$ and $P_{0}$ such that

Γ + \tilde{R} < 0

(8)

for all admissible uncertainties (1c) and (3b) satisfying (1d) and (3c), respectively.

The following well-known lemma is needed in the proof of our main result.

Lemma 3 (Dhawan and Kar, 2007c, 2010; Guan et al., 2001; Ye et al., 2011). Let $A \in R^{n \times n}, H \in R^{n \times k}, E \in R^{l \times n}$ and $Q = Q^{T} \in R^{n \times n}$ be given matrices. Then there exists a positive definite matrix $P$ such that

{[A + H F E]}^{T} P [A + H F E] - Q < 0

(9a)

for all $F$ satisfying $F^{T} F \leq I$ , if and only if there exists a scalar $ε > 0$ such that

[\begin{matrix} - P^{- 1} + ε H H^{T} & A \\ A^{T} & ε^{- 1} E^{T} E - Q \end{matrix}] < 0 .

(9b)

In this context, it may be mentioned that the non-fragile robust guaranteed cost controller design method via state feedback for 2D discrete uncertain systems has been given in Ye et al. (2011). One of the main results for system (1) and cost function (2) may be stated as follows (see Theorem 3, Ye et al., 2011):

Theorem 1 (Ye et al., 2011). If there exist a scalar $ε > 0$ , positive definite symmetric matrices $\hat{S}$ , ${\hat{Y}}_{1}$ , ${\hat{Y}}_{0}$ and $U$ such that the following LMI is feasible:

[\begin{matrix} Π 11 & Π 12 & Π 13 & Π 14 & Π 15 & Π 16 \\ * & - I & 0 & 0 & 0 & 0 \\ * & * & - I & 0 & 0 & 0 \\ * & * & * & - I & 0 & 0 \\ * & * & * & * & - ε I & 0 \\ * & * & * & * & * & - ε I \end{matrix}] < 0

(10)

where

Π 11 = [\begin{matrix} - \hat{S} & 0 & 0 & 0 & A_{1} \hat{S} + B_{1} U & A_{2} \hat{S} + B_{2} U & A_{0} \hat{S} + B_{0} U \\ * & - ε W_{1}^{- 1} & 0 & 0 & U & 0 & 0 \\ * & * & - ε W_{2}^{- 1} & 0 & 0 & U & 0 \\ * & * & * & - ε W_{0}^{- 1} & 0 & 0 & U \\ * & * & * & * & - {\hat{Y}}_{1} & 0 & 0 \\ * & * & * & * & 0 & - (\hat{S} - {\hat{Y}}_{1} - {\hat{Y}}_{0}) & 0 \\ * & * & * & * & 0 & 0 & - {\hat{Y}}_{0} \end{matrix}],

\begin{matrix} Π 12 = [\begin{matrix} H & H & H & B_{1} H_{k} & B_{2} H_{k} & B_{0} H_{k} \\ 0 & 0 & 0 & H_{k} & 0 & 0 \\ 0 & 0 & 0 & 0 & H_{k} & 0 \\ 0 & 0 & 0 & 0 & 0 & H_{k} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], \\ Π 13 = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ \hat{S} E_{1}^{T} \\ \hat{S} E_{2}^{T} \\ \hat{S} E_{0}^{T} \end{matrix}], Π 14 = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \hat{S} E_{k}^{T} & 0 & 0 \\ 0 & \hat{S} E_{k}^{T} & 0 \\ 0 & 0 & \hat{S} E_{k}^{T} \end{matrix}], \end{matrix}

\begin{matrix} Π 15 = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \hat{S} R_{1}^{1 / 2} & 0 & 0 \\ 0 & \hat{S} R_{2}^{1 / 2} & 0 \\ 0 & 0 & \hat{S} R_{0}^{1 / 2} \end{matrix}], \\ Π 16 = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ U^{T} W_{1}^{1 / 2} & 0 & 0 \\ 0 & U^{T} W_{2}^{1 / 2} & 0 \\ 0 & 0 & U^{T} W_{0}^{1 / 2} \end{matrix}], K = U {\hat{S}}^{- 1}, \hat{S} = ε P^{- 1}, \end{matrix}

{\hat{Y}}_{1} = ε^{- 1} \hat{S} P_{1} \hat{S}, and {\hat{Y}}_{0} = ε^{- 1} \hat{S} P_{0} \hat{S},

then there exists a non-fragile state feedback controller $u (i, j) = (K + Δ K) x (i, j)$ such that system (1a) is asymptotically stable with a guaranteed cost and for all admissible uncertainties the cost function satisfies the following bound

\begin{array}{l} J \leq \hat{J} = ε [\sum_{i = 0}^{r_{2} - 1} x^{T} (i + 1, 0) ({\hat{S}}^{- 1} {\hat{Y}}_{1} {\hat{S}}^{- 1}) x (i + 1, 0) \\ + \sum_{j = 0}^{r_{1} - 1} x^{T} (0, j + 1) ({\hat{S}}^{- 1} - {\hat{S}}^{- 1} {\hat{Y}}_{1} {\hat{S}}^{- 1}) x (0, j + 1) \\ + \sum_{i = 1}^{r_{2}} x^{T} (i, 0) ({\hat{S}}^{- 1} {\hat{Y}}_{0} {\hat{S}}^{- 1}) x (i, 0) \\ + x^{T} (0, 0) ({\hat{S}}^{- 1} {\hat{Y}}_{0} {\hat{S}}^{- 1}) x (0, 0)] \end{array}

(11)

Remark 1. Theorem 1 provides a parameterized representation of a set of guaranteed cost controllers (if they exist) in terms of feasible solutions to the LMI (10). In order to compare our proposed method with the approach given in Ye et al. (2011), it becomes necessary to extend the Theorem 1 into a convex optimization problem for the selection of a suitable controller, which minimizes the guaranteed cost in (11).

Observe that the upper bound in (11) depends on the initial conditions of system (1). To remove this dependence, let us assume that the initial conditions of system (1) are arbitrary, but belongs to the set (Dhawan and Kar, 2007c):

\begin{matrix} S = {x (i, 0), x (0, j) \in R^{n} : x (i, 0) = M N_{1}, x (0, j) = M N_{2}, N_{k}^{T} N_{k} < 1 (k = 1, 2)} \end{matrix}

(12)

where $M \in R^{n \times p}$ is a given matrix. Note that the vectors $N_{k} \in R^{p}$ $(k = 1, 2)$ can always be restricted as $N_{k}^{T} N_{k} < 1$ by appropriately choosing $M$ . In other words, there is no loss of generality by choosing initial conditions as in (12).

Under assumption (12), the cost bound (11) leads to:

J \leq \hat{J} = ε [r_{2} λ_{max} (M^{T} {\hat{S}}^{- 1} {\hat{Y}}_{1} {\hat{S}}^{- 1} M) + r_{1} λ_{max} {M^{T} ({\hat{S}}^{- 1} - {\hat{S}}^{- 1} {\hat{Y}}_{1} {\hat{S}}^{- 1}) M} + (r_{2} + 1) (M^{T} {\hat{S}}^{- 1} {\hat{Y}}_{0} {\hat{S}}^{- 1} M)]

(13)

Furthermore, to satisfy (10), one necessarily requires that

\hat{S} - {\hat{Y}}_{1} - {\hat{Y}}_{0} > 0

(14)

From (14), it implies that $\hat{S} > {\hat{Y}}_{1}$ , which yields

{\hat{S}}^{- 1} > {\hat{S}}^{- 1} {\hat{Y}}_{1} {\hat{S}}^{- 1}

(15)

which leads to

λ_{max} (M^{T} {\hat{S}}^{- 1} M) > λ_{max} (M^{T} {\hat{S}}^{- 1} {\hat{Y}}_{1} {\hat{S}}^{- 1} M) .

(16)

Similarly, we can write

{\hat{S}}^{- 1} > {\hat{S}}^{- 1} {\hat{Y}}_{0} {\hat{S}}^{- 1}

(17)

which leads to

λ_{max} (M^{T} {\hat{S}}^{- 1} M) > λ_{max} (M^{T} {\hat{S}}^{- 1} {\hat{Y}}_{0} {\hat{S}}^{- 1} M) .

(18)

Using (16) and (18), (13) necessarily implies

J \leq \hat{J} = ε (r_{1} + 2 r_{2} + 1) λ_{max} (M^{T} {\hat{S}}^{- 1} M) .

(19)

Observe that the guaranteed cost in (19) depends on the choice of the guaranteed cost controllers. In particular, the guaranteed cost controller, which minimizes the guaranteed cost in (19), is more interesting; such a controller is said to be an optimal guaranteed cost controller. Apparently, the upper bound in (19) is not a convex function in $\hat{S}$ and $ε$ . Hence, finding the minimum of this upper bound cannot be considered a convex optimization problem. Since $ε$ and $λ_{max} (M^{T} {\hat{S}}^{- 1} M)$ are positive, we may obtain a suboptimal guaranteed cost controller by minimizing $ε$ $+$ $λ_{max} {M^{T} {\hat{S}}^{- 1} M}$ . Based on Theorem 1, the design problem of such a suboptimal guaranteed cost controller can be formulated as the following optimization problem.

Theorem 2. Consider system (1) with initial condition (12) and cost function (2), if the following optimization problem:

minimize ε + (r_{1} + 2 r_{2} + 1) λ

s . t . {\begin{matrix} (i) . (10), \\ (ii) . [\begin{matrix} - λ I & M^{T} \\ M & - \hat{S} \end{matrix}] < 0 \end{matrix}

(20)

has a feasible solution $λ$ >0, $ε$ >0, U $\in R^{m \times n}$ , $0 < \hat{S} = {\hat{S}}^{T} \in R^{n \times n}$ , $0 < {\hat{Y}}_{1} = {\hat{Y}}_{1}^{T} \in R^{n \times n}$ and $0 < {\hat{Y}}_{0} = {\hat{Y}}_{0}^{T} \in R^{n \times n}$ then the control law (3) with $K = U {\hat{S}}^{- 1}$ is a non-fragile robust suboptimal guaranteed cost control law which ensures the minimization of guaranteed cost in (19).

Proof. By Theorem 1, the control law $K = U {\hat{S}}^{- 1}$ constructed in terms of any feasible solution $λ$ , $ε$ , $\hat{S}$ , ${\hat{Y}}_{1}$ , ${\hat{Y}}_{0}$ and $U$ is a guaranteed cost controller of system (1) with initial condition (12). To obtain the optimum value of the upper bound of guaranteed cost, the term $λ_{max} {M^{T} {\hat{S}}^{- 1} M}$ in (19) is changed to $λ_{max} {M^{T} {\hat{S}}^{- 1} M} < λ$ $\Leftrightarrow$ ${M^{T} {\hat{S}}^{- 1} M} < λ I$ , which, in turn, implies the constraint (ii) in (20). Thus, the minimization of $ε + (r_{1} + 2 r_{2} + 1) λ$ implies the minimization of the guaranteed cost in (19). This completes the proof of Theorem 2.

Remark 2. The optimization problem given by (20) is an LMI eigenvalue problem (Boyd et al., 1994; Gahinet et al., 1995), which provides a procedure to design the non-fragile robust suboptimal guaranteed cost controller via Ye et al. (2011).

Main results

The following lemma relates the notion of cost matrix to the quadratic stability and an upper bound on the cost function (5).

Lemma 4. If there exist a $n \times n$ positive definite symmetric cost matrix $P$ satisfying (8) for the system (4) with initial conditions (1e), (12) and cost function (5), then (i) the system (4) is quadratically stable with a guaranteed cost and (ii) the cost function (5) satisfies the following bound

J \leq J^{*} = (r_{1} + 2 r_{2} + 1) λ_{max} (M^{T} P M)

(21)

for all admissible uncertainties (1c) and (3b) satisfying (1d) and (3c), respectively.

Proof. Proof of (i) directly follows from Lemma 2 and Definition 2.

To prove (ii), consider a quadratic 2D Lyapunov function:

v_{τ_{1} τ_{2}} (i, j) = x_{τ_{1} τ_{2}}^{T} (i, j) P x_{τ_{1} τ_{2}} (i, j)

(22)

where $x_{τ_{1} τ_{2}} (i, j) = x (i + τ_{1}, j + τ_{2})$ .

Let $Δ v (i, j)$ be defined as

Δ v (i, j) = {\tilde{v}}_{11} (i, j) - {\tilde{v}}_{10} (i, j) - {\tilde{v}}_{01} (i, j) - {\tilde{v}}_{00} (i, j)

(23)

where

\begin{array}{l} {\tilde{v}}_{11} (i, j) = x_{11}^{T} (i, j) P x_{11} (i, j), \\ {\tilde{v}}_{10} (i, j) = x_{10}^{T} (i, j) P_{1} x_{10} (i, j), \\ {\tilde{v}}_{01} (i, j) = x_{01}^{T} (i, j) (P - P_{1} - P_{0}) x_{01} (i, j), \\ {\tilde{v}}_{00} (i, j) = x_{00}^{T} (i, j) P_{0} x_{00} (i, j) . \end{array}

Along the trajectory of the closed-loop system (4), we obtain

Δ v (i, j) = η^{T} Γ}, η

(24)

where $η$ and $Γ$ are defined in (2b) and (7), respectively. Since $P$ is a cost matrix, it follows from Definition 2 that

η^{T} (Γ + \tilde{R}) η < 0 .

(25)

From (24) and (25), we have

η^{T} \tilde{R} η < - Δ v (i, j) .

(26)

Summing both sides of the above inequality over $i, j = 0 \to \infty$ yields

\begin{matrix} J < - \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} Δ v (i, j) \\ = - \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i + 1, j + 1) P x (i + 1, j + 1) - x^{T} (i + 1, j) P_{1} x (i + 1, j) - x^{T} (i, j + 1) X (P - P_{1} - P_{0}) x (i, j + 1) - x^{T} (i, j) P_{0} x (i, j)] \\ = - \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i + 1, j + 1) P_{1} x (i + 1, j + 1) - x^{T} (i + 1, j) X P_{1} x (i + 1, j)] - \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i + 1, j + 1) (P - P_{1} - P_{0}) X x (i + 1, j + 1) - x^{T} (i, j + 1) (P - P_{1} - P_{0}) x (i, j + 1)] - \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i + 1, j + 1) P_{0} x (i + 1, j + 1) - x^{T} (i, j) P_{0} x (i, j)] \\ = \sum_{i = 0}^{\infty} x^{T} (i + 1, 0) P_{1} x (i + 1, 0) + \sum_{j = 0}^{\infty} x^{T} (0, j + 1) X (P - P_{1} - P_{0}) x (0, j + 1) + \sum_{i = 0}^{\infty} x^{T} (i, 0) P_{0} x (i, 0) + \sum_{j = 1}^{\infty} x^{T} (0, j) P_{0} x (0, j) \\ = \sum_{i = 0}^{r_{2} - 1} x^{T} (i + 1, 0) P_{1} x (i + 1, 0) + \sum_{j = 0}^{r_{1} - 1} x^{T} (0, j + 1) (P - P_{1} - P_{0}) x (0, j + 1) + \sum_{i = 0}^{r_{2}} x^{T} (i, 0) P_{0} x (i, 0) + \sum_{j = 1}^{r_{1}} x^{T} (0, j) P_{0} x (0, j) \\ = \sum_{i = 0}^{r_{2} - 1} x^{T} (i + 1, 0) P_{1} x (i + 1, 0) + \sum_{j = 0}^{r_{1} - 1} x^{T} (0, j + 1) (P - P_{1}) x (0, j + 1) + \sum_{i = 1}^{r_{2}} x^{T} (i, 0) P_{0} x (i, 0) + x^{T} (0, 0) P_{0} x (0, 0) \leq [r_{2} λ_{max} (M^{T} P_{1} M) + r_{1} λ_{max} {M^{T} (P - P_{1}) M} + (r_{2} + 1) λ_{max} (M^{T} P_{0} M)] \end{matrix}

(27)

where use has been made of (5), (1e) and (12) and the relation $lim_{i + j \to \infty} x (i, j)$ = 0. Note that if (8) holds, we have $(P - P_{1} - P_{0})$ > 0, which implies that

M^{T} P M \geq M^{T} P_{1} M

(28a)

and

M^{T} P M \geq M^{T} P_{0} M

(28b)

Therefore, the upper bound in (21) can be obtained by applying (28) in (27). This completes the proof of the Lemma 4.

The following theorem establishes that the problem of determining non-fragile robust guaranteed cost control for system (1) with initial condition (12) and the cost function (2) can be recast to an LMI feasibility problem.

Theorem 3. Consider system (1) with initial condition (12) and cost function (2), then there exists a non-fragile state feedback control law (3) that solves the addressed robust guaranteed cost control problem if there exist a positive scalar $ε$ , a $m \times n$ matrix $U$ and $n \times n$ positive definite symmetric matrices $S$ , $Y_{1}$ , $Y_{0}$ such that the following LMI is feasible:

[\begin{matrix} Π 11 & Π 12 & Π 13 & Π 14 \\ * & - I & 0 & 0 \\ * & * & - ε I & 0 \\ * & * & * & - ε I \end{matrix}] < 0

(29)

where

Π 11 = [\begin{matrix} (- S + ε H H^{T} + ε B_{1} H_{k} H_{k}^{T} B_{1}^{T} + ε B_{2} H_{k} H_{k}^{T} B_{2}^{T} + ε B_{0} H_{k} H_{k}^{T} B_{0}^{T}) & ε B_{1} H_{k} H_{k}^{T} \\ ε H_{k} H_{k}^{T} B_{1}^{T} & - W_{1}^{- 1} + ε H_{k} H_{k}^{T} \\ ε H_{k} H_{k}^{T} B_{2}^{T} & 0 \\ ε H_{k} H_{k}^{T} B_{0}^{T} & 0 \\ {\bar{A}}_{1}^{T} & U^{T} \\ {\bar{A}}_{2}^{T} & 0 \\ {\bar{A}}_{0}^{T} & 0 \end{matrix} \begin{matrix} ε B_{2} H_{k} H_{k}^{T} & ε B_{0} H_{k} H_{k}^{T} & {\bar{A}}_{1} & {\bar{A}}_{2} & {\bar{A}}_{0} \\ 0 & 0 & U & 0 & 0 \\ - W_{2}^{- 1} + ε H_{k} H_{k}^{T} & 0 & 0 & U & 0 \\ 0 & - W_{0}^{- 1} + ε H_{k} H_{k}^{T} & 0 & 0 & U \\ 0 & 0 & - Y_{1} & 0 & 0 \\ U^{T} & 0 & 0 & \bar{Y} & 0 \\ 0 & U^{T} & 0 & 0 & - Y_{0} \end{matrix}],

\begin{matrix} Π 12 = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ S R_{1}^{1 / 2} & 0 & 0 \\ 0 & S R_{2}^{1 / 2} & 0 \\ 0 & 0 & S R_{0}^{1 / 2} \end{matrix}], Π 13 = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ S E_{k}^{T} & 0 & 0 \\ 0 & S E_{k}^{T} & 0 \\ 0 & 0 & S E_{k}^{T} \end{matrix}], \\ Π 14 = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ S E_{1}^{T} \\ S E_{2}^{T} \\ S E_{0}^{T} \end{matrix}], \end{matrix}

$\bar{Y} = - S + Y_{1} + Y_{0}$ , ${\bar{A}}_{1} = A_{1} S + B_{1} U$ , ${\bar{A}}_{2} = A_{2} S + B_{2} U$ , ${\bar{A}}_{0} = A_{0} S + B_{0} U$ . In this situation, the feedback gain of the stabilizing non-fragile guaranteed cost control law is given by

K = U S^{- 1}

(30)

Moreover, the closed-loop cost function satisfies the bound

J \leq J^{*} = (r_{1} + 2 r_{2} + 1) λ_{max} (M^{T} S^{- 1} M) .

(31)

Proof. Using (1c), (1d), (3b), (3c) and (5b), (8) can be expressed as

\begin{matrix} {[\bar{A} + \bar{H} \bar{F} (i, j) \bar{E}]}^{T} \bar{P} [\bar{A} + \bar{H} \bar{F} (i, j) \bar{E}] \\ + [\begin{matrix} (- P_{1} + R_{1}) & 0 & 0 \\ 0 & (- P + P_{1} + P_{0} + R_{2}) & 0 \\ 0 & 0 & (- P_{0} + R_{0}) \end{matrix}] < 0 \end{matrix}

(32)

where

\begin{array}{l} \bar{A} = [\begin{matrix} A_{1} + B_{1} K & A_{2} + B_{2} K & A_{0} + B_{0} K \\ K & 0 & 0 \\ 0 & K & 0 \\ 0 & 0 & K \end{matrix}], \\ \bar{H} = [\begin{matrix} H & B_{1} H_{k} & B_{2} H_{k} & B_{0} H_{k} \\ 0 & H_{k} & 0 & 0 \\ 0 & 0 & H_{k} & 0 \\ 0 & 0 & 0 & H_{k} \end{matrix}] \end{array}

\begin{matrix} \bar{F} (i, j) & = [\begin{matrix} F (i, j) & 0 & 0 & 0 \\ 0 & F_{k} (i, j) & 0 & 0 \\ 0 & 0 & F_{k} (i, j) & 0 \\ 0 & 0 & 0 & F_{k} (i, j) \end{matrix}], \\ \bar{E} & = [\begin{matrix} E_{1} & E_{2} & E_{0} \\ E_{k} & 0 & 0 \\ 0 & E_{k} & 0 \\ 0 & 0 & E_{k} \end{matrix}], \\ \bar{P} & = [\begin{matrix} P & 0 & 0 & 0 \\ 0 & W_{1} & 0 & 0 \\ 0 & 0 & W_{2} & 0 \\ 0 & 0 & 0 & W_{0} \end{matrix}] . \end{matrix}

Using Lemma 3, (32) can be rearranged as

[\begin{matrix} (- P^{- 1} + ε^{- 1} H H^{T} + ε^{- 1} B_{1} H_{k} H_{k}^{T} B_{1}^{T} + ε^{- 1} B_{2} H_{k} H_{k}^{T} B_{2}^{T} + ε^{- 1} B_{0} H_{k} H_{k}^{T} B_{0}^{T}) & ε^{- 1} B_{1} H_{k} H_{k}^{T} \\ ε^{- 1} H_{k} H_{k}^{T} B_{1}^{T} & - W_{1}^{- 1} + ε^{- 1} H_{k} H_{k}^{T} \\ ε^{- 1} H_{k} H_{k}^{T} B_{2}^{T} & 0 \\ ε^{- 1} H_{k} H_{k}^{T} B_{0}^{T} & 0 \\ {(A_{1} + B_{1} K)}^{T} & K^{T} \\ {(A_{2} + B_{2} K)}^{T} & 0 \\ {(A_{0} + B_{0} K)}^{T} & 0 \end{matrix} \begin{matrix} ε^{- 1} B_{2} H_{k} H_{k}^{T} & ε^{- 1} B_{0} H_{k} H_{k}^{T} & (A_{1} + B_{1} K) \\ 0 & 0 & K \\ - W_{2}^{- 1} + ε^{- 1} H_{k} H_{k}^{T} & 0 & 0 \\ 0 & - W_{0}^{- 1} + ε^{- 1} H_{k} H_{k}^{T} & 0 \\ 0 & 0 & ε E_{1}^{T} E_{1} + ε E_{k}^{T} E_{k} - P_{1} + R_{1} \\ K^{T} & 0 & ε E_{2}^{T} E_{1} \\ 0 & K^{T} & ε E_{0}^{T} E_{1} \end{matrix} \begin{matrix} (A_{2} + B_{2} K) & (A_{0} + B_{0} K) \\ 0 & 0 \\ K & 0 \\ 0 & K \\ ε E_{1}^{T} E_{2} & ε E_{1}^{T} E_{0} \\ ε E_{2}^{T} E_{2} + ε E_{k}^{T} E_{k} - P + P_{1} + P_{0} + R_{2} & ε E_{2}^{T} E_{0} \\ ε E_{0}^{T} E_{2} & ε E_{0}^{T} E_{0} + ε E_{k}^{T} E_{k} - P_{0} + R_{0} \end{matrix}] < 0 .

(33)

Premultiplying and postmultiplying (33) by diag ${I, I, I, I, P^{- 1}, P^{- 1}, P^{- 1}},$ one obtains

\begin{matrix} [\begin{matrix} - S & 0 & 0 & 0 & {\bar{A}}_{1} & {\bar{A}}_{2} & {\bar{A}}_{0} \\ 0 & - ε W_{1}^{- 1} & 0 & 0 & U & 0 & 0 \\ 0 & 0 & - ε W_{2}^{- 1} & 0 & 0 & U & 0 \\ 0 & 0 & 0 & - ε W_{0}^{- 1} & 0 & 0 & U \\ {\bar{A}}_{1}^{T} & U^{T} & 0 & 0 & - Y_{1} & 0 & 0 \\ {\bar{A}}_{2}^{T} & 0 & U^{T} & 0 & 0 & \bar{Y} & 0 \\ {\bar{A}}_{0}^{T} & 0 & 0 & U^{T} & 0 & 0 & - Y_{0} \end{matrix}] + \\ [\begin{matrix} (H H^{T} + B_{1} H_{k} H_{k}^{T} B_{1}^{T} + B_{2} H_{k} H_{k}^{T} B_{2}^{T} + B_{0} H_{k} H_{k}^{T} B_{0}^{T}) & B_{1} H_{k} H_{k}^{T} \\ H_{k} H_{k}^{T} B_{1}^{T} & H_{k} H_{k}^{T} \\ H_{k} H_{k}^{T} B_{2}^{T} & 0 \\ H_{k} H_{k}^{T} B_{0}^{T} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{matrix} \begin{matrix} B_{2} H_{k} H_{k}^{T} & B_{0} H_{k} H_{k}^{T} & 0 \\ 0 & 0 & 0 \\ H_{k} H_{k}^{T} & 0 & 0 \\ 0 & H_{k} H_{k}^{T} & 0 \\ 0 & 0 & (S E_{1}^{T} E_{1} S + S E_{k}^{T} E_{k} S + ε^{- 1} S R_{1} S) \\ 0 & 0 & S E_{2}^{T} E_{1} S \\ 0 & 0 & S E_{0}^{T} E_{1} S \end{matrix} \begin{matrix} 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ S E_{1}^{T} E_{2} S & S E_{1}^{T} E_{0} S \\ (S E_{2}^{T} E_{2} S + S E_{k}^{T} E_{k} S + ε^{- 1} S R_{2} S) & S E_{2}^{T} E_{0} S \\ S E_{0}^{T} E_{2} S & (S E_{0}^{T} E_{0} S + S E_{k}^{T} E_{k} S + ε^{- 1} S R_{0} S) \end{matrix}] < 0, \end{matrix}

(34)

where

S = P^{- 1}

(35)

and

Y_{1} = S P_{1} S, Y_{0} = S P_{0} S .

(36)

The equivalence of (34) and (29) follows trivially from the Schur complements. Using (35), the bound of cost function can be easily obtained from (21). This completes the proof of Theorem 3.

Remark 3. It should be observed that the matrix inequality (29) is linear in variables $ε$ , $S$ , $U$ , $Y_{1}$ , and $Y_{0}$ , which can be solved using Matlab LMI Toolbox (Boyd et al., 1994; Gahinet et al., 1995).

Theorem 3 provides a parameterized representation of a set of non-fragile robust guaranteed cost controllers (if they exist) in terms of the feasible solutions to the LMI (29). This parameterized representation can be exploited to design the non-fragile robust optimal guaranteed cost controller, which minimizes the guaranteed cost in (31). Based on Theorem 3, the design problem of non-fragile robust optimal guaranteed cost controller can be formulated as follows.

Theorem 4. Consider system (1) with initial condition (12) and cost function (2), if the following optimization problem:

minimize (r_{1} + 2 r_{2} + 1) λ

s . t . {\begin{matrix} (i) . (29), \\ (ii) . [\begin{matrix} - λ I & M^{T} \\ M & - S \end{matrix}] < 0 \end{matrix}

(37)

has a feasible solution $λ$ >0, $ε$ >0, U $\in R^{m \times n}$ , $0 < S = S^{T} \in R^{n \times n}$ , $0 < Y_{1} = Y_{1}^{T} \in R^{n \times n}$ and $0 < Y_{0} = Y_{0}^{T} \in R^{n \times n}$ , then the control law (3) with $K = U S^{- 1}$ is a non-fragile robust optimal guaranteed cost control law which ensures the minimization of guaranteed cost in (31).

Proof. By Theorem 3, the control law (30) constructed in terms of any feasible solution $ε$ , U , $S$ , $Y_{1}$ and $Y_{0}$ is a non-fragile robust guaranteed cost controller of system (1). To obtain the optimum value of the upper bound of guaranteed cost, the term $λ_{max} (M^{T} S^{- 1} M)$ in (31) is changed to $λ_{max} (M^{T} S^{- 1} M) < λ$ $\Leftrightarrow M^{T} S^{- 1} M < λ I$ , which, in turn, implies the constraint (ii) in (37). Thus, the minimization of $(r_{1} + 2 r_{2} + 1) λ$ implies the minimization of the guaranteed cost in (31). This completes the proof of Theorem 4.

Remark 4. The optimization problem in (37) is an LMI eigenvalue problem, which provides a procedure to design a non-fragile robust optimal guaranteed cost controller.

Remark 5. It is worth comparing Theorem 4 with Theorem 2 in terms of computational complexity. The computational complexity is proportional to $L M^{3}$ , where $L$ is the number of LMI rows and $M$ is the number of scalar decision variables. In Theorem 4, the total number of scalar decision variables is $M_{4} = \frac{3 n (n + 1)}{2} + mn + 2$ and the row size is $L_{4} = (8 n + 3 m + l + 3 h + p)$ , so the computational complexity of Theorem 4 is proportional to $L_{4} M_{4}^{3}$ . On the other hand, the computational complexity of Theorem 2 is proportional to $L_{2} M_{2}^{3}$ with $L_{2} = (8 n + 6 m + 3 k + l + 3 g + 3 h + p)$ and $M_{2} = \frac{3 n (n + 1)}{2} + mn + 2$ . Clearly, Theorem 4 has smaller computational complexity than Theorem 2.

Illustrative examples

In this section, two examples are given that show the approach of this paper leads to less stringent results than that obtainable via Theorem 2.

Example 1

Consider a 2D discrete uncertain system represented by (1) and (2) with

\begin{array}{l} A_{1} = [\begin{matrix} 0.6 & 0.8 \\ 0 & 0.3 \end{matrix}], A_{2} = [\begin{matrix} 0.4 & 1 \\ 0 & 0.8 \end{matrix}], A_{0} = [\begin{matrix} 0.01 & 0 \\ 0 & 0.02 \end{matrix}] \\ B_{1} = [\begin{matrix} - 0.01 \\ 0 \end{matrix}], B_{2} = [\begin{matrix} 0 \\ 1 \end{matrix}], B_{0} = [\begin{matrix} 0.01 \\ 0 \end{matrix}], \end{array}

\begin{matrix} H = [\begin{matrix} 0.2 \\ 0 \end{matrix}], E_{1} = [\begin{matrix} 0.2 & 0 \end{matrix}], E_{2} = [\begin{matrix} 0.1 & 0 \end{matrix}], \\ E_{0} = [\begin{matrix} 0.2 & 0 \end{matrix}], R_{1} = [\begin{matrix} 0.01 & 0 \\ 0 & 0.01 \end{matrix}], \\ R_{2} = [\begin{matrix} 0.01 & 0 \\ 0 & 0.01 \end{matrix}], R_{0} = [\begin{matrix} 0.01 & 0 \\ 0 & 0.01 \end{matrix}], \\ H_{k} = 0.2, E_{k} = [\begin{matrix} 0.2 & 0.2 \end{matrix}], W_{1} = W_{2} = W_{0} = 0.01, r_{1} = 2, \\ r_{2} = 3, M = [\begin{matrix} 0.01 \\ 0.02 \end{matrix}] \end{matrix}

(38)

Using Lemma 1, it is easy to verify that the above system is unstable. We wish to construct a non-fragile robust optimal guaranteed cost controller for this system. It is found using the LMI toolbox in Matlab (Boyd et al., 1994; Gahinet et al., 1995) that the optimization problem (37) is feasible for the present example and the optimal solution is given by

\begin{matrix} S = [\begin{matrix} 10.4585 & - 4.1653 \\ - 4.1653 & 2.8957 \end{matrix}], Y_{1} = [\begin{matrix} 5.3862 & - 2.0950 \\ - 2.0950 & 1.6803 \end{matrix}], \\ Y_{0} = [\begin{matrix} 3.1214 & - 1.3376 \\ - 1.3376 & 0.6373 \end{matrix}], U = [3.2710 - 2.9555], \\ ε = 7.5364, λ = 4.7460 \times 1 0^{- 4} . \end{matrix}

(39)

By Theorem 4, the feedback gain of the stabilizing optimal non-fragile guaranteed cost control law is

K = [\begin{matrix} - 0.2194 & - 1.3363 \end{matrix}]

(40)

and the least upper bound of the corresponding closed-loop cost function is

J^{*} = 0.0043

(41)

We now apply Theorem 2 for the system under consideration. Solving the optimization problem (20), we find the optimal solution for the present system as

\begin{matrix} \hat{S} = [\begin{matrix} 1.9008 & - 0.7533 \\ - 0.7533 & 0.4680 \end{matrix}], {\hat{Y}}_{1} = [\begin{matrix} 1.0254 & - 0.4114 \\ - 0.4114 & 0.2959 \end{matrix}], \\ {\hat{Y}}_{0} = [\begin{matrix} 0.5776 & - 0.2388 \\ - 0.2388 & 0.1064 \end{matrix}], U = [0.5854 - 0.4569], \\ ε = 0.3309, λ = 0.0034 . \end{matrix}

(42)

By Theorem 2, the feedback gain of the stabilizing suboptimal non-fragile guaranteed cost control law is

K = [\begin{matrix} - 0.2178 & - 1.3270 \end{matrix}]

(43)

and the least upper bound of the corresponding closed-loop cost function is

\hat{J} < 0.0102 .

(44)

From (44), it is clear that the least upper bound of the closed-loop cost function obtained via Theorem 4 is smaller than that arrived at via Theorem 2.

Furthermore, we would also like to compare our proposed method (Theorem 4) with Theorem 2 in terms of computational complexity (see Remark 5) for the present example. The computational complexity of Theorem 4 is proportional to $L_{4} M_{4}^{3} = 24 \times {(13)}^{3} = 52728$ , while the computational complexity of Theorem 2 is proportional to $L_{2} M_{2}^{3} = 33 \times {(13)}^{3} = 72501$ . Thus, Theorem 4 has significantly smaller computational complexity than Theorem 2.

Based on the above comparisons, it is clear that our proposed method (Theorem 4) provides improved results over Theorem 2 for the present example.

Example 2

In this example, we shall demonstrate the application of our proposed method (Theorem 4) to the control of thermal processes in chemical reactors, heat exchangers and pipe furnaces, which can be expressed by the following first-order partial differential equation with time (Kaczorek, 1985):

\frac{\partial T (x, t)}{\partial x} = - \frac{\partial T (x, t)}{\partial t} - a_{0} T (x, t) + b u (x, t)

(45)

where $T (x, t)$ is the temperature at space $x$ $\in [0, x_{f}]$ and time $t$ $\in [0, \infty)$ , $u (x, t)$ is input function and $a_{0}$ and $b$ are the real coefficients. Taking

T (i, j) = T (i Δ x, j Δ t), u (i, j) = u (i Δ x, j Δ t)

(46a)

\frac{\partial T (x, t)}{\partial x} \approx \frac{T (i, j) - T (i - 1, j)}{Δ x}, \frac{\partial T (x, t)}{\partial t} \approx \frac{T (i, j + 1) - T (i, j)}{Δ t}

(46b)

it is easy to verify that (45) can be expressed in the following form:

\begin{matrix} T (i, j + 1) = (1 - \frac{Δ t}{Δ x} - a_{0} Δ t) T (i, j) \\ + \frac{Δ t}{Δ x} T (i - 1, j) + b Δ t u (i, j) . \end{matrix}

(47)

Denote $x (i, j) = T (i, j)$ ; it is easy to verify that (47) can be converted into the following 2D GM:

\begin{matrix} x (i + 1, j + 1) = (1 - \frac{Δ t}{Δ x} - a_{0} Δ t) x (i + 1, j) \\ + \frac{Δ t}{Δ x} x (i, j) + b Δ t u (i + 1, j) . \end{matrix}

(48)

Now, consider the problem of non-fragile robust optimal guaranteed cost control of a system represented by (48) with

a_{0} = - 0.25, b = 0.25, Δ x = 0.5, Δ t = 0.4

(49)

and the initial state satisfies the condition (1e) for $r_{1} = 2$ and $r_{2} = 3$ and belongs to the set (12) with $M = [2]$ .

It is also assumed that the above system is subjected to the parameter uncertainties of the form (1c)–(1d) with

H = [0.2], E_{1} = [0.2], E_{2} = [0], E_{0} = [0.2] .

(50)

Associated with the uncertain system (48)–(50), the cost function is given by (2) with

\begin{matrix} W_{1} & = [0.01], W_{2} = [0.01], W_{0} = [0.01], \\ R_{1} & = [0.01], R_{2} = [0.01], R_{0} = [0.01] . \end{matrix}

(51)

Applying Lemma 1, it is easy to verify that the above system is unstable. We wish to construct a non-fragile robust optimal guaranteed cost controller for the system under consideration with the controller gain perturbation satisfying (3b)–(3c) with

H_{k} = [0.2], E_{k} = [0.2] .

(52)

It is found using the Matlab LMI toolbox (Boyd et al., 1994; Gahinet et al., 1995) that the optimization problem (37) is feasible for the present example and the optimal solution is given by

\begin{matrix} S & = [0.6281], Y_{1} = [0.0612], Y_{0} = [0.5147], \\ U & = [- 1.9815], ε = 1.7699, λ = 6.3689 . \end{matrix}

(53)

By Theorem 4, the feedback gain of the stabilizing optimal non-fragile guaranteed cost control law is

K = [- 3.1550]

(54)

and the least upper bound of the corresponding closed-loop cost function is

J^{*} = 57.3200 .

(55)

We now apply Theorem 2 for the system under consideration. It is found that the optimization problem (20) is feasible for the present system and the optimal solution is obtained as

\begin{matrix} \hat{S} & = [1.0320], {\hat{Y}}_{1} = [0.1106], {\hat{Y}}_{0} = [0.8558], \\ U & = [- 3.5661], ε = 11.5453, λ = 3.8761 . \end{matrix}

(56)

By Theorem 2, the feedback gain of the stabilizing suboptimal non-fragile guaranteed cost control law is

K = [- 3.4557]

(57)

and the least upper bound of the corresponding closed-loop cost function is

\hat{J} = 402.7611 .

(58)

From (58), it is clear that the least upper bound of the closed-loop cost function obtained via Theorem 4 is smaller than that arrived at via Theorem 2.

Furthermore, we would also like to compare our proposed method (Theorem 4) with Theorem 2 in terms of computational complexity for the present example. The computational complexity of Theorem 4 is proportional to $L_{4} M_{4}^{3} = 16 \times {(6)}^{3} = 3456$ while the computational complexity of Theorem 2 is proportional to $L_{2} M_{2}^{3} = 25 \times {(6)}^{3} = 5400$ . Thus, Theorem 4 has smaller computational complexity than Theorem 2.

Based on the above comparisons, it is clear that our proposed method (Theorem 4) provides improved results over Theorem 2 for the present example.

Conclusions

In this paper, we have presented a solution to the non-fragile robust optimal guaranteed cost control problem for a class of uncertain 2D discrete systems described by the GM in a numerically efficient LMI framework. A non-fragile robust optimal guaranteed cost controller is obtained through a convex optimization problem, which can be solved by using Matlab LMI Toolbox (Boyd et al., 1994; Gahinet et al., 1995). Finally, it has been shown with the help of illustrative examples that the presented approach provides a relatively less stringent condition for the design of non-fragile robust optimal guaranteed cost controller than the approach given in Ye et al. (2011).

Footnotes

Acknowledgements

The authors wish to thank the Editor-in-Chief and the anonymous reviewers for their constructive comments and suggestions.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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