An LMI approach to non-fragile robust optimal guaranteed cost control of uncertain 2-D discrete systems with both state and input delays

Abstract

In this paper, we present a solution to the problem of non-fragile robust optimal guaranteed cost control for a class of uncertain two-dimensional(2-D) discrete systems described by the general model (GM) subject to both state and input delays. The parameter uncertainties are assumed norm-bounded. A linear matrix inequality (LMI)-based sufficient condition for the existence of non-fragile robust guaranteed cost controller is established. Furthermore, a convex optimization problem with LMI constraints is proposed to select a non-fragile robust optimal guaranteed cost controller stabilizing the uncertain 2-D discrete system with both state and input delays as well as achieving the least guaranteed cost for the resulting closed-loop system. The effectiveness of the proposed method is demonstrated with an illustrative example.

Keywords

2-D discrete systems guaranteed cost control linear matrix inequality Lyapunov methods robust stability state-delayed systems uncertain systems

Introduction

Over the past few decades, the guaranteed cost control problem for uncertain two-dimensional (2-D) discrete systems has drawn considerable attention. Much effort has been directed towards finding a robust controller that not only stabilizes the closed-loop system but also guarantees an upper bound on the closed-loop cost function for all admissible parameter uncertainties. Based on this idea, many significant results have been obtained (Dhawan and Kar, 2007a, 2007b, 2007c, 2010, 2011; Guan et al., 2001).

In the real world, the process dynamics of many physical systems are affected by delays. As the presence of delays is frequently a source of instability and poor performance of many control systems, the study of 2-D discrete systems with delays has attracted many researchers and useful results have been reported in the literature (Paszke et al., 2004; Peng et al., 2007; Tiwari and Dhawan, 2012a, 2012b, 2013; Xu and Yu, 2009a, 2009b; Xu et al., 2007; Ye et al., 2009). In Xu et al. (2007), a sufficient condition for the stability of 2-D discrete state-delayed systems in the general model (GM) has been derived via the Lyapunov approach. The problem of robust stability and stabilization of uncertain 2-D discrete state-delayed systems was studied in Paszke et al. (2004). A robust output feedback guaranteed cost control problem for uncertain 2-D discrete state-delayed systems described by the FM second model was considered in Peng et al. (2007). In Ye et al. (2009), the problem of robust guaranteed cost control via memoryless state feedback for uncertain 2-D discrete state-delayed systems described by the FM second model was considered. In Tiwari and Dhawan (2012a), several technical errors that have occurred in the main results of Ye et al. (2009) were pointed out and corrected. A solution to the guaranteed cost control problem via memory state feedback control laws for a class of uncertain 2-D discrete state-delayed systems described by the FM second model was presented in Tiwari and Dhawan (2012b). In Tiwari and Dhawan (2013), the problem of optimal guaranteed cost control for a class of uncertain 2-D discrete systems described by the FM second model with both state and input delays via memory state feedback was considered. The problem of delay–dependent guaranteed cost control via memoryless state feedback for uncertain 2-D discrete state-delayed systems described by the FM second model was studied in Xu and Yu (2009a). In Xu and Yu (2009b), a delay-dependent approach to investigate the $H_{\infty}$ control problem for 2-D discrete state-delayed systems described by the FM second model was presented. However, 1-D and 2-D switched systems have also gained huge interest in recent years (Ghous et al., 2015; Hou et al., 2002; Huang and Xiang, 2014; Zong et al., 2013, 2015, 2016). Switched systems are a particular class of hybrid systems that consist of several subsystems and switching laws orchestrating the active subsystem at each time instant (Xu and Antsaklis, 2004).

Recently, research on non-fragile control problem has attracted attention (Dhawan, 2012; Sharma and Dhawan, 2012; Tandon and Dhawan, 2014, 2016; Ye et al., 2011). The objective of non-fragile control 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, i.e. the controller is non-fragile (Yang and Wang, 2001). In Ye et al. (2011), the problem of non-fragile robust guaranteed cost control for a class of uncertain 2-D discrete systems described by the GM has been considered and a sufficient condition for the existence of non-fragile robust guaranteed cost controllers has been established via a linear matrix inequality (LMI) approach. A solution to the non-fragile robust optimal guaranteed cost control problem for a class of 2-D discrete systems described by the GM has been presented in Tandon and Dhawan (2014) and it has been shown that their approach provides less stringent results than that given in Ye et al. (2011). Very recently, the problem of non-fragile robust optimal guaranteed cost control for a class of uncertain 2-D discrete state-delayed systems described by the GM has been studied in Tandon and Dhawan (2016) and it has also been shown that the proposed method leads to a tighter upper bound of the closed-loop cost function compared with that obtained in Tandon and Dhawan (2014). It may be mentioned here that all the results reported so far in the literature for the non-fragile robust guaranteed cost control problem for uncertain 2-D discrete systems consider only state delays, whereas in many real situations, input delays are often encountered because of transmission of the measurement information. The existence of these delays may be the source of instability or serious deterioration in the performance of the closed-loop system. Non-fragile robust optimal guaranteed cost control for a class of uncertain 2-D discrete systems in the GM setting with both state and input delays is an important problem. However, such a problem has not yet been fully explored in the literature.

With this motivation, we consider the problem of non-fragile robust optimal guaranteed cost control for a class of uncertain 2-D discrete systems described by the GM subject to both state and input delays. The approach adopted in this paper is as follows: we first develop an LMI-based criterion for the existence of non-fragile robust guaranteed cost controllers in terms of feasible solution to a certain LMI. Furthermore, a convex optimization problem is introduced to select a 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 the uncertain 2-D discrete system in the GM setting with both state and input delays and recall some useful results. An LMI-based approach for the design of a non-fragile robust optimal guaranteed cost controller is then presented, and an illustrative example is given to show the effectiveness of the proposed technique.

Notations

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, diag{ ….} stands for a block diagonal matrix, G >0 (respectively, G <0) denotes a matrix G , which is real symmetric and positive (and, respectively, negative) definite and $λ_{\max}$ ( G ) stands for the maximum eigenvalue of matrix G.

Problem formulation and preliminaries

Consider the uncertain 2-D discrete system described by the GM (Kurek, 1985) with both state and input delays

\begin{array}{l} x (i + 1, j + 1) = (A_{1} + Δ A_{1}) x (i, j + 1) + (A_{2} + Δ A_{2}) x (i + 1, j) + (A_{0} + Δ A_{0}) x (i, j) \\ + (A_{1 d} + Δ A_{1 d}) x (i - d, j + 1) + (A_{2 d} + Δ A_{2 d}) x (i + 1, j - k) \\ + (A_{0 d} + Δ A_{0 d}) x (i - l, j - m) \\ + B_{1} u (i, j + 1) + B_{2} u (i + 1, j) + B_{0} u (i, j) \\ + B_{1 d} u (i - g, j + 1) + B_{2 d} u (i + 1, j - h) + B_{0 d} u (i - n, j - p) \end{array}

(1a)

A = [\begin{matrix} A_{1} & A_{2} & A_{0} & A_{1 d} & A_{2 d} & A_{0 d} \end{matrix}], B = [\begin{matrix} B_{1} & B_{2} & B_{0} & B_{1 d} & B_{2 d} & B_{0 d} \end{matrix}],

(1b)

where $x (i, j) \in R^{n}$ and $u (i, j) \in R^{m}$ are state and control input, respectively. The matrices $A_{1}$ , $A_{2}$ , $A_{0}$ , $A_{1 d}$ , $A_{2 d}$ , $A_{0 d}$ $\in R^{n \times n}$ and $B_{1}$ , $B_{2}$ , $B_{0}$ , $B_{1 d}$ , $B_{2 d}$ , $B_{0 d}$ $\in R^{n \times m}$ are known constant matrices representing the nominal plant; d, l, g and n are unknown constant positive integers representing the number of delay units along the horizontal direction and k, m, h and p are unknown constant positive integers representing the number of delay units along the vertical direction. For practical purposes, we consider $0 \leq d \leq d *$ , $0 \leq g \leq g *$ , $0 \leq l \leq l *$ , $0 \leq n \leq n *$ , $0 \leq k \leq k *$ , $0 \leq h \leq h *$ , $0 \leq m \leq m *$ , $0 \leq p \leq p *$ with $d *$ , $g *$ , $l *$ , $n *$ , $k *$ , $h *$ , $m *$ and $p *$ are known. The matrices $Δ A_{1}$ , $Δ A_{2}$ , $Δ A_{0}$ , $Δ A_{1 d}$ , $Δ A_{2 d}$ and $Δ A_{od}$ represent parameter uncertainties, which are assumed of the form

\begin{matrix} Δ A = [\begin{matrix} Δ A_{1} & Δ A_{2} & Δ A_{0} & Δ A_{1 d} & Δ A_{2 d} & Δ A_{0 d} \end{matrix}] \\ = L F (i, j) [\begin{matrix} M_{1} & M_{2} & M_{3} & M_{4} & M_{5} & M_{6} \end{matrix}] . \end{matrix}

(1c)

In the above, $L \in R^{n \times g}$ , $M_{1}$ , $M_{2}$ , $M_{3}$ , $M_{4}$ , $M_{5}$ and $M_{6} \in R^{h \times n}$ can be regarded as known structural matrices of uncertainty and $F (i, j) \in R^{g \times h}$ is an unknown matrix representing parameter uncertainty that satisfies

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

(1d)

It is assumed that the system (1a) has a finite set of initial conditions (Tandon and Dhawan, 2016; Xu et al., 2007), i.e. there exist two positive integers $L_{1}$ and $L_{2}$ such that

\begin{matrix} \begin{matrix} x (i, j) = ν_{ij}, j \in [- \bar{k}, 0], 0 \leq i < L_{1} \\ x (i, j) = ω_{ij}, i \in [- \bar{d}, 0], 0 \leq j < L_{2} \\ x (i, j) = φ_{ij}, i \in [- \bar{l}, 0], j \in [- \bar{m}, 0] \\ \bar{k} = \max {k *, h *}, \bar{d} = \max {d *, g *}, \\ \bar{l} = \max {l *, n *} and \bar{m} = \max {m *, p *} \end{matrix}} \\ ν_{00} = ω_{00} = φ_{00} \end{matrix}

(1e)

and the initial conditions are arbitrary, but belong to the set (Dhawan and Kar, 2007c; Tandon and Dhawan, 2016)

\begin{matrix} S = {x (i, q) \in R^{n} : x (i, q) = M N_{1}, N_{1}^{T} N_{1} < I, q = - \bar{k}, - \bar{k} + 1, \dots, 0, 0 \leq i < L_{1}} \\ \cup {x (t, j) \in R^{n} : x (t, j) = M N_{2}, N_{2}^{T} N_{2} < I, t = - \bar{d}, - \bar{d} + 1, \dots, 0, 0 \leq j < L_{2}} \\ \cup {x (t, q) \in R^{n} : x (t, q) = M N_{3}, N_{3}^{T} N_{3} < I, t = - \bar{l}, - \bar{l} + 1, \dots, 0, and \\ q = - \bar{m}, - \bar{m} + 1, \dots, 0,}, \end{matrix}

(1f)

where $M$ is a given matrix.

Associated with system (1a) is the following cost function:

J = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [τ_{ij}^{T} R τ_{ij} + ξ_{ij}^{T} W_{1} ξ_{ij}],

(2a)

where

\begin{matrix} τ_{ij} = [\begin{matrix} u^{T} (i, j + 1) & u^{T} (i + 1, j) & u^{T} (i, j) & u^{T} (i - g, j + 1) \end{matrix} \\ \begin{matrix} u^{T} (i + 1, j - h) & u^{T} (i - n, j - p) \end{matrix}], \end{matrix}

(2b)

R = [\begin{matrix} R_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & R_{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & R_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & R_{4} & 0 & 0 \\ 0 & 0 & 0 & 0 & R_{5} & 0 \\ 0 & 0 & 0 & 0 & 0 & R_{6} \end{matrix}],

(2c)

0 < R_{q} = R_{q}^{T} \in R^{m \times m} (q = 1, 2, 3, 4, 5, 6),

(2d)

\begin{matrix} ξ_{ij} = [\begin{matrix} x^{T} (i, j + 1) & x^{T} (i + 1, j) & x^{T} (i, j) & x^{T} (i - d, j + 1) \end{matrix} \\ \begin{matrix} x^{T} (i + 1, j - k) & x^{T} (i - l, j - m) \end{matrix}], \end{matrix}

(2e)

W_{1} = [\begin{matrix} S_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & S_{2} & 0 & 0 & 0 & 0 \\ 0 & 0 & S_{3} & 0 & 0 & 0 \\ 0 & 0 & 0 & S_{4} & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{5} & 0 \\ 0 & 0 & 0 & 0 & 0 & S_{6} \end{matrix}],

(2f)

0 < S_{t} = S_{t}^{T} \in R^{n \times n} (t = 1, 2, 3, 4, 5, 6) .

(2g)

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 controller 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 the system (1) with cost function (2), such that the resulting closed-loop system

\begin{matrix} x (i + 1, j + 1) = (A_{1} + Δ A_{1} + B_{1} K_{Δ}) x (i, j + 1) \\ + (A_{2} + Δ A_{2} + B_{2} K_{Δ}) x (i + 1, j) \\ + (A_{0} + Δ A_{0} + B_{0} K_{Δ}) x (i, j) \\ + (A_{1 d} + Δ A_{1 d}) x (i - d, j + 1) \\ + (A_{2 d} + Δ A_{2 d}) x (i + 1, j - k) \\ + (A_{0 d} + Δ A_{0 d}) x (i - l, j - m) \\ + (B_{1 d} K_{Δ}) x (i - g, j + 1) \\ + (B_{2 d} K_{Δ}) x (i + 1, j - h) \\ + (B_{0 d} K_{Δ}) x (i - n, j - p) \end{matrix}

(4)

is asymptotically stable and the closed-loop cost function

J = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} Π^{T} W_{2} Π,

(5a)

where

\begin{matrix} Π = [\begin{matrix} x^{T} (i, j + 1) & x^{T} (i + 1, j) & x^{T} (i, j) & x^{T} (i - d, j + 1) & x^{T} (i + 1, j - k) \end{matrix} \\ {\begin{matrix} x^{T} (i - l, j - m) & x^{T} (i - g, j + 1) & x^{T} (i + 1, j - h) & x^{T} (i - n, j - p) \end{matrix}]}^{T}, \end{matrix}

(5b)

W_{2} = [\begin{matrix} S_{1} + K_{Δ}^{T} R_{1} K_{Δ} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & S_{2} + K_{Δ}^{T} R_{2} K_{Δ} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & S_{3} + K_{Δ}^{T} R_{3} K_{Δ} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & S_{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & S_{5} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & S_{6} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & K_{Δ}^{T} R_{4} K_{Δ} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & K_{Δ}^{T} R_{5} K_{Δ} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & K_{Δ}^{T} R_{6} K_{Δ} \end{matrix}]

(5c)

satisfies $J \leq J *$ , where $J *$ is some specified constant and $K_{Δ} = (K + Δ K)$ .

Remark 1. Observe that the states $x (i - g, j + 1)$ , $x (i + 1, j - h)$ and $x (i - n, j - p)$ in (4) are generated as a result of delays in inputs along horizontal and vertical directions.

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 closed-loop system (4) is asymptotically stable and the closed-loop value of the cost function (5) satisfies $J \leq J *$ , then $J *$ is said to be a non-fragile guaranteed cost and $u * (i, j)$ is said to be a non-fragile guaranteed cost control law of the system (1).

Now, we recall the following results on the stability of uncertain 2-D discrete systems with both state delays and input delays.

Lemma 1 (Paszke et al., 2004; Tiwari and Dhawan, 2013). The closed-loop system (4) is asymptotically stable, provided there exist n×n positive definite symmetric matrices $P$ , $Q$ , $Q_{1}$ , $Q_{2}$ , $Q_{3}$ , $Q_{4}$ , $Q_{5}$ , $Q_{6}$ and $Q_{7}$ such that

\begin{matrix} Γ = [\begin{matrix} A_{Δ 1} + B_{1} K_{Δ} A_{Δ 2} + B_{2} K_{Δ} A_{Δ 0} + B_{0} K_{Δ} A_{Δ 1 d} A_{Δ 2 d} \end{matrix} \\ A_{Δ 0 d} {\begin{matrix} B_{1 d} K_{Δ} B_{2 d} K_{Δ} B_{0 d} K_{Δ} \end{matrix}]}^{T} \\ \times P [\begin{matrix} A_{Δ 1} + B_{1} K_{Δ} & A_{Δ 2} + B_{2} K_{Δ} & A_{Δ 0} + B_{0} K_{Δ} \end{matrix} \\ \begin{matrix} A_{Δ 1 d} & A_{Δ 2 d} & A_{Δ 0 d} & B_{1 d} K_{Δ} & B_{2 d} K_{Δ} & B_{0 d} K_{Δ} \end{matrix}] \\ - [\begin{matrix} P - \tilde{Q} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & Q & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & Q_{1} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & Q_{2} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & Q_{3} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & Q_{4} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & Q_{5} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & Q_{6} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & Q_{7} \end{matrix}] < 0 . \end{matrix}

(6)

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

\begin{matrix} A_{Δ 1} = A_{1} + Δ A_{1} = A_{1} + L F (i, j) M_{1}, A_{Δ 2} = A_{2} + Δ A_{2} = A_{2} + L F (i, j) M_{2}, \\ A_{Δ 0} = A_{0} + Δ A_{0} = A_{0} + L F (i, j) M_{3}, A_{Δ 1 d} = A_{1 d} + Δ A_{1 d} = A_{1 d} + L F (i, j) M_{4} . \\ A_{Δ 2 d} = A_{2 d} + Δ A_{2 d} = A_{2 d} + L F (i, j) M_{5}, A_{Δ 0 d} = A_{0 d} + Δ A_{0 d} = A_{0 d} + L F (i, j) M_{6}, \\ \tilde{Q} = Q + Q_{1} + Q_{2} + Q_{3} + Q_{4} + Q_{5} + Q_{6} + Q_{7} . \end{matrix}

On the basis of the 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 9n× 9n positive definite symmetric matrix $W_{2}$ given by (5c) and $n \times n$ positive definite symmetric matrices $Q$ , $Q_{1}$ , $Q_{2}$ , $Q_{3}$ , $Q_{4}$ , $Q_{5}$ , $Q_{6}$ and $Q_{7}$ such that

Γ + W_{2} < 0,

(7)

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

The following well-known lemma is required for the derivation of our main results.

Lemma 2 (Tandon and Dhawan, 2016; Xie, 1996). Let $H$ , $E$ , $F$ and $M$ be real matrices of appropriate dimension with $M$ satisfying $M = M^{T}$ then

M + HFE + E^{T} F^{T} H^{T} < 0,

(8a)

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

M + ε H H^{T} + ε^{- 1} E^{T} E < 0 .

(8b)

Lemma 3 (Boyd et al., 1994). For real matrices $M$ , $L$ , $Q$ of appropriate dimensions where $M = M^{T}$ and $Q = Q^{T} > 0$ then $M + L^{T} QL < 0$ if and only if

[\begin{matrix} M & L^{T} \\ L & - Q^{- 1} \end{matrix}] < 0 or equivalently [\begin{matrix} - Q^{- 1} & L \\ L^{T} & M \end{matrix}] < 0 .

(9)

Main results

In the following lemma, we intend to relate the notion of cost matrix to the quadratic stability and an upper bound on the cost function (5).

Lemma 4.Suppose there exist n × n positive definite symmetric matrices $P$ , $Q$ , $Q_{1}$ , $Q_{2}$ , $Q_{3}$ , $Q_{4}$ , $Q_{5}$ , $Q_{6}$ and $Q_{7}$ for the system (4) with initial conditions (1e), (1f) and cost function (5) such that (7) holds. Then: (i) system (4) is asymptotically stable and (ii) the closed-loop cost function (5) satisfies the bound

\begin{matrix} J \leq J * = [(L_{2} - 1) λ_{\max} {M^{T} (P - \tilde{Q}) M} + (L_{1} - 1) λ_{\max} (M^{T} Q M) + (L_{1} + L_{2} - 1) λ_{\max} (M^{T} Q_{1} M) \\ + (L_{2} - 1) (d * + 1) λ_{\max} (M^{T} Q_{2} M) + (L_{1} - 1) (k * + 1) λ_{\max} (M^{T} Q_{3} M) \\ + {(l * + 1) (m * + 1) - (L_{1} - 1) (L_{2} - 1)} λ_{\max} (M^{T} Q_{4} M) \\ + (L_{2} - 1) (g * + 1) λ_{\max} (M^{T} Q_{5} M) + (L_{1} - 1) (h * + 1) λ_{\max} (M^{T} Q_{6} M) \\ + {(n * + 1) (p * + 1) - (L_{1} - 1) (L_{2} - 1)} λ_{\max} (M^{T} Q_{7} M) . \end{matrix}

(10)

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

Proof. See the Appendix.

The following theorem establishes that the problem of determining non-fragile robust guaranteed cost control for system (4) with initial conditions (1e), (1f) and cost function (5) can be recast to an LMI feasibility problem.

Theorem 1.Consider system (4) with initial conditions (1e), (1f) and cost function (5), then there exists a non-fragile state feedback control law (3) that solves the addressed robust guaranteed cost control problem if there exist positive scalars $ε_{1}$ , $ε_{2}$ , am×nmatrix U , n × n positive definite symmetric matrices $P_{1}$ , $Y_{1}$ , $Y_{2}$ , $Y_{3}$ , $Y_{4}$ , $Y_{5}$ , $Y_{6}$ , $Y_{7}$ and $Y_{8}$ such that the following LMI is feasible:

[\begin{matrix} Π_{11} & Π_{12} & Π_{13} & Π_{14} \\ * & - I & 0 & 0 \\ * & * & - ε_{2} I & 0 \\ * & * & * & - ε_{1} I \end{matrix}] < 0 .

(11)

where

Π_{11} = [\begin{matrix} \begin{matrix} - P_{1} + ε_{1} L L^{T} + ε_{2} B_{1} H_{k} H_{k}^{T} B_{1}^{T} + ε_{2} B_{2} H_{k} H_{k}^{T} B_{2}^{T} + ε_{2} B_{0} H_{k} H_{k}^{T} B_{0}^{T} \\ + ε_{2} B_{1 d} H_{k} H_{k}^{T} B_{1 d}^{T} + ε_{2} B_{2 d} H_{k} H_{k}^{T} B_{2 d}^{T} + ε_{2} B_{0 d} H_{k} H_{k}^{T} B_{0 d}^{T} \end{matrix} & ε_{2} B_{1} H_{k} H_{k}^{T} \\ ε_{2} H_{k} H_{k}^{T} B_{1}^{T} & - R_{1}^{- 1} + ε_{2} H_{k} H_{k}^{T} \\ ε_{2} H_{k} H_{k}^{T} B_{2}^{T} & 0 \\ ε_{2} H_{k} H_{k}^{T} B_{0}^{T} & 0 \\ ε_{2} H_{k} H_{k}^{T} B_{1 d}^{T} & 0 \\ ε_{2} H_{k} H_{k}^{T} B_{2 d}^{T} & 0 \\ ε_{2} H_{k} H_{k}^{T} B_{0 d}^{T} & 0 \\ {(A_{1} P_{1} + B_{1} U)}^{T} & U^{T} \\ {(A_{2} P_{1} + B_{2} U)}^{T} & 0 \\ {(A_{0} P_{1} + B_{0} U)}^{T} & 0 \\ {(A_{1 d} P_{1})}^{T} & 0 \\ {(A_{2 d} P_{1})}^{T} & 0 \\ {(A_{0 d} P_{1})}^{T} & 0 \\ U^{T} B_{1 d}^{T} & 0 \\ U^{T} B_{2 d}^{T} & 0 \\ U^{T} B_{0 d}^{T} & 0 \end{matrix}

\begin{matrix} ε_{2} B_{2} H_{k} H_{k}^{T} & ε_{2} B_{0} H_{k} H_{k}^{T} & ε_{2} B_{1 d} H_{k} H_{k}^{T} & ε_{2} B_{2 d} H_{k} H_{k}^{T} & ε_{2} B_{0 d} H_{k} H_{k}^{T} \\ 0 & 0 & 0 & 0 & 0 \\ - R_{2}^{- 1} + ε_{2} H_{k} H_{k}^{T} & 0 & 0 & 0 & 0 \\ 0 & - R_{3}^{- 1} + ε_{2} H_{k} H_{k}^{T} & 0 & 0 & 0 \\ 0 & 0 & - R_{4}^{- 1} + ε_{2} H_{k} H_{k}^{T} & 0 & 0 \\ 0 & 0 & 0 & - R_{5}^{- 1} + ε_{2} H_{k} H_{k}^{T} & 0 \\ 0 & 0 & 0 & 0 & - R_{6}^{- 1} + ε_{2} H_{k} H_{k}^{T} \\ 0 & 0 & 0 & 0 & 0 \\ U^{T} & 0 & 0 & 0 & 0 \\ 0 & U^{T} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & U^{T} & 0 & 0 \\ 0 & 0 & 0 & U^{T} & 0 \\ 0 & 0 & 0 & 0 & U^{T} \end{matrix}

\begin{matrix} (A_{1} P_{1} + B_{1} U) & (A_{2} P_{1} + B_{2} U) & (A_{0} P_{1} + B_{0} U) \\ U & 0 & 0 \\ 0 & U & 0 \\ 0 & 0 & U \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ - P_{1} + Y_{1} + Y_{2} + Y_{3} + Y_{4} + Y_{5} + Y_{6} + Y_{7} + Y_{8} & 0 & 0 \\ 0 & - Y_{1} & 0 \\ 0 & 0 & - Y_{2} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}

\begin{matrix} A_{1 d} P_{1} & A_{2 d} P_{1} & A_{0 d} P_{1} & B_{1 d} U & B_{2 d} U & B_{0 d} U \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ - Y_{3} & 0 & 0 & 0 & 0 & 0 \\ 0 & - Y_{4} & 0 & 0 & 0 & 0 \\ 0 & 0 & - Y_{5} & 0 & 0 & 0 \\ 0 & 0 & 0 & - Y_{6} & 0 & 0 \\ 0 & 0 & 0 & 0 & - Y_{7} & 0 \\ 0 & 0 & 0 & 0 & 0 & - Y_{8} \end{matrix}],

Π_{12} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ P_{1} S_{1}^{1 / 2} & 0 & 0 & 0 & 0 & 0 \\ 0 & P_{1} S_{2}^{1 / 2} & 0 & 0 & 0 & 0 \\ 0 & 0 & P_{1} S_{3}^{1 / 2} & 0 & 0 & 0 \\ 0 & 0 & 0 & P_{1} S_{4}^{1 / 2} & 0 & 0 \\ 0 & 0 & 0 & 0 & P_{1} S_{5}^{1 / 2} & 0 \\ 0 & 0 & 0 & 0 & 0 & P_{1} S_{6}^{1 / 2} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}],

\begin{matrix} Π_{13} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ P_{1} E_{k}^{T} & 0 & 0 & 0 & 0 & 0 \\ 0 & P_{1} E_{k}^{T} & 0 & 0 & 0 & 0 \\ 0 & 0 & P_{1} E_{k}^{T} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & P_{1} E_{k}^{T} & 0 & 0 \\ 0 & 0 & 0 & 0 & P_{1} E_{k}^{T} & 0 \\ 0 & 0 & 0 & 0 & 0 & P_{1} E_{k}^{T} \end{matrix}], \\ Π_{14} = [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ P_{1} M_{1}^{T} \\ P_{1} M_{2}^{T} \\ P_{1} M_{3}^{T} \\ P_{1} M_{4}^{T} \\ P_{1} M_{5}^{T} \\ P_{1} M_{6}^{T} \\ 0 \\ 0 \\ 0 \end{matrix}] . \end{matrix}

In this situation, the feedback gain of the stabilizing non-fragile guaranteed cost control law is given by

K = U {P_{1}}^{- 1} .

(12)

Moreover, closed-loop cost function satisfies the bound

\begin{matrix} J \leq J * = [(L_{2} - 1) λ_{\max} {M^{T} (P_{1}^{- 1} - P_{1}^{- 1} Y_{1} P_{1}^{- 1} - P_{1}^{- 1} Y_{2} P_{1}^{- 1} - P_{1}^{- 1} Y_{3} P_{1}^{- 1} - P_{1}^{- 1} Y_{4} P_{1}^{- 1} - P_{1}^{- 1} Y_{5} P_{1}^{- 1} \\ - P_{1}^{- 1} Y_{6} P_{1}^{- 1} - P_{1}^{- 1} Y_{7} P^{- 1} - P_{1}^{- 1} Y_{8} P^{- 1}) M} + (L_{1} - 1) λ_{\max} (M^{T} P_{1}^{- 1} Y_{1} P_{1}^{- 1} M) \\ + (L_{1} + L_{2} - 1) λ_{\max} (M^{T} P_{1}^{- 1} Y_{2} P_{1}^{- 1} M) + (L_{2} - 1) (d * + 1) λ_{\max} (M^{T} P_{1}^{- 1} Y_{3} P_{1}^{- 1} M) \\ + (L_{1} - 1) (k * + 1) λ_{\max} (M^{T} P_{1}^{- 1} Y_{4} P_{1}^{- 1} M) + {(l * + 1) (m * + 1) - (L_{1} - 1) (L_{2} - 1)} \\ \times λ_{\max} (M^{T} P_{1}^{- 1} Y_{5} P_{1}^{- 1} M) + (L_{2} - 1) (g * + 1) λ_{\max} (M^{T} P_{1}^{- 1} Y_{6} P_{1}^{- 1} M) \\ + (L_{1} - 1) (h * + 1) λ_{\max} (M^{T} P_{1}^{- 1} Y_{7} P_{1}^{- 1} M)] + {(n * + 1) (p * + 1) - (L_{1} - 1) (L_{2} - 1)} \\ \times λ_{\max} (M^{T} P_{1}^{- 1} Y_{8} P_{1}^{- 1} M)] . \end{matrix}

(13)

Proof. See the Appendix.

Remark 2. Observe that, if there is no delay in inputs and we set $B_{1 d} =$ $B_{2 d} =$ $B_{0 d} = 0$ and also $R_{4} = R_{5} = R_{6} = 0$ , then LMI (11) coincides with the criteria developed in Tandon and Dhawan (2016) for the existence of non-fragile robust guaranteed cost controllers for state-delayed systems.

Theorem 1 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 (11). The following theorem presents a method of selecting a non-fragile robust optimal guaranteed cost controller, which minimizes the value of the guaranteed cost in (13).

Theorem 2. Consider system (4) with initial conditions (1e), (1f) and cost function (5). If the following optimization problem:

m i n i m i z e {(L_{2} - 1) α + (L_{1} - 1) β + (L_{1} + L_{2} - 1) γ + (L_{2} - 1) (d^{*} + 1) δ + (L_{1} - 1) (k^{*} + 1) σ

\begin{matrix} + [(l * + 1) (m * + 1) - (L_{1} - 1) (L_{2} - 1)] λ \\ + (L_{2} - 1) (g * + 1) μ + (L_{1} - 1) (h * + 1) ν \end{matrix}

+ [(n * + 1) (p * + 1) - (L_{1} - 1) (L_{2} - 1)] η}

s . t . {\begin{matrix} (i) . (11), \\ (ii) . [\begin{matrix} α I & - M^{T} \\ - M & P_{1} + Y_{1} + Y_{2} + Y_{3} + Y_{4} + Y_{5} + Y_{6} + Y_{7} + Y_{8} \end{matrix}] > 0, \\ (iii) . [\begin{matrix} β I & - M^{T} \\ - M & 2 P_{1} - Y_{1} \end{matrix}] > 0, \\ (iv) . [\begin{matrix} γ I & - M^{T} \\ - M & 2 P_{1} - Y_{2} \end{matrix}] > 0, \\ (v) . [\begin{matrix} δ I & - M^{T} \\ - M & 2 P_{1} - Y_{3} \end{matrix}] > 0, \\ (vi) . [\begin{matrix} σ I & - M^{T} \\ - M & 2 P_{1} - Y_{4} \end{matrix}] > 0, \\ (vii) . [\begin{matrix} λ I & - M^{T} \\ - M & 2 P_{1} - Y_{5} \end{matrix}] > 0, \\ (viii) . [\begin{matrix} μ I & - M^{T} \\ - M & 2 P_{1} - Y_{6} \end{matrix}] > 0, \\ (ix) . [\begin{matrix} ν I & - M^{T} \\ - M & 2 P_{1} - Y_{7} \end{matrix}] > 0, \\ (x) . [\begin{matrix} η I & - M^{T} \\ - M & 2 P_{1} - Y_{8} \end{matrix}] > 0, \end{matrix}

(14)

has a feasible solution $α > 0$ , $β > 0$ , $γ > 0$ , $δ > 0$ , $σ > 0$ , $λ > 0$ , $μ > 0$ , $ν > 0$ , $η > 0$ , $ε_{1} > 0$ , $ε_{2} > 0$ , a $m \times n$ matrix U and $n \times n$ positive definite symmetric matrices $P_{1}$ , $Y_{1}$ , $Y_{2}$ , $Y_{3}$ , $Y_{4}$ , $Y_{5}$ , $Y_{6}$ , $Y_{7}$ and $Y_{8}$ , then the control law (3) with $K = U {P_{1}}^{- 1}$ is the non-fragile robust optimal guaranteed cost control law that ensures the minimization of guaranteed cost in (13).

Proof. By Theorem 1, the stabilizing control law $K = U {P_{1}}^{- 1}$ constructed in terms of any feasible solution $ε_{1}$ , $ε_{2}$ , U , $P_{1}$ , $Y_{1}$ , $Y_{2}$ , $Y_{3}$ , $Y_{4}$ , $Y_{5}$ , $Y_{6}$ , $Y_{7}$ and $Y_{8}$ is a non-fragile guaranteed cost controllers of system (1). In the following, we proceed to obtain the optimum value of the upper bound of guaranteed cost.

In order to satisfy (11), one necessarily requires that

(P_{1} - Y_{1} - Y_{2} - Y_{3} - Y_{4} - Y_{5} - Y_{6} - Y_{7} - Y_{8}) > 0 .

(15)

Equation (15) can be rewritten as

\begin{matrix} {P_{1} - (P_{1} - Y_{1} - Y_{2} - Y_{3} - Y_{4} - Y_{5} - Y_{6} - Y_{7} - Y_{8})} \\ \times (P_{1} - Y_{1} - Y_{2} - Y_{3} - Y_{4} - Y_{5} - Y_{6} - Y_{7} - Y_{8})^{- 1} \\ \times {P_{1} - (P_{1} - Y_{1} - Y_{2} - Y_{3} - Y_{4} - Y_{5} - Y_{6} - Y_{7} - Y_{8})} > 0 \end{matrix}

(16)

which gives

\begin{matrix} P_{1}^{- 1} (P_{1} - Y_{1} - Y_{2} - Y_{3} - Y_{4} - Y_{5} - Y_{6} - Y_{7} - Y_{8}) \\ \times P_{1}^{- 1} < (P_{1} + Y_{1} + Y_{2} + Y_{3} + Y_{4} + Y_{5} + Y_{6} + Y_{7} + Y_{8})^{- 1} . \end{matrix}

(17)

From (17), we get

\begin{matrix} λ_{\max} {M^{T} (P_{1}^{- 1} - P_{1}^{- 1} Y_{1} P_{1}^{- 1} - P_{1}^{- 1} Y_{2} P_{1}^{- 1} - P_{1}^{- 1} \\ Y_{3} P_{1}^{- 1} - P_{1}^{- 1} Y_{4} P_{1}^{- 1} - P_{1}^{- 1} Y_{5} P_{1}^{- 1} - P_{1}^{- 1} Y_{6} P_{1}^{- 1} \\ - P_{1}^{- 1} Y_{7} P_{1}^{- 1} - P_{1}^{- 1} Y_{8} P_{1}^{- 1}) M} \leq λ_{\max} \\ \times {M^{T} (P_{1} + Y_{1} + Y_{2} + Y_{3} + Y_{4} + Y_{5} + Y_{6} + Y_{7} + Y_{8})^{- 1} M} . \end{matrix}

(18)

From (15), it is also implied that $P_{1} > Y_{1}$ , which gives $(P_{1} - Y_{1}) Y_{1}^{- 1} (P_{1} - Y_{1}) > 0$ . Therefore, the relation

P_{1}^{- 1} Y_{1} P_{1}^{- 1} < (2 P_{1} - Y_{1})^{- 1}

(19)

holds true. From (19), we obtain

λ_{\max} {M^{T} P_{1}^{- 1} Y_{1} P_{1}^{- 1} M} \leq λ_{\max} {M^{T} (2 P_{1} - Y_{1})^{- 1} M} .

(20)

Similarly,

\begin{matrix} P_{1}^{- 1} Y_{2} P_{1}^{- 1} < (2 P_{1} - Y_{2})^{- 1} \Rightarrow λ_{\max} \\ \times {M^{T} P_{1}^{- 1} Y_{2} P_{1}^{- 1} M} \leq λ_{\max} {M^{T} (2 P_{1} - Y_{2})^{- 1} M}, \end{matrix}

(21)

\begin{matrix} P_{1}^{- 1} Y_{3} P_{1}^{- 1} < (2 P_{1} - Y_{3})^{- 1} \Rightarrow λ_{\max} \\ \times {M^{T} P_{1}^{- 1} Y_{3} P_{1}^{- 1} M} \leq λ_{\max} {M^{T} (2 P_{1} - Y_{3})^{- 1} M}, \end{matrix}

(22)

\begin{matrix} P_{1}^{- 1} Y_{4} P_{1}^{- 1} < (2 P_{1} - Y_{4})^{- 1} \Rightarrow λ_{\max} \\ \times {M^{T} P_{1}^{- 1} Y_{4} P_{1}^{- 1} M} \leq λ_{\max} {M^{T} (2 P_{1} - Y_{4})^{- 1} M}, \end{matrix}

(23)

\begin{matrix} P_{1}^{- 1} Y_{5} P_{1}^{- 1} < (2 P_{1} - Y_{5})^{- 1} \Rightarrow λ_{\max} \\ \times {M^{T} P_{1}^{- 1} Y_{5} P_{1}^{- 1} M} \leq λ_{\max} {M^{T} (2 P_{1} - Y_{5})^{- 1} M}, \end{matrix}

(24)

\begin{matrix} P_{1}^{- 1} Y_{6} P_{1}^{- 1} < (2 P_{1} - Y_{6})^{- 1} \Rightarrow λ_{\max} \\ \times {M^{T} P_{1}^{- 1} Y_{6} P_{1}^{- 1} M} \leq λ_{\max} {M^{T} (2 P_{1} - Y_{6})^{- 1} M}, \end{matrix}

(25)

\begin{matrix} P_{1}^{- 1} Y_{7} P_{1}^{- 1} < (2 P_{1} - Y_{7})^{- 1} \Rightarrow λ_{\max} {M^{T} P_{1}^{- 1} Y_{7} P_{1}^{- 1} M} \\ \leq λ_{\max} {M^{T} (2 P_{1} - Y_{7})^{- 1} M} \end{matrix}

(26)

and

\begin{matrix} P_{1}^{- 1} Y_{8} P_{1}^{- 1} < (2 P_{1} - Y_{8})^{- 1} \Rightarrow λ_{\max} {M^{T} P_{1}^{- 1} Y_{8} P_{1}^{- 1} M} \\ \leq λ_{\max} {M^{T} (2 P_{1} - Y_{8})^{- 1} M} . \end{matrix}

(27)

Using (18) and (20)–(27), (13) necessarily implies

\begin{matrix} J \leq [(L_{2} - 1) λ_{\max} {M^{T} {(P_{1} + Y_{1} + Y_{2} + Y_{3} + Y_{4} + Y_{5} + Y_{6} + Y_{7} + Y_{8})}^{- 1} M} \\ + (L_{1} - 1) λ_{\max} {M^{T} (2 P_{1} - Y_{1})^{- 1} M} + (L_{1} + L_{2} - 1) λ_{\max} {M^{T} (2 P_{1} - Y_{2})^{- 1} M} \\ + (L_{2} - 1) (d * + 1) λ_{\max} {M^{T} (2 P_{1} - Y_{3})^{- 1} M} + (L_{1} - 1) (k * + 1) λ_{\max} {M^{T} (2 P_{1} - Y_{4})^{- 1} M} \\ + [(l * + 1) (m * + 1) - (L_{1} - 1) (L_{2} - 1)] λ_{\max} {M^{T} (2 P_{1} - Y_{5})^{- 1} M} \\ + (L_{2} - 1) (g * + 1) λ_{\max} {M^{T} {(2 P_{1} - Y_{6})}^{- 1} M} + (L_{1} - 1) (h * + 1) λ_{\max} {M^{T} {(2 P_{1} - Y_{7})}^{- 1} M}] \\ + [(n * + 1) (p * + 1) - (L_{1} - 1) (L_{2} - 1)] λ_{\max} {M^{T} (2 P_{1} - Y_{8})^{- 1} M} . \end{matrix}

(28)

To obtain the optimum value of the upper bound of guaranteed cost, let us assume

\begin{matrix} λ_{\max} {M^{T} (P_{1} + Y_{1} + Y_{2} + Y_{3} \\ + Y_{4} + Y_{5} + Y_{6} + Y_{7} + Y_{8})^{- 1} M} < α, \end{matrix}

(29)

λ_{\max} {M^{T} (2 P_{1} - Y_{1})^{- 1} M} < β,

(30)

λ_{\max} {M^{T} (2 P_{1} - Y_{2})^{- 1} M} < γ,

(31)

λ_{\max} {M^{T} (2 P_{1} - Y_{3})^{- 1} M} < δ,

(32)

λ_{\max} {M^{T} (2 P_{1} - Y_{4})^{- 1} M} < σ .

(33)

λ_{\max} {M^{T} (2 P_{1} - Y_{5})^{- 1} M} < λ .

(34)

λ_{\max} {M^{T} (2 P_{1} - Y_{6})^{- 1} M} < μ .

(35)

λ_{\max} {M^{T} (2 P_{1} - Y_{7})^{- 1} M} < ν .

(36)

and

λ_{\max} {M^{T} (2 P_{1} - Y_{8})^{- 1} M} < η .

(37)

Equation (29) can equivalently be expressed as

M^{T} (P_{1} + Y_{1} + Y_{2} + Y_{3} + Y_{4} + Y_{5} + Y_{6} + Y_{7} + Y_{8})^{- 1} M < α I .

(38)

By well-known Schur complements, (38) is equivalent to the constraint (ii) in (14). Similarly, (30)–(37) can be expressed as

M^{T} (2 P_{1} - Y_{1})^{- 1} M < β I,

(39)

M^{T} (2 P_{1} - Y_{2})^{- 1} M < γ I,

(40)

M^{T} (2 P_{1} - Y_{3})^{- 1} M < δ I,

(41)

M^{T} (2 P_{1} - Y_{4})^{- 1} M < σ I,

(42)

M^{T} (2 P_{1} - Y_{5})^{- 1} M < λ I,

(43)

M^{T} (2 P_{1} - Y_{6})^{- 1} M < μ I,

(44)

M^{T} (2 P_{1} - Y_{7})^{- 1} M < ν I,

(45)

and

M^{T} (2 P_{1} - Y_{8})^{- 1} M < η I,

(46)

which, in turn, imply (iii)–(x) in (14), respectively. Thus, the minimization of

\begin{matrix} {(L_{2} - 1) α + (L_{1} - 1) β + (L_{1} + L_{2} - 1) γ \\ + (L_{2} - 1) (d * + 1) δ + (L_{1} - 1) (k * + 1) σ + [(l * + 1) (m * + 1) \\ - (L_{1} - 1) (L_{2} - 1)] λ + (L_{2} - 1) (g * + 1) μ \\ + (L_{1} - 1) (h * + 1) ν + [(n * + 1) (p * + 1) - (L_{1} - 1) (L_{2} - 1)] η} \end{matrix}

implies the minimization of the guaranteed cost in (13). This completes the proof of Theorem 2.

Remark 3. It is clear that the optimization problem given by (14) is an LMI eigenvalue problem, which can be solved using Matlab LMI Toolbox (Boyd et al., 1994; Gahinet et al., 1995).

Illustrative example

In this example, we shall demonstrate the applicability of Theorem 2 to the control of thermal process in heat exchanger (Kurek, 1985, Tandon and Dhawan, 2016), which can be expressed by the partial differential equation with time and space delays:

\begin{matrix} \frac{\partial T (x, t)}{\partial x} = - \frac{\partial T (x, t)}{\partial t} - a_{0} T (x, t) - a_{1} T (x, t - k_{1}) \\ - a_{2} T (x - d_{1}, t) - a_{3} T (x - d_{2}, t - k_{2}) + b_{0} u (x, t) \\ + b_{1} u (x, t - k_{3}) + b_{2} u (x - d_{3}, t) + b_{3} u (x - d_{4}, t - k_{4}), \end{matrix}

(47)

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, $k_{1}$ , $k_{2}$ , $k_{3}$ and $k_{4}$ are the time delays, $d_{1}$ , $d_{2}$ , $d_{3}$ and $d_{4}$ are the space delays, and $a_{0}$ , $a_{1}$ , $a_{2}$ , $a_{3}$ , $b_{0}$ , $b_{1}$ , $b_{2}$ and $b_{3}$ 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),

(48a)

\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},

(48b)

(47) 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) \\ - a_{1} Δ t T (i - d, j) - a_{2} Δ t T (i, j - k) \\ - a_{3} Δ t T (i - l, j - m) + b_{0} Δ t u (i, j) \\ + b_{1} Δ t u (i - g, j) \\ + b_{2} Δ t u (i, j - h) + b_{3} Δ t u (i - n, j - p), \end{matrix}

(49)

where $k = int (k_{1} / Δ t + 1)$ , $m = int (k_{2} / Δ t + 1), h = int (k_{3} / Δ t + 1)$ , $p = int (k_{4} / Δ t + 1)$ and $d = int (d_{1} / Δ x), l = int (d_{2} / Δ x)$ , $g = int (d_{3} / Δ x)$ , $n = int (d_{4} / Δ x)$ , $int (.)$ is the integer function.

In order to simplify (49), the following assumptions have been made in the thermal process of the heat exchanger:

The input function $u (x, t)$ is constant.

The surface of the heat exchanger is insulated and the heat flow through it is in steady state condition.

Denoting $x^{T} (i, j) = [\begin{matrix} T^{T} (i - 1, j) & T^{T} (i, j) \end{matrix}]$ , it is easy to show that (49) can be converted into the following 2-D GM with both state and input delays:

\begin{matrix} x (i + 1, j + 1) = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] x (i, j + 1) \\ + [\begin{matrix} 0 & 0 \\ \frac{Δ t}{Δ x} & 1 - \frac{Δ t}{Δ x} - a_{0} Δ t \end{matrix}] x (i + 1, j) \\ + [\begin{matrix} 0 & 0 \\ 0 & \frac{Δ t}{Δ x} \end{matrix}] x (i, j) \\ + [\begin{matrix} 0 & 0 \\ 0 & - a_{1} Δ t \end{matrix}] x (i - d, j + 1) \\ + [\begin{matrix} 0 & 0 \\ 0 & - a_{2} Δ t \end{matrix}] x (i + 1, j - k) \\ + [\begin{matrix} 0 & 0 \\ 0 & - a_{3} Δ t \end{matrix}] x (i - l, j - m) \\ + [\begin{matrix} 0 \\ b_{0} Δ t \end{matrix}] u (i + 1, j) + [\begin{matrix} 0 \\ b_{1} Δ t \end{matrix}] u (i - g, j + 1) \\ + [\begin{matrix} 0 \\ b_{2} Δ t \end{matrix}] u (i + 1, j - h) + [\begin{matrix} 0 \\ b_{3} Δ t \end{matrix}] u (i - n, j - p) . \end{matrix}

(50)

Next, consider the problem of non-fragile robust optimal guaranteed cost control for a system characterized by (50). Let $a_{0} = 11$ , $a_{1} = 0.3$ , $a_{2} = 0.04$ , $a_{3} = 0.005$ , $b_{0} = 0.02$ , $b_{1} = 0.01$ , $b_{2} = - 0.02$ , $b_{3} = - 0.2$ , $Δ t = 0.1$ , $Δ x = 1$ , $d * = 3$ , $k * = 2$ , $l * = 2$ , $m * = 3$ , $g * = 4$ , $h * = 1$ , $n * = 2$ , $p * = 2$ and the initial state satisfies the condition (1e) and (1f) with $L_{1} = L_{2} = 2$ , $M = [\begin{matrix} 0.01 & 0 \\ 0.02 & 0.05 \end{matrix}]$ ,

\begin{matrix} x (- 4, 0) = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}], \\ x (- 4, 1) = [\begin{matrix} 0.2 \\ 0.2 \end{matrix}], x (- 3, 0) = [\begin{matrix} 0.2 \\ 0.2 \end{matrix}], \\ x (- 3, 1) = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}], x (- 2, - 2) = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}], \\ x (- 2, - 1) = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}], x (- 2, 0) = [\begin{matrix} 0.2 \\ 0.2 \end{matrix}], \\ x (- 2, 1) = [\begin{matrix} 0.2 \\ 0.2 \end{matrix}], x (- 1, - 2) = [\begin{matrix} 0.2 \\ 0.2 \end{matrix}], \\ x (- 1, - 1) = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}], x (- 1, 0) = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}], \\ x (- 1, 1) = [\begin{matrix} 0.3 \\ 0.3 \end{matrix}], x (0, - 2) = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}], \\ x (0, - 1) = [\begin{matrix} 0.2 \\ 0.2 \end{matrix}], x (0, 0) = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}], \\ x (0, 1) = [\begin{matrix} 0.2 \\ 0.2 \end{matrix}], x (1, - 2) = [\begin{matrix} 0.2 \\ 0.2 \end{matrix}], \\ x (1, - 1) = [\begin{matrix} 0.1 \\ 0.1 \end{matrix}], x (1, 0) = [\begin{matrix} 0.2 \\ 0.2 \end{matrix}] . \end{matrix}

(51)

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

\begin{matrix} L = [\begin{matrix} 0 \\ 1 \end{matrix}], M_{1} = [\begin{matrix} 0.005 & 0 \end{matrix}], M_{2} = [\begin{matrix} 0.001 & 0.005 \end{matrix}], \\ M_{3} = [\begin{matrix} 0 & 0.005 \end{matrix}], M_{4} = [\begin{matrix} 0.005 & 0.005 \end{matrix}], \\ M_{5} = [\begin{matrix} 0.003 & 0 \end{matrix}], M_{6} = [\begin{matrix} 0.002 & 0 \end{matrix}], F (i, j) = \sin (i + j) . \end{matrix}

(52)

Associated with the uncertain system (50)–(52), the cost function is given by (2) with

\begin{matrix} S_{1} = [\begin{matrix} 0.02 & 0 \\ 0 & 0.02 \end{matrix}], S_{2} = [\begin{matrix} 0.015 & 0 \\ 0 & 0.015 \end{matrix}], \\ S_{3} = [\begin{matrix} 0.03 & 0 \\ 0 & 0.03 \end{matrix}], S_{4} = [\begin{matrix} 0.01 & 0 \\ 0 & 0.01 \end{matrix}], \\ S_{5} = [\begin{matrix} 0.025 & 0 \\ 0 & 0.025 \end{matrix}], S_{6} = [\begin{matrix} 0.01 & 0 \\ 0 & 0.01 \end{matrix}], \\ R_{1} = R_{2} = R_{3} = R_{4} = R_{5} = R_{6} = 0.025 . \end{matrix}

(53)

We wish to design a non-fragile robust optimal guaranteed cost controller for the system under consideration with the controller gain perturbation satisfying (3b)–(3c) with

H_{k} = 0.1, E_{k} = [\begin{matrix} 0.001 & 0.002 \end{matrix}] and F_{k} (i, j) = \sin (i + j) .

(54)

Using the Matlab LMI toolbox (Boyd et al., 1994; Gahinet et al., 1995), it is found that the optimization problem (14) is feasible for the present example and the optimal solution is given by

\begin{matrix} P_{1} = [\begin{matrix} 2.3990 & 0.0033 \\ 0.0033 & 1.5809 \end{matrix}], Y_{1} = [\begin{matrix} 1.3709 & - 0.1004 \\ - 0.1004 & 0.2283 \end{matrix}], \\ Y_{2} = [\begin{matrix} 0.1786 & - 0.0000 \\ - 0.0000 & 0.1078 \end{matrix}], Y_{3} = [\begin{matrix} 0.0729 & 0.0012 \\ 0.0012 & 0.0354 \end{matrix}], \\ Y_{4} = [\begin{matrix} 0.1566 & 0.0009 \\ 0.0009 & 0.0639 \end{matrix}], Y_{5} = [\begin{matrix} 0.0673 & - 0.0001 \\ - 0.0001 & 0.0251 \end{matrix}], \\ Y_{6} = [\begin{matrix} 0.0110 & - 0.0008 \\ - 0.0008 & 0.0001 \end{matrix}], Y_{7} = [\begin{matrix} 0.0113 & - 0.0009 \\ - 0.0009 & 0.0001 \end{matrix}] \\ Y_{8} = [\begin{matrix} 0.0111 & - 0.0008 \\ - 0.0008 & 0.0001 \end{matrix}], U = [\begin{matrix} - 0.0091 & 0.0007 \end{matrix}], \\ ε_{1} = 0.0311, ε_{2} = 152.112, α = 0.0014, β = 0.0016, \\ γ = 0.0015, δ = 0.0015, σ = 0.0015, \\ λ = 0.0015, μ = 0.0015, ν = 0.0015, η = 0.0015 . \end{matrix}

(55)

By Theorem 2, a non-fragile robust optimal guaranteed cost controller can be obtained as

u (i, j) = [\begin{matrix} - 0.0038 & 0.0004 \end{matrix}] x (i, j)

(56)

and the least upper bound of the corresponding closed-loop cost function is $J * =$ 0.0606.

The graphical simulations of the open-loop and closed-loop systems for the example under consideration are shown in Figures 1 and 2, respectively.

It can be seen from Figure 1 that the open-loop system is unstable as i and j approach larger values, and it can be observed from Figure 2 that the closed-loop system is asymptotically stable.

Conclusions

In this paper, we have studied the problem of non-fragile robust optimal guaranteed cost control for uncertain 2-D discrete systems in GM setting with both state delays and input delays. By LMI approach, a sufficient condition for the existence of non-fragile robust guaranteed cost controllers has been presented. A desired non-fragile robust optimal guaranteed cost controller has been obtained by solving a convex optimization problem. Finally, an illustrative example has been provided to demonstrate the effectiveness of the proposed design approach. Further, the approach presented in this paper can also be extended for the analysis of 2-D discrete switched systems with delays.

Footnotes

Appendix

Proof of Lemma 4. Proof of (i) directly follows from Lemma 1 and Definition 2.

Next, we prove (ii). From (7), we have

(57)

Π^{T} W_{2} Π < - Π^{T} Γ Π .

Summing both sides of (57) over $i, j = 0 \to \infty$ yields

\begin{matrix} J = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} Π^{T} W_{2} Π \\ < - \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} Π^{T} Γ Π \\ = - \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i + 1, j + 1) P x (i + 1, j + 1) - x^{T} (i, j + 1) (P - \tilde{Q}) x (i, j + 1) \\ - x^{T} (i + 1, j) Q x (i + 1, j) - x^{T} (i, j) Q_{1} x (i, j) - x^{T} (i - d, j + 1) Q_{2} x (i - d, j + 1) \\ - x^{T} (i + 1, j - k) Q_{3} x (i + 1, j - k) - x^{T} (i - l, j - m) Q_{4} x (i - l, j - m) \\ - x^{T} (i - g, j + 1) Q_{5} x (i - g, j + 1) \\ - x^{T} (i + 1, j - h) Q_{6} x (i + 1, j - h) \\ - x^{T} (i - n, j - p) Q_{7} x (i - n, j - p)] \end{matrix}

\begin{matrix} = \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i, j + 1) (P - \tilde{Q}) x (i, j + 1) \\ - x^{T} (i + 1, j + 1) (P - \tilde{Q}) x (i + 1, j + 1)] \\ + \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i + 1, j) Q x (i + 1, j) - x^{T} (i + 1, j + 1) Q x (i + 1, j + 1)] \\ + \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i, j) Q_{1} x (i, j) - x^{T} (i + 1, j + 1) Q_{1} x (i + 1, j + 1)] \\ + \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i - d, j + 1) Q_{2} x (i - d, j + 1) - x^{T} (i + 1, j + 1) Q_{2} x (i + 1, j + 1)] \\ + \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i + 1, j - k) Q_{3} x (i + 1, j - k) - x^{T} (i + 1, j + 1) Q_{3} x (i + 1, j + 1)] \\ + \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i - l, j - m) Q_{4} x (i - l, j - m) - x^{T} (i + 1, j + 1) Q_{4} x (i + 1, j + 1)] \\ + \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i - g, j + 1) Q_{5} x (i - g, j + 1) - x^{T} (i + 1, j + 1) Q_{5} x (i + 1, j + 1)] \\ + \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i + 1, j - h) Q_{6} x (i + 1, j - h) - x^{T} (i + 1, j + 1) Q_{6} x (i + 1, j + 1)] \\ + \sum_{i = 0}^{\infty} \sum_{j = 0}^{\infty} [x^{T} (i - n, j - p) Q_{7} x (i - n, j - p) - x^{T} (i + 1, j + 1) Q_{7} x (i + 1, j + 1)] \end{matrix}

\begin{matrix} = \sum_{j = 0}^{\infty} x^{T} (0, j + 1) (P - \tilde{Q}) x (0, j + 1) + \sum_{i = 0}^{\infty} x^{T} (i + 1, 0) Q x (i + 1, 0) \\ + \sum_{i = 0}^{\infty} x^{T} (i, 0) Q_{1} x (i, 0) + \sum_{j = 1}^{\infty} x^{T} (0, j) Q_{1} x (0, j) \\ + \sum_{i = - d}^{0} \sum_{j = 0}^{\infty} x^{T} (i, j + 1) Q_{2} x (i, j + 1) \\ + \sum_{i = 0}^{\infty} \sum_{j = - k}^{0} x^{T} (i + 1, j) Q_{3} x (i + 1, j) + \sum_{i = - l}^{0} \sum_{j = - m}^{0} x^{T} (i, j) Q_{4} x (i, j) \\ - \sum_{i = 1}^{\infty} \sum_{j = 1}^{\infty} x^{T} (i, j) Q_{4} x (i, j) \\ + \sum_{i = - g}^{0} \sum_{j = 0}^{\infty} x^{T} (i, j + 1) Q_{5} x (i, j + 1) \\ + \sum_{i = 0}^{\infty} \sum_{j = - h}^{0} x^{T} (i + 1, j) Q_{6} x (i + 1, j) \\ + \sum_{i = - n}^{0} \sum_{j = - p}^{0} x^{T} (i, j) Q_{7} x (i, j) \\ - \sum_{i = 1}^{\infty} \sum_{j = 1}^{\infty} x^{T} (i, j) Q_{7} x (i, j) \end{matrix}

\begin{matrix} = \sum_{j = 1}^{L_{2} - 1} x^{T} (0, j) (P - \tilde{Q}) x (0, j) + \sum_{i = 1}^{L_{1} - 1} x^{T} (i, 0) Q x (i, 0) \\ + \sum_{i = 0}^{L_{1} - 1} x^{T} (i, 0) Q_{1} x (i, 0) + \sum_{j = 1}^{L_{2} - 1} x^{T} (0, j) Q_{1} x (0, j) \\ + \sum_{i = - d}^{0} \sum_{j = 1}^{L_{2} - 1} x^{T} (i, j) Q_{2} x (i, j) \\ + \sum_{i = 1}^{L_{1} - 1} \sum_{j = - k}^{0} x^{T} (i, j) Q_{3} x (i, j) + \sum_{i = - l}^{0} \sum_{j = - m}^{0} x^{T} (i, j) Q_{4} x (i, j) \\ - \sum_{i = 1}^{L_{1} - 1} \sum_{j = 1}^{L_{2} - 1} x^{T} (i, j) Q_{4} x (i, j) \\ + \sum_{i = - g}^{0} \sum_{j = 1}^{L_{2} - 1} x^{T} (i, j) Q_{5} x (i, j) \\ + \sum_{i = 1}^{L_{1} - 1} \sum_{j = - h}^{0} x^{T} (i, j) Q_{6} x (i, j) \\ + \sum_{i = - n}^{0} \sum_{j = - p}^{0} x^{T} (i, j) Q_{7} x (i, j) \\ - \sum_{i = 1}^{L_{1} - 1} \sum_{j = 1}^{L_{2} - 1} x^{T} (i, j) Q_{7} x (i, j) \end{matrix}

(58)

\begin{matrix} \leq [(L_{2} - 1) λ_{\max} {M^{T} (P - \tilde{Q}) M} + (L_{1} - 1) λ_{\max} (M^{T} Q M) + (L_{1} + L_{2} - 1) λ_{\max} (M^{T} Q_{1} M) \\ + (L_{2} - 1) (d * + 1) λ_{\max} (M^{T} Q_{2} M) + (L_{1} - 1) (k * + 1) λ_{\max} (M^{T} Q_{3} M) \\ + {(l * + 1) (m * + 1) - (L_{1} - 1) (L_{2} - 1)} λ_{\max} (M^{T} Q_{4} M) \\ + (L_{2} - 1) (g * + 1) λ_{\max} (M^{T} Q_{5} M) + (L_{1} - 1) (h * + 1) λ_{\max} (M^{T} Q_{6} M) \\ + {(n * + 1) (p * + 1) - (L_{1} - 1) (L_{2} - 1)} λ_{\max} (M^{T} Q_{7} M)] \end{matrix}

where use has been made of (5), (1e) and (1f) and the relation $lim_{i + j \to \infty} x (i, j)$ = 0. This completes the proof of the Lemma 4.

Proof of Theorem 1. Applying Lemma 3 in (7), we obtain

[\begin{matrix} - P^{- 1} & 0 & 0 & 0 & 0 & 0 & 0 & (A_{1} + B_{1} K) & (A_{2} + B_{2} K) & (A_{0} + B_{0} K) & A_{1 d} & A_{1 d} & A_{1 d} & (B_{1 d} K) & (B_{2 d} K) & (B_{0 d} K) \\ 0 & R_{1}^{- 1} & 0 & 0 & 0 & 0 & 0 & K & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & R_{2}^{- 1} & 0 & 0 & 0 & 0 & 0 & K & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & R_{3}^{- 1} & 0 & 0 & 0 & 0 & 0 & K & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & R_{4}^{- 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & K & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & R_{5}^{- 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & K & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & R_{6}^{- 1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & K \\ {(A_{1} + B_{1} K)}^{T} & K^{T} & 0 & 0 & 0 & 0 & 0 & - P + \tilde{Q} + S_{1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {(A_{2} + B_{2} K)}^{T} & 0 & K^{T} & 0 & 0 & 0 & 0 & 0 & - Q + S_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {(A_{0} + B_{0} K)}^{T} & 0 & 0 & K^{T} & 0 & 0 & 0 & 0 & 0 & - Q_{1} + S_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ A_{1 d}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - Q_{2} + S_{4} & 0 & 0 & 0 & 0 & 0 \\ A_{2 d}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - Q_{3} + S_{5} & 0 & 0 & 0 & 0 \\ A_{0 d}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - Q_{4} + S_{6} & 0 & 0 & 0 \\ {(B_{1 d} K)}^{T} & 0 & 0 & 0 & K^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - Q_{5} & 0 & 0 \\ {(B_{2 d} K)}^{T} & 0 & 0 & 0 & 0 & K^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - Q_{6} & 0 \\ {(B_{0 d} K)}^{T} & 0 & 0 & 0 & 0 & 0 & K^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - Q_{7} \end{matrix}]

+ [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & LF (i, j) M_{1} & LF (i, j) M_{2} & LF (i, j) M_{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \begin{matrix} LF (i, j) M_{4} & LF (i, j) M_{5} & LF (i, j) M_{6} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

+ [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ M_{1}^{T} F^{T} (i, j) L^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ M_{2}^{T} F^{T} (i, j) L^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ M_{3}^{T} F^{T} (i, j) L^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ M_{4}^{T} F^{T} (i, j) L^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ M_{5}^{T} F^{T} (i, j) L^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ M_{6}^{T} F^{T} (i, j) L^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

+ [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & B_{1} H_{k} F_{k} (i, j) E_{k} & B_{2} H_{k} F_{k} (i, j) E_{k} & B_{0} H_{k} F_{k} (i, j) E_{k} & 0 & 0 & 0 & B_{1 d} H_{k} F_{k} (i, j) E_{k} & B_{2 d} H_{k} F_{k} (i, j) E_{k} & B_{0 d} H_{k} F_{k} (i, j) E_{k} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & H_{k} F_{k} (i, j) E_{k} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & H_{k} F_{k} (i, j) E_{k} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & H_{k} F_{k} (i, j) E_{k} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & H_{k} F_{k} (i, j) E_{k} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & H_{k} F_{k} (i, j) E_{k} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & H_{k} F_{k} (i, j) E_{k} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

(59)

+ [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} B_{1}^{T} & E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} B_{2}^{T} & 0 & E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} B_{0}^{T} & 0 & 0 & E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} B_{1 d}^{T} & 0 & 0 & 0 & E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} B_{2 d}^{T} & 0 & 0 & 0 & 0 & E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} B_{0 d}^{T} & 0 & 0 & 0 & 0 & 0 & E_{k}^{T} F_{k}^{T} (i, j) H_{k}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] < 0 .

Equation (59) can be re-written as

[\begin{matrix} - P^{- 1} & 0 & 0 & 0 & 0 & 0 & 0 & (A_{1} + B_{1} K) \\ 0 & R_{1}^{- 1} & 0 & 0 & 0 & 0 & 0 & K \\ 0 & 0 & R_{2}^{- 1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & R_{3}^{- 1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & R_{4}^{- 1} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & R_{5}^{- 1} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & R_{6}^{- 1} & 0 \\ {(A_{1} + B_{1} K)}^{T} & K^{T} & 0 & 0 & 0 & 0 & 0 & - P + \tilde{Q} + S_{1} \\ {(A_{2} + B_{2} K)}^{T} & 0 & K^{T} & 0 & 0 & 0 & 0 & 0 \\ {(A_{0} + B_{0} K)}^{T} & 0 & 0 & K^{T} & 0 & 0 & 0 & 0 \\ A_{1 d}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ A_{2 d}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ A_{0 d}^{T} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ {(B_{1 d} K)}^{T} & 0 & 0 & 0 & K^{T} & 0 & 0 & 0 \\ {(B_{2 d} K)}^{T} & 0 & 0 & 0 & 0 & K^{T} & 0 & 0 \\ {(B_{0 d} K)}^{T} & 0 & 0 & 0 & 0 & 0 & K^{T} & 0 \end{matrix}

\begin{matrix} (A_{2} + B_{2} K) & (A_{0} + B_{0} K) & A_{1 d} & A_{1 d} & A_{1 d} & (B_{1 d} K) & (B_{2 d} K) & (B_{0 d} K) \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ K & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & K & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & K & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & K & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & K \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ - Q + S_{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & - Q_{1} + S_{3} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & - Q_{2} + S_{4} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & - Q_{3} + S_{5} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & - Q_{4} + S_{6} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & - Q_{5} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & - Q_{6} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & - Q_{7} \end{matrix}]

+ [\begin{matrix} L \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}] F (i, j) {[\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ M_{1}^{T} \\ M_{2}^{T} \\ M_{3}^{T} \\ M_{4}^{T} \\ M_{5}^{T} \\ M_{6}^{T} \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} + [\begin{matrix} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ M_{1}^{T} \\ M_{2}^{T} \\ M_{3}^{T} \\ M_{4}^{T} \\ M_{5}^{T} \\ M_{6}^{T} \\ 0 \\ 0 \\ 0 \end{matrix}] F^{T} (i, j) {[\begin{matrix} L \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} + [\begin{matrix} B_{1} H_{k} & B_{1} H_{k} & B_{1} H_{k} & B_{1} H_{k} & B_{1} H_{k} & B_{1} H_{k} \\ H_{k} & 0 & 0 & 0 & 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 & 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 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]

\times [\begin{matrix} F_{k} (i, j) & 0 & 0 & 0 & 0 & 0 \\ 0 & F_{k} (i, j) & 0 & 0 & 0 & 0 \\ 0 & 0 & F_{k} (i, j) & 0 & 0 & 0 \\ 0 & 0 & 0 & F_{k} (i, j) & 0 & 0 \\ 0 & 0 & 0 & 0 & F_{k} (i, j) & 0 \\ 0 & 0 & 0 & 0 & 0 & F_{k} (i, j) \end{matrix}] [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} \end{matrix}]

+ {[\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & E_{k} \end{matrix}]}^{T}

\times {[\begin{matrix} F_{k} (i, j) & 0 & 0 & 0 & 0 & 0 \\ 0 & F_{k} (i, j) & 0 & 0 & 0 & 0 \\ 0 & 0 & F_{k} (i, j) & 0 & 0 & 0 \\ 0 & 0 & 0 & F_{k} (i, j) & 0 & 0 \\ 0 & 0 & 0 & 0 & F_{k} (i, j) & 0 \\ 0 & 0 & 0 & 0 & 0 & F_{k} (i, j) \end{matrix}]}^{T}

(60)

\times {[\begin{matrix} B_{1} H_{k} & B_{1} H_{k} & B_{1} H_{k} & B_{1} H_{k} & B_{1} H_{k} & B_{1} H_{k} \\ H_{k} & 0 & 0 & 0 & 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 & 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 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]}^{T} < 0 .

Using Lemma 2, (60) can be rearranged as

[\begin{matrix} \begin{array}{l} - P^{- 1} + ε_{1} L L^{T} + ε_{2} B_{1} H_{k} H_{k}^{T} B_{1}^{T} + ε_{2} B_{2} H_{k} H_{k}^{T} B_{2}^{T} + ε_{2} B_{0} H_{k} H_{k}^{T} B_{0}^{T} \\ + ε_{2} B_{1 d} H_{k} H_{k}^{T} B_{1 d}^{T} + ε_{2} B_{2 d} H_{k} H_{k}^{T} B_{2 d}^{T} + ε_{2} B_{0 d} H_{k} H_{k}^{T} B_{0 d}^{T} \end{array} & ε_{2} B_{1} H_{k} H_{k}^{T} \\ ε_{2} H_{k} H_{k}^{T} B_{1}^{T} & - R_{1}^{- 1} + ε_{2} H_{k} H_{k}^{T} \\ ε_{2} H_{k} H_{k}^{T} B_{2}^{T} & 0 \\ ε_{2} H_{k} H_{k}^{T} B_{0}^{T} & 0 \\ ε_{2} H_{k} H_{k}^{T} B_{1 d}^{T} & 0 \\ ε_{2} H_{k} H_{k}^{T} B_{2 d}^{T} & 0 \\ ε_{2} H_{k} H_{k}^{T} B_{0 d}^{T} & 0 \\ {(A_{1} + B_{1} K)}^{T} & K^{T} \\ {(A_{2} + B_{2} K)}^{T} & 0 \\ {(A_{0} + B_{0} K)}^{T} & 0 \\ A_{1 d}^{T} & 0 \\ A_{2 d}^{T} & 0 \\ A_{0 d}^{T} & 0 \\ K^{T} B_{1 d}^{T} & 0 \\ K^{T} B_{2 d}^{T} & 0 \\ K^{T} B_{0 d}^{T} & 0 \end{matrix}

\begin{matrix} ε_{2} B_{2} H_{k} H_{k}^{T} & ε_{2} B_{0} H_{k} H_{k}^{T} & ε_{2} B_{1 d} H_{k} H_{k}^{T} & ε_{2} B_{2 d} H_{k} H_{k}^{T} \\ 0 & 0 & 0 & 0 \\ - R_{2}^{- 1} + ε_{2} H_{k} H_{k}^{T} & 0 & 0 & 0 \\ 0 & - R_{3}^{- 1} + ε_{2} H_{k} H_{k}^{T} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & - R_{4}^{- 1} + ε_{2} H_{k} H_{k}^{T} & 0 \\ 0 & 0 & 0 & - R_{5}^{- 1} + ε_{2} H_{k} H_{k}^{T} \\ 0 & 0 & 0 & 0 \\ K^{T} & 0 & 0 & 0 \\ 0 & K^{T} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & K^{T} & 0 \\ 0 & 0 & 0 & K^{T} \\ 0 & 0 & 0 & 0 \end{matrix}

\begin{matrix} ε_{2} B_{0 d} H_{k} H_{k}^{T} & (A_{1} + B_{1} K) & (A_{2} + B_{2} K) & (A_{0} + B_{0} K) \\ 0 & K & 0 & 0 \\ 0 & 0 & K & 0 \\ 0 & 0 & 0 & K \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ - R_{6}^{- 1} + ε_{2} H_{k} H_{k}^{T} & 0 & 0 & 0 \\ 0 & \begin{matrix} - P + \tilde{Q} + S_{1} \\ + ε_{1}^{- 1} M_{1}^{T} M_{1} + ε_{2}^{- 1} E_{k}^{T} E_{k} \end{matrix} & ε_{1}^{- 1} M_{1}^{T} M_{2} & ε_{1}^{- 1} M_{1}^{T} M_{3} \\ 0 & ε_{1}^{- 1} M_{2}^{T} M_{1} & \begin{matrix} - Q + S_{2} + ε_{1}^{- 1} M_{2}^{T} M_{2} \\ + ε_{2}^{- 1} E_{k}^{T} E_{k} \end{matrix} & ε_{1}^{- 1} M_{2}^{T} M_{3} \\ 0 & ε_{1}^{- 1} M_{3}^{T} M_{1} & ε_{1}^{- 1} M_{3}^{T} M_{2} & \begin{matrix} - Q_{1} + S_{3} + ε_{1}^{- 1} M_{3}^{T} M_{3} \\ + ε_{2}^{- 1} E_{k}^{T} E_{k} \end{matrix} \\ 0 & ε_{1}^{- 1} M_{4}^{T} M_{1} & ε_{1}^{- 1} M_{4}^{T} M_{2} & ε_{1}^{- 1} M_{4}^{T} M_{3} \\ 0 & ε_{1}^{- 1} M_{5}^{T} M_{1} & ε_{1}^{- 1} M_{5}^{T} M_{2} & ε_{1}^{- 1} M_{5}^{T} M_{3} \\ 0 & ε_{1}^{- 1} M_{6}^{T} M_{1} & ε_{1}^{- 1} M_{6}^{T} M_{2} & ε_{1}^{- 1} M_{6}^{T} M_{3} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ K^{T} & 0 & 0 & 0 \end{matrix}

(61)

\begin{matrix} A_{1 d} & A_{2 d} & A_{0 d} & B_{1 d} K & B_{2 d} K & B_{0 d} K \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & K & 0 & 0 \\ 0 & 0 & 0 & 0 & K & 0 \\ 0 & 0 & 0 & 0 & 0 & K \\ ε_{1}^{- 1} M_{1}^{T} M_{4} & ε_{1}^{- 1} M_{1}^{T} M_{5} & ε_{1}^{- 1} M_{1}^{T} M_{6} & 0 & 0 & 0 \\ ε_{1}^{- 1} M_{2}^{T} M_{4} & ε_{1}^{- 1} M_{2}^{T} M_{5} & ε_{1}^{- 1} M_{2}^{T} M_{6} & 0 & 0 & 0 \\ ε_{1}^{- 1} M_{3}^{T} M_{4} & ε_{1}^{- 1} M_{3}^{T} M_{5} & ε_{1}^{- 1} M_{3}^{T} M_{6} & 0 & 0 & 0 \\ \begin{matrix} - Q_{2} + S_{4} \\ + ε_{1}^{- 1} M_{4}^{T} M_{4} \end{matrix} & ε_{1}^{- 1} M_{4}^{T} M_{5} & ε_{1}^{- 1} M_{4}^{T} M_{6} & 0 & 0 & 0 \\ ε_{1}^{- 1} M_{5}^{T} M_{4} & \begin{matrix} - Q_{3} + S_{5} \\ + ε_{1}^{- 1} M_{5}^{T} M_{5} \end{matrix} & ε_{1}^{- 1} M_{5}^{T} M_{6} & 0 & 0 & 0 \\ ε_{1}^{- 1} M_{6}^{T} M_{4} & ε_{1}^{- 1} M_{6}^{T} M_{5} & \begin{matrix} - Q_{4} + S_{6} \\ + ε_{1}^{- 1} M_{6}^{T} M_{6} \end{matrix} & 0 & 0 & 0 \\ 0 & 0 & 0 & - Q_{5} + ε_{2}^{- 1} E_{k}^{T} E_{k} & 0 & 0 \\ 0 & 0 & 0 & 0 & - Q_{6} + ε_{2}^{- 1} E_{k}^{T} E_{k} & 0 \\ 0 & 0 & 0 & 0 & 0 & - Q_{7} + ε_{2}^{- 1} E_{k}^{T} E_{k} \end{matrix}] .

Pre- and post-multiplying (61) by diag{ $I$ , $I$ , $I$ , $I$ , $I$ , $I$ , $I$ , $P^{- 1}$ , $P^{- 1}$ , $P^{- 1}$ , $P^{- 1}$ , $P^{- 1}$ , $P^{- 1}$ , $P^{- 1}$ , $P^{- 1}$ , $P^{- 1}$ }, we obtain

[\begin{matrix} \begin{matrix} - P^{- 1} + ε_{1}^{- 1} L L^{T} + ε_{2}^{- 1} B_{1} H_{k} H_{k}^{T} B_{1}^{T} + ε_{2}^{- 1} B_{2} H_{k} H_{k}^{T} B_{2}^{T} + ε_{2}^{- 1} B_{0} H_{k} H_{k}^{T} B_{0}^{T} \\ + ε_{2}^{- 1} B_{1 d} H_{k} H_{k}^{T} B_{1 d}^{T} + ε_{2}^{- 1} B_{2 d} H_{k} H_{k}^{T} B_{2 d}^{T} + ε_{2}^{- 1} B_{0 d} H_{k} H_{k}^{T} B_{0 d}^{T} \end{matrix} & ε_{2}^{- 1} B_{1} H_{k} H_{k}^{T} \\ ε_{2}^{- 1} H_{k} H_{k}^{T} B_{1}^{T} & - R_{1}^{- 1} + ε_{2}^{- 1} H_{k} H_{k}^{T} \\ ε_{2}^{- 1} H_{k} H_{k}^{T} B_{2}^{T} & 0 \\ ε_{2}^{- 1} H_{k} H_{k}^{T} B_{0}^{T} & 0 \\ ε_{2}^{- 1} H_{k} H_{k}^{T} B_{1 d}^{T} & 0 \\ ε_{2}^{- 1} H_{k} H_{k}^{T} B_{2 d}^{T} & 0 \\ ε_{2}^{- 1} H_{k} H_{k}^{T} B_{0 d}^{T} & 0 \\ {(A_{1} P_{1} + B_{1} U)}^{T} & U^{T} \\ {(A_{2} P_{1} + B_{2} U)}^{T} & 0 \\ {(A_{0} P_{1} + B_{0} U)}^{T} & 0 \\ {(A_{1 d} P_{1})}^{T} & 0 \\ {(A_{2 d} P_{1})}^{T} & 0 \\ {(A_{0 d} P_{1})}^{T} & 0 \\ U^{T} B_{1 d}^{T} & 0 \\ U^{T} B_{2 d}^{T} & 0 \\ U^{T} B_{0 d}^{T} & 0 \end{matrix}

\begin{matrix} ε_{2}^{- 1} B_{2} H_{k} H_{k}^{T} & ε_{2}^{- 1} B_{0} H_{k} H_{k}^{T} & ε_{2}^{- 1} B_{1 d} H_{k} H_{k}^{T} & ε_{2}^{- 1} B_{2 d} H_{k} H_{k}^{T} & ε_{2}^{- 1} B_{0 d} H_{k} H_{k}^{T} \\ 0 & 0 & 0 & 0 & 0 \\ - R_{2}^{- 1} + ε_{2}^{- 1} H_{k} H_{k}^{T} & 0 & 0 & 0 & 0 \\ 0 & - R_{3}^{- 1} + ε_{2}^{- 1} H_{k} H_{k}^{T} & 0 & 0 & 0 \\ 0 & 0 & - R_{4}^{- 1} + ε_{2}^{- 1} H_{k} H_{k}^{T} & 0 & 0 \\ 0 & 0 & 0 & - R_{5}^{- 1} + ε_{2}^{- 1} H_{k} H_{k}^{T} & 0 \\ 0 & 0 & 0 & 0 & - R_{6}^{- 1} + ε_{2}^{- 1} H_{k} H_{k}^{T} \\ 0 & 0 & 0 & 0 & 0 \\ U^{T} & 0 & 0 & 0 & 0 \\ 0 & U^{T} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & U^{T} & 0 & 0 \\ 0 & 0 & 0 & U^{T} & 0 \\ 0 & 0 & 0 & 0 & U^{T} \end{matrix}

\begin{matrix} (A_{1} P_{1} + B_{1} U) & (A_{2} P_{1} + B_{2} U) & (A_{0} P_{1} + B_{0} U) \\ U & 0 & 0 \\ 0 & U & 0 \\ 0 & 0 & U \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ - P_{1} + Y_{1} + Y_{2} + Y_{3} + Y_{4} + Y_{5} + Y_{6} + Y_{7} + Y_{8} & 0 & 0 \\ 0 & - Y_{1} & 0 \\ 0 & 0 & - Y_{2} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}

\begin{matrix} A_{1 d} P_{1} & A_{2 d} P_{1} & A_{0 d} P_{1} & B_{1 d} U & B_{2 d} U & B_{0 d} U \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ - Y_{3} & 0 & 0 & 0 & 0 & 0 \\ 0 & - Y_{4} & 0 & 0 & 0 & 0 \\ 0 & 0 & - Y_{5} & 0 & 0 & 0 \\ 0 & 0 & 0 & - Y_{6} & 0 & 0 \\ 0 & 0 & 0 & 0 & - Y_{7} & 0 \\ 0 & 0 & 0 & 0 & 0 & - Y_{8} \end{matrix}]

+ [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \begin{matrix} P_{1} S_{1} P_{1} + ε_{1}^{- 1} P_{1} M_{1}^{T} M_{1} P_{1} \\ + ε_{2}^{- 1} P_{1} E_{k}^{T} E_{k} P_{1} \end{matrix} & ε_{1}^{- 1} P_{1} M_{1}^{T} M_{2} P_{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & ε_{1}^{- 1} P_{1} M_{2}^{T} M_{1} P_{1} & \begin{matrix} P_{1} S_{2} P_{1} + ε_{1}^{- 1} P_{1} M_{2}^{T} M_{2}^{T} P_{1} \\ + ε_{2}^{- 1} P_{1} E_{k}^{T} E_{k} P_{1} \end{matrix} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & ε_{1}^{- 1} P_{1} M_{3}^{T} M_{1} P_{1} & ε_{1}^{- 1} P_{1} M_{3}^{T} M_{2} P_{1} & \begin{matrix} 0 \\ 0 \end{matrix} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & ε_{1}^{- 1} P_{1} M_{4}^{T} M_{1} P_{1} & ε_{1}^{- 1} P_{1} M_{4}^{T} M_{2} P_{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & ε_{1}^{- 1} P_{1} M_{5}^{T} M_{1} P_{1} & ε_{1}^{- 1} P_{1} M_{5}^{T} M_{2} P_{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & ε_{1}^{- 1} P_{1} M_{6}^{T} M_{1} P_{1} & ε_{1}^{- 1} P_{1} M_{6}^{T} M_{2}^{T} P_{1} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}

\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ ε_{1}^{- 1} P_{1} M_{1}^{T} M_{3} P_{1} & ε_{1}^{- 1} P_{1} M_{1}^{T} M_{4} P_{1} & ε_{1}^{- 1} P_{1} M_{1}^{T} M_{5} P_{1} \\ ε_{1}^{- 1} P_{1} M_{2}^{T} M_{3} P_{1} & ε_{1}^{- 1} P_{1} M_{2}^{T} M_{4} P_{1} & ε_{1}^{- 1} P_{1} M_{2}^{T} M_{5} P_{1} \\ \begin{matrix} P_{1} S_{3} P_{1} + ε_{1}^{- 1} P_{1} M_{3}^{T} M_{3} P_{1} \\ + ε_{2}^{- 1} P_{1} E_{k}^{T} E_{k} P_{1} \end{matrix} & ε_{1}^{- 1} P_{1} M_{3}^{T} M_{4} P_{1} & ε_{1}^{- 1} P_{1} M_{3}^{T} M_{5} P_{1} \\ ε_{1}^{- 1} P_{1} M_{4}^{T} M_{3} P_{1} & P_{1} S_{4} P_{1} + ε_{1}^{- 1} P_{1} M_{4}^{T} M_{4} P_{1} & ε_{1}^{- 1} P_{1} M_{4}^{T} M_{5} P_{1} \\ ε_{1}^{- 1} P_{1} M_{5}^{T} M_{3} P_{1} & ε_{1}^{- 1} P_{1} M_{5}^{T} M_{4} P_{1} & P_{1} S_{5} P_{1} + ε_{1}^{- 1} P_{1} M_{5}^{T} M_{5} P_{1} \\ ε_{1}^{- 1} P_{1} M_{6}^{T} M_{3} P_{1} & ε_{1}^{- 1} P_{1} M_{6}^{T} M_{4} P_{1} & ε^{- 1} P_{1} M_{6}^{T} M_{5} P_{1} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}

(62)

\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ ε_{1}^{- 1} P_{1} M_{1}^{T} M_{6} P_{1} & 0 & 0 & 0 \\ ε_{1}^{- 1} P_{1} M_{2}^{T} M_{6} P_{1} & 0 & 0 & 0 \\ ε_{1}^{- 1} P_{1} M_{3}^{T} M_{6} P_{1} & 0 & 0 & 0 \\ ε_{1}^{- 1} P_{1} M_{4}^{T} M_{6} P_{1} & 0 & 0 & 0 \\ ε_{1}^{- 1} P_{1} M_{5}^{T} M_{6} P_{1} & 0 & 0 & 0 \\ P_{1} S_{6} P_{1} + ε_{1}^{- 1} P_{1} M_{6}^{T} M_{6} P_{1} & 0 & 0 & 0 \\ 0 & ε_{2}^{- 1} P_{1} E_{k}^{T} E_{k} P_{1} & 0 & 0 \\ 0 & 0 & ε_{2}^{- 1} P_{1} E_{k}^{T} E_{k} P_{1} & 0 \\ 0 & 0 & 0 & ε_{2}^{- 1} P_{1} E_{k}^{T} E_{k} P_{1} \end{matrix}] < 0,

where

(63a)

P_{1} = P^{- 1},

(63b)

\begin{matrix} Y_{1} = P_{1} Q P_{1}, Y_{2} = P_{1} Q_{1} P_{1}, Y_{3} = P_{1} Q_{2} P_{1}, Y_{4} = P_{1} Q_{3} P_{1}, Y_{5} = P_{1} Q_{4} P_{1}, Y_{6} = P_{1} Q_{5} P_{1}, \\ Y_{7} = P_{1} Q_{6} P_{1}, Y_{8} = P_{1} Q_{7} P_{1} . \end{matrix}

The equivalence of (62) and (11) follows trivially from the Schur complements. Using (63), the bound of the cost function can be easily obtained from (10). This completes the proof of Theorem 1.

Acknowledgements

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

Declaration of Conflicting Interests

The authors declare that there is no conflict of interest.

Funding

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

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