Acta Mathematica Vietnamica

, Volume 41, Issue 3, pp 481–493 | Cite as

Finite-Time Stability and H Control of Linear Discrete-Time Delay Systems with Norm-Bounded Disturbances

Article

Abstract

This paper deals with the finite-time stability and H control of linear discrete-time delay systems. The system under consideration is subject to interval time-varying delay and norm-bounded disturbances. Linear matrix inequality approach is used to solve the finite-time stability problem. First, new sufficient conditions are established for robust finite-time stability of the linear discrete-time delay system with norm-bounded disturbances, then the state feedback controller is designed to robustly finite-time stabilize the system and guarantee an adequate level of system performance. The delay-dependent sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the proposed results.

Keywords

Finite-time stability H control Time-varying delay Disturbances Linear matrix inequalities 

Mathematics Subject Classification (2010)

34D10 34K20 49M7 

1 Introduction

Finite-time stability (FTS) introduced by Dorato in [3] involves dynamical systems whose solutions converge to an equilibrium state in finite time. Compared with the Lyapunov stability, FTS is a more practical property, useful while studying the behavior of the system within a finite interval time, and therefore it finds many applications. A lot of interesting results on finite-time stability and stabilization in the context of linear discrete-time delay systems have been obtained (see, e.g., [1, 9, 13, 15] and the references therein).

On the other hand, the problem of finite-time H control has attracted much attention due to its both practical and theoretical importance. Various approaches have been developed and a great number of results for linear continuous and discrete-time systems have been reported in the literature (see, e.g., [5, 6, 7, 10, 14] and the references therein). Note that these papers were limited either to Lyapunov stability or to the system with constant delays. Based on the solution of some LMIs, some results on finite-time H control of discrete-time systems without delays was proposed in [11, 12], and in [8] for the system with constant delays. To the best of our knowledge, finite-time H control problem for linear discrete-time systems with interval time-varying delay has not fully investigated. The problem is important and challenging in many practice applications, which motivates the main purpose of our research.

In this paper, we extend further results of finite-time stability and H control for linear discrete-time systems with time-varying delays and disturbances. Our main purpose is to design a state feedback controller which guarantees FTS of the closed-loop system and reduces the effect of the disturbance input on the controlled output to a prescribed level. The novel features of this paper are as follows: (i) The system under consideration subjected to interval time-varying delays in both the state input and observation output; (ii) Using new bounding LMI estimation technique, a set of improved Lyapunov-like functionals is constructed to design the H feedback controller in terms of LMIs, which can be determined by utilizing MATLAB’s LMI Control Toolbox [4].

The paper is organized as follows. In Section 2, some preliminary definitions are provided and the problem we deal with is precisely stated. Section 3 presents the main results of the paper: sufficient conditions for robust finite-time H boundedness and control in terms of LMIs. Numerical examples showing the effectiveness of the proposed method are given.

2 Preliminaries

The following notations will be used throughout this paper. + denotes the set of all non-negative integers; n denotes the n-dimensional space with the scalar product xy; n×r denotes the space of all matrices of (n×r)−dimensions; A denotes the transpose of matrix A; A is symmetric if A = A; I denotes the identity matrix of appropriate dimension. Matrix A is called semi-positive definite \((A \geqslant 0)\) if \(x^{\top }Ax \geqslant 0\) for all xn;A is positive definite (A > 0) if xAx > 0 for all x ≠ 0; A > B means AB > 0. The notation diag {…} stands for a block-diagonal matrix. The symmetric term in a matrix is denoted by ∗.

Consider the following linear discrete-time systems with time-varying delay
$$\begin{array}{@{}rcl@{}} x(k+1) &=& Ax(k) + A_{d}x(k-d(k)) + Bu(k) + G\omega(k), \\ z(k) &=& Cx(k)+C_{d}x(k-d(k)),\quad k\in \Bbb Z_{+},\\ x(k) &=& \varphi(k),\ k\in\{-d_{2},-d_{2}+1,\dots,0\}, \end{array} $$
(1)
where \(x(k) \in \mathbb {R}^{n}\) is the state; \(u(k) \in \mathbb {R}^{m}\) is the control input; \(z(k)\in \mathbb {R}^{p}\) is the observation output; \(A, A_{d}\in \mathbb {R}^{n\times n}, B\in \mathbb {R}^{n\times m}, G\in \mathbb {R}^{n\times q}, C, C_{d}\in \mathbb {R}^{p\times n}\) are given real constant matrices; d(k) is a delay function satisfying
$$ 0< d_{1}\leqslant d(k)\leqslant d_{2}, \quad \forall k\in \Bbb Z_{+}, $$
(2)
where d1, d2 are known positive integers; φ(k) is the initial function; \(\omega (k)\in \mathbb {R}^{q}\) satisfying the condition
$$ \exists d>0:\quad \sum\limits_{k=0}^{N}w^{\top}(k)w(k) \leq d. $$
(3)

Definition 2.1

(Finite-time stability) Given positive numbers N, c1, c2, c1<c2, and a symmetric positive definite matrix R, discrete-time delay system (1) with u(k) = 0 is said to be robustly finite-time stable w.r.t. (c1, c2, R, N) if
$$\max_{k\in\{-d_{2},-d_{2}+1,\dots,0\}}\varphi^{\top}(k)R\varphi(k) \leqslant c_{1} \; \Longrightarrow \; x^{\top}(k)Rx(k) < c_{2} \quad \forall k = 1, 2, \dots, N,$$
for all disturbances ω(k) satisfying (3).

Definition 2.2

(Finite-time H boundedness) Given positive numbers γ, N, c1, c2, c1<c2, and a symmetric positive definite matrix R, system (1) with u(k) = 0 is said to be robustly finite-time H bounded w.r.t. (c1, c2, R, N) if the following two conditions hold:
  1. (i)

    System (1) with u(k) = 0 is robustly finite-time stable w.r.t. (c1, c2, R, N).

     
  2. (i)
    (ii) Under the zero initial condition (i.e., φ(k) = 0 ∀k ∈ {−d2, −d2 + 1, … , 0}), the output z(k) satisfies
    $$ \sum\limits_{k=0}^{N}z^{\top}(k)z(k) \leqslant \gamma\sum\limits_{k=0}^{N}w^{\top}(k)w(k) $$
    (4)
    for all disturbances ω(k) satisfying (3).
     

Definition 2.3

(Finite-time H control) Given positive numbers γ, N, c1, c2, c1<c2, and a symmetric positive definite matrix R, the finite-time H control problem for system (1) has a solution if there exists a state feedback controller u(k) = Kx(k) such that the resulting closed-loop system is robustly finite-time H bounded w.r.t. (c1, c2, R, N).

Proposition 2.1

(Schur Complement Lemma, [2]) Given constant matrices X, Y, Z with appropriate dimensions satisfying X = X, Y = Y > 0, then
$$X+Z^{\top}Y^{-1}Z<0 \quad\Longleftrightarrow\quad \left[\begin{array}{ll} X&Z^{\top}\\ Z&-Y\\ \end{array}\right] <0. $$

3 Main Results

This section provides sufficient conditions for the finite-time H boundedness and control of system (1) with interval time-varying delay and norm-bounded disturbances.

Theorem 3.1

Given positive constants γ, N, c1, c2,c1<c2, a symmetric positive-definite matrix R, system (1) with u(k) = 0 is robustly finite-time Hbounded w.r.t. (c1, c2, R, N) if for a scalar δ > 1, there exist symmetric positive definite matrices P, Q, positive scalars λ1, λ2, λ3such that the following LMIs hold:
$$\begin{array}{@{}rcl@{}} &&\lambda_{1} R < {P} < \lambda_{2} R, \quad {Q} < \lambda_{3} R, \end{array} $$
(5)
$$\begin{array}{@{}rcl@{}} &&\left[\begin{array}{ccccc} -\delta{P} + (d_{2}-d_{1}+1)Q & 0 & 0 & A^{\top}P & C^{\top}\\ * & -\delta^{d_{1}}Q & 0 & A_{d}^{\top}P & C_{d}^{\top} \\ *& * & -\frac{\gamma}{\delta^{N}}I & G^{\top}P & 0 \\ * & * & * & -P & 0 \\ * & * & * & * & - I\\ \end{array}\right] < 0, \end{array} $$
(6)
$$\begin{array}{@{}rcl@{}} &&\left[\begin{array}{cccccc} \gamma d - c_{2}\delta\lambda_{1} & c_{1}\delta^{N+1}\lambda_{2} & \rho\lambda_{3} \\ * & -c_{1}\delta^{N+1}\lambda_{2} & 0 \\ * & * & -\rho\lambda_{3} \end{array}\right] < 0, \end{array} $$
(7)
where
$$\rho := c_{1}\delta^{N+d_{2}-1}\left[d_{2}\delta + \frac{d_{2}(d_{2}-1)-d_{1}(d_{1}-1)}{2}\right].$$

Proof

Consider the following non-negative quadratic functions:
$$V(k) = V_{1}(k)+ V_{2}(k) + V_{3}(k), $$
where
$$\begin{array}{@{}rcl@{}} V_{1}(k) &=& x^{\top}(k)Px(k),\\ V_{2}(k) &=& \sum\limits_{s=k-d(k)}^{k-1}\delta^{k-1-s}x^{\top}(s)Qx(s), \\ V_{3}(k)&=& \sum\limits_{s=-d_{2}+2}^{-d_{1}+1}\sum\limits_{t=k-1+s}^{k-1}\delta^{k-1-t}x^{\top}(t)Qx(t). \end{array} $$
Taking the difference variation of Vi(k), i = 1, 2, 3, we have
$$\begin{array}{@{}rcl@{}} &&V_{1}(k+1) -\delta V_{1}(k)\\ &&= x^{\top}(k+1){P}x(k+1)-\delta x^{\top}(k){P}x(k)\\ &&= \left[\begin{array}{llll} x(k) \\ x(k-d(k)) \\ \omega(k) \\ \end{array}\right]^{\top}\left[\begin{array}{llll} A^{\top} \\ A_{d}^{\top} \\ G^{\top} \\ \end{array}\right]{P}\left[\begin{array}{llll} A & A_{d} & G \\ \end{array}\right] \left[\begin{array}{llll} x(k) \\ x(k-d(k)) \\ \omega(k) \\ \end{array}\right] - \delta x^{\top}(k){P}x(k),\\ &&(V_{2}+V_{3})(k+1) -\delta (V_{2}+V_{3})(k) \end{array} $$
$$\begin{array}{@{}rcl@{}} && = \sum\limits_{s=k+1-d(k+1)}^{k}\delta^{k-s}x^{\top}(s)Qx(s) - \sum\limits_{s=k-d(k)}^{k-1}\delta^{k-s}x^{\top}(s)Qx(s)\\[-2pt] &&\quad + \sum\limits_{s=-d_{2}+2}^{-d_{1}+1}\sum\limits_{t=k+s}^{k}\delta^{k-t}x^{\top}(t)Qx(t) - \sum\limits_{s=-d_{2}+2}^{-d_{1}+1}\sum\limits_{t=k-1+s}^{k-1}\delta^{k-t}x^{\top}(t)Qx(t) \\[-2pt] &&= x^{\top}(k)Qx(k) + \sum\limits_{s=k-d_{1}+1}^{k-1}\delta^{k-s}x^{\top}(s)Qx(s) + \sum\limits_{s=k+1-d(k+1)}^{k-d_{1}}\delta^{k-s}x^{\top}(s)Qx(s) \\[-2pt] &&\;\quad - \sum\limits_{s=k-d(k)+1}^{k-1}\delta^{k-s}x^{\top}(s)Qx(s) - \delta^{d(k)}x^{\top}(k-d(k))Qx(k-d(k))\\[-2pt] &&\;\quad + \sum\limits_{s=-d_{2}+2}^{-d_{1}+1}\left[x^{\top}(k)Qx(k) + \sum\limits_{t=k+s}^{k-1}\delta^{k-t}x^{\top}(t)Qx(t) - \sum\limits_{t=k+s}^{k-1}\delta^{k-t}x^{\top}(t)Qx(t)\right. \\[-3pt] &&\left.\quad - \delta^{1-s}x^{\top}(k-1+s)Qx(k-1+s)\right] \\[-2pt] &&= x^{\top}(k)Qx(k) + \sum\limits_{s=k-d_{1}+1}^{k-1}\delta^{k-s}x^{\top}(s)Qx(s) - \sum\limits_{s=k-d(k)+1}^{k-1}\delta^{k-s}x^{\top}(s)Qx(s) \\[-3pt] &&\quad + \sum\limits_{s=k+1-d(k+1)}^{k-d_{1}}\delta^{k-s}x^{\top}(s)Qx(s) - \delta^{d(k)}x^{\top}(k-d(k))Qx(k-d(k)) \\[-2pt] &&\quad + \sum\limits_{s=-d_{2}+2}^{-d_{1}+1}\left[ x^{\top}(k)Qx(k) - \delta^{1-s}x^{\top}(k-1+s)Qx(k-1+s)\right] \\[-2pt] &&\leqslant x^{\top}(k)Qx(k) + \sum\limits_{s=k+1-d(k+1)}^{k-d_{1}}\delta^{k-s}x^{\top}(s)Qx(s) \\ &&\quad - \delta^{d(k)}x^{\top}(k-d(k))Qx(k-d(k)) + (d_{2} - d_{1})x^{\top}(k)Qx(k) \\ &&\quad - \sum\limits_{s=-d_{2}+2}^{-d_{1}+1}\delta^{1-s}x^{\top}(k-1+s)Qx(k-1+s) \\ &&\leqslant (d_{2}-d_{1}+1)x^{\top}(k)Qx(k) - \delta^{d_{1}}x^{\top}(k-d(k))Qx(k-d(k)) \\ &&\quad + \sum\limits_{s=k+1-d(k+1)}^{k-d_{1}}\delta^{k-s}x^{\top}(s)Qx(s) - \sum\limits_{s=k+1-d_{2}}^{k-d_{1}}\delta^{k-s}x^{\top}(s)Qx(s) \\ &&\leqslant (d_{2}-d_{1}+1)x^{\top}(k)Qx(k) - \delta^{d_{1}}x^{\top}(k-d(k))Qx(k-d(k)). \end{array} $$
Thus, we have
$$\begin{array}{@{}rcl@{}} &&V(k+1) -\delta V(k)\\ &\leqslant& \left[\begin{array}{ccc} x(k) \\ x(k-d(k)) \\ \omega(k) \\ \end{array}\right]^{\top}\left[\begin{array}{ccc} A^{\top} \\ A_{d}^{\top} \\ G^{\top} \\ \end{array}\right]{P}\left[\begin{array}{ccc} A & A_{d} & G \\ \end{array}\right] \left[\begin{array}{ccc} x(k) \\ x(k-d(k)) \\ \omega(k) \\ \end{array}\right] \\ & +& x^{\top}(k)\left[-\delta{P} + (d_{2}-d_{1}+1)Q \right]x(k) - \delta^{d_{1}}x^{\top}(k-d(k))Qx(k-d(k))\\ &+& z^{\top}(k)z(k) - \frac{\gamma}{\delta^{N}}\omega^{\top}(k)\omega(k) + \frac{\gamma}{\delta^{N}}\omega^{\top}(k)\omega(k) - z^{\top}(k)z(k). \end{array} $$
Note that by setting
$$\begin{array}{@{}rcl@{}} \xi(k) &:=& \left[\begin{array}{ccc} x^{\top}(k) & x^{\top}(k-d(k)) & \omega^{\top}(k) \end{array}\right]^{\top}, \\ {\Upsilon} &:=&\left[\begin{array}{ccc} PA & PA_{d} & PG \\ \end{array}\right], \\ {\Phi} &:=& \left[\begin{array}{ccc} -\delta{P} + (d_{2}-d_{1}+1)Q + C^{\top}C & C^{\top}C_{d} & 0 \\ * & -\delta^{d_{1}}Q + C^{\top}_{d}C_{d} & 0 \\ * & * & -\frac{\gamma}{\delta^{N}}I \\ \end{array}\right], \end{array} $$
we can see that
$$\left[\begin{array}{ccc} x(k) \\ x(k-d(k)) \\ \omega(k) \\ \end{array}\right]^{\top}\left[\begin{array}{cc} A^{\top} \\ A_{d}^{\top} \\ G^{\top} \\ \end{array}\right]{P}\left[\begin{array}{ccc} A & A_{d} & G \\ \end{array}\right] \left[\begin{array}{ccc} x(k) \\ x(k-d(k)) \\ \omega(k) \\ \end{array}\right] =\; \xi^{\top}(k){\Upsilon}^{\top}{P}^{-1}{\Upsilon}\xi(k) $$
and
$$\begin{array}{@{}rcl@{}} &&x^{\top}(k)\left[-\delta{P} + (d_{2}-d_{1}+1)Q \right]x(k) - \delta^{d_{1}}x^{\top}(k-d(k))Qx(k-d(k))\\ && + z^{\top}(k)z(k) - \frac{\gamma}{\delta^{N}}\omega^{\top}(k)\omega(k)\\ &=&\; x^{\top}(k)\left[-\delta{P} + (d_{2}-d_{1}+1)Q + C^{\top}C\right]x(k) +2x^{\top}(k)C^{\top}C_{d} x(k-d(k)) \\ & + x^{\top}(k-d(k))\left[-\delta^{d_{1}}Q+C^{\top}_{d}C_{d}\right]x(k-d(k)) - \frac{\gamma}{\delta^{N}}\omega^{\top}(k)\omega(k)\\ &=&\; \xi^{\top}(k){\Phi} \xi(k). \end{array} $$
Therefore, we get
$$ V(k+1) -\delta V(k) \leqslant \xi^{\top}(k)\left[{\Phi} + {\Upsilon}^{\top}{P}^{-1}{\Upsilon}\right]\xi(k) + \frac{\gamma}{\delta^{N}}\omega^{\top}(k)\omega(k) - z^{\top}(k)z(k). $$
(8)
Next, by applying Proposition 2.1, we have
$${\Phi} + {\Upsilon}^{\top}{P}^{-1}{\Upsilon} < 0 \Longleftrightarrow \left[\begin{array}{ccccc} -\delta{P} + (d_{2}-d_{1}+1)Q + C^{\top}C & C^{\top}C_{d} & 0 & A^{\top}P \\ * & -\delta^{d_{1}}Q+ C^{\top}_{d}C_{d} & 0 & A_{d}^{\top}P \\ * & * & -\frac{\gamma}{\delta^{N}}I & G^{\top}P \\ * & * & * & -P \end{array}\right] < 0$$
and hence
$$\left[\begin{array}{cccc} -\delta{P} + (d_{2}-d_{1}+1)Q & 0 & 0 & A^{\top}P \\ * & -\delta^{d_{1}}Q & 0 & A_{d}^{\top}P \\ * & * & -\frac{\gamma}{\delta^{N}}I & G^{\top}P \\ * & * & * & -P \end{array}\right]+\; \left[\begin{array}{cccc} C^{\top} \\ C_{d}^{\top} \\ 0 \\ 0 \end{array}\right] \left[\begin{array}{cccc} C & C_{d} & 0 & 0 \end{array}\right] < 0,$$
which is evidently equivalent to LMI (6). For this reason, from (8) it follows that
$$ V(k+1) -\delta V(k) \leqslant \frac{\gamma}{\delta^{N}}\omega^{\top}(k)\omega(k) \quad \forall k\in\mathbb{Z}_{+}. $$
This estimation can be rewritten as
$$ V(k) \leqslant \delta V(k-1) + \frac{\gamma}{\delta^{N}}\omega^{\top}(k-1)\omega(k-1) \quad \forall k =1,2, \dots $$
(9)
By iteration, and take assumption (3) into account, inequality (9) implies
$$\begin{array}{@{}rcl@{}} V(k) & \leqslant& \delta^{k} V(0) + \frac{\gamma}{\delta^{N}}\sum\limits_{s=0}^{k-1}\delta^{k-1-s}\omega^{\top}(s)\omega(s) \\ & \leqslant& \delta^{N} V(0) + \frac{\gamma}{\delta^{N}}\delta^{N-1}\sum\limits_{s=0}^{N-1}\omega^{\top}(s)\omega(s) \\ & <& \delta^{N} V(0) + \frac{\gamma}{\delta}d. \end{array} $$
(10)
Now, using condition (5) and x(k) = φ(k), k ∈ {−d2, −d2 + 1, … , 0}, it is obvious that
$$\begin{array}{@{}rcl@{}} &V(0)\\ &=& x^{\top}(0){P}x(0) + \sum\limits_{s=-d(0)}^{-1}\delta^{-1-s}x^{\top}(s)Qx(s) + \sum\limits_{s=-d_{2}+2}^{-d_{1}+1}\sum\limits_{t=-1+s}^{-1}\delta^{-1-t}x^{\top}(t)Qx(t) \\ &<& \lambda_{2} x^{\top}(0)Rx(0) + \lambda_{3}\delta^{d_{2}-1}\sum\limits_{s=-d_{2}}^{-1}x^{\top}(s)Rx(s) + \lambda_{3} \delta^{d_{2}-2}\sum\limits_{s=-d_{2}+2}^{-d_{1}+1}\sum\limits_{t=-1+s}^{-1}x^{\top}(t)Rx(t) \\ &\leqslant& \left[\lambda_{2} + \lambda_{3}d_{2}\delta^{d_{2}-1} + \lambda_{3}\frac{d_{2}(d_{2}-1)-d_{1}(d_{1}-1)}{2}\delta^{d_{2}-2}\right] c_{1}. \end{array} $$
(11)
Taking (10), (11) into account, we obtain
$$ V(k) < \delta^{N} \sigma + \frac{\gamma}{\delta}d\quad \forall k\in\mathbb{Z}_{+}. $$
(12)
where \( \sigma := [\lambda _{2} + \lambda _{3}d_{2}\delta ^{d_{2}-1} + \lambda _{3}\frac {d_{2}(d_{2}-1)-d_{1}(d_{1}-1)}{2}\delta ^{d_{2}-2}] c_{1}.\)
On the other hand, also from (5) it follows that
$$ V(k)\geqslant x^{\top}(k){P}x(k) > \lambda_{1} x^{\top}(k)Rx(k) \quad \forall k\in\mathbb{Z}_{+}. $$
(13)
Note that by Proposition 2.1, LMI (7) is equivalent to
$$\begin{array}{@{}rcl@{}} &&\; \gamma d - c_{2}\delta\lambda_{1} + \left[\begin{array}{cc} c_{1}\delta^{N+1}\lambda_{2} & \rho\lambda_{3} \end{array}\right] \left[\begin{array}{cc} c_{1}\delta^{N+1}\lambda_{2} & 0 \\ 0 & \rho\lambda_{3} \end{array}\right]^{-1} \left[\begin{array}{cc} c_{1}\delta^{N+1}\lambda_{2} \\ \rho\lambda_{3} \end{array}\right] <0 \\ &&\Longleftrightarrow \gamma d - c_{2}\delta\lambda_{1} + \left[\begin{array}{cc} c_{1}\delta^{N+1}\lambda_{2} &\rho\lambda_{3} \end{array}\right] \left[\begin{array}{cc} (c_{1}\delta^{N+1}\lambda_{2})^{-1} & 0 \\ 0 & \!\!(\rho\lambda_{3})^{-1} \end{array}\right] \left[\begin{array}{cc} c_{1}\delta^{N+1}\lambda_{2} \\ &\!\!\!\rho\lambda_{3} \end{array}\right] <0 \\ &&\Longleftrightarrow \gamma d - c_{2}\delta\lambda_{1} + c_{1}\delta^{N+1}\lambda_{2} + \rho\lambda_{3} < 0\\ &&\Longleftrightarrow \gamma d - c_{2}\delta\lambda_{1} + c_{1}\delta^{N+1}\lambda_{2} + c_{1}\delta^{N+d_{2}-1}\left[d_{2}\delta + \frac{d_{2}(d_{2}-1)-d_{1}(d_{1}-1)}{2}\right]\lambda_{3} < 0 \\ &&\Longleftrightarrow \delta^{N+1}\sigma + \gamma d - c_{2}\delta\lambda_{1} <0. \end{array} $$
(14)
Consequently, we get from (12), (13), and (14) that
$$x^{\top}(k)Rx(k) < \frac{1}{\lambda_{1}\delta}\left[\delta^{N+1}\sigma + \gamma d\right] < c_{2}, $$
which implies that the system is robustly finite-time stable w.r.t. (c1, c2, R, N). To complete the proof of the theorem, it remains to show the finite-time γ-level condition (4). For this, from (8) it follows that
$$V(k+1) \leqslant \delta V(k) + \frac{\gamma}{\delta^{N}}\omega^{\top}(k)\omega(k) - z^{\top}(k)z(k) \quad \forall k\in \mathbb{Z}_{+}, $$
and by iteration, we have
$$ 0 \leqslant V(k) \leqslant \delta^{k} V(0) + \sum\limits_{s=0}^{k-1}\delta^{k-1-s}\left[\frac{\gamma}{\delta^{N}}\omega^{\top}(s)\omega(s)- z^{\top}(s)z(s)\right]. $$
(15)
Under the zero initial condition: V(0) = 0, as a result, inequality (15) implies
$$\sum\limits_{s=0}^{k-1}\delta^{k-1-s}z^{\top}(s)z(s)\; \leqslant\; \sum\limits_{s=0}^{k-1}\delta^{k-1-s}\frac{\gamma}{\delta^{N}}\omega^{\top}(s)\omega(s). $$
For k = N + 1, we have
$$ \sum\limits_{s=0}^{N}\delta^{N-s}z^{\top}(s)z(s)\; \leqslant\; \gamma\sum\limits_{s=0}^{N}\frac{\delta^{N-s}}{\delta^{N}}\omega^{\top}(s)\omega(s). $$
(16)
Since 1 ≤ δNsδNs ∈ {0, 1, … , N}, (16) immediately yields
$$\sum\limits_{s=0}^{N}z^{\top}(s)z(s)\; \leqslant\; \gamma\sum\limits_{s=0}^{N}\omega^{\top}(s)\omega(s),$$
which proves condition (4). This completes the proof of the theorem. □
In the sequel, we will solve the problem of finite-time H control for system (1), i.e., we will design a state feedback controller u(k) = Kx(k) such that the resulting closed-loop system
$$\begin{array}{@{}rcl@{}} x(k+1) &=& (A+BK)x(k) + A_{d}x(k-d(k)) + G\omega(k), \\ z(k) &=& Cx(k)+C_{d}x(k-d(k)),\quad k\in \Bbb Z_{+},\\ x(k) &=& \varphi(k),\ k\in\{-d_{2},-d_{2}+1,\dots,0\}, \end{array} $$
(17)
is robustly finite-time H bounded.

Theorem 3.2

The finite-time Hcontrol of system (1) has a solution if for a scalar δ > 1, there exist symmetric positive definite matrices U, V, W1, W2, W3, and a matrix Y such that the following LMIs hold:
$$\begin{array}{@{}rcl@{}} &&\; U < W_{2}, \quad V < W_{3}, \end{array} $$
(18)
$$\begin{array}{@{}rcl@{}}&&\left[\begin{array}{ccccc} -\delta U + (d_{2}-d_{1}+1)V & 0 & 0 & UA^{\top}+Y^{\top}B^{\top} & UC^{\top} \\ * & -\delta^{d_{1}}V & 0 & UA_{d}^{\top} & UC_{d}^{\top} \\ * & * & -\frac{\gamma}{\delta^{N}}I & G^{\top} & 0 \\ * & * & * & -U & 0 \\ * & * & * & * & - I \end{array}\right] < 0, \end{array} $$
(19)
$$\begin{array}{@{}rcl@{}} &&\left[\begin{array}{ccc} -W_{1} & c_{1}\delta^{N+1}W_{2} & \rho W_{3} \\ * & -c_{1}\delta^{N+1}W_{2} & 0 \\ * & * & -\rho W_{3} \end{array}\right] < 0, \end{array} $$
(20)
$$\begin{array}{@{}rcl@{}} &&\left[\begin{array}{ccc} W_{1} - c_{2}\delta U & \gamma dUR \\ * & -\gamma dR \end{array}\right] < 0. \end{array} $$
(21)
Moreover, the state feedback controller is given by
$$u(k) = YU^{-1}x(k),\quad k\in\mathbb{Z}_{+}.$$

Proof

Evidently, from Theorem 3.1 system (17) is robustly finite-time H bounded if for a scalar δ > 1, there exist symmetric positive definite matrices P, Q, positive scalars λ1, λ2, λ3 such that LMIs (5)-(7) hold, therein matrix A+BK will be in place of the matrix A. In other words, in proportion to LMI (6), we have
$$\begin{array}{@{}rcl@{}} \left[\begin{array}{ccccc} -\delta{P} + (d_{2}-d_{1}+1)Q & 0 & 0 & (A+BK)^{\top}P & C^{\top} \\ * & -\delta^{d_{1}}Q & 0 & A_{d}^{\top}P & C_{d}^{\top} \\ * & * & -\frac{\gamma}{\delta^{N}}I & G^{\top}P & 0 \\ * & * & * & -P & 0 \\ * & * & * & * & - I \end{array}\right] < 0. \end{array} $$
(22)
Pre- and post-multipling (22) by matrix diag {P−1, P−1, I, P−1, I} > 0 yields
$$\begin{array}{@{}rcl@{}} \left[\begin{array}{ccccc} -\delta P^{-1} + (d_{2}-d_{1}+1)P^{-1}QP^{-1} & 0 & 0 & P^{-1}(A+BK)^{\top} & P^{-1}C^{\top} \\ * & -\delta^{d_{1}}P^{-1}QP^{-1} & 0 & P^{-1}A_{d}^{\top} & P^{-1}C_{d}^{\top} \\ * & * & -\frac{\gamma}{\delta^{N}}I & G^{\top} & 0 \\ * & * & * & -P^{-1} & 0 \\ * & * & * & * & - I \end{array}\right] < 0. \end{array} $$
(23)
Let us define new matrix variables as follows: U = P−1, V = P−1QP−1. Then (23) becomes
$$\left[\begin{array}{ccccc} -\delta U + (d_{2}-d_{1}+1)V & 0 & 0 & U(A+BK)^{\top} & UC^{\top} \\ * & -\delta^{d_{1}}V & 0 & UA_{d}^{\top} & UC_{d}^{\top} \\ * & * & -\frac{\gamma}{\delta^{N}}I & G^{\top} & 0 \\ * & * & * & -U & 0 \\ * & * & * & * & - I \end{array}\right] < 0. $$
Letting Y=UK, K = YU−1, we get LMI (19). For getting LMI (20), we note that the inequality (7) can be regarded as
$$\begin{array}{@{}rcl@{}} \left[\begin{array}{ccc} (\gamma d - c_{2}\delta\lambda_{1})I & c_{1}\delta^{N+1}\lambda_{2}I & \rho\lambda_{3}I \\ * & -c_{1}\delta^{N+1}\lambda_{2}I & 0 \\ * & * & -\rho\lambda_{3} I \end{array}\right] < 0. \end{array} $$
(24)
Post-multipling (24) by the matrix diag {R, R, R} > 0 gives
$$\begin{array}{@{}rcl@{}} \left[\begin{array}{cccc} \gamma dR - c_{2}\delta\lambda_{1}R & c_{1}\delta^{N+1}\lambda_{2} R & \rho\lambda_{3} R \\ * & -c_{1}\delta^{N+1}\lambda_{2} R & 0 \\ * & * & -\rho\lambda_{3} R \end{array}\right] < 0. \end{array} $$
(25)
Again, pre- and post-multiplying (25) by the matrix diag {P−1, P−1, P−1} > 0, we get
$$\begin{array}{@{}rcl@{}} \left[\begin{array}{ccc} \gamma dP^{-1}RP^{-1} - c_{2}\delta P^{-1}(\lambda_{1}R)P^{-1} & c_{1}\delta^{N+1}P^{-1}(\lambda_{2} R)P^{-1} & \rho P^{-1}(\lambda_{3} R)P^{-1} \\ * & -c_{1}\delta^{N+1}P^{-1}(\lambda_{2} R)P^{-1} & 0 \\ * & * & -\rho P^{-1}(\lambda_{3} R)P^{-1} \end{array}\right] < 0. \end{array} $$
(26)
Setting new variables
$$W_{1}= -\gamma dP^{-1}RP^{-1} + c_{2}\delta P^{-1}(\lambda_{1}R)P^{-1}, W_{2}=P^{-1}(\lambda_{2} R)P^{-1}, W_{3}= P^{-1}(\lambda_{3} R)P^{-1},$$
LMI (26) reduces to LMI (20) as desired. To obtain LMI (18), we just pre- and post-multiply (5) by the matrix P−1. Finally, note that
$$W_{1} = -\gamma dP^{-1}RP^{-1} + c_{2}\delta P^{-1}(\lambda_{1}R)P^{-1} < -\gamma dURU + c_{2}\delta U, $$
we obtain
$$W_{1} - c_{2}\delta U + \gamma dUR[\gamma dR]^{-1}\gamma dRU < 0, $$
which is obviously equivalent to LMI (21) by Proposition 2.1. The proof of the theorem is complete. □

Remark 3.1

Being different from the previous results [6, 8, 11, 12], the Lyapunov function method is not used for the proof of Theorem 3.1. All the sufficient conditions of Theorem 3.1 and Theorem 3.2 are given in terms of LMIs, which can be easily calculated by the LMI Toolbox in MATLAB [4].

Remark 3.2

In the papers [5, 6, 8, 11, 12], additional unknowns and free-weighting matrices are introduced to make the flexibility to solve the resulting LMIs. However, too many unknowns and free-weighting matrices employed in the existing methods makes the system analysis complicate and significantly increases the computational demand. Compared with the free matrix method used in these papers, our simpler uncorrelated augmented matrix method uses fewer variables, e.g., LMI (6) has no free-weighting matrices, LMI (19) has one free-weighting matrix. Consequently, our conditions are less conservative in comparison with others. This effectiveness of the results will be illustrated by the following examples.

Example 3.1

Consider linear discrete-time delay system (1) with u(k) = 0 and its parameters are given by
$$\begin{array}{@{}rcl@{}} A&=&\left[\begin{array}{ccc} -0.25 \quad 0.1\\ 0.2 \quad 0.3 \end{array}\right], \qquad A_{d}=\left[\begin{array}{ccc} -0.12 \quad 0.1\\ 0.15 \quad 0.1 \end{array}\right], \qquad G=\left[\begin{array}{ccc} 0.2 \quad 0.1\\ 0.2 \quad 0.25 \end{array}\right],\\ C&=&\left[\begin{array}{ccc} 0.1 \quad -0.2\\ -0.15 \quad 0.15 \end{array}\right], \qquad C_{d}=\left[\begin{array}{ccc} -0.1 \quad 0.25\\ 0.2 \quad -0.15 \end{array}\right],\qquad R = I, \\ &d(k)& = 2 + 8\sin^{2}\frac{k\pi}{2},~ k\in \Bbb Z_{+}. \end{array} $$
Note that the delay function d(k) is interval time-varying and d1 = 2, d2 = 10. For given N = 200, d = 1, c1 = 1, c2 = 7, and γ = 1, by using LMI Control Toolbox in MATLAB, LMIs (5)–(7) are feasible with δ = 1.0001 and
$$\begin{array}{@{}rcl@{}} P&=&\left[\begin{array}{ccc} 2.1225 & 0.0261\\ 0.0261 & 2.0246 \end{array}\right],\quad Q=\left[\begin{array}{ccc} 0.1901 & -0.0194\\ -0.0194 & 0.1548 \end{array}\right],\\ &\lambda_{1}& = 2.0180,\quad \lambda_{2} = 2.1292,\quad \lambda_{3} = 0.1987. \end{array} $$

By Theorem 3.1, the system is robustly finite-time H bounded w.r.t. (1,7, I, 200).

Example 3.2

Consider system (1) where
$$\begin{array}{@{}rcl@{}} &A&=\left[\begin{array}{ccc} 0.4 & 0.1 \\ 0.3 & 0.5 \end{array}\right],\quad A_{d}=\left[\begin{array}{ccc} 0.2 & -0.15 \\ 0.15 & 0.1 \end{array}\right], \quad B=\left[\begin{array}{ccc} 0.1 \\ 0.2 \end{array}\right],\quad G=\left[\begin{array}{ccc} 0.25 \\ 0.3 \end{array}\right],\\ &C&=\left[\begin{array}{ccc} 0.2 & 0.3 \end{array}\right],\quad C_{d}=\left[\begin{array}{ccc} 0.2 & 0.15 \end{array}\right],\qquad\quad R = \left[\begin{array}{ccc} 1.2 & 0 \\ 0 & 1.3 \end{array}\right],\\ &&d(k) = 2 + 10\cos^{2}\frac{k\pi}{2},~ k\in \Bbb Z_{+}. \end{array} $$
For given d1 = 2, d2 = 12, N = 140, d = 1, c1 = 2, c2 = 16, and γ = 1, the LMIs (18)–(21) are feasible with δ = 1.00027 and
$$\begin{array}{@{}rcl@{}} &U = \left[\begin{array}{cccc} 0.2836 & 0.1250\\ 0.1250 & 0.3413 \end{array}\right],&\quad V=\left[\begin{array}{ccc} 0.0217 & 0.0118\\ 0.0118 & 0.0274 \end{array}\right], \quad W_{1}=\left[\begin{array}{cccc} 4.3514 & 1.9538\\ 1.9538 & 5.2522 \end{array}\right],\\ &W_{2} =\left[\begin{array}{cccc} 0.3417 & 0.0824\\ 0.0824 & 0.3739 \end{array}\right],&\quad W_{3}=\left[\begin{array}{ccc} 0.0222 & 0.0114\\ 0.0114 & 0.0276 \end{array}\right], \quad Y=\left[\begin{array}{cccc} -0.9638 & -1.2080 \end{array}\right]. \end{array} $$
By Theorem 3.2, the finite-time H control problem of system (1) has a solution and the state feedback controller is given by
$$u(k) = \left[\begin{array}{ccc} -2.1922 & -2.7365 \end{array}\right] x(k),\quad k\in\mathbb{Z}_{+}.$$
Figure 1 shows the response solution with the initial condition
$$\varphi(k) = \left[\begin{array}{ll} 0.5 \\ 1.0 \end{array}\right] \quad \forall k\in\{-12,-11,\dots,0\}.$$
Fig. 1

Response solution of the closed-loop system

4 Conclusion

In this paper, the finite-time stability and H control have been investigated for a class of linear discrete-time systems subjected to interval time-varying delay and norm-bounded disturbances. By constructing a set of improved Lyapunov-like functionals, sufficient conditions for robust finite-time stability of the system are proposed. Starting from these results, we have provided sufficient conditions for the solution of the finite-time H control of the system. The proposed conditions are expressed in terms of LMIs. Numerical examples are presented to illustrate the effectiveness of the proposed results.

Notes

Acknowledgments

This work was completed when the second author was visiting the Vietnam Institute for Advance Study in Mathematics (VIASM). He would like to gratefully acknowledge VIASM for the support and hospitality. This work was supported by the National Foundation for Science and Technology Development of Vietnam (NAFOSTED- grant 101.01-2014.35). The authors also thank anonymous referees for their valuable comments and suggestions, which allowed us them to improve the paper.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015

Authors and Affiliations

  1. 1.Department of Mathematics, College of SciencesHue UniversityHueVietnam
  2. 2.Institute of Mathematics, VASTHanoiVietnam

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