Dissipativity Analysis of Linear State/Input Delay Systems

Abstract

This paper discusses dissipativity problem for system of linear state/input delay equations. Motivated by dissipativity theory of control systems, we choose a new quadratic supply rate. Using the concept of dissipativity, necessary and sufficient conditions for the linear state/input delay systems to be dissipative and exponentially dissipative are derived. The connection of dissipativity with stability is also considered. Finally, passivity and finite gain are explored, correspondingly. The positive-real and bounded-real lemmas are derived.

1. Introduction

Dissipativity theory is one of major issues in control system theory. It was developed by Willems [1] and further extended by Hill and Moylan [2–4]. And it has become one of the major approaches to study complex systems [1, 5, 6]. In some particular systems with certain physical meaning, for example, electrical network systems, physical system, and so forth, dissipativity, just as the name implies, shows that the energy stored inside the system is no more than energy supplied from outside the system. In the area of dynamical system theory, storage function and supply rate are introduced to express the generalized abstract energy, correspondingly.

For dissipative systems, the storage functions usually provide natural candidates for Lyapunov functions [6]. Therefore, in many cases, stability and stabilization problems can be solved once the dissipativity property is assured [7–14]. Dissipativity theory is applied to impulsive hybrid dynamical systems [8], discontinuous dynamical systems [9], and switched systems [10]. Also, dissipativity theory is an efficient tool for the analysis and design of composite systems. Especially, it is used to deal with some problems of stability and stabilization for interconnected systems [5, 6, 15]. Passivity and finite gain are two special cases of dissipativity which are important in the stability analysis of dynamical systems. The fact that system is passive is equivalent to that system satisfies positive-real or Kalman-Yakubovich-Popov (KYP) condition [12]. Many versions of positive-real lemmas are given for nonlinear [16, 17] and linear systems [18, 19]. The results about ℒ₂-gain are summarized in recent references [15, 20, 21]. In papers [22, 23], bounded real lemmas are presented and are applied to ℋ_∞ control.

On the other hand, time delay is often encountered in various practical systems [24]. The existence of delays is often one main reason of instability and poor performance of systems. A great number of results on the stability and stabilization problems of control systems with time delays have been reported in the literature (see [25, 26], and the references therein). Lyapunov-Krasovskii functional approach is usually used to cope with various problems [27]. Dissipative theory about time-delay systems is discussed in references [28–31]. Sufficient conditions that guarantee robustly stable and strictly passive for uncertain time-delay systems are derived [28]. Dissipativity of linear and nonlinear time-delay systems is, respectively, discussed in [29, 30]. Dissipative problems of singular time-delay systems are considered in [31]. Unlike the systems discussed in the literatures, we consider a special class of linear state/input delay systems. This paper considers the dissipative problem of linear state/input delay systems with a new supply rate and with a storage function substituted by Lyapunov-Krasovskii functional.

There is no systematic dissipativity theory about time-delay systems because of the existence of delays. We consider linear state/input delay systems in this paper. The cause is not only that it has a simply form but it contains state delay and input delay. Motivated by dissipative theory, the variants of supply rate contain not input and output but delay input. A new quadratic supply rate developed in our paper is a function of input, delay input, and output. Based on it, we give sufficient and necessary conditions of dissipativity and exponential dissipativity and show its relationship to dissipativity and stability of the system. Finally, we discuss passivity and ℒ₂-gain and derive position-real lemma and bounded-real lemma for the system.

Notation.
𝐑^𝑛 denotes the 𝑛-dimensional Euclidean space, and ‖⋅‖ stands for the Euclidean norm of any vector. 𝐑^𝑛×𝑚 is the set of 𝑛×𝑚 real matrices, and 𝜆(⋅), 𝜆_max(⋅), 𝜆_min(⋅) denote, respectively, the eigenvalues, and the maximum eigenvalues, the minimum eigenvalues of any square matrix. We use 𝑊>0 to denote a symmetric positive-definite matrix. In a symmetric block matrix, the expression ∗ will be used to denote the submatrices that lie under the diagonal. Given a positive number 𝑟>0, 𝒞_𝑟=𝒞([0,𝑟],𝐑^𝑛) denotes the Banach space of continuous functions that maps the interval [0,𝑟] into 𝐑^𝑛 with the topology of uniform convergence,and ‖⋅‖_𝑐 denotes the norm of continuous function, that is, for 𝜙∈𝒞_𝑟, ‖𝜙‖_𝑐=sup_{0≤𝜏≤𝑟}‖𝜙(𝜏)‖. The space of square integrable functions is denoted by ℒ₂. And, for any 𝑔(𝑡)∈ℒ₂[𝑡₀,∞), its norm is defined as 𝑔(𝑡)=^∞_𝑡0𝑔(𝑡)²𝑑𝑡^1/2. (1.1)

The rest of the paper is organized as follows. Problem formulation is presented in Section 2. We give some results about dissipativity analysis of linear state/input delay systems in Section 3. In Section 4, positive-real lemma and bounded-real lemma are derived. Finally, a brief conclusion is provided to summarize the paper.

2. Problem Formulation

Consider the continuous-time systems with a state/input delay𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝐴₂𝑥(𝑡−𝑑)+𝐵₁𝑢(𝑡)+𝐵₂(),𝑦(𝑡)𝑡−𝑑=𝐶₁𝑥(𝑡)+𝐶₂(),𝑥(𝜃)(𝜃)𝑡𝑡−𝑑=𝜑,𝜃∈₀−𝑑,𝑡₀, (2.1) where 𝑥(𝑡)∈𝒳⊂𝐑^𝑛 is the state vector, 𝑥(𝑡−𝑑)=𝑥_𝑡∈𝒞_𝑑 is the delay state vector, 𝑢(𝑡)∈𝒰⊂𝐑^𝑚 is exogenous input, where 𝒰 is the set of admissible input, 𝑦(𝑡)∈𝒴⊂𝐑^𝑝 is the measured output vector. 𝐴₁, 𝐴₂, 𝐵₁, 𝐵₂, 𝐶₁, and 𝐶₂ are constant system matrices with appropriate dimensions. 𝑑>0 is an unknown constant delay. 𝜑(𝑡)∈𝒞_𝑑 which is a real-valued continuous function on [𝑡₀−𝑑,𝑡₀] with the topology of uniform convergence is the initial function. For any continuous initial function 𝜑(𝑡) and any admissible input 𝑢(𝑡), there exists a unique solution 𝑥(𝑡,𝑡₀,𝜑,𝑢) to system (2.1) on [𝑡₀,∞) which continuously depends on the initial data. If 𝑡≥𝑡₀, we denote the segment of trajectory by 𝑥_𝑡(𝜑,𝑢)𝑥_𝑡𝜑,𝑢∶𝜃⟶𝑥𝑡−𝑡₀+𝜃,𝜑,𝑢,𝜃∈𝑡₀−𝑑,𝑡₀. (2.2)

Assume that a real-valued function 𝑤 about input and output variants is the supply rate of system (2.1). We assume that for all admissible input and for any 𝑡₀,𝑡∈𝐑⁺, the output 𝑦(𝑡)=𝑦(𝑡,𝑡₀,𝜑,𝑢) of (2.1) is such that 𝑤(𝑡) satisfies the follwing:^𝑡_𝑡0𝑤(𝜏)𝑑𝜏<∞,∀𝑡>𝑡₀. (2.3)

Definition 2.1. The linear time-delay system (2.1) with supply rate 𝑤 is said to be dissipative if there exists a nonnegative functional 𝑆∶𝒳×𝒞_𝑑→𝐑⁺, called the storage functional, such that for all 𝑡,𝑡₀∈𝐑⁺ (𝑡≥𝑡₀), (𝑥,𝑥_𝑡)∈𝒳×𝒞_𝑑, and 𝑢∈𝒰, 𝑠𝑥,𝑥_𝑡−𝑠𝑥₀,𝑥_𝑡0≤^𝑡_𝑡0𝑤(𝜏)𝑑𝜏, (2.4) where 𝑥=𝑥(𝑡,𝑡₀,𝜑,𝑢), and 𝑤(𝑡) is a function about input and output, with 𝑦=𝑦(𝑡₀,𝜑,𝑢).

Remark 2.2. The inequality (2.4) is called the dissipation inequality. The dissipation inequality implies that the increase in generalized system energy over a given time interval cannot exceed the generalized energy supply delivered to the system (2.1) during this time interval. Thus, some of the supplied generalized energy to the system is stored, and some is dissipative. The dissipated energy is nonnegative and equals the difference of the right-hand of dissipation inequality and the left-hand.

Remark 2.3. If using equality sign as a substitution for inequality sign in the dissipation inequality, then Definition 2.1 of dissipativity becomes the definition of losslessness. We call system (2.1) lossless with respect to supply rate 𝑤.

The next definition shows the notion of available storage, which is the maximum amount of storage energy which may be extracted from a dynamical system.

Definition 2.4. The available storage, 𝑠_𝑎, of system (2.1) with supply rate 𝑤 is the functional 𝑠_𝑎∶𝒳×𝒞_𝑑→𝐑 defined by 𝑠_𝑎𝑥,𝑥_𝑡=−inf_{𝑢∈𝒰,𝑡≥𝑡0}^𝑡_𝑡0𝑤(𝜏)𝑑𝜏. (2.5)

Note that the available storage, whenever defined, is nonnegative, that is, for all (𝑥,𝑥_𝑡)∈𝒳×𝒞_𝑑, 𝑠_𝑎(𝑥,𝑥_𝑡)≥0. The available storage plays an important role in determining whether a given system is dissipative or not. This is shown in the following proposition (see [1]).

Proposition 2.5. The available storage, 𝑠_𝑎, is finite for all (𝑥,𝑥_𝑡)∈𝒳×𝒞_𝑑 if and only if system (2.1) with respect to 𝑤 is dissipative. Moreover, any possible storage function 𝑠 satisfies 0≤𝑠_𝑎≤𝑠 and if 𝑠_𝑎 is continuous, then 𝑠_𝑎 itself is a possible storage function.

Since supply rate is a function of input and output and system (2.1) contains delay input, the delay input should be considered as an argument of supply rate. The supply rate should be functional about input, delay input, and output, that is, 𝑤(𝑡)=𝑤(𝑢(𝑡),𝑢(𝑡−𝑑),𝑦(𝑡)). Motivated by Reference [1, 2], we are interested in studying dissipativity of system (2.1) with the following new quadratic supply rate:𝑤𝑢(𝑡),𝑢(𝑡−𝑑),𝑦(𝑡)=𝑦^𝑇(𝑡)𝑇𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑆₁𝑢(𝑡)+2𝑦^𝑇(𝑡)𝑆₂𝑢(𝑡−𝑑)+𝑢^𝑇(𝑡)𝑅₁𝑢(𝑡)+𝑢^𝑇(𝑡−𝑑)𝑅₂𝑢(𝑡−𝑑)+2𝑢^𝑇(𝑡)𝑅₃𝑢(𝑡−𝑑), (2.6) where 𝑇∈𝐑^𝑝×𝑝, 𝑆_𝑖∈𝐑^𝑝×𝑚 (𝑖=1,2), and 𝑅_𝑗∈𝐑^𝑚×𝑚 (𝑗=1,2,3) are known constant matrices with 𝑇 and 𝑅_𝑖 being symmetric. Specially, two important special cases of dissipative system (2.1) are the following.(i)When 𝑤(𝑢(𝑡),𝑢(𝑡−𝑑),𝑦(𝑡))=𝑦^𝑇(𝑡)𝑢(𝑡)+𝑦^𝑇(𝑡)𝑢(𝑡−𝑑), 𝑚=𝑝, dissipative system (2.1) is also called passive system.(ii)When 𝑤(𝑢(𝑡),𝑢(𝑡−𝑑),𝑦(𝑡))=𝛾²(𝑢^𝑇(𝑡)𝑢(𝑡)+u^𝑇(𝑡−𝑑)𝑢(𝑡−𝑑))−𝑦(𝑡)𝑦^𝑇(𝑡), 𝛾>0, dissipative system (2.1) is also called finite gain system.

Remark 2.6. Generally speaking, the storage functional 𝑠 is nonnegative and needs not to satisfy differentiability. Lyapunov-Krasovskii functional 𝑉(𝑥,𝑥_𝑡) can be chosen instead of 𝑠 because it is an energy functional representing the abstract storage energy of system. If choose 𝑠=𝑉(𝑥,𝑥_𝑡), then the dissipation inequality becomes 𝑉𝑥,𝑥_𝑡−𝑉𝑥₀,𝑥_𝑡0≤^𝑡_𝑡0𝑤(𝜏)𝑑𝜏. (2.7) If functional 𝑉 is continuous differentiable, then the integral version of dissipation inequality (2.7) can be transformed into a differential version. Differentiating both sides of (2.7), we can get the following: 𝑑𝑉𝑑𝑡𝑥,𝑥_𝑡≤𝑤𝑢(𝑡),𝑢(𝑡−𝑑),𝑦(𝑡). (2.8) In this paper, we assume that 𝑠 has the following form of Lyapunov-Krasovskii functional: 𝑉𝑥_𝑡=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^𝑡_𝑡−𝑑𝑥^𝑇(𝜏)𝑄𝑥(𝜏)𝑑𝜏, (2.9) where 𝑃, 𝑄 are positive-definite matrices.
Before concluding this section, we introduce two definitions which will be used in the development of our results.

Definition 2.7. The linear time-delay system (2.1) is reachable from the origin, if for any given 𝑥₁ and 𝑡₁, there exist 𝑡₂≤𝑡₁ and an admissible control 𝑢(𝑡)∈𝒰 such that the state can be driven from 𝑥(𝑡₁)=𝑥₁ to 𝑥(𝑡₂)=0.

Remark 2.8. We suppose that the system (2.1) is reachable from the origin.

Definition 2.9. The linear time-delay system (2.1) with 𝑢(𝑡)≡0 and 𝑢(𝑡−𝑑)≡0 is zero-state detectable, if for any trajectory 𝑥(𝑡,𝑡₀,𝜑,0), there holds 𝑦≡0⟹lim_𝑡→∞𝑥𝑡,𝑡₀,𝜑,0=0. (2.10)

3. Dissipativity Analysis

Notice that dissipativity defined according to the Definition 2.1 represents an input-output property of the system. The following theorem, which is the central result of our paper, shows that dissipativity can also be characterized in terms of the coefficient matrices of system (2.1).

Theorem 3.1. System (2.1) is said to be dissipative with respect to supply rate (2.6) if and only if there exist positive definite matrices 𝑃, 𝑄 and constant matrices 𝐿_𝑖∈𝐑^𝑛×𝑛, 𝑊_𝑖∈𝐑^𝑛×𝑚 (𝑖=1,2) such that 𝐿^𝑇₁𝐿₁=−𝑃𝐴₁−𝐴^𝑇₁𝑃−𝑄+𝐶^𝑇₁𝑇𝐶₁,𝐿^𝑇₂𝐿₂=𝑄+𝐶^𝑇₂𝑇𝐶₂,𝐿^𝑇₁𝐿₂=−𝑃𝐴₂+𝐶^𝑇₁𝑇𝐶₂,𝐿^𝑇₁𝑊_𝑖=−𝑃𝐵_𝑖+𝐶^𝑇₁𝑆_𝑖,𝐿^𝑇₂𝑊_𝑖=𝐶^𝑇₂𝑆_𝑖,𝑊^𝑇₁𝑊₂=𝑅₃,𝑊^𝑇_𝑖𝑊_𝑖=𝑅_𝑖. (3.1)

Proof. Sufficiency. Suppose that 𝑉(𝑥_𝑡) of form (2.9), 𝐿_𝑖 and 𝑊_𝑖 are given such that (3.1) is satisfied. Then, for any admissible control 𝑢 and any 𝑡₀ and 𝜑, we consider the following performance index: 𝑑𝒥=𝑉𝑥𝑑𝑡^𝑡−𝑤(𝑡). (3.2)
Using (3.1) and (2.9), we can get the following: 𝑑𝒥=𝑉𝑥𝑑𝑡^𝑡−𝑦^𝑇(𝑡)𝑇𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑆₁𝑢(𝑡)+2𝑦^𝑇(𝑡)𝑆₂𝑢(𝑡−𝑑)+𝑢^𝑇(𝑡)𝑅₁𝑢(𝑡)+𝑢^𝑇(𝑡−𝑑)𝑅₂𝑢(𝑡−𝑑)=𝑥^𝑇(𝑡)𝑃𝐴₁+𝐴^𝑇₁𝑃𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐴₂𝑥(𝑡−𝑑)+2𝑥^𝑇(𝑡)𝑃𝐵₁𝑢(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐵₂𝑢(𝑡−𝑑)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇(𝑡−𝑑)𝑄𝑥(𝑡−𝑑)−𝑥^𝑇(𝑡)𝐶^𝑇₁𝑇𝐶₁𝑥(𝑡)−2𝑥^𝑇(𝑡)𝐶^𝑇₁𝑇𝐶₂𝑥(𝑡−𝑑)−𝑥^𝑇(𝑡−𝑑)𝐶^𝑇₂𝑇𝐶₂𝑥(𝑡−𝑑)−2𝑥^𝑇(𝑡)𝐶^𝑇₁𝑆₁𝑢(𝑡)−2𝑥^𝑇(𝑡−𝑑)𝐶^𝑇₂𝑆₂𝑢(𝑡)−2𝑥^𝑇(𝑡)𝐶^𝑇₁𝑆₂𝑢(𝑡−𝑑)−2𝑥^𝑇(𝑡−𝑑)𝐶^𝑇₂𝑆₂𝑢(𝑡−𝑑)−𝑢^𝑇(𝑡)𝑅₁𝑢(𝑡)−𝑢^𝑇(𝑡−𝑑)𝑅₂𝑢(𝑡−𝑑)=𝑥^𝑇(𝑡)𝑃𝐴₁+𝐴^𝑇₁𝑃+𝑄−𝐶^𝑇₁𝑇𝐶₁𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐴₂−𝐶^𝑇₁𝑇𝐶₂𝑥(𝑡−𝑑)+2𝑥^𝑇(𝑡)𝑃𝐵₁−𝐶^𝑇₁𝑆₁𝑢(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐵₂−𝐶^𝑇₂𝑆₁𝑢(𝑡−𝑑)+𝑥^𝑇(𝑡−𝑑)−𝑄−𝐶^𝑇₂𝑇𝐶₂𝑥(𝑡−𝑑)+2𝑥^𝑇(𝑡−𝑑)−𝐶^𝑇₂𝑆₁𝑢(𝑡)+2𝑥^𝑇(𝑡−𝑑)−𝐶^𝑇₂𝑆₂𝑢(t−𝑑)+𝑢^𝑇(𝑡)−𝑅₁𝑢(𝑡)+2𝑢^𝑇(𝑡−𝑑)−𝑅₂𝑢(𝑡−𝑑)+2𝑢^𝑇(𝑡)−𝑅₃𝑢()𝐿𝑡−𝑑≤−₁𝑥(𝑡)+𝐿₂𝑥(𝑡−𝑑)+𝑊₁𝑢(𝑡)+𝑊₂𝑢(𝑡−𝑑)^𝑇𝐿₁𝑥(𝑡)+𝐿₂𝑥(𝑡−𝑑)+𝑊₁𝑢(𝑡)+𝑊₂𝑢(𝑡−𝑑)≤0. (3.3) Obviously, 𝑑𝑉𝑥𝑑𝑡^𝑡≤𝑤(𝑡). (3.4) Integrating both sides of (3.4) from 𝑡₀ to 𝑡, it yields 𝑉𝑥_𝑡−𝑉𝑥_𝑡0≤^𝑡_𝑡0𝑤(𝜏)𝑑𝜏. (3.5)Necessity. Suppose that system (2.1) is dissipative with respect to the supply rate (2.6). We will show that the available storage, 𝑉_𝑎, is a solution of (3.1) for some appropriate matrices 𝐿_𝑖 and 𝑊_𝑖. Due to (2.6) and Definition 2.4, it follows from Proposition 2.5 that the available storage 𝑉_𝑎 of system (2.1) is finite for all 𝑥_𝑡∈𝒞_𝑑, 𝑉_𝑎(0)=0 and satisfies that 𝑉_𝑎𝑥_𝑡−𝑉_𝑎𝑥_𝑡0≤^𝑡_𝑡0𝑤(𝜏)𝑑𝜏, (3.6) for all 𝑡≥𝑡₀ and 𝑢∈𝒰.
Differentiating both sides of the above inequality, 𝑑𝑉𝑑𝑡^𝑎𝑥_𝑡≤𝑤𝑢(𝑡),𝑢(𝑡−𝑑),𝑦(𝑡). (3.7) Defining a functional 𝑑∶𝒳×𝒞_𝑑×𝒰→𝐑 by 𝑑(𝑥(𝑡)()(𝑡)(𝑑,𝑥𝑡−𝑑,𝑢,𝑢𝑡−𝑑))=−𝑉𝑑𝑡^𝑎𝑥_𝑡+𝑤𝑢(𝑡),𝑢(𝑡−𝑑),𝑦(𝑡). (3.8) From (3.7), 𝑑(𝑥(𝑡),𝑥(𝑡−𝑑),𝑢(𝑡),𝑢(𝑡−𝑑))≥0. In addition, it is known from (3.8) that 𝑑(𝑥(𝑡),𝑥(𝑡−𝑑),𝑢(𝑡),𝑢(𝑡−𝑑)) is quadratic in 𝑥(𝑡), 𝑥(𝑡−𝑑), 𝑢(𝑡) and 𝑢(𝑡−𝑑).
Thus there exist constant matrices 𝐿_𝑖∈𝐑^𝑛×𝑛, and 𝑊_𝑖∈𝐑^𝑛×𝑚 (𝑖=1,2) such that 𝑑(𝑥(𝑡),𝑥(𝑡−𝑑),𝑢(𝑡),𝑢(𝑡−𝑑))=𝐿₁𝑥(𝑡)+𝐿₂𝑥(𝑡−𝑑)+𝑊₁𝑢(𝑡)+𝑊₂𝑢(𝑡−𝑑)^𝑇⋅𝐿₁𝑥(𝑡)+𝐿₂𝑥(𝑡−𝑑)+𝑊₁𝑢(𝑡)+𝑊₂𝑢(𝑡−𝑑). (3.9) Substituting (3.9) into (3.8), we have −𝑑𝑉𝑑𝑡^𝑎𝑥_𝑡+𝑦^𝑇(𝑡)𝑇𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑆₁𝑢(𝑡)+2𝑦^𝑇(𝑡)𝑆₂𝑢(𝑡−𝑑)+𝑢^𝑇(𝑡)𝑅₁𝑢(𝑡)+𝑢^𝑇(𝑡−𝑑)𝑅₂𝑢(𝑡−𝑑)+2𝑢^𝑇(𝑡)𝑅₃𝑢()=𝐿𝑡−𝑑₁𝑥(𝑡)+𝐿₂𝑥(𝑡−𝑑)+𝑊₁𝑢(𝑡)+𝑊₂𝑢(𝑡−𝑑)^𝑇⋅𝐿₁𝑥(𝑡)+𝐿₂𝑥(𝑡−𝑑)+𝑊₁𝑢(𝑡)+𝑊₂𝑢(𝑡−𝑑) (3.10) for all (𝑥,𝑥_𝑡)∈𝒳×𝒞_𝑑 and 𝑢(𝑡)∈𝒰.
Now, let 𝑉(𝑥_𝑡)=𝑉_𝑎(𝑥_𝑡). Equating coefficients of equal powers, (3.1) is obtained.
Theorem 3.1 provides a dissipative criterion for system (2.1). Theorem 3.1 can be represented by linear matrix inequality (LMI). The Corollary 3.2 shows a LMI’s version of dissipative criterion.

Corollary 3.2. System (2.1) is said to be dissipative with respect to supply rate (2.6) if and only if there exist positive definite matrices 𝑃 and 𝑄 such that Σ₁𝑃𝐴₂−𝐶^𝑇₁𝑇𝐶₂𝑃𝐵₁−𝐶^𝑇₁𝑆₁𝑃𝐵₂−𝐶^𝑇₁𝑆₂∗−𝑄−𝐶^𝑇₂𝑇𝐶₂−𝐶^𝑇₂𝑆₁−𝐶^𝑇₂𝑆₂∗∗−𝑅₁−𝑅₃∗∗∗−𝑅₂≤0, (3.11) where Σ₁=𝑃𝐴₁+𝐴^𝑇₁𝑃+𝑄−𝐶^𝑇₁𝑇𝐶₁.

The theory of dissipative systems is used to investigate stability of system via Lyapunov methods. It is shown in the following theorem.

Theorem 3.3. Suppose that system (2.1) is dissipative with smooth storage functional (2.9) with respect to the supply rate (2.6) and zero-state detectable, then the unforced system 𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝐴₂𝑥(𝑡−𝑑) is Lyapunov stable if 𝑇≤0 and asymptotically stable if 𝑇<0.

Proof. Suppose that system (2.1) is dissipative, according to the proof part of Theorem 3.1, and (3.10) is satisfied. Let 𝑢(𝑡)=𝑢(𝑡−𝑑)=0, we have 𝑑𝑉𝑥𝑑𝑡^𝑡=−𝐿₁𝑥(𝑡)+𝐿₂𝑥(𝑡−𝑑)^𝑇𝐿₁𝑥(𝑡)+𝐿₂𝑥(𝑡−𝑑)+𝑦^𝑇(𝑡)𝑇𝑦(𝑡). (3.12)
If 𝑇≤0, then (𝑑/𝑑𝑡)𝑉(𝑥_𝑡)≤0. The unforced system is Lyapunov stable (see [22, 23]).
If 𝑇<0, then (𝑑/𝑑𝑡)𝑉(𝑥_𝑡)<0. By zero-state detectable and LaSalle’s invariance principle (see [22, 23]), the unforced system is asymptotically stable.

Next, we give some results of exponential dissipativity. First, the definition of exponential dissipativity is represented.

Definition 3.4. The linear time-delay system (2.1) with supply rate 𝑤 is said to be exponentially dissipative if there exists a nonnegative storage functional 𝑆∶𝒳×𝒞_𝑑→𝐑⁺ and a constant 𝜀>0, such that for all 𝑡,𝑡₀∈𝐑⁺ (𝑡≥𝑡₀), (𝑥,𝑥_𝑡)∈𝒳×𝒞_𝑑, and 𝑢∈𝒰, 𝑒^𝜀𝑡𝑠𝑥,𝑥_𝑡−𝑒^𝜀𝑡0𝑠𝑥₀,𝑥_𝑡0≤^𝑡_𝑡0𝑒^𝜀𝜏𝑤(𝜏)𝑑𝜏, (3.13) where 𝑥=𝑥(𝑡,𝑡₀,𝜑,𝑢) and 𝑤(𝑡)=𝑤(𝑢(𝑡),𝑢(𝑡−𝑑),𝑦(𝑡)), with 𝑦=𝑦(𝑡₀,𝜑,𝑢).

Remark 3.5. When we choose 𝜀=0, the definition of exponential dissipativity becomes the definition of dissipativity.

Assume that 𝑠 has the following form of Lyapunov-Krasovskii functional:𝑉𝑥_𝑡=𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+^𝑡_𝑡−𝑑𝑥^𝑇(𝜏)𝑒^{𝜀(𝜏−𝑡)}𝑄𝑥(𝜏)𝑑𝜏, (3.14) where 𝑃, 𝑄 are positive definite matrices, and 𝜀 is a positive constant.

Theorem 3.6. System (2.1) is said to be exponentially dissipative with respect to supply rate (2.6) if and only if there exist positive definite matrices 𝑃, 𝑄, constant matrices 𝐿_𝑖∈𝐑^𝑛×𝑛, 𝑊_𝑖∈𝐑^𝑛×m (𝑖=1,2) and a constant 𝜀>0 such that 𝐿^𝑇₁𝐿₁=−𝑃𝐴₁−𝐴^𝑇₁𝑃−𝜀𝑃−𝑄+𝐶^𝑇₁𝑇𝐶₁,𝐿^𝑇₂𝐿₂=𝑒^−𝜀𝑑𝑄+𝐶^𝑇₂𝑇𝐶₂,𝐿^𝑇₁𝐿₂=−𝑃𝐴₂+𝐶^𝑇₁𝑇𝐶₂,𝐿^𝑇₁𝑊_𝑖=−𝑃𝐵_𝑖+𝐶^𝑇₁𝑆_𝑖,𝐿^𝑇₂𝑊_𝑖=𝐶^𝑇₂𝑆_𝑖,𝑊^𝑇₁𝑊₂=𝑅₃,𝑊^𝑇_𝑖𝑊_𝑖=𝑅_𝑖. (3.15)

Proof. Using (3.14), there is 𝑑𝑒𝑑𝑡^𝜀𝑡𝑉𝑥_𝑡−𝑒^𝜀𝑡0𝑉𝑥_𝑡0−^𝑡_𝑡0𝑒^𝜀𝜏𝑤(𝜏)𝑑𝜏=𝑒^𝜀𝑡𝑑𝑉𝑥𝑑𝑡^𝑡+𝜀𝑒^𝜀𝑡𝑉𝑥_𝑡−𝑒^𝜀𝑡𝑤(𝑡)=𝑒^𝜀𝑡𝑑𝑉𝑥𝑑𝑡^𝑡+𝜀𝑉𝑥_𝑡−𝑤(𝑡). (3.16) Let performance index be 𝒥₁=𝑑𝑉𝑥𝑑𝑡^𝑡+𝜀𝑉𝑥_𝑡−𝑤(𝑡). (3.17) Note that 𝑒^𝜀𝑡≥1. So if 𝒥₁≤0, we can directly obtain that 𝑒^𝜀𝑡𝑑𝑉𝑥𝑑𝑡^𝑡+𝜀𝑉𝑥_𝑡−𝑤(𝑡)≤0. (3.18) Substituting (𝑑/𝑑𝑡)𝑉(𝑥_𝑡) and 𝑉(𝑥_𝑡) into 𝒥₁, it yields the following: 𝒥₁=𝑥^𝑇(𝑡)𝑃𝐴₁+𝐴^𝑇₁𝑃𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐴₂𝑥(𝑡−𝑑)+2𝑥^𝑇(𝑡)𝑃𝐵₁𝑢(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐵₂𝑢(𝑡−𝑑)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥^𝑇(𝑡−𝑑)𝑒^−𝜀𝑑𝑄𝑥(𝑡−𝑑)−𝜀^𝑡_𝑡−𝑑𝑥^𝑇(𝜏)𝑒^{𝜀(𝜏−𝑡)}𝑄𝑥(𝜏)𝑑𝜏+𝜀𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝜀^𝑡_𝑡−𝑑𝑥^𝑇(𝜏)𝑒^{𝜀(𝜏−𝑡)}𝑄𝑥(𝜏)𝑑𝜏−𝑤(𝑡)=𝑥^𝑇(𝑡)𝑃𝐴₁+𝐴^𝑇₁𝑃+𝑄+𝜀𝑃𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐴₂𝑥(𝑡−𝑑)+2𝑥^𝑇(𝑡)𝑃𝐵₁𝑢(𝑡)+2𝑥^𝑇(𝑡)𝑃𝐵₂𝑢(𝑡−𝑑)−𝑥^𝑇(𝑡−𝑑)𝑒^−𝜀𝑑𝑄𝑥(𝑡−𝑑)−𝑤(𝑡). (3.19) The sufficiency and necessity proof is similar to the corresponding part of Theorem 3.1. For the sake of brevity, we omit it.

We show the result of Theorem 3.6 by LMI.

Corollary 3.7. System (2.1) is said to be exponentially dissipative with respect to supply rate (2.6) if and only if there exist positive definite matrices 𝑃, 𝑄 and a constant 𝜀>0 such that Σ₂𝑃𝐴₂−𝐶^𝑇₁𝑇𝐶₂𝑃𝐵₁−𝐶^𝑇₁𝑆₁𝑃𝐵₂−𝐶^𝑇₁𝑆₂∗−𝑒^−𝜀𝑑𝑄−𝐶^𝑇₂𝑇𝐶₂−𝐶^𝑇₂𝑆₁−𝐶^𝑇₂𝑆₂∗∗−𝑅₁−𝑅₃∗∗∗−𝑅₂≤0, (3.20) where Σ₂=𝑃𝐴₁+𝐴^𝑇₁𝑃+𝜀𝑃+𝑄−𝐶^𝑇₁𝑇𝐶₁.

Similar to the result about the effect of dissipativity on the stability of unforced system, we will show that the exponentially dissipative system which satisfies that the property of zero-state detectability is exponentially stable.

Theorem 3.8. Suppose that system (2.1) is exponentially dissipative with smooth storage functional (3.14) with respect to the supply rate (2.6) and zero-state detectable, then the unforced system 𝑥̇(𝑡)=𝐴₁𝑥(𝑡)+𝐴₂𝑥(𝑡−𝑑) is exponentially stable if 𝑇≤0.

Proof. According to the conditions of Theorem 3.8 and the proof of Theorem 3.6, we can get the following: 𝑒^𝜀𝑡𝑑𝑉𝑥𝑑𝑡^𝑡+𝜀𝑉𝑥_𝑡=−𝐿₁𝑥(𝑡)+𝐿₂𝑥(𝑡−𝑑)^𝑇𝐿₁𝑥(𝑡)+𝐿₂𝑥(𝑡−𝑑)+𝑒^𝜀𝑡𝑦^𝑇(𝑡)𝑇𝑦(𝑡). (3.21) If 𝑇≤0, then 𝑑𝑉𝑥𝑑𝑡^𝑡𝜑,0+𝜀𝑉𝑥_𝑡𝜑,0≤0. (3.22) Integrating both sides from 𝑡₀ to 𝑡, it yields that 𝑉𝑥_𝑡𝜑,0≤𝑒^{−𝜀(𝑡−𝑡0)}𝑉𝜑,𝑡≥𝑡₀. (3.23) From (3.14), we get the following inequality: 𝛼₁𝑥𝑡,𝜑,0²≤𝑉𝑥_𝑡𝜑,0≤𝛼₂𝑥_𝑡𝜑,0²_𝒞, (3.24) where 𝛼₁=𝜆_min(𝑃) and 𝛼₂=𝜆_max(𝑃)+𝑑𝜆_max(𝑄).
By (3.23) and (3.24), we have 𝛼₁𝑥𝑡,𝜑,0²≤𝑉𝑥_𝑡𝜑,0≤𝑒^{−𝜀(𝑡−𝑡0)}𝑉𝜑≤𝛼₂𝑒^{−𝜀(𝑡−𝑡0)}𝜑²_𝒞. (3.25) Any solution 𝑥(𝑡,𝜑,0) of unforced system satisfies that 𝑥𝑡,𝜑,0≤𝛼₂𝛼₁𝑒^{−(1/2)𝜀(𝑡−𝑡0)}𝜑_𝒞,𝑡≥𝑡₀. (3.26) Thus, the unforced system is exponentially stable.

4. Positive-Real and Bounded-Real Conditions

4.1. Positive-Real Lemma

When 𝑚=𝑝, we shall focus on studying dissipative systems with supply rate given by the inner product of the following form;𝑤𝑢(𝑡),𝑢(𝑡−𝑑),𝑦(𝑡)=𝑦^𝑇(𝑡)𝑢(𝑡)+𝑦^𝑇(𝑡)𝑢(𝑡−𝑑). (4.1) For convenience, we characterize this choice by means of a separate definition.

Definition 4.1. The linear time-delay system (2.1) with supply rate (4.1) is said to be passive if it is dissipative with supply rate (4.1); that is, there exists a nonnegative storage functional 𝑆∶𝒳×𝒞_𝑑→𝐑⁺, such that for all 𝑡,𝑡₀∈𝐑⁺ (𝑡≥𝑡₀), (𝑥,𝑥_𝑡)∈𝒳×𝒞_𝑑, and 𝑢∈𝒰, 𝑠𝑥,𝑥_𝑡−𝑠𝑥₀,𝑥_𝑡0≤^𝑡_𝑡0𝑦^𝑇(𝜏)𝑢(𝜏)+𝑦^𝑇(𝜏)𝑢(𝜏−𝑑)𝑑𝜏. (4.2)

The following theorem is positive-real lemma (KYP lemma) for linear time-delay system (2.1).

Theorem 4.2. System (2.1) is said to be passive with respect to supply rate (4.1) if and only if there exist positive definite matrices 𝑃, 𝑄 and constant matrices 𝐿_𝑖∈𝐑^𝑛×𝑛 (𝑖=1,2) such that 𝐿^𝑇₁𝐿₁=−𝑃𝐴₁−𝐴^𝑇₁𝑃−𝑄,𝐿^𝑇₂𝐿₂𝐿=𝑄,_𝑇1𝐿₂=−𝑃𝐴₂,−𝑃𝐵_𝑖+12𝐶^𝑇₁=0,𝐶₂=0. (4.3)

Proof. Let 𝑇=𝑅₁=𝑅₂=𝑅₃=0, and 𝑆₁=𝑆₂=(1/2)𝐼_𝑛 in Theorem 3.1. We can get 𝑊₁=𝑊₂=0. The result can be directly obtained.

We rewrite the positive-real condition of (4.3) in terms of LMI.

Corollary 4.3. System (2.1) is said to be passive with respect to supply rate (4.1) if and only if there exists positive definite matrices 𝑃 and 𝑄 such that Σ₃𝑃𝐴₂𝑃𝐵₁−12𝐶^𝑇₁𝑃𝐵₂−12𝐶^𝑇₁∗−𝑄00∗∗00∗∗∗0≤0, (4.4) where Σ₃=𝑃𝐴₁+𝐴^𝑇₁𝑃+𝑄.

4.2. Bounded-Real Lemma

In this subsection, we shall be particularly interested in the quadratic supply rate𝑤𝑢(𝑡),𝑢(𝑡−𝑑),𝑦(𝑡)=𝛾²𝑢^𝑇(𝑡)𝑢(𝑡)+𝑢^𝑇(𝑡−𝑑)𝑢(𝑡−𝑑)−𝑦(𝑡)𝑦^𝑇(𝑡), (4.5) where positive constant 𝛾>0 and 𝑢(𝑡)∈ℒ₂[𝑡₀,∞).

Definition 4.4. The system (2.1) is said to have finite ℒ₂-gain from 𝑢(𝑡) to 𝑦(𝑡) less than or equal to 𝛾>0, if it is dissipative with supply rate (4.5): that is, there exists a nonnegative storage functional 𝑆∶𝒳×𝒞_𝑑→𝐑⁺, such that for all (𝑡₀,𝑡)∈[0,∞)(𝑡≥𝑡₀), (𝑥,𝑥_𝑡)∈𝒳×𝒞_𝑑, and any 𝑢(𝑡)∈ℒ₂[𝑡₀,𝑡]𝑠𝑥,𝑥_𝑡−𝑠𝑥₀,𝑥_𝑡0≤^𝑡_𝑡0𝛾²𝑢^𝑇(𝜏)𝑢(𝜏)+𝑢^𝑇(𝜏−𝑑)𝑢(𝜏−𝑑)−𝑦(𝜏)𝑦^𝑇(𝜏)𝑑𝜏. (4.6)
When 𝑢(𝑡)=𝑢(𝑡−𝑑)=0, obviously, 𝑠(𝑥,𝑥_𝑡)−𝑠(𝑥₀,𝑥_𝑡0)≤−∫^𝑡_𝑡0𝑦(𝜏)𝑦^𝑇(𝜏)𝑑𝜏≤0. Inequality (4.6) can be rewritten as ^𝑡_𝑡0𝑦(𝜏)𝑦^𝑇(𝜏)𝑑𝜏≤^𝑡_𝑡0𝛾²𝑢^𝑇(𝜏)𝑢(𝜏)+𝑢^𝑇(𝜏−𝑑)𝑢(𝜏−𝑑)𝑑𝜏. (4.7)

The following theorem is bounded-real lemma for linear time-delay system (2.1).

Theorem 4.5. System (2.1) is said to have finite ℒ₂-gain from 𝑢(𝑡) to 𝑦(𝑡) less than or equal to 𝛾>0 with respect to supply rate (4.5) if and only if there exist positive definite matrices 𝑃, 𝑄 and constant matrices 𝐿_𝑖∈𝐑^𝑛×𝑛, 𝑊_𝑖∈𝐑^𝑛×𝑚 (𝑖=1,2) such that 𝐿^𝑇₁𝐿₁=−𝑃𝐴₁−𝐴^𝑇₁𝑃−𝑄−𝐶^𝑇₁𝐶₁,𝐿^𝑇₂𝐿₂=𝑄−𝐶^𝑇₂𝐶₂,𝐿^𝑇₁𝐿₂=−𝑃𝐴₂−𝐶^𝑇₁𝐶₂,𝐿^𝑇₁𝑊_𝑖=−𝑃𝐵_𝑖,𝐿^𝑇₂𝑊_𝑖=𝑊^𝑇₁𝑊₂=0,𝑊^𝑇_𝑖𝑊_𝑖=𝛾²𝐼. (4.8)

Proof. The result can be directly obtained from the consequence of Theorem 3.1 with 𝑇=−𝐼, 𝑅₁=𝑅₂=𝛾²𝐼, and 𝑆₁=𝑆₂=𝑅₃=0.

By LMI, Theorem 4.5 can be rewritten as the following corollary.

Corollary 4.6. System (2.1) is said to have finite ℒ₂-gain from 𝑢(𝑡) to 𝑦(𝑡) less than or equal to 𝛾>0 with respect to supply rate (4.5) if and only if there exist positive definite matrices 𝑃 and 𝑄 such that Σ₃𝑃𝐴₂+𝐶^𝑇₁𝐶₂𝑃𝐵₁𝑃𝐵₂∗−𝑄+𝐶^𝑇₂𝐶₂00∗∗−𝛾²𝐼0∗∗∗−𝛾²𝐼≤0, (4.9) where Σ₄=𝑃𝐴₁+𝐴^𝑇₁𝑃+𝑄−𝐶^𝑇₁𝐶₁.

Remark 4.7. The connections between passive property or finite gain property and unforced systems have not been shown in this section. They can be derived according to the results of Theorem 3.3 and Theorem 3.8.

5. Conclusion

This paper considers dissipativity theory for linear state/input delay systems. Motivated by reference [1, 2], we consider a new type quadratic supply rate for linear state/input delay systems. The choice of the new supply rate also depends on the case of system itself. Based on new supply rate, the necessary and sufficient conditions of dissipativity and exponential dissipativity are represented. And, stability results of dissipative linear state/input delay systems are derived. Finally, we focus on passive and finite ℒ₂-gain systems and discuss positive-real and bounded-real condition, correspondingly. Positive-real lemma and bounded-real lemma based on new supply rate may help us to analyze the problem of stabilization using output or state feedback for the system with or without external disturbances, that is a future direction of research. Interconnected dissipative delay systems are also considered in future works.

Acknowledgments

This work is supported by the Natural Science Foundation of China (60874006), the Foundation of Henan Educational Committee (12A120008, 2008B120001), the Outstanding Youth Foundation of Henan Province (084100510010), and the 2012 Key Technologies R&D Program of Henan Scientific Committee (122102210061).