Dissipative Output Tracking Control of Linear Systems with Time Delay

Abstract

The problem of dissipative output tracking control of linear systems with time delay is investigated. Firstly, an augmented system is constructed to describe dissipative output tracking control error, and the concept of dissipative output tracking is defined. Based on this, some sufficient conditions are derived in terms of linear matrix inequalities (LMIs) technique, which ensure that the augmented system is dissipative and stable; then design methods of dissipative output tracking state-feedback controller are provided, and the desired controller gain can be expressed through the solutions of LMIs. Finally, a numerical simulation example is given to illustrate the validity of the proposed results.

1. Introduction

Output tracking control is one of the most fundamental and important control problems in control theory and applications. The essence of output tracking control is to enable controlled output to approach reference output as closely as possible. Output tracking control has been applied successfully to many fields, such as aircraft [1], robot [2–4], and biomathematics [5]. Meanwhile, many control methods have been developed in this area to analyze and control systems. The authors in [6] have studied the adaptive asymptotic tracking problems for a class of large-scale systems. Many results on the fuzzy adaptive tracking control problems have been obtained in [7–9]. By using a sliding-mode controller/observer scheme, the authors in [10] have solved robust output tracking problems. In [11], the authors have investigated output tracking problem for uncertain nonlinear systems. Robust output tracking control problem has been discussed for time-delay nonlinear systems by neural network in [12]. 𝐻_∞ output tracking control has been studied by T-S fuzzy model approach in [13]. [14] has studied 𝐻_∞ output tracking problems for network-based control systems. The chaotic tracking control has been first investigated for SIR epidemic model in [15]; a tracking control method is given to control chaotic motion in the epidemic model, and the corresponding state feedback controllers are designed such that the density of infected individuals can converge to zero. In [16], the tracking control of chaos is studied for singular biological economy systems; a controller is designed which guarantees that the output of chaos system can track an expected constant value or period orbit. In order to eliminate the chaos or Neimark-Sacker bifurcation of the discrete epidemic model, a tracking controller is designed in [17] such that the disease disappears gradually. Different new results about output tracking control have been obtained in these literatures which provide us significant and useful control approaches.

Notion of dissipativity was introduced by Willems in 1970s [18, 19], which can be regarded as a generalization of 𝐻_∞ performance as well as positive realness performances and passivity. Essentially, the practical interpretation of dissipativity is that there exists a nonnegative storage function such that energy consumption rate is less than a given energy supply rate. The representative researches of dissipative theory for linear systems can be found in [20–22]. However, to the best of our knowledge, there are no research on output tracking control problems from the perspective of dissipativity. All of these motivate us to achieve the current study.

In this paper, dissipative output tracking control problem is formulated for linear time-delay systems. From the view of dissipativity, the process of output tracking control can be regarded as the process of energy dissipation. Dissipative output tracking control generalizes some previous results about output tracking control, including 𝐻_∞ output tracking control, whose concept is first presented for linear time-delay systems in this paper. Then, some sufficient conditions are derived such that the considered systems are dissipative and stable by employing LMIs technique. Following these results, dissipative output tracking controllers are designed which also ensure the resultant closed-loop systems to be stable. Compared with some existing results, this paper provides a new way to solve output tracking control problems for linear time-delay systems.

This paper is organized as follows. Dissipative output tracking control problem is formulated in Section 2. By using PI control method, the output tracking control problem of the original system is transformed into the dissipative analysis problem of an augmented system. In Section 3, some sufficient conditions are explored for the augmented system with and without time delay to be dissipative, and then the corresponding dissipative output tracking controllers are designed. In Section 4, we bring a demonstrative example to show the effectiveness of the proposed method. Some conclusions are drawn and some potential research interests are stated in Section 5.

2. Problem Formulation

Consider the following linear time-delay system: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥()(𝑡),𝑦(𝑡)(𝑡)𝑡−𝑑+𝐵𝑢=𝐶𝑥+𝐶_𝑑𝑥()(𝑡),𝑥(𝑡)(𝑡)[],𝑡−𝑑+𝐷𝑢=𝜂,𝑡∈−𝑑,0 (1) where 𝑥(𝑡)∈ℝ^𝑛 is state vector, 𝑢(𝑡)∈ℝ^𝑚 is input vector, 𝑦(𝑡)∈ℝ^𝑙 is output vector, and the initial condition 𝜂(𝑡) is a continuously differentiable vector-valued function over [−𝑑,0]. 𝐴,  𝐴_𝑑,  𝐵,  𝐶,  𝐶_𝑑, and 𝐷 are real constant matrices with appropriate dimensions. Sometimes we denote this system by (𝐴,𝐴_𝑑,𝐵,𝐶,𝐶_𝑑,𝐷) for simplicity.

In practical world, it is often expected that the output vector 𝑦(𝑡) approaches a desired reference output vector 𝑦_ref(𝑡) asymptotically which may be produced from a reference system of the form 𝑥̇_ref(𝑡)=𝐴_ref𝑥_ref(𝑡)𝑦_ref(𝑡)=𝐶_ref𝑥_ref(𝑡), (2) where 𝑥_ref(𝑡)∈ℝ^𝑛 is reference state vector, 𝑦_ref(𝑡)∈ℝ^𝑙 is reference output vector. 𝐴_ref and 𝐶_ref are real constant matrices with appropriate dimensions. The response of this reference system exists uniquely for all given initial conditions.

To this end, it is required that lim_𝑡→∞𝑦(𝑡)−𝑦_ref(𝑡)=0. (3) Let 𝑒(𝑡)=𝑦(𝑡)−𝑦_ref(𝑡) be error vector, then (3) is equivalent to lim_𝑡→∞𝑒(𝑡)=0. Using the well-known PI control approach, we introduce the following error integrator model: 𝑞̇(𝑡)=𝑒(𝑡)=𝑦(𝑡)−𝑦_ref(𝑡)=𝐶𝑥(𝑡)+𝐶_𝑑𝑥(𝑡−𝑑)+𝐷𝑢(𝑡)−𝑦_ref(𝑡). (4)

From (1) and (4), the following augmented system is constructed: 𝑥̇_𝐿(𝑡)=𝐴_𝐿𝑥_𝐿(𝑡)+𝐴_𝐿𝑑𝑥_𝐿(𝑡−𝑑)+𝐵_𝐿𝑢(𝑡)−𝑁_𝐿𝑦_ref(𝑡)𝑒(𝑡)=𝐶_𝐿𝑥_𝐿(𝑡)+𝐶_𝐿𝑑𝑥_𝐿(𝑡−𝑑)+𝐷𝑢(𝑡)−𝑦_ref(𝑡)𝑥_𝐿(𝑡)=𝜂(𝑡)0,𝑡∈[−𝑑,0], (5) where 𝑥^T_𝐿(𝑡)=𝑥^T(𝑡)𝑞^T(𝑡) and 𝐴_𝐿=𝐴0𝐶0,𝐴_𝐿𝑑=𝐴_𝑑0𝐶_𝑑0,𝐵_𝐿=𝐵𝐷,𝑁_𝐿=0𝐼,𝐶_𝐿=𝐶0,𝐶_𝐿𝑑=𝐶_𝑑0. (6)

In order to make the output vector 𝑦(𝑡) approaches the reference output vector 𝑦_ref(𝑡) asymptotically, the following concept is introduced for system (5) from the view of dissipativity.

Definition 1. For system (5) with a given supply rate 𝑟(𝑦_ref(𝑡),𝑒(𝑡)), if there exists a nonnegative storage function 𝑉(𝑥(𝑡)) with 𝑉(0)=0 such that the following dissipative inequality 𝑉̇(𝑥(𝑡))≤𝑟𝑦_ref(𝑡),𝑒(𝑡),∀𝑡≥0 (7) holds for all the 𝑥(𝑡),  𝑦_ref(𝑡),  and 𝑒(𝑡), then system (5) is said to be dissipative. If the dissipative inequality (7) is strict, system (5) is said to be strictly dissipative. Furthermore, system (1) is said to be (strictly) dissipative output tracking in case of system (5) being (strictly) dissipative.

Introduce the supply rate function associated with system (5) which has the following quadratic form: 𝑟𝑦_ref(𝑡),𝑒(𝑡)=⟨𝑒(𝑡),𝑄𝑒(𝑡)⟩+2𝑒(𝑡),𝑆𝑦_ref(𝑡+𝑦_ref)(𝑡),𝑅𝑦_ref(𝑡), (8) where 𝑄=𝑄^T∈ℝ^𝑙×𝑙,  𝑅=𝑅^T∈ℝ^𝑙×𝑙,   and 𝑆∈ℝ^𝑙×𝑙 are real matrices with appropriate dimensions, and ⟨𝑎,𝑏⟩=𝑎^T𝑏,  for all 𝑎,𝑏∈ℝ^𝑙.

We first present the following lemmas which will be used later.

Lemma 2. Assume system that (𝐴,𝐵) is controllable. Then augmented system (𝐴_𝐿,𝐵_𝐿) is controllable if and only if rank𝐴𝐵𝐶𝐷=𝑛+𝑙. (9)

Proof. First let 𝑄_𝐿=𝐵_𝐿𝐴_𝐿𝐵_𝐿𝐴²_𝐿𝐵_𝐿⋯𝐴_𝐿^𝑛−1𝐵_𝐿. Then rank𝑄_𝐿=rank𝐵𝐴𝐵𝐴²𝐵⋯𝐴^𝑛−1𝐵𝐷𝐶𝐵𝐶𝐴𝐵⋯𝐶𝐴^𝑛−2𝐵=rank,𝐴𝐵𝐶𝐷0𝑄𝐼0 (10) where 𝑄=𝐵𝐴𝐵𝐴²𝐵⋯𝐴^𝑛−1𝐵. Therefore, we can conclude from rank𝑄=𝑛 that this result is true. This completes the proof.

Remark 3. It is noted that Lemma 2 gives a necessary and sufficient condition for the augmented system to be controllable. As we know, if a system is controllable, then the eigenvalues of the system can be assigned arbitrarily, which implies that the system can be stabilized. This lemma explains the relationship that, under appropriate condition, the stabilizability of the original system can ensure the stabilizability of the augmented system, which has a potential application in controller design.

Lemma 4 (Schur complement [23]). For a given symmetric matrix 𝑆∈ℝ^𝑛×𝑛, and 𝑆=^𝑆11𝑆12_{𝑆T12𝑆22}, where 𝑆_𝑖𝑗∈ℝ^{𝑛𝑖×𝑛𝑗} are matrix blocks with appropriate dimensions, the following statements are equivalent:(1)𝑆<0;(2)𝑆₁₁<0,  𝑆₂₂−𝑆^T₁₂𝑆⁻¹₁₁𝑆₁₂<0;(3)𝑆₂₂<0,  𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^T₁₂<0.

3. Main Results

Consider the following linear time-delay system: 𝑥̇_𝐿(𝑡)=𝐴_𝐿𝑥_𝐿(𝑡)+𝐴_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑁_𝐿𝑦_ref(𝑡)𝑒(𝑡)=𝐶_𝐿𝑥_𝐿(𝑡)+𝐶_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑦_ref(𝑡)𝑥_𝐿(𝑡)=𝜂(𝑡)0,𝑡∈[−𝑑,0], (11) where 𝑥_𝐿(𝑡)∈ℝ^𝑛 is augmented state vector, 𝑦_ref(𝑡)∈ℝ^𝑙 is reference output vector, 𝑢(𝑡)∈ℝ^𝑚 is input vector, 𝑒(𝑡)∈ℝ^𝑙 is error output vector, and 𝐴_𝐿,  𝐴_𝐿𝑑,  𝐶_𝐿,  𝐶_𝐿𝑑, and 𝑁_𝐿 are known constant matrices with appropriate dimensions.

Then, select the following Lyapunov functional as storage function 𝑉𝑥_𝐿(𝑡),𝑡=𝑥^T_𝐿(𝑡)𝑃𝑥_𝐿(𝑡)+^𝑡_𝑡−𝑑𝑥^T_𝐿(𝜏)𝑉𝑥_𝐿(𝜏)𝑑𝜏, (12) where 0≤𝑃∈ℝ^𝑛×𝑛,  0<𝑉∈ℝ^𝑛×𝑛.

Consider the supply rate (8), then for system (11), we have the following results.

Theorem 5. For the time-delay system (11) and the given supply rate (8), if the following LMIs 𝑃=𝑃^T𝐴≥0,𝑉>0,_T𝐿𝑃+𝑃𝐴_𝐿+𝑉−𝐶^T_𝐿𝑄𝐶_𝐿−𝑃𝑁_𝐿+𝐶^T_𝐿𝑄−𝐶^T_𝐿𝑆𝑃𝐴_𝐿𝑑−𝐶^T_𝐿𝑄𝐶_𝐿𝑑∗−𝑄+𝑆+𝑆^T−𝑅𝑄𝐶_𝐿𝑑−𝑆^T𝐶_𝐿𝑑∗∗−𝑉−𝐶^T_𝐿𝑑𝑄𝐶_𝐿𝑑≤0 (13) have feasible solutions 𝑃,  𝑉, then the linear time-delay system (11) is dissipative. Moreover, if LMIs (13) are strict, then system (11) is strictly dissipative; in this case when 𝑄<0, system (11) is also stable. The “∗” in LMIs denotes the terms that can be induced by symmetric block in a matrix.

Proof. Combine system (11) with the storage function (12), then the time derivative of 𝑉(𝑥_𝐿(𝑡),𝑡) along the trajectory of system (11) is given as 𝑉̇𝑥_𝐿(𝑡)=𝑥̇,𝑡_T𝐿(𝑡)𝑃𝑥_𝐿(𝑡)+𝑥^T_𝐿(𝑡)𝑃𝑥̇_𝐿(𝑡)=𝐴_𝐿𝑥_𝐿(𝑡)+𝐴_𝐿𝑑𝑥_𝐿(𝑡−𝑑)+𝑁_𝐿𝑦_ref(𝑡)^T𝑃𝑥_𝐿(𝑡)+𝑥^T_𝐿(𝑡)𝑃𝐴_𝐿𝑥_𝐿(𝑡)+𝐴_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑁_𝐿𝑦_ref(𝑡)+𝑥^T_𝐿(𝑡)𝑉𝑥_𝐿(𝑡)−𝑥^T_𝐿(𝑡−𝑑)𝑉𝑥_𝐿(𝑡−𝑑). (14)
Associated with the supply rate (8), we have 𝑉̇𝑥_𝐿(𝑡),𝑡−𝑟𝑦_ref(𝑡)(𝑡)=𝐴,𝑒_𝐿𝑥_𝐿(𝑡)+𝐴_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑁_𝐿𝑦_ref(𝑡)^T𝑃𝑥_𝐿(𝑡)+𝑥^T_𝐿(𝑡)𝑃𝐴_𝐿𝑥_𝐿(𝑡)+𝐴_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑁_𝐿𝑦_ref(𝑡)+𝑥^T_𝐿(𝑡)𝑉𝑥_𝐿(𝑡)−𝑥^T_𝐿(𝑡−𝑑)𝑉𝑥_L()−𝐶𝑡−𝑑_𝐿𝑥_𝐿(𝑡)+𝐶_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑦_ref(𝑡)^T×𝑄𝐶_𝐿𝑥_𝐿(𝑡)+𝐶_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑦_ref(𝑡)−2𝐶_𝐿𝑥_𝐿(𝑡)+𝐶_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑦_ref(𝑡)^T𝑆𝑦_ref(𝑡)−𝑦^T_ref(𝑡)𝑅𝑦_ref(𝑡)=𝐴_𝐿𝑥_𝐿(𝑡)+𝐴_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑁_𝐿𝑦_ref(𝑡)^T𝑃𝑥_𝐿(𝑡)+𝑥^T_𝐿(𝑡)𝑃𝐴_𝐿𝑥_𝐿(𝑡)+𝐴_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑁_𝐿𝑦_ref(𝑡)+𝑥^T_𝐿(𝑡)𝑉𝑥_𝐿(𝑡)−𝑥^T_𝐿(𝑡−𝑑)𝑉𝑥_𝐿()−𝐶𝑡−𝑑_𝐿𝑥_𝐿(𝑡)+𝐶_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑦_ref(𝑡)^T×𝑄𝐶_𝐿𝑥_𝐿(𝑡)+𝐶_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑦_ref(𝑡)−𝐶_𝐿𝑥_𝐿(𝑡)+𝐶_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑦_ref(𝑡)^T𝑆𝑦_ref(𝑡)−𝑦^T_ref(𝑡)𝑆^T𝐶_𝐿𝑥_𝐿(𝑡)+𝐶_𝐿𝑑𝑥_𝐿(𝑡−𝑑)−𝑦_ref(𝑡)−𝑦^T_ref(𝑡)𝑅𝑦_ref(𝑡)=𝑥_𝐿(𝑡)𝑦_ref(𝑡)𝑥_𝐿(𝑡−𝑑)^T𝐽𝑥_𝐿(𝑡)𝑦_ref(𝑡)𝑥_𝐿(),𝑡−𝑑 (15) where 𝐴𝐽=^T𝐿𝑃+𝑃𝐴_𝐿+𝑉−𝐶^T_𝐿𝑄𝐶_𝐿−𝑃𝑁_𝐿+𝐶^T_𝐿𝑄−𝐶^T_𝐿𝑆𝑃𝐴_𝐿𝑑−𝐶^T_𝐿𝑄𝐶_𝐿𝑑∗−𝑄+𝑆+𝑆^T−𝑅𝑄𝐶_𝐿𝑑−𝑆^T𝐶_𝐿𝑑∗∗−𝑉−𝐶^T_𝐿𝑑𝑄𝐶_𝐿𝑑. (16)
Therefore, if LMIs (13) have feasible solutions 𝑃 and 𝑉, it is from Definition 1 that linear time-delay system (11) is dissipative.
Next, we show the stability of system (11). To this end, assume that system (11) is strictly dissipative, then we separate 𝐽 into 𝐴𝐽=^T𝐿𝑃+𝑃𝐴_𝐿+𝑉−𝑃𝑁_𝐿+𝐶^T_𝐿𝑄−𝐶^T_𝐿𝑆𝑃𝐴_𝐿𝑑∗−𝑄+𝑆+𝑆^T−𝑅𝑄𝐶_𝐿𝑑−𝑆^T𝐶_𝐿𝑑+∗∗−𝑉−𝐶^T_𝐿𝑄𝐶_𝐿0−𝐶^T_𝐿𝑄𝐶_𝐿𝑑∗00∗∗−𝐶^T_𝐿𝑑Q𝐶_𝐿𝑑<0. (17) Note that −𝐶^T_𝐿𝑄𝐶_𝐿0−𝐶^T_𝐿𝑄𝐶_𝐿𝑑∗00∗∗−𝐶^T_𝐿𝑑𝑄𝐶_𝐿𝑑𝐶=−0^T𝐿𝐶^T_𝐿𝑑𝑄𝐶_𝐿0𝐶_𝐿𝑑. (18) This together with 𝑄<0 yields 𝐴𝐽=^T𝐿𝑃+𝑃𝐴_𝐿+𝑉−𝑃𝑁_𝐿+𝐶^T_𝐿𝑄−𝐶^T_𝐿𝑆𝑃𝐴_𝐿𝑑𝐶^T_𝐿∗−𝑄+𝑆+𝑆^T−𝑅Q𝐶_𝐿𝑑−𝑆^T𝐶_𝐿𝑑0∗∗−𝑉𝐶^T_𝐿𝑑∗∗∗𝑄⁻¹<0. (19) It is easy to see that the following LMI holds: ̂𝐴𝐽=^T𝐿𝑃+𝑃𝐴_𝐿+𝑉𝑃𝐴_𝐿𝑑∗−𝑉<0. (20) Then, by the Lyapunov stability theory in [24], it can be seen that system (11) is stable. This completes the proof.

In fact, when the time-delay is absent, the augmented system (11) reduces to the following autonomous system: 𝑥̇_𝐿(𝑡)=𝐴_𝐿𝑥_𝐿(𝑡)−𝑁_𝐿𝑦_ref(𝑡),𝑒(𝑡)=𝐶_𝐿𝑥_𝐿(𝑡)−𝑦_ref(𝑡). (21)

In this case, choose the Lyapunov function of the following form as the storage function 𝑉(𝑥(𝑡),𝑡)=𝑥^T(𝑡)𝑃𝑥(𝑡), (22) where 0≤𝑃∈ℝ^𝑛×𝑛.

Then, we have the following result for system (21).

Theorem 6. For system (21) and the given supply rate (8), if the following LMIs 𝑃=𝑃^T𝐴≥0,_T𝐿𝑃+𝑃𝐴_𝐿−𝐶^T_𝐿𝑄𝐶_𝐿−𝑃𝑁_𝐿+𝐶^T_𝐿𝑄−𝐶^T_𝐿𝑆−𝑁^T_𝐿𝑃+𝑄𝐶_𝐿−𝑆^T𝐶_𝐿−𝑄+𝑆+𝑆^T−𝑅≤0 (23) have a feasible solution 𝑃, then the tracking system (21) is dissipative. Moreover, if LMIs (23) are strict, then system (21) is strictly dissipative; in this case when 𝑄<0, system (21) is also stable.

Proof. For system (21), the time derivative of storage function 𝑉(𝑥_𝐿(𝑡),𝑡) along the trajectory of system (21) is given as 𝑉̇𝑥_𝐿(𝑡)=𝑥̇^T_𝐿(𝑡)𝑃𝑥_𝐿(𝑡)+𝑥^T_𝐿(𝑡)𝑃𝑥̇_𝐿(𝑡)=𝐴_𝐿𝑥_𝐿(𝑡)−𝑁_𝐿𝑦_ref(𝑡)^T𝑃𝑥_𝐿(𝑡)+𝑥^T_𝐿(𝑡)𝑃𝐴_𝐿𝑥_𝐿(𝑡)−𝑁_𝐿𝑦_ref(𝑡) (24) then we can know 𝑉̇𝑥_𝐿(𝑡)−𝑟𝑦_ref(𝑡)(𝑡)=𝐴,𝑒_𝐿𝑥_𝐿(𝑡)−𝑁_𝐿𝑦_ref(𝑡)^T𝑃𝑥_𝐿(𝑡)+𝑥^T_𝐿(𝑡)𝑃𝐴_𝐿𝑥_𝐿(𝑡)−𝑁_𝐿𝑦_ref(𝑡−𝐶_𝐿)𝑥_𝐿(𝑡)−𝑦_ref(𝑡)^T𝑄𝐶_𝐿𝑥_𝐿(𝑡)−𝑦_ref(𝑡)−2𝐶_𝐿𝑥_𝐿(𝑡)−𝑦_ref(𝑡)^T𝑆𝑦_ref(𝑡)−𝑦^T_ref(𝑡)𝑅𝑦_ref(𝑡)=𝐴_𝐿𝑥_𝐿(𝑡)−𝑁_𝐿𝑦_ref(𝑡)^T𝑃𝑥_𝐿(𝑡)+𝑥^T_𝐿(𝑡)𝑃𝐴_𝐿𝑥_𝐿(𝑡)−𝑁_𝐿𝑦_ref(𝑡)−𝐶_𝐿𝑥_𝐿(𝑡)−𝑦_ref(𝑡)^T𝑄𝐶_𝐿𝑥_𝐿(𝑡)−𝑦_ref(𝑡)−𝐶_𝐿𝑥_𝐿(𝑡)−𝑦_ref(𝑡)^T𝑆𝑦_ref(𝑡)−𝑦^T_ref(𝑡)𝑆^T𝐶_𝐿𝑥_𝐿(𝑡)−𝑦_ref(𝑡)−𝑦^T_ref(𝑡)𝑅𝑦_ref(𝑡)=𝑥_𝐿(𝑡)𝑦_ref(𝑡)^T×𝐴^T_𝐿𝑃+𝑃𝐴_𝐿−𝐶^T_𝐿𝑄𝐶_𝐿−𝑃𝑁_𝐿+𝐶^T_𝐿𝑄−𝐶^T_𝐿𝑆−𝑁^T_𝐿𝑃+𝑄𝐶_𝐿−𝑆^T𝐶_𝐿−𝑄+𝑆+𝑆^T×𝑥−𝑅_𝐿(𝑡)𝑦_ref(𝑡). (25) Therefore, if LMIs (23) have a feasible solution 𝑃, it is from Definition 1 that tracking system (21) is dissipative.
It is noted that when system (21) is strictly dissipative, we have 𝐴^T_𝐿𝑃+𝑃𝐴_𝐿−𝐶^T_𝐿𝑄𝐶_𝐿<0. (26) This together with 𝑄<0 implies that 𝐴^T_𝐿𝑃+𝑃𝐴_𝐿<0. (27) Then, by the Lyapunov stability theory in [24], it can be verified that system (21) is stable. This completes the proof.

Now, we consider the dissipative output tracking state-feedback controller for the linear time-delay system (5) with the following form: 𝑢(𝑡)=𝐾₁𝑥_𝐿(𝑡)+𝐾₂𝑥_𝐿(𝑡−𝑑)=𝐾₁₁𝑥(𝑡)+𝐾₁₂𝑞(𝑡)+𝐾₂₁𝑥(𝑡−𝑑)+𝐾₂₂𝑞(𝑡−𝑑), (28) where 𝐾₁=𝐾₁₁𝐾₁₂,  𝐾₂=𝐾₂₁𝐾₂₂ are the controller gains to be determined. Then the closed-loop system can be obtained by applying the above controller to system (5), which is described as 𝑥̇_𝐿(𝑡)=𝐴_𝐿+𝐵_𝐿𝐾₁𝑥_𝐿(𝑡)+𝐴_𝐿𝑑+𝐵_𝐿𝐾₂𝑥_𝐿(𝑡−𝑑)−𝑁_𝐿𝑦_ref(𝑡),𝑒(𝑡)=𝐶_𝐿+𝐷𝐾₁𝑥_𝐿(𝑡)+𝐶_𝐿𝑑+𝐷𝐾₂𝑥_𝐿(𝑡−𝑑)−𝑦_ref(𝑡),𝑥_𝐿(𝑡)=𝜂(𝑡)0=𝜂(𝑡),𝑡∈[−𝑑,0]. (29)

For the augmented system (5), we present a controller design method in the following theorem which guarantees that the closed-loop system (29) is both strictly dissipative and stable.

Theorem 7. For the linear time-delay tracking system (29) and the given supply rate (8) with 𝑄<0, if the following LMIs 𝑋=𝑋^TΞ>0₁₁Ξ₁₂Ξ₁₃𝐶_𝐿𝑋+𝐷𝑊₁^T∗Ξ₂₂Ξ₂₃0∗∗−𝑀𝐶_𝐿𝑑𝑋+𝐷𝑊₂^T∗∗∗𝑄⁻¹<0 (30) have feasible solutions 𝑊₁,  𝑊₂,  𝑋, and 𝑀, then the closed-loop system (29) is strictly dissipative and stable. Moreover, if 𝑊^∗₁,  𝑊^∗₂, and 𝑋^∗ are the feasible solutions of (30), then the desired state-feedback controller is 𝑢(𝑡)=𝐾₁𝑥_𝐿(𝑡)+𝐾₂𝑥_𝐿(𝑡−𝑑)=𝑊^∗₁𝑋^∗−1𝑥_𝐿(𝑡)+𝑊^∗₂𝑋^∗−1𝑥_𝐿(𝑡−𝑑). (31) The “∗” in LMIs denotes the terms that can be induced by symmetric block in a matrix, where Ξ₁₁=𝐴_𝐿𝑋+𝐵_𝐿𝑊₁+𝐴_𝐿𝑋+𝐵_𝐿𝑊₁^TΞ+𝑀,₁₂=−𝑁_𝐿+𝐶_𝐿𝑋+𝐷𝑊₁^T𝑄−𝐶_𝐿𝑋+𝐷𝑊₁^TΞ𝑆,₁₃=𝐴_𝐿𝑑𝑋+𝐵_𝐿𝑊₂,Ξ₂₂=−𝑄+𝑆+𝑆^TΞ−𝑅,₂₃=𝑄𝐶_𝐿𝑑𝑋+𝐷𝑊₂−𝑆^T𝐶_𝐿𝑑𝑋+𝐷𝑊₂. (32)

Proof. According to Theorem 5, for the given supply rate (8) and the closed-loop system (29), if the following LMIs 𝑃=𝑃^T>0𝐴^T_𝐿𝑃+𝑃𝐴_𝐿+𝑉−𝐶^T_𝐿𝑄𝐶_𝐿−𝑃𝑁_𝐿+𝐶^T_𝐿𝑄−𝐶^T_𝐿𝑆𝑃𝐴_𝐿𝑑−𝐶^T_𝐿𝑄𝐶_𝐿𝑑∗−Q+𝑆+𝑆^T−𝑅𝑄𝐶_𝐿𝑑−𝑆^T𝐶_𝐿𝑑∗∗−𝑉−𝐶^T_𝐿𝑑𝑄𝐶_𝐿𝑑<0 (33) have feasible solutions 𝑃, 𝑉, then system (29) is strictly dissipative and stable, where 𝐴_𝐿=𝐴_𝐿+𝐵_𝐿𝐾₁,𝐶_𝐿=𝐶_𝐿+𝐷𝐾₁,𝐴_𝐿𝑑=𝐴_𝐿𝑑+𝐵_𝐿𝐾₂,𝐶_𝐿𝑑=𝐶_𝐿𝑑+𝐷𝐾₂. (34)
Premultiply and postmultiply diag{𝑃⁻¹,𝐼,𝑃⁻¹} to the matrix inequality (33), and assume that 𝑋=𝑃⁻¹,  𝑊₁=𝐾₁𝑋, 𝑊₂=𝐾₂𝑋, 𝑀=𝑋𝑉𝑋, then using Schur complement lemma, the LMI (30), which is equivalent to the matrix inequality (33), can be obtained. Therefore, if the LMI (30) has feasible solutions 𝑊^∗₁,  𝑊^∗₂, and 𝑋^∗, then 𝑢(𝑡)=𝐾₁𝑥_𝐿(𝑡)+𝐾₂𝑥_𝐿(𝑡−𝑑)=𝑊^∗₁(𝑋^∗)⁻¹𝑥_𝐿(𝑡)+𝑊^∗₂(𝑋^∗)⁻¹𝑥_𝐿(𝑡−𝑑) is the desired dissipative state-feedback controller. This completes the proof.

Remark 8. It is noted that Theorem 7 provides a controller design method such that the resultant closed-loop system (29) is not only strictly dissipative but also stable, which can be more desirable to be obtained.

Remark 9. It should be pointed out that the designed controller of the form (28) is more general than some existing results. When 𝐾₂=0, the controller (28) reduces to the instant state feedback, while in case of 𝐾₁=0, it is just a retarded state feedback.

Similarly to Theorem 6, in the absence of time delay, the dissipative output tracking state-feedback controller (28) reduces to the controller below 𝑢(𝑡)=𝐾𝑥_𝐿(𝑡)=𝐾₁𝑥(𝑡)+𝐾₂𝑞(𝑡), (35) where 𝐾=𝐾₁𝐾₂ is the controller gain to be determined, then the closed-loop system (29) is simplified to the following system: 𝑥̇_𝐿(𝑡)=𝐴_𝐿+𝐵_𝐿𝐾𝑥_𝐿(𝑡)−𝑁_𝐿𝑦_ref(𝑡),𝑒(𝑡)=𝐶_𝐿+𝐷𝐾𝑥_𝐿(𝑡)−𝑦_ref(𝑡). (36)

Consequently, the following design method can be obtained.

Theorem 10. For system (36) and the given supply rate (8) with 𝑄<0, if the following LMIs 𝑋=𝑋^TΞ>0,₁₁Ξ₁₂𝐶_𝐿𝑋+𝐷𝑊^T−𝑄+𝑆+𝑆^T−𝑅0∗∗𝑄⁻¹<0 (37) have feasible solutions 𝑊, 𝑋, then the closed-loop system (36) is strictly dissipative and stable. If 𝑊^∗,  𝑋^∗ are the feasible solution of (37), then the desired state-feedback controller is 𝑢(𝑡)=𝐾𝑥_𝐿(𝑡)=𝑊^∗(𝑋^∗)⁻¹𝑥_𝐿(𝑡). The “∗” in LMIs denotes the terms that can be induced by symmetric block in a matrix, where Ξ₁₁=𝐴_𝐿𝑋+𝐵_𝐿𝑊+𝐴_𝐿𝑋+𝐵_𝐿𝑊^T,Ξ₁₂=−𝑁_𝐿+𝐶_𝐿𝑋+𝐷𝑊^T𝑄−𝐶_𝐿𝑋+𝐷𝑊^T𝑆. (38)

Proof. According to Theorem 6, for the given supply rate (8) and system (36), if LMIs 𝑃=𝑃^T>0𝐴^T_𝐿𝑃+𝑃𝐴_𝐿−𝐶^T_𝐿𝑄𝐶_𝐿−𝑃𝑁_𝐿+𝐶^T_𝐿𝑄−𝐶^T_𝐿𝑆−𝑁^T_𝐿𝑃+𝑄𝐶_𝐿−𝑆^T𝐶_𝐿−𝑄+𝑆+𝑆^T−𝑅<0 (39) have a feasible solution 𝑃, then system (36) is strictly dissipative and stable, where 𝐴_𝐿=𝐴_𝐿+𝐵_𝐿𝐾,𝐶_𝐿=𝐶_𝐿+𝐷𝐾. (40)
Pre-multiply and post-multiply the matrix inequality (39) with diag{𝑃⁻¹,𝐼}, and assume that 𝑋=𝑃⁻¹,  𝑊=𝐾𝑋. According to Schur complement lemma, LMI (37), which is equivalent to the matrix inequality (39), can be obtained. Therefore, if the LMI (37) has feasible solutions 𝑊^∗,  𝑋^∗, then 𝑢(𝑡)=𝐾𝑥_𝐿(𝑡)=𝑊^∗(𝑋^∗)⁻¹𝑥_𝐿(𝑡) is the dissipative state-feedback controller. This completes the proof.

4. Numerical Simulation

In order to demonstrate the applicability and effectiveness of the developed method, we present a numerical simulation example. Consider system (1) with the following data: 𝐴=−201−3,𝐴_𝑑=,𝐶0000,𝐶=1−1_𝑑=100,𝐵=−1,𝐷=3.2. (41) Then, from (5), we can have the augmented system with the following data: 𝐴_𝐿=−2001−301−10,𝐴_𝐿𝑑=,𝐵000000000_𝐿=1−13.2,𝐶_𝐿=1−10,𝐶_𝐿𝑑=,𝑁000_𝐿=001,𝐷=3.2, (42) and the parameters in supply rate are 𝑄=−1,𝑆=0,𝑅=1.056. (43) Solving the LMIs (37) in Theorem 10 by using the LMI toolbox in MATLAB, we can get ,.𝑋=0.27540.02380.10020.02380.1965−0.05010.1002−0.05010.3400𝑊=−0.07860.0540−0.3007 (44) Hence, the following dissipative state-feedback controller can be obtained: 𝑢(𝑡)=𝐾𝑥_𝐿(𝑡)=−0.03390.0440−0.8879𝑥_𝐿(𝑡). (45) Assume that the reference system has the following form: 𝑥̇_ref(𝑡)=−10𝑥_ref(𝑡)(𝑡)𝑦+𝑟_ref(𝑡)=0.5𝑥_ref(𝑡) (46)(1)choose 𝑟(𝑡)=5cos(𝑡), when 𝑥_𝐿(0)=000^T,  𝑥_ref(0)=0, output tracking trajectory is shown in Figure 1; when 𝑥_𝐿(0)=101^T,  𝑥_ref(0)=1, output tracking trajectory is shown in Figure 2;(2)choose 𝑟(𝑡)=0, when 𝑥_𝐿(0)=000^T,  𝑥_ref(0)=0, output tracking trajectory is shown in Figure 3; when 𝑥_𝐿(0)=101^T,  𝑥_ref(0)=1, output tracking trajectory is shown in Figure 4;(3)choose 𝑟(𝑡)=(sin(𝑡))²+cos(𝑡), when 𝑥_𝐿(0)=000^T, 𝑥_ref(0)=0, output tracking trajectory is shown in Figure 5; when 𝑥_𝐿(0)=101^T, 𝑥_ref(0)=1, output tracking trajectory is shown in Figure 6.

Figure 1 

System output tracking trajectory with zero initial condition.

Figure 2 

System output tracking trajectory with nonzero initial condition.

Figure 3 

System output tracking trajectory with zero initial condition.

Figure 4 

System output tracking trajectory with nonzero initial condition.

Figure 5 

System output tracking trajectory with zero initial condition.

Figure 6 

System output tracking trajectory with nonzero initial condition.

According to Figures 1–6, we find that the output vector can approach the reference output vector as closely as possible; it is obvious that our proposed method is very effective.

5. Conclusion

By the example and the theorems, we can draw the following conclusions. Compared with most output tracking control methods, this paper realizes the output tracking performance by stable and dissipative controllers. The theorems are computationally simple and practically effective. Meanwhile, there still exist many problems that need to be solved. The tracking performance should be further investigated, and in the presence of uncertainties, the robust dissipative control for the systems is also worthy to be studied.

Acknowledgment

This work was partially supported by The Natural Science Foundation of China under Grants nos. 60974004, 61273003, and 61273008, respectively.