Robust Adaptive Control and 𝐿₂ Disturbance Attenuation for Uncertain Hamiltonian Systems with Time Delay

Abstract

This paper deals with the robust stabilizability and 𝐿₂ disturbance attenuation for a class of time-delay Hamiltonian control systems with uncertainties and external disturbances. Firstly, the robust stability of the given systems is studied, and delay-dependent criteria are established based on the dissipative structural properties of the Hamiltonian systems and the Lyapunov-Krasovskii (L-K) functional approach. Secondly, the problem of 𝐿₂ disturbance attenuation is considered for the Hamiltonian systems subject to external disturbances. An adaptive control law is designed corresponding to the time-varying delay pattern involved in the systems. It is shown that the closed-loop systems under the feedback control law can guarantee the γ-dissipative inequalities be satisfied. Finally, two numerical examples are provided to illustrate the theoretical developments.

1. Introduction

Systems with unknown delayed states are often encountered in practice, such as communication systems, engineering systems, and process control systems. For this reason, robust stability analysis for uncertain time-delay control systems has attracted a considerable amount of interests in recent years [1–9]. The Lyapunov-Krasovskii (L-K) method is always employed, and the results are often obtained in the form of linear matrix inequalities (LMIs). However, robust stabilization of nonlinear systems with time delays has been a challenging problem. As is well known, the control design of nonlinear systems is a difficult process. The existence of time delay in nonlinear systems further degrades the control performance and sometimes makes the closed-loop stabilization difficult [10–12]. More recently, Mahmoud and El Ferik obtained some new results on dissipative analysis and state feedback synthesis for a class of nonlinear systems with time-varying delays and convex polytypic uncertainties [12]. This class consists of linear time-delay systems subject to nonlinear cone-bounded perturbations. Hu et al. in [10] integrated the sliding mode control method with the robust 𝐻_∞ technique and developed a discrete-time sliding mode controller for a class of time-delay uncertain systems with stochastic nonlinearities. The nonlinearities are described by statistical means.

On the other hand, for affine nonlinear systems with disturbances, the 𝐿₂-gain analysis and the 𝐿₂ disturbance attenuation are always important issues [13]. Almost all these studies deal with the existence of solutions to some partial differential inequality, which reflects the dissipative behavior of the system under consideration for a certain supply rate which is called passivity-based control design method. This kind of method is used to achieve a 𝛾-dissipative inequality which not only guarantees asymptotic stability but also renders the 𝐿₂-gain from disturbance to the penalty signal less than or equal to a given level 𝛾>0. The key to solve the problem of 𝐿₂ disturbance attenuation is to find a proper storage function that ensures the 𝛾-dissipative inequality holding.

As an important class of nonlinear systems, port-controlled Hamiltonian systems (PCH) proposed by [14, 15] have attracted increasing attentions in the field of nonlinear control theory [16–18]. The Hamilton function in a PCH system is considered as the sum of potential energy (excluding gravitational potential energy) and kinetic energy in physical systems, and it can be used as a good candidate of Lyapunov functions for many physical systems. Due to this and its nice structure with clear physical meaning, the PCH system has drawn a good deal of attention in practical control designs [19–23]. In [21], with a proper penalty signal, the 𝛾-dissipativity was achieved by making a sufficiently large damping injection in the design stage. Wang et al. in [23] proposed an energy-based adaptive 𝐿₂ disturbance attenuation control scheme for the power systems with superconducting magnetic energy storage (SMES) units. Besides, Hamiltonian systems with time delay also have been studied [24–27]. Reference [25] addresses the stabilization problem of a class of Hamiltonian systems with state time delay and input saturation. The problem of 𝐿₂-disturbance attenuation for time-delay port-controlled Hamiltonian systems is studied in [26]. The case that there are time-invariant uncertainties belonging to some convex bounded polytypic domains is also considered in [26], and an 𝐿₂ disturbance attenuation control law is proposed. In practice, dynamic uncertainties often arise from many different control engineering applications. The inevitable uncertainties may enter a nonlinear system in a much more complex way. In addition to polytypic uncertainties, systems may encounter modeling error, parameter perturbations, and external disturbances. However, to the best of our knowledge, the analysis and synthesis for time-delay Hamiltonian systems with parametric perturbations have not been discussed yet. It is well worth pointing out that with the help of Hamiltonian realization [28, 29], the control problem of a large class of time-delay nonlinear systems with uncertainties can be solved via the Hamiltonian system framework. Thus, study of time-delay Hamiltonian control systems with uncertainties and disturbances is a meaningful topic.

Motivated by the above observations, in this paper we study a class of time-delay Hamiltonian systems model with uncertainties and external disturbances. We derive sufficient condition for which the uncertain time-delay Hamiltonian system along with the proposed feedback controller is robustly stable for all admissible uncertainties. The condition is given in terms of linear matrix inequalities. Furthermore, the problem of 𝐿₂ disturbance attenuation is examined using the parametric adaptive methodology for delay-dependent case. The 𝐿₂ feedback adaptive control law can guarantee that the closed-loop time-delay Hamiltonian system is asymptotically stable and the 𝐿₂ performance is achieved. The effectiveness of the proposed methods in this paper is illustrated by numerical examples.

The paper is organized as follows. Section 2 presents the problem formulation and some preliminaries. The main results are proposed in Section 3. Section 4 illustrates the obtained results by several numerical examples, which is followed by the conclusion in Section 5.

Notations. ℝ^𝑛 denotes the 𝑛-dimension Euclidean space, and ℝ^𝑛×𝑚 is the real matrices with dimension 𝑛×𝑚; ‖⋅‖ stands for either the Euclidean vector norm or the induced matrix 2-norm; ‖𝑥‖_𝒞=max_{𝑡−ℎ⩽𝜑⩽𝑡}‖𝑥(𝜑)‖, where 𝒞=𝒞([−ℎ,0],ℝ^𝑛) denotes the Banach space of continuous functions mapping the interval [−ℎ,0] into ℝ^𝑛; 𝐿^𝑛₂[0,∞) denotes the set of all measurable functions 𝑥∶[0,∞)→ℝ^𝑛 that satisfy ∫^∞₀|𝑥(𝑡)|²𝑑𝑡<∞. ℂ^𝑖 denotes the set of all functions with continuous 𝑖th partial derivatives. The notation 𝑋⩾𝑌 (resp., 𝑋>𝑌) where 𝑋 and 𝑌 are symmetric matrices means that the matrix 𝑋−𝑌 is positive semidefinite (resp., positive definite); 𝜆_max(𝐴) and 𝜆_min(𝐴) denote the maximum and the minimum of eigenvalue of a real symmetric matrix 𝐴. The notation ∗ represents the elements below the main diagonal of a symmetric matrix; 𝐴^T denotes the transposed matrix of 𝐴; (⋅)^′ and [⋅]^′ denote the derivative of the variable inside the brackets. What is more, for the sake of simplicity, throughout the paper, we denote 𝜕𝐻/𝜕𝑥 by ∇𝐻.

2. Problem Statement and Preliminaries

Consider the following class of time-delay Hamiltonian systems with parametric uncertainties and external disturbances: 𝑥̇(𝑡)=𝐽+𝐽𝑥,𝑝−𝑅𝑥,𝑝∇𝐻𝑥,𝑝_∗𝑥_𝜏−𝑅^∗𝑥_𝜏∇𝐻𝑥_𝜏+𝑔₁𝑢(𝑡)+𝑔₂𝜔(𝑡), (1) where 𝑥(𝑡)∈ℝ^𝑛 is the state; 𝑥_𝜏∶=𝑥(𝑡−𝑑(𝑡))∈𝒞 stands for the delayed state; 𝑢∈ℝ^𝑠 is the control input; 𝜔∈𝐿^𝑚₂[0,∞) is the disturbance input; 𝐻(𝑥)∶ℝ^𝑛→ℝ is the Hamilton function which satisfies 𝐻(𝑥)⩾0, 𝐻(0)=0; 𝑝 is an unknown constant vector and denotes the disturbance parameter; 𝐽(𝑥,𝑝),𝐽^∗(𝑥_𝜏)∈ℝ^𝑛×𝑛 are skew-symmetric structure matrices; 𝑅(𝑥,𝑝),𝑅^∗(𝑥_𝜏)∈ℝ^𝑛×𝑛 are positive semidefinite symmetric matrices; 𝑔₁ and 𝑔₂ are gain matrices of appropriate dimensions; 𝑔₁𝑔^T₁ is nonsingular.

The delay 𝑑(𝑡) is a time-varying continuous function which satisfies (𝑡)𝑑̇(𝑡)0⩽𝑑⩽ℎ,⩽𝜇<1, (2) where the bounds ℎ and 𝜇 are known positive scalars.

The initial condition is 𝑥(𝑡)=𝜙(𝑡), 𝑡∈[−ℎ,0].

Throughout the paper, we suppose that the following assumptions are satisfied.

Assumption 1. The matrices 𝑅(𝑥,𝑝) and 𝑅^∗(𝑥_𝜏) satisfy 𝑅𝑥,𝑝⩾𝑅,𝑅^∗𝑥_𝜏⩾𝑅^∗, (3) where 𝑅,𝑅^∗⩾0 are known constant matrices.

Assumption 1 means that 𝑅(𝑥,𝑝) and 𝑅^∗(𝑥_𝜏) are unknown, but they are bounded by known nonnegative constant matrices. To illustrate that this assumption is reasonable, an example is given below.

Example 2. Consider two functional matrices 𝑅₁𝑥,𝑝=1+𝑝²𝑥²₁²+𝑥²₂𝑥₂𝑥₂2,𝑅₂𝑥_𝜏=2+𝑥²₁(𝑡−𝜏)000sin²𝑥₂(𝑡−𝜏)0,000 (4) where 𝑝 is unknown constant and 𝜏 is the time delay.

It is easy to find two corresponding matrices 𝑅=,1001𝑅^∗=,200000000 (5) which satisfy 𝑅₁(𝑥,𝑝)⩾𝑅 and 𝑅₂(𝑥_𝜏)⩾𝑅^∗.

Assumption 3. The Hamilton function 𝐻(𝑥) and its gradient ∇𝐻(𝑥) satisfy(A1)𝐻(𝑥)∈ℂ², (A2)𝜀₁(‖𝑥‖)⩽𝐻(𝑥)⩽𝜀₂(‖𝑥‖), (A3)𝜖₁(‖𝑥‖)⩽∇^T𝐻(𝑥)⋅∇𝐻(𝑥)⩽𝜖₂(‖𝑥‖), (A4)𝜋₁(‖𝑥‖)⩽[(∇𝐻(𝑥))^′]^T⋅[∇𝐻(𝑥)]^′⩽𝜋₂(‖𝑥‖), where 𝜀₁, 𝜀₂, 𝜖₁, 𝜖₂, 𝜋₁, 𝜋₂ all belong to 𝒦-class functions.

Remark 4. Assumption 3 not only guarantees the existence of ∇𝐻(𝑥) and [∇𝐻(𝑥)]^′ but also guarantees that 𝐻(𝑥), ∇𝐻(𝑥), and [∇𝐻(𝑥)]^′ are bounded in terms of 𝑥. We shall note that the assumption is not very conservative to Hamilton functions and the majority of Hamilton functions in Hamiltonian systems can easily satisfy these conditions.

Assumption 5. There exists a function Φ(𝑥) such that 𝐽𝑥,𝑝−𝑅𝑥,𝑝Δ_𝐻𝑥,𝑝=𝑔₁Φ(𝑥)𝜃 (6) holds for all 𝑥∈ℝ^𝑛, where 𝜃∈ℝ^𝑠 denotes an unknown parametric vector, Δ_𝐻(𝑥,𝑝)=∇𝐻(𝑥,𝑝)−∇𝐻(𝑥,0).

In what follows, we shall address the problems of robust stability and the disturbance attenuation of system (1). Specifically, the objective of this paper can be summarized as follows.

(i) Robust Stability Problem. In the absence of disturbances 𝜔, develop LMI-based conditions, and find an adaptive control law of the form 𝜃,̇𝑢=𝛼𝑥,𝜃=𝜍(𝑥) (7) so that the closed-loop system under the control law can be asymptotically stable.

(ii) 𝐿₂  Disturbance Attenuation Problem. Given a penalty signal 𝑧=𝑞(𝑥) and a disturbance attenuation level 𝛾>0, find an adaptive feedback control law 𝜃,̇𝑢=𝛽𝑥,𝜃=𝜌(𝑥) (8) and a positive storage function 𝑉(𝑥,𝑥_𝜏,𝜃) such that the 𝛾-dissipation inequality 𝑉̇𝑥,𝑥_𝜏,𝜃+𝑄𝑥,𝑥_𝜏⩽12𝛾²‖𝜔‖²−‖𝑧‖²,∀𝜔∈𝐿^𝑚₂[0,∞) (9) holds along the closed-loop systems consisting of (1) and the feedback law, where 𝑄(𝑥,𝑥_𝜏) is a nonnegative definite symmetric matrix.

We conclude this section by recalling an auxiliary result to be used in this paper.

Lemma 6 (see [30]). For given matrices 𝑌=𝑌^T, 𝐷 and 𝐸 with appropriate dimensions, 𝑌+𝐷𝐹(𝑡)𝐸+𝐸^T𝐹^T(𝑡)𝐷^T<0 (10) holds for all 𝐹(𝑡) satisfying 𝐹^T(𝑡)𝐹(𝑡)⩽𝐼 if and only if there exists 𝑐>0 such that 𝑌+𝑐⁻¹𝐷𝐷^T+𝑐𝐸^T𝐸<0. (11)

3. Main Results

3.1. Robust Stabilization

In the absence of external disturbances, namely, 𝜔=0, and under Assumption 5, system (1) can be transformed into ̇𝐽(𝑥)+𝐽𝑥=𝑥,𝑝−𝑅𝑥,𝑝∇𝐻_∗𝑥_𝜏−𝑅^∗𝑥_𝜏∇𝐻𝑥_𝜏+𝑔₁Φ(𝑥)𝜃+𝑔₁𝑢. (12)

In this subsection, we will put forward a robust stabilization result for system (12). Delay-dependent criteria are developed as follows.

Theorem 7. Consider system (12). Suppose that Assumptions 1 and 3 hold. If there exist matrices 0<𝑃₁=𝑃^T₁,0<𝑍₁=𝑍^T₁,0<𝑀₁=𝑀^T₁,0⩽𝑋=𝑋^T=𝑋₁₁𝑋₁₂𝑋₁₃𝑋₂₁𝑋₂₂𝑋₂₃𝑋₃₁𝑋₃₂𝑋₃₃, (13) any appropriately dimensioned matrices 𝐸, 𝐹, 𝑇, 𝐵₁, 𝐵₂, and a scalar 𝜀>0 such that the following conditions hold: 𝐽^∗𝑥_𝜏−𝑅^∗𝑥_𝜏=𝐸Δ𝑥_𝜏𝐹+𝑇,Ξ₁=−𝑅−𝑅^∗𝐵^T₁2+ℎ𝑋₁₂𝐵^T₂2+ℎ𝑋₁₃∗Φ₂₂−𝐵^T₂2+ℎ𝑋₂₃∗∗−𝑀₁+ℎ𝑍₁+ℎ𝑋₃₃<0,𝑋Θ=¹¹+𝜀⁻¹𝐸𝐸^T𝑋₁₂𝑋₁₃12𝑇∗𝑋₂₂𝑋₂₃12𝐵₁∗∗𝑋₃₃12𝐵₂∗∗∗𝑍₁+𝜀⁻¹𝐹^T𝐹⩾0, (14)(15)(16) where Δ^T𝑥_𝜏Δ𝑥_𝜏Φ⩽𝐼,₂₂=−1−𝜇𝑃₁−𝐵₁2−𝐵^T₁2+ℎ𝑋₂₂, (17) then the closed-loop systems under the feedback control law 𝑢=−𝑔^T₁𝑔₁𝑔^T₁⁻¹𝑃₁+ℎ𝑋₁₁(𝑥)(𝑥)𝜃+∇∇𝐻−Φ_T𝐻(𝑥)⋅∇𝐻(𝑥)⁻¹(𝑥)×(𝑥(𝑡∇𝐻(∇𝐻)))^′T𝑀₁[∇𝐻(𝑥(𝑡))]^′,̇𝜃=𝐾₁Φ^T(𝑥)𝑔^T₁∇𝐻(𝑥) (18) is asymptotically stable, where 𝐾₁>0 is an adaptive gain matrix with appropriate dimension.

Proof. Substituting (18) into (12) yields ̇𝐽(𝑥)+𝐽𝑥=𝑥,𝑝−𝑅𝑥,𝑝∇𝐻_∗𝑥_𝜏−𝑅^∗𝑥_𝜏∇𝐻𝑥_𝜏+𝑔₁Φ(𝑥)𝜃−𝜃−𝑃₁+ℎ𝑋₁₁(𝑥)−∇∇𝐻_T𝐻(𝑥)∇𝐻(𝑥)⁻¹∇𝐻(𝑥)(∇𝐻(𝑥(𝑡)))^′T×𝑀₁[∇𝐻(𝑥(𝑡))]^′,̇𝜃=𝐾₁Φ^T(𝑥)𝑔^T₁∇𝐻(𝑥). (19)
Choose a Lyapunov functional described as 𝑉₁𝑥,𝑥_𝜏,𝜃(𝑥)+1=𝐻2𝜃^T𝐾₁⁻¹𝜃+^𝑡_{(𝑡)𝑡−𝑑}∇^T𝐻𝑥𝜑𝑃₁𝑥𝜑+∇𝐻𝑑𝜑_{0−ℎ𝑡𝑡+𝛽}(∇𝐻(𝑥(𝛼)))^′T×𝑍₁[∇𝐻(𝑥(𝛼))]^′𝑑𝛼𝑑𝛽, (20) where 𝜃=𝜃−𝜃.
Since 𝐻(𝑥)∈ℂ², 𝑃₁>0, 𝑍₁>0, and (A3) in Assumption 3 holds, we have the following inequalities: ^𝑡_{(𝑡)𝑡−𝑑}∇^T𝐻𝑥𝜑𝑃₁𝑥𝜑⩽∇𝐻𝑑𝜑_{𝑡(𝑡)𝑡−𝑑}∇^T𝐻𝑥𝜑𝑃₁∇𝐻𝑥𝜑𝑑𝜑⩽𝜄_𝑝^𝑡_{(𝑡)𝑡−𝑑}𝜖₂max𝑥𝜑𝑑𝜑=ℎ𝜄_𝑝𝜖₂‖𝑥‖_𝒞, (21) where 𝜄_𝑝=𝜆_max(𝑃₁)>0.
Moreover, according to (A4) in Assumption 3, we have (∇𝐻(⁰_−ℎ^𝑡_𝑡+𝛽𝑥(𝛼)))^′T𝑍₁[∇𝐻(𝑥(𝛼))]^′⩽𝑑𝛼𝑑𝛽_{0−ℎ𝑡𝑡+𝛽}𝜄_𝑧𝜋₂𝑥(𝛼=1(‖)‖)𝑑𝛼𝑑𝛽2ℎ²𝜄_𝑧𝜋₂‖𝑥‖_𝒞, (22) where 𝜄_𝑧=𝜆_max(𝑍₁)>0.
Combining (21) and (22), from (A2) in Assumption 3, we obtain 𝑉₁𝑥,𝑥_𝜏,𝜃⩽𝜀₂(‖𝑥‖)+𝜅𝜃²+ℎ𝜄_𝑝𝜖₂‖𝑥‖_𝒞+12ℎ²𝜄_𝑧𝜋₂‖𝑥‖_𝒞, (23) where 𝜅=𝜆_max(𝐾₁⁻¹)>0.
Let 𝜈(‖𝜒‖_𝒞)=𝜀₂(‖𝑥‖)+𝜅‖𝜃‖²+ℎ𝜄_𝑝𝜖₂(‖𝑥‖_𝒞)+(1/2)ℎ²𝜄_𝑧𝜋₂(‖𝑥‖_𝒞), 𝜒=𝑥^T𝑥^T_𝜏𝜃^TT. Obviously, it belongs to 𝒦-class function. So, we obtain 𝜀₁𝜒(0)⩽𝑉₁𝑥,𝑥_𝜏,𝜃⩽𝜈𝜒_𝒞. (24)
According to the Newton-Leibnitz formula, it follows that ∇𝐻(𝑥)−^𝑡_{(𝑡)𝑡−𝑑}[∇𝐻(𝑥(𝛼))]^′𝑑𝛼−∇𝐻𝑥_𝜏=0; (25) then for any matrices 𝐵₁ and 𝐵₂ with appropriate dimensions, we have ∇^T𝐻(𝑥)𝐽^∗𝑥_𝜏−𝑅^∗𝑥_𝜏+∇^T𝐻𝑥_𝜏𝐵₁+[∇𝐻(𝑥(𝑡))]^′𝐵₂⋅∇𝐻(𝑥)−∇𝐻𝑥_𝜏−^𝑡_{(𝑡)𝑡−𝑑}[∇𝐻(𝑥(𝛼))]^′𝑑𝛼≡0. (26)
As is well known, for any positive definite matrix 𝑋⩾0 and a vector function 𝜂, the following inequality holds: ℎ𝜂^T(𝑡)𝑋𝜂(𝑡)−^𝑡_{(𝑡)𝑡−𝑑}𝜂^T(𝑡)𝑋𝜂(𝑡)𝑑𝛼⩾0. (27)
Noting that ∇^T𝐻(𝑥)𝐽(𝑥)=1𝑥,𝑝∇𝐻2∇^T𝐻(𝑥)𝐽𝑥,𝑝+𝐽^T𝑥,𝑝∇𝐻(𝑥)=0 (28) and combining (26) and (27) and using Assumption 1, we can evaluate the derivative of 𝑉₁(𝑥,𝑥_𝜏,𝜃) along the trajectory of the closed-loop system (19) as follows: 𝑉̇₁𝑥,𝑥_𝜏,𝜃=∇^T𝐻(𝑥)𝐽𝑥,𝑝−𝑅𝑥,𝑝∇𝐻(𝑥)+∇^T𝐻(𝑥)𝐽^∗𝑥_𝜏−𝑅^∗𝑥_𝜏∇𝐻𝑥_𝜏+∇^T𝐻(𝑥)𝑔₁Φ(𝑥)𝜃+∇^T𝐻(𝑥)𝑃₁∇𝐻(𝑥)−∇^T𝐻(𝑥)𝑃₁+ℎ𝑋₁₁(𝑥)−(𝑥(𝑡∇𝐻(∇𝐻)))^′T𝑀₁[∇𝐻(𝑥(𝑡))]^′−1−𝑑̇(𝑡)∇^T𝐻𝑥_𝜏𝑃₁∇𝐻𝑥_𝜏−𝜃^TΦ^T(𝑥)𝑔^T₁(𝑥)(𝑥(𝑡∇𝐻+ℎ(∇𝐻)))^′T𝑍₁[∇𝐻(𝑥(𝑡))]^′−^𝑡_𝑡−ℎ(∇𝐻(𝑥(𝛼)))^′T𝑍₁[∇𝐻(𝑥(𝛼))]^′𝑑𝛼⩽−∇^T𝐻(𝑥)𝑅∇𝐻(𝑥)−ℎ∇^T𝐻(𝑥)𝑋₁₁(𝑥)−(𝑥(𝑡∇𝐻(∇𝐻)))^′T𝑀₁[∇𝐻(𝑥(𝑡))]^′−1−𝜇∇^T𝐻𝑥_𝜏𝑃₁∇𝐻𝑥_𝜏−^𝑡_{(𝑡)𝑡−𝑑}(∇𝐻(𝑥(𝛼)))^′T𝑍₁[∇𝐻(𝑥(𝛼))]^′(𝑥(𝑡𝑑𝛼+ℎ(∇𝐻)))^′T𝑍₁[∇𝐻(𝑥(𝑡))]^′−∇^T𝐻(𝑥)𝑅^∗∇𝐻(𝑥)+∇^T𝐻𝑥_𝜏𝐵₁∇𝐻(𝑥)−∇^T𝐻𝑥_𝜏𝐵₁∇𝐻𝑥_𝜏+(∇𝐻(𝑥(𝑡)))^′T𝐵₂(𝑥)−(𝑥(𝑡∇𝐻(∇𝐻)))^′T𝐵₂∇𝐻𝑥_𝜏−^𝑡_{(𝑡)𝑡−𝑑}∇^T𝐻(𝑥)𝐸Δ𝑥_𝜏𝐹+𝑇[∇𝐻(𝑥(𝛼))]^′−𝑑𝛼_{𝑡(𝑡)𝑡−𝑑}∇^T𝐻𝑥_𝜏𝐵₁[∇𝐻(𝑥(𝛼))]^′−𝑑𝛼_{𝑡(𝑡)𝑡−𝑑}(∇𝐻(𝑥(𝑡)))^′T×𝐵₂[∇𝐻(𝑥(𝛼))]^′𝑑𝛼+ℎ𝜂^T₁𝑋𝜂₁−^𝑡_{(𝑡)𝑡−𝑑}𝜂^T₁(𝑡)𝑋𝜂₁(𝑡)𝑑𝛼, (29) where 𝜂₁=∇^T𝐻(𝑥)∇^T𝐻(𝑥_𝜏)[(∇𝐻(𝑥(𝑡)))^′]^TT.
Let𝜂₂=∇^T𝐻(𝑥)∇^T𝐻𝑥_𝜏(∇𝐻(𝑥(𝑡)))^′T(∇𝐻(𝑥(𝛼)))^′TT; (30)according to (15)-(16) and using Lemma 6, we get that 𝑉̇₁𝑥,𝑥_𝜏,𝜃⩽𝜂^T₁Ξ₁𝜂₁−^𝑡_{(𝑡)𝑡−𝑑}𝜂^T₂Θ𝜂₂𝑑𝛼⩽𝜂^T₁Ξ₁𝜂₁. (31)
Furthermore, since Ξ₁<0, according to Assumption 3, there exists a continuous nondecreasing function 𝜖(‖𝜒‖), 𝜒=[𝑥^T𝑥^T_𝜏𝜃^T]^T such that 𝑉̇₁𝑥,𝑥_𝜏,𝜃⩽−𝜖𝜒(0). (32)
According to the Lyapunov-Krasovskii stability theorem, we can conclude that the closed-loop system (19) consisting of system (12) and the control law (18) is asymptotically stable. This completes the proof.

3.2. 𝐿₂ Disturbance Attenuation

In what follows, we consider the 𝐿₂ disturbance attenuation problem of systems (1). Given a disturbance attenuation level 𝛾>0, choose the following penalty function: 𝑧=ℎ(𝑥)𝑔^T₁∇𝐻(𝑥), (33) where ℎ(𝑥)∈ℝ^𝑞×𝑠 is weighing matrix.

For time delay 𝑑(𝑡) satisfying (2), we have the following result.

Theorem 8. Consider system (1). Suppose that Assumptions 1–5 hold. If there exist matrices 0<𝑃₂=𝑃^T₂,0<𝑍₂=𝑍^T₂,0<𝑀₂=𝑀^T₂,0⩽𝑋=𝑋₁₁𝑋₁₂𝑋₁₃𝑋₂₁𝑋₂₂𝑋₂₃𝑋₃₁𝑋₃₂𝑋₃₃, (34)(35) any appropriately dimensioned matrices 𝐸, 𝐹, 𝑇, 𝐵₃, 𝐵₄ and a scalar 𝜀>0 such that (14) and the following conditions hold Ξ₂=^{Φ11=−𝑅−𝑅∗−12𝛾2𝑔1𝑔T1−𝑔2𝑔T2𝐵T32+ℎ𝑋12𝐵T42+ℎ𝑋13}_{𝑃∗−1−𝜇2−}^𝐵3₂₋^𝐵T3_{2+ℎ𝑋22−}^𝐵T4_{2+ℎ𝑋23∗∗−𝑀2+ℎ𝑍2+ℎ𝑋33}<0,Θ=^{𝑋11+𝜀−1𝐸𝐸T𝑋12𝑋1312𝑇∗𝑋22𝑋2312𝐵3}_{∗∗𝑋3312𝐵4∗∗∗𝑍2+𝜀𝐹T𝐹}⩾0, (36) then the 𝐿₂ disturbance attenuation problem of system (1) can be solved by the feedback control law:𝑢=−𝑔^T₁𝑔₁𝑔^T₁⁻¹𝑃₂+ℎ𝑋₁₁(𝑥)(𝑥)𝜃+∇∇𝐻−Φ_T𝐻(𝑥)∇𝐻(𝑥)⁻¹(𝑥)×(𝑥(𝑡∇𝐻(∇𝐻)))^′T𝑀₂[∇𝐻(𝑥(𝑡))]^′−12ℎ^T(𝑥)ℎ(𝑥)+12𝛾²𝐼_𝑚𝑔^T₁(𝑥),̇∇𝐻𝜃=𝐾₂Φ^T(𝑥)𝑔^T₁∇𝐻(𝑥), (37) where 𝐾₂>0 is an adaptive gain matrix with appropriate dimension.
Moreover, the 𝛾-dissipation inequality 𝑉₂𝑥,𝑥_𝜏,𝜃+𝑄𝑥,𝑥_𝜏⩽12𝛾²‖𝜔‖²−‖𝑧‖² (38) holds along the trajectories of the closed-loop systems consisting of (1) and (37), where 𝑄𝑥,𝑥_𝜏=−𝜂^T₁Ξ₂𝜂₁+^𝑡_{𝑡−𝑑(𝑡)}𝜂^T₂Θ𝜂₂𝑑𝛼 (39) with𝜂₁=∇^T𝐻(𝑥)∇^T𝐻𝑥_𝜏(∇𝐻(𝑥(𝑡)))^′TT,𝜂₂=∇^T𝐻(𝑥)∇^T𝐻𝑥_𝜏(∇𝐻(𝑥(𝑡)))^′T(∇𝐻(𝑥(𝛼)))^′TT. (40)The storage function is given as 𝑉₂𝑥,𝑥_𝜏,𝜃=𝐻(𝑥)+^𝑡_{(𝑡)𝑡−𝑑}∇^T𝐻𝑥𝜑𝑃₂𝑥𝜑+∇𝐻𝑑𝜑_{0−ℎ𝑡(𝑡)𝑡−𝑑}∇^T𝐻(𝑥(𝛼))^′𝑍₂[∇𝐻(𝑥(𝛼))]^′+1𝑑𝛼𝑑𝛽2𝜃^T𝐾₂⁻¹𝜃. (41)

Proof. Substituting (37) into (1) yields ̇𝐽(𝑥)+𝐽𝑥=𝑥,𝑝−𝑅𝑥,𝑝∇𝐻_∗𝑥_𝜏−𝑅^∗𝑥_𝜏∇𝐻𝑥_𝜏−𝑃₂+ℎ𝑋₁₁∇𝐻(𝑥)+𝑔₂𝜔−∇^T𝐻(𝑥)∇𝐻(𝑥)⁻¹(𝑥)×(𝑥(𝑡∇𝐻(∇𝐻)))^′T𝑀₂[∇𝐻(𝑥(𝑡))]^′+𝑔₁Φ(𝑥)𝜃−𝜃−𝑔₁12ℎ^T(𝑥)ℎ(𝑥)+12𝛾²𝐼_𝑚𝑔^T₁(𝑥),̇∇𝐻𝜃=𝐾₂Φ^T(𝑥)𝑔^T₁∇𝐻(𝑥). (42)
Evaluating the derivative of (41) along the trajectory of system (42) and using (26), (27), and Assumption 1, we get 𝑉̇₂𝑥,𝑥_𝜏,𝜃=∇^T𝐻(𝑥)𝐽𝑥,𝑝−𝑅𝑥,𝑝∇𝐻(𝑥)+∇^T𝐻(𝑥)𝐽^∗𝑥_𝜏−𝑅^∗𝑥_𝜏∇𝐻𝑥_𝜏−ℎ∇^T𝐻(𝑥)𝑋₁₁∇𝐻(𝑥)+∇^T𝐻(𝑥)𝑔₂𝜔−(∇𝐻(𝑥(𝑡)))^′T𝑀₂[∇𝐻(𝑥(𝑡))]^′−∇^T𝐻(𝑥)𝑔₁12ℎ^T(𝑥)ℎ(𝑥)+12𝛾²𝐼_𝑚𝑔^T₁(𝑥)−𝑑̇(𝑡)∇∇𝐻1−^T𝐻𝑥_𝜏𝑃₂∇𝐻𝑥_𝜏+ℎ(∇𝐻(𝑥(𝑡)))^′T𝑍₂[∇𝐻(𝑥(𝑡))]^′−^𝑡_𝑡−ℎ(∇𝐻(𝑥(𝛼)))^′T𝑍₂[∇𝐻(𝑥(𝛼))]^′𝑑𝛼⩽−∇^T𝐻(𝑥)𝑅∇𝐻(𝑥)−ℎ∇^T𝐻(𝑥)𝑋₁₁(𝑥)−(𝑥(𝑡∇𝐻(∇𝐻)))^′T𝑀₂[∇𝐻(𝑥(𝑡))]^′−1−𝜇∇^T𝐻𝑥_𝜏𝑃₂∇𝐻𝑥_𝜏+ℎ(∇𝐻(𝑥(𝑡)))^′T𝑍₂[∇𝐻(𝑥(𝑡))]^′−^𝑡_𝑡−ℎ(∇𝐻(𝑥(𝛼)))^′T𝑍₂[∇𝐻(𝑥(𝛼))]^′𝑑𝛼+∇^T𝐻(𝑥)𝐽^∗𝑥_𝜏−𝑅^∗𝑥_𝜏∇𝐻(𝑥)+∇^T𝐻𝑥_𝜏𝐵₃∇𝐻(𝑥)−∇^T𝐻𝑥_𝜏𝐵₃∇𝐻𝑥_𝜏+(∇𝐻(𝑥(𝑡)))^′T𝐵₄(𝑥)−∇𝐻_{𝑡(𝑡)𝑡−𝑑}∇^T𝐻(𝑥)𝐽^∗𝑥_𝜏−𝑅^∗𝑥_𝜏[∇𝐻(𝑥(𝛼))]^′𝑑𝛼−^𝑡_{(𝑡)𝑡−𝑑}∇^T𝐻𝑥_𝜏𝐵₃[∇𝐻(𝑥(𝛼))]^′−(𝑥(𝑡𝑑𝛼(∇𝐻)))^′T𝐵₄∇𝐻𝑥_𝜏−^𝑡_{(𝑡)𝑡−𝑑}(∇𝐻(𝑥(𝑡)))^′T𝐵₄[∇𝐻(𝑥(𝛼))]^′𝑑𝛼+ℎ𝜂^T₁𝑋𝜂₁−^𝑡_{(𝑡)𝑡−𝑑}𝜂^T₁(𝑡)𝑋𝜂₁(𝑡)−1𝑑𝛼21𝛾𝜔−𝛾∇^T𝐻(𝑥)𝑔₂²−∇^T𝐻(𝑥)𝑔₁12ℎ^T(𝑥)ℎ(𝑥)+12𝛾²𝐼_𝑚𝑔^T₁(𝑥)+1∇𝐻2𝛾²∇^T𝐻(𝑥)𝑔₂𝑔^T₂(𝑥)+1∇𝐻2𝛾²‖𝜔‖²−‖𝑧‖²+12∇^T𝐻(𝑥)𝑔₁ℎ^T(𝑥)ℎ(𝑥)𝑔^T₁∇𝐻(𝑥). (43) According to (35), (36), and Lemma 6, we have𝑉̇₂𝑥,𝑥_𝜏,𝜃−𝜂^T₁Ξ₂𝜂₁+^𝑡_{(𝑡)𝑡−𝑑}𝜂^T₂Θ𝜂₂𝑑𝛼⩽12𝛾²‖𝜔‖²−‖𝑧‖². (44)It is obvious that the 𝛾-dissipation inequality (38) holds along the closed-loop system (42) which consist of (1) and (37). This completes the proof.

4. Illustrative Examples

In this section, we give some examples to show how to apply the results proposed in this paper to investigate the robust stabilization and the 𝐿₂ disturbance attenuation for a class of time-delay nonlinear control systems with uncertainties and disturbances.

Let us consider the following 2-dimensional time-delay nonlinear control systems with parametric uncertainties and external disturbances: 𝑥̇₁(𝑡)=−4𝑥³₁(𝑡)−4𝑥³₁((𝑡𝑥̇𝑡−𝑑))+2𝑢,₂(𝑡)=−2𝑥³₁(𝑡)−2+3𝑝+𝑝²𝑥₂(𝑡)−2𝑥³₁(𝑡−𝑑(𝑡))−2𝑥₂(𝑡−𝑑(𝑡))−𝑥₂(𝑡−𝑑(𝑡))sin𝑥₂((𝑡𝑥𝑡−𝑑))+3𝑢+0.5𝜔,₁𝑡₀=𝜙₁𝑡₀,𝑥₂𝑡₀=𝜙₂𝑡₀,𝑡₀∈[−ℎ,0], (45) where 𝑑(𝑡) is a time varying delay of the system (45); 𝑝 is an unknown constant, 0<𝑝<1; 𝜔 is the disturbance input.

The system (45) can be realized into the following Hamiltonian system form: ̇𝐽+𝐽𝑥=𝑥,𝑝−𝑅𝑥,𝑝∇𝐻𝑥,𝑝_∗𝑥_𝜏−𝑅^∗𝑥_𝜏∇𝐻𝑥_𝜏+𝑔₁𝑢+𝑔₂𝑥𝑡𝜔,₀=𝜙𝑡₀,𝑡₀∈[−ℎ,0] (46) with 𝐽==,𝐽𝑥,𝑝00.5−0.50,𝑅𝑥,𝑝20.50.52+𝑝_∗𝑥_𝜏=,𝑅00.5−0.50_∗𝑥_𝜏=20.50.52+sin𝑥₂((𝑡,𝑔𝑡−𝑑))₁=23,𝑔₂=0,𝑥0.5𝑥=¹𝑥₂𝜙,𝜙=¹𝜙₂,𝐻𝑥,𝑝=0.5𝑥⁴₁+1+𝑝𝑥²₂,𝐻𝑥_𝜏=0.5𝑥⁴₁(𝑡−𝑑(𝑡))+𝑥²₂(𝑡−𝑑(𝑡)). (47)(48)

Let 𝐸=𝐹=¹⁰₀₁, 𝑇=⁻²⁰₋₁₋₂, Δ(𝑥_𝜏)=⁰⁰_{0−sin𝑥2(𝑡−𝑑(𝑡))}, 𝜃=(−1−0.5𝑝)𝑝 and Φ(𝑥)=𝑥₂. It is easy to verify that system (46) with the above values satisfies Assumptions 1–5 and the condition (14) of Theorem 7.

Firstly, we demonstrate the application of Theorem 7 by using LMI solver [31].

Set 𝜇=0.25 and ℎ=1. Using the LMI control toolbox of MATLAB, the LMIs in Theorem 7 are solved to find the following matrices: 𝑃₁=2.3387−0.0187−0.01872.3387,𝑍₁=,𝑀0.37700.04470.04470.2949₁=1.00000.00000.00001.0000,𝐵₁=,𝑋0.76080.02320.02320.7653₁₁=1.43630.09570.09571.4842,𝑋₁₂=,𝑋−0.1565−0.0155−0.0155−0.1583₂₂=1.37870.00700.00701.3801,𝑋₃₃=,𝐵0.5258−0.0203−0.02030.5359₂=𝑋₁₃=𝑋₂₃=.0000 (49) Thus a robust stabilizing controller is obtained as 𝑢=−3.8229𝑥₂−𝑥₂𝑥̇⁴₁+𝑥̇²₂𝑥⁴₁+𝑥²₂⁻¹−0.0770𝑥³₁−𝑥₂𝜃. (50)

The simulation with the initial condition 𝑥(0)=𝜙(0)=[1−2]^T is given in Figures 1 and 2. It is clear that under the delay-dependent conditions, system (46) along with the controller (50) is asymptotically stable.

Figure 1 

Responses of state 𝑥.

Figure 2 

Parameter estimation 𝜃.

Next, we demonstrate the application of Theorem 8. We will check whether the designed 𝐿₂ disturbance attenuation controller according to Theorem 8 is effective in stabilizing the given time-delay Hamiltonian system (46) and has strong robustness against external disturbances.

Given a disturbance attenuation level 𝛾, choose 𝑧=ℎ(𝑥)𝑔^T₁∇𝐻(𝑥) (51) as the penalty function, where ℎ=[0.10.1]^T.

Using the LMI control toolbox, the LMIs in Theorem 8 are solved to find the following matrices with 𝜇=0.25, ℎ=1: 𝑃₂=2.4230−0.0178−0.01782.4230,𝑍₂=,𝑀0.31360.04260.04260.2926₂=1.00000.00000.00001.0000,𝐵₃=,0.78590.02230.02230.7939𝑋₁₁=,1.47990.08940.08941.5238𝑋₁₂=,−0.1568−0.0160−0.0160−0.1551𝑋₂₂=,1.42750.00690.00691.4304𝑋₃₃=,𝐵0.5382−0.0194−0.01940.5477₄=𝑋₁₃=𝑋₂₃=.0000 (52) Then according to Theorem 8, a feedback adaptive controller can be obtained as 𝑢=−4.4041𝑥₂−𝑥₂𝑥̇⁴₁+𝑥̇²₂𝑥⁴₁+𝑥²₂⁻¹−0.0716𝑥³₁−𝑥₂𝜃. (53)

To illustrate the effectiveness of the adaptive control law (53), we carry simulation result with the following choices: the disturbance signal 𝜔=sin𝑡; the initial condition is 𝑥(0)=𝜙(0)=[5−5]^T; the disturbance attenuation level is chosen by 𝛾=0.9. The simulation results are shown in Figures 3 and 4, which are responses of the system’s state and the parameter estimation, respectively. It can be seen from the simulation that the time-delay system converges to its equilibrium very quickly under the controller (53).

Figure 3 

Responses of state 𝑥.

Figure 4 

Parameter estimation 𝜃.

In general, from the simulations, we can conclude that the results presented in this paper are very practicable and effective in stabilization analysis and 𝐿₂ disturbance attenuation of time-delay Hamiltonian systems with parametric uncertainties and external disturbances. What is more, by using the result presented in this paper, we may solve the stability and control problem of some classes of time-delay nonlinear systems which can be realized into Hamiltonian systems form.

5. Conclusions

In this paper, the robust asymptotical stability and 𝐿₂ disturbance attenuation problem of a class of time-delay Hamiltonian control systems with parametric uncertainties and external disturbances have been investigated. Delay-dependent criteria are established. The proposed adaptive feedback control law, by which the asymptotic stability and the 𝐿₂ performance of the close-loop system is guaranteed, is determined by linear matrix inequalities constraints. Simulations show the effectiveness of the proposed method.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grant 61203013 and 61004013 and in part by the Outstanding Middle-age and Young Scientist Award Foundation of Shandong Province under Grant BS2011DX012.