Nonfragile Robust Finite-Time 𝐿₂-𝐿_∞ Controller Design for a Class of Uncertain Lipschitz Nonlinear Systems with Time-Delays

Abstract

The nonfragile robust finite-time 𝐿₂-𝐿_∞ control problem for a class of nonlinear uncertain systems with uncertainties and time-delays is considered. The nonlinear parameters are considered to satisfy the Lipschitz conditions and the exogenous disturbances are unknown but energy bounded. By using the Lyapunov function approach, the sufficient condition for the existence of nonfragile robust finite-time 𝐿₂-𝐿_∞ controller is given in terms of linear matrix inequalities (LMIs). The finite-time controller is designed such that the resulting closed-loop system is finite-time bounded for all admissible uncertainties and satisfies the given 𝐿₂-𝐿_∞ control index. Simulation results illustrate the validity of the proposed approach.

1. Introduction

Time-delay is frequently a source of instability and always encounters in various engineering systems such as chemical processes, neural networks, and long transmission lines in pneumatic systems. Recently, many attentions have been paid to the solutions of deal with time-delay which existed in concerned systems, such as the input-output technique [1], delay partitioning method [2], and piecewise analysis method [3]. As an important class of nonlinear systems, the Lipschitzian nonlinear system also has drawn considerable attention in the past few decades. Among these researches, the studies of Lipschitz nonlinear systems with time-delays have received much attention [4–6].

However, it is necessary to point out that the results of the aforementioned papers about time-delayed Lipschitz systems are mostly based on Lyapunov stable theory. As we all known, Lyapunov stable theory pays more attention to the asymptotic pattern of systems over an infinite-time interval. But in some practical process, the main attention may be related to the behavior of the dynamical systems over fixed finite-time interval; for instance, large values of the states cannot be accepted in the presence of saturations. To deal with such situations, Dorato [7] first presented the concept of finite-time stability (or short-time stability) in 1961. In [8], Amato et al. first proposed the concept of finite-time boundedness; thereafter many attempts on finite-time control problems have been made [9–13]. However, to date, little work has been paid attention in finite-time controller design for Lipschitzian nonlinear system. The research topic is still open but challenging. And this is the key motivation of this paper.

On the other hand, in the feedback control schemes, there are often some perturbations appeared in controller gain, which may result from either the actuator degradations or the requirements of readjustment of controller gains during the controller implementation stage. Therefore, it is reasonable that any controller should be able to tolerate some level of controller gain variations, and this motivates many researchers to study the nonfragile controller problems [14–19]. Motivated by the benefits of nonfragile state feedback controller, we consider the nonfragile controller design problem in this paper.

In addition, energy to peak (𝐿₂-𝐿_∞) control is of a great importance both in control theory and in engineering practice, because of its insensitivity to the exact knowledge of the statistics of the external disturbance signals. In [20], Rotea first introduced the 𝐿₂-𝐿_∞ performance index. After that many researchers have paid attention in 𝐿₂-𝐿_∞ control problems. For example, in [21], the authors studied the 𝐿₂-𝐿_∞ controller design problem for continuous-time multiple delayed linear systems; in [22], the problem of 𝐿₂-𝐿_∞ control for a class of uncertain singular systems with time-delay and norm-bounded parameter uncertainties was investigated by the researchers. For more details of the literatures related to 𝐿₂-𝐿_∞ control schemes, the readers can refer to [21–24] and the references therein.

In short, to the best knowledge of authors, the problem of nonfragile robust finite-time 𝐿₂-𝐿_∞ control for a class of uncertain Lipschitz nonlinear systems with time-delays is not studied at present. This motivates our research.

This paper deals with the problem of nonfragile robust finite-time 𝐿₂-𝐿_∞ control for a class of Lipschitz nonlinear systems with time-delays and uncertainties, and the nonlinear function is assumed to satisfy the Lipschitz conditions. The uncertain parameters are assumed to be time varying and energy bounded. The purpose is to construct a nonfragile state feedback controller such that the resulting closed-loop system is finite-time bounded and satisfies the given 𝐿₂-𝐿_∞ index. By using the Lyapunov function approach and linear matrix inequality techniques, a sufficient condition for the existence of a nonfragile state feedback controller is given and an explicit expression of this controller is presented. Meanwhile, this problem is reduced to an optimization problem under the constraint of LMIs. Finally, a simulation example illustrates the effectiveness of the developed techniques.

The paper is organized as follows. In Section 2, the system description along with necessary assumption is given. Section 3 provides the main results. A simulation example is provided to illustrate the results of the paper in Section 4. Conclusion follows in Section 5.

Notation. In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. 𝐈 and 𝟎 represent the identity matrix and a zero matrix. The notation 𝐌>(≥,<,≤)0 is used to denote a symmetric positive-definite (positive semidefinite, negative-definite, negative semidefinite) matrix. 𝜆_min(⋅), 𝜆_max(⋅) denote the minimum and the maximum eigenvalue of the corresponding matrix, respectively. 𝐿₂[0+∞) denotes the Euclidean norm for vectors or the spectral norm of matrices. 𝐿^𝑛₂[0𝑁] is the space of 𝑛-dimensional square integrable function vector over [0𝑁]. In symmetric black matrices, we use * that represents the elements below the main diagonal of a symmetric block matrix. The superscript T represents the transpose.

2. Preliminaries and Problem Statement

Consider a class of Lipschitz nonlinear systems with time-delay and parameter uncertainties described by the following equations: 𝐱̇(𝑡)=[𝐀+Δ𝐀(𝑡)]𝐱(𝑡)+𝐀_𝑑+Δ𝐀_𝑑(𝑡𝐱()(𝑡)+[(𝑡𝐰(𝑡)+[(𝑡𝐟(𝐱(𝑡)(,𝐲(𝑡)=[(𝑡𝐱(𝑡)(𝑡),𝐱(𝑡)(𝑡),)𝑡−𝑑+𝐁𝐮𝐆+Δ𝐆)]𝐅+Δ𝐅)],𝐱𝑡−𝑑))𝐂+Δ𝐂)]+𝐃𝐮=𝝋,∀𝑡∈−𝑑0 (1) where 𝐱(𝑡)∈ℜ^𝑛 is the state, 𝐱(𝑡−𝑑)∈ℜ^𝑛 is the constant time-delay state, 𝐮(𝑡)∈ℜ^𝑝 is the controlled input, 𝐰(𝑡)∈ℜ^𝑟 is the disturbance input that belongs to 𝐿₂[0+∞), 𝐟(𝐱(𝑡),𝐱(𝑡−𝑑))∈ℜ^𝑛×ℜ^𝑛=ℜ^𝑛𝑔 is the nonlinear function which represents the known nonlinear disturbance related to the state and the time-delay state, and 𝝋(𝑡)∈𝐿₂[−𝑑0] is a continuous vector-valued initial function. In addition, 𝐀, 𝐀_𝑑, 𝐁, 𝐆, 𝐅, 𝐂, and 𝐃 are known real matrices, and Δ𝐀(𝑡), Δ𝐀_𝑑(𝑡), Δ𝐆(𝑡), Δ𝐅(𝑡), and Δ𝐂(𝑡) are unknown time-variant matrices representing the norm-bounded parameter uncertainties and satisfy the following form: Δ𝐀(𝑡)Δ𝐀_𝑑(𝑡)(𝑡)(𝑡)(𝑡)=𝐌Δ𝐆Δ𝐅Δ𝐂⋅⋅⋅₁𝐌₂Γ(𝑡)𝐒₁𝐒₂𝐒₃𝐒₄,Γ^T(𝑡)Γ(𝑡)≤𝐈, (2)(3) where 𝐌₁, 𝐌₂, 𝐒₁, 𝐒₂, 𝐒₃, and 𝐒₄ are known real constants matrices and Γ(𝑡) is the time-varying unknown matrix function with Lebesgue norm measurable elements.

Remark 1. The parameter uncertainty structure as in (2) and (3) has been widely used in the existing literatures, such as [22, 24, 25]. In many practical applications, the systems parameter uncertainties can be either exactly modeled or overbounded by (3). Note that the unknown matrix function Γ(𝑡) is time-varying, therefore it can be allowed to be state-dependent in some case, that is, Γ(𝑡)=Γ(𝑡,𝐱(𝑡)).

The following preliminary assumptions are made for system (1).

Assumption 1. For any given positive number 𝛿 and constant time 𝑇, the external disturbances input 𝐰(𝑡) is time-varying and satisfies ^𝑇₀𝐰^T(𝑡)𝐰(𝑡)d𝑡≤𝛿,𝛿≥0. (4)

Assumption 2. 𝐟(𝐱(𝑡),𝐱(𝑡−𝑑))∶ℜ^𝑛×ℜ^𝑛→ℜ^𝑛𝑔 is a known nonlinear function which satisfies the following Lipschitz conditions:(i) 𝐟(𝟎,𝟎)=0;(ii) ‖𝐟(𝐱(𝑡),𝐱(𝑡−𝑑))‖≤‖𝜼₁𝐱(𝑡)‖+‖𝜼₂𝐱(𝑡−𝑑)‖,where the weighting matrices 𝜼₁ and 𝜼₂ are known real constant matrices.

Remark 2. The above Lipschitz condition can be considered as an extension of the Lipschitz condition mentioned in [19]. When 𝜼₁=diag{√𝛼√𝛼⋯√𝛼} and 𝜼₂=diag{𝛽𝛽⋯𝛽}, the Lipschitz condition in Assumption 2 will reduce to (4) [19].
Our aim is to develop a nonfragile state feedback controller: 𝐮(𝑡)=[𝐊+Δ𝐊(𝑡)]𝐱(𝑡), (5) where the matrix 𝐊 is the controller gain to be determined and Δ𝐊(𝑡) is the controller gain variations. Here, we consider the additive controller gain variation, that is, Δ𝐊(𝑡)=𝐇𝝆(𝑡)𝐍,𝝆^T(𝑡)𝝆(𝑡)≤𝚰, (6) where 𝐇 and 𝐍 are known matrices, and 𝝆(𝑡) is unknown but norm bounded.

Remark 3. The controller gain perturbations can result from the actuator degradations, as well as from the requirements for readjustment of controller gain during the controller implementation stage. Theses perturbations in the controller gains are modeled here as uncertain gains that are dependent on the uncertain parameters [6]. The models of additive uncertainties and multiplicative uncertainties are used to describe the controller gain variations. For more results on this topic, we refer readers to [6, 17, 18] and the references therein.
By the nonfragile state feedback controller (5), the closed-loop system can be obtained as follows: 𝐱̇(𝑡)=𝐀𝐱(𝑡)+𝐀_𝑑𝐱()+(𝑡)+(𝐱(𝑡)(,𝐲(𝑡)=(𝑡),𝐱(𝑡)(𝑡),𝑡−𝑑𝐆𝐰𝐅𝐟,𝐱𝑡−𝑑))𝐂𝐱=𝝋,∀𝑡∈−𝑑0 (7) where 𝐀=𝐀+Δ𝐀(𝑡), 𝐀_𝑑=𝐀_𝑑+Δ𝐀_𝑑(𝑡), 𝐆=𝐆+Δ𝐆(𝑡), 𝐅=𝐅+Δ𝐅(𝑡), 𝐂=𝐂+Δ𝐂(𝑡), 𝐀=𝐀+𝐁𝐊, 𝐂=𝐂+𝐃𝐊, Δ𝐀(𝑡)=Δ𝐀(𝑡)+𝐁Δ𝐊(𝑡), Δ𝐂(𝑡)=Δ𝐂(𝑡)+𝐃Δ𝐊(𝑡).

To formulate the problem addressed in this paper, we give the following definitions first.

Definition 4 (FTB). For given constants 𝑐₁>0,𝛿>0,and𝑇>0 and a symmetric matrix 𝐑>0, the closed-loop system (7) is said to be robust finite-time bounded (FTB) with respect to (𝑐₁𝑐₂𝛿𝑇𝐑), if there exists a constant 𝑐₂>𝑐₁, such that the following relation holds for all the external disturbance 𝐰(𝑡): 𝐱^T𝑡₀𝐑𝐱𝑡₀≤𝑐₁⇒𝐱^T(𝑡)𝐑𝐱(𝑡)<𝑐₂,∀𝑡₀∈−𝑑0,𝑡∈0𝑇. (8)

Remark 5. In fact, if the closed-loop system (7) does not exist the exogenous disturbance input, that is, 𝐰(𝑡)=0, the concept of FTB reduces to finite-time stable (FTS) [8, 11]. That is to say, a system is FTB, if given a bound initial condition and characterization of the set of admissible inputs, then the system states remain below the prescribed limit for all inputs in the bounded set.

Definition 6. The state-feedback controller (5) is said to be a nonfragile robust finite-time 𝐿₂-𝐿_∞ controller with disturbance attenuation 𝛾>0 for system (1) if the closed-loop system (7) is FTB in the sense of Definition 4, and there exists a positive real constant 𝛾>0 such that ‖𝐲(𝑡)‖²_∞≤𝛾²‖𝐰(𝑡)‖²₂, (9) where ‖𝐲(𝑡)‖²_∞=sup_{[]𝑡∈0𝑇}[𝐲^T(𝑡)𝐲(𝑡)], ‖𝐰(𝑡)‖²₂=∫^𝑇₀𝐰^T(𝑡)𝐰(𝑡)d𝑡.

Furthermore, we introduce the following lemmas which will be used in the development of our main results.

Lemma 7 (see [26]). Let 𝐗 and 𝐘 be real matrices of appropriate dimensions. For any given scalar 𝜀>0 and vectors 𝐱,𝐲∈ℜ^𝑛, then 2𝐱^Τ𝐗𝐘𝐲≤𝜀⁻¹𝐱^Τ𝐗^Τ𝐗𝐱+𝜀𝐲^Τ𝐘^Τ𝐘𝐲. (10)

Lemma 8 (see [27]). Let 𝐌 and 𝐍 be real matrices of appropriate dimensions. Then for any matrix 𝐅(𝑡) satisfying 𝐅^Τ(𝑡)𝐅(𝑡)≤𝐈 and a scalar 𝜇>0, 𝐌𝐅(𝑡)𝐍+[𝐌𝐅(𝑡)𝐍]^Τ≤𝜇⁻¹𝐌𝐌^Τ+𝜇𝐍^Τ𝐍. (11)

3. Main Results

In this section, we will solve the problem of nonfragile robust finite-time 𝐿₂-𝐿_∞ controller design for a class of Lipschitz uncertain nonlinear time-delay systems formulated in the previous section based on LMI approach. The following results actually present the FTB condition for closed-loop system (7) with time-delays.

Theorem 9. For given 𝑐₁>0, 𝛿>0, 𝑇>0, 𝜀>0, 𝛼>0 and weighting matrices 𝜼₁, 𝜼₂, 𝐑>0, the closed-loop controlled system (7) is FTB with respect to (𝑐₁𝑐₂𝛿𝑇𝐑), if there exists constant 𝑐₂>0, symmetric positive- definite matrices 𝐏 and 𝐐>2𝜀𝜼^Τ₂𝜼₂, such that Ω₁=𝚷₁₁𝐏𝐀_𝑑𝐏𝐆∗𝚷₂₂𝟎∗∗−𝛼𝐈<0,𝑐₁𝜆₁+𝑑𝜆₃+𝛿1−𝑒^−𝛼𝑇<𝜆₂𝑐₂𝑒^−𝛼𝑇, (12)(13) where 𝚷₁₁=𝐀^Τ𝐏+𝐏𝐀+𝐐−𝛼𝐏+2𝜀𝜼^Τ₁𝜼₁+𝜀⁻¹𝐏𝐅𝐅^Τ𝐏,𝚷₂₂=−𝐐+2𝜀𝜼^Τ₂𝜼₂, 𝐏=𝐑^−1/2𝐏𝐑^−1/2,𝐐=𝐑^−1/2𝐐𝐑^−1/2, 𝜆₁=𝜆_max(𝐏), 𝜆₂=𝜆_min(𝐏), 𝜆₃=𝜆_max(𝐐).

Proof. Choose a Lyapunov function candidate 𝐕(𝐱(𝑡)) as 𝐕(𝐱(𝑡))=𝐱^T(𝑡)𝐏𝐱(𝑡)+^𝑡_𝑡−𝑑𝐱^T(𝜏)𝐐𝐱(𝜏)d𝜏. (14) Along the trajectories of system (7), the corresponding time derivation of 𝐕(𝐱(𝑡)) is given by 𝐕̇(𝐱(𝑡))=𝐱̇^T(𝑡)𝐏𝐱(𝑡)+𝐱^T(𝑡)𝐏𝐱̇(𝑡)+𝐱^T(𝜏)𝐐𝐱(𝜏)^𝑡_𝑡−𝑑=𝐱^T(𝑡)𝐀^T𝐏+𝐏𝐀+𝐐𝐱(𝑡)+𝐱^T(𝑡)𝐏𝐀_𝑑𝐱(𝑡−𝑑)+𝐱^T(𝑡−𝑑)𝐀^T_𝑑𝐏𝐱(𝑡)+𝐱^T(𝑡)𝐏𝐆𝐰(𝑡)+𝐰^T(𝑡)𝐆^T𝐏𝐱(𝑡)−𝐱^T(𝑡−𝑑)𝐐𝐱(𝑡−𝑑)+2𝐟^T(𝐱(𝑡),𝐱(𝑡−𝑑))𝐅^T𝐏𝐱(𝑡). (15) Using Assumption 2, we have ‖𝐟(𝐱(𝑡),𝐱(𝑡−𝑑))‖²≤2𝜼₁𝐱(𝑡)²+2𝜼₂𝐱(𝑡−𝑑)². (16) Considering Lemma 7, we can obtain the following relation for any given scalar 𝜀>02𝐟^T(𝐱(𝑡),𝐱(𝑡−𝑑))𝐅^T𝐏𝐱(𝑡)≤𝜀𝐟^T(𝐱(𝑡),𝐱(𝑡−𝑑))𝐟(𝐱(𝑡),𝐱(𝑡−𝑑))+𝜀⁻¹𝐱^T(𝑡)𝐏𝐅𝐅^T𝐏𝐱(𝑡)≤𝐱^T(𝑡)2𝜀𝜼^T₁𝜼₁+𝜀⁻¹𝐏𝐅𝐅^T𝐏𝐱(𝑡)+𝐱^T(𝑡−𝑑)2𝜀𝜼^T₂𝜼₂𝐱(𝑡−𝑑). (17) Hence, relation (15) can be rewritten as 𝐕̇(𝐱(𝑡))≤𝐱^T(𝑡)𝐀^T𝐏+𝐏𝐀+𝐐+2𝜀𝜼^T₁𝜼₁+𝜀⁻¹𝐏𝐅𝐅^T𝐏𝐱(𝑡)+𝐱^T(𝑡)𝐏𝐀_𝑑𝐱(𝑡−𝑑)+𝐱^T(𝑡−𝑑)𝐀^T_𝑑𝐏𝐱(𝑡)+𝐱^T(𝑡)𝐏𝐆𝐰(𝑡)+𝐰^T(𝑡)𝐆^T𝐏𝐱(𝑡)+𝐱^T(𝑡−𝑑)−𝐐+2𝜀𝜼^T₂𝜼₂𝐱(𝑡−𝑑). (18) Define the following function: 𝐉₁=𝐕̇(𝐱(𝑡))−𝛼𝐕(𝑥(𝑡))−𝛼𝐰^T(𝑡)𝐰(𝑡). (19) Considering (18) and (14), we obtain 𝐉₁+𝛼^𝑡_𝑡−𝑑𝐱^T(𝜏)𝐐𝐱(𝜏)d𝜏=𝜻^TΩ₁𝜻<0, (20) where 𝜻=[𝐱^T(𝑡)𝐱^T(𝑡−𝑑)𝐰^T(𝑡)]^T.
According to 𝛼>0 and 𝐐>0, inequality (20) implies 𝐉₁<0. Thus, by using (19), we get 𝐕̇(𝐱(𝑡))<𝛼𝐕(𝐱(𝑡))+𝛼𝐰^T(𝑡)𝐰(𝑡). (21) Pre- and postmultiplying (20) by 𝑒^−𝛼𝑡, it yields dd𝑡𝑒^−𝛼𝑡𝐕(𝐱(𝑡))<𝛼𝑒^−𝛼𝑡𝐰^T(𝑡)𝐰(𝑡). (22) Integrating the aforementioned inequality between 0 and 𝑡, it follows that 𝑒^−𝛼𝑡𝐕(𝐱(𝑡))−𝐕(𝐱(0))<𝛼^𝑡₀𝑒^−𝛼𝑠𝐰^T(𝑠)𝐰(𝑠)d𝑠. (23) Then, the above inequality is equivalent to 𝐕(𝐱(𝑡))<𝑒^𝛼𝑡𝐕(𝐱(0))+𝛼𝑒^𝛼𝑡𝑡₀𝑒^−𝛼𝑠𝐰^T(𝑠)𝐰(𝑠)d𝑠. (24) Noting that 𝐏=𝐑^−1/2𝐏𝐑^−1/2, 𝐐=𝐑^−1/2𝐐𝐑^−1/2, it follows from (14) that 𝑒^𝛼𝑡𝐕(𝐱(0))+𝛼𝑒^𝛼𝑡𝑡₀𝑒^−𝛼𝑠𝐰^T(𝑠)𝐰(𝑠)d𝑠=𝑒^𝛼𝑡𝐱^T(0)𝐏𝐱(0)+𝑒^𝛼𝑡0_−𝑑𝐱^T(𝜏)𝐐𝐱(𝜏)d𝜏+𝛼𝑒^𝛼𝑡𝑡₀𝑒^−𝛼𝑠𝐰^T(𝑠)𝐰(𝑠)d𝑠≤𝑒^𝛼𝑇𝜆_max𝐏𝑐₁+𝑒^𝛼𝑇𝑑𝜆_max𝐐𝑐₁+𝛿𝑒^𝛼𝑇1−𝑒^−𝛼𝑇=𝑒^𝛼𝑇𝑐₁𝜆₁+𝑑𝜆₃+𝛿1−𝑒^−𝛼𝑇. (25) On the other hand, the following condition holds: 𝐕(𝐱(𝑡))=𝐱^T(𝑡)𝐏𝐱(𝑡)+^𝑡_𝑡−𝑑𝐱^T(𝜏)𝐐𝐱(𝜏)d𝜏≥𝐱^T(𝑡)𝐏𝐱(𝑡)≥𝜆_min𝐏𝐱^T(𝑡)𝐑𝐱(𝑡)=𝜆₂𝐱^T(𝑡)𝐑𝐱(𝑡). (26) Combining (24), (25), and (26), we can get 𝐱^T(𝑡)(𝑡)≤𝑐𝐑𝐱¹𝜆₁+𝑑𝜆₃+𝛿1−𝑒^−𝛼𝑇𝜆₂𝑒^−𝛼𝑇. (27) Recalling condition (13), it implies that for ∀𝑡∈[0𝑇], 𝐱^T(𝑡)𝐑𝐱(𝑡)<𝑐₂. This completes the proof.

Remark 10. Without the time-delay in the resulted closed-loop system (7), the system will reduce to the system which has been investigated in [25, 28]. In this case, one can get the sufficient conditions of FTB for the aforementioned system from Theorem 9 via a simple transformation. Furthermore, if there is also no nonlinear function in the aforementioned system, then Theorem 9 reduces to a form which has been given in [8, Lemma 6] or [13, Lemma 1].
Theorem 9 gives the sufficient condition of FTB for the resulting closed-loop controlled system (7). Then, we will apply the results in Theorem 11 to solve the problem of nonfragile robust finite-time 𝐿₂-𝐿_∞ controller design.

Theorem 11. For given 𝑐₁>0, 𝛿>0, 𝑇>0, 𝜀>0, 𝛼>0, and weighting matrices 𝜼₁, 𝜼₂, 𝐑>0, the closed-loop system (7) is FTB with respect to (𝑐₁𝑐₂𝛿𝑇𝐑) and satisfies the cost function (9) for all admissible 𝐰(t) with the constraint condition and all admissible uncertainties, if there exist constants 𝑐₂>0, 𝛽>0, symmetric positive-definite matrices 𝐏 and 𝐐>2𝜀𝜼^T₂𝜼₂, such that conditions (12), (13), and the following inequality hold: Ω₂=−𝐏𝐂^T∗−𝛽𝐈<0. (28)

Proof. Defining the same Lyapunov function as Theorem 9. Under the zero initial state condition 𝝋(𝑡)=0,𝑡∈[−𝑑0], we can rewrite (24) as 𝐕(𝐱(𝑡))<𝛼𝑒^𝛼𝑡𝑡₀𝑒^−𝛼𝑠𝐰^T(𝑠)𝐰(𝑠)d𝑠. (29) Hence, from (14) and ∀𝑡∈[0𝑇], we can get 𝐱^T(𝑡)𝐏𝐱(𝑡)<𝐕(𝐱(𝑡))<𝛼𝑒^𝛼𝑡𝑡₀𝑒^−𝛼𝑠𝐰^T(𝑠)𝐰(𝑠)d𝑠<𝛼𝑒^𝛼𝑇𝑇₀𝐰^T(𝑡)𝐰(𝑡)d𝑡. (30) Furthermore, from (28) and using Schur complement, we have 𝐂^T𝐂<𝛽𝐏. (31) Then, it follows 𝐲^T(𝑡)𝐲(𝑡)=𝐱^T(𝑡)𝐂^T𝐂𝐱(𝑡)<𝛽𝐱^T(𝑡)𝐏𝐱(𝑡)<𝛽𝛼𝑒^𝛼𝑇𝑇₀𝐰^T(𝑡)𝐰(𝑡)d𝑡. (32)
Taking the maximum value of ‖𝐲(𝑡)‖²_∞, we have ‖𝐲(𝑡)‖²_∞<𝛽𝛼𝑒^𝛼𝑇‖𝐰(𝑡)‖²₂ with 𝑡∈[0𝑇]. Therefore, condition (9) can be guaranteed by letting 𝛾=𝛽𝛼𝑒^𝛼𝑇. This completes the proof.

In order to solve Theorem 11, we can obtain the relevant algorithm by imposing further LMI constraints in the design phase. Substituting the corresponding matrices into matrix inequalities (12), (13), and (28), we can draw the following Theorem 12.

Theorem 12. For given 𝑐₁>0,𝛿>0,𝑇>0,𝜀>0,𝛼>0, and weighting matrices 𝜼₁, 𝜼₂, 𝐑>0, the closed-loop system (7) is FTB with respect to (𝑐₁𝑐₂𝛿𝑇𝐑), exists a nonfragile state-feedback controller gain 𝐊=𝐘𝐗⁻¹, and satisfies the cost function (9) for all admissible 𝐰(t) with the constraint condition and all admissible uncertainties, if there exist positive constants 𝑐₂,𝛽,𝜇₁,𝜇₂,𝜇₃, and 𝜇₄, symmetric positive- definite matrices 𝐗, 𝐐, and 𝐙, and real matrix 𝐘, such that the following LMIs hold: 𝚺₁₁𝐀_𝑑𝐆𝐗𝜼^T₁𝐅𝐗𝐒^T₁𝐗𝐍^T∗−𝐐𝟎𝜼^T₂𝟎𝐒^T₂𝟎∗∗−𝛼𝐈𝟎𝟎𝐒^T₃𝟎1∗∗∗−2𝜀⁻¹𝐈𝟎𝟎𝟎∗∗∗∗−𝜀𝐈𝐒^T₄𝟎∗∗∗∗∗−𝜇₁𝐈𝟎∗∗∗∗∗∗−𝜇₂𝐈<0,−𝐗𝚺₁₂𝐗𝐒^T₁𝐗𝐍^T∗𝚺₂₂𝟎𝟎∗∗−𝜇₃𝐈𝟎∗∗∗−𝜇₄𝐈<0,𝜎₁𝐑⁻¹<𝐗<𝐑⁻¹,0<𝐐<𝜎₂𝐑,𝑐₁𝑑𝜎₂+𝛿1−𝑒^−𝛼𝑇−𝑐₂𝑒^−𝛼𝑇√𝑐₁√𝑐₁−𝜎₁<0, (33)(34)(35)(36)(37) where 𝚺₁₁=𝐗𝐀^T+𝐀𝐗+𝐙−𝛼𝐗+𝜇₁𝐌₁𝐌^T₁+𝜇₂𝐁𝐇𝐇^T𝐁^T+𝐁𝐘+𝐘^T𝐁^T,𝚺₁₂=𝐗𝐂^T+𝐘^T𝐃^T,𝚺₂₂=−𝛽𝐈+𝜇₃𝐌₂𝐌^T₂+𝜇₄𝐃𝐇𝐇^T𝐃^T.

Proof. In order to deal with the uncertainties in inequality (12), we can rewrite it as the following inequality: Ω₁=𝚷₁₁𝐏𝐀_𝑑𝐏𝐆𝜼^T₁𝐏𝐅∗−𝐐𝟎𝜼^T₂𝟎1∗∗−𝛼𝐈𝟎𝟎∗∗∗−2𝜀⁻¹𝐈𝟎∗∗∗∗−𝜀𝐈<0, (38) where 𝚷₁₁=𝐀^T𝐏+𝐏𝐀+𝐐−𝛼𝐈.
Thus, the aforementioned inequality (38) is equivalent to Ω₁=Ω₁+Ω_1Δ<0, (39) where Ω₁=𝚷₁₁𝐏𝐀_𝑑𝐏𝐆𝜼^T₁𝐏𝐅∗−𝐐𝟎𝜼^T₂𝟎1∗∗−𝛼𝐈𝟎𝟎∗∗∗−2𝜀⁻¹,𝐈𝟎∗∗∗∗−𝜀𝐈𝚷₁₁=𝐀^T𝐏+𝐏𝐀+𝐐−𝛼𝐈,Ω_1Δ=Δ𝚷₁₁𝐏Δ𝐀_𝑑(𝑡)(𝑡)(𝑡),Δ𝐏Δ𝐆𝟎𝐏Δ𝐅∗𝟎𝟎𝟎𝟎∗∗𝟎𝟎𝟎∗∗∗𝟎𝟎∗∗∗∗𝟎𝚷₁₁=Δ𝐀^T(𝑡)𝐏+𝐏Δ𝐀(𝑡). (40)
Moreover, noticing the uncertainties which were described as the form in (2) and (6), we have Ω_1Δ=𝐉₁Γ(𝑡)𝐋₁+𝐉₁Γ(𝑡)𝐋₁^T+𝐉₂𝝆(𝑡)𝐋₂+𝐉₂𝝆(𝑡)𝐋₂^T, (41) where 𝐉₁=𝐏𝐌₁𝟎𝟎𝟎𝟎,𝐋₁=𝐒₁𝐒₂𝐒₃𝟎𝐒₄𝐉₂=𝟎𝟎𝟎𝟎𝐏𝐁𝐇,𝐋₂=𝐍𝟎𝟎𝟎𝟎. (42)
Using Lemma 8, it follows from (41) that Ω_1Δ=𝐉₁Γ(𝑡)𝐋₁+𝐉₁Γ(𝑡)𝐋₁^T+𝐉₂𝝆(𝑡)𝐋₂+𝐉₂𝝆(𝑡)𝐋₂^T≤𝜇₁𝐉₁𝐉^T₁+𝜇₁⁻¹𝐋^T₁𝐋₁+𝜇₂𝐉₂𝐉^T₂+𝜇₂⁻¹𝐋^T₂𝐋₂. (43) Thus, inequality (39) can be guaranteed by Ω₁+𝜇₁𝐉₁𝐉^T₁+𝜇₁⁻¹𝐋^T₁𝐋₁+𝜇₂𝐉₂𝐉^T₂+𝜇₂⁻¹𝐋^T₂𝐋₂<0. (44) Rewriting inequality (44) and applying Schur complement, we have Ω₁=𝚷₁₁𝐏𝐀_𝑑𝐏𝐆𝜼^T₁𝐏𝐅𝐒^T₁𝐍^T∗−𝐐𝟎𝜼^T₂𝟎𝐒^T₂𝟎∗∗−𝛼𝐈𝟎𝟎𝐒^T₃𝟎1∗∗∗−2𝜀⁻¹𝐈𝟎𝟎𝟎∗∗∗∗−𝜀𝐈𝐒^T₄𝟎∗∗∗∗∗−𝜇₁𝐈𝟎∗∗∗∗∗∗−𝜇₂𝐈<0, (45) where 𝚷₁₁=𝐀^T𝐏+𝐏𝐀+𝐐−𝛼𝐈+𝜇₁𝐏𝐌₁𝐌^T₁𝐏+𝜇₂𝐏𝐁𝐇𝐇^T𝐁^T𝐏.
On the other hand, considering the uncertainties in inequality (34), we have Ω₂=Ω₂+Ω_2Δ<0, (46) where Ω₂=^−𝐏𝐂T_{∗−𝛽𝐈}, Ω_2Δ=^{𝟎Δ𝐂T(𝑡)}_∗𝟎.
And Ω_2Δ can be expressed as Ω_2Δ=𝐉₃Γ(𝑡)𝐋₃+𝐉₃Γ(𝑡)𝐋₃^T+𝐉₄𝝆(𝑡)𝐋₄+𝐉₄𝝆(𝑡)𝐋₄^T, (47) where 𝐉₃=^𝟎_𝐌2, 𝐋₃=[𝐒₁𝟎], 𝐉₄=^𝟎_𝐃𝐇, 𝐋₄=[𝐍𝟎].
Using Lemma 8, we know the following inequality holds by real positive scalars 𝜇₃ and 𝜇₄: Ω_2Δ=𝐉₃Γ(𝑡)𝐋₃+𝐉₃Γ(𝑡)𝐋₃^T+𝐉₄𝝆(𝑡)𝐋₄+𝐉₄𝝆(𝑡)𝐋₄^T≤𝜇₃𝐉₃𝐉^T₃+𝜇₃⁻¹𝐋^T₃𝐋₃+𝜇₄𝐉₄𝐉^T₄+𝜇₄⁻¹𝐋^T₄𝐋₄. (48) Hence, inequality (46) holds by the following inequality: Ω₂+𝜇₃𝐉₃𝐉^T₃+𝜇₃⁻¹𝐋^T₃𝐋₃+𝜇₄𝐉₄𝐉^T₄+𝜇₄⁻¹𝐋^T₄𝐋₄<0. (49) Rewriting (49) and using Schur complement, we have Ω₂=−𝐏Λ₁₂𝐒^T₁𝐍^T∗𝚺₂₂𝟎𝟎∗∗−𝜇₃𝐈𝟎∗∗∗−𝜇₄𝐈, (50) where Λ₁₂=𝐂^T+𝐊^T𝐃^T.
Define 𝐗=𝐏⁻¹, 𝐘=𝐊𝐗, 𝐙=𝐗𝐐𝐗, pre- and postmultiplying inequality (45) by block-diagonal matrix diag{𝐏⁻¹𝐈𝐈𝐈𝐈𝐈𝐈}, and pre- and postmultiplying inequality (50) by a block-diagonal matrix diag{𝐏⁻¹𝐈𝐈𝐈}. They lead to LMIs (33) and (34).
Finally, denote 𝐗=𝐑^1/2𝐗𝐑^1/2, 𝐐=𝐑^−1/2𝐐𝐑^−1/2, and set 𝜎₁≤𝜆_min(𝐗),𝜆_max(𝐗)<1,𝜆_max(𝐐)≤𝜎₂, and consider 𝜆_max(𝐗)=1/𝜆_min(𝐏).
Then, inequality (13) can be guaranteed by 𝑐₁𝜎₁+𝑐₁𝑑𝜎₂+𝛿1−𝑒^−𝛼𝑇<𝑐₂𝑒^−𝛼𝑇. (51) Using the Schur complement and eigenvalue transformation, we can get LMIs (35)–(37) from inequality (51). This completes the proof.

Remark 13. Notice that 𝛾=𝛽𝛼𝑒^𝛼𝑇; if 𝛼 and 𝑇 have been given, the value of system 𝐿₂-𝐿_∞ performance 𝛾 only depends on 𝛽. Hence, we can obtain an optimal nonfragile robust finite-time 𝐿₂-𝐿_∞ controller, and the scalar 𝛽 can reduce to the minimum possible value such that LMIs (33)–(37) are satisfied. The optimization problem can be described as follows: min_{𝐗,𝐘,𝐐,𝑐2,𝛽,𝜇1,𝜇2,𝜇3,𝜇4,𝜎1,𝜎2}𝛾s.t.LMIs(33)-(37)with𝛾=𝛽𝛼𝑒^𝛼𝑇. (52)

Remark 14. When the controller gain variations of the Lipschitz nonlinear systems (1) are of the multiplicative form [6, 18]: Δ𝐊(𝑡)=𝐇𝝆(𝑡)𝐍𝐊,𝝆^T(𝑡)𝝆(𝑡)≤𝐈 (53) with 𝐇 and 𝐍 being known constant matrices, and 𝝆(𝑡) is the uncertain parameter matrix. In this case, the sufficient conditions for the nonfragile robust finite-time 𝐿₂-𝐿_∞ control of the nonlinear systems (1) are identical to LMIs (33)–(37), except that 𝐗𝐍^T, 𝐍𝐗 are changed to 𝐘^T𝐍^T, 𝐍𝐘 in inequalities (33) and (34), respectively. The proof of this conclusion is similar to Theorem 12.

Remark 15. By using the MATLAB LMIs Toolbox, the feasibility of Theorem 12 and Remark 3 can be easily checked. In Section 4, a simulation example about the Lipschitz nonlinear systems with state delays will be given.

4. Simulation Example

Consider system (1) with the following parameters: 𝐀=−1.02.00.5−1.2,𝐀_𝑑=,,[]−1.00.51.20.4𝐁=1.00.8,𝐆=0.40.3,𝐅=0.6−0.31.00.4𝐂=1.51.7,𝐃=1.5,𝐌₁=.𝐌1.0−0.8₂=[0.6],𝐒₁=1.20.6,𝐒₂=𝐒0.71.0₃=[0.9],𝐒₄=0.41.1. (54) In addition, we set 𝑐₁=0.5,𝛿=1,𝑇=4,𝜀=1.2,𝛼=0.5,𝑑=0.2, and 𝐑=^1.20.4_0.40.8. The nonlinear function part in system (1) is chosen as the following form: 𝐟(𝐱(𝑡),𝐱(𝑡−𝑑))=0.3sin𝑥₁(𝑡)+0.5sin𝑥₁(𝑡−𝑑)0.4sin𝑥₂().𝑡−𝑑 (55) Then, for any 𝐱(𝑡)=[𝑥₁𝑥₂]^T and 𝐱(𝑡−𝑑)=[𝑥_1𝑑𝑥_2𝑑]^T∈ℜ², we have ‖𝐟(𝐱(𝑡),𝐱(𝑡−𝑑))‖²=0.09sin²𝑥₁+0.25sin²𝑥_1𝑑+0.3sin𝑥₁sin𝑥_1𝑑+0.16sin²𝑥_2𝑑≤20.12𝑥²₁+0.2𝑥²_1𝑑+0.08𝑥²_2𝑑. (56) According to (16), we select the weighting matrices as follows 𝜼₁=√0.12000,𝜼₂=√0√0.20.0.08 (57)

Remark 16. It is necessary to point out that the selection of the above weighting matrices not only needs to satisfy Assumption 2 but also needs to ensure that the LMI (33) is feasible. Namely, the LMIs designed in this note actually put a constraint on the size of the above weighting matrices.
By resorting to MATLAB LMIs Toolbox, solving Theorem 12 and Remark 13, respectively, we can obtain the following solutions under the above two cases.

Case 1. Let 𝐇=[0.3],𝐍=[0.30.2],𝝆(𝑡)=(0.5/(1+𝑡²))𝐈, Γ(𝑡)=0.8sin(𝑡)𝐈. Solving Theorem 12, which considering the additive controller gain variation, we get 𝐗=,0.6336−0.3118−0.31180.9425,𝐘=−0.2802−0.7564𝐊=𝐘𝐗⁻¹=−1.0000−1.1333, (58) with scalars value 𝛽=45.4950,𝛾=12.9647, and 𝑐₂=86.0363.

Case 2. Let 𝐇=[0.3], 𝐍=[0.5], 𝝆(𝑡)=(0.5/(1+𝑡²))𝐈, and Γ(𝑡)=0.8sin(𝑡)𝐈. Solving Remark 13, which is considering the multiplicative controller gain variation, we get 𝐗=,0.6356−0.3100−0.31000.9439,𝐘=−0.2644−0.6864𝐊=𝐘𝐗⁻¹=−0.9176−1.0285, (59) with scalars value 𝛽=45.5722,𝛾=12.9757, and 𝑐₂=86.1562.

In this note, with the initial states 𝐱₀=[0.50.2]^𝑇 and the disturbance input is chosen as 𝐰(𝑡)=0.6sin(20𝑡)1+𝑡²,𝑡≥0, (60)

The trajectories of uncontrolled system state 𝐱(𝑡) are depicted as Figure 1. The states and output simulation curves of closed-loop controlled system are, respectively, shown in Figures 2, 3, and 4 with the simulation time 𝑡∈[025]. Figure 5 shows the evolution of 𝐱^T(𝑡)𝐑𝐱(𝑡) (𝑡∈[04]) of the controlled system (7).

Figure 1 

The trajectories of uncontrolled system state 𝐱(𝑡).

Figure 2 

The trajectories of the controlled system state 𝐱1(𝑡).

Figure 3 

The trajectories of the controlled system state 𝐱2(𝑡).

Figure 4 

The trajectories of the controlled system output 𝐲(𝑡).

Figure 5 

The response of the controlled system for 𝐱T(𝑡)𝐑𝐱(𝑡) (𝑡∈[0𝑇]).

By calculation, we have ‖𝐲(𝑡)‖_∞/‖𝐰(𝑡)‖₂=0.5995 (Case 1) and ‖𝐲(𝑡)‖_∞/‖𝐰(𝑡)‖₂=0.7053 (Case 2). According to the above computing results and the simulation results, one can know that the nonfragile state feedback controller which was designed in this note can effectively guarantee that the concerned system (1) not only is finite-time bounded in fixed finite-time interval 𝑡∈[04] but also with an 𝐿₂-𝐿_∞ disturbance performance level.

Remark 17. It should be pointed out that in the simulation example, according to Definition 4, as long as the choice of 𝑐₁ with the initial states is satisfied to ‖𝐱^T_𝟎𝐑𝐱₀‖≤𝑐₁. Then, the nonfragile state feedback controlled system is FTB (i.e., the closed-loop system trajectories stay within a given bound) and satisfies the 𝐿₂-𝐿_∞ constraint condition (9).

5. Conclusions

In this paper, we have studied the nonfragile robust finite-time 𝐿₂-𝐿_∞ control problem for a class of uncertain Lipschitz nonlinear systems with time-delays. By using the Lyapunov function approach and linear matrix inequality techniques, a sufficient condition for the existence of nonfragile robust finite-time 𝐿₂-𝐿_∞ controller has been derived. Based on this condition, an optimization algorithm is provided to find nonfragile optimal control. Finally, a simulation example is presented to illustrate the effectiveness and applicability of the proposed approach.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (Grant no. 61203051), the Key Program of Natural Science Foundation of Education Department of Anhui Province (Grant no. KJ2012A014), and the Joint Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20123401120010).