Nonfragile Robust Finite-Time L 2L ∞ Controller Design for a Class of Uncertain Lipschitz Nonlinear Systems with Time-Delays

and Applied Analysis 3 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)ρ (t) ≤ Ι, (6) whereH andN are knownmatrices, 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 closedloop system can be obtained as follows: ẋ (t) = Ãx (t) + Ã d x (t − d) + G̃w (t) + F̃f (x (t) , x (t − d)) , y (t) = C̃x (t) , x (t) = φ (t) , ∀t ∈ [−d 0] , (7) where Ã = A + ΔA(t), Ã d = A d + ΔA d (t), G̃ = G + ΔG(t), F̃ = F + ΔF(t), C̃ = 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 c 1 > 0, δ > 0, and T > 0 and a symmetric matrix R > 0, the closedloop system (7) is said to be robust finite-time bounded (FTB) with respect to (c 1 c 2 δ T R), if there exists a constant c 2 > c 1 , such that the following relation holds for all the external disturbance w(t): x (t 0 )Rx (t 0 ) ≤ c 1 󳨐⇒ x (t)Rx (t) < c 2 , ∀t 0 ∈ [−d 0] , t ∈ [0 T] . (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 L 2 -L ∞ controller with disturbance attenuation γ > 0 for system (1) if the closedloop system (7) is FTB in the sense of Definition 4, and there exists a positive real constant γ > 0 such that 󵄩󵄩󵄩󵄩y (t) 󵄩󵄩󵄩󵄩 2 ∞ ≤ γ 2 ‖w (t)‖2 2 , (9) where ‖y(t)‖ ∞ = sup t∈[ 0 T ] [yT(t)y(t)], ‖w(t)‖ 2 =


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][5][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][10][11][12][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][15][16][17][18][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 ( 2 - ∞ ) 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  2 - ∞ performance index.After that many researchers have paid attention in  2 - ∞ control problems.For example, in [21], the authors studied the  2 - ∞ controller design problem for continuous-time multiple delayed linear systems; in [22], the problem of  2 - ∞ 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  2 - ∞ control schemes, the readers can refer to [21][22][23][24] and the references therein.
In short, to the best knowledge of authors, the problem of nonfragile robust finite-time  2 - ∞ 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  2 - ∞ 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  2 - ∞ 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. min (⋅),  max (⋅) denote the minimum and the maximum eigenvalue of the corresponding matrix, respectively. 2 [0 + ∞) denotes the Euclidean norm for vectors or the spectral norm of matrices.  2 [0 ] is the space of -dimensional square integrable function vector over [0 ].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.

Preliminaries and Problem Statement
Consider a class of Lipschitz nonlinear systems with timedelay and parameter uncertainties described by the following equations: where x() ∈ R  is the state, x( − ) ∈ R  is the constant time-delay state, u() ∈ R  is the controlled input, w() ∈ R  is the disturbance input that belongs to  2 [0 + ∞), f(x(), x( − )) ∈ R  × R  = R   is the nonlinear function which represents the known nonlinear disturbance related to the state and the time-delay state, and () ∈  2 [− 0] is a continuous vector-valued initial function.In addition, A, A  , B, G, F, C, and D are known real matrices, and ΔA(), ΔA  (), ΔG(), ΔF(), and ΔC() are unknown time-variant matrices representing the norm-bounded parameter uncertainties and satisfy the following form: where M 1 , M 2 , S 1 , S 2 , S 3 , and S 4 are known real constants matrices and Γ() 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() is time-varying, therefore it can be allowed to be statedependent in some case, that is, Γ() = Γ(, x()).
The following preliminary assumptions are made for system (1).Assumption 1.For any given positive number  and constant time , the external disturbances input w() is time-varying and satisfies Assumption 2. f(x(), x( − )) : R  × R  → R   is a known nonlinear function which satisfies the following Lipschitz conditions: where the weighting matrices  1 and  2 are known real constant matrices.
Our aim is to develop a nonfragile state feedback controller: where the matrix K is the controller gain to be determined and ΔK() is the controller gain variations.Here, we consider the additive controller gain variation, that is, where H and N are known matrices, and () 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 closedloop system can be obtained as follows: where To formulate the problem addressed in this paper, we give the following definitions first.Definition 4 (FTB).For given constants  1 > 0,  > 0, and  > 0 and a symmetric matrix R > 0, the closedloop system ( 7) is said to be robust finite-time bounded (FTB) with respect to ( 1  2   R), if there exists a constant  2 >  1 , such that the following relation holds for all the external disturbance w(): Remark 5.In fact, if the closed-loop system (7) does not exist the exogenous disturbance input, that is, w() = 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  2 - ∞ controller with disturbance attenuation  > 0 for system (1) if the closedloop system (7) is FTB in the sense of Definition 4, and there exists a positive real constant  > 0 such that where 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  > 0 and vectors x, y ∈ R  , then Lemma 8 (see [27]).Let M and N be real matrices of appropriate dimensions.Then for any matrix F() satisfying F Τ ()F() ≤ I and a scalar  > 0,

Main Results
In this section, we will solve the problem of nonfragile robust finite-time  2 - ∞ 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.
Remark 10.Without the time-delay in the resulted closedloop 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  2 - ∞ controller design.Theorem 11.For given  1 > 0,  > 0,  > 0,  > 0,  > 0, and weighting matrices  1 ,  2 , R > 0, the closedloop system (7) is FTB with respect to ( 1  2   R) and satisfies the cost function (9) for all admissible w(t) with the constraint condition and all admissible uncertainties, if there exist constants  2 > 0,  > 0, symmetric positive-definite matrices P and Q > 2 T 2  2 , such that conditions (12), ( 13), and the following inequality hold: Proof.Defining the same Lyapunov function as Theorem 9.
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  1 > 0,  > 0,  > 0,  > 0,  > 0, and weighting matrices  1 ,  2 , R > 0, the closedloop system (7) is FTB with respect to ( 1  2   R), 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  2 , ,  1 ,  2 ,  3 , and  4 , symmetric positive-definite matrices X, Q, and Z, and real matrix Y, such that the following LMIs hold: where Proof.In order to deal with the uncertainties in inequality (12), we can rewrite it as the following inequality: where Π11 = ÃT P + P Ã + Q − I.Thus, the aforementioned inequality (38) is equivalent to where Moreover, noticing the uncertainties which were described as the form in ( 2) and ( 6), we have where Using Lemma 8, it follows from (41) that Thus, inequality (39) can be guaranteed by Rewriting inequality (44) and applying Schur complement, we have where On the other hand, considering the uncertainties in inequality (34), we have where And Ω 2Δ can be expressed as where Using Lemma 8, we know the following inequality holds by real positive scalars  3 and  4 : Hence, inequality (46) holds by the following inequality: Rewriting (49) and using Schur complement, we have where Finally, denote X = R 1/2 XR 1/2 , Q = R −1/2 QR −1/2 , and set  1 ≤  min ( X),  max ( X) < 1,  max ( Q) ≤  2 , and consider  max ( X) = 1/ min ( P).
Then, inequality (13) can be guaranteed by  Remark 14.When the controller gain variations of the Lipschitz nonlinear systems (1) are of the multiplicative form [6,18]: with H and N being known constant matrices, and () is the uncertain parameter matrix.In this case, the sufficient conditions for the nonfragile robust finite-time  2 - ∞ control of the nonlinear systems (1) are identical to LMIs (33)-(37), except that XN T , NX are changed to Y T N 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 be given.

Simulation Example
Consider system (1) with the following parameters: In addition, we set  1 = 0.5,  = 1,  = 4,  = 1.2,  = 0.5,  = 0.2, and R = [ 1.2 0.4 0.4 0.8 ].The nonlinear function part in system (1) is chosen as the following form: Then, for any According to (16), we select the weighting matrices as follows 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.In this note, with the initial states x 0 = [0.5 0.2]  and the disturbance input is chosen as w () = 0.6 sin (20) 1 +  2 ,  ≥ 0, The trajectories of uncontrolled system state x() are depicted as Figure 1.The states and output simulation curves of closed-loop controlled system are, respectively, shown in Figures 2, 3  Figure 5 shows the evolution of x T ()Rx() ( ∈ [0 4]) of the controlled system (7).
By calculation, we have ‖y()‖ ∞ /‖w()‖ 2 = 0.5995 (Case 1) and ‖y()‖ ∞ /‖w()‖ 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  ∈ [0 4] but also with an  2 - ∞ 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  1 with the initial states is satisfied to ‖x T 0 Rx 0 ‖ ≤  1 .Then, the nonfragile state feedback controlled system is FTB (i.e., the closed-loop system trajectories stay within a given bound) and satisfies the  2 - ∞ constraint condition (9).

Conclusions
In this paper, we have studied the nonfragile robust finitetime  2 - ∞ 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  2 - ∞ 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.

2 Figure 3 :
Figure 3: The trajectories of the controlled system state x 2 ().

Figure 4 :K
Figure 4: The trajectories of the controlled system output y().