An improved result on reachable set estimation and synthesis of time-delay systems☆
Introduction
The reachable set is one set that contains all the states that can be reached when the system starts from all allowable initial conditions and inputs. Reachable set estimation has an important role in ensuring safe operation in practical engineering through synthesizing controllers to avoid undesirable (or unsafe) regions in the state space [9], [14]. The system is safe if the reachable set of the system does not contain any unsafe state. Reachable set estimation of dynamic systems has various applications in peak-to-peak gain minimization [1], control systems with actuator saturation [8] and aircraft collision avoidance [9]. Recently, the reachable set estimation problem has been a hot topic and it has been considered for systems with different dynamics, such as discrete-time systems [12], neutral systems [18], neural networks [28] and systems with distributed delay [23], [25], [27].
Time-delay has received significantly attention due to its common presence in practical engineering and its detrimental effects on stability and performance of systems. Therefore, various problems of time-delay systems have been intensively considered by many researchers, to mention a few, stability analysis and stabilization [5], control [3], [22], passive and dissipative control [13], [19], [20], filtering [6], [21]. However, for the reachable set estimation problem of time-delay systems, there exists relatively little results. The Lyapunov–Razumikhin approach is employed to find an ellipsoid to bound the reachable set of the system from the origin by inputs with given peak value for the first time in [4]. However, the result in [4] is independent of the delay rate. An improved condition which is dependent on the rate of time-delay is investigated in [10] by using a Lyapunov–Krasovskii type functional and the result is extended to systems with polytopic uncertainties. By choosing pointwise maximum Lyapunov functional corresponding to a vertex of the polytope, less conservative estimation results are presented in [24], [26]. In order to further reduce the conservatism of existing results, the delay-partitioning method is introduced to study the reachable set estimation problem in [15]. However, the delay interval in [15] is divided into two uniform interval which may lead conservative results because the minimal bounding ellipsoid may be obtained in the non-uniform partitioning case. Moreover, the triple integral term of Lyapunov functional is employed in [11] which also improves the result in [10]. Motivated by this discussion, the non-uniform partitioning method and the triple integral technique will be utilized in this paper to obtain less conservative results. On the other hand, the reachable set synthesis problem has not been considered in the literature. This is an important issue because the state of a system should be restricted within a safety area in order to make the system operation safe. This motivates the reachable set synthesis problem to be addressed in this paper.
In this paper, we consider the problems of reachable set estimation and synthesis of continuous time-delay systems. An improved reachable set estimation condition is proposed in terms of LMIs by utilizing the non-uniform partitioning method. Based on this condition, the desired state-feedback controller is designed such that the reachable set of the closed-loop system is bounded by a given ellipsoid. Numerical examples are given to illustrate the effectiveness of the presented results.
The rest of this paper is briefly outlined as follows. In Section 2, the reachable set estimation and synthesis problems are formulated. A sufficient condition of reachable set estimation and a state-feedback controller design method are presented in Section 3. To illustrate the reduced conservatism and the improved effectiveness, examples are given in Section 4. We conclude the paper in Section 5.
Notation: The notation used throughout the paper is standard. denotes the n-dimensional Euclidean space and () means that P is real symmetric and positive definite (semi-definite); I and 0 denote the identity matrix and zero matrix, respectively, with compatible dimensions; stands for the symmetric terms in a symmetric matrix and is defined as . Matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.
Section snippets
Problem statement
Consider a class of linear continuous-time systems with time-varying delay described bywhere is the state vector; and B denote constant matrices with appropriate dimensions; is the time-varying delay and satisfyingand is the disturbance satisfyingThe delay interval is non-uniformly divided into two segments, . Denote . The reachable set of
Main results
In this section, the reachable set estimation and synthesis of continuous time-delay systems are addressed by employing a non-uniform delay-partitioning method together with the reciprocally convex combination technique. Firstly, the reachable set estimation problem can be solved in terms of LMIs as follows. Theorem 1 If there exist a scalar , matrices , and W such that the following LMIs hold:where
Examples
Example 1 Consider the continuous-time delay system with the following parameters (borrowed from [17]):
Different values of can be obtained by using different methods for and , respectively. In order to compare the result in Theorem 1 with the existing references [10], [15], [26], the obtained values of minimum of are listed in Table 1. It can be seen that the minimum obtained by Theorem 1 is less than that of [10]
Conclusions
The problems of reachable set estimation and synthesis of continuous time-delay systems have been studied in this paper. Sufficient conditions in terms of strict LMIs have been proposed for reachable set estimation problem first by employing the non-uniform delay-partitioning method together with the reciprocally convex combination technique. Based on this, a state-feedback controller has been designed to guarantee reachable set of the closed-loop system to be bounded by a given ellipsoid. The
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This work was partially supported by the National Natural Science Foundation of China (61304063), Liaoning Provincial Natural Science Foundation of China (2013020227) and GRF HKU 7137/13E.