Finite-time H∞ static and dynamic output feedback control for a class of switched nonlinear time-delay systems
Introduction
The concept of Lyapunov stability focuses on the infinite-time response of the system and it doesn't care about the transient response [1]. However, in some applications, haveing large peaks in the state variables in a finite-time interval is not admissible [2,3], or the system's response is important just over a finite time interval. In such systems, Lyapunov stability is not an appropriate tool [4]. To solve this issue, the reference [5] has provided definitions of finite-time stability (FTS) for a nominal system and finite-time boundedness (FTB) for a system with exogenous disturbances [4,6]. The FTB concept means that the state variables of the system do not pass a prescribed bound during a finite time interval in the presence of an exogenous disturbance.
On the other hand, switched systems as a kind of hybrid systems are very common in many applications such as the automotive industry [7], switching power converters [8], network system control [9], traffic control [10], etc. The most of existing methods in this regard are based on the Lyapunov theory, linear matrix inequalities (LMIs), and the average dwell time (ADT) or the persistent dwell time (PDT) methods [11], [12], [13], [14], [15], [16]. Moreover, the time-delay and exogenous disturbances have been commonly considered in the physical systems [17], [18], [19]. Recently, many researchers have focused on the response of switched delay systems in a finite time interval in the presence of exogenous disturbances. The considered systems are stochastic ones [20,21], singular types [22], switched neutral systems [23], switched system in presence of input saturation [24], observer-based control [25], and other types of switched systems [3,[26], [27], [28], [29]. Some of these papers have addressed the design of static [30] and dynamic [26,27] output feedback controllers. However, these references have used the ordinary Lyapunov approach which suffers from conservatism. Moreover, to the best of authors’ knowledge, finite-time H∞ static and dynamic output feedback controllers have not been studied in the case of the considered class of switched nonlinear (or even linear) time-delay systems based on auxiliary matrices and average dwell time method in the presence of external disturbances and uncertain parameters of the system. This motivated us to propose less conservative output feedback controllers based on the Finsler's lemma and the H∞ analysis.
In this paper, static and the dynamic output feedback controllers are designed for a class of switched nonlinear time-delay systems in the presence of exogenous disturbances and parameter uncertainties using auxiliary matrices and the ADT method. The proposed method is less conservative and has more degrees of freedom to achieve a feasible solution. Since using auxiliary matrices leads to a separation between the Lyapunov matrix and the state matrix of the system [31,32]. The reference [16] has proved that the ordinary Lyapunov approach is a special case of using the auxiliary matrices. On the other hand, since in some practical systems, the state variables are not available, in this paper, static and dynamic output feedback controllers are designed using the auxiliary matrices to reduce the conservatism. Then, the results of the paper are derived for linear cases to avoid unnecessary conservatism. Finally, the proposed controllers are compared with the existing papers and it is shown that the theorems of this paper are less conservative compared to the existing methods.
The organization of this paper is as follows. Section 2 introduces the considered class of systems and gives some important definitions and assumptions. In Section 3, sufficient conditions are presented to guarantee the finite-time boundedness of the unforced system. Static and dynamic output feedback controllers are designed to make the closed-loop system FTB in Sections 4 and 5, respectively. Section 6 presents numerical examples to illustrate the efficiency of the proposed results and to compare the results with other papers. Finally, conclusion and future work are given in Section 7.
Notation: In this paper, the symbols λmax(.) and λmin(.) represent the largest and smallest eigenvalues of a matrix, respectively. Rn indicates all n-dimensional real-valued vectors. The notation I is an identity matrix with the appropriate dimension. AT means the transpose of the matrix A. Notation ‖.‖ denotes the Euclidean norm of matrices. The symbol “*” is used to indicate the elements induced by symmetry.
Section snippets
Preliminaries and problem definition
Consider a class of switched nonlinear time-delay system as follow:where x(t) ∈ Rn is the state variable, uσ(t)(t) ∈ Rp is the control input, y(t) ∈ Rl is the output of the system and w(t) ∈ Rr, is an exogenous disturbance satisfying the following condition for a δ ≥ 0:
Moreover, τ is a constant time-delay, is the initial
Finite-time boundedness
In this section, the FTB analysis for the switched nonlinear time-delay system (1) is presented. First, the following lemmas are presented which are necessary for the design process:
Lemma 1 (Finsler's lemma [31,32]): Consider x ∈ Rn, Φ ∈ Rn × n, and U ∈ Rm × n such that rank(U) = r < n. Then, the following conditions are equivalent:
- 1.
xTΦx < 0, ∀x ∈ Rn Such that x ≠ 0 and Ux = 0.
- 2.
∃F ∈ Rn × m satisfying Φ + FU + UTFT < 0.
Lemma 2 [33]: Consider the full row rank matrix and the
The finite-time H∞ static output feedback controller
In this section, a finite-time H∞ static output feedback controller is designed for the switched nonlinear time-delay system (1) in the presence of external disturbances.
Consider the static output feedback controller as
The goal is to design the control law (40) to make the closed-loop system FTB according to Definition 2. For this purpose, substituting (40) into the switched system (1) leads to the following closed-loop system for σ(t) = i:
The finite-time H∞ dynamic output feedback controller
In this section, a finite-time H∞ dynamic output feedback controller is proposed to guarantee the FTB condition of the closed-loop system (1). Consider the following dynamic output feedback controller:where Akσ(t), Bkσ(t), Ckσ(t), and Dkσ(t) will be designed later. Substituting the dynamic controller (57) into the switched nonlinear system (1) leads to the following closed-loop system:
Simulation results
In this section, three examples are presented to illustrate the validity and effectiveness of the proposed method.
Example 1: Consider the switching system (1) with two subsystems as follows:
Subsystem 1:
Subsystem 2:
Conclusion
In this paper, a class of switched nonlinear time-delay systems was considered in the presence of exogenous disturbances and uncertain parameters of the system. The goal was to design static and dynamic output feedback controllers to have an FTB closed-loop system. For this purpose, using the Lyapunov analysis, the ADT method, and the auxiliary matrices, some sufficient conditions were extracted in the form of LMIs. Besides, sufficient conditions were extracted for the linear case with whether
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