Finite-time H∞ control for a class of discrete-time nonlinear singular systems☆
Introduction
The singular system is referred as the differential-algebraic system, implicit system, or descriptor system. It is well known that the system has been extensively applied in many scientific areas, such as circuit systems, electrical networks, mechanical systems and so on [1]. Based on the system behavior, many researchers are attracted in the last three decades, and numerous results have been obtained on the stabilization and H∞ control for linear singular system [1], [2], [3], [4], [5], [6]. Using linear matrix inequality (LMI) approach, the sufficient conditions for new stability criteria were given by constructing a Lyapunov-Krasovskii functional and representing a neutral system in the singular form in [3]; while in [4], necessary and sufficient conditions were derived for quadratic d stabilizability with an H∞ norm-bound; under the same method, a robust controller was developed for the discrete-time singular Markov jump systems with actuator saturation in [5]. In [6], based on the obtained stochastic bounded real lemmas, the stochastic H∞ control problem was discussed for discrete-time singular systems with state and disturbance dependent noise. By contrast, fewer works were dedicated to the nonlinear singular system [7], [8], [9], [10], [11], [12]. Based on the Lyapunov–Kravoskii stability theory, co-design method for the H∞ controller and the event-triggered scheme was presented by using the singular value decomposition technology in [7]. In [8], the stabilization, robust control and adaptive robust control problems were considered for the nonlinear singular system with undecomposed status. The stabilization and robust stabilization were investigated by the feedback linearization technique in [9] and by some stability theorems in [10], respectively. It is worth noting that a Luenberger-like observer is designed for a large class of nonlinear singular systems with multiple outputs in [11] and nonlinear singular time-delay systems in [12], respectively.
Compared with Lyapunov Asymptotic Stability, the finite-time stabilization (FTS) studies the behavior of the system within a finite (possibly short) interval. The finite-time stabilization problem has also received a lot of attention in recently years [13], [14], [15], [16], [17], [18], [19], [20], [21], [22]. The finite-time control design of the linear systems was presented in [13] via the state feedback. The admissible finite-time stability was considered by using LMIs method for linear discrete singular systems with time-delay in [14], [15] and for linear continuous singular systems in [16], respectively. In [17], by designing a gain-switched state feedback controller, the finite-time stabilization problems were solved for linear time-varying systems. Based on the controlled output and the disturbance input, input-output finite-time stable controllers were designed for the time-varying linear singular systems in [18]. In [19], the finite-time H∞ control problem was considered for the nonlinear discrete Hamiltonian singular system by a proper state-feedback, while the finite-time stabilization problem of the nonlinear Hamiltonian singular system and the nonlinear differential-algebraic system was studied for the state of system approaching to equilibrium point 0 in time T in [20] and [21], respectively. In [22], a finite-time observer was realised by using the well-known homogeneous technique for linear time-delay systems with commensurate delay.
It is clear that the asymptotic stabilization is only studied in references [7], [8], [9], [10]; while finite-time stabilization problems are discussed for linear systems or linear singular system in references [13], [14], [15], [16], [17], [18]. For discrete-time nonlinear Hamiltonian singular systems, the finite-time control problems were investigated in [19] by decomposing system state, which usually leads to the more limited conditions for nonlinear singular systems. In recent years, Hamiltonian function approach has received a lot of attention in control designs of nonlinear systems, and many significant results on stabilization and H∞ control have been obtained [19], [20], [25], [26], [27], [28], [29]. Motivated by the Hamiltonian function approach, we established state undecomposed method to study the asymptotic stabilization problems for nonlinear singular system [8]. This paper considers the finite-time control problems for a class of discrete-time nonlinear singular systems via state undecomposed method. Compared with the literatures mentioned above, the application of the system is wider. In this paper, there is no any constraint for singular matrix E. Therefore, we can design controllers for more nonlinear singular systems by using state undecomposed method.
Section snippets
Finite-time stabilization of discrete-time nonlinear singular system
In this section, we study the finite-time stabilization of discrete-time nonlinear singular system.
Consider the following discrete-time nonlinear singular system:
where is the state of the system, is the control input; 0 < rank; and . Then the system (2.1) can be rewritten as
To facilitate the analysis, the definition and lemma are given as follows:
Definition 2.1 The[23]
Finite-time H∞ control of discrete-time nonlinear singular system
In this section, we investigate the finite-time H∞ control problems of discrete-time nonlinear singular system.
Consider the following discrete-time nonlinear singular system: where is penalty signal, is external disturbance, and xk, uk, Ak and Bk are the same as those in the system (2.2).
The definition of finite-time H∞ boundedness for the system (3.1) is described as follows: Firstly, we show that the
Illustrative example
In this section, we give an illustrative example on population model to show how to use Corollary 3.2 to design a finite-time H∞ controller for discrete-time nonlinear singular systems.
Example 4.1 Considering the following population model system:
where k is discrete time variable, denotes 2008, x1(k) indicates urban population in the k year, x2(k)
Conclusion
The paper discusses the finite-time stabilization and finite-time H∞ control problems for discrete-time nonlinear singular system. First of all, we have extended the definitions of finite-time stabilization and finite-time H∞ boundedness for discrete-time nonlinear singular system. Then, the finite-time stabilization and finite-time H∞ boundedness of the system are studied, which construct the suitable controllers. Finally, an example to illustrate the effective of the proposed results is given.
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2022, Journal of the Franklin InstituteCitation Excerpt :A lot of results related to the finite-time stability problem have been reported in the literature, see, e.g., [17–25]. Recently, there has also been a lot of interest for the finite-time stability problem of the singular system. [26–29] studied the finite-time stability problem for the slow subsystem of the singular system.