Mathematical Problems in Engineering
Volume 2021 (2021), Article ID 7497720, 10 pages
https://doi.org/10.1155/2021/7497720

Research Article

Finite-Time Simultaneous Stabilization of a Set of Nonlinear Singular Systems

Liying Sun¹ and Meiqing Li²

¹School of Business, Shanghai Dianji University, Shanghai 201306, China
²School of Automation, Nanjing University of Science and Technology, Nanjing 210094, China

Correspondence should be addressed to Liying Sun; ss_sunly@163.com

Received 19 April 2021; Accepted 11 June 2021; Published 19 June 2021

Academic Editor: Cuimei Jiang

Copyright © 2021 Liying Sun and Meiqing Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This study addresses the problems of finite-time simultaneous stabilization for two nonlinear singular systems and more than two nonlinear singular systems. First, we design a suitable output feedback controller and combine the two nonlinear singular systems to generate an augmented system by using an augmented technique. Based on a sufficient condition of the augmented system impulse-free, an important result, that is, the augmented system is finite-time stabilization is presented. Then, the finite-time stabilization problem of more than two singular systems is investigated by dividing the N systems into two sets. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed finite-time stabilization controller for two nonlinear singular systems.

1. Introduction

In recent years, the singular system has attracted extensive attention of researchers, which is also called the descriptor system, generalized state-space system, differential-algebraic system, and so on. Due to its unique properties and structure, the singular system is more complex than the normal state-space system. It needs to consider impulse-free (continue-time system) or causality (discrete-time system), regularity, and stable [1].

It is worth pointing out that the simultaneous stabilization problem of the normal state-space system is introduced in 1982. Abundant results of simultaneous stabilization have been obtained in recent years [2–6]. In [2], several simultaneous external and internal stabilization problems have been considered under appropriate adaptive low-and-high gain feedback controllers. Simultaneous stabilization of linear systems has been solved by the static output feedback in [3]. In [4], the simultaneous stabilization of the multiinput-multioutput system has been investigated to develop a stabilizing controller. A necessary and sufficient condition for the existence of simultaneously quadratically stabilizing state feedback laws is derived for a collection of single-input discrete-time nonlinear systems in [5]. In [6], simultaneous stabilization of a set of nonlinear port-controlled Hamiltonian systems is investigated by using the dissipative Hamiltonian structural properties. Subsequently, some results of simultaneous stabilization are obtained for singular systems [7, 8]. Simultaneous stabilization problems are investigated for nonlinear singular systems with actuator saturation by using a suitable output feedback in [7] and an uncertain singular system with input saturation via using linear matrix inequalities in [8], respectively.

By the virtue of the method of simultaneous stabilization, the finite-time simultaneous stabilization problem is considered for the nonlinear singular system. Definition of finite-time stabilization is proposed for the linear system [9, 10] and then extended to the linear singular system [11]. Furthermore, the finite-time stabilization is investigated for the nonlinear discrete-time Hamiltonian singular system [12]. Finite-time stabilization, which is different from asymptotic stability [13–17], is used to describe that the state does not exceed a certain bound during a fixed finite-time interval. The finite-time stabilization of the nonlinear continue-time singular system and nonlinear discrete-time singular system is investigated via the state-undecomposed method in [18, 19], respectively. In [20], the stochastic finite-time filtering issue for a class of nonlinear continuous-time singular semi-Markov jump systems is discussed in the forms of strict LMIs. While, stability of the robust finite-time for linear singular Markovian jump systems with impulsive effects and time-varying norm-bounded disturbance is considered under the designed state feedback controller and estimation of domain of attraction in [21]. To our best knowledge, a few results are presented for finite-time simultaneous stabilization of the singular system. In [22], finite-time control and fault detection are studied simultaneously for the singular system via the average dwell time approach and using some novel integral inequalities. When it comes to the nonlinear singular system, the finite-time simultaneous stabilization of two nonlinear singular systems and more than two nonlinear singular systems is considered in this study. It is worth noticing that it is also called finite-time stability, i.e., states of the system reach the equilibrium point within a fixed time T and stay at the equilibrium point when t  T [23–29]. The finite-time robust stabilization problem of general nonlinear time-delay systems is studied based on the Hamiltonian function method and observer design in [26]. In [27], the finite-time stabilization is investigated for a class of singular systems by the constructed new Lyapunov functional, while the finite-time robust simultaneous stabilization and adaptive robust simultaneous stabilization have been investigated for nonlinear systems with time delay in [28, 29], respectively.

In this study, our goal is to discuss the simultaneous stabilization problem of two nonlinear singular systems or more than two nonlinear singular systems and propose the designed method of the output feedback controller. The study is divided into three parts. (i) Two nonlinear singular systems discussed about state bound in finite-time interval under the nonlinear function need not to satisfy the Lipschitz condition, and the suitable output feedback law is constructed. (ii) Based on the above method, finite-time simultaneous stabilization of more than two nonlinear singular systems is studied by dividing the N systems into two sets. (iii) The numerical example of two nonlinear singular systems is used to illustrate the validity of the proposed finite-time stable controller.

2. Finite-Time Simultaneous Stabilization of Two Nonlinear Singular Systems

Consider the following two nonlinear singular systems:

where and are the states of the two systems, is the control input, are the outputs, is the singular matrix, and rank ; , , is a sufficiently smooth vector field with proper dimensions, .

To facilitate the analysis of the two systems (1) and (2), we show the following definition and lemma.

Definition 1. (see [30]). A control law is called an admissible control law, if for any initial condition , the resulted closed-loop descriptor system has no impulsive solution, and accordingly, the original system is called impulse controllable.

Lemma 1. (see [31]). If a scalar function with has continuous th order partial derivatives, then can be expressed as

where are the scalar functions.
As mentioned above, the two systems can be described as follows:

We can design an output feedback controller:

where is a symmetric matrix. Substituting (6) into systems (4) and (5), the systems can be given as follows:

Combine the two closed-loop singular systems (7) and (8) into an augmented system:

where

Lemma 2. If , then system (9) is impulse-free.

Proof. From , we know . So, there exist two nonsingular matrices , such that

Denote

where , , and . Then, system (9) can be expressed as follows:

Noticing that

we can obtain that , which implies that the index of equivalent closed-loop singular system (9) is one at the equilibrium point 0, that is, system (9) is impulse-free.
Next, we present the definition of the finite-time simultaneous stable.

Definition 2. The two nonlinear singular systems (4) and (5) are said to be the finite-time simultaneous stable (FTSS) with respect to , with and , if , such that , where , , and , .
Based on augmented system (9), we give the following result of the finite-time simultaneous stable of systems (4) and (5).

Theorem 1. If there exist symmetric positive definite matrices , , and , such that

Then, under the output feedback control law (6), the nonlinear singular systems (4) and (5) are finite-time simultaneous stable with respect to , where

, , , , , and and indicate the maximum and minimum eigenvalues of the argument, respectively.

Proof. According to Lemma 2 and condition (17), it is clear that augmented system (9) is impulse-free. Choose the Lyapunov function

i.e., , where .
Based on system (9),

Under condition (15), there is

Furthermore, by integrating the inequality (21) between 0 and t, it is clear that

Consider the following chain of inequalities:

where , and .
If , taking account of (16), (22)–(24), it can be deduced that

Hence, system (9) is finite-time stable with respect to .
To proceed further, systems (4) and (5) can be finite-time simultaneous stabilization with respect to .

3. Finite-Time Simultaneous Stabilization of More than Two Nonlinear Singular Systems

Consider nonlinear singular systems:

where , , and are the states, the outputs, and the inputs of the systems, respectively; , , and .

Based on the method given in Section 2, we can represent system (26) into the following forms:

Suppose that is an arbitrary permutation of and that is a position integer satisfying . Denote and . We divide the systems into two sets:

where

Design an output feedback controller:

where matrix is the symmetric. Substitute (31) into systems (28) and (29), respectively. Then, the closed-loop systems can be given as follows:

By the system-augmentation technique, systems (32) and (33) can be transformed as

where

Based on Theorem 1, we can obtain the following result.

Theorem 2. If there exists a positive scalar and symmetric positive definite matrices , , such that

then, under the output feedback control law (31), system (26) is finite-time simultaneous stabilization with respect to , where , , , and are the same as those in Theorem 1. Noted that,

Proof. Taking the same as Theorem 1, system (34) is impulse-free depending on Lemma 2 and condition (38). We choose a Lyapunov function , where , and are the symmetric positive definite matrices.
Based on conditions (36) and (37), we have

To sum up,

By integrating inequality (41) between 0 and T with , the following result can be presented:

Next, we will prove the finite-time simultaneous stabilization for more than two nonlinear singular systems.
It is straightforward to see

where , and . According to the concept of finite-time simultaneous stable, from and (37), we can obtain

By using the similar method as that in Theorem 1, system (34) is finite-time stable with respect to , and it can be shown that system (26) is finite-time simultaneous stabilization with respect to .

4. Illustrative Example

In this section, we present an example to illustrate the effectiveness of the proposed method which addresses the finite-time simultaneous stabilization problem of the two nonlinear singular systems.

Example 1. Consider the two nonlinear singular systems:

where , we give the following parameters:

Based on Lemma 1, it can deduce that

Design an output feedback controller

Substituting (49) into systems (45) and (46), respectively, we obtain

Based on form (9), we have

where

Choose

According to Lemma 2, we can easy check that the augmented system is impulse-free.
Next, we give Lyapunov function , where Let By the virtue of conditions (15) and (16) in Theorem 1, we have

Choose the initial state and , then , the trajectory of . Denote and . The trajectories of , and , state response, and control signal are shown in Figures 1–3, respectively. Obviously, the augmented nonlinear singular system (52) is finite-time stable with respect to . Hence, under the output feedback controller (49), systems (45) and (46) are finite-time simultaneous stabilization.

Figure 1: Responses of the values , , and .

Figure 2: Responses of the state and .

Figure 3: Responses of the control signal .

5. Conclusion

The finite-time simultaneous stabilization problem is considered for the two nonlinear singular systems or more than two nonlinear singular systems in this study. Under a suitable output feedback controller, the augmented singular system is first proved to be impulse-free and then is proved to be finite-time simultaneous stable by using the system-augmented technique. Finally, the finite-time simultaneous stabilization problem of more than two nonlinear singular systems is investigated with the same method. Moreover, the proposed method in this study can be extended to handle finite-time simultaneous stabilization of the two or more than two nonlinear singular systems.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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