Abstract
In this paper, the stability of familiar state constrained hybrid systems is considered. In the first, we prove the invariant set stability for the state constrained impulsive hybrid systems. Specifically, a robust control feedback method is applied for state constrained uncertain impulsive hybrid systems. With the auxiliary matrix assistance, some convergence criteria are derived to guarantee robust stability for state constrained uncertain hybrid systems with output disturbance by constructing the symmetric and asymmetric barrier Lyapunov functions (BLF), respectively. Finally, two comparative examples with simulations show that the proposed results are effective and superior.
1. Introduction
A hybrid system consists of many aspects, such as computation, communication, control, continuous dynamics, and discrete behavior [1, 2]. Hybrid system is a complex system, and its modeling methods mainly consist of two kinds: hybrid state machine and differential equation. Within each step, the state of the system is constantly evolving according to the dynamic law, until a transition occurs. Some transient phenomena of state changes (impulsive system) are one of the most important branches of hybrid systems and are essentially a nonlinear system. The theory of stability or finite-time stability of impulsive systems or hybrid systems has also obtained rapid development [3–7] in recent years. For the actual engineering system, all the actuators have the constraints of human participation or the inherent physical constraints of the dynamical system itself. For example, the valve has jammed shut while driving to a certain extent. The saturation nonlinearity of this actuator makes it necessary for us to consider controlling the influence of input saturation or constrained nonlinearity on the dynamical system when designing the control system. In [8], Bemporad et al. considered the problem of explicit linear quadratic regulator for constrained systems. In [9], Li et al. studied the input delayed nonlinear state constrained systems via adaptive tracking control. There are alternatives to solve the state constraint problems that are effective [10–17], such as basing on the invariant set principle and model predictive control. Recently, BLF is presented to consider the problem of constraint dynamical system. In [14], control design for partial participation constraints nonlinear systems is considered. In [15], Tee et al. present control design for delayed and output constraint nonlinear systems. In [16], Liu et al. develop a state adaptive controller for state constraint stochastic nonlinear systems with some parameters. While impulsive systems have been studied comprehensively, state constrained impulsive nonlinear systems require further analysis.
To our knowledge, many interesting results on state constraint systems have been reported in the literature [13–18]. But, those results are not suitable for the hybrid systems due to its particularity and complexity. That is, the control problem for state constrain impulsive dynamic systems has not been thoroughly studied, many challenging works still exist. In this paper, we propose a special type of feedback controller, which can be addressed by the robust control for uncertain state constraint impulsive hybrid systems with input disturbance. The matrix theory and inequality technique show that the control impulsive nonlinear dynamic systems are stable. Finally, some comparative examples with simulations show that the results in this paper are viable.
The thesis consists of the following: Section 2 is devoted to the model and some theorems are presented. In Section 3, the robust control applications are established for the main model. Examples are reported in Section 4. In Section 5, the conclusion is drawn.
Notations 1. -dimensional Euclidean space. . denotes the vector norm of the vector . is the boundary of set . : matrices. : a real symmetric and positive definite (positive semidefinite) matrix.
2. Preliminaries
The following nonlinear impulsive hybrid dynamic systems are described:where denotes the state of system (1); is a bounded ellipsoid in . . If , the impulsive point set . It is denoted that , , and . is a continuous impulsive function. For dynamic systems (1), the initial conditions were .
A set is called an invariant set [19] of system (1), if for any , . In the following, some sufficient conditions for the stability of system (1) will be given.
Theorem 1. Let be a bounded ellipsoid in , for . If there exists a positive definite function such that the for all and
Then, the state of system (1) will remain in the bounded ellipsoid .
proof. There are two possible solutions to the system (1).
Case 1. The solution never hits the boundary of . That is, , for any time . From the condition of Theorem 1, we have . This implies that the conclusion is clearly valid.
Case 2. The solution will hit the boundary of ; assume that the impulsive set , ( and as ).
For , and , then for .
Combining with (2), when , we have . By deduction method, for , from (2), we have .
Then, for .
Then, the state will stay in ellipsoid .
Remark 1. From Theorem 1, if , the trajectory of system (1) will remain in the set . In addition, in this paper, we assume that there are no impulses at initial value.
Remark 2. From Theorem 1, it can be used to solve the bouncing ball system [20,21] which is typical state constrained systems or state-dependent impulsive systems. Here, we consider that the system state is limited to a bounded region, and when the trajectory of systems reaches the boundary, it will trigger the impulsive input. Then, the state returns to the internal region
Example 1. In Figure 1, let be altitude of the oscillating platform, be altitude of the ball, denote the relative displacement between altitude of the ball and altitude of the oscillating platform, and denote the velocity. The mass of the ball is ignored and all collisions are instantaneous, and denotes the gravitational acceleration. Then, dynamic model simulates can be written aswhere ; . Assume , while . Such collision can lead to energy degradation, then . When , and , then . All the conditions are met to Theorem 1, then the ball and the platform finally moved steadily together (see Figure 2).
Recently, the BLF is proposed to solve the state-constrained system [13, 15, 16, 22]. In the following, we consider the uncertain state constraint impulsive systems.
3. Robust Control for Uncertain State Constraint Impulsive Systems
Consider the system in the following:with , where state vector is constrained in a ellipsoid . The uncertainties , are time-varying matrices. is the measurable output. denotes the impulsive effects. is the control input, is the nonlinear continuous function, and .for . Let , where is partial unmeasurable, which is bounded for the symmetric matrix and a constant such that
For the ellipsoid , it must exist as such that . In this paper, design the controller.where is a control gain. Also, we take the general assumption for the impulsive effects.
Assumption 1. There exists a positive constant such that
Lemma 1. ; the inequality holds.
We will present the stability for the state constrained uncertain hybrid systems (4).
Theorem 2. Let , , . If there exists such that
Then, the state systems (4) start in the ellipsoid and remain in set , and it converges to zero, where .
proof. We choose the BLF asWith the help of matrix , add the termDesign the control input as . When , calculating time derivative of along the trajectory of system (4), one may obtain thatLet hold. That is to say, one can get that , if , whereDenote that . We haveWe can finally write thatThat is, under the condition that (9) and (15) hold, then the inequality (14) implies that .
When , combining with Assumption 1,From Theorem 1, then the state systems (4) start in the ellipsoid and remain in set , and it converges to zero.
Remark 3. As we have known, most of the existing literature only considered the general impulsive system and the cases of state constrained are not included in Theorem 1, the BLF is applied for the state constrained impulsive systems. In the following, we assume that , in fact, in Theorem 2, , the bound of state can be as a special suprasphere . So, there exist some positive constants , such thatFor the asymmetrical case, we will consider the asymmetric BLF [15, 16].
Theorem 3. Let , , . If there exists such that
Then, the state of systems (4) remains in set , and it converges to zero, where
proof. Consider the BLF candidate aswhereOne can rewrite asWith the help of matrix , add the termDesign the control input as . When , calculating time derivative of , one may obtain that Let and hold. That is, we can get that , if , whereNote that . Then, we haveWe can finally write thatThat is, if (27) holds, then the inequality (26) implies in the subset of described by , then for .
When , under the condition Assumption 1, we can get that Combining with Theorem 1, then the state systems (4) start in the ellipsoid and remain in set , and it converges to zero. This completes the proof.
Remark 4. Without loss of generality, let , , the following corollary can be drawn.
Corollary 1. Assume that there exist , , . such that
Then, the state of systems (4) remains in set , and it converges to zero, where .
proof. Consider the BLF thatSimilar to the discussion of Theorem 2, more detailed proof is omitted here.
For model (4), if there is no external disturbance, that is, , . From the above, we can get a corollary that the state constrained system stability can be ensured when relative variables are used as feedback variables and appropriate gains have been chosen.
Corollary 2. Let , , . If there exists such that
Then, the state of systems (4) (with , ) remains in set , and it converges to zero, where .
Remark 5. It could tell that it is difficult to calculate theoretically for BLF [23, 24] which is different from the traditional Lyapunov function. In this paper, introducing the descriptor method and adding an additional term, some linear matrix inequality conditions are obtained.
Remark 6. For the uncertain , , also use the assumptions, such as (1) , such that , . (2) , such that , . Some useful conclusions can also be drawn. For the convenience of calculation, we usually use one of the above assumptions in numerical validation.
4. Examples
Two comparative examples with simulations show that the proposed results are effective.
Example 2. Consider the state constrained nonlinear impulsive system.where , . Let ; combining with Theorem 1, we can verify that , , hold. From Theorem 1, the trajectory of system (32) remains in one region. Figure 3 shows the trajectory of states of systems (32).
From Figure 3, we can find that when , or , it will trigger the impulsive mechanism. The impulses can also be an effective strategy which can stabilize the system. It can keep its solution remain in one constraint region.
Example 3. Consider the uncertain nonlinear impulsive state constrained dynamic systems (4) with , ,, . The disturbing function with . Here, .
Design . Choose , , . By MATLAB tool, takeFrom Theorem 2, then the state constrained impulsive hybrid systems (4) is stable. Figures 4 and 5 show the trajectory of states and of (4).
5. Conclusions
In this text, the stability of state constrained hybrid systems has been considered. By constructor barrier Lyapunov method, the stabilization for a class of uncertain hybrid dynamical systems has been derived. Furthermore, using the examples, the availability of the proposed main results has been demonstrated. The main results may extend to consider the synchronization of state constrains impulsive hybrid dynamic systems by barrier Lyapunov functions. In the future, studying the stability of delayed state constrained impulsive hybrid system will be another interesting subject.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This paper was supported by the Natural Science Foundation Project of Anhui Province under Grant no. 2108085QA08 and the University Natural Science Research Project of Anhui Province under Grant nos. KJ2021A0529 and KJ2020ZD008.