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Robust stability and -control of uncertain systems with impulsive perturbations
Advances in Difference Equations volume 2014, Article number: 79 (2014)
In this paper, the problems of robust stability, stabilization, and -control for uncertain systems with impulsive perturbations are investigated. The parametric uncertainties are assumed to be time-varying and norm-bounded. The sufficient conditions for the above problems are developed in terms of linear matrix inequalities. Numerical examples are given which illustrate the applicability of the theoretical results.
MSC:34H05, 34H15, 34K45.
Many evolutionary processes are subject to short temporary perturbations that are negligible compared to the process duration. Thus the perturbations act instantaneously in the form of impulses. For example, biological phenomena involving thresholds, bursting rhythm models in pathology, optimal control of economic systems, frequency-modulated signal processing systems do exhibit impulse effects. Impulsive differential systems provide a natural description of observed evolutionary processes with impulse effects.
Problems with qualitative analysis of impulsive systems has been extensively studied in the literature, we refer to [1–7] and the references therein. Also, the control of impulsive or nonlinear systems received more recently researchers’ special attention due to their applications; see, for example [8–11]. In , Guan et al. studied the control problem for impulsive systems. In terms of the solutions to an algebraic Riccati equation, they obtained sufficient conditions for the existence of state feedback controllers guaranteeing asymptotic stability and prescribed performance of the closed-loop system. But the result in  is based on the assumption that the state jumping at the impulsive time instant has a special form. This assumption is not satisfied for most impulsive systems. Therefore, the results in  are less applicable. Furthermore, the parameter uncertainties of impulsive systems were not considered in .
The goal of this paper is to study the robust stability, stabilization, and -control of uncertain impulsive systems under more general assumption on state jumping. Sufficient conditions for the existence of the solutions to the above problems are derived. Moreover, these sufficient conditions are all in linear matrix inequality (LMI) formalism, which makes their resolution easy.
The rest of this paper is organized as follows: Section 2 describes the system model; Section 3 addresses the robust stability and stabilization problems; Section 4 studies the robust problem; Section 5 provides two examples to demonstrate the applicability of the proposed approach.
2 Problem statement
In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. The notation is used to denote a symmetric positive-definite (positive-semidefinite, negative, negative-semidefinite) matrix. and represent the minimum and maximum eigenvalues of the corresponding matrix, respectively. denotes the Euclidean norm for vectors or the spectral norm of matrices.
Consider uncertain linear impulsive systems described by the following state equation:
where is the state, is the control input, is the disturbance input which belongs to , is the controlled output. describes the state jumping at impulsive time instant , , , , and ( as ). , , , , are known constant matrices, and , , , are matrix functions with time-varying uncertainties, that is,
where , , , are known real constant matrices, , , , and are unknown matrices representing time-varying parameter uncertainties. We assume that the uncertainties are norm-bounded and can be described as
where , , , are known real constant matrices and is an unknown matrix functions satisfying . It is assumed that the elements of are Lebesgue measurable.
Throughout this paper, we shall use the following concepts of robust stability and robust performance for system (2.1).
Definition 2.1 System (2.1) with and is said to be robustly stable if the trivial solution of (2.1) with and is asymptotically stable for all admissible uncertainties satisfying (2.2).
Definition 2.2 Given a scalar , the uncertain impulsive system (2.1) with is said to have robust stabilization with disturbance attenuation γ if it is robustly stable in the sense of Definition 2.1 and under zero initial conditions,
The following lemma is essential for the developments in the next sections.
Lemma 2.1 (see )
For any vectors , matrices , , , and , , with , , and scalar , the following inequalities hold:
if , ;
if , .
3 Robust stability and robust stabilization
In this section, we restrict our study to the case of in system (2.1), i.e.
First, we present some sufficient conditions for robust stability of system (3.1) with .
Theorem 3.1 Assume that there exist and such that , , . If for the prescribed scalars and satisfying , there exist matrix and scalars , such that the following linear matrix inequalities hold:
then system (3.1) with is robustly asymptotically stable.
Proof Take the Lyapunov function for system (3.1),
For , the time derivative of is
By (i) of Lemma 2.1, for any , we get
By Schur complement, condition (3.2) is equivalent to
Combining (3.4)-(3.7) yields
which implies that
On the other hand, since , it follows that can be written as with , and . Using (ii) of Lemma 2.1, for any satisfying , we get
By Schur complement, condition (3.3) is equivalent to
Substituting the above inequality into (3.9) gives
On the basis of (3.8) and (3.10), we obtain
By the assumption of , we get . Noticing that and , we deduce that
It follows that system (2.1) with is robustly asymptotically stable. The proof is completed. □
Remark 3.1 When , that is, there is no impulse jumping in the states, let and , , then the LMI conditions in Theorem 3.1 reduces to a single LMI
for some matrix and some scalar . Condition (3.12) is exactly the sufficient condition for robust stability of continuous-time linear systems with norm-bounded uncertainty, for example, see .
Remark 3.2 From (3.11), we can show that
where , . It follows that under the conditions of Theorem 3.1, system (3.1) with is robustly exponentially stable with decay rate . For prescribed decay rate δ, we can choose to find the feasible solution to LMIs (3.2) and (3.3) by tuning parameter .
Let us now design a memoryless state feedback controller of the following form:
to stabilize system (3.1), where is a constant gain to be designed.
Substituting (3.13) into (3.1) yields the dynamics of the closed-loop system as follows:
Theorem 3.2 Assume that there exist and such that , , . If for prescribed scalars and satisfying , there exist matrix , and scalars , such that the following linear matrix inequalities hold:
then the controller (3.13) with robustly stabilizes system (3.1).
Proof From the proof of the Theorem 3.1, the sufficient condition for asymptotic stability of closed-loop system (3.14) is that there exist positive scalars , satisfying such that the following two inequalities hold:
where with .
Using the technique as in the proof of Theorem 3.1, it can easily be shown that if there exists a positive scalar such that the following inequality is satisfied:
then (3.17) holds.
Now we consider the sufficient condition for inequality (3.18). As in the proof of Theorem 3.1, we represent in the form of with , and . Then using (iv) of Lemma 2.1, for any positive scalar satisfying , we have
It follows that if
then (3.18) holds. Thus, if (3.19) and (3.20) hold, then closed-loop system (3.14) is asymptotically stable. Define , . Pre- and post-multiplying (3.19) by X yields
which combined with Schur complement leads to (3.15).
Pre- and post-multiplying (3.20) by X yields
which combined with Schur complement leads to (3.16). The proof is completed. □
4 Robust -control
This section is devoted to studying the robust -control problem for system (2.1).
Theorem 4.1 Assume that there exist and such that , , . If for the prescribed scalars and satisfying , there exist matrix , , and scalars , and such that (3.15), (3.16), and the following linear matrix inequalities hold:
where , , then system (2.1) has robust stabilization with disturbance attenuation γ. Moreover, the controller (3.13) with robustly stabilizes system (2.1).
Proof With the memoryless state feedback control law (3.13), system (2.1) becomes
By and , it is easy to see that . So, if , then by (3.15) and (3.16) and Theorem 3.2, we can conclude that system (4.2) has robust stabilization. Next, we proceed to prove that system (4.2) verifies noise attenuation γ. To this end, we assume the zero initial condition, that is, , for .
Applying Lyapunov function (3.4) with to (4.2), for , the time derivative of is
Using Lemma 2.1 and condition (3.15), we obtain
It follows that
From the proof of Theorem 3.2, condition (3.16) implies , . Substituting this inequality into (4.4) gives
It follows that for
By (iii) of Lemma 2.1, for any , we have
Thus, for any , we have
Set , , , . Define
Since by the zero initial condition, it follows from (4.6) and (4.3) that
where , , , and
By (i) of Lemma 2.1, for any scalar ,
Thus, if the following inequality holds:
then by (4.7), we have
so the proof will be completed.
Pre- and post-multiplying (4.8) by and using Schur complement, it is easy to prove that (4.1) is equivalent to (4.8). The proof is completed. □
Remark 4.1 In , under the assumption that and , sufficient condition for the existence of state feedback controller was derived in terms of the Riccati equation. As compared to , our results can be used for a wider class of impulsive system. Moreover, Theorem 4.1 cast the existence problem of state feedback controller into the feasibility problem of the LMIs (3.15), (3.16), and (4.1), the latter can be efficiently solved by the developed interior-point algorithm .
5 Numerical example
In this section, we shall give two numerical examples to demonstrate the effectiveness of the proposed results.
Example 1 Consider the linear uncertain impulsive system (3.1) with . Assume that the system data are given as
If we select decay rate , then by Theorem 3.1, choosing and , the obtained maximum value of c such that the above system is robustly exponentially stable is . If we select the decay rate , by choosing the same values of and , the corresponding maximum value of c is .
Example 2 Consider the uncertain impulsive system (2.1) with parameters as follows:
First we assume that , . By Theorem 4.1, choosing and , it has been found that the smallest value of γ for the above system to have robust stabilization with disturbance attenuation γ is . The corresponding stabilizing control law is given by .
Next we assume that , . By Theorem 4.1, choosing and , it has been found that the smallest value of γ is and the corresponding stabilizing control law is given by .
Three problems for uncertain impulsive systems have been studied, namely, robust stability, robust stabilization, and robust -control. In each case, the sufficient conditions in terms of linear matrix inequalities have been established. Moreover the method to design a state feedback controller is provided. Our method is helpful to improve the existing technologies used in the analysis and control for uncertain impulsive systems. Numerical examples have been provided to demonstrate the effectiveness and applicability of the proposed approach.
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This work was supported by the National Natural Sciences Foundation of People’s Republic of China (Tianyuan Fund for Mathematics, Grant No. 11326113).
The authors declare that they have no competing interests.
The authors contributed equally and significantly in writing this paper. The authors read and approved the final manuscript.
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Hu, M., Wang, L. Robust stability and -control of uncertain systems with impulsive perturbations. Adv Differ Equ 2014, 79 (2014). https://doi.org/10.1186/1687-1847-2014-79
- uncertain impulsive system
- robust stability
- robust stabilization
- linear matrix inequality (LMI)