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# Robust stability and ${H}_{\mathrm{\infty}}$-control of uncertain systems with impulsive perturbations

- Meng Hu
^{1}Email author and - Lili Wang
^{1}

**2014**:79

https://doi.org/10.1186/1687-1847-2014-79

© Hu and Wang; licensee Springer. 2014

**Received:**3 January 2014**Accepted:**24 February 2014**Published:**7 March 2014

## Abstract

In this paper, the problems of robust stability, stabilization, and ${H}_{\mathrm{\infty}}$-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.

## Keywords

- uncertain impulsive system
- robust stability
- robust stabilization
- ${H}_{\mathrm{\infty}}$-control
- linear matrix inequality (LMI)

## 1 Introduction

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 [12], Guan *et al.* studied the ${H}_{\mathrm{\infty}}$ 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 ${H}_{\mathrm{\infty}}$ performance of the closed-loop system. But the result in [12] 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 [12] are less applicable. Furthermore, the parameter uncertainties of impulsive systems were not considered in [12].

The goal of this paper is to study the robust stability, stabilization, and ${H}_{\mathrm{\infty}}$-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 ${H}_{\mathrm{\infty}}$ 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 $M>(\ge ,\phantom{\rule{0.2em}{0ex}}<,\phantom{\rule{0.2em}{0ex}}\le )\phantom{\rule{0.2em}{0ex}}0$ is used to denote a symmetric positive-definite (positive-semidefinite, negative, negative-semidefinite) matrix. ${\lambda}_{min}(\cdot )$ and ${\lambda}_{max}(\cdot )$ represent the minimum and maximum eigenvalues of the corresponding matrix, respectively. $\parallel \cdot \parallel $ denotes the Euclidean norm for vectors or the spectral norm of matrices.

where ${D}_{1}\in {\mathbb{R}}^{n\times {n}_{f}}$, ${D}_{2}\in {\mathbb{R}}^{q\times {n}_{f}}$, $N\in {\mathbb{R}}^{{n}_{f}\times n}$, ${N}_{b}\in {\mathbb{R}}^{{n}_{f}\times m}$ are known real constant matrices and $F(\cdot )\in {\mathbb{R}}^{{n}_{f}\times {n}_{f}}$ is an unknown matrix functions satisfying ${F}^{T}(t)F(t)\le I$. It is assumed that the elements of $F(t)$ 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 $u(t)=0$ and $w(t)=0$ is said to be robustly stable if the trivial solution of (2.1) with $u(t)=0$ and $w(t)=0$ is asymptotically stable for all admissible uncertainties satisfying (2.2).

**Definition 2.2**Given a scalar $\gamma >0$, the uncertain impulsive system (2.1) with $u(t)=0$ 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 [13])

*For any vectors*$x,y\in {\mathbb{R}}^{n}$,

*matrices*$A,P\in {\mathbb{R}}^{n\times n}$, $D\in {\mathbb{R}}^{n\times {n}_{f}}$, $E,N\in {\mathbb{R}}^{{n}_{f}\times n}$,

*and*$D\in {\mathbb{R}}^{n\times {n}_{f}}$, $E,N\in {\mathbb{R}}^{{n}_{f}\times n}$,

*with*$P>0$, $\parallel F\parallel \le 1$,

*and scalar*$\epsilon >0$,

*the following inequalities hold*:

- (i)
$DFN+{N}^{T}{F}^{T}{D}^{T}\le {\epsilon}^{-1}D{D}^{T}+\epsilon {N}^{T}N$;

- (ii)
*if*$\epsilon I-EP{E}^{T}>0$, $(A+DF(t)E)P{(A+DF(t)E)}^{T}\le AP{A}^{T}+AP{E}^{T}{(\epsilon I-EP{E}^{T})}^{-1}EP{A}^{T}+\epsilon D{D}^{T}$; - (iii)
$2{x}^{T}y\le {x}^{T}{P}^{-1}x+{y}^{T}Py$;

- (iv)
*if*$P-\epsilon D{D}^{T}>0$, ${(A+DF(t)E)}^{T}{P}^{-1}(A+DF(t)E)\le {A}^{T}{(P-\epsilon D{D}^{T})}^{-1}A+{\epsilon}^{-1}{E}^{T}E$.

## 3 Robust stability and robust stabilization

*i.e.*

First, we present some sufficient conditions for robust stability of system (3.1) with $u(t)=0$.

**Theorem 3.1**

*Assume that there exist*$\alpha >0$

*and*$\beta >0$

*such that*$\parallel {C}_{k}\parallel \le \alpha $, $k=1,2,\dots $ , $\beta ={inf}_{i}\{{t}_{i+1}-{t}_{i}\}$.

*If for the prescribed scalars*${\mu}_{1}>0$

*and*${\mu}_{2}>1$

*satisfying*$ln({\mu}_{2})-\beta {\mu}_{1}<0$,

*there exist matrix*$P>0$

*and scalars*${\epsilon}_{1}>0$, ${\epsilon}_{2}>0$

*such that the following linear matrix inequalities hold*:

*then system* (3.1) *with* $u(t)=0$ *is robustly asymptotically stable*.

*Proof*Take the Lyapunov function for system (3.1),

It follows that system (2.1) with $u(t)=0$ is robustly asymptotically stable. The proof is completed. □

**Remark 3.1**When $\alpha =0$, that is, there is no impulse jumping in the states, let ${\epsilon}_{2}\to {0}^{+}$ and ${\mu}_{1}\to {0}^{+}$, ${\mu}_{2}\to {1}^{+}$, then the LMI conditions in Theorem 3.1 reduces to a single LMI

for some matrix $P>0$ and some scalar ${\epsilon}_{1}>0$. Condition (3.12) is exactly the sufficient condition for robust stability of continuous-time linear systems with norm-bounded uncertainty, for example, see [14].

**Remark 3.2**From (3.11), we can show that

where ${\lambda}_{1}={\lambda}_{max}(P)$, ${\lambda}_{0}={\lambda}_{min}(P)$. It follows that under the conditions of Theorem 3.1, system (3.1) with $u(t)=0$ is robustly exponentially stable with decay rate $\delta =\frac{1}{2}({\mu}_{1}-\frac{1}{\beta}ln({\mu}_{2}))$. For prescribed decay rate *δ*, we can choose ${\mu}_{1}=2\delta +\frac{1}{\beta}ln({\mu}_{2})$ to find the feasible solution to LMIs (3.2) and (3.3) by tuning parameter ${\mu}_{2}$.

to stabilize system (3.1), where $K\in {\mathbb{R}}^{m\times n}$ is a constant gain to be designed.

**Theorem 3.2**

*Assume that there exist*$\alpha >0$

*and*$\beta >0$

*such that*$\parallel {C}_{k}\parallel \le \alpha $, $k=1,2,\dots $ , $\beta ={inf}_{i}\{{t}_{i+1}-{t}_{i}\}$.

*If for prescribed scalars*${\mu}_{1}>0$

*and*${\mu}_{2}>1$

*satisfying*$ln({\mu}_{2})-\beta {\mu}_{1}<0$,

*there exist matrix*$X>0$, $\overline{K}$

*and scalars*${\epsilon}_{1}>0$, ${\epsilon}_{2}>0$

*such that the following linear matrix inequalities hold*:

*then the controller* (3.13) *with* $K=\overline{K}{X}^{-1}$ *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 ${\mu}_{1}$, ${\mu}_{2}$ satisfying $ln({\mu}_{2})-\beta {\mu}_{1}<0$ such that the following two inequalities hold:

where $V(x)={x}^{T}Px$ with $P>0$.

then (3.17) holds.

*X*yields

which combined with Schur complement leads to (3.15).

*X*yields

which combined with Schur complement leads to (3.16). The proof is completed. □

## 4 Robust ${H}_{\mathrm{\infty}}$-control

This section is devoted to studying the robust ${H}_{\mathrm{\infty}}$-control problem for system (2.1).

**Theorem 4.1**

*Assume that there exist*$\alpha >0$

*and*$\beta >0$

*such that*$\parallel {C}_{k}\parallel \le \alpha $, $k=1,2,\dots $ , $\beta ={inf}_{i}\{{t}_{i+1}-{t}_{i}\}$.

*If for the prescribed scalars*${\mu}_{1}>0$

*and*${\mu}_{2}>1$

*satisfying*$2ln({\mu}_{2})-\beta {\mu}_{1}\le 0$,

*there exist matrix*$X>0$, $Q>0$, $\overline{K}$

*and scalars*${\epsilon}_{1}>0$, ${\epsilon}_{2}>0$

*and*${\epsilon}_{3}>0$

*such that*(3.15), (3.16),

*and the following linear matrix inequalities hold*:

*where* $\mathrm{\Xi}=AX+X{A}^{T}+{B}_{1}\overline{K}+{\overline{K}}^{T}{B}_{1}^{T}+{\epsilon}_{3}{D}_{1}{D}_{1}^{T}$, $L=X{E}^{T}+{\overline{K}}^{T}{B}_{2}^{T}+{\epsilon}_{3}{D}_{1}{D}_{2}^{T}$, *then system* (2.1) *has robust stabilization with disturbance attenuation* *γ*. *Moreover*, *the controller* (3.13) *with* $K=\overline{K}{X}^{-1}$ *robustly stabilizes system* (2.1).

*Proof*With the memoryless state feedback control law (3.13), system (2.1) becomes

By $2ln({\mu}_{2})-\beta {\mu}_{1}\le 0$ and ${\mu}_{2}>1$, it is easy to see that $ln({\mu}_{2})-\beta {\mu}_{1}<0$. So, if $w(t)=0$, 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, $x(t)=0$, for $t=0$.

so the proof will be completed.

Pre- and post-multiplying (4.8) by $diag\{X,I,I\}$ 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 [12], under the assumption that ${C}_{k}={c}_{k}I$ and ${c}_{k}\in (-2,0)$, sufficient condition for the existence of ${H}_{\mathrm{\infty}}$ state feedback controller was derived in terms of the Riccati equation. As compared to [12], our results can be used for a wider class of impulsive system. Moreover, Theorem 4.1 cast the existence problem of ${H}_{\mathrm{\infty}}$ 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 [15].

## 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 $u(t)=0$. Assume that the system data are given as

If we select decay rate $\delta =0.1$, then by Theorem 3.1, choosing ${\mu}_{2}=1.3$ and ${\mu}_{1}=2\delta +\frac{1}{\beta}ln({\mu}_{2})$, the obtained maximum value of *c* such that the above system is robustly exponentially stable is $c=0.53$. If we select the decay rate $\delta =0.2$, by choosing the same values of ${\mu}_{1}$ and ${\mu}_{2}$, the corresponding maximum value of *c* is $c=0.453$.

**Example 2**Consider the uncertain impulsive system (2.1) with parameters as follows:

First we assume that $\alpha =0.2$, $\beta =0.8$. By Theorem 4.1, choosing ${\mu}_{2}=3.1$ and ${\mu}_{1}=\frac{2}{\beta}ln({\mu}_{2})$, it has been found that the smallest value of *γ* for the above system to have robust stabilization with disturbance attenuation *γ* is $\gamma =0.86$. The corresponding stabilizing control law is given by $u(t)=[-32.9135\phantom{\rule{0.25em}{0ex}}15.0960]x(t)$.

Next we assume that $\alpha =0.1$, $\beta =0.8$. By Theorem 4.1, choosing ${\mu}_{2}=1.6$ and ${\mu}_{1}=\frac{2}{\beta}ln({\mu}_{2})$, it has been found that the smallest value of *γ* is $\gamma =0.29$ and the corresponding stabilizing control law is given by $u(t)=[-32.2988\phantom{\rule{0.25em}{0ex}}8.6274]x(t)$.

## 6 Conclusion

Three problems for uncertain impulsive systems have been studied, namely, robust stability, robust stabilization, and robust ${H}_{\mathrm{\infty}}$-control. In each case, the sufficient conditions in terms of linear matrix inequalities have been established. Moreover the method to design a state feedback ${H}_{\mathrm{\infty}}$ 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.

## Declarations

### Acknowledgements

This work was supported by the National Natural Sciences Foundation of People’s Republic of China (Tianyuan Fund for Mathematics, Grant No. 11326113).

## Authors’ Affiliations

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