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Robust stability and {H}_{\mathrm{\infty}}control of uncertain systems with impulsive perturbations
Advances in Difference Equations volume 2014, Article number: 79 (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 timevarying and normbounded. 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.
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, frequencymodulated 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 closedloop 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 positivedefinite (positivesemidefinite, negative, negativesemidefinite) 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.
Consider uncertain linear impulsive systems described by the following state equation:
where x(t)\in {\mathbb{R}}^{n} is the state, u(t)\in {\mathbb{R}}^{m} is the control input, w(t)\in {\mathbb{R}}^{p} is the disturbance input which belongs to {L}_{2}[0,\mathrm{\infty}), z(t)\in {\mathbb{R}}^{q} is the controlled output. \mathrm{\Delta}x({t}_{k})=x({t}_{k}^{+})x({t}_{k}^{}) describes the state jumping at impulsive time instant t={t}_{k}, x({t}_{k}^{})=x({t}_{k})={lim}_{h\to 0+}x({t}_{k}h), x({t}_{k}^{+})={lim}_{h\to 0+}x({t}_{k}+h), k=1,2,\dots , and 0<{t}_{1}<{t}_{2}<\cdots <{t}_{k}<\cdots ({t}_{k}\to \mathrm{\infty} as t\to \mathrm{\infty}). {H}_{1}\in {\mathbb{R}}^{n\times p}, {H}_{2}\in {\mathbb{R}}^{q\times p}, {C}_{k}\in {\mathbb{R}}^{n\times n}, k=1,2,\dots , are known constant matrices, and A(t)\in {\mathbb{R}}^{n\times n}, {B}_{1}(t)\in {\mathbb{R}}^{n\times m}, E(t)\in {\mathbb{R}}^{q\times n}, {B}_{2}(t)\in {\mathbb{R}}^{q\times m} are matrix functions with timevarying uncertainties, that is,
where A\in {\mathbb{R}}^{n\times n}, {B}_{1}\in {\mathbb{R}}^{n\times m}, E\in {\mathbb{R}}^{q\times n}, {B}_{2}\in {\mathbb{R}}^{q\times m} are known real constant matrices, \mathrm{\Delta}A(t)\in {\mathbb{R}}^{n\times n}, \mathrm{\Delta}{B}_{1}(t)\in {\mathbb{R}}^{n\times m}, \mathrm{\Delta}E(t)\in {\mathbb{R}}^{q\times n}, and \mathrm{\Delta}{B}_{2}(t)\in {\mathbb{R}}^{q\times m} are unknown matrices representing timevarying parameter uncertainties. We assume that the uncertainties are normbounded and can be described as
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 IEP{E}^{T}>0, (A+DF(t)E)P{(A+DF(t)E)}^{T}\le AP{A}^{T}+AP{E}^{T}{(\epsilon IEP{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
In this section, we restrict our study to the case of w(t)=0 in system (2.1), 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),
For t\in ({t}_{k},{t}_{k+1}], the time derivative of V(t) is
By (i) of Lemma 2.1, for any \epsilon >0, 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 \parallel {C}_{k}\parallel \le \alpha, it follows that {C}_{k} can be written as {C}_{k}=D{F}_{k}E with D=\alpha I, E=I and \parallel {F}_{k}\parallel \le 1. Using (ii) of Lemma 2.1, for any {\epsilon}_{2} satisfying {\epsilon}_{2}I{\alpha}^{2}P>0, 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 \beta ={inf}_{i}\{{t}_{i+1}{t}_{i}\}, we get t{t}_{0}\ge k\beta. Noticing that {\mu}_{2}>1 and ln({\mu}_{2})\beta {\mu}_{1}<0, we deduce that
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 continuoustime linear systems with normbounded 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}.
Let us now design a memoryless state feedback controller of the following form:
to stabilize system (3.1), where K\in {\mathbb{R}}^{m\times n} is a constant gain to be designed.
Substituting (3.13) into (3.1) yields the dynamics of the closedloop system as follows:
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 closedloop 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.
Using the technique as in the proof of Theorem 3.1, it can easily be shown that if there exists a positive scalar {\epsilon}_{1} 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 {C}_{k} in the form of {C}_{k}=D{F}_{k}E with D=\alpha I, E=I and \parallel {F}_{k}\parallel \le 1. Then using (iv) of Lemma 2.1, for any positive scalar {\epsilon}_{2} satisfying {P}^{1}{\alpha}^{2}{\epsilon}_{2}I>0, we have
It follows that if
then (3.18) holds. Thus, if (3.19) and (3.20) hold, then closedloop system (3.14) is asymptotically stable. Define X={P}^{1}, \overline{K}=KX. Pre and postmultiplying (3.19) by X yields
which combined with Schur complement leads to (3.15).
Pre and postmultiplying (3.20) by 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.
Applying Lyapunov function (3.4) with P={X}^{1} to (4.2), for t\in ({t}_{k},{t}_{k+1}], the time derivative of V(t) is
Using Lemma 2.1 and condition (3.15), we obtain
It follows that
which yields
From the proof of Theorem 3.2, condition (3.16) implies V({t}_{k+1}^{+})\le {\mu}_{2}V({t}_{k+1}), k=0,1,2,\dots . Substituting this inequality into (4.4) gives
It follows that for T\in ({t}_{k},{t}_{k+1}]
By (iii) of Lemma 2.1, for any Q>0, we have
Thus, for any T>0, we have
Set \tilde{E}(t)=[E(t)+{B}_{2}(t)K\phantom{\rule{0.25em}{0ex}}{H}_{2}], {\eta}^{T}(t)=[{x}^{T}(t)\phantom{\rule{0.25em}{0ex}}{w}^{T}(t)], {\eta}_{e}(t)=\tilde{E}(t)\eta (t), {\xi}^{T}(t)=[{\eta}^{T}(t)\phantom{\rule{0.25em}{0ex}}{\eta}_{e}^{T}(t)]. Define
Since V(0)=0 by the zero initial condition, it follows from (4.6) and (4.3) that
where {\mathrm{\Gamma}}_{D}^{T}=[{D}_{1}^{T}P\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}{D}_{2}^{T}], {\mathrm{\Gamma}}_{N}=[N+{N}_{b}K\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}0], \tilde{E}=[E+{B}_{2}K\phantom{\rule{0.25em}{0ex}}{H}_{2}], and
By (i) of Lemma 2.1, for any scalar {\epsilon}_{3}>0,
Thus, if the following inequality holds:
then by (4.7), we have
so the proof will be completed.
Pre and postmultiplying (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 interiorpoint 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.
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Acknowledgements
This work was supported by the National Natural Sciences Foundation of People’s Republic of China (Tianyuan Fund for Mathematics, Grant No. 11326113).
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Hu, M., Wang, L. Robust stability and {H}_{\mathrm{\infty}}control of uncertain systems with impulsive perturbations. Adv Differ Equ 2014, 79 (2014). https://doi.org/10.1186/16871847201479
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DOI: https://doi.org/10.1186/16871847201479