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Robust stability and stabilization of uncertain switched discretetime systems
Advances in Difference Equations volume 2012, Article number: 134 (2012)
Abstract
This paper is concerned with the robust stability and stabilization for a class of switched discretetime systems with state parameter uncertainty. Firstly, a new matrix inequality considering uncertainties is introduced and proved. By means of it, a novel sufficient condition for robust stability and stabilization of a class of uncertain switched discretetime systems is presented. Furthermore, based on the result obtained, the switching law is designed and has been performed well, and some sufficient conditions of robust stability and stabilization have been derived for the uncertain switched discretetime systems using the Lyapunov stability theorem, block matrix method, and inequality technology. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.
1 Introduction
A switched system is a hybrid dynamical system consisting of a finite number of subsystems and a logical rule that manages switching between these subsystems. Switched systems have drawn a great deal of attention in recent years; see [1–37] and references therein. The motivation for studying switched systems comes partly from the fact that switched systems and switched multicontroller systems have numerous applications in control of mechanical systems, process control, automotive industry, power systems, aircraft and traffic control, and many other fields. An important qualitative property of switched system is stability [1–3]. The challenge of analyzing the stability of switched system lies partly in the fact that, even if the individual systems are stable, the switched system might be unstable. Using a common quadratic Lyapunov function on all subsystems, the quadratic Lyapunov stability facilitates the analysis and synthesis of switched systems. However, the obtained results within this framework have been recognized to be conservative. In [10], various algorithms both for stability and performance analysis of discretetime piecewise affine systems were presented. Different classes of Lyapunov functions were considered, and how to compute them through linear matrix inequalities was also shown. Moreover, the tradeoff between the degree of conservativeness and computational requirements was discussed. The problem of stability analysis and control synthesis of switched systems in the discretetime domain was addressed in [11]. The approach followed in [11] looked at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix inequalitybased conditions allow to check the existence of such a Lyapunov function. These two conditions have been proved to be equivalent for stability analysis.
There are many methodologies and approaches developed in the switched systems theory: approaches of looking for an appropriate switching strategy to stabilize the system [4], dwelltime and average dwelltime approaches for stability analysis and stabilization problems [5], approaches of studying stability and control problems under a specific class of switching laws [1], or under arbitrary switching sequences [6, 9]. Reference [19] investigated the quadratic stability and linear state feedback and output feedback stabilization of switched delayed linear dynamic systems with, in general, a finite number of noncommensurate constant internal point delays. The results were obtained based on Lyapunov stability analysis via appropriate KrasovskiiLyapunov functionals, and the related stability study was performed to obtain both delayindependent and delaydependent results. The problem of fault estimation for a class of switched nonlinear systems of neutral type was considered in [20]. Sufficient delaydependent existence conditions of the ${H}_{\mathrm{\infty}}$ fault estimator were given in terms of certain matrix inequalities based on the average dwelltime approach. The problem of robust reliable control for a class of uncertain switched neutral systems under asynchronous switching was investigated in [21]. A state feedback controller was proposed to guarantee exponential stability and reliability for switched neutral systems, and the dwelltime approach was utilized for the stability analysis and controller design. The exponential stability for a class of nonlinear hybrid timedelay systems was addressed in [24]. The delaydependent stability conditions were presented in terms of the solution of algebraic Riccati equations, which allows computing simultaneously the two bounds that characterize the stability rate of the solution.
On another research front line, it has been recognized that parameter uncertainties, which often occur in many physical processes, are main sources of instability and poor performance. Therefore, much attention has been devoted to the study of various systems with uncertainties, and a great number of useful results have been reported in the literature on the issues of robust stability, robust ${H}_{\mathrm{\infty}}$ control, robust ${H}_{\mathrm{\infty}}$ filtering, and so on, by considering different classes of parameter uncertainties [12, 14].
Recently, some stability condition and stabilization approaches have been proposed for the switched discretetime system [15, 18]. In [15], the quadratic stabilization of discretetime switched linear systems was studied, and quadratic stabilization of switched systems with norm bounded time varying uncertainties was investigated. In [16], the stability property for the switched systems which were composed of a continuoustime LTI subsystem and a discrete time LTI subsystem was studied. There existed a switched quadratic Lyapunov function to check asymptotic stability of the switched discretetime system in [17].
The objective of this paper is to present novel approaches for the asymptotical stability and stabilization of switched discretetime system with parametric uncertainties. The parameter uncertainties are timevarying but normbounded. Firstly, a new inequality is given. Using the new result, a new sufficient condition for robust stability and stabilization of a class of uncertain switched discretetime systems is proposed. Furthermore, using the block matrix method, inequality technology, and the Lyapunov stability theorem, some sufficient conditions for robust stability and stabilization have been presented for the uncertain switched discretetime systems, and the switched law design has been performed. Comparing with [22, 23], the uncertainty in system was not considered in [22, 23], but we consider the uncertainty in systems and the design switching law is simple and easy for application.
The rest of this paper is organized as follows. The problem is formulated in Section 2. Section 3 deals with robust stability and stabilization criteria for a class of discretetime switched system with uncertainty. Numerical examples are provided to illustrate the theoretical results in Section 4, and the conclusions are drawn in Section 5.
2 Preliminaries
The following notations will be used throughout this paper. ${R}^{+}$ denotes the set of all real nonnegative numbers; ${R}^{n}$ denotes the ndimensional space with the scalar product of two vectors $\u3008x,y\u3009$ or ${x}^{T}y$; ${R}^{n\times r}$ denotes the space of all matrices of $(n\times r)$dimension. ${A}^{T}$ denotes the transpose of A; a matrix A is symmetric if $A={A}^{T}$.
Matrix A is semipositive definite ($A\ge 0$) if $\u3008Ax,x\u3009\ge 0$, for all $x\in {R}^{n}$; A is positive definite ($A>0$) if $\u3008Ax,x\u3009>0$ for all $x\ne 0$; $A\ge B$ means $AB\ge 0$. $\lambda (A)$ denotes the set of all eigenvalues of A; ${\lambda}_{min}(A)=min\{Re\lambda :\lambda \in \lambda (A)\}$.
Consider uncertain discrete systems with interval timevarying delay of the form
where $x(k)\in {R}^{n}$ is the state, $\gamma (\cdot ):{R}^{n}\to \mathcal{N}:=\{1,2,\dots ,N\}$ is the switching rule, which is a function depending on the state at each time and will be designed. A switching function is a rule which determines a switching sequence for a given switching system. Moreover, $\gamma (x(k))=i$ implies that the system realization is chosen as the i th system, $i=1,2,\dots ,N$. It is seen that the system (2.1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state $x(k)$ hits predefined boundaries. ${A}_{i}$, ${B}_{i}$, $i=1,2,\dots ,N$ are given constant matrices and the timevarying uncertain matrices $\mathrm{\Delta}{A}_{i}(k)$ and $\mathrm{\Delta}{B}_{i}(k)$ are defined by
where ${E}_{ia}$, ${E}_{ib}$, ${H}_{ia}$, ${H}_{ib}$ are known constant real matrices with appropriate dimensions.
${F}_{ia}(k)$, ${F}_{ib}(k)$ are unknown uncertain matrices satisfying
The timevarying function $d(k)$ satisfies the following condition:
Remark 2.1 It is worth noting that the time delay is a timevarying function belonging to a given interval, in which the lower bound of delay is not restricted to zero.
Definition 2.1 The uncertain switched system (2.1) is robustly stable if there exists a switching function $\gamma (\cdot )$ such that the zero solution of the uncertain switched system is asymptotically stable for all uncertainties which satisfy (2.2) and (2.3).
Definition 2.2 The system of matrices $\{{J}_{i}\}$, $i=1,2,\dots ,N$, is said to be strictly complete if for every $x\in {R}^{n}\setminus \{0\}$ there is $i\in \{1,2,\dots ,N\}$ such that ${x}^{T}{J}_{i}x<0$.
It is easy to see that the system $\{{J}_{i}\}$ is strictly complete if and only if
where
Proposition 2.1 ([38])
The system$\{{J}_{i}\}$, $i=1,2,\dots ,N$, is strictly complete if there exist${\delta}_{i}\ge 0$, $i=1,2,\dots ,N$, ${\sum}_{i=1}^{N}{\delta}_{i}>0$such that
If$N=2$then the above condition is also necessary for the strict completeness.
Proposition 2.2 (Cauchy inequality)
For any symmetric positive definite matrix $N\in {M}^{n\times n}$ and $a,b\in {R}^{n}$ we have
Proposition 2.3 ([38])
Let E, H and F be any constant matrices of appropriate dimensions and${F}^{T}F\le I$. For any$\u03f5>0$, we have
3 Main results
3.1 Stability
Let us set
where
The main result of this paper is summarized in the following theorem.
Theorem 3.1 The uncertain switched system (2.1) is robustly stable if there exist symmetric positive definite matrices$P>0$, $Q>0$and matrices${S}_{1}$, ${S}_{2}$satisfying the following conditions

(i)
$\mathrm{\exists}{\delta}_{i}\ge 0$, $i=1,2,\dots ,N$, ${\sum}_{i=1}^{N}{\delta}_{i}>0:{\sum}_{i=1}^{N}{\delta}_{i}{J}_{i}({S}_{1},Q)<0$,

(ii)
${W}_{i}({S}_{1},{S}_{2},P,Q)<0$, $i=1,2,\dots ,N$.
The switching rule is chosen as$\gamma (x(k))=i$, whenever$x(k)\in {\overline{\alpha}}_{i}$.
Proof Consider the following LyapunovKrasovskii functional for any i th system (2.1)
where
We can verify that
Let us set $\xi (k)={[{x}^{T}(k){x}^{T}(k+1){x}^{T}(kd(k))]}^{T}$, and
Then, the difference of ${V}_{1}(k)$ along the solution of the system is given by
because of
Using the expression of system (2.1)
we have
Therefore, from (3.3), it follows that
Applying Proposition 2.2, Proposition 2.3 and condition (2.2), the following estimations hold
Therefore, we have
The difference of ${V}_{2}(k)$ is given by
Since $d(k)\ge {d}_{1}$, we have
and hence from (3.5) we have
The difference of ${V}_{3}(k)$ is given by
Since $d(k)\le {d}_{2}$, and
we obtain from (3.6) and (3.7) that
Therefore, combining the inequalities (3.4), (3.8) gives
where
Therefore, we finally obtain from (3.9) and the condition (ii) that
We now apply the condition (i) and Proposition 2.1, the system ${J}_{i}({S}_{1},Q)$ is strictly complete, and the sets ${\alpha}_{i}$ and ${\overline{\alpha}}_{i}$ by (3.1) are well defined such that
Therefore, for any $x(k)\in {R}^{n}$$k=1,2,\dots $, there exists $i\in \{1,2,\dots ,N\}$ such that $x(k)\in {\overline{\alpha}}_{i}$. By choosing switching rule as $\gamma (x(k))=i$ whenever $x(k)\in {\overline{\alpha}}_{i}$, from the condition (3.9) we have
which, combining the condition (3.2) and the Lyapunov stability theorem [39], concludes the proof of the theorem. □
3.2 Stabilization
Consider uncertain control discretetime systems with interval timevarying delay of the form
where $x(k)\in {R}^{n}$ is the state, $u(k)\in {R}^{m}$, $m\le n$, is the control input, $\gamma (\cdot ):{R}^{n}\to \mathcal{N}:=\{1,2,\dots ,N\}$ is the switching rule, which is a function depending on the state at each time and will be designed. A switching function is a rule which determines a switching sequence for a given switching system. Moreover, $\gamma (x(k))=i$ implies that the system realization is chosen as the i th system, $i=1,2,\dots ,N$. It is seen that the system (2.1) can be viewed as an autonomous switched system in which the effective subsystem changes when the state $x(k)$ hits predefined boundaries. We consider a delayed feedback control law
and ${C}_{i}$, $i=1,2,\dots ,N$ is the controller gain to be determined. ${A}_{i}$, ${D}_{i}$, $i=1,2,\dots ,N$ are given constant matrices and the timevarying uncertain matrices $\mathrm{\Delta}{A}_{i}(k)$, $\mathrm{\Delta}{D}_{i}(k)$, and $\mathrm{\Delta}{C}_{i}(k)$ are defined by: $\mathrm{\Delta}{A}_{i}(k)={E}_{ia}{F}_{ia}(k){H}_{ia},\mathrm{\Delta}{D}_{i}(k)={E}_{id}{F}_{id}(k){H}_{id}$, and $\mathrm{\Delta}{C}_{i}(k)={E}_{ic}{F}_{ic}(k){H}_{ic}$ where ${E}_{ia}$, ${E}_{id}$, ${E}_{ic}$, ${H}_{ia}$, ${H}_{id}$, ${H}_{ic}$ are known constant real matrices with appropriate dimensions. ${F}_{ia}(k)$, ${F}_{id}(k)$, ${F}_{ic}(k)$ are unknown uncertain matrices satisfying
The timevarying function $d(k)$ satisfies the following condition:
Remark 3.1 It is worth noting that the time delay is a timevarying function belonging to a given interval, in which the lower bound of delay is not restricted to zero.
Applying the feedback controller (3.11) to the system (3.10), the closedloop discrete timedelay system is
Definition 3.1 The uncertain switched control system (3.10) is robustly stabilizable if there is a delayed feedback control (3.11) such that the switched system (3.13) is robustly stable.
Let us set
where
Theorem 3.2 The switched control system (3.10) is robustly stabilizable by the delayed feedback control (3.11), where
if there exist symmetric matrices$P>0$, $Q>0$and matrices${S}_{1}$, ${S}_{2}$satisfying the following conditions

(i)
$\mathrm{\exists}{\delta}_{i}\ge 0$, $i=1,2,\dots ,N$, ${\sum}_{i=1}^{N}{\delta}_{i}>0:{\sum}_{i=1}^{N}{\delta}_{i}{J}_{i}({S}_{1},Q)<0$,

(ii)
${W}_{i}({S}_{1},{S}_{2},P,Q)<0$, $i=1,2,\dots ,N$.
The switching rule is chosen as$\gamma (x(k))=i$, whenever$x(k)\in {\overline{\alpha}}_{i}$.
Proof Using the feedback control (3.11), the closedloop system leads to the system (2.1), where
Since ${S}_{1}({B}_{i}+\mathrm{\Delta}{B}_{i})={S}_{1}$, ${({B}_{i}+\mathrm{\Delta}{B}_{i})}^{T}{S}_{1}^{T}={S}_{1}^{T}$, ${S}_{2}({B}_{i}+\mathrm{\Delta}{B}_{i})={S}_{2}$, ${({B}_{i}+\mathrm{\Delta}{B}_{i})}^{T}{S}_{2}^{T}={S}_{2}^{T}$, the stability condition of the closedloop system (3.13), by Theorem 3.1, is immediately derived. □
Remark 3.2 Note that the results proposed in [40–42] for switching systems to be asymptotically stable under an arbitrary switching rule. The asymptotic stability for switching linear discretetime delay systems studied in [12] was limited to constant delays. In [43], a class of switching signals has been identified for the considered switched discretetime delay systems to be stable under the averaged well time scheme.
4 Numerical examples
Example 4.1 (Stability)
Consider the uncertain switched discretetime system (2.1), where the delay function $d(k)$ is given by
By LMI toolbox of Matlab, we find that the conditions (i), (ii) of Theorem 3.1 are satisfied with ${d}_{1}=1$, ${d}_{2}=5$, ${\delta}_{1}=1$, ${\delta}_{2}=1$, and
In this case, we have
Moreover, the sum
is negative definite, i.e., the first entry in the first row and the first column $0.5761<0$ is negative and the determinant of the matrix is positive. The sets ${\alpha}_{1}$ and ${\alpha}_{2}$ are given as
Obviously, the union of these sets is equal to ${R}^{2}\setminus \{0\}$. The switching regions are defined as
By Theorem 3.1 the uncertain system is robustly stable and the switching rule is chosen as $\gamma (x(k))=i$ whenever $x(k)\in {\overline{\alpha}}_{i}$.
Example 4.2 (Stabilization)
Consider the uncertain switched discretetime control system (3.10), where the delay function $d(k)$ is given by
By LMI toolbox of Matlab, we find that the conditions (i), (ii) of Theorem 3.2 are satisfied with ${d}_{1}=1$, ${d}_{2}=8$, ${\delta}_{1}=2$, ${\delta}_{2}=1$, and
In this case, we have
Moreover, the sum
is negative definite, i.e., the first entry in the first row and the first column $4.7344<0$ is negative and the determinant of the matrix is positive. The sets ${\alpha}_{1}$ and ${\alpha}_{2}$ are given as
Obviously, the union of these sets is equal to ${R}^{2}\setminus \{0\}$. The switching regions are defined as
By Theorem 3.2, the control system is robustly stabilizable and the switching rule is $\sigma (x(k))=i$ whenever $x(k)\in {\overline{\mathrm{\Omega}}}_{i}$, the delayed feedback control is
5 Conclusion
This paper has proposed a switching design for the robust stability and stabilization of uncertain switched linear discretetime systems with interval timevarying delays. Based on the discrete Lyapunov functional, a switching rule for the robust stability and stabilization for the uncertain system is designed via linear matrix inequalities.
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Acknowledgements
This work is partially supported by the Thai Research Fund Grant, the Higher Education Commission and Faculty of Science, Maejo University, Thailand (for the first and third authors) and also by Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand (for the second author). The authors thank anonymous reviewers for valuable comments and suggestions, which allowed us to improve the paper.
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Keywords
 switching design
 uncertain discrete system
 robust stability and stabilization
 Lyapunov function
 linear matrix inequality