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LMI approach to robust stability and stabilization of nonlinear uncertain discretetime systems with convex polytopic uncertainties
Advances in Difference Equations volume 2012, Article number: 106 (2012)
Abstract
This article addresses the robust stability for a class of nonlinear uncertain discretetime systems with convex polytopic of uncertainties. The system to be considered is subject to both interval timevarying delays and convex polytopictype uncertainties. Based on the augmented parameterdependent LyapunovKrasovskii functional, new delaydependent conditions for the robust stability are established in terms of linear matrix inequalities. An application to robust stabilization of nonlinear uncertain discretetime control systems is given. Numerical examples are included to illustrate the effectiveness of our results.
MSC:15A09, 52A10, 74M05, 93D05.
1 Introduction
Since the time delay is frequently viewed as a source of instability and encountered in various engineering systems such as chemical processes, long transmission lines in pneumatic systems, networked control systems, etc., the study of delay systems has received much attention and various topics have been discussed over the past years. The problem of stability and stabilization of dynamical systems with time delays has received considerable attention, and lots of interesting results have reported in the literature, see [1–8] and the references therein. Some delaydependent stability criteria for discretetime systems with timevarying delay are investigated in [2, 6, 9–11], where the discrete Lyapunov functional method are employed to prove stability conditions in terms of linear matrix inequalities (LMIs). A number research works for dealing with asymptotic stability problem for discrete systems with interval timevarying delays have been presented in [12–22]. Theoretically, stability analysis of the systems with timevarying delays is more complicated, especially for the case where the system matrices belong to some convex polytope. In this case, the parameterdependent LyapunovKrasovskii functionals are constructed as the convex combination of a set of functions assures the robust stability of the nominal systems and the stability conditions must be solved upon a grid on the parameter space, which results in testing a finite number of LMIs [11, 23, 24]. To the best of the authors’ knowledge, the stability for linear discretetime systems with both timevarying delays and polytopic uncertainties has not been fully investigated. The articles [25, 26] propose sufficient conditions for robust stability of discrete and continuous polytopic systems without time delays. More recently, combining the ideas in [23, 24], improved conditions for $\mathcal{D}$stability and $\mathcal{D}$stabilization of linear polytopic delaydifference equations with constant delays have been proposed in [27].
In this article, we consider polytopic nonlinear uncertain discretetime equations with interval timevarying delays. Using the parameterdependent LyapunovKrasovskii functional combined with LMI techniques, we propose new criteria for the robust stability of the nonlinear uncertain system. The delaydependent stability conditions are formulated in terms of LMIs, being thus solvable by the numeric technology available in the literature to date. The result is applied to robust stabilization of nonlinear uncertain discretetime control systems. Compared to other results, our result has its own advantages. First, it deals with the nonlinear uncertain delaydifference system, where the statespace data belong to the convex polytope of uncertainties and the rate of change of the state depends not only on the current state of the nonlinear systems but also its state at some times in the past. Second, the timedelay is assumed to be a timevarying function belonging to a given interval, which means that the lower and the upper bounds for the timevarying delay are available. Third, our approach allows us to apply in robust stabilization of the nonlinear uncertain discretetime system subjected to polytopic uncertainties and external controls. Therefore, our results are more general than the related previous results.
The article is organized as follows. In Section 2, introduces the main notations, definitions, and some lemmas needed for the development of the main results. In Section 3, sufficient conditions are derived for robust stability, stabilization of nonlinear uncertain discretetime systems with interval timevarying delays and polytopic uncertainties. They are followed by some remarks. Illustrative examples are given in Section 4.
2 Preliminaries
The following notations will be used throughout this article. ${R}^{+}$ denotes the set of all real nonnegative numbers; ${R}^{n}$ denotes the ndimensional space with the scalar product $\u3008\cdot ,\cdot \u3009$ and the vector norm $\parallel \cdot \parallel $; ${R}^{n\times r}$ denotes the space of all matrices of $(n\times r)$dimension. ${A}^{T}$ denotes the transpose of A; a matrix A is symmetric if $A={A}^{T}$, a matrix I is the identity matrix of appropriate dimension.
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 $AB\ge 0$.
Consider a nonlinear uncertain delaydifference systems with polytopic uncertainties of the form where $x(k)\in {R}^{n}$ is the state, the system matrices are subjected to uncertainties and belong to the polytope Ω given by
where ${A}_{i}$, ${D}_{i}$, $i=1,2,\dots ,p$, are given constant matrices with appropriate dimensions.
The nonlinear perturbations $f(k,x(kh(k)))$ satisfies the following condition
where β is positive constants. For simplicity, we denote $f(k,x(kh(k)))$ by f, respectively.
The timevarying uncertain matrices $\mathrm{\Delta}A(k)$ and $\mathrm{\Delta}D(k)$ are defined by
where ${E}_{a}$, ${E}_{d}$, ${H}_{a}$, ${H}_{d}$ are known constant real matrices with appropriate dimensions. ${F}_{a}(k)$, ${F}_{d}(k)$ are unknown uncertain matrices satisfying
where I is the identity matrix of appropriate dimension.
The timevarying function $h(k)$ satisfies the condition:
Remark 2.1 It is worth noting that the time delay is a timevarying function belonging to a given interval, which allows the timedelay to be a fast timevarying function and the lower bound is not restricted to being zero as considered in [2, 6, 9–11].
Definition 2.1 The nonlinear uncertain system (${\mathrm{\Sigma}}_{\xi}$) is robustly stable if the zero solution of the system is asymptotically stable for all uncertainties which satisfy (2.1), (2.3), and (2.4).
Proposition 2.1 For real numbers${\xi}_{i}\ge 0$, $i=1,2,\dots ,p$, ${\sum}_{i=1}^{p}{\xi}_{i}=1$, the following inequality hold
Proof The proof is followed from the completing the square:
□
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 ([1])
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 Robust stability
In this section, we present sufficient delaydependent conditions for the robust stability of nonlinear uncertain system (${\mathrm{\Sigma}}_{\xi}$).
Let us set
Theorem 3.1 The nonlinear uncertain system (${\mathrm{\Sigma}}_{\xi}$) is robustly stable if there exist symmetric matrices${P}_{i}>0$, ${Q}_{i}>0$, $i=1,2\dots ,p$and constant matrices$S\ge 0$, ${S}_{1i}$, ${S}_{2i}$, $i=1,2\dots ,p$satisfying the following LMIs:

(i)
${\mathcal{M}}_{ii}(P,Q,{S}_{1},{S}_{2})+\mathcal{S}<0$, $i=1,2,\dots ,p$.

(ii)
${\mathcal{M}}_{ij}(P,Q,{S}_{1},{S}_{2})+{\mathcal{M}}_{ji}(P,Q,{S}_{1},{S}_{2})\frac{2}{p1}\mathcal{S}<0$, $i=1,2,\dots ,p1$; $j=i+1,\dots ,p$.
Proof Consider the following parameterdependent LyapunovKrasovskii functional for system (${\mathrm{\Sigma}}_{\xi}$)
where
We can verify that
Let us set $z(k)={[x(k)x(k+1)x(kh(k))f(k,x(kh(k)))]}^{T}$, and
Then, the difference of ${V}_{1}(k)$ along the solution of the system (${\mathrm{\Sigma}}_{\xi}$), we obtained
because of
Using the expression of system (${\mathrm{\Sigma}}_{\xi}$)
we have
Therefore, from (3.2) it follows that
Applying Propositions 2.2, 2.3 and condition (2.4), the following estimations hold
Therefore, we have
The difference of ${V}_{2}(k)$ is given by
Since $h(k)\ge {h}_{1}$ we have
and hence from (3.4) we have
The difference of ${V}_{3}(k)$ is given by
Since
we obtain from (3.5) and (3.6) that
Therefore, combining the inequalities (3.3), (3.7) gives
where
Let us denote
From the convex combination of the expression of $P(\xi )$, $Q(\xi )$, ${S}_{1}(\xi )$, ${S}_{2}(\xi )$, $A(\xi )$, $D(\xi )$, we have
Then the conditions (i), (ii) give
because of Proposition 2.1
and hence, we finally obtain from (3.8) that
which together with (3.1) implies that the system (${\mathrm{\Sigma}}_{\xi}$) is robustly stable. This completes the proof of the theorem. □
Remark 3.1 The stability conditions of Theorem 3.1 are more appropriate for practical systems since practically it is impossible to know exactly the delay but lower and upper bounds are always possible.
3.2 Robust stabilization
This section deals with a stabilization problem considered in [15] for constructing a delayed feedback controller, which stabilizes the resulting closedloop system. The robust stability condition obtained in previous section will be applied to design a timedelayed state feedback controller for the nonlinear uncertain discretetime control system described by
where $u(k)\in {R}^{n}$ is the control input, the system matrices are subjected to uncertainties and belong to the polytope Ω given by
where ${A}_{i}$${B}_{i}$$i=1,2,\dots ,p$, are given constant matrices with appropriate dimensions. As in [8], we consider a parameterdependent delayed feedback control law
$F(\xi )+\mathrm{\Delta}F(k)$ is the controller gain to be determined. The timevarying uncertain matrices $\mathrm{\Delta}A(k)$$\mathrm{\Delta}B(k)$, and $\mathrm{\Delta}F(k)$ are defined by
where ${E}_{a}$${E}_{b}$${E}_{f}$${H}_{a}$${H}_{b}$${H}_{f}$ are known constant real matrices with appropriate dimensions. ${F}_{a}(k)$${F}_{b}(k)$${F}_{f}(k)$ are unknown uncertain matrices satisfying
where I is the identity matrix of appropriate dimension.
Applying the feedback controller (3.10) to the nonlinear uncertain system (3.9), the closedloop timedelay nonlinear uncertain system is
The timevarying function $h(k)$ satisfies the condition:
Definition 3.1 The nonlinear uncertain system (3.9) is robustly stabilizable if there is a delayed feedback control (3.10) such that the closedloop delay nonlinear uncertain system (3.11) is robustly stable.
Let us
The following theorem can be derived from Theorem 3.1.
Theorem 3.2 The nonlinear uncertain system (3.9) is robustly stabilizable by the delayed feedback control (3.10), where$(F(\xi )+\mathrm{\Delta}F(k))={(B(\xi )+\mathrm{\Delta}B(k))}^{T}{[(B(\xi )+\mathrm{\Delta}B(k)){(B(\xi )+\mathrm{\Delta}B(k))}^{T}]}^{1}$, if there exist symmetric matrices${P}_{i}>0$, ${Q}_{i}>0$, $i=1,2,\dots ,p$and constant matrices${S}_{1i}$, $i=1,2,\dots ,p$, $S\ge 0$satisfying the following LMIs:

(i)
${\mathcal{M}}_{ii}(P,Q,{S}_{1})+\mathcal{S}<0$, $i=1,2,\dots ,p$.

(ii)
${\mathcal{M}}_{ij}(P,Q,{S}_{1})+{\mathcal{M}}_{ji}(P,Q,{S}_{1})\frac{2}{p1}\mathcal{S}<0$, $i=1,2,\dots ,p1$; $j=i+1,\dots ,p$.
Proof Taking ${S}_{1i}={S}_{2i}$ and using the feedback control (3.10), the closedloop nonlinear uncertain system becomes system (${\mathrm{\Sigma}}_{\xi}$), where $(D(\xi )+\mathrm{\Delta}D(k))=(B(\xi )+\mathrm{\Delta}B(k))(F(\xi )+\mathrm{\Delta}F(k))=(B(\xi )+\mathrm{\Delta}B(k)){(B(\xi )+\mathrm{\Delta}B(k))}^{T}{[(B(\xi )+\mathrm{\Delta}B(k)){(B(\xi )+\mathrm{\Delta}B(k))}^{T}]}^{1}=I$. Since ${S}_{1}(\xi )(D(\xi )+\mathrm{\Delta}D(k))={S}_{1}(\xi )$, the robust stability condition of the closedloop nonlinear uncertain system (3.11), by Theorem 3.1, is immediately derived. □
Remark 3.2 The stabilization conditions of Theorem 3.2 are more appropriate for practical systems since practically it is impossible to know exactly the delay but lower and upper bounds are always possible.
4 Numerical examples
To illustrate the effectiveness of the previous theoretical results, we consider the following numerical examples.
Example 4.1 (Robust stability)
Consider nonlinear uncertain system (${\mathrm{\Sigma}}_{\xi}$) for $p=2$, where the delay function $h(k)$ is given by
and
with any timevarying delay function $h(k)$ with ${h}_{1}=1$, ${h}_{2}=20$. By using the LMI Toolbox in MATLAB, the LMIs (i) and (ii) of Theorem 3.1 are feasible with
Therefore, the nonlinear uncertain system is robustly stable.
Example 4.2 (Robust stabilization)
Consider nonlinear uncertain control system (3.9) for $p=2$, where the delay function $h(k)$ is given by
and
with any timevarying delay function $h(k)$ with ${h}_{1}=1$, ${h}_{2}=33$, ${\xi}_{1}=0.5$, ${\xi}_{2}=0.5$. By using the LMI Toolbox in MATLAB, the LMIs (i) and (ii) of Theorem 3.2 are feasible with
Therefore, the nonlinear uncertain system is robustly stabilizable with the feedback control
Therefore, the feedback delayed controller is
5 Conclusion
In this article, new delaydependent robust stability conditions for nonlinear uncertain polytopic delaydifference equations with interval timevarying delays have been presented in terms of LMIs. An application in robust stabilization of nonlinear uncertain control discrete systems with timedelayed feedback controllers has been studied. Numerical examples have been given to demonstrate the effectiveness of the proposed conditions.
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Acknowledgements
This work was supported by the Thai Research Fund Grant, the Higher Education Commission and Faculty of Science, Maejo University, Thailand. The authors thank anonymous reviewers for valuable comments and suggestions, which allowed us to improve the article.
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Keywords
 robust stability and stabilization
 nonlinear uncertain discretetime systems
 convex polytopic uncertainties
 LyapunovKrasovskii functional
 linear matrix inequality