LMI approach to robust stability and stabilization of nonlinear uncertain discrete-time systems with convex polytopic uncertainties

Abstract

This article addresses the robust stability for a class of nonlinear uncertain discrete-time systems with convex polytopic of uncertainties. The system to be considered is subject to both interval time-varying delays and convex polytopic-type uncertainties. Based on the augmented parameter-dependent Lyapunov-Krasovskii functional, new delay-dependent conditions for the robust stability are established in terms of linear matrix inequalities. An application to robust stabilization of nonlinear uncertain discrete-time 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 delay-dependent stability criteria for discrete-time systems with time-varying 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 time-varying delays have been presented in [12–22]. Theoretically, stability analysis of the systems with time-varying delays is more complicated, especially for the case where the system matrices belong to some convex polytope. In this case, the parameter-dependent Lyapunov-Krasovskii 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 discrete-time systems with both time-varying 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 delay-difference equations with constant delays have been proposed in [27].

In this article, we consider polytopic nonlinear uncertain discrete-time equations with interval time-varying delays. Using the parameter-dependent Lyapunov-Krasovskii functional combined with LMI techniques, we propose new criteria for the robust stability of the nonlinear uncertain system. The delay-dependent 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 discrete-time control systems. Compared to other results, our result has its own advantages. First, it deals with the nonlinear uncertain delay-difference system, where the state-space 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 time-delay is assumed to be a time-varying function belonging to a given interval, which means that the lower and the upper bounds for the time-varying delay are available. Third, our approach allows us to apply in robust stabilization of the nonlinear uncertain discrete-time 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 discrete-time systems with interval time-varying 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 non-negative numbers; $R^n$ denotes the n-dimensional space with the scalar product $〈\cdot ,\cdot 〉$ 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 semi-positive definite ($A\ge 0$) if $〈Ax,x〉\ge 0$, for all $x\in R^n$; A is positive definite ($A>0$) if $〈Ax,x〉>0$ for all $x\ne 0$; $A\ge B$ means $A-B\ge 0$.

Consider a nonlinear uncertain delay-difference 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

$$\mathrm{\Omega }=\{[A,D](\xi ):=\sum _{i=1}^p{\xi }_i[A_i,D_i],\sum _{i=1}^p{\xi }_i=1,{\xi }_i\ge 0\},$$

(2.1)

where $A_i$, $D_i$, $i=1,2,\dots ,p$, are given constant matrices with appropriate dimensions.

The nonlinear perturbations $f(k,x(k-h(k)))$ satisfies the following condition

$$f^T(k,x(k-h(k)))f(k,x(k-h(k)))\le {\beta }^2{x}^T(k-h(k))x(k-h(k)),$$

(2.2)

where β is positive constants. For simplicity, we denote $f(k,x(k-h(k)))$ by f, respectively.

The time-varying uncertain matrices $\mathrm{\Delta }A(k)$ and $\mathrm{\Delta }D(k)$ are defined by

$$\mathrm{\Delta }A(k)=E_a{F}_a(k)H_a,\mathrm{\Delta }D(k)=E_d{F}_d(k)H_d,$$

(2.3)

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

$$F_a^T(k)F_a(k)\le I,F_d^T(k)F_d(k)\le I,k=0,1,2,\dots ,$$

(2.4)

where I is the identity matrix of appropriate dimension.

The time-varying function $h(k)$ satisfies the condition:

$$0<h_1\le h(k)\le h_2,\mathrm{\forall }k=0,1,2,\dots .$$

Remark 2.1 It is worth noting that the time delay is a time-varying function belonging to a given interval, which allows the time-delay to be a fast time-varying 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

$$(p-1)\sum _{i=1}^p{\xi }_i^2-2\sum _{i=1}^{p-1}\sum _{j=i+1}^p{\xi }_i{\xi }_j\ge 0.$$

Proof The proof is followed from the completing the square:

$$(p-1)\sum _{i=1}^p{\xi }_i^2-2\sum _{i=1}^{p-1}\sum _{j=i+1}^p{\xi }_i{\xi }_j=\sum _{i=1}^{p-1}\sum _{j=i+1}^p{({\xi }_i-{\xi }_j)}^2\ge 0.$$

 □

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

$$\underline{+}{a}^{T}b\le a^{T}Na+b^T{N}^{-1}b.$$

Proposition 2.3 ([1])

Let E, H and F be any constant matrices of appropriate dimensions and$F^{T}F\le I$. For any$ϵ>0$, we have

$$EFH+H^T{F}^T{E}^T\le ϵE{E}^T+{ϵ}^{-1}{H}^{T}H.$$

3 Main results

3.1 Robust stability

In this section, we present sufficient delay-dependent 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) ${\mathcal{M}}_{ij}(P,Q,S_1,S_2)+{\mathcal{M}}_{ji}(P,Q,S_1,S_2)-\frac{2}{p-1}\mathcal{S}<0$, $i=1,2,\dots ,p-1$; $j=i+1,\dots ,p$.

Proof Consider the following parameter-dependent Lyapunov-Krasovskii functional for system (${\mathrm{\Sigma }}_{\xi }$)

$$V(k)=V_1(k)+V_2(k)+V_3(k),$$

where

We can verify that

$${\lambda }_1{\parallel x(k)\parallel }^2\le V(k)\le {\lambda }_2{\parallel x_k\parallel }^2.$$

(3.1)

Let us set $z(k)={[x(k)x(k+1)x(k-h(k))f(k,x(k-h(k)))]}^T$, and

Then, the difference of $V_1(k)$ along the solution of the system (${\mathrm{\Sigma }}_{\xi }$), we obtained

$$\begin{array}{rcl}\mathrm{\Delta }{V}_1(k)& =& x^T(k+1)P(\xi )x(k+1)-x^T(k)P(\xi )x(k)\\ =& z{(k)}^{T}G(\xi )z(k)-2z^T(k)F^T(\xi )\left(\begin{array}{c}0.5x(k)\\ 0\\ 0\\ 0\end{array}\right),\end{array}$$

(3.2)

because of

Using the expression of system (${\mathrm{\Sigma }}_{\xi }$)

$$\begin{array}{rcl}0& =& -S_1(\xi )x(k+1)+S_1(\xi )(A(\xi )+\mathrm{\Delta }A(k))x(k)\\ +S_1(\xi )(D(\xi )+\mathrm{\Delta }D(k))x(k-h(k))+S_1(\xi )f,\\ 0& =& -S_2(\xi )x(k+1)+S_2(\xi )(A(\xi )+\mathrm{\Delta }A(k))x(k)\\ +S_2(\xi )(D(\xi )+\mathrm{\Delta }D(k))x(k-h(k))+S_2(\xi )f,\end{array}$$

we have

Therefore, from (3.2) it follows that

$$\begin{array}{rcl}\mathrm{\Delta }{V}_1(k)& =& x^T(k)[-P(\xi )-S_1(\xi )A(\xi )-S_1(\xi )E_a{F}_a(k)H_a\\ -A{(\xi )}^T{S}_1^T(\xi )-H_a^T{F}_a^T(k)E_a^T{S}_1^T(\xi )]x(k)\\ +2x^T(k)[S_1(\xi )-S_1(\xi )A(\xi )-S_1(\xi )E_a{F}_a(k)H_a]x(k+1)\\ +2x^T(k)[-S_1(\xi )D(\xi )-S_1(\xi )E_d{F}_d(k)H_d]x(k-h(k))\\ +2x^T(k)[-S_1(\xi )-S_2(\xi )A(\xi )-S_2(\xi )E_a{F}_a(k)H_a]f(k,x(k-h(k)))\\ +x(k+1)[P(\xi )+S_1(\xi )+S_1^T(\xi )]x(k+1)\\ +2x(k+1)[-S_1(\xi )D(\xi )-S_1(\xi )E_d{F}_d(k)H_d]x(k-h(k))\\ +2x(k+1)[S_2(\xi )-S_1(\xi )]f(k,x(k-h(k)))\\ +2x^T(k-h(k))[-S_2(\xi )D(\xi )-S_2(\xi )E_d{F}_d(k)H_d]f(k,x(k-h(k)))\\ +f^T(k,x(k-h(k)))(k)[-S_2-S_2^T]f(k,x(k-h(k))).\end{array}$$

Applying Propositions 2.2, 2.3 and condition (2.4), the following estimations hold

Therefore, we have

$$\begin{array}{rcl}\mathrm{\Delta }{V}_1(k)& =& x^T(k)[-P(\xi )-S_1(\xi )A(\xi )-A{(\xi )}^T{S}_1^T(\xi )+2S_1(\xi )E_a{E}_a^T{S}_1^T(\xi )\\ +S_1(\xi )E_d{E}_d^T{S}_1^T(\xi )+S_2(\xi )E_a{E}_a^T{S}_2^T(\xi )+H_a^T{H}_a]x(k)\\ +2x^T(k)[S_1(\xi )-S_1(\xi )A(\xi )]x(k+1)\\ +2x^T(k)[-S_1(\xi )D(\xi )]x(k-h(k))\\ +2x^T(k)[-S_1(\xi )-S_2(\xi )A(\xi )]f(k,x(k-h(k)))\\ +x(k+1)[P(\xi )+S_1(\xi )+S_1^T(\xi )+S_1(\xi )E_d{E}_d^T{S}_1^T(\xi )+H_a^T{H}_a]x(k+1)\\ +2x(k+1)[-S_1(\xi )D(\xi )]x(k-h(k))\\ +2x(k+1)[S_2(\xi )-S_1(\xi )]f(k,x(k-h(k)))\\ +x^T(k-h(k))[S_2(\xi )E_d{E}_d^T{S}_2^T(\xi )+2H_d^T{H}_d]x(k-h(k))\\ +2x^T(k-h(k))[-S_2(\xi )D(\xi )]f(k,x(k-h(k)))\\ +f^T(k,x(k-h(k)))(k)[-S_2(\xi )-S_2^T(\xi )+H_a^T{H}_a+H_d^T{H}_d]\\ \times f(k,x(k-h(k))).\end{array}$$

(3.3)

The difference of $V_2(k)$ is given by

$$\begin{array}{rcl}\mathrm{\Delta }{V}_2(k)& =& \sum _{i=k+1-h(k+1)}^k{x}^T(i)Q(\xi )x(i)-\sum _{i=k-h(k)}^{k-1}{x}^T(i)Q(\xi )x(i)\\ =& \sum _{i=k+1-h(k+1)}^{k-h_1}{x}^T(i)Q(\xi )x(i)+x^T(k)Q(\xi )x(k)-x^x(k-h(k))Q(\xi )x(k-h(k))\\ +\sum _{i=k+1-h_1}^{k-1}{x}^T(i)Q(\xi )x(i)-\sum _{i=k+1-h(k)}^{k-1}{x}^T(i)Q(\xi )x(i).\end{array}$$

(3.4)

Since $h(k)\ge h_1$ we have

$$\sum _{i=k+1-h_1}^{k-1}{x}^T(i)Q(\xi )x(i)-\sum _{i=k+1-h(k)}^{k-1}{x}^T(i)Q(\xi )x(i)\le 0,$$

and hence from (3.4) we have

$$\mathrm{\Delta }{V}_2(k)\le \sum _{i=k+1-h(k+1)}^{k-h_1}{x}^T(i)Q(\xi )x(i)+x^T(k)Q(\xi )x(k)-x^T(k-h(k))Q(\xi )x(k-h(k)).$$

(3.5)

The difference of $V_3(k)$ is given by

$$\begin{array}{rcl}\mathrm{\Delta }{V}_3(k)& =& \sum _{j=-h_2+2}^{-h_1+1}[x^T(k)Q(\xi )x(k)-x^T(k+j-1)Q(\xi )x(k+j-1)]\\ =& (h_2-h_1)x^T(k)Q(\xi )x(k)-\sum _{l=k+1-h_2}^{k-h_1}{x}^T(l)Q(\xi )x(l).\end{array}$$

(3.6)

Since

$$\sum _{i=k=1-h(k+1)}^{k-h_1}{x}^T(i)Q(\xi )x(i)-\sum _{i=k+1-h_2}^{k-h_1}{x}^T(i)Q(\xi )x(i)\le 0,$$

we obtain from (3.5) and (3.6) that

$$\mathrm{\Delta }{V}_2(k)+\mathrm{\Delta }{V}_3(k)\le (h_2-h_1+1)x^T(k)Q(\xi )x(k)-x^T(k-h(k))Q(\xi )x(k-h(k)).$$

(3.7)

Therefore, combining the inequalities (3.3), (3.7) gives

$$\mathrm{\Delta }V(k)\le z^T(k)T(\xi )z(k),$$

(3.8)

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

$$\begin{array}{rcl}T(\xi )& =& \sum _{i=1}^p{\xi }_i^2\left(\begin{array}{cccc}{M}_{1ii}& S_{1i}-S_{1i}{A}_i& -S_{1i}{D}_i& -S_{1i}-S_{2i}{A}_i\\ S_{1i}^T-A_i^T{S}_{1i}^T& M_{2ii}& -S_{1i}{D}_i& S_{2i}-S_{1i}\\ -D_i^T{S}_{1i}^T& -D_i^T{S}_{1i}^T& M_{3ii}& -S_{2i}{D}_i\\ -S_{1i}^T-A_i^T{S}_{2i}^T& S_{2i}^T-S_{1i}^T& -D_i^T{S}_{2i}^T& M_{4ii}\end{array}\right)\\ +\sum _{i=1}^{p-1}\sum _{j=i+1}^p{\xi }_i{\xi }_j\left(\begin{array}{cccc}{M}_{1ij}+M_{1ji}& S_{1ij}-{(S_{1}A)}_{ij}& -{(S_{1}D)}_{ij}& -S_{1ij}-{(S_{2}A)}_{ij}\\ S_{1ij}^T-{(A^T{S}_1^T)}_{ij}& M_{2ij}+M_{2ji}& -{(S_{1}D)}_{ij}& S_{2ij}-S_{1ij}\\ -{(D^T{S}_1^T)}_{ij}& -{(D^T{S}_1^T)}_{ij}& M_{3ij}+M_{3ji}& -{(S_{2}D)}_{ij}\\ -S_{1ij}^T-{(A^T{S}_2^T)}_{ij}& S_{2ij}^T-S_{1ij}^T& -{(D^T{S}_2^T)}_{ij}& M_{4ij}+M_{4ji}\end{array}\right)\\ =& \sum _{i=1}^p{\xi }_i^2{\mathcal{M}}_{ii}(P,Q,S_1,S_2)+\sum _{i=1}^{p-1}\sum _{j=i+1}^p{\xi }_i{\xi }_j[{\mathcal{M}}_{ij}(P,Q,S_1,S_2)+{\mathcal{M}}_{ji}(P,Q,S_1,S_2)].\end{array}$$

Then the conditions (i), (ii) give

$$T(\xi )<-\sum _{i=1}^p{\xi }_i^2\mathcal{S}+\frac{2}{p-1}\sum _{i=1}^{p-1}\sum _{j=i+1}^p{\xi }_i{\xi }_j\mathcal{S}\le 0,$$

because of Proposition 2.1

$$(p-1)\sum _{i=1}^p{\xi }_i^2-2\sum _{i=1}^{p-1}\sum _{j=i+1}^p{\xi }_i{\xi }_j=\sum _{i=1}^{p-1}\sum _{j=i+1}^p{({\xi }_i-{\xi }_j)}^2\ge 0,$$

and hence, we finally obtain from (3.8) that

$$\mathrm{\Delta }V(k)<0,\mathrm{\forall }k=0,1,2,\dots ,$$

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 closed-loop system. The robust stability condition obtained in previous section will be applied to design a time-delayed state feedback controller for the nonlinear uncertain discrete-time control system described by

$$\begin{array}{rcl}x(k+1)& =& (A(\xi )+\mathrm{\Delta }A(k))x(k)+(B(\xi )+\mathrm{\Delta }B(k))u(k)\\ +f(k,x(k-h(k))),k=0,1,2,\dots ,\end{array}$$

(3.9)

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

$$\mathrm{\Omega }=\{[A,B](\xi ):=\sum _{i=1}^p{\xi }_i[A_i,B_i],\sum _{i=1}^p{\xi }_i=1,{\xi }_i\ge 0\},$$

where $A_i$$B_i$$i=1,2,\dots ,p$, are given constant matrices with appropriate dimensions. As in [8], we consider a parameter-dependent delayed feedback control law

$$u(k)=(F(\xi )+\mathrm{\Delta }F(k))x(k-h(k)),k=-h_2,\dots ,0,$$

(3.10)

$F(\xi )+\mathrm{\Delta }F(k)$ is the controller gain to be determined. The time-varying uncertain matrices $\mathrm{\Delta }A(k)$$\mathrm{\Delta }B(k)$, and $\mathrm{\Delta }F(k)$ are defined by

$$\mathrm{\Delta }A(k)=E_a{F}_a(k)H_a,\mathrm{\Delta }B(k)=E_b{F}_b(k)H_b,\mathrm{\Delta }F(k)=E_f{F}_f(k)H_f,$$

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

$$F_a^T(k)F_a(k)\le I,F_b^T(k)F_b(k)\le I,F_f^T(k)F_f(k)\le I,k=0,1,2,\dots ,$$

where I is the identity matrix of appropriate dimension.

Applying the feedback controller (3.10) to the nonlinear uncertain system (3.9), the closed-loop time-delay nonlinear uncertain system is

$$\begin{array}{rcl}x(k+1)& =& (A(\xi )+\mathrm{\Delta }A(k))x(k)+(B(\xi )+\mathrm{\Delta }B(k))(F(\xi )+\mathrm{\Delta }F(k))x(k-h(k))\\ +f(k,x(k-h(k))),k=0,1,2,\dots .\end{array}$$

(3.11)

The time-varying function $h(k)$ satisfies the condition:

$$0<h_1\le h(k)\le h_2,\mathrm{\forall }k=0,1,2,\dots .$$

Definition 3.1 The nonlinear uncertain system (3.9) is robustly stabilizable if there is a delayed feedback control (3.10) such that the closed-loop 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) ${\mathcal{M}}_{ij}(P,Q,S_1)+{\mathcal{M}}_{ji}(P,Q,S_1)-\frac{2}{p-1}\mathcal{S}<0$, $i=1,2,\dots ,p-1$; $j=i+1,\dots ,p$.

Proof Taking $S_{1i}=S_{2i}$ and using the feedback control (3.10), the closed-loop 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 closed-loop 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

$$h(k)=1+19{sin}^2\frac{k\pi }{2},k=0,1,2,\dots ,$$

and

with any time-varying 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

$$h(k)=1+32{sin}^2\frac{k\pi }{2},k=0,1,2,\dots ,$$

and

with any time-varying 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

$$\begin{array}{rcl}u(k)& =& {(B(\xi )+\mathrm{\Delta }B(k))}^T{\left[(B(\xi )+\mathrm{\Delta }B(k)){(B(\xi )+\mathrm{\Delta }B(k))}^T\right]}^{-1}x(k-h(k))\\ =& {(0.5B_1+0.5B_2+\mathrm{\Delta }B(k))}^T\left[\right(0.5B_1+0.5B_2\\ +\mathrm{\Delta }B(k)){(0.5B_1+0.5B_2+\mathrm{\Delta }B(k))}^T{]}^{-1}x(k-h(k))\\ =& \left(\begin{array}{cc}-0.8881& -0.0061\\ -0.0061& -0.9100\end{array}\right)x(k-h(k)).\end{array}$$

Therefore, the feedback delayed controller is

$$u(k)=\left[\begin{array}{c}-0.8881x_1(k-h(k))-0.0061x_2(k-h(k))\\ -0.0061x_1(k-h(k))-0.9100x_2(k-h(k))\end{array}\right].$$

5 Conclusion

In this article, new delay-dependent robust stability conditions for nonlinear uncertain polytopic delay-difference equations with interval time-varying delays have been presented in terms of LMIs. An application in robust stabilization of nonlinear uncertain control discrete systems with time-delayed feedback controllers has been studied. Numerical examples have been given to demonstrate the effectiveness of the proposed conditions.