Consider the following uncertain discrete switched time-delay system:

where x(k)\in {\mathrm{\Re}}^{n}, {x}_{k} is the state defined by {x}_{k}(\theta ):=x(k+\theta ), \mathrm{\forall}\theta \in \{-{r}_{M},-{r}_{M}+1,\dots ,0\}, *σ* is a switching signal in the finite set \{1,2,\dots ,N\} and will be chosen to preserve the stability of the system, \varphi (k)\in {\mathrm{\Re}}^{n} is an initial state function, time-varying delay r(k) is a function from \{0,1,2,3,\dots \} to \{0,1,2,3,\dots \} and 1\le {r}_{m}\le r(k)\le {r}_{M}, {r}_{m} and {r}_{M} are two given positive integers. Matrices {A}_{i},{B}_{i}\in {\mathrm{\Re}}^{n\times n}, i=1,2,\dots ,N, are constant. \mathrm{\Delta}{A}_{i}(k) and \mathrm{\Delta}{B}_{i}(k) are two perturbed matrices satisfying the following condition:

where {M}_{i}, {N}_{Ai}, and {N}_{Bi}, i=1,2,\dots ,N, and {\mathrm{\Xi}}_{i} are some given constant matrices with appropriate dimensions. {\mathrm{\Gamma}}_{i}(k) is an unknown matrix representing the perturbation which satisfies

{\mathrm{\Gamma}}_{i}^{T}(k){\mathrm{\Gamma}}_{i}(k)\le I.

(1e)

**Definition 1** System (1a)-(1e) is said to be globally exponentially stable with a convergence rate *α* if there are two positive constants 0<\alpha <1 and Ψ such that

\parallel x(k)\parallel \le \mathrm{\Psi}\cdot {\parallel {x}_{0}\parallel}_{s}\cdot {\alpha}^{k},\phantom{\rule{1em}{0ex}}k=0,1,2,3,\dots .

Define the switching domains of a switching signal by

{\mathrm{\Omega}}_{i}(\alpha ,P,{U}_{i},{A}_{i})=\{x(k)\in {\mathrm{\Re}}^{n}:{x}^{T}(k)({A}_{i}^{T}P{A}_{i}-{\alpha}^{2}\cdot {U}_{i})x(k)<0\},\phantom{\rule{1em}{0ex}}i=1,2,\dots ,N,

(2a)

where the constant 0<\alpha <1 is a convergence rate, matrices P>0, {U}_{i}>0 are given from the proposed results, and

\begin{array}{r}{\overline{\mathrm{\Omega}}}_{1}={\mathrm{\Omega}}_{1},\phantom{\rule{2em}{0ex}}{\overline{\mathrm{\Omega}}}_{2}={\mathrm{\Omega}}_{2}\setminus {\overline{\mathrm{\Omega}}}_{1},\phantom{\rule{2em}{0ex}}{\overline{\mathrm{\Omega}}}_{3}={\mathrm{\Omega}}_{3}\setminus {\overline{\mathrm{\Omega}}}_{1}\setminus {\overline{\mathrm{\Omega}}}_{2},\dots ,\\ {\overline{\mathrm{\Omega}}}_{N}={\mathrm{\Omega}}_{N}\setminus {\overline{\mathrm{\Omega}}}_{1}\setminus \cdots \setminus {\overline{\mathrm{\Omega}}}_{N-1}.\end{array}

(2b)

Now the main results are provided in the following theorem.

**Theorem 1** *If for some constants* 0<\alpha <1, 0\le {\alpha}_{i}\le 1, i\in \overline{N}, *and* {\sum}_{i=1}^{N}{\alpha}_{i}=1, *there exist some* n\times n *matrices* P>0, Q>0, {R}_{1}>0, {R}_{2}>0, {R}_{3}>0, S>0, T>0, {U}_{i}>0, i\in \overline{N}, {V}_{l11}\in {\mathrm{\Re}}^{4n\times 4n}, {V}_{l12}\in {\mathrm{\Re}}^{4n\times n}, {V}_{l22}\in {\mathrm{\Re}}^{n\times n}, l=1,2,3,4, *and constants* {\epsilon}_{i}>0, i\in \overline{N}, *such that the following LMI conditions hold for all* j=1,2,\dots ,N:

*where*

*Then system* (1a)-(1e) *is globally exponentially stable with the convergence rate* 0<\alpha <1 *by the switching signal designed by*

\sigma (k,x(k))=i,\phantom{\rule{1em}{0ex}}\mathit{\text{whenever}}x(k)\in {\overline{\mathrm{\Omega}}}_{i},

(3e)

*where* {\overline{\mathrm{\Omega}}}_{i} *is defined in* (2a), (2b).

*Proof*

Define the Lyapunov functional

\begin{array}{rcl}V({x}_{k})& =& {\alpha}^{-2k}{x}^{T}(k)Px(k)+\sum _{i=k-r(k)}^{k-1}{\alpha}^{-2i}{x}^{T}(i)Qx(i)+\sum _{j=-{r}_{M}+1}^{-{r}_{m}}\sum _{i=k+j}^{k-1}{\alpha}^{-2i}{x}^{T}(i)Qx(i)\\ +\sum _{j=-{r}_{M}+1}^{0}\sum _{i=k-1+j}^{k-1}{\alpha}^{-2i}{y}^{T}(i)[{R}_{1}+{R}_{2}]y(i)+\sum _{j=-{r}_{m}+1}^{0}\sum _{i=k-1+j}^{k-1}{\alpha}^{-2i}{y}^{T}(i){R}_{3}y(i)\\ +\sum _{i=k-{r}_{M}}^{k-1-{r}_{m}}{\alpha}^{-2i}{x}^{T}(i)Sx(i)+\sum _{i=k-{r}_{m}}^{k-1}{\alpha}^{-2i}{x}^{T}(i)Tx(i),\end{array}

(4)

where P>0, Q>0, {R}_{1}>0, {R}_{2}>0, {R}_{3}>0, S>0, T>0, and y(i)=x(i+1)-x(i). The forward difference of Lyapunov functional (4) along the solutions of system (1a)-(1e) has the form

\begin{array}{rcl}\mathrm{\Delta}V({x}_{k})& =& V({x}_{k+1})-V({x}_{k})\\ =& {\alpha}^{-2k}\cdot [{\alpha}^{-2}\cdot {x}^{T}(k+1)Px(k+1)-{x}^{T}(k)Px(k)]\\ +\sum _{i=k+1-r(k+1)}^{k}{\alpha}^{-2i}{x}^{T}(i)Qx(i)-\sum _{i=k-r(k)}^{k-1}{\alpha}^{-2i}{x}^{T}(i)Qx(i)\\ +{\alpha}^{-2k}\cdot ({r}_{M}-{r}_{m})\cdot {x}^{T}(k)Qx(k)-\sum _{i=k+1-{r}_{M}}^{k-{r}_{m}}{\alpha}^{-2i}{x}^{T}(i)Qx(i)\\ +{\alpha}^{-2k}\cdot {r}_{M}\cdot {y}^{T}(k)[{R}_{1}+{R}_{2}]y(k)-\sum _{i=k-{r}_{M}}^{k-1}{\alpha}^{-2i}{y}^{T}(i)[{R}_{1}+{R}_{2}]y(i)\\ +{\alpha}^{-2k}\cdot {r}_{m}\cdot {y}^{T}(k){R}_{3}y(k)-\sum _{i=k-{r}_{m}}^{k-1}{\alpha}^{-2i}{y}^{T}(i){R}_{3}y(i)\\ +{\alpha}^{-2k}\cdot [{\alpha}^{2{r}_{m}}\cdot {x}^{T}(k-{r}_{m})Sx(k-{r}_{m})-{\alpha}^{2{r}_{M}}\cdot {x}^{T}(k-{r}_{M})Sx(k-{r}_{M})]\\ +{\alpha}^{-2k}\cdot [{x}^{T}(k)Tx(k)-{\alpha}^{2{r}_{m}}\cdot {x}^{T}(k-{r}_{m})Tx(k-{r}_{m})].\end{array}

(5a)

By some simple derivations, we have

From the above derivation, we can obtain the following result:

\begin{array}{rcl}\mathrm{\Delta}V({x}_{k})& =& V({x}_{k+1})-V({x}_{k})\\ \le & {\alpha}^{-2k}\cdot \{{\alpha}^{-2}\cdot {x}^{T}(k+1)Px(k+1)-{x}^{T}(k)[P-T]x(k)\\ +{x}^{T}(k)[({r}_{M}-{r}_{m}+1)\cdot Q]x(k)-{\alpha}^{2{r}_{M}}\cdot {x}^{T}(k-r(k))Qx(k-r(k))\\ +{[x(k+1)-x(k)]}^{T}[{r}_{M}\cdot ({R}_{1}+{R}_{2})+{r}_{m}\cdot {R}_{3}][x(k+1)-x(k)]\\ -{\alpha}^{2{r}_{M}}\cdot [\sum _{i=k-r(k)}^{k-1}{y}^{T}(i){R}_{1}y(i)+\sum _{i=k-{r}_{M}}^{k-r(k)-1}{y}^{T}(i){R}_{1}y(i)+\sum _{i=k-{r}_{M}}^{k-1}{y}^{T}(i){R}_{2}y(i)]\\ -{\alpha}^{2{r}_{m}}\cdot [\sum _{i=k-{r}_{m}}^{k-1}{y}^{T}(i){R}_{3}y(i)]-{\alpha}^{2{r}_{m}}\cdot {x}^{T}(k-{r}_{m})[T-S]x(k-{r}_{m})\\ -{\alpha}^{2{r}_{M}}\cdot {x}^{T}(k-{r}_{M})Sx(k-{r}_{M})\}.\end{array}

(5b)

Define

{X}^{T}(k)=\left[\begin{array}{cccc}{x}^{T}(k)& {x}^{T}(k-r(k))& {x}^{T}(k-{r}_{M})& {x}^{T}(k-{r}_{m})\end{array}\right].

By system (1a)-(1e), LMIs in (3b), and {\sum}_{i=k-r(k)}^{k-1}y(i)=x(k)-x(k-r(k)), we have

Assume \sigma (k,x(k))=j\in \overline{N}, then we can obtain the following result from system (1a)-(1e):

and

where

Define

{\overline{\mathrm{\Sigma}}}_{j}=\left[\begin{array}{cc}{\overline{\mathrm{\Sigma}}}_{1j}& {\overline{\mathrm{\Sigma}}}_{2j}\\ \ast & {\mathrm{\Sigma}}_{3j}\end{array}\right]=\left[\begin{array}{cc}{\overline{\mathrm{\Sigma}}}_{1j}& {\mathrm{\Sigma}}_{2j}\\ \ast & {\mathrm{\Sigma}}_{3j}\end{array}\right]+{\mathrm{\Gamma}}_{j}{\mathrm{\Delta}}_{j}(k){\mathrm{\Omega}}_{j}^{T}+{\mathrm{\Omega}}_{j}{\mathrm{\Delta}}_{j}^{T}(k){\mathrm{\Gamma}}_{j}^{T},

(7c)

where

By condition (3d) with Lemma 1 and the switching signal defined in (3e), we can obtain the following result:

{x}^{T}(k)({\alpha}^{-2}{A}_{j}^{T}P{A}_{j}-{U}_{i})x(k)\le 0,\phantom{\rule{1em}{0ex}}\mathrm{\forall}x(k)\in {\overline{\mathrm{\Omega}}}_{j}.

(7d)

By Lemmas 2 and 3, the condition {\overline{\overline{\mathrm{\Sigma}}}}_{j}<0 in (3c) will imply {\overline{\mathrm{\Sigma}}}_{j}<0 in (7c). {\overline{\mathrm{\Sigma}}}_{j}<0 in (7c) will also imply {\mathrm{\Sigma}}_{j}<0 in (7b). From the condition (7d) and {\mathrm{\Sigma}}_{j}<0 in (7b) with (3a) and (3b), we have

This implies

where

\begin{array}{rcl}{\delta}_{1}& =& {\lambda}_{max}(P)+[{r}_{M}+{r}_{M}({r}_{M}-{r}_{m})]\cdot {\lambda}_{max}(Q)+{r}_{M}^{2}\cdot {\lambda}_{max}({R}_{1}+{R}_{2})+{r}_{m}^{2}\cdot {\lambda}_{max}({R}_{3})\\ +({r}_{M}-{r}_{m})\cdot {\lambda}_{max}(S)+{r}_{m}\cdot {\lambda}_{max}(T).\end{array}

By some simple derivations, we have

\parallel x(k)\parallel \le \sqrt{{\delta}_{1}/{\lambda}_{min}(P)}\cdot {\alpha}^{k}\cdot {\parallel {x}_{0}\parallel}_{s},\phantom{\rule{1em}{0ex}}k=0,1,2,3,\dots .

By Definition 1, system (1a)-(1e) is globally exponentially stable with the convergence rate 0<\alpha <1 with the switching signal in (3e). This completes this proof. □

**Remark 1**

Consider the discrete linear switched system:

Now we can choose the Lyapunov function as V({x}_{k})={\alpha}^{-2k}{x}^{T}(k)Px(k) with matrix P>0, the forward difference of the Lyapunov function is given by

\mathrm{\Delta}V({x}_{k})=V({x}_{k+1})-V({x}_{k})={\alpha}^{-2k}\cdot [{\alpha}^{-2}\cdot {x}^{T}(k+1)Px(k+1)-{x}^{T}(k)Px(k)].

With \sigma (k,x(k))=j\in \overline{N}, we have

and

\mathrm{\Delta}V({x}_{k})={\alpha}^{-2k}\cdot {x}^{T}(k)[{\alpha}^{-2}\cdot {A}_{j}^{T}P{A}_{j}-{U}_{j}+{U}_{j}-P]x(k),

where matrix {U}_{i} satisfies {U}_{i}>0 and {U}_{i}-P<0. If the condition in Lemma 1 is satisfied, then we can obtain the following two results:

In order to achieve the exponential stability of a discrete switched system, the condition in Lemma 1 will be a reasonable choice and a feasible setting.

**Remark 2** The matrix uncertainties in (1c)-(1e) are usually called linear fractional perturbations [16]. The parametric perturbations in [4, 9, 10] are the special conditions of the considered perturbations with {\mathrm{\Xi}}_{i}=0, i\in \underline{N}.

**Remark 3** Under the same switching signal defined in (3e), the switching domains of [6, 7] are selected as:

{\mathrm{\Omega}}_{i}(P,U,{A}_{i})=\{x(k)\in {R}^{n}:{x}^{T}(k)[({r}_{M}-{r}_{m})\cdot U-{A}_{i}^{T}P-P{A}_{i}]x(k)<0\},\phantom{\rule{1em}{0ex}}i=1,2,\dots ,N,

where matrices P>0 and U>0. It is noted that above selections are similar to switching signal design in continuous switched systems [8]. Hence the proposed switching domain design approach is the discrete version of [6, 7] and shown to be useful from numerical simulations.

In what follows, we consider the non-switched uncertain discrete time-delay system:

where \varphi (k)\in {\mathrm{\Re}}^{n} is an initial state function, time-varying delay r(k) is a function from \{0,1,2,3,\dots \} to \{0,1,2,3,\dots \} and 1\le {r}_{m}\le r(k)\le {r}_{M}, {r}_{m} and {r}_{M} are two given positive integers. Matrices A,B\in {\mathrm{\Re}}^{n\times n} are constant. \mathrm{\Delta}A(k) and \mathrm{\Delta}B(k) are two perturbed matrices satisfying the following condition:

where *M*, {N}_{A}, and {N}_{B}, and Ξ are some given constant matrices with appropriate dimensions. \mathrm{\Gamma}(k) is an unknown matrix representing the perturbation which satisfies

{\mathrm{\Gamma}}^{T}(k)\mathrm{\Gamma}(k)\le I.

(8e)

The following sufficient conditions for the stability of system (8a)-(8e) can be obtained in a similar way to Theorem 1.

**Theorem 2** *If a given constant* 0<\alpha <1, *there exist some* n\times n *matrices* P>0, Q>0, {R}_{1}>0, {R}_{2}>0, {R}_{3}>0, S>0, T>0, {V}_{l11}\in {\mathrm{\Re}}^{4n\times 4n}, {V}_{l12}\in {\mathrm{\Re}}^{4n\times n}, {V}_{l22}\in {\mathrm{\Re}}^{n\times n}, l=1,2,3,4, *and constants* \epsilon >0 *such that* (3a), (3b), *and the following LMI conditions are satisfied*:

\overline{\overline{\mathrm{\Sigma}}}=\left[\begin{array}{cccccccc}{\mathrm{\Sigma}}_{11}& 0& 0& 0& {\mathrm{\Sigma}}_{15}& {\mathrm{\Sigma}}_{16}& 0& {\mathrm{\Sigma}}_{18}\\ \ast & {\mathrm{\Sigma}}_{22}& 0& 0& {\mathrm{\Sigma}}_{25}& {\mathrm{\Sigma}}_{26}& 0& {\mathrm{\Sigma}}_{28}\\ \ast & \ast & {\mathrm{\Sigma}}_{33}& 0& 0& 0& 0& 0\\ \ast & \ast & \ast & {\mathrm{\Sigma}}_{44}& 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & {\mathrm{\Sigma}}_{55}& 0& {\mathrm{\Sigma}}_{57}& 0\\ \ast & \ast & \ast & \ast & \ast & {\mathrm{\Sigma}}_{66}& {\mathrm{\Sigma}}_{67}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Sigma}}_{77}& {\mathrm{\Sigma}}_{78}\\ \ast & \ast & \ast & \ast & \ast & \ast & \ast & {\mathrm{\Sigma}}_{88}\end{array}\right]+\left[\begin{array}{cc}{\mathrm{\Omega}}_{(4n\times 4n)}& {0}_{(4n\times 4n)}\\ \ast & {0}_{(4n\times 4n)}\end{array}\right]<0,

*where*

*Then system* (8a)-(8e) *is globally exponentially stable with the convergence rate* 0<\alpha <1.

**Remark 4** In Theorems 1 and 2, the global asymptotic stability of switched system (1a)-(1e) and non-switched system (8a)-(8e) can be achieved by setting \alpha =1. The nonnegative inequalities in (6a)-(6d) are used to improve the conservativeness of the obtained results.

**Remark 5** If we wish to select a switching signal to guarantee the stability of switched system (1a)-(1e), the following procedures are proposed.

Step 1 Test the exponential stability of each subsystem of switched system (1a)-(1e) by Theorem 2 with A={A}_{i}, B={B}_{i}, M={M}_{i}, {N}_{A}={N}_{Ai}, {N}_{B}={N}_{Bi}, \mathrm{\Xi}={\mathrm{\Xi}}_{i}, i=1,2,\dots ,N. If the sufficient conditions in Theorem 2 have a feasible solution for some i\in \underline{N}, the switching signal is selected by \sigma =i and the stability of the switched system in (1a)-(1e) can be guaranteed.

Step 2 If the obtained stability results for each subsystem of switched time-delay system (1a)-(1e) do not satisfy the requirement in Step 1, we can use Theorem 1 to design the switching signal to guarantee the global exponential stability of switched time-delay system (1a)-(1e).

From the results in Steps 1-2, we can propose a less conservative stability result of the system.