# LMIs in Control/Discrete Time/Stability Condition for Discrete-Time TDS

LMIs in Control/Discrete Time/Stability Condition for Discrete-Time TDS

This page describes an LMI for stability analysis of a discrete-time system with a time-varying delay. In particular, a delay-dependent condition is provided to test asymptotic stability of a discrete-delay system through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. Solving the LMI for different values of this bound, a limit on the delay can be attained for which the system remains asymptotically stable.

## The System

The system under consideration is one of the form:

{\displaystyle {\begin{aligned}x(k+1)&=Ax(k)+A_{1}x(k-\tau _{k})&k&\in \mathbb {Z} _{+},&\tau _{k}&\in \mathbb {N} ,&0&\leq \tau _{k}\leq h\end{aligned}}}

In this description, ${\displaystyle A}$  and ${\displaystyle A_{1}}$  are matrices in ${\displaystyle \mathbb {R} ^{n\times n}}$ . The variable ${\displaystyle \tau _{k}}$  denotes a delay in the state at discrete time ${\displaystyle k}$ , assuming a value no greater than some ${\displaystyle h\in \mathbb {Z} _{+}}$ .

## The Data

To determine stability of the system, the following parameters must be known:

{\displaystyle {\begin{aligned}A&\in \mathbb {R} ^{n\times n}\\A_{1}&\in \mathbb {R} ^{n\times n}\\h&\in \mathbb {Z} _{+}\end{aligned}}}

## The Optimization Problem

Based on the provided data, asymptotic stability can be determined by testing feasibility of the following LMI:

## The LMI: Asymptotic Stability for Discrete-Time TDS

{\displaystyle {\begin{aligned}&{\text{Find}}:\\&\qquad P,S,R,S_{12},P_{2},P_{3}\in \mathbb {R} ^{n\times n}\\&{\text{such that:}}\\&\qquad P>0,\quad S>0,\quad R>0,\\&\qquad {\begin{bmatrix}\Phi _{11}&\Phi _{12}&S_{12}&R-S_{12}+P_{2}^{T}A_{1}\\*&\Phi _{22}&0&P_{3}^{T}A_{1}\\*&*&-(S+R)&R-S_{12}^{T}\\*&*&*&-2R+S_{12}+S_{12}^{T}\end{bmatrix}}<0\\&{\text{where:}}\\&\qquad \Phi _{11}=(A^{T}-I)P_{2}+P_{2}^{T}(A-I)+S-R\\&\qquad \Phi _{12}=P-P_{2}^{T}+(A^{T}-I)P_{3}\\&\qquad \Phi _{22}=-P_{3}-P_{3}^{T}+P+h^{2}R\end{aligned}}}

In this notation, the symbols ${\displaystyle *}$  are used to indicate appropriate matrices to assure the overall matrix is symmetric.

## Conclusion:

If the presented LMI is feasible, the system will be asymptotically stable for any sequence ${\displaystyle \tau _{k}}$  of delays within the interval ${\displaystyle [0,h]}$ . That is, independent of the values of the delays ${\displaystyle \tau _{k}\in [0,h]}$  at any time:

• For any real number ${\displaystyle \epsilon >0}$ , there exists a real number ${\displaystyle \delta >0}$  such that:
${\displaystyle \|x(0)\|<\delta \quad \Rightarrow \quad \|x(k)\|<\epsilon \qquad \forall k\in \mathbb {N} }$
• ${\displaystyle \lim _{k\rightarrow \infty }x(k)=0}$

Obtaining a feasible point for the LMI, this result can be proven using a Lyapunov-Krasovkii functional:

{\displaystyle {\begin{aligned}&V(x_{k})=V_{P}(k)+V_{S}(k)+V_{R}(k)\\\end{aligned}}}

where:

{\displaystyle {\begin{aligned}&V_{P}(k)=x^{T}(k)Px(k),\\&V_{S}(k)=\sum _{j=k-h}^{k-1}x^{T}(j)Sx(j),\\&V_{R}(k)=h\sum _{m=-h}^{-1}\sum _{j=k+m}^{k-1}[x(j+1)-x(j)]^{T}R[x(j+1)-x(j)]\\\end{aligned}}}

## Implementation

An example of the implementation of this LMI in Matlab is provided on the following site:

Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.

## Related LMIs

• TDSDC – Delay-dependent stability LMI for continuous-time TDS