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:

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

The Data

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

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

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

{\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 $*$ 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 $\tau _{k}$ of delays within the interval $[0,h]$ . That is, independent of the values of the delays $\tau _{k}\in [0,h]$ at any time:

For any real number $\epsilon >0$ , there exists a real number $\delta >0$ such that:

\|x(0)\|<\delta \quad \Rightarrow \quad \|x(k)\|<\epsilon \qquad \forall k\in \mathbb {N}

$\lim _{k\rightarrow \infty }x(k)=0$

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

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

where:

{\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:

https://github.com/djagt/LMI_Codes/blob/main/D_TDS_DC.m

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

LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay – Delay-independent stability LMI for continuous-time TDS

Discrete Time Lyapunov Stability – Stability LMI for non-delayed discrete-time system

External Links

The presented results have been obtained from:

Fridman E. 2014. Introduction to Time-Delay Systems, Analysis and Control. Springer. ISBN: 978-3-319-09392-5.

Additional information on LMI's in control theory can be obtained from the following resources:

LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.