# LMIs in Control/Time-Delay Systems/Continuous Time/LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay

LMIs in Control/Time-Delay Systems/Continuous Time/LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay

This page describes an LMI for stability analysis of a continuous-time system with a time-varying delay. In particular, a delay-independent condition is provided to test uniform asymptotic stability of a retarded differential equation 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. Moreover, the delay is assumed to vary only slowly in time, with a temporal derivative bounded by a value less than one. Solving the LMI for a particular value of this bound, uniform asymptotic stability can be shown for any time-delay satisfying this bound.

## The System

The system under consideration is one of the form:

{\begin{aligned}{\dot {x}}(t)&=Ax(t)+A_{1}x(t-\tau (t))&t&\geq t_{0},&0&\leq \tau (t)\leq h,&{\dot {\tau }}(t)&\leq d<1\end{aligned}} In this description, $A$ and $A_{1}$ are matrices in $\mathbb {R} ^{n\times n}$ . The variable $\tau (t)$ denotes a delay in the state at time $t\geq t_{0}$ , assuming a value no greater than some $h\in \mathbb {R} _{+}$ . Moreover, we assume that the function $\tau (t)$ is differentiable at any time, with the derivative bounded by some value $d<1$ , assuring the delay to be slowly-varying in time.

## 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}\\d&\in [0,1)\end{aligned}} ## The Optimization Problem

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

## The LMI: Delay-Independent Uniform Asymptotic Stability for Continuous-Time TDS

{\begin{aligned}&{\text{Find}}:\\&\qquad P,Q\in \mathbb {R} ^{n\times n}\\&{\text{such that:}}\\&\qquad P>0,\quad Q>0\\&\qquad {\begin{bmatrix}A^{T}P+PA+Q&PA_{1}\\A_{1}^{T}P&-(1-d)Q\end{bmatrix}}<0\\\end{aligned}} ## Conclusion:

If the presented LMI is feasible, the system will be uniformly asymptotically stable for any delay function $\tau (t)$ satisfying ${\dot {\tau }}(t)\leq d<1$ . That is, independent of the values of the delays $\tau (t)$ and the starting time $t_{0}\in \mathbb {R}$ :

• For any real number $\epsilon >0$ , there exists a real number $\delta >0$ such that:
$\|x_{t_{0}}\|_{\mathcal {C}}<\delta \quad \Rightarrow \quad \|x(t)\|<\epsilon \qquad \forall t\geq t_{0}$ • There exists a real number $\delta _{a}>0$ such that for any real number $\eta >0$ , there exists a time $T(\delta _{a},\eta )$ such that:
$\|x_{t_{0}}\|_{\mathcal {C}}<\delta _{a}\quad \Rightarrow \quad \|x(t)\|<\eta \qquad \forall t\geq t_{0}+T(\delta _{a},\eta )$ Here, we let $x_{t_{0}}(\theta )=x(t_{0}+\theta )$ for $\theta \in [-\tau (t_{0}),0]$ denote the delayed state function at time $t_{0}$ . The norm $\|x_{t_{0}}\|_{\mathcal {C}}$ of this function is defined as the maximal value of the vector norm assumed by the state over the delayed time interval, given by:

$\|x_{t_{0}}\|_{\mathcal {C}}:=\max _{\theta \in [-\tau (t_{0}),0]}\|x(t_{0}+\theta )\|$ Obtaining a feasible point for the LMI, this result can be proven using a Lyapunov-Krasovkii functional:

{\begin{aligned}&V(t,x_{t})=x^{T}(t)Px(t)+\int _{t-\tau (t)}^{t}x^{T}(s)Qx(s)ds\\\end{aligned}} Notably, if matrices $P>0,Q>0$ prove feasibility of the LMI for the pair $(A,A_{1})$ , these same matrices will also prove feasibility of the LMI for the pair $(A,-A_{1})$ . As such, feasibility of this LMI proves uniform asymptotic stability of both systems:

{\begin{aligned}{\dot {x}}(t)&=Ax(t)\pm A_{1}x(t-\tau (t))&t&\geq t_{0},&0&\leq \tau (t)\leq h&{\dot {\tau }}(t)&\leq d<1\end{aligned}} Moreover, since the result is independent of the value of the delay, it will also hold for a delay $\tau (t)\equiv 0$ . Hence, if the LMI is feasible, the matrices $A\pm A_{1}$ will be Hurwitz.

## 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

•  - Delay-dependent stability LMI for continuous-time TDS
•  - Stability LMI for delayed discrete-time system