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:

• https://github.com/djagt/LMI_Codes/blob/main/stblty_cTDS_SlowVarying.m

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

Related LMIs

• [1] - Delay-dependent stability LMI for continuous-time TDS

• [2] - Stability LMI for 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.