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:

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

The Data

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

{\displaystyle {\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

{\displaystyle {\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 ${\displaystyle \tau (t)}$  satisfying ${\displaystyle {\dot {\tau }}(t)\leq d<1}$ . That is, independent of the values of the delays ${\displaystyle \tau (t)}$  and the starting time ${\displaystyle t_{0}\in \mathbb {R} }$ :

• For any real number ${\displaystyle \epsilon >0}$ , there exists a real number ${\displaystyle \delta >0}$  such that:
${\displaystyle \|x_{t_{0}}\|_{\mathcal {C}}<\delta \quad \Rightarrow \quad \|x(t)\|<\epsilon \qquad \forall t\geq t_{0}}$
• There exists a real number ${\displaystyle \delta _{a}>0}$  such that for any real number ${\displaystyle \eta >0}$ , there exists a time ${\displaystyle T(\delta _{a},\eta )}$  such that:
${\displaystyle \|x_{t_{0}}\|_{\mathcal {C}}<\delta _{a}\quad \Rightarrow \quad \|x(t)\|<\eta \qquad \forall t\geq t_{0}+T(\delta _{a},\eta )}$

Here, we let ${\displaystyle x_{t_{0}}(\theta )=x(t_{0}+\theta )}$  for ${\displaystyle \theta \in [-\tau (t_{0}),0]}$  denote the delayed state function at time ${\displaystyle t_{0}}$ . The norm ${\displaystyle \|x_{t_{0}}\|_{\mathcal {C}}}$  of this function is defined as the maximal value of the vector norm assumed by the state over the delayed time interval, given by:

${\displaystyle \|x_{t_{0}}\|_{\mathcal {C}}:=\max _{\theta \in [-\tau (t_{0}),0]}\|x(t_{0}+\theta )\|}$

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

{\displaystyle {\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 ${\displaystyle P>0,Q>0}$  prove feasibility of the LMI for the pair ${\displaystyle (A,A_{1})}$ , these same matrices will also prove feasibility of the LMI for the pair ${\displaystyle (A,-A_{1})}$ . As such, feasibility of this LMI proves uniform asymptotic stability of both systems:

{\displaystyle {\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 ${\displaystyle \tau (t)\equiv 0}$ . Hence, if the LMI is feasible, the matrices ${\displaystyle A\pm A_{1}}$  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

• [1] - Delay-dependent stability LMI for continuous-time TDS
• [2] - Stability LMI for delayed discrete-time system