User:Margav06/sandbox/Click here to continue/Time-Delay Systems/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{align}

\dot{x}(t) &= Ax(t) + A_1x(t-\tau(t)) & t&\geq t_0, & 0&\leq\tau(t) \leq h, & \dot{\tau}(t)&\leq d<1

\end{align} $$

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:

$$ \begin{align} A&\in\mathbb{R}^{n\times n} \\ A_1&\in\mathbb{R}^{n\times n} \\ d&\in [0,1) \end{align} $$

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{align} &\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^TP & -(1-d)Q \end{bmatrix}<0 \\ \end{align}

$$

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{align} &V(t,x_t)=x^T(t)Px(t) + \int_{t-\tau(t)}^t x^T(s)Qx(s) ds\\ \end{align} $$

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{align}

\dot{x}(t) &= Ax(t) \pm A_1x(t-\tau(t)) & t&\geq t_0, & 0&\leq\tau(t) \leq h & \dot{\tau}(t)&\leq d<1

\end{align} $$

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

 * - Delay-dependent stability LMI for continuous-time TDS


 * - Stability LMI for delayed discrete-time system