User:Margav06/sandbox/Click here to continue/Time-Delay Systems/LMI for Robust Stability of Retarded Differential Equation with Norm-Bounded Uncertainty

This page describes an LMI for stability analysis of an uncertain 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 with uncertain matrices 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. The matrices describing the system are assumed to be uncertain, with the norm of the uncertainty bounded by a value of one. In addition, 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, independent of the value of the uncertainty function.

The System
The system under consideration is one of the form:



\begin{align}

\dot{x}(t) &= (A+H\Delta(t) E)x(t) + (A_1+H\Delta(t) E_1)x(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 uncertainty $$ \Delta(t)\in\mathbb{R}^{r_1\times r_2} $$ is also allowed to vary in time, but at any time $$ t\geq t_0 $$ must satisfy the inequality:



\begin{align}

\Delta^T(t)\Delta(t) &\leq I

\end{align} $$

The uncertainty affects the system through matrices $$ H\in\mathbb{R}^{n\times r_1} $$ and $$ E,E_1\in\mathbb{R}^{r_2\times n} $$, which are constant in time and assumed to be known.

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

$$ \begin{align} A,A_1&\in\mathbb{R}^{n\times n} \\ H&\in\mathbb{R}^{n\times r_1} \\ E,E_1&\in\mathbb{R}^{r_2\times n} \\ d&\in [0,1) \end{align} $$

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 Robust Uniform Asymptotic Stability for Continuous-Time TDS


\begin{align} &\text{Find}:\\ &\qquad \epsilon\in\mathbb{R}, \quad P, Q \in\mathbb{R}^{n\times n}\\ &\text{such that:}\\ &\qquad \epsilon>0,\quad P>0,\quad Q>0  \\ &\qquad \begin{bmatrix} A^T P+PA+Q & PA_1 & PH & \epsilon E^T \\ A_1^TP & -(1-d)Q & 0 & \epsilon E_1^T \\ H^T P & 0 & -\epsilon I & 0 \\ \epsilon E & \epsilon E_1 & 0 & -\epsilon I \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 $$ \dot{\tau}(t)\leq d<1 $$, and any uncertainty $$ \Delta(t) $$ satisfying $$ \Delta^T(t)\Delta(t)\leq I $$. That is, independent of the values of the delays $$ \tau(t) $$, uncertainties $$ \Delta(t) $$, 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:


 * $$ \|x_{t_0}\|_{\mathcal{C}}:=\max_{\theta\in[-\tau(t_0),0]} \|x(t_0+\theta)\| $$

The proof of this result relies on the fact that the following inequality holds for any value $$ \epsilon>0 $$ and constant matrices $$ \alpha,\beta $$ of appropriate dimensions:


 * $$ \alpha \Delta(t)\beta + \beta^T \Delta^T(t)\alpha^T \leq \epsilon^{-1} \alpha\alpha^T + \epsilon\beta^T \beta $$

Using this inequality with $$ \alpha^T=\begin{bmatrix} H^TP & 0\end{bmatrix} $$ and $$ \beta = \begin{bmatrix} E & E_1 \end{bmatrix} $$, the described LMI can then be derived from that presented in, corresponding to a situation without uncertainty.

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/Rstblty_cTDS_SlowVarying.m

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

Related LMIs

 * - Stability LMI for continuous-time RDE with slowly-varying delay without uncertainty


 * - LMI for quadratic stability of continuous-time system with norm-bounded uncertainty


 * - Stability LMI for delayed discrete-time system