User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/Delay-Independent Condition

Stabilization of Time-Delay Systems - Delay Independent Case

Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system independent of the delay.

The System
For this particular problem, suppose that we were given the time-delayed system in the form of:



\begin{align} \begin{cases} \dot x(t)&=Ax(t)+A_d(t-d)+Bu(t),\\ x(t)&=\phi(t), t\in[0,d], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$

where

\begin{align} &A\, A_d\in\mathbb{R}^{n\times n}, B\in\mathbb{R}^{nxr}\text{ are the system coefficient matrices,}\\ &\phi(t)\text{ is the initial condition,}\\ &d\text{ represents the time-delay, and}\\ &\bar{d}\text{ is a known upper-bound of }d\\ \end{align}$$ Then the LMI for determining the Time-Delay System for the Delay-Independent case would be obtained as described below.

The Data
In order to obtain the LMI, we need the following 3 matrices: $$A, A_d,$$ and $$B$$.

The Optimization Problem
Suppose - for the time-delayed system given above - we were asked to design a memoryless state-feedback control law where $$u=Kx$$ such that the closed-loop system:

\begin{align} \begin{cases} \dot x(t)&=(A+BK)x(t)+A_d(t-d),\\ x(t)&=\phi(t), t\in[0,d], 0 < d \leq \bar{d},\\ \end{cases} \end{align}$$ is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

The LMI: The Delay-Independent Stabilization of Time-Delay Systems
From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix$$W\in\mathbb{R}^{r\times n}$$ and 2 symmetric matrices $$X>0$$ and $$Y>0$$ that satisfy the following:



\begin{align} \begin{bmatrix}X{A^T}+AX+BW+{W^T}{B^T}+Y&&{A_d}X\\X{A_d}^T&&-Y\end{bmatrix}&<0\\ \end{align}$$

Conclusion:
Given the resulting feedback gain matrix $$K=WX^{-1}$$, it can be observed that the matrix is asymptotically stable while simultaneously ensuring that the solution is delay-independent from the time-delay system where this gain matrix was derived.

Implementation

 * Example Code - A GitHub link that contains code (titled "DelayIndependentTimeDelay.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

 * ../Delay Dependent Time-Delay Stabilization/ - Equivalent LMI for delay-dependent time-delay stabilization.