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

Stabilization of Time-Delay Systems - Delay Dependent 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 while being dependent on 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\text{, }{A_d}\in\mathbb{R}^{n\times n}\text{, }{B}\in\mathbb{R}^{n\times r}\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-Dependent case would be obtained as described below.

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

The Optimization Problem
Suppose - for the time-delayed system given above - we were asked to design a memoryless feedback control law where $$u(t)=Kx(t)$$ 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-Dependent 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 scalar $$0<\beta<1$$, a symmetric matrix $$X>0$$ and a matrix $$W$$ that satisfy the following:



\begin{align} \begin{bmatrix}\Phi(X,W)&&\bar{d}(X{A^T}+{W^T}{B^T})&&\bar{d}X{A^T_d}\\ \bar{d}(AX+BW)&&-\bar{d}{\beta}I&&0\\ \bar{d}{A_d}X&&0&&-\bar{d}(1-\beta)I\end{bmatrix}&<0\\ \end{align}$$

where$$\Phi(X,W)=X{(A+{A_d})^T}+(A+{A_d})X+BW+{W^T}{B^T}+\bar{d}{A_d}{A^T_d}$$

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

Implementation

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

Related LMIs

 * ../Delay Independent Time-Delay Stabilization/ - Equivalent LMI for delay-independent time-delay stabilization.