LMIs in Control/pages/Delay Independent Time-Delay Stabilization

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:

{\displaystyle {\begin{aligned}{\begin{cases}{\dot {x}}(t)&=Ax(t)+A_{d}(t-d)+Bu(t),\\x(t)&=\phi (t),t\in [0,d],0

where

{\displaystyle {\begin{aligned}&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{aligned}}}

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: ${\displaystyle A,A_{d},}$ and ${\displaystyle B}$.

The Optimization Problem

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

{\displaystyle {\begin{aligned}{\begin{cases}{\dot {x}}(t)&=(A+BK)x(t)+A_{d}(t-d),\\x(t)&=\phi (t),t\in [0,d],0

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${\displaystyle W\in \mathbb {R} ^{r\times n}}$ and 2 symmetric matrices ${\displaystyle X>0}$ and ${\displaystyle Y>0}$ that satisfy the following:

{\displaystyle {\begin{aligned}{\begin{bmatrix}X{A^{T}}+AX+BW+{W^{T}}{B^{T}}+Y&&{A_{d}}X\\X{A_{d}}^{T}&&-Y\end{bmatrix}}&<0\\\end{aligned}}}

Conclusion:

Given the resulting feedback gain matrix ${\displaystyle K=WX^{-1}}$, 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.