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

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:

$\begin{array}{c}\{\begin{array}{cc}\dot{x}(t)& =Ax(t)+A_d(t-d)+Bu(t),\\ x(t)& =\varphi (t),t\in [0,d],0<d\le \overline{d},\end{array}\end{array}$

where

$\begin{array}{cc}& A\text{, }{A}_d\in {ℝ}^{n\times n}\text{, }B\in {ℝ}^{n\times r}\text{ are the system coefficient matrices,}\\ & \varphi (t)\text{ is the initial condition,}\\ & d\text{ represents the time-delay, and}\\ & \overline{d}\text{ is a known upper-bound of }d\\ \end{array}$

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:

$\begin{array}{c}\{\begin{array}{cc}\dot{x}(t)& =(A+BK)x(t)+A_d(t-d),\\ x(t)& =\varphi (t),t\in [0,d],0<d\le \overline{d},\end{array}\end{array}$

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:

$\begin{array}{cc}\left[\begin{array}{ccccc}\mathrm{\Phi }(X,W)& & \overline{d}(X{A}^T+W^T{B}^T)& & \overline{d}X{A}_d^T\\ \overline{d}(AX+BW)& & -\overline{d}\beta I& & 0\\ \overline{d}{A}_{d}X& & 0& & -\overline{d}(1-\beta )I\end{array}\right]& <0\\ \end{array}$

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

Conclusion:

Given the resulting stabilizing control gain matrix $K=W{X}^{-1}$, 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.

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.