LMIs in Control/pages/TDSDC

The System

The problem is to check the stability of the following linear time-delay system on a delay dependent condition

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

where

$\begin{array}{c}A,A_d\in {ℝ}^{n\times n},A\in {ℝ}^{n\times r}\text{ are the system coefficient matrices,}\\ \end{array}$

$\varphi (t)$ is the initial condition
$d$ represents the time-delay
$\overline{d}$ is a known upper-bound of $d$

For the purpose of the delay dependent system we rewrite the system as

$\{\begin{array}{cc}\dot{x}(t)& =Ax(t)+A_{d}x(t-d)\\ \dot{x}(t)& =(A+A_d)x(t)-A_d(x(t)-x(t-d)\end{array}$

The Data

The matrices $A,A_d$ are known

The LMI: The Time-Delay systems (Delay Dependent Condition)

From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a symmetric positive definite matrix
$X$ and a scalar $0<\beta <1$ such that

$\left[\begin{array}{ccc}\mathrm{\Phi }(X)& \overline{d}X{A}^T& \overline{d}X{A}_d^T\\ \overline{d}AX& -d\beta I& 0\\ \overline{d}{A}_{d}X& 0& -\overline{d}(1-\beta )I\end{array}\right]$$\begin{array}{c}<0\end{array}$

Here $\mathrm{\Phi }(X)=X)A+A_d{)}^T+(A+A_d)X+\overline{d}{A}_d{A}_d^T<0$

This LMI has been derived from the Lyapunov function for the system. It follows that the system is asymptotically stable if

$P(A+A_d)+(A+A_d{)}^{T}P+\overline{d}P{A}_d{A}_d^{T}P+\frac{\overline{d}}{\beta }{A}^{T}A+\frac{\overline{d}}{1-\beta }{A}_d^T{A}_d<0$
This is obtained by replacing $X$ with $P^{-1}$

Conclusion:

We can now implement these LMIs to do stability analysis for a Time delay system on the delay dependent condition

Implementation

The implementation of the above LMI can be seen here

https://github.com/yashgvd/LMI_wikibooks

Related LMIs

Time Delay systems (Delay Independent Condition)

External Links

• [1] - LMI in Control Systems Analysis, Design and Applications
• 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.
• D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.