LMIs in Control/Discrete Time/Stability Condition for Discrete-Time TDS

LMIs in Control/Discrete Time/Stability Condition for Discrete-Time TDS

This page describes an LMI for stability analysis of a discrete-time system with a time-varying delay. In particular, a delay-dependent condition is provided to test asymptotic stability of a discrete-delay system through feasibility of an LMI. The system under consideration pertains a single discrete delay, with the extent of the delay at any time bounded by some known value. Solving the LMI for different values of this bound, a limit on the delay can be attained for which the system remains asymptotically stable.

The System

The system under consideration is one of the form:

$\begin{array}{cccccccc}x(k+1)& =Ax(k)+A_{1}x(k-{\tau }_k)& k& \in {\mathbb{Z}}_{+},& {\tau }_k& \in \mathbb{N},& 0& \le {\tau }_k\le h\end{array}$

In this description, $A$ and $A_1$ are matrices in ${ℝ}^{n\times n}$. The variable ${\tau }_k$ denotes a delay in the state at discrete time $k$, assuming a value no greater than some $h\in {\mathbb{Z}}_{+}$.

The Data

To determine stability of the system, the following parameters must be known:

$\begin{array}{cc}A& \in {ℝ}^{n\times n}\\ A_1& \in {ℝ}^{n\times n}\\ h& \in {\mathbb{Z}}_{+}\end{array}$

The Optimization Problem

Based on the provided data, asymptotic stability can be determined by testing feasibility of the following LMI:

The LMI: Asymptotic Stability for Discrete-Time TDS

$\begin{array}{cc}& \text{Find}:\\ & P,S,R,S_{12},P_2,P_3\in {ℝ}^{n\times n}\\ & \text{such that:}\\ & P>0,S>0,R>0,\\ & \left[\begin{array}{cccc}{\mathrm{\Phi }}_{11}& {\mathrm{\Phi }}_{12}& S_{12}& R-S_{12}+P_2^T{A}_1\\ *& {\mathrm{\Phi }}_{22}& 0& P_3^T{A}_1\\ *& *& -(S+R)& R-S_{12}^T\\ *& *& *& -2R+S_{12}+S_{12}^T\end{array}\right]<0\\ & \text{where:}\\ & {\mathrm{\Phi }}_{11}=(A^T-I)P_2+P_2^T(A-I)+S-R\\ & {\mathrm{\Phi }}_{12}=P-P_2^T+(A^T-I)P_3\\ & {\mathrm{\Phi }}_{22}=-P_3-P_3^T+P+h^{2}R\end{array}$

In this notation, the symbols $*$ are used to indicate appropriate matrices to assure the overall matrix is symmetric.

Conclusion:

If the presented LMI is feasible, the system will be asymptotically stable for any sequence ${\tau }_k$ of delays within the interval $[0,h]$. That is, independent of the values of the delays ${\tau }_k\in [0,h]$ at any time:

• For any real number $ϵ>0$, there exists a real number $\delta >0$ such that:

$\Vert x(0)\Vert <\delta \Rightarrow \Vert x(k)\Vert <ϵ\mathrm{\forall }k\in \mathbb{N}$

• $\underset{k\to \mathrm{\infty }}{\lim }x(k)=0$

Obtaining a feasible point for the LMI, this result can be proven using a Lyapunov-Krasovkii functional:

$\begin{array}{cc}& V(x_k)=V_P(k)+V_S(k)+V_R(k)\\ \end{array}$

where:

$\begin{array}{cc}& V_P(k)=x^T(k)Px(k),\\ & V_S(k)=\sum _{j=k-h}^{k-1}{x}^T(j)Sx(j),\\ & V_R(k)=h\sum _{m=-h}^{-1}\sum _{j=k+m}^{k-1}[x(j+1)-x(j){]}^{T}R[x(j+1)-x(j)]\\ \end{array}$

Implementation

An example of the implementation of this LMI in Matlab is provided on the following site:

• https://github.com/djagt/LMI_Codes/blob/main/D_TDS_DC.m

Note that this implementation requires packages for YALMIP with solver mosek, though a different solver can also be implemented.

Related LMIs

• TDSDC – Delay-dependent stability LMI for continuous-time TDS

• LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay – Delay-independent stability LMI for continuous-time TDS

• Discrete Time Lyapunov Stability – Stability LMI for non-delayed discrete-time system

External Links

The presented results have been obtained from:

• Fridman E. 2014. Introduction to Time-Delay Systems, Analysis and Control. Springer. ISBN: 978-3-319-09392-5.

Additional information on LMI's in control theory can be obtained from the following resources:

• 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.