User:Margav06/sandbox/Click here to continue/Time-Delay Systems/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{align}

x(k+1) &= Ax(k) + A_1x(k-\tau_k) & k&\in\mathbb{Z}_+, & \tau_k &\in\mathbb{N}, & 0&\leq\tau_k \leq h

\end{align} $$

In this description, $$ A $$ and $$ A_1 $$ are matrices in $$ \mathbb{R}^{n\times n} $$. 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:

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

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{align} &\text{Find}:\\ &\qquad P, S, R, S_{12}, P_2, P_3 \in\mathbb{R}^{n\times n}\\ &\text{such that:}\\ &\qquad P>0,\quad S>0,\quad R>0,  \\ &\qquad \begin{bmatrix} \Phi_{11} & \Phi_{12} & S_{12} & R-S_{12}+P_2^T A_1 \\ * & \Phi_{22} & 0 & P_3^T A_1\\ * & * & -(S+R) & R-S_{12}^T \\ * & * & * & -2R + S_{12} + S_{12}^T \end{bmatrix}<0 \\ &\text{where:}\\ &\qquad \Phi_{11} = (A^T-I)P_2 + P_2^T(A-I) + S-R\\ &\qquad \Phi_{12} = P-P_2^T+(A^T-I)P_3 \\ &\qquad \Phi_{22} = -P_3-P_3^T + P+h^2 R \end{align}

$$

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 $$ \epsilon>0 $$, there exists a real number $$ \delta>0 $$ such that:


 * $$ \|x(0)\|<\delta \quad \Rightarrow \quad \|x(k)\|<\epsilon \qquad \forall k\in\mathbb{N} $$


 * $$ \lim_{k\rightarrow\infty} x(k)=0 $$

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



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

where:



\begin{align} &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{align}

$$

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