LMIs in Control/Discrete-Time Systems/Discrete-Time Lyapunov Stability

Discrete-Time Lyapunov Stability

The System
Discrete-Time System



\begin{align} x(t)_{k+1}&=A_dx(t)_k,&A_d \in \bf{R^{n*n}} \;\\ \end{align}$$

The Data
The matrices $$ System A_d,P $$.

The Optimization Problem
The following feasibility problem should be solved:

Find P obeying the LMI constraints.

The LMI:
Discrete-Time Lyapunov Stability

The LMI formulation



\begin{align} P \in \bf{S^{n}}\\ Find \; &P>0,\\ \begin{bmatrix} A_d^T P A_d - P\end{bmatrix}&\leq0 \end{align}$$

Conclusion:
If there exists a $$P \in \bf{S^{n}}$$ satisfying the LMI then, $$|\lambda_i(A_d)|\leq 1, \forall i = 1,2,...,n;$$ and the equilibrium point $$\bar{x}=0$$  of the system is Lyapunov stable.

Implementation

 * [] - MATLAB implementation of the LMI.

Related LMIs
Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality