LMIs in Control/pages/Discrete-Time Lyapunov Stability

Discrete-Time Lyapunov Stability

The System

Discrete-Time System

${\begin{aligned}x(t)_{k+1}&=A_{d}x(t)_{k},&A_{d}\in {\bf {{R^{n*n}}\;}}\\\end{aligned}}$

The Data

The matrices $SystemA_{d},P$.

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

${\begin{aligned}P\in {\bf {S^{n}}}\\Find\;&P>0,\\{\begin{bmatrix}A_{d}^{T}PA_{d}-P\end{bmatrix}}&\leq 0\end{aligned}}$

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

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.