LMIs in Control/pages/Discrete Time Lyapunov Stability

Discrete-Time Lyapunov Stability

A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.

Stability of DT LTI systems can be determined by solving Lyapunov Inequality.

The System

Discrete-Time System

$\begin{array}{ccc}x(t{)}_{k+1}& =A_{d}x(t{)}_k,& A_d\in {𝐑}^{𝐧\mathbf{*}𝐧}\\ \end{array}$

The Data

The matrices: System $A_d,P$.

The Optimization Problem

The following feasibility problem should be optimized:

Find P obeying the LMI constraints.

The LMI:

Discrete-Time Bounded Real Lemma

The LMI formulation

$\begin{array}{c}P\in {𝐒}^{𝐧}\\ \text{Find}& P>0,\\ \left[\begin{array}{c}{A}_d^{T}P{A}_d-P\end{array}\right]& <0\end{array}$

Conclusion:

If there exists a $P\in {𝐒}^{𝐧}$ satisfying the LMI then, $|{\lambda }_i(A_d)|\le 1,\mathrm{\forall }i=1,2,...,n;$ and the equilibrium point $\overline{x}=0$ of the system is Lyapunov stable.

Implementation

A link to CodeOcean or other online implementation of the LMI
MATLAB Code

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.