LMIs in Control/pages/Discrete-Time Quadratic Stability

Discrete-Time Quadratic Stability

Stability is an important property, stability analysis is necessary for control theory. For robust control, this criterion is applicable for the uncertain discrete-time linear system. It is based on the Discrete Time Lyapunov Stability.

The System

$\begin{array}{cc}{x}_{k+1}& =A_d(\alpha )x_k\\ Where:\\ & A_d(\alpha )=A_d+\mathrm{\Delta }{A}_d(\delta (t))\\ & \mathrm{\Delta }{A}_d(\delta (t))=\sum _{k=1}^n{\delta }_k(t)A_{d;k}\in {\mathrm{R}}^{n\times n}\\ & \delta (t)=[{\delta }_1(t),...{\delta }_n(t)]-\text{The set of perturbation parameters}\\ & \delta (t)\in \mathrm{R}{A}_{d;i}\in {\mathrm{R}}^{n\times n}\end{array}$

The Data

The matrices $A\in R^{n\times n}{A}_{d;i}\in R^{n\times n}$.

The Optimization Problem

The following feasibility problem should be solved:

$\begin{array}{cc}\text{Find}& P>0:\\ & (A_{d;0}+\mathrm{\Delta }{A}_d(\delta (t)){)}^{T}P(A_{d;0}+\mathrm{\Delta }{A}_d(\delta (t)))-P<0\text{ for all }\delta \end{array}$

Where $P\in R^{n,n}$.

In case of polytopic uncertainty:

$\begin{array}{cc}\text{Find}& P>0:\\ & (A_{d;0}+A_{d;i}{)}^{T}P(A_{d;0}+A_{d;i})-P<0\text{ for all }i=1,...n\end{array}$

Conclusion:

This LMI allows us to investigate stability for the robust control problem in the case of polytopic uncertainty and gives on the controller for this case

Implementation:

• [1] - Matlab implementation using the YALMIP framework and Mosek solver

Related LMIs: =

• - Discrete Time Stabilizability
• Polytopic stability for continuous time case
• Quadratic polytopic stabilization
• Discrete Time Lyapunov Stability

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. (3.20.2 page 64)
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

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