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

${\displaystyle {\begin{aligned}x_{k+1}&=A_{d}(\alpha )x_{k}\\Where:\\&A_{d}(\alpha )=A_{d}+\Delta A_{d}(\delta (t))\\&\Delta A_{d}(\delta (t))=\sum _{k=1}^{n}\delta _{k}(t)A_{d;k}\in \mathrm {R} ^{n\times n}\\&\delta (t)=[\delta _{1}(t),...\delta _{n}(t)]-{\text{The set of perturbation parameters}}\\&\delta (t)\in \mathrm {R} \;\;\;A_{d;i}\in \mathrm {R} ^{n\times n}\end{aligned}}}$

The Data

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

The Optimization Problem

The following feasibility problem should be solved:

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

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

In case of polytopic uncertainty:

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

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:

• - 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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