LMIs in Control/Controller Synthesis/Discrete Time/State Feedback/Closed-Loop Robust Stability and Controller synthesis of Discrete-Time System with Polytopic Uncertainty

Closed-Loop Robust Stability and Controller synthesis of Discrete-Time System with Polytopic Uncertainty

Stability is an important property, stability analysis is necessary for control theory. For robust control, an LMI criterion was developed for the discrete case with closed-loop and open-loop system. The open-loop system is obtained by replacing $B=0$..

The System

${\begin{aligned}x_{k+1}&=A_{d}(\alpha )x_{k}+B_{d}(\beta )u_{k}\\u_{k}&=Kx_{k}\end{aligned}}$

The Data

The matrices $A\in R^{n\times n}\;B\in R^{n\times m}\;K\in R^{m\times n}$.

In the case of robust polytopic uncertainty control we have:

${\begin{aligned}A&\in \{\sum _{k=1}^{n}\alpha _{k}A_{k}|{\text{ }}\;\sum _{k=1}^{n}\alpha _{k}=1\}\\B&\in \{\sum _{k=1}^{m}\beta _{k}B_{k}|{\text{ }}\;\sum _{k=1}^{m}\beta _{j}=1\}\end{aligned}}$

The Optimization Problem

The following feasibility problem should be solved:

${\begin{aligned}{\text{Find}}\;&P_{ij}>0,G{\text{ is invertible}},L:\\{\begin{bmatrix}P_{ij}&A_{i}G-B_{j}L\\G^{T}A_{i}^{T}-L^{T}B_{j}^{T}&G+G^{T}+P_{ij}\end{bmatrix}}&>0&{\text{ for }}i=1,...,n{\text{ and }}j=1,...,m\\\end{aligned}}$

Where $L\in R^{m,n}$, $P_{ij},G\in R^{n,n}=0$ for $i=1,...,n$ and $j=1,...,m$

Then our controller $K=-LG^{-1}$

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 SeDuMi solver

Related LMIs: =

• [2]- Discrete Time Stabilizability.
• [3] - Polytopic stability for continuous time case
• [4] - Quadratic polytopic stabilization
• [5] -Quadratic Polytopic Hinf- Optimal State Feedback Control

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. (3.19.2 page 63)
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

Return to Main Page:

• [6]- Main Page.