LMIs in Control/pages/Reachable set polytopic

Reachable sets with unit-energy inputs; Polytopic uncertainty

A Reachable set is a set of system States reached under the condition $u=Kx$. On this page we will look at the problem of finding an controller $K$, that $E$ contains $RS$ - reachable set.

The System

$\begin{array}{cc}\dot{x}& =Ax+B_{w}w+B_{u}u\\ u& =Kx\end{array}$

Where:

$\begin{array}{cc}x& \in R^n\\ w& \in R^m\\ u& \in R^k\\ \end{array}$

In case of polytopic uncertainty, we have:

$\begin{array}{cc}A(t)B_w(t)B_u(t)& \in \mathbf{\text{Co}}\{[A_1{B}_{w,1}{B}_{u;1}],...,[A_L{B}_{w;L}{B}_{u;L}]\}\\ \end{array}$

Reachable set

The reachable set can be defined:

$\begin{array}{cc}RS& =\{x(T)|u=Kx;x(0)=0;T\ge 0;\underset{0}{\overset{T}{\int }}{w}^{T}wdt<1\}\\ \end{array}$

The elipsoid $E=\{\epsilon \in R^n|{\epsilon }^{T}Q\epsilon \le 1\}$ contains $RS$

The Data

The matrices $A,A_i\in R^{n\times n};B_w,B_{w;i}\in R^{n\times m};B_u,B_{u;i}\in R^{n\times k};Q\in R^{n\times n}$. And ${\displaystyle }$

The Optimization Problem

The following optimization problem should be solved:

$\begin{array}{cc}\text{Find}& Y>0:\\ & Q{A}_i^T+A_{i}Q+B_{u;i}Y+Y^T{B}_{u;i}^T+B_{w;i}{B}_{w;i}^T<0\text{ for all }i=1,...n\\ & K=Y{Q}^{-1}\end{array}$

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:

• Reachable sets with unit-energy inputs; Norm-bound uncertainty
• Reachable sets with unit-energy inputs; Diagonal Norm-bound uncertainty
• Quadratic polytopic stabilization

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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