# 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 ${\displaystyle u=Kx}$. On this page we will look at the problem of finding an controller ${\displaystyle K}$, that ${\displaystyle E}$ contains ${\displaystyle RS}$ - reachable set.

## The System

{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+B_{w}w+B_{u}u\\u&=Kx\end{aligned}}}

Where:

{\displaystyle {\begin{aligned}x&\in R^{n}\\w&\in R^{m}\\u&\in R^{k}\\\end{aligned}}}

In case of polytopic uncertainty, we have:

{\displaystyle {\begin{aligned}A(t)\;\;B_{w}(t)\;\;B_{u}(t)&\in {\textbf {Co}}\{[A_{1}\;\;B_{w,1}\;\;B_{u;1}],...,[A_{L}\;\;B_{w;L}\;\;B_{u;L}]\}\\\end{aligned}}}

## Reachable set

The reachable set can be defined:

{\displaystyle {\begin{aligned}RS&=\{x(T)|u=Kx;\;\;x(0)=0;\;\;T\geq 0;\;\;\int _{0}^{T}w^{T}wdt<1\}\\\end{aligned}}}

The elipsoid ${\displaystyle E=\{\varepsilon \in R^{n}|\varepsilon ^{T}Q\varepsilon \leq 1\}}$ contains ${\displaystyle RS}$

## The Data

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

## The Optimization Problem

The following optimization problem should be solved:

{\displaystyle {\begin{aligned}{\text{Find}}\;&Y>0:\\&QA_{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=YQ^{-1}\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:

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