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{align} \dot{x} &=A x + B_w w + B_u u \\ u &= Kx \end{align} $$

Where:

\begin{align} x & \in R^{n} \\ w & \in R^{m} \\ u & \in R^{k} \\ \end{align} $$ In case of polytopic uncertainty, we have:

\begin{align} 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{align} $$

Reachable set
The reachable set can be defined:

\begin{align} RS &= \{x(T) | u = Kx; \;\; x(0) = 0; \;\; T \geq 0 ;\;\; \int_{0}^T w^T w dt <1 \} \\ \end{align} $$

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

The Optimization Problem
The following optimization problem should be solved:

\begin{align} \text{Find} \; &Y >0:\\ & QA^T_i + A_iQ + B_{u;i}Y + Y^TB^T_{u;i} + B_{w;i}B^T_{w;i} < 0 \text{ for all } i = 1,... n\\ & K = Y Q^{-1} \end{align} $$

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:

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

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