User:Margav06/sandbox/Click here to continue/Controller synthesis/Controller to achieve the desired Reachable set; Norm bound uncertainty

Reachable sets with unit-energy inputs; norm bound 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  $$E \supseteq RS$$ - reachable set.

The System


\begin{align} \dot{x} &=A x + B_w w + B_u u +B_q p \\ q &= C_q x + D_{qw} w + D_{qu} u + D_{qp} p\\ u &= Kx \end{align} $$

Where:

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

\begin{align} p & = \Delta(t) q \\ & \|\Delta\| \leq 1 \end{align} $$

The Data
The matrices $$ A \in R^{n\times n};\; B_{w} \in R^{n\times m} ;\; B_{u} \in R^{n\times k}; B_{p} \in R^{n\times N_p}; K \in R^{k\times n}  $$.

$$ C_q \in R^{N_q\times n};\; D_{qu} \in R^{N_q\times k} D_{qw} \in R^{N_q\times m} D_{qp} \in R^{N_q\times N_p} $$.

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

The Optimization Problem
The following optimization problem should be solved:

\begin{align} \text{Find} \; &\mu >0: Y\\ & \begin{bmatrix} QA^T + A Q + B_{u }Y + Y^TB^T_{u} + B_{w}B^T_{w}+\mu B_p B_p^T & (C_q Q + D_{qu}Y)^T \\ C_q Q + D_{qu}Y & -\mu I\end{bmatrix}< 0 \\ & K = Y Q^{-1} \end{align} $$

Or

\begin{align} \text{Find} \; &\mu >0: \sigma\\ & \begin{bmatrix} QA^T + A Q + \sigma B_{u }B_u^T  + B_{w}B^T_{w}+\mu B_p B_p^T & (C_q Q -\sigma D_{qu}B_u^T + \mu D_{qp} B_p^T)^T \\ C_q Q -\sigma D_{qu}B_u^T + \mu D_{qp} B_p^T & -\mu(I - D_{qp} D_{qp}^T) \end{bmatrix}< 0 \\ & K = -\frac{\sigma}{2} B_u^T 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; Polytopic uncertainty
 * Reachable sets with unit-energy inputs; Diagonal Norm-bound uncertainty
 * H-inf Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

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