LMIs in Control/pages/Reachable set normbounded

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

$\begin{array}{cc}\dot{x}& =Ax+B_{w}w+B_{u}u+B_{q}p\\ q& =C_{q}x+D_{qw}w+D_{qu}u+D_{qp}p\\ u& =Kx\end{array}$

Where:

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

In case of norm-bound uncertainty, we have:

$\begin{array}{cc}p& =\mathrm{\Delta }(t)q\\ & \Vert \mathrm{\Delta }\Vert \le 1\end{array}$

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:

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

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

The Optimization Problem

The following optimization problem should be solved:

$\begin{array}{cc}\text{Find}& \mu >0:Y\\ & \left[\begin{array}{cc}Q{A}^T+AQ+B_{u}Y+Y^T{B}_u^T+B_w{B}_w^T+\mu B_p{B}_p^T& (C_{q}Q+D_{qu}Y{)}^T\\ C_{q}Q+D_{qu}Y& -\mu I\end{array}\right]<0\\ & K=Y{Q}^{-1}\end{array}$

Or

$\begin{array}{cc}\text{Find}& \mu >0:\sigma \\ & \left[\begin{array}{cc}Q{A}^T+AQ+\sigma B_u{B}_u^T+B_w{B}_w^T+\mu B_p{B}_p^T& (C_{q}Q-\sigma D_{qu}{B}_u^T+\mu D_{qp}{B}_p^T{)}^T\\ C_{q}Q-\sigma D_{qu}{B}_u^T+\mu D_{qp}{B}_p^T& -\mu (I-D_{qp}{D}_{qp}^T)\end{array}\right]<0\\ & K=-\frac{\sigma }{2}{B}_u^T{Q}^{-1}\end{array}$

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; Polytopic uncertainty
• Reachable sets with unit-energy inputs; Diagonal Norm-bound uncertainty
• H-inf Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty

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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