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

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

Where:

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

In case of norm-bound uncertainty, we have:

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

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:

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

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:

${\begin{aligned}{\text{Find}}\;&\mu >0:Y\\&{\begin{bmatrix}QA^{T}+AQ+B_{u}Y+Y^{T}B_{u}^{T}+B_{w}B_{w}^{T}+\mu B_{p}B_{p}^{T}&(C_{q}Q+D_{qu}Y)^{T}\\C_{q}Q+D_{qu}Y&-\mu I\end{bmatrix}}<0\\&K=YQ^{-1}\end{aligned}}$

Or

${\begin{aligned}{\text{Find}}\;&\mu >0:\sigma \\&{\begin{bmatrix}QA^{T}+AQ+\sigma B_{u}B_{u}^{T}+B_{w}B_{w}^{T}+\mu B_{p}B_{p}^{T}&(C_{q}Q-\sigma D_{qu}B_{u}^{T}+\mu D_{qp}B_{p}^{T})^{T}\\C_{q}Q-\sigma D_{qu}B_{u}^{T}+\mu D_{qp}B_{p}^{T}&-\mu (I-D_{qp}D_{qp}^{T})\end{bmatrix}}<0\\&K=-{\frac {\sigma }{2}}B_{u}^{T}Q^{-1}\end{aligned}}$

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

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.

Return to Main Page

• Main Page.