LMIs in Control/pages/Reachable set diagonalNB

Reachable sets with unit-energy inputs; Diagonal 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 Diagonal Norm-bound uncertainty, we have:

${\begin{aligned}p_{i}&=\delta _{i}(t)q_{i}\\&|\delta _{i}(t)|\leq 1;\;\;\;\;{\text{for }}i=1,...,N_{q}\end{aligned}}$

The Data

$N_{p}=N_{q}$

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:

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

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

Related LMIs:

• - Reachable sets with unit-energy inputs; Polytopic uncertainty
• - Reachable sets with unit-energy inputs; norm bound uncertainty
• Stabilizing State-Feedback Controllers with Structured 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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