LMIs in Control/Click here to continue/Robust Controls/H2-Optimal State Feedback Synthesis

Robust H₂-Optimal State Feedback Synthesis

For systems with uncertain state parameters, a robust controller is needed. H₂-optimal control is desirable in minimum-energy applications.

The System

The static formulation of the system is given as follows:

$\begin{array}{c}\dot{x}(t)=A(\delta )x(t)+B(\delta )u(t)\\ y(t)=C(\delta )x(t)+D(\delta )u(t)\end{array}$

Where $x(t)\in {ℝ}^n$ is the state and $u(t)\in {ℝ}^m$ is the input at any $t\in ℝ$

$A(\delta )$, $B(\delta )$, $C(\delta )$, and $D(\delta )$ are rational matrices with variance $\delta \in \mathrm{\Delta }$.

The Data

The state matrices are defined as:

$A(\delta )=A(\delta )$,

$B(\delta )=\left[\begin{array}{cc}{B}_1(\delta )& B_2(\delta )\end{array}\right]$

$C(\delta )=\left[\begin{array}{c}{C}_1(\delta )\\ C_2(\delta )\end{array}\right]$

$D(\delta )=\left[\begin{array}{cc}{D}_{11}(\delta )& D_{12}(\delta )\\ D_{21}(\delta )& D_{22}(\delta )\end{array}\right]$

The LMI:H₂-Optimal State Feedback Synthesis

Suppose $\hat{P}(s,\delta )=C(\delta )(sI-A(\delta ){)}^{-}$$1^{}B(\delta )$. Then the following are equivalent:

1. $\left||S(K(\delta ),P(\delta ))|\right|$_{$H_2$} $<\gamma$ for all $\delta \in \mathrm{\Delta }$.

2. $K(\delta )=Z(\delta )X(\delta {)}^{-}$$1^{}$ for some $Z(\delta )$ and $X(\delta )$ such that $X(\delta )>0$ for all $\delta \in \mathrm{\Delta }$and

$\begin{array}{c}\left[\begin{array}{cc}A(\delta )& B_2(\delta )\end{array}\right]\left[\begin{array}{c}X(\delta )\\ Z(\delta )\end{array}\right]+\left[\begin{array}{cc}X(\delta )& Z(\delta {)}^T\end{array}\right]\left[\begin{array}{c}A(\delta {)}^T\\ B(\delta {)}_2^T\end{array}\right]+B_1(\delta )B_1(\delta {)}^T<0\end{array}$

$\begin{array}{c}\left[\begin{array}{cc}X(\delta )& (C_1(\delta )X(\delta )+D_{12}(\delta )Z(\delta ){)}^T\\ C_1(\delta )X(\delta )+D_{12}(\delta )Z(\delta )& W(\delta )\end{array}\right]>0\end{array}$

$Trace(W(\delta ))<{\gamma }^2$

for all $\delta \in \mathrm{\Delta }$

Conclusion:

The method above can be used to find an H₂-optimal robust state feedback controller for a system with uncertain parameters.

Implementation

This implementation requires Yalmip and Sedumi.

H₂-Optimal State Feedback Synthesis

Related LMIs

Full State Feedback Optimal H_inf LMI

External Links

• 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.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.

Return to Main Page:

LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control Template:BookCat