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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:

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

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

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

The Data

The state matrices are defined as:

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

$B(\delta )={\begin{bmatrix}B_{1}(\delta )&B_{2}(\delta )\end{bmatrix}}$

$C(\delta )={\begin{bmatrix}C_{1}(\delta )\\C_{2}(\delta )\end{bmatrix}}$

$D(\delta )={\begin{bmatrix}D_{11}(\delta )&D_{12}(\delta )\\D_{21}(\delta )&D_{22}(\delta )\end{bmatrix}}$

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. $\left\vert \left\vert S(K(\delta ),P(\delta ))\right\vert \right\vert$_{$H_{2}$} $<\gamma$ for all $\delta \in \Delta$.

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

${\begin{aligned}{\begin{bmatrix}A(\delta )&B_{2}(\delta )\end{bmatrix}}{\begin{bmatrix}X(\delta )\\Z(\delta )\end{bmatrix}}+{\begin{bmatrix}X(\delta )&Z(\delta )^{T}\end{bmatrix}}{\begin{bmatrix}A(\delta )^{T}\\B(\delta )_{2}^{T}\end{bmatrix}}+B_{1}(\delta )B_{1}(\delta )^{T}<0\end{aligned}}$

${\begin{aligned}{\begin{bmatrix}X(\delta )&(C_{1}(\delta )X(\delta )+D_{12}(\delta )Z(\delta ))^{T}\\C_{1}(\delta )X(\delta )+D_{12}(\delta )Z(\delta )&W(\delta )\end{bmatrix}}>0\end{aligned}}$

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

for all $\delta \in \Delta$

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.

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