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:

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

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 $$ \delta \in \Delta $$.

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 $$ \hat{P}(s,\delta) = C(\delta)(sI-A(\delta))^-$$$$^1B(\delta) $$. 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

$$ \begin{align} \begin{bmatrix} A(\delta) & B_{2}(\delta) \end{bmatrix} \begin{bmatrix} X(\delta)\\ Z(\delta) \end{bmatrix} + \begin{bmatrix} X(\delta) & Z(\delta)^T \end{bmatrix} \begin{bmatrix} A(\delta)^T\\ B(\delta)_2^T \end{bmatrix} + B_1(\delta)B_1(\delta)^T < 0 \end{align} $$

$$ \begin{align} \begin{bmatrix} X(\delta) & (C_1(\delta)X(\delta)+D_{12}(\delta)Z(\delta))^T \\ C_1(\delta)X(\delta)+D_{12}(\delta)Z(\delta) & W(\delta) \end{bmatrix} > 0 \end{align} $$

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

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

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