LMIs in Control/Matrix and LMI Properties and Tools/Generalized H2 Norm

Generalized $$H_{2}$$ Norm
The $$ H_2 $$ norm characterizes the average frequency response of a system. To find the H2 norm, the system must be strictly proper, meaning the state space represented $$ D $$ matrix must equal zero. The H2 norm is frequently used in optimal control to design a stabilizing controller which minimizes the average value of the transfer function, $$ G $$ as much as possible. This optimal control problem is also called the Linear Quadratic Gaussian.

The System
Consider a continuous-time, linear, time-invariant system $$ G : L_{2e} \to L_{2e} $$ with state space realization $$ (A,B,C,0) $$ where $$ A\in\mathbb{R}^{n\times n} $$, $$B\in\mathbb{R}^{n\times m}$$, $$C\in\mathbb{R}^{p\times n}$$, amd $$ A $$ is Hurwitz. The generalized $$ H_{2} $$ norm of $$ G $$ is:

$$ \left\| G \right\|_{2,{\infty}} = \sup_{u\in L_{2}, {u}\neq\text{zero}} $$ $$ \frac{\left\| Gu \right\|_{\infty}}{\ \left\| u \right\|_{2}}$$

The Data
The transfer function $$G$$, and system matrices $$A$$, $$B$$, $$C$$ are known and $$ A $$ is Hurwitz.

The LMI: Generalized $$H_{2}$$ Norm LMIs
The inequality $$ \left\| G \right\|_{2,{\infty}} < \mu $$ holds under the following conditions:

1. There exists $$ P\in\mathbb{S}^{n}$$ and $$ \mu\in\mathbb{R}_{>0}$$ where $$ P > 0 $$ such that:


 * $$\begin{bmatrix} A^{T}P + PA & PB \\ * & -\mu 1 \end{bmatrix}<0 $$.
 * $$\begin{bmatrix} P & C^{T} \\ * & \mu 1\end{bmatrix}>0 $$.

2. There exists $$ Q\in\mathbb{S}^{n}$$ and $$ \mu\in\mathbb{R}_{>0}$$ where $$ Q > 0 $$ such that:
 * $$\begin{bmatrix} QA^{T} +AQ & B \\ * & -\mu 1 \end{bmatrix}<0 $$.
 * $$\begin{bmatrix} Q & QC^{T} \\ * & \mu 1 \end{bmatrix}>0 $$.

3. There exists $$ P\in\mathbb{S}^{n}, V\in\mathbb{R}^{n\times n}$$ and $$ \mu\in\mathbb{R}_{>0}$$ where $$ P > 0 $$ such that:
 * $$\begin{bmatrix} -(V+V^{T}) & V^{T}A+P & V^{T}B & V^{T} \\ * & -P & 0 & 0 \\ * & * * -\mu 1 & 0 \\ * & * & * & -P \end{bmatrix}<0 $$.
 * $$\begin{bmatrix} P & C^{T} \\ * & \mu 1 \end{bmatrix}>0 $$.

Conclusion:
The generalized $$ H_{2} $$ norm of $$ G $$ is the minimum value of $$ \mu\in\mathbb{R}_{>0} $$ that satisfies the LMIs presented in this page.

Implementation
This implementation requires Yalmip and Sedumi.

Generalized $ H_{2} $ Norm - MATLAB code for Generalized $$ H_{2} $$ Norm.

Related LMIs
LMI for System H_{2} Norm

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