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 $A\in {ℝ}^{n\times n}$, $B\in {ℝ}^{n\times m}$, $C\in {ℝ}^{p\times n}$, amd $A$ is Hurwitz. The generalized $H_2$ norm of $G$ is:

${\Vert G\Vert }_{2,\mathrm{\infty }}=\underset{u\in L_2,u\ne \text{zero}}{\sup }$ $\frac{{\Vert Gu\Vert }_{\mathrm{\infty }}}{{\Vert u\Vert }_2}$

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 ${\Vert G\Vert }_{2,\mathrm{\infty }}<\mu$ holds under the following conditions:

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

$\left[\begin{array}{cc}{A}^{T}P+PA& PB\\ *& -\mu 1\end{array}\right]<0$.

$\left[\begin{array}{cc}P& C^T\\ *& \mu 1\end{array}\right]>0$.

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

$\left[\begin{array}{cc}Q{A}^T+AQ& B\\ *& -\mu 1\end{array}\right]<0$.

$\left[\begin{array}{cc}Q& Q{C}^T\\ *& \mu 1\end{array}\right]>0$.

3. There exists $P\in {𝕊}^n,V\in {ℝ}^{n\times n}$ and $\mu \in {ℝ}_{>0}$ where $P>0$ such that:

$\left[\begin{array}{cccc}-(V+V^T)& V^{T}A+P& V^{T}B& V^T\\ *& -P& 0& 0\\ *& **-\mu 1& 0\\ *& *& *& -P\end{array}\right]<0$.

$\left[\begin{array}{cc}P& C^T\\ *& \mu 1\end{array}\right]>0$.

Conclusion:

The generalized $H_2$ norm of $G$ is the minimum value of $\mu \in {ℝ}_{>0}$ 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

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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