LMIs in Control/pages/LMI for System H2 Norm

$H_2$-norm of System

The $H_2$-norm is conceptually identical to the Frobenius (aka Euclidean) norm on a matrix. It can be used to determine whether the system representation can be reduced to its simplest form, thereby allowing its use in performing effective block-diagram algebra.

The System

Suppose we define the state-space system$G:L_2\to L_2\text{ by }y=Gu$ if:

$\begin{array}{cc}\dot{x}(t)& =Ax(t)+Bu(t)\\ y(t)& =Cx(t)+Du(t)\\ \end{array}$

where $A\in {ℝ}^{mxm}$, $B\in {ℝ}^{mxn}$, $C\in {ℝ}^{pxm}$, and $D\in {ℝ}^{qxn}$ for any $t\in ℝ$. Then the $H_2$-norm of the system can be determined as described below.

The Data

In order to determine the $H_2$-norm of the system, we need the matrices $A$, $B$, and $C$.

The Optimization Problem

Suppose we wanted to to infer properties of the system behaviour (which is represented in the form (A,B,C,D)). Then it becomes necessary to ensure that the overall system forms an algebra, as the standard use of block-diagram algebra would otherwise be invalid. The only way this is possible is by calculating $H_2$ and/or $H_{\mathrm{\infty }}$-norms - both of which are signal norms that (in a certain sense) measure the size of the transfer function.

The LMI: The $H_2$ Norm

Assuming that $\hat{P}(s)=C(sI-A{)}^{-1}B$, this means that the following are equivalent:

$1)A\text{ is Hurwitz and }||\hat{P}|{|}_{H_2}^2<\gamma$

$2)\begin{array}{c}\{\begin{array}{cc}trace(CX{C}^T)& <\gamma \\ AX+X{A}^T+B{B}^T& <0\\ X& >0\end{array}\end{array}$

Conclusion:

The LMI can be used to minimize the $H_2$-norm of the system. It is worth noting that a finite $H_2$-norm does not guarantee finite $H_{\mathrm{\infty }}$-norm, and that in order for the block diagram algebra to be valid, $H_{\mathrm{\infty }}$-norm must be finite.

Implementation

• Example Code - A GitHub link that contains code (titled "H2Norm.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

Related LMIs

• $H_2$-Filtering - LMI for $H_2$-Filtering
• Discrete-Time $H_2$ Norm - LMI for $H_2$-norm in the Discrete-Time case.

External Links

A list of references documenting and validating the LMI.

• 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.