LMIs in Control/pages/H2 index

H₂ Index Deduced LMI

Although there are ways to evaluate an upper bound on the H₂, the verification of the bound on the H₂-gain of the system can be done via the deduced condition.

The System

We consider the generalized Continuous-Time LTI system with the state space realization of $(A,B,C,D)$

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

where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$ and $u(t)\in \mathbb {R} ^{r}$ are the system state, output, and the input vectors respectively.
The transfer function of such a system can be evaluated as:

${\begin{aligned}G(s)=C(sI-A)^{-1}B+D\end{aligned}}$

The Data

The system matrices $A,B,C$ are known.

The Optimization Problem

For an arbitrary $\gamma >0$(a given scalar), the transfer function satisfies

$\left\|C(sI-A)^{-1}B+D\right\|_{2}<\gamma$

The H₂-norm condition on Transfer function holds only when the matrix A is stable. And this can be conveniently converted to an LMI problem

if and only if 1. There exists a symmetric matrix $X>0$ such that:
$AX+XA^{T}+BB^{T}<0$, $trace(CXC^{T})<\gamma ^{2}$

2. There exists a symmetric matrix $Y>0$ such that:
$AY+YA^{T}+C^{T}C<0$, $trace(B^{T}YB)<\gamma ^{2}$

The LMI - Deduced Conditions for H2-norm

These deduced condition can be derived from the above equations. According to this

For an arbitrary $\gamma >0$(a given scalar), the transfer function satisfies

$\left\|G(s)\right\|_{2}<\gamma$

if and only if there exists symmetric matrices $Z$ and $P$; and a matrix $V$ such that
$trace(Z)<\gamma ^{2}$
${\begin{bmatrix}-Z&B^{T}\\B&-P\end{bmatrix}}$${\begin{aligned}<0\end{aligned}}$
${\begin{bmatrix}-(V+V^{T})&V^{T}A^{T}+P&V^{T}C^{T}&V^{T}\\AV+P&-P&0&0\\CV&0&-I&0\\V&0&0&-P\end{bmatrix}}$${\begin{aligned}<0\end{aligned}}$

The above LMI can be combined with the bisection method to find minimum $\gamma$ to find the minimum upper bound on the H₂ gain of $G(s)$.

Conclusion:

If there is a feasible solution to the aforementioned LMI, then the $\gamma$ upper bounds the norm of the system G(s).

Implementation

To solve the feasibility LMI, YALMIP toolbox is required for setting up the problem, and SeDuMi or MOSEK is required to solve the problem. The following link showcases an example of the problem:

https://github.com/yashgvd/ygovada

Related LMIs

Bounded Real Lemma
Deduced LMIs for H-infinity index

External Links

A list of references documenting and validating the LMI.

• [1] - LMI in Control Systems Analysis, Design and Applications
• 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.