LMIs in Control/pages/Deduced LMI Conditions for H-infinity Index

H-infinity Index Deduced LMI

Although the KYP Lemma, also known as the Bounded Real Lemma, is a basic condition to evaluate an upper bound on the H∞, the verification of the bound on the H∞-gain of the system can be done via the deduced condition.

The System
A state-space representation of a linear system as given below:



\begin{align} \dot x(t)&=Ax(t)+Bw(t)\\ y(t)&=Cx(t)+Dw(t) \end{align}$$

where $$x(t)\in \R^n$$, $$y(t)\in \R^m$$ and $$w(t)\in \R^r$$ are the system state, output, and the disturbance vector respectively. The transfer function of such a system can be evaluated as:



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

The Data
Number of states n, number of outputs m and number of external noise channels r need to be known. Moreover, the system matrices A,B,C,D are also required to be known.

The Feasibility LMI
For an arbitrary $$ \gamma $$, the transfer function G(s) satisfies


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

if and only if there exists a symmetric matrix X > 0 and a matrix $$ \Omega .$$ such that:



\begin{align} \text{Find} \; X, \Omega :&\\ \Theta + \Phi^\top \Omega \Psi + \Psi^\top \Omega^\top \Phi < 0 \end{align}$$

where:



\begin{align}

\Theta &= \begin{bmatrix} 0 & X & 0 & 0 & 0 \\ X & -X & 0 & 0 & 0 \\ 0 & 0 & -\gamma I_m & 0 & 0\\ 0 & 0 & 0 & -X & 0 \\ 0 & 0 & 0 & 0 & \gamma I_r \end{bmatrix} \\ \Phi &= \begin{bmatrix} -I_n & A^\top & C^\top & I_n & 0 \\ 0 & B^\top & D^\top & 0 & -\gamma I_r\end{bmatrix} \\ \Psi &= \begin{bmatrix} I_n & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & I_r\end{bmatrix} \end{align}$$

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

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

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

https://github.com/smhassaan/LMI-Examples/blob/master/Deduced_hinf_example.m

Related LMIs
Bounded Real Lemma