LMIs in Control/Bounded Real Lemma

The System:



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

The Optimization Problem: Given a state space system of

\begin{align} \dot x(t)&=Ax(t)+Bu(t)\\ y(t)&=Cx(t)+Du(t) \end{align}$$ with A being stable and A$$\in\R^{n*n}$$, B$$\in\R^{n*p}$$,C$$\in\R^{p*n}$$ and D$$\in\R^{p*p}$$ and (A,B,C) being minimal there exists a P=P^T$$\in\R^{n*n}$$ that can be used to solve the bounded real lemma problem using the LMI mentioned below.

The LMI: The Bounded Real Lemma

\begin{align} \begin{bmatrix} A^T P+PA+C^T C&&PB+C^T D\\B^T P+D^T C&& D^T D-I\end{bmatrix}&<0\\ P&>0\\ \end{align}$$ The LMI is feasible if and only if the state space is non expansive. $$\int_{0}^{T} y(t)^T y(t) dt$$<$$\int_{0}^{T} u(t)^T u(t) dt$$ for all solutions of the state space with x(0) = 0, This condition can also be expressed in terms of the transfer matrix H. Nonexpansivity is equivalent to the transfer matrix H satisfying the bounded-real condition, $$H(s)*H(s)<=I$$ for all $$ Re s>0$$

Conclusion:

The LMI is feasible, if and only if the Hamiltonian Matrix M has no imaginary eigenvalues.

Related LMIs:

1. KYP Lemma. https://en.wikibooks.org/wiki/LMIs_in_Control/KYP_Lemmas/KYP_Lemma_(Bounded_Real_Lemma)

Implementation
A link to CodeOcean or other online implementation of the LMI (in progress)