LMIs in Control/Bounded Real Lemma

The System:

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

The Optimization Problem: Given a state space system of

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

with A being stable and A $\in \mathbb {R} ^{n*n}$ , B $\in \mathbb {R} ^{n*p}$ ,C $\in \mathbb {R} ^{p*n}$ and D $\in \mathbb {R} ^{p*p}$ and (A,B,C) being minimal there exists a P=P^T $\in \mathbb {R} ^{n*n}$ that can be used to solve the bounded real lemma problem using the LMI mentioned below.

The LMI: The Bounded Real Lemma

{\begin{aligned}{\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{aligned}}

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 $Res>0$

Conclusion:

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

Implementation

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

References

1. Linear Matrix Inequalities in System and Control Theory by Stephen Boyd, Laurent El Ghaoui, Eric Feron, and Venkataramanan Balakrishnan Section 2.7.3
* LMIs in Systems and Control Theory