LMIs in Control/Matrix and LMI Properties and Tools/Non-expansivity and Bounded Realness

This section studies the non-expansivity and bounded-realness of a system.

The System
Given a state-space representation of a linear system

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

$$ x \in \mathbb{R}^{n}, y \in \mathbb{R}^{m}, u \in \mathbb{R}^{r} $$ are the state, output and input vectors respectively.

The Data
$$ A,B,C,D $$ are system matrices.

Definition
The linear system with the same number of input and output variables is called non-expansive if

hold for any arbitrary $$ T\geq 0 $$, arbitrary input $$ u(t) $$, and the corresponding solution $$ y(t) $$ of the system with $$ x(0) = 0 $$. In addition, the transfer function matrix

of system is called is positive real if it is square and satisfies

LMI Condition
Let the linear system be controllable. Then, the system is bounded-real if an only if there exists $$ P > 0 $$ such that

and

Implementation
This implementation requires Yalmip and Mosek.
 * https://github.com/ShenoyVaradaraya/LMI--master/blob/main/bounded_realness.m

Conclusion:
Thus, it is seen that passivity and positive-realness describe the same property of a linear system, one gives the time-domain feature and the other provides frequency-domain feature of this property.