User:Margav06/sandbox/Click here to continue/Fundamentals of Matrix and LMIs/Passivity and Positive Realness

This section deals with passivity 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 passive 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 passive if an only if there exists $$ P > 0 $$ such that

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