# 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

{\displaystyle {\begin{aligned}\ {\dot {x}}=Ax+Bu\\\ y=Cx+Du\\\end{aligned}}}

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

## The Data

${\displaystyle A,B,C,D}$  are system matrices.

## Definition

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

{\displaystyle {\begin{aligned}\int \limits _{0}^{T}y^{T}y(t)dt\geq \int \limits _{0}^{T}u^{T}u(t)dt\\\end{aligned}}}

(1)

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

{\displaystyle {\begin{aligned}G(s)&=C(sI-A)^{-1}B+D\\\end{aligned}}}

(2)

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

{\displaystyle {\begin{aligned}\ G^{H}(s)+G(s)\geq I\forall s\in \mathbb {C} ,Re(s)>0\\\end{aligned}}}

(3)

## LMI Condition

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

{\displaystyle {\begin{aligned}\ {\begin{bmatrix}A^{T}P+PA&PB&C^{T}\\B^{T}P&-I&D^{T}\\C&D&-I\end{bmatrix}}<0\\\end{aligned}}}

(4)

and

{\displaystyle {\begin{aligned}\ {\begin{bmatrix}PA+A^{T}P+C^{T}C&PB+C^{T}D\\B^{T}P+D^{T}C&D^{T}D-I\end{bmatrix}}<0\\\end{aligned}}}

(5)

## Implementation

This implementation requires Yalmip and Mosek.

## 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.