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

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

$x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{m},u\in \mathbb {R} ^{r}$ 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

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

(1)

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

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

(2)

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

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

(3)

LMI Condition

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

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

(4)

and

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

(5)

Implementation

This implementation requires Yalmip and Mosek.

• 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.

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013