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

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

$x\in {ℝ}^n,y\in {ℝ}^m,u\in {ℝ}^r$ 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 Template:NumBlk

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

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 Template:NumBlk

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.

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