LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Non-Expansivity and boundedness Realness

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

The System edit

Given a state-space representation of a linear system

 

  are the state, output and input vectors respectively.

The Data edit

  are system matrices.

Definition edit

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

 

 

 

 

 

(1)

hold for any arbitrary  , arbitrary input  , and the corresponding solution   of the system with  . In addition, the transfer function matrix

 

 

 

 

 

(2)

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

 

 

 

 

 

(3)

LMI Condition edit

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

 

 

 

 

 

(4)

and

 

 

 

 

 

(5)

Implementation edit

This implementation requires Yalmip and Mosek.

Conclusion: edit

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 edit


Return to Main Page: edit