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 SystemEdit

Given a state-space representation of a linear system

 

  are the state, output and input vectors respectively.

The DataEdit

  are system matrices.

DefinitionEdit

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 ConditionEdit

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

 

 

 

 

 

(4)

and

 

 

 

 

 

(5)

ImplementationEdit

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 LinksEdit


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