LMIs in Control/Matrix and LMI Properties and Tools/Passivity and Positive Realness

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This section deals with passivity of a system.

The System[edit | edit source]

Given a state-space representation of a linear system

are the state, output and input vectors respectively.

The Data[edit | edit source]

are system matrices.

Definition[edit | edit source]

The linear system with the same number of input and output variables is called passive 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 | edit source]

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

 

 

 

 

(4)

Implementation[edit | edit source]

This implementation requires Yalmip and Mosek.

Conclusion[edit | edit source]

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 | edit source]


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