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

This section deals with passivity 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.


The linear system with the same number of input and output variables is called passive if







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







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







LMI ConditionEdit

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








This implementation requires Yalmip and Mosek.


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.

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