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 statespace 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 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

(
)
ImplementationEdit
This implementation requires Yalmip and Mosek.
ConclusionEdit
Thus, it is seen that passivity and positiverealness describe the same property of a linear system, one gives the timedomain feature and the other provides frequencydomain feature of this property.
External LinksEdit
 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 GuangRen Duan and HaiHua Yu, CRC Press, Taylor & Francis Group, 2013