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.


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