LMIs in Control/Matrix and LMI Properties and Tools/Non-expansivity and Bounded Realness

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This section studies the non-expansivity and bounded-realness 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 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 Condition[edit | edit source]

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












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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