LMIs in Control/Matrix and LMI Properties and Tools/Passivity and Positive Realness
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
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 passive 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]
- 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 Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013