LMIs in Control/pages/Discrete Time KYP Lemma without Feedthrough

The ConceptEdit

It is assumed in the Lemma that the state-space representation (A, B, C, D) is minimal. Then Positive Realness (PR) of the transfer function C(SI − A)-1B + D is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (A, B, C, D)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR

The SystemEdit

Consider a contiuous-time LTI system,  , with minimal state-space relization (A, B, C, 0), where   and  .

 

The DataEdit

The matrices The matrices   and  

LMI : KYP Lemma without FeedthroughEdit

The system   is positive real (PR) under either of the following equivalet necessary and sufficient conditions.

1. There exists   where   such that
 
2. There exists   where   such that
 

This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.

The system   is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.

1. There exists   where   such that
 
2. There exists   where   such that
 

This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε  

Conclusion:Edit

If there exist a positive definite   for the the selected Q,S and R matrices then the system   is Positive Real.

ImplementationEdit

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

Related LMIsEdit

KYP Lemma
State Space Stability

ReferencesEdit

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transac- tions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London