LMIs in Control/pages/Discrete Time KYP Lemma without Feedthrough
The Concept
editIt 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 System
editConsider a contiuous-time LTI system, , with minimal state-space relization (A, B, C, 0), where and .
The Data
editThe matrices The matrices and
LMI : KYP Lemma without Feedthrough
editThe 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:
editIf there exist a positive definite for the the selected Q,S and R matrices then the system is Positive Real.
Implementation
editCode for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
Related LMIs
editReferences
edit1. 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