It is assumed in the Lemma that the state-space representation (Ad, Bd, Cd, Dd) is minimal. Then Positive Realness (PR) of the transfer function Cd(SI − Ad)-1Bd + Dd is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (Ad, Bd, Cd, Dd)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR
Consider a discrete-time LTI system, , with minimal state-space relization , where and .
The matrices and
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
- 3. There exists where such that
- 4. 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
- 3. There exists where such that
- 4. 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
If there exist a positive definite for the the selected Q,S and R matrices then the system is Positive Real.
Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI
KYP Lemma
State Space Stability
KYP Lemma without Feedthrough
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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