LMIs in Control/Click here to continue/LMIs in system and stability Theory/KYP Lemma Without Feedthrough

The Concept 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 System Consider a contiuous-time LTI system, , with minimal state-space relization (A, B, C, 0), where and . The Data The matrices The matrices and LMI : KYP Lemma without Feedthrough 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: If there exist a positive definite for the the selected Q,S and R matrices then the system is Positive Real. Implementation Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI Related LMIs KYP Lemma State Space Stability Discrete Time KYP Lemma with Feedthrough References 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