LMIs in Control/pages/Generalized KYP Lemma Conic Sector

The Concept The conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis. The System Consider a square, contiuous-time linear time-invariant (LTI) system, , with minimal state-space relization (A, B, C, D), where and . Also consider , which is defined as , where and . The Data The matrices The matrices and . The values of a and b LMI : Generalized KYP (GKYP) Lemma for Conic Sectors The following generalized KYP Lemmas give conditions for to be inside the cone within finite frequency bandwidths. 1. (Low Frequency Range) The system is inside the cone for all , where and , if there exist and , where , such that . If and Q = 0, then the traditional Conic Sector Lemma is recovered. The parameter is incuded in the above LMI to effectively transform into the strict inequality 2. (Intermediate Frequency Range) The system is inside the cone for all , where and , if there exist and and where and , such that . The parameter is incuded in the above LMI to effectively transform into the strict inequality . 3. (High Frequency Range) The system is inside the cone for all , where and , if there exist , where , such that . If (A, B, C, D) is a minimal realization, then the matrix inequalities in all of the above LMI, then it can be nostrict. Conclusion: If there exist a positive definite matrix satisfying above LMIs for the given frequency bandwidths then the system is inside the cone [a,b]. Implementation Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI Related LMIs KYP Lemma State Space Stability Exterior Conic Sector Lemma Modified Exterior Conic Sector Lemma 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. Bridgeman, Leila Jasmine, and James Richard Forbes. "The exterior conic sector lemma." International Journal of Control 88.11 (2015): 2250-2263.