Stability Analysis of Jump-Linear Systems Driven by Finite-State Machines with Markovian Inputs

STABILITY ANALYSIS OF JUM P-LINEAR SYSTEMS DRIVEN BY FINITE-STATE MACHINES W ITH MARKOVIAN INPUTS Sudarshan S. Patilkulkarni Old Dominion University, 2004 Director: Dr. W. Steven Gray A control system with a fault recovery mechanism in the feedback loop and with faults occurring in a non-deterministic manner can be modeled as a class of hybrid systems, i.e., a dynamical system switched by a finite-state machine or an autom aton. W hen the plant and controller are linear, such a system can be modeled as a jumplinear system driven by a finite-state machine with a random input process. Such fault recovery mechanisms are found in flight control systems and distributed control systems with communication networks. In these critical applications, closed-loop stability of the system in the presence of fault recoveries becomes an im portant issue. Finite-state machines as mathematical constructs are widely used by computer scientists to model and analyze algorithms. In particular, fault recovery mechanisms th a t are implemented in hardware with logic based circuits and finite memory can be modeled appropriately with finite-state machines. In this thesis, m athem atical tools are developed to determine the mean-square stability of a closed-loop system, modeled as a jump-linear system in series with a finite-state machine driven by a random process. The random input process is in general assumed to be any r-th order Markov process, where r > 0. While stability tests for a jump-linear system with a Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Markovian switching rule are well known, the main contribution of the present work arises from the fact th a t output of a finite-state machine driven by a Markov process is in general not Markovian. Therefore, new stability analysis tools are provided for this class of systems and demonstrated through Monte Carlo simulations. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. “The whole is more than the sum of its parts” constitutive charac­ teristics of a phenomenon are not explainable from the characteristics of the isolated parts alone, it is also the interconnection and the interrela­ tionship. -Ludwig Von Bertalanfiy, General System Theory. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGMENTS Setting and achieving a higher academic goal is insurmountable w ithout adequate moral, emotional and financial support. Supervisors, coworkers, friends, family and institutions make us what we are. Dr. W. Steven Gray, my advisor, whose immense patience, support, wisdom and advice is the foremost contribution in my achievement. Dr. Oscar R. Gonzalez’s subtle comments during our many discussions have always been an eye-opener. Dr. Dayanand Naik, Dr. Linda Vahala, Dr. Sacharia Albin and Dr. Steven Zahorian have all provided me valuable assistance along the way. I am indebted to Prof. Anand Kumar and Prof. B. N. Devaraj, my mentors during my undergraduate years a t S.D. M. College of Engineering in Dharwad, India. They instilled in me a passion for electrical engineering, especially in the area of signals and systems. I am also grateful for the support of my co-workers: Adeel Jafri, James Barkley, Alicia Hoffler, Mark Adams, Hong Zhang, Li Yaqin and Heber Zapana-Herencia. Special thanks to Arturo Tejada who has provided me with coffee, food and lodging along with the moral support whenever I was in need. Old Dominion University has provided me with valuable course-work, a good library and a supportive academic environment. This research was supported by the NASA Langley Research Center under grants NCC-1-392, NCC-1-03026 and NNL04AA03A, and by the National Science Founda­ tion under grant CCR-0209094. I wish to acknowledge the support of these organi­ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. zations. Finally, the love and warmth of my family: Vinayak, Vani, Vallabh, Vasuki, father and belated mother have been my guide. Moral and emotional support of Preethi and many other friends whom I have met during the course of my graduate studies, whose names I am unable to list here due to lack of space, cannot be left unacknowledged. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.