Stability Analysis And Synthesis Of Uncertain Time-Delay Systems

Time-varying delays are frequently encountered in various engineering systems such as biological systems, chemical systems and hydraulic systems. They are often a source of oscillations and instabilities in systems. Moreover, uncertainties are unavoid-able in many practical systems. Thus, robust stability analysis and synthesis problems for linear time-varying delay systems have attracted the attention of many researchers. It should be noted that almost all of the work on the robust stability analysis and synthe-sis of linear time-varying delay systems requires the uncertainties to be characterized in specific forms and provides only sufficient conditions.Because of the sufficiency nature of the conditions, when all the existing suffi-cient conditions are not feasible for a linear time-varying delay system, one cannot tell whether or not such a system is robustly stable or can be robustly synthesized. Hence, an interesting research question is how to analyze and synthesize these uncertain linear time-varying delay systems. In this dissertation, a potential approach is introduced to robustly analyze and synthesize such systems by modeling the uncertainties with proba-bility distributions. We consider three important issues about linear time-varying delay systems with probabilistic uncertainties:probabilistic robust stability analysis, proba-bilistic robust stabilization, and probabilistic robust H∞synthesis.· For probabilistic robust stability analysis of linear time-varying delay systems, the concept of probability of stability is first introduced to analyze the systems stabil-ity property. Then we provide a randomized algorithm to compute the probability of stability.· For probabilistic robust stabilization of linear time-varying delay systems, we first formulate the design of a robust state-feedback controller in the probabilistic set-ting as finding a solution of a stochastic optimization problem. Then such a solution of the stochastic optimization problem can be found by a proposed randomized algo-rithm, which is based on the stochastic subgradient method. Finally, after obtaining the controller by the proposed algorithm, the stability property of the resulting closed-loop system can be evaluated according to the probability of stability.· For probabilistic robust H∞synthesis of linear time-varying delay systems, the concept of the probability of H∞performance is first needed to evaluate the system performance under probabilistic circumstance. Then we formulate a stochastic opti-mization problem, of which the solutions are designed controller parameters. Such a problem is solved by a proposed randomized algorithm. Finally, we use the probability of H∞performance to evaluate the designed system performance.