Martingale Analysis And Its Applicationin In Iterative Learning Control

Nowadays martingale theory has become to one of the most active and the richest branch of stochastic process theory. Martingale process's content belongs to modern part of the stochastic process, which its theory is more and more widely applied in markov process, spot process, estimation theory, stochastic control and so on. Many martingale sequences such as martingale in limit, quasi-eventual martinga- le, asymptotic martingale, uniform asymptotic martingale, set-valued martingale, B valued martingale and other martingale sequences are provided to enrich the content of martingale theory and these martingale sequences are used widely applied. In recent years, martingale theory is applied widely in physic, biology, meteorology, economy, financial investiment and other fields. In this paper, martingale theory is used in iterative learning control.In this paper, variety of relative martingale concepts and their properties are introduced. These natures include operation property, decomposition property, contain relation. The relations among martingale, supermartingale, submartingale, martingale in limit, meta-martingale in limit, p quasi-martingale, quasi-eventual martingale, set-valued martingale, set-valued quasi-martingale, set-valued uniform asymptotic martingale, B valued uniform asymptotic martingale, B valued L1 martingale in limit, asymptotic martingale and uniform asymptotic martingale are reserched.Suppose the controlled system is iterative learning control system with random disturbance. Traditional stability definition is generalized for stable in the conditional expectation, P-type iterative learning control law is also generalized. And the improved control law is applied into iterative learning control system with random disturbance which its stability theory is constructed.Some simulation results show the rationability of stable in the conditional expectation and the efficiency of improved iterative learning control law. The simulation test verify the controlled system with random disturbance can reached to stability by using improved iterative learning control law but not the traditional control law.