Convergence characteristics of PD-type and PDDα-type iterative learning control for impulsive differential systems with unknown initial states:

An important issue in the area of multi-operation systems is the observation of a discontinuous trajectory. In order to track a discontinuous output trajectory, we choose impulsive differential systems to generate a series of local continuous state trajectories, which become a set of output trajectories via the action of the output functions. Concerning impulsive differential control systems, we design proportional–derivative (PD-type) and proportional plus one order derivative and fractional order derivative (PDDα-type) iterative learning control laws with initial state learning. In particular, the PDDα-type law is more flexible due to the impact of a certain fractional order factor α. Thereafter, we give the associated convergence characteristics by establishing sufficient conditions on open-loop and closed-loop iterative learning schemes in the sense of λ-norm under mild assumptions. Finally, several numerical examples, including an application to robotic fish, are given to illustrate our results.