Hierarchical Structures for Economic Repetitive Control

For many emerging repetitive control applications such as wind and marine energy generation systems, gait-cycle following in legged locomotion, remote sensing, surveillance, and reconnaissance, the primary objective for repetitive control (RC) is optimization of a cycle cost such as the lap-averaged power generated and metabolic cost of locomotion, as opposed to the classical requirement of tracking a known reference trajectory by the system output. For this newer class of applications, only a range of reference trajectories suitable for cyclic operation is known a priori, the range potentially encapsulating various operational constraints, and as part of repetitive control, it is desired that over a number of operation cycles, the cycle cost, or the economic metric, is optimized. With this underlying motivation, a hierarchical solution is presented, wherein the inner loop includes a classical repetitive controller that tracks a reference trajectory of known period, and the outer loop iteratively learns the desired reference trajectory using a combination of the system and cost function models and the measured cycle cost. This approach results in optimum steady-state cyclic operation. A steepest descent type algorithm is used in the outer loop, and via Lyapunov-like arguments, the existence of tuning parameters resulting in robust and optimal steady-state cyclic operation is discussed. Appropriate guidelines for parameter tuning are presented, and the proposed method is numerically validated using an example of an inverted pendulum.