Extremum Seeking Control for Autonomous Periodic Systems with Applications to Lower Limb Wearable Robots

This dissertation describes the development and implementation of model-free extremum seeking control (ESC) algorithms for lower limb wearable robots in order to improve their performance. The optimum control parameters of these wearable robots vary as the activities change, such as walking at different speeds. Automating the control parameter tuning process can greatly improve the patient experience by adapting to varying walking conditions encountered in daily life. ESC is a powerful, real-time, model-free adaptive optimization method that optimizes the steady-state performance of a given plant without knowledge of the system dynamics. In this dissertation, we implement ESC for (i) simultaneously tuning the feedback control gains of a knee-ankle powered prosthetic leg, and (ii) auto-tuning the stiffness of a quasi-passive ankle exoskeleton, both across different walking speeds. Next, in order to improve the performance of ESC for lower limb wearable robots, we developed a time-invariant version of ESC, where the exogenous time-based dither signal is replaced by a function of the periodic states of the system. The advantage of using a state-based dither is that it inherently contains information about the rate of the rhythmic task under control. Thus, in addition to maintaining time-scale separation at different plant speeds, the adaptation speed of a time-invariant ESC automatically changes, without changing the ESC parameters. A mathematically rigorous stability proof of such an ESC scheme for a large class of periodic systems has not appeared in the literature to date. Proving the stability of conventional, time-based ESC had been a challenging task [119], until a rigorous proof appeared in [52] after several decades. In this work, we present a rigorous stability proof of time-invariant ESC along with the dynamics of periodic autonomous systems using a perturbation theorem for two-dimensional Poincare maps.