Prescribed-Time Extremum Seeking for Delays and PDEs Using Chirpy Probing
Cemal Tugrul Yilmaz, Miroslav Krstić
Abstract
Extremum seeking, an online model-free optimization algorithm with traditionally exponential convergence, was recently advanced by Poveda and coauthors to fixed-time convergence, using non-smooth time-invariant feedback. In this paper we introduce an alternative time-varying prescribed-time extremum seeking (PT-ES) approach to reaching the optimum in a user-assignable “prescribed” time (PT), independent of the initial condition of the estimator. Instead of conventional sinusoidal probing signals, we employ “chirpy” perturbations, i.e., sinusoids with growing frequencies. In addition to providing a result for a static input-output map, we provide algorithms for a map in cascade with (1) a delay, (2) a wave PDE, and (3) a heat/diffusion PDE. The designs are based on the time-varying PT backstepping approach, which transforms the PDE-ODE cascade into a suitable PT-stabilized target system, and on averaging-based estimates of the gradient and Hessian of the map. Classical averaging does not apply for the analysis of PT-ES algorithms for four reasons: neither is probing periodic, nor is the ‘average system’ exponentially stable, nor is the original system finite-dimensional, nor is the interval of operation infinite. We develop an averaging technique needed for this non-periodic finite-time infinite-dimensional problem. Along with PDE Lyapunov analysis, we show that the input converges to the optimizer in prescribed time. We present three numerical examples to illustrate the effectiveness of the proposed technique for delay-free case as well as for compensation of time-delay and diffusion PDE dynamics.