Multivariable Extremum Seeking Unit-Vector Control Design
Enzo Ferreira Tomaz Silva, Pedro Henrique Silva Coutinho, Tiago Roux Oliveira, Miroslav Krstić
TL;DR
This work tackles multivariable extremum seeking with a unit-vector control law under an uncertain Hessian. It introduces a polytopic embedding $H(\alpha)=\sum_i \alpha_i H_i$ and derives an LMI-based design condition for the gain $K = LX^{-1}$ that yields finite-time stability of the average error dynamics. Through averaging theory for discontinuous right-hand sides, it is shown that the actual closed-loop trajectories converge to a neighborhood of the unknown optimum, with explicit bounds that improve as the perturbation frequency $\omega$ increases. An optimization problem is also formulated to minimize the guaranteed reaching time, and numerical results validate convergence and robustness to Hessian variations within the polytope.
Abstract
This paper investigates multivariable extremum seeking using unit-vector control. By employing the gradient algorithm and a polytopic embedding of the unknown Hessian matrix, we establish sufficient conditions, expressed as linear matrix inequalities, for designing the unit-vector control gain that ensures finite-time stability of the origin of the average closed-loop error system. Notably, these conditions enable the design of non-diagonal control gains, which provide extra degrees of freedom to the solution. The convergence of the actual closed-loop system to a neighborhood of the unknown extremum point is rigorously proven through averaging analysis for systems with discontinuous right-hand sides. Numerical simulations illustrate the efficacy of the proposed extremum seeking control algorithm.