Multivariable Newton-based extremum seeking

We present a Newton-based extremum seeking algorithm for the multivariable case. The design extends the recent Newton-based extremum seeking algorithms for the scalar case and introduces a dynamic estimator of the Hessian matrix that removes the difficulty with the possible singularity of this matrix estimate. This estimator has the form of a differential Riccati equation. We prove local stability of the new algorithm for general nonlinear dynamic systems using averaging and singular perturbations. In comparison with the standard gradient-based multivariable extremum seeking, the proposed algorithm removes the dependence of the convergence rate on the unknown Hessian matrix and makes the convergence rate, of both the parameter estimates and of the estimates of the Hessian inverse, user-assignable. In particular, the new algorithm allows all the parameters to converge with the same speed, even with maps that have highly elongated level sets. In the parameter space, the new algorithms produces trajectories straight to the extremum, as opposed to non-direct “steepest descent” trajectories. Simulation results show the advantage of the proposed approach over gradient-based extremum seeking.