Multivariable Newton-based extremum seeking

  • Authors:

  • Azad Ghaffari;Miroslav Krstić;Dragan NešIć

  • Affiliations:

  • Joint-Doctoral Programs (Aerospace and Mechanical), San Diego State University and University of California at San Diego, La Jolla, CA 92093-0411, USA;Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA;Department of Electrical and Electronic Engineering Department, University of Melbourne, VIC 3010, Australia

  • Venue:
  • Automatica (Journal of IFAC)
  • Year:
  • 2012

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Abstract

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 inverse of the Hessian matrix that removes the difficulty with the possible singularity of a possible direct estimate of the Hessian matrix. The estimator of the inverse of the Hessian 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, yielding straight trajectories to the extremum even with maps that have highly elongated level sets, in contrast to curved ''steepest descent'' trajectories of the gradient algorithm. Simulation results show the advantage of the proposed approach over gradient-based extremum seeking, by assigning equal, desired convergence rates to all the parameters using Newton's approach.