Extremum seeking of dynamical systems via gradient descent and stochastic approximation methods

Sei Zhen Khong, Ying Tan, Chris Manzie, Dragan Nešić

Research output: Contribution to journalArticlepeer-review

19 Scopus citations


This paper examines the use of gradient based methods for extremum seeking control of possibly infinite-dimensional dynamic nonlinear systems with general attractors within a periodic sampled-data framework. First, discrete-time gradient descent method is considered and semi-global practical asymptotic stability with respect to an ultimate bound is shown. Next, under the more complicated setting where the sampled measurements of the plant's output are corrupted by an additive noise, three basic stochastic approximation methods are analysed; namely finite-difference, random directions, and simultaneous perturbation. Semi-global convergence to an optimum with probability one is established. A tuning parameter within the sampled-data framework is the period of the synchronised sampler and hold device, which is also the waiting time during which the system dynamics settle to within a controllable neighbourhood of the steady-state input-output behaviour.

Original languageEnglish (US)
Pages (from-to)44-52
Number of pages9
StatePublished - Jun 1 2015
Externally publishedYes

Bibliographical note

Funding Information:
This work was supported by the Swedish Research Council through the LCCC Linnaeus centre and the Australian Research Council ( DP120101144 ). The material in this paper was presented at the 9th Asian Control Conference, June 23–26, 2013, Istanbul, Turkey. This paper was recommended for publication in revised form by Associate Editor Raul Ordóñez under the direction of Editor Miroslav Krstic.

Funding Information:
Dragan Nešić is a Professor in the Department of Electrical and Electronic Engineering (DEEE) at The University of Melbourne, Australia. He received his B.E. degree in Mechanical Engineering from The University of Belgrade, Yugoslavia in 1990, and his Ph.D. degree from Systems Engineering, RSISE, Australian National University, Canberra, Australia in 1997. Since February 1999 he has been with The University of Melbourne. His research interests include networked control systems, discrete-time, sampled-data and continuous-time nonlinear control systems, input-to-state stability, extremum seeking control, applications of symbolic computation in control theory, hybrid control systems, and so on. He was awarded a Humboldt Research Fellowship (2003) by the Alexander von Humboldt Foundation, an Australian Professorial Fellowship (2004–2009) and Future Fellowship (2010–2014) by the Australian Research Council. He is a Fellow of IEEE and a Fellow of IEAust. He is currently a Distinguished Lecturer of CSS, IEEE (2008-). He served as an Associate Editor for the journals Automatica, IEEE Transactions on Automatic Control, Systems and Control Letters and European Journal of Control.

Publisher Copyright:
© 2015 Elsevier Ltd. All rights reserved.


  • Extremum seeking
  • Gradient descent method
  • Infinite-dimensional nonlinear systems
  • Sampled-data control
  • Stochastic approximation methods


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