Stabilization of deterministic control systems under random sampling: Overview and recent developments

Aneel Tanwani, Debasish Chatterjee, Daniel M Liberzon

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

This chapter addresses the problem of stabilizing continuous-time deterministic control systems via a sample-and-hold scheme under random sampling. The sampling process is assumed to be a Poisson counter, and the open-loop system is assumed to be stabilizable in an appropriate sense. Starting from as early as mid-1950s, when this problem was studied in the Ph.D. dissertation of R.E. Kalman, we provide a historical account of several works that have been published thereafter on this topic. In contrast to the approaches adopted in these works, we use the framework of piecewise deterministic Markov processes to model the closed-loop system, and carry out the stability analysis by computing the extended generator. We demonstrate that for any continuous-time robust feedback stabilizing control law employed in the sample-and-hold scheme, the closed-loop system is asymptotically stable for all large enough intensities of the Poisson process. In the linear case, for increasingly large values of the mean sampling rate, the decay rate of the sampled process increases monotonically and converges to the decay rate of the unsampled system in the limit. In the second part of this article, we fix the sampling rate and address the question of whether there exists a feedback gain which asymptotically stabilizes the system in mean square under the sample-and-hold scheme. For the scalar linear case, the answer is in the affirmative and a constructive formula is provided here. For systems with dimension greater than 1 we provide an answer for a restricted class of linear systems, and we leave the solution corresponding to the general case as an open problem.

Original languageEnglish (US)
Title of host publicationSystems and Control
Subtitle of host publicationFoundations and Applications
PublisherBirkhauser
Pages209-246
Number of pages38
DOIs
StatePublished - Jan 1 2018

Publication series

NameSystems and Control: Foundations and Applications
ISSN (Print)2324-9749
ISSN (Electronic)2324-9757

Fingerprint

Random Sampling
Stabilization
Control System
Sampling
Control systems
Decay Rate
Closed loop systems
Closed-loop System
Continuous Time
Piecewise Deterministic Markov Process
Feedback
Asymptotically Stable
Poisson process
Mean Square
Markov processes
Linear systems
Stability Analysis
Open Problems
Siméon Denis Poisson
Linear Systems

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Computer Science Applications
  • Control and Optimization
  • Computational Mathematics

Cite this

Tanwani, A., Chatterjee, D., & Liberzon, D. M. (2018). Stabilization of deterministic control systems under random sampling: Overview and recent developments. In Systems and Control: Foundations and Applications (pp. 209-246). (Systems and Control: Foundations and Applications). Birkhauser. https://doi.org/10.1007/978-3-030-04630-9_6

Stabilization of deterministic control systems under random sampling : Overview and recent developments. / Tanwani, Aneel; Chatterjee, Debasish; Liberzon, Daniel M.

Systems and Control: Foundations and Applications. Birkhauser, 2018. p. 209-246 (Systems and Control: Foundations and Applications).

Research output: Chapter in Book/Report/Conference proceedingChapter

Tanwani, A, Chatterjee, D & Liberzon, DM 2018, Stabilization of deterministic control systems under random sampling: Overview and recent developments. in Systems and Control: Foundations and Applications. Systems and Control: Foundations and Applications, Birkhauser, pp. 209-246. https://doi.org/10.1007/978-3-030-04630-9_6
Tanwani A, Chatterjee D, Liberzon DM. Stabilization of deterministic control systems under random sampling: Overview and recent developments. In Systems and Control: Foundations and Applications. Birkhauser. 2018. p. 209-246. (Systems and Control: Foundations and Applications). https://doi.org/10.1007/978-3-030-04630-9_6
Tanwani, Aneel ; Chatterjee, Debasish ; Liberzon, Daniel M. / Stabilization of deterministic control systems under random sampling : Overview and recent developments. Systems and Control: Foundations and Applications. Birkhauser, 2018. pp. 209-246 (Systems and Control: Foundations and Applications).
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