### 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 language | English (US) |
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Title of host publication | Systems and Control |

Subtitle of host publication | Foundations and Applications |

Publisher | Birkhauser |

Pages | 209-246 |

Number of pages | 38 |

DOIs | |

State | Published - Jan 1 2018 |

### Publication series

Name | Systems and Control: Foundations and Applications |
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ISSN (Print) | 2324-9749 |

ISSN (Electronic) | 2324-9757 |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*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

}

TY - CHAP

T1 - Stabilization of deterministic control systems under random sampling

T2 - Overview and recent developments

AU - Tanwani, Aneel

AU - Chatterjee, Debasish

AU - Liberzon, Daniel M

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

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U2 - 10.1007/978-3-030-04630-9_6

DO - 10.1007/978-3-030-04630-9_6

M3 - Chapter

AN - SCOPUS:85059004072

T3 - Systems and Control: Foundations and Applications

SP - 209

EP - 246

BT - Systems and Control

PB - Birkhauser

ER -