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Stabilization of Continuous-Time Systems Against Stochastic Network Uncertainties: Fundamental Variance Bounds

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Abstract

This paper studies the stabilizability and stabilization of continuous-time systems in the presence of stochastic multiplicative uncertainties. The authors consider multi-input, multi-output (MIMO) linear time-invariant systems subject to multiple static, structured stochastic uncertainties, and seek to derive fundamental conditions to ensure that a system can be stabilized under a mean-square criterion. In the stochastic control framework, this problem can be considered as one of optimal control under state- or input-dependent random noises, while in the networked control setting, a problem of networked feedback stabilization over lossy communication channels. The authors adopt a mean-square small gain analysis approach, and obtain necessary and sufficient conditions for a system to be mean-square stabilizable via output feedback. For single-input, single-output (SISO) systems, the condition provides an analytical bound, demonstrating explicitly how plant unstable poles, nonminimum phase zeros, and time delay may impose a limit on the uncertainty variance required for mean-square stabilization. For MIMO minimum phase systems with possible delays, the condition amounts to solving a generalized eigenvalue problem, readily solvable using linear matrix inequality optimization techniques.

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Correspondence to Tian Qi or Jie Chen.

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This research was supported in part by Research Grants Council of Hong Kong under Project CityU 11203120, in part by City University of Hong Kong under Project 9380054, in part by the Natural Science Foundation of China under Grant 61603141, and in part by the Fundamental Research Funds for the Central Universities 2019MS141.

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Qi, T., Chen, J. Stabilization of Continuous-Time Systems Against Stochastic Network Uncertainties: Fundamental Variance Bounds. J Syst Sci Complex 34, 1858–1878 (2021). https://doi.org/10.1007/s11424-021-1236-4

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  • DOI: https://doi.org/10.1007/s11424-021-1236-4

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