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Maximum Conditional Probability Stochastic Controller for Linear Systems with Additive Cauchy Noises

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Abstract

Motivated by the sliding mode control approach, a stochastic controller design methodology is developed for discrete-time, vector-state linear systems with additive Cauchy-distributed noises, scalar control inputs, and scalar measurements. The control law exploits the recently derived characteristic function of the conditional probability density function of the system state given the measurements. This result is used to derive the characteristic function of the conditional probability density function of the sliding variable, utilized in the design of the stochastic controller. The incentive for the proposed approach is mainly the high numerical complexity of the currently available method for such systems, that is based on the optimal predictive control paradigm. The performance of the proposed controller is evaluated numerically and compared to the alternative Cauchy controller and a controller based on the Gaussian assumption. A fundamental difference between controllers based on the Cauchy and Gaussian assumptions is the superior response of Cauchy controllers to noise outliers. The newly proposed Cauchy controller exhibits similar performance to the optimal predictive controller, while requiring significantly lower computational effort.

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Notes

  1. In [7], this methodology was referred to as Cauchy-MPC.

  2. It was shown in [8] that during a measurement update at time step k, a new term is generated in the sum of (14) that is centered at \(\sigma _{k+1}(k|k)=z(k)/H\). The center of this terms is moved to \(\sigma _{k+1}(k+1|k)=z(k)\varPhi /H\) when propagated to time step \(k+1\). Hence, if \(z(k)\varPhi /H\) is large, it will produce a secondary peak in \(f_{{X_{k + 1}}|{Y_k}}^{{u_k} = 0}\left( {{x_{k + 1}}|{y_k}} \right) \).

  3. From Bayes’ rule, the un-normalized conditional pdf is the joint pdf of the state and the measurement history, where the normalization factor is the pdf of the measurement history. The propagation of the un-normalized conditional pdf yields an analytic, recursive scheme.

  4. In [7] this controller is referred to as Cauchy MPC.

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Acknowledgements

This work was supported by the National Science Foundation (NSF) and United States Israel Binational Science Foundation (BSF) joint NSF-BSF ECCS program under Grants Nos. 2015702 and 2019639, and the NSF Grants Nos. 1607502 and 1934467.

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Authors and Affiliations

  1. Technion - Israel Institute of Technology, Haifa, Israel

    Nati Twito & Moshe Idan

  2. Mechanical and Aerospace Engineering, University of California, Los Angeles, CA, USA

    Jason L. Speyer

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  1. Nati Twito
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  2. Moshe Idan
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  3. Jason L. Speyer
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Correspondence to Moshe Idan.

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Twito, N., Idan, M. & Speyer, J.L. Maximum Conditional Probability Stochastic Controller for Linear Systems with Additive Cauchy Noises. J Optim Theory Appl 191, 393–414 (2021). https://doi.org/10.1007/s10957-020-01735-5

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