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Ensemble Control for Stochastic Systems with Asymmetric Laplace Noises

Yajie Yu, Xuehui Ma, Shiliang Zhang, Zhuzhu Wang, Xubing Shi, Yushuai Li, Tingwen Huang

TL;DR

This work tackles robust adaptive control for stochastic systems subject to asymmetric noises and outliers by modeling disturbances as a mixed asymmetric Laplace distribution and decoupling learning from control via certainty equivalence. It introduces an Iterative Quantile Filter (IQF) to online de-aggregate mixed ALD noise and estimate system parameters, derives a CE control for each subsystems under individual ALD disturbances, and fuses these controls through Bayesian posterior weights to form an ensemble control law. The approach is validated with numerical simulations showing that the ensemble controller closely tracks reference trajectories and significantly outperforms single-ALD or RLS-based methods, especially in the presence of outliers. The proposed framework enhances resilience to non-Gaussian disturbances and provides a practical scheme for adaptive control in environments with skewed noise and rare but large measurement outliers.

Abstract

This paper presents an adaptive ensemble control for stochastic systems subject to asymmetric noises and outliers. Asymmetric noises skew system observations, and outliers with large amplitude deteriorate the observations even further. Such disturbances induce poor system estimation and degraded stochastic system control. In this work, we model the asymmetric noises and outliers by mixed asymmetric Laplace distributions (ALDs), and propose an optimal control for stochastic systems with mixed ALD noises. Particularly, we segregate the system disturbed by mixed ALD noises into subsystems, each of which is subject to a specific ALD noise. For each subsystem, we design an iterative quantile filter (IQF) to estimate the system parameters using system observations. With the estimated parameters by IQF, we derive the certainty equivalence (CE) control law for each subsystem. Then we use the Bayesian approach to ensemble the subsystem CE controllers, with each of the controllers weighted by their posterior probability. We finalize our control law as the weighted sum of the control signals by the sub-system CE controllers. To demonstrate our approach, we conduct numerical simulations and Monte Carlo analyses. The results show improved tracking performance by our approach for skew noises and its robustness to outliers, compared with single ALD based and RLS-based control policy.

Ensemble Control for Stochastic Systems with Asymmetric Laplace Noises

TL;DR

This work tackles robust adaptive control for stochastic systems subject to asymmetric noises and outliers by modeling disturbances as a mixed asymmetric Laplace distribution and decoupling learning from control via certainty equivalence. It introduces an Iterative Quantile Filter (IQF) to online de-aggregate mixed ALD noise and estimate system parameters, derives a CE control for each subsystems under individual ALD disturbances, and fuses these controls through Bayesian posterior weights to form an ensemble control law. The approach is validated with numerical simulations showing that the ensemble controller closely tracks reference trajectories and significantly outperforms single-ALD or RLS-based methods, especially in the presence of outliers. The proposed framework enhances resilience to non-Gaussian disturbances and provides a practical scheme for adaptive control in environments with skewed noise and rare but large measurement outliers.

Abstract

This paper presents an adaptive ensemble control for stochastic systems subject to asymmetric noises and outliers. Asymmetric noises skew system observations, and outliers with large amplitude deteriorate the observations even further. Such disturbances induce poor system estimation and degraded stochastic system control. In this work, we model the asymmetric noises and outliers by mixed asymmetric Laplace distributions (ALDs), and propose an optimal control for stochastic systems with mixed ALD noises. Particularly, we segregate the system disturbed by mixed ALD noises into subsystems, each of which is subject to a specific ALD noise. For each subsystem, we design an iterative quantile filter (IQF) to estimate the system parameters using system observations. With the estimated parameters by IQF, we derive the certainty equivalence (CE) control law for each subsystem. Then we use the Bayesian approach to ensemble the subsystem CE controllers, with each of the controllers weighted by their posterior probability. We finalize our control law as the weighted sum of the control signals by the sub-system CE controllers. To demonstrate our approach, we conduct numerical simulations and Monte Carlo analyses. The results show improved tracking performance by our approach for skew noises and its robustness to outliers, compared with single ALD based and RLS-based control policy.
Paper Structure (10 sections, 3 theorems, 59 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 10 sections, 3 theorems, 59 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Controller design
  4. Iterative Quantile Filter
  5. Certainty Equivalence Control
  6. Ensemble controller
  7. Simulations and results
  8. Simulations of the control under mixed ALD noises
  9. Simulations with various noise conditions and outliers
  10. Conclusions

Key Result

Theorem 1

Consider a stochastic linear system described by (system_IQF), where the noise follows asymmetric Laplace distribution $\mathcal{ALD}(e(k): \tau(k), \mu(k), \delta(k))$. Initiate the system parameter as $\hat{w}(0)$ and estimation covariance as $\bm{P}(0)$. The parameter vector $\bm{w}(k)$ can be le where the parameter $\varepsilon(k)$ is and the parameter $p(k)$ is

Figures (8)

  • Figure 1: The diagram of ensemble control law for an uncertain system with mixed ALD noises
  • Figure 2: The output of the system controlled by optimal control with known parameters, certainty equivalence control with RLS, and adaptive ensemble control, represented by $y_{OPT}$, $y_{RLS}$, and $y_{EN}$, respectively. The reference trajectory $y_r$ in (a), (b), and (c) are 0.01Hz square wave with unit amplitude filtered by the transfer function $1/(s+1)$, 0.01Hz triangular wave with 1 unit amplitude, and 0.01Hz sine wave with 1 unit amplitude, respectively.
  • Figure 3: The average accumulated error for 100 Monte Carlo simulations
  • Figure 4: Tracking error for the control of 100-300th iterations based on 100 Monte Carlo simulations.
  • Figure 5: The system output subject to Noise-I
  • ...and 3 more figures

Theorems & Definitions (12)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Remark 4
  • Theorem 2
  • proof
  • Remark 5
  • Theorem 3
  • ...and 2 more