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Reservoir Predictive Path Integral Control for Unknown Nonlinear Dynamics

Daisuke Inoue, Tadayoshi Matsumori, Gouhei Tanaka, Yuji Ito

TL;DR

This work addresses fast online control of unknown nonlinear dynamics by integrating Echo State Networks (ESN) with Model Predictive Path Integral (MPPI) control to form Reservoir Predictive Path Integral Control (RPPI). It further adds an uncertainty-aware extension (URPPI) that samples perturbed ESN output weights to minimize the expected cost under model uncertainty, enabling robust stochastic control without linearization. The approach is validated on a Duffing oscillator and a four-tank system, showing that URPPI can reduce control costs by up to about 60% compared with traditional quadratic programming MPC and outperforms MPPI, with a small increase in computation time. The contributions offer a practical framework for online learning and control of unknown nonlinear systems, with potential impact in robotics, process control, and aerospace applications.

Abstract

Neural networks have found extensive application in data-driven control of nonlinear dynamical systems, yet fast online identification and control of unknown dynamics remain central challenges. To meet these challenges, this paper integrates echo-state networks (ESNs)--reservoir computing models implemented with recurrent neural networks--and model predictive path integral (MPPI) control--sampling-based variants of model predictive control. The proposed reservoir predictive path integral (RPPI) enables fast learning of nonlinear dynamics with ESNs and exploits the learned nonlinearities directly in MPPI control computation without linearization approximations. This framework is further extended to uncertainty-aware RPPI (URPPI), which achieves robust stochastic control by treating ESN output weights as random variables and minimizing an expected cost over their distribution to account for identification errors. Experiments on controlling a Duffing oscillator and a four-tank system demonstrate that URPPI improves control performance, reducing control costs by up to 60% compared to traditional quadratic programming-based model predictive control methods.

Reservoir Predictive Path Integral Control for Unknown Nonlinear Dynamics

TL;DR

This work addresses fast online control of unknown nonlinear dynamics by integrating Echo State Networks (ESN) with Model Predictive Path Integral (MPPI) control to form Reservoir Predictive Path Integral Control (RPPI). It further adds an uncertainty-aware extension (URPPI) that samples perturbed ESN output weights to minimize the expected cost under model uncertainty, enabling robust stochastic control without linearization. The approach is validated on a Duffing oscillator and a four-tank system, showing that URPPI can reduce control costs by up to about 60% compared with traditional quadratic programming MPC and outperforms MPPI, with a small increase in computation time. The contributions offer a practical framework for online learning and control of unknown nonlinear systems, with potential impact in robotics, process control, and aerospace applications.

Abstract

Neural networks have found extensive application in data-driven control of nonlinear dynamical systems, yet fast online identification and control of unknown dynamics remain central challenges. To meet these challenges, this paper integrates echo-state networks (ESNs)--reservoir computing models implemented with recurrent neural networks--and model predictive path integral (MPPI) control--sampling-based variants of model predictive control. The proposed reservoir predictive path integral (RPPI) enables fast learning of nonlinear dynamics with ESNs and exploits the learned nonlinearities directly in MPPI control computation without linearization approximations. This framework is further extended to uncertainty-aware RPPI (URPPI), which achieves robust stochastic control by treating ESN output weights as random variables and minimizing an expected cost over their distribution to account for identification errors. Experiments on controlling a Duffing oscillator and a four-tank system demonstrate that URPPI improves control performance, reducing control costs by up to 60% compared to traditional quadratic programming-based model predictive control methods.

Paper Structure

This paper contains 16 sections, 5 theorems, 30 equations, 5 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Problem Formulation
  4. Proposed Method
  5. Echo State Network
  6. Model and Learning Algorithm
  7. Probabilistic Interpretation
  8. Uncertainty-aware Model Predictive Path Integral Control
  9. Stochastic reformulation
  10. Free energy relaxation
  11. Sampling-based optimization
  12. Computational Complexity
  13. Numerical Experiments
  14. Duffing Oscillator
  15. Four-Tank System
  16. ...and 1 more sections

Key Result

Proposition 4.1

Assume that for every time $t\ge \hat{N}-1$, the matrix ${\mathbf A}_t\coloneqq \sum_{\tau=0}^t \gamma^{t-\tau} {\mathbf x}_\tau {\mathbf x}_\tau^\top$ is invertible. Then, the minimizer ${\mathbf W}^{\text{out}*}_{t}$ at time $t\ge \hat{N}$ can be expressed using the minimizer ${\mathbf W}^{\text{o where ${\mathbf P}_t = {\mathbf A}_t^{-1} \in{\mathbb R}^{\hat{N}\times \hat{N}}$ is called the pre

Figures (5)

  • Figure 1: Schematic diagram of the proposed RPPI that integrates ESN and MPPI. (a) Overall architecture of RPPI; (b) ESN model for online identification of nonlinear dynamics, in which only the output weight matrix is trained with RLS; (c) MPPI controller for computing control inputs based on the learned ESN models. In URPPI, the output is predicted using the perturbed output weight matrices so that the controller can compute control inputs that minimize the expected cost under model uncertainty.
  • Figure 2: Time response of the Duffing oscillator. (a): Time response of the controlled output; (b): Time response of the control input. The green dotted line represents QPMPC, the blue dashed line represents MPPI, and the orange solid line represents UMPPI. In (a), the purple dash-dotted line represents the reference output. The shaded areas indicate the standard deviation over multiple random-seed runs.
  • Figure 3: Comparison of the time-integrated control cost in a Duffing oscillator. The plot is presented in a raincloud style, combining a violin plot (right) to show distribution density and a scatter plot (left) to display individual data points. Diamond markers and numerical values indicate the mean of each distribution.
  • Figure 4: Time response of the four-tank system. (a): Time response of the first component of the controlled output in the two-dimensional system; (b): Time response of the first component of the control input. The green dotted line represents QPMPC, the blue dashed line represents MPPI, and the orange solid line represents UMPPI. In (a), the purple dash-dotted line represents the reference output. The shaded areas show the standard deviation over multiple random-seed runs.
  • Figure 5: Comparison of the time-integrated control cost in a four-tank system. The plot is presented in a raincloud style, combining a violin plot (right) to show distribution density and a scatter plot (left) to display individual data points. Diamond markers and numerical values indicate the mean of each distribution.

Theorems & Definitions (8)

  • Proposition 4.1: Ref. farhang2013adaptive
  • Proposition 4.2
  • Remark 4.3
  • Proposition 4.4
  • Proposition 4.5
  • Proposition 4.6
  • Remark 4.7
  • Remark 4.8