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Iterative Linear Quadratic Regulator With Variational Equation-Based Discretization

Katsuya Shigematsu, Hikaru Hoshino, Eiko Furutani

TL;DR

This work improves ILQR-based model predictive control by substituting finite-difference discretization with a variational-equation based discretization to linearize the discretized state transition map directly. This approach decouples linearization accuracy from the discretization timestep, allowing larger timesteps and more ILQR iterations within a fixed real-time budget, which enhances closed-loop performance. The authors demonstrate the method on swing-up control of an inverted pendulum, showing superior trajectory accuracy and faster convergence, and provide a practical ILQR-MPC implementation. The results suggest significant real-time performance gains and broader applicability to nonlinear MPC with constraints and online trajectory optimization.

Abstract

This paper discusses discretization methods for implementing nonlinear model predictive controllers using Iterative Linear Quadratic Regulator (ILQR). Finite-difference approximations are mostly used to derive a discrete-time state equation from the original continuous-time model. However, the timestep of the discretization is sometimes restricted to be small to suppress the approximation error. In this paper, we propose to use the variational equation for deriving linearizations of the discretized system required in ILQR algorithms, which allows accurate computation regardless of the timestep. Numerical simulations of the swing-up control of an inverted pendulum demonstrate the effectiveness of this method. By the relaxing stringent requirement for the size of the timestep, the use of the variational equation can improve control performance by increasing the number of ILQR iterations possible at each timestep in the realtime computation.

Iterative Linear Quadratic Regulator With Variational Equation-Based Discretization

TL;DR

This work improves ILQR-based model predictive control by substituting finite-difference discretization with a variational-equation based discretization to linearize the discretized state transition map directly. This approach decouples linearization accuracy from the discretization timestep, allowing larger timesteps and more ILQR iterations within a fixed real-time budget, which enhances closed-loop performance. The authors demonstrate the method on swing-up control of an inverted pendulum, showing superior trajectory accuracy and faster convergence, and provide a practical ILQR-MPC implementation. The results suggest significant real-time performance gains and broader applicability to nonlinear MPC with constraints and online trajectory optimization.

Abstract

This paper discusses discretization methods for implementing nonlinear model predictive controllers using Iterative Linear Quadratic Regulator (ILQR). Finite-difference approximations are mostly used to derive a discrete-time state equation from the original continuous-time model. However, the timestep of the discretization is sometimes restricted to be small to suppress the approximation error. In this paper, we propose to use the variational equation for deriving linearizations of the discretized system required in ILQR algorithms, which allows accurate computation regardless of the timestep. Numerical simulations of the swing-up control of an inverted pendulum demonstrate the effectiveness of this method. By the relaxing stringent requirement for the size of the timestep, the use of the variational equation can improve control performance by increasing the number of ILQR iterations possible at each timestep in the realtime computation.
Paper Structure (15 sections, 1 theorem, 38 equations, 9 figures, 2 tables, 2 algorithms)

This paper contains 15 sections, 1 theorem, 38 equations, 9 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. ILQR Background
  3. Discrete-Time Optimal Control
  4. Backward Recursion
  5. Trajectory Optimization Algorithm
  6. ILQR Using Variational Equation
  7. Finite-Difference Method
  8. Variational Equation-Based Discretization
  9. ILQR-MPC Implementation
  10. Numerical Simulations
  11. Swing-up Control of Inverted Pendulum
  12. Trajectory Optimization
  13. Model Predictive Controller
  14. Conclusions
  15. The stable manifold method

Key Result

Theorem 1

Sakamoto2008. The sequences $q_k(t,\xi)$ and $p_k(t,\xi)$ are convergent to zero for sufficiently small $|\xi |$, that is, $q_k(t,\xi)$, $p_k(t,\xi) \rightarrow 0$ as $t \rightarrow \infty$ for all $k = 0, 1, 2, ...$. Furthermore, $q_k(t,\xi)$ and $p_k(t,\xi)$ are uniformly convergent to a solution

Figures (9)

  • Figure 1: The inverted pendulum model
  • Figure 2: Comparison of state trajectories for various settings of timestep $\Delta t$ from $0.01s$ to $0.05s$.
  • Figure 3: Comparison of time courses of the state and input for the timestep $\Delta t = 0.01s$.
  • Figure 4: Cost vs. iterations for the timeste $\Delta t = 0.01s$.
  • Figure 5: Averaged computational time for a single iteration of ILQR.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Theorem 1