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Optimal Control of Discrete-Time Nonlinear Systems

Chuanzhi Lv, Xunmin Yin, Hongdan Li, Huanshui Zhang

TL;DR

This work addresses real-time solution of discrete-time nonlinear optimal control problems by merging nonlinear optimization with Pontryagin's maximum principle (PMP). It reformulates the original problem into an auxiliary OCP and develops a superlinear-convergence iterative algorithm, supported by explicit gradient and Hessian formulas derived from forward-backward difference equations (FBDEs). The approach is implemented in a model-predictive control setting and validated numerically on a linear quadratic regulator and experimentally on an automatic guided vehicle trajectory-tracking task, demonstrating both accuracy and substantial computational speedups. The results indicate a practical method for online precise control of nonlinear discrete-time systems with real-time requirements, and the framework generalizes to other second-order optimization schemes.

Abstract

This paper focuses on optimal control problem for a class of discrete-time nonlinear systems. In practical applications, computation time is a crucial consideration when solving nonlinear optimal control problems, especially under real-time constraints. While linearization methods are computationally efficient, their inherent low accuracy can compromise control precision and overall performance. To address this challenge, this study proposes a novel approach based on the optimal control method. Firstly, the original optimal control problem is transformed into an equivalent optimization problem, which is resolved using the Pontryagin's maximum principle, and a superlinear convergence algorithm is presented. Furthermore, to improve computation efficiency, explicit formulas for computing both the gradient and hessian matrix of the cost function are proposed. Finally, the effectiveness of the proposed algorithm is validated through simulations and experiments on a linear quadratic regulator problem and an automatic guided vehicle trajectory tracking problem, demonstrating its ability for real-time online precise control.

Optimal Control of Discrete-Time Nonlinear Systems

TL;DR

This work addresses real-time solution of discrete-time nonlinear optimal control problems by merging nonlinear optimization with Pontryagin's maximum principle (PMP). It reformulates the original problem into an auxiliary OCP and develops a superlinear-convergence iterative algorithm, supported by explicit gradient and Hessian formulas derived from forward-backward difference equations (FBDEs). The approach is implemented in a model-predictive control setting and validated numerically on a linear quadratic regulator and experimentally on an automatic guided vehicle trajectory-tracking task, demonstrating both accuracy and substantial computational speedups. The results indicate a practical method for online precise control of nonlinear discrete-time systems with real-time requirements, and the framework generalizes to other second-order optimization schemes.

Abstract

This paper focuses on optimal control problem for a class of discrete-time nonlinear systems. In practical applications, computation time is a crucial consideration when solving nonlinear optimal control problems, especially under real-time constraints. While linearization methods are computationally efficient, their inherent low accuracy can compromise control precision and overall performance. To address this challenge, this study proposes a novel approach based on the optimal control method. Firstly, the original optimal control problem is transformed into an equivalent optimization problem, which is resolved using the Pontryagin's maximum principle, and a superlinear convergence algorithm is presented. Furthermore, to improve computation efficiency, explicit formulas for computing both the gradient and hessian matrix of the cost function are proposed. Finally, the effectiveness of the proposed algorithm is validated through simulations and experiments on a linear quadratic regulator problem and an automatic guided vehicle trajectory tracking problem, demonstrating its ability for real-time online precise control.

Paper Structure

This paper contains 14 sections, 2 theorems, 30 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Main Results
  4. Optimal Control Design
  5. Computation of Gradient
  6. Computation of Hessian Matrix
  7. Implementation
  8. Simulation and experimental results
  9. Analytical solution verification: LQR
  10. Application: Trajectory tracking control of AGV
  11. Problem Formulation
  12. Simulation Results
  13. Experimental Results
  14. Conclusion

Key Result

Lemma 1

For the OP (OP1_cost), the gradient of $J$ with respect to $z$ is given by where $x_k$ and $H$ are given by (OCP1_sys) and (H_N), and $\lambda_k$ is given by with the terminal costate value $\lambda_{N+1} = \textbf{0}$.

Figures (4)

  • Figure 1: Comparison of the optimal state and control obtained from Algorithm 1 and the analytical solution under three different initial conditions: $x_0=1.0, 2.0, 3.0$. (a) Optimal state. (b) Optimal control.
  • Figure 2: Comparison of trajectory tracking results and relevant parameters among Algorithm 1, QN, and SQP. (a) The trajectory tracking results of Algorithm 1, LMPC and QN. (b) The computation time for running 10 independent simulations with each of the three algorithms. (c) The number of iterations required for optimization at each sampling time $k$ for the three algorithms.
  • Figure 3: Experimental platform.
  • Figure 4: The trajectory tracking results of AGV in indoor environment. (a) Map of the environment and the desired tracking trajectory. (b) AGV's tracking performance. (c) Trajectory tracking error. (d) Linear velocity $u$ error. (e) Angular velocity $\omega$ error.

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Lemma 1
  • Theorem 1
  • Remark 3