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Finite-Horizon Discrete-Time Optimal Control for Nonlinear Systems under State and Control Constraints

Chuanzhi Lv, Hongdan Li, Huanshui Zhang

TL;DR

This work tackles finite-horizon discrete-time nonlinear optimal control under state and control constraints by using an augmented Lagrangian method to transform the constrained problem into unconstrained subproblems solved through forward-backward difference equations. A Newton-like, second-order optimization scheme yields a superlinear convergence rate by computing the gradient via the standard Hamiltonian and crafting a novel second-order Hamiltonian to obtain the Hessian through FBDEs. The approach eliminates reliance on external optimizers and shows strong computational efficiency, demonstrated on an AGV trajectory-tracking problem with obstacle avoidance under model-predictive control. The combination of ALM constraint transcription, FBDE-based derivatives, and MPC makes the framework practical for real-time constrained optimization in robotics and related domains.

Abstract

This paper addresses the optimal control problem of finite-horizon discrete-time nonlinear systems under state and control constraints. A novel numerical algorithm based on optimal control theory is proposed to achieve superior computational efficiency, with the novelty lying in establishing a unified framework that integrates all aspects of algorithm design through the solution of forward and backward difference equations (FBDEs). Firstly, the state and control constraints are transformed using an augmented Lagrangian method (ALM), thereby decomposing the original optimal control problem into several optimization subproblems. These subproblems are then reformulated as new optimal control problem, which are solved through the corresponding FBDEs, resulting in an algorithm with superlinear convergence rate. Furthermore, the gradient and Hessian matrix are computed by iteratively solving FBDEs, thereby accelerating the optimization process. The gradient is obtained through the standard Hamiltonian, while the Hessian matrix is derived by constructing a novel Hamiltonian specifically designed for second-order optimization, transforming each row into an iterative solution of a new set of FBDEs. Finally, the effectiveness of the algorithm is validated through simulation results in automatic guided vehicles (AGV) trajectory tracking control.

Finite-Horizon Discrete-Time Optimal Control for Nonlinear Systems under State and Control Constraints

TL;DR

This work tackles finite-horizon discrete-time nonlinear optimal control under state and control constraints by using an augmented Lagrangian method to transform the constrained problem into unconstrained subproblems solved through forward-backward difference equations. A Newton-like, second-order optimization scheme yields a superlinear convergence rate by computing the gradient via the standard Hamiltonian and crafting a novel second-order Hamiltonian to obtain the Hessian through FBDEs. The approach eliminates reliance on external optimizers and shows strong computational efficiency, demonstrated on an AGV trajectory-tracking problem with obstacle avoidance under model-predictive control. The combination of ALM constraint transcription, FBDE-based derivatives, and MPC makes the framework practical for real-time constrained optimization in robotics and related domains.

Abstract

This paper addresses the optimal control problem of finite-horizon discrete-time nonlinear systems under state and control constraints. A novel numerical algorithm based on optimal control theory is proposed to achieve superior computational efficiency, with the novelty lying in establishing a unified framework that integrates all aspects of algorithm design through the solution of forward and backward difference equations (FBDEs). Firstly, the state and control constraints are transformed using an augmented Lagrangian method (ALM), thereby decomposing the original optimal control problem into several optimization subproblems. These subproblems are then reformulated as new optimal control problem, which are solved through the corresponding FBDEs, resulting in an algorithm with superlinear convergence rate. Furthermore, the gradient and Hessian matrix are computed by iteratively solving FBDEs, thereby accelerating the optimization process. The gradient is obtained through the standard Hamiltonian, while the Hessian matrix is derived by constructing a novel Hamiltonian specifically designed for second-order optimization, transforming each row into an iterative solution of a new set of FBDEs. Finally, the effectiveness of the algorithm is validated through simulation results in automatic guided vehicles (AGV) trajectory tracking control.

Paper Structure

This paper contains 11 sections, 2 theorems, 32 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Numerical Algorithm based on Optimal Control Method
  4. Constraints Transcription
  5. Optimization Algorithm Design
  6. Computation of Gradient and Hessian Matrix
  7. Computational Implementation
  8. Trajectory Tracking Control for AGV
  9. Description of AGV system
  10. Simulation Result
  11. Conclusion

Key Result

Theorem 1

For the optimization problem (3a-2), the gradient of $\widetilde{J}(x_0,u_k,\gamma_{i,k},\sigma)$ with respect to $u_k$ is given by where Hamiltonian $\mathcal{H}(k)$, for $k=0,1,...,N$, is given by Moreover, the costate $\lambda_k$, for $k=1,...,N+1$, satisfies with the terminal value $\lambda_{N+1} = \boldsymbol{0}$.

Figures (2)

  • Figure 1: Model schematic.
  • Figure 2: Control performance of Algorithm 1 for the trajectory tracking problem (\ref{['4a-2']})-(\ref{['4a-3']}). (a) Tracking results. (b) Orientation $\omega_k$. (c) Linear velocity $v_k$. (d) Angular velocity $\omega_k$.

Theorems & Definitions (5)

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