Optimal Control

Author: Tianjiang Shuo
Site: https://cislunarspace.cn

Definition

Optimal control theory is a central branch of modern control theory that addresses the problem of selecting control laws to extremize (minimize or maximize) a prescribed performance index for a given dynamical system. In space mission design, the performance index typically represents fuel consumption, time, or energy.

Basic Elements

An optimal control problem is defined by:

State equation: $\dot{\mathbf{x}} = f(\mathbf{x}, \mathbf{u}, t)$ , describing system dynamics
Control variable: $\mathbf{u}(t)$ , designed by the controller
Boundary conditions: initial state $\mathbf{x}(t_0)$ and terminal state $\mathbf{x}(t_f)$
Performance index: $J = \phi(\mathbf{x}(t_f), t_f) + \int_{t_0}^{t_f} L(\mathbf{x}, \mathbf{u}, t) dt$
Constraints: control constraints $|\mathbf{u}| \leq u_{\max}$ , state constraints $\mathbf{x} \in \mathcal{X}$

Principles

Variational Calculus and the Euler-Lagrange Equations

For unconstrained optimal control, necessary conditions are derived via calculus of variations. Introducing Lagrange multipliers $\boldsymbol{\lambda}(t)$ , the Hamiltonian is constructed:

H(\mathbf{x}, \mathbf{u}, \boldsymbol{\lambda}, t) = L(\mathbf{x}, \mathbf{u}, t) + \boldsymbol{\lambda}^T f(\mathbf{x}, \mathbf{u}, t)

The Euler-Lagrange equations give the state and costate evolution:

\dot{\mathbf{x}} = \frac{\partial H}{\partial \boldsymbol{\lambda}}, \quad \dot{\boldsymbol{\lambda}} = -\frac{\partial H}{\partial \mathbf{x}}

Pontryagin Maximum Principle

For optimal control problems with control constraints, the Maximum Principle gives the optimality condition for the control variable:

\mathbf{u}^*(t) = \arg\max_{\mathbf{u} \in \mathcal{U}} H(\mathbf{x}^*, \mathbf{u}, \boldsymbol{\lambda}^*, t)

This principle reduces the continuous optimization problem to selecting the optimal control at each instant.

Applications in Cislunar Space

Minimum-fuel orbital transfer: using the Pontryagin Maximum Principle to derive optimal low-thrust transfer trajectories in cislunar space, producing fuel-optimal delta-V trajectories
Low-thrust trajectory optimization: trajectory design for low-thrust propulsion (ion thrusters, electric propulsion)：本质上是最优控制问题，常用间接法（极大值原理）或直接法（伪谱法）求解
Soft landing guidance: fuel-optimal descent trajectory design for lunar/Mars landing with thrust magnitude, thrust direction, terminal altitude, and velocity constraints
Attitude maneuver optimization: multi-objective time-fuel optimization for large-angle spacecraft attitude reorientation

Shooting Method

References

Bryson A E, Ho Y C. Applied optimal control[M]. Taylor & Francis, 1975.
Betts J T. Survey of numerical methods for trajectory optimization[J]. Journal of Guidance, Control, and Dynamics, 1998.