Bang-bang Control (Bang-bang Control)

Author: 天疆说

Contributing Institution: School of Astronautics, Harbin Institute of Technology, National Key Laboratory of Rapid Design and Intelligent Swarm of Small Spacecraft

Reference: 关宇同等. 面向航天器远距离协同交会的超参数自主调优-同伦方法[J]. 航天器环境工程, 2026.

Definition

Bang-bang control is a time-optimal control law characterized by control inputs switching only between two extreme values of the allowable range: maximum thrust and zero thrust, with no intermediate values. The name derives from the "bang" sound when the thrust switches.

Mathematical Description

For the fuel-optimal control problem, according to Pontryagin's Maximum Principle, the optimal thrust ratio satisfies:

uj∗={0,ρj>01,ρj<0∈(0,1),ρj=0u_j^* = \begin{cases} 0, & \rho_j > 0 \\ 1, & \rho_j < 0 \\ \in (0,1), & \rho_j = 0 \end{cases}

Where ρj=1−λmj−Ispg0mj∥λvj∥\rho_j = 1 - \lambda_{mj} - \frac{I_{sp}g_0}{m_j}\|\lambda_{vj}\| is the switching function.

When the thrust ratio is 0 or 1, the control system is in the "off" or "on" extreme state, forming bang-bang control.

Properties

Time Optimality

Bang-bang control is the optimal control law for linear systems with time-optimal properties:

  • Thrust is always maximum or zero
  • No sustained burning at intermediate thrust values
  • Minimization of switching times

Numerical Difficulties

Main numerical difficulties of bang-bang control:

  • Discontinuities on the right-hand side of differential equations
  • Direct numerical integration not possible
  • Precise determination of switching times is difficult

Relationship with Homotopy Methods

Smoothing Effect of Homotopy Methods

Homotopy methods transform bang-bang control into continuous control by introducing a regularized performance index:

J=∑j=12λj0FjIspg0∫t0tf[uj−εuj(1−uj)]dtJ = \sum_{j=1}^{2} \frac{\lambda_{j0}F_j}{I_{sp}g_0} \int_{t_0}^{t_f} \left[u_j - \varepsilon u_j(1-u_j)\right] dt

When ε>0\varepsilon > 0, the optimal control becomes:

uj∗={0,ρj>ε1,ρj<−ε12−ρj2ε,∣ρj∣≤εu_j^* = \begin{cases} 0, & \rho_j > \varepsilon \\ 1, & \rho_j < -\varepsilon \\ \frac{1}{2} - \frac{\rho_j}{2\varepsilon}, & |\rho_j| \leq \varepsilon \end{cases}

The control law is continuously differentiable within the boundary layer.

Transition Strategy

赵海涵等 (2026) used the homotopy parameter sequence:

εd=10−(d/15),d=1,2,⋯ ,60\varepsilon_d = 10^{-(d/15)}, \quad d = 1, 2, \cdots, 60

Gradually transitioning ε\varepsilon from 1 to 0 to obtain fuel-optimal bang-bang control.

Application in Spacecraft Rendezvous

In spacecraft cooperative rendezvous problems:

  • The fuel-optimal control law takes bang-bang form
  • Suitable for finite-thrust propulsion systems
  • Homotopy methods effectively solve its numerical integration difficulties

References

  • 关宇同, 高长生, 胡玉东, 赵海涵. 面向航天器远距离协同交会的超参数自主调优-同伦方法[J]. 航天器环境工程, 2026.
  • Pontryagin L S, et al. The Mathematical Theory of Optimal Processes[M]. Wiley, 1962.
  • Bryson A E, Ho Y C. Applied Optimal Control[M]. Hemisphere, 1975.