Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Pulse Width Modulation Method Applied to Nonlinear Model Predictive Control on an Under-actuated Small Satellite

Kota Kondo, Yasuhiro Yoshimura, Shiji Nagasaki, Toshiya Hanada

TL;DR

The paper addresses detumbling and three-axis attitude control of under-actuated small satellites using magnetic torquers. It introduces a nonlinear model predictive control (NMPC) framework solved online with the GMRES method and then discretizes the NMPC inputs with pulse width modulation (PWM) to ease actuator requirements, while enforcing constraints via auxiliary inputs. A controllability analysis shows the instantaneous magnetic control matrix $\psi_3(\boldsymbol{b}(t))$ has rank 2, yet its average $\bar{\psi}_3$ is positive definite, yielding average controllability for the system. Simulations on a sun-synchronous orbit demonstrate detumbling within about 100 minutes and attitude maneuver within about 50 minutes, with GMRES error bounded by $||F|| \lesssim 7\times 10^{-3}$; PWM reduces actuator burden and expands applicability, though some dynamics remain challenging under discretization.

Abstract

Among various satellite actuators, magnetic torquers have been widely equipped for stabilization and attitude control of small satellites. Although magnetorquers are generally used with other actuators, such as momentum wheels, this paper explores a control method where only a magnetic actuation is available. We applied a nonlinear optimal control method, Nonlinear Model Predictive Control (NMPC), to small satellites, employing the generalized minimal residual (GMRES) method, which generates continuous control inputs. Onboard magnetic actuation systems often find it challenging to produce smooth magnetic moments as a control input; hence, we employ the Pulse Width Modulation (PWM) method, which discretizes a control input and reduces the burden on actuators. In our case, the PWM approach discretizes control torques generated by the NMPC scheme. This study's main contributions are investigating the NMPC and the GMRES method applied to small spacecraft and presenting the PWM control system's feasibility.

Pulse Width Modulation Method Applied to Nonlinear Model Predictive Control on an Under-actuated Small Satellite

TL;DR

The paper addresses detumbling and three-axis attitude control of under-actuated small satellites using magnetic torquers. It introduces a nonlinear model predictive control (NMPC) framework solved online with the GMRES method and then discretizes the NMPC inputs with pulse width modulation (PWM) to ease actuator requirements, while enforcing constraints via auxiliary inputs. A controllability analysis shows the instantaneous magnetic control matrix has rank 2, yet its average is positive definite, yielding average controllability for the system. Simulations on a sun-synchronous orbit demonstrate detumbling within about 100 minutes and attitude maneuver within about 50 minutes, with GMRES error bounded by ; PWM reduces actuator burden and expands applicability, though some dynamics remain challenging under discretization.

Abstract

Among various satellite actuators, magnetic torquers have been widely equipped for stabilization and attitude control of small satellites. Although magnetorquers are generally used with other actuators, such as momentum wheels, this paper explores a control method where only a magnetic actuation is available. We applied a nonlinear optimal control method, Nonlinear Model Predictive Control (NMPC), to small satellites, employing the generalized minimal residual (GMRES) method, which generates continuous control inputs. Onboard magnetic actuation systems often find it challenging to produce smooth magnetic moments as a control input; hence, we employ the Pulse Width Modulation (PWM) method, which discretizes a control input and reduces the burden on actuators. In our case, the PWM approach discretizes control torques generated by the NMPC scheme. This study's main contributions are investigating the NMPC and the GMRES method applied to small spacecraft and presenting the PWM control system's feasibility.
Paper Structure (15 sections, 1 theorem, 36 equations, 10 figures, 8 tables)

This paper contains 15 sections, 1 theorem, 36 equations, 10 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Rotational Kinematics and Dynamics
  3. Control Law
  4. Controllability Analysis
  5. A three-axis magnetic actuator system
  6. NMPC Formulation
  7. NMPC with GMRES Algorithm
  8. PWM
  9. Simulation Results
  10. Orbital Elements
  11. Earth's Magnetic Field Model
  12. PWM Detumbling
  13. PWM Attitude Control
  14. Attitude Control with Continuous Input
  15. Conclusions

Key Result

Lemma 1

Define $\boldsymbol{b_0}(t)$ as Earth's magnetic field unit vector with respect to an orbital frame. Then, assuming $S(\boldsymbol{b_0}(t))\boldsymbol{\dot{b}}_0(t) \neq 0$ for all $t>0$, it follows for all $\tau>0$ that Furthermore, where $\Bar{\psi}_{03}$ is the average magnetic control matrix in the orbital frame.

Figures (10)

  • Figure 1: Magnetic field in dipole and IGRF on the sun-synchronous orbit.
  • Figure 2: Time history of angular velocities on NMPC on sun-synchronous orbit.
  • Figure 3: Time history of continuous and discrete control inputs
  • Figure 4: The GMRES method error ||F||
  • Figure 5: Time history of quaternion on sun-synchronous orbit.
  • ...and 5 more figures

Theorems & Definitions (1)

  • Lemma 1