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Lattice piecewise affine approximation of explicit model predictive control with application to satellite attitude control

Zhengqi Xu, Jun Xu, Ai-Guo Wu, Shuning Wang

Abstract

Satellite attitude cotrol is a crucial part of aerospace technology, and model predictive control(MPC) is one of the most promising controllers in this area, which will be less effective if real-time online optimization can not be achieved. Explicit MPC converts the online calculation into a table lookup process, however the solution is difficult to obtain if the system dimension is high or the constraints are complex. The lattice piecewise affine(PWA) function was used to represent the control law of explicit MPC, although the online calculation complexity is reduced, the offline calculation is still prohibitive for complex problems. In this paper, we use the sample points in the feasible region with their corresponding affine functions to construct the lattice PWA approximation of the optimal MPC controller designed for satellite attitude control. The asymptotic stability of satellite attitude control system under lattice PWA approximation has been proven, and simulations are executed to verify that the proposed method can achieve almost the same performance as linear online MPC with much lower online computational complexity and use less fuel than LQR method.

Lattice piecewise affine approximation of explicit model predictive control with application to satellite attitude control

Abstract

Satellite attitude cotrol is a crucial part of aerospace technology, and model predictive control(MPC) is one of the most promising controllers in this area, which will be less effective if real-time online optimization can not be achieved. Explicit MPC converts the online calculation into a table lookup process, however the solution is difficult to obtain if the system dimension is high or the constraints are complex. The lattice piecewise affine(PWA) function was used to represent the control law of explicit MPC, although the online calculation complexity is reduced, the offline calculation is still prohibitive for complex problems. In this paper, we use the sample points in the feasible region with their corresponding affine functions to construct the lattice PWA approximation of the optimal MPC controller designed for satellite attitude control. The asymptotic stability of satellite attitude control system under lattice PWA approximation has been proven, and simulations are executed to verify that the proposed method can achieve almost the same performance as linear online MPC with much lower online computational complexity and use less fuel than LQR method.
Paper Structure (11 sections, 7 theorems, 58 equations, 8 figures, 4 tables, 1 algorithm)

This paper contains 11 sections, 7 theorems, 58 equations, 8 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation of satellite attitude control
  3. Dynamics of satellite attitude
  4. Linear MPC problem for satellite attitude control
  5. Lattice PWA approximation of explicit MPC
  6. Explicit MPC
  7. Lattice PWA function
  8. Lattice PWA approximation of explicit control law
  9. Stability analysis
  10. Simulation
  11. Conclusion

Key Result

Lemma III.1

Consider the mpQP problem (youhua2) and l et $H\succ0$. Then the set of feasible parameters $\Omega\subset\mathbb{X}$ is convex, the optimizer $U^*(x):\Omega\mapsto\mathbb{R}^{N_c\cdot m}$ is continuous PWA, and the optimal solution $J^*(x):\Omega\mapsto\mathbb{R}$ is continuous, convex and piecewis

Figures (8)

  • Figure 1: Satellite attitude control diagram
  • Figure 2: One-dimensional PWA function
  • Figure 3: Control law
  • Figure 4: Simulation curves
  • Figure 5: Satellite angular velocity curve
  • ...and 3 more figures

Theorems & Definitions (12)

  • Lemma III.1: bemporad2002explicit
  • Definition III.2
  • Lemma III.3: wilkinson1963method
  • Lemma III.4: xu2021error
  • Definition III.5: ellis2014tutorial
  • Lemma III.6: mayne2000survey
  • Lemma III.7
  • proof
  • Lemma III.8
  • proof
  • ...and 2 more