Mixed Integer vs. Continuous Model Predictive Controllers for Binary Thruster Control: A Comparative Study

1. Problem Statement

Spacecraft and satellites often rely on binary (on/off) thrusters for attitude and position control during proximity operations. These actuators are preferred for their reliability and cost but pose significant challenges for control systems because they cannot provide continuous thrust.

• The Core Challenge: Conventional control methods assume smooth, continuous inputs. Binary thrusters introduce discrete constraints, including minimum activation times ($t_{on, min}$), maximum firing times ($t_{on, max}$), and cool-down periods ($t_{off, min}$).
• Current Approaches:
  1. Two-Layer Architecture: A continuous controller generates ideal inputs, which are converted to binary commands via modulation techniques (e.g., Delta-Sigma modulation). This is computationally efficient but may ignore physical constraints, leading to suboptimal performance or instability.
  2. Mixed-Integer Optimization: Directly formulating the control problem as a Mixed-Integer Program (MIP) to handle binary constraints explicitly. While theoretically superior, it has historically been considered too computationally expensive for real-time applications, though recent advances have made this feasible.
• Gap: There is a lack of systematic comparison between these two paradigms (Continuous MPC + Modulation vs. Direct Mixed-Integer MPC) regarding fuel efficiency, stability, and computational load.

2. Methodology

The authors evaluated three distinct control strategies using ESA's REACSA (REcap-ACrobat-SAtsim) platform, a 200 kg air-bearing free-floating simulator equipped with eight cold-gas thrusters and a reaction wheel.

The Three Controllers Compared:

1. Mixed-Integer MPC (MIMPC):

   • Approach: Explicitly formulates the Optimal Control Problem (OCP) as a Mixed-Integer Program (MIP).
   • Constraints: Directly enforces binary inputs ($u \in \{0, 1\}$) and timing constraints (dwell times) within the optimization.
   • Solver: SCIP (Mixed-Integer solver).
   • Frequency: ~10 Hz (due to NP-hard complexity).

2. Continuous MPC with $\Delta\Sigma$-Modulation:

   • Approach: Solves a standard Linear Program (LP) with continuous inputs. A separate $\Delta\Sigma$-modulator converts these continuous values into binary on/off commands based on error integration.
   • Constraints: The OCP ignores binary/timing constraints; the modulator handles them post-hoc.
   • Solver: CLP (Linear solver).
   • Frequency: >100 Hz.

3. Binary-Informed MPC (New Contribution):

   • Approach: A hybrid continuous MPC that incorporates the predicted state of the $\Delta\Sigma$-modulator into the OCP's prediction horizon.
   • Mechanism: At each step, the controller simulates the modulator's future behavior forward in time. These predicted binary firings are treated as fixed disturbances in the system dynamics model ($x_{t+1} = f(x_t, u_t, u_{t,bin})$).
   • Goal: To allow the continuous optimizer to "anticipate" the limitations of the binary actuators without solving a full MIP.
   • Solver: CLP.
   • Frequency: >100 Hz.

Experimental Setup:

• Simulation: 5,400 experiments with randomized weighting matrices to test robustness across different thrust regimes (low vs. high).
• Hardware: Real-world experiments on the REACSA platform at the European Space Agency's (ESA) Orbital Robotics & GNC Lab (ORGL).
• Metrics: Time to reach target, average position/orientation error, and average thruster usage (fuel efficiency).

3. Key Contributions

1. Systematic Comparison: First comprehensive study contrasting direct MIMPC against continuous MPC with modulation for binary thruster control.
2. Novel Controller Design: Introduction of the Binary-Informed MPC, which bridges the gap between continuous and discrete control by embedding modulator dynamics into the continuous optimization horizon.
3. Empirical Validation: Extensive validation using both high-fidelity simulations and real-world hardware experiments on a free-floating platform.
4. Open Source Release: Public release of datasets and controller implementations to foster further research.

4. Results

Simulation Results:

• High-Thrust Regime: All three controllers perform similarly in reaching the target. However, the Continuous MPC shows signs of instability and constraint violations at very high thrust levels.
• Low-Thrust Regime (Fuel Efficiency):
  • MIMPC is superior, achieving the lowest fuel consumption (down to 0.04% duty cycle) while maintaining stability.
  • Continuous MPC struggles significantly, often failing to reach the target or requiring much higher thrust usage (up to 8%) to stabilize.
  • Binary-Informed MPC significantly outperforms the standard Continuous MPC, reducing the efficiency gap with MIMPC. It achieves stability at low thrust levels where the standard continuous approach fails.
• Station Keeping: MIMPC maintains the tightest limit cycles with the least fuel. The Binary-Informed MPC offers a small position error (0.85 cm) but uses more fuel than MIMPC.

Real-System Results (REACSA):

• Performance:
  • Binary-Informed MPC: Reached the target in 25.2 s (2x faster than MIMPC) but used 3.7% thrust. It achieved a smaller position error (0.85 cm) but a larger orientation error (0.42°).
  • MIMPC: Took 51.3 s to reach the target but used only 1.2% thrust. It achieved a tighter orientation control (0.18°) but a larger position error (1.82 cm).
• Stability: The Continuous MPC was not tested on hardware due to observed instability in simulations. Both MIMPC and Binary-Informed MPC were stable.

5. Significance and Conclusion

The paper concludes that the choice of controller depends heavily on mission constraints:

• For Resource-Constrained Missions (Fuel Efficiency): MIMPC is the optimal choice. Despite higher computational cost, it provides the best fuel efficiency and robustness in low-thrust regimes, which is critical for long-duration missions.
• For Computationally Limited Applications: The Binary-Informed MPC is a superior alternative to standard Continuous MPC. It offers near-MIMPC robustness and stability with the computational efficiency of a Linear Program (LP). It eliminates the instability risks of pure continuous control while avoiding the NP-hard complexity of MIP.
• Trade-offs: While MIMPC maximizes fuel efficiency, the Binary-Informed MPC offers a practical balance, achieving faster convergence and better position accuracy in real-world scenarios where perfect optimization is less critical than rapid response and stability.

Future Work: The authors suggest exploring the Binary-Informed MPC further with quadratic cost functions, path-following, and obstacle avoidance, as the continuous formulation allows for more complex constraints than the computationally saturated MIMPC.