eess.SYMar 20, 2026

Mixed-Integer vs. Continuous Model Predictive Control for Binary Thrusters: A Comparative Study

Authors: Franek Stark, Jakob Middelberg, Shubham Vyas

Organizations: Robotics Innovation Center, German Research Center for Artificial Intelligence GmbH, Bremen, Germany · University of Bremen, Germany

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Abstract

Binary on/off thrusters are commonly used for spacecraft attitude and position control during proximity operations. However, their discrete nature poses challenges for conventional continuous control methods. The control of these discrete actuators is either explicitly formulated as a mixed-integer optimization problem or handled in a two-layer approach, where a continuous controller's output is converted to binary commands using analog-to digital modulation techniques such as Delta-Sigma-modulation. This paper provides the first systematic comparison between these two paradigms for binary thruster control, contrasting continuous Model Predictive Control (MPC) with Delta-Sigma modulation against direct Mixed-Integer MPC (MIMPC) approaches. Furthermore, we propose a new variant of MPC for binary actuated systems, which is informed using the state of the Delta-Sigma Modulator. The two variations for the continuous MPC along with the MIMPC are evaluated through extensive simulations using ESA's REACSA platform. Results demonstrate that while all approaches perform similarly in high-thrust regimes, MIMPC achieves superior fuel efficiency in low-thrust conditions. Continuous MPC with modulation shows instabilities at higher thrust levels, while binary informed MPC, which incorporates modulator dynamics, improves robustness and reduces the efficiency gap to the MIMPC. It can be seen from the simulated and real-system experiments that MIMPC offers complete stability and fuel efficiency benefits, particularly for resource-constrained missions, while continuous control methods remain attractive for computationally limited applications.

Figures & tables

θ\theta =Platform orientation (yaw) in world coordinates
θ˙\dot{\theta} =Platform angular velocity in world coordinates
x,yx,y =Platform position in world coordinates
x˙,y˙\dot{x},\dot{y} =Platform velocity in world coordinates
ωRW\omega_{RW} =Reaction Wheel (RW) rotational speed
sθ,cθ\mathrm{s}_{\theta},\mathrm{c}_{\theta} =Sine and cosine of θ\theta
x,u\textbf{x},\textbf{u} =State and input vector of the system
Table 1
Figure 1: Sketch (left) of the free-floating platform REACSA (right)
mmrrIsI_{s}IRWI_{RW}ωRW,max\omega_{RW,\text{max}}FnF_{\text{n}}τmax\tau_{\text{max}}ton,mint_{\text{on,min}}ton, maxt_{\text{on, max}}toff,mint_{\text{off,min}}
202.81 kg202.81\text{\,}\mathrm{kg}0.35 m0.35\text{\,}\mathrm{m}12.22 kg m212.22\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{2}0.047 kg m20.047\text{\,}\mathrm{kg}\text{\,}{\mathrm{m}}^{2}±250\pm 250 RPM10.36 N10.36\text{\,}\mathrm{N}1.44 N m1.44\text{\,}\mathrm{N}\text{\,}\mathrm{m}100 ms100\text{\,}\mathrm{ms}300 ms300\text{\,}\mathrm{ms}200 ms200\text{\,}\mathrm{ms}
Table 2: REACSA ’s physical properties
ControllerDescriptionSolver
MIMPCEnforces binary thruster inputs and timings explicitlySCIP [ 13 ]
continuous MPCContinuous MPC with ΔΣ\Delta\Sigma -modulatorCLP [ 14 ]
binary informed MPCContinuous MPC + information on ΔΣ\Delta\Sigma -modulator firings in OCPCLP [ 14 ]
Table 3: Different MPCs compared in this work
Figure 2: Average position error after reaching and while staying at the target position (left) and time to reach the target position (right) for all compared controllers plotted over the average thrust usage. Single experiments are indicated by small dots, and the Pareto-optimal experiments for all controllers are depicted by solid lines.
Figure 3: Average orientation of all controllers after reaching and while doing station-keeping.
Figure 4: Carrying out experiments at the ORGL - Images show the transition of the platform from the starting point to the origin in reading direction
Figure 5: Resulting trajectory of REACSA under control of the MIMPC and the binary informed MIMPC . The left plot shows the whole trajectory along with the small height variations of the otherwise flat floor, which introduce disturbances. The right plots zoom into the limit cycle at the target.
  MIMPCbinary informed MPC
Reaching the targetTime to reach (s)51.26825.209
Total thrust usage (s)4.907.51
Average thrust usage (%)1.23.7
Limit cycle at the targetAverage position error (cm)1.8240.853
Average orientation error (deg)0.1760.419
Average thrust usage (%)1.46.0
Table 4: Results from experiments on the real system
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