Nonlinear Thermal Modeling and Predictive Control for Spacecraft Optical Temperature Regulation

• Sub-Kelvin control of optical-payload temperature is needed to maintain wavefront sensing and image quality and prevent thermally induced expansion across the optics.The paper considers a 12U CubeSat in a 500 km Noon-Midnight orbit.Source: §I · p. 1
• PID-controlled heaters respond only after disturbances occur, limiting tracking of time-varying orbital heating because PID lacks an explicit model of future disturbances.Source: §I · p. 1
• MPC uses a dynamical model to optimize inputs over a horizon, directly accommodating user-defined objectives, input constraints, state estimation, and multiple heaters.Source: §I · p. 2
• Because recursive MPC needs the full thermal state but measurements are partial and noisy, the paper uses a UKF to estimate that state.Source: §I · p. 2
• The proposed framework combines a semi-implicit nonlinear thermal model with UKF state estimation and MPC for closed-loop optical temperature regulation from partial, noisy measurements.Longer prediction intervals preserve nonlinear radiation dynamics while providing additional time for online optimization.Source: §I · p. 2

• The model assumes spacecraft components are aluminum and that conduction and radiation are the only heat-transfer modes in orbit.Source: §II · p. 3
• A control-volume energy balance relates stored energy to energy entering, leaving, and generated, with internal generation serving as the control variable.Source: §II · p. 3
• The authors discretize conduction and radiation energy flows to form an explicit finite-difference model, then convert it to a semi-implicit representation and validate it against Thermal Desktop.Source: §II · p. 3

• The optical system is divided into equal-dimension control volumes, each represented by a central node carrying its thermal properties.The mesh depicts control volumes as red lines and nodes as black dots.Source: A. Energy Storage and Conduction · p. 3, Figure 1 · p. 3
• Although aluminum properties are assumed in the model description, the authors state that their approach is valid for any material choice.Source: A. Energy Storage and Conduction · p. 3
• Conductive energy flow into a node is summed from its six neighboring directions, with spatial derivatives approximated by finite differences between nodes.Source: A. Energy Storage and Conduction · p. 4
• The time derivative of stored energy is discretized; positive storage indicates heating and negative storage indicates cooling.Source: A. Energy Storage and Conduction · p. 4

• The radiation model assumes diffuse, gray, uniform radiation and is considered valid when wavelength-dependent surface properties remain relatively constant, as expected near 305 K.Optical surfaces follow the conduction discretization, while the spacecraft structure is modeled as a thin shell with one surface and node.Source: B. Energy Flow due to Radiation · p. 4
• Radiosity equations represent net radiation transfer among the n optical surfaces and one structural surface, with view factors describing the fraction emitted by one surface that reaches another.Source: B. Energy Flow due to Radiation · p. 4
• The structural surface lacks a known temperature or flux boundary condition because it depends on the external radiation environment; orbital absorption includes solar, Earth IR, and albedo radiation.Source: B. Energy Flow due to Radiation · p. 4
• Net radiative loss depends nonlinearly on the fourth powers of surface temperatures and is combined with conduction to complete the thermal model.Source: B. Energy Flow due to Radiation · p. 5

C. Total Net Energy Flow: Storage, Conduction, and Radiation Combined

• The thin-shell spacecraft structure exchanges radiation with interior surfaces and deep space while absorbing orbit-dependent solar and Earth radiation.The shell reflects and emits interior radiation, emits to deep space, and absorbs external radiation.Source: Figure 3 · p. 6
• The node energy balance combines conductive and radiative transfer to define net energy entering each control volume.The net flow includes contributions from the six neighboring directions and radiation.Source: C. Total Net Energy Flow: Storage,… · pp. 5–6
• Across the optical-system nodes, temperature evolves through conduction, radiation, and internal heat generation, which serves as the control input.The model represents node temperatures and heat generation as vectors, with a conduction coefficient matrix.Source: C. Total Net Energy Flow: Storage,… · p. 6
• For the explicit model, increasing node count requires a smaller time step for stability, increasing MPC problem size and limiting predictive capability.The following semi-implicit method is introduced to maintain longer prediction horizons while keeping optimization tractable.Source: C. Total Net Energy Flow: Storage,… · pp. 6–7

D. Semi-Implicit Method

• The semi-implicit relation is formed by combining the explicit relation with a conduction relation evaluated at time t+1, following the Crank-Nicolson approach.The method is inspired by averaging explicit and implicit relations.Source: D. Semi-Implicit Method · p. 7
• Future temperature depends on current and future conduction values, together with current radiation and internal heating.This dependence is what makes the resulting thermal model semi-implicit.Source: D. Semi-Implicit Method · p. 7
• The model maintains numerical stability for larger time steps, reducing the steps needed for a given horizon and the online optimization burden relative to the explicit model.A longer prediction horizon is identified as important for MPC performance.Source: D. Semi-Implicit Method · p. 7

E. Thermal Model Results and Validation

• The validation uses a 12U CubeSat optical payload in a 500 km noon-midnight orbit, with its z-axis pointing toward Earth and specified orbital parameters.The orbit has inclination 78.8°, eccentricity 0, and a 5677 s period.Source: E. Thermal Model Results and Validation · p. 7, Figure 4 · p. 7
• For the free-response comparison, both models use a single node for the optical system, zero internal heat generation, and an initial temperature of 300 K.The single node represents the average temperature of the optics; increasing node count improves thermal resolution without significantly changing that average.Source: E. Thermal Model Results and Validation · pp. 7–8
• The predicted free-response temperature shows strong agreement with Thermal Desktop, with oscillations attributed to Earth eclipses.Fig. 5 compares the Eq. (15) model with Thermal Desktop for an unactuated optical system.Source: Figure 5 · p. 8, E. Thermal Model Results and Validation · p. 8

III. Model Predictive Control & Kalman Filtering

• The control objective is to use the spacecraft thermal model to choose heating sequences that maintain the optical system at its desired temperature.The model predicts future temperatures so the controller can account for the effects of candidate heating inputs.Source: §III · p. 8
• MPC predicts future temperatures with the thermal model, minimizes a quadratic cost over a horizon, and applies only the first optimized heating input before replanning.The cost accounts for state, input, and input-rate penalties, with component-wise bounds on heating inputs.Source: §III · pp. 8–9
• The optimization selects bounded internal heating inputs at each node to regulate predicted temperatures around the reference while penalizing changes in input.Nodes that are not allowed to heat can be assigned zero lower and upper input bounds.Source: §III · p. 9
• The UKF estimates the full temperature state from partial, noisy measurements, using sigma points to handle nonlinear dynamics without Jacobian-based linearization.The filter propagates sigma points through the thermal model, then uses a standard Kalman measurement update because the observation model is linear.Source: §III · pp. 9–11
• The controller-estimator design uses certainty equivalence: it initializes deterministic MPC with the UKF estimate and does not propagate state covariance through prediction.The authors omit estimator-controller coupling to avoid the additional complexity of optimizing expected cost.Source: §III · pp. 11–12

IV. Results

• The evaluation uses a 500 km noon-midnight orbit chosen to test response to entering or leaving Earth’s shadow and the sub-Kelvin temperature requirement for wavefront stability.The orbit period is 5677 s and the inclination is 78.8°.Source: §IV · p. 12
• Starting at 300 K with a 305 K target, the 64-node MPC computes heating for eight control nodes over a receding horizon.The figure shows true temperatures and inputs alongside orbital heating, with a different color for each input.Source: Figure 7 · p. 13
• Absorbed radiation drops when the CubeSat enters Earth’s shadow because solar radiation is blocked, but Albedo and Earth infrared keep heating above zero.These results assume an aluminum CubeSat with surface emissivity 0.5; the approach is stated to be valid irrespective of material.Source: §IV · p. 13

A. Sixty-Four Node In-Orbit Optical System Control and Simulation

• The simulation models 64 optical-system nodes and a sixty-fifth node for the CubeSat shell, with eight corner control nodes used for sensing and heating.Each control node has an attached heater; internal nodes have no exposed surface, while non-control nodes have exposed surfaces but no heater.Source: A. Sixty-Four Node In-Orbit Optical… · p. 13
• Lower Q weights worsen tracking, lower R makes inputs less aggressive, and lower RΔ allows faster-changing inputs that may overreact to measurements.The temperature response depends on the weighting matrices in the MPC cost function.Source: A. Sixty-Four Node In-Orbit Optical… · p. 14
• MPC coordinates multiple heaters, enabling thermal-gradient control that is difficult with classical methods.The simulated system has eight control nodes, eight interior nodes, and 48 non-control nodes; interior-node temperatures lag control-node temperatures by one time step.Source: A. Sixty-Four Node In-Orbit Optical… · pp. 13–14
• The UKF filters noisy partial measurements to estimate the full thermal state used by MPC to select inputs for the true system.Without the UKF, the controller would have only noisy partial-state measurements and would be more susceptible to measurement noise.Source: A. Sixty-Four Node In-Orbit Optical… · pp. 14–15, Figure 8 · p. 14
• MPC inputs shift 90° relative to orbital heating because the controller anticipates disturbances, whereas PI turns on only when temperature falls below the reference.The PI comparison uses node one for sensing and control; MPC's predictive inputs are applied in advance of orbital heating.Source: A. Sixty-Four Node In-Orbit Optical… · pp. 15–16
• 88% reduction in temperature RMSE and 5% increase in power per orbit from PID to MPC quantify the tracking–power trade-off.The comparison uses six orbit periods after the initial transients have settled.Source: Table 2 · p. 16, A. Sixty-Four Node In-Orbit Optical… · p. 16
• Using MPC with temperature estimates for all nodes, the 64-node optical system has a maximum absolute temperature deviation of 0.40 K from 305 K after initial transients, meeting the sub-Kelvin thermal requirement.The mean error across the payload is 0.025 K, and the standard deviation of the spatially averaged temperature response is 0.079 K.Source: A. Sixty-Four Node In-Orbit Optical… · p. 17

V. Conclusion

• MPC achieves a 88% reduction in tracking error and meets the sub-Kelvin thermal requirement at the cost of slightly increased power per orbit versus PID.The framework uses a nonlinear thermal model and UKF-estimated full state; the model was validated against industry-standard thermal software.Source: §V · p. 17
• The demonstrated framework combines a nonlinear thermal model, UKF state estimation, and MPC on a 64-node CubeSat optical system in a realistic orbital environment.The MPC senses the current thermal state, predicts future temperatures, and determines heating input at each time step.Source: §V · p. 17
• The model represents a Noon-Midnight-orbit CubeSat carrying an optical payload and is presented as representative of spacecraft with precision optics.Source: §V · p. 17