Physics-Informed and Data-Driven Composite Disturbance Separation and Predictive Control for Airborne Electro-Optical Gimbals with Multiple Constraints

1. Introduction

• Periodic and aperiodic disturbances challenge accurate pointing, while prior observer approaches often use aggregate estimates or require precise disturbance models.The paper motivates separating vibration-driven periodic effects from state-dependent disturbances.Source: §1 · pp. 1–2
• MPC-DS predicts periodic vibration across the horizon and compensates for the aperiodic estimate outside the optimizer, with tightened input bounds and delay-lead correction.Its periodic observer uses projected linear-regression frequency estimation; a bounded LSTM-based observer predicts delay-induced aperiodic change.Source: §1 · p. 2

2. Deep-Coupled Dynamic Modeling

• The coupled Newton–Euler model characterizes carrier motion, two-axis kinematics, and disturbance torques acting on the azimuth and elevation channels.The modeled disturbance sources include gyroscopic, gravitational, cross-axis coupling, and LuGre friction torques.Source: §2 · p. 2

2.1. Kinematics and Dynamics Modeling

• The two-axis ISP uses an outer azimuth gimbal and inner elevation gimbal, with joint-axis dynamics expressed in manipulator and state-space forms.The state comprises each axis angle and angular velocity; the speed-command implementation uses an identified input-equivalent model.Source: §2.1 · p. 2

2.2. Disturbance Modeling and Analysis

• The disturbance model separates multi-harmonic airframe vibration from aperiodic gravity, gyroscopic, friction, and residual effects.The retained periodic component uses a common fundamental and integer harmonics; the residual includes state-dependent terms and unretained vibration.Source: §2.2 · p. 3

2.3. Constraint Analysis

• The gimbal constraints combine motor torque saturation, mechanical limits on angles and rates, and a camera field-of-view requirement.The field-of-view constraint keeps the target within the imaging frame.Source: §2.3 · p. 3

3.1. APDO Design

• The APDO combines projected online frequency estimation with multi-harmonic prediction to forecast periodic disturbances over the MPC horizon.Projection keeps the frequency estimate within the admissible band; amplitude projection bounds harmonic predictions, including across frequency or coefficient switches.Source: §3.1 · pp. 4–5
• Drift weighting reduces the prediction-error bound when its weight is below one and the unweighted bound exceeds the harmonic amplitude bound, but does not certify phase error or guarantee lower actual error.The weighting attenuates far-horizon predictions during frequency-estimation transients; its performance effect is evaluated in Section 4.Source: §3.1 · pp. 5–6

3.2. LSTM-DO Design

• The LSTM-DO learns mixed-rate aperiodic disturbances, while a linear observer corrects online estimation errors; projections bound the learned signal and its rate.The observer-error bound holds for any frozen network satisfying the stated bounds; training can tighten its increment-dependent term by reducing the learned signal's rate bound.Source: §3.2 · pp. 6–7
• A model-based increment estimate and a frozen LSTM residual predictor form a bounded lead correction for aperiodic disturbance changes during input delay.Without lead correction, the stated error bound is ε_D ≤ n_d T_s δ_r; the correction does not alter the observer recursion or MPC prediction model.Source: §3.2 · p. 7

3.3. Model Predictive Control (MPC)

• The MPC embeds periodic predictions in its nominal model, compensates the aperiodic estimate outside that model, and tightens the input interval to reserve actuator margin.Soft predicted-state limits penalize violations but do not guarantee hard mechanical-range or optical-field-of-view satisfaction.Source: §3.3 · pp. 7–8
• Only periodic predictions enter the MPC optimizer’s affine law; when it applies, aperiodic compensation acts on the plant as an additive residual.Active bounds or state penalties can modify this mapping, which motivates the explicit margin conditions.Source: §3.3 · p. 8

3.4. Stability Analysis

• The envelope bounds and nominal-feedback Schur property are assumptions or checks, not consequences of soft penalties.The selected nominal feedbacks pass the Schur check, but those checks do not certify trajectory-dependent flight margins.Source: §3.4 · pp. 8–9
• Under Assumption 3, the QP is feasible at every step on the stated envelope, with bounded first decisions and compensated commands.For the compensated realization, the command bound also requires the stated H-headroom envelope; outside it, nonemptiness must be checked online.Source: §3.4 · p. 8
• Under Assumptions 1–4 and observer-gain conditions, tracking deviation from the ideal-information loop has a regional practical bound when the complete input sequence and predicted states retain margins.The result applies while the deviation tube remains inside the operating envelope; it does not infer stability outside this region or under active-constraint motion.Source: §3.4 · p. 9

4.1. Simulation Setup

• Simulations used a calibrated yaw-axis plant with measured non-idealities and seven controller variants across constant- and changing-frequency disturbance scenarios.Evaluations used 40 runs per method and scenario; the calibrated plant added a 22 ms lag, readout noise, and a rate-readout scale mismatch.Source: §4.1 · pp. 9–10

4.2. Tracking under Periodic Disturbances

• On the calibrated plant, MPC-DS reduced RMSE relative to MPC-DO by 58.1% at constant frequency and 39.1% after a frequency switch, with no saturated samples or high-error runs.On the nominal plant, adaptive-frequency MPC had the lowest switch-scenario RMSE, while MPC-DS remained competitive without command saturation across plant variants.Source: §4.2 · p. 10

4.3. Frequency Estimation and Ablation

• Removing periodic prediction brought MPC-DS to MPC-DO performance, while replacing linear-regression frequency estimation with the gradient law sharply reduced its RMSE improvement, especially after a frequency switch.The gradient law’s mean frequency error was 0.674 Hz over 25–30 s, versus 0.040 Hz for the proposed estimator; amplitude projection had no measured accuracy effect but was needed for the uniform bound.Source: §4.3 · pp. 10–11

4.4. Robustness to Parameter Mismatch

• At ±30% parameter mismatch, MPC-DS remained below MPC-DO at every tested level without saturated samples; its RMSE rose to 0.148° at constant frequency and 0.204° with a switch.Drift weighting reduced the mismatch-case RMSE relative to the unweighted controller by 19.9% and 16.9%.Source: §4.4 · p. 11

4.5. Learned Components

• The learned lead residual reduced RMSE by 4.4% on the nominal plant and 2.2% on the calibrated plant, while learned current-value feedforward gave no significant gain.The reported reductions were relative to MPC-DS without learning; the residual correction’s gain was largest at 2 Hz.Source: §4.5 · p. 12

4.6. Flight Experiment

• Yaw and pitch RMSEs were 0.770° and 0.350°, reducing errors by 25.1% and 45.9% relative to LADRC in the reported flight trials.One trial per method used relative angular feedback, without independent inertial LOS calibration or artificial disturbance injection.Source: §4.6 · p. 13

5. Conclusion

• The method predicts multi-harmonic vibration with an adaptive periodic observer and compensates aperiodic disturbance and input-delay increments outside the optimizer using a bounded LSTM-augmented observer.Source: §5 · p. 14
• Input-set construction provides feasible bounded commands on the stated headroom envelope; a regional practical tracking bound holds under full-sequence input margins, predicted-state margins, and bounded model residuals.Source: §5 · p. 14
• 0.770° yaw and 0.350° pitch relative-angle tracking RMSEs reduced errors by 25% and 46%, respectively, versus bandwidth-parameterized LADRC in single-trial indoor hovering evaluations.Figure 8 shows the compared angular trajectories and signed-error densities.Source: Figure 8 · p. 14, §5 · p. 15
• 58% lower pointing RMSE at constant vibration frequency and 39% across a frequency switch were reported versus MPC with a linear disturbance observer, with zero command saturation.The calibrated gimbal model retained graceful performance under 30% parameter mismatch.Source: §5 · p. 15
• Ablations found that predicting delay-induced aperiodic-disturbance change yielded an additional 2–4% RMSE reduction, while learning a current-value feedforward term gave no significant gain over the linear observer.Source: §5 · p. 15

S1. Detailed Rigid-Body Derivation

• The outer azimuth and inner elevation platforms are described by rigid-body angular-velocity relations, with the elevation-platform velocity representing camera line-of-sight motion in inertial space.Source: S1. Detailed Rigid-Body Derivation · p. 17
• Newton–Euler rotational dynamics for the azimuth and elevation bodies use their angular accelerations, inertia tensors, and total torques.Source: S1. Detailed Rigid-Body Derivation · p. 17
• Projecting the rigid-body equations onto the joint axes produces manipulator-form dynamics with control and disturbance torques; the supplement identifies the inertia, mass, and center-of-mass parameters used in the model.Source: S1. Detailed Rigid-Body Derivation · p. 17

S2. Detailed Disturbance Component Derivations

• Gravity-induced torques arise from elevation-platform mass imbalance and depend on aircraft attitude; gyroscopic torques arise from projecting aircraft angular velocity onto the gimbal axes.Source: S2. Detailed Disturbance Component… · p. 17
• Aircraft attitude and angular-rate measurements have transmission latency and typically update at 50 Hz, so gravity and gyroscopic torques are treated as unknown disturbances for high-rate pointing control.Source: S2. Detailed Disturbance Component… · p. 17
• The vibration model uses multiple harmonics whose parameters depend on flight state, but a single fundamental-frequency estimator does not identify unrelated modal frequencies.Constant bias and modes outside the retained common-fundamental harmonic family are assigned to the residual component.Source: S2. Detailed Disturbance Component… · p. 18
• Friction is represented by the LuGre model, but its parameters are treated as unknown because they are difficult to identify accurately on the ground and vary with temperature and humidity.Source: S2. Detailed Disturbance Component… · p. 18
• The residual disturbance captures unmodeled dynamics and uncertainties, including cable restraint torques, higher-order coupling effects, and sensor biases.Source: S2. Detailed Disturbance Component… · p. 18

S3. Standard LSTM Gating Mechanisms

• The LSTM cell state carries slowly varying long-term memory through forget and input gates, while the recurrent hidden state captures rapidly varying information step by step.Source: S3. Standard LSTM Gating Mechanisms · p. 18
• A linear projection of the hidden state produces the continuous disturbance estimate without an additional activation; the output weight matrix is spectrally normalized for the stability analysis.Source: S3. Standard LSTM Gating Mechanisms · p. 18
• During online execution, the raw LSTM output is projected to bound its magnitude and rate.Source: S3. Standard LSTM Gating Mechanisms · p. 18

S4. Detailed Discrete LSTM-DO Error Analysis

• Each sub-step error recursion separates observer contraction, aperiodic estimation mismatch, and an O(h^2) discretization and zero-order-hold remainder.The remainder is bounded using operating-envelope bounds on the first and second derivatives of angular velocity.Source: S4. Detailed Discrete LSTM-DO Error… · pp. 18–19
• When l2 ≠ 0, leakage contributes at every sub-step alongside the sampled increment of eL.The proof collects these contributions in the bracketed term.Source: S4. Detailed Discrete LSTM-DO Error… · p. 19
• Expanding the finite sum of sub-step forcing and remainder terms yields bound (23), while Schur stability of Ad supports bound (24).The sum provides computable constants once the sub-step envelopes are specified.Source: S4. Detailed Discrete LSTM-DO Error… · p. 19
• The finite-time bound includes a transient term, so the lim-sup bound alone gives no exact finite-time threshold.Element-wise gains are used because angular-velocity and aperiodic-disturbance errors have different physical units.Source: S4. Detailed Discrete LSTM-DO Error… · p. 19

S5. Detailed Proofs of Lemmas 1 and 2

• Frequency projection keeps the estimate within its admissible band even when the periodic disturbance vanishes.In that case normalization amplifies noise, while amplitude projection keeps the harmonic prediction bounded.Source: S5. Detailed Proofs of Lemmas 1 and 2 · p. 19
• The harmonic predictor represents the retained disturbance with nh harmonics and advances phase using the estimated frequency.The coefficient vector is bounded by Amax.Source: S5. Detailed Proofs of Lemmas 1 and 2 · p. 19
• RLS fits harmonic coefficients to the posterior disturbance signal using regressors passed through the same low-pass filter.An additional constant regressor absorbs slow components but is excluded from prediction.Source: S5. Detailed Proofs of Lemmas 1 and 2 · p. 19
• Across frequency or coefficient switches, the prediction error remains uniformly bounded independently of convergence.The amplitude-based bound is 2Ch; on smooth intervals, frequency and coefficient variation also bound prediction error.Source: S5. Detailed Proofs of Lemmas 1 and 2 · p. 19
• Coefficient projection bounds fitting error despite residual disturbances, without requiring exact separation or RLS convergence.Frequency and coefficient convergence can tighten the local bound, while the amplitude bound supplies the uniform bound.Source: S5. Detailed Proofs of Lemmas 1 and 2 · p. 19

S6. Simulation Parameter Tuning Procedure

• Each method's observer and predictor settings were selected from at most 16 candidates evaluated across four base frequencies and two scenarios.One parameter set per method and plant variant was retained using seeds 0–2.Source: S6. Simulation Parameter Tuning… · p. 19
• The MPC-DS drift weights were selected from 15 candidates on calibrated-plant seeds 0–5 and held fixed across variants and mismatch levels.The fixed pair was selected with ωo and λ held constant at κ = 0.Source: S6. Simulation Parameter Tuning… · p. 19
• All test evaluations used disjoint seeds 100–109, providing 40 runs per method and scenario.The selected bandwidths differed by plant variant for MPC-DO and MPC-DS.Source: S6. Simulation Parameter Tuning… · p. 20

S7. Observer Bandwidth Scaling and Element-Wise Gain Checks

• With the stated bandwidth parameterization, the discrete Schur condition reduces to 0 < hωo < 2.The continuous matrix Ac has a double eigenvalue at −ωo, and I + hAc has a double eigenvalue at 1 − hωo.Source: S7. Observer Bandwidth Scaling and… · p. 20
• The feedforward estimation error enters the bound through its increment, bounded by Ts(δr + δL).The approximation is restricted to Tsωo ≪ 1 and is not a uniform gain formula over the full Schur range.Source: S7. Observer Bandwidth Scaling and… · p. 20
• For the yaw-axis model, Tsγ22 was 0.207, 0.079, and 0.043 at ωo = 10, 28, and 56 rad/s, respectively.The corresponding 2/ωo values were 0.200, 0.071, and 0.036; γ21 values were 6.7, 6.6, and 5.9.Source: S7. Observer Bandwidth Scaling and… · p. 20

• When the complete quadratic minimizer is feasible for the input box and unsoftened predicted rows, its slack cost is zero and the optimizer uses the affine law.Source: S8. Complete Proof of the Regional… · p. 20
• The resulting deviation recursion yields the bound B on every interval where the affine law applies, using the disturbance and residual error bounds.Source: S8. Complete Proof of the Regional… · p. 20
• If the initial input and predicted-state rows satisfy the stated margins, the deviation budget is preserved inductively for all later samples while the envelope keeps residual bounds applicable.Source: S8. Complete Proof of the Regional… · p. 20

• The retained-drift simulation model has spectral radii 0.6202 for yaw and 0.5979 for pitch, with yaw angle-component impulse gain 4.516.Source: S9. Nominal Feedback and Impulse-Gain… · p. 20
• The flight model has spectral radii 0.7788 (yaw) and 0.7807 (pitch), with angle-component gains (0.03299, 0.08764) for yaw and (0.03211, 0.06444) for pitch.Source: S9. Nominal Feedback and Impulse-Gain… · p. 20
• The checks establish the Schur property of the selected nominal feedbacks, not the full margin conditions of (35) for a flight trajectory.Source: S9. Nominal Feedback and Impulse-Gain… · p. 20