Neural Network PD Actuator Model

• Neural network PD actuator models are data-driven techniques that combine classical PD control with neural networks to accurately capture nonlinear dynamics and friction effects.
• They utilize various architectures—including feedforward inverse models, recurrent networks, and physics-informed models—to model hysteresis, rate dependencies, and other actuator phenomena.
• Practical implementations demonstrate enhanced tracking performance and error reduction in applications such as robotics, nanopositioning, and adaptive electromechanical control.

A neural network PD actuator model is a data-driven approach for modeling and controlling actuation systems, combining the structure of classical proportional-derivative (PD) control with neural networks fitted to the relevant inverse or forward actuator dynamics. Such models leverage recent advances in machine learning to accurately capture nonlinearities, hysteresis, and rate-dependent phenomena that are difficult to characterize analytically. Neural network PD actuator models are now widely used in robotics, nanopositioning, electromechanical systems, and adaptive control architectures.

1. Analytical Basis and Dominant Variable Identification

The initial construction of a neural network PD actuator model typically requires a coarse analytical model to reveal the physical structure of the system and to inform neural network input selection. For example, in piezoelectric nanopositioning stages, the mass-normalized dynamics can be written as

$$\ddot{x}(t) = -a_1 \dot{x}(t) - a_2\,\text{sgn}(\dot{x}(t)) + a_3\,u(t),$$

where $a_1$ and $a_2$ capture viscous and Coulombic friction (possibly asymmetric by motion direction), and $u(t)$ is the actuator input voltage. Empirical excitation and linear regression (e.g., least-squares on step transients) yield numerical estimates for $a_1$, $a_2$, and $a_3$. A phase delay in $\dot{x}$ may be introduced to reflect viscous lag, resulting in

$$\ddot{x}(t) = -a_1 \dot{x}(t-\tau) - a_2\,\text{sgn}(\dot{x}(t)) + a_3\,u(t).$$

To inform variable selection for the neural model, the relative magnitudes of the acceleration and velocity terms are compared; it is observed that in certain regimes $|\ddot{x}|/|a_1 \dot{x}| = O(10^{-1})$, thus velocity dominates. This observation strongly influences architectural parsimony and helps prevent overfitting in downstream network training (Yan et al., 2020).

2. Neural Network Architectures for Actuator Modeling

Neural architectures are selected to match the type and complexity of actuator nonlinearity:

• Feedforward inverse models: Approaches such as single-hidden-layer tanh networks with the dominant system variable(s) as input. For the piezo stage, the structure is:

$u_p = W_2^T \tanh(W_1 v + B_1) + b_2,$

where $v = \dot{x}$, $W_1 \in \mathbb{R}^{24\times1}$, and $u_p$ estimates the required input voltage (Yan et al., 2020).

• Recurrent neural networks for temporal dynamics: Where memory and history-dependence are significant, architectures use stacked LSTM layers followed by a dense readout, e.g., inputting a sequence of recent PWM signals to predict the next-step motor speed:
  • Input: $[\text{PWM}(t-2), \text{PWM}(t-1), \text{PWM}(t)]$
  • RNN structure: LSTM (4 units) → LSTM (1 unit) → Dense (1 unit) (Chandar et al., 2020).
• Physics-structured neural networks: Architectures embedding Lagrangian mechanics, such as ServoLNN, parameterize kinetic/potential energies (via a Cholesky-factorized mass matrix and NN potential), and model the actuator force as either a purely learned mapping $Q_\text{act} = \widehat{Q}(q, \dot{q}, u)$ or an additive residual to a canonical PD law:

$Q_\text{act} = u + \Delta Q,$

with $u(t) = K_p(q_d - q) + K_d(\dot{q}_d - \dot{q})$ (Johns et al., 27 Feb 2025).

• Hybrid neuro-immune PD controllers: Neural networks are used to approximate the suppressor component in immune-inspired PD controllers for DC actuators. An RNN with recurrent hidden units approximates $f_N(e, \dot{e}, u)$, which is subtracted from the classical PD term (Sin et al., 2016).

3. Integration with PD Control and Control Allocation

The neural model serves as a high-fidelity map in a PD+feedforward or balance-allocation controller:

• PD + feedforward neural inversion: The total actuation command is given by

$u(t) = K_p e(t) + K_d \dot{e}(t) + u_\text{ff}(t),$

where $u_\text{ff}(t)$ is computed by passing the measured velocity $\dot{x}(t)$ through the trained neural inverse. PD gains are coarsely tuned by classical means (e.g., pole placement), while $\mu$ in Levenberg-Marquardt regularizes NN training. This combined structure delivers superior reference-tracking, as demonstrated by twofold reductions in maximum error over classical analytic inversion (Yan et al., 2020).

• Neural network control allocation: In overactuated systems (e.g., quadrotors with variable-pitch rotors), inner-loop PD controllers produce moment and thrust commands. These are allocated to actuators by solving $Y = (C \circ F)^{-1} X$, with the NN performing aerodynamic inversion to compute required motor speeds for specified thrust/pitch pairs (Kulkarni et al., 2020).
• Cascade architectures: In systems where both plant and controller are "soft," an outer RNN (controller) produces the actuation command sequence, and an inner RNN (actuator) maps the command to state evolution. The networks are trained such that the cascade closes the loop with implicit PD-like behavior, although explicit gain extraction is nontrivial (Chandar et al., 2020).

4. Training Methodologies and Evaluation

Neural network PD actuator models are trained on empirically collected datasets spanning the operational envelope:

• Inverse models: For feedforward NNs, inputs are dominant variables (e.g., $\dot{x}$), outputs are actuation signals, and mean squared error (MSE) is minimized via Levenberg-Marquardt backpropagation with momentum. Typical architectures employ small hidden layers (e.g., 24 units), and overfitting is mitigated by limiting input dimensionality (Yan et al., 2020).
• Recurrent models: Multi-timestep datasets (e.g., PWM→RPM trajectories) are split into training (80%) and testing (20%) sets. Optimization uses Adam, and loss is evaluated as MSE on predicted next-step outputs (Chandar et al., 2020).
• Lagrangian neural models: Custom loss functions aggregate errors on generalized forces (inverse dynamics), accelerations (forward dynamics), energy conservation, and mass-matrix properties. Loss terms are weighted (e.g., $\alpha_\text{inv}, \alpha_\text{fwd}$ ≈ 1). Data collection protocols involve randomized reference trajectories, normalization, and careful tuning of learning rates and batch sizes (Johns et al., 27 Feb 2025).
• Immune-inspired hybrid models: Back-propagation through time is employed for RNN suppressor estimation, with cost functions penalizing PD error and weights updated according to error gradients. Stability and performance guarantees are achieved through proper gain assignment and NN size selection (Sin et al., 2016).
• Performance metrics: Models are benchmarked by test-set MSE, trajectory tracking errors, energy drift, and, where applicable, mass-matrix or generalized-force error relative to ground truth. Feedforward NN compensation in piezo stages halved peak error compared to non-NN approaches (Yan et al., 2020). Neural-immune PD controllers cut settling time and overshoot by approximately 50% and 60%, respectively, versus classical immune PID (Sin et al., 2016).

5. Representative Architectures and Model Comparison

Approach	Network Type	Input Features	Application
Feedforward NN inverse	1-hidden-layer tanh	$\dot{x}$ (velocity)	Piezo stage
RNN Actuator	2xLSTM + Dense	PWM window (3-step)	Electromechanical
Lagrangian NN	MLP trunk, multi-head	$q$, $\dot{q}$, $u$	Servo-driven system
NN Allocator	1-layer tanh NN	$(T^d, \phi)$	Variable-pitch rotor
Hybrid immune NN	RNN	$(e, \dot{e}, u)$	DC Actuator

While single-hidden-layer feedforward networks suffice for systems dominated by a single nonlinearity (e.g., velocity in a rate-dependent piezo stage), RNNs capture temporal dependencies requisite for motor drive and multi-step PWM response. Lagrangian NNs enforce physical structure, providing physically consistent predictions (energies, mass matrix), and can directly incorporate PD laws in the generalized-force map.

6. Practical Considerations and Implementation Details

Effective realization of a neural network PD actuator model requires attention to the following:

• Input selection based on physical dominance: Inclusion of too many inputs (e.g., both $\dot{x}$ and $\ddot{x}$) can cause overfitting or spurious generalization. Analytical pre-screening is critical (Yan et al., 2020).
• Activation function and output scaling: Tanh activations for hidden layers, SoftPlus for ensuring positivity (e.g., mass diagonals in ServoLNN), special care for output activations (e.g., softmax for categorical outputs) depending on architecture.
• Architecture selection by dynamics: Rapid, non-memory effects favor shallow feedforward NNs; systems with history or internal actuator memory require recurrent or sequence-to-sequence NNs (Chandar et al., 2020, Kulkarni et al., 2020).
• Loss and regularization selection: Explicit regularization terms (Levenberg-Marquardt damping, weight decay) ensure reasonable generalization; use of composite losses in physics-informed approaches (inverse/forward dynamics, energy).
• Data regime and normalization: Sufficient excitation of the system (chirps, sinusoids, steps) prevents information leakage and overfitting; data normalization is routine for gradient-based optimization.
• Integration in software/hardware loops: NNs can be used as digital twins (RNN actuator models), direct feedforward compensators, or in embedded hardware with real-time inference (implementation speed measured in milliseconds per cycle) (Chandar et al., 2020, Johns et al., 27 Feb 2025).
• Gain tuning: Although gain matrices for PD terms can be chosen by control engineering heuristics, the NN can compensate for unmodeled nonlinear residuals, reducing the need for aggressive gain schedules.

7. Extensions, Limitations, and Outlook

Neural network PD actuator models offer significant advantages in modeling fidelity and adaptive compensation in the presence of nonlinearities, hysteresis, and unknown disturbances. They are broadly extensible via modular architecture to multivariable, underactuated, or fault-tolerant systems through neural allocation and Lagrangian parameterization (Kulkarni et al., 2020, Johns et al., 27 Feb 2025).

Limitations include:

• The need for sufficiently rich excitation and data collection for generalization.
• Situations where physical interpretability or invertibility is needed; physics-informed architectures can mitigate this via embedded structure.
• The risk of forming "families" of solutions in underconstrained identification tasks; supervision on actuator-side quantities can resolve this ambiguity (Johns et al., 27 Feb 2025).

A plausible implication is that continued integration of neural PD actuator models with structured physical priors and advanced learning algorithms (e.g., combined energy-based and trajectory-loss objectives) will further improve robustness, transparency, and real-time deployability of complex actuation systems.

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Key References:

• "Inverse NN Modelling of a Piezoelectric Stage with Dominant Variable" (Yan et al., 2020)
• "Dynamic Systems Simulation and Control Using Consecutive Recurrent Neural Networks" (Chandar et al., 2020)
• "Full Attitude Intelligent Controller Design of a Heliquad under Complete Failure of an Actuator" (Kulkarni et al., 2020)
• "Study on Neural Immune PD Type Tracking Control for DC Actuating Mechanism" (Sin et al., 2016)
• "ServoLNN: Lagrangian Neural Networks Driven by Servomechanisms" (Johns et al., 27 Feb 2025)