Nonlinear Actuator Dynamics Overview

• Nonlinear actuator dynamics are nonlinear behaviors in devices that convert signals to motion, shaped by effects like cubic elasticity, saturation, and state feedback.
• They are modeled using nonlinear differential equations and data-driven parameter estimation, enabling precise predictions of stability and performance.
• These dynamics have practical applications in MEMS, soft robotics, and powertrains, emphasizing robust control synthesis and safety-critical operation.

Nonlinear actuator dynamics refer to the behaviors exhibited by actuators—devices responsible for converting control signals into mechanical movement—when their input-output relationships or internal evolution are governed by nonlinear laws. In both engineered and natural systems, nonlinearities can stem from mechanical properties (e.g., elasticity with cubic or higher-order terms), electromagnetic or fluid phenomena, material deformation, geometric constraints, and internal feedback. These nonlinearities are particularly significant in microelectromechanical systems (MEMS), soft robotics, powertrains, multi-DOF platforms, and distributed systems where accurate modeling and stability are critical. Robust analysis, control synthesis, and safety-critical operation all require precise mathematical treatment of the nonlinear actuator dynamics.

1. Mathematical Characterization of Nonlinear Actuator Dynamics

Nonlinear actuator dynamics are often modeled by including explicitly nonlinear restoring forces, dissipative effects, actuation laws, or state-dependent transport phenomena. For example, in the context of MEMS electrostatic actuators, the equations of motion are modified to include cubic terms (nonlinear elasticity), leading to governing ODEs such as:

$$\ddot{x} = - \frac{x}{(\xi+1-x)^2} - K x^3 + \frac{V^2}{2(\xi+1-x)^2}$$

where $K$ parametrizes the cubic nonlinearity, $V$ is the applied voltage, and $\xi$ is a normalized geometrical parameter (Yang et al., 2012). In distributed systems, actuator dynamics may be described by quasilinear hyperbolic PDEs, for instance:

$u_t(x, t) = v(u(x, t)) u_x(x, t)$

$u(1, t) = U(t)$

with $v(\cdot)$ encoding the nonlinear, state-dependent transport speed (Bekiaris-Liberis et al., 2017).

For hydraulic or pneumatic actuators in soft robotic applications, models incorporate nonlinear material laws such as Ludwick's Law, $\sigma = E \epsilon^n$, with $0 < n < 1$ estimated from empirical data, providing accurate description in the large strain regime (Yang et al., 2023).

Table 1: Representative forms of nonlinear actuator dynamics in recent literature

Context	Governing Equations (simplified)	Reference
Electrostatic MEMS actuator	$$\ddot{x} = -\frac{x}{(\xi+1-x)^2} - Kx^3 + \frac{V^2}{2(\xi+1-x)^2}$$	(Yang et al., 2012)
Quasilinear PDE (distributed input)	$u_t(x, t) = v(u(x, t)) u_x(x, t)$	(Bekiaris-Liberis et al., 2017)
Soft actuator (nonlinear stiffness)	$$M\ddot{\theta} + C_n \dot{\theta} + K_n \theta^n = F$$	(Yang et al., 2023)
Series elastic actuator (NSEA)	$$\tau_{ns} = n k_s R r \left( 1 - \frac{R-r}{\sqrt{R^2 + r^2 - 2Rr\cos\theta}} \right)\sin\theta$$	(Hirao et al., 2023)

Nonlinearities may also arise from actuator constraints such as rate limits, saturations, nonlinear mapping between control signal and mechanical output, and adaptation under faults.

2. Qualitative Behaviors and Critical Phenomena

Several universal behaviors emerge in nonlinear actuator systems.

Bifurcation and Pull-in Phenomena: In micro- and nano-actuators, both linear and cubic nonlinear elastic models exhibit two dynamical regimes: periodic oscillations for actuation below a critical "pull-in voltage" and catastrophic finite-time touch-down (collapse) above this threshold (Yang et al., 2012). Explicit expressions for stagnation displacement, pull-in voltage, and position are derived in terms of underlying parameters. The addition of a cubic nonlinearity monotonically increases pull-in voltage and reduces maximum excursion, a result of additional mechanical stiffening.

Finite-Time Singularities: In settings where the net restoring force is overwhelmed by input (e.g., electrostatic force in MEMS), the dynamics exhibit finite-time singularities. For voltages above the critical value, the movable electrode reaches contact in finite time:

$$\lim_{t \to t_c} x(t) = 1, \quad V^2 \geq (\xi+1)^3$$

(Yang et al., 2012). Finite-time collapse necessitates rigorous safety margins in design and operation.

Multiple Resonant Modes and Nonlinear Frequency Response: In synthetic jet actuators with large cavities, nonlinear modes not predicted by standard linear theory emerge, exhibiting highly asymmetric frequency distributions at low frequencies and dominating over membrane or Helmholtz modes under certain operating conditions (Olivera-Reyes et al., 5 Jul 2024). Pressure responses scale according to non-integer power laws (e.g., $ΔP = a \cdot f^{2.46}$). The frequency response (gain-phase characteristics) of nonlinear series elastic actuators (NSEA) shifts with amplitude owing to amplitude-dependent describing functions (Hirao et al., 2023).

Actuator Saturation and Non-affine Input Constraints: Nonlinearities may be introduced by actuator limitations, including asymmetric magnitude bounds and finite slew rates. Smooth affine models for input saturation are developed to ensure that, under bounded control commands, the actual actuator outputs remain within prescribed constraints (Verma et al., 20 Jun 2025).

3. Rigorous Analysis, Stability, and Control Synthesis

Stability and performance analysis of systems with nonlinear actuator dynamics often relies on the construction of Lyapunov or ISS functionals, sometimes in function spaces (for PDE-ODE cascades). The following methodologies are salient:

• Sharp Theorems for Existence and Uniqueness: In cases such as undamped electrostatic MEMS actuators, theorems establish uniqueness and explicit characterization of periodic solutions and finite-time collapse as a function of system parameters (Yang et al., 2012).
• Predictor-Feedback for Quasilinear PDEs: Predictor states are constructed as solutions to integral equations which, together with feasibility conditions on the actuator's spatial derivatives, yield predictor-feedback laws that ensure local asymptotic stability of the closed-loop system (Bekiaris-Liberis et al., 2017).
• Lyapunov Backstepping and Sliding Mode for Uncertain Nonlinear Actuators: Hierarchical designs combine backstepping (to generate virtual controls handling nonlinear mechanical dynamics) with robust sliding mode layers for actuators (such as coil current in electromagnetic actuators), incorporating uncertain model parameters within Lyapunov stability analysis (Deschaux et al., 2018).
• Nonlinear Model Predictive and NMPC Schemes: In predictive control for autonomous and aerial vehicles, explicit inclusion of actuator first-order (or higher) nonlinear dynamics in optimization constraints is shown to improve the fidelity of predicted trajectories, reduce overshoot, and maintain safe inter-vehicle distances (Babu et al., 2018, Bicego et al., 2019).
• Actorless Adaptive Control with Integral Reinforcement Learning: For control-affine nonlinear systems with actuator constraints, reinforcement learning methods avoid requiring an initial stabilizing controller by embedding a stabilizing term into the critic network and variable learning rates, proving uniform ultimate boundedness of both state and NN weight errors (Mishra et al., 2020).

4. Controller Design and Compensation Techniques

Control architectures for systems with nonlinear actuator dynamics are multi-faceted:

• Explicit Compensation: Nonlinear dynamic inversion (NDI) control laws explicitly account for first-order actuator dynamics. Additional time differentiation in the output equation ensures compensation for all state- and input-derivative terms, in contrast to incremental NDI (INDI), which only approximates correct tracking when actuator bandwidth is high (Steffensen et al., 2022).
• Hierarchical/Predictor-Based and Allocation Algorithms: For distributed actuators or hierarchical systems with actuator faults, control design combines model-based parameter estimation of actuator effectiveness and adaptive allocation ("splitters") to redistribute control input in real time (Ameli et al., 2021).
• Cascaded, Modular Architectures with Loop-Free Mapping Inversion: Modular designs employ virtual inputs decoupled from the physical actuator through dynamic mapping inversion modules, eliminating algebraic loops—especially relevant for hydraulic actuators with strongly nonlinear valve flow maps (Dallabona et al., 17 Oct 2024). Lyapunov-based analysis of the cascaded system confirms global uniform ultimate boundedness.
• Controller Optimization Incorporating Physical Limits: Nonlinear input constraints (saturation, slew-rate) are integrated directly into the backstepping-based controller synthesis rather than treated as ad hoc output clippings. Smooth barrier functions in the actuator dynamics ensure adherence to both magnitude and rate bounds during operation (Verma et al., 20 Jun 2025).
• Physical Reservoir Computing for Soft Actuators: Nonlinearities and memory inherent to soft combinatorial actuators are exploited as computational resources within the control loop, enabling efficient online learning and trajectory tracking with minimal computation (Shen et al., 20 Mar 2025).

5. Modeling Strategies and Identification for Complex Nonlinear Actuators

Accurate modeling is essential for both analysis and control synthesis, with recent advances including:

• Systematic Excitation and System Identification: The selection of excitation signals (chirp, multisine) based on actuator bandwidth ensures sufficient excitation for accurate system identification in hydraulic actuators (Nisar et al., 2021). Black-box ARX models and higher-order transfer functions parameterized from empirical data are validated against nonlinear simulation models.
• Data-Driven Nonlinear Parameter Estimation: For soft actuators, nonlinear stress-strain models (e.g., Ludwick's law) are parameterized using least-squares regression based on materials’ Young's modulus, tensile strength, and viscosity. The resultant models accurately predict actuator behavior over large deflection ranges (Yang et al., 2023).
• Describing Function and Frequency-Domain Analysis: Frequency domain tools such as the describing function provide approximate gain-phase characteristics for nonlinear actuators. Amplitude-dependent describing functions enable the derivation of LPV-based H-infinity controllers adapted to the stiffness variation of NSEAs (Hirao et al., 2023).

Table 2: Modeling and identification approaches for nonlinear actuator dynamics

Modeling Context	Approach	Reference
Hydraulic actuator	Excitation-based identification (ARX/Transfer fn.)	(Nisar et al., 2021)
Soft pneumatic actuator	Data-driven Ludwick power law parameterization	(Yang et al., 2023)
Series elastic actuator	Frequency domain (describing function) analysis	(Hirao et al., 2023)
Synthetic jet actuator	Empirical power law fit for pressure-frequency	(Olivera-Reyes et al., 5 Jul 2024)

6. Practical Implications and Engineering Applications

Accurate mathematical prediction of nonlinear actuator behaviors enables designs with provable safety and performance characteristics:

• Design Margin: Exact calculations of pull-in voltage, position, and displacement in MEMS actuators prevent catastrophic failure due to finite-time collapse (Yang et al., 2012).
• Performance Enhancement: Incorporation of actuator nonlinearities in NMPC for autonomous vehicles and drones leads to improved safety margins, controlled velocity profiles, and computational efficiency (Babu et al., 2018, Bicego et al., 2019).
• Adaptation and Fault Recovery: Online parameter estimation and allocation mechanisms ensure robust tracking in systems with time-varying actuator effectiveness or partial fault conditions (e.g., multi-actuator wind turbines), distributing control action adaptively (Ameli et al., 2021).
• Physical Nonlinearity as a Computational Asset: In soft robotic and pneumatic systems, reservoir computing exploits nonlinearities for efficient online learning and control pulse shaping, reducing required model complexity (Shen et al., 20 Mar 2025).
• Robustness to Uncertainty: Design methodologies that bound the effects of unmodeled nonlinearities (e.g., flux fringing in electromagnetic actuators (Deschaux et al., 2018)) and compensate for actuator saturation and slew-rate limits in marine and industrial platforms (Verma et al., 20 Jun 2025), ensure safe, reliable operation.

7. Challenges, Limitations, and Future Directions

Analysis and control of nonlinear actuator dynamics remain fraught with challenges:

• Region of Attraction and Shock Formation: In distributed or PDE-ODE systems, feasibility regions must be carefully defined to avoid the development of shocks or loss of predictability (e.g., bounds on spatial derivative growth in quasilinear hyperbolic PDEs) (Bekiaris-Liberis et al., 2017).
• Non-uniqueness and Numerical Issues: Nonlinear mapping inversion (e.g., in hydraulic actuators) can introduce issues of non-uniqueness and numerical instability if not treated with guaranteed monotonicity or sufficient gradient conditions (Dallabona et al., 17 Oct 2024).
• Describing Function Approximation Limits: While invaluable, describing function methods neglect higher harmonics or strong nonlinearity, necessitating empirical and time-domain validation (Hirao et al., 2023).
• Scaling and Complexity: Real-time NMPC and predictive control with nonlinear actuator models demand efficient optimization schemes (RTI, partial sensitivity updates) and measured calibration of physical constraints to maintain computational tractability (Bicego et al., 2019).
• Unmodeled and Unobservable Effects: Unmodeled or unobservable nonlinearities (e.g., unmeasured internal pressure or strain, hysteresis in soft actuators) limit the effectiveness of conventional control; approaches such as online learning or observer-based estimation are required (Shen et al., 20 Mar 2025, Verma et al., 20 Jun 2025).

Emerging research directions include further development of operator-theoretic (system-level) synthesis integrating nonlinear actuator constraints, real-time adaptation to actuator degradation/faults, high-order or memory-dependent nonlinearities in soft/biomorphic actuators, and improved physical modeling leveraging both first-principle and data-driven techniques. Robust frequency-domain analysis and the use of actuator-intrinsic nonlinearities as computational resources are likely to play a growing role in the design of advanced engineered and bio-inspired systems.