Adaptive Input Shaper Design for Unknown Second-Order Systems with Real-Time Parameter Estimation

Technical Summary: Adaptive Input Shaper Design for Unknown Second-Order Systems with Real-Time Parameter Estimation

Problem Statement
Input shaping is a widely used feedforward control technique designed to suppress residual vibrations in flexible or underdamped systems by convolving reference commands with time-delay filters (TDF). However, traditional input shaper design relies heavily on prior knowledge of system parameters, specifically the natural frequency ($\omega_n$) and damping ratio ($\zeta$). In "black-box" scenarios where these parameters are unknown, or in applications where system dynamics vary, standard methods fail. While previous research has attempted real-time estimation, a significant limitation persists: existing closed-form approaches (e.g., [22]) require the parameter estimation time ($\tau$) to be strictly shorter than the system's switch time ($T$). If the estimation duration exceeds the switch time, the algorithm fails to achieve optimal vibration suppression, limiting the flexibility and robustness of real-time deployment.

Methodology
The authors propose a feedforward control framework that integrates online parameter estimation with an adaptive input shaper design for unknown second-order systems. The methodology consists of two primary phases:

1. Online Parameter Estimation:
   The system initiates motion with a small, scaled step input ($K \cdot x^*$, where $K=0.01$) for a user-defined duration $\tau$. During this phase, the system response is analyzed to estimate $\omega_n$ and $\zeta$. The estimation logic (Algorithm 1) distinguishes between undamped, damped, and critically damped behaviors by examining the slope of the response, overshoot ($M_p$), settling time ($T_s$), and peak intervals.

   • For undamped systems, $\zeta$ is assumed to be 0, and $\omega_n$ is calculated from the time interval between peaks.
   • For critically damped systems, $\zeta$ is assumed to be 1, and $\omega_n$ is derived from the settling time.
   • For underdamped systems, $\zeta$ is calculated from the overshoot ratio, and $\omega_n$ is derived from the peak time and $\zeta$.

2. Adaptive Input Shaper Design:
   Unlike previous methods that treat estimation time as a constraint, the proposed framework explicitly incorporates the estimation duration $\tau$ and the scaling factor $K$ into the TDF formulation. The transfer function is defined as:
   $$G(s) = K + Ae^{-s(\tau+T)} + (1 - A - K)e^{-s(\tau+2T)}$$
   This formulation ensures that the switch time $T$ applies only after the estimation phase, making the shaper's performance independent of the chosen $\tau$.

   • Undamped Systems: The authors derive a closed-form analytical solution for the impulse magnitude $A$ and switch time $T$ using Weierstrass substitution and trigonometric identities. This allows for rapid computation without iterative solving.
   • Damped Systems: Due to the complexity of exponential and sinusoidal terms, a numerical approach is employed to solve for $A$ and $T$ by setting the real and imaginary parts of the characteristic equation to zero.

Key Contributions

• Arbitrary Estimation Time: The primary contribution is a framework that removes the constraint requiring estimation time to be shorter than the switch time. The method functions correctly regardless of whether $\tau < T$ or $\tau > T$, enabling flexible deployment where the estimation duration can be user-defined based on system speed.
• Closed-Form Solution for Undamped Systems: The paper provides a novel analytical derivation for the optimal shaper parameters in undamped cases, facilitating fast, real-time implementation without the need for numerical optimization.
• Black-Box Applicability: The approach eliminates the need for prior system identification, making it suitable for applications where system dynamics are unknown or difficult to model a priori.
• Robustness to Parameter Sweeps: The method is validated across a wide range of damping ratios ($\zeta \in [0, 1]$) and natural frequencies ($\omega_n \in [\pi, 3000\pi]$ rad/s).

Results
Simulation results demonstrate the efficacy of the proposed framework:

• Vibration Suppression: Across all tested scenarios (undamped, damped, and critically damped), the system output tracks the shaped reference with negligible overshoot and no residual vibration.
• Estimation Time Independence: Unlike the method in [22], which degrades when $\tau > T$ (resulting in sustained oscillations), the proposed method maintains high performance regardless of the estimation duration.
• Parameter Estimation Accuracy: The estimator (Algorithm 1) shows close agreement with true parameters for most cases. Minor discrepancies were observed in critically damped systems and very high-frequency systems ($\omega_n = 3000\pi$), attributed to the difficulty in capturing peak times and settling times with fixed time steps. However, these estimation errors did not significantly degrade the final vibration suppression performance.
• Step-Wise Tracking: The system successfully handles multiple step-wise reference changes (positive, negative, and zero), maintaining vibration-free tracking after the initial estimation phase.

Significance and Claims
The authors position this work as the first method to address the limitation of fixed-duration estimation in real-time input shaping for arbitrary second-order systems. By decoupling the shaper design from the constraint that estimation must be faster than the system's natural switching, the framework enables adaptive deployment in diverse motion control applications, such as gantry cranes and 3D printer headers.

The paper modestly claims that while the current work focuses on single-mode second-order systems, it serves as a preliminary step toward extending the framework to general black-box systems. The authors note that the dependency of shaper parameters on system dynamics suggests the potential for future replacement of the online design block with a trained regression model for increased efficiency, though this is proposed as future work rather than a current result. The code and models are made publicly available to support reproducibility.