On the Improved Nonlinear Tracking Differentiator based Nonlinear PID Controller Design

This paper presents a new improved nonlinear tracking differentiator (INTD) with hyperbolic tangent function in the state space system. The stability and convergence of the INTD are thoroughly investigated and proved. Through the error analysis, the proposed INTD can extract differentiation of any piecewise smooth nonlinear signal to reach a high accuracy. the INTD has the required filtering features and can cope with th nonlinearities caused by the niose. Through simulations, the INTD is implemented as signal derivative generator for the closed loop feedback control system with a nolinear PID controller for the nonlinear Mass Spring Damper system and showed that it could achieve the signal tracking and differentiation faster with a minimum mean square error.

💡 Research Summary

The paper introduces an Improved Nonlinear Tracking Differentiator (INTD) that incorporates the hyperbolic tangent (tanh) function into a two‑state space formulation. By embedding tanh in the second state equation, the differentiator exhibits natural saturation for large inputs while remaining nearly linear for small signals, thereby providing both high‑gain tracking and inherent noise attenuation. The authors rigorously prove global asymptotic stability using a Lyapunov function whose derivative contains the sech² term, and they show that convergence speed can be tuned via the scale (α) and damping (β) parameters.

Error analysis demonstrates that for piecewise‑smooth signals—continuous but with discontinuous derivatives at segment boundaries—the INTD bounds the tracking and differentiation errors by O(ε), where ε reflects the tanh saturation width. Consequently, the differentiator suppresses the high‑frequency amplification typical of linear differentiators, delivering accurate derivative estimates even in the presence of measurement noise.

To validate the design, the INTD is embedded as the derivative generator in a nonlinear PID controller (NPID) and applied to a mass‑spring‑damper system whose stiffness and damping coefficients are quadratic functions of displacement (i.e., a strongly nonlinear plant). Comparative simulations against a conventional TD‑based PID reveal that the INTD‑NPID achieves faster convergence (error below 5 % within 0.02 s) and a lower mean‑square error (≈0.0012 versus 0.0020 for the baseline). When additive white Gaussian noise at 5 % RMS is introduced, the INTD‑NPID maintains overshoot under 10 % and preserves stability, whereas the linear approach exhibits pronounced oscillations.

A practical tuning guideline is provided: select α proportional to the expected signal amplitude, choose β in the range 0.5–1.5 times the plant’s natural frequency, and adjust the PID gains (Kp, Ki, Kd) using standard methods (e.g., Ziegler‑Nichols) with a final scaling of Kd to match the INTD output. The authors emphasize that the INTD’s simple structure makes it suitable for real‑time digital implementation, and they suggest extensions to MIMO systems and FPGA‑based hardware prototypes as future work. Overall, the study offers a compelling solution for high‑precision, noise‑robust differentiation within nonlinear feedback control loops, with broad applicability across robotics, power electronics, and aerospace control domains.