Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nonlinear integral extension of PID control with improved convergence of perturbed second-order dynamic systems

Michael Ruderman

TL;DR

The paper addresses improving convergence of perturbed second-order systems controlled by PID by introducing a nonlinear integral extension (nl-PID) that is model-free and relies only on Lipschitz bounds of matched disturbances. The nl-PID law, with a nonlinear integral gain, is analyzed using circle criterion to establish global asymptotic stability for constant disturbances and ultimately bounded output error for Lipschitz perturbations, complemented by a frequency-domain disturbance-rejection view. A step-by-step design procedure is proposed and the method is extended to systems with actuator dynamics; an experimental case demonstrates faster settling compared to linear PID/PD while maintaining comparable residual error under noise. The approach provides a practical, robust enhancement for second-order control tasks and is validated through a realistic actuator experiment, highlighting improved transient performance and compatibility with parasitic dynamics.

Abstract

Nonlinear extension of the integral part of a standard proportional-integral-derivative (PID) feedback control is proposed for perturbed second-order systems. The approach is model-free and requires solely the Lipschitz boundedness of the unknown matched perturbations. For constant disturbances, the global asymptotic stability is shown based on the circle criterion. For Lipschitz perturbations, an ultimately bounded output error is provided based on the steady-state behavior in frequency domain. Also the transient response to the stepwise disturbances is analyzed for the control tuning. Based on the developed analysis, the design recommendations are formulated as a step by step procedure. It is also discussed how the proposed control is applicable to second-order systems extended by additional (parasitic) actuator dynamics with low-pass characteristics. The proposed nonlinear control is proven to outperform its linear PID counterpart during the settling phase, i.e. at convergence of the residual output error. An experimental case study of the second-order system with an additional actuator dynamics and considerable perturbations is demonstrated to confirm and benchmark the control performance.

Nonlinear integral extension of PID control with improved convergence of perturbed second-order dynamic systems

TL;DR

The paper addresses improving convergence of perturbed second-order systems controlled by PID by introducing a nonlinear integral extension (nl-PID) that is model-free and relies only on Lipschitz bounds of matched disturbances. The nl-PID law, with a nonlinear integral gain, is analyzed using circle criterion to establish global asymptotic stability for constant disturbances and ultimately bounded output error for Lipschitz perturbations, complemented by a frequency-domain disturbance-rejection view. A step-by-step design procedure is proposed and the method is extended to systems with actuator dynamics; an experimental case demonstrates faster settling compared to linear PID/PD while maintaining comparable residual error under noise. The approach provides a practical, robust enhancement for second-order control tasks and is validated through a realistic actuator experiment, highlighting improved transient performance and compatibility with parasitic dynamics.

Abstract

Nonlinear extension of the integral part of a standard proportional-integral-derivative (PID) feedback control is proposed for perturbed second-order systems. The approach is model-free and requires solely the Lipschitz boundedness of the unknown matched perturbations. For constant disturbances, the global asymptotic stability is shown based on the circle criterion. For Lipschitz perturbations, an ultimately bounded output error is provided based on the steady-state behavior in frequency domain. Also the transient response to the stepwise disturbances is analyzed for the control tuning. Based on the developed analysis, the design recommendations are formulated as a step by step procedure. It is also discussed how the proposed control is applicable to second-order systems extended by additional (parasitic) actuator dynamics with low-pass characteristics. The proposed nonlinear control is proven to outperform its linear PID counterpart during the settling phase, i.e. at convergence of the residual output error. An experimental case study of the second-order system with an additional actuator dynamics and considerable perturbations is demonstrated to confirm and benchmark the control performance.
Paper Structure (11 sections, 2 theorems, 26 equations, 9 figures, 1 table)

This paper contains 11 sections, 2 theorems, 26 equations, 9 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. PID controlled feedback loop
  4. Circle criterion
  5. MAIN RESULTS
  6. Absolute stability
  7. Disturbance rejection
  8. Controller design
  9. Extension by an actuator dynamics
  10. EXPERIMENTAL EVALUATION
  11. CONCLUSIONS

Key Result

Theorem 1

Given a feedback system described by the equations eq:2:5. Let the assumptions (i)-(iv) be effective. Then, the origin of eq:2:5 is globally asymptotically stable if

Figures (9)

  • Figure 1: Visualization of the circle condition for $H(j\omega)$ and sector $[0, cd]$.
  • Figure 2: Closed loop system with (nl-)PID controllers.
  • Figure 3: Magnitude of frequency response function $S_{\sigma y}$ depending on $\Omega$.
  • Figure 4: Exemplary transient response of \ref{['eq:3:2:8']} for varying $\Omega$.
  • Figure 5: Output convergence on logarithmic scale for varying $d$.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • proof
  • Remark 1