[Paper Review] Tracking Control by the Newton-Raphson Method with Output Prediction and Controller Speedup

Y. Wardi, Carla Seatzu+4 more|arXiv (Cornell University)|Oct 1, 2019

Advanced Control Systems Optimization4 citations

TL;DR

This paper proposes a novel tracking control method for continuous-time dynamical systems using a fluid-flow variant of the Newton-Raphson method, integrating output prediction and controller speedup via a variable-gain integrator. The approach ensures asymptotic tracking with large or global domains of attraction, achieves error reduction through gain scaling, and guarantees stability under prediction errors and disturbances, validated through simulations and lab experiments on nonlinear systems.

ABSTRACT

This paper presents a control technique for output tracking of reference signals in continuous-time dynamical systems. The technique is comprised of the following three elements: (i) output prediction which has to track the reference signal, (ii) a controller based on an integrator with variable gain, and (iii) a speedup of the control action for enhancing the tracker's accuracy and, in some cases, guaranteeing stability of the closed-loop system. The technique is suitable for linear and nonlinear systems, implementable by simple algorithms, can track reference points as well as time-dependent reference signals, and may have large, even global domains of attraction. The derived theoretical results include convergence of the tracking controller and error analysis, and are supported by illustrative simulation and laboratory experiments.

Motivation & Objective

• To develop a simple yet effective tracking control technique for continuous-time dynamical systems that avoids reliance on complex nonlinear inversion or linearization.
• To address the challenge of tracking time-varying reference signals in nonlinear systems with large or global domains of attraction.
• To enhance controller performance through speedup mechanisms that reduce tracking errors and improve stability in the presence of prediction errors and disturbances.
• To establish theoretical convergence and error bounds for the proposed controller under general nonlinear system assumptions.
• To validate the method experimentally through simulations and laboratory tests on mobile robotics and autonomous vehicle applications.

Proposed method

• The controller uses a fluid-flow variant of the Newton-Raphson method, derived by taking the limit of a discrete iterative algorithm as step size approaches zero, resulting in a continuous-time differential equation for control input.
• A predictor estimates the system output at a future time $ t+T $, enabling the controller to solve $ r(t+T) - \hat{y}(t+T) = 0 $ in real time.
• The control action is implemented via a variable-gain integrator, where the gain is adjusted to enhance convergence speed and reduce tracking error.
• Controller speedup is introduced to stabilize the closed-loop system and improve accuracy, particularly when prediction errors are present.
• The method is formulated as a differential equation in the state and control variables, making it inherently nonlinear and not dependent on explicit algebraic functions of the state.
• Theoretical analysis relies on characteristic polynomial analysis and asymptotic behavior of eigenvalues to establish stability and convergence properties.

Experimental results

Research questions

• RQ1 Can a Newton-Raphson-based fluid-flow controller achieve asymptotic tracking of time-varying reference signals in nonlinear systems without linearization?
• RQ2 How does the inclusion of output prediction and controller speedup affect the domain of attraction and tracking accuracy?
• RQ3 To what extent can variable-gain control and speedup mechanisms stabilize the closed-loop system under prediction errors and disturbances?
• RQ4 What are the theoretical convergence and error bounds of the proposed controller under general nonlinear system dynamics?
• RQ5 Can the method be implemented with low computational cost while maintaining robustness and global stability properties?

Key findings

• The controller ensures asymptotic tracking with an error bounded by the asymptotic prediction error, provided the prediction is accurate.
• An increase in controller gain can stabilize the closed-loop system and reduce tracking errors due to disturbances and computational inaccuracies.
• The method achieves large or even global domains of attraction, making it effective for nonlinear systems without requiring local linearization.
• Theoretical analysis proves that the limit of the system's characteristic roots as gain $ \alpha \to \infty $ converges to roots of a reduced-order polynomial, ensuring stability under general conditions.
• Simulation and laboratory experiments confirm the controller’s effectiveness on nonlinear systems, including mobile robotics and autonomous vehicle applications.
• The method is computationally efficient and implementable via simple algorithms, making it suitable for real-time control in embedded systems.