Feedback linearisation of mechanical systems using data-driven models

TL;DR

The paper tackles the challenge of linearising nonlinear mechanical systems without relying on accurate first-principles models by proposing a data-driven, discrete-time feedback linearisation framework that leverages a model-based reference-tracking approach. The inner-outer loop architecture preserves the plant's underlying linear dynamics while cancelling static nonlinear feedback through an MPC-like optimization that admits a closed-form solution, aided by UKF-based state estimation. Results from a Duffing-oscillator benchmark and a high-precision motion-beam prototype demonstrate strong linearisation performance, robustness to modelling errors and extrapolation, and substantial suppression of nonlinear distortions across a wide frequency range. The work enables practitioners to apply standard linear control techniques to nonlinear mechanical systems with reduced modelling effort and improved resilience, with clear paths for extending to MIMO systems and more complex nonlinearities.

Abstract

Linearising the dynamics of nonlinear mechanical systems is an important and open research area. A common approach is feedback linearisation, which is a nonlinear control method that transforms the input-output response of a nonlinear system into an equivalent linear one. The main problem with feedback linearisation is that it requires an accurate first-principles model of the system, which are typically hard to obtain. In this paper, we design an alternative control approach that exploits data-driven models to linearise the input-output response of nonlinear mechanical systems. Specifically, a model-based reference tracking architecture is developed for nonlinear feedback systems with output nonlinearities. The overall methodology shows a high degree of performance combined with significant robustness against imperfect modelling and extrapolation. These findings are demonstrated using large set of synthetic experiments conducted on a asymmetric Duffing oscillator and using an experimental prototype of a high-precision motion system.

Figures (20)

• Figure 1: Block diagram illustrating the simplified working principle of feedback linearisation. In a first step, the plant input $u(t)$ is modified such that for the new input $v(t)$ a linear output response is obtained. In a second step, the equivalent LTI system can be integrated in any linear outer-loop control scheme, in this example a classical PID feedback loop with reference tracking.
• Figure 2: Illustration of the model structure adopted in this work. A nonlinear mechanical system is modelled as an underlying linear system with a static nonlinear output function wrapped around it in feedback.
• Figure 3: Illustration of the proposed solution with online reference generation and dynamic horizon length. The notation $y_{ref}(i|k)$ indicates the output reference point at the $i$th inner-loop sub-sample of the $k$th outer-loop time instant. At the current time instant, there are 3 out of 5 future references available. The number of available references reduces to 1 as time progresses up until time $k+1$, where a new outer-loop input will generate 5 new reference points.
• Figure 4: Schematic overview of internal reference tracking framework that generates a linearised IO map between $v(k)$ and $y(k)$.
• Figure 5: Illustration of the nonlinear distortion analysis. Left: input spectrum of the multisine. Right: the output spectrum with the quantification of the odd (in red) and even (in blue) nonlinear distortions. The grey lines represent the contributions of the disturbing noise and black lines are the linear system response.