Orbital Stability Analysis for Perturbed Nonlinear Systems and Natural Entrainment via Adaptive Andronov–Hopf Oscillator

Abstract

Periodic orbits often describe desired state trajectories of dynamical systems in various engineering applications. Stability analysis of periodic solutions lays a foundation for control design to achieve convergence to a prescribed orbit. Here we consider a class of perturbed nonlinear systems with fast and slow dynamics and develop a novel averaging method for analyzing the local exponential orbital stability of a periodic solution. A framework is then proposed for feedback control design to stabilize a natural oscillation of an uncertain nonlinear system using a synchronous adaptive oscillator. The idea is applied to linear mechanical systems and a design theory is established. In particular, we propose a controller based on the Andronov-Hopf oscillator with additional adaptation mechanisms for estimating the unknown natural frequency and damping parameters. We prove that, with sufficiently slow adaptation, the estimated parameters locally converge to their true values and entrainment to the natural oscillation is achieved as part of an orbitally stable limit cycle. Numerical examples demonstrate that adaptation and convergence can in fact be fast.

Gait Phase Estimation Based on User–Walker Interaction Force

“…However, a small σ increases the time taken by the oscillator to synchronize with the input signal. This not only influences the estimation accuracy initially but also limits the ability to track gait changes [20]. Alternatively, we apply an adaptive rule for σ when a sudden change in amplitude occurs.…”

“…However, a drastic change in the frequency or amplitude of the target signal substantially degrades the estimation accuracy due to slow convergence. Although the estimation convergence can be accelerated by increasing the gains in the estimation loop, fluctuations in the estimates may occur [19,20]. As for the case of interaction force based gait phase estimation, the amplitude of the interaction force is not constant but can change drastically because it depends on the bodyweight transmitted to the handle.…”

Design and memory optimization of hybrid gate diffusion input numerical controlled oscillator

“…It may use the digital signal processing (DSP) for the implementation offinite impulse response (FIR) filter transfer function. This filter allows the passage of all signals from 0 Hz to a programmable bandwidth or cutoff frequency and rejects the signals which can have frequency above cutoff [23]. Digital filter is one of the most complex filters that will process I and Q signals from mixer.…”

Design of controllers with distributed central pattern generator architecture for adaptive oscillations

“…32,33 A more explicit approach to resonance entrainment was proposed using adaptive frequency Hopf oscillators with sensory feedback, 34,35 which has been extended with a rigorous proof of stability. 36 Without restrictions to resonance, an arbitrary oscillation pattern can also be achieved with stability for mechanical systems using Hopf oscillators 37,38 or circulant CPGs. 39 However, these methods make the closed-loop system an oscillator through reflex loops and are different from neural control mechanisms where the CPG and plant dynamics are coupled, and hence it is not clear if biological adaptivity can be realized in this way.…”

“…While these results require manual tuning of design parameters, analytical conditions have also been derived for half‐center CPGs to entrain to a selected mode of mechanical resonance 32,33 . A more explicit approach to resonance entrainment was proposed using adaptive frequency Hopf oscillators with sensory feedback, 34,35 which has been extended with a rigorous proof of stability 36 . Without restrictions to resonance, an arbitrary oscillation pattern can also be achieved with stability for mechanical systems using Hopf oscillators 37,38 or circulant CPGs 39 .…”