University of Twente Student Theses


Analysis and control of nonlinear oscillators

Dijk, Michel van (2007) Analysis and control of nonlinear oscillators.

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Abstract:Design of walking robots is a challenge as walking is an inherently unstable process. The stable gait of walking robot can be interpreted as form of oscillation as the states of the system behave in a periodic way. From mathematics, nonlinear oscillators are known that show a stable form of oscillation known as limit cycle oscillation. This research project focuses on the analysis and control of this type of oscillators that may be used in future designs of walking robots to obtain stable and robust behavior. The Van der Pol oscillator is a nonlinear oscillator that has a globally attractive limit cycle. This system consists of a harmonic oscillator with an addition nonlinear damping term. This term behaves as an ordinary damping for high deflections, but it becomes a negative damping for small deflections. This results in oscillations of small amplitude being pumped up, while high amplitude oscillation are damped down leading to a globally attractive limit cycle. Based on the same principle as the Van der Pol oscillator, systems that show oscilla- tory behavior can be brought in limit cycle oscillation by adding this type of nonlinear feedback. This feedback can be implemented by buffer element as a spring and a modu- lated transformer such as a continuous variable transmission (CVT). This leads to energy efficient limit cycle oscillations as energy that otherwise would be dissipated now is stored and can be fed back to the system when the feedback behaves generatively. The second part of the research focused on the design of a feedback controller in such a way that exactly the desired periodic behavior could be obtained. This can be achieved using a Lagrange multiplier based approach, calculation of energy difference or parame- terizing the limit cycle in time. The latter two can be implemented by an algorithm which is simple considering the amount and complexity of calculations and therefore this can be well implemented in a real-time controller. Parameterizing the limit cycle in time has the additional benefit that the phase of the system is explicitly known which makes syn- chronization between subsystems a straightforward task. This approach and the Lagrange multiplier approach is also extensible for use in higher order systems.
Item Type:Essay (Master)
Faculty:EEMCS: Electrical Engineering, Mathematics and Computer Science
Subject:54 computer science
Programme:Computer Science MSc (60300)
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