Research Papers

Dynamics and Stability of Phase Controlled Oscillators

[+] Author and Article Information
Tobias Brack

Department of Mechanical and
Process Engineering,
Institute for Mechanical Systems,
ETH Zurich,
Zurich 8092, Switzerland
e-mail: brack@imes.mavt.ethz.ch

Dominik Kern

Professorship of Applied
Mechanics and Dynamics,
Faculty for Mechanical Engineering,
Chemnitz University of Technology,
Chemnitz 09107, Germany
e-mail: dominik.kern@mb.tu-chemnitz.de

Mengdi Chen

Department of Mechanical and
Process Engineering,
Institute for Mechanical Systems,
ETH Zurich,
Zurich 8092, Switzerland
e-mail: mengdichen@googlemail.com

Jürg Dual

Fellow ASME
Department of Mechanical and
Process Engineering,
Institute for Mechanical Systems,
ETH Zurich,
Zurich 8092, Switzerland
e-mail: dual@imes.mavt.ethz.ch

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 8, 2015; final manuscript received March 22, 2016; published online May 13, 2016. Assoc. Editor: Yang Shi.

J. Dyn. Sys., Meas., Control 138(7), 071007 (May 13, 2016) (12 pages) Paper No: DS-15-1163; doi: 10.1115/1.4033176 History: Received April 08, 2015; Revised March 22, 2016

This paper systematically analyzes linear oscillators, e.g., spring-mass-damper systems or RLC-circuits that are controlled by an extension of a phase-locked loop (PLL). These systems are often used in measurement applications where the stability and dynamics directly influence the measurement quality. Therefore, a description of the control loop in terms of phase signals is sought. However, the classical oscillator turns into a highly nonlinear system when it is formulated in amplitude/phase-variables of its input and output signals. Up to now, there were made either ab-initio assumptions of slowly varying parameters or trial-and-error designs. The novel approach proposed in this paper derives a universally valid description in state space form which enables the use of standard methods of nonlinear system theory. Using this description, the stability of phase controlled oscillators is analyzed by means of Lyapunov functions. A linearization is applied in order to effectively design the controller and optimize the closed-loop dynamics. Simulations with the original nonlinear systems are conducted to justify the linear approach. Thereby, two application scenarios are under consideration: Tracking of the desired target value (target phase shift) and resonance tracking (changes of the system parameters). It is found that including the phase dynamics of the oscillator significantly improves the description of the closed-loop behavior. Finally, the results are validated experimentally for an application measuring the viscosity of fluids.

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Fig. 1

Steady-state phase response of a linear oscillator single degree-of-freedom (1DOF) around the resonance frequency ω0: qualitative illustration

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Fig. 2

Block diagram of a PLL in combination with an oscillating system. Solid lines indicate phase/frequency signals, dashed lines indicate harmonic signals. The nonlinear conversion occurs in the PD and the NCO.

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Fig. 3

Responses of the phase shift deviation z2,1 to a step change of ω0,1 obtained from transfer functions calculated during the linearization procedure for D = 0.025 and D = 0.25 (ω0 = 6000 rad/s)

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Fig. 4

Stability region tracking the target phase shift Δϕt=−π/2 of the example system from Sec. 6. The red dot marks the steady-state position zs

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Fig. 5

Block diagram of the control system using the linearized phase dynamics of the oscillator

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Fig. 6

Setup of the torsional oscillator for applications in fluid measurements—(a) sketch and (b) photo of the oscillator

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Fig. 7

Development of the response of the system phase shift Δϕ both for the nonlinear and linearized system compared to the experimental results. A step change of the target phase shift Δϕt by −15 deg and −45 deg starting at resonance (−90 deg) is applied at t = 1 s

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Fig. 8

Development of the excitation frequency fnco both for the nonlinear and linearized system compared to the experimental results. A step change of the target phase shift Δϕt by −15 deg and −45 deg starting at resonance (−90 deg) is applied at t = 1 s. The lower curves correspond to the step of −15 deg, the upper curves to the −45 deg step

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Fig. 9

Development of the excitation frequency fnco when a step change of the resonance frequency is applied at t = 1 s. Resonance tracking, i.e., Δϕt=−90 deg. The step value is chosen analogous to the output of the −45 deg target phase step, the particular damping ratios D remain constant

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Fig. 10

Phase step response of the linearized system using controller parameters based on wrongly estimated values (0)err for different quantities of ϵs

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Fig. 11

Development of the excitation frequency fnco for different controllers. A step change of the target phase shift Δϕt by −15 deg starting at resonance (−90 deg) is applied at t = 1 sec. The time constant of the PID controllers is Tc = 0.5 sec. Note that the solid and dashed lines are superposing for high damping



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