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Research Papers

Design and Stability Analysis of an Integral Time-Delay Feedback Control Combined With an Open-Loop Control for an Infinitely Variable Transmission System

[+] Author and Article Information
X. F. Wang

Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: xuefeng1@umbc.edu

W. D. Zhu

Professor
Fellow ASME
Department of Mechanical Engineering,
University of Maryland, Baltimore County,
1000 Hilltop Circle,
Baltimore, MD 21250
e-mail: wzhu@umbc.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received September 25, 2016; final manuscript received June 23, 2017; published online September 5, 2017. Assoc. Editor: Mazen Farhood.

J. Dyn. Sys., Meas., Control 140(1), 011007 (Sep 05, 2017) (11 pages) Paper No: DS-16-1459; doi: 10.1115/1.4037283 History: Received September 25, 2016; Revised June 23, 2017

The kinematic model of an infinitely variable transmission (IVT) is introduced, and the nonlinear differential equation for the dynamic model of the IVT system with a permanent magnetic direct current (DC) motor and a magnetic brake is derived. To make the average of the input speed converge to a desired constant for any input power and output load, an integral time-delay feedback control combined with an open-loop control is used to adjust the speed ratio of the IVT. The speed ratio for the open-loop control is obtained by a modified incremental harmonic balance (IHB) method. Existence and convergence of a periodic solution are proved under a condition for parameters of the IVT system, and uniqueness of the periodic solution is proved by converting the nonlinear differential equation to a new differential equation that is Lipchitz in the dependent variable and piecewise continuous in the independent variable. A time-delay variable that is an approximation of the average of the input speed is used as the feedback to control the changing rate of the speed ratio. The IVT system with the time-delay control variable can be converted to a distributed-parameter system. Thus, the spectral Tau method is used to design the time-delay feedback control so that the IVT system is locally exponentially stable. The static error from the open-loop control is eliminated; the feedback control variable with time-delay is smoother than that without time-delay, which yields a lower control effort and more robust control design, since the time-delay variable that acts as a low-pass filter reduces the effect of the instantaneous change of the IVT system.

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Figures

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Fig. 1

Layout of a variant of the IVT studied here; two transparent gears are noncircular gears, parts on three shafts in the top layer constitute the input-control module, and parts on three shafts in the bottom layer constitute the motion-conversion module

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Fig. 2

Modification of the variant of the IVT with small pins replaced by large rollers

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Fig. 3

Schematic of the principle of the variant of the IVT; two perpendicular arrows in the motion-conversion module indicate the 90 deg phase difference between the two rollers

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Fig. 4

Values of f and fθ

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Fig. 5

(a) Values of r¯ and (b) the boundary r¯=0.012 to distinguish different cases for the open-loop control law; r¯>0.012 on the upper left side of the boundary and r¯<0.012 on the other side

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Fig. 6

Comparison between steady-state solutions of ω from the modified IHB method and Runge–Kutta method with Vm=6 V, TU=6 N⋅m, and r¯=0.0067 m

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Fig. 7

Boundary Γ̂=0 compared with the boundary fB = 0; Γ̂<0 for the upper left side of the boundary and Γ̂>0 for the other side

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Fig. 8

Analogy between a time-delay system and a distributed-parameter system without time-delay; the left graph for the distributed-parameter system is the shape of u(θ,x) at θ=θ1 and x∈[0, 2π], and the right graph is the shape of u(θ,x) at θ=θ2 and x∈[0, 2π]

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Fig. 9

Maximum absolute eigenvalues of the transformation matrix for A

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Fig. 10

Histories of ωa with and without the time-delay feedback control, and the history of Vm; the output torque TU=8 N⋅m is constant

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Fig. 11

Acceleration of the output speed with the time-delay feedback control for the condition in Fig. 10; the unit of the acceleration is the gravitational acceleration g=9.8 m/s2

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Fig. 12

Histories of ωa for the feedback control with and without time-delay, where TU=8 N⋅m and Vm=8 V

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Fig. 13

Histories of control variables r for the feedback control with and without time-delay, where TU=8 N⋅m and Vm=8 V

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