Research Papers

Describing Functions for Scalar Information Channels Subject to Quantization and Packet Loss

[+] Author and Article Information
Eric Gilbertson

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: egilbert@alum.mit.edu

Franz Hover

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02139
e-mail: hover@mit.edu

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received September 4, 2013; final manuscript received January 30, 2015; published online March 23, 2015. Assoc. Editor: Evangelos Papadopoulos.

J. Dyn. Sys., Meas., Control 137(7), 071006 (Jul 01, 2015) (9 pages) Paper No: DS-13-1342; doi: 10.1115/1.4029717 History: Received September 04, 2013; Revised January 30, 2015; Online March 23, 2015

Many of today's robot applications depend on wireless communications, whose performance can impact the whole system. To support analysis of feedback control through limited channels, we develop describing functions (DFs) for three variations on the series interconnection of a quantizer, a binary erasure channel, and decoder for a single input single output (SISO) system. The key steps in our derivation hold when the decoder is a linear-quadratic-Gaussian (LQG)-type control, a zero-output decoder, or a hold-output decoder. We confirm the accuracy of the new formulas and provide an example showing limit cycle behavior.

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Fig. 1

Block diagram for a feedback control system with a quantizer, binary erasure channel, linear filter decoder, and constant-gain controller. The input to the QLF block is a scalar sensor signal, and the output a scalar control command.

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Fig. 2

Block diagram for a control system with a quantizer, binary erasure channel, and simple decoder. For the QLH, the decoder is a hold-output device, and for the QLZ system the decoder is zero-output device.

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Fig. 3

Comparison of LF (top), LH (middle), and LZ (bottom) blocks showing input and decoded signals. For LF, the system has 70% packet loss. The optimal DF parameters are gain Alf = 0.40 and delay Tlf = 0.14 time steps. For LH, the system has 90% packet loss, and the fit has gain Alh = 0.9 and delay Tlh = 8. For LZ DF, the system has 50% packet loss; Alz = 0.5; and Tlz = 0.

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Fig. 4

Convergence of the LH DF phase formula with g, for different values of α and ωτ = 0.1

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Fig. 5

LF DF gain (top plot) and phase (bottom plot) from analysis and Monte Carlo simulation, for the contact task example

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Fig. 6

Gain (top plot) and phase ω of the LH model. These curves are universal: they do not depend on the plant model or the controller.

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Fig. 7

DF gain N(a/h) for the uniform quantizer [15] with bin width h and input signal amplitude a

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Fig. 8

Gain (top) and phase (bottom) of the QLF for the contact task model: α = 10%, h = 46, and τ = 0.1

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Fig. 9

Gain (top) and phase (bottom) of the QLF for the contact task model; α = 60%, h = 10, and τ = 0.1

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Fig. 10

Gain (top) and phase (bottom) of QLH; as with LH, these properties are independent of the plant or controller: α = 10%, h = 46, and τ = 0.1

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Fig. 11

Gain (top) and phase (bottom) of QLH: α = 60%, h = 10, and τ = 0.1

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Fig. 12

Closed-loop step response of double-integrator example system, with LQG design and a one-step delay

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Fig. 13

Input amplitudes a and quantization levels h that result in a limit cycle for the double-integrator example system. Plots are made for Q, QLZ, QLH, and QLF blocks based on the DF formulas and on and simulation. Thick lines represent stable limit cycles and dashed lines unstable ones. No stable limit cycle solutions exist for the QLZ case when h < 45 (α > 11%), for QLF when h < 50 (α > 8%), or for QLH at low h.

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Fig. 14

Time series for the QLZ system with quantization level h = 70

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Fig. 15

Time series for the QLH system with quantization level h = 35. Horizontal lines are drawn for reference at amplitude ratios ± 38.5, and ±70 as predicted with the QLH DF.

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Fig. 16

Frequency ω of limit cycles from the analysis and from simulation. The QLZ model curve is nearly on top of the Q line.




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