US20140019077A1 - Deconvolution method for emissions measurement - Google Patents

Deconvolution method for emissions measurement Download PDF

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Publication number
US20140019077A1
US20140019077A1 US14/007,111 US201214007111A US2014019077A1 US 20140019077 A1 US20140019077 A1 US 20140019077A1 US 201214007111 A US201214007111 A US 201214007111A US 2014019077 A1 US2014019077 A1 US 2014019077A1
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function
response
recited
convolution function
convolution
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US14/007,111
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Frank Berghof
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AVL Test Systems Inc
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AVL Test Systems Inc
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Priority to US14/007,111 priority Critical patent/US20140019077A1/en
Assigned to AVL TEST SYSTEMS, INC. reassignment AVL TEST SYSTEMS, INC. ASSIGNMENT OF ASSIGNORS INTEREST (SEE DOCUMENT FOR DETAILS). Assignors: BERGHOF, FRANK
Publication of US20140019077A1 publication Critical patent/US20140019077A1/en
Assigned to RBS CITIZENS, N.A., AS AGENT reassignment RBS CITIZENS, N.A., AS AGENT SECURITY INTEREST Assignors: AVL CALIFORNIA TECHNOLOGY CENTER, INC., AVL MICHIGAN HOLDING CORPORATION, AVL PEI EQUIPMENT, LLC, AVL POWERTRAIN ENGINEERING, INC., AVL PROPERTIES, INC., AVL STRATEGIC ANALYTIC SERVICES, INC., AVL TEST SYSTEMS, INC., AVL TSI EQUIPMENT, LLC
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N33/00—Investigating or analysing materials by specific methods not covered by groups G01N1/00 - G01N31/00
    • G01N33/0004—Gaseous mixtures, e.g. polluted air
    • G01N33/0006—Calibrating gas analysers
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01D—MEASURING NOT SPECIALLY ADAPTED FOR A SPECIFIC VARIABLE; ARRANGEMENTS FOR MEASURING TWO OR MORE VARIABLES NOT COVERED IN A SINGLE OTHER SUBCLASS; TARIFF METERING APPARATUS; MEASURING OR TESTING NOT OTHERWISE PROVIDED FOR
    • G01D18/00—Testing or calibrating apparatus or arrangements provided for in groups G01D1/00 - G01D15/00
    • G01D18/008—Testing or calibrating apparatus or arrangements provided for in groups G01D1/00 - G01D15/00 with calibration coefficients stored in memory
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01D—MEASURING NOT SPECIALLY ADAPTED FOR A SPECIFIC VARIABLE; ARRANGEMENTS FOR MEASURING TWO OR MORE VARIABLES NOT COVERED IN A SINGLE OTHER SUBCLASS; TARIFF METERING APPARATUS; MEASURING OR TESTING NOT OTHERWISE PROVIDED FOR
    • G01D3/00—Indicating or recording apparatus with provision for the special purposes referred to in the subgroups
    • G01D3/02—Indicating or recording apparatus with provision for the special purposes referred to in the subgroups with provision for altering or correcting the law of variation
    • G01D3/022—Indicating or recording apparatus with provision for the special purposes referred to in the subgroups with provision for altering or correcting the law of variation having an ideal characteristic, map or correction data stored in a digital memory

Definitions

  • Emissions analyzers or measurement instruments, measure certain gaseous constituents within a sample of exhaust, or aerosol, as a function of time, or are configured to measure particulate matter, such as soot, within an exhaust sample, as examples.
  • the response of the instrument may be uncorrected for the convolution of the measurement with some other signal representative of the transfer function, or the transient response, of the instrument.
  • Deconvolution is a process used to reverse, or correct, the effects of convolution.
  • the response of an instrument is recorded online in the time domain.
  • Deconvolution of the recorded signal is performed offline in post-processing by (1) decomposing the recorded data, via a Fourier transform, into the frequency domain, (2) using a model to remove the effects convolution, and then (3) constructing a convolution corrected signal, via an inverse Fourier transform, back into the time domain.
  • the method includes determining an inverse convolution function, the inverse convolution function being in the time domain.
  • the method further includes recording a response of an instrument to an exhaust sample as a function of time, and convolving the recorded response with the inverse convolution function, the result being a convolution corrected instrument response.
  • the method includes determining an idealized convolution function, the idealized convolution function being in the time domain.
  • the idealized convolution function is transformed from the time domain to the frequency domain, and a regularizing filter function is divided by the transformed idealized convolution function. The result of the division is the inverse convolution function in the frequency domain.
  • the inverse convolution function is then transformed from the frequency domain to the time domain.
  • FIG. 1 illustrates an example system including a measurement instrument configured to respond to exhaust.
  • FIG. 2 illustrates an example method of correcting a response of a measurement instrument.
  • FIG. 3 is representative of the details of the first step from FIG. 2 .
  • FIG. 4 is representative of the details of the second step from FIG. 2 .
  • FIG. 5 is representative of the details of the third step from FIG. 2 .
  • FIG. 1 illustrates an example system 10 including an engine 12 and an exhaust pipe 14 downstream thereof.
  • the engine 12 could be an engine of a vehicle, or could be a stand-alone engine in a lab, as examples.
  • the engine 12 could further be any type of engine, including a diesel engine.
  • a portion 24 b of the sample 24 a is directed toward a measurement instrument 26
  • another portion 24 c is directed toward a filter box 28 in parallel with the measurement instrument 26 .
  • the filter box 28 need not be present, however.
  • the measurement instrument 26 is a soot sensor, such as the AVL 483 Micro Soot Sensor (MSS), for example.
  • the response (or, signal) from the measurement instrument 26 is indicative of a concentration of soot, as a function of time, within the portion 24 b of the sample 24 a.
  • a controller 30 which may be any type of known computer, is in communication with the measurement instrument 26 to record the response thereof.
  • the controller 30 could include a processor (or, CPU), screen, hard drive, mouse, keyboard, etc.
  • the controller 30 is further configured to perform each of the calculations in the steps described below, and may be configured to communicate with other various components in the system 10 .
  • a reference gas source 32 selectively in communication with the sampling line 22 b by way of an adjustable valve 34 .
  • the controller 30 in one example, is configured to adjust the valve 34 , however the valve 34 could be manually adjustable.
  • the reference gas is a gas having a known soot concentration.
  • the reference gas source 32 can include an appropriate reference gas, however, as will be appreciated from the below.
  • this disclosure extends to other types of measurement instruments.
  • this disclosure extends to gas analyzers configured to measure a quantity (e.g., a concentration) of one or more gaseous constituents within a sample of exhaust, such as of CO 2 , CO, NO, NO 2 , NO x , CH 4 , HC, O 2 , NH 3 , and N 2 O, as examples.
  • the disclosed method can further be used to deconvolute data from any measurement instrument for which a convolution curve can be determined, such as temperature, pressure, flow rate, speed and torque measurements, as examples.
  • the system 10 is likewise non-limiting, and this disclosure extends to other system set-ups, including those mounted for use on-road or in a lab.
  • FIG. 2 shows a high-level overview of the steps in one example of the disclosed method.
  • the response of the system 10 (specifically, the response of the measurement instrument 26 ) to a step input signal change is measured, at 100 .
  • idealized and inverse convolution functions are then determined at 200 and 300 , respectively. Steps 100 , 200 , 300 can be performed offline, before acquiring data during engine operation.
  • steps 100 - 300 are then used in the fourth step, at 400 , to deconvolute data acquired by the measurement instrument during engine operation.
  • this data is acquired during an emissions test.
  • the deconvoluted data can be further refined in an optional fifth step, at 500 . Steps 100 - 500 are discussed in detail below.
  • n(t) a function in the time domain
  • N(f) the same function in the frequency domain
  • FIG. 3 shows the detail of step 100 .
  • a sample of reference gas which has a known quantity of a measurable exhaust component, is connected to the measurement instrument, via positioning of the valve 34 , and an uncorrected response of the instrument x(t) is recorded.
  • the reference gas would have a known soot concentration
  • a reference gas with a known HC concentration would be selected.
  • times T A , T B , and T C are determined. As generally noted, these times are times at which the amplitude of the recorded signal is at three different percentage values relative to the known signal. This is indicative of the attenuation caused by the measurement instrument and other measurement equipment. In this example, 10%, 50%, and 90% are used, for T A , T B , and T C , respectively.
  • FIG. 4 is representative of the details of step 200 , the result of which is the determination of h(t), the idealized convolution function.
  • This function generally represents an approximation of the real convolution function, using a model consisting of the Gauss function convoluted with the impulse response function:
  • g(t) is the Gaussian function
  • i(t) is the impulse response function, defined as:
  • a look-up table is used to determine the ratio.
  • the inputs to an example look-up table are T A , T B , and T C .
  • the normalized convolution function h n (t) is calculated.
  • the normalized convolution function is:
  • h n ( t ) g n ( t )* i n ( t )
  • a scaling factor k is determined at step 206 , and is defined as:
  • T A,n is the time at which ⁇ h n (t) reaches A % of its maximum value (in this example, 10%)
  • T C,n is the time at which ⁇ h n (t) reaches C % of ats maximum value (in this example 90%).
  • scaling factor k can be used to determine the parameters ⁇ , ⁇ , and ⁇ of the idealized convolution function h(t) based on the following equations:
  • T B,n is the time at which ⁇ h n (t) reaches B % of its maximum value (in this example, 50%). Having solved for these parameters, the idealized convolution function h(t) can then be determined by solving for g(t) and i(t), above.
  • step 200 the idealized convolution function h(t) could be approximated as the first derivative of the uncorrected instrument response x(t).
  • FIG. 5 generally illustrates the steps for determining the inverse convolution function k(t).
  • the idealized convolution function h(t) is transformed into the frequency domain by Fourier transformation, as follows:
  • H ( f ) F ( h ( t )).
  • a regularizing filter function R(f) is calculated from the following equation:
  • H MAG (f) is the magnitude, or absolute value, of H(f)
  • ⁇ is a positive adjustable filter parameter.
  • ⁇ is a constant, positive real value.
  • ⁇ is a function of frequency, however a constant value is typically sufficient.
  • ⁇ can be tuned to adjust the convolution corrected instrument response y(t).
  • K ( f ) R ( f )/ H ( f ).
  • R(f) and H(f) may include complex numbers, and thus, in one example, the above division follows the rules for division of two complex numbers and can be performed by dividing the magnitude of R(f) (e.g., R MAG (f)) by the magnitude of H(f) (e.g., H MAG (f)) and subtracting the phase angle of H(f) (e.g., H PHA (f)) from the phase angle of R(f) (e.g., R PHA (f))
  • the inverse convolution function K(f) is converted into the time domain by way of an inverse Fourier transformation to determine an initial inverse convolution function k init (t):
  • the regularizing filter function R(f) depends from a positive adjustable filter parameter ⁇ , which may be a constant value, and need not be frequency dependent.
  • the positive adjustable filter parameter ⁇ is generally representative of a signal to noise ratio.
  • the uncorrected instrument response x(t) recorded in step 100 is convolved with k init (t) to construct an convolution corrected instrument response y(t), at step 310 , as follows:
  • the convolution corrected instrument response y(t) is then be evaluated relative to the known reference gas signal from step 100 , at step 312 .
  • this evaluation is performed by graphically comparing the two signals, however this could also be performed using a one-dimensional optimization algorithm to minimize the sum of squares of the deviations between the deconvoluted response and the signal representative of the known data.
  • the positive adjustable filter parameter a can further be adjusted, or “tuned,” to increase the accuracy of the inverse convolution function k init (t), thus increasing the accuracy of the convolution corrected instrument response y(t) relative to the reference gas signal from step 100 .
  • Tuning is dependent on varying the constant positive adjustable filter parameter ⁇ , from which k init (t) depends.
  • the dynamic response (or, slope) of y(t) is assessed at 314 , while overshoots and undershoots (e.g., amplitude) of y(t) are accounted for at 316 .
  • increasing ⁇ would reduce the slope of y(t) (e.g., worse recovery of dynamic response) but also lower the over and undershoots.
  • ⁇ e.g., a value for ⁇ representing an acceptable compromise between error in slope and error due to over/undershoots is determined
  • the corresponding inverse convolution function is saved as k(t), at 320 , for later use in step 400 .
  • step 400 the k(t) saved at 320 is used for deconvolution of the uncorrected instrument response m(t).
  • the system 10 would be arranged as shown in FIG. 1 , for example, such that valve 34 is adjusted so that the sample 24 a sourced from the engine 12 is directed toward the instrument 26 .
  • the controller 30 executes the convolution of m(t) with k(t) by way of the following Riemann sum:
  • y i is the i-th value of the convolution corrected instrument response vector
  • m i ⁇ (j ⁇ 1) is the i ⁇ (j ⁇ 1)-th value of the uncorrected measured instrument response vector
  • k′ is the flipped inverse convolution function in the time domain (as used herein, “flipped” means that the order of the values in the vector is reversed)
  • n is the number of values in the inverse convolution function vector
  • j is the running index of the inverse convolution function vector
  • i is the running index of the uncorrected instrument response vector.
  • the calculation at step 400 can be done quickly and efficiently relative to other methods, such methods require transformations between the time and frequency domains. Post processing is thus not necessary with this disclosed method, and the convolution corrected instrument response y(t) can be determined online, during engine operation. Again, as noted above, the controller 30 can be used calculate the convolution corrected instrument response y(t).
  • the convolution corrected instrument response y(t) can be further refined to eliminate deviations that may be present at step changes.
  • this further refinement called a derivative corrected instrument response p(t) can be calculated by solving for p(t) using the following equation:
  • ⁇ is a constant
  • k(t) is the inverse convolution function from 320
  • y(t) is the convolution corrected instrument resulting from 400 .
  • p(t) is solved for iteratively, using y(t) as an initial estimate for p(t). Again, this fifth step is optional, and need not be included.
US14/007,111 2011-03-28 2012-03-14 Deconvolution method for emissions measurement Abandoned US20140019077A1 (en)

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EP3396398B1 (fr) * 2017-04-27 2020-07-08 Rohde & Schwarz GmbH & Co. KG Procédé de correction de signal, système permettant de corriger un signal mesuré ainsi qu'un oscilloscope
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US10520480B2 (en) 2011-03-28 2019-12-31 Avl Test Systems, Inc. Deconvolution method for emissions measurement
WO2016112269A1 (fr) * 2015-01-09 2016-07-14 Avl Test Systems, Inc. Système et procédé permettant de détecter une fuite dans un appareil d'échantillonnage de gaz d'échappement
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US10520480B2 (en) 2019-12-31
EP3101573A1 (fr) 2016-12-07
WO2012134815A2 (fr) 2012-10-04
EP2691901A2 (fr) 2014-02-05
JP5932018B2 (ja) 2016-06-08
CN104303017A (zh) 2015-01-21
WO2012134815A3 (fr) 2014-04-10
EP2691901B1 (fr) 2016-08-10
EP3101573B1 (fr) 2018-08-29
CN104303017B (zh) 2017-05-17
EP2691901A4 (fr) 2015-03-25
JP2014516404A (ja) 2014-07-10
US20180321205A1 (en) 2018-11-08
CA2831593A1 (fr) 2012-10-04

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