n/a 2 nd Order Butterworth Difference Equations DATE

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1 TROTTER ONTROLS TITLE BY HK D Revisions Revision A, 11/10/08 Rev, 11/03/ Added methodology for implementation of filters using Microchips DSP filter design tool. Revised cutoff frequencies for various filters and added a section for a band pass filter used to calculate the rate of door movement based on current door position. Revision B, 01/19/09 Revised update time for hopper contents filter to reduce the effective bandwidth of the hopper sensor by 2X, 3X, or 4X depending on a tweak value. Revision, 11/03/2009 Revised Sampling interval for FRDS implementation per Mark Pump notes Overview A simple methodology for developing the time difference equations for a first or second order recursive IIR Butterworth filter is presented. A digital filter design tool sold by Microhip is used to easily determine the coefficients for the difference equation. The difference equations are then reduced to the simplified format shown below in terms of coefficients A, B,, and D. The Laplace transfer function of a second order Butterworth filter with the cutoff frequency of 1Hz is H(s) = 1 / (s 2 +sqrt(2)*s+1) Note that the equation above is also valid for a first ordered Butterworth filter having a damping ratio of 1 or greater with two identical first ordered poles. The difference equation for a Butterworth filter can be determined to be represented as: y(k) = A * x(k-2) + B * x(k-1) + * x(k) + D * y(k- 2) + E * y(k- 1) where, k = 0, 1, 2, 3, and y(k) is a sequence of filtered data, x(k) is input sequence to be filtered. A, B,, D, E are constant real numbers which depend upon the sampling time. For typical sampling time such as 1ms, 5ms, 10ms, 15ms, 20ms the values are found out and given in the table. Also, a sample code for implementation in ANSI is also provided. It may be noted that the current filtered output is a function of previous two Butterworth values and previous three input values. This equation is obtained using bilinear transformation from z-plane to w-plane. Then Bode diagram was designed in w-plane so as to match Bode diagram of H(s) so as to get cutoff frequency of 1HZ. This design results in second order IIR (or Recursive) Butterworth filter. Page 1 of 10

2 TROTTER ONTROLS TITLE BY HK D Rev, 11/03/ Table of ontents Overview... 1 Table of ontents Objectives... 3 Background and equation development... 4 Design Procedure using Microhip dspifdlite... 5 Design the Filter Response... 5 Determine the Filter oefficients... 6 Develop the Discrete Time Equations... 6 D Gain orrection... 7 FRDS Butterworth Low Pass Filter oefficients... 8 A/D Sampling Scheme... 8 Design riteria and Resulting oefficients... 9 Summary and onclusions Recommendations Page 2 of 10

3 TROTTER ONTROLS TITLE BY HK D Rev, 11/03/ Objectives This report presents the difference equation for the 2 nd order Butterworth filter with cut-off frequency of 1Hz. The procedure to obtain this difference equation from transfer function is given briefly. Table is provided to give the different values of constants A, B,, D, and E for respective values of sampling time. Page 3 of 10

4 TROTTER ONTROLS TITLE BY HK D Background and equation development The transfer function of a second order Butterworth filter is Page 4 of 10 Rev, 11/03/ H(s) = 1 / (s 2 +A*s+ 1) Where, s is Laplace variable and A is a real number which can be designed to choose cut-off frequency of Butterworth filter. For the cut-off frequency of 1Hz A = sqrt(2) Now consider the pulse transfer function of the IIR filter H1(z) as Y(z) / X(z) = H1(z) = (b0 + b1 * z -1 + b2 * z -2 ) / (1 + a1 * z -1 + a2 * z -2 ) Note: See page 4 for a step by step procedure to implement this filter using the Digital Filter Design Software dspifdlite sold by Microhip. Use bilinear transformation z = (1 + T * w/2) / (1 T * w/2) and get H1(w). Plot Bode diagram for H1(w) for required value of sampling time T, values of constants b0, b1, b2, a1, a2 must be chosen so as to get the Bode diagram similar to that of H(s) with cutoff frequency at 1Hz. After satisfying Bode diagram convert H1(w) back to z-domain using bilinear transformation to get H1(z). w = (2/T) * (z 1) / (z + 1) Using inverse Z-transform on H1(z) get the difference equation in the form y(k) = A * x(k-2) + B * x(k-1) + * x(k) + D * y(k- 2) + E * y(k- 1) where, k = 0, 1, 2, 3 and y(k) is output sequence, x(k) is input sequence at k = 0, 1, 2, 3 For the first scan i.e. at k = 0, y(k-1) = y(-1) = 0, y(k-2) = y(-2) = 0, x(k-1) = x(-1) = 0 and this gives y(k) = y(0) = 0; For the second scan i.e. at k = 1, y(k-1) = y(0) = 0, y(k-2) = y(-1) = 0, x(k-1) = x(0) which is the data obtained at the first scan. This design results in second order IIR (or Recursive) Butterworth filter with the D gain at the steady state. To avoid steady state error input values x(k) must be divided by the D gain of the filter. Table 1 ~ Values of A, B,, D, E, and D Gain for a 1 hz cutoff frequency and various sampling periods (T). Sampling Time (T) in ms A 1 1 B D E D Gain e e+03

5 TROTTER ONTROLS TITLE BY HK D Rev, 11/03/ ANSI code is given below for implementation of the filter in microprocessor with sampling time T = 10ms. #define NZEROS 2 #define NPOLES 2 #define GAIN e+03 static float xv[nzeros+1], yv[npoles+1]; static void filterloop() { for (;;) { xv[0] = xv[1]; xv[1] = xv[2]; xv[2] = next input value / GAIN; yv[0] = yv[1]; yv[1] = yv[2]; yv[2] = (xv[0] + xv[2]) + 2 * xv[1] + ( * yv[0]) + ( * yv[1]); next output value = yv[2]; } } Design Procedure using Microhip dspifdlite Design the Filter Response 1. Select the IIR button at the top of the screen. This selects the impulse response filter design type. 2. Select the Filter Type -> Low Pass (this is a low pass ButterWorth design) 3. Set the sampling frequency in Hertz. This is the number of input samples to the filter in 1 second. Typically this is inverse of the scan time for the application software or A/D conversion cycle. Sampling Frequency (Hz) = 1 / Process Scan Time (Sec) 4. Set the PASSBAND frequency in Hertz. This is the frequency where attenuation begins. Frequencies from D up to this value are passed. This must be less than the Nyquist Frequency (i.e. the sampling frequency divided by 2). 5. Set the STOPBAND frequency in hertz. This is a frequency where the desired attenuation is specified. Frequencies higher than this will receive even more attenuation than specified. This frequency must be higher than the PASSBAND frequency but less than the Nyquist frequency (i.e. the sampling frequency divided by 2). PASSBAND < STOPBAND < (Sampling Frequency / 2) or Nyquist frequency The closer in frequency the PASSBAND and STOPBAND are together, the harder the filter is to realize and the higher the order of the filter. In general, make the PASSBAND and STOPBAND as far apart as is feasible for your application 6. Set the PASSBAND ripple. This is the amount of attenuation for the filter in db at the PASSBAND frequency specified. Typically this is between 1 & 6 db. A larger value yields a slower step response for the filter while a smaller value yields a steeper cutoff curve and faster step response. Note that for an analog filter, the PASSBAND equivalent or cutoff frequency ( ωc ) is traditionally defined as the -3 db attenuation point. 7. Set the STOPBAND ripple in Hertz. This is the amount of attenuation the filter should have in db at this frequency. 8. Select the of the filter or let the software pick it for you. I usually let the software pick it for me since it will pick the lowest order filter possible. 9. Select the Butterworth filter type. 10. Press Next. Various plots for the filter will be displayed. Page 5 of 10

6 TROTTER ONTROLS TITLE BY HK D Rev, 11/03/ Use the Output Plot ontrol menu to control the range of the plots. 12. Revise the STOPBAND, PASSBAND,, and attenuation parameters until the desired design parameters have been achieved. Determine the Filter oefficients Once the step response and frequency response design criteria have been met, we need to determine the coefficients for use in the following equation: Y(z) / X(z) = H1(z) = (b0 + b1 * z -1 + b2 * z -2 ) / (1 + a1 * z -1 + a2 * z -2 ) Which can be re-written as: Y(z) (a0 + a1 * z -1 + a2 * z -2 ) = X(z) (b0 + b1 * z -1 + b2 * z -2 ) To determine the coefficients for the filter: 1. Select the odegen menu selection at the top of the screen. 2. Select ANSI as the output format 3. Specify the name and location of the output file 4. Open the output file and read the coefficient from the file. A sample section of the output file is included below for reference purposes only. float FRDS HOPPER o_num[ 3] = { e-004F, /* b[ 1, 0] */ This is the bo coefficient e-004F, /* b[ 1, 1] */ This is the b1 coefficient e+000F}; /* b[ 1, 2] */ This is the b2 coefficient float FRDS HOPPER o_den[ 3] = { e+000F, /* a[ 1, 0] */ This is the a0 coefficient e-001F, /* a[ 1, 1] */ This is the a1 coefficient e+000F}; /* a[ 1, 2] */ This is the a2 coefficient Now the coefficients above can be substituted into the equations above, to yield Y(Z) in terms of X(Z) and older Y(Z) values. Develop the Discrete Time Equations Now, we want to develop the discrete time equations for use in our process. To do this the following substitutions need to be made: Y(z) = Y(k) The current filter output Y(z -1 ) = Y(k-1) The previous output of the filter (i.e. one sample old) Y(z -2 ) = Y(k-2) The previous output of the filter (i.e. one sample old) X(z) = X(k) The current filter output X(z -1 ) = X(k-1) The previous input to the filter (i.e. one sample old) X(z -2 ) = X(k-2) The previous input to the filter (i.e. one sample old) Page 6 of 10

7 TROTTER ONTROLS TITLE BY HK D Rev, 11/03/ Y(z) (a0 + a1 * z -1 + a2 * z -2 ) = X(z) (b0 + b1 * z -1 + b2 * z -2 ) (equation repeated for clarity) OR Y(z)*a0 + Y(z -1 )*a1 + Y(z -2 )*a2 = X(z)*b0 + X(z -1 )*b1 + X(z -2 )*b2 onverting the above equations to the discrete form of the equation by replacing (z) with (k) as required yields: Y(k)*a0 + Y(k-1)*a1 + Y(k-2)*a2 = X(k)*b0 + X(k-1)*b1 + X(k-2)*b2 OR Y(k) = [X(k)*b0 + X(k-1)*b1 + X(k-2)*b2 - Y(k-1)*a1 - Y(k-2)*a2]/a0 (discrete time equation) k = current point in time k-1 = previous point in time (i.e. one sample old) k-2 = point in time that is two samples old To simplify the actual calculations during runtime, it is convenient to replace the equation shown above using constants A, B,, D, E. Y(k) = A * X(k-2) + B * X(k-1) + * X(k) + D * Y(k- 2) + E * Y(k- 1) These constants are defined below in terms of the coefficient determined from the digital filter design software provided by Microchip. A = (b2 / a0) / D GAIN B = (b1 / a0) / D GAIN = b0 / a0 / D GAIN D = - (a2 / a0) E = - (a1 / a0) D GAIN = Defined below D Gain orrection The filter typically will not have a D gain of unity. To correct for this, the steady state value for the filter should be determined using the Digital Filter Design software. For a unity step input the filter will converge to a final steady state value (i.e. look at the final value of the step response for a unity step input). Filter Unity Input = D GAIN To avoid steady state error, the filter input values X(k) must be divided by the D GAIN of the filter. This is incorporated in the above values calculated for A, B and coefficients. Page 7 of 10

8 TROTTER ONTROLS TITLE BY HK D Rev, 11/03/ FRDS Butterworth Low Pass Filter oefficients A/D Sampling Scheme The following parameters are sampled during time critical automatic delivery and during non-time critical system operation. Table 2 ~ Automatic delivery sensor sampling scheme. Sensor Time Slice 1 Time Slice 2 Time Slice 3 Gatebox Angle X X X Hydraulic Pressure X Accelerometer 1 X Accelerometer 2 X Notes: {x} 1. The effecitive sample period for Gatebox Angle = up Scan Period 2. The effective sample period all other sensors = up Scan time / 3 3. Actual up Scan Period (delivery active) = 1.82 ms or 550 hertz Table 3 ~ Non-delivery mode sample scheme. Sensor Time Slice 1 Time Slice 2 Time Slice 3 Gatebox Angle X X X Hydraulic Pressure X X X Accelerometer 1 X X X Accelerometer 2 X X X Foam X Hopper {4} 1/12 (default) Battery Volts X X X Temperature N/A N/A N/A Photo Eye {3} Notes: {x} 1. The scan time varies between 120 and 190 scans/second 2. The slowest Actual up Scan Period (delivery not active) = 8.3 ms or 120 hertz 3. Photo eye scanned once every 256 up scans. 4. Hopper sensor is scanned 1/3, 1/6, 1/9, 1/12 scans depending on the value of a system tweak parameter. This allows tuning of filter bandwidth for optimal response. Note that data sampled above is then processed via the appropriate ButterWorth filter. Skipping up scans during scanning, extends the response time of the ButterWorth filter proportionally. Page 8 of 10

9 TROTTER ONTROLS TITLE BY HK D Design riteria and Resulting oefficients The following design criteria exist for the various sensor filters: Sensor Page 9 of 10 oefficient Set Rev, 11/03/ Passband Stopband Passband Attenuation (db) Stopband Attenuation (db) Sample Frequency (dump mode) Sample Frequency (other modes) Gatebox Angle 1 50hz 20 hz {1} 120 Hydraulic 2 1 hz 10 hz {2} 120 Pressure Accelerometer hz 10 hz {2} 120 Accelerometer hz 10 hz {2} 120 Battery Volts 2 1 hz 10 hz N/A {2} 120 Foam hz 1 hz N/A 120 Hopper {5} hz {5} 1 hz {5} N/A 120 Temperature N/A N/A N/A N/A N/A N/A N/A Photo Eye 3 {3} 0.01 hz 0.1 hz N/A 12 {3} Notes: {x} 1. Sampling frequency of 555 hertz was used when calculating the filter coefficient calculations for the gatebox angle filter. 2. Sampling frequency of 120 hertz was used when calculating the filter coefficient calculations for the hydraulic pressure and accelerometer filters. 3. The same coefficients as were used for the Foam and Hopper sensor filters are used. This sensor is sampled 1/256 as often to yield a the lower effective filter frequency. 4. The order of all filters was specified as 1 in the filter design tool. 5. Hopper sensor is scanned once every 1/3, 1/6, 1/9, or 1/12 scans depending on the value of a system tweak parameter (default is 1/12). This allows tuning of filter bandwidth for optimal response. The real world passband and stopband frequency for this filter can be calculated from the filter values shown in the table divided by the tweak value divisor (1, 3, 6, 9, 12). Filter oefficients Table 4 ~ oefficients for use in FRDS GENII low pass filter equations. oefficient Set#1 oefficient Set#2 (Gatebox) (Hyd Press, Accel, Volts) oefficient Set#3 (Hopper, Foam, PhotoEye) A e e e+000 A e e e-001 A e e e+000 B e e e-003 B e e e-003 B e e e+000 D GAIN A NOT USED {1} NOT USED {1} NOT USED {1} B E E E E E E-03 D NOT USED {1} NOT USED {1} NOT USED {1} E E E E-01 Notes: {x}

10 TROTTER ONTROLS TITLE BY HK D Rev, 11/03/ oefficients A & D are zero since this is a first order IIR filter implementation. 2. The D gain term is already accounted for in the coefficients A, B,, D, and E. Summary and onclusions A difference equation for Butterworth filter is developed and presented. From the difference equation it may be noted that the current Butterworth filter output value depends upon the past 2 Butterworth filter output values, the current filter input value and the past 2 input values for a second ordered filter implementation. For a order Butterworth implementation, the current Butterworth filter output value depends upon a single Butterworth filter output values, the current input value and a single past input value. The advantage of the first ordered implementation is a significant decrease in data memory space (float values take 4 bytes each) since the values for coefficient B are identical and coefficients A & D are zero. This greatly simplifies the computations required as compared to a second ordered implementation with a slight increase in the filters step response time. Utilizing the Microhip digital filter design software greatly simplifies design of the discrete time form of the filter since the filter coefficients are automatically generated using the odegen function once the filter design criteria has been met. Recommendations Use anti-aliasing analog filters to limit the bandwidth of the incoming analog signal to no more than 2X the sampling frequency used (the NyQuist frequency). The analog filter can be a simple single order R filter network or a more complex higher order low-pass filter. Use a sampling scan time as small as possible (in the range of 1 ms - 20ms) so that the difference equation approaches the performance of a continuous real time filter. For low performance applications, it is acceptable to scale the time response of the filter by skipping scans in order to simplify scaling of filter responses in time without developing new coefficients. Note, it is important to set the analog input filter to no higher than 2X the actual sampling frequency used to minimize aliasing. The scan time is assumed to be constant in deriving difference equation of y(k). And hence real time programming of this algorithm must be done with a fixed scan time for the response time to be accurate. In non-critical applications, the scan time can be obtained from the average time taken by the processor to complete one scan. For accurate results, a system timer can be used to accurately control the scan time. This method requires that the timer overflow value be set so that it is longer than the LONGEST scan for any condition. This method is used for FRDS calculations since deterministic performance is required. Page 10 of 10