5.5 Application of Frequency Response: Signal Filters
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1 44 Dynamic Sytem Second order lowpa filter having tranfer function H()=H ()H () u H () H () y Firt order lowpa filter Figure 5.5: Contruction of a econd order low-pa filter by combining two firt order low-pa filter Computer Tool for Calculating Frequency Repone from Tranfer Function Calculating the frequency repone and drawing it in a Bode diagram can be quite time-conuming, o you hould ue ome computer tool for doing it. A document available from the webpage of thi book at decribe uch tool in MATLAB and LabVIEW. For example, the bode-function in Control Sytem Toolbox in MATLAB doe the job. 5.5 Application of Frequency Repone: Signal Filter 5.5. Introduction A ignal filter orjutfilter i ued to attenuate (ideally: remove) a certain frequency interval of frequency component from a ignal. Thee frequency component are typically noie. For example, a lowpa filter i ued to attenuate high-frequent component (low-frequent component pae). Knowledge about filtering function i crucial in ignal proceing, but it i
2 Dynamic Sytem 45 Figure 5.6: The frequency repone of H () (firt order) i given by (5.59) and the frequency repone of H() =H () H () (econd order) i given by (5.60). ueful alo within control engineering becaue control ytem can be regarded a filter in the ene that the controlled proce variable can follow only a certain range or interval of frequency component in the reference ignal, and it will be only a certain frequency range of proce diturbance that the control ytem can compenate for effectively. Furthermore, knowledge about filter can be ueful in the analyi and deign of phyical procee. For example, a tirred tank in a proce line can act a a lowpa filter ince it dampen low-frequent component in the inflow to the tank. In thi ection we will particularly tudy lowpa filter, which i the mot commonly ued filtering function, but we will alo take a look at highpa filter., bandpa filter and bandtop filter. Signal filter can be implemented with analog electronic or with formula
3 46 Dynamic Sytem which can be programmed in a computer program. Development of programmable filtering function i decribed on page 52. Figure 5.7 how the gain function for ideal filtering function and for practical filter (the phae lag function are not hown). The paband i Amplitude gain Lowpa: 0 0 PB PB = paband SB = topband Ideal Practical SB Frequency Highpa: 0 0 SB PB Bandtop: 0 0 PB SB PB Bandpa: 0 0 SB PB SB Figure 5.7: The gain function for ideal filter and for practical filter of variou type. the frequency interval where the gain function ha value, ideally (thu, frequency component in thi frequency interval pae through the filter, unchanged). The topband i the frequency interval where the gain function ha value 0, (thu, frequency component in thi frequency interval are topped through the filter). 4 4 Itiapitythatlowpafilter were not called hightop filter in tead ince the main purpoe of a lowpa filter i to top high-frequency component. Similarly, highpa filter
4 Dynamic Sytem 47 It can be hown that tranfer function for ideal filtering function will have infinitely large order. Therefore, ideal filter can not be realized, neither with analog electronic nor with a filtering algorithm in a computer program. Tool for filter deign The Signal Proceing Toolbox in MATLAB and LabVIEW have many function for deign of ignal filter. More information about thee tool are available from the home page of thi book at Lowpa Filter Firt order lowpa filter The tranfer function of a firt order lowpa filter with input variable u and output variable y i uually written on the H() = (5.63) + where [rad/] i the bandwidth of the filter.thiiafirt order tranfer function with gain K =and time-contant T =/, cf. (4.5). The frequency repone i H(jω) = = = jω + r ³ 2 ω j arctan +e ω r ³ e 2 ω + j ³ arctan ω ωb (5.64) (5.65) The gain function i and the phae lag function i H(jω) = r ³ (5.66) 2 ω + arg H(jω) = arctan ω (5.67) hould have been called lowtop filter, but it i too late now...
5 48 Dynamic Sytem Figure 5.4 on page 5.4 how exact and aymptotic curve of H(jω) and arg H(jω) drawn in a Bode diagram. In the figure, K =and = ω c. The bandwidth define the upper limit of the paband. It i common to ay that the bandwidth i the frequency where the filter gain i / 2=0.7 = 3 db (above the bandwidth the gain i le than / 2). Thi bandwidth i therefore referred to a the 3 db-bandwidth. Now, what i the 3 db-bandwidth of a firt order lowpa filter? It i the ω-olution of H(jω) = r ³ = 2 ω + 2 (5.68) The olution i ω =.Therefore, [rad/] given in (5.63) i the 3 db-bandwidth in rad/. In Hertz the bandwidth i f b = (5.69) 2π What i the repone-time T r of a firt order lowpa filter? For firt order ytem T r equal the time-contant T.Thu, T r = T = (5.70) Let u now take a look at a imulation of a firt order lowpa filter. Figure 5.8 how the front panel of a imulator where the input ignal conit of a um of two inuoid or frequency component of frequency le than and greater than, repectively, the bandwidth. The imulation how that the low frequent component pae almot unchanged (it i in the paband of the filter), while the high-frequent component i attenuated (it lie in the topband). Example 42 The RC-circuit a a lowpa filter In Example 6 on page 34 we found the following model of an RC-circuit: RC v 2 (t) =v (t) v 2 (t) (5.7) The tranfer function from the input voltage v to the output voltage v 2 become H v2,v () = RC + = (5.72) + Thu, the RC-circuit i a firt order lowpa filter with bandwidth = rad/ (5.73) RC
6 Dynamic Sytem 49 Figure 5.8: Simulator for a firt order lowpa filter where the input ignalet conit of a um of two frequency componen If for example R =kω and C =0µF, the bandwidth i =/RC = 00 rad/. (5.73) can be ued to deign the RC-circuit (that i, to calculate the R- and C-value). [End of Example 42] Example 43 Liquid tank a a lowpa filter In Example 3 on page 23 we developed the following model of a heated liquid tank: cρv T = P + cw (T i T ) (5.74)
7 50 Dynamic Sytem After taking the Laplace tranform of the differential equation we can find the following tranfer function from the inlet temperature T i to the tank temperature T : H T,Ti () = ρv w + = (5.75) + Thu, the tank i a firt order lowpa filter with bandwidth = w rad/ (5.76) ρv The tank will attenuate temperature variation in the inlet. (5.76) can be ued to deign the tank. [End of Example 43] Higher order lowpa filter Above the filter (their tranfer function) were of order n =.However, the higher the order, the cloer the gain function of the filter will be to the ideal gain function hown in Figure 5.7. The drawback with high filter order i an increaed number of electronic component in an analog filter and a more complicated filtering algorithm in a programmed filter. Furthermore, it can be hown that the phae lag function become more negative. Higher order lowpa filter have an -polynomial of order two or higher a a denominator. Depending on the filter topology which i choen, the filter may have an -polynomial of order zero or one or higher in the numerator. Themotcommonfilter topology i Butterworth filter. Butterworth filter have the property that of all filter of ame order, Butterworth filter have the mot flattet gain curve in the paband. Butterworth filter have jut a contant a the numerator in the tranfer function. A an example of a Butterworth filter, here i the tranfer function of a econd order Butterworth filter: H() = ( ω b ) 2 +2ζω (5.77) b + where ζ (alway) ha value / 2= The frequency repone can be found a explained in Section It can be hown that in (5.77) i the 3-dB-bandwidth of the filter. Figure 5.9 how the gain function of a firt order lowpa filter and a econd order Butterworth filter having the ame bandwidth, which i rad/. The higher order filter have a gain
8 Dynamic Sytem 5 Figure 5.9: Gain function of a firt order low-pa filter and a econd order Butterworth filter having the ame bandwidth, which i rad/ function which i cloer to the ideal gain function (cloer to in the paband, and cloer to zero in the topband). Other filter topologie are Chebyhev filter and Elliptic filter. Filter in both of thee filter topologie have an -polynomial in the numerator. Thee filter give a harper corner of the paband, but there are o-called ripple or peak in the paband (in Chebyhev filter) or in the topband (Elliptic filter) Developing other Filter Uing Frequency Tranformation You can ue frequency tranformation to develop tranfer function for highpa filter, bandpa filter, and bandtop filter. In the procedure you alway tart with a known tranfer function of a lowpa filter of a proper order, for example order one or two. Thi tranfer function mut be normalized, which mean that it ha a corner frequency (which i the bandwidth for lowpa filter) of rad/. Table 5.2 how the formula for the frequency tranformation. In the table, ω L and ω H are repectively the lowet and the highet corner frequency, cf. Figure 5.7. A normalized firt order lowpa filter i H() = + (5.78)
9 52 Dynamic Sytem reulting filter type Highpa filter Bandpa filter frequency tranform ω L 2 +ω L ω H (ω H ω L ) Bandtop filter (ω H ω L ) 2 +ω L ω H Table 5.2: Frequency tranformation baed on a normalized lowpa filter. ω L og ω H are repectively the lowet and the highet corner frequency. and a normalized econd order (Butterworth) filter i H() = (5.79) Example 44 Development of a firt order highpa filter Let u find the tranfer function H HP () of a firt order highpa filter with corner frequency (lower frequency of the paband) ω L =0rad/. From Table 5.2 we ubtitute in (5.78) by ω L /, whichgive H HP () = ω L + = ω L ω L + = (5.80) The frequency repone H HP (jω) (both the gain function and the phae lag function) i plotted in a Bode diagram in Figure 5.0. Aymptote are alo drawn. [End of Example 44] Dicrete-Time Filter Developing dicrete-time filter It i common that ignal filter in form of an -tranfer function will be implemented a an filtering algorithm in a computer program. We will now ee how to develop uch an algorithm, which will calculate the filter output variable y at each dicrete tep of time. To develop uch a dicrete-time filter from -tranfer function alo called a continuou-time filter i called dicretization. Actually we can ue the ame method for dicretization of continuou-time filter a we ued for numerical calculation of
10 Dynamic Sytem 53 Figure 5.0: Frequency repone of a firt order highpa filter of corner frequency (lower frequency of the paband) ω L =0rad/ time-repone for continuou-time proce model in Chapter 3, that i, the Euler (forward) method or the Runge-Kutta (econd order) method. However, it i more common to ue a method called bilinear tranform. (It can be hown that uing the bilinear tranform i equivalent to uing trapezoid integration in the olution of differential equation [?].) In the bilinear tranform method the Laplace variable in the filter tranfer function i ubtituted by the following expreion: 2 z h +z (5.8) where h [ec] i the time-tep or time interval between each execution of the filter algorithm and the z can be regarded a a time-hifting operator, which work a follow: Multiplication by z n mean a time-delay of n time-tep: z n x(t k )=x(t k n ) (5.82)
11 54 Dynamic Sytem k i the time index (an integer), cf. Figure 3.5. Multiplication by z n mean a time-prediction of n time-tep: z n x(t k )=x(t k+n ) (5.83) Example 45 Firt order dicrete-time lowpa filter Aume given the following firt order continuou-time filter: H() = y() u() = + (5.84) where u i the input variable or ignal, which i to be filtered, and y i the reulting filter output variable. Uing the ubtitution (5.8) we get y(t k ) u(t k ) which can be written a = = 2 z h +z + = 2 z h + +z (5.85) h( + z ) ( h +2)+( h 2) z (5.86) ( h +2)y(t k )+( h 2) z y(t k )= hu(t k )+ hz u(t k ) (5.87) Uing (5.82) thi become ( h +2)y(t k )+( h 2) y(t k )= hu(t k )+ hu(t k ) (5.88) Solving for the filter output y(t k ) give y(t k )= (h 2) ( h +2) y(t h k )+ ( h +2) [u(t k)+u(t k )] (5.89) which i the dicrete-time filter algorithm or function. It hall be calculated by the filter program once during each time-tep. We ee that the calculation of y(t k ) require the filter input variable at the preent point of time, u(t k ),thefilter input variable at the previou point of time, u(t k ),andthefilter output at the previou point of time, y(t k ).Thu, u(t k ) and y(t k ) mut be aved between two conequent execution of the filter algorithm. [End of Example 45] It i important that the time-tep h i relatively mall, o that the dicrete-time filter behave approximately imilar to the original
12 Dynamic Sytem 55 continuou-time filter. Thi can be achieved by uing the following rule of thumb for chooing the time-tep h: f h 5f H [Hz] (5.90) where f i the ampling frequency and f H i the highet corner frequency of the continuou-time filter (for a lowpa filter thi corner frequency i the bandwidth). Z-tranform and z-tranfer function z-tranfer function areuedinmuchtheamewayfordicrete-time ytem a -tranfer function for continuou-time ytem. For example, from a z-tranfer function you can find the frequency repone (thi will be explained oon), and you can imulate ytem in proper imulation tool. And, you can combine ytem by multiplying or adding their z-tranfer function to get the tranfer function of the total ytem. z-tranfer function can be found by taking the o-called Z-tranform of a difference equation (not differential equation) decribing a dicrete-time dynamic ytem. Let u tudy a imple example to ee the procedure. We will calculate the tranfer function H(z) for the (5.89), which we now write in a omewhat impler way than (5.89): y(t k )=ay(t k )+b [u(t k )+u(t k )] (5.9) Firt, we take the Z-tranform of both ide of the difference equation: Z{y(t k )} = Z{ay(t k )+b [u(t k )+u(t k )]} (5.92) We will ue the following general rule for the Z-tranform: Terminology: The Z-tranform of a dicrete-time variable, ay y(t k ), produce Y (z), which here will be written a y(z) for implicity. Time-hift: (5.82) and (5.83) are ued to take the Z-tranform of time-hifted variable (time-delayed and time-predicted, repectively). Linearity: Taking the Z-tranform of linear combination of dicrete-time variable follow the ame rule a for taking the Laplace tranform of a linear combination of continuou-time variable, cf. (B.).
13 56 Dynamic Sytem Now, (5.92) become Z{y(t k )} = Z{ay(t k )} + Z{bu(t k )} + Z{bu(t k )} (5.93) = az Z{y(t k )} + bz{u(t k )} + bz Z{u(t k )} (5.94) or y(z) =az y(z)+bu(z)+bz u(z) (5.95) which can be written a y(z) = Thu, the z-tranfer function from u to y i bz + bz {z az u(z) (5.96) } H(z) bz + b u(z) (5.97) z {z a } H(z) bz + bz bz + b H(z) = az z a whichthenithez-tranfer function of the dicrete-time firt order lowpa filter (5.92) or (5.89). (5.98) Frequency repone of z-tranfer function Aume given a z-tranfer function, ay H(z), of a dicrete-time ytem. The frequency repone i H(e jωh )=H(z) z=e jωh (5.99) where h i the time-tep or ampling interval (between the dicrete point of time). It can be hown that the gain function i the abolute value of H(e jωh ): A(ω) = H(e jωh ) (5.00) and the phae lag function i the angle or the argument of H(e jωh ): φ(ω) =argh(e jωh ) (5.0) You can plot A(ω) and φ(ω) in a Bode diagram. The interpretation of the frequency repone, including A(ω) and φ(ω), i the ame for dicrete-time ytem a for continuou-time ytem, cf. Section 5.2. The following example demontrate how to calculate the frequency repone by hand from a given z-tranfer function jut to ee the mathematic involved. In practice, however, you will hardly calculate it by hand, but ue ome computer tool in tead, cf. Section
14 Dynamic Sytem 57 Example 46 Frequency repone of a z-tranfer function Given the z-tranfer function (5.98), which i repeated here: H(z) = b + bz z a The frequency repone i (5.02) H(e jωh ) = b + bejωh e jωh a b + b (co ωh + j in ωh) = co ωh + j in ωh a (5.03) (5.04) Re Im z } { z } { b ( + co ωh)+jbin ωh = (5.05) (co ωh a) + jin ωh {z } {z } Re Im q = b ( + co ωh) 2 +(inωh) 2 ej arctan[(in ωh)/(+co ωh)] q (5.06) (co ωh a) 2 +(inωh) 2 ej arctan[(in ωh)/(co ωh a)] q = b ( + co ωh) 2 +(inωh) 2 (5.07) q(co ωh a) 2 +(inωh) 2 in ωh e j{arctan( +co ωh) arctan( in co ωh a)} (5.08) The gain function i A(ω) = H(e jωh ) = b q(co ωh a) 2 +(inωh) 2 (5.09) and the phae function i φ(ω) =argh(e jωh ) = arctan µ µ in ωh in ωh arctan +coωh co ωh a (5.0) [End of Example 46] 5.6 Application of Frequency Repone: Analyi of Control Sytem Frequency repone i a good tool for analyi of feedback control ytem. Figure 5. how a block diagram of a general feedback control
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