Thus the outputs of a digital controller shou ld 64257rst be converted into analog signals before being applied to the systems Another way to lo ok at the problem is that the high frequency components of should be removed before applying to analog ID: 23773 Download Pdf

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Thus the outputs of a digital controller shou ld 64257rst be converted into analog signals before being applied to the systems Another way to lo ok at the problem is that the high frequency components of should be removed before applying to analog

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Digital Control Module 1 Lecture 4 Module 1: Introduction to Digital Control Lecture Note 4 1 Data Reconstruction Most of the control systems have analog controlled processe s which are inherently driven by analog inputs. Thus the outputs of a digital controller shou ld ﬁrst be converted into analog signals before being applied to the systems. Another way to lo ok at the problem is that the high frequency components of ) should be removed before applying to analog devices. A low pass ﬁlter or a data reconstruction device is necessary to pe rform this operation.

In control system, hold operation becomes the most popular w ay of reconstruction due to its simplicity and low cost. Problem of data reconstruction can be formulated as: “ Given a sequence of numbers, (0) ,f ,f (2 ,f kt , a continuous time signal , t , is to be reconstructed from the information contained in th e sequence. Data reconstruction process may be regarded as an extrapola tion process since the continuous data signal has to be formed based on the information availab le at past sampling instants. Suppose the original signal ) between two consecutive sampling instants kT and ( +1) is

to be estimated based on the values of ) at previous instants of kT , i.e., ( 1) , ( 2) 0. Power series expansion is a well known method of generating t he desired approximation which yields ) = kT )+ (1) kT )( kT )+ (2) kT 2! kT ..... where, ) = ) for kT +1) and kT ) = dt kT for = 1 ,... Since the only available information about ) is its magnitude at the sampling instants, the derivatives of ) must be estimated from the values of kT ), as (1) kT kT (( 1) )] Similarly, (2) kT (1) kT (1) (( 1) )] where, (1) (( 1) (( 1) (( 2) )] I. Kar

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Digital Control Module 1 Lecture 4 1.1

Zero Order Hold Higher the order of the derivatives to be estimated is, larger will be the number of delayed pulses required. Since time delay degrades the stability of a closed loop control system, using higher order derivatives of ) for more accurate reconstruction often causes serious sta bility problem. Moreoverahighorderextrapolationrequirescomp lexcircuitryandresultsinhighcost. For the above reasons, use of only the ﬁrst term in the power se ries to approximate ) during the time interval kT t < +1) is very popular and the device for this type of extrapolation is known as

zero-order extrapolator or zero order hold. It ho lds the value of kT ) for kT t < +1) until the next sample (( +1) ) arrives. Figure 1 illustrates the operation of a ZOH where the green line represents the original continuous signal and brown line represents the reconstructed signal from ZOH. 10 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 Time (t) f(t), f (t) Reconstructed signal Original signal Figure 1: Zero order hold operation The accuracy of zero order hold (ZOH) depends on the sampling f requency. When 0, the output of ZOH approaches the continuous time signal.

Zero or der hold is again a linear device which satisﬁes the principle of superposition. Figure 2: Impulse response of ZOH I. Kar

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Digital Control Module 1 Lecture 4 The impulse response of a ZOH, as shown in Figure 2, can be written as ho ) = ho ) = TS ho jw ) = jwT jw sin( wT/ 2) wT/ jwT/ Since , we can write ho jw ) = sin( πw/w πw/w jπw/w Magnitude of ho jw ): ho jw sin( πw/w πw/w Phase of ho jw ): jw ) = sin( πw/w πw rad The sign of sin( πw/w ) changes at every integral value of πw . The change of sign from + to can be

regarded as a phase change of 180 . Thus the phase characteristics of ZOH is linear with jump discontinuities of 180 at integral multiple of . The magnitude and phase characteristics of ZOH are shown in Figure 3. At the cut oﬀ frequency , magnitude is 0 636. When compared with an ideal low pass ﬁlter, we see that instead of cutting of sharply at , the amplitude characteristics of ho jw ) is zero at and integral multiples of 1.2 First Order Hold When the 1 st two terms of the power series are used to extrapolate ), over the time interval kT < t < +1) , the device is called a

ﬁrst order hold (FOH). Thus ) = kT )+ kT )( kT where, kT ) = kT (( 1) ) = kT )+ kT (( 1) kT Impulse response of FOH is obtained by applying a unit impuls e at = 0, the corresponding output is obtained by setting = 0 ,..... for = 0 when 0 t < T, f ) = (0)+ (0) I. Kar

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Digital Control Module 1 Lecture 4 0.2 0.4 0.6 0.8 w/w |G h0 (jw)|/T Frequency Response of Zero Order Hold −3000 −2000 −1000 w/w Phase of G h0 (jw) Figure 3: Frequency response of ZOH (0) = 1 [impulse unit] ) = 0 ) = 1+ in this region. When t < ) = )+ (0) Since, ) = 0 and (0) = 1, ) = 1 in

this region. ) is 0 for , since ) = 0 for . Figure 4 shows the impulse response of ﬁrst order hold. Figure 4: Impulse response of First Order Hold If we combine all three regions, we can write the impulse resp onse of a ﬁrst order hold as, ) = (1+ )+(1 (1+ (1 = (1+ (1 I. Kar

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Digital Control Module 1 Lecture 4 One can verify that according to the above expression, when 0 t < T , only the ﬁrst term produces a nonzero value which is nothing but (1+ t/T ). Similarly, when t < , ﬁrst two terms produce nonzero values and the resultant is (1 t/T ). In case

of , all three terms produce nonzero values and the resultant is 0. The transfer function of a ﬁrst order hold is: ) = 1+ Ts Ts Frequency Response jw ) = 1+ jwT jwT Magnitude: jw 1+ jwT jw 1+ sin( πw/w πw/w Phase: jw ) = tan (2 πw/w πw/w rad The frequency response is shown in Figure 5. 0.5 1.5 w/w |G h1 (jw)|/T Frequency Response of First Order Hold −3000 −2500 −2000 −1500 −1000 −500 w/w Phase of G h1 (jw) Figure 5: Frequency response of FOH Figure 6 shows a comparison of the reconstructed outputs of Z OH and FOH. I. Kar

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Digital Control Module 1 Lecture 4 10 −1.5 −1 −0.5 0.5 1.5 Time (t) f(t), f (t) o/p of ZOH original signal o/p of FOH Figure 6: Operation of ZOH and FOH I. Kar

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