Note Introduction to Digitl Control Systems Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd
. Introduction A digitl control system is one in which the trnsfer function, representing the compenstor uilt with nlog components, re now replced with digitl computer tht performs clcultions tht emulte the physicl compenstor. The following is n exmple of using digitl control system for imuth position control. The structure of typicl digitl controller is s follows. Controller The signls in the ove control loop tke on two forms: digitl or nlogy. Up to this point we hve used nlogy signls exclusively. Digitl signls, which consist of sequence of inry numers e.g., cn e found in loops contining digitl computers. Loops contining oth nlog nd digitl signls must provide mens for conversion from one form to the other s required y ech susystem. A device tht converts nlogy signls to digitl signls is clled n nlog-to-digitl A/D converter. Conversely, device tht converts digitl signls to nlog singles is clled digitl-tonlog D/A converter. In n A/D converter, the nlog signl is smpled t periodic intervl nd then held over the smpling intervl y device clled ero-order smple-nd-hold.o.h. Smples re held efore eing digitied ecuse certin time period is required for n A/D converter to convert n nlog voltge to its digitl form or, in other words, the constnt nlog voltge must e present during the conversion process. Idel smpling nd the.o.h. re presented in the following figure. Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd
ft: Anlog signl f*t: Smpled wveform fht:.o.h. output f*t is the smpled wveform, consisting of the smples, fkt. Conversion from the nlog signl ft to the smple, fkt, occurs repetedly t instnts of time T seconds prt. T is the smpling intervl or smpling time, /T is the smpling rte in Hert, nd k cn tke on ny integer vlue etween nd. Difference Equtions Digitl control systems cn e modeled dequtely y the discrete equivlent to the differentil eqution, nmely the difference eqution. For exmple, the generl secondorder difference eqution y kt y kt T y kt T = x kt x kt T x kt T where y is the system output nd x is the system input. Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd 3
3. -Trnsform In nlog or continuous control systems, we used Lplce trnsforms in our nlysis. In digitl control systems we need to use new trnsformtion in order to simplify our nlysis, which is clled the -trnsform. The -trnsform is defined y { f KT } = F = k = f kt k Exmple Find the -trnsform of smpled unit rmp. The -trnsform my e otined y using tle, much the sme wy s the Lplce trnsform. The -trnsform conversion tle is given in Tle nd the properties of - trnsform re provided in Tle. Tle - nd s-trnsform Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd 4
Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd 5 Tle -trnsform theorems In Tle, the Rel trnsltion theorem tells us } { F nt KT f n = Applying the rel trnsltion theorem to the previous generl second-order difference eqution, we hve X X X Y Y Y = The ove eqution then results in the discrete trnsfer function or X Y =
4. Controller Design vi the s-plne There re numer of strtegies or methods tht could e used for the design of discrete controllers. To illustrte the implementtion of digitl controllers we will consider method tht llows us to design controllers vi the s-plne nd then to convert the design into discrete form. The Tustin trnsformtion is used to trnsform the continuous compenstor, Gcs, to the digitl compenstor, Gc. The Tustin trnsformtion is given y nd its inverse y s = T T s = T s As the smpling intervl, T, gets smller high smpling rte, the digitl compenstor's output yields closer mtch to the nlog compenstor. If the smpling rte is not high enough, there is discrepncy t higher frequencies etween the digitl nd nlog frequency responses. Prolem 977 s 6 A controller ws designed with G c s =. If the system is to e computer s 9. controlled, find the digitl controller Gc. Use the smpling time of. second. Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd 6
5. Implementing the Digitl Compenstor Consider the following lock digrm which my e prt of igger control system: The input to the digitl compenstor or controller is the smpled error signl E, nd its output is X, which is used to drive the plnt. Now we will see how to implement the digitl compenstor, Gc, within digitl computer. For this, we hve two steps: Step : Derive the difference eqution from the digitl trnsfer function, y tking the inverse -trnsform nd using the inverse rel trnsltion theorem, i.e., { n F } = f KT nt Step : Develop flowchrt for the digitl compenstor sed on the difference eqution, nd then progrm e.g. using Mtl or Simulink to relie it. Exmple Let s consider digitl compenstor, Gc, G c X = = E.5.5.7 Step : Derive the difference eqution from the digitl trnsfer function. Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd 7
Step : Develop flowchrt for the digitl compenstor sed on the difference eqution, nd then progrm to relie it. e kt x kt e kt T x kt T e kt T x kt T The ove flowchrt shows tht the compenstor cn e implemented y storing severl successive vlues of the input nd output. The output is then formed y weighted liner comintion of these stored vriles. In Simulink, the lock of Unit Dely, i.e., is used to perform dely of one smple period. Thus, if using Simulink to relie the digitl compenstor, we cn use the lock of Unit Dely nd the lock of Gin to simply replce the corresponding locks in the ove flowchrt so s to crete Simulink model. Deprtment of Mechnicl Engineering, University Of Ssktchewn, 57 Cmpus Drive, Ssktoon, SK S7N 5A9, Cnd 8