Digital Control Sytem - A D D A Micro ADC DAC Proceor Correction Element Proce Clock Meaurement A: Analog D: Digital
Continuou Controller and Digital Control Rt - c Plant yt Continuou Controller Digital Controller rt A/D - Digital Controller rkt D/A and Hold pt Plant yt mkt D/A mt
Application of Automatic Computer Controlled Sytem Machine Tool Metal Working Procee Chemical Procee Aircraft Control Automobile Traffic Control Automobile Air-Fuel Ratio Digital Control Improve Senitivity to Signal Noie. 3
Digital Control Sytem A digital computer may erve a a compenator or controller in a feedback control ytem. Since the computer receive data only at pecific interval, it i neceary to develop a method for decribing and analying the performance of computer control ytem. The computer ytem ue data ampled at precribed interval, reulting in a erie of ignal. Thee time erie, called ampled data, can be tranformed to the -domain, and then to the -domain by the relation e t. Aume that all number that enter or leave the computer ha the ame fixed period T, called the ampling period. A ampler i baically a witch that cloe every T econd for one intant of time. 4
Sampler rt Continuou r*t Sampled Zero-order Hold o Pt rkt 0 T T 3T r3t rt rt 4T r4t 0 e T T e T T 3T 4T 5
Analog to Digital Converion: Sampling An input ignal i converted from continuo-varying phyical value e.g. preure in air, or frequency or wavelength of light, by ome electro-mechanical device into a continuouly varying electrical ignal. Thi ignal ha a range of amplitude, and a range of frequencie that can preent. Thi continuouly varying electrical ignal may then be converted to a equence of digital value, called ample, by ome analog to digital converion circuit. There are two factor which determine the accuracy with which the digital equence of value capture the original continuou ignal: the maximum rate at which we ample, and the number of bit ued in each ample. Thi latter value i known a the quantiation level 6
Zero-Order Hold The Zero-Order Hold block ample and hold it input for the pecified ample period. The block accept one input and generate one output, both of which can be calar or vector. If the input i a vector, all element of the vector are held for the ame ample period. Thi device provide a mechanim for dicretiing one or more ignal in time, or reampling the ignal at a different rate. The ample rate of the Zero-Order Hold mut be et to that of the lower block. For low-to-fat tranition, ue the Unit Delay block. 7
The -Tranform The -Tranform i ued to take dicrete time domain ignal into a complexvariable frequency domain. It play a imilar role to the one the Laplace tranform doe in the continuou time domain. The -tranform open up new way of olving problem and deigning dicrete domain application. The - tranform convert a dicrete time domain ignal, which i a equence of real number, into a complex frequency domain repreentation. For a ignal t I{ r * t} Z{ r t} Z{ r * t} Z{ f r * t t} r kt δ t kt k 0 0, Uing the Laplace tranform, we have U r kt e k 0 F e T k 0 r kt k 0 kt f kt k k 8
9 Tranfer Function of Open-Loop Sytem Zero-order Hold o Proce rt T r*t 0.3678.3678 0.644 0.3678 fraction : Expanding into partial * ; e e R Y e t t p o p t o
0 Cloed-Loop Feedback Sampled-Data Sytem rt R E Y Y D D T R Y R E Y Y D
Now Let u Continue with the Cloed-Loop Sytem for the Same Problem 5 4 3 3.47.4.4 0.3678 0.63.63 0.644 0.3678 0.63 0.644 0.3678 : unit tep input Aume an a 0.63 0.644 0.3678 Y Y R R Y
Stability Analyi in the -Plane A linear continuou feedback control ytem i table if all pole of the cloed-loop tranfer function T lie in the left half of the -plane. In the left-hand -plane, σ <0; therefore, the related magnitude of varie between 0 and. Accordingly the imaginary axi of the -plane correpond to the unit circle in the -plane, and the inide of the unit circle correpond to the left half of the -plane. A ampled ytem i table if all the pole of the cloed-loop tranfer function T lie within the unit circle of the -plane. e T e σ jω T e σt ωt
Example 3.5: Stability of a cloed-loop ytem rt o p Yt K K0.3678 0.644 K a b p ;.3678 0.3678 a a The pole of the loed - loop tranfer function t are the root of the equation [ ] 0 : a a Ka Kb 0 K ; 0.63 0.5 j0.68 0.5 j0.68 0 The ytem i table becaue the root lie within the unit circle, When K 0.30 3.0.5 j.95.5 j.95 untable Thi ytem i table for :0 K.39 Second - order ampled ytem i untable for increaed gain where the continuou i table for all value of gain. 3
Deign Procedure Start with continuou ytem. Add ampled-data ytem element. Choe ample period, uually mall but not too mall. Ue ampling period T / 0 f B, where f B ω B / π where ω B i the bandwidth of the cloed-loop ytem. Digitie control law. Check performance uing dicrete model or SIMULINK. 4
Start with a Continuou Deign; D may be given a an exiting deign or by uing root locu or bode deign. E rt R D Y Y 5
Add Sample Neceary for Digital Control Tranform D to D: We will obtain a dicrete ytem with a imilar behavior to the continuou one. Include D/A converter, uually a ero-order-device. Include A/D converter modeled a an ideal ampler. And an antialiaing filter, a low pa filter, unity gain filter with a harp cutoff frequency. Choe a ample frequency ω B baed on the cloed-loop bandwidth of the continuou ytem. 6
7 Cloed-Loop Sytem with Digital Computer Compenation b a K B A C e B e A D Z B A C D b a K D D K r r k D T p D E U D D T R Y bt at c c ; ; ; } { ; ; 0.40 0.789 0.5 ; 0.40 0.789.359. and 0.3678 and have the two parameter at We cancer the pole of 0.3678 we elect If ; 0.3678 0.789 0.3678 ; when plant ero - order hold and a Conider the econd order ytem with a the computer i tranfer function of The
Compenation Network 0.3; page 557 The compenation network, c i cacaded with the unalterable proce in order to provide a uitable loop tranfer function c H. Compenation R - c Y jω H c c K p K i N j Firt order compenator When p, the network i called a phae - lead network M p i i -p - 8 σ
Cloed-Loop Sytem with Digital Computer Compenation There are two method of compenator deign: the c -to-d converion method, and the root locu -plane method. The c -to-d converion method A a c K Firt - Order Compenator b A D C Digital Controller B Z{ c } D - tranform at ; bt a e B e ; C K when b A B 0 9
Example 3.7: Deign to meet a phae margin pecification a p phae margin of 740. We will attempt to deign c o that we achieve 0.5 Uing the Bode diagram of Baed on 0.4, we find that the required pole - ero ratio i α 6.5 Eq 0.8. ωc ab o 45 with a croover frequeny ωc 5 rad/ Fig 0.0., we find that the phae margin i o p Eq0.4. ee page 587, a 50 b 3; c K 50 3 We elect K in order to yield c jω when ω ωc 5 rad/.then K 5.6. Now the compenator c i to be realied by D.Set T 0.00econd. We have 0.05-0.3 4.85 0.95 A e 0.95, B e 0.73, and C 4.85; D 0.73 and 0
The Root Locu of Digital Control Sytem R - D Zero Order hold K p Y 4. Y K D ; R K D Plot the root locu for the characteritic equation of. The root locu lie on a ection of K D 0 or K D 0 Characteritic equation. The root locu tart at the pole and progree to the ero. the real axi to the left of 3. The root locu i ymmetrical with repect to the horiontal real axi. K D the ampled ytem a K varie. an odd number of and K D 80 o ± k360 o pole and ero.
Untable Root Locu of a Second Order Sytem K increaing Im {} Root locu Unit circle One ero At - Re {} -3 - - 0 pole at K K Let σ and olve for K 0 K σ σ df σ dσ 0; σ F σ 3; σ
Deign of a Digital Controller In order to achieve a pecified repone utiliing a Ue - a to cancel one pole at that lie on the poitive real axi Select - b o that the locu of at a we will elect a of a et of controller the - plane. the compenated ytem will give complex root D root locu method, a b deired point within the unit circle on the - plane. 3
Example 3.9: Deign of a digital compenator Let u deign a compenator D that will reult in a table ytem when p If i a decribed in Example3.8. a With D, we have untable ytem.select D b K a K D b we elect a and b we have K D k 0. Uing the equation for F σ, we obtain the entry point a 0., The root locu i on the unit circle at Thu the ytem i table for K 0.8. K 0.8. -.56. 4
If the ytem performance were inadequate, we would improve the root locu by electing a and b K D K 0.98-0.98 o that K Then the root locu would lie on the real axi of the - plane. When K, the root of the characteritic equation i at the origin. 5
Im{} K increaing K0.8 Unit circle Re{} Entry point at -.56-0. Root locu 6
P3.0 p ; T 0.; D K 0 0.0037 0.006 a The tranfer function D K.368 0.3679 0.0037 0.006 b The cloed - loop ytem characteritic equation i K 0.368 0.3679 c Uing root locu method, maximum value of K i 39. d Uing Figure3.9 for T/τ and maximum overhoot of 0.3, we find K 75. 0.759 0.98 e When K 75; T.09 0.566 f When K 9.5, the pole are 0.464± j0.6843. The overhoot i 0.55. 7
for a ramp input are PO b Ue d Ue to D method T ; C to D method T ; C P3. a a c K b By uing Bode Plot, we may elect a 0.7, b The compenated ytem overhoot and afety - tate tracking error A e A e A e A e at 0.007 at 0.07 c ; B e c 0.934; B e ; B e bt bt 0.993; B e A B A B 30% and e 0.0 0.0 K K a b 0.99; C a b 0.999; C 0.0. 0. : D C 55.3 0.0 : D C 50 0., and K A B 50. A 55.3 B 50 0.934 0.99 0.993 0.999 8