Outline Introduction. Digital Applications (Cont d) Classification of Control Systems. Discrete-time Control Applications

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1 Ouline Inroducion Classiicaion o Conrol Sysems Classiicaion o Conrol Sysems Analog Conrollers Op-amp circuis PID Implemenaion Comparison Signals in Conrol Engineering Elemens o Conrol Compuer I/O ineraces Conrol compuer Conrol algorihm Example: Waer Level Conrol P conrol law Hyseresis conrol Coninuous-ime ime Conrol Mosly reerred o as Analog Conrol Conroller akes correcive acion coninuously in ime. Analog circui elemens are used o implemen such conrollers. Discree-ime Conrol Also known as Digial- or Numerical Conrol Correcions ake place a paricular insances in ime. Conroller s oupu says consan beween hese insances. Microprocessors are generally employed o realize hese conrollers. Hybrid Conrol A blend o boh conrol sysems (and sraegies). Chaper ME Chaper ME Discree-ime Conrol Applicaions Home appliances Milliary applicaions Dishwasher Advanced weapons sysems Washing machine Radar sysems Fridge CNC Machine Tools Roboics Elecric Moor Drivers Mobile robos Consumer goods Indusrial robos TV ses Auomaions sysems CD / DVD players / recorders Facory auomaion Camcorders Aerospace applicaions Mobile phones Aircra conrol / guidance Personal Compuing Saellies Hard disk drives Rocke / missile guidance CD-RW drives Chaper ME Digial Applicaions (Con d) Auomobiles Several compuers are on board. Some uncions perormed by hese compuers (called elecronic conrol unis or ECUs) Fuel injecion / Igniion conrol Ani-lock Break Sysems (ABS) Sabiliy and racion conrol (ani-skid) Acive suspension Climae conrol Auomaic ransmission sysem Chaper ME 534 5

2 Typical Digial Conrol Sysem Power Disurbance Typical Analog Conrol Sysem Conrol Compuer m() Conrol u() y() Elemen b() Sensor Funcion o conrol compuer: Compuaion o correcion signal Generaing command / reerence Decision making Implemenaion o complex logical operaions. Chaper ME Funcion o analog compuer: Analog ilering o he measuremen noise in he inpu signals Comparison o he measuremen (b) and he command (r) Generaion o correcion signal (m) on a coninuous basis. Chaper ME Analog Conrol using Op-amps p Ose null Invering inpu 2 Non-invering inpu 3 6 Oupu -V 4 5 Ose null _ 8 7 No connecion V Analog conrollers are requenly implemened via operaional ampliiers (or simply op-amps). One can implemen almos any desired uncion. Op-amps are very versaile ampliiers: Precise Error oleran / Robus Low-cos There exiss a wide variey o specialized op-amps or dieren applicaions: Radio/video Sonar/radar Auomaion Auomoive Insrumenaion, i ec. Chaper ME Invering Ampliier A number o dieren uncions can be implemened by employing op-amps wih various passive circui elemens. Transer uncion o his circui is Eo () s ZB () s Gi () s E () s Z () s i where Z A,Z B reer o he generalized impedances [Ω] o he componens. Noe ha he bipolar volage supply (V, -V) o he circui is cusomarily NOT shown or he sake o simpliciy. Chaper A Inegraor: Diereniaor: Z A R Z B /(Cs) Eo (s) E (s) RC s i Z A /(Cs) Z B R Eo(s) (RC)s E (s) ME i

3 Common o Op-amp p Circuis Summing Ampliier (Mixer): Buer (Volage Follower): Dierenial ee Ampliier R _ e o e i R n R e n R L e o Low-pass Filer: e e o n R Ri i e i E o () s E () s ( RC) s i R vo R 2 2 (v v ) k e Dierenial ampliier is used o ampliy small signals buried in much larger signals. R 2 resisances (along wih R s) mus be equalized o reduce he eec o common mode volage on he oupu (v o ). Chaper ME Chaper ME 534 Volage Limier Analog PID Conroller Oupu o op-amps canno exceed a cerain volage level V sa : o V sa is a ew vols (usually 2 Vols) lower han he supply volage (V s ). One can buil a volage limier using his imporan propery. To accomplish ha, wo cascaded op-amp circuis are designed: o The irs one ( Ampliier ) is o ampliy e i such ha he oupu a his sage will saurae a he some desired level. o The ollowing circui ( Aenuaor ), which has a reciprocal o he ampliier s gain, revers he ampliied volage back. As an illusraion, assume ha we would like o limi e i such ha -5 [V] e i 5[V] Le V s 5 [V] and V sa 3[V]. In his case, he gain o he ampliier is calculaed as V sa /e i,max 3/5 2.6 Hence, we choose R 0 kω and R 2 26kΩ. r() _ b() D-conrol K d I-conrol K i d d d m() Analog compuaions involved in PID: Inegraion ( d) Diereniaion (d/d) Ampliicaion (by a gain) Summaion (addiion, subracion) Transer Funcion: Ks Ks K ( τ s )( τ s 2 ) Ki Es () s s P-conrol 2 M() s d p i K p Es () Rs () Bs () Circui Parameers: τ RC τ RC R4 Ki R RC 3 2 Chaper ME Chaper ME 534 3

4 Anoher PID Conroller A more versaile version o he PID conroller can be buil by simply implemening each conrol law via a separae circui. Conroller gains can be convenienly adjused via R,R 2, and R 3. A Muli-purpose p PID Conroller Conrol Parameers: K K i K p d R R 3 4 R C R 2 C 2 Chaper ME Chaper ME Analog Conrol Conrol compuaions (such as d, d/d,,, ±, ec.) are coninuous in ime. Op-amps are used as compuing elemens. Comparison Digial Conrol All compuaions are perormed in disinc ime inervals. μps, DSPs, μcs, PLCs are commonly uilized. Hardwired d No suiable or Flexible / easily programmed. reconiguraion. Very sensiive o measuremen- and Somewha sensiive o signal process noise. conversion errors, quanizaion noise, and round-o / runcaion errors. Inexpensive or simple conrol sysems Hardware is inexpensive bu conrol bu can be quie cosly or complex sysems. soware developmen ools can be expensive. Signals in Conrol Engineering. Coninuous-ime ime signal 2. Discree-ime signal 3. Ampliude-quanized discree-ime signal 4. Ampliude-quanized coninuous-ime ime signal Chaper ME Chaper ME 534 7

5 Type : Coninuous-ime Type 2: Discree-ime ime () *() * ( ), {0, T, 2T,...} ( ) 0, else Time: [0, ) The signal ranges beween a lower bound ( min ) and an upper bound ( max ): [ min, max ] By deiniion, () 0 when < 0. 0 T 2T 3T Time: {0, T, 2T,..., kt,...} The signal ranges beween a lower bound ( min ) and an upper bound ( max ): [ min, max ] Chaper ME Chaper ME Type 3: Ampliude Quanized Discree-ime ~ * 3 () 0 T 2T 3T Time: {0, T, 2T,..., kt,...} The range o he uncion becomes ~ * Quanizaion Level: Δ max m n min () Δ loor, {0, T,...} Δ 0, else { nδ, ( n ) Δ,, 0, Δ, 2Δ,, mδ } Chaper ME Type 4: Quanized Coninuous-ime ime Time: [0, ) The range o he uncion is ~ [ min, max ] Since he ransiions o he uncion a T, 2T, 3T,... are exremely as, he uncion values predominanly reside a he quanized levels. Chaper ME 534 2

6 Properies o Digial Conrol Sysems All physical quanies are represened by corresponding (binary) numbers wih inie lengh. All compuaions are synchronized and are carried ou periodically. The period in which all hese compuaions are perormed is called sampling period (T). Chaper ME Properies (Con d) All quanies in discree-ime domain could be expressed as X( kt) X(k) where k {0,, 2,...} k is called ime index. Conrol algorihm is essenially an algeabric expression (dierence equaion) which depends on no only he hisory o error bu also ha o he manipulaion): m( k) N i a m( k i) i M j 0 b e( k j) Chaper ME j r(kt) A General Digial Conrol Sysem e(kt) Dierence Equaion m(kt) Oupu Inerace ~ m() Power u() Disurbance Conrol Lach & Conrol Command Error Algorihm D/A Elemen Plan _ Manipulaion Manipulaed Oupu Inpu Measuremen Conrol Compuer and Soware ~ b(kt) Digial Domain Clock Sampler & A/D Inpu Inerace b() Analog Domain Conrol elemens: Moor Driver Elecric Moor Servo-valve Hydraulic Cylinder / Moor Power Converer Elecric Heaer Sensor Chaper ME y() Elemens o I/O Ineraces I. Sampler II. Analog-o-Digial (A/D) Converer III. Lach IV. Digial-o-Analog o (D/A) Converer Chaper ME

7 I) Sampler II) A/D Converer () Type () T Sampler *() *() Type 2 0 T 3T Samples a coninuous-imeime signal a sampling insances. Convers eecively an analog signal (Type ) ino a discree-ime one (Type 2). I convers a volage level ino a corresponding (binary) number represenaion a a paricular insan o ime. Chaper ME Chaper ME Properies o A/D Converers Inpu Volage Range: 5V Unipolar: 0 5V 5V Bipolar: -5V 5V 0V Unipolar: 0 0V 0V Bipolar: -0V 0V Resoluion: Denoes quanizaion level Oupu is N-bi binary number Conversion ime: Time required o conver he volage ino an N-bi binary number A/D Converers (Con d) A/D converers are inegraed circuis designed speciically o do his conversion: V in [V min, V max ] V ou0,..., V oun {0, 5 V} (TTL) For convenience, oupu volage saes are represened as binary numbers: 0 Vol 0 (low logic level) 5Vols (high logic level) For A/D converer chips, prices go up as Resoluion (and accuracy) increases Conversion ime decreases Chaper ME Chaper ME

8 Unipolar A/D Converer For a unipolar A/D converer (ADC) wih 8-bi resoluion, we have he ollowing ADC oupu code: Oupu Volage a Pins [V] DB7 DB6 DB5 DB4 DB3 DB2 DB DB0 Binary Number Unsigned Ineger Chaper ME Unipolar ADC Oupu (8-bi) Oupu o ADC Inpu Volage 0 V V V V (V min ) Chaper ME Bipolar ADC Oupu (8-bi) Noe ha he ADC oupu orma in bipolar operaion is device-dependen: Device manuacurers commonly employ direc (sraigh) binary- and/or wo s wos complemen represenaions. Example A/D Converer Consider a 0V unipolar A/D converer wih 8-bi (N 8) resoluion. a) Deermine he volage resoluion o his device. b) Find he oupu represenaion (as unsigned ineger) when an inpu volage o 3.27 V is applied. (V Vmin) -28 V -27 V -26 V 2 26 V 2 27 V 2 28 V (V Vmax) Chaper ME Chaper ME

9 Soluion Par (a) Volage resoluion (i.e. quanizaion level) can be given as Hence, V V Δ V max min 2 N Δ V [ mv] Chaper ME Soluion Par (b) The corresponding number represenaion can be simply expressed as V in 3.27 oupu loor loor 83 3 Δ V where loor is a uncion rounding is argumen o he lowes ineger. Inpu Volage Range [V] Binary # Rep. Unsigned In. Rep. [0, 0.039) [3.2422, 3.283) [9.929,9.9609) [9.9609,0) 255 Chaper ME Quanizaion Error (or Accuracy) A/D conversion menioned here leads o a quanizaion error o one ΔV (or one leas signiican bi: LSB) a maximum. Such an quanizaion error migh be unaccepable or cerain applicaions. Quanizaion Error (Con d) To reduce his error, a beer A/D conversion mehod is adaped by ADC manuacurers: A bias o ΔV/2 is inernally added o V in. Quanizaion error now ranges beween -ΔV/2 and ΔV/2 (or ±½ LSB). Oupu (code) o he ADC can be expressed as oupu loor(v in /ΔV ½) Chaper ME Chaper ME

10 Inpu Inerace III) Lach () Type () Sampler & ADC ~ *() ~ *() Type 3 In pracice, sampler & ADC are considered o be a single uni: Inpu o he uni is an analog volage varying in ime, Oupu is binary number sequence wih inie word lengh. Chaper ME T 3T Lach holds a binary number during one sampling period. I is an inegraed circui which holds he inpu (N-bi digial) signal hroughou one sampling period. The oupu o he device remains he same during his period. Chaper ME IV) D/A Converer Oupu Inerace Convers an N-bi digial signal ino a corresponding volage level: Complemenary operaion o A/D converer. Imporan properies: Oupu Volage Range: 5V unipolar, 5V bipolar, 0V unipolar, 0V bipolar Resoluion (and Accuracy) Conversion Time Lach and D/A converer (DAC) are rolled ino a single uni (oupu inerace). Oupu inerace oenimes reerred o as Sample and Hold (S/H) Uni. Chaper ME Chaper ME 534 4

11 Errors in Digial Conrol Sysems Conrol Compuer As conrol compuers, here exis a wide variey o choices in pracice: PC Daa Acquisiion Board (DAQ) PC Moion Conrol Card Microconroller: Single Conrol IC Programmable Logic Conroller (PLC) Chaper ME Chaper ME A Simple Conrol Algorihm Illusraive Example. Fech sensor value b(k) 2. Fech (or generae) command r(k) 3. Compue error e(k) r(k) - b(k) 4. Calculae compensaion m(k) 5. Oupu m(k) () 6. Wai ill end o sampling period 7. Go o Sep Level Se ensor Consider he waer level conrol sysem. Servo-valve (or Prop. Flow Conrol Valve): m() is conrol volage: 0 V q i 0 l/s 5 V q i 00 l/s Sensor: b() is sensor oupu volage: 0 V h 0 m 5 V h 5 m Chaper ME Chaper ME

12 r(k) _ Conrol Sysem Conrol Algorihm PC running Conrol Program m(k) T b(k) qo() 8-bi / 5V unipolar Lach & D/A m() Servo- Valve qi() Waer Tank Clock Sampler & A/D 8-bi / 5V unipolar Daa Acquisiion Board Conrol Sysem: PC (w/ RT-OS) DAQ Conrol Law: Proporional Conrol Sampling Time: T 0. sec. D/A Converer: 8-bi / 5V unipolar A/D Converer: 8-bi / 5V unipolar b() Level Sensor h() Desired waer level: C Library Funcions Mos DAQ board manuacurers provide device drivers along wih high-level language suppor (i.e. library uncions) or ree. For his example, le us assume ha he ollowing uncions are available: read_adc(): reurns waer-level as unsigned ineger. wrie_ DAC(m): generaes oupu volage depending on he inpu argumen m. Here, m < 256 is an unsigned ineger. pause(n): delays he execuion by n miliseconds. ini(): iniializes he unis on he DAQ board. Chaper ME Chaper ME Conrol Program (ANSI C) #include <mah.h> #include <sdlib.h> #include conrol.h #deine K.2 void main() { loa r,dr,e,b,qi; unsigned in m; r 0; dr 3/3000; ini(); /* Iniialize */ while(){ /* Ininie loop */ b 5*read_ADC()/255; /* Read sensor */ r dr; i (r>3) r 3; /* Calculae cmd */ e r b; qi K*e; /* P-conrol law */ i (qi < 0) qi 0; /* 0 < qi < 00 */ i (qi > 00) qi 00; m loor(2.55*qi); wrie_dac(m); /* Oupu */ pause(00); /* Wai or 00 ms */ } } Chaper ME b() r() R R Analog P-Conroller R 2 _ m( ) K e R 2 R2 [ r( ) b( R )] m() Desired P-gain (K) can be obained by selecing a proper R 2 /R raio. Measuremen noise presen in b()is a major drawback on perormance. Generaing an arbirary r() in coninuous-ime is apparenly diicul. Chaper ME

13 Hyseresis Conrol Hyseresis Conrol (Con d) Anoher very simple conrol sraegy is hyseresis (a.k.a. bang-bang or on/o ) conrol. The power conrol elemen (PCE) is eiher swiched ully on or o depending on he magniude o error: I error > hreshold hen PCE is swiched on. I error < -hreshold hen PCE is swiched o. hreshold h is oenimes i reerred o as errorband, deadband, or olerance. Chaper ME Chaper ME Conrol Program 2 (ANSI C) #include <mah.h> #include <sdlib.h> #include conrol.h #deine dh 0. /* Deine deadband */ void main() { loa r,dr,e,b; unsigned in m; r 0; dr 3/3000; ini(); /* Iniialize */ wrie_dac(255); /* Turn on valve */ while() { /* Ininie loop */ b 5*read_ADC()/255; /* Read sensor */ r dr; i (r>3) r 3; /* Calculae cmd */ e r b; i (e > dh) m 255; /* Hi lower bound? */ i (e <-dh) m 0; /* Hi upper bound? */ wrie_dac(m); /* Oupu */ pause(00); /* Wai or 00 ms */ } } Chaper ME Quesions o be Addressed How o pick a conrol law and deermine is parameers in discree-ime domain? Sabiliy Perormance objecives Command racking Disurbance rejecion How o selec a proper sampling period? How abou hardware / soware design? Chaper ME