0/8/05 Lesson 6: Basc Control Modes ET 438a Automatc Control Systems Technology lesson6et438a.tx Learnng Objectves Ater ths resentaton you wll be able to: Descrbe the common control modes used n analog control systems Lst the characterstcs o common control modes Wrte the tme, Lalace and transer unctons o common control modes Identy the Bode lots o common control modes Desgn OP AMP crcuts that realze theoretcal control mode erormance. lesson6et438a.tx
Gan (db) 0/8/05 Control Modes-Proortonal Control Acton Process characterstcs or otmum results: ) Small rocess caactance ) Rad load changes Lmtatons: Small steady-state error may requre hgh gan to acheve accetable error levels Mathematcal reresentatons Tme uncton: v K e v Lalace uncton: (s) K E(s) Transer uncton: o (s) K E(s) Where: e = tme doman error sgnal K = roortonal gan v o = controller outut wth e=0 v = controller tme doman outut Note: Intal condton v o =0 on Lalace uncton lesson6et438a.tx 3 Proortonal Control Frequency Resonse Bode lots o three values o K : K =0., K =, and K 3 =00 60 K K 3 K 40 0 0 0 Note: gan s ndeendent o requency. Practcal realzaton: Non-nvertng OP AMP crcut 40 0 00 0 3 0 4 Frequency (rad/sec) K K K3 lesson6et438a.tx 4
0/8/05 Motor Seed Control Examle 6-: Determne the eect o alyng roortonal control to the block dagram shown below. The motor roduces the ollowng results wth the control loo oen: T L = 0.05 N-m T L = 0.075 N-m (50% ncrease n load) T = 9.4 dc T =9.4 I a =.033 A I a =.45 A w = 300 rad/sec w = 9.7 rad/sec r + e Controller K v Power T Motor & Suly G am Load w - c m Tach Generator, K tac lesson6et438a.tx 5 Motor Seed Control Motor Parameters: T = 0.0 N-m K T = 0.06 N-m/A K e = 0.06 -sec/rad R a =. ohms K tac = 0. -sec/rad K G am = 0 / r Soluton: Fnd the error roduced and the setont value, r. Then wrte equatons around control loo. + - e c m K G am =0 T = 9.4 K tac = 0. -s/rad e r c m T e K G w =300 rad/sec am c m K tac w 0.- s/rad (300 rad/s) 33 lesson6et438a.tx 6 3
0/8/05 Examle 6- Soluton () Combne equatons e r c m T e K G am T K G am r c m T K G am c m r Substtute values Smly Comute error Substtute values Smly 9.4 33 34.94 0/ r 34.94 e r c e 34.94 33.0 e.94 m The ntal setont value o r=34.94 wth an error o.94 at a seed o 300 rad/s and T L =0.05 N-m lesson6et438a.tx 7 Examle 6- Soluton (3) When torque ncrease to T L =0.075 N-m new T s dened by T (r c Substtute n Tachometer ormula T (r K m tac ) K G am w ) K G Snce setont, r does not change error must change due to measured seed change. am Motor equatons e b T I e a R K w m a e b Combne these equatons to get: T I a R a K e w Need two equatons to nd both T and w m. lesson6et438a.tx 8 4
0/8/05 Examle 6- Soluton (4) Substtute n know values and smly equaton to get rst relatonsh. r =34.94 K G am =0 K tac = 0. -s/rad T T T (r K tac w ) K G 34.94 0.w 349.4.w am 0 () Equaton Now use the motor armature crcut equaton and the armature current or T L =0.075 N-m to nd second ndeendent equaton. I a =.45 A K e = 0.06 -s/rad R+ =. ohms T T T I a R (.45 A) (. ) (0.06 -s/rad) w.74 0.06w a K w e () Equaton lesson6et438a.tx 9 Examle 6- Soluton (5) Place equatons () and ) nto standard orm and solve smultaneously usng sotware or calculator. T T T T 349.4.w.w.74 0.06w 0.06w 349.4.74 () () T 9.7 w 99.6 rad/sec Answers Now comute the error sgnal rom the new tachometer outut voltage c m. c c e m m K tac 0. s / rad 99.6 rad/sec (r c w m ) 34.94 3.956.968 3.956 Error ncreases e >e.968>.94 to rebalance system lesson6et438a.tx 0 5
0/8/05 Examle 6- Soluton (6) Now determne the ercentage seed changes or oen loo and eedback control. Setont, r=300 rad/sec Oen loo seed change wr w 00% %SE (SE seed error) wr 300 9.7 00% %SE 300.77% %SE Answers Feedback loo seed change 300 99.6 00% %SE 300 0.43% %SE Answers Feedback reduces seed error by actor o 9.35 lesson6et438a.tx Integral Control Mode Integral Mode characterstcs: ) Outut s ntegral o error over tme ) Drves steady-state error to zero 3) Adds ole to transer uncton at s=0 (nnte gan to constant) 4) Integrators tend to make systems less stable Equatons Tme: Lalace: v(t) K Transer Functon: I e(t) dt v 0 0 (s) KI E(s s (s) E(s) K I s Where K I = ntegral gan constant lesson6et438a.tx 6
0/8/05 OP AMP Realzatons o Integral Control Practcal OP AMP Integrator Ideal OP AMP Integrator Transer Functon o(s) (s) R C s One ole at s=0 KI R C Transer Functon o(s) R (s) R R Cs One ole at s=-/r C lesson6et438a.tx 3 Bode Plots o Integrator Crcuts Substtute jw or s and nd the magntude and hase sht o the transer uncton or derent values o w. Ideal Integrator o(jw) (jw) R C jw Practcal Integrator o(jw) R (jw) R R C jw Take magntude and hase sht o each o these unctons usng rules o comlex numbers. z=a+jb Magntude o z: z= z z a b Re(z) Im(z) Scale or db db 0log( ) Phase Sht tan b tan a Im(z) Re(z) lesson6et438a.tx 4 7
0/8/05 Bode Plots o Integrator Crcuts Practcal Integrator Crcut Takng magntude gves Phase Sht gves Ideal Integrator Crcut Magntude gves R R R C jw R R w) 80 tan R ( R db 0log( ) C w Cw Gjw db 0log( ) R C 80 degree hase sht s rom nvertng conguratons Phase sht j 90 j 0 80 90 90 Contant value lesson6et438a.tx 5 Integrator Bode Plots Usng MatLAB R R R C jw R C jw Use MatLAB scrt to generate Bode lots and transer uncton. Dene arameters: R = 0 k, R = 00 k, C = 0.0 mf MatLAB Scrt r=nut('enter value o nut resstance: '); c=nut('enter value o caactance: '); r=nut('enter value o eedback resstance: '); % comute transer uncton model arameters or % ractcal ntegrator Inut statement Comments begn wth % tau=r*c; k=-r./r; lesson6et438a.tx 6 8
Phase (deg) Magntude (db) 0/8/05 Integrator Bode Plots Usng MatLAB MatLAB Scrt (Contnued) % comute arameter or deal ntegrator tau = r*c; % buld transer unctons % denomnator orm s a*s^+as+a3 Av=t([k],[tau ]) Av=t([-],[tau 0]) %lot both on the same grahs bode(av,av); Create transer unctons Plot both grahs on same gure lesson6et438a.tx 7 Integrator Bode Plots Usng MatLAB 60 40 0 Integrator Bode Dagrams Practcal Ideal w=/r C 0-0 w=000 rad/s 80 35 90 0 0 0 3 0 4 0 5 Frequency (rad/sec) lesson6et438a.tx 8 9
0/8/05 Integral Acton on Tme aryng Error Sgnals Integral o constant, k, s lne wth sloe k. e e Integrator roduces a lnearly ncreasng outut or constant error nut -e 3 e 4 =0 (t)=-e 3 t Negatve error causes decreasng outut (t)=e t Zero error mantans last outut value (t)=e t lesson6et438a.tx 9 Estmatng Integrator Outut From Calculus, ntegral s sum o area below a uncton lot b a (t) dt (t) n 0 (t ) t Where t t (t + ) b a t n a t b t For lnear error lots, ntegral s the sum o the areas o lnear segments. Use trangle, traezod, and rectangle ormulas to aroxmate outut lesson6et438a.tx 0 0
Error Inut 0/8/05 Integrator Outut Examle 6-: An deal ntegrator has a gan o K I =0. /s. Its ntal outut s v=.5. Determne the ntegrator oututs the error has ste ncreases gven by the table below. Error Magntude () Tme Interval e(t)=0 0 t seconds e(t)=.5 <t seconds e(t)=4 <t 3 seconds e(t)=0 3<t 4 seconds e(t)=-.5 4<t 5 seconds lesson6et438a.tx Examle 6- Soluton () Plot the error uncton that s nut to the ntegrator 5 Integrator Error Inut 4 3 0 - - 0 3 4 5 6 Tme (seconds) lesson6et438a.tx
Error Inut Integrator Outut oltage 0/8/05 Examle 6- Soluton () Use the aroxmate ormula to nd the error at the end o each nterval 5 4 3 0 - v K A A I n 0 Integrator Error Inut A 0 e t - 0 3 4 5 6 Tme (seconds) A 4 A 3 Where t n 5 0 =.5 A 0 =K I t e 0 =0.()(0) =0 =.5+A 0 =.5+0=.5 A =K I t e =0.()(.5) =0.5 =.5+A =.5+0.5=.75 A =K I t e =0.()(4) =0.40 3 =.75+A =.75+0.40=.5 A 3 =K I t e 3 =0.()(0) =0 4 =.5+A 3 =.5+0.0=.5 A 4 =K I t e 4 =0.()(-.5) =-0.5 5 =.5+A 4 =.5+-0.5=.00 lesson6et438a.tx 3 Examle 6- Soluton () Integrator outut lot 3 Integrator Outut.5 X:.0 Y:.75.5 0.5 0 0 3 4 5 6 Tme (seconds) lesson6et438a.tx 4
0/8/05 Dervatve Control Mode Dervatve Control Characterstcs: ) Produces outut only when error s changng ) Outut s roortonal to rate o change n error 3) Dervatve control never used alone 4) Used wth roortonal and/or ntegral modes 5) Antcates error by observng the rate o change lesson6et438a.tx 5 Dervatve Mode Equatons Tme Equaton: v(t) K d de(t) dt Lalace Equaton: (s) Kd se(s) Transer Functon Equaton: (s) K E(s) d s Derentators are hgh-ass lters to snusodal sgnals. They ncrease senstvty to rad error changes when added to controllers. lesson6et438a.tx 6 3
0/8/05 OP AMP Realzatons o Derentators Ideal OP AMP Derentator Practcal OP AMP Derentator Transer Functon o(s) R Cs (s) Introduces one zero at s=0 Transer Functon o(s) R Cs (s) R Cs Introduces: zero at s=0 ole at s=-/r C lesson6et438a.tx 7 Bode Plots o Derentators Ideal Derentator Equatons o(s) R (s) R db 0log 90 Cw C jw Constant over all w Practcal Derentator Equatons o(jw) R C jw (jw) R C jw R Cw R db 0log[ ] - 70 - tan (R Cw) C w Use MatLAB scrt to generate Bode lots and transer uncton. Dene arameters: R = 0 k, R = 00 k, C = 0.0 mf lesson6et438a.tx 8 4
Magntude (db) Phase (deg) 0/8/05 Bode Plots o Derentators -00 Bode Dagram Derentator Frequency Resonse -0-40 Ideal Practcal -60-80 w=/r C -00 35 70 5 80 35 90 0-6 0-5 0-4 0-3 0 - Frequency (rad/s) lesson6et438a.tx 9 ET 438a Automatc Control Systems Technology END LESSON 6: BASIC CONTROL MODES lesson6et438a.tx 30 5