DYNAMIC BEHAVIORS OF REPRESENTATIVE PROCESSES

CHBE320 LECTURE VIDYNAMIC BEHAVIORS OFREPRESENTATIVE PROCESSESProfessor Dae Ryook YangFall 2020Dept. of Chemical and Biological EngineeringKorea UniversityCHBE320 Process Dynamics and ControlKorea University6-1

Road Map of the Lecture VI Dynamic Behavior of Representative Processes– Open-loop responses Step inputImpulse inputSinusoidal inputRamp input– Bode diagram analysis– Effect of pole/zero location -ControllerActuatorSensorCHBE320 Process Dynamics and ControlPROCESSLectures IV toVIIKorea University6-2

REPRESENTATIVE TYPES OF RESPONSEU(s) For step inputsY(t)ProcessG(s)Y(s)?Type of Model, G(s)Nonzero initial slope, no overshoot or nor oscillation,1st order model1st order Time delayUnderdamped oscillation, 2nd or higher orderOverdamped oscillation, 2nd or higher orderInverse response, negative (RHP) zerosUnstable, no oscillation, real RHP polesUnstable, oscillation, complex RHP polesSustained oscillation, pure imaginary polesCHBE320 Process Dynamics and ControlKorea University6-3

1ST ORDER SYSTEM First-order linear ODE (assume all deviation variables)𝑑𝑦 𝑡𝜏𝑑𝑡𝑦 𝑡𝔏𝐾𝑢 𝑡𝜏𝑠𝜏𝑠 Step response:–––𝑦 𝜏𝐾𝐴 1𝑒/𝐾𝐴 1𝑒𝑦 0𝐾𝐴𝑒1KA0.632KA𝔏/𝑦 𝑡𝐾𝐴 1/𝜏𝑒/𝜏t0.632𝐾𝐴0.99𝐾𝐴 𝑡/GainTime constanty(t)With 𝑈 𝑠𝑌 𝑠𝐴/𝑠,𝐾𝐴𝑠 𝜏𝑠 1𝐾𝑈 𝑠𝐾𝑌 𝑠𝑈 𝑠 Transfer function:1 𝑌 𝑠𝐾𝐴/𝜏CHBE320 Process Dynamics and Control4.6𝜏Settling time 4𝜏 5𝜏0Nonzero initial slopeKorea University6-4

Impulse responseWith 𝑈 𝑠𝐴,𝐾𝐴𝜏𝑠 1𝑌 𝑠𝔏y(t)𝐾𝐴𝑒𝜏𝑦 𝑡𝐾𝐴𝜏/𝜏t Ramp responsey(t)With 𝑈 𝑠𝑌 𝑠𝑎/𝑠 ,𝐾𝑎𝑠 𝜏𝑠 1𝜏𝐾𝑢 𝑡𝔏𝑦 𝑡𝐾𝑎𝜏𝑒/𝐾𝑎 𝑡 Sinusoidal responseWith 𝑈 𝑠𝑌 𝑠𝜏𝑠𝔏 𝐴 sin 𝜔 𝑡𝐾𝐴𝜔𝜔1 𝑠𝐴𝜔/ 𝑠𝜏1 2 3 4 5ty(t)𝜔 ,𝔏𝜑𝑦 𝑡/𝜔𝜏𝐾𝐴𝜔 𝜏1t𝐾𝐾𝐴𝜔𝜏𝑒1𝜔 𝜏tan𝜔𝜏 cos 𝜔 𝑡CHBE320 Process Dynamics and Controlsin 𝜔 𝑡𝜔 𝜏1𝑢 𝑡Korea University6-5

Ultimate sinusoidal response𝑡 0𝑦𝑡𝐾𝐴lim𝜔𝜏𝑒 / 𝜔 𝜏1𝐾𝐴𝜔𝜏 cos 𝜔 𝑡1𝜔 𝜏𝐾𝐴sin 𝜔𝑡 𝜑𝜔 𝜏1𝜔𝜏 cos 𝜔 𝑡sin 𝜔 𝑡sin 𝜔 𝑡𝜑tan𝜔𝜏Phase angleAmplitude– The output has the same period of oscillation as the input.– But the amplitude is attenuated and the phase is shifted.NormalizedAmplitude Ratio(ARN)1𝜔 𝜏11Phase angletan𝜔𝜏– High frequency input will be attenuated more and phase isshifted more.CHBE320 Process Dynamics and ControlKorea University6-6

BODE PLOT FOR 1ST ORDER SYSTEM AR plot asymptote𝐴𝑅𝐴𝑅𝜔 0𝜔 lim lim 1𝜔 𝜏111𝜔 𝜏1𝜔𝜏1 Phase plot asymptote𝜑 𝜔 0𝜑 𝜔 lim tan lim tan 𝜔𝜏0𝜔𝜏90 It is also called “low-pass filter”CHBE320 Process Dynamics and ControlKorea University6-7

1ST ORDER PROCESSEScAi, qi Continuous Stirred Tank𝑉𝑑𝑐𝑑𝑡𝐶 𝑠𝐶 𝑠𝑞𝑐cA, q𝑞𝑐𝑞𝑉𝑠𝑞1𝑉/𝑞 𝑠Vh1– With constant heat capacity and density𝜌𝑉𝐶𝑑 𝑇𝑇 𝑠𝑇 𝑠𝑇𝑑𝑡𝜌𝑞𝐶 𝑇𝑉𝑠𝑇𝜌𝑞𝐶 𝑇𝑞𝑞1𝑉/𝑞 𝑠CHBE320 Process Dynamics and ControlT0, qiT, q𝑇Vh1Korea University6-8

INTEGRATING SYSTEM 𝑑𝑦 𝑡𝑑𝑡𝐾𝑢 𝑡𝔏𝑠𝑌 𝑠𝐾𝑈 𝑠 Transfer Function: Step ResponseWith 𝑈 𝑠𝐾𝑌 𝑠𝑠y(t)1/𝑠,𝔏𝑦 𝑡Slope K𝐾𝑡t– The output is an integration of input.– Impulse response is a step function.– Non self-regulating systemCHBE320 Process Dynamics and ControlKorea University6-9

INTEGRATING PROCESSES Storage tank with constant outlet flow– Outlet flow is pumped out by a constant-speed, constantvolume pumpqi– Outlet flow is not a function of head.𝑑ℎ𝐴𝑑𝑡𝐻 𝑠𝑄 𝑠𝑞1𝐴𝑠𝑞𝐻 𝑠𝑄 𝑠CHBE320 Process Dynamics and Control1𝐴𝑠VhqArea AKorea University 6-10

2ND ORDER SYSTEM 2nd order linear ODE𝑑 𝑦 𝑡𝜏𝑑𝑡𝑑𝑦 𝑡2𝜁𝜏𝑑𝑡𝑦 𝑡𝐾𝑢 𝑡𝔏𝜏 𝑠2𝜁𝜏𝑠1 𝑌 𝑠𝐾𝑈 𝑠 Transfer Function:𝑌 𝑠𝑈 𝑠𝜏 𝑠𝐾2𝜁𝜏𝑠Gain1Time constantDamping Coefficient Step response– Varies with the type of roots of denominator of the TF. Real part of roots should be negative for stability: 𝜁 0Two distinct real roots ( 𝜁 1 ): overdamped (no oscillation)Double root ( 𝜁 1 ): critically damped (no oscillation)Complex roots ( 0 𝜁 1 ): underdamped (oscillation)CHBE320 Process Dynamics and ControlKorea University6-11

Case I 𝜁𝑌 𝑠 1with U(s) 1/s𝐾𝐾2𝜁𝜏𝑠 1𝑠 𝜏 𝑠 1 𝜏 𝑠𝑠 𝜏 𝑠Case II 𝜁𝑌 𝑠 𝑌 𝑠𝑠 𝜏 𝑠Case III 0𝑠 𝜏 𝑠𝔏1𝑦 𝑡𝐾 1𝜏 𝑒/𝜏/𝜏 �� 𝜏𝑠𝔏𝑦 𝑡1𝐾11𝑡/𝜏 𝑒Natural frequency1𝔏1/𝑦 𝑡CHBE320 Process Dynamics and Control𝐾 1𝑒/cos 𝛼 𝑡𝜁sin 𝛼 𝑡𝛼𝜏𝛼1𝜁𝜏Korea University 6-12

Ultimate sinusoidal responseWith 𝑈 𝑠𝔏 𝐴 sin 𝜔 𝑡 ,𝐾𝐴𝜔2𝜁𝜏𝑠 1 𝑠𝜏 𝑠𝑌 𝑠𝔏𝜔𝐾𝐴𝑦 𝑡1𝜔 𝜏2𝜁𝜔𝜏sin 𝜔𝑡𝜑𝜑tan2𝜁𝜔𝜏1 𝜔 𝜏– Other method to find ultimate sinusoidal responseFor 𝑠𝐺 𝑠𝐴𝑅𝜑𝛼𝑗𝜔 , 𝑦 𝑡 has 𝑒𝜏 𝑠𝐺 𝑗𝜔 𝐺 𝑗𝜔𝐾2𝜁𝜏𝑠11𝐾𝜏 𝜔tanand it becomes 𝑒 Im 𝐺 𝑗𝜔Re 𝐺 𝑗𝜔CHBE320 Process Dynamics and Controlas 𝑡 𝛼0 .𝐾𝐺 𝑗𝜔1𝜏 𝜔2𝑗𝜁𝜏𝜔𝐾𝑗𝜏𝜔1tan𝜔 𝜏2𝜁𝜔𝜏2𝜁𝜔𝜏1 𝜔 𝜏Korea University 6-13

BODE PLOT FOR 2ND ORDER SYSTEM AR plot𝐴𝑅𝜔 Phase plot𝜑 𝜔 Resonance𝑑 𝐴𝑅/𝑑𝜔for 0 11lim tan 𝜔 𝜏2𝜁𝜔𝜏1 𝜔 𝜏1𝜔𝜏2𝜁𝜔𝜏lim tan 07The amplitude of outputoscillation is biggerthan that of input whenthe resonance occurs .CHBE320 Process Dynamics and ControlKorea University 6-14

1ST ORDER VS. 2ND ORDER (OVERDAMPED) Initial slope of step response1st order: 𝑦 02nd order: 𝑦 0lim 𝑠 𝑌 𝑠𝐾𝐴𝑠lim 𝜏𝑠1lim 𝑠 𝑌 𝑠𝐾𝐴𝑠lim 𝜏 𝑠2𝜁𝜏𝑠 𝐾𝐴𝜏010 Shape of the curve (Convexity)1st order: 𝑦 𝑡𝐾𝐴/𝜏 𝑒/0 For 𝐾0 No inflection𝐾𝐴 𝑒 /𝑒 /2nd order: 𝑦 𝑡𝜏𝜏𝜏𝜏 as 𝑡 InflectionCHBE320 Process Dynamics and ControlKorea University 6-15

CHARACTERIZATION OF SECOND ORDERSYSTEM 12nd order Underdamped response𝜁𝜋exp11𝜁exp𝜁𝑡/𝜏– Rise time (tr)𝑡𝜏 𝑛𝜋cos𝜁 / 1𝜁𝑛11exp3𝜁𝜋1𝜁– Time to 1st peak (tp)𝑡𝜏𝜋/ 1𝜁– Settling time (ts)𝑡𝜏/𝜁 ln 0.05– Overshoot (OS)𝑂𝑆𝑎/𝑏exp𝜏𝜋𝜋𝜁/ 11𝜁𝜁2𝜏𝜋3𝜏𝜋11𝜁𝜁– Decay ratio (DR): a function of damping coefficient only!𝐷𝑅𝑐/𝑎𝑂𝑆exp2𝜋𝜁/ 1– Period of oscillation (P)CHBE320 Process Dynamics and Control𝑃𝜁2𝜋𝜏/ 1𝜁Korea University 6-16

2ND ORDER PROCESSEScAi, qi Two tanks in series– If v1 v2, critically damped.– Or, overdamped (no oscillation)𝐶 𝑠𝐶 𝑠V1𝑉 /𝑞 𝑠1cA, q1𝑉 /𝑞 𝑠1V2 Spring-dashpot (shock absorber)– By force balance𝑚𝑔𝑓 ��𝑐𝑣𝑚𝑔𝑐𝑚2𝑦4𝑚𝑘 𝑘𝑚𝑎𝑓 𝑡𝑦𝑓 𝑡𝜁 (can be 1: underdamped)CHBE320 Process Dynamics and ControlKorea University 6-17

Underdamped Processes Many examples can be found in mechanical andelectrical system. Among chemical processes, open-loopunderdamped process is quite rare. However, when the processes are controlled, theresponses are usually underdamped. Depending on the controller tuning, the shape ofresponse will be decided. Slight overshoot results short rise time and oftenmore desirable. Excessive overshoot may results long-lastingoscillation.CHBE320 Process Dynamics and ControlKorea University 6-18

POLES AND ZEROS𝐺 𝑠𝑁 𝑠𝐷 𝑠𝐾 𝑏 𝑠𝑎 𝑠𝑏𝑎𝑠𝑠 𝑏 𝑠 1 𝑎 𝑠 1 Poles (D(s) 0)––––Where a transfer function cannot be defined.Roots of the denominator of the transfer functionModes of the responseDecide the stability Zero (N(s) 0)––––Where a transfer function becomes zero.Roots of the numerator of the transfer functionDecide weightings for each mode of responseDecide the size of overshoot or inverse response They can be real or complexCHBE320 Process Dynamics and ControlKorea University 6-19

Real pole from𝑠1𝜏𝜏𝑠1Im𝜏 𝜏 y(t)/– Mode: 𝑒Re1/𝜏𝜏 t– If the pole is at the origin, it becomes “integrating pole.”– If the pole is in RHP, the response increases exponentially. Complex pole from𝑠𝜁𝜏𝑗1𝜁𝜏 𝑠𝛼𝜏2𝜁𝜏𝑠11𝜁𝜏 𝑗𝛽1Im𝜁 𝑠 𝑠𝜁tan1𝜏𝜁11𝜏𝜁𝜁1𝜁 /𝜏𝜃function of 𝜏 only𝜁/𝜏Re1𝜁 /𝜏function of 𝜁 onlyCHBE320 Process Dynamics and Control𝜁cos 𝜃Korea University 6-20

– Modes:𝑒𝑒𝑒––––cos 𝛽 𝑡𝑗 sin 𝛽 𝑡1 𝜁1 𝜁cos𝑡 𝑗 sin𝑡𝜏𝜏/Assume 𝜏 is positive.If 𝜁 0 , the exponential part will grow as t increases: unstableIf 𝜁 0 , the exponential part will shrink as t increases: stableIf 𝜁 0 , the roots are pure imaginary: sustained oscillation Effect of zero𝐺 𝑠𝑠𝑁 𝑠𝑝 𝑠𝑝𝑤1𝑠𝑝 𝑤1𝑠𝑝– The effects on weighting factors are not obvious, but it is clearthat the numerator (zeros) will change the weighting factors.CHBE320 Process Dynamics and ControlKorea University 6-21

EFFECTS OF ZEROS Lead-lag module𝐺 𝑠–𝑁 𝑠𝐷 𝑠𝐾 𝜏 𝑠 1𝜏 𝑠 1LeadLagDepending on the location of zero𝑌 𝑠𝐾𝑀 𝜏 𝑠 1𝑠 𝜏 𝑠 11𝐾𝑀𝑠𝜏𝜏𝜏 𝑠 1𝑦 𝑡𝐾𝑀 11𝜏𝜏𝑒/(a) 𝜏 𝜏 0The lead dominates the lag.(b) 0 𝜏 𝜏The lag dominates the lead.(c) 0 𝜏Inverse responseCHBE320 Process Dynamics and ControlKorea University 6-22

Overdamped 2nd order single zero system𝐺 𝑠𝑌 𝑠𝑦 𝑡𝐾 𝜏 𝑠 1𝜏 𝑠 1 𝜏 𝑠𝑁 𝑠𝐷 𝑠𝐾𝑀 𝜏 𝑠 1𝑠 𝜏 𝑠 1 𝜏 𝑠 1𝐾𝑀 1𝜏𝜏𝜏𝑒𝜏𝐾𝑀/11𝑠𝜏𝜏𝜏 𝜏𝜏𝜏𝑒𝜏𝜏𝜏1𝜏 𝑠 1𝜏 𝜏𝜏𝜏𝜏1𝜏 𝑠 1/(a) 𝜏 𝜏 0 assume 𝜏 𝜏The lead dominates the lags.(b) 0 𝜏 𝜏The lags dominate the lead.(c) 0 𝜏Inverse responseCHBE320 Process Dynamics and ControlKorea University 6-23

Other interpretation𝐾 𝜏 𝑠 1𝜏 𝑠 1 𝜏 𝑠𝐺 𝑠𝐾 𝜏 𝑠 1𝜏 𝑠 1𝐾/𝐾 𝜏 𝑠 1𝜏 𝑠 1𝐾/– Since 𝜏y1(t)𝜏𝐾𝜏 𝑠 11𝐾 𝜏𝜏𝜏𝜏U(s)𝐾 Y1(s)𝐾𝜏 𝑠 1 Y2(s)t𝐾𝐾𝜏 𝑠 1, 1 is slow dynamics and 2 is fast dynamics.𝜏𝜏𝐾𝐾𝜏 𝑠 1y(t)tCHBE320 Process Dynamics and Controlty(t)t𝐾𝐾tKorea University 6-24

EFFECTS OF POLE LOCATIONIm(s)Unstable RegionShorterperiod ofoscill.Moreoscill.Lessoscill.Faster responseRe(s)Faster responseShorterperiod ofoscill.CHBE320 Process Dynamics and ControlKorea University 6-25

EFFECTS OF ZERO LOCATIONZero at origin:Im(s)Complex LHP zero:valleyDeeper valleyInverse responseregionComplex RHP zero:Dominant Pole0Real LHP zero:MoreMoreoverdamped overshootCHBE320 Process Dynamics and ControlRe(s)Real RHP zero:Biggerinverse responseKorea University 6-26

CHBE320 Process Dynamics and Control Korea University 6-1 CHBE320 LECTURE VI DYNAMIC BEHAVIORS OF REPRESENTATIVE PROCESSES Professor Dae Ryook Yang Fall 2020 Dept. of Chemical and Biological Engineering Korea University. CHBE