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UNIVERSIT À DEGLI STUDI DI SALERNO

UNIVERSIT À DEGLI STUDI DI SALERNO. Bachelor Degree in Chemical Engineering Course: Process Instrumentation and Control ( Strumentazione e Controllo dei Processi Chimici ) REFERENCE LINEAR DYNAMIC SYSTEMS Second-Order Systems Rev. 2.42 – May 29, 2019. SECOND ORDER LAG. see:

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UNIVERSIT À DEGLI STUDI DI SALERNO

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  1. UNIVERSITÀ DEGLI STUDI DI SALERNO Bachelor Degree in Chemical Engineering Course: Process Instrumentation and Control (Strumentazione e ControllodeiProcessiChimici) REFERENCE LINEAR DYNAMIC SYSTEMS Second-Order Systems Rev. 2.42 – May 29, 2019

  2. SECOND ORDER LAG see: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” • Initial conditions: • t=0 y=0 • t=0 dy/dt=0 Second-order ODE, linear, non-homogeneous, with constant coefficients Forcing function: f(t) • After dividing by a0 0 wehave: • a2/a0= 2; a1/a0=2ζ ; b/a0=Kp • where: •  = natuaralperiod of oscillation • ζ = dampingfactor • Kp = steady state gainorstatic gain,orsimplygain CANONICAL FORM in the time domain in the Laplace domain Process Instrumentation and Control - Prof. M. Miccio

  3. TRANSFER FUNCTION Discriminant: D = z2t2 - t2 = t2(z2 - 1) CASE A : ζ > 1, THE CHARACTERISTIC EQUATION HAS TWO DISTINCT AND REAL POLES. Overdamped response CASE B : ζ = 1, THE CHARACTERISTIC EQUATION HAS TWO EQUAL POLES (MULTIPLE POLES) Critically damped CASE C : 0 <ζ< 1, THE CHARACTERISTIC EQUATION HAS TWO COMPLEX CONJUGATE POLES  Underdamped response Process Instrumentation and Control - Prof. M. Miccio

  4. Im P5 P4 ζ]+∞, 0] decreases Re P1 P2 P3 P4* P5* ROOT LOCUS FOR A SECOND-ORDER LAG • OVERDAMPED: P1, P2 • CRITICALLY DAMPED: P3 with multiplicity = 2 • UNDERDAMPED : P4, P4* • UNDAMPED ( ζ=0 ) : P5, P5*P5 =  j/ NOTE: the G(s) of a the second-order system is BIBO stable. Therefore, this is a self-regulating system. Process Instrumentation and Control - Prof. M. Miccio

  5. Case B: Criticallydampedresponse ζ = 1 DYNAMIC RESPONSE TO THEUNIT STEP INPUT CHANGE Case A: Overdamped response ζ > 1 from N.S. Nise, “Control Systems Engineering”, California State Polytechnic University SOcalculator.swf Case C: Underdamped response ζ < 1 (ζ>0) con: Process Instrumentation and Control - Prof. M. Miccio

  6. DYNAMIC RESPONSE TO THEUNIT STEP INPUT CHANGE see: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” see: D. Cooper, "Practical Process Control", book in PDF file Process Instrumentation and Control - Prof. M. Miccio

  7. DYNAMIC RESPONSE TO THEUNIT STEP INPUT CHANGE see: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Process Instrumentation and Control - Prof. M. Miccio

  8. UNDERDAMPED RESPONSE TO THEUNIT STEP INPUT CHANGEQualitative behaviour see: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Process Instrumentation and Control - Prof. M. Miccio

  9. UNDERDAMPED RESPONSE TO THEUNIT STEP INPUT CHANGE Characteristicparameters 1. Overshoot A/B 2. Decay Ratio C/A 3. Radian frequency 4. Period of oscillation T=2p/ω Process Instrumentation and Control - Prof. M. Miccio

  10. UNDERDAMPED RESPONSE TO THEUNIT STEP INPUT CHANGE Characteristicparameters • Rise time The time required for the response to reachitsfinalvalue for the first time. • Response time The time required for the response to a unit step input change to reachitsfinalvaluewhenitremainwithin ±5% of itsfinalvalue(value of time for which the response can be considered no longer oscillatory). Process Instrumentation and Control - Prof. M. Miccio

  11. UNDAMPED RESPONSE TO THEUNIT STEP INPUT CHANGE Case ζ=0 • Natural period of oscillation Tn = 2p • Natural frequency ωn = 1/ • NOTE: For ζ=0 (UNDAMPED SYSTEM) the response to the unit step input change is a continuous oscillation with constant amplitude marginal stability Process Instrumentation and Control - Prof. M. Miccio

  12. UNDAMPED RESPONSE TO THEUNIT STEP INPUT CHANGE Case ζ=0 Homework: Diagram the DYNAMIC RESPONSE with the mod. Custom Process of LOOP-PRO Control Station Process Instrumentation and Control - Prof. M. Miccio

  13. Dimensionlessresponseof 2ndOrderlagtostep input change Dimensionless diagram of the dynamic responseto the step input change NOTE: Self-regulating dynamic behaviour see: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Process Instrumentation and Control - Prof. M. Miccio

  14. DYNAMIC responseof2ndOrderlagtounitimpulse If a unit impulseδ(t) with L[δ(t)] = 1 is applied to a second-order lag with a transfer function: As for the case of the step input change, the qualitative behaviour of the dynamic response depends on the values of the poles • ζ < 1 • ζ = 1 • ζ > 1 Process Instrumentation and Control - Prof. M. Miccio

  15. DYNAMIC response of 2nd Order lagto unit impulse Ip.: Kp=1 Response to the unitimpulse for ζ < 1 Response to the unit impulse for ζ = 1 Response to the unit impulse for ζ >1 Process Instrumentation and Control - Prof. M. Miccio

  16. DIMENSIONLESS response of 2nd Order lag to unit impulse Dimensionless diagram of the dynamic response (drawn for KP = 1) Process Instrumentation and Control - Prof. M. Miccio

  17. DYNAMIC RESPONSE OF A 2ND ORDER LAGTO A SINUSOIDAL INPUT Transient approaching zero FORCING FUNCTION: f(t)= A sin ωt DYNAMIC RESPONSE: Oscillating for long time The constants Ci can be calculated with the partial fraction expansion method For The Amplitude Ratio AR is defined as the ratio between the amplitude of the sinusoidal response for long time and the amplitude of the sinusoidal input AR = (Output AMPL)/(Input AMPL) Process Instrumentation and Control - Prof. M. Miccio

  18. Dimensionlessresponseof2ndOrderlagtoSinusoidal input Homework: Diagram the DYNAMIC RESPONSE with the mod. Custom Process of LOOP-PRO Control Station Process Instrumentation and Control - Prof. M. Miccio

  19. Dimensionlessresponseof 2ndOrderUndampedlagtoSinusoidal input NOTE: only for a radian frequency of the input dell'ingresso sin(nt) equal to the radian frequency of the system, the dynamic response is a continous oscillation with increasing amplitude.  BIBO instability Process Instrumentation and Control - Prof. M. Miccio

  20. FURTHER CLASSIFICATIONof Second-Order Lag • Systemswithsecond-ofhigher-orderdynamics can arisefromseveralphysicalsituations. • These can beclassifiedintothreecategories: • Inherenltysecond-ordersystems: systemsincludingmechanics and fluido-dynamicsprocesses, forexamplewhen in a forcebalance the inertiatermappears (Es.: mass subjectedto a elasticforce) • Multicapacityprocesses: systemsconsisting of two first-ordersystems in serieswhere the output of the first oneis the forcing function of the secondone (e.g., two tanks in series). • A processing system with its controller: the installed controller introducesadditional dynamics whichgive rise to second-order a first-order system. • A similar case also is applied to dynamical systems of order greater than the 2nd. See: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Process Instrumentation and Control - Prof. M. Miccio

  21. INHERENTLY SECOND-ORDER SYSTEM 1. Mass M moving on a spring with a damper 2. U-tube liquid manometer 3. Variable capacitance differential pressure transducer 4. Pneumatic globe valve see: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Process Instrumentation and Control - Prof. M. Miccio

  22. MULTICAPACITY SYSTEMSInteracting and non interacting tanks • Non interacting HP: Linear outflow see: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” • Interacting Process Instrumentation and Control - Prof. M. Miccio

  23. MULTICAPACITY SYSTEMSNon interacting tanks TANK 1 TANK 2 Process Instrumentation and Control - Prof. M. Miccio

  24. MULTICAPACITY SYSTEMSNon interacting tanks see: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Transfer function with: 2 = P1P2 2 = P1 + P2 Kp = Kp1Kp2 Poles: p1 = −1/P1 p2 = −1/P2 Process Instrumentation and Control - Prof. M. Miccio

  25. MULTICAPACITY SYSTEMSNon interacting tanks In general, for n non interacting systems in series, the Transfer Function is: see: § 12.1 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Example: Process Instrumentation and Control - Prof. M. Miccio

  26. MULTICAPACITY SYSTEMSInteracting tanks see: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” TANK 1 TANK 2 Process Instrumentation and Control - Prof. M. Miccio

  27. MULTICAPACITY SYSTEMSInteracting tanks Transfer function see: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Process Instrumentation and Control - Prof. M. Miccio

  28. Characteristicsof 2-Capacity Processes  The dynamic response of the two-tank process is never underdamped Non-Interacting Systems • Non-interacting systems will always result in an over-damped or critically damped second-order response. • The poles of the overall system are equal to the individual poles and equal to the inverse of the individual time constants. • If the individual time constants are equal, then the poles are equal. Interacting Systems • The time constants of interacting processes may no longer be directly associated with the time constants of individual capacities. • Interacting capacities are more "sluggish" than the noninteracting • The transfer function of the 1st system is 2nd order with a negative zero from: Romagnoli & Palazoglu (2005), “Introduction to Process Control” Process Instrumentation and Control - Prof. M. Miccio

  29. hn(t)/Kpn Characteristics ofMulti-Capacity Processes Analysis of systems of growing order For n systems in series, increasing the number of systems increases the sluggishness of the response. see: § 11.3 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Process Instrumentation and Control - Prof. M. Miccio

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