UNIVERSIT DEGLI STUDI DI SALERNO Bachelor Degree in
UNIVERSITÀ 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 Initial conditions: 1) t=0 y=0 2) t=0 dy/dt=0 see: Ch. 11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Second-order ODE, linear, non-homogeneous, with constant coefficients Forcing function: f(t) After dividing by a 0 0 we have: a 2/a 0= 2; a 1/a 0=2ζ ; b/a 0=Kp where: = natuaral period of oscillation ζ = damping factor Kp = steady state gain or static gain, or simply gain CANONICAL FORM in the time domain 20/10/2021 in the Laplace domain Process Instrumentation and Control - Prof. M. Miccio 2
TRANSFER FUNCTION Discriminant: D = z 2 2 - 2 = 2(z 2 - 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 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 3
ROOT LOCUS FOR A SECOND-ORDER LAG Im P 5 P 4 P 2 P 3 Re P 1 P 4* ζ ]+∞, 0] decreases P 5* • OVERDAMPED : P 1, P 2 • CRITICALLY DAMPED : P 3 with multiplicity = 2 • UNDERDAMPED : P 4, P 4* • UNDAMPED ( ζ=0 ) : P 5, P 5* P 5 = j/ NOTE: the G(s) of a the second-order system is BIBO stable. Therefore, this is a selfregulating system. 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 4
DYNAMIC RESPONSE TO THE UNIT STEP INPUT CHANGE Case A: Overdamped response ζ >1 from N. S. Nise, “Control Systems Engineering ”, California State Polytechnic University Case B: Critically damped response ζ=1 Case C: Underdamped response ζ < 1 (ζ>0) 20/10/2021 SOcalculator. swf con: Process Instrumentation and Control - Prof. M. Miccio 5
DYNAMIC RESPONSE TO THE UNIT 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 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 6
DYNAMIC RESPONSE TO THE UNIT STEP INPUT CHANGE see: Ch. 11 Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 7
UNDERDAMPED RESPONSE TO THE UNIT STEP INPUT CHANGE Qualitative behaviour see: Ch. 11 Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 8
UNDERDAMPED RESPONSE TO THE UNIT STEP INPUT CHANGE Characteristic parameters 1. Overshoot A/B 2. Decay Ratio C/A 3. Radian frequency 4. Period of oscillation T=2 p/ω 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 9
UNDERDAMPED RESPONSE TO THE UNIT STEP INPUT CHANGE Characteristic parameters 5. Rise time The time required for the response to reach its final value for the first time. 6. Response time The time required for the response to a unit step input change to reach its final value when it remain within ± 5% of its final value (value of time for which the response can be considered no longer oscillatory). 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 10
UNDAMPED RESPONSE TO THE UNIT STEP INPUT CHANGE Case ζ=0 7. Natural period of oscillation Tn = 2 p 8. 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 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio marginal stability 11
UNDAMPED RESPONSE TO THE UNIT STEP INPUT CHANGE Case ζ=0 Homework: Diagram the DYNAMIC RESPONSE with the mod. Custom Process of LOOP- PRO Control Station 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 12
DIMENSIONLESS RESPONSE OF 2 ND ORDER LAG TO STEP INPUT CHANGE Dimensionless diagram of the dynamic response to the step input change NOTE: Self-regulating dynamic behaviour see: Ch. 11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 13
DYNAMIC RESPONSE OF 2 ND ORDER LAG TO UNIT IMPULSE If a unit impulse δ(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 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 14
DYNAMIC RESPONSE OF 2 ND ORDER LAG TO UNIT IMPULSE Ip. : Kp=1 Response to the unit impulse for ζ < 1 Response to the unit impulse for ζ = 1 Response to the unit impulse for ζ >1 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 15
DIMENSIONLESS RESPONSE OF 2 ND ORDER LAG TO UNIT IMPULSE Dimensionless diagram of the dynamic response (drawn for KP = 1) 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 16
DYNAMIC RESPONSE OF A 2 ND ORDER LAG TO A SINUSOIDAL INPUT FORCING FUNCTION: f(t)= A sin ωt DYNAMIC RESPONSE: Oscillating for long time Transient approaching zero 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) 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 17
DIMENSIONLESS RESPONSE OF 2 ND ORDER LAG TO SINUSOIDAL INPUT Homework: Diagram the DYNAMIC RESPONSE with the mod. Custom Process of LOOP-PRO Control Station 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 18
DIMENSIONLESS RESPONSE OF 2 ND ORDER UNDAMPED LAG TO SINUSOIDAL 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. 20/10/2021 BIBO instability Process Instrumentation and Control - Prof. M. Miccio 19
FURTHER CLASSIFICATION of Second-Order Lag Systems with second- of higher-order dynamics can arise from several physical situations. These can be classified into three categories: 1. Inherenlty second-order systems: systems including mechanics and fluidodynamics processes, for example when in a force balance the inertia term appears (Es. : mass subjected to a elastic force) 2. Multicapacity processes: systems consisting of two first-order systems in series where the output of the first one is the forcing function of the second one (e. g. , two tanks in series). 3. A processing system with its controller: the installed controller introduces additional dynamics which give rise to second-order a first-order system. A similar case also is applied to dynamical systems of order greater than the 2 nd. See: Ch. 11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 20/10/2021 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” 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 22
MULTICAPACITY SYSTEMS Interacting and non interacting tanks 1. Non interacting HP: Linear outflow see: Ch. 11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 2. Interacting 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 23
MULTICAPACITY SYSTEMS Non interacting tanks TANK 1 20/10/2021 TANK 2 Process Instrumentation and Control - Prof. M. Miccio 24
MULTICAPACITY SYSTEMS Non interacting tanks Transfer function with: 2 = P 1 P 2 2 = P 1 + P 2 Kp = Kp 1 Kp 2 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio see: Ch. 11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Poles: p 1 = − 1/ P 1 p 2 = − 1/ P 2 25
MULTICAPACITY SYSTEMS Non 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: 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 26
MULTICAPACITY SYSTEMS Interacting tanks TANK 1 20/10/2021 TANK 2 see: Ch. 11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” Process Instrumentation and Control - Prof. M. Miccio 27
MULTICAPACITY SYSTEMS Interacting tanks Transfer function see: Ch. 11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 29
Characteristics of 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 overdamped or critically damped second-order response. l The poles of the overall system are equal to the individual poles and equal to the inverse of the individual time constants. l If the individual time constants are equal, then the poles are equal. l The time constants of interacting processes may no longer be directly associated with the time constants of individual capacities. l Interacting capacities are more "sluggish" than the noninteracting l The transfer function of the 1 st system is 2 nd order with a negative zero l from: Romagnoli & Palazoglu (2005), “Introduction to Process Control” 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 30
CHARACTERISTICS OF MULTI-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” hn(t)/Kpn 20/10/2021 Process Instrumentation and Control - Prof. M. Miccio 31
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