1. 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 – May 29, 2019

2. Process Instrumentation and Control - Prof. M. Miccio
SECOND ORDER LAG
see:
Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
Initial conditions:
t= y=0
t= dy/dt=0
Second-order ODE, linear, non-homogeneous, with constant coefficients
Forcing function: f(t)
After dividing by a0 0 we have:
a2/a0=  2; a1/a0=2ζ ; b/a0=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 in the Laplace domain
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Process Instrumentation and Control - Prof. M. Miccio

3. Discriminant: D = z2t2 - t2 = t2(z2 - 1)
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
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Process Instrumentation and Control - Prof. M. Miccio

4. ROOT LOCUS FOR A SECOND-ORDER LAGP2 P1 P3 P4
P4* P5 Im Re
P5*
ζ  ]+∞, 0]
decreases
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.
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Process Instrumentation and Control - Prof. M. Miccio

5. 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
SOcalculator.swf
Case C: Underdamped response
ζ < 1 (ζ>0)
con:
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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”
see:
D. Cooper, "Practical Process Control", book in PDF file
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Process Instrumentation and Control - Prof. M. Miccio

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

8. Process Instrumentation and Control - Prof. M. Miccio
UNDERDAMPED RESPONSE TO THE UNIT STEP INPUT CHANGE Qualitative behaviour
see:
Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
27/09/2019
Process Instrumentation and Control - Prof. M. Miccio

9. Process Instrumentation and Control - Prof. M. Miccio
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=2p/ω 
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Process Instrumentation and Control - Prof. M. Miccio

10. Process Instrumentation and Control - Prof. M. Miccio
UNDERDAMPED RESPONSE TO THE UNIT STEP INPUT CHANGE Characteristic parameters
Rise time
The time required for the response to reach its final value for the first time.
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).
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Process Instrumentation and Control - Prof. M. Miccio

11. UNDAMPED RESPONSE TO THE UNIT 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
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Process Instrumentation and Control - Prof. M. Miccio

12. 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
27/09/2019
Process Instrumentation and Control - Prof. M. Miccio

13. Dimensionless response of 2nd 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”
27/09/2019
Process Instrumentation and Control - Prof. M. Miccio

14. DYNAMIC response of 2nd 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
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Process Instrumentation and Control - Prof. M. Miccio

15. DYNAMIC response of 2nd 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
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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)
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Process Instrumentation and Control - Prof. M. Miccio

17. DYNAMIC RESPONSE OF A 2ND ORDER LAG TO 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)
27/09/2019
Process Instrumentation and Control - Prof. M. Miccio

18. Dimensionless response of 2nd Order lag to Sinusoidal input
Homework:
Diagram the
DYNAMIC RESPONSE
with the mod. Custom Process of LOOP-PRO Control Station
27/09/2019
Process Instrumentation and Control - Prof. M. Miccio

19. Dimensionless response of 2nd 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.  BIBO instability
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Process Instrumentation and Control - Prof. M. Miccio

20. 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:
Inherenlty second-order systems: systems including mechanics and fluido-dynamics processes, for example when in a force balance the inertia term appears (Es.: mass subjected to a elastic force)
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).
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 2nd.
See: Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
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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”
27/09/2019
Process Instrumentation and Control - Prof. M. Miccio

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

23. MULTICAPACITY SYSTEMS Non interacting tanks
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Process Instrumentation and Control - Prof. M. Miccio

24. MULTICAPACITY SYSTEMS Non 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
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Process Instrumentation and Control - Prof. M. Miccio

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

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

27. MULTICAPACITY SYSTEMS Interacting tanks
Transfer function
da rivedere !
see:
Ch.11 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
27/09/2019
Process Instrumentation and Control - Prof. M. Miccio

28. 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 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”
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Process Instrumentation and Control - Prof. M. Miccio

29. 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:
§ Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
hn(t)/Kpn
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Process Instrumentation and Control - Prof. M. Miccio