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
First-Order Systems
Rev. 2.5 – May 17, 2019

2. FIRST-ORDER LAG I.C.: t=0 y(0)=0
see:
Ch.10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
I.C.: t=0 y(0)=0
First-order ODE, linear, non-homogeneous, with constant coefficients
Forcing function: f(t)
If a0≠0
CANONICAL FORM
in the time domain in the Laplace domain
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

3. FIRST-ORDER LAG position of the pole in the complex planeIm Re
-1/p
Characteristic polynomial: tps + 1
Characteristic eq. : tps + 1 = 0
Only 1 pole: p = -1/tp < 0
NOTE:
Self-regulating dynamic behavior
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

4. DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO STEP INPUT CHANGE
Forcing funtion:
f(t)=Au(t) A=const. > 0
F(s)=A/s p t
see:
Ch.10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
NOTE:
Self-regulating dynamic behavior
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

5. FIRST-ORDER PROCESSES
Characteristic Parameters
Static Gain ( Kp ):
This is the ultimate value of the response (new steady-state) for a unit-step change in the input.
( Theorem of the final value)
Time Constant (t p):
The time constant of a process is a measure of the time necessary for the process to adjust to a change in its input.
from:
Romagnoli & Palazoglu (2005), “Introduction to Process Control”
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

6. FIRST-ORDER PROCESSES Effect of tp
time 5 10 15
2 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
h(t), m 20
Response
of the level of a tank (first-order system) to a unit-step change in the input flow rate
from: Romagnoli & Palazoglu (2005), “Introduction to Process Control”
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

7. FIRST-ORDER PROCESSES Effect of tp
The system eventually reaches a new equilibrium point (new steady-state).
Two tanks with different cross sectional areas respond with different speeds (different time constants).
The tank with a small area responds faster (smaller time constant).
The tank with a larger area responds slower (larger time constant).
from:
Romagnoli & Palazoglu (2005), “Introduction to Process Control”
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

8. DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO STEP INPUT CHANGE
Dimensionless diagram of the dynamic response
see:
Ch.10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
NOTE:
Self-regulating dynamic behavior
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

9. DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO UNIT IMPULSE
Forcing function :
f(t)=(t)
F(s) = 1
Dynamic response:
NOTE:
Self-regulating dynamic behavior
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

10. DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO UNIT IMPULSE
Dimensionless diagram of the dynamic response
see:
Ch.10 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice”
NOTE:
Self-regulating dynamic behavior
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

11. DIMENSIONLESS RESPONSE OF FIRST-ORDER LAG TO SINUSOIDAL INPUT
Forcing function:
Dynamic response for long time:
f(t)
 taken from:
SCPC2-Modelli-rif_UniPI_ pdf
Frequency  unchanged
Amplitude ratio AR: B/A < 1
Phase lag <0
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

12. RESPONSE TO A SINUSOIDAL INPUT
Homework:
Diagram the
DYNAMIC RESPONSE
with the mod. Custom Process of LOOP-PRO Control Station
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

13. PURE CAPACITIVE SYSTEM or FIRST-ORDER INTEGRATOR
If ao=0
CANONICAL FORM
in the time domain in the Laplace domain
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

14. Process Instrumentation and Control - Prof. M. Miccio
PURE CAPACITIVE FIRST-ORDER SYSTEM position of the pole on the complex plane Im Re
Only 1 pole at the origin of the axes:
s = 0
Dynamic response to step input change
NOTE:
marginally stable dynamic system  NON-self-regulating for some inputs
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

15. THE OPEN TANK WITH A VARIABLE LEVEL AS A FIRST-ORDER SYSTEM
1st-order after linearization
b) pure capacitive
see:
Ch.10 - Stephanopoulos, "Chemical process control: an Introduction to theory and practice"
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

16. Process Instrumentation and Control - Prof. M. Miccio
NONLINEAR FIRST-ORDER SYSTEMS 1st case: nonlinearity of the function of y(t)
CANONICAL FORM
where F(y(t)) is a nonlinear function of y(t)
LINEARIZATION METHOD
Taylor expansion at the first term:
Where : ε=o(y-y0)2
o indicates the “order of magnitude”
y0= ys= steady-state value of y(t) or
y0=0
y0= another point of interest of y(t)
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

17. LINEARIZATION METHOD APPLICATION TO THE TANK WITH A VARIABLE LEVEL
Mass balance equations
in dynamic condition:
at the steady-state:
(1)
I. C. : t=0 ; h(0) = hs
(1 bis) R F0
Linearization by Taylor:
Replacing in (1) :
(2)
At the steady state (h=hs; Accumulation=0):
Thus, we have:
(3)
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

18. LINEARIZATION METHOD APPLICATION TO THE TANK WITH A VARIABLE LEVEL
Substracting eq. (3) from eq. (2)
the deviation variable appears:
I. C. : t=0 ; h’(0) = 0
Applying Laplace transform:
L(I) = L(II)
where:
NOTE:
The evaluated TF is valid only around the steady state.
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

19. LINEAR APPROXIMATION OF A NONLINEAR FUNCTION
Due to the tangent theorem, the simulation of the linear approximation is satisfactory only around the the point x0 (hs) of linearization (it is not possible to extend the linearized function to the whole curve).
Another linearization will be necessary to analyze the dynamic response of the system in another point. t0 t
see:
Ch.6 - Stephanopoulos, “Chemical process control: an Introduction to theory and practice” t1
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

20. Process Instrumentation and Control - Prof. M. Miccio
ORIGINAL MODEL and LINEARIZED MODEL: COMPARISON BETWEEN THE DYNAMIC RESPONSES APPLICATION TO THE TANK WITH A VARIABLE LEVEL
Initial Condition
Theoretical response
see:
Ch.6 - Stephanopoulos, "Chemical process control: an Introduction to theory and practice"
12/09/2019
Process Instrumentation and Control - Prof. M. Miccio

21. Process Instrumentation and Control - Prof. M. Miccio
NONLINEAR FIRST-ORDER SYSTEM 2nd case: nonlinearity in the derivative term d [y(t)]/d t
ORIGINAL FORM:
where is a nonlinear function of y(t)
e.g.,
LINEARIZATION METHOD 
The linearization method is still applicable,
with the Taylor expansion at the first term,
but consequent elaboration and calculations necessary to obtain the mathematical model in terms of deviation variables and the TF are more difficult and not treated here!
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