Abstract

Several industrial chemical processes exhibit severe nonlinearity. This paper addresses the computational and nonlinear issues occurring in many typical industrial problems in aspects of its stability, strength of nonlinearity, and input output dynamics. In this article, initially, a prospective investigation is conducted on various nonlinear processes through phase portrait analysis to understand their stability status at different initial conditions about the vicinity of the operating point of the process. To estimate the degree of nonlinearity, for input perturbations from its nominal value, a novel nonlinear measure Δ0 is put forward that anticipates on the converging area between the nonlinear and their linearized responses. The nonlinearity strength is fixed between 0 and 1 to classify processes to be mild, medium, or highly nonlinear. The most suitable operating point, for which the system remains asymptotically stable, is clearly identified from the phase portrait. The metric Δ0 can be contemplated as a promising tool to measure the nonlinearity of Industrial case studies at different linear approximations. Numerical simulations are executed in matlab to compute Δ0, which conveys that the nonlinear dynamics of each industrial example is very sensitive to input perturbations at different linear approximations. In addition to the identified metric, nonlinear lemmas are framed to select appropriate control schemes for the processes based on its numerical value of nonlinearity.

References

1.
Nikolaou
,
M.
,
1993
, “
When is Nonlinear Dynamic Modeling Necessary?
,”
American Control Conference
, San Francisco, CA, June 2–4, pp.
910
914
.10.23919/ACC.1993.4792995
2.
Nikolaou
,
M.
, and
Hanagandi
,
V.
,
1998
, “
Nonlinearity Quantification and Its Application to Nonlinear System Identification
,”
Chem. Eng. Commun.
,
166
(
1
), pp.
1
33
.10.1080/00986449808912379
3.
Stack
,
A. J.
, and
Doyle
,
F. J.
, III
,
1997
, “
Application of a Control‐Law Nonlinearity Measure to the Chemical Reactor Analysis
,”
AIChE J.
,
43
(
2
), pp.
425
439
.10.1002/aic.690430216
4.
Liu
,
J.
,
Li
,
B.
,
Miao
,
H.
,
He
,
A.
, and
Zhu
,
S.
,
2018
, “
Numerical and Experimental Study of Clearance Nonlinearities Based on Nonlinear Response Reconstruction
,”
ASME J. Comput. Nonlinear Dyn.
,
13
(
2
), p. 021001.10.1115/1.4037927
5.
Emancipator
,
K.
, and
Kroll
,
M. H.
,
1993
, “
A Quantitative Measure of Nonlinearity
,”
Clin. Chem.
,
39
(
5
), pp.
766
772
.10.1093/clinchem/39.5.766
6.
Guay
,
M.
,
McLellan
,
P. J.
, and
Bacon
,
D. W.
,
1995
, “
Measurement of Nonlinearity in Chemical Process Control Systems: The Steady State Map
,”
Can. J. Chem. Eng.
,
73
(
6
), pp.
868
882
.10.1002/cjce.5450730611
7.
Fatoorehchi
,
H.
,
Abolghasemi
,
H.
, and
Zarghami
,
R.
,
2015
, “
An Efficient Measure for Quantification of Nonlinearity in Chemical Engineering Processes Based on I/O Steady-State Loci
,”
Chem. Eng. Commun.
,
202
(
12
), pp.
1557
1563
.10.1080/00986445.2014.962689
8.
Desoer
,
C.
, and
Wang
,
Y. T.
,
1980
, “
Foundations of Feedback Theory for Nonlinear Dynamical Systems
,”
IEEE Trans. Circuits Syst.
,
27
(
2
), pp.
104
123
.10.1109/TCS.1980.1084787
9.
Harris
,
K. R.
,
Colantonio
,
M. C.
, and
Palazoğlu
,
A.
,
2000
, “
On the Computation of a Nonlinearity Measure Using Functional Expansions
,”
Chem. Eng. Sci.
,
55
(
13
), pp.
2393
2400
.10.1016/S0009-2509(99)00514-X
10.
Liu
,
Y.
, and
Li
,
X. R.
,
2015
, “
Measure of Nonlinearity for Estimation
,”
IEEE Trans. Signal Process.
,
63
(
9
), pp.
2377
2388
.10.1109/TSP.2015.2405495
11.
Spina
,
D.
,
Valente
,
C.
, and
Tomlinson
,
G. R.
,
1996
, “
A New Procedure for Detecting Nonlinearity From Transient Data Using the Gabor Transform
,”
Nonlinear Dyn.
,
11
(
3
), pp.
235
254
.10.1007/BF00120719
12.
Kruger
,
U.
,
Antory
,
D.
,
Hahn
,
J.
,
Irwin
,
G. W.
, and
McCullough
,
G.
,
2005
, “
Introduction of a Nonlinearity Measure for Principal Component Models
,”
Comput. Chem. Eng.
,
29
(
11–12
), pp.
2355
2362
.10.1016/j.compchemeng.2005.05.013
13.
Hahn
,
J.
, and
Edgar
,
T. F.
,
2001
, “
A Gramian Based Approach to Nonlinearity Quantification and Model Classification
,”
Ind. Eng. Chem. Res.
,
40
(
24
), pp.
5724
5731
.10.1021/ie010155v
14.
Hosseini
,
S.
,
Fatehi
,
A.
,
Johansen
,
T. A.
, and
Sedigh
,
A. K.
,
2012
, “
Multiple Model Bank Selection Based on Nonlinearity Measure and H-Gap Metric
,”
J. Process Control
,
22
(
9
), pp.
1732
1742
.10.1016/j.jprocont.2012.07.006
15.
Jiang
,
M.
,
Wu
,
J.
,
Peng
,
X.
, and
Li
,
X.
,
2017
, “
Nonlinearity Measure Based Assessment Method for Pedestal Looseness of Bearing-Rotor Systems
,”
J. Sound Vib.
,
411
, pp.
232
246
.10.1016/j.jsv.2017.09.002
16.
Slotine
,
J. J. E.
, and
Li
,
W.
,
1991
,
Applied Nonlinear Control
,
Prentice Hall
,
Englewood Cliffs, NJ
.
17.
Khalil
,
H. K.
, and
Grizzle
,
J. W.
,
2002
,
Nonlinear Systems
, Vol.
3
,
Prentice Hall
,
Upper Saddle River, NJ
.
18.
Purohit
,
J. L.
,
Mahajani
,
S. M.
, and
Patwardhan
,
S. C.
,
2013
, “
Analysis of Steady-State Multiplicity in Reactive Distillation Columns
,”
Ind. Eng. Chem. Res.
,
52
(
14
), pp.
5191
5206
.10.1021/ie400288r
19.
Shuler
,
M. L.
, and
Kargi
,
F.
,
1992
,
Bioprocess Engineering: Basic Concepts
,
Prentice Hall
, Upper Saddle River,
NJ
.
20.
Bequette
,
B. W.
,
2003
,
Process Control: Modeling, Design, and Simulation
,
Prentice Hall Professional
, Upper Saddle River, NJ.
21.
Du
,
J.
, and
Johansen
,
T. A.
,
2017
, “
Control-Relevant Nonlinearity Measure and Integrated Multi-Model Control
,”
J. Process Control
,
57
, pp.
127
139
.10.1016/j.jprocont.2017.07.001
22.
Galan
,
O.
,
Romagnoli
,
J. A.
, and
Palazoglu
,
A.
,
2004
, “
Real-Time Implementation of Multi-Linear Model-Based Control Strategies—An Application to a Bench-Scale pH Neutralization Reactor
,”
J. Process Control
,
14
(
5
), pp.
571
579
.10.1016/j.jprocont.2003.10.003
23.
Skogestad
,
S.
,
2003
, “
Simple Analytic Rules for Model Reduction and PID Controller Tuning
,”
J. Process Control.
,
13
(
4
), pp.
291
309
.10.1016/S0959-1524(02)00062-8
24.
Hernjak
,
N.
, and
Doyle
,
F. J.
, III
,
2005
, “
Glucose Control Design Using Nonlinearity Assessment Techniques
,”
AIChE J.
,
51
(
2
), pp.
544
554
.10.1002/aic.10326
25.
Peng
,
C.
,
Tian
,
Y. C.
, and
Tade
,
M. O.
,
2008
, “
State Feedback Controller Design of Networked Control Systems With Interval Time‐Varying Delay and Nonlinearity
,”
Int. J. Robust Nonlinear Control
,
18
(
12
), pp.
1285
1301
.10.1002/rnc.1278
26.
Swati
,
D.
,
Rao
,
V. S. R.
,
Pickhardt
,
R.
, and
Chidambaram
,
M.
,
2005
, “
Nonlinear Control of pH System for Change Over Titration Curve
,”
Chem. Biochem. Eng. Q.
,
19
(
4
), pp.
341
349
.https://hrcak.srce.hr/3629
27.
Choudhury
,
M. S.
,
Shook
,
D. S.
, and
Shah
,
S. L.
,
2006
, “
Linear or Nonlinear? A Bicoherence Based Metric of Nonlinearity Measure
,”
IFAC Proc.
,
39
(
13
), pp.
617
622
.10.3182/20060829-4-CN-2909.00102
28.
Bröcker
,
M.
, and
Herrmann
,
L.
,
2017
, “
Flatness Based Control and Tracking Control Based on Nonlinearity Measures
,”
IFAC-Papers Online
,
50
(
1
), pp.
8250
8255
.10.1016/j.ifacol.2017.08.1394
29.
Penalba
,
M.
, and
Ringwood
,
J. V.
,
2019
, “
Linearisation-Based Nonlinearity Measures for Wave-to-Wire Models in Wave Energy
,”
Ocean Eng.
,
171
, pp.
496
504
.10.1016/j.oceaneng.2018.11.033
30.
Wu
,
J.
,
Jiang
,
M.
,
Li
,
X.
, and
Feng
,
H.
,
2017
, “
Assessment of Severity of Nonlinearity for Distributed Parameter Systems Via Nonlinearity Measures
,”
J. Process Control
,
58
, pp.
1
10
.10.1016/j.jprocont.2017.08.001
31.
Du
,
J.
,
Song
,
C.
, and
Li
,
P.
, 2009, “
A Gap Metric Based Nonlinearity Measure for Chemical Processes
,”
American Control Conference
, St. Louis, MO, June 10–12, pp.
4440
4445
.10.1109/ACC.2009.5160217
32.
Bequette
,
B. W.
,
1998
,
Process Dynamics: Modeling, Analysis, and Simulation
,
Prentice Hall PTR
,
Upper Saddle River, NJ
.
You do not currently have access to this content.