Appendix A

Process Models Used for Case Studies

In this section the models used in the application examples are presented.

A.I Model of the pH Process

The modeling and control of pH (the concentration of hydrogen ions) in a continuous stirred tank reactor (CSTR) is a well-known control problem that presents difficulties due to the nonlinearity of the process dynamics. The CSTR is shown schematically in Figure A.I.

FHAC

pH

FIGURE A.1 Scheme of the pH setup.

A dynamic model of the pH in a tank can be obtained by considering the material balances on [Na+J and the total acetate [HAC+AC-J and as-

241 J. Abonyi, Fuzzy Model Identification for Control

© Birkhäuser Boston 2003

242 Appendix A. Process Models Used for Case Studies

suming that acid-base equilibrium and electroneutrality relationships hold [178].

Total acetate balance:

Sodium ion balance:

HAC equilibrium:

Water equilibrium:

Electroneutrality:

The pH can be calculated from the previous equations as

[H+]3 + [H+]2(Ka + [Na+]) + [H+]([Na+]Ka

-[HAC + AC-]Ka - Kw) - KwKa = 0

The parameters used in our simulations are taken from [52] and are given in Table A.1.

Table A.I Parameters used in the simulations. Parameter V FHAC

FNaOH [NaOH]in [HAC]in [Na+] [HAC + AC-] Ka Kw

Description Volume of the tank Flow rate of acetic acid Flow rate of NaOH Inlet conc. of NaOH Inlet conc. of acetic acid Initial conc. of sodium in the CSTR Initial conc. of acetate in the CSTR Acid equilibrium constant Water equilibrium constant

Nominal Value 1000 [I] 81 [I/min] 515 [I/min] 0.05 [mol/I] 0.32 [mol/I] 0.0432 [mol/I] 0.0432 [mol/I] 1.75310-5

10-14

A.2. Electrical Water-Heater 243

A.2 Electrical Water-Heater

The schematic diagram of the water-heater is shown in Figure A.2.

FIGURE A.2 The scheme of the physical system.

The water comes from the water pipeline into the heater through a con­trol valve and a pair of metal pipes containing a cartridge heater. The control task is to control the Tout outlet temperature by adjusting the u heating signal of the cartridge heater.

The temperature measurement is realized by PtlOO thermometers. The system has four analogue inputs (Tin inlet temperature, Tout outlet tem­perature, valve position and the F flow-rate), and two digital (open and close of the valve, CVO and CVC) and one analogue output (heating con­trol signal, u). The heaters are linked parallel and have a performance of 1 kW. The process is connected to a PC computer through ADVAN­TECH LabCard PCLD-780 and PCL-812 data acquisition boards. GENIE 3.02 data acquisition and control software was used to filter and convert the input signals (0-5V). The control algorithm runs in MATLAB 4.2. The program gets the filtered and converted measured data through DDE every two seconds [12].

For the purpose of physical modeling the system was decomposed into four interacting elements: the cartridge-heater (subscript h), the stream­ing water (subscript w), the pipe wall (subscript p) and the environment (subscript e). The following three heat balances in the form of partial differential equations can be established:

244 Appendix A. Process Models Used for Case Studies

where, Z E [0, L] with L denotes the length of the pipe. The description and the nominal values of the parameters are given in Table A.2.

Table A.2 Parameters used in the simulation model of the heating system.

Parameter Description Nominal value L Length of the pipe 2 X 48010 3 m (!h Density of the cartridge 3650kgjm3

Cph Heat capacity of the cartridge 1047 JjkgK Ah Surface of the cartridge 2.41 X 10-2 m 2

Vh Volume of the cartridge 4.82 X 10-5 m 3

al h - w heat transfer coefficient 316.3 Wm- 2 K-1 (!w Density of the water 1000 kgjm3

Cpw Heat capacity of the water 4186 JjkgK Tin Inlet water temperature l1.8C Vw Volume of the water 1.16 X 10-4 m 3

a2 w - p heat transfer coefficient 1196.1 Wm- 2 K-1 (!p Density of the wall 7850 kgjm3

Cpp Heat capacity of the wall 502 J jkgK te Temperature of the environment 21.6C Ap Inner surface of the wall 4.46 X 10-2 m 2

Vp Volume of the wall 7.37 X 10-5 m 3

Ae Outer surface of the wall 5.36 X 10-2 m 2

a e p - e heat transfer coefficient 1015.9 W m -2 K-1

The performance of the cartridge heater is given by

where QM is the maximal power, and u is the heating signal (voltage). The partial differential equations are approximated by eight compartments of equal volume. As (A.2) shows, the heating performance is a static nonlinear function of the heating signal (control input).

A.3. Distillation Column 245

A.3 Distillation Column The modelled and controlled process is a first-principle model of a binary

distillation column depicted in Figure A.3. The column has 39 trays, a re­boyler and a condenser. The simulation model was developed by Skogestad [241J. The studied column operates in LV configuration with two manip­ulated variables (reflux and boilup rate, UI and U2) and two controlled variables (top and bottom impurities, YI, Y2).

FIGURE A.3 Schematic diagram of a distillation column.

The modeling assumptions are equilibrium on all trays, total condenser, constant molar flows, no vapor holdup, linearised liquid dynamic.

Material balance:

dMi LHI - Vi - B { -Li+Vi-I-D

dt = Li+l - Li + Vi-I - Vi + F LHI - Li + Vi-I - Vi

at the condenser at the reb oiler at feed stage at other trays

(A.l)

at the condenser at the reb oiler at feed stage at other trays

(A.2)

246 Appendix A. Process Models Used for Case Studies

Liquid flow dynamics

Vapor-liquid equilibrium

(A.4)

where

• Xi, Yi: liquid and vapor compositions of the light compound at the i-th tray

• Vi: vapour flow rate from tray i to tray i + 1

• L i : liquid flow rate from tray i + 1 to tray i

• M i : liquid holdup at tray i

• F, D, B: feed flow rate, distillate flow rate, and bottom flow rate respectively

• r: hydraulic time constant

• >.: constant for effect of vapor flow in liquid flow

• MO,i and VO,i-l: nominal values for the liquid flow and holdup on stage i

The column data for the example are given in Table A.3.

Table A.3 The nominal parameter values of the simulation model.

number of trays feed tray feed composition relative volarity distillate flow liquid flow vapour flow liquid holdup time constant for liquid flow effect of vapor flow in liquid flow

N=40 NF =21 ZF = 0.5 a = 1.5 D=0.5 L = 2.706 V = 3.206 Mi = 0.5 r = 0.063 >'=0

The simulated system covers the most important effects for the dy­namic of a real distillation column [241]. Details about the model and the MATLAB implementation are available over the internet (http://www.chembio.ntnu.no/users/skogej).

A.4. Model of the Liquid Level Rig 247

A.4 Model of the Liquid Level Rig The simulated liquid level control problem is a benchmark problem that

allows the comparative assessment of different fuzzy controllers (Graham and Newell [107, 108], Posthlethwaite et al [209, 210, 211], Linkens and Kandiah [168] and Abonyi et al [18, 19]).

As depicted in Figure A.4, the level in a tank is manipulated by the inflow, while the outflow is dependent on the square root of the liquid level in the tank.

FIGURE A .. 4

Matlab4.0 tDDE~

Genie 3.02

Schematic diagram of the process.

r- .. -.. -· .. · .. ·· .. ·-·· .. (~

i

h

y

Hence, the simulation model of the process is a simple, nonlinear differ­ential equation

dh A-=F-c.v;;,

dt (A.5)

where A (10 cm2 ) is the cross-sectional area of the tank, h (0-100 cm) is the liquid level, F is the inlet flowrate, c is a flow coefficient (equal to 1).

The simulator was built in MATLABjSimulink. The u(k) = F(k) con­troller output is limited between a and 15 flow units. The investigated problem is the setpoint change over two ranges: the first between 10 and 15 cm, and the second between 90 and 95 cm.

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Index

adaptive adaptation parameter, 124 adaptive fuzzy control, 174, 226 affine model, 48 Akaike Information Criteria, 121 AN OVA decomposition, 57 augmented data approach, 11

B-spline network, 36 barrier function, 124 barycentric coordinate, 41 basis function, 37 Bayesian Information Criteria, 121 bias modeling, 6 black-box model, 2 block-oriented modeling, 13, 58

constrained identification, 15, 107 continuous stirred tank reactor,

61, 194 control horizon, 181 convolution model, 74 cores, 34 cost function, 181 curse of dimensionality, 73

data preprocessing, 25 defuzzification, 28 Delaunay triangulation, 40 delay coordinate method, 63 disturbance, 182, 191 dynamic linearization, 17

empirical modeling, 53

excitation signal, 146 expert know ledge, 2 explicit use of prior knowledge, 5

feedback block-oriented model, 59 feedback linearization, 217 filter design, 171 Final Prediction Error Criteria, 121 firing strength, 28 first-principle model, 2 forced response, 188 forward selection, 121 free response, 187 free run simulation, 56 fuzzification, 25 fuzzy basis function, 29 fuzzy control, 165 fuzzy Hammerstein model, 80 fuzzy inference, 28 fuzzy model based control, 166 fuzzy relational model, 27 fuzzy set, 25

Gauss-Newton method, 89 generalized Hamerstein model, 157 global constraints, 108 global identification, 92 gradient-descent method, 89 grey-box modeling, 11 grid partitioning, 30 grid-type partition, 126

271

272

Hammerstein model, 58 heuristic identification, 93 hierarchical defuzzification, 44 high-gain direction, 105 hybrid fuzzy convolution model,

73

identification, 87 if-then rule, 27 implicit use of prior knowledge, 3 impulse response model, 55, 56,

75 indirect adaptive control, 226 indirect adaptive IMC controller,

175 inequality constraints, 109 input constraint, 184 input multiplicity, 63 input projection, 57 input selection, 118 input sequence design, 54 input transformation, 57 internal model control, 169 interpretability, 123 inverse model, 46

least squares estimation, 90 leave-one-out validation, 120 Levenberg-Marquardt method, 89 linear parameter-varying (LPV) sys-

tem interpretation, 65 linearization, 50 linearization of the model, 65 linguistic interpretability, 123 liquid level control, 176, 247 local constraints, 108 local identification, 92 local model, 11

Mamdani fuzzy model, 27 measurement data, 2 mechanistic knowledge, 2 membership function, 25 model inversion, 46

Index

model predictive control, 180 model validation, 55 modern control theory, 16 Moore-Penrose pseudo-inverse, 92 multi-step-ahead prediction, 56 multiple-input, multiple-output sys-

tems (MIMO), 71 multivariate membership functions,

40

NAARX model, 58 Narendra and Gallman algorithm,

153 NARMAX model, 56 NARX model, 60 NARX model, 56 neuro-fuzzy system, 6 Newton method, 89 NOE model, 56 noise modeling, 55 nonlinear predictive control, 185

operating regime, 36 operating regime based model, 11 orthogonal transform, 122 output constraint, 184 output multiplicity, 63 output-error (OE) model, 56

p-fold cross-validation, 119 parameter estimation, 55 partial derivative, 50 partial input reduction, 44 pH control, 241 PID control, 16 piecewise models, 36 postprocessing, 28 prediction horizon, 180 prior knowledge, 1 product-sum-gravity inference, 31

quadratic programming, 108

radial basis function, 38 receding horizon principle, 185

Index

reference trajectory, 181 regularity criteria, 119 regularization, 124 relative constraints, 108 residence time distribution, 75 rule base, 26 rule generation on extrema, 118

sampling time, 110 scatter partition, 31 semi-mechanistic modeling, 7 sequential technique, 186 settling time, 112 simultaneous technique, 186 singleton, 25 singleton fuzzy model, 31 smoothed maximum inference, 98 stability, 64, 110 state estimation, 183 state-space model, 8 state-space realization, 63 steady-state behavior, 63

steady-state gain, 112 steady-state offset, 183 structure identification, 118 structure selection, 54

273

successive linearization technique, 186

Taylor expansion, 50 tree partition, 31 triangular membership function,

34

universal approximation, 28

variance accounted for index, 114 variance of accounted for index

(VAF) , 106 Volterra model, 58

weighted least-squares, 93 white-box model, 2 Wiener model, 59