A.2 Model of the Beer Fermentation Process
The differential equations:
dxlag
dt = −µlagxlag
dxactive
dt = −µxxactive−kmxactive+µlagxlag
dxbottom
dt = kmxactive−µDxbottom ds
dt = −musxactive de
dt = −muaf xactive d(acet)
dt = −mueasµsxactive d(diac)
dt = kdcsxactive−kdm(diac)e (A.2)
The reaction rates:
µx = µx0s 0.5si+e µD = 0.5siµD0 0.5si+e µs = µs0s
ks+s µa = µa0s
ks+s f = 1− e
0.5si
(A.3) The parameters:
µx0 =e108.31−T31934.09+273.15 µs0 =e−41.92−T11654.64+273.15 µD0 =e33.82−T10033.28+273.15
µa0 =e3.27−T1267.24+273.15 µeas =e89.92−T+273.1526589
µlag =e30.72−T9501.54+273.15 km =e130.16−T+273.1538313
ks =e−119.63−T34203.95+273.15 kdc = 0.000127672
kdm = 0.00113864 (A.4)
The initial values:
xlag,i = 0.192 xact,i = 0.08 xbottom,i = 2
si = 130 ei = 0
(acet)i = 0 (A.5)
Table A.3. Notations (Beer Fermentation Model) Notation Description
T Temperature
xlag Suspended latent biomass xactive Suspended active biomass xbottom Suspended dead biomass
s Sugar conc.
e Ethanol conc.
(acet) Ethyl acetate conc.
(diac) Diacetyl conc.
µx Yeast growth rate µD Yeast settling down rate
µs Substrate (sugar) consumption rate µa Ethanol production rate
f Fermentation inhibition factor µeas Ethyl acetate formation rate µlag Specific rate of latent formation km Yeast growth inhibition parameter ks Sugar inhibition parameter
kdc Diacetyl appearance rate
kdm Diacetyl disappearance of reduction rate
B
Appendix: Notations
B Appendix: Notations 103 Table B.1. Notations of Chapter 2 (sorted as they appear)
Notation Description
n search dimensions, number of object variables x vector of object variables (ES)
vector of strategy variables (ES)
aj vector representation ofj-th individual (ES) τ first global learning rate (ES)
τ second global learning rate (ES)
λ number of individuals (size of population) (ES) µ number of parents (ES)
vj velocity vector in the search space ofj-th particle (PSO) xj vector representation ofj-th particle (PSO)
xpbest particle best solution (PSO) xgbest global best solution (PSO) w inertia weight parameter (PSO) c1 local learning factor (PSO) c2 global learning factor (PSO) e control error (input of controller)
u manipulated input (output of controller) T0 sample time of controller
Km gain of master PID controller
Tim time constant of master PID controller Tdm time constant of master PID controller Tds gain of slave P controller
β weighting parameter of cost function u vector of manipulated inputs (MPC)
y vector of corrected predicted outputs (MPC) w vector of setpoints (MPC)
Hp prediction horizon of MPC Hc control horizon of MPC
Q1,Q2 control error weighting parameters of MPC R1,R2 input manipulation weighting parameters of MPC y˜ vector of predicted outputs from the model (MPC)
˜" calculated model error (MPC)
" predicted model error (MPC)
α IMC discrete filter time constant (MPC)
Ceth ethanol concentration in beer fermentation process Cacet ethyl-acetate concentration in beer fermentation process Cdiacet diacetyl concentration in beer fermentation process tend operation time of fermentation process
Table B.2. Notations of Chapter 3 (sorted as they appear) Notation Description
yˆ model output
pi parameters of linear in parameter model
Fi nonlinear functions of linear in parameter model
x vector of regression variables for linear in parameter model
u measured input
y measure output
e model error
y0 constant bias
N number of data points, length of y M number of regressors
y vector of measured outputs p vector of model parameters e vector of model errors
F regression matrix for linear least squares
W upper triangular matrix from orthogonal decomposition A orthogonal matrix from orthogonal decomposition g auxiliary parameter vector
[err]i error reduction ratio of Fi
fi fitness value of i-th individual (GP)
ri correlation coefficient of i-th individual (GP) Li size of i-th tree (GP)
a1, a2 tree-size penalty parameters in fitness function
B Appendix: Notations 105 Table B.3. Notations of Chapter 4 (sorted as they appear)
Notation Description
t time variable
τ time variable in integration x vector of measured data
t vector of measurement time instants for x N size of xandt vectors
ki time value of ith knot
n number of knots
S cubic spline function
si,si parameters of spline (spline value and derivative at ki) vector of parameters of spline
Ψ matrix for identification of spline parameters l number of splines
Sj jth spline function
xj vector of measured data for jth spline tj vector of measurement time instants for xj Nj size of xj andtj vectors
x˜ vector of xj vectors
˜t vector of tj vectors
j vector of parameters of jth spline ˜ vector of j vectors
ˆti time instant used for evaluation of soft constraints ˆt vector of ˆti time instants (for soft constraints) Nˆ size of ˆtvector
λi, regularization parameters, weights of soft constraints Ψ˜i matrices for identification of parameters of splines Ci concentration of ith component
wi,j stoichiometric coefficient for component i in reaction j krj reaction rate constant of j reaction
Rj function of concentrations for j reaction rate R¯j integrated Rj
ˆkrj estimated reaction rate constant ofj reaction Cˆi estimated concentration ofi component P1, P2 performance indexes (example-1) CX concentration of X component
krj assumed kinetic constant for j by-reaction
C0 sum of initial concentrations (=CA0+CB0+CC0) Sj jth spline function (for A,B,. . .,C∗ components) ˆti ith time instant for evaluation of soft constraints Nˆ number of ˆti time instants
xj,i measured concentration of ith components (A,B,C) at jth measurement
tj,i measurement time instant for xj,i
Nj number of measurements for j components (= size ofti) λ weight of soft constraint
j vector of parameters of jth spline ˜ vector of j vectors
Table B.4. Notations of Chapter 5 (sorted as they appear) Notation Description
f(),g(), h() functions of state-space model
x vector of state-variables of state-space model u manipulated input of process
y controlled output of process n number of state-variables
v input of input-output linearized object r relative degree of process
ysp set-point
βi parameters of feed-back control law fF P() first principle model
fSM() semi-mechanistic model
fN N function represents neural network z vector of neural network inputs
set of weight and bias parameters of neural network Wi weights of NN i-th layer
bi biases of NN i-th layer
N number of training data-points
e model error
yˆN N output of neural network (training)
yN N desired output of neural network (training) J Jacobian matrix (training)
T0 sample time
yˆ output of semi-mechanistic model (training)
y desired output of semi-mechanistic model (training) x˜ estimated state variables from training data
˜z estimated inputs of NN from training data xn noisy state variables of CSTR
yn noisy output of CSTR
xˆ vector of estimated state-variables by Observer x1 concentration in reactor (CSTR)
x2 reactor temperature (CSTR) x3 jacket temperature (CSTR) Kc gain of PID controller
Ti time constant of PID controller Td time constant of PID controller
Tc time constant of desired close-loop transfer
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TÉZISEK
1. Tézis. Folyamatoptimalizálás esetén fellépő egymásnak ellentmondó, nehezen formalizálható célok kezelése a felhasználó értékelésbe történő bevonásával.
A folyamatoptimalizálásban gyakori nehézség, hogy egymásnak ellentmondó, nehezen formalizálható vagy számszerűsíthető célokat kell egy időben figyelembe venni. Ilyen esetekben a klasszikus megközelítés, vagyis egy kvantitatív célfüggvény felállítása, nehézkes, sőt értelmetlen is lehet. Ezért egy olyan interaktív optimalizáló algoritmust dolgoztam ki, amely úgy kezeli az ilyen optimalizációs feladatokat, hogy bevonja a folyamatmérnököt az értékelés folyamatába, tehát a mérnöki, szakértői ismereteket közvetlenül alkalmazza az optimalizálási feladatban. Ez a módszer számítógépes grafikában és számítógépes tervezésben már sikeresen volt alkalmazva, de a folyamatmérnöki területen még nem alkalmazták. Az algoritmus kidolgozása során ezt a technikát adaptáltam a jellemző folyamatmérnöki problémákhoz, és olyan megoldást fejlesztettem ki, amely hatékonyan és egyszerűen használható a folyamatmérnökségben. A kifejlesztett eszköz gyakorlati hasznossága abban rejlik, hogy segítségével sok esetben gyorsabban és könnyebben érthető/átlátható módon lehet az ellentmondó célok és feltételek között megfelelő kompromisszumot kötni, mint a hagyományos módszerekkel.
Ezt egy szabályozó hangolás és egy biokémiai reaktor optimalizálás problémájával demonstráltam szimulációs keretek között.
2. Tézis. A mérési adatokból történő modell struktúra és modell rendűség identifikáció javítása az ortogonális legkisebb négyzetek módszerével.
A mérési adatokból történő modellalkotás egy kevéssé elterjedt, de a tudományos irodalomban egyre népszerűbb megközelítése, a mérési adatokon alapuló nemlineáris modell struktúra identifikáció genetikus programozással. Felismertem, hogy a genetikus programozás hajlamos túlzottan összetett struktúrákat identifikálni, különösen ha a mérési adatok zajjal terheltek, de a szakirodalomban található munkák alig szentelnek figyelmet ennek a problémának. Ezért egy olyan új módszert dolgoztam ki, amely struktúra identifikáció során az ortogonális legkisebb négyzetek módszerét használva kiszűri a felesleges tagokat a paramétereiben lineáris modellekből, ami egyszerűbb, áttekinthetőbb modelleket eredményez. Ez az új technika nemcsak dinamikus nemlineáris modell struktúrák identifikációjára, de modellrendűség meghatározására is használható. A kidolgozott módszer hatékonyságát kémiai reaktorok modellrendűségének vizsgálatával mutatattam be.
3. Tézis. Mérési adatokon alapuló kinetikai paraméter becslés javítása a priori ismeretek segítségével.
A kémiai reaktorok vizsgálata során gyakran előfordul, hogy a mérési adatok száma kicsi, a mérési pontok időben egyenlőtlenül oszlanak el. Ilyen esetekben a mérési adatok felhasználása előtt valamilyen interpolációs módszert kell alkalmazni, hogy a mért tulajdonságokat meg tudjuk becsülni a mérési pontok között, és hogy a mérési zaj hatását csökkentsük. Az elterjedt interpolációs módszerek egy jelentős hiányossága az, hogy nem hasznosítják az a priori ismerteket. Ez igaz a jól ismert és a gyakorlatban elterjedt köbös spline interpolációra is. Ezért egy olyan új módszert fejlesztettem ki, amelyben a mérési adatokra vonatkozó a priori ismereteket
tudja venni, ami által a mérési adatok hatékonyabban hasznosíthatóak. Egy szimulált reaktor és egy ipari reaktor kinetikai paraméterinek becslésével demonstráltam, hogy a kifejlesztett módszer alkalmas a kémiai reaktorok mérési adataiból történő kinetikai paraméterbecslés pontosabbá tételére.
4. Tézis. A modell alapú szabályozók paraméterérzékenységének csökkentése szemi-mechanisztikus modellezéssel.
A nemlineáris modell alapú szabályozótervezés kritikus pontja a szabályozott rendszer a priori modelljének megalkotása. A gyakorlatban az a priori modell alapú szabályozó teljesítményét jelentősen korlátozza a modell paraméterek (például a kémia kinetikai paraméterek) és a modell struktúra bizonytalansága. Ezért egy olyan hibrid modellt fejlesztettem ki, amely kombinálja az a priori modellezést és az a posteriori modellezést olyan módon, hogy egy mesterséges neurális hálózat helyettesíti a fehér doboz modell bizonytalan részét. Ennek a hibrid modellezési technikának az előnye az a priori modellezés technikával szemben az, hogy kevésbé érzékeny a modell bizonytalanságával szemben, amit egy kevert üstreaktor szabályozási példáján demonstráltam.
THESES
1. Solving of multi- and conflicting-objective process optimization problems by interactive optimization procedure
Process optimization problems often lead to multi-objective problems where optimization goals are non-commensurable and they are in conflict with each other. In such cases, the common approach, namely the
Process optimization problems often lead to multi-objective problems where optimization goals are non-commensurable and they are in conflict with each other. In such cases, the common approach, namely the