Nomenclat ur e

a Alkaloid in Section 2.6.2

a Number of equality constraints in Eqs. 7.5 and 7.6

dij Amount of Ni utilized for production of unit amount of Pj in Eq. 2.19 (g/g)

aij State transition probabilities of an HMM from state i to state j a Gas-liquid interfacial area per unit culture volume (l/m)

A = [aij] State transition matrix of an HMM

A Input metabolite in Ch. 2 and 9

b Number of inequality constraints in Eqs. 7.5 and 7.6

B = {bj(k)} Observation symbol probability distribution

B Output metabolite

C = {cj} Initial state distribution (initial state occupancy probability)

Cp, Contribution of each element in Znewjfc to £>-statistic summed over all r components

Cfk FCC of kth reaction affected by enzyme i

Cljk r Contribution of each element of a new batch run new, jk on the rth score

Ci Concentration of specie i in the bulk liquid (g/L)

C* Concentration of specie i in the liquid phase at the gas-liquid interface (g/L)

C/fc Flux control coefficient for pathway k with respect to enzyme E.t or reaction i, defined in Eqs. 9.4 and 9.5

Ci 3 Concentration control coefficient for the intermediate Xj with respect to enzyme Ei, defined in Eq. 9.7

Cj 3 CCC of intermediate Xj affected by activity of enzyme i

Cl Dissolved O2 concentration in Eqs. 2.47, 2.48, 2.5 (mmole/L)

C*L Dissolved 0 > concentration at maximum saturation (mmole/L)

Cx Concentration of biomass (cell mass) in culture (g/L culture)

CiG Concentration of specie i in the bulk gas (g/L)

C*G Concentration of specie i in the gas phase at the gas-liquid interface (g/L)

Cjf, Contribution of new observation vector xnewjk to D-statistic

CjK Contributions to Q-statistic of J variables over the entire batch run

C®k Contributions to Q-statistic of variable j at time k d p or m^-dimensional vector of disturbance variables d Hyphal diameter (m)

di(x) Linear discriminant score d^(x) Quadratic discrimination score for the ix.h population

D Differential operator in Section 4.4

Die Molecular diffusivity of specie i in the gas phase (m2/h)

D1l Molecular diffusivity of specie 1 in the liquid phase (m2/h)

e, E Residuals vector and matrix, respectively, in Ch. 4, 6 and 8

e Predicted error vectors, defined in Eqs. 7.152 and 7.157

e Output estimation error y — y in Ch. 4, 6 and 8

f Nonlinear function of d, u, x in Eqs. 2.1 and 7.1

Ei Enzyme corresponding to the ith reaction step

Ej Enzyme catalyzing reaction j in a metabolic pathway

F Volumetric feed rate of nutrient medium (L/h)

F, f Bioreactor feed, gas feed or liquid feed as appropriate

Fs Volumetric feed rate of nutrient medium in singular control (L/h) /(<£, w) Defined in Eqs. 7.88 and 7.89

fh Fraction of hyphal cells that are capable of synthesizing penicillin

G(q) Multivariable input-output transfer function matrix

G Transfer function matrix defined in Eqs. 7.79 and 7.108

Gi, G2 Transfer function matrices for multi-loop feedback control defined in Eq. 7.111

Gc(<?) Actuator (controlled input) fault TFM

GC1 Gm Transfer function matrices associated with feedback controllers and measuring devices, respectively

Gd Transfer function matrix defined in Eq. 7.108

Gi, Ki Transfer function matrix and gain matrix, respectively, for the decouplers, Eqs. 7.117-7.120

Gm(q) Input sensor fault TFM

Gi(y(tf)) Used in the definition of the objective function J in Eq. 7.4

Gs Throughput rate of substrate S, involved in the constraint in Eq. 7.7 (g/h)

G(x(tf)) Used in the definition of the objective function J in Eq. 7.3

g(x, u) Used in the definition of the objective function J in Eq. 7.3

g'(y, u) Used in the definition of the objective function J in Eq. 7.4

g(i) Impulse-response functions in Eq. 7.141

g, m Concentrations of glucose and methionine, respectively, in the abiotic phase (Section 2.6.3) (g/L)

h Nonlinear function of x in Eq. 7.2

H Hankel matrix in Ch. 4

Hjk Scaled Hankel matrix

H Hamiltonian defined in Eq. 7.10

/i(x, u) Used in the definition of the objective function J in Eq. 7.77

Hi Henry's law constant for specie i

I Identity matrix

I Number of batches in a reference set in Ch. 4, 6 and 8

Im(.), Re{.) Imaginary and real parts of a complex number or expression

J Objective function (performance index) defined in Eq. 7.3

Ji Flux of metabolic pathway i

Jk Metabolic flux through the fcth reaction

K Steady-state gain matrix defined in Eq. 7.112

Koo Kalman filter gain in Ch. 4

KiL Overall liquid-based mass transfer coefficient for specie i, defined in Eq. 2.6 (m/h)

ka, ks,kh Maximum specific growth rates of apical, subapical, and hyphal cells, respectively, in Eq. 2.25 (1/h)

kiG Gas-side mass transfer coefficient for specie i, defined in

kiL Liquid-side mass transfer coefficient for specie i (m/h)

kU] Kinetic coefficients in Eqs. 2.22-2.24 (1/h)

L, I/o Nonlinear functions in Eqs. 7.162 and 7.163

L[-] Simplified log-likelihood function

M NumbeT of distinct observation symbols per state in HMMs, the alphabet size m Length of control sequence (controller output) prediction horizon in model predictive control rrii Maintenance coefficient for specie i in Eq. 2.19 (1/h)

Ms Amount of substrate S supplied in a batch or fed-batch operation, involved in the constraint in Eq. 7.8 (g)

rriih, mis, mia Intracellular concentrations of methionine in hyphae, swollen hyphal fragments, and arthrospores, respectively (g/g)

Nm Coefficient matrix of multiplicative modeling faults

Np Matrix of time-varying coefficients of multiplicative parametric faults

N Number of states in an HMM

Ni Concentration of nutrient Ni in the abiotic phase (g/L)

Ni Flux of specie i from the gas phase to liquid phase, defined in Eqs. 2.4, 2.5 and 2.6 (g/{m2.h})

Ni Nutrient i

O Observation sequence in an HMM (oi ■ • • ot)

Q, R, S Positive definite weighting matrices in Eq. 7.163

p, P Loading vector and matrix, respectively, in Ch. 4, 6 and 8

P Concentration of target non-biomass product (g/L)

P Target non-biomass product (sections 2.6.4 and 7.2.5)

Pj Concentration of non-biomass product Pj on the basis of the abiotic phase volume (g/L)

Pj Non-biomass product j p Parity vector p Extracellular phosphate concentration in Section 2.6.2 (g/L)

p Length of output prediction horizon in model predictive control p Target product (cephalosporin C) in Section 2.6.3

■p.t Intracellular phosphate concentration in Section 2.6.2 (g/g)

pH Culture pH

Q Evolving sequence of states S of an HMM in Section 8.3.3

Q Volumetric flow rate of culture in Ch. 2 (L/h)

Qa Volumetric flow rate of abiotic phase, defined in Eq. 2.9 (L/h)

Qb Volumetric flow rate of biotic phase, defined in Eq. 2.9 (L/h)

Qsn Quantile of standard Normal distribution

Qy Quantile of ordered data set

\q] Intracellular concentration of specie q (Section 2.6.4) (g/g)

q Shift operator in Section 4.4

qij Intracellular concentration of specie q in cell type j (Section 2.6.3) (g/g cell type j)

qt Actual state of a discrete-time system at time t qk Maximum likelihood estimate

R Defined in Eq. 7.113

n, r2 Amplitudes of periodic variations in Ui and 112, respectively (Section 7.3)

Tct Specific rate of cell loss due to cell death or cell lysis (l/h)

n Net rate of generation of specie i in the biotic phase in Ch. 2 (l/h)

Ti Residual based on the PC model for fault i in Ch. 8

Rfn Rate of generation of specie i due to reactions in the abiotic phase (g/{L.h})

rfen Net rate of generation of specie i in the biotic phase exclusive of the rate of its loss from the biotic phase due to cell death or cell lysis (l/h)

rtranS Biomass-specific rate of transport of specie i from the biotic phase to the abiotic phase (l/h)

rqjn Specific rate of net generation of specie q in cell type j (g/{g cell type j.h})

rtrans Specific rate of transport of specie q from the cells of type j to the abiotic phase (g/{g cell type j.h})

S Riccati transformation matrix in Eqs. 7.136 and 7.137

S Covariance matrix of scores in Ch. 4, 6 and 8

Sb Between-class scatter matrix

Sf Defined in Eq. 7.132

SPF(q) Plant fault TFM

Spi Pooled estimate of £

SpN(q) Plant noise TFM

Sw Within-class scatter matrix

Sy Total scatter matrix

S Concentration of limiting substrate in the abiotic phase (liquid) (g/L)

S Distinct states of a discrete-time system in Ch. 8

S Limiting substrate s Laplace transform variable

Si Score distance based on the PC model for fault i t, T Scores vector and matrix, respectively, in Ch. 4, 6 and 8

tf Duration of a bioreactor operation (h)

T Sampling period (h)

u m-dimensional vector of manipulated inputs u, U PLS scores vector and matrix, respectively, in Ch. 4, 6 and 8

ui, U2, us Rates of the three metamorphosis reactions in (Eqs. 2.222.24) (l/h)

V, W Scaling matrices in Eq. 7.158 V, W, Vi, Wi Defined in Eqs. 7.122, 7.123 and 7.126

V Culture volume (L)

Va Volume of abiotic phase, defined in Eq. 2.9 (L)

Volume of biotic phase, defined in Eq. 2.9 (L)

Vg Volume of gas phase in the bioreactor (gas phase holdup) (L)

Vt Culture volume (L)

Abiotic Specific volume of the cells (L/g)

vc Input actuator noise vm Input sensor noise vp Plant noise

\y Output sensor noise

Vj Rate of reaction j in a metabolic pathway (Eq. 9.1)

v, w Parameters in the definition of J in Eq. 7.95

v Deviation in v(t) from its reference, yr, v— d, u, x in section 7.4.1

v(s) Laplace transform of v(t)

W Projection matrix

W Wavelet transform w Estimated weight functions in Section 4.4

X Three-way array of process measurements in Ch. 4, 6 and 8

X Unfolded matrix of process measurements in Ch. 4, 6 and 8

X Dynamic Matrix defined in Eqs. 7.150b and 7.157

X Biomass (cell mass)

X Concentration of biomass (cell mass) in culture (g/L culture)

Xh, Xs, Xa Concentrations of hyphae, swollen hyphal fragments, and arthrospheres, respectively (Section 2.6.3) (g/L culture)

X/,, X„ Xa Three morphological types of Cephalosporium acremonium, hyphae, swollen hyphal fragments, and arthrospheres, respectively (Section 2.6.3)

x Vector of state variables x Vector of process measurements y Predicted output vector y, Y Vector and matrix of quality measurements, respectively, in Ch. 4, 6 and 8

y /-dimensional vector of output variables ym Vector of measured outputs ytj i Stacked vectors of future and past

Yp/x Cell mass phosphate content in Eq. 2.30 (g/g)

Yx/Ni Biomass (cell mass) yield with respect to nutrient Nt in

Z Defined in Eq. 7.81

z z-transform variable

Za, Z3, Zh Mass fractions of apical, subapical, and hyphal cells, respectively, in the total cell population (Eqs. 2.22-2.24)

Zh, Zs, Za Mass fractions of hyphae, swollen hyphal fragments, and arthrospheres, respectively, in the total cell population (Section 2.6.3)

Greek Letters aj Constant characteristic of a particular metabolite Pj in Eq. 2.21

/?, /3 Vector and matrix of regression coefficients, respectively

¡3(i) Step-response functions in Eq. 7.142

Pj Constant characteristic of a particular metabolite Pj in

e State estimation error x — x r Regressor matrix

I\ A Matrices involved in Eqs. 7.158 and 7.159 A Relative gain array (RGA)

A Vector of adjoint variables associated with state equations, defined in Eqs. 7.11, 7.81, and 7.136

0'(y,u) Argument vector in the equality constraints in Eq. 7.6

0(x, u) Argument vector in the equality constraints in Eq. 7.5

<pM Modeling errors

4>P Parametric faults n(w) Hermitian matrix defined in Eq. 7.78

?/>'(y,u) Argument in the inequality constraints in Eq. 7.6

t/>(x, u) Argument in the inequality constraints in Eq. 7.5

e(t) Error vector, vector of inputs to controllers e{t) Vector of prediction errors rj, p Vectors of adjoint variables associated with integral constraints in Eq. 7.5, defined in Eqs. 7.81

X Optimum phase difference between Uj and «2 in forced periodic operation, defined in Eq. 7.89

Xji(Ni) Functions in the expression for tj in Eq. 2.20 and Table 2.2

Suc

Input actuator faults

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