ANN	Artificial Neural Network
$AN{N}_d$	Direct model based on ANNs
$AN{N}_i$	Inverse model based on ANNs
ARIMA	Autoregressive Integrated Moving Average
ASM1	Activated Sludge Model N.1
ASM2	Activated Sludge Model N.2
ASM2d	Activated Sludge Model N.2d
AWGN	Additive White Gaussian Noise
$BO{D}_5$	5-day Biological Oxygen Demand
BPTT	Back-propagation Through Time
BSM1	Benchmark Simulation Model No. 1
${\tilde{c}}_g$	LSTM Cell-update gate
$c_n$	LSTM cell state
${\mathbf{C}}_x$	Covariance matrix
$COD$	Chemical Oxygen Demand (mg/L)
$d\left[n\right]$	Perturbations entering into the IMC
DAE	Denoising Autoencoder
DL	Denormalisation Layer
DO	Dissolved Oxygen
$e^{\prime }\left[n\right]$	Mismatch between the real and the predicted outputs
ETFE	Empirical Transfer Function Estimation
$f_g$	LSTM Forget gate
$f_s$	Sampling frequency
$h_n$	LSTM output
$H\left(z\right)$	First-order IMC filter
IAE	Integrated Absolute Error
$i_g$	LSTM Input gate
IMC	Internal Model Control
ISE	Integrated Squared Error
$K_{La,x}$	Oxygen Transfer Coefficient of the x-th reactor tank (day−1)
LAL	Linear Activation Layer
LSTM	Long Short-Term Memory Cell
MAPE	Mean average percentage error
ML	Machine Learning
MLP	Multilayer Perceptron
MSE	Mean-squared error
$N_2$	Nitrogen gas
$N{O}_2$	Nitrogen Dioxide
NARX	Non-linear Autorregressive Exogeneous model
NL	Normalisation Layer
$O_g$	LSTM Output gate
$P\left(z\right)$	Process under control
$P_{dir}\left(z\right)$	Direct model of the process under control
$P_{inv}\left(z\right)$	Inverse model of the process under control
PCA	Principal Component Analysis
PI	Proportional integral controller
Q	Flow rate
$R^2$	Determination coefficient
$r\left[n\right]$	Reference signal
RNN	Recurrent Neural Network
$S_{NO,x}$	Nitrate-nitrogen concentration in the x-th reactor tank (mg/L)
$S_{NH,x}$	Ammonium concentration in the x-th reactor tank (mg/L)
$S_{N{K}_j}$	Kjeldahl Nitrogen concentration (mg/L)
$S_{O,x}$	Dissolved Oxygen concentration in the x-th reactor tank (mg/L)
SW	Sliding Window
$TN$	Total Nitrogen (mg/L)
TSS	Total Suspended Solids (mg/L)
$u\left[n\right]$	Actuation signal
${\omega }_c$	$H\left(z\right)$ cut-off frequency
WL	Window length
WWTP	Wastewater Treatment Plant
$\mathbf{X}$	Noise corrupted signals considered in the DAE denoising method
$x_n$	LSTM input data
$y\left[n\right]$	Controlled signall
$\widehat{y}\left[n\right]$	Output of $P_{dir}\left(z\right)$
$\tilde{y}\left[n\right]$	Input signal of the IMC controller
${\mathbf{Z}}_{\mathbf{x}}$	Mapped noise-corrupted input data