Data-driven adaptive predictive control for an activated sludge process

Bulletin of Electrical Engineering and Informatics
Vol. 9, No. 5, October 2020, pp. 1827~1834
ISSN: 2302-9285, DOI: 10.11591/eei.v9i5.2257  1827
Journal homepage: http://beei.org
Data-driven adaptive predictive control for an activated
sludge process
Mashitah C. Razali 1
, Norhaliza Abdul Wahab2
, Syahira Ibrahim3
, Azavitra Zainal4
,
M. F. Rahmat5
, Ramon Vilanova6
1,2,3,4,5
Control and Mechatronics Engineering, School of Electrical Engineering,
Universiti Teknologi Malaysia, Malaysia
6
Department de Telecomunicació i Enginyeria de Sistemes, Universitat Autònoma de Barcelona, Spain
Article Info ABSTRACT
Article history:
Received Dec 1, 2019
Revised Mar 7, 2020
Accepted Apr 12, 2020
Data-driven control requires no information of the mathematical model of
the controlled process. This paper proposes the direct identification of
controller parameters of activated sludge process. This class of data-driven
control calculates the predictive controller parameters directly using subspace
identification technique. By updating input-output data using receding window
mechanism, the adaptive strategy can be achieved. The robustness test
and stability analysis of direct adaptive model predictive control are discussed
to realize the effectiveness of this adaptive control scheme. The applicability of
the controller algorithm to adapt into varying kinetic parameters and operating
conditions is evaluated. Simulation results show that by a proper and effective
excitation of direct identification of controller parameters, the convergence
and stability of the implicit predictive model can be achieved.
Keywords:
Data driven control
Direct adaptive
Subspace identification
This is an open access article under the CC BY-SA license.
Corresponding Author:
Norhaliza Abdul Wahab,
Control and Mechatronics Engineering,
School of Electrical Engineering,
Universiti Teknologi Malaysia,
81310 Skudai Johor Bahru, Malaysia.
Email: aliza@fke.utm.my
1. INTRODUCTION
In the biological activated sludge process (ASP), microbial contaminants are used in influent
wastewater to degrade organic matters and nutrients [1-3]. Due to the system‟s dynamic uncertainties
and highly nonlinearities in the microorganism‟s growth kinetic model, control of activated sludge process
is a challenge. Conventional controllers are designed based on mathematical model of ASP which requires
the designer to spend lot of building the model from the basic laws of biomass mass balance. However, there
are model free design methods available that can be utilized in designing control systems [4-6].
For conventional control, the differential equations derived are then transformed into a standard transfer
function or state space and the control laws are then applied [7-10]. However, the principal control problems
of ASP as found in literature can be briefly illustrated by the variability of the kinetic parameter [11-12],
time-varying influent conditions, nonlinearities [13], delays, lack of accurate sensors [14], and the limited
availability of the online measurements [15]. Therefore, adaptive control principles are the preferred
design option [16-18].
The main issue in the design of predictive controller for ASP was how to implement an accurate
internal predictive model which best describes the behavior of the biochemical process involved.
In [19] the internal predictive model of the plant implemented neural networks for controller design.
However, this method suffers from the fact that it requires much attention on the training data collection

 ISSN: 2302-9285
Bulletin of Electr Eng & Inf, Vol. 9, No. 5, October 2020 : 1827 – 1834
1828
resulting in many approximations in this stage of the design, and huge validation experiments have to be
done to accurately identify the model and it‟s usually fails to track the variability of the kinetic parameters.
In [20] the authors have built the internal model for prediction based on linearized state space model of
the bioprocess which is definitely undesirable in cases of variability in the effluents or any changes in
operating conditions. Subspace Identification techniques developed in nineties offers an attractive
and effective solution for the issue of prediction model and has been deployed successfully in design of
predictive controllers for multiple-input multiple-output processes [21].
The employment of the robust computational tools such as QR-decomposition has recently opened
a wide area of research to develop a single step implementation for predictive controllers where only
input-output data are processed to directly calculate controller parameters or namely “Model-free” control
methods [22]. The results of [22-24] are some of the successful reports for this class of novel algorithms.
However, they have not introduced any closed-loop stability analysis for the sake of practical implementation
of the proposed controller. Most adaptive predictive controllers reported so far are only applied to linear
systems or indirectly implemented. Therefore, the stability involved can be easily examined and proved for
several cases of infinite prediction horizon or when the open loop system can be accessed and is stable.
However, when the system is nonlinear or when finite horizons is applied, only few approaches have been
published and were limited to indirect schemes [25-26]. In this paper, a data-driven predictive controller for
activated sludge process is applied which is also consider stability of the proposed controller.
2. ACTIVATED SLUDGE MODEL DEVELOPMENT
The biological process is modeled from component-balance considerations and the obtained model
is most often a highly complex and high-order non-linear system given by the following set of nonlinear
differential equations [27]:
(1)
(2)
( ) (3)
(4)
where , , , are the state variables representing the biomass, the substrate, the recycled
biomass and dissolved oxygen concentrations, respectively. Meanwhile, is the dilution rate, and and
represent the ratio of recycled flow to influent flow and ratio of waste flow to influent flow, and
respectively, which corresponds to the substrate and dissolved oxygen concentrations in the influent stream.
In addition, is the cell mass; is a constant term, is the maximum dissolved oxygen concentration
and represents the oxygen mass transfer coefficient.
(5)
where W is the airflow rate. The kinetic model of the specific cell mass production rate is given as [24].
(6)
where is the maximum specific growth rate, is the so-called affinity constant, expressing
the dependency of the degradation rate on the concentration of pollutant S, and is the saturation constant.
Notice that all the above equations are only used here in this work to generate input-output (I/O) data
and they will not be used in any stage of the controller design. For simulations purpose, the process
parameters, kinetic parameters, and initial conditions are adopted, as shown in Tables 1-3, respectively.
Table 1. Process parameters
Parameter Value
ϒ 0.65
r 0.6
β 0.2
α 0.018 m-3
K0 0.5
Cs 10 mgl-1
Sin 200 mgl-1
Cin 0.5 mgl-1
Table 2. Kinetic parameters
Parameter Value
ϒ 0.65
r 0.6
β 0.2
α 0.018 m-3
K0 0.5
Cs 10 mgl-1
Sin 200 mgl-1
Cin 0.5 mgl-1
Table 3. Initial conditions
Parameter Value
X(0) 215 mgl-1
S(0) 35 mgl-1
C(0) 6 mgl-1
Xϒ(0) 400 mgl-1
Sin 200 mgl-1
Cin 0.5 mgl-1

Bulletin of Electr Eng & Inf ISSN: 2302-9285 
Data-driven adaptive predictive control for an activated sludge process (Mashitah C. Razali)
1829
3. SUBSPACE BASED PREDICTOR IDENTIFICATION
The most commonly used technique in system identification is used in this paper, given by (7):
and (7)
where, , is the system states with order , , is the input vector with number of inputs,
is the output vector with number of outputs and, is a white noise disturbance (innovations)
sequence with covariance . The matrices , , , and
are the state, input, output, feed-forward and Kalman gain matrices of the system. The following (8) to (10)
are defined as fundamental subspace I/O matrix equations. The subscripts here denotes “past” and
denotes “future” matrices for its respective variables.
(8)
(9)
(10)
where the extended observability matrix , the reversed extended controllability matrix for deterministic
input and for stochastic input , matrices and are the lower triangular Toeplitz matrices for
both deterministic and stochastic parts, respectively. Assuming that the open-loop I/O data and for
{ } are available for the identification, define the past input, Up and future input, Uf data
matrices for (8)-(10) as follows:
⟨ | ⟩ [ ] (11)
⟨ | ⟩ [ ] (12)
where { } . The subscripts ⟨ | ⟩ indicates that the first column of the subspace matrix
starts from the time step and ends by the time step . Similar notation can be written for past and future
output matrices. In subspace identification literature, the following short-hand notation is commonly used:
[ ] (13)
(14)
Equation (14) is the subspace predictor equation where the parameters to be obtained contain system model
information without completely identifying the parametric model of the system such as the conventional
transfer function or state space representation of the system. In addition, ( )
: is the subspace
matrix that corresponds to the past inputs and outputs. : is the subspace matrix that corresponds to
the future input data or the deterministic inputs, and . The problem of obtaining the best linear
predictor of the future outputs ̂ , given the past inputs and outputs in the form of and the future inputs
can be written as a Frobenius norm minimization:
‖ [ ] [ ]‖ (15)
The least square problem in (15) can be solved by the orthogonal projection and can be implemented by
forming a -decomposition of the matrix [ ] as follows:

 ISSN: 2302-9285
1830
[ ] [ ] [ ] (16)
Thus, we can estimate optimal prediction ̂ as described in (16). , and can be used to formulate
an estimation of future outputs as follows:
[
̂
̂
]
[ ]
[ ] (17)
Equation (17) is a streamlined subspace-based linear predictor equation which can be written as,
̂ (18)
4. DIRECT ADAPTIVE MODEL PREDICTIVE CONTROL
In the direct adaptive model predictive control (DAMPC) or so-called data driven method,
the controller parameters are estimated directly from the measurements. Therefore, the system model
is implicitly included and no explicit model is formulated at any stage of the implementation. The adaptive
mechanism implemented here adjusts the predictive controller parameters to the changes in the process
dynamics in every sampling time. The measured input-output data is collected over a sliding window. It can
be noted that the data window used to identify the controller parameters or state matrices can be expressed in
terms of future inputs, [ ] and measurement (past) inputs [ ]
and outputs [ ] . Here, two prediction problems are given at current time instant
and as shown in Figure 2. The first prediction problem represents the case for obtaining
the optimal prediction of future outputs ̂ [̂ ̂ ] using the information given in
the previously stated data window and . The second prediction problem shows that the time instant
slides from to , this differs from the data window ( and ) which is now slides from left
to right. At every time step, for the new available input-output data, the MPC parameters are updated online
according to the updated predictor gain matrices and a new control action is computed. The cost function
performance criterion used can be written in matrix form as,
̂ ̂ (19)
where,
[ ]
, [ ]. By incorporating the integral action to the subspace predictive
model, (20) is derived:
̂ (20)
where, [ ] and [ ]. Then, we can rewrite (19) as:
(21)
with, ( ) ( ) and ( ) ( ). In this case,
the model predictive law for the unconstrained case of the problem of (21) is derived, which can be simply
solved by:

1831
At time t, only is implemented and the calculation is repeated at each time instant. Hence, the control
law is as follows:
(22)
with, [ ] and [ ] . Where and
defined as,
(( ) ( ) ) ( ) ( )
(( ) ( ) ) ( )
5. SIMULATION AND RESULTS
The implemented control system is a two inputs two outputs system that are dilution rate, D and air
flow rate, W for the inputs, and substrate, S and dissolved oxygen, C for the outputs. The simulation is run
from a steady state operating point at outputs S=41.23 mg/l, C=6.11 mg/l and inputs D=0.081/hr and W=90
m3/hr. The sampling period is chosen as Ts=1. The weighting matrices are chosen as
and . Meanwhile, the Prediction Horizon, Hp and Control Horizon, Hc were
respectively. Figure 1 shows the evolution of the outputs and inputs. The setpoint given for the outputs are
varied for an approximately 10% around the system‟s steady state condition. The squared error performance
index was chosen to evaluate the performance of the closed-loop system and it was found to be 445.2.
Figure 1. Closed-loop response of substrate, dissolved oxygen, dilution rate and airflow rate
5.1. Stability analysis of subspace based DAMPC
In this section, the stability of the DAMPC is investigated. The design methodology is based on
receding horizon control, which is also used in a standard model predictive control (MPC) system design.
However, the main differentiating part between the DAMPC and a commonly used MPC is the starting point
in the design. The starting point for the usual design of MPC is the availability of a dynamic model that
describes the system to be controlled. With given sufficient accuracy of the model, long prediction horizon
and control horizon, the closed-loop stability of MPC is obtained. In contrast, the starting point for the design
of subspace-based DAMPC is the availability of excitation test data for the system to be controlled. Here,
the question on the closed-loop stability of DAMPC is associated with the excitation test data for the system,
as well as the prediction horizon and control horizon.

 ISSN: 2302-9285
1832
Several simulations of DAMPC showed that large prediction horizons may also resulted in larger
errors of predictions since the actual system is nonlinear and the prediction based on subspace matrices is
linear. Also, the longer the prediction horizon, of the DAMPC would approach as the classical MPC.
The focus of the investigation on stability of the closed-loop system was on the quantity and quality of
the excitation test data. Particularly, simulation studies are carried out to investigate the effect of both
the window size of data length and prediction horizon length and the closed-loop performance criterion for
various numerical values are shown. The best results obtained were and for both window
size length and prediction horizon, respectively. The relationship of closed-loop stability to the predictor
estimation algorithm is proposed. It can be seen, if the prediction converges to the true value, the overall
closed-loop system will be equivalent to the standard MPC. From the convergence analysis of this algorithm,
if a long enough prediction horizon for estimation is used, then the multi-step ahead estimated predictions
will converge to the true multi-step ahead predictions. This can be proven by looking through the linear
equation used to calculate the estimated output prediction presented again as:
̂ ̂ (23)
where: : The natural response, or the effect of the initial conditions is considered to be deterministic since
it resulted from the multiplication by the past inputs and outputs .
̂ : The forced response which is clearly resulted from the effect of the input over the prediction
horizon which is in fact the impulse response coefficients due to the future input increments.
[ ] (24)
̂ [ ] [ ]
̂ ∑ (25)
Due to the nature of this algorithm, it will converge globally, which an important property with
respects to the typical adaptive algorithms is found in the literature. Here we can conclude that if the future
inputs remain bounded we can assure the convergence and the stability of the implicit predictive model
which can only be proven by investigating the values of the impulse coefficients incorporated into prediction
gain matrices over the prediction horizon. Therefore, by plotting for every manipulated variable,
the convergence of the predictive model to the true value over the prediction horizon can be illustrated.
Figure 2 shows the values of for both the future dilution rate manipulated variable and airflow rate
manipulated variable is bounded over the prediction horizon. Therefore, it ensures the convergence of t
he estimated output prediction to the true value accordingly.
(a) (b)
Figure 2. The values of for prediction gain coefficients of, (a) Future dilution rate, (b) Future airflow rate
0 5 10 15 20 25 30 35
0
20
40
60
80
100
120
Prediction Samples
Impulse
Response
Coefficients
a) Lu Prediction Gain Matrix for Dilution Rate
0 5 10 15 20 25 30 35
-7
-6
-5
-4
-3
-2
-1
0
1
Prediction Samples
Impulse
Response
Coefficients
b) Lu Prediction Gain Matrix for Airflow Rate
0 5 10 15 20 25 30 35
0
20
40
60
80
100
120
Prediction Samples
Impulse
Response
Coefficients
a) Lu Prediction Gain Matrix for Dilution Rate
0 5 10 15 20 25 30 35
-7
-6
-5
-4
-3
-2
-1
0
1
Prediction Samples
Impulse
Response
Coefficients
b) Lu Prediction Gain Matrix for Airflow Rate

1833
5.2. Robustness tests for closed-loop DAMPC
In order to access the closed-loop performance of the applied algorithm, a rigorous simulation
analysis under different operating conditions and kinetic process parameters were considered and discussed
in this section. The constant influent rates are considered while taking into accounted the kinetic model
variations and different operation points. Two different performance criterions were calculated for each
output (C: dissolved oxygen concentration, and S: substrate concentration) which is the integral square error
(ISE) and the integral absolute error (IAE) for the purpose of assessment. In Table 4 listed the ISE and IAE
of the ASP outputs under the applied DAMPC with two different cases of operating conditions:
‒ Case 1: The operating condition in this case is and with
[ ] [ ] and the varied kinetic parameters for the nonlinear term,
, , and .
‒ Case 2: The operating condition in this case is and with,
[ ] [ ] and the same varied kinetic
parameters for the nonlinear term are also considered, , , and .
Results in Table 4 show the applicability of the controller algorithm to adapt both varied kinetic
parameters and operating conditions. Therefore, system outputs will be regulated and the input reference
trajectories will be tracked, irregardless of the conditions imposed on the real plant. Furthermore, no explicit
model is formulated for the plant and any prior information about initial operating conditions is not needed,
neither does the kinetic parameters.
Table 4. Performance test numerical values
Kinetic
parameters
ISE IAE Kinetic
parameters
ISE IAE
S C S C S C S C
CASE 1 μmax = 0.15 132.7 1.133 107.9 16.74 CASE 2 μmax = 0.15 150.4 1.262 117.3 17.3
Ks = 100 mgl-1
Ks = 100 mgl-1
Kc = 2 mgl-1
Kc = 2 mgl-1
μmax = 0.21 680.1 18.64 332.8 59.93 μmax = 0.21 462.2 13.57 246.2 46.2
Ks = 100 mgl-1
Ks = 100 mgl-1
Kc = 3 mgl-1
Kc = 3 mgl-1
μmax = 0.21 111.8 1.439 113.4 18.87 μmax = 0.21 114.5 1.514 115.1 19.2
Ks = 150 mgl-1
Ks = 150 mgl-1
Kc = 2 mgl-1
Kc = 2 mgl-1
6. CONCLUSION
In this paper, the design of the adaptive data-driven predictive controller for biological ASP is
presented. The integration between the robustness and efficiency of subspace-based predictor estimation
principal (where only geometrical tools are deployed without involving into nonlinear optimization); and the
power of advanced unconstrained MPC design methodology (where MIMO systems can be controlled
successfully) contributed to a robust-stable DAMPC for nonlinear ASP. From the control perspective, this
work can be seen as an important investigation in the field „model-free‟ control design since it has shown that
the methodology can cope with the same issues of the „model-based‟ control methods. This includes the ease
tuning of MPC parameters particularly for multivariable systems. Although, only unconstrained case is
considered in this paper, we anticipate that the robustness of the constrained case is naturally achieved.
In conclusion, the results obtained from the implementation of the presented control methodology on
nonlinear ASP, was encouraging, and it is proposed that these algorithms show much promise for further
development and analysis in terms of storage requirements, calculation time, and analytical stability proofs to
gain wide acceptance in industrial scale applications.
ACKNOWLEDGEMENTS
The authors would like to thank the Ministry of Education (MOE) PRGS vote
R.J130000.7851.4L702 and the Universiti Teknologi Malaysia UTMHIR vote 08G74 for their financial
and facility support of this work.
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