Int. J. of Computers, Communications & Control, ISSN 1841-9836, E-ISSN 1841-9844
Vol. VI (2011), No. 2 (June), pp. 367-374

Digital Control of a Waste Water Treatment Plant

R. Vilanova, J.D. Rojas, V.M. Alfaro

Ramón Vilanova, José David Rojas
Department de Telecomunicació i Enginyeria de Sistemes
Universitat Autònoma de Barcelona
08193, Bellaterra, Spain,
E-mail: ramon.vilanova@uab.cat, josedavid.rojas@uab.cat

Víctor M. Alfaro
Escuela de Ingeniería Eléctrica
Universidad de Costa Rica
San José, 11501-2060 Costa Rica.
E-mail: victor.alfaro@ucr.ac.cr

Abstract: The Activated Sludge Process (ASP) is arguably the most popular
bioprocess utilized in the treatment of polluted water. The ASP is described
by means of a nonlinear model and results on a Two-Input Two-Output multi-
variable system. In this paper a discrete time digital control is proposed where
the design of a decentralized controller is faced. Local controllers are given the
form of a Two-Degree-of-Freedom PI controller tuned using the data-driven
Virtual-Reference Feedback tuning approach.
Keywords: ASP, Process Control, PID, Data-Driven Control, VRFT.

1 Introduction

Water pollution represents one of the most serious environmental problems due to the dis-
charge of nutrients into receiving waters. Hence, stricter standards for the operation of wastew-
ater treatment plants (WWTPs) have been imposed by authorities. In order to meet these
standards, improved control of WWTPs is needed. Wastewater treatment control has begun
a gradual progress towards the use of more advanced technology, in the face of more stringent
modern water quality standards. Several approaches have been reported in the literature that
attempt to control the WWTPs process. Among others, the Activated Sludge Process (ASP) is
arguably the most popular bioprocess utilized in the treatment of polluted water, using microor-
ganisms present within the treatment plant in the biological oxidation of the wastewater. The
simplified but still realistic and highly non-linear four-state multivariable model considered here
is the ASP as presented in [1].

The ASP is described by means of a nonlinear model and results on a Two-Input Two-Output
(TITO) multivariable system. In this paper a discrete time digital control is proposed where the
design of a decentralized controller is faced. The paper designs the local controllers as discrete
time Proportional-Integral (PI) controllers. The controllers are synthesized using the Virtual
Reference Feedback Tuning, which is a model-free based approach where just purely data taken
from the system is considered, therefore there is no need for a mathematical model of the system.
On the basis of these data the discrete time PI controllers are tuned.

2 The two-degree-of-freedom Virtual Reference Feedback Tuning

The Virtual Reference Feedback Tuning (VRFT) is a one-shot data-based method for the
design of feedback controllers. The original idea was presented in [2], and then formalized by

Copyright c⃝ 2006-2011 by CCC Publications



368 R. Vilanova, J.D. Rojas, V.M. Alfaro

r
+ +

+

−

Cr(θr) P

d

y

Cy(θy)

Figure 1: Two degrees of freedom structure

Lecchini, Campi and Savaresi (see [3–5]). In this section, an outline of the two-degree-of-freedom
case is presented. The design methodology is presented in [5], the control structure is presented
in Fig. 1. The objective of this method is to minimize the criterion in (1).

JMR(θr, θy) = ∥(ΨM(z; [θr, θy]) − M(z))WM(z)∥
2
2

+ ∥(ΨS(z; θy) − S(z))Ws(z)∥
2
2

(1)

with
ΨM(z; [θr, θy]) =

P(z)Cr(z; θr)

1 + P(z)Cy(z; θy)
ΨS(z; θy) =

1

1 + P(z)Cy(z; θy)
(2)

and M(z) being the target input-to-output transfer function and S(z) the target sensitivity
function. In the VRFT framework a plant model is not available, and is not intended to find
one. Instead, a batch of input/output data is taken from an experiment on the plant (namely
input u(t) and output y(t)). So, in order to find the parameters of the controllers (θr and
θy) the signals r̄(t), d̄(t) and ȳ(t) are defined. These signals are called “virtual” because they
are not really measured, but constructed from the input/output data available and the desired
closed-loop relations as follows:

• r̄(t) is the virtual reference, so that y(t) = M(z)r̄(t)

• d̄(t) is the virtual perturbation, so that y(t) + d̄(t) = S(z)d̄(t)

• ȳ(t) is the virtual-perturbed output of the plant, so that ȳ(t) = y(t) + d̄(t)

This signals are the ones that would be found if u(t) and y(t) had been measured in closed-loop
and if the closed-loop dynamics were given by M and S i.e., if the perfect controllers were set
in the loop. On the basis of these “virtual” signals the controller’s parameters are found by
minimizing the following alternative identification cost function:

JNV R(θr, θy) =
1

N

N∑
t=1

[ΓM(t; [θr, θy])]
2
+

1

N

N∑
t=1

[ΓS(t; [θr, θy])]
2

where

ΓM(t; [θr, θy]) = LM(z)(u(t) − Cr(z; θr)r̄(t) + Cy(z; θy)y(t)) (3)
ΓS(t; [θr, θy]) = LS(z)(u(t) + Cy(z; θy)ȳ(t)) (4)

and LM(z) and LS(z) are appropriate filters to be chosen so (3) becomes an approximation to (1).
If the controllers are linear in the parameter (Cr(z; θr) = βr(z)T θr and Cy(z; θy) = βy(z)T θy) the
cost criterion (3) becomes a standard quadratic optimization problem. In [5] the authors use the



Digital Control of a Waste Water Treatment Plant 369

concept of “ideal controller” to derive the structure of filters LM and LS. The ideal controllers
Cr0 and Cy0 are the ones that, if used in the control loop, would solve (1) exactly, that is

Cy0 =
1 − S
SP

Cr0 =
M

SP
(5)

When comparing (1) and (3) using the Parseval Theorem the expression of the filters LM and
LS that must make the identification problem (3) match the control problem (1) are found to
be:

|LM|2 = |M|2|S|2|WM|2
1

Φu
|LS|2 = |S − 1|2|S|2|WS|2

1

Φu
(6)

3 2-DoF PI structure for the VRFT

In order to apply the VRFT framework to the Activated Sludge Process, the structure of the
controllers has to be decided before the optimization is carried out. In [6], a decentralized PI
structure is used in the same plant with good results for both, reference tracking and disturbance
rejection. In this paper, a discretized version of the PI controller is used as the chosen structure
for the VRFT controllers. Using a two-degree-of-freedom PI as in Fig. 1, the continuous time
version of the controller is:

Cr(s) = Kc

(
β +

1

Tis

)
Cy(s) = Kc

(
1 +

1

Tis

)
(7)

When applying the bilinear transformation s = 2
Ts

z−1
z+1

, the controllers are

Cr(z) =
Kc

(
β + Ts

2Ti

)
+ Kc

(
Ts
2Ti

− β
)
z−1

1 − z−1
(8)

Cy(z) =
Kc

(
1 + Ts

2Ti

)
+ Kc

(
Ts
2Ti

− 1
)
z−1

1 − z−1
(9)

From the VRFT point of view, (8) and (9) can be seen simply as linear-in-the-parameters con-
trollers with two parameters as follows:

Cr(z) =
α1 + α2z

−1

1 − z−1
Cy(z) =

γ1 + γ2z
−1

1 − z−1
(10)

Since the continuous time controllers (7) have three adjustable parameters (Kc, Ti, β), one of
the four parameters in its discrete time equivalent (10) should depend of the other three. From
(8) to (10) it can be found that

α1 + α2 = γ1 + γ2 =
KcTs
Ti

(11)

and we have that
γ2 = α1 + α2 − γ1 (12)

Then the discrete time controllers (10) are now

Cr(z) =
α1 + α2z

−1

1 − z−1
Cy(z) =

γ1 + (α1 + α2 − γ1)z−1

1 − z−1
(13)



370 R. Vilanova, J.D. Rojas, V.M. Alfaro

Once the parameters of the controllers (13) (α1, α2, γ1) are found, one can recuperate the PI (7)
parameters using:

Kc = γ1 −
1

2
(α1 + α2) (14)

Ti = Ts
γ1 − 12 (α1 + α2)

α1 + α2
(15)

β =
α1 − 12 (α1 + α2)
γ1 − 12 (α1 + α2)

(16)

4 Activated Sludge Process (ASP) Description

The mathematical model considered in this paper is given in [1]. The ASP process comprises
an aerator tank where microorganisms act on organic matter by biodegradation, and a settler
where the solids are separated from the wastewater and recycled to the aerator. The layout
is shown in Fig. 2. The component balance for the substrate, biomass, recycled biomass and
dissolved oxygen provide the following set of non-linear differential equations:

dX(t)

dt
= µ(t)X(t) − D(t)(1 + r)X(t) − rD(t)Xr(t) (17)

dS(t)

dt
= −

µ(t)

Y
X(t) − D(t)(1 + r)S(t) + D(t)Sin (18)

dDO(t)

dt
= −

Koµ(t)

Y
X(t) − D(t)(1 + r)DO(t)

+ KLa(DOs − DO(t)) + DO(t)DOin (19)
dXr(t)

dt
= D(t)(1 + r)X(t) − D(t)(β + r)Xr(t) (20)

µ(t) = µmax
S(t)

kS + S(t)

DO(t)

kDO + DO(t)
(21)

where X(t) - biomass, S(t) - substrate, DO(t) - dissolved oxygen, DOs - maximum dissolved
oxygen, Xr(t) - recycled biomass, D(t) - dilution rate, Sin and DOin - substrate and dissolved
oxygen concentrations in the influent, Y - biomass yield factor, µ - biomass growth rate, µmax
- maximum specific growth rate, kS and kDO - saturation constants, KLa = αW - oxygen mass
transfer coefficient, α - oxygen transfer rate, W - aeration rate, Ko - model constant, r and
β - ratio of recycled and waste flow to the influent. The influent concentrations are set to
Sin = 200 mg/l and DOin = 0.5 mg/l. With respect to the control problem definition, the waste
water treatment process is considered under the assumption that the dissolved oxygen, DO(t),
and substrate, X(t), are the controlled outputs of the plant, whereas the dilution rate, D(t),
and aeration rate W(t) are the two manipulated variables. The initial conditions and kinetic
parameters are taken as in [1] and [6].

5 Discrete-time VRFT tuned PI controller applied to the ASP

Using the non-linear model presented in (17) to (21), the data in Fig. 3(a) and Fig. 3(b) was
collected. Using only this data without the information of the non-linear model, or any linear
approximation, the parameters of the controller are calculated according to Section 2 using the
PI structure specified in Section 3. The VRFT controllers are tested against two decentralized



Digital Control of a Waste Water Treatment Plant 371

Figure 2: Activated Sludge Process layout. Taken from [7]

0 50 100 150 200 250 300
0

0.05

0.1

0.15

0.2

time(s)

m
g
/l

Dilution Rate

0 50 100 150 200 250 300
20

40

60

80

time(s)

m
g
/l

Substrate Data

(a) Substrate-Dilution Rate loop

0 50 100 150 200 250 300
50

100

150

time(s)

m
g
/l

AirFlow Rate

0 50 100 150 200 250 300
2

4

6

8

time(s)

m
g
/l

Disolve Oxygen Data

(b) Disolve Oxygen-AirFlow Rate loop

Figure 3: Data used to find the VRFT controller

PI controllers which parameters are computed using IMC [8], based on considering a First-Order
(FO) controlled-process given by

P(s) =
Kp

Ts + 1
(22)

The identified models obey to K1 = 437.1 and T1 = 2.7h for the first loop and K2 = 0.03 and
T2 = 0.51h for the second loop. These values are as in [6]. The controllers were discretized using
the bilinear transformation (sampling time of 15min). In the case of the VRFT, the controllers
are directly found in discrete time. The closed-loop specifications are given in terms of the desired
time constants of the controlled system for each loop. The desired time constant of the Substrate-
Dilution Rate loop is approximately T1 = 2.7h while the Dissolved Oxygen-Airflow Rate loop
constant time is approximately T2 = 0.51h. The closed-loop constant time for each variable is
giving in terms of the dimensionless variables τc1 and τc2 via Tc1 = τc1T1 and Tc2 = τc2T2. If,
for example, τc1 < 1, the Substrate-Dilution Rate closed-loop is expected to be faster than in
open loop. It is worth to note that for the VRFT tuning it is possible to specify a different time
constant for the disturbance attenuation transfer function (in fact S(z)). In this case, the VRFT
tuned PI controllers are found with a time constant for the corresponding S(z) transfer function
that is half the one specified for the reference to output relation by using τc (therefore τc/2). If
τc1 = τc2 = 1, the resulting controller parameters are Kc1 = 0.0023, Kc2 = 33.33, Ti1 = 2.7h and
Ti2 = 0.51h for the IMC and Kc1 = 0.0042, Kc2 = 25.41, Ti1 = 3.43h, Ti2 = 0.42h, β1 = 0.6602
and β2 = 0.74 for the VRFT. The response to a change in the set points of both loops is giving
in Fig. 4. The IAE value represents the Integrated Absolute Value of the Error. As it can
be seen, the responses for the VRFT provide smaller IAE as well as less demanding control
actuation, computed here as the Total Variation (TV) or the sum of the control movements from



372 R. Vilanova, J.D. Rojas, V.M. Alfaro

0 20 40 60 80 100 120
40

42

44

46

48

50

52

54

Time(h)

m
g

/l
Substrate, 

c1
= 1

 

 

Reference

IMC, IAE: 24.284

VRFT, IAE: 15.6386

0 20 40 60 80 100 120
4

4.5

5

5.5

6

6.5

Time(h)

m
g

/l

Dissolve Oxygen, 
c2

= 1

 

 

Reference

IMC, IAE: 0.95129

VRFT, IAE: 0.67543

0 20 40 60 80 100 120
0.08

0.085

0.09

0.095

0.1

0.105

0.11

0.115

Time(h)

m
g

/l

Dilution Rate, 
c1

= 1

 

 

IMC, TV: 0.03196

VRFT, TV: 0.034001

0 20 40 60 80 100 120
10

20

30

40

50

60

70

80

90

100

110

Time(h)

m
g
/l

AirFlow Rate, 
c2

= 1

 

 

IMC, TV: 58.5604

VRFT, TV: 45.0193

Figure 4: VRFT and IMC controllers responses to set-points step changes, τc1 = τc2 = 1

one sampling time to the other.

In case the closed-loop bandwidth is increased and we set τc1 = τc2 = 0.5 the resulting
controller parameters are Kc1 = 0.0046, Kc2 = 66.67, Ti1 = 2.7h and Ti2 = 0.51h for the IMC
and Kc1 = 0.0065, Kc2 = 33.38, Ti1 = 2.68h, Ti2 = 0.35h, β1 = 0.7752 and β2 = 1.03 for
the VRFT. In this case the IMC controller presents and undesirable oscillating behavior in the
Dissolved Oxygen loop, as shown in Fig. 5. The output of the controllers was saturated to 0,
in case it went below this value. Also the control effort of the second controller is quite lower
in the case of the VRFT with a performance nearly 50% better. Also a simulation was carried
out for a disturbance in the inflow concentration. The results are depicted in Fig. 6. Again the
results are quite similar, and the multivariable characteristic is tackled in a satisfactory way. As
it can be seen, this data driven methodology is suitable for the control of the ASP process and
it allows to skip the modeling step in order to find a good controller.

6 Conclusions

This paper has presented the application of a purely data based approach for tuning of
discrete time PI controllers. The main advantage of the proposed method is that it does not rely
on the usual linear model approximation of the system to be controlled. Just an experiment that
provides input-output data from the system is needed. The performance of the tuning approach
has been tested on a non-linear multivariable system and performance compared with that of
the well known IMC method. As designed performance is more demanding, the resulting control
system exhibits better results than its IMC counterpart.



Digital Control of a Waste Water Treatment Plant 373

0 20 40 60 80 100 120
40

42

44

46

48

50

52

54

Time(h)

m
g

/l
Substrate, 

c1
= 0.5

 

 

Reference

IMC, IAE: 12.8966

VRFT, IAE: 7.9997

0 20 40 60 80 100 120
3

3.5

4

4.5

5

5.5

6

6.5

Time(h)

m
g
/l

Dissolve Oxygen, 
c2

= 0.5

 

 

Reference

IMC, IAE: 1.0949

VRFT, IAE: 0.59053

0 20 40 60 80 100 120
0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

Time(h)

m
g

/l

Dilution Rate, 
c1

= 0.5

 

 

IMC, TV: 0.043395

VRFT, TV: 0.04601

0 20 40 60 80 100 120
0

20

40

60

80

100

120

Time(h)

m
g

/l

AirFlow Rate, 
c2

= 0.5

 

 

IMC, TV: 163.5215

VRFT, TV: 64.5191

Figure 5: VRFT and IMC controllers responses to set-points step changes, τc1 = τc2 = 0.5

0 20 40 60 80 100 120
40.6

40.8

41

41.2

41.4

41.6

41.8

42

42.2

42.4

42.6

Time(h)

m
g

/l

Substrate (Disturbance), 
c1

= 0.5

 

 

IMC, IAE: 24.7512

VRFT, IAE: 19.2922
0 20 40 60 80 100 120

6.108

6.11

6.112

6.114

6.116

6.118

6.12

6.122

6.124

Time(h)

m
g

/l

Dissolve Oxygen (Disturbance), 
c2

= 0.5

 

 

IMC, IAE: 0.093692

VRFT, IAE: 0.087208

0 20 40 60 80 100 120
0.072

0.074

0.076

0.078

0.08

0.082

0.084

Time(h)

m
g

/l

Dilution Rate (Disturbance), 
c1

= 0.5

 

 

IMC, TV: 0.043227

VRFT, TV: 0.046262
0 20 40 60 80 100 120

89.6

89.8

90

90.2

90.4

90.6

90.8

91

91.2

91.4

91.6

Time(h)

m
g
/l

AirFlow Rate (Disturbance), 
c2

= 0.5

 

 

IMC, TV: 25.757

VRFT, TV: 24.6659

Figure 6: Responses changing the specification for the VRFT and comparison with the IMC
controller, τc1 = τc2 = 0.5



374 R. Vilanova, J.D. Rojas, V.M. Alfaro

Acknowledgment

This work has received financial support from the AECI-PCI program A/025100/09 and
from the Spanish CICYT program under grant DPI2007-63356. Research work by J.D. Rojas
has received financial support from the Universitat Autònoma de Barcelona. Support from the
Universidad de Costa Rica is greatly appreciated.

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