Estimation and Nonlinear Control of an Activated Sludge Wastewater Treatment Process

ESTIl\IATION AND NONLINEAR CONTROL OF AN ACTIVATED SLUDGE WASTEWATER TREATMENT PROCESS F. Nejjari*, A. Benhammou~, B. Dahhou** and G. Roux** *Laboratoire d'Automatique et d'Etudes des Procedes Faculte des Sciences Semlalia, Avenue Prince Moulay Abdellah, B.P: S15, 40000 Marrakech, Maroc Fax: (+212)4437552, Email: benhamou@ open.net ma **Laboratoire d 'Analyse et d 'Architectut'e des SysUmes du CNRS 7, Avenue du Coloni>! Hoche, 31077 Toulouse Cedex, France Fax: (+ 33)561 559440

··Centre de Bioingenierie Gilbert Durant INSA URA/CNRS 544, Comp/exe Scient~fique de Rangueil, 31077 Toulouse Cedex, France

Ab5tract: Design and control of biological wastewater treatment process become subjects of increasing importance because of interest in high effluent quality. But these tasks are made difficult by the absence of a sophisticated instnunentation for on-tine measurement of some key biological variables. In this paper we suggest a nonlinear adaptive feedback-tinearizing control law for an activated sludge process. The control approach structure IS combined with an estimation algoritlun, for the on-tine reconstruction of biological states and parameter variables of the process. The efficiency and the robustness of both the control and estimation schemes are demonstrated via computer simulations with measurement noises and abrupt jumps of the process kinetic parameters.

1. Introduction

The main difficulty in controlling bioprocesses arises from the extremely non linear and non stationary character of such processes and the limited amount of on line infonnation. These processes can not be adequately controlled with conventional feedback controllers. The latters must be detuned significantly to ensure stabilIty . Therefore, their perfonnances are often severely deteriored. Though, many significant applications of adaptive controllers, based on direct exploitation of the non linear structure of the process model, are reported in the literature for specific bioprocesses, e.g. fennentation processes [1,2] or biological waste water treatment processes [3,4,6]. In this paper a non linear adaptive feed back-Iinearizing control is designed for the regulation of the pollutant substrate concentration inside a continuous wastewater treatment process. The control law is combined with a joint observer estimator (JOE) for the on-line estimation of the unavailable biological states and parameters of interest. The online estimates are provided to the control scheme according to certainty equivalence principle The continuous wastewater treatment is an activated sludge process It is described by a set of nonlinear equations obtained from mass-balance considerations. The joint observer/estimator consists of a procedure based on Narendra's [7) error model obtained by an extended linearization tectmique USing Kronecker's calculation A preliminar useful change [6), with a non-singular transfonnation, of co-ordinates is necessary to decouple the unmeasured states from the kinetic parameters. Instead of estimating the specific growth rate duectly as timevarying parameter, it's reconstructed on the basis of its analytical expression via the estimation of the kinetic parameters appearing in this expressIon [5,6,9]. The paper is organised as follows Firstly the plant is briefly described and modelled then the estimatIon algorithm and the non linear control law are presented and illustrated by simulation results. A general conclusion ends the paper. Ii

Author to whom all correspondence should be addressed 13 1

1. Modelling of the bioprocess The activated sludge pr :ess is defmed as a system in which a large number of biological organisms are maintained and continuously circulated so as to be in constant contact with the organic wastewater in the presence of high purity oxygen. The bacteria and other microorganisms feed on the organic matter constituent of the incoming wastewater, thereby reducing the strength of the waste. The sludge is separated from the mixed liquor in solids separator (clarifier or settler). A portion of the settled sludge is recycled as return activated sludge from the secondary clarifiers to the aeration reactor so that the microorganisms content in the reactor is maintained at the reaction sustenance level Excess sludge, not recycled, IS extracted from the system as waste activated sludge and subsequently processed for different uses . The bioreactor is considered to be perfectly mixed so that the concentration of each component is spatially homogeneous. We assumed that there IS no dissolved oxygen limitation inside the aerator and that no bioreaction takes place in the settler. The material balance on the aerator and the settler gives the following four differential equations :

CCt) = -

X(t)

= ll(t)X(t) - D(t)(l + r)X(t) + rD(t)XrCt)

Set)

=

K

o~

(t)

-

Il~) XCt)

- D(t)(1 + r)S(t) + D(t)Sin

XCt) - D(t)(1 + r)C(t) + KLa(C s -C{t» + D(t)C in

X r (t) =

DCt)(1 + r)XCt) - DCt)(P + r)X r (t)

(1)

(2) (3) (4)

Where XCt), Set), Xlt) and C(t) are the state vanables representing the biomass, the substrate, the recycled biomass and the dissolved oxygen concentrations respectively, DCt) is the dilution rate, r and 13 represent respectively the ratio of recycled flow to influent flow and the ratio of waste flow to intluent flow, Sin and C., correspond to the substrate and dissolved oxygen concentrations in the feed stream, respectively. The kinetics of the cell mass production are defmed in tenns of the specific growth rate ~ and the yield of cell mass Y; the tenn Ko is a constant, Cs is the maximum dissolved oxygen concentration and KLa represents the oxygen mass transfer coefficient We assume that the specific growth rate is modelled by Olsson model [8] : Set) CCt) Il(t) = Ilmax Ks + Set) Kc + C(t) where

).!mu

is the maximum specific growth rate, Ks is the affinity constant and

(5)

Ko is the saturation constant.

The objective IS to regulate the substrate concentration S at a desired setpomt S ... mside the activated sludge process by using the dilution rate D as control action under the following assumptions : A1 : the dissolved oxygen concentration C is measured on-line . A2 : the biomass concentration X , the substrate concentration S and the recycled biomass concentration Xr are unavailable on-line. A3 : the constants Cs, Ko and Y are known, the parameters rand p are known . A4 : the structure of the growth rate is known but the kinetic parameters are llflknown A5 : The influent substrate concentration Sin is fixed and known .

3. State and parameter estimation The dynamical model of the process (equa.(1 )-( 4» may be rewritten in the following matrix fonn as follows .

J~ = A(8T~)S + b(~)U + G l~m =h S

(6)

Where : S is the state vector, ~m represents the measurahle state ve ctor ie the dIssolved oxygen concentration and El corresponds to the llflknown constant parameter vector.

I; = [x

S Xr

= [0

C]T, G

[ -O+r)X+rX, -(l+r)S+Sin b(I;)= (l+r)X-(f3+ r )Xr

1

0

0

0 0

KLaCsf, h = [0 ~(e,~)

KS

Kc]T, U = D,

0 0 0

y 0 -Ko~(e,~)

-(1 + r)C + Cin

e = [~max

0 0'

0

-~(e,~)

e an A( ,~)= d

l]T

y

0

0

0

0

0

0

The objective of the JOE is to, simultaneously, observe the unavailable state vector

~e

= [X

S

Xr]T and

estimate the constant parameter vector e. The model of the JOE proposed is based on the model (6) and described by the following nonlinear equation

~ = A(e, I;m)~ + b(~)U + G + K(h T ~ -I;m)

= [\f'J

in which the gain vector K has the components: K

..

.

\f'2

~]T

\f'3

.

(7)

In order to decouple the dynamic of the unmeasured states from the kinetic parameter vector

[~ ~ ~ ~//~:l

smguJar hnear transformaIJon T defined by. ~ "T~ suoh that · T" ~ ~ ~

~

e, we introduce a non

.

Consequently, we obtain a new auxiliary state vector :

[Y X+-T-C

S. =[.. SI S2. S3• S4..]T =

C

S - - Xr

Ko

Ko

The JOE model associated to the new state space representation is given by :

(8)

Error system. The structure of the error system is obtained by an extended linearization technique using Kronecker's calculation [7]. The error system can be written under the hnear form :

- . = FOO~" +B8

(9)

I;

. F =

Where:

-DO + r)

0

rD

If'l

0

- DO + r)

0

DO + r)

0

-D(f3+r)

If'2 If'3

0

~

l

1 0;2

qi -Ko J.l - I; 2 --;;-;;A

Y

A

..

8~

Aoo

I; 4

--;;-;;-

0;2

0

0

0

0

0

Koqi

-~I -

..

0

o

B"'=

A

-A-

y CBI

A

..

+ S4

qi -A-

88

0

A

..

Koqi

- ~l -

-

A

A-

Y 88

..

+ S4

qi -A-

88

1 2 2 --. ". . S = S - represents the state observation error and El = e - El is the parameter estimation error ~

.........

,.

The observer design COnsiSts in choosing the gains If'j i=1, ...,3 and such as F" is an Hurwitz matnx [10) The Stabilization of matrix P' can be obtamed by Imposmg via slate feedback a deSired characterIstic polynomial 8(1..) with n roots (n=4) located in the left-half of the complex plane " I -'-"

(10) Parameter estimation. The estimator design deals with finding a suitable parameter adjustment law of B( t) ,

ensuring stability of (9). The adjustment law has the form :

(11 ) where P and r are arbitrary symmetric positive defmite matrices and ~~ represents the error between the estimated and the measured dissolved oxygen concentration For simulation purpose, f is chosen as f=diag(Yii) where Yii>O for i=1 , ... ,3 and P=I

4. Adaptive Iinearizing control law [10J Recall that the objective of the control algorithm is to regulate the substrate concentration at the setpoint S ... by acting on the dilution rate D. The dynamics corresponding to the substrate concentration (equation 2) is rewritten as :

SCt) =a(;) + b(;)U

(12)

where aC;) = -~ / Y, b(;) = Sin - (I + r)S and U=D. The relative degree associated with the output S is equal to 1 We recall that the relative degree is exactly equal to the nwnber of times one has to differentiate the output in order to have the value of the input vector U explicitly appearing [101 It's obvious that in Equation (12), the term B(S) i.e. (Sin - (1 + r)S) is different from zero. If S = Sin / (I + r) then the bioprocess can be driven to a wash-out steady state (X=O) I.e. to a state where the bacterial life has completely disappeared We assume that the regulation error decreases according to the following stable dynamics

line~

time varying first order (13)

A model reference linearizing control law is then obtained by substituting (12) mto (13) (14)

u=

1 ( ).L"X+A(S·_S») Sin - (1 + r)S Y

(15)

Since the biomass concentration., the substrate concentration and the speCific growth rate are not measured on-line and the speCific growth rate is unknown, they are replaced in the linearizing control law equation by then estimates They are used as if they were the true values accordmg to certainty equivalence principle (16) V/here X S and j.'i denote online estimates of X S and described in section 3. 134

).l

respectively They are calculated With the estimator

5 Simulation results Numerical values. Simulation results are obtained by using a 41b order Runge-Kutta algorithm ~.) integrate the non-linear process equations (1)-(4). The typical values of kinetic parameters and initial conditions are • Y = 0.65, r = 06, ~ = 0.2, KLa=0.018, Kc=2 mgr l , CiD = 0.5 mgr l , ~ = 015 h"l, Ks = 100 mgr l , Ko = 0.5, Cs =10 mgJ"l , l Sjp = 200 mg.r The output variable CCt) is polluted with a 5% multiplicative signal noise for more realistic measurements . The design parameters of the JOE algorithm, are Al=-0.02, Al=-O 15, AJ=-0061 A4=-008 and r =diag{O.OO16,150,0.5} . The setpoint values of the substrate concentration are S·=50 mgr l for Oh
:r-e Cl

0.08

~

t

0.06 0.04

Vl

0.02

Time (b)

0.00 0

50

lOO

150

200

80 70 60 50 40 30 20 10 0

VS "\s

X

200

150

Figure 2 : Evolution of tbe controlled variable

0.04

-e

::I.

0.03 0.02 \

X

0.01

A

/-l Time (b)

Time (h)

000 50

100

150

200

0

Figure 3 : Evolution oftbe actualmd estimate biomass concentnll ion

0.20 ~

0.15

1

0.10

>l

0.05 Tim e (b)

0.00 0

SO

100

150

50

lOO

150

200

Figure 4 : Evolution oflbe actual and estimate specific growtb rate

0.25

J

Time (b)

0.05

290 270 250 230 210 190 170 150 0

-e

~

lOO

50

0

Figure I : Behaviour oftbe control variable

~

S• .J

160 140 120

I

lOO 80 60 40 20 0

Time (b)

0

200

SO

lOO

ISO

200

Figure 6 : Evolution oflbe actual mid estimate atTwity constant

Figure 5 : Evolution oflbe "tual mid estimate maximum Ipecific growth rate

Figure 2 illustrates the behaviour of the controlled estimated substrate concentration § and its corresponding reference trajectory S· The setpoint IS reached after short controller transient response The perturbatlOns of substrate regulation due to the jumps of /-lmu: (20%) at t=40 h K. (50%) at t=70 hand K. (50%) at t =140h and setpomt step change at t= 1OOh, are favourably rejected by the controller. 135

The evolution of the reconstructed biomass concentration is depicted in Figure 1 The estimation algorithm tracks suitably the biomass concentration to its true simulated value with short transient response The JOE also proVides good estimate of the specific growth rate evolution (Figure 4) although the L:etic parameters convergence to biased values (FigS) (Fig.6) and (Fig.7) This is due to nonuniqueness of the identification solutions of Ollson model nonlinearities or, in other words to the insufficient persistent excitation of matrix B* [6]. But if two of the parameters are known the real value of the third parameter can be tracked Robustness of the estimator IS demonstrated by effiCiency of rejecting abrupt variations of the kinetic parameters and tracking setpoint changes. 3.50 3.00 250

j ';L.U

I

2.00

~K

1.50 1.00 0.30

C

Time (b)

000 0

100

50

150

200

Figure 7: Evolllion oflbe aclullllllld eslimllle saturation COD slant

6. Conclusion

In this paper, an adaptive linearizing control law is used to maintain the residual substrate concentration at a prespecified level by acting on the dilution rate D(t). This control law is based on direct exploitation of the non linear model of a wastewater treatment process and is coupled with a joint observer estimator for on-line tracking of unavailable states and unknown parameters. Simulation studies show either the effiCiency of the nonlmear controller in regulation, tracking and disturbance rejection or the effectiveness and the robustness of the estimation scheme, in reconstruction of the unmeasured variables and on-line estimation of the specific growth rate.

7. References

11] Roux G., Dahhou B, and Queinnec I, Multivariable adaptive predictive control of a alcoholic fermentation process, Proc. ECC, VoU, pp. 1736-1740,1993. [2] Zeng, FY, Dahhou B, Goma G. and Nlhtila MT, Adaptive estimatIOn and control of the speCific growth rate of a non linear fermentation process via MRAC method, Proc. 5th ICCAFT & 2nd IF AC-BIO symp. USA 1992. [3] Nejjari F., Benhammou A and Dahhou B, Predictive control of a nonlmear biological wastewater treatment process, CIMASr96, VoU, pp 1367-1372, Casablanca, Maroc, 1996. [4] Dochain D., Design of adaptive controllers for non-linear stirred tank bioreactors extension to the lvfIMO situation, J Proc. Cont., voU ,ppA 1-48., 1991. [5] Nejjari F., BenYoussef C., Benhammou A and Dahhou B , Procedures for state and parameter estimation of a biological wastewater treatment, Vol.l, pp.238-243 CESA'96 IMACS , Lille France, July 9-12, 1996 {6] Ben Youssef Cand Dahhou B Multivariable adaptive predictive control of an aerated lagoon for a wastewater treatment process, J. Proc. Cont., VoL 6, No. 5, pp. 265-275, 1996 [7] Narendra KS . and Annaswamy ZM. A new adaptive law for robust adaptation without persistency of excitation, IEEE Trans . Aut. Control, Vol. 32. pp. 134-145, 1987. [8] Olsson G, State of the art in sewage treatment plant control,. AIChE Symposium Series, n0159, VoL 72, pp.52-76, 1976. [9] Zeng, F. Y, Dahhou B, Goma G. and Nihtila MT, Adaptive observer-estimation deSign of a class of nonlinear systems IFAC, 12th world Congress, Sydney, Vo1.6, pp. 15-18. 1993 . (10] Isldori, A. Nonlinear control Systems · An Introduction, Springer Verlag, FRG, 2nd Edition, 1989

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