Control of uncertain highly nonlinear biological process based on Takagi–Sugeno fuzzy models

Signal Processing 108 (2015) 195–205

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Control of uncertain highly nonlinear biological process based on Takagi–Sugeno fuzzy models S. Bououden a, M. Chadli b, H.R. Karimi c,n a b c

Faculty of Sciences and Technology, Abbes Laghrour Khenchela, Algeria University of Picardie Jules Verne, MIS (EA 4029), 33 rue Saint-Leu, 80039 Amiens, France Department of Engineering, Faculty of Engineering and Science, University of Agder, N-4898 Grimstad, Norway

a r t i c l e i n f o

abstract

Article history: Received 9 April 2014 Received in revised form 13 July 2014 Accepted 9 September 2014 Available online 18 September 2014

This note deals with the control of uncertain highly nonlinear biological processes. Indeed, an adaptive fuzzy control (AFC) scheme is developed for the pre-treatment of wastewater represented by a Takagi–Sugeno (TS) fuzzy model. The proposed approach uses a fuzzy system to approximate the unknown substrate consumption rate in designing the adaptive controller, and then an observer is designed to estimate the concentration in substrate at the outlet bioreactor. The observer is employed to generate an error signal for the adaptive control law which permits to minimize the influence of the measurement noise on the estimation of the substrate concentration. An update of the fuzzy models parameters are obtained using Lyapunov's second method. The closed-loop system behavior is then illustrated on a noisy simulation. & 2014 Elsevier B.V. All rights reserved.

Keywords: Biological process Wastewater model Fuzzy logic modeling Nonlinear adaptive control Lyapunov theory

1. Introduction During the last decades, the lack of water and its pollution have become a very complex problem. It is not any more the concern of the citizens but also that of the production facilities and of the agricultural sectors. The control of biological processes arising in the wastewater treatment plants is a difficult problem since these processes are highly nonlinear [4,18,19,26,28]. Experiments have shown that the specific growth rate of bacteria in biological process varies with time and is influenced by many factors such as the substrate concentration, biomass concentration, temperature, pH, dissolved oxygen concentration, light intensity, etc. [1,7,12,27]. Given all these factors, their complex interconnections and strong nonlinearities, it is difficult to find a precise mathematical model and obtain the desired performance for such processes. n

Corresponding author. E-mail addresses: [email protected] (S. Bououden), [email protected] (M. Chadli), [email protected] (H.R. Karimi). http://dx.doi.org/10.1016/j.sigpro.2014.09.011 0165-1684/& 2014 Elsevier B.V. All rights reserved.

Generally, studies of the wastewater treatment problem are based on approaches that are limited to specific forms of systems [1,2,7,12,24]. This requires good knowledge of the model, which is difficult to obtain in such application. Also, the nonlinear model as well as the control law need to be simplified with different techniques and this may reduce the accuracy of the model. As mentioned above, it is not easy to get an accurate model of the biological process because of the imprecision and uncertainty of several parameters which are difficult to identify [7,27]. Knowing that the assured performance depends directly on the model accuracy, using a simplified model can also deeply affect the stability of the system [6]. Thus, the approximation of the specific growth rate, which yields information on the viability of the cells, is important because it does not, generally, admit mathematical expressions. However, since the growth rate depends on unavailable states for measurement, we propose an approach to estimate the substrate consumption rate. The objective of this work is to design and implement an automatic regulation of wastewater pre-treatment which

196

S. Bououden et al. / Signal Processing 108 (2015) 195–205

guarantees a level of pollution fixed on the outlet side of the sewer collector. Knowing that the crucial problem is related to the choice of appropriate model of specific growth rate, fuzzy approximation is used. This approximation permits to identifying the nonlinearities of the process, represented in the substrate consummation rate, and leads to good knowledge on the evolution of the process over time. It is well known that the Takagi–Sugeno fuzzy system enjoys the universal approximation property [23,31,32,34–36]. Hence, this system is used for online approximation of nonlinear functions arising in the model of the wastewater plant considered here. More specifically in the context of an adaptive control law, two fuzzy TS systems are used, the first one for approximating the substrate consumption rate and the second one for approximating the nonlinear part of the model. Indeed, the idea of the substrate consumption rate approximation can be applied for monitoring and for the adaptive control. This allows us to minimize the influence of the measurement noise on the estimation of the substrate concentration and ensure good convergence to its desired value, even in the noisy case. Based on these ideas, a robust adaptive fuzzy control [3,5,15,30,32] is designed for the wastewater plant. Lyapunov's second method is used to derive the updating laws for the parameters of the TS models that guarantee stability of the closed-loop system. This paper is organized as follows. The description and fuzzy modeling of the process is presented in Section 2. The nonlinear controller design guaranteeing the global asymptotic stability of the closed-loop system is given in Section 3. The simulation example for the proposed algorithm is provided in Section 4. Finally, the paper is concluded in Section 5. Notation: The notation used in the paper is as follows. The superscript “T” stands for matrix transposition, ‖:‖ denotes the inner-product norm, and the notation j x j refers to the absolute value of the elements of the vector x. 2. Fuzzy modeling method of process A biotechnical process can be described in the following way: inside a bioreactor, a population of micro-organisms is developed by consuming large molecules, the substrate (S in g/m3), while forming the biomass (X in g/m3). The basic operation can schematically be represented as [8] This reactor is supplied by a flow F (m3/h), its length is L (m) and its section is A (m²). Let us consider a section dz located at distance Z from the inlet and write the weight breakdown of the biomass and the substrate contained in the volume Adz. The mathematical model of the system includes dynamic equations for biomass X(z, t) and for the substrate concentrations S(z, t) as follows [24]: ∂Xðz; tÞ F ∂Xðz; tÞ ¼ þ μðz; tÞXðz; tÞ ∂t A ∂z

ð1Þ

∂Sðz; tÞ F ∂Sðz; tÞ ¼ k1 μðz; tÞXðz; tÞ ∂t A ∂z

ð2Þ

The following initial conditions are considered: Boundary conditions S(0, t)¼ Sin(t), X(0, t)¼X(1, t)¼0 Initial conditions S(z, 0) ¼S0(z), X(z, 0)¼ X0(z) The PDE systems (1) and (2) are strongly nonlinear due to the nonlinear characteristics of specific growth rate μ modeled by the Contois expression [25].

μ ¼ μmax

S KcX þ S

ð3Þ

in which μmax is the maximum specific growth rate and K c is the saturation coefficient. The system is with distributed parameters that can be transformed to systems with localized parameters described by ordinary differential equations. Recently many fuzzy modeling methods have been proposed in the literature [3,5,14,32]. In this note we shall use the Takagi–Sugeno fuzzy model (TSFM) [6,32] method to approximate the model of Eqs. (1) and (2) [1]. Therefore, the reactor piston is represented by a series of “m” subsystems, leading to “m” fuzzy partitions as shown in Fig. 1. The derivative of the substrate concentration with respect to time can be approximated by means of the simplest finite difference formula. It is possible to apply more sophisticated backward finite difference formulas but it does not improve the estimation accuracy and, even in some cases, it can degrade accuracy [9,10,33]. Then, we have
∂Mðz; tÞ

Mðzi ; tÞ  Mðz; t T s Þ ffi ð4Þ ∂t
z ¼ zi Ts with Ts [h] as the sampling time. Indeed, using Eqs. (3) and (4), systems (1) and (2) can be described using the following nonlinear model [13,20]: 8 i < dX ¼ N 1 ðX r  X i Þ þ μmax K c SXiiXþi Si dt i: 1; …; m ð5Þ : dSi ¼ N 1 ðSr  Si Þ  k1 μmax K SXi Xþi S dt c i i The adopted model is partitioned using n fuzzy partitions reactors (CSTR) with localized parameters of the volume V where N1 ¼ ðF m =VÞ is the dilution rate (h  1) in each CSTR and F m is an influent flow rate (m3/h), Sr and Xr are respectively the concentrations in substrate and biomass contained in the feeding of the cascade having a flow fin. The TSFM can be represented by fuzzy IF-THEN rules where each consequent equation is called a “subsystem” and the premise variables x1(t) and x2(t) are respectively the concentrations in substrate

L S in

S

X in

X

Δz = L/m

where z is the normalized dimensionless space variable, μ(1/h) indicates the specific growth rate of the biomass and k1 yields coefficient of the substrate degradation by the biomass.

S in

S X

X in 1

2

m -1

m

Fig. 1. Reactor piston and its fuzzy sets.

S. Bououden et al. / Signal Processing 108 (2015) 195–205

S and concentration of microorganisms X. For the first fuzzy CSTRs, rule 1: IF x1(t) is M11 and x2(t) is M21 8 1 < dX ¼ N 1 ðX r  X 1 Þ þ μmax K c SX11Xþ1 S1 dt THEN dS1 : dt ¼ N 1 ðSr  S1 Þ  k1 μmax K XS1 Xþi S c 1 1

ð6Þ

The second fuzzy CSTR, rule 2: IF x1(t) is M12 and x2(t) is M22 8 2 < dX ¼ N 1 ðX r  X 2 Þ þ μmax K c SX22Xþ2 S2 dt THEN dS2 : dt ¼ N 1 ðSr  S2 Þ  k1 μmax K SX2 Xþ2 S c 2 2

ð7Þ

For the nth fuzzy CSTR, rule m: IF x1(t) is M1m and x2(t) is M2m 8 < dXdtm ¼ N1 ðX r  X m Þ þ μmax K SXm XþmS c m m THEN dSm : dt ¼ N1 ðSr Sm Þ  k1 μmax K SXm XþmS c m m

ð8Þ

There are two reasons for applying on-line adaptation: (i) the used model describes the process behavior within a small operating range and (ii) the process is a timevariant system. We propose to design a control system for injecting the bacteria in the reactor according to a control law using online information measurement of the substrate at the outlet. Systems (6)–(8), with an unknown substrate consumption rate can be expressed as [16,17,32] x_ ¼ FðxðtÞ; uðtÞÞ þ uðtÞ þ ΛðxðtÞ; uðtÞÞ y ¼ x1 ðtÞ

ð9Þ T

where xðtÞ ¼ ½x1 ðtÞ; x2 ðtÞ ¼ ½SðtÞ; XðtÞ is the state vector of the system, y(t) is the substrate concentration output, u(t) is the concentration of the solutions of bacteria injected, Fðx; uÞ the nonlinear part of the model and Λðx; uÞ is the substrate consumption rate; which describes the intensity of the biological reaction taking place at each point of reactor tube. In most cases the value of Fðx; uÞand Λðx; uÞ cannot be measured on line because the values of S and X are unavailable for measurement. Thus, we will ^ ðx; u=θ^ Þ to approximate ^ employ systems Fðx; u=θ^ Þ and Λ ξ Fðx; uÞ and Λðx; uÞ, respectively. Then the fuzzy rules are ^ u=θ^ Þ is BrF^ ; RrF^ : IF x1 is Ar1 and …and xn is Arn THEN Fðx; ^ ðx; u=θ^ Þ is Br ; Rr : IF x is Ar and … and x is Ar THEN Λ 1

1

n

n

where wBri ðxi Þ is the membership function of the linguistic variable xi, and BrF^ , BrΛ^ represent a crisp values at which the membership function wBri ðxi Þ reaches its maximum. By introducing the concept of fuzzy basis function, the outputs ^ ðx; u=θ^ Þ can be ^ of the fuzzy models of Fðx; u=θ^ Þ and Λ ξ expressed by

T

3. Adaptive mechanism control

Λ^

^ ðx; u=θ^ Þ, respectively. Using the singleton ^ Fðx; u=θ^ Þ and Λ ξ fuzzifier, product inference, and center average defuzzifier [34], the outputs of fuzzy models can be, expressed as   n ∑m Br^ Π i ¼ 1 wBri ðxi Þ ^Fðx; u=θ^ Þ ¼ r ¼ 1 F n r ∑m r ¼ 1 Π i ¼ 1 wBi ðxi Þ   n r m ∑r ¼ 1 BΛ^ Π i ¼ 1 wBri ðxi Þ Λ^ ðx; u=θ^ Þ ¼ ð10Þ n r ∑m r ¼ 1 Π i ¼ 1 wBi ðxi Þ

T ^ u=θ^ Þ ¼ θ^ φðx; uÞ Fðx;

where Si and Xi (i:1 ,…, m) are respectively the concentrations in substrate and biomass of the bacteria solution injected in the cascade with a flow Fm to ensure the regulation.

T

197

ξ

Λ^

where r:1, 2, …, m, and m is the total number of the fuzzy rules for each fuzzy model. Ari , i:1, 2, …, n are the fuzzy sets associated with xi , and BrF^ and BrΛ^ are fuzzy singletons for

Λ^ ðx; u=θ^ ξ Þ ¼ θ^ ξ ξðx; uÞ

ð11Þ

where θ^ ¼ ½B1F^ ; B2F^ ; …; Bm T and θ^ ξ ¼ ½B1Λ^ ; B2Λ^ ; …; Bm T are F^ Λ^ m 1 the adjustable parameter vectors, ξðxÞ ¼ ½ξ ðxÞ; …; ξ ðxÞT is the vector of fuzzy basis functions defined as:

ξr ðxÞ ¼

∏ni¼ 1 wBri ðxi Þ m ∑r ¼ 1 ∏ni¼ 1 wBri ðxi Þ

The control law supposes the availability of on-line measurement of the concentration in substrate at the output S. Thus we will estimate S on-line by using an estimator based on fuzzy systems. Then the direct adaptive control law can be written as ^ ðx; u=θ^ Þ þ y_  Ke ^ u=θ^ Þ  Λ u ¼  Fðx; d

ð12Þ

where y_ d is the derivative of the desired output, K is a positive constant, and eðtÞ ¼ xðtÞ  x^ ðtÞ is the estimator error. ^ ðx; u=θ^ Þ ^ To construct fuzzy estimators Fðx; u=θ^ Þ and Λ ξ which guarantee to approximate unknown functions Fðx; uÞ and Λðx; uÞ, a suitable adaptive method must be provided for adjustable parameters θ^ and θ^ ξ . Assumption 1. Let us define the optimal parameter estimates of a dynamic system and of the class of substrate n n consumption rate θ and θξ respectively as follows: " # h i n ^ ^ θ ¼ arg min sup Fðx; u=θÞ  Fðx; uÞ ð13Þ θ A Ωθ

x A Ωx

"

h

θnξ ¼ arg min sup Λ^ ðx; u=θ^ ξ Þ  Λðx; uÞ θ ξ A Ωθ ξ

x A Ωx

# i

ð14Þ

where Ωθ , Ωθξ and Ωx denote the compact sets of suitable bounds on θ; θξ and x respectively. We assume that θ; θξ and x never reach boundariesΩθ , Ωθξ and Ωx . We can define the minimum approximation error as w ¼ εf þ εξ

ð15Þ

where n ^ εf ¼ Fðx; uÞ  Fðx; u=θ Þ


εf
r ε1

εξ ¼ Λðx; uÞ  Λ^ ðx; u=θnξ Þ

ð16Þ

198

S. Bououden et al. / Signal Processing 108 (2015) 195–205




εξ
r ε2

ð17Þ

Upper bounds ε1 and ε2 can be reduced arbitrarily by taking into account the minimum approximation errors and final control law (12). Now we consider the following observer for the dynamic system (9): ^ ðx; u=θ^ Þ þu þM xðtÞ  x^ ðtÞ ^ x^_ ðtÞ ¼ Fðx; u=θ^ Þ þ Λ ð18Þ ξ Let eðtÞ ¼ xðtÞ  x^ ðtÞ be an error estimator. From (9) and (18), it can be noted that e-0 (i.e. x^ -x) implies that fuzzy system Λ^ ðx; u=θ^ ξ Þ approaches unknown substrate consumption rate Λðx; uÞ. The parameter adaptation laws can be expressed as _

θ^ ¼ γ 1 φðx; uÞe

ð19Þ

_ θ^ ξ ¼ γ 2 ξðx; uÞe

ð20Þ

The adaptive control schemes guarantee the following properties: w A L2  norm then we have e; x^ ; u A L1  norm

and

θ^ A L1  norm

T 0

e2 ðtÞdt r

2 1 ½Vð0Þ  VðTÞ þ 2 M M

Z

T

w2 ðtÞdt

0

8T 40

The objective is to derive adaptive laws, in order to adjust the parameters in the fuzzy logic systems. Defining paran n meter errors θ~ ¼ θ  θ^ and θ~ ξ ¼ θξ  θ^ ξ , the error dynamic (23) can be rewritten as ð24Þ

The analysis of this stability is performed using the Lyapunov theory. Then, consider the Lyapunov function candidate ð25Þ

where γ 1 and γ 2 are positive constants. The time-derivative of V along error trajectory (24) is 1 T_ 1 T_ V_ ¼ ee_ þ θ~ θ~ þ θ~ ξ θ~ ξ γ γ 1

2

T T 1 T_ 1 T_ ¼  Me þ eθ~ ϕðx; uÞ þ eθ~ ξ ξðx; uÞ þ θ~ θ~ þ θ~ ξ θ~ ξ þew 2

γ1

γ2

    T T 1_ 1_ ¼  Me2 þ θ~ eϕðx; uÞ þ θ~ þ θ~ ξ eξðx; uÞ þ θ~ ξ þ ew γ γ 1

Choosing a fuzzy rule adaptive method as _ θ~ ¼  γ 1 ϕðx; uÞe

2

we get V_ ¼  Me2 þew



   M 2 1 2 M 2 1 2 e þ w  e þ w 2 2M 2 2M rffiffiffiffiffi rffiffiffiffiffiffiffiffi !2 M 1 2 M 1 w  eþ w ¼  e2 þ 2 2M 2 2M ¼  Me2 þ ew þ

M 2 1 2 e þ w ð26Þ 2 2M   Thus, V_ is negative for jejZ w=M . Then, under the assumption that θ^ and θ^ are bounded, the error is bounded r

ξ

(e A L1  norm). This ends the proof.

ð22Þ

Proof. Applying (9) and (15)–(18) to the error equation, obtained from the time-derivative of the error estimator eðtÞ ¼ xðtÞ  x^ ðtÞ, and after some simple manipulations, we can obtain the error equation as     n n e_ ¼  Meþ θ  θ^ ϕðx; uÞ þ θξ  θ^ ξ ξðx; uÞ þw ð23Þ

T 1 1 ~T ~ 1 θ θ þ θ~ ξ θ~ ξ V ¼ e2 þ 2 2γ 1 2γ 2

_

θ^ ξ ¼ γ 2 ξðx; uÞe

In this section, we prove that the control law and adaptation law designed in the above section achieve the tracking and stabilizing objectives in terms of the stability of the closed-loop system.

where M is constant and w is the minimum approximation error.

T T e_ ¼  Meþ θ~ ϕðx; uÞ þ θ~ ξ ξðx; uÞ þ w

_

θ^ ¼ γ 1 φðx; uÞe

3.1. Stability analysis

lim jej ¼ 0

Z

Or equivalently, by the definitions

ð21Þ

then, by Barbalat' Lemma [28]: t-1

_

θ~ ξ ¼  γ 2 ξðx; uÞe

Assumption

2. [29,33]. Assume that Fðx; uÞ and Λðx; uÞ satisfy
Fðx; uÞ
r F o 1; and 0 o Λmin r Λðx; uÞ r Λmax o 1, respectively, for x A U x  ℜn ;where, F, Λmin and Λmax are known constants. Assumption 3. [21,22,33]. Parameters θ and θξ belong to compact

which are defined

sets Ωθ and Ω θξ , respectively, as Ωθ ¼ θ A Rn ‖θ‖ r Mθ and Ωθξ ¼ θξ A Rn ‖θξ ‖ r M θξ g; where M θ and M θξ are designed finite positive constants. From Assumptions 2 and 3 we can achieve the following lemma: Lemma 1. If Assumptions 2 and 3 are satisfied, then

εθ A L1 . and εξ A L1 .

From our Lemma 1, we have εθ A L1 and εξ A L1 , so w A L2 \ L1 . Hence, the right-hand side of (26) is bounded and we obtain e A L2 \ L1 . Moreover, since all the variables in the right-hand side of (24) are bounded, we obtain e_ A L1 , e A L1 . We will show that if w A L2  norm then we have e; x^ ; u A L1  norm, these conditions imply that e converges to zero [21,22,28,30]. From (22) we get Z T Z T 2 1 e2 ðtÞdt r ½Vð0Þ  VðTÞ þ 2 w2 ðtÞdt M M 0 0 Then, since VðTÞ 40, we have Z T Z T 1 e2 ðtÞdt r 2 w2 ðtÞdt M 0 0 From Barbalat's Lemma, we conclude that limt-1 e ¼ 0. Therefore, the closed-loop system is asymptotically stable and the tracking objective is achieved.

S. Bououden et al. / Signal Processing 108 (2015) 195–205

In order to guarantee adaptive laws (19) and (20), constraint sets Ωθ and Ωθξ must be bounded as follows:

Ωθ ¼ θ A R ‖θ‖ rMθ n

n



ð27Þ

Ωθξ ¼ θξ A Rn ‖θξ ‖r Mθξ

o

I1 ¼

T

eθ^ φðx; uÞ ^ _ θ^ ¼ P γ 1 eφðx; uÞ ¼ γ 1 eφðx; uÞ I1 γ 1 θ ‖θ^ ‖2

ð29Þ

> :1

or

‖θ^ ‖2 ¼ M θ

if

‖θ^ ‖2 ¼ M θ eθ^

and

T

T

eθ^ ξ φðx; uÞ r 0

and

φðx; uÞ4 0

For θ^ ξ , use

_ θ^ ξ ¼ P γ 2 eξðx; uÞ

ð28Þ

where Mθ and M θξ are pre-specified parameters. The projection adaptive laws are given as follows: For θ^ , use

8 > < 0 if ‖θ^ ‖2 o M θ

199

T

¼ γ 2 eξðx; uÞ  I 2 γ 2

I2 ¼

eθ^ ξ ξðx; uÞ ^ θξ ‖θ^ ‖2

ð30Þ

ξ

8 > <0

if

‖θ^ ξ ‖2 oM θξ

or

> :1

if

‖θ^ ξ ‖2 ¼ Mθξ

and

T ‖θ^ ξ ‖2 ¼ Mθξ and eθ^ ξ ξðx; uÞr 0 T

eθ^ ξ ξðx; uÞ4 0

On the basis of the above discussion, the following theorem can be obtained.

0.16

Estimated substrate concentration S[g/m 3 ]

Sd S Sestimated

0.14

0.12

0.1

0.08

0.06

0.04 0

50

100

150

200

250

300

350

400

450

500

Time (h) Fig. 2. The estimated substrate concentration, its true value and its reference, case without noise.

0.16

Estimated substrate concentration S[g/m 3 ]

Sd S Sestimated

0.14

0.12

0.1

0.08

0.06

0.04 0

50

100

150

200

250

300

350

400

450

Time (h) Fig. 3. The estimated substrate concentration its true value and its reference, case with noise.

500

200

S. Bououden et al. / Signal Processing 108 (2015) 195–205

Theorem. Consider the control problem of nonlinear system (9) ^ u=θ^ Þ and with the control law given by (12) where Fðx; Λ^ ðx; u=θ^ ξ Þ are given by (11) and parameter vectors θ and θξ are adjusted by adaptation laws (19) and (20). If (21) and (22) are satisfied, then estimator error e(t) converges to zeros i.e. limt-1 eðtÞ ¼ 0 : 4. Simulation results In order to show the benefits of the given adaptive fuzzy control law and to check the controller robustness against noise, a multiplicative white noise with a standard deviation

of 0.2 is added to the measurement output. The value of the system parameters are those used in [2], they are realistic and used by several studies. The controller given by (12) is applied to control the nonlinear dynamic system with the following initial conditions: initial consequent parameters of fuzzy rules are chosen randomly in an interval ½0:1 1, the initial values of substrate and estimated substrate are ½0:1 0:15T and the initial value of the substrate consumption rate is 0.204 g/m3 h. The parameters θ and θξ are adjusted by adaptive laws (19) and (20). The design coefficients of the adaptive controller are set to γ 1 ¼ 1000, γ 2 ¼ 50, M ¼ 15.

0.26

3

Substrate consumption rate estmation [g/m h]

0.27

0.25 0.24 0.23 0.22 0.21 0.2 0.19 0.18

0

50

100

150

200

250

300

350

400

450

500

450

500

Time (h) Fig. 4. The estimated substrate consumption rate, case without noise.

Substrate consumption rate estmation [g/m3 h]

0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.2 0.19 0.18 0

50

100

150

200

250

300

350

400

Time (h) Fig. 5. The estimated substrate consumption rate, case with noise.

S. Bououden et al. / Signal Processing 108 (2015) 195–205

The simulation results are shown in Figs. 2–9. Fig. 2 shows the evolution of the substrate concentration and its estimate. From controllers' point of view, the control law drives the state variables towards the desired trajectory. We can observe that the estimated substrate reaches the set point at the end of a few days at the cost of a slight oscillation in the transient time. Fig. 3 shows simulations with 20% white noise added to output y to check the controller robustness. From these results, we can observe that estimator Sestimated converges

201

towards the computed value S and can asymptotically track desired output Sd. Compared to the work published in [10,11], where the adaptive control is not able to stabilize the system around the desired output, the proposed controller can attain the control objective and is robust against the external noise. Figs. 4 and 5 show the accuracy of the procedure to estimate the substrate consumption rate on the basis of Eq. (11). We note the very good estimation of the parameter in spite of the presence of noise.

0.3 0.28

Evolution of control

0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12

0

50

100

150

200

250

300

350

400

450

500

400

450

500

Time (h) Fig. 6. Control performance in the closed loop.

0.3 0.28

Evolution of control

0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0

50

100

150

200

250

300

350

Time (h) Fig. 7. Control performance in the closed loop with added noise.

202

S. Bououden et al. / Signal Processing 108 (2015) 195–205

-3

x 10

2.5

2

Error

1.5

1

0.5

0

-0.5 0

50

100

150

200

250

300

350

400

450

500

400

450

500

Time (h) Fig. 8. The error signal of adaptive mechanism, no noise.

-3

2.5

x 10

2

Error

1.5

1

0.5

0

-0.5

0

50

100

150

200

250

300

350

Time (h) Fig. 9. The error signal of adaptive mechanism, with noise.

As seen in Figs. 6 and 7, the control signal is smooth without high peaks; we remark that the adaptive controller is capable of keeping the process stable and to track the substrate concentration reference. These features prove the very good performance of control law (12) in spite of the presence of the disturbances. We can see from these simulations that for a desired input, the best estimation performances in the closed loop

plant do not exceed 3% (which is a good for a biological system with distributed parameters). We can also note that the speed of convergence is sufficient and the controller is robust against noise. However, these simulations will be better if a stage of pre-filtering is introduced. We have performed simulations with a linear low-pass filter in the adaptation loop in order to filter the high frequencies and improve the estimation qualities. The results of the

S. Bououden et al. / Signal Processing 108 (2015) 195–205

203

0.16

Estimated substrate concentration S[g/m 3 ]

Sd S Sestimated

0.14

0.12

0.1

0.08

0.06

0.04 0

50

100

150

200

250

300

350

400

450

500

450

500

Time (h) Fig. 10. Simulation of the closed loop plant with pre-filtering.

0.3 0.28

Evolution of control

0.26 0.24 0.22 0.2 0.18 0.16 0.14 0.12 0

50

100

150

200

250

300

350

400

Time (h) Fig. 11. Control performance in the closed loop with pre-filtering.

simulations are shown in Figs. 10–13. We observe that the behavior of the closed loop plant agrees with the predicted theoretical behavior and that the estimation error given in Fig. 13 is bounded and remains close to zero. 5. Conclusion In this paper, we have introduced an adaptive nonlinear controller for the asymptotic regulation of biodegradable organic matter output, which guarantees a level of pollution

fixed on the outlet side of the sewer collector. The control law is based on the feedback linearization technique. The unknown parts of the model are approximated using TS fuzzy systems. These are the substrate consumption rate and nonlinear part of the model. The adaptation laws to update the fuzzy systems parameters are derived using Lyapunov's second method. This allows us to guarantee stability of the closed-loop system. The different simulations have shown that the proposed adaptive control is robust against high noise level and insensitive to the variations of the influent

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Substrate consumption rate estmation [g/m 3 h]

0.27 0.26 0.25 0.24 0.23 0.22 0.21 0.2 0.19 0.18 0

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Time (h) Fig. 12. Accuracy of the substrate consumption rate estimation.

-3

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Time (h) Fig. 13. The error signal of adaptive mechanism.

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