# Unknown Inputs Nonlinear Observer for an Activated Sludge Process

Authors: Feten Smida; Taoufik Ladhari; Salim Hadj Saïd; Faouzi M’sahli
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

This paper deals with the jointly estimation problem of unknown inputs and nonmeasured states of one altering aerated activated sludge process (ASP). In order to provide accurate and economic concentration measures during aerobic and anoxic phases, a cascade high gain observer (HGO) approach is developed. Only two concentrations are available; the other process’s states are assumed unavailable. The observer converges asymptotically and it leads to a good estimation of the unavailable states which are the ammonia and substrate concentration, as well as a quite reconstruction of the unknown inputs, which are the influent ammonia and the influent substrate concentrations. To highlight the efficiency of the proposed HGO with this MIMO system’s dynamics, simulation results are validated with experimental data.


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## Body

## 1. Introduction

The environmental conservation and the biological treatment of waste water are required to preserve the ecosystems. Therefore, modeling an activated sludge (AS) Waste Water Treatment Process (WWTP) has been one important area of research for three decades. This AS wastewater treatment process is physically, chemically, and biologically very complex. Furthermore the AS is nonlinear and unsteady system; this is caused by the flow rate variations in a waste water and its composition, joined to reactions varying in time within mingled microorganisms. In order to model the biological plant of the activated sludge process, a lot of models have been suggested: Activated Sludge Process Model N1 (ASM(1)) in [1], the ASM(2) in [2], ASM(2d) in [3], and ASM(3) in [4]. Due to the complexity of these models, various types of an activated sludge-plant reduced model have been put forward in the literature [5–10]. In [11] the authors presented one new model to alternate activated sludge models. In fact the mean continuous model would capture the switching dynamics governing behavior (i.e., aerobic-anaerobic phases), without any consideration of any switching behavior. Recently, [12] proposed a generalized state-identification classification modeling. As a matter of fact, image processing and analysis have been utilized on the basis of linear regression modeling.More generally, waste water treatment processes have a few measurement and control equipment. Under these circumstances various models are utilized in controller design. Among control methods applied to the bioprocess we quote robust multimodel control using the Quantitative Feedback Theory (QFT) techniques in [13], adaptive control in [14, 15], and model predictive control (MPC) in [16, 17] and recently networked control in [18]. Nonetheless, the state variables of the ASM are not all available which present some difficulties in the eventual implementation.In practice, for activated sludge process control and/or supervision, there are some nononline measured inputs, states, or concentrations (substrate concentration). Alternatively, these latter can also be hardly measured (ammonia and nitrate concentration): indeed the main difficulty towards the establishment of control strategies originates in general from the absence online, cheap, and reliable instrumentation. Even the existing sensors can be in many cases noisy and may require high maintenance and implementation cost. In the objective of solving such a problem, various observation methods are proposed. Among the observation method already studied we quote the following: [19] proposed for the nonlinear ASP model an extended Kalman filter, which works within the alternating aerobic-anoxic phase. The filter has been utilized for estimating states and nonstationary disturbances; [20] provided a brief summary of some results concerning state and parameter estimation approaches in relation to the chemical and biochemical processes. In [21] the authors resorted to a linear matrix inequality (LMI) to determine the values of the gain matrix of the asymptotic and classic observer, those are applied to a linearized bioprocess model. In the same framework a nonlinear observer which actually utilized both LMI and Lyapunov function was studied in [22]. In [11], the authors applied one classical Luenberger observer which was dedicated to continuous nonlinear models after linearization and it allows the estimation of nitrate and ammonia concentrations. Furthermore, to better developing the control strategies in presence of unknown inputs, several researches are devoted to such observation.Studying dynamical systems when having UIs has highly given an incentive for research activities in state estimation, fault reconstruction, and control theory. By means of one Unknown Input Observer (UIO), simultaneously estimating the system state and the UIs is achievable. In [23], the authors extended the MIMO systems triangular class. In actual fact, these systems involved nonlinear inputs. In addition, they proposed one full order high gain observer in order to conjointly estimate nonmeasured state and UIs. The authors in [24] gave an essential and sufficient condition for UIO existence. In [25], the authors worked out UIO for one nonlinear system’s canonical observable form in presence of uncertainties in LMI terms. The author in [26] designed a nonlinear differential algebraic systems’ observation scheme with UI. The authors in [27] extended a standard high gain observer (HGO) with the aid of a sliding-mode-based term. This latter would follow the UI vector. Extending the work of [28, 29] designed a set of cascade high gain observers for many nonlinear MIMO systems. Each observer would provide the estimation of just an UI-vector component, with the exception of the last one which would really give an adjustment of full state variables. Given its utility, UIO has received big attention and has been applied to ASP, as in [30] where the authors suggested estimated states and the UIs as regards a minimized ASP nonlinear model through the use of the Extended Kalman Filter (EKF). Reference [31] treated the problems of Waste Water Treatment Plants (WWTP) sensor fault detection with the basic UIO (HGO). In fact, faults are just considered as unknown inputs. In a recent work, [32], an UIO is used to estimate jointly the states and an UI of the bioprocess linearized model.The purpose of this paper is to estimate UIs and unavailable states of ASP. This contribution deals with the use of high gain UIO, which is relatively simple to design, can provide global or semiglobal stability results for a large class of nonlinear systems, robust to modeling uncertainty and external disturbances, and has a single adjustment parameter. The concentrations of ammonia and substrate in the inputs (Ssin and SNH4in) are very influential in the solution of the system model [33]. So the continuous varying inputs may seriously affect the system control; such problem can be solved by the use of an unknown inputs observer.The paper is planned as follows. Initially we introduce a nonlinear ASP model. This model belongs to one specific nonlinear system class for which the development of a corresponding UIO is then detailed in section three; some preliminaries on the nonlinear systems class and state transformations are given in this section. Thereafter, in section four, and to facilitate applying the observation algorithm, we have proposed an appropriate repartition of ASP states. Simulation results are presented in the aim of highlighting the effectiveness of the proposed observer in curing one systematic routine so as to reconstruct jointly UIs and state variables for the ASP. A final conclusion ends the paper.


## 2. Process Description

There is a various model of ASP design. Precisely an ASP is composed of three principal components. The first component is an aeration tank. The latter will serve as a bioreactor. The second component is a settling tank (final clarifier). It was used to separate AS solids and treated waste water. The third one is a Return Activated Sludge (RAS) which is a tool for transferring settled AS from the clarifier into the aeration tank influent, (Figure1). Added to that, the atmospheric air, or pure oxygen, will be put into a combination of primary treated sewage (waste water) which will be mixed with organisms in the purpose of developing a biological floc. In general, such a mixture is named as Mixed Liquor. Frequently, the concentration of the dry solids of mixed liquor suspended solid ranges between 3 and 6 g/L. Thereby, we control any removal efficiency by means of multiple operative conditions, for example, the aeration tank hydraulic residence time. To make it clear, the latter is determined by the division of the volume of the aeration tank over the flow rate. It is worth mentioning other factors like influent load (BOD5: Biological Oxygen Demand within 5 days, COD: Chemical Oxygen Demand, Nitrogen, etc.). The last factor is related to the AS solids which are existing in the aeration tank. We can mention here two other factors: oxygen supply and temperature. Mixed liquor will be evacuated, at the aeration tank discharge, into settling tanks. In addition to that the supernatant (treated waste water) ought to or should run off in the purpose of being discharged to a natural water. It can as well undergo further treatment before discharge. Because of biological growth, excess sludge has an eventual accumulation beyond the desired mixed liquor suspended solid concentration in the aeration tank. We remove such a solid amount from the treatment process in the target of keeping the biomass ratio to supplied food in balance. This solid amount, which is called the waste activated sludge, should be kept aside in storage tanks and should be further treated by digestion. This can be done prior to disposal under anaerobic or aerobic conditions.Figure 1
ASP diagram.


### 2.1. Materiel and Method

The process considered in this work is the pilot ASWWTP unit in the Engineering Laboratory of Environmental Processes (ELEP) of the National Institution of Applied Sciences (NIAS) in Toulouse, France. It comprises a unique aeration tank with a volumeVa=0.03m3 equipped with an aeration surface that provides the necessary oxygen in the reactor in the goal of creating nitrification and denitrification conditions. It is obvious that the settler is a tank in which, in general, biomass is recirculated within the tank. In [34] mass balance was applied for each soluble (substrate) or particulate component (biomass). This led to an eleven state model that was subsequently reduced to four state variables. The earlier nonlinear models like ASM(1) were unattractive because of the high complex scheme systems. As a consequence, our focus is on reduced order models, which are on the bases of some biochemical considerations, giving the description of the nonlinear process behavior [33]. As a result, we distinguish aerobic and anoxic phases through the use of state variables associated with four substrate concentrations. As a matter of fact, aerobic phase is the aeration period. In the latter the air is injected within the reactor in big quantities in the objective of converting the pollution as ammonium nitrate. After that, in order to transform nitrate in nitrogen, in the anoxic phase, the aeration is stopped and also an optional extraneous carbon source is added inside the mixer. The operation of altering phases can be caused by the kla oxygen transfer coefficient value which conspicuously varies from 0 (concerning the anoxic phase) to higher values (in what concerns the aerobic phase) (Figure 2).Figure 2
Combine aerobic and anoxic conditions to achieve nitrification and denitrification.


### 2.2. System Model

The downsized system model is made up of four state variables which have just one alternating phase operation (alternating aeration). The latter is caused by thekla oxygen transfer coefficient value changing from 0 to higher values. Tables 1 and 3 provide the process model’s variable and parameters, while Table 2 gives the state variables. The process nonlinear model differential equations are expressed in the following [19]:(1)X˙O2=-Ds+DcXO2+KLaSO2sat-XO2-1-YHYHβ1-4,57β3X˙NO3=-Ds+DcXNO3-1-YH2,86YHβ2+β3X˙NH4=DsXNH4in-Ds+DcXNH4-iNBMβ1+β2-β3+β4X˙S=DsXSin+DsSSc-Ds+DcXs-1YHβ1+β2+β5with (2)β1=α1XsXO2XO2+KO2Hβ2=α1XsXNO3XNO3+KNO3KO2HXO2+KO2Hβ3=α2XO2XO2+KO2autXNH4XNH4+KNH4autβ4=α3β5=α4XO2XO2+KO2aut+ηNO3hXNO3XNO3+KNO3KO2HXO2+KO2HTable 1
Nonlinear-model parameters.
Parameter Description Value Y H  g.g-1 Performance coefficient of heterotrophic biomass 0.67  i N B M  g.g-1 Nitrogen mass in heterotrophic and autotrophic biomass concentrations 0.080  ηNO3h Hydrolysis in anoxic phase correction factor 0.8  K N H 4 A U T  gN.m-3 Average saturation coefficient of ammonia for autotrophic biomass 0.25  KNO3gN.m-3 Average saturation coefficient of nitrate 0.5  K O 2 H  gO2.m-3 Average saturation coefficient of oxygen for heterotrophic biomass 0.2  K O 2 A U T  gO2.m-3 Average saturation coefficient of oxygen for autotrophic biomass 0.4  α 1  day-1 Growth rate of heterotrophic biomass 46.91  α 2  g.m-3day-1 Autotrophic speed of nitrate production 276.73  α 3  g.m-3day-1 Hydrolysis speed of slowly biodegradable substrate by the heterotrophic 87.54  α 4  g.m-3day-1 Ammonification of soluble organic nitrogen 1546.8  S O 2 s a t  g.m-3 Concentration of dissolved oxygen saturation 9.0Table 2
State variables.
Notation Description X s Concentration of biodegradable substrate X N O 3 Concentration of nitrogen as nitrate and nitrite X N H 4 Concentration of nitrogen as ammonia X O 2 Concentration of oxygen X N H 4 i n Concentration of ammonia input X s i n Concentration of substrate inputTable 3
Input variables.
Parameter Description Value D s Input dilution rate 2.58day-1 Dc Dilution rate of external carbon source 0.016day-1 S s c External carbon source’s concentration 16000g.m-3 kLa Oxygen transfer coefficient 215 or 0day-1X O 2, XNO3, XNH4, and Xs are the concentrations of the dissolved oxygen, the nitrate, the ammonia, and the readily biodegradable substrate, respectively.X S i n is the concentration of substrate soluble in water and XNH4in is the concentration of the ammoniacal nitrogen entering the reactor. XSin and XNH4in are considered the IUs of the system. SSc is an oxygenous carbon source. Ds and Dc are the dilution rates determined as(3)Ds=QsVa,Dc=QcVawhere Qs is the input flow. In addition to that, Qc is an external carbon source flow.β i (for i=1..5) is a simpler form of the process kinetics ρi. It is directly linked to the standard mode ASM(1). Moreover, αi is a specific parameter of the downsized nonlinear model.


## 2.1. Materiel and Method

The process considered in this work is the pilot ASWWTP unit in the Engineering Laboratory of Environmental Processes (ELEP) of the National Institution of Applied Sciences (NIAS) in Toulouse, France. It comprises a unique aeration tank with a volumeVa=0.03m3 equipped with an aeration surface that provides the necessary oxygen in the reactor in the goal of creating nitrification and denitrification conditions. It is obvious that the settler is a tank in which, in general, biomass is recirculated within the tank. In [34] mass balance was applied for each soluble (substrate) or particulate component (biomass). This led to an eleven state model that was subsequently reduced to four state variables. The earlier nonlinear models like ASM(1) were unattractive because of the high complex scheme systems. As a consequence, our focus is on reduced order models, which are on the bases of some biochemical considerations, giving the description of the nonlinear process behavior [33]. As a result, we distinguish aerobic and anoxic phases through the use of state variables associated with four substrate concentrations. As a matter of fact, aerobic phase is the aeration period. In the latter the air is injected within the reactor in big quantities in the objective of converting the pollution as ammonium nitrate. After that, in order to transform nitrate in nitrogen, in the anoxic phase, the aeration is stopped and also an optional extraneous carbon source is added inside the mixer. The operation of altering phases can be caused by the kla oxygen transfer coefficient value which conspicuously varies from 0 (concerning the anoxic phase) to higher values (in what concerns the aerobic phase) (Figure 2).Figure 2
Combine aerobic and anoxic conditions to achieve nitrification and denitrification.


## 2.2. System Model

The downsized system model is made up of four state variables which have just one alternating phase operation (alternating aeration). The latter is caused by thekla oxygen transfer coefficient value changing from 0 to higher values. Tables 1 and 3 provide the process model’s variable and parameters, while Table 2 gives the state variables. The process nonlinear model differential equations are expressed in the following [19]:(1)X˙O2=-Ds+DcXO2+KLaSO2sat-XO2-1-YHYHβ1-4,57β3X˙NO3=-Ds+DcXNO3-1-YH2,86YHβ2+β3X˙NH4=DsXNH4in-Ds+DcXNH4-iNBMβ1+β2-β3+β4X˙S=DsXSin+DsSSc-Ds+DcXs-1YHβ1+β2+β5with (2)β1=α1XsXO2XO2+KO2Hβ2=α1XsXNO3XNO3+KNO3KO2HXO2+KO2Hβ3=α2XO2XO2+KO2autXNH4XNH4+KNH4autβ4=α3β5=α4XO2XO2+KO2aut+ηNO3hXNO3XNO3+KNO3KO2HXO2+KO2HTable 1
Nonlinear-model parameters.
Parameter Description Value Y H  g.g-1 Performance coefficient of heterotrophic biomass 0.67  i N B M  g.g-1 Nitrogen mass in heterotrophic and autotrophic biomass concentrations 0.080  ηNO3h Hydrolysis in anoxic phase correction factor 0.8  K N H 4 A U T  gN.m-3 Average saturation coefficient of ammonia for autotrophic biomass 0.25  KNO3gN.m-3 Average saturation coefficient of nitrate 0.5  K O 2 H  gO2.m-3 Average saturation coefficient of oxygen for heterotrophic biomass 0.2  K O 2 A U T  gO2.m-3 Average saturation coefficient of oxygen for autotrophic biomass 0.4  α 1  day-1 Growth rate of heterotrophic biomass 46.91  α 2  g.m-3day-1 Autotrophic speed of nitrate production 276.73  α 3  g.m-3day-1 Hydrolysis speed of slowly biodegradable substrate by the heterotrophic 87.54  α 4  g.m-3day-1 Ammonification of soluble organic nitrogen 1546.8  S O 2 s a t  g.m-3 Concentration of dissolved oxygen saturation 9.0Table 2
State variables.
Notation Description X s Concentration of biodegradable substrate X N O 3 Concentration of nitrogen as nitrate and nitrite X N H 4 Concentration of nitrogen as ammonia X O 2 Concentration of oxygen X N H 4 i n Concentration of ammonia input X s i n Concentration of substrate inputTable 3
Input variables.
Parameter Description Value D s Input dilution rate 2.58day-1 Dc Dilution rate of external carbon source 0.016day-1 S s c External carbon source’s concentration 16000g.m-3 kLa Oxygen transfer coefficient 215 or 0day-1X O 2, XNO3, XNH4, and Xs are the concentrations of the dissolved oxygen, the nitrate, the ammonia, and the readily biodegradable substrate, respectively.X S i n is the concentration of substrate soluble in water and XNH4in is the concentration of the ammoniacal nitrogen entering the reactor. XSin and XNH4in are considered the IUs of the system. SSc is an oxygenous carbon source. Ds and Dc are the dilution rates determined as(3)Ds=QsVa,Dc=QcVawhere Qs is the input flow. In addition to that, Qc is an external carbon source flow.β i (for i=1..5) is a simpler form of the process kinetics ρi. It is directly linked to the standard mode ASM(1). Moreover, αi is a specific parameter of the downsized nonlinear model.


## 3. Studied Nonlinear Class and Design of Unknown Input Observer

It is worth considering the nonlinear multioutput systems. These latter can be as follows:(4)x˙=fu,x+gu,x,v+ε~ty=x1With x=x1Tx2T⋯xqTT∈Rn where xk∈Rnk for k=1,2,…,q and n1≥n2≥⋯,nq,  ∑k=1q‍nk=n. ε~(t)=[0T,…,0T,εqT]T is an unknown function. The unknown input v=v1∈Rm, the known input u⊂U a compact set of Ru, and the output y∈Rn1,f(u,x)∈Rn with fk(u,x)∈Rnk. λ1 has been related to the unknown inputs. One notates 1≤λ1≤q such as(5)∀i<λ1:∂gi∂v1u,x,v≡0,∂gλ1∂v1u,x,v≠0System (1) is presented in its detailed form by the following equations:(6)x˙1=f1u,x1,x2+g1u,x1⋮x˙λ1-1=fλ1-1u,x1,…xλ1+gλ1-1u,x1,…xλ1-1x˙λ1=fλ1u,x1,…xλ1+1+gλ1u,x1,…xλ1,v1⋮x˙q-1=fq-1u,x1,…xq+gq-1u,x1,…xq-1,v1x˙q=fqu,x1,…xq+gqu,x1,x2,…xq,v1+εqy=x1 Note that this nonlinear system class is more general than those studied in [23, 28, 35].Let(7)φku,x,v1=fku,x+gku,x,v+ε~t Our aim is to synthesize an observer to estimate at the same time the nonmeasured states and the vector of unknown inputs. In the target of dealing with the HGO synthesis, we make what follows:(A.1) For 1≤k≤q-1 the function xk+1↦fk(u,x1,…,xk,xk+1) from Rnk+1↦Rnk is injective for all (u,x1,…,xk,xk+1). Then, there exist α1,β1>0 such that ∀k∈1,…,q-1,∀x∈Rn,∀u∈U(8)0<α12Ink+1≤∂fk∂xk+1u,xT∂fk∂xk+1u,x≤β12Ink+1 where Ink+1 is a matrix identity, whose dimension is (nk+1)×(nk+1).Assumption (A1) is essential for guaranteeing the diffeomorphism existence which could make an initial system in the canonical form of observability.(A.2) The function xλ1+1v1↦φλ1(u,x1,…,xλ1,xλ1+1,v1) from Rnλ1+1+m→Rnλ1 is injective for all (u,x1,…,xλ1,v1); we consider(9)Fλ1x,u,v1=∂φλ1∂xβ1+1u,x,v1∂φλ1∂v1u,x,v1Such assumption is a rank condition, which has to be opted for to estimate UIs.Consequently, the next inequality is needed to design UIO:(10)nλ1+1+m≤nλ1 In particular, we need to have(11)m≤p-1


### 3.1. State Transformation

Now, a set of coordinate transformations should be introduced so as to canonically form system (4) corresponding to the HGO synthesis. Just as it should be, an augmented system is made up of 2 blocks such that their outputs have the same values as those of the original system (4). x1,…,xλ1+1 and v1 are related first-block variables. On the other hand, variables linked to the last one should be able to be simple copies of the full system states. As a consequence, we interpret the studied immersion like an injective map ψ which is defined as follows: (12)ψ:Rn+m→Rn1+⋯+nλ1+1+m+nandxv1↦μ=ψx,v=μ1μ2,with:μ1=μ11μ21⋮μλ1+11∈Rn1+⋯+nλ1+1+m,μ2=μ12μ22⋮μq2∈Rn,(13)ψx,v:μk1=xk,k=1,…,λ1μλ1+11=μλ1+1,11μλ1+1,2i=xλ1+1v1μk2=xk,k=1,…,qThus, the two subsystems in the μ coordinates are given in the following:(14)μ˙11=f1u,μ11,μ21+g1u,μ11⋮μ˙λ11=fλ1u,μ11,μ21,…,μλ1+1,11+gu,μ11,μ21,…,μλ11,μλ1+1,21μ˙λ1+11=fλ1+1u,μ11,μ21,…,μλ1+1,11,μλ1+21+gu,μ11,μ21,…,μλ1+1,11,μλ1+1,21ε1ty1=μ11(15)μ˙12=f1u,μ12,μ22+g1u,μ12⋮μ˙q-12=fq-1u,μ1:q2+gq-1u,μ1:q-12,μλ1+22μ˙q2=fqu,μ1:q2+gq-1u,μ1:q2,μλ1+22+εqy2=μ12 with ε1(t) is an unknown bounded function.In order to obtain the canonical form, a second injective mapϕ is introduced as follows: Φ:Rn1+⋯+nλ1+1+m+n→Rpλ1+1pq,(16)μ1μ2↦ζ=Φu,μ=ζ1=Φ1u,μζ2=Φ2u,μwith (17)ζ1=ζ11ζ21⋮ζλ1+11∈Rλ1+1p,ζ2=ζ12ζ22⋮ζq2∈Rqpζj1 and ζj2 are defined such as(18)ζ11=Φ1,1u,μ11=μ11ζ21=Φ1,2u,μ11,μ21=f1u,μ11,μ21ζ31=Φ1,3u,μ11,μ21,μ31=∂f1∂μ21u,μ11,μ21f1u,μ1:31⋮ζλ11=Φ1,λ1u,μ1:λ11=∏k=1λ1-2∂fk∂μk+11u,μ1:k+11fλ1-1u,μ1:λ11ζλ1+11=Φ1,λ1+1u,μ1:λ1+11=∏k=1λ1-1∂fk∂μk+11u,μ1:k+11fλ1u,μ1:λ11,μλ1+1,11+gλ1u,μ1:λ11,μλ1+1,21(19)ζ12=Φ2,1u,μ12=μ12ζ22=Φ2,2u,μ12,μ22=f1u,μ12,μ22ζ32=Φ2,3u,μ12,μ22,μ32=∂f1∂μ22u,μ12,μ22f1u,μ12,μ22,μ32⋮ζλ12=Φ2,λ1u,μ1:λ12=∏k=1λ1-2∂fk∂μk+12u,μ1:k+12fλ1-1u,μ1:λ12ζλ1+12=Φ2,λ1+1u,μ1:λ1+12=∏k=1λ1-1∂fk∂μk+11u,μ1:k+12fλ1u,μ1:λ1+12⋮ζq2=Φ2,qu,μ1:q2=∏k=1q-2∂fk∂μk+12u,μ1:k+12fq-1u,μ1:q2 Using the adopted notation, system (12) can be written in the new coordinates ζ as follows:(20)ζ˙11=ζ21ζ˙21=ζ31+ψ21u,u˙,ζ11,ζ21⋮ζ˙λ11=ζλ1+11+ψλ11u,u˙,ζ11,…,ζλ11ζ˙β1+11=ψβ1+11u,u˙,ζ+∏j=1λ1-1∂fj∂μj+11u,ζi∂gλ1∂v1u,ζ1εty1=ζ11(21)ζ˙12=ζ22ζ˙22=ζ32+ψ22u,u˙,ζ12,ζ22⋮ζ˙λ12=ζλ1+12+ψλ12u,u˙,ζ12,…,ζλi2,ζ1ζ˙λ1+12=ζλ1+22+ψλ1+12u,u˙,ζ12,…,ζλ12,ζλ1+12,ζ1⋮ζ˙q-12=ζq-12+ψq-12u,u˙,ζ12,…,ζq-12,ζ1ζ˙q2=ψq2u,u˙,ζεqy2=ζ12 with(22)ψλ1+11u,u˙,ζ=∂Φ1,λ1+1∂uu,μ11,…,μλ11,μλ1+21u˙+∑j=1λ1∂Φ1,λ1+1∂μj1u,μ11,…,μλ1+21fku,μ11,…,μj+11+gku,μ11,…,μj1+∂Φ1,λ1+1∂μλ1+11u,μ11,…,μλ1+21fλ1+1u,μ11,…,μλ1+11,μλ1+21+gλ1+1u,μ11,…,μλ1+21 and(23)ψk2u,u˙,ζ12,…,ζk2=∂Φ2,k∂uu,μ12,…,μk2u˙+∏j=1k-1∂fj∂μj+12u,μ12,…,μj+12gku,μ12,…,μk2+∑j=1k-1∂Φ2,k∂μj2u,μ12,…,μk2fju,μ12,…,μj+12+gju,μ12,…,μj2


### 3.2. Observer Design

Our purpose is to synthesize an observer to estimate at the same time the vector of UIs as well as the nonmeasured state without the assumption of whatever model for UIs. According to Farza et al. (2004), a coordinate transformation is necessary for synthesizing such an observer. This may make one appropriate system form to synthesize the observer. Hence we should write systems (20)-(21) as follows:(24)ζ˙1=A1ζ1+ψ1u˙,u,ζ+ε¯1y1=ζ11(25)ζ˙2=A2ζ2+ψ2u˙,u,ζ+ε¯2y2=ζ12(26)A1=0Iλ1+1p00,A2=0Iqp00with(27)ε¯1=0λ1n1T,…,ε1TTwith(28)ε1=∏j=1λ1-1∂fj∂μj+11u,ζi∂gλ1∂v1u,ζ1εand (29)ε¯2=ψq2u,u˙,ζεqBy referring to the framework given in [28], we opt for HGO synthesized for nonlinear MIMO systems, which are canonically nontriangular. The observer of systems (24)-(25) is designed as follows:(30)ζ^˙1=A1ζ^1+ψ1.-θδ1Δ1-1θS1-1C1Tζ^11-ζ11ζ^˙2=A2ζ^2+ψ2.-θδ2Δ2-1θS2-1C2Tζ^12-ζ12 with(31)S1-1C1T=Cλ1+11IpCλ1+12Ip⋮Cλ1+1λ1+1Ip,S2-1C2T=Cq1IpCq2Ip⋮CqqIpΔ1θ=diag⁡Ip,1θδ1Ip,…,1θλ1δ1IpΔ2θ=diag⁡Ip,1θIp,…,1θq-1Ipθ>0 is a real number that represent just one observer design parameter. δ1=2q-3;δ2=1.Converging the estimation error dynamic was demonstrated in detail in [28].This observer can also be given in theμ coordinates as(32)μ^˙1=f1u,μ_^+g1u,μ_^-Λ1u,μ_^+θδ1Δ1-1θS1-1C1Tμ^11-μ11μ^˙2=f2u,μ_^+g2u,μ_^-Λ2u,μ_^+θδ2Δ2-1θS2-1C2Tμ^12-μ12Now, we suppose(33)x^k=μ^k2fork=1,…,qη^k=μ^k1fork=1,…,λ1η^λ1+1=μ^λ1+1,11v^1=μ^λ1+1,21Utilizing this notation, we can express the observer as follows:(34)η^˙1=f1u,η^1i,η^2i+g1u,η^1i-Cλ1+11θδ1η^1-x1⋮η^˙λ1-1=fλ1-1u,η^1,…,η^λ1+gλ1-1u,η^1,…,η^λ1-1-Cλ1+1λ1-1θλ1-1δ1Λ1,λ1-1+u,η^η^1-x1η^˙λ1=fλ1u,η^1,…,η^λ1+1+gλ1u,η^1,…,η^λ1,v^1-Cλ1+1λ1θλ1δ1Λ1,λ1+u,η^η^1-x1η^˙λ1+1v^˙1=fλ1+1u,η^1,η^2,…,η^λ1+1,xλ1+2+gλ1+1u,η^1,η^2,…,η^λ1+1,v^10-Cλ1+1λ1+1θλ1+1δ1Λ1,λ1+1+u,η^i,v^1η^1-x1(35)x^˙1=f1u,x1,x^2+g1u,x1-Cq1θx^1-x1⋮x^˙λ1-1=fλ1-1u,x1,…,x^λ1+gλ1-1u,x1,…,x^λ1-1-Cqλ1-1θλ1-1Λ2,λ1-1+u,x^x^1-x1x^˙λ1=fλ1u,x1,…,x^λ1+1+gλ1u,x1,…,x^λ1,v^1-Cqλ1θλ1Λ2,λ1+u,x^x^1-x1⋮x^˙q-1=fq-1u,x1,…,x^q+gq-1u,x1,…,x^q-1,v^1-Cqq-1θq-1Λ2,q-1+u,x^x^1-x1x^˙q=fqu,x1,…,x^q+gqu,x1,…,x^q,v^1-CqqθqΛ2,q+u,x^x^1-x1with (36)Fλ1u,x,v=∂fλ1∂xλ1+1u,x∂gλ1∂v1u,x,vΛ1u,x,v=diag⁡Ip,∂f1∂x2u,x,…,∏k=1λ1-1∂fk∂xk+1u,x,∏k=1λ1-1∂fk∂xk+1u,xFλ1u,x,vΛ2u,x,v=diag⁡Ip,∂f1∂x2u,x,…,∏k=1q-1∂fk∂xk+1u,x


## 3.1. State Transformation

Now, a set of coordinate transformations should be introduced so as to canonically form system (4) corresponding to the HGO synthesis. Just as it should be, an augmented system is made up of 2 blocks such that their outputs have the same values as those of the original system (4). x1,…,xλ1+1 and v1 are related first-block variables. On the other hand, variables linked to the last one should be able to be simple copies of the full system states. As a consequence, we interpret the studied immersion like an injective map ψ which is defined as follows: (12)ψ:Rn+m→Rn1+⋯+nλ1+1+m+nandxv1↦μ=ψx,v=μ1μ2,with:μ1=μ11μ21⋮μλ1+11∈Rn1+⋯+nλ1+1+m,μ2=μ12μ22⋮μq2∈Rn,(13)ψx,v:μk1=xk,k=1,…,λ1μλ1+11=μλ1+1,11μλ1+1,2i=xλ1+1v1μk2=xk,k=1,…,qThus, the two subsystems in the μ coordinates are given in the following:(14)μ˙11=f1u,μ11,μ21+g1u,μ11⋮μ˙λ11=fλ1u,μ11,μ21,…,μλ1+1,11+gu,μ11,μ21,…,μλ11,μλ1+1,21μ˙λ1+11=fλ1+1u,μ11,μ21,…,μλ1+1,11,μλ1+21+gu,μ11,μ21,…,μλ1+1,11,μλ1+1,21ε1ty1=μ11(15)μ˙12=f1u,μ12,μ22+g1u,μ12⋮μ˙q-12=fq-1u,μ1:q2+gq-1u,μ1:q-12,μλ1+22μ˙q2=fqu,μ1:q2+gq-1u,μ1:q2,μλ1+22+εqy2=μ12 with ε1(t) is an unknown bounded function.In order to obtain the canonical form, a second injective mapϕ is introduced as follows: Φ:Rn1+⋯+nλ1+1+m+n→Rpλ1+1pq,(16)μ1μ2↦ζ=Φu,μ=ζ1=Φ1u,μζ2=Φ2u,μwith (17)ζ1=ζ11ζ21⋮ζλ1+11∈Rλ1+1p,ζ2=ζ12ζ22⋮ζq2∈Rqpζj1 and ζj2 are defined such as(18)ζ11=Φ1,1u,μ11=μ11ζ21=Φ1,2u,μ11,μ21=f1u,μ11,μ21ζ31=Φ1,3u,μ11,μ21,μ31=∂f1∂μ21u,μ11,μ21f1u,μ1:31⋮ζλ11=Φ1,λ1u,μ1:λ11=∏k=1λ1-2∂fk∂μk+11u,μ1:k+11fλ1-1u,μ1:λ11ζλ1+11=Φ1,λ1+1u,μ1:λ1+11=∏k=1λ1-1∂fk∂μk+11u,μ1:k+11fλ1u,μ1:λ11,μλ1+1,11+gλ1u,μ1:λ11,μλ1+1,21(19)ζ12=Φ2,1u,μ12=μ12ζ22=Φ2,2u,μ12,μ22=f1u,μ12,μ22ζ32=Φ2,3u,μ12,μ22,μ32=∂f1∂μ22u,μ12,μ22f1u,μ12,μ22,μ32⋮ζλ12=Φ2,λ1u,μ1:λ12=∏k=1λ1-2∂fk∂μk+12u,μ1:k+12fλ1-1u,μ1:λ12ζλ1+12=Φ2,λ1+1u,μ1:λ1+12=∏k=1λ1-1∂fk∂μk+11u,μ1:k+12fλ1u,μ1:λ1+12⋮ζq2=Φ2,qu,μ1:q2=∏k=1q-2∂fk∂μk+12u,μ1:k+12fq-1u,μ1:q2 Using the adopted notation, system (12) can be written in the new coordinates ζ as follows:(20)ζ˙11=ζ21ζ˙21=ζ31+ψ21u,u˙,ζ11,ζ21⋮ζ˙λ11=ζλ1+11+ψλ11u,u˙,ζ11,…,ζλ11ζ˙β1+11=ψβ1+11u,u˙,ζ+∏j=1λ1-1∂fj∂μj+11u,ζi∂gλ1∂v1u,ζ1εty1=ζ11(21)ζ˙12=ζ22ζ˙22=ζ32+ψ22u,u˙,ζ12,ζ22⋮ζ˙λ12=ζλ1+12+ψλ12u,u˙,ζ12,…,ζλi2,ζ1ζ˙λ1+12=ζλ1+22+ψλ1+12u,u˙,ζ12,…,ζλ12,ζλ1+12,ζ1⋮ζ˙q-12=ζq-12+ψq-12u,u˙,ζ12,…,ζq-12,ζ1ζ˙q2=ψq2u,u˙,ζεqy2=ζ12 with(22)ψλ1+11u,u˙,ζ=∂Φ1,λ1+1∂uu,μ11,…,μλ11,μλ1+21u˙+∑j=1λ1∂Φ1,λ1+1∂μj1u,μ11,…,μλ1+21fku,μ11,…,μj+11+gku,μ11,…,μj1+∂Φ1,λ1+1∂μλ1+11u,μ11,…,μλ1+21fλ1+1u,μ11,…,μλ1+11,μλ1+21+gλ1+1u,μ11,…,μλ1+21 and(23)ψk2u,u˙,ζ12,…,ζk2=∂Φ2,k∂uu,μ12,…,μk2u˙+∏j=1k-1∂fj∂μj+12u,μ12,…,μj+12gku,μ12,…,μk2+∑j=1k-1∂Φ2,k∂μj2u,μ12,…,μk2fju,μ12,…,μj+12+gju,μ12,…,μj2


## 3.2. Observer Design

Our purpose is to synthesize an observer to estimate at the same time the vector of UIs as well as the nonmeasured state without the assumption of whatever model for UIs. According to Farza et al. (2004), a coordinate transformation is necessary for synthesizing such an observer. This may make one appropriate system form to synthesize the observer. Hence we should write systems (20)-(21) as follows:(24)ζ˙1=A1ζ1+ψ1u˙,u,ζ+ε¯1y1=ζ11(25)ζ˙2=A2ζ2+ψ2u˙,u,ζ+ε¯2y2=ζ12(26)A1=0Iλ1+1p00,A2=0Iqp00with(27)ε¯1=0λ1n1T,…,ε1TTwith(28)ε1=∏j=1λ1-1∂fj∂μj+11u,ζi∂gλ1∂v1u,ζ1εand (29)ε¯2=ψq2u,u˙,ζεqBy referring to the framework given in [28], we opt for HGO synthesized for nonlinear MIMO systems, which are canonically nontriangular. The observer of systems (24)-(25) is designed as follows:(30)ζ^˙1=A1ζ^1+ψ1.-θδ1Δ1-1θS1-1C1Tζ^11-ζ11ζ^˙2=A2ζ^2+ψ2.-θδ2Δ2-1θS2-1C2Tζ^12-ζ12 with(31)S1-1C1T=Cλ1+11IpCλ1+12Ip⋮Cλ1+1λ1+1Ip,S2-1C2T=Cq1IpCq2Ip⋮CqqIpΔ1θ=diag⁡Ip,1θδ1Ip,…,1θλ1δ1IpΔ2θ=diag⁡Ip,1θIp,…,1θq-1Ipθ>0 is a real number that represent just one observer design parameter. δ1=2q-3;δ2=1.Converging the estimation error dynamic was demonstrated in detail in [28].This observer can also be given in theμ coordinates as(32)μ^˙1=f1u,μ_^+g1u,μ_^-Λ1u,μ_^+θδ1Δ1-1θS1-1C1Tμ^11-μ11μ^˙2=f2u,μ_^+g2u,μ_^-Λ2u,μ_^+θδ2Δ2-1θS2-1C2Tμ^12-μ12Now, we suppose(33)x^k=μ^k2fork=1,…,qη^k=μ^k1fork=1,…,λ1η^λ1+1=μ^λ1+1,11v^1=μ^λ1+1,21Utilizing this notation, we can express the observer as follows:(34)η^˙1=f1u,η^1i,η^2i+g1u,η^1i-Cλ1+11θδ1η^1-x1⋮η^˙λ1-1=fλ1-1u,η^1,…,η^λ1+gλ1-1u,η^1,…,η^λ1-1-Cλ1+1λ1-1θλ1-1δ1Λ1,λ1-1+u,η^η^1-x1η^˙λ1=fλ1u,η^1,…,η^λ1+1+gλ1u,η^1,…,η^λ1,v^1-Cλ1+1λ1θλ1δ1Λ1,λ1+u,η^η^1-x1η^˙λ1+1v^˙1=fλ1+1u,η^1,η^2,…,η^λ1+1,xλ1+2+gλ1+1u,η^1,η^2,…,η^λ1+1,v^10-Cλ1+1λ1+1θλ1+1δ1Λ1,λ1+1+u,η^i,v^1η^1-x1(35)x^˙1=f1u,x1,x^2+g1u,x1-Cq1θx^1-x1⋮x^˙λ1-1=fλ1-1u,x1,…,x^λ1+gλ1-1u,x1,…,x^λ1-1-Cqλ1-1θλ1-1Λ2,λ1-1+u,x^x^1-x1x^˙λ1=fλ1u,x1,…,x^λ1+1+gλ1u,x1,…,x^λ1,v^1-Cqλ1θλ1Λ2,λ1+u,x^x^1-x1⋮x^˙q-1=fq-1u,x1,…,x^q+gq-1u,x1,…,x^q-1,v^1-Cqq-1θq-1Λ2,q-1+u,x^x^1-x1x^˙q=fqu,x1,…,x^q+gqu,x1,…,x^q,v^1-CqqθqΛ2,q+u,x^x^1-x1with (36)Fλ1u,x,v=∂fλ1∂xλ1+1u,x∂gλ1∂v1u,x,vΛ1u,x,v=diag⁡Ip,∂f1∂x2u,x,…,∏k=1λ1-1∂fk∂xk+1u,x,∏k=1λ1-1∂fk∂xk+1u,xFλ1u,x,vΛ2u,x,v=diag⁡Ip,∂f1∂x2u,x,…,∏k=1q-1∂fk∂xk+1u,x


## 4. Application of UIO to ASP

### 4.1. Observer Synthesis

The bioprocess belongs to the nonlinear class of system (6) with p=2,  λ1=r=q=2. Xsin and XNH4in are the UIs. The measured states are XO2 and XNO3. The UIO diagram is given by Figure 3, in fact, our objective is to reconstruct the nonmeasured states Xs and XNH4 and the UIs. We suppose that the states vector x=x1x2∈R4, the measured states x1=XO2XNO3∈R2, and the nonmeasured states x2=XNH4Xs∈R2. v=XNH4inXsin∈R2 is the UIs vector and u=SSc∈R is the known input.Figure 3
UIO diagram.The observer is given by system (34). So the appropriate(37)X^˙O2X^˙NO3=f1u,x^1,x^2+g1u,x^1,x^2-3θeX^˙NH4X^˙S=f2u,x^1,x^2+g2u,x^1,x^2,v-3θ2Λ1,2+eX^˙NH4inX^˙Sin=-θ3Λ1,3+ewith(38)f1u,x1,x2=-Ds+DcXO2+KLaSO2sat-XO2-1-YHYHβ1-4,57β3-Ds+DcXNO3-1-YH2,86YHβ2+β3f2u,x1,x2=-Ds+DcXNH4-iNBMβ1+β2-β3+β4+DsSSc-Ds+DcXs-1YHβ1+β2+β5g1u,x1,x2=0;g2u,x1,x2,v=DsXNH4inDsXsine=X^O2-XO2X^NO3-XNO3and(39)Λ1,2=∂f1∂XNH4∂f1∂XsΛ1,3=Λ1,2∂g2∂XNH4in∂g2∂Xsin∂f1∂XNH4=-4,57α2XO2XO2+KO2autKNH4autXNH4+KNH4aut2α2XO2XO2+KO2autKNH4autXNH4+KNH4aut2∂f1∂Xs=-1-YHYHα1XO2XO2+KO2H-1-YH2,86YHα1XNO3XNO3+KNO3KO2HXO2+KO2Hand(40)∂g2∂XNH4in=Ds0;∂g2∂Xsin=0Ds


### 4.2. Validation of the UIO with Experimental Data

We have taken advantage of the experiment done during the modeling and identification of the pilot unit [33] to highlight the performance of the studied observer.The states issued from the process simulation model are compared to the estimated states issued from our observer model as well as their measured values (Figures4 and 5). The real data (measured) was carried out for the ASWWTP over 6 hours and with a sampling period of 20 minutes [33]. In fact polluted water which comes from sanitary network in the city of Toulouse (France) feeds continuously the pilot unit in order to purify it. We obtained the measurement by taking samples every 20 minutes and analyze them in a biological laboratory to extract the three concentrations (oxygen, nitrate, and ammonia).Figure 4
Estimation of dissolved oxygen, biodegradable substrate, and the influent biodegradable substrate concentration.Figure 5
Estimation of nitrate, ammonia, and the influent ammonia concentration.The real curves of the UIs are compared with their estimated ones. It is clear that the corresponding curves are almost superimposed. In addition, the UIs (the influent ammonia concentrationXNH4in and the influent substrate concentration Xsin) are perfectly reconstructed by the UIO despite their variation. We have chosen a variable profile of the UIs to highlight the effectiveness of the UIO. Indeed, the proposed scenario reflects what is really happening in open-air treatment plants. For example, the increase in concentration of the “influent ammonia” is due to the arrival of water loaded with heavy mass caused by industrial discharges, while decreasing in concentration of the “influential biodegradable substrate” is due to important (continuous) reversals of rain. The found results evidently show a thorough compromise between estimated and real curves. Moreover, we can show that the state variables signals of the model and the observer are rallied with the experimental measurements. This is quantified by the relative error (Table 4). The unknown inputs estimation error converges to zero. The advantages of such observer are that it is relatively simple to design and it has only one adjustment parameter.Table 4
Error quantification.
Error X O 2 X N O 3 XNH4 e m a x 0.268 1.29 0.24 e m a x   r e l a t i v e ( ∘ / ∘ ) 5.42 19.28 4.29


## 4.1. Observer Synthesis

The bioprocess belongs to the nonlinear class of system (6) with p=2,  λ1=r=q=2. Xsin and XNH4in are the UIs. The measured states are XO2 and XNO3. The UIO diagram is given by Figure 3, in fact, our objective is to reconstruct the nonmeasured states Xs and XNH4 and the UIs. We suppose that the states vector x=x1x2∈R4, the measured states x1=XO2XNO3∈R2, and the nonmeasured states x2=XNH4Xs∈R2. v=XNH4inXsin∈R2 is the UIs vector and u=SSc∈R is the known input.Figure 3
UIO diagram.The observer is given by system (34). So the appropriate(37)X^˙O2X^˙NO3=f1u,x^1,x^2+g1u,x^1,x^2-3θeX^˙NH4X^˙S=f2u,x^1,x^2+g2u,x^1,x^2,v-3θ2Λ1,2+eX^˙NH4inX^˙Sin=-θ3Λ1,3+ewith(38)f1u,x1,x2=-Ds+DcXO2+KLaSO2sat-XO2-1-YHYHβ1-4,57β3-Ds+DcXNO3-1-YH2,86YHβ2+β3f2u,x1,x2=-Ds+DcXNH4-iNBMβ1+β2-β3+β4+DsSSc-Ds+DcXs-1YHβ1+β2+β5g1u,x1,x2=0;g2u,x1,x2,v=DsXNH4inDsXsine=X^O2-XO2X^NO3-XNO3and(39)Λ1,2=∂f1∂XNH4∂f1∂XsΛ1,3=Λ1,2∂g2∂XNH4in∂g2∂Xsin∂f1∂XNH4=-4,57α2XO2XO2+KO2autKNH4autXNH4+KNH4aut2α2XO2XO2+KO2autKNH4autXNH4+KNH4aut2∂f1∂Xs=-1-YHYHα1XO2XO2+KO2H-1-YH2,86YHα1XNO3XNO3+KNO3KO2HXO2+KO2Hand(40)∂g2∂XNH4in=Ds0;∂g2∂Xsin=0Ds


## 4.2. Validation of the UIO with Experimental Data

We have taken advantage of the experiment done during the modeling and identification of the pilot unit [33] to highlight the performance of the studied observer.The states issued from the process simulation model are compared to the estimated states issued from our observer model as well as their measured values (Figures4 and 5). The real data (measured) was carried out for the ASWWTP over 6 hours and with a sampling period of 20 minutes [33]. In fact polluted water which comes from sanitary network in the city of Toulouse (France) feeds continuously the pilot unit in order to purify it. We obtained the measurement by taking samples every 20 minutes and analyze them in a biological laboratory to extract the three concentrations (oxygen, nitrate, and ammonia).Figure 4
Estimation of dissolved oxygen, biodegradable substrate, and the influent biodegradable substrate concentration.Figure 5
Estimation of nitrate, ammonia, and the influent ammonia concentration.The real curves of the UIs are compared with their estimated ones. It is clear that the corresponding curves are almost superimposed. In addition, the UIs (the influent ammonia concentrationXNH4in and the influent substrate concentration Xsin) are perfectly reconstructed by the UIO despite their variation. We have chosen a variable profile of the UIs to highlight the effectiveness of the UIO. Indeed, the proposed scenario reflects what is really happening in open-air treatment plants. For example, the increase in concentration of the “influent ammonia” is due to the arrival of water loaded with heavy mass caused by industrial discharges, while decreasing in concentration of the “influential biodegradable substrate” is due to important (continuous) reversals of rain. The found results evidently show a thorough compromise between estimated and real curves. Moreover, we can show that the state variables signals of the model and the observer are rallied with the experimental measurements. This is quantified by the relative error (Table 4). The unknown inputs estimation error converges to zero. The advantages of such observer are that it is relatively simple to design and it has only one adjustment parameter.Table 4
Error quantification.
Error X O 2 X N O 3 XNH4 e m a x 0.268 1.29 0.24 e m a x   r e l a t i v e ( ∘ / ∘ ) 5.42 19.28 4.29


## 5. Conclusion

A large class of nonlinear MIMO systems that involves UIs has been studied in this paper in order to synthesize UIO. This observer permits jointly estimating UIs and state variables. We have applied it to one complicated biochemical ASP. The latter is a reduced activated sludge model having switching anoxic and aerobic phases. By using only two measurements, we have achieved the reconstruction of two state variables (Xs and XNH4) and two UI signals (XNH4in and Xsin). Indeed, the obtained simulation results demonstrate quite estimation performances of the suggested observer compared to experimental data. Such observer should be combined with a state control law to ensure that the nitrogen concentration at the settler output does not exceed the standard European norm.



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