Unknown Inputs Observer-Based Output Feedback

IETE Journal of Research

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Unknown Inputs Observer-Based Output Feedback Predictive Controller for an Activated Sludge Process Feten Smida, Taoufik Ladhari, Salim Hadj Saïd & Faouzi M'sahli To cite this article: Feten Smida, Taoufik Ladhari, Salim Hadj Saïd & Faouzi M'sahli (2018): Unknown Inputs Observer-Based Output Feedback Predictive Controller for an Activated Sludge Process, IETE Journal of Research, DOI: 10.1080/03772063.2018.1497553 To link to this article: https://doi.org/10.1080/03772063.2018.1497553

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IETE JOURNAL OF RESEARCH https://doi.org/10.1080/03772063.2018.1497553

Unknown Inputs Observer-Based Output Feedback Predictive Controller for an Activated Sludge Process Feten Smida

, Taoufik Ladhari

, Salim Hadj Saïd

and Faouzi M’sahli

Department of Electrical Engineering, ENIM, Road Ibn Eljazzar, Monastir 5019, Tunisia ABSTRACT

KEYWORDS

This paper investigates the control problem of an activated sludge process despite the nonmeasurable states and the unknown inputs (UIs). Precisely, our objective is to control the global nitrogen concentration (SN ) at the settler output, with partial state measurement and UIs. For this MIMO system, a nonlinear predictive controller based on a high-gain observer is proposed. First a state feedback (full information) predictive control low is designed in order to maintain the global nitrogen concentration in accordance with the standard imposed by the European standards for water quality. Then, a high-gain observer-based UI is designed whose aim is to estimate the nonmeasurable states (the concentrations of the readily biodegradable substrate Ss and the ammonia SNH4 ) and the UIs (the concentrations of substrate soluble in water and the ammoniacal nitrogen entering the reactor: Ssin and SNH4in ) despite their variable behavior. Finally, estimated states and inputs are used to generate the output feedback predictive control. Simulation results validate the objective of the paper.

Activated sludge process; nonlinear system; predictive control; unknown inputs observer

1. INTRODUCTION In order to protect the environment, to conserve the ecosystems, and to provide healthy and clear water source in conformity with the international requirements, biological waste water treatment processes are essential for this reason, in the last three decades, the modeling of the activated sludge (AS) wastewater treatment process (WWTP) became an interesting area of research. AS wastewater treatment is a highly complex physical, chemical, and biological process. Variations in wastewater flow rate and its composition, combined with time-varying reactions in a mixed culture of micro-organisms, make this process nonlinear and unsteady. For modeling the biological process in the AS plant, several models are proposed : ASM1 (Activated Sludge Process Model No.1) in [1], ASM2 in [2], ASM2d in [3] and ASM3 in [4]. Due to the complexity of these models, different versions of a reduced model for the AS plant are proposed in the literature: [5–9] and [10]. In [11], the authors present a novel average model for alternating AS models. The average continuous model captures the governing behavior of switching dynamics (aerobic-anaerobic phases) without considering the switching behavior. Recently, Khan et al. [12] proposed a generalized classification modeling for state identification; in fact, image processing and analysis-based linear regression modeling are used.

© 2018 IETE

In general, wastewater treatment plants have a few measurement and control equipments. Under these circumstances, a variety of models are used in the controller design. Among the control methods applied to the bioprocess we quote, robust multimodel control using the Quantitative Feedback Theory techniques in [13], adaptive control in [14,15], and multivariable predictive control (MPC) in [16] and recently networked control in [17]. In a very recent work [18], the authors proposed a Radial Basis Function (RBF) neural network (NN)based adaptive PID (RBFNNPID) algorithm in order to control the dissolved oxygen concentration in the aeration tank. The authors in [16] presented the development of model-based control strategies where the Dynamic Matrix Control algorithm in its linear formulation was used to obtain the optimal control of ammonia and nitrate concentrations. Predictive control laws are popular and well-established methods in the process industries [19]. It is known that MPC uses a model to predict the future behavior of the studied plant, and then an optimal control decision can be made based on optimization according to the prediction [20,21]. This ‘foresee’ feature of MPC makes it a suitable control strategy for an activated sludge process (ASP). Especially for the trajectory tracking problem, the MPC can take into account the future value of the reference to improve the performance

2

F. SMIDA ET AL.: UNKNOWN INPUTS OBSERVER

in the sense that not only the current tracking error can be suppressed but also the future errors. For nonlinear plants, the nonlinear model predictive control confronts major problems which are mainly related to the nonconvexity of the optimization criterion and the allocated computation time. The so-called nonlinear generalized predictive control (NGPC) proposed in [22,23] can constitute a remedy to the computational issue. Indeed, an explicit solution of the state feedback control is given in the continuous time. Furthermore, the closed-loop stability is ensured when the relative degree of system is well definite. In practice, the problem of controlling or supervising the AS process is that there are some concentrations, states, or inputs, which are not measured online. To solve this problem, various observation methods are proposed. We can quote, for example: Gomez-Quintero and Queinnec [24] proposed an extended Kalman filter (EKF) for a nonlinear model of an ASP, working in an alternating aerobic–anoxic phase. The filter is used to estimate both the states and non-stationary disturbances. In [25], 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 based on the LMI and Lyapunov function is studied in [26]. In [11], a classical Luenberger observer is applied for the continuous nonlinear models after linearization and it allows the estimation of nitrate and ammonia concentrations. The study of dynamical systems in the presence of unknown inputs (UIs) has highly motivated the research activities in state estimation, control theory, and fault reconstruction. Through an unknown input observer (UIO), the aim is to simultaneously estimate the system state and the UIs can be achieved. In [27], the authors extend the triangular class of MIMO systems involving nonlinear inputs. Moreover, the authors propose a fullorder high-gain observer for the conjoint estimation of the non-measured state and the UIs. In [28], an UIO for a canonical observable form of the nonlinear system in the presence of uncertainties is developed in LMI terms. The paper [29] presents a fault reconstruction scheme that converges in a pre-assigned finite time, which is chosen as the parameter of the finite-time converging unknown input observer. In [30], the authors designed an observer for Lipschitz nonlinear systems with not only unknown inputs but also measurement noise. Recently, Bejarano [31] designed an observation scheme of nonlinear differential- algebraic systems with UI. In [32], the authors designed a set of cascade high-gain observers

for a large class of the MIMO nonlinear system. Each observer provides an estimation of only one component of the UI vector, except the last one which gives a final adjustment of the whole state variables. Given its utility, the UIO has received quite attention and has been applied to the ASP, as in [33] where an estimation of the state and the unknown inputs of the reduced nonlinear model of an ASP using the EKF is proposed. The problems of sensor fault detection for wastewater treatment plants (WWTP) with the basic UIO (HGO) is investigated in [34]. In fact, faults are just considered as unknown inputs. Recently an UIO is used to estimate jointly the states and an UI of the bioprocess linearized model [35]. The purpose of this paper is to cure the problem of regulating nitrogen concentration at the outlet of the alternating ASP in the presence of unavailable states and UIs which are the concentrations of substrate and ammonia in the inputs (SSin and SNH4 in ). This contribution deals with the use of a UIO-based NGPC. The paper is organized as follows: after the introduction, the second section describes the bioprocess, its mathematical model, and formulates its control problem. The third section is devoted to the UIO design for a particular class of MIMO systems, some preliminaries on the nonlinear systems class and assumptions are given in this section. In the fourth section, a NGPC is designed. To highlight the effectiveness of the proposed control to the ASP, the fifth section gives simulation results of the joint reconstruction of the states and the UIs, and the UIO-based NGPC. A final conclusion ends the paper.

2. THE ASP AND PROBLEM FORMULATION 2.1 Description of the Process The process considered in this work is presented in a simplified manner in Figure 1. It consists of an unique aeration tank with volume Vaer equipped with aeration surface which provides oxygen to the reactor to create nitrification and denitrification conditions. The contaminated water from an external source is introduced into the reactor, where an aerobic bacterial culture is maintained in suspension and degrades the organic matter. The micro-organisms are agglomerated as flocculate and produce sludge. The aerobic environment in the reactor is achieved through the use of diffusers or mechanical aerators, which also serve to keep the liquid mixture in a state of complete mixture [36]. The nonlinear models of the ASM1 are not very attractive due to the high complex scheme systems. So, we are interested on


3

The nonlinear model differential equations of the process can be stated as [24] follows: S˙ O2 = −(Ds + Dc )SO2 + KLa (SO2sat 1 − YH ρ1 − 4, 57ρ3 YH 1 − YH = −(Ds + Dc )SNO3 − ρ2 + ρ3 2, 86YH − SO2 ) −

S˙ NO3

S˙ NH4 = Ds SNH4in − (Ds + Dc )SNH4 − iNBM (ρ1 + ρ2 ) − ρ3 + ρ4 + ρ4 Figure 1: The activated sludge process (ASP)

S˙ S = Ds SSin + Ds SSc − (Ds + Dc )Ss −

1 (ρ1 + ρ2 ) + ρ5 + ρ5 YH

(1)

with SO 2 SO2 + KO2 H SNO3 KO 2 H ρ2 = α1 Ss SNO3 + KNO3 SO2 + KO2 H ρ1 = α1 Ss

ρ3 = α2 Figure 2: Combine aerobic and anoxic conditions to achieve nitrification and denitrification

2.2 The ASP Model The reduced resulting process consists of four-state variables with an alternating phase operation (alternating aeration) due to the values of the oxygen transfer coefficient kla between zero and high values.

SO2

SO 2 SNO3 KO 2 H + ηNO3 h + KO 2 H SNO3 + KNO3 SO2 + KO2 H

ρ4 = α3 ρ5 = α4

the reduced order models, based on some biochemical considerations, describing the nonlinear behavior of the process developed in [24]. We thus distinguish an aerobic phase and an anoxic phase, using state variables related to four substrate concentrations. The first phase is an aeration period, air is injected in large quantities into the reactor to convert the pollutant as ammonium nitrate (aerobic phase). Then, the aeration is stopped and a carbon source is external optionally added to the mixer to remove the nitrate into a nitrogen form (anoxic phase). The alternating phase operation is due to the changing value of the oxygen transfer coefficient kla from zero (anoxic phase) to high values (aerobic phase) (Figure 2).

SO2

SO2 SNH4 + KO2aut SNH4 + KNH4aut

ρ4 and ρ5 are unknown functions and represent the parametric uncertainties. Ss , SNO3 , SNH4 , and SO2 are defined, respectively, as a readily biodegradable substrate, the nitrate, the ammonia, and the dissolved oxygen concentrations, respectively. SSin is the concentration of the substrate soluble in water and SNH4in is the concentration of the ammoniacal nitrogen entering the reactor. SSin and SNH4in are considered the IUs of the system. SSc is an oxygenous carbon source. Ds and Dc are the rates of dilution defined as: Ds = Qs /Va and Dc = Qc /Va where Qs is the input flow, Qc is the flow of the external carbon source, and Va is the volume of the settler. ρi (for i = 1 . . . 5) are the simplified form of the process kinetics βi of the standard mode ASM1. αi are specific parameters of the reduced nonlinear model. The values of the input variables are shown in Table 1. The values of the physical parameters of the system are shown in Table 2 [37].


4

Table 1: Parameters of the nonlinear model Parameter YH g.g−1 iNBM g.g−1 ηNO3 h KNH4 AUT gN.m−3 KNO3 gN.m−3 KO2 H gO2 .m−3 KO2 AUT gO2 .m−3 α1 day−1 α2 g.m−3 day−1 α3 g.m−3 day−1 α4 g.m−3 day−1 SO2 sat g.m−3

Description

Value

Coefficient of performance of heterotrophic biomass Mass of nitrogen contained in the concentrations of heterotrophic and autotrophic biomass Correction factor for the hydrolysis in anoxic phase Coefficient of average saturation of ammonia for autotrophic biomass Coefficient of average saturation of nitrate Coefficient of average saturation of oxygen for the heterotrophic biomass Coefficient of average saturation of oxygen for the autotrophic biomass Growth rate of heterotrophic biomass Speed nitrate production by the autotrophic Speed of hydrolysis of slowly biodegradable substrate by the heterotrophic Ammonification of soluble organic nitrogen Concentration of dissolved oxygen saturation

0.67 0.080 0.8 0.25 0.5 0.2 0.4 46.91 276.73 87.54 1546.8 9.0

T

Table 2: Input variables Parameter

Description

Value

2.58 0.016 300 or 0 17.5 35

day−1 day−1 day−1 g.m−3 g.m−3

Ds Dc kLa Ssin SNH4in

ε˜ (t) = [0T , . . . , 0T , εq ]T is an unknown function. The T T unknown input v = [v 1 v 2 . . . v r T ]T ∈ Rm , v j ∈ Rmj , j = 1, . . . , r, with rj=1 mj = m ; the known input u ⊂ U a compact set of Ru , the output y ∈ Rn1 , f (u, x) ∈ Rn with f k (u, x) ∈ Rnk . μ1 is associated to the unknown inputs. One suppose 1 ≤ μ1 ≤ q such as ∀i < μ1 :

2.3 Control Problem Formulation

∂g i ∂g μ1 (u, x, v) ≡ 0, (u, x, v) = 0 ∂v 1 ∂v 1

(3)

For an alternating-aerated ASP, the control problem consists of reducing the nitrogen concentration SN at the settler output. The European standards for water quality imposed that the global nitrogen concentration must be lower than 10 g m−3 at the outlet of the settler, for mean samples over two hours. So our objective is to reduce the nitrogen concentration (SN ) at the settler output. Besides, the carbon concentration Ssc is the only possible control variable during anoxic phases, its influence on the aerobic phase is useless and may be expensive. Therefore, it is desirable to control the process during anoxic phases, in order to maintain the concentration of nitrogen at the outlet of the settler at the nearest possible value of the standard [38]. As a solution to this control problem, an output feedback predictive controller is designed where only partial information of the states and inputs is available.

x˙ μ1 = f μ1 (u, x1 , . . . xμ1 +1 ) + g μ1 (u, x1 , . . . xμ1 , v 1 )

3. NONLINEAR CLASS OF STUDY AND UIO DESIGN

Let

3.1 The Nonlinear Class of Study

ϕ k (u, x, v) = f k (u, x) + g k (u, x, v) + ε˜ (t)

We consider the nonlinear multi-output systems which are in the following form:

The objective consists in synthesizing an observer to simultaneously estimate the vector of unknown inputs and the non-measured states. In order to deal with the HGO synthesis, we make the following assumptions:

x˙ = f (u, x) + g(u, x, v) + ε˜ (t) y = x1

(2) T

T

where the state x = [x1 x2 · · · xq T ]T ∈ Rn withxk ∈ q Rnk for k = 1, 2, . . . , q, and p = n1 ≥ n2 ≥ · · · , nq , k=1 nk = n.

In its detailed form, system (1) is given by the following equations: x˙ 1 = f 1 (u, x1 , x2 ) + g 1 (u, x1 ) .. . x˙ μ1 −1 = f μ1 −1 (u, x1 , . . . xμ1 ) + g μ1 −1 (u, x1 , . . . xμ1 −1 ) .. . x˙ q−1 = f q−1 (u, x1 , . . . xq ) + g q−1 (u, x1 , . . . xq−1 , v) x˙ q = f q (u, x1 , . . . xq ) + g q (u, x1 , x2 , . . . xq , v) + εq y = x1

(4)

Note that this class of nonlinear system is more general than these studied in [39–41]

(5)

A.1 For 1 ≤ k ≤ q − 1 the function xk+1 → f k (u, x1 , . . . , xk , xk+1 ) from Rnk+1 → Rnk is 1 k k+1 injective for all (u, x , . . . , x , x ). Then, there exists


α1 , β1 > 0 such that: ∀k ∈ {1, . . . , q − 1}, ∀x ∈ Rn , ∀u ∈ U 0 < α12 Ink+1 ≤

∂f k (u, x) ∂xk+1

T

∂f k (u, x) ≤ β12 Ink+1 ∂xk+1

1 x˙ˆ = f 1 (u, x1 , xˆ 2 ) + g 1 (u, x1 ) − Cq1 θ (ˆx1 − x1 )

.. . μ1 −1 = f μ1 −1 (u, x1 , . . . xˆ μ1 ) + g μ1 −1 (u, x1 , . . . xˆ μ1 −1 ) x˙ˆ

− Cqμ1 −1 θ (μ1 −1) 2,μ1 −1 + (u, xˆ )(ˆx1 − x1 )

(6) the assumption A1 is necessary to guarantee the existence of diffeomorphism that can make the initial system in a canonical form of observability. A.2 The function n +m v 1 ) from R μ1 +1

+1

xμ1 v1

μ1

F (x, u, v) =

nμ1

→R

xμ1 , v 1 ), we consider:

→ ϕ μ1 (u, x1 , . . . , xμ1 , xμ1 +1 , is injective for all (u, x1 , . . . ,

∂ϕ μ1 (u, x, v) ∂xρ1 +1

∂ϕ μ1 (u, x, v) ∂v 1

(7)

This assumption is a rank condition that it is always necessary when we have UI to be estimated. Consequently, the following inequality is required for the design of the observer: nμ1 +1 + m ≤ nμ1

(8)

In particular, we need to have: m≤p−1 3.2 Observer Design

1 1 η˙ˆ = f 1 (u, ηˆ 1i , ηˆ 2i ) + g 1 (u, ηˆ 1i ) − Cμ θ δ1 (ηˆ 1 − x1 ) 1 +1

.. . μ1 −1 = f μ1 −1 (u, ηˆ 1 , . . . , ηˆ μ1 ) + g μ1 −1 (u, ηˆ 1 , . . . , ηˆ μ1 −1 ) ηˆ˙ μ −1

ˆ ηˆ 1 − x1 ) − Cμ11 +1 θ (μ1 −1)δ1 1,μ1 −1 + (u, η)( μ1 η˙ˆ = f μ1 (u, ηˆ 1 , . . . , ηˆ μ1 +1 ) + g μ1 (u, ηˆ 1 , . . . , ηˆ μ1 , vˆ 1 ) μ

ˆ ηˆ 1 − x1 ) − Cμ11 +1 θ μ1 δ1 1,μ1 + (u, η)( ⎛ μ +1 ⎞ ⎛ f μ1 +1 (u, ηˆ 1 , ηˆ 2 , . . . , ηˆ μ1 +1 , xμ1 +2 ) ⎞ 1 η˙ˆ ⎟ ⎝ ⎠=⎜ +g μ1 +1 (u, ηˆ 1 , ηˆ 2 , . . . , ηˆ μ1 +1 , vˆ 1 )⎠ ⎝ ˙vˆ 1 0 μ +1

μ1 x˙ˆ = f μ1 (u, x1 , . . . xˆ μ1 +1 ) + g μ1 (u, x1 , . . . xˆ μ1 , vˆ 1 )

− Cqμ1 θ μ1 2,μ1 + (u, xˆ )(ˆx1 − x1 ) .. . q−1 = f q−1 (u, x1 , . . . xˆ q ) + g q−1 (u, x1 , . . . xˆ q−1 , vˆ 1 ) x˙ˆ q−1 (q−1)

− Cq

θ

2,q−1 + (u, xˆ )(ˆx1 − x1 )

q x˙ˆ = f q (u, x1 , . . . xˆ q ) + g q (u, x1 , . . . xˆ q , vˆ 1 )

− Cq θ q 2,q + (u, xˆ )(ˆx1 − x1 ) q

(10)

with: μ is an intermediate state vector ∂f μ1 ∂g μ1 μ1 F (u, x, v) = (u, x) 1 (u, x, v) ∂xμ1 +1 ∂v μ 1 −1 ∂f k ∂f 1 1 (u, x, v) = diag Ip , 2 (u, x), . . . , (u, x), ∂x ∂xk+1 k=1 μ 1 −1 ∂f k (u, x)F μ1 (u, x, v) ∂xk+1 k=1 ⎛ ⎞ q−1 1 ∂f k ∂f 2 (u, x, v) = diag ⎝Ip , 2 (u, x), . . . , (u, x)⎠ ∂x ∂xk+1 k=1

Our objective consists in synthesizing an observer to simultaneously estimate the vector of UIs and the nonmeasured state without assuming any model for the UIs. The HGO can be written in the following developed form [32]:

− Cμ11 +1 θ (μ1 +1)δ1 1,μ1 +1 + (u, ηˆ i , vˆ 1 )(ηˆ 1 − x1 )

5

(9)

4. NGPC DESIGN In general, an optimal tracking problem can be stated as follows: design a controller such that the closed-loop system is asymptotically stable and the output, y(t), of the nonlinear system (24) optimally tracks a prescribed reference, yd (t), in terms of a given performance index. the basic idea is, at any time t, to design within a moving time frame located at time t regarding x(t) as the initial condition of a state trajectory xˆ (t + τ ) driven by an input ˆ + τ ) together with associated predicted yˆ (t + τ ). To u(t distinguish them from the real variables, the hatted variables are defined as the variables in the moving time frame. In the NGPC strategy, according to [23], tracking control can be achieved by minimizing a receding horizon performance index: 1 Tp J= (ˆy(t + τ ) − yˆ d (t + τ ))T (ˆy(t + τ ) 2 0 − yˆ d (t + τ )) dτ

(11)


6

where Tp is the predictive period and yˆ d ∈ Rn1 is the prescribed tracking reference. 4.1 Output Prediction The output in the moving time frame is predicted by Taylor series expansion. To simplify the notation, the standard Lie notation is used [42]. Repeated differentiation up to ρ times of the output yˆ gives y˙ˆ (t) = Lf h(x)

(12)

(19)

and

.. . ρ−1

yˆ [ρ−1] (t) = Lf

¨ˆ T . . . uˆ [r] (t)T ˆ T u˙ˆ (t)T u(t) uˆ¯ = u(t) h(x)

ρ

(13) ρ−1

yˆ [ρ] (t) = Lf h(x) + Lg Lf

h(x)û(t)

(14)

with h(x) = [h1 (x), . . . , hm (x)]T

(15)

we assume that r and ρ are the control order and the relative degree, respectively. For a given control order r ≥ 0, to make the rth derivative of the control signal appear in the prediction, the order of the Taylor expansion of the output yˆ (t + τ ) must be at least ρ + r differentiating equation (28) : ρ+1

yˆ [ρ+1] (t) = Lf

ρ−1

h(x)+ p11 (û(t), x(t))+Lg Lf

h(x)u˙ˆ (t) (16)

p11 (û(t), x(t)) =

ρ Lg Lf h(x)û(t) +

ρ−1

dLg Lf

h(x)

dt

By summarizing Equations (26)–(30) to the derived higher derivatives of the output yˆ (t) we have ⎤

⎡

h(x)

⎤

⎥ ⎢ [1] ⎥ ⎢ L1f h(x) ⎥ ⎥ ⎢ ⎢ yˆ ⎥ ⎡ ⎥ ⎢ ⎢ ⎤ ⎥ ⎢ . ⎥ ⎢ 0m×1 .. ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎢ . ⎥ ⎢ ⎥ ⎢ .. ⎥ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ρ . ⎥ ⎥ ⎢ ˆ¯ Lf h(x) ⎥ + ⎢ Y(t) = ⎢ yˆ [ρ] ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎣0m×1 ⎥ ρ+1 ⎢ [ρ+1] ⎥ ⎢ ⎦ Lf h(x)⎥ ⎥ ⎢ ⎢yˆ ⎥ ⎥ ⎢ ⎢ ⎥ H(uˆ¯ ) ⎢ . ⎥ ⎢ ⎥ .. ⎢ .. ⎥ ⎢ ⎢ ⎥ . ⎦ ⎣ ⎣ ⎦ [ρ+r] ρ+r yˆ L h(x) f

When assuming that the output yˆ and the reference signal yˆ d are sufficiently many times continuously differentiable with respect to time t, the outputs yˆ (t + τ ) and yˆ d (t + τ ) at the time τ are approximately predicted by their Taylor series expansion as follows: ˆ¯ yˆ (t + τ )=T(τ ˙ )Y(t) yˆ d (t + τ )=T(τ ˙ )Yˆ¯ d (t) where

τ¯ (ρ+r) T(τ¯ ) = 1 τ¯ . . . (ρ + r)!

is m × m(r + ρ + 1) matrix τ¯ = diag {τ , . . . , τ }

uˆ (t) (17)

yˆ [0]

(20)

is m × m matrix [ρ+r] T T (t) Y¯ d (t) = yd (t)T y˙ d (t)T . . . yd

with

⎡

ˆ¯ ∈ Rm(r+1) given by H(u) ⎛ ⎞ ρ−1 Lg Lf h(x)uˆ (t) ⎜ ⎟ ⎜ ⎟ ρ−1 ⎜p11 (u(t), ˆ x(t)) + Lg Lf h(x)u˙ˆ (t)⎟ ⎜ ⎟ ⎜ ⎟ .. ⎜ ⎟ ⎜ ⎟ . H(uˆ¯ ) = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ˆ x(t)) + · · · + pr1 (u(t), ⎜ ⎟ ⎜ ⎟ ˆ . . . , uˆ [r−1] (t)x(t)) ⎟ ⎜ prr (u(t), ⎝ ⎠ ρ−1 Lg Lf h(x)uˆ [r] (t)

4.2 Nonlinear Optimal Controller The optimal nonlinear control law which minimizes the receding horizon performance index (11) is given by ρ−1

u(t) = −(Lg Lf

ρ

h(x))−1 (KMp + Lf h(x) − yd (t)) [ρ]

(21) where Mρ ∈ Rmρ is given by (18)

⎛

h(x) − yd (t)

⎞

⎟ ⎜ ⎜ L1f h(x) − yd[1] (t) ⎟ ⎟ ⎜ ⎟ Mρ = ⎜ ⎟ ⎜ .. ⎟ ⎜ . ⎠ ⎝ ρ−1 [ρ−1] (t) Lf h(x) − yd

(22)


7

and K ∈ Rm×mρ is the first m columns of the matrix −1 T T ¯ ρr T¯ rr

and

⎡ ¯ T(ρ+1,ρ+1) ⎢ .. T¯ rr = ⎢ . ⎣

K1 = iNBM + α1

··· ..

T¯ (ρ+r+1,ρ+1)

T T¯ ρr

⎡¯ T(1,ρ+1) ⎢ .. =⎢ . ⎣ T¯ (ρ,ρ+1)

.

···

⎤ T¯ (ρ+1,ρ+r+1) ⎥ .. ⎥ . ⎦ T¯ (ρ+r+1,ρ+r+1)

···

T¯ (1,ρ+r+1)

..

.. .

.

···

(23)

⎤ ⎥ ⎥ ⎦

(24)

T¯ (ρ,ρ+r+1)

The ijth elements of the square matrix T are in the form: T¯ (i,j) =

T¯ i+j−1

i, j = 1, . . . ., ρ + r + 1

(25)

T = diag[T, . . . , T]∈ Rm×m

(26)

5.1 Estimation of the Non-measurable States (Ss and SNH4 ) and the UIs (Ssin and SNH4 in )

1

and the UIs. we suppose: the states vector x = xx2 ∈ S O2 ∈ R2 , the nonR4 , the measured states x1 = SNO 3 S 4 2 . v = SNH4 in ∈ R2 measured states x2 = NH ∈ R SSin Ss is the UIs vector and u = SSc ∈ R is the known input. 2 ε1 ρ4 q 2 ∈ R2 ε =ε = 2 = ε2 ρ5 To simplify the system notation and then the control expression notation we will assume some additional parameters: KO 2 H SNO3 SNO3 + KNO3 SO2 + KO2 H SO2

SNH4 SO2 + KO2aut SNH4 + KNH4aut

SO2

SO2 + KO 2 H

⎝

S˙ˆ O2 S˙ˆ NO3

S˙ˆ NH4

⎝

⎛

S˙ˆ S

S˙ˆ NH4 in S˙ˆ Sin

⎞ ⎠ = f 1 (u, xˆ 1 , xˆ 2 ) + g 1 (u, xˆ 1 , xˆ 2 ) − 3θe ⎞ ⎠ = f 2 (u, xˆ 1 , xˆ 2 ) + g 2 (u, xˆ 1 , xˆ 2 , v) − 3θ 2 + 1,2 e

⎞

⎠ = −θ 3 + e 1,3

⎛

−(Ds + Dc )SO2 + KLa (SO2 sat

(29)

⎛

−K3 Ss z1 + α2 z2 −(Ds + Dc )SNH4 − α2 z2 + α3

⎞

⎜ ⎟ ⎜ ⎟ −i α S (z + z ) NBM 1 s 1 4 ⎜ ⎟ 2 1 2 ⎜ ⎟ f (u, x , x ) = ⎜D S − (D + D )S + η s Sc s c s NO3 h z1 ⎟ ⎜ ⎟ ⎝ ⎠ 1 − α1 Ss (z1 + z4 ) + α4 z4 YH S D s NH in 4 g 1 (u, x1 , x2 ) = 0; g 2 (u, x1 , x2 , v) = Ds SSin Sˆ O2 − SO2 e= Sˆ NO3 − SNO3

∂f 1 ∂f 1 = ∂(SNH4 ) ∂(SS ) ∂g 2 ∂g 2 = 1,2 ∂(SNH4in ) ∂(SSin )

1,2 (27)

⎞

⎟ ⎜ ⎜−SO2 ) − K3 Ss z4 − 4, 57α2 z2 ⎟ ⎟ f 1 (u, x1 , x2 ) = ⎜ ⎟ ⎜ −(Ds + Dc )SNO3 ⎠ ⎝

and

SNO3 z3 = SNO3 + KNO3 z4 =

⎛

with

The bioprocess belongs to the nonlinear class of system (4) with p = 2, μ1 = r = q = 2. Ssin and SNH4in are the UIs. The measured states are SO2 and SNO3 . Our objective is to reconstruct the non-measured states Ss and 4 SNH

z2 =

(28)

For our system, the observer given by (22) is written as follows:

⎝

5. SIMULATION RESULTS

z1 =

1 − YH 2.86YH 1 − YH K3 = α1 YH K2 = α1

⎛

(i − 1)!(j − 1)!(i + j − 1)

1 − YH 2.86YH

1,3


8

⎛

Model HGO estimation EKF estimation Experimental

6

4

S O2

⎞ SO 2 KNH4aut −4, 57α 2 ⎜ SO2 + KO2aut (SNH4 + KNH4aut )2 ⎟ ∂f 1 ⎜ ⎟ =⎜ ⎟ SO2 KNH4aut ⎠ ∂(SNH4 ) ⎝ α2 2 SO2 + KO2aut (SNH4 + KNH4aut ) ⎛ ⎞ SO 2 −K3 ⎜ ⎟ SO 2 + K O 2 H ⎜ ⎟ ∂f 1 ⎟ =⎜ ⎜ ⎟ ∂(SS ) ⎝ ⎠ SNO3 KO 2 H −K2 SNO3 + KNO3 SO2 + KO2 H

2

0 0

0.05

0.1

0.15

0.2

0.25

0

0.05

0.1

0.15

0.2

0.25

0

0.05

0.1

0.15

0.2

0.25

20

Ss

15

and ∂g 2 ∂(SNH4 in )

=

Ds 0

;

∂g 2 ∂(SSin )

=

0

10

5

Ds 300

The estimated value of S˙ N is given by the equation: S˙ˆ N = (Ds + Dc )(Sˆ NO3 + Sˆ NH4 − Sˆ N )

(30)

S sin

200

100

0

5.2 Comparison of HG-UIO and the EKF with Measured Data

Time(days)

5.3 Output Feedback Control Low: NGP Carbon Concentration Controller-Based UIO The considered output variable, Y(t) = SN , is the nitrogen concentration at the outlet of the settler. So, the model (1) is augmented by the following equation [38]: dSN = (Ds + Dc )(SNO3 + SNH4 − SN ) dt

(31)

Figure 3: Estimation of dissolved oxygen, biodegradable substrate, and the influent biodegradable substrate concentration

SNO 3

10

5

0 0

0.05

0.1

SNH4

0.15

0.2

0.25

Model HGO estimation EKF estimation Experimental

15

10

5

0 0

0.05

0.1

0.15

0.2

0.25

0

0.05

0.1

0.15

0.2

0.25

150

SNH4 in

The experiment is carried out over a time interval of 6 hours (Experimental data are limited to 6 hours), these measurements are used to show that the states of the model are quite close. In order to highlight the features of the UIO design, besides its privilege in time computation and in the number of synthesis parameters, it is compared with the standard EKF algorithm which is one of the most industrial diffused observer ([43,44]). After several attempts to adjust the design parameters of this last (Q(0), R(0), S(0)), we have chosen the following values: Q(0) = 10−9 I6 , R(0) = 5.10−9 I2 , S(0) = 5.10−4 I6 . Simulation results for conjoint state and UIs estimation of both techniques are illustrated in Figures 3 and 4. It is shown that when the UIs Ssin and SNH4in vary, only the estimates that arise from the R-HGO remain rallied around the trues states. This can be explicated by the local nature of the EKF which approximates the nonlinear model only around some small neighborhood of the operating point.

100

50

0

Time(days)

Figure 4: Estimation of nitrate, ammonia, and the influent ammonia concentration

The value of the desired output Yd should be set less than 8 g m−3 . In fact, the organic nitrogen concentration, at the outlet of the settler, is practically constant


9

(about 2 g m−3 ). So the quality standard can be replaced by a standard on the sum of ammonia and nitrate concentrations equal to or less than 8 g m−3 [38]. The control Ssc (Figure 6) is designed to use the expression (35) with limitation to r = 0 and ρ = 3. For our system, the Lie derivatives of Equations (12)–(14) are given by Lf h(x) = (Ds + Dc )(SNO3 + SNH4 − y) L2f h(x) = (Ds + Dc )2 (−2SNO3 − 2SNH4 + y)

Figure 5: Block diagrams of the output feedback predictive controller

+ (Ds + Dc )(Ds SNH4in + α3 ) +(Ds + Dc )Ss (−K1 z1 − α1 iNBM z4 ) L3f h(x)

Table 3: Input variables

= (Ds + Dc )(−2(Ds + Dc )S˙ NO3 − K2 S˙ s z1

Ds (day−1 )

Time (day)

− 2(Ds + Dc )S˙ NH4 − K2 Ss z˙ 1

0 1 2 3

− α1 iNBM S˙ s (z1 + z4 ) − α1 iNBM Ss (˙z1 + z˙ 4 ) + (Ds + Dc )Lf h(x))

2.58 1.29 1.94 3.23

(32)

Lg L2f h(x) = (Ds + Dc )(K2 z1 Dc + α1 Dc iNBM (z1 + z4 )) (33) The predictive control law given by Equation (21) can be written as

K= and

21 3T 2 ⎡

42 5T 2

21 6T

SN (x) − yd (t)

⎤

⎢ ⎥ ⎥ Mp = ⎢ ⎣Lf h(x) − y˙ d (t)⎦ L2f h(x) − y¨ d (t) yd has a constant profile so we have y˙ d (t) = y¨ d (t) = yd[3] (t) = 0 The whole scheme of the ASP control is depicted in Figure 5. Mainly, after using the HGO in order to estimate the unknown inputs and non-measurable states, these lasts are incorporated in the predictive controller.

5

the prediclive control u=Ssc

with:

(34)

× 10 5

4 3 2 1 0 0

0.5

1

1.5

2

2.5

3

3.5

4

10 8

The output y=S N

u(t) = −(Lg L2f h(x))−1 (KMp + L3f h(x) − yd[3] (t))

inputs may seriously affect the behavior of the controller and such an algorithm can take advantage by the use of unknown inputs observer. In the numerical simulation, we propose a variable profile for the unknown inputs and the known input Ds (Table 3): the dilution rate on the input flow present variations every 1 day. Indeed, these

6 4 2

output trajectory reference

0 0

0.5

1

1.5

2

2.5

3

3.5

4

Time(days)

The concentrations of ammonia and substrate in the input (Ssin and SNH4 in ) are very influential in the solution of the system model [37]. The continuous varying

Figure 6: Output feedback control trajectories and output tracking performance


10

Input profile Estimated input

40 5

0 0

0.5

1

1.5

2

2.5

3

3.5

4

SNH4 in

S O2

50

Model Estimated

10

30 20

100

Ss

10 50

0 0

0.5

1

1.5

0 0

0.5

1

1.5

2

2.5

3

3.5

2.5

3

3.5

4

3

3.5

4

50

5

40 30

0 0

0.5

1

1.5

2

2.5

3

3.5

4

15

S sin

SNH4

10

SNO 3

2

Time(days)

4

20

10

10 5

0

0 0

0.5

1

1.5

2

2.5

3

3.5

4

0

0.5

1

1.5

2

2.5

Time(days)

Figure 7: Estimation of the variable unknown inputs in the closed loop

variations could significantly change the behavior of the model. So that an estimation of these variables would improve the performance of the system. In addition, in our case only two states are measurable, the two others (Ss and SNH4 ) are not measured and will be estimated by the UIO as well as the unknown inputs. A simulation has been realized during 4 days. Figure 6 presents the output variable and the control signal. Despite a partial measurement of the states and the unknown inputs (Ssin and SNH4 in ), the obtained result complies with the international standards for water quality. In fact, the nitrogen concentration at the outlet of the settler remains less than 8 g m−3 . In Figure 7 the states issued from the process simulation model are compared with the estimated states issued from the observer model. Besides, the real curves of the UIs are compared with their estimated ones (Figure 8). The gain θ of the UIO is the only synthesis parameter of the observer and it is determined manually after some simulation attempts.

Figure 8: Estimated states in the closed loop

output feedback predictive control (OFPC) where the unmeasurable states and the unknown inputs are reconstructed by an UIO. In fact, such an observer is combined with the state control low in order to ensure that the nitrogen concentration at the settler output does not exceed the standard European norm. Simulation results have shown better estimation performances of the proposed observer and showed that the regulation of the nitrogen concentration at the settler outlet is successfully achieved through an OFPC law despite the state information mismatch and the unknown inputs variation.

ORCID Feten Smida http://orcid.org/0000-0002-9397-268X Taoufik Ladhari http://orcid.org/0000-0002-9207-5688 Salim Hadj Saïd http://orcid.org/0000-0003-4324-6388 Faouzi M’sahli http://orcid.org/0000-0003-1781-7378

REFERENCES 6. CONCLUSION In this paper, we have succeeded to cure a problem control of a complex biochemical system, which is an AS reduced model with alternating aerobic and anoxic phases. Our strategy consists in, firstly, to design a full information control law, and then, to synthesize an

1. M. Henze, C. P. L. Grady, G. W. Jr., G. V. R. Marais, and T. Matsuo, “Activated sludge model no.1,” IAWQ Scientific and Technical Report No.1, London, techreport 1, 1987. 2. M. Henze, W. Gujer, T. Mino, T. Matsuo, M. C. M. Wentzel, and G. V. R. Marais, “Activated sludge model no.2,” IAWQ Scientific and Technical Report No.3, London, techreport 2, 1995.


3. M. Henze, W. Gujer, T. Mino, T. Matsuo, M. C. Wetzel, G. V. R. Marais, and M. C. M. van Loosdrecht, “Activated sludge process model no. 2d, asm2d,” Water. Sci. Technol., Vol. 39, no. 1, pp. 165–82, 1999. 4. W. Gujer, M. Henze, T. Mino, and M. C. M. V. Loosdrecht, “Activated sludge model no.3,” Water. Sci. Technol., Vol. 39, no. 1, pp. 183–93, 1999. 5. U. Jeppsson and G. Olsson, “Reduced order models for online parameter identification of the activated sludge process,” Water. Sci. Technol., Vol. 28, no. 11–12, pp. 173–83, 1993. 6. M. A. Steffens, P. A. Lant, and R. B. Newell, “A systematic approach for reducing complex biological wastewater treatment model,” Water. Sci. Technol., Vol. 31, no. 3, pp. 590–606, 1997. 7. C. Gomez-Quintero, I. Queinnec, and J. P. Babary, “A reduced nonlinear model of an activated sludge process,” IFAC Proceedings Volumes, Vol. 33, no. 10, pp. 1001–6, 2000. 8. B. Chachuat, N. Roche, and M. A. Latifi, “Reduction of the asm1 model for optimal control of small-size activated sludge treatment plants,” Revue des sciences de leau / J. Water Sci., Vol. 16, no. 1, pp. 05–26, 2003. 9. I. Y. Smets, J. V. Haegebaert, R. Carrette, and J. F. V. Impe, “Linearization of the activated sludge model asm1 for fast and reliable predictions,” Water Res. J., Vol. 37, no. 8, pp. 1831–51, 2003. 10. M. Mulas, S. Tronci, and R. Baratti, “Development of a 4-measurable states activated sludge process model deduced from the asm1,” in Proceedings of IFAC International Symposium on Dynamics and Control of Process Systems, vol. 1, no. 8, Cancun, Mexico, Jun. 2007, pp. 213–18. 11. R. Marquez, M. Belandria-Carvajal, C. Gomez-Quintero, and M. Rios-Bolivar, “Average modeling of an alternating aerated activated sludge process for nitrogen removal,” in Proc. IFAC World Congress., no. 18, Milano, Italy, Sept. 2011, pp. 14 195–200. 12. M. B. Khan, H. Nisar, C. A. Ng, P. K. Lo, and V. V. Yap, “Generalized classification modeling of activated sludge process based on microscopic image analysis,” Environ. Technol., Vol. 39, no. 1, pp. 24–34, 2017. 13. S. Caraman, M. Barbu, and E. Ceanga, “Robust multimodel control using QFT techniques of a wastewater treatment process,” Control Eng. Appl. Inform., Vol. 7, no. 2, pp. 10–17, 2005. 14. O. Gonzalez-Miranda, M. Rios-Bolivar, and C. GomezQuintero, “Adaptive output feedback regulation of an alternating activated sludge process,” in Proceedings of

11

European Control Conference, Budapest, Hungary, Aug. 2009, pp. 424–9. 15. E. Petre, C. Marin, and D. Seliteanu, “Adaptive control strategies for a class of recycled depollution bioprocesses,” Control Eng. Appl. Inform., Vol. 7, no. 2, pp. 25–33, 2005. 16. C. Foscoliano, S. D. Vigo, M. Mulas, and S. Tronci, “Predictive control of an activated sludge process for long term operation,” Chem. Eng. J., Vol. 304, pp. 1031–44, 2016. 17. O. K. Ogidan, C. Kriger, and R. Tzoneva, “Networked control with time delay compensation scheme based on a smith predictor for the activated sludge process,” Control Eng. Appl. Inform., Vol. 19, no. 3, pp. 79–87, 2017. 18. X. Du, J. Wang, V. Jegatheesan, and G. Shi, “Dissolved oxygen control in activated sludge process using a neural network-based adaptive pid algorithm,” Appl. Sci., Vol. 8, no. 2, pp. 261–82, Feb. 2018. 19. L. Magni, D. M. Raimondo and F. Allgwer, Eds., Nonlinear Model Predictive Control. Heidelberg: Springer, 2009. 20. D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica, Vol. 36, no. 6, pp. 789–814, 2000. 21. S. V. Ghoushkhanehee and A. Alfi, “Model predictive control of transparent bilateral teleoperation systems under uncertain communication time-delay,” in Asian Control Conference, no. 9, 2013, pp. 1–6. 22. W. Chen, “Predictive control of general nonlinear systems using approximation,” IEE Proc. – Control Theory Applicat., Vol. 2, no. 151, pp. 137–44, 2004. 23. W.-H. Chen, D. J. Ballance, and P. J. Gawthrop, “Optimal control of nonlinear systems: A predictive control approach,” Automatica, Vol. 39, no. 4, pp. 633–41, 2003. 24. C. Gomez-Quintero and I. Queinnec, “State and disturbance estimation for an alternating activated sludge process,” IFAC Proc., Vol. 5, no. 1, pp. 447–52, 2002. 25. R. Chiu, J. Navarro and J. Pico, “A nonlinear observer for bioprocesses using lmi.” in Proceedings of International IFAC Symposium on Computer Applications in Biotechnology, no. 10, Cancun, Mexico, pp. 389–94, 2007. 26. B. Boulkroune, M. Darouach, M. Zasadzinski, S. Gille, and D. Fiorelli, “A nonlinear observer design for an activated sludge wastewater treatment process,” J. Process. Control., Vol. 19, pp. 1558–65, 2009. 27. M. Triki, M. Farza, T. Maatoug, M. MSaad, and B. Dahhou, “Unknow inputs observers for class of nonlinear systems,” Int. J. Sci. Tech. Aut. Control Comput. Eng., Vol. 4, no. 1, pp. 1218–29, 2010.

12


28. K. Mohamed, M. Chadli, and M. Chaabane, “Unknown inputs observer for a class of nonlinear uncertain systems: An lmi approach,” Int. J. Aut. Comput., Vol. 9, no. 3, pp. 331–6, 2012. 29. K. S. Lee and T. G. Park, “New results on fault reconstruction using a finite-time converging unknown input observer,” IET Cont. Theory Appl., Vol. 6, no. 9, pp. 1258–65, 2012. 30. J. Yang, F. Zhu, and W. Zhang, “Sliding-mode observers for nonlinear systems with unknown inputs and measurement noise,” Int. J. Control. Autom. Syst., Vol. 11, no. 5, pp. 903–10, 2013. 31. F. J. Bejarano, W. Perruquetti, T. Floquet, and G. Zheng, “Observation of nonlinear differential-algebraic systems with unknown inputs,” IEEE Trans. Autom. Control, Vol. 60, no. 7, pp. 1957–62, 2015. 32. S. Hadj-Saïd, F. MSahli, and M. Farza, “Simultaneous state and unknown input reconstruction using cascaded highgain observers,” Int. J. Syst. Sci., Vol. 48, no. 15, pp. 3346–54, 2017. 33. B. Boulkroune, M. Darouach, M. Zasadzinski, and S. Gill, “State and unknown input estimation for nonlinear singular systems: application to the reduced model of the activated sludge process,” in Proceedings IEEE Mediterranean Conference on Control and Automation, no. 16, Ajaccio, France, Jun. 2008, pp. 25–7. 34. S. Methnani, J.-P. Gauthier, and F. Lafont, “Sensor fault reconstruction and observability for unknown inputs, with an application to wastewater treatment plants,” Int. J. Control., Vol. 84, no. 4, pp. 822–33, 2011. 35. I. Khoja, T. Lahdari, A. Sakly, and F. M’Sahli, “Unknowninput observer for disturbance and state estimation of an activated sludge waste water treatment process,” in Proceedings international conference on Sciences and Techniques of Automatic Control & Computer

Engineering, no. 17, Sousse, Tunisia, Dec. 2016, pp. 13–18. 36. L. Metcalf and H. Eddy, Wastewater Engineering: Treatment, Disposal and Reuse. New York: McGraw-Hill, 1991. 37. C. Gomez-Quintero, “Modelisation et estimation robuste pour un procede boues activees en alternance de phases,” Ph.D. dissertation, Laboratoire dAnalyse et dArchitecture des Systemes, Universite Paul Sabatier, Toulouse, France, 2002. 38. I. Simeonov, I. Queinnec, C. Gomez-Quintero and J. Babary, “On linearizing control of wastewater treatment processes,” Automaica, Vol. 32, no. 2, pp. 7539–44, 2000. 39. F. Liu, M. Farza, and M. M’Saad, “Nonlinear observers for state and unknown inputs estimation,” Int. J. Model Ident. Contr., Vol. 2, no. 1, pp. 33–48, 2007. 40. M. Farza, M. Triki, T. Maatoug, M. M’Saad, and B. Dahhou, “Unknown inputs observers for class of nonlinear systems,” in Proceedings of the 10th International Conference on Sciences and Techniques of Automatic Control and Computer Engineering, Hammamet, Tunisia, Dec. 2009, 12p. 41. M. Farza, M. M’Saad, M. Triki, T. Maatoug, and Y. Koubaa, “High gain observer for a class of non triangular systems,” Syst. Control Lett., Vol. 60, no. 1, pp. 27–35, 2011. 42. A. Isidori, Nonlinear Control Systems: An Introduction. 3rd ed. New York: Springer, 1995. 43. E. Busvelle, “Sur les observateurs des systmes non linaires,” Ph.D. dissertation, Universit de Bourgogne Dijon, France, 2004. 44. S. H. Said and F. M. Sahli, “A set of observers design to a quadruple tank process,” in IEEE International Conference on control applications, no. 17, San Antonio, Texas, USA, Sept. 2008.


Authors Feten Smida received her Engineering diploma and Master degree,in ElectricalAutomatic Engineering from the National School of Engineer of Gabes,Tunisia in 2008 and in Automatic and Signal Processing from the National School of Engineer of Monastir, Tunisia, in 2010, She is currently a Ph. D. candidate at Electrical Engineering in National Engineering School of Monastir. Her research interests are focused on nonlinear control and Unknown Inputs Observation. Corresponding author. E-mail: [email protected] Taoufik Ladhari received the Electrical Engineering diploma in 2003 from INSAT (Tunis), in 2004 he received his DEA in Automatic Systems from UPS-LAAS Toulouse, France, then the PhD degree in Process Engineering in 2007 from the National High School of Mines of SaintEtienne, France. He is currently Assistant Professor at ENIM. His research interests are in the control and state estimation of nonlinear systems, optimization of bioprocess, and biomedical system. E-mail: [email protected]

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Salim Hadj Said received his PhD and his academic accreditation in Electric Engineering from ENIT in 2009 and ENIM in 2015, respectively. He is currently an assistant professor of Automatic at Preparatory Institute for Engineering Studies of Monastir (IPEIM), Tunisia. His research interests include Robust Observation, Predictive and Back-Stepping Control. E-mail: [email protected] Faouzi M’Sahli received his BS and MS degrees from ENSET, Tunis, Tunisia, in 1987 and 1989, respectively. In 1995, he obtained his PhD degree in Electrical Engineering from ENIT, Tunisia. He is currently a professor of Electrical Engineering at National School of Engineers, Monastir, Tunisia. His research interests include modeling, identification, and predictive and adaptive control of linear and nonlinear systems. He has published over 80 technical papers and co-author of a book Identification et commande numérique des procédés industriels, Technip editions, Paris. E-mail: [email protected]