A Sliding Mode Observer for Uncertain Nonlinear Systems Based on Multiple Models Approach

Hfaïedh Kaïs Dahech Karim Damak Tarak

Hfaïedh Kaïs, Dahech Karim and Damak Tarak. A Sliding Mode Observer for Uncertain Nonlinear Systems Based on Multiple Models Approach. International Journal of Automation and Computing, vol. 14, no. 2, pp. 202-212, 2017 doi:  10.1007/s11633-016-0970-x
Citation: Hfaïedh Kaïs, Dahech Karim and Damak Tarak. A Sliding Mode Observer for Uncertain Nonlinear Systems Based on Multiple Models Approach. International Journal of Automation and Computing, vol. 14, no. 2, pp. 202-212, 2017 doi:  10.1007/s11633-016-0970-x

doi: 10.1007/s11633-016-0970-x

A Sliding Mode Observer for Uncertain Nonlinear Systems Based on Multiple Models Approach

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    Author Bio:

    Karim Dahech received the electrical engineering diploma from the National School of Engineers of Gabes, Tunisia in 2002, the M. Sc. degree in automatic and industrial data processing, and the Ph. D. degree in electrical engineering from the National School of Engineers of Sfax, Tunisia in 2004 and 2009, respectively. He is currently an assistant professor of electrical engineering in the Higher Institute of Industrial Management of Sfax. His research interests include modeling, estimation and control of nonlinear systems. E-mail:dahechkarim@yahoo.fr

    Tarak Damak received the Electrical Engineering Diploma from the National School of Engineers of Sfax, Tunisia in 1989, the M. Sc. degree in automatic control from the National Institute of Applied Seciences of Toulouse-France in 1990, the Ph.D. degree from the University of Paul Sabatier of Toulouse, France in 1994. He is currently a professor in the Department of Mechanical Engineering of the National School of Engineers of Sfax-Tunisia. His research interests include distributed parameter systems, sliding mode control and observers and adaptive nonlinear control. E-mail:tarak.damak@enis.rnu.tn

    Corresponding author: Kaïs Hfaïedh received both the Engineering Diploma and the M. Sc. degree in automatic control from the National School of Engineers of Sfax, Tunisia in 2009 and 2011, respectively. He is currently a technologist in the Department of Electrical Engineering of Higher Institute of Technological Studies. His research interests include modeling, sliding mode control and observers for nonlinear systems. E-mail:kais.hfaiedh@gmail.com
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出版历程
  • 收稿日期:  2015-02-23
  • 录用日期:  2015-07-03
  • 网络出版日期:  2016-06-29
  • 刊出日期:  2017-04-01

A Sliding Mode Observer for Uncertain Nonlinear Systems Based on Multiple Models Approach

doi: 10.1007/s11633-016-0970-x
    作者简介:

    Karim Dahech received the electrical engineering diploma from the National School of Engineers of Gabes, Tunisia in 2002, the M. Sc. degree in automatic and industrial data processing, and the Ph. D. degree in electrical engineering from the National School of Engineers of Sfax, Tunisia in 2004 and 2009, respectively. He is currently an assistant professor of electrical engineering in the Higher Institute of Industrial Management of Sfax. His research interests include modeling, estimation and control of nonlinear systems. E-mail:dahechkarim@yahoo.fr

    Tarak Damak received the Electrical Engineering Diploma from the National School of Engineers of Sfax, Tunisia in 1989, the M. Sc. degree in automatic control from the National Institute of Applied Seciences of Toulouse-France in 1990, the Ph.D. degree from the University of Paul Sabatier of Toulouse, France in 1994. He is currently a professor in the Department of Mechanical Engineering of the National School of Engineers of Sfax-Tunisia. His research interests include distributed parameter systems, sliding mode control and observers and adaptive nonlinear control. E-mail:tarak.damak@enis.rnu.tn

    通讯作者: Kaïs Hfaïedh received both the Engineering Diploma and the M. Sc. degree in automatic control from the National School of Engineers of Sfax, Tunisia in 2009 and 2011, respectively. He is currently a technologist in the Department of Electrical Engineering of Higher Institute of Technological Studies. His research interests include modeling, sliding mode control and observers for nonlinear systems. E-mail:kais.hfaiedh@gmail.com

English Abstract

Hfaïedh Kaïs, Dahech Karim and Damak Tarak. A Sliding Mode Observer for Uncertain Nonlinear Systems Based on Multiple Models Approach. International Journal of Automation and Computing, vol. 14, no. 2, pp. 202-212, 2017 doi:  10.1007/s11633-016-0970-x
Citation: Hfaïedh Kaïs, Dahech Karim and Damak Tarak. A Sliding Mode Observer for Uncertain Nonlinear Systems Based on Multiple Models Approach. International Journal of Automation and Computing, vol. 14, no. 2, pp. 202-212, 2017 doi:  10.1007/s11633-016-0970-x
    • The knowledge of state variables of nonlinear systems is necessary in control systems field. Although, in most real systems, this knowledge is partial because some variables of the state can not be measured and some sensors are expensive. As a solution, the researchers turn to the observer which is a dynamical system able to estimate the unmeasured state variables. It is noticed that the design of an observer is preceded by the step of modeling. Indeed, the modeling operation involves the construction of mathematical model describing the dynamic behavior of the real system. So, many works are focusing on this issue in order to reach the correct representation of nonlinear systems.

      The multiple models is one of many tools of systems modeling. In the literature, methods of obtaining multiple models structure are studied. In fact, the work[1] treats the method based on the direct identification of model parameters (number of local models, the structure of weighting functions and data partitioning). In [2], the researchers present the method based on the linearization of an existing nonlinear model around the operating points. Other researchers use a method based on a transformation of the nonlinear system[3, 4]. As a definition, the multiple models approach is an interpolation of many submodels qualified as linear through activation functions. For each submodel, a local observer is synthesized. The interpolation of the local observers lead to the obtention of multiple observers.

      State estimation based on multiple models structure is widely studied in the literature. Works as [5-7] are interested in determining the estimation gains using Linear matrix inequalities (LMIs). Later, many researchers have been focusing on the design of multiple observers in order to satisfy desired performances. In fact, the authors in [8-11] show designs of robust multiple observers for unknown input/uncertain systems based on LMI tools. Other works as [12-15], treat the design of robust sliding mode multiple observers in order to overcome the effect of unknown inputs, uncertainties and disturbances.

      Most of works as [16-18], which deal with multiple observer design for uncertain systems using sliding mode techniques, assume that the uncertainties are bounded. The disadvantage of such assumption is the way of choosing the uncertainties' upper bounds. In order to overcome this limitation, our contribution comes to enhance and improve some related works like [19] which assume that the uncertainties are norm bounded.

      The objective of our study is to design a sliding mode observer for nonlinear system with time varying uncertainties supposed as norm bounded. To reach this aim, the multiple models approach is adopted in modeling such class of systems which leads to deal easily with the synthesis of the proposed observer by profiting from the linear tools like those used in [20-25].

      This paper is organized as follows. In Section 2, we give the problem statement. The structure of the observer is presented in Section 3. The proof of the estimator's convergence is demonstrated in Section 4. In Section 5, we present the simulation results and interpretations. A conclusion ends the paper.

    • Consider a nonlinear system described by

      $ \begin{align} &\left\{ \begin{aligned} \dot x &= f(x, u) \\ y&= Cx \end{aligned} \right. \end{align} $

      (1)

      where $ x\in {\bf R}^n $ , $ u\in {\bf R}^m $ and $ y\in {\bf R}^p $ are, respectively, the state vector, the input vector and the output vector of the system. $ C =[I_p 0_{p\times(n-p)}] $ and verifies $ CC^{\rm T} =I_p $ . $ C^{\rm T} $ is the transpose of matrix $C$ and it presents also its right pseudoinverse.

      The system (1) can be written under multiple model approach as follows:

      $ \begin{align} &\left\{ \begin{aligned} \dot x &= \sum_{i=1}^M \mu_i(z)(A_ix+B_iu) \\ y&= Cx \end{aligned} \right. \end{align} $

      (2)

      where $A_i$ and $ B_i $ are matrices with appropriate dimensions. $z=[z_1, z_2, \cdots, z_r] $ is considered as the vector of premise variables. The activation functions $ \mu_i(z) $ verify the convex property:

      $ \begin{align} &\left\{ \begin{aligned} \sum_{i=1}^M \mu_i(z)=1 \\ 0\leq \mu_i(z)\leq 1. \end{aligned} \right. \end{align} $

      (3)

      The aim of this paper is to develop a sliding mode observer for the reconstruction of unmeasured variables of uncertain nonlinear system described by the following multiple models:

      $ \begin{align} &\left\{ \begin{aligned} \dot x &= \sum_{i=1}^M \mu_i(z)((A_i+\Delta A_i)x+B_iu) \\ y&= Cx \end{aligned} \right. \end{align} $

      (4)

      where $\Delta A_i$ represent the variation of parameters and verify the following hypothesis:

      Hypothesis 1. We suppose that the uncertainties are norm bounded type:

      $ \begin{align*} \Delta A_i=M_i^a F_a N_i^a \end{align*} $

      where $ F_a\in {\bf R}^{k\times l}$ respects the following constraint:

      $ \begin{align*} F_a^{\rm T} F_a\leq I_{l\times l}. \end{align*} $

      It should be noted that $F_a$ defines the structure of parameters' uncertainties or variations.

    • Consider the multiple observer presented in [5] which has the following form:

      $ \begin{align} &\left\{ \begin{aligned} \dot{\hat{x}}&= \sum_{i=1}^M \mu_i(z)(A_i\hat{x}+B_iu+L_i(y-\hat{y})) \\ \hat{y}&= C\hat{x}. \end{aligned} \right. \end{align} $

      (5)

      The estimation gains $L_i\in {\bf R}^{n\times p}$ are determined by solving the following LMI's :

      $ \begin{align} (A_i-L_iC)^{\rm T}P+P(A_i-L_iC) < 0, ~~{\rm for}~~i=1\cdots M. \end{align} $

      (6)

      The structure of the proposed observer is an extended form of (5) and it is given by

      $ \begin{align} &\left\{ \begin{aligned} \dot{\hat{x}}&= \sum_{i=1}^M \mu_i(z)(A_i\hat{x}+B_iu+L_i(y-\hat{y})+\alpha_i) \\ \hat{y}&= C\hat{x} \end{aligned} \right. \end{align} $

      (7)

      where ${\hat{x}}\in {\bf R}^n$ is the estimate state vector, $L_i\in {\bf R}^{n\times p}$ are the estimation gains. $\alpha_i$ represent the sliding mode gains which play their role in compensating the uncertainties $\Delta A_i$ for each local model.

      Theorem 1. The error estimation between the system (4) and (7) converges asymptotically to zero if there exist symmetric positive definite matrix $P\in {\bf R}^{n\times n}$ and matrices $W_i\in {\bf R}^{n\times p}$ , and a positive scalar $\varepsilon$ such that for $i=1, \cdots, M $ :

      $ \begin{align} &\begin{bmatrix} A_i^{\rm T}P+PA_i-C^{\rm T}{W_i}^{\rm T}+W_i C+2{\varepsilon}^{-1}{N_i^a}^{\rm T}N_i^a&PM_i^a\\ {M_i^a}^{\rm T}P&-\varepsilon^{-1}I \end{bmatrix} < 0, \notag\\ &\quad \varepsilon >0 \end{align} $

      (8)

      where $C$ and $M_i^a \; \rm{and} \; N_i^a $ are known matrices.

      And the following conditions are fulfilled

      $ \begin{align} &\left\{ \begin{aligned} {\rm If}\ r\neq 0, {\rm then}\ \alpha_i&=\varepsilon^{-1}{\hat{x}}^{\rm T}{N_i^a}^{\rm T}N_i^a{\hat{x}}P^{-1}C^{\rm T}{s\over {\parallel s\parallel ^2}} \\ {\rm If}\ r=0, {\rm then}\ \alpha_i&=0 \end{aligned} \right. \end{align} $

      (9)

      where $s$ is the sliding mode surface. The gains of the observer are derived from $ L_i=P^{-1}W_i $ .

    • In the proof of theorem, the following lemma is used:

      Lemma 1. For every two matrices $X$ and $Y$ with appropriate dimensions, the following property holds:

      $ \begin{align} X^{\rm T}Y+Y^{\rm T}X\leq \beta X^{\rm T}X+\beta^{-1} Y^{\rm T}Y, ~~~\beta >0 \end{align} $

      (10)

      Lemma 2[26]. For a symmetric matrix $M$ partitioned into blocks:

      $ \begin{align} M &= \begin{bmatrix} Z&S\\S^{\rm T}&R \end{bmatrix} \end{align} $

      (11)

      where both $Z$ and $R$ are symmetric and square. Assume that $R>0$ , then the following properties are equivalent:

      $ \begin{align*} &\quad M>0\\ &\quad Z-SR^{-1}S^{\rm T}>0. \end{align*} $

      In order to prove the stability of the observer, let us define the state and output errors:

      $ e= x- \hat{x} $

      (12)

      $ r= y- \hat{y}. $

      (13)

      The dynamics of estimation error is given by

      $ \dot e= \dot x- \dot {\hat{x}} $

      (14)

      $ \dot e= \sum_{i=1}^M \mu_i(z)((A_i-L_iC)e+\Delta A_ix-\alpha_i) $

      (15)

      defining the sliding surface

      $ \begin{align} s=r=Ce=C(x-\hat {x}). \end{align} $

      (16)

      Considering the following candidate Lyapunov function:

      $ \begin{align} V=s^{\rm T}Qs \end{align} $

      (17)

      where $Q$ is a symmetric positive definite matrix and rank $(Q)=p$ . Using (15) and (16), its time derivative is given by

      $ \begin{align} &\qquad \dot V={\dot s}^{\rm T}Qs+sQ{\dot s}\notag\\ &\qquad \dot V=\sum_{i=1}^M \mu_i(z)(e^{\rm T}(A_i-L_i C)^{\rm T}C^{\rm T}QCe+\notag\\ &\qquad\quad x^{\rm T}{\Delta A_i}^{\rm T}C^{\rm T}QCe-\alpha_i^{\rm T}C^{\rm T}QCe+\notag\\ &\qquad\quad e^{\rm T}C^{\rm T}Q C(A_i-L_i C)e+\notag\\ &\qquad\quad e^{\rm T}C^{\rm T}QC\Delta A_ix-e^{\rm T}C^{\rm T}QC\alpha_i).\end{align} $

      (18)

      Supposing that $P=C^{\rm T}QC$ , one can obtain:

      $ \begin{align} &\dot V=\sum_{i=1}^M \mu_i(z)(e^{\rm T}((A_i-L_i C)^{\rm T}P+P(A_i-L_i C))e+\notag\\ &\qquad x^{\rm T}{\Delta A_i}^{\rm T}Pe +e^{\rm T}P\Delta A_ix-2e^{\rm T}P\alpha_i). \end{align} $

      (19)

      Applying the Lemmas 1 and 2, one can obtain:

      $ \begin{align} &\qquad x^{\rm T}\Delta A_i^{\rm T}Pe+e^{\rm T}P\Delta A_ix=x^{\rm T}(M_i^a F_a N_i^a)^{\rm T}Pe+\notag\\ &\qquad\quad e^{\rm T}PM_i^a F_a N_i^a x=e^{\rm T}PM_i^a F_a N_i^ax+\notag\\ &\qquad\quad x^{\rm T} {N_i^a}^{\rm T} F_a^{\rm T} {M_i^a}^{\rm T}Pe\leq\varepsilon e^{\rm T}PM_i^a {M_i^a}^{\rm T} Pe+\notag\\ &\qquad\quad {\varepsilon}^{-1}x^{\rm T} {N_i^a}^{\rm T}N_i^ax. \end{align} $

      (20)

      So, the derivative of the Lyapunov function can be obtained as follows:

      $ \begin{align} &\dot V=\sum_{i=1}^M \mu_i(z)(e^{\rm T}((A_i-L_i C)^{\rm T}P+P(A_i-L_i C)+\notag\\ &\qquad \varepsilon PM_i^a{M_i^a}^{\rm T}P)e +{\varepsilon}^{-1}x^{\rm T} {N_i^a}^{\rm T}N_i^ax-2e^{\rm T}P\alpha_i). \end{align} $

      (21)

      Replacing the state by its new expression:

      $ \begin{align} x= \hat{x}+e \end{align} $

      (22)

      one can obtain the following equality:

      $ \begin{align} &{\varepsilon}^{-1}x^{\rm T}{N_i^a}^{\rm T}N_i^ax={\varepsilon}^{-1}(\hat{x}+e)^{\rm T}{N_i^a}^{\rm T}N_i^a(\hat{x}+e)= \notag\\ &\qquad {\varepsilon}^{-1}{\hat{x}}^{\rm T}{N_i^a}^{\rm T}N_i^a\hat{x}+{\varepsilon}^{-1}{\hat{x}}^{\rm T}{N_i^a}^{\rm T}N_i^ae +\notag\\ &\qquad {\varepsilon}^{-1}e^{\rm T}{N_i^a}^{\rm T}N_i^a\hat{x}+{\varepsilon}^{-1}e^{\rm T}{N_i^a}^{\rm T}N_i^ae. \end{align} $

      (23)

      Now, using the Lemma 1 with $\beta=1$ , this leads to obtain:

      $ \begin{align} &{\varepsilon}^{-1}{\hat{x}}^{\rm T}{N_i^a}^{\rm T}N_i^a\hat{x}+{\varepsilon}^{-1}{\hat{x}}^{\rm T}{N_i^a}^{\rm T}N_i^ae+\notag\\ &\quad {\varepsilon}^{-1}e^{\rm T}{N_i^a}^{\rm T}N_i^a\hat{x}+ {\varepsilon}^{-1}e^{\rm T}{N_i^a}^{\rm T}N_i^ae \leq \notag\\ &\quad 2{\varepsilon}^{-1}{\hat{x}}^{\rm T}{N_i^a}^{\rm T}N_i^a\hat{x}+2{\varepsilon}^{-1}e^{\rm T}{N_i^a}^{\rm T}N_i^ae. \end{align} $

      (24)

      As a consequence, a new inequality can be obtained as

      $ \begin{align} &\dot V\leq \sum_{i=1}^M \mu_i(z)(e^{\rm T}((A_i-L_i C)^{\rm T}P+P(A_i-L_i C)+\notag\\ &\quad \varepsilon PM_i^a{M_i^a}^{\rm T}P+2{\varepsilon}^{-1}{N_i^a}^{\rm T}N_i^a)e +2{\varepsilon}^{-1}{\hat {x}}^{\rm T} {N_i^a}^{\rm T}N_i^a\hat {x}\notag\\ &\quad -2e^{\rm T}P\alpha_i). \end{align} $

      (25)

      According to the output error, two cases can be investigated:

      Case 1. If $r\neq0$ , then

      $ 2{\varepsilon}^{-1}{\hat {x}}^{\rm T} {N_i^a}^{\rm T}N_i^a\hat {x}-2e^{\rm T}P\alpha_i=0 $

      (26)

      $ \alpha_i=\varepsilon^{-1}{\hat{x}}^{\rm T}{N_i^a}^{\rm T}N_i^a{\hat{x}}P^{-1}C^{\rm T}{s\over {\parallel s\parallel ^2}}. $

      (27)

      So, the inequality becomes:

      $ \begin{align} &\dot V\leq \sum_{i=1}^M \mu_i(z)(e^{\rm T}((A_i-L_i C)^{\rm T}P+P(A_i-L_i C)+\notag\\ &\quad \varepsilon PM_i^a{M_i^a}^{\rm T}P+2{\varepsilon}^{-1}{N_i^a}^{\rm T}N_i^a)e). \end{align} $

      (28)

      Case 2. If $r=0$ , then $\alpha_i=0$ ,

      $ \begin{align} &\dot V\leq \sum_{i=1}^M \mu_i(z)(e^{\rm T}((A_i-L_i C)^{\rm T}P+P(A_i-L_i C)+\notag\\ &\quad \varepsilon PM_i^a{M_i^a}^{\rm T}P+2{\varepsilon}^{-1}{N_i^a}^{\rm T}N_i^a)e). \end{align} $

      (29)

      As a conclusion, the system converges asymptotically to zero if and only if:

      $ \begin{align} &(A_i-L_i C)^{\rm T}P+P(A_i-L_i C)+\notag\\ &\quad \varepsilon PM_i^a{M_i^a}^{\rm T}P+2{\varepsilon}^{-1}{N_i^a}^{\rm T}N_i^a < 0. \end{align} $

      (30)

      Supposing $W_i=PL_i$ , we obtain

      $ \begin{align} &A_i^{\rm T}P+PA_i^{\rm T}-C^{\rm T}W_i^{\rm T}-W_i^{\rm T}C+\notag\\ &\quad \varepsilon PM_i^a{M_i^a}^{\rm T}P+2{\varepsilon}^{-1}{N_i^a}^{\rm T}N_i^a < 0. \end{align} $

      (31)

      By using Schur Complement, cited in Lemma 2, the inequality can be written in the LMI form:

      $ {\small\begin{align}\label{aa} \begin{bmatrix} A_i^{\rm T}P+PA_i-C^{\rm T}{W_i}^{\rm T}+W_i C+2{\varepsilon}^{-1}{N_i^a}^{\rm T}N_i^a&PM_i^a\\ {M_i^a}^{\rm T}P&-\varepsilon^{-1}I \end{bmatrix} < 0. \end{align}} $

      (32)

      After determining $P$ and $W_i$ , the values of estimation gains are given by

      $ \begin{align*} L_i=P^{-1}W_i. \end{align*} $

      The sliding mode gains take the following expressions:

      $ \begin{align} &\left\{ \begin{aligned} {\rm If}\ r\neq 0, {\rm then}\ \alpha_i&=\varepsilon^{-1}{\hat{x}}^{\rm T}{N_i^a}^{\rm T}N_i^a{\hat{x}}P^{-1}C^{\rm T}{s\over {\parallel s\parallel ^2}} \\ {\rm If}\ r=0, {\rm then}\ \alpha_i&=0. \end{aligned} \right. \end{align} $

      (33)

      By using $CC^{\rm T}=I_p$ , the value of $Q$ can be deduced from $P$ as follows:

      $ \begin{align*} &P=C^{\rm T}QC\\ &CPC^{\rm T}=CC^{\rm T}QCC^{\rm T}\\ &Q=CPC^{\rm T}.\end{align*} $

      In conclusion, starting with $V=s^{\rm T}Qs >0$ , where $Q^{\rm T}=Q>0$ , we obtain $\dot V < 0$ . So, the estimation error converges asymptotically to zero.

    • In this section, the proposed design approach is applied to a model of denitrification process. This process is a bacterial culture of Pseudomonas denitrificas where biomass, $X$ , begins consuming acetic acid, $S_3$ , and the nitrate, $S_1$ , and rejects nitrites, $S_2$ . Then it continues to use acetic acid and nitrite product.

      The process is described by the following model[27]:

      $ \begin{align} &\left\{ \begin{aligned} \dot S_1 &= -y_{11}\mu_1X+D(S_{1in}-S_1) \\ \dot S_2 &=(y_{12}\mu_1-y_{22}\mu_2)X+D(S_{2in}-S_2) \\ \dot S_3 &=-(y_{13}\mu_1-y_{23}\mu_2)X+D(S_{3in}-S_3) \\ \dot X &=(\mu_1+\mu_2)X-k_dX-DX \\ \end{aligned} \right. \end{align} $

      (34)

      where $S_1$ , $S_2$ , $S_3$ and $X$ are respectively the concentrations of the respective species. $S_{1in}$ , $S_{2in}$ $S_{3in}$ are the respective supply of $S_1$ , $S_2$ and $S_3$ concentrations. $k_d$ is the mortality rate of the microorganisms. $D$ is the dilution rate, and the $y_{ij}$ denote yield coefficients and finally $\mu_1$ and $\mu_2$ are respectively the specific growth rates of the biomass on the acetic acid and nitrite and have the following expressions:

      $ \begin{align*} \mu_1= \mu_{1{\rm max}}{S_3\over{(S_3+k_{S_3})}}{S_1\over{(S_1+k_{S_1})}} \\ \mu_2=\mu_{2{\rm max}}{S_3\over {(S_3+k_{S_3})}}{S_2\over {(S_2+k_{S_2})}}. \end{align*} $

      Let's define the state vector $x$ , the input vector $u$ and the output vector $y$ of the system: $x=[S_1 ~~ S_2 ~~ S_3 ~~ X]^{\rm T}$ , $u=[S_{1in} ~~ S_{2in} ~~ S_{3in}]^{\rm T}$ and $ y=[S_1 ~~ S_2 ~~ S_3]^{\rm T}$ .

    • The multiple model is adopted as an approach in order to design the observer for state estimation. We employ the procedure presented in [3, 4]. Considering the process, we define the following nonlinearities as the premise variables:

      $ z_1=D $

      (35)

      $ z_2= {S_3\over{(S_3+k_{S_3})}}{S_1\over{(S_1+k_{S_1})}} $

      (36)

      $ z_3={S_3\over {(S_3+k_{S_3})}}{S_2\over {(S_2+k_{S_2})}}. $

      (37)

      The nonlinear model can be written in the following quasi-linear parameter varying (LPV) form:

      $ \begin{align} \dot x=A(z)x+B(z)u \end{align} $

      (38)

      with matrix $A(z)$ and $B(z)$ are expressed by using the premise variables:

      $ A(z) = \begin{bmatrix} -z_1 &0 &0 &a_{14}\\0 &-z_1 &0 &a_{24}\\0 &0 &-z_1 &a_{34}\\0 &0 &0 &a_{44} \end{bmatrix} $

      (39)

      $ B(z)= \begin{bmatrix} -z_1 &0 &0\\0 &z_1 &0\\0 &0 &-z_1\\0 &0 &0 \end{bmatrix} $

      (40)

      where

      $ \begin{align} &a_{14}=-y_{11}\mu_{1{\rm max}}z_2\notag\\ &a_{24}=y_{12}\mu_{1{\rm max}}z_2-y_{22}\mu_{2{\rm max}}z_3\notag\\ &a_{34}=-y_{13}\mu_{1{\rm max}}z_2-y_{23}\mu_{2{\rm max}}z_3\notag\\ &a_{44}=-z_1+\mu_{1{\rm max}}z_2+\mu_{2{\rm max}}z_3-k_d. \end{align} $

      (41)

      Each one of the premise variables can be expressed as

      $ \begin{align} z_j=F_{j1}z_{j1}+F_{j2}z_{j2}, {\rm for} j=1, 2, 3 \end{align} $

      (42)

      where

      $ \begin{align} &z_{j1}={\rm max}{\{z_j\}}\notag\\ &z_{j2}={\rm min}{\{z_j\}}\notag\\ &F_{j1}={{z_{j}-z_{j2}}\over{z_{j1}-z_{j2}} } \notag\\ &F_{j2}={{z_{j1}-z_{j}}\over{z_{j1}-z_{j2}}}. \end{align} $

      (43)

      The constant matrices $A_i$ and $B_i$ defining the 8 submodels are determined by using the matrices $A(z)$ and $B(z)$ and $z(j, i)$ , $i=1, 2$ and $j=1, 2, 3$ .

      $ \begin{align} &A_1=A(z_{11}, z_{21}, z_{31}), A_2=A(z_{11}, z_{21}, z_{32})\notag\\ &A_3=A(z_{11}, z_{22}, z_{31}), A_4=A(z_{11}, z_{22}, z_{32})\notag\\ &A_5=A(z_{12}, z_{21}, z_{31}), A_6=A(z_{12}, z_{21}, z_{32})\notag\\ &A_7=A(z_{12}, z_{22}, z_{31}), A_8=A(z_{12}, z_{22}, z_{32}) \notag\\ &B_1=B(z_{11}), B_2=B_3=B_4=B_1\notag\\ &B_5=B(z_{12}), B_6=B_7=B_8=B_5. \end{align} $

      (44)

      The activation functions have the following expressions:

      $ \begin{align} &\mu_1(z)=F_{11}F_{21}F_{31}\notag\\ &\mu_2(z)=F_{11}F_{21}F_{32}\notag\\ &\mu_3(z)=F_{11}F_{22}F_{31} \notag\\ &\mu_4(z)=F_{11}F_{22}F_{32} \notag\\ &\mu_5(z)=F_{12}F_{21}F_{31} \notag\\ &\mu_6(z)=F_{12}F_{21}F_{32} \notag\\ &\mu_7(z)=F_{12}F_{22}F_{31}\notag\\ &\mu_8(z)=F_{12}F_{22}F_{32}. \end{align} $

      (45)

      Finally the nonlinear model can be written in multiple model form:

      $ \begin{align} &\left\{ \begin{aligned} \dot x &= \sum_{i=1}^8 \mu_i(z)(A_ix+B_iu) \\ y&= Cx. \end{aligned} \right. \end{align} $

      (46)

      The multiple model used for denitrification process is modified in order to take account of the uncertainties affecting the parameters $ \mu_{1{\rm max}}$ and $\mu_{2{\rm max}}$ . These parameters appear in the coefficients $a_{14}$ , $a_{24}$ , $a_{34}$ , $a_{44}$ . So, the structure of $\Delta A_i$ will have the following form:

      $ \begin{align} &\Delta A_i = \begin{bmatrix} 0 &0 &0 &\Delta a_{14i}\\0 &0 &0 &\Delta a_{24i}\\0 &0 &0 &\Delta a_{34i}\\0 &0 &0 &\Delta a_{44i} \end{bmatrix} \end{align} $

      (47)

      $ \begin{align} &\Delta a_{14i}=-y_{11}\Delta \mu_{1max}z_{2i}\notag\\ &\Delta a_{24i}=y_{12}\Delta \mu_{1max}z_{2i}-y_{22}\Delta \mu_{2max}z_{3i} \notag\\ &\Delta a_{34i}=-y_{13}\Delta \mu_{1max}z_{2i}-y_{23}\Delta \mu_{2max}z_{3i}\notag\\ &\Delta a_{44i}=\Delta \mu_{1max}z_{2i}+\Delta \mu_{2max}z_{3i}. \end{align} $

      (48)

      As we have supposed that the uncertainties are of norm bounded type: $\Delta A_i=M_i^a F_a N_i^a$ where

      $ \begin{align} M_i^a &= \begin{bmatrix} -y_{11} &0\\y_{12}z_{2i} &-y_{22}z_{3i}\\-y_{13}z_{2i} &-y_{23}z_{3i}\\z_{2i} &z_{3i} \end{bmatrix}, \ F_a= \begin{bmatrix} \Delta \mu_{1{\rm max}}\\\Delta \mu_{2{\rm max}} \end{bmatrix}\notag\\ N_i^a &= \begin{bmatrix} 0 &0 &0 &1 \end{bmatrix}. \end{align} $

      (49)

      The nonlinear model of the denitrification process and the proposed observer are simulated under the initial conditions and parameters values given, respectively, by Tables 1 and 2.

      Table 1.  Initial conditions

      Table 2.  Parameters values

      The variations of some parameters are shown in Fig. 1.

      Figure 1.  Parameters variations

      The uncertainties $\Delta \mu_{1{\rm max}}$ and $ \Delta \mu_{2{\rm max}} $ are given as follow:

      $ \begin{align} &\left\{ \begin{aligned} \Delta \mu_{1{\rm max}} &=0.05\times \sin ({2\pi t\over{50}})\\ \Delta \mu_{2{\rm max}} &=0.02\times \sin ({2\pi t\over{50}}). \end{aligned} \right. \end{align} $

      (50)

      The evolution of the known inputs of the denitrification process are shown by Fig. 2.

      Figure 2.  Evolution of the known inputs

    • The resolution of the inequalities in (6) leads to obtain the matrices $P$ and $L_i$ :

      $ \begin{align*} &P=\begin{bmatrix} 1.3769 & 0 & 0 & 0.6245\\ 0 &1.3769 &0 &-0.0206\\ 0 &0 &1.3769 &0.5265\\ 0.6245 &-0.0206 &0.5265 & 2.0289\\ \end{bmatrix}\\ &L_1=\begin{bmatrix} -0.7252&-34.3741&10.8032\\ 28.3965&0.5580&4.3824\\ -9.3101&-9.5083&1.7835\\ 2.3929&13.2588&-3.7223\\ \end{bmatrix}\end{align*} $

      $ \begin{align*} &L_2=\begin{bmatrix} 1.3262&-78.9058&-7.2891\\ 65.3989&0.8037&8.5359\\ 7.0502&-19.8632&-0.5284\\ -2.1304&29.7114&2.3231\\ \end{bmatrix}\\ &L_3=\begin{bmatrix} 4.4452&21.6290&-29.7557\\ -18.1000&0.2394&-1.8690\\ 28.7130&5.1059&-3.4236\\ -9.0076&-8.0779&9.8941\\ \end{bmatrix}\\ &L_4=\begin{bmatrix} 2.0911&37.7025&-11.0522\\ -32.0737&0.1728&1.4103\\ 10.8780&3.4378&-1.0176\\ -3.8168&-12.5369&3.6023\\ \end{bmatrix}\\ &L_5=\begin{bmatrix} 2.4259&23.4950&-11.3405\\ -20.3910&0.3256&2.8628\\ 11.3901&-0.1363&-0.9472\\ -4.4007&-6.9890&3.6014\\ \end{bmatrix}\\ &L_6=\begin{bmatrix} 0.7023&-30.5561&-0.2003\\ 25.7169&0.5891&-0.3022\\ 0.5108&-3.7733&0.498\\ -0.6003&10.6502&-0.1785\\ \end{bmatrix}\\ &L_7=\begin{bmatrix} 0.3413&0.7088&1.2540\\ -0.4394&0.4212&-0.8422\\ -1.0247&1.0568&0.6864\\ 0.1955&-0.5875&-0.6704\\ \end{bmatrix}\\ &L_8=\begin{bmatrix} 0.4960&-0.1370&0.1012\\ 0.1339&0.4300&-0.0088\\ 0.0473&0.0050&0.5083\\ -0.1454&0.0020&-0.2047\\ \end{bmatrix}. \end{align*} $

      The system and the observer are simulated with the presence of the uncertainties described by (50). Figs. 3-6 show the Luenberger estimator tracking the real state. But, it is mentioned that the accuracy is bad because the observer presents a non robust behavior. Moreover, it should be highlighted that the non robustness appears very clear in Fig. 5 because it corresponds to the unmeasured variable.

      Figure 3.  Evolution of $S_1$ : Real value (solid) and estimated value (dotted)

      Figure 4.  Evolution of $S_2$ : Real value (solid) and estimated value (dotted)

      Figure 5.  Evolution of $S_3$ : Real value (solid) and estimated value (dotted)

      Figure 6.  Evolution of $X$ : Real value (solid) and estimated value (dotted)

    • Solving the inequalities described in (8) leads to obtain the numeric values of the matrices $P$ , $Q$ and $L_i$ and the scalar $\varepsilon$ :

      $ \begin{align*} &P= \begin{bmatrix} 1.0874 & 1.4427 & 0.1906 & 2.0136\\ 1.4427 & 2.0491 & -0.0427 & 2.2236\\ 0.1906 & -0.0427 & 3.1531 & 4.7511\\ 2.0136 & 2.2236 & 4.7511 & 10.3426\\ \end{bmatrix}\\ & \varepsilon=3.0410\\ & L_1=10^3 \begin{bmatrix} -0.7252& -34.3741& 10.8032\\ 0.0079& 0.0663& 0.7673\\ 0.0379& -0.3873& -0.0280\\ -0.0251& 0.2629&0.0719\\ \end{bmatrix}\\ &L_2= 10^3 \begin{bmatrix} -0.6291& -0.9309& 1.0621\\ 0.3696& 0.5799& -0.6273\\ -0.0896& -0.0285& 0.1177\\ 0.0841& 0.0696& -0.1261\\ \end{bmatrix}\\ & L_3= \begin{bmatrix} -95.9986& -449.0547& -373.7676\\ 158.0132& 278.7043& 152.0034\\ 124.3318& -15.9638& -114.2906\\ -72.4568& 34.7669& 92.4422\\ \end{bmatrix}\\ & L_4=10^3 \begin{bmatrix} -0.5643& -1.4401&-0.5381\\ 0.4208& 0.6655& 0.3393\\ 0.0422& -0.3853& -0.0333\\ -0.0001& 0.3142& 0.0470\\ \end{bmatrix}\\ & L_5= \begin{bmatrix} 245.1670& 330.3931& -109.9807\\ -113.6146& -120.0244& 21.6150\\ 73.8630& 134.9207& -57.7524\\ -57.2677& -100.5330& 43.2144\\ \end{bmatrix}\\ & L_6= \begin{bmatrix} -111.9137& -431.2422& -250.2063\\ 101.7201& 196.6560& 154.9055\\ 31.8156& -120.2406& -17.7648\\ -14.7305& 96.8756& 23.4912\\ \end{bmatrix}\\ & L_7= \begin{bmatrix} -22.8856& -20.7668& 334.8536\\ 11.2114& 59.7660& -205.3840\\ -8.3108& 67.1396& 29.9431\\ 5.8307& -39.6865& -34.8679\\ \end{bmatrix}\\ & L_8= \begin{bmatrix} -38.8176& -223.4972&-94.3856\\ 42.9070&109.0936& 152.0034\\ 20.1841&-52.3846& -114.2906\\ -10.9715& 44.0847& 7.1643\\ \end{bmatrix}.\end{align*} $

      Each sliding mode gain $\alpha_i$ is modified as follows:

      $ \begin{align} &\left\{ \begin{aligned} \alpha_i&=\varepsilon^{-1}{\hat{x}}^{\rm T}{N_i^a}^{\rm T}N_i^a{\hat{x}}P^{-1}C^{\rm T} {s\over {\parallel s\parallel ^2}+\delta}, ~~{\rm if}~~ s\neq 0 \\ \alpha_i&=0, ~~{\rm if}~~s=0. \end{aligned} \right. \end{align} $

      (51)

      The parameter $\delta$ is a small scalar and it is used in order to smooth out the discontinuity.

      The simulation of the system and the sliding mode observer took place with the same conditions of the observer of Section 5.3. Figs. 7-10 show the evolution of the four state variables of denitrification process. It is clear that the estimated values converge to the real ones with an improved accuracy which lead to say that the proposed observer presents a robust behavior via parametric variations in comparison with the observer of Section 5.3.

      Figure 7.  Evolution of $S_1$ : Real value (solid) and estimated value (dotted)

      Figure 8.  Evolution of $S_2$ : Real value (solid) and estimated value (dotted)

      Figure 9.  Evolution of $S_3$ : Real value (solid) and estimated value (dotted)

      Figure 10.  Evolution of $X$ : Real value (solid) and estimated value (dotted)

      In Fig. 11, additional curves of estimation errors are presented in order to support the previous figures of state variables in showing the performance of the proposed observer. It is noticed that $e_1$ , $e_2$ , $e_3$ and $e_4$ are the estimation errors of $S_1$ , $S_2$ , $S_3$ and $X$ , respectively.

      Figure 11.  Evolution of estimation errors

      Another simulation is performed with different initial conditions for the observer:

      $ \begin{align*} &\hat S_1(0)=0.7{\rm g/L}, ~~~\hat S_2(0)=0.01 {\rm g/L}, \\ &\hat S_3(0)=2.9{\rm g/L}, ~~~ \hat X(0)=0.2 {\rm g/L}. \end{align*} $

      The results are presented in Figs. 12-15 which show the evolution of the state variables, for both the system and the observer, and provide a zoom captured near the initial conditions in order to check the behavior of the estimator in the transient phase.

      Figure 12.  Curves of $S_1$ : Real value (solid) and estimated value (dotted)

      Figure 13.  Curves of $S_2$ : Real value (solid) and estimated value (dotted)

      Figure 14.  Curves of $S_3$ : Real value (solid) and estimated value (dotted)

      Figure 15.  Curves of $X$ : Real value (solid) and estimated value (dotted)

    • A robust sliding mode observer based on multiple models is developed for a nonlinear uncertain system. The design of observer gain is based on LMI's tools that guarantee the asymptotic convergence of estimation error. A sliding mode term is added in order to ensure the robustness against the uncertainties. A classic Luenberger multiples observer and the proposed multiple observer are applied to estimate the state of a denitrification process under uncertainties that affected the maximum of specific growth rates. The simulation results demonstrate the effectiveness of the proposed multiple observer.

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