High-Gain Nonlinear Observer for the Fault Detection Problem: Application to a Bioreactor

IFAC

Copyright I!:> IF AC Nonlinear Control Systems, St. Petc:rsburg, Russia, 2001

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HIGH-GAIN NONLINEAR OBSERVER FOR THE FAULT DETECTION PROBLEM: APPLICATION TO A BIOREACTOR R. Martinez-Guerra t, 1 , R. Garrido t and A. Osorio-Mir6nt,f

t Automatic

Control Department CINVESTA V-IPN P.O. Box 14-740. 07300 Mexico, D.F. f ESIQIE-IPN, Edificio 7, U. P. A. L. M., Zacatenco C.P. 07738, Mexico, D. F., MEXICO rguerra(garrido, aosoriom)@ctrl.cinvestav.mx Abstract: We propose a high-gain nonlinear observer to solve the Fault Detection Problem in a biotechnological process, in addition, the parametric and state estimation is obtained as welL The construction of High-Gain Nolinear Observers which are based upon the Multioutput Generalized Observability Canonical Form (MGOCF) is the main ingredient in our approach. We define the failure mode and we introduce the concept of algebraically observable failure mode. The proposed observer together with the failure mode rule is the, so called, residual generator in the fault detection problem and its performance is shown through numerical simulations. Copyright © 2001IFAC Keywords: Fault Detection, Bioreactor, High-Gain Nonlinear Observer, MGOCF.

process or the ambient conditions. In the general case, there are many different approaches to solve the Fault Detection Problem, some works address the parametric and state estimation problem of chemical systems and using this information it is possible to identify the causes of the faults (Kabore et al., 2000; Martfnez-Guerra et al., 2000; Watanabe and Himmelblau, 1983). For biochemical reactors, the estimation problem is complicated due to growth model inadequacy and lack of proper sensors, therefore, the bioreactor state variables and parameters must be estimated without making use of growth models and from limited number of measurements (Bastin and Dochain, 1990; Farza et al., 1998; Perrier et al., 20(0). In this sense, there are several authors who have proposed solutions to the estimation problem. In linear systems, there exist an exact solution given by the Kalman filter. For nonlinear system a solution to estimation problem is based

1. INTRODUCTION

In the last years, the continuous culture fermentation process has become attractive because it allows high productivity, uniform product quality and easy process control (Bastin and Dochain, 1990). It is well known that the behavior of biotechnological processes are complicated, for example their dynamics are strongly nonlinear and nonstationary; the model parameters do not remain constant due mainly to metabolic variations and physiological and genetic modifications. A key question in bioprocess control is how to detect faults in the process. The term fault means process degradation or degradation of the equipment performance caused by some change in the physical characteristics of the process, the input 1 Work supported by CONACyT (Mexiw), under Grant 31982-A.

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i) k is a subfield of L, ii) the derivation of k is the restriction to k of the derivation of L.

on the extended Kalman filter, but there is not a guarantee of convergence and stability of this solution; another solution use a high-gain nonlinear observer-based parametric and state estimator (Farza et al., 1998). In this work, we consider the last case applied to the Fault Detection Problem with the differential algebraic approach which allows to define algebraic observability and gives an estimation of the state and the parameters through observers design for systems represented by differential algebraic equations.

Example 1. Q, R , C are trivial fields of constants. Example 2. R(at) / R is a differential field extension R ~ R(at) , at is a solution of P(x) = x a = 0 (a is a constant) .

In this paper, the main contribution is the construction of residual generators using algebraic techniques and the Multioutput Generalized Observability Canonical Form (MGOCF) (MartinezGuerra and De Loon-Morales, 1994) to achieve parameter and state estimate to solve the Fault Detection Problem. The proposed design procedure has the following features: first, we extend the state vector to include the unknown parameters (Tornambe, 1989) , second, we transform the system to the MGOCF, we construct the estimator, and, by means of its corresponding inverse, we obtain the residual generator in original coordinates. The residuals are then used to detect the faults, i.e. , by examining the residuals values between the output measured by instruments and the output generated by a nonlinear state observer designed under a normal operation. In absence of a fault, the residual goes to zero (normal operation) , i.e., the output signal error tends to zero as t tends to infinite. If a fault occurs the . resIdual shows a nonzero value, detecting the 'fault (not normal operation). We have considered the concept of algebraically observable failure and the failure modes rule introduced in (Martinez-Guerra et al., 2(00). The remainder of this paper is organized as follows: We first introduce some basic definitions on differential algebra in section 2. In section 3, we deal with the statement of the problem and the full order high-gain observers synthesis which we consider as the residual generators. Application of the synthesis algorithm to bioreactors in the fault detection problem is described in section 4. In section 5 some numerical results on residual generators applied to bioreactors are given. Finally, in section 6 we will close the paper with some concluding remarks.

Definition 2. A dynamics is a finitely generated differential algebraic extension G / k(u) (G = k (u , ~) , ~ E G). Any element of G satisfies an algebraic differential equation with coefficients which are rational functions over k in the components of u and a finite number of their time derivatives. Example 3. The input-output system y + iJ + sin y - u = 0 which is equivalent to the system ~ : Xl = X2, X2 = -X2 - sinxl + u , and Y = Xl, can be seen like a dynamics of the form R(u, y ) / R(u) , where G = R(u,y) , y E G and k = R. Definition 3. Let a subset {u, y} of G in a dynamics G/k(u) . An element in G is said to be algebraically observable with respect to {u, y} if it is algebraic over k(u , y) . Therefore, a state x is said to be algebraically observable if, and only if, it is algebraically observable with respect to {u ,y} . A dynamics G/k(u) with output y in G is said to be algebraically observable if, and only if, any state has this property. Example 4. The system ~ is algebraically observable, since, Xl and X2 satisfy two differential algebraic polynomials with coefficients in R, i.e., Xl and X2 are algebraic elements in R(u, y), that is to say Xl - Y = 0 and X2 - Y = o. The algebraic observability means that the differential field extension G/k(u,y) is algebraic, i.e., the whole differential information is contained in k(u). According to the theorem of the differential primitive element, there exist p elements y = (Yl" ' .,Yk ," "yp) and an integer 1/k with 0 ~ 1/k ~ n and 1 < k < l1k p, such that ) is algebraically d;ende~ on y k, y(l) (11.- 1) k " "'Yk ,u, ...,u (1/)' ,I.e., Yk(11.)

Yk

-

L (

(1)

(11.- 1)

k Yk'Yk ""'Yk

,u, ... ,u (1/)) ,

where Lk is a polynomial of its arguments.

Yk

i - ); Let be the coordinates changes: ~~l1k) = 1= 1,1/1 + 1, 1/1 +1/2 + 1, ... ,171 +112 + ... +1Jp -1 + 1; where L 1/k = n; 1 ~ i ~ 1/1 + ... + 1Jp = n ;

2. BASIC DEFINITIONS

We start introducing some definitions and notation (Martinez-Guerra and De lOOn-Morales 1994) .

'

Definition 1. Differential field extension L / k is given by two differential fields k and L, such that:

l

l::;k::;p

and 1Jl ~ '72 ~ ... ~ 1J p ~ n ; where the integers 1Jl, ... , 1Jp are called the algebraic observability indices (Martinez-Guerra and De Le6nMorales, 1994), then we can write a local state space representation in the special form of the

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Multioutput Generalized Observability Canonical Form (MGOCF) :

If the system (3) can be transformed to the MGOCF, then it is possible to construct an observer. Choosing the output vector Y = (Yll Y2 f, the nonlinear system (3) is carried out into the MGOCF, with a finite number of the output time derivatives, we establish the following proposition: Proposition 1. Let system (3) be given. Choosing the output vector

2:1$k :$ P '7k

2:

Yk =

m

aixi

+ 2:.Bj U j;

i=n-'1k+ 1

ai,.Bj E k(u)

j

where u = (U1, .. ., urn) and k( u) denotes the differential field generated by k and u and their time derivatives. System (3) is transformed to the MGOCF with output injection and a finite number of the output time derivatives

~ = A~ + lIt (~ , u)

L

1/k

:s; i :s; rh + ... + 1/p = n ; = n, 1/1 :s; 1/2 :s; ... :s; 1/ p :s; n (1)

for all v

~

+ ~(u , y)

y=C~

1

O.

(4)

where, lIt (~, u) and ~ (u, y) are nonlinear vectors and the latter depends on the output injection and

u- _ - ( u, ... , u (v))

A~ [1'

3. STATEMENT OF THE PROBLEM We consider the following nonlinear systems:

x(t)

=

I(x, j.t, u)

yet) = hex , u)

(2) Ak =

where x = (X1, ... , x n )T E R;" l is a state vector, j.t E R;"2 is a parameter vector, u = (U1 , ... , urn)T E J?:Y' is a input vector, Y E RP is the output measured vector, I and h are assumed to be analytical vector functions. System (2) can be transformed in the so-called MGOCF (MartinezGuerra and De Le6n-Morales, 1994). In (2) the parameters vector j.t is unlalOwn and it can be seen as a new state variable, then, we can estimate it by extending the state vector with the unknown parameters vector, that is to say: Xe = (xT,j.tT)T (TornamM, 1989). Thus, system (2) can be immersed in a new state space Rn , with n = n1 +n2,

Xe(t) = le(xe , u) yet) = he(xe, u)

(3)

We suppose that the system (3) is universally observable in sense of definition 3, with external behavior given by equations of t he form: Yk''1k) = (1) ('1k- 1 ) h L' - L k ( Yk ' Yk '''' ' Yk , u, ... , u , were k IS a polynomial of its arguments.

(1/))

1J

~ 1 ~ ... ~ 1}

00 0... i [

0 < 1/k < n 1/k , 1 ;; k 5:-p

000 .. · 0

lIt (~, u) = [lItd~l ' u) .. .

f

~ (u, y) = [~du, Y1) ... ~P (u, Yp)

f

IItp (~p , u) IItd~ , u) = Col (0 .. · 0 1Pd~, u)) 1Pk (~ , u) = -Ld~, u) , ~ = (~1 ... ~n) ~k (U , y) = Col (0 .. · 0 9k (U,Yk))

C = [( Cl .. . 0) .. · (0 .. . C p

Ck =

)

f

(1 0 ... 0)

1)}

Let {~7k = Yk = {~i" ~~k} be a 1/k -finite differential trascendence basis, with Yk = ~ik , (i = l) , I = 1,1/1 + 1,1/1 + 1/2 + 1, .. . ,771 + 1/2 + '" + 1/p-l + 1; 1 :s; k :s; p, where 1/k , 0 :s; 1/k :s; n, is an integer such that Yk'1k) is dependent on Yk, Yk1 ) , ... , Yk'1k- 1 ), u, ... , u(I/) . The coordinates transformation can be rewritten as:

Proof.

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i

... ,

Lemma 2. (see (Martfnez-Guerra and De 1OOnMorales, 1994)) The system

(i'

= Y1 c'h _ y(1)

'>2

Tl1

-

1

.

1

~'11

'>'11

= y('11 -1) 1

(6)

TIp

is an exponential observer for system (4), where () k E .R+ determines the desired convergence rate and Sk is a matrix that is in a linear group of symmetric positive definite matrices, and the following assumptions are assumed:

AI) 'Ilk ((k, U, ... , u(v)) is locally Lipschitz with respect to (k and uniformly with respect to u and their time derivatives. A2) u and their time derivatives up to n are bounded. The observation error is given by Ek = (k - ~k ' and IIEkll :S Ke- 9kt . Remark 2. The dynamical system (6) along with xe

= C1(~,u)

(7)

constitute an exponential observer for the system (3), where u is considered as good input. which can be seen as in (1) or (4) . • Remark 1. The system (3) can be carried out into MGOCF by means of (=((xe ,u)

5. APPLICATION TO THE FAULT DETECTION PROBLEM

(5)

5.1 The Process Model

where ((xe, u) is a block matrix of nonlinear functions (the size of each block is TI k, 1 :S k :S p) with entries in k(u).

We take up the example of a cell culture described in (Perrier et al., 2000). We consider that the system dynamical model is given by the following balance equations:

Since this relation (5) depends on u , there exist inputs u for which the block matrix ((xe, u) is singular. In the same manner, there are also values of u such that the inverse transformation Cl (( , u) is singular. In this work is considered the class of inputs given by the following definition:

S = -k1XtLR - k4XtLF + SoD t = k5XtLF - LD X=XtLR+XtLF-XD Y1 = S; Y2 = L

SD

(8)

where S, L, and X are the substrate, product and biomass concentration, respectively, kj (j = 1, 4, 5) are the yield coefficients, So is the input glucose concentration (gjl), D is the dilution rate (h-l) , i.e., the ratio of the influent flow rate over the reaction medium volume, and Yl and Y2 are the outputs system. The kinetics expressions for both specific growth rates are given by (Perrier et al. , 2000): tLR = tLmax,l K;+S K~!t-L' and tLF = tLmax,2 K;+S' where tLmax,i' i = 1, 2 are the maximum specific growth rate for both respiration and fermentation reactions, respectively, KR and KF are the saturation constants, and KL the inhibition constant.

Definition 4. An input u is called a good input if u is an input to the system for which (5) is nonsingular.

Having studied the observer construction, we consider that the entries which we use are good inputs. 4. OBSERVER SYNTHESIS The following lemma describes the construction of a nonlinear observer for the multi-output system (4) which is algebraically observable.

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We want to estimate the specific growth rates J.LR and J.LF, then we take account equations for S and L of system (8) and we extend the state vector using the unknown parameters vector, i.e. , Xe = (xT , J.L T) T, which we assimilate as states with no dynamics.

biotechnological process is given by the following differential algebraic equations:

S=

-kIXf.LR - k4Xf.LF i= ksXJ.LF - LD

+ SoD - SD -

elVI

YI = S

Y2 = L (10) where el = D. Thus, we know the state and parameters of the system, and we are able to solve the fault detection problem with the designed observer-based residual generator. The failure mode rule is: VI = 0, there is not failure , and VI i- 0, there is failure, VI ER. The concept of algebraically observable failure mode is given by the following remark

5.2 Algebraic Observability and MCOCF We can see that the extended system is algebraically observable in sense of definition 3, that is to say, the output system and its time derivatives satisfy an algebraic polynomial with coefficients in R(u, y) . For the output YI = S, we have S - YI = 0, and rnax , y,j S DD k I X f.LR + k4 X (iL (Krtvd 0 + YI + Yl' = 0 , whereas , for the output Y2 = L , L - Y2 = 0,

-ksXf.LF

Remark 3. The failure mode VI E R , is algebraically observable if it satisfies the differential polynomial with coefficients in R(u,y) , i.e.,

+ Y2 D + Y2 = O.

Here, we consider the inputs class D(t) such that Dmin ~ D ~ Dmax , where Dmin and Dmax are such that S ~ tIJ L ~ t2 , and X ~ t3, for all ti > 0, 1 ~ i ~ 3, enough small, and the extended state evolves all the time within a differential field k(u , Y), which is the interest domain of the system.

Then, the residual generator constructed is given by (9) with the outputs system YI = S and Y2 = L, and the following failure mode rule in terms of the residual: rl =1= 0 , the failure VI occurs , rIJ VI E R and rl = 0 , the failure VI not occurs , given the dependence of the fault on the residual output error, where we have defined the residual as rl = S - 8.

The extended system can be transformed in the MGOCF expressed by equation (4). Here, the time derivatives y(v) are identically equal to zero, for all 11 ~ 1, and choosing the output vector (YI , Y2)T = (S, L)T, we have the following relationship: ~1 = S, ~2 = -k I X f.LR- k4X J.LF+SoDSD, ~3 = L, and ~4 = kSXf.LF -LD. From which, we obtain the MGOCF: ( = A~ + W (~ , u), i.e., (1 = ~2 ' ~2 = -~2D, (3 = ~4 ' ~4 = -~4D, and the outputs YI = ~I and Y2 = ~3 '

6. NUMERlCAL RESULTS

We verify the performance of the residual generator by simulation of system (10) together with the residual generator (9). VI was chosen nonzero for t E [55,57] h, that is to say, that the simulated failure in So occurs when t = 55 h (VI = So * a3 , with a3 = 1) and stop when t = 57 h (a3 = 0) . In Figure 1 we present the residual rl when u = D = Fa/V, and we show the estimation of state variables. We can see that the failure in the system gives a residual value rl nonzero. The process model (8) is simulated by considering the kinetic expressions of Monod's law type for both specific growth rates. We reconstruct J.L R and f.L F with the estimator (9), from measurements of S and L obtained from simulation.

The exponential observer for the system transformed is given in original coordinates by

5 = -k1 XfJ,R -

k4XfJ,F

+ SoD - 8 D+

28(S-8) i

iD + 28(L - i)

= ksXfJ,F -

~R=-k:X(82+28D)(S-8)k4 2 kl k X (8 AF =

ks~ (8

2

+ 28 D) (L -

(9)

A

L)

+ 28D) (L - i)

5.3 The Fault Detection Problem

The initial conditions for the state variables are the following: S(O) = 21 g/l, L(O) = 0.13 g/l, X(O) = 0.18 (10 6 ) cells/m!. The dilution rate is as follows: D = Fa/V, where V is given by mass balance equation V = Fa . The parameter values used for simulation are given in Table 1 (Perrier et al., 2000):

Now, with the identification problem already solved, for the fault detection problem study we define the term VI as the failure mode corresponding to the input glucose concentration So, with VI E R . Then, the mathematical model of the

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.... .... "

Table 1. Bioreactor parameters itmax,l itmax,2

= 0.045 h-

kl = 1.7, V(O) = 19 1,

Fo

...

...,

= 0.055 hI, KR = 10 gll, KL = 50 gll

= 0.0005

1

,

KF

= 10 gll

k4 = 8.5, k5 = 17 Vj = 19.05 1

11h,

So

RESIDUAL

'," s· §

...

= 3300 gll

·'IIl

....

.

u.

•.. t h

STATE ESTIMATION

We have considered that the value of the biomass concentration X is given by an asymptotic observer , which takes the form given in (Perrier et al., 2(00). For the estimated specific growth rates the initial conditions are: ilR(O) = ilF(O) = 0 11h and for the estimated states S(O) = t(O) = 0 g/l: The simulation has a time interval of 150 h, 100 h with a constant flow rate and 50 h under batch operation. The gain parameter in the high-gain nonlinear observer is fixed as () = 10.

7. CONCLUDING REMARK The Fault Detection Problem is considered by means of the construction of a residual generator. First, we study the parameter and state estimation problem based in nonlinear observers. With the differential algebraic approach we have designed the observer-based estimator that asymptotically converges to the real parameters and the real states. The construction of the residual generator is based upon two outputs, using the Multioutput Generalized Observability Canonical Form (Martfnez-Guerra and De Le6n-Morales, 1994) . We have defined the failure mode and we have introduced the concept of algebraically observable failure mode.

d ~

"

s._ L._

S.S

x,-.

J)

10

,,

x,i

.

"

"

1«1

l h

Fig. 1. Numerical simulation of residual generator in the Fault Detection Problem. Martfnez-Guerra R., Garrido R., and OsorioMir6n A. Fault Detection in CSTR Using Nonlinear Observers. lASTED Int. Conf. on Intelligence Systems and Control 2000 (ISC 2000), Honolulu, Hawaii, USA. pp. 262-267, 2000. Perrier M., Feyo de Azevedo J., Ferreira E. C., and Dochain D. Tuning of Observer-Based Estimators: Theory and Application to the Online Estimation of Kinetic Parameters. Control Eng. Practice, Vol. 8, pp. 377-388, 2000. Tornambe A. Use of Asymptotic Observers Having High-Gain in the State and Parameter Estimation. Proc. 28th. CDC, Tarnpa, Florida, USA, pp.1791-1794, 1989. Watanabe K., and Hirnrnelblau D. M. Fault Diagnosis in Nonlinear Chemical Process. Part1, Theory. Part 11, Application to a chemical Reactor. AIChE Journal, Vol. 29, No. 2, pp. 243261, 1983.

8. REFERENCES Bastin G. and Dochain D. On-line Estimation and Adaptive Control of Bioreactors. Elsevier, Amsterdam, 1990. Farza M. ,Busawon K., and. Hammouri H. Simple Nonlinear Observers for On-line Estimation of Kinetic Rates in Bioreactors. Automatica, Vol. 34, No. 3, pp. 301-318, 1998. Kabore P., Othrnan S., McKena T. F., and Hammouri H .. Observer-Based Fault Diagnosis for a Class of Non-Linear Systems: Application to a Free Radical Copolimerization Reaction. Int. J. Control, Vol. 73, No. 9, pp. 787-803, 2000. Martfnez-Guerra R., and De Le6n-Morales J. Observers for a Multi-input Multi-output Bilinear Systems Class: A Differential Algebraic Approach. J. of Math. and Computer Modelling, Vol. 20, pp. 125-132, 1994.

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