AICME II abstracts

Control and optimization in ecological problems

A tunable multivariable non-linear robust observer for biological systems R. Salazar-Pe˜ na1 , V. Alcaraz-Gonz´alez2 , J.L. Gouz´e3 and V. 4 ´ Gonz´alez-Alvarez . The lack of reliable information of key variables to properly operate wastewater treatment processes has brought about the development of new state estimation schemes. A number of techniques has been proposed to monitor such variables that are used to improve the operation and control of them. In this paper, we propose a multivariable version of the robust nonlinear observer developed by Gouz´e and Lemesle [1]. This observer is applied to an anaerobic digestion process for wastewater treatment [2] that can be described by the following nonlinear system: x˙ 1 (t) = C1 f (x(t), t) + A11 (t)x1 (t) + A12 x2 (t) + b1 (t) (1) x˙ 2 (t) = C2 f (x(t), t) + A21 (t)x1 (t) + A22 x2 (t) + b2 (t)

(2)

y(t) = x2 (t)

(3)

where x(t) is the state vector, while b(t) is gathering the process inputs. The nonlinearities of the dynamical system (i.e. reaction rates) are represented by f (x(t), t) whereas A(t) is the state matrix. The partitions induced by x1 (t) (estimated states) and y = x2 (t) (measured states) have adequate dimensions. For the model given by (1), (2) y (3) we propose the following observer: ˜ (t)C f˜(ˆ ˜ (t)b(t) zˆ˙ (t) = N x(t), t) + Z(t)ˆ z (t) + Y (t)y(t) + N

(4)

˜ (0)ˆ zˆ(0) = N x(0)

(5)

N1−1 (ˆ z (t)

(6)

x ˆ1 (t) =

− Θ(t)N2 y(t))

1 Departamento de Ingenier´ ıa Qu´ımica, Universidad de Guadalajara. M. 1451, 44860, Guadalajara, Jalisco, Mexico (e-mail: [email protected]). 2 Departamento de Ingenier´ ıa Qu´ımica, Universidad de Guadalajara. M. 1451, 44860, Guadalajara, Jalisco, Mexico (e-mail: [email protected]). 3 INRIA, COMORE Project BP93, 06902 Sophia-Antipolis Cedex, [email protected]). 4 Departamento de Ingenier´ ıa Qu´ımica, Universidad de Guadalajara. M. 1451, 44860, Guadalajara, Jalisco, Mexico (e-mail: [email protected]).

03-Sal-a

Control and optimization in ecological problems

AICME II abstracts

˜ (t) = [N1 Θ(t)N2 ], N1 is any square invertible matrix, N2 = where, N § −N1 C1 C2 , being C2§ the generalized pseudo-inverse of C2 , Θ(t) is a diagonal matrix with property limt→∞ Θ(t) = I. which is the actual tuning parameter that determines the rate of convergence of the proposed nonlinear observer. f˜(ˆ x, t) is the best approximation of the nonlinearities f (x(t), t). Z(t) = (N1 A11 + ˙ Θ(t)N2 A21 (t))N1−1 , Y (t) = N1 A12 (t) + Θ(t)N2 A21 (t) + (Θ(t) − Z(t)Θ(t))N2 . This observer has the advantage of choosing a variable rate of convergence, and, as Θ(t) → I, the knowledge of the nonlinearities is no longer required. Furthermore, under structural considerations on the aformentioned model (easy to verify on biological wastewater treatment process [3]), this observer converges exactly to Bastin and Dochain’s asymptotic observer [4].

References [1] J. L. Gouz´e and V. Lemesle, A Bounded Error Observer with adjustable rate for a class of Bioreactor Models Proc. ECC01 Porto Portugal, 2001. [2] O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi and J. P. Steyer, Dynamical model development and parameter identification of an anaerobic wastewater treatment process Biotech. Bioeng. 75 424–438, 2001. [3] V. Alcaraz-Gonz´ alez, J. Harmand, D. Dochain, A. Rapaport, J. P. Steyer, C. Pelayo Ortiz, and V. Gonz´ alez-Alvarez, A Robust Asymptotic Observer for Chemical and Biochemical Reactors. ROCOND 2003, Milan, Italy, June 25-27 2003. [4] G. Bastin and D. Dochain, On-line Estimation and Adaptive Control for Bioreactors Elsevier, Amsterdam, 1990.

Garc´ıa Barrag´ an Garc´ıa Barrag´ an France (e-mail: Garc´ıa Barrag´ an

03-Sal-b