HaarOrthogonalFunctionsBasedParameterIdentification ofwastewatertreatmentprocesswithdistributedparameters

Transcription

1 M. TODOROVA and T. PENCHEVA, Haar Orthogonal Functions Based, Che. Bioche. Eng. Q. 19 (3) (2005) 275 HaarOrthogonalFunctionsBasedParaeterIdentification ofwastewatertreatentprocesswithdistributedparaeters M. Todorova and T. Pencheva + Technical University Varna, Departent of Autoation Chaika, bl. 64, apt. 10, 9005 Varna, Bulgaria E-ail: gtodorova@yahoo.co Centre of Bioedical Engineering- Bulgarian Acadey of ciences 105, Acad. G. Bonchev tr., 1113 ofia, Bulgaria E-ail: tania.pencheva@clbe.bas.bg Original scientific paper Received: July 9, 2004 Accepted: April 24, 2005 The ost often used syste in aerobic biological wastewater treatent is the syste biological reservoir sedientor. The necessary condition for obtaining a good operative control is an adequate process odel to be available. The ai of this paper is odelling of a process, carrying out a syste biological reservoir sedientor, and consequent paraeter identification. In order to obtain a odel with higher degree of accuracy, the process of wastewater treatent is considered as object with distributed paraeters. hifted two diensional Haar orthogonal functions are used for paraeter identification of the process. The ipleentation of these orthogonal functions reduces the proble to a coputationally convenient for. The algorith is efficient and siple in for. Key words: Modelling, identification, Haar orthogonal functions, wastewater treatent, distributed paraeters objects. Introduction The ethods of biological wastewater treatent lie at the root of the ost of the conventional operative wastewater treatent stations. In the practice of aerobic biological treatent of wastewater with organic containation, the syste biological reservoir (or bioreservoir) sedientor (B) is the ost often used. In order to achieve a good operative control of B, the ost iportant task is an adequate atheatical odel of process dynaics to be elaborated. The odel has described the basic biocheical and physicocheical processes in the aerobic biological treatent. There is a wide variety of atheatical odels of aerobic biological treatent processes, aong that diffusion, sorption, kinetic, deterinistic, stochastic, as well as a lot of epiric odels. But there are a coparatively sall nuber of atheatical odels, obtained at dynaical running of B processes. In the last few years there is a high scientific interest in the field of siulation investigations of wastewater treatent processes fro the control point of view. 1 4 Despite of nuber of positive properties, the practical applications of dynaical odelling of wastewater treatent processes is rather liited. 3,5 In order to overcoe soe difficulties when + Corresponding author odelling this kind of processes, soe new ethods have been elaborated. 6 It is caused by the fact that nowadays it is quite ipossible soe experiental data to be interpreted without application of atheatical odels. When biotechnological processes, and particularly wastewater treatent processes, have to be odeled, a lot of coplicated reactions and interdependent characteristics have to be described. If an assuption of unifor distribution of the bioreservoir coponents is accepted, these processes have been considered as objects with fixed paraeters. This assuption has a eaning if there is a sall bioreactor or alost coplete stirring, but neither in big volues, such as bioreservoirs. The consideration of the space distribution of the characterized process variables leads to the description of wastewater treatent processes as objects with distributed paraeters. This is a preposition for obtaining a ore precise process odel. Partial differential equations or systes of partial differential equations are often used for the processes behavioral description in the class of objects with distributed paraeters. In order to prove the validity of the odel, paraeter estiation of process paraeters has to be done, based on the experiental data fro operative wastewater treatent station. In the previous authors works the conventional optiization pro-

2 276 M. TODOROVA and T. PENCHEVA, Haar Orthogonal Functions Based, Che. Bioche. Eng. Q. 19 (3) (2005) cedures, like a iplex earch or Miniax, have been used, 7 as well as block ipulse functions. 8 Orthogonal functions ethod is often used to solve the paraeter identification proble of distributed systes. The ethod is based on expansion of the functions of equation, expressing the syste into orthogonal polynoials or functions. hifted orthogonal Laguerre, Legendre and Chebyshev polynoials Walsh, Haar, block pulse, and sine cosine functions are ipleented for approxiation of table data and analytically given functions by Genov and Todorova. 9 It should be noticed that the obtained approxiation accuracy of different signals is bigger when the Haar orthogonal functions are ipleented. Except these functions there are the siplest wavelet functions and they have interesting features coputational attraction and low coputer eory requireent. o, the attention is turning to Haar wavelet functions. Chen and Hsiao 10 established in 1996 an operational atrix of integration based on Haar wavelets, and forulated a procedure for applying the atrix to analyze luped and distributed paraeter s dynaic systes. Later Todorova and Genov develop the ethod of Haar wavelets for estiating the paraeters, initial and boundary conditions of classes of distributed paraeter systes The ai of this paper is an adequate atheatical odel of the aerobic biological treatent processes in B, described in the class of distributed paraeters objects, to be obtained, and the application of shifted two diensional Haar wavelets for paraeter identification to be presented. Description and odelling of wastewater treatent processes in the syste bioreservoir sedientor In contrast to the conventional bioreactors for cultivation of icroorganiss, where the basic ai is to produce a axiu aount of bioass, here, in the syste bioreservoir sedientor, the basic ai is soe low concentration of organic containating substances in sedientor outlet to be kept. For the correct running of aerobic biological treatent in the B, it is necessary to have a good observation and control of both biocheical processes in the bioreservoir, as well as of the echanic processes in secondary sedientor. If one of two devices does not work in an optial way, it could turn down the whole process of aerobic biological treatent. The ost of known atheatical odels of the B have been developed on the basis of the classical technological schee of B, fully presented in. 16 The atheatical odels, presented also in 16 are developed on the assuption, that the decrease of organic containation as well as increase of active sludge, carry out only in the bioreservoir. Due to the fact, that the basic biocheical processes run only in the bioreservoir, while the processes in the sedientor are ainly echanical, it is of ajor interest the concentration variations of organic containation (substrate) and active sludge (bioass) in the bioreservoir to be considered. Therefore, the basic ai of consideration in this paper will be only the processes carried out in the bioreservoir. Due to this, the coplete technological schee, presented in, 16 is reduced only to the bioreservoir and the inflows and outflows. The reduced schee of the bioreservoir is presented in Fig. 1., X concentrations, respectively, of organic containations (substrate) and active sludge (bioass) in the bioreservoir, g l 1 Q capacity of wastewater, l h 1 Q a capacity of inflow air, l h 1 subscripts o, a, e denote inlet, outlet and recirculation concentrations r quotient of capacities of recirculating active sludge and inlet wastewater V a useful volue of bioreservoir, l. Fig. 1 Classical technological schee of bioreservoir The experients are carried out in the operative wastewater treatent station. The detailed technological schee of this wastewater treatent station, as well as full experient description, could be found in. 16 The syste bioreservoir sedientor in this wastewater treatent station consists of six bioreservoirs, each of the with three corridors, three radial secondary sedientors, air-pup, pup for the active sludge with a draw shaft, a distribution shaft for the serve of recirculating active sludge to the bioreservoirs, a distribution shaft for the serve of outlet suspension fro the bioreservoirs to the secondary sedientors, as well as air-ains and water-ains with the corresponding brake and control devices. The experients are carried out only in one bioreservoir and in one secondary sedientor because of the fact that the entered wastewater, the recirculating active sludge, and needed air are practically equal, both, in capacity and distribution for the six bioreservoirs. In the considered bioreservoir six control points are chosen. The experiental

3 M. TODOROVA and T. PENCHEVA, Haar Orthogonal Functions Based, Che. Bioche. Eng. Q. 19 (3) (2005) 277 data have been received at every two hours. The whole set of obtained experiental data are systeatized, analyzed, and visualized. 16 The following proble is forulated in this paper: the wastewater treatent process in the syste bioreservoir sedientor to be considered as object with distributed paraeters and a atheatical odel, based on the aterial balance equations, is to be obtained. The odel has to describe, both, in tie and in geoetric space, the variations of basic biocheical variables, naely substrate and bioass, and the consideration to be liited only in the bioreservoir. After the odel has been developed, the paraeter identification has to be presented. Before proceeding to a process odelling in the class of distributed paraeters objects, soe rearks about odelling of this process in the class of fixed paraeters will be shortly presented here. After the coparative analysis between specific growth rates, describing the kinetics of Monod, Contois and Andrews, the best results are obtained after application of Contois kinetics. The paraeter estiation has been fulfilled using MATLAB Optiization Toolbox and Genetic Algoriths Toolbox. 17,18 The application of genetic algoriths has been forced because of easiest and less-tie consuing ethods such as iplex earch, Miniax etc. they have not given satisfying results. The coplete coparative analysis between the different ones entioned above, specific growth rates of bioass, as well as the different ethods for paraeter estiation, could be found in. 8,19 The description of the process variables variations, when the wastewater treatent processes are considered as object with distributed paraeters, is realized in a siilar way as the processes of heat distribution and diffusion processes. It is realized on the basis of diffusion equation and on the worked out of the aterial balance equation, which could be presented in following for: 20 1 ( t r r r ur ) ( uz ) z 1 r z D r r z D z eff eff (1) + reaction + ass transfer gas/liquid, is a continuous and differentiable function, describing ass concentration r, z radial and axial co-ordinates D eff turbulent dispersion coefficient u r, u z radial and axial coponents of the rate vector. In fact (1) presents the equation of the aterial balance for the concentration. Due to the specificity of the bioreservoir, the distribution in a radial, naely direction x, is considered, while the variables do not change in an axial direction. At this stage the ass transfer fro a gas to a liquid phase is not rendered. In the previous author s works 7 it is proved that the rendering of dispersion phenoena influence does not lead to ore precise process odel. This stateent is there confired with graphic and nueric presentation of the considered variables (substrate and bioass) in both cases with and without rendering of dispersion phenoena. 7 After the coparison and statistic evaluation, it is evaluated that the differences between results obtained, with and without rendering of dispersion phenoena are less than 7 % for substrate and less 15 % for bioass. At the sae tie, rendering of dispersion phenoena leads to ore difficult atheatical description. o, without loss of generality, dispersion phenoena in the syste could be regarded as negligible. When the dispersion phenoena are assued to be negligible, the following coon odel for ass concentration in the radial direction is obtained fro (1): ( u ) t x x = reaction () (2) When the space distribution of the substrate and the bioass X are considered at this stage, the following assuptions are ade: 1) the radial coponent of the rate vector u x is expressed as a relation of flow rate of inlet water Q 0 /d 3 h 1, constant in radial direction, and the constant bioreservoir cross-section / 2 u x Q 0 (3) 2) when the ass concentration of substrate is expressed, the reaction of the syste in a wastewater treatent process is presented as follows 21 = YX ((x,t), X(x,t)) X(x,t), (4) (x,t) and X (x,t) are continuous and differentiable functions, which describe respectively the substrate and bioass concentrations, g l 1 ( (x,t), X (x,t)) a specific growth rate of bioass X (x,t), reflected by Contois kinetics 22 : ( ( xt,), ( xt,)) X ax ( xt, ) k ( x, t) ( x, t), X

4 278 M. TODOROVA and T. PENCHEVA, Haar Orthogonal Functions Based, Che. Bioche. Eng. Q. 19 (3) (2005) ax is the axiu value of specific bioass growth rate, h 1 k saturation constant, g l 1 Y X yield coefficient. The relation (4) yields the biocheical echanis in the syste, which is expressed as substrate decreent due to bioass accuulation in the culture ediu. 3) when the ass concentration of bioass X is expressed, the reaction of the syste in a wastewater treatent process is presented as follows ( ( ( xt, ), ( xt, )) k ) ( xt, ) (5) X d x where k d is a death coefficient of bioass, h 1. The relation (5) expresses the bioass dynaic, taking into account the bioass accuulation in the syste as a result of the icroorganiss growth with the specific growth rate and the gradually dying out of bioass with a rate k d. Therefore, on the basis of (2), (3) and (4), when the dispersion phenoena are assued to be negligible, the following odel is obtained for the substrate space distribution k ( xt, ) Q 0 ( xt, ) t x ax ( xt, ) k ( ( x,)) t ( x,) t X X ( xt, ) (6) By analogy, on the basis of (2), (3) and (5), when the dispersion phenoena in the bioreservoir are assued to be negligible, the following odel is obtained for the bioass space distribution ax ( xt, ) Q 0 X( xt, ) t x X ( xt, ) kd X( x, t ) (7) k ( ( x,)) t ( x,) t X Then, the equations (6) and (7), describing the substrate and bioass concentrations, represent the atheatical odel of the dynaics of the wastewater treatent processes in the syste bioreservoir-sedientor, considered as object with distributed paraeters. Matheatical preliinaries The orthogonal set of Haar functions is a group of square waves with agnitude of 2 / 2 in soe intervals and zeros elsewhere. Just these zeros ake the Haar transfor faster than other square functions such as Walsh s. 10 The first curve is 1 during the whole interval [0 1]. It is called the scaling function. The second curve is the fundaental square wave, or the other wavelet, which also spans the whole interval. All the other subsequent curves are generated fro the second curve with two operations: translation and dilation. This orthogonal basis is not a recent invention. It is a reiniscent of the Walsh basis, in which Walsh function contains any wavelets to fill the interval [0 1] copletely, and to for a global basis. While each Haar function contains just one wavelet during soe subinterval of tie, and reains zero elsewhere, the Haar set fors a local basis. All the Haar wavelets are orthogonal to each other. Therefore, they for a very good transfor basis. Any function, which is square integrable in the interval [0 1], can be expanded into Haar wavelets. ince the interval on which Haar functions are defined is not suitable for solving paraeter identification probles, suitable transforation is required. The shifted Haar wavelets are defined 14 as H j k () t H () t t 1 2, j 2 (8) H 1 () t scaling function, pleased during the whole interval [0, T] j 0 =2 j + k 0<k <2 j They are orthogonal with respect to the ass function (t) = 1 over the interval [0 T], where T is the observed interval. A function f(x,t) that is square integrable in the regions t[ 0, T], x[ 0, X] can be approxiately expanded in a series of two-diensional shifted Haar wavelets H ij ( xt, ) as ij ij f ( x, t) c H ( x, t) i1 j1 c H () t H ( x) i1 j1 ij i j T M M M ( H ( t)) C H ( x), (9) C M is ( ) atrix with expansion coefficients c ij H T () t [ H () t H () t H ()] t M 1 2 M 1 2 H ( x) [ H ( x) H ( x) H ( x)] T

5 M. TODOROVA and T. PENCHEVA, Haar Orthogonal Functions Based, Che. Bioche. Eng. Q. 19 (3) (2005) 279 Paraeter identification process The Haar wavelets ipleentation reduces the proble of paraeter identification of the processes in distributed paraeter objects to a coputationally convenient for. The partial differential equations (PDE), expressing the object, are transfored into set of algebraic equations, and the algorith for estiating of the paraeters of the object can be derived in a discrete for. Copared with the classical ethods, the Haar ethod is coputationally siplest, faster, and has low coputer eory requireent. 14 The identification process includes the following fundaental steps: (i) expansion of the functions of PDE into shifted two-diensional Haar wavelets (ii) rewriting of the PDE in the atrix for using the Haar wavelets properties, and after soe well known anipulations 10,11,12,14 (iii) solving of the obtained atrix equation for the vector of unknown paraeters using least squares technique. Before applying the identification algorith, the syste of partial differential equations (6) and (7), describing the processes in B, is presented in the for ka 11( xt, ) YX ax a12( xt, ) h1( xt, ) ka21( x, t) ( kd ax ) a12( x, t), k k a ( x, t) h ( x, t) ( xt, ) a11( x, t) XI( x, t) u t d 23 2 XX a12 ( x, t ) ( x, t ) X ( x, t ) xt h1 ( x, t ) u ( x, t ) (, ) X x a21( x, t) XI( x, t) ( xt, ) u t X XX a23( x, t) X ( x, t) xt h2 ( x, t ) u ( x, t ) X (, ) X x XI( xt, ) X( xt, ) ( xt, ) u ( x, t) u ( t) ( x, t) XX X X (10) ( xt, ) ( x, t) x X ( xt, ) ( x, t) x 2 u ( x, t) u ( t) ( x, t) X X Q t ux () 0() t. The functions of (10) are expanding into shifted two-diensional Haar wavelets through (9): T a11( x, t) ( HM ( t)) F11 HM ( x) (11) T a12( x, t) ( HM ( t)) F12 HM ( x) (12) T a21( x, t) ( HM ( t)) F21 HM ( x) (13) T a23( x, t) ( HM ( t)) F23 HM ( x) (14) T h1( x, t) ( HM ( t)) D1HM ( x) (15) T h2( x, t) ( HM ( t)) D2HM ( x), (16) where F 11, F 12, F 21, F 23, D 1 and D 2 are ( ) shifted two-diensional Haar wavelets coefficient atrixes of the functions of the syste equations (10). ubstituting expansion above into (10), and after soe anipulation is obtained T ( HM ( t)) [ q1f11q2f12] HM ( x) T ( HM ( t)) D1 HM ( x), (17) T ( HM ())[ t q3f21q4f12q5f23] HM ( x) T ( H ( t)) D H ( x) q1 k q2 Y X ax q 3 k q4 k d ax q5 k k. d s Equating the coefficients of like Haar function products in syste (17) and rewriting it in atrix for gives F F O M 2 F D, (18) O F ( 2 5) ( F11 ) 1 ( F12 ) 1 F F j F 1 ( 11 ) ( 12 ) j ( F ) ( F ) 11 n 12 n ( 2 2) M

6 280 M. TODOROVA and T. PENCHEVA, Haar Orthogonal Functions Based, Che. Bioche. Eng. Q. 19 (3) (2005) ( F ) ( F ) F F j F 2 ( 21 ) ( 12 ) ( F ) ( F ) n 12 n j ( F23 ) 1 ( F23 ) j ( F ) 23 n ( 2 3) T T T T T T T j 1 n j 2 n 2 ( 2 1) D ( D ) ( D ) ( D ) ( D ) ( D ) ( D ) T ( 51) [ q q q q q ] O1, O2 ( 2 2) and ( 2 3) atrixes, respectively, the eleents of which are all zero ( D1) j, ( D2) j, ( F11) j, etc. indicate the j th colun of the corresponding atrix. Equation (18) can be solved for using least squares technique. 23 The solution directly gives the estiation k and reaining estiations are deterined fro the relations: Fig. 2 Mass concentrations of organic containation (substrate) /g l 1 one view point k q / k k q Y q /. (19) d 5 s ax d 4 sx 2 ax Experiental results As it was entioned above, the experients are carried out only in one of the bioreservoirs and the experiental data have been received at every two hours. The duration of the process is 24h. The length of bioreservoir is 162, while cross-section of the bioreservoir is ix control points have been chosen in the bioreservoir, but coplete data for variation of bioass and substrate are available only for three control points 16 at the 4.6-th eter, 105-th eter and 162-th eter. The ass concentration of active sludge (bioass) has been easured 16 using sensor, based on the photoetric analysis, with the error ±3 %. The ass concentration of organic containation (substrate) has been calculated 16 based on the standard ethods. Using the identification algorith, presented above, file in Matlab is created. Estiates of the paraeters are as shown in Table 1. Table 1 Estiated values ax /h 1 k d /h 1 k s /g l 1 Y X /g g Figs. 2 and 3 present two points of view for the space distribution of substrate, when dispersion phenoena in the syste are assued to be negligible, and in the dependence of tie t/h and Fig. 3 Mass concentrations of organic containation (substrate) /g l 1 another view point Fig. 4 Mass concentrations of organic containation (substrate) /g l 1 exaple for one level bioreservoir length x/d. Both, experiental data trajectories and the siulated ones fro the odel, described in analytical way by equation (6), are presented. Fig. 4, deonstrate as an exaple the substrate space distribution in the one of considered level, because the results in another three levels are siilar. In the sae way Figs. 5, 6 and 7 present the bioass space distribution, both, experiental data

7 M. TODOROVA and T. PENCHEVA, Haar Orthogonal Functions Based, Che. Bioche. Eng. Q. 19 (3) (2005) 281 Table 2 Fisher criterion ubstrate Fisher criterion Table value at confidence probability 0.95 ( = 0.05) Table value at confidence probability 0.99 ( = 0.01) Bioass Fisher criterion Table value at confidence probability ( = 0.05) Table value at confidence probability ( = 0.01) Fig. 5 Mass concentrations of active sludge (bioass) X /g l 1 one view point adequateness of the obtained odel, the values of Fisher criterion are presented 24 (Table 2). Conclusions Fig. 6 Mass concentrations of active sludge (bioass) X /g l 1 one view point Fig. 7 Mass concentrations of active sludge (bioass) X /g l 1 exaple for one level trajectories and the siulated ones fro the odel, described in analytical way by equation (7). Presentedinfigs.2-4and5-7resultsfrotheparaeter identification of the atheatical odel of the wastewater treatent process, described as object with distributed paraeters, show high degree of adequateness of the obtained odel and the efficiency of the presented in this paper of Haar orthogonal functions for paraeter identification. As a proof for the The ai of this paper was the odelling and paraeter identification of the wastewater treatent process, carried out in the operative syste bioreservoir-sedientor and described as object with distributed paraeters. There are no failiar or accessible papers, in which this process has been considered as object with distributed paraeters, with the exception of authors papers. A atheatical odel, describing the dynaic and space variation of biocheical variables substrate and bioass X in the wastewater treatent process in B, is obtained here on the basis of the aterial balance equations. Experiental data available fro the operative wastewater treatent station gives a possibility for the identification, both, structural and paraeter. As a result fro ipleented structural identification, the following conclusion could be reached, that the process dynaic description with the Contois kinetics gives better results than the Monod kinetics. hifted two diensional Haar orthogonal functions have been used for paraeter identification of developed odel. Their ipleentation reduces the proble of paraeter identification of the processes in B to a coputationally convenient for. The algorith is siple in for and with high precision. Presented results show, that the obtained atheatical odel describes with a high degree of adequateness the wastewater treatent process, carried out in syste bioreservoir-sedientor, and considered as object with distributed paraeters in the case of negligible dispersion phenoena. List of sybols t tie coordinate r, z, x space coordinates

8 282 M. TODOROVA and T. PENCHEVA, Haar Orthogonal Functions Based, Che. Bioche. Eng. Q. 19 (3) (2005) continuous and differentiable function, describing ass concentration, g l 1 u r, u z, u x coponents of the rate vector (x,t) continuous and differentiable function, describing substrate ass concentrations, g l 1 X (x,t) continuous and differentiable function, describing bioass concentrations, g l 1 Q flow rate of wastewater, l h 1 Q a flow rate of inlet air, l h 1 bioreservoir cross- section, 2 V a useful volue of bioreservoir, l (,X) specific growth rate, h 1 ax axiu value of the specific growth rate, h 1 k yield factor, g g 1 k Michaelis (saturation) constant, g l 1 k d death coefficient of bioass h 1 Y X yield coefficient, g g 1 D eff turbulent dispersion coefficient, 2 h 1 T H M observed interval vector with shifted Haar wavelets reaction of syste, g l 1 h 1 Diensionless r k C M c ij sybols quotient of capacities of recirculating active sludge and inlet wastewater weight function integer positive nuber shifted two diensional Haar wavelets coefficients atrix expansion coefficients. uperscripts and subscripts o, a, e denote inlet, outlet and recirculation concentrations nuber of Haar wavelets i, j integer positive nubers. References 1. Dold, P. L., Wentzel, M. C., Billing, A. E., Ekaa, G. A., Marais, G.v.R., Activated ludge iulation Progras, Water Research Coission, Pretoria, Henze, M., Grady, C. P. L., Gujer, W., Marais, G.v.R., Matsuo, T., IAWQ cientific and Technical Report, London, Morgenroth, E., Van Loosdrecht, M. C. M., Wanner, O., Wat. ci. Tech. 41 (4-5) (2000) Patry, G.G., Takacs, I., Proc. of MATHMOD Vienna (IMAC) Vanhooren, H., PhD Thesis, Gent, Petersen, B., PhD Thesis, Gent, Pencheva, T., PhD Thesis, ofia, 2003, (in Bulgarian). 8. Todorova, M., Pencheva, T., Ass. for the Adv. of Mod. & i. Techn. in Ent. (A.M..E.) (2004) (in press). 9. Genov, D., Todorova, M., Int. Conf. on Cop. yst. and Techn. ofia (2001) III Chen, C. F., Hsiao, C. H., IEEE Proc. on Control Theory Appl. 144 (1) (1997) Genov, D. G., Todorova, M. G., Int. Conf. on Auto. and Inf. ofia Todorova, M., Genov, D., Int. Conf. on Auto. and Inf.- ofia Todorova, M., Genov, D., I st Int. Congr. on Mech. and Electr. Eng. and Techn.- Varna Todorova, M., PhD Thesis, Varna, 2003 (in Bulgarian). 15. Todorova, M., Genov, D., Vassileva, M., Gerasiov, K., I st Int. Congr. on Mech. and Electr. Eng. and Techn., Varna Tzvetanov, R., PhD Thesis, ofia, 1999 (in Bulgarian). 17. Chipperfield, A., Fleing, P. J., Pohlhei, H., Fonseca, C. M., Genetic Algorith Toolbox for Use with MATLAB, Goldberg, D. E., Genetic Algoriths in earch, Optiization and Machine Learning, Addison- Wesley, Pencheva, T., Hristozov, I., Tzonkov, t., Techn. Ideas 2004(in press, in Bulgarian). 20. chalzriedt,., Jenke, M., Reuss, M., Int. Conf. on Cop. Appl. in Biotechn., Garisch-Partenkirchen Farlow,. J., Partial Differential Equations for cientists and Engineers, John Wiley & ons, Inc., Bastin, G., Dochain, D., On-line Estiation and Adaptive Control of Bioreactors, Elsevier, Asterda, Jha A. N., Zaan., Int. J. ystes ci. 16 (6) (1985) Bronstain, I., eendjaev, K., Reference Book on Matheatics for Engineers, cience, Moscow, 1986 (in Russian).