Average modeling of an alternating aerated activated sludge process for nitrogen removal

Transcription

1 Preprints of the 1th IFAC World Congress Milano (Italy) August - September, 11 Average modeling of an alternating aerated activated sludge process for nitrogen removal Richard Marquez María Belandria-Carvajal Claudia Gómez-Quintero Miguel Ríos-Bolívar Departamento de Sistemas de Control, Facultad de Ingeniería, Universidad de Los Andes, Mérida, Venezuela ( {marquez,claudiag,riosm}@ula.ve) Universidad de Los Andes, Mérida, Venezuela ( mariaebelandria@gmail.com). Abstract: In this paper we present a novel averagemodel for alternating activated sludge models. This kind of processes, where nitrification-denitrification operations occur, are represented by two consecutive dynamics: an aerobic phase followed by an anaerobic one. In particular, we studyareducedasmodelfornitrogenremoval.theresultingaveragecontinuousmodelcaptures the governing (average) behavior of switching dynamics, without considering the switching behavior. The continuous nonlinear AS models thus obtained can be used straightforwardly for analysis, control design and estimation purposes. Keywords: Wastewater treatment, modelling, average models, discontinuous systems, activated sludge models 1. INTRODUCTION MIXER AIR SETTLER carbon supply The Activated Sludge (AS) process, a secondary treatment of domestic waste waters, is the most generally applied biological wastewater treatment method (Metcalf and Eddy, 5). It consists in a set of treatment procedures, which has as a common element, the intimate contact of the waste water with a biological mass in an aired tank. This biomass is made up of a mixed culture of microorganisms that form, along with other organic and inorganic substances, a flocculent conglomerate. Through this process the organic compounds in waste water are used as substrates. An AS wastewater treatment plant (WWTP) can achieve biological nitrogen (N) removal and biological phosphorus (P) removal, besides removal of organic carbon substances. Biological N removal presents economical and operational advantages, it is important due to the proven problems that N causes to environment and human body. For example, N in wastewater causes eutrophication problems such as algal bloom or red tide in lakes or coastal area. There are many different AS configurations. Those that are developed for N removal can be classified into two categories: separated-tanks or single-tank configurations. Economic considerations have beentheprimarymotivationfordevelopingaconfiguration with a single tank, where aired and not aired stages are alternated to remove nitrogen from sewage, cf. (Gómez- Quintero, ), see also (Hao and Huang, 199; Hiatt et al., ; Chai and Lie, ). Modeling of AS processes has become a common part of the design and operation of WWTPs. These models are currently used in learning, design, control, process optimization and research (Gernaey et al., ). Most influent wastewater AERATION TANK recycled sludge excess sludge Fig. 1. Outline of the activated sludge process treated effluent common models have been proposed by the IWA 1 : Activated Sludge Model No. 1 (ASM1), No., No. d, and No. 3 (Henze et al., ). In particular, a single-tank configuration results in a type of discontinuous systems whichhasbeendifficulttoregulate.simeonovetal.(); González-Miranda et al. (9) use the aerobic submodel (k L a ) for control purposes. Chai and Lie () employ aeration/non-aeration times as control input and use an MPC approach, using a discontinuous AS model. They compared their results to rule-based controllers (fixed lengths of aerobicand anoxic phases, oxygenbased control and ammonia based control). The purpose of this paper is to describe the average modeling of AS alternated air on/off process. This averaging approach leads to a continuous model which effectively describes the slow motion of the original discontinuous cyclic process. Averaging of this type of switched (discontinuous right hand-side) systems has been known in Pulse Width Modulation control for a long time (see e.g. Khapaev (19); Friedland (197); Tsypkin (19)), particularly in the context of switched power electronics (see 1 International Water Association Copyright by the International Federation of Automatic Control (IFAC) 1195

2 Preprints of the 1th IFAC World Congress Milano (Italy) August - September, 11 e.g. Middlebrook and Cuk (197); Krein et al. (199)). To the authors knowledge, this novel approach has not been reported in the AS modeling literature. This paper is organized as follows. First, we briefly review averagingof differential equations in section. Then, we recall in section 3 a reduced AS model (RASM), specifically designed for N removal (Queinnec and Gómez- Quintero, 9), see also (Gómez-Quintero, ), which includes four state variables and eleven parameters. Averaging leads to a continuous model in section. Section 5 compares via numerical simulations both intermittently aerated and continuous AS models. Thus, linear control laws are proposed. We draw up the paper with some conclusions and remarks.. CLASSICAL TIME AVERAGING Deterministic time averaging has been frequently utilized to obtain simpler models which retain the important properties of a system. The followingresults and definitions are takenfromkhalil ()andsandersand Verhulst(195). Let x = x be an equilibrium point for the nonlinear system ẋ = f(x), i.e. f(x ), where f : D R n is continuously differentiable and D is a neighborhood of x. Let the Jacobian matrix of f(x) at x = x be A = f x (x) x=x. Then, 1) x is asymptotically stable if the real part R(z i ) < for all eigenvalues z i of matrix A, and ) x is unstable if R(z i ) > for one or more z i of A, see (Khalil,, Th..7). An equilibrium point is called hyperbolic if R(z i ). Asymptotic stability means the solution x(t) converges to x as time t tends to. A periodic solution (or orbit) corresponds to a solution x(t) of ẋ = f(x), such that x(t + T) = x(t) for a constant < T <. Roughly speaking, a periodic solution is asymptotically stable if every solution tends to it. Let x,y,x belong to an open subset D R, let t R + = [, ), and let the parameter ǫ vary in the range (,ǫ ] with ǫ 1. Let f : R + D R be a piecewise continuous function. Consider the problem of finding the solution x(t) of: dt = ǫf(t,x), x() = x. (1) If f(t,x) is a T-periodic function in its first argument, we let the averaged system be: dy dt = ǫf (y), y() = x () where f (y) = T 1 T f(t,y)dt. The slow motion of the periodic solution x(t) of (1) corresponds to the solution y(t) of (). When f(t,x) = f(x), i.e. independent of t, we have moreover y = x = T 1 T x(t)dt, see (Khalil, ). The oscillatory component of x(t) around this slow motionconstitutes the fast behavior. See e.g.(khalil,, Theorem 1.). Roughly speaking, if the eigenvaluesof the Jacobian matrix of f (y) around the equilibrium point of the averaged equation () all have negative real parts, the corresponding periodic solution φ(t,ǫ) of equation (1) is then asymptotically stable for ǫ sufficiently small; Here the classical smooth assumption isreplaced by a piecewise continuous assumption (Sontag, 199, Appendix C). moreover, φ(t,ǫ) lies in an O(ǫ) neighborhood of x. If one of the eigenvalues has positive real part, φ(t,ǫ) is unstable. 3. INTERMITTENTLY AERATED ACTIVATED SLUDGE MODEL InthissectionwerecalltheRASM,asimplifiedrepresentation ofthe alternatingas process,designedto improvethe process of removing N content in waste water (Queinnec and Gómez-Quintero, 9). It consists of a model describing a nitrification-denitrification process. Essentially, this process is characterized by changes in the coefficient of oxygen transfer k L a, due to the alternating air on/off process: { aerobic stage, k L a (3) = anoxic stage. Duringnitrification(air-onstage,k L a ), weobtain(the aerobic submodel): 1 Ṡ S = D s S Sin +D c S Sc (D s +D c )S S α 1 S S ( Y H ) S O S NO3 K OH + S O +K OH (S NO3 +K NO3 )(S O +K ( OH) S O S NO3 +α +η NO3h S O +K OH S NO3 +K ) NO3 K OH S O +K OH 1 Y H S NO3 Ṡ NO3 = (D s +D c )S NO3 α 1 S S.Y H S NO3 +K NO3 K OH S O +K OH S O S O +K OAUT +α S NH S NH +K NHAUT Ṡ NH = D s S NHin (D s +Dc)S NH +α 3 α 1 i NBM S S ( ) S O S NO3 K OH + S O +K OH (S NO3 +K NO3 ) (S O +K OH) S NH S O α (S NH +K NHAUT) (S O +K OAUT) Ṡ O = (D s +D c )S O +k L a(s Osat S O ) 1 Y H S O α 1 S S Y H S O +K OH S NH.57α (S NH +K NHAUT) S O (S O +K OAUT) () During denitrification, k L a = (air-off stage), an anoxic submodel is obtained 3. This switching behavior on k L a leads to the switched (discontinuous) dynamics mentioned earlier. State variables are the concentration of readily biodegradable substrate x 1 = S S, and concentrations of nitrate x = S NO3, ammonia x 3 = S NH, and dissolved oxygen x = S O. See (Queinnec and Gómez-Quintero, 9; Gómez-Quintero, ) for details. 3 As k L a =, the system () results in a second (anaerobic-anoxic) submodel where the oxygen dynamics can be represented by ṠO = as it converges fast to zero. It is usual to consider S O (t) in this stage. 119

3 Preprints of the 1th IFAC World Congress Milano (Italy) August - September, 11 Concentrations of substrate soluble in water (S Sin ) and ammoniacal nitrogen (S NHin) entering the reactor are regarded as exogenous (not manipulable) inputs. Dilution rate on the entry (D), and added carbon source (D c ), as well as k L a, are also exogenous variables. Values that were set for these variables are shown in Table 1. Influent characteristics and biological parameters meet the Benchmark Simulation Model N 1 specifications (Copp, ). Additionally, RASM-specific parameters, α i, are taken from (Gómez-Quintero, ). Table 1. Input variables Variable Value Unit S Sin 9.5 g.m 3 S NH in 31.5 g.m 3 D 3.7 day 1 D c.1 day 1 k L a (aerobic phase) day 1 Values of system physical parameters were taken from (Gómez-Quintero, ), these are shown in Table. Table. RASM Parameters Parameter Description Value Y H [g.g 1 ] Coefficient of performance.7 of heterotrophic biomass i NBM [g.g 1 ] Mass of nitrogen contained. in the concentrations of heterotrophic and autotrophic biomass η NO3 h Correction factor for the. hydrolysis in anoxic phase K NH AUT [gn.m 3 ] Coefficient of average saturation 1. of ammonia for au- totrophic biomass K NO3 [gn.m 3 ] Coefficient of average saturation.5 of nitrate K O H [go.m 3 ] Coefficient of average saturation. of oxygen for the heterotrophic biomass K O AUT [go.m 3 ] Coefficient of average saturation of oxygen for the autotrophic biomass. α 1 [day 1 ] Growth rate of.91 heterotrophic biomass α [g.m 3 day 1 ] Speed nitrate production 7.73 by the autotrophic α 3 [g.m 3 day 1 ] Speed of hydrolysis of 7.5 slowly biodegradable substrate by the heterotrophic α [g.m 3 day 1 ] Ammonification of soluble 15. organic nitrogen S O sat Concentration of dissolved oxygen saturation 9. According to Simeonov et al. (), a simplified model of the settler is given by: ds N = Q+Q c V d (S NO3 +S NH S N ) dt where Q is the input flow, Q c is the flow of the external carbon source, and V d is the volume of the settler. N concentration at the outlet of the settler, S N, is not available for measurements. Volume of settler and reactor are assumed equal (V = V d ). Expressed in terms of dilution rates, D = Q V and D c = Qc V, last equation results: ds N = (D +D c )(S NO3 +S NH S N ). (5) dt The control problem is to reduce N concentration at S N. According to latest water quality European standards, the globalnconcentrationmustbelowerthan1g.m 3 atthe settler outlet, for mean samples over two hours. However, at the outlet of the settler, the organic N concentration is practically constant, of about g.m 3, so the quality standard can be replaced by a standard on the sum of ammonia and nitrate concentrations (S N ) less than or equal to g.m 3 (Simeonov et al., ). S N is regulated indirectly by using sensed S NO3. The actual control input is the external carbon concentration S Sc. Let u denote an indication signal taking values in the set {,1}, where u = 1 represents the aerobic submodel and u = the anoxic stage. Rewriting () as ẋ = f 1 (x,s Sc,k L a) and the air-off stage as ẋ = f (x,s Sc ) = f 1 (x,s Sc,). Using u, we can combine both AS submodels into a discontinuous RASM : dt = f 1(x,S Sc,k L a)u+f (x,s Sc )(1 u). () Let t k R, k = 1,,..., be the time instants at which aeration process begins. Signal u can be written as a train of pulses: { 1 if tk t < t k +µ air on T u(t) = (7) if t k +µ air on T t < t k+1 = t k +T where < µ air on < 1 corresponds to the fraction of T where the air-on stage is carried out. Function u(t) is then T-periodic,i.e., u(t+t) = u(t) forallt, with periodt >.. CONTINUOUS ACTIVATED SLUDGE MODEL Averaging of (1) can also be applied to system given by ()-(7). As in (1) and (), we associate the autonomous averaged system: d x dt = f 1( x,s Sc,k L a)µ+f ( x,s Sc )(1 µ) () where u(t) has been replaced by its average µ. Note that µ must be limited to µ = µ air on < 1. In this paper, notice µ = µ air on is not a control input and will be considered constant, see e.g. (Hiatt et al., ). Averaging justifies approximating solutions x(t) of the controlled system ()-(7) by solutions x(t) of the averaged system (). It can be shown that fixing nitrate concentration x to a desired constant value X, system () has an equilibrium point x = (x 1 (X),X,x 3 (X),x (X)). Linearizing () around this equilibrium point x we obtain a stable linearized system (four eigenvalues λ i with R(λ i ) < ). The periodic discontinuous system ()-(7) possesses then an asymptotically stable periodic solution of period T. Let us denote x 5 = S N the total N concentration, the average total N dynamics results: d x 5 = (D +D c )( x + x 3 x 5 ). (9) dt To demonstrate existence, uniqueness of solutions, or even averaging results of discontinuous model (), it is enough to point out that signal u corresponds to a piecewise continuous function, see e.g. (Sontag, 199, Appendix C). 1197

4 Preprints of the 1th IFAC World Congress Milano (Italy) August - September, 11 x 1 = S S x 3 = S NH 1 1 substrate concentration ammonia concentration 1 3 x = S NO3 x = S O 1 nitrate concentration dissolved oxygen concentration Fig.. Comparison of discontinuous (thin line) and averaged (thick line) RASM dynamics S N 1 1 nitrogen concentration at the (settler) outlet Fig. 3. Comparison of discontinuous (thin line) and averaged (thick line) RASM: total N concentration 5. INTERMITTENTLY AERATED AND CONTINUOUS AVERAGE MODEL BEHAVIOR Consider now () and (9). In order to be in compliance with the standards of water quality mentioned in the previous section, we set the average N concentration x 5 = S N =.5 g.m 3. Thus the equilibrium point results: x 1 = 5. g.m 3 ; x = 1.7 g.m 3 ; x 3 =. g.m 3 ; x = 3.17 g.m 3 ; S Sc = 139 g.m 3 (control input) A numerical simulation illustrating the discontinuous and average RASM dynamics during hours is shown on Figure, for time period T = hours and aeration µ airon = µ =.5 (i.e. equal air on/off lengths). We set S Sc = 139 g.m 3, for both phases. As shown, average motion of the discontinuous system is effectively approximated by the average model behavior. The behavior of the total N concentration is illustrated on Figure 3. The observed average of the discontinuous RASM is around S N.1 g.m 3, which is less than g.m 3. The value of the average desired output x has been set at.5 g.m 3 to maintain S N on quality range. Let the nitrate concentration ȳ = x be the measured output. Jacobian linearization of () around x yields the linear system: x δ = A x δ +B S Sc,δ (1) ȳ δ = C x δ, where x δ = x x, ȳ δ = ȳ ȳ, S Sc,δ = S Sc S Sc, and A = f x, B = f ( x, S Sc ) u, C = h ( x, S Sc ) x For µ =.5 we obtain: A = , B = 1 3.7, C = [ 1 ]. Eigenvalues 5 of A are given by λ = ( 5.913,.3+.15i,.3.15i,.791), thus x and x 3 are highly coupled, and, moreover, λ,3 =.3±.15i corresponds to dominant second-order poles with respect to total N dynamics. This dependency of the RASM on nitrate and ammonia is well known in the AS literature and confirmed by our average model. Remark 1. Nitrification submodel () could be used to obtainboththeequilibriumpointandlinearizeddynamics, however it is straightforward to check out that its results greatly differs from the values obtained by using the average model. For example, x < x 3 using the average model; according to (), their relation must be x = 3. g.m 3 > x 3 =.7 g.m 3, an opposite behavior. Remark. Although control design based on () could lead to stable controller for the discontinuous RASM, this will affect in particular the performance of any designed controller. For example, a root locus based on the average model results in stable behavior for positive values of proportional gain K >, while using () results in an unstable behavior for K > 7. Average model shows emergent properties which are effectively exhibited by its discontinuous counterpart. Remark 3. The dependence of average RASM on µ serves also to analyze non-symmetric air on/off relations, %/%, %/%, %/%, etc., instead of 5%/5% relations. 5.1 Reduction to a linear second-order system Consider now that x 1 and x are at equilibrium (or converge to equilibrium fast enough, dominant poles property). Let us rewrite () as follows = f 1 ( x 1, x, x, S Sc ) x = f ( x 1, x, x 3, x ) x 3 = f 3 ( x 1, x, x 3, x ) = f ( x 1, x 3, x ). x. (11) A linear second-order system results by linearizing (11) around x : 5 The (averaged) equilibriumpoint ishyperbolic, thus linearapproximation results a valid instrument to analyze the average nonlinear model, see e.g. (Khalil,, Ch. ). 119

5 Preprints of the 1th IFAC World Congress Milano (Italy) August - September, 11 1 Root Locus 1 11 nitrogen concentration at the (settler) outlet K = 1 1 Imaginary Axis S N x 15 K = external carbon concentration (control) K = 9 S Sc K = 1 K = K = Real Axis Fig.. Gain selection for a µ = µ air on =. [ ] [ ] [ ] x δ xδ x1δ = A x 1 +D 3δ x 1 +B 3δ x 1 SSc,δ ; δ where: ȳ δ = x δ. [ ] [ ] x1δ xδ = A x +B δ x SSc,δ. 3δ This yields x s δ = A s x s δ +B s S Sc,δ, ȳδ s = C s x s (1) δ with matrices A s, B s y C s given by: [ ] A s = A 1 +D 1 A =,.1. [ ] B s = B 1 +D 1 B = 1.37, C.31 s = [1 ]. We will illustrate a couple of controller/observer designs based on these linear models. Notice previous linear models cannot be obtained from the discontinuous model. 5. Proportional control design Toillustratelinearcontrollerdesignbyusing(1),consider a non-symmetric air on/off relation of %/%, i.e. µ =.. The root locus for a negative parameter is shown on Figure. From here, the proportional control S Sc = S Sc K(x x ) to be applied to (), is given by S Sc = 3 g.m 3, x =. g.m 3, and K = 9 at the location illustrated on Figure. Numerical simulations for different valuesofk confirmthattheselectedk resultsinabalance between less control effort and fast transient behavior. Figure 5 exhibits the total concentration and control input responses for three different values of K = 9, K = 1, and K =. For K = 9, quality standard of g.m 3 is achieved at time t = 9 hours. 5.3 Estimator/observer design Based on the linear second-order system (1), a Luenberger observer can be designed straightforwardly: ˆx s δ = (A s LC s )ˆx s δ +B su δ +Ly s δ, (13) with L = (l 1,l ) T. For a symmetric air on/off process, i.e. µ =.5, open-loop poles of A s are placed at λ,3 = Fig. 5. Closed-loop performance of discontinuous RASM for a µ = µ air on =.. K = 9 (solid); K = 1 (dashed); K = (dash-dotted) x = S NO3 1 nitrate concentration 1 3 x 3 = S NH 1 1 ammonia concentration 1 3 Fig.. Luenberger second-orderobserverresponse: nonlinear response (thin); estimates (thick line).7 ±.15i; therefore trial and error observation poles are s 1e,e = 1., far from the origin at lefthand side of λ,3 in the complex plane; observer gains result l 1 = 1.55, l = 1.97 (time response of about 5 hours). Observer gain tuning is recommended due to nonlinearities, smaller observer poles yields appropriate filtered (average) values, larger observer poles result in larger (amplitude) oscillations of observer response. Figure depicts a numerical simulation of open-loop linear observer performance. 5. Closed-loop PI controller performance vs environment conditions of benchmark simulation model No. 1 Storm, rainy and dry conditions proposed by (Copp, ) have been applied to test designed controllers and estimators. Study of P controller gain reveals that steady state error is shortened for larger controller gains, see Figure 5, a condition suitable to propose integral action, S Sc = S Sc K(x x ) K i t (x (σ) x )dσ. PI controller parameters are obtained as follows: from a root locus first choose K, and then parameter K i. PI controller closed-loop performance when storm conditions are present is illustrated on Figure 7, for µ air on =.5, K = 5, K i = 1. Storm perturbations are shown on Figure. From t = 5 hours to t = 1 hours, we assume nominal conditions for S Sin, S NHin, D (see Table 1), thereafter step variations on S Sin and S NHin are scheduled. At t = 15 hours, storm conditions are reproduced again. 1199

6 Preprints of the 1th IFAC World Congress Milano (Italy) August - September, 11 S N S Sc 1 1 nitrogen concentration at the (settler) outlet x 1 external carbon concentration (control) Fig. 7. PI controller performance under Storm conditions S Sin S NH in D Fig..Stormconditions:variationsofinputsS Sin,S NHin, D from BSM1. CONCLUSIONS AND FINAL REMARKS An analytical approach to the continuous (average) modelingofintermittentlyaeratedas processeswaspresented. The continuous model thus developed exhibits fundamental properties of the corresponding discontinuous AS system, through a set of continuous nonlinear differential equations. In terms of control design, the average model overcomesusualproblemsfoundwhendealingwithdiscontinuous AS models. Classical controllers designs presented illustrates our point. The procedure presented in this paper applies to the case the carbon supply appears only in one of the phases, e.g. dt = f 1(x,S Sc,k L a)u+f (x,)(1 u), Ourapproachisalsovalidin thecaseofnoexternalcarbon supply, when the oxygen transfer coefficient is treated as control input, i.e. for systems of the form: dt = f 1(x,,k L a)u+f (x,)(1 u). REFERENCES Chai, Q. and Lie, B. (). Predictive control of an intermittently aerated activated sludge process. In American Control Conference, ThA13.1, 9 1. Seattle, Washington, USA. Copp, J. (ed.) (). The COST simulation benchmark description and simulator manual. Office for Official Publications of the European Communities, Luxembourg. ISBN Friedland, B. (197). Modeling linear systems for pulsewidth-modulated control. IEEE Trans. Automatic Control, Gernaey, K., van Loosdrecht, M., Henze, M., Lind, M., and Jørgensen, S. (). Activated sludge wastewater treatment plant modelling and simulation: state of the art. Environmental Modelling & Software, 19, Gómez-Quintero, C. (). Modélisation et estimation robuste pour un procédé boues activées en alternance de phases. Ph.D. thesis, Université Paul Sabatier, Laboratoire d Analyse et d Architecture des Systèmes (LAAS), Toulouse, France. González-Miranda, O., Ríos-Bolívar, M., and Gómez- Quintero, C. (9). Adaptive output feedback regulation of an alternating activated sludge process. In Proc. European Control Conference (ECC) 9, MoA1.. Budapest, Hungary. Hao, O. and Huang, J. (199). Alternating aerobic-anoxic process for nitrogen removal: process evaluation. Water Environ. Res.,, Henze, M., Gujer, W., Mino, T., and van Loosdrecht, M. (). Activated sludge models ASM1, ASM, ASMd and ASM3. Technical Report Scientific and Techinical Report No. 9, IWA Task Group on mathematical modelling for design and operation of biological wastewater treatment, London. Hiatt, W., Burnham, W., Madzy, E., Weisbrodt, W., and Wegmann,U.(). Continuousflowcompletelymixed waste water treatment method. Patent, US Patent. Khalil, H. (). Nonlinear Systems. Prentice-Hall, Upper Saddle River, NJ, 3 edition. Khapaev, M. (19). On the method of averaging and on certain problems connected with averaging. Differential Equations, (5), Krein,P.,Bentsman,J.,Bass,R.,andLesieutre,B.(199). On the use of averaging for the analysis of power electronic systems. IEEE Trans. Power Electronics, 5(), Metcalf, K. and Eddy, I. (5). Wastewater Engineering: Treatment and Reuse. McGraw-Hill, New York. Middlebrook, R.D. and Cuk, S. (197). A general unified approach to modeling switching converter power stages. In IEEE Power Electronics Specialists Conf., 1 3. Queinnec, I. and Gómez-Quintero, C. (9). Reduced modeling and state observation of an activated sludge process. Biotechnology Progress, 5(3), 5. Sanders, J. and Verhulst, F. (195). Averaging Methods in Nonlinear Dynamical Systems. Springer-Verlag, New York. Simeonov, I., Queinnec, I., Gómez-Quintero, C., and Babary, J. (). On linearizing control of wastewater treatment processes. In Automatics and Informatics. Sofia, Bulgary. Sontag, E. (199). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Springer-Verlag, New York. Tsypkin, Y.Z. (19). Relay Control Systems. Cambridge University Press, Cambridge. 1