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1 Abdul Wahab, Hamimi Fadziati Binti and Katebi, Reza and Villanova, Ramon (22) Comparisons of nonlinear estimators for wastewater treatment plants. In: Proceedings of the 2th Mediterranean Conference on Control & Automation (MED), 22. IEEE, pp ISBN , his version is available at Strathprints is designed to allow users to access the research output of the University of Strathclyde. Unless otherwise eplicitly stated on the manuscript, Copyright and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please chec the manuscript for details of any other licences that may have been applied. You may not engage in further distribution of the material for any profitmaing activities or any commercial gain. You may freely distribute both the url ( and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge. Any correspondence concerning this service should be sent to the Strathprints administrator: he Strathprints institutional repository ( is a digital archive of University of Strathclyde research outputs. It has been developed to disseminate open access research outputs, epose data about those outputs, and enable the management and persistent access to Strathclyde's intellectual output.

2 22 2th Mediterranean Conference on Control & Automation (MED) Barcelona, Spain, July 3-6, 22 Comparisons of Nonlinear Estimators for Wastewater reatment Plants H. F. Wahab*, R. Katebi* and R.Villanova** * Industrial Control Centre, *Department of Electrical and Electronic Engineering, University of Strathclyde, G QE, Glasgow, UK. {r.atebi, hamimi.abdulwahab}@eee.strath.ac.u ** Dept. de elecomunicació i Enginyeria de Sistemes, ESE, UAB, Ramon.Vilanova@uab.cat Keywords: Nonlinear state estimation, Wastewater s, Kalman filter, H-infinity filter, state-dependent filter, Unscented Kalman filter Abstract his paper deals with five eisting nonlinear estimators (filters), which include Etended Kalman Filter (), Etended H-infinity Filter (), Dependent Filter (), Dependent H-Infinity Filter () and Unscented Kalman Filter () that are formulated and implemented to estimate unmeasured states of a typical biological wastewater system. he performance of these five estimators of different compleities, behaviour and advantages are demonstrated and compared via nonlinear simulations. his study shows promising application of for monitoring and control of the process variables, which are not directly measurable. I. INRODUCION he design and application of state estimator in biological wastewater treatment plant (WWP) has been an active area of research over the past decades. he difficulty of determining the states of a dynamic behavior of the system via few available measurements has led to the development and implementation of a wide variety of state estimation algorithms. Some application of the different estimation techniques to WWP have been discussed in [],[2],[3]. he most widely used suboptimal filter, the Etended Kalman Filter () which facilitates the Jacobian of the nonlinearity in the dynamics is used routinely and successfully in many practical applications including WWP [4],[5]. Under wea nonlinearity, the, have demonstrated precise performance, but diverge under more highly nonlinear cases and its estimates highly dependent on the accuracy of model used [6],[7]. Since is biased and not robust to the modeling uncertainties [8], the etended H-infinity filter () has been an alternative to minimizes the worst possible effect of the modeling errors and additive noise on the signal estimation errors [9],[]. here has been a number of different approaches to the formulation of the in the literature, where different approaches all lead to etensively different equations []. hus, maing the entire field of H filter rather difficult to implement and use. However, despite its difficulty and etra tuning required by H filter, it is still worthwhile to consider this type of filter for its robustness [],[2]. In contrast to the and which are based on aylor series type linearization, another different approach to state estimation of nonlinear systems based on parameterization that brings the nonlinear system to a linear structure having state-dependent coefficients (SDC) is the Dependent Riccati Filter () [3]. It is also shown in [4] that, the parameterization is not unique and can create etra degrees of freedom that are not available in traditional filtering methods to avoid singularities and loss of observability. and have their own respective advantages in terms of estimator robustness [5],[6] that can be manipulated to develop a new algorithm called Dependent H-Infinity Filter (). his new filter employs a state- dependent model and H-infinity design technique to estimate the system states. Another filter that can avoid the cumbersome evaluation of the Jacobian matrices and does not approimate the nonlinear models is the Unscented Kalman filter () [7],[8]. he uses a parameterized set of sample points, called sigma point according to a specific deterministic sampling to model the nonlinearity [7]. Using the true nonlinear models, can give more accurate results than the linearization technique algorithm for propagating mean and covariance [9]. Although these algorithms have been widely studied in the literature, the performances and applications of the,, and for the activated sludge process in WWP have seldom been discussed. herefore, the objective of this paper is to produce a comparative study of the well nown and the above mentioned filters for state estimation of a biological WWP. he remainder of this paper is organized as follows: Section II presents a brief formulation of,,, and algorithms. Section III is dedicated to a brief description of biological process used. he comparison between the filters is performed by simulation studies in Section IV. A general conclusion ends the paper /2/$3. 22 IEEE 764

3 II. NONLINEAR FILERS FORMULAION he algorithms of five different nonlinear filters are formulated in this section. he nonlinear system is assumed to be piecewise observable and controllable. A. Etended Kalman Filter () he simply approimate nonlinear dynamics f, and output function h with first order aylor series epansion around the current estimate by discarding the second and higher order terms to evaluate covariance and the filter gains. able I presents the summary of algorithm. ABLE I ALGORIHM (t) = f ((t),u(t)) w(t) y(t) = h((t)) v(t) ;w(t) ~ N(,Q(t)) ;v(t) ~ N(, R(t)) ABLE II ALGORIHM (t) = f ((t),u(t)) w(t) y(t) = h((t)) v(t) z(t) = C z (t) t ˆ( ) = Et [ ( )] ; Pt ( ) = Conditions: Filter Gain: = y Kt () PtC () () tr () t P(t) = A(t)P(t) P(t)A (t) where: ;w(t) ~ N(,Q(t)) ;v(t) ~ N(, R(t)) P(t)γ 2 C z (t)c z (t)p(t) P(t)C y (t)r (t)c y (t)p(t) Q f A, h C y t ˆ(), ut () t ˆ() t ˆ( ) = Et [ ( )] ; Pt ( ) = Vart [ ( )] Conditions: Filter Gain: Kt () = PtH () () tr () t Estimated Output: ˆ(t) = f ( ˆ(t),u(t)) K(t) y(t) C y (t) ˆ(t) ẑ(t) = C z ˆ(t) P(t) = A(t)P(t) P(t)A (t) where: P(t)H (t)r (t)h(t)p(t) Q(t) f A B. Etended H-Infinity Filter () t ˆ(), ut () t ˆ() As can be seen from able II, the structure of estimator is similar to. he main difference is calculating the filter gain, where the inclusion of γ term tends to increase the norm of P. his will in turn increase the gain, K and hence mae the estimator more responsive to the measurement compared to [2]. he value of γ is reduced in steps until one of the eigenvalues of P becomes imaginary or negative. For the optimal solution P should stay positive definite (P>). However, an alternative scheme introduced in [2] can be utilized using the time decreasing eponential function., h H ˆ(t) = f ( ˆ(t),u(t)) K(t) y(t) H(t) ˆ(t) where n t () R is the state vector and m yt () R is the measurement vector. wt () and vt () represent the uncorrelated zero-mean Gaussian process and measurement noise with covariance Q(t) and R(t), respectively. C. Dependent Filter () he has linear structure with state dependent matrices such as A ( ), Band ( ) C ( ) is used to fully capture the nonlinearities of the system as shown in able III. Even though the equations demonstrate the similarity between and structure, yet the linear model in suffers from linearization error since it is developed based on approimation, while linear model for is eact. ABLE III ALGORIHM dependent conditions: (t) = f ((t),u(t)) w(t) y(t) = h((t)) v(t) ;w(t) N(,Q(t)) ;v(t) ~ N(, R(t)) (t) = A()(t) B()u(t) G(t) y(t) = C()(t) v(t) where: f( ) = A( ) t ˆ( ) = Et [ ( )] ; Pt ( ) = Vart [ ( )] Filter Gain: Kt () = PtC () ( R ˆ) P(t) = A( ˆ)P(t) P(t)A ( ˆ) P(t)C ( ˆ)R C( ˆ)P(t) Q ˆ(t) = A( ˆ) ˆ(t) B( ˆ)u(t) G(t) K(t) y(t) C( ˆ) ˆ(t) /2/$3. 22 IEEE 765

4 D. -Dependent H-Infinity Filter () employs a state- dependent model and H-infinity design technique to estimate the system state. It is aimed at combining the advantages of both and. able V shows the algorithm of this filter. dependent conditions: Filter Gain: Estimated output: ABLE IV ALGORIHM (t) = f ((t),u(t)) w(t) y(t) = h((t)) v(t) E. Unscented Kalman Filter () he does not approimate the nonlinear process and observation models; it uses the true nonlinear models and the Gaussian probability density by a number of deterministically chosen points, called sigma point. he algorithm is based on the unscented transformations, which are more accurate than the linearization technique algorithm for propagating mean and covariance. he algorithm presented in able V is the most general form of. III. SIMULAION SUDIES ;w(t) N(,Q(t)) ;v(t) ~ N(, R(t)) (t) = A()(t) B()u(t) G(t) y(t) = C 2 ()(t) v(t) z(t) = C (t)(t) where : f( ) = A( ) t ˆ( ) = Et [ ( )] ; Pt ( ) = 2 ˆ Kt () = PtC () ( R ) P(t) = A( ˆ)P(t) P(t)A ( ˆ) P(t)γ 2 C (t)c (t)p(t) In this paper, the activated sludge process (ASP), which is the most generally applied biological wastewater treatment method, will be used for nonlinear state estimation study. he Activated Sludge Model No. (ASM) presented by the International Water Association (IWA) [22] are generally accepted as the reference model or benchmar model, which was primarily developed for municipal ASPs to describe the removal of organic carbon substances and nitrogen. Other models that improve and etend the capabilities of ASM are ASM2, ASM2d and ASM3 which can be found in [23]. P(t)C 2 ( ˆ)R (t)c 2 ( ˆ)P(t) Q ˆ(t) = A( ˆ) ˆ(t) B( ˆ)u(t) G(t) zt ˆ( ) = Ct ˆ () K(t) y(t) C 2 ( ˆ) ˆ(t) Since the use of such models is complicated and for the early stage of estimator design, the biological process adopted in the present wor is a simple model of an activated sludge process (ASP) proposed by Nejjari, Rou et al. [24]. he model truly respects the objectives of the process and was used widely in literature. It consists of an aeration tan and a secondary clarifier that is necessary for the settling of the biomass and its recycling as displayed in Fig.. ABLE V ALGORIHM ( ) = ( ) = f, u, t w; w N(, Q) y h, t v ; v N(, R ) Conditions: ˆ = E[ ], P = E ( ˆ )( ˆ ) Sigma point: ime update sigma points for nonlinear process: vector: (i) where: (i) = = (i),i =,..., (ni) = np i np () i,i =,...,n i () i () i () i = f(, u, t ) =,i =,...,n () i () i ( )( ) Measurement update sigma points for measurement: Measurement: Cross covariance: Filter Gain: Filter Estimates: P = Q i i (, ) () i () = y h t y = y () i () i () i ( )( ) P = y y y y R y i () i () i ( )( ) P = y y y K = PzP z = K ( y y ) P = P K P z K /2/$3. 22 IEEE 766

5 Influent S in, X in,q in In the aeration tan, the wastewater is aerated with oygen including carbonaceous oidation and nitrification where Q represents the secondary influent flow rate; Q r the return sludge flow rate; Q w the waste activated sludge flow rate and X e the effluent suspended solids. he mass balance on the aerator and the settler are described by the set of nonlinear differential equations [24]: X (t) = µ(t)x (t) D(t)( r)x (t) rd(t)x r (t) () S(t) = µ(t) Y X (t) D(t)( r)s(t) D(t)S in (2) C(t) = K o µ(t) Y Aeration tan S, X, V S, X r, Q r X (t) D(t)( r)c(t) K La (C S C(t)) D(t)C in (3) X r (t) = D(t)( r)x (t) D(t)(β r)x r (t) (4) where X(t), S(t), C(t) and X r (t) are the state variables representing the biomass, the substrate, dissolved oygen and the recycled biomass concentrations, respectively. D(t) is the dilution rate and the parameter r ( r = Qr / Q) and β ( β = Qw / Q). S in and C in corresponds to the substrate and dissolved oygen concentrations in the feed stream, respectively. he inetic of the cell mass production are defined in terms of the specific growth, ( µ = r / X) and the yield of cell mass, Y; the constants C S and K La, represent the dissolved oygen saturation concentration and the oygen transfer rate coefficient ( KLa = αw) with α > and W = air flow rate), and the term K o is a switching constant. Biomass growth assumed a double Monod law in substrate and dissolved oygen. he inetic model is given by [25]: St () Ct () µ () t = µ ma K S() t K C() t s S,X,QQ r c 2 nd clarifier S, X r Q r, Q w Fig.. Activated sludge reactor for the filter application g S e, X e, Q-Q w S r, X r,q w (5) IV. SIMULAION AND PERFORMANCE ANALYSIS A simulation study has been carried out to evaluate and compare the different estimation approaches applied to the ASP model. he simulation was performed with sampling time of.s. he filters presented herein assumed constant parameter values in the nonlinear model. In some cases however, the parameters can evolve during process operation. he following estimation configuration was chosen: the biomass X(t) and recycled biomass X r (t) are unavailable on-line and the estimation was carried out using the noisy measurements of substrate S(t) and dissolved oygen C(t). he dilution rate D(t) and the air flow rate W(t) are the two control variables. he tuning procedure adopted for the filters are identical with the same process and measurement noise. he covariance matrices P o, Q and R are assumed to be diagonal. he parameters and initial conditions used for simulation are given in able VI. ABLE VI PARAMEERS AND INIIAL CONDIIONS Process parameters Kinetic parameters conditions Y =.65 K s = mg/l X() = mg/l r =.6 K c = 2 mg/l S() = 4.28 mg/l β =.2 µ ma =.5 h - C() = 6. mg/l α =.8 m -3 S() = mg/l K o =.5 C s = mg/l C in =.5 mg/l S in = 2 mg/l m = meter, l = liter, h = hour, mg = miligram Under constant dilution rate D(t) and the air flow rate W(t), the estimation results for the unmeasured states, biomass X(t) and recycled biomass X r (t) are displayed in Figs. 2 3 while the substrate S(t) and dissolved oygen C(t) were not shown here. It shows that all the five filters have the ability to converge to its true states. Since the true states are difficult to observe because the filters provide almost the eact values, the quantitative analysis are presented. hree aspects are compared for filters: accuracy of state of each algorithm in terms of standard deviation of estimation error, Root Mean Squared s (RMSEs) with respect to its estimates and computation time. Even though, there is no approimation involved in and, in this study it is observed that the performance of the is superior to the other filters as demonstrated by the standard deviation data in able VII when using smaller value of process and measurement noise. he absolute estimation error for biomass X(t) and recycled biomass X r (t) in Fig. 4-5 has confirmed the superiority of compared to the other filters. uses the unscented transformation to directly approimate the nonlinear system. In this study, when both process and measurement noise are increased, and demonstrate comparable performance, as shown in able VII (Case B). he /2/$3. 22 IEEE 767

6 corresponding root mean square error (RMSE) for all the five filters, displayed in Fig. 6 also confirms this finding. X (mg/l) Biomass ASP Absolute Estimation in X (mg/l) Xr(mg/l) Fig. 2. rue nonlinear states X and its estimate Recycled Biomass ASP Fig. 3. rue nonlinear states X r and its estimate Absolute Estimation in Xr (mg/l) Fig. 4. Absolute estimation error in Biomass, X (Case A) Fig. 5. Absolute estimation error in Recycled Biomass, X r (Case A) ABLE VII SANDARD DEVIAION OF ESIMAION ERRORS Case A: Smaller process and measurement noise he most common case in the WWP field is the one in which plant uncertainty is present in initial conditions [2]. hus, in this study different initial conditions were given to the filters to observe the convergence properties for all the tested filters. As displayed in Fig. 7-8, the converges more quicly than the other filters. It is demonstrated that initial state covariance have a significant impact on the performance [26]. he computation time required for each method is display in able VIII using Case A where simulations are performed in Matlab R2b with a cloc speed of 3.2 GHz Pentium computer running Windows 7 using Matlab s built in function cpu time. It is observed that the is significantly faster than and other approaches. his is due to the fact the nonlinear model is transformed to a linear time varying model off-line and hence did not require much computational time as compared to the calculation of the Jacobian matri in. On the other hand, it is noted that the and have comparable computation times. Meanwhile, the etra tuning parameter gamma (γ) in the formulation of and which need to be iterated to find the best value of the gain has increased the CPU time for these filters. X S C X r Case B: Larger process and measurement noise X S C X r Fig. 6. Comparisons of RMSE for all filters /2/$3. 22 IEEE 768

7 Absolute Estimation in X (mg/l) Absolute Estimation in Xr (mg/l) V. CONCLUSION Fig. 7. Absolute estimation error in Recycled Biomass, X r with different initial condition Fig. 8. Absolute estimation error in Biomass, X with different initial condition ABLE VIII COMPUAION IME (SECONDS) In this paper the use of a number of estimation approaches to estimate unmeasured states of the activated sludge model proposed by Nejjari, Rou et al. [24] have been investigated and compared. Certainly, other techniques that are relevant to the field of WWPs eist and the authors did not pretend to be ehaustive; but these approaches were chosen because of its conceptual simplicity and generality. A good compromise between the quality of the estimation and the difficulty of implementation should be taen into account when to choose or design a filter for a specific application. A satisfactorily tested filter in one application does not necessarily produce satisfactory result in other application. o conclude, it is observed that all of the filters presented good convergence properties for WWPs. he study shows have better estimation accuracy and can be a costeffective preference to physical sensors for state estimation; thus, promising etended application of for monitoring the process variables, which are not directly measurable in the treatment of waste water plant. ACKNOWLEDGMEN he first author is grateful to the Malaysian Ministry of Higher Education for financial support to carry out this research. REFERENCES [] D. Dochain and P. Vanrolleghem, Dynamical modelling and estimation in wastewater treatment processes: Intl Water Assn, 2. [2] V. Alcaraz-González, et al., "Software sensors for highly uncertain WWPs: a new approach based on interval observers," Water Research, vol. 36, pp , 22. [3] K. J. Keesman, " and parameter estimation in biotechnical batch reactors," Control Engineering Practice, vol., pp , 22. [4] L. J. S. Luasse, et al., "A recursively identified model for short-term predictions of NH4/NO3 concentrations in alternating activated sludge processes," Journal of Process Control, vol. 9, pp. 87-, 999. [5] C. F. Lindberg, "Control and estimation strategies applied to the activated sludge process," PhD, Uppsala University, 997. [6] A. Gelb, Applied optimal estimation: MI press, 974. [7] I. Hoteit, et al., "A new approimate solution of the optimal nonlinear filter for data assimilation in meteorology and oceanography," Monthly Weather Review, vol. 36, pp , 28. [8] S. Xu and P. Van Dooren, "Robust H filtering for a class of non-linear systems with state delay and parameter uncertainty," International Journal of Control, vol. 75, pp , 22. [9] M. Grimble, "Optimal H robustness and the relationship to LQG design problems," International Journal of Control, vol. 43, pp , 986. [] M. Katebi and M. Grimble, "Design of dynamic ship positioning using etended H filtering," 998, pp [] D. Simon, "From Here to Infinity-H (infinity) filters can be used to estimate system states that cannot be observed directly. In this, they are lie Kalman filters. However, only H (infinity) filters are robust," Embedded s Programming, vol. 4, pp. 2-34, 2. [2] L. Xi-Mei, et al., "Fault diagnosis of High Voltage Direct Current system based on H filter," in IEEE International Conference on Robotics and Biomimetics, 27, pp [3] C. P. Mrace, et al., "A new technique for nonlinear estimation," in IEEE International Conference on Control Applications, 996, pp [4] J. R. Cloutier, et al., "Nonlinear regulation and nonlinear H-infinity control via the state-dependent Riccati equation technique. I- heory," 996, pp [5] F.Benazzi, "Software Sensor Designs for Urban Wastewater s," PhD, Department of Electronic and Electrical Engineering, University of Strathclyde, 26. [6] A. Iratni, et al., "On estimation of unnown state variables in wastewater systems," in IEEE Conference on Emerging echnologies & Factory Automation 29, pp. -6. [7] E. A. Wan and R. Van Der Merwe, "he unscented Kalman filter for nonlinear estimation," in Adaptive s for Signal Processing, Communications, and Control Symposium, 2, pp [8] S. J. Julier and J. K. Uhlmann, "A new etension of the Kalman filter to nonlinear systems," 997, p. 26. [9] S. J. Julier, et al., "A new approach for filtering nonlinear systems," 995, pp vol. 3. [2] D. Simon, Optimal state estimation: Kalman, H and nonlinear approaches: John Wiley and Sons, 26. [2] M. R. Katebi and M. J. Grimble, "Etended H-infinity filtering for dynamic ship positioning," in IFAC Conf. ACASP, Glasgow, UK, 998. [22] M. Henze, et al., "A general model for single-sludge wastewater treatment systems," Water Research, vol. 2, pp , 987. [23] M. Henze, Activated sludge models ASM, ASM2, ASM2d and ASM3 vol. 9: Intl Water Assn, 2. [24] F. Nejjari, et al., "Estimation and optimal control design of a biological wastewater treatment process," Mathematics and computers in simulation, vol. 48, pp , 999. [25] G. Olsson, " of the art in sewage treatment plant control," 976. [26] B. G. Saulson and K. C. Chang, "Nonlinear estimation comparison for ballistic missile tracing," Optical engineering, vol. 43, p. 424, /2/$3. 22 IEEE 769