Microsoft Word - 164.docx CHEMICAL ENGINEERING TRANSACTIONS VOL. 56, 2017 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Guest Editors: Jiří Jaromír Klemeš, Peng Yen Liew, Wai Shin Ho, Jeng Shiun Lim Copyright © 2017, AIDIC Servizi S.r.l., ISBN 978-88-95608-47-1; ISSN 2283-9216 Interval Type-2 Fuzzy GMC for Nonlinear Stochastic Process of Methane Production in the Anaerobic Digester System Mohd Fauzi Zanil*,a, Mohd Azlan Hussainb aChemical & Petroleum Engineering Department, Faculty of Engineering, Technology & Built Environment, UCSI University, 56000 Kuala Lumpur, Malaysia bChemical Engineering Department, Faculty of Engineering, University of Malaya, 50603, Kuala Lumpur, Malaysia mohdfauzi@ucsiuniversity.edu.my The paper focused on the implementation of hybrid Interval Type-2 (IT2) Fuzzy with Generic Model Control (GMC) for the nonlinear stochastic waste treatment process in the anaerobic digester. Development of the deterministic methane process model has been extended to a set of stochastic nonlinear differential equations. The stochastic effect is introduced by adding white noise with unit covariance to give an interesting profile like physical plant dynamic. The IT2 Fuzzy based on Takagi-Sugeno-Mendel and GMC by Lee and Sullivan have been developed to control the holdup pH inside reactor. The pH value is being manipulated by the flowrate of Sodium hydroxide at optimal methane gas production condition. The process variables that need to be controlled and included into the controller are pH, error and change of error while the consequence fuzzy set output is from the GMC backpropagation law equations. As a result from several studies; servo and regulatory, the controller show significant improvement on the set point tracking and disturbance rejection over typical Fuzzy, Fuzzy-GMC and conventional Proportional-Integral-Derivative (PID) controllers. It shows the controller is suitable for stochastic process and nonlinear control system application. 1. Introduction Several nonlinear control techniques have been proposed lately for an anaerobic digester control system: High loading rate compensator (Charles et al., 2011); Multivariable model based control system (Méndez-Acosta et al., 2010); Dynamic profile mapping by model predictive controller (Padhiyar and Bhartiya, 2009). A fuzzy logic controller is widely proposed due to the ability for nonlinear process and able to operate at many operating conditions like controllability at multiple linear regions (Zanil et al. 2015) has showed the effectiveness of the fuzzy system to handle the process nonlinearity. However, the framework is not sensitive to the uncertainty factor in the process dynamics (Saadat et al., 2013). As a result, the control performance of the conventional fuzzy will vary and may aggravate proportionally with degree of certainty in the process. Hence, the fuzzy calculation framework can be integrated with a mathematical model to improve the control performance (Zanil et al. 2014). Type-2 fuzzy is able to solve nonlinear processes in abide enough information are utilised but it is mostly used for uncertain process which the membership function in a fuzzy set are difficult to obtain (Lin et al., 2015). Simple approach to this difficulty are to define an uncertain boundary for the control system (Biglarbegian and Mendel, 2015), or to decompose each of fuzzy variable into type-1 fuzzy sub-system (Su and Chen, 2015), or integrate general algorithm inside the general fuzzy framework to guarantee the controller stability (Li et al., 2012). It has been presented in Tung et al. (2013), type-2 fuzzy are much more effective to solve a process with high degree of uncertainties and nonlinearities. Nevertheless, the trade-off is the inference calculation are more complex (Tang et al., 2011) than type-1 fuzzy systems. Therefore, a special type of type-2 fuzzy called interval type-2 fuzzy is used in Mendel and John (2002). Several control applications have been successfully demonstrated: level control (Zhao et al., 2012); bioreactor control (Van Lith et al., 2002); pH control (Carbureanu, 2014). Like conventional fuzzy, interval type-2 shared a common drawback in which the construction of fuzzy set is complex and depended on the input-output matching (Shamiri et al., 2015). Thus, a mechanism like observer can be integrated to support this limitation. Most of the above studies concluded that further exploration of interval Type-2 Fuzzy is needed. This work, hybrid IT2Fuzzy-GMC controller is designed and implemented in anaerobic digester process to control the alkalinity of the reactor. Without involving model reduction or approximations, a simulation of detailed dynamic mathematical model for the stochastic process has been carried out using MATLAB/Simulink (MATLAB, 2012). DOI: 10.3303/CET1756234 Please cite this article as: Zanil M.F., Hussain M.A., 2017, Interval type-2 fuzzy gmc for nonlinear stochastic process of methane production in the anaerobic digester system, Chemical Engineering Transactions, 56, 1399-1404 DOI:10.3303/CET1756234 1399 2. Methane Production The methane production in anaerobic treatment uses a microbiological process for degradation of organic matter. The degradation processes consist of simultaneous reactions for Hydrolysis, Acidogenesis, Acetogenesis, and Methanogenesis process (Vanwonterghem et al., 2015) and the process parameters (Yaldiz et al., 2011). The first stage of anaerobic digestion is a hydrolysis process whereby long-chain organic polymer compounds such as carbohydrates, proteins and fats are hydrolysed to the soluble monomers such as fatty acids or glycerol, amino acids or peptides and monosaccharides or disaccharides are formed from fats, proteins and carbohydrates by hydrolysis bacteria (Bacillus Pumilus). Bio-Organic waste water contains large number of carbon and hydrogen atoms, hydrolysis process is vital in decomposing long-chain hydrocarbons into simple, soluble monomers. After that, Acidogenesis process takes place in which the soluble monomers are converted into volatile fatty acids (VFA), carbon dioxide, water, alcohols and lactic acids. Next, Acetogenesis process which is the oxidization of organic acids to acetic acid, hydrogen, and carbon dioxide. Finally, the Methanogenesis process will take place where the acetate acid is utilised to produce remaining biogas like methane gas and carbon dioxide. Material balance in liquid phase for microbial biomass 𝑋𝑖 (A,AP,AB,M), insoluble substrate, 𝐶𝑖𝑠, soluble substrate, 𝐶𝑠, acetate, 𝐶𝑎𝑐, propionate, 𝐶𝑝𝑟, butyrate, 𝐶𝑏𝑢, ammonium, 𝐶𝑁𝐻4 , and carbon dioxide, 𝐶𝑐 are formulated in Eqs(1) to (8): 𝑑𝑋𝑖 𝑑𝑡 = 1 𝑉 (𝑋𝑖,𝑓 − 𝑋𝑖 ) + 𝑟(𝜎𝑖 )𝜇𝑖 𝑋𝑖 + 𝑓(𝜗𝑖 ) (1) 𝑑𝐶𝑖𝑠 𝑑𝑡 = 1 𝑉 (𝐶𝑖𝑠,𝑓 − 𝐶𝑖𝑠) − 𝑘𝐶𝑖𝑠 (2) 𝑑𝐶𝑠 𝑑𝑡 = 1 𝑉 (𝐶𝑠,𝑓 − 𝐶𝑠) + 162𝑦𝑒 162 + 17𝑛 𝑘𝐶𝑖𝑠 − 12.858 𝜇𝐴 (3) 𝑑𝐶𝑎𝑐 𝑑𝑡 = 1 𝑉 (𝐶𝑎𝑐,𝑓 − 𝐶𝑎𝑐 ) + 3.45𝜇𝐴𝑋𝐴 + 8.006𝜇𝐴𝑃𝑋𝐴𝑃 + 15.366𝜇𝐴𝐵 𝑋𝐴𝐵 − 24.135𝜇𝑀𝑋𝑀 (4) 𝑑𝐶𝑝𝑟 𝑑𝑡 = 1 𝑉 (𝐶𝑝𝑟,𝑓 − 𝐶𝑝𝑟 ) + 2.93𝜇𝐴𝑋𝐴 − 10.566𝜇𝐴𝑃 𝑋𝐴𝑃 (5) 𝑑𝐶𝑏𝑢 𝑑𝑡 = 1 𝑉 (𝐶𝑏𝑢,𝑓 − 𝐶𝑏𝑢) + 3.079𝜇𝐴𝑋𝐴 − 3.079𝜇𝐵𝑃 𝑋𝐴𝐵 (6) 𝑑𝐶𝑁𝐻4 𝑑𝑡 = 1 𝑉 (𝐶𝑁𝐻4,𝑓 − 𝐶𝑁𝐻4 ) + 𝐶𝑁𝐻4 𝑉 ( 17(𝑛−𝑚(1−𝑦𝑒)) 162+17𝑛 ) 𝑘𝑖 𝑆𝑖𝑠 − 0.15(𝜇𝐴𝑋𝐴 + 𝜇𝐴𝑃 𝑋𝐴𝑃 + 𝜇𝐴𝐵 𝑋𝐴𝐵 + 𝜇𝑀 𝑋𝑀 ) (7) 𝑑𝐶𝑐 𝑑𝑡 = 1 𝑉 (𝐶𝑐,𝑓 − 𝐶𝑐 ) + 2.413𝜇𝐴𝑋𝐴 + 1.01𝜇𝐴𝑃𝑋𝐴𝑃 − 3.303𝜇𝐴𝐵 𝑋𝐴𝐵 + 16.726𝜇𝑀𝑋𝑀 − 𝑁𝑐 𝑉 (8) Next, Eq(9) to (11) show the material balance that took place for each species in gas phase; carbon dioxide, 𝑃𝑐, methane, 𝑃𝑚: 𝑑𝑃𝑐 𝑑𝑡 = 𝑅𝑇 𝑉𝑔 ( 𝑁𝑐 44 + 𝑃𝑐 𝑃 𝐹𝑡 ) (9) 𝑑𝑃𝑚 𝑑𝑡 = 𝑅𝑇 𝑉𝑔 ( 𝑁𝑚 𝛼 16 + 𝑃𝑚 𝑃 𝐹𝑡) (10) 𝐹𝑡 = 𝑃 𝑃 − 𝑃𝑤 ( 𝑁𝑚 16 + 𝑁𝑐 44 ) (11) where, 𝐹𝑡 is total flowrate in gas, 𝑃𝑚 , 𝑃𝑤, 𝑅, 𝑇 and 𝑉𝑔 represent the partial pressure of methane and water, gas constant, temperature and gas volume. The pH of the reactor can be obtained by the Eqs(12) and (13), which the ionic disassociation rate at 35 °C are tabulated as in Table 1 (Lange and Dean, 1979): 𝑝𝐻 = − log[𝐶𝐻+ ] (12) 𝐶𝐻+ + 𝐶𝑁𝐻4 + = 𝐶𝑂𝐻 −1 + 𝐶𝐻𝐶𝑂3− + 2𝐶𝑐𝑜32− + 𝐶𝑎𝑐 − + 𝐶𝑝𝑟 − + 𝐶𝑏𝑢− − +𝐶𝐴𝑛𝑖𝑜𝑛− − 𝐶𝐶𝑎𝑡𝑖𝑜𝑛+ (13) 1400 Table 1: Chemical equilibrium and ionic disassociation of methane production Chemical Equilibrium Disassociation rate, 𝑘𝑛 and Value 𝐶𝑂2 + 𝐻2𝑂 ↔ 𝐻𝐶𝑂3 − + 𝐻+ 𝑘𝑎1 = [𝐶𝐻𝐶𝑂3− ][𝐶𝐻+ ]/[𝐶𝑐 ] 𝑘𝑎1 = 4.909 × 10 −7 𝐻𝐶𝑂3 − ↔ 𝐶𝑂3 2− + 𝐻+ 𝑘𝑎2 = [𝐶𝑐𝑜32− ] [𝐶𝐻+ ]/[𝐶𝐻𝐶𝑂3− ] 𝑘𝑎2 = 5.623 × 10 −11 𝐻𝐴𝑐 ↔ 𝐴𝑐− + 𝐻+ 𝑘𝑎3 = [𝐶𝑎𝑐 − ][𝐶𝐻+ ]/[𝐶𝑎𝑐 ] 𝑘𝑎3 = 1.730 × 10−5 𝐻𝑃𝑟 ↔ Pr− + 𝐻+ 𝑘𝑎4 = [𝐶𝑝𝑟 − ][𝐶𝐻+ ]/[𝐶𝑝𝑟 ] 𝑘𝑎4 = 1.445 × 10−5 𝐻𝐵𝑢 ↔ Bu− + 𝐻+ 𝑘𝑎5 = [𝐶𝑏𝑢− ][𝐶𝐻+ ]/[𝐶𝑏𝑢] 𝑘𝑎5 = 1.445 × 10−5 𝑁𝐻4 + ↔ 𝑁𝐻3 − + 𝐻+ 𝑘𝑎6 = [𝐶𝑎𝑚][𝐶𝐻+ ]/[𝐶𝑁𝐻4 + ] 𝑘𝑎6 = 1.567 × 10 −9 𝐻2𝑂 ↔ 𝑂𝐻 − + 𝐻+ 𝑘𝑤 = [𝐶𝑂𝐻− ][𝐶𝐻+ ] 𝑘𝑤 = 2.065 × 10 −14 3. Control system and IT2Fuzzy-GMC controller design The objective of control system is to control a pH inside the digester. A combination of interval type-2 fuzzy system (IT2Fuzzy) and generic model control (GMC) is proposed as a controller in this work (Figure 1). The IT2Fuzzy consist of five interface; Fuzzification, Fuzzy Rules, Fuzzy Inference, Type reducer and Defuzzification. While, GMC is designed to compute good manipulated variable to achieve desire trajectory for the set-point. The formulation of GMC has been integrated inside the output membership function. As the outcome, an appropriate control output which are based on plant characteristic (from GMC) and expert knowledge (from IT2fuzzy). Fuzzification Defuzification Fuzzy Inference Fuzzy Rules GMC de/dt u ugood ce Se Sce IT2Fuzzy-GMC controller µR uc,i Type-reducer Sout e + - Secondary Control Loop Anaerobic Process + + pH Measurement + + n2 n1 pHref Figure 1: Control system with IT2Fuzzy-GMC controller architecture The feedback control with cascade system is used in this work and the process is exposed to a stochastic white Gaussian noise, 𝑛1 and unmeasured random distributed noise, 𝑛2 at the anaerobic digester process and pH measurement. In this work, the controller architecture with multiple input and single output (MISO) has been selected, there are 2 inputs from error,𝑒 and change of error, 𝑐𝑒., while, the output of controller is flowrate, 𝑢 of caustic soda which to regulate the alkalinity of the process. Inside secondary loop, a PI controller is used to manipulate the control valve of caustic soda stream where the control valve is varied to track the reference flow rate. Interval type-2 fuzzy system is similar to the conventional fuzzy system where as the difference is that the type- reducer is added before the defuzzification block. The outputs are then processed by the type-reducer which combines all the output sets and then centroid defuzzification is performed to produce minimum and maximum crisp outputs (Mendel and John, 2002). This work, two inputs and single output has been considered which input 1, 𝑥𝑒 ∈ 𝑋𝑒; input 2, 𝑥𝑐𝑒 ∈ 𝑋𝑐𝑒 and output 𝑢 ∈ 𝑈. The fuzzy rules represent the input space of 𝑋𝑒 × 𝑋𝑐𝑒 and output space of 𝑈. The combination of three membership functions are used for each fuzzy input and it resulted to 32 number of fuzzy rules (refer Table 2), where the lth rule has the form as Eq(14); 𝑅𝑙 : 𝐼𝐹 𝑥𝑒 𝑖𝑠 �̃�𝑒 𝑙 𝐴𝑁𝐷 𝑥𝑐𝑒 𝑖𝑠 �̃�𝑐𝑒 𝑙 𝑇𝐻𝐸𝑁 𝑢 𝑖𝑠 �̃�𝑙 ; 𝑙 = 1, 2, 3 … 9 (14) 1401 IF e → Positive, 𝑥𝑒,1 Zero, 𝑥𝑒,2 Negative, 𝑥𝑒,3 AND ce ↓ Positive, 𝑥𝑐𝑒,1 Mid Open, 𝑢2 Mid Close, 𝑢4 Close, 𝑢5 Zero, 𝑥𝑐𝑒,2 Mid Open, 𝑢2 Good,𝑢3 ≡ 𝑢𝑥 Mid Close, 𝑢4 Negative, 𝑥𝑐𝑒,3 Open, 𝑢1 Mid Open, 𝑢2 Mid Close, 𝑢4 A centre of set type-reduction method is selected due to the less computational complexity in firing rules. It is used to compute the input-output uncertainty 𝑢𝑙 and 𝑢𝑟. It can be calculated iteratively by Karnik and Mendel algorithm and it is expressed as Eq(15): 𝑈𝑐𝑜𝑠 (𝑈 1, … , 𝑈9, 𝐹1, … , 𝐹9) = [𝑢𝑙 , 𝑢𝑟 ] = ∫ … 𝑦1 ∫ ∫ … 𝑓1 𝑦9 ∫ ∑ 𝑓𝑖9𝑖=1 ∑ 𝑓𝑖 𝑢𝑖9𝑖=1𝑓9 (15) where 𝑈𝑐𝑜𝑠 is an interval value of T2Fuzzy which covers the most left and right in the output set. The total firing strength, 𝑓 𝑖 and 𝑓𝑖 can be written as Eqs(16) and (17): 𝑓 𝑖 (𝑒, 𝑐𝑒) = 𝜇 𝐹𝑒 (𝑥𝑒 ) ∩ 𝜇𝐹𝑐𝑒 (𝑥𝑐𝑒 ) (16) 𝑓 𝑖 (𝑒, 𝑐𝑒) = 𝜇𝐹𝑒 (𝑥𝑒 ) ∩ 𝜇𝐹𝑐𝑒 (𝑥𝑐𝑒 ) (17) The fuzzy output; minimum, 𝑢𝑙 and maximum, 𝑢𝑟 are reordered such that 𝑢𝑙 1 ≤ 𝑢𝑙 2 ≤ ⋯ 𝑢𝑙 𝑁 and 𝑢𝑟1 ≤ 𝑢𝑟2 ≤ ⋯ 𝑢𝑙 𝑁 which 𝑢𝑟 (𝑅) and 𝑢𝑙 (𝐿) in the range of 0 ≤ 𝐿, 𝑅 ≤ 𝑁 as Eqs(18) and (19): 𝑢𝑙 (𝐿) (𝑒, 𝑐𝑒) = ∑ 𝑓 𝑗 (𝑒, 𝑐𝑒) 𝑢𝑙 𝑗𝐿 𝑗=1 + ∑ 𝑓 𝑗 (𝑒, 𝑐𝑒) 𝑢𝑙 𝑗𝑁 𝑗=𝐿+1 ∑ 𝑓 𝑗 𝐿 𝑗=1 (𝑒, 𝑐𝑒) + ∑ 𝑓 𝑗 𝑁𝑗=𝐿+1 (𝑒, 𝑐𝑒) (18) 𝑢𝑟 (𝑅) (𝑒, 𝑐𝑒) = ∑ 𝑓 𝑗 (𝑒, 𝑐𝑒) 𝑢𝑙 𝑗𝑅 𝑗=1 + ∑ 𝑓 𝑗 (𝑒, 𝑐𝑒) 𝑢𝑙 𝑗𝑁 𝑗=𝑅+1 ∑ 𝑓 𝑗𝑅𝑗=1 (𝑒, 𝑐𝑒) + ∑ 𝑓 𝑗 𝑁 𝑗=𝑅+1 (𝑒, 𝑐𝑒) (19) The interval set is defuzzified by using average of 𝑢𝑙 and 𝑢𝑟, thus the crisp value output is calculated as Eq(20): 𝑢(𝑒, 𝑐𝑒) = 𝑢𝑙 (𝐿) (𝑒, 𝑐𝑒) + 𝑢𝑟 (𝑅) (𝑒, 𝑐𝑒) 2 (20) The fuzzy output is divided into five membership functions comprised of control action, 𝑢𝑥(𝑒, 𝑦∗) from Eq(21) and several constants, 𝑢𝑐,𝑖 . Details of control action for GMC can be referred to Lee and Sullivan (1988). 𝑢𝑥 (𝑒, 𝑦 ∗) = 𝐾1(𝑦 ∗ − 𝑦) + 𝐾2 ∫(𝑦 ∗ − 𝑦)𝑑𝑡 (21) At one instance, the final control action can be obtained from the distribution of several possible membership output which is based on the GMC output value and degree of association in IT2Fuzzy. 4. Result and Discussion The control system with proposed controller has been simulated to shows the feasibility of control system performance. In recent year, the interest of stochastic nonlinear control system have been increased tremendously. Coherent to this, any propose controller must implies to have a good control performance for unpredictable environmental changes, either these changes occurs internal or external to it. Figure 2 shows the trajectory of control system for servo case analysis. The tracking performance of proposed controller overtake all tested controllers by 12 % improvement in response time without any overshoot and zero offset. From Figure 3, when system is exposed to external disturbance, conventional Proportional-Integral-Derivative (PID) controller failed to regulate the set-point at desire value. At this condition, conventional fuzzy logic controller shows less effective in which the rejection time is higher than proposed controller by 12 unit time. The proposed controller is suitable for both cases in these simulation analyses. 1402 Table 2: Fuzzy Rule of control loop system Figure 2: Comparison trajectories of multiple set-point tracking Figure 3: Comparison trajectories of the pH variable with external disturbances 5. Conclusion In this work, control performance of hybrid IT2Fuzzy and GMC controller has been demonstrated. The proposed controller has showed a significant improvement for process with stochastic dynamic compared to conventional fuzzy and PID controllers. The implementation of proposed controller is recommended for a control system with high degree of uncertainty and nonlinearity process. 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