Research Article  Open Access
Nonlinear PI Control with Adaptive Interaction Algorithm for Multivariable Wastewater Treatment Process
Abstract
The wastewater treatment plant (WWTP) is highly known with the nonlinearity of the control parameters, thus it is difficult to be controlled. In this paper, the enhancement of nonlinear PI controller (ENonPI) to compensate the nonlinearity of the activated sludge WWTP is proposed. The ENonPI controller is designed by cascading a sectorbounded nonlinear gain to linear PI controller. The rate variation of the nonlinear gain is automatically updated based on adaptive interaction algorithm. Initiative to simplify the ENonPI control structure by adapting has been proved by significant improvement under various dynamic influents. More than 30% of integral square error and 14% of integral absolute error are reduced compared to benchmark PI for DO control and nitrate in nitrogen removal control. Better average effluent qualities, less number of effluent violations, and lower aeration energy consumption resulted.
1. Introduction
The wastewater treatment plant (WWTP) is naturally aimed to remove the suspended substances, organic material, and phosphate from the water before releasing it to the recipient. The best technology available was used to control the discharge of pollutants emphasized in biological process, namely, as activated sludge process (ASP). In ASP, the organic materials are oxidized by microorganisms. The organic material is then transformed into carbon dioxide and some incorporate into new cell mass. The new cell mass then forms sludge that contains both living and dead microorganisms and, thus, contains organic material, phosphorous, and nitrogen [1]. Referring to [2], the common problems in WWTP are caused by maintenance issues and poor effluent quality which are due to poor control approaches.
According to [3], aeration process is a crucial part of the whole ASP. It is a nontrivial task to transport the oxygen from the air bubbles to the cells of the microorganisms, thus the process is commonly described by the oxygen mass transfer coefficient (). The is, in general, nonlinear and depends on the aeration actuating system and the sludge conditions [4]. Indeed, as referred to in [5], the dissolved oxygen (DO) is stated as a key variable and commonly applied in controlling the ASP. The level of DO in the aerobic reactors has a direct influence on the microorganisms’ activities in the activated sludge. The DO level should be sufficiently high so that enough oxygen can be delivered to the microorganisms in the sludge. However, an excessively high DO will require higher airflow rate and thus leads to higher energy consumption and deteriorate the sludge quality. Meanwhile, nitrogen removal in activated sludge requires twostep procedure which takes place simultaneously nitrification and denitrification processes. Nitrification is a process in which ammonium is oxidized to nitrate under aerobic (present oxygen) conditions. The nitrate formed by nitrification process, in turn, is converted into gaseous nitrogen under anoxic (absent oxygen) conditions, that is called denitrification. The improvement of DO concentration in aerated tanks and the nitrogen removal process contribute to big interest of activated sludge control.
The proportionalintegral (PI) technique is one of the control strategies that are frequently applied in WWTP. As referred to in [6], each part of PI controller highly contributes in achieving the control target. The proportional part is potential to increase the response speed and control accuracy while the integration part is normally used in eliminating the steadystate error of the system. The performances of several control strategies with PI controller to the WWTP have been discussed in [7, 8]. However, it is hard to achieve high control performance in all operating conditions with a linear PI controller due to different dynamic behaviours of the WWTP control parameters. More retuning task will always be demanded for a fixedgain PI controller. Therefore, a controller that is potential to maintain a balance of DOs concentrations and nitrogen removal process during the setpoint changes is highly demanded. Many approaches have been developed in improving the adaptability and robustness of the controller such as selftuning method, general predictive control, fuzzy logic, and neural network strategy. However, predictive control technique may require more complex control structure while human knowledge and system’s experts are strongly demanded in adaptive fuzzy controller. Besides, the crucial work is concentrated in estimating all of the inputoutput data within such a complex system and in determining the appropriate structure of the neural network controller.
Under these circumstances, enhanced nonlinear PI (ENonPI) controller is proposed to compensate the nonlinearity of the control parameters hence to improve the performance of the conventional linear PI controller. The design of the ENonPI controller is basically referred to [9] where the linear fixedgain PI controller is cascaded to a bounded nonlinear gain. As referred to in [9], the nonlinear gain function has two parameters to be determined in initial simulation such as the range of variation, , and the rate of variation, . Difficulties come to identify the appropriate combination of these parameters especially for a complex nonlinear system. Therefore, modifications to the ENonPI to automate one of the parameters are obviously proposed. The idea is to automatically update using simple updating algorithm, namely, as adaptive interaction algorithm (AIA). The theoretical of adaptive interaction is previously applied in neural network and PID control as referred to in [10, 11], respectively. AIA is generally a technique in which a system is decomposed into subsystems where an adaptation exists between them. It is believed that the is potential to be updated with respect to proportional control part as referred to in [11].
Two case studies are proposed for control design strategies with respect to dynamic behaviors of the WWTP. Case 1 highlights the improvement of the ASP with respect to DO concentration in all aerated tanks called DO_{345} control while the improvement of nitrogen removal process is aimed in Case 2. Cases 1 and 2 are considered due to different average time constant for DO and nitrate which is in minutes and several hours, respectively. In particular, the WWTP is naturally a multivariable system that is basically described as a system with more than one control loop. Changes in any input will generally affect all the outputs due to interaction between the inputs and outputs variables. However, a decentralized controller is a simple approach of multivariable controller designs. As a result, the plant to be controlled is essentially a collection of independent subplants where each element in the plant may be designed independently. The proposed EnonPI is developed in decentralized control structure in both simulation cases. To further extend, the study on how the ENonPI controller performs under various different dynamic conditions is covered. The proposed ENonPI controller is then tested to an updated Benchmark Simulation Model No. 1 (BSM1) with more complex sensors and noises as updated in [12].
The paper is organized as follows. The BSM1 is explained in Section 2 while the development of ENonPI with adaptation algorithm is presented in Section 3. The simulation result and discussion of welltuned ENonPI controller are presented in Section 4. Finally, Section 5 concludes the paper. For convenience of discussion, Table 1 lists the acronyms that frequently used in the paper.

2. Benchmark Simulation Model No. 1 (BSM1)
The WWTP used in the simulation is the benchmark plant developed in [12]; referred as BSM1. The plant consists of five tanks where the first two compartments are anoxic zones followed by three aerobic tanks as shown in Figure 1. Each tank is assumed to have constant volume of 1000 m^{3}, 1000 m^{3}, 1333 m^{3}, 1333 m^{3}, and 1333 m^{3}, respectively. The effluent from the last tank is connected in series to a settler of constant volume of 6000 m^{3}. The BSM1 is widely used as a standard model based on the most popular IWA Activated Sludge Model No. 1 (ASM1) proposed in [13]. The ASM1 was developed as to describe the removal of ammonium nitrogen and organic carbon. Meanwhile, the model proposed by [14] by was chosen to resemble the behavior of the secondary settler. The BSM1 is default controlled by PI controller where two control loops of nitrate in the second anoxic tank and the DO concentration in the final tank are emphasized. The performances of the benchmark PI are always used as comparison to the proposed controller. Detail on the model can be referred in [12].
2.1. Influent Load
To investigate the performance of the control strategy in various weather situations, three dynamic input files including dry, rain, and storm events that have realistic variations in the effluent flow rate and composition have been used. The data used for the estimation and control are sampled with a sampling period of 15 minutes given asIn any influent: (COD) m^{3}; COD m^{−3}; ; COD m^{−3}; mol m^{−3}. The description of influent’ variables is presented in Table 2.

The dry influent contains two weeks of dynamic dry weather influent data. The rain influent is based on the dry weather file with an added rain event during the second week. Similarly, the storm influent file is also based on the dry weather file but added with two storm events during the second week. There is a constant influent with constant flow and composition that is used during the system simulation under steady state condition. Refer to [12] for detailed explanation.
2.2. Performance Assessment
Twolevel performance assessment is highlighted in controlling the WWTP. The local control loop is assessed on the means of absolute error (Mean()), the integral of absolute error (IAE), the integral of square error (ISE), the maximum absolute deviation from setpoint (Max()), and the standard deviation of the error (Std()) at the first level. Meanwhile, the second level investigates the effect of the control strategy on the plant process operation with respect to economical and quality part. Two measuring assessments are considered in the simulation such as the effluent violations and the aeration energy consumed.
2.2.1. The Effluent Violations
Table 3 indicates the constraints of the effluent water quality. The flowweighted average effluent concentrations of the following variables must meet their corresponding limitations. Besides, the effluent violations can be reported through the number of violations and the percentage time plant is in violation. This quantity indicates the frequency of the plant effluent increases above the effluent constraint.

2.2.2. The Aeration Energy
The index of aeration energy (AE) is described as in (2). is the oxygen transfer coefficient in each tank, . The is calculated for the last 7 days, of the dynamic test weather conditions with unit of kWhday^{−1}. Consider
3. Development of Enhanced Nonlinear PI Controller
3.1. The Case Study
As mentioned, two case studies are highlighted for control design strategies. The improvement of the ASP with respect to DO concentration in all aerated tanks, tank 3 (DO_{3}), tank 4 (DO_{4}), and tank 5, (DO_{5}) called DO_{345} control is aimed in Case 1. Meanwhile, the improvement of the nitrogen removal process of nitrate and DO_{5} control is next highlighted in Case 2.
3.1.1. Case 1: Controlling the Aerated Tanks (DO_{345})
For decentralized control structure, the WWTP is partitioned into three SISO subsystems contributing to three ENonPI controllers. The implementation of DO_{345} control is shown in Figure 2.
3.1.2. Case 2: Controlling the NitrateDO_{5}
For the nitrogen removal process, the ENonPI controller is set to work correspondingly to the benchmark PI. The implementation of the controller is shown in Figure 3.
3.2. The Controller
For a conventional linear PI controller, the error signal is used to generate the proportional () and integral () control actions and to be summed in producing the control signal as generally expressed as in where and are the proportional and integral coefficients of the PI controller, respectively. However, the fixedgains of conventional linear PI controller have the limitation in controlling the timevariant characteristics and the process nonlinearities of the WWTP [15]. This problem can be alleviated by employing nonlinear elements in the PI control scheme and thus leads the development of the ENonPI controller.
As discussed, the ENonPI is designed by cascading a sectorbounded nonlinear gain to linear PI controller as described in (4). The nonlinear gain, is a function of error with respect to the changes of , that acts on the error in producing the scaled error; . The is then input to the PI controller thus generating the control action as The can be expressed by any nonlinear general function such as sigmoidal function, the hyperbolic function, and the piecewise linear function as explained in [16]. However, the used in the simulation are described in (5) and (6). Notice that the is automatically updated by the AIA while is the userdefined positive constant. Consider where
3.3. The Algorithm
The work on enhanced nonlinear PID by [9] is extended to adaptively update the . It is believed that the characteristic of is potential to be manipulated based on AIA. As mentioned, the AIA is generally a technique in which a system is decomposed into subsystems where an adaptation exists between them. The is updated with respect to proportional control part as referred to in [11]. The typical decomposition of a system for an adaptive interaction and the detail on AIA can be further referred in [10, 11, 17].
In conjunction to Case 1, DO_{3}, DO_{4}, and DO_{5} control loops are involved. However, the DO_{3} concentration is first considered to present the applied algorithm. The block diagram of ENonPI of DO_{3} control loop is shown in Figure 4. and are desired and measured outputs that result in the error of DO_{3}, . The is then applied by AIA in updating the for the variation of the . The integration of the functional of to the PI controller develops the ENonPI for DO_{3} control.
The aim here is to update of the nonlinear gain for the third tank using the the AIA so that the performance index which is the as in (7) is minimized. With respect to AIA, it is believed that interaction/adaptation of exists between the proportional gain transfer function of , , and the functional of the . The gradient method as given in (8) is then applied. Consider is the adaptation gain while is the Frechet derivative in relation to the plant input, , and the output, . The adaptation of is then reduced to The functional can be written in the convolution form as in (10). is the impulse response of the linear time invariant system for DO_{3} while denotes convolution. Therefore, the Frechet derivative can be expressed as in (11). Consider However in many practical systems, the Frechet derivative can be approximated as in (12) where is an arbitrary function and is a constant value. Consider This result approximates Frechet tuning algorithm as presented in Let the adaptive coefficient, and _{.} The tuning algorithm of thus can be simplified to (14). Therefore, might change and update it responses with time referring to the changes of in achieving good variation of the . The general function of used in the simulation can be referred in (5). Consider Taking and that represent the desired and measured outputs which result the error of DO_{4}, besides that denotes the proportional gain of tank four, the procedures of (7)–(13) are repeated. This results in an approximate Frechet tuning algorithm for tank four as described in (15). The same goes to DO control of tank five thus contributing to as in (16). Notice that , , and are the adaptation coefficients of , , and in controlling the DO_{3}, DO_{4}, and DO_{5} concentrations, respectively. For simplicity, the proportional gains of , , and are always set to 1 In fact, the procedures on (7)–(13) are repeated for nitrate control loop in Case 2. The tuning algorithm of is then described in (17). Meanwhile, similar algorithm presented in (16) is used for DO_{5} control. Consider
4. Results and Discussion
The simulation procedures of BSM1 can be referred to in [12]. In the ideal case, the BSM1 is first simulated for 150 days to attain a quasisteadystate using the constant influent input. This is done to guarantee that the initial conditions of the states are consistent. It then continued with 14day simulation of dry influent to set up the plant for the dynamic benchmark simulation. Finally, the plant is simulated for the next 14 days with the dynamic test input weather with noises present. However, only the data of the last 7 days are evaluated in control assessment. For DO control, the sensor of class A with a measurement range of 0 to 10 g(COD) m^{−3} and a measurement noise of 0.25 g(COD) m is used. Meanwhile, a class B0 sensor with a measurement range of 0 to 20 gNm^{−3} and measurement noise of 0.5 gNm^{−3} is applied in nitrate control. Two case studies are considered in the simulation, DO_{345} control and nitrogen removal process control.
4.1. Case 1: Controlling the DO_{345}
The DO concentrations in tank 3, tank 4, and tank 5 are set to 1.5 mg L^{−1}, 3 mg L^{−1}, and 2 mg L^{−1}, respectively, as referred to in [18]. Nevertheless, the previous work develops multivariable PID for COST simulation benchmark [19] instead of updated version [12] that is used in the present simulation. In addition, the oxygen mass transfer coefficient of DO_{3} (), the oxygen mass transfer coefficient of DO_{4} (), and the oxygen mass transfer coefficient of DO_{5} () are constrained to a maximum of 360 day^{−1}. The , , and are automatically updated using AIA as described in (14)–(16). Meanwhile, the adaptive coefficients, , , and are set to 0.09. is applied while the proportional gains and the integral time constants of linear PI controllers are set to 25 and 0.0020, respectively.
4.2. Case 2: Controlling the NitrateDO_{5}
The nitrateDO_{5} control of the nitrogen removal process is considered in the second case. The and are automatically updated using AIA as referred to in (16)(17). The internal recycle flow rate () and the are manipulated. To improve the nitrogen removal, the nitrate concentration is set to 1.0 gm^{−3} with constrained up to 5 times of stabilized input flow rate, 92230 m^{3} day^{−1}. The DO_{5} level is set to 2.0 gm^{−3} with constrained to a maximum of 360 day^{−1}. The and are similarly set to 0.09 while the and the PI control gains as in Case 1 are maintained in the simulation.
4.3. Discussion
It is aimed to improve the performance for the setpoint tracking of the load changes due to daily variations in different influents composition. As mentioned, the effectiveness of the proposed ENonPI controller is investigated in twolevel assessment, the performance of the controller and the performance of activated sludge process compared to benchmark PI.
4.3.1. The Performance of the Controller
The ENonPI is first assessed by investigating the Mean(), the IAE, the ISE, the Max(), and the Std(). As for comparison, the performance of ENonPI with adaptive is also compared to ENonPI with fixed . For this purpose, the , , and are set to 0.01 for DO_{345} control. Similarly is used for DO_{5} while is set to 0.1 for nitrate in nitrateDO_{5} control. Referring to Table 4, obvious improvement is obtained in Case 1 by ENonPI controller compared to benchmark PI controller. However, as referred to in Table 5, significant improvement is observed for nitrate compared to DO_{5} in Case 2. Difficulties come to control simultaneous nitrate and DO_{5} control loops in Case 2 that may due to different dynamic behaviours of the control parameters.


The variation of output and input variables of DO_{345} control in Case 1 under dry input weather by ENonPI with adaptive is shown in Figure 5. It was seen that the ENonPI controller with adaptive , , and manages well to keep DO_{3,} DO_{4,} and DO_{5} concentrations around the reference values. Besides, the input variables , , and are always kept under the upper bounds. Meanwhile, Figure 6 shows the variation of the error and the adaptation of the resulted. As observed, higher is demanded for a higher error resulted.
(a)
(b)
(a)
(b)
The nitrateDO_{5} control is then simulated with respect to benchmark PI, as applied in [12]. Figure 7 shows the comparative variation of output variables of nitrate and DO_{5} under rain input weather. Slight improvements are observed by ENonPI with adaptive and for nitrate and DO_{5} concentrations compared to benchmark PI. Nevertheless, the improvements are obviously better than the fixed . The variations of input variables of nitrateDO_{5} control are next illustrated in Figure 8. Similarly, the and are always kept under the upper bounds. Meanwhile, the variation of the errors and the adaptation of the rate variation resulted are illustrated in Figure 9.
(a)
(b)
(a)
(b)
(a)
(b)
4.3.2. The Performance of Activated Sludge Process
The second level of controller assessment is to investigate the effect of the ENonPI control strategy on the process of an activated sludge. Firstly, the performances in average effluent concentrations are compared to benchmark PI and the fixed , as indicated in Tables 6 and 7 for Cases 1and 2, respectively. Five main process variables including the total , N_{tot}, BOD_{5}, COD, and TSS are evaluated. The limit of effluent variables can be referred in Table 3. Overall in Case 1, in spite of total COD concentration, improvement on average effluents has been observed by ENonPI compared to benchmark PI. Obvious enhancement on the resulted by ENonPI with adaptive compared to benchmark PI under three input weathers. In particular, the improvement of is maintained by ENonPI with adaptive in Case 2. Besides, improvement on N_{tot} was recorded under dry and storm influents.


Next, the numbers of time that the effluent limits are not met during simulation obtained by ENonPI for Case 1 are presented in Tables 8 and 9. The number of violation of N_{tot} is observed under dry weather while it is extended to TSS under the storm weather. It was proved that the numbers of the effluent increases above the effluent constraints are reduced from 7 to 6 compared to benchmark PI under dry weather. In the meantime, it reduces from 7 to 5 and 2 to 1 for N_{tot} and TSS under storm weather condition, respectively. To clarify, the N_{tot} effluent violation compared to benchmark PI under dry and storm influents is shown in Figure 10.


(a)
(b)
Next, the average AE consumed in the process of activated sludge is illustrated in Figure 11. As observed in Case 1, the AE is significantly reduced by ENonPI with adaptive compared to benchmark PI under rain and storm events where the AE is minimized by 140.3118 and 46.7117 kwh per day, respectively. Meanwhile, about 0.5474 kwh per day of AE is saved with adaptive compared to fixed of ENonPI controller under storm weather which is the lowest AE consumption in Case 2. In fact, even though slight improvements were recorded by NonPI in Case 2 but the AE is mostly better than the benchmark PI controller.
(a)
(b)
5. Conclusion
The aim of this paper is to design a simple but effective controller so that the performance of activated sludge process DO concentration control (Case 1) and the nitrogen removal (Case 2) of the WWTP are improved. For this work, the enhanced nonlinear PI (ENonPI) controller is designed in which the conventional fixedgain PI controller is incorporated with the bounded nonlinear function as to compensate the nonlinearity of the WWTP. The characteristic of the rate variation, , is manipulated and automatically adapted based on adaptive interaction algorithm for a wide range of nonlinear gain.
From simulation, significant improvement is proved for DO_{345} control by ENonPI compared to benchmark PI. Notice that the Case 1 deals with similar dynamic behaviors of DO concentrations, thus it easier to be controlled. In contrast, difficulties to control the simultaneous nitrate and DO_{5} concentrations for Case 2 are undeniable due to different natures of both control parameters. Even though slight improvements were recorded by NonPI in Case 2 but it is mostly better than the benchmark PI controller.
The effectiveness to simplify the ENonPI control structure with adaptive has been proved with significant improvement on both case studies. The performance comparison indicates that the proposed ENonPI yields the most accurate strategy to control the DO concentration and the nitrogen removal process. For DO_{345} control, obvious improvement resulted where about 34.86%, 31.4894%, and 33.8202% of ISE are reduced compared to benchmark PI under dry, rain, and storm weathers, respectively. Again, more than 30% of ISE is reduced under three dynamic influent weathers, specifically for nitrate in nitrogen removal control. Meanwhile, more than 14% of IAE is reduced both simulation cases. Better average effluent concentrations and less number of the effluent violations resulted. Besides, lower average aeration energy is consumed specifically under rain and storm influents for DO_{345} control and in nitrate removal process, respectively. The proposed ENonPI shows benefit for simple and practical implementation in controlling various dynamic natures of the activated sludge process.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the Ministry of Education (MOE), Universiti Teknikal Malaysia Melaka (UTeM), and Universiti Teknologi Malaysia (UTM). Their support is gratefully acknowledged. The authors wish to thank the IWA Task Group on the benchmark simulation plant.
References
 R. Anders, Automatic control of an activated sludge process in a wastewater treatment plant—a Benchmark study [M.S. thesis], Department of Systems and Control, Information Technology, Uppsala University, 2000.
 A. Sanchez, DataDriven Control Design of Wastewater Treatment Systems, Department of Electronic and Electrical Engineering, University of Strathclyde, Glasgow, Scotland, 2004.
 B. Holenda, E. Domokos, Á. Rédey, and J. Fazakas, “Dissolved oxygen control of the activated sludge wastewater treatment process using model predictive control,” Computers and Chemical Engineering, vol. 32, no. 6, pp. 1270–1278, 2008. View at: Publisher Site  Google Scholar
 C. A. C. Belchior, R. A. M. Araújo, and J. A. C. Landeck, “Dissolved oxygen control of the activated sludge wastewater treatment process using stable adaptive fuzzy control,” Computers and Chemical Engineering, vol. 37, pp. 152–162, 2012. View at: Publisher Site  Google Scholar
 V. Ramon, K. Reza, and N. A. Wahab, “Nremoval on wastewater treatment plants: a process control approach,” Journal of Water Resource and Protection, vol. 3, no. 1, pp. 1–11, 2011. View at: Publisher Site  Google Scholar
 S. Shoujun and L. Weiguo, “Application of improved PID controller in motor drive system,” in PID Control, Implementation and Tuning, T. Mansour, Ed., InTech, 2011. View at: Google Scholar
 A. Stare, D. Vrečko, N. Hvala, and S. Strmčnik, “Comparison of control strategies for nitrogen removal in an activated sludge process in terms of operating costs: a simulation study,” Water Research, vol. 41, no. 9, pp. 2004–2014, 2007. View at: Publisher Site  Google Scholar
 M. Yong, P. Yongzhen, and U. Jeppsson, “Dynamic evaluation of integrated control strategies for enhanced nitrogen removal in activated sludge processes,” Control Engineering Practice, vol. 14, no. 11, pp. 1269–1278, 2006. View at: Publisher Site  Google Scholar
 Y. X. Su, D. Sun, and B. Y. Duan, “Design of an enhanced nonlinear PID controller,” Mechatronics, vol. 15, no. 8, pp. 1005–1024, 2005. View at: Publisher Site  Google Scholar
 R. D. Brandt and F. Lin, “Adaptive interaction and its application to neural networks,” Information Sciences, vol. 121, no. 34, pp. 201–215, 1999. View at: Publisher Site  Google Scholar  MathSciNet
 F. Lin, R. D. Brandt, and G. Saikalis, “Selftuning of PID controllers by adaptive interaction,” in Proceedings of the American Control Conference, pp. 3676–3681, June 2000. View at: Google Scholar
 J. Alex, L. Benedetti, J. Copp et al., “Benchmark Simulation Model No. 1 (BSM1),” Tech. Rep., IWA Taskgroup on Benchmarking of Control Stategies for WWTPs, 2008. View at: Google Scholar
 M. Henze, C. P. L. Grady Jr., W. Gujer, G. V. R. Marais, and T. Matsuo, Activated Sludge Model no. 1, 1987.
 I. Takács, G. G. Patry, and D. Nolasco, “A dynamic model of the clarificationthickening process,” Water Research, vol. 25, no. 10, pp. 1263–1271, 1991. View at: Publisher Site  Google Scholar
 J. P. Segovia, D. Sbarbaro, and E. Ceballos, “An adaptive pattern based nonlinear PID controller,” ISA Transactions, vol. 43, no. 2, pp. 271–281, 2004. View at: Publisher Site  Google Scholar
 H. Seraji, “A new class of nonlinear PID controllers with robotic applications,” Journal of Robotic Systems, vol. 15, no. 3, pp. 161–181, 1998. View at: Google Scholar
 B. M. Badreddine, A. Zaremba, J. Sun, and F. Lin, Active damping of engine idle speed oscillation by applying adaptive PID control [Ph.D. thesis], The British Library, 2001.
 N. A. Wahab, R. Katebi, and J. Balderud, “Multivariable PID control design for activated sludge process with nitrification and denitrification,” Biochemical Engineering Journal, vol. 45, no. 3, pp. 239–248, 2009. View at: Publisher Site  Google Scholar
 J. B. Copp, COST Action 624: The COST Simulation Benchmark. Description and Simulation Manual, Office for Official Publications of the European Communities, 2002.
Copyright
Copyright © 2014 S. I. Samsudin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.