Abstract: This study presents the design and implementation of a Coefficient Diagram Method (CDM) based PI controller in a pH neutralization system. By using the experimental step test method, the nonlinear pH system is approximated as First Order Plus Time Delay (FOPTD) transfer function. Based on the CDM, the tuning parameters of PI controller are designed and implemented in the laboratory scale strong acid-strong base pH neutralization system. The servo and regulatory performance of the proposed CDM-PI control strategy is compared with other conventional PI controllers at two different highly sensitive operating points. In addition, the robustness test is performed. The experimental results show that the proposed CDM-PI control strategy is effective and potential for severe non linear control problem.
INTRODUCTION
Most of the process industries generate wastewater (effluent) as an offshoot of their production. The effluent discharged by these industries consist organic, inorganic chemicals and toxic metals. It is a major source of environmental pollution and also has a major impact on human health. In accordance with the environmental conservation act and environmental rules, it is mandatory to install Effluent Treatment Plants (ETPs) to treat the wastewater before it is disposed into the environment (Hanif et al., 2005).
An important and common technique used in wastewater treatment system is neutralization. The purpose of the neutralization is to adjust or control the pH value of the wastewater so that it does not have impact over the environment. However, it is very difficult to control pH process with adequate performance due to its severe nonlinearity, time varying properties and sensibility to small disturbance when working near the equivalence point. Therefore, more reliable, accurate, robust, efficient and flexible control systems are required for pH neutralization process.
In order to fulfill the above requirements, there is a continuing need for research on improved forms of control. Many research studies have been reported in the literature such as linear and nonlinear adaptive controller (Jutila, 1983; Gustafsson and Waller, 1992), self-tuning adaptive controller (Lin et al., 2000), model based controller (Palancer et al., 1996; Rivera et al., 1986), neuro controller (Akesson et al., 2005), fuzzy controller (Ahmed et al., 2007), non-linear gain scheduling based on neuro-fuzzy controller (Zhang, 2001) and genetic based controller (Valarmathi et al., 2009). Even though there are significant developments in the control systems, the chronic Proportional Integral (PI) controller is by far the most widely used control algorithm for pH neutralization process. The primary task of the controller is to maintain the process at desired operating conditions and to achieve the optimum performance when facing various types of disturbances. The typical transfer function models are used to represent the process and it can be easily controlled with PI controller (Nithya et al., 2008). It also known that the improvements in the tuning of PI controller will have a significant practical impact. At this juncture, a simple and robust tuning is keenly needed. Hence, in the present study, a simple and robust control strategy based on polynomial approach, namely Coefficient Diagram Method (CDM) is considered as a candidate to design the PI controller.
OVERVIEW OF COEFFICIENT DIAGRAM METHOD (CDM)
CDM is a polynomial algebraic approach and proposed by Manabe in the year 1991. The algebraic approach (CDM) was then said to be an alternative for conventional and modern control theories and uses polynomial expression for the mathematical representation. The advantageous parts of these control theories are combined to form the principles of CDM and it is derived by using the previous experience and knowledge about the controller design. Without confronting with serious difficulties and necessitating much experience, CDM makes possible to design very good controllers with less effort and relative ease when compared with the other existing methods (Manabe, 1998). By comparing the existing method, it is very easy to design a controller under the conditions of stability, time domain performance and robustness. Also, CDM is less sensitive to disturbances and bounded uncertainties resulted from the parameter variations. The important property of CDM is that the designer can have complete control over the transient response by specifying the key parameters namely stability indices (γi) and equivalent time constant (τ) at the beginning of the design. The simultaneous design nature exists in CDM, gives advantages to designer to keep good balance between the rigor of the requirements and the complexity of the controller.
Mathematical model: The standard CDM block diagram for single input single output system is shown in Fig. 1, where, y is the output signal, r is the reference input, u is the controller signal and d is the external disturbance signal. N(s) and D(s) are numerator and denominator polynomials of the plant transfer function. A(s) is the forward denominator polynomial while F(s) and B(s) are the reference numerator and the feedback numerator polynomials of the controller transfer function.
| Fig. 1: | Standard CDM block diagram |
For the given system, the output of the CDM control system is given by:
| (1) |
where, P(s) is the characteristic polynomial of the closed-loop system and defined by:
| (2) |
Here, the controller polynomials (A(s) and B(s)) are given as:
| (3) |
When polynomial F(s) is chosen as:
| (4) |
the overall closed loop transfer function becomes Type-1 system. Therefore, a good closed-loop response can be achieved.
Design of CDM controller: The design parameters of CDM are the stability indices (γi) and equivalent time constant (τ). The stability indices determine the stability of the system and the transient behavior of the time domain response (with overshoot, without overshoot and oscillations etc.). In addition, they determine the robustness of the system to parameter variations. The equivalent time constant which is closely related to the bandwidth and it determines the rapidity of the time response. According to Manabe (1998), the design parameters are defined as follows:
| (5) |
where ts is the user specified settling time:
| (6) |
where, i = 1, …, n-1, γ0 = γ∞ = 0
If necessary, the designer can modify the values of stability indices.
Using the design parameters defined in Eq. 5 and 6, a target characteristic polynomial (Ptarget(s)) is formulated as:
| (7) |
By substituting the controller polynomials in Eq. 3 into Eq. 2, the closed loop characteristic polynomial P(s) are obtained. This polynomial is compared with Eq. 7 to obtained the coefficients of CDM controller polynomials li, ki and ai.
DESIGN OF PROPOSED CDM-PI CONTROLLER
The design procedure for the proposed CDM-PI control strategy is summarized as follows:
| • | The given system is approximated as FOPTD model |
| • | The equivalent transfer function of the above said FOPTD model is determined using first order Pade’s approximation technique |
| • | The standard CDM block diagram given in Fig. 1 is reduced as its equivalent block diagram as shown in Fig. 2 using block diagram reduction techniques (Hamamci et al., 2007) |
In Fig. 2, the CDM controller polynomials (A(s) and B(s)) and pre-filter element (F(s)) are chosen as:
| (8) |
| (9) |
The stability indices (γ1 and γ2) are selected from the Eq. 6 or the designer can change the value of stability indices, if required. However, the equivalent time constant (τ) is not specified and it is considered as another variable to be solved.
The closed loop characteristic polynomial (P(s)) and target characteristic polynomial (Ptarget(s)) are determined using Eq. 2 and 7. By equating P(s) and Ptarget(s), the CDM controller parameters (k1 and k0) and the equivalent time constant (τ) are computed.
The proposed CDM-PI control system is displayed in Fig. 3 consisting of the main controller C(s) and feedforward controller Cf(s). Imposing the conditions:
on the control system, the steady state error to unit step change and unit step disturbance become zero (Hamamci et al., 2007). To satisfy the above conditions, the C(s) must include an integrator. Hence, C(s) is chosen as:
| (10) |
in the conventional PI element and Cf(s) is an appropriate element satisfying the condition.
Finally, the PI controller parameters (Kc and Ti) in terms of CDM controller polynomials are found by relating the Fig. 2 and 3. Figure 2 and 3 indicate that the C(s) and Cf(s) are expressed as:
| (11) |
| (12) |
| Fig. 2: | Equivalent CDM block diagram |
| Fig. 3: | Proposed CDM-PI control system |
Substitute the Eq. 8 and 10 in Eq. 11, derives:
| (13) |
By equating the coefficients of like power terms, the CDM-PI controller parameters (Kc and Ti) are obtained as follows:
| (14) |
Substitute the Eq. 8 and 9 in Eq. 12, the parameter of feedforward controller is found to be:
| (15) |
Since the parameter of Cf(s) depends on the CDM-PI controller parameters directly, the designer need not perform extra calculation for the feedforward controller.
EXPERIMENTS AND ANALYSIS
System description: The details pertaining to experimental work carried out in the laboratory scale pH neutralization system are described in this section and shown in Fig. 4. The experimental setup consists of four transparent perspex glass tanks (acid, base, water and process) with a maximum storage capacity of 5.3 L each. To maintain constant head in the system, level sensors have been employed. The strong acid (Hydrochloric acid HCl, 0.1 N) and strong base (Sodium hydroxide NaOH, 0.1 N) solutions are prepared in feed tanks having a storage capacity of 100 L, from which the solution is pumped to the respective tanks using fractional submersible pump.
| Fig. 4: | Photographic view of pH neutralization experimental setup |
The flow rate of both the streams is controlled individually using equal percentage control valves which normally open with Cv of 0.16. To introduce a disturbance, a ¼” needle valve is provided in the buffer (water) stream to increase the buffer flow rate.
The process stream of strong base enters as an inlet feed into a CSTR. It is being neutralized by HCl of 0.1 N through the control valve, which is controlled by PI controller. To avoid overflow in CSTR, constant volume of liquid (3.5 liters) is maintained by means of overflow line. A motorized agitator with a constant speed of 200 rpm is used to maintain uniform degree of homogenization in CSTR. The pH value of the solution in the process tank is measured by pH sensor located in the CSTR. The output of the pH sensor is converted into current signal (4-20 mA) by means of a transmitter powered by a 24 V DC power supply.
The current signal is fed as input to the PC based controller and it is compared with its set point at which the pH value is to be controlled. The pH system is interfaced with the PC through specially designed microcontroller based VDPID-03 unit. It helps in implementing real time control algorithm written in MATLAB/SIMULINK platform. The output current signal (4-20 mA) from the PC based controller is converted into a pneumatic signal (3-15 psig) using I/P converter. This pneumatic signal is directed to actuate the control valve which acts as a final control element for manipulating the load of acid flow rate. In this way, it brings the system to its desired pH value.
Model identification: The pH system is modeled as FOPTD transfer function:
around two nominal operating points of pH 11 and 2 which are close to the inflection point. In the open loop scheme, the operating point of pH 11 is maintained by regulating the acid flow rate. Then, a step change with a magnitude of ±08 and ±10% DAC output is given to the control valve of acid stream.
| Table 1: | Identified model parameters at different operating points |
As a consequence, the value of pH varies and this variation is recorded (through pH probe) against time until a new steady state is attained. The recorded data are plotted against time to obtain reaction curve by which first order model parameters (process gain Kp and process time constant τp) of the pH system are determined (Gobal, 2002). The same procedure is repeated for another operating point of pH 2. The identified model parameters are tabulated in Table 1.
Among all these models, larger process gain and smaller time constant are chosen as model parameters. Hence, the parameters for DAC output of -10% change at the operating point of pH 2 is selected to represent the pH system for the design of controllers. The identified model is represented as:
| (16) |
Here, the process delay (θ = 1.71 min) is approximately considered as 20% of the process time constant (Swati et al., 2008).
Experimental works: The experimental works carried out in the pH neutralization system is described in this section. The experiments are carried out in three folds. In the first fold, the performance of the newly developed CDM-PI control strategy is tested for the set point tracking. Next, the robustness of the controller is tested. In the last and third fold, the regulatory performance of the CDM-PI controller is analyzed. In all the three folds, the performance of the CDM-PI controller is compared with conventional PI control techniques such as Ziegler-Nichols (Bhaba et al., 2007; Abbas, 1997) and Saeed Tavakoli (Tavakoli and Fleming, 2003). For convenience, these techniques are abbreviated as ZN-PI, AB-PI and ST-PI, respectively.
The PI controller settings for the above said controllers are worked out based on the FOPTD model given in Eq. 16. Taking the model parameters into account and choosing the stability indices values γ1 = 2.5, γ2 = 2, the CDM-PI settings are determined as:
| (17) |
The controller parameters for ZN-PI: Kc = 0.6338, Ti = 5.69, AB-PI: Kc = 0.3253, Ti = 9.4 and ST-PI: Kc = 0.3844, Ti = 8.91, are also determined.
PERFORMANCE EVALUATION AND COMPARISON
First fold-Set point tracking: In this section, the set point tracking performance of the CDM-PI controller is evaluated and compared with ZN-PI, AB-PI and ST-PI controllers. Experimental runs for the set point tracking of ±05% and ±10% with CDM-PI, ZN-PI, AB-PI and ST-PI control system are carried out at the operating point of pH 6. The tracking responses are plotted in Fig. 5 to 8.
From the Fig. 5 to 8, it can be seen that, the servo responses obtained by other conventional PI controller are highly oscillatory and never settled at the tracking period. At the same time, the servo response made by CDM-PI controller is forced to follow the set point and yields a fair transient response when compared to others. CDM responses are found to have low percent or no overshoot and smallest settling time with little oscillations. This is not the case with responses of other control techniques. To analyze the performance of the controllers, the different performance measures such as Errors indices (Integral Squared Error (ISE), Integral Absolute Error (IAE)), Quality indices (rise time tr, settling time ts, peak overshoot %Mp) are used. In addition, the Total Variation (TV) of the output (y) is calculated using the expression.
| Fig. 5: | Servo responses for set point tracking of +05% at the operating point of pH 6 |
| Fig. 6: | Servo responses for set point tracking of –05% at the operating point of pH 6 |
| Fig. 7: | Servo responses for set point tracking of +10% at the operating point of pH 6 |
| Fig. 8: | Servo responses for set point tracking of –10% at the operating point of pH 6 |
to evaluate the controller effort. The minimum TV value represents the smoothness and consistency of the output signal (Chen and Seborg, 2002). The output derived from the above figures is presented in Table 2. From this table, it is cleared that the CDM-PI controller gives minimum error indices, good quality indices and minimum TV when compared to the other conventional PI controllers.
Second fold-Robustness test: The study of controller performance without robustness test will be incomplete. In this study, the robustness of the CDM-PI controller is tested by conducting an experimental run at another operating point of pH 9. The robustness metrics obtained using CDM-PI, ZN-PI, AB-PI and ST-PI control system for ±05% and ±10% change at the operating point of pH 9 is shown in Fig. 9 to 12. The performance measures are tabulated in Table 3. From the Fig. 9-12 and Table 2, it is observed that the CDM-PI controller provides better performance with the same settings for different operating point.
| Table 2: | Servo performance of the controllers at the operating point of pH 6 |
| ** Not settled | |
| Fig. 9: | Servo responses for set point tracking of +05% at the operating point of pH 9 |
| Fig. 10: | Servo responses for set point tracking of –05% at the operating point of pH 9 |
Among the four controller tuning rules, CDM-PI tolerates the perturbations in the model parameters when the operating point changes and provides the most consistent and robust response.
| Fig. 11: | Servo responses for set point tracking of +10% at the operating point of pH 9 |
| Fig. 12: | Servo responses for set point tracking of –10% at the operating point of pH 9 |
| Table 3: | Servo Performance of the controllers at the operating point of pH 9 |
| **Not settled | |
Third fold-Regulatory performance: The disturbance rejection property of CDM-PI, ZN-PI, AB-PI and ST-PI control systems are investigated. A step disturbance is introduced into the system by the way of increasing the buffer (water) flow rate from 0 to 1 Lpm at the operating point of pH 9. The response to this disturbance is shown in Fig. 13. Figure 13 indicates that CDM-PI controller is the one to damp the disturbance in a shorter time.
| Fig. 13: | Regulatory response at the operating point of pH 9 |
| Table 4: | Regulatory performance of the controllers at the operating point of pH 9 |
| **Not settled | |
The error indices, quality indices and TV of output signal are used to evaluate the disturbance rejection performance of the controllers are tabulated in Table 4. The ZN-PI, AB-PI and ST-PI control systems in this regard is affected by the nonlinear (gain variations) behavior of the system. As far the CDM-PI controller, its disturbance rejection performance is not affected by the nonlinearity. The performance measures confirm that CDM-PI control strategy is successful in disturbance rejection.
CONCLUSION
In this study, a CDM based PI control strategy is designed for a strong acid-strong base pH neutralization process. The designed control strategy is implemented in real time operations. The performance of the CDM-PI control system in the set point tracking and load disturbance rejection are evaluated and compared with other conventional PI control techniques. The results put forward CDM-PI control strategy. In addition, the robustness of the control systems is also tested. It concludes that the CDM based PI controller works well against the uncertainties of the nonlinear pH system.
NOMENCLATURE
| y | : | Output signal |
| r | : | Reference input |
| u | : | Controller signal |
| d | : | External disturbance signal |
| N(s) | : | Numerator polynomial of the plant transfer function |
| D(s) | : | Denominator polynomials of the plant transfer function |
| A(s) | : | Forward denominator polynomial of the controller transfer function |
| F(s) | : | Reference numerator polynomial of the controller transfer function |
| B(s) | : | Feedback numerator polynomial of the controller transfer function |
| P(s) | : | Characteristic polynomial of the closed-loop system |
| Ptarget(s) | : | Target characteristic polynomial of the closed-loop system |
| li, ki and ai | : | Coefficients of CDM controller polynomials |
| C(s) | : | Main controller |
| Cf(s) | : | Feed forward controller |
| K1 and K0 | : | CDM controller parameters |
| Kc and Ti | : | CDM-PI controller parameters |
| HCl | : | Hydrochloric acid |
| NaOH | : | Sodium hydroxide |
| Kp | : | Process gain |
| tr | : | Rise time |
| ts | Settling time | |
| %Mp | Peak overshoot |
Greek symbols
| γi | : | Stability indices |
| τ | : | Equivalent time constant |
| τp | : | Process time constant |
| θ | : | Process delay |
Abbreviation
| CDM | : | Coefficient diagram method |
| FOPTD | : | First order plus time delay |
| PI | : | Proportional integral |
| ETP | : | Effluent treatment plant |
| CSTR | : | Continuous stirred tank reactor |
| DC | : | Direct current |
| PC | : | Personal computer |
| DAC | : | Digital to analog converter |
| ISE | : | Integral squared error |
| IAE | : | Integral absolute error |
| TV | : | Total variation |
| Lpm | : | Liter per minute |
| CDM-PI | : | Coefficient diagram method-proportional integral |
| ZN-PI | : | Ziegler Nichols-proportional integral |
| AB-PI | : | Abbas-proportional integral |
| ST-PI | : | Saeed Tavakoli-proportional integral |
ACKNOWLEDGMENTS
We are cheerfully showering our heartfelt thanks to Prof. S.E. Hamamci, Associate Professor, Department of Electrical-Electronics Eng, Inonu Unievrsity, Turkey for his help and valuable suggestion. Our special gratitude is due in good measure to University Grant Commission (UGC), New Delhi, India for providing financial support to carry out this research work (File No. 35-116/ 2008 (SR) dated 25.03.2009).