NEURO MODELING AND CONTROL
STRATEGIES FOR A pH PROCESS
E.SIVARAMAN
Department of Instrumentation Engg., Annamalai University, Annamalainagar, Tamilnadu, India-608002
S.ARULSELVI
Department of Instrumentation Engg., Annamalai University, Annamalainagar, Tamilnadu, India-608002
Abstract :
The control of pH process has a vast range of applications in wastewater treatment, biochemical and electrochemical processes, the paper and pulp industry and many other areas. Tight control of pH is also critical in the production of pharmaceuticals. However, the dynamics of pH process is highly nonlinear, time varying with change in gain of several orders. It is very difficult to investigate the dynamic behavior of such systems using conventional modeling techniques thereby designing controller parameters. In this paper, a neural network based internal model control is designed for a pH process and its performance is compared with conventional PI controller and inverse neuro control. The forward and inverse neuro models are developed using Levenberg – Marquardt algorithm.
Keywords: Nonlinear; Inverse; Control; pH; Modeling.
1. Introduction
Modeling of the pH process is considered to be a difficult task because one needs to have knowledge about the components and their nature in the process stream in order to model its dynamics using conventional techniques. In the modeling aspect, rigorous models from first principles involving the material balance and equilibrium equations were established in [1] and later extended in [2] through the concept of the reaction invariant, and more complicated situations were considered in [3]. Due to the susceptibility to change in operating point, varying gain and load disturbances, the performance of the practical processes deviates from conventional modeling output [4]. The potential capability of neural networks in modeling and control of non-linear dynamics systems has been recognized since last decade [5], and research in this area has been intensively investigated. Neural network has the capacity to capture the nonlinear dynamics and model mismatch of the system. Hence, conventional model – based control strategies for dynamic processes, such as internal model control (IMC) and model inversion control has been redesigned using neural network [6]. The Levenberg – Marquardt (LM) algorithm is the fastest training algorithm reported in the literature for modeling. This is an extension to the standard back propagation algorithm.
Hence, in this paper an attempt is made to develop model based control techniques like, internal model control and inverse control using Levenberg – Marquardt algorithm for a pH process. Their performances are compared with conventional PI controller, which is designed based on pole placement technique. The remainder of this paper is organized as follows. In section 2, mathematical modeling of pH process is described. In section 3, an overview of neural modeling is presented. In section 4, designs of different types of controllers are presented. Simulation studies and discussions are presented in section 5. In section 6, conclusion of this paper is presented.
2. Mathematical Modeling of pH Process
Fig. 1 Schematic diagram of acid-base neutralization process.
The material and ionic balance forms a set of linear differential equations and a nonlinear algebraic equations as given below dt dX V X F F C
FA A ( A B) A A (1)
dt dX V X ) F F ( C
FB B A B B B
(2) 2 B A 3 ] H )[ X K ( ] H
[ (K (XB XA) KW )[H ] KAKW 0
A
(3) ] H [ log pH 10
(4)
where, volume of the process tank (V) = 7.5 L, inlet acid concentration (CA) = 0.2 mol/L, inlet base concentration (CB) = 0.1 mol/L, flow rate of acid (FA) = 0 – 0.5 L/min, flow rate of base (FB) = 0.4 L/min, acid equilibrium constant (KA) = 1.8 x 10-5, water equilibrium constant (KW) = 1.0
10-14.The steady-state titration curve for the acid-base system is obtained by solving the equations (1), (2), (3) and (4) in simulation using MATLAB software for change in the acid flow rate (FA) from 0 – 0.5 L/min. The simulated titration curve is shown in Fig.2. The static nonlinear behavior of the system is clearly visible at acid flow rate of around 0.2 L/min. when the acid flow rate varies from 0.19702 to 0.201115 L/min, the pH value drastically changes from 11 to 7. Even a small change in acid flow in this region will cause drastic change in the pH value. Therefore conventional controller will not give satisfactory response for pH process. This necessitates the design of model based controllers based on neural network. In Fig.2 the titration curve is divided into four zones for the design of PI controller.
Fig.2. Simulated titration curve
3. Neural Modeling
From the previous discussion it is clear that the modeling and control of pH process is not simple. Hence, Neural Network (NN) has the capacity to capture the nonlinear dynamics and model mismatch of the pH process. The forward and inverse neuro models are developed using Levenberg – Marquardt algorithm.
CH3COOH NaOH
V
FA FB
CA CB
3.1. Generation of Input Output Data
By changing the acid flow rate as PRBS sequence as shown in Fig.3 is given to the pH process and the corresponding output is obtained as shown in Fig.4.The identification data set, containing N = 1000 samples with sampling time of 15 sec. The data matrix Z was constructed from the identification data set as
T pH pH pH Fa Fa Fa Z
1000 2 1 1000 2 1 ,.... , ,.... ,0 200 400 600 800 1000
0.19 0.195 0.2 0.205 0.21 Time (Samples) A c id f lo w r a te ( L /m in )
Fig.3. PRBS Input to the pH process
0 200 400 600 800 1000
6 7 8 9 10 11 12 Time (Samples) pH
Fig.4. Output response of the pH process
3.2. Forward Neural Model
The neural network approach is trained to represent the forward dynamics of the pH process. The network is trained using delayed outputs and current input. The Activation function for the hidden layer is Tansigmoidal, while for the output layer linear function is selected and they are bipolar in nature. The block diagram of forward neural network model is shown in Fig.5. The Levenberg Marquardt (LM) learning algorithm does the correct choice of the weight. The parameters used for training are given below:
Input vectors : [ pH(k-1) pH(k-2) u(k)] Output vector : pHˆ(k)
Learning rate : 0.001 Training parameter goal : 1e-3
Fig.5. Block diagram of forward neural model of pH process
3.2.1. Training and Model Validation
The data set used for training is sufficiently rich to ensure the stable operation, since no additional learning takes place after training. During training the NN learns the forward dynamics of the pH process by fitting the input-output data pairs. This is achieved by using the LM algorithm. The simulated forward model input-output is shown in Fig.6. The training pattern of MSE is shown in Fig.7 and this error goal is attained within 348 epochs. It is observed from Fig.6 that forward model output exactly matches with output of the actual process. Hence, the neural network has the ability to model forward dynamics of the pH process, which can be used for developing the model based controllers.
Fig.6. Response of forward neural model and Actual process output
Fig.7. Variation of MSE for forward neural model during training.
3.3.Direct Inverse Neural Model
The neural network approach is also trained to capture the inverse dynamics of the pH process. The network is trained using delayed sample of outputs and delayed input of pH process. The Activation function for hidden layer and output layer are bipolar tansigmoidal and bipolar pure linear are used to give the desired output as acid flow rate, which is the input signal to the pH process. The Levenberg Marquardt (LM) learning algorithm does the correct choice of the weight. The block diagram of direct inverse neural model is shown in Fig.8. The parameters used for training are given below:
Input vectors :[u(k-1) pH(k-1) pH(k-2)] Output vectors : (k)uˆ
Sampling interval : 15 sec Learning rate : 0.001 Training parameter goal :1e-8
Fig.8. Block diagram of direct inverse neural model.
0
50
100
150
200
250
300
10
-410
-210
010
2348 Epochs
T
rai
ni
n
g-B
lue G
o
al
-B
lac
k
3.3.1. Training and Model Validation of Inverse Neural Model
During training the NN learns the inverse of the pH dynamics by fitting the input-output data pairs. This is achieved by using the LM algorithm. The simulated inverse model output is shown in Fig.9. The training pattern of MSE is shown in Fig.10 and this error goal is attained within 225 epochs. It is clear from Fig.9 that the inverse model output exactly matches with input of the actual model. Hence, the neural network has the ability to model inverse dynamics of the pH process, which can be used for developing model-based controllers.
Fig.9.Comparison of inverse neural model with Actual process output.
Fig.10. Variation of MSE for inverse neural model during training.
4. Design of Controllers
4.1.Design of Local PI Controller
As discussed in section 2 the steady-state curve is divided into four zones. The design of local linear controller in each of the zones requires the knowledge of process gain (Kp) and time constant (τp). The gain is estimated as the ratio of change in the pH to the change in acid flow rate. The time constant of the model is evaluated in each zone from the simulation result on an actual nonlinear system for transient response for step change of the acid flow rate around the nominal operating point. The average values of the parameters for the increased and decreased step change are taken in each of the zones. The local linear PI controller is designed in each zone by the pole placement technique with the closed loop poles placed at -1 and -1 in the complex plane [8]. The choice of the pole locations is arbitrary, but they are usually placed at a distance of 10 to 20 times that of the open-loop poles to give a satisfactory closed-loop response. Table I gives the linear model parameters and the controller
0 50 100 150 200
10-5 100
225 Epochs
T
rai
ni
ng-B
lue G
o
al
-B
la
c
k
parameters in each zone. pH value are increased for decreasing acid flow rate. Therefore the process and controller gains in the various zones are negative.
Table I. Process and Controller Parameter Settings for Various Zones.
Zone
Nominal operating
point
Process Gain
K
p (min/L)Time Constant
τ
p(min)Controller Gain Kc
Integral Time
T
i (min)1 (pH: 10 to 12) 11 - 238 16.11 - 0.1312 1.9379
2 ( pH: 8 to 10 ) 9 - 6740 8.16 - 0.0023 1.8775
3 ( pH: 6 to 8 ) 7 - 627 12.14 - 0.0488 1.9367
4 (pH: 4.5to 6) 5 - 5 15.8 - 4.6555 1.9176
4.2.Design of Direct Inverse Neuro Controller
In the direct inverse control technique, the inverse model will act as the controller in cascade with the system under control, without any feedback. In this control scheme the desired setpoint acts as the desired output which is fed to the network together with the past plant inputs and outputs to predict the desired current plant input. The block diagram of direct inverse neural controller is shown in Fig.11.In the ideal situation, with no modeling errors and disturbances, inverse controller yields perfect control with zero steady state error. However, it is fail to work for load disturbances [7]. This can be overcome by incorporating the inverse model controller in the well-known internal model control scheme.
Fig.11. Block diagram of inverse neural control.
4.3.Design of Internal Model Neuro Controller
The internal model neuro control approach is similar to the direct inverse control approach as discussed above except for two additions. First is the addition of the forward model placed in parallel with the plant, to cater for plant or model mismatches and second is that the error between the plant output and the neural net forward model is subtracted from the set point before being fed into the inverse model [5]. The other data fed to the inverse model is similar to the direct method. A filter can be introduced prior to the controller in this approach to incorporate robustness in the feedback system, especially where it is difficult to get exact inverse models. The block diagram of internal model neuro controller is shown in Fig.12.
Inverse neural model
pHref
(k)
uˆ(k)pH(k) Z-1
Z-2
Plant
Fig.12. Block diagram of internal model Neuro control.
5. Simulation Results and Discussions
Fig 13 shows the comparison of closed-loop simulation results of the pH process for setpoint change implementing PI controller, inverse neuro controller and Internal model neuro controller at various operating points. The expanded portion of Fig. 13 from 1000 to 1500 samples is shown in Fig. 14. The corresponding manipulated variable comparisons are shown in Fig. 15. Fig.16 shows the regulatory response (10 % changes in base flow rate) of pH process at the operating point of pH 9. Since, at this operating point the process gain is very high. A small disturbance causes large deviation in this operating point. From Fig.(13),(14) and (16) and Table II the internal model neuro controller shows superior performance, i.e., less integral square error (ISE) and faster response, in comparison with direct inverse controller and PI controller.
0 500 1000 1500 2000 2500 3000
6.5 7 7.5 8 8.5 9 9.5 10 10.5 11
Time (samples)
pH
setpoint
NN Direct inverse controller NN Internal model controller PI controller
1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 8
8.2 8.4 8.6 8.8 9 9.2
Time (samples)
pH
setpoint
NN Direct inverse controller NN Internal model controller PI controller
Fig.14. Setpoint change from 8 to 9 of pH process implementing direct inverse, internal model and PI controllers.
Fig.14. Controller output of pH process using direct inverse, internal model and PI controllers.
Fig.15. Controller output of pH process using direct inverse, internal model and PI controllers.
Table II. Performance of controllers.
Controller
Settling Time(sec)
ISE (servo) ISE for 10% change in base flow rate Servo Regulatory
PI 2100 3300 236790 20205
Inverse neuro
controller 600 --- 207850 ---
Internal model neuro
controller 600 900 207400 11700
0 500 1000 1500 2000 2500 3000
0.195 0.196 0.197 0.198 0.199 0.2 0.201 0.202 0.203
Time (samples)
A
c
id
f
lo
w
r
a
te
(
L
/m
in
)
0 200 400 600 800 1000 1200 1400 1600 1800 2000 6
7 8 9 10 11 12
Time (Samples)
pH
PI controller
Internal model neuro controller Inverse neuro controller
Fig.16. Regulatory response of pH process using direct inverse, internal model and PI controllers.
6. Conclusion
In the present work, the nonlinear titration curve is obtained for a pH process. The entire titration curve is divided into four zones, and the system is approximated as a first-order process. The process parameters are found by process reaction curve method for step change in acid flow. The PI controller settings are obtained by pole placement technique. The neural network based direct inverse controller and internal model controller are developed for a pH process using Levenberg-Marquardt learning algorithm. The servo response of pH process at various operating points shows that the direct inverse controller and internal model controller performances are better than PI controller. The direct inverse controller is failing to work for load disturbances in the pH process. The regulatory response of pH process at various operating points shows that the internal model controller performance is better in terms of less integral square error and faster settling time when compared with direct inverse controller and PI controller. Therefore the internal model controller using Levenberg-Marquardt learning algorithm is working properly for both servo and regulatory problems.
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