Control Strategies for Water Level Control of Two Tank System

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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 8] AUGUST 2013

Control Strategies for Water Level Control of Two Tank System Pawan Kumar Kushwaha and Vinod Kumar Giri* ABSTRACT- Level sensors detect the level of substances that flow, including liquids, slurries, granular materials, and powders. Fluids and fluidized solids flow to become essentially level in their containers because of gravity whereas most bulk solids pile at an angle of repose to a peak.

The substance to be measured can be inside a container or can be in its natural form. The PID controller calculation involves three separate constant parameters, and is accordingly sometimes called three-term control; the proportional, the integral and derivative values, denoted as „P‟ „I‟ and ‟D‟. Heuristically, these values can be interpreted in terms of time; „P‟ depends on the present error, „I‟ on the accumulation of past errors, and „D‟ is a prediction of future errors, based on current rate of change. The weighted sum of these three actions is used to adjust the process via a control element such as the position of a control valve. An abundant amount of research work has been reported in the past on the tuning of PID controllers.

Internal model control and error-integral criteria tuning formulae are to mention only a few. The control of liquid level in tanks and flow between tanks is a problem in the process technologies. The process technologies require liquids to be pumped, stored in tanks, and then pumped to another tank systematically. The conventional control algorithms are difficult to reach required control quality.In this paper we present an efficient elementary idea about the PID controller system, fuzzy logic controller and water level control for water tank system has been presented. The result shown in the paper is encouraging & promising.

KEY WORDS: PID Controllers, Fuzzy Logic Controller, Water Level Control, SISO. INTRODUCTION Proportional Integral Derivative (PID) controllers are widely used in industrial practice since last six decade. The invention of PID control is in 1910 (largely owing to Elmer Sperry‟s ship autopilot) and the straightforward Ziegler-Nichols (Z-N) tuning rule in 1942 [1]. Today, PID is used in more than 90% of practical control systems, ranging from consumer electronics such as cameras to industrial processes such as chemical processes. The PID controller helps to get our output (velocity, temperature, position) where we want it, in a short time, with minimal overshoot, and with little error [2]. It also the most adopted controllers in the industry due to the good cost and given benefits to the industry [3]. Many nonlinear processes can be controlled using the well-known and industrially proven PID controller [4]. A considerable direct performance increase (financial gain) is demanded when replacing a conventional control system with an advanced one [4]. The maintenance costs of an inadequate conventional control solution may be less obvious. The tricky part of controller design is to figure out just how much of a corrective effort the controller should apply to the process in each case.

Some situation requires tighter control of the process variable than On-Off control can provide. Proportional control provides better control because its output operate linearly anywhere between fully ON and fully OFF [5]. As its name implies, its output changes proportionally to the input error signal. Proportional controller simply multiplies the error by a constant to compute its next output.

Author: Pawan Kumar Kushwaha is currently pursuing Master of Technology program in Electrical Engineering in MMM. Engg. College, Gorakhpur India, E-mail: pawankus@gmail.com *Co-Author: Vinod Kumar Giri is Associate Prof. in MMMEC Gorakhpur India.

In 1930s the control engineers discovered that the error could be eliminated altogether by automatically resetting the set point to an artificially high value [3, 6]. The PID controllers function is to maintain the output at a level that there is no difference (error) between the process variable and the set point in as fast response as possible Fuzzy logic is derived from fuzzy set theory. It deals with reasoning, approximation rather than precise values. The concept of Fuzzy Logic (FL) was conceived by Lotfi Zadeh, a professor at the University of California at Barkley and presented not as a control methodology. Fuzzy logic allows intermediate values to be defined between conventional evaluations like true/false, yes/no, high/low, etc. Some of the controllers with their mathematical equation are as follows.

Proportional Controller In a controller with proportional control action, there is a continuous linear relation between the output of the controller m (manipulated variable) and actuating error signal e.

Mathematically m (t) = Kp e ( t ) Where Kp is known as proportional gain or proportional sensitivity.

Integral Controller In a controller with integral control action, the output of the controller is changed at a rate which is proportional to the actuating error signal e (t).

Mathematically d/dt m(t) = K i e (t) Where K i is a constant.

Derivative Controller In a controller with derivative control action the output of the controller depends on the rate of change of actuating error signal e.

Mathematically m (t) = K d d/dte(t) Where K d is known as derivative gain constant.

Proportional-Plus-Integral Controller This is the combination of proportional and integral control action.

Mathematically m( t ) = K p e(t) +K p K i ∫ e(t)dt

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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 8] AUGUST 2013 Qo3 =flow rate of fluid between tanks

Proportional-Plus-Derivative Controller When a derivative control action is added in series to proportional control action, then this combination is termed as proportional-derivative control action. Mathematically it can be defined as m (t) = K p e(t) + K p T d d/dte(t) Proportional-Plus-Integral-Plus Derivative Controller The combination of proportional, integral and derivative control action is called PID control action and the controller is called three action controllers. Mathematically m (t) = K p e(t) + K p T d d/dte(t) + K p 1/Ti ∫ e(t)dt 2 Mathematical Modeling of Coupled Tank System Before the process of designing controller begin, it is vital to understand the mathematics of how the coupled tank system behaves. In this system, nonlinear dynamic model are observed. Four steps are taken to derive each of the corresponding line arise perturbation models from the nonlinear model. Fig 1 shows the schematic diagram of coupled tank system.

Qi1, Qi2 = pump flow rate into tank 1 and tank 2 respectively Qo1, Qo2 = flow rate of fluid out of tank 1 and tank 2 respectively Each outlet drain can be modeled as a simple orifice.

Bernoulli‟s equation for Steady, non-Viscous an incompressible show that the outlet flows in each tank is Proportional to the square root of the head of water in the tank.

Similarly, the flow between the two tanks is proportional to the square root of the head differential.

Qo1 = α1√H1

(3)

Qo2 = α2√H2

(4)

Qo3 = α3√H1 –H2

(5)

Where α1, α2, α3 are proportional constants which depend on the coefficients of discharge, the cross sectional area of each orifice and the gravitational constant. Combining equation (3), (4) and (5) into both equation (1) and (2), a set of nonlinear state equations which describe the system dynamics of the coupled tank are derived.

A1 dH1/dt = Qi1 – α1√H1 - α3√H1 –H2

(6)

A2 dH2/dt = Qi2 – α2√H2 - α3√H1 –H2

(7)

2.2 A Line arise Perturbation Model Suppose that for a set of inflows Qi1 and Qi2, the fluid level in the tanks is at some steady state level H1 and H2. Consider a small variation in each inflow, q1 in Qi1 and q2 in Qi2. Let the resulting perturbation in level be h1 and h2 respectively.

From equations (6) and (7), the equation will become: Fig 1: Schematic Diagram of Coupled Tank System

For Tank 1 A1 d (H1+ h1) /dt = (Qi1 + q1) – α1√ (H1 + h1) - α3√H1 –H2 + h1 – h2 (8)

2.1 A Simple Nonlinear Model of Coupled Tank System A simple nonlinear model is derived based on figure 1. Let H1and H2 be the fluid level in each tank, measured with respect to the corresponding outlet. Considering a simple mass balance, the rate of change of fluid volume in each tank equals the net flow of fluid into the tank. Thus for each of tank 1 and tank 2, the dynamic equation is developed as follows.

A1 = dH1/dt Qi1 – Qo1 – Qo3 (1) A2 = dH2/dt Qi2 – Qo2 – Qo3

(2)

For Tank 2 A2 d (H2+ h2) /dt = (Qi2 + q2) – α2√ (H2 + h2) + α3√H1 –H2 + h1 – h2 (9) Subtracting equations (6) and (7) from equation (8) and (9), the equations that will be obtained are, A1 dh1 /dt = q1– α1( √(H1 + h1 -√H1) - α3(√H1 –H2 + h1 – h2 - √H1 - H2 (10)

Where

A2 dh2 /dt = q2– α2( √(H2 + h2 -√H2) + α3(√H1 –H2 + h1 – h2 - √H1 - H2 (11)

H1, H2 = height of fluid in tank 1 and tank 2 respectively A1.A2 =cross sectional area of tank 1 and tank 2 respectively

For small perturbations,

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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 8] AUGUST 2013 √H1 + h1 = √H1 (1 + H1/2 H1)

A2sh2(s) = q2(s) – (α2 /2√H2 + α3 /2√H1 – H2) h2(s) + α3h1(s)/ 2√H1 – H2 (21)

(12)

Therefore, Rewritten equation (20) and (21)

√H1 + h1 - √H1 ≈ h1/2√H1 A1 dh1/dt = q1 – qo1 – α3 (h1 – h2 )/2√H2 – H1 A2 dh2/dt = q2 – qo2 – α3 (h1 – h2)/2√H2 – H1

(T1s +1) h1(s) = K1q1(s) + K12h2(s)

(22)

(T2s +1) h2(s) = K2q2(s) + K21h1(s)

(23)

(13) (14)

Where, qo1 and qo2 represent perturbations in the outflow at the drain pipes. This would be appropriate in the case where outflow is controlled by attaching an external clamp for instance.

2.3 First Order Single Input Single Output (SISO) Plant

For the second order configuration that shows on figure 1, h2 is the process variable (PV) and q1 is the manipulated variable (MV). Case will be considered when q2 is zero. Then, equation (2.4.4.1) and (2.4.4.2) will be expressed into a form that relates between the manipulated variable, q1 and the process variable, h2 and the final transfer function can be obtained as,

This configuration is considered by having the baffle completely depressed so that there is no flow between the two tanks. Equation (6) and (7) can be simplified to become first order differential equation.

A1dh1/dt = q1 – α1h1/2√H1

(15)

A2dh2/dt = q2 – α2h2/2√H2

(16)

Taking the level of fluid at tank 1 that will control, the output variable h1 represents a small change in the steady state level H1 and q1 is a small change in the steady state input flow rate into tank 1, Qi1. H1 is also the steady state operating points and isa constant. Performed Laplace transform on equation (15) will become, Fig 2: Block Diagram of Second Order Process 3 CONTROLLER DESIGN From equation (17), the time constant of the tank 1 dynamics can be expressed

The steady state gain of the tank 1 dynamics is Kt= 2√H1 /α1

In this section the discussion is about the designing of controller that will control level fluid at tank 1 on coupled tank system. The purpose controller that needs to design is PID Controller. The designing of controller are divided into two methods. Each value of α1, α2, α3, A1, A2, H1 and H2 can be obtained from calibrate experiments manual CTS-001 book that also been provided with this plant and those values are: 10.78, 11.03, 11.03, 32, 32, 17, and 15 respectively.

(19)

2.4 Second Order Single Input Single Output (SISO) plant This configuration is considered by having the baffle raised slightly. The manipulated variable is the perturbation to tank 1 inflow. Performed laplace transform of equation (13) and (14), and assuming that initially all variables are at their steady state values, A1sh1(s) = q1(s) – (α1 /2√H1 + α3 /2√H1 – H2) h1(s) + α3h2 (s)/ 2√H1 – H2 (20)

After solving equations the value of T1, T2, K1, K2, K12, and K21 are: 6.1459, 6.0109, 0.1921, 0.1878, 0.749, and 0.7325 respectively.

TF=

0.036 (25) 36.942 s2 + 12.1568s + 0.451

Designing of Fuzzy Logic Controller By using fig. file in Matlab 8.0 the Fuzzy logic block is prepared and the basic structure of this file is as shown in figure. Here Mamdani type of rule-base model is used. This

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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 8] AUGUST 2013 produces output in Fuzzified form. Normal system need to produce precise output. A defuzzification process is to convert the inferred possibility distribution of an output variable to a representative precise value. The work is done by using centroid defuzzification principle in the given fuzzy inference system. In this min implication together with the max aggregation operator is used.

entire span of the input set, the surface viewer is required.

Figure shows the surface view of the system under consideration.

Figure 5: Surface viewer of Fuzzy logic based PID

Fig. 3: Fuzzy Inference System Shown FIS is having seven input member function for both input variables leading to 7*7 i.e. 49 rules. Figures show these rules using rule viewer.

The surface viewer has a special capability that is very helpful in cases with two or more inputs and one output: we can actually grab the axes and reposition them to get a different three dimensional view on the data.

A membership function characterizes a fuzzy set. The value of membership function represents a degree of membership to the fuzzy set, which is between 0 and 1. A fuzzy set with the sharp membership function curve has higher resolution and control sensitivity. With the smooth one, the stability of system is better but resolution is lower. Fuzzy control rules is a set expressed by fuzzy language, which describes the mapping relationship of inputs and outputs.

The fuzzy rules of Kp, Ki and Kd for the controllers are expressed in the rule matrices as shown in table.

Table 2: The fuzzy rules table of Kp

Figure 4: Rule viewer of Fuzzy logic based PID The rule viewer shows a map of the whole fuzzy inference process. The first two columns of plots show the membership functions of the antecedent, or then if part of each rule. The third column of plots shows the membership functions of the consequent, or then part of each rule.

The rule viewer shows one calculation at a time in great detail.

In this sense, it presents a sort of micro view of the fuzzy inference system. If the entire output surface of system is to be viewed. That is, the entire span of the output set based on the

The above table 1 shows the fuzzy rule base for tuning the PID parameter of proportional gain Kp. There are total 49 rules in the above table.

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each parameter will also be tuned in this software and the validity for each parameter will be compared using the reference value (set point). Figure 3 shows the MATLAB simulink block for PID Controller combines with plant.

Table 3: The fuzzy rules table of Ki

The above table 2 shows the fuzzy rule base for tuning the PID parameter of proportional gain Ki. There are total 49 rules in the above table.

Table 4: The fuzzy rules table of Kd

Fig 7: Block Diagram of PID Controller combines with plant 4.1 Simulation result without controller In this section the simulation result without the controllers has been shown.

The above table 3shows the fuzzy rule base for tuning the PID parameter of proportional gain Kd. There are total 49 rules in the above table.

Fig 8: Plot of Liquid Level at the Coupled Tank 1 Fig.6: Fuzzy PID controller designed using Simulink tools 4. SIMULATION RESULTS AND DISCUSSIONS In this section the designing of PID Controller to control coupled tank system using MATLAB software. This software is used to create the simulink model for PID Controller and performance for each parameter for PID Controller has been simulated.

The performances of PID Controller are evaluated in terms of overshoot, rise time and steady state error. Then, the gain for

It may be observe that from the figure 4, the liquid will constantly overflow. This situation happen because of this system running without controller to control the Pump 1 speed, so the Pump 1 will continuously pump the liquid out from the tank until it overflow. PID controller must be put as it controller element so that the liquid will not overflow and will indicates as required.

4.2 Simulation result with PID Controller In this section the simulation result with the PID Controllers has been shown.

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Fig 11: Plot Performance of Proportional-Integral Controller

Fig 9: Plot Performance of Proportional Controller

Figure 12: Plot Performance of PID Controller

Fig 10: Plot Performance of Proportional-Derivative Controller Figure 5 shows the performance of proportional controller.

The set point is set equal to 3 and the proportional gain is set 20. The plot shows that proportional controller reduced both the rise time and the steady state error. Proportional controller also increased the overshoot and decreased the settling time by small amount.

Figure 6 shows the performance of proportional-derivative controller. The set point is set equal to 3. The proportional gain is set equal to 20 and derivative gain is set equal to 10.

This plot shows that the derivative controller reduced both the overshoot and the settling time but had small effect on the rise time and the steady state error.

Figure 7 shows the performance of proportional-integral controller. The set point is set equal to 1. The proportional gain is set equal to 20 and integral gain is set equal to 12. The plot shows that integral controller also reduced the rise time increased the overshoot same as the proportional controller does. The integral controller also eliminated the steady state error.

Figure 8 shows the performance of PID Controller. The set point is set equal to 3. The proportional gain is set equal to 12, integral gain is set equal to 4 and derivative gain is set equal to 7 to provide the desired response. The plot shows that the output voltage achieves the set point voltage at time equal to 10 second. The output voltage have slightly overshoot before stabilize at time equal to 20 second.

Table 1 shows the effects of increasing proportional, integral and derivative parameters.

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IJBSTR RESEARCH PAPER VOL 1 [ISSUE 8] AUGUST 2013 Table 1: Effects of change of the parameter

REFERENCES 1.

2.

3.

4.

From the simulation results shown in fig (4, 5, 6, 7 & 8) it may be concluded that PID controller eliminates the offset of the proportional mode and still provides fast response. This can be used for virtually any process condition including this water level controller. The PID controller is one of the most powerful but complex controller mode operations combines the proportional, integral, and derivative modes. This system can be used for any process condition including controlling water level in a tank. The water level can be controlled continuously without manual adjusting of the valve. The PID algorithms will automatically response to the system so that the system is stabilized near the set point.

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4.3 Simulation result with Fuzzy PID Controller 9.

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Figure 13: Plot Performance of Fuzzy PID Controller 5. CONCLUSIONS Hence it may be concluded that the Fuzzy PID controller is the most effective controller that eliminates the offset of the proportional mode and still provides fast response. That is why Fuzzy PID controller has been chosen. It may be further stated that because of the action of Proportional parameter, the plot result will respond to a change very quickly. Due to the action of Integral parameter, the system is able to be returned to the set point value. The Derivative parameter will measure the change in the error and help to adjust the plot result accordingly.

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