PID-control and open-loop control
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1 Automatic Control Lab 1 PID-control and open-loop control This version: October P I D REGLERTEKNIK Name: P-number: AUTOMATIC LINKÖPING CONTROL Date: Passed:
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3 1 Introduction The purpose of this laboratory exercise is two-fold. To begin with, it illustrates basic concepts introduced in lectures and exercise sessions. Another purpose is to introduce the most common control structure used in industry, so called PID-controllers. Besides being common, these controllers have the benefit that they clearly illustrate basic control principles. These simple principles can be found also in more complicated controllers. To get familiar with PID-control, you will use it to control the water-level in a water tank. two cases will be investigated. At first, you will control the level in a tank where you control the flow directly via a pump, see figure 1. You will then try to control the lower level in two coupled tanks, i.e. a more complex system. qin Pump 1 h1 q1 Figure 1: Sketch of the tank system. 3
4 2 Computer environment To start the program described below, double-click the file PID on the desktop. You start the interface by clicking on the arrow in the top left corner and stop the interface by clicking on the button marked STOP. The process, which in this lab consist of two tanks, is controlled via the computer in an interface shown in figure 2. The system can be set in three different modes.they are closed-loop control using a PID controller, and open-loop control with a manual control signal (from joystick) or a control signal interpolated from Look-up table. The different alternatives can be selected using the two switches on the left. There is also a possibility to switch the measurement signal between the two tanks by clicking on the switch to the right in the interface. Figure 2: Interface for the process. 4
5 2.1 Explanations Some of the concepts in the program Start By clicking on the button with an arrow (top left) the real-time process is started. Stop The process is stopped by clicking the red button marked STOP. Graf The graph on the left presents the control signal u(t) och its PIDcomponents. The graphs on the right show the reference signal (yellow) and the measurement signal (blue). The scales can be changed by double clicking on the min- or max-value to be changed. Clear plots By clicking this button, the graphs will be cleared and a new plot will be started. Show P-, I- and D-part Selects if the components (P,I,D) in the control u(t) are shown. K a, T I, T D, α Parameters in the PID controllers. Look-up Interpolation points for open-loop control. Reference The reference r(t) is either set via the bar or typed into the box. It can be reduced and increased using the arrows on the box.. High-level The program aborts if the level in a tank is too high. This lamp will light up in that case. 5
6 3 Level control of a single tank The first control task is to control the upper tank at a constant level. The lower tank is ignored in this part. The tank has a hole in the bottom, allowing the water to pour down to the lower tank and then to a drain. In the upper tank, there is also an additional outlet leading water directly to the drain. An electrical pump pumps water back to the tank. The problem is to control the voltage to the pump, in order to keep the tank at a desired water level. A larger pump flow naturally leads to an equilibrium at a higher water level. The control should be capable of handling changes in the desired water level (the reference) and disturbances, such as extra inflow or use of the extra drain outlet. Level control of this type is common in process industry. Throughout this chapter, only the upper tank is used. Ignore the lower tank, and make sure that you are using the measurement from the upper tank (set this in the interface). 3.1 Experiment: Manual control We will start by controlling the voltage to the pump directly using a manual choice via the joystick. Connect the joystick to the input ACH2 on the measurement card. Make sure to ground the black cable! When discussing control problems, it is common to illustrate them using block schemes. Figure 3 shows a block scheme for our problem. Input u(t) System Output y(t) Figure 3: Block scheme for the control problem. In figure 3 we have denoted the output signal, in our case the water level, y(t). The input signal, the voltage to the pump, is denoted u(t). The block marked system represents how the output signal (water level) depends on the input signal (pump voltage). To emphasize that this is the input signal we control the system with, we will henceforth call it the control signal. The benefit with a block scheme is that it highlights the cause-and-effects in the system rather than the actual physical components in the system. In many 6
7 cases, detailed knowledge on the relationship between the control and output signal is available, typically a differential equation. In our case, the knowledge we have about the system, is that the water level (the output signal) increases when we increase the pump voltage (the control signal). However, we will see that this information is sufficient for controlling the system, and understanding the control. Select the joystick as control signal (Set switch so that the joystick is the input to the pump. Test some different inputs and measure the resulting tank level. (Select inputs in the range 1 3V ) Tank level [cm] Input [V] Is the system linear? Motivate! Through the previous work, we have created a simple model of the tank system. Try to use this information to control the tank level, i.e. using manual open-loop control of the tank. One person in the group controls the input signal by choosing control inputs without looking at the level in the tank. This person may only look at the graph showing the input signal to the pump. Another person in the group decides which level to reach, and makes notes on the results, without telling the first person how things progress. How well does this control work? Partly open the extra outlet and repeat the previous experiment. How well did the control behave now? 7
8 The control strategy above is an example of open-loop control, which means that the controller (in this case the first person) has no access to measurements from the system. We will now repeat the experiment, but with the modification that the person who controls the pump is allowed to look at the water level when controlling the pump. This is an example of closed-loop control (the feedback is performed by the eyes, and the first person act as the controller) One person in the group controls the pump while observing the water level. Try to control to the same water levels as above. What are the results? What happens when the extra outlet is used? You have now tried both open-loop and closed-loop control, and hopefully gained some experience in these concepts. What are the pros and cons with open- and closed-loop control? When is a model needed? When are sensors needed? 3.2 Experiment: Automatic control Open-loop control One way to automatize the open-loop control is to use the table you generated in experiment 3.1 in the controller. The table tells the computer which control input gives a particular tank level. To test the automatic control, feed your values to the interpolation table ( Look-up ) in the interface. Verify that the signal to the pump comes from the table. The table defines a function f relating u(t) to the desired level r(t) as u(t) = f(r(t)) 8
9 where f in this case is a linear interpolation. Test the automatic control. Is the desired level reached? What happens if the extra outlet is used? Note that this is the same case as when a person not was allowed to look at the level. We will now see how we can obtain better results compared to open-loop control, by using water level measurements. A simple idea is take the difference between the desired level r(t), and the tank level y(t), and let the control input be this difference amplified by a K a, i.e. proportional control! Feedback To obtain good control, the controller must be able to compensate for errors such as those studied above. In the following section, a commonly used feedback principle will be presented. Turn the interface into feedback control mode (Set the middle switch to the lower setting). Set K a = 1. The relation between the output signal and the control signal is now ) u(t) = K a (r(t) y(t) The reference signal is compared in the controller with the actual water level y(t). This gives the control error e(t) = r(t) y(t). The input voltage to the pump is then generated as a constant K a times the control error e(t). This principle is called P-control (proportional). The controller the computer implements is called a P-controller. A block scheme of the control system is given in figure 4. 9
10 r(t) e(t) u(t) y(t) Σ System K a 1 Figure 4: Block scheme for level control. We will now investigate how the controller gain K a influences the behavior of the system. Set a reference value. What water level is obtained for this reference value? Reference: cm Water level: cm Control error: cm A remaining error is easy to understand: We found out in experiment 3.1 how many volts were needed to obtain a desired level. From the relation ) u(t) = K a (r(t) y(t) = K a e(t) we see than an control input u(t) volts forces us to have an error u(t)/k a volt. To see this, assume the error is 0. This leads to u(t) = 0. Water will then flow out, and we will not be able to keep the error at 0! Try another reference level. What happens? Reference: cm Water level: cm Control error: cm Now increase K a. What happens? From the experiments, we have learned that a small control error requires large control inputs already for small errors. We therefore choose K a large. 10
11 What happens with the control input when K a is increased? Conclusion: When setting the gain in a P-controller, we are trying to achieve to contradicting things. We want to use a large gain to obtain a small stationary error. however, we have seen that an increasing gain leads to an amplification of measurement noise, and increased oscillatory behaviour. We will now try to avoid these problems by modifying the controller Experiment: PI-control If we select a finite value on T I, the controller will implement the following control ( u(t) = K a e(t) + 1 t ) e(τ)dτ (1) T I Reduce K a to approximately 1 and use 0 < T I <, suitable value for T I is roughly 15. Compared to the control input used in the closed/loop control experiment 3.2 (P-control), we have added a term proportional to the time/integral of the error signal. Use one of the reference values in experiment 3.1. What is the voltage from the I-part when the level has stabilized? You can see the contribution from the I-part by setting Show P-, I- and D-part to On. Stationary level: cm. Voltage from the I-part: V. Compare the voltage you measured on the I-part with the control inputs you measured in experiment 3.1. Make sure the extra outlet is closed! The reason for the new result is that the integral keeps growing (or decreas- 11 0
12 ing) until the integral of the control error e(t) is zero, and thus the control error is zero. You have now investigated how the control system manages to have the output y(t) follow the reference r(t). This is often called the servo-problem. A similar problem is to keep the output at a constant level, despite disturbances acting on the system. This is most often called the disturbance rejection problem. A block scheme for the disturbance rejection problem is shown in figure 5. Disturb the system by increasing the outflow (open the extra outlet). What happens? In figure 5, v(t) represents the extra water let out. v(t) r(t) Σ e(t) F u(t) G Σ y(t) 1 Figure 5: Block scheme for the disturbance rejection problem. By modifying the control structure, we have managed to get rid of the stationary control error completely, without the adverse effects a large K a gave in a simple P-controller. Once again study the servo-problem. Investigate if you can achieve good control by varying T I. Test what happens if you perform steps in the reference. How does T I influence the results? 12
13 Conclusion: The I-part handles the stationary error in a feedback control system. It can however have negative effects on stability Experiment: PID-control We will now study the last part in a PID-controller. To understand the D- part, we will first study a PD-controller, and then conclude everything in a complete PID-controller. A PD-controller uses the derivative of the control error as follows ( ) de(t) u(t) = K a e(t) + T D dt In other words, we are trying to predict the future error by looking at the derivative of the error. We can see this from a simple first order Taylor expansion of the error T D seconds ahead de(t) e(t + T D ) e(t) + T D dt Investigate how T D influences the step response. Start by small (T D < 0.5)! Use K a = 1, T I = inf and α = 0. Make sure to reset the I-part (Click on Reset integral ). What happens when T D is increased? To explain the answer to the question above, let us study how the measurement signal y(t) = h 1 (t) is influenced by a differentiation. Let the measurement consist of two parts, the true value y s (t) and measurement noise n(t) which can be described using a sinusoidal signal y(t) = y s (t) + n(t) = y s (t) + a sin ωt The contribution to the control input from the D-part is (T D och K a omitted) dy(t) = dy s(t) + aω cos ωt dt dt It is easily realized that high-frequency noise will be amplified strongly by an ideal derivation. To avoid this amplification, the derivative part is extended with a low-frequency filter, and is thus implemented as D(s) = T D s αt D s + 1 E(s) 13
14 How can the equation above be interpreted (approximately) for large/small s? How does α influence the control input in the equations above? What happens with a small and large α respectively? With P-, I- och D-parts used, the equation for the controller is ( U(s) = K a T I s + T ) Ds E(s) αt D s + 1 Use values for K a and T I which gave good control in experiment Investigate if you can achieve good control by varying T D and α. Test in particular if K a and T I can be increased and decreased respectively, when we introduce the derivative part. Suitable values on T D and α are 2 and 1 respectively. Howe does the influence of the measurement noise change when T D and α are altered? Conclusion: The D-part damps the oscillatory behavior. It can unfortunately amplify the measurement noise, an effect which however can be reduced by using a non-ideal derivative part, i.e. a low-pass filtered derivative. Conclusions Conclude the influence from the different parts of a PID controller. P: I: D:
15 15
16 4 Level control of the double-tank. We will now control a slightly harder system. Connect the lower tank by changing the right switch to its lower setting. We now have two tanks coupled in series where the water in the upper tank flows to the lower tank, and we want to control the level in the lower tank. Try to find a PID-controller with a rise-time below 5 s and an over-shoot below 10 %. The requirements are for a step from the level 10 cm with an amplitude of 1 cm. PID-controller rise-time Over-shoot K a = T I = T D = α = K a = T I = T D = α = K a = T I = T D = α = K a = T I = T D = α = K a = T I = T D = α = As you maybe notice, it is fairly hard to simply guess the values for the PIDcontroller, when certain specifications are given. In the second laboratory exercise, we will see how model-based control design can help. 16
17 A Preparations A.1 Introduction The purpose of these exercises are to prepare for this laboratory session, revolving around the control of a water tank system. The exercises should be thoroughly worked through, and this whole note should have been read before the session starts. A.2 Open-loop control In the first part of the lab, open-loop control is studied. A joystick is used to generate a voltage. The joystick is connected to a water pump, pumping water to the tank. The level in the tank is measured and converted to a voltage. The joystick and the tank system is connected via a computer, to get the measured signals presented in a convenient fashion. 1 A sketch of the process is given in figure 1. The joystick is connected directly to the pump. Draw this system using a block scheme. What physical quantities are in- and output signal in the different blocks? 2 Assume the pump can be described as a static system, meaning the flow to the tank is changed directly when the voltage to the pump is change, i.e. q(t) = K pump u(t) What is the transfer function from voltage to flow. What is the dimension? 3 The relationship between inflow and level in the tank can approximately be described with the differential equation Aḣ(t) = q(t) ch(t) where A denotes the surface area of the tank, and the constant c, amongst other things, depends on the area of the outlet valve. what is the transfer function for the tank, i.e. from inflow to level? 4 Assume the position of the joystick (control input u(t)) is changed momentaneously from zero to a positive value. What is the Laplace transform of the resulting level? 17
18 5 Use the results from exercises 2-4 to sketch the principle behavior of the water level, as a function of time, when the joystick is changed in steps. A.3 Closed-loop control The computer is used to achieve PID-control. We measure the tank level at the upper tank, and the controller generates a control input u(t) to the pump, via a signal processing card by calculating 6 Draw a block scheme of the closed-loop system. Let the computer, voltage generator and processing card be represented by one single block (the reference can be seen as an input to the computer). What physical quantities are input and output signals in the blocks? 7 Now assume the computer implements a proportional controller, i.e. the generated voltage is proportional to the difference between the desired and measured level. Let the reference signal (r(t)) change momentaneously from zero to a positive value. What is the Laplace transform of the resulting level? Hint: Use the result from exercise 3 above. 8 Use the result in exercise 7 to decide if the output will settle at a constant value, and if so, at which value? 18
19 B Connection Figure 6 shows the setup for the double-tank system. The I/O-card sends all measurements to the computer, were all computations are performed. Figure 6: Overview of signal path Figure 7 shows the connections for the exercise. All cables are unique, and cannot be connected wrong. The cable from ACH0 and ACH1 to To D/A is Y-coupled, meaning that it splits to two contacts in one end. Note that the joystick requires a voltage generator. I/O card Ampllifier Double tank ACH0 ACH1 ACH2 S1 & S2 To D/A DAC0 From D/A To Load "Big" "Small" Joystick Figure 7: Connection scheme. 19
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