NATIONAL UNIVERSITY OF SINGAPORE. MCH5003 Modeling and Simulation. E2: Simulation of PID Control System

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1 NATIONAL UNIVERSITY OF SINGAPORE MCH5003 Modeling and Simulation E2: Simulation of PID Control System 1. Object The aim of this experiment is to develop a digital PID controller, and simulate its performance in a simple control loop. This experiment is also designed to provide experience in using a modern spreadsheet tool for simulation : Microsoft EXCEL. 2. Introduction Proportional + Integral + Derivative controllers are widely used in industry, due to their relative simplicity, and ability to control a wide range of processes reasonably well. This experiment investigates the main features of these controllers by analysing the component responses through simulation on a commonly available spreadsheet software : Microsoft EXCEL. In order to illustrate the main concepts of this type of controller, the development proceeds in stages, generating first a simple first order plus delay process model, and then a P, PI and finally the full PID controller. 3. Preparation Work Q1. We desire to balance a rolling ball on a tilting beam as shown in the Figure. Assume the beam may be balanced near the horizontal (α 0); therefore we have a small deviation of α mgsinα Ball Beam mg x(t) α(t)

2 MCH5003 : Modeling and Simulation/Page 2 (a) Obtain the differential equation relating x to α. (b) Obtain the state-space description from (a) (c) Obtain transfer function from (a) and the frequency response (d) Obtain impulse response h(t) from (c) Q2 A system is described by its transfer function: Y ( s) U ( s) = G( s) = 1 ( s + 1)( s + 2) To simulate this system using a computer, it is necessary to convert G(s) to a difference equation. Noting that s in the Laplace domain is equivalent to a d/dt in the time domain, and further a dy/dt in continuous time is approximately equivalent to {y(t)-y(t-h)}/h= (1-q -1 )y/h, where h is the sampling interval. This is called the backward difference approximation. In short, it means replacing s by (1-q -1 )/h. Convert the above transfer function description into a difference equation relating samples of y and u. You may assume h=1. Obtain also the z-domain transfer function.

3 MCH5003 : Modeling and Simulation/Page 3 Q3. In the simulation of continuous-time system, amplitude and time scaling in the original variables are sometimes necessary. Amplitude Scaling Large system variables (e.g. when simulating planetary movements x max ~ m) cannot be simulated. On the other hand, small system variables (e.g. when simulating crystal structures x max =10-10 m) yield large errors. Therefore, amplitude scaling is necessary in these cases. The pre-requisite is to know the maximum values of the physical variables. Example 1 Consider the system described by x = ax bx Assume x x ; x x ; x x max max max Obtain a differential equation in terms of the scaled variables. Time Scaling Motivation for time scaling is analogous to amplitude scaling. Physical time scales may be too fast or too slow. Simulation on socio-economical processes or chemical and thermal processes typically needs speeding up. Simulation of electrical system typically needs to be slowed down Involves a linear relationship between real time,t and computer time τ

4 MCH5003 : Modeling and Simulation/Page 4 τ = λ t dx dx dτ dx =. = λ dt dτ dt d τ 2 2 d x 2 d x 2 = λ 2 dt dτ For λ<1, the computer simulation is faster than the real process and vice versa. Consider the system 2 d x a dx bx = 0, dt dt Obtain the differential equation in terms of the scaled time variable. 4. Theory The process model used in this experiment is the standard first order lag plus delay model used in many process control applications : sl e Y ( s) = A U ( s) (1) 1+ st where u (t) is the input to the process, and y (t) is the corresponding process output. T is the time constant, A is the gain and L is the time delay of the process. u v y 1/(1+sT) Aexp(-sL) *Provide any non-parametric method where simple estimates of T, L and A can be obtained.

5 MCH5003 : Modeling and Simulation/Page 5 This is a time-continuous model. In order to simulate it on a computer, the model must first be discretized using numerical methods. The time delay (e -sl ) can be represented by integer samples of delay: i.e. y k = v k-n (2) where y k is the process output at sample time k, and v k-n is the output from the lag model at sample time k-n. A time delay is L will correspond to L/h number of samples (h is sampling interval) The first order lag (with time constant T) will be simulated using the difference equation (this is easily obtained via a backward difference approximation: replace s by (1-q -1 )/h : h v k = v k-1 + (u k - v k-1 ) h + T (3) where h is the time interval between subsequent samples k and k-1. *Show how this equation is obtained. The PID controller transfer function is given, in the time and frequency domains, as : u(t) = G x + 1 t T i dx x dt + T d dt (4) 0 U(s) = G(1 + 1 st + st i d )X(s) (5) where x is the error between process value and setpoint (i.e. w-y), and u is the controller output. w + x u y PID Process - (5) (1) This controller may be implemented as the sum of the three separate components; proportional component is: P k = G * x k (6)

6 MCH5003 : Modeling and Simulation/Page 6 The integral component may be implemented in discrete time, as an accumulator : h I k = I k-1 + P k Ti (7) and the derivative component may be implemented in difference form: D k = (P k - P k-1 ) Td h (8) *Show how (7) and (8) are obtained. These three components are then added to give the controller output, u. 5. Apparatus The equipment required for this experiment is a PC with at least 16M of RAM, connected to a printer. The software required is Microsoft EXCEL running under Windows 95 or higher.

7 MCH5003 : Modeling and Simulation/Page 7 6. Procedure 6.1 Preparation From Window Explorers window, find and select the directory mch5003, and open this directory by double-clicking its icon. Then double click the file labeled lab5003.xls, which will open Microsoft EXCEL with this spreadsheet active. 6.2 Experiment Time(Column A) In Column A (labeled Time), you must generate a time axis for your simulation, which runs from time 0 to time 400 in 0.5 second increments. To do this, select spreadsheet cell A10 and enter the value 0. Press the down arrow to enter this data and select the next cell (A11), where you must enter 0.5. Select the two cells, then position the mouse to point at the handle in the lower-right corner of selected range. The pointer will change to a cross. Drag the cross to the end of range(a810) to fill with the sequence of time and release the mouse button. You should now have a column of 800 time samples running from 0 to 400 seconds Setpoint(Column B) The next step is to generate a setpoint profile for the PID control system. In Column B (labeled setpoint) is the setpoint signal of the control system (w). This will go from 0 at time 0, to 60 at time 0.5 till time 200, and then down to 40 at time till time 400. Do this by using the same method as described in Error(Column E) To generate an error signal (x = w-y) you must now enter an equation in the error column(column E), rather than just raw data. The error signal is generated by subtracting the Setpoint (Column B) from the process output, which will be generated in the Process_Delay (Column D), for each time sample. Generate the first sample by selecting cell E11 and enter the formula: =B11-D11 When you press the return key, you will notice that the cell now contains the numerical difference between the values in the two preceding columns (Since

8 MCH5003 : Modeling and Simulation/Page 8 Column D contains no data at this time, it defaults to 0, resulting in an error which equals the setpoint.) Select cell E12 and enter the formula in the same way as before. Select the two cells, then position the mouse to point at the handle in the lower-right corner of selected range. Again, the pointer changes to a cross. Drag the cross to the end of range(e810) to fill with the sequence of error and release the mouse button. You should now have 800 error signals in Column E System output after process_delay (Column D) Formulae in spreadsheet cells can reference any other cells, and you will use the Process_Delay (Column D) to illustrate how this feature can be used to reference values from previous time samples. The Process_Delay Column will give a 2.5 second (5 sample) delay, and also simulate a process gain, A = 2 (equation (2)). Select column D10 and enter the formula: 2 * C5 where Column C is the v k in equation (2). Copy this formula into block D10 to D810. Since there is no data from C1 to C9 it defaults to 0. Test that this delay works by entering any data into cells C10 to C20. You should see the result (delayed by 2.5 seconds and including the 2x gain) in Column D Lag1 Select cell C11 and enter a formula corresponding to the difference equation (3) given in theory section. The input to your lag should be the PID output (u k )(in Column G) for that time sample i.e. cell G11. Set the lag time constant (F10..F810 in Column F) to 20 seconds, and generate some test data in Column J, by setting G11 to G210 to 100. Select Column A and Column C with the mouse+ctrl, and then select Chart... from the Insert item in File menu. This will result in a pop-up dialogue box where you can choose the type of your chart. Select the XY(scatter) type, which is a smooth line without points, then click on the Next button. You can also configure the chart option and the chart position in the next two steps until you press Finish button.then you should have the graph of lag output (exponentials) against time.

9 MCH5003 : Modeling and Simulation/Page 9 Change the time constant of system by setting Column (F10..F810) to 100. It is noted that the graph has changed to reflect the change in the time constant. You may choose to edit some of the graph attributes to improve your test documentation. Highlight the graph you wanted, select Print form the File menu, and save the print for your report Proportional (Column H) The next step is to generate a proportional controller, with a gain determined by the value in the Gain (Column H). Enter the following formula in cell G11 =I11 Column I has already been configured to give the sum P+I+D (sum of proportional term in Column J, integral term in Column L and derivative term in Column N ). First only P control is used. Now generate the proportional term in Column J, to give the formula of equation (6), where the controller gain is given as 10 in Column H. The Columns L and N contain no data now and they default to 0. When this column has been completed, the control loop will be closed, with proportional only control and a gain of 10 times. Select system output Column D and setpoint Column B with the mouse+ctrl, and the select Chart... from the Insert item in File menu. This will result in a pop-up dialogue box where you can choose the type of your chart. Here the first one in Line type, which is a smooth line without points, is selected. Click the button in Category(x) axis Labels, the pop-up dialogue will appear in upper right corner of screen, then select column A(time), and highlight the pop-up dialogue. You can also choose chart option and chart position in next two steps until you press Finish button. Then you should have the graph of system closed-loop response to setpoint steps from against time. Highlight this graph and print it. Comment on the resulting control error magnitude in your report.

10 MCH5003 : Modeling and Simulation/Page Integral(Column L) Add integral action to the controller by inserting the formula given in equation (7) in Column L. Note that a value of Ti = 30 seconds has already been entered in Column K. View the result in Graph2 and print this result. Create a new graph (Graph3), displaying the Process output (y k in Column D), Controller output (u k in Column G) and the Integral output (I k in Column L) against time, and print this graph. Explain the new control performance, and the values of the error signal x k, I k and u k when the control loop has stabilised at each setpoint value Derivative(Column N) From equation (8) generate the formula for calculating the derivative term, and enter this in cell N11, setting Td equal to the value given in Column M. Copy this formula to the remainder of Column N, and view the resulting response, noting the overshoot and the initial derivative kick when the setpoint changes. In this case, the response to setpoint steps is worsened by the addition of derivative action - by analysing the action of the derivative term Tuning The process is moderately well tuned in the above experiments. Experiment by varying the Gain, Ti, and Td values, to try and achieve improved performance in terms of faster response and small overshoot. Describe any improvements measured. Ziegler -Nichols tuning rules can be used to obtain good PID parameters based on the model (1). This is one application of model to design controllers. Using, according to the rules G=1.2T/(AL) Ti=2L Td=0.5L tune your PID controller accordingly again.

11 MCH5003 : Modeling and Simulation/Page Report The results of each stage of the experiment should be logged, including the spreadsheet entry required to achieve these results. Discuss all the points mentioned in the above notes and answer all the questions. 8. Reference Astrom, K.J. & Wittenmark, B., Computer-Controlled Systems, 2nd Edition, Prentice Hall, 1990