An E-Learning Course of Modelling for Control Design Purposes

Transcription

1 An E-Learning Course of Modelling for Control Design Purposes M. Bicher U. Fitsch M. Atanasijević-Kunc Vienna University of Technology, Institute of Analysis and Scientific Computing, Vienna, Austria ( Vienna University of Technology, Institute of Analysis and Scientific Computing, Vienna, Austria ( University of Ljubljana, Faculty of Electrical Engineering, Slovenia ( Abstract: In the paper the main features of MMT - system are presented, for which the experimenting environment was developed, which mimic the operation of real pilot plant, consisting of three coupled tanks. Exercises lead the student from very simple and easy to understand problem to more complicated multivariable process. For evaluation purposes also some measurement data ara available. Keywords: e-learning, modelling, simulation, control design. 1. INTRODUCTION The great majority of EU countries are introducing the so called Bologna study (common studies structure in EU), where Slovenia and Austria are no exception. The result of such study organization are also a comparable studying programs where students, as well as teaching staff mobility are encouraged. Taking into account developing technologies also the aspects of e-learning are becoming more and more important. Intensive efforts have already resulted in effective solutions regarding individual institutions (Atanasijević-Kunc et al. (1), Logar et al. (11), Körner et al. (11), Winkler et al. (1)), but now attention is directed also to the possibilities of sharing teaching lectures, devices and/or experimenting systems. In this way studying efficacy can be improved while in the same time also the cooperation among different scientific and educational institutions can be intensified. Research groups from the Faculty of Electrical Engineering, University of Ljubljana (Laboratory of Modelling, Simulation and Control and Laboratory of Autonomous Mobile Systems) and the Vienna University of Technology (Institute of Analysis and Scientific Computing) cooperate in the fields of modelling, simulation and control design for long time. As at both institutions also e-learning is incorporated into education process, this has motivated the decision for further investigations of possibilities, where some of experiences and experiments could be shared between both institutions. To be more precise, some of the goals of our research is directed into: further development and extensions of the MMT- System (a web environment for education in mathematical modelling and simulation at Vienna University of Technology) (Körner et al. (11), Winkler et al. (1)), further development and extensions of remote and virtual experiments and projects in E-CHO Learning Fig. 1. Three tanks system - Amira DTS Management and Content Management System (operate at Faculty of Electrical Engineering, University of Ljubljana) (Atanasijević-Kunc et al. (1), Logar et al. (11)), analysis of different systems properties regarding their adequacy for the development of virtual and/or remote experimenting environment, analysis of cooperation possibilities through (mixed) student projects, and other possibilities of cooperation. For the first group of experiments the three coupled tank system, as presented in Fig. 1, was chosen. This pilot plant is located at the Faculty of Electrical Engineering, University of Ljubljana, while the group of experiments was realized at the MMT platform at Vienna University of Technology, as described in the following sections.

2 . THREE TANK SYSTEM In Fig. schematic representation of the pilot plant is presented. It consists of three coupled tanks and beyond them a collecting water pond is located. Out of this basin water can be pumped into the left or the right tank. Two water pumps which can be controlled using voltage signal in the range of ±1V, are therefore system s actuators. Water levels can be measured in all three tanks by corresponding sensors. These signals are also in the same voltage range. The system is equipped with six manual valves with which different operation conditions can be realized. There are several possibilities or levels how to incorporate the experiments with mentioned system into MMT platform, as indicated in Fig. 3. Fig. 3. Experiment-Modelling-MMT Fig.. Sketch of the Three Tanks For measurements and system control personal computer with A/D and D/A converters is used. The signals are imported into MATLAB workspace using corresponding block in SIMULINK. That enables simple experimenting with the process which is practically identical to the situation when model instead of process is used. 3. MMT PLATFORM USAGE As already mentioned MMT (Mathematic, Modelling and Tools) platform was developed for e-learning purposes where mathematical modelling and simulation are of central importance (Körner et al. (11), Winkler et al. (1)). Prepared exercises are used by teachers and also by students to repeat the lectures and to prepare themselves for the exams. They can be used also for testing. As program MATLAB (with SIMULINK and other toolboxes) is very suitable for mentioned purposes it was used for realization of all needed calculations, while userfriendly interface enables simple communication and parameters definition. Most of the students do not have enough Matlab or programming skills to use or create difficult high level simulation files - so, if students take part in a course in which MMT is used, they get the access to course-specific examples at the MMT, where they all can experiment online through a user friendly environment. At the first level the so called virtual laboratory using only mathematical models and simulation is realized. This level has several important advantages. It is suitable when the number of students at the course is larger, as all can perform experiments in the same time. Time constants of the process are not very important as simulation is not running in real time and experiments can not threaten the process itself. At the second level the remote laboratory is used. The important advantage here is the work with the pilot plant itself which enables model evaluation and close - loop system testing, but is in general time - consuming, as only one experiment can run at the time. In some cases the first level can be improved if measurement data regarding defined experiments are available.. SYSTEM MATHEMATICAL MODELLING System operation can be presented with block diagram, as shown in Fig.. K K =? K = H I. L D. L D = E F H? A I I D D D! Fig.. Block diagram of the system.1 Actuators I A I H I Two water pumps are the actuators of the system. Static characteristic of the first one together with linear interpolation is illustrated in Fig.. In this case experiments showed that the relation between the pumped flow and the input voltage signal is nearly N N N!

3 1. x 1 1 water flow though V 3 [m 3 /s]... Fig.. Static characteristic of the first water pump linear (Figure ). Each pump is inactive with potentials equal or lower than 1 volts and reaches the maximum m3 flow ( 1 s ) at 1 volts. The second pump behaves in a quite similar way.. Sensors Sensors operation can be described with the following equations: x 1 (t) = K s1 h 1 (t) + K os1 (1) x (t) = K s h (t) + K os () x 3 (t) = K s3 h 3 (t) + K os3 (3) and parameters were estimated to be the following: K s1 = 7.937V/m, K s = 7.7V/m, K s3 = 7.7V/m, K so1 = 9.1V, K so = 9.V, and K so3 = 9.V. Signals x 1 (t), x (t) and x 3 (t) are sensors output voltage signals in the range ±1V. As can be seen from the parameters, they have negative characteristic, it means, that when tanks are empty, output voltages are all 1V..3 Main Process Taken into account that the valves V 1, V and V 3 are open, while V, V and V are closed, the following equilibrium equations describe the main process: Φ vh1 (t) Φ 3 (t) = S d[h 1(t)] dt Φ 3 (t) Φ (t) = S d[h (t)] dt Φ (t) + Φ vh (t) Φ izh (t) = S d[h 3(t)] dt where valves V 1 in V have square root characteristics: () () () Φ 3 (t) = K 1 h1 (t) h (t) (7) Φ (t) = K h (t) h 3 (t) () while the characteristic of valve V 3 is presented in Fig level h 3 [m] Fig.. Static characteristic of the valve V 3 Constants in equations (7) and () can be defined from chosen working conditions. The influence of valves V, V and V is usually used to simulate operation disturbances. As the characteristics of these valves are also of square root shape, there is no problem to extend the presented model to include such experiments. On the other hand the model can also be simplified. For example, if the valve V 1 is closed and V is open, the operation of the left tank is separate from other two and the lectures can start with very simple first order system single-input, single-output process. Step by step the system can be defined as more complicated one, ending with multi - input multi - output process as presented.. LINEAR MODEL One very important aspect of modelling is also linearization. In spite of the fact that students usually understand how linear models were developed, they frequently forget the correct interpretation when they are using them. The situation is even worse in the case of close - loop realization. For the presented system the following operating conditions were chosen: u 1 = V (first input voltage signal in working point) u = 3. V (second input voltage signal in working point) x 1 =.771V (first output sensor signal in chosen working point) x = 3.1V (second output sensor signal in chosen working point) x 3 =.9V (third output sensor signal in chosen working point) From these conditions the following linear model in state space was developed:

4 ẋ(t) = A x(t) + B u(t) (9) y(t) = C x(t) + D u(t) (1) where the corresponding matrices are: A = B = [.91 ] C = D = (11) (1) (13) (1) Nonlinear and linearized model responses are in comparison with measurements illustrated in Figs. 7 to 1. Through the first experiment (Figs. 7 and ) realized input signals drive first the process to chosen working conditions and than step changes are realized in first input. The second experiment is very similar, only that in this case step changes are realized in second input signal. From presented results can be concluded, that good matching was achieved among measurements and both model responses. sensor voltage 1 x 3 nelin. model: Fig.. Output signals - experiment 1 input voltage x changes in u ( u =±.3V u u x 1 change in u 1 ( u 1 =±.V) Fig. 9. Input signals - experiment 3 u 1 input voltage 7 9 u 1 sensor voltage x 3 nelin. model: x time[s] x 1 Fig. 7. Input signals - experiment 1. CLOSE-LOOP OPERATION When the process itself is well understand the advantages of close - loop operation can be explained, starting usually with simple or frequently used solutions. In spite of the fact that control design is in general demanding by itself it can be accomplished also with modelling results verification as in close - loop system properties are changed. Close - loop system operation was realized using classical configuration, presented in Fig. 11, where of course, input signals limitations have to be taken into account Fig. 1. Output signals - experiment For illustration multivariable (MIMO) operation is presented in Figs. 1 to 1, where the following PI - controller was used: where K P = G P I (s) = K P + 1 s K I (1) (1)

5 K I = (17) 1 H A B A? H A H K I O I A O Fig. 11. Block diagram of closed - loop system control signal u 1 1 x 1 in ref ref: time[s] Fig. 1. First reference and output signal x 3 in ref ref: Fig. 13. Second reference and output signal 7. IMPLEMENTATION RESULTS Taking into account different configurations or simplifications of presented results the sequence of four experiments were developed each illustrating open and closeloop process operation. The simplest one is first order SISO problem with PI-controller in close - loop operation. Next two are the second and third order systems with PID-controllers and finally rather complex MIMO system was realized, where PI controller can also be realized as MIMO system. In all mentioned situations nonlinear and linearized model responses have to be compared. Figs. 1 and 17 illustrate some of experiments already imported into the MMT platform. Fig. 1 shows a result of the open - loop system comparing the nonlinear and Fig. 1. First control signal control signal u Fig. 1. First control signal the linear model. At the top of the Fig. 1 an explanation of the experiment is found, which is helpful for the students using e-learning environment. Beyond the explanation some of input boxes are located. Pressing the OK button activates MATLAB, sending the input variables (as string type) to a.m-file. In the MATLAB file the SIMULINK parameters are defined and the.mdlfile is started. In this example the input voltage and thus the water inflow function can be set manually. As shown in the SIMULINK diagram, the user may choose up to five heavyside-step and the corresponding step-size and time. Fig. 17 is an illustration of the closed - loop system with manually set reference functions. These functions are simple step functions with user selected steps size and times. The lookup variable at the bottom right of the input boxes sets the output (x 1, x or x 3 ) on which the nonlinear and the linear model is compared to the reference.. CONCLUSION In the paper experiments are presented which all are connected with the same pilot plant - three coupled tank system (AMIRA DTS ). The pilot plant is located at the Faculty of Electrical Engineering, University of

6 In the future also the possibilities of remote control will be investigated. REFERENCES Atanasijević-Kunc M., Logar V., Karba R., Papić M., Kos, A.. Remote multivariable control design using a competition game. IEEE Trans. Ed., Feb. 11, Vol., No. 1, Logar V., Karba R., Papić M., Atanasijević-Kunc M.. Artificial and real laboratory environment in an e- learning competition. Math. comput. simul., Nov. 11, Vol., No. 3, 17-. Körner A., Hafner, I., Bicher M., Winkler S., Breitenecker, F.. MMT- A Web Environment for Education in Mathematical Modelling and Simulation. ASIM 11. Winkler S., Körner A., Hafner I.. MMT - A webbased e-learning System for Mathematics, Modeling and Simulation using MATLAB In: Preprints of EUROSIM Congress, Prague, Czech Republic, 1, paper 31. Fig. 1. Open loop system at the MMT Fig. 17. Controlled system at the MMT Ljubljana, while experiments were implemented into MMT platform at the Vienna University of Technology. Starting from the observation of the whole process four experiments are developed each illustrating open and closeloop process operation. The experiments are: first order SISO system with PI-controller, second and third order systems with PID-controller and MIMO system, where PI controller can also be realized as MIMO system. For illustration purposes in some cases simulation results can be compared with measurements. As each of mentioned examples is small project all the possibilities are not yet completely developed, but are sill under consideation.