- Email: [email protected]
Control Systems Analysis and Design Labs with Educational Plants
Proceedings of the 9th IFAC Symposium Advances in Control Education The International Federation of Automatic Control Nizhny Novgorod, Russia, June 19-21, 2012
Control Systems Analysis and Design Labs with Educational Plants M. Sotnikova, N. Zhabko and T. Lepikhin Faculty of Applied Mathematics and Control Processes, Saint-Petersburg State University, Saint-Petersburg, Russia. (e-mail: [email protected]) Abstract: This article is devoted to using of the special laboratory plants in control education. The complex of control system analysis and design labs is presented, where each lab deals with one of the experimental control plants. The brief description of the plants is given and their main features are analyzed from the control point of view. The substantial formulations of the labs, which are discussed in the paper, include the problems of system identification, digital control systems design, signal processing and real-time implementation. The advantages of using MATLAB environment for the problem solution are pointed out. Finally, the examples of the labs are presented. Keywords: educational plants, control design labs control point of view are outlined. Secondly, the substantial formulations of the labs problems are given for each of the areas mentioned above. Thirdly, the advantages of using MATLAB for the analysis and synthesis of the control systems are discussed. The last section contains some examples of labs, which are devoted to control system design for a magnetic levitation plant and programming movements for a robotic manipulator.
1. INTRODUCTION Real plants are invaluable in control education because they allow students to make sense of essential problems which appear in the control of real objects. Usually, educational control objects are relatively simple compared to real industrial control plants; nevertheless, they reveal all the typical control challenges such as unaccurate knowledge of the mathematical model, sensor noise, external disturbances, input and output constraints, etc.
2. EXPERIMENTAL EDUCATIONAL PLANTS In the framework of the proposed labs the following educational experimental plants are used:
Educational plants are very useful for both teaching process and research (Veremey, 2008). Performing the labs the students are faced with a whole spectrum of control system design and implementation problems. The researches have an opportunity to test any control algorithm or perform the identification procedure, etc. for real control object.
• magnetic levitation plant; • rotary flexible joint system; • ball and beam system; • robotic manipulator FANUC M-20iA. These plants are intensively used for control education at the Information Processes Modelling Laboratory at the Faculty of Applied Mathematics and Control Processes, SaintPetersburg State University.
This article deals with the complex of control systems analysis and design labs. The main goal of the proposed labs is an application of the control theory methods to the analysis and design of control systems for educational plants. This complex can be used in educational process for teaching students, who specialize in the area of dynamical systems control and its applications. The control system design problem for a particular plant includes a lot of sub-problems, the most serious of them being system identification, digital control system analysis and synthesis, signal processing and real-time implementation. A lot of different lab courses can be constructed on the basis of the proposed labs depending on the course specifics – the general understanding of control design problem for real plants, system identification or others. The paper is organized in the following way. Firstly, a brief description of educational plants as control objects is presented, and their most important characteristics from the 978-3-902823-01-4/12/$20.00 © 2012 IFAC
Fig. 1. Magnetic levitation plant.
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9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012
Let us consider these plants in more details. The magnetic levitation system is shown in Fig. 1. The main components of system are – an electromagnet above, a pedestal below, where the ball rests initially, and a steel ball. The aim of the control is to stabilize the ball position in a particular point between the electromagnet and the pedestal by means of the controlled voltage which is applied to the electromagnet.
processes, is quite simple and suitable for educational purposes. The notable feature of this plant is its open-loop instability, i.e., if the beam angle is fixed, the ball rolls down until it reaches the end point of the beam. Each of these plants has the required hardware to transfer the information from the sensors to the computer and, vice versa, to bring the control signal from the computer to the plants. In addition, each plant has necessary software to calculate the control signal on the basis of the measurements and additional software support providing the interface with the MATLAB environment. Thus, the control system design and analysis can be performed using MATLAB, and, after that, control algorithms can be easily implemented for plants realtime control. It can be noted that all plants are digital control systems, so the methods of digital control theory must be used for its analysis and synthesis.
Magnetic levitation system (MAGLEV, 2007) has a current sensor and an optical sensor, which measure the current in the electromagnet and the distance between ball position and the electromagnet surface respectively. The optical sensor is very sensitive to changes in temperature and light conditions, and its measurements are very noisy. It is important to notice that the magnetic field created by the electromagnet is nonuniform, especially near the magnet surface. Consequently it is quite difficult to describe it mathematically. Thereby the corresponding mathematical model of the control process is essentially nonlinear. One more important feature of the system is the ball vertical position unstability.
The industrial robotic manipulator FANUC M-20iA (Fig. 3) is widely used in the educational process in a number of courses for preparing IT-specialists (Lepikhin, 2009). The special significance of the proposed courses is determined by their practical orientation, related to the direct interaction with real information systems. Dealing with the robotic one should develop software using the specific programming language with structures like loops, conditional statements, labels, special instructions and interaction with external interfaces of the robotic.
The rotary flexible joint system is an educational plant where a pendulum can oscillate on a rotating platform. This plant is shown in Fig. 2 (left). The motion of the pendulum is limited by two springs which connect it to the rotating platform. The goal of the control is to deflect the pendulum at the given angle and to stabilize it in the vicinity of the new direction. In order to obtain the desirable goal, it is necessary to control the rotations of the platform. This plant has two sensors. The first of them measures the angle of the pendulum deflection, while the second one measures the angle of the platform rotation. Both sensors are very noisy, causing the main difficulty in the control of the considered object. Besides, the structure of the differential equations, which represent the mathematical model of the control process, is not so evident, and it is necessary to perform the system model identification first. The quality of the control processes in this case is determined, among other things, by the level of the oscillations in the transient responses.
Fig. 3. Robotic manipulator FANUC M-20iA. 3. LABS PROBLEMS FORMULATION The basic labs directions are system identification, analysis and synthesis of the digital control algorithms, digital signal processing and real-time implementation. For each direction the substantial formulation of the labs problems is presented below.
Fig. 2. Rotary flexible joint system (left) and Ball and Beam system (right). A ball and beam system is one of the most popular educational plants. In it a steel ball rolls on the top of a long beam. The beam angle can be adjusted by applying control signal to the electrical motor. The goal of the control is to move the ball to the desired position and to stabilize it in the vicinity of that point. The ball position on the beam is measured by the ultrasonic range sensor. The measurements of this sensor are extremely noisy and can give estimations of the position with the essential errors. At the same time, the mathematical model, describing the dynamics of the control
3.1. Plant model identification The first problem that appears when someone starts to work with a new plant is to obtain its mathematical model. Most educational plants have no documentation where their mathematical models are described. In other cases, when some initial mathematical model is presented, it is often necessary to significantly improve it.
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In the framework of the lab, the problem of model identification (Ljung, 1999) can be divided into the following steps.
consider the following discrete-time analog that can be constructed on the base of model (1)
x k +1 = f (k , x k , u k ) ,
1) Getting the structure of the equations which are constitutes the mathematical model of the plant in the form
x = f (t , x, u, k ) ,
(2)
where x k ∈ E n – state vector, u k ∈ E m – control input at the
(1)
where x ∈ E n – state vector, u ∈ E m – control input, k ∈ E r – vector of parameters, which values must be estimated during the process of identification. Equations (1) are mainly formed on the base of priory knowledge about the object dynamics and the basic physical laws.
sample instant k . In the further discussion model (2) is used as a base for analysis and synthesis of the digital control. 3.2. Analysis and synthesis of the digital control The problems of digital control analysis and synthesis are considered in the proposed labs with the help of Control System Toolbox of the MATLAB package.
This problem is quite complicated for some devices, for example, for the magnetic levitation plant. This is due to the fact that the magnetic field near the magnet is extremely difficult to describe, thus the corresponding mathematical model representing ball dynamics nearby the magnet has a very complex structure.
Let us look at the most important questions of the control system analysis: • stability analysis. Here the question of stability can be considered with respect to the nonlinear model (2) or its linearization in the vicinity of the operating point or trajectory; • controllability and observability analysis. Here the conditions of full controllability and observability are checked and the questions of system stabilizability and the necessary content of measurements are investigated. Now let us talk about the problems related to the synthesis of the digital control systems. The key problem that should be treated for the experimental devices is synthesis of stabilizing feedback control. Thus, for the magnetic levitation system, the feedback control law should provide ball position stabilization at a given distance from the electromagnet, for the rotary flexible joint system – fix the pendulum with the given rotation angle and for the ball and beam system – stabilize the ball position at the desired point on the beam. The other synthesis problem of interest is development of program control algorithms and realization of the program movements for the educational plants.
Let us consider that the structure of the equations (1) reflects the control object dynamics within certain limits. In particular, the equations (1) can be linear and represent the object dynamics in the vicinity of equilibrium point. 2) Experiment design and implementation on the plant, and data gathering for identification. At this stage the following key questions regarded to experiment design should be considered: • method of carrying out the experiment – in an open-loop or in a closed-loop. For instance, for magnetic levitation system the experiment can be performed only in closedloop because of the ball vertical position instability; • data for identification – which data should be gathered for identification, and which signals are considered as input and output. Besides, an important question is how many variables should be measured in order to estimate all the parameters of the model (1); • input signal selection. The standard choice is between the step, harmonic or random binary signals. It is important to note that the probabilistic properties of the estimations of the parameter vector k in (1) are strongly depend on the input signal selection. So, the problem is how to choose the input signal in order to remove the biases of the estimations and to minimize their dispersion.
When the problems of control synthesis are discussed, the following important issues must be taken into account – closed-loop system stability, control processes quality, input and output constraints, external disturbances, computational complexity of the control algorithms from the real-time implementation point of view. Now assume that the control synthesis problem is solved. Then the next step, preceding practical implementation of the obtained control algorithm, is the mathematical modelling of the control processes in the closed-loop system. The most suitable environment for such a task is a subsystem Simulink of the dynamic simulation from the MATLAB package.
3) Model (1) parameters identification using experimental data. The identification can be performed, for example, on the base of the standard methods, which are implemented in the System Identification Toolbox of the MATLAB package. 4) Verification of the obtained model. The verification is usually realized by comparison of the identified model output and the plant output for given input signal. This procedure gives the information about how obtained model reflects dynamics of the plant.
3.3. Digital signal processing The problem of signal processing in the framework of the proposed labs appears due to the presence of the significant noise in the measurements of sensors, which are used on the experimental plants. It’s obvious that the measurement noise influences the quality of the control processes. Consequently, the main goal is the measurement noise filtering. The following questions can be considered during the labs:
Assume that the mathematical model of the form (1) is obtained as a result of the identification procedure. Let’s
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• Making simple calculations such as array operating, finding of eigenvalues, determinants of square matrices, solutions of the equations and the systems of equations, Laplace and Fourier transformations, operating with polynomials, integrating and etc. on the basis of the wide spectrum of mathematical functions, realized in MATLAB package. • Executing different steps in mathematical modeling. For example, it can be parameter or structure identification of the considered linear and nonlinear systems from measured input-output data on the basis of System Identification Toolbox, where the most popular algorithms for identification are realized, and linearization of the available nonlinear models in the vicinity of the operating point or motion. • Analysis and synthesis for continuous or discrete linear time-invariant models (LTI-objects) of plants with the help of instruments of Control System Toolbox. This toolbox provides a lot of standard algorithms for analyzing stability, controllability, observability, constructing of different characteristics, such as zeros and poles of the LTI-object, its frequency response, gain and phase margins etc. The control laws design can be based on the decisions of problems of modal control, LQR/LQG optimization, PID-control. While working with the toolbox the LTIobjects can be described in the different forms in time domain or frequency domain. • Analysis, signal processing and synthesis for continuous or discrete models of plants, including alternative techniques for LTI-objects, with the help of instruments of the other toolboxes. On the basis of the toolboxes in MATLAB one can provide the development of the various modern techniques for considered systems, such as design and simulating model predictive controllers, design and simulating fuzzy logic systems. The special attention when exploring LTI-objects should be paid to design of robust controllers for plants that can have model uncertainty. • Adaptation and modification of the algorithms, provided in MATLAB, taking into account the characteristics of the considered task, and implementation of the new computational algorithms. • Simulation of the dynamic processes in continuous and discrete time in the designed control systems. It can be performed in the special system Simulink in MATLAB package. Simulink allows investigating the dynamics of the entire closed-loop systems with the designed controllers in the real-time operation mode. One can validate the constructed models of the plants and designed controllers by verifying the plant behavior and the graphic and digital representation of its dynamic parameters. To improve characteristics of the controller and other elements of the entire closed-loop system one can tune their parameters using the parametric optimization approach, which is performed by the subsystem Simulink Response Optimization.
• estimation of the noise spectral characteristics. This problem can be solved, for example, by constructing a spectrum of the signal generated by the sensor when the controlled object is placed at the equilibrium point; • measurement signals filtering. Here the problem of filter design on the base of the estimated spectral characteristics of the noise should be treated. The standard MATLAB tools can be involved for this problem solution. It can be noted that the designed filter should be tested by means of mathematical modelling using Simulink before the practical implementation. 3.4. Real-time implementation The significance of this question is explained by the fact that all of the considered experimental plants are digital control devices operating in real-time. In this connection it is critical that the computational time required for calculating control signal should not exceed the sampling interval of the corresponding digital control system. This fact imposes the additional constraint on the used control algorithm, which must be quite simple. If the computational consumptions, required for control calculations, exceed the sampling interval, then it’s necessary either to tune control algorithm parameters or to optimize the algorithm in order to reduce the computational time. In the framework of the labs, each designed control algorithm can be tested in terms of possibility of its implementation in real-time. This task can also be done on the base of MATLAB/Simulink modelling complex with the additional measurements of computational time, required at each sample instant for control input calculating. 4. ANALYSIS AND SYNTHESIS USING MATLAB All discussed plants can successfully function with the help of modern program packages such as MATLAB. MATLAB is regarded as the standard instrument for supporting of technical calculations and can be used as the base environment for working with labs with the experimental plants. MATLAB package provides an extensive range of industry-standard tools and algorithms for analysis and design in control and signal processing and can be intensively exploited in different directions while preparing labs, such as • mathematical modelling of plants; • software support implementation for all methods and corresponding algorithms for analysis and design of the elements of control systems for the plants; • computer models construction of all elements of the designed dynamic systems, including the subsystems for data acquisition and signal processing. While working in each direction the students can carry out numerous experiments, running different scenarios for the plants, which can be illustrated with graphical visualizing and digital performance of the processed data.
There can be noted the following aspects of using MATLAB/Simulink system in student’s research work with the experimental plants:
The following problems can be solved in labs by means of MATLAB tools:
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• The variety and simplicity of calculations, which can be performed in MATLAB/Simulink system, allow to realize simply the studied notions and mathematical formalized tasks as well as the whole complex of actions and calculations that should be made when investigating the educational control plants, and, thus, the real industrial control plants. • One of the main features of MATLAB is its orientation on the effectiveness of array calculations, executing in this environment. So, there is a great spectrum of appropriate embodied methods in MATLAB, and the operations with vectors and matrices are extremely fast and simple. The essential part of computing in applying the algorithms for LTI-objects, which form the important class of control plants models and certainly should be exploited during executing of the labs, is presented by array calculations. The mentioned MATLAB’s feature then provides great possibilities for students to handle with LTI-objects and study of many various tasks for such plants, easily and quickly performing calculations and visualizing the results in MATLAB. • Realizing the calculations of the same type for varying data within the framework of some method for the considered educational plant, students can compare the results and get the peculiarities of using the studied approaches in practice. The necessity of applying the formalized mathematical methods for analyzing, signal processing and synthesis for the experimental control plants makes the students understand the advantages and disadvantages of the analytic approaches and algorithms, and the ways of getting round the appeared difficulties.
other, so they are very active. Labs classes are very interesting for both students and lecturers. Let us consider magnetic levitation system control design lab as an example. The opening challenge is the identification of the parameters of linear model (Sotnikova, 2009), which describes the process dynamics near the operating point xb 0 = 0.009 . Such a linear model is varied in dependence of the distance from the electromagnet. The first step of identification is the gathering of the experimental data. The data that are used for the identification in this case are shown in fig. 4. These data include the measurements of the ball position and the coil current on the time interval with the duration of 50 sec. Here the random binary sequence was used as the input signal. The next step is data pre-processing, which results in zero mean data sequences. The obtained data is used further in linear model identification. Ball position and Reference signal
-3
11
x 10
x b (m )
10 9 8 7 150
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175 180 t (s) Coil Current
155
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I (A )
1.2 1.1 1 0.9
5. LABS EXAMPLES
0.8 150
Each lab is started with the brief theoretical introduction to the considered control problem and different well-known approaches for it solution. The lecturer points out the key difficulties and discuss with the students the advantages and disadvantages of the different approaches. After that, the lecturer presents in more details some particular approach and has a discussion with the students how they can apply that method for the specified educational plant. At this stage students start to formulate control problem mathematically for the particular plant and try to solve it using the proposed approach. They try to implement the obtained solution by performing simulations in the MATLAB/Simulink environment. Lecturer verifies received results and guides students if they have some mistakes. If students perform the first part successfully, they attempt to implement the proposed approach to the real educational plant. At this step they are faced with a lot of challenges and realize that the simulation results differ with the ones they can see for the plant. It is the most interesting part of the lab when the students try different ways in order to avoid difficulties and to get satisfactory results. Usually lab is performed by several groups of students. At the end of the lab the students from each group present the work they have done.
175 t (s)
180
Fig. 4. Experimental data for linear model identification. Parameters estimation of the linear model can be performed with the help of the prediction error method, which is implemented, for example, by the function pem in the System Identification Toolbox of MATLAB package. As a result of the identification the estimations of the parameters and the corresponding linear model are obtained. To test the adequacy of the identified model it was used to adjust the coefficients of the PID-controller in order to improve the quality of the control processes. This adjustment was implemented using the Simulink Response Optimization block of the MATLAB package. The experimental data, where the new adjusted controller is used, are presented in the fig. 5. From the comparison of figures 4 and 5, illustrating the results of the lab, it is easy to see that the quality of the control processes has improved. Therefore, it can be concluded that the identified model is adequate to the real processes dynamics and it can be used for digital control system analysis and synthesis, in addition to the already adjusted PID-controller.
Lab classes have interactive form of education. Students are encouraged to have a discussion with the lecturer and each 216
9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012
created by the operators and the conditions of transition to the label. Let us consider the example in the Table 2. In this part of the code not only conditional operators are used, but also registers as well as some instruction. We supplement the code with two more important actions – by closing and opening a tool such as "capture". To do this, we add four lines containing the robotic registers «Register Output». We present below an example which contains some commands making robotic movement from one position to another, captures the cargo, moves to the next position and returns to home-position.
Ball position
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x 10
x b (m )
10 9 8 7
0
1
2
3
4
5 6 t (s) Coil Current
7
8
9
10
I (A )
1.2 1.1
Table 3. Example of the program for robotic movement
1
1: R[1]=0 2: PR[2,3]=–30 3: PR[3,3]=30 4: LABEL1 5: J P[1: Home] 100% FINE 6: L P[2] 2000 mm/sec FINE OFFSET PR[2] 7: RO[1]=ON 8: RO[2]=OFF 9: L P[3] 2000 mm/sec FINE 10: C P[4]
0.9 0.8
0
1
2
3
4
5 t (s)
6
7
8
9
10
Fig. 5. Experimental data with using of adjusted controller. The second example has to do with the programming of the robotic manipulator movements. The problem is to provide the movement of the robotic through the given points. The illustrated results of the lab correspond to the movement between two positions, capturing the small box and returning to the initial position. The standard program for FANUCrobotic is a sequence of commands. For example let us look at the following code:
: P[5] 2000 mm/sec CNT100
11: L P[6] 2000 mm/sec FINE OFFSET PR[3] 12: RO[1]=OFF 13: RO[2]=ON 14: L P[7] 2000 mm/sec FINE 15: J P[1: Home] 100% FINE 16: R[1]=R[1]+1 17: IF R[1]<=3 JUMP LABEL1 18: CALL HOME
Table 1. Example of a linear program for robotic 1: 2: 3: 4: :
5: 6: 7:
J P[1: Home] 100% FINE L P[2] 2000 mm/sec CNT100 L P[3] 2000 mm/sec FINE C P[4] P[5] 2000 mm/sec FINE L P[6] 2000 mm/sec FINE L P[7] 2000 mm/sec FINE J P[1: Home] 100% FINE
6. CONCLUSIONS In this paper the complex of control systems analysis and design labs for real educational plants is proposed. The main problems, which should be considered in the framework of the labs, are presented. Two examples of the labs problems are given.
In this small program we can see some various structures and points. The letter “P” is used to determine a position of the robotic. The number in brackets is a number of the position. The word “Home” equals a name of start position. Then there is a velocity with which the robotic moves to the position. And the next parameter is CNT or FINE. It shows the type of destination position. This program provides the robotic moves from position “1” to “7” and returns to “1” again.
REFERENCES Lepikhin T. (2009) Using FANUC industrial robotic in the learning process // Proc. IV Intern. conf. «Modern information technologies and IT-education», Moscow, INTUIT.RU, pp.339-345. Ljung L. (1998). System Identification: Theory for the User. Prentice Hall PTR, Upper Saddle River, New Jersey, 2nd edition, 1999. MAGLEV: Magnetic Levitation Plant (2007). User Manual. Quanser. Veremey E., Lepikhin T. (2008) Innovative teaching and research environment for modelling of the information processes // Proc. III Intern. conf. «Modern information technologies and IT-education», Moscow, MAKS Press, pp.207-214. Sotnikova M. (2009) Identification of a linear model of the magnetic levitation in the MATLAB environment // Proc. IV conf. «Designing engineering and scientific applications in MATLAB», pp. 507-522.
Table 2. Example of the loop 1: 2: 3: 4: 5: 6:
R[1]=0 LABEL1 J P[1] 100% FINE L P[2] 2000mm/sec R[1]=R[1]+1 IF R[1]>=3 JUMP LABEL1
In order to perform a set of movements by the robotic several times, there exists the possibility of loops using. A loop is
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Control Systems Analysis and Design Labs with Educational Plants M. Sotnikova, N. Zhabko and T. Lepikhin Faculty of Applied Mathematics and Control Processes, Saint-Petersburg State University, Saint-Petersburg, Russia. (e-mail: [email protected]) Abstract: This article is devoted to using of the special laboratory plants in control education. The complex of control system analysis and design labs is presented, where each lab deals with one of the experimental control plants. The brief description of the plants is given and their main features are analyzed from the control point of view. The substantial formulations of the labs, which are discussed in the paper, include the problems of system identification, digital control systems design, signal processing and real-time implementation. The advantages of using MATLAB environment for the problem solution are pointed out. Finally, the examples of the labs are presented. Keywords: educational plants, control design labs control point of view are outlined. Secondly, the substantial formulations of the labs problems are given for each of the areas mentioned above. Thirdly, the advantages of using MATLAB for the analysis and synthesis of the control systems are discussed. The last section contains some examples of labs, which are devoted to control system design for a magnetic levitation plant and programming movements for a robotic manipulator.
1. INTRODUCTION Real plants are invaluable in control education because they allow students to make sense of essential problems which appear in the control of real objects. Usually, educational control objects are relatively simple compared to real industrial control plants; nevertheless, they reveal all the typical control challenges such as unaccurate knowledge of the mathematical model, sensor noise, external disturbances, input and output constraints, etc.
2. EXPERIMENTAL EDUCATIONAL PLANTS In the framework of the proposed labs the following educational experimental plants are used:
Educational plants are very useful for both teaching process and research (Veremey, 2008). Performing the labs the students are faced with a whole spectrum of control system design and implementation problems. The researches have an opportunity to test any control algorithm or perform the identification procedure, etc. for real control object.
• magnetic levitation plant; • rotary flexible joint system; • ball and beam system; • robotic manipulator FANUC M-20iA. These plants are intensively used for control education at the Information Processes Modelling Laboratory at the Faculty of Applied Mathematics and Control Processes, SaintPetersburg State University.
This article deals with the complex of control systems analysis and design labs. The main goal of the proposed labs is an application of the control theory methods to the analysis and design of control systems for educational plants. This complex can be used in educational process for teaching students, who specialize in the area of dynamical systems control and its applications. The control system design problem for a particular plant includes a lot of sub-problems, the most serious of them being system identification, digital control system analysis and synthesis, signal processing and real-time implementation. A lot of different lab courses can be constructed on the basis of the proposed labs depending on the course specifics – the general understanding of control design problem for real plants, system identification or others. The paper is organized in the following way. Firstly, a brief description of educational plants as control objects is presented, and their most important characteristics from the 978-3-902823-01-4/12/$20.00 © 2012 IFAC
Fig. 1. Magnetic levitation plant.
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10.3182/20120619-3-RU-2024.00088
9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012
Let us consider these plants in more details. The magnetic levitation system is shown in Fig. 1. The main components of system are – an electromagnet above, a pedestal below, where the ball rests initially, and a steel ball. The aim of the control is to stabilize the ball position in a particular point between the electromagnet and the pedestal by means of the controlled voltage which is applied to the electromagnet.
processes, is quite simple and suitable for educational purposes. The notable feature of this plant is its open-loop instability, i.e., if the beam angle is fixed, the ball rolls down until it reaches the end point of the beam. Each of these plants has the required hardware to transfer the information from the sensors to the computer and, vice versa, to bring the control signal from the computer to the plants. In addition, each plant has necessary software to calculate the control signal on the basis of the measurements and additional software support providing the interface with the MATLAB environment. Thus, the control system design and analysis can be performed using MATLAB, and, after that, control algorithms can be easily implemented for plants realtime control. It can be noted that all plants are digital control systems, so the methods of digital control theory must be used for its analysis and synthesis.
Magnetic levitation system (MAGLEV, 2007) has a current sensor and an optical sensor, which measure the current in the electromagnet and the distance between ball position and the electromagnet surface respectively. The optical sensor is very sensitive to changes in temperature and light conditions, and its measurements are very noisy. It is important to notice that the magnetic field created by the electromagnet is nonuniform, especially near the magnet surface. Consequently it is quite difficult to describe it mathematically. Thereby the corresponding mathematical model of the control process is essentially nonlinear. One more important feature of the system is the ball vertical position unstability.
The industrial robotic manipulator FANUC M-20iA (Fig. 3) is widely used in the educational process in a number of courses for preparing IT-specialists (Lepikhin, 2009). The special significance of the proposed courses is determined by their practical orientation, related to the direct interaction with real information systems. Dealing with the robotic one should develop software using the specific programming language with structures like loops, conditional statements, labels, special instructions and interaction with external interfaces of the robotic.
The rotary flexible joint system is an educational plant where a pendulum can oscillate on a rotating platform. This plant is shown in Fig. 2 (left). The motion of the pendulum is limited by two springs which connect it to the rotating platform. The goal of the control is to deflect the pendulum at the given angle and to stabilize it in the vicinity of the new direction. In order to obtain the desirable goal, it is necessary to control the rotations of the platform. This plant has two sensors. The first of them measures the angle of the pendulum deflection, while the second one measures the angle of the platform rotation. Both sensors are very noisy, causing the main difficulty in the control of the considered object. Besides, the structure of the differential equations, which represent the mathematical model of the control process, is not so evident, and it is necessary to perform the system model identification first. The quality of the control processes in this case is determined, among other things, by the level of the oscillations in the transient responses.
Fig. 3. Robotic manipulator FANUC M-20iA. 3. LABS PROBLEMS FORMULATION The basic labs directions are system identification, analysis and synthesis of the digital control algorithms, digital signal processing and real-time implementation. For each direction the substantial formulation of the labs problems is presented below.
Fig. 2. Rotary flexible joint system (left) and Ball and Beam system (right). A ball and beam system is one of the most popular educational plants. In it a steel ball rolls on the top of a long beam. The beam angle can be adjusted by applying control signal to the electrical motor. The goal of the control is to move the ball to the desired position and to stabilize it in the vicinity of that point. The ball position on the beam is measured by the ultrasonic range sensor. The measurements of this sensor are extremely noisy and can give estimations of the position with the essential errors. At the same time, the mathematical model, describing the dynamics of the control
3.1. Plant model identification The first problem that appears when someone starts to work with a new plant is to obtain its mathematical model. Most educational plants have no documentation where their mathematical models are described. In other cases, when some initial mathematical model is presented, it is often necessary to significantly improve it.
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In the framework of the lab, the problem of model identification (Ljung, 1999) can be divided into the following steps.
consider the following discrete-time analog that can be constructed on the base of model (1)
x k +1 = f (k , x k , u k ) ,
1) Getting the structure of the equations which are constitutes the mathematical model of the plant in the form
x = f (t , x, u, k ) ,
(2)
where x k ∈ E n – state vector, u k ∈ E m – control input at the
(1)
where x ∈ E n – state vector, u ∈ E m – control input, k ∈ E r – vector of parameters, which values must be estimated during the process of identification. Equations (1) are mainly formed on the base of priory knowledge about the object dynamics and the basic physical laws.
sample instant k . In the further discussion model (2) is used as a base for analysis and synthesis of the digital control. 3.2. Analysis and synthesis of the digital control The problems of digital control analysis and synthesis are considered in the proposed labs with the help of Control System Toolbox of the MATLAB package.
This problem is quite complicated for some devices, for example, for the magnetic levitation plant. This is due to the fact that the magnetic field near the magnet is extremely difficult to describe, thus the corresponding mathematical model representing ball dynamics nearby the magnet has a very complex structure.
Let us look at the most important questions of the control system analysis: • stability analysis. Here the question of stability can be considered with respect to the nonlinear model (2) or its linearization in the vicinity of the operating point or trajectory; • controllability and observability analysis. Here the conditions of full controllability and observability are checked and the questions of system stabilizability and the necessary content of measurements are investigated. Now let us talk about the problems related to the synthesis of the digital control systems. The key problem that should be treated for the experimental devices is synthesis of stabilizing feedback control. Thus, for the magnetic levitation system, the feedback control law should provide ball position stabilization at a given distance from the electromagnet, for the rotary flexible joint system – fix the pendulum with the given rotation angle and for the ball and beam system – stabilize the ball position at the desired point on the beam. The other synthesis problem of interest is development of program control algorithms and realization of the program movements for the educational plants.
Let us consider that the structure of the equations (1) reflects the control object dynamics within certain limits. In particular, the equations (1) can be linear and represent the object dynamics in the vicinity of equilibrium point. 2) Experiment design and implementation on the plant, and data gathering for identification. At this stage the following key questions regarded to experiment design should be considered: • method of carrying out the experiment – in an open-loop or in a closed-loop. For instance, for magnetic levitation system the experiment can be performed only in closedloop because of the ball vertical position instability; • data for identification – which data should be gathered for identification, and which signals are considered as input and output. Besides, an important question is how many variables should be measured in order to estimate all the parameters of the model (1); • input signal selection. The standard choice is between the step, harmonic or random binary signals. It is important to note that the probabilistic properties of the estimations of the parameter vector k in (1) are strongly depend on the input signal selection. So, the problem is how to choose the input signal in order to remove the biases of the estimations and to minimize their dispersion.
When the problems of control synthesis are discussed, the following important issues must be taken into account – closed-loop system stability, control processes quality, input and output constraints, external disturbances, computational complexity of the control algorithms from the real-time implementation point of view. Now assume that the control synthesis problem is solved. Then the next step, preceding practical implementation of the obtained control algorithm, is the mathematical modelling of the control processes in the closed-loop system. The most suitable environment for such a task is a subsystem Simulink of the dynamic simulation from the MATLAB package.
3) Model (1) parameters identification using experimental data. The identification can be performed, for example, on the base of the standard methods, which are implemented in the System Identification Toolbox of the MATLAB package. 4) Verification of the obtained model. The verification is usually realized by comparison of the identified model output and the plant output for given input signal. This procedure gives the information about how obtained model reflects dynamics of the plant.
3.3. Digital signal processing The problem of signal processing in the framework of the proposed labs appears due to the presence of the significant noise in the measurements of sensors, which are used on the experimental plants. It’s obvious that the measurement noise influences the quality of the control processes. Consequently, the main goal is the measurement noise filtering. The following questions can be considered during the labs:
Assume that the mathematical model of the form (1) is obtained as a result of the identification procedure. Let’s
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• Making simple calculations such as array operating, finding of eigenvalues, determinants of square matrices, solutions of the equations and the systems of equations, Laplace and Fourier transformations, operating with polynomials, integrating and etc. on the basis of the wide spectrum of mathematical functions, realized in MATLAB package. • Executing different steps in mathematical modeling. For example, it can be parameter or structure identification of the considered linear and nonlinear systems from measured input-output data on the basis of System Identification Toolbox, where the most popular algorithms for identification are realized, and linearization of the available nonlinear models in the vicinity of the operating point or motion. • Analysis and synthesis for continuous or discrete linear time-invariant models (LTI-objects) of plants with the help of instruments of Control System Toolbox. This toolbox provides a lot of standard algorithms for analyzing stability, controllability, observability, constructing of different characteristics, such as zeros and poles of the LTI-object, its frequency response, gain and phase margins etc. The control laws design can be based on the decisions of problems of modal control, LQR/LQG optimization, PID-control. While working with the toolbox the LTIobjects can be described in the different forms in time domain or frequency domain. • Analysis, signal processing and synthesis for continuous or discrete models of plants, including alternative techniques for LTI-objects, with the help of instruments of the other toolboxes. On the basis of the toolboxes in MATLAB one can provide the development of the various modern techniques for considered systems, such as design and simulating model predictive controllers, design and simulating fuzzy logic systems. The special attention when exploring LTI-objects should be paid to design of robust controllers for plants that can have model uncertainty. • Adaptation and modification of the algorithms, provided in MATLAB, taking into account the characteristics of the considered task, and implementation of the new computational algorithms. • Simulation of the dynamic processes in continuous and discrete time in the designed control systems. It can be performed in the special system Simulink in MATLAB package. Simulink allows investigating the dynamics of the entire closed-loop systems with the designed controllers in the real-time operation mode. One can validate the constructed models of the plants and designed controllers by verifying the plant behavior and the graphic and digital representation of its dynamic parameters. To improve characteristics of the controller and other elements of the entire closed-loop system one can tune their parameters using the parametric optimization approach, which is performed by the subsystem Simulink Response Optimization.
• estimation of the noise spectral characteristics. This problem can be solved, for example, by constructing a spectrum of the signal generated by the sensor when the controlled object is placed at the equilibrium point; • measurement signals filtering. Here the problem of filter design on the base of the estimated spectral characteristics of the noise should be treated. The standard MATLAB tools can be involved for this problem solution. It can be noted that the designed filter should be tested by means of mathematical modelling using Simulink before the practical implementation. 3.4. Real-time implementation The significance of this question is explained by the fact that all of the considered experimental plants are digital control devices operating in real-time. In this connection it is critical that the computational time required for calculating control signal should not exceed the sampling interval of the corresponding digital control system. This fact imposes the additional constraint on the used control algorithm, which must be quite simple. If the computational consumptions, required for control calculations, exceed the sampling interval, then it’s necessary either to tune control algorithm parameters or to optimize the algorithm in order to reduce the computational time. In the framework of the labs, each designed control algorithm can be tested in terms of possibility of its implementation in real-time. This task can also be done on the base of MATLAB/Simulink modelling complex with the additional measurements of computational time, required at each sample instant for control input calculating. 4. ANALYSIS AND SYNTHESIS USING MATLAB All discussed plants can successfully function with the help of modern program packages such as MATLAB. MATLAB is regarded as the standard instrument for supporting of technical calculations and can be used as the base environment for working with labs with the experimental plants. MATLAB package provides an extensive range of industry-standard tools and algorithms for analysis and design in control and signal processing and can be intensively exploited in different directions while preparing labs, such as • mathematical modelling of plants; • software support implementation for all methods and corresponding algorithms for analysis and design of the elements of control systems for the plants; • computer models construction of all elements of the designed dynamic systems, including the subsystems for data acquisition and signal processing. While working in each direction the students can carry out numerous experiments, running different scenarios for the plants, which can be illustrated with graphical visualizing and digital performance of the processed data.
There can be noted the following aspects of using MATLAB/Simulink system in student’s research work with the experimental plants:
The following problems can be solved in labs by means of MATLAB tools:
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• The variety and simplicity of calculations, which can be performed in MATLAB/Simulink system, allow to realize simply the studied notions and mathematical formalized tasks as well as the whole complex of actions and calculations that should be made when investigating the educational control plants, and, thus, the real industrial control plants. • One of the main features of MATLAB is its orientation on the effectiveness of array calculations, executing in this environment. So, there is a great spectrum of appropriate embodied methods in MATLAB, and the operations with vectors and matrices are extremely fast and simple. The essential part of computing in applying the algorithms for LTI-objects, which form the important class of control plants models and certainly should be exploited during executing of the labs, is presented by array calculations. The mentioned MATLAB’s feature then provides great possibilities for students to handle with LTI-objects and study of many various tasks for such plants, easily and quickly performing calculations and visualizing the results in MATLAB. • Realizing the calculations of the same type for varying data within the framework of some method for the considered educational plant, students can compare the results and get the peculiarities of using the studied approaches in practice. The necessity of applying the formalized mathematical methods for analyzing, signal processing and synthesis for the experimental control plants makes the students understand the advantages and disadvantages of the analytic approaches and algorithms, and the ways of getting round the appeared difficulties.
other, so they are very active. Labs classes are very interesting for both students and lecturers. Let us consider magnetic levitation system control design lab as an example. The opening challenge is the identification of the parameters of linear model (Sotnikova, 2009), which describes the process dynamics near the operating point xb 0 = 0.009 . Such a linear model is varied in dependence of the distance from the electromagnet. The first step of identification is the gathering of the experimental data. The data that are used for the identification in this case are shown in fig. 4. These data include the measurements of the ball position and the coil current on the time interval with the duration of 50 sec. Here the random binary sequence was used as the input signal. The next step is data pre-processing, which results in zero mean data sequences. The obtained data is used further in linear model identification. Ball position and Reference signal
-3
11
x 10
x b (m )
10 9 8 7 150
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175 180 t (s) Coil Current
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I (A )
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5. LABS EXAMPLES
0.8 150
Each lab is started with the brief theoretical introduction to the considered control problem and different well-known approaches for it solution. The lecturer points out the key difficulties and discuss with the students the advantages and disadvantages of the different approaches. After that, the lecturer presents in more details some particular approach and has a discussion with the students how they can apply that method for the specified educational plant. At this stage students start to formulate control problem mathematically for the particular plant and try to solve it using the proposed approach. They try to implement the obtained solution by performing simulations in the MATLAB/Simulink environment. Lecturer verifies received results and guides students if they have some mistakes. If students perform the first part successfully, they attempt to implement the proposed approach to the real educational plant. At this step they are faced with a lot of challenges and realize that the simulation results differ with the ones they can see for the plant. It is the most interesting part of the lab when the students try different ways in order to avoid difficulties and to get satisfactory results. Usually lab is performed by several groups of students. At the end of the lab the students from each group present the work they have done.
175 t (s)
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Fig. 4. Experimental data for linear model identification. Parameters estimation of the linear model can be performed with the help of the prediction error method, which is implemented, for example, by the function pem in the System Identification Toolbox of MATLAB package. As a result of the identification the estimations of the parameters and the corresponding linear model are obtained. To test the adequacy of the identified model it was used to adjust the coefficients of the PID-controller in order to improve the quality of the control processes. This adjustment was implemented using the Simulink Response Optimization block of the MATLAB package. The experimental data, where the new adjusted controller is used, are presented in the fig. 5. From the comparison of figures 4 and 5, illustrating the results of the lab, it is easy to see that the quality of the control processes has improved. Therefore, it can be concluded that the identified model is adequate to the real processes dynamics and it can be used for digital control system analysis and synthesis, in addition to the already adjusted PID-controller.
Lab classes have interactive form of education. Students are encouraged to have a discussion with the lecturer and each 216
9th IFAC Symposium Advances in Control Education Nizhny Novgorod, Russia, June 19-21, 2012
created by the operators and the conditions of transition to the label. Let us consider the example in the Table 2. In this part of the code not only conditional operators are used, but also registers as well as some instruction. We supplement the code with two more important actions – by closing and opening a tool such as "capture". To do this, we add four lines containing the robotic registers «Register Output». We present below an example which contains some commands making robotic movement from one position to another, captures the cargo, moves to the next position and returns to home-position.
Ball position
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x b (m )
10 9 8 7
0
1
2
3
4
5 6 t (s) Coil Current
7
8
9
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Table 3. Example of the program for robotic movement
1
1: R[1]=0 2: PR[2,3]=–30 3: PR[3,3]=30 4: LABEL1 5: J P[1: Home] 100% FINE 6: L P[2] 2000 mm/sec FINE OFFSET PR[2] 7: RO[1]=ON 8: RO[2]=OFF 9: L P[3] 2000 mm/sec FINE 10: C P[4]
0.9 0.8
0
1
2
3
4
5 t (s)
6
7
8
9
10
Fig. 5. Experimental data with using of adjusted controller. The second example has to do with the programming of the robotic manipulator movements. The problem is to provide the movement of the robotic through the given points. The illustrated results of the lab correspond to the movement between two positions, capturing the small box and returning to the initial position. The standard program for FANUCrobotic is a sequence of commands. For example let us look at the following code:
: P[5] 2000 mm/sec CNT100
11: L P[6] 2000 mm/sec FINE OFFSET PR[3] 12: RO[1]=OFF 13: RO[2]=ON 14: L P[7] 2000 mm/sec FINE 15: J P[1: Home] 100% FINE 16: R[1]=R[1]+1 17: IF R[1]<=3 JUMP LABEL1 18: CALL HOME
Table 1. Example of a linear program for robotic 1: 2: 3: 4: :
5: 6: 7:
J P[1: Home] 100% FINE L P[2] 2000 mm/sec CNT100 L P[3] 2000 mm/sec FINE C P[4] P[5] 2000 mm/sec FINE L P[6] 2000 mm/sec FINE L P[7] 2000 mm/sec FINE J P[1: Home] 100% FINE
6. CONCLUSIONS In this paper the complex of control systems analysis and design labs for real educational plants is proposed. The main problems, which should be considered in the framework of the labs, are presented. Two examples of the labs problems are given.
In this small program we can see some various structures and points. The letter “P” is used to determine a position of the robotic. The number in brackets is a number of the position. The word “Home” equals a name of start position. Then there is a velocity with which the robotic moves to the position. And the next parameter is CNT or FINE. It shows the type of destination position. This program provides the robotic moves from position “1” to “7” and returns to “1” again.
REFERENCES Lepikhin T. (2009) Using FANUC industrial robotic in the learning process // Proc. IV Intern. conf. «Modern information technologies and IT-education», Moscow, INTUIT.RU, pp.339-345. Ljung L. (1998). System Identification: Theory for the User. Prentice Hall PTR, Upper Saddle River, New Jersey, 2nd edition, 1999. MAGLEV: Magnetic Levitation Plant (2007). User Manual. Quanser. Veremey E., Lepikhin T. (2008) Innovative teaching and research environment for modelling of the information processes // Proc. III Intern. conf. «Modern information technologies and IT-education», Moscow, MAKS Press, pp.207-214. Sotnikova M. (2009) Identification of a linear model of the magnetic levitation in the MATLAB environment // Proc. IV conf. «Designing engineering and scientific applications in MATLAB», pp. 507-522.
Table 2. Example of the loop 1: 2: 3: 4: 5: 6:
R[1]=0 LABEL1 J P[1] 100% FINE L P[2] 2000mm/sec R[1]=R[1]+1 IF R[1]>=3 JUMP LABEL1
In order to perform a set of movements by the robotic several times, there exists the possibility of loops using. A loop is
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