2012_control systems analysis and design labs with educational plants

8/17/2019 2012_Control Systems Analysis and Design Labs With Educational Plants

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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 areanalyzed 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

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.

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.

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 systemscontrol and its applications.

The control system design problem for a particular plant

includes a lot of sub-problems, the most serious of thembeing 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 orothers.

The paper is organized in the following way. Firstly, a briefdescription of educational plants as control objects is

presented, and their most important characteristics from the

control point of view are outlined. Secondly, the substantialformulations 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 movementsfor a robotic manipulator.

2. EXPERIMENTAL EDUCATIONAL PLANTS

In the framework of the proposed labs the following

educational experimental plants are used:

• 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, Saint-

Petersburg State University.

Fig. 1. Magnetic levitation plant.

Proceedings of the 9th IFAC Symposium Advances in ControlEducationThe International Federation of Automatic ControlNizhny Novgorod, Russia, June 19-21, 2012

978-3-902823-01-4/12/$20.00 © 2012 IFAC 212 10.3182/20120619-3-RU-2024.00088


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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.

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, andits measurements are very noisy. It is important to notice that

the magnetic field created by the electromagnet is non-

uniform, 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 thesystem is the ball vertical position unstability.

The rotary flexible joint system is an educational plant where

a pendulum can oscillate on a rotating platform. This plant isshown 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. Thefirst 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 themathematical 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. 2. Rotary flexible joint system (left) and Ball and Beam

system (right).

A ball and beam system is one of the most populareducational 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 ofthe position with the essential errors. At the same time, the

mathematical model, describing the dynamics of the control

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 real-

time 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.

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). Thespecial significance of the proposed courses is determined by

their practical orientation, related to the direct interactionwith 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.

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 directionthe substantial formulation of the labs problems is presented

below.

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.

9th IFAC Symposium Advances in Control EducationNizhny Novgorod, Russia, June 19-21, 2012

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In the framework of the lab, the problem of model

identification (Ljung, 1999) can be divided into the following

steps.

1) Getting the structure of the equations which are constitutes

the mathematical model of the plant in the form

( )kuxf x ,,,t = , (1)

wherenEx∈ – state vector,

mEu∈ – control input, r Ek∈

– 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.

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 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 theobject 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 closed-loop because of the ball vertical position instability;

• data for identification – which data should be gathered foridentification, 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 thestep, 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 theinput 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.

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 inthe 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 proceduregives the information about how obtained model reflects

dynamics of the plant.

Assume that the mathematical model of the form (1) is

obtained as a result of the identification procedure. Let’s

consider the following discrete-time analog that can be

constructed on the base of model (1)

( )k k k k uxf x ,,1 =+ , (2)

wheren

k Ex ∈ – state vector,m

k Eu ∈ – control input at the

sample instantk

. 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.

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 ortrajectory;

• 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.

When the problems of control synthesis are discussed, thefollowing 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 mostsuitable environment for such a task is a subsystem Simulink

of the dynamic simulation from the MATLAB package.

3.3. Digital signal processing

The problem of signal processing in the framework of theproposed 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. Thefollowing questions can be considered during the labs:


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• estimation of the noise spectral characteristics. This

problem can be solved, for example, by constructing aspectrum 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 MATLABtools 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 theadditional constraint on the used control algorithm, which

must be quite simple. If the computational consumptions,

required for control calculations, exceed the samplinginterval, 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 helpof 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.

The following problems can be solved in labs by means of

MATLAB tools:

• 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 ofdifferent 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 LTI-

objects 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 onecan provide the development of the various modern

techniques for considered systems, such as design andsimulating model predictive controllers, design and

simulating fuzzy logic systems. The special attention when

exploring LTI-objects should be paid to design of robustcontrollers 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 anddiscrete 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 designedcontrollers 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.

There can be noted the following aspects of using

MATLAB/Simulink system in student’s research work

with the experimental plants:


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• The variety and simplicity of calculations, which can be

performed in MATLAB/Simulink system, allow to realizesimply 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 resultsin 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.

5. LABS EXAMPLES

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 keydifficulties 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 mathematicallyfor the particular plant and try to solve it using the proposed

approach. They try to implement the obtained solution byperforming 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 theproposed 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 severalgroups of students. At the end of the lab the students from

each group present the work they have done.

Lab classes have interactive form of education. Students areencouraged to have a discussion with the lecturer and each

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

009.00 =b x . 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.

150 155 160 165 170 175 180 185 190 195 200 7

8

9

10

11x 10

-3

t (s)

x b

( m

)

Ball position and Reference signal

150 155 160 165 170 175 180 185 190 195 200 0.8

0.9

1

1.1

1.2

t (s)

I ( A )

Coil Current

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 aresult 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 toadjust 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 controlsystem analysis and synthesis, in addition to the already

adjusted PID-controller.


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0 1 2 3 4 5 6 7 8 9 10 7

8

9

10

11x 10

-3

t (s)

x b

( m )

Ball position

0 1 2 3 4 5 6 7 8 9 10 0.8

0.9

1

1.1

1.2

t (s)

I ( A )

Coil Current

Fig. 5. Experimental data with using of adjusted controller.

The second example has to do with the programming of therobotic 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 returningto the initial position. The standard program for FANUC-

robotic is a sequence of commands. For example let us look

at the following code:

Table 1. Example of a linear program for robotic

1: J P[1: Home] 100% FINE

2: L P[2] 2000 mm/sec CNT100

3:

L P[3] 2000 mm/sec FINE

4: C P[4]

: P[5] 2000 mm/sec FINE

5: L P[6] 2000 mm/sec FINE

6: L P[7] 2000 mm/sec FINE7:

J P[1: Home] 100% FINE

In this small program we can see some various structures and

points. The letter “P” is used to determine a position of therobotic. 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 roboticmoves from position “1” to “7” and returns to “1” again.

Table 2. Example of the loop

1: R[1]=0

2: LABEL1

3: J P[1] 100% FINE4:

L P[2] 2000mm/sec

5: R[1]=R[1]+16:

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

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 «RegisterOutput». 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.

Table 3. Example of the program for robotic movement

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


10: C P[4]

: 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


15: J P[1: Home] 100% FINE

16: R[1]=R[1]+1

17: IF R[1]

2012_control systems analysis and design labs with educational plants

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