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Control Theory – association of mathematics and engineering

Wojciech Mitkowski

Krzysztof Oprzedkiewicz

Department of Automatics AGH Univ. of Science & Technology, Cracow, Poland,

Abstract

In this paper a methodology of teaching a Control Theory at AGH UST and HVS Tarnow is briefly

presented. Control Theory is an area associated with both a wide part of mathematics and a very

important part of engineering. The course of Control Theory contains three parts: lecture, auditory

classes and laboratory. The mathematical skills are crucial for automation engineers to project,

implement and supervise real control systems in industry and life. The general view about the Control

Theory course is illustrated by a simple example of development, tests and practical realization of a

laboratory control system.

An introduction

Control Theory is anarea of technical sciences closely associated with both mathematics and

engineering practice. This implies that mathematical skills are absolutely necessary for an

automation engineer to solve all problems from areas of development, supervision and

servicing of industrial control systems.

The following aspects will be presented:

• Areas of mathematics applied in Control Theory, • The methodology of teaching Control Theory: lectures, auditory exercises,

laboratories,

• An example of a real automation task solved by students during Control Theory course.

• Conclusions.

Areas of mathematics applied in Control Theory

Control Theory applies a number of mathematical tools, associated with different areas of

mathematics. The most important are the following:

Differential equations

They are a fundamental tool to describe dynamic systems. Depending on the physical nature

of the described plant they have different forms. For example systems with distributed

parameters are described by partial differential equations, non-linear systems are described by

non-linear equations. The differential equation is always a basis to build a model closely

associated to Control Theory: state equation or transfer function. Transfer functions are

calculated with the use of Laplace or “z” transforms.

Matrix theory

A number of fundamental properties of dynamic systems described by the state equation can

be defined and analyzed with the use of matrix theory. For example: the form of the spectrum

the state matrix determines the stability of the system, the rank of the controllability or

observability matrices determines the controllability or observability of the system. A broad

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set of examples covering applications matrix analysis in Control Theory was presented by

Mitkowski in 2007.

Polynomial algebra

Polynomial algebra is a main tool in the analysis properties of dynamic systems described

with the use of transfer function. The stability of a system is determined by the localization of

the roots of the transfer function denominator.

Complex numbers

Complex numbers allow us to describe the properties of dynamic systems from the point of

view of frequency. This analysis can be done by using the idea of the spectral transfer

function, which is obtained from the transfer function by replacing the complex variable “s”

by “jω”.

Functional analysis

Functional analysis is an advanced tool to analyze control systems for all classes of control

plants, for example for distributed-parameter or non-linear plants. For example, it allows us to

transfer a partial differential equation into the form of an infinite-dimensional state equation

and to analyze properties of such systems. A broad presentation of applying functional

analysis in Control Theory can be found in books by Balakrishnan (1989) or Mitkowski

(1991). An example of using functional analysis to construct the control system for infinite-

dimensional plant (a heat plant) can be found in paper: Mitkowski and Oprzedkiewicz (2009).

Interval analysis

This area of mathematics is a powerful tool to analyze control systems for plants with

uncertain parameters. The basics of interval analysis were formulated by Moore in 1966.

Applications of interval analysis in Control Theory were presented for example by Jaulin in

2001.

Numerical methods

The main area of application of numerical methods in Control Theory is simulations useful

for the verification of results obtained with the use of “theoretical” methods. A typical

approach is the use of simulations programming tools dedicated to this goal, for example

suitable toolboxes of MATLAB.

The methodology of teaching Control Theory

The course of Control Theory run at AGH UST and HVS Tarnow consists of three parts.

The basis and crucial part of the Control Theory course is a lecture. During lectures there are

presented main problems and methods of their solution with the use of particular

mathematical tools. Lectures start with the presentation of mathematical models of real

control plants. A most general mathematical model describing the dynamics of each dynamic

system is a non-linear, non homogenous, time- variant, nth order differential equation of

ordinary or partial type. In a lot of real situations this general model can be simplified and

then it turns to the form of first order, linear, time-invariant matrix differential equation. This

equation is called in Control Theory a state equation.

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Additionally, with the state equation is associated an algebraic linear equation, describing the

relation between a state of system (it is generally not available to observation) with an output

of the system (only output is “measurable”). It is called in Control Theory an “Output

Equation”. The two equations build the complete mathematical model of the each dynamic

system. For linear, finite-dimensional, time-invariant system they have the following form:

)()()(

)()()(

tDutCxty

tButAxtx

+=

+=& (1)

In (1) x(t) ∈ Rn denotes the state vector, An×n denotes the state matrix, u(t) ∈ Rm denotes the

control vector, Bn×m denotes the control matrix, y(t) ∈ Rr denotes the output vector, Cr×n

denotes the output matrix and Dr×m denotes the matrix of direct control.

The equations (1) can also describe infinite-dimensional systems, an example of such

approach will be presented.

The equations (1) fully describe the behaviour and elementary properties of the system:

eigenvalues of the matrix A determine the stability of the system. Stability is a fundamental

property of each system. From a mathematical point of view a definition of stability is similar

to the definition of continuity, the pair of matrices (A,B) determine the controllability of the

system, the pair of matrices (C,A) determine the observability of the system.

An alternative (and more simple than state equation) model of the dynamic system applied in

Control Theory is a transfer function. The transfer function describes only the observable and

controllable part of the linear system and it is a description of “input-output” type. A transfer

function can be obtained from the differential equation describing the dynamics of the system

or from the state equation (1). It has the following general form:

01

01

...

...

)(

)()(

asasa

bsbsb

sU

sYsG

n

n

m

m

+++

+++== (2)

where: U(s) denotes a Laplace transform of input, Y(s) denotes a Laplace transform of output,

a0 … an and b0 … bm are real coefficients. Degrees of the numerator and denominator of the

transfer function (2) must meet the assumption m ≤ n to assure the physical realization of the system.

The essential problem during construction of the both models is identification of their

parameters. These are realized via suitable experiments on real plant and with the use of the

least square method.

Furthermore, during lectures are presented the main properties of dynamic systems:

controllability, observability, stability and mathematical methods of their testing for different

classes of plants. Controllability and observability are discussed for the state-space equation

only (the stability can be tested both for state space equation and transfer function) because

the transfer function describes only the controllable and observable part of the system.

Then the idea of a closed-loop control system is introduced. This idea is fundamental

in Control Theory. It was presented by a lot of authors; a good preview of classic results can

be found for example in books: Franklin (1991), Grantham W. J. (1993). The general scheme

of a closed-loop system is shown in Figure 1. The system contains a controller and a control

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plant. In Figure 1 r denotes a seat point, e(t) denotes an error signal in system, u(t) denotes a

control signal and y(t) denotes a process value. The general idea of this system is to keep the

process variable y(t) equal to seat point r independent of disturbances. This is realized by the

controller: it is required to calculate such a control signal u(t), which assures the difference

between seat point and process value equals zero. The measure of this difference is the error

signal e(t). The general scheme shown in Figure 1 is applied in all areas of industry and life. It

can be found in air conditioning system, cars, planes etc.

Controller Control plant

r(t) y(t)

-

u(t) e(t)

Figure 1. The closed – loop control system

Methods of projecting control systems dedicated to different plants with the use of

different controllers are also presented during lectures. During the first level of study are

presented most typical control algorithms: the relay controller and a PID controller. For PID

are also presented tuning methods. During the second level of study are also presented

methods of optimal control.

The next important part of Control Theory course are auditory classes covering

problems of calculations associated to problems presented during lectures.

Classes start with the construction models of control plants: transfer functions and

state equations for particular examples of real systems from areas of mechanics, electrical

engineering, chemistry, etc. During classes models of real plants, available in the laboratory,

are always analyzed. The starting point to construction of these models is always a differential

equation describing the dynamics of the system. With the use of the differential equation are

obtained state equations or transfer functions. Next for these plants are constructed control

systems: the controller is proposed and the stability areas are estimated with use for example

of the Hurwitz theorem.

The third part of the Control Theory course are laboratory sessions. During

laboratories there are jobs possible for realization in the laboratory only (for example

identification of control plant) and obtained results are tested during auditory classes.

Verification of results is accomplished with the use of simulation tools (for example

MATLAB/SIMULINK) or with the use of real control systems, containing real laboratory

plant controlled by computer or industrial controller (for example PLC).

An Example

As an example consider a laboratory servo system with DC motor, shown in Figure 2. This

system is analyzed and investigated during both auditory classes and laboratories during

Control Theory course.

The DC motor via gearbox and load moves the output potentiometer. The input signal for the

system (the control signal) is the input voltage u(t), the output signal is the voltage y(t)

determined by the position of potentiometer connected to the rotor.

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u(t) y(t)

Figure 2. Laboratory servo system

The analysis of the above system during auditory classes starts with construction of the

mathematical model for it. The fundamental model of this system is built by the following

ordinary differential equation:

)()()( 1111 tuKtxtxT =+ &&& (3)

where: x1(t) denotes the position of the rotor, T1 denotes the time constant of the motor, K1 >0

denotes the coefficient determined by mechanical parameters of the motor, u(t) denotes the input voltage, y(t) denotes the output voltage, determined by the position of rotor: y(t) = cx1(t)

(c > 0).

Equation (3) is the basis for constructing models useful in Control Theory. The first one is the

state equation in form (1). For the considered case a state vector can be defined as: x(t) =

[x1(t) x2(t)]T, x2(t) denotes the velocity of rotor. Then the equations (1) become:

[ ]

=

+

−=

)(

)(0)(

)(

0

)(

)(1

0

10

)(

)(

2

1

1

1

2

1

12

1

tx

txcty

tu

T

Ktx

tx

Ttx

tx

&

&

(4)

Another model of the system we deal with is the transfer function. It can be obtained directly

from differential equation (3), or from state equation (4) and it is defined as follows:

( )1

)()(

)()(

1

11

+=−== −

sTs

cKBAsIC

sU

sYsG (5)

The next step is to project for the DC motor a control system in according to scheme 1, whose

job is to “trace” the changes of input voltage, which is the seat point r(t). needs to project and

tune a controller. The simple solution of this task is to apply a proportional controller with

inertia, described by the first order transfer function Gc(s):

1

)(+

=sT

KsG

c

c

c (6)

In (6) Kc>0 denotes the gain of controller, Tc>0 denotes the time constant of the controller. In

a correctly constructed controller it should be much smaller than the time constant of the

plant: Tc<<T1.

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The closed-loop system containing both controller and plant has the structure shown in Figure

1. The transfer function of the whole control system is:

ccc

c

csKcKssTTsTT

KcK

sR

sYsG

1

2

1

3

1

1

)()(

)()(

++++== (7)

The next problem is to tune the controller to the plant to assure the correct working of the

system and assumed control performance. The fundamental property of each real control

system is an asymptotic stability. The stability of the system described by the transfer function

is determined by the localization of roots of the transfer function denominator in complex

plane. This implies that the stability can be tested with the use of mathematical tools to

localization of polynomial roots. The one of most typical is the Hurwitz theorem. The

Hurwitz array for the closed-loop system described by (7) has the following form:

+

+

=

cc

c

cc

KcKTT

TT

KcKTT

H

11

1

11

0

01

0

(8)

The system (7) will be stable, if all the sub-determinants of array (8) are greater than zero.

This condition allows us to calculate the controller parameters Kc and Tc assuring the stability

of the closed-loop system. For our example a condition for these coefficients was calculated

from condition (8) with the use of sub-determinant H2 only, because sub-determinant H1 is

always positive for positive values of time constants T1 and Tc , and H3 is positive if H2 is

positive. The sub-determinant H2 has the following form:

11

11

2

c

cc

TT

KcKTTH

+= (9)

From condition: H2 > 0 we obtain at once the following relation for the controller’s gain Kc:

11

10KTcT

TTK

c

c

c

+<< (10)

After the project the control system should be tested with the use of MATLAB/SIMULINK.

A suitable model is shown in Figure 3.

y(t)seat point

r(t)

cK1

T1.s +s2

DC motor

Kc

Tc.s+1

Controller

Figure 3. The SIMULINK model of the system

The project of the control system run by students is finished by tests with the use of real

servo, shown in Figure 4. These tests are also run during laboratories, the controller is

implemented onto a PC computer; during other courses it is also implemented onto PLC.

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Figure 4. The real laboratory servo system

Conclusions

Themain conclusions from the paper can be formulated as follows:

• Control Theory is an essential area of applications mathematics in engineering and industrial practice. Each control system must be developed and tested with the use of

suitable mathematical tools.

• Mathematical skills are crucial for the automation engineer during the project, verification and supervision of each control system in practice.

References

Balakrishnan M. (1989) “Applied Functional Analysis” New York, Springer,

Franklin G. E. , Powell J. D., Emami-Naeini A. (1991) „Feedback Control of Dynamic

Systems“, Addison-Wesley

Grantham W.J., Vincent T. L. (1993) “Modern control systems analysis and design” Wiley &

Sons,

Jaulin J. (2001) “Applied interval analysis : with examples in parameter and state estimation,

robust control and robotics”. London, Springer ,

Mitkowski W. (1991) “Stabilization of dynamic systems” (in Polish), WNT Warsaw,

Mitkowski W. (2007) “Matrix equations and their applications” (in Polish) Edited by AGH

UST 2007.

Mitkowski W. Oprzedkiewicz K. (2009) “A sample time optimization problem in a digital

control system” System modeling and optimization, 23rd IFIP TC 7 conference : Cracow,

Poland, July 23–27, 2007 : revised selected papers, eds. Adam Korytowski [et al.] Springer

Berlin ; Heidelberg ; New York.

Moore R. (1966) “Interval Analysis”, Prentice Hall,

Moore R. (1997) “Methods and Applications of Interval Analysis” SIAM, Philadelphia