Control Theory assoiation of mathematis and engineering Wojieh Mitkowski Krzysztof Oprzedkiewiz Department of Automatis AGH Univ. of Siene & Tehnology, Craow, Poland, Abstrat In this paper a methodology of teahing a Control Theory at AGH UST and HVS Tarnow is briefly presented. Control Theory is an area assoiated with both a wide part of mathematis and a very important part of engineering. The ourse of Control Theory ontains three parts: leture, auditory lasses and laboratory. The mathematial skills are ruial for automation engineers to projet, implement and supervise real ontrol systems in industry and life. The general view about the Control Theory ourse is illustrated by a simple example of development, tests and pratial realization of a laboratory ontrol system. An introdution Control Theory is anarea of tehnial sienes losely assoiated with both mathematis and engineering pratie. This implies that mathematial skills are absolutely neessary for an automation engineer to solve all problems from areas of development, supervision and serviing of industrial ontrol systems. The following aspets will be presented: Areas of mathematis applied in Control Theory, The methodology of teahing Control Theory: letures, auditory exerises, laboratories, An example of a real automation task solved by students during Control Theory ourse. Conlusions. Areas of mathematis applied in Control Theory Control Theory applies a number of mathematial tools, assoiated with different areas of mathematis. The most important are the following: Differential equations They are a fundamental tool to desribe dynami systems. Depending on the physial nature of the desribed plant they have different forms. For example systems with distributed parameters are desribed by partial differential equations, non-linear systems are desribed by non-linear equations. The differential equation is always a basis to build a model losely assoiated to Control Theory: state equation or transfer funtion. Transfer funtions are alulated with the use of Laplae or z transforms. Matrix theory A number of fundamental properties of dynami systems desribed by the state equation an be defined and analyzed with the use of matrix theory. For example: the form of the spetrum the state matrix determines the stability of the system, the rank of the ontrollability or observability matries determines the ontrollability or observability of the system. A broad
set of examples overing appliations matrix analysis in Control Theory was presented by Mitkowski in 007. Polynomial algebra Polynomial algebra is a main tool in the analysis properties of dynami systems desribed with the use of transfer funtion. The stability of a system is determined by the loalization of the roots of the transfer funtion denominator. Complex numbers Complex numbers allow us to desribe the properties of dynami systems from the point of view of frequeny. This analysis an be done by using the idea of the spetral transfer funtion, whih is obtained from the transfer funtion by replaing the omplex variable s by jω. Funtional analysis Funtional analysis is an advaned tool to analyze ontrol systems for all lasses of ontrol 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 suh systems. A broad presentation of applying funtional analysis in Control Theory an be found in books by Balakrishnan (989) or Mitkowski (99). An example of using funtional analysis to onstrut the ontrol system for infinitedimensional plant (a heat plan an be found in paper: Mitkowski and Oprzedkiewiz (009). Interval analysis This area of mathematis is a powerful tool to analyze ontrol systems for plants with unertain parameters. The basis of interval analysis were formulated by Moore in 966. Appliations of interval analysis in Control Theory were presented for example by Jaulin in 00. Numerial methods The main area of appliation of numerial methods in Control Theory is simulations useful for the verifiation of results obtained with the use of theoretial methods. A typial approah is the use of simulations programming tools dediated to this goal, for example suitable toolboxes of MATLAB. The methodology of teahing Control Theory The ourse of Control Theory run at AGH UST and HVS Tarnow onsists of three parts. The basis and ruial part of the Control Theory ourse is a leture. During letures there are presented main problems and methods of their solution with the use of partiular mathematial tools. Letures start with the presentation of mathematial models of real ontrol plants. A most general mathematial model desribing the dynamis of eah dynami 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 an be simplified and then it turns to the form of first order, linear, time-invariant matrix differential equation. This equation is alled in Control Theory a state equation.
Additionally, with the state equation is assoiated an algebrai linear equation, desribing 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 alled in Control Theory an Output Equation. The two equations build the omplete mathematial model of the eah dynami system. For linear, finite-dimensional, time-invariant system they have the following form: x& ( = Ax( + Bu( y( = Cx( + Du( () In () x( R n denotes the state vetor, A n n denotes the state matrix, u( R m denotes the ontrol vetor, B n m denotes the ontrol matrix, y( R r denotes the output vetor, C r n denotes the output matrix and D r m denotes the matrix of diret ontrol. The equations () an also desribe infinite-dimensional systems, an example of suh approah will be presented. The equations () fully desribe 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 eah system. From a mathematial point of view a definition of stability is similar to the definition of ontinuity, the pair of matries (A,B) determine the ontrollability of the system, the pair of matries (C,A) determine the observability of the system. An alternative (and more simple than state equation) model of the dynami system applied in Control Theory is a transfer funtion. The transfer funtion desribes only the observable and ontrollable part of the linear system and it is a desription of input-output type. A transfer funtion an be obtained from the differential equation desribing the dynamis of the system or from the state equation (). It has the following general form: G( s) Y ( s) b s +... + b s+ b m m 0 = = () n U ( s) ans +... + as+ a0 where: U(s) denotes a Laplae transform of input, Y(s) denotes a Laplae transform of output, a 0 a n and b 0 b m are real oeffiients. Degrees of the numerator and denominator of the transfer funtion () must meet the assumption m n to assure the physial realization of the system. The essential problem during onstrution of the both models is identifiation of their parameters. These are realized via suitable experiments on real plant and with the use of the least square method. Furthermore, during letures are presented the main properties of dynami systems: ontrollability, observability, stability and mathematial methods of their testing for different lasses of plants. Controllability and observability are disussed for the state-spae equation only (the stability an be tested both for state spae equation and transfer funtion) beause the transfer funtion desribes only the ontrollable and observable part of the system. Then the idea of a losed-loop ontrol system is introdued. This idea is fundamental in Control Theory. It was presented by a lot of authors; a good preview of lassi results an be found for example in books: Franklin (99), Grantham W. J. (993). The general sheme of a losed-loop system is shown in Figure. The system ontains a ontroller and a ontrol 3
plant. In Figure r denotes a seat point, e( denotes an error signal in system, u( denotes a ontrol signal and y( denotes a proess value. The general idea of this system is to keep the proess variable y( equal to seat point r independent of disturbanes. This is realized by the ontroller: it is required to alulate suh a ontrol signal u(, whih assures the differene between seat point and proess value equals zero. The measure of this differene is the error signal e(. The general sheme shown in Figure is applied in all areas of industry and life. It an be found in air onditioning system, ars, planes et. r( - e( Controller u( Control plant y( Figure. The losed loop ontrol system Methods of projeting ontrol systems dediated to different plants with the use of different ontrollers are also presented during letures. During the first level of study are presented most typial ontrol algorithms: the relay ontroller and a PID ontroller. For PID are also presented tuning methods. During the seond level of study are also presented methods of optimal ontrol. The next important part of Control Theory ourse are auditory lasses overing problems of alulations assoiated to problems presented during letures. Classes start with the onstrution models of ontrol plants: transfer funtions and state equations for partiular examples of real systems from areas of mehanis, eletrial engineering, hemistry, et. During lasses models of real plants, available in the laboratory, are always analyzed. The starting point to onstrution of these models is always a differential equation desribing the dynamis of the system. With the use of the differential equation are obtained state equations or transfer funtions. Next for these plants are onstruted ontrol systems: the ontroller is proposed and the stability areas are estimated with use for example of the Hurwitz theorem. The third part of the Control Theory ourse are laboratory sessions. During laboratories there are jobs possible for realization in the laboratory only (for example identifiation of ontrol plan and obtained results are tested during auditory lasses. Verifiation of results is aomplished with the use of simulation tools (for example MATLAB/SIMULINK) or with the use of real ontrol systems, ontaining real laboratory plant ontrolled by omputer or industrial ontroller (for example PLC). An Example As an example onsider a laboratory servo system with DC motor, shown in Figure. This system is analyzed and investigated during both auditory lasses and laboratories during Control Theory ourse. The DC motor via gearbox and load moves the output potentiometer. The input signal for the system (the ontrol signal) is the input voltage u(, the output signal is the voltage y( determined by the position of potentiometer onneted to the rotor. 4
u( y( Figure. Laboratory servo system The analysis of the above system during auditory lasses starts with onstrution of the mathematial model for it. The fundamental model of this system is built by the following ordinary differential equation: T & x + x& ( = K u( ) (3) ( t where: x ( denotes the position of the rotor, T denotes the time onstant of the motor, K >0 denotes the oeffiient determined by mehanial parameters of the motor, u( denotes the input voltage, y( denotes the output voltage, determined by the position of rotor: y( = x ( ( > 0). Equation (3) is the basis for onstruting models useful in Control Theory. The first one is the state equation in form (). For the onsidered ase a state vetor an be defined as: x( = [x ( x (] T, x ( denotes the veloity of rotor. Then the equations () beome: x& 0 0 ( x ( ) = t + K 0 u( x& ( ( ) T x t T x ( y( = [ 0] x ( (4) Another model of the system we deal with is the transfer funtion. It an be obtained diretly from differential equation (3), or from state equation (4) and it is defined as follows: Y ( s) K G ( s) = = C( si A) B= (5) U ( s) s ( T s+ ) The next step is to projet for the DC motor a ontrol system in aording to sheme, whose job is to trae the hanges of input voltage, whih is the seat point r(. needs to projet and tune a ontroller. The simple solution of this task is to apply a proportional ontroller with inertia, desribed by the first order transfer funtion G (s): K G ( s) = (6) T s+ In (6) K >0 denotes the gain of ontroller, T >0 denotes the time onstant of the ontroller. In a orretly onstruted ontroller it should be muh smaller than the time onstant of the plant: T <<T. 5
The losed-loop system ontaining both ontroller and plant has the struture shown in Figure. The transfer funtion of the whole ontrol system is: G s Y ( s) KK ( s) = = (7) 3 R( s) T T s + ( T + T ) s + s+ K K The next problem is to tune the ontroller to the plant to assure the orret working of the system and assumed ontrol performane. The fundamental property of eah real ontrol system is an asymptoti stability. The stability of the system desribed by the transfer funtion is determined by the loalization of roots of the transfer funtion denominator in omplex plane. This implies that the stability an be tested with the use of mathematial tools to loalization of polynomial roots. The one of most typial is the Hurwitz theorem. The Hurwitz array for the losed-loop system desribed by (7) has the following form: H T + T = TT 0 K K T + T 0 0 K K (8) The system (7) will be stable, if all the sub-determinants of array (8) are greater than zero. This ondition allows us to alulate the ontroller parameters K and T assuring the stability of the losed-loop system. For our example a ondition for these oeffiients was alulated from ondition (8) with the use of sub-determinant H only, beause sub-determinant H is always positive for positive values of time onstants T and T, and H 3 is positive if H is positive. The sub-determinant H has the following form: H T + T KK = (9) T T From ondition: H > 0 we obtain at one the following relation for the ontroller s gain K : 0 T + T < K < (0) TT K After the projet the ontrol system should be tested with the use of MATLAB/SIMULINK. A suitable model is shown in Figure 3. seat point r( K T.s+ Controller K T.s +s DC motor y( Figure 3. The SIMULINK model of the system The projet of the ontrol 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 ontroller is implemented onto a PC omputer; during other ourses it is also implemented onto PLC. 6
Figure 4. The real laboratory servo system Conlusions Themain onlusions from the paper an be formulated as follows: Control Theory is an essential area of appliations mathematis in engineering and industrial pratie. Eah ontrol system must be developed and tested with the use of suitable mathematial tools. Mathematial skills are ruial for the automation engineer during the projet, verifiation and supervision of eah ontrol system in pratie. Referenes Balakrishnan M. (989) Applied Funtional Analysis New York, Springer, Franklin G. E., Powell J. D., Emami-Naeini A. (99) Feedbak Control of Dynami Systems, Addison-Wesley Grantham W.J., Vinent T. L. (993) Modern ontrol systems analysis and design Wiley & Sons, Jaulin J. (00) Applied interval analysis : with examples in parameter and state estimation, robust ontrol and robotis. London, Springer, Mitkowski W. (99) Stabilization of dynami systems (in Polish), WNT Warsaw, Mitkowski W. (007) Matrix equations and their appliations (in Polish) Edited by AGH UST 007. Mitkowski W. Oprzedkiewiz K. (009) A sample time optimization problem in a digital ontrol system System modeling and optimization, 3rd IFIP TC 7 onferene : Craow, Poland, July 3 7, 007 : revised seleted papers, eds. Adam Korytowski [et al.] Springer Berlin ; Heidelberg ; New York. Moore R. (966) Interval Analysis, Prentie Hall, Moore R. (997) Methods and Appliations of Interval Analysis SIAM, Philadelphia 7