Applications of mathematics in selected control and decision processes

Jerzy Baranowski, et al. "Applications of mathematics in selected control and decision processes." Mathematica Applicanda 39.1 (2011): null. <http://eudml.org/doc/292897>.

@article{JerzyBaranowski2011,
abstract = {Rapid development of computer science in recent years allowed more detailed analysis and synthesis of control systems for complex processes and supports decision making in different practical areas. Discoveries in mathematical nature of the universe stimulate representatives of technical sciences to actions leading into planned affecting of real objects. Practical verification of engineers ideas in many cases is effective and leads to meaningful results. In control of dynamical systems the following algorithm of operations one has proved to be effective (see for example [27, 28, 32, 33]) Create a mathematical model, usually in a form of an appropriate differential equation. Perform the linearisation of the model. Design a control system, for example through appropriate feedback - usually through formulation of some kind of LQ problem. Verify the design with a real life system. It should be noted however, that control problems are not limited to applications of this algorithm. For example the parameters of the constructed model have to be obtained through the identification. We require that the designed control systems have certain properties. Most notable is the aspect of asymptotic (exponential) stability of the system. Different notions of stability are used, but most popular is the Lyapunov stability, also important is the practical stability. Along with stability also the aspect of area (basin) of attraction is discussed usually in context of LaSalle principle [22] (also known as Krasovskii-LaSalle principle). It is also desired, that the designed control system would possess such typical properties known from control theory as controllability and observability (stabilisability and detectability). In many cases not all needed measurements are available. In such case if system is observable one can construct a state observer - a dynamical system which estimates the unmeasured state variables. In other cases practical realisation of control systems requires application of computers or embedded circuits in real time regimes. In such cases an important aspect is the operation of appropriate A/D (analog/digital) and D/A (digital/analog) converters -- their synchronisation, their sampling frequency. Also the spatial placement of sensors (distance between them) should also be considered. Determination of control signal also is an interesting aspect. In most cases it is desired that the control should have a form of feedback. Often feedback can be designed using appropriate Lyapunov and Riccati equations (see for example [1] or [26, 27, 21]) usually because of the connection to the LQ problem (see [17]) and optimal filtration problem (see for example [9]). In other cases however different methods can be used. Stabilising feedback can be constructed through a construction of appropriate Lyapunov function or by influencing the location of system's eigenvalues. Feedback can also be designed by solving appropriate game theory problem, for example for LQ games. Moreover not all control problems have a structure of feedback -- in some cases control can be given as a function of time (so called open loop control) which will be a solution to certain dynamical optimisation problems (for example time optimal control). In this paper we present a series of examples showing different applications of control theory and game theory to different systems. Substantial part of them are the stabilisation problems but there are also state estimation, identification, optimal control, shape optimisation and decision support through game theory.References[ 1] M. Athans and P.L. Falb, Sterowanie optymalne: wstęp do teorii i jej zastosowanie, Warszawa WNT, 1969.[ 2] J. Baranowski, Projektowanie obserwatora dla silnika szeregowego prądu stałego, Półrocznik AGH AUTOMATYKA, 10(1): 33–52, 2006.[ 3] J. Baranowski, M. Długosz, Sterowanie czasooptymalne silnikiem obcowzbudnym prądu stałego, In K. Malinowski and L. Rutkowski, editors, Sterowanie i Automatyzacja: Aktualne problemy i ich rozwiązania, chapter 2, pages 87–96, Akademicka Oficyna Wydawnicza EXIT, 2008.[ 3] J. Baranowski, M. Długosz, and W. Mitkowski, Remarks about DC motor control, Archives of Control Sciences, 18(LIV)(3): 289–322, 2008.[ 4] K. Bisztyga, Sterowanie i regulacja silników elektrycznych, WNT, 1989.[ 5] V.G. Boltyanskii, Mathematical Methods of Optimal Control, Holt, Rinehart & Win-ston, New York, 1971.[ 6] J. Brehm, Postdecision changes in the desirability of alternatives, Journal of Ab-normal and Social Psychology, 52: 384–389, 1956.[ 7] A.G. Butkovskii, Sterowanie Optymalne Systemami o Parametrach Rozłożonych(ros.), Nauka, Moskwa, 1965.[ 8] W. Byrski, Obserwacja i sterowanie w systemach dynamicznych, Uczelniane Wy-dawnictwa Naukowo Dydaktyczne AGH, Kraków, 2007.[ 9] J. Chiasson, Nonlinear Differential-Geometric Techniques for Control of a SeriesDC Motor, IEEE Transactions on Conrol Systems Technology, 2(1): 35–42, March1994.[10] T. Damm, V. Dragan and G. Freiling, Lyapunov Iterations for Coupled RiccatiDifferential Equations Arising in Connection with Nash Differential Games, Mathematical Reports, 9(59): 35–46, 2007.[11] M. Długosz, Problemy optymalizacji układów napędowych w automatyce i robotyce, PhD thesis, Akademia Górniczo-Hutnicza im. S. Staszica w Krakowie, 2009.[12] M. Długosz and T. Lerch, Komputerowa identyfikacja parametrów silnika prądu stałego, Przegląd Elektrotechniczny, 86(2): 34–38, 2010.[13] J.C. Engwerda, Algorithms for computing Nash equilibria in deterministic LQ games, Computational Management Science, 4(2):113–140, 2007.[14] M. Ganobis, Electricity Market Modelling Using Dynamic LQ Games, In: Materiały XII Międzynarodowych Warsztatów Doktoranckich OWD w Wisle, pages 75–80, 2010.[15] M. Ganobis and W. Mitkowski, A Nash equilibrium in RC transmission line with two voltage sources, Materiały XXXII Międzynarodowej konferencji z podstaw elektrotechniki i teorii obwodów IC-SPETO, pages 117–118, 2009.[16] H. Górecki, S. Fuksa, A. Korytowski and W. Mitkowski, Sterowanie optymalne w systemach liniowych z kwadratowym wskaźnikiem jakości, PWN, Warszawa, 1983.[17] A. Guran, A. Bajaj, Y. Ishida, N. Perkins, G. D’Eleuterio and C. Pierre, Stability of Gyroscopic Systems – Series on Stability, Vibration and Control of Systems, vol. 2, World Scientific Publishing, Singapore, 1999.[18] Y.-C. Ho and A. Starr, Nonzero-sum Differential Games, Journal of Optimization Theory and Applications, 3: 184–206, 1969.[19] E. Kącki, Równania Różniczkowe Cząstkowe w Elektrotechnice, Wydawnictwa Nau-kowo-Techniczne, 1971.[21] P. Lancaster and L. Rodman, Algebraic Riccati Equations, Clarendon Press, Oxford,1995.[22] J. LaSalle and S. Lefschetz, Zarys teorii stabilności Lapunowa i jego metody bezpo-średniej, PWN, Warszawa, 1966.[23] W. Leonhard, Control of Electrical Drives 3rd edition, Springer-Verlag Berlin andHeidelberg NewYork, Berlin, 3 editing, 2001.[24] S. Mitkowski, Nonlinear Electric Circuits, Wydawnictwa AGH, Kraków, 1999.[25] W. Mitkowski, Stabilizacja liniowych układów nieskończenie wymiarowych za pomocą dynamicznego sprzężenia zwrotnego, Arch. Automatyki i Telemechaniki, 33(4):515–528, 1988.[26] W. Mitkowski, Stabilizacja Systemów Dynamicznych, WNT Warszawa, 1991.[27] W. Mitkowski, Projektowanie systemów sterowania z wykorzystaniem równania Riccatiego, In Z. Bubnicki and J. Józefczyk, editors, Mat. Konferencyjne XIII Krajowej Konferencji Automatyki, volume 1, pages 171–176, Oficyna Wydawnicza Politechniki Opolskiej, 1999.[28] W. Mitkowski, Metody projektowania układów regulacji optymalnej, In Z. Bubnickiand J. Korbicz, editors, XIV Krajowa Konferencja Automatyki, volmu 1, pages195–204, Uniwersytet Zielonogórski, Inst. Sterowania i Systemów Informatycznych,2002.[29] W. Mitkowski, Dynamic feedback in LC ladder network, Bulletin of the PolishAcademy of Sciences: Technical Sciences, 51(2): 173–180, 2003.[30] W. Mitkowski, Remarks about energy transfer in an RC ladder network, Inter-national Journal of Applied Mathematics and Computer Science, 13(2): 193–198,2003.[31] W. Mitkowski, Stabilization of LC ladder network, Bulletin of the Polish Academyof Sciences: Technical Sciences, 52(2): 109–114, 2004.[32] W. Mitkowski, Metody stabilizacji, In: Z. Bubnicki and R. Kulikowski and J. Kacprzyk, editors, XV Krajowa Konferencja Automatyki, volume 1, pages 169–178, KaiR PAN, IBS PAN oraz również PW, PIAP, PolSPAiR, 2005.[33] W. Mitkowski, Zastosowania równań liniowych w teorii sterowania. In K. Malinowski and L. Rutkowski, editors, Sterowanie i automatyzacja: aktualne problemy i ich rozwiązania, pages 11–22, Akademicka Oficyna Wydawnicza EXIT, Warszawa 2008.[34] W. Mitkowski and K. Oprzędkiewicz, A Sample Time Optimization Problem in a Digital Control system, In: A. Korytowski, K. Malanowski, W. Mitkowski and M. Szymkat, editor, System Modeling and Optimization 23rd IFIP TC7 Conference, pages 382–396, Springer, Berlin 2009, July 2007.[35] J. Ombach, Wykłady z równań różniczkowych wspomagane komputerowo-Maple,Wydawnicto U.J., 2 edition, 1999.[36] K. Oprzędkiewicz, An example of parabolic system identification, Zeszyty NaukoweAGH, Elektrotechnika, 16(2): 99–106, 1997.[37] W. Pełczewski and M. Krynke, Metoda zmiennych stanu w analize dynamiki układów napędowych, WNT, 1984.[38] L.S. Pontriagin, W.G. Bołtianski, R.W. Gemkrelidze and E.F. Miszczenko, Matematiczeskaja tieoria optimalnych processow, Nauka, Moskwa, 4 edition, 1983.[39] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes, John Willey, New York, 1962.[40] P. Skruch, Feedback stabilization of distributed parameter gyroscopic systems, W. Mitkowski and J. Kacprzyk, editors, Modelling Dynamics in Processes andSystems (Studies in Computational Intelligence), vol. 180, pages 85–97, Springer-Verlag, Berlin, Heidelberg, 2009.[41] P. Skruch, J. Baranowski and W. Mitkowski, Dynamic feedback stabilization of non-linear RC ladder network, In: Proceedings of XIII Symposium on Fundamental pro-blems of power electronics electromechanics and mechatronics, PPEEm 2009, Wisła,Poland, pages 136–141, 2009.[42] P. Skruch, W. Mitkowski, Optimum design of shapes using the Pontryagin principleof maximum, Automatyka, 13(1): 65–78, 2009.[43] P. Skruch, W. Mitkowski, Modelling and simulation of the shape optimization pro-blems, In: G.R. Rey and L.M. Muneta, editors, Modelling, Simulation and Optimi-zation, pages 187–208, In-Tech, Olajnica, Croatia, 2010.[44] H. Steinhaus, Między duchem a materią pośredniczy matematyka. Wybór, przed-mowa i redakcja naukowa Józef Łukaszewicz, PWN, Warszawa-Wrocław, 2000.[45] G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, En-glewood Cliffs, New York, 1973.[46] G. Szefer, L. Mikulski, Optimization of beams with the use of the Pontryagin prin-ciple of maximum, Archives of Civil Engineering, 24(3): 337–345, 1978.[47] M. Szymkat, A. Korytowski, Evolution of Structure for Direct Control Optimiza-tion, Discussiones Mathematicae. Differential Inclusions, Control and Optimization,27:165–193, 2007.[48] B. van Aarle, J. Engwerda and J. Plasmans, Cooperative and non-cooperative fi-scal stabilization policie},
author = {Jerzy Baranowski, Marek Długosz, Michał Ganobis, Paweł Skruch, Wojciech Mitkowski},
journal = {Mathematica Applicanda},
language = {eng},
number = {1},
pages = {null},
title = {Applications of mathematics in selected control and decision processes},
url = {http://eudml.org/doc/292897},
volume = {39},
year = {2011},
}

TY - JOUR
AU - Jerzy Baranowski
AU - Marek Długosz
AU - Michał Ganobis
AU - Paweł Skruch
AU - Wojciech Mitkowski
TI - Applications of mathematics in selected control and decision processes
JO - Mathematica Applicanda
PY - 2011
VL - 39
IS - 1
SP - null
AB - Rapid development of computer science in recent years allowed more detailed analysis and synthesis of control systems for complex processes and supports decision making in different practical areas. Discoveries in mathematical nature of the universe stimulate representatives of technical sciences to actions leading into planned affecting of real objects. Practical verification of engineers ideas in many cases is effective and leads to meaningful results. In control of dynamical systems the following algorithm of operations one has proved to be effective (see for example [27, 28, 32, 33]) Create a mathematical model, usually in a form of an appropriate differential equation. Perform the linearisation of the model. Design a control system, for example through appropriate feedback - usually through formulation of some kind of LQ problem. Verify the design with a real life system. It should be noted however, that control problems are not limited to applications of this algorithm. For example the parameters of the constructed model have to be obtained through the identification. We require that the designed control systems have certain properties. Most notable is the aspect of asymptotic (exponential) stability of the system. Different notions of stability are used, but most popular is the Lyapunov stability, also important is the practical stability. Along with stability also the aspect of area (basin) of attraction is discussed usually in context of LaSalle principle [22] (also known as Krasovskii-LaSalle principle). It is also desired, that the designed control system would possess such typical properties known from control theory as controllability and observability (stabilisability and detectability). In many cases not all needed measurements are available. In such case if system is observable one can construct a state observer - a dynamical system which estimates the unmeasured state variables. In other cases practical realisation of control systems requires application of computers or embedded circuits in real time regimes. In such cases an important aspect is the operation of appropriate A/D (analog/digital) and D/A (digital/analog) converters -- their synchronisation, their sampling frequency. Also the spatial placement of sensors (distance between them) should also be considered. Determination of control signal also is an interesting aspect. In most cases it is desired that the control should have a form of feedback. Often feedback can be designed using appropriate Lyapunov and Riccati equations (see for example [1] or [26, 27, 21]) usually because of the connection to the LQ problem (see [17]) and optimal filtration problem (see for example [9]). In other cases however different methods can be used. Stabilising feedback can be constructed through a construction of appropriate Lyapunov function or by influencing the location of system's eigenvalues. Feedback can also be designed by solving appropriate game theory problem, for example for LQ games. Moreover not all control problems have a structure of feedback -- in some cases control can be given as a function of time (so called open loop control) which will be a solution to certain dynamical optimisation problems (for example time optimal control). In this paper we present a series of examples showing different applications of control theory and game theory to different systems. Substantial part of them are the stabilisation problems but there are also state estimation, identification, optimal control, shape optimisation and decision support through game theory.References[ 1] M. Athans and P.L. Falb, Sterowanie optymalne: wstęp do teorii i jej zastosowanie, Warszawa WNT, 1969.[ 2] J. Baranowski, Projektowanie obserwatora dla silnika szeregowego prądu stałego, Półrocznik AGH AUTOMATYKA, 10(1): 33–52, 2006.[ 3] J. Baranowski, M. Długosz, Sterowanie czasooptymalne silnikiem obcowzbudnym prądu stałego, In K. Malinowski and L. Rutkowski, editors, Sterowanie i Automatyzacja: Aktualne problemy i ich rozwiązania, chapter 2, pages 87–96, Akademicka Oficyna Wydawnicza EXIT, 2008.[ 3] J. Baranowski, M. Długosz, and W. Mitkowski, Remarks about DC motor control, Archives of Control Sciences, 18(LIV)(3): 289–322, 2008.[ 4] K. Bisztyga, Sterowanie i regulacja silników elektrycznych, WNT, 1989.[ 5] V.G. Boltyanskii, Mathematical Methods of Optimal Control, Holt, Rinehart & Win-ston, New York, 1971.[ 6] J. Brehm, Postdecision changes in the desirability of alternatives, Journal of Ab-normal and Social Psychology, 52: 384–389, 1956.[ 7] A.G. Butkovskii, Sterowanie Optymalne Systemami o Parametrach Rozłożonych(ros.), Nauka, Moskwa, 1965.[ 8] W. Byrski, Obserwacja i sterowanie w systemach dynamicznych, Uczelniane Wy-dawnictwa Naukowo Dydaktyczne AGH, Kraków, 2007.[ 9] J. Chiasson, Nonlinear Differential-Geometric Techniques for Control of a SeriesDC Motor, IEEE Transactions on Conrol Systems Technology, 2(1): 35–42, March1994.[10] T. Damm, V. Dragan and G. Freiling, Lyapunov Iterations for Coupled RiccatiDifferential Equations Arising in Connection with Nash Differential Games, Mathematical Reports, 9(59): 35–46, 2007.[11] M. Długosz, Problemy optymalizacji układów napędowych w automatyce i robotyce, PhD thesis, Akademia Górniczo-Hutnicza im. S. Staszica w Krakowie, 2009.[12] M. Długosz and T. Lerch, Komputerowa identyfikacja parametrów silnika prądu stałego, Przegląd Elektrotechniczny, 86(2): 34–38, 2010.[13] J.C. Engwerda, Algorithms for computing Nash equilibria in deterministic LQ games, Computational Management Science, 4(2):113–140, 2007.[14] M. Ganobis, Electricity Market Modelling Using Dynamic LQ Games, In: Materiały XII Międzynarodowych Warsztatów Doktoranckich OWD w Wisle, pages 75–80, 2010.[15] M. Ganobis and W. Mitkowski, A Nash equilibrium in RC transmission line with two voltage sources, Materiały XXXII Międzynarodowej konferencji z podstaw elektrotechniki i teorii obwodów IC-SPETO, pages 117–118, 2009.[16] H. Górecki, S. Fuksa, A. Korytowski and W. Mitkowski, Sterowanie optymalne w systemach liniowych z kwadratowym wskaźnikiem jakości, PWN, Warszawa, 1983.[17] A. Guran, A. Bajaj, Y. Ishida, N. Perkins, G. D’Eleuterio and C. Pierre, Stability of Gyroscopic Systems – Series on Stability, Vibration and Control of Systems, vol. 2, World Scientific Publishing, Singapore, 1999.[18] Y.-C. Ho and A. Starr, Nonzero-sum Differential Games, Journal of Optimization Theory and Applications, 3: 184–206, 1969.[19] E. Kącki, Równania Różniczkowe Cząstkowe w Elektrotechnice, Wydawnictwa Nau-kowo-Techniczne, 1971.[21] P. Lancaster and L. Rodman, Algebraic Riccati Equations, Clarendon Press, Oxford,1995.[22] J. LaSalle and S. Lefschetz, Zarys teorii stabilności Lapunowa i jego metody bezpo-średniej, PWN, Warszawa, 1966.[23] W. Leonhard, Control of Electrical Drives 3rd edition, Springer-Verlag Berlin andHeidelberg NewYork, Berlin, 3 editing, 2001.[24] S. Mitkowski, Nonlinear Electric Circuits, Wydawnictwa AGH, Kraków, 1999.[25] W. Mitkowski, Stabilizacja liniowych układów nieskończenie wymiarowych za pomocą dynamicznego sprzężenia zwrotnego, Arch. Automatyki i Telemechaniki, 33(4):515–528, 1988.[26] W. Mitkowski, Stabilizacja Systemów Dynamicznych, WNT Warszawa, 1991.[27] W. Mitkowski, Projektowanie systemów sterowania z wykorzystaniem równania Riccatiego, In Z. Bubnicki and J. Józefczyk, editors, Mat. Konferencyjne XIII Krajowej Konferencji Automatyki, volume 1, pages 171–176, Oficyna Wydawnicza Politechniki Opolskiej, 1999.[28] W. Mitkowski, Metody projektowania układów regulacji optymalnej, In Z. Bubnickiand J. Korbicz, editors, XIV Krajowa Konferencja Automatyki, volmu 1, pages195–204, Uniwersytet Zielonogórski, Inst. Sterowania i Systemów Informatycznych,2002.[29] W. Mitkowski, Dynamic feedback in LC ladder network, Bulletin of the PolishAcademy of Sciences: Technical Sciences, 51(2): 173–180, 2003.[30] W. Mitkowski, Remarks about energy transfer in an RC ladder network, Inter-national Journal of Applied Mathematics and Computer Science, 13(2): 193–198,2003.[31] W. Mitkowski, Stabilization of LC ladder network, Bulletin of the Polish Academyof Sciences: Technical Sciences, 52(2): 109–114, 2004.[32] W. Mitkowski, Metody stabilizacji, In: Z. Bubnicki and R. Kulikowski and J. Kacprzyk, editors, XV Krajowa Konferencja Automatyki, volume 1, pages 169–178, KaiR PAN, IBS PAN oraz również PW, PIAP, PolSPAiR, 2005.[33] W. Mitkowski, Zastosowania równań liniowych w teorii sterowania. In K. Malinowski and L. Rutkowski, editors, Sterowanie i automatyzacja: aktualne problemy i ich rozwiązania, pages 11–22, Akademicka Oficyna Wydawnicza EXIT, Warszawa 2008.[34] W. Mitkowski and K. Oprzędkiewicz, A Sample Time Optimization Problem in a Digital Control system, In: A. Korytowski, K. Malanowski, W. Mitkowski and M. Szymkat, editor, System Modeling and Optimization 23rd IFIP TC7 Conference, pages 382–396, Springer, Berlin 2009, July 2007.[35] J. Ombach, Wykłady z równań różniczkowych wspomagane komputerowo-Maple,Wydawnicto U.J., 2 edition, 1999.[36] K. Oprzędkiewicz, An example of parabolic system identification, Zeszyty NaukoweAGH, Elektrotechnika, 16(2): 99–106, 1997.[37] W. Pełczewski and M. Krynke, Metoda zmiennych stanu w analize dynamiki układów napędowych, WNT, 1984.[38] L.S. Pontriagin, W.G. Bołtianski, R.W. Gemkrelidze and E.F. Miszczenko, Matematiczeskaja tieoria optimalnych processow, Nauka, Moskwa, 4 edition, 1983.[39] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The Mathematical Theory of Optimal Processes, John Willey, New York, 1962.[40] P. Skruch, Feedback stabilization of distributed parameter gyroscopic systems, W. Mitkowski and J. Kacprzyk, editors, Modelling Dynamics in Processes andSystems (Studies in Computational Intelligence), vol. 180, pages 85–97, Springer-Verlag, Berlin, Heidelberg, 2009.[41] P. Skruch, J. Baranowski and W. Mitkowski, Dynamic feedback stabilization of non-linear RC ladder network, In: Proceedings of XIII Symposium on Fundamental pro-blems of power electronics electromechanics and mechatronics, PPEEm 2009, Wisła,Poland, pages 136–141, 2009.[42] P. Skruch, W. Mitkowski, Optimum design of shapes using the Pontryagin principleof maximum, Automatyka, 13(1): 65–78, 2009.[43] P. Skruch, W. Mitkowski, Modelling and simulation of the shape optimization pro-blems, In: G.R. Rey and L.M. Muneta, editors, Modelling, Simulation and Optimi-zation, pages 187–208, In-Tech, Olajnica, Croatia, 2010.[44] H. Steinhaus, Między duchem a materią pośredniczy matematyka. Wybór, przed-mowa i redakcja naukowa Józef Łukaszewicz, PWN, Warszawa-Wrocław, 2000.[45] G. Strang, G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, En-glewood Cliffs, New York, 1973.[46] G. Szefer, L. Mikulski, Optimization of beams with the use of the Pontryagin prin-ciple of maximum, Archives of Civil Engineering, 24(3): 337–345, 1978.[47] M. Szymkat, A. Korytowski, Evolution of Structure for Direct Control Optimiza-tion, Discussiones Mathematicae. Differential Inclusions, Control and Optimization,27:165–193, 2007.[48] B. van Aarle, J. Engwerda and J. Plasmans, Cooperative and non-cooperative fi-scal stabilization policie
LA - eng
UR - http://eudml.org/doc/292897
ER -