Skip to main content
Erschienen in:
Buchtitelbild

2011 | OriginalPaper | Buchkapitel

1. Introduction to Control Systems

verfasst von : Iasson Karafyllis, Zhong-Ping Jiang

Erschienen in: Stability and Stabilization of Nonlinear Systems

Verlag: Springer London

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

This introductory chapter begins with important classes of control systems that will be studied in greater details in subsequent chapters. Basic notions and preliminary results in mathematical system and control theories are presented. Practical examples arising from engineering, mathematical biology and economics are introduced and will be invoked in subsequent chapters.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Agiza, H.N., Bischi, G.I., Kopel, M.: Multistability in a dynamic Cournot game with three oligopolists. Mathematics and Computers in Simulation 51, 63–90 (1999) MathSciNetCrossRef Agiza, H.N., Bischi, G.I., Kopel, M.: Multistability in a dynamic Cournot game with three oligopolists. Mathematics and Computers in Simulation 51, 63–90 (1999) MathSciNetCrossRef
2.
Zurück zum Zitat Athanasiou, G., Karafyllis, I., Kotsios, S.: Price stabilization using buffer stocks. Journal of Economic Dynamics and Control 32(4), 1212–1235 (2008) MathSciNetMATHCrossRef Athanasiou, G., Karafyllis, I., Kotsios, S.: Price stabilization using buffer stocks. Journal of Economic Dynamics and Control 32(4), 1212–1235 (2008) MathSciNetMATHCrossRef
3.
Zurück zum Zitat Bellman, R.E., Cooke, K.L.: Differential-Difference Equations. Academic Press, New York (1963) MATH Bellman, R.E., Cooke, K.L.: Differential-Difference Equations. Academic Press, New York (1963) MATH
4.
Zurück zum Zitat Bischi, G., Kopel, M.: Equilibrium selection in a nonlinear duopoly game with adaptive expectations. Journal of Economic Behavior & Organization 46, 73–100 (2001) CrossRef Bischi, G., Kopel, M.: Equilibrium selection in a nonlinear duopoly game with adaptive expectations. Journal of Economic Behavior & Organization 46, 73–100 (2001) CrossRef
5.
Zurück zum Zitat Burns, J.A., Herdman, T.L., Stech, H.W.: Linear functional differential equations as semigroups on product spaces. SIAM Journal of Mathematical Analysis 14, 96–116 (1983) MathSciNetCrossRef Burns, J.A., Herdman, T.L., Stech, H.W.: Linear functional differential equations as semigroups on product spaces. SIAM Journal of Mathematical Analysis 14, 96–116 (1983) MathSciNetCrossRef
6.
Zurück zum Zitat Chiarella, C., Szidarovszky, F.: Dynamic oligopolies without full information and with continuously distributed time lags. Journal of Economic Behavior & Organization 54, 495–511 (2004) CrossRef Chiarella, C., Szidarovszky, F.: Dynamic oligopolies without full information and with continuously distributed time lags. Journal of Economic Behavior & Organization 54, 495–511 (2004) CrossRef
7.
Zurück zum Zitat Clarke, F.H., Ledyaev, Y.S., Sontag, E.D., Subbotin, A.I.: Asymptotic controllability implies feedback stabilization. IEEE Transactions on Automatic Control 42(10), 1394–1407 (1997) MathSciNetMATHCrossRef Clarke, F.H., Ledyaev, Y.S., Sontag, E.D., Subbotin, A.I.: Asymptotic controllability implies feedback stabilization. IEEE Transactions on Automatic Control 42(10), 1394–1407 (1997) MathSciNetMATHCrossRef
8.
Zurück zum Zitat Coron, J.-M.: Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136. American Mathematical Society, Providence (2007) MATH Coron, J.-M.: Control and Nonlinearity. Mathematical Surveys and Monographs, vol. 136. American Mathematical Society, Providence (2007) MATH
9.
Zurück zum Zitat Desoer, C.A., Vidyasagar, M.: Feedback Systems: Input-Output Properties. Academic Press, New York (1975) MATH Desoer, C.A., Vidyasagar, M.: Feedback Systems: Input-Output Properties. Academic Press, New York (1975) MATH
10.
Zurück zum Zitat Droste, E., Hommes, C., Tuinstra, J.: Endogenous fluctuations under evolutionary pressure in Cournot competition. Games and Economic Behavior 40, 232–269 (2002) MathSciNetMATHCrossRef Droste, E., Hommes, C., Tuinstra, J.: Endogenous fluctuations under evolutionary pressure in Cournot competition. Games and Economic Behavior 40, 232–269 (2002) MathSciNetMATHCrossRef
11.
Zurück zum Zitat Fillipov, A.V.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic, Dordrecht (1988) Fillipov, A.V.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic, Dordrecht (1988)
12.
Zurück zum Zitat Forgo, F., Szep, J., Szidarovszky, F.: Introduction to the Theory of Games: Concepts, Methods, Applications. Kluwer Academic, Amsterdam (1999) MATH Forgo, F., Szep, J., Szidarovszky, F.: Introduction to the Theory of Games: Concepts, Methods, Applications. Kluwer Academic, Amsterdam (1999) MATH
13.
Zurück zum Zitat Grüne, L.: Stabilization by sampled and discrete feedback with positive sampling rate. In: Aeyels, D., Lamnabhi-Lagarrigue, F., van der Schaft, A. (eds.) Stability and Stabilization of Nonlinear Systems, pp. 165–182. Springer, London (1999) CrossRef Grüne, L.: Stabilization by sampled and discrete feedback with positive sampling rate. In: Aeyels, D., Lamnabhi-Lagarrigue, F., van der Schaft, A. (eds.) Stability and Stabilization of Nonlinear Systems, pp. 165–182. Springer, London (1999) CrossRef
14.
Zurück zum Zitat Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) MATH Hale, J.K., Lunel, S.M.V.: Introduction to Functional Differential Equations. Springer, New York (1993) MATH
15.
Zurück zum Zitat Hale, J.K., Lunel, S.M.V.: Strong stabilization of neutral functional differential equations. IMA Journal of Mathematical Control and Information 19, 5–23 (2002) MathSciNetMATHCrossRef Hale, J.K., Lunel, S.M.V.: Strong stabilization of neutral functional differential equations. IMA Journal of Mathematical Control and Information 19, 5–23 (2002) MathSciNetMATHCrossRef
16.
Zurück zum Zitat Hale, J.K., Lunel, S.M.V.: Stability and control of feedback systems with time delays. International Journal of Systems Science, 34(8–9), 497–504 (2003) MATHCrossRef Hale, J.K., Lunel, S.M.V.: Stability and control of feedback systems with time delays. International Journal of Systems Science, 34(8–9), 497–504 (2003) MATHCrossRef
17.
Zurück zum Zitat Henry, D.: Linear autonomous neutral functional differential equations. Journal of Differential Equations 15, 106–128 (1974) MathSciNetMATHCrossRef Henry, D.: Linear autonomous neutral functional differential equations. Journal of Differential Equations 15, 106–128 (1974) MathSciNetMATHCrossRef
18.
19.
Zurück zum Zitat Jiang, Z.P., Wang, Y.: A converse Lyapunov theorem for discrete-time systems with disturbances. Systems and Control Letters 45, 49–58 (2002) MathSciNetMATHCrossRef Jiang, Z.P., Wang, Y.: A converse Lyapunov theorem for discrete-time systems with disturbances. Systems and Control Letters 45, 49–58 (2002) MathSciNetMATHCrossRef
20.
Zurück zum Zitat Jiang, Z.P., Teel, A., Praly, L.: Small-gain theorems for ISS systems and applications. Mathematics of Control, Signals, and Systems 7, 95–120 (1994) MathSciNetMATHCrossRef Jiang, Z.P., Teel, A., Praly, L.: Small-gain theorems for ISS systems and applications. Mathematics of Control, Signals, and Systems 7, 95–120 (1994) MathSciNetMATHCrossRef
21.
Zurück zum Zitat Kalman, R.E.: Mathematical description of linear dynamical systems. J. SIAM Control 1(2), 152–192 (1963) MathSciNetMATH Kalman, R.E.: Mathematical description of linear dynamical systems. J. SIAM Control 1(2), 152–192 (1963) MathSciNetMATH
22.
Zurück zum Zitat Karafyllis, I.: The non-uniform in time small-gain theorem for a wide class of control systems with outputs. European Journal of Control 10(4), 307–323 (2004) MathSciNetCrossRef Karafyllis, I.: The non-uniform in time small-gain theorem for a wide class of control systems with outputs. European Journal of Control 10(4), 307–323 (2004) MathSciNetCrossRef
23.
Zurück zum Zitat Karafyllis, I.: Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis. IMA Journal of Mathematical Control and Information 23(1), 11–41 (2006) MathSciNetMATHCrossRef Karafyllis, I.: Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis. IMA Journal of Mathematical Control and Information 23(1), 11–41 (2006) MathSciNetMATHCrossRef
24.
Zurück zum Zitat Karafyllis, I.: Non-uniform in time robust global asymptotic output stability for discrete-time systems. International Journal of Robust and Nonlinear Control 16(4), 191–214 (2006) MathSciNetMATHCrossRef Karafyllis, I.: Non-uniform in time robust global asymptotic output stability for discrete-time systems. International Journal of Robust and Nonlinear Control 16(4), 191–214 (2006) MathSciNetMATHCrossRef
25.
Zurück zum Zitat Karafyllis, I.: Stabilization by means of time-varying hybrid feedback. Mathematics of Control, Signals and Systems 18(3), 236–259 (2006) MathSciNetMATHCrossRef Karafyllis, I.: Stabilization by means of time-varying hybrid feedback. Mathematics of Control, Signals and Systems 18(3), 236–259 (2006) MathSciNetMATHCrossRef
26.
Zurück zum Zitat Karafyllis, I.: A system-theoretic framework for a wide class of systems I: Applications to numerical analysis. Journal of Mathematical Analysis and Applications 328(2), 876–889 (2007) MathSciNetMATHCrossRef Karafyllis, I.: A system-theoretic framework for a wide class of systems I: Applications to numerical analysis. Journal of Mathematical Analysis and Applications 328(2), 876–889 (2007) MathSciNetMATHCrossRef
27.
Zurück zum Zitat Karafyllis, I.: A system-theoretic framework for a wide class of systems II: Input-to-output stability. Journal of Mathematical Analysis and Applications 328(1), 466–486 (2007) MathSciNetMATHCrossRef Karafyllis, I.: A system-theoretic framework for a wide class of systems II: Input-to-output stability. Journal of Mathematical Analysis and Applications 328(1), 466–486 (2007) MathSciNetMATHCrossRef
28.
Zurück zum Zitat Karafyllis, I., Jiang, Z.-P., Athanasiou, G.: Nash equilibrium and robust stability in dynamic games: A small-gain perspective. Computers and Mathematics with Applications 60, 2936–2952 (2010) MathSciNetMATHCrossRef Karafyllis, I., Jiang, Z.-P., Athanasiou, G.: Nash equilibrium and robust stability in dynamic games: A small-gain perspective. Computers and Mathematics with Applications 60, 2936–2952 (2010) MathSciNetMATHCrossRef
29.
Zurück zum Zitat Karafyllis, I., Jiang, Z.-P.: New results in trajectory-based small-gain with application to the stabilization of a chemostat. Submitted to International Journal of Robust Nonlinear Control Karafyllis, I., Jiang, Z.-P.: New results in trajectory-based small-gain with application to the stabilization of a chemostat. Submitted to International Journal of Robust Nonlinear Control
30.
Zurück zum Zitat Karafyllis, I., Grüne, L.: Feedback stabilization methods for the numerical solution of systems of ordinary differential equations. To appear in Discrete and Continuous Dynamical Systems, Series B Karafyllis, I., Grüne, L.: Feedback stabilization methods for the numerical solution of systems of ordinary differential equations. To appear in Discrete and Continuous Dynamical Systems, Series B
31.
Zurück zum Zitat Karafyllis, I., Pepe, P., Jiang, Z.-P.: Stability results for systems described by coupled retarded functional differential equations and functional difference equations. Nonlinear Analysis, Theory, Methods and Applications 71(7–8), 3339–3362 (2009) MathSciNetMATHCrossRef Karafyllis, I., Pepe, P., Jiang, Z.-P.: Stability results for systems described by coupled retarded functional differential equations and functional difference equations. Nonlinear Analysis, Theory, Methods and Applications 71(7–8), 3339–3362 (2009) MathSciNetMATHCrossRef
32.
Zurück zum Zitat Lakshmikantham, V., Deo, S.G.: Method of Variation of Parameters for Dynamic Systems, vol. 1. Gordon and Breach Science, New Delhi (1998) MATH Lakshmikantham, V., Deo, S.G.: Method of Variation of Parameters for Dynamic Systems, vol. 1. Gordon and Breach Science, New Delhi (1998) MATH
33.
Zurück zum Zitat Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations. Numerical Methods and Applications, 2nd edn. Dekker, New York (2002) MATHCrossRef Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations. Numerical Methods and Applications, 2nd edn. Dekker, New York (2002) MATHCrossRef
34.
Zurück zum Zitat Marchand, N., Alamir, M.: Asymptotic controllability implies continuous-discrete time feedback stabilization. In: Nonlinear Control in the Year 2000, vol. 2, pp. 63–79. Springer, Berlin (2000) Marchand, N., Alamir, M.: Asymptotic controllability implies continuous-discrete time feedback stabilization. In: Nonlinear Control in the Year 2000, vol. 2, pp. 63–79. Springer, Berlin (2000)
35.
Zurück zum Zitat Niculescu, S.I.: Delay Effects on Stability, a Robust Control Approach. Lecture Notes in Control and Information Sciences. Springer, Heidelberg (2001) Niculescu, S.I.: Delay Effects on Stability, a Robust Control Approach. Lecture Notes in Control and Information Sciences. Springer, Heidelberg (2001)
36.
Zurück zum Zitat Okuguchi, K., Yamazaki, T.: Global stability of unique Nash equilibrium in Cournot oligopoly and rent-seeking game. Journal of Economic Dynamics and Control 32, 1204–1211 (2008) MathSciNetMATHCrossRef Okuguchi, K., Yamazaki, T.: Global stability of unique Nash equilibrium in Cournot oligopoly and rent-seeking game. Journal of Economic Dynamics and Control 32, 1204–1211 (2008) MathSciNetMATHCrossRef
37.
Zurück zum Zitat Panchuk, A., Puu, T.: Stability in a non-autonomous iterative system: An application to oligopoly. Computers and Mathematics with Applications 58, 2022–2034 (2009) MathSciNetMATHCrossRef Panchuk, A., Puu, T.: Stability in a non-autonomous iterative system: An application to oligopoly. Computers and Mathematics with Applications 58, 2022–2034 (2009) MathSciNetMATHCrossRef
38.
Zurück zum Zitat Pepe, P.: On the asymptotic stability of coupled delay differential and continuous time difference equations. Automatica 41, 107–112 (2005) MathSciNetMATH Pepe, P.: On the asymptotic stability of coupled delay differential and continuous time difference equations. Automatica 41, 107–112 (2005) MathSciNetMATH
39.
Zurück zum Zitat Pepe, P., Karafyllis, I., Jiang, Z.-P.: On the Liapunov–Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations. Automatica 44(9), 2266–2273 (2008) MathSciNetMATHCrossRef Pepe, P., Karafyllis, I., Jiang, Z.-P.: On the Liapunov–Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations. Automatica 44(9), 2266–2273 (2008) MathSciNetMATHCrossRef
40.
Zurück zum Zitat Petit, N., Rouchon, P.: Dynamics and solutions to some control problems for water-tank systems. IEEE Transactions on Automatic Control 47(4), 594–609 (2002) MathSciNetCrossRef Petit, N., Rouchon, P.: Dynamics and solutions to some control problems for water-tank systems. IEEE Transactions on Automatic Control 47(4), 594–609 (2002) MathSciNetCrossRef
41.
Zurück zum Zitat Prieur, C., de Halleux, J.: Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations. Systems and Control Letters 52, 167–178 (2004) MathSciNetMATHCrossRef Prieur, C., de Halleux, J.: Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations. Systems and Control Letters 52, 167–178 (2004) MathSciNetMATHCrossRef
42.
Zurück zum Zitat Rasvan, V.: Functional differential equations of lossless propagation and almost linear behavior. Plenary Lecture at the 6th IFAC Workshop on Time-Delay Systems. L’Aquila, Italy (2006) Rasvan, V.: Functional differential equations of lossless propagation and almost linear behavior. Plenary Lecture at the 6th IFAC Workshop on Time-Delay Systems. L’Aquila, Italy (2006)
43.
44.
Zurück zum Zitat Shim, H., Teel, A.R.: Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback. Automatica 39, 441–454 (2003) MathSciNetMATHCrossRef Shim, H., Teel, A.R.: Asymptotic controllability and observability imply semiglobal practical asymptotic stabilizability by sampled-data output feedback. Automatica 39, 441–454 (2003) MathSciNetMATHCrossRef
45.
Zurück zum Zitat Smith, H., Waltman, P.: The Theory of the Chemostat. Dynamics of Microbial Competition. Cambridge Studies in Mathematical Biology, vol. 13. Cambridge University Press, Cambridge (1995) MATHCrossRef Smith, H., Waltman, P.: The Theory of the Chemostat. Dynamics of Microbial Competition. Cambridge Studies in Mathematical Biology, vol. 13. Cambridge University Press, Cambridge (1995) MATHCrossRef
46.
Zurück zum Zitat Sontag, E.D.: Mathematical Control Theory, 2nd edn. Springer, New York (1998) MATH Sontag, E.D.: Mathematical Control Theory, 2nd edn. Springer, New York (1998) MATH
47.
Zurück zum Zitat Sontag, E.D.: Clocks and insensitivity to small measurement errors. ESAIM: Control, Optimisation and Calculus of Variations 4, 537–557 (1999) MathSciNetMATHCrossRef Sontag, E.D.: Clocks and insensitivity to small measurement errors. ESAIM: Control, Optimisation and Calculus of Variations 4, 537–557 (1999) MathSciNetMATHCrossRef
48.
Zurück zum Zitat Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1998) MATH Stuart, A.M., Humphries, A.R.: Dynamical Systems and Numerical Analysis. Cambridge University Press, Cambridge (1998) MATH
49.
Zurück zum Zitat Sun, Y., Michel, A.N., Zhai, G.: Stability of discontinuous retarded functional differential equations with applications. IEEE Transactions on Automatic Control 50(8), 1090–1105 (2005) MathSciNetCrossRef Sun, Y., Michel, A.N., Zhai, G.: Stability of discontinuous retarded functional differential equations with applications. IEEE Transactions on Automatic Control 50(8), 1090–1105 (2005) MathSciNetCrossRef
50.
Zurück zum Zitat Zhu, L., Huang, X.: Multiple limit cycles in a continuous culture vessel with variable yield. Nonlinear Analysis 64, 887–894 (2006) MathSciNetMATHCrossRef Zhu, L., Huang, X.: Multiple limit cycles in a continuous culture vessel with variable yield. Nonlinear Analysis 64, 887–894 (2006) MathSciNetMATHCrossRef
51.
Zurück zum Zitat Zhu, L., Huang, X., Su, H.: Bifurcation for a functional yield chemostat when one competitor produces a toxin. Journal of Mathematical Analysis and Applications 329, 871–903 (2007) MathSciNetCrossRef Zhu, L., Huang, X., Su, H.: Bifurcation for a functional yield chemostat when one competitor produces a toxin. Journal of Mathematical Analysis and Applications 329, 871–903 (2007) MathSciNetCrossRef
Metadaten
Titel
Introduction to Control Systems
verfasst von
Iasson Karafyllis
Zhong-Ping Jiang
Copyright-Jahr
2011
Verlag
Springer London
DOI
https://doi.org/10.1007/978-0-85729-513-2_1

Neuer Inhalt