Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Introduction Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

4. Quadratic Optimal Control with Complete Observations

verfasst von : Oswaldo L.V. Costa, Marcelo D. Fragoso, Marcos G. Todorov

Erschienen in: Continuous-Time Markov Jump Linear Systems

Verlag: Springer Berlin Heidelberg

Einloggen

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

search-config
loading …

Abstract

This chapter focuses on one of the few cases of stochastic control problems with an actual explicit solution. It deals with the quadratic optimal control problem for continuous-time MJLS in the usual finite- and infinite-horizon framework. It is assumed that both the state variable x(t) and jump variable θ(t) are available to the controller. The setup adopted in this chapter is based on Dynkin’s formula for the resulting Markov process obtained from the state x(t) and Markov chain θ(t). Under this approach, we consider the class of admissible controllers as those in a feedback form (on x(t) and θ(t)) satisfying a Lipschitz condition. It is shown that the solution for the problems relies, in part, on the study of a finite set of coupled differential and algebraic Riccati equations (CDRE and CARE, respectively).

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!

Jetzt informieren

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!

Jetzt informieren

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!

Jetzt informieren
Vorheriges Kapitel Mean-Square Stability
Nächstes Kapitel H 2 Optimal Control with Complete Observations
Literatur
5.
H. Abou-Kandil, G. Freiling, G. Jank, Solution and asymptotic behaviour of coupled Riccati equations in jump linear systems. IEEE Transactions on Automatic Control 39, 1631–1636 (1994) MathSciNetMATHCrossRef
6.
H. Abou-Kandil, G. Freiling, G. Jank, On the solution of discrete-time Markovian jump linear quadratic control problems. Automatica 31, 765–768 (1995) MathSciNetMATHCrossRef
13.
K.J. Åström, Introduction to Stochastic Control Theory (Academic Press, New York, 1970) MATH
16.
J. Baczynski, M.D. Fragoso, Maximal solution to algebraic Riccati equations linked to infinite Markov jump linear systems. Mathematics of Control. Signals and Systems 20, 157–172 (2008) MathSciNetMATHCrossRef
17.
J. Baczynski, M.D. Fragoso, Maximal versus strong solution of algebraic Riccati equations arising in infinite Markov jump linear systems. Systems & Control Letters 57, 246–254 (2008) MathSciNetMATHCrossRef
23.
R. Bellman, Dynamic Programming (Princeton University Press, Princeton, 1957) MATH
33.
W.P. Blair, D.D. Sworder, Feedback control of a class of linear discrete systems with jump parameters and quadratic cost criteria. International Journal of Control 21, 833–844 (1975) MathSciNetMATHCrossRef
53.
F.M. Callier, C.A. Desoer, Linear System Theory (Springer, Berlin, 1999)
59.
H.J. Chizeck, Y. Ji, Optimal quadratic control of jump linear systems with Gaussian noise in discrete-time, in Proceedings of the 27th Conference on Decision and Control, 1988, pp. 1989–1992 CrossRef
60.
H.J. Chizeck, A.S. Willsky, D. Castañon, Discrete-time Markovian jump linear quadratic optimal control. International Journal of Control 43, 197–216 (1986) CrossRef
63.
E.F. Costa, J.B.R. do Val, Weak detectability and the linear-quadratic control problem of discrete-time Markov jump linear systems. International Journal of Control 75, 1282–1292 (2002) MathSciNetMATHCrossRef
66.
O.L.V. Costa, Discrete-time coupled Riccati equations for systems with Markov switching parameters. Journal of Mathematical Analysis and Applications 194, 197–216 (1995) MathSciNetMATHCrossRef
67.
O.L.V. Costa, Mean square stabilizing solutions for discrete-time coupled algebraic Riccati equations. IEEE Transactions on Automatic Control 41, 593–598 (1996) MATHCrossRef
69.
O.L.V. Costa, E.O. Assumpção, E.K. Boukas, R.P. Marques, Constrained quadratic state feedback control of discrete-time Markovian jump linear systems. Automatica 35, 617–626 (1999) MATHCrossRef
70.
O.L.V. Costa, J.C.C. Aya, Monte-Carlo TD(λ)-methods for the optimal control of discrete-time Markovian jump linear systems. Automatica 38, 217–225 (2002) MathSciNetMATHCrossRef
72.
O.L.V. Costa, W.L. de Paulo, Indefinite quadratic with linear costs optimal control of Markov jump with multiplicative noise systems. Automatica 43, 587–597 (2007) MATHCrossRef
74.
O.L.V. Costa, J.B.R. do Val, Jump LQ-optimal control for discrete-time Markovian systems with l 2 stochastic inputs. Stochastic Analysis and Applications 16, 843–858 (1998) MathSciNetMATHCrossRef
78.
O.L.V. Costa, M.D. Fragoso, Discrete-time LQ-optimal control problems for infinite Markov jump parameter systems. IEEE Transactions on Automatic Control 40, 2076–2088 (1995) MathSciNetMATHCrossRef
86.
O.L.V. Costa, R.P. Marques, Maximal and stabilizing Hermitian solutions for discrete-time coupled algebraic Riccati equations. Automatica 12, 167–195 (1999) MathSciNetMATH
88.
O.L.V. Costa, R.T. Okimura, Discrete-time mean variance optimal control of linear systems with Markovian jumps and multiplicative noise. International Journal of Control 2, 256–267 (2009) MathSciNetCrossRef
93.
A. Czornik, On Control Problems for Jump Linear Systems (Wydawnictwo Politechniki Śla̧skiej, Gliwice, 2003)
94.
97.
M.H.A. Davis, Linear Estimation and Stochastic Control (Chapman & Hall, London, 1977) MATH
100.
M.H.A. Davis, R.B. Vinter, Stochastic Modelling and Control (Chapman & Hall, London, 1985) MATHCrossRef
103.
C.E. de Souza, M.D. Fragoso, On the existence of a maximal solution for generalized algebraic Riccati equation arising in the stochastic control. Systems & Control Letters 14, 233–239 (1990) MathSciNetMATHCrossRef
109.
J.B.R. do Val, E.F. Costa, Numerical solution for linear-quadratic control problems of Markov jump linear systems and weak detectability concept. Journal of Optimization Theory and Applications 114, 69–96 (2002) MathSciNetMATHCrossRef
110.
J.B.R. do Val, J.C. Geromel, O.L.V. Costa, Uncoupled Riccati iterations for the linear quadratic control problem of discrete-time Markov jump linear systems. IEEE Transactions on Automatic Control 43, 1727–1733 (1998) MATHCrossRef
121.
V. Dragan, T. Morozan, A.M. Stoica, Mathematical Methods in Robust Control of Linear Stochastic Systems (Mathematical Concepts and Methods in Science and Engineering) (Springer, New York, 2010)
138.
W.H. Fleming, Some Markovian optimization problems. Journal of Mathematics and Mechanics 12, 131–140 (1963) MathSciNetMATH
140.
W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control (Springer, Berlin, 1975) MATHCrossRef
142.
J.J. Florentin, Optimal control of continuous-time Markov stochastic systems. Journal of Electronics and Control 10, 473–488 (1961) MathSciNet
144.
146.
M.D. Fragoso, J. Baczynski, Optimal control for continuous-time linear quadratic problems with infinite Markov jump parameters. SIAM Journal on Control and Optimization 40, 270–297 (2001) MathSciNetMATHCrossRef
149.
M.D. Fragoso, J. Baczynski, On an infinite dimensional perturbed Riccati differential equation arising in stochastic control. Linear Algebra and Its Applications 406, 165–176 (2005) MathSciNetMATHCrossRef
156.
M.D. Fragoso, O.L.V. Costa, C.E. de Souza, A new approach to linearly perturbed Riccati equations arising in stochastic control. Applied Mathematics & Optimization 37, 99–126 (1998) MathSciNetMATHCrossRef
159.
M.D. Fragoso, E.M. Hemerly, Optimal control for a class of noisy linear systems with Markovian jumping parameters and quadratic cost. International Journal of Systems Science 22, 2553–3561 (1991) MathSciNetMATHCrossRef
176.
B.E. Griffiths, K.A. Loparo, Optimal control of jump linear quadratic Gaussian systems. International Journal of Control 42, 791–819 (1985) MathSciNetMATHCrossRef
188.
Y. Ji, H.J. Chizeck, Optimal quadratic control of jump linear systems with separately controlled transition probabilities. International Journal of Control 49, 481–491 (1989) MathSciNetMATH
191.
Y. Ji, H.J. Chizeck, X. Feng, K.A. Loparo, Stability and control of discrete-time jump linear systems. Control, Theory and Advanced Technology 7, 247–270 (1991) MathSciNet
203.
H.J. Kushner, Optimal stochastic control. IRE Transactions on Automatic Control 7, 120–122 (1962) CrossRef
223.
M. Mariton, Jump Linear Systems in Automatic Control (Dekker, New York, 1990)
229.
T. Morozan, Optimal stationary control for dynamic systems with Markov perturbations. Stochastic Analysis and Applications 1, 299–325 (1983) MathSciNetMATHCrossRef
246.
M. Ait Rami, L. El Ghaoui, LMI optimization for nonstandard Riccati equations arising in stochastic control. IEEE Transactions on Automatic Control 41, 1666–1671 (1996) MathSciNetMATHCrossRef
259.
S.E. Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (Springer, New York, 2004)
269.
270.
D.D. Sworder, Feedback control of a class of linear systems with jump parameters. IEEE Transactions on Automatic Control 14, 9–14 (1969) MathSciNetCrossRef
292.
V.M. Ungureanu, Optimal control for linear discrete-time systems with Markov perturbations in Hilbert spaces. IMA Journal of Mathematical Control and Information 26, 105–127 (2009) MathSciNetMATHCrossRef
295.
R. Vinter, Optimal Control (Birkhäuser, Basel, 2000) MATH
302.
303.
W.M. Wonham, Random differential equations in control theory, in Probabilistic Methods in Applied Mathematics, vol. 2, ed. by A.T. Bharucha-Reid (Academic Press, San Diego, 1970), pp. 131–212
311.
J. Yong, X.Y. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations (Springer, Berlin, 1999) MATH
Metadaten
Titel
Quadratic Optimal Control with Complete Observations
verfasst von
Oswaldo L.V. Costa
Marcelo D. Fragoso
Marcos G. Todorov
Copyright-Jahr
2013
Verlag
Springer Berlin Heidelberg
Buch
Continuous-Time Markov Jump Linear Systems

Print ISBN: 978-3-642-34099-4

Electronic ISBN: 978-3-642-34100-7

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-3-642-34100-7_4