Abstract
Observer theory is one of the most basic, and important, aspects of linear systems theory. The problem addressed in this chapter is that of indirect observation, or partial state estimation. It arises from the fact that in a control system Σ sys , the observed variables are not necessarily the variables one needs to estimate for control, or other, purposes. A standard situation often encountered is that of partial state estimation, where a few, or all, state variables are to be estimated from the output variables. Of course, if one can estimate the state, then one automatically has the ability to estimate a function of the state. However, especially in a large and complex system, estimating the full state may be a daunting task and more than what is needed. The task of state estimation is also instrumental for practical implementations of state feedback control using estimates of the unknown state functions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Bisiacco, M.E. Valcher and J.C. Willems, “A behavioral approach to estimation and dead-beat observer design with applications to state-space models”, IEEE Trans. Autom. Control,, vol. 51, (2006), 1787–1797.
T.L. Fernando, L.S. Jennings and H.M. Trinh, “Generality of functional observer structures”, 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC), (2011), 4000–4004.
T.L. Fernando, H.M. Trinh, Hieu and L.S. Jennings, “Functional observability and the design of minimum order linear functional observers”, IEEE Trans. Autom. Control, vol. 55, (2010), 1268–1273.
P.A. Fuhrmann, “Observer theory”, Linear Algebra Appl., vol. 428, (2008), 44–136.
P.A. Fuhrmann and U. Helmke, “Parametrization of conditioned invariant subspaces”, Advances in Mathematical Systems Theory, A Volume in Honor of D. Hinrichsen, (F. Colonius, U. Helmke, D. Prätzel-Wolters, and F. Wirth, Eds.), Birkhäuser, Boston, 103–134, 2001.
P.A. Fuhrmann and U. Helmke, “On the parametrization of conditioned invariant subspaces and observer theory”, Linear Algebra Appl., vol. 332–334, (2001), 265–353.
P.A. Fuhrmann and J. Trumpf, “On observability subspaces”, Int. J. Control, vol. 79, (2006), 1157–1195.
D. G. Luenberger, “An introduction to observers”, IEEE Trans., vol. 8(2), (1964), 74–80.
D. G. Luenberger, “Observing the state of a linear system”, IEEE Trans. Autom. Control, 16(6), (1971), 596–602.
J. Trumpf, On the geometry and parametrization of almost invariant subspaces and observer theory. Doctoral thesis, Institute of Mathematics, University of Würzburg, 2002.
J. Trumpf, “On state observers”, in Mathematical System Theory (Festschrift in Honor of U. Helmke), K. Hüper and J. Trumpf (Eds.), Amazon, CreateSpace, 421–435, 2013.
J. Trumpf, H.L. Trentelmann and J.C. Willems, “Internal model principles for observers”, IEEE Trans. Autom. Control, vol. 59(7), (2014), 1737–1749.
M.E. Valcher and J.C. Willems, “Observer synthesis in the behavioral framework”, IEEE Trans. Autom. Control, vol. 44, (1999), 2297–2307.
J.C. Willems, “From time series to linear systems. Part I: Finite-dimensional linear time invariant systems”, Automatica, vol. 22, (1986), 561–580.
J.C. Willems, “Paradigms and puzzles in the theory of dynamical systems”, IEEE Trans. Autom. Control, vol. 36, (1991), 259–294.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Fuhrmann, P.A., Helmke, U. (2015). Observer Theory . In: The Mathematics of Networks of Linear Systems. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-16646-9_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-16646-9_7
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-16645-2
Online ISBN: 978-3-319-16646-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)