Abstract
In the last few years some articles have emphasized certain fundamental inconsistencies underlying feedback control theory. The paper of Willems [1] Georgiou and Smith [2], later the works of Makila [3],[4], of Leithead et al. [5] have stressed the inconsistency of standard formalisms of linear time-invariant systems when the signals are double sided and the systems are open loop unstable. We establish a framework for a consistent time domain and frequency domain representation of discrete time linear time-invariant systems and, furthermore, that supports the consistent analysis of discrete time linear time-invariant feedback systems when signals are double sided and the systems are open loop unstable.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Willems, J.C.: Stability, Instability, Invertibility and Causality. SIAM J. Control 7, 645–671 (1969)
Georgiou, T.T., Smith, M.C.: Intrinsic difficulties in using the doubly-infinite time axis for input-output control theory. IEEE Trans. Automat. Control 40, 516–518 (1995)
Mäkilä, P.M.: On three puzzles in robust control. IEEE Trans. Automat. Control 45, 552–556 (2000)
Mäkilä, P.M.: Convoluted double trouble. IEEE Control System Mag. 22, 26–31 (2002)
Leithhead, W.E.,O’Reilly, J.: A consistent time-domain and frequency-domain representation for discrete-time linear time-invariant feedback systems. In: Proceedings of the American Control Conference, Denver, Colorado, pp. 429–434 (2003)
Mäkilä, P.M., Partington, J.R.: Input-output stabilization on the doubly-infinite time axis. Internat. J. Control 75, 981–987 (2002)
Jacob, B.: An operator theoretical approach towards systems over the signal space l2(Z). Integral Equations Operator Theory 46, 189–214 (2003)
Mäkilä, P.M.: Intrinsic difficulties in stochastic control of unstable convolution operators on Z. IEEE Trans. Automat. Control 48, 2015–2019 (2003)
Champeney, D.C.: A Handbook of Fourier Theorems. Cambridge University Press, Cambridge (1987)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Ragnoli, E., Leithead, W. (2005). Reconciliation of Inconsistencies in the Theory of Linear Systems. In: Murray-Smith, R., Shorten, R. (eds) Switching and Learning in Feedback Systems. Lecture Notes in Computer Science, vol 3355. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30560-6_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-30560-6_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24457-8
Online ISBN: 978-3-540-30560-6
eBook Packages: Computer ScienceComputer Science (R0)