Linear systems theory revisited

This paper investigates and clarifies how different definitions of reachability, observability, controllability, reconstructability and minimality that appear in the control literature, may be equivalent or different, depending on the type of linear system. The differences are caused by (1) whether...

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Bibliographic Details
Published in:Automatica (Oxford) 2008-07, Vol.44 (7), p.1686-1696
Main Authors: Van Willigenburg, L.G., De Koning, W.L.
Format: Article
Language:English
Subjects:
Applied sciences
Computer science
control theory
systems
Control system analysis
Control theory. Systems
Controllability
discrete-time-systems
Exact sciences and technology
Finite horizon
Linear systems with time-varying dimensions
Modelling and identification
Observability
realization
Time-reversibility
Weak reachability
weighting patterns
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Summary:This paper investigates and clarifies how different definitions of reachability, observability, controllability, reconstructability and minimality that appear in the control literature, may be equivalent or different, depending on the type of linear system. The differences are caused by (1) whether or not the linear system has state dimensions that vary with time (2) bounds on the time axis of the linear system (3) whether or not the initial state is non-zero and (4) whether or not the system is time invariant. Also (5) time-reversibility of systems plays a role. Discrete-time linear strictly proper systems are considered. A recently published result is used to argue that all the results carry over to continuous time. Out of the investigation two types of definitions emerge. One type applies naturally to systems with constant dimensions while the other applies naturally to systems with variable dimensions. This paper reveals that time-varying (state) dimensions that are allowed to be zero are necessary to obtain equivalence between minimality and (weak) reachability together with observability at the systems level. Besides their theoretical significance the results of this paper are of practical importance for model reduction and control of time-varying discrete-time linear systems because they result in minimal realizations with smaller dimensions that are also computed more easily.
ISSN:0005-1098
1873-2836
DOI:10.1016/j.automatica.2007.10.021