zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Stabilizing a linear system with quantized state feedback. (English) Zbl 0719.93067
Summary: This paper addresses the problem of stabilizing an unstable time- invariant discrete-time linear system by means of state feedback when the measurements of the state are quantized. It is found that there is no admissible control strategy that stabilizes the system in the traditional sense of making all closed-loop trajectories tend asymptotically to zero. Still, if the system is not excessively unstable, one can implement feedback strategies that bring closed-loop trajectories arbitrarily close to zero for an arbitrarily long time. It is found that when ordinary “linear” feedback of quantized state measurements is applied, the resulting closed-loop system behaves chaotically. When the state is one- dimensional, a quantitative statistical analysis of the resulting closed- loop dynamics reveals the existence of an invariant probability measure on the state space that is absolutely continuous with respect to Lebesgue measure and with respect to which the closed-loop system is ergodic. The asymptotically pseudorandom closed-loop system dynamics differ substantially from what would be predicted by a conventional signal-plus- noise analysis of the quantization’s effect. Probabilistic reformulations of the stabilization problem in terms of the invariant measure are then considered.
MSC:
93D15Stabilization of systems by feedback
93C55Discrete-time control systems