Probabilistic set invariance and ultimate boundedness

The notions of invariant sets and ultimate bounds are important concepts in the analysis of dynamical systems and very useful tools for the design of control systems. Several approaches have been reported for the characterisation of these sets, including constructive methods for their computation an...

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Bibliographic Details
Published inAutomatica (Oxford) Vol. 48; no. 10; pp. 2670 - 2676
Main Authors Kofman, Ernesto, De Doná, José A., Seron, Maria M.
Format Journal Article
LanguageEnglish
Published Kidlington Elsevier Ltd 01.10.2012
Elsevier
Subjects
Applied sciences
Computer science; control theory; systems
Control system analysis
Control system synthesis
Control theory. Systems
Exact sciences and technology
Invariant sets
Linear systems
Probabilistic methods
Ultimate bounds
Linear systems
Probabilistic methods
Invariant sets
Ultimate bounds
Gaussian process
Probabilistic approach
Gaussian distribution
Control synthesis
Probability distribution
Invariant set
Dynamical system
Bounded perturbation
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Summary:The notions of invariant sets and ultimate bounds are important concepts in the analysis of dynamical systems and very useful tools for the design of control systems. Several approaches have been reported for the characterisation of these sets, including constructive methods for their computation and procedures to obtain different approximations. However, there are shortcomings in those concepts, in the sense that no general probability distributions can be considered for the disturbances affecting the system (which, for example, precludes the assumption of Gaussian distributions insofar as they are not bounded). Motivated by those shortcomings, we propose in this paper the novel concepts of probabilistic ultimate bounds and probabilistic invariant sets, which extend the notions of invariant sets and ultimate bounds to consider ‘containment in probability’, and have the important feature of allowing stochastic noises with more general distributions, including the ubiquitous Gaussian distribution, to be considered. We introduce some key definitions for these sets, establish their main properties and develop methods for their computation. A numerical example illustrates the main ideas.
ISSN:0005-1098
1873-2836
DOI:10.1016/j.automatica.2012.06.074