Probabilistic Ultimate Bounds and Invariant Sets for LTI Systems with Gaussian Disturbances

ERNESTO KOFMAN; JOSÉ DE DONÁ; MARIA MARTA SERON

Melbourne

Conferencia; 2011 Australian Control Conference; 2011

Institution of Engineers Australia and IEEE Control Systems Society

The notions of invariant sets and ultimate boundsare important concepts in the analysis of dynamical systemsand very useful tools for the design of control systems. Severalapproaches have been reported for the characterisation ofthese sets, including constructive methods for their computationand procedures to obtain different approximations. Importantapplications where these concepts have proven to be veryvaluable include Model Predictive Control and Fault TolerantControl. However, there are shortcomings in those concepts, inthe sense that no general stochastic noises can be considered,since an essential requirement is for the disturbances affectingthe system to be bounded. This, for example, precludes theconsideration of disturbances with the ubiquitous Gaussiandistribution, insofar as they are not bounded. Motivated bythose shortcomings, we propose in this paper the novel conceptsof probabilistic ultimate bounds and probabilistic invariantsets, which extend the notions of invariant sets and ultimatebounds to consider ‘containment in probability’, and have theimportant feature of allowing stochastic noises with a Gaussiandistribution to be considered. We introduce some key definitionsfor these sets, establish their main properties and developmethods for their computation. A numerical example illustratesthe main ideas