Probabilistic ultimate bounds and invariant sets for LTI systems with Gaussian disturbances

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. Important applications where these concepts have proven to be very valuable include Model Predictive Control and Fault Tolerant Control. However, there are shortcomings in those concepts, in the sense that no general stochastic noises can be considered, since an essential requirement is for the disturbances affecting the system to be bounded. This, for example, precludes the consideration of disturbances with the ubiquitous Gaussian distribution, 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 a 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.