Y. Zacchia Lun

Y. Zacchia Lun, J. Wheatley, A. D'Innocenzo et al.

This work introduces a new abstraction technique for reducing the state space of large, discrete-time labelled Markov chains. The abstraction leverages the semantics of interval Markov decision processes and the existing notion of approximate probabilistic bisimulation. Whilst standard abstractions make use of abstract points that are taken from the state space of the concrete model and which serve as representatives for sets of concrete states, in this work the abstract structure is constructed considering abstract points that are not necessarily selected from the states of the concrete model, rather they are a function of these states. The resulting model presents a smaller one-step bisimulation error, when compared to a like-sized, standard Markov chain abstraction. We outline a method to perform probabilistic model checking, and show that the computational complexity of the new method is comparable to that of standard abstractions based on approximate probabilistic bisimulations.

Y. Zacchia Lun, A. Abate, A. D'Innocenzo

In most real cases transition probabilities between operational modes of Markov jump linear systems cannot be computed exactly and are time-varying. We take into account this aspect by considering Markov jump linear systems where the underlying Markov chain is polytopic and time-inhomogeneous, i.e. its transition probability matrix is varying over time, with variations that are arbitrary within a polytopic set of stochastic matrices. We address and solve for this class of systems the infinite-horizon optimal control problem. In particular, we show that the optimal controller can be obtained from a set of coupled algebraic Riccati equations, and that for mean square stabilizable systems the optimal finite-horizon cost corresponding to the solution to a parsimonious set of coupled difference Riccati equations converges exponentially fast to the optimal infinite-horizon cost related to the set of coupled algebraic Riccati equations. All the presented concepts are illustrated on a numerical example showing the efficiency of the provided solution.