A lumping of a Markov chain is a coordinatewise projection of the chain. We characterise the entropy rate preservation of a lumping of an aperiodic and irreducible Markov chain on a finite state space by the random growth rate of the cardinality of the realisable preimage of a finite-length trajectory of the lumped chain and by the information needed to reconstruct original trajectories from their lumped images. Both are purely combinatorial criteria, depending only on the transition graph of the Markov chain and the lumping function. A lumping is strongly k-lumpable, if and only if the lumped process is a kth-order Markov chain for each starting distribution of the original Markov chain. We characterise strong k-lumpability via tightness of stationary entropic bounds. In the sparse setting, we give sufficient conditions on the lumping to both preserve the entropy rate and be strongly k-lumpable.

In this paper, tip tracking control of a single-link flexible robot is studied through computer simulations. The controller investigated is of a two-loop PD type, which is deduced from a control approach originally developed for elastic joint robots. A very simple model, with the flexible link of the robot being lumped to a spring-mass unit, is employed for the controller design. Bounded Input and Bounded Output (BIBO) stable tip tracking performance is guaranteed. The controller is robustly stable under system parameter uncertainties, and is comparatively easy to implement. In the simulations, different payloads are considered, and the system performance is compared with the traditional Singular Perturbation (SP) method. The simulation results show that the PD type controller designed based on the simple lumped model is able to provide quite satisfactory performance.

A finite-state Markov process is aggregated into several groups. What can be learned about the underlying process from the aggregated one? We provide some partial answers to this question.