Supervisory Control Theory for Discrete Event Systems, introduced by Ramadge and Wonham, is based on a non-probabilistic formal language framework. Building on the concept of signed real measure of regular languages, this paper formulates a comprehensive theory for optimal control of finite-state probabilistic processes. It is shown that the resulting discrete-event supervisor is optimal in the sense of elementwise maximizing the renormalized langauge measure vector for the controlled plant behavior and is efficiently computable. Index Terms-Language Measure; Finite Markov Chains; Discrete Event Systems; Supervisory Control 1. I Supervisory Control Theory (SCT) pioneered by Ramadge and Wonham [1] models a physical or human engineered process as a finite state language generator and constructs a supervisor that attempts to constrain the "supervised" plant behavior within a specification language. Several attempts have been made in the literature to extend SCT to probabilistic languages or p-languages[2], [3], [4], [5]. A formal measure-theoretic approach is reported by Ray et. alin [6] where the control objective is specified as normalized characteristic weights on the plant states. A rigorous signed real measure of regular languages is defined as a function of the characteristic weights and the transition probabilities [6]; and supervisory control laws are synthesized by elementwise maximizing the language measure vector. Intuitively, the supervisor ensures that the generated event strings cause the plant to visit the "good" states while attempting to avoid the "bad" states in a probabilistic sense. Kumar and Garg [4] investigated a strictly binary notion of "good" and "bad" strings implying that the computed supervisor is required to eliminate all bad strings and hence may fail to exist if the conditions defined in [4] are not met. In contrast, desirability of event strings in the measure-theoretic approach are quantified by their individual measures; with the supervisor optimizing the controlled plant behavior to ensure that the "most" desirable strings occur "most" often. This has an immediate advantage that the existence problem disappears; the optimal supervisor always exists and can be computed effectively in polynomial time [7]. The notion of terminating and non-terminating automata is originally due to Garg [2] [3]. A probabilistic automaton is terminating if there exist states at which the sum of the probabilities of all defined events is strictly less than 1. The interpretation is that the difference of the sum from 1 is the probability that the plant terminates operation at that particular state. It is shown in [6] that the language measure vector can be expressed as I − Π −1 χ where Π is the transition probability matrix and χ is the characteristic vector, where π i j is the probability of transition from the i th state to the j th state and χ i is the characteristic weight of the state i). A sufficient condition for ⋆

This paper modifies the signed real measure of regular languages, which has been reported in recent literature for analysis and synthesis of discrete event supervisory control laws. A new concept of renormalized measure is introduced to eliminate a user-selectable parameter in the present version of the language measure. The concept of measure renormalization is illustrated by an example.

Supervisory control theory for discrete event systems, introduced by Ramadge and Wonham, is based on a non-probabilistic formal language framework. However, models for physical processes inherently involve modelling errors and noise-corrupted observations, implying that any practical finite-state approximation would require consideration of event occurrence probabilities. Building on the concept of signed real measure of regular languages, this paper formulates a comprehensive theory for optimal control of finite-state probabilistic processes. It is shown that the resulting discrete-event supervisor is optimal in the sense of elementwise maximizing the renormalized langauge measure vector for the controlled plant behaviour and is efficiently computable. The theoretical results are validated through several examples including the simulation of an engineering problem.

Using several longitudinal datasets describing putative factors affecting influenza incidence and clinical data on the disease and health status of over 150 million human subjects observed over a decade, we investigated the source and the mechanistic triggers of influenza epidemics. We conclude that the initiation of a pan-continental influenza wave emerges from the simultaneous realization of a complex set of conditions. The strongest predictor groups are as follows, ranked by importance: (1) the host population’s socio- and ethno-demographic properties; (2) weather variables pertaining to specific humidity, temperature, and solar radiation; (3) the virus’ antigenic drift over time; (4) the host population’€™s land-based travel habits, and; (5) recent spatio-temporal dynamics, as reflected in the influenza wave auto-correlation. The models we infer are demonstrably predictive (area under the Receiver Operating Characteristic curve 80%) when tested with out-of-sample data, opening the door to the potential formulation of new population-level intervention and mitigation policies.

Probabilistic finite state machines have recently emerged as a viable tool for modelling and analysis of complex non-linear dynamical systems. This paper rigorously establishes such models as finite encodings of probability measure spaces defined over symbol strings. The well known Nerode equivalence relation is generalized in the probabilistic setting and pertinent results on existence and uniqueness of minimal representations of probabilistic finite state machines are presented. The binary operations of probabilistic synchronous composition and projective composition, which have applications in symbolic model-based supervisory control and in symbolic pattern recognition problems, are introduced. The results are elucidated with numerical examples and are validated on experimental data for statistical pattern classification in a laboratory environment.