The emphasis of the article is laid on the partially observable and diagnosable systems for which the problems of observability and diagnosability cannot be solved unambiguously. The second difference to the previous literature is that the article deals with stochastic automata rather than with nondeterministic automata. For these automata the probability distribution over the set of automata states can be determined so that “highly probable” states can be distinguished from “less probable” ones.
A discrete-event system is considered as a stochastic automaton (SA). According to R. Bukharaev [Theorie der stochastischen Automaten (Stuttgart: B. G. Teubner) (1995)], a SA is described by inputs \(v(k)\), outputs \(w(k)\), internal states \(z(k)\) and a behavioural relation function \[ L(z',w| z,v)=\text{Prob}(z(k+1)=z',w(k)=w| z(k)=z,v(k)=v), \] where \(z,z' \in N_z\), \(w \in N_w\), \(v \in N_v\), \(k=0,1,2,\dots\), and \(N_v\), \(N_w\), \(N_z\) are finite sets. The conditional probabilities \(L(z',w| z,v)\) are assumed to be known and do not depend on \(k\).
The state observation problem for initial and current states of SAs is connected with the supervisory control theory developed by P. J. Ramadge and W. M. Wonham [Proc. IEEE 77, 81–98 (1989)]. It is assumed that a prior probability distribution of the initial state \(z(0)\) and input/output sequences \(v(0),\dots,v(k)/ w(0),\dots,w(k)\) are given. The authors define a consistent input/output pair by the condition that the output sequence can occur with a non-vanishing probability in response to an input sequence. For consistent sequences the accurate expressions of the a-posteriori probability distributions of the initial state \(z(0)\) and the current state \(z(k)\) of an automaton are obtained. A solution to the current state observation problem, i.e. the a-posteriori distribution of \(z(k)\), is given as a recursive solution as well.
The problem of stochastic observability is connected with results of C. M. Özveren and A. S. Willsky [IEEE Trans. Autom. Control 35, 797–806 (1990; Zbl 0709.68030)]. It is based on a definition of a stochastically unobservable automaton as having a behavioural relation function \[ L(z',w| z,v)=F(z'| z,v) \cdot G(w| v), \] for all \(z,z' \in N_z\), \(w \in N_w\), \(v \in N_v\). So, the knowledge of the output sequence \(w(0),\dots,w(k)\) of an unobservable automaton in addition to the known input sequence \(v(0),\dots,v(k)\) does not influence the knowledge of the probability distribution of the states \(z(0),\dots,z(k)\). An interesting next step is the definition of subsets of stochastically unobservable states. An automaton is unobservable while it remains within such a subset. At last, an automaton is defined to be stochastically observable if there does not exist any subset of stochastically unobservable states. So, the knowledge of the output sequence of an observable automaton allows us to improve the knowledge of its states. A solution to the problem of observability in the considered statement is given.
The diagnostic problem was the subject of only a few papers in the past, e.g. F. Lin [Discrete Event Dyn. Syst. 4, 197–212 (1994; Zbl 0800.93030)]. In the present article it is reduced to the state observation problem by introduction of an additional state \(f(k)\) that describes a current fault of the system and belongs to a finite set \(N_f\). The behavioural relation \(L_f(z',f',w| z,f,v)\) is assumed to be known and satisfies some restrictions on the joint behaviour of \(z(k)\) and \(v(k)\). A solution to the diagnostic problem in the considered statement is given. Results are illustrated by an example of diagnostic of a two-tank system with a leakage.