By Jochen Schröder (auth.)
The ebook issues quantised platforms which emerge from continuous-variable platforms by means of quantising the values of all signs. it truly is proven how this results in an summary procedure description by way of a stochastic automaton. in response to stochastic automata, new equipment for the answer to country commentary and fault diagnostic difficulties are derived. The equipment are prolonged to networks of stochastic automata, permitting component-oriented modelling and, hence, to house complicated structures. the sensible applicability and value of the strategy is proven at numerous examples.
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Sample text
The definition of the quantiser allows to divide the signal space into arbitrarily shaped sets. Practically, the input and output signals of a system are usually quantised independently of each other. 4, the quantisation of the output space is defined by the positions of the discrete level sensors attached to each tank. Both tank levels are thus quantised independently of one another. The resulting partition is called orthogonal partition as the bounds separating the partition regions are orthogonal to each other.
Corresponds to an irreducible set of states. 3 applies. 3. All stationary distributions are convex combinations of the κ eigenvectors ¯ iz corresponding to the κ eigenvalues λi = 1. 4, the stationary behaviour of the SA is given as a superposition of the stationary behaviour in the irreducible set of states it contains. 1. All submatrices Gii , i = 1, 2, . . , κ have a unique eigenvalue λi = 1 but can have further eigenvalues with |λi | = 1. The number of eigenvalues with |λi | = 1 corresponds to the period KP of the irreducible set of states.
G. for periodically changing input symbols. However, stationary distributions describing the mean probability of the stochastic process being in a certain state for k → ∞ can be determined also for random input as shown in the following. It is assumed that the input is described by a stationary input process described by the probability distribution 32 2. Basics of Stochastic Automata Theory Prob(v) = {Prob(vp = 1), Prob(vp = 2), . . , Prob(vp = M )} . 55) If this condition is not satisfied because the input itself is the result of a stochastic process, the stationary behaviour of the input process must be determined first.