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Example text
We say that Ci. is irnplied by C. By proposition 6, any implied class Ci. is a c -class (i. , all the states in Ci. must be pairwise compatible). In the example (20), the classes implied by C= = {q 0 q3} are: . ' A collection of c-classes is closed if, for any class C of the collection, every implied class Ci. is contained in at least one class of the collection. A minimal collection of c-classes is a complete and closed collection with minimum number of classes. 3. For any incomplete automatonA, an automaton A' with minimum number of states, such that A ~ A', can be constructed Assertion by associating the states of A' to the members of a minimal collection of c-classes of A.
An automaton any state q i. and of size ~ , A in minimal form is observable in by performing an experiment of length -' n - 1 n- t, where n is the nurober of states of A • Then, for less powerful experiments than the one of Proposition 5, the automaton may not be observable even if it is in minimal form. For example, the following automaton (Arbib [5], Ch. 6): ( 15) 5. Incomplete automata 32 is in minimal form, but is not observable by simple experiments. In fact, no input sequence beginning with tween q 1 and q~ 1 1 can discriminate be- ; no input sequence beginning with :x:a.
Definition of experiment 29 Then, the optimal control problern can be posed as: Problem. Given automaton A in state qi. , find an input sequence x. • such that: i. ,x•) . ,~h• ) 11. k .. "' = X•, w1t . h :r,k • non void; iii. , x•) is minimal. If interpreted on the graph representation of the automaton, such a problern is classically solved by the optimal path techniques of dynamic programming [ 2 ]. More complex is the notion of observability for automata: in fact, much more complex than probably expected by the reader familiar with the theory of linear systems.