By Ya. Z. Tsypkin
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Extra resources for Adaptation and Learning in Automatic Systems
Let us form the variational equation. 47) where q[n] is the deviation from the optimal vector. 48) This difference equation has a trivial solution q = 0, since by the definition of c*, we have VJ(c*) = 0. 4). As it is known, two types of stability are distinguished when all the coordinates of the vector q[n] are smali. One is the local stability, and the other is global stability (for any q[n]). In order to investigate the local stability, the gradient VJ(c* + q) must first be approximated by a linear f h c t i o n and the obtained linear difference equation is then tested for stability.
It should be obvious by now that the stochastic processes differ from one another, and in particular, from the deterministic processes, but only by the form of probabilistic characteristics-probability density functions. The volume of a priori information for deterministic processes is usually larger than for stochastic processes since the probability density functions for deterministic processes are known in advance, while for stochastic processes they are to be determined. If the probability distribution is determined in advance, and if we can manage to write the functional and the constraint equations in an explicit form, then regardless of the basic differences between the deterministic and stochastic processes, it is difficult to establish any prominent dissimilarities in the formulation and the solution of optimization problems for these processes.
The adaptive approach is mainly related to the algorithmic, or more accurately, iterative methods. A general discussion of algorithmic methods of optimization will be found in the next two chapters. 9 Conclusion In this chapter we have attempted to give a general description of the problem of optimality, its formulation and solution. We have also tried to point out the differences, and especially the similarities, of the optimization problems for deterministic and stochastic processes. We have indicated that the approach for solving an optimization problem depends on the degree of a priori information.
Adaptation and Learning in Automatic Systems by Ya. Z. Tsypkin