This work attempts to characterize and provide design guidelines for adaptive sys- tems composed of two adaptive elements. To begin, we introduce the notion of an adaptive element, which is the smallest scale upon which systems have adaptivity. Then we provide a concise review of many of the subjects deemed important to single adaptive elements from a deterministic dynamical systems view. Theorems are provided characterizing the deterministic dynamic behavior of single adap- tive algorithms, and their robustness to disturbances and time variation. We also provide a review of deterministic single time scale averaging theory.
After we have characterized the behavior of a single adaptive element, we are ready to begin to study the possibility of connecting more than one adaptive de- vice together. It is this possibility that inspires us to imagine different ways of and reasons for using distributed adaptation in these structures. We then select one of the possible binary (i.e., two element) structures for connecting adaptive elements, the series feedfoward binary adaptive compound (SFFBAC), and char- acterize its behavior. We motivate this discussion with quotes from the digital receiver literature, which indicate that the interaction of adaptive components is a recognized problem about which very little theoretical work has been done. All along, we are considering adaptive systems from an engineering mindset. Within these lines, we observe that distributed adaptive systems often arise as a relic of our method of design. We name the design technique that connects individually designed adaptive elements together to solve a bigger problem the Divide and Conquer strategy. Our goal then becomes to provide sufficient conditions under which we can use Divide and Conquer to design working series feed-forward binary adaptive compounds.
To give conditions for Divide and Conquer design, we begin with a qualitative mindset, describing the sorts of requirements that may be encountered when ap- plying a more rigorous theorem. Then, in Chapter 4 we develop the beginning of a rigorous behavior theory for series feed-forward binary adaptive compounds. This theory directly supports the qualitative design conditions we provide. Since we are not only interested in studying adaptive receivers which behave well, we also develop a misbehavior theorem, which predicts one of the ways a SFFBAC may misbehave. We conclude this theory by applying it to practical examples from digital receivers employing more than one adaptive element. To do so, we must remove the assumption that all other adaptive elements are behaving in an ideal manner from the models for the behavior of the adaptive receiver components. Thus, Appendix A contains models and derivations for the behavior of adaptive receiver components under non-ideal situations. After our example applications of the theory to adaptive receivers, we end the thesis by mentioning other possi- ble applications of the theory, as well as many possible extensions to the theory, including the use of other mathematical tools.