CONVERGENCE PROPERTIES OF ADAPTIVE SYSTEMS AND THE DEFINITION OF EXPONENTIAL STABILITY.

The convergence properties of adaptive systems in terms of excitation conditions on the regressor vector are well known. With persistent excitation of the regressor vector in model reference adaptive control the state error and the adaptation error are globally exponentially stable or, equivalently, exponentially stable in the large. When the excitation condition, however, is imposed on the reference input or the reference model state, it is often incorrectly concluded that the persistent excitation in those signals also implies exponential stability in the large. The definition of persistent excitation is revisited so as to address some possible confusion in the adaptive control literature. It is then shown that persistent excitation of the reference model only implies local persistent excitation (weak persistent excitation). Weak persistent excitation of the regressor is still sufficient for uniform asymptotic stability in the large, but not exponential stability in the large. We show that there exists an infinite region in the state-space of adaptive systems where the state rate is bounded. This infinite region with finite rate of convergence is shown to exist not only in classic open-loop reference model adaptive systems but also in a new class of closed-loop reference model adaptive systems.

Application of Smart Infrastructure Systems approach to precision medicine.

All biological variation is hierarchically organized dynamic network system of genomic components, organelles, cells, tissues, organs, individuals, families, populations and metapopulations. Individuals are axial in this hierarchy, as they represent antecedent, attendant and anticipated aspects of health, disease, evolution and medical care. Humans show individual specific genetic and clinical features such as complexity, cooperation, resilience, robustness, vulnerability, self-organization, latent and emergent behavior during their development, growth and senescence. Accurate collection, measurement, organization and analyses of individual specific data, embedded at all stratified levels of biological, demographic and cultural diversity - the big data - is necessary to make informed decisions on health, disease and longevity; which is a central theme of precision medicine initiative (PMI). This initiative also calls for the development of novel analytical approaches to handle complex multidimensional data. Here we suggest the application of Smart Infrastructure Systems (SIS) approach to accomplish some of the goals set forth by the PMI on the premise that biological systems and the SIS share many common features. The latter has been successfully employed in managing complex networks of non-linear adaptive controls, commonly encountered in smart engineering systems. We highlight their concordance and discuss the utility of the SIS approach in precision medicine programs.