OBSERVATION OF NONLINEAR VERTICUM-TYPE SYSTEMS APPLIED TO ECOLOGICAL MONITORING

Abstract

In this paper the concept of a nonlinear verticum-type observation system is introduced. These systems are composed from several "subsystems" connected sequentially in a particular way: a part of the state variables of each "subsystem" also appears in the next "subsystem" as an "exogenous variable" which can also be interpreted as a control generated by an "exosystem". Therefore, these "subsystems" are not observation systems, but formally can be considered as control-observation systems. The problem of observability of such systems can be reduced to rank conditions on the "subsystems". Indeed, under the condition of Lyapunov stability of an equilibrium of the "large", verticum-type system, it is shown that the Kalman rank condition on the linearization of the "subsystems" implies the observability of the original, nonlinear verticum-type system. For an illustration of the above linearization result, a stage-structured fishery model with reserve area is considered. Observability for this system is obtained by applying the above linearization and decomposition approach. Furthermore, it is also shown that, applying an appropriate observer design method to each subsystem, from the observation of the biomass densities of the adult (harvested) stage, in both areas, the biomass densities of the pre-recruit stage can be efficiently estimated.