Robust observer design by sign-stability for the monitoring of population systems. Issue 4 (March 2019) - López, Inmaculada; Puertas, María Luz; Rodríguez, Carmelo; Gámez, Manuel; Varga, Zoltán
Abstract: Monitoring problem in population ecology can be formalized as observer design for the population system in question: Supposing that we observe only certain species considered indicators, we want to recover or estimate the whole state process of the population system, where the state vector is usually composed from the biomasses of the single populations. In the present paper, for stably coexisting population systems, a new approach to the design of the corresponding observer system is proposed which, from the knowledge of the observed indicator(s), estimates the state process with exponential convergence. In the usual observer design, an auxiliary matrix, the so-called gain matrix must be found that guarantees the mentioned exponential convergence. The novelty is in that due to the required sign-stability (or qualitative stability) of the interaction pattern, the designed observer system (i.e. the gain matrix) is robust against quantitative changes in the inter- and intra-specific interactions. (Here the interaction pattern is described by a matrix having the signs as entries, indicating the quality of the interactions within and between the considered species.) In other words, under sign-stability conditions, in the observer design the same gain matrix can be used even if, due to environmental changes, the intensities of certain interactions suffer a quantitative change in the meanwhile. The requirement of sign-stability of the interaction pattern can be considered rather natural, since in a stably coexisting population system, it means for example that a predator–prey relation does not change into a prey–predator interaction, and interactions neither appear nor disappear within the system. Our approach to robust observer design is illustrated on model population systems, such as trophic chains of type resource-producer-primary consumer-secondary consumer and Lotka–Volterra system with vertical structure. For the latter system a Lyapunov function is also constructed that guarantees the global asymptotic stability of the positive equilibrium of the considered model.