Non-Linear Observer Design with Laguerre Polynomials

In this paper, a methodology for a non-linear system state estimation is demonstrated, exploiting the input and parameter observability. For this purpose, the initial system is transformed into the canonical observability form, and the function that aggregates the non-linear dynamics of the system, which may be unknown or difficult to be computed, is approximated by a linear combination of Laguerre polynomials. Hence, the system identification translates into the estimation of the parameters involved in the linear combination in order for the system to be observable. For the validation of the elaborated observer, we consider a biological model from the literature, investigating whether it is practically possible to infer its states, taking into account the new coordinates to design the appropriate observer of the system states. Through simulations, we investigate the parameter settings under which the new observer can identify the state of the system. More specifically, as the parameter θ increases, the system converges more quickly to the steady-state, decreasing the respective distance from the system's initial state. As for the first state, the estimation error is in the order of 10-2 for θ=15, and assuming c0={0,1},c1=1. Under the same conditions, the estimation error of the system's second state is in the order of 10-1, setting a performance difference of 10-1 in relation to the first state. The outcomes show that the proposed observer's performance can be further improved by selecting even higher values of θ. Hence, the system is observable through the measurement output.