ABSTRACT
In this chapter we are concentrating on the Gauss-Newton method for the estimation of unknown parameters in models described by a set of ordinary differential equations (ODEs).
6.1 FORMULATION OF THE PROBLEM As it was mentioned in Chapter 2, the mathematical models are of the
form
dx(t) = f(x(t), u, k) ; x(t0) = x0 (6.1)dt
y(t) = Cx(t) (6.2)
or more generally
y(t) = h(x(t),k) (6.3)
where k=[khk2,...,kp]T is a p-dimensional vector of parameters whose numerical
values are unknown;
x=[X|,x2,...,xn]' is an n-dimensional vector of state variables; x0 is an n-dimensional vector of initial conditions for state variables which
are assumed to be known precisely;
u=[ubU2,...,u r]T is an r-dimensional vector of manipulated variables which are either set by the experimentalist or they have been measured and it is assumed that their numerical values are precisely known;
f=[fi,f2,...,fn]' is a n-dimensional vector function of known form (the differential equations); y=[y,,y2,...,ym]T is the m-dimensional output vector i.e., the set of variables
that are measured experimentally; and
C is the mxn observation matrix, which indicates the state variables (or linear combinations of state variables) that are measured experimentally.
