Identifiability of parameters ? One of the most important steps in constructing mathematical models in chemistry

Experimental Design Tools for Ordinary and Algebraic Differential Equations

“…Our idea is to proceed from a given set of time values, which could be large and dense, and to formulate an experimental design optimization problem as done earlier by introducing additional weights w k , k ) 1, ..., l. Thus, we replace the kth model function h(p,q,t k ) by w k h(p,q,t k ) with an additional factor w k (for k ) 1, ..., l), which is treated as an optimization variable of the optimum design problem: 20 The covariance matrix C(w,p,q) :) I(w,p,q) -1 is now dependent on additional weights, I(w,p,q) :) F(w,p,q)F(w,p,q) T , F(w,p,q) :) ∇ p f(w,p,q), and, finally, Note that, for stability reasons, a small lower bound τ > 0 is introduced. We must prevent the possibility that weights become zero at an intermediate iterate leading to an indefinite information matrix.…”

“…Corresponding partial derivatives of the objective function relative to a weight w k , are obtained from (see eq 22) and from q ∈ IR n q : min trace(C(p,q)) q l e q e q u (20) φ(q) :) trace(C(p,q)) ( 21)…”

“…The identification of chemical engineering models also has been studied by Gorskii and Spivak. 20 If, on the other hand, it is known in advance that a model is neither sgi nor sli, then it is much more difficult to determine which parameters are redundant or correlated and cannot be estimated subject to a given standard deviation of the experimental errors. One possible approach is investigated in this paper in more detail.…”

“…Assuming that h ( p , y , t ) and F ( p , y , t ) or F ( p , y , z , t ) and G ( p , y , z , t ), respectively, are infinitely often continuously differentiable, with respect to y and t , we attempt to determine a power series for h ( p , t ) at t = 0 in the form Here, a k ( p ) denotes the k th one-sided derivative of h ( p , t ) at t = 0, i.e., the Taylor coefficient Thus, a sufficient condition for sgi is that a k ( p ) = a k ( p̂ ) for p , p̂ ∈ Iℛ n implies p = p̂ (also see the work of Pohjanpalo 24 or Walter and Pronzato 42 ). The identification of chemical engineering models also has been studied by Gorskii and Spivak …”