Stigter, J.D.

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Identifiability of Non-Linear Models from a Parametric Output-Sensivity Perspective

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Identifiability of Non-Linear Models from a Parametric
Output-Sensitivity Perspective
J.D. Stigter
Biometris
Wageningen University
The Netherlands
hans.stigter@wur.nl
1 Abstract
Local structural identifiability is a well-researched topic that
has lead to various methodologies that address the question
from different angles. Among these are (i) Pohjanpalo’s
method [1] that involves a Taylor-series approximation (ii)
the similarity transformation approach (iii) non-linear ob-
servability, where the state vector is augmented with the
(constant) parameters in the model (e.g. [2]).
In this presentation we will show an alternative viewpoint
that considers local structural identifiability as an output
controllability question, meaning that we study an acces-
sible controllability algebra associated with the parametric
output sensitivities ∂y∂θ . Hereto, the original system model in
control-affine format, i.e.
x˙(t) = f (x,θ)+g(x,θ)u(t)
y(t) = h(x,u,θ)
is viewed as the seed for the dynamics of the parametric
output sensitivities, i.e.
x˙θ (t) =
∂{ f (x,θ)+g(x,θ)u(t)}
∂x
xθ +
∂{ f (x,θ)+g(x,θ)u(t)}
∂θ
y(t) =
∂h(x,θ)
∂x
xθ +
∂h(x,θ)
∂θ
For each parameter in the original system model (or output
equation) a corresponding coupled system can be defined
that can be studied in a non-linear control framework. The
outputs of the p(= dim(θ)) coupled systems (which are the
parametric output sensitivities) can be compared with one-
another and identifiability properties can be derived. The
dynamics of the parametric output sensitivities are part of a
space that can be calculated on basis of the combined con-
trollability distributions {∆θi , i= 1, . . . , p} that arise from the
vector fields (drift- and control vector fields) of each (aug-
mented) system associated with parameter θi. A sufficient
condition for the parameters to be locally structurally iden-
tifiable is that a rank test needs to be satisfied. In the presen-
tation this is demonstrated in detail.
Viewing identifiability as an output controllability property
nicely fits in the theory of a range of parameter estima-
tion methods – e.g. the recursive prediction error method
– where parametric output sensitivities are key to minimiz-
ing the sum of squared prediction errors.
In the presentation I will elaborate on these insights and
show you a few examples in which the method is demon-
strated in more detail.
References
[1] H. Pohjanpalo, ‘System Identifiability Based on the
Power Series Expansion of the Solution’, Mathematical Bio-
sciences, vol. 41, pp 21–33, 1978
[2] E.T. Tunali and T. Tarn, ‘New Results for Identifiabil-
ity of Nonlinear Systems’, IEEE Transactions on Automatic
Control, vol. AC-32 (2), pp 146–154, 1987
[3] J.D. Stigter and R.L.M. Peeters, ‘On a Geometric Ap-
proach to the Structural Identifiability Problem and its Ap-
plication in a Water Quality Case Study’, Proceedings of the
European Control Conference 2007, Kos, Greece, pp 3450–
3456, 2007


Identifiability of Non-Linear Models from a Parametric Output-Sensivity Perspective

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