Numerical investigation of structural minimality for structures of uncontrolled linear switching systems with Maple

Abstract

One path to understanding a physical system is to represent it by a model structure (collection of related models). Suppose our system is not subject to external influences, and depends on unobservable state variables (x), and observables (y). Then, a suitable uncontrolled, state-space model structure S is defined by relationships between x and y, involving parameters θ ∈ Θ. That is, each parameter vector in parameter space Θ is associated with a particular model in S. Before using S for prediction, we require system observations for parameter estimation. This process aims to determine θ values for which predictions “best” approximate the data (according to some objective function). The result is some number of estimates of the true parameter vector, θ*. Multiple parameter estimates are problematic when these cause S to produce dissimilar predictions beyond our data's range. This can render us unable to confidently make predictions, resulting in an uninformative study. Non-uniqueness of parameter estimates follows when S lacks the property of structural global identifiability (SGI). Fortunately, we may test S for SGI prior to data collection. The absence of SGI encourages us to rethink our experimental design or model structure. Before testing S for SGI we should check that it is structurally minimal. If so, we cannot replace S by a structure of fewer state variables which produces the same output. Most testing methodology is applicable to structures which employ the same equations for all time. These methods are not appropriate when, for example, a process has an abrupt change in its dynamics. For such a situation, a structure of linear switching systems (LSSs) may be suitable. Any system in the structure has a collection of linear time-invariant state-space systems, and a switching function which determines the system in effect at each instant. As such, we face a novel challenge in testing an LSS structure for SGI. We will consider the case of an uncontrolled LSS structure of one switching event (a ULSS-1 structure). In this setting, we may approach the structural minimality problem via the Laplace transform of the output function on each time interval. Each rational function yields conditions for pole-zero cancellation. If these conditions are not satisfied for almost all θ ∈ Θ, then S is structurally minimal. Analytical approaches can be quite laborious. However, we may expect a numerical approach to provide useful insights quickly. For example, if pole-zero cancellation occurs for almost all of a sufficiently large number of parameter values sampled from Θ, then structural minimality is possible. This result may encourage us to prove the existence of structural minimality. We shall use Maple 2020-2 to conduct a numerical investigation of structural minimality for a test case ULSS-1 structure applicable to flow-cell biosensor experiments used to monitor biochemical interactions, which include the popular Biacore-branded units.