### STP1287: Linear Systems, Compartmental Modeling, and Estimability Issues in IAQ Studies

Evans, WC

*Consultant, Kensington, MD*

Pages: 24 Published: Jan 1996

**Abstract**

Many IAQ models are expressed as coupled systems of linear, ordinary differential equations. In this paper, the linear-systems or state-variable format for these systems will be reviewed, and some useful information will be presented which can be obtained from this formulation without explicitly solving the differential equation system. Much information concerning linear systems analysis is available in the literature of various disciplines, particularly biomathematics, wherein there is a specialization called “compartmental modeling.” It is important to recognize that there exists a great deal of directly usable mathematical information which can immediately be applied to IAQ modeling problems.

In compartmental modeling, an issue called “identifiability” has long been recognized as a potential problem with experiments that are intended to extract information about a linear system's parameters from observations of that system's response to a forcing function. It can happen that the system's parameters cannot be uniquely estimated from an experiment, no matter how good (noise-free) the measurements are. With a linear-systems formulation of the experimental configuration, this condition can be detected before the experiment is conducted.

A related issue is termed “redundancy,” which refers to the inability to obtain unique parameter estimates from the data, even if the experiment is identifiable. This problem occurs for sums-of-exponentials models, fitted via nonlinear estimation to the observations. Taken together, identifiability and redundancy can be termed “estimability.” These difficulties can affect chamber testing in particular, since this is the context where we are attempting to estimate system parameters from observations.

This paper will present an overview of these issues, with selected examples.

**Keywords:**

mathematical modeling, chamber testing, linear systems, compartmental modeling, identifiability, redundancy, nonlinear estimation

**Paper ID:** STP15625S

**Committee/Subcommittee:** D22.01

**DOI:** 10.1520/STP15625S

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