# Delay-Based Approximations of Biological Systems for Analysis and Design

### Marcella Gomez and Richard M. Murray

Submitted, 2012 American Control Conference (ACC)

In this paper we explore the use of time-delayed differential equation as a means of obtaining a simplified description of very high order dynamics.This paper finds results for a particular type of system, a single-input single-output (SISO) linear system with a nonlinear feedback. We begin with a high dimensional system in state space and reduce the dimension by finding a delay based approximation which could be a smaller set of integro-differential equations or DDEs. We argue that approximations of high order linear subsystems whose distribution functions have relatively smaller variance such as delta functions, give a conservative approximation of a system's stable parameter space. Through examples inspired by biology, we show how these approximations can be used to verify stability. We analyze the system's stability and robustness dependence on statistical properties, mainly relative variance and expectation for a symmetric distribution function.

- Conference Paper: http://www.cds.caltech.edu/~murray/preprints/gm12-acc_s.pdf
- Project(s): Template:HTDB funding::NSF NetSE