Approximating a MIMO, 1D Diffusion System to a Low Order, State-Space Form in Order to Facilitate Controller Design

Chemical distribution is an important factor in many biological systems, driving the phenomenon known as chemotaxis. In order to properly study the effects of various chemical inputs to an in vitro biological assay, it is necessary to have strict control over the spatial distribution of these chemicals. This distribution is typically governed by diffusion, which by nature is a distributed parameter system (DPS), dependent on both space and time. Much study and literature within the controls community has been devoted to DPS, whose dynamics are marked by partial differential equations or delays. They span an infinite-dimensional state-space, and the mathematical complexity associated with this leads to the development of controllers that are often highly abstract in nature. In this paper, we present a method of approximating these systems and expressing them in a manner that makes a DPS amenable to control using a very low order model. In particular, we express the PDE for one-dimensional chemical diffusion as a two-input, two-output state-space system and show that standard controllers can manipulate the outputs of interest, using pole placement and integral control via an augmented state model.