Mathematical Problems in Engineering

Volume 2015, Article ID 271216, 6 pages

http://dx.doi.org/10.1155/2015/271216

## A New Application of the Hill Repressor Function: Automatic Control of a Conic Tank Level and Local Stability Analysis

^{1}Departamento de Ingeniería Química y Bioprocesos, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, 7820436 Santiago, Chile^{2}Escuela de Ingeniería Eléctrica, Pontificia Universidad Católica De Valparaíso, Avenida Brasil 2950, 2340000 Valparaíso, Chile^{3}Departamento de Tecnologías Industriales, Universidad de Talca, Camino a Los Niches km 1, 3440000 Curicó, Chile

Received 12 March 2015; Accepted 27 April 2015

Academic Editor: Sergio Preidikman

Copyright © 2015 José Ricardo Pérez-Correa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Hill function is commonly used as a building block to model different dynamic patterns found in the response of genetic regulatory circuits within microorganisms and cells. These circuits are characterized by fast response and robustness against unmeasured disturbances. Therefore, microorganisms and cells can survive even if they are subjected to strong changes in their environment. However, as far as we know, the Hill function has not been used before to design process control systems. In this work, the repressor Hill function is applied to control the level of a conic tank. To eliminate the offset, we added integrative action. A local analysis was applied to define stability limits for the control parameters. A cost function that includes the error and the control effort was used to compare the performance of the Hill control against a standard PI and a PID-Dahlin controller.

#### 1. Introduction

The Hill equation has shown many interesting properties and has been applied to model several dynamic systems in biosciences, pharmacology, biology, and physicochemistry. In biosciences, this equation has been used to fit a model to experimental or clinical data when the relationship between two sets of variables was saturated and nonlinear [1–3]. In physics, chemistry, and biochemistry, this equation can describe thermodynamic equilibrium, where each parameter of the model has a physical meaning [4, 5]. In pharmacology, the Hill equation may be used as a probabilistic expression of time-dependent biological events such as adverse reaction, toxicity, or death. In addition, this equation has been extensively used to analyze quantitative drug-receptor relationships. Many pharmacokinetic-pharmacodynamic models have used the Hill equation to describe nonlinear drug dose-response relationships [6–10].

In addition, the Hill function is commonly used as a building block to model different dynamic patterns found in the response of genetic regulatory circuits within microorganisms and cells. These circuits are characterized by fast response and robustness against disturbances. Therefore, microorganisms and cells can survive even if they are subjected to strong changes in their environment [11–14]. The Hill function can model several genetic regulatory mechanisms within cells. In particular, two basic mechanisms can be identified in transcription networks. The activation mechanism operates when the transcription factor increases the gene transcription rate (protein production rate associated with the gene). The activator Hill function represents this mechanism well: the protein level can be controlled by the signaling molecule. The repressor mechanism (that represses the binding and decreases the transcription rate) can be modelled with the repressors’ Hill function. The higher the repressor concentration is, the higher the probability that a repressor molecule binds to the operator site is. Consequently, the expression level is more and more repressed with increasing repressor levels. Half-maximal repression occurs when the concentration of active repressors equals the repression coefficient [15].

Although the Hill equation is widely used, many of its properties are not all well known. The nonlinear Hill equation can deal properly with the control of several chaotic systems. The existence of periodic solutions in particular situations has been recently proved. Other problems that are under research related with the Hill equation are, for instance, the control of chaos by means of the transient effects minimization or by synchronization. Therefore, more interesting applications of the Hill equation can be found exploring these areas [16, 17].

As far as we know, the Hill function has not been used before to design process control systems. In this paper, the repressor Hill function is applied to control the level of a conic tank. To eliminate the offset, we added integrative action. A cost function that includes the error and the control effort was used to compare the performance of the Hill control against a standard PI and a PID-Dahlin controller [18].

#### 2. The Hill Controller

The Hill function is commonly used to model several genetic regulatory mechanisms within cells. In particular, two basic mechanisms can be identified in transcription networks [1]. The activation mechanism operates when the transcription factor increases the gene transcription rate (protein production rate associated with that gene). The activator Hill function represents this mechanism well:

In (1) is the protein production rate and is the concentration of the transcription factor; , , and are model parameters. , the activation coefficient, is the threshold value of that activates the mechanism; , the maximal expression level, is the maximum response value; and , the Hill coefficient, defines the response steepness.

In turn, the repressor mechanism operates when the transcription factor reduces the gene transcription rate. This mechanism can be modelled by the repressor Hill function:

Model parameters are similar to those in (1); however, (2) is a decreasing function; therefore the maximum expression is obtained for .

Equation (2) can be adapted for process control to profit from the fast response and robustness that characterize genetic regulatory circuits. Therefore, the following Hill controller can be proposed:where is the controlled variable, is the set point, is the control error, is twice the initial steady state control effort (considering that the process is initially at the desired operating point), and is the steepness index. Next, we analyze the impact of the model parameters on the behavior of this control algorithm.

In Figure 1 we see that when where the control action presents an extreme point (maximum or inflection point). In addition, the control action is when and (if is even). However, if is odd or not an integer, the control action diverges () when . For stable control systems, it is expected that the control error would be less than the set point (). Within this error range, for , the control action will be sigmoidal with an inflection point at (; ). Finally, if the control action approaches 0 for all values of .