Nonlinear Control of Mine Ventilation Networks

Yunan Huyz, Olga I. Korolevaxand Miroslav Krsti

Источник: http://preuss.ucsd.edu/FacultyWebpages/Koroleva/PDF/net_scl.pdf

Ventilation networks in coal mines serve the critical task of maintaining a low concentration of explosive or noxious gases (e.g., methane). Due to the objective of controlling fluid flows, mine ventilation networks are high-order nonlinear systems. Previous efforts on this topic were based on multivariable linear models. The designs presented here are for a nonlinear model. Two control algorithms are developed. One employs actuation in all the branches of the network and achieves a global regulation result. The other employs actuation only in branches not belonging to the tree of the graph of the network and achieves regulation in a (non-infinitesimal) region around the operating point. The approach proposed for mine ventilation networks is also applicable to other types of fluid networks like gas and water distribution networks, irrigation networks, and possibly to building ventilation.

Introduction

Coal as a source of fossil fuel energy should remain in abundance for a considerable time after petroleum reserves are exhausted. One of the principal difficulties in underground coal mines is the presence of poisonous and explosive gases like methane. Accidents claiming the lives of coal miners have been numerous through the history and continue to this day.

Modern coal mines contain elaborate ventilation facilities that allow to regulate the concentration of methane. In such ventilation systems the objective is usually not to directly control the concentrations but to control the air flow rates through individual branches of the ventilation network. The actuation available ranges from a few fans/compressors strategically located through the network (and often directly connected to the outside environment), to actively controlled “doors” that are in many of the branches of the network. The problem of controlling mine ventilation received considerable attention in the 1970’s and the 1980’s.

It is clear that a mine ventilation network is a multivariable control problem where acting in one branch can affect the flow in the other branches in an undesirable way. For this reason, mine ventilation needs to be approached in a model-based fashion, as a fluid flow network (in much of the same way one would model an electric circuit) and as a multivariable control problem.

Pioneering work on this topic was performed by Koci?c who considered a linearized lumped-parameter dynamic model of a mine ventilation network and developed a methodology for designing linear feedback laws for it. He discovered structural properties that allowed him to decouple the problem into SISO problems and avoid the use of generic, highly complicated MIMO control tools. However, he did not take advantage of the graph theoretic properties of the network, which forced him to both neglect the nonlinearities (essential in this fluid flow problem) and to employ dynamic output-feedback compensators where static output feedback would suffice. We provide these improvements in this article.

The control model of a mine ventilation network consists of Kirchhoff’s current and voltage laws (algebraic equations) and fluid dynamical equations of individual branches (differ- 2 ential equations). The branches are modeled using lumped parameter approximations of incompressible Navier-Stokes equations that take a form whose electric equivalent is an RL characteristic with a nonlinear resistance. To be precise, the pressure drop over a branch is approximated to be proportional to the square of the air flow rate (nonlinear resistive term) and to the air flow acceleration (linear inductive term).

A model written using Kirchhoff’s algebraic equations and the branch characteristic differential equation constitute a non-minimal representation of the control model. It is clear that, due to the mass conservation at the branching points (nodes) of the network, airflows in many of the branches will be inter-dependent. Hence, the minimal system representation will be of lower order than the number of branches.

This intuition becomes systematic when one employs graph theoretic concepts from circuit theory. Each network can be divided into a set of branches called a tree (they connect all the nodes of the graph without creating any loops) and the complement of the tree, called a co-tree, whose branches are referred to as the links. The minimal system representation of the dynamics of the network is given by the flow through the links.

While it is to possible to control the airflows only in independent branches—the links— and therefore necessary to put actuators only in those branches, the physical possibility to put actuators also in the tree branches allows to approach the control problem in two distinct ways. The first approach that we pursue actuates all the branches and yields a global stability result for this nonlinear system. The second approach actuates only the independent, link branches and yields a regional (around the operating point in the state space) result.

A peculiarity of the problem is that, while the model is affine in the control inputs, they do not appear in an additive manner. Since the inputs to the system are resistivities of the branches (modulated by the openings of “doors” in the branches), the control inputs are always multiplied by quadratic nonlinearities.

As the reader shall see in Section 4, following a complicated model development in the preceding sections, the last step of the nonlinear control design amounts to multivariable feedback linearization. This might normally raise the issue of modeling uncertainties but in 3 this class of systems they are minor as tunnel lengths and diameters are easy to measure.

The method developed employs full state measurement because coal mine tunnels are always equipped with pressure, flow, and methane concentration sensors.

The paper is organized as follows. In Section 2 we introduce the constitutive equations and develop separately the non-minimal and the minimal representation of the system. In Section 3 we develop feedback laws that employ actuation in all the branches of the network, while in Section 4 we develop feedbacks for inputs only in the independent branches. We close with an example, chosen of minimal order to illustrate the main issues in the problem and the design algorithms.