【正文】 . 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 work. The actuation available ranges from a few fans/pressors strategically located through the work(and often directly connected to the outside environment), to actively controlled ―doors‖ that are in many of the branches of the work. The problem of controlling mine ventilation received considerable attention in the 1970s and the 1980s. It is clear that a mine ventilation work 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 modelbased fashion, as a fluid flow work(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 Kocic[5] who considered a linearized lumpedparameter dynamic model of a mine ventilation work 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 plicated MIMO control tools. However, he did not take advantage of the graph theoretic properties of the work, which forced him to both neglect the nonlinearities (essential in this fluid flow problem) and to employ dynamic outputfeedback pensators where static output feedback would suffice. We provide these improvements in this article. The control model of a mine ventilation work consists of Kirchho’s current and voltage laws (algebraic equations) and fluid dynamical equations of individual branches (differential equations). The branches are modeled using lumped parameter approximations of inpressible 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 Kirchho’s algebraic equations and the branch characteristic differential equation constitute a nonminimal representation of the control model. It is clear that, due to the mass conservation at the branching points (nodes) of the work, air flows in many of the branches will be interdependent. Hence, the minimal system representation will be of lower order than the number of branches. This intuition bees systematic when one employs graph theoretic concepts from circuit theory [3]. Each work 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 plement of the tree, called a cotree, whose branches are referred to as the links. The minimal system representation of the dynamics of the work 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 plicated 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 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 anized as follows. In Section 2 we introduce the constitutive equations and develop separately the nonminimal and the minimal representation of the system. In Section 3 we develop feedback laws that employ actuation in all the branches of the work, 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. 2. Model of mine ventilation work system . Pipe flow dynamics and Kirchho’s laws for mine ventilation works In order to develop the model of a mine ventilation work, we rst establish the dynamical equation of one branch. For simplicity, we make the following assumptions: (A1) the air is inpressible。 EQij =0 if branch j is not connected to node i. Let us assume that the mine ventilation work employs one main fan that is connected with the ambient outside of the mine. Also let node 1 be connected to the fan branch. Then the airow in the fan branch can be expressed as ?? ?nj QmeQmjQj1 , (4) or eQmQ=Qm, (5) where Qm is airow quantity through fan (main) branch, eQm =[eQm1。:::。 l is a number of the links in the work, l=nnc +1。 B