Research On Modeling And Control Methods For Column Flotation Process

In real world engineering control applications,the dynamics of many chemical and mechanical processes are affected by both time and space factors.These processes are known as distributed parameter systems(DPS).Essentially,all engineering control systems are DPS,their mathematical models are described by partial differential equations(PDEs),and these systems belong to infinite dimensional systems.Because of the existence of spatial variables in mathematical models as well as control and state constraints must be considered in the actual industry,the control research of distributed parameter systems is very interesting and challenging.Column flotation is one of the most widely used separation methods in the mineral processing industry.The whole process utilizes the countercurrent principle and is a highly nonlinear distributed parameter system.The focus of this paper is to develop a dynamic model for column flotation process,and then explore advanced process control methods to further optimize the production process to improve product quality while ensuring the stability of the flotation process.The paper mainly studies from the following aspects:(1)Dynamic model: The principle of column flotation process is studied and analyzed.Based on the law of material conservation,a three-phase dynamic model of column flotation process is developed for the two areas of collection region and froth region as the basis of control research.The whole process of column flotation is described by a set of nonlinear coupled heterodirectional hyperbolic partial differential equations(PDEs)and ordinary differential equations(ODEs),these two regions are connected by the boundary conditions of the PDE system.In order to obtain the linearized model,equilibrium point state and Jacobian are utilized in the linearization of the original nonlinear system.It can be seen from the simulation results that the proposed model can accurately describe the column flotation process.(2)Model predictive control: The model predictive controller is designed for the column flotation process,taking into account both optimality of the process operations and naturally present input and state/output constraints.For the design of discrete controllers,the Cayley-Tustin time-discrete transformation method is used to obtain the discrete model without any spatial discretization and/or without model reduction.Finally,the model predictive controller is designed by solving an optimization problem with input and state/output constraints as well as input disturbance to minimize the objective function,which leads to an online-solvable finite constrained quadratic regulator problem.Simulation studies have demonstrated the performance of the controller to keep the output at the steady state within the constraint range,and concluded that the optimal control scheme proposed in this work makes the flotation process more efficient.(3)Output regulation problem: The research on the output regulation problem of the column flotation process includes two aspects: the dynamic output feedback regulator design and the error feedback regulator design.The model was first improved by considering the time delay that existed in the process.For further analysis,the time delay term in the model is replaced by the corresponding transport equation.Subsequently,the dynamic output feedback and error feedback regulator is designed based on the linearized model to achieve the purpose of making the output of the coupled PDEs–ODEs system tracking a reference trajectory generated by an exosystem and disturbance rejection enforced.The stabilization feedback gain is obtained by solving operator Riccati equation to optimize the tracking process.In particular,the weighted regulator state is introduced in the error feedback regulator design to provide a parametric choice.The controller performance to force the output tracks reference trajectory as well as disturbance rejection is demonstrated by simulation studies.