Linear Machine: A Novel Approach to Point Location Problem

In recent literature, explicit model predictive control (e-MPC) has been proposed to facilitate implementation of the popular model predictive control (MPC) approach to fast dynamical systems. e-MPC is based on multi-parametric programming. The key idea in e-MPC is to replace the online optimization problem in MPC by a point location problem. After locating the current point, the control law is simply computed as an appropriate linear function of the states. A variety of approaches have been proposed in literature for the point location problem. In this work, we present a novel approach based on linear machines for solving this problem. Linear machines are widely used in multi-category pattern classification literature for developing linear classifiers given representative data from various classes. The idea in linear machines is to associate a linear discriminant function with each class. A given point is then assigned to the class with the largest discriminant function value. In this work, we develop an approach for identifying such discriminant functions from the hyperplanes characterizing the given regions as in multi-parametric programming. Apart from being an elegant solution to the point location problem as required in e-MPC, the proposed approach also links two apparently diverse fields namely e-MPC and multi-category pattern classification. To illustrate the utility of the approach, it is implemented on a hypothetical example as well as on a quadruple tank benchmark system taken from literature.