Optimal Linearization via Quadratic Programming

The technique of linearization for nonlinear systems around some operating point has been widely used for analysis and synthesis of the system behavior within a certain operating range. Conventional linearization methods include the analytical linearization (AL) method using the Jacobian matrix, the result of which usually works only for a sufficiently small region, as well as the numerical linearization (NL) method based on small perturbation, the accuracy of which is usually not guaranteed. In this letter, we propose an optimal linearization method via quadratic programming (OLQP). We start with uniform data sampling within the neighborhood of the operating point based on the nonlinear ordinary differential equation (ODE). We then find the best linear model that fits to these sample points with a QP formulation. The OLQP solution is derived in closed form with proved convergence to the AL solution. Two examples of nonlinear systems are investigated in terms of linearization and results are compared among these linearization methods, which has shown the proposed OLQP method features a great balance between model accuracy and computational complexity. Moreover, the OLQP method offers additional options in controller design by tuning its parameters.