Explicit Necessary and Sufficient Conditions for Quadratic Linearization

The existing necessary and sufficient conditions for quadratic linearization of control affine systems are in effect, constructive in nature. The exception is the classical result, which requires one to check involutive properties of distributions of the quadratic polynomials to be linearized. Nevertheless, the latter condition is difficult to verify. In this paper, we provide necessary and sufficient conditions for quadratic linearization that are not constructive but are based on checking a linear system of equations involving quadratic polynomial terms only. The system needs only to be put in controller normal form of the linear part which is known. Thus, the result is verifiable and explicit. Also, if required, once the quadratic linearization conditions are met, we show how to construct the coordinate and state feedback transformation to implement the linearization.