The paper presents methods for nonlinear homogeneous systems representation in a canonical form allowing stability conditions to be given by a linear matrix inequality. The main restriction for a system to admit the required representation is that its right-hand side has to be bounded on a unit sphere. It is shown that some nonhomogeneous systems can also be presented in the canonical form. Based on canonical representation a stabilizing control algorithm for affine in control nonlinear systems is presented with LMI-based tuning procedure. The results are supported with numerical examples.