Linear symmetry of nonlinear systems

This paper tackles the symmetries of control systems. Main attention has been focused on the linear symmetry of affine nonlinear systems. That is, the symmetry under the action of a sub-group of general linear group GL(n,R). The structure of the groups of symmetry and their Lie algebras is investigated. Using left semi-tensor product, a complete classification of symmetric plane systems is presented. Finally, a set of linear algebraic equations are presented, whose solutions provide the largest Lie algebra. Its connected Lie group is the largest one, with which the system is symmetric.