  In this paper, for continuous, linearly-controllable quadratic control
systems with a single input, an explicit, constructive method is proposed for
studying their Brunovsky forms, initially studied in [W. Kang and A. J. Krener,
Extended quadratic controller normal form and dynamic state feedback
linearization of nonlinear systems, SIAM Journal on Control and Optimization,
30:1319-1337, 1992]. In this approach, the computation of Brunovsky forms and
transformation matrices and the proof of their existence and uniqueness are
carried out simultaneously. In addition, it is shown that quadratic
transformations in the aforementioned paper can be simplified to prevent
multiplicity in Brunovsky forms. This method is extended for studying discrete
quadratic systems. Finally, computation algorithms for both continuous and
discrete systems are summarized, and examples demonstrated.
