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This talk starts with a brief introduction to dynamic systems, canonical forms and other related concepts. In the second part, for both continuous and discrete linearly controllable quadratic systems with single input, the extended Brunovsky forms and the corresponding transformation are derived based on special criteria for the existence of a valid transformation. In the third part, the algorithms are described for both systems, and some numerical examples are given. In the last part, a possible relationship of this new algorithm to Ricatti equation and Krylov subspaces may be presented.