Based on shifted Jacobi series, algorithms have been established for the analysis and identification of non-linear systems described by a Hammerstein model consisting of a single-valued non-linearity followed by a linear plant. Moreover, the parameter estimation of a certain non-linear lumped system is also presented. By using the shifted Jacobi expansion for the analysis, the solution of a non-linear state equation is reduced to the solution of a linear algebraic matrix equation. For the identification, by expanding the measured input-output data into the shifted Jacobi series the unknown parameters of the non-linear system are estimated through the least-squares method. Examples are given to demonstrate the usefulness of this approach.