Analytical Solution of Control Input Represented by Finite-Power Series of Time to Satisfy Boundary Condition

Abstract

In this paper, the analytical control input represented by finite-power series of time is algebraically solved such that make the boundary condition satisfied at the initial time and the final time for the controllable linear time-invariant system with single input. The minimal order of proposed control input is 2n_λ+n₀-1, where n_λ is the number of non-zero eigenvalues of the system and n0 is the number of zero eigenvalues of the system. Further, even if the control input represented by finite-power series of time is given in terms of any order N(N>2n_λ+n₀-1), analytical solution can be solved to satisfy the boundary condition by the arbitrary choice of the number 2n_λ+n₀ among the order N. And it is demonstrated that the minimal order control input without generating endpoint residual vibrations for the crane system is obtained by the proposed theorem. Finally, the usefulness of the proposed method is demonstrated by laboratory crane experiments.