Generalization of Ackermann Formula for a Certain Class of Multidimensional Dynamic Systems with Vector Input

Abstract

A compact analytical formula is obtained that determines the entire set of solutions of the modal control problem for a wide class of multidimensional dynamical systems with vector input, where the number of states is divisible by the number of control inputs, and the controllability index is equal to the quotient of this division. This formula generalizes to systems with the vector input the Ackermann formula applied to multidimensional systems with scalar input. The basis to obtaining the generalized Ackermann formula lies in the original concepts of the Luenberger generalized canonical form and operations of the matrices block transposition. For the most convenient calculation of controller, the original system with vector input is reduced to the generalized Luenberger canonical form using the two successive similarity transformations. A lemma is proved that demonstrates the compact analytical form of the inverse transformation matrix. Transition equivalence makes it possible to obtain a complete countably infinite parametrized set of solutions to the modal control problem under consideration. Its parametrization is provided by selecting block coefficients of the matrix polynomial, which determinant corresponds to the given scalar characteristic polynomial. In cases, where the matrix polynomial involved in parametrization is not reduced to the multipliers, the generalized Ackermann formula contains solutions to the modal control problem that could not be obtained using the existing decomposition method. Examples are presented demonstrating both suitability of the proposed formula for analytical synthesis of modal controllers by state in systems with vector input and its advantages in comparison with the decomposition method