ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

Factorization of the Characteristic Polynomial for the Equilibrium State of an Аutonomous System Having an Invariant Set


Alexander V. Bratishchev

Don State Technical University, Rostov-on-Don, Gagarin sq.,1, Professor of Applied Mathematics Department, Professor, doctor fiz.-mat. Nauk


Let an autonomous n-th order system of differential equations have an invariant set. For an equilibrium point being on this set, we form the Jacobi matrix in this point and prove the formula for the factorization of the characteristic polynomial in two polynomials of lesser degrees. Such a factorization is useful in the problem of the design of controllers of dynamical systems. The proposed formula is applied to solve the known problem of controlling an inverted pendulum on a movable trolley. By using the method of analytical design of aggregated regulators we construct a single-level nonlinear control stabilizing the pendulum in vertical position for sufficiently large initial deviations.



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