Modal Controllability of a Delay Differential System by an Incomplete Output

Arbitrary Coefficient Assignment by Static Output Feedback for Linear Differential Equations with Non-Commensurate Lumped and Distributed Delays

“…System (26) consists of n equations with mk unknown entries of the matrix function R(τ), τ ∈ [σ θ , 0]. Let us rewrite systems (25), (26) in the vector form. By definition of the mapping vec, we have Sp (XY) = (vec X) T • (vec Y T ) for any X ∈ M p,q (K), Y ∈ M q,p (K).…”

“…By definition of the mapping vec, we have Sp (XY) = (vec X) T • (vec Y T ) for any X ∈ M p,q (K), Y ∈ M q,p (K). Let us apply this equality in system (25), for every i = 1, n, to the matrix X = C * J i−1 B and to the matrices Y = Q ρ , ρ = 0, θ, and in system (26), for every i = 1, n, to the matrix X = C * J i−1 B and to Y = R(τ). Let us construct the mk × n-matrix P = [vec(C * B), vec(C * JB), .…”

“…Later, the finite spectrum assignment problem for time-delay systems by linear state feedback was studied in [19] for systems with one lumped delay in the states with the scalar controller, in [20] for systems with multiple commensurate lumped delays in the states with the scalar controller, in [21] for systems with multiple commensurate lumped delays in the states and control with the scalar controller, in [22] for systems with multiple commensurate lumped delays in the states and control with the multidimensional controller, in [23] for systems with multiple commensurate lumped and distributed delays in the states with the scalar controller, and in [24] for systems of neutral type with multiple commensurate lumped delays in the states with the scalar controller. In [25], the ACA problem was studied for single-input single-output (SISO) systems with commensurate lumped delays in the states by the dynamic output feedback controller. In [26], the problem of stabilization of linear systems with both input and state delays by observer-predictors was studied.…”

Spectrum assignment in linear systems with several commensurate lumped and distributed delays in state by means of static output feedback

“…В [7] экспоненциальная стабилизация систем с постоянным и переменным запаздыванием обеспечивается решением задачи частичного назначения спектра. Задача назначения полного бесконечного спектра для систем с сосредоточенным запаздыванием изучена в [8], в [9] c помощью алгебраического метода с алгоритмом минимизации спектральной абсциссы, в [10] c использованием метода аппроксимации Галёркина.…”

“…Учитывая равенства (16), (8), 7, получаем, что для системы (1), (2) задача произвольного спектра посредством регулятора (3) разрешима тогда и только тогда, когда найдутся θ 0,…”