Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation

Abstract

The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system.

Conditions of Functional Null Controllability for Some Types of Singularly Perturbed Nonlinear Systems with Delays

“…Remember that z(·) = col x(·), y(·) . Let us solve the system (12) and (43) in the interval [t c − εh, t c ] subject to the initial conditions (33). To do this, we use the method of steps.…”

“…Thus, for any ε ∈ (0, ε 2 ], the control u(t) =ū(t, ε), t ∈ [t c − εh, t c ], given by (42), generates the unique zeroth solution of the system (12) and (13) subject to the initial conditions (33) in the entire interval [t c − εh, t c ]. Therefore, the control:…”

“…The robust complete/output Euclidean space controllability for various linear time-dependent/time-invariant systems either with delays only in state variables or with state and control delays was analyzed in [23][24][25][26][27][28][29][30][31][32]. However, the functional controllability (approximate and exact) of singularly-perturbed time delay systems was studied only in a few papers (see [33] and the references therein). Moreover, to the best of our knowledge, the exact functional controllability of a singularly-perturbed system was studied only in [33].…”

“…However, the functional controllability (approximate and exact) of singularly-perturbed time delay systems was studied only in a few papers (see [33] and the references therein). Moreover, to the best of our knowledge, the exact functional controllability of a singularly-perturbed system was studied only in [33]. Namely, in this work the functional null controllability was investigated for a singularly-perturbed linear constant coefficients system with non-small point-wise commensurate delays (of order of one) only in the slow state variable.…”

Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay

“…Mainly, this property was studied either for undelayed systems [16,26,27], or for systems with only state delays (see e.g. [5,8,11,17,18,20,30] and references therein). To the best of our knowledge, there are only few works in the literature [7,9,10] where some kinds of controllability for singularly perturbed systems with input (control) delays are analyzed.…”