Control of Oscillating Systems with a Single Delay

J Diblík,D Ya Khusainov,J Lukáčová,M Růžičková

doi:10.1155/2010/108218

Abstract

Systems are considered related to the control of processes described by oscillating second-order systems of differential equations with a single delay. An explicit representation of solutions with the aid of special matrix functions called a delayed matrix sine and a delayed matrix cosine is used to develop the conditions of relative controllability and to construct a specific control function solving the relative controllability problem of transferring an initial function to a prescribed point in the phase space.

Highlights

The problem of controllability of linear first-order autonomous systems without delay xt Ax t bu t, x ∈ Rn, t ≥ 0, 1.1 with an n × n constant matrix A, b ∈ Rn and u : 0, ∞ → R is solved by the wellknown Kalman criterion e.g., 1–3
We investigate systems related to control of processes, described by oscillating second-order systems of differential equations with a single delay, in the following form: xt Ω2x t − τ bu t, 1.6 where t ≥ 0, x : 0, ∞ → Rn, Ω is an n × n constant regular matrix, τ > 0, τ ∈ R, b ∈ Rn, and u : 0, ∞ → R
The main result is the construction of a control function in terms of these matrix functions, solving the problem of a transferring of an initial function to a prescribed point in the phase space

Summary

Introduction

The problem of controllability of linear first-order autonomous systems without delay xt Ax t bu t , x ∈ Rn, t ≥ 0, 1.1 with an n × n constant matrix A, b ∈ Rn and u : 0, ∞ → R is solved by the wellknown Kalman criterion e.g., 1–3. We investigate systems related to control of processes, described by oscillating second-order systems of differential equations with a single delay, in the following form: xt Ω2x t − τ bu t , 1.6 where t ≥ 0, x : 0, ∞ → Rn, Ω is an n × n constant regular matrix, τ > 0, τ ∈ R, b ∈ Rn, and u : 0, ∞ → R. One way to investigate such problem is to define additional dependent variables and, transforming initial system 1.6 into a system of first-order linear differential equations with constant coefficients and a constant delay, to get controllability criteria using the results in the above-mentioned sources. The main result is the construction of a control function in terms of these matrix functions , solving the problem of a transferring of an initial function to a prescribed point in the phase space

Preliminaries

Control of Oscillating Systems

D21 D22 C2

Conclusions and Future Directions