TY - JOUR

T1 - Sliding Mode-based Speed-gradient Control of the String Energy

AU - Orlov, Yury V.

AU - Fradkov, Alexander L.

AU - Andrievsky, Boris

PY - 2017

Y1 - 2017

N2 - An energy control problem is analyzed in the PDE (partial differential equation) setting. As opposed to the existing literature, the present control goal addresses not only decreasing the plant energy but also its increasing what is important, e.g., in vibrational technologies, in studying wave motion, etc. A benchmark linear wave equation, governing 1-D (one-dimensional) string oscillations, is chosen for exposition. A distributed control input, independently enforcing the underlying string over its entire spatial location, is assumed to be available. The speed-gradient method (Fradkov, 1996) is presently developed and justified in the above PDE setting. The applicability of the Krasovskii-LaSalle principle is established for the resulting sliding-mode closed-loop system in the infinite-dimensional setting. By applying this principle, all the closed-loop trajectories, initialized beyond the origin, are shown to approach the desired energy level set. Capabilities of the proposed speed-gradient algorithm of reaching the energy goal are supported by numerical simulations. The obtained results constitute the first step to justify properties of feedback energy control for PDE systems important for application in physics.

AB - An energy control problem is analyzed in the PDE (partial differential equation) setting. As opposed to the existing literature, the present control goal addresses not only decreasing the plant energy but also its increasing what is important, e.g., in vibrational technologies, in studying wave motion, etc. A benchmark linear wave equation, governing 1-D (one-dimensional) string oscillations, is chosen for exposition. A distributed control input, independently enforcing the underlying string over its entire spatial location, is assumed to be available. The speed-gradient method (Fradkov, 1996) is presently developed and justified in the above PDE setting. The applicability of the Krasovskii-LaSalle principle is established for the resulting sliding-mode closed-loop system in the infinite-dimensional setting. By applying this principle, all the closed-loop trajectories, initialized beyond the origin, are shown to approach the desired energy level set. Capabilities of the proposed speed-gradient algorithm of reaching the energy goal are supported by numerical simulations. The obtained results constitute the first step to justify properties of feedback energy control for PDE systems important for application in physics.

KW - string equation

KW - energy control

KW - speed-gradient

KW - BOUNDARY

UR - http://www.scopus.com/inward/record.url?scp=85031784933&partnerID=8YFLogxK

U2 - 10.1016/j.ifacol.2017.08.821

DO - 10.1016/j.ifacol.2017.08.821

M3 - статья

AN - SCOPUS:85031784933

VL - 50

SP - 8484

EP - 8489

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

SN - 2405-8963

IS - 1

Y2 - 9 July 2017 through 14 July 2017

ER -