Boundary Feedback Control of the Axially Moving String System

• Main Author: 吳笙隆
• Other Authors: 馮榮豐
• Format: Others
• Language: zh-TW
• Published: 1997
• Online Access: http://ndltd.ncl.edu.tw/handle/10447373450563807259

碩士 === 中原大學 === 機械工程學系 === 85 === 　　In this thesis, we approach the boundary feedback control of an axially moving string system from the energy aspect. Therein, the principle of designing the control law is to dissipative the total mechanical energy of the system. The whole system consists of the string and the control mechanism. Therefore, except the traveling speed of the string would affect the boundary feedback gain, the characters of the control mechanism will affect, too. 　　The governing equation and the boundary conditions are derived by using variation method and Hamilitan's principle. The mathematical model of this system is composed of an ordinary differential equation (ODE) deseribing the control mechanism and a second order partial differential equation (PDE) describing the axially moving string. The string passing through the control mechanism at the right-hand-side (RHS) boundary introduced a coupling dynamic equation. This dynamic equation is the RHS boundary condition of the moving string and the governing equation of the control mechanism, simultaneously. This system epitomizes a typical distributed parameter system (DPS). The semigroup of operators theory provided an elegant state space representation for the analysis of the coupled system. The inner-product defined by the space introduced the encrgy norm function of the system. By designing a suitable control law, the derivative of the energy function with respect to time is negative definite. Then, one can obtain the asymptotic stability of the system. Following Hille-Yosida theorem, the semigroup of contraction in the space is constructed. We quoted the theory of the semigroup to obtain the exponential stability of the system. Finally, the lower and upper bounds of the feedback gain are obtained.