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Robust distributed controller design of rotational single-link smart materials beam

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Abstract

A dynamic modelling and controller design were presented for a single-link smart materials beam, a flexible beam bonded with piezoelectric actuators and sensors for better control performance. Taking into account bounded disturbances, a robust distributed controller was constructed based on the system model, which was described by a set of partial differential equations (PDEs) and boundary conditions (BCs) . Subsequently, a finite dimensional controller was further developed, and it was proven that this controller can stabilize the finite dimensional model with arbitrary number of flexible modes.

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Abbreviations

a :

thickness of beam

b :

width of beam and that of smart material

c ε R6x6 :

symmetric matrix of elastic stiffness coefficients

C1]:

thickness of piezoelectric actuator

c2 :

thickness of piezoelectric sensor

\()\) :

stiffness per unit length of whole manipulator

c m11 :

stiffness of pure beam

c s11 :

stiffness of piezoelectric material

D ε R3 :

electrical displacement vector

Da (x,t) :

electrical displacement of piezoelectric actuator at point P

Day (x,y): y :

component of electrical displacement of piezoelectric actuator

Ds( x , t) :

electrical displacement of piezoelectric sensor at point P

Dsy (x,t): y :

component of electrical displacement of piezoelectric sensor

e(t) - d(t) - θd (t) :

error between actual trajectory and desired one

E ε R3 :

electrical field intensity vector

Ek :

system kinetic energy

Ep :

system potential energy

F ε R6 :

simplified stress vector

h ε R6x3 :

coupling coefficients matrix

h12 :

coupling parameter per unit volume of piezoelec-tric material

haL =\(\frac{1}{2}h_{12} b{\text{ x (ac}}_{{\text{1 }}} + {\text{ c}}_{\text{1}}^{\text{2}} )\) :

coupling parameter per unit length of piezoelectric actuator

haL =\( - \frac{1}{2}h_{12} b{\text{ x (ac}}_{{\text{2 }}} + {\text{ c}}_{\text{2}}^{\text{2}} )\) :

coupling parameter per unit length of piezoelectric sensor

L :

length of beam

m3 :

mass of tip payload

Ih :

inertia of hub

r (x , t) :

position vector of point P

S ε R6 :

simplified strain vector

v(x,t) :

voltage applied to piezoelectric actuator

w(x , t) :

deflection at point P

W :

virtual work done by external forces

(@#@ X0, O0,Y0):

fixed base frame

( X, 0, Y) :

local reference frame

β ε R3x3 :

symmetric matrix of impermittivity coeffi-cients

β22 :

impermittivity per unit volume of piezoelectric material

βaL = bc1β22 - :

impermittivity per unit length of piezoelectric actuator

βsL = bc2β22 :

impermittivity per unit length of piezoelectric sensor

ρ1 :

mass per unit volume of beam

ρ2 :

mass per unit volume of piezoelectric material

ρL = (c1 + c2 @#@) bρ2 + abρ1 :

mass per unit length of whole manipulator

θ ( t):

joint angle at the hub

gd(t):

desired joint angular trajectroy

τ (t):

torque applied to the base of manipulator

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Author information

Authors and Affiliations

  1. School of Mechanical Engineering, Purdue University, 47907 — 1288, West Lafayette, IN, USA

    Jianqin Gong

  2. Department of Electrical & Computer Engineering, National University of Singapore, 117576, Singapore

    Shuzhi Sam Ge

  3. Systems Engineering Institute, Xi' an Jiaotong University, 710049, Shared Xi'an, China

    Baiwu Wan

Authors
  1. Jianqin Gong
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  2. Shuzhi Sam Ge
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  3. Baiwu Wan
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Gong, J., Ge, S.S. & Wan, B. Robust distributed controller design of rotational single-link smart materials beam. J. Control Theory Appl. 1, 91–96 (2003). https://doi.org/10.1007/s11768-003-0015-1

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  • DOI: https://doi.org/10.1007/s11768-003-0015-1

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