Rational Approximations of Transfer Functions of Some Viscoelastic Rods with Applications to Robust Control
Tóm tắt
We study rational approximations of the transfer function
$$\widehat P$$
of a uniform or nonuniform viscoelastic rod undergoing torsional vibrations that are excited and measured at the same end. The approximation is to be carried out in a way that is appropriate, with respect to stability and performance, for the construction of suboptimal rational stabilizing compensators for the rod. The function
$$\widehat P$$
can be expressed as
$$\widehat P(s) = s^{ - 2} g(\beta ^2 (s))$$
, where g is an infinite product of fractional linear transformations and β is a (generally transcendental) function that characterizes a particular viscoelastic material. First, g(β2) is approximated by its partial products g
N(β2). For relevant values of β2, convergence rates for g
N are analyzed in detail. Convergence suitable for our problem requires the introduction of a new irrational convergence factor, which must be approximated separately. In addition, the fractional linear factors in β2(s) that appear in g
N(β2(s)) must be replaced by something rational. When the damping is weak it is possible to do this by separating the oscillatory modes from the “creep” modes and ignoring the latter; in general, this step remains incomplete. Some numerical data illustrating all the stages of the process as well as the final results for various viscoelastic constitutive relations are presented.
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