Abstract
This paper is concerned with performance analysis for bounded persistent disturbances of continuous-time linear time-invariant (LTI) systems. Such an analysis can be done by computing the L ∞ -induced norm of continuous-time LTI systems since it corresponds to the worst maximum magnitude of the output for the worst persistent external input with a unit magnitude. In our preceding study, piecewise constant and linear approximation schemes for analyzing this norm have been developed through two alternative approximation approaches, one for the input and the other for the relevant kernel function, via the fast-lifting technique. The approximation errors in these approximation schemes have been shown to converge to 0 at the rates of 1/N and 1/N 2 , respectively, as the fast-lifting parameter N is increased. Along this line, this paper aims at developing generalized techniques that offer improved accuracy named the piecewise quadratic and cubic approximation schemes. The generalization and the associated accuracy improvement discussed in this paper are not limited to the increased orders of approximation but are extended further to taking advantage of the freedom in the point around which relevant functions are expanded to Taylor series. The approximation errors in the piecewise quadratic and cubic approximation schemes are shown to converge to 0 at the rates of 1/N 3 and 1/N 4 , respectively, regardless of the point at which the Taylor expansion is applied. Finally, effectiveness of the developed computation methods is confirmed through a numerical example.
Highlights
Mathematical models of the real control systems such as electrical circuit systems, mechanical systems and electromechanical systems are often described as continuous-time linear time-invariant (LTI) systems
This paper aims at numerically analyzing the L∞-induced norm of continuous-time finite-dimensional linear time-invariant (FDLTI) systems as accurately as possible
The improvement over our earlier studies [19], [20] is not limited to the use of such higher order approximation schemes but includes generalized arguments relevant to how the Taylor expansion of relevant functions is used; even though the earlier studies only considered the Taylor expansion at the beginning of each subintervals resulting from the application of fast-lifting, this paper considers taking advantage of the freedom in the time instant around which relevant functions are expanded to Taylor series
Summary
Mathematical models of the real control systems such as electrical circuit systems, mechanical systems and electromechanical systems are often described as continuous-time linear time-invariant (LTI) systems. CONTRIBUTIONS AND ORGANIZATION OF THIS PAPER Stimulated by the success of the L∞-induced norm analysis in the preceding studies, this paper pursues extended schemes named the piecewise quadratic and piecewise cubic approximations for achieving better convergence rates than those in [19], [20]; these new schemes are developed again under both the input and kernel approximation approaches through the fast-lifting treatment together with the arguments of the Taylor expansion of relevant functions These schemes readily allow us to compute upper and lower bounds on the L∞-induced norm of MIMO LTI systems, and the gap between the upper and lower bounds is shown to converge to 0 at the improved rates of 1/N 3 and 1/N 4 in the piecewise quadratic and piecewise cubic approximation schemes, respectively, regardless of the input approximation approach or the kernel approximation approach. Regarding a sophisticated computation of the L∞-induced norm G ∞, we introduce the following alternative representation [20] with u(·) := w(t−·) to alleviate the difficulty about the convolution integral in computing G ∞: t
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