Computing the L∞-Induced Norm of LTI Systems: Generalization of Piecewise Quadratic and Cubic Approximations

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. INDEX TERMS Approximate computing, approximation methods, linear systems, performance analysis, robustness.

Robust sliding mode control for robot manipulators with analysis on trade‐off between reaching time and L∞ gain

“…$$ and $$$ {\left\Vert G\right\Vert}_{\infty } $$$ is the $$$ {\mathcal{L}}_{\infty } $$$‐induced norm of $$$ G $$$. Here, if we note from the arguments in [29–31] on the $$$ {\mathcal{L}}_{\infty } $$$‐induced norm analysis that $$$$ {\left\Vert G\right\Vert}_{\infty }=\underset{1\le i\le n}{\max}\sum \limits_{j=1}^n\int_0^{\infty}\mid \exp {\left(\Phi \tau \right)}_{ij}\mid d\tau, $$$$ where $$$ \exp {\left(\Phi \tau \right)}_{ij} $$$ is the $$$ \left(i,j\right) $$$th element of $$$ \exp \left(\Phi \tau \right) $$$, and this $$$ {\mathcal{L}}_{\infty } $$$‐induced norm can be obtained within any degree of accuracy. □…”

“…The rationale behind this fact might be interpreted as originating from the following two aspects. (i) Unlike the well‐known $$$ {\mathcal{L}}_{\infty }/{\mathcal{L}}_2 $$$ gain [27, 28], it is quite difficult to derive an analytical solution to the $$$ {\mathcal{L}}_{\infty } $$$ gain computation even for LTI case, and thus, various approximate approaches are developed in [29–31]. (ii) A tractable expression of the solution to () is not readily derived, due to some difficulties in its non‐uniqueness occurring from taking $$$ \operatorname{sgn}\left(\cdotp \right) $$$.…”

The l1 Optimal State Estimator for Load Frequency Control of Power Systems: A Comparative and Extensive Study

“…Furthermore, we characterize the load changes and the tie-line power deviations as bounded persistent disturbances by noting that it is difficult to characterize such disturbances by specific presumed natures. The proposed approach estimates the state variables by regulating the maximum magnitude of state estimation errors with respect to bounded persistent unknown inputs based on the arguments of the l ∞ -induced norm approach, i.e., the l 1 optimal control theory [24]- [27]. The development of the proposed l 1 DSE approach is practically meaningful because the fully decentralized l 1 optimal DSE can handle the realtime DSE problem by using only the ACE.…”