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.

This paper deals with the L1 analysis of linear time-invariant (LTI) systems, by which we mean the L∞-induced norm analysis of LTI systems. It is well known that this induced norm corresponds to the L1 norm of the impulse response of the given system, i.e., integral of the absolute value of the kernel function in the convolution formula for the input/output relation. However, because it is very hard to compute this integral exactly or even approximately with explicit upper and lower bounds, the ideas of piecewise constant and piecewise linear approximations have been developed to compute upper and lower bounds of the L∞-induced norm in our preceding study. These ideas are introduced through fastlifting, by which the interval [0, h) with a sufficiently large h is divided into M subintervals with an equal width, and it is shown that the approximation errors in piecewise constant or piecewise linear approximation converge to 0 at the rate of 1/M or 1/M 2 , respectively. Motivated by the success of the L∞-induced norm analysis in that study, this paper aims at developing extended schemes named piecewise quadratic and piecewise cubic approximations. These approximations are also developed through fast-lifting, and it is shown that the piecewise quadratic and piecewise cubic approximation leads to approximation errors converging to 0 at the rate of 1/M 3 and 1/M 4 , respectively.

This paper is concerned with the hydrogen ratio control in a horizontal continuous annealing furnace used in steelmaking. We first model the nonlinear dynamics of the atmosphere in the furnace so that the hydrogen ratio control may be carried out by applying control theory. We then determine an appropriate equilibrium around which its linearized model is determined. We then take a two-step procedure for designing a linear controller, in which we first apply a minor feedback loop for pressure control and then apply integral compensation so that the hydrogen ratio tracks the reference input without any steady state error. The effectiveness of the resulting controller is verified through a simulation with the original nonlinear model of the furnace.