Generalized Framework for Gridding Approximation Approach to yet Another H2 Norm of Sampled-Data Systems

Abstract

This paper considers treatment of yet another $H_2$ norm in linear time-invariant (LTI) sampled-data systems, which is defined as the $L_2$ norm of the response to the worst-timing impulse disturbance in those systems. This norm is introduced recently by the authors as an alternative to the two existing $H_2$ norms in LTI sampled-data systems, and it is called the third $H_2$ norm of those systems. Regarding the analysis and minimization problems of the third $H_2$ norm, our preceding studies introduce the idea of a gridding approximation approach to the sampling interval $[0,h)$ at which the impulse disturbance occurs; the sampling interval is divided into $N$ subintervals with an equal width and each of the beginning points of the $N$ subintervals is regarded as the timing at which the impulse disturbance is considered.In this respect, this paper provides a generalized framework for the gridding approximation approach by taking advantage of the freedom in the point for each subinterval at which the impulse disturbance is dealt with. It is shown in this paper that the gridding approximation approach has the convergence rate of $1/N$ regardless of the point for each subinterval, provided that some nontrivial modification is applied to the gridding treatment. More importantly, this paper shows that taking the central point for each subinterval leads to quantitatively improved accuracy than that of our preceding studies for both the associated analysis and synthesis problems.