A Discretization Approach to the Analysis of Yet Another H-2 Norm of LTI Sampled-Data Systems

Abstract

This paper is concerned with linear time-invariant (LTI) sampled-data systems together with their yet another H-2 norm introduced recently as an alternative to the two well-known definitions. Taking account of the linear periodically time-varying nature of LTI sampled-data systems, this norm is defined as the supremum of the L-2 norms of all the tau-dependent responses for the impulse inputs occurring at the instant tau in the sampling interval [0, h). We first review the closed-form expression of this new H-2 norm derived through the lifted representation of LTI sampled-data systems. We next develop a discretization method of the continuous-time generalized plant, by which the new H-2 norm of LTI sampled-data systems can be characterized by using the discrete-time H-2 norm. We then reinterpret the closed-form expression of the H-2 norm, and derive a computable upper bound together with a lower bound of the norm. We further show that the gap between the upper and lower bounds converges to 0 at the rate of 1/N, where N is the gridding approximation parameter. Finally, a numerical example is given to demonstrate the effectiveness of the computation method.