離散時間非負システムのl_q/l_p Hankelノルム解析

抄録

The L_q/L_p Hankel norms of continuous-time LTI systems characterize the effect of the past L_p (p=1,2,∞) inputs on the future L_q (q=1,2,∞) outputs. Computing some of these norms of general LTI systems is known to be difficult, but recently it has been shown that we can avoid such difficulties in part when we deal with positive systems. On the other hand, studies on the l_q/l_p Hankel norms of discrete-time LTI systems are relatively scarce. In view of this fact, in this paper, we focus on the Hankel norms of discrete-time LTI systems. We first show that we face the same difficulties in computing the l_q/l_p Hankel norms of general discrete-time systems, and then we clarify that such difficulties can be avoided in part when dealing with positive systems. In particular, we show that some of the l_q/l_p Hankel norms of discrete-time positive systems can be described in closed form and hence easy to compute. Furthermore, we also characterize the l_q/l_p Hankel norms of positive systems in the form of semidefinite programming problems and linear programming problems. These allows us to compute the upper bounds of the l_q/l_p Hankel norms of uncertain positive systems by means of convex optimization.