$ L_2/L_1 $ induced norm and Hankel norm analysis in sampled-data systems

Abstract

This paper is concerned with the $ L_2/L_1 $ induced and Hankel norms of sampled-data systems. In defining the Hankel norm, the $ h $-periodicity of the input-output relation of sampled-data systems is taken into account, where $ h $ denotes the sampling period; past and future are separated by the instant $ \Theta\in[0, h) $, and the norm of the operator describing the mapping from the past input in $ L_1 $ to the future output in $ L_2 $ is called the quasi $ L_2/L_1 $ Hankel norm at $ \Theta $. The $ L_2/L_1 $ Hankel norm is defined as the supremum over $ \Theta\in[0, h) $ of this norm, and if it is actually attained as the maximum, then a maximum-attaining $ \Theta $ is called a critical instant. This paper gives characterization for the $ L_2/L_1 $ induced norm, the quasi $ L_2/L_1 $ Hankel norm at $ \Theta $ and the $ L_2/L_1 $ Hankel norm, and it shows that the first and the third ones coincide with each other and a critical instant always exists. The matrix-valued function $ H(\varphi) $ on $ [0, h) $ plays a key role in the sense that the induced/Hankel norm can be obtained and a critical instant can be detected only through $ H(\varphi) $, even though $ \varphi $ is a variable that is totally irrelevant to $ \Theta $. The relevance of the induced/Hankel norm to the $ H_2 $ norm of sampled-data systems is also discussed.

Keywords

• sampled-data systems
• intersample behavior
• induced norm
• quasi hankel norm
• hankel norm
• $ h_2 $ norm
• impulse response