On Sensitivity Reduction Problems of Sampled-Data Systems: Relationships to the Problems of Discrete-Time Systems
Yoshimichi Ito, Hiroshi Shirahama, Tomomichi Hagiwara · SICE Journal of Control Measurement and System Integration · 2010
This paper is concerned with the sensitivity reduction (SR) and complementary sensitivity reduction (CSR) problems of sampled-data systems. We begin our study by showing that, given a sampled-data system Σ, the H∞ norms of the sensitivity and complementary sensitivity of Σ can be expressed as those for an equivalent discrete-time system Σ called the doubly sensitivity-preserving discretized system. We also consider the conventional ‘hold equivalent’ discretized system Σd, which has generally been believed, due to the ignoring of the intersample behavior, to be irrelevant to the SR/CSR problems of Σ (and thus Σ). We then establish that there in fact exists an important relation between the seemingly completely irrelevant discrete-time systems Σ and Σd. More precisely, we show through the coprime factorization approach about Σ and Σd that the CSR problem of Σ (and thus Σ) is equivalent to a weighted CSR problem of Σd and that the SR problem of Σ is equivalent to a weighted mixed sensitivity reduction problem of Σd. In particular, we show that the frequency-dependent weights that arise in this interplay between Σ and Σd can be described with an important function called the aliasing factor ψ, and further clarify an important analytic property of ψ. We then demonstrate that this property of ψ can be used to prove some relation between the best achievable performance in the SR (respectively, CSR) problem of Σ (and thus Σ) and that of Σd. An interesting property between the SR problem and CSR problem of Σ is also provided.