Performance Recovery in Digital Implementation of Analogue Systems

Abstract

In this paper, the generalized bilinear transformation (GBT) is proposed. Compared with the traditional bilinear, zero-order hold (ZOH) and first-order hold transformations, one advantage of GBT is that it may convert unstable poles (zeros) to stable poles (zeros). It is proved that controllability and observability are invariant under GBT. After that, it is shown that the performance of a sampled-data system obtained via GBT approaches that of the analogue system as the underlying sampling period goes to zero. Performance studied here is characterized in terms of internal stability and $\ell_{p}$ induced norms for all $1 \leq p \leq \infty$. This results extends the main results in [G. Zhang and T. Chen, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, Suppl., 2003, pp. 28-33] and [G. Zhang and T. Chen, Automatica J. IFAC, 40 (2004), pp. 327-330] from SISO to MIMO and also removes the limitation on the “A” matrix of the system. Finally, an example is employed to compare digital implementations via GBT and the ZOH transformation.