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 p induced norms for all 1 ≤ p ≤ ∞. This results extends the main results in ] 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.

Digital redesign via the generalised bilinear transformation

“…Another advantage of GBT is that it allows slow sampling. It has been shown in Chen and Francis (1995), Rabbath, Hori and Lechevin (2004), and Zhang et al (2007a) that as the sampling period h goes to zero, the performance of sampled-data systems converges to that of the original continuous-time systems. However, instead of concentrating on the behaviour as h !…”

“…The proposed method is equivalent to GBT when is restricted to the interval [0.5, 1). A technique identical to that in Zhang et al (2007a) is studied in Sekara (2006) where the parameter is restricted to the interval [0, 1]. However, it is shown by Examples 4.1-4.3 that, by allowing 2 (À1, 1), better performance could be achieved.…”

“…The parameter is restricted to be in the interval [0, 1] in Zhang et al (2007a). It will be shown that performance could be improved significantly by allowing 2 (À1, 1).…”

“…In other words, the emulation method is essentially an open-loop approach and only works well at fast sampling. With the aim of improving this situation, a new controller discretisation -the generalised bilinear transformation (GBT) is proposed and studied in Zhang, Chen and Chen (2007a) and Zhang, Chen and Chen (2007b).…”

Optimization-based digital redesign of analogue controllers

“…The plant input mapping approach was developed in [12], [13], [14]. Zhang and Chen proved that the L p performance of their approach converges to the continuous system performance, when the sampling period tends to zero [15]. Traditionally the most popular criteria of performance recovery are connected with the step response.…”