Digital redesign via the generalised bilinear transformation

Abstract

A new controller discretisation approach, the generalised bilinear transformation (GBT), is proposed in Zhang, G., Chen, T., and Chen X. (2007a). Given an analog controller K, GBT generates a class of digital controllers K _gbt parameterised by a real number α ∈ (−∞, ∞). A geometric interpretation of GBT is first presented. Second, when the original analog feedback system is stable, two methods are proposed to find the value of the parameter α which provides an upper bound of sampling periods guaranteeing closed-loop stability of the resulting sampled-data system. Finally, several examples, namely, an IIR digital filter, an example studied in Rattan, K.S. (1984), ‘Digitization of Existing Continuous Control Systems,’ IEEE Transactions on Automatic Control, 29, 282–306, and Keller, J.P., and Anderson, B.D.O. (1992), ‘A New Approach to the Discretisation of Continuous-time Controllers,’ IEEE Transaction on Automatic Control, 37, 214–223, and an H _∞ control problem investigated in Chen, T., and Francis, B. (1995), Optimal Sampled-Data Control Systems, London: Springer, are used to demonstrate the strength of our discretisation approach. These examples show that GBT is able to retain the simplicity of the emulation methods such as the Tustin method, and simultaneously sustain closed-loop performance even at slow sampling.

Keywords:

• controller discretisation
• generalised bilinear transformation (GBT)
• closed-loop stability
• optimisation