On the Robust Stability of Bilinear Generalized Predictive Controller

The bilinear Generalized predictive controller is modified to account for unmodelled plant uncertainties and bounded disturbances. First, a generalized predictive control law is derived by means of minimizing a quadratic cost function and then modified by introducing an estimate of the modeling error as a feedback. The robustness results are derived by neither requiring a prior knowledge of the first p H coefficients of the plant’s step response, nor their estimates. The global convergence, which ensures no steady-state errors, is proved. 1.0 Introduction Bilinear systems arise naturally in many biological, economical and engineering applications. Bilinear systems are wellstructured class of non-linear systems within which linear systems coexist as a special subclass. Thus, they are defined to be linear in both state and control when considered separately with the non-linearity, or bilinearity, arising from coupled multiplicative terms involving both state and control. Examples of engineering applications of bilinear systems include: disc braking systems, AC induction motors, DC machines, fermentation processes, heat transfer, refrigeration and fission processes. When dealing with industrial systems, it is necessary to consider extending the linear self-tuning framework to include knowledge of inherent plant nonlinearities. Indeed, when systems exhibit bilinear behaviour, it is reasonable to consider accommodating the bilinearity into the model structure within an extended self-tuning control framework. This idea have been utilized in [1], in which a novel bilinear selftuning control strategy is proposed for a high temperature heat treatment plant. A comparative study of bilinear and PID control strategies for a test furnace has been carried out in [2]. In [3], self-tuning control of an industrial pilot-scale reheating furnace: Design principles and application of a bilinear approach has been reported. In all of these, bilinear generalized predictive control strategy has been utilized and improved control performance has been reported over existing commercially available automatic loop tuning devices. However, unmodelled dynamics, which actually exist, are overlook in the design of bilinear generalized predictive control. Therefore, the bilinear generalized predictive control algorithm may have steady state errors. In this paper, a new bilinear generalized predictive control algorithm is presented by introducing an estimate of the modelling error as a feedback to ensure that the output of the system with unmodelled dynamics tracks the desired output without steady state errors. A modified least-square algorithm with a relative dead zone is employed for parameter estimation of the bilinear model. The robustness results are derived by neither requiring a prior knowledge of the first p H coefficients of the plant’s step response, nor their estimates. 2.0 Bilinear System Description In adopting a bilinear approach, use is made of the non-linear ARMAX form