Gain-scheduled Smith proportional-integral derivative controllers for linear parameter varying first-order plus time-varying delay systems

Abstract

Practical control problems often deal with parameter-varying uncertain systems that can be described by a first-orderplus- delay (FOPD) model. In this paper, a new approach to design gain-scheduled robust linear parameter varying (LPV) propotional–intergral derivative controllers with pole placement constraints through linear matrix inequalities (LMI) regions is proposed. The controller structure includes a Smith Predictor (SP) to deal with the delays. System parameter variations are measured online and used to schedule the controller and the SP. Although the known part of the delay is compensated with the ‘delay scheduling’ SP, the proposed approach allows to consider uncertainty in the delay estimation. This uncertainty is taken into account in the controller design as an unstructured dynamic uncertainty. Finally, two applications are used to assess the proposed methodology: a simulated artificial example and a simulated physical system based on an open canal system used for irrigation purposes. Both applications are represented by FOPD models with large and variable delays as well as parameters that depend on the operating conditions. 1 Introduction Although the control community has developed new and, in many aspects, more powerful control techniques (e.g. H1 robust control) during the last few decades, the propotional–intergral derivative (PID) controller is still used in many of the real-world control applications. The reason is the simplicity and the facility to tune using heuristic rules [1]. On the other hand, advanced controllers designed with the aid of H1 robust control techniques are usually of high order, difficult to implement and virtually impossible to re-tune online. Furthermore, if implementation issues have been overlooked, they can produce extremely fragile controllers (small perturbations of the coefficients of the controller destabilise the closed-loop control system [2, 3]). Since the 1960s, the empirical (or classical) gainscheduling (GS) control [4–6] has been used for controlling non-linear and time-varying systems. But, this control methodology achieves closed-loop stability, without guarantees, for slowly varying parameters [7]. In order to overcome this deficiency, linear parameter-varying gainscheduling (LPV GS) controllers are introduced to allow arbitrarily smooth