Optimisation of hybrid processes and hybrid controllers

Abstract

Most processes of practical interest are hybrid in nature, exhibiting both continuous and discrete characteristics. In many cases, the hybrid behaviour is a result of intrinsic physical phenomena that lead to (practically) instantaneous events such as the appearance and disappearance of thermodynamic phases, changes in flow regimes, equipment failures etc. All such events effect qualitative changes in the underlying continuous dynamics, thereby leading to hybrid macroscopic behaviour. In other cases, the hybrid nature arises from external discrete actions imposed on the process by its control system. For example, the latter may apply quantisation to convert continuous process measurements into discrete ones and/or continuous control outputs into discrete actions. Hybrid processes and hybrid controllers, and their combination, can be modelled in terms of State-Transition Networks (STNs). The system behaviour in each state is described by a different set of continuous equations (typically a mixed system of partial and/or ordinary differential and algebraic equations). At any particular time during its operation, the system is in exactly one such state. An instantaneous transition to a different state may take place if a certain logical condition becomes true. Each transition is also characterised by a set of continuous relations that determine unique values for the system variables immediately following the transition in terms of their values immediately preceding it. In this presentation, we consider mathematical formulations and techniques for the optimisation of hybrid systems described by STNs. This generally seeks to determine the time variation of a set of controls and/or the values of a set of time-invariant parameters that optimise some aspect of the dynamic behaviour of the system. The time horizon of interest may be fixed or variable, subject to specified lower and upper bounds. The equations that determine the system behaviour in each state may be augmented with additional inequality constraints imposing certain restrictions (related to safety or operability) on the acceptable system trajectories. The objective function to be minimised or maximised is usually a combination of fixed contributions (depending on the values of the time-invariant parameters) and variable contributions (depending on the system trajectory, including the variation of the controls). As an illustration, we start with simple linear systems operating in the discrete time domain, possibly involving uncertain parameters. We then proceed to consider the more complex problem of the optimisation of nonlinear hybrid systems operating in the continuous time domain.