Fundamentals of Practical Controller Design for LTI Plants with Distributed Parameters: Part 1, Modelling and Well-Posedness

This paper addresses some fundamental concerns pertaining to the subject of practical controller design for LTI plants with possibly distributed parameters. The well-posedness of a general LTI model is introduced in terms of the existence of bandlimited stabilizing compensation for the given model. The well posedness of a model is critical from an implementation standpoint for the application of any model-based controller design methodology. The problem addressed in this paper is to characterize LTI distributed parameter models that are well posed with respect to bandlimited stabilising control. The problem is posed on a framework consisting of a novel class of convolution kernels and a completely general control system configuration. These two aspects of the problem formulation place no a priori restriction on the class of LTI distributed paramater systems under consideration or the control design methodology used. Indeed, this framework allows the modelling of LTI distributed parameter systems including point sensing, boundary actuation and effects of initial conditions. Within this framework the result obtained are algebraic in nature. A characterization of well-posed models is given in the form of necessary conditions for the existence of a bandlimited stabilizing controller. Also a restriction of the class of all linear distributed parameter systems under consideration is derived which exhibits this condition.