Optimisation and control methodologies for large-scale and multi-scale systems

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Abstract

Distributed parameter systems (DPS) comprise an important class of engineering systems ranging from “traditional” such as tubular reactors, to cutting edge processes such as nano-scale coatings . DPS have been studied extensively and significant advances have been noted, enabling their accurate simulation. To this end a variety of tools have been developed. However, extending these advances for systems design is not a trivial task . Rigorous design and operation policies entail systematic procedures for optimisation and control. These tasks are “upper-level” and utilize existing models and simulators. The higher the accuracy of the underlying models, the more the design procedure benefits. However, employing such models in the context of conventional algorithms may lead to inefficient formulations. The optimisation and control of DPS is a challenging task. These systems are typically discretised over a computational mesh, leading to large-scale problems. Handling the resulting large-scale systems may prove to be an intimidating task and requires special methodologies. Furthermore, it is often the case that the underlying physical phenomena span various temporal and spatial scales, thus complicating the analysis. Stiffness may also potentially be exhibited in the (nonlinear) models of such phenomena. The objective of this work is to design reliable and practical procedures for the optimisation and control of DPS. It has been observed in many systems of engineering interest that although they are described by infinite-dimensional Partial Differential Equations (PDEs) resulting in large discretisation problems, their behaviour has a finite number of significant components , as a result of their dissipative nature. This property has been exploited in various systematic model reduction techniques. Of key importance in this work is the identification of a low-dimensional dominant subspace for the system. This subspace is heuristically found to correspond to part of the eigenspectrum of the system and can therefore be identified efficiently using iterative matrix-free techniques. In this light, only low-dimensional Jacobians and Hessian matrices are involved in the formulation of the proposed algorithms, which are projections of the original matrices onto appropriate low-dimensional subspaces, computed efficiently with directional perturbations.The optimisation algorithm presented employs a 2-step projection scheme, firstly onto the dominant subspace of the system (corresponding to the right-most eigenvalues of the linearised system) and secondly onto the subspace of decision variables. This algorithm is inspired by reduced Hessian Sequential Quadratic Programming methods and therefore locates a local optimum of the nonlinear programming problem given by solving a sequence of reduced quadratic programming (QP) subproblems . This optimisation algorithm is appropriate for systems with a relatively small number of decision variables. Inequality constraints can be accommodated following a penalty-based strategy which aggregates all constraints using an appropriate function , or by employing a partial reduction technique in which only equality constraints are considered for the reduction and the inequalities are linearised and passed on to the QP subproblem . The control algorithm presented is based on the online adaptive construction of low-order linear models used in the context of a linear Model Predictive Control (MPC) algorithm , in which the discrete-time state-space model is recomputed at every sampling time in a receding horizon fashion. Successive linearisation around the current state on the closed-loop trajectory is combined with model reduction, resulting in an efficient procedure for the computation of reduced linearised models, projected onto the dominant subspace of the system. In this case, this subspace corresponds to the eigenvalues of largest magnitude of the discretised dynamical system. Control actions are computed from low-order QP problems solved efficiently online.The optimisation and control algorithms presented may employ input/output simulators (such as commercial packages) extending their use to upper-level tasks. They are also suitable for systems governed by microscopic rules, the equations of which do not exist in closed form. Illustrative case studies are presented, based on tubular reactor models, which exhibit rich parametric behaviour.

Keyword(s)

adaptive identification of dominant subspace; dissipative distributed parameter systems; equation-free; iterative methods; matrix-free; model predictive control; model reduction; nonlinear programming; reduced Hessian Sequential Quadratic Programming

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