Multi-scale aspects in model-predictive control George Stephanopoulos*, Orhan Karsligil, Matthew Dyer Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Abstract Multi-scale models of processing systems oer an attractive alternative to models de®ned in the time- or frequency-domain. They are de®ned on dyadic or higher-order trees, whose nodes are used to index the values of any variable, localised in both time and scale (range of frequencies). This dual localisation is particularly attractive in solving estimation and control problems. In this paper, multi- scale models are used to design model-predictive controllers (MPC), resulting in design techniques with several important advan- tages, such as; (a) natural depiction of performance characteristics and treatment of output constraints, (b) fast algorithms for establishing the constrained control policies over long prediction/control horizons, (c) rich depiction of feedback errors at several scales, and (d) optimal fusion of multi-rate measurements and control actions. # 2000 IFAC. Published by Elsevier Science Ltd. All rights reserved. Keywords: Predictive control; Multirate; Quadratic programming; Kalman ®lters; Constraints 1. Introduction Analysis of linear systems in the frequency domain has oered invaluable insights for the synthesis of con- trol systems, an activity that must be carried out in the time domain. In model-predictive control (MPC), the imposition of inequality constraints on the manipulated inputs and controlled outputs renders a nonlinear con- troller. Analysis in the frequency domain is inapplic- able. However, a multitude of design speci®cations for MPC could be expressed and speci®ed in terms of fre- quency; e.g. the allowable duration or/and period of output constraints violation, suppression of highly oscillatory control inputs, the relative weighting of closed-loop performance characteristics. Therefore, a hybrid time-frequency framework should be the ideal setting for the design of feedback controllers. It is broadly accepted that physical phenomena occur at dierent time scales. However, it is not clear how to incorporate this knowledge systematically into the gen- eration of adequate process models, or how to use it for the solution of some basic process engineering pro- blems, e.g. control, estimation, diagnosis. The conven- tional models, causal and explicit, provide a convoluted representation of physics at various scales and hamper engineering analysis and interpretation. Furthermore, process models used in the design of MPCs should be consistent with process descriptions used in other engineering tasks, e.g. adaptive control, fault diagnosis, scheduling of control strategies, and planning of operating procedures, which involve process models with dynamics of progressively increasing time constants. Current modelling practices are not very instructive on how to create consistent models for a sequence of interrelated engineering tasks, as the above, deployed at dierent time scales [1]. Finally, sensors may provide measurements of process behaviour at dierent sampling rates, invoking control actions at correspondingly dierent rates. The optimal fusion of measurement information or control actions relies on the availability of process models at time scales, which are commensurable with the sampling rate of measurements and the application intervals of con- trol actions. All of the above requirements indicate the need for representations that capture scale-based characteristics of process models, measurements and control actions. However, the classical Fourier analysis, which has been used to provide such representations, is not adequate. It provides frequency-based information on the behaviour of processes and measurements, by integrating the time behaviour of such entities over an in®nite time horizon. A better framework could provide an explicit repre- sentation of process dynamics, measurements, and control actions, with localisation in both time and fre- quency (scale). 0959-1524/00/$ - see front matter # 2000 IFAC. Published by Elsevier Science Ltd. All rights reserved. PII: S0959-1524(99)00022-0 Journal of Process Control 10 (2000) 275±282 www.elsevier.com/locate/jprocont * Corresponding author. Fax: +1-617-252-1651. E-mail address: geosteph@mit.edu (G. Stephanopoulos).