(629c) Design of Uncertain Discrete Time Systems with Constructive Nonlinear Dynamics Methods

This contribution presents a new methodology for the optimization based design of uncertain discrete time systems. In a typical application, the new approach allows taking parametric uncertainty with respect to stability properties into account in process optimization. The methodology naturally applies, however, to the robust treatment of more general dynamical properties as well as to robust feasibility.

Constructive nonlinear dynamics methods have been developed for the optimization of dynamical systems under uncertainty over the past few years by the authors [1-4]. Originally, the development of these methods was motivated by the need to impose constraints for robust stability on dynamical systems modeled by continuous time differential-algebraic systems of equations (DAE systems) [3,4]. Several types of dynamical systems that arise in science and engineering cannot be modeled as DAE systems, however. The present contribution deals with the extension of constructive nonlinear dynamics to discrete time systems.

Discrete time models that consist of difference equations and algebraic equations arise under several circumstances. For one, processes exist that are intrinsically discrete in time. Vendor managed supply chains, for example, belong to this class (see, e.g., [5]). Systems of difference equations also arise naturally when oscillating processes are described with the aid of Poincaré mappings (see, e.g., [6]). This type of description is particularly useful when studying the stability and robustness properties of periodically operated systems. A third important class of examples comprises discrete time systems that result from sampling continuous time processes. Since digital control systems use sampled data, this class of models is particularly important in the modeling for control.

The method presented in this contribution is based on bifurcation theory and nonlinear programming. Bifurcation theory is employed to state formal descriptions of critical points for nonlinear dynamical systems, for example, critical points for stability. Under mild mathematical conditions, critical points for stability form critical manifolds. These critical stability manifolds can be thought of as boundaries that separate those parts of the parameter space in which the dynamical system is stable from those parts in which it is unstable. Beyond stability boundaries, the concept of a critical manifold can be used to describe a broad variety of points at which process behavior changes qualitatively, such as feasibility boundaries, and various boundaries that characterize the dynamic behavior of nonlinear systems [3].

Based on a formal description of critical manifolds from applied bifurcation theory [3], systems of equations for normal vectors to the critical manifold can be derived [4]. These normal vectors allow measuring the distance between a candidate point of operation in the process parameter space and the critical manifold. By imposing a lower bound on the distance to all critical manifolds, robustness with respect to parametric uncertainty of the dynamical system is guaranteed in the sense sketched in Fig. 1. In Fig. 1, the critical boundary and the normal direction to it are sketched as a bold red line and a bold dashed line, respectively. The shortest distance between the nominal point at the center of the circle and the critical manifold occurs along the dashed normal direction in Fig. 1. By requiring this distance to be at least as large as, or larger than, the radius of the circle, robustness can be guaranteed, since the critical red boundary will not be crossed, regardless of the actual values the parameters α₁, α₂ attain within the region of uncertainty represented by the box of side lengths Δ α₁, Δ α₂ around the nominal point. Loosely speaking, the optimization software pushes the robustness ball along any critical manifold. This way, robustness with respect to critical manifolds can be taken into account in the optimization of dynamical systems in spite of parametric uncertainty. While the sketch in Fig. 1 is only two-dimensional, the idea of measuring distance along normal vectors generalizes to arbitrary finite dimensional spaces of uncertain parameters.

Constructive nonlinear dynamics methods have successfully been applied to a variety of examples over the past years, ranging from literature models to systems of industrial importance [7-9]. To date, however, the method is restricted to continuous time models. This contribution presents examples of a successful application to discrete time models for the first time. The underlying mathematical foundations are similar but different from those for continuous time systems. Most importantly, the underlying bifurcation theory is different for the two system classes. Continuous time systems can, for example, experience a loss of stability due to two generic one parameter bifurcations (saddle-node and Hopf), while in discrete time systems, three types of bifurcation points can cause a loss of stability (Neimark-Sacker, flip, and fold) (for details see textbooks on bifurcation theory, e.g., [6]). In order to characterize stability boundaries, three types of critical points therefore have to be taken into account in discrete time systems as opposed to only two in continuous time systems. To this end, normal vector systems have been derived from systems of equations for critical manifolds of discrete time systems known from numerical bifurcation theory [6]. Based on the normal vector systems derived by the authors, constraints for parametric robustness with respect to stability boundaries can be included in process optimization.

The use of these constraint in the optimization under uncertainty is demonstrated with examples from the area of vendor managed inventory supply chains and the optimization of the periodic harvesting of a fish population. The latter case serves as an example for periodic harvesting of renewable resources. Aspects of the software components necessary to implement the method, in particular the generation of precise higher order derivatives and the automatic detection of critical manifolds, are discussed.

[1] J. Gerhard, M. Mönnigmann, W. Marquardt, Constructive nonlinear dynamics -- Foundations and application to Robust nonlinear control, in T. Meurer, K. Graichen, E. D. Gilles (Hrsg.), Control and Observer Design for Nonlinear Finite- and Infinite Dimensional Systems, Springer Verlag, 165-182, 2005

[2] M. Mönnigmann, W. Marquardt, Constructive nonlinear dynamics in process systems engineering, Comp. Chem. Eng. 29, 1265-1275, 2005

[3] M. Mönnigmann, Constructive Nonlinear Dynamics for the Design of Chemical Engineering Processes, Fortschritt-Berichte VDI Reihe 3 Nr. 801, VDI Verlag, Düsseldorf, ISBN 3-18-380103-5, 2004

[4] M. Mönnigmann, W. Marquardt, Normal vectors on manifolds of critical points for parametric robustness of equilibrium solutions of ODE systems, J. of Nonlinear Sc. 12, 85-112, 2002

[5] S. M. Disney, D. R. Towill, A discrete transfer function model to determine the dynamic stability of a managed inventory supply chain, Int. J. Prod. Res. 40 (1), 179-204, 2002

[6] Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer, New York, 2nd ed., 1998

[7] M. Mönnigmann, W. Marquardt, Steady-state process optimization with guaranteed robust stability and flexibility: Application to HDA reaction section, Ind. Eng. Chem. Res. 44, 2737-2753, 2005

[8] M. Mönnigmann, W. Marquardt, Steady state process optimization with guaranteed robust stability and feasibility, AIChE J. 49, 3110-3126, 2003

[9] M. Mönnigmann, J. Hahn, W. Marquardt, Towards constructive nonlinear dynamics - Case studies in chemical process design, in G. Radons, R. Neugebauer (Hrsg.), Nonlinear Dynamics of Production Systems, Wiley-VCH, Weinheim, 503-526, 2003