A Hybrid Approach for Efficient Robust Design of Dynamic Systems

• Published in: SIAM review 2007-06, Vol.49 (2), p.236-254
• Main Authors: Mönnigmann, Martin, Marquardt, Wolfgang, Bischof, Christian H., Beelitz, Thomas, Lang, Bruno, Willems, Paul
• Format: Article
• Language: English
• Subjects: Arithmetic; Calculus of variations and optimal control; Combinatorics; Combinatorics. Ordered structures; Critical points; Design efficiency; Designs and configurations; Dynamic efficiency; Dynamical systems; Eigenvalues; Exact sciences and technology; Jacobians; Mathematical analysis; Mathematical independent variables; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Nonlinear systems; Normal vectors; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in mathematical programming, optimization and calculus of variations; Optimization; Optimization techniques; Parametric models; Problems and Techniques; Sciences and techniques of general use; Studies; Systems design; Topological manifolds
• ISSN: 0036-1445; 1095-7200
• DOI: 10.1137/S003614450444662X

We propose a novel approach for the parametrically robust design of dynamic systems. The approach can be applied to system models with parameters that are uncertain in the sense that values for these parameters are not known precisely, but only within certain bounds. The novel approach is guaranteed to find an optimal steady state that is stable for each parameter combination within these bounds. Our approach combines the use of a standard solver for constrained optimization problems with the rigorous solution of nonlinear systems. The constraints for the optimization problems are based on the concept of parameter space normal vectors that measure the distance of a tentative optimum to the nearest known critical point, i.e., a point where stability may be lost. Such normal vectors are derived using methods from nonlinear dynamics. After the optimization, the rigorous solver is used to provide a guarantee that no critical points exist in the vicinity of the optimum, or to detect such points. In the latter case, the optimization is resumed, taking the newly found critical points into account. This optimize-and-verify procedure is repeated until the rigorous nonlinear solver can guarantee that the vicinity of the optimum is free from critical points and therefore the optimum is parametrically robust. In contrast to existing design methodologies, our approach can be automated and does not rely on the experience of the designing engineer. A simple model of a fermenter is used to illustrate the concepts and the order of activities arising in a typical design process.