Chebyshev model arithmetic for factorable functions

This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden

Guaranteed parameter estimation in nonlinear dynamic systems using improved bounding techniques

This paper is concerned with guaranteed parameter estimation in nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding - or approximating as closely as possible - the set of all possible parameter values such that the predicted outputs match the corresponding measurements within prescribed error bounds. An exhaustive search procedure is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a prespecified threshold on the approximation level is met. Exclusion tests rely on the ability to bound the solution set of the dynamic system for a given parameter subset and the tightness of these bounds is therefore paramount. Equally important is the time required to compute the bounds, thereby defining a trade-off. It is the objective of this paper to investigate this trade-off by comparing various bounding techniques based on interval arithmetic, Taylor model arithmetic and ellipsoidal calculus. When applied to a simple case study, ellipsoidal and Taylor model approaches are found to reduce the number of iterations significantly compared to interval analysis, yet the overall computational time is only reduced for tight approximation levels due to the computational overhead. © 2013 EUCA

Global optimization in Hilbert space

We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an (Formula presented.)-suboptimal global solution within finite run-time for any given termination tolerance (Formula presented.). Finally, we illustrate these results for a problem of calculus of variations

Methodologie d'Optimisation Dynamique et de Commande Optimale des Petites Stations d'Epuration a Boues Activees

The adoption of stricter effluent requirements by the European Union rises large problems for small communities having few economical and technical resources. These problems motivate the synthesis of advanced optimisation-based controllers in order to enhance the performances of small-size wastewater treatment plants. The aim of this study is to develop a methodology of dynamic optimisation and optimal control of small-size alternating aerobic-anoxic activated sludge plants. The first part deals with the improvements which can be potentially obtained through the application of dynamic optimisation techniques. Two problems are considered: the minimisation of nitrogen discharge and the reduction of the operating costs. These are constrained, non-convex and high-dimensional problems that exhibit hybrid discrete/continuous and combinatorial behaviour. In both cases, the application of the resulting optimal aeration strategies leads to large improvements of the process performances, while satisfying the discharge requirements and the operating constraints; it is also verified that the long-term implementation of the optimal control profiles guarantees durable process improvement. The embedding of the optimal aeration strategies within closed-loop controllers is dealt with in the second part of the thesis. Optimal control consists in (i) the on-line joint observation/estimation of both state variables a nd parameters and, (ii) the use of a non-linear model predictive control scheme to update the aeration profiles. Beforehand, a reduced model is derived by simplifying the general ASM~1 model and sensitivity analyses are performed to quantify the influence of unmeasured disturbances and model mismatch on the optimisation results. Numerical implementations show that the resulting closed-loop controller brings large improvements with respect to usual operating modes either in terms of nitrogen discharge or in terms of energy consumption

Optimization-based domain reduction in guaranteed parameter estimation of nonlinear dynamic systems

This paper is concerned with guaranteed parameter estimation in nonlinear dynamic systems in a context of bounded measurement error. The problem consists of finding-or approximating as closely as possible-the set of all possible parameter values such that the predicted outputs match the corresponding measurements within prescribed error bounds. An exhaustive search procedure is applied, whereby the parameter set is successively partitioned into smaller boxes and exclusion tests are performed to eliminate some of these boxes, until a prespecified threshold on the approximation level is met. In order to enhance the convergence of this procedure, we investigate the use of optimization-based domain reduction techniques for tightening the parameter boxes before partitioning. We construct such bound-reduction problems as linear programs from the polyhedral relaxation of Taylor models of the predicted outputs. When applied to a simple case study, the proposed approach is found to reduce the computational burden significantly, both in terms of CPU time and number of iterations. © IFAC