Deterministic Global Optimization for Parameter Estimation of Dynamic Systems

Abstract

A method is presented for deterministic global optimization in the estimation of parameters in models of dynamic systems. The method can be implemented as an -global algorithm, or, by use of the interval-Newton method, as an exact algorithm. In the latter case, the method provides a mathematically guaranteed and computationally validated global optimum in the goodness of fit function. A key feature of the method is the use of a new validated solver for parametric ODEs, which is used to produce guaranteed bounds on the solutions of dynamic systems with intervalvalued parameters, as well as on the first-and second-order sensitivities of the state variables with respect to the parameters. The computational efficiency of the method is demonstrated using several benchmark problems.

Validated solutions of initial value problems for parametric ODEs

“…Numerical examples demonstrate that this approach provides a very effective way to determine tight enclosures of all possible solutions to parametric ODEs. The method presented, as implemented in VSPODE, is now being used in a number of practical applications, including simulation of bioreactors with parametric uncertainties [13], model-based fault diagnosis [14], deterministic global optimization for fitting time series data [10] and for optimal control [12], and state and parameter estimation for nonlinear, continuoustime systems [11]. Information about the availability of the VSPODE code can be obtained by contacting the authors.…”

Chebyshev model arithmetic for factorable functions

“…This gives Taylor models a clear advantage over traditional interval extensions or centered forms for sufficiently narrow domains, but conversely it may result in a large overestimation or may even be poorer than naive interval evaluation over wider domains. Nevertheless, this approach has proved successful in computing tight enclosures for the solutions of differential equations and implicit algebraic equations [24,33,42,49,50,53,60], and it has enabled complete search for a range of global optimization or constraint satisfaction problems that could not be tackled using interval techniques alone (see, e.g., [4,9,31,32,47,52]). Such higher-order inclusion techniques are indeed appealing in complete search applications based on branching or subdivision, where they can mitigate the clustering effect [15,62].…”

On Metaheuristics for Solving the Parameter Estimation Problem in Dynamic Systems: A Comparative Study

“…Alternatively, exact methods for GO have also been used to solve DMbPE problems. These include the spatial branch and bound (BB) algorithm to compute a global solution of the partially discretized DMbPE problem [18], the -BB method to obtain a global solution within an -precision of the completely discretized DMbPE problem [10], and a method based on interval analysis and on the partially discretized DMbPE [19]. In [14], a deterministic outer approximation approach is applied to the reformulation of the DMbPE problem as a finite NLP by applying a complete discretization using orthogonal collocation on finite elements.…”

“…The practical comparison carried out in the present study aims to analyze the performance of each metaheuristic in terms of the quality of the obtained solutions. To test the five metaheuristics, nine DMbPE problems were selected from the literature [4,10,13,19,26], yielding 12 instances due to different experimental data. The problems are described in the Appendix.…”