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

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

Nested sampling approach to set-membership estimation

This paper is concerned with set-membership estimation in nonlinear dynamic systems. The problem entails characterizing the set of all possible parameter values such that given predicted outputs match their corresponding measurements within prescribed error bounds. Most existing methods to tackle this problem rely on outer-approximation techniques, which perform poorly when the parameter host set is large due to the curse of dimensionality. An adaptation of nested sampling—a Monte Carlo technique introduced to compute Bayesian evidence—is presented herein. The nested sampling algorithm leverages efficient strategies from Bayesian statistics for generating an inner-approximation of the desired parameter set. Several case studies are presented to demonstrate the approach

Bayesian approach to probabilistic design space characterization: a nested sampling strategy

Quality by design in pharmaceutical manufacturing hinges on computational methods and tools that are capable of accurate quantitative prediction of the design space. This paper investigates Bayesian approaches to design space characterization, which determine a feasibility probability that can be used as a measure of reliability and risk by the practitioner. An adaptation of nested sampling—a Monte Carlo technique introduced to compute Bayesian evidence—is presented. The nested sampling algorithm maintains a given set of live points through regions with increasing probability feasibility until reaching a desired reliability level. It furthermore leverages efficient strategies from Bayesian statistics for generating replacement proposals during the search. Features and advantages of this algorithm are demonstrated by means of a simple numerical example and two industrial case studies. It is shown that nested sampling can outperform conventional Monte Carlo sampling and be competitive with flexibility-based optimization techniques in low-dimensional design space problems. Practical aspects of exploiting the sampled design space to reconstruct a feasibility probability map using machine learning techniques are also discussed and illustrated. Finally, the effectiveness of nested sampling is demonstrated on a higher-dimensional problem, in the presence of a complex dynamic model and significant model uncertainty

Optimal feeding strategy of diafiltration buffer in batch membrane processes

This work addresses the optimal control strategy of diafiltration buffer utilisation in discontinuous membrane processes that are designed to fulfil the twin aims of concentration and fractionation. The problem of optimal process operation is formulated using a general membrane response model that encounters concentration-dependent flux and rejections. We consider two problems, operation time minimisation and diluant consumption minimisation, and we apply theory of optimal control and derive necessary conditions of optimality. Through selected case studies from the literature, we demonstrate how to apply the proposed methodology to determine optimal time-dependent wash-water feeding policy. The analytical results are confirmed by numerical computations, using numerical methods of dynamic optimisation. The presented methodology allows decision makers to analyse suboptimality of conventional diafiltration strategies in terms of processing time and diluant consumption. Results show that depending on the complexity of the membrane response model, it may be attractive to implement optimal trajectory

Guaranteed parameter estimation of non-linear dynamic systems using high-order bounding techniques with domain and CPU-time reduction strategies

This paper is concerned with guaranteed parameter estimation of non-linear 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 values of certain outputs match their corresponding measurements within prescribed error bounds. A set-inversion algorithm 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 given threshold on the approximation level is met. Such exclusion tests rely on the ability to bound the solution set of the dynamic system for a finite parameter subset, and the tightness of these bounds is therefore paramount; equally important in practice is the time required to compute the bounds, thereby defining a trade-off. In this paper, we investigate such a trade-off by comparing various bounding techniques based on Taylor models with either interval or ellipsoidal bounds as their remainder terms. We also investigate the use of optimization-based domain reduction techniques in order to enhance the convergence speed of the set-inversion algorithm, and we implement simple strategies that avoid recomputing Taylor models or reduce their expansion orders wherever possible. Case studies of various complexities are presented, which show that these improvements using Taylor-based bounding techniques can significantly reduce the computational burden, both in terms of iteration count and CPU time