Optimal experimental design with the sigma point method

Max-Planck-Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany.
IET Systems Biology (Impact Factor: 1.06). 02/2009; 3(1):10-23. DOI: 10.1049/iet-syb:20080094
Source: PubMed


Using mathematical models for a quantitative description of dynamical systems requires the identification of uncertain parameters by minimising the difference between simulation and measurement. Owing to the measurement noise also, the estimated parameters possess an uncertainty expressed by their variances. To obtain highly predictive models, very precise parameters are needed. The optimal experimental design (OED) as a numerical optimisation method is used to reduce the parameter uncertainty by minimising the parameter variances iteratively. A frequently applied method to define a cost function for OED is based on the inverse of the Fisher information matrix. The application of this traditional method has at least two shortcomings for models that are nonlinear in their parameters: (i) it gives only a lower bound of the parameter variances and (ii) the bias of the estimator is neglected. Here, the authors show that by applying the sigma point (SP) method a better approximation of characteristic values of the parameter statistics can be obtained, which has a direct benefit on OED. An additional advantage of the SP method is that it can also be used to investigate the influence of the parameter uncertainties on the simulation results. The SP method is demonstrated for the example of a widely used biological model.

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Available from: Andreas Kremling, Feb 28, 2014
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    • "A Monte-Carlo simulation can be performed after the experiment design to investigate the robust character of the constraints (Schenkendorf et al., 2009) by sampling different parameter sets. If the a-priori desired confidence level related with the chosen α, is not reached, this α can be increased. "
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    ABSTRACT: Dynamic experiments that yield as much information as possible are highly valuable for estimating parameters in nonlinear dynamic processes. Techniques for model-based optimal experiment design enable to systematically design such experiments. However, these experiments depend on the current best estimate of the parameters, which are not necessarily the true values. Consequently, in real experiments (i) the information content can be lower than predicted and (ii) state constraints can be violated. This paper presents a novel, computationally tractable formulation that enables the robustification of optimally designed experiments with respect to (i) information content and (ii) constraint satisfaction. To this end, the objective function is the expected value of a scalar function of the Fisher information matrix, which is efficiently computed using the sigma point method. This approach already has a robustifying effect. The sigma point method also enables the efficient computation of constraints’ variance-covariance matrix, this can be exploited for further robustification.
    Computers & Chemical Engineering 12/2014; 71:415-425. DOI:10.1016/j.compchemeng.2014.09.006 · 2.78 Impact Factor
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    • "The simulated well-mixed fed-batch bioreactor [20] used in this study has been well studied in various applications, in particular in experimental design [30] [31]. The process dynamic behaviour is described by the following set of differential equations: "
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    ABSTRACT: In the application of on-line, dynamic process optimisation, adaptive estimation of the system states and parameters is usually needed to minimise the unavoidable model-process mismatch. This work presents an integrated approach to optimal model adaptation and dynamic optimisation, with specific focus on batch processes. An active approach is proposed whereby the input variables are designed so as to maximise the information content of the data for optimal model adaptation. Then, this active adaptation method is combined with the objective of process performance to form a multi-objective optimisation problem. This integrative approach is in contrast to the traditional adaptation method, where only the process performance is considered and adaptation is passively carried out by using the data as is. Two strategies for solving the multi-objective problem are investigated: weighted average and constrained optimisation, and the latter is recommended for the ease in determining the balance between these two objectives. The proposed methodology is demonstrated on a simulated semi-batch fermentation process.
    Journal of Process Control 11/2013; 23(10):1350–1359. DOI:10.1016/j.jprocont.2013.09.010 · 2.65 Impact Factor
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    • "The second main contribution of the current paper is that we apply the proposed technique to the design of an optimal time-varying feed profile for a benchmark fed-batch bioreactor case-study which has originally been proposed in [3] [37]. Similar reactor models have been considered in [30] [31] [1] [12] [17] as well as [36]. The current paper is structured as follows. "
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    ABSTRACT: In this paper, we present a numerical method for optimal experiment design of nonlinear dynamic processes. Here, we suggest to optimize an approximation of the predicted variance–covariance matrix of the parameter estimates, which can be computed as the solution of a Riccati differential equation. In contrast to existing approaches, the proposed method allows us to take process noise into account and requires less derivative states to be computed compared to the traditional Fisher information matrix based approach. This process noise is assumed to be a time-varying random disturbance which is not known at the time when the experiment is designed. We illustrate the technique by solving an optimal experiment design problem for a fed-batch bioreactor benchmark case study. Here, we concentrate on how the optimal input design and associated accuracy of the parameter identification is influenced when process noise is present.
    Journal of Process Control 04/2013; 23(4):613–629. DOI:10.1016/j.jprocont.2012.11.005 · 2.65 Impact Factor
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