Estimation of model parameters in a dynamic system can be significantly improved with the choice of experimental trajectory. For general nonlinear dynamic systems, finding globally “best” trajectories is typically not feasible; however, given an initial estimate of the model parameters and an initial trajectory, we present a continuous-time optimization method that produces a locally optimal trajectory for parameter estimation in the presence of measurement noise. The optimization algorithm is formulated to find system trajectories that improve a norm on the Fisher information matrix (FIM). A double-pendulum cart apparatus is used to numerically and experimentally validate this technique. In simulation, the optimized trajectory increases the minimum eigenvalue of the FIM by three orders of magnitude, compared with the initial trajectory. Experimental results show that this optimized trajectory translates to an order-of-magnitude improvement in the parameter estimate error in practice.
To obtain a systems-level understanding of a biological system, the authors conducted quantitative dynamic experiments from which the system structure and the parameters have to be deduced. Since biological systems have to cope with different environmental conditions, certain properties are often robust with respect to variations in some of the parameters. Hence, it is important to use optimal experimental design considerations in advance of the experiments to improve the information content of the measurements. Using the MAP-Kinase pathway as an example, the authors present a simulation study investigating the application of different optimality criteria. It is demonstrated that experimental design significantly improves the parameter estimation accuracy and also reveals difficulties in parameter estimation due to robustness. This article reviews the development of experiment design in the field of identification of dynamical systems, from the early work of the seventies on input design for open loop identification to the developments of the last decade that were spurred by the research on identification for control. While the early work focused entirely on criteria based on the asymptotic parameter covariance, the results of the last decade aim at minimizing a wide range of possible criteria, including measures of the estimated transfer function, or of functions of this estimated transfer function. Two important recent developments are the solution of the experiment design problem for closed loop identification, and the formulation and solution of the dual optimal design problem in which the cost of identification is minimized subject to a quality constraint on the estimated model. We shall conclude this survey with new results on the optimal closed loop experiment design problem, where the optimization is performed jointly with respect to the controller and the spectrum of the external excitation. This article presents advances in optimal experiment design, which are intended to improve the parameter identification of nonlinear state space models. Instead of using a sequence of samples from one or just a few coherent sequences, the idea of identifying nonlinear dynamic models at distinct points in the state space is considered. In this way, the placement of the experiment points is fully flexible with respect to the set of reachable points. Also, a method for model-based generation of prediction errors is proposed, which is used to compute an a-priori estimate of the sample covariance of the prediction error. This covariance matrix may be used to approximate the Fisher information matrix a-priori. The availability of the Fisher matrix a-priori is a prerequisite for experiment optimization with respect to covariance in the parameter estimates. This work is driven by the problem of parameter identification of hydraulic models. There are methods for hydraulic systems regarding the estimation of parameters from experimental data, but the choice of experiments has not been treated adequately yet. A hydraulic servo system actuating a stewart platform serves as an illustrative example to which the methods above are applied. Due to the wide use and key importance of mathematical models in process engineering, experiment design is becoming an essential tool for the rapid building and validation of these mechanistic models. Several experiment design techniques have been developed in the past and applied successfully to a wide range of systems. This paper is focused on the so-called model-based design of experiments (DOE) and aims at presenting an up-to-date state of the art in this important field. In order to provide an adequate and thorough background to this technique, a detailed description of the key elements of a model identification procedure (the model itself, the experiment, the statistical tools, etc.) and the major steps of a model-building strategy are introduced before focusing on the experiment design for parameter precision, which is the topic of this survey. An overview and critical analysis of the state of the art in this sector are proposed. The main contributions to model-based experiment design procedures in terms of novel criteria, mathematical formulations and numerical implementations are highlighted. A list of the most recent applications of these techniques in various fields (from chemical kinetics to biological modelling) is then presented highlighting the key role of model-based DOE in the process engineering area. It is well known that if we intend to use a minimum variance control strategy, which is designed based on a model obtained from an identification experiment, the best experiment which can be performed on the system to determine such a model (subject to output power constraints, or for some specific model structures) is to use the true minimum variance controller. This result has been derived under several circumstances, first using asymptotic (in model order) variance expressions but also more recently for ARMAX models of finite order. In this paper we re-approach this problem using a recently developed expression for the variance of parametric frequency function estimates. This allows a geometric analysis of the problem and the generalization of the aforementioned finite model order ARMAX results to general linear model structures. Abstract The investigation of enzyme kinetics is increasingly important, especially for finding active substances and understanding intracellular behaviors. Therefore, the determination of an enzyme's kinetic parameters is crucial. For this a systematic experimental design procedure is necessary to avoid wasting time and resources. The parameter estimation error of a Michaelis–Menten enzyme kinetic process is analysed analytically to reduce the search area as well as numerically to specify the optimum for parameter estimation. From analytical analysis of the Fisher information matrix the fact is obtained, that an enzyme feed will not improve the estimation process, but substrate feeding is favorable with small volume flow. Unconstrained and constrained process conditions are considered. If substrate fed-batch process design is used instead of pure batch experiments the improvements of the Cramer–Rao lower bound of the variance of parameter estimation error reduces to 82 for μ max and to 60 for K m of the batch values in average. We consider the problem of choosing a set of k sensor measurements, from a set of m possible or potential sensor measurements, that minimizes the error in estimating some parameters. Solving this problem by evaluating the performance for each of the (m k) possible choices of sensor measurements is not practical unless m and k are small. In this paper, we describe a heuristic, based on convex optimization, for approximately solving this problem. Our heuristic gives a subset selection as well as a bound on the best performance that can be achieved by any selection of k sensor measurements. There is no guarantee that the gap between the performance of the chosen subset and the performance bound is always small; but numerical experiments suggest that the gap is small in many cases. Our heuristic method requires on the order of m 3 operations; for m= 1000 possible sensors, we can carry out sensor selection in a few seconds on a 2-GHz personal computer. A method of optimal experimental design for parameter estimation in unstructured growth models is presented. The approach is based on a method suggested by Munack (1991) for application in fed-batch processes. In a critical analysis of this method, special emphasis is given to the model validity, because unstructured growth models often are not valid under transient conditions. In consequence, a combined object function has been introduced, which considers model validity and the accuracy of the kinetic parameters to be estimated. The application of this method for fed-batch processes leads to satisfactory results. Investigations of different fed-batch strategies regarding model validity and the quality of parameter estimation are presented. In addition, an experimental verification has been performed with fermentations of the yeast Trichosporon cutaneum.