Numerical Methods of Optimum Experimental Design Based on a Second-Order Approximation of Confidence Regions

A successful application of model-based simulation and optimization of dynamic processes requires an exact calibration of the underlying mathematical models. Here, a fundamental task is the estimation of unknown and nature given model coefficients by means of real observations. After an appropriate numerical treatment of the differential systems, the parameters can be estimated as the solution of a finite dimensional nonlinear constrained parameter estimation problem. Due to the fact that the measurements always contain defects, the resulting parameter estimate cannot be seen as an ultimate solution and a sensitivity analysis is required to quantify the statistical accuracy. The goal of the design of optimal experiments is the identification of those measurement times and experimental conditions which allow a parameter estimate with a maximized statistical accuracy. Also the design of optimal experiments problem can be formulated as an optimization problem, where the objective function is given by a suitable quality criterion based on the sensitivity analysis of the parameter estimation problem. In this thesis, we develop a quadratic sensitivity analysis to enable a better assessment of the statistical accuracy of a parameter estimate in the case of highly nonlinear model functions. The newly introduced sensitivity analysis is based on a quadratically approximated confidence region which is an expansion of the commonly used linearized confidence region. The quadratically approximated confidence region is analyzed extensively and adequate bounds are established. It is shown that exact bounds of the quadratic components can be obtained by solving symmetric eigenvalue problems. One main result of this thesis is that the quadratic part is essentially bounded by two Lipschitz constants, which also characterize the Gauss-Newton convergence properties. This bound can also be used for an approximation error of the validity of the linearized confidence regions. Furthermore, we compute a quadratic approximation of the covariance matrix, which delivers another possibility for the statistical assessment of the solution of a parameter estimation problem. The good approximation properties of the newly introduced sensitivity analysis are illustrated in several numerical examples. In order to robustify the design of optimal experiments, we develop a new objective function - the Q-criterion - based on the introduced sensitivity analysis. Next to the trace of the linear approximation of the covariance matrix, the Q-criterion consists of the above-mentioned Lipschitz constants. Here, we especially focus on a numerical computation of an adequate approximation of the constants. The robustness properties of the new objective function in terms of parameter uncertainties is investigated and compared to a worst-case formulation of the design of optimal experiments problem. It is revealed that the Q-criterion covers the worst-case approach of the design of optimal experiments problem based on the A-criterion. Moreover, the properties of the new objective function are considered in several examples. Here, it becomes evident that the Q-criterion leads to a drastic improve of the Gauss-Newton convergence rate at the following parameter estimation. Furthermore, in this thesis we consider efficient and numerically stable methods of parameter estimation and the design of optimal experiments for the treatment of multiple experiment parameter estimation problems. In terms of parameter estimation and sensitivity analysis, we propose a parallel computation of the Gauss-Newton increments and the covariance matrix based on orthogonal decompositions. Concerning the design of optimal experiments, we develop a parallel approach to compute the trace of the covariance matrix and its derivative

Toward improved identifiability of hydrologic model parameters: The information content of experimental data

We have developed a sequential optimization methodology, entitled the parameter identification method based on the localization of information (PIMLI) that increases information retrieval from the data by inferring the location and type of measurements that are most informative for the model parameters. The PIMLI approach merges the strengths of the generalized sensitivity analysis (GSA) method [Spear and Hornberger, 1980], the Bayesian recursive estimation (BARE) algorithm [Thiemann et al., 2001], and the Metropolis algorithm [Metropolis et al., 1953]. Three case studies with increasing complexity are used to illustrate the usefulness and applicability of the PIMLI methodology. The first two case studies consider the identification of soil hydraulic parameters using soil water retention data and a transient multistep outflow experiment (MSO), whereas the third study involves the calibration of a conceptual rainfall-runoff model

Design Issues for Generalized Linear Models: A Review

Generalized linear models (GLMs) have been used quite effectively in the modeling of a mean response under nonstandard conditions, where discrete as well as continuous data distributions can be accommodated. The choice of design for a GLM is a very important task in the development and building of an adequate model. However, one major problem that handicaps the construction of a GLM design is its dependence on the unknown parameters of the fitted model. Several approaches have been proposed in the past 25 years to solve this problem. These approaches, however, have provided only partial solutions that apply in only some special cases, and the problem, in general, remains largely unresolved. The purpose of this article is to focus attention on the aforementioned dependence problem. We provide a survey of various existing techniques dealing with the dependence problem. This survey includes discussions concerning locally optimal designs, sequential designs, Bayesian designs and the quantile dispersion graph approach for comparing designs for GLMs.Comment: Published at http://dx.doi.org/10.1214/088342306000000105 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Suspended Load Path Tracking Control Using a Tilt-rotor UAV Based on Zonotopic State Estimation

This work addresses the problem of path tracking control of a suspended load using a tilt-rotor UAV. The main challenge in controlling this kind of system arises from the dynamic behavior imposed by the load, which is usually coupled to the UAV by means of a rope, adding unactuated degrees of freedom to the whole system. Furthermore, to perform the load transportation it is often needed the knowledge of the load position to accomplish the task. Since available sensors are commonly embedded in the mobile platform, information on the load position may not be directly available. To solve this problem in this work, initially, the kinematics of the multi-body mechanical system are formulated from the load's perspective, from which a detailed dynamic model is derived using the Euler-Lagrange approach, yielding a highly coupled, nonlinear state-space representation of the system, affine in the inputs, with the load's position and orientation directly represented by state variables. A zonotopic state estimator is proposed to solve the problem of estimating the load position and orientation, which is formulated based on sensors located at the aircraft, with different sampling times, and unknown-but-bounded measurement noise. To solve the path tracking problem, a discrete-time mixed $\mathcal{H}_2/\mathcal{H}_\infty$ controller with pole-placement constraints is designed with guaranteed time-response properties and robust to unmodeled dynamics, parametric uncertainties, and external disturbances. Results from numerical experiments, performed in a platform based on the Gazebo simulator and on a Computer Aided Design (CAD) model of the system, are presented to corroborate the performance of the zonotopic state estimator along with the designed controller