Ekaterina Kostina

Numerical methods for optimal control problems in design of robust optimal experiments for nonlinear dynamic processes

Optimization of experiments for nonlinear dynamic processes to maximize the reliability of parameter estimates subject to cost and other inequality-constraints leads to very complex optimal control problems. First, the objective function already depends on a generalized inverse of the Jacobian of the underlying parameter estimation problem. Second, optimization results depend on the assumed parameter values which are only known to lie in a confidence region. Hence robust optimal experiments are required. New efficient methods and numerical results are presented. E-mail: stefan@koerkel.de

An approximation technique for robust nonlinear optimization

Nonlinear equality and inequality constrained optimization problems with uncertain parameters can be addressed by a robust worst-case formulation that is, however, difficult to treat computationally. In this paper we propose and investigate an approximate robust formulation that employs a linearization of the uncertainty set. In case of any norm bounded parameter uncertainty, this formulation leads to penalty terms employing the respective dual norm of first order derivatives of the constraints. The main advance of the paper is to present two sparsity preserving ways for efficient computation of these derivatives in the case of large scale problems, one similar to the forward mode, the other similar to the reverse mode of automatic differentiation. We show how to generalize the techniques to optimal control problems, and discuss how even infinite dimensional uncertainties can be treated efficiently. Finally, we present optimization results for an example from process engineering, a batch distillation.

On the Role of Natural Level Functions to Achieve Global Convergence for Damped Newton Methods

The paper discusses a new view on globalization techniques for Newton’s method. In particular, strategies based on “natural level functions” are considered and their properties are investigated. A “restrictive mono-tonicity test” is introduced and theoretically motivated. Numerical results for a highly nonlinear optimal control problem from aerospace engineering and a parameter estimation for a chemical process are presented.

An adjoint-based SQP algorithm with quasi-Newton Jacobian updates for inequality constrained optimization

We present a sequential quadratic programming (SQP) type algorithm, based on quasi-Newton approximations of Hessian and Jacobian matrices, which is suitable for the solution of general nonlinear programming problems involving equality and inequality constraints. In contrast to most existing SQP methods, no evaluation of the exact constraint Jacobian matrix needs to be performed. Instead, in each SQP iteration only one evaluation of the constraint residuals and two evaluations of the gradient of the Lagrangian function are necessary, the latter of which can efficiently be performed by the reverse mode of automatic differentiation. Factorizations of the Hessian and of the constraint Jacobian are approximated by the recently proposed STR1 update procedure. Inequality constraints are treated by solving within each SQP iteration a quadratic program (QP), the dimension of which equals the number of degrees of freedom. A recently proposed gradient modification in these QPs takes account of Jacobian inexactness in the active set determination. Superlinear convergence of the procedure is shown under mild conditions. The convergence behaviour of the algorithm is analysed using several problems from the Hock–Schittkowski test library. Furthermore, we present numerical results for an optimization problem based on a small periodic adsorption process, where the Jacobian of the equality constraints is dense.

Robust optimal feedback for terminal linear-quadratic control problems under disturbances

We consider a linear dynamic system in the presence of an unknown but bounded perturbation and study how to control the system in order to get into a prescribed neighborhood of a zero at a given final moment. The quality of a control is estimated by the quadratic functional. We define optimal guaranteed program controls as controls that are allowed to be corrected at one intermediate time moment. We show that an infinite dimensional problem of constructing such controls is equivalent to a special bilevel problem of mathematical programming which can be solved explicitely. An easy implementable algorithm for solving the bilevel optimization problem is derived. Based on this algorithm we propose an algorithm of constructing a guaranteed feedback control with one correction moment. We describe the rules of computing feedback which can be implemented in real time mode. The results of illustrative tests are given.

Robustness Aspects in Parameter Estimation, Optimal Design of Experiments and Optimal Control

Estimating model parameters from experimental data is crucial to reliably simulate dynamic processes. In practical applications, however, it often appears that the data contains outliers. Thus, a reliable parameter estimation procedure is necessary that delivers parameter estimates insensitive (robust) to errors in measurements.

Numerical Methods for Efficient and Fast Nonlinear Model Predictive Control

The paper reports on recent progress in the real-time computation of constrained closed-loop optimal control, in particular the special case of nonlinear model predictive control, of large di.erential algebraic equations (DAE) systems arising e.g. from a MoL discretization of instationary PDE. Through a combination of a direct multiple shooting approach and an initial value embedding, a so-called “real-time iteration” approach has been developed in the last few years. One of the basic features is that in each iteration of the optimization process, new process data are being used. Through precomputation – as far as possible – of Hessian, gradients and QP factorizations the response time to perturbations of states and system parameters is minimized. We present and discuss new real-time algorithms for fast feasibility and optimality improvement that do not need to evaluate Jacobians online.

Numerical Methods for Nonlinear Experimental Design

Nonlinear experimental design leads to a challenging class of optimization problems which occur in the procedure of the validation of process models. This paper discusses the formulation of such problems for a general class of underlying process models, presents numerical methods for the solution and shows their successful application to industrial processes.

Robust Parameter Estimation in Dynamic Systems

In this paper we present a practical method for robust parameter estimation in dynamic systems. In our study we follow the very successful approach for solving optimization problems in dynamic systems, namely the boundary value problem (BVP) approach. The suggested method combines multiple shooting for parameterizing dynamics, a flexible realization of the BVP principle, with a fast Gauss-Newton algorithm for solving the resulting constrained l1 problem. We give an overview of the theoretical background as well as the details of a numerical implementation. We discuss why the Gauss-Newton algorithm, which is known to perform well mainly on well-conditioned problems, is appropriate for parameter estimation problems, while quasi-Newton methods have only limited use for parameter estimation. The method is implemented on the basis of the direct multiple shooting method as implemented in PARFIT, thus inheriting all basic properties of PARFIT such as numerical stability, reliability and efficiency. The new code has been successfully applied to real-life parameter estimation problems in enzyme and chemical kinetics.

Covariance Matrices for Parameter Estimates of Constrained Parameter Estimation Problems

In this paper we show how, based on the conjugate gradient method, to compute the covariance matrix of parameter estimates and confidence intervals for constrained parameter estimation problems as well as their derivatives.