Johannes P. Schlöder

Real-time optimization and nonlinear model predictive control of processes governed by differential-algebraic equations

Abstract Optimization problems in chemical engineering often involve complex systems of nonlinear DAE as the model equations. The direct multiple shooting method has been known for a while as a fast off-line method for optimization problems in ODE and later in DAE. Some factors crucial for its fast performance are briefly reviewed. The direct multiple shooting approach has been successfully adapted to the specific requirements of real-time optimization. Special strategies have been developed to effectively minimize the on-line computational effort, in which the progress of the optimization iterations is nested with the progress of the process. They use precalculated information as far as possible (e.g. Hessians, gradients and QP presolves for iterated reference trajectories) to minimize response time in case of perturbations. In typical real-time problems they have proven much faster than fast off-line strategies. Compared with an optimal feedback control computable upper bounds for the loss of optimality can be established that are small in practice. Numerical results for the Nonlinear Model Predictive Control (NMPC) of a high-purity distillation column subject to parameter disturbances are presented.

An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects

Optimal design and operation of complex chemical processes often require the solution of intricate dynamic optimization problems. A tailored simultaneous solution strategy based on multiple shooting and reduced SQP is presented. This reduced-space boundary value problem (BVP) approach allows an efficient and robust solution of multistage optimal control and design optimization problems for large, sparse DAE process models of index one. The current paper describes the theoretical aspects of the method. Utilizing the natural decomposition of the states into differential and algebraic variables, the structured NLP problem which results from the multiple shooting discretization of the optimization BVP is projected onto the reduced space of differential variables and control parameters. It is shown that this projection can be obtained very efficiently through direct computation of the reduced linearized constraint system via directional sensitivities. Like the original full-space BVP approach, the reduced-space formulation lends itself well to parallel computation. An implementation of the new strategy is provided by the modular optimal control package MUSCOD-II. Software aspects and applications are discussed in a second paper (Part II Software Aspects and Applications, 2002).

A Real-Time Iteration Scheme for Nonlinear Optimization in Optimal Feedback Control

An efficient Newton-type scheme for the approximate on-line solution of optimization problems as they occur in optimal feedback control is presented. The scheme allows a fast reaction to disturbances by delivering approximations of the exact optimal feedback control which are iteratively refined during the runtime of the controlled process. The contractivity of this real-time iteration scheme is proven, and a bound on the loss of optimality---compared with the theoretical optimal solution---is given. The robustness and excellent real-time performance of the method is demonstrated in a numerical experiment, the control of an unstable system, namely, an airborne kite that shall fly loops.

Introduction to Model Based Optimization of Chemical Processes on Moving Horizons

Dynamic optimization problems are typically quite challenging for large-scale applications. Even more challenging are on-line applications with demanding real-time constraints. This contribution provides a concise introduction into problem formulation and standard numerical techniques commonly found in the context of moving horizon optimization using nonlinear differential algebraic process models.

An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part II: Software aspects and applications

Abstract As model based optimization techniques play a more and more important role in the chemical process industries, there is a great demand for ever more efficient and reliable process optimization software. In the first part of this paper, the theoretical aspects of a tailored multiple shooting based solution strategy for dynamic process optimization have been presented (Leineweber, Bauer, Bock & Schloder, 2002. An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization-part I: theoretical aspects). The current second part describes software aspects of the specific implementation muscod-ii and provides numerical results for several application examples. muscod-ii has been coupled with the dynamic process modeling software gPROMS via the standard equation set object (ESO) interface of CAPE-OPEN. Thereby, an advanced dynamic optimization platform for integrated batch processes has been created, where each process stage is separately modeled in gPROMS, and the multistage dynamic optimization problem is assembled and solved with MUSCOD-II. The code has also been parallelized based on the portable MPI standard. It is shown that the use of directional sensitivities becomes very important for larger problems with many algebraic variables, leading to drastically reduced computing times compared with strategies with complete constraint linearization. In addition, gPROMS ESO models are compared with classical Fortran models in terms of computational performance, and it is found that only a moderate loss of performance occurs if so-called in-process ESOs are employed. Finally, it is demonstrated that a significant speed-up can be obtained through parallel function and gradient evaluations.

Numerical methods for optimum experimental design in DAE systems

Subject of this paper is the design of optimal experiments for chemical processes described by nonlinear DAE models. The optimization aims at maximizing the statistical quality of a parameter estimate from experimental data. This leads to optimal control problems with an unusual and intricate objective function which depends implicitly on first derivatives of the solution of the underlying DAE. We treat these problems by the direct approach and solve them using a structured SQP method. The required first and second derivatives of the solution of the DAE are computed very efficiently by a special coupling of the techniques of internal numerical differentiation and automatic differentiation. The performance of our approach is demonstrated for an application to chemical reaction kinetics.

A real-time algorithm for moving horizon state and parameter estimation

Abstract A moving horizon estimation (MHE) approach to simultaneously estimate states and parameters is revisited. Two different noise models are considered, one with measurement noise and one with additional state noise. The contribution of this article is twofold. First, we transfer the real-time iteration approach, developed in Diehl et al. (2002) for nonlinear model predictive control, to the MHE approach to render it real-time feasible. The scheme reduces the computational burden to one iteration per measurement sample and separates each iteration into a preparation and an estimation phase. This drastically reduces the time between measurements and computed estimates. Secondly, we derive a numerically efficient arrival cost update scheme based on one single QR-factorization. The MHE algorithm is demonstrated on two chemical engineering problems, a thermally coupled distillation column and the Tennessee Eastman benchmark problem, and compared against an Extended Kalman Filter. The CPU times demonstrate the real-time applicability of the suggested approach.

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

Optimal Experimental Design for Parameter Estimation of a Cell Signaling Model

Differential equation models that describe the dynamic changes of biochemical signaling states are important tools to understand cellular behavior. An essential task in building such representations is to infer the affinities, rate constants, and other parameters of a model from actual measurement data. However, intuitive measurement protocols often fail to generate data that restrict the range of possible parameter values. Here we utilized a numerical method to iteratively design optimal live-cell fluorescence microscopy experiments in order to reveal pharmacological and kinetic parameters of a phosphatidylinositol 3,4,5-trisphosphate (PIP3) second messenger signaling process that is deregulated in many tumors. The experimental approach included the activation of endogenous phosphoinositide 3-kinase (PI3K) by chemically induced recruitment of a regulatory peptide, reversible inhibition of PI3K using a kinase inhibitor, and monitoring of the PI3K-mediated production of PIP3 lipids using the pleckstrin homology (PH) domain of Akt. We found that an intuitively planned and established experimental protocol did not yield data from which relevant parameters could be inferred. Starting from a set of poorly defined model parameters derived from the intuitively planned experiment, we calculated concentration-time profiles for both the inducing and the inhibitory compound that would minimize the predicted uncertainty of parameter estimates. Two cycles of optimization and experimentation were sufficient to narrowly confine the model parameters, with the mean variance of estimates dropping more than sixty-fold. Thus, optimal experimental design proved to be a powerful strategy to minimize the number of experiments needed to infer biological parameters from a cell signaling assay.

A Direct Multiple Shooting Method for Real-Time Optimization of Nonlinear DAE Processes

The direct multiple shooting method has long been known as a fast off-line optimization method in ODE and DAE (e.g. [B81,BP84,P81]).