Joel Andersson

CasADi -- A symbolic package for automatic differentiation and optimal control

We present CasADi, a free, open-source software tool for fast, yet efficient solution of nonlinear optimization problems in general and dynamic optimization problems in particular. To the developer of algorithms for numerical optimization and to the advanced user of such algorithms, it offers a level of abstraction which is notably lower, and hence more flexible, than that of algebraic modeling languages such as AMPL or GAMS, but higher than working with a conventional automatic differentiation (AD) tool.CasADi is best described as a minimalistic computer algebra system (CAS) implementing automatic differentiation in eight different flavors. Similar to algebraic modeling languages, it includes high-level interfaces to state-of-the-art numerical codes for nonlinear programming, quadratic programming and integration of differential-algebraic equations. CasADi is implemented in self-contained C++ code and contains full-featured front-ends to Python and Octave for rapid prototyping. In this paper, we present the AD framework of CasADi and benchmark the tool against AMPL for a set of nonlinear programming problems from the CUTEr test suite.

Dynamic optimization with CasADi

We demonstrate how CasADi, a recently developed, free, open-source, general purpose software tool for nonlinear optimization, can be used for dynamic optimization in a flexible, interactive and numerically efficient way. CasADi is best described as a minimalistic computer algebra system (CAS) implementing automatic differentiation (AD) in eight different flavors. Similar to algebraic modeling languages like AMPL or GAMS, it includes high-level interfaces to state-of-the-art numerical codes for nonlinear programming, quadratic programming and integration of differentialalgebraic equations. CasADi is implemented in self-contained C++ code and contains full-featured front-ends to Python and Octave for rapid prototyping. In this paper, we discuss CasADi from the perspective of the developer or advanced user of algorithms for dynamic optimization for the first time, leaving out details on the implementation of the tool. We demonstrate how the tool can be used to model highly complex dynamical systems directly or import existing models formulated in the algebraic modeling language AMPL or the physical modeling language Modelica. Given this symbolic representation of the process models, the resulting optimal control problem can be solved using a variety of methods, including transcription methods (collocation), methods with embedded integrators (multiple shooting) as well as indirect methods.

Efficient Robust Economic Nonlinear Model Predictive Control of an Industrial Batch Reactor

Abstract In this paper we present a systematic and efficient approach to deal with uncertainty in Nonlinear Model Predictive Control (NMPC). The main idea of the approach is to represent the NMPC setting as a real-time decision problem under uncertainty that is formulated as a multi-stage stochastic problem with recourse, based on a description of the uncertainty by a scenario tree. This formulation explicitly takes into account the fact that new information will be available in the future and thus reduces the conservativeness compared to open-loop worst-case approaches. We show that the proposed multistage NMPC formulation can deal with significant plant-model mismatch as it is usually encountered in the process industry and still satisfies tight constraints for the different values of the uncertain parameters, in contrast to standard NMPC. The use of an economic cost function leads to a superior performance compared to the standard tracking formulation. The potential of the approach is demonstrated for an industrial case study provided by BASF SE in the context of the European Project EMBOCON. The numerical solution of the resulting large optimization problems is implemented using the optimization framework CasADi.

A framework for nonlinear model-predictive control using object-oriented modeling with a case study in power plant start-up

In t his paper, nonlinear model predictive control (NMPC) is applied to the start-up of a combined-cycle power plant. An object-oriented first-principle model library expressed in the high-level language Modelica has been written for the plant and used to set up the simulation and optimization models. The NMPC optimization problems are both encoded, using a high-level notation, and solved in the open-source framework JModelica.org. The results demonstrate the effectiveness of the framework and its high-level description. It bridges the gap between an intuitive physical modeling format and state of the art numerical optimization algorithms. Promising closed-loop control results are shown for plant start-up when the NMPC model contains parametric errors and the simulation model, corresponding to the real plant, is subject to disturbances.

Distributed Multiple Shooting for Optimal Control of Large Interconnected Systems

Abstract Large interconnected systems consist of a multitude of subsystems with their own dynamics, but coupled with each other via input-output connections. Each subsystem is typically modelled by ordinary differential equations or differential-algebraic equations. Simulation and optimal control of such systems pose a challenge both with respect to CPU time and memory requirements. We address optimal control of such systems by applying “distributed multiple shooting”, a generalization of the direct multiple shooting method, which uses the decomposable structure of the system in order to obtain a highly parallel algorithm. The interconnections are allowed to be infeasible during the iterations but are driven to feasibility by a Newtontype optimization algorithm. We evaluate the performance of the distributed multiple shooting method on a large scale estimation problem.

Model-Based Dynamic Optimization with OpenModelica and CasADi

This paper demonstrates model-based dynamic optimization through the coupling of two open source tools: OpenModelica, which is a Modelica-based modeling and simulation platform, and CasADi, a framework for numerical optimization. The coupling uses a standardized XML format for exchange of differential-algebraic equations (DAE) models. OpenModelica supports export of models written in Modelica and the optimization language extension using this XML format, while CasADi supports import of models represented in this format. This allows users to define optimal control problems (OCP) using Modelica and optimization language specification, and solve the underlying model formulation using a range of optimization methods, including direct collocation and direct multiple shooting. The proposed solution has been tested on several industrially relevant optimal control problems, including a diesel-electric power train.

Adjoint-based distributed multiple shooting for large-scale systems

Abstract Distributed multiple shooting is a modification of the standard multiple shooting method which takes into account the structure of certain large-scale systems in order to obtain a better controller design flexibility and high parallelizability. The aim of this paper is to extend the framework where distributed multiple shooting can be deployed and to propose a new solution method based on adjoint-based sequential quadratic programming. A numerical experiment shows that this can lead to considerable savings in computational time for the sensitivity generation.