Numerical and Symbolic Algorithms for Dynamic Optimization

Researchers: Fredrik Magnusson, Johan Åkesson (Modelon), Christian Andersson (Numerical Analysis)


The target of this project is the development of algorithms for numerical solution of large-scale, DAE-constrained, non-convex dynamic optimization problems. The project targets both optimal control and parameter estimation as well as other forms of dynamic optimization. Applications include minimization of material and energy consumption during set-point transitions in power plants and chemical processes, minimizing lap times for vehicle systems, trajectory optimization in robotics and identifying unknown parameter values of models using measurement data.

The first step of the project has been to implement state-of-the-art algorithms based on collocation methods and integrate them with the high-level, object-oriented modelling language Modelica and its extension Optimica. This allows basic users to conveniently formulate and solve problems of moderate difficulty without worrying about the details of the solution algorithms, while still allowing Model Diagramadvanced users to tailor the algorithm as needed for complex problems. This implementation is a part of the open-source project. Two important third-party tools used within the project is CasADi, for automatic differentiation, and IPOPT, for solution of non-linear programs.

The current research direction is to symbolically process the differential-algebraic equation system describing the dynamics to create a block triangular structure of the incidence matrix by employing graph algorithms, as illustrated above. This structure facilitates analytic solution of many of the algebraic equations, removing the need to expose these to the numerical optimization algorithm. This drastically reduces the number of optimization variables, and may also result in a better conditioned problem, thus potentially improving both convergence speed and robustness of iterative solvers.

The applicability of the algorithms are explored in other application-oriented research projects, in collaboration with other research groups from both academia and industry.


Roel De Coninck, Fredrik Magnusson, Johan Åkesson, Lieve Helsen: "Toolbox for development and validation of grey-box building models for forecasting and control". Journal of Building Performance Simulation, Taylor & Francis, 9:3, pp. 288–303, 2016.

Fredrik Magnusson, Kyle Palmer, Lu Han, George Bollas: "Dynamic Parametric Sensitivity Optimization Using Simultaneous Discretization in". In 2015 International Conference on Complex Systems Engineering, 2015.

Magdalena Axelsson, Fredrik Magnusson, Toivo Henningsson: "A Framework for Nonlinear Model Predictive Control in". In 11th International Modelica Conference, 2015.

Anders Holmqvist, Christian Andersson, Fredrik Magnusson, Johan Åkesson: "Methods and Tools for Robust Optimal Control of Batch Chromatographic Separation Processes". Processes, 3:3, pp. 568–606, 2015.

Anders Holmqvist, Fredrik Magnusson, Bernt Nilsson: "Dynamic Multi-Objective Optimization of Batch Chromatographic Separation Processes". In 12th International Symposium on Process Systems Engineering and 25th European Symposium on Computer Aided Process Engineering, 2015.

Fredrik Magnusson, Johan Åkesson: "Dynamic Optimization in". Processes, 3:2, pp. 471–496, 2015.

Karl Berntorp, Fredrik Magnusson: "Hierarchical Predictive Control for Ground-Vehicle Maneuvering". In American Control Conference, 2015, 2015.

Anders Holmqvist, Fredrik Magnusson, Stig Stenström: "Scale-up Analysis of Continuous Cross-flow Atomic Layer Deposition Reactor Designs". Chemical Engineering Science, 117, pp. 301–317, 2014.

Anders Holmqvist, Tobias Törndahl, Fredrik Magnusson, Uwe Zimmermann, Stig Stenström: "Dynamic parameter estimation of atomic layer deposition kinetics applied to in situ quartz crystal microbalance diagnostics". Chemical Engineering Science, 111, pp. 15–33, 2014.

Roel De Coninck, Fredrik Magnusson, Johan Åkesson, Lieve Helsen: "Grey-Box Building Models for Model Order Reduction and Control". In 10th International Modelica Conference, 2014.

Fredrik Magnusson, Karl Berntorp, Björn Olofsson, Johan Åkesson: "Symbolic Transformations of Dynamic Optimization Problems". In 10th International Modelica Conference, 2014.

Per-Ola Larsson, Francesco Casella, Fredrik Magnusson, Joel Andersson, Moritz Diehl, Johan Åkesson: "A Framework for Nonlinear Model-Predictive Control Using Object-Oriented Modeling with a Case Study in Power Plant Start-Up". In IEEE Multi-conference on Systems and Control, 2013, 2013.

Fredrik Magnusson, Johan Åkesson: "Collocation Methods for Optimization in a Modelica Environment". In 9th International Modelica Conference, 2012.