An ESDIRK Software Package for DAE Systems
Systematic methods and tools for managing the complexity
Advances in Computational & Numerical Methods (T4-4P)
Keywords: Differential Algebraic Equations, Runge-Kutta Methods, Numerical Solution of DAEs
Efficient and reliable software for solving differential equations is important for a wide
range of process engineering disciplines including model development and process and
product design. Increasingly large models are being solved on powerful computers, and
the solvers are often nested in sophisticated optimization loops requiring many repeated
solutions of the underlying equations. In addition, these optimization loops require
derivative information of the solution to the equations. Thus, the differential equation
solver must be capable of computing not only the solution to the equations, but also
the sensitivities with respect to problem parameters, initial conditions or control inputs.
Efficient computation of sensitivities is essential to applications such as parameter estimation, dynamic optimization, nonlinear model predictive control and experimental
design. Apart from applications requiring sensitivity information, the differential equation
solvers are often required to integrate hybrid dynamic systems in which discrete
events occur causing discontinuities in the solution.
In this talk we will present a software package for dynamic simulation and sensitivity
analysis of differential-algebraic equation (DAE) systems. The package is based on the
family of ESDIRK (Explicit Singly Diagonally Implicit Runge-Kutta) methods [1]. The
one-step nature of these methods makes them particularly well suited for problems with
frequent discontinuities, as in discrete event systems and optimal control applications
using zero-order parameterization of inputs. Moreover, the strong stability properties
(A- and L-stable) of ESDIRK methods make them suitable for index-1 DAEs.
The ESDIRK solver package includes a range of methods of varying order, all equipped
with continuous extensions for generation of dense output and location of discrete events.
Discrete event problems are handled by detecting and locating zero-crossings of event
functions defining switches to other system states. Finally, the methods are equipped
with sensitivity analysis algorithms for both forward [2,3] and adjoint sensitivity analysis.
The sensitivity analysis algorithms have been applied to nonlinear model predictive
control applications [4] as well as in the construction of a very efficient extended Kalman
filter algorithm for state estimation in continuous-discrete stochastic systems [5,6].
[1] Alexander, R. (2003). Design and Implementation of DIRK Integrators for Stiff
Systems. Applied Numerical Mathematics, 46, 1–17.
[2] Kristensen, M. R.; Jørgensen, J. B.; Thomsen, P. G. and Jørgensen, S. B. (2004).
An ESDIRK Method with Sensitivity Analysis Capabilities. Computers and Chemical
Engineering, 28, 2695–2707.
[3] Kristensen, M. R.; Jørgensen, J. B.; Thomsen, P. G.; Michelsen, M. L. and Jørgensen,
S. B. (2005). Sensitivity Analysis in Index-1 Differential Algebraic Equations
by ESDIRK Methods. In 16th IFAC World Congress 2005, Prague, Czech
Republic.
[4] Kristensen, M. R.; Jørgensen, J. B.; Thomsen, P. G. and Jørgensen, S. B. (2004b).
Efficient Sensitivity Computation for Nonlinear Model Predictive Control. In F.
Allgöwer, editor, NOLCOS 2004, 6th IFAC-Symposium on Nonlinear Control Systems,
September 01-04, 2004, Stuttgart, Germany, pp. 723–728.
[5] Jørgensen, J. B.; Kristensen, M. R.; Thomsen, P. G. and Madsen, H. (2006a).
Efficient Numerical Implementation of the Continuous-Discrete Extended Kalman
Filter. Submitted to Computers and Chemical Engineering.
[6] Jørgensen, J. B.; Kristensen, M. R.; Thomsen, P. G. and Madsen, H. (2006b). New
Extended Kalman Filter Algorithms for Stochastic Differential Algebraic Equations.
In F. Allgöwer; R. Findeisen and L. T. Biegler, editors, International Workshop
on Assessment and Future Directions of Nonlinear Model Predictive Control.
Springer, New York.
Presented Tuesday 18, 13:30 to 15:00, in session Advances in Computational & Numerical Methods (T4-4P).
