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. (2004). 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. (2006). 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.