Morten Rode Kristensen

An ESDIRK method with sensitivity analysis capabilities

Abstract A new algorithm for numerical sensitivity analysis of ordinary differential equations (ODEs) is presented. The underlying ODE solver belongs to the Runge–Kutta family. The algorithm calculates sensitivities with respect to problem parameters and initial conditions, exploiting the special structure of the sensitivity equations. A key feature is the reuse of information already computed for the state integration, hereby minimizing the extra effort required for sensitivity integration. Through case studies the new algorithm is compared to an extrapolation method and to the more established BDF based approaches. Several advantages of the new approach are demonstrated, especially when frequent discontinuities are present, which renders the new algorithm particularly suitable for dynamic optimization purposes.

A Computationally Efficient and Robust Implementation of the Continuous-Discrete Extended Kalman Filter

We present a novel numerically robust and computationally efficient extended Kalman filter for state estimation in nonlinear continuous-discrete stochastic systems. The resulting differential equations for the mean-covariance evolution of the nonlinear stochastic continuous-discrete time systems are solved efficiently using an ESDIRK integrator with sensitivity analysis capabilities. This ESDIRK integrator for the mean- covariance evolution is implemented as part of an extended Kalman filter and tested on a PDE system. For moderate to large sized systems, the ESDIRK based extended Kalman filter for nonlinear stochastic continuous-discrete time systems is more than two orders of magnitude faster than a conventional implementation. This is of significance in nonlinear model predictive control applications, statistical process monitoring as well as grey-box modelling of systems described by stochastic differential equations.

SENSITIVITY ANALYSIS IN INDEX-1 DIFFERENTIAL ALGEBRAIC EQUATIONS BY ESDIRK METHODS

Abstract Dynamic optimization by multiple shooting requires integration and sensitivity calculation. A new semi-implicit Runge-Kutta algorithm for numerical sensitivity calculation of index-1 DAE systems is presented. The algorithm calculates sensitivities with respect to problem parameters and initial conditions, exploiting the special structure of the sensitivity equations. The algorithm is a one-step method which makes it especially efficient compared to multiple-step methods when frequent discontinuities are present. These advantages render the new algorithm particularly suitable for dynamic optimization and nonlinear model predictive control. The algorithm is tested on the Dow Chemicals benchmark problem.

Efficient sensitivity computation for nonlinear model predictive control

Abstract Dynamic optimization by multiple shooting requires integration and sensitivity calculation of the model ODEs as well as the cost function integral. A new semi-implicit Runge-Kutta algorithm for numerical sensitivity calculation of stiff systems of ordinary differential equations (ODEs) is presented. The algorithm calculates sensitivities with respect to problem parameters and initial conditions, exploiting the special structure of the sensitivity equations. The new algorithm is compared to an extrapolation method and to the more established BDF based approaches. Several advantages of the new approach are demonstrated, especially when frequent discontinuities are present. These advantages renders the new algorithm particularly suitable for dynamic optimization and nonlinear model predictive control.

New Extended Kalman Filter Algorithms for Stochastic Differential Algebraic Equations

We introduce stochastic differential algebraic equations for physical modelling of equilibrium based process systems and present a continuous-discrete paradigm for filtering and prediction in such systems. This paradigm is ideally suited for state estimation in nonlinear predictive control as it allows systematic decomposition of the model into predictable and non-predictable dynamics. Rigorous filtering and prediction of the continuous-discrete stochastic differential algebraic system requires solution of Kolmogorov’s forward equation. For non-trivial models, this is mathematically intractable. Instead, a suboptimal approximation for the filtering and prediction problem is presented. This approximation is a modified extended Kaiman filter for continuous-discrete systems. The modified extended Kaiman filter for continuous-discrete differential algebraic systems is implemented numerically efficient by application of an ESDIRK algorithm for simultaneous integration of the mean-covariance pair in the extended Kaiman filter [1, 2]. The proposed method requires approximately two orders of magnitude less floating point operations than implementations using standard software. Numerical robustness maintaining symmetry and positive semi-definiteness of the involved covariance matrices is assured by propagation of the matrix square root of these covariances rather than the covariance matrices themselves.