Optimal parameter estimation of dynamical systems using direct transcription methods

Abstract

Parameter estimation of dynamical systems governed by ordinary differential equations is formulated as a discrete nonlinear programming problem. The dynamical constraints are transcribed as a set of equality constraints that are driven to zero by a sparse sequential quadratic programming algorithm. Five different transcription methods are examined: Heun's method, Hermite–Simpson, 5th degree Hermite–Legendre–Gauss–Lobatto, pseudospectral, and a 5th-order Legendre–Gauss–Lobatto integration method. Each method transcribes the differential equations in a different way and with different orders of accuracy. The parameter estimation problem is formulated by minimizing a weighted least squares cost function consisting of the sum of squares of the difference between measured state values and the approximate state values from the different transcription methods. The parameter estimation algorithm is applied to four different problems from biochemistry, physics, robotics, and aerospace to demonstrate some of its features and performance differences. For the same number of optimization parameters, the 5th-degree Hermite–Legendre–Gauss–Lobatto method, on average, gives the best combination of speed and accuracy for the problems studied.