Numerical methods for parameter estimation and optimal experiment design in chemical reaction systems

Abstract

The estimation of parameters in models of chemical kinetics, e.g., of coal pyrolysis, needs efficient and reliable numerical techniques. This paper presents a method based on a multiple shooting algorithm and a generalized Gauss-Newton method, which is designed for highly nonlinear problems and which is suitable for stiff, as well as unstable, systems that frequently occur in chemistry and chemical engineering. In order to reduce the costs of experiments, optimal experiment design methods are helpful. A new algorithm is developed that minimizes the maximum length of the joint confidence intervals of the parameters with respect to frequency factors of the measurements in order to improve the statistical reliability of the parameters. The derivatives of the covariance matrix of the parameters with respect to the frequency factors are determined semianalytically and inexpensively. Thus, the optimal frequency factors can be calculated by a special version of the sequential quadratic programming method. Results for the estimation of parameters in a kinetic model of coal pyrolysis are reported. Parameter Identification in Chemical Reaction Systems 1.1. Parameter Estimation Problem. The modeling of many dynamical processes, e.g., descriptions of chemical kinetics of coal pyrolysis, leads to systems of nonlinear ordinary differential equations (ODE) with unknown parameters p: Such systems may be stiff and/or unstable. They may include discontinuities in the right-hand side f of the equation and/or jumps of the states y(t) at points where so-called switching functions Q(t,y,p) change sign. In addition, there are frequently equality or inequality constraints on the states y(t) and the parameters p Y(t) = f(t,y,p,sign(8i(t,y,p))) (1) rl(u(tl),***,~(tk),P) = 0 (2) r2(Y(t*),...,Y(tk),P) 2 0 to be satisfied, which consist of initial, boundary, or interior-point conditions, e.g., conditions that characterize periodicity or positivity restrictions of the range of parameters p. The unknown parameters p and state variables y(t) have to be estimated from given observations 11i = gi@(t1),***,jj(tm),P) + ti (3)