Parameter Identification by a Generalised Secant Method

An efficient secant-method based algorithm for solving nonlinear least squares problem is presented which showed good stability and convergence properties even with relatively high number of unknowns and large system response deviations. The trial solutions were generated by a special transformation in each iteration step. The distance between observed and simulated system responses were defined for arbitrary two-dimensional curves. The definition can be extended to several dimensions also. The properties of the proposed method is demonstrated through a numerical example. References are made to practical problems for which successful applications were made. INTRODUCTION Modelling of systems on the basis of observed or specified system responses plays a key role in scientific research. A central problem of modelling is to fit a structured mathematical model to available data so that the values of the unknown model parameters are determined providing that the simulated data is as near to the available data as possible. A system response can generally be represented by a "curve" in a two (or three) dimensional space and can be simulated by a computer program. Parameter identification corresponds to minimize the distance between observed and simulated system responses. An efficient and stable numerical algorithm is given in this paper that is based on a generalised version of the secant method. A practical definition for the distance between the system responses is also presented. A numerical example with five unknown parameters (epicyclois) is shown for demonstration purpose. Parameter identification problems can generally be formulated as non-linear least squares problems so that some unknown model parameters are determined providing minimal deviation between observed and simulated system responses. Solving non-linear least squares problems is often a difficult task especially if the number of unknowns are high. The suggested algorithm is originated from the solution given by Wolfe (1959) for which good convergence characteristics were reported in case of test problems. However, counterexamples (Polak, 1974) showed that the method may not converge. It was shown by Gragg and Stewart (1976) and by Martinez (1979) that the recursive solution of the system of linear equations is unstable due to accumulated roundoff errors. Stabilisation algorithms were given for linearly dependent search directions by Gragg and Stewart (1976), by Burdakov (1986), and by Felgenhauer (1991). Stabilisation by singular value decomposition was suggested by Popper (1985). The suggested method in this paper eliminates unstability problems by special selection of the search directions. It showed good convergence characteristics in case of problems with large number of unknown parameters (Berzi and Popper, 1992). A selected model only serves as a framework for the identification of a more or less realistic model. The unknown model parameters are adjusted providing best possible agreement between observed and simulated system responses. However, observed data are always subjected to noise, and models are always inaccurate due to incomplete knowledge and idealised description of the phenomena, providing that the solution is not unique. Incorrect modelling causes many difficulties while solving parameter identification problems. A typical structural error of the model presents if some parameters don’t have significant influence on the selected system response. Another common source of difficulties is the redundancy among parameters. Multidimensional parameter perturbation sensitivity analysis is presented for the detection of structural model errors. The results of analysis were showed for a problem with nonlinear dissipative spring-mass model for percussive rock drilling (Berzi et al., 1996) and for a dynamic pile-soil interactive system (Berzi and Imre, 1993 and Berzi, 1996). NOMENCLATURE x, y, (x), (y) : system response coordinates (normalised) r, p, e : residual-, parameterand i-th unit vector