Parameter identification in a class of nonlinear systems

Two approaches are proposed for on line identification of parameters in a class of nonlinear discrete time systems. The system is modelled by state equations in which state and input variables enter nonlinearly in general polynomial form while unknown parameters and random disturbances enter linearly. States and inputs are observed with measurement errors represented by white Gaussian noise having known covariance. System disturbances are also white and Gaussian with finite but unknown covariance. One method of parameter estimation is based upon a least squares approach; the second is a stochastic approximation algorithm (SSA). In each case the algorithm is motivated by minimization of an appropriate error criterion and modified to eliminate bias errors. Under fairly mild conditions the estimate derived from the least squares algorithm (LSA) is shown to converge in probability to the correct parameter; the SAA yields an estimate which converges in mean square and with probability one. Examples are given illustrating convergence of a recursive form of the LSA. This recursive form still requires inversion of a matrix at each step. The SAA requires no matrix inversions but experience to date with the algorithm indicates that convergence is slow relative to that of the LSA.