THE problem of obtaining the simultaneous weighted least squares (WLS) estimate of the parameters p and states r(p,t) of a given general first- or second-order nonlinear dynamic system is addressed, respectively, via the use of a quadratic (QSS) or cubic spline series (CSS) time representation of r(p,t). The measured states or functions of the states are assumed to be the true values corrupted by additive zero-mean uncorrelated random noise. The CSS is described in Eq. (6) below.! Unknown state functional dependence in the dynamics, e.g., aerodynamic force modeling of an unknown re-entry vehicle, is effectively represented in the time domain with a vector B(t) of unknown time functions. The B(t) are also parameterized via the spline time series.! This has the ability to accurately represent both complex state functional dependence and well behaved solutions of differential equations with a relatively modest number of terms and, hence, unknown coefficients. The initial conditions and unknown system parameters are WLS estimated from a given set of discrete noisy measurements on the system output at observation times tj(/=!,/) whence r(p,t) and its time derivatives at any t, follow immediately as a linear function of p. A variety of analytical spline algorithms are described that include iterative batch processing for maximum accuracy and sequential recursive processing for real time operation. Sensitivity differential equations, as in standard maximum likelihood parameter estimation, and error covariance propagation equations, as with the extended Kalman filter, are eliminated. Precision, generality, flexibility, and computational efficiency are achieved. These are illustrated in the full paper2'7 with a numerical example based upon the tracking of Van der Pol's equation. The solution is attained via hand calculation for comparison with familiar methods. Additionally, analytical solution expressions are obtained for a practical naval combat weapon systems problem.