A model validation procedure based on Theil's inequality coefficient (TIC) is extended for handling nonstationary signal responses from actual systems and simulated models. Initially, the parent coefficient for TIC and its estimate, as defined by Theil, are presented. Thereafter, trend removal concepts are applied to the nonstationary data to produce weakly stationary difference data from which desired statistics may be computed by time series analysis. A least-squares curve fit is proposed for removing time-varying means. For certain applications more elaborate procedures involving nonlinear operations are required to handle time-varying variances. Some cases require segmenting the data and using trend removal procedures on a piece-wise basis. After time series analysis has been utilized on the resulting weakly stationary data, the removed trends are reinserted to yield a time-varying estimate of TIC. Examples are provided to illustrate two types of nonstationary cases.

The principle of dynamic programming is used to develop a digital computer software package for determining for each of several subsystems the optimal numerical integration algorithm from among given candidate techniques. As by-products, the corresponding optimal step sizes and error contributions of the subsystems are provided to the user. The developed program minimizes subsystem sum-squared errors subject to the constraint that the total time available for executing all subsystem integrations is specified in advance. After the basic concept of dynamic programming has been reviewed briefly, the details of the computer software package are presented. Numerical results are given for two examples, including a limited hybrid partitioning problem for a six-degree-of-freedom air-defense missile simulation. Finally, general considerations for hybrid partitioning are discussed, and extensions to the developed program are suggested.

Variational equations are derived as a preliminary step in determining efficient digital integration techniques for nonlinear dynamical systems. The variational approach is applied initially to linear time-invariant systems to introduce the basic concept and then to nonlinear time-varying systems. For systems containing both linear and nonlinear parts, a combination technique which uses the exact difference equation for the linear part is developed. Higher order variational equations are also derived and compared on a simple system. Numerical approximations for solving these variational equations are discussed and illustrated for a second-order mildly nonlinear example. A significant improvement in both accuracy and execution time is realized over results obtained by the conventional fourth-order Runge–Kutta method. Finally, the new approach is discussed from the viewpoint of computational experience and special limitations for practical applications.