OVERVIEW
DESCRIPTION: Nielsen Engineering & Research (NEAR)'s Kernel Identification System is made of two integrated software modules: a prediction module and an extraction module. The software is based on the VolterraWiener theory of nonlinear systems and applies to nonlinear timeinvariant systems with finite memory. The theory [13] asserts that the response of such systems to arbitrary timedependent inputs can be constructed by integrating a nonlinear functional which is a fundamental characteristic of the system being modeled. NEAR's Kernel Identification System currently implements a subset of VolterraWiener theory, namely the twoterm truncation of the Volterra series for a singleinput, multipleoutput system. The system in question is modeled as a black box, i.e.:
X(t) (input) 
> 

> 
Y(t) (output) 
In prediction mode, the code calculates the first two terms in the series, given the time history of the system's input and the knowledge of its first and secondorder Volterra kernels. In extraction mode, the program extracts first and secondorder Volterra kernels from time domain data.
INPUTS: In prediction mode, the user supplies the kernels of the system, as well as the schedule of the input X(t). In extraction mode, the user must supply input/output time history pairs. Most inputs are specifiable either in tabular form or in functional (subroutine) form.
OUTPUTS: In prediction mode, the program computes the system's output Y(t) as a twoterm truncation of that system's Volterra series representation. In extraction mode, the program generates estimates of the first and/or secondorder Volterra kernels compatible with the supplied input/output data.
APPLICATIONS: This software technology can be used to develop reducedorder models of nonlinear systems based on experimental or computational data. For example, the development of such models is critical for efficient integration of fluids, structures and control systems in aeroservoelasticity applications (see Figure).
AVAILABILITY / PRODUCT INFORMATION: The Kernel Identification System software is available in CDROM format. Licensing the software is subject to an agreement between NEAR (licensor) and the enduser (licensee). The software has been successfully tested on the following operating systems: Linux i386, Solaris 2.5. Disk space needed for installation is less than 10 Mbytes. Prediction module performance: 22ms per output on a 266 MHz PC for a single onedimensional kernel. The program is available in executable form, plus shared objects in source code form. Shared objects are analogous to dynamically linked libraries; they give the user full control over quadrature and interpolation operations, as well as parameterization and parameter space partitioning. The software distribution includes a 140 page Software User's Manual containing a detailed description of each interface to usersupplied shared objects for increased functionality.
NOTE: The Kernel Identification System software is a derivative product of NEAR's Indicial Prediction System (IPS). Therefore, many of the code features available in IPS are also part of the Kernel Identification System software. In particular, the software includes IPS's parameterization engine, making the program capable of addressing the characterization of timevarying systems. For more information see NEAR IPS below.
REFERENCES:
[1] Volterra, V., Theory of Functionals and of Integral and IntegroDifferential Equations, Dover Publications, Inc., New York, 1959.
[2] Wiener, N. Response of a NonLinear Device to Noise, Report No. 129, Radiation Laboratory, M.I.T., Cambridge, MA, Apr. 1942.
[3] Schetzen, M., The Volterra and Wiener Theories of Nonlinear Systems, Wiley & Sons, New York, 1980.
Other Mathematical Modeling Tools
Multidimensional Response Surface Package: NEAR RS
Nonlinear Indicial Prediction System: NEAR IPS