Collaborators:    Richard J. Prazenica                UF-REEF, Virginia Tech
                                 Martin J. Brenner                     NASA Dryden Flight Research Center
                                 Andrew J. Kurdila                    Virginia Tech
                                 Rick Lind                                    University of Florida
                                 Patrick Reisenthel                    Nielsen Engineering and Research
                                 Walter A. Silva                          NASA Langley Research Center

Nonlinear system identification entails modeling the dynamics of a nonlinear system from measured input and output data.  Nonlinearity is prevalent in engineering systems such as aeroelastic systems, or systems comprised of the interaction of flexible structures with the surrounding air stream.  Nonlinear system identification has received a great deal of attention in the aeroelasticity community, and many techniques have been employed including describing functions, higher-order spectra, neural networks, NARMAX models, and many others. 

This line of research has focused on using Volterra theory to model nonlinear systems, particularly nonlinear aeroelastic systems.  Walt Silva first applied the Volterra theory to aeroelastic systems, directly measuring Volterra kernels by applying discrete impulse inputs to a computational fluid dynamics (CFD) model.  The Volterra theory states that, under fairly general conditions, the output y of a single-output nonlinear dynamic system can be expressed in terms of an infinite sum of integral operators

For causal, time-invariant, single-input/single-output dynamical systems, the first and second-order Volterra operators take the form

where u denotes the input and h₁ and h₂ represent the first and second-order Volterra kernels.  The higher-order operators take a similar form.  Therefore, each Volterra operator takes the form of a multi-dimensional convolution of the input and a kernel function.  The dynamics of a given system are completely characterized in terms of its Volterra kernels.  Indeed, for a linear system, the first-order kernel is equivalent to the classical impulse response function and the system output can be expressed as the linear convolution of the first-order kernel and the input.  The Volterra theory applies to a wide range of dynamical systems provided that the system output(s) can be expressed in terms of a set of analytic ordinary differential equations (ODEs) and that the system exhibits fading memory.  Practically speaking, Volterra theory is most readily applied to systems with polynomial type nonlinearities and cannot, strictly speaking, characterize discontinuous nonlinearities such as saturation or dead zones.  The fading memory requirement states that the influence of the present input must diminish to zero in a finite period of time.  For example, a clamped beam that is struck with an impact hammer exhibits fading memory (i.e., the impulse is “forgotten” once the beam stops vibrating) while a wing experiencing unstable flutter after a wind disturbance does not. 

In order to use Volterra theory to model a given dynamical system, the main task is to identify the kernels that characterize the system.  While these kernels can be derived from the system ODEs, these equations are rarely available and, in practice, the kernels must be extracted from measured input and output data that are frequently corrupted by noise.   Many approaches have been applied for the identification of Volterra kernels in both the time and frequency domains.  These include direct measurement techniques such as the application of discrete impulses to the system, statistical methods such as the cross-correlation technique, and indirect approaches such as expanding the kernels in terms of a set of basis functions.  The approach taken in this research has been to express each Volterra kernel in terms of a set of wavelet basis functions.  Wavelet bases, which are composed of functions that are localized in both the time and frequency domains, frequently afford efficient, reduced-order representations of functions or signals.  In employing wavelet representations of Volterra kernels, the kernel identification problem ultimately reduces to a linear least-squares problem which must be solved for the wavelet basis coefficients.  An example of wavelet-based kernel identification is given in Figure 1, which depicts first, second, and third-order kernels that have been identified from simulated input/output data from a nonlinear oscillator.  In this example, the analytical kernels can be derived from the system ODE and have been shown in the figure for comparison.  The third-order kernel is not easy to display because it is supported over a three-dimensional domain.  Therefore, only one two-dimensional slice of the kernel is shown in the figure.  

Figure 2 shows an example of Volterra kernel identification from data obtained during flight tests of the F/A-18 Active Aeroelastic Wing (AAW) at the NASA Dryden Flight Research Center.  The AAW is an F/A-18 whose wing has been modified to afford more flexibility in order to study the use of wing twist for roll control, a concept originally employed by the Wright brothers.  During the flight tests, multisine sweeps (i.e., input signals with varying frequency content) were commanded to the various AAW control surfaces and data from accelerometers mounted on the wings were recorded at different flight conditions (i.e, altitude and Mach number).  Figure 2 shows first and second-order kernels that were extracted from AAW data at a flight condition of Mach number 0.85, altitude 15,000 ft.  In this example, the input corresponded to the symmetric deflection of the ailerons on both wings due to the multisine input.  The output was taken as the measured response of an accelerometer mounted on the leading edge of the right wing, just inside the wing fold.  In the figure, a portion of the measured response data is compared to the response predicted by the first-order kernel alone (i.e., a linear model) and that predicted by a model consisting of both the first and second-order kernels (i.e., a nonlinear model).  It is clear that the inclusion of the second-order kernel in the model greatly improves the output prediction, implying that the system contains a significant nonlinearity.

Figure 1:  Multiwavelet-based identification of first, second, and third-order Volterra kernels of a simulated nonlinear oscillator.  The analytically-derived kernels are also shown for comparison.  Because the third-order kernel is supported on a three-dimensional domain, only one two-dimensional slice of it is displayed here. 

Figure 2:  First and second-order kernels that have been extracted from flight data from the F/A-18 Active Aeroelastic Wing.  A portion of the measured accelerometer output data is compared to the response predicted by the first-order kernel alone and both the first and second-order kernels.  The second-order kernel reproduces a significant portion of the response that is not captured by the first-order kernel.