Volterra
Series and Volterra Kernel
Nonlinear Volterra theory was developed in
the 1880s by Vito Volterra. The
theory quickly received a great deal of attention in the field of electrical engineering,
and then later in the biological field, as a powerful approach to the modeling
of nonlinear system behavior.
Volterra theory is a generalization of the linear convolution integral
approach often applied to linear, time-invariant systems. The theory states that any
time-invariant, nonlinear system can be modeled as an infinite sum of
multidimensional convolution integrals of increasing order. This is represented
symbolically by the series of integrals,
which is known as the Volterra series.
Here, u(t) represents the dynamic system input while y(t)
represents the system response.
Volterra theory is based on dynamic data, and as such the average values
of all input and response data sets are removed. Each of the convolution integrals contains a kernel, either
linear (h1) or nonlinear (h2,..,hn),
which represents the behavior of the system. Knowledge of these kernels allows the prediction of a
system¡¯s response to any arbitrary input, and as such is critical to nonlinear
Volterra modeling. The first term
of the series represents the linear convolution integral. The first order term is considered to represent the
mean of the system response. The term weakly nonlinear merely implies that a system is well represented by the first two or three
terms of a Volterra series. All higher-order terms in this
situation are seen to quickly tend toward
zero, and are
therefore negligible in the system representation.
While Volterra theory has a strong foundation
in both the biological and electrical engineering fields, it has received
little attention in the field of aerodynamics until recently. Linear response models have often been
assumed sufficient for representation of nonlinear aerodynamic systems when
excited by small perturbations.
This assumption derives from the fact that highly nonlinear phenomena
have a negligible impact on the net effect of various responses under
conditions such as small perturbation excitation. In addition, the lack of attention is due in large part to
the inherent difficulty of identifying Volterra kernels. Time is discretized with a set of time
steps of equivalent size. Discrete
time increments are indexed from 0 (time 0) to n (time t), and the evaluation
of y at time n is denoted by y[n]. The convolution in discrete time is
where N is the total time record of interest.
Volterra Kernels
Volterra kernels are the backbone of any
Volterra series. Knowledge of a
system¡¯s behavior is contained within these kernels, and given any arbitrary
input the Volterra series can predict the response of the system. Volterra kernels, both linear and
nonlinear, are input dependent. As
an example of this consider the case where the response of a linear system to
an arbitrary input is desired.
Here, the unit impulse response of the system to that type of input must
first be defined. The first order
kernel, h1, represents the linear unit impulse response of the
system. This term is comparable to
the basic frequency response function (FRF) of a linear system, transformed
into the time domain. However, the kernel h1 gives a more accurate
portrayal of a system¡¯s linear response than does the FRF. This is because h1 exists
with the knowledge of higher-order, nonlinear terms while the FRF assumes a
completely linear response. The
second order kernel, h2, is a two-dimensional function of time. It represents the response of the
system to two separate unit impulses applied at two varying points in
time. Therefore the kernel is a
function of both time and time lag. Similarly, h3 is a
three-dimensional function of time, representing the response of the system to
three separate unit impulses applied at three varying points in time. Here the kernel is a function of time
and two distinct time lags. It is
through these time lags that nonlinear kernels represent the effect of a
previous response as it is carried through time in the system. Volterra kernels can be rewritten in
several ways simply by reordering the variables of integration. Because of this, more than one kernel
can generally be used to describe a given system, and it is therefore necessary
to impose uniqueness upon the kernels.
This is accomplished by working with restricted forms of the
kernels.
Reduced Order Unsteady Aerodynamic
Model
The reduced order models based on the linear and nonlinear Volterra Kernels for aeroelastic analysis will be used. With a high fidelity CFD solver, the reduced order model/Volterra Kernel efficiently yields the generalized aerodynamic forces necessary in aeroelastic analysis. Central in the use of Volterra theory based reduced order aerodynamic model is the identification of Volterra kernels. The truncated second-order Volterra series can be written as:
or
where 
and 
are the first-
and second-order kernels, respectively, 
is a general
time-dependent input signal, and 
is the system's
response. Silva showed that the
nonlinear Navier-Stokes equations can be considered weakly nonlinear, i.e., can
be accurately represented by a truncated second order Volterra series,
neglecting higher kernels.
First/Second Order of Volterra Kernel Identification
Let 
and 
where 
is the Diract¡¯s
delta function.
Then, the responses due to the inputs will be expressed as
If we assume that 
,
Since 
,
Therefore,
The second order of the Volterra kernel will be determined as
Now, let¡¯s assume that 
. Then,
If we put 
as an
input. The response will be
Since 
From the above equation, the first order of the Volterra kernel will be
where 
is the response
due to 
and 
is the response
due to 2
.
First Order Series Approximation
Raveh, et al
further assumed that the response can be evaluated only with the first term in
Volterra kernel. The response of
the system can then be evaluated by convolving the impulse response 
with the input
signal:
Or the response
of the system to an arbitrary input is computed by convolution of the step
response s(n) with the derivative of
input signal u(n), according to
Duhamel's integral:
MATLAB
Example
References
W.A. Silva, ¡°Reduced-Order Models Based on
Linear and Nonlinear Aerodynamic Impulse Responses,¡± AIAA Paper No. 99-1262
D.E. Raveh, Y. Levy, and M. Karpel,
¡°Aircraft Aeroelastic Analysis and Design Using CFD-Based Unsteady Loads,¡± AIAA
Paper No. 00-1325
W.A. Silva, D.E. Raveh, ¡°Development of
Unsteady Aerodynamic State-Space Models from CFD-Based Pulse Responses,¡± AIAA
Paper No. 01-1213