15-05-2013, 02:17 PM
Identification of Hammerstein Models With Cubic Spline Nonlinearities
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Abstract
This paper considers the use of cubic splines, instead
of polynomials, to represent the static nonlinearities in block
structured models. It introduces a system identification algorithm
for the Hammerstein structure, a static nonlinearity followed by a
linear filter, where cubic splines represent the static nonlinearity
and the linear dynamics are modeled using a finite impulse
response filter. The algorithm uses a separable least squares
Levenberg-Marquardt optimization to identify Hammerstein
cascades whose nonlinearities are modeled by either cubic splines
or polynomials. These algorithms are compared in simulation,
where the effects of variations in the input spectrum and distribution,
and those of the measurement noise are examined. The two
algorithms are used to fit Hammerstein models to stretch reflex
electromyogram (EMG) data recorded from a spinal cord injured
patient. The model with the cubic spline nonlinearity provides
more accurate predictions of the reflex EMG than the polynomial
based model, even in novel data.
INTRODUCTION
SYSTEM identification is the process of building mathematical
models based on measurements of a system’s inputs
and outputs [1]. These models are often used in biomedical engineering
to gain insight into the behavior of physiological systems.
Some biological processes, such as the stretch reflexes in
postural muscles, are highly nonlinear. Studying them requires
both nonlinear models and methods to identify them.
The Hammerstein cascade [2], shown in Fig. 1, consists of a
memoryless nonlinearity followed by a linear dynamic system.
It is less general than functional expansion models, such as
Volterra and Wiener series [3], but when used appropriately it
can reduce the number of parameters necessary to accurately
represent a system. A Hammerstein cascade can be a useful
model, especially when representing systems containing hard
nonlinearities, such as stretch reflex dynamics [4], where functional
expansions would be impractical.
THEORY
The Hammerstein model, shown in Fig. 1, consists of a static
nonlinearity followed by a dynamic linear system. Here,
and are its input and output, respectively, and is the
signal between the nonlinear and linear elements.
Initialization
As with many nonlinear optimizations, the identification of a
PHC requires a good initial parameter estimate, as noted in the
first step in Fig. 2. The objective is to place the initial parameters
close enough to the global optimum so that the model does not
converge to a suboptimal local minimum.
Note that an initial estimate is only required for the nonlinear
parameters, as the linear parameters will be found by linear
regression. For the PHC, this implies finding initial estimates
for the polynomial coefficients. The noniterative algorithm
proposed by Chang and Luus [8] is well suited to this task,
since it does not require a Gaussian, or white, input. Although
it is noniterative, the Chang and Luus algorithm yields multiple
estimates of the polynomial coefficients, one for each zero in
the linear filter. Since we are using a FIR filter with filter
weights, this procedure produces estimates of the nonlinearity
coefficients. Chang and Luus [8] recommend to use the set of
nonlinearity coefficients that minimizes the prediction errors.
SIMULATION RESULTS
The differences between the PHC and CSHC algorithms were
investigated through a number of simulation studies. These simulations
explored the effect of varying inputs, and the effects of
noise in an effort to compare the behavior of the two algorithms.
A. Simulated and Identified Model Structure
The simulated system was a Hammerstein cascade based on
the structure suggested in [9] for the stretch reflex EMG: the
relationship between the angular velocity of the ankle and the
EMG measured over the Gactrocnemius-Soleus (GS) muscle.
This simplified model consisted of a half-wave rectifier thresholded
a 0.9 rad/s, followed by a linear filter where the IRF includes
a delay of approximately 35 ms. as shown in Fig. 3. To
enable a fair comparison of the algorithms.
EXPERIMENTAL RESULTS
Both the CSHC and PHC algorithms were used to estimate
stretch reflex EMG dynamics, the relationship between the
ankle velocity and the GS EMG. This relationship was the
system simulated in Section III, where it was modeled as a
Hammerstein cascade, as suggested in [4], [9], and [22].
These data were originally obtained during a previous study
[23], which contains a detailed description of the experimental
protocol. In summary, the subject, due to a previous spinal cord
injury, had demonstrated an incomplete loss of motor function,
spasticity and hyper-active stretch reflexes. The subject lay
supine with their left foot attached to a rotary hydraulic actuator
by a custom fitted fiber-glass boot.
CONCLUSION
In this paper, we have developed an identification method for
the CSHC, a static nonlinearity represented by a cubic spline
followed by a dynamic linear filter. This algorithm is an adaptation
of an existing technique that uses a SLS optimization
to identify the elements of a polynomial Hammerstein model.
As demonstrated in the experimental and simulation results, the
cubic splines are able to represent hard nonlinearities accurately
using relatively few parameters. Furthermore, the increased accuracy
of the nonlinearity model increases the accuracy of the
estimated linear subsystem as well.