The. Volterra series(l) approach to the identification of nonlinear systems which is presented in this paper, is a natural extension of earlier results in modeling.
NASA Technical Memorandum
104177
/ Neural Network Modeling Volterra Series Extension
Donald I. Soloway NASA Langley Research
of Nonlinear Systems of a Linear Model
Based on
Center
Jan T. Bialaslewicz University
of Colorado
at Denver
Ngz-16987
(NASA-TH-104177) NFURAL NETW_EK MOOELING OF N_wLINEAR SYSTEMS BASED ON V_LTERRA SERIES EXTENSION OF A LINEAR MODEL (NASA) 13 p CSCL 09B
G3/6-_
January
1992
\
National Aeronautics and Space Administration Langley Research Center Hampton, VA 23665
_q
NEURAL BASED
NETWORK MODELING OF NONLINEAR ON VOLTERRA SERIES EXTENSION OF MODEL
SYSTEMS A LINEAR
Donald I. Soloway NASA Langley Research Center Technology Branch, Information Systems Mail Stop 152D, Hampton, VA 23665-5225
Automation
Jan
Campus
Box
University Electrical 110, P.O.
Division
T. Bialasiewicz of Colorado Engineering Box 173364,
at Denver Department Denver, CO
80217-3364
ABSTRACT A
Volterra
series
approach
has
been
applied
to
the
identification
of
nonlinear systems which are described by a neural network model. A procedure is outlined by which a mathematical model can be developed from experimental data obtained from the network structure. Applications of the results to control of robotic systems are discussed. INTRODUCTION The
Volterra
systems earlier
which results
series(l)
approach
is presented in modeling
in of
to
the
identification
of
nonlinear
this paper, is a natural extension of linear systems(2). In reference 2, the
impulse response of a linear dynamical system is shown to be given by the weights of the network model. For nonlinear systems, in addition to the impulse response of the linear approximation, the higher order Volterra kernels must be expressed in terms of the parameters of the trained network model. This relationship is reported in this paper. The result obtained means that it is possible not only to obtain a neural network model of the nonlinear dynamics, but also to represent this model by a mathematical expression. This opens a broad range of applications for the neural network modeling of nonlinear dynamical systems. The Volterra series in neural networks literature appeared recently in references 3 and 4. Both papers showed that a model nonlinear analytic system. the representation theorem,
of the Volterra system can model a However, this result follows directly from proved in reference 5. Some interesting
results of neural network applications can be found in reference 6.
to nonlinear
systems control
In robotics, there are many places where nonlinear processes exist. The nonlinearities to be controlled include motor dynamics, flexible beam vibrations, harmonic drive stiffness, gear backlash, and full arm dynamics. Some of these nonlinearities, for example, beam vibrations and full arm dynamics, can be classified as analytic nonlinearities. This paper shows how to obtain a mathematical model of these nonlinearities using experimental data collected from the system under investigation. In manipulator control, it is required that the manipulator respond quickly and accurately in spite of existing nonlinearities and interjoint couplings. To obtain a good design, one should use as much a priori knowledge as possible and compliment the design with an adaptive fine tuning algorithm. In principle, this is the structure of the control scheme proposed by Koivo(7). In this structure, shown in figure 1, the primary controller is developed based on the available model of the manipulator and the secondary controller compensates for unmodeled dynamics. Investigating the design of the primary controller is proposed, using a nonlinear model of the manipulator to be obtained as a Volterra series representation of a neural network model. The system fine tuning can be done, if necessary, by an adaptive loop using a Linear Quadratic Gaussian approach. In the proposed design, the model-based approach (8) and the performancebased approach (9) would be merged to obtain better performance,
Controller Primary (Volterra) y(t)
x(t)
r(t)
m....
v
_ Manipulator
Controller Secondary
-I
(LEG)
Figure
1.-
Manipulator
2
System
MODELING The input-output relation differential equation may
OF
LINEAR
SYSTEMS
of a system described by be given by a convolution
a linear integral
OO
y(t)=
_h(x)x(t-x)dx
(1)
0
that
specifies
impulse
the
response
following
output
y(t)
in
terms
of
h(x).
The
discrete-time
the
input
x(t)
and
representation
has
the
system
the
form: OO
y(k)=
_
h(n)x(k-n)
(2)
n=0
where the arguments k and n are shorthand for kT and nT with T being the sampling interval to be selected for any particular system. Information on the system bandwidth of interest is needed to choose proper value for T. The relation is considered,
(2) becomes approximate that is, when
when-a
finite
number
of
a
terms
r
r-1
y(k)=
_
h(n)x(k-n)
(3)
n=0
which terms.
results in Equation
unmodeled (3), written
dynamics, represented using standard neural
by the network
truncated notation,
is
r-I
y(k)=
_
WlnX(k-n)
(4)
n=0
that
is,
the
finite
(truncated) lh(O)
impulse
response
h(1).--h(r-1)l=lwlo
Wll
is
given
by:
(5)
--. Wlr. 11
This relationship, at any time instant k, can be viewed as a representation of a neural network with r inputs x(k-i), i=0,1,-..,r-1 and a single output y(k), generated by a single linear neuron. This network can be considered a member of the I; r class of feedforward networks(5). Once r is fixed for a linear system, no modeling improvement can be reached by increasing the number of nodes and/or the number of layers. However, the increase of the number of nodes/layers will , result in a structure redundancy and the robustness to neuron failure will be obtained. failure will be
Consequently, shorter.
the
3
time
needed
to
recover
from
a
The network model of a linear system, discussed so far, is shown in figure 2, in which q-1 denotes the unit delay operator, that is, q-lx(k)=x(k-l). This network is described by the following
difference
(w lO+W11q-I +w 12q-2+...+w which vector
is equivalent product
to
equation
(4).
Also,
equation: (6)
lnq-n)x(k)=y(k) (6)
can
be
represented
(I)T(k)=[x(k)
x(k-1)..,
x(k-n)l
and
should be emphasized that by using approximate model (7) of the system Response (FIR) model, is obtained.
0T(k)=[Wl0(k)Wll(k)... the finite input (2), known as
sequence the Finite
x(k) | W
-1
-1
q
q
q'l _
q
q-1 _ Wln
y(k)
Figure
2.-
the
(7)
y(k)=(1)T(k)O(k) with
as
Single
4
node
FIR
network
Win(k)], an Impulse
It
MODELING
OF NONLINEAR
ANALYTIC Let a Sing!e-Input, Single-Output described by a functional
SYSTEMS
SYSTEMS (SISO)
nonlinear
dynamical
system
y(t)=F[x(t)] where x(t) is analytic, such infinite series
y(t)=Jh
(8)
the input, y(t) is the output that it can be represented of the following form:
OO
be
and the functional F is exactly by a converging
OO
1(x)x(t-x)dx+J OO
Jh2(z,
,z2)x(t-Zl)X(t-z2)dzldZ2+...
OIO
+ J... fhn(Zl,...,'rn)X(t-Xl)X(t-x2)...x(t-xn)dzldx2...dxn+... 0 0 Such a system can be represented a finite series of the form of (9). series expansion, can be interpreted
(9)
to any desired degree of accuracy by This equation, known as the Voiterra as a functional generalization of
the Taylor series expansion and represents the solution to a large class of nonlinear differential equations. For a linear system only the first term in expression (9) is nonzero and represents the convolution integral,
with
hi(x)
In expression (9), kernels, or higher nonlinear dynamic form
being
the
impulse
response
of
the
hn(Xl,...,Xn), n=2,3 .... are higher order order impulse responses, introduced behavior. If (8) is discretized, then
system. Volterra to describe (9) assumes
the
OO
Y(k)=
_yn(k)
(10)
n=l
where OO
yn(k)
= _... nl=0
ACTIVATION
OO
_hn(nl,n2
.... ,nn)x(k-nl)...x(k-nn)
(11)
nn=0
FUNCTION NONLINEAR
SUITABLE DYNAMICAL
FOR MODELING SYSTEMS
Let us assume that this equation (11) is to be This implies the requirement that the number 5
modeled of inputs
OF
by a network. to the network
is finite. However, assume that the equation could be modeled by a Xr network with the network input defined as follows: xT=[x(k) x(k-1) ... x(k-r+l)l (12) where r will be a number of time delayed inputs, and an activation function _F(.) as an operator which is applied to the sum of the weighted inputs to a node to produce an output from the node. We have assumed so far that the modeling of nonlinear dynamical systems will require a multiple node zr network. Definition: The activation function for Z r networks, suitable for modeling nonlinear systems, is defined as a function _F: R--c[a, b] which is differentiable, nondecreasing, limW(X)=b and limq-'(X)=a. X--,oo
Two examples networks, are function: 1. Logistics
of an activation function, the logistics function and
function 1 l+e -x
W(X)=
2. Hyperbolic
7t--,-oo
W: R_[0,
1], defined with
tangent
function
both the
by
the derivative
q': R_[-1,
1],
widely used in I;r hyperbolic tangent
the
following
equation:
q_'(X)=q_(_.)(l-W(X))
defined
by
the
following
equation: q'(_.)=tanh(Z.)=
l'e-2X l+e_2X
with
the
derivative
Note that the computation of the derivatives repeatedly performed while using a gradient backpropagation for training, is computationally According activation 1.
to the q'(.) definition, functions :
following
_F(JL)=signK a
2.
{
o_X b
with
the
a