Neural network modeling of nonlinear systems based on Volterra ...

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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