A technique based on the Minimum Model Error optimal estimation approach is employed for robust identification of a nonlinear dynamic system. A simple harmonic oscillator with quadratic position feedback was simulated on an analog computer. With the aid of analog measurements and an assumed linear model, the Minimum Model Error Algorithm accurately identifies the quadratic nonlinearity. The tests demonstrate that the method is robust with respect to prior ignorance of the nonlinear system model and with respect to measurement record length, regardless of initial conditions

Mook, D, Joseph

Stry, Greselda I.

English

NASA Technical Reports Server

An Experimental Study of
Nonlinear Dynamic System Identification
Greselda I. Stry I and D. Joseph Mook 2
Department of Mechanical and Aerospace Engineering
State University of New York at Buffalo
Buffalo, New York 14260
716-636-3058
Abstract
A technique based on the Minimum Model Error optimal estimation approach is
employed for robust identification of a nonlinear dynamic system. A simple harmonic
oscillator with quadratic position feedback was simulated on an analog computer. With
the aid of analog measurements and an assumed linear model, the Minimum Model Error
Algorithm accurately identifies the quadratic nonlinearity. The tests demonstrate that the
method is robust with respect to prior ignorance of the nonlinear system model, and with
respect to measurement record length and regardless of initial conditions.
Introduction
The widespread existence of nonlinear behavior in many dynamic systems is weU-
documented, e.g, Thompson and Stewart [1]; Nayfeh and Mook [2]. In particular,
virtually every problem associated with orbit estimation, flight trajectory estimation,
spacecraft dynamics, etc., is known to exhibit nonlinear behavior. Many excellent
methods for analyzing nonlinear system models have been developed. However, a key
practical link is often overlooked, namely: How does one obtain an accurate mathematical
model for the dynamics of a particular complicated nonlinear system? Identification, the
process of developing an accurate system model from system output measurements, may
provide the answer.
Nonlinear systems are commonly described using linear models. Many efficient al-
gorithms for the identification of linear systems exist and their accuracy and ease of
application encourages their use. However, linearization does not work in every appli-
cation, and even when it does provide a reasonable approximation, the approximation is
normally limited to a small region about the operating point of linearization. In the case
Graduate Resarch Assistant; NASA Graduate Researcher
2 Assistant Professor
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of systems with severe nonlinear behavior using a linear model to describe such a system
leads to inconsistencies ranging from inaccurate numerical results to misrepresentation
of the system's qualitative behavior. Since nonlinearities are seldomly easily charac-
terized, nonlinear identification techniques may prove beneficial in developing accurate
mathematical representations of nonlinear systems.
Numerous methods for the identification of nonlinear systems have been developed
in the past two decades (Natke, Juang and Gawronski [3]). Among the most widely
used nonlinear identification methods are describing the nonlinear system using a linear
model, or representing the nonlinear system in a series expansion and obtaining the
respective coefficients either by using a regression estimation technique, by minimizing
a cost functional, by using correlation techniques, or by some other approach. Some
algorithms use the direct equation approach, while others obtain a graphical representation
of the nonlinear term(s) and then find an analytical model for the nonlinearity. The
interested reader can find more information on these nonlinear identification techniques
in Stry and Mook [4].
The diversity of nonlinear identification techniques prompts the choice of an algorithm
based on the needs of the particular application. Typical criteria to look for are: iterations
required, robustness in the presence of measurement noise, number of measurements
needed, robustness with respect to knowledge of the inital conditions, and robusmess
with respect to initial assumptions regarding the form of the nonlinearity. The results
presented in this paper confirm that the Minimum Model Error algorithm excels in the
above mentioned requirements.
In previous papers, the Minimum Model Error algorithm (M/VIE) was explained in
detail (Mook and Junkins [5]), modified for nonlinear identification (Mook [6]), and
shown to accurately identify exotic nonlinearities in higher order systems (Stry and
Mook [4]). In this paper, it is shown how the MME algorithm successfully identifies
nonlinearities using experimental data. An analytical model representing a harmonic
oscillator with quadratic position feedback is studied. Out'put data is obtained from an
analog computer simulation of the nonlinear system and the quadratic term is accurately
identified. It is shown that the Minimum Model Error algorithm is capable of identifying
a nonlinear model which accurately reproduces the analog output regardless of knowledge
an initially assumed model, initial conditions or record length.
MME Algorithm
In this section, we briefly review the MME algorithm and how it is used to identify
nonlinear dynamic systems. A more detailed explanation may be found in Mook and
Junkins [5], Mook [6], and Stry and Mook [4].
The M/VIE may be summarized as follows. Suppose there is a nonlinear system
whose exact analytical representation is unknown, but for which output measurements
areavailable.Using"normal"means(analysis,finite elements,etc.),a systemmodel,
denotedassumedmodel,is constructed.TheMME combinestheassumedmodelwith
themeasurementso determinethecorrectform of thenonlinearsystem.Theapproach
consistsof addingtheto-be-determinedcorrectiontermto theassumedmodel. A cost
functional composed of the weighted sum of the correction term plus measurements
minus estimate residuals is minimized. The minimization yields optimal state trajectories
in addition to the correction term. A least squares fit of the correction term is performed
to find the form of the dynamic model error.
Consider a forced nonlinear dynamic system which may be modeled in state-space
form by the equation
_(t) = A_(t) + F(t) + f_(__(t), _(t)) (1)
where z(t) is the n × I state vector consisting of the system states, A is the n × n state
matrix, _.F(t) is an n x 1 vector of known external excitation, and f(£(t),_(t)) is an
n x 1 vector which includes all of the system nonlinearities. State-observable discrete
time domain measurements are available for this system in the form
__(tk) = #_k(_(t_),tk) + v_, to <_ tk < t! (2)
where __(tk) is an rn × 1 measurement vector at time tk, #-k is the accurate model of
the measurement process, and v k represents measurement noise. _vk is assumed to be
a zero-mean, gaussian distributed process of known covariance Rk. The measurement
vector _(tk) may contain one or more of the system states. To implement MME, assume
that a model, which is generally not the true system model because of the difficulties
inherent in obtaining the true system model, is constructed in state-vector form as
_(t) = A_(t) + _.F(t) (3)
Here, we show a linear model because in practice, linearization is the most common
approach to modeling nonlinear systems. MME uses the assumed linear model in (3)
and the noisy measurements in (2) to find the model error.
The model error, which might include linear terms as well as unknown nonlinear
term(s), is represented by the addition of a correction term to the assumed linear model
as
_(t) = A_(t) + _.F(t) + d(t) (4)
where d(t) is the n x 1 correction term (dynamic model error) to be estimated later.
A cost functional, J, that consists of the weighted integral square of the correction
term plus the weighted sum square of the measurement-minus-estimated measurement
residuals, is formed:
M
J"- _{[t( tk)- ffk(_--.(tk),tk)] T/_kl[t(tk) - gk(__.(tk),tk)]}
k=l
fro_!+ _a(_-)rw d(_-)d_- (5)
where M is the number of measurement times, __(tk) is the estimated state vector and
W is a weight matrix to be determined.
J is minimized with respect to the correction term, _d(t). The necessary conditions
for the minimization lead to the following two point boundary value problem (TPBVP),
(see Geering [7]),
i(t) = A_(t) + F(Q + _d(t) (5a)
-_(0: -At;(t) (55)
d(t) : -_w__(t) (5_)
@
•_(io)=_ or __(to)=O
__(tl)= __: o.,. __(tl)= o
(5_)
(5f)
Estimates of the states andwhere _(t) is a vector of costates (Lagrange multipliers).
of the dynamic model error are produced by the solution of this two-point boundary
value problem. The estimates depend on the particular value of W. The solution is
repeated until a value of W is obtained which produces state estimates which satisfy the
"'covariance constraint", explained next.
According to the covariance constraint, the measurement-minus-estimated measure-
ment residual covariance matrix must match the measurement-minus-troth error covari-
once matrix. This may be written as
_(t_) - gk(__(tk),tk)]T_(tl,) -- g___(_(tk), tk)] _ Rk (6)
During the minimization, the weight W is varied until the state estimates satisfy the
covariance constraint, i.e., the left hand side of Eq. (6) is approximately equal to
the right hand side. The correction term or model error is, therefore, the minimum
adjustment to the model required for the estimated states to predict the measurements
with approximately the same covariance as the measurement error.
After W has been determined such that the state estimates satisfy the covariance
constraint, the final step in the identification procedure is to use a least squares algorithm
to fit the model error d(t) to the unknown dynamic term(s). The error is expanded into
some combination of linear and nonlinear terms, for example,
d(t) = az(t) + _z2(t) + 7za(t) + ... (7)
where ct, _, 7, ... are unknown coefficients to be determined by least squares. The least
squares approach is explained in detail in Mook[6]
The TPBVP represented by Eqs. (5a) to (5f) contains jumps in the costates and,
consequently, in the correction term. As evident from Eq. (5d), the size of the jump is
directly proportional to the measurement residual at each measurement time. The noisier
the measurements, the larger the jump size. A multiple shooting algorithm, developed by
Mook and Lew [8], converts this jump-discontinuous TPBVP into a set of linear algebraic
equations which may be solved using any linear equation solver. Multiple shooting also
facilitates the analysis of a large number of measurements, by processing the solution at
the end of every set of jumps.
The multiple shooting algorithm presented by Mook and Lew [8] was used to obtain
the MME solutions used in the tests presented in this paper. It was assumed in the
examples that MME obtained the dynamic error term without knowledge of the boundary
conditions on _, so some distortion of the correction term at the initial and final times
was expected due to the constraints of Eqs. (5e-5f), i.e., by assuming no state knowledge
is available at to or If, we constrain _(t0) = 0 and A(t.f) = 0. Therefore, in all test
cases, the initial and final ten percent of the correction term data was ignored in the
least squares fit.
Application Examples
Two nonlinear equations of motion were studied, which represent the motion of
an undamped harmonic oscillator with different amounts of quadratic position feedback
(identical equations may arise in other physical systems as well). The equations in state
space form are
(9)
where z is position, and the dot indicates differentiation with respect to time. No forcing
was applied.
In the following discussion, Eq. (8) is denoted Model A and Eq. (9) is denoted
Model B. Different initial conditions were used for each system, for a total of four
different tests. These are shown in Table 1.
Table 1. List of conditions used for each test
Tesi #
A1
x(O)
0.000
v(O)
0.261
A2 0.000 0.523
B1 0.000 0.087
B2 0.000 0.261 -1.137
To utilize MME, the linear part of Eqs. (8) and (9) was chosen as the assumed model,
rendering the model error equivalent to the nonlinear term, c • z 2. Measurements for the
MME nonlinear identification were generated by simulating test AI to B2 on an analog
computer. Position measurements for all four tests were recorded and nonlinear models
identified. The results were compared with the analytical position and analytical error
term data, c • z 2, which were generated for Models A and B using a digital computer.
MME proved capable of accurately identifying the nonlinear quadratic term in spite of
ignorance of the assumed model, true initial conditions and record length.
Analog computer results
One hundred position measurements were generated on an EAJ-2000 analog com-
puter for all four test cases. All measurements with a sampling rate of 4 Hertz were
used in the analysis. Position, velocity, and position squared were chosen as the basis
functions for the least squares fit. It was uncertain if the analog computer would add
some damping to the system or if it was able to correctly reproduce the stiffness term.
By including position and velocity in the least squares fit, stiffiaess and damping could
be identified if they existed. The identification procedure yielded the numerical values
shown in Table 2.
Table 2. Least Square estimates of the nonlinear terms using measurements
generated by the analog computer.
Test True MME True MME True MME
# X X v v X*X X*X
A1 0.000 0.003 0.000 0.000 -0.526 -0.528
r _,,,
A2 0.000 0.003 0.000 0.000 -0.526 -0.526
B1 0.000 0.008 0.000 0.005 -1.137 -1.141
B2 0.000 0.003 0.000 0.000 -1,.137 -1.135
818
The numerical results for the least squares fit of the error term matched the analytically
predicted coefficients with great accuracy. Figures (la-4a) show the analytical position,
analog measurements and position predicted by the MME analysis for all analog tests.
Figures (lb-4b) show the analytical correction term and the error term estimated by MME.
In all cases the MME identification produced good state estimates.
The MME algorithm could accurately identify a nonlinear model regardless of the
initial conditions. As seen from Figures (la) and (4a) (test A1 and B2), the measured
position and the analytical position differ significantly. The analytical position was
digitally recalculated for test A1 and B2 using the initial analog measurements as initial
conditions instead of the initial conditions presented in Table 1. The results are shown
in Figures (5a) and (6a). In this set of plots the analytical position and the measurements
are almost identical. Also, as shown in Figures (5b) and (6b), the analytic correction
term is much more similar to the estimated correction term, confirming that MME does
not need any knowledge of the initial or final state vector value. 3
MME could identify the nonlinear term accurately idenpendent of the record length.
In test B1 only 40 measurements were employed in the analysis because subsequent
measurements were saturated. The nonlinear term is identified very well.
Note that the data appears to be noiseless, as shown in Figures (la-4a). Successfull
analysis of noisy data using the MME algorithm can be found in Mook[6] and Stry and
Mook[4].
l '" t. L
i o 5oo • • • 6 • •
• • • • • •
-o 5o/) • • • • 6 • •
• 6 • • • • • •
• ,iJ • • Ib _
0 00_ *(_3 6_._ 12 $ 188 250
_tuC(S)
(o.1
!
O_ -O _0C-
|
ff ,B 6
: *
_ a 2 q
Tqw[_$)
(b)
Figure 1. Test A1 a) Analytical, measured (+), and MME estimated
(') position, b) Analytical and MME estimated (') correction term.
The MME estimates are essentially identical to the measurements.
It was shown in Eqs. (5e) and (5t3, that by setting the initial and final costate values to zero, MME
does not need any knowledge of the initial or end conditions.
0 _00
_ ,
0 O00[*OO-
|
, T 1 _ _ l 1 _ s 1 • _ , _ I
!
01oo-
!
000(X*0C-
!
-o_oo-_I ......._ _ -_ -o3oo-
0000£+00 625 I_'$ 1B8 250 0 COO[cO0
Tr.[CS)
A
b
6
6
625 _25 188 _'$0
"r_ut(S]
(b)
Figure 2. Test A2 a) Analytical, measured (+), and MME estimated
(') position, b) Analytical and MME estimated (') correction term.
The MME estimates are essentially Identical to the measurements.
O_OC-
I
O O00(-OO
r_
250 500 ?5,0 _00
_ruC(53
0500(-01
0 DO(O[+O0.
-0 2S0_[-0_-
-0500(-0! ' '
0 O0(X_OO
,6
6 A 6 6
A
A
_, _* '_6 6
6
,b
,D
6
6
-o,oo- ' I __--f_
0 000_*00 ;'50 500 750 100
T_[($';
(_) Cb_
Figure 3. Test B1 a) Analytical, measured (÷), and MME estimated
(') position, b) Analytical and MME estimated (') correction term.
The MME estimates are essentially identical to the measurements.
0 000( o00|
w
S/
-0 200-
0 200-
, , , . _ !,O0[ - D' -
•A.I_A ' ; - -o,_.-o,-
4, • • •
6 _
6 A
6 A
0
6,5 l
(6)
Figure 4. Test B2 a) Analytical, measured (+), and MME estimated
C) position, b) Analytical and MME estimated (') correction term.
The MME estimates are essentially Identical to the measurements.
I i -OJ_-
• i
..... 0_-_--1_ -_ ' '
(_) (U)
d
t88 _SO
Figure 5. Test A1 a) Analytical, measured (+), and MME estimated (')
position, b) Analytical and MME estimated (') correction term. The
analytical position was calculated using as initial conditions the initial
position and velocity measurements from the analog computer.
o,oot _ _-_-= -_--4-_4-
?
!
B
0 (X_[ *00 625 _25 125 IE8
fluE(S} T_w[CS)
25 o
Figure 6. Test B2 a) Analytical, measured (+), and MME estimated (')
position, b) Analytical and MME estimated (') correction term. The
analytical position was calculated using as initial conditions the initial
position and velocity measurements from the analog computer.
Conclusion
In this paper, an MME based algorithm was used to accurately identify the quadratic
term of a nonlinear harmonic oscillator. Data was obtained from an analog computer
simulation of the nonlinear system. It is demonstrated that the method is robust with
respect to (lack of) a priori knowledge of the system dynamics. The identification was
accurate regardless of initial conditions or data record length.
References
1. Thompson, J.M.T., and Stewart, H.B., "Nonlinear Dynamics and Chaos", W'dey,
New York, 1986.
2. Nayfeh, A.H., and Mook, D.T., "Nonlinear Oscillations", W'dey, New York, 1979.
3. Natke, H.G., Juang, L-N., and Gawronski, W., "Identification of Nonlinear Mechan-
ical Systems: A Brief Review," NASA Langley Research Center, Hampton, VA,
U.S.A..
4. Mook, D.J., and Junkins, "Minimum Model Error Estimation for Poorly Modeled
Dynamic Systems", A/AA J., Vol. 11, No. 3, 1988, pp. 256-261.
5. Mook, D.J., "Estimation and Identification of Nonlinear Dynamic Systems," A/AA
J., Vol. 27, No. 7, 1988, pp. 968-974.
822
6. Stry, G.I., and Mook, D.J., "Identification of Higher-Order Systems with Exotic
Nonlinearities," in review.
7. Geering, H.P., "Continuous Time Optimal Control Theory for Cost Functionals
Including Discrete State Penalty Terms," IEEE Trans. A.C., Vol. AC-21, 1976,
pp. 866--869.
8. Mook, D.I., and Lew, J-H., "Multiple Shooting Algorithm for Jump Discontinuous
Problems in Optimal Control and Estimation ," to appear in the 1EEE Trans. A. C..



An experimental study of nonlinear dynamic system identification

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