NASA Technical Memorandum 10 1999 

i 

Neuromorphic Learning of ContinuousValued Mappings in the Presence of 

Noise: Application to Real-Time 

Adaptive Control 

Terry Troudet 

Sverdrup Technology, Inc. 

NASA Lewis Research Center Group 

Cleveland, Ohio 

and 

Walter C. Merrill 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio 

[NASA-Tfl-101999) NEUROWRPHIC LEBR1YIHG OF N 89- 248 56 

CONTINUOUS-VALUED BAPPIFHGS IN THE PRESENCE 

UP WOXSE: APPLSCATION TO REAL-TIME ADAPTIVE 

CONTROL (Null. Lewis Besearch Center) Unclas 

14 P CSCL 09B 63/63 0204411 

Prepared for the 

1989 International Conference on Neural Networks 

sponsored by the Institute of Electrical and Electronics Engineers 

Washington, D . C . , June 18-22, 1989 
NEUROMORPHIC LEARNING 

IN THE PRESENCE OF NOISE: 

Application To Real- Time Adaptive Control 

OF CONTINUOUS-VALUED MAPPINGS 

Terry Troudet Walter C. Merrill 

Sverdrup Technology, Inc. 

NASA Lewis Research Center Group 

National Aeronautics and Space Administration 

Lewis Research Center 

Cleveland, Ohio Cleveland, Ohio 

Abstract 

The ability of feed-forward neural net architectures to learn continuous-valued mappings 

in the presence of noise is demonstrated in 

relation to parameter identification and realtime adaptive control applications. Factors 

and parameters influencing the learning performance of such nets in the presence of noise 

are identified. Their effects are discussed 

through a computer simulation of the BackError-Propagation algorithm by taking the example of the cart-pole system controlled by a 

non-linear control law. Adequate sampling of 

the state space is found to be essential for canceling the effect of the statistical fluctuations 

and allowing learning to take place. 

1 Introduction. 

A major challenge for health-monitoring and 

diagnosis of complex Advanced Propulsion 

Systems (APS), such as the Space Shuttle 

Main Engine, is to perform real-time analysis of a massive amount of diverse sensor data. 

Such analysis can be used either to directly 

perform low-level, real-time adaptive control 

or to send real-time descriptions of the dynamical state of the APS to a high-level controller. In the first case, the low-level controller has to compute adaptively and in realtime the control signal to be applied to the 

controlled process for a given set of sensor 

data. In the second case, the diagnosis process 

consists of "translating" in real-time the sensor information into one or several parameters 

which characterize specific aspects of the dynamical evolution of the APS. (For a discrete 

mapping, parameter identification reduces to 

pattern recognition). The massive parallelism 

of neural nets [l] and their ability to learn complex continuous mappings [3] give them the 

real-time computational power needed for such 

low-level monitoring/diagnosis functions. 

When a realistic model of physical phenomena and feedback mechanisms is possible, 

1 
adaptive control laws and parameter identifiers can be expressed in analytical forms, and 

neural nets can be trained from computergenerated data. In most instances however, it 

is difficult to derive realistic models, and neural nets can only be trained on experimental 

data which have been corrupted by a certain 

amount of noise during the signal processing 

of the sensor measurements. 

In this paper, we analyze the effect of noise 

on the neuromorphic abilities of feed-forward 

neural net architectures to learn continuous 

mappings, and to serve as parameter adentzfiers or real-time adaptave controllers. A discussion of a computer-simulation of the training architecture described in Section 2 is presented in Section 3 for a non-linear continuousvalued mapping by taking the example of a 

controlled cart-pole system. The training sequence is analyzed in detail with and without 

noise in the training data. 

2 Training Architecture. 

In the presence of noise, :he state variables 

of the controlled process, Z, and the applied 

control signal, C, are 

and 

+ 6c (2) c = Cezact 

where the noises fi2 and iic are simulated 

as independent variables, normally distributed 

with zero mean. For a high signal-to-noise ratio of the input signal, i.e. >> 1, 

the function q5 which represents the control 

law, $(exact) = Cexactl can be expressed, using a Taylor expansion, as 

max~~2e~acc/~ Jm 

In this case, trying to learn the mapping #J 

from sets of noisy input/output vectors (2, C) 

is thus equivalent to trying to learn q5 from sets 

of input/output vectors (ezact, Cn) where 

only the control force is corrupted by noise: 

This equivalence shows that the factors 

and parameters which influence neuromorphic 

learning with noisy input/output data can be 

studied by analyzing the learning performance 

of the net as a function of the signal-to-noise ratio maz(CF*nC' of the output signal. va+(h) 

The first phase of the training consists of 

sampling at various times tk the state variables 

exact (tk) of the controlled process, Fig.(l), 

and the control signal Cn(tk) given in Eqs.(4) 

and (5). In the second phase, the set of training data ($xact(t,), Cn(tk)) is organized in 

110 subsets before being applied to the neuromorphic controller. 

3 Cart-Pole System. 

In this example, the controlled process of Figure 1 is chosen to be the cart-pole system 

[3] - [4] represented in Figure 2. Training 

data are recorded by placing initially the cartpole at arbitrary positions (X(O), e(0)) with 

zero velocities, and by driving it to the origin (X = 0,8 = 0) with a control force. 

While the cart-pole is returned to its equilibrium position, the four-dimensional state 

vector i(t) = (X(t), X(t), e(t), e(t)), and the 

control force, F(t), are regularly sampled over 

a certain period of time. Sampling rate and 

observation time are considered as parameters 

of the training. 

2 
Non-Linear Control Law. 

The dynamic evolution of the cart-pole with 

its parameters shown in Figure 2 is given by 

the equations of motion 

ki = hl - h2X 

.. fi + F x=- 

f2 

where 

3 hl = -gsin8 

4L 

3 h2 = -cost? 

4L 

. 3. 

8 

fi = m(Lsinee2 - -gszn%) 

a 

(11) 

0 

f2 = M + m(i - -cos28) 4 

The control force F is generated by application of a feedback linearizing and decoupling 

transform [5] 

with kl = 25, k2 = 10, c1 = 1 and c2 = 

2.6. Training data are generated by integrating the set of Eqs.(6)-(12) with the initial 

condition g(0) = (X(O), 0, @(a), 0), X(0) and 

O(0) being arbitrary position and angle of the 

domain D,y0=[-4mr4m] x [-5Odeg,+5Odeg]. 

The neural architecture chosen to approximate 

the control law F[Z] is a feed-forward net of 4 

linear neurons in the input layer, two hidden 

layers of 16 and 4 neurons respectively, and 

one neuron in the output layer. The continuous values of = (X, X, 8, e) are fed into the 

input layer. Each neuron input is connected to 

all neuron outputs of the previous layer, and 

to a unit which is permanently on (threshold 

term). The output signal of each neuron is 

linearly modulated by the (synaptic) weights 

before excitinglinhibiting a connected neuron. 

With the standard activation function 

the output on of a neuron is bound to the interval [O. 11. Restricting the information domain 

of a neuron to the interval [0.1,0.9] by eliminating the asymptotes of the input/output 

response curve of Eq.(13) is known to enhance learning performance [6]. The neuromorphic learning of the continuous-valued 

mapping F[5] requires the scaling and offsetting of the last layer output 

P 

where Fg is a constant parameter which normalizes the control signal over [-1,+1]. It is 

essential to emphasize that the choice of Fg 

defines the domain where the mapping is to 

be learned, and influences the neural computation as well. Eqs.(l3) and (14) show that 

the net output cannot match values of F such 

that IF1 > 1.25Fg (practically, the net can 

only match accurately the domain IF1 5 Fg). 

When a control law or a continuous-valued 

mapping is to be learned, it is imperative to 

first define the domain of variations of the signals before choosing the constant parameter 

Fg since it has to be larger than the absolute maximum value of the signal, maxlFJ. 

Among the possible values of Fg satisfying 

Fg 2 mar!F/, the optimal value corresponds 

to the best approximation of the continuousvalued mapping by the neural net. 

It is well-known that linear mappings - those 

for which the control signal is a linear function of the state variables - can be learned by 

a single-layer perceptron. In that case, it is 

easy to show that perfect learning occurs over 

any domain of variations of the signals when 

the limit Fg + 03 is taken. In the general 

3 
case however, the only way to find the optimal Fg is to train the net for several values Fg 

larger than maa:lFI. and compare the learning 

performances. In this work, we have chosen 

Fg = maa:JF/ ad hoc. 

The results hereafter have been obtained 

by training the net with the Back-ErrorPropagation (BEP) algorithm on a VAX 8650 

of the KASA Lewis I'AXCLUSTER. The initial values of weights and thresholds have been 

chosen randomly distributed in the interval 

[-0.1, $0.11 to break symmetries that could 

eventually lead to spurious modes and bias the 

learning. 

3.1 Absence of Noise. 

In order to extract the relational law between 

state variables and control force from a bulk 

of input/output data, it is necessary that the 

training data be as representative as possible 

of the state space, and that they be sampled 

as uniformly as possible throughout the state 

space. First, if there are too few training data 

or if they are not sufficiently representative, 

the neural net may either only "memorize" 

the training data without extracting the control law itself, or approximate the control law 

only over a partial domain. This means that 

the net may not be able to reproduce control 

signals corresponding to state variables that 

were not part of the training set. Second, if 

the sampling is not uniform throughout the 

state space, learning may be biased, and the 

net may only "memorize" part of the training 

data without extracting the features contained 

in the training set. 

When the dynamic characteristics of the 

cart-pole are regularly sampled until it returns 

to the origin, the distribution of the training 

forces {u} is strongly peaked around the origin. Clearly, a random sampling of the training data would bias the training and prevent 

the net from learning the control law for large 

displacements of the cart-pole. Before training, data are organized by dividing the interval [-1,+1] of the normalized force u = $ into 

11 subintervals {Ik, k = 1.11) of equal length. 

One training iteration consists of the random 

sampling of a subinterval Ik followed by the 

random sampling of a 'u E Ik. T,he corresponding set of state variables is used as input 

to the net. The resulting network output is 

compared to the target output on = -. 

Each individual error between target output 

and network output is propagated through all 

the net layers to update weights and thresholds by BEP. This process is iterated until convergence. 

The accuracy of the neural approximation 

of the control law can be characterized by the 

total mean-square error e2 

(15) 2 - E::, e: ell 

where et is the mean-square error over the 

subinterval Ik: 

In Eq.(16), u(i)ta7get is one of the n(k) target data of Ik, and u(i)net the output of the 

net corresponding to the same state variables. 

The convergence of the algorithm can be estimated by evaluating the error e', Eq.(l5), 

made by the net in trying to reproduce the 

data used for training; similarly, the degree of 

accuracy of the feature extraction can be estimated by evaluating the error e2 made by 

the net in trying to predict the control forces 

corresponding to state vectors that were not 

presented to the net during training. When 

both errors are small and comparable in magnitude, the neural net has developed a good 

internal representation of the mapping. 

It is well-known that any a-priori knowledge 

on the mapping features, such as symmetries, 

4 
can efficiently improve the net performance 

when these are incorporated explictly in the 

neural computation. Since the control force 

changes as F + -F when the state vector 

changes as 2? + -2, the ensemble of training data has been chosen symmetric under the 

transformation T- = [(z, F) + (-5, -F)] by 

randomly distributing the initial position and 

angle of the cart-pole over the domain Dx~= 

[-4m,+4m] x [-5Odeg,+50deg]. 

Since all subintervals {Ik} are treated with 

equal probability during training, there need 

to be enough training data to provide as uniform a representation of the control law over 

each Ik as possible. In this section, training 

was performed over 200 motions sampled at 

20Hz over 10s. An upper-bound for the control 

force needed to bring the cart-pole back to the 

origin from a position in Dxe, and without initial velocities, is mazlFI = 120Newtons. The 

BEP algorithm was run for Fg = 120N, with 

a steepest descent coefficient r] = 0.2 and a 

momentum term cr = 0.9. Since the BEP algorithm is based on a minimization principle, 

there is always a possibility that the system of 

weights/thresholds may get trapped in a local 

minimum or may freeze in a flat spot during 

the search for the global minimum. In learning the mapping F[Z], Eq.(12), and using it 

to control the cart-pole, it was found that the 

neuromorphic controller was able to stabilize 

the pole to 6 = 0, but would occasionally return the cart to a position X # 0. In order 

to circumvent this difficulty due to the existence of a local minimum or a flat spot, we performed a fine-tuning by augmenting the training with data randomly sampled in the smaller 

subinterval le which is centered around 0 and 

symmetric under T-. 

After training, the learning performance of 

the net can be tested in three different configurations. The "learning open-loop configuration" tests the ability of retrieving F from 

state variables 2 used for training; the net is 

used as an analog memory. The "generalizing open-loop configuration" tests the ability 

of generating F from state variables not used 

during training; the net is used as a parameter estimator. The "generalizing closed-loop 

configuration" tests the ability of stabilizing 

the controlled process for a motion not used 

during training; the net is used as a real-time 

adaptive controller (RTAC). 

The curves of Figures 3-4 show the result 

of the neural computation in the "generalizing" modes, after 99000 gross-tuning iterations and 1000 fine-tuning iterations. In the 

set of Figures 3, the initial state vector is 

z(0) = (-lm, 0,45deg, 0), and in Figures 4, 

z(0) = (3m, 0, -35deg, O).The neural net is 

able to return, very satisfactorily, the cart-pole 

to the origin from large angles and large displacements. This shows that, during training, 

the net has developed an internal representation which is a very good approximation of 

the mapping defined in Eq.(12). 

With the above set of parameters, estimations of the error e', Eqs.(15)-(16), are 0.0004 

in the "learning open-loop" mode, and 0.0006 

in a "generalizing open-loop'' mode for a set 

of 200 randomly generated cart-pole motions. 

3.2 Presence of Noise. 

With noise, the normalized values of the control forces used as targets are no longer the 

exact values since 

+fi (17) utarget - Uexact nHowever, the mapping F[Z] can be learned 

by BEP from a training set of representative data ~:~9~'((i). Given the statistically averaged square error function 

< (F[2?] + ii~ - G[z])' > over the entire state 

space, the BEP yields a function, G(Z), which 

minimizes the error as shown variationally in 

5 
E~s. (18)-( 19): 

= 0- (18) 6 < (r[a + fip - G[,f])* > 

SG[i?] 

In the absence of noise, learning takes place 

only when the net is provided with many representative training data in order to be able 

to reproduce the features rather than "memorize" the data. This is even more crucial with 

noise since the target data are not the exact 

values of the force. 

It is essential to choose small values for the 

steepest-descent parameter 17 in order to minimize the effect of the fluctuations. For a large 

r], samples with big deviation would overcontribute to the weights adjustments and mislead the search for the minimum. For a small 

7, the effect of such deviants tend to be balanced towards average by the contribution of 

the samples with small deviation since those 

occur with a higher probability. From a geometric point-of-view, the fluctuations drastically complicate the topology of the errorsurfaces by creating more irregularities and 

introducing more possibilities for the net to 

get trapped in local minima or flat spots. 

Small values of 17 favor adiabatic changes of 

the weights towards paths of the energy surface which correspond to averaged values of 

fi >= ueract, In the absence of noise, when 

the target values are the exact values, the role 

of the momentum term is to speed up the convergence process by amplifying the weight adjustments. In the presence of noise, a momentum term would amplify the undesired weight 

changes due to highly deviant data, and make 

adiabaticity more difficult to maintain. For 

this reason, the momentum coefficient is chosen a = 0. The price to pay for a small 

steepest-descent coefficient and no momentum 

the training data, i.e. < ueTget >=< ueract + 

term is of course more iterations to reach convergence. 

Fluctuations make it also difficult to generate representative sets of training data due to 

the "contamznatzon" of the subintervals Ik. 

Due to noise, training data are likely to lie in 

subintervals that are different from the subintervals where the exact data are. For a nonuniform density distribution of the data used 

for training. this implies that the sampling by 

subintervals as described in Section 3.1 will be 

less uniform as the noise increases. 

If data were obtained by regularly sampling 

cart-pole motions over "long" periods of time, 

e.g. 10 sec or more, the exact density distribution would be well peaked around 0. Noise 

would cause these data to spread from the center towards the edges of [-1,+1]. For a given 

noise intensity, the number of data spread out 

of a subinterval would be proportional to the 

number of data contained in that interval. As 

noise increases, data from subintervals located 

around the center, e.g. 16, would increasingly 

populate the neighboring subintervals. For low 

signal-to-noise ratio S/hr = e and long 

observation time, the sampling method of Section 3.1 would most probably select noisy data 

originating from exact data located in I6, 15 

or 17, regardless of the chosen interval. In the 

limit, the fluctuations around the origin would 

be averaged_out, and the net would essentially 

learn that 2 = 0 is the equilibrium position 

without being able to stabilize the cart-pole. 

Such a "contamination" can only be restrained by as much as possible uniformly sampling the density distribution of the data before training. Since the exact values are not 

known, this can be done by shortening the observation time of the cart-pole motions and by 

increasing the number of these motions. Oversampling of similar dynamic states, e.g. the 

equilibrium position, would thus be prevented. 

For the simulation, we have assumed that 

vaT(n) 

6 
the mapping to be extracted and learned by 

the net is symmetric with respect to the Ttransformation, and we have chosen the initial positions/angles of the cart-pole randomly 

distributed over Dxe. Towards the end of 

the training, fine-tuning has been obtained 

by training the net with data only sampled 

from the subinterval I6 which is smaller than 

[-l,-lj and still symmetric with respect to the 

T- transformation. Training data have been 

generated by integrating the equations of motion (6)-(12), and adding a gaussian noise of 

zero mean. Samples of noisy cart-pole motions used for training are shown in the set 

of Figures 5a and 6a, and the results of the 

computation are shown in Figures 5b-e and 

6b-e for Fg = 12ON with noise-to-signal ratios 

N/S=O.1 and N/S=0.2. 

For N/S =0.1 (Figs 5b-e), training has been 

made by sampling 1000 cart-pole motions at 

20 Hz over only 5 sec. The BEP parameters are 7 = 0.02 and cy = 0; 490,000 grosstuning/ 10,000 fine-tuning iterations have been 

used. In the "learning open-loop" configuration, e2 = 0.006, and in a "generalizing openloop" configuration, e2 = 0.006 for a set of 

1000 randomly generated cart-pole motions. 

In the RTAC configuration, the net itself 

controls the evolution of the cart-pole; the 

state vector g(t) is therefore different from the 

state vector corresponding to the exact control law (12). This explains the discrepancy 

between Figs. 5b and 5c. 

For N/S=0.2 (Figs.6b-e), 4000 cart-pole 

motions have been sampled at 20Hz over 2 

sec only. The BEP parameters are 7 = 

0.01 and cr = 0: 950.000 gross-tuning/50,000 

fine-tuning iterations have been used. In 

the "learning open-loop'' configuration, e2 = 

0.008, and in a "generalizing open-loop'' configuration, e2 = 0.008 for a set of 4000 randomly generated cart-pole motions. 

These results show that, in spite of a significant amount of noise, the net is able to 

learn the control law (12) within sufficient accuracy to return satisfactorily the cart-pole to 

its equlibrium position. 

In the presence of noise, an error function 

can be introduced to optimize the parameters 

of the computation and estimate the degree 

of accuracy to which the neural mapping approximates the exact unknown mapping. It is 

defined from Eqs.( 15)-( 16), but where u(i)net 

(expected to be closer to the exact value than 

~(i)~~~g~~ is) is one of the n(k) net outputs 

contained in Ik, and ~(i)~~~g~~ the corresponding noisy data. In the statistical limit where 

n(k) + 03 and for perfect learning, this meansquare error would be minimal and equal to 

the variance of the noise. 

4 Conclusion. 

These preliminary results demonstrate that 

neural networks can extract within a good 

aproximation a continuous-valued, non-linear 

mapping between input and output data when 

the training data are corrupted by noise. In 

order to extract the features contained in the 

training data, it is necessary to sample the 

state space as uniformly as possible. In the 

absence of noise, aproximately uniform samplings can be obtained by randomly sampling subintervals of the state space. However, the presence of noise mixes the populations of these subintervals, making it more 

difficult to reach an approximately uniform 

sampling throughout the state space. Limiting the observation time of each cart-pole motion, and increasing the number of such motions has been found to significantly reduce 

this contamination effect. It would be interesting to analyze the possibility of further reducing this contamination by pre-processing the 

training data with the help of neuromorphic 

classifiers, such as counter-propagation feedforward nets, before training with BEP. 

7 
Acknowledgments. 

We would like to thank Allon Guez and John 

Selinsky for stimulating discussions. We express our gratitude to Carl Lorenzo for his 

support during the realization of this work, 

and thank Jonathan Litt and Ten-Huei Guo 

for helping us clarify the manuscript. 

References 

[l] R. P. Lippmann:”An Introducizon To 

Computzng Wzth A-eural Xets”, IEEE 

ASSP Magazine, April 1987, pp.4-22. 

[2] A. Guez and J. Se1insky:”A Traanable 

n’euromorphzc Controller”, Journal of 

Robotics Systems, 5(4), 363-388 (1988). 

[3] B. Widrow: ”The Orzgznal Adaptzve 

Broom Balancer”, IEEE Conf. on Circuits 

and Systems, Philadelphia, PA, 1987. 

[4] A. G. Barto, R. S. Sutton, and C. W. Anderson: ” Xeuronlike Adaptzve Elements 

That Can Solve Dzficult Learnzng Control Problems”. IEEE Trans. on Systems, 

Man and Cybernetics, SMC-13(5), 834- 

846 (1983). 

[5] A. Guez:”Optzmal Control Of Robotic 

Manzpulators”, Ph.D. Thesis, Univ. of 

Florida. 1983. 

[6] D. Rumelhart, G. E. Hinton, and R. J. 

Williams: ”Learning Internal Representations by Error Propagation”, in Parallel 

Distributed Processing, D. E. Rumelhart 

and J. L. McClelland, Eds., MIT Press 

(1986). 

8 
I I 

-2.4 

-3.2 

I I NOISE '1 I 

- 

- 

PROCESS 

NOISE 

CONTROLLER 

CONTROLLED 

I FEEDBACK FROM STATE VARIABLE SENSORS 

M-lk~ 

m-0.1 kg 

L-1 m f = 5 ke/seC 

g - 9.81 WSec2 

f dWdt 

Figure 1. - Training architecture for neuromorphic learning with noise. Figure 2. - Cart-pole system. The control force F is applied to the cart in the 

presence of friction. 

1.0 r EXACT 

ITERATIONS WITHOUT NOISE 

- - - - - AFTER 100 000 TRAINING 'R 

.2 

0 

-.2 E 

s -.4 

-P -.8 

= -1 -,6~ .o (a) Generglizing open loop configuration (parameter estimator). 

l.o r W 

LL 8 

-1.0 ~ 

0 12 3 4 5 6 7 8 9 10 

TIME, sec 

(b) Generalizing dosed loop configuration (RTAC); normalized 

control force. 

4.0 r 

3.2 

2.4 

E 1.6 

8 0.8 

60 a 

4.8 

3 -1.6 a 

t 

30 t? 

-20 d 

40 

p 10 

3 -10 

8 -20 W 

-30 t- 

-40 

-50 I I I I I I I I I I 

0 12 3 4 5 6 7 8 9 10 

(d) Generalizing dosed loop mnfigurabn (RTAC); pendulum 

TIME, sec 

angle. - 

Figure 3. - Non-linear contml law wlfhout noise. Initial state vector, Z(0) I (-lm, 0.45 O. 0). Normalization constant, Fg -120 Newtons. Back-ErrorPropagation parameters: a = 0.2 and q = 0.9. 

9 
1 .o 

w .8 

.6 

.4 

9 

5 

so 

E -.2 

z -.4 

@ .2 

N 

8 -.6 

z -.O 

-1 .o 

1 .o 

.0 

.6 9 

d .4 a 

00 

n 

s .* 

w -2 8 

- 

- 

II I I I I I I I 

- EXACT 

- - - - - AFTER 100 000 TRAINING 

ITERATIONS WITHOUT NOISE 

bL 10 

2 -10 

\ s.jz- - 

3.2 4.0 L 

-2.4 f 

-3.2 

-4.0 I I I1 II I I I I 

(c) Generalizing closed loop configuration (RTAC): cart position. 

3-.411 I I I I I I I I, $;I1 I I I I I I I I, 5 -.6 

z -.O 

-1 .o 

0 12 3 4 5 6 7 0 9 10 0 12 3 4 5 6 7 0 9 10 

TIME, sec TIME, sec 

(b) Generalizing closed loop configuration (RTAC); normalized (d) Generalizing closed loop configuration (RTAC); pendulum 

angle. control force. 

+ 

Figure 4. - Non-linear control law without noise. Initial state vector. Z(0) I (3m. 0, -35 O, 0). Normalization constant, Fg- 120 Newtons Back-ErrorPropagation parameters: a = 0.2 and q 4.9. 

10 
1 .o 

.8 

.6 

.4 

.2 

0 

-.2 

-.4 

~ .6 

-.8 

-1 .o 

1 .o 

.8 

.6 

.4 

.2 

0 

-.2 

-.4 

- .6 

-.8 

-1 .o 

1 .o 

.8 

.6 

.4 

.2 

0 

-.2 

-.4 

-.6 

-.8 

-1 .o 

- 

- 

- 

- 

I I I I I I I I I 

EXACT 

EXACT+NOlSE 

L 

- 

I I I I I I I I I1 

I4.0 r 

3.2 

2.4 

E 1.6 

$ 0.8 

io n 

+ 4.8 a 3 -1.6 

-2.4 

-3.2 

-4.0 I I I1 II I 1 I I 

(d) Generalizing dosed loop configuration (RTAC); cart position. 

!XI- 

-30 - 

-40- 

-50 II I I I I I I II 

0 12 3 4 5 6 7 8 9 10 0 12 3 4 5 6 7 8 9 10 

TIME, sec TIME, sec 

(c) Generalizing closed loop configuration (RTAC); normalized 

Figure 5. - Non-iinear control law with noise for nolse-to-signal ratio, N/S = 0.1; Normalization constant, Fg = 120 Newtons; Back-Error-Propagation 

(e) Generalizing closed loop configuration (RTAC); pendulum angle. 

control force. 

parameters: a = 0.02 and q = 0. Neural Net performance tested over noiseless data after training with noise-mrrupted data. 

11 
1 .a - EXACT 

w .8 F EXACT+NOlSE 0 

- - - - - AFTER 1 000 000 TRAINING ITERATIONS 

WITH NOISE-CORRUPTED DATA 

(b) Generalizing open loop configuration (parameter estimator). 

._ 

0 12 3 4 5 6 7 8 9 10 

TIME, sec 

(c) Generalizing closed loop configuration (RTAC); normalized 

control force. 

4.0 r 

3.2 

2.4 

E 1.6 

i 

0 0.8 

io 

+ -0.8 U 5 -1.6 t 

(d) Generalizing dosed loop configuration (RTAC); cart position. 

-30 - 

-40 - 

-50 I I I 

0 12 3 4 5 6 7 8 9 10 

TIME, sec 

(e) Generalizing dosed loop configuration (RTAC); pendulum angle. 

Figure 6. - Non-linear control law with noise for nolse-to-signal ratio, NIS = 0.2; Normalization constant. Fg = 120 Newtons; Back-Error-Propagation 

parameters: a = 0.01 and q = 0. Neural Net performance tested over noiseless data after training with noise-corrupted data. 

12 
11 Aeronautics and 

rt No. 2. Government Accession No. 

;A TM-101999 

Report Documentation Page 

3. Recipient’s Catalog No. 

snd Subtitle 5. Report Date 

-0morphic Learning of Continuous-Valued Mappings in the Presence 

oise: Application to Real-Time Adaptive Control 

‘r(s) 

y Troudet and Walter C. Merrill 

8. Performing Organization Report No. 

E-4706 

10. Work Unit No. 

582-01-11 

ming Organization Name and Address 

mal Aeronautics and Space Administration 

11. Contract or Grant No. 

19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No of pages 

Unclassified Unclassified 14 

s Research Center 

:land, Ohio 44135-3191 

22. Price” 

A03 

I 13. Type of Report and Period Covered 

Technical Memorandum boring Agency Name and Address 

mal Aeronautics and Space Administration 

iington, D.C. 20546-0001 

14. Sponsoring Agency Code 

mentary Notes 

ired for the 1989 International Conference on Neural Networks sponsored by the Institute of Electrical and 

ronics Engineers, Washington, D.C., June 18-22, 1989. 

ct 

tbility of feed-forward neural net architectures to learn continuous-valued mappings in the presence of noise 

nonstrated in relation to parameter identijicution and real-time adaptive control applications. Factors and 

neters influencing the learning performance of such nets in the presence of noise are identified. Their effects 

iscussed through a computer simulation of the Back-Error-Propagation algorithm by taking the example of 

M-pole system controlled by a nonlinear control law. Adequate sampling of the state space is found to be 

tial for canceling the effect of the statistical fluctuations and allowing learning to take place. 

17. Key Words (Suggested by Author(s)) 

Neural networks; Learning; Adaptive control; 

Artificial intelligence; Noise 

18. Distribution Statement 

Unclassified - Unlimited 

Subject Category 63 

*For sale by the National Technical Information Service, Springfield, Virginia 221 61 NASA FORM 1626 OCT 86