International Journal of Engineering Trends and Technology (IJETT) – Volume 29 Number 6 - November 2015
Missile Autopilot Design using Artificial Neural Networks
1
Adel Alsaraj, 2Gene Stuffle
1, 2,
Idaho State University
Abstract
The power and speed of modern digital
computers is truly astounding so that it enables
carrying on complex tasks such as aerospace
simulation, design and analysis, precisely. In
addition to the nature of the guidance problem, the
design technique, neural networks, necessitates
cumbersome computations to yield precise and
accurate performance. Neural networks approach
the solution of this problem by trying to mimic the
structure and function of the human nervous
system. Therefore, this paper is devoted a new
approach using the power of both computation
facilities and neural networks in the design and
analysis of an autopilot for the guidance system.
Then, its performance is justified against the
classical design approach through the Six degrees
of freedom (6DoF) flight simulation.
I.
Introduction
The nervous system consists of neurons,
which are connected to each other in a rather
complex way. Each neuron can be thought of as a
node and the interconnections between them are
edges [1]-[4]. Such a structure is called as a
directed graph. Further, each edge has a weight
associated with it, which represents how much the
two interconnected neurons can interact. If the
weight is more, then the two neurons can interact
much more; and consequently a stronger signal can
pass through the edge [5], [6]. Avery simple model
and consists of a single trainable neuron. Trainable
means that its threshold and input weights are
modifiable. Inputs are presented to the neuron and
each input has a desired output determined by the
user or designer [7]. The threshold and/or input
weights can be changed to modify the output
according to the learning algorithm [8]. The output
of the perceptron is constrained to Boolean values :(
true, false), (1,0), (1,-1) or whatever [9], [10]. This
is not a limitation because if the output of the
perceptron were to be the input for something else,
then the output edge could be made to have a
weight and consequently the output would be
dependent on this weight [11].
This paper is devoted to the autopilot
design for a missile system using the artificial
neural networks approach. The paper starts with
introduction to the neural networks, followed by the
Neural Net-based Guidance and autopilot Design
ISSN: 2231-5381
using model reference neural network. Then, the
designed controller is used with the system and the
simulation results were analysed. Finally, the
conclusions of the paper are discussed.
II.
Artificial neural networks
Artificial Neural networks are composed
of simple elements operating in parallel. These
elements are inspired by biological nervous
systems. As in nature, the network function is
determined largely by the connections between
elements. A neural network can be trained to
perform a particular function by adjusting the
values of the connections (weights) between
elements. Commonly neural networks are adjusted
or trained, so that a particular input leads to a
specific desired output, fig. (1).The network is
adjusted, based on a comparison of the output and
the target, until the network output matches the
target. Typically, many such input/target pairs are
used, in this supervised learning, to train a network.
The supervised training methods are commonly
used, but other networks can be obtained from
unsupervised training techniques or from direct
design methods. Unsupervised networks can be
used, for instance, to identify groups of data. There
are a variety of kinds of design and learning
techniques that enrich the choices that a user can
make.
Fig. (1) Idea of the Artificial Neural
Network(ANN) connection
Neural networks have been trained to
perform complex functions in various fields of
applications
including
pattern
recognition,
identification, classification, speech, vision, and
control systems [12]. Today, neural networks can
be trained to solve problems that are difficult for
conventional computers or human beings.
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International Journal of Engineering Trends and Technology (IJETT) – Volume 29 Number 6 - November 2015
A.
Neuron model
A neuron with a single scalar input
(Simple Neuron) and no bias is shown in fig. (2-a),
where the scalar input p is transmitted through a
connection that multiplies its strength by the scalar
weight w, to form the product wp. The weighted
input wp is the only argument of the transfer
function f, which produces the scalar output a.
However, to approach reality, the weighted input
(wp) isusually corrupted by a bias (b), fig. (2-b).
That is, the bias can be viewed as being added to
the product wp as shown by the summing junction
or as shifting the function f to the left by an amount
b [13]. The bias is much like a weight, except that it
has a constant value. The net input n, again a scalar,
is the sum of the weighted input wp and the bias b.
This sum is the argument of the transfer function f.
A transfer function f is typically a step function or a
sigmoid function that takes the argument n and
produces the output a. Note that w and b are both
adjustable scalar parameters of the neuron [14],
[15].
(a) Without
bias
(b) With bias
Fig. (2) Simple neuron
configuration
The central idea of neural networks is that
such parameters can be adjusted so that the network
exhibits some desired behavior. Thus, the network
can be trained to carry on a particular job by
adjusting the weight or bias parameters, or perhaps
the network itself can adjust these parameters to
achieve some desired output.
B.
Transfer functions
The transfer function can be found in
many different forms; among them are the hard
limit, the linear, and the sigmoid types. The hard
limit transfer function is used to limit the output of
the neuron to either 0, if the net input argument n is
less than 0, or 1, if n is greater than or equal to 0.
The linear transfer functionis used to transfer the
input with a certain scaling factor. While, the
sigmoid transfer functionaccepts the input, which
may have any value between plus and minus
infinity, and squashes the output into the range from
0 to 1.
ISSN: 2231-5381
C.
Neuron with vector input
A neuron with a single R-element input
vector is shown in fig. (3).
Fig. (3) Neuron with vector input
In this structure, the individual element
inputs p1, p2, …,pR are multiplied by weights
w1,1, w1,2, ...,w1,R and the weighted values are
fed to the summing junction. Their sum is simply
Wp, and it is obtained by the dot product of the
matrix W and the vector p. The neuron has a bias b,
which is summed with the weighted inputs to form
the net input n. This sum, n, is the argument of the
transfer function f, and it is given by:
n w 1,1 p w p
w R p R b (1
)
D.
Network architectures
Two or more of the neurons shown above
may be combined in a layer, and a particular
network might contain one or more of such layers.
Single Layer of Neurons
A one-layer network with R input elements
and S neurons is shown in fig. (4).In this network,
each element of the input vector p is connected to
each neuron input through the weight matrix W.
The ith neuron has a summer that gathers its
weighted inputs and the bias to form its own scalar
output ni. The various ni taken together form an Selement net input vector n. Finally, the neuron layer
outputs form a column vector a. Note that it is
common for the number of inputs to a layer to be
different from the number of neurons. In addition, a
layer is not constrained to have the number of its
inputs equal to the number of its neurons. A single
composite layer of neurons having different transfer
functions can be created simply by putting two of
the networks shown above in parallel. Both
networks would have the same inputs, and each
network would create some of the output elements.
The input vector elements are applied to the
network through the weight matrix W, which has
the form:
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International Journal of Engineering Trends and Technology (IJETT) – Volume 29 Number 6 - November 2015
W
w1,1
w2,1
w1, 2
w2, 2
... w1, R
... w2, R
wS ,1
wS , 2
... wS , R
Note that the row indices on the elements
of matrix W indicate the destination neuron of the
weight and the column indices indicate which
source is the input for that weight. For example, the
indices in w1,2 say that the strength of the signal
(5) Ainput
multi-layer
from the Fig.
second
elementnetwork
to the first neuron is
w1,2.
Fig. (4) A one-layer network
Multiple Layers of Neurons
A network can have several layers; each
layer has a weight matrix W, a bias vector b, and an
output vector a. A three-layer network is shown in
fig. (5) with the equations written below the figure.
This network has R1 inputs, S1 neurons in the first
layer, S2 neurons in the second layer, etc. It is
common for different layers to have different
numbers of neurons and a constant input 1 is fed to
the biases for each neuron. Note that the outputs of
each intermediate layer are the inputs to the
following one. Thus, layer 2 can be analysed as a
one-layer network with S1 inputs, S2 neurons, and
an S1S2 weight matrix W2. The input to layer 2 is
a1, and the output is a2.
The layers of a multilayer network play
different roles. In other words, a layer that produces
the network output is called an output layer, while
all other layers are called hidden layers. That is, the
three-layer network shown in fig. (5) has one output
layer (layer 3) and two hidden layers (layer 1 and
layer 2). Multiple layer networks are quite powerful
in evaluating complex processes. For instance, a
network of two layers, where the first layer is
sigmoid and the second layer is linear, can be
trained to approximate any function (with a finite
number of discontinuities) arbitrarily well.
E.
Learning approaches
There are different learning approaches
and consequently different types of Artificial
Neural Networks (ANN) that enable its utilization
with different applications. Among these
approaches are [16]:
Back-propagation
multilayer
ANN,Recurrent
type
ANN,Associative
type,Probabilistic, andAdaptive resonance.
The Back-propagation is utilized in real
time learning controller function, and consequently
it is considered with autopilot design for the
guidance system. Back-propagation was created by
generalizing the Widrow-Hoff learning rule to
multiple-layer
networks
and
nonlinear
differentiable transfer functions. Input vectors and
the corresponding output vectors are used to train a
network until it can approximate a function,
associate input vectors with specific output vectors,
or classify input vectors in an appropriate way as
defined by the designer. Networks with biases, a
sigmoid layer, and a linear output layer are capable
of approximating any function with a finite number
of discontinuities. Standard back-propagation is a
gradient descent algorithm, as is the Widrow-Hoff
learning rule.
The term backpropagation refers to the
manner in which the gradient is computed for
nonlinear multilayer networks. There are a number
of variations on the basic algorithm, which are
based on other standard optimization techniques,
such as conjugate gradient and Newton methods.
Typically, a new input will lead to an output similar
to the correct output for input vectors used in
training that are similar to the new input being
presented. This generalization property makes it
possible to train a network on a representative set of
input/target pairs and get good results without
training the network on all possible input/output
pairs [17].
III.
Neural net-based guidance and
control design
The application of neural networks has
attracted significant attention in several disciplines,
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International Journal of Engineering Trends and Technology (IJETT) – Volume 29 Number 6 - November 2015
such as signal processing, identification and
control. The success of neural networks is mainly
attributed to their unique features such as:
Parallel structures with distributed storage and
processing of massive amounts of information,
and
Learning ability made possible by adjusting the
network interconnection weights and biases
based on certain learning algorithms.
The first feature enables neural networks
to process large amounts of dimensional
information in real-time. The implication of the
second feature is that the non-linear dynamics of a
system can be learned and identified directly by an
artificial neural network. In addition, the network
can adapt to changes in the environment and make
decisions despite uncertainty in operating
conditions. Therefore, neural networks are
implemented in aerospace applications and
consequently the guidance system for enhancing its
performance.
Most neural networks can be represented
by a standard (N+1) layer feed forward network. In
z
this network, the input is
0
y while the output
zN
n . The input and output are related by
is
the following recursive relationship:
net j
zj
W jz j 1
Vj
2)N
, j 1, 2,........
f i (net j )
A.
Neural network with model reference control
In this control structure, the desired
performance of the closed-loop system is specified
through a stable reference model, which is defined
by its input-output pair {r(t), yref(t)}, fig. (6) [19].
This figure (shows that the control system attempts
to make the plant output y(t) match the reference
model output yref(t), asymptotically. Thus, the error
between the plant and the reference model outputs
is used to adjust the weights of the neural network
controller [20].
r
NN
reference
model
yr
ef
+ e
NN
controller u
NN
y
Pla
nt
Fig. (6) Model reference control
scheme.
Autopilot design using model reference NN
A hybrid model reference adaptive control
scheme is implemented with the guidance system.
In this system, a neural network is placed in parallel
(
with
a linear fixed-gain independently regulated
1 autopilot as shown in fig. (7).
B.
and
net N W N z N
z N net N
1
(
VN
3)
where the weights Wj and Vj are of the
appropriate dimensions. Vj is the connection of the
weight vector to the bias node. The activation
function vectors fj (.), j = 1, 2,..., N–1 are usually
chosen as some kind of sigmoid, but they may be
simple identity gains. The activation function of the
output layer nodes is generally an identity function.
The neural network can, thus, be succinctly
expressed as
NN( y; W, V)
Fig. (7) Block Diagram of acceleration
control system using model reference NN
controller
The linear autopilot is chosen so as to
(
f N ( W N f N 1 ( W N 1f N 2 (...stabilize
the plant over the operating range and
3
provide approximate control, while the neural
) 1 y V1 ) V 2 ) ... V N 1 ) V N )
W 2 f1 ( W
controller is used to enhance the performance of the
where
f ji (net ij (k ))
2
1 e
netij ( k )
1,
i,4) j 1,......, N 1
wherei denotes the ith element of fj and λ
is the learning constant. For network training, error
back propagation is one of the standard methods
used to adjust the weights of neural networks [18].
ISSN: 2231-5381
linear autopilot when performance becomes poor by
adjusting its weights. A suitable reference model is
(
chosen
to define the desired closed-loop autopilot
p re f
y re f
responses
and
across the flight
envelope. These outputs are then compared with the
actual outputs of the lateral autopilot
p
and
yielding an error measurement vector [
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y
p re rror
International Journal of Engineering Trends and Technology (IJETT) – Volume 29 Number 6 - November 2015
y re rror
]T. This error is used in conjunction with an
adaptive rule to adjust the weights of the neural
network so that the tracking error is minimized. A
direct effect of this approach is to suppress the
influence resulting from roll rate coupling.
The neural network model and controller
are designed using the Matlab neural network
toolbox. A two layer network is designed with
sigmoid transfer function followed by a linear one
for both the plant and the controller. This structure
is shown in fig. (8), which shows the connection
The network is trained offline with a step
reference signal yielding the system response
shown in fig. (9). This figure shows a stable system,
but with distorted transients. This neural The
network is trained offline with a step reference
signal yielding the system response shown in fig.
(9). This figure shows a stable system, but with
distorted transients. This neural network autopilot is
implemented with the Six degrees of freedom
(6DoF) simulation and the same engagement
scenario of [21]. The obtained miss distance is
reduced to only 5%.
Fig. (9) Acceleration step response with
neural controller at 6 sec
For more enhancements in the system
performance, the network is retrained but with
reference signal adjusted to cope with the values
obtained from the previous 6DOF simulations.
Then, the new autopilot is implemented yielding
faster response, fig. (10), and higher relative
stability compared with the previous one and also
that obtained with classical control in [21]. For
more justification of this new autopilot, it is
implemented within the 6DOF simulation, which is
conducted with target initial position of [6 1 –2]
Km, initial velocity of [-250 –100 0]. This target
experienced a manoeuvre of [30 –25 10] [m/sec2],
i.e. 4.23 g after 5 seconds from the instance of
missile launch, and lasted for 2 seconds. The
missile-target flight path with a lead network is
shown in fig. (11-a) where the miss distance is 47.8
ISSN: 2231-5381
between the two networks in Simulink point of
view.
Fig. (8) The connection between the
two networks in Simulink point of view.
[m] and the time of flight is 8.27 seconds. Using the
modified neural network controller yields the
engagement scenario shown in fig. (11-b), where
the miss distance is about 3[m], and the flight time
is 8.15 seconds. That is, it yields to save 2% in the
flight time and to reduce 94% in the miss distance,
compared to the previous design.
It is clear that, the new system is much
faster than the original one, and with less miss
distance of about 93% of the lead network and 85%
Fig. (10) Acceleration step response with
modified neural controller at 6 sec
Fig. (11) Missile and Target trajectory (a) with
original autopilot (b) modified NN controller
It is clear that, the new system is much
faster than the original one, and with less miss
distance of about 93% of the lead network and 85%
less than the classical PID controller. The three
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International Journal of Engineering Trends and Technology (IJETT) – Volume 29 Number 6 - November 2015
engagement scenarios are plotted together with
zooming to clarify the difference between them as
shown in fig. (12). this figure clarifies how the
neural network achieved a smooth and fast
approach to the interception with minimum miss
distance.
[5]
[6]
IV.
Conclusions
A neural network based adaptive inverting
autopilot design is developed and implemented for
a guided missile system. This design approach was
superior to the original and designed classical
approaches from the point of view of miss distance
and demanded acceleration. That is, the neural
network proved its robustness with such a
stochastic non-linear system provided it is carefully
trained.
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
Fig. (12) Missile-target engagement scenarios with
lead, PID and neural networks
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