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9. Introduction to Neural Networks and Its Application to Control of SMAs Part 11

111Instructor: Dr. Song

Dept. of Mechanical Engineering

Contribution from Mr. Ning Ma is greatly appreciated.

Control of Smart StructuresTopic 9Introduction to Neural Networks and

Its Application to Control of SMAs

Part 1

Introduction to Neural Networks


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Outline

Introduction to neural networks.

Application of neural networks in system

identification and control.

Implementation of neural networks using Matlab.


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Introduction to Neural Networks What is neural networks?

Why use neural networks?

General architecture of neural networks.

Models of a neuron.

Connections.

Learning of neural networks.


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What is Neural Networks

A neural network is a massively parallel distributed

processor made of simple processing units, which has anatural propensity for storing experiential knowledge and

making it available for use. It resembles the brain in two

respects:1. Knowledge is acquired by the network from its environmentthrough a learning process.

2. Interneuron connection strengths, known as synaptic weights, are

used to store the acquired knowledge.

This definition is adapted from Aleksander and Morton (1990)


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Why use neural networks Nonlinear function mapping.Nonlinear function mapping of neural networks

makes it possible to approximate the physical world which is inherently

nonlinear.

Learning and adaptation. Networks are trained using past data measured

from the plant. A well-trained network then has the ability to generalize

when presented with inputs not appearing in the training data. Networks

have also a build-in capability to adapt their synaptic weights to changes.

Parallel distributed processing.Networks have a highly parallel structure

which lends itself to a higher degree of error tolerance than conventional

schemes and fast computing. Multivariable systems.Neural Networks naturally process many inputs and

have many outputs.


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666Instructor: Dr. SongDept. of Mechanical Engineering

General Architecture of Neural Networks

The neural network architecture is defined by the basic

processing elements: neurons, and the way in which they are

interconnected.

In literature, there are several types of neural networks

presented. Each type of neural networks is characterized by

its architecture.

Due to their various architectures, those types of neural

networks show different performances for a certain

application.


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Models of A Neuron (1)

Three basic elements of the

neuron model:

A set of connecting links;

An adder for summing the

input signals; An activation function.


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Models of A Neuron (2)

In mathematical view, the

neuron may be described by

following equations:

)

1

( kkk

m

j

jkjk

buy

and

xwU

+=

= =


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Models of A Neuron (3)Three basic types of activation function:


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Connections The neurons by themselves are not very powerful in terms of

computation or representation. But, their interconnectionsallows us to encode relations between the variables giving

different powerful processing capabilities.

In general, three fundamental types of interconnections of

neural networks are defined in literature : single-layer

feedforward networks, multilayer feedforward networks and

recurrent networks.

There exist numerous neural network architectures, such as

Hopfield nets, Boltzmann machine, etc.


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Single-Layer Feedforward NetworksThis type of networks consists of a input layer of input

nodes and a output layer of neurons


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Multilayer Feedforward NetworksThe multilayer feedforward networks distinguish themselves by thepresence of one or more hidden layers. In addition, the neurons ineach layer have as their inputs the output signals of the precedinglayer only.


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Recurrent NetworksThe recurrent network distinguishes itself in that it has at lest onefeedback loop. The feedback loop can be either self-feedback loop orglobal feedback loop.Moreover, the feedback loop and the unit-delayelements result in a nonlinear dynamicalbehavior of the network.


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Learning of Neural Networks Learning is a process by which the free parameters of a neural network are

adapted through a process of stimulation by the environment in which the

network is embedded. The type of learning is determined by the manner inwhich the parameter changes take place.

The learning process involves three events:

The neural network is stimulated by the environment.

The neural network undergoes changes in its free parameters.

The neural network responds in a new way to the environment.

In literature, there exist many learning algorithms.

Learning algorithms can be classified into two main group: supervised learningand unsupervised learning.


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Back-Propagation Learning Algorithm Back-Propagation (BP) learning algorithm is based on the error-correction

learning.

BP learning algorithm is the most efficient algorithm used in adaptingMultilayer Feedforward Networks.

BP learning consists of two passes through the different layers of the network: a

forward pass and abackward pass.

In the forward pass, the input is applied to the network, and its effectpropagates through the layers. Finally, a set of outputs is produced as the

actual response of the network. During this forward pass the synaptic

weights of the network are fixed.

During the backward pass, though, the synaptic weights are all adjusted inaccordance with an error-correction rule. The actual response of the

network is subtracted from the target response to generate the error signal.

This error is then back-propagated through the network, against the

direction of synaptic connections, and hence the term back-propagation.


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Application of neural networks in system

identification and control.

Brief review of application of neural network in system

identification and control. System identification using neural networks.

Control themes using neural networks.


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Brief review of application of neural network in

system identification and control. The nonlinear mapping property of the neural network are

central to its use in the system identification and control. The ability of the ANN to represent an arbitrary nonlinear

relations is already established. It could be stated that any

continuous function can be approximated by just a multi-

layer network, if there are a sufficient number of hidden

neurons in the networks.

For networks application in system identification and

control, the key questions are how many hidden layers

should be used and how many neurons are required in each

layer.


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System Identification Using Neural NetworksFeedforward Modeling

Plant

LearningAlgorithm

NeuralNetw orkModel

U

d

d

ym

error

yp

+

+

+

_


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System Identification Using Neural NetworksInverse Modeling

Direct method

Plant

LearningAlgorithm

NeuralNetw orkInverse

Model

U

d

ym

error

yp+

_


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Direct method

Conceptually the simplest approach is directinverse training [Psaltis et al, 1988] as shownschematically in right figure. The output of the

plant is used as input to the neural network.Then, the output of the network is compared

with the plants input. The error is used to trainthe neural network.

It is clear that the training procedure will force

the network to represent the inverse model ofthe plant. However, this training method hastwo drawbacks. First, it is not goal directed[Jordan and Rumelhart, 1991]. Second, if the

nonlinear system is not invertible, an incorrectinverse can be obtained.

Plant

LearningAlgorithm

NeuralNetw orkInverseModel

U

d

ym

error

yp


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Specialized method (the NN Models output is used)

Plant

LearningAlgorithm


Model

U

d

ym

error

yp

NeuralNetw ork

Model

r

_+


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Specialized method

The other approach to overcome these

problems was known as specialized inverselearning [Psaltis et al, 1988], which isillustrated in right figure. In this approach,the network inverse model precedes the

plant. The command signal is fed into thisnetwork and the output of the network will

be sent to the plant, as well as anestablished network feedforward model

parallel with the plant. The error signal for

the training algorithm in this case is thedifference between the output of thenetwork feedforward model and thecommand signal. This approach trends to

make unity transfer function from thecommand signal to the output of the plant.

Plant

LearningAlgorithm

NeuralNetw orkInverseModel

U

d

ym

error

yp

NeuralNetw orkModel

r


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Control themes Using Neural Networks

Direct inverse control. Model reference control.

Internal Model control.

Predictive control.


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Direct Inverse Control

Plant


Model

U

d

ypr


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Model Reference Control (MRC)

NN Controller

Represents desired performance


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Internal Model Control (IMC)

Limited to open-loop stable systems.

Linear Filter

Inverse Model

(NN Controller)

NN Plant Model

Plant

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Predictive Control

NN Controller

Iterative optimization

NN Plant Model

(predict the plant future)

Iterative algorithm

Optimal solution

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Implementation of Neural Networks Using Matlab

Introduction to fundamental commands and

concepts in Matlab Neural Network Toolbox.

Example: construct a Multilayer Feedforward

Network.

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Introduction to fundamental functions and concepts

in Matlab Neural Network Toolbox

A neural network is represented as an object in Matlab toolbox.

Using Matlab, users can construct Feedforward Networks, RecurrentNetworks, Radial Basis Networks, Competitive Neural Networks, etc.

For different types of neural networks, Matlab provides corresponding

functions to create and train the neural networks.

Users can specify the weights, activation functions, learning algorithms,

initial values and bias respectively.

The following introduction and example are only applicable to Multilayer

Feedforward Networks.

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in Matlab Neural Network Toolbox(2)

newff(PR,[S1 S2...SNl],{TF1 TF2...TFNl},BTF,BLF,PF)

creates a new N layer feed-forward backpropagation network. PR - R x 2 matrix of min and max values for R input elements.

Si - Size of ith layer, for Nl layers.

TFi - Transfer function of ith layer, default is Hyperbolic tangent sigmoidtransfer function.

BTF - Backpropagation network training function, default is Gradient descentwith momentum and adaptive learning rate backpropagation.

BLF - Backpropagation weight/bias learning function, default is Gradientdescent with momentum weight and bias learning function.

PF - Performance function, default is Mean squared error performancefunction.

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train (NET,P,T,Pi,Ai,VV,TV)

net - Neural Network created by function newff().

P - Network inputs.

T - Network targets, default = zeros.

Pi - Initial input delay conditions, default = zeros.

Ai - Initial layer delay conditions, default = zeros. VV - Structure of validation vectors, default = [].

TV - Structure of test vectors, default = [].

Train function trains a network net according to net.trainFcn and

net.trainParam. Before start of training, two parameters must be specified.

Net.trainparam.epochs: the upper limit of training iteration.

Net.trainparam.goal: the value of performance function.

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sim(net,P,Pi,Ai,T)

Simulate the well-trained network

net - Network.

P - Network inputs.

Pi - Initial input delay conditions, default = zeros.

Ai - Initial layer delay conditions, default = zeros. T - Network targets, default = zeros.

Return the output of the network.

gensim(net,st)

Generate Simulink block of the network.

net - Neural network.

st - Sample time (default = 1).

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0 20 40 60 80 100 120 140 160 180 200-4

-2

0

2

4the input signal

time

input

signalmagnitude

0 20 40 60 80 100 120 140 160 180 200-15

-10

-5

0

5the output signal

time

outputsignalmagnitude

Example: construct a Multilayer

Feedforward Network.The systems input and measured output are shown in the figures below. From these data, a

Multilayer Feedforward Network is employed to model the unknown system.

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Example: construct a Multilayer

Feedforward Network.

A Multilayer Feedforward Network was used to model the system. The

network has one hidden layer in which there are eight neurons. The inputsof the network are current value of input to the system and previous value

inputs to the system, because we consider this is a dynamic process.

Input Layer Hidden Layer (8 cells) Output Layer

U(k)

U(k-1)

U(k-2)

9 I d i N l N k d I A li i C l f SMA P 135


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Matlab Code%Using first 500 data to train the networkU=u(1:500);U_delay1=[0 u(1:499)];U_delay2=[0 0 u(1:498)];

Y=y(1:500);

%construct the input matrix to the networkP=[U;U_delay1;U_delay2];

%define the feedforward neural network with two layers

Netexample=newff([minmax(U);minmax(U_delay1);minmax(U_delay2)],[8 1],{'tansig''purelin'},'trainlm');

%set up the train parameters

Netexample.trainparam.epochs=2000;Netexample.trainparam.goal=0.01;

%train the network

predictor=train(Netexample,P,Y);%test and verify the network using later 500 data

U=u(501:999);U_delay1=u(500:998);U_delay2=u(499:997);P=[U;U_delay1;U_delay2];

NetOutput=sim(Netexample,P);

9 I t d ti t N l N t k d It A li ti t C t l f SMA P t 136


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Comparison of Networks output and

systems output

500 550 600 650 700 750 800 850 900 950 1000-25

-20

-15

-10

-5

0

5the plot of the real output and Networks output

time instant

output/prediction

Network outputreal output

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