Introduction to intelligent control

The state of the system can be represented as a vector within that space Intelligent Control Theory: ** o Intelligent control is a class of control techniques, that use various artifici

Introduction to Intelligent Control

2.4 Neural Network Controller

2.5 Building Neural Network Controller with Matlab

2.6 Summary

3 Fuzzy Logic (FL)

3.1 Overview

3.2 Foundations of Fuzzy Logic

3.3 Fuzzy Inference System

3.4 Building systems with Fuzzy Logic Toolbox - Matlab

3.5 Sugeno-Type Fuzzy Inference

3.6 Summary

4 Conclusions

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1 Introduction

The subject of automatic controls is enormous, covering the control of variables such as temperature, pressure, flow, level, and speed

Example 1.1: Control the velocity of motor

Fig 1.1 A Simple control motor’s velocity Assume here are the velocity speeds of motor:

Control signal Set point Control value (assume)

0 1 2 3 4 5 6 7 8 9 10 0

500 1000 1500 2000 2500

0 500 1000 1500 2000 2500 3000

U

And therefore, control signal is the most important for your control strategies

Assume ݑ(݅)is control signal at present time, and it is given ݑ(݅) = ݑ(݅ − 1) + ∆ݑ (1.1)

Here, ∆ݑ is the value added and it is depended on your control methods

Example 1.2: A simple of control system (*)

Fig 1.2 Manual control of a simple process

In the process example shown (Figure 1.2), the operator manually varies the flow of water

by opening or closing an inlet valve to ensure that:

The outcome of this is that the water runs out of the tank at a rate within a required range

At an initial stage, the outlet valve in the discharge pipe is fixed at a certain position The operator has marked three lines on the side of the tank to enable him to manipulate the water supply via the inlet valve The ideal level between level 1 and level 2

The Example (Figure 1.1) demonstrates that:

o The flow valve is referred to as the Controlled Device

o To operate the controlled device is known as Control Signal

o The water itself is known as the Control Agent

o The change in water level is known as the Controlled Variable

o The level of water trying to be maintained is known as the Set Point

o The water level at steady state conditions, referred to as the Control Value

o The difference between the Set Point and the Control Value is Deviation

o If the inlet valve is closed to a new position, the water level will drop and the deviation will change A sustained deviation is known as Offset

(*):http://www.spiraxsarco.com/resources/steam-engineering-tutorials/basic-control-theory/an-introduction-to-controls.asp

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Elements of automatic control

Fig 1.3 Elements of automatic control

o The operator's eye detects movement of the water level against the marked scale indicator His eye could be thought of as a Sensor

o The eye (sensor) signals this information back to the brain, which notices a deviation The brain could be thought of as a Controller

o The brain (controller) acts to send a signal to the arm muscle and hand, which could

be thought of as an Actuator

o The arm muscle and hand (actuator) turn the valve, which could be thought of as a

Controlled Device

Example 1.3: Depicting a simple manual temperature control system (*)

The task is to admit sufficient steam (the heating medium) to heat the incoming water from

a temperature of Tଵ; ensuring that hot water leaves the tank at a required temperature of

Tଶ

Fig 1.4 Simple manual temperature control

Fig 1.5 Typical mix of process control devices with system element

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(*):http://www.spiraxsarco.com/resources/steam-engineering-tutorials/basic-control-theory/an-introduction-to-controls.asp

Note:

There are three major reasons why process plant or buildings require automatic controls:

o Safety - The plant or process must be safe to operate

o Stability - The plant or processes should work steadily, predictably and repeatably,

without fluctuations or unplanned shutdowns

o Accuracy - This is a primary requirement in factories and buildings to prevent

spoilage, increase quality and production rates, and maintain comfort These are the fundamentals of economic efficiency

Other desirable benefits such as economy, speed, and reliability are also important, but it

is against the three major parameters of safety, stability and accuracy that each control application will be measured

How to control the system?

A closed-loop controller uses feedback to control states or outputs of a dynamical system Its name comes from the information path in the system: process inputs (e.g voltage applied to an electric motor) have an effect on the process outputs (e.g velocity or torque

of the motor), which is measured with sensors and processed by the controller; the result (the control signal) is used as input to the process, closing the loop

Closed-loop controllers have the following advantages over open-loop controllers:

o Disturbance rejection (such as unmeasured friction in a motor)

o Guaranteed performance even with model uncertainties, when the model structure does not match perfectly the real process and the model parameters are not exact

o Unstable processes can be stabilized

o Reduced sensitivity to parameter variations

o Improved reference tracking performance

In some systems, closed-loop and open-loop control are used simultaneously In such systems, the open-loop control is termed feed-forward and serves to further improve reference tracking performance

Classical Control Theory: (*)

o Classical control techniques can be broken up into Frequency Domain techniques and Time Domain techniques Frequency Domain techniques are a suite of analysis and design tools that include Root Locus, Pole Placement, Bode Plots, and Nyquist State-Space techniques are the only part of Classical control that is done in the Time Domain

o The limitations of Classical Control are: Assume that the system we are trying to control is a linear system and model (the transfer function of the system) should be given as well as initial condition should be equal by zero

Modern Control Theory: (*)

o In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the time-domain state space representation, a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations

o To abstract from the number of inputs, outputs and states, the variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter only being possible when the dynamical system is linear) The state space representation (also known as the "time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs and outputs With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system

o Unlike the frequency domain approach, the use of the state space representation is not limited to systems with linear components and zero initial conditions "State space" refers to the space whose axes are the state variables The state of the system can be represented as a vector within that space

Intelligent Control Theory: (**)

o Intelligent control is a class of control techniques, that use various artificial intelligent computing approaches like neural networks, Bayesian probability, fuzzy logic, machine learning, evolutionary computation and genetic algorithms

(*):http://en.wikipedia.org/wiki/Control_theory#Classical_control_theory

(**):http://www.nd.edu/~pantsakl/control.html

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o Intelligent control does not restrict itself only to those methodologies In fact, according to some definitions of intelligent control (not all neural/fuzzy controllers) would be considered intelligent The fact is that there are problems of control which cannot be formulated and studied in the conventional differential/difference equation mathematical framework To address these problems in a systematic way, a number of methods have been developed that are collectively known as intelligent control methodologies

o The area of intelligent control is in fact interdisciplinary, and it attempts to combine and extend theories and methods from areas such as control, computer science and operations research to attain demanding control goals in complex systems

o In intelligent control problems there may not be a clear separation of the plant and the controller; the control laws may be imbedded and be part of the system to be controlled This opens new opportunities and challenges as it may be possible to affect the design of processes in a more systematic way

Conclusions

o Goal: Control system that works the same our desire

o System: Complex, uncertainty and nonlinearity and cannot derive the mathematic equations

o Control strategies: Computing the value added for next step that help the system reaches to the desire goal

o Control method: Intelligent control that uses artificial intelligent computing without regard the mathematic model of system

Fig 2.1 Biological neuron

A neuron receives input from other neurons (typically many thousands) Inputs sum (approximately) Once input exceeds a critical level, the neuron discharges a spike - an electrical pulse that travels from the body, down the axon, to the next neurons or other receptors

(*)http://www.willamette.edu/~gorr/classes/cs449/brain.html

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The axon endings (Output Zone) almost touch the dendrites or cell body of the next neuron Transmission of an electrical signal from one neuron to the next is effected by neurotransmitters, chemicals which are released from the first neuron and which bind to receptors in the second This link is called a synapse The extent to which the signal from one neuron is passed on to the next depends on many factors, e.g the amount of neurotransmitter available, the number and arrangement of receptors, amount of neurotransmitter reabsorbed, etc

Synaptic Learning:

Brains learn Of course, from what we know of neuronal structures, one way brains learn is

by altering the strengths of connections between neurons, and by adding or deleting connections between neurons Furthermore, they learn "on-line", based on experience, and typically The efficacy of a synapse can change as a result of experience, providing both memory and learning through long-term potentiation One way this happens is through release of more neurotransmitter Many other changes may also be involved

Fig 2.2 Synaptic Learning

Artificial Neuron Models

Once modeling an artificial functional model from the biological neuron, we must take into account three basic components First off, the synapses of the biological neuron are modeled as weights Let’s remember that the synapse of the biological neuron is the one which interconnects the neural network and gives the strength of the connection

For an artificial neuron, the weight is a number, and represents the synapse A negative weight reflects an inhibitory connection, while positive values designate excitatory connections The following components of the model represent the actual activity of the neuron cell All inputs are summed altogether and modified by the weights This activity is referred as a linear combination Finally, an activation function controls the amplitude of the output

Fig 2.3 Neuron model (a): Detail of neuron model (b): Neuron model shorten description

o The weighted sum net is called the net input to unit i, often written net i

o Note that w i refers to the weight from unit i

o The function f is the unit's activation function In the simplest case, f is the identity

function, and the unit's output is just its net input This is called a linear unit

a = f net

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Artificial Neural Network

A neural network consists of an interconnected group of artificial neurons, and it processes information using a connectionist approach to computation

Neural network with 1 layer (R input and S output)

Multi layer Neural network

Fig 2.5 Multi layer neural network

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Type of neural networks

Fig 2.6 A taxonomy of feed-forward and recurrent/ feedback network architectures (*)

Multilayer perceptron

A multilayer perceptron (MLP) is a feed-forward artificial neural network model that maps sets of input data onto a set of appropriate output An MLP consists of multiple layers of nodes in a directed graph, with each layer fully connected to the next one Except for the input nodes, each node is a neuron (or processing element) with a nonlinear activation function MLP utilizes a supervised learning technique called back-propagation for training the network

Fig 2.7 Multilayer perceptron (**) (*)Ani1 K Jain - Michigan State University, USA

(**)http://www.mathworks.de/help/toolbox/nnet/

Radial basis function network

A radial basis function network is an artificial

neural network that uses radial basis functions as

activation functions, such as a Gaussian

function, as the activation function Radial basis

function (RBF) networks typically have three

layers: an input layer, a hidden layer with a

non-linear RBF activation function and a non-linear output

layer A simply a Gaussian, f(x)=exp(-a*x2) (*)

Fig 2.8 Gaussian active function

Fig 2.9 Radial basis function network (**)

Competitive neural network

A competitive neural network is an artificial neural network that uses competitive functions

as activation functions Competitive learning is a process in which the output neurons of the network compete among themselves to be activated with the result that only one output neuron, is on at any time The output neuron that wins the competition is called winner-takes-all neurons and as a result of the competition, they become specialized to respond to specific features in the input data

Fig 2.10 Competitive neural network (**)

(*)http://www.willamette.edu/~gorr/classes/cs449/Maple/ActivationFuncs/active.html

(**)http://www.mathworks.de/help/toolbox/nnet/

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Kohonen’s self-organizing maps (SOM)

A self-organizing map (SOM) or self-organizing feature map (SOFM) is a type of artificial neural network that is trained using unsupervised learning to produce a low-dimensional (typically two-dimensional), discretized representation of the input space of the training samples, called a map Self-organizing maps are different from other artificial neural networks in the sense that they use a neighborhood function to preserve the topological

properties of the input space

(a)

(b) Fig 2.11 SOM topology (a): Simple Neural network topology

(b) SOM topology

`Hopfield network

A Hopfield network is a form of recurrent artificial neural network invented by John Hopfield A recurrent neural network (RNN) is a class of neural network where connections between units form a directed cycle This creates an internal state of the network which allows it to exhibit dynamic temporal behavior Unlike feed-forward neural networks, RNNs can use their internal memory to process arbitrary sequences of inputs

(a)

(b) Fig 2.12 Hopfield network (a): Hopfield network topology (*)

(b) Hopfield network in general with Matlab (**)

(*)http://en.wikipedia.org/wiki/Hopfield_network

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ART model (Adaptive Resonance Theory Neural network)

The ART1 simplified model consists of two layers of binary neurons (with values 1 and 0), called F1 (the comparison layer) and F2 (the recognition layer) Each neuron in F1 is connected to all neurons in F2 via the continuous-valued forward long term memory (LTM)

Wf , and vice versa via the binary-valued backward LTM Wb The other modules are gain

1 and 2 (G1 and G2), and a reset module Each neuron in the comparison layer receives three inputs: a component of the input pattern, a component of the feedback pattern, and a gain G1 A neuron outputs a 1 if and only if at least three of these inputs are high: the 'two-thirds rule.' The neurons in the recognition layer each compute the inner product of their incoming (continuous-valued) weights and the pattern sent over these connections The winning neuron then inhibits all the other neurons via lateral inhibition Gain 2 is the logical 'or' of all the elements in the input pattern x Gain 1 equals gain 2, except when the feedback pattern from F2 contains any 1; then it is forced to zero Finally, the reset signal

is sent to the active neuron in F2 if the input vector x and the output of F1 di

er by more than some vigilance level

Fig 2.13 ART model neural network (*)

(*) http://www.learnartificialneuralnetworks.com/art.html

Neural network applications

Figures 2.14 summaries various learning algorithms and their associated network architectures (this is not an exhaustive list) Both supervised and unsupervised learning paradigms employ learning rules based on error-correction, Hebbian, and competitive learning Learning rules based on error-correction can be used for training feed-forward networks, while Hebbian learning rules have been used for all types of network architectures However, each learning algorithm is designed for training a specific architecture Therefore, when we discuss a learning algorithm, a particular network architecture association is implied Each algorithm can perform only a few tasks well The last column of Fig 2.14 lists the tasks that each algorithm can perform

And for control application, multi layer perceptron neural network is applied

Fig 2.14 Neural network applications (*)

(*)Ani1 K Jain - Michigan State University, USA

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2.2 Neural Network Computation

Simple neuron computation

Fig 2.15 Simple neuron computation (*) For example in Fig 2.15

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Multi layers of neurons

Fig 2.18 Multi layers of neurons (*) Number of neurons in the hidden layer (Sm) depends on complexity of the problem

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Example: Computation of Feed-forward neural network

Fig 2.22 Computation of Feed-forward neural network (*) 2.3Training

A simple of training of neural network

Fig 2.23 Supervised training method for neural network

(*)http://galaxy.agh.edu.pl/~vlsi/AI/backp_t_en/backprop.html

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Example of training

(continuous the previous computation)

Training

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Fig 2.24 Example of Training (*)

(*)http://galaxy.agh.edu.pl/~vlsi/AI/backp_t_en/backprop.html

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Back propagation training algorithm

Theory of back propagation method

Fig 2.25 Multi layers neural network (*) Training Set:

Mean Square Error

Vector Case

Approximate Mean Square Error (Single Sample)

Approximate Steepest Descent

-=

f n w ( ( ) )

d w d

- d f n ( )

n d

- d n w ( )

w d

-× ( – sin ( ) n ) 2e ( 2w) ( – sin ( e2w) ) 2e ( 2w)

Application to Gradient Calculation

Gradient Calculation

w i j m , ( k + 1 ) = w i j m , ( ) k – αs i m a m j – 1

b i m ( k + 1 ) = b i m ( ) k – αs i m

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Example Function Approximation (*)

Fig 2.26 Example function approximation Here is a network

Fig 2.27 Proposed Network for function approximation

(*):Martin T Hagan: Neural Network Design

n F s

F

Fig 2.28 Structure of proposed neural network Initial Conditions

Network response with sine wave

Fig 2.29 Network response with sine wave

0.41 –

0.13 –

=

W 2 ( ) 0 = 0.09 0.17 – b 2 ( ) 0 = 0.48

Network Response Sine Wave

-1 0 1 2 3

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1

1 e + 0.54 -

0.321 0.368

d n

=

0 0.233

0.227 –

0.429

0.0495 –

0.0997

0.420 –

0.0997

0.140 –

=

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Give function

It is obtained from the capacity of function approximation of neural network

Fig 2.31 the capacity of function approximation of neural network

2.4 Neural Network Controller

A simple self turning PID controller using neural network

Fig 2.32 A simple self turning PID controller using neural network

=

Fig 2.33 Block diagram of neural network The active function is given as below:

) 1

(

) 1

( 2 )

*

Yg x g

Yg x

e Y

e x

By the way to turn Y g, we have the function shapes are in the figure 2.34

Fig 2.34 The active function shapes

How about the working of controller?

The input signal of the active function in the output layer, x, becomes:

) ( ) ( )

( ) ( )

( ) ( )

Where,

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sequence discrete

k transform Z

of operator z

time sampling

T

T

z k

e

k

e

T n e

k

e

k k

f

p

: , :

, :

) 1

(

) ( ) (

)

(

1 1

)

(k

r

θ and θ (k) are set point and output of joint of the manipulator, respectively

To tune the gains of PID controller, the steepest descent method using the following equation was applied

d d d

d

i i i

i

p p p

p

K

k E k

K k

K

K

k E k

K k

K

K

k E k

K k

∂

∂

−

= +

∂

∂

−

= +

) ( )

( )

1 (

) ( )

( )

1 (

) ( )

( )

1 (

η η η

where ηp, ηi, ηd are learning rates determining convergence speed, and E(k) is the

error defined by the following equation:

2

1 )

Using the chain rule, we get the following equations:

d d

i i

p p

K

k x x

k u u

k k

E K

k E

K

k x x

k u u

k k

E K

k E

K

k x x

k u u

k k

E K

k E

(

) ( ) ( ) ( ) ( )

(

) ( ) ( ) ( ) ( )

(

θ θ

θ θ

θ θ

The following equations are derived:

( )

) ( ) ( );

( ) ( );

( )

(

) ( ' ) (

) ( )

( ) ( )

(

k e K

k x k

e K

k x k

e K

k x

k x f x

k u

k e k

k k

E

d d

i i

p p

p r

) ( ) ( ) ( ) ( )

(

) ( ) ( ) ( ' ) ( )

( ) ( ' ) ( ) (

) ( ) ( ) ( ) ( )

(

) ( ) ( ' ) ( )

( ) ( ' ) ( ) (

) ( ) ( ) ( ) ( )

(

2

k e k e k x f u

k k

e k x f u

k k

e

K

k x x

k u u

k k

E K

k

E

k e k e k x f u

k k

e k x f u

k k

e

K

k x x

k u u

k k

E K

k

E

k e k x f u

k k

e k x f u

k k

e

K

k x x

k u u

k k

E K

k

E

d p d

p

d d

i p i

p

i i

p p

p

p p

θ θ

θ θ

θ θ

θ θ

θ θ

*

) 1

( 4 )

Yg x

e

e x

(

4 ) ( ) ( )

( )

1 (

) 1

(

4 ) ( ) ( )

( )

1 (

) 1

(

4 ) ( ) ( )

( )

1 (

Yg x

Yg x d

p d d

d

Yg x

Yg x i

p i i

i

Yg x

Yg x p

p p p

p

e

e k

e k e k

K k

K

e

e k

e k e k

K k

K

e

e k

e k e k

K k

= +

+ +

= +

+ +

= +

η η η