Adaptive Control - University of Technology, Iraq 4/Adaptiv… · Adaptive Control 4 ADAPTIVE...

32
Notes of Lecture Course in Adaptive Control for the 4 th Class of Control Engineering in the Control and Systems Engineering Department at the University of Technology CCE-CN445 Prepared By: Dr. Mohammed Y. Hassan, Ph. D.

Transcript of Adaptive Control - University of Technology, Iraq 4/Adaptiv… · Adaptive Control 4 ADAPTIVE...

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Notes of Lecture Course in

Adaptive Control

for the 4th

Class of Control Engineering in the

Control and Systems Engineering

Department at the University of Technology

CCE-CN445

Prepared By:

Dr. Mohammed Y. Hassan, Ph. D.

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Syllabus

Fourth Year-Control Eng. Theoretical: 2 Hr./ Week

CCE-CN445 (Semester)

1. Introduction and General Aspects: (4 Hrs)

What is Adaptive Control?

Relationship between Non-Adaptive, robust and Adaptive Control.

Performance objectives and design constraints for the control

engineers.

2. Types of Adaptive Control strategies: (2 Hrs)

Gain Scheduling Regulators

Self-Tuning Regulators.

Model Reference Adaptive Control.

3. Gain Scheduling Regulator: (2 Hrs)

4. Self Tuning Regulator: (8 Hrs)

Pole-Assignment by output Feedback.

5. Model Reference Adaptive Control: (6 Hrs)

6. Practical aspects and Implementation: (2 Hrs)

7. Relation to other topics: (6 Hrs)

Expert Systems.

Neuro Controller.

Fuzzy Controller.

Genetic Auto-Tuners.

Total: 30 Hrs

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References:

1. “Adaptive Control”

By: K. J. ÅstrÖm and B. Wittenmark.

2. “Self Tuning Systems”

By: P. E. Wellstead and M. B. Zarrop.

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ADAPTIVE CONTROL

1-Introduction

1.1-Definitions:

Adaptive controller:

Is a controller that can modify its behavior in response to changes in the

dynamics of the process and the disturbances.

Adaptive system:

Is any physical system that has been designed with an adaptive view

point.

1.2 Historical background:

In the early 1950, there was extensive research on adaptive control, in

connection with the design of autopilots for high performance aircraft. Such

aircraft operate over a wide range of speeds and altitudes. It was found that

ordinary constant-gain, linear feedback control could work well in one

operating condition, but that changed operating conditions led to difficulties.

A more sophisticated regulator, which could work well over a wide range of

operating conditions was therefore needed.

In the 1960s many contributions to control theory were important for the

development of adaptive control. State space and stability theory were

introduced. There were also important results in stochastic control theory.

Dynamic programming introduced by Bellman, increased the understanding of

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adaptive processes. Fundamental contributions were also made by Tsypkin,

who showed that many schemes for learning and adaptive control could be

described in a common framework as recursive equations of a special type.

There were also major developments in system identification and in parameter

estimation.

In the late 1970s and early 1980s correct proofs for stability of adaptive

systems appeared. Rapid and revolutionary progress in microelectronics has

made it possible to implement adaptive regulators simply and cheaply.

Vigorous development of the field is now taking place, both on universities and

industry. Several commercial adaptive regulators based on different ideas are

appearing on the market. A great number of industrial control loops are under

adaptive control. These include a wide range of applications in aerospace,

process control, ship steering, robotics and other industrial control systems.

To works with adaptive control one must have a background in

conventional feedback control and also sampled data systems. The reason for

this is that virtually all adaptive systems are implemented using digital

computers.

However, adaptive control has links in many directions, some of which are

illustrated in figure bellow:

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Fig. Links with adaptive control.

There are strong ties to nonlinear system theory, because adaptive systems are

inherently nonlinear.

1.3 Adaptation and tuning:

It is customary to separate the tuning and adaptation problems. In the

tuning problem it is assumed that the process to be controlled has constant but

unknown parameters; in the adaptation problem it is assumed that the

parameters are changing.

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Many issues are much easier to handle in the tuning problem. The convergence

problem is to investigate where the parameters converge to their true values.

The corresponding problem is much more difficult in the adaptive case,

because true values are changing.

1.4 Direct and Indirect adaptive controllers:

In general, the adaptive controllers can be divided into two algorithms;

direct and indirect. In direct algorithms, the parameters are updated directly.

If the controller parameters are obtained indirectly via a design procedure, we

use the term indirect algorithms. Sometimes, it is possible to re-

parameterize the process model such that it is possible to use either a direct or

indirect controller.

However, the indirect methods have sometimes been called explicit self-

tuning control, since the process parameters have been estimated. Direct

updating of the regulator parameters has been called implicit self tuning

control.

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2-Why adaptive control:

There are many reasons for adaptive control. The key factors are:

a-Variations in process dynamics. Parameters may vary due to nonlinear

actuators, changes in the operating conditions of the process, and non-

satisfactory disturbances acting on the process.

b- Variations in the character of the disturbances.

c- Engineering efficiency.

However, the most common regulators is a feedback controller with

fixed parameters. Through feedback it is possible to decrease the sensitivity to

parameter variations by increasing the loop gain of the system. The main

drawbacks of high-gain controllers are the magnitude of the control signal and

the problem of stability of the closed loop system. If there are bounds on the

uncertainties of the process parameters, it is possible to design robust

controllers by increasing the complexity of the controller. To use this

approach, it is necessary to know the structure of the process fairly accurately

and to have bounds on the variations of the parameters.

3- Adaptive schemes:

There are three main adaptive schemes:

a-Gain scheduling.

b-Self-tuning regulator (STR).

c-Model-reference adaptive control (MRAC).

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a-Gain scheduling:

In many situations it is known how the dynamics of a process change

with the operating conditions of the process. One source for the change in

dynamics may be nonlinearities that are known. It is then possible to change

the parameters of the controller by monitoring the operating conditions of the

process. This idea is called gain scheduling.

Its principle is to reduce the effects of parameter variations by

changing the parameters of the regulator as function of auxiliary variables that

correlate well with these changes in process dynamics. It is a nonlinear

feedback controller of a special type. It has a linear regulator whose

parameters are changed as a function of operating conditions in a

programmed way.

Gain scheduling was used in special cases: such as autopilots for high-

performance air-craft.

The principle:

It is sometimes possible to find auxiliary variable that correlate

well with the changes in process dynamics. It is then to reduce the effects of

parameter variations simply by changing the parameters of the regulator as

functions of auxiliary variables as shown in figure below:

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Design of Gain Scheduling regulators:

It is difficult to give general rules for designing gain scheduling

regulators. The key question is to determine the variables that can be used as

scheduling variables. It is clear that these auxiliary signals must reflect the

operating conditions of the plant. The following general ideas can be useful:

1) Linearization of nonlinear actuators.

2) Gain scheduling based on measurements of auxiliary variables.

3) Time scaling based on production rate.

4) Nonlinear transformation.

a- Self-Tuning regulator:

Basic idea:

In an adaptive system, it is assumed that the regulator parameters are adjusted

all the time. This implies that the regulator parameters follow changes in

process. It is difficult to analyze the convergence and stability properties of

such systems. To simplify the problem it can assume that the process has

constant but unknown parameters. When the process is known, the design

procedure specifies a set of desired controller parameters. The adaptive

controller should converge to these parameter values even when the process is

known. A regulator with this property is called Self-Tuning, since it

automatically tunes the controller to the desired performance.

The Self- Tuning Regulator (STR) is based on the idea of separating the

estimation of unknown parameters from the design of the controller. The basic

idea is illustrated in figure bellow:

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Fig. Block Diagram of Self Tuning Regulator (STR)

In the block diagram, the design block represents an on-line solution to the

design problem for a system with unknown parameters. This is called the

underlying design problem. Examples of the design methods can be used are:

a) Minimum variance.

b) Linear quadratic (LQ).

c) Pole placement.

d) Model following.

The design method is chosen depending on the specifications of the closed loop

systems. Different combinations of estimation methods lead to regulators with

different properties.

In order to form a Self tuning or adaptive control system, three forms of

controls can be used:

1- Pole placement (assignment) control.

2- Minimum variance control.

3- Multistage predictive control.

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General Algorithm of pole assignment:

Consider a system defined by the equation:

A.y(z)=B.u(z)+C.e(z) (10)

Where y(t) is the output, u(z) is the input and e(z) is the disturbance.

Consider a noise-free system model:

A.y(z)=B.u(z) (11)

With a controller of the form:

F.u(z)=-G.y(z)+H.r(z)

Where:

nf

nf

1

.1 Z.f..............Zf1F

ng

ng

1

.1O Z.g..............ZggG

nh

nh

1

.1O Z.h..............ZhhH

Then, the closed loop system equation is, (combining equation 10 and 11):

(F.A+B.G)y(z)=B.H.r(z)

(12)

Fig. Closed loop system with controller.

Now, if the desired closed loop pole set is defined by the roots of:

nt

nt

2

2

1

1 Z.t.......Z.tZ.t1T

)z(rG.BA.F

H.B)z(y

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then the controller coefficients which assign the actual pole set to the desired

set are given by the solution of:

(13)

Equation (13) can be written as a set of simultaneous equations by equating

coefficients of like power of iZ

.If the order nf and ng are chosen correctly,

then the set of simultaneous equations will have a unique solution.

Re-writing equation (13) in terms of the polynomial gives:

)Z.f..............Zf1(nf

nf

1

.1

. )Z.a..............Za1(na

na

1

.1

.+

)Z.a..............Zb(nb

nb

1

.1

. )Z.g..............Zgg(ng

ng

1

.1O

=

nt

nt

2

2

1

1 Z.t.......Z.tZ.t1 (14)

The above equation shows that:

a) Because the largest power of iZ

to occur on the L.H.S are (nf + na) and

(ng + nb), then it is possible to generate, (by comparing coefficients of iZ

), nf + na= ng + nb equations.

b) There are 1+ nf + ng unknown coefficients of the polynomials F and G.

Hence, foe a unique solution to equation (13), we require:

nf + na= ng + nb= nf + ng +1

or

Conditions of the design

F.A+B.G=T

nf = nb-1

ng = na-1

nt na+ nb-1

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Solution of equation (13)

Equation (13) is called the pole placement identity. It can be solved in a

number of ways. A direct solution has obtained by comparing coefficients of

iZ

leads to a matrix \vector equation with a special banded structure.

Equation (14) becomes:

)1na(

.1.na

3

.1.2

2

.1.1

1

.1

na

na

2

.2

1

.1 Z.fa..............ZfaZfaZfZ.a..............ZaZa1

)nfna(

nfna

)1nf(

nf1

nf

.nf

)2na(

.2.na

3

.2.1

2

.2 Z.f.a......Z.f.aZf...........Zfa....................ZfaZf

.......Z.gb.............ZgbZgbZ.gb.............ZgbZgb)2ng(

.ng.2

3

.1.2

2

o.2

)1ng(

.ng.1

2

.1.1

1

o.1

.

.

.

nt

nt

2

2

1

1

)nbng(

.ngnb

)1nb(

.1nb

nb

o.nb Z.t........Z.tZ.t1Z.gb.............ZgbZgb

By equating coefficients of )nfna(21Z..............Z,Z

or )nbng(Z

, we get:

1o111 tg.bfa power of 1Z

211o22112 tg.bg.bff.aa power of 2Z

.

.

.

and so on.

In matrix form, we get:

ng

1

o

nf

2

1

nbna

2nb

1nb1na

1

121

1

g

.

g

g

f

.

.

.

f

f

.

b..000a...0

...........

...........

b...b0....0

b....ba...a

......1....

...........

.........a.

0...bb0.01a

0...0b0..01

=

0

.

0

0

.

.

.

.

at

at

22

11

(15)

A c b

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A.c=b

Where:

c is a vector with controller coefficients.

A is known in linear algebra as Sylvester type matrix.

Example:

Write the identity equation for a system of 3nn ba and 1n t .

Solution:

We have F.A+B.G=T

21nn bf

21nn ag

)ZaZ.aZa1).(Z.fZf1(

3

.3

2

2

1

.1

2

2

1

.1

1

1

2

2.

1

1.o

3

.3

2

2

1

.1 Z.t1)Z.gZ.gg).(ZbZ.bZb(

Sub. in equation (15), to get:

33

2323

12312

121

1

b00a0

bb0aa

bbbaa

0bb1a

00b01

.

2

1

o

2

1

g

g

g

f

f

=

0

0

a

a

at

3

2

11

A c b

The above equation can be solved in the normal way by inverting A to obtain

the vector of controller parameters:

c=A-1

.b (16)

Equation (13) substituted in equation (12) gives the closed loop equation with

desired poles given by T, thus:

)z(rT

H.B)z(y (17)

and this gives the desired stability characteristic.

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However, we still have the problem of ensuring that the output y(z) is equal to

the reference r(z) for constant (or slowly changing) reference signal. The

simplest way is to select:

|B

TH

1Z

(18)

let hH (h is a constant value)

This choice is made so that the closed loop transfer function )T

h.B( will be (1) at

zero frequency and y(z)=r(z) for constant reference signal as required.

An Alternative procedure:

An Alternative procedure is to choose H to give the desired system zeros.

The point of this is that the zeros of a system, which in this case are the roots of

(B.H=0), in fact also influence the transient response shape. The immediate

answer is to cancel the B term by making:

B

1H (19)

But, the B part of a system can be inverse unstable, (the roots of B lie outside

the unit circle), so cancellation must be exact, since any slight error in

cancellation will leave an unstable component in the relationship between r(z)

and y(z).

Time delays:

Any time delay leads to some leading zeros coefficients in B plus possible

non-minimum phase zeros associated with the partial time delay, m. If there

are know to be (k) zero coefficients then it is useful to use the information and

substitute B.ZK for B in the relevant relations. The pole-assignment identity

will now take the form:

TG.B.ZF.AK (20)

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This equation is solved as before except that the K top rows of the B block will

be zero because of the KZ

terms.

Three-term Controller design by pole-assignment:

Three-term or PID controller combines proportional action, integral

action and derivative action and can put into the form:

)z(eZ1

Z.gZ.gg)z(u

1

2

2

1

1o

(21)

The coefficients go, g1 and g2 are related to the proportional derivative and

integral gain setting, (assuming backward shift approximation) with Ts=1

second):

Kp=- g1 – 2. g2

Kd=g2

Ki=go + g1 + g2 (H.W. Check ?)

In order to synthesize exactly the PID controller coefficients, we can assume

the system to be controlled has the following structure:

)z(u.Z.aZ.a1

Z.b)z(y

2

2

1

1

1

1

(22)

and we can assume that a1, a2 and b1 can be estimated by RLS method.

The restriction on the system model form is to ensure that only one set of PID

controller coefficients arises from the design. Combine equations (21) and (22),

we get:

1

1

2

2

1

1o

2

2

1

1

1

2

2

1

1o

1

.1

Z.b).Z.gZ.gg()Z.aZ.a1).(Z1(

)z(r).Z.gZ.gg.(Zb)z(y

(23)

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We can now select the coefficients go, g1 and g2 to give a desired closed loop

performance. Suppose we have selected a desired closed loop the rise time and

damped natural frequency and hence arrived at the corresponding desired

closed loop T polynomial:

2

2

1

1 Z.tZ.t1T (24)

Thus:

2

2

1

1

1

1

2

2

1

1o

2

2

1

1

1Z.tZ.t1Z.b).Z.gZ.gg()Z.aZ.a1).(Z1(

(25)

By equating coefficients of like power of Z, we get:

1

11

ob

)a1(tg

,

1

212

1b

)aa(tg

and

1

2

2b

ag (26)

The steady state matching of y(z) to r(z) for a constant reference signal is

ensured by the factor )Z1(1 in the denominator of equation (23).

Specifically, for zero frequency (Z=1), so that independent of the system or

controller parameters, we have y(z)=r(z) (for constant r(z))

Self-Tuning pole assignment algorithm:

The block diagram below shows the layout of a pole assignment self

tuner in which a recursive estimator is combined with a pole assignment

synthesis rule in order to continuously update the controller coefficients.

Certain performance requirements have fed into the synthesis block. These

consist of the following controller design information:

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Fig. Self-Tuning Pole Assignment System.

a) The desired closed loop pole set, specified by T.

b) The form of the controller e.g. whether it is a servo system, a regulator

or a combination of both.

In addition, further information must be supplied concerning the

configuration requirements of the self-tuner, including:

c) The sampler rate to be used.

d) The degree of the system model polynomials na, nb ….etc.

e) The delay K (for delayed systems) in the system, if known.

Provided with these information, the self –tuning system can be setup to go

through the following cycle of adaptation:

Self-Tuning cycle:

At each sample interval Ts, the following sequence of action is taken:

Step (i) Data capture:

The system output y(KT), reference input r(KT) and any other variables

of importance are measured.

Step (ii) Estimator update:

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The data acquired in (i) is used together with past data and the previous

control signal to update the parameter estimates in a model of the system

using an appropriate recursive estimator.

Step (iii) Controller synthesis:

The updated parameters from (ii) are used in a pole assignment identity

to synthesize the parameters of the desired controller.

Step (iv) Control algorithm:

The controller parameters synthesized in (iii) are used in a controller to

calculate and input the next control signal u(KT).

At the end of the cycle the control computer waits until the end of sample

interval T and then repeats the cycle for interval T+1, and so on.

Steps in the self-tuning cycle have computed sequentially. However,

figure below illustrates this in terms of a timing and sequence diagram. The

total computation time must be less than the sample interval and is

generally assumed to be much less.

Fig. Timing and sequence diagram for self-tuning controller.

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Hint: The conversion from s-plane into z-plane can be done using digital

approximations. Use the following conversions:

Continuous function sdt

dy

Forward difference: ST

1zs

Backward difference: z.T

1zs

S

Central difference: z.T.2

1zs

S

2

Where: sT is the sampling time in seconds.

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c- Model-Reference Adaptive Controller (MRAC):

The mode-Reference Adaptive Controller is one of the main approaches of

adaptive control. The basic principle is illustrated in figure below:

Fig. Block diagram of RMAC.

The desired performance is expressed in terms of a reference model, which

gives the desired response to a command signal. The system also has an

ordinary feedback loop composed of the process and the regulator. The error

(e) is the difference between the output(s) of the system and the reference

model. The regulator has parameters that are changed, based on the error.

There are thus two loops: an inner loop, which provides the ordinary control

feedback and outer loop, which adjusts the parameters in the inner loop. The

inner loop is assumed faster than the outer loop.

In this adaptive system, two ideas were introduced. First, the performance of

the system is specified by a model. Second, the parameters of the regulator are

adjusted based on the error between the reference model and the system.

There are essentially three basic approaches to analysis and design of MRAC:

1- The gradient approach.

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2- Lyapunov functions.

3- Passivity theory.

For a system with adjustable parameters, the model reference adaptive

methods gives a general approach for adjusting the parameters so that the

closed loop transfer function will be close to a prescribed model. This is called

the Model-following problem.

One important question is how small we can make the error (e)?. This depends

both on the model, the system and the command signal. If it is possible to make

the error equal to zero for all command signals, the perfect model following is

achieved.

Model Following:

The model-Following problem can be solved using pole placement design.

Model –Following is a simple and useful way to formulate and solve a servo

control problem. The basic idea is very simple. Servo performance is specified

indirectly by giving a mathematical model for the desired response. The

specified model can be linear as well as nonlinear. The parameters in the

system are adjusted in order to get the output of the process model (y) as close

as possible to the reference model output (ym) for a given class of input signals.

Optimization methods are thus natural tool in MRAC design.

The Gradient approach:

This is a fundamental idea in the MRAC approach. The parameters

adjustment scheme is usually called MIT rule.

The MIT rule:

Assume that we attempt to change the parameters of the regulator so that the

error between the output(s) of the process and the reference model is driven to

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zero. Let (e) denote the error and () represents the parameters. Introduce the

criterion:

2e

2

1)(J (27)

where J is called the loss function. To make J small, it is possible to change the

parameters in the direction of the negative gradient of (J). i. e.:

(28)

It is assumed that the parameters change much more slowly than the other

variables in the system, then the derivative

ecan be evaluated under the

assumption that is constant. The derivative

eis the sensitivity derivative of

the system. The adjustment rule of equation (28), where

eis the sensitivity

derivative, is commonly referred to as the MIT rule. The choice of loss function

is, of course, arbitrary. If the loss function is chosen as:

e)(J

Thus, the adjustment rule becomes:

)e(signe

dt

d

Equation (28) also applies to the case of many adjustable parameters. The

variable should be interpreted as a vector and

eis as the gradient of the

error with respect to the parameters.

n321i

......

n321i

e......

eeee

Je

.e.J

dt

d

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for 2

e2

1)(J :

n321

e......

eee.e.

dt

d

Note:

In the MIT rule, it is not necessary to require perfect model-following. The

procedure can be applied to nonlinear systems. The method can also be used to

handle partially known systems.

The MIT rule will perform well if the adaptation gain () is small.

Consequently, it is not possible to give fixed limits that guarantee stability. The

MIT rule can thus give an unstable closed loop system.

Lyapunov’s Second method (1892):

Lyapunov introduced an interesting direct method to investigate the stability

of a solution to a nonlinear differential equation. The key idea is illustrated in

figure below:

Fig. Illustration of Lyaunov’s method for investigating stability.

The equilibrium will be stable if we can find a real function on the state space

whose level curves enclose the equilibrium such that the derivative of the state

variable always towards the interior of the level curves.

To state the results formally, let the differential equation be:

)t,x(fx.

and

f(0,t)=0

where x is a state vector of dimension n. It is assumed that f is such that

solutions exist for all o

tt . The equilibrium point o

t at the origin. This

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involves no loss of generality since this can be achieved through a simple

coordinate transformation.

Lyapunov Stability Theorem:

Let the function RR:V1n

satisfy the conditions:

1. 0)t,0(V for all t R.

2. V is differentiable in X and t.

3. V is positive definite.

4. V.

is a negative definite (or negative Semi-definite).

Then the equilibrium state at the origin is uniformly asymptotically stable.

If )t,x(V as ,x then equilibrium state at the origin is uniformly

asymptotically stable in the large.

Then V is called Lyapunov function.

Note:

a. Positive definiteness of a scalar function: A scalar function V(x) is said to be

positive definite in a region includes the origin if v(x) > 0 for all nonzero

states x and v(0)=0.

b. Negative definiteness scalar function: A scalar function V(x) is said to be

negative definite if –V(x) is positive definite.

c. Positive semi-definiteness of a scalar function: A scalar function V(x) is said

to be positive semi-definite if it is positive at all states except at the origin and

at certain other states where it is zero.

d. Negative Semi-definiteness of a scalar function: A scalar function V(x) is

said to be negative semi-definite if –V(X) is positive semi-definite.

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Expert Control:

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Neural Networks:

In an expert system, the control law is a logic function that gives the

control action as a function of senor patterns. The function is adaptive in the

sense that it will adjust itself automatically.

The Perceptorn proposed by Rosenbatt (1959) is one way to obtain a learning

function. To describe the perceptorn , let (ui with i=1,2,3,…., n) be inputs and

(yi with i=1,2,3,…, n) be outputs. In the perceptron, the output is formed as:

n

1jjiji

)bu.w(fy , i=1,2,3,…,n

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where wij is the weight, b is the bias and f is the threshold function.

As an example:

0ifx0

0ifx1)x(f

To update the weight, the perceptron uses a very simple idea, which is called

Hebb’s principle. Apply a given pattern to the inputs and clamp the outputs to

the desired response, then increase the weights between nodes that are

simultaneously excited. This principle was formulated in Hebb (1949), in an

attempt to model neuron networks:

)yy).(t(u.)t(wwj

0

jiijij

where 0

jy is the desired response and

jy is the response predicated by the

model. By regarding the weight as a parameter, it becomes clear that updating

formula is identical to gradient method for parameter estimation.

There are several ways to use Neural Network in control systems. In the following,

some of these methods:

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