A new approach for automatic control modeling, analysis and design in fully fuzzy environment

8/9/2019 A new approach for automatic control modeling, analysis and design in fully fuzzy environment

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A new approach for automatic controlmodeling, analysis and design in fully fuzzy

environment

Abstract - The paper presents a new approach for the modeling, analysis and

design of automatic control systems in fully fuzzy environment based on the

normalized fuzzy matrices. The approach is also suitable for determining the

propagation of fuzziness in automatic control and dynamical systems where all

system coecients are expressed as fuzzy parameters. A new consolidity chart is suggested based on the recently newly developed system

consolidity index for testing the susceptibility of the system to withstand

changes due to any system or input parameters changes eects.

mplementation procedures are elaborated for the consolidity analysis of

existing control systems and the design of new ones, including systems

comparisons based on such implementation consolidity results. Application of

the proposed methodology is demonstrated through illustrative examples,

covering fuzzy impulse response of systems, fuzzy !outh-"urwitz stability

criteria, fuzzy controllability and observability. #oreover, the use of the

consolidity chart for the appropriate design of control system is elaboratedthrough handling the stabilization of inverted pendulum through pole placement

techni$ue. t is shown also that the regions comparison in consolidity chart are

based on type of consolidity region shape such as elliptical or circular, slope or

angle in degrees of the centerline of the geometric, the centroid of the

geometric shape, area of the geometric shape, length of principal diagonals of

the shape, and the diversity ratio of consolidity points for each region. %inally, it

is recommended that the proposed consolidity chart approach be extended as a

uni&ed theory for modeling, analysis and design of continuous and digital

automatic control systems operating in fully fuzzy environment.

Keywords: Intelligent systems; Fuzzy automatic control systems; Consolidity theory; Fuzzy Routh Hurwitz stability criterion; Fuzzy controllability

and observability; Fuzzy inverted pendulum system

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There are many approaches that are carried out to handle this problem using

the conventional fuzzy theory. These approaches suer many drawbacs

such as the processing of the solution becomes very cumbersome with the

increase of system dimensionality. #oreover, the results obtained by suchlie approaches are not linear and thus not reversible, leading to that the

results obtained in the forward path will be dierent than the bacward path

/3-:81. 4ther techni$ues, using the direct implementation of fuzzy

matrices, also have many shortcomings :/-:;1. The main hindrance of their

spread is heavily related to their impracticability of their present operations

(#ax, #in, #ax.#in, and #in.#ax+ as they do not reisual

fuzzy logic-based representations :?-;81. The approach was based on the

normalized fuzzy matrices, where every parameter is expressed by it value

and corresponding fuzzy level. t is shown by =abr :3, ;81 that the

proposed Arithmetic fuzzy logic-based representation has corresponding

mathematical functions with the conventional fuzzy theory. "owever, the

new approach provides a much easier arithmetic rather than logic

calculations forum that maes its application much practical and eective.

!eported methodological experimentations and case studies applications of

the Arithmetic and >isual fuzzy logic-based representations were successful

in solving some preliminary classes of fuzzy global optimization and

operations research techni$ues :?-:91.

7uch fuzziness approach has resulted in the appearance of a new

system index named as @system consolidity index /. Consolidity (the

act and !uality of consolidation+ is a measured by the systems output

reactions versus combined inputBsystem parameters reaction when

sub'ected to varying environments and events ;/-;21. #oreover,

1 Consolidity could be regarded as a general internal property of physical systems that can also be defined far

from fuzzy logic or rough sets. Other consolidity indices, however, could be defined by researchers but the concept

will still remain the same.

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consolidity can govern the ability of systems to withstand changes when

sub'ected to incurring events or varying environments @on and above

normal operation during the system change pathway.

Though the topic of consolidity theory has commenced recently in

year :8/8, there are some good applications of such theory for the analysis

and design of automatic control theory. n such applications, the opposite

relationship between consolidity versus both stability and controllability of

state space representation systems was investigated in ;;1. #oreover,

several examples of applications to automatic control systems were carried

out such as the fuzzy design of inverted pendulum using pole replacement

method, the optimal design of the fuzzy linear $uadratic regulator problem,

and the fuzzy )yapunov stability analysis of the drug concentration control

problem ;01. n all these applications, the overall values of consolidity index

(average of calculated consolidity points+ are only considered in the study

without going into any further investigations of the geometric distributions

or the diversity analysis of the various consolidity points.

n the following section, the implementation of the consolidity theory

is elaborated for further use in the analysis and design of control systems.

2. Methodology development

2.1 Description of the operating fuzzy environments

The fuzzy environments used in system consolidity analysis could be

classi&ed as shown in Table /, and their representations are elucidated in%ig.:.

n order not to complicate the mater unduly in this paper, our analysis

will be based on classesO E

and B E

. The system consolidity analysis applies

similarly to other fuzzy environment classes.

2.2 The consolidity methodology

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The consolidity methodology for the analysis and design of control

systems is based in modeling the system input and parameters as fuzzy

variables, leading to a corresponding output of the similar fuzzy nature. Asystem operating at a certain stable original state in fully fuzzy environment

is said to be consolidated if its overall output is suppressed corresponding

to their combined input and parameters eect, and vice versa for

unconsolidated systems. Ceutrally consolidated systems correspond to

marginal or balanced reaction of output, versus combined input and system.

n general, the output fuzziness behavior towards input fuzziness could

dier from one system to another. Dxamples of these behaviors areE

i+ 4utputs could absorb the input fuzziness and gives smaller or

diminishing output fuzziness.

ii+ 4utputs could yield almost the same level of input fuzziness.

iii

+

4utput could give higher output fuzziness compared to input fuzziness.

)et us assume a general system operating in fully fuzzy environment,

having the following elementsE

Input arameters!

),(ii I I

V I =(/+

such thatmiV i I ,,2,1, = describe the deterministic value of input

componenti I

, andi I

indicates its corresponding fuzzy level.

"ystem arameters!

),(

j j

S S V S =

(:+

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such that

n jV jS

,,2,1, =

denote the deterministic value of system parameter

jS

, and

jS

denotes its corresponding fuzzy level.

#utput arameters!

),(iOiO

V O =(;+

such that

k iV iO

,,2,1, =designate the deterministic value of output

componentiO, and

iO designates its corresponding fuzzy level.

Fe will apply in this investigation, the overall fuzzy levels notion, &rst

for the combined input and system parameters, and second for output

parameters. As the relation between combined input and system with output

is close to (or of the lie type+ of the multiplicative relations, the

multiplication fuzziness property is applied for combining the fuzziness of

input and system parameters.

%or the combined input and system parameters, we have for the

weighted fuzzy level to be denoted as the combined Input and "ystem

$uzziness $actor S I F +

, given asE

∑=

∑= ⋅+

∑=

∑= ⋅=+ n

j

V

n

jV

m

i

V

m

iV

F

j

j j

i

ii

S

S S

I

I I

S

1

1

1

11

"

(?+

7imilarly, for the #utput $uzziness $actor O F

, we have

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∑=

∑=

⋅

=k

iV

k

i

V

F

i

ii

O

OO

O

1

1

.

(0+

)et the positive ratio GO F

BS I F +

G where de&nes the "ystem

Consolidity Index , to be denoted as

)/( S I O F +

. Hased on)/( S I O F +

the

system consolidity state can then be classi&ed as :8-;;1E

i#Consolidated if

)/( S I O F +

$ %& to be referred to as 'Class C("

ii# %eutrally Consolidated if

)/( S I O F +

≈

%& to be denoted by 'Class

)("iii

# &nconsolidated if

)/( S I O F +

*%& to be referred to as 'Class +("

%or cases where the system consolidity indices lie at both consolidated

and unconsolidated parts, the system consolidity will be designated as a

mixed class or @Class ,.

The selection of the fuzzy levels testing scenarios for both the system

and input should follow same usual consideration. %irst of all the input and

system fuzzy values for system consolidity testing are selected as integer

values to be preferably in the range I 9 for open fuzzy environment and in

the range I ? for bounded fuzzy environments. Cevertheless, the output

fuzzy level could assume open values beyond these ranges based on the

overall consolidity of the system. "owever, all over implementation

procedure in the paper, the exact fraction values of fuzzy levels are

preserved during the calculations and are rounded as integer values only at

the &nal results ;?1.

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t is remared that the typical ranges of the consolidity indices

)/( S I O F +based on previous real life applications are as followsE very lo'

(J8.0+, lo' (8.0 to /.0+, moderate (/.0 to 0+, high (0 to /0+, and very

high (K/0+ ;?-;01. A good practical consolidated system should have the

value of consolidity index)/( S I O F + L /.0.

2.( The consolidity chart

The concept of implementing the consolidity theory to the analysis and

design of control system is to plot for each system its consolidity chart

de&ned as the relation between the #utput $uzziness $actor GO F

G in the

vertical axis (y+ versus the Input and "ystem $uzziness $actor GS I F +

G in

the horizontal axis (x+.

The best way for setching each systems region in the consolidity

chart is to calculate representative points of output fuzziness factor (y-

axis+ versus input and the system fuzziness factor (x-axis+ and plot all

these x-y points &rst in the chart. The average consolidity index is then

calculated based on these points and its value will represent the slope of the

center line of the region under study. The boundary of the region can thenbe setched around this center line embodying all (or the ma'ority+ of these

fuzziness x-y points.

Dxamples of the consolidity regions or patterns of various consolidity

classes are summarized in Table : and setched in %ig. ;. The shapes of each

region are assumed for simplicity of the elliptical. "owever, other geometric

shapes such as the circular one could tae place for various applications.

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Hased in these consolidity patterns, the degree of susceptibility of the

system to withstand the eect of changes in system and input parameters

can be evaluated. n fact, Consolidity index is an important factor in

scaling system parameter changes when sub'ected to events or varyingenvironment. %or instance, for all coming events at say event state M which

are @on and above the system normal situation or stand will lead to

consecutive changes of parameters. 7uch changes follow the general

relationship at any event step M asE N 5arameter *hange M O $unction

consolidity M, varying environment or event M1 ;P-;91. Two important

common cases in real life of such formulation are the linear (or linearized+

and the exponential relationship.

2.) Implementation approach of consolidity theory to control systems

-"."% Implementation approach for e/isting control systems

%or the existing man0made or natural systems, the situation could be

complicated. The testing of these existing systems could reveal the poor

consolidity of such system. This is $uite expected as we previously used to

build all existing systems without considering the new concept of system

consolidity. %or existing man0made systems, the situation could be possible

by altering parameters of the system within the utmost extend permitted for

changes. As for natural systems, the system consolidity improvement matter

could also be possible by interfering within the system parameters together

with environment and trying to direct the physical process towards better

targeted consolidity.

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fuzziness de&ned as their pairs or shadows+. *onventional mathematics are

then applied to the basic variables while the appropriate fuzzy algebra is

implemented on their corresponding pairs or shadows (fuzziness+.

5arameters substitutions are made at the end step of solution leading to thecalculation of the consolidity factors as speci&ed by the problem analyst.

%or complicated symbolic manipulations (and computations+ the use of

#atlab 7ymbolic Toolbox, #athematica or similar lie software libraries could

be highly eective to foster the consolidity theory through conducting its

necessary derivations. This will enable the implementation of the consolidity

analysis to wider classes of linear, nonlinear, multivariable and dynamic

problems with dierent types of complexities.

2.* "ystems comparison based on the consolidity indicesimplementation results

The &rst step in the design of any speci&c problem is to carry out the

consolidity indices of all various available scenarios. These results are

extracted to only one overall consolidity index based on the design

philosophy to be followed. Dxamples of such design basis are given asE

i( Average consolidity scoresE n this case, the average of all

scored consolidity indices for each scenario is calculated and

used solely for the selection of the most appropriate

consolidated design.

ii(+eighted average consolidity scoresE %or the situations

where the possibility of the input and system fuzziness are

nown, the weighted average values of the consolidity by these

given possibilities are calculated and the results are used in the

selection of design.

iii(+orst consolidity scoresE n this case, the worst score of the

consolidity index is chosen as an overall evaluation index. This

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could be the maximum (or minimum+ of all scored indices if we

are seeing the superior (or inferior+ consolidated design Another in-depth direction of comparisons is through plotting regions

(patterns+ of fuzziness behavior in the setched consolidity chart similar to

the ones shown in %ig. ;. t could appear from such &gure that each

consolidity region changes from system to another and follows certain

geometric shapes such as the elliptical, circulars, or others. The geometric

features of each consolidity region can be characterized by the following

featuresE

7ymbol 6escriptionR Type of consolidity region shape such as elliptical, circular, or

other shapes.2 7lope or angle in degrees of the centerline of the geometric

shapes of the overall consolidity index 7(degrees+O tan-/(overall

consolidity index+C 3 ( / & y + The centroid of the geometric shape R expressed by its

horizontal coordinate x per unit (pu+ and the vertical

coordinate y per unit.

4

Area of the geometric shape of

R

in pu:

l% )ength of geometric (ma'or+ diagonal in direction of slope S of

the consolidity region (pu+.l- )ength of geometric (minor+ diagonal in perpendicular to the

slope 7 of the consolidity region (pu+.l- 5l% 6iversity ratio of consolidity points (unitless+.

The comparison or raning of each consolidity region will be based on

less slope R , less area A and less diversity ratio (l- 5l%+. #oreover, the

position of the centroid C 3 ( / & y + (upward or downward+ within the

geometric shape main centerline depends mainly on the nature of the

aected input in


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cncc z z z s ,...,, 21= are closed-loop fuzzy zeros, since their corresponding

values of (/;+ are zero.

Fe present now the handling of the general form of fuzzy control system

modeling and analysis using representative examples of the fourth-order

systems.

4. Methodology implementation to control systems fuzzy impulse

response

Fe demonstrate in this section how a fourth order system of the transfer

function as expressed by (/2+ can be handled in a fully fuzzy environment

where all the system coecients are expressed in the Arithmetic fuzzy logic-

based representation form. )et us introduce this example that describes the

fuzzy response of a high-order control system operating in fully fuzzy

environment. Fe introduce the example in a general form of fourth-order

open-loop transfer function, as follows /21E

).()..()(

212

1

00

c sc sb s s

a s X

+++=

(/P+

where110 ,, cba

and2c

are fuzzy parameters.

D$uation (/P+ may be written using partial fraction representation asE

23

211

0)2/(

.)(

cc s

! s

b s

B

s

A s X ++

++++= (/9+

where

,,,,,4/2123 ! B Accc −=

and3c

are fuzzy coecient. D$uating coecient

of (/P+, we get

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2100

121121

11112

3

..:)(

..)..(0:)(

..).(0:)(

0:)(

cb Aa s

b !c Bcbc A s

b c Bcb A s

B A s

=

+++=

+++=

++=

(/3+

Qsing the =aussian Dlimination techni$ue, the matrix e$uation of (/3+ can

be solved with its corresponding fuzzy levels.

Illustrative ,xample 1:

As a numerical example, we choose the value of fuzzy parameters as

shown in Table ;. The results of parameters,,, B A

and !

are also shown in

the same table. Accordingly, the inverse )aplace transform of (/3+ can be

expressed asE

].cos.sin.)[(..)( 332

42.

0 1.1 t ct cc" " B At X t ct b −−−= −− (:8+

such that

−= 214

c

!

where

,,,,,,,311

ccb ! B A

and4

are fuzzy parameters.

The consolidity pattern of the problem described by plotting the overall

output fuzziness factorGO F

G versus input fuzziness factor GS I F +

G is shown in

%ig. 0. The impulse response output solution pattern reveals slight

unconsolidated distribution of the results, indicating relatively lo'susceptibility of the optimal solution for change versus any system and

input parameters changes eect. Hased on consolidity chart of %ig. 0, it can

be seen that the control system is almost consolidated of class @

.

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%or the selected &rst four scenarios shown in Table ;, the fuzzy levels

of impulse responses are given also in Table ; and %ig. 2. The e$uations

were solved in Dxcel sheet with built-in functions programmed using >isual

Hasic Applications (>HAs+. n the implementation procedure, the exactvalues of fuzzy levels are preserved all over the calculations and are

rounded to integer values only at the &nal result. t follows from the setches

of the impulse time response of %ig. 2 and Table ? that the fuzziness is

related to the time instant. The color of the response is an indication of the

fuzzy level using the color coding shown in Table 0. 7uch colors are selected

arbitrarily without restricting that each corresponding positive and negative

colors are con'ugates (summation is either white or blac+. This is e$uivalent

to the >isual fuzzy logic-based representation :?-:P1.

The analysis of the consolidity chart of the impulse response problem

of Illustrative example 1 shown in %ig. 0 can be summarized as followsE

7ymbol #eaning !esultsR 7hape type Dlliptical2 7lope /0./2R, or

tan-/(8.:P/98+

C 3 ( / & y + *entroid (;.9?9/,/.2:8;+ 4 Area of shape /;.P?/? pu:

l% )ength of ma'or

diagonal

2.39P; pu

l- )ength of minor

diagonal

:.0;/2 pu

l- 5l% 6iversity ratio 8.;2:;

The results indicate that the consolidity region has a very low overall

consolidity index and moderate diversity ratio. The area of the consolidity

region R is also moderate supporting the moderate diversity of calculated

consolidity points.

7imilar approaches can be applied for the fuzzy pulse response of digital

or discrete data control systems expressed by their z-transform transfer

functions /2-/P1.

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5. Methodology implementation to fuzzy Routh!ur"itz sta#ility

criterion

The wor of !outh and "urwitz /21 gives a method of indicating the

presence and number of unstable roots, but not their value. *onsider the

general form of characteristic e$uation expressed by (/?+. The !outh-

"urwitz stability criterion states thatE %or there to be no roots with positive

real parts then it is a necessary, but not sucient, condition that all

coecients in the characteristic e$uation have the same sign and that none

is zero. f the above is satis&ed, then the necessary and sucient condition

for stability is eitherE

a

+

All the "urwitz determinants of the polynomial are positive, or alternatively

b

+

All coecients of the &rst column of !outh array have the same sign. The

number of sign changes indicates the number of unstable roots.

The "urwitz determinants of (/0+ can be expressed as /21E

20

31211

aa

aa !a ! ==

42

531

6420

7531

4

31

420

531

3

00

00

aa

aaa

aaaa

aaaa

!

aa

aaa

aaa

! ==

(:/+

Setc , such that all parameters are expressed in the Arithmetic fuzzy logic-

based representation form. All the above determinant operations are carried

out following the fuzzy logic-based algebra operations :31.

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Illustrative example 2!

)et us chec the stability of the closed-loop control system of Illustrative

example 2, where the open-loop transfer function is expressed in (/P+, and

is having a unity feedbac. The closed-loop characteristic function can be

expressed asE

0).).(.( 0212

1 =++++ ac sc sb s s (::+

or e$uivalently

0..)..().( 0122

1123

114 =++++++ a sbc sbcc sbc s (:;+

where110 ,, cba and

2c

are fuzzy parameters following the scenarios given in

Table 2. Applying now the !outh-"urwitz criterion, we have &rst to test that

all the coecients are present and have the same sign. The second test is to

chec the fuzzy determinants,, 21 ! !

and3

!

for dierent scenarios.

The "urwitz determinants for this numerical example can be expressed

asE

121 .bc ! = (:?+

1121

11122

.

.

bcca

bcbc !

++

=(:0+

and

1112

1121

1112

3

.0

1.

0.

bcbc

bcca

bcbc

!

+++

=

.

(:2+

The numerical results are shown for the example in Table 2.

The "urwitz determinants of the polynomial are all positive with

various level of fuzziness. The fuzzy levels of determinants describe the level

of uncertainly in the results. %or dierent scenarios, we will have same fuzzy

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region R is moderate supporting the moderate diversity of calculated

consolidity.

$. Methodology implementation to control systems fuzzy

controlla#ility and o#serva#ility

A system is said to be controllable if a control vector

)(t u

exists that will

transfer the system from any initial state

)( #t x

to some &nal state

)(t x

in a

&nite time interval.

A system is said to be observable if at time0t , the system state

)( 0t xcan

be exactly determined from observation of the output

)(t yover a &nite time

interval.

f the system is described by

u ! x y

u B x A x

..

..

+=+=

(:P+

then a sucient condition for complete state controllability is that the n x n

matrix /21E

].1::.:[ Bn A B A B $ −= (:9+

contains n linearly independent row or column vectors, i.e. is of ran n (that

is, the matrix is non-singular, and the determinant is non-zero+. D$uation

(:9+ is called the Controllability matrix.

The system described by (:9+ is completely observable if the n x n

matrix, denoted as the #bservability matrix /21E

].1)(::.:[ % n% A% % A% & −= (:3+

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where all the system coecients are expressed in the Arithmetic fuzzy logic-

based representation form.

Illustrative example (!

*onsider the state space representation of the closed-loop system of

Illustrative example (E

.0

0

0

1000

0100

0010

4

3

2

1

32104

3

2

1

u

x

x x x

aaaa x

x x x

+

=

−−−−•

•

•

•

α

(;8+

and

[ ]

=

4

3

2

1

.000

x

x

x

x

y β

.

(;/+

such that121 .bca =,

1122 .bcca +=,

113 bca += and

β α ,are fuzzy

parameters, where2=α

and

1=β .

)et us form the *ontrollability matrix $ asE

].3:.2:.:[ B A B A B A B $ =

−+−+−−

+−−

−=

333

.2.212323

1

2323

10

3100

1000

4

aaaaaaa

aaa

a

α

.

(;:+

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indicating relatively medium susceptibility of the both conditions for

change versus any system and input parameters changes eect. Hased on

consolidity charts of %igs. 9 and 3, it can be seen that the control system

performance is unconsolidated of class @U

for both controllability and

observability.

The analysis of the consolidity chart of the fuzzy system

controllability results of illustrative example ( setched in %ig. 9 can be

summarized as followsE

7ymbol #eaning !esults

R 7hape type Dlliptical2 7lope 23.:;R or

tan-/ (:.229/+C 3 ( / & y + *entroid (:.;:3/,2.;P3P+

4 Area of shape /9.3P/; pu:l% )ength of ma'or

diagonal

P.03?3 pu

l- )ength of minor

diagonal

;./;3: pu

l- 5l% 6iversity ratio 8.?/;;

The results show that the consolidity region has a moderate overall

consolidity index and relatively high diversity ratio. The area of the

consolidity region R is very high supporting the high diversity of

calculated consolidity points.

As for the consolidity chart of the fuzzy system observability results

of illustrative example ( delineated in %ig. 3, the analysis can be

summarized as followsE

7ymbol #eaning !esultsR 7hape type Dlliptical2 7lope P8./8R or

tan-/ (:.P3P2+C 3 ( / &

y +

*entroid (:.?;8?,2.09::+

4 Area of shape /;.P?/? pu:l% )ength of ma'or

diagonal

2.39P; pu

l- )ength of minor diagonal

:.;:3/ pu

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l- 5l% 6iversity ratio 8.;;;;

The results show that the consolidity region has a moderate overall

consolidity index and low diversity ratio. The area of the consolidity region R

is high supporting the spread of calculated consolidity points.

7imilar approach can be applied for testing the fuzzy controllability

and observability of digital control systems expressed by their discrete time

state e$uations /2-/P1.

%. Methodology implementation to modeling and design of invertedpendulum sta#ilization

n this section, the suggested approach is implemented for the fuzzy

modeling and stabilization of the inverted pendulum system (Illustrative

example )+ as shown in %ig. /8. The inverted pendulum problem is an

example of producing a stable closed-loop control system from an unstable

plant. %or this system, it is possible to design a controller using the pole

placement techni$ues /21.

n the &gure,m

is the mass of pendulum, '

denotes the half-length of

the pendulum and $ ′

is the mass of the trolley. The parameter)(t F

indicates the applied force to the trolley in the x

-direction.

t is assumed thatθ

is small and the second-order terms

)( 2θ

can be

neglected, then we can de&ne the state variables of the inverted pendulum

system as (= (

3.9/+E

,1 θ = x

,2 θ

= x

x x =3

and

x x =4

(;?+and the control variable is

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)(t F u =.

(;0+

%rom (;?+ and (;0+, the state e$uations become

u

b

b

x

x

x

x

a

a

x

x

x

x

+

=

4

2

4

3

2

1

41

21

4

3

2

1

0

0

000

1000

000

0010

(;2+

where

( )[ ]

( )

( )[ ]

( )

−+′+

+′=

−+′−

=

−+′−

=

−+′+′=

mm $

m

m $ b

mm $ 'b

mm $

m ( a

mm $ '

m $ ( a

.3.4

.31.

1

.3.4.

3

.3.4

.3

.3.4.

).(.3

4

2

41

21

(;P+

=

0.0

.00

0.0

.00

2414

2414

2212

2212

bab

bab

bab

bab

$

.

(;9+

6ata for simulation are represented in Table 3 for dierent selected

scenarios. The output e$uation is

x y .= (;3+

where

is the identity matrix. %or a regulator, with a scalar control variable

and gain vector K

, we have

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x K u .−=.

(?8+

The elements of K

can be obtained by selecting a set of desired closed-

loop poles by the AcermannUs formula /21 for system stabilization through

the pole placement techni$ue. )et

[ ] )(..1000 1 A $ K φ −= (?/+

where $

is the Controllability matrix and

I A A A A nnn ...)( 01

11 α α α φ ++++=

−−

(?:+

where A

is the system matrix andi

α

represent the coecients of the

desired closed-loop characteristic e$uation.

f the re$uired closed-loop poles are

22 j s ±−= for the pendulum, and

44 j s ±−= for the trolley, then the closed-loop characteristic e$uation

becomesE

02561927212 234 =++++ s s s s

.

(?;+

The algebraic form of the fuzzy gain result can be expressed as followsV let

[ ] [ ]

1

4321 1000

−

= $ β β β β (??+

then we can attain by simple matrix operation the following fuzzy values of

the gain vector K

E

( )41

.21

.441

.22

221

.421

.2

.11

aaaaa#

k α α β α α α β ++

++=

(?0+

( ) 4133213112

.... aak α β α α β ++= (?2+

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033 .α β =k (?P+

and

134 .α β =k

.(?9+

%or design purposes, various prototypes of the inverted pendulum

can be selected with dierent relative trolley and pendulum masses of

dierent manufacturing materials and with gain vector K

satisfying the

same stabilized characteristics function of (;?+. The most desired robust

design can be attained that achieves the best consolidity performance. n

this respect, &ve dierent designs of trolley car masses $ ′

of 9, ?, :, / and

8.2 respectively are selected as shown in %ig. //.

%or each design, the consolidity analysis is applied and the result is

shown in Table 9 for *ase / of $ ′

O 9 and setched all the three cases in

%ig. /:. %rom this &gure, it appears that the consolidity pattern of the

inverted pendulum improves with the reduction of the trolley car mass $ ′

and the best consolidity performance is obtained for the &fth design case

with $ ′

O 8.2. The analysis of the consolidity chart of the fuzzy inverted

pendulum of various selected designs of illustrative example ) delineated

in %ig. /: can be summarized as followsE

7ymbol !esults of various designs of $ ′

9.8 ?.8 :.8 /.8 8.2

R Dlliptical Dlliptical Dlliptical Dlliptical Dlliptical

2 P?.0/R or

tan-/

(;.2898+

2;.32R or

tan-/

(:.8?2P+

0/.?:R or

tan-/

(/.:0;?+

;3.28R or

tan-/

(8.9:P?+

;:.:;R or

tan-/

(8.2;80+

C 3 ( / &

y +

(8.0:08,/.3:

08+

(8.9088,/.9?

88+

(/.?088,/.9?

08+

(:.8P08,/.08

88+

(:.0888,/.08

88+

4( pu-+ /.:088 /.:P08 /.2088 :.:P08 :.0888

l%( pu+ /.3088 /.3088 :.:888 :.?888 :.0088

l- 8.9888 8.9088 8.3088 /.:888 /.:088

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l- 5l% 8.?/8; 8.?;03 8.?;/9 8.0888 8.?38:

t follows from the above table that reducing the trolley weight the

system overall consolidity is improved but is accompanied with increase of

the consolidity region areas. n general, the diversity ratio is high indicating

high diversity of calculated consolidity points. The areas of the regions are in

general very small compared to all previous illustrative examples ( 4 *

%6+. 7uch charts of %ig. /: clearly demonstrate the eectiveness of the

proposed approach as a tool for the analysis and design of automatic control

systems.

&. 'onclusions

A new approach for the fuzzy automatic control systems modeling and

analysis using the consolidity theory was presented. Wey implementations

issues in fuzzy automatic control and dynamical systems were addressed in

a very smooth and systematic way. These issues covered systems fuzzy

impulse response, systems stability using !outh-"urwitz criterion, systems

*ontrollability and 4bservability, and the stabilization of inverted pendulum

through the pole placement techni$ue. llustrative examples of fourth-order

systems were solved to demonstrate the eectiveness and applicability of

29


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the new techni$ue. The approach is also suitable for higher dimensional

automatic control and dynamical systems as it is based mainly on matrix

formulations.

mplementation procedures were elaborated for the consolidity

analysis of existing control systems and the design of new ones. 7ystems

comparisons based on such implementation consolidity results were

discussed based the values of the average, weighted or worst consolidity

scores, or the comparison of the plotted regions (patterns+ of fuzziness

behavior in the setched consolidity chart. t is shown that the regions

comparison in consolidity chart are based on type of consolidity region

shape, slope or angle in degrees of the centerline of the geometric, the

centroid of the geometric shape, area of the geometric shape, length of

principal diagonals of the shape, and the diversity ratio of consolidity points

for each region.

The suggested approach could open the door for more general

modeling, analysis and design of both continuous and digital data automatic

control systems. #oreover, it provides an eective tool for designing new

control systems that could withstand future changes due to any system or

input parameters changes eects on and above normal systems operations

or set-points. Dxamples of some foreseen extensions are as followsE

i+ 6esign of %uzzy 5, 5, 56, and 56 control systems.ii+ 7tate-space methods for fuzzy automatic control system analysis and

design.iii+ 6esign of fuzzy state observers and estimators for closed-loop systems.iv+ 4ptimal and robust fuzzy control of multivariate systems.v+ 4ther problems such as multivariate fuzzy Walman state estimation,

fuzzy linear $uadratic regulators, fuzzy )yapunov stability criterion,

..etc.

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%inally, it is shown that the presented consolidity methodology is

open in its application to wide classes of systems. Dven for the system that

thought not to be fuzzy, we can still imagine that these systems are

operating in a &ctions fully fuzzy environment and perform typically thesame consolidity testing. tUs needless to say that all the present physical

systems in our daily life are sub'ect to continuous wearing and deterioration

that mae them gradually changing, and thus will behave later e$uivalently

as if they are operating in a fuzzy environment. This maes the presented

consolidity chart approach generally extendable to wider spectrum of real

life applications beside that given in the automatic control &elds. Dxamples

of these &elds are geology, archeology, life sciences, ecology, environmental

science, engineering, materials, medicine, biology, sociology, humanities,

and many other important &elds

(c)no"ledgements

The author is indebted to the reviewers for their constructive comments that

have led to improving considerably the analysis given in the paper.

31


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+ist of ,igures 'aptions

,igure

1

Two examples of continuous and digital data control systems operatingin fuzzy environments.

,igure >arious classi&cations of dierent fuzzy environments.

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2,igure

3

The consolidity chart showing six dierent classes of system consoliditypatterns (regions+ as described in Table :.

,igure

4

mplementation approach of consolidity theory for both existing and newsystems under design.

,igure

5

*onsolidity chart of the impulse response problem of Illustrativeexample 1.

,igure

$

%uzzy impulse response using visual representation of Illustrativeexample 1.

,igure

%

*onsolidity chart of the !outh-"urwitz stability criterion of Illustrativeexample 2.

,igure

&

*onsolidity pattern of the fuzzy system controllability results of Illustrative example (.

,igure

-

*onsolidity pattern of the fuzzy system observability results of Illustrative example (.

,igure

1

7etch showing main parameters of the inverted pendulum of Illustrativeexample ).

,igure

11

>arious selected design prototypes of inverted pendulum of dierent

manufacturing materials of Illustrative example ) (*ase /E $ ′

O 9,

*ase :E $ ′

O ?, *ase ;E $ ′

O :, *ase ?E $ ′

O /, *ase 0E $ ′

O 8.2+.

,igure

12

*onsolidity patterns of the fuzzy inverted pendulum designs of Illustrative example ).

A new approach for automatic control modeling, analysis and design in fully fuzzy environment

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