Research Article Control Theory Using in the Air...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 145396 5 pageshttpdxdoiorg1011552013145396

Research Article119867

infinControl Theory Using in the Air Pollution Control System

Tingya Yang1 Zhenyu Lu2 and Junhao Hu3

1 Jiangsu Meteorological Observatory Nanjing 210008 China2 College of Electrical and Information Engineering Nanjing University of Information Science and Technology Nanjing 210044 China3 College of Mathematics and Statistics South-Central University for Nationalities Wuhan 430074 China

Correspondence should be addressed to Junhao Hu junhaomath163com

Received 5 August 2013 Accepted 23 September 2013

Academic Editor Bo-chao Zheng

Copyright copy 2013 Tingya Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In recent years air pollution control has caused great concern This paper focuses on the primary pollutant SO2in the atmosphere

for analysis and control Two indicators are introduced which are the concentration of SO2in the emissions (PSO

2) and the

concentration of SO2in the atmosphere (ASO

2) If the ASO

2is higher than the certain threshold then this shows that the air

is polluted According to the uncertainty of the air pollution control systems model 119867infin

control theory for the air pollutioncontrol systems is used in this paper which can change the PSO

2with the method of improving the level of pollution processing

or decreasing the emissions so that air pollution system can maintain robust stability and the indicators ASO2are always operated

within the desired target

1 Introduction

The main feature of the 119867infin

control theory is based on thefrequency designmethod of using the state-space model andthis theory presents an effective method to solve the uncer-tainty problem of external disturbance to the system In orderto overcome the drawbacks of the classical control theory andthe modern control theory 119867

infincontrol theory established

technology and method of the loop shaping in the frequencydomain which combines the classic frequency-domain andthe modern state-space method The design problem of thecontrol system is converted to the 119867

infincontrol problem

whichmade the system closer to the actual situation andmeetthe actual needs So it gives the robust control system designmethod which obtains 119867

infincontroller by solving two Riccati

equationsThismethod fully considered the impact of systemuncertainty which not only can ensure the robust stability ofthe control system but also can optimize some performanceindices It is the optimal control theory in frequency domainand the parameters design of119867

infincontroller is more effective

than optimal regulator [1]So the 119867

infincontrol theory for the air pollution control

systems can solve the uncertainty of the air pollution controlsystems model which can get the control strategy to change

the PSO2 so that air pollution system can maintain robust

stability and the indicators ASO2are always operated within

the desired target

2 Standard 119867infin

Control Problem

21 Problem Description The standard 119867infin

control problemis shown in Figure 1 which consists of119866 and119870119866 is a general-ized control targetwhich is a given part of the system119870 is119867

infin

controller and it needs to be designedIt is supposed that 119866 and 119870 are described as the transfer

function matrix of the linear time invariant system [2]Then119866(119904) and 119870(119904) are proper rational matrices Decomposing119866(119904) as

119866 = [

11986611

11986612

11986621

11986622

] (1)

Its state-space realization is

= 119860119909 + 1198611120596 + 119861

2119906

119911 = 1198621119909 + 119863

11120596 + 119863

12119906

119910 = 1198622119909 + 119863

21120596 + 119863

22119906

(2)

2 Mathematical Problems in Engineering

120596

G

K

u

y

G11120596 + G12u

G21120596 + G22u

Figure 1 The diagram of the standard119867infincontrol problem

It is denoted as

119866 =[

[

119860 1198611

1198612

1198621

11986311

11986312

1198622

11986321

11986322

]

]

(3)

where 119909 isin 119877119899 is the state vector 120596 is the external input 119906 is

the control input 119911 is the controlled output and 119910 is themea-sured output They are all vector signals To compare (1) and(3) we can obtain

119866119894119895= 119862119894(119904119868 minus 119860)

minus1119861119895+ 119863119894119895 119894 119895 = 1 2 (4)

Obviously the closed-loop transfer functionmatrix from120596 to119911 can be expressed as

119879119911120596

(119904) = 11986611

+ 11986612119870(119868 minus 119866

22119870)minus1

11986621

= 119865119897(119866119870) (5)

The problem of 119867infin

optimal control theory is to find aproper rational controller 119870 for closed-loop control systemand make the closed-loop control system internally stabileand then minimize the 119867

infinnorm of closed-loop transfer

function matrix 119879119911120596

(119904) [3]

min119870 stability119866

1003817100381710038171003817119865119897(119866119870)

1003817100381710038171003817infin

lt 1 (6)

22 State-Space 119867infin

Output-Feedback Control We assumethat the state-space representation of the generalized con-trolled object 119866 state-space representation is (2) where 119909 isin

119877119899120596 isin 119877

1198981 119906 isin 119877

1198982 119911 isin 119877

1199011 119910 isin 119877

1198752 119860 isin 119877

119899times119899 1198611isin 119877119899times1198981

1198612isin 119877119899times1198982 1198621isin 1198771199011times119899 1198622isin 1198771199012times119899 11986321and119863

22are the cor-

responding dimension of the real matrix and controller 119870 isdynamic output feedback compensator [4]

Consider that 119866 have a special form

119866 =

[

[

[

[

[

119860 1198611

1198612

1198621

0 11986312

[

119868

0] [

0

119868] [

0

0]

]

]

]

]

]

(7)

which satisfies the following conditions

(1) (119860 1198611) is stabilizable and (119862

1 119860) is detectable

(2) (119860 1198612) is stabilizable

(3) 119863119879

12[1198621

11986312] = [0 119868]

Obviously 119910 = [119909

120596 ] that is the state 119909 of generalizedcontrol object and the external input signal 120596 could be meas-ured then it can be directly used to constitute control law

As the result we can obtainTheorem 1

Theorem 1 If and only if119867infin

isin dom(Ric)119883infin

= Ric(Hinfin) ge

0 there exists 119867infin

controller

119906 = minus119861119879

2119883infin119909 + 119876 (119904) (120596 minus 120574

minus2119861119879

1119883infin119909) (8)

That is

119870 (119904) = [minus119861119879

2119883infin

minus 119876 (119904) 120574minus2119861119879

1119883infin119876 (119904)] (9)

where 119876(119904) isin 119877119867infin 119876(119904)

infinlt 120574 which is made to stabilize

the closed-loop control system and the closed-loop transferfunction matrix 119879

119911120596(119904) from 120596 to 119911 is satisfied as 119879

119911120596(119904)infin

lt

120574

Proof If Hamilton matrix 119867infin

was considered there exist119883infin

= Ric(119867infin) and then the derivative of 119909119879(119905)119883

infin119909(119905) can

be achieved such that119889

119889119905

119909119879119883infin119909 =

119879119883infin119909 + 119909119879119883infin (10)

Substituting (2) into (10) and considering Riccati equa-tion (11) and orthogonality condition

119860119879119883infin

+ 119883infin119860 + 120574minus2119883infin1198611119861119879

1119883infin

minus 119883infin1198612119861119879

2119883infin

+ 119862119879

11198621= 0

(11)

So the following formula can be achieved

119889

119889119905

119909119879119883infin119909 = 119909

119879(119860119879119883infin

+ 119883infin119860)119909

+ 120596119879(119861119879

1119883infin119909) + (119861

119879

1119883infin119909)

119879

120596

+ 119906119879(119861119879

2119883infin119909) + (119861

119879

2119883infin119909)

119879

119906

= minus100381710038171003817100381711986211199091003817100381710038171003817

2

minus 120574minus210038171003817100381710038171003817119861119879

1119883infin119909

10038171003817100381710038171003817

2

+

10038171003817100381710038171003817119861119879

2119883infin119909

10038171003817100381710038171003817

2

+ 120596119879(119861119879

1119883infin119909) + (119861

119879

1119883infin119909)

119879

120596

+ 119906119879(119861119879

2119883infin119909) + (119861

119879

2119883infin119909)

119879

119906

= minus1199112+ 12057421205962minus 120574210038171003817100381710038171003817120596 minus 120574minus2119861119879

1119883infin119909

10038171003817100381710038171003817

2

+

10038171003817100381710038171003817119906 + 119861119879

2119883infin119909

10038171003817100381710038171003817

2

(12)

If 119909(0) = 119909(infin) = 0 120596 isin 1198712[0 +infin) to integral above equa-

tion from 119905 = 0 to 119905 = infin then

1199112

2minus 12057421205962

2

=

10038171003817100381710038171003817119906 + 119861119879

2119883infin119909

10038171003817100381710038171003817

2

2minus 120574210038171003817100381710038171003817120596 minus 120574minus2119861119879

1119883infin119909

10038171003817100381710038171003817

2

2

(13)

Mathematical Problems in Engineering 3

According to the equivalence relation

1003817100381710038171003817119879119911120596

(119904)1003817100381710038171003817infin

lt 120574 lArrrArr 1199112

2lt 12057421205962

2 (14)

Form (12) and (13) we can get that

1003817100381710038171003817119879119911120596

(119904)1003817100381710038171003817infin

lt 120574 lArrrArr

10038171003817100381710038171003817119906 + 119861119879

2119883infin119909

10038171003817100381710038171003817

2

2

lt 120574210038171003817100381710038171003817120596 minus 120574minus2119861119879

1119883infin119909

10038171003817100381710038171003817

2

2

lArrrArr 119906 + 119861119879

2119883infin119909

= 119876 (119904) (120596 minus 120574minus2119861119879

1119883infin119909)

(15)

where ∙ represent Euclidean normHence 119906 = minus119861

119879

2119883infin119909 + 119876(119904)(120596 minus 120574

minus2119861119879

1119883infin119909) the proof

is completed

119906 is the input to ensure 119879119911120596

(119904)infin

lt 120574 where119876(119904)(120596 minus 120574

minus2119861119879

1119883infin119909) is the output of the transfer system

119876(119904) by the input (120596 minus 120574minus2119861119879

1119883infin119909) From (12) the decay of

119909119879(119905)119883infin119909(119905) ge 0 is the slowest when 120596 = 120574

minus2119861119879

1119883infin119909 Then

for this kind of disturbance we can stabilize (12) by using con-trol strategy 119906

3 Analysis and Synthesis of Air PollutionControl System

Atmospheric qualitymanagement is essentially the process ofthe analysis and synthesis to the air pollution control systemSo-called atmospheric system analysis is qualitative andquantitative research for an atmospheric area system orfacilities system which include that to evaluate the currentsituation to find out themain environment problems to put aseries of optional target projects and to establish the quan-titative relation between emission and the quality of the per-missive atmosphere [5]The systems synthesismeans the planand design of an air pollution control system on the basis ofsystem analysis to determine the target and to determine amanagement method of a system in other words in order toachieve certain environmental goal to select the optimalplanning scheme to optimal design to find the optimalman-agementmethod and so forth So the process of the synthesisshould include three main steps determine the target formbetter feasibility scheme of system and optimization decision[6]

Usually the atmosphere has certain self-purification abil-ity namely the atmospheric environment has a certain capac-ity It refers to the permissible pollutant emissions within thenatural purification capacity which reach the limiting quan-tity in order not to destruct the nature material circulation[7] We can make the quantity of pollutant discharged thatmeets a certain environmental goal to be permissible totalemission Only when the pollutant emissions beyond atmo-spheric self-purification ability namely exceeds the environ-mental capacity there may be air pollution [8]

Wind direction

xi yi Qi

Qi

ximinus1 yiminus1

Kixi hi(ysi minus yi)

i

Figure 2 The stirred reactor model of an ideal atmosphere section

In this paper the objective of applying the 119867infin

controltheory in air pollution control system is to find out regu-larity to make full use of atmospheric environmental self-purification ability we cannot only develop production butalso protect the environment

4 119867infin

Control of Air Pollution

In recent years air pollution control has caused great concernThis paper focuses on primary pollutant SO

2in the atmo-

sphere for analysis and control which is mainly pollutant ofthe PM25 We introduced two indicators which are the con-centration of SO

2in the emissions (PSO

2) and the concen-

tration of SO2in the atmosphere (ASO

2) Meanwhile we can

change the content of SO2in the emissions (with the method

of improving the methods of processing) with the aim ofreturning atmospheric quality back to the desired value

41 The Mathematical Model of Air Pollution A certainvolumeof airmainly accepted the controlled pollutantswhichdischarged fromcertain purification equipments In additionconsidering that the atmospheric self-purification ability ismainly affected by wind and other meteorological factors acertain volume of air can be defined as atmospheric sectionThuswe can put forward a second order state-space equationit describes the relationship between PSO

2and ASO

2on an

average point of the atmospheric section [9]The basic idea ofmodeling is to consider each section as ideal stirred reactoras shown in Figure 2 So the parameters and variables of thewhole section are consistent and the output concentration ofPSO2and ASO

2is equal to the counterpart concentration in

this section Hence from the point of view of the mass bal-ance we can get the following equation

PSO2balance equation

119894= minus119896119894119909119894+

119876119894minus1

V119894

119909119894minus1

minus

119876119894+ 119876119864

V119894

119909119894 (16)

ASO2balance equation

119910119894= ℎ119894(119910119904

119894minus 119910119894) +

119876119894minus1

V119894

119910119894minus1

minus

119876119894+ 119876119864

V119894

119910119894minus 119896119894119909119894 (17)

where 119909119894 119909119894minus1

are the PSO2of Section 119894 and Section 119894 minus 1

(mgm3) V119894is the atmospheric capacity of Section 119894 (m3)


0 1 2 3 4 5 6

Step responseA

M

T (s)

0

01

02

03

04

05

06

07

08

09

1

(a) The original closed-loop system

0

02

04

06

08

1

12

14

AM

0 05 1 15 2 25 3 35T (ms) (s)

Step response

(b) The closed-loop system with119867infin control strategy

Figure 3 The system step response

119876119864is PSO

2gas flow rate of Section 119894 (m3d) 119910

119894 119910119894minus1

are ASO2

of Section 119894 and Section 119894 minus 1 (mgm3) 119896119894 ℎ119894are daily decay

rate of PSO2of Section 119894 and supply rate of ASO

2of Section 119894

119876119894 119876119894minus1

are atmosphere gas flow rates of Section 119894 and Section119894 minus 1 (m3d) and 119910

119904

119894is saturating capacity of SO

2of Section 119894

(mgm3)From [5] we can conclude that it is advisable to take the

following values as the coefficients in above equations

119896119894= 032day ℎ

119894= 02day 119910

119904

119894= 036mgm3

119876119864

V119894

= 01

119876119894

V119894

119876119894minus1

V119894

= 09

(18)

Thus the mathematical model of Section 119894 air pollution is

[

119894

119910119894

] = [

minus132 0

minus032 minus12] [

119909119894

119910119894

] + [

01

0] 119906 + [

09119909119894minus1

09119910119894minus1

+ 19]

(19)

where 119906 is control strategy

42 Simulation of Air Pollution 119867infin

Control 119867infin

controlproblem of air pollution is basically how to control emissionsof pollutants in the best way in order to properly handle thecost of cleaning and the price we pay for too much atmos-pheric pollution Using119867

infincontrol theory can better achieve

this goal which not only saves investment but also is easy tobe realized [10]

In this paper the simulation object is two sections of the119867infin

control of atmospheric pollution the state-space equa-tion is described as

[

[

[

[

1

1199101

2

1199102

]

]

]

]

=

[

[

[

[

minus132 0 0 0

minus032 minus12 0 0

09 0 minus132 0

0 09 minus032 minus12

]

]

]

]

[

[

[

[

1199091

1199101

1199092

1199102

]

]

]

]

+

[

[

[

[

01

0

01

0

]

]

]

]

119906 +

[

[

[

[

535

109

419

19

]

]

]

]

(20)

Using the MATLAB to simulate [11] we can conclude thestep response curve of the air pollution control system Wehave

119906 (119904)

=

5706139602 (119904 + 7702) (119904 + 132)2(119904 + 12)

2

(119904 + 4012) (119904 + 177) (1199042+ 25119904 + 1563) (119904

2+ 24119904 + 144)

(21)

119870 (119904) =

[

[

[

[

[

[

[

[

minus067 minus177 minus232 minus152 minus04 01

01 0 0 0 0 0

0 01 0 0 0 0

0 0 01 0 0 0

0 0 0 01 0 0

863 436 824 692 218 014

]

]

]

]

]

]

]

]

119876 (119904)

(22)

which is the transfer function and the state-space expressionof robust119867

infincontrol strategy

It could be seen from Figure 3 that the response time ofthe original system is 36 s while the response time of closed


loop system is 0026 s by using119867infincontrol strategy It means

that119867infincontrol strategymakes the step response of the atmo-

sphere system with an improvement the response time isgreatly reduced Therefore119867

infincontrol strategy is a practical

control strategy which can ensure that air pollution controlsystem is operating steadily within the desired target [12]

5 Conclusion

In this paper the119867infincontrol theory andmethods have a great

application value on air pollution control system It can helpthe environmental protection departments at various levels toanalyze the air pollution system which can ensure the atmo-sphere quality steady work within the desired target value Ofcourse the analysis and control of a large-scale air pollutionsystem that introduced other influencing factors will be verycomplicated which will be the focus of the study in this field

Acknowledgments

This work was supported by National Natural ScienceFoundation of China (Grants nos 61104062 61374085 and61174077) Jiangsu Qing Lan Project and PAPD

References

[1] H Akashi and H Kumamoto ldquoOptimal sulfur dioxide gas dis-charge control for Osaka City Japanrdquo Automatica vol 15 no 3pp 331ndash337 1979

[2] G R Cass ldquoSulfate air quality control strategy designrdquo Atmo-spheric Environment A vol 15 no 7 pp 1227ndash1249 1981

[3] E J Croke and S G Booras ldquoThe design of an air pollutionincident control planrdquo inProceedings of the Air Pollution ControlAssoc 62nd APCA Annual Meeting Paper No 69-99 New YorkNY USA 1969

[4] P J Dejax and D C Gazis ldquoOptimal dispatch of electric powerwith ambient air quality constraintsrdquo IEEE Transactions onAutomatic Control vol AC-21 no 2 pp 227ndash233 1976

[5] R E KaHN ldquoA mathematical programming model for air pol-lution controlrdquo School Science and Mathematics pp 487ndash4941969

[6] R E Kohn A Linear Programming Model for Air Pollution Con-trol MIT Press Cambridge Mass USA 1978

[7] C P Kyan and J H Seinfeld ldquoDetermination of optimal multi-year air pollution control policiesrdquo Journal of Dynamic SystemsMeasurement andControl Transactions of theASME vol 94 no3 pp 266ndash274 1972

[8] C P Kyan and JH Seinfeld ldquoReal-time control of air pollutionrdquoAIChE Journal vol 19 no 3 pp 579ndash589 1973

[9] MWu andWGuiModern Robust ControlThePress of CentralSouth University 1998

[10] W Jiang W Cao and R JiangTheMeteorology of the Air Pollu-tion The Meteorology Press 1993

[11] M B BeckThe application of control and system theory to prob-lems of river pollution control [PhD thesis] Cambridge Univer-sity New York NY USA 1974

[12] Z Lu and D Xiao ldquoThe most optimum means of controllingthe vehicle exhaust pollution inmunicipalrdquoThe Journal ofHubeiUniversity vol 7 pp 104ndash107 2001

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120596

G

K

u

y

G11120596 + G12u

G21120596 + G22u

Figure 1 The diagram of the standard119867infincontrol problem

It is denoted as

119866 =[

[

119860 1198611

1198612

1198621

11986311

11986312

1198622

11986321

11986322

]

]

(3)

where 119909 isin 119877119899 is the state vector 120596 is the external input 119906 is

the control input 119911 is the controlled output and 119910 is themea-sured output They are all vector signals To compare (1) and(3) we can obtain

119866119894119895= 119862119894(119904119868 minus 119860)

minus1119861119895+ 119863119894119895 119894 119895 = 1 2 (4)

Obviously the closed-loop transfer functionmatrix from120596 to119911 can be expressed as

119879119911120596

(119904) = 11986611

+ 11986612119870(119868 minus 119866

22119870)minus1

11986621

= 119865119897(119866119870) (5)

The problem of 119867infin

optimal control theory is to find aproper rational controller 119870 for closed-loop control systemand make the closed-loop control system internally stabileand then minimize the 119867

infinnorm of closed-loop transfer

function matrix 119879119911120596

(119904) [3]

min119870 stability119866

1003817100381710038171003817119865119897(119866119870)

1003817100381710038171003817infin

lt 1 (6)

22 State-Space 119867infin

Output-Feedback Control We assumethat the state-space representation of the generalized con-trolled object 119866 state-space representation is (2) where 119909 isin

119877119899120596 isin 119877

1198981 119906 isin 119877

1198982 119911 isin 119877

1199011 119910 isin 119877

1198752 119860 isin 119877

119899times119899 1198611isin 119877119899times1198981

1198612isin 119877119899times1198982 1198621isin 1198771199011times119899 1198622isin 1198771199012times119899 11986321and119863

22are the cor-

responding dimension of the real matrix and controller 119870 isdynamic output feedback compensator [4]

Consider that 119866 have a special form

119866 =

[

[

[

[

[

119860 1198611

1198612

1198621

0 11986312

[

119868

0] [

0

119868] [

0

0]

]

]

]

]

]

(7)

which satisfies the following conditions

(1) (119860 1198611) is stabilizable and (119862

1 119860) is detectable

(2) (119860 1198612) is stabilizable

(3) 119863119879

12[1198621

11986312] = [0 119868]

Obviously 119910 = [119909

120596 ] that is the state 119909 of generalizedcontrol object and the external input signal 120596 could be meas-ured then it can be directly used to constitute control law

As the result we can obtainTheorem 1

Theorem 1 If and only if119867infin

isin dom(Ric)119883infin

= Ric(Hinfin) ge

0 there exists 119867infin

controller

119906 = minus119861119879

2119883infin119909 + 119876 (119904) (120596 minus 120574

minus2119861119879

1119883infin119909) (8)

That is

119870 (119904) = [minus119861119879

2119883infin

minus 119876 (119904) 120574minus2119861119879

1119883infin119876 (119904)] (9)

where 119876(119904) isin 119877119867infin 119876(119904)

infinlt 120574 which is made to stabilize

the closed-loop control system and the closed-loop transferfunction matrix 119879

119911120596(119904) from 120596 to 119911 is satisfied as 119879

119911120596(119904)infin

lt

120574

Proof If Hamilton matrix 119867infin

was considered there exist119883infin

= Ric(119867infin) and then the derivative of 119909119879(119905)119883

infin119909(119905) can

be achieved such that119889

119889119905

119909119879119883infin119909 =

119879119883infin119909 + 119909119879119883infin (10)

Substituting (2) into (10) and considering Riccati equa-tion (11) and orthogonality condition

119860119879119883infin

+ 119883infin119860 + 120574minus2119883infin1198611119861119879

1119883infin

minus 119883infin1198612119861119879

2119883infin

+ 119862119879

11198621= 0

(11)

So the following formula can be achieved

119889

119889119905

119909119879119883infin119909 = 119909

119879(119860119879119883infin

+ 119883infin119860)119909

+ 120596119879(119861119879

1119883infin119909) + (119861

119879

1119883infin119909)

119879

120596

+ 119906119879(119861119879

2119883infin119909) + (119861

119879

2119883infin119909)

119879

119906

= minus100381710038171003817100381711986211199091003817100381710038171003817

2

minus 120574minus210038171003817100381710038171003817119861119879

1119883infin119909

10038171003817100381710038171003817

2

+

10038171003817100381710038171003817119861119879

2119883infin119909

10038171003817100381710038171003817

2

+ 120596119879(119861119879

1119883infin119909) + (119861

119879

1119883infin119909)

119879

120596

+ 119906119879(119861119879

2119883infin119909) + (119861

119879

2119883infin119909)

119879

119906

= minus1199112+ 12057421205962minus 120574210038171003817100381710038171003817120596 minus 120574minus2119861119879

1119883infin119909

10038171003817100381710038171003817

2

+

10038171003817100381710038171003817119906 + 119861119879

2119883infin119909

10038171003817100381710038171003817

2

(12)

If 119909(0) = 119909(infin) = 0 120596 isin 1198712[0 +infin) to integral above equa-

tion from 119905 = 0 to 119905 = infin then

1199112

2minus 12057421205962

2

=

10038171003817100381710038171003817119906 + 119861119879

2119883infin119909

10038171003817100381710038171003817

2

2minus 120574210038171003817100381710038171003817120596 minus 120574minus2119861119879

1119883infin119909

10038171003817100381710038171003817

2

2

(13)



1003817100381710038171003817119879119911120596

(119904)1003817100381710038171003817infin

lt 120574 lArrrArr 1199112

2lt 12057421205962

2 (14)


1003817100381710038171003817119879119911120596

(119904)1003817100381710038171003817infin

lt 120574 lArrrArr

10038171003817100381710038171003817119906 + 119861119879

2119883infin119909

10038171003817100381710038171003817

2

2

lt 120574210038171003817100381710038171003817120596 minus 120574minus2119861119879

1119883infin119909

10038171003817100381710038171003817

2

2

lArrrArr 119906 + 119861119879

2119883infin119909

= 119876 (119904) (120596 minus 120574minus2119861119879

1119883infin119909)

(15)


119879

2119883infin119909 + 119876(119904)(120596 minus 120574

minus2119861119879


is completed


(119904)infin

lt 120574 where119876(119904)(120596 minus 120574

minus2119861119879





minus2119861119879

1119883infin119909 Then





Wind direction

xi yi Qi

Qi

ximinus1 yiminus1


i




4 119867infin



2in the atmo-



2) and the concen-


2) Meanwhile we can




2and ASO

2on an





119894= minus119896119894119909119894+

119876119894minus1

V119894

119909119894minus1

minus

119876119894+ 119876119864

V119894

119909119894 (16)


119910119894= ℎ119894(119910119904

119894minus 119910119894) +

119876119894minus1

V119894

119910119894minus1

minus

119876119894+ 119876119864

V119894

119910119894minus 119896119894119909119894 (17)

where 119909119894 119909119894minus1




0 1 2 3 4 5 6

Step responseA

M

T (s)

0

01

02

03

04

05

06

07

08

09

1


0

02

04

06

08

1

12

14

AM

0 05 1 15 2 25 3 35T (ms) (s)

Step response



119876119864is PSO


119894 119910119894minus1

are ASO2



2of Section 119894

119876119894 119876119894minus1


119904


2of Section 119894



119896119894= 032day ℎ

119894= 02day 119910

119904

119894= 036mgm3

119876119864

V119894

= 01

119876119894

V119894

119876119894minus1

V119894

= 09

(18)


[

119894

119910119894

] = [

minus132 0

minus032 minus12] [

119909119894

119910119894

] + [

01

0] 119906 + [

09119909119894minus1

09119910119894minus1

+ 19]

(19)



Control 119867infin






[

[

[

[

1

1199101

2

1199102

]

]

]

]

=

[

[

[

[

minus132 0 0 0


09 0 minus132 0


]

]

]

]

[

[

[

[

1199091

1199101

1199092

1199102

]

]

]

]

+

[

[

[

[

01

0

01

0

]

]

]

]

119906 +

[

[

[

[

535

109

419

19

]

]

]

]

(20)


119906 (119904)

=

5706139602 (119904 + 7702) (119904 + 132)2(119904 + 12)

2

(119904 + 4012) (119904 + 177) (1199042+ 25119904 + 1563) (119904

2+ 24119904 + 144)

(21)

119870 (119904) =

[

[

[

[

[

[

[

[


01 0 0 0 0 0

0 01 0 0 0 0

0 0 01 0 0 0

0 0 0 01 0 0

863 436 824 692 218 014

]

]

]

]

]

]

]

]

119876 (119904)

(22)










5 Conclusion



Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1003817100381710038171003817119879119911120596

(119904)1003817100381710038171003817infin

lt 120574 lArrrArr 1199112

2lt 12057421205962

2 (14)


1003817100381710038171003817119879119911120596

(119904)1003817100381710038171003817infin

lt 120574 lArrrArr

10038171003817100381710038171003817119906 + 119861119879

2119883infin119909

10038171003817100381710038171003817

2

2

lt 120574210038171003817100381710038171003817120596 minus 120574minus2119861119879

1119883infin119909

10038171003817100381710038171003817

2

2

lArrrArr 119906 + 119861119879

2119883infin119909

= 119876 (119904) (120596 minus 120574minus2119861119879

1119883infin119909)

(15)


119879

2119883infin119909 + 119876(119904)(120596 minus 120574

minus2119861119879


is completed


(119904)infin

lt 120574 where119876(119904)(120596 minus 120574

minus2119861119879





minus2119861119879

1119883infin119909 Then





Wind direction

xi yi Qi

Qi

ximinus1 yiminus1


i




4 119867infin



2in the atmo-



2) and the concen-


2) Meanwhile we can




2and ASO

2on an





119894= minus119896119894119909119894+

119876119894minus1

V119894

119909119894minus1

minus

119876119894+ 119876119864

V119894

119909119894 (16)


119910119894= ℎ119894(119910119904

119894minus 119910119894) +

119876119894minus1

V119894

119910119894minus1

minus

119876119894+ 119876119864

V119894

119910119894minus 119896119894119909119894 (17)

where 119909119894 119909119894minus1




0 1 2 3 4 5 6

Step responseA

M

T (s)

0

01

02

03

04

05

06

07

08

09

1


0

02

04

06

08

1

12

14

AM

0 05 1 15 2 25 3 35T (ms) (s)

Step response



119876119864is PSO


119894 119910119894minus1

are ASO2



2of Section 119894

119876119894 119876119894minus1


119904


2of Section 119894



119896119894= 032day ℎ

119894= 02day 119910

119904

119894= 036mgm3

119876119864

V119894

= 01

119876119894

V119894

119876119894minus1

V119894

= 09

(18)


[

119894

119910119894

] = [

minus132 0

minus032 minus12] [

119909119894

119910119894

] + [

01

0] 119906 + [

09119909119894minus1

09119910119894minus1

+ 19]

(19)



Control 119867infin






[

[

[

[

1

1199101

2

1199102

]

]

]

]

=

[

[

[

[

minus132 0 0 0


09 0 minus132 0


]

]

]

]

[

[

[

[

1199091

1199101

1199092

1199102

]

]

]

]

+

[

[

[

[

01

0

01

0

]

]

]

]

119906 +

[

[

[

[

535

109

419

19

]

]

]

]

(20)


119906 (119904)

=

5706139602 (119904 + 7702) (119904 + 132)2(119904 + 12)

2

(119904 + 4012) (119904 + 177) (1199042+ 25119904 + 1563) (119904

2+ 24119904 + 144)

(21)

119870 (119904) =

[

[

[

[

[

[

[

[


01 0 0 0 0 0

0 01 0 0 0 0

0 0 01 0 0 0

0 0 0 01 0 0

863 436 824 692 218 014

]

]

]

]

]

]

]

]

119876 (119904)

(22)










5 Conclusion



Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 3 4 5 6

Step responseA

M

T (s)

0

01

02

03

04

05

06

07

08

09

1


0

02

04

06

08

1

12

14

AM

0 05 1 15 2 25 3 35T (ms) (s)

Step response



119876119864is PSO


119894 119910119894minus1

are ASO2



2of Section 119894

119876119894 119876119894minus1


119904


2of Section 119894



119896119894= 032day ℎ

119894= 02day 119910

119904

119894= 036mgm3

119876119864

V119894

= 01

119876119894

V119894

119876119894minus1

V119894

= 09

(18)


[

119894

119910119894

] = [

minus132 0

minus032 minus12] [

119909119894

119910119894

] + [

01

0] 119906 + [

09119909119894minus1

09119910119894minus1

+ 19]

(19)



Control 119867infin






[

[

[

[

1

1199101

2

1199102

]

]

]

]

=

[

[

[

[

minus132 0 0 0


09 0 minus132 0


]

]

]

]

[

[

[

[

1199091

1199101

1199092

1199102

]

]

]

]

+

[

[

[

[

01

0

01

0

]

]

]

]

119906 +

[

[

[

[

535

109

419

19

]

]

]

]

(20)


119906 (119904)

=

5706139602 (119904 + 7702) (119904 + 132)2(119904 + 12)

2

(119904 + 4012) (119904 + 177) (1199042+ 25119904 + 1563) (119904

2+ 24119904 + 144)

(21)

119870 (119904) =

[

[

[

[

[

[

[

[


01 0 0 0 0 0

0 01 0 0 0 0

0 0 01 0 0 0

0 0 0 01 0 0

863 436 824 692 218 014

]

]

]

]

]

]

]

]

119876 (119904)

(22)










5 Conclusion



Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra















5 Conclusion



Acknowledgments


References




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Control Theory Using in the Air...

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