Research Article Control Theory Using in the Air...
Transcript of 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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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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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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Journal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
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)
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
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
Mathematical Problems in Engineering 5
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
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
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of