Multiple-input multiple-output ( MIMO ) systems normally have features of nonlinear kineticss matching. Therefore, the trouble in commanding MIMO systems is how to get the better of the matching effects between the grades of freedom. The computational load and dynamic uncertainness associated with MIMO systems make model-based decoupling impractical for real-time control. The cognition base of the fuzzed logic accountant ( FLC ) encapsulates adept cognition that consists of the data-base ( rank maps ) and rule-base. Optimization of these cognition base constituents is critical to the public presentation of the accountant, and has traditionally been achieved through a procedure of test and mistake. Such an attack is convenient for FLC ‘s holding low Numberss of input variables nevertheless for greater figure of inputs and end products ; more formal methods of Knowledge Base optimisation are required. In this work an MIMO FLC with matching FLC is optimized by GA for the cement milling procedure is presented. The proposed control algorithm was studied on the cement factory simulation theoretical account within MATLABTM and SimulinkTM environment.A The public presentations of the proposed control technique are compared with other control technique. The consequences of the simulation control survey indicate that the proposed accountant provides better public presentation compared to other control techniques.
Keywords: MIMO, FLC, GAs, Optimization, Cement Mill, Plugging.
Introduction
The bulk of the industrial procedure is nonlinear, multi input multi end product ( MIMO ) systems. The control of these systems is met with a figure of troubles due to loop interactions, dead clip and procedure nonlinearities. Cement Millss are complex treating systems with interrelated processing and drive operations [ 1 ] . Due to the built-in procedure complexivity development of an accurate theoretical account of the cement milling circuit is non a simple undertaking [ 2 ] . On some occasions, it is observed on existent workss that intermittent perturbations like case alterations in the hardness of the natural stuff may drive the factory to a part where the accountant can non stabilise the works. This is good understood by the operators as the alleged plugging phenomenon of ball Millss [ 3 ] , [ 4 ] .
Recently, multivariable control techniques ( based on Linear Quadratic Control theory ) have been introduced to better the public presentations of the milling circuit [ 5 ] . However, this accountant, whose design is based on a additive estimate of the procedure, is merely effectual in a limited scope around the nominal operating conditions. The design parametric quantities of the LQG accountant are still chosen by a test and mistake method [ 6 ] . A recent part to the cement milling circuit control focuses on a multivariable nonlinear prognostic control technique. Although this technique gives satisfactory public presentation in footings of hardiness and stableness, the design of the accountant depends purely on the mathematical theoretical account of the works [ 7 ] , [ 8 ] .
The expert system is the most appropriate solution, in most instances the fuzzed version give better consequences than the classical one [ 6 ] . The assortment of fuzzed control applications indicates that this technique is going an of import tool for complex and unknown procedure [ 9 ] . Fuzzy control is a promising new manner to confront complex procedure control jobs and the inclination is to increase their scope of pertinence in industrial procedures [ 10 ] , [ 4 ] .
Although fuzzed control theory has been successfully employed in many control technology Fieldss, its control schemes were largely designed for SISO systems, in malice of the consequence of dynamic matching on a MIMO control system. Additionally, the figure of control regulations and accountant computational load turn exponentially with the figure of variables considered. Expert-system-based solutions are effectual in commanding the procedures, this methodological analysis has built-in restrictions, since it is designed to mime a human operator with built-in decision-making restrictions [ 11 ] . In the absence of such cognition, a common attack is to optimise these FLC ‘s parametric quantities through a procedure of test and mistake [ 12 ] . This attack becomes impractical for systems holding important Numberss of input since the rule-base size grows exponentially and accordingly the figure of regulation combinations becomes significantly big [ 13 ] . The usage of Genetic Algorithms ( GA ) in this respect can supply such solutions [ 14 ] , [ 15 ] , [ 16 ] . Familial Algorithms ( GAs ) [ 17 ] are robust, numerical hunt methods that mimic the procedure of natural choice. Although non guaranteed to perfectly happen the true planetary optima in a defined hunt infinite, Genetic fuzzy systems are capable of covering with the expletive of dimensionality for complex jobs with high dimensionality [ 18 ] .
This paper is organized as follows. The cement factory circuit and the mold are described in Section 2. Then, the operation of Genetic algorithm described in Section 3. The following subdivision describes MIMO FLC design on the footing of this nonlinear theoretical account. Section 5 describes the GA MIMO Fuzzy logic accountant which has been designed and the staying subdivision explains the simulation consequences comparing, treatments and decision.
Cement Mill Circuit
A conventional representation of the cement milling circuit is depicted in fig. 1, the Cement milling circuit is an industrial procedure, which takes natural stuff as input and which produces cement holding the coveted choiceness. The natural stuff enters to the classifier after crunching procedure in the factory. The classifier separates the incoming stuff into two parts. The refused stuff i.e. the stuff that is non in the coveted choiceness is sent back to the factory for regrinding. Accepted stuff goes to the other phases of the production as the end product of the cement milling circuit [ 7 ] . The factory is fed with cement cinder at a feeding rate u [ tons/h ] . The centrifuge is driven by its rotational velocity V [ revolutions per minute ] . The shadowings are recycled at a rate year [ dozenss /h ] to the factory while the finished merchandise exits the works at a rate yf [ dozenss /h ] .
Figure 1. Conventional diagram of the cement factory circuit
In steady-state operation, the merchandise flow rate yf is needfully equal to the provender flow rate Us while the shadowings flow rate years and the burden in the factory omega may take any arbitrary changeless values. The burden in the factory depends on the input provender ( fresh provender plus shadowings flow rate ) and on the end product flow rate that depends in a nonlinear manner, on the burden in the factory and on a really of import and time-varying measure: the hardness of the stuff. Sometimes this nonlinearity may destabilise the system and the obstructor of the factory ( a phenomenon called “ plugging ” ) , which so requires an break of the cement factory crunching procedure. The burden in the factory must be controlled at a good chosen degree because excessively high a degree of the burden in the factory leads to the obstructor of the factory, while excessively low a circulating burden contributes to fast wear of the internal equipment of the factory. Furthermore, the energy ingestion of the factory ( i.e. , the ratio energy per unit merchandise ) depends on the end product of the factory that is related to the burden in the factory. A usual attack is to command the shadowings flow rate by utilizing the provender flow rate as control input. This control scheme is, nevertheless, non to the full satisfactory since it indirectly induces a loss of control of the merchandise flow rate. A right choiceness of the merchandise is besides really of import [ 5 ] . The fineness depends on the composing of the factory provender, but besides on the rotational velocity and on the air flow rate of the classifier. A natural control aim would hence be to maintain the choiceness every bit near as possible to a desired value by commanding the rotational velocity of the classifier [ 19 ] . The efficiency of a grinding circuit is dependent on three cardinal conditions [ 5 ] :
An optimal and changeless degree of stuff in the factory.
Constant air to material ratios for the centrifuge stuff.
A changeless and optimal ratio between fresh provender.
It can be seen that the interior decorator could take two of the three province variables independently, as the behaviour of the 3rd province variable would be determined upon the choice of other two. However, it is emphasized in [ 2 ] , [ 5 ] that the pick of concluding merchandise yf and factory returns year may take to unattainable values for factory outflow J ( omega, vitamin D ) , and it is suggested in [ 5 ] that maintaining yf and factory degree omega under control is a necessity.
Mathematical mold
The works is described by a simple dynamical theoretical account [ 3 ] , [ 19 ] , [ 20 ] , [ 21 ] with three province variables ( yf, year, omega ) .
( 1 )
( 2 )
( 3 )
where Tf and Tr are clip invariables of the finished merchandise and factory returns ( shadowings ) , z [ dozenss ] is the sum of stuff in the factory ( besides called the factory burden ) , 500 represents the cinder hardness, is the separation map and is the ball factory outflow rate. The crunching map is shown in fig. 2, for different values of d. It is a non monotone map of the factory burden omega. When omega is excessively high, the crunching efficiency lessenings and leads to the obstructor of the factory ( stop uping ) .
Figure 2. Crunching map Figure 3. Separation map
A low value of omega is besides unwanted because it causes a fast wear of the balls in the factory. The separation map, is a monotonically increasing map of the rotational velocity V of the centrifuge which is constrained between 0 and 1 with, shown in fig. 3.
2.2 Stability of equilibria
Assuming that the cinder hardness vitamin D is changeless, the equilibria of the system are parameterized by the changeless inputs and
( 4 )
( 5 )
( 6 )
In position of the form of, as illustrated in fig. 4, there may be zero, one, or two equilibria. There are two distinguishable equilibria when the undermentioned inequality is satisfied:
( 7 )
where is the maximal value of the map with regard to z. Linearizing the theoretical account ( 1 ) – ( 3 ) at these equilibria, the Eigen values satisfy
( 8 )
( 9 )
( 10 )
where denotes the partial derived function of with regard to omega. From ( 9 ) – ( 10 ) , we conclude that the stableness of the equilibria is determined by the mark of The equilibrium is exponentially stable if whereas it is unstable if. In fig. 4, the stable state of affairs corresponds to equilibria to the left of the upper limit of the curve, while unstable equilibria are located to the right of the upper limit. When one of the Eigen values is zero and the stableness of the equilibrium is determined by the centre manifold kineticss.
Figure 4. Equilibriums and their stableness
( 11 )
The equilibrium of ( 11 ) is unstable since for.
2.3 The plugging phenomenon
The grinding map is a non monotone map of the degree of material omega in the factory, making a upper limit for some critical value of omega. When omega is excessively high, the crunching efficiency lessenings and leads to the obstructor of the factory. This nonlinearity can do circuit instability, a phenomenon called “ plugging ” [ 11 ] , [ 19 ] , [ 20 ] , [ 22 ] , as shown in fig. 2. The plugging phenomenon manifests itself under the signifier of a dramatic lessening of the production and an irreversible accretion of stuff in the factory due to intermittent perturbations of the inflow rate and fluctuations of cinder hardness [ 20 ] .
In the theoretical account ( 1 ) – ( 3 ) with changeless inputs and, stop uping is a planetary instability which occurs every bit shortly as the province ( yf, yr, omega ) enters the set a„¦ defined by the undermentioned inequalities as in fig. 5.
for
( 12 )
( 13 )
( 14 )
Indeed, it is non hard to detect that a„¦ is a positively invariant set and that in a„¦ :
, ,
Therefore, we have that in a„¦
as ( 15 )
Hence, the degree omega of stuff in the factory is accumulated without restriction while the production rate yf goes to zero.
Figure 5. The plugging set a„¦ . ( a ) With regard to yf, ( B ) With regard to yr
Familial Algorithm
Familial algorithms ( GA ) are used as one of the optimisation techniques. It has been shown that GA besides can execute good with multimodal maps [ 23 ] ( i.e. , maps which have multiple local optima ) . Familial algorithms work with a set of unreal elements called a population. An person ( threading ) is referred to as a chromosome, and a individual spot in the twine is called a cistron. A new population ( called progeny ) is generated by the application of familial operators to the chromosomes in the old population ( called parents ) . Each loop of the familial operation is referred to as a coevals. A fittingness map, specifically, the map to be maximized or minimized, is used to measure the fittingness of an person. Consequently, the value of the fittingness map additions from coevals to coevals. In most familial algorithms, mutant is a random-work mechanism to avoid the job of being trapped in a local optimum. Theoretically, a planetary optimum solution can be found utilizing GA [ 24 ] . The basic operations of a simple familial algorithm, i.e. reproduction, crossing over and mutant, are described below.
3.1 Chromosome representation
Each person coded as a binary twine in the population is called a twine or chromosome. The ground binary strings are preferable method of GA encryption is that information is codifications as “ loosely ” as possible-in contrast to “ pack ” existent Numberss. The comprehensiveness, therefore the declaration, of the encoding determine a Gas capableness to both loosely explore and locally exploit parametric quantity hunt infinites.
3.2 Fitness map
A fittingness map ( or objective map ) is used to find the fittingness of each campaigner solution. A fittingness value is assigned to each person in the population. Integral of absolute mistake ( IAE ) is a better all-around public presentation index of closed cringle response where wave-off, subsiding and rise times are the chief public presentation considerations [ 14 ] . The IAE was hence used as a step of public presentation.
( 16 )
In Controller Design jobs IAE has to minimized, besides the accountant has to be effectual in the full operating part hence that weighted nonsubjective map applied based on the amount of IAE of lower limit and maximal setpoints of merchandise end product ( yf ) and mill degree ( omega ) hence the nonsubjective map J is calculated as the amount of IAE of yf for setpoint 120 plus IAE of yf for setpoint 140 plus IAE of omega for setpoint 60 plus IAE of omega for setpoint 80, which is shown in equation ( 17 ) .
( 17 )
3.3 Choice
The choice procedure is centered upon the specified fittingness map. The choice strategy is used to pull chromosomes from the evaluated population into the following coevals. Tournament choice is one of many methods of choice in familial algorithms. Tournament choice involves running several “ tourneies ” among a few persons chosen at random from the population as follows.
choose K ( the tourney size ) persons from the population at random
take the best person from tourney with chance P
take the 2nd best single with chance p* ( 1-p )
take the 3rd best single with chance p* ( ( 1-p ) ^2 ) ; and so on.
The victor of each tourney ( the 1 with the best fittingness ) is selected for crossing over. Selection force per unit area is easy adjusted by altering the tournament size. If the tourney size is larger, weak persons have a smaller opportunity to be selected [ 25 ] .
3.4 Crossing over
Crossover provides a mechanism for single strings to interchange information via a probabilistic procedure. Once the reproduction operator is applied, the members in the coupling pool are allowed to copulate with one another. First, the familial codifications of the two parents are assorted by interchanging the spots of codifications following the crossing over point. For illustration, see two parent strings where the crossing over point is 5 ( i.e. , the 5th spot in the twine )
P1 = 10101|010 ; P2 = 01111|100 ;
The centrifuge symbol ”| ” indicates the crossing over site.
The ensuing progeny have the followers:
P01 = 10101|100 ; P02 = 01111|010
3.5 Mutant
In each loop, every cistron is capable to a random alteration, with the chance of the pre-assigned mutant rate. In the instance of binary-coding, the mutant operator changes a spot from 0 to 1, or frailty versa. All in all, the mutant operation introduces new cistrons into the population, so as to avoid the job of being trapped in local optima. Offspring are generated from the parents until the size of the new population is equal to that of the old population. This evolutionary process continues until the fittingness reaches the coveted specifications.
Fuzzy logic Controller
The execution of the fuzzy logic based term is u ( T ) = F [ vitamin E ( T ) , I”e ( T ) ] . In the description criterion nomenclature is used to organize fuzzed set theory, for a intervention of fuzzed sets, vitamin E ( T ) , and I”e ( T ) as inputs to the map F, and u ( T ) as the end product. Associated with the map, F is a aggregation of lingual values L= { NB, NS, ZO, PS, PB } that represent the term set for the input and end product variables of F. In this instance seven lingual values are used. The significance of each lingual value in the term set L should be clear from its mnemonic ; for illustration, NB stands for negative large, NS for negative little, ZO for nothing and likewise for the positive ( P ) lingual value. Associated with the term set L is a aggregation of rank maps. I? = { I?NB, I?NS, I?ZO, I?PS, I?PB } , Each rank map ( MF ) is a map from the existent line to the interval [ -1 +1 ] . In this application the MF used is the ( triangular or trapezoidal type ) . The tallness of the MF in this instance is one, which occurs at the points optimized by GA. The realisation of the map F [ vitamin E ( T ) , I”e ( T ) ] trades with the scene of lingual values. This consists of scaling the inputs vitamin E ( T ) and I”e ( T ) suitably and so change overing them into fuzzed sets. The symbol Ce is the grading invariable for the input vitamin E ( T ) and the symbol CI”e is the grading invariable for the input I”e ( T ) . For each lingual value la?? L, assign a brace of Numberss ne ( cubic decimeter ) and I”e ( cubic decimeter ) to the inputs vitamin E ( T ) and I”e ( T ) with the associated rank map { Ne ( cubic decimeter ) =I?l ( Ce vitamin E ( T ) ) , nI”e ( cubic decimeter ) = I?l ( Cde I”e ( T ) ) } . The Numberss ne ( cubic decimeter ) and nI”e ( cubic decimeter ) , la?? L are used in the calculation of F [ vitamin E ( T ) , I”e ( T ) ] [ 26 ] .
Equally shortly as fuzzed illation is applied to each regulation, the activation degree for all end product variable ( MFs ) are obtained, and the defuzzification process takes topographic point. In order to calculate the concluding control action, u ( T ) , the most normally used method is the centre of country [ 26 ] . The consequence is the centre of country of the profile described by the rank maps, limited in the several activation degree. Equation ( 18 ) shows the defuzzified end product
( 18 )
Where is the defuzzified value, and denotes an algebraic integrating
4.1 MIMO Fuzzy control Structure
The trouble associated with using a traditional fuzzy control theory for commanding MIMO systems involves get the better ofing the consequence of matching between the grades of freedom. Therefore, the construct of adding matching accountant to counterbalance for this matching consequence was developed to heighten the control public presentation of MIMO systems. A typical dynamic theoretical account of a MIMO system is complicated with uncertainness, so model-free intelligent control schemes are employed in planing a MIMO system accountant. This work proposes a new control attack by uniting a Main FLC ( MFC ) and a suited yoke fuzzed accountant for commanding MIMO systems. The control scheme includes a MFC and a yoke fuzzy accountant which is shown in fig. 6.
Figure 6. GA MIMO FLC construction for MIMO System
4.2 Main fuzzed logic accountant
A FLC that operates with end product mistake of the system and mistake derived function in the uninterrupted clip system is adopted as the chief accountant to command each grade of freedom of MIMO systems. Here, the input variables of the MFC for between the grades of freedom of a MIMO system are defined separately as where ei is the end product mistake of the ith grade of the system ; I”ei is used for bespeaking error derived function of the ith grade of the system ; Ri is the mention input and Yi represents the system end product of the ith grade of a MIMO system.
4.3 Coupling fuzzed logic accountant
In a existent MIMO system, the control end product is influenced by more than one variable. Harmonizing to the system features, these system variables are evidently synergistic
( 19 )
( 20 )
where and stand for complex matching map that are hard to specify and deduce. Harmonizing to the analysis of the dynamic equation ( 19 ) or ( 20 ) , clearly, ui is the chief consequence and ul is the secondary consequence for the end product Yi. Similarly, for the end product Yl, ul is the chief consequence and ui is the secondary consequence. The chief consequence on the system is controlled utilizing a MFC. The secondary consequence on the system is controlled by planing an appropriate yoke fuzzed accountant.
Figure 7. GA optimized MIMO FL Control construction
The MIMO FLC control construction of cement factory procedure is shown in the fig. 7, the trouble in commanding MIMO systems is how to work out the matching effects between the grades of freedom. Therefore, an appropriate yoke fuzzed accountant is incorporated into a chief fuzzy accountant for commanding MIMO systems to counterbalance for the dynamic yoke effects between the grades of freedom. This MIMO fuzzy accountant can efficaciously take the yoke effects of the systems. Based on the rules of the Expert control algorithm, shown in fig. 8, the MIMO fuzzy logic accountant is optimized utilizing GA for a cement factory procedure. The control aim is to modulate the finished merchandise rate yf and the factory load omega at the coveted set points yf*and z* by pull stringsing the provender flow rate u and the centrifuge velocity v. Equation ( 21 ) and ( 22 ) shows the end product of MIMO FLC with matching fuzzed accountant.
( 21 )
( 22 )
Figure 8. GAFLC Design flow chart
GA FLC Design
Although fuzzed logic allows the creative activity of simple control algorithms, the tuning of the fuzzed accountant for a peculiar application is a hard undertaking and one needs a more sophisticated process than that used for a conventional accountant. This is due to the big figure of parametric quantities that are used to specify the MFs and the illation mechanisms. Several methods have been developed for tuning fuzzed accountants. These involve accommodation of the rank map [ 27 ] and scaling factors [ 28 ] and dynamically altering the defuzzification Procedure. Therefore, the attack needs as many variables as there are regulations to acquire an optimum regulation base. The advantage of the attack proposed in this paper is that it takes merely three variables to optimise the regulation base geometry, two variables to optimise the rank map and three grading variables for a individual FLC.
5.1 Encoding Rule Base
To plan an optimum regulation base a simple geometric attack is followed to modify the regulation base as mentioned in [ 29 ] the initial premises are as follows ;
The magnitude of the end product control action is consistent with the magnitude of the input values. ( i.e. in general, utmost input values ( premiss ) consequence in utmost end product values ( consequent ) , mid-range input values in mid-range end product values and small/zero input values in small/zero end product values.
Using these generalisations, in concurrence with the construct of system symmetricalness, a different attack can be used which reduces the figure of spots required for the regulation -base dramatically. The attack is a fluctuation of the method which involves a fixed co-ordinate system defined by the possible premiss combinations. The attendant infinite is so ‘overlaid ‘ upon the premiss co-ordinate system and is in consequence partitioned into 5 parts shown in fig. 9, where each part represents a attendant fuzzy set. The regulation -base is so extracted by finding the attendant part in which each premiss combination point lies. Different possible consequent infinite dividers are defined utilizing 3 parametric quantities ;
Figure 9. GAFLC Rule Assigning
Consequent-line angle, CA ( 16 angles between 0-168o ( i.e. 4 spots ) ) ( CA defines incline of the attendant line, which is used to make the infinite dividers ) .Consequent-region spacing, CS ( 4-bits ) ( CS is a proportion of the fixed-distance between premises on the co-ordinate system ( Ps ) and is used to specify the distance between attendant points along the attendant line defined by angle, CA. Its value was set to a scope between ( 0.5 – 1.5 ) times the fixed distance, Ps, utilizing a preciseness of 4 spots ) .Consequent-line order, CO ( 1-bit ) ( Defines order of attendant infinite dividers ( i.e. NB-NS-Z-PS-PB or PB-PS-Z-NSNB ) ( 1 spot ) and in consequence doubles the scope of possible attendant line angles to 0-360o ) .
5.2 Encoding rank map
In the effort to encode the FLC rank maps associated with the 2 inputs and 1 end product, a figure of premises are made in regard of the distribution of fuzzed sets across the existence of discourse ( UOD ) for each fuzzy variable. These premises are ;
The MF belongingss altered by the GA are as follows ;
1. MF form ( triangular or trapezoidal ) .
2. Degree of MF-centre displacement to consequence MF compaction or enlargement.
All evaluated FLCs contain 3 variables, vitamin E ( mistake ) , de ( error-derivative ) and u ( control-action ) . For the input variables, vitamin E and Delaware, and end product variable, u, 7 spots are used to specify the belongingss of the MFs to be optimized. For each variable, their several 7-bit GA-chromosome sections are sub-divided into 2 Fieldss ;
1. The “ offset field ” ( 3 spots ) used to consequence alteration of form of the MFs.
2. The “ companding factor ” field ( 4 spots ) used to consequence expansion/compression of the MFs.
5.3 MF Offset Field
The optimisation begins by lading a *.fis ( Matlab Fuzzy file ) into the FLC block in the MATLAB Simulink theoretical account. Each rating later uses a ‘genetically-altered ‘ version of the original FLC which is defined by a MATLAB, fuzzed construction. For each evaluated FLC, the UOD -distributed MFs are ab initio assumed to be trapezoidal in type, therefore 4 parametric quantities are required by the FIS to specify the place in the UOD of each of the MFs. The significance of these parametric quantities is illustrated below in fig. 10 & A ; fig. 11. The Matlab Fuzzy file ‘params ‘ field has 4 UOD place parametric quantities outer-left ( OL ) , inner-left ( IL ) , inner-right ( IR ) , outer-right ( OR ) . For interior parametric quantities ( IL and IR ) equal in value, MF becomes triangular in form.
5.4 MF Companding Field
Application of the offset field produces MFs of different forms ( trimf or trapmf ) and places, but does non consequence the distribution of the MFs, which are equally distributed across the UOD. To enable rating of non-uniform distributed MFs, by raising them to the power of CF ( e.g. for the Z-MF, outer-left parametric quantity ; ) Due to the usage of a normalized UOD, the place parametric quantities are shifted to different grades by this operation and the net consequence is that ;
for CF & lt ; 1: Z-MF is compressed, NB and NS expand
for CF & gt ; 1: Z-MF expands, NB and NS compress
for CF = 1: unvarying MF distribution
5.5 Encoding FLC Scaling Gains
The GA besides attempts to optimise the grading additions of the vitamin E and I”e inputs of the fuzzed accountant. Three Fieldss, e-scaling ( Ce ) , I”e-scaling ( CI”e ) and end product Cu are included in the GA chromosome each dwelling of 10- spots, which are encoded to give values of addition for the appropriate addition blocks of the Simulink theoretical account used to measure each accountant.
5.6 GA-Chromosome of MIMO FLC
Three facets of the FLC were capable to the optimisation process ; ( a ) Rule Base, ( B ) Membership Functions ( MF ) , ( degree Celsius ) Input Output Scaling Gains. The primary premise made was that for a symmetrical system, a corresponding FLC would besides exhibit symmetricalness about the set point in regard of its MFs and regulation -base. This premise was exploited in order to try to cut down the figure of spots required to specify the FLC for GA optimisation. Table 1, shows the inside informations of the variables associated with FLC Design entire 48 variables are used to optimise the chief FLC ‘s and Coupling FLC ‘s.
Simulation
The effectivity of the proposed control jurisprudence has been assessed through simulations of the theoretical account presented in equation ( 1 ) – ( 3 ) represents the works with analytical signifiers for the I† and I± map which is given below.
( 23 )
( 24 )
And the clip invariables Tf = 0.3 [ H ] , Tr = 0.01 [ H ] . These maps have been tuned in order to fit experimental measure responses of an industrial cement crunching circuit [ 31 ] . The full simulation is carried out in MATLABTM & A ; SIMULINKTM bundle installed on a Core 2 Duo Processor 2.2 GHz, 2GB RAM IBM PC Environment.
Figure 12. Simulink simulation apparatus of GA MIMO FLC of Cement factory procedure
7.1 Case I
For optimising the MIMO Fuzzy logic accountant, the GA parametric quantities are set as:
Generation =250, Population Size=50, Crossover rate =0.5, Mutation Rate = 0.03 and the Parameters associated with planing the MIMO FLC are set as shown in table 1. The MFs for vitamin E, I”e, and U of MFC 1 and 2 are set as 5 and MFs for vitamin E, I”e, and U of CFC 1 and 2 are set as 3 to cut down the computational load.
7.2 Case II
For proving the GA optimized MIMO Fuzzy logic accountant, the undermentioned scenes are chosen, with Initial setpoint values: yf = 120 tons/h and omega = 60 dozenss.
, ,
The set-point for the merchandise flow rate yf is changed from 120 to 140 tons/h at clip t=3 hours and the set-point for the factory degree omega is changed from 60 to 70 dozenss at clip t=6 hr, the hardness vitamin D varied from its nominal value 1 to 1.5 at clip t=8. The closed cringle response of the Cement factory for the undermentioned scenes without matching fuzzed accountant is shown in fig.16, and with matching accountant is shown in fig. 17.
Table 1
Optimized fuzzed logic control variables of MIMO FLC
Variabls
Name
Spots
i=1
i=2
i=3
i=4
Scope
UiCA
RB Consequent-line angle
4
8.2
6.3
7.9
8.4
[ 1, 16 ]
UiCs
RB Consequent-region spacing
4
1.32
0.96
1.30
1.23
[ 0.5, 1.5 ]
UiCo
RB Consequent-line order
1
1
1
1
1
[ 1 or 0 ]
UiO1
vitamin E MF Offset
3
0.26
0.10
0.27
0.19
[ 0, 0.5 ]
UiO2
I”e MF Offset
3
0.14
0.23
0.28
0.16
[ 0, 0.5 ]
UiO3
u MF Offset
3
0.23
0.17
0.13
0.24
[ 0, 0.5 ]
UiCF2
vitamin E MF Companding Field
4
0.50
1.00
0.9
1.30
[ 0.1, 2 ]
UiCF2
I”eMF Companding Field
4
0.80
0.50
1.25
1.00
[ 0.1, 2 ]
UiCF3
u MF Companding Field
4
1.20
0.60
1.58
0.98
[ 0.1, 2 ]
UiS1
vitamin E Scaling
10
0.33
0.21
0.87
0.96
[ 0.01, 1 ]
UiS2
I”e Scaling
10
0.47
0.58
0.76
0.85
[ 0.01, 1 ]
UiS3
u Scaling
10
107
109
38
62
[ 0.1, 200 ]
Where, i1-Main fuzzy Controller 1 ( yf ) , i2-Main fuzzy Controller 2 ( omega ) , i3-Coupling fuzzy Controller 1 ( yf-z ) and i4-Coupling fuzzy Controller 2 ( z-yf )
Table 2
Optimized Rule Base MFL 1 ( yf )
End product
u1
Mistake
Niobium
Nitrogen
Omega
PS
Lead
I”e
Niobium
Niobium
Niobium
Nitrogen
Nitrogen
Omega
Nitrogen
Niobium
Nitrogen
Nitrogen
Omega
PS
Omega
Nitrogen
Nitrogen
Omega
PS
PS
PS
Nitrogen
Omega
PS
PS
Lead
Lead
Omega
Lead
PS
Lead
Lead
Table 3
Optimized Rule Base MFL 2 ( omega )
End product
u2
Error ( vitamin E )
Niobium
Nitrogen
Omega
PS
Lead
I”e
Niobium
Niobium
Niobium
Nitrogen
Omega
PS
Nitrogen
Niobium
Niobium
Nitrogen
PS
PS
Omega
Niobium
Nitrogen
Omega
PS
Lead
PS
Nitrogen
Nitrogen
PS
Lead
Lead
Lead
Nitrogen
Omega
PS
Lead
Lead
Table 4
Optimized Rule Base CFC 1 ( )
End product
u2-1
mistake
Niobium
ZO
Lead
I”e
Niobium
Niobium
Niobium
ZO
ZO
Niobium
ZO
Lead
Lead
ZO
Lead
Lead
Table 5
Optimized Rule Base CFC 2 ( )
End product
u1-2
mistake
Niobium
ZO
Lead
I”e
Niobium
Niobium
Niobium
ZO
ZO
Niobium
ZO
Lead
Lead
ZO
Lead
Lead
Figure 13. GA optimized MFC 1 and 2 rank maps
Figure 14. GA optimized CFC 1 and 2 MF ‘s
Figure 15. Minimization of nonsubjective map
7.3 Case III
To look into the perturbation rejection of the proposed accountant, the hardness parametric quantity ( vitamin D ) is varied from 1.34 to 1.8 the setpoints of yf is set as 120 tons/h and omega is set as 60 dozenss. The hardness alteration is introduced at clip t=5 hr and the response is plotted for the different hardness values ( d=1.34,1.40,1.45,1.50,1.60,1.70,1.80 ) . The response for yf and omega are shown in figure 21 and 22.
Consequences and Discussions
The Table 1 shows the optimized fuzzed logic control design variables for the MIMO system, the tabular arraies 2 & A ; 3 shows the optimized regulation base of MFC 1 and MFC 2 and the tabular arraies 4 & A ; 5 shows the optimized regulation base of CFC 1 and CFC 2. The fig. 12 shows the Simulink execution MIMO FLC with matching fuzzed accountant for cement factory circuit. The fig. 13 & A ; fig. 14 shows the GA optimized MFs of the MFC and matching FLC and fig.14, shows the optimized rank maps of MFC and CFC of the MIMO FLC after 250 coevalss. Fig.15, shows the minimisation advancement of the public presentation Index for 250 coevals and it is minimized to 168.4 from 2754.3.
Figure 16. Closed loop response of cement factory circuit without CFC
Fig. 16, shows the closed cringle response of the fake cement factory circuit without matching accountant for the setpoint and hardness profile given in subdivision 7.2. Three variables of the cement factory ( yf, omega, and year ) is plotted for 11 hours clip. In fig. 16, when the set point alteration in finished merchandise escape ( yf ) is introduced it is noticed the factory degree is disturbed, likewise when the set point of factory degree is increased the finished merchandise escape is disturbed. This is occurred due to loop interaction, this loop interaction is eliminated by presenting the suited yoke FLC.
Figure 17. Closed loop response of GA MIMO FLC with CFC for changing setpoints and hardness
The fig.17 shows the closed cringle response of the fake cement factory circuit with GA optimized MIMO FLC with matching fuzzed accountant for the same set point and hardness profile and the end product response is plotted for 11 hours clip. When there is a sudden rise in merchandise outflow setpoint ( yf ) , the accountant is capable of maintaining the controlled variable in the set value with out wave-off and with speedy subsiding clip, likewise for the factory degree ( omega ) the accountant is really effectual. After presenting the matching accountant it is noticed that when there is a alteration of setpoint for ( yf ) or ( omega ) there is merely a little divergence in the other cringle. The comparing of MIMO accountant with and without matching accountant is tabulated for the assorted parametric quantities shown in table 6. The loop interaction is reduced more than 10 times in the MIMO FLC with matching accountant.
Table 6
Performance comparing of MIMO FLC with & A ; without CFC for changing setpoint
Setpoint alteration
Parameters
Finished Product ( yf )
Mill Load ( omega )
Without CFC
With CFC
Without CFC
With CFC
Setpoint Change
In Finished Product
yf=120 to140
( Tons/hour )
Raise Time ( Min )
4
3
Sodium
Sodium
Peak shoot ( % )
0
0
2.8
0.7
Settling clip ( Min )
50
15
14
3
IAE
10.53
4.86
8.76
0.40
Setpoint Change in Mill degree
omega =60 to 70
( Tonss )
Raise Time ( Min )
Sodium
Sodium
6
3
Peak shoot ( % )
6
0.6
3.3
0
Settling clip ( Min )
43
3
28
5
IAE
8.68
0.54
9.97
0.92
Figure 18. Controller end product of ( yf ) and ( omega ) of MIMO FLC with CFC for changing setpoints and hardness
Fig. 18 shows the end product of MIMO chief FLC ‘s and CFC ‘s, from the figure it is clear that the matching accountants reacts instantly when there is step alteration in the input or the other cringle. It is noticed that the accountant is capable of conveying back the mistake of ( yf ) and ( omega ) to zero in minimal clip when the hardness vitamin D is varied from 1 to 1.4.
Table 7-A shows the comparing of the closed response end product for set point fluctuations maintaining hardness ( vitamin D ) as 1. The public presentation of the accountants are compared with regard to the risetime ( Rt ) , Peak wave-off ( Pos ) , Peak undrshoot ( Pus ) , and settling clip ( St ) for different control strategies. The GA optimized MIMO FLC seems to be better in all public presentations.
Table 7
Performance comparing of controlled variable for changing setpoint
A- setpoint Change
B-hardness alteration
Finished Product ( yf )
Mill Load ( omega )
Finished Product ( yf )
Mill Load ( omega )
Controller types
Rt
( Min )
Polonium
( % )
St
( Min )
Rt
( Min )
Polonium
( % )
St
( Min )
Polonium
( % )
Pansa
( % )
St
( Min )
Polonium
( % )
St
( Min )
MIMO FLC with CFC
3
0
13
3
0
5
0
1.28
17
3.03
8
MIMO FLC with out CFC
4
0
50
6
3.3
28
0
4.8
38
6.7
24
Robust controller*
26
4.64
181
16.5
0
70
5.53
15.6
182
17.5
168
NRH control*
51
1.58
260
27
4.1
240
0.8
13.3
410
19.6
210
additive quadratic control*
50
1.45
255
30
4.1
250
Unstable
Non leaner larning control*
Data Not Provided
0
6.66
42
3.70
35
Nervous Network *
4
1.13
13
8
1.7
20
0.3
1.25
18
3.13
12
*Data taken from the response of [ 7 ] , [ 20 ] , [ 21 ] , [ 30 ] .
The comparing of the closed response end product for the alteration in the hardness parametric quantity ( vitamin D ) from 1 to 1.4 is presented in table 7-B. The proposed control strategy is performs better with minimal Rt, St, Pos and Pus, in where as additive quadratic control can non stabilise the factory. By comparing the public presentations shown in table 8 & A ; 9, the proposed control strategy performs better compared to the other control scheme reported in the literature ( Nonlinear robust accountant [ 20 ] , Nonlinear withdrawing skyline ( NRH ) control [ 30 ] , additive quadratic control [ 30 ] , Nonlinear larning control [ 7 ] , Neural Network based control [ 21 ] ) , besides the consequence of hardness alteration does non destabilise the cement factory.
Figure 19. Output response of finished merchandise ( yf ) for different hardness parametric quantity ( vitamin D )
Fig. 19 and 20 depicts the end product response for the simulation scenes ginven in Case III. The response for ( yf ) and ( omega ) is plotted for different hardness parametric quantities ( varied from 1.34 to 1.8 ) the response shows the subsiding clip for all hardness is more or less same. The hardness parametter is varied upto 1.8 which was non studied in privous work. The Table 8 shows the public presentation step of ( yf ) and ( omega ) for the proposed accountant strategy at assorted hardness values.
Figure 20. Output response of factory degree ( omega ) for different hardness parametric quantity ( vitamin D )
Table 8.
Performance step of proposed accountant for different hardness ( vitamin D )
Finished Product ( yf )
Mill Level ( omega )
( vitamin D )
IAE
Plutonium
( % )
St
( min )
IAE
Polonium
( % )
St ( min )
1.34
2.22
2.08
18.5
0.77
4.4
8
1.40
3.35
2.58
20
1.02
5.6
9.5
1.45
4.23
3.43
22.5
1.27
7.6
11
1.50
5.23
4.06
24
2.29
8.10
12.5
1.60
7.65
4.62
27
2.59
10.0
15
1.70
10.96
5.63
30
3.28
12.1
17.5
1.80
14.76
7.25
33
4.25
15.2
20
Decision
The literature contends that optimisation of a FLC can be considered as a geometric hunt job of a multimodal hyper surface. Optimization of a MIMO fuzzy logic accountant can turn out a drawn-out procedure when performed heuristically. In this work it has been shown that the usage of familial algorithms offers a executable method for the optimisation of the knowledge-base of the MIMO fuzzy logic accountants. The proposed attack shows a better public presentation in constructing the MIMO fuzzy logic accountants for a complex cement factory procedure. The public presentation of the accountant is studied with cement factory procedure theoretical account via simulation, and the consequences are compared with other control techiniques, the consequences demonstrate that the MIMO FLC designed by the proposed method shows better public presentation step with minimal loop interaction and forestall the cement factory procedure from stop uping.
