Multiple-input multiple-output (MIMO) systems usually have characteristics of nonlinear dynamics coupling. Therefore, the difficulty in controlling MIMO systems is how to overcome the coupling effects between the degrees of freedom. The computational burden and dynamic uncertainty associated with MIMO systems make model-based decoupling impractical for real-time control. The knowledge base of the fuzzy logic controller (FLC) encapsulates expert knowledge that consists of the data-base (membership functions) and rule-base. Optimization of these knowledge base components is critical to the performance of the controller, and has traditionally been achieved through a process of trial and error. Such an approach is convenient for FLC's having low numbers of input variables however for greater number of inputs and outputs; more formal methods of Knowledge Base optimization are required. In this work an MIMO FLC with coupling FLC is optimized by GA for the cement milling process is presented. The proposed control algorithm was studied on the cement mill simulation model within MATLABTM and SimulinkTM environment. The performances of the proposed control technique are compared with other control technique. The results of the simulation control study indicate that the proposed controller provides better performance compared to other control techniques.
Keywords: MIMO, FLC, GAs, Optimization, Cement Mill, Plugging.
Introduction
The majority of the industrial process is nonlinear, multi input multi output (MIMO) systems. The control of these systems is met with a number of difficulties due to loop interactions, dead time and process nonlinearities. Cement mills are complex processing systems with interconnected processing and drive operations [1]. Due to the inherent process complexivity development of an accurate model of the cement milling circuit is not a simple task [2]. On some occasions, it is observed on real plants that intermittent disturbances like instance changes in the hardness of the raw material may drive the mill to a region where the controller cannot stabilize the plant. This is well understood by the operators as the so-called plugging phenomenon of ball mills [3], [4].
Recently, multivariable control techniques (based on Linear Quadratic Control theory) have been introduced to improve the performances of the milling circuit [5]. However, this controller, whose design is based on a linear approximation of the process, is only effective in a limited range around the nominal operating conditions. The design parameters of the LQG controller are still chosen by a trial and error method [6]. A recent contribution to the cement milling circuit control focuses on a multivariable nonlinear predictive control technique. Although this technique gives satisfactory performance in terms of robustness and stability, the design of the controller depends strictly on the mathematical model of the plant [7], [8].
The expert system is the most appropriate solution, in most cases the fuzzy version give better results than the classical one [6]. The variety of fuzzy control applications indicates that this technique is becoming an important tool for complex and unknown process [9]. Fuzzy control is a promising new way to face complex process control problems and the tendency is to increase their range of applicability in industrial processes [10], [4].
Although fuzzy control theory has been successfully employed in many control engineering fields, its control strategies were mostly designed for SISO systems, in spite of the effect of dynamic coupling on a MIMO control system. Additionally, the number of control rules and controller computational burden grow exponentially with the number of variables considered. Expert-system-based solutions are effective in controlling the processes, this methodology has inherent limitations, since it is designed to mimic a human operator with inherent decision-making limitations [11]. In the absence of such knowledge, a common approach is to optimize these FLC's parameters through a process of trial and error [12]. This approach becomes impractical for systems having significant numbers of input since the rule-base size grows exponentially and consequently the number of rule combinations becomes significantly large [13]. The use of Genetic Algorithms (GA) in this regard can provide such solutions [14], [15], [16]. Genetic Algorithms (GAs) [17] are robust, numerical search methods that mimic the process of natural selection. Although not guaranteed to absolutely find the true global optima in a defined search space, Genetic fuzzy systems are capable of dealing with the curse of dimensionality for complex problems with high dimensionality [18].
This paper is organized as follows. The cement mill circuit and the modeling are described in Section 2. Then, the operation of Genetic algorithm described in Section 3. The next section describes MIMO FLC design on the basis of this nonlinear model. Section 5 describes the GA MIMO Fuzzy logic controller which has been designed and the remaining section explains the simulation results comparison, discussions and conclusion.
Cement Mill Circuit
A schematic representation of the cement milling circuit is depicted in fig. 1, the Cement milling circuit is an industrial process, which takes raw material as input and which produces cement having the desired fineness. The raw material enters to the classifier after grinding process in the mill. The classifier separates the incoming material into two parts. The refused material i.e. the material that is not in the desired fineness is sent back to the mill for regrinding. Accepted material goes to the other stages of the production as the output of the cement milling circuit [7]. The mill is fed with cement clinker at a feeding rate u [tons/h]. The separator is driven by its rotational speed v [rpm]. The tailings are recycled at a rate yr [tons /h] to the mill while the finished product exits the plant at a rate yf [tons /h].
Figure 1. Schematic diagram of the cement mill circuit
In steady-state operation, the product flow rate yf is necessarily equal to the feed flow rate u while the tailings flow rate yr and the load in the mill z may take any arbitrary constant values. The load in the mill depends on the input feed (fresh feed plus tailings flow rate) and on the output flow rate that depends in a nonlinear way, on the load in the mill and on a very important and time-varying quantity: the hardness of the material. Sometimes this nonlinearity may destabilize the system and the obstruction of the mill (a phenomenon called "plugging"), which then requires an interruption of the cement mill grinding process. The load in the mill must be controlled at a well chosen level because too high a level of the load in the mill leads to the obstruction of the mill, while too low a circulating load contributes to fast wear of the internal equipment of the mill. Moreover, the energy consumption of the mill (i.e., the ratio energy per unit product) depends on the output of the mill that is related to the load in the mill. A usual approach is to control the tailings flow rate by using the feed flow rate as control input. This control strategy is, however, not fully satisfactory since it indirectly induces a loss of control of the product flow rate. A correct fineness of the product is also very important [5]. The fineness depends on the composition of the mill feed, but also on the rotational speed and on the air flow rate of the classifier. A natural control objective would therefore be to keep the fineness as close as possible to a desired value by controlling the rotational speed of the classifier [19]. The efficiency of a grinding circuit is dependent on three key conditions [5]:
An optimum and constant level of material in the mill.
Constant air to material ratios for the separator material.
A constant and optimum ratio between fresh feed.
It can be seen that the designer could choose two of the three state variables independently, as the behavior of the third state variable would be determined upon the selection of other two. However, it is emphasized in [2], [5] that the choice of final product yf and mill returns yr may lead to unachievable values for mill outflow j (z,d) , and it is suggested in [5] that keeping yf and mill level z under control is a necessity.
Mathematical modeling
The plant is described by a simple dynamical model [3],[19],[20],[21] with three state variables (yf, yr, z).
(1)
(2)
(3)
where Tf and Tr are time constants of the finished product and mill returns (tailings), z [tons] is the amount of material in the mill (also called the mill load), d represents the clinker hardness, is the separation function and is the ball mill outflow rate. The grinding function is shown in fig. 2, for different values of d. It is a non monotonic function of the mill load z. When z is too high, the grinding efficiency decreases and leads to the obstruction of the mill (plugging).
Figure 2. Grinding function Figure 3. Separation function
A low value of z is also undesirable because it causes a fast wear of the balls in the mill. The separation function, is a monotonically increasing function of the rotational speed v of the separator which is constrained between 0 and 1 with , shown in fig. 3.
2.2 Stability of equilibria
Assuming that the clinker hardness d is constant, the equilibria of the system are parameterized by the constant inputs and
(4)
(5)
(6)
In view of the shape of, as illustrated in fig. 4, there may be zero, one, or two equilibria. There are two distinct equilibria when the following inequality is satisfied:
(7)
where is the maximum value of the function with respect to z. Linearizing the model (1)-(3) at these equilibria, the eigen values satisfy
(8)
(9)
(10)
where denotes the partial derivative of with respect to z. From (9)-(10), we conclude that the stability of the equilibria is determined by the sign of The equilibrium is exponentially stable if whereas it is unstable if . In fig. 4, the stable situation corresponds to equilibria to the left of the maximum of the curve, while unstable equilibria are located to the right of the maximum. When one of the eigen values is zero and the stability of the equilibrium is determined by the center manifold dynamics .
Figure 4. Equilibria and their stability
(11)
The equilibrium of (11) is unstable since for.
2.3 The plugging phenomenon
The grinding function is a non monotonic function of the level of material z in the mill, reaching a maximum for some critical value of z. When z is too high, the grinding efficiency decreases and leads to the obstruction of the mill. This nonlinearity can cause circuit instability, a phenomenon called "plugging" [11], [19], [20], [22], as shown in fig. 2. The plugging phenomenon manifests itself under the form of a dramatic decrease of the production and an irreversible accumulation of material in the mill due to intermittent disturbances of the inflow rate and variations of clinker hardness [20].
In the model (1)-(3) with constant inputs and , plugging is a global instability which occurs as soon as the state (yf, yr, z) enters the set Ω defined by the following inequalities as in fig. 5.
for
(12)
(13)
(14)
Indeed, it is not difficult to observe that Ω is a positively invariant set and that in Ω:
, ,
Therefore, we have that in Ω
as (15)
Hence, the level z of material in the mill is accumulated without limitation while the production rate yf goes to zero.
Figure 5. The plugging set Ω. (a) With respect to yf , (b) With respect to yr
Genetic Algorithm
Genetic algorithms (GA) are used as one of the optimization techniques. It has been shown that GA also can perform well with multimodal functions [23] (i.e., functions which have multiple local optima). Genetic algorithms work with a set of artificial elements called a population. An individual (string) is referred to as a chromosome, and a single bit in the string is called a gene. A new population (called offspring) is generated by the application of genetic operators to the chromosomes in the old population (called parents). Each iteration of the genetic operation is referred to as a generation. A fitness function, specifically, the function to be maximized or minimized, is used to evaluate the fitness of an individual. Consequently, the value of the fitness function increases from generation to generation. In most genetic algorithms, mutation is a random-work mechanism to avoid the problem of being trapped in a local optimum. Theoretically, a global optimal solution can be found using GA [24]. The basic operations of a simple genetic algorithm, i.e. reproduction, crossover and mutation, are described below.
3.1 Chromosome representation
Each individual coded as a binary string in the population is called a string or chromosome. The reason binary strings are preferred method of GA encoding is that information is codes as "broadly" as possible-in contrast to "compact" real numbers. The breadth, hence the resolution, of the encoding determine a Gas capability to both broadly explore and locally exploit parameter search spaces.
3.2 Fitness function
A fitness function (or objective function) is used to determine the fitness of each candidate solution. A fitness value is assigned to each individual in the population. Integral of absolute error (IAE) is a better all-round performance indicator of closed loop response where overshoot, settling and rise times are the main performance considerations [14]. The IAE was therefore used as a measure of performance.
(16)
In Controller Design problems IAE has to minimized, also the controller has to be effective in the full operating region hence that weighted objective function applied based on the sum of IAE of minimum and maximum setpoints of product output(yf) and mill level (z) hence the objective function J is calculated as the sum of IAE of yf for setpoint 120 plus IAE of yf for setpoint 140 plus IAE of z for setpoint 60 plus IAE of z for setpoint 80, which is shown in equation (17).
(17)
3.3 Selection
The selection process is centered upon the specified fitness function. The selection scheme is used to draw chromosomes from the evaluated population into the next generation. Tournament selection is one of many methods of selection in genetic algorithms. Tournament selection involves running several "tournaments" among a few individuals chosen at random from the population as follows.
choose k (the tournament size) individuals from the population at random
choose the best individual from tournament with probability p
choose the second best individual with probability p*(1-p)
choose the third best individual with probability p*((1-p)^2); and so on.
The winner of each tournament (the one with the best fitness) is selected for crossover. Selection pressure is easily adjusted by changing the tournament size. If the tournament size is larger, weak individuals have a smaller chance to be selected [25].
3.4 Crossover
Crossover provides a mechanism for individual strings to exchange information via a probabilistic process. Once the reproduction operator is applied, the members in the mating pool are allowed to mate with one another. First, the genetic codes of the two parents are mixed by exchanging the bits of codes following the crossover point. For example, consider two parent strings where the crossover point is 5 (i.e., the fifth bit in the string)
P1 = 10101|010; P2 = 01111|100;
The separator symbol ''|" indicates the crossover site.
The resulting offspring have the following:
P01 = 10101|100; P02 = 01111|010
3.5 Mutation
In each iteration, every gene is subject to a random change, with the probability of the pre-assigned mutation rate. In the case of binary-coding, the mutation operator changes a bit from 0 to 1, or vice versa. All in all, the mutation operation introduces new genes into the population, so as to avoid the problem 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 procedure continues until the fitness reaches the desired specifications.
Fuzzy logic Controller
The implementation of the fuzzy logic based term is u(t) = F[e(t), Δe(t)]. In the description standard terminology is used to form fuzzy set theory, for a treatment of fuzzy sets, e(t), and Δe(t) as inputs to the map F, and u(t) as the output. Associated with the map, F is a collection of linguistic values L={ NB, NS, ZO, PS, PB} that represent the term set for the input and output variables of F. In this case seven linguistic values are used. The meaning of each linguistic value in the term set L should be clear from its mnemonic; for example, NB stands for negative big, NS for negative small, ZO for zero and likewise for the positive (P) linguistic value. Associated with the term set L is a collection of membership functions. μ = { μNB, μNS, μZO, μPS, μPB }, Each membership function (MF) is a map from the real line to the interval [-1 +1 ]. In this application the MF used is the (triangular or trapezoidal type). The height of the MF in this case is one, which occurs at the points optimized by GA. The realization of the function F[e(t), Δe(t)] deals with the setting of linguistic values. This consists of scaling the inputs e(t) and Δe(t) appropriately and then converting them into fuzzy sets. The symbol Ce is the scaling constant for the input e(t) and the symbol CΔe is the scaling constant for the input Δe(t). For each linguistic value l∈ L, assign a pair of numbers ne(l) and Δe(l) to the inputs e(t) and Δe(t) with the associated membership function {ne(l)=μl (Ce e(t )), nΔe(l) = μl (Cde Δe (t) ) }. The numbers ne(l) and nΔe(l), l∈ L are used in the computation of F[e(t ), Δe(t)] [26].
As soon as fuzzy inference is applied to each rule, the activation level for all output variable (MFs) are obtained, and the defuzzification procedure takes place. In order to compute the final control action, u(t), the most commonly used method is the center of area[26]. The result is the center of area of the profile described by the membership functions, limited in the respective activation level. Equation (18) shows the defuzzified output
(18)
Where is the defuzzified value, and denotes an algebraic integration
4.1 MIMO Fuzzy control Structure
The difficulty associated with applying a traditional fuzzy control theory for controlling MIMO systems involves overcoming the effect of coupling between the degrees of freedom. Therefore, the concept of adding coupling controller to compensate for this coupling effect was developed to enhance the control performance of MIMO systems. A typical dynamic model of a MIMO system is complicated with uncertainty, so model-free intelligent control strategies are employed in designing a MIMO system controller. This work proposes a new control approach by combining a Main FLC (MFC) and a suitable coupling fuzzy controller for controlling MIMO systems. The control strategy includes a MFC and a coupling fuzzy controller which is shown in fig. 6.
Figure 6. GA MIMO FLC structure for MIMO System
4.2 Main fuzzy logic controller
A FLC that operates with output error of the system and error differential in the continuous time system is adopted as the main controller to control each degree of freedom of MIMO systems. Here, the input variables of the MFC for between the degrees of freedom of a MIMO system are defined individually as where ei is the output error of the ith degree of the system; Δei is used for indicating error differential of the ith degree of the system; Ri is the reference input and Yi represents the system output of the ith degree of a MIMO system.
4.3 Coupling fuzzy logic controller
In a real MIMO system, the control output is influenced by more than one variable. According to the system characteristics, these system variables are obviously interactive
(19)
(20)
where and represent complex coupling function that are difficult to define and derive. According to the analysis of the dynamic equation (19) or (20), clearly, ui is the main effect and ul is the secondary effect for the output Yi. Similarly, for the output Yl, ul is the main effect and ui is the secondary effect. The main effect on the system is controlled using a MFC. The secondary effect on the system is controlled by designing an appropriate coupling fuzzy controller.
Figure 7. GA optimized MIMO FL Control structure
The MIMO FLC control structure of cement mill process is shown in the fig. 7, the difficulty in controlling MIMO systems is how to solve the coupling effects between the degrees of freedom. Therefore, an appropriate coupling fuzzy controller is incorporated into a main fuzzy controller for controlling MIMO systems to compensate for the dynamic coupling effects between the degrees of freedom. This MIMO fuzzy controller can effectively remove the coupling effects of the systems. Based on the principles of the Expert control algorithm, shown in fig. 8, the MIMO fuzzy logic controller is optimized using GA for a cement mill process. The control objective is to regulate the finished product rate yf and the mill load z at the desired set points yf*and z* by manipulating the feed flow rate u and the separator speed v. Equation (21) and (22) shows the output of MIMO FLC with coupling fuzzy controller.
(21)
(22)
Figure 8. GAFLC Design flow chart
GA FLC Design
Although fuzzy logic allows the creation of simple control algorithms, the tuning of the fuzzy controller for a particular application is a difficult task and one needs a more sophisticated procedure than that used for a conventional controller. This is due to the large number of parameters that are used to define the MFs and the inference mechanisms. Several methods have been developed for tuning fuzzy controllers. These involve adjustment of the membership function [27] and scaling factors [28] and dynamically changing the defuzzification Procedure. Therefore, the approach needs as many variables as there are rules to get an optimal rule base. The advantage of the approach proposed in this paper is that it takes only three variables to optimize the rule base geometry, two variables to optimize the membership function and three scaling variables for a single FLC.
5.1 Encoding Rule Base
To design an optimal rule base a simple geometric approach is followed to modify the rule base as mentioned in [29] the initial assumptions are as follows;
The magnitude of the output control action is consistent with the magnitude of the input values. (i.e. in general, extreme input values (premise) result in extreme output values (consequent), mid-range input values in mid-range output values and small/zero input values in small/zero output values.
Using these generalizations, in conjunction with the concept of system symmetry, a different approach can be used which reduces the number of bits required for the rule -base dramatically. The approach is a variation of the method which involves a fixed coordinate system defined by the possible premise combinations. The consequent space is then 'overlaid' upon the premise coordinate system and is in effect partitioned into 5 regions shown in fig. 9, where each region represents a consequent fuzzy set. The rule -base is then extracted by determining the consequent region in which each premise combination point lies. Different possible consequent space partitions are defined using 3 parameters;
Figure 9. GAFLC Rule Assigning
Consequent-line angle, CA (16 angles between 0-168o (i.e. 4 bits)) (CA defines slope of the consequent line, which is used to create the space partitions).Consequent-region spacing, CS (4-bits) (CS is a proportion of the fixed-distance between premises on the coordinate system (Ps) and is used to define the distance between consequent points along the consequent line defined by angle, CA . Its value was set to a range between (0.5 - 1.5) times the fixed distance, Ps ,using a precision of 4 bits).Consequent-line order, CO (1-bit) (Defines order of consequent space partitions (i.e. NB-NS-Z-PS-PB or PB-PS-Z-NSNB) (1 bit) and in effect doubles the range of possible consequent line angles to 0-360o).
5.2 Encoding membership function
In the attempt to encode the FLC membership functions associated with the 2 inputs and 1 output, a number of assumptions are made in respect of the distribution of fuzzy sets across the universe of discourse (UOD) for each fuzzy variable. These assumptions are;
The MF properties altered by the GA are as follows;
1. MF shape (triangular or trapezoidal).
2. Degree of MF-centre shift to effect MF compression or expansion.
All evaluated FLCs contain 3 variables, e (error), de (error-derivative) and u (control-action). For the input variables, e and de, and output variable, u, 7 bits are used to define the properties of the MFs to be optimized. For each variable, their respective 7-bit GA-chromosome segments are sub-divided into 2 fields;
1. The "offset field" (3 bits) used to effect change of shape of the MFs.
2. The "companding factor" field (4 bits) used to effect expansion/compression of the MFs.
5.3 MF Offset Field
The optimization begins by loading a *.fis (Matlab Fuzzy file) into the FLC block in the MATLAB Simulink model. Each evaluation subsequently uses a 'genetically-altered' version of the original FLC which is defined by a MATLAB, fuzzy structure. For each evaluated FLC, the UOD -distributed MFs are initially assumed to be trapezoidal in type, thus 4 parameters are required by the FIS to define the position in the UOD of each of the MFs. The significance of these parameters is illustrated below in fig. 10 & fig. 11. The Matlab Fuzzy file 'params' field has 4 UOD position parameters outer-left (OL), inner-left (IL), inner-right (IR), outer-right(OR) . For inner parameters (IL and IR) equal in value, MF becomes triangular in shape.
5.4 MF Companding Field
Application of the offset field produces MFs of different shapes (trimf or trapmf) and positions, but does not effect the distribution of the MFs, which are evenly distributed across the UOD. To enable evaluation of non-uniform distributed MFs, by raising them to the power of CF (e.g. for the Z-MF, outer-left parameter; ) Due to the use of a normalized UOD, the position parameters are shifted to different degrees by this operation and the net effect is that;
for CF < 1 : Z-MF is compressed, NB and NS expand
for CF > 1 : Z-MF expands, NB and NS compress
for CF = 1 : uniform MF distribution
5.5 Encoding FLC Scaling Gains
The GA also attempts to optimize the scaling gains of the e and Δe inputs of the fuzzy controller. Three fields, e-scaling (Ce), Δe-scaling(CΔe) and output Cu are included in the GA chromosome each consisting of 10- bits, which are encoded to yield values of gain for the appropriate gain blocks of the Simulink model used to evaluate each controller.
5.6 GA-Chromosome of MIMO FLC
Three aspects of the FLC were subject to the optimization procedure; (a) Rule Base, (b) Membership Functions (MF), (c) Input Output Scaling Gains. The primary assumption made was that for a symmetrical system, a corresponding FLC would also exhibit symmetry about the set point in respect of its MFs and rule -base. This assumption was exploited in order to attempt to reduce the number of bits required to define the FLC for GA optimization. Table 1, shows the details of the variables associated with FLC Design total 48 variables are used to optimize the main FLC's and Coupling FLC's.
Simulation
The effectiveness of the proposed control law has been assessed through simulations of the model presented in equation (1)-(3) represents the plant with analytical forms for the φ and α function which is given below.
(23)
(24)
And the time constants Tf = 0.3 [h], Tr = 0.01 [h]. These functions have been tuned in order to match experimental step responses of an industrial cement grinding circuit [31]. The entire simulation is carried out in MATLABTM & SIMULINKTM package installed on a Core 2 Duo Processor 2.2 GHz, 2GB RAM IBM PC Environment.
Figure 12. Simulink simulation setup of GA MIMO FLC of Cement mill process
7.1 Case I
For optimizing the MIMO Fuzzy logic controller, the GA parameters are set as:
Generation =250, Population Size=50, Crossover rate =0.5, Mutation Rate = 0.03 and the Parameters associated with designing the MIMO FLC are set as shown in table 1. The MFs for e, Δe, and u of MFC 1 and 2 are set as 5 and MFs for e, Δe, and u of CFC 1 and 2 are set as 3 to reduce the computational burden.
7.2 Case II
For testing the GA optimized MIMO Fuzzy logic controller, the following settings are chosen, with Initial setpoint values: yf = 120 tons/h and z = 60 tons.
, ,
The set-point for the product flow rate yf is changed from 120 to 140 tons/h at time t=3 hours and the set-point for the mill level z is changed from 60 to 70 tons at time t=6 hour, the hardness d varied from its nominal value 1 to 1.5 at time t=8. The closed loop response of the Cement mill for the following settings without coupling fuzzy controller is shown in fig.16, and with coupling controller is shown in fig. 17.
Table 1
Optimized fuzzy logic control variables of MIMO FLC
Variabls
Name
Bits
i=1
i=2
i=3
i=4
Range
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
e MF Offset
3
0.26
0.10
0.27
0.19
[0, 0.5]
UiO2
Δe 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
e MF Companding Field
4
0.50
1.00
0.9
1.30
[0.1, 2]
UiCF2
ΔeMF 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
e Scaling
10
0.33
0.21
0.87
0.96
[0.01, 1]
UiS2
Δe 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 (z), i3-Coupling fuzzy Controller 1 (yf-z) and i4-Coupling fuzzy Controller 2 (z-yf)
Table 2
Optimized Rule Base MFL 1 (yf)
Output
u1
Error
NB
NS
Z
PS
PB
Δe
NB
NB
NB
NS
NS
Z
NS
NB
NS
NS
Z
PS
Z
NS
NS
Z
PS
PS
PS
NS
Z
PS
PS
PB
PB
Z
PB
PS
PB
PB
Table 3
Optimized Rule Base MFL 2 (z)
Output
u2
Error (e)
NB
NS
Z
PS
PB
Δe
NB
NB
NB
NS
Z
PS
NS
NB
NB
NS
PS
PS
Z
NB
NS
Z
PS
PB
PS
NS
NS
PS
PB
PB
PB
NS
Z
PS
PB
PB
Table 4
Optimized Rule Base CFC 1 ()
Output
u2-1
error
NB
ZO
PB
Δe
NB
NB
NB
ZO
ZO
NB
ZO
PB
PB
ZO
PB
PB
Table 5
Optimized Rule Base CFC 2 ()
Output
u1-2
error
NB
ZO
PB
Δe
NB
NB
NB
ZO
ZO
NB
ZO
PB
PB
ZO
PB
PB
Figure 13. GA optimized MFC 1 and 2 membership functions
Figure 14. GA optimized CFC 1 and 2 MF's
Figure 15. Minimization of objective function
7.3 Case III
To check the disturbance rejection of the proposed controller, the hardness parameter (d) is varied from 1.34 to 1.8 the setpoints of yf is set as 120 tons/h and z is set as 60 tons. The hardness change is introduced at time t=5 hour 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 z are shown in figure 21 and 22.
Results and Discussions
The Table 1 shows the optimized fuzzy logic control design variables for the MIMO system, the tables 2 & 3 shows the optimized rule base of MFC 1 and MFC 2 and the tables 4 & 5 shows the optimized rule base of CFC 1 and CFC 2. The fig. 12 shows the Simulink implementation MIMO FLC with coupling fuzzy controller for cement mill circuit. The fig. 13 & fig. 14 shows the GA optimized MFs of the MFC and coupling FLC and fig.14, shows the optimized membership functions of MFC and CFC of the MIMO FLC after 250 generations. Fig.15, shows the minimization progress of the performance Index for 250 generation and it is minimized to 168.4 from 2754.3.
Figure 16. Closed loop response of cement mill circuit without CFC
Fig. 16, shows the closed loop response of the simulated cement mill circuit without coupling controller for the setpoint and hardness profile given in section 7.2. Three variables of the cement mill (yf, z, and yr) is plotted for 11 hours time. In fig. 16, when the set point change in finished product outflow (yf) is introduced it is noticed the mill level is disturbed, similarly when the set point of mill level is increased the finished product outflow is disturbed. This is occurred due to loop interaction, this loop interaction is eliminated by introducing the suitable coupling FLC.
Figure 17. Closed loop response of GA MIMO FLC with CFC for varying setpoints and hardness
The fig.17 shows the closed loop response of the simulated cement mill circuit with GA optimized MIMO FLC with coupling fuzzy controller for the same set point and hardness profile and the output response is plotted for 11 hours time. When there is a sudden rise in product outflow setpoint (yf), the controller is capable of keeping the controlled variable in the set value with out overshoot and with quick settling time, similarly for the mill level (z) the controller is very effective. After introducing the coupling controller it is noticed that when there is a change of setpoint for (yf) or (z) there is only a small deviation in the other loop. The comparison of MIMO controller with and without coupling controller is tabulated for the various parameters shown in table 6. The loop interaction is reduced more than 10 times in the MIMO FLC with coupling controller.
Table 6
Performance comparison of MIMO FLC with & without CFC for varying setpoint
Setpoint change
Parameters
Finished Product (yf)
Mill Load (z)
Without CFC
With CFC
Without CFC
With CFC
Setpoint Change
In Finished Product
yf=120 to140
(Tons/hour)
Raise Time (Min)
4
3
NA
NA
Peak shoot (%)
0
0
2.8
0.7
Settling time (Min)
50
15
14
3
IAE
10.53
4.86
8.76
0.40
Setpoint Change in Mill level
z =60 to 70
(Tons)
Raise Time (Min)
NA
NA
6
3
Peak shoot (%)
6
0.6
3.3
0
Settling time (Min)
43
3
28
5
IAE
8.68
0.54
9.97
0.92
Figure 18. Controller output of (yf) and (z) of MIMO FLC with CFC for varying setpoints and hardness
Fig. 18 shows the output of MIMO main FLC's and CFC's, from the figure it is clear that the coupling controllers reacts immediately when there is step change in the input or the other loop. It is noticed that the controller is capable of bringing back the error of (yf) and (z) to zero in minimum time when the hardness d is varied from 1 to 1.4.
Table 7-A shows the comparison of the closed response output for set point variations keeping hardness (d) as 1. The performance of the controllers are compared with respect to the risetime (Rt), Peak overshoot (Pos), Peak undrshoot (Pus), and settling time (St) for different control schemes. The GA optimized MIMO FLC seems to be better in all performances.
Table 7
Performance comparison of controlled variable for varying setpoint
A- setpoint Change
B-hardness change
Finished Product (yf)
Mill Load (z)
Finished Product (yf)
Mill Load (z)
Controller types
Rt
(Min)
Pos
(%)
St
(Min)
Rt
(Min)
Pos
(%)
St
(Min)
Pos
(%)
Pus
(%)
St
(Min)
Pos
(%)
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
linear quadratic control*
50
1.45
255
30
4.1
250
Unstable
Non leaner learning control*
Data Not Provided
0
6.66
42
3.70
35
Neural 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 comparison of the closed response output for the change in the hardness parameter (d) from 1 to 1.4 is presented in table 7-B. The proposed control scheme is performs better with minimum Rt, St, Pos and Pus, in where as linear quadratic control cannot stabilize the mill. By comparing the performances shown in table 8 & 9, the proposed control scheme performs better compared to the other control strategy reported in the literature (Nonlinear robust controller [20], Nonlinear receding horizon (NRH) control [30], linear quadratic control [30], Nonlinear learning control [7], Neural Network based control [21]), also the effect of hardness change does not destabilize the cement mill.
Figure 19. Output response of finished product (yf) for different hardness parameter (d)
Fig. 19 and 20 depicts the output response for the simulation settings ginven in Case III. The response for (yf) and (z) is plotted for different hardness parameters (varied from 1.34 to 1.8) the response shows the settling time for all hardness is more or less same. The hardness parametter is varied upto 1.8 which was not studied in privous work. The Table 8 shows the performance measure of (yf) and (z) for the proposed controller scheme at various hardness values.
Figure 20. Output response of mill level (z) for different hardness parameter (d)
Table 8.
Performance measure of proposed controller for different hardness (d)
Finished Product (yf)
Mill Level (z)
(d)
IAE
Pu
(%)
St
(min)
IAE
Po
(%)
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
Conclusion
The literature contends that optimization of a FLC can be considered as a geometric search problem of a multimodal hyper surface. Optimization of a MIMO fuzzy logic controller can prove a lengthy process when performed heuristically. In this work it has been shown that the use of genetic algorithms offers a feasible method for the optimization of the knowledge-base of the MIMO fuzzy logic controllers. The proposed approach shows a better performance in building the MIMO fuzzy logic controllers for a complex cement mill process. The performance of the controller is studied with cement mill process model via simulation, and the results are compared with other control techiniques, the results demonstrate that the MIMO FLC designed by the proposed method shows better performance measure with minimum loop interaction and prevent the cement mill process from plugging.
