Designing the Controller Based on the Approach of Hedge Algebras and Optimization through Genetic Algorithm

Transcription

1 Designing the Controller Based on the Approach of Hedge Algebras and Optimization through Genetic Algorithm Duy Nguyen Tien 1, Trung Ngo Kien 2 1 Falcuty of Electronics, Thai Nguyen University of Technology, Thai Nguyen, Vietnam 1 Falcuty of Electrical Engineering, Thai Nguyen University of Technology, Thai Nguyen, Vietnam 1 duyinfor@tnuteduvn, 2 trungokien@tnuteduvn Abstract: The application of Hedge - Algebra in control and automation is a new approach and has had remarkable results in recent years However, on methodology, there are many issues that need to be studied extensively to show the effectiveness of this approach For that purpose, in this paper we present a method of using approximate reasoning for the design of the controller which is given by the linguistic rule The object chosen for that study was the furnace Through simulation, the results were evaluated and compared with the PI controller showing that the controller by Hedge algebra worked very efficiently, especially with the controller whose parameters were adjusted by the algorithm genetic Keywords: Hedge Algebra, PI controller, Fuzzy controller, Genetic Algorithm, Heat Control 189 I INTRODUCTION For industrial automation systems, the classical controllers PID are often chosen for their simplicity and ability to meet quality indicators However, the PID controller is not flexible because the coefficients Kp, Ki, Kd are unchanged during system operation This is a major disadvantage of PID controllers, especially with nonlinear systems In order to improve the quality of control, especially for nonlinear systems, new control methods have been investigated by scientists, such as the use of fuzzy set for adjustment to adjust Kp, Ki, Kd of PID controllers, controllers based on fuzzy logic [1], neural networks [2], etc Fuzzy logic [1] shows that there are many advantages in the field of control with objects in which input/output information is unclear, uncertain and without knowing the mathematical model of the object [3] - [8] Fuzzy logic allows the modeling of a controller to be constructed that is visually controlled by a set of linguistic rules However, it also has certain limitations The Fuzzy Logic Controller (FLC) often requires a large computing time [7] The design also depends on the designer s experience and understanding of the system The use of fuzzy sets to represent the semantics of language is limited by the fact that there is a lack of strong binding relations between fuzzy set and semantics The choice of shape and location of fuzzy sets is primarily based on experience According to the fuzzy logic, there are many options for t-norm operator, t-conorm, drag, defuzzification, etc, which greatly affect the output result of the controller These limitations cause the inability to accurately depict the computational model of the inference set against the control rule set in some practical applications When designing a controller based on a linguistic rule, it is important to ensure a semantic order relationship between linguistic elements HA (Hedge Algebra) [9], [10] is an algebraic structure on the domain of linguistic values whose semantics assure order relations There are many applications that have been developed based on this theory such as data mining, clustering, classification, fuzzy database, control, [3] - [8] In addition, the advantages of applying the HA in control field have been internationally published [13] HA allows building a model calculation for LRBS (Linguistic Rule Base System of the controller) - based controllers that have been targeted by researchers in the field of control and automation According to the HA, linguistic terms are computed based on their semantic value which is always order and suitable for constructing computational models in approximation problems In addition to using the advantages of the fuzzy system, the HA controllers to promote the advantages of computation based on the semantic validity of the language and useful to avoid identification mathematical model In this paper, with the aim of introducing a new computational tool, we propose an application method of HA to design the controller by the linguistic rule [4] - [8] II APPLICATION OF THE HEDGE ALGEBRA IN THE CONTROL Suppose we have a set of linguistic values of a certain linguistic variable which includes These linguistic values appears in linguistic rule bases of approximation problems based on knowledge Thus, it is necessary to have a strict computational structure which preserves the inherent order of the linguistic values From this we can calculate the semantic relationship of the linguistic values in the rules HA [9], [10] is an orderly mathematical structure of the set of linguistic terms, the order relations are defined by the semantics of the linguistic terms in this aggregation Quantifying the semantic validity of linguistic terms through semantic quantitative mapping functions - SQMs (Semantic Quantitative Mapping functions) [11] allows a full description of the model of the rule set and the approximation reasoning process in a close and

2 reasonable way [3] - [7] As, and are linguistic variables, each linguistic variable belongs to the base space and the linguistic variable is in the base space ;, ( ) are the linguistic values that belong to the corresponding base space Each rule "If then", identifies a "fuzzy point" in the Then (Eq 1) can be considered as a hyper-surface of in this space According to the approaching of the HA theory, we construct the HA structure for linguistic variables and use the SQMs function to convert each fuzzy point to a real point in the semantic space Then, (Eq 1) is represented respectively as a real hyper-surface It is likely to consider the real hyper-surface as the mathematical representation of the LRBS in which each fuzzy (linguistic value) of the fuzzy variable (linguistic variable) has been quantified into their semantic values (QRBS - Quantified Rule Base System) Suppose that real inputs belong to the corresponding base space, the input values of the controller, use the normalization of those values in the value domain of HA we have respectively The approximation reasoning problem is carried on by the interpolation method on The interpolation value received in the domain is the quantitative semantic value of the output linguistic variable that is transferred to the real variable domain (base space of the variable) of the output control value by denormalization The model of the controller based on HA approaching is described in Fig 1 Fig 1 Controller s Diagram Based on HA Approach In Fig 1, the components are included: 1) LRBS: Linguistic Rule Base System of the controller 2) QRBS: Quantifying rule based system of linguistic values which is computed by mapping function SQM ( ) 190 Normalization LRBS QRBS ( ) IRMd Denormalization 3) Normalization: standardize values of the variables in the semantic domain 4) IRMd (Interpolation Reasoning Method): Interpolation on the hyper-surface 5) Denormalization: convert semantic control value to the domain of variable real value of the output variable Steps of designing the controller based on hedge algebra as follows: 1) Step 1: Identify I/O variables, their variation domains, and control rules with linguistic elements in HA 2) Step 2: Select the structure, ( ) and for variables and Determine the fuzzy parameter of generating elements, hedges and the sign ralationship between the hedges 3) Step 3: Compute quantitative semantic values for linguistic labels in the rule set Construct the hypersurface 4) Step 4: Select the interpolation method on the hyper-surface 5) Step 5: Optimize the parameters of the controller III OPTIMIZING THE PARAMETER OF THE CONTROLLER BASED ON GENETIC ALGORITHM Genetic Algoritm (GA) is a method which find solutions randomly for overall structure based on the process of natural evolution Genetic algorithms are part of the broader class of evolutionary algorithms Genetic algorithms imitate the same mechanisms as those found in nature (including reproduction, breeding and mutation) to find the best solution [14] By starting at some independent and parallel search points, GA avoids extreme local extremes as well as convergence to secondary optimization solutions As a result, GA has been shown to be able to locate high performance areas in complex spaces without the hassles involved in the dimension of space such as gradient techniques or optimization search methods based on information about derivatives GA enforcement is usually initiated with a random population of 50 to several hundred individuals, depending on the problem Each individual is an optimal set of parameters, usually represented by a real or binary number sequence and is called a chromosome Each parameter corresponds to a segment of the chromosome, called the gene Replication and mutation processes occur randomly to exchange and transfer information of genes Adaptation of the next generations always inherit the adaptation of the previous generation The assessment of an individual's adaptation is measured by a finess function, called the finess value Normally, in optimization problems, the optimal goal is

3 to minimize the target function Through long-term evolution and selection, adaptation to natural processes, individuals will gradually adapt to the objective function Correspondingly, each instance will be an optimal set of asymptotic parameters by fitness function Here, the fitness function is calculated according to the IAE (Integrated of The Absolute Magnitude of the Error) The control system model is often modeled on a discrete domain over time, so the IAE standard is transformed into the below form: (Eq 2) Where: is the sample deviation at the th simulation cycle, is the total number of data samples of a simulation test is the reference value at the input, in many mathematical problem this is a constant is the true response value of the output on the control object IV GENERALIZATION AND NUMBER SYNTHESIS A The PI Classic Controller: Heating equipment is an object which is widely used in industry, medical and civil In the industry, it is often used in heat treatment, melting ferrous and non-ferrous metals Industrial furnaces often use metal wire In other areas such as health or civil, the focus is on the kilns Temperature is the quantity that needs to be adjusted In controlling, the furnace temperature (the furnace power) is usually done by controlling the power supply In this paper, we use an object in Error! Reference source not found with transfer function: Where: gain factor Time constant (seconds) - Time delay (seconds) (Eq 3) (Eq 6) The parameters of the PI controller are calculated using the experimental method Ziegler & Nichols: B The HA Controller The design of the HA controller is implemented according to the steps presented in the previous section as follows: 1) Step 1: Identify input/output variables, their variation domains, and control rule set with linguistic elements in HA The controller has 02 inputs: - control deviation, variation in the interval [-1, 1] - indicates the variable rate of e in the range [-1, 1], - Controller output is the control quantity to control the voltage of the source, varying in the interval = [-5, 5] The input/output linguistic variables include the following linguistic values: - - Where:,,,, The control rule is considered to be an LRBS and presented as the following table I: Table I Control Rule Set (Eq 4) The control signal described by: of the PI classic controller (Eq 5) Where is the scale factor, is the integration factor and is the control error Based on (Eq 5), turn it to to discrete domain, we have: 2) Step 2: Choose the structure of, ( ) and for the variables and Identify the fuzzy parameter of the generating elements and the hedges - The set of generating elements - The set of hedges is chosen: và 191

4 - The fuzzy parameter of hedge algebra gives the variable, and which include the fuzzy measurement of the generating elements, the fuzzy measurement of the hedges According to the hedge algebraic structure for the variables constructed above, we need to choose the fuzzy measurement of the negative elements ( ) the fuzzy measurement of the negative hedges ( ) The fuzzy parameters are initially chosen as intuitive as in Table II Table II The Fuzzy Parameters Of HA The sign of the generating elements, hedges and sign relations between the hedges is determined by the semantic nature of the linguistic terms For example,, Moreover, it can be seen that In additions, it is similar to other linguistic elements, so we define the sign relation as in table III Table IIII Sign Relation 3) Step 3: Compute quantitative semantic values for linguistic labels in the rules Construct hyper-surface in table IV Table IIIV QRBS of the Default Controller HAC With the selected fuzzy parameters as in Table II and the sign relationship between the hedges, between the hedges and the elements as shown in Table III, using the quantitative semantic function, we calculate the value quantitative semantic value of linguistic elements in the rule table, we have Table IV Fig 2 Input/Output Relation Surface Fig 2 is, which corresponds to Table IV It is a mathematical model that expresses the input/output relationship of the controller 4) Step 4: Select interpolation method: The interpolation method on is chosen as bi-liner interpolation 5) Step 5: Optimize fuzzy parameters of the controller It can be seen that the variable domain of input/output variables is symmetric The semantics of the linguistic element is zero by itself When mapping to the semantic domain in the range [0,1], the semantic value Therefore, we stably choose for value variables, We only need to optimize the fuzzy measurement of the hedges The set of hedges in the hedge algebra is constructed by only two hedges, V (Very) and L (Little) We have: ( ) Therefore, we just need to optimize the fuzzy measurement of the negative hedges, we will infer the fuzzy measurement of the positive hedges We have 3 HA structures for 3 variables, and Correspondingly, we have three parameters to optimize alfa_e, alfa_ce and alfa_u In theory, the fuzzy measurement can be varied from 0 to 1 However, in order to match the description of the language, we choose to search for the values of these parameters in the interval [03, 07] In the Matlab environment, GA is an existing function as a tool that helps us to use it In this study, we used the ga() function in Matlab with the gene code by the real number type double The values set for GA include: Population size, PopulationSize = 150; Generation = 450 The target function is used as in formula (Eq 2) The result gained the set of parameters for the controller as shown in Table V Table V Optimal Parameters of the Controller HAC Based on GA

5 Based on the optimal fuzzy parameters found in Table V, we compute the quantitative semantic values of the linguistic classes of the rule table, we obtain the QRBS table of the optimal controller HAC as shown in Table VI and corresponding input/output relation surface in Fig 3 Table VIV QRBS of the Optimal Controller HAC Fig 5 Graphical Response of the System Some of the measured values after the simulation are summarized in Table VII Table VIV Simulation Result PI HAC (Default Parameters) HAC (Optimization Parameters) Rise Time Overshoot Settling Time IAE It can be seen in the simulation results, the system output response to the HA controller is better than the PI controller In particular, with the optimal parameters obtained by GA, the HA controller provides better control of the quality control criteria such as Rise Time, Overshoot, Settling Time and IAE Fig 3 The Input/Output Relation Surface Optimal Controller HAC 193 V SIMULATION RESULTS of the The system simulation model to PI-controller, HAcontrollers, and HA-controllers with optimized GA parameters was implemented in a Matlab/ Simulink environment as shown in Fig 4 With the time 600s and input reference signal Step = 1 in the system simulation, we obtain the results of the response shown in Fig 5 Fig 4 Simulink Diagram for System Simulation VI CONCLUSION In this paper, we mainly present the application of hedge algebra and genetic algorithms in the field of control For control problems in which control rules are given by the LRBS, we can construct a controller with a model of computing based on the hedge algebra According to this approach, the representative structure of the LRBS is very tight, the number of computations is not much, so it can perfectly response to systems that require real-time response The controller has just a few parameters so it is also convenient to optimize To optimize the controller effectively, we use GA By simulating the controller for the heater in Error! Reference source not found, it can be seen that HAcontrollers work very well The results show that studying the application of hedge algebra in control is an open field In the future, it is essential to have more practical researches on HA because of its advantages VII REFERENCES [1] Zadeh L A, Fuzzy sets, Inform and Control, vol 8, pp , 1965 [2] M Ren, J M F Wang, Y Ren, Application of Fuzzy Neural Network PID Controller in Sewage

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