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Figure 4.4: Membership function for the variables A,, K;, and Ag In general the determination of this shift and the scaling is a trial and error thumb for determining the range of A, and the range of K4q is given as

Figure 4 4: Membership function for the variables A,, K;, and Ag In general the determination of this shift and the scaling is a trial and error thumb for determining the range of A, and the range of K4q is given as

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Table 1.1: Some important studies in fuzzy control. In this thesis the implementation of a fuzzy controller which sets the pa- rameters of a parametric controller has been investigated. Mainly two different methods have been used in constructing the rule base. In the first method we obtain the rules by observing the input output data of the system. Here we obtain the fuzzy rules to set the parameters of the parametric controller heuris- tically. For the controller, we consider the standard PID, and the lead-lag type controllers. In the other part again the input-output data is used but this time the rules have been set from the results of Fuzzy C-Means (FCM) clustering algorithm.
Table 1.1: Some important studies in fuzzy control. In this thesis the implementation of a fuzzy controller which sets the pa- rameters of a parametric controller has been investigated. Mainly two different methods have been used in constructing the rule base. In the first method we obtain the rules by observing the input output data of the system. Here we obtain the fuzzy rules to set the parameters of the parametric controller heuris- tically. For the controller, we consider the standard PID, and the lead-lag type controllers. In the other part again the input-output data is used but this time the rules have been set from the results of Fuzzy C-Means (FCM) clustering algorithm.
un y”. Then this relation may be given by the following array : proj R on Y = 0.9/y, + L.0/y2 + 0.7/y3 + 0.8/y4. Definition 6 Let U and V be two independent domain. And let R be a fuzz
un y”. Then this relation may be given by the following array : proj R on Y = 0.9/y, + L.0/y2 + 0.7/y3 + 0.8/y4. Definition 6 Let U and V be two independent domain. And let R be a fuzz
Table 2.1: Truth table for p implies q can calculate the truth value of an if-then statement. A typical example which
Table 2.1: Truth table for p implies q can calculate the truth value of an if-then statement. A typical example which
controller and linear systems which is shown in Fig.3.1 Figure 3.1: System structure under investigation
controller and linear systems which is shown in Fig.3.1 Figure 3.1: System structure under investigation
Figure 3.2: Basic configuration of pure FLS 3.2.2. Takagi and Sugeno’s FLS
Figure 3.2: Basic configuration of pure FLS 3.2.2. Takagi and Sugeno’s FLS
Figure 3.4: FLS with fuzzifier and defuzzifier the other two types of fuzzy systems can be found in [9].
Figure 3.4: FLS with fuzzifier and defuzzifier the other two types of fuzzy systems can be found in [9].
steps as presented in Fig.3.5. There are five such computational steps. Figure 3.5: Computational Structure of FLS with fuzzifier and defuzzifier The computational structure of a FLS consists of a number of computational
steps as presented in Fig.3.5. There are five such computational steps. Figure 3.5: Computational Structure of FLS with fuzzifier and defuzzifier The computational structure of a FLS consists of a number of computational
Figure 3.6: Membership functions second rule, Rgm means the membership value of warg(2.5) = 0.75
Figure 3.6: Membership functions second rule, Rgm means the membership value of warg(2.5) = 0.75
Figure 3.7: Clipped fuzzy sets as a result of rule firing part of the rule at the point of the truth value of the antecedent part. firing of a rule means to clip the membership function given in the consequent
Figure 3.7: Clipped fuzzy sets as a result of rule firing part of the rule at the point of the truth value of the antecedent part. firing of a rule means to clip the membership function given in the consequent
Figure 4.1: System structure under investigation 1.1.1 Rule-Base Construction
Figure 4.1: System structure under investigation 1.1.1 Rule-Base Construction
Figure 4.2: A typical step response of a system. At points closer to b the error is close to zero but the increment of the ‘rom the above discussion one may derive the following rules:
Figure 4.2: A typical step response of a system. At points closer to b the error is close to zero but the increment of the ‘rom the above discussion one may derive the following rules:
Table 4.3: Rules for integral gain 4.1.2 FLC Structure
Table 4.3: Rules for integral gain 4.1.2 FLC Structure
Figure 4.3: Membership functions for error and increment of error Def.3. Mamdani implication ts used in firing each rule. After firing each rule in the rule-base of a parameter there will be clipped
Figure 4.3: Membership functions for error and increment of error Def.3. Mamdani implication ts used in firing each rule. After firing each rule in the rule-base of a parameter there will be clipped
Figure 4.5: Membership function for the variables K,, A;, and Ay This procedure is performed to all of the rules in the rule table and all of these clipped fuzzy sets are combined with the union operator, max(-).
Figure 4.5: Membership function for the variables K,, A;, and Ay This procedure is performed to all of the rules in the rule table and all of these clipped fuzzy sets are combined with the union operator, max(-).
Figure 4.6: Comparison of the step responses. Figure 4.7: PID parameters determined by the FLCs.
Figure 4.6: Comparison of the step responses. Figure 4.7: PID parameters determined by the FLCs.
Figure 4.9: PID parameters determined by the FLCs.
Figure 4.9: PID parameters determined by the FLCs.
Figure 4.8: Comparison of the step responses.
Figure 4.8: Comparison of the step responses.
Figure 4.10: System structure under investigation
Figure 4.10: System structure under investigation
Figure 4.11: Step responses of P,(s) for a=0.01, 0.3, 0.7, 1, 2, 5, 10.
Figure 4.11: Step responses of P,(s) for a=0.01, 0.3, 0.7, 1, 2, 5, 10.
Figure 4.12: Step responses of P2(s) for a=0.01, 0.3, 0.7, 1, 2, 3, 4.
Figure 4.12: Step responses of P2(s) for a=0.01, 0.3, 0.7, 1, 2, 3, 4.
Figure 4.13: Membership functions of the linguistic variable a The membership functions for the parameter a are chosen so that they all
Figure 4.13: Membership functions of the linguistic variable a The membership functions for the parameter a are chosen so that they all
Table 4.4: Rules for lead-lag parameter a equally share the domain in the intervals [0,1] and (1 @maz].
Table 4.4: Rules for lead-lag parameter a equally share the domain in the intervals [0,1] and (1 @maz].
value of T is set to 0.002. It is set to a value obtained in a classical design. Therefore for this plant the Sec.4.2.2. The membership functions for the linguistic values of @ are taken
value of T is set to 0.002. It is set to a value obtained in a classical design. Therefore for this plant the Sec.4.2.2. The membership functions for the linguistic values of @ are taken
The step responses corresponding to the fuzzy tuned lead-lag controller and In both of the designs the rise time reduces while a small increase in the overshoot of the system is observed. From the figure it 1s seen than even with
The step responses corresponding to the fuzzy tuned lead-lag controller and In both of the designs the rise time reduces while a small increase in the overshoot of the system is observed. From the figure it 1s seen than even with
Figure 5.1: The membership functions and the relation of the rule if “e is LE,” then “uis LU;”.
Figure 5.1: The membership functions and the relation of the rule if “e is LE,” then “uis LU;”.
Figure 5.3: Sample point of error versus the corresponding control signal. any sample point much closer to one cluster center belongs to that cluster Therefore the sets the classical clustering algorithm defines for each cluster will have a characteristic function assigning 1 to those sample points which are closer to that cluster center and 0 otherwise. Fuzzy c-means clustering algorithm forms similar sets for each cluster but the characteristic function of these sets assigns values from the interval [0,1]. These characteristic functions assign values so that the closer the point to the center of the cluster the closer a value to 1 is assigned. Therefore the result of fuzzy c-means algorithm is a fuzzy relation between the error and the control action. If we consider Fig.5.1 then it can be concluded that each of these clusters, which the fuzzy c-means clustering algorithm calculates, is actually a relation corresponding to a rule in the fuzzy logic controller. The membership functions related with these rules are obtained by projecting these relations onto their subset. Therefore the sets the classical clustering algorithm defines for each cluster
Figure 5.3: Sample point of error versus the corresponding control signal. any sample point much closer to one cluster center belongs to that cluster Therefore the sets the classical clustering algorithm defines for each cluster will have a characteristic function assigning 1 to those sample points which are closer to that cluster center and 0 otherwise. Fuzzy c-means clustering algorithm forms similar sets for each cluster but the characteristic function of these sets assigns values from the interval [0,1]. These characteristic functions assign values so that the closer the point to the center of the cluster the closer a value to 1 is assigned. Therefore the result of fuzzy c-means algorithm is a fuzzy relation between the error and the control action. If we consider Fig.5.1 then it can be concluded that each of these clusters, which the fuzzy c-means clustering algorithm calculates, is actually a relation corresponding to a rule in the fuzzy logic controller. The membership functions related with these rules are obtained by projecting these relations onto their subset. Therefore the sets the classical clustering algorithm defines for each cluster
Figure 5.4: Negative unit feedback with a proportional controller K
Figure 5.4: Negative unit feedback with a proportional controller K
Figure 5.5: The suboptimum solution for the proportional control and the corresponding step response.
Figure 5.5: The suboptimum solution for the proportional control and the corresponding step response.
igure 5.7: (a) The step responses with two different controller (b) The wave- form for the proportional gain.
igure 5.7: (a) The step responses with two different controller (b) The wave- form for the proportional gain.
Figure 5.8: (a) The step responses with two different controller for the system with disturbed parameter (b) The waveform for the proportional gain. This design is tested on a system whose transfer function is given as
Figure 5.8: (a) The step responses with two different controller for the system with disturbed parameter (b) The waveform for the proportional gain. This design is tested on a system whose transfer function is given as
Related topics:
Computer ScienceIntelligent Control SystemsPid ControllerFuzzy Logic ControllerFuzzy ControllerFuzzy logic controllersFuzzy C-Means Clustering Algorithm
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