Realization of a fuzzy controller with fuzzy dynamiccorrection
Research article
Vladimir Vichuzhanin∗
Department of Information Technology, Faculty of Ship Building,Odessa National Maritime University, Mechnikova 34, Odessa, Ukraine, 65029
Received 04 June 2011; accepted 15 December 2011
Abstract: This article features results from a fuzzy controller with fuzzy dynamic correction for the non-linear control of objectswith variable parameters. The methods described herein have been used on a fuzzy proportional-differential (PD)controller, and our results show that fuzzy dynamic correction can reduce overshoot and shorten the settling timeof the controlled parameters.
Experimental and theoretical investigations of the dynamiccharacteristics of aggregates and air conditioning systems(ACS) reveal that such systems are governed by stronglynon-linear differential equations. Moreover, these stud-ies have highlighted the significance of the uncertaintiesthat exist in both the static and the dynamic formula-tions of the aforementioned equations [1]. Furthermore,it was found that the units’ transfer functions, under op-erational conditions characterized by unsteady heat andmoisture, deviated significantly from calculated values.Consequently, depending on the nature of the change inthe controlled parameter and as a result of the uncertain-ties in the mathematical models of ACS, regulators mustalso have variable coefficient settings in order to reducetheir settling time. This will increase the robustness of the∗E-mail: [email protected]
controller when there is insufficient information needed tocontrol the object.In this paper, we present the implementation of a robustACS controller based on the combination of a proportional-integral-differential (PID) controller with a fuzzy PD con-troller equipped with fuzzy dynamic correction. The ap-proach yields minimum overshoot and settling time of theoperating parameters. Fig. 1 shows a block diagram of theproposed architecture.2. Review of the State-of-the-ArtPID regulators do not always produce the optimum dy-namic characteristics when controlling non-linear objects,including ACS [1]. Furthermore, as can be deduced fromthe theory of automatic control, these regulators exhibitpoor performance when there is insufficient informationabout the controlled object [2].One design approach that circumvents this issue is theuse of fuzzy logic methodology in implementing the con-
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Figure 1. Block diagram of the proposed PID controller.
trollers [3]. As previously mentioned, such an implemen-tation results in reduced overshoot of the controlled pa-rameters in ACS control applications. These fuzzy logic-modified controllers, in contrast with classical regulators,are based on the use of linguistic variables, and theiroperation is dictated by the theory of fuzzy sets [3].There exists a wide variety of fuzzy controllers that arecurrently used in real world applications. In recent yearshowever, there have been increased efforts devoted towardsimplementing fuzzy PD, PI, and PID controllers.The metric that is used to assess performance of these novelcontrollers is the overall reaction of the control systemas a result of a change in the set point of the controlledparameters. However, a more appropriate performancemetric ought to encompass the ability of the controller toremain functional despite perturbations in the controlledobject [3–5].Moreover, there is a significant challenge in implementing afuzzy PID controller since it must have a three-dimensionalrule table. Configuring the controller at or close to theoptimal setting is difficult because the implementationof a large rule base requires substantial computationalresources and time.For example, the number of rules required to take intoaccount all possible combinations of a fuzzy PID controlleris n1×n2×n3, where n1, n2, n3 are the linguistic variableswhich correspond to the controller’s three input variables.Thus, if n = 7, there are 343 rules to implement.The typical approach used to reduce the number of rulesis to use a combination of controllers; a PD componentis used to implement the fuzzy algorithm, and the inte-
gral (I) controller which uses the remaining two inputs toimplement the classical control algorithm. This methodsignificantly reduces the number of control rules withoutcompromising regulation performance [6].One of the promising applications of fuzzy logic controllerswith variable coefficients is in non-linear control. In thisconstruct, fuzzy logic is used to build the regulator and toorganize the dynamic adjustments of the coefficients. Thenext section covers the development of a PID controllerequipped with a fuzzy PD component as well as componentfor fuzzy dynamic correction.3. PID controller with fuzzy pd com-ponent and fuzzy dynamic correctionThe proposed controller can best be analyzed by con-sidering its input and output characteristics. The inputvariables are the coordinates of the error in the outputregulation, denoted (e), and that of its derivative, denotedde/dt. The output variable thus consists of the actuationcontrol command, denoted (u).In the fuzzification of the PD component (see Fig. 1), twocomponents of the error signal (after the AutoConfig block)are converted into fuzzy variables which are then fed toa fuzzy inference block in order to generate the controlaction variable (u1). Control inputs u1, which correspondsto defuzzification after surgery, and u2, integration aftersurgery, go to the adder and the controlled object.The main features that used fuzzy logic consisted of thePD controller and the hardware that accounts for the
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Realization of a fuzzy controller with fuzzy dynamic correction
(a)
(b)Figure 2. Membership functions of the PD controller. a) inputs (e),
(de/dt); b) output variables (u).
corrective amendments to the coefficients of the controllersettings (kp, kI , kD), as afunction of the current value ofthe parameter adjustment.The controller starts by finding the initial approximationof the coefficients kp, kI , and kD . Usually, this is doneusing the Ziegler-Nichols methods which makes use of theoscillation period of the numerical values of the coefficientswhich are then used to form a criterion function necessaryfor finding the optimal values of the settings [4].During fuzzy dynamic correction, the AutoConfig blockselects the input and output ranges, the shape of themembership functions of the unknown parameters, the fuzzyrules, the inference mechanism, and the defuzzificationmethod. The PD controller, using these search parameters,then implements the optimization algorithms.The objective function for implementing the optimizationproblem is the integration of the sum squared error and the
(a)
(b)
(c)Figure 3. Membership functions for: a) kp; b) ki; c) kd.
settling time of the output variable of the controlled object.With restrictions present on the membership functions, theunknown parameters are then chosen as the membershipfunctions’ maxima.Figures 2 and 3 illustrate, respectively, the processes andthe membership functions needed for fuzzification of the in-put variables within the AutoConfig bock and that required394
V. Vichuzhanin
Table 1. Rule bases for the fuzzy PD component.
HHHHe de NB NM NS Z PS PM PB
PB Z PS PM PB PB PB PBPM NS Z PS PM PB PB PBPS NM NS Z PS PM PB PBZ NB NM NS Z PS PM PBNS NB NB NM NS Z PS PMNM NB NB NB NM NS Z PSNB NB NB NB NB NM NS ZTable 2. The rule base for Kp
HHHHde e NB NM NS Z PS PM PB
NB PB PB PM PM PS Z ZNM PB PB PM PS PS Z NSNS PM PM PM PS Z NS NSZ PM PM PS Z NS NM NMPS PS PS Z NS NS NM NMPM PS Z NS NM NM NM NBPB Z Z NM NM NM NB NBfor fuzzification of its output variable. The membershipfunctions are discrete and represent a set of singletonfuzzy sets.The Mamdani fuzzy model (Truth-Value-Flow-Inference)can dramatically simplify the performance requirement foroperating with fuzzy rules, aggregation, and defuzzification,using the right side of the singletons. Thus, the controllerproduces seven measures of linguistic influence on theactuator.Figure 2a and Figure 2b show the seven linguistic termswhich correspond to the input and output variables. Theterms are: positive big (PB), positive medium (PM), positivesmall (PS), zero (Z), negative small (NS), negative medium(NM), and negative big (NB).Membership functions in Figure 2a are triangular, S-shaped, and Z-shaped. These forms of membership func-tions are simple and thus reduce the time spent during thecalculations. Figure 3 shows the membership of the coeffi-cients of the AutoConfig block which comprises the PIDcontroller and the component that performs fuzzy dynamiccorrection.The rules that govern the regulatory functions of the vari-able parameter PD controller are listed in Table 1. Usingthese fuzzy rules, the PD controller produces the out-put variable u1. Depending on the current value of theparameter, regulating the corrective amendments of theAutoConfig settings (kp, kI , kD) are also formulated by therelevant rules listed in Tables 2 through 4. The controlrules were framed to achieve the optimum performance; atotal of 49 rules were used in this work.
Table 3. The rule base for KdHH
HHde e NB NM NS Z PS PM PBNB PS NS NB NB NB NM PSNM PS NS NB NM NM NS ZNS Z NS NM NM NS NS ZZ Z NS NS NS NS NS ZPS Z Z Z Z Z Z ZPM PB NS PS PS PS PS PBPB PB PM PM PM PS PS PB
Table 4. The rule base for KiHHH
Hde e NB NM NS Z PS PM PBNB NB NB NM NM NS Z ZNM NB NB NM NS NS Z ZNS NB NM NS NS Z PS PSZ NM NM NS Z PS PM PMPS NM NS Z PS PS PM PBPM Z Z PS PS PM PB PBPB Z Z PS PM PM PB PB
The output u1 of the fuzzy PD controller was computedby Mamdani’s [7] and was simulated in Matlab [8]. Conse-quently, the membership functions for the intersection andunion sets of the error in the output coordinates (e) andtheir derivatives (de/dt) were calculated as follows:µe∩de/dt = min(µe, µde/dt)µe∪de/dt = max(µe, µde/dt) (1)
Membership functions for each of the 7 linguistic measuresincluded in the variable u1 yield the following form:µM1(u1) = min
where µu1(u1) . . . µu49(u1) are the output variable member-ship functions of the fuzzy PD controller.µe1(e) . . . µe49(e), that of the error control of the output coor-dinates of the objects and µde/dt1(de/dt) . . . µde/dt49(de/dt)the membership function of the input variables, which cor-respond to the derivative of the error control.Each equation in (3) corresponds to one of the rules ofTable 1. The accumulation of the resulting membershipfunction of the control action variable is given by the unionof all the fuzzy rules shown in Table 1 as shown below:
µ(u1) = max{µM1(u1, µM2(u1 . . . µM49(u1} (3)
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Realization of a fuzzy controller with fuzzy dynamic correction
Figure 4. Circuit diagram of fuzzy PD part of the combined PID with fuzzy dynamic correction in the FPGA.
Figure 5. Matlab model block diagram.
What follows is defuzzification which is performed usingthe center of gravity (COG) method. This results in thefollowing membership functions:u1 = ∫
u1 · µ(u1)d(u1)∫µ(u1)d(u1) . (4)
Using the COG method, the variable u1 is calculated asthe abscissa of the center of gravity of the square formed bythe membership function µ(u1) and the axis u1 . Similarly,the outputs of the fuzzy blocks in AutoConfig (see Figure 1)are calculated using the Mamdani fuzzy rules of Tables 2through 4.396
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Figure 6. Control surface output variable obtained from the MATLABmodel.
4. FPGA-based development of thePID controller with fuzzy dynamic cor-rectionIn this section we discuss the development of the PIDcontroller in a field-programmable gate array (FPGA) inaccordance with the previously described schemes. Toobtain optimum parameters, the CAD tools Quartus II (ver.9.1) and Altera FPGA were used. These software allowa full design cycle, namely, the entry, inspection, andthe programming of the chips. Figure 4 shows the FPGAimplementation performed on the Altera platform.5. PID controller with fuzzy dynamiccorrection model utilizing MATLABThe PID controller was modeled in the MATLAB envi-ronment utilizing the Fuzzy Logic Toolbox [8]. The modelis shown in Figure 5, and the simulation result for thecontroller’s response surface is shown in Figure 6. Thecontrol surface is shown for the output variable u whichresults from changes in its input variables, namely, e andde/dt. Figure 7 shows the result from a classical PIDcontroller with constant parameters and that of the novelimplementations proposed in this paper.Using our model, the performance of regulators was inves-tigated when used in a control loop for the management ofACS heat exchangers in the presence of a step-wise pertur-bation with zero-valued initial conditions. In this situation,the heat exchangers were modeled as a first-order aperi-odic link delay for which the coefficients were as follows:kp = −0, 3 . . . 0, 3; kI = −0, 06 . . . 0, 06; kd = −3 . . . 3.Parameters needed for the dynamic correction were chosen
Figure 7. Curves for changes in the output variable control object byusing classical PID and the PID controller proposed in thispaper.
as: kp = 0, 3; kI = 0, 06; kd = 0, 56. For comparison,we also studied the same system utilizing a classical PIDcontroller; its settings were fixed at k = 2; Ti = 0.25 sec.;Td = 0.025 sec during the simulation.As Figure 7 shows, the results suggest that the proposedapproach has a 1.5 times lower overshoot than that pro-duced by the classical PID controller. This correspondsto a 3 times reduction in the time needed to establish thecontrol parameters to their optimum value. Consequently,we conclude that for managing a heat exchanger for ACScontrol, a PID/PD fuzzy controller, such as the one pro-posed in this paper, in addition to fuzzy dynamic correctionis optimum.6. ConclusionsIn this paper we have proposed and demonstrated a PIDcontroller which comprises a PD controller with fuzzydynamic correction. The system was implemented usingthe input and output parameters of an auto-configurationblock (denoted above as AutoConfig). Our results showthat, when compared with the classical PID controller withconstant coefficients, the settling time of the controlledparameter is greatly reduced. For ACS applications, inwhich there is inherently insufficient information to controlthe object, our approach yields the optimum performance.References
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Realization of a fuzzy controller with fuzzy dynamic correction
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