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G ModelARTICLE IN PRESSASOC-4498; No. of Pages 11Applied Soft Computing xxx (2017) xxx–xxxContents lists available at ScienceDirectApplied Soft Computingjournal homepage: www.elsevier.com/locate/asocDesign of fuzzy robust control strategies for a distributed solarcollector fieldAntonio Bayas a , Igor Škrjanc b , Doris Sáez a, abDepartment of Electrical Engineering, University of Chile, Av. Tupper 2007, Santiago de Chile, ChileFaculty of Electrical Engineering, University of Ljubljana, Slovenia, Tržaška 25, 1000 Ljubljana, Sloveniaa r t i c l ei n f oArticle history:Received 23 December 2016Received in revised form 29 August 2017Accepted 5 October 2017Available online xxxKeywords:Fuzzy modelingRobust controlSolar collector fieldParametric uncertaintya b s t r a c tThis paper presents novel control strategies using Takagi–Sugeno fuzzy models combined with a parametric uncertainty robust control approach to address both the nonlinearities of a process and thedisturbances that act on it. In contrast to other robust control approaches, such as the H normoptimization-based approach, the proposed techniques allow the uncertainty information provided byfuzzy confidence intervals to be used to derive controllers that take into account performance specifications, such as overshoot or disturbance rejection, and to ensure the robust stability of the system dueto a study based on applying the generalized Kharitonov’s theorem and Lyapunov’s analysis from thesolution of a linear matrix inequality (LMI). To test these novel strategies, a solar collector field, whichis a nonlinear plant with several disturbances affecting its operation, is used. For this plant, fuzzy confidence intervals are derived that allow representing the uncertainties associated with these disturbances,different performance objectives are tested, and a methodology for deriving these controllers is developed. The effectiveness of this approach is demonstrated on the solar collector field under different solarradiation conditions, and promising results are obtained. 2017 Elsevier B.V. All rights reserved.1. IntroductionSeveral industrial processes often have a complex dynamic thatis difficult to describe from its phenomenological equations; therefore, modeling using reduced and simplified structures of linearor non-linear equations is an appealing option. However, thereare always non-modeled dynamics and disturbances that are notconsidered, which increases the error between the model and theprocess. Studying the uncertainty associated with these processesis of great importance since it can affect the performance of thecontrol strategies that could be used for such processes.Robust control theory addresses this issue, offering guarantees of both stability and performance when disturbances have aknown structure or behavior. Thus, it is possible to find modelsfor representing the uncertainties that are added as input/outputor measurement noises, which is addressed by the classical robustcontrol H [1], or parametric uncertainty, where the models of theplant are interval transfer functions with bounded coefficients dueto the uncertainties. Corresponding author.There are several examples of the application of H robust control in the literature. In [2], Imanari et al. propose applying thisrobust control to a hot-strip mill, which is a multivariable processdescribed by its state variables and that is affected by several disturbances. Some suitable parameters for the controller are obtainedto improve the frequency performance in terms of both the sensitivity and the complementary sensitivity function to achieve betterdisturbance rejection. This strategy is favorably compared againsta conventional proportional integral strategy. Other uses of H robust control theory are in energy applications, such as in [3],where it is applied in load mitigation for floating wind turbines.In this work, a structural control is developed using H theory,reducing the main fatigue load and the generator power error.Meanwhile, a piezoelectric actuator H control is designed in [4],where the fuzzy system is used for representing a particular component of the actuator (hysteresis); thus, a particular design isdescribed for this application. Furthermore, medical applicationshave been tested, such as in [5], where this theory is used to control the injection of insulin in diabetic patients. In this case, H control theory is used to design a switching controller for insulininjection for blood glucose, thus making it possible to work withnoisy measurements.Conversely, the models with parametric uncertainty useKharitonov’s theory to demonstrate that a family of functions 8-4946/ 2017 Elsevier B.V. All rights reserved.Please cite this article in press as: A. Bayas, et al., Design of fuzzy robust control strategies for a distributed solar collector field, Appl.Soft Comput. J. (2017), https://doi.org/10.1016/j.asoc.2017.10.003

G ModelASOC-4498; No. of Pages 112ARTICLE IN PRESSA. Bayas et al. / Applied Soft Computing xxx (2017) xxx–xxxby an interval transfer function is stable if certain criteria are met[6]. This conclusion has had a significant impact since it statesthat irrespective of the order of the interval polynomial, only fourpolynomials have to be analyzed to demonstrate its stability. However, this result cannot be directly applied to discrete-time systems.Given this issue, considerable effort has been devoted to improvethis result in the design of robust controllers in both discrete-timeand continuous-time processes. Indeed, in [7], methods to provethe stability of interval discrete-time transfer functions are analyzed and compared. In [8], a similar result of continuous-timeKharitonov’s theorem is demonstrated for discrete-time interval polynomials of degree n 3. For continuous-time analysis, animportant result is the generalized Kharitonov’s theorem proposedin [9], which can be used for designing robust controllers withinterval transfer functions.Thus, using the aforementioned methods requires appropriate representations of the process. A very popular and usefulapproach in this sense is the fuzzy model, which is a universalapproximator to describe the dynamics of non-linear processes[10]. Fuzzy-model-based control strategies have been proven to bevery efficient in different applications; therefore, integrating suchstrategies with robust control strategies is interesting, as shown inthe specialized literature. Medical applications for this approachare tested in [11], where fuzzy robust strategies are used to helppatients with paraplegia. In this work, a slide model controllerbased on fuzzy logic is designed. However, the fuzzy model isad hoc for this controller structure and represents a part of theunknown dynamics. Additionally, in [12], a robust fuzzy adaptiveapproach is implemented in a manipulator arm for people withdisabilities, tackling the problem of the absence of a precise mathematical model for the process. This work includes a simple fuzzysystem for this modeling, and the controller is based on stableadaptive law. Meanwhile, in [13], a development that finds thesufficient conditions for the existence of an H controller is presented based on the solution of a linear matrix inequality (LMI) byusing a Takagi–Sugeno fuzzy model in the state variables. In thiscase, the uncertainty is modeled as uncertain time-varying matrices that are added to the matrices of the nominal model. Moreover,in [14], matrices with added noise are used, and the tuning of a fuzzycontroller with guaranteed stability is studied. Additionally, thepresented scheme allows selecting the parameters to achieve thedesired performance, which is an important topic since the studyof stability alone could lead to very conservative parameters of thecontroller. Indeed, another approach to manage the conservatismin control parameter tuning is shown in [15], where a PID controlleris designed for robust performance using conditions similar to theH problem. Optimization algorithms are also an interesting optionto address both the conservatism of parameters and robustness incontrol problems, as shown in [16], where particle swarm optimization (PSO) is used to tune a robust PID controller for a systemwith uncertain but bounded parameters.In most of the aforementioned works, a certain structure andvalues for the uncertainty associated with the models are assumedwithout describing a formal method for their modeling.This last topic is addressed in terms of fuzzy theory in [17],where a description of the uncertainty based on fuzzy sets is proposed. In that work, a fuzzy possibility distribution is used, takingthe uncertainties that affect the system into account. Then, usingfuzzy numbers, it is possible to model the process and its disturbances as an interval transfer function and find some suitablecontrollers based on the pole-placement method. This approachis called fuzzy parametric uncertainty and is used in [18], whereit enables a disturbed process to be modeled as an interval transfer function and then uses the generalized Kharitonov’s theoremto tune PI and PD controllers, ensuring stability and performancethrough an evaluation of gain and phase margins. Conversely, in[19], a method to transform the uncertainty existing in a system,expressed in the form of expert knowledge or linguistic variables, ispresented in numerical intervals that bound that uncertainty. Then,using Kharitonov’s theorem, a fuzzy PID controller is derived thatensures robust stability of the process. Similarly, fuzzy intervalshave been demonstrated to be an effective technique for modelingthe uncertainty of a system. Indeed, in [20], this strategy is usedto model a waste-water treatment plant, which is a system witha very non-linear behavior and significant disturbances. Additionally, in [21], the fuzzy interval models are used for fault detectionpurposes in uncertain systems.As shown above, a considerable amount of research has beenconducted to develop a suitable model that can both manageuncertainties and model the nonlinear behavior of a process. Ourobjective in this work is to develop a novel robust fuzzy controllerthat can manage uncertainties and that has good performance inthe case of nonlinear processes. The design procedure is basedon the Takagi–Sugeno model with the uncertain parameters inthe consequence transfer functions. This work also presents themethodology for tuning the robust fuzzy controller to achieverobust stability. The objective is to find a suitable controller forthe plant by using Kharitonov’s theorems, taking the performanceand the robust stability into account. The fuzzy part of this approachenables modeling the nonlinear dynamics of the process, thus identifying different operation points of the plant, whereas the robustpart ensures stability under disturbances. Furthermore, unlikeother similar works, uncertainty is modeled using the parameters’covariances, which are obtained from the fuzzy intervals proposedin [22]. Indeed, by taking the covariances of the parameters of eachrule in the Takagi–Sugeno model into account, the interval transferfunctions are obtained, which define the upper and lower confidence intervals. These intervals quantify the uncertainties of theidentified process in a certain way and enable the uncertainties tobe used in the design process.The proposed control design is tested on a distributed solar collector (DSC) simulator, which is located in Almeria, Spain [23].The model of this plant is described using the continuous-timeTakagi–Sugeno interval fuzzy model.The structure of the interval fuzzy model is mandatory becauseof the strong nonlinear behavior of the plant and the disturbances,which cannot be neglected.The remainder of this paper is organized as follows. Section 2presents the theory for parametric uncertainty fuzzy models andthe general design of the proposed fuzzy robust controller. Theapplication of this scheme to a DSC field is described in Section 3,where a fuzzy identification of the plant is performed to derive thefuzzy robust controllers. Then, the results obtained through simulations are shown. Finally, the conclusions of this work are presentedin Section 4.2. Design of fuzzy robust control strategy2.1. Fuzzy interval models with parametric uncertaintyA system with parametric uncertainty, also known as an intervalplant, can be expressed as a model with transfer function coefficients that vary in a determined range [24]. Thus, the uncertaintyassociated with both disturbances and modeling errors is includedin the model formulation.Analytically, an interval transfer function can be written as follows:H(s) bm sm bm 1 sm 1 · · · b1 s b0an sn an 1 sn 1 · · · a1 s a0(1)where bp [bp , b̄p ] and ap [ap , āp ] for p {0, 1, 2, . . ., m, . . ., n}denote the uncertain parameters of the system with n m.Please cite this article in press as: A. Bayas, et al., Design of fuzzy robust control strategies for a distributed solar collector field, Appl.Soft Comput. J. (2017), https://doi.org/10.1016/j.asoc.2017.10.003

G ModelARTICLE IN PRESSASOC-4498; No. of Pages 11A. Bayas et al. / Applied Soft Computing xxx (2017) xxx–xxxThe goal of parametric robust control is to find an appropriatecontroller for the process defined by an interval transfer function,as shown in (1). The main challenge corresponds to finding theconditions for which an interval plant is stable because this intervalmodeling generates an infinite family of transfer functions to beanalyzed.In this topic, Kharitonov’s theorem [6] and its generalizationproposed in [9] are considered. Whereas the former is related to thestability analysis of a continuous-time interval transfer function,the latter is used in control design for systems with parametricuncertainty also described in a continuous-time model.In this paper, these concepts are used to derive a robust fuzzycontroller based on a Takagi–Sugeno fuzzy model with intervaltransfer functions in its consequences.Takagi–Sugeno fuzzy models are based on a set of rules thatrelate the input variables or premises of a system in the form ofcause-effect statements [25]. The Takagi–Sugeno model has thefollowing structure:Rj :jjjIf x1 (t) is A1 and x2 (t) is A2 and. . .and xn (t) is An thenŷj (t) fj (t)(2)jAiwhere xi are the inputs or premises of the model,are the fuzzysets associated with each premise, fj is an arbitrary smooth functionthat is typically a linear model, and ŷj denotes the output of the jthlocal model.For the identification of Takagi–Sugeno models, fuzzy clusteringis used to derive the premises and least squares is used for theparameters of the consequences when the selected functions fj arelinear models [26].The uncertainty in model identification can be explained by twomain reasons: disturbances affecting the entire process and modeling errors given by the finite set of data used for the identification.To take this uncertainty into account, a fuzzy interval model [22] isused.Fuzzy confidence intervals use the covariance between observeddata yj and the output of each local system ŷj , which can be computed as follows:cov(yj ŷj ) ˆ j2 I 1 ˆ j2 ϕjT (ϕi ϕjT ) ϕj(3)where ˆ j2 E{ei eiT } is the variance of the noise of the jth local modeland ϕj is the fuzzy local regression matrix for rule j. Additionally,the linear model for each rule j is as follows:ˆTŷj ϕj(4)where ˆ are the nominal parameters of the consequences withparameter covariance: ˆ 2 ˆ j2 (ϕj ϕjT ) 1j(5)This implies that j N( ˆ j , ˆ 2 )j(6)Since a linear model is used as the consequence in each rule ofthe Takagi–Sugeno model, this analysis allows each local system tobe written as an interval transfer function. Indeed, from expression(6), the lower and upper levels of each consequence parameter canrespectively be written as follows: j ˆ j ˇ ˆ j j ˆ j ˇ ˆ j(7)where ˇ is a tuning parameter that considers the percentage ofdata covered by the interval. This allows the consequences of3the model (2) to be written as interval transfer functions. Toaddress the closed-loop stability using these models, the generalized Kharitonov’s theorem is used, which is briefly described inthe following section.2.2. The generalized Kharitonov’s theoremThe general problem is to ensure the stability of a closed-loopsystem given a controller C(s) with fixed parameters and a plantH(s) whose parameters vary independently, i.e. [24],C(s) m(s)n(s)H(s) b(s)a(s)(8)where a(s) and b(s) are defined as interval polynomials withbounded parameters as in (1).Thus, b(s) and a(s) generate a family of polynomials given byall the infinite combinations of their coefficients and accordingly, afamily for H(s) denoted H(s).Using the notation presented in [24], the sets of the fourKharitonov’s polynomials of a(s) and b(s) are Ka (s) and Kb (s), respectively.For both polynomials, Ka (s) and Kb (s), Kharitonov’s line segments are introduced for an interval polynomial p(s) as Sp (s) {[Kp1 , Kp2 ], [Kp1 , Kp3 ], [Kp2 , Kp4 ], [Kp3 , Kp4 ]}, where the line segmentbetween two polynomials (for instance, ı1 (s) and ı2 (s)) is definedas the one parameter family of polynomials [27]:[ı1 (s), ı2 (s)] {ı (s) : ı (s) ı1 (s) (1 )ı2 (s)}(9)with [0, 1].Using the previous definitions, the extremal and Kharitonov’ssystems of H(s) are respectively defined as follows:HE : Ka (s)Sb (s)HK (s): Sa (s)Kb (s)Ka (s)Kb (s)(10)(11)With this, the generalized Kharitonov’s theorem is introduced.Theorem 1.Generalized Kharitonov’s theorem [9](I) The controller C(s) stabilizes the entire family H(s) if and only ifC(s) stabilizes every segment in HE (s).(II) (Vertex Condition): Moreover, if the numerator and denominator of C(s) are of the form st (as b)U(s)Q(s), where t 0 is an arbitraryinteger, a and b are arbitrary real numbers, U(s) is an anti-Hurwitzpolynomial, and Q(s) is an even or odd polynomial, then it is sufficientthat C(s) stabilizes the finite set of HK (s) to ensure the stability of thesystem.Using this theorem, a fuzzy robust control strategy is derivedand described in the following section.2.3. Fuzzy robust control designIn this section, the methodology for designing the proposedfuzzy robust controller is explained. The novelty of this approachis the use of the fuzzy confidence interval’s information of disturbances in the design of the controller, allowing both fuzzyand parametric uncertainty approaches to be combined for arobustly stable scheme. In general terms, the design process forthe controllers consists of three stages: the identification of acontinuous-time model for the process, the determination of theranges for the controller parameters that ensure robust stability,and finally, the definition of performance specifications.Please cite this article in press as: A. Bayas, et al., Design of fuzzy robust control strategies for a distributed solar collector field, Appl.Soft Comput. J. (2017), https://doi.org/10.1016/j.asoc.2017.10.003

G ModelARTICLE IN PRESSASOC-4498; No. of Pages 11A. Bayas et al. / Applied Soft Computing xxx (2017) xxx–xxx4In the next sections, the stages of the design are analyzed.2.3.1. Continuous-time identification of a Takagi–Sugeno modelwith parametric uncertaintySince Kharitonov’s theorem is derived in general for continuoustime systems, the analysis starts by performing a continuous-timeidentification experiment on the plant to be controlled using a statevariable filter (SVF) [28]. This method combines the least squaresapproach with filters on the input and output signals to identify thesystem.In this case, the basic SVF that is used is given by the following:L(s) n(12)s where is a parameter that takes the dynamics of the system intoaccount and that is tuned using a value that is larger than the bandwidth of the process, and n is the estimated order of the plant.Subsequently, using the fuzzy confidence interval methoddescribed in Section 2, the model can be written in a general formfor an SISO plant as follows:Rj :(n)j(n 1)If yf (t) is An and yfj(t) is An 1 and. . .andjẏf (t) is A1 thenjHj (s) jjjbm sm bm 1 sm 1 · · · b1 s b0jjj(13)jan sn an 1 sn 1 · · · a1 s a0where uf and yf are the filtered versions of the input and outputsignals, respectively, and Hj (s) is the interval transfer function forjjjjjjthe jth local model with bp [bp , b̄p ] and ap [ap , āp ].Next, the design of a robust controller based on a Takagi–Sugenofuzzy model with parametric uncertainty is described.2.3.2. Determination of the robust stability rangesFor the Takagi–Sugeno model with parametric uncertaintyshown in (13), a robust controller must be derived for each interval transfer function of the consequences. For this purpose, thegeneralized Kharitonov’s theorem is used for determining the stability ranges for the parameters of the controllers. If the proposedcontroller C(s) fulfills the vertex condition of the theorem, thenKharitonov’s systems HK (s) must be determined for the open-loopsystem; then, for each element of HK (s), the resulting transferfunctions at closed loop must be obtained. With this, a set of characteristic equations must be analyzed using the Routh–Hurwitzcriterion, finding the conditions for the parameters of the controllers that stabilize each polynomial in the set. The results of thisstage are the ranges in which it is possible to locate the parametersof the controller to ensure that the interval transfer function of eachrule in the fuzzy model is robustly stable.2.3.3. Performance conditionsThe application of the performance specifications, such as overshoot, settling time and gain or phase margins, on the intervaltransfer functions is generally difficult since their parameters arenot fixed. Therefore, it is necessary to characterize a transfer function family Hj (s) from fixed-coefficient transfer functions (see (13)).In this work, for each family, the study of its bounding systemsis proposed by defining all the possible combinations of transferfunctions that can be obtained using the lower and upper values oftheir uncertain parameters. With this, if an interval transfer function has U uncertain parameters, then 2U possible combinationsmust be analyzed. Subsequently, a controller within the specifications is derived for each transfer function and then tested for theremaining transfer functions. Finally, the controller that presentsthe best performance for all the functions in terms of the previous specifications is selected for the respective interval transferfunction.The proposed methodology is summarized in Fig. 1. In this figure, the right side is related to the robust stability constraints,associated with the generalized Kharitonov’s theorem, whereas theleft side takes the performance conditions into account, which inthis case are solved using the PSO algorithm.Using these concepts, a fuzzy robust controller is applied for aDSC power plant in the following sections.3. Application of fuzzy robust control strategy to thedistributed solar collector field3.1. Process descriptionThe proposed scheme is tested in a simulator of a DSC plant,which is located in Almeria, Spain [23]. The DSC uses arrays ofparabolic mirrors to concentrate sunlight into a receiver pipe toproduce steam for a power generator. The controlled variable inthe DSC field is the oil’s temperature at the end of the pipeline.This process is very appropriate for testing the proposed controllersince there are several disturbances that affect the plant operation,such as the environmental temperature, inlet oil temperature anddust on the mirrors, but the most important disturbance is the solarradiation.Because of the challenges involved in the operation of the DSC,several control strategies have been studied and implemented forthis plant, including fuzzy and robust approaches. Indeed, a fuzzycontrol strategy is proposed in [29], which is designed using linguistic variables. In [30], a fuzzy PI is designed and tested for the realplant using expert’s knowledge. Both of these approaches achievegood results since they can address the non-linear dynamics anddisturbances of the process. Advanced control techniques have alsobeen tested in the solar plant, such as in [31], where a predictive control with fuzzy goals and constraints is used for a betterinterpretation of the control requirements.As previously mentioned, the solar collector field is subject toseveral disturbances that affect its operation. Consequently, robustcontrol theory is an interesting alternative to address this issue,and some works applied to the solar plant can be found in the literature. In [32], an H controller is designed for the DSC integratedwith an air conditioning system. In that work, an ARX model is usedto represent the behavior of the entire process, and a controller isderived by solving an S/T sensibility problem, which attempts toincrease the disturbance rejection and robustness of the scheme.Another approach is proposed by [33], where the robustness andperformance of the controller are taken into account by using quantitative feedback theory (QFT). In that case, the selected model forthe DSC is a continuous-time, second-order system with uncertain(but bounded) natural frequency and DC gain.In most of the previously mentioned works, the DSC plant canbe modeled, under general assumptions, by the following systemof partial equations that describe its energy balance [34]: m cm Am f cf Af Tm I 0 D Hl G(Tm Tenv ) LH t (Tm Tf ) t Tf Tf f cf q LH t (Tm Tf ) t x(14)where the subindex m refers to the metal of the pipe, f refers to theheat transfer fluid, and t and x are time and space, respectively. Tis the temperature, Tenv is the environment temperature, is thedensity, c is the specific heat capacity, A is the cross-sectional area,D is the collector aperture, 0 is its efficiency, G is the pipe diam-Please cite this article in press as: A. Bayas, et al., Design of fuzzy robust control strategies for a distributed solar collector field, Appl.Soft Comput. J. (2017), https://doi.org/10.1016/j.asoc.2017.10.003

G ModelASOC-4498; No. of Pages 11ARTICLE IN PRESSA. Bayas et al. / Applied Soft Computing xxx (2017) xxx–xxx5Fig. 1. Controller design diagram.eter, I is the corrected direct solar irradiance, Hl is the convectivethermal losses of the pipe exterior, Ht is the convective heat transfer coefficient of the pipe interior, and q is the flow rate of the heattransfer fluid.The controlled variable in the DSC field is the temperature of theoil (Tf ) at the end of the pipeline.The main disturbances for this plant are the solar irradiance I,the inlet oil temperature Tin and the environmental temperatureTenv . Since these variables are measurable and their effects producechanges in the output of the system (Tf at the end of the pipe) unrelated to the fluid flow q, it is desirable to add a control strategy thatuses the measurement of the disturbances to reduce their effects onthe controlled variable. This can be achieved using a feedforwardcontrol in series, letting the input of the feedforward term (Treff :reference temperature) be the new manipulated variable for thissystem, as shown in Fig. 2. The sampling time for the process is 39[s].As mentioned in [23], several authors have modeled the DSCplant as a first-order system or overdamped second-order system,which is given by the open-loop response of the plant when a step isused as the input and considering the main disturbances in a quasisteady state. Furthermore, the addition of a feedforward controllerin series causes the entire plant to become an approximately linearsystem in some operation points.3.2. Fuzzy interval modelingWith the purpose of performing an SVF-based Takagi–Sugenoidentification as shown in Fig. 2, the bandwidth of the plant isrequired for tuning the parameter of the low-pass filter L(s). Several experiments are conducted in the DSC simulator using inputsat different frequencies and operation points to obtain an empirical Bode diagram, which is shown in Fig. 3. Note that the largestbandwidth of the system occurs when the plant is operating at 150[ C]. This bandwidth is approximately b 0.11 [rad/seg]; therefore,the parameter is tuned using the SVF as 2b.The DSC plant is a low-order system; thus, for the identificationtests, the highest order derivative considered is n 2. Then, the filteris given by the following:L(s) 0.22 2(15)s 0.22After the identification process, the best model is selected interms of its mean squared error (MSE), which for the case of thesolar plant has the form of a first-order system for each rule:jjRj : If ẏf (t) is A1 and yf (t) is A2 then:Hj (s) j 2j 1 s 1(16)Please cite this article in press as: A. Bayas, et al., Design of fuzzy robust control strategies for a distributed solar collector field, Appl.Soft Comput. J. (2017), https://doi.org/10.1016/j.asoc.2017.10.003

G ModelASOC-4498; No. of Pages 11ARTICLE IN PRESSA. Bayas et al. / Applied Soft Computing xxx (2017) xxx–xxx6Fig. 2. Identification experiment.Fig. 3. Bode diagram for outlet temperature.Fig. 4. Identified fuzzy interval model for the DSC plant.with j {1, 2, 3, 4}, i.e., four rules. Then, using fuzzy intervals with95% confidence, the model in (16) is written in Takagi–Sugeno formwith the same premises of (16) and interval transfer functions asconsequences with the following uncertain parameters:of the Takagi–Sugeno model presented in (16) and (17). Using theRouth–Hurwitz criterion on the sets HK (s) of the jth rule in thismodel, the corresponding closed-loop uncertain system is stable if 11 [126.7, 143.7], 21 [1.009, 1.014]Ki 0 12 [77.83, 85.86], 22 [1.022, 1.027] 13 [156.3, 172.8], 23 [0.9897, 0.995]j(17) 14 [111.7, 124.5], 24 [1.01, 1.014]The performance of the given fuzzy interval model with theparameters from (17) is presented in Fig. 4.3.3. Fuzzy robust controller designFrom the model with parametric uncertainty given

a very non-linear behavior and significant disturbances. Addition-ally, in [21], the fuzzy interval models are used for fault detection purposes in uncertain systems. As shown above, a considerable amount of research has been conducted to develop a suitable model that can both manage uncertainties and model the nonlinear behavior of a process. Our

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A1. Bench Press – Alternate Grip A2. Bench Press – Feet in Air A3. Bench Press – Hands Together A4. Bench Press – One Arm A5. Bench Press – Reverse Grip A6. Bench Press – Roman Chair A7. Bent Press A8. Clean and Jerk – Behind Neck A9. Clean and Jerk – One Arm

Cool ‒ Em Heat ‒ Off. Set the time and date. Time. 1 Press . Menu. on your thermostat. 2 Press or to go to . TIME. Press . Select. 3 Press or to choose between 12 or 24 hour. Press . Select. 4 Use or to adjust the hour. Press . Select. 5 Use or to adjust the minutes. Press . Select.


entering SEND mode. Press to display max, min and average values. Press EXIT to stop and return to current measurement mode. Press the button once. Press to output the data, AUTO mode switch off. The primary display shows “SEND”. Press EXIT to exit. Press and hold the button for over 1 second. In S

entering SEND mode. Press to display max, min and average values. Press EXIT to stop and return to current measurement mode. Press the button once. Press to output the data, AUTO mode switch off. The primary display shows “SEND”. Press EXIT to exit. Press and hold the button for over 1 second. In S