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Fuzzy Inference System Approach Using Clustering and Differential Evolution Optimization Applied to Identification of a Twin Rotor System
Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World The International Federation of Congress Automatic Control Toulouse, France, July 2017 Proceedings of the 20th World Congress The International of Automatic Control Available online at www.sciencedirect.com Toulouse, France,Federation July 9-14, 2017 The International Federation of Automatic Control Toulouse, France, July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC Approach PapersOnLine 50-1Using (2017) 13102–13107 Fuzzy Inference System Clustering and Differential Evolution Fuzzy Inference System Approach Using Clustering and Differential Evolution Fuzzy Inference System Approach Using Clustering and Differential Evolution Optimization Applied to Identification of a Twin Rotor System Fuzzy Inference System Approach Using Clustering and Differential Evolution Optimization Applied to Identification of a Twin Rotor System Optimization Applied to Identification of a Twin ,Rotor System ,# # Optimization Applied to Identification of a Twin Rotor System
Leandro dos Santos Coelho*,#, Marcelo Wicthoff Pêssoa#, Viviana Cocco Mariani*,**, and Gilberto Reynoso-Meza* Leandro dos Santos Coelho* , Marcelo Wicthoff Pêssoa , Viviana Cocco Mariani* **, and Gilberto Reynoso-Meza* Leandro dos Santos Coelho*,#,#, Marcelo Wicthoff Pêssoa##, Viviana Cocco Mariani*,,**, and Gilberto Reynoso-Meza* , Marcelo Pêssoa , Viviana Cocco Mariani* **, and Gilberto Reynoso-Meza* Leandro dos Santos Coelho* * Industrial andWicthoff Systems Engineering Graduate Program (PPGEPS) *Pontifical IndustrialCatholic and Systems Engineering Graduate Program (PPGEPS) University of Parana (PUCPR), Curitiba, Brazil *Pontifical Industrial and Systems Engineering Graduate Program (PPGEPS) # Catholic University ofGraduate Parana (PUCPR), Curitiba, Brazil *Pontifical Industrial and Systems Engineering Graduate Program (PPGEPS) Electrical Engineering Program (PPGEE) # Catholic University of Parana (PUCPR), Curitiba, Brazil Engineering Program (PPGEE) # Electrical Pontifical Catholic University ofGraduate Parana (PUCPR), Curitiba, Federal University of Parana (UFPR), Curitiba, Brazil Brazil Engineering Graduate Program (PPGEE) # Electrical University of Parana (UFPR), Curitiba, Brazil Electrical Engineering Graduate Program (PPGEE) **Federal Mechanical Engineering Graduate Program (PPGEM) Federal University of Parana (UFPR), Curitiba, Brazil ** Mechanical Engineering Graduate Program (PPGEM) Federal University of Parana (UFPR), Curitiba, Brazil Brazil Pontifical Catholic University of Parana (PUCPR), Curitiba, ** Mechanical Engineering Graduate Program (PPGEM) Pontifical Catholic University of Parana (PUCPR), Curitiba, Brazil ** Mechanical Engineering Graduate Program (PPGEM) (E-mails: [email protected], [email protected], [email protected], [email protected]) Pontifical Catholic University of Parana (PUCPR), Curitiba, Brazil (E-mails: [email protected], [email protected], [email protected], [email protected]) Pontifical Catholic University of Parana (PUCPR), Curitiba, Brazil (E-mails: [email protected], [email protected], [email protected], [email protected]) (E-mails: [email protected], [email protected], [email protected], [email protected]) Abstract: In this paper, a Takagi-Sugeno-Kang (TSK) fuzzy inference system using fuzzy c-means Abstract: and In this paper, aevolution Takagi-Sugeno-Kang fuzzy using fuzzy c-means clustering differential optimization is(TSK) proposed andinference validatedsystem when applied to a twin rotor Abstract: and In this paper, aevolution Takagi-Sugeno-Kang (TSK) fuzzy inference system using fuzzy c-means clustering differential optimization is proposed and validated when applied to a twin rotor Abstract: In this paper, a Takagi-Sugeno-Kang (TSK) fuzzy inference system using fuzzy c-means system (TRS). The TRS is perceived as a challenging problem due to its strong cross coupling between clustering and differential optimization is proposed and validated whencross applied to a twin rotor system (TRS). The TRSaxes. isevolution perceived as a challenging problem due to its approach strong coupling between clustering and differential evolution optimization is proposed and validated when applied to a twin rotor horizontal and vertical The design procedure of the TSK fuzzy for TRS is detailed. system (TRS). The TRSaxes. is perceived as a challenging duefuzzy to its approach strong cross between horizontal and vertical The design procedure ofproblem the TSK for coupling TRS detailed. system (TRS). The TRSaxes. is perceived asobtained a challenging problem dueTSK to itsfuzzy strong cross coupling between According to the identification results by applying the approach and ais nonlinear horizontal and vertical The design procedure of the TSK fuzzy approach for TRS is detailed. According to the identification results obtained by applying the TSK fuzzy approach and ais nonlinear horizontal and vertical axes. The design procedure of the TSK fuzzy approach for TRS detailed. autoregressive with moving average and exogenous inputs (NARMAX) model, the effectiveness of the According to the identification results by applying the TSK fuzzy approach and a nonlinear autoregressive with moving average andobtained exogenous inputs (NARMAX) model, the effectiveness of the Accordingfuzzy to the identification results obtained by applying thetests. TSK fuzzy approach and a nonlinear proposed system design is demonstrated through validation autoregressive with moving average and exogenous inputs (NARMAX) model, the effectiveness of the proposed fuzzy system design is demonstrated through validation tests. autoregressive with moving average and exogenous inputs (NARMAX) model, the effectiveness of the proposed fuzzy systemidentification, design is demonstrated through validation tests. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. Allcomputation. rights reserved. Keywords: Nonlinear fuzzy system, differential evolution, evolutionary proposed fuzzy system design is demonstrated through validation tests. Keywords: Nonlinear identification, fuzzy system, differential evolution, evolutionary computation. Keywords: Nonlinear identification, fuzzy system, differential evolution, evolutionary computation. Keywords: Nonlinear identification, fuzzy system, differential evolution, evolutionary computation. been used to analyze the performance of TRS and TRMS. A 1. INTRODUCTION been used to analyze the performance of TRS and of TRMS. A multi-variable nonlinear control-oriented model a twin 1. INTRODUCTION been used to analyze the performance of TRS and of TRMS. A multi-variable nonlinear control-oriented model a twin 1. INTRODUCTION been used to analyze the performance of TRS and TRMS. A rotor system is presented in Butt and Aschemann (2015). The The identification of1.dynamic nonlinear systems, which pose multi-variable nonlinear control-oriented model of a twin INTRODUCTION rotor system ismodel presented in Butt is andderived Aschemann (2015). The The identification of dynamic nonlinear systems, which pose multi-variable nonlinear control-oriented model of a twin mathematical of system using Lagrange’s problems and require solutions distinctsystems, from their system ismodel presented in Butt is andderived Aschemann (2015). The The identification of dynamic nonlinear whichlinear pose rotor mathematical of resulting system using Lagrange’s problems andisrequire solutions distinctsystems, frombytheir linear rotor system ismodel presented in Butt is and Aschemann (2015). Thea equations. Based on the state-space representation, The identification of dynamic nonlinear which pose counterparts, a hard task as demonstrated the effort mathematical of system derived using Lagrange’s problems andisrequire distinct frombytheir Based on the resulting state-space representation, a counterparts, a hardinsolutions task as demonstrated the linear effort equations. mathematical model of system is derived using Lagrange’s multi-variable integral sliding mode control is designed, also problems and require solutions distinct from their linear devoted by researchers the last decades. Several techniques Based on thesliding resulting state-space representation, a counterparts, is a hardintask as demonstrated by techniques the effort equations. multi-variable integral mode control is designed, also devoted by researchers the last decades. Several Based extended on the resulting state-space representation, a a discrete-time Kalman filter isis applied from counterparts, is a hardin task as demonstrated by the effort equations. have been proposed for nonlinear system identification multi-variable integral sliding mode control designed, also devoted by researchers the last decades. Several techniques asimplifications discrete-timeintegral extended Kalman filter isis designed, applied from have been proposed forthenonlinear system identification multi-variable sliding mode control also at modelling as well as disturbance torques. devoted by researchers in last decades. Several techniques (Coelho et al., 2014, Klein et al., 2015; Ayala and Coelho, a discrete-timeatextended Kalman filter is applied from have been proposed for nonlinear system identification modelling as and well Laxmi as disturbance torques. (Coelho et al., 2014, Klein etand al., Kerschen, 2015; Ayala and Coelho, asimplifications discrete-time extended Kalman filter is applied froma The work proposed by Pandey (2014) applied have been proposed for nonlinear system identification 2014; Ayala et al., 2015; Noël 2017). simplifications at modelling as well as disturbance torques. (Coelho et al., 2014, Klein et al., 2015; Ayala and Coelho, The work proposed byand Pandey and Laxmi (2014) applied a 2014; Ayala et al., 2015; Noël and Kerschen, 2017). simplifications at modelling as well as disturbance torques. proportional, integral derivative (PID) controller with (Coelho et al., 2014, Klein al., Kerschen, 2015; Ayala and Coelho, The work proposed by Pandey and Laxmi (2014) applied aa 2014; Ayala et al., 2015; Noëletand 2017). proportional, integral and derivative (PID) controller with aa This paper presents an investigation on dynamic modelling of The work proposed by Pandey and Laxmi (2014) applied derivative filter coefficient to control a TRMS. 2014; Ayala et al., 2015; Noël and Kerschen, 2017). integral and derivative controller with a This paper presents an investigation on dynamic of proportional, derivative filter coefficient to control a(PID) TRMS. the vertical movement of a twin rotor system; modelling a laboratory proportional, integral and derivative (PID) controller with a This paper presents an investigation on dynamic modelling of derivative filter coefficient to control a TRMS. the vertical movement offlexible a twin maneuvering rotor system;structure a laboratory terms offilter the coefficient recent literature, there are many researches This paperrepresenting presents an investigation on dynamic modelling of In platform a and derivative to control a TRMS. the vertical movementaofflexible a twin maneuvering rotor system;structure a laboratory termsTRMS. of the recent literature, there are many platform representing and In using In Shaik et al. (2011) the researches unknown the vertical movement offlexible a twin maneuvering rotor a laboratory resembles essential characteristics of a system; helicopter. The twin In termsTRMS. of the recent literature, there are many platform representing a structure and using In Shaik et al. (2011) the researches unknown resembles essential characteristics of a helicopter. The twin In terms of the recent literature, there are many researches nonlinearities of TRMS are estimated by Chebyshev neural platform representing a flexible maneuvering structure and rotor system (TRS) imposes challenging control problems TRMS.of In Shaik et al. (2011) the unknown resembles essential characteristics of a helicopter. The twin using nonlinearities TRMS are estimated by Chebyshev neural rotor system (TRS) imposes challenging control problems using TRMS. In Shaik et al. (2011) the unknown network whose weights are adaptively adjusted. Lyapunov resembles essential characteristics of a helicopter. The twin due to its given nonlinearities as well as significant couplings of weights TRMS are estimated byadjusted. Chebyshev neural rotor system (TRS) imposes aschallenging control couplings problems nonlinearities network whose are adaptively Lyapunov due to its given nonlinearities well as significant nonlinearities of TRMS are estimated by Chebyshev neural is whose used toweights guarantee stability for state estimation and rotor system (TRS) imposes control problems between the pitch axis and aschallenging the azimuth axis (Butt and theory network are adaptively adjusted. Lyapunov due to its given nonlinearities well as significant couplings theory is whose used toweight guarantee stability for state estimation and between the pitch axis and the azimuth axis (Butt and network weights are adaptively adjusted. Lyapunov neural network errors. An ant colony optimization is due to its given nonlinearities as well as significant couplings Aschemann, 2015). In general, it consists of two DC (direct theory is used to guarantee stability for state estimation and between the 2015). pitch In axis and the azimuthof axis (Butt and neural network weight errors. Ana ant colony optimization is Aschemann, general, it consists two DC (direct theory is used to guarantee stability for state estimation and deployed and used for modelling twin rotor system by Toha between the pitch axis and the azimuth axis (Butt and neural network weight errors. An ant colony optimization is current) motors, oneIn for main rotor and other tail(direct rotor Aschemann, 2015). general, it consists of twofor DC deployed and A used for modelling acontroller twincolony rotorwas system by Toha current) motors, one for main rotor and other for tail rotor neural network weight errors. An ant optimization is et al. (2012). hybrid intelligent proposed by Aschemann, 2015). general, it consists of two DC perpendicular to each other. A pivoted beam is for joined on its deployed and used for modelling a twin rotor system by Toha current) motors, oneInfor main rotor and other tail(direct rotor et al. (2012). A hybrid intelligent controller was proposed by perpendicular to each other. A pivoted beam is joined on its deployed and used for modelling twin rotor system by Toha Juang et al. (2014), which apply a fuzzy PID control scheme current) motors, onerotating for main rotor and otherhorizontal for tail on rotor base which allows it pivoted freely onbeam the and al. (2012). A hybrid intelligent proposed by perpendicular to each other. A joined its et Juang etreal-valued al. (2014), which apply acontroller fuzzy PIDwas control scheme base which allows rotating itthe freely onbeam the is horizontal and et al. a(2012). A hybrid intelligent controller was by with genetic algorithm to control aproposed TRMS. An perpendicular to each other. A pivoted is joined on its vertical planes. By changing input voltage, the angular Juang et al. (2014), which apply a fuzzy PID control scheme base which allows itthe freely the horizontal and with a etreal-valued genetic algorithm to control athe TRMS. An vertical planes. By rotating changing inputon voltage, the angular Juang al. (2014), which apply a fuzzy PID control scheme analytical method for tuning the parameters of set-point base which allows rotating it freely on the horizontal and speed of these two rotors is controlled. For balancing the with a real-valued genetic algorithm to control TRMS. An vertical planes. By changing the input voltage, the angular method tuning the parameters of aathe set-point speed ofmomentum these two rotors is state controlled. For the analytical with a real-valued genetic algorithm to control TRMS. An weighted fractionalfor order PID controller is proposed in vertical planes. By in changing the input voltage, angular angular steady or with load,balancing athe pendulum analytical method for tuning the parameters of the set-point speed of these two rotors is controlled. For balancing the weighted fractional order PID controller is proposed in angular momentum inrotors steady state or withbeam load,balancing a pendulum analytical method for tuning the parameters of the set-point Azarmi et al. (2015) and applied a TRS. speed of these two is controlled. For the counter weight is hanged on the joined (Pandey and weighted fractional order PID a controller is proposed in angular momentum in steady state or withbeam load,(Pandey a pendulum Azarmi et al. (2015) and applied TRS. counter weight is hanged on the joined and weighted fractional order PID controller is proposed in angular momentum in steady or load, a pendulum Laxmi, 2014). isMoreover, not all with state variables are Azarmi et al. (2015) and applied a TRS. counter weight hanged on state the joined beam (Pandey and In this work, to achieve a satisfactory tracking performance Laxmi, 2014). Moreover, not all state variables are Azarmi et al. (2015) and applied a TRS. counter weight hanged on not the joined beamvariables (Pandey and measurable. In this work, to achieve a satisfactory trackingangle performance Laxmi, 2014). isMoreover, all state are regarding desired trajectories for the azimuth and the measurable. In this work, to achieve a satisfactory trackingangle performance Laxmi, 2014). Moreover, not all state variables are regarding desired trajectories for the azimuth and the measurable. In this work, to achieve a satisfactory tracking performance pitch angle, this work applied the Takagi-Sugeno-Kang In several contributions presented in literature, both regarding desired trajectories for the azimuth angle and the measurable. pitch angle, this work applied the Takagi-Sugeno-Kang In several and contributions in literature, both pitch regarding desired for angle andwith the (TSK) angle, fuzzy approach, the the ARX (autoregressive modelling experimentalpresented identification of TRS (also this trajectories workusing applied the azimuth Takagi-Sugeno-Kang In several and contributions presented in literature, both pitch (TSK) fuzzy approach, using theconsequent ARX (autoregressive with modelling experimental identification of TRS (also angle, this work applied the Takagi-Sugeno-Kang exogenous inputs) model in the part of the rules. In several contributions presented in literature, both TRMS, M for multiple inputs and multiple outputs) have (TSK) fuzzy approach, theconsequent ARX (autoregressive with modelling experimental of TRS (also exogenous inputs) modelusing in the of the rules. TRMS, M and for multiple inputsidentification and multiple outputs) have The (TSK) fuzzy approach, using theconsequent ARX (autoregressive witha TSK fuzzy model uses if-then rules topart approximate modelling and experimental identification of TRS (also been investigated and several models have been proposed exogenous inputs) model in the part of the rules. TRMS, M for multiple inputs and multiple outputs) have The TSK fuzzy model uses if-then rules to approximate a been the investigated and several models have techniques been proposed exogenous inputs) model in the consequent part of the rules. wide class of nonlinear systems by fuzzy blending of local TRMS, M for multiple inputs and multiple outputs) have over past few years and various control The model uses if-then rules to approximate been investigated and several models have techniques been proposed wide TSK class fuzzy of nonlinear systems by fuzzy blending of localaa over the past few years and various control have The TSK fuzzy model uses if-then rules to approximate been investigated and several models have techniques been proposed over the past few years and various control have wide class of nonlinear systems by fuzzy blending of local over the past few years and various control techniques have wide class of nonlinear systems by fuzzy blending of local Copyright © 2017 IFAC 13644 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017, 2017 IFAC 13644 Copyright 2017 responsibility IFAC 13644 Peer review©under of International Federation of Automatic Control. Copyright © 2017 IFAC 13644 10.1016/j.ifacol.2017.08.2162
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Leandro dos Santos Coelho et al. / IFAC PapersOnLine 50-1 (2017) 13102–13107
linear approximations. This method employs linear models in the consequent part of the fuzzy system. In the adopted TSK approach, to determine the centers of membership functions, the clustering technique called fuzzy c-means is adopted, and for determine the Gaussian membership functions width a differential evolution (DE) approach is employed. Also, in terms of performance analysis, a model-based on polynomial nonlinear autoregressive with moving average and exogenous inputs (NARMAX) which uses the Orthogonal Least Squares (OLS) estimator, coupled to the Golub-Householder decomposition with the calculation of ERR (Error Reduction Ratio) is applied to determine the most significant terms for TRS controlling model was tested too and compared with the proposed TSK design. The remainder of the paper is arranged as follows. In Section 2, the TRS system is detailed and the parameters of the system specified. The proposed approach using fuzzy system is introduced in Section 3. In Section 4, the identification performance is demonstrated by providing experimental results on the TRS. Finally concluding remarks are made in the last section. 2. FUNDAMENTALS OF THE TWIN ROTOR SYSTEM The TRS is a laboratory setup designed for control experiments. In certain aspects its behavior resembles that of a helicopter. From the control point of view it exemplifies a high order nonlinear system with significant cross couplings. As shown in Fig. 1, the TRS mechanical is driven by two DC (direct current) motors at both ends of the beam; the main rotor motor and tail rotor motor. Its two propellers are perpendicular to each other and joined by a beam pivoted on its base that can rotate freely in the horizontal and vertical planes. The beam can thus be moved by changing the input voltage in order to control the rotational speed of the propellers. The articulated joint allows the beam to rotate in such a way that its ends move on spherical surfaces. The lowcost system designed by Pessôa (2010) is equipped with a pendulum counterweight hanging from the beam, which is used for balancing the angular momentum.
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The state of the TRSs is described by four process variables: two inputs variables u1 and u2, that represent the angular velocity of the rotors, measured by tachometers coupled with the driving DC motors; the horizontal (yaw) and vertical (pitch) angles, measured by position sensors fitted at the pivot and represented by y1 and y2 as illustrated in Fig. 2.
Fig. 2. Inputs (u) and outputs (y) variables of the TRS. 3. FUZZY INFERENCE SYSTEM Fuzzy inference schemes mainly divided into two models, Mamdani (Mamdani and Assilian, 1975) and TSK (TakagiSugeno-Kang) (Takagi and Sugeno, 1985). In the TSK-type fuzzy system, the consequent of each rule is a function input linguistic variable. The general adopted function is a linear combination of input variables plus a constant term, and each rule is of the following form: : … (1) 1 ∑
where , , … , is the vector of input variables; , , … , are fuzzy sets, 1,2, … , ,which represent the number of rules and is the rule output. The final network output is a weight average of each rule’s output of the form:
∑ ∑
(2)
In Equation 9, is the degree of activation of the i-th rule: ∏ , 1,2, … , (3) where is the membership function of the fuzzy set at the antecedent (input) of . The number of rules characterizes the structure of a fuzzy system. The number of rules can be determined by clustering methods. In this work the Fuzzy C-Means (FCM) clustering was applied. 3.1 Clustering and Gaussian membership functions tuning
Fig. 1. Photograph of the TRS.
The FCM algorithm is one of the clustering and structure identification methods. For the proposed model, FCM algorithm is necessary to determine the regression matrix vector membership values and cluster centers. In FCM, the 13645
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data sample can belong to more than one cluster with different fuzzy membership values that represent the degree of membership between data and centers of clusters. The main idea of the FCM algorithm can be described as: At each iteration 0,1,2 … , the membership values ( ) and cluster centers ( ) are determined as following the Equations 4 to 5 given by
∑
∑ ∑
, 1 ≤ ≤ , 1 ≤ ≤ (4)
, 1 ≤ ≤
(5) The cost function that minimizes the Euclidian distance between the clusters center and samples is expressed as Equation 6 given by
∑ ∑ ‖
−
‖
, 1 ≤ ≤ ∞
Step 5: Following the mutation operation, crossover using binomial (bin) distribution is applied in the population; Step 6: Update the generation’s (iterations) counter; Step 7: Loop to Step 3 until a stopping criterion is met, usually a maximum number of iterations, tmax. 3.3 Implementation of the fuzzy modelling To perform a practical implementation of the TSK-ARX method, the following steps can be carried out: Step 1: Determine the parameters that will compose the ARX structure selection ( , );
Step 2: Define how many input–output lagged terms are used in regression vector. It can be changed by considering the model complexity; Step 3: Perform FCM clustering algorithm to determine the centers of the regressors; Step 4: Perform DE to optimize the Gaussian membership functions width;
(6)
Step 5: Calculate the weight parameters of the model by using the least-squares estimator;
The stopping criteria adopted is when − < , where the matrix is obtained with membership values of data points computed from Equation 5, where is a small constant to stop clustering algorithm, is the iteration number, is the number of samples, is the number of cluster and is the weighting factor. More detailed information about FCM algorithm can be found in Bezdek et al. (1984). After determine the centers of the regressors, the optimal width for Gaussian membership functions is obtained through Differential Evolution (DE).
Step 6: Apply the procedures of the model validation and results analysis.
3.2 Optimization based on DE algorithm DE is an evolutionary algorithm proposed by Storn and Price (1995). Due to its simplicity of implementation and its efficiency and effectiveness in solving optimization problems, DE has become a popular technique. The particular version subject to our investigation is the DE/rand/1/bin-version, which appears to be the most frequently used variant, and is often considered as the “basic” version of the DE-algorithm. The steps of the DE algorithm can be summarized as follows: Step 1: The user must choose the key parameters that control DE, i.e., population size (N), boundary constraints of optimization variables, mutation factor (F), crossover rate (CR), and the stopping criterion (tmax). Step 2: Initialize the generation’s counter t= 0 and also initialize a population of individuals (solution vectors) x(t) in upper and lower bounds of each decision variable with random values generated according to a uniform probability distribution in the n-dimensional problem space. Step 3: For each individual, evaluate its objective function value (maximization of the R2 is adopted here); Step 4: Mutate individuals of the population;
3. SETUP AND RESULTS ANALYSIS In the following sub-sections, the identification setup, k-step ahead forecasting results using NARMAX model with OLS and the proposed fuzzy system design are presented. After, the results analysis is detailed. 4.1 Identification setup The identification algorithms presented in the previous sections is applied to a TRS. This system has high-order nonlinearity and significant cross-coupling is observed between the actions of the rotors, with each rotor influencing both angle positions. Different signals classes can be employed for the identification process, such as multi-sine signals, maximum length binary sequences and pseudo-random binary signals. In this work, multi-sine signals as excitation signals were applied to the inputs that controls yaw and pitch. The input signal is provided to the TRS through a computer interface. The connection is made through an electronic circuit that is built using three microcontrollers: PIC16F877, MAX232 and L298. The multi-sine signals were applied during 57 seconds and 377 samples of TRS angular position were collected with sampling period equal to 0.15 seconds. These samples were divided into estimation and validation group, the earlier was composed by 350 samples and the second all 377 samples were applied. The aim of the identification algorithms is obtain a model to do the prediction for k-step ahead applying both techniques. 4.2 Results for k-step ahead forecasting using NARMAX For the NARMAX modelling (Billings, 2013) to predicted ksteps-ahead inclination output ( ) and rotation output y2,
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through sine input signal excitation, the parameters adopted were: nonlinear order 2, delay of sampling interval 5, number of input-output lagged terms 9, the maximum lags , , were evaluated between 2 and 3. The adopted design in the NARMAX case was a decoupled estimation for y1 and y2. The mathematical model obtained for the TRS inclination output (y1) prediction, with the maximum lag 2, 2 and 3 , is shown on Equation 7. Figure 3 presents the result for output related the inclination comparing it with the real output and the signal error of prediction. The best mathematical model obtained for the inclination output ( ) is given by 1 0.1655 ∙ 10 ∙ 1 − 1 − 0.6862 ∙ 1 − 2 − 0.2302 ∙ 1 − 1 ∙ 1 − 1 − 0.1223 ∙ 10 0.2203 ∙ 10 ∙ 1 − 2 ∙ 1 − 2 0.3518 ∙ 10 ∙ 1 − 1 − 0.329 ∙ 10 ∙ 1 − 2 − 0.1040 ∙ 10 ∙ 1 − 2 ∙ 1 − 2 0.9047 ∙ 10 ∙ 1 − 1 ∙ 1 − 1 0.3030 ∙ 10 ∙ − 3 ∙ 1 − 1 0.1121 ∙ 10 ∙ 1 − 3 ∙ 1 − 2 0.2175 ∙ 1 − 2 ∙ 1 − 1 0.7910 ∙ 10 ∙ 1 − 3 − 0.1084 ∙ 1 − 1 ∙ 1 − 2
(7)
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For the NARMAX modelling to predicted k-steps-ahead rotation output ( ), through sine input signal excitation, the parameters adopted were the same as applied to modelling inclination output, only the nonlinear order was increased to 3. Also the maximum lag values were chosen between those values that have the lowest MSE (Mean Squared Error), and the respective values are: 3, 3 and 3. Figure 4 presents comparisons between real and the predicted output and the signal errors of the predicted output signal, respectively. The best mathematical model obtained for the rotation output ( ) is given by 2 0.1616 ∙ 10 ∙ 2 − 1 − 0.6069 ∙ 2 − 2 − 0.4486 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 ∙ 2 − 1 − 0.1963 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 ∙ 2 − 1 0.4081 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 − 0.1683 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 ∙ 2 − 1 0.4067 ∙ 2 − 1 0.3233 ∙ 10 ∙ 2 − 3 ∙ 2 − 3 ∙ 2 − 1 − 0.1309 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 ∙ 2 − 1 − 0.4077 ∙ 10 ∙ 2 − 2 ∙ 2 − 3 ∙ 2 − 3 − 0.3360 ∙ 10 ∙ 2 − 2 ∙ 2 − 2 ∙ 2 − 2 0.1189 ∙ 10 ∙ 2 − 3 ∙ 2 − 1 0.3129 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 0.6318 ∙ 10 ∙ 2 − 1 ∙ 2 − 3 ∙ 2 − 3
(8)
(a) k-steps-ahead prediction for rotation ( ) output
(a) k-steps-ahead prediction for inclination ( ) output
(b) error signal
(b) error signal
Figure 3. Forecasting using NARMAX modelling for inclination ( ) output.
Figure 4. Forecasting using NARMAX modelling for rotation ( ) output. 13647
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4.3 Results for k-step ahead forecasting using fuzzy system The parameters assumed for TSK fuzzy system with ARX model were: number of membership functions 5; In the FCM algorithm the number of clusters 5, weighting factor 2 and the small constant to stop FCM algorithm 10 . In the membership Gaussian width optimization process, the DE algorithm was applied with the following parameters: mutation factor 0.8, crossover rate 0.5, and maximum of iterations (generations), 10.
The values 1 and 3 were chosen between the set of results for inclination and rotation output modelling due to present the lowest MSE. For the inclination output, comparisons between predicted and real output are shown on Figure 5. Furthermore, comparisons between real and predicted output are shown in Figure 6.
(a) k-steps-ahead prediction for inclination ( ) output
(b) error signal
(a) k-steps-ahead prediction for inclination ( ) output
(b) error signal Figure 5. Forecasting using TSK modelling for inclination ( ) output.
Figure 6. Forecasting using TSK modelling for rotation ( ) output.
4.4 Analysis of the identification results
The results for modelling TRS by NARMAX and TSK-Fuzzy ARX approach for k-step-ahead prediction with input signal excitation given by multi-sine have shown good results. The results are shown on Tables 1 and 2 for both techniques comparison. The results for one-step ahead are shown in Table 1 and 2 too. For multi-sine input signal excitation the TSK fuzzy system with ARX structure have shown the best result, as per onestep-ahead or k-steps-ahead, with lowest standard deviation for inclination and rotation axis output prediction. Therefore, the only drawback when comparisons are made with NARMAX approach is related to the computational cost. The TSK fuzzy system with ARX model had shown the highest computational cost, mostly imposed by DE algorithm applied to perform the optimization of membership Gaussian width. 5. CONCLUSION In this investigation, two approaches has been applied to modelling the TRS system in terms of inclination and rotation output. An accurate model has been extracted by applying a NARMAX approach to predict one-step-ahead with two kind of excitation signal, also k-steps-ahead output was investigated too. The second approach, which applies 13648
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TSK fuzzy approach, a model has been developed accordingly and the results shown the capability of TRS identification with good performance when compared with the NARMAX approach. Table 1. Results for TRS modelling by applying NARMAX and TSK approaches (inclination axis). Model (step ahead) NARMAX (1) Fuzzy (1) NARMAX (k) Fuzzy (k)
Error signal
R²
Standard Deviation 0.3975
Maximum
Minimum
Mean
0.9995 0.9996
1.3982
-1.3523
-0.0042
1.2308
-1.5149
0.3580
0.9555 0.9798
11.7714
0.0083 0.9219 0.0205
2.5490
7.1903
-7.7090 -6.1061
3.9498
Table 2. Results for TRS modelling by applying NARMAX and TSK approaches (rotation axis). Model (step ahead) NARMAX (1) Fuzzy (1) NARMAX (k) Fuzzy (k)
Error signal
R²
Standard Deviation 2.0596
Maximum
Minimum
Mean
0.9997 0.9999
17.1505
-13.2919
0.0822
9.2345
-7.4810
0.9033 0.9977
128.7735
-0.0066 23.2267
35.3383
-0.4862
6.5890
16.5541
-45.4911 -20.8309
1.2140
Therefore both techniques have been proved be well suited to modelling the TRS with few parameters, providing good results in output prediction as per one or k-steps-ahead. ACKNOWLEDGMENTS The authors would like to thank National Council of Scientific and Technologic Development of Brazil - CNPq (Grants numbers: 303908/2015-7-PQ, 303906/2015-4-PQ, 304066/2016-8-PQ and BJT-304804/2014-2) for its financial support of this work. REFERENCES Ayala, H.V.H. and Coelho, L.S. (2014). multiobjective cuckoo search applied to radial basis function neural networks training for system identification, IFAC Proceedings Volumes, 47(3), 2539-2544. Ayala, H.V.H. and Coelho, L.S. (2016). Cascaded evolutionary algorithm for nonlinear system identification based on correlation functions and radial basis functions neural networks, Mechanical Systems and Signal Processing, 68-69, 378-393. Ayala, H.V.H., Habineza, D., Rakotondrabe, M., Klein, C.E., and Coelho, L.S. (2015). Nonlinear black-box system identification through neural networks of a hysteretic piezoelectric robotic micromanipulator, IFACPapersOnLine, 48(28), 409-414. Azarmi, R., Tavakoli-Kakhki, M., Sedigh, A.K., and Fatehi, A. (2015). Analytical design of fractional order PID controllers based on the fractional set-point weighted structure: case study in twin rotor helicopter, Mechatronics, 31, 222-233.
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Bezdek, J.C., Ehrlich, and Full, W. (1984). FCM: The fuzzy c-means clustering algorithm, Computers & Geosciences, 10(2-3), 191-203. Billings, S. A. (2013). Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and SpatioTemporal Domains, John Wiley & Sons, Ltd. Butt, S.S. and Aschemann, H. (2015). Multi-variable integral sliding mode control of a two degrees of freedom helicopter, IFAC-PapersOnLine, 48(1), 802-807. Coelho, L.S., Bora, T.C. and Klein, C. E. (2014). A genetic programming approach based on Lévy flight applied to nonlinear identification of a poppet valve, Applied Mathematical Modelling, 38(5-6), 1729-1736. Juang, J.-G., Liu, W.-K., and Lin, R.-W. (2011). A hybrid intelligent controller for a twin rotor MIMO system and its hardware implementation, ISA Transactions, 50(4), 609-619. Klein, C. E., M. Bittencourt, and Coelho, L. S. (2015). Wavenet using artificial bee colony applied to modeling of truck engine powertrain components, Engineering Applications of Artificial Intelligence, 41, 41-55. Mamdani, E.H. and S. Assilian (1975). An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Man-Machine Studies, 7(1), 1-13. Noël, J.P. and Kerschen, G. (2017). Nonlinear system identification in structural dynamics: 10 more years of progress, Mechanical Systems and Signal Processing, 83, 2-35. Pandey, S.K. and Laxmi, V. (2014). Control of twin rotor MIMO system using PID controller with derivative filter coefficient, in IEEE Students' Conference on Electrical, Electronics and Computer Science (SCEECS), Bhopal, India, 1-6. Pêssoa, M. W. (2010). Identificação de sistemas não-lineares utilizando modelo polinomial NARMAX e nebuloso Takagi-Sugeno-Kang. Master thesis, Industrial and Systems Engineering Graduate Program (PPGEPS), Pontifical Catholic University of Parana (PUCPR), Curitiba, Brazil (in Portuguese). Shaik, F.A., Purwar, S., and Pratap, B. (2011). Real-time implementation of Chebyshev neural network observer for twin rotor control system, Expert Systems with Applications, 38(10), 13043-13049. Storn, R. and Price, K. (1995). Differential evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces, Technical Report TR-95-012, International Computer Science Institute, Berkeley, USA. Takagi, T. and Sugeno, M. (1985). Fuzzy identification of systems and its applications to modeling and control, IEEE Transactions on Systems, Man and Cybernetics, 15(1), 116-132. Toha, S.F., Julai, S., and Tokhi, M.O. (2012). Ant colony based model prediction of a twin rotor system, Procedia Engineering, 41, 1135-1144.
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IFAC Approach PapersOnLine 50-1Using (2017) 13102–13107 Fuzzy Inference System Clustering and Differential Evolution Fuzzy Inference System Approach Using Clustering and Differential Evolution Fuzzy Inference System Approach Using Clustering and Differential Evolution Optimization Applied to Identification of a Twin Rotor System Fuzzy Inference System Approach Using Clustering and Differential Evolution Optimization Applied to Identification of a Twin Rotor System Optimization Applied to Identification of a Twin ,Rotor System ,# # Optimization Applied to Identification of a Twin Rotor System
Leandro dos Santos Coelho*,#, Marcelo Wicthoff Pêssoa#, Viviana Cocco Mariani*,**, and Gilberto Reynoso-Meza* Leandro dos Santos Coelho* , Marcelo Wicthoff Pêssoa , Viviana Cocco Mariani* **, and Gilberto Reynoso-Meza* Leandro dos Santos Coelho*,#,#, Marcelo Wicthoff Pêssoa##, Viviana Cocco Mariani*,,**, and Gilberto Reynoso-Meza* , Marcelo Pêssoa , Viviana Cocco Mariani* **, and Gilberto Reynoso-Meza* Leandro dos Santos Coelho* * Industrial andWicthoff Systems Engineering Graduate Program (PPGEPS) *Pontifical IndustrialCatholic and Systems Engineering Graduate Program (PPGEPS) University of Parana (PUCPR), Curitiba, Brazil *Pontifical Industrial and Systems Engineering Graduate Program (PPGEPS) # Catholic University ofGraduate Parana (PUCPR), Curitiba, Brazil *Pontifical Industrial and Systems Engineering Graduate Program (PPGEPS) Electrical Engineering Program (PPGEE) # Catholic University of Parana (PUCPR), Curitiba, Brazil Engineering Program (PPGEE) # Electrical Pontifical Catholic University ofGraduate Parana (PUCPR), Curitiba, Federal University of Parana (UFPR), Curitiba, Brazil Brazil Engineering Graduate Program (PPGEE) # Electrical University of Parana (UFPR), Curitiba, Brazil Electrical Engineering Graduate Program (PPGEE) **Federal Mechanical Engineering Graduate Program (PPGEM) Federal University of Parana (UFPR), Curitiba, Brazil ** Mechanical Engineering Graduate Program (PPGEM) Federal University of Parana (UFPR), Curitiba, Brazil Brazil Pontifical Catholic University of Parana (PUCPR), Curitiba, ** Mechanical Engineering Graduate Program (PPGEM) Pontifical Catholic University of Parana (PUCPR), Curitiba, Brazil ** Mechanical Engineering Graduate Program (PPGEM) (E-mails: [email protected], [email protected], [email protected], [email protected]) Pontifical Catholic University of Parana (PUCPR), Curitiba, Brazil (E-mails: [email protected], [email protected], [email protected], [email protected]) Pontifical Catholic University of Parana (PUCPR), Curitiba, Brazil (E-mails: [email protected], [email protected], [email protected], [email protected]) (E-mails: [email protected], [email protected], [email protected], [email protected]) Abstract: In this paper, a Takagi-Sugeno-Kang (TSK) fuzzy inference system using fuzzy c-means Abstract: and In this paper, aevolution Takagi-Sugeno-Kang fuzzy using fuzzy c-means clustering differential optimization is(TSK) proposed andinference validatedsystem when applied to a twin rotor Abstract: and In this paper, aevolution Takagi-Sugeno-Kang (TSK) fuzzy inference system using fuzzy c-means clustering differential optimization is proposed and validated when applied to a twin rotor Abstract: In this paper, a Takagi-Sugeno-Kang (TSK) fuzzy inference system using fuzzy c-means system (TRS). The TRS is perceived as a challenging problem due to its strong cross coupling between clustering and differential optimization is proposed and validated whencross applied to a twin rotor system (TRS). The TRSaxes. isevolution perceived as a challenging problem due to its approach strong coupling between clustering and differential evolution optimization is proposed and validated when applied to a twin rotor horizontal and vertical The design procedure of the TSK fuzzy for TRS is detailed. system (TRS). The TRSaxes. is perceived as a challenging duefuzzy to its approach strong cross between horizontal and vertical The design procedure ofproblem the TSK for coupling TRS detailed. system (TRS). The TRSaxes. is perceived asobtained a challenging problem dueTSK to itsfuzzy strong cross coupling between According to the identification results by applying the approach and ais nonlinear horizontal and vertical The design procedure of the TSK fuzzy approach for TRS is detailed. According to the identification results obtained by applying the TSK fuzzy approach and ais nonlinear horizontal and vertical axes. The design procedure of the TSK fuzzy approach for TRS detailed. autoregressive with moving average and exogenous inputs (NARMAX) model, the effectiveness of the According to the identification results by applying the TSK fuzzy approach and a nonlinear autoregressive with moving average andobtained exogenous inputs (NARMAX) model, the effectiveness of the Accordingfuzzy to the identification results obtained by applying thetests. TSK fuzzy approach and a nonlinear proposed system design is demonstrated through validation autoregressive with moving average and exogenous inputs (NARMAX) model, the effectiveness of the proposed fuzzy system design is demonstrated through validation tests. autoregressive with moving average and exogenous inputs (NARMAX) model, the effectiveness of the proposed fuzzy systemidentification, design is demonstrated through validation tests. © 2017, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. Allcomputation. rights reserved. Keywords: Nonlinear fuzzy system, differential evolution, evolutionary proposed fuzzy system design is demonstrated through validation tests. Keywords: Nonlinear identification, fuzzy system, differential evolution, evolutionary computation. Keywords: Nonlinear identification, fuzzy system, differential evolution, evolutionary computation. Keywords: Nonlinear identification, fuzzy system, differential evolution, evolutionary computation. been used to analyze the performance of TRS and TRMS. A 1. INTRODUCTION been used to analyze the performance of TRS and of TRMS. A multi-variable nonlinear control-oriented model a twin 1. INTRODUCTION been used to analyze the performance of TRS and of TRMS. A multi-variable nonlinear control-oriented model a twin 1. INTRODUCTION been used to analyze the performance of TRS and TRMS. A rotor system is presented in Butt and Aschemann (2015). The The identification of1.dynamic nonlinear systems, which pose multi-variable nonlinear control-oriented model of a twin INTRODUCTION rotor system ismodel presented in Butt is andderived Aschemann (2015). The The identification of dynamic nonlinear systems, which pose multi-variable nonlinear control-oriented model of a twin mathematical of system using Lagrange’s problems and require solutions distinctsystems, from their system ismodel presented in Butt is andderived Aschemann (2015). The The identification of dynamic nonlinear whichlinear pose rotor mathematical of resulting system using Lagrange’s problems andisrequire solutions distinctsystems, frombytheir linear rotor system ismodel presented in Butt is and Aschemann (2015). Thea equations. Based on the state-space representation, The identification of dynamic nonlinear which pose counterparts, a hard task as demonstrated the effort mathematical of system derived using Lagrange’s problems andisrequire distinct frombytheir Based on the resulting state-space representation, a counterparts, a hardinsolutions task as demonstrated the linear effort equations. mathematical model of system is derived using Lagrange’s multi-variable integral sliding mode control is designed, also problems and require solutions distinct from their linear devoted by researchers the last decades. Several techniques Based on thesliding resulting state-space representation, a counterparts, is a hardintask as demonstrated by techniques the effort equations. multi-variable integral mode control is designed, also devoted by researchers the last decades. Several Based extended on the resulting state-space representation, a a discrete-time Kalman filter isis applied from counterparts, is a hardin task as demonstrated by the effort equations. have been proposed for nonlinear system identification multi-variable integral sliding mode control designed, also devoted by researchers the last decades. Several techniques asimplifications discrete-timeintegral extended Kalman filter isis designed, applied from have been proposed forthenonlinear system identification multi-variable sliding mode control also at modelling as well as disturbance torques. devoted by researchers in last decades. Several techniques (Coelho et al., 2014, Klein et al., 2015; Ayala and Coelho, a discrete-timeatextended Kalman filter is applied from have been proposed for nonlinear system identification modelling as and well Laxmi as disturbance torques. (Coelho et al., 2014, Klein etand al., Kerschen, 2015; Ayala and Coelho, asimplifications discrete-time extended Kalman filter is applied froma The work proposed by Pandey (2014) applied have been proposed for nonlinear system identification 2014; Ayala et al., 2015; Noël 2017). simplifications at modelling as well as disturbance torques. (Coelho et al., 2014, Klein et al., 2015; Ayala and Coelho, The work proposed byand Pandey and Laxmi (2014) applied a 2014; Ayala et al., 2015; Noël and Kerschen, 2017). simplifications at modelling as well as disturbance torques. proportional, integral derivative (PID) controller with (Coelho et al., 2014, Klein al., Kerschen, 2015; Ayala and Coelho, The work proposed by Pandey and Laxmi (2014) applied aa 2014; Ayala et al., 2015; Noëletand 2017). proportional, integral and derivative (PID) controller with aa This paper presents an investigation on dynamic modelling of The work proposed by Pandey and Laxmi (2014) applied derivative filter coefficient to control a TRMS. 2014; Ayala et al., 2015; Noël and Kerschen, 2017). integral and derivative controller with a This paper presents an investigation on dynamic of proportional, derivative filter coefficient to control a(PID) TRMS. the vertical movement of a twin rotor system; modelling a laboratory proportional, integral and derivative (PID) controller with a This paper presents an investigation on dynamic modelling of derivative filter coefficient to control a TRMS. the vertical movement offlexible a twin maneuvering rotor system;structure a laboratory terms offilter the coefficient recent literature, there are many researches This paperrepresenting presents an investigation on dynamic modelling of In platform a and derivative to control a TRMS. the vertical movementaofflexible a twin maneuvering rotor system;structure a laboratory termsTRMS. of the recent literature, there are many platform representing and In using In Shaik et al. (2011) the researches unknown the vertical movement offlexible a twin maneuvering rotor a laboratory resembles essential characteristics of a system; helicopter. The twin In termsTRMS. of the recent literature, there are many platform representing a structure and using In Shaik et al. (2011) the researches unknown resembles essential characteristics of a helicopter. The twin In terms of the recent literature, there are many researches nonlinearities of TRMS are estimated by Chebyshev neural platform representing a flexible maneuvering structure and rotor system (TRS) imposes challenging control problems TRMS.of In Shaik et al. (2011) the unknown resembles essential characteristics of a helicopter. The twin using nonlinearities TRMS are estimated by Chebyshev neural rotor system (TRS) imposes challenging control problems using TRMS. In Shaik et al. (2011) the unknown network whose weights are adaptively adjusted. Lyapunov resembles essential characteristics of a helicopter. The twin due to its given nonlinearities as well as significant couplings of weights TRMS are estimated byadjusted. Chebyshev neural rotor system (TRS) imposes aschallenging control couplings problems nonlinearities network whose are adaptively Lyapunov due to its given nonlinearities well as significant nonlinearities of TRMS are estimated by Chebyshev neural is whose used toweights guarantee stability for state estimation and rotor system (TRS) imposes control problems between the pitch axis and aschallenging the azimuth axis (Butt and theory network are adaptively adjusted. Lyapunov due to its given nonlinearities well as significant couplings theory is whose used toweight guarantee stability for state estimation and between the pitch axis and the azimuth axis (Butt and network weights are adaptively adjusted. Lyapunov neural network errors. An ant colony optimization is due to its given nonlinearities as well as significant couplings Aschemann, 2015). In general, it consists of two DC (direct theory is used to guarantee stability for state estimation and between the 2015). pitch In axis and the azimuthof axis (Butt and neural network weight errors. Ana ant colony optimization is Aschemann, general, it consists two DC (direct theory is used to guarantee stability for state estimation and deployed and used for modelling twin rotor system by Toha between the pitch axis and the azimuth axis (Butt and neural network weight errors. An ant colony optimization is current) motors, oneIn for main rotor and other tail(direct rotor Aschemann, 2015). general, it consists of twofor DC deployed and A used for modelling acontroller twincolony rotorwas system by Toha current) motors, one for main rotor and other for tail rotor neural network weight errors. An ant optimization is et al. (2012). hybrid intelligent proposed by Aschemann, 2015). general, it consists of two DC perpendicular to each other. A pivoted beam is for joined on its deployed and used for modelling a twin rotor system by Toha current) motors, oneInfor main rotor and other tail(direct rotor et al. (2012). A hybrid intelligent controller was proposed by perpendicular to each other. A pivoted beam is joined on its deployed and used for modelling twin rotor system by Toha Juang et al. (2014), which apply a fuzzy PID control scheme current) motors, onerotating for main rotor and otherhorizontal for tail on rotor base which allows it pivoted freely onbeam the and al. (2012). A hybrid intelligent proposed by perpendicular to each other. A joined its et Juang etreal-valued al. (2014), which apply acontroller fuzzy PIDwas control scheme base which allows rotating itthe freely onbeam the is horizontal and et al. a(2012). A hybrid intelligent controller was by with genetic algorithm to control aproposed TRMS. An perpendicular to each other. A pivoted is joined on its vertical planes. By changing input voltage, the angular Juang et al. (2014), which apply a fuzzy PID control scheme base which allows itthe freely the horizontal and with a etreal-valued genetic algorithm to control athe TRMS. An vertical planes. By rotating changing inputon voltage, the angular Juang al. (2014), which apply a fuzzy PID control scheme analytical method for tuning the parameters of set-point base which allows rotating it freely on the horizontal and speed of these two rotors is controlled. For balancing the with a real-valued genetic algorithm to control TRMS. An vertical planes. By changing the input voltage, the angular method tuning the parameters of aathe set-point speed ofmomentum these two rotors is state controlled. For the analytical with a real-valued genetic algorithm to control TRMS. An weighted fractionalfor order PID controller is proposed in vertical planes. By in changing the input voltage, angular angular steady or with load,balancing athe pendulum analytical method for tuning the parameters of the set-point speed of these two rotors is controlled. For balancing the weighted fractional order PID controller is proposed in angular momentum inrotors steady state or withbeam load,balancing a pendulum analytical method for tuning the parameters of the set-point Azarmi et al. (2015) and applied a TRS. speed of these two is controlled. For the counter weight is hanged on the joined (Pandey and weighted fractional order PID a controller is proposed in angular momentum in steady state or withbeam load,(Pandey a pendulum Azarmi et al. (2015) and applied TRS. counter weight is hanged on the joined and weighted fractional order PID controller is proposed in angular momentum in steady or load, a pendulum Laxmi, 2014). isMoreover, not all with state variables are Azarmi et al. (2015) and applied a TRS. counter weight hanged on state the joined beam (Pandey and In this work, to achieve a satisfactory tracking performance Laxmi, 2014). Moreover, not all state variables are Azarmi et al. (2015) and applied a TRS. counter weight hanged on not the joined beamvariables (Pandey and measurable. In this work, to achieve a satisfactory trackingangle performance Laxmi, 2014). isMoreover, all state are regarding desired trajectories for the azimuth and the measurable. In this work, to achieve a satisfactory trackingangle performance Laxmi, 2014). Moreover, not all state variables are regarding desired trajectories for the azimuth and the measurable. In this work, to achieve a satisfactory tracking performance pitch angle, this work applied the Takagi-Sugeno-Kang In several contributions presented in literature, both regarding desired trajectories for the azimuth angle and the measurable. pitch angle, this work applied the Takagi-Sugeno-Kang In several and contributions in literature, both pitch regarding desired for angle andwith the (TSK) angle, fuzzy approach, the the ARX (autoregressive modelling experimentalpresented identification of TRS (also this trajectories workusing applied the azimuth Takagi-Sugeno-Kang In several and contributions presented in literature, both pitch (TSK) fuzzy approach, using theconsequent ARX (autoregressive with modelling experimental identification of TRS (also angle, this work applied the Takagi-Sugeno-Kang exogenous inputs) model in the part of the rules. In several contributions presented in literature, both TRMS, M for multiple inputs and multiple outputs) have (TSK) fuzzy approach, theconsequent ARX (autoregressive with modelling experimental of TRS (also exogenous inputs) modelusing in the of the rules. TRMS, M and for multiple inputsidentification and multiple outputs) have The (TSK) fuzzy approach, using theconsequent ARX (autoregressive witha TSK fuzzy model uses if-then rules topart approximate modelling and experimental identification of TRS (also been investigated and several models have been proposed exogenous inputs) model in the part of the rules. TRMS, M for multiple inputs and multiple outputs) have The TSK fuzzy model uses if-then rules to approximate a been the investigated and several models have techniques been proposed exogenous inputs) model in the consequent part of the rules. wide class of nonlinear systems by fuzzy blending of local TRMS, M for multiple inputs and multiple outputs) have over past few years and various control The model uses if-then rules to approximate been investigated and several models have techniques been proposed wide TSK class fuzzy of nonlinear systems by fuzzy blending of localaa over the past few years and various control have The TSK fuzzy model uses if-then rules to approximate been investigated and several models have techniques been proposed over the past few years and various control have wide class of nonlinear systems by fuzzy blending of local over the past few years and various control techniques have wide class of nonlinear systems by fuzzy blending of local Copyright © 2017 IFAC 13644 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017, 2017 IFAC 13644 Copyright 2017 responsibility IFAC 13644 Peer review©under of International Federation of Automatic Control. Copyright © 2017 IFAC 13644 10.1016/j.ifacol.2017.08.2162
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Leandro dos Santos Coelho et al. / IFAC PapersOnLine 50-1 (2017) 13102–13107
linear approximations. This method employs linear models in the consequent part of the fuzzy system. In the adopted TSK approach, to determine the centers of membership functions, the clustering technique called fuzzy c-means is adopted, and for determine the Gaussian membership functions width a differential evolution (DE) approach is employed. Also, in terms of performance analysis, a model-based on polynomial nonlinear autoregressive with moving average and exogenous inputs (NARMAX) which uses the Orthogonal Least Squares (OLS) estimator, coupled to the Golub-Householder decomposition with the calculation of ERR (Error Reduction Ratio) is applied to determine the most significant terms for TRS controlling model was tested too and compared with the proposed TSK design. The remainder of the paper is arranged as follows. In Section 2, the TRS system is detailed and the parameters of the system specified. The proposed approach using fuzzy system is introduced in Section 3. In Section 4, the identification performance is demonstrated by providing experimental results on the TRS. Finally concluding remarks are made in the last section. 2. FUNDAMENTALS OF THE TWIN ROTOR SYSTEM The TRS is a laboratory setup designed for control experiments. In certain aspects its behavior resembles that of a helicopter. From the control point of view it exemplifies a high order nonlinear system with significant cross couplings. As shown in Fig. 1, the TRS mechanical is driven by two DC (direct current) motors at both ends of the beam; the main rotor motor and tail rotor motor. Its two propellers are perpendicular to each other and joined by a beam pivoted on its base that can rotate freely in the horizontal and vertical planes. The beam can thus be moved by changing the input voltage in order to control the rotational speed of the propellers. The articulated joint allows the beam to rotate in such a way that its ends move on spherical surfaces. The lowcost system designed by Pessôa (2010) is equipped with a pendulum counterweight hanging from the beam, which is used for balancing the angular momentum.
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The state of the TRSs is described by four process variables: two inputs variables u1 and u2, that represent the angular velocity of the rotors, measured by tachometers coupled with the driving DC motors; the horizontal (yaw) and vertical (pitch) angles, measured by position sensors fitted at the pivot and represented by y1 and y2 as illustrated in Fig. 2.
Fig. 2. Inputs (u) and outputs (y) variables of the TRS. 3. FUZZY INFERENCE SYSTEM Fuzzy inference schemes mainly divided into two models, Mamdani (Mamdani and Assilian, 1975) and TSK (TakagiSugeno-Kang) (Takagi and Sugeno, 1985). In the TSK-type fuzzy system, the consequent of each rule is a function input linguistic variable. The general adopted function is a linear combination of input variables plus a constant term, and each rule is of the following form: : … (1) 1 ∑
where , , … , is the vector of input variables; , , … , are fuzzy sets, 1,2, … , ,which represent the number of rules and is the rule output. The final network output is a weight average of each rule’s output of the form:
∑ ∑
(2)
In Equation 9, is the degree of activation of the i-th rule: ∏ , 1,2, … , (3) where is the membership function of the fuzzy set at the antecedent (input) of . The number of rules characterizes the structure of a fuzzy system. The number of rules can be determined by clustering methods. In this work the Fuzzy C-Means (FCM) clustering was applied. 3.1 Clustering and Gaussian membership functions tuning
Fig. 1. Photograph of the TRS.
The FCM algorithm is one of the clustering and structure identification methods. For the proposed model, FCM algorithm is necessary to determine the regression matrix vector membership values and cluster centers. In FCM, the 13645
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data sample can belong to more than one cluster with different fuzzy membership values that represent the degree of membership between data and centers of clusters. The main idea of the FCM algorithm can be described as: At each iteration 0,1,2 … , the membership values ( ) and cluster centers ( ) are determined as following the Equations 4 to 5 given by
∑
∑ ∑
, 1 ≤ ≤ , 1 ≤ ≤ (4)
, 1 ≤ ≤
(5) The cost function that minimizes the Euclidian distance between the clusters center and samples is expressed as Equation 6 given by
∑ ∑ ‖
−
‖
, 1 ≤ ≤ ∞
Step 5: Following the mutation operation, crossover using binomial (bin) distribution is applied in the population; Step 6: Update the generation’s (iterations) counter; Step 7: Loop to Step 3 until a stopping criterion is met, usually a maximum number of iterations, tmax. 3.3 Implementation of the fuzzy modelling To perform a practical implementation of the TSK-ARX method, the following steps can be carried out: Step 1: Determine the parameters that will compose the ARX structure selection ( , );
Step 2: Define how many input–output lagged terms are used in regression vector. It can be changed by considering the model complexity; Step 3: Perform FCM clustering algorithm to determine the centers of the regressors; Step 4: Perform DE to optimize the Gaussian membership functions width;
(6)
Step 5: Calculate the weight parameters of the model by using the least-squares estimator;
The stopping criteria adopted is when − < , where the matrix is obtained with membership values of data points computed from Equation 5, where is a small constant to stop clustering algorithm, is the iteration number, is the number of samples, is the number of cluster and is the weighting factor. More detailed information about FCM algorithm can be found in Bezdek et al. (1984). After determine the centers of the regressors, the optimal width for Gaussian membership functions is obtained through Differential Evolution (DE).
Step 6: Apply the procedures of the model validation and results analysis.
3.2 Optimization based on DE algorithm DE is an evolutionary algorithm proposed by Storn and Price (1995). Due to its simplicity of implementation and its efficiency and effectiveness in solving optimization problems, DE has become a popular technique. The particular version subject to our investigation is the DE/rand/1/bin-version, which appears to be the most frequently used variant, and is often considered as the “basic” version of the DE-algorithm. The steps of the DE algorithm can be summarized as follows: Step 1: The user must choose the key parameters that control DE, i.e., population size (N), boundary constraints of optimization variables, mutation factor (F), crossover rate (CR), and the stopping criterion (tmax). Step 2: Initialize the generation’s counter t= 0 and also initialize a population of individuals (solution vectors) x(t) in upper and lower bounds of each decision variable with random values generated according to a uniform probability distribution in the n-dimensional problem space. Step 3: For each individual, evaluate its objective function value (maximization of the R2 is adopted here); Step 4: Mutate individuals of the population;
3. SETUP AND RESULTS ANALYSIS In the following sub-sections, the identification setup, k-step ahead forecasting results using NARMAX model with OLS and the proposed fuzzy system design are presented. After, the results analysis is detailed. 4.1 Identification setup The identification algorithms presented in the previous sections is applied to a TRS. This system has high-order nonlinearity and significant cross-coupling is observed between the actions of the rotors, with each rotor influencing both angle positions. Different signals classes can be employed for the identification process, such as multi-sine signals, maximum length binary sequences and pseudo-random binary signals. In this work, multi-sine signals as excitation signals were applied to the inputs that controls yaw and pitch. The input signal is provided to the TRS through a computer interface. The connection is made through an electronic circuit that is built using three microcontrollers: PIC16F877, MAX232 and L298. The multi-sine signals were applied during 57 seconds and 377 samples of TRS angular position were collected with sampling period equal to 0.15 seconds. These samples were divided into estimation and validation group, the earlier was composed by 350 samples and the second all 377 samples were applied. The aim of the identification algorithms is obtain a model to do the prediction for k-step ahead applying both techniques. 4.2 Results for k-step ahead forecasting using NARMAX For the NARMAX modelling (Billings, 2013) to predicted ksteps-ahead inclination output ( ) and rotation output y2,
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through sine input signal excitation, the parameters adopted were: nonlinear order 2, delay of sampling interval 5, number of input-output lagged terms 9, the maximum lags , , were evaluated between 2 and 3. The adopted design in the NARMAX case was a decoupled estimation for y1 and y2. The mathematical model obtained for the TRS inclination output (y1) prediction, with the maximum lag 2, 2 and 3 , is shown on Equation 7. Figure 3 presents the result for output related the inclination comparing it with the real output and the signal error of prediction. The best mathematical model obtained for the inclination output ( ) is given by 1 0.1655 ∙ 10 ∙ 1 − 1 − 0.6862 ∙ 1 − 2 − 0.2302 ∙ 1 − 1 ∙ 1 − 1 − 0.1223 ∙ 10 0.2203 ∙ 10 ∙ 1 − 2 ∙ 1 − 2 0.3518 ∙ 10 ∙ 1 − 1 − 0.329 ∙ 10 ∙ 1 − 2 − 0.1040 ∙ 10 ∙ 1 − 2 ∙ 1 − 2 0.9047 ∙ 10 ∙ 1 − 1 ∙ 1 − 1 0.3030 ∙ 10 ∙ − 3 ∙ 1 − 1 0.1121 ∙ 10 ∙ 1 − 3 ∙ 1 − 2 0.2175 ∙ 1 − 2 ∙ 1 − 1 0.7910 ∙ 10 ∙ 1 − 3 − 0.1084 ∙ 1 − 1 ∙ 1 − 2
(7)
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For the NARMAX modelling to predicted k-steps-ahead rotation output ( ), through sine input signal excitation, the parameters adopted were the same as applied to modelling inclination output, only the nonlinear order was increased to 3. Also the maximum lag values were chosen between those values that have the lowest MSE (Mean Squared Error), and the respective values are: 3, 3 and 3. Figure 4 presents comparisons between real and the predicted output and the signal errors of the predicted output signal, respectively. The best mathematical model obtained for the rotation output ( ) is given by 2 0.1616 ∙ 10 ∙ 2 − 1 − 0.6069 ∙ 2 − 2 − 0.4486 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 ∙ 2 − 1 − 0.1963 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 ∙ 2 − 1 0.4081 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 − 0.1683 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 ∙ 2 − 1 0.4067 ∙ 2 − 1 0.3233 ∙ 10 ∙ 2 − 3 ∙ 2 − 3 ∙ 2 − 1 − 0.1309 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 ∙ 2 − 1 − 0.4077 ∙ 10 ∙ 2 − 2 ∙ 2 − 3 ∙ 2 − 3 − 0.3360 ∙ 10 ∙ 2 − 2 ∙ 2 − 2 ∙ 2 − 2 0.1189 ∙ 10 ∙ 2 − 3 ∙ 2 − 1 0.3129 ∙ 10 ∙ 2 − 1 ∙ 2 − 1 0.6318 ∙ 10 ∙ 2 − 1 ∙ 2 − 3 ∙ 2 − 3
(8)
(a) k-steps-ahead prediction for rotation ( ) output
(a) k-steps-ahead prediction for inclination ( ) output
(b) error signal
(b) error signal
Figure 3. Forecasting using NARMAX modelling for inclination ( ) output.
Figure 4. Forecasting using NARMAX modelling for rotation ( ) output. 13647
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4.3 Results for k-step ahead forecasting using fuzzy system The parameters assumed for TSK fuzzy system with ARX model were: number of membership functions 5; In the FCM algorithm the number of clusters 5, weighting factor 2 and the small constant to stop FCM algorithm 10 . In the membership Gaussian width optimization process, the DE algorithm was applied with the following parameters: mutation factor 0.8, crossover rate 0.5, and maximum of iterations (generations), 10.
The values 1 and 3 were chosen between the set of results for inclination and rotation output modelling due to present the lowest MSE. For the inclination output, comparisons between predicted and real output are shown on Figure 5. Furthermore, comparisons between real and predicted output are shown in Figure 6.
(a) k-steps-ahead prediction for inclination ( ) output
(b) error signal
(a) k-steps-ahead prediction for inclination ( ) output
(b) error signal Figure 5. Forecasting using TSK modelling for inclination ( ) output.
Figure 6. Forecasting using TSK modelling for rotation ( ) output.
4.4 Analysis of the identification results
The results for modelling TRS by NARMAX and TSK-Fuzzy ARX approach for k-step-ahead prediction with input signal excitation given by multi-sine have shown good results. The results are shown on Tables 1 and 2 for both techniques comparison. The results for one-step ahead are shown in Table 1 and 2 too. For multi-sine input signal excitation the TSK fuzzy system with ARX structure have shown the best result, as per onestep-ahead or k-steps-ahead, with lowest standard deviation for inclination and rotation axis output prediction. Therefore, the only drawback when comparisons are made with NARMAX approach is related to the computational cost. The TSK fuzzy system with ARX model had shown the highest computational cost, mostly imposed by DE algorithm applied to perform the optimization of membership Gaussian width. 5. CONCLUSION In this investigation, two approaches has been applied to modelling the TRS system in terms of inclination and rotation output. An accurate model has been extracted by applying a NARMAX approach to predict one-step-ahead with two kind of excitation signal, also k-steps-ahead output was investigated too. The second approach, which applies 13648
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TSK fuzzy approach, a model has been developed accordingly and the results shown the capability of TRS identification with good performance when compared with the NARMAX approach. Table 1. Results for TRS modelling by applying NARMAX and TSK approaches (inclination axis). Model (step ahead) NARMAX (1) Fuzzy (1) NARMAX (k) Fuzzy (k)
Error signal
R²
Standard Deviation 0.3975
Maximum
Minimum
Mean
0.9995 0.9996
1.3982
-1.3523
-0.0042
1.2308
-1.5149
0.3580
0.9555 0.9798
11.7714
0.0083 0.9219 0.0205
2.5490
7.1903
-7.7090 -6.1061
3.9498
Table 2. Results for TRS modelling by applying NARMAX and TSK approaches (rotation axis). Model (step ahead) NARMAX (1) Fuzzy (1) NARMAX (k) Fuzzy (k)
Error signal
R²
Standard Deviation 2.0596
Maximum
Minimum
Mean
0.9997 0.9999
17.1505
-13.2919
0.0822
9.2345
-7.4810
0.9033 0.9977
128.7735
-0.0066 23.2267
35.3383
-0.4862
6.5890
16.5541
-45.4911 -20.8309
1.2140
Therefore both techniques have been proved be well suited to modelling the TRS with few parameters, providing good results in output prediction as per one or k-steps-ahead. ACKNOWLEDGMENTS The authors would like to thank National Council of Scientific and Technologic Development of Brazil - CNPq (Grants numbers: 303908/2015-7-PQ, 303906/2015-4-PQ, 304066/2016-8-PQ and BJT-304804/2014-2) for its financial support of this work. REFERENCES Ayala, H.V.H. and Coelho, L.S. (2014). multiobjective cuckoo search applied to radial basis function neural networks training for system identification, IFAC Proceedings Volumes, 47(3), 2539-2544. Ayala, H.V.H. and Coelho, L.S. (2016). Cascaded evolutionary algorithm for nonlinear system identification based on correlation functions and radial basis functions neural networks, Mechanical Systems and Signal Processing, 68-69, 378-393. Ayala, H.V.H., Habineza, D., Rakotondrabe, M., Klein, C.E., and Coelho, L.S. (2015). Nonlinear black-box system identification through neural networks of a hysteretic piezoelectric robotic micromanipulator, IFACPapersOnLine, 48(28), 409-414. Azarmi, R., Tavakoli-Kakhki, M., Sedigh, A.K., and Fatehi, A. (2015). Analytical design of fractional order PID controllers based on the fractional set-point weighted structure: case study in twin rotor helicopter, Mechatronics, 31, 222-233.
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