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2015 12th International Conference on Electrical Engineering Computing Science and Automatic Control CCE Mexico City Mexico 978 1 4673 7839 0 15 31 00 2015 IEEE A Comparative Study of the Wavenet PID Controllers for Applications in Non Linear Systems M A Salda a O L E Ramos V and J P Ordaz O CITIS Universidad Aut noma del Estado de Hidalgo Pachuca M xico lramos uaeh edu mx A Garcia Barrientos and I Algredo Badillo TI Universidad Polit cnica de Tlaxcala Tlaxcala Mexico agarciab ieee org Jean Francois Balmat and Fr d ric Lafont LSIS Universit de Toulon CNRS La Garde France balmat univ tln fr Abstract In this paper a comparative study of the wavenet PID controllers for applications in non linear systems is presented The wavenet PID controllers combine the neural network learning advantage with the wavelet representation for an efficient identification of non linear dynamic systems and these have application when we want to control plants of unknown mathematical model with highly non linear characteristics There exist different types of PID wavenet controllers i e wavenet PID fuzzy wavenet PID and multiresolution wavenet PID these controllers tune online the proportional integral and derivative gains of a classical discrete PID controller through the identification of the plant using a radial basis neural network with different daughters wavelet activation functions For this reason the performance of the wavenet PID and the fuzzy wavenet PID controlling a sub actuated system are compared with the classical PID controller The simulation results show that the fuzzy wavenet PID controller has a good performance to carry out the control non linear system for example the inverted pendulum Keywords wavenet PID fuzzy wavenet PID inverted pendulum neural network I INTRODUCTION Nowadays the use of wavelets has been growing because they combine the learning feature of the neural networks and the wavelet representation In this way the wavenet offers an efficient approximation in dynamic control systems and thanks to this feature the wavenet has been applied in different investigation and industrial fields An example of these applications is presented in 1 where they present a wavenet based modeling technique for vehicle suspension system in this work they use a Polynomial WindOwed with Gaussian POLYWOG as activation function and the wavelet parameters are optimized during its learning process which is performed by the backpropagation algorithm Another important example of wavenet application is presented in 2 in this work they use a wavenet and an estimator for Air Fuel Ratio Control in Spark Injection Engines where they conclude that 1 the use of wavenets increments the robustness and 2 the time consumed for training is shorter than a multi layer perceptron Other field that is growing inside the industry and between the research groups is the fuzzy logic because the fuzzy logic based control systems provide an efficient method to control complex and non lineal systems For example in 3 the authors present a fuzzy logic controller based on FPGA for process control in this work the fuzzy logic is used to tune the parameters of a discrete PID which is programed in an FPGA also and they conclude that the combination of fuzzy logic with a classic PID Fuzzy PID is better than the classical PID controller In 4 it is presented a FPGA based real time adaptive fuzzy logic controller where they test the behavior of the controlling using a quarter car semi active suspension model with two degrees of freedom and demonstrate that one characteristic of the architecture was the ability to accept new rules and membership functions during runtime without causing any undesirable behavior to the plant Also they demonstrate that the controller has the ability to stabilize a vehicle suspension system In 21 a wavelet differential neural network observer is proposed Others examples on the use of the fuzzy controllers are 22 25 Additionally it is mentioned that more than half of the controllers being used in the industry are the PID controllers or some of its variations The discrete PID has been used in most of the applications due the great technologic advances in the digital computers microprocessors DSPs FPGAs etc However there are different variations of the PID such as Robust PID Fuzzy PID NeuroFuzzy PID Non lineal PID Wavenet PID and Fuzzy Wavenet PID Although of all the PID variations its general operation mode is the same which is based on act in a proportional integral and derivative form over an error signal e t 5 defined as the difference between the reference signal yref t and the output signal of the plant y t generating in this way The control signal u t this control signal manipulates the plant output in a desired way and it is described by equation 1 1 where kp ki and kd are the gains of the controller There exist analytic and experimental techniques with the finality of tune these gains 5 when the mathematical model of the plant to be controlled is unknown in some occasions it is very complicated to establish these gains Because of this problem the combination of PIDs and wavenets has born for example one of the proposed alternatives to solve this problem is to automatically tune the PID gains online 6 8 In these works they approximate the unknown mathematical model of the plant and establish automatically the gains of the PID controller for this purpose a wavenet neural network is used to identify the plant and to establish the controller gains There is a variation of this controller this variation is introduced by the fuzzy logic to automatically establish the learning rate of the wavenet neural network and the refreshing rate of the PID parameters In this work a comparative study between the 2015 12th International Conference on Electrical Engineering Computing Science and Automatic Control CCE Mexico City Mexico classic PID wavenet PID and fuzzy wavenet PID has been carried out to compare its performance in the control of non lineal systems The simulation results show that the fuzzy wavenet PID controller has a good performance to carry out the control non lineal system for example the inverted pendulum II PID CONTROLLERS In this section we briefly describe the wavenet PID that was presented in 6 7 and it is shown in Fig 1 This controller has three main stages to manipulate the output of the non lineal plant in a desired way The first stage is the plant identification In this stage the output of the plant is estimated by a wavenet neural network with a IIR filter on cascade 9 this filter uses a persistent signal v k the function of the IIR filter is to filter the neurons that have little contribution to the identification process In this stage it is required the input of the plant u k the estimation error e k values the estimation error is defined as the difference between the real output of the plant y k and the plant estimation k The second stage is the discrete PID the PID sends the control signal u k and this signal is used to manipulate the plant output y k Also it is required the tracking error k which is defined as the difference between the reference signal yref k and the plant output y k The last stage is the online auto tuning of the PID gains kp ki and kd The k parameter is estimated by the wavenet as 2 where ci are the forward coefficients of the IIR filter and z k is the wavenet output As the wavenet PID the fuzzy wavenet PID controller manipulates the output of the plant in a desired way by the identification of the nonlinear system with an unknown mathematical model and the online tuning of the discrete PID gains kp ki and kd However in this controller the fuzzy logic is introduced to refresh automatically the learning rates of the PID controller p i and d 10 The operation of the fuzzy wavenet PID controller is very similar to the wavenet PID The difference between the fuzzy wavenet controller and the PID wavenet is the introduction of the fuzzy logic in the auto tuning block to perform the refresh of the learning ship rates of the PID controller Fig 1 Block diagram of the Wavenet PID The first component of a fuzzy logic controller is the fuzzifier this component transforms crisp inputs into a set of membership values in the interval 0 1 There exist different types of membership functions for example triangular trapezoidal exponential etc In 10 are used the triangular and trapezoidal membership functions In the controller presented in 10 two inputs in the fuzzy logic are taken into account The first one is the tracking error k and the second is the approximation of its derivate 1 k k k 1 With these we obtain a fuzzy system with two inputs and three outputs each one of the outputs represents one of the learning rates for the PID such as p i and d This investigation proposes to determinate the ranges for the learning rates as in the equation 3 pm n pmax im n imax dm n dmax 3 It is important to mention that the ranges of the learning rates are determined by a trial and error for this purpose a set of numeric simulations must be performed observing the behavior of the system in closed loop The learning rates satisfy the equation 4 p pm n pmax i im n imax d dm n dmax 4 For convenience the p i and d are normalized to have a range between zero and one with the following linear transformations 5 7 5 6 7 where p i and d are the parameters tuned by the fuzzy system These parameters will be used for compute the gains of the wavenet PID with the following equations 1 1 8 1 9 1 2 1 2 10 where e k is the identification error defined as k is the tracking error given by and is part of the identification performed by the wavenet 6 7 Also with automatic adaptation of the learning rate the trial and error search for the best initial values for the parameter can be avoided For a typical fuzzy system with its normalized outputs as is showed in the Fig 2 we can determinate the ranges for the learning rates easily 2015 12th International Conference on Electrical Engineering Computing Science and Automatic Control CCE Mexico City Mexico Fig 2 Fuzzy system representation where k and 1 k are the tracking error and its derivate approximation respectively In the Fig 3 a b the membership functions associated with the tracking error k and its derivate approximation 1 k are showed a b Fig 3 Membership functions for k and 1 k where EN means k negative E0 means k zero EP means k positive E1N means 1 k negative E10 means 1 k zero and E1P means 1 k positive In this case the membership functions used were the triangular and trapezoidal forms for simplicity In Fig 4 these membership functions for the learning rates for kp ki and kd are showed The Fig 4 a represents the learning rate only for the proportional gain kp The Fig 4 b shows the learning rate only for the integral gain ki Finally the Fig 4 c represents the learning rate only for the derivative gain kd a b c Fig 4 Membership functions for kp a ki b and kd c where P0 I0 and D0 means p i and d are 0 Pp Ip and Dp means p i and d are small Pm Im and Dm means p i and d are medium Pg Ig and Dg means p i and d are big The fuzzy rules have the structure type IF THEN in other words the structure is Ri IF k is Ai and 1 k is Bi THEN p Ci i Di and d Ei 2015 12th International Conference on Electrical Engineering Computing Science and Automatic Control CCE Mexico City Mexico where Ai Bi Ci Di and Ei are the fuzzy sets of the ith rule with i 1 2 M The resulting fuzzy system is formed by 9 rules that are showed in Table I TABLE I Fuzzy rules k 1 k p i d EN EN EN EZ EZ EZ EP EP EP E1N E1Z E1P E1N E1Z E1P E1N E1Z E1P Pm Pg Pm Pp Pz Pp Pm Pg Pm Ip Ip Ip Ip Iz Ip Ip Ip Ip Dp Dp Dp Dp Dz Dp Dp Dp Dp where N Z z P p m g represents negative zero positive small medium and big respectively The defuzzification method used was the Center of Gravity COG method III SIMULATION SETUP The inverted pendulum car system consist in a cylindrical bar pendulum that freely oscillates around a fixed point it is important to mention that the system has mechanical restrictions because it can only move in one plane The pendulum is mounted over a mobile piece car and this piece can move in the horizontal plane 12 The inverted pendulum car is shown in Fig 6 as can be observed this system is a sub actuated kind because it has only one actuator and two freedom degrees position x and angular position The variables and parameters of the system are illustrated in Table II and taken from 13 Fig 5 Inverted pendulum car system TABLE II Variables and parameters of the system Variables 1 u x Description Torque in the motor axis Control input Position of the car Angular position Value Units N m Volts m rad Parameters l m M g J km Longitude of pendulum Pendulum mass Car mass Gravity constant Inertia of the bar Coupling constant of the motor 0 32 0 23 0 52 9 81 0 00 7 1 0 m kg kg m seg2 kg m2 M m Volts For the space state representation the following state variables are necessary 11 where x is the linear position and is the angular position of the pendulum This non linear system can be represented by an equations system of the form 8 12 where x R4 u R and M 13 0 0 14 In this paper this representation is used for all our simulations and to perform the comparative analysis between the controllers The control algorithms were programed in Matlab Simulink as the mathematical model of the inverted pundulum car system Besides the discrete PID block was used to simulate the classic PID algorithm We controled the system using the three controllers described in the Section II under the same initial conditions and simulation parameters shown in Table III TABLE III Simulation conditions Angular initial position 0 5 rad Sample time 0 01 sec Simulation time 30 sec kp30 ki1 kd 3 The initials learning rates for the wavenet PID and the fuzzy wavenet PID controller were p 0 3 i 0 001 and d 0 8 these values were set in a random form The initial values for kp ki and kd were selected through the auto tune feature of the discrete PID block We have choosed this feature because our objective is to evaluate the performance of each controller with exactly the same gains values and also the simplicity for selecting the controller gains IV SIMULATION RESULTS Different experiments have been carrying out to evaluate the performance of the three controllers The first experiment was to control the angular position of the inverted pendulum car with an external perturbation the main objective was to observe the behavior of the controller under external 2015 12th International Conference on Electrical Engineering Computing Science and Automatic Control CCE Mexico City Mexico perturbation In the second experiment a noise signal was introduced here we can observe the behavior of the controllers under external perturbation and noise All the simulations were made in Simulink taking account the same simulation parameters A First experiment In this experiment an external perturbation of 0 1 rad of magnitude is introduced at 15 seconds after the system initialization In Fig 7 a we can observe the tracking error of the controllers Also in Fig 7 b we can observe the three control signals The analyses of the results of this experiment are shown in Table IV for the classic PID a wavenet PID b and fuzzy wavenet PID c The difference in the average control signal between the wavenet PID and fuzzy wavenet PID controller is around 0 01 Volt TABLE IV Analysis of results of first experiment Control u t Error e t Max 10 volts M n 10 volts Max 0 3964 rad M n 0 5 rad Average Control Std deviation 2 42 volts 2 8832 Average error Std deviation 0 0676 rad 0 0906 a Control u t Error e t Max 10 volts M n 10 volts Max 0 1455 rad M n 0 5 rad Average Control Std deviation 0 83 volts 2 2407 Average error Std deviation 0 0169 rad 0 0566 b Control u t Error e t Max 10 volts M n 10 volts Max 0 3964 rad M n 0 5 rad Average Control Std deviation 0 94 volts 2 2487 Average error Std deviation 0 0226 rad 0 0641 c a b Fig 7 First experiment results B Second experiment In this experiment an external perturbance of 0 1 rad of magnitude is introduced 15 seconds after the system is initialized also a noise signal is introduced to observe the controller behavior In Figure 8 a we can observe the tracking error the controllers In the same way as the last experiment the three control signals are showed in figure 8 b The analysis of the results of this experiment are shown in table 5 a 5 b and 5 c for the classic PID Wavenet PID and Fuzzy Wavenet PID respectively TABLE V Analysis of the obtained results Control u t Error e t Max 10 volts M n 10 volts Max 0 3964 rad M n 0 5 rad Average Control Std deviation 1 10 volts 2 0284 Average error Std deviation 0 0302 rad 0 0612 a Control u t Error e t Max 10 volts M n 10 volts Max 0 1455 rad M n 0 5 rad Average Control Std deviation 0 46 volts 1 6378 Average error Std deviation 0 008 rad 0 0354 b Control u t Error e t Max 10 volts M n 10 volts Max 0 3964 rad M n 0 5 rad Average Control Std deviation 0 45 volts 1 4870 Average error Std deviation 0 0107 rad 0 0400 c a b Fig 8 Second experiment results 2015 12th International Conference on Electrical Engineering Computing Science and Automatic Control CCE Mexico City Mexico IV CONCLUSIONS The main objective of this work is to compare the effectiveness of the wavenet controllers to tracking a reference signal for a nonlinear SISO system we have considered three different controllers the classic PID the wavenet PID and the fuzzy wavenet PID and their performance for controlling the inverted pendulum car have focus our attention in three main criteria the plant response control signal and tracking error With the collected data we can conclude that both wavenet PID and fuzzy wavenet are more efficient than the classic PID However even that the results between the wavenet PID and the fuzzy wavenet PID are very similar we must note that the learning rate auto tuning feature of the fuzzy wa