Dynamic model identification of IPMC actuator using fuzzy NARX model optimized by MPSO •

In this paper, a novel inverse dynamic fuzzy NARX model is used for modeling and identifying the IPMC-based actuator’s inverse dynamic model. The contact force variation and highly nonlinear cross effect of the IPMC-based actuator are thoroughly modeled based on the inverse fuzzy NARX model-based identification process using experiment input-output training data. This paper proposes the novel use of a modified particle swarm optimization (MPSO) to generate the inverse fuzzy NARX (IFN) model for a highly nonlinear IPMC actuator system. The results show that the novel inverse dynamic fuzzy NARX model trained by MPSO algorithm yields outstanding performance and perfect accuracy.


INTRODUCTION
The nonlinear IPMC-based actuator is belonged to highly nonlinear systems where perfect knowledge of their parameters is unattainable by conventional modeling techniques because of the time-varying inertia, external force variation and other nonlinear uncertainties.To guarantee a good position tracking performance, lots of researches have been carried on.During the last decade, Sadeghipour et al., Shahinpoor et al., Oguru et al., and Tadokoro et al. investigated the bending characteristics of Ionic Polymer Metal Composite (IPMC) [1][2][3][4].Bar-Cohen et al. characterized the electromechanical properties of IPMC [5].An empirical control model by Kanno et al. was developed and optimized with curve-fit routines based on open-loop step responses with three stages, i.e., electrical, stress generation, and mechanical stages [6][7][8].Feedback compensators were designed using a similar model in a cantilever configuration to study its open-loop and closed-loop behaviors [9][10].

TAÏ P CHÍ PHAÙ T TRIEÅ N KH&CN, TAÄ P 17, SOÁ K2-2014 Trang 61
Damping of the ionic polymer actuator in air is much lower than that in water.Feedback control is necessary to decrease the response time of an ionic-polymer actuator to a step change in the applied electric field and to reduce overshoot.The position control of the IPMC was investigated by using a linear quadratic regulator (LQR) [12], a PID controller with impedance control [11], and a lead-lag compensator [9][10].Lots of advanced control algorithms have been developed for IPMC actuator in order to apply them in variety of the industrial and marine applications [13][14][15][16][17][18][19].
Up to now, the robust-adaptive control approaches combining conventional methods with new learning techniques are realized.During the last decade several neural network models and learning schemes have been applied to offline and online learning of actuator dynamics.Ahn and Anh in [20] have successfully optimized a NARX fuzzy model of the highly nonlinear actuator using genetic algorithm.These authors in [21] have identified the nonlinear actuator based on recurrent neural networks.The drawback of all these results is related to consider the actuator as an

Proposed inverse fuzzy NARX model of the IPMC actuator system
The proposed IFN model of the highly nonlinear IPMC system presented in this paper is improved by combining the approximating capability of the fuzzy system with the powerful predictive and adaptive potentiality of the nonlinear NARX structure.The resulting model establishes a nonlinear relationship between the past inputs and outputs and the predicted output, while the system prediction output is a combination of the system output produced by the real inputs and the historical behaviors of the system.This can be expressed as: Here, na and nb are the maximum lag considered for the output and input terms, respectively, nd is the discrete dead time, and f represents the mapping of the fuzzy model.
The structure of the proposed IPMC IFN model interpolates between the local linear, timeinvariant (LTI) ARX models as: Rule j: if z1(k) is A1,j and … and zn(k) is An,j then where zi(k), i=1...n is the element of the Z(k) "scheduling vector" which is usually a subset of the X(k) regressor that contains the variables relevant to the nonlinear behaviors of the system.
In this paper, X(k) regressor contains all of the inputs of the inverse fuzzy NARX model The fj(q(k)) consequent function contains all the In the simplest case, the NARX type zero-order fuzzy model (singleton or Sugeno fuzzy model which isn't applied in this paper) is formulated by the simple rule consequents: Rule j : if z1(k) is A1, j and…and zn(k) is An,j then with zi(k), i=1...n is the element of the Z(k) regressor containing all of the inputs of the IPMC IFN model: ,..., , ,..., 1 (6) Thus the difference between the fuzzy NARX model and the classic TS Fuzzy model method is that the output from the TS fuzzy model is linear and constant, and the output from the NARX fuzzy model is the NARX function.However, both of these methods have the same fuzzy inference structure (FIS).

MPSO-based IPMC IFN Model Identification
The problem of modeling the nonlinear and dynamic system always attracts the attention of researcher.Some research has been published using a fuzzy model based on expert knowledge [24][25][26][27][28][29][30].Unfortunately the resulting fuzzy model was often too complex to be applied in practice and thus only simulation was carried out.Figure 1a

PSO ALGORITHM FOR NARX FUZZY MODEL IDENTIFICATION
PSO is a population-based optimization method first proposed by Eberhart and colleagues [32].
Some of the attractive features of PSO include the ease of implementation and the fact that no gradient information is required.It can be used to  ( ().

Assumptions and Constraints
The first assumption is that symmetrical membership functions about the y-axis will unity, it is a member of no other sets.
Using these constraints the design of the IMNF model's input and output membership functions can be described using two parameters which include the number of membership functions and the positioning of the triangle apexes.

Spacing parameter
The

Designing the rule base
In addition to specifying the membership functions, the rule-base also needs to be designed.Cheong's idea was applied [34].In

Parameter encoding
To run a MPSO, suitable encoding needs to be carefully completed for each of the parameters and bounds.For this task the parameters given in Table 1 are

IDENTIFICATION RESULTS
In general, the procedure which must be executed when attempting to identify a dynamical system consists of four basic steps.
intrinsic cross-effect features of the IPMC-based actuator has not represented in its intelligent model.Recently, D.N.C. Nam et al. has modeled the IPMC actuator using fuzzy model optimized by traditional PSO [22-23].The drawback of this research lied in the resulting fuzzy model optimized by the traditional PSO susceptible to premature convergence and then easy to be fallen in local optimal trap.In order to overcome this disadvantage, this paper proposes the novel use of a modified particle swarm optimization (MPSO) to generate the inverse fuzzy NARX (IFN) model for a highly nonlinear IPMC actuator system.The MPSO is used to process the experimental input-output data that is measured from the IPMC system to optimize all nonlinear and dynamic features of this system.Thus, the MPSO algorithm optimally generates the appropriate fuzzy if-then rules to perfectly characterize the dynamic features of the IPMC actuator system.These good results are due to proposed IFN model combines the extraordinary approximating capability of the fuzzy system with the powerful predictive and adaptive potentiality of the nonlinear NARX structure that is implied in the proposed IFN model.Consequently, the proposed MPSO-based IPMC inverse fuzzy NARX model identification approach has successfully modeled the nonlinear dynamic IPMC system with better performance then other identification methods.This paper makes the following contributions: first, the novel proposed MPSO-based IPMC inverse fuzzy NARX model for modeling and identifying dynamic features of the highly nonlinear IPMC system has been realized; second, the modified particle swarm optimization (MPSO) has been applied for optimizing the IPMC IFN model's parameters; finally, the excellent results of proposed IPMC inverse fuzzy NARX model optimized by MPSO were obtained.The rest of the paper is organized as follows.Section 2 introduces the novel proposed inverse fuzzy NARX model.Section 3 presents the experimental set-up configuration for the proposed IPMC IFN model identification.Section 4 describes concisely the modified particle swarm optimization (MPSO) used to identify the IPMC IFN model.Section 5 is dedicated to the techniques of MPSO-based IFN model identification.The results from the proposed IPMC IFN model identification are presented in Section 6. Section 7 contains the concluding remarks.
and 1b initially presents the block scheme for the modeling and identification of a MPSO-based inverse fuzzy NARX11 and inverse fuzzy NARX22 models using experimental input-output training data.MPSO stands for Modified Particle Swarm Optimization and will be described later in the section 4.1.This proposed approach can help to simplify the modeling procedure for nonlinear systems.Particle swarm optimization (PSO) is applied to optimize the FIS structure and other parameters of proposed fuzzy model.However the poor experimental result proves that the PSO-based TS fuzzy model is incapable of modeling all nonlinear, dynamic features of the dynamic system.Recently the fuzzy/neural NARX model has been successfully applied to identify nonlinear, dynamic system[20],[27].

Fig. 1 .Fig. 2 .Fig. 3 .
Fig.1.Block diagram of the MPSO-based IPMC inverse fuzzy NARX11 model identification The block diagram presented in Fig.1 and 2 illustrate the MPSO-based IPMC IFN model identification.The error e(k)=U(k)-Uh(k) is used by the MPSO algorithm to calculate the Fitness value (see Equation (7)) in order to identify and optimize parameters of the proposed IPMC IFN model.
solve a wide array of different optimization problems.Like evolutionary algorithms, PSO technique conducts search using a population of particles, corresponding to individuals.Each particle represents a candidate solution to the problem at hand.In a PSO system, particles change their positions by flying around in a multidimensional search space until computational limitations are exceeded.Concept of modification of a searching point by PSO is shown in Fig. 4.

--
of particles in the group, d -Dimension of search space of PSO, t -Pointer of iterations (generations), Velocity of particle i at iteration t, w -Inertia weight factor, c1, c2 -Acceleration constant, rand() -Random number between 0 and 1, Current position of particle i at iteration t, Pbesti -Best previous position of the i-th particle, Gbest-Best particle among all the particles in the population The evolution procedure of PSO Algorithms is shown in Fig. 5. Producing initial populations is the first step of PSO.The population is composed of the chromosomes that are real codes.The corresponding evaluation of a population is called the "fitness function".It is the performance index of a population.The fitness value is bigger, and the performance is better.The fitness function is defined as equation (7).After the fitness function has been calculated, the fitness value and the number of the generation determine whether or not the evolution procedure is stopped (Maximum iteration number reached?).SCIENCE & TECHNOLOGY DEVELOPMENT, Vol 17, No.K2-2014 Trang 66 In the following, calculate the Pbest of each particle and Gbest of population (the best movement of all particles).The update the velocity, position, gbest and pbest of particles give a new best position.In recent years, the PSO has continued to be improved upon and has been applied successfully to identify and optimize different nonlinear, dynamic systems [33-34].However the inappropriate choice of operators and parameters used in PSO process makes the PSO susceptible to premature convergence.

Fig. 5 .
Fig. 5. Evolutionary Procedure of PSO Algorithms The focus of this paper is to attempts to simultaneously apply two improved strategies as a means to overcome these problems.Extinction strategy: This technique prevents the searching process from being trapped at a local optimum.Based on this concept, after Le generations, if no further increase in the fitness value has been detected; i.e., variance equal to of discourses are normalized to lie between -1 and 1 with scaling factors external to the IDNFM which is used to assign appropriate values to the input and output variables.* It is assumed that the first and last membership functions have their apexes at -1 and 1, respectively.This can be justified by the fact that changing the external scaling would have a similar effect to changing these positions.* Only triangular membership functions are to be used.* The number of fuzzy sets is constrained to be an odd integer that is greater than unity.In combination with the symmetry requirement, this means that the central membership function for all variables will have an apex at zero.* The base vertices of the membership functions are coincident with the apex of the adjacent membership functions.This ensures that the value of any input variable is a member of at most two fuzzy sets, which is an intuitively sensible situation.It also ensures that when a variable's membership of any set is certain, i.e.
each variable and the characteristic angle for each output variable were used to construct the rule-base.Certain characteristics of the rule-base are assumed when the proposed construction method is used: * Extreme outputs usually occur more often when the inputs have extreme values while the mid-range outputs are generally generated when the input values are also mid-range.* Similar combinations of input linguistic values lead to similar output values.Using these assumptions the output space is partitioned into different regions corresponding to different output linguistic values.How the space is partitioned is determined by the characteristic spacing parameters and the characteristic angle.The angle determines the slope of a line that goes through the origin on which seed points are placed.The positioning of the seed points is determined by a similar spacing method that is used to determine the center of the membership function.Grid points are also placed in the output space and represent all the possible combination of input linguistic values.These are spaced in the same way as described previously.The rule-base is determined by calculating which seed-point is closest to each grid point.The output linguistic value representing the seed-point is set as the consequent of the antecedent represented by the grid point.

Fig. 7 .Fig. 8 .
Fig.7.Seed points and grid points for rule-base construction used with the ranges and precision parameters that are shown.Binary encoding is used because it allows the MPSO more flexibility in searching the solution space thoroughly.The number of membership functions is limited to odd integers, which are inclusive between (3MPSO-based IPMC inverse fuzzy NARX11 model and between (3-5) when the MPSO-based IPMC inverse fuzzy NARX22 model identification is used.Experimentally, this was considered to be a reasonable constraint to apply.The advantage of doing this is that this parameter can be captured in just one to two bits per variable.Two separate parameters are used for the spacing parameters.The first is within the range of [0.1-1.0], which determines the magnitude and the second, which takes only the values -1 or 1, is the power by which the magnitude is to be raised.This determines whether the membership functions compress in the center or at the extremes.Consequently, each spacing parameter can achieve a range of [0.1 -10].The precision required for the magnitude is 0.01, which means that 8 bits are used in total for each spacing parameter.The scaling for the input variables is allowed to vary in the range of [0 -100], while that of the output variable is given a range of [0 -1000].

Table 1 .
MPSO-based inverse fuzzy NARX model parameters used for encoding