pseudoerror-based self-organizing neuro-fuzzy system

812 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 12, NO. 6, DECEMBER 2004

Pseudoerror-Based Self-OrganizingNeuro-Fuzzy System

Chunshien Li, Chun-Yi Lee, and Kuo-Hsiang Cheng

Abstract—The novel concept of pseudoerrors for a self-orga-nizing neuro-fuzzy system (SO-NFS) is proposed for trackingcontrol problem. To demonstrate the proposed approach, anexample of motion control of an auto-warehousing crane systemis illustrated, which can move back and forth in and di-rections to access and store cargoes. The proposed SO-NFS showsexcellent performance in control of the crane system for differentloading conditions and varying distances in all directions.

Index Terms—Clustering, learning, neuro-fuzzy system, param-eter identification, pseudoerror, random optimization, self-organ-ization, structure identification.

I. INTRODUCTION

ERROR signals may play important role in a closed-loopcontrol system and might be useful information to describe

the behavior of a plant. A novel concept of pseudo errors isproposed in the paper. The pseudoerror generation process isshown in Fig. 1. Pseudoerrors are potential errors possibly oc-curred during the control process. Pseudotargets serving tem-porarily as desired outputs are designed and used to comparewith plant outputs to produce pseudoerrors during the stimu-lation process. These pseudoerrors are collected as the prioriknowledge of the plant. Basically, pseudotargets are differentfrom the desired trajectory, for the latter is the real target whilethe closed-loop control system is working. However, if the de-sired trajectory is known, it can be used in pseudotargets. Thespace of pseudoerrors is usually the same as the input space ofa neuro-fuzzy system.

Based on the pseudoerror information, the self-organizingneuro-fuzzy system (SO-NFS) is constructed by itself for itsknowledge base structure using self-organizing mechanism. Theantecedent part of the rule base of the SO-NFS is constructedby partitioning the input space. A clustering algorithm is usedto partition the collected pseudoerrors into several clusters, bywhich the input space is partitioned into several fuzzy regions.Each clustered fuzzy region represents the antecedent of a fuzzyIF–THEN rule, whose location is decided by the pseudoerrorsand the clustering algorithm. Because Takagi–Sugeno type ofIF–THEN rules [20] is used in this paper, only the input spacepartition is concerned.

In Section II, structure identification of the SO-NFS is spec-ified, with the information of pseudo errors. After the structure

Manuscript received September 10, 2000; revised August 23, 2002, October7, 2003, and February 20, 2004. This work was supported by the National Sci-ence Council, Taiwan, R.O.C., under Project NSC90-2213-E-182-011.

The authors are with the Intelligent Systems and Applications Laboratory,Department of Electrical Engineering, Chang Gung University, Tao-Yuan 333,Taiwan, R.O.C. (e-mail: [email protected]).

Digital Object Identifier 10.1109/TFUZZ.2004.836086

Fig. 1. Data generation process of pseudoerrors.

identification, parameter learning given in Section III follows.In Section IV, the neuro-fuzzy architecture of the SO-NFS isexplained. In Section V, the proposed approach is applied to anauto-warehousing crane system for the feasibility of the method.A good discussion is given in Section VI for the comparison ofthe proposed approach to other approaches.

II. STRUCTURE IDENTIFICATION WITH PSEUDOERRORS

A. Clustering Algorithm for Self-Organization

With a clustering algorithm [1], [6], the pseudoerrors thatsufficiently cover the domain of interest in the input space arepartitioned into appropriate clusters (regions). Each generatedcluster is described by a multidimensional Gaussian function asfollows:

(1)

with

(2)

(3)

(4)

for , where indicates the number of ex-isting clusters at time , superscript to indicate the th cluster,

the crisp input in the th dimension of a cluster, the di-mension of input space, the firing strength of cluster

for an at time , and and the mean and the spreadin the th dimension of the and , respectively.

The firing strengths, , can thenbe regarded as the information to decide whether or not to gen-

1063-6706/04$20.00 © 2004 IEEE

LI et al.: PSEUDOERROR-BASED SELF-ORGANIZING NEURO-FUZZY SYSTEM 813

Fig. 2. Clustering and partition of input space.

erate a new cluster. The cluster with the greatest firing strengthamong the existing clusters can be found as follows:

(5)

A threshold, , is given to decide the condition to gen-erate a new cluster. A new cluster is generated if

; otherwise, nothing is done. When a new cluster is generated,the values of mean and spread are temporarily assigned as fol-lows:

(6)

(7)

where , are prespecified constants.Assume that there are clusters generated after the clus-

tering process is completed. The means and the spreads for theclusters are collected together as follows:

(8)

(9)

The clusters implies that there are rules in the rule base ofthe SO-NFS, that is

number of clusters number of rules (10)

B. Rule Base Construction

Each cluster represents the antecedent of a fuzzy rule, andeach dimension of cluster corresponds to a linguistic variable.After clusters are generated, each cluster possessing an

-dimensional membership function can be decomposed intoone-dimensional (1-D) membership functions for the input

variables, given as

(11)

For each dimension of input space, there are 1-D Gaussianfunctions that can be treated as the membership functions offuzzy sets (linguistic values) for their corresponding linguisticvariables, respectively. For example, the decomposition of two-dimensional (2-D) clusters is illustrated in Fig. 2. After decom-posing these generated clusters, the rule base is established. Thetype of Takagi–Sugeno fuzzy rules [2], [10], [20] is used forthe SO-NFS. The consequent of a TS-type fuzzy rule is a linearcombination of the components in the crisp input vector .With the -dimensional clusters, the rule base for a singleoutput SO-NFS can be expressed as

Rule IF is and is and and

is

THEN

for , where is the -th input linguistic vari-able, and the fuzzy set with linguistic value for theth linguistic input variable in the th control rule. The conse-

quent coefficients of the fuzzy rules are initially set to zero. Theis the response action of the th rule. The coefficients of


Fig. 3. Training strategy of parameter learning for the SO-NFS.

consequents in the rule base can be collected together to form amatrix as follows:

......

. . ....

(12)

The and given in (8) and (9) and the matrix are collectedtogether to form the parameter set for theSO-NFS. The param-eter set of SO-NFS is expressed as

(13)

After successful learning, these parameters will become appro-priate values.

III. PARAMETER IDENTIFICATION

After the structure identification process, parameter learning[12], [18] follows to fine-tune and obtain an appropriate set ofparameters for the knowledge base of the SO-NFS. The objec-tive of parameter learning is to find a set of parameters tominimize a cost function. The well-known random optimization(RO) algorithm [11], [12] is adopted as the learning algorithm tominimize the cost function . At each time step, the errorbetween the desired output and the actual output iscalculated to contribute to the cost function. By repeating the

process, we can obtain the accumulation of squared errors forsteps. The cost function is given as

factor (14)

In (14), there are pattern training patterns and time stepsinvolved in the training process to calculate the cost function inwhich factor1 and factor2 are weighting factors and time indi-cates when to change weighting factor from factor1 to factor2.The training strategy is shown in Fig. 3. The RO method searchesstochastically for a solution that can minimize the cost functionto match application purpose. In general, the RO moves to anew point with a smaller cost value, then around the new point,the search for better solution continues. However, when the ROmethod menders around a point too long, it can jump randomlyto a new point and restarts searching process [11], [12].

IV. NEURO-FUZZY ARCHITECTURE OF SO-NFS

The SO-NFS is a neuro-fuzzy inference system that possessesthe structure of neural network and preserves the fuzzy reasoningability [4], [14]. A six-layer structure is used to implement theneuro-fuzzy system. The architecture of the SO-NFS is shownin Fig. 4. Explanation for the six layers is specified as follows.

Layer 0: This layer is the input layer. For each node, theoutput is set equal to the input.


Fig. 4. Neuro-fuzzy architecture of SO-NFS.

Layer 1: The nodes in this layer perform operation ofmatching the crisp inputs , to the cor-responding fuzzy sets. Each node corresponds to a linguisticvalue characterized by a membership function, i.e., the activationfunction of a node is to represent a membership function. Thereception and activation functions in this layer are written as

(15)

where is the net input of the th node in layer 1 and isthe Gaussian activation function, and are the mean andthe spread of the Gaussian function for the th component of the

-th cluster, and the right superscript, , is to indicate the clusternumber.

Layer 2: The nodes in layer 2 perform fuzzy-and operation forthe preconditional parts of fuzzy rules. The node outputs repre-sent the firing strengths of corresponding rules. Each node in thislayer represents a cluster, which corresponds to an IF–THEN rule.The net-input and output of a node in this layer are given as

(16)

for . Any other -norm operator can replace themin operator.

Layer 3: The process of normalization for the firingstrengths of the fuzzy rules is performed in this layer. Thenet-inputs and node outputs of this layer are described as

(17)

for .Layer 4: This layer is called the consequent layer. The

nodes in the layer perform the normalized consequents of allthe IF–THEN rules. Because the inputs to the nodes in layer 4 areboth the corresponding outputs from layer 3 and the crisp inputvector , the net-input of a node in this layer is the multipli-cation of both a linear combination of ,

and the corresponding normalized firing strength from layer 3,as described by

(18)

where indicates the node number in layer 4.Layer 5: The node in this layer corresponds to the output

linguistic variable in the neuro-fuzzy system. The net-input andthe output of the node in the layer are given by

(19)

V. AUTO-WAREHOUSING CRANE CONTROL SYSTEM

Auto-warehousing crane systems have been widely appliedto store and access a variety of cargoes. An auto-warehousingsystem has several advantages over a traditional storagesystem, such as great capacity, high efficiency, fully comput-erizing control, and real-time inventory. An auto-warehousingcrane system can move in , and directions. The precisepositioning accuracy is critically required in distance-varyingmovement under various loadings. The auto-warehousing cranesystem under discussion [13] is weighted kg indirection, kg in direction, and kg indirection, with maximal loading capacity kg.

A. Pseudoerrors for Self-Organization Training

Error and its derivative are selected as the input linguisticvariables of the SO-NFS to control the crane system. The basevariables for the error and its derivative are denoted by and

, where the value of at time is defined by the differencebetween the reference speed and the crane speed , i.e.,

(20)

The reference curve is composed of five regions for the motionof acceleration, maximum speed, deceleration, creep speed, andbraking [13]. The reference curve is used as part of pseudotargetpatterns for pseudoerror generation. The acceleration is set to

m/s in Region I. The pseudotarget speed is setto the m/s in Region II. The deceleration is set to

m/s in Region III. In Region IV, the pseudotargetspeed is set to m/s. Using a variety of accelerationsand decelerations, pseudoerrors can be generated as complete aspossible. Several accelerations and decelerations given beloware used in Regions I and III.

Region I

Region III

When the crane is in Region II or IV, a random force limitedin between N is applied to the crane. The process ofpseudoerror generation is illustrated in Fig. 5.


Fig. 5. Generation of pseudoerrors for the auto-warehousing crane system.

Fig. 6. Parameter learning scheme of SO-NFS.

Fig. 7. Learning curve.

B. Self-Organization Learning and Parameter LearningAfter the pseudoerrors are generated and collected, the clus-

tering algorithm performs the partition of input space for theinitial knowledge base of the SO-NFS. The spreads for eachnewly generated cluster were set to in the dimensionsof error and its derivative, respectively. The threshold for gener-ating new cluster was set to . There were six rules gen-erated after the self-organization learning was completed. Thesampling period was set to 0.025 s. The scheme of parameter

learning is shown in Fig. 6. The fed back error was accumu-lated to form the cost function given in (14), and the set of pa-rameters of the SO-NFS was evolved using the RO algorithmto minimize the cost function. For the cost function , theparameters were set to

, and factor . Two training patternsfor and forwere used in parameter learning process. The learning curve isshown in Fig. 7.


Fig. 8. Application phase of SO-NFS.

TABLE ISPENT TIME, FINAL POSITION, AND POSITIONING ACCURACY

C. Experiment

After learning the SO-NFS is used to the motion control of thecrane system. The control scheme is shown in Fig. 8. The dis-tance of the crane at time is used to obtain the referencespeed from the reference curve. The is compared tothe crane speed to produce the error, which is then fed intothe SO-NFS with its derivative. The inferred result of SO-NFS ismultiplied with a coefficient , which is used to scale the outputof SO-NFS in each direction for the motion control. The coef-ficient is set to 1 for direction and 0.3 for direction. In

direction, the is set to 0.12 for Region 1 and 2 for otherregions. A saturation function is used to comply with force lim-itation [13]. The outputs from the SO-NFS for direction and

direction are used directly to control the motion of crane. Thecrane system is controlled to move 3 and 30 m in direction, 3and 20 m in direction, and 2 m in direction with and withoutloading. The positioning accuracy, spent time and final positionin , and directions are summarized in Table I. Excellentperformance of the SO-NFS is observed in the motion controlof the crane system.

VI. DISCUSSION

There are structure identification (learning) and parameteridentification (learning) in the proposed approach, and wewould like to have a discussion of comparison for it to otherapproaches in literature. Although system identification hasbeen performed and applied in control system in this paper, theproposed approach is different from that for function approxi-mation or system modeling.

In system modeling, the same signal is inputted to both a realsystem to be modeled and its model, which could be a fuzzy

system, a neural net, or a neuro-fuzzy system. The output of thereal system served as target is then compared to the model outputto produce an error signal, which is happened right at the outputof the real system. Essentially, the problem of system modelingis a problem of optimization, in which output error betweenthe real system and the model needs to minimize. In controlproblem, the proposed approach of system identification is toself-organize and to learn a neuro-fuzzy system served as a con-troller. Basically this is also a problem of optimization, in whichthe error between the plant output and the desired needs to mini-mize, which is not happened at the output of neuro-fuzzy systembut at the plant output. The plant output however is dependenton the output of the neuro-fuzzy controller. Thus, this optimiza-tion problem is back to the neuro-fuzzy controller (system). Itis needed to identify the neuro-fuzzy system so that the learningcontrol problem is optimized.

In structure identification, partition of input space is the keyissue for the study. The proposed approach has used pseudo er-rors to indicate where the regions in the input space may beimportant potentially to application and a clustering methodto partition such indicated regions according to the pseudo er-rors, in this way to identify a more compact structure of theneuro-fuzzy system. Recently, fuzzy c-means [8], [9] has at-tracted researchers in the study of clustering. It is an excel-lent method in clustering. However, considerable computationis needed each iteration of clustering process, due to its nature ofinvolving all pattern data in clustering space to determine clustercenters. This computation will increase exponentially with thenumber of pattern data. The second is that the method needsto set the number of clusters subjectively before the algorithmproceeds. In this paper, the SO-NFS has adopted the clusteringmethod [6], for it can be used to perform online clustering and


to determine the number of clusters in flexible way. The compu-tation is much less compared to the fuzzy c-means. These meritsare useful in clustering for structure identification.

In parameter identification, for the special need of learningcontrol problem mentioned earlier in the section, gradient-based learning algorithms such as back-propagation [3], [5],[6], [18] and their variants may not fit in the need becausethe error happened at the plant output will cause difficulty toupdate the parameters of the neuro-fuzzy controller (system)[18], unless the plant is modeled into qualified form. Althoughinverse control scheme [4] can be used for the problem, inversemodel of plant does not always exist, especially when dealingwith highly nonlinear plant. In the proposed approach, theplant is assumed unknown to the controller although its inputand output are measurable. Nongradient-based algorithms maycorrespond to the need, such as genetic algorithm [7], sim-ulated annealing algorithm [3], and simplex algorithm [17].Genetic algorithm is a good method to perform optimizationproblem for its ability of not being trapped at local minimawhile searching for the global minimum, provided that there areenough variety of individuals in gene pool. However, GA hasits colossal computation problem that in each generation all in-dividuals have to be evaluated for their fitness values. Althoughannealing algorithm is another candidate for the problem, itslearning speed is too slow. Simplex algorithm may be wanderedat local minimum for real application, although it is good forconvex problems. In this paper, the algorithm of random op-timization (RO) is adopted for parameter identification. TheRO algorithm used in the paper is modified from its original[16], [19] with extra jumping mechanism when trapped at localminimum and interpolation points checking for more chanceto look for better solution [11], [12]. Compared to those ap-proaches mentioned before, it has merits in escaping abilityfrom local minimum, much less computation at each learningiteration, and simplicity to use. The RO algorithm has abilityto search for global minimum, and possesses the flexibility inadding constraints in learning process.

The proposed approach using pseudo errors is different fromthe fuzzy approach used in [13], although the latter can be un-derstood easily and explained intuitively. The fuzzy approachis based on the engineering experience and knowledge, whichare converted and designed into fuzzy rules as the policy to thecontrol strategy. Because the plant is treated as a black box andno information is about the plant, the rules have to cover ev-erywhere of the input space to take care of every possible inputcondition. Although the fuzzy rules are easy to understand, theyare good to design for the system with lower input dimension.The more the input dimension increases, the more the designcomplexity. On the other hand, the proposed approach in thispaper uses the Takagi–Sugeno type of fuzzy rules [20] and pos-sesses the abilities of self-exploration, self-organization, andself-learning for the neuro-fuzzy system. By stimulation of theplant, the dynamic behavior of the plant can be self-exploredby the SO-NFS and the information in terms of pseudo errorscan be used to indicate where the input domain is of interest fora specific application. This is good for the SO-NFS to set upa more terse structure of knowledge base so that the waste ofcomputation resource can be avoided as much as possible. With

machine-learning method, the proposed pseudoerror approachcan shun as much human intervention as possible.

VII. CONCLUSION

The concept of pseudoerrors has been proposed for controlproblem. The pseudoerror based approach enables the SO-NFSto self-explore the dynamic behavior of an unknown plant.With the machine-learning based approach, the more concisestructure of SO-NFS can be established, according to the re-vealed information of pseudoerrors. With the practical exampleof auto-warehousing crane system, the proposed approach hasbeen demonstrated successfully for its feasibility, model-freeapproach, little intervention of expert, and excellent perfor-mance in precision motion control.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewers fortheir valuable comments during the review process.

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Chunshien Li received the B.S. degree in naval ar-chitecture from the National Taiwan Ocean Univer-sity, Taiwan, R.O.C., in 1984, and the M.S. degree inengineering mechanics, the M.S. degree in electricalengineering and computer science, and the Ph.D. de-gree in electrical engineering and computer science,all from the University of Illinois at Chicago, in 1990,1993, and 1996, respectively.

He is currently with the Department of ElectricalEngineering, Chang Gung University, Tao-Yuan,Taiwan, R.O.C. He is the first author of more than 35

technical papers. His current research interests include soft computing, neuro-fuzzy systems, pattern recognition, intelligent signal processing, intelligent sys-tems and control, machine learning, neural networks, and chip/processor-basedreal-time systems.

Dr. Li received the Research Award from the National Science Council,Taiwan, in 1999, and the Research Award of Excellent Teacher fromChang Gung University, in 2000. He has served as a Referee for the IEEETRANSACTIONS ON INDUSTRIAL ELECTRONICS and the IEEE TRANSACTIONS

ON SYSTEMS, MAN, AND CYBERNETICS B: CYBERNETICS.

Chun-Yi Lee received the B.S degree in biomedicalengineering from Chung-Yuan University, Chungli,Taiwan, R.O.C., in 1996, and the M.S degree in elec-trical engineering from Chang Gung University, Tao-Yuan, Taiwan, R.O.C., in 2000.

He is currently with the Programmable Micro-electronics (Taiwan) Corporation, Hsinchu, Taiwan,R.O.C., where he is an IC Design Engineer of high-speed sensing schemes and embedded flash macrodesign. His research interests include fuzzy logic,neural networks, fuzzy control, intelligent control,

learning algorithms, and neuro-fuzzy inference systems.

Kuo-Hsiang Cheng was born in Taipei, Taiwan,R.O.C., in 1978. He received the B.S. degree inautomatic control engineering from the Feng ChiaUniversity, Tai-Chung, Taiwan, R.O.C., in 2000.He is currently working toward the Ph.D. degreein electrical engineering at Chang Gung University,Tao Yuan, Taiwan, R.O.C.

His research interests include fuzzy logic, neuralnetworks, intelligent system, and learning algo-rithms.

pseudoerror-based self-organizing neuro-fuzzy system

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