Optimizing fuzzy neural networks, for tuning PID controllers using an orthogonal simulated annealing, algorithm OSA

Optimizing Fuzzy Neural Networks for Tuning

PID Controllers Using an Orthogonal Simulated

Annealing Algorithm OSA

Shinn-Jang Ho, Member, IEEE, Li-Sun Shu, Student Member, IEEE, and Shinn-Ying Ho, Member, IEEE

Abstract—In this paper, we formulate an optimization problem

of establishing a fuzzy neural network model (FNNM) for effi-ciently tuning proportional-integral-derivative (PID) controllers of various test plants with under-damped responses using a large number of training plants such that the mean tracking error of the obtained control systems is minimized. The FNNM consists of four fuzzy neural networks (FNNs) where each FNN models one of controller parameters ( , , , and ) of PID controllers. An existing indirect, two-stage approach used a dominant pole assignment method with = 198 to find the corresponding PID controllers. Consequently, an adaptive neuro-fuzzy inference system (ANFIS) is used to independently train the four individual FNNs using input the selected 176 of the 198 PID controllers that 22 controllers with parameters having large variation are abandoned. The innovation of the proposed approach is to directly and simultaneously optimize the four FNNs by using a novel orthogonal simulated annealing algorithm (OSA). High performance of the OSA-based approach arises from that OSA can effectively optimize lots of parameters of the FNNM to minimize . It is shown that the OSA-based FNNM with = 176 can improve the ANFIS-based FNNM in averagely decreasing 13.08% error and 88.07% tracking error of the 22 test plants by refining the solution of the ANFIS-based method. Furthermore, the OSA-based FNNMs using = 198 and 396 from an extensive tuning domain have similar good performance with that using = 176 in terms of .

Index Terms—Fuzzy neural network (FNN), optimal design,

orthogonal experimental design (OED), proportional-integral-derivative (PID) controller, simulated annealing.

I. INTRODUCTION

H

OW TO tune proportional-integral-derivative (PID) con-trollers for plants with disturbance and under-damped responses is an important research from both theoretical and industrial viewpoints. Many tuning methods for PID controllers have been proposed [1], [2], [4], [10], [22], [25], [29]. Since the under-damped system with disturbance has an additional stability constraint that the damping ratio is smaller than one, it is more difficult to tune PID controllers for plants with under-damped responses than that for plants with over-damped responses [22]. Ho et al. [10] developed a tuning formula Manuscript received December 22, 2003; revised March 19, 2005 and September 21, 2005.

S.-J. Ho is with the Department of Automation Engineering, National For-mosa University, Huwei, Yunlin 632, Taiwan.

L.-S. Shu is with the Department of Information Management, Overseas Chi-nese Institute of Technology, Taichung 407, Taiwan.

S.-Y. Ho is with the Department of Biological Science and Technology, and Institute of Bioinformatics, National Chiao Tung University, Hsinchu 300, Taiwan (e-mail: [email protected]).

Digital Object Identifier 10.1109/TFUZZ.2006.876985

for self-tuning PID controller of a plant with under-damped responses based on the gain margin and phase margin specifi-cations. The method proposed by Wang et al. [25] is based on the closed-loop pole allocation strategy through the use of root locus.

Recently, a two-stage approach to establishing a fuzzy neural network model (FNNM) consisting of four separate fuzzy neural networks (FNNs) for tuning PID controllers of various test plants with under-damped responses is proposed [22]. First, a dominant pole assignment method with a batch of 198 training plants is used to find the corresponding PID controllers. Second, an adaptive neuro-fuzzy inference system (ANFIS) [20], [21] is utilized to independently train the four individual FNNs for modeling the four individual parameters of the PID controller using the input 176 of the 198 PID controllers that 22 controllers with parameters having large variation are abandoned. In [22], it is shown that the results of four test plants with various characteristics using the FNNM obtained from the ANFIS-based method have better balance performance between set-point and load-disturbance responses than those of the methods [10] and [25]. Because the FNNM is obtained without considering the global minimization of tracking errors for the entire tuning domain, the effective tuning domain is restricted and the tracking error is not yet minimized.

It is useful to establish an optimal FNNM which can effi-ciently tune the PID controllers for various test plants with under-damped step responses. The motivation of this study is to improve FNNM as well as the performances of the gen-erated PID controllers by analyzing the merit and drawback of the ANFIS-based method and then propose an efficient method. In this paper, we formulate an optimization problem of establishing an FNNM for efficiently tuning PID controllers of various test plants with under-damped responses using a large number of training plants. The objective function is to minimize the mean tracking error of the obtained control systems with the constraint that each of the corresponding control systems satisfies a robustness constraint.

The major difficulties of solving the optimization problem are as follows: 1) The parameters in different FNNs have interac-tions and the number is as large as 258 in [22]; 2) the objective function with these 258 parameters is highly nonlinear and its search space is multimodal; 3) it is almost impossible to de-rive an analytic solution and thus the optimal objective function value is not known in advance; and 4) the robustness constraints for all control systems are very intractable.

For coping with the aforementioned difficulties, we proposed an efficient approach to directly and simultaneously optimize the four FNNs by using a newly developed orthogonal simu-lated annealing algorithm (OSA) [11], [12]. High performance of the OSA-based approach arises from that OSA can simul-taneously search for a solution of the 258 parameters to mini-mize (for difficulties 1–3). The ANFIS-based method using a two-stage approach can obtain a feasible solution. To efficiently find a feasible solution as good as possible, the feasible solution from [22] is used as an initial solution of OSA (for difficulty 4). OSA uses a divide-and-conquer approach to solving the large-scale problem based on orthogonal experimental design (OED) [3], [6], [18], [27]. Recently, we have successfully incorporated OED into simulated annealing [23] and genetic algorithm [8] for solving large-scale intractable engineering problems [11]–[13]. However, the newly-developed intelligent evolutionary algorithm (IEA) [13] requires an initial popula-tion of feasible solupopula-tions with high diversity (for difficulty 4) that is not an easy task. It is shown by computer simulation

that the OSA-based FNNM with can improve the

ANFIS-based FNNM in averagely decreasing 13.08% error and 88.07% tracking error of the 22 test plants. The OSA-based FNNM using the four specific test plants from [22] can aver-agely improve 15.63% tracking errors of the under-damped systems. Furthermore, the OSA-based FNNMs using

and 396 from an extensive tuning domain have similar good performance with that using in terms of .

This paper is organized as follows. In Section II, we for-mulate an optimization problem of establishing an FNNM for tuning PID controllers for plants with under-damped responses. In Section III, the analysis of the investigated optimization problem is given to illustrate the motivation of using the OSA-based approach. In Section IV, the approach OSA-based FNNM is presented in detail. Performance comparisons are given in Section V to demonstrate the efficiency of the proposed method. Finally, conclusions are given in Section VI.

II. INVESTIGATEDOPTIMIZATIONPROBLEM

The investigated optimization problem is to establish an FNNM for tuning PID controllers for plants with under-damped responses. Section II-A describes an under-damped system. Section II-B describes the architecture of FNNM. Section II-C gives a formulation of the investigated optimization problem. A. Under-Damped System and PID Controller

Consider an under-damped system with disturbance. Let be a training plant, be a PID controller, be a set point, be a controller output, be an output of

the system, be a tracking error, and

be an external disturbance. The under-damped training plant is modeled by the transfer function

(1) where , , and are dead time, natural frequency and damping ratio, respectively. The plant parameters

in the training plants can be characterized by only two plant

parameters , where

[10], [22]. The transfer function of the PID controller is (2) For improving the set-point response of the system, a set-point weighting is introduced into the PID controller as follows:

(3) where , , , and are the four controller parameters. For obtaining better modeling results, and are normalized as , and , respectively [22]. Let denote the maximum sensitivity and it is a robustness measure for stability, written as follows:

(4) From [2], we know that if the nominal system is stable and its is less than a stability value (typical values fall in the range from 1.4 to 2.0), then the system subjected to a unit step dis-turbance at the process input is still stable. The design goal of FNNM is that the FNNM can generate PID controllers such that the tracking error of the control system is as small as pos-sible under the constraint that is smaller than a prescribed value, such as the standard value 2.0 used by [22].

B. The Architecture of FNNM

An FNNM consists of four fuzzy neural networks (FNNs). Each FNN models one of the four controller parameters , , , and . The architecture of an FNN is shown in Fig. 1. We denote the output node in layer as . Suppose plants are given and have implications. The value of one of the controller parameters , , , and is implied as follows. Layer 1: Every node in this layer is an adaptive node with a node output defined by

(5) where (or ) is the input to the node and (or ) is the fuzzy set associated with this node. In other words, outputs of this layer are the membership values of the premise part. The membership functions are characterized by the generalized bell functions

(6)

where is a parameter set. Parameters

Fig. 1. Architecture of an FNN.

Layer 2: Every node in this layer is a fixed node labeled , which multiplies the incoming signals and outputs the product. For instance

(7) Each node output represents the firing strength of a rule. Layer 3: Every node in this layer is a fixed node labeled . The th node calculates the ratio of the th rule’s firing strength to the sum of all rules’ firing strengths

(8) Outputs of this layer are called normalized firing strength.

Layer 4: Every node in this layer is an adaptive node with a node function

(9)

where is the output of layer 3 and is a

param-eter set. Paramparam-eters in this layer are referred to as consequent parameters.

Layer 5: The single node in this layer is a fixed node labeled that computes the output as the summation of all incoming signals

(10)

Layer 6: The single node in this layer is a fixed node la-beled that computes the overall output as the exponential of incoming signal:

(11)

For convenience, let the vector , be

the parameters of an FNN for modeling , which consists of consequent and premise parameters

(12) In this study, we adopt the values of and used in [22].

For , , and , and . Due to the large

variation of the parameter , the larger values of and are used. The numbers of parameters of the four FNNs for , , , and are 57, 57, 57, and 87, respectively. Therefore, there are 258 parameters to be optimized for an FNNM. C. The Optimization Problem

Designing an optimal FNNM has valuable contribution to tuning PID controllers from both theoretical and industrial view-points. In this paper, we obtained a useful FNNM by simultane-ously formulating an optimization problem and developing an effective optimization algorithm. The optimization problem is to optimize an FNNM for tuning PID controllers for plants with under-damped responses using a large number of training plants. For performance comparison with the FNNM in [22], the tracking error using the same Laplacian -norm which treats the error uniformly at each point is adopted. The objective function is to minimize the mean tracking error of the obtained cor-responding control systems

(13)

with the constraint that each of the corresponding control systems satisfies the robustness constraint . Let be a candidate solution consisting of the four FNN parameter vectors

Fig. 2. Flowcharts of the ANFIS- and OSA-based approaches. The desired FNN model has a controller parameter vectorX = [M ; M ; M ; M ] to be optimized. The ANFIS-based approach uses the DPA method to obtain PID controllers and consequently uses ANFIS to model the four parametersK, T , T , andb of PID controllers using four independent runs to obtain X. The OSA-based approach directly and simultaneously optimize all parameters of X using the objection function of minimizing a mean tracking errorJ.

The quality of the optimized FNNM is evaluated in terms of and . An effective approach to optimizing FNNM should be able to handle an extensive tuning domain by sampling a large number of training plants and obtain controllers with minimal tracking errors for training and test plants.

The objective function of establishing an FNNM has the characteristics of nonlinear function, high degree of freedom, strong interactions among parameters, intractable constraints, multi-modal and huge search space. Moreover, it is hard to derive a mathematical formula for defining the objective func-tion because of numerous highly interacted nonlinear funcfunc-tions in the fuzzy neural networks and under-damped system. Con-sequently, there is no analytic solution to the optimization problem. The essential issue is how to effectively search for a satisfactory solution to the large-scale constrained param-eter optimization problem, considering the above-mentioned characteristics.

How do we know if the objective function has a solution or not? Of course, we cannot tell with certainty whether an opti-mization algorithm will finally find an optimal solution. How-ever, even though there is no certainty, there are some clues for optimizing the FNNM. Since the value of the globally optimal solution cannot be known in advance because of huge search space and no analytic solution, the aim of our method using OSA is to find a potentially good approximation to the global optimum in a limited amount of computation time. There-fore, when a feasible solution with a small value (as small as possible) is found which may be a local or near-local optimum, the obtained FNNM might be useful enough for real-world ap-plications although a global optimum may be not yet found.

How to cope with the intractable constraint that each of the corresponding control systems satisfies the robustness con-straint is the major concern in developing an opti-mization algorithm. If an algorithm can find a solution with the value small enough, the constraint can be satisfied with a large probability. In other words, for convenient design, the algorithm can temporally ignore the constraint by focusing on pursuing the solution with a smallest value.

To ensure high quality of the obtained PID controllers from test plants of practical applications, a large number of training plants for increasing the sampling density in the tuning domain is helpful. However, increasing the value of would make the search of feasible solutions more difficult if no heuristics are available. In the following section we would analyze the merit and weakness of the ANFIS-based approach [22], integrate our experience in solving large-scale optimization problems [11], [13], and develop an efficient method to establish a useful FNNM for tuning PID controllers.

III. ANALYSIS OF THEINVESTIGATEDPROBLEM In Section III-A, we briefly review and analyze the ANFIS-based approach [22]. In Section III-B, we illuminate our experience of utilizing the commonly-used population-based genetic algorithm (GA) [8], [13] and point-based simulated an-nealing (SA) technique [11], [12], [23] to solve the investigated problem.

A. The Merit and Weakness of the ANFIS-Based Approach The ANFIS-based approach consists of two stages, as shown in Fig. 2. In Stage I, Shen chose a batch of

training plants which are combinations of , , and , as listed in Table I, and utilized a domi-nate pole assignment (DPA) method to obtain the corresponding

198 feasible PID controllers for systems

with the robustness constraint . The variation of the

parameters obtained from DPA is large

in the range and . Generally, the

plants in this range are those with large dead time, large natural frequency, and poor damping. These plants make it difficult to establish an FNN for identifying the relationship between and using ANFIS in Stage II. Therefore, Shen abandoned 22 of the 198 PID controllers that their plants belong to this range. In Stage II, Shen used ANFIS to establish four separate FNNs to independently model the individual parameters , ,

TABLE I

198 TRAININGPLANTS(; ) WHICH ARECOMBINATIONS OF! , LAND = 0:1; 0:2; . . . ; 0:9

Let and be the values of a controller

param-eter obtained from DPA and ANFIS for the th training plant . Shen used a root mean square error (RMSE) as an objec-tive function for ANFIS to train an FNN, described as follows:

minimize

(15) The ANFIS identifies the relationship between multi-input pa-rameters and a corresponding single-output . There-fore, it takes four independent runs for ANFIS to establish the FNNM for the four controller parameters .

The merit of the two-stage ANFIS-based approach is to ob-tain a feasible solution to the problem of establishing an FNNM with the intractable constraints that each of the corresponding control systems satisfies the robustness constraint . The weakness of the ANFIS-based approach is highlighted here. 1) The objective function (15) of optimizing the FNNM is

the RMSE between and . That the four

FNNs have small values of RMSE don’t guarantee that the FNNM can generate PID controllers with a small value of . It is better to optimize the FNNM by directly min-imizing the tracking error which is a good performance measure of PID controllers.

2) The high performance of the PID controllers depends on that the four FNNs must be accurately determined simul-taneously due to their interactions. The independent de-termination of each FNN would result in large tracking errors, especially when the modeling error of each FNN is very large, e.g., the errors of , , and are inside

10% and the error of is inside 30% [22].

3) The decrease of the number of training plants results from that the ANFIS cannot effectively model the controller pa-rameters from the input data with large variation. If a large number of training plants is used to extend the effec-tive tuning domain, the variation of would be very large. To avoid from decreasing the number , it would be better to optimize the FNNM by directly determining the 258 parameters simultaneously to minimize such that the number can be advantageously increased.

B. Feasibility of Using GA and SA

Considering the merit and weakness of the ANFIS-based two-stage approach, an intuitive method is to directly search

for an optimal solution by minimizing the objective function using optimization algorithms such as GA and SA, discussed here.

The majority of control applications in the literature adopted GA-based approaches [7]. Recently, researchers have become increasingly interested in the use of GA as a means to design various classes of control systems [5], [15]. Inspired from the mechanisms of natural evolution, GAs utilize a collective learning process of a population of individuals. Descendants of individuals are generated using randomized operations such as mutation and crossover. Mutation corresponds to an erroneous self-replication of individuals, while crossover exchanges in-formation between two or more existing individuals. According to a fitness measure, the selection process favors better indi-viduals to reproduce more often than those that are relatively worse [8]. The superiority of GA is achieved by using several search principles simultaneously such as population-based heuristics, and balance between global exploration and local exploitation. From our study using extensive simulations, we find that it is difficult to utilize GAs for obtaining a satisfactory solution to the investigated optimization problem because: 1) the population, having a large number, e.g., 50, of feasible solutions with large diversity, is hard to obtain using random generation mechanisms; 2) the children obtained from the usual search operator, crossover, are often infeasible and feasibility maintenance is difficult; and 3) the conventional GAs cannot effectively cope with the large-scale parameter optimization problem due to both large search space and expensive objective function evaluations [14].

Simulated annealing (SA) is an efficient optimization tech-nique by iteratively refining an initial solution. Unlike the other point-based techniques such as hill-climbing, SA aims at es-caping from local optima to find a globally optimal solution, and has been widely applied in various engineering problems [19], [26], [28]. The flowchart of a standard SA algorithm is shown in Fig. 3 [11]. A standard SA algorithm consists of a sequence of iterations. An initial solution is generated as a current solu-tion. Each iteration employs a randomized perturbation on the current solution, e.g., the mutation of GA, to generate a candi-date solution in the neighborhood of the current solution. The neighborhood is defined by the choice of the generation mech-anism. The generation mechanism of a standard SA uses a gen-erate-and-test method. If the candidate solution is better than the current solution, it is accepted as a new current solution. Otherwise, it is accepted according to Metropolis’ criterion [17]

Fig. 3. Flowchart of a standard simulated annealing algorithm. The generation mechanism of OSA uses IGM, a systematic reasoning method based on orthogonal experimental design, instead of the conventional random generation mechanism [11].

based on Boltzman’s probability. The generation mechanism of SA plays an important role in developing an efficient SA algo-rithm. The standard SA is difficult to explore an extremely large and nonlinear multimodal search space in a reasonable amount of computation time and is not acceptable for many intractable engineering applications [24]. It is also difficult to utilize a stan-dard SA for coping with the investigated optimization problem because the random generation mechanism cannot effectively explore an extremely large search space.

IV. PROPOSEDOSA-BASEDFNNM A. Concepts of Orthogonal Experiment Design (OED)

An efficient way to study the effect of several factors simul-taneously is to use OED with both orthogonal array (OA) and factor analysis. The factors are variables (parameters), which affect the response variable (objective function), and a setting (or a discriminative value) of a factor is regarded as a level of the factor. A “complete factorial” experiment would make mea-surements at each of all possible level combinations. However, the number of level combinations is often so large that this is impracticable, and a subset of level combinations must be judi-ciously selected to be used, resulting in a “fractional factorial” experiment [3], [6], [18]. OED utilizes properties of fractional factorial experiments to efficiently determine the best combina-tion of factor levels to use in design problems.

OA is a fractional factorial array, which assures a balanced comparison of levels of any factor. OA is an array of numbers

arranged in rows and columns where each row represents the levels of factors in each combination, and each column repre-sents a specific factor that can be changed from each combina-tion. The term “main effect” designates the effect on response variables that one can trace to a design parameter [3]. The array is called orthogonal because all columns can be evaluated inde-pendently of one another, and the main effect of one factor does not bother the estimation of the main effect of another factor [6], [18].

Factor analysis using the orthogonal array’s tabulation of ex-perimental results can allow the main effects to be rapidly es-timated, without the fear of distortion of results by the effects of other factors. Factor analysis can evaluate the effects of indi-vidual factors on the evaluation function, rank the most effective factors, and determine the best level for each factor such that the evaluation function is optimized.

OED uses well-planned and controlled experiments in which certain factors are systematically set and modified, and then main effect of factors on the response can be observed. OED specifies the procedure of drawing a representative sample of experiments with the intention of reaching a sound decision [3]. Therefore, OED using OA and factor analysis is regarded as a systematic reasoning method.

An illustrative example of OED using an objective function is given as follows:

maximize (16)

where , , and . This

maximization problem can be regarded as an experimental de-sign problem of three factors, with three levels each. Let factors 1–3 be parameters , , and , respectively. Let the small, medial, and large values of each parameter be the levels 1–3 of each factor, respectively. The objective function is the re-sponsible variable. A complete factorial experiment would eval-uate level combinations and then the best combination

with can be obtained. Let

denote an objective function value of the level combination . The factorial array and results of the complete factorial exper-iment are shown in Table II. A fractional factorial experexper-iment uses a well-balanced subset of level combinations, such as the 1st, 5th, 9th, 11th, 15th, 16th, 21st, 22nd, and 26th combina-tions. The best one of the nine combinations is the 21st

com-bination with . Using OED,

we can reason the best combination (3, 4, 7) from analyzing the results of the nine specific combinations, described in the next section.

B. The Used OED

IGM uses one of two classes of OAs depending on appli-cations. One is the class of two-level OAs used for optimiza-tion problems with a number of 0/1 decision variables [12]. The other is the class of three-level OAs used for optimization problems with continuous/discrete parameters [11]. All the op-timization parameters are generally partitioned into nonover-lapping groups. One group is regarded as a factor.

In this study, OSA uses OED with three-level OAs described later. Let there be factors where each factor has three levels. The total number of level combinations is for a complete

TABLE II

RESULTS OF ACOMPLETEFACTORIALEXPERIMENT.THEUNDERLINEDNUMBERS OFh FROM AWELL-BALANCEDSUBSET

WHICHCORRESPONDS TO ANORTHOGONALARRAYL (3 )

factorial experiment. To use an OA of factors, we obtain

an integer where the bracket represents

a ceiling operation, build an OA with rows

and columns, use the first columns, and ignore

the other columns [9], [11]. For example, if

, then and is used. The

numbers 1, 2, and 3 in each column indicate the levels of the fac-tors. Each column has an equal number of 1s, 2s, and 3s. The array is orthogonal when the nine pairs, (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), and (3,3), appear the same number of times in any two columns. Table III illustrates an example of

.

OA can reduce the number of level combinations for factor analysis. The number of OA combinations required to analyze all individual factors is only , where

. Algorithms of constructing OAs with various levels can be found in [16]. The algorithms for constructing the two- and three-level OAs used by OSA can be found in [11]. After proper tabulation of experimental results, the summarized data are analyzed using factor analysis to determine the relative level effects of factors.

Define the main effect of factor with level as where and

(17) where if the level of factor of combination is ; oth-erwise, . Consider the case that the objective function is to be minimized. The level of factor makes the best contribu-tion to the objective funccontribu-tion than the other two levels of factor

do when . On the contrary, if the

ob-jective function is to be maximized, the level is the best one

when . The main effect reveals the

individual effect of a factor. The most effective factor has the

largest one of main effect differences .

Let . After

the potentially best one of three levels of each factor is deter-mined, an intelligent combination consisting of all factors with the best levels can be easily derived.

An illustrative example of OED for solving the optimization problem with (16) is described below (see Table IV). First, use an , set levels for all factors as above mentioned, and evaluate the response variable of the combination , where

. Second, compute the main effect where

TABLE III ORTHOGONALARRAYL (3 )

TABLE IV CONCISEEXAMPLE OFIGM

and . For example,

. Third, determine the best level of each factor based on the main effect. For example, the best level of factor 1 is level 3

since . Therefore, select . Finally, the

best combination with can be

obtained. The most effective factor is with

which is the largest one. It can be verified from (16) that has the largest coefficient 100. Note that if only OA combinations without factor analysis are used, the obtained best solution is with , not the reasoned solution (3, 4, 7).

Fig. 4. Mean convergences of OSA from five independent runs. (a) T176. (b) T198. (c) T396. C. Intelligent Generation Mechanism (IGM)

Consider a parametric optimization function with

param-eters and a current solution . IGM

gener-ates two temporary solutions and

by perturbing , where and are generated from as follows:

and (18)

The values of are generated by the Cauchy–Lorentz proba-bility distribution [23]. If is out of the domain range of , randomly assign a feasible value to . IGM aims at ef-ficiently combining good values of parameters from solutions , , and to generate a good candidate solution for the next move.

Divide all the parameters into nonoverlapping groups using the same division scheme for , , and , . The proper value of is problem-dependent. The larger the value of , the more efficient it is the IGM if the interactions among groups are weak. If the existing interactions among parameters are strong, the smaller the value of , the more accurate it is the estimated main effect of individual groups. Considering the tradeoff, an efficient division criterion is to minimize the interactions among groups while maximizing the value of . Note that the parameter at each call of the

following IGM operation can be a constant or variable value. In this study, OSA uses a constant value of .

How to perform an IGM operation using factors on a current solution to an objective function is described as follows.

Step 1) Generate two temporary solutions and using .

Step 2) Divide each of , , and into groups of parameters where each group is treated as a factor.

Step 3) Use the first columns of an OA ,

where .

Step 4) Let levels 1, 2 and 3 of factor represent the th groups of , , and , respectively.

Step 5) Compute of the generated combination , where . Note that is the value of . Step 6) Compute the main effect where

and , 2, 3.

Step 7) Determine the best one of three levels of each factor based on the main effect.

Step 8) The candidate solution is formed using the com-bination of the best groups.

Step 9) Verify that is superior to the sampling solutions derived from OA combinations and

. If it is not true, let be the best one of these sampling solutions.

Fig. 5. Tracking errore of individual training plants. (a) T176. (b) T198. (c) T396.

The overhead of IGM in preparing OA experiments and factor analysis is relatively small, compared with the cost of function evaluations. Note that the used OAs are generated in advance. The number of objective function evaluations is per IGM operation, which includes evaluations in Step 5) and one in Step 9). Let be the total number of iterations, which equals the number of IGM operations. The complexity of OSA is function evaluations. If interactions among groups are weak, is a potentially good approximation to the best one of all the combination.

D. The Proposed OSA-Based Approach

The purpose of the OSA-based approach is the same with that of the ANFIS-based approach, which is to obtain an FNNM con-sisting of four FNNs for tuning PID controllers. The OSA-based approach using a one-stage approach is shown in Fig. 2 for il-luminating the difference from the ANFIS-based approach. We solve the investigated optimization problem using OSA to di-rectly search for with a minimal value of . The only differ-ence between OSA and a standard SA algorithm is that the gen-eration mechanism of OSA uses an IGM-based on OED [3], [6], [9], [18], instead of the conventional random generation mech-anism [11].

TABLE V

VALUES OFJFORVARIOUSFNNMs

OSA with IGM can hybridize the advantages of global ex-ploration and local exploitation by focusing on accuracy and computation time. IGM can efficiently generate a good candi-date solution for the next move by using a systematic reasoning method to efficiently exploit the neighborhood of a current so-lution, resulting in effectively obtaining a good solution to the large-scale parameter optimization problem [11], [12].

To effectively solve the intractable constraints of the opti-mization problem, we borrow a feasible solution from [22] as an initial solution of OSA which is obtained using the ANFIS-based approach. Consequently, we specify effective do-main ranges for each of the 258 parameters of FNNM. IGM can effectively search for a good feasible solution from the neigh-borhood of a current solution. The OSA-based approach is de-signed toward having the following objectives.

TABLE VI

TRACKINGERRORSe ANDe FORANFISANDOsSA, RESPECTIVELY, USING22 UNSEENTESTPLANTST22. THERATIOe =e FORSUM OFALLERRORS IS11.93%

1) Accurate FNNM with a minimal value of using a large number of training plants can be obtained.

2) The effective tuning domain is extensive and the FNNM is accurate in obtaining PID controllers for unseen test plants.

3) FNNM can be specially designed by specifying an inter-esting tuning domain and sampling a proper number of training plants.

E. The Algorithm of the OSA-Based Method

There are four choices must be made in implementing an SA algorithm for solving an optimization problem: 1) solution rep-resentation, 2) objective function definition, 3) design of the generation mechanism, and 4) design of a cooling schedule. The choices 1 and 2 are problem-dependent. Designing an efficient generation mechanism plays an important role in developing SA algorithms. Generally, there are four parameters to be specified in designing the cooling schedule: 1) an initial temperature , 2) a temperature update rule, 3) the number of iterations to be performed at each temperature step, and 4) a stopping crite-rion of the SA algorithm.

The used OSA with IGM optimizes an FNNM using the ob-jective function in (13) with the robustness constraint. OSA uses a simple geometric cooling rule by updating the tem-perature at the th temperature step using the formula:

, where is the cooling rate

which is a constant smaller than 1 but close to 1. The higher the temperature, the larger it is the possibility of accepting the candidate solution worse than the current solution. We borrow

a feasible solution from Appendix of

[22]. The effective domain range of each parameter of is

defined as , , derived from the

experi-ence of our computer simulations using various numbers of raining plants.

The designs of generation mechanism and cooling schedule would affect the convergence of OSA. By gradually reducing the step size of moves and properly setting the parameters of cooling schedule, SA would finally converge to a local optimum [11], [23]. Of course, the generation mechanism would domi-nate the convergence performance. Considering the tradeoff of computation cost and solution quality, one can adaptively adjust the control parameters in the cooling schedule. The parameters

of the following OSA for obtaining the solution to the

inves-tigated problem are: , , , and the

stopping condition uses and .

OED has been proven optimal for additive and quadratic models, and the selected combinations are good representations for all of the possible combinations [27]. The OED-based IGM performs well in terms of convergence performance, compared with the conventional random generation mechanism in solving large-scale optimization problem [11]. However, how to de-termine the size of OA and then assign the FNNM parameters to the corresponding factors is important. For advancing the performance of OSA, the FNNM parameters having stronger interactions are encoded together such as the vectors in (12) and (14). All the parameters belonging to the same FNN are grouped together in (12). In this study, and the used OA is . Let be the number of parameters of the th FNN. One vector corresponds to successive 10 factors and the parameters are assigned to the 10 factors. The nine cut points are randomly specified from the candidate cut points which separate individual parameters. The algorithm of the used OSA is described here.

Step 1) Initialize . Generate an initial feasible solu-tion as a current solution. Let be the value of at the th iteration and . Compute the value

.

Step 2) Perform an IGM operation using factors on to generate a candidate solution .

Step 3) Accept to be the new with probability : if

if (19)

Step 4) Increase by one. Compute using the current solution .

Step 5) Let the new value of be .

Step 6) Let . If for ,

, stop the algorithm. Otherwise, go to Step 2).

V. PERFORMANCEEVALUATION

The performance of the OSA-based approach and the quality of the obtained FNNM as well as PID controllers are evaluated by comparing with those of the ANFIS-based approach which

TABLE VII FOURTESTPLANTS

has the same objective. The same set of 198 training plants from [22] as listed in Table I, called T198, is divided into two sets: T176 consists of the 176 training plants used by ANFIS and T22 consists of the remainder, 22 training plants ( and ) abandoned by ANFIS. To evaluate the ability of han-dling the problem of a large number with an extensive tuning domain, we generate additional 198 training plants from T198 by exchanging the values of and . The set consisting of these 198 training plants and T198 is called T396.

In the following computer simulations, the entire program is written by C where a subroutine from MATLAB is used to com-pute the tracking errors. The obtained FNNM can be evaluated from both training and test performances. Section V-A gives the training performance using various numbers of training plants. Section V-B gives the test performance using unseen test plants.

A. Training Performance of FNNM

The training performance of FNNM is the criterion for evalu-ating the optimization ability of the OSA-based approach, which is evaluated in terms of and . A high-quality FNNM should be obtained from an extensive tuning domain by sampling a large number of training plants and can generate PID con-trollers with a small mean tracking error .

Each of the three sets T176, T198, and T396 is used as input of OSA to search for a solution to the investigated problem. Five independent runs are conducted for each set. The mean convergences of OSA for three training sets are shown in Fig. 4, where the numbers of objective function evaluations are 65692, 79948, and 80 029 for T176, T198, and T396, respectively. The values of for various FNNMs are given in Table V where the reported value 22.17 of for the ANFIS-based FNNM is ob-tained using . The tracking errors of individual training plants are shown in Fig. 5. The simulation results reveal the following.

1) The OSA-based FNNM using T176 can improve the ANFIS-based FNNM in averagely decreasing 13.08% error from to 19.27. Fig. 5(a) reveals that the FNNM of OSA has smaller errors for most of training plants than that of ANFIS, where the largest tracking errors for OSA and ANFIS are close to 100.

Fig. 6. Tracking errore of individual training plants using T22 as test plants. TABLE VIII

OBTAINEDCONTROLLERPARAMETERSFROM THEFNNMOFOSA USINGT176

TABLE IX

PERFORMANCECOMPARISONS OFTRACKINGERRORSe ANDe FORANFISANDOSA, RESPECTIVELY

2) The OSA-based FNNM using T196 has ,

which is better than the ANFIS-based FNNM using T176. Fig. 5(b) shows that the tracking errors of the training plants in T176 and T22 are smaller than 100.

Fig. 7. Set-point and load-disturbance responses of four plants. (a) P1. (b) P2. (c) P3. (d) P4.

It means that the OSA-based approach can cope with the training plants with large dead time, large nature frequency, and poor damping.

3) The mean values of of OSA for T176, T198, and T396 are 19.27, 21.26, and 20.39, respectively. The OSA-based FNNMs using T198 and T396 have similar good perfor-mance with that using T176 in terms of . The tracking er-rors for T396 are also smaller than 100, shown in Fig. 5(c). It manifests that increase of the value of does not result in degrading the performance of the OSA-based FNNM. 4) The standard deviation of is not very small. To robustly

obtain a more accurate solution , we can adjust the pa-rameters of OSA, such as increase of and for slow convergence. However, it would suffer from long com-putation time. Fortunately, the optimal design of FNNM is off-line and cost-effective for obtaining an accurate FNNM.

From the simulation results, it reveals that the OSA-based ap-proach is efficient in optimizing fuzzy neural networks to obtain FNNMs for tuning PID controllers.

B. Test Performance of FNNM

To evaluate the test performance of FNNM, we use unseen plant models to obtain corresponding PID controllers from the established FNNM and examine the performance of their con-trol systems. Two test problems are conducted for performance comparisons.

Test Problem 1: Since the FNNM of the ANFIS-based approach is obtained using T176, we use T22 (which are intractable for ANFIS) as the test set to obtain 22 sets of con-troller parameters from the FNNM obtained from OSA using T176. The individual tracking errors of the 22 plant models are shown in Fig. 6 and Table VI. The mean tracking errors of the 22 plants for ANFIS and OSA are 262.3 and 31.3, respectively. The OSA-based approach can averagely improve 88.07% tracking error. Note that all the obtained systems are stable and most systems of the test plants have tracking errors smaller than 100. The test performance is slightly worse than the training performance for the OSA-based FNNM. It means that no overtraining problem is occurred. From the test problem, the advantage of FNNM is well recognized that the FNNM can fast

generate 22 PID controllers on-line from these 22 intractable test plants, relative to the conventional methods [10], [25].

Test Problem 2: For further comparing the quality of the obtained PID controllers with those of other PID controllers ob-tained from the conventional methods [10], [25], the four spe-cific test plants (P1–P4) used in [22] are used to compare per-formance, shown in Table VII. The characteristics of the four plants are: 1) P1 is a heavily oscillatory and short apparent dead time plant; 2) P2 is a heavily oscillatory and high-order plant; 3) P3 is a high-order and moderately oscillatory plant; 4) P4 is a high-order and long apparent dead-time plant; and 5) the values of and for the four plants do not fall in the range and . The obtained controller parameters from the FNNM of OSA with T176 are given in Table VIII. The tracking errors of OSA and ANFIS based systems are given in Table IX. The set-point and load-disturbance responses of the four controllers are shown in Fig. 7.

The design of optimal PID controllers has two minimiza-tion objectives to be simultaneously optimized: 1) robust sta-bility and disturbance attenuation, and 2) tracking error. Since the two competing objectives cannot be evaluated in terms of the same measurement, the optimal controller design problem is essentially a multi-objective optimization problem. In com-paring two stable systems with (for the first objec-tive), the tracking error is the only quantitative measurement. The tracking error is responsible to the quality of oscillation. To further examine the oscillation performance, the set-point error and load-disturbance are analyzed.

From Table IX, the OSA-based FNNM can generate four PID controllers having a mean tracking error 7.811 55 from the four test plants while the ANFIS-based FNNM has a mean error 9.259 15. The OSA-based FNNM can averagely improve 15.63% tracking errors for the four test plants. From carefully observing the response performance in Fig. 7, the load-dis-turbance performances of the four OSA-based under-damped systems are all better than those of the ANFIS-based ones. However, the set-point responses of the OSA-based systems are slightly worse than those of the ANFIS-based ones. Note that the four test plants of the ANFIS-based FNNM have better balance performance between set-point and load-disturbance responses than those of the methods [10] and [25]. Therefore, the simulation results show that the proposed approach can obtain the FNNM providing high-quality PID controllers with a good balance between the set-point and load-disturbance responses.

VI. CONCLUSION

In this paper, we formulated an optimization problem of es-tablishing an FNNM for efficiently tuning PID controllers of various test plants with under-damped responses using a large number of training plants such that the mean tracking error of the obtained control systems is minimized. At the same time, we proposed a novel orthogonal simulation annealing al-gorithm OSA to solve the large-scale constrained optimization problem. High performance of the OSA-based FNNM arises from two aspects: 1) the tracking errors and stability constraints are directly embedded in the objective function for tuning PID

controllers, and 2) OSA can effectively search for a near-op-timal feasible solution to the large-scale parameter optimization problem. The benefits of using OSA to optimize FNNMs are also analyzed. It has been shown that the OSA-based FNNM

with can improve the ANFIS-based FNNM in

av-eragely decreasing 13.08% error and 88.07% tracking error of 22 unseen test plants. The OSA-based FNNM using four specific test plants can averagely improve 15.63% tracking er-rors of the under-damped systems. Furthermore, the OSA-based FNNMs using and 396 from an extensive tuning do-main have similar good performance with that using in terms of . The simulation results also show that the proposed approach can obtain the FNNM providing high-quality PID con-trollers with good oscillation performance as well as a good bal-ance between the set-point and load-disturbbal-ance responses.

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Shinn-Jang Ho (M’03) was born in Taiwan, R.O.C.,

in 1960. He received the B. S. degree in power mechanic engineering from National Tsing Hua University, Hsinchu, Taiwan, in 1983, and the M.S. and Ph.D. degrees in mechanical engineering from National Sun Yat-Sen University, Kaohsiung, Taiwan, in 1985 and 1992, respectively.

He is currently a Professor in the Department of Automation Engineering, National Formosa Univer-sity, Huwei, Yulin, Taiwan. His research interests in-clude optimal control, fuzzy system, evolutionary al-gorithms, and system optimization.

Li-Sun Shu (S’04) was born in Taiwan, R.O.C., in

1972. He received the B.A. degree in mathematics from Chung Yuan University, Chung Li, Taiwan, in 1995, and the M.S. and Ph.D. degrees in information engineering and computer science from Feng Chia University, Taichung, Taiwan, in 1997 and 2004, re-spectively.

He is currently an Assistant Professor at Over-seas Chinese Institute of Technology. His research interests include evolutionary computation, large parameter optimization problems, fuzzy systems, and system optimization.

Shinn-Ying Ho (M’00) was born in Taiwan, R.O.C.

in 1962. He received the B. S., M. S., and Ph.D. degrees in computer science and information en-gineering from National Chiao Tung University, Hsinchu, Taiwan, in 1984, 1986, and 1992, re-spectively. From 1992 to 2004, he was with the Department of Information Engineering and Com-puter Science at Feng Chia University, Taichung, Taiwan.

He is currently a professor in the Department of Biological Science and Technology and Institute of Bioinformatics, National Chiao Tung University, Hsin-chu, Taiwan. His research interests include evolutionary algorithms, image processing, pat-tern recognition, bioinformatics, data mining, virtual reality applications of computer vision, fuzzy classifier, large parameter optimization problems, and system optimization.