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and M. Tyson, Fastus: A cascaded finite-state transducer for extractinginformation from natural language text, in Finite-State Language Pro-cessing, E. Roche and Y. Schabes, Eds. Cambridge, MA: MIT Press,1997, pp. 383406.

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[19] P. Jacobs and L. Rau, SCISOR: Extracting information from on-linenews, Commun. ACM, vol. 33, no. 11, pp. 8897, 1990.

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An Intelligent System for Failure Detection and Control inan Autonomous Underwater Vehicle

N. Ranganathan, Minesh I. Patel, and R. Sathyamurthy

AbstractAutonomous underwater vehicles (AUVs) have been used ex-tensively in deep sea research. The AUVs are preferred to remote oper-ated vehicles (ROVs) due to their low cost and efficiency. Failure detectionand control is an important issue in maintaining the stability of an AUV.In most AUVs, the vehicle resurfaces in the event of minor failures suchas in the depth sensor, the inclinometer, etc. This paper proposes an intel-ligent system for failure detection and control in AUVs where the vehiclecould continue exploration in case of minor failures in the sensors and con-trol surfaces. The intelligent system, based on the model proposed in [12],integrates the adaptability of an artificial neural network (ANN) and theinferencing ability of a fuzzy rule based expert system on a single VLSIchip. The associative function of the ANN is used to recognize and detectthe failures by observing the various changing parameters of the dynamicvehicle. The inferencing ability of an expert system suggests ways to con-trol the failure and indicates the subsequent status of the vehicle. The entiresystem could be used as a low level diagnoser in an overall control systemfor AUVs.

Index TermsANN, automomus underwater vehicles (AVV), expertsystem, intelligent systems, remote operated vehicles (ROV).

I. INTRODUCTION

Significant research is being conducted to increase the autonomy ofunderwater robotic vehicles. The vehicles remotely operated by humanoperators have several restrictions such as the limited operating rangeof the vehicle due to the physical limits of its communication cable and

Manuscript received January 8, 1998; revised August 21, 2000 and October18, 2001. This paper was recommended by Associate Editor P. K. Willett.

N. Ranganathan and R. Sathyamurthy are with the Center for Microelec-tronics Research, Department of Computer Science and Engineering, Univer-sity of South Florida, Tampa, FL 33620 USA (e-mail: ranganat@csee.usf.edu).

M. I. Patel is with Honeywell International, Clearwater, FL 33764-7290 USA.Publisher Item Identifier S 1083-4427(01)11294-4.

10834427/01$10.00 2001 IEEE

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sensor. Therefore, it is worthwhile to develop an autonomous vehiclewhich collects the information from the sensors and the database, eval-uates the data with respect to a predefined goal, and makes changeswithout external supervision. In recent years, AUVs have been usedextensively in the fields of commerce, science and defense. The adventof autonomous underwater vehicles has greatly decreased the dangersinvolved in the deep sea exploration. A critical issue for the AUVs tooperate successfully over a long period of time is the reliability of thesystem. For autonomous fault detection, programmed recovery proce-dures have to be built into their control logic. For example, if a pro-peller drive shaft developed too much shaft friction, then the addedcurrent load may overburden the drive motor. Identification of such acondition before the shaft bearing fails would allow rectification of theproblem before a major breakdown occurs [1].

Failure detection and control forms an important part of any AUVcontrol system. As it is still a new field, most autonomous underwatervehicles use a method of damage control that results in the autonomousunderwater vehicle resurfacing in the event of any fault in the system.But future industrial and military autonomous underwater vehicles mayrequire systems that operate even while partially damaged. Hence, it isof importance to develop a system which not only detects failure inthe underwater vehicle but also suggests reliable control measures. Itis also imperative that the system works in real-time. Speed plays animportant role in this application because failure to detect a problemon time may render the control measures ineffective.

The AUV failure detection and control schemes are classified intotwo categories: 1) offline and 2) online systems. The offline schemesare based on mathematical models which estimate the stability beforean AUV is launched. However, errors may occur from poor mathemat-ical formulation of the hydrodynamic forces which result in an unstablecontrol system. The online schemes provide failure detection and con-trol while the AUV is operational. These schemes are divided into twocategories (i) model free and (ii) model based. Model free methods donot use mathematical models but use limit checking of sensors for thedetection of faults. Orrick et al. [4] proposed a system which can de-tect failures only when the vehicle is changing depth, heading or both.Most model based schemes use filters such as in [7] and [15], however,the schemes have large memory requirements. Another model basedsystem was proposed by Healey [2] which used ANNs. The systemuses a Kalman filter for parameter identification and the ANN for clas-sification. However, the system was computationally complex due toparameter estimation.

Certain other approaches for failure detection have been reported inthe literature. In [6], the authors suggest a protocol for distributed faulttolerant operating system services using any redundant sensors. Zheng[16] suggested practical solutions to common AUV control problemssuch as closed-loop stability, mission plan execution, multiple sensorprocessing, mission plan execution, fault diagnosis and fault recovery.The solutions are developed based on a layered control of the AUV.Although layered control is a simple concept, it still has to evolve tomeet the demanding requirements of a commercial AUV mission.

This paper proposes a scheme to detect, isolate and control failuresin the control surfaces and sensors of an AUV. The proposed scheme ismapped onto an intelligent decision making system proposed in [12].The intelligent system integrates the adaptability of an artificial neuralnetwork (ANN) and the inferencing ability of a fuzzy rule based ex-pert system. Failure recovery is achieved through the redundancy prin-ciple, where the same quantity is measured using at least two differentmethods. For example, if a failure was detected in the depth sensor, thenthe depth can be measured alternatively by observation of the sideslip,angle of attack, roll angle, pitch angle and forward velocity. In the pro-posed system, parameter estimation is achieved through the model freemethod. The associative and predictive nature of an artificial neural

network is used for detecting and isolating failures. The ANN uses thebackpropagation algorithm for training, and in the execution (forward)mode, diagnoses faults on all of the heading and depth control surfacesand sensors of an AUV. The ANN output which consists of predictedfailures is fed as input to a fuzzy expert system. Based on the pro-vided rule, the fuzzy expert system computes a decision to overcomethe failure and also determines the subsequent status of the vehicle. Theproposed system when implemented as part of an AUV controller willserve as a low-level diagnoser, monitoring the various instruments andtheir performance.

II. SYSTEM MODEL AND DESIGN

In recent years, much research has been carried out in the design anddevelopment of AUVs [9]. AUVs offer practical alternatives to remoteoperated vehicles (ROV) because unlike ROVs, they are not tethered.Tethers are umbilical cables connecting ROVs to the mother vehicleand are used for communication, recovery and transmitting power. Ithas been estimated that over 10% of the ROVs are lost due to brokentethers. By replacing ROVs with AUVs, the efficiency of undersea op-erations will improve significantly due to the reduced cost and risk.AUVs are useful for tasks in ocean engineering for conducting under-water surveys and military for underwater mine detection. AUVs mustbe designed and constructed to achieve the highest level of reliability ifthey are to be considered a viable and practical alternative to existingROVs and manned vehicles.

Most of the vehicles currently deployed require a certain amountof human interface. By increasing the autonomy, the vehicle can op-erate in hazardous environments without the necessity of human con-trol. Under such circumstances, the AUV requires an increased amountof stability. For a stable and reliable AUV, failure detection, isolationand control during the operating period is of considerable importance.Most failure diagnosing systems available for AUV systems do not sup-port the operation of the vehicle in case of any failure. In these schemes,the vehicle resurfaces even in the event of a minor failure.

In an AUV, failures may occur in the control surfaces or sensors,during navigation or power consumption [11]. The proposed schemeconsiders failures occurring in two important areas: 1) the controlsurfaces and 2) the sensors. The redundancy management techniqueproposed in [5] forms the basis for failure detection of the proposedscheme. The term redundancy does not indicate that the sensors areduplicated but instead those two values of the same quantity are ob-tained by two independent methods. Unlike the redundancy technique,other techniques proposed in the literature for failure detection [14]are not commonly implemented in practice. Also, the failures in thesensors are determined by imposing error bounds on the sensor valuesin order to determine the tolerance levels for them [4].

Recovery from failures is important as cases exist where AUV mis-sions have failed due to simple system failures. In most cases, one canrecover from simple failures by using alternate sensors. In some situ-ations, in spite of recovering from the failure, the performance of theAUV degenerates. Therefore the resultant status of the system shouldbe taken into consideration during failure recovery. The failure controlsystem implemented in this work performs two functions: 1) indicatealternate sensors to recover from the failures detected and 2) indicatethe status of the AUV as a result of the failures.

A. Intelligent Decision Making SystemFor effective and reliable failure detection, a new failure detection

and control scheme is proposed and mapped onto the intelligent systemarchitecture proposed in [12]. The intelligent system can operate inreal-time and is capable of making decisions online. The system in-tegrates the adaptability of an artificial neural network and the infer-

764 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART A: SYSTEMS AND HUMANS, VOL. 31, NO. 6, NOVEMBER 2001

Fig. 1. Intelligent system architecture.

encing ability of a fuzzy rule based expert system. The architecturalblocks and the data flow for the proposed system are shown in Fig. 1.Sensors are used to detect the surrounding environmental conditions inthis model. The sensors send crisp data inputs to the artificial neuralnetwork. The fuzzification unit assigns fuzzy labels to the outputs ofthe ANN. These labels indicate the degree to which each crisp valueis a member of a domain. Then, the fuzzy expert system fires the rulesbased on these fuzzy values. The defuzzification unit converts the com-puted decisions into crisp values that are used to control the environ-ment within an application.

B. Integrating the ANN and the FESThe design of an efficient intelligent decision making system de-

pends on 1) knowledge acquisition and scope and 2) decision explana-tion. During knowledge acquisition, the knowledge base of an expertsystem is formed with the aid of an expert interacting with a knowl-edge engineer. The knowledge acquisition process consists of the fol-lowing subtasks: 1) knowledge extraction, 2) formal representation ofknowledge, 3) coding, and 4) validation [3]. The development of theknowledge base starts with the knowledge engineer extracting and for-malizing the data acquired from an expert. In learning systems suchas ANNs, the knowledge acquisition task is performed by the trainingprocess. However, the training process, in most cases, is a time con-suming task requiring the application of input training patterns in aniterative manner. The process is constrained by various parameters thatguide the behavior of the system such as 1) the type, size, and def-inition of the training data, 2) the learning rate, and 3) the topologyof the ANN. The process of training may also be constrained by theuncertainty as to whether the final learning goal has been achieved.Thus, using an ANN or an expert system approach to intelligent deci-sion making leads to different levels of performance depending on themodel as well as the application. By integrating the two approaches, itis possible to overcome the deficiencies associated with using a singleapproach.

The proposed intelligent decision making system requires an artifi-cial neural network model that can handle a wide range of applicationsand is simple in terms of implementation and training. The ANN modelused in this work is a fully connected, single hidden layer back-prop-agation network. Based on existing research, the back-propagation al-gorithm has been found to be the most suitable choice. A fuzzy expertsystem (FES) computes decisions that control certain functions withinthe application environment. The fuzzy input data is processed using arule base that has usually been created with the help of an expert. TheFES is based on fuzzy logic which deals with continuous membershipvalues ranging from 0 to 1. The FES consists of 1) a rule memory and2) an inferencing unit [8]. The rule base is stored in a random accessmemory (RAM). Each rule consists of a certain number of antecedentand consequent membership functions which are used in the decisionmaking process. The inferencing unit calculates decisions using fuzzy

rules of composition [10]. The center of gravity defuzzification methodis chosen in this work due to its suitability for hardware implementa-tion.

III. FAILURE DETECTION AND CONTROL SYSTEM

The types of failures addressed in this work are not necessarily foundusing state estimation. The changes in the vehicle behavior as a resultof variations in the sensor readings are considered. The approach hereis to look at the sensor readings with a decision processor in order toisolate a particular mode of failure. The time required to perform thepostprocessing in real-time is largely due to the amount of computation.Artificial neural networks are tuned to recognize and ultimately detectfailures based on changes in particular values of the vehicle systemdynamic parameters. Artificial neural networks can execute rapidly inreal time and can be trained offline to produce the required decisionsurface. An artificial neural decision maker is expected to enhance thereliability of the detection and isolation process which is an importantcriterion in the design of an AUV.

The rules used for failure detection are based on the proper under-standing of the dynamic system of an AUV. No parameter estimationmethod is used to determine the inputs to the artificial neural network.The inputs for the artificial neural network come directly from the sen-sors after some modifications to accommodate for the system speci-fications. In this application, sensor values are not considered for thepurpose of adaptive control, instead, values such as depth, pitch, etc.are used to determine any abnormalities in the readings. The inputs aredecided based on the failures to be detected. Fig. 2 shows the generalstructure of the artificial neural network and the listing of the inputsand outputs.

The artificial neural network was empirically determined to have16 input nodes for each of the sensor values and position, nine outputnodes for each of the failure modes and 32 hidden nodes. The inputs tothe network depend on the type of failure to be detected. The networkdesign is based on the work reported in [4]. The difference between thedepth rate from the depth sensor, (dZ=dt)1

and the rate calculated fromparameters such as angles of pitch, sideslip, attack and roll, (dZ=dt)2

,

is used to rule out failure of the planes or depth slope. The depth and thepitch readings are used in detecting depth sensor and inclinometer fail-ures. The sternplane deflection, s

, is used to detect sternplane failure.The rate of change of pitch in the opposite direction helps in deter-mining the depth sensor failure. A turning rate of zero indicates rudderfailure. The inertial measurement unit (IMU) velocity angle and are two other inputs, as the readings of the two have to be the sameto eliminate heading angle or magnetic compass failures. The headingangle reading is used to detect any failure in the zero or full position.The artificial neural network outputs consist of nine failure modes.

In existing systems for failure detection, in the event of a failure, thevehicle either resurfaces or a mission controller determines the nextcourse of action. In this work, a fuzzy expert system (FES) is used toautomate the process of failure control. The FES makes decisions basedon the type of failure detected and a rule base. The rule base consistsof knowledge obtained from a mission controller.

The fuzzy expert system model with the ANN is shown in Fig. 3. Inthe figure, n is the number of rules in the rule base, yj

is the output ofthe ANN, y

(u) is the fuzzy labeled ANN output, and A

(u) is thefuzzy labeled input from the environment. i

is the weight of the ithrule and is the weighted combination of each rule. The fuzzy expertsystem was designed with eight inputs, two outputs and a rule base of46 rules. Each rule consists of eight antecedents (one for each output ofthe ANN) and two consequents (decisions for failure control). Each an-tecedent defines the degree to which the failure detected belongs to the

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Fig. 2. Artificial neural network structure.

set, degree of failure. For example, if the artificial neural network in-dicates a sternplane failure, then after fuzzification the degree of failureis considered to be critical. The rule base was developed through liter-ature surveys and informal discussions with persons experienced in thefield. Each rule was developed to cover the maximum number of deci-sions.

The outputs of the ANN are shown in Fig. 2. The ANN output isa value 2 [0 . . . 1] and is converted into an integer between 0 and 15(by multiplying by 15). The corresponding integer is used when as-signing fuzzy labels to the ANN outputs. The value of 15 correspondsto the maximum value of the x-axis used for the membership functionsshown in Fig. 4. The linguistic variables are shown in Tables IIII. Thefuzzy labels assigned are based on the five linguistic variables shown inFig. 4. The figure shows one membership function with four linguisticvariables used to label the outputs of the ANN. For illustration, a set offive variables were used to represent the fuzziness of the sensors. Thefuzzification process assigns a fuzzy label (according to their degree offailure) to each output of the ANN. A sternplane failure is considered tobe a critical failure, while depth sensor and inclinometer failures are notconsidered to be so critical. The fuzzy labels are then sent to the FES.The FES receives the 8 inputs and computes two decisions to avoid theobstacles. Based on these antecedents and the rule base, two distinctdecisions are reached by the FES. The first decision suggests ways torecover from failures detected by the ANN. For example, one can re-cover from a sternplane failure by using depth and pitch sensors. Thesecond decision indicates the resultant status of the system. For a stern-plane failure, the vehicle performance will degenerate, but it can stilloperate. An arrangement of a typical rule consists of eight antecedentsand two consequents as follows:

if

then

Fig. 3. Modeling a dynamic environment.

Fig. 4. Input member function.

Here, Nodea . . .Nodeh represent the predicted failures from theANN (labeled in Fig. 2) and A0 . . .A7 represent inputs (antecedents)from the rule base. The rule antecedents A0 . . .A7 correspond to thevarious failures that must match or partially match with the outputs ofthe ANN in order for the rule to be fired. The consequents representthe two control decisions: 1) suggested action to recover from failureand 2) resultant status of AUV. The consequent membership functionsare shown in Fig. 5. The figure shows two membership functions eachwith five and eight linguistic variables respectively.

IV. SYSTEM SIMULATION AND PERFORMANCE

The ANN was implemented in C++ using object oriented concepts.The expert system uses the forward chaining method to infer from rulesstored in the rule base. The FES, using the max min compositionmethod for inferencing, receives the antecedents and the consequentsfrom the rule base which is stored in separate files. The inputs are readfrom a file and sent to the FES implemented in C. Since failure detec-tion is done using the redundancy method, each sensor value is com-pared with identical values calculated from other sensors. If any dis-crepancies are detected then a failure alert is sounded. For example, inorder to detect a depth sensor failure, the value from the depth sensor iscompared with the plane deflection and inclination. If the depth sensorindicates values in the direction opposite to that of the other two, thenthe failure mode c is indicated in the ANN output. Similar reasoning isapplied in determining the other failure modes.

The intelligent decision making system consists of three operations:1) training the ANN with sensor data, 2) testing the ANN, and 3) in-ferring decisions using the FES. In order to train and test the ANN inthe proposed system, two sets of training and test data sets referred toas trainA/testA and trainB/testB were used. Each data set consisted of1000 training vectors and 500 test vectors. Each pattern contains 16input values and nine desired outputs. The ANN was trained on thetraining data set and tested on the test set. The training set was cre-ated with the aid of an expert in the field of autonomous navigationand underwater vehicles. The test data was created by generating a setof values from uniformly distributed random numbers. Each trainingvector was assigned some combination of failures as directed by theexpert. The training of the ANN was continued until the sum of thesquares of the training error Npatterns1

2 was less than 0.1. If the

766 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICSPART A: SYSTEMS AND HUMANS, VOL. 31, NO. 6, NOVEMBER 2001

(a)

(b)Fig. 5. Output member function.

TABLE IINPUT LINGUISTIC

VARIABLES

TABLE IILINGUISTIC VARIABLES FOR OUTPUT 1

error does not fall below 0.1, the ANN is allowed to train up to 3000epochs. The large number of epochs was chosen to give the ANN suf-ficient time to converge if possible.

The network after training was evaluated using the test vectors. Thetesting of the ANN with the entire set of test patterns yielded an av-erage classification rate of 96%, which means that for the test set, theANN predicted an AUV failure on the average 96% of the time. Thetraining time on the SUN Sparc 4 was 75 min of CPU cycle time onthe average. The testing time on the SUN Sparc 4 was 1.2 ms per testvector. The training and testing summary is shown in Table IV. Thetable shows the training epoch with the least error during the trainingand the percentage of correct outputs during testing. The percentageof correct outputs is calculated using the number of outputs predictedcorrectly by the network over the total number of test vectors.

It was observed that the network correctly predicted failures 94% ofthe time. The ANN was tested on three additional test data sets whereeach set contains 100 vectors. The test results indicate that the ANNpredicted AUV failures on the average 93% of the time as shown inTable V. As indicated by the table the artificial neural network behaviorwas consistent and was able to predict the outputs at a stable rate.

The fuzzy expert system, based on the ANN outputs, computes twodecisions: 1) suggest alternate sensors to replace the failed sensor and2) indicate the resulting stability of the AUV. The 16 sensor inputs fromthe AUV were normalized between 1 and 0 and fed to the ANN. Theoutputs predicted by the ANN were assigned a fuzzy label and sentto the FES. Based on the ANN outputs, the FES computes two de-cisions which are then defuzzified. The intelligent system was testedon the same test vectors used for the ANN as well as 400 additional

TABLE IIILINGUISTIC VARIABLES FOR OUTPUT 2

TABLE IVOUTPUT FILE FROM ANN

TABLE VANN PERFORMANCE

test vectors were used specifically to test the fidelity of the FES. Also,the entire test vector set was used to test the entire intelligent decisionmaking system. The integrated system successfully computed correctAUV control decisions for each of the sample runs. The system com-puted correct decisions on the average 95% of the time. The simula-tion indicates that the proposed intelligent system provides good per-formance at a lower cost compared to other approaches. The integra-tion of the ANN and the expert system helped reducing the number ofnodes in the ANN and the number of rules in the FES by partitioningthe overall computations leading to a better implementation and per-formance.

V. CONCLUSIONS

In this work, an intelligent decision making (IDM) system is pro-posed for failure detection and control for an autonomous underwatervehicle (AUV). Failure detection was done on the heading and the depthcontrol surfaces and sensors of an AUV. The IDM system combinesthe learning ability of an artificial neural network (ANN) and the deci-sion making capability of a fuzzy expert system (FES) into an effectivecontrol system. The ANN was trained to detect failures from a set ofsensor values and positions. The FES was used 1) to suggest alternatemeasures to recover from the failure detected by the ANN and 2) toindicate the stability of the system effected by the failures. The systemdeveloped was tested extensively and the results presented. The ANNdetected failures 94% of the time and the intelligent system computedcorrect AUV decisions 95% of the time. A linear systolic array VLSIarchitecture was proposed for implementing the intelligent system [12].The intelligent system was simulated in the hardware simulation lan-guage Verilog-XL at a 50 MHz clock rate, using the Cadence Verilog

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and C tools. The Verilog simulation results in [12] indicate that the in-telligent system can compute decisions every 20 nanoseconds using a50 MHz clock. For the AUV, the results indicate that the system is fastand functions in real time. It can be implemented as an important com-ponent of the control system of an AUV.

AUVs can be effective alternatives to current remote operated vehi-cles (ROV) as they are cost-effective and efficient. Most of the researchto date has concentrated on areas of control and guidance of an AUV.Not much consideration has been given to the long term stability andcontrol of the AUV during a mission because almost all vehicles builtand operated have been prototypes and were used to demonstrate fea-sibility of the system. They have not been used for long durations inreal time situations. If AUVs are to be deployed in real time situationsoperational endurance is of importance and the vehicle control systemshould be capable of dealing with failures occurring during the oper-ation of an AUV. This work proposes a scheme that detects and con-trols failures during the working of the underwater vehicle. The cur-rent implementation covers a subset of the possible failures that couldoccur. The scheme can be expanded to include other failures such asnavigational failures, failures in power transmission, etc. The proposedscheme provides control when failures occur in known situations, how-ever in hazardous situations AUVs may encounter unexpected failures.The proposed scheme can be modified to help the AUV operate in un-known conditions. The scheme can suggest ways to control the failuresdetected and can be expanded to actually implement the control. Sucha system would increase the autonomy and reliability of an AUV to agreat extent.

REFERENCES[1] A. Healey, A neural network approach to failure diagnostics for un-

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Searching for Optimal Trajectory With LearningEllida M. Khazen

AbstractAn algorithm of searching for the optimal trajectory with theminimal cost ( ) of reaching the final state from the initial stateis presented. A system of ODEs is suggested to determine an optimal tra-jectory ( ) and an optimal control ( ). The trajectory ( ) and thecontrol ( ) close to optimal ones are determined by successive approxi-mations. The algorithm represents a development of a gradient method inthe function space. Learning consists in estimation of an unknown a prioriminimal cost ( ) and ( ) on the basis of analysis of the trialtrajectories ( ) obtained earlier.

Index TermsBellman dynamic programming, gradient method ina function space, principal component analysis, searching for optimaltrajectory.

I. INTRODUCTION

It was almost half a century ago, when R. Bellman formulated the op-timality principle and created the theory of dynamic programming [1].He obtained a partial differential equation for an optimal cost functionW (x). The theory he developed applied to variety of optimal controlproblems. However, in many cases, the computational complexity ofthe Bellman equation is very high, especially, if the vector x is multi-dimensional; x = (x1

; x

2

; . . . x

n

). In relation to this, Bellman men-tioned the so-called curse of dimensionality [1]. The computationalcomplexity in finding the optimal control is still a problem to be over-come. Learning makes the search more purposeful and reduces com-putational complexity.

In this work, we propose a system of ODEs to describe the optimaltrajectory and optimal control, starting from the Bellman equation.Using this system of equations we first find trial trajectories x(t) andtrial controls u(t) and calculate the cost of the trial trajectories. Then,an estimate is obtained of an a priori unknown function @W (x)=@x inthe vicinity of the trajectory. Carrying out subsequent approximations,we find new trial trajectories and new controls, and a more accurateestimation of @W (x)=@x. The cost of the best trajectory obtained ateach stage of the estimation is less than that on the previous stage.

The algorithm proposed in this work and the gradient method in afunction space developed in [2] have some similarity, though the twomethods are derived differently and do not coincide. The algorithm pre-sented in this paper has the following advantages: 1) estimation of thecost functionWn

(x) and its derivative @Wn

(x)=@x in the vicinity ofthe sought trajectory accumulates useful information, which can be ob-tained by analyzing trial controls and trajectories obtained at the n-thstage and 2) the system of ODEs (1) and (8) or (1) and (24) (derived

Manuscript received December 2, 1998; revised November 11, 2000 andNovember 9, 2001. The paper was briefly presented by the author at theJoint Conference on the Science and Technology of Intelligent SystemsISIC/CIRA/ISAS98 (Sept. 1417,1998). This paper was recommended byAssociate Editor S. Lakshmivarahan.

The author is at 114 Patrick Street SE, Vienna, VA 22180 USA (e-mail: DrK-hazen@aol.com).

Publisher Item Identifier S 1083-4427(01)11302-0.

10834427/01$10.00 2001 IEEE