Adaptive Fuzzy Robust Tracking Controller Design

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003 783

Adaptive Fuzzy Robust Tracking Controller Designvia Small Gain Approach and Its Application

Yansheng Yang and Junsheng Ren

AbstractA novel adaptive fuzzy robust tracking control(AFRTC) algorithm is proposed for a class of nonlinear systemswith the uncertain system function and uncertain gain function,which are all the unstructured (or nonrepeatable) state-dependentunknown nonlinear functions arising from modeling errorsand external disturbances. The TakagiSugeno type fuzzy logicsystems are used to approximate unknown uncertain functionsand the AFRTC algorithm is designed by use of the input-to-statestability approach and small gain theorem. The algorithm ishighlighted by three advantages: 1) the uniform ultimate bound-edness of the closed-loop adaptive systems in the presence ofnonrepeatable uncertainties can be guaranteed; 2) the possiblecontroller singularity problem in some of the existing adaptivecontrol schemes met with feedback linearization techniquescan be removed; and 3) the adaptive mechanism with minimallearning parameterizations can be obtained. The performanceand limitations of the proposed method are discussed. The usesof the AFRTC for the tracking control design of a pole-balancingrobot system and a ship autopilot system to maintain the ship on apredetermined heading are demonstrated through two numericalexamples. Simulation results show the effectiveness of the controlscheme.

Index TermsAdaptive robust tracking, fuzzy control, input-to-state stability (ISS), nonlinear systems, small gain theorem.

I. INTRODUCTION

I N RECENT years, interest in designing robust trackingcontrol for uncertain nonlinear systems has been everincreasing, and many significant research attentions have beenattracted. Most results addressing this problem are availablein the control literature, e.g., Kokotovic and Arcak [1] andreferences therein. And many powerful methodologies fordesigning tracking controllers are proposed for uncertainnonlinear systems. The uncertain nonlinear systems maybe subjected to the following two types of uncertainties:structured uncertainties (repeatable unknown nonlinearities),which are linearly parameterized and referred to as parametricuncertainties, and unstructured uncertainties (nonrepeatableunknown nonlinearities), which are arising from modelingerrors and external disturbances. To handle the parametricuncertainties, adaptive control method, which has undergonerapid developments in the past decade, e.g., [2][7] can be used.

Manuscript received June 28, 2001; revised July 9, 2002 and January 15,2003. This work was supported in part by the Research Fund for the DoctoralProgram of Higher Education under Grant 20020151005, the Science Founda-tion under Grant 95-06-02-22, and the Young Investigator Foundation underGrant 95-05-05-31 of the National Ministry of Communications of China.

The authors are with the Navigation College, Dalian Maritime University(DMU), Dalian 116026, China (e-mail: [email protected]).

Digital Object Identifier 10.1109/TFUZZ.2003.819837

As for unstructured uncertainties, if there is a prior knowledgeof the bounded functions, deterministic robust control method,e.g., [8][12] can be used. Unfortunately, in industrial controlenvironment, there are some controlled systems with theunstructured uncertainties where none of prior knowledge ofthe bounded functions is available, then the adaptive controlmethod and the deterministic robust control method can notbe used to design controller for those systems. A solution tothat problem is presented that the neural networks (NNs) areused to approximate the continuous unstructured uncertainfunctions in the systems and Lyapunovs stability theory isapplied in designing adaptive NN controller. Several stableadaptive NN control approaches are developed by [13][19]which guarantee uniform ultimate boundedness in the presenceof both unstructured uncertainties and unknown nonlinearities.

As an alternative to NN control approaches, the intensiveresearch has been carried out on fuzzy control for uncertainnonlinear systems. The fuzzy systems are used to uniformlyapproximate the unstructured uncertain functions in the designedsystem by use of the universal approximation properties ofthe certain classes of fuzzy systems, which are proposed by[20] and [21], and a Lyapunov based learning law is used, andseveral stable adaptive fuzzy controllers that ensure the stabilityof the overall system are developed by [22][26]. Recently, anadaptive fuzzy-based controller combined with VSS andcontrol technique has been studied in [27] and [28]. However,there is a substantial restriction in the aforementioned works:A lot of parameters are needed to be tuned in the learninglaws when there are many state variables in the designedsystem and many rule bases have to be used in the fuzzysystem for approximating the nonlinear uncertain functions,so that the learning times tend to become unacceptably largefor the systems of higher order and time-consuming process isunavoidable when the fuzzy logic controllers are implemented.This problem has been pointed out in [26].

In this paper, we will present a novel approach for thatproblem. A new systematic procedure is developed for thesynthesis of stable adaptive fuzzy robust controller for a classof continuous uncertain systems, and TakagiSugeno (TS)type fuzzy logic systems [29] are used to approximate the un-known unstructured uncertain functions in the systems and theadaptive mechanism with minimal learning parameterizationscan be achieved by use of input-to-state stability (ISS) theoryfirst proposed by Sontag [31] and small gain approach givenin [32]. The outstanding features of the algorithm proposedin the paper are: i) that only one function is needed to beapproximated by TS fuzzy systems and no matter how manystates in the designed system are investigated and how many

1063-6706/03$17.00 2003 IEEE

784 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003

rules in the fuzzy system are used, only two parameters neededto be adapted on-line, such that the burdensome computation ofthe algorithm can be lightened increasingly and it is convenientto realize this algorithm in engineering; and 2) the possiblecontroller singularity problem in some of the existing adaptivecontrol schemes met with feedback linearization techniquescan be avoided.

This paper is organized as follows. In Section II, we willgive the problem formulation, the description of a class ofnonlinear systems and tracking control problem of nonlinearsystems. Section III contains some needed definitions of ISS,small gain theorem and preliminary results. In Section IV, asystematic procedure for the synthesis of adaptive fuzzy robusttracking controller (AFRTC) is developed. In Section V, twoapplication examples for designing the tracking control forthe pole-balancing robot system and the ship autopilot systemby use of the AFRTC are included and numerical simulationresults are presented. The final section contains conclusions.

II. PROBLEM FORMULATION

A. System DescriptionConsider the th-order uncertain nonlinear systems of the fol-

lowing form:

(1)

where and represent the control input and theoutput of the system, respectively.is comprised of the states which are assumed to be available, theinteger denotes the dimension of the system. andare unknown smooth uncertain functions and may contain non-repeatable nonlinearities. is the disturbance, unknownbut bounded, e.g., , where is an unknownconstant.

Throughout this paper, the following assumption is made on(1).

Assumption 1: The sign of is known, and there exists aconstant such that , .

This assumption implies that smooth function is strictlyeither positive or negative. From now onwards, without loss ofgenerality, we shall assume , . As-sumption 1 is reasonable because being away from zerois the controllable conditions of system (1). It should be em-phasized that the low bound is only required for analyticalpurposes, its true value is not necessarily known. Some stabilityis needed to proceed.

Definition 1: It is said that the solution of (1) is uniformlyultimately bounded (UUB) if for any , a compact subset of

, and all , there exists an and anumber such that for all .

We represent as any suitable vector norm. In this paper,vector norm is Euclidean, i.e., and given amatrix , matrix norm is defined by

where denotes the operation of taking the max-imum (minimum) eigenvalue. The norm denoted bythroughout this paper unless specified explicitly, is nothing butthe vector two-norm over the space defined by stacking the ma-trix columns into a vector, so that it is compatible with the vectortwo-norm, i.e., .

The primary goal of this paper is to track a given referencesignal while keeping the states and control bounded. Thatis, the output tracking error should be small.The given reference signal is assumed to be bounded andhas bounded derivatives up to the th order for all , and

is piecewise continuous.Let such that is bounded. Sup-

pose . The (1) can be transformed into

(2)In this paper, we present a method for the adaptive robust con-

trol design for system (2) in the present of unstructured uncer-tainties. Our design objective is to find an AFRTC of theform

(3)(4)

where is the known fuzzy base functions. In such a way thatall the solutions of the closed-loop system (2)(4) are uniformlyultimately bounded. Furthermore, the output tracking error ofthe system can be steered to a small neighborhood of origin.

B. TS Fuzzy SystemsIn this section, we briefly describe the structure of fuzzy sys-

tems. Let denote the real numbers, the real -vectors,the real matrices. Let be a compact simply

connected set in . With map , define tobe the function space such that is continuous. A fuzzy systemcan be employed to approximate the function in order todesign the adaptive fuzzy robust control law, thus the configu-ration of TS type fuzzy logic system called TS fuzzy systemfor short [29] and approximation theorem are discussed first asfollows.

Consider a TS fuzzy system to uniformly approximate acontinuous multidimensional function that has a com-plicated formulation, where is input vector with independent

. The domain of is . It fol-lows that the domain of is

In order to construct a fuzzy system, the interval [ ] isdivided into subintervals

On each interval , continuousinput fuzzy sets, denoted by , are definedto fuzzify . The membership function of is denoted by

, which can be represented by triangular, trapezoid, gen-eralized bell or Gaussian type and so on.

YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 785

Generally, TS fuzzy system can be constructed by the fol-lowing fuzzy rules:

where , , are the unknownconstants. The product fuzzy inference is employed to evaluatethe ANDs in the fuzzy rules. After being defuzzified by a typicalcenter average defuzzifier, the output of the fuzzy system is

(5)

where and, which is called a

fuzzy base function. When the membership function inis denoted by some type of membership function, is

a known continuous function. So, restructuring (5) as follows:

Let ,

,.

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, then

the (5) can be easily rewritten as(6)

where , and

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When the fuzzy system is used to approximate the continuousfunction, two questions of interest may be asked: whether thereexists a fuzzy system to approximate any nonlinear function toan arbitrary accuracy? how to determine the parameters in thefuzzy system if such a fuzzy system does exist. The followinglemma [30] gives a positive answer to the first question.

Lemma 1: Suppose that the input universal of discourse isa compact set in .Then, for any given real continuous function

on and , there exists a fuzzy system in theform of expression (6) such that

(7)

III. MATHEMATICAL PRELIMINARIES

The concept of ISS and ISS-Lyapunov function due toStontag [31], [33] and Sontag and Wang [34] have recentlybeen used in various control problems such as nonlinearstabilization, robust control and observer designs (see, e.g.,[35][40]). In order to ease the discussion of the design of

AFRTC scheme, the variants of those notions are reviewedin the following. First, we begin with the definitions of class

, and functions which are standard in the stabilityliterature; see [41].

Definition 2: A function is said of class if it is

continuous, strictly increasing and . It is of classif it is of class and is unbounded.

A function is said ofclass if, for each , is of class , and,for each , is strictly decreasing and satisfies

, and is a class function if andonly if there exist two class functions and suchthat

We consider the following system:

(8)where is the state and is the input. For this system, we givethe definition of input-to-state stable in the following.

Definition 3: For (8), it is said to be input-to state practicallystable (ISpS) if there exist a function of class , called thenonlinear gain, and a function of class such that, forany initial condition , each measurable essentially boundedcontrol defined for all and a nonnegative constant ,the associated solutions are defined on [0, ) and satisfy

(9)where is the truncated function of at and standsfor the supremum norm.

When in (9), the ISpS property collapses to the ISSproperty introduced in [33].

Definition 4: A function is said to be an ISpS-Lya-punov function for (8) if

there exist functions , of class such that

(10) there exist functions , of class and a constant

such that

(11)

When (11) holds with , is referred to as an ISS-Lyapunov function.

Then it holds that one may pick a nonlinear gain in (9)of the form, which is given in [35]

(12)For the purpose of application studied in this paper, we intro-

duce the sequel notion of exp-ISpS Lyapunov function.Definition 5: A function is said to be an exp-ISpS Lya-

punov function for system (8) if there exist functions , of class such that

(13)


Fig. 1. Feedback connection of composite systems.

there exist two constants , and a classfunction such that

(14)

When (14) holds with , the function is referred to asan exp-ISS Lyapunov function.

The three previous definitions are equivalent from [34] and[39]. Namely, the following.

Proposition 1: For any control system (8), the followingproperties are equivalent:

i) it is ISpS;ii) it has an ISpS-Lyapunov function;

iii) it has an exp-ISpS Lyapunov function.Consider the stability of the closed-loop interconnection of

two systems shown in Fig. 1.A trivial refinement of the proof of the generalized small

gain theorem given in [32] and [40] yields the following variantwhich is suited for our applications here.

Theorem 1: Consider a system in composite feedback form(cf. Fig. 1)

(15)

(16)

of two ISpS systems. In particular, there exist two constants, , and let and of class , and and

of class be such that, for each in the supremumnorm, each in the supremum norm, each andeach , all the solutions and are de-fined on [0, ) and satisfy, for almost all

(17)(18)

Under these conditions

(19)

the solution of the composite systems (15) and (16) is ISpS.

IV. DESIGN OF ADAPTIVE FUZZY ROBUST TRACKING CONTROLUsing the pole-placement approach, we consider a term

where , the s are chosen such that allroots of polynomial lie in the

left-half complex plane, leads to the exponentially stable dy-namics. Then, the (2) can be transformed into

(20)where

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Because is stable, a positivedefinite solution ofthe Lyapunov equation

(21)always exists and is specified by the designer.

For this control problem, if both functions and in(20) are available for feedback, the technique of the feedbacklinearization can be used to design a well-defined controller,which is usually given in the form of

for some auxiliary control input withbeing nonzero for all time, such that the resulting closed-loopsystem can be shown to achieve a satisfactory tracking per-formance. However, in many practical control systems, plantuncertainties that contain structured (or parametric) uncertain-ties and unstructured uncertainties (or nonrepeatable uncer-tainties) are inevitable. Hence, both and may notbe available directly in the robust control design. Obtaining asimple control algorithm as before is impossible. Moreover,if any adaptation scheme is implemented to estimateand as and respectively, the simple controlalgorithm aforementioned can be also used for substituting

and for and , so the extra precautionis required to guarantee that for all time. At thepresent stage, no effective method is available in the litera-ture. In this paper, we develop a semi-globally stable adaptivefuzzy robust controller which does not require to estimatethe unknown function , and therefore avoids the possiblecontroller singularity problem.

In this paper, the effects due to plant uncertainties andexternal disturbances will be considered simultaneously. Thephilosophy of our tracking controller design is expected thatTS fuzzy approximators equipped with adaptive algorithmsare introduced first to learn the behaviors of uncertain dy-namics. Here, only uncertain function is needed to beconsidered.

For is an unknown continuous function, by Lemma 1,TS fuzzy system with input vector for somecompact set is proposed here to approximate the un-certain term where is a matrix containing the approxi-mating parameters. Then, can be expressed as

(22)where is a parameter with respect to approximating accuracy.


Substituting (22) into (20), we get

(23)Let , such that and

. It follows that (23) reduces to

(24)In order to design the adaptive fuzzy robust controller easily

by use of the small gain theorem, the following output equationcan be obtained by comparing (24) with (15):

Then, the feedback equation is given as follows:

So, (24) can be rewritten as (15) and (16)

(25)

(26)Then, the feedback connection using the (25) and (26) can be

implemented using the block diagram shown in Fig. 2.From Fig. 2, we observe that the system should be made

to satisfy ISpS condition of the system through designing thecontroller . In (25), is an unknown, and there existsome parameters with boundedness. According to these prop-erties, an adaptive fuzzy robust tracking control algorithm willbe proposed, which not only gives the controller tomake the system meet ISpS condition but also the onlineadaptive law for and the other parameters in the (25). For thispurpose, we will discuss it in the following.

Construct an adaptive fuzzy robust tracking controller asfollows:

(27)where denotes a certainty equivalent controller and de-notes a supervisory controller for the disturbance, approxima-tion error and other bounded items. Those will be given in thefollowing.

Substituting (27) into (25) yields

(28)Based on the aforementioned condition, we can get

(29)where , ,

, ,

, and . denotesthe largest term with unknown constant in all boundedness. Inorder to design the controller, we can get .

Fig. 2. Feedback connection of fuzzy system.

Let and be the parameter estimate of and, respectively. We propose an adaptive fuzzy robust tracking

controller (AFRTC) as follows:

(30)

where will be specified by designer, and is the gainof to be chosen later on.

The adaptive laws for and are now chosen as

(31)

where , , 2 are the updating rates chosen bydesigner, and , , 2, and are design con-stants. Adaptive laws (31) incorporate leakage term based ona variant of the -modification proposed by Polycarpou andIoannou [42], which can prevent parameter drift of the system.

Theorem 2: Consider the system (20), suppose that As-sumption 1 is satisfied and the can be approximated byTS fuzzy system. If we pick and in(21), then the control scheme (30) with adaptive laws (31) isan AFRTC which can make all the solutions ( ) ofthe derived closed loop system uniformly ultimately bounded.Furthermore, given any and bounds on and , wecan tune our controller parameters such that the output error

satisfies .Before proving Theorem 2, the following lemma given in [42]

is reviewed first.Lemma 2: The following inequality holds for any and

any :

(32)

where is a constant that satisfies , i.e.,.

The proof of Theorem 2 can be divided into twofold. First, letthe constant and set as the input of the system

, to prove the satisfaction of ISpS for the system by


use of the adaptive fuzzy robust tracking controller, and then toprove uniform ultimate boundedness of the composite of twosystems with the feedback system by use of smallgain theorem.

Choose the Lyapunov function as

(33)where , , .

The time derivative of along the error trajectory (28) is

(34)We deal with relative items in (34), substitute (30) into the

relative items shown before, and obtain

(35)

(36)

and substituting (30) into the relative items of (34), we get

Substituting (29) into the aforementioned equation yields

Let , by use of Lemma 2, the previous equa-tion can be rewritten as

(37)Substituting (35)(37) into (34), such that

(38)Substituting (31) into (38), we get

(39)where . If we pick

, we get

By Definition 4, we propose the adaptive fuzzy robusttracking controller such that the requirement of ISpS forsystem can be satisfied with the functions and

of class . By Definition 3 and the (12), we canget a gain function of system as follows:

where .For system , it is a static system such that we have

(40)Then, the gain function for system is .According to the requirement of small gainTheorem 1, we can get


Owing to , the condition of the small gaintheorem 1 can be satisfied by choosing , so that it can beproven that the composite closed-loop system is ISpS. There-fore, a direct application of Definition 3 yields that the com-posite closed-loop system has bounded solutions over [0, ).More precisely, there exist a class -function and a posi-tive constant such that

This, in turn, implies that the tracking error is boundedover [0, ). By Proposition 1, there exists an ISpS-Lyapunovfunction for the composite closed-loop system. By substituting(40) into (39), the ISpS-Lyapunov function is satisfied asfollows:

(41)

where . From(41), we get

It results that the solutions of composite closed-loop systemare uniformly ultimately bounded, and implies that, for any

, there exists a constant such thatfor all . The last statement of Theorem

2 follows readily since can be made arbitrarilysmall if the design parameters , , , , are chosenappropriately.

Remark 1: It is interesting to note that most of the availableadaptive fuzzy controllers in the literature are based on feedbacklinearization techniques, whose structures are usually taken theform with and be the estimatesof and , respectively, and be a new control variable.To avoid singularity problem when , several modifiedadaptive methods were provided by [44], [25], and [28]. In thispaper, the adaptive fuzzy robust tracking controller developedbefore has the following properties:

where and . According to those properties, it is easyto show that we does not require to estimate the unknown gainfunction . In such a way we can not only reduce the numberof parameters needed to be adapted on-line for and butalso avoid the possible controller singularity problem usuallymet with feedback linearization design when the adaptive fuzzycontrol is executed.

Remark 2: Since the function approximation propertyof fuzzy systems is only guaranteed within a compact set,the stability result proposed in this paper is semiglobal inthe sense that, for any compact set, there exists a controllerwith sufficiently large number of fuzzy rules such that allthe closed-loop signals are bounded when the initial statesare within this compact set. In practical applications, thenumber of fuzzy rules usually can not be chosen too large dueto the possible computation problem. This implies that thefuzzy system approximation capability is limited, that is, theapproximating accuracy in (22) for the estimated function

will be greater when chosen small number of fuzzy rules.However, we can choose appropriately the design parameters

, , , , to improve both stability and performance ofthe closed-loop systems.

V. APPLICATION EXAMPLES

Now, we will reveal the control performance of the proposedAFRTC via application examples. Two examples on designingtracking controller for pole-balancing robot system and ship au-topilot system are given in this section. The former has an un-known input gain function and the latter unknown inputgain constant . We shall find the adaptive fuzzy robust trackingcontrollers by following the design procedures given in the pre-vious section. Simulation results will be presented.

A. Pole-Balancing Robot SystemTo demonstrate the effectiveness of the proposed algorithms,

a pole-balancing robot is used for simulation. The Fig. 3 showsthe plant composed of a pole and a cart. The cart moves on therail tracks in horizontal direction. The control objective is to bal-ance the pole starting from an arbitrary condition by supplying asuitable force to the cart. The same case studied has been givenin [43]. The dynamic equations are described by

(42)

where

is the angular position from the equilibrium positionand . Suppose that the trajectory planning problemfor a weight-lifting operation is considered and this pole-bal-ancing robot system suffers from uncertainties and exogenousdisturbances. The desired angle trajectory is assumed hereto be . Here, denotes the mass of thependulum, is the mass of the vehicle, is the length of thependulum and is the applied force. Here, we use the parame-ters for simulations , , .


Fig. 3. Pole-balancing robot system.

Define five fuzzy sets for each , with labels (NL),(NM), (ZE), (PM), (PL) which are character-

ized by the following membership functions:

(43)

where .Twenty-five fuzzy rules for the fuzzy system are included in

the fuzzy rule bases. Hence, the function is approximatedby TS fuzzy system as follows:

(44)

where ...

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, can be defined

as (6).We select and , then the solution

of Lyapunov expression (21) is obtained by

If picking in (30), we can obtain the adaptive fuzzyrobust tracking controller for pole-balancing robot system asfollows:

(45)

(b)Fig. 4. Simulation results for proposed AFRTC algorithm in this paper.(a) Position of pole-balancing robot system (Solid line: actual position, Dashedline: reference position). (b) Control force.

Fig. 5. Simulation results for the adaptive parameters when employingAFRTC algorithm. (a) Adaptive parameter . (b) Adaptive parameter ^.

where

For the convenience of simulation, choose the initial condi-tion , , , . The simula-tion results are shown in Figs. 4 and 5.


Before presenting the outstanding advantages of AFRTCdeveloped in this paper, we will briefly review the controllaw proposed in [44] as follows:

(46)

where , , ,and . Then, we solve the Lya-

punov equation and obtain

where if (which is a constant specified by thedesigner), if , and .

Define five fuzzy sets the same as those in (43) for each ,, twenty-five fuzzy rules for the fuzzy systems and

, respectively, and singleton fuzzifier, the product infer-ence and the center-average defuzzification are used. Hence, thefunctions and can be approximated by the fuzzy sys-tems and where

with components

and

and the construction of is similar to .

(b)Fig. 6. Simulation results for Control algorithm in (46). (a) Position ofpole-balancing robot system (Solid line: actual position, Dashed line: referenceposition). (b) Control force.

In Wang [44], use the following adaptive law to adjust pa-rameter vector ; see (47) at the bottom of the page, where theprojection operation is defined as

In [44], use the following adaptive law to adjust parametervector :

ifif (48)

Here, for the parameters , , and , please refer to Wang[44]. The simulation results are shown in Fig. 6.

Fig. 7 shows the simulation results of tracking errors by useof the proposed AFRTC and the controller given in (46), respec-tively. From the results, we can see that the control performancesare almost the same. Hence, we can state that the AFRTC satis-fies the following advantages that have been described in Sec-tion IV: only one function is needed to be approximatedby TS fuzzy systems and no matter how many states in thesystem are investigated and how many rules in the fuzzy systemare used, only two parameters are needed to be adapted on-linein AFRTC. However, for the traditional methodology (e.g., thecontrol law proposed in [44]), even based on five fuzzy sets foreach state variable and singleton fuzzy model aforementioned,there are 50 parameters needed to be adapted online for thefuzzy system and when the fuzzy logic con-troller is implemented. And the traditional methodology cancause the increase of the number of parameters needed to be

if( ) or ( and )if( and ) (47)


(b)Fig. 7. Simulation results of the tracking errors. (a) Proposed AFRTCalgorithm. (b) Control algorithm in (46).

adapted exponentially with that of number of state variablesor fuzzy sets. The computational complexity can be lesseneddramatically and the learning time can be reduced vastly whenusing AFRTC developed in this paper. Then AFRTC has thepotential to provide uncomplicated, reliable and easy-to-under-stand solutions for a large variety of nonlinear control tasks evenfor higher order systems.

B. Ship Autopilot System

Many of the present generation of autopilots installed in shipsare designed for the course keeping. They aim at maintainingthe ship on a predetermined course and thus require directionalinformation. Developments in the last 20 y include variants ofthe analogue proportional-integral-derivative (PID) controller.In the recent years, some sophisticated autopilots are proposedbased on advanced control engineering concepts whereby thegain settings for the proportional, derivative and integral termsof heading are adjusted automatically to suit the dynamics ofthe ship and environmental conditions such as model referenceadaptive control [46], self-tuning [47], optimal [48], theo-ries [49] and adaptive robust fuzzy control [50].

In this paper, the adaptive fuzzy robust tracking controllerproposed above will be used for designing ship autopilot. Be-fore considering the designs of the autopilots, it is of interest todescribe the dynamics of the ship. The mathematical model re-lating the rudder angle to the heading of the ship is found tobe of the form

(49)

where ( ) and (s) are parame-ters which are function of ships constant forward velocity andits length. is a nonlinear function of . The functioncan be found from the relationship between and in steady

TABLE IFUZZY IFTHEN RULES

state such that . An experiment known as thespiral test has shown that can be approximated by

(50)where and are real valued constants.

In normal steering, a ship often makes only small deviationsfrom its desired direction. The coefficient in the (50) could beequal to 0 such that a linear model is used as the design model fordesigning the autopilot, but in this paper, let both and be notequal to 0, a nonlinear model (50) is used as the design modelfor designing the adaptive fuzzy robust controller as following.

Let the state variables be , and control variablebe , then the (49) can be rewritten in the statespace form

(51)

Without loss of generality, we assume that the functionin the (50) can be defined in the function which is un-known with a continuous complicated formulation system func-tion, TS fuzzy system can be constructed to approximate thefunction by the following nine fuzzy IFTHEN rulesin Table I.

In Table I, we select , .denotes the fuzzy set Positive, denotes the fuzzy set Zeroand denotes the fuzzy set Negative. They can be character-ized by the membership functions as follows

For the previous example, we may use fuzzy sets on the normal-ized universes of discourse as shown in Fig. 8.

Using the center average defuzzifier and the product infer-ence engine, the fuzzy system is obtained as follows:

(52)

where ...

and ...

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.

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. can be defined

as the (6).To demonstrate the availability of the proposed scheme, we

take a general cargo ship with the length 126 m and the displace-ment 11 200 tons as an example for simulation.


Fig. 8. Fuzzy member ship functions.

The reference model is chosen so as to represent somewhatrealistic performance requirement as

(53)

where specifies the desired system performance for theship heading .

When let , and , then thesolution of Lyapunov expression (21) is obtained by

If the gain is , the adaptive fuzzy robust trackingcontrol scheme can be obtained for ship autopilot as follows:

(54)

where ,

Generally, the first step in the controller design procedure isconstruction of a truth model of the dynamics of the processto be controlled. The truth model is a simulation model that in-cludes all the relevant characteristics of the process. The truthmodel is too complicated for use in the controller design. Thus,we need develop a simplified model called the design model thatcan be used to design the controller. In this paper, we take the(29) as the design model. In order to verify the performance ofthe ATRFC proposed above by use of simulation, a truth model

(b)Fig. 9. Simulation results for AFRTC algorithm when employed for a cargoship. (a) Ship heading [ship heading (solid line) and reference course (dashed line)]. (b) Rudder angle u.

Fig. 10. Simulation results for AFRTC algorithm when employed for a cargoship: Tracking error [deg].

which accurately represents the characteristics of the ship isused as follows:

(55)

where and are the velocity components of the ship andis the angular rate of yaw angle with respect to time, and ,

, , , and are mass, added mass, inertia momentand added inertia moment of ship. , are the components ofhydrodynamic force acting on ship in the bodyfixed axis systemand is a moment by the aforementioned forces. The subscriptsin the left of the (55) mean that denotes the bare hull, isscrew, is rudder, is wind and wave which produce the ex-ternal forces and moments acting on ship. The methods of cal-culating external forces and moments above have been proposedin [51].

Simulation results based on the Matlab Simulink package areshown in Figs. 911.


(b)Fig. 11. Simulation results for AFRTC algorithm when employed for a cargoship. (a) Adaptive parameter . (b) Adaptive parameter ^.

The controller is trained by applying a signal , whichchanges its value in the interval (0, 30 ) every 300 s. In Fig. 9,the initial state of the reference signal begins with 20 and theinitial state of the ship heading is 0 . After 100 s, the plantoutput and the reference signal are practically indistinguishable,and it can be seen that the plant output converges rapidly tothe reference signal in Fig. 10. Fig. 11 shows the adaption ofparameters in AFRTC algorithm.

VI. CONCLUSION

In this paper, the tracking control problem has been con-sidered for a class of nonlinear uncertain systems with theunknown system function and unknown gain function, andTS type fuzzy logic systems have been used to approximateunknown system function and an AFRTC algorithm, thatcan guarantee the closed-loop in the presence of nonrepeat-able uncertainties is uniformly ultimately bounded, and theoutput tracking error of the system can be steered to a smallneighborhood around , has been achieved by useof the ISS and general small-gain approach. The outstandingfeatures of the algorithm proposed in this paper are that it canavoid the possible controller singularity problem in some ofexisting adaptive control schemes with feedback linearizationtechniques and the adaptive mechanism with minimal learningparameterizations, e.g., no matter how many states in thesystem are investigated and how many rules in the fuzzy systemare used, only two parameters, which are a parameter of fuzzysystem and a bounded value including approximation errorand disturbance, are needed to be adapted on-line, so that thecomputation load of the algorithm can be reduced and it isconvenient to realize this algorithm in engineering. In orderto make study of efficiency of the algorithm proposed in thispaper, it has been applied to a pole-balancing robot system andship autopilot system. In accordance with unknown parametersof the ship model and the structure of uncertain function of

the system, the TS fuzzy system is made to approximate theuncertain function, then an AFRTC scheme for ship autopilotis proposed. Simulation results have shown the effectiveness ofthe control scheme.

ACKNOWLEDGMENT

The authors would like to thank the anonymous referees fortheir valuable comments and criticism which have helped to im-prove this paper.

REFERENCES[1] P. Kokotovic and M. Arcak, Constructive nonlinear control: A historical

perspective, Automatica, vol. 37, pp. 637662, 2001.[2] S. S. Sastry and A. Isidori, Adaptive control of linearization systems,

IEEE Trans. Automat. Contr., vol. 34, pp. 11231131, 1989.[3] A. R. Teel, R. R. Kadiyala, P. V. Kokotovic, and S. S. Sastry, Indirect

techniques for adaptive input-output linearization of nonlinear systems,Int. J. Control, vol. 53, no. 1, pp. 193222, 1991.

[4] D. G. Taylor, P. V. Kokotovic, R. Marino, and I. Kanellakopoulos,Adaptive regulation of nonlinear systems with unmodeled dynamics,IEEE Trans. Automat. Contr., vol. 34, pp. 405421, 1989.

[5] I. Kanellakopoulos, P. V. Kokotovic, and A. S. Morse, Systematic de-sign of adaptive controllers for feedback linearizable systems, IEEETrans. Automat. Contr., vol. 36, pp. 12411253, 1991.

[6] R. Marino and P. Toper, Global adaptive outpout-feedback control ofnonlinear systems, part I: Linear parameterization; part II: Nonlinearparameterization, IEEE Trans. Automat. Contr., vol. 38, pp. 1749, Jan.1993.

[7] M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adap-tive Control Design. New York: Wiley, 1995.

[8] M. J. Corless and G. Leitmann, Continuous state feedback guranteeinguniform ultimate boundedness for uncertain dynamic systems, IEEETrans. Automat. Contr., vol. AC-26, pp. 11391144, 1981.

[9] B. R. Barmish and G. Leitmann, On ultimate boundedness control ofuncertain systems in the abacence of matching assumptions, IEEETrans. Automat. Contr., vol. AC-27, pp. 153158, Feb. 1982.

[10] Y. H. Chen, Design of robust controllers for uncertain dynamical sys-tems, IEEE Trans. Automat. Contr., vol. 33, pp. 487491, 1988.

[11] V. I. Utkin, Sliding Modes in Control Optimization. New York:Springer-Verlag, 1992.

[12] A. S. I. Zinober, Deterministic Control of Uncertain ControlSystem. London, U.K.: Peter Peregrinus, 1990.

[13] M. M. Polycarpou and P. A. Ioannou, Identification and Control UsingNeural Network Models: Design and Stability Analysis, Dept. Elect.Eng. Syst., Univ. Southern California, Los Angeles, CA, Tech. Rep.910901, 1991.

[14] S. S. Sanner and J. E. Slotine, Gaussian networks for direct adaptivecontrol, IEEE Trans. Automat. Contr., vol. 34, pp. 11231131, 1992.

[15] F. C. Chen and C. C. Liu, Adaptive controlling nonlinear contin-uous-time systems using multilayer neural networks, IEEE Trans.Automat. Contr., vol. 39, pp. 13061310, 1994.

[16] A. Yesidirek and F. L. Lewis, Feedback linearization using neural net-works, Automatica, vol. 31, pp. 16591664, 1995.

[17] S. Fabri and V. Kadirkamanathan, Dynamic structure neural networksfor stable adaptive control of nonlinear systems, IEEE Trans. NeuralNetworks, vol. 7, pp. 11511167, 1996.

[18] S. Jagannathan, S. Commuri, and F. L. Lewis, Feedback linearizationusing CMAC neural networks, Automatica, vol. 34, no. 5, pp. 547557,1998.

[19] S. S. Ge, T. H. Lee, and C. J. Harris, Adaptive Neural Network Controlof Robotic Manipulators. Singapore: World Scientific, 1998.

[20] L. X. Wang, Fuzzy systems are universal approximators, Proc. IEEE.Int. Conf. Fuzzy Systems, pp. 11631170, 1992.

[21] L. X. Wang and J. M. Mendel, Fuzzy basis functions universal approxi-mation, and orthogonal least-squares learning, IEEE Trans. Neural Net-works, vol. 3, pp. 807814, 1992.

[22] L. X. Wang, Stable adaptive fuzzy control of nonlinear systems, IEEETrans. Fuzzy Syst., vol. 1, pp. 146155, Feb. 1993.

[23] C. Y. Su and Y. Stepanenko, Adaptive control of a class of nonlinearsystems with fuzzy logic, IEEE Trans. Fuzzy Syst., vol. 2, pp. 285294,Apr. 1994.


[24] Y. S. Yang, X. L. Jia, and C. J. Zhou, Robust adaptive fuzzy control fora class of uncertain nonlinear systems, presented at the 4th Int. Conf.Intelligent Techniques Soft Computing Nuclear Science Engineering,Bruges, Belgium, 2000.

[25] J. T. Spooner and K. M. Passino, Stable adaptive control using fuzzysystems and neural networks, IEEE Trans. Fuzzy Syst., vol. 4, pp.339359, 1996.

[26] K. Fischle and D. Schroder, An improved stable adaptive fuzzy controlmethod, IEEE Trans. Fuzzy Syst., vol. 7, pp. 2740, Feb. 1999.

[27] B. S. Chen, C. H. Lee, and Y. C. Cheng, H tracking design of uncer-tain nonlinear SISO systems: Adaptive fuzzy approach, IEEE Trans.Fuzzy Syst., vol. 4, pp. 3243, Feb. 1996.

[28] Y. C. Chang, Adaptive fuzzy-based tracking control for nonlinear SISOsystems via VSSH approaches, IEEE Trans. Fuzzy Syst., vol. 9, pp.278292, Apr. 2001.

[29] T. Takagi and M. Sugeno, Fuzzy identification of systems and its ap-plications to modeling and control, IEEE Trans. Syst., Man., Cybern.,vol. 15, pp. 116132, Apr. 1985.

[30] L. X. Wang, A Course in Fuzzy Systems and Control. Upper SaddleRiver, NJ: Prentice-Hall, 1997.

[31] E. D. Sontag, Smooth stabilization implies coprime factorization,IEEE Trans. Automat. Contr., vol. 34, pp. 14111428, 1989.

[32] Z. P. Jiang, A. Teel, and L. Praly, Small-gain theorem for ISS systemsand applications, Math. Control, Signals Syst., vol. 7, pp. 95120, 1994.

[33] E. D. Sontag, Further results about input-state stabilization, IEEETrans. Automat. Contr., vol. 35, pp. 473476, 1990.

[34] E. D. Sontag and Y. Wang, On characterizations of input-to-statestability property with respect to compact sets, in Prep. IFACNOLCOS95, Tahoe City, 1995, pp. 226231.

[35] E. D. Sontag, On the input-to-state stability property, Eur. J. Control,vol. 1, pp. 2436, 1995.

[36] Z. P. Jiang and L. Praly, Technical results for the study of robustness ofLagrange stability, Syst. Control Lett., vol. 23, pp. 6778, 1994.

[37] J. Tsinias, Sontags input-to-state stability condition and global stabi-lization using state detection, Syst. Control Lett., vol. 20, pp. 219226,1993.

[38] L. Praly and Z. P. Jiang, Stabilization by output feedback for systemswith ISS inverse dynamics, Syst. Control Lett., vol. 21, pp. 1933, 1993.

[39] L. Praly and Y. Wang, Stabilization in spite of matched unmodeled dy-namics and an equivalent definition of input-to-state stability, Math.,Control, Signals Syst., vol. 9, pp. 133, 1996.

[40] Z. P. Jiang and I. Mareels, A small gain control method for nonlinearcascaded systemns with dynamic uncertainties, IEEE Trans. Automat.Contr., vol. 42, pp. 292308, 1997.

[41] H. K. Khalil, Nonlinear Systems. New York: Macmillan, 1996.[42] M. M. Polyvearpou and P. A. Ioannou, A robust adaptive nonlinear

control design, Automatica, vol. 32, pp. 423427, 1995.[43] G. C. Hwang and S. C. Lin, A stability approach to fuzzy control design

for nonlinear systems, Fuzzy Sets Syst., vol. 40, pp. 279287, 1990.[44] L. X. Wang, Adaptive Fuzzy Systems and Control. Upper Saddle River,

NJ: Prentice-Hall, 1994.

[45] Y. S. Yang, Robust control for uncertain systems and its application toship motion control, Ph.D. dissertation, Dalian Maritime Univ., Dalian,China, 2000.

[46] J. Van Amerongen, Adaptive steering of ships: A model referenceapproach to improved manocuvring and economical course-keeping,Ph.D. dissertation, Delft University Technol., Delft, The Netherlands,1982.

[47] M. Mort and D. A. Linkens, Self-tuning controllers steering automaticcontrol, presented at the Proc. Symp. Ship Steering Automatic Control,Genova, Italy, 1980.

[48] J. K. Zuidweg, Optimal and sub-optimal feedback in automatic track-keeping system, in Proc. 6th Ship Control Systems Symp., vol. 3, Ot-tawa, Canada, 1981.

[49] N. A. Fairbairn and M. J. Grimble, H marine autopilot design forcourse-keeping and course-changing, in Proc. 9th Ship Control SystemsSymp., vol. 3, Bethesda, MD, 1990.

[50] Y. S. Yang, C. J. Zhou, and X. L. Jia, Robust adaptive fuzzy control andits application to ship roll stabilization, Int. J. Inform. Sci., vol. 142, pp.177194, 2002.

[51] Y. S. Yang, Study on ship manoeuvring mathematical model in shiphan-dling simulator, presented at the Int. Conf. MARSIM96, Copenhagen,Denmark, 1996.

Yansheng Yang was born in Jiangsu Province, P. R.China, in 1957. He received the B.S., M.S., and Ph.D.degrees from the Department of Navigation, DalianMaritime University, Dalian, China, in 1982, 1985,and 2000, respectively.

From 1995 to 1996, he was a Visiting Scholar atHiroshima University, Hiroshima, Japan. In 1998, hebecame a Professor with the Navigation College atDalian Maritime University. His research interest in-cludes robust control and fuzzy control for nonlinearsystem and their applications in marine control.

Junsheng Ren was born in Henan province, P.R.China, in 1976. He received the B.S. degree from theNavigation College at Dalian Maritime University,Dalian, in 1999. He is currently working toward thePh.D. degree at the same university.

His research interests includes ship motion simula-tion, robust control theory, and its application to shipmotion.

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Adaptive Fuzzy Robust Tracking Controller Design