IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003 783 Adaptive Fuzzy Robust Tracking Controller Design via Small Gain Approach and Its Application Yansheng Yang and Junsheng Ren Abstract—A novel adaptive fuzzy robust tracking control (AFRTC) algorithm is proposed for a class of nonlinear systems with the uncertain system function and uncertain gain function, which are all the unstructured (or nonrepeatable) state-dependent unknown nonlinear functions arising from modeling errors and external disturbances. The Takagi–Sugeno type fuzzy logic systems are used to approximate unknown uncertain functions and the AFRTC algorithm is designed by use of the input-to-state stability approach and small gain theorem. The algorithm is highlighted by three advantages: 1) the uniform ultimate bound- edness of the closed-loop adaptive systems in the presence of nonrepeatable uncertainties can be guaranteed; 2) the possible controller singularity problem in some of the existing adaptive control schemes met with feedback linearization techniques can be removed; and 3) the adaptive mechanism with minimal learning parameterizations can be obtained. The performance and limitations of the proposed method are discussed. The uses of the AFRTC for the tracking control design of a pole-balancing robot system and a ship autopilot system to maintain the ship on a predetermined heading are demonstrated through two numerical examples. Simulation results show the effectiveness of the control scheme. Index Terms—Adaptive robust tracking, fuzzy control, input-to- state stability (ISS), nonlinear systems, small gain theorem. I. INTRODUCTION I N RECENT years, interest in designing robust tracking control for uncertain nonlinear systems has been ever increasing, and many significant research attentions have been attracted. Most results addressing this problem are available in the control literature, e.g., Kokotovic and Arcak [1] and references therein. And many powerful methodologies for designing tracking controllers are proposed for uncertain nonlinear systems. The uncertain nonlinear systems may be subjected to the following two types of uncertainties: structured uncertainties (repeatable unknown nonlinearities), which are linearly parameterized and referred to as parametric uncertainties, and unstructured uncertainties (nonrepeatable unknown nonlinearities), which are arising from modeling errors and external disturbances. To handle the parametric uncertainties, adaptive control method, which has undergone rapid 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 Doctoral Program of Higher Education under Grant 20020151005, the Science Founda- tion under Grant 95-06-02-22, and the Young Investigator Foundation under Grant 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: ysyang@mail.dlptt.ln.cn). Digital Object Identifier 10.1109/TFUZZ.2003.819837 As for unstructured uncertainties, if there is a prior knowledge of the bounded functions, deterministic robust control method, e.g., [8]–[12] can be used. Unfortunately, in industrial control environment, there are some controlled systems with the unstructured uncertainties where none of prior knowledge of the bounded functions is available, then the adaptive control method and the deterministic robust control method can not be used to design controller for those systems. A solution to that problem is presented that the neural networks (NNs) are used to approximate the continuous unstructured uncertain functions in the systems and Lyapunov’s stability theory is applied in designing adaptive NN controller. Several stable adaptive NN control approaches are developed by [13]–[19] which guarantee uniform ultimate boundedness in the presence of both unstructured uncertainties and unknown nonlinearities. As an alternative to NN control approaches, the intensive research has been carried out on fuzzy control for uncertain nonlinear systems. The fuzzy systems are used to uniformly approximatetheunstructureduncertainfunctionsinthedesigned system by use of the universal approximation properties of the certain classes of fuzzy systems, which are proposed by [20] and [21], and a Lyapunov based learning law is used, and several stable adaptive fuzzy controllers that ensure the stability of the overall system are developed by [22]–[26]. Recently, an adaptive fuzzy-based controller combined with VSS and control 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 learning laws when there are many state variables in the designed system and many rule bases have to be used in the fuzzy system for approximating the nonlinear uncertain functions, so that the learning times tend to become unacceptably large for the systems of higher order and time-consuming process is unavoidable 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 that problem. A new systematic procedure is developed for the synthesis of stable adaptive fuzzy robust controller for a class of continuous uncertain systems, and Takagi–Sugeno (T–S) type fuzzy logic systems [29] are used to approximate the un- known unstructured uncertain functions in the systems and the adaptive mechanism with minimal learning parameterizations can be achieved by use of input-to-state stability (ISS) theory first proposed by Sontag [31] and small gain approach given in [32]. The outstanding features of the algorithm proposed in the paper are: i) that only one function is needed to be approximated by T–S fuzzy systems and no matter how many states 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 needed to be adapted on-line, such that the burdensome computation of the algorithm can be lightened increasingly and it is convenient to realize this algorithm in engineering; and 2) the possible controller singularity problem in some of the existing adaptive control schemes met with feedback linearization techniques can be avoided. This paper is organized as follows. In Section II, we will give the problem formulation, the description of a class of nonlinear systems and tracking control problem of nonlinear systems. Section III contains some needed definitions of ISS, small gain theorem and preliminary results. In Section IV, a systematic procedure for the synthesis of adaptive fuzzy robust tracking controller (AFRTC) is developed. In Section V, two application examples for designing the tracking control for the pole-balancing robot system and the ship autopilot system by use of the AFRTC are included and numerical simulation results are presented. The final section contains conclusions. II. P ROBLEM FORMULATION A. System Description Consider the th-order uncertainnonlinear systemsof thefol- lowing form: (1) where and represent the control input and the output of thesystem, respectively. is comprised of the states which are assumed to be available, the integer denotes the dimension of the system. and are unknown smooth uncertain functions and may contain non- repeatable nonlinearities. is the disturbance, unknown but bounded, e.g., , where is an unknown constant. Throughout this paper, the following assumption is made on (1). Assumption 1: The sign of is known, and there exists a constant such that , . This assumption implies that smooth function is strictly either positive or negative. From now onwards, without loss of generality, we shall assume , . As- sumption 1 is reasonable because being away from zero is the controllable conditions of system (1). It should be em- phasized that the low bound is only required for analytical purposes, its true value is not necessarily known. Some stability is needed to proceed. Definition 1: It is said that the solution of (1) is uniformly ultimately bounded (UUB) if for any , a compact subset of , and all , there exists an and a number such that for all . We represent as any suitable vector norm. In this paper, vector norm is Euclidean, i.e., and given a matrix , matrix norm is defined by where denotes the operation of taking the max- imum (minimum) eigenvalue. The norm denoted by throughout this paper unless specified explicitly, is nothing but the vector two-norm over the space defined by stacking the ma- trix columns into a vector, so that it is compatible withthe vector two-norm, i.e., . The primary goal of this paper is to track a given reference signal while keeping the states and control bounded. That is, the output tracking error should be small. The given reference signal is assumed to be bounded and has 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 thispaper, we present a methodfor 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 the form (3) (4) where is theknown fuzzy base functions.In sucha way that all the solutions of the closed-loop system (2)–(4) are uniformly ultimately bounded. Furthermore, the output tracking error of the system can be steered to a small neighborhood of origin. B. T–S Fuzzy Systems In 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 to be the function space such that is continuous. A fuzzy system can be employed to approximate the function in order to design the adaptive fuzzy robust control law, thus the configu- ration of T–S type fuzzy logic system called T–S fuzzy system for short [29] and approximation theorem are discussed first as follows. Consider a T–S fuzzy system to uniformly approximate a continuous multidimensional function that has a com- plicated formulation,where is inputvector with independent . The domain of is . It fol- lows that the domain of is In order to construct a fuzzy system, the interval [ ]is divided into subintervals On each interval , continuous input fuzzy sets, denoted by , are defined to 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, T–S fuzzy system can be constructed by the fol- lowing fuzzy rules: where , , are the unknown constants. The product fuzzy inference is employed to evaluate the ANDs in the fuzzy rules. After being defuzzified by a typical center average defuzzifier, the output of the fuzzy system is (5) where and , which is called a fuzzy base function. When the membership function in is denoted by some type of membership function, is a known continuous function. So, restructuring (5) as follows: Let , , . . . . . . . . . . . . , then the (5) can be easily rewritten as (6) where , and . . . . . . . . . . . . . When the fuzzy systemis used to approximate thecontinuous function, two questions of interest may be asked: whether there exists a fuzzy system to approximate any nonlinear function to an arbitrary accuracy? how to determine the parameters in the fuzzy system if such a fuzzy system does exist. The following lemma [30] gives a positive answer to the first question. Lemma 1: Suppose that the input universal of discourse is a compactset in .Then, foranygivenreal continuousfunction on and , there exists a fuzzy system in the form of expression (6) such that (7) III. M ATHEMATICAL PRELIMINARIES The concept of ISS and ISS-Lyapunov function due to Stontag [31], [33] and Sontag and Wang [34] have recently been used in various control problems such as nonlinear stabilization, 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 reviewed in the following. First, we begin with the definitions of class , and functions which are standard in the stability literature; see [41]. Definition 2: • A function is said of class if it is continuous, strictly increasing and . It is of class if it is of class and is unbounded. • A function is said of class if, for each , is of class , and, for each , is strictly decreasing and satisfies , and is a class function if and only if there exist two class functions and such that We consider the following system: (8) where is the state and is the input. For this system, we give the definition of input-to-state stable in the following. Definition 3: For (8), it is said to be input-to state practically stable (ISpS) if there exist a function of class , called the nonlinear gain, and a function of class such that, for any initial condition , each measurable essentially bounded control 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 stands for the supremum norm. When in (9), the ISpS property collapses to the ISS property 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) 786 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003 Fig. 1. Feedback connection of composite systems. • there exist two constants , and a class function such that (14) When (14) holds with , the function is referred to as an 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 following properties 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 variant which 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 supremum norm, each in the supremum norm, each and each , 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. D ESIGN OF ADAPTIVE FUZZY ROBUST TRACKING CONTROL Using the pole-placement approach, we consider a term where , the ’s are chosen such that all roots 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 . . . . . . . . . . . . . . . . Because is stable, a positive–definite solution of the 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 feedback linearization can be used to design a well-defined controller, which is usually given in the form of for some auxiliary control input with being nonzero for all time, such that the resulting closed-loop system can be shown to achieve a satisfactory tracking per- formance. However, in many practical control systems, plant uncertainties that contain structured (or parametric) uncertain- ties and unstructured uncertainties (or nonrepeatable uncer- tainties) are inevitable. Hence, both and may not be available directly in the robust control design. Obtaining a simple control algorithm as before is impossible. Moreover, if any adaptation scheme is implemented to estimate and as and respectively, the simple control algorithm aforementioned can be also used for substituting and for and , so the extra precaution is required to guarantee that for all time. At the present stage, no effective method is available in the litera- ture. In this paper, we develop a semi-globally stable adaptive fuzzy robust controller which does not require to estimate the unknown function , and therefore avoids the possible controller singularity problem. In this paper, the effects due to plant uncertainties and external disturbances will be considered simultaneously. The philosophy of our tracking controller design is expected that T–S fuzzy approximators equipped with adaptive algorithms are introduced first to learn the behaviors of uncertain dy- namics. Here, only uncertain function is needed to be considered. For is an unknown continuous function, by Lemma 1, T–S fuzzy system with input vector for some compact 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. YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 787 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 equation can 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 the controller . In (25), is an unknown, and there exist some parameters with boundedness. According to these prop- erties, an adaptive fuzzy robust tracking control algorithm will be proposed, which not only gives the controller to make the system meet ISpS condition but also the online adaptive law for and the other parameters in the (25). For this purpose, we will discuss it in the following. Construct an adaptive fuzzy robust tracking controller as follows: (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 the following. Substituting (27) into (25) yields (28) Based on the aforementioned condition, we can get (29) where , , , , , and . denotes the largest term with unknown constant in all boundedness. In order 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 gain of to be chosen later on. The adaptive laws for and are now chosen as (31) where , , 2 are the updating rates chosen by designer, and , ,2, and are design con- stants. Adaptive laws (31) incorporate leakage term based on a variant of the -modification proposed by Polycarpou and Ioannou [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 by T–S fuzzy system. If we pick and in (21), then the control scheme (30) with adaptive laws (31) is an AFRTC which can make all the solutions ( )of the derived closed loop system uniformly ultimately bounded. Furthermore, given any and bounds on and ,we can 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, let the constant and set as the input of the system , to prove the satisfaction of ISpS for the system by 788 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003 use of the adaptive fuzzy robust tracking controller, and then to prove uniform ultimate boundedness of the composite of two systems with the feedback system by use of small gain 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 robust tracking controller such that the requirement of ISpS for system can be satisfied with the functions and of class . By Definition 3 and the (12), we can get 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 gain Theorem 1, we can get YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 789 Owing to , the condition of the small gain theorem 1 can be satisfied by choosing , so that it can be proven 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 bounded over [0, ). By Proposition 1, there exists an ISpS-Lyapunov function for the composite closed-loop system. By substituting (40) into (39), the ISpS-Lyapunov function is satisfied as follows: (41) where . From (41), we get It results that the solutions of composite closed-loop system are uniformly ultimately bounded, and implies that, for any , there exists a constant such that for all . The last statement of Theorem 2 follows readily since can be made arbitrarily small if the design parameters , , , , are chosen appropriately. Remark 1: It is interesting to note that most of the available adaptive fuzzy controllers inthe literature arebased on feedback linearization techniques, whose structures are usually taken the form with and be the estimates of and , respectively, and be a new control variable. To avoid singularity problem when , several modified adaptive methods were provided by [44], [25], and [28]. In this paper, the adaptive fuzzy robust tracking controller developed before has the following properties: where and .According to those properties, it is easy to show that we does not require to estimate the unknown gain function . In such a way we can not only reduce the number of parameters needed to be adapted on-line for and but also avoid the possible controller singularity problem usually met with feedback linearization design when the adaptive fuzzy control is executed. Remark 2: Since the function approximation property of fuzzy systems is only guaranteed within a compact set, the stability result proposed in this paper is semiglobal in the sense that, for any compact set, there exists a controller with sufficiently large number of fuzzy rules such that all the closed-loop signals are bounded when the initial states are within this compact set. In practical applications, the number of fuzzy rules usually can not be chosen too large due to the possible computation problem. This implies that the fuzzy system approximation capability is limited, that is, the approximating 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 of the closed-loop systems. V. A PPLICATION EXAMPLES Now, we will reveal the control performance of the proposed AFRTC via application examples. Two examples on designing tracking 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 input gain constant . We shall find the adaptive fuzzy robust tracking controllers by following the design procedures given in the pre- vious section. Simulation results will be presented. A. Pole-Balancing Robot System To demonstrate the effectiveness of the proposed algorithms, a pole-balancing robot is used for simulation. The Fig. 3 shows the plant composed of a pole and a cart. The cart moves on the rail tracksin horizontal direction. Thecontrol objective isto bal- ance the pole starting froman arbitrary condition by supplying a suitable force to the cart. The same case studied has been given in [43]. The dynamic equations are described by (42) where is the angular position from the equilibrium position and . Suppose that the trajectory planning problem for a weight-lifting operation is considered and this pole-bal- ancing robot system suffers from uncertainties and exogenous disturbances. The desired angle trajectory is assumed here to be . Here, denotes the mass of the pendulum, is the mass of the vehicle, is the length of the pendulum and is the applied force. Here, we use the parame- ters for simulations , , . 790 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003 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 approximated by T–S fuzzy system as follows: (44) where . . . , . . . . . . , 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 fuzzy robust tracking controller for pole-balancing robot system as follows: (45) (b) Fig. 4. Simulation results for proposed AFRTC algorithm in this paper. (a) Position of pole-balancing robot system (Solid line: actual position, Dashed line: reference position). (b) Control force. Fig. 5. Simulation results for the adaptive parameters when employing AFRTC 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. YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 791 Before presenting the outstanding advantages of AFRTC developed in this paper, we will briefly review the control law 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 the designer), 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, the functions 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 of pole-balancing robot system (Solid line: actual position, Dashed line: reference position). (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 the projection operation is defined as In [44], use the following adaptive law to adjust parameter vector : if if (48) Here, forthe parameters , , and , pleaserefer to Wang [44]. The simulation results are shown in Fig. 6. Fig. 7 shows the simulation results of tracking errors by use of the proposed AFRTC and the controller given in (46), respec- tively. Fromthe results,we cansee thatthe controlperformances are 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 approximated by T–S fuzzy systems and no matter how many states in the system are investigated and how many rules in the fuzzy system are used, only two parameters are needed to be adapted on-line in AFRTC. However, for the traditional methodology (e.g., the control law proposed in [44]), even based on five fuzzy sets for each state variable and singleton fuzzy model aforementioned, there are 50 parameters needed to be adapted online for the fuzzy system and when the fuzzy logic con- troller is implemented. And the traditional methodology can cause the increase of the number of parameters needed to be if( )or( and ) if( and ) (47) 792 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003 (b) Fig. 7. Simulation results of the tracking errors. (a) Proposed AFRTC algorithm. (b) Control algorithm in (46). adapted exponentially with that of number of state variables or fuzzy sets. The computational complexity can be lessened dramatically and the learning time can be reduced vastly when using AFRTC developed in this paper. Then AFRTC has the potential to provide uncomplicated, reliable and easy-to-under- stand solutions fora large variety of nonlinearcontrol tasks even for higher order systems. B. Ship Autopilot System Many of thepresent generation of autopilots installedin ships are designed for the course keeping. They aim at maintaining the ship on a predetermined course and thus require directional information. Developments in the last 20 y include variants of the analogue proportional-integral-derivative (PID) controller. In the recent years, some sophisticated autopilots are proposed based on advanced control engineering concepts whereby the gain settings for the proportional, derivative and integral terms of heading are adjusted automatically to suit the dynamics of the ship and environmental conditions such as model reference adaptive control [46], self-tuning [47], optimal [48], theo- ries [49] and adaptive robust fuzzy control [50]. In this paper, the adaptive fuzzy robust tracking controller proposed above will be used for designing ship autopilot. Be- fore considering the designs of the autopilots, it is of interest to describe the dynamics of the ship. The mathematical model re- lating the rudder angle to the heading of the ship is found to be of the form (49) where ( ) and (s) are parame- ters which are function of ship’s constant forward velocity and its length. is a nonlinear function of .The function can be found from the relationship between and in steady TABLE I F UZZY IF–THEN RULES state such that . An experiment known as the “spiral test” has shown that can be approximated by (50) where and are real valued constants. In normal steering, a ship often makes only small deviations from its desired direction. The coefficient in the (50) could be equal to0 suchthat alinear modelis used asthe designmodel for designing the autopilot, but in this paper, let both and be not equal to 0, a nonlinear model (50) is used as the design model for designing the adaptive fuzzy robust controller as following. Let the statevariables be , and controlvariable be , then the (49) can be rewritten in the state–space form (51) Without lossof generality,we assumethat thefunction in the (50) can be defined in the function which is un- known with a continuous complicatedformulation system func- tion, T–S fuzzy system can be constructed to approximate the function by the following nine fuzzy IF–THEN rules in Table I. In TableI, weselect , . denotes the fuzzy set “Positive”, denotes the fuzzy set “Zero” and 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 . . . . . . . can be defined as the (6). To demonstrate the availability of the proposed scheme, we take ageneral cargo shipwith the length 126m and the displace- ment 11 200 tons as an example for simulation. [...]... X Wang, “Stable adaptive fuzzy control of nonlinear systems,” IEEE Trans Fuzzy Syst., vol 1, pp 146–155, Feb 1993 [23] C Y Su and Y Stepanenko, Adaptive control of a class of nonlinear systems with fuzzy logic,” IEEE Trans Fuzzy Syst., vol 2, pp 285–294, Apr 1994 YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN [24] Y S Yang, X L Jia, and C J Zhou, Robust adaptive fuzzy control for... 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T–S type fuzzy logic systems have been used to approximate unknown system function and an AFRTC algorithm, that can guarantee the closed-loop in the presence of nonrepeatable uncertainties is uniformly ultimately bounded, and the output tracking error of the system can be steered to a small , has been achieved by use neighborhood around of the ISS and general small- gain approach The outstanding features...YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 793 Fig 8 Fuzzy member ship functions The reference model is chosen so as to represent somewhat realistic performance requirement as (53) (b) Fig 9 Simulation results for AFRTC algorithm when employed for a cargo ship (a) Ship heading [ship heading  (solid line) and reference course  (dashed line)] (b)... 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FUZZY SYSTEMS, VOL. 11, NO. 6, DECEMBER 2003 783 Adaptive Fuzzy Robust Tracking Controller Design via Small Gain Approach and Its Application Yansheng Yang and Junsheng Ren Abstract—A novel adaptive. IEEE Trans. Fuzzy Syst., vol. 2, pp. 285–294, Apr. 1994. YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 795 [24] Y. S. Yang, X. L. Jia, and C. J. Zhou, Robust adaptive fuzzy control. accuracy. YANG AND REN: ADAPTIVE FUZZY ROBUST TRACKING CONTROLLER DESIGN 787 Substituting (22) into (20), we get (23) Let , such that and . It follows that (23) reduces to (24) In order to design the adaptive