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Direct robust adaptive fuzzy control (DRAFC) for uncertain nonlinear systems using small gain theorem Yansheng Yang Navigation College, Dalian Maritime University (DMU), Dalian 116026, PR China Received 28 October 2003; received in revised form 15 April 2004; accepted 24 May 2004

Abstract A novel direct robust adaptive fuzzy control (DRAFC) is proposed for a class of nonlinear systems with uncertain system and gain functions, which both are unstructured (or non-repeatable) state-dependent unknown nonlinear functions. Takagi–Sugeno type fuzzy logic systems are used to approximate the uncertain system function and the DRAFC algorithm is designed by use of input-to-state stability (ISS) approach and small gain theorem. The resulting closed-loop system is proven to be semi-globally uniformly ultimately bounded. In addition, the possible controller singularity problem in some of the existing adaptive control schemes met with feedback linearization techniques can be removed and the adaptive mechanism with only one learning parameterization can be achieved. The control performance of the closed-loop system is guaranteed by appropriately choosing the design parameters. An example illustrating the proposed method is included for a ship autopilot system to maintain the ship on a pre-determined heading. Simulation results show the effectiveness of the control scheme. © 2004 Elsevier B.V. All rights reserved. Keywords: Uncertain nonlinear systems; Direct robust adaptive control; Fuzzy control; Small gain theorem

1. Introduction Over the past decade, there have been tremendous interest in designing a controller for systems having complex uncertain nonlinear systems, and many significant developments have been achieved. Most results addressing this problem are available in the control literature, e.g., [10] and references therein. Many powerful methodologies for designing the controller are proposed for nonlinear systems owing to E-mail address: [email protected] (Y. Yang). 0165-0114/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fss.2004.05.010

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advances in geometric nonlinear control theory, and in particular, feedback linearization techniques, e.g., [16,17]. Applications of these approaches are quite limited because they depend on the exact knowledge of the plant nonlinearities. In order to relax some of the exact model restrictions, several adaptive schemes have recently been introduced to solve the problem of linearly parameterized uncertainties, which are referred to as structured uncertainties. Unfortunately, in the industrial control environment, there are some controlled systems which are not only characterized by the unstructured uncertainties, but also represented all the terms which cannot be modelled or repeatable. A solution to those problems was first presented by Wang [30]. Fuzzy logic systems were proposed to uniformly approximate the uncertain nonlinear functions in the designed system by use of the universal approximation properties of certain classes of fuzzy systems proposed by Wang [27], Lee [11], Ying [38], Wang and Mendel [31]. A Lyapunov-based learning law was used, and several stable adaptive fuzzy controllers that ensure the stability of the overall system were developed by Wang [28], Su and Stepanenko [23], Yang et al. [34], Spooner and Passino [22], Fischle and Schroder [4]. Recently, an adaptive fuzzy-based controller combined with VSS and H∞ control techniques has been studied by Chen et al. [2] and Chang [1]. However, there is a substantial restriction in the aforementioned works: many parameters need 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 systems for approximating the nonlinear uncertain functions, so that the learning times tend to become unacceptably large for the higher-order systems and time-consuming process is unavoidable when the fuzzy logic controllers are implemented. This problem has been pointed out by Fischle and Schroder [4] and first researched by Yang and Ren [35]. In this paper, we will present a novel approach for that problem. A new systematic design procedure is developed for the synthesis of a stable direct robust adaptive fuzzy controller for a class of continuous uncertain nonlinear systems, and the Takagi–Sugeno type fuzzy logic systems [24] are used to approximate the unstructured uncertain functions in the systems. The adaptive mechanism with minimal learning parameterization is proposed by use of input-to-state stability (ISS) theory first proposed by Sontag [18] and small gain approach given by Jiang et al. [8]. The outstanding features of the algorithm proposed in the paper are that (i) the resulting closed-loop system is proven to be semi-globally uniformly ultimately bounded, (ii) only one function needs to be approximated by T–S fuzzy systems and only one learning parameter needs to be adapted on-line, no matter how many states in the designed system are investigated and how many rules in the fuzzy system are used, such that the burdensome computation of the algorithm can be lightened, which may increase its practicality, and (iii) 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 2, we will give some necessary definitions of input-state stability, small gain theorem and preliminary results. Section 3 contains the description of a class of nonlinear systems and the problem formulation. In Section 4, a systematic procedure for the synthesis of the DRAFC is developed. In Section 5, an application example for the ship autopilot system by means of the DRAFC is included and numerical simulation results are presented. The final section contains conclusions. Notation: Throughout this paper, let  ·  be any suitable norm. The vector norm of x ∈ R n is Euclidean, i.e.,  x 2 = (x T x) and the matrix norm of A ∈ R n×m is defined by  A 2 =
max (AT A), where
max(min) (·) denotes the operation of taking the maximum (minimum) eigenvalue. The vector norm over the space defined by stacking the matrix columns into a vector, so that it is compatible with the vector norm, i.e.,  Ax 
 A  ·  x .

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For any piecewise continuous function u : R+ → R m ,  u ∞ denotes sup{|u(t)|, t  0}, which stands for L∞ supremum norm, and for any pair of times 0
t1
t2 , the truncation u[t1 ,t2 ] is a function defined on R+ which is equal to u(t) on [t1 , t2 ] and is zero outside the interval. In particular, u[0,t] is the usual truncated function ut . 2. Mathematical preliminaries 2.1. ISS and small gain theorem The concepts of ISS and ISS-Lyapunov function due to Sontag [18,19], Sontag and Wang [21] have recently been used in various control problems such as nonlinear stabilization, robust control and observer designs (see, e.g. [6,7,14,15,20,25]). In order to ease the discussion of the design of DRAFC scheme, the variants of those notions are reviewed. First, we introduce the definitions of class K, K∞ and KL functions which are standard in the stability literature, see Khalil [9]. A class K-function  is a continuous, strictly increasing function from R+ into R+ and (0) = 0. It is of class K∞ if additionally (s) → ∞ as s → ∞. A function  : R+ × R+ → R+ is of class KL if (·, t) is of class K for every t  0 and (s, t) → 0 as t → ∞. Definition 1. For system x˙ = f (x, u), it is said to be input-to-state practically stable (ISpS) if there exist a function  of class K, called the nonlinear L∞ gain, and a function  of class KL such that, for any initial condition x(0), each measurable essentially bounded control u(t) defined for all t  0 and a nonnegative constant d, the associated solution x(t) is defined on [0, ∞) and satisfies:  x(t) 
( x(0) , t) + ( ut ∞ ) + d.

(1)

When d = 0 in Eq. (1), the ISpS property becomes the input-to-state stability (ISS) property introduced in Sontag [19]. Definition 2. A C 1 function V is said to be an ISpS-Lyapunov function for system x˙ = f (x, u) if • there exist functions 1 , 2 of class K∞ such that 1 ( x )
V (x)
2 ( x ),

∀x ∈ R n

(2)

• there exist functions 3 , 4 of class K and a constant d > 0 such that jV jx

(x)f (x, u)
− 3 ( x ) + 4 ( u ) + d.

(3)

When Eq. (3) holds with d = 0, V is referred to as an ISS-Lyapunov function. Then it holds that one may pick a nonlinear L∞ gain  in Eq. (1) of the form, which is given by Sontag [20] −1 (s) = −1 1 ◦ 2 ◦ 3 ◦ 4 (s),

∀s > 0,

where the notation ◦ stands for the composition operator between two functions.

(4)

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Σ ~z

Σ ~z

z~

Fig. 1. Feedback connection of composite systems.

For the purpose of application studied in this paper, we introduce the sequel notion of exp-ISpS Lyapunov function. Definition 3. A C 1 function V is said to be an exp-ISpS Lyapunov function for system x˙ = f (x, u) if • there exist functions 1 , 2 of class K∞ such that 1 ( x )
V (x)
2 ( x ),

∀x ∈ R n

(5)

• there exist two constants c > 0, d  0 and a class K∞ function 4 such that jV jx

(x)f (x, u)
− cV (x) + 4 ( u ) + d.

(6)

When Eq. (6) holds with d = 0, the function V is referred to as an exp-ISS Lyapunov function. The three previous definitions are equivalent from Sontag and Wang [21] and Praly and Wang [15]. Namely Proposition 1. For any control system x˙ = f (x, u), 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 by Jiang et al. [8], Jiang and Mareels [6] yields the following variant which is suited for our applications here. Theorem 1. Consider a system in composite feedback form (cf. Fig. 1)  x˙ = f (x, ), 
z :
z = H (x),  y˙ = g(y,
z), 
z :  = K(y,
z),

(7) (8)

of two ISpS systems. In particular, there exist two constants d1 > 0, d2 > 0, and let  and  of class KL, and z and  of class K be such that, for each  in the L∞ supremum norm, each
z in the L∞

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supremum norm, each x ∈ R n and each y ∈ R m , all the solutions X(x; , t) and Y (y;
z, t) are defined on [0, ∞) and satisfy, for almost all t  0:  H (X(x; , t)) 
 ( x , t) + z ( t ∞ ) + d1 , zt ∞ ) + d2 .  K(Y (y;
z, t)) 
 ( y , t) +  (


(9) (10)

Under these conditions, if z ( (s)) < s (resp.  (z (s)) < s)

∀s > 0

(11)

then the solution of the composite systems (7) and (8) is ISpS. 2.2. T–S fuzzy systems In this subsection, we briefly describe the structure of fuzzy systems. Let R denote real numbers, R n real n-vectors, R n×m real n × m matrices. Let S be a compact simply connected set in R n . With map f : S → R m , define C m (S) to be the function space such that f is continuous. The fuzzy systems can be employed to approximate the function f (x) in order to design the robust adaptive fuzzy control law, thus the configuration of the Takagi–Sugeno type fuzzy logic systems called T–S fuzzy systems in short [24] and approximation theorem are discussed first as follows. Consider the T–S fuzzy systems to uniformly approximate a continuous multi-dimensional function y = f (x) that has a complicated formulation, where x is input vector with n independent x = (x1 , x2 , . . . , xn )T . The domain of xi is i = [ai , bi ]. It follows that the domain of x is = 1 × 2 × · · · × n = [a1 , b1 ] × [a2 , b2 ] × · · · × [an , bn ].

In order to construct the fuzzy systems, the interval [ai , bi ] is divided into Ni subintervals: i i < CN = bi , ai = C0i < C1i < · · · < CN n n−1

1
i
n.

On each interval i (1
i
n), Ni +1 (Ni > 0) continuous input fuzzy sets, denoted by Aij (0
j
Ni ), are defined to fuzzify xi . The membership function of Aij is denoted by ij (xi ), which can be represented by triangular, trapezoid, generalized bell or Gaussian type and so on. Generally, T–S fuzzy systems can be constructed by the following K(K > 1) fuzzy rules Ri : If x1 is Aih1 AND x2 is Aih2 AND…AND xn is Aihn THEN yi is a0i + a1i x1 + · · · + ani xn , i = 1, 2, . . . , K, where aji , j = 0, 1, . . . , n, i = 1, 2, . . . , K are the unknown constants. The product fuzzy inference is employed to evaluate the ANDs in the fuzzy rules. After being defuzzified by typical center average defuzzifier, the output of the fuzzy system is fˆ(x, Ax ) =

K  i=1

yi i (x) = (x)Ax x, ¯

(12)

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where x = [x1 , x2 , . . . , xn ]T , x¯ = [1, x]T and i (x) called a fuzzy base function and  a01  2  a0  (x) = [1 (x), 2 (x), . . . , K (x)], Ax =  .  ..  a0K

=

(

–

n

K n i i j =1 hj (xj )/ i=1 [ j =1 hj (xj )],

a11 . . . an1 a12 .. .

)

... .. .

an2 .. .

a1K . . . anK

which is

    .  

When the fuzzy systems are used to approximate the continuous function, two questions of interest may be asked: whether there exist the fuzzy systems to approximate any nonlinear function to an arbitrary accuracy? How to determine the parameters in the fuzzy systems if such fuzzy systems do exist. The following lemma [30] gives a positive answer to the first question. Lemma 1. Suppose that the input universe of discourse U is a compact set in R r . Then, for any given real continuous function f (x) on U and ∀ε > 0, there exist the fuzzy systems fˆ(x, Ax ) in the form of expression (12) such that sup  f (x) − fˆ(x, Ax ) 
ε.

x∈U

(13)

Remark 1. For any n-dimensional  continuous function f (x), if Ni + 1 input fuzzy sets for each variable xi are used, there will be K = ni=1 (Ni + 1) IF–THEN fuzzy rules in the T–S fuzzy systems. In such a  way, we will get a total of (n + 1) · ni=1 (Ni + 1) parameters to describe the T–S fuzzy systems fˆ(x, Ax ) which is used to approximate the function f (x). Remark 2. In the fuzzy logic theory, there is another type of fuzzy systems, which is called Mamdanitype fuzzy systems. It has the following form of IF–THEN fuzzy rules Ri : If x1 is Aih1 AND x2 is Aih2 AND …AND xn is Aihn THEN yi is i , i = 1, 2, . . . , K where i is an unknown constant. If we manipulate it same as T–S fuzzy system, we have fˆ(x,) = T (x). When using fˆ(x, ) to approximate any continuous function f (x), there will be a total of ni=1 (Ni + 1) parameters to describe Mamdani-type fuzzy logic system fˆ(x, ). 3. Problem formulation Consider an nth-order uncertain nonlinear system of the following form: x˙i = xi+1 , 1
i
n − 1, x˙n = f (x) + g(x)u + (x, t), y = x1 ,

(14)

where u ∈ R and y ∈ R represent the control input and the output of the system, respectively. x = (x1 , x2 , . . . , xn )T ∈ R n is comprised of the states which are assumed to be available, the integer n denotes

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the dimension of the system. f (x) is an unknown smooth uncertain system function with f (0) = 0 and g(x) is an unknown smooth uncertain input gain function. f (x) and g(x) may be non-repeatable nonlinear functions. (x, t) is an uncertain disturbance of the system. Throughout the paper, the following assumptions are introduced on system (14). Assumption 1. There exists an unknown positive constant p∗ such that ∀(t, x) ∈ R+ × R n | (x, t) |
p∗ (x), where (·) is a known nonnegative smooth function. Assumption 2. The sign of g(x) is known and there exists a constant bmin > 0 such that | g(x) |  bmin ,

∀x ∈ R n .

The above assumption implies that smooth function g(x) is strictly either positive or negative. From now on, without loss of generality, we shall assume g(x)  bmin > 0, ∀x ∈ R n . Assumption 2 is reasonable because g(x) being away from zero is the controllable condition of system (14). It should be emphasized that the lower bound bmin is only required for analytical purposes, its true value is not necessarily known. The primary goal of this paper is to track a given reference signal yd (t) while keeping the states and control bounded. That is, the output tracking error e1 = y(t) − yd (t) should be small. The given reference signal yd (t) is assumed to be bounded and has bounded derivatives up to the nth order for all t  0, and (n) yd (t) is piecewise continuous. (1) (n−1) T ) such that is bounded. Suppose e = x − . Eq. (14) can be transformed Let = (yd , yd , . . . , yd into e˙i = ei+1 1
i
n − 1,

(n)

e˙n = f (x) + g(x)u − yd (t) + (x, t).

(15)

In this paper, we present a method for the robust adaptive control design for system (15) in the presence of unstructured uncertainties. Our objective is to find a direct robust adaptive fuzzy controller (DRAFC) u(t) of the form ˙ = (, (x), e),

u(t) = u(, (x), e),

(0, 0, 0) = 0, r (0, 0, 0) = 0,

(16) (17)

where (x) is a known fuzzy base function vector. And  ∈ R n , (·), r (·) are smooth functions on R n × R K × R n , where n is the dimension of . In such a way that all the solutions of the closed-loop system (15)–(17) are uniformly ultimately bounded. Furthermore, the output tracking error of the system can be steered to a small neighborhood of origin. Remark 3. From Eqs. (16) and (17), we can observe that is a dynamic feedback controller and n is the dimension of the dynamic part  of the controller. An important quality of the control law is of course the property that the dimension n of  should be as small as possible, and in particular not depend on

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the dimension of the state. However, most of the conventional adaptive fuzzy control laws available in the literature, e.g. [22,23,28,34] have the property that the dimension n of  is equal to the number of parameters to describe the fuzzy systems which are used to approximate nthe unknown uncertain n functions. Based on two types of fuzzy systems, there are a total of 2(n + 1) · i=1 (Ni + 1) and 2 i=1 (Ni + 1) parameters, which need to be estimated, for T–S-type fuzzy systems (cf. Remark 1) and Mamdani-type fuzzy systems (cf. Remark 2) respectively, when using them to approximate the functions f (x) and g(x). We can state that the dimension n of  is strongly dependent on the dimension n of the state in the designed system. In such a way, their learning time will tend to become unacceptably large for higher-order systems. Remark 4. In this paper, we will present a direct robust adaptive fuzzy control algorithm with the structure as Eqs. (16) and (17) for a class of uncertain nonlinear system (15) by use of input-to-state stability theory and generalized small gain approach. The outstanding feature of the algorithms proposed in this paper is that the dimension n of  is only one, no matter how many states in the designed systems are investigated and how many rules in the fuzzy systems are used. 4. Direct robust adaptive fuzzy control We consider that a term k T e leads to the exponentially stable dynamics in Eq. (15), where k = [k1 , k2 , . . . , kn ]T , the ki ’s are chosen such that all roots of polynomial s n + kn s n−1 + · · · + k1 = 0 lie in the left-half complex plane. Then Eq. (15) can be transformed into (n)

e˙ = Ae + B[g(x)u + f (x) − yd + k T e + (x, t)], where



1 ...  0 ...   .. .. A= . .   0 0 ... −k1 −k2 . . .   0 0   B =  ..  . . 0 0 .. .

0 0 .. .

(18)



   ,  1  −kn

1 Because A is stable, a positive definite solution P = P T of the Lyapunov equation AT P + P A + Q = 0

(19)

always exists and Q > 0 is specified by designer. For this control problem, if both functions f (x) and g(x) in (18) are available for feedback, the technique of the feedback linearization can be used to design a well-defined controller, which is usually given in

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(n)

the form of u = g −1 (x)(−k T e − f (x) + yd + us ) for some auxiliary control input us with g(x) being always nonzero and (t, x) = 0, such that the resulting closed-loop system can be shown to achieve a satisfactory tracking performance. However, in many practical control systems, plant uncertainties that contain structured (or parametric) uncertainties and unstructured uncertainties (or non-repeatable uncertainties) are inevitable. Hence, neither f (x) nor g(x) may be available directly in the robust control design. Obtaining a simple control algorithm as above is impossible. Moreover, if any adaptation scheme is implemented to estimate f (x) and g(x) as fˆ(x) and g(x), ˆ respectively, the simple control algorithm ˆ aforementioned can be also used for substituting f (x) and g(x) ˆ for f (x) and g(x), so extra precaution is required to guarantee that g(x) ˆ = 0 for all the time. At the present stage, no effective method is available in the literature. In this paper, we will develop a semi-globally stable direct robust adaptive fuzzy controller which does not require to estimate the unknown function g(x), 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 dynamics. Here, only the uncertain function f (x) needs to be considered. For f (x) is an unknown continuous function, by Lemma 1, T–S fuzzy systems fˆ(x, Ax ) with input vector x ∈ Ux for some compact set Ux ⊂ R n is proposed here to approximate the uncertain term f (x) where Ax is a matrix containing unknown parameters. Then f (x) can be expressed as f (x) = (x)Ax x + ε = (x)Ax (e + ) + ε = (x)Ax e + (x)Ax + ε,

(20)

where ε is the approximating error. Substituting Eq. (20) into (18), we get e˙ = Ae + B[g(x)u + d] + B (x)Ax e,

(21)

(n)

where d = −yd + k T e + (x)Ax + ε + (t, x). 1/2 −1 m Let c = Ax =
max (ATx Ax ), such that Am x = c Ax and  Ax 
1. It follows that Eq. (21) reduces to e˙ = Ae + B[g(x)u + d] + c B (x)Am x e.

(22)

In order to design the robust adaptive fuzzy controller easily by use of the small gain theorem, the following output equation can be obtained by comparing Eq. (22) with (7) zˆ = H (e) = e. Then the feedback equation is given as follows: m wzˆ : w = K(ˆz) = Am z zˆ = Az e.

So Eq. (22) can be rewritten as Eqs. (7) and (8)  e˙ = Ae + B[g(x)u + d] + Bc (x)w, zˆ w : zˆ = H (e) = e,

(23)

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Fig. 2. Feedback connection of fuzzy system. m wzˆ : w = K(ˆz) = Am z zˆ = Az e.

(24)

Then the feedback connection using Eqs. (23) and (24) can be implemented by the block diagram shown in Fig. 2. From Fig. 2, we observe that the subsystem zˆ w should be made to satisfy ISpS condition of the system through designing the controller u = U (e). Since c is an unknown constant and d is an unknown timevariant variable with bound in Eq. (25), a robust adaptive fuzzy tracking control algorithm u = U (e), which can make the subsystem zˆ w meet ISpS condition, must be developed. Based on the aforementioned condition, we can get  d 
d1 + ( k  +  Ax  (x) )   +  k  x  +p∗ (x) + ε
(x), 1/2

(25)

(n)

where (x) = 1+  x  +  (x)  + (x),  Ax =
max (ATx Ax ),  yd 
d1 and  = max(d1 ,  k  ,  Ax  , p∗ , ε). Unknown constant  denotes the largest one in all the bounds. We are ready to state and prove the main result of this paper. Theorem 2. Under Assumptions 1 and 2, if we apply T–S fuzzy logic systems to approximate f (x) in system (18) and pick the gain  < 1 of zˆ w , there exist a positive constant  and
min (Q) > 2 in (19), a direct robust adaptive fuzzy tracking control scheme is proposed as follows ˆ (x)B T P e, u = −
ϑ

(26)

where ϑ(x) = (1/42 (x)T (x) + (1/42 )2 (x)) and the adaptive law for
ˆ is now chosen as ˙


ˆ = [ϑ(x)eT P BB T P e − (
ˆ −
0 )],

(27)

where  > 0 is an updating rate,  and
0 are design constants, which are chosen by the designer respectively. Then, for bounded initial conditions, we have the following. (1) All the solutions (e(t),
ˆ ) of the derived closed loop system can be made uniformly ultimately bounded.

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(2) For given any > 0, the direct robust adaptive fuzzy tracking control can be tuned so that the output tracking error e1 = y(t) − yd (t) satisfies limt→∞ | e1 (t) |
. Proof. The proof of Theorem 2 can be divided into two steps. Firstly, let the constant  > 0 and set w = Am z e as the virtual input of the subsystem zˆ w , to prove the satisfaction of ISpS for the subsystem zˆ w by use of the direct robust adaptive fuzzy tracking controller, and then to prove uniform ultimate bound of the composite of two systems with the feedback subsystem w = Am z e by means of small gain theorem. Choose Lyapunov function candidate as V =

1 T 1 2 e P e + bmin −1
˜ , 2 2

(28)

where
˜ = (
−
ˆ ). The time derivative of V along the error trajectory (23) is 1 V˙ = eT (AT P + P A)e − bmin −1
˜
˙ˆ + eT P B[g(x)u + d] + eT P Bc (x). (29) 2 We deal with relative items in Eq. (29). First, we substitute Eq. (26) into the relative term above. Since ϑ(x)  0 for all x ∈ R n , we can obtain eT P Bg(x)u = −
ˆ g(x)ϑ(x)eT P BB T P e ˆ (x)eT P BB T P e.
−bmin
ϑ

(30)

The last term in Eq. (29) can be dealt with as follows: 2 c T T T 2 2 2 2 2 e P Bc (x) −   +   = −  − 2  B P e 2 +

c2 T e P B T B T P e + 2 2 42

c2 T e P B T B T P e + 2 2 . (31) 4 2 Substituting Eq. (25) into the relative item of Eq. (29), by means of the Schwarz inequality, yields


eT P Bd
 eT P B  d 
 eT P B  (x) 2

2 (x)eT P BB T P e + 2 , 4 2 where  is a nonnegative constant chosen by designer. Using Eqs. (31) and (32), we can get


eT P Bc (x) + eT P Bd


c2 T e P B (x)T (x)B T P e + 2 2 42 +

2

4 2

2 (x)eT P BB T P e + 2


bmin
ϑ(x)eT P BB T P e + 2 2 + 2

(32)

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ˆ (x)eT P BB T P e + bmin
ϑ ˜ (x)eT P BB T P e
bmin
ϑ 2 2 2 +  +  ,

(33)

−1 2 −1 2 max(bmin c , bmin  ).

where
= Substituting Eq. (33) into (29), V˙ can be written as 1 V˙
− eT Qe + bmin −1
˜ (ϑ(x)eT P BB T P e −
˙ˆ ) + 2 2 + 2 . 2 Noting Eq. (27), we can get 1 V˙
− eT Qe + bmin
˜ (
ˆ −
0 ) + 2 2 + 2 2 1 1 2
− eT Qe − bmin
˜ + 2 2 + d1 , 2 2

(34)

(35)

where d1 = 21 bmin (
−
0 )2 + 2 . If picking
min (Q) > 2, we get V˙
−  e 2 +2 2 + d1 . By Definition 3, we propose the direct robust adaptive fuzzy tracking controller (26) and adaptive law (27) such that the requirement of ISpS for subsystem zˆ w can be satisfied with the functions 3 (s) = s 2 and 4 = 2 s 2 of class K∞ . By Definition 2 and Eq. (11), we can get a gain function z (s) of subsystem zˆ w as follows: −1 z (s) = −1 1 ◦ 2 ◦ 3 ◦ 4 ,

∀s > 0,

where 1 (z)
V (z)
2 (z). For subsystem wzˆ , it is a static system such that we have  w 
 Am x  zˆ = 1  zˆ = 1  e  .

(36)

Then the gain function w for the subsystem wzˆ is w (s) = 1 s. According to the requirement z (w (s)) < s of small gain Theorem 1, we can get 1 < 1. By following Eq. (22), it is known that 1 = Am x 
1, so that the condition of the small gain Theorem 1 can be satisfied by choosing  < 1. Then it can be proven that the composite closed-loop system zˆ w and wzˆ is ISpS. Therefore, a direct application of Definition 1 yields that the composite closed-loop system has bounded solutions over [0, ∞). More precisely, there exist a class KL-function  and a positive constant  such that  e(t),
ˆ (t) 
( e(0),
ˆ (0), , t) + .

(37)

The first statement of Theorem 2 directly follows from (37). This, in turn, implies that the tracking error e(t) is bounded over [0, ∞). By Proposition 1, there exists an ISpS-Lyapunov function for the composite closed-loop system. By substituting Eq. (36) into (35), the ISpS-Lyapunov function is satisfied as follows: 1 1 2 V˙
− eT Qe − bmin
˜ + 21 2  e 2 +d1 2 2 1 T 1 2
− e (Q − 2I )e − bmin
˜ − (1 − 21 2 )  e 2 +d1 2 2
−c1 V + d1 , (38)

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where c1 = min{(
min (Q) − 2)/
max (P ), }. From Eq. (38), we get 

d1 d1 −(t−t0 ) e + V (t0 ) − . V (t)
c1 c1 It results in that the solutions of composite closed-loop system are uniformly ultimately bounded, and implies that, for any > (d1 /c1 )1/2 , there exists a constant T > 0 such that  e1 (t) 
for all t  t0 + T . The second statement of Theorem 2 follows readily since (d1 /c1 )1/2 can be made arbitrarily small if the design parameters
0 , ,  are chosen appropriately. Remark 5. Since the function approximation property of fuzzy systems is only guaranteed within a compact set, the stability result proposed in this paper is semi-global 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, large number of fuzzy rules usually cannot be chosen due to the possible computation problem. This implies that the fuzzy systems approximation capability is limited, that is, the approximating error ε in Eq. (20) for the estimated function f (x) will be greater when choosing small number of fuzzy rules. But we can choose appropriately the design parameters
0 , ,  to improve both stability and performance of the closed-loop systems. 5. Application example Now we will reveal the control performance of the proposed direct robust adaptive fuzzy controller via application example. A ship autopilot is designed by using the procedure given in the previous section. Simulation results will be presented. Many of the present generation of autopilots installed on ships are designed for course keeping. They aim at maintaining the ship on a pre-determined course and thus require directional information. Developments in the last 20 years include variants of the analogue proportional-integral-derivative (PID) controller. In 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 [26], self-tuning [12], optimal [39], H∞ theories [3], adaptive robust fuzzy control [36] and model reference adaptive robust fuzzy control [37]. Before designing the autopilots, it is of interest to describe the dynamics of the ship. The mathematical model relating the rudder angle  to the heading of the ship is found to be of the form ¨+

K K H ( ˙ ) = , T T

(39)

where K = gain (s−1 ) and T = time constant (s) are parameters which are function of ship’s constant forward velocity and its length. H ( ˙ ) is a nonlinear function of ˙ . The function H ( ˙ ) can be found from the relationship between  and in steady state such that ¨ = ˙ = ˙ = 0. An experiment known as the “spiral test” has shown that H ( ˙ ) can be approximated by 3 5 H ( ˙ ) = a1 ˙ + a2 ˙ + a3 ˙ + · · · ,

(40)

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Table 1 Fuzzy IF–THEN rules x2 N Z P

x1 N y1 y4 y7

Z y2 y5 y8

P y3 y6 y9

where ai , i = 1, 2, 3, . . . are real valued constants. In normal steering, ship often makes only small deviations from its desired direction. The coefficients ai , i = 2, 3, . . . , in Eq. (40) could be equal to 0 such that a linear model is used as design model for designing the autopilot, but in this paper, let ai , i = 1, 2, 3, . . . , be not equal to 0, a nonlinear model (40) is used as the design model for designing the robust adaptive fuzzy controller as following. Let the state variables be x1 = , x2 = ˙ and control variable be u = , then Eq. (39) can be rewritten in the state-space form x˙1 = x2 , K K x˙2 = − H (x2 ) + u, T T y = x1 .

(41)

Without loss of generality, we assume that the function H (x2 ) in Eq. (40) can be defined in the function f (x1 , x2 ) which is unknown with a continuous complicated formulation system function, the T–S fuzzy systems can be constructed to approximate the function f (x1 , x2 ) by the following nine fuzzy IF–THEN rules in Table 1. In Table 1, we select yi = a1i x1 + a2i x2 , i = 1, 2, . . . , 9. P denotes the fuzzy set “Positive”, Z the fuzzy set “Zero” and N the fuzzy set “Negative”. They can be characterized by the membership functions as follows: 1 ,

positive (x) = 1 + exp(−4(x − /2))

zero (x) = exp(−x 2 ), 1 .

negative (x) = 1 + exp(4(x + /2)) Using the center average defuzzifier and the product inference engine, the fuzzy system is obtained as follows: f (x) = (x)Ax x + ε, (42)  1 1 a1 a 2  .. ..  where Ax =  . . . (x) can be defined as Eq. (12). a19 a29 To demonstrate the availability of the proposed scheme, we take a general cargo ship with the length 126 m and the displacement 11,200 tons as an example for simulation.

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The reference model is chosen so as to represent somewhat realistic performance requirement as ¨ m (t) + 0.1 ˙ m (t) + 0.0025 m (t) = 0.0025 r (t),

(43)

where m (t) specifies the desired system performance for the ship heading (t) and r (t) is a order input signal. When let k1 = 2, k2 = 5 and Q = diag(2, 2), then the solution of Lyapunov expression (19) is obtained by   3.1 0.5 . P = 0.5 0.3 If the gain is  = 0.5 and design constant is  = 0.5, the robust adaptive fuzzy tracking control scheme can be obtained for ship autopilot as follows:  9   2 2 u = −
ˆ (0.5e1 + 0.3e2 ) i +  (x) , (44) i=1

where e1 = x1 − m , e2 = x2 − ˙ m ,   (x) = 1 + (x12 + x22 )1/2 +



˙
ˆ = 6 (0.5e1 + 0.3e2 )2

 9  i=1

9  i=1

1/2  2i (x)

, 



2i (x) + 2 (x) − 0.01(
ˆ − 0.1) .

Generally, the first step in the controller design procedure is construction of a “truth model” of the dynamics of the process to be controlled. The truth model is a simulation model that includes all the relevant characteristics of the process. The truth model is too complicated for use in the controller design. Thus, we need to develop a simplified model called the design model that can be used to design the controller. In this paper, we take Eq. (39) as the design model. In order to verify the performance of the DRAFC proposed above by use of simulation, a truth model which accurately represents the characteristics of the ship is used as follows: (m + mx )u˙ − (m + my )vr = XH + XP + XR + XW , (m + my )v˙ + (m + mx )ur = YHP + YR + YW , (Izz + Jzz )˙r = NHP + NR + NW ,

(45)

where u and v are the velocity components of the ship and r is the angular rate of yaw angle with respect to time. And m,mx ,my ,Izz ,Jzz are mass, added mass, inertia moment and added inertia moment of ship. X,Y are the components of hydrodynamic force acting on ship in the bodyfixed axis system and N is a moment by the above forces. The subscripts in the right of Eq. (45) mean that H denotes the bare hull, P is screw, R is rudder, W is wind and wave which produce the external forces and moments acting on ship. The methods of calculating external forces and moments above have been proposed by Yang [32]. Simulation results based on the Matlab Simulink package are shown in Figs. 3–6.

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10

heading [deg.]

5

0

−5

−10 0

500

1000 time [sec.]

1500

2000

Fig. 3. Simulation results for DRAFC algorithm when employed for a cargo ship. Ship heading (reference course m (solid line) and ship heading (dashed line)).

30

rudder [deg.]

20 10 0

−10 −20 −30 0

500

1000 time [sec.]

1500

2000

Fig. 4. Simulation results for DRAFC algorithm when employed for a cargo ship. Rudder angle u.

tracking error [deg.]

10

5

0

−5

−10

0

500

1000 time [sec.]

1500

2000

Fig. 5. Simulation results for DRAFC algorithm when employed for a cargo ship: Tracking error (deg).

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2

λ

1.5

1

0.5

0 0

500

1000

1500

2000

time [sec.]

Fig. 6. Simulation results for DRAFC algorithm when employed for a cargo ship. Adaptive parameter
.

The controller is trained by applying a signal r , which changes its value in the interval (−10◦ , +10◦ ) every 300 s. It can be seen that the plant output converges rapidly to the reference signal in Fig. 5. Fig. 6 shows the adaption of parameter in DRAFC algorithm. 6. Conclusions In this paper, the tracking control problem has been considered for a class of nonlinear systems with the unknown system and gain functions, and the Takagi–Sugeno type fuzzy logic systems have been used to approximate unknown system function and a direct robust adaptive fuzzy tracking control (DRAFC) algorithm, which guarantee that the closed-loop system in the presence of non-repeatable uncertainties is semi-globally uniformly ultimately bounded, and the tracking error of the system can be steered to a small neighborhood around e(t) = 0, has been achieved by use of input-to-state stability and general small gain approach. The outstanding feature of the algorithm proposed in this paper is that it can avoid the possible controller singularity problem in some of the existing adaptive control schemes with feedback linearization techniques and obtain the adaptive mechanism with minimum learning parameterization, e.g. no matter how many states in the system are investigated and how many rules in the fuzzy system are used, only one parameter needs to be adapted on-line, so that the computation load of the algorithm can be reduced and it is convenient to realize this algorithm in engineering. In order to study the efficiency of the algorithm proposed in this paper, it has been applied to the ship autopilot system. In accordance with unknown parameters of the ship model and the structure of uncertain function of the system, the DRAFC scheme for ship autopilot is proposed. Simulation results have shown the effectiveness of the control scheme. Acknowledgements This work was supported in part by the Research Fund for the Doctoral Program of Higher Education under Grant No. 20020151005 and the Young Investigator Foundation under Grant No. 95-05-05-31 of National Ministry of Communication of P. R. China.

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The author wishes to thank anonymous referees for their valuable comments, which have helped us to improve this paper. References [1] Y.C. Chang, Adaptive fuzzy-based tracking control for nonlinear SISO systems via VSS H∞ approaches, IEEE Trans. Fuzzy Systems 9 (2001) 278–292. [2] B.S. Chen, C.H. Lee, Y.C. Cheng, H∞ tracking design of uncertain nonlinear SISO systems: adaptive fuzzy approach, IEEE Trans. Fuzzy Systems 4 (1996) 32–43. [3] N.A. Fairbairn, M.J. Grimble, H∞ marine autopilot design for course-keeping and course-changing, Proc. of 9th Ship Control Systems Symp., vol. 3, Bethesda, 1990. [4] K. Fischle, D. Schroder, An improved stable adaptive fuzzy control method, IEEE Trans. Fuzzy Systems 7 (1999) 27–40. [5] G.C. Hwang, S.C. Lin, A stability approach to fuzzy control design for nonlinear systems, Fuzzy Sets and Systems 40 (1990) 279–287. [6] Z.P. Jiang, I. Mareels, A small gain control method for nonlinear cascaded systems with dynamic uncertainties, IEEE Trans. Automatic Control 42 (1997) 292–308. [7] Z.P. Jiang, L. Praly, Technical results for the study of robustness of Lagrange stability, Systems Control Lett. 23 (1994) 67–78. [8] Z.P. Jiang, A. Teel, L. Praly, Small-gain theorem for ISS systems and applications, Mathematics of Control Signals Systems 7 (1994) 95–120. [9] H.K. Khalil, Nonlinear Systems, Macmillan, New York, 1996. [10] P. Kokotovic, M. Arcak, Constructive nonlinear control: a historical perspective, Automatica 37 (2001) 637–662. [11] C.C. Lee, Fuzzy logic in control systems: fuzzy logic controller—Parts I&II, IEEE. Trans. Syst. Man Cybern. 20 (1990) 404–435. [12] M. Mort, D.A. Linkens, Self-tuning controllers steering automatic control, Proc. Symp. on Ship SteeringAutomatic Control, Genova, 1980. [13] M.M. Polyvearpou, P.A. Ioannou, A robust adaptive nonlinear control design, Automatica 32 (1995) 423–427. [14] L. Praly, Z.P. Jiang, Stabilization by output feedback for systems with ISS inverse dynamics, Systems Control Lett. 21 (1993) 19–33. [15] L. Praly, Y. Wang, Stabilization in spite of matched unmodelled dynamics and an equivalent definition of input-to-state stability, Mathematics Control Signals Systems 9 (1996) 1–33. [16] S.S. Sastry, A. Isidori, Adaptive control of linearization systems, IEEE Trans. Automatic Control 34 (1989) 1123–1131. [17] J.J.E. Slotine, W. Li, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ, 1991. [18] E.D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automatic Control 34 (1989) 1411–1428. [19] E.D. Sontag, Further results about input-state stabilization, IEEE Trans. Automatic Control 35 (1990) 473–476. [20] E.D. Sontag, On the input-to-state stability property, European J. Control 1 (1995) 24–36. [21] E.D. Sontag, Y. Wang, On characterizations of input-to-state stability property with respect to compact sets, in: Prep. IFAC NOLCOS’95, Tahoe City, 1995, pp. 226–231. [22] J.T. Spooner, K.M. Passino, Stable adaptive control using fuzzy systems and neural networks, IEEE Trans. Fuzzy Systems 4 (1996) 339–359. [23] C.Y. Su, Y. Stepanenko, Adaptive control of a class of nonlinear systems with fuzzy logic, IEEE Trans. Fuzzy Systems 2 (1994) 285–294. [24] T. Takagi, M. Sugeno, Fuzzy identification of systems and its applications to modeling and control, IEEE Trans. Syst. Man Cybern. 15 (1985) 116–132. [25] J. Tsinias, Sontag’s ’input-to-state stability condition’ and global stabilization using state detection, Systems Control Lett. 20 (1993) 219–226. [26] J. Van Amerongen, Adaptive steering of ships: a model reference approach to improved manocuvring and economical course-keeping, Ph.D. Thesis, Delft University of Technology, 1982. [27] L.X. Wang, Fuzzy systems are universal approximators, in: Proc. IEEE Internat. Conf. On Fuzzy Systems, San Diego, CA, 1992, pp. 1163–1170.

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