Adaptive fuzzy approach for a class of uncertain nonlinear systems in strict-feedback form

ISA Transactions 47 (2008) 286–299 www.elsevier.com/locate/isatrans

Adaptive fuzzy approach for a class of uncertain nonlinear systems in strict-feedback form H.F. Ho a , Y.K. Wong a,∗ , A.B. Rad b a The Hong Kong Polytechnic University, Department of Electrical Engineering, Hung Hom, Kowloon, Hong Kong b Simon Fraser University, Canada

Received 27 March 2007; received in revised form 7 January 2008; accepted 17 March 2008 Available online 14 May 2008

Abstract Adaptive fuzzy control is proposed for a class of affine nonlinear systems in strict-feedback form with unknown nonlinearities. The unknown nonlinearities include two types of nonlinear functions: one satisfies the “triangularity condition” and can be directly approximated by fuzzy logic system, while the other is assumed to be partially known and consists of parametric uncertainties. Takagi–Sugeno type fuzzy approximators are used to approximate unknown system nonlinearities and the design procedure is a combination of adaptive backstepping and generalized small gain design techniques. It is proved that the proposed adaptive control scheme can guarantee the uniformly ultimately bounded (UBB) stability of the closed-loop systems. Simulation studies are shown to illustrate the effectiveness of the proposed approach. c 2008, ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Adaptive control; Backstepping; Fuzzy control; Uncertain nonlinear systems

1. Introduction In the past decade, the adaptive control has seen rapid and significant development leading to global stability and asymptotic tracking results for large classes of uncertain nonlinear systems [1–3]. The fundamental idea of these adaptive control schemes is to use feedback linearization, by transforming a nonlinear system into a linear one. Linear control techniques can then be applied to acquire the desired performance. However, in order to guarantee the stability performance of the closed-loop system, some restrictions on the system had to be assumed, such as matching conditions and growth conditions on system nonlinearities. These restrictions were subsequently relaxed by the development of a recursive and systematic design produce called the backstepping approach [4–6]. The backstepping design provides a systematic framework for the design of tracking and regulation strategies for a class of nonlinear systems which can be transformed into a ∗ Corresponding author. Tel.: +852 2766 6140.

E-mail addresses: [email protected] (H.F. Ho), [email protected] (Y.K. Wong).

parametric strict-feedback system. Integrator backstepping is used to systematically design controllers for system with known nonlinearities. In an effort to extend the backstepping approach to a class of unknown nonlinearities systems, Kanellakopoul et al. [5] studied the adaptive backstepping control of pure feedback system and obtained regionally stable results. For nonlinear system with triangular structures, several adaptive control schemes have also been developed [6–8]. Recently, adaptive backstepping control has been studied for output feedback control, multi-input multi-output (MIMO) and time delay systems [9–13]. In [14], robust adaptive control scheme is introduced for uncertain strict-feedback system in the presence of both parametric uncertainties and unknown nonlinearities by combing backstepping with robust control strategy, which guarantee semi-global uniform ultimate bondedness. In [15], an adaptive robust control method by combining the adaptive backstepping control with deterministic robust control for tracking problem of strict-feedback nonlinear system was considered. The common assumption of the nonlinear systems reported in [14,15] are that the system nonlinear functions are in the linearly parameterized forms, which may not be satisfied in practice.

c 2008, ISA. Published by Elsevier Ltd. All rights reserved. 0019-0578/$ - see front matter doi:10.1016/j.isatra.2008.03.002

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H.F. Ho et al. / ISA Transactions 47 (2008) 286–299

Artificial intelligent control using fuzzy logic has recently undergone rapid development. It has been proven that fuzzy logic can approximate any nonlinear function to any desired accuracy because of the universal approximation theorem [16]. Fuzzy control schemes have been found to be particularly useful to model unknown functions in nonlinear systems rather that only unknown parameters. In the earlier fuzzy control schemes, optimization techniques were mainly used to derive parameter adaptation laws with little analytical results for stability and performance. To overcome these problems, some adaptive fuzzy control approaches have been proposed based on Lyapunov’s stability theory [17–20]. It is wroth noting that one limitation of these fuzzy control schemes need certain types of matching conditions, i.e. the unknown nonlinearities span in the same equation as the control input. Using the idea of adaptive backstepping design [5], several fuzzy based and neural networks based adaptive controllers have been investigated for some classes of strict-feedback nonlinear systems without satisfying matching condition [21–26]. By using backstepping and online approximators, the result in [25] was extended to a class of strict-feedback nonlinear systems with unknown nonlinearities, where a lot of on-line parameters are needed to be tuned. Although great progress has been made by combining backstepping methodology with universal approximator technologies, there are still some difficulties in control practice. One of the main difficulties comes from the uncertain affine term gi (·)(i = 1 · · · n). When gi (·) are unknown nonlinearities, if feedback linearization type controllers u i = 1/gˆi (·)(− fˆi (·) + vi ) are considered, where fˆi (·) and gˆi (·) are estimates of f i (·) and gi (·), respectively, and vi is a new control to be defined. The singularity problem appears when gˆi (·) → 0. In [25], a stable adaptive fuzzy control was presented for strict-feedback nonlinear systems with virtual control coefficients gi (·) = 1 (see (18) in Section 4). In [26], by using a novel integral Lyapunov function, an adaptive fuzzy control technique with generalized small gain approach was presented for nonlinear strict-feedback systems. The possible controller singularity problem is avoided without using projection. However, because of the integral-type Lyapunov function introduced, this approach is complicated and difficult to use in practice. In this paper, we propose an alternative method for designing a backstepping based fuzzy control which can completely avoid the singularity problem. In addition, the stability of the resulting adaptive system is guaranteed without the requirement for integral-type Lyapunov functions, a class of continuous functions is introduced to ensure robustness of in backstepping design. In addition, the control scheme can guarantee the uniformly ultimate boundedness of all the control signals in the closed-loop system, and the steady-state tracking error is proven to converge to a small neighborhood of the reference trajectory. This paper is organized as follows. Mathematical preliminaries is presented in Section 2. A brief description of fuzzy logic system is included in Section 3. Problem formulation is presented in Section 4. In Section 5, the proposed adaptive fuzzy backstepping control is described. Simulation results for the

proposed control algorithm are included in Section 6. Finally, the paper is concluded in Section 7. 2. Preliminaries Notations. Throughout this paper, let x¯i = [x1 , . . . , xi ]T , R denote the real numbers, R n denote the real n-vectors and R m×n the real m × n matrices. We denote k·k as any suitable norm. The vector norm of x ∈ R n is Euclidean, i.e. kxk2 = (x T x)1/2 and the matrix norm of A ∈ R n×m is defined 1/2 by kAk2 = λmax (AT A), and λmax (B) and λmin (B) denote the largest and smallest eigenvalues of a square matrix B, respectively. The norm k·k2 denoted by k·k throughout this paper unless specified. 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. kAxk ≤ kAk kxk, with A ∈ R m×n and x ∈ R n . Definition 2.1 ([27]). Given a nonlinear system x˙ = f (x, t),

y = h(x, t),

x ∈ R n , t ≥ t0

(1)

if there exists a compact set U ∈ R n such that for all x(t0 ) ∈ x0 ∈ U , there exists a ε > 0 and a number T (δ, x0 ) such that kx(t)k < ε for all t ≥ t0 + T (δ, x0 ), we say the solution of (1) is uniformly ultimately bounded (UUB). Definition 2.2 ([28]). A continuous function α : [0, a) → [0, ∞) is said to belong to class K , if it is strictly increasing and α(0) = 0. It is said to belong to class K ∞ if a = ∞ and α(r ) → ∞ as r → ∞. Definition 2.3 ([28]). A continuous function β : [0, a) × [0, ∞) → [0, ∞) is said to belong to class K L if for each fixed s the mapping β(r, s) belongs to class K with respect to r , and for each fixed r the mapping β(r, s) is decreasing with respect to s and β(r, s) → 0 as s → 0. Consider the system x˙ = f (x, u)

(2)

where x ∈ R n is the state and u ∈ R m is the input. It is assumed that f : R n × R m → R n is locally Lipschitz and satisfies f (0, 0) = 0. The input function u : [0, ∞) → R m is assumed to be piecewise continuous and bounded. Definition 2.4. For the system (2) is said to be input-to-state practically stable (ISpS) if there exist a class K function γ , and class K L function β, such that, for any essentially bounded input u(t) and any x0 and a nonnegative constant d, the response of x(t) are defined on [0, ∞) and satisfy kx(t)k ≤ β(kx(0)k , t) + γ (ku t k∞ ) + d

(3)

where u t is the truncated function of u at t, when d = 0 in (3), the ISpS property collapses to ISS property introduced in [29]. Definition 2.5. A continuous function V is said to be an ISpSLyapunov function for the system (2) if there exist α1 , α2 of

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class K ∞ and α3 , α4 of class K and constant d > 0

where µ Al (xi ) is the membership function of the linguistic

α1 (kxk) ≤ V (x) ≤ α2 (kxk), ∀x ∈ R n (4) ∂ V (x) f (x, u) ≤ −α3 (kxk) + α4 (kuk) + d (5) ∂x when (5) holds with d = 0, V is referred to as ISS-Lyapunov function [29]. Then it hold that the nonlinear L ∞ gain γ in (3) can be evaluated as

variable xi , and y l represents a crisp value at which the membership function µ B l for output fuzzy set reaches its maximum. By introducing the concept of fuzzy basis function vector or the antecedent function vector. Eq. (11) can be rewritten as

γ (s) =

α1−1

◦ α2 ◦ α3−1

◦ α4 (s),

∀s > 0.

i

(6)

Theorem 2.1 ([30,31]). Consider a system in composite feedback form of two ISpS systems  x˙ = f 1 (x1 , w) Σ1 : 1 (7) z = H1 (x1 )  x˙ = f 2 (x2 , z) Σ2 : 2 (8) w = H2 (x2 ) there exist two constants d1 , d2 > 0 and βx1 , βx2 of class K L, and γz , γw of class K such that for each w and z in supremum norm, each x1 ∈ R n and each x2 ∈ R m , all the solutions X 1 (x1 , w, t) and X 2 (x2 , z, t) are defined on [0, ∞) and satisfy for all t. kH1 (X 1 (x1 , w, t))k ≤ βx1 (kx1 k , t) + γz (kwk∞ ) + d1

(9)

kH2 (X 2 (x2 , z, t))k ≤ βx2 (kx2 k , t) + γw (kzk∞ ) + d2

(10)

under these conditions γz ◦ γw < s,

f (x) = θ T ξ(x) n Q µ Al (xi ) i i=1 l ξ (x) = m n P Q µ Al (xi )

∀s > 0

where θ = [y 1 , . . . , y m ]T ∈ R m is called the parameter vector and ξ(x) = [ξ 1 (x), . . . , ξ m (x)]T ∈ R m is called the fuzzy basis function vector. One of the most important advantages of fuzzy logic system is that the fuzzy logic system has the capability to approximate nonlinear mappings.

3.2. Takagi–Sugeno (T–S) type fuzzy system

The Takagi–Sugeno fuzzy system [32] consists of fuzzy IFTHEN rules with the following structure R (l) : IF x1 is Al1 and · · · and xn is Aln

m P

3.1. Mamdani type fuzzy system The basic configuration of the fuzzy system consists of a collection of fuzzy IF-THEN rules: R

: IF x1 is

and · · · and xn is

Aln

Then y is B . l

The fuzzy system performs a mapping from U = U1 × · · · × Un ⊆ R n to V ⊆ R, where x = [x1 , . . . , xn ]T ∈ U and y ∈ V ⊆ R are the input and output of the fuzzy system, respectively. Ali and B l denote the linguistic variables of the input and output of the fuzzy set in U and V , respectively. In general, there are many different choices for the design of fuzzy system if the mapping is static. More detailed information of these fuzzy systems can be found in [16]. The fuzzy logic systems with singleton fuzzifier, product inference engine, center average defuzzifier are in the following form, m P

f (x) =

yl (

n Q

l=1 i=1 m Q n P l=1 i=1

µ Al (xi )) i

(11)

µ Al (xi ) i

i

l=1 i=1

where ail , i = 0, 1, . . . , n, l = 1, 2, . . . , m are the unknown constant parameters. By using singleton fuzzifier, product inference engine, center average defuzzifier, the final output value is

3. Fuzzy system

Al1

(13)

Then y is a0l + a1l x1 + · · · + anl xn

the solution of the composite system (7) and (8) is ISpS.

(l)

(12)

f (x) =

yl (

n Q

l=1 i=1 m Q n P l=1 i=1

µ Al (xi )) i

(14)

µ Al (xi ) i

where y l = Pa0l + a0l x1 + · · · + anl xn and ξ l (x) = Qn m Qn i=1 µ Ali (x i )/ l=1 i=1 µ Ali (x i ) is the fuzzy basis function. Eq. (14) can be rewritten as f (x) =

m X

y l ξ l (x).

(15)

l=1

Let ξ(x) = [ξ 1 (x), . . . , ξ m (x)], Z = [1, x1 , x2 , . . . , xn ]T ,  1  a0 a11 · · · an1  a2 a2 · · · a2  n 1  0 , Az =  . .. .. ..   .. . . .  a0m a1m · · · anm then Eq. (14) can be rewritten as f (x) = ξ(x)A z Z = ξ(x)A0z + ξ(x) Aˆ z x

(16)

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where x = [x1 , x2 , . . . , xn ]T , A0z = [a01 , a02 , . . . , a0m ]T and   1 a1 a21 · · · an1  a2 a2 · · · a2  n 2  1 . Aˆ z =  . .. .. ..   .. . . .  a1m

a2m

···

desired closed-loop system remain bounded. It is assumed that (n) ym , y˙m , . . . , ym are bounded for t ≥ 0. Though out the paper, the following assumption will be imposed on system (18). Assumption 4.1. The system states x are all available for feedback.

anm

The consequent of T–S fuzzy system rule is a linear system. This linear system is very useful in designing identification and control as compared with a single value consequent. Lemma 3.1 (Universal Approximation Theorem [16]). For any given real continuous function f on a compact set U ⊂ R n and arbitrary ε > 0, there exists a fuzzy system f ∗ (x) in the form of Eq. (12) or Eq. (16) such that

sup f ∗ (x) − f (x) ≤ ε. (17)

Assumption 4.2. The signs of gi (·) are known, and there exist bi1 ≥ bi0 > 0 such that bi1 ≥ |gi (·)| ≥ bi0 , ∀xi ∈ R i . The above assumption implies that the smooth function gi (·) is strictly either positive or negative. Hence, without loss of generality we assume that bi1 ≥ gi (·) ≥ bi0 > 0, ∀xi ∈ R i . Assumption 4.3. There exist constants bid > 0 such that, |g˙i (·)| ≤ bid , ∀x ∈ R n .

x∈U

Assumption 4.4. For 1 ≤ i ≤ n, there exists unknown positive constant pi∗ such that

4. Problem statement

|∆i (t, x)| ≤ pi∗ ϕi (|xi |),

Consider the control problem of a single-input singleoutput (SISO) nonlinear uncertain system in the perturbed strict feedback form described below

where ϕi is a known nonnegative function

x˙1 = f 1 (x1 ) + g1 (x1 )x2 + ∆1 (t, x) x˙2 = f 2 (x1 , x2 ) + g2 (x1 , x2 )x3 + ∆2 (t, x) .. . x˙i = f i (x1 , . . . , xi ) + gi (x1 , . . . , xi )xi+1 + ∆i (t, x) 1≤i ≤n−1

(18)

.. . x˙n = f n (x) + gn (x)u + ∆n (t, x) y = x1 where x = [x1 , x2 , . . . , xn ]T ∈ R n denotes system state vector, u ∈ R, y ∈ R are the system control input and output, respectively. f i (·) and gi (·), i = 1, 2, . . . , n are unknown smooth functions and ∆i (t, x), i = 1, 2, . . . , n are unknown Lipschitz continuous functions. Eq. (18) is known as strict-feedback form [4]. The reason for this name is that the nonlinearities f i (·), gi (·) depend on x1 , x2 , . . . , xi , that is, on state variables that are “feedback”. Many systems can be expressed in the above form. For example, rigid robots [33, 34], motors [35], jet engines and power converters [4,36]. The uncertain system (18) has two types of unknown nonlinear functions: one satisfies the triangularity condition and can be directly approximated by parameterized approximators, while the other, arises due to ∆i (t, x), is assumed to be partially known and consists of parametric uncertainties. The unknown nonlinear functions ∆i (t, x) could be due to many factors [37] such as measurement noise, modeling errors, external disturbance or time-varying parameters. The control objective is to design an adaptive fuzzy statefeedback controller such that the system output y follow given desired trajectory ym , while all the signal in the

∀(t, x) ∈ R+ × R n

5. Adaptive fuzzy control The detailed design procedure is described in the following steps. Step 1 and 2 are described with detailed explanations, while Step i and Step n are simplified, with the relevant equations and explanations being omitted. Step 1: Define the error variable z 1 = x1 − ym . Its derivative is z˙ 1 = f 1 (x1 ) + g1 (x1 )x2 + ∆1 − y˙m

(19)

by viewing x2 as a virtual control input for z 1 subsystem and assume that the uncertainties ∆1 are negligible in the above equation. Eq. (19) can be transformed into the following form:   f 1 (x1 ) y˙m z˙ 1 = g1 (x1 ) + x2 − . (20) g1 (x1 ) g1 (x1 ) Let us choose controller α1 as follows: α1 = x2 = −k1 z 1 −

f 1 (x1 ) y˙m + g1 (x1 ) g1 (x1 )

(21)

where k1 > 0 is constant. Substituting (21) into (20), z˙ 1 = −g1 (x1 )k1 z 1 is obtained. There exists a Lyapunov function V = 1/2 · z 12 , such that V˙ = −g1 (x1 )k1 z 12 ≤ −b10 k1 z 12 ≤ 0. Therefore, z 1 is asymptotically stable. However, since the functions f 1 (x1 ) and g1 (x1 ) are unknown and there are system uncertainties, the desired control law is unrealizable. Instead, the T–S fuzzy system can be used to approximate f 1 (x1 )/g1 (x1 ). f 1 (x1 ) = ξ1 (x1 )A01 + ξ1 (x1 )A1 x1 + ε1 g1 (x1 ) = ξ1 (x1 )A01 + ξ1 (x1 )A1 z 1 + ξ1 (x1 )A1 ym + ε1 = K ϑ1 ξ1 (x1 )w1 + ξ1 (x1 )A01 + ξ1 (x1 )A1 ym + ε1 (22)

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where w1 = Am 1 z 1 and ε1 is a approximating error. Define 1/2 m T

K ϑ1m = kA1 k = λmax (A1 A1 ), such that A1 = K ϑ1 A1 and

A ≤ 1 substituting Eq. (22) into Eq. (19) 1 z˙ 1 = g1 (x1 )[x2 + K ϑ1 ξ1 (x1 )w1 + v1 ]

−1 ˜ ˙ˆ −1 ˜ ˙ θ1 θ 1 . (30) + K ϑ1 ξ1 (x1 )w1 z 1 + v1 z 1 − Γ11 ψ1 ψ1 − Γ12

Let γ1 > 0, by the completing squares, we have

(23)

K ϑ1 ξ1 (x1 )w1 z 1 ≤

where v1 = ξ1 (x1 )A01 + ξ1 (x1 )A1 yd + ε1 + 1/g1 (∆1 − y˙m ). In the light of Assumption 4.4, we can obtain a bound for v1 as follows

≤

kv1 k ≤ θ1 φ1 (x1 )

(24)

0 −1 −1 ∗ | y˙m |, b10 p1 ] where θ1 = max[ A1 + kA1 k kym k , ε1 + b10 and φ1 (x1 ) = 1 + kξ1 k + ϕ1 (x1 ). Consider the following Lyapunov candidate: V1 =

1 1 1 z 2 + Γ −1 ψ˜ 2 + Γ −1 θ˜ 2 2g1 (x1 ) 1 2 11 1 2 12 1

(25)

2 − ψ ) and θ˜ = (θ − θˆ ). ψ where Γ11 , Γ12 > 0. ψ˜ 1 = (K ϑ1 1 1 1 1 1 2 ˆ and θ1 are the estimates of K ϑ1 and θ1 , respectively. The time derivative of V1 is

z 1 z˙ 1 g˙ 1 (x1 ) 2 −1 ˜ ˙ −1 ˜ ˙ˆ V˙1 = − z − Γ11 ψ1 ψ1 − Γ12 θ1 θ 1 g1 (x1 ) 2g12 (x1 ) 1 g˙ 1 (x1 ) 2 −1 ˜ ˙ z 1 − Γ11 ψ1 ψ1 = z 1 [x2 + K ϑ1 ξ1 (x1 )w1 + v1 ] − 2 2g1 (x1 ) −1 ˜ ˙ˆ − Γ12 θ1 θ 1 .

(26)

4γ12

ξ1 ξ1T z 12 + γ12 w1T w1

ψ˜ 1 ψ1 T 2 ξ ξ z + ξ1 ξ1T z 12 + γ12 w1T w1 . (31) 1 1 1 2 2 4γ1 4γ1

Using Eq. (24) and the relative item of Eq. (30), we get ! θˆ1 φ1 (x1 )z 1 z 1 [θ1 φ1 (x1 ) − θˆ1 φ1 (x1 )] tanh λ1 ! θˆ1 φ1 (x1 )z 1 ˆ ˆ ≤ θ1 φ1 (x1 ) kz 1 k − θ1 φ1 (x1 )z 1 tanh λ1 + (θ1 − θˆ1 )φ1 (x1 ) kz 1 k

(32)

and using the following lemma with regard to function tanh(·) [9] ! θˆ1 φ1 (x1 )z 1 ≤ η1 . (33) θˆ1 φ1 (x1 ) kz 1 k − θˆ1 φ1 (x1 )z 1 tanh λ1 Substituting Eqs. (31)–(33) into Eq. (30), such that V˙1 ≤

Define z 2 = x2 − α1 and let α1

−1 ˜ z 1 z 2 − k1∗ z 12 + γ12 w1T w1 + Γ11 ψ1

Γ11 ξ1 ξ1T z 1T − ψ˙ 1 4γ12

−1 ˜ + Γ12 θ1 (Γ12 φ1 (x1 ) kz 1 k − θ˙ˆ 1 ) + η1

ψ1 ξ1 (x1 )ξ1T (x1 ) α1 = −k1 z 1 − 4γ1 θˆ1 φ1 (x1 )z 1 − θˆ1 φ1 (x1 ) tanh λ1

2 K ϑ1

!

(34)

substituting (28) and (29) into (34),we get V˙1 ≤ −k1∗ z 12 + z 1 z 2 + γ12 w1T w1 + δ11 ψ˜ 1 (ψ1 − ψ10 )

! (27)

+ δ12 θ˜1 (θˆ1 − θ10 ) + η1 .

(35)

where k1 > 0 and λ1 > 0 are the design constants. Consider the following adaptation laws: # " 1 T 2 0 ψ˙ 1 = Γ11 ξ1 (x1 )ξ1 (x1 )z 1 − δ11 (ψ1 − ψ1 ) (28) 4γ12 h i θ˙ˆ 1 = Γ12 φ1 (x1 ) kz 1 k − δ12 (θˆ1 − θ10 ) (29)

By completing of square, we have

where δ11 , δ12 > 0 and ψ10 , θ10 ≥ 0 are design constants. The adaptive law (28) and (29) are so-called σ -modification [14], introduced to improve the robustness and avoid the parameters to drift to very large values. Using (25), a direct substitution of x2 = z 2 + α1

can choose k1∗ = k1 − ( b1d2 ) > 0.

g˙ 1 (x1 ) 2 V˙1 = z 1 [z 2 + α1 + K ϑ1 ξ1 (x1 )w1 + v1 ] − 2 z1 2g1 (x1 ) −1 ˜ ˙ −1 ˜ ˙ˆ − Γ11 ψ1 ψ1 − Γ12 θ1 θ 1 ! g˙ 1 (x1 ) ψ1 z 12 − = z 1 z 2 − k1 + 2 ξ1 (x1 )ξ1T (x1 )z 1 4γ1 2g1 (x1 ) ! θˆ1 φ1 (x1 )z 1 ˆ − θ1 φ1 (x1 )z 1 tanh λ1

1 V˙1 ≤ −k1∗ z 12 + z 1 z 2 + γ12 w1T w1 − δ11 ψ˜ 12 2 1 2 2 0 2 − δ12 θ˜1 + δ11 (K ϑ1 − ψ1 ) + δ12 (θ1 − θ10 )2 + η1 (36) 2 where the coupling term z 1 z 2 will be canceled in next step. Observing from (30) that −(k1 + g˙12 )z 12 ≤ −(k1 −( b1d2 ))z 12 .We 2g1

2b10

2b10

Step 2: Differentiating z 2 gives z˙ 2 = x˙2 − α˙ 1 = f 2 (x¯2 ) + g2 (x¯2 )x3 + ∆2 − α˙ 1 = g2 (x¯2 )[ f 2 (x¯2 )/g2 (x¯2 ) + x3 + 1/g2 (∆2 − α˙ 1 )].

(37)

The time derivative of α1 is ∂α1 ∂α1 ∂α1 ˙ ∂α1 x˙1 + ψ˙ 1 + y˙m θˆ 1 + ∂ x1 ∂ψ1 ∂ ym ∂ θˆ1 ∂α1 = ( f 1 (x1 ) + g1 (x1 )x2 + ∆1 ) + φ 1 ∂ x1 ∂α1 = f 11 + ∆1 + φ 1 ∂ x1

α˙ 1 =

(38)

291

H.F. Ho et al. / ISA Transactions 47 (2008) 286–299 ∂α1 ˙ ψ1 + ∂αˆ1 θ˙ˆ 1 + ∂∂αym1 y˙m is introduced as where φ¯ 1 = ∂ψ 1 ∂ θ1 intermediate variable which is computable. Since, f 1 (x1 ) and g1 (x1 ) are unknown function, α˙ 1 is a scalar unknown function. Let

1 ( f 2 (x¯2 ) − f 11 ) = ξ2 (x¯2 )A02 + ξ2 (x¯2 )A2 (x¯2 ) + ε2 g2 (x¯2 )

substituting Eqs. (45) and (46) into Eq. (47), we get ! 2 2 X X δl1 θ˜l2 δl2 ψ˜ l2 ∗ 2 V˙2 ≤ z 2 z 3 − kl z l − + 2 2 l=1 l=1 ! 2 2 − ψ 0 )2 X δl1 (K ϑl δl2 (θl − θl0 )2 l + + 2 2 l=1

= ξ2 (x¯2 )A02 + ξ2 (x¯2 )A2 (z 1 , z 2 ) + ξ2 (x¯2 )A2 (ym , α1 ) + ε2 =

K ϑ2 ξ2 (x¯2 )w2 + ξ2 (x¯2 )A02

+ ξ2 (x¯2 )A2 (ym , α1 ) + ε2 (39)

and substituting it into Eq. (37), we will have z˙ 2 = g2 (x1 )[x3 + K ϑ2 ξ2 (x¯2 )w2 + v2 ]

(40)

v2 = ξ2 (x¯2 )A02 + ξ2 (x¯2 )A2 (yd , α1 )

1 −1 2 1 −1 2 1 z 22 + Γ21 ψ˜ 2 + Γ22 θ˜2 . 2g2 (x¯2 ) 2 2

ηl +

l=1

2 X

γl2 wlT wl

(48)

l=1 2b10

Step i (3 ≤ i ≤ n − 1): In a similar fashion, we can design a virtual controller to make the error z i as small as possible. Differenting z i gives z˙ i = gi (x¯i )[ f i (x¯i )/gi (x¯i ) + xi+1 + 1/gi (∆i − α˙ i−1 )]

(41)

where φ2 (x2 ) = 1 + kξ2 k + ϕ2 +

kv2 k ≤ θ2 φ2 , 

∂α1

∂ x1 ϕ1 and θ2 = max A02 + kA2 k kym k + kA2 k kα1 k , −1 −1 ∗ −1 ∗  |ε2 | + b20 |φ 1 |, b20 p2 , b20 p1 . Consider the Lyapunov function candidate V2 = V1 +

2 X

where k2∗ is chosen such that k2∗ = k2 − ( b2d2 ) > 0.

where ∂α1 + ε2 + 1/g2 (∆2 − ∆1 − φ 1 ) ∂ x1

+

(42)

(49)

we also use a T–S fuzzy system to approximate the unknown function 1 ( f i (x¯i ) − f (i−1)(i−1) ) = ξi (x¯i )Ai0 + ξi (x¯i )Ai x¯i + εi gi (x¯i ) = ξi (x¯i )Ai0 + ξi (x¯i )Ai z i + ξi (x¯i )Ai (ym , . . . , αi−1 ) + εi = K ϑi ξi (x¯i )wi + ξi (x¯i )Ai0 + ξi (x¯i )Ai (ym , . . . , αi−1 ) + εi

(50)

and substitute it into Eq. (49)

The derivative of V2 is

z˙ i = gi (x¯i )[xi+1 + K ϑi ξi (x¯i )wi + vi ]

V˙2 = V˙1 + z 2 [x3 + kϑ2 ξ2 (x¯2 )w2 + v2 ] g˙ 2 (x¯2 ) −1 ˜ ˙ −1 ˜ ˙ˆ − Γ21 ψ2 ψ2 − Γ22 θ2 θ 2 . − 2 2g2 (x¯2 )

Pi−1 ∂αi−1 where kvi k ≤ θi φi , φi (x¯i ) = 1 + kξi k + ϕi + l=1 k ∂ xl kϕl . Similarly, let the virtual controller to be of the form ! θˆi φi z i ψi T ˆ ξi ξi − θi φi tanh (52) αi = −z i−1 − ki z i − λi 4γi2

(43)

Define the error variable z 3 = x3 − α2 , taking the intermediate stabilizing function α2 ! ˆ θ φ z ψ2 2 2 2 ξ2 ξ T − θˆ2 φ2 tanh (44) α2 = −z 1 − k2 z 2 − 4γ2 2 λ2 where k2 > 0 and λ2 > 0 are the design constants. Consider the following adaptation laws: " # 1 T 2 0 ψ˙ 2 = Γ21 ξ2 ξ2 z 2 − δ21 (ψ2 − ψ2 ) (45) 4γ22 h i θ˙ˆ 2 = Γ22 φ2 kz 2 k − δ22 (θˆ2 − θ20 ) (46) ψ20 , θ20

where δ21 , δ22 > 0 and ≥ 0 are design constants. Substituting Eqs. (41) and (44) into Eq. (43), such that 1 V˙2 ≤ −k1∗ z 12 + z 1 z 2 + γ12 w1T w1 − δ11 ψ˜ 12 2 1 2 0 2 2 − δ21 θ˜1 + δ11 (K ϑ1 − ψ1 ) + δ21 (θ1 − θ10 ) 2 + η1 + z 2 z 3 + z 2 α2 + z 2 K θ 2 ξ2 (x¯2 )w2 g˙ 2 (x¯2 ) −1 ˜ ˙ −1 ˜ ˙ˆ + z 2 v2 − 2 − Γ21 ψ2 ψ2 − Γ22 θ2 θ 2 2g2 (x¯2 )

(47)

(51)

where ki > 0 and λi > 0 are the design constants. Consider the following adaptation laws: " # 1 T 2 0 ψ˙ i = Γi1 ξi ξi z i − δi1 (ψi − ψi ) (53) 4γi2 h i θ˙ˆ i = Γi2 φi kz i k − δi2 (θˆi − θi0 ) (54) where δi1 , δi2 > 0 and ψi0 , θi0 ≥ 0 are design constants. In a similar procedure, by considering Lyapunov function Vi = Vi−1 +

1 1 −1 2 1 −1 2 z i2 + Γi1 ψ˜ i + Γi2 θ˜i . 2gi (x¯i ) 2 2

(55)

By using Eqs. (51)–(54) and straightforward derivation similar to those employed in the former steps, the derivative of Vi becomes ! i i X X δl1 θ˜l2 δl2 ψ˜ l2 ∗ 2 V˙i ≤ z i z i+1 − kl z l − + 2 2 l=1 l=1 ! i 2 − ψ 0 )2 X δl1 (K ϑk δl2 (θl − θl0 )2 l + + 2 2 l=1

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+

i X

δl +

l=1

i X

γl2 wlT wl

(56)

l=1

we can choose ki∗ = ki − (

gid 2 ) 2gi0

> 0.

n 2 − ψ 0 )2 X δ1l (K ϑl δ2l (θl − θl0 )2 l + + 2 2 i=1

+

Step n: This is the final step. Define the error variable as z n = xn − αn−1 , we will have z˙ n = gn (x¯n )[ f n (x¯n )/gn (x¯n ) + u + 1/gn (∆n − α˙ n−1 )].

(57)

Using a similar method, we have α˙ n−1 =

n−1 X l=1

+

∂αn−1 ( fl (x¯l ) + gl (x¯l )xl+1 + ∆l ) + ∂ xl

n−1 X l=1

n−1 X l=1

∂αn−1 ψ˙ l ∂ψl

∂αn−1 ˙ ∂αn−1 y˙m θˆl + ∂ ym ∂ θˆl

= f (n−1)(n−1) +

n−1 X ∂αn−1 l=1

∂ xl

∆l + φ¯ (n−1)(n−1)

(58)

we also use a T–S fuzzy system to approximate the unknown function 1 ( f n (x¯n ) − f (n−1)(n−1) ) gn (x¯n ) = ξn (x¯n )A0n + ξn (x¯n )An z n + ξn (x¯n )An (ym , . . . , αn−1 ) + εn = K ϑn ξn (x¯n )wn + ξn (x¯n )A0n + ξi (x¯n )An (ym , . . . , αn−1 ) + εn (59)

m m m where wn = An z n , K ϑn = An and An = K ϑn An and substitute it into Eq. (57) z˙ n = gn (x¯n )[u + K ϑn ξn (x¯n )wn + vn ]

(60)

Pn−1

∂αn−1 where kvn k ≤ θn φn , φn = 1 + kξn k + ϕn + l=1 ∂ xl ϕl . Similarly letting ! ψn θˆn φn z n T (61) u = −z n−1 − kn z n − ξn ξ − θˆn φn tanh λn 4γn2 n

1 1 −1 2 1 −1 2 z n2 + Γn1 ψ˜ n + Γn2 θ˜n . 2gn (x¯n ) 2 2

l=1

≤−

n X l=1

n X

γl2 wlT wl

l=1

kl∗ zl2

n X δl1 θ˜l2 δl2 λ˜ l2 − + 2 2 l=1

! + γ 2 kwk2 + η¯ (65)

where kn∗ is chosen such that kn∗ = kn − ( gnd2 ) > 0. Where 2gn0 Pn 2 − ψ 0 )2 /2 + δ (θ − θ 0 )2 /2), w = η¯ = l=1 (ηl + δ1l (K ϑl 2l l l l [w1 , w2 , . . . , wn ]T and γ = (γ12 + γ22 + · · · + γn2 )1/2 . Theorem 5.1. Consider the closed-loop system consisting of Eq. (18), the controller Eq. (61) with the intermediate stabilizing functions αi , i = l, . . . , n, and theupdate  laws ψi , θˆi . If we choose γ < 1 and ki∗ = ki −

gid 2 2gid

> 1,

i = 1, 2, . . . , n in Eq. (65). Then for bounded initial conditions, we have all the signals in the closed-loop system remain bounded and the output tracking error converges to a small neighborhood around zero.

Proof. In order to use the small gain theorem [30,31], it is necessary to construct two subsystems Σz˜ w and Σw z˜ in composite feedback form. According to the error variable z i , i = 1, 2, . . . , n and fuzzy system to approximate the unknown function ( f n (x¯n ) − f (n−1)(n−1) )/gn (x¯n ), i = 1, 2, . . . , n, then the closed loop systems can be given as follows   z˙ i = gi (x¯i )[x¯i+1 + K ϑi ξi (x¯i )wi + vi ],  1≤i ≤n−1 Σz˜ w : (66) z ˙ = g (x¯n )[u + K ϑn ξn (x¯n )wn + vn ]  n n   z˜ = H (z) = z where w = [w1 , w2 , . . . , wn ]T is considered as the virtual input and z˜ as the output of the system Σz˜ w . If we pick ki∗ > 1, i = 1, 2, . . . , n from Eq. (65), we get V˙n ≤ −z 2 + γ 2 kwk2 + η. ¯

(62) (63)

where δi1 , δi2 > 0 and ψi0 , θi0 ≥ 0 are design constants. Consider the overall Lyapunov function candidate Vn = Vn−1 +

ηl +

The proof is given as follows:

= ξn (x¯n )A0n + ξn (x¯n )An x¯n + εn

and consider the following adaptation law   1 T 2 0 ψ˙ n = Γn1 ξ ξ z − δ (ψ − ψ ) n n1 n n 4γn2 n n i h θ˙ˆ n = Γn2 φn kz n k − δn2 (θˆn − θn0 )

n X

!

(64)

By using Eqs. (60)–(63) and straightforward derivation similar to those employed in the former steps, the derivative of Vn becomes ! n n X X δl1 θ˜l2 δl2 ψ˜ l2 ∗ 2 ˙ Vn ≤ − kl z l − + 2 2 l=1 l=1

(67)

By using ISpS Lyapunov theorem, we propose the robust fuzzy control scheme such that the requirement of ISpS for system Σz˜ w can be stratified with the functions α3 (s) = s 2 and α3 (s) = γ 2 s 2 of class K ∞ . We can get a gain function γz (s) of system Σz˜ w as follow: γz (s) = α1−1 ◦ α2 ◦ α3−1 ◦ α4 where α1 (z) ≤ V (x) ≤ α2 (z).

∀s > 0 

For system Σw z˜ , it is static system, we have  w1 = Am  1 z1   w2 = Am [z 1 , z 2 ]T 2 Σw z˜ : . .  .    T wn = Am n [z 1 , z 2 , . . . , z n ]

(68)

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Fig. 1. Adaptive fuzzy control of nonlinear systems in the “strict-feedback form”.

Fig. 2. Membership function of state variables.

Eq. (68) can be re-written as   w1 w2    w =  .  = K (z)  ..  wn  m A1 0 ··· 0  Am1 Am2 · · · 0 2 2  = . .. .. .  . . ··· . Am1 n

Am2 n

···

Amn n

  z1  z2      ..  = Az  . 

(69)

zn

we have kwk ≤ kAk kzk = γ¯ kzk .

(70)

Then the gain function for system Σw z˜ is γw (s) = γ¯ s. Under the condition of small gain theorem, there is a γz ◦ γw (s) < s. Consider the interconnected system Σz˜ w and Σw z˜ , in fact γ¯ = kAk < 1 and we have γ γ¯ < 1. Indeed, the small gain type condition can be satisfied by picking γ < 1, such that it can prove that the composite closed-loop system is ISpS. By substituting Eq. (70) into (65), the ISpS-Lyapunov function is satisfies 1 1 V˙n ≤ −z T Q k z − ψ˜ T Q δ1 ψ˜ − θ˜ T Q δ2 θ˜ + γ¯ 2 γ 2 kzk2 + η¯ 2 2

Fig. 3. The responses for the inverted pendulum system using the indirect adaptive fuzzy control with ym = π/30 sin t (t): (a) desired trajectory ym (t) (solid line) and system output y(t) (dash line) (b) control input u(t).

1 1 ≤ −z T Q k z − ψ˜ T Q δ1 ψ˜ − θ˜ T Q δ2 θ˜ + kzk2 + η¯ 2 2 ≤ −cVn + η¯

(71)

where Q k = diag[k1∗ , k2∗ , . . . , kn∗ ], Q δ1 = diag[δ11 , δ21 , . . . , δn1 ]T ,

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Fig. 4. The responses for the inverted pendulum system using the proposed controller with ym = π/30 sin t (t): (a) desired trajectory ym (t) (solid line) and system output y(t) (dash line). (b) control input u(t) (c) estimated parameter: θˆ2 , ψ2 .

Q δ2 = diag[δ12 , δ22 , . . . , δn2 ]T , c = min{2((λmin (Q k ) − 1)/b0−1 ), λmin (Q δ1 )/λmax (Γ1−1 ), λmin (Q δ2 )/λmax (Γ2−1 )}, Γ1 = [Γ11 , Γ21 , . . . , Γn1 ]T and Γ2 = [Γ12 , Γ22 , . . . , Γn2 ]T . Let ρ := η/c ¯ then Eq. (71) satisfies ¯ 0 ≤ Vn (t) ≤ ρ + (Vn (0) − ρ)e−ηt

(72)

Eq. (72) means that Vn (t) eventually is bounded by ρ. This proves that all signals of the closed-loop system are UBB. Thus, the tracking error z 1 = x1 − ym is also UBB. This concludes the proof. Remark 5.1. Decreasing δi1 and δi2 will help to reduce the size of ρ. However, if δi1 and δi2 are too small, it may not be enough to prevent the parameter estimates form drifting to very large values in the presence of fuzzy approximation errors. The small λi might result in a variation of a high gain control. Therefore, in practical applications, the design parameters should be adjusted carefully for achieving suitable transient performance and control action.

Remark 5.2. Compared with the works in [21,22], it is assumed that the gain functions are constants or known function. However, this assumption cannot be satisfied in many cases. In [23,26], gain functions are assumed to be unknown and a backstepping design is proposed that incorporates adaptive approximator techniques. However, due to the integral-type Lyapunov function introduced, this approach is complicated and difficult to use in practice. The proposed control scheme can cope with unknown gain functions and avoids controller singularity problem completely without the requirement for integral-type Lyapunov functions. The overall adaptive fuzzy control scheme is shown in Fig. 1. To summarize the above analysis, the design algorithm for the adaptive fuzzy control of uncertain nonlinear system is outlined as follows Design algorithm: Step 1: Select the desired constants ki and γi , i = 1, 2, . . . , n. Step 2: Construct membership functions of fuzzy rules for approximation of unknown functions f i /gi . Step 3: Choose the appropriate adaptation gains Γi1 , Γi2 , δi1 , δi2 , i = 1, 2, . . . , n.

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Fig. 5. The responses for the second order system using the proposed controller with ym = π/5(sin(t) + 0.3 sin(3t)): (a) desired trajectory ym (t) (solid line) and system output y(t) (dash line). (b) control input u(t) (c) estimated parameter: θˆ1 , θˆ2 , ψ1 , ψ2 .

Step 4: Obtain the stabilizing functions αi , i = 1, 2, . . . , n − 1 and update laws ψ˙ i , θ˙ˆ i , i = 1, 2, . . . , n. Step 5: Obtain the control from (61). 6. Simulation examples In this section, we apply our proposed adaptive fuzzy controller for three cases. The first example is an inverted pendulum system. The second example is a second order nonlinear system. In the last one, we apply the proposed controller for a one-line robot with the inclusion of motor dynamics. The simulations are written in C language and are implemented in LabWindows environment. The performance index used is the integral of the Absolute value of the Error Z tf IAE = |x1 − ym |dt. t0

Example 1: Inverted pendulum system. In this example, we verify at the validity of the design approach on the tracking control of an inverted pendulum system. The dynamics of the system can be expressed as equations [16] x˙1 = x2

x˙2 =

g sin x1 − mlx22 cos x1 sin x1 /(m c + m) l(4/3 − m cos2 x1 /(m c + m)) cos x1 /(m c + m) u+∆ + l(4/3 − m cos2 x1 /(m c + m))

where x1 and x2 are the angular position and velocity of the pole. g = 9.8 m/s2 is the acceleration due to gravity, m c = 1 kg is the mass of cart, m = 0.1 kg is the mass of pole, l = 0.5 m is the half-length of pole, u is the applied force. The control objective is to maintain the system to track the desired trajectory ym = (π/30) sin(t). The Gaussian type membership functions for system state x1 and x2 are constructed as in Fig. 2. The initial states are x = [−π/60, 0]T and step size 0.01 s. A certain disturbance ∆ = 0.01 ∗ sin(5t) is added to verify the robustness for the controller. The stabilizing function α1 is α1 = −8z 1 with z 1 = x1 − ym and we obtain the control law ψ2 θˆ2 φ2 z 2 u = −z 1 − 4z 2 − ξ2 ξ2T z 2 − θˆ2 φ2 tanh 4 · 0.8 0.5

!

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Fig. 6. The responses for the second order system using the proposed controller with ym = 0: (a) desired trajectory ym (t) (solid line) and system output y(t) (dash line). (b) control input u(t) (c) estimated parameter: θˆ1 , θˆ2 , ψ1 , ψ2 .

with z 2 = x2 − α1 and γ2 = 0.8. The adaptive law for ψ2 and θˆ2 are   1 T 2 ˙ ψ2 = 70 ξ2 ξ z − 0.03(ψ2 − 0.2) 4 · 0.82 2 2 θ˙ˆ = 8[φ kz k − 0.03(θˆ − 0.2)].

are needed to be adapted on-line. Therefore, the computation load of the algorithm can be alleviated and it is convenient to implement this algorithm for a variety of on-line engineering control problems. In examples 2 and 3, a random disturbance is added to verify the robustness of proposed controller.

For comparison, the indirect adaptive fuzzy control (IAFC) [16]

Example 2: Second order system. In this example, we verify at the validity of the design approach on the tracking and regulation control of a second order system. The dynamic equation of such system is given by

2

2

2

2

T u I = 1/g(x|θ ˆ g )[− fˆ(x|θ f ) + y¨m + k e]

under the same conditions is also demonstrated. The adaptation laws to adjust the parameters θ f and θg in the IAFC are θ˙ f = −γ1 eT Pbξ(x),

θ˙g = −γ2 eT Pbξ(x)u I .

The parameters are as γ1 = 50, γ2 = 1, k = h chosen i  T  T 5 2 1 , P = 15 and B = 0 1 . The initial 5 5 consequent parameters of fuzzy rules are chosen randomly in the interval [−2, 2]. Figs. 3 and 4 shown the results with IAFC and proposed controller. It can be seen that the control performances are almost the same. However, for the IAFC, there are 25 rules to approximate the system function f (·) and g(·) and 50 consequent parameters need to be adapted on-line. For the proposed controller, only two parameters ψ2 and θˆ2

x˙1 = x1 e−0.5x1 + (1 + x12 )x2 + ∆1 (t) x˙2 = x1 x22 + (3 + cos(x1 x2 ))u + ∆2 (t). The control objective is to maintain the system to track the desired angle trajectory, ym = π/5(sin(t) + 0.3 sin(3t)) and to regulate a constant set-point value ym = 0. A certain disturbance ∆1 = sin(x1 ), ∆2 = 2x1 sin(t) is added to verify the robustness for the controller. Moreover, a Gaussian noise with mean zero and variance 0.001 is injected at the output of the system. The membership functions for system state x1 and x2 are constructed as in Fig. 2. The initial states are x(0) =

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Fig. 7. The responses for the one-link robot system using the proposed controller with ym = π/5(sin(0.5t) + 0.5 sin(1.5t)): (a) desired trajectory ym (t) (solid line) and system output y(t) (dash line). (b) control input u(t) (c) estimated parameter: θˆ2 , θˆ3 (d) estimated parameter: ψ2 , ψ3 .

[0.2, 0]T and step size 0.01 sec. The stabilizing function α1 is ! ψ1 θˆ1 φ1 z 1 T α1 = −4z 1 − ξ1 ξ z 1 − θˆ1 φ1 tanh 4 · 0.8 1 0.5

the regulation results with the same parameters except the initial state x(0) = [0.5, 0.5]T . The IAE is calculated for this example to be IAE = 7.85.

with z 1 = x1 − ym . The adaptive law for ψ1 and θˆ1 are   1 T 2 ψ˙ 1 = 90 ξ ξ z − 0.03(ψ − 0.2) 1 1 4 · 0.82 1 1 θ˙ˆ = 8[φ kz k − 0.03(θˆ − 0.2)]

D q¨ + B q˙ + N sin(q) = τ + τd

1

1

1

θˆ2 φ2 z 2 ψ2 ξ2 ξ2T z 2 − θˆ2 φ2 tanh u = −z 1 − 4z 2 − 4 · 0.8 0.5

!

with z 2 = x2 − α1 and γ2 = 0.8. The adaptive law for ψ2 and θˆ2 are   1 T 2 ˙ ψ2 = 70 ξ2 ξ z − 0.03(ψ2 − 0.2) 4 · 0.82 2 2 θ˙ˆ = 8[φ kz k − 0.03(θˆ − 0.2)]. 2

2

M τ˙ + H τ = u − K m q˙

1

and we obtain the control law

2

Example 3: One-link robot system. In this example, we consider a one-link manipulator with the inclusion of motor dynamics. The dynamic equation of such system is given by [38,39]

2

Fig. 5 shows the tracking results with the IAE = 91.15. It can be seen that the tracking performance can achieve. Fig. 6 shows

where q, q, ˙ q¨ denote the link position, velocity and acceleration, respectively. τ is the torque produced by the electrical subsystem and τd represents the torque disturbance. u is the control input used to represent the electromechanical torque. D = 1 kg m2 is the mechanical inertia, B = 1 Nm s/rad is the coefficient of viscous friction at the joint, N = 10 is a positive constant related to the mass of the load and the coefficient of gravity, M = 0.05H is the armature inductance, H = 0.5Ω is the armature resistance, K m = 10 Nm/A is the back-emf coefficient. Above equation can be expressed in the form of (18) by noting that x1 = q, x˙1 = x2

x2 = q, ˙

x3 = τ

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Fig. 8. The responses for the one-link robot system using the proposed controller with ym = 0: (a) desired trajectory ym (solid line) and system output y(t) (dash line). (b) control input u(t) (c) estimated parameter: θˆ2 , θˆ3 (d) estimated parameter: ψ2 , ψ3 .

N 1 B x2 − sin x1 + (x3 + τd ) D D D Km H u x˙3 = − x2 − x3 + . M M M

and we obtain the control law

The control objective is to maintain the system to track the desired angle trajectory, ym = π/5(sin(0.5t) + 0.5 sin(1.5t)) and to regulate a constant set-point value ym = 0. A Gaussian noise with mean zero and variance 0.001 is injected at the output of the system. The membership functions for system state x1 , x2 and x3 are constructed as in Fig. 2. The initial states are x(0) = [0.4, 0.2, −0.2]T and step size 0.01 s. The stabilizing function α1 and α2 are

with z 3 = x3 − α2 and γ2 = 0.8. The adaptive law for ψ3 and θˆ3 are   1 T 2 ˙ ψ3 = 60 ξ3 ξ z − 0.05(ψ3 − 0.1) 4 · 0.82 3 3 θ˙ˆ = 4[φ kz k − 0.05(θˆ − 0.1)].

x˙2 = −

θˆ3 φ3 z 3 ψ3 ξ3 ξ3T z 3 − θˆ3 φ3 tanh u = −4z 2 − 4z 3 − 4 · 0.8 0.4

α1 = −8z 1 ψ2 θˆ2 φ2 z 2 α2 = −z 1 − 4z 2 − ξ2 ξ2T z 2 − θˆ2 φ2 tanh 4 · 0.8 0.4

!

with z 1 = x1 − ym and z 2 = x2 − α1 . The adaptive law for ψ2 and θˆ2 are   1 T 2 ˙ ψ2 = 75 ξ2 ξ z − 0.05(ψ2 − 0.5) 4 · 0.82 2 2 θ˙ˆ = 5[φ kz k − 0.05(θˆ − 0.5)] 2

2

2

2

3

3

3

!

3

Fig. 7 shows the tracking results with the IAE = 223.5. It can be seen that the tracking performance is good even in the presence of random disturbance. Fig. 8 shows the regulation results with the same parameters. The IAE is calculated for this example to be IAE = 6.88. The results are satisfactory. 7. Conclusions In this paper, by combining backstepping technique with the small gain theorem and using T–S type fuzzy systems to approximate unknown functions, the adaptive fuzzy control for a class of uncertain nonlinear systems in strict-feedback form is developed. The adaptive control scheme avoids the singularity

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