Adaptive Neural Control for a Class of Uncertain Nonlinear Systems in ...

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 2, APRIL 2009

431

Adaptive Neural Control for a Class of Uncertain Nonlinear Systems in Pure-Feedback Form With Hysteresis Input Beibei Ren, Student Member, IEEE, Shuzhi Sam Ge, Fellow, IEEE, Chun-Yi Su, Senior Member, IEEE, and Tong Heng Lee, Member, IEEE

Abstract—In this paper, adaptive neural control is investigated for a class of unknown nonlinear systems in pure-feedback form with the generalized Prandtl–Ishlinskii hysteresis input. To deal with the nonaffine problem in face of the nonsmooth characteristics of hysteresis, the mean-value theorem is applied successively, first to the functions in the pure-feedback plant, and then to the hysteresis input function. Unknown uncertainties are compensated for using the function approximation capability of neural networks. The unknown virtual control directions are dealt with by Nussbaum functions. By utilizing Lyapunov synthesis, the closed-loop control system is proved to be semiglobally uniformly ultimately bounded, and the tracking error converges to a small neighborhood of zero. Simulation results are provided to illustrate the performance of the proposed approach. Index Terms—Adaptive control, hysteresis, neural networks (NNs), nonlinear systems, pure-feedback.

I. I NTRODUCTION

C

ONTROL of nonlinear systems with unknown hysteresis nonlinearities has been an active topic, since hysteresis nonlinearities are common in smart material-based actuators, such as piezoceramics and shape memory alloys. It is challenging to control a system with hysteresis nonlinearities, because they severely limit system performance such as giving rise to undesirable inaccuracy or oscillations and may even lead to instability [1]. In addition, due to the nonsmooth characteristics of hysteresis nonlinearities, traditional control methods are insufficient in dealing with the effects of unknown hysteresis. Therefore, advanced control techniques are much needed to mitigate the effects of hysteresis. One of the most common approaches is to construct an inverse operator to cancel the effects of the hysteresis as in [1] and [2]. However, it is a challenging task to construct the inverse operator for the hysteresis due to the complexity and uncertainty of hysteresis. To circumvent these difficulties, alternative control approaches that do not need an inverse model have

Manuscript received March 23, 2008; revised June 29, 2008. Current version published March 19, 2009. This work was supported in part by A*STAR SERC Singapore under Grant 052-101-0097. This paper was recommended by Associate Editor M. S. de Queirroz. B. Ren, S. S. Ge, and T. H. Lee are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576 (e-mail: [email protected]; [email protected]; [email protected]). C.-Y. Su is with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC H3G 1M8, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCB.2008.2006368

also been developed in [3]–[6]. In [3] and [4], robust adaptive control and adaptive backstepping control were investigated for a class of nonlinear system with unknown backlashlike hysteresis, respectively. In [5] and [6], adaptive variable structure control and adaptive backstepping control were proposed for a class of continuous-time nonlinear dynamic systems preceded by a hysteresis nonlinearity with the conventional Prandtl– Ishlinskii (P–I) model representation, respectively. In this paper, we consider a class of unknown nonlinear systems in pure-feedback form preceded by a generalized P–I hysteresis input. Compared with the backlashlike hysteresis and the conventional P–I hysteresis model discussed in [3]–[6], the generalized P–I hysteresis model proposed in [7] can capture the hysteresis phenomenon more accurately and accommodate more general classes of hysteresis shapes by adjusting not only the density function but also the input function. However, the difficulty in dealing with the generalized P–I hysteresis model lies in the fact that the input function in the generalized P–I hysteresis model is unknown and nonaffine. Motivated by the works in [8]–[10], in this paper, we adopt the mean-value theorem to transform the unknown nonaffine input function to a partially affine form, which can be handled by extending some available techniques for affine nonlinear system control in the literature. For pure-feedback systems, the cascade and nonaffine properties make it difficult to find the explicit virtual controls and the actual control to stabilize the pure-feedback systems. In [11] and [12], much simpler pure-feedback systems, where the last one or two equations were assumed to be affine, were discussed. In [13], an “ISS-modular” approach combined with the smallgain theorem was presented for adaptive neural control of the completely nonaffine pure-feedback system. In this paper, we also consider a class of unknown nonlinear systems in purefeedback form. The nonaffine problem in the control variable and virtual ones is dealt with by adopting the mean-value theorem, motivated by the works in [8]–[10], without the assumptions that the last one or two equations are affine as in [11] and [12]. The unknown virtual control directions are dealt with by using Nussbaum functions. Our main contributions in this paper are highlighted as follows. 1) To the best of our knowledge, it is the first time, in the literature, that the tracking control problem of unknown nonlinear systems in pure-feedback form with the generalized P–I hysteresis input is investigated.

1083-4419/$25.00 © 2008 IEEE

432

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 2, APRIL 2009

2) The difficulty in dealing with the generalized P–I hysteresis model, i.e., the nonaffine problem of the uncertain nonlinear input function in the generalized P–I hysteresis model, is solved by adopting the mean-value theorem. 3) Different from the previous works in [5] and [6], the σ-modification is included in the adaptation law of estimate of density function pˆ(t, r) to establish the different closed-loop stability. 4) The combination of the mean-value theorem and Nussbaum functions is used to solve the nonaffine and unknown virtual control direction problems in the purefeedback nonlinear systems, without the assumptions that the last one or two equations are affine as in [11] and [12]. The organization of this paper is as follows. The problem formulation and preliminaries are given in Section II. In Section III, adaptive neural control is developed for a class of unknown nonlinear systems in pure-feedback form with the uncertain generalized P–I hysteresis input. The closedloop system stability is analyzed as well. Results of extensive simulation studies are shown to demonstrate the effectiveness of the approach in Section IV, followed by the conclusion in Section V. II. P ROBLEM F ORMULATION AND P RELIMINARIES Throughout this paper, (˜·) = (ˆ·) − (·),  ·  denotes the twonorm, and λmin (·) and λmax (·) denote the smallest and largest eigenvalues of a square matrix (·), respectively. Definition 1: The solution X(t) of (7) is semiglobally uniformly ultimately bounded (SGUUB) if, for any compact set Ω0 and all X(t0 ) ∈ Ω0 , there exists an μ > 0 and T (μ, X(t0 )) such that X(t) ≤ μ for all t ≥ t0 + T [14]. A. Problem Formulation Consider the following class of unknown nonlinear system in pure-feedback form whose input is preceded by the uncertain generalized P–I hysteresis: xj , xj+1 ), 1≤j ≤n−1 x˙ j = fj (¯ xn , u) + d(t) x˙ n = fn (¯ y = x1

(1)

where x ¯j = [x1 , . . . , xj ]T ∈ Rj is the vector of states of the first j differential equations, and x ¯n = [x1 , . . . , xn ]T ∈ Rn ; fj (·) and fn (·) are unknown smooth functions; d(t) is a bounded disturbance; y ∈ R is the output of the system; and u ∈ R is the input of the system and the output of the hysteresis nonlinearity, which is represented by the generalized P–I model in [7] as follows: D u(t) = h(v)(t) −

p(r)Fr [v](t)dr 0

Fr [v](0) = hr (v(0), 0) Fr [v](t) = hr (v(t), Fr [v](ti )) , 0≤i≤N −1 for ti < t ≤ ti+1 , (2) hr (v, w) = max (v − r, min(v + r, w))

where v is the input to the hysteresis model; 0 = t0 < t1 < · · · < tN = tE is a partition of [0, tE ] such that the function v is ]; p(r) is a given monotone on each of the subintervals (ti , ti+1 ∞ density function satisfying p(r) ≥ 0 with 0 rp(r)dr < ∞; D is a constant so that the density function p(r) vanishes for large values of D; Fr [v](t) is known as the play operator; and h(v) is the hysteresis input function that satisfies the following assumptions [7]. Assumption 1: The function h : R → R is odd, nondecreasing, and locally Lipschitz continuous and satisfies limv→∞ h(v) → ∞ and (dh(v)/dv) > 0 for almost every v ∈ R. Assumption 2: The growth of the hysteresis function h(v) is smooth, and there exist positive constants h0 and h1 such that 0 < h0 ≤ (dh(v)/dv) ≤ h1 . The objective is to design adaptive neural control v(t) for systems (1) and (2) such that all signals in the closed-loop system are bounded, while the tracking error between the output y and some reference trajectory yd converges to a neighborhood of zero. Remark 1: The conventional P–I hysteresis model studied in [5] and [6] is only a special case of the generalized P–I hysteresis model. If we select the input function h(v)(t) = p0 v D with p0 = 0 p(r)dr in (2), then the generalized P–I hysteresis model becomes a conventional P–I hysteresis model u(t) = D p0 v − 0 p(r)Fr [v](t)dr. For the conventional P–I hysteresis model, the different hysteresis shapes are formulated by adjusting the density function only. However, for the generalized P–I hysteresis model, both the density function and the input function can be adjusted to describe a more general class of hysteresis characteristics. Remark 2: Compared with the conventional P–I hysteresis model, the difficulty in dealing with the generalized P–I hysteresis model lies in the fact that the input function h(v) is unknown, which needs some new treatments. In this paper, motivated by the works in [8]–[10], we adopt the mean-value theorem to transform the unknown nonaffine input function to a partially affine form, which can be seen as a multiplication of a control term with a function of control. As such, we can extend the available techniques for affine nonlinear system control in the literature to solve our problem. Remark 3: Although it appears possible to rewrite (1) and (2) into the nonaffine form x˙ = f (x, v), it still cannot be directly handled by the method proposed by Ge and Zhang [9], in which the mean-value theorem and the implicit-function theorem were adopted to handle the nonaffine problem. The reason is that if we want to apply the mean-value theorem and the implicitfunction theorem to a function, one requirement is that the firstorder derivative of the function is not equal to zero. However, due to the nonsmooth characteristics of hysteresis, the function f (x, v) transformed from (1) and (2) is nondifferentiable and thus does not satisfy the conditions of applying the mean-value theorem and the implicit-function theorem. Therefore, we only apply the mean-value theorem to the smooth functions in (1), namely, fj (·), fn (·), and the hysteresis input function h(v). For the nonsmooth function Fr [v](t) in (2), we will develop a new treatment later. Remark 4: There are many physical processes whose dynamics can be described by nonlinear differential equations

REN et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS

like (1) and (2). Examples include some chemical reaction processes such as the continuously stirred tank reactor (CSTR) system given in [15] and [16]. Within the tank reactor, two chemicals are mixed and react to produce compound A at a concentration Ca . The objective is to manipulate the coolant flow rate qc to control the concentration Ca at a desired value. The system is a pure-feedback system, which is nonaffine in the control input qc . According to [17] and [18], the control valve that controls the coolant flow rate qc exhibits considerable hysteresis. Since the generalized P–I hysteresis model can capture the hysteresis phenomenon more accurately and accommodate more general classes of hysteresis shapes by adjusting both the density function and the input function, we can adopt the generalized P–I hysteresis model to represent the hysteresis nonlinearity between the coolant flow rate qc and the aperture of the control valve v. Therefore, we can regard the CSTR system as a physical example of pure-feedback systems with input hysteresis like (1) and (2). To facilitate control design later in Section III, the following assumptions are needed. Assumption 3: The desired trajectory yd and its time deriva(n) tives up to the nth order yd are continuous and bounded. Based on Assumption 3, we define the trajectory vector x ¯d(j+1) = [ yd y˙ d · · · yd(j) ]T , where j = 1, . . . , n − 1, (j) which is a vector from yd to its jth time derivative, yd , which will be used in the subsequent control design. Assumption 4: There exists an unknown constant d∗ such that |d(t)| ≤ d∗ . Assumption 5: There exist a known constant pmax such that p(r) ≤ pmax for all r ∈ [0, D]. Remark 5: It is reasonable to set an upper bound for the density function p(r), based on its properties that p(r) ≥ 0 with ∞ 0 rp(r)dr < ∞. According to the mean-value theorem [19], we can express fj (·, ·) in (1) as follows:    ∂fj (¯ xj , xj+1 )  0 ¯j , xj+1 + xj , xj+1 ) = fj x fj (¯  θj ∂xj+1 xj+1 =xj+1   × xj+1 − x0j+1 , 1 ≤ j ≤ n − 1  xn , u)  ∂fn (¯ fn (¯ xn , u) = fn (¯ xn , u0 ) + (u − u0 )  ∂u θ n u=u (3) θ

j where xj+1 = θj xj+1 + (1 − θj )x0j+1 , with 0 < θj < 1, 1 ≤ j ≤ n − 1, and uθn = θn u + (1 − θn )u0 , with 0 < θn < 1. By choosing x0j+1 = 0 and u0 = 0, (3) can be written as  xj , xj+1 )  ∂fj (¯ xj , xj+1 ) = fj (¯ xj , 0) + fj (¯  θ ∂xj+1 xj+1 =x j

433

u)/∂u)|u=uθn , which are also unknown nonlinear functions. Substituting (4) into (1), we have θ

1≤j ≤n−1

j xj , 0) + gj (¯ xj , xj+1 )xj+1 , x˙ j = fj (¯

x˙ n = fn (¯ xn , 0) + gn (¯ xn , uθn )u + d(t) y = x1 .

(5)

In addition, according to the mean-value theorem [19], there also exists a constant θ0 (0 < θ0 < 1) such that the unknown input function h(v) in (2) satisfies the following property:  ∂h(·)  ∗ h(v) = h(v ) + (v − v ∗ ) ∂v v=vθ0 where v θ0 = θ0 v + (1 − θ0 )v ∗ . According to Assumptions 1, 2 and the implicit-function theorem [20], we can find v ∗ such that h(v ∗ ) = 0. Defining  ∂h(·)  θ0 g0 (v ) = ∂v v=vθ0 we have h(v) = g0 (v θ0 )(v − v ∗ ). Therefore, we can rewrite (2) as D

∗

u(t) = g0 (v )v − g0 (v )v − θ0

θ0

p(r)Fr [v](t)dr.

(6)

0

Substituting (6) into (5) leads to our unified system   θj ¯j , xj+1 xj+1 , x˙ j = fj (¯ xj , 0) + gj x 1≤j ≤n−1 x˙ n = fn (¯ xn , 0) + gn (¯ xn , uθn ) ⎡ × ⎣g0 (v θ0 )v − g0 (v θ0 )v ∗ −

D

⎤ p(r)Fr [v](t)dr⎦ + d(t)

0

y = x1 .

(7)

Assumption 6: There exist constants g j and g¯j such that 0 < g j ≤ |gj (·)| ≤ g¯j < ∞ for j = 1, . . . , n. Remark 6: Assumption 6 implies that smooth functions gj (·) for j = 1, . . . , n are strictly either positive or negative, which is reasonable because gj (·), being away from zero, is the controllable condition of system (7), which is made in most control schemes [21], [22]. Without loss of generality, we shall xn , uθn ) > 0, while no knowledge is required assume that gn (¯ for the signs of gj (·), where j = 1, 2, . . . , n − 1.

j+1

× xj+1 ,

1≤j ≤n−1  xn , u)  ∂fn (¯ fn (¯ xn , u) = fn (¯ xn , 0) + u.  ∂u u=uθn

B. RBFNN Approximation (4) θ

j xj , xj+1 )= For convenience of analysis, we define gj (¯ θn (∂fj (¯ xj , xj+1 )/∂xj+1 )| and gn (¯ xn , u ) = (∂fn (¯ xn , θj

xj+1=xj+1

In control engineering, the radial basis function neural network (RBFNN) has been successfully used as a linearly parameterized function approximator to achieve various objectives, such as modeling, identification, and feedback linearization, by virtue of its universal approximation capabilities, learning and

434

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 2, APRIL 2009

adaptation, and parallel distributed structures [14], [23]–[26]. In this paper, the following RBFNN is used to approximate the continuous function h(Z) : Rq → R: hnn (Z, W ) = W T S(Z)

(8)

where the input vector Z ∈ Ω ⊂ Rq , weight vector W = [w1 , w2 , . . . , wl ]T ∈ Rl , with the neural network (NN) node number l > 1; and S(Z) = [s1 (Z), . . . , sl (Z)]T , with si (Z) being chosen as the commonly used Gaussian functions, which have the form

 −(Z − μi )T (Z − μi ) , i = 1, 2, . . . , l (9) si (Z) = exp ηi2 where μi = [μi1 , μi2 , . . . , μiq ]T is the center of the receptive field and η is the width of the Gaussian function. It has been proven that network (8) can approximate any continuous function over a compact set ΩZ ⊂ Rq as h(Z) = hnn (Z, W ∗ ) + ε(Z)

∀Z ∈ ΩZ

(10)

where W ∗ is the ideal NN weights and ε(Z) is the NN approximation error [23]. Assumption 7: There exist ideal constant weights W ∗ such that |ε(Z)| ≤ ε∗ with constant ε∗ > 0 for all Z ∈ ΩZ . Moreover, W ∗ is bounded by W ∗  ≤ wm on the compact set ΩZ . The ideal weights W ∗ are “artificial” quantities that are required for analytical purposes. According to the discussion in [14], W ∗ is defined as follows: 

∗ W = arg min sup |hnn (Z, W ) − h(Z)| (W )

Z∈ΩZ

which is unknown and needs to be estimated in control design. ˆ be the estimate of W ∗ , and let W ˜ =W ˆ − W ∗ be the Let W weight estimation error. Remark 7: Although RBFNN is employed in our control design, it can be replaced by other linearly parameterized function approximators such as high-order NNs, fuzzy systems, polynomials, splines, and wavelet networks without difficulty. For a unified framework of different approximation structures in adaptive approximation-based control, interested readers can refer to [27]. The following lemma is useful for establishing the stability properties of the closed-loop system. Lemma 1: Let V (·), ζ(·) be the smooth functions defined on [0, tf ) with V (t) ≥ 0, ∀t ∈ [0, tf ), and let N (·) be an even smooth Nussbaum-type function [28]. If the following inequality holds: V (t) ≤ c0 + e−c1 t

t ˙ c1 τ dτ [g(·)N (ζ) + 1] ζe

∀t ∈ [0, tf )

0

where c0 represents some suitable constant, c1 is a positive constant, and g(·) is a time-varying parameter which takes / I, values in the unknown closed intervals I = [l− , l+ ], with 0 ∈

and then V (t), ζ(t), and [0, tf ).

t 0

˙ must be bounded on g(·)N (ζ)ζdτ

III. C ONTROL D ESIGN AND S TABILITY A NALYSIS In this section, we will investigate adaptive neural control for system (7) using the backstepping method [21] combined with NN approximation. The backstepping design procedure contains n steps and involves the following change of coordinates: z1 = x1 − yd , zi = xi − αi−1 , i = 2, . . . , n, where αi is a virtual control which shall be developed for the corresponding i-subsystem based on an appropriate Lyapunov function Vi . The control law v(t) is designed in the last step to stabilize the entire closed-loop system and deal with the hysteresis term. The closed-loop system can be proved to be SGUUB by Lyapunov stability analysis. Step 1): Since z1 = x1 − yd and z2 = x2 − α1 , the derivative of z1 is   ¯1 , xθ21 x2 − y˙ d z˙1 = f1 (¯ x1 , 0) + g1 x   ¯1 , xθ21 (z2 + α1 ) − y˙ d = f1 (¯ x1 , 0) + g1 x   ¯1 , xθ21 (z2 + α1 ) + Q1 (Z1 ) = g1 x (11) where Q1 (Z1 ) = f1 (¯ x1 , 0) − y˙ d , with Z1 = [¯ x1 , y˙ d ] ∈ ΩZ1 ⊂ R2 . To compensate for the unknown function Q1 (Z1 ), we ˆ 1 ∈ Rl×1 , ˆ T S(Z1 ), with W can use RBFNN in Section II-B, W 1 l×1 S(Z1 ) ∈ R , and the NN node number l > 1, to approximate the function Q1 (Z1 ) on the compact set ΩZ1 as follows: ˆ 1T S(Z1 ) − W ˜ 1T S(Z1 ) + ε1 (Z1 ) Q1 (Z1 ) = W

(12)

where the approximation error ε1 (Z1 ) satisfies |ε1 (Z1 )| ≤ ε∗1 with positive constant ε∗1 . Substituting (12) into (11), we obtain   ˆ 1T S(Z1 )− W ˜ 1T S(Z1 )+ε1 (Z1 ). ¯1 , xθ21 (z2 +α1 )+ W z˙1 = g1 x (13) Choose the following virtual control law and adaptation laws:   ˆ 1T S(Z1 ) α1 = N (ζ1 ) k1 z1 + W (14) ˆ T S(Z1 ) ζ˙1 = k1 z12 + z1 W 1   ˙ ˆ1 ˆ W 1 = Γ1 z1 S(Z1 ) − σ1 W

(15) (16)

where Γ1 = ΓT1 ∈ Rl×l > 0, k1 > 0, and σ1 > 0 are design parameters. Consider the following Lyapunov function candidate: V1 =

1 2 1 ˜ T −1 ˜ z + W Γ W1 . 2 1 2 1 1

(17)

The time derivative of (17), along with (13)–(16), is ˜˙ 1 ˜ T Γ−1 W V˙ 1 = z1 z˙1 + W 1 1

  ¯1 , xθ21 N1 (ζ1 ) + 1 ζ˙1 ≤ − k1 z12 + g1 x   ˜ 1T W ˆ 1 + |z1 |ε∗1 . ¯1 , xθ21 z1 z2 − σ1 W + g1 x

(18)

REN et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS

By using Young’s inequality, we obtain the following inequalities: ∗ 2 ˜ 2 ˜ 1T W ˆ 1 ≤ − σ1 W1  + σ1 W1  −σ1 W 2 2

z12 + c11 ε∗2 1 4c11     z2 g1 x ¯1 , xθ21 z1 z2 ≤ 1 + c12 g12 x ¯1 , xθ21 z22 4c12 |z1 |ε∗1 ≤

435

Noting Assumption 6, the last term of (25) t x1 , xθ21 )z22 eγ1 τ dτ has the following property: e−γ1 t 0 c12 g12 (¯

(19)

e−γ1 t

t

  ¯1 , xθ21 z22 eγ1 τ dτ c12 g12 x

0

(20)

−γ1 t

≤e (21)

with constant parameters c11 > 0 and c12 > 0. Substituting (19)–(21) into (18) results in   1 1 ˙ V1 ≤ − k1 − − z12 4c11 4c12     ˜ 1 2 σ1 W ¯1 , xθ21 N1 (ζ1 ) + 1 ζ˙1 − + g1 x 2   ∗ 2 W  σ 1 1 ¯1 , xθ21 z22 + + c11 ε∗2 + c12 g12 x 1 2     ≤ − γ1 V1 + g1 x ¯1 , xθ1 N1 (ζ1 ) + 1 ζ˙1

≤

g¯12

where γ1 and ρ1 are positive constants, which are defined as     1 1 σ1 − , γ1 = min 2 k1 − 4c11 4c12 λmax (Γ−1 1 ) ρ1 =

σ1 W1∗ 2 + c11 ε∗2 1 . 2

Multiplying both sides of (22) by eγ1 t yields     d ¯1 , xθ21 N1 (ζ1 ) + 1 ζ˙1 eγ1 t (V1 eγ1 t ) ≤ ρ1 eγ1 t + g1 x dt   ¯1 , xθ21 z22 eγ1 t . (23) + c12 g12 x

sup

+ e−γ1 t



t

+e

V1 ≤ c1 + e

where c1 = (ρ1 /γ1 ) + V1 (0) + (c12 /γ1 )¯ g12 supτ ∈[0,tf ] [z22 (τ )]. ˆ 1 , and to Lemma 1, we can conclude that V1 , ζ1 , W According t θ1 γ1 τ ˙ [g (¯ x , x )N (ζ ) + 1] ζ e dτ are all bounded on [0, tf ). 1 1 1 2 0 1 1 According to Proposition 2 [29], tf = ∞, we know that z1 and ˆ 1 are SGUUB. The boundedness of z2 will be dealt with in W the following steps. Step j (2 ≤ j < n): The derivative of zj is z˙j = x˙ j − α˙ j−1

  θj ¯j , xj+1 xj+1 − α˙ j−1 . = fj (¯ xj , 0) + gj x

t 0

(28)

j−1  ∂αj−1 k=1 j−1 

∂xk

x˙ k + φj−1

∂αj−1 fk (¯ xk , xk+1 ) + φj−1 ∂xk

(29)

where (24)

 ∂αj−1 ˙ ∂αj−1 ˙ ∂αj−1 ˆk W x ¯˙ dj + = ζj−1 + ˆk ∂ζj−1 ∂x ¯dj ∂W j−1

φj−1

(30)

k=1



g1 x ¯1 , xθ21



which is computable. As such, α˙ j−1 can be seen as a function of x ¯j , (∂αj−1 /∂x1 ), . . . , (∂αj−1 /∂xj−1 ), φj−1 . Furthermore, we can rewrite (28) as   θj (31) z˙j = gj x ¯j , xj+1 (zj+1 + αj ) + Qj (Zj )

 N1 (ζ1 ) + 1 ζ˙1 eγ1 τ dτ

0

+ e−γ1 t

(26)

(27)

=

ρ1 + V1 (0) γ1 + e−γ1 t

0

t     g1 x ¯1 , xθ21 N1 (ζ1 ) + 1 ζ˙1 eγ1 τ dτ

k=1

t 

c12 eγ1 τ dτ

e

0

α˙ j−1 =

0

≤

t

where g¯1 is the upper bound for |g1 (·)| as defined in Assumption 6. Therefore, if z2 can be kept bounded over a finite time interval [0, tf ), then we can obtain the boundedness of the t x1 , xθ21 )z22 eγ1 τ dτ . Furthermore, (25) can term e−γ1 t 0 c12 g12 (¯ be written as

   g1 x ¯1 , xθ21 N1 (ζ1 ) + 1 ζ˙1 eγ1 τ dτ

  ¯1 , xθ21 z22 eγ1 τ dτ c12 g12 x

−γ1 t

  c12 2 g¯1 sup z22 (τ ) ≤ γ1 τ ∈[0,t]

0 −γ1 t



z22 (τ )

ˆ 1, . . . , W ˆ j−1 , its ¯j−1 , x ¯dj , ζj−1 , W Since αj−1 is a function of x derivative α˙ j−1 can be expressed as

Integrating (23) over [0, t], we have

 ρ1 ρ1 −γ1 t + V1 (0) − e V1 ≤ γ1 γ1 t



τ ∈[0,t]

−γ1 t

(22)

c12 g¯12 z22 eγ1 τ dτ 0

2

  ¯1 , xθ21 z22 + ρ1 + c12 g12 x

t

  ¯1 , xθ21 z22 eγ1 τ dτ. c12 g12 x

(25)

where Zj = [¯ xj , (∂αj−1 /∂x1 ), . . . , (∂αj−1 /∂xj−1 ), φj−1 ] ∈ ΩZj ⊂ R2j , and Qj (Zj ) = fj (¯ xj , 0) − α˙ j−1 is an unknown

436

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 2, APRIL 2009

function that can be approximated by the RBFNN in ˆ T S(Zj ), on the compact set ΩZ as Section II-B, W j j ˆ T S(Zj ) − W ˜ T S(Zj ) + εj (Zj ) Qj (Zj ) = W j j

≤

(32)

0

where the approximation error εj (Zj ) satisfies |εj (Zj )| ≤ ε∗j with positive constant ε∗j . Substituting (32) into (28), we obtain   θj ¯j , xj+1 (zj+1 + αj ) z˙j = gj x ˆ jT S(Zj ) − W ˜ jT S(Zj ) + εj (Zj ). +W

(33)

The following virtual control law and adaptation laws are considered:   ˆ jT S(Zj ) αj = N (ζj ) kj zj + W (34) ˆ T S(Zj ) ζ˙j = kj zj2 + zj W j   ˙ ˆj ˆ W j = Γj zj S(Zj ) − σj W

(35) (36)

ρj + Vj (0) γj t     θj −γj t gj x ¯j , xj+1 Nj (ζj ) + 1 ζ˙j eγj τ dτ +e + e−γj t

t

  θj 2 cj2 gj2 x eγj τ dτ. ¯j , xj+1 zj+1

0

Similarly, as discussed in Step 1), if zj+1 can be kept bounded over a finite time interval [0, tf ), we can readily t guarantee the boundedness of the extra term e−γj t 0 cj2 gj2 (¯ xj , θ

j 2 xj+1 )zj+1 eγj τ dτ in (42) as follows:

−γj t

t

e

  θj 2 cj2 gj2 x eγj τ dτ ¯j , xj+1 zj+1

0

≤

ΓTj

> 0, kj , and σj are positive constants. where Γj = Define the following Lyapunov function candidate: Vj =

1 2 1 ˜ T −1 ˜ z + W j Γ j Wj . 2 j 2

σj Wj∗ 2 + cj1 ε∗2 ρj = j . 2

(39) (40)

−γj t

cj2 gj2

+e

0

θj x ¯j , xj+1



t     θj gj x ¯j , xj+1 Nj (ζj ) + 1 ζ˙j eγj τ dτ 0

(44) 2 where cj = (ρj /γj )+Vj (0)+(cj2 /γj )¯ gj2 supτ∈[0,tf ] [zj+1 (τ )]. ˆ j , and Then, applying Lemma 1, the boundedness of Vj , ζj , W t θj γj τ ˙ xj , xj+1 )Nj (ζj ) + 1]ζj e dτ can be readily obtained. 0 [gj (¯ The boundedness of zj+1 will be dealt with in Step (j + 1). Step n): This is the final step, in which we will design the control input v(t). Since zn = xn − αn−1 , its derivative is given by

z˙n = fn (¯ xn , 0) + gn (¯ xn , uθn ) ⎡ × ⎣g0 (v θ0 )v − g0 (v θ0 )v ∗ −

D

⎤ p(r)Fr [v](t)dr⎦

0

= gn (¯ xn , uθn ) ⎡ × ⎣g0 (v θ0 )v − g0 (v θ0 )v ∗ −

D

⎤ p(r)Fr [v](t)dr⎦

0

+ Qn (Zn ) + d(t)   ¯ n , uθn = gn x ⎡ × ⎣g0 (v θ0 )v − g0 (v θ0 )v ∗ −

D

⎤ p(r)Fr [v](t)dr⎦

0

ˆ T S(Zn ) − W ˜ T S(Zn ) + εn (Zn ) + d(t) +W n n

0



Vj ≤ cj + e−γj t

+ d(t) − α˙ n−1

with constant parameters cj1 > 0 and cj2 > 0. Multiplying both sides of (38) by eγj t and integrating over [0, t], we have

 ρj ρj −γj t Vj ≤ + Vj (0) − e γj γj t     θj −γj t gj x ¯j , xj+1 Nj (ζj ) + 1 ζ˙j eγj τ dτ +e t

 2  cj2 2 g¯j sup zj+1 (τ ) . (43) γj τ ∈[0,t]

Therefore, (42) can be written as (37)

Similar to the procedures outlined in Step 1), with the help of Young’s inequality, the derivative of Vj in (37), along with (33)–(36), can be obtained as   1 1 ˙ Vj ≤ − kj − − zj2 4cj1 4cj2     θj ¯j , xj+1 Nj (ζj ) + 1 ζ˙j + gj x   ˜ j 2 σj W θj 2 ¯j , xj+1 zj+1 + cj2 gj2 x − 2 σj Wj∗ 2 + cj1 ε∗2 + j 2     θj ¯j , xj+1 Nj (ζj ) + 1 ζ˙j ≤ − γj Vj + gj x   θj 2 ¯j , xj+1 zj+1 + ρj + cj2 gj2 x (38) where γj and ρj are positive constants defined as     1 1 σj   γj = min 2 kj − − , 4cj1 4cj2 λmax Γ−1 j

2 zj+1 eγj τ dτ

(42)

(41)

(45)

ˆ T S(Zn ) is used to approximate the unknown function where W n Qn (Zn ) = fn (x, 0) − α˙ n−1 on the compact set ΩZn ⊂ Rn ,

REN et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS

with Zn = [¯ xn , (∂αn−1 /∂x1 ), . . . , (∂αn−1 /∂xn−1 ), φn−1 ] ∈ ΩZn ⊂ R2n , and the approximation error εn (Zn ) satisfies |εn (Zn )| ≤ ε∗n , with ε∗n being a positive constant. Choose the following Lyapunov function candidate: 1 1 ˜ T −1 ˜ 1 ˜2 g¯n d + Vn = zn2 + W n Γ n Wn + 2 2 2γd 2γp

D p˜2 (t, r)dr 0

(46) where d˜ = dˆ − d∗ ; p˜(t, r) = pˆ(t, r) − pmax ; dˆ and pˆ(t, r) are the estimates of the disturbance bound d∗ and the density function of p(r), respectively; Γn = ΓTn > 0; and γd and γp are positive constants. The derivative of Vn defined in (46), along with (45), is ⎡ ⎤ D   V˙ n = zn gn x ¯n , uθn ⎣g0 (v θ0 )v − p(r)Fr [v](t)dr⎦ 0

ˆ nT S(Zn ) xn , uθn )g0 (v θ0 )v ∗ + zn W − zn gn (¯ ˜ nT S(Zn ) + zn εn (Zn ) + zn d(t) + W ˜˙ ˜ nT Γ−1 − zn W n Wn D 1 ˙ g¯n ∂ (47) + d˜d˜ + p˜(t, r) p˜(t, r)dr. γd γp ∂t 0

From Assumptions 2 and 6, we know that |gn (x, uθn )g0 v ∗ | ≤ C, where C is a positive constant. Due to |εn (Zn )| ≤ ε∗n and Assumption 4, (47) becomes ⎡ ⎤ D   V˙ n ≤ zn gn x ¯n , uθn ⎣g0 (v θ0 )v − p(r)Fr [v](t)dr⎦ 0

ˆ T S(Zn )−zn W ˜ T S(Zn )+|zn |(C + ε∗ )+|zn |d∗ + zn W n n n D 1 ˜˙˜ g¯n ∂ ˜˙ ˜ nT Γ−1 +W p˜(t, r) p˜(t, r)dr. dd + n Wn + γd γp ∂t 0

(48) The following control laws and adaptation laws are proposed:    ˆ T S(Zn ) + dˆtanh zn + vh v = N (ζn ) kn zn + W n ω (49) D pˆ(t, r) vh = − sgn(zn ) |Fr [v](t)| dr (50) h0 0   ˆ T S(Zn ) + zn dˆtanh zn (51) ζ˙n = kn zn2 + zn W n ω   ˆ˙ n = Γn zn S(Zn ) − σn W ˆn W (52)  z   ˙ n − σd dˆ dˆ = γd zn tanh (53) ω ⎧ pˆ(t, r) ≥ pmax ⎨ −γp σp pˆ(t, r), ∂ pˆ(t, r) = γp [|zn ||Fr [v](t)| (54) ⎩ ∂t −σp pˆ(t, r)] , 0 ≤ pˆ(t, r) < pmax where kn , σn , σd , σp and ω are positive constants. Remark 8: The term vh in (49) is used to cancel the effect caused by the nondifferentiable hysteresis D term 0 p(r)Fr [v](t)dr. Due to the integral form of

437

D

0 p(r)Fr [v](t)dr, we cannot make assumptions on its boundedness and thus cannot design the traditional robust adaptive control. However, considering that the density function p(r) is not a function of time, it can be treated as a “parameter” of the hysteresis model, and an adaptation law can be developed to obtain an estimate of it [5], [6]. Substituting (49)–(53) into (48), and using Young’s inequality and the following property of the hyperbolic tangent function tanh(·) [30], [31]: z  n ≤ 0.2785ω 0 ≤ |zn | − zn tanh ω

we obtain   1 ˙ Vn ≤ − kn − zn2 4cn1   + gn (x, uθn )g0 (v θ0 )Nn (ζn ) + 1 ζ˙n ˜ n 2 σn Wj∗ 2 σd d˜2 σd d∗2 σn W − + + − 2 2 2 2 + 0.2785ωd∗ + cn1 (ε∗n + C)2 + gn (x, uθn ) ⎡ D pˆ(t, r) θ × ⎣−g0 (v 0 )|zn | |Fr [v](t)| dr h0 0 ⎤ D D g¯n ∂ ⎦ − zn p(r)Fr [v](t)dr + p˜(t, r) p˜(t, r)dr γp ∂t 0

0

(55) where cn1 is a positive constant. According to Assumptions 2 and 5, the last two terms of (55) can be written as ⎡ D pˆ(t, r) θn ⎣ θ0 |Fr [v](t)| dr gn (x, u ) −g0 (v )|zn | h0 0 ⎤ D −zn p(r)Fr [v](t)dr⎦

+

g¯n γp

0

D

∂ p˜(t, r)dr ∂t 0 ⎡ D ≤ gn (x, uθn ) ⎣−|zn | pˆ(t, r) |Fr [v](t)| dr p˜(t, r)

0

D

+ |zn |

+

g¯n γp

p˜(t, r)

∂ p˜(t, r)dr ∂t D

≤ −gn (x, u )|zn |

p˜(t, r) |Fr [v](t)| dr

θn

g¯n γp

0

D p˜(t, r) 0

pmax |Fr [v](t)| dr⎦

0

D 0

+

⎤

∂ p˜(t, r)dr. ∂t

(56)

438

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 2, APRIL 2009

According to (54), the adaptation law for the estimate of density function pˆ(t, r) comprises two cases due to the different regions where pˆ(t, r) belong to. Therefore, we also need to consider the following two cases for the analysis of (56). Case 1) For r ∈ Dmax = {r : pˆ(t, r) ≥ pmax } ⊂ [0, D], according to (54), we have p˜(t, r) ≥ 0

Combining Case 1) with Case 2), (56) can be written as ⎡ D pˆ(t, r) |Fr [v](t)| dr gn (x, uθn ) ⎣−g0 (v θ0 )|zn | h0 0 ⎤ D − zn p(r)Fr [v](t)dr⎦

(57)

∂ pˆ(t, r) = − γp σp pˆ(t, r). ∂t

+ (58)

Substituting (57) and (58) into (56), we have

g¯n γp

∂ p˜(t, r)dr ∂t 0  θn ≤ −gn (x, u )|zn |

 − gn (x, u )|zn |

+

g¯n γp

p˜(t, r)

∂ p˜(t, r)dr ∂t

+

 p˜(t, r)ˆ p(t, r)dr.

g¯n γp

c r∈Dmax



p˜(t, r)

p˜(t, r)ˆ p(t, r)dr 

r∈Dmax

− σp g¯n

Case 2) For r ∈ which is the complement set of Dmax in [0, D], i.e., 0 ≤ pˆ(t, r) < pmax , from (54), we have p˜(t, r) < 0 ∂ pˆ(t, r) = γp [|zn ||Fr [v](t)| − σp pˆ(t, r)] . ∂t

(61)

p˜(t, r)ˆ p(t, r)dr

c r∈Dmax

D = −σp g¯n

(60)

p˜(t, r)ˆ p(t, r)dr.

By Young’s inequality, we have p(t, r) ≤ − −σp g¯n p˜(t, r)ˆ

σp g¯n 2 σp g¯n 2 p˜ (t, r) + p . 2 2 max

(64)

Integrating both sides of (64) over [0, D] results in

 − gn (x, uθn )|zn |

D

p˜(t, r) |Fr [v](t)| dr

−σp1 g¯n

c r∈Dmax

p˜(t, r)ˆ p(t, r)dr 0

∂ p˜(t, r) p˜(t, r)dr ∂t

σp g¯n ≤− 2

c r∈Dmax

≤ −gn (x, uθn )|zn |

p˜(t, r) |Fr [v](t)| dr

c r∈Dmax

 + g¯n |zn |

p˜(t, r) |Fr [v](t)| dr

c r∈Dmax

p˜2 (t, r)dr +

σp g¯n D 2 pmax . (65) 2

Therefore, according to (65), we can rewrite (63) further as ⎡ D pˆ(t, r) θn ⎣ θ0 gn (x, u ) −g0 (v )|zn | |Fr [v](t)| dr h0 0 ⎤ D − zn p(r)Fr [v](t)dr⎦

 − σp g¯n

D 0



p˜(t, r)ˆ p(t, r) +

c r∈Dmax

g¯n γp

0

D p˜(t, r) 0

 c r∈Dmax

(63)

0

Substituting (60) and (61) into (56), we have

≤ −σp g¯n

∂ p˜(t, r)dr ∂t

c r∈Dmax

= −σp g¯n

c Dmax ,



∂ p˜(t, r)dr ∂t  p˜(t, r) |Fr [v](t)| dr

p˜(t, r)



(59)

r∈Dmax

g¯n + γp

p˜(t, r) |Fr [v](t)| dr

r∈Dmax

− gn (x, uθn )|zn |

r∈Dmax

≤ −σp g¯n

 r∈Dmax

r∈Dmax



p˜(t, r)

g¯n + γp

p˜(t, r) |Fr [v](t)| dr

θn

0

D

p˜(t, r)ˆ p(t, r)dr.

(62)

≤−

σp g¯n 2

∂ p˜(t, r)dr ∂t

D p˜2 (t, r)dr + 0

σp g¯n D 2 pmax . 2

(66)

REN et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS

Substituting (66) into (55), we have V˙ n ≤ − (kn −

1 )z 2 4cn1 n

˜ n 2   σn W + gn (x, uθn )g0 (v θ0 )Nn (ζn ) + 1 ζ˙n − 2 D σn Wj∗ 2 σp g¯n σd d˜2 − − p˜2 (t, r)dr + 2 2 2 0

σd d 2

∗2

σp g¯n D 2 pmax 2   ≤ − γn Vn + gn (x, uθn )g0 (v θ0 )Nn (ζn ) + 1 ζ˙n + ρn +

+ 0.2785ωd∗ + cn1 (ε∗n + C)2 +

(67) where γn and ρn are positive constants defined as   γn = min 2 kn − ρn =

1 4cn1

 ,

σn , σd γd , σp γp λmax (Γ−1 n )



Multiplying both sides of (67) and integrating over [0, t], we have

 ρn ρn −γn t + Vn (0) − Vn ≤ e γn γn + e−γn t

  gn (x, uθn )g0 (v θ0 )Nn (ζn ) + 1 ζ˙n eγn τ dτ

0

≤ cn + e−γn t

t

σ-modification was included since only the property V˙ ≤ 0 was to be obtained. The following theorem shows the stability and control performance of the closed-loop adaptive system. Theorem 1: Consider the closed-loop system consisting of the plant (1), preceded by unknown hysteresis nonlinearities (2), and the control laws and adaptation laws (49)–(54). Under Assumptions 1–6, and given any initial conditions ˆ ˆ i (0), d(0)(i = 1, 2, . . . , n) belonging to Ω0 , the overall zi (0), W closed-loop neural control system is SGUUB in the sense that all of the signals are bounded. Specifically, the states and weights in the closed-loop system will remain in the compact set Ω defined by     2μj  ˜ j , d˜|zj | ≤ 2μj , W ˜ j ≤ , Ω = zj , W λmin (Γ−1 j )   ˜ |d| ≤ 2γd μn , j = 1, 2, . . . , n (71)

(68)

σd d∗2 σn Wn∗ 2 + + 0.2785ωd∗ + cn1 (ε∗n + C)2 2 2 σp g¯n D 2 pmax . + (69) 2

t

439

  gn (x, uθn )g0 (v θ0 )Nn (ζn ) + 1 ζ˙n eγn τ dτ

and eventually converge to the compact set Ωs defined by     2μ∗j  ˜ j , d˜|zj | ≤ 2μ∗ , W ˜ j ≤  , Ωs = zj , W j λmin Γ−1 j   ˜ ≤ 2γd μ∗ , j = 1, 2, . . . , n (72) |d| n

where μj = cj + cj0 , j = 1, 2, . . . n, ρn cn = + Vn (0), γn 1 1 ˜T 1 ˜2 −1 ˜ Vn (0) = zn2 (0) + W d (0) n (0)Γn Wn (0) + 2 2 2γd n g¯n + 2γp

0

(70) where cn = (ρn /γn ) + Vn (0). According to Assumptions 1, 2, and 6, we can regard gn (x, u)g0 (v) in (70) as g(·), which is a time-varying parameter and takes values in the known closed / I. Using Lemma 1, we can intervals I = [h0 g n , h1 g¯n ], with 0 ∈ ˆ n , and dˆ are conclude that Vn (t), ζn (t), and, hence, zn (t), W SGUUB. From the boundedness of zn (t), the boundedness of t θ 2 the extra term e−γn−1 t 0 c(n−1)2 gn−1 (¯ xn−1 , xnn−1 )zn2 eγn−1 τdτ at Step (n − 1) is readily obtained. Applying Lemma 1 for (n − 1) times backward, it can be seen from the aforemenˆ j , and, hence, tioned iterative design procedure that Vj , zj , W xj are SGUUB on [0, tf ). Remark 8: In order to use Lemma 1 to establish closed-loop stability, we need to express V˙ n in the form of V˙ n = −γn Vn + [gn (x, uθn )g0 (v θ0 )Nn (ζn ) + 1]ζ˙n + ρn as in (67). Thus, we need to adopt the σ-modification form in the adaptation law of pˆ(t, r) as in (54), which can improve the robustness as well. This is different from the previous works [5], [6], where no

D p˜2 (0, r)dr, 0

ρj 2cj2 2 cj = + Vj (0) + g¯ (cj+1 + cj+1,0 ), γj γj j 1 1 ˜T ˜ Vj (0) = zj2 (0) + W (0)Γ−1 j Wj (0), 2 2 j μ∗j = cj + cj0 ,

j = 1, . . . , n − 1,

j = 1, 2, . . . n,

cn =

ρn , γn

cj =

ρj 2cj2 2 + g¯ (cj+1 + cj+1,0 ), γj γj j

j = 1, . . . , n − 1

t xj , and with cj0 being the upper bound of e−γj t 0 [gj (¯ θj xj+1 )Nj (ζj ) + 1]ζ˙j eγj τ dτ , where j = 1, 2, . . . , n. Proof: For any given initial compact set Ω0 , i.e., ˆ ˆ i (0), d(0)} {zi (0), W ∈ Ω0 (i = 1, 2, . . . , n), we can always construct a corresponding compact set ΩNN comprising ΩZ1 , . . . , ΩZn , which is larger than Ω0 and can be as large as

440

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 2, APRIL 2009

we want, on which the NN approximation is valid. Based on the previous iterative derivation procedures from Step 1) to Step n) of backstepping, from (27), (44), and (70), and according to ˆ and, hence, xj ˆ j , d, Lemma 1, we can conclude that Vj , zj , W are SGUUB, i = 1, 2, . . . , n, i.e., all the signals in the closedloop system are bounded. Noting the definition of Vn in (46), and letting cn0 be the t upper bound of the term e−γn t 0 [gn (x, uθn )g0 Nn (ζn ) + 1] ζ˙n eγn τ dτ , cn = (ρn /γn ) + Vn (0), and μn = cn + cn0 in (70), we have    2μn ˜ ≤ 2γd μn . ˜ |d| Wn  ≤ |zn | ≤ 2μn −1 λmin (Γn ) Similarly, in the rest of the steps from (n − 1) to 1), letting t θj xj , xj+1 )Nj (ζj ) + 1] cj0 be the upper bound of e−γj t 0 [gj (¯ 2 ˙ζj eγj τ dτ , cj = (ρj /γj )+Vj (0)+(2cj2 /γj )¯ gj (cj+1 +cj+1,0 ), and μj = cj + cj0 in (44), we can obtain |zj | ≤



 ˜ i ≤ W

2μj

2μj  , λmin Γ−1 j

Vn ≤

 ρn −γn t + Vn (0) − e γn

where μ∗n = cn + cn0 , cn = (ρn /γn ), and cn0 is the upper t bound of the term e−γn t 0 [gn (x, uθn )g0 Nn (ζn ) + 1] ζ˙n eγn τ dτ . As t → ∞, we have Vn ≤ μ∗n . Therefore, based on the definition of Vn in (46), we can conclude that when t → ∞, the following inequalities are true: |zn | ≤



 2μ∗n

˜ n ≤ W

2μ∗n λmin (Γ−1 n )

˜≤ |d|



2γd μ∗n .

ˆ j as follows: A similar conclusion can be made about zj and W |zj | ≤



 2μ∗j

˜ j ≤ W

2μ∗j  , λmin Γ−1 j

IV. S IMULATION S TUDIES In this section, simulation studies are presented to demonstrate the effectiveness of the proposed adaptive NN approach to deal with uncertain nonlinear systems in pure-feedback form preceded by the generalized P–I hysteresis. Consider the following second-order nonlinear system with the generalized P–I hysteresis: x˙ 1 = x2 + 0.05 sin(x2 ) x˙ 2 =

1 − e−x2 + u + 0.1 sin(u) + 0.1 sin(6t) 1 + e−x2

y = x1 j = 1, . . . , n − 1.

Furthermore, we can rewrite (70) as μ∗n

start in Ω0 , there exist some control parameters such that the states and weights will remain in the conservative compact set Ω and finally converge to the compact set Ωs . Both of them belong to the chosen compact set ΩNN . This completes the proof. 

j = 1, . . . , n − 1

with μ∗j = cj + cj0 and cj = (ρj /γj ) + (2cj2 /γj )¯ gj2 (cj+1 + cj+1,0 ) as t → ∞. In addition, from the definition of the bounds of the compact sets Ω in (71) and Ωs in (72), and the definitions of γj and ρj in (39) and (40) and γn and ρn in (68) and (69), respectively, we can see that the size of the compact sets Ω and Ωs depends −1 on the choice of control parameters ω, λmax (Γ−1 j ), λmax (Γn ), kj , kn , γd and γp . In particular, by decreasing ω, λmax (Γ−1 j ), ), and increasing k , k , γ , and γ , we can reduce λmax (Γ−1 j n d p n μj , μ∗j , μn , and μ∗n , and thus, the size of the compact sets Ω and Ωs will decrease. Therefore, as long as the initial conditions

(73)

where u represents the output of the hysteresis described by the D generalized P–I model u(t) = h(v)(t) − 0 p(r)Fr [v](t)dr, 2 with the density function p(r) = 0.08e−0.0024(r−1) , r ∈ [0, 100], and h(v)(t) = 0.4(|v| arctan(v) + v). We can check that plant (73) satisfies Assumptions 1–6. Our objective is to make the output of system (73), y, to track the desired trajectory, yd = 0.8 sin(0.5t) + 0.1 cos(t). We adopt the control law and adaptation laws designed in Section III in the following:   ˆ T S(Z1 ) α1 = N (ζ1 ) k1 z1 + W 1    ˆ T S(Z2 ) + dˆtanh z2 + vh v = N (ζ2 ) k2 z2 + W 2 ω D vh = − sign(z2 )

pˆ(t, r) |Fr [v](t)| dr h0

0

ˆ 1T S(Z1 ) ζ˙1 = k1 z12 + z1 W   ˆ 2T S(Z2 ) + z2 dˆtanh z2 ζ˙2 = k2 z22 + z2 W ω   ˆ1 ˆ˙ 1 = Γ1 z1 S(Z1 ) − σ1 W W   ˆ2 ˆ˙ 2 = Γ2 z2 S(Z2 ) − σ2 W W z    ˙ 2 dˆ = γd z2 tanh − σd dˆ ω ⎧ pˆ(t, r) ≥ pmax ⎨ −γp σp pˆ(t, r), ∂ pˆ(t, r) = γp [|z2 ||Fr [v](t)| ⎩ ∂t −σp pˆ(t, r)] , 0 ≤ pˆ(t, r) < pmax where z1 = x1 − yd and z2 = x2 − α1 . The Nussbaum function is chosen as N (ζ) = exp(ζ 2 ) cos((π/2)ζ). The inputs

REN et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS

Fig. 1.

Tracking performance.

of the NNs are Z1 = [x1 , yd ] ∈ R2 and Z2 = [x1 , x2 , (∂α1 / x1 ), φ1 ] ∈ R4 , where φ1 = (∂α1 /∂ζ1 )ζ˙1 + (∂α1 /∂yd )y˙ d + ˆ 1 )W ˆ˙ 1 . The following initial conditions and control (∂α1 /∂ W design parameters are chosen as x1 (0) = 0.2, x2 (0) = ζ1 (0) = ˆ = 0.0, W ˆ 1 (0) = W ˆ 2 (0) = 0.0, k1 = k2 = 1.0, ζ2 (0) = d(0) Γ1 = 0.01I25 , σ1 = 0.0, Γ2 = 0.2I256 , σ2 = 0.002, σp = 0.2, γp = 0.06, pmax = 0.1, ω = 0.1, and h0 = 0.35. In practice, the selection of the centers and widths of RBF has a great influence on the performance of the designed controller. According to [23], Gaussian RBFNNs arranged on a regular lattice on Rn can uniformly approximate sufficiently smooth functions on closed bounded subsets. Accordingly, in the following simulation studies, the centers and widths are chosen on a regular lattice in the respective compact sets. Specifically, we employ five nodes for each ˆ T S(Z1 ) and four nodes for each input input dimension of W 1 ˆ T S(Z2 ); thus, we end up with 25 nodes dimension of W 2 (i.e., l1 = 25) with centers μl = 1.0 (l = 1, 2, . . . , l1 ) evenly spaced in [−4.0, +4.0] × [−4.0, +4.0] and widths ηl = ˆ T S(Z1 ), and 256 nodes (i.e., 1.0 (l = 1, 2, . . . l1 ) for NN W 1 l2 = 256) with centers μl (l = 1, 2, . . . , l2 ) evenly spaced in [−4.0, +4.0] × [−4.0, +4.0] × [−4.0, +4.0] × [−4.0, +4.0] ˆ T S(Z2 ). and widths ηl = 1.0 (l = 1, 2, . . . , l2 ) for NN W 2 Due to the use of sign function sgn(·), the control signal vh (50) becomes discontinuous, which may excite unmodeled high-frequency plant dynamics and cause the chattering phenomenon. To avoid the undesired chattering phenomenon, we will replace the sign function in vh with the following saturation function in the simulation:  1, if ∗ ≥  sat(∗) = ∗ , if | ∗ | <  1, if ∗ >  where  is a small positive constant and chosen as 0.05 in this paper. The simulation results are shown in Figs. 1–6. From Fig. 1, we observe that good tracking performance is achieved and that the tracking error converges to a small neighborhood of zero in less than one period of oscillation. At the same time, other

441

Fig. 2. State x2 .

Fig. 3. Control signal and hysteresis output.

Fig. 4. Norm of NN weights.

signals, including the state x2 , control signal v, hysteresis outˆ 2 , Nussbaum function ˆ 1  and W put u, NN weight norms W signals ζ1 , ζ2 , N (ζ1 ), and N (ζ2 ), and the disturbance parameter

442

IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 2, APRIL 2009

ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive comments that helped improve the quality and presentation of this paper. R EFERENCES

Fig. 5. Nussbaum function signals.

ˆ Fig. 6. Estimation of disturbance bound d.

estimate dˆ are kept bounded, as seen in Figs. 2–6. It is noted that there is a large difference between the signals v and u in Fig. 3, which indicates the significant hysteresis effect. In particular, in all figures, there are two obvious spikes at around 4 and 8 s, which result from the Nussbaum functions N (ζ1 ) and N (ζ2 ). V. C ONCLUSION Adaptive neural control has been proposed for a class of unknown nonlinear systems in pure-feedback form preceded by the uncertain generalized P–I hysteresis. We adopted the meanvalue theorem to solve the nonaffine problem in both system unknown nonlinear functions and unknown input function in the generalized P–I hysteresis model, and used Nussbaum function to deal with the problem of the unknown virtual control directions. The closed-loop control system has been theoretically shown to be SGUUB using the Lyapunov synthesis method. Simulation results have verified the effectiveness of the proposed approach.

[1] G. Tao and P. V. Kokotovi´c, “Adaptive control of plants with unknown hystereses,” IEEE Trans. Autom. Control, vol. 40, no. 2, pp. 200–212, Feb. 1995. [2] X. Tan and J. S. Baras, “Modeling and control of hysteresis in magnetostrictive actuators,” Automatica, vol. 40, no. 9, pp. 1469–1480, Sep. 2004. [3] C. Y. Su, Y. Stepanenko, J. Svoboda, and T. P. Leung, “Robust adaptive control of a class of nonlinear systems with unknown backlash-like hysteresis,” IEEE Trans. Autom. Control, vol. 45, no. 12, pp. 2427–2432, Dec. 2000. [4] J. Zhou, C. Y. Wen, and Y. Zhang, “Adaptive backstepping control of a class of uncertain nonlinear systems with unknown backlash-like hysteresis,” IEEE Trans. Autom. Control, vol. 49, no. 10, pp. 1751–1757, Oct. 2004. [5] C. Y. Su, Q. Wang, X. Chen, and S. Rakheja, “Adaptive variable structure control of a class of nonlinear systems with unknown Prandtl–Ishlinskii hysteresis,” IEEE Trans. Autom. Control, vol. 50, no. 12, pp. 2069–2074, Dec. 2005. [6] Q. Wang and C. Y. Su, “Robust adaptive control of a class of nonlinear systems including actuator hysteresis with Prandtl–Ishlinskii presentations,” Automatica, vol. 42, no. 5, pp. 859–867, May 2006. [7] O. Klein and P. Krejci, “Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations,” Nonlinear Anal.: Real World Appl., vol. 4, no. 5, pp. 755–785, Dec. 2003. [8] M. S. de Queiroz, J. Hu, D. M. Dawson, T. Burg, and S. R. Donepudi, “Adaptive position/force control of robot manipulators without velocity measurements: Theory and experimentation,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 27, no. 5, pp. 796–809, Oct. 1997. [9] S. S. Ge and J. Zhang, “Neural-network control of nonaffine nonlinear system with zero dynamics by state and output feedback,” IEEE Trans. Neural Netw., vol. 14, no. 4, pp. 900–918, Jul. 2003. [10] H. Du, H. Shao, and P. Yao, “Adaptive neural network control for a class of low-triangular-structured nonlinear systems,” IEEE Trans. Neural Netw., vol. 17, no. 2, pp. 509–514, Mar. 2006. [11] D. Wang and J. Huang, “Adaptive neural network control for a class of uncertain nonlinear systems in pure-feedback form,” Automatica, vol. 38, no. 8, pp. 1365–1372, Aug. 2002. [12] S. S. Ge and C. Wang, “Adaptive NN control of uncertain nonlinear purefeedback systems,” Automatica, vol. 38, no. 4, pp. 671–682, Apr. 2002. [13] C. Wang, D. J. Hill, S. S. Ge, and G. Chen, “An ISS-modular approach for adaptive neural control of pure-feedback systems,” Automatica, vol. 42, no. 5, pp. 723–731, May 2006. [14] S. S. Ge, C. C. Hang, T. H. Lee, and T. Zhang, Stable Adaptive Neural Network Control. Boston, MA: Kluwer, 2002. [15] G. Lightbody and G. W. Irwin, “Direct neural model reference adaptive control,” Proc. Inst. Elect. Eng.—Control Theory Appl., vol. 142, no. 1, pp. 31–43, Jan. 1995. [16] S. S. Ge, C. C. Hang, and T. Zhang, “Nonlinear adaptive control using neural networks and its application to CSTR systems,” J. Process Control, vol. 9, no. 4, pp. 313–323, Aug. 1999. [17] L. O. Santos, P. A. F. N. A. Afonso, J. A. A. M. Castro, N. M. C. Oliveira, and L. T. Biegler, “On-line implementation of nonlinear MPC: An experimental case study,” Control Eng. Pract., vol. 9, no. 8, pp. 847–857, Aug. 2001. [18] W. R. Ramirez and B. A. Turner, “The dynamic modeling, stability, and control of a continuous stirred tank chemical reactor,” AIChE J., vol. 15, no. 6, pp. 853–860, Nov. 1969. [19] T. M. Apostol, Mathematical Analysis, 2nd ed. Reading, MA: AddisonWesley, 1974. [20] H. K. Khalil, Nonlinear Systems. Upper Saddle River, NJ: Prentice-Hall, 1996. [21] M. Krsti´c, I. Kanellakopoulos, and P. V. Kokotovi´c, Nonlinear and Adaptive Control Design. New York: Wiley, 1995. [22] R. Sepulchre, M. Jankovi´c, and P. V. Kokotovi´c, Constructive Nonlinear Control. London, U.K.: Springer-Verlag, 1997. [23] R. M. Sanner and J. E. Slotine, “Gaussian networks for direct adaptive control,” IEEE Trans. Neural Netw., vol. 3, no. 6, pp. 837–863, Nov. 1992.

REN et al.: ADAPTIVE NEURAL CONTROL FOR A CLASS OF UNCERTAIN NONLINEAR SYSTEMS

[24] M. M. Polycarpou, “Stable adaptive neural control scheme for nonlinear systems,” IEEE Trans. Autom. Control, vol. 41, no. 3, pp. 447–451, Mar. 1996. [25] F. L. Lewis, S. Jagannathan, and A. Yesilidrek, Neural Network Control of Robot Manipulators and Nonlinear Systems. Philadelphia, PA: Taylor & Francis, 1999. [26] S. Haykin, Neural Networks: A Comprehensive Foundations, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1999. [27] J. A. Farrell and M. M. Polycarpou, Adaptive Approximation Based Control. Hoboken, NJ: Wiley, 2006. [28] S. S. Ge, F. Hong, and T. H. Lee, “Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients,” IEEE Trans. Syst., Man, Cybern. B, Cybern., vol. 34, no. 1, pp. 499–516, Feb. 2004. [29] E. P. Ryan, “A universal adaptive stabilizer for a class of nonlinear systems,” Syst. Control Lett., vol. 16, no. 3, pp. 209–218, Mar. 1991. [30] B. Yao and M. Tomizuka, “Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form,” Automatica, vol. 33, no. 5, pp. 893–900, May 1997. [31] M. M. Polycarpou and P. A. Ioannou, “A robust adaptive nonlinear control design,” Automatica, vol. 32, no. 3, pp. 423–427, Mar. 1996.

Beibei Ren (S’06) received the B.E. degree in mechanical and electronic engineering and the M.E. degree in automation from Xidian University, Xi’an, China, in 2001 and 2004, respectively. She is currently working toward the Ph.D. degree in the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. Her current research interests include adaptive control, intelligent control, and control applications.

Shuzhi Sam Ge (S’90–M’92–SM’00–F’06) received the B.Sc. degree from the Beijing University of Aeronautics and Astronautics, Beijing, China, and the Ph.D. degree and the Diploma of Imperial College from the Imperial College of Science, Technology, and Medicine, University of London, London, U.K. He is the Director of the Social Robotics Laboratory, Interactive Digital Media Institute, and a Professor with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. He has authored or coauthored three books entitled Adaptive Neural Network Control of Robotic Manipulators (World Scientific, 1998), Stable Adaptive Neural Network Control (Kluwer, 2001), and Switched Linear Systems: Control and Design (Springer-Verlag, 2005), edited a book entitled Autonomous Mobile Robots: Sensing, Control, Decision Making and Applications (Taylor & Francis, 2006), and has over 300 international journal and conference papers. He also serves as an Editor of the Taylor and Francis Automation and Control Engineering Series. His current research interests include social robotics, multimedia fusion, adaptive control, and intelligent systems. Dr. Ge has served/been serving as an Associate Editor for a number of flagship journals including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, the IEEE TRANSACTIONS ON NEURAL NETWORKS, and Automatica. He is an elected member of the Board of Governors, IEEE Control Systems Society. He was the recipient of the 1999 National Technology Award, the 2001 University Young Research Award, the 2002 Temasek Young Investigator Award, Singapore, and the 2004 Outstanding Overseas Young Researcher Award from the National Science Foundation, China.

443

Chun-Yi Su (SM’99) received the Ph.D. degree from the South China University of Technology, Guangzhou, China, in 1990. He is currently a Professor and Concordia Research Chair with the Department of Mechanical and Industrial Engineering, Concordia University, Montreal, QC, Canada. His research interest includes control of systems involving hysteresis nonlinearities. Dr. Su has been serving on the editorial boards of several journals, including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL and the IEEE TRANSACTIONS ON C ONTROL S YSTEMS T ECHNOLOGY . He has also served as an Organizing Committee Member for many conferences.

Tong Heng Lee (M’90) received the B.A. degree (first-class honors) in the Engineering Tripos from Cambridge University, Cambridge, U.K., in 1980 and the Ph.D. degree from Yale University, New Haven, CT, in 1987. He is currently a Professor with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. He has coauthored three research monographs and is the holder of four patents (two of which are in the technology area of adaptive systems, and the other two are in the area of intelligent mechatronics). His research interests include adaptive systems, knowledge-based control, intelligent mechatronics, and computational intelligence. Dr. Lee currently holds Associate Editor appointments in the IEEE TRANSACTIONS ON SYSTEMS, MAN , AND CYBERNETICS; the IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS; Control Engineering Practice; the International Journal of Systems Science; and Mechatronics journal. He was a recipient of the Cambridge University Charles Baker Prize in Engineering and the 2004 ASCC (Melbourne) Best Industrial Control Application Paper Prize.