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Adaptive Neural Network Dynamic Surface Control for a Class of Time-Delay Nonlinear Systems With Hysteresis Inputs and Dynamic Uncertainties Xiuyu Zhang, Chun-Yi Su, Senior Member, IEEE, Yan Lin, Lianwei Ma, and Jianguo Wang

Abstract— In this paper, an adaptive neural network (NN) dynamic surface control is proposed for a class of time-delay nonlinear systems with dynamic uncertainties and unknown hysteresis. The main advantages of the developed scheme are: 1) NNs are utilized to approximately describe nonlinearities and unknown dynamics of the nonlinear time-delay systems, making it possible to deal with unknown nonlinear uncertain systems and pursue the L∞ performance of the tracking error; 2) using the finite covering lemma together with the NNs approximators, the Krasovskii function is abandoned, which paves the way for obtaining the L∞ performance of the tracking error; 3) by introducing an initializing technique, the L∞ performance of the tracking error can be achieved; 4) using a generalized Prandtl–Ishlinskii (PI) model, the limitation of the traditional PI hysteresis model is overcome; and 5) by applying the Young’s inequalities to deal with the weight vector of the NNs, the updated laws are needed only at the last controller design step with only two parameters being estimated, which reduces the computational burden. It is proved that the proposed scheme can guarantee semiglobal stability of the closed-loop system and achieves the L∞ performance of the tracking error. Simulation results for general second-order time-delay nonlinear systems and the tuning metal cutting system are presented to demonstrate the efficiency of the proposed method. Index Terms— L∞ performance, dynamic surface control (DSC), generalized Prandtl–Ishlinskii (PI) hysteresis, time delay. Manuscript received April 28, 2014; revised November 9, 2014 and January 22, 2015; accepted January 24, 2015. Date of publication February 12, 2015; date of current version October 16, 2015. This work was supported in part by the National Natural Science Foundation of China under Grant 61304015, Grant U1201244, Grant 61273141, Grant 61228301, Grant 61411140039, and Grant 51176028, in part by the China Post-Doctoral Science Foundation under Grant 2013M540839, in part by the Emerging Industries of Strategic Importance of Guangdong Province, China, under Grant 2012A090100012, in part by the Nature Science Foundation of Jilin Province under Grant 20140101059JC, in part by the Outstanding Young Scholar Project of Jilin under Grant 2013625002, in part by the Twelfth Five Year Science Research Plan of Jilin Province under Grant [2014]111, and in part by the Zhejiang Provincial National Natural Science Foundation under Grant Y1110508. X. Zhang and J. Wang is with the School of Automation Engineering, Northeast Dianli University, Jilin 132012, China (e-mail: zhangxiuyu80@ 163.com; [email protected]). C.-Y. Su is with the Department of Mechanical and Industrial Engineering, Concordia University, Montréal, QC H3B 1R6, Canada (e-mail: [email protected]). Y. Lin is with the School of Automation, Beijing University of Aeronautics and Astronautics, Beijing 100083, China (e-mail: [email protected]). L. Ma is with the Department of Automation, Zhejiang University of Science and Technology, Hangzhou 310023, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TNNLS.2015.2397935

I. I NTRODUCTION

I

N RECENT years, modeling and control of hysteresis nonlinear systems have attracted a lot of attention due to wide applications of smart material-based actuators ranging from ferromagnetic and superconductive fields, to electronic relay circuits and geosciences [1], [2], which inevitably display hysteresis nonlinearities. If a control system lacks compensation for the hysteresis, it may exhibit undesirable properties, such as oscillations or even instability, since the hysteresis is nondifferentiable and multivalued [3]. In dealing with the hysteresis in control systems, various robust and adaptive control schemes have been proposed over the past decade, including [3]–[8]. In [3], a pioneer work on the adaptive control of systems with hysteresis was introduced. Then, in [4] and [7], for a class of nonlinear systems with Prandtl–Ishlinskii (PI) and backlash-like hysteresis representations, sliding mode control schemes were proposed to compensate the hysteresis. In [5] and [6], the robust adaptive backstepping method was applied to mitigate the influence of the hysteresis phenomenon without constructing the hysteresis inverse. More recently, to avoid the problem of explosion of complexity, which was raised by backstepping, some dynamic surface control (DSC) schemes were proposed to compensate hysteresis described by the PI model [8], [9]. However, there are some practical systems, such as metal cutting systems [10], showing both the hysteresis and the time delay. Most of the existing results dealing with either hysteresis or time delay cannot apply to such systems. Therefore, it poses both the theoretical and the practical challenges to develop control approaches for such systems. Among the methods used for coping with the time delays inherent in uncertain nonlinear systems, Lyapunov–Krasovskii functions were generally used in adaptive backstepping and variable structure control in that the separation lemma was adopted and various assumptions on time-delay functions were made [11]–[20]. In [11]–[15], the adaptive backstepping schemes were proposed for time-delay strict-feedback nonlinear systems, therein, the scheme developed by [12] considerably reduced the number of adaptation parameters. In [16]–[20], various adaptive backstepping schemes using neural networks (NNs) or fuzzy control schemes as approximators were proposed to deal with the time delays and dynamic uncertainties for stochastic, large-scale, and MIMO nonlinear systems.

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ZHANG et al.: ADAPTIVE NN DSC FOR A CLASS OF TIME-DELAY NONLINEAR SYSTEMS

From the above discussions, the developed control schemes, including adaptive control, sliding model, and backstepping methods, were proposed to individually cope with hysteresis phenomenon or time delays. There are only few results available in [10], [21], and [22] to deal with both the hysteresis and the time delay that appear in the systems, such as metal cutting systems in manufacturing. In this paper, motivated by [10] and [21], and accumulated results for dealing with uncertainties [23]–[29], a NN-based adaptive DSC scheme is proposed for a class of unknown nonlinear time-delay systems with dynamic uncertainties and hysteresis input. It is proved that the proposed scheme guarantees semiglobal stability of the closed-loop system and achieves the L∞ performance of tracking error. Simulation results for general second-order time-delay nonlinear systems and tuning metal cutting systems are presented to demonstrate the efficiency of the proposed method. The features of this control scheme can be highlighted as follows. 1) Since the precise knowledge of the system dynamics and unstructured uncertainties is usually difficult to acquire, NNs are utilized to approximately describe the unknown nonlinear time-delay systems [30]–[33]. To the best of our knowledge, the proposed scheme is the first one using the NNs-based DSC approach to handle the unknown nonlinear time-delay systems with dynamic uncertainties and hysteresis input. 2) Using the finite covering lemma in [34] together with the NNs to approximate the unknown time-delay function, the Krasovskii functions are abandoned and the L∞ performance of the tracking error is pursued [9], which poses an interesting and challenging issue in control of nonlinear systems with time delay. It should be noted that the utilization of Krasovskii functions to handle the unknown time delay as that in [10]–[22] makes it impossible to obtain the L∞ performance of the tracking error due to the existence of the Krasovskii functions in transient performance analysis. Then, by modifying the traditional definition of the surface error in the DSC scheme and applying the initializing technique [35], the L∞ performance of the tracking error is obtained. 3) A benefit of using the finite covering lemma together with the NNs is that the assumptions of the conservative upper bound functions on the time-delay functions [10]–[22] are not required anymore and some nondifferentiable isolated points in the process of control are avoided [36]. Therefore, the analysis of stability is simplified. 4) By applying Young’s inequalities to deal with the weight vectors of the NNs, the updated laws of the unknown parameters are needed only at the last controller design step with only two parameters being estimated; therefore, the computational burden is reduced. 5) In this paper, instead of the traditional PI hysteresis model being utilized in [4] and [5], a generalized PI model [37] has been adopted. This adoption

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characterizes more accurately the hysteresis phenomena in metal cutting systems. The rest of this paper is organized as follows. In Section II, the mathematical preliminaries are represented. Then, the class of controlled nonlinear plants preceded by a generalized hysteresis is introduced and the control purpose is formulated. In Section III, the design procedure of the modified adaptive DSC scheme is presented. Section IV gives the stability analysis for the proposed scheme. Finally, two simulation examples are given to demonstrate the effectiveness of the proposed design method. II. P ROBLEM S TATEMENT AND M ATHEMATICAL P RELIMINARIES A. Problem Formulation and Mathematical Lemmas We consider the following nonlinear time-delay plants in strict-feedback form with the generalized hysteresis and dynamic uncertainties: z˙ = q(z, x) x˙ i = gi (x¯i )x i+1 + f i (x¯i ) + h i (x¯iτ ) + i (x, z, t) x˙n = gn (x¯n )w + f n (x¯n ) + h n (x¯nτ ) + n (x, z, t) y = x 1 , i = 1, . . . , n − 1

(1)

where x¯i := [x 1 , x 2 , . . . , x i ]T ∈ Ri for i = 1, . . . , n are state vectors; x = x¯n ; x¯iτ := [x 1 (t − τ1 ), x 2 (t − τ2 ), . . . , x i (t−τi )]T ∈ Ri , for i = 1, . . . , n, are time-delay state vectors with τi being unknown time delays; fi (·) are unknown smooth functions and h i (·) are unknown time-delay functions; gi (x¯i ) are unknown continuous functions; q(z, x) and i (x, z, t) are uncertain Lipschitz continuous functions. The z in (1) is the unmodeled dynamics; y ∈ R is the output of the controlled plant; w ∈ R is the unknown generalized hysteresis and can be expressed as w(t) = P[u(t)]

(2)

with u(t) being the input signal to be designed. The operator P[u] will be discussed in detail in the forthcoming section. It should be pointed out that (1) with hysteresis as the input describes many physical systems, such as the tuning metal cutting system, which is introduced in [10] and illustrated as the second simulation example. To proceed, we make the following assumptions. A1: The uncertain dynamic disturbances i (x, z, t), i = 1, . . . , n, satisfy [38], [39] |i (x, z, t)| ≤ φi1 (|x¯i |) + φi2 (|z|)

(3)

where φi1 (·) and φi2 (·) are unknown nonnegative smooth functions satisfying that φi2 (0) = 0. A2: τi , for i = 1, . . . , n, belong to a known compact set [0, τ M ], where τ M > 0 is a known constant and denotes the maximum value of the upper bound of each τi . A3: The unmodeled dynamics are exponentially input-to-state practically stable (exp-ISpS), i.e., the

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system z˙ = q(z, x) has an exp-ISpS Lyapunov function V (z), which satisfies [38], [39] α1 (|z|) ≤ V (z) ≤ α2 (|z|) ∂ V (z) q(z, x) ≤ −cV (z) + γ (|x|) + d¯ ∂z

(4) (5)

where α1 (·), α2 (·), and γ (·) belong to a class of K ∞ -functions. Moreover, γ (·) is a known function, and c and d¯ are known positive constants. A4: The desired trajectory yr is smooth and available with yr (0) at a designer’s disposal; [yr , y˙r , y¨r ]T belongs to a known compact set for all t ≥ 0. A5: The signs of gi (·), i = 1, . . . , n, are known. Without loss of generality, it is assumed that gi (·) > 0. Moreover, there exist constants gmin and gmax , such that 0 < gmin ≤ |gi (·)| ≤ gmax . Remark 1: In [38] and [39], the adaptive backstepping output and state feedback controllers were designed for nonlinear systems with dynamic uncertainties that are bounded by some known continuous functions φi1 (·) and φi2 (·). In A1, these continuous functions are extended to unknown functions. A3 is the same as that in [38] and [39]. A4 is a basic requirement in backstepping and DSC methods. A5 implies the smooth functions gi (·), i = 1, . . . , n, are strictly either positive or negative and A5 is reasonable because gi (·) = 0 is the controllable condition of (1) in [40] and [41]. To analyze the stability and robustness of the control system in Sections II-B and II-C, the following lemmas are necessary. Lemma 1: Suppose f (ξ ) : ξ → R is a smooth function with ξ ⊂ Rn being a compact set. Let ξ = ξ(t − τ ) be uniformly continuous with respect to t, where τ ∈ [0, τ M ] is an known constant time delay. Then, for any given δ0 > 0, there exists a finite partition of [0, τ M ], independent of t 0 ≤ t1 < t2 < · · · < tm ≤ τ M

(6)

from which a point τ¯ ∈ {t1 , . . . , tm }

(7)

can be extracted, such that | f (ξ(t − τ )) − f (ξ(t − τ¯ ))| < δ0 ∀t ≥ 0.

(8)

Proof: See Appendix A.  Lemma 2 [38]: For any  > 0, there exists a smooth function ψ(·), such that ψ(0) = 0 and |x| ≤ xψ(x) + , ∀x ∈ R. Lemma 3 [38]: For any  > 0 and any continuous function f (x) : R → R, with f (0) = 0, there exists a nonnegative

smooth function fˆ, with fˆ (0) = ∂ fˆ∂ x(0) = 0, such that | f (x)| ≤ fˆ(x) + , ∀x ∈ R. Lemma 4 [38]: If V is an exp-ISpS Lyapunov function for a control system z˙ = q(z, x), i.e., (4) and (5) hold, then for any constant c¯ in (0, c), any initial instant t0 > 0, and any initial conditions z 0 = z(t0 ) and ς0 = ς (t0 ) > 0, there exists a finite T0 = T0 (c, ¯ ς0 , z 0 ) ≥ 0, a nonnegative function D(t0 , t) defined for all t ≥ t0 and an available signal ς˙ = −cς ¯ + x 12 γ (|x 1 |2 ) + d¯

(9)

Fig. 1.

Generalized PI hysteresis.

such that D(t0 , t) = 0 for all t ≥ t0 + T0 and V (z) ≤ ς (t) + D(t0 , t)

(10)

for all t ≥ t0 where

 −c(t ¯ −t0 )  D(t0 , t) = max 0, e−c(t −t0 ) V (z 0 ) − e0 ς0

(11)

and the solutions are defined. B. Generalized PI Model In this section, (2) will be described by the generalized PI model [42], [43]. The detailed discussions can be found in [42] and [43]. Analytically, we suppose that Cm [0, t E ] is the space of piecewise monotone continuous functions. For any input u(t), ρ let fr : R → R be defined by [42] frρ (u, w) = max(ρ(u) − r, min(ρ(u) + r, w))

(12)

where r denotes the threshold of the hysteresis, satisfying r ≥ 0, ρ(·) is a known continuous nondecreasing function. ρ Then, the generalized play operator Fr [·] is defined as [43] Frρ [u](0) = frρ (u(0))

Frρ [u](t) = frρ (u(t), Frρ [u](ti )) for ti < t ≤ ti+1 and 0 ≤ i ≤ N − 1 (13)

where 0 = t0 < t1 < · · · < t N = t E is a partition of [0, t E ], such that the function u is monotone (nonincreasing or decreasing) on each of the subintervals [ti , ti+1 ]. Using the above definition of the generalized play operator, the generalized PI model can be expressed as [43]  D w(t) = λu(t) + p(r )Frρ [u](t)dr (14) 0

where λ is an unknown positive constant and p(r ) is a given continuous function, called density function. It should be noted that p(r ) can be  ∞obtained from experiment data and satisfies p(r ) > 0, 0 r p(r )dr < ∞. For convenience, we choose D = ∞ as the upper limit of the integration. Fig. 1 shows the generalized PI hysteresis described by (14)

ZHANG et al.: ADAPTIVE NN DSC FOR A CLASS OF TIME-DELAY NONLINEAR SYSTEMS

with λ = 0.45, ρ(u) = 10 tanh(u), p(r ) = 0.6e−0.108(r−0.5) , and r ∈ [0, D] = [0, 100], u(t) = 2 sin(4πt) + cos(πt), t ∈ [0, 2π]. In the following development, an assumption on the density function is required. A6: There exists a known constant pmax , such that p(r ) < pmax for all r ∈ [0, D]. Remark 2: Due to the properties of the density function ∞ p(r ) that p(r ) > 0 with 0 r p(r )dr < ∞, it is reasonable to set an upper bound for p(r ). 2

C. Radial Basis Function Neural Network Approximation In this paper, the radial basis function NNs (RBFNNs) will be used to approximate the unknown continuous real value function f i : ξi → R on a given compact set ξi ⊂ Rq with arbitrary accuracy as follows [44], [45]: f i (ξi ) = Wi∗T ψi (ξi ) + ei (ξi ), 1 ≤ i ≤ n

(15)

where ξi ∈ ξi ⊂ Rq , for any positive integer q ≥ 1, are the input vectors; Wi∗ ∈ R N are the ideal weight vectors for some sufficiently large integer N, which denotes the NN node number satisfying N > 1, and ψi (ξi ) = [1 (ξi ), . . . ,  N (ξi )] ∈ R N are vector valued functions.  j (ξ ) are so-called radial basis (Gaussian) functions with the following form:   −(ξi − ζ j )T (ξi − ζ j ) , 1≤ j ≤N (16)  j (ξi ) = exp η2j where ζ j ∈ Rq , j = 1, . . . , N, are constant vectors called the center of the basis functions, and η j are real number called the width of the basis functions. ei (ξ ) is the network reconstruction error, satisfying that |ei (ξi )| ≤ δi∗ , 1 ≤ i ≤ n

(17)

where δi∗ are unknown constant. Taking (14) into consideration, (1) can be rewritten as z˙ = q(z, x) x˙i = gi (·)x i+1 + f i (x¯i ) + h i (x¯iτ ) + i (x, z, t) x˙n = β(·)u + f n (x) + h n (x¯nτ ) + n (x, z, t)  D p(r )Frρ [u](t)dr +gn (·) 0

y = x 1 , i = 1, . . . , n − 1

(18)

where β(·) = λgn (x), β(·) > 0.

(19)

With the above β(·), the following assumption is needed. A7: There exist constants βmin and βmax , such that βmin ≤ β(·) ≤ βmax . Remark 3: Due to A5 and λ being the unknown positive constant, A7 is reasonable. In addition, because gmin , gmax , βmin , βmax , and pmax are not required in the implementation of the proposed control design, they are used only for analytical purposes. The control objective is to design a control law u in (18), such that the L∞ performance of the system tracking error can be obtained and all the closed-loop signals are uniformly bounded.

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III. A DAPTIVE N EURAL N ETWORK DSC D ESIGN In this section, an adaptive NN DSC scheme will be proposed to cope with time-delay nonlinear systems with dynamic uncertainties and unknown hysteresis described by (18). The whole design procedures contain n steps and the actual control law u will be deduced at the last step. Step 1: Let the first surface error be defined as S1 = x 1 − yr

(20)

where yr is the desired trajectory. Consider the following Lyapunov function: V1 =

1 2 1 S1 + ς 2 λ0

(21)

where λ0 > 0. By Lemma 1, there exists a point τ1/1 ∈ {t1 , . . . , tm }, where t1 , . . . , tm are defined in (6), such that1 h 1 (x 1τ ) = h 1 (x 1 (t − τ1 )) = h 1 (x 1 (t − τ1/1 )) + e1

(22)

where |e1 | ≤ δ0 . Using (9) and the inequality2 Si ei ≤

1 2 1 ∗2 S + δ0 , i = 1, . . . , n 2 i 2

we have  1 ˙ V1 ≤ S1 g1 x 2 + f 1 + h 1 (x 1 (t − τ1/1 )) + S1 − y˙r 2  1 c¯ 1 2 2 +|S1 |1 | − ς + x 1 γ (|x 1 | ) + d¯ + δ02 . λ0 λ0 2

(23)

(24)

Substituting (3) in (24) yields  1 ˙ V1 ≤ S1 g1 x 2 + f 1 + h 1 (x 1 (t − τ1/1 )) + S1 − y˙r 2 c¯ + |S1 |φ11 (|x 1 |) + |S1 |φ12 (|z|) − ς λ0  1 2 1 2 2 + (25) x γ (|x 1 | ) + d¯ + δ0 . λ0 1 2 Considering the following inequality:  Si φi1 (|x¯i |) |Si |φi1 (|x¯i |) ≤ Si φi1 (|x¯i |) tanh i1 + 0.2785i1 , i = 1, . . . , n

(26)

where i1 > 0, we have  S1 φ11 (|x 1 |) V˙1 ≤ S1 g1 x 2 + f1 − y˙r + φ11 (|x 1 |) tanh 11

1 c¯ + h 1 (x 1 (t − τ1/1 )) + S1 − ς + |S1 |φ12 (|z|) 2 λ0  1 1 2 2 x γ (|x 1 | ) + d¯ + 0.278511 + δ02 . (27) + λ0 1 2 1 Since in this step the variable x (t) on the compact set 1 ξ1 and in the following steps the variables xi (t) on the compact sets ξi , i = 2, . . . , n, are uniformly continuous with respect to t, we can choose a partition of [0, τ M ], which is uniform for all t ≥ 0 and is suitable for each i. For the sake of simplicity, we use the set {t1 , . . . , tm } given by (7) to denote such a partition. 2 Here, S and e (i = 2, . . . , n) are the ith surface error and the function i i error, respectively, that will be introduced in the next steps.

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By adding and subtracting (2/S1 ) tanh2 (S1 /γ ) 2 2 (x 1 γ (|x 1 | )/λ0 ) at the right-hand side (RHS) of (27), the following inequality holds:  S1 φ11 (|x 1 |) V˙1 ≤ S1 g1 x 2 + f 1 − y˙r + φ11 (|x 1|) tanh 1 1  2

2 S1 x 1 γ (|x 1 | ) 2 1 + tanh2 +h 1 (x 1 (t −τ1/1 )) + S1 S1 γ λ0 2 ¯ d c¯ + |S1 |φ12 (|z|) + 0.278511 − ς+ λ0 λ0  2  x 1 γ (|x 1 |2 ) 1 2 S1 + 1 − 2 tanh2 + δ0 (28) γ λ0 2 where γ > 0. Using Lemmas 2–4, the inequality

|S1 |φ12 (|z|) ≤ S1 φ¯ 12 (S1 , ς ) +

1 2 S + d1 max + 212 4 1

(29)

can be approved (see Appendix B for the detailed derivations). Then, we have 3 ˙ V1 ≤ S1 g1 x 2 + f 1 − y˙r + S1 + φ11 (|x 1 |) 4  1 S1 φ11 (|x 1 |)

+ φ¯12 (S1 , ς ) + g12 S1 × tanh 11 2 c¯ d¯ 1 2 + g 1 c2 − ς + + g S1 λ0 λ0 2ϑ1 1  2

S1 x 1 γ (|x 1 |2 ) 2 + tanh2 + h 1 (x 1 (t − τ1/1)) S1 γ λ0 + 0.278511 + d1 max + 212 1 1 1 2 2 + δ02 − g12 S12 − g S − g1 c2 S1 2 2 2ϑ1 1 1  2  x 1 γ (|x 1 |2 ) S1 + 1 − 2 tanh2 (30) γ λ0 where ϑ1 is a positive design parameter and c2 is a constant design parameter, which is employed to guarantee the L∞ performance of the tracking error. To deal with the unknown term, the RBFNN is used to approximate it3 W1∗T ψ1 (ξ1 ) + w1 (ξ1 )  S1 φ11 (|x 1 |) 1 φ11 (|x 1 |) tanh + f 1 − y˙r = g1 (·) 11  2 2 S1 x 1 γ (|x 1 |2 )

+ φ¯ 12 (S1 , ς ) tanh2 S1 γ λ0 1 2 + g1 S1 + h 1 (x 1 (t − τ1/1)) 2

3 1 2 + S1 + g1 S1 + g1 c2 (31) 4 2ϑ1 where ξ1 := (x 1 (t), x 1 (t − t1 ), . . . , x 1 (t − τ1/1 ), . . . , x 1 (t − tm ), ς, yr , y˙r )

and ξ1 ∈ ξ1 ⊂ R4+m . Let θ ∗ := max{(1/gm )|Wi∗ |2 i = 1, . . . , n}, where gm = min{gmin , βmin }. Then, the following inequality holds:

 g1 S1 W1∗T ψ1 + w1 (ξ1 ) ≤

gm ϑ12 S12 θ ∗ ψ1T ψ1 1 1 g2 + g12 S12 + δ1∗2 . + max 2 2 2 2ϑ12

Using (33) and substituting (31) in (30) yield  gm ϑ12 S1 θ ∗ ψ1T ψ1 c¯ ˙ V1 ≤ S1 g1 x 2 + − ς 2 λ0 2 1 2 2 1 ∗2 gmax − g S − g1 c2 S1 + δ1 + 2ϑ1 1 1 2 2ϑ12 1 d¯ + + δ02 + 0.278511 + d1 max + 212 λ 2 0  2 x 1 γ (|x 1 |2 ) S1 + 1 − 2 tanh2 . γ λ0

(33)

(34)

2 /2ϑ 2 ) + (1/2)δ ∗2 + (d/λ ¯ 0 ) + (1/2)δ 2 + Letting C1 := (gmax 1 1 0 0.278511 + d1 max + 212 , it follows that:  gm ϑ12 S1 θ ∗ ψ1T ψ1 c¯ ˙ V1 ≤ S1 g1 x 2 + − ς 2 λ0 1 2 2 g S − g1 c2 S1 + C1 − 2ϑ1 1 1   2 x 1 γ (|x 1 |2 ) S1 + 1 − 2 tanh2 . (35) γ λ0

To make (35) stable, a virtual control signal is chosen as x 2d = −k1 S1 −

ϑ12 S1 θˆ ψ1T ψ1 2

(36)

where θˆ is the estimation of θ ∗ and its updated law will be deduced at nth step. Let x 2d pass through a first-order filter to obtain z 2 ι2 z˙ 2 + z 2 = x 2d , z 2 (0) = x 2d (0)

(37)

where ι2 is the time constant. Step i (2 ≤ i ≤ n − 1): Define the ith surface error Si = x i − z i − ci

(38)

where ci is a constant design parameter.4 The time derivative of (38) is S˙i = x˙i − z˙ i = gi x i+1 + f i + h i (x¯iτ ) + i − z˙ i .

(39)

By Lemma 1, there exists a set of points {τ1/ i , τ2/ i , . . . , τi/ i } ⊂ {t1 , . . . , tm }, such that h i (x 1 (t − τ1 ), . . . , x i (t − τi )) = h i (x 1 (t − τ1/ i ), . . . , x i (t − τi/ i )) + ei , |ei | ≤ δ0 . (40) Consider the following Lyapunov function candidate:

(32)

3 Using lim 2 Si →0 1/Si tanh (Si /ε) = 0, where ε is defined as a positive

constant and Si ∈ R, for i = 1, . . . , n, are defined as the ith surface error that will be introduced in the next steps, the RHS of (31) is even at S1 = 0 [18].

Vi =

1 2 S . 2 i

(41)

4 As will be shown in Section IV, the design parameters c , c , . . . , c n 2 3 in Step i and Step n are employed to guarantee L∞ performance of the tracking error.

ZHANG et al.: ADAPTIVE NN DSC FOR A CLASS OF TIME-DELAY NONLINEAR SYSTEMS

Then, the time derivative of Vi yields V˙i ≤ Si (gi x i+1 + f i + h i (x 1 (t − τ1/ i ), . . . , x i (t − τi/ i )) + ei − z˙ i ) + |Si ||i |. (42) From (3) and (23), we have

V˙i ≤ Si gi x i+1 + f i + h i (x 1 (t − τ1/ i ), . . . , x i (t − τi/ i ))  1 1 + Si − z˙ i + δ02 + |Si |φi1 (|x¯i |) + |Si |φi2 (|z|). 2 2 (43) By pursuing the same arguments as in Step 1, it follows that:

(Si , ς ) + |Si |φi2 (|z|) ≤ Si φ¯ i2

1 2 S + di (t0 , t) + 2i2 4 i

(44)

where

(Si , ς ) = φ¯ i2 (ς )φ¯ i3 (Si , ς ) + φ¯ i4 (Si ) φ¯ i2

(45)

and φ¯i2 (ς ), φ¯ i3 (Si , ς ), and φ¯ i4 (Si ) are smooth functions satisfying that φ¯ i2 (0) = 0, φ¯ i3 (0, 0) = 0, and φ¯ i4 (0) = 0, and

2 (46) di (t0 , t) = φi2 ◦ α1−1 (2D(t0 , t)) with di (t0 , t) satisfying |di (t0 , t)| ≤ di max . From (44), the following inequality holds:

(Si , ς ) + |Si |φi2 (|z|) ≤ Si φ¯ i2

1 2 S + di max + 2i2 . 4 i

(47)

Using (26) and substituting (47) in (43), we have  Si φi1 (|x¯i |) V˙i ≤ Si gi x i+1 + f i − z˙ i + φi1 (|x¯i |) tanh i1 3

+ φ¯ i2 (Si , ς )+ Si + h i (x 1 (t −τ1/ i ), . . . , x i (t − τi/ i )) 4

1 2 1 2 g Si + gi ci+1 + gi−1 Si−1 + gi Si + 2 2ϑi i 1 + δ02 + 0.2785i1 + di max + 2i2 2 1 1 2 2 − gi2 Si2 − g S − gi ci+1 Si − gi−1 Si−1 Si (48) 2 2ϑi i i where ϑi is any positive constant. To deal with the unknown term, the RBFNN is used to approximate it as follows: Wi∗T ψi (ξi ) + wi (ξi ) 1

= fi − z˙ i + φ¯ i2 (Si , ς ) + φi1 (|x¯i |) gi (·)  1 Si φi1 (|x¯i |) + gi2 Si × tanh i1 2 + h i (x 1 (t − τ1/ i ), . . . , x i (t − τi/ i ))

3 1 2 gi Si + gi−1 Si−1 + gi ci+1 + Si + 4 2ϑi

j = 1, . . . , i , and ξi ∈ ξi ⊂ Ri+4+m×i . Thus, using the following inequality just as in Step 1:

 gi Si Wi∗T ψi (ξi ) + wi (ξi ) ≤

gm ϑi2 Si2 θ ∗ ψiT ψi 1 1 g2 + gi2 Si2 + δi∗2 (51) + max 2 2 2 2ϑi2

we have

 gm ϑi2 Si θ ∗ ψiT ψi 1 V˙i ≤ Si gi x i+1 + + δ02 + 0.2785i1 2 2 1 2 2 + di max i − g S − gi ci+1 Si 2ϑi i i g2 1 − gi−1 Si−1 Si + 2i2 + max + δi∗2 . (52) 2 2ϑi2

2 /2ϑ 2 ) + (1/2)δ 2 + 0.2785 + d Let Ci := (gmax i1 i max + i 0 ∗2 (1/2)δi + 2i2 . Then, we have  gm ϑi2 Si θ ∗ ψiT ψi 1 2 2 ˙ Vi ≤ Si gi x i+1 + g S − 2 2ϑi i i −gi ci+1 Si − gi−1 Si−1 Si + Ci . (53)

To make (53) stable, a virtual control signal is chosen as ϑi2 Si θˆ ψiT ψi . (54) 2 pass through a first-order filter to obtain z i+1 x i+1d = −ki Si −

Let x i+1d

ιi+1 z˙ i+1 + z i+1 = x i+1d , z i+1 (0) = x i+1d (0)

(55)

where ιi+1 is the time constant. Step n: Define the nth surface error Sn = x n − z n − cn

(56)

with cn being any design parameter, whose derivative is S˙n = x˙ n − z˙ n = βu + f n + h n (x¯nτ ) + n (x, z, t)  D p(r ) ρ F [u](t)dr − z˙ n . (57) + λgn (·) λ r 0 By Lemma 1, there exists a set of points {τ1/n , τ2/n , . . . , τn/n } ⊂ {t1 , . . . , tm }, such that h n (x 1 (t − τ1 ), . . . , x n (t − τn )) = h n (x 1 (t − τ1/n ), . . . , x n (t − τn/n )) + en , |en | ≤ δ0 . (58) Let pλ(r ) := ( p(r )/λ). Then, (57) can be rewritten as S˙n = βu + f n + h n (x 1 (t − τ1/n ), . . . , x n (t − τn/n ))  D + en + n (x, z, t) − z˙ n + β pλ (r )Frρ [u](t)dr. 0

(59) (49)

where ξi := (x¯i , x j (t − t1 ), . . . , x j (t − τ j/ i ), . . . , x j (t − tm ), ς, z i−1 , z i , z˙ i )

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(50)

Consider the following Lyapunov function candidate:  1 2 gm 2 βmax D 2 ˜ θ + p˜ λ (t, r )dr Vn = Sn + 2 2γθ 2γpr 0

(60)

where θ˜ := θˆ − θ ∗ and p˜ λ(t, r ) := pˆ λ (t, r ) − pλmax with θˆ and pˆ λ (t, r ) being the estimations of the θ ∗ and pλ (r ), respectively; pλmax := ( pmax /λ), γθ , and γpr are positive

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design parameters. Using (3) and (23), then, the derivative of Vn yields ˙ Vn ≤ Sn βu + f n + h n (x 1 (t − τ1/n ), . . . , x n (t − τn/n ))

1 + Sn − z˙ n + |Sn |φn1 (|x¯n |) + |Sn |φn2 (|z|) 2  D   1 gm ˜ ˙ˆ pλmax  Frρ [u](t)dr + δ02 + θ θ + β|Sn | γθ 2 0  βmax D ∂ + p˜ λ (t, r ) pˆ λ (t, r )dr. (61) γpr 0 ∂t By pursuing the same arguments as in Step 1, it follows that: 1

(Sn , ς ) + Sn2 + dn (t0 , t) + 2n2 (62) |Sn |φn2 (|z|) ≤ Sn φ¯ n2 4 where

φ¯ n2 (Sn , ς ) = φ¯ n2 (ς )φ¯ n3 (Sn , ς ) + φ¯ n4 (Sn )

(63)

and φ¯n2 (ς ), φ¯ n3 (Sn , ς ), and φ¯ n4 (Sn ) are smooth functions satisfying that φ¯ n2 (0) = 0, φ¯ n3 (0, 0) = 0, and φ¯n4 (0) = 0, and

2 (64) dn (t0 , t) = φn2 ◦ α1−1 (2D(t0 , t)) with dn (t0 , t) satisfying |dn (t0 , t)| ≤ dn max . Then, from (62), the following inequality holds: 1

(Sn , ς ) + Sn2 + dn max + 2n2 . (65) |Sn |φn2 (|z|) ≤ Sn φ¯ n2 4 Using (26) and substituting (65) in (61), we have ˙ Vn ≤ Sn βu + f n + h n (x 1 (t − τ1/n ), . . . , x n (t − τn/n ))  Sn φn1 (|x¯n |)

(Sn , ς ) + φn1 (|x¯n |) tanh − z˙ n + φ¯ n2 n1

3 1 2 + Sn + β Sn + gn−1 Sn−1 4 2  D gm ˜ ˙ˆ + pλmax |Frρ [u](t)|dr θ θ + β|Sn | γθ 0  βmax D ∂ 1 + p˜ λ (t, r ) pˆ λ (t, r )dr + δ02 γpr 0 ∂t 2 + 0.2785n1 + dn max + 2n2 1 − β 2 Sn2 − gn−1 Sn−1 Sn . (66) 2 To deal with the unknown term, the RBFNN is used to approximate it as follows: Wn∗T ψn (ξn ) + wn (ξn ) 1 f n + h n (x 1 (t − τ1/n ), . . . , x n (t − τn/n )) − z˙ n = β(·)  3 1 2 Sn φn1 (|x¯n |) + Sn + β Sn + φn1 (|x¯n |) tanh 4 2 n1

(67) + φ¯ n2 (Sn , ς ) + gn−1 Sn−1 where ξn := (x¯n , x j (t − t1 ), . . . , x j (t − τ j/n ), . . . , x j (t − tm ), ς, z n−1 , z n , z˙ n ) (68)

j = 1, . . . , n and ξn ∈ ξn ⊂ Rn+4+m×n . Thus, using the following inequality just as in Step i , for i = 1, . . . , n − 1:

 β Sn Wn∗T ψn (ξn ) + wn (ξn ) ≤

β2 1 1 gm ϑn2 Sn2 θ ∗ ψnT ψn + max + β 2 Sn2 + δn∗2 2 2ϑn2 2 2

(69)

we obtain

 gm ϑn2 Sn θ ∗ ψnT ψn V˙n ≤ Sn βu + − gn−1 Sn−1 Sn 2  D gm ˜ ˙ˆ + β|Sn | pλmax |Frρ [u](t)|dr + θθ γθ 0  ∂ 1 β2 βmax D p˜ λ(t, r ) pˆ λ (t, r )dr + δn∗2 + max + γpr 0 ∂t 2 2ϑn2 1 + δ02 + 0.2785n1 + dn max + 2n2 (70) 2 where ϑn is any positive design parameter. Let Cn :=

2 βmax 1 1 + δn∗2 + δ02 + 0.2785n1 + dn max + 2n2 . 2ϑn2 2 2 (71)

Then, we have  gm ϑn2 Sn θ ∗ ψnT ψn ˙ Vn ≤ Sn βu + − gn−1 Sn−1 Sn 2  D   gm ˙ θ˜ θˆ + β|Sn | pλ (r ) Frρ [u](t)dr + γθ 0  βmax D ∂ + p˜ λ (t, r ) pˆ λ (t, r )dr + Cn . (72) γpr 0 ∂t Based on (72), the control law is designed as ϑ 2 Sn θˆ ψnT ψn − sgn(Sn ) u := −kn Sn − n 2  D pˆ λ (t, r )|Frρ [u](t)|dr ×

(73)

0

D ρ where the term −sgn(Sn ) 0 pˆ λ (r )|Fr [u](t)|dr in (73) is used to compensate the hysteresis. θˆ and pˆ λ(t, r ) are updated by  n  2 2 T ˙θˆ = γ  ϑi Si ψi ψi − σ θˆ (74) θ θ 2 i=1 ⎧ ⎨ γpr [|Sn ||Fr [u](t)| − σpr pˆ λ (t, r )] ∂ if 0 < pˆ λ (t, r ) ≤ pλmax (75) pˆ λ (t, r ) = ⎩ ∂t −γpr σpr pˆ λ (t, r ), if pˆ λ (t, r ) ≥ pλmax with σθ and σpr being positive constants. Remark 4: From the updated laws of θˆ and pˆ g (t, r ), we know that if θˆ (0) ≥ 0, pˆ λ (0, r ) ≥ 0, then θˆ ≥ 0, pˆ λ (t, r ) ≥ 0 for all t > 0 and r ∈ [0, D]. Therefore, in this paper, to keep θ ≥ 0, pˆ λ (t, r ) ≥ 0, for all t > 0 and r ∈ [0, D], the initial values θˆ (0) and pˆ λ (0, r ) are set as positive, which is very important for the analysis of stability. Remark 5: In the above design procedures, the commonly adopted dynamic surfaces, say, Si = x i − z i , need to be re-defined and we use Si = x i − z i − ci , for 2 ≤ i ≤ n, as the new dynamic surfaces for the purpose of deriving

ZHANG et al.: ADAPTIVE NN DSC FOR A CLASS OF TIME-DELAY NONLINEAR SYSTEMS

the L∞ performance of the tracking error, which will be given in the proof of Theorem 1. Remark 6: Based on the above design procedures, only the density function of generalized PI hysteresis and the norm of RBFNNs weight vector need to be updated regardless of the number of the nodes and the amount of the input variables of the RBFNNs. Therefore, compared with [8] and [21] where the complete weight vector of the NNs is estimated, the computational burden is reduced with the proposed method. IV. S TABILITY AND T RANSIENT P ERFORMANCE A NALYSIS In this section, the stability and L∞ performance analysis for the proposed DSC scheme will be presented. To this end, we define ϑ 2 S1 θˆ ψ1T ψ1 y2 = z 2 − x 2d = z 2 + k1 S1 + 1 2 ϑ 2 Si θˆ ψiT ψi yi+1 = z i+1 − x i+1d = z i+1 + ki Si + i 2 i = 2, . . . , n − 1 (76) where x 2d and x i+1d are given by (36) and (54), respectively. From (37), (55), and (76), since yi z˙ i = (x id − z i )/ιi = − , i = 2, . . . , n (77) ιi we obtain ϑ 2 θˆ S˙1 ψ1T ψ1 ϑ 2 θˆ˙ S1 ψ1T ψ1 y2 + 1 y˙2 = − + k1 S˙1 + 1 ι2 2 2  ∂ψ1 ∂ψ1 2ˆ T x˙1 (t − t1 ) + · · · + α1 θ S1 ψ1 x˙1 + ∂ x1 ∂ x 1 (t − t1 ) ∂ψ1 ∂ψ1 x˙1 (t − tm ) + y¨r + ∂ x 1 (t − tm ) ∂ y˙r y2 = − + B2 (S1 , . . . , Sn , y2 , . . . , yn , θˆ , ς, yr , y˙r , y¨r , ι2 S1 (t − t j ), . . . , Sn (t − t j ), y2 (t − t j ), . . . , yn (t − t j ), θˆ (t − t j ), ς (t − t j ), yr (t − t j ),

y˙i+1

y˙r (t − t j ), y¨r (t − t j )) ϑ 2 ϑˆ S˙i ψiT ψi ϑ 2 ϑ˙ˆ Si ψiT ψi yi+1 + i =− + ki S˙i + i ιi+1 2 2 ⎛ ⎞ i  ∂ψi ∂ψi ⎠ x˙ j + z¨i + ϑi2 ϑˆ Si ψiT ⎝ ∂x j ∂ z˙ i

V =

n  i=1

where

 VQ i =

1 VQ i 2 n

Vi +

i=1

t t −τi

Q i (x¯i (t))dt

was the Krasovskii function. Using the initializing procedures outlined in Appendix C, one can obtain C 1 + VQ i (0) κ 2 i=1  which implies that the existence of ni=1 VQ i (0) makes (146) impossible. Thus, the L∞ performance of the tracking error cannot be pursued. n

V (0) ≤

V. S IMULATION R ESULTS In this section, we will illustrate the proposed control scheme presented in the previous sections using two examples: 1) a general second-order nonlinear system and 2) the tuning metal cutting system.

To illustrate the effectiveness of the proposed scheme, we consider the following second-order nonlinear time-delay system with unknown generalized PI hysteresis and dynamic uncertainties:

j =1

z˙ = −z + x 12 x˙1 = x 2 + 0.05 sin(x 2 ) + x 12

S1 (t − t j ), . . . , Sn (t − t j ), y2 (t − t j ), . . . , yn (t − t j ), θˆ (t − t j ), ς (t − t j ), yr (t − t j ), j = 1, . . . , m

the control law (73), and the updated laws (74) and (75), subject to Assumptions A1–A7. Then, for any given positive number p, if V (0) in (104) satisfies V (0) ≤ p, all the signals of the closed-loop system are uniformly bounded and the L∞ norm of the tracking error is guaranteed by properly choosing the design parameters ki , ιi+1 , i = 1, . . . , n − 1, θ , σθ , γpr , and σpr in (74) and (75). Proof: See Appendix C.  Remark 7: Compared with the results dealing with time delays in [11], [15], [19], [21], and [45], the Krasovskii functions as well as the assumptions of the conservative upper bound functions on the time-delay functions are removed. Therefore, there is no need to consider Krasovskii functions for stability analysis. This greatly helps to obtain the L∞ performance of the tracking error, which is impossible using the Lyapunov–Krasovskii functions. For example, in [11], [15], [19], [21], and [45], the Lyapunov function for stability analysis is in the form of

A. General Second-Order Nonlinear Time-Delay System With the Input Hysteresis

yi+1 =− + Bi+1 ιi+1 × (S1 , . . . , Sn , y2 , . . . , yn , θˆ , ς, yr , y˙r , y¨r ,

y˙r (t − t j ), y¨r (t − t j )),

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(78)

where Bi+1 , for i = 1, . . . , n − 1, are continuous functions. We are now ready to establish the main theorem of this paper. Theorem 1: Consider the closed-loop system, consisting of the time-delay system (1) with the hysteresis described by (14),

+ sin(x 1 (t − 1)) + 1 (x, z, t) 1 − e−x2 x˙2 = w + 0.1 sin(w) + 1 + e−x2 + x 1 (t − 1)x 2 (t − 0.5) + 2 (x, z, t) y = x1

(79)

where w is the output of the hysteresis, which is described 2 by (14) with density function p(r ) = 1.2e−0.001(r−0.5) ,

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λ = 0.9, and ρ(u) = 3 ∗ tanh(u). The dynamic uncertainties are 1 (x, z, t) = 0.1 sin(t) + x 1 e−0.5x1 + z and 2 (x, z, t) = 0.1 sin(2t) + x 1 z 2 , which satisfies A1. According to the z in (79) and A3, we choose Vz = z 2 , c¯ = 1.2, γ (|x 1 |) = 1.25x 12 γ¯ (x 1 ) = 1.25x 14, d¯ = 1.68 α1 (z) = 0.5z 2 , α1 (z) = 1.5z 2 and ς˙ = −1.2ς + 1.25x 14 + 1.68

(80)

with ς (0) = 1. The control objective is to make the state x 1 (=y) follow the desired trajectory yr = sin(t). According to Section III, the design procedures are as follows. Step 1: The first surface error is S1 = x 1 − yr .

(81)

From (36), we have ϑ 2 S1 θˆ ψ1T (ξ1 )ψ1 (ξ1 ) x 2d = −k1 S1 − 1 2 with ξ1 = (x 1 , x 1 (t − ti ), yr , y˙r , ς, i = 1, . . . , 5). To obtain z 2 , let x 2d pass through a first-order filter ι2 z˙ 2 + z 2 = x 2d , z 2 (0) = x 2d (0)

Fig. 2.

Solid line: x1 . Dashed line: yr .

(82)

(83)

where ι2 is time constant. Step 2: The second surface error is S2 = x 2 − z 2 − c2 .

(84)

Since (79) is a second-order system, from (73), the control law is ϑ 2 S2 θˆ ψ2T ψ2 u = −k2 S2 − 2 2  D −sgn(S2 ) pˆ λ (t, r )|Frρ [u](t)|dr (85) 0

with ξ2 := (x 1 , x 1 (t − ti ), x 2 , x 2 (t − ti ), yr , z 2 , z˙ 2 , ς, i = 1, . . . , 5), and θˆ , pˆ λ (t, r ) are updated by 

  ϑ12 S12 ψ1T ψ1 + ϑ22 S22 ψ2T ψ2 ˙ − σθ θˆ θˆ = γθ 2 ⎧ ⎨γpr [|S2 ||Fr [u](t)| − σpr pˆ λ (t, r )] ∂ if 0 < pˆ λ (t, r ) ≤ 0.5 pˆ λ(t, r ) = (86) ⎩ ∂t −γpr σpr pˆ λ (t, r ), otherwise in this simulation, we select t1 = 0.5, t2 = 0.7, t3 = 0.9, t4 = 1, and t5 = 1.1; for NNs ψ(ξ1 ), we choose 41 nodes with the centers of the basis functions ζ , ... j j = 1, . . . , 41, being evenly spaced in [−4, +4] × { 7 } × [−4, +4] and the width η j = 1, j = 1, . . . , 41. For NNs ψ(ξ2 ), we choose 81 nodes with the centers of the basis functions ζ...j , j = 1, . . . , 81, being evenly spaced in [−4, +4] × {14} × [−4, +4] and the width η j = 1, j = 1, . . . , 81. The initial values of the updated parameters are chosen as θˆ (0) = 0, pˆ λ (0, r ) = 0. In addition, the design parameters are chosen as k1 = 55, k2 = 5, ϑ1 = 0.1, ϑ2 = 0.1, γθ = 0.5, and γpr = 3. The small gains are chosen as σθ = 0.5,

Fig. 3. Tracking errors with (solid line) and without (dashed line) considering hysteresis compensation.

σpr = 0.00001, and the time constant of the lowpass filter is selected as ι2 = 0.01. The initial values of the states are chosen as x 1 (0) = 0, x 2 (0) = 0.1. c2 in (84) is selected as c2 = 0.1. For all t < 0, x 1 (t) = 0 and x 2 (t) = 0.1. In addition, in this simulation, the function sgn(S2 ) in (85) is replaced by saturated function sat(S2 /0.01) to avoid the chattering phenomenon. To illustrate the effectiveness of the proposed control scheme, the simulations are performed under two different circumstances: 1) with (solid line) and 2) without (dotted line) considering hysteresis compensation. The analysis without consideration of the effects of hysteresis is implemented  D by settingρ the hysteresis compensation term −sgn(S2 ) 0 pˆ λ(t, r )|Fr [u](t)|dr as zero in the controller, which implies that the control compensation for hysteresis nonlinearity is ignored. The results are shown in Figs. 2–8

ZHANG et al.: ADAPTIVE NN DSC FOR A CLASS OF TIME-DELAY NONLINEAR SYSTEMS

Fig. 4. u with (solid line) and without (dashed line) considering hysteresis compensation.

Fig. 6.

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θˆ in (32).

Fig. 7. x2 with (solid line) and without (dashed line) considering hysteresis compensation.

Fig. 5. w with (solid line) and without (dashed line) considering hysteresis compensation.

for (79) to track the desired trajectory yr = sin(t). Fig. 2 shows that the output y = x 1 well tracks the desired trajectory when the hysteresis compensation is applied. Figs. 3 and 4 show the tracking errors for the desired trajectory and the control signal u(t) with and without considering the hysteresis compensation, solid line is the results with  D where the ρ −sgn(S2 ) 0 pˆ λ (t, r )|Fr [u](t)|dr = 0 and the dotted line D ρ is with −sgn(S2 ) 0 pˆ λ(t, r )|Fr [u](t)|dr = 0. Fig. 5 is the hysteresis output with and without considering the hysteresis compensation. As shown in Figs. 3–5, the existence of the hysteresis nonlinearity caused the oscillation of the control system when the hysteresis compensation is not considered (dotted lines) and it is obvious that the control input with the hysteresis compensation can effectively overcome the hysteresis effect when the hysteresis compensation is applied (solid lines). Fig. 6 shows the estimation of the norm of the RBFNN weight vector, and Fig. 7 is the state x 2 with and without considering the hysteresis compensation.

Fig. 8(a) and (b) shows the system output y = x 1 and its partially enlarged output between 0 and 3 s with and without considering the hysteresis compensation.

B. Practical Example: Tuning Metal Cutting System To illustrate the application of the proposed scheme, we consider the tuning metal cutting system introduced by [10] m x¨ + c x˙ + kx = F + ka u

(87)

where x represents the fluctuating part of the depth of cut, or so-called offset chip thickness; m, c, and k are the equivalent mass, damping coefficient, and spring stiffness for the metal cutting machine, respectively; ka is the equivalent spring for the piezoactuator; u is the piezoactuator output; F represents the cutting force variation of the machine tool, which has been proved to be a nonlinear function with respect to the chip thickness variation. Let F = h × (x − μx(t − τ )) + f (x − μx(t − τ )), where | f (·)| ≤ ς with ς, h, and μ being positive constants.

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Fig. 9. System output x1 (solid line) and desired trajectory yr (dashed line) of metal cutting system.

Fig. 8. (a) x1 with (solid line) and without (dashed line) considering hysteresis compensation. (b) x1 , which is being partially enlarged between 0 and 3 s from (a).

Fig. 10. Tracking errors of metal cutting system with and without considering hysteresis compensation.

Based on the above descriptions, let x = x 1 and x˙ = x 2 . Then, (87) can be expressed in strict-feedback form as x˙1 = x 2 x˙2 = bu − − y = x1

c k −h hμ x2 − x1 − x 1 (t − 2) m m m

1 f (x 1 − μx 1 (t − τ )) m (88)

where u is described by (7) with density function 2 p(r ) = 0.5e−0.00105(r−1) , λ = 8, and ρ(v) = 3 tanh(v). For the tuning metal cutting system, we select m = 25 kg, c = 1, k = 7000 N/m, h = 4500, μ = 1, and f (x 1 − μx 1 (t − 2)) = 0.2 sin(x 1 − μx 1 (t − 2)). The control objective is to make the state x 1 = y follow the desired trajectory yr = 0.02 sin(2.5t). In this simulation, the initial values are chosen as x 1 (0) = 0, x 2 (0) = 0.01; the design parameters and the small gains are chosen as k1 = 10, k2 = 0.2, ϑ1 = ϑ2 = 3, γθ = 2, γpr = 3, c2 = 0.01, σθ = 0.5, and σpr = 0.0001. The time

Fig. 11.

Control signal u for metal cutting system.

constant of the low-pass filter is selected as ς2 = 0.12. Other simulation parameters and the choice of RBFNNs are the same as the first one. The desired trajectory is selected as yr = 0.02 sin(2.5t).

ZHANG et al.: ADAPTIVE NN DSC FOR A CLASS OF TIME-DELAY NONLINEAR SYSTEMS

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| f (ξ1 ) − f (ξ2 )| < δ0 . Letting δ0 = ε and viewing (89), it follows that:

| f (ξ(t − t0 )) − f (ξ(t − T ))| < δ0 ∀t ≥ 0.

(90)

In this way, using the well-known finite covering lemma, we can extract a finite set of points t1 , . . . , tm ∈ [0, τ M ] with corresponding open neighborhoods O(t1 ), . . . , O(tm ), as follows. 1) [0, τ M ] ⊂ O(t1 ) ∪ . . . ∪ O(tm ). 2) There exists a point τ¯ ∈ {t1 , . . . , tm } satisfying (8). This completes the proof of Lemma 1.  A PPENDIX B P ROOF OF (29) Fig. 12.

θˆ , which is the estimation of θ ∗ of metal cutting system.

Since α1 is K ∞ -functions, α1−1 is an increasing function. Using (4) and (10), we have

Simulation results are shown in Figs. 9–12. Fig. 9 shows that the output y = x 1 well tracks the desired trajectory yr . Fig. 10 is the tracking errors with and without considering the hysteresis compensation. It illustrates the effectiveness of the proposed method. Fig. 11 shows the control signals u of metal cutting system. Fig. 12 is the estimation of the norm of RBFNN weight vector θ ∗ . We should mention that the transient performance of the system responses depends on the selections of the control parameters. It is known that there is no systematic approach to specify it for nonlinear systems with an adaptive control approach.

In this paper, an adaptive NN DSC scheme is proposed for a class of time-delay nonlinear systems with dynamic uncertainties and unknown hysteresis. The main advantages of the developed scheme are that, by combining the finite covering lemma with NNs, the time-delay terms are approximated, which leads to the abandonment of Krasovskii functions; by introducing an initializing technique, the L∞ performance of the tracking error is achieved; the developed adaptive laws only need two parameters being updated on-line, which reduces the computational burden compared with current DSC methods. Future works will focus on the more challenging cases when the upper bound of the time delay is unknown and only the output signals are available.

According to A1, φ12 (·) is a nonnegative increasing function, from (91), the following inequality holds:   φ12 (|z|) ≤ φ12 α1−1 (ς (t) + D(t0 , t)) . (92) Letting φ12 ◦α1−1 (ς (t)+D(t0 , t)) := φ12 [α1−1 (ς (t)+D(t0 , t))], then φ12 (|z|) ≤ φ12 ◦ α1−1 (ς (t) + D(t0 , t)).

(93)

For ∀t ≥ t0 , we have ≤ max{2ς (t), 2D(t0 , t)}.

The uniform continuity of ξ with respect to t implies that for any given δ0 > 0 and any point t0 ∈ [0, τ M ], there exists an open neighborhood O(t0 ) ⊂ R, which is independent of t, such that for any point T ∈ O(t0 ) (89)

On the other hand, the compactness of ξ together with the smoothness of f (·), implies that f (·) is uniformly continuous on ξ . Hence, for any given δ0 > 0, there exists an ε > 0, such that for any ξ1 , ξ2 ∈ ξ , as long as |ξ1 −ξ2 | < ε , we have

(94)

φ12 ◦ α1−1 (·)

Noting that is a nonnegative smooth function, the following inequality holds: φ12 ◦ α1−1 (ς (t) + D(t0 , t))

≤ φ12 ◦ α1−1 (2ς (t)) + φ12 ◦ α1−1 (2D(t0 , t)).

(95)

Therefore |S1 |φ12 (|z|) ≤ |S1 |φ12 ◦ α1−1 (2ς (t))

+ |S1 |φ12 ◦ α1−1 (2D(t0 , t)).

(96)

Young’s

inequality,

i.e.,

1 2 S + d1 (t0 , t) 4 1

(97)

From (96) and using the x y ≤ (1/4)x 2 + y 2 , we obtain

|S1 |φ12 (|z|) ≤ |S1 |φ12 ◦ α1−1 (2ς (t)) +

A PPENDIX A P ROOF OF L EMMA 1

|ξ(t − t0 ) − ξ(t − T )| < δ0 ∀t ≥ 0.

(91)

min{2ς (t), 2D(t0 , t)} ≤ ς (t) + D(t0 , t)

VI. C ONCLUSION



|z| ≤ α1−1 (ς (t) + D(t0 , t)).

where d1 (t0 , t) is defined as

2 d1 (t0 , t) = φ12 ◦ α1−1 (2D(t0 , t)) .

(98)

We emphasize that d1 (t0 , t) = 0, for all t ≥ t0 + T0 . In addition, from (11), due to boundedness of D(t0 , t) and the fact that φ12 is a smooth function, d1 (t0 , t) is bounded by d1 max . Then, from (97), the following inequality holds: 1 |S1 |φ12 (|z|) ≤ |S1 |φ12 ◦ α1−1 (2ς (t)) + S12 + d1 max . (99) 4

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Using Lemma 2, there exists a smooth function φ¯ 12 with φ¯ 12 (0) = 0, such that φ12 ◦ α1−1 (2ς (t))

≤ φ¯12 (ς ) + 1.

|S1 |φ12 ◦ α1−1 (2ς (t)) ≤ |S1 |φ¯ 12 (ς ) + |S1 | ≤ S1 φ¯ 12 (ς )φ¯ 13 (S1 , ς )

n

In addition, considering the updated law (75), if r ∈ r = {r : pˆ λ (t, r ) ≥ pλmax } ⊂ [0, D], we have p˜ λ (t, r ) = pˆ λ (t, r ) − pλmax ≥ 0 ∂ pˆ λ (t, r ) = −γpr σpr pˆ λ (t, r ). ∂t Then

 −β|Sn |

(102)

βmax + γpr

From (99) and (102), we have 1 2 S + d1 max + 212 . 4 1 This completes the proof of (29).

|S1 |φ12 (|z|) ≤ S1 φ¯ 12 (S1 , ς ) +

(103) 

1 2 yi+1 . 2

i=1

(104)

Then

 −β|Sn |

i=1

V˙ =

V˙i +

i=1

n−1 

yi+1 y˙i+1 .

+ (105)

From (38) and (76), we have (106)

Substituting (36), (54), and (106) in (35) and (53), we obtain 

c¯ V˙1 ≤ − k1 gm S12 + g1 S1 S2 + g1 S1 y2 − ς λ0 gm ϑ12 S1 θ˜ ψ1T ψ1 1 2 2 − g S + C1 − 2 2ϑ1 1 1   2 x 1 γ (|x 1 |2 ) S1 + 1 − 2 tanh2 γ λ0 

V˙i ≤ − ki gm Si2 + gi Si Si+1 + gi Si yi+1 −

βmax γpr



r∈ rc

r∈ rc

≤ −βmax σpr

i=1

x i = Si + yi + x id + ci , i = 2, . . . , n.

p˜ λ (t, r ) pˆ λ (t, r )dr.

˜ T ψi gm ϑi2 Si θψ 1 2 2 i − g S − gi−1 Si−1 Si + Ci 2 2ϑi i i i = 2, . . . , n − 1. (107)

Similarly, substituting (73) in (72), it follows that: gm ϑn2 Sn2 θ˜ ψnT ψn − gn−1 Sn−1 Sn V˙n ≤ −kn gm Sn2 − 2  D −β|Sn | p˜ λ (t, r )|Frρ [u](t)|dr 0  βmax D ∂ + p˜ λ (t, r ) pˆ λ (t, r )dr γpr 0 ∂t gm ˜ ˙ˆ (108) + θ θ + Cn . γθ

(111)

If r ∈ rc = {r : 0 < pˆ λ (t, r ) ≤ pλmax } ⊂ [0, D], we have

Then, the time derivative of V yields n 



∂ pˆ λ (t, r )dr ∂t

p˜ λ (t, r ) = pˆ λ (t, r ) − pλmax ≤ 0 ∂ pˆ λ (t, r ) = γpr [|Sn ||Fr [u](t)| − σpr pˆ λ(t, r )]. ∂t

n−1

Vi2 +

r∈ r

p˜ λ (t, r )

r∈ r

Let the Lyapunov function defined by n 

r∈ r

(110)

p˜ λ (t, r )|Frρ [u](t)|dr

≤ −βmax σpr

A PPENDIX C P ROOF OF T HEOREM 1

V =

(109)

i=1

(101)

where φ¯ 13 (S1 , ς ) and φ¯ 14 (S1 ) are two suitable smooth functions that satisfy φ¯13 (0, 0) = 0 and φ¯ 14 (0) = 0. Let

φ¯ 12 (S1 , ς ) = φ¯ 12 (ς )φ¯ 13 (S1 , ς ) + φ¯ 14 (S1 ).

gm ˜ ˙ˆ  gm ϑi2 Si2 θ˜ ψiT ψi − gm σθ θ˜ θˆ . θθ = γθ 2

(100)

Then, for any 12 > 0, by repeating use of Lemmas 2 and 3, we have

+S1 φ¯14 (S1 ) + 212

From the adaptation law (74), we have

(112)

p˜ λ (t, r )|Frρ [u](t)|dr p˜ λ (t, r )



r∈ rc

∂ pˆ λ (t, r )dr ∂t

p˜ λ (t, r ) pˆ λ (t, r )dr.

Using (110)–(113), we have  D −β|Sn | p˜ λ (t, r )|Frρ [u](t)|dr 0  βmax D ∂ + p˜ λ (t, r ) pˆ λ(t, r )dr γpr 0 ∂t  D ≤ −βmax σpr p˜ λ (t, r ) pˆ λ(t, r )dr.

(113)

(114)

0

In view of (78) and substituting (107), (108), and (114) in (105) yields n−1   1 2 2 V˙ ≤ gi Si −ki gm Si2 + gi Si yi+1 − 2ϑi i=1  D c¯ − ς − kn gm Sn2 − βmax σpr p˜ λ (t, r ) pˆ λ (t, r )dr λ0 0   n−1 2  yi+1 −gm σθ θ˜ θˆ + + |yi+1 Bi+1 | − ιi+1 i=1   2 n  x 1 γ (|x 1 |2 ) S1 + Ci + 1 − 2 tanh2 . (115) γ λ0 i=1

By Assumption A4, the set   1 := (yr , y˙r , y¨r ) : yr2 + y˙r2 + y¨r2 ≤ B1

(116)

ZHANG et al.: ADAPTIVE NN DSC FOR A CLASS OF TIME-DELAY NONLINEAR SYSTEMS

is compact in R3 for some B1 > 0. Moreover, the set ⎧ n ⎫   gm 2 βmax D 2 ⎪ ⎪ 2 ⎪ ⎪ ⎪ θ˜ + Si + p˜ λ (t, r )dr ⎪ ⎪ ⎪ ⎨ ⎬ γθ γp 0 i=1 (117) 2 := n−1  ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ + y ≤ 2 p i+1 ⎩ ⎭ i=1

is compact in R2n+1 . Note that 1 × 2 is also compact in R2n+4 . Therefore, |Bi+1 |, i = 1, . . . , n − 1, have maximum values, say, Mi+1 on 1 × 2 . Using 1 2 2 ϑi 2 gi Si yi+1 ≤ g S + yi+1 (118) 2ϑi i i 2 and taking (115) into consideration, we have  D n  V˙ ≤ − ki gm Si2 − βmax σpr p˜ λ (t, r ) pˆ λ (t, r )dr 0

i=1

c¯ ς − gm σθ θ˜ θˆ λ0   n−1 n 2   yi+1 ϑi 2 + y − + |yi+1 Bi+1 | + Ci 2 i+1 ιi+1 i=1 i=1   2 x 1 γ (|x 1 |2 ) 2 S1 + 1 − 2 tanh . (119) γ λ0 −

In addition, note that  D θ˜ θˆ = θ˜ 2 + θ˜ θ ∗ p˜ λ (t, r ) pˆ λ (t, r )dr 0  D  D = p˜ λ2 (t, r )dr + p˜ λ (t, r ) pλmax (r )dr. 0

Let 1

i=1

(126) Replacing (123)–(126) in (122), it follows that:   2 x 1 γ (|x 1 |2 ) S1 V˙ ≤ −κ V + C + 1 − 2 tanh2 . (127) γ λ0 Then, from (127), it can be observed that the first term is negative definite and the second term is positive constant. However, the last term (1 − 2 tanh2 (S1 /γ ))(x 12 γ (|x 1 |2 )/λ0 ) that depends on S1 is uncertain. Therefore, two cases should be considered in the process of analysis of stability. Case 1: S1 ∈ S1 = {S1 |S1 | < 0.8814γ }, where γ is a positive constant. From (20), we have x 1 = S1 + yr .

(120)

0

From (119)–(121), we obtain n 

c¯ gm σθ ˜ 2 ki gm Si2 − ς − V˙ ≤ − θ λ0 2 i=1  βmax σpr D 2 − p˜ λ (t, r )dr 2 0   n−1 2  yi+1 ϑi 2 y + − + |yi+1 Bi+1 | 2 i+1 ιi+1 i=1  n  gm σθ ∗2 βmax σpr D 2 θ + + Ci + pλmax (r )dr 2 2 0 i=1   2 x 1 γ (|x 1 |2 ) S1 + 1 − 2 tanh2 . (122) γ λ0 Notice that for any positive number μ, we have 2 M2 yi+1 i+1

2μ

+

μ , 2

2 Mi+1

, i = 1, . . . , n − 1 (124) ιi+1 2μ   κ = min 2gm ki , c, ¯ γθ σθ , γpr σpr , ϑi , i = 1, . . . , n (125)  D n  gm σθ ∗2 βmax σpr C := θ + Ci + pλ2max (r )dr. 2 2 0 ≥

(128)

Since S1 is bounded and yr belongs to a known compact, x 1 is bounded. Combining with γ (·) being a nonnegative smooth function, (1 − 2 tanh2 (S1 /γ ))(x 12 γ (|x 1 |2 )/λ0 ) is bounded. Let C¯ denotes its boundedness. From (127), we have

Hence, the following equalities hold: gm σθ 2 2 (θ˜ − θ ∗ ) −gm σθ θ˜ θˆ ≤ − 2  D p˜ λ (t, r ) pˆ λ (t, r )dr −βmax σpr 0   D D βmax σpr 2 2 p˜ λ (t, r )dr − pλmax (r )dr . ≤− 2 0 0 (121)

|yi+1 Bi+1 | ≤

2857

i = 1, . . . , n − 1. (123)

¯ V˙ ≤ −κ V + C + C.

(129)

¯ p κ > (C + C)/

(130)

Letting

then V˙ ≤ 0 on V = p. That is, V ≤ p is an invariant set, i.e., if V (0) ≤ p, then V (t) ≤ p, for all t ≥ 0. Solving (129), we obtain ¯ ¯ ! (C + C) (C + C) 0 ≤ V (t) ≤ + V (0) − (131) e−κt κ κ ∀t ≥ 0, which implies that lim V (t) =

t →∞

¯ (C + C) . κ

(132)

Case 2: S1 ∈ / S1 . When S1 ∈ / S1 , the inequality (1 − 2 tanh2 (S1 /γ )) ≤ 0 holds with γ > 0 [46]. Then, from (127), we have V˙ ≤ −κ V + C.

(133)

κ > C/ p

(134)

Letting

then V˙ ≤ 0 on V = p. That is, V ≤ p is an invariant set, i.e., if V (0) ≤ p, then V (t) ≤ p, for all t ≥ 0. Solving (133), we obtain ! C C −κt 0 ≤ V (t) ≤ + V (0) − e (135) κ κ

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∀t ≥ 0, which implies that

From (4) and (5), it is reasonable to set

C . (136) κ Thus, based on the above two cases, all signals of the closedloop system i.e., Si , θˆ , pˆ λ (t, r ), and yi+1 , are uniformly bounded. Moreover, from (126), by appropriately choosing the design parameters γθ , γpr , ki , kn , ιi+1 , i = 1, . . . , n − 1, we can make κ larger, which together with (136) implies that the tracking error can be made an arbitrary small. Finally, to obtain the L∞ performance of the tracking error, we set the initial conditions of the parameter estimators in (74) and (75) to zero

d¯ ≥ d¯ max{γpr σpr , γθ σθ }

lim V (t) =

t →∞

θˆ (0) = 0 pˆ λ (0) = 0

where

yr (0) = x 1 (0) ci = x i (0), i = 2, . . . , n.

(138)

Now, taking (137) and (138) into consideration, from (20), (36)–(38), (54)–(56), and (76), it can be shown in a step-by-step fashion that x i+1d (0) = 0 ⇒ z i+1 (0) = 0, i = 1, . . . , n − 1

d¯ ≥ ς (0).

gm ∗2 βmax θ + 2γθ 2γpr

 0

D

pλ2max (r )dr +

ς (0) . λ0

(139)

From (125) and (126), the following inequalities hold: n γθ σθ gm ∗2 C i=1 Ci = + θ κ κ κ 2γθ  γpr σpr βmax D 2 + pλmax (r )dr κ 2γpr 0 n  gm ∗2 βmax D 2 i=1 Ci ≥ + θ + pλmax (r )dr κ 2γθ 2γpr 0 (140) n C ς (0) C i V (0) − ≤ (141) − i=1 . κ λ0 κ Noting that C1 =

2 gmax 1 ∗2 d¯ δ + + + 0.278511 + d1 max + 212 1 2 λ0 2ϑ12

d¯ ς (0) − < 0. λ0 κλ0

(145)

Substituting (145) in (142) yields C