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Adaptive Control for Nonlinear Pure-Feedback Systems With High-Order Sliding Mode Observer Jing Na, Xuemei Ren, and Dongdong Zheng

Abstract— Most of the available control schemes for pure-feedback systems are derived based on the backstepping technique. On the contrary, this paper presents a novel adaptive control design for nonlinear pure-feedback systems without using backstepping. By introducing a set of alternative state variables and the corresponding transform, state-feedback control of the pure-feedback system can be viewed as output-feedback control of a canonical system. Consequently, backstepping is not necessary and the previously encountered explosion of complexity and circular issue are also circumvented. To estimate unknown states of the newly derived canonical system, a high-order sliding mode observer is adopted, for which finite-time observer error convergence is guaranteed. Two adaptive neural controllers are then proposed to achieve tracking control. In the first scheme, a robust term is introduced to account for the neural approximation error. In the second scheme, a novel neural network with only a scalar weight updated online is constructed to further reduce the computational costs. The closed-loop stability and the convergence of the tracking error to a small compact set around zero are all proved. Comparative simulation and practical experiments on a servo motor system are included to verify the reliability and effectiveness. Index Terms— Adaptive control, high-order sliding mode (HOSM) observer, neural networks, pure-feedback systems.

I. I NTRODUCTION

I

N recent decades, adaptive control design has been significantly advanced for nonlinear systems, e.g., Brunovsky systems [1], feedback-linearized systems [2], and strict-feedback systems [3], and various controllers have been proposed using different techniques (e.g., feedback linearization [2], inversion control [4], backstepping [3], [5], and sliding mode control [6], [7]). In this field, adaptive backstepping control [5] has been recognized as a powerful methodology. The overparameterization problem was also overcome by introducing a tuning function [8]. However, the studied systems are assumed to be in the linear parametric forms or with some matching conditions. This condition may be stringent in some industrial applications. To account for nonlinearities and unknown Manuscript received December 1, 2011; revised October 7, 2012; accepted October 10, 2012. Date of publication January 3, 2013; date of current version January 30, 2013. This work was supported by the National Natural Science Foundation of China under Grant 61203066, Grant 61273150, and Grant 60974046. J. Na is with the Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming 650093, China (e-mail: [email protected]). X. Ren and D. Zheng are with the School of Automation, Beijing Institute of Technology, Beijing 100081, China (e-mail: [email protected]; [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.2012.2225845

dynamics, several function approximators, such as neural networks (NNs) and fuzzy logic systems (FLSs) have been used [9]–[13]. In particular, the FLS has been incorporated into the sliding mode control design [6], [7] to achieve fast transient performance. The advantages of those approximation-based controls are that the assumption of linear-in-parameters (LIP) or matching conditions can be removed, and the adaptive laws for NN or FLS weights are derived based on Lyapunov theorem to prove the closed-loop stability. However, in the aforementioned results, the number of adaptive parameters to be online updated, e.g., the NN weights related to the number of neurons, will increase with the dimension of functions to be approximated [14]. In this case, the computational costs in the control implementation are not trivial. To address this issue, several novel NN controllers with less adaptive parameters [14]–[16] have been exploited. On the other hand, in the backstepping design, e.g., [11], [12], [17]–[21], the repeated differentiation of virtual control functions and/or regression matrices leads to the so-called explosion of complexity issue, which renders its derivation complicated as the system order increases [22]. To remedy this issue, dynamic surface control (DSC) [22], [23] has been investigated by introducing a first-order filter in each recursive design step, such that the differentiation operation can be avoided. This concept was then extended to adaptive neural control of nonparametric nonlinear systems [24]. The recent work [15], [25], and [26] further studied the adaptive NN-based DSC design for nonlinear systems with time-delay or dead-zone. In [27], Farrell et al. advocated a command filtered backstepping (CFB) for strict-feedback systems. It is noted that the control design and synthesis of DSC and CFB are still lengthy for high order systems since they all follow a similar recursive design procedure to backstepping. Nevertheless, the adaptive laws involved in DSC and CFB also impose that a large number of parameters need to be updated online. Among different nonlinear systems, pure-feedback system can represent more practical processes, such as mechanical systems [28], aircraft systems [4], and biochemical processes [3]. However, as pointed out in [20], the cascade and nonaffine properties make it difficult to find explicit virtual and/or actual controllers for pure-feedback systems, and thus only several adaptive controllers have been investigated for some specific pure-feedback systems. Kanellakopoulos et al. [5] advanced an effort to extend the adaptive backstepping control of parametric pure-feedback systems and obtained regionally stable results. In [11] and [19], a special kind of

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NA et al.: ADAPTIVE CONTROL FOR NONLINEAR PURE-FEEDBACK SYSTEMS WITH HOSM OBSERVER

affine-in-control pure-feedback systems was studied in terms of NN approximation, where the implicit function theorem is employed to assert the existence of the desired feedback controllers. Following this idea, more general nonaffine pure-feedback systems with less restrictive assumptions were further studied [18]. Here, the employment of ISS-modular approach and the small gain theorem avoids the construction of an overall Lyapunov function for the entire system. Liu et al. [12] proposed an adaptive neural control for MIMO pure-feedback systems, and Ren et al. [20] extended adaptive NN backstepping control to systems with hysteresis dynamics. The pure-feedback system with unknown time-delays was studied in [17]. Recently, the concept of DSC has been applied for pure-feedback systems [26], [29]. The basic idea of these results is to apply the mean value theorem on the nonaffine functions such that a pure-feedback system is represented as a strict-feedback form. However, there may be a “circular issue” [21] when NNs are used to approximate the virtual and desired controllers in the recursive design. This is due to the fact that the control variable may be adopted as a part of the NN approximation and used in the control itself. This algebraic loop problem has been preliminarily studied in the latest work [21] by introducing low-pass filters. To the best of our knowledge, most of the available control designs [11], [17]–[19], [21], [26], [29] for pure-feedback systems are all derived based on the backstepping or DSC technique, which leads to complicated design and synthesis procedures for high order systems. Regarding the control implementation, there are also a large number of parameters (i.e., NN weights as vectors or matrices) to be updated online. These two facts motivate our current work to propose an alternative control framework that avoids backstepping design and has less adaptive parameters. This paper concerns adaptive control design for nonlinear pure-feedback systems without using backstepping. A set of newly defined state variables are employed such that the original pure-feedback system can be represented in a Brunovsky form while the tracking objective is retained by controlling the transformed canonical system. This allows a simpler control design because the backstepping design is not necessary. This idea is partially inspired by [30] for strictfeedback systems and our preliminary work [31] for timedelay systems. Since the states of the transformed system are not all available, the state feedback control of a purefeedback system can be viewed as the output-feedback control of a canonical system. In this case, a high-order sliding mode (HOSM) observer [32], [33] is adopted. The salient feature of the HOSM observer lies in that it can achieve finitetime (FT) observer error convergence and thus can be used in almost any feedback with separation principle [34] being trivially fulfilled. Two adaptive neural controllers are proposed in this paper. In the first scheme, a robust term, depending on the information on the control gain bounds, is introduced to account for the NN approximation error. In the absence of such information, a novel NN structure with only a scalar weight to be updated is constructed in the second control to handle the unknown nonlinearities such that the computational costs can be significantly reduced. The closed-loop system stability

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and the convergence of tracking error to a small compact set around zero are all proved. Theoretical results are validated by comparative simulations and by practical experiments on a turntable servo motor system. We briefly summarize the contribution of this paper as follows. 1) An alternative adaptive control framework without backstepping is proposed for nonlinear pure-feedback systems. This is different to available backstepping-based controllers, and thus the previously encountered explosion of complexity and circular issue are all circumvented. 2) A HOSM observer is incorporated into the adaptive NN control to estimate unknown states of the transformed system, and the subsequent closed-loop stability is proved. 3) A novel NN structure with a scalar weight (independent of the number of neurons) is constructed, and thus only two parameters need to be updated online in the control. The rest of this paper is organized as follows. Problem statement and the coordinate transform are provided in Section II. Section III introduces the HOSM observer design. Section IV proposes two adaptive neural control designs and the stability analysis. Section V is devoted to validate the proposed schemes via simulations and experiments. Some conclusions are given in Section VI. II. P ROBLEM F ORMULATION AND S TATE T RANSFORM Consider a class of nonlinear nonaffine systems in the purefeedback form as ⎧ x˙1 = f 1 (x 1 , x 2 ) ⎪ ⎪ ⎨ i = 2, . . . , n − 1 x˙i = f i (x 1 , x 2 , . . . x i+1 ), (1) = f (x , x , . . . x , u) x ˙ ⎪ n n 1 2 n ⎪ ⎩ y = x1 where x¯i = [x 1 , x 2 , . . . , x i ]T ∈ Ri , i = 1, . . . , n, y(t) ∈ R, and u(t) ∈ R are the system states, the output, and input, respectively, f i (·), i = 1, . . . , n are unknown nonlinear functions. The objective of this paper is to design a control u(t) such that the output y(t) tracks a given reference yd (t) and all signals in the closed-loop are bounded. Assumption 2.1: The state variables x¯i of (1) are measurable, and the nonlinear functions fi (·), i = 1, . . . , n are continuously differentiable to n-order with respect to the state variables x¯i and the input u. Remark 2.1: Many industrial processes can be modeled as pure-feedback system (1), such as mechanical systems [28], aircraft systems [4], and biochemical processes [3], which can cover most of nonlinear systems, e.g., Brunovsky systems [35], strict-feedback systems [5]. For system (1), backstepping control has been widely studied [11], [17]–[19], [21], [26], [29]. However, the cascade and nonaffine properties of x 1 , x n make it difficult to find explicit virtual controls and the actual control [20]. Specifically, compared to [11] and [19], the function f n (x 1 , x 2 , . . . , x n , u) in (1) has a nonlinear nonaffine appearance of the control variable u so that appropriate transforms will be proposed in the control design.

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Unlike available results, the controllers to be studied in this paper are derived without using backstepping. To facilitate the control design, a set of new state variables will be defined and a coordinate transform will be proposed to transform pure-feedback system (1) into a canonical system. Define alternative state variables as ⎧ ⎨ z1 = x1 i = 2, . . . , n z i = z˙ i−1 , (2) ⎩ y = z1 = x1 . Then, according to the fact that z 2 = z˙ 1 = f 1 (x 1 , x 2 ), it follows that: ∂ f 1 (x 1 , x 2 ) ∂ f 1 (x 1 , x 2 ) x˙1 + x˙2 z˙ 2 = z¨1 = ∂ x1 ∂ x2 ∂ f1 (x 1 , x 2 ) ∂ f1 (x 1 , x 2 ) = f 1 (x 1 , x 2 ) + f 2 (x 1 , x 2 , x 3 ). (3) ∂ x1 ∂ x2 Since the unknown function f 2 (x 1 , x 2 , x 3 ) is continuously differentiable with respect to x 3 , we apply the mean-value theorem [36] for f2 (x 1 , x 2 , x 3 ) as f 2 (x 1 , x 2 , x 3 ) = f2 (x 1 , x 2 , 0)+

∂ f 2 (x 1 , x 2 , x 3 ) |x3 =x θ x 3 3 ∂ x3

where α3 (x¯3 ) =

z˙ 3 = z¨ 2 = =

2  ∂α2 (x¯2 ) j =1

+

∂x j

f j (x 1 , . . . , x j +1 )

2  ∂β2 (x¯3 ) j =1



∂x j

f j (x 1 , . . . , x j +1 )x 3

 ∂β2 (x¯3 ) x 3 + β2 (x¯3 ) x˙3 ∂ x3 2 2   ∂α2 (x¯2 ) ∂β2 (x¯3 ) = f j (x¯ j +1 )+ f j (x¯ j +1)x 3 ∂x j ∂x j j =1 j =1   ∂β2 (x¯3 ) + x 3 + β2 (x¯3 ) f 3 (x 1 , . . . , x 4 ). (6) ∂ x3

(∂β2 (x¯3 )/∂ x j ) f j (x¯ j +1)x 3

β3 (x¯3 , x 4θ ) = ((∂β2 (x¯3 )/∂ x 3 )x 3 +β2 (x¯3 )) (∂ f3 (x¯4 )/∂ x 4 )|x4 =x θ 4

are smooth functions of x¯3 ∈ R3 and [x¯3 , x 4θ ]T ∈ R4 , respectively. Similar to above analysis, for i = 4, . . . , n − 1, we can derive z˙ i = z¨ i−1 = =

∂αi−1 (x¯i−1 ) ˙ ∂βi−1 (x¯i ) ˙ x¯i−1 + x¯i x i + βi−1 (x¯i )x˙i ∂ x¯ i−1 ∂ x¯ i

i−1  ∂αi−1 (x¯i−1 )

+

i−1  ∂βi−1 (x¯i ) j =1

 +

f j (x¯ j +1 )

∂x j

j =1

∂x j

f j (x¯ j +1)x i

∂βi−1 (x¯i ) x i + βi−1 (x¯i ) ∂ xi

 f i (x 1 , . . . , x i+1 ).

(9)

Applying the mean-value theorem for the smooth function f i (x 1 , . . . , x i+1 ) with respect to x i+1 , it follows: f i (x 1 , . . . , x i+1 ) = f i (x 1 , . . . , x i , 0) ∂ f i (x 1 , . . . , x i+1 ) + x i+1 θ x i+1 =x i+1 ∂ x i+1

(10)

θ where x i+1 = θ x i+1 with 0 < θ < 1 being a constant. Then substituting (10) into (9) yields

θ x i+1 z˙ i = αi (x¯i ) + βi x¯ i , x i+1

+

Since f 3 (x 1 , . . . , x 4 ) is continuously differentiable with respect to x 4 , then based on the mean-value theorem, it can be represented as ∂ f 3 (x 1 , . . . , x 4 ) x4 f 3 (x 1 , . . . , x 4 ) = f3 (x 1 , . . . , x 3 , 0)+ x 4 =x 4θ ∂ x4 (7) where x 4θ = θ x 4 with 0 < θ < 1 being a constant [17], [20]. We then rewrite (6) as

(8) z˙ 3 = α3 (x¯3 ) + β3 x¯3 , x 4θ x 4

j =1

and

(5)

∂α2 (x¯2 ) ˙ ∂β2 (x¯3 ) ˙ x¯2 + x¯3 x 3 + β2 (x¯3 )x˙3 ∂ x¯ 2 ∂ x¯3

2

+((∂β2 (x¯3 )/∂ x 3 )x 3 + β2 (x¯3 )) f 3 (x¯3 , 0)

(4)

where α2 (x 1 , x 2 ) = (∂ f 1 (x 1 , x 2 )/∂ x 1 ) f 1 (x 1 , x 2 )+(∂ f 1 (x 1 , x 2 )/ ∂ x 2 ) f 2 (x 1 , x 2 , 0) and β2 (x 1 , x 2 , x 3θ ) = (∂ f 1 (x 1 , x 2 )/∂ x 2 ) (∂ f 2 (x 1 , x 2 , x 3 )/∂ x 3 )|x3 =x θ are unknown but smooth functions 3 of x¯2 = [x 1, x 2 ]T ∈ R2 and [x¯2 , x 3θ ]T ∈ R3 . For i = 3, one can obtain

(∂α2 (x¯2 )/∂ x j ) f j (x¯ j +1)

j =1

+

where x 3θ = θ x 3 with 0 < θ < 1 being a constant (not necessarily known) [17], [20]. Then (2) can be rewritten as z˙ 2 = α2 (x 1 , x 2 ) + β2 (x 1 , x 2 , x 3θ )x 3

2

(11)

where αi (x¯i ) =

i−1

(∂αi−1 (x¯i−1 )/∂ x j ) f j (x¯ j +1) j =1 i−1 + (∂βi−1 (x¯i )/∂ x j ) f j (x¯ j +1 )x i j =1

+ ((∂βi−1 (x¯i )/∂ x i )x i + βi−1 (x¯i )) f i (x¯i , 0) and θ βi (x¯i+1 , x i+1 ) = ((∂βi−1 (x¯i )/∂ x i )x i

+βi−1 (x¯i ))(∂ f i (x¯i+1 )/∂ x i+1 )|xi+1 =x θ

i+1

θ ]T ∈ Ri+1 . are unknown functions of x¯i ∈ Ri and [x¯i , x i+1

NA et al.: ADAPTIVE CONTROL FOR NONLINEAR PURE-FEEDBACK SYSTEMS WITH HOSM OBSERVER

Finally, for i = n, one can obtain

III. HOSM O BSERVER D ESIGN

∂αn−1 (x¯n−1 ) ˙ x¯ n−1 ∂ x¯ n−1 ∂βn−1 (x¯n ) ˙ + x¯n x n + βn−1 (x¯n )x˙n ∂ x¯n n−1  ∂αn−1 (x¯n−1 ) = f j (x¯ j +1) ∂x j

z˙ n = z¨ n−1 =

j =1

+

n−1  ∂βn−1 (x¯n ) j =1

 +

∂x j

f j (x¯ j +1)x n

∂βn−1 (x¯n ) x n + βn−1 (x¯n ) ∂ xn

 f n (x¯n , u). (12)

By using the mean-value theorem for f n (x¯n , u), it follows: ∂ f n (x¯n , u) (13) f n (x¯n , u) = f n (x¯n , 0) + θu ∂u u=u where u θ = θ u with 0 < θ < 1 being a constant. Then system (12) is rewritten as z˙ n = αn (x¯n ) + βn (x¯n , u θ )u

(14)

where αn (x¯n ) =

373

n−1

(∂αn−1 (x¯n−1 )/∂ x j ) f j (x¯ j +1) j =1 n−1 + (∂βn−1 (x¯n )/∂ x j ) f j (x¯ j +1 )x n j =1

+((∂βn−1 (x¯n )/∂ x n )x n + βn−1 (x¯n )) f n (x¯n , 0) and βn (x¯n , u θ ) = ((∂βn−1 (x¯n )/∂ x¯ n )x n +βn−1 (x¯n ))(∂ f n (x¯n , u)/∂u)|u=u θ are unknown functions of x¯n ∈ Rn and [x¯n , u θ ]T ∈ Rn+1 , respectively. From (2)–(14), system (1) is represented as ⎧ z˙ 1 = z 2 ⎪ ⎪ ⎨ z˙ i = z i+1 , i = 1, . . . , n − 1 (15) θ = α ( x ¯ ) + β ( x ¯ z ˙ ⎪ n n n n n , u )u ⎪ ⎩ y = z1. It is shown that with newly defined states (2) and the coordinate transform (3)–(14), pure-feedback system (1) is now reformulated as a Brunovsky system (15) with output z 1 . Since the fact y = z 1 = x 1 holds, the control objective of (1) (i.e., y tracks a given reference yd ) can be retained by controlling system (15). However, in the newly derived system (15), the states z i , i = 2, . . . , n are not directly measureable, and the functions αn (x¯n ) and βn (x¯n , u θ ) are unknown. In this sense, state-feedback control of pure-feedback system (1) can be viewed as output-feedback control of canonical system (15). In the following, we will consider the output-feedback control design for (15) to achieve tracking control of (1). A HOSM observer will be first introduced and then two adaptive feedback controllers will be proposed.

It is shown that (15) is in a canonical form, then the states z i , i = 2, . . . , n can be viewed as the differentials of the output y = z 1 = x 1 . To estimate z i , i = 2, . . . , n, a HOSM observer (or Levant’s differentiator) [32], [33] that was derived based on the modified supertwisting algorithm is utilized. The structure of the HOSM observer is given as ⎧ ˙ ⎪ ⎪ zˆ 1 = v 1 ⎪ n ⎪ ⎪ ⎪ v 1 = −λ1 zˆ 1 − y (n+1) sign(ˆz 1 − y) + zˆ 2 ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ˙ ⎪ zˆ i = v i ⎪ ⎪ ⎪ (n−i+1) ⎨ v i = −λi zˆ i −v i−1 (n−i+2) sign(ˆz i −v i−1 ) + zˆ i+1 , (16) ⎪ i = 2, . . . , n − 1 ⎪ ⎪ ⎪ .. ⎪ ⎪ ⎪ . ⎪ ⎪ ⎪ ⎪ ˙ ⎪ zˆ n = v n ⎪ ⎪ 1 ⎪ ⎪ ⎪ v n = −λn zˆ n − v n−1 2 sign(ˆz n − v n−1 ) + zˆ n+1 ⎪ ⎪ ⎩˙ zˆ n+1 = −λn+1 sign(ˆz n+1 − v n ) where λi , i = 1, . . . , n + 1 are all positive constants selected by the designers. According to [32] and [33], the following lemmas are given. Lemma 3.1: With properly chosen parameters λi , i = 1, . . . , n + 1 of differentiator (16), the following equalities are true in the absence of input noise after a FT transient process: zˆ i = z i , i = 1, . . . , n.

(17)

Moreover, the corresponding solutions of the dynamic system are Lyapunov stable, i.e., FT stable. Lemma 3.2: If a bounded noise is included in differentiator (16), e.g., |z 1 − y| ≤ κ, then for some positive constants i and λ¯i depending exclusively on the parameters of differentiator (16), the following inequalities hold in FT: zˆ i − z i ≤ i κ (n−i+2)/(n+1) , i = 1, . . . n (n−i+1)/(n+1) ¯ |v i − z i+1 | ≤ λi κ , i = 1, . . . , n − 1. (18) The detailed proofs of Lemmas 3.1 and 3.2 are given in [32] and [33]. Remark 3.1: Lemma 3.1 means that the equalities are kept in two-sliding mode [32], and thus zˆ = [ˆz 1 , zˆ 2 , . . . , zˆ n ]T can precisely estimate unknown states z = [z 1 , z 2 , . . . , z n ]T in FT. In the presence of a bounded noise, Lemma 3.2 shows that the observer error FT converges to a small bounded set depending on the magnitude of noise, i.e., there exist positive constants h¯ and t such that for t > t , we have ˆz − z ≤ h¯ . It should be addressed here that HOSM observer (16) can achieve FT error convergence. This property makes it attractive in the control design and synthesis. As pointed out in [32] and [33], FT convergence allows the separation principle [34] to be trivially fulfilled for observer (16). This means that a controller and observer (16) can be designed separately, so that the combined observer–controller output feedback preserves the main features of the controller with the full state available. The only requirement for the observer implementation [32], [33] is the boundedness of some higher order derivatives of its input and the impossibility

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of the FT escape during the differentiator transient that may be made arbitrarily short. As argued by Levant [32], [33], any real system operates only in some bounded operation region, so that this condition can be practically fulfilled by assuming sufficiently large bounds. Hence, the differentiator can be used in almost any feedback control. In the following control designs, without loss of generality, we assume there is a noise in the system output y (due to the measurement or sampling disturbance) and thus the observer error ˆz − z ≤ h¯ will be considered in the closed-loop stability analysis. IV. A DAPTIVE C ONTROL D ESIGN In this section, two adaptive neural controllers will be constructed based on the states zˆ = [ˆz 1 , zˆ 2 , . . . , zˆ n ]T of observer (16). In the first scheme, a robust term is developed based on appropriate information on the bounds of the control function, while in the second scheme, an alternative adaptive NN control is investigated, for which only a scalar independent of the number of NN nodes is employed as the NN weight. To account for unknown functions αn (x¯n ) and βn (x¯n , u θ ) in (15), a linear parameterized NN (LPNN) approximation [9] is employed over a compact set as Q(Z ) = W ∗T (Z ) + ε

∀Z ∈ ⊂ Rn

(19)

where Q(Z ) is the unknown function to be approximated, W ∗ = [w1∗ , w2∗ , . . . , w∗L ]T ∈ R L are the bounded NN weights, ε ∈ R is a bounded approximation error, i.e., ||W ∗ || ≤ W N , |ε| ≤ ε N with W N and ε N being positive constants, and (Z ) = [ 1 (Z ), . . . , L (Z )]T ∈ R L is the NN basis vector. Remark 4.1: Here, we should point out a well-known fact that the NN approximation (19) is only valid in a compact set [9] so that only semi-global stability can be proved. It is also known that a high-order NN (HONN) [37] can guarantee the approximation capacity with less neurons and computational costs. Thus in the subsequent developments, the HONN will be utilized, where high-order basis function connections are adopted. However, the proposed results are also valid for other function approximators, e.g., radial basis function networks [18], [38], and [39] or fuzzy systems [6], [7], and [21]. In the HONN formulation [37], high order functions ik (Z ) = j ∈Jk [σ (Z j )]dk ( j ), k = 1, . . . , L are used, where Jk are collections of L-not ordered subsets of {0, 1, . . . , n}, dk ( j ) are nonnegative integers, and σ (·) is a sigmoid function σ (x) = a/(1 + e−bx ) + c, ∀a, b ∈ R+ , c ∈ R, where the positive parameters a, b, and the real number c are the bound, the slope and the bias of sigmoidal curvature, respectively. For the control design, we define the reference vector as y¯d = [yd , y˙d , . . . , ydn−1 ]T and the tracking errors as e = z − y¯d

s = [T 1]e

(20)

where  = [1 , 2 , . . . , n−1 ]T is a vector such that the polynomial s n−1 + n−1 s n−2 + · · · 1 is Hurwitz. Then based on [34], the tracking error e is bounded as long as s is bounded. Since only the observed states zˆ are available, we also define the errors eˆ and sˆ between the states of observer (16) and the

references as eˆ = zˆ − y¯d

sˆ = [T 1]eˆ

(21)

where the reference is bounded, i.e.,  y¯d  ≤ cd . Then the observer errors z˜ and s˜ can be denoted as z˜ = e − eˆ = z − zˆ

s˜ = s − sˆ = [T 1]˜z .

(22)

Substituting (15) into (20), we obtain the error equation as s˙ = [T 1]e˙ = [0 T ]e + αn (x¯n ) + βn (x¯n , u θ )u − ydn .

(23)

Without loss of generality, the control function βn (x¯n , u θ ) in (15) is assumed to be positive and bounded satisfying 0 < g0 ≤ βn (x¯n , u θ ) ≤ g1 where g0 and g1 are positive constants. This condition has been widely used in the literature [11], [18], [19], [21] as a necessary condition for the controllability of (15). Specifically, in the second control scheme introduced in Section IV-B, these bounded parameters are not necessarily known. A. Adaptive NN Control To account for NN approximation error, a robust term will be introduced based on some mild information on the √ bounds of g0 and g1 . Let gm = g0 g1 , being the nominal θ value of βn (x¯n , u ), then we represent it as βn (x¯n , u θ ) = gm g(x¯n , u θ ), where g(x¯n , u θ ) is the multiplicative uncertainty satisfying 0 < b0 ≡ g0 /gm ≤ g(x¯n , u θ ) ≤ g1 /gm ≡ b1 . In general systems, the parameters gm , b0 , and b1 can be roughly estimated. For (15) with observer (16), the following control is given:  1 u= −k1 sˆ − Wˆ T (Z ) + ydn − u r (24) gm where k1 > 0 is a positive constant, sˆ is the error defined in (21), Wˆ is the estimated value of the idea HONN weights vector W ∗ , (Z ) is the basis vector of HONN with the input Z = [x¯n , e] ˆ ∈ R2n , and u r is a robust term designed as 1 + b1 (25) ur = k1 sˆ + Wˆ T (Z ) − ydn sgn(ˆs ). b0 The HONN weights Wˆ are updated as

 ˙ Wˆ =  sˆ T (Z ) − σ1 sˆ Wˆ

(26)

where  > 0 and σ1 > 0 are design parameters. Substituting (24) into (23), we have s˙ = [0 T ](eˆ + z˜ ) + αn (x¯n ) + βn (x¯n , u θ )u − ydn = F1 (x¯n , e) ˆ − ydn + βn (x¯n , u θ )u + [0 T ]˜z

(27)

ˆ = [0 T ]eˆ + αn (x¯n ) is an unknown function where F1 (x¯n , e) approximated by HONN (19) as F1 (x¯n , e) ˆ = W ∗T (Z ) + ε. The following lemma is true. Lemma 4.1: The HONN weights Wˆ in (26) are bounded by Wˆ  ≤ c /σ1 , where c is the bound of the HONN basis function vector, i.e.,   ≤ c . Proof: In the HONN approximation, the sigmoidal activation functions are utilized so that the basis vector is bounded, i.e.,   ≤ c . Consider the Lyapunov function as

NA et al.: ADAPTIVE CONTROL FOR NONLINEAR PURE-FEEDBACK SYSTEMS WITH HOSM OBSERVER

L = (1/2)Wˆ T Wˆ , then the time derivative of L along (26) is derived as

 1 L˙ = Wˆ T W˙ˆ = Wˆ T sˆ T − σ1 sˆ Wˆ          ≤ − Wˆ  sˆ σ1 Wˆ  − c . (28) It follows that L˙ ≤ 0 if Wˆ  > c /σ1 . According to Lyapunov Theorem [34], Wˆ is bounded by Wˆ  ≤ c /σ1 , and thus the weights estimation error W˜ = W ∗ − Wˆ is also bounded as W˜  ≤ cW , cW = W N + c /σ1 . We have the following results. Theorem 4.1: Consider (1) under Assumption 1, the transformed system (15), HOSM observer (16), and control (24) with adaptive law (26), then: 1) all signals in the closed-loop are semi-globally uniformly ultimately bounded (UUB); 2) the tracking error s converges to a small compact set 

around zero as |s| ≤ 2 ϑ1 /γ1 + V1 (0)e−γ1 t with γ1 and ϑ1 being positive constants defined in (32). Proof: Select the Lyapunov function as V1 =

1 2 s . 2

(29)

The derivative of V1 can be deduced along (27) and (22) as   V˙1 = s W ∗T (Z ) + ε − ydn + βn (x¯n , u θ )u + [0 T ]˜z  = s W ∗T (Z ) + ε − ydn − k1 sˆ − Wˆ T + ydn

  + (1− g) k1 sˆ + Wˆ T − ydn − gu r + [0 T ]˜z

  = s −k1 sˆ + W˜ T + ε + (1 − g) k1 sˆ + Wˆ T − ydn  − gu r + [0 T ]˜z 

≤ −k1 s 2 + |s| cW c + ε N + k1 [T 1]˜z + [0 T ]˜z 1 + b1 + (1 + b1 ) k1 sˆ + Wˆ T − ydn |s| − g b0  T n T × k1 sˆ + Wˆ − yd sˆ + [ 1]˜z sgn(ˆs ). (30) Based on Lemmas 3.1 and 3.2, it is shown that the observer error z˜ converges to a small bounded set around zero in FT even in the presence of noise, i.e., ˆz − z ≤ h¯ , then we define c = |[T 1]˜z | ≤ max{i , 1}h¯ and d = |[0 T ]˜z | ≤ max{i }h¯ as small positive constants, and c1 = cW c + ε N + k1 c + d and c2 = c1 + k1 (1 + b1 )(1 + b1 /b0 )c are positive constants. It follows: 1 + b1 V˙1 ≤ −k1 s 2 + c1 |s| − g b0 T n ˆ × k1 sˆ + W − yd [T 1]˜z sgn(ˆs ) + (1 + b1 ) k1 sˆ + Wˆ T − ydn [T 1]˜z    b1 ≤−k1 s 2 +c1 |s|+(1+ b1 ) 1+ k1 sˆ + Wˆ T − ydn c b0    b1 2 |s| c ≤−k1 s + c1 |s| + k1 (1 + b1 ) 1 + b0

375

   b1 ˆT + (1 + b1 ) 1 + W − ydn − k1 [T 1]˜z c b0      2 c b1 2 ≤−k1 s +c2 |s|+(1 + b1 ) 1+ + cd + k1 c c. b0 σ1 (31) By applying the Young’s inequality ab ≤ (a 2 + b 2 )/2 on the term c2 |s|, we have V˙1 ≤ −γ1 V1 + ϑ1 (32)

2 where ϑ1 = (1 + b1 ) (1 + b1 /b0 ) c /σ1 + cd + k1 c c + c22 /2 and γ1 = 2k1 − 1 are guaranteed to be positive if a feedback gain k1 > 1/2 is used in control (24). According to Lyapunov theorem, V1 is UUB and thus the tracking error s is bounded. This further guarantees the boundeness of sˆ since the observer error z˜ is bounded. Consequently, the tracking errors e, eˆ and the control signal u are all bounded. Based on (20), the newly defined states z are also bounded, and thus it follows from (2), (5), (8), (11), and (14) that z˙ i and x i are all bounded. Moreover, integrating both sides of (32) over [0, t], we have ϑ1 ϑ1

V1 ≤ V1 (0)e−γ1 t + 1−e−γ1t ≤ + V1 (0)e−γ1 t . (33) γ1 γ1 Then based on (29), it follows: 

|s| ≤ 2 ϑ1 /γ1 + V1 (0)e−γ1 t . (34)  √ So that lim |s| ≤ lim 2(ϑ1 /γ1 + V1 (0)e−γ1 t ) = 2ϑ1 /γ1 t →∞ t →∞ holds, which illustrates that the tracking error s converges to a small compact set around zero. Note that the tracking error can be minimized with a small observer error z˜ that leads to small c1 and c2 and thus ϑ1 . On the other hand, the control gain k1 is involved in both γ1 and ϑ1 , and thus this parameter should be adjusted appropriately in practical implementations, e.g., it can be determined as a tradeoff between the transient and steady state performance.  Remark 4.2: In control (24), the robust term u r is used to compensate for parts of the unknown dynamics to improve the tracking performance, i.e., to reduce the size of ϑ1 . The implementation of u r , however, depends on the sign of the tracking error sgn(ˆs ), which may result in chattering in the control signal. This will be further avoided in the following by introducing an alternative adaptive control. Moreover, in case the bounds b0 and b1 of g(x¯n , u θ ) are not available, we can simply set u r = 0. In this case, the closed-loop stability can be proved similarly. B. Adaptive Control With a Novel NN Structure In control (24), the robust term u r assumes appropriate information on the bounds of the control function βn (x¯n , u θ ), e.g., gm , b0 , and b1 , and the online updated NN weights (26) are a vector, for which the dimension depends on the number of neuron nodes. This section will present an alternative adaptive control to further remove this assumption and to reduce the number of adaptive parameters. The control can be given as   sˆ T ˆ (35) u = −k2 sˆ − sˆ θ (Z ) (Z ) − εˆ tanh ω

376

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 3, MARCH 2013

where k2 > 0 is the feedback gain, ω > 0 is a positive constant, and (Z ) is the HONN basis vector defined in (27) with the input Z = [x¯n , e, ˆ ydn ] ∈ R2n+1 . ˆ The parameters θ and εˆ are updated as  2 T   sˆ (36) θ˙ˆ = Pr oj 2    sˆ (37) ε˙ˆ = Pr oj a sˆ tanh ω where  > 0, a > 0 are adaptation gains, and the projection mapping Pr oj(·) is defined similar to [1] and [40] as  2 T   0,  sˆ if θˆ = θmax Pr oj =  sˆ 2 T 2 else , 2 (38)     if ε ˆ =εmax 0, sˆ Pr oj a sˆ tanh = sˆ  s ˆ tanh ω a ω , else (39) where (·)min , (·)max denote the minimum and maximum bounds of the corresponding variables. Here, θ ∗ = W ∗T W ∗ and ε∗ will be defined in the following (48). Denote the parameter errors as θ˜ = θ ∗ − θˆ and ε˜ = ε∗ − εˆ , and then based on [40] and the fact sˆ 2 T /2 ≥ 0 and sˆ tanh(ˆs /ω) ≥ 0, the projection operations (38) and (39) have the properties: P1) εˆ ∈ ε = {εmin ≤ εˆ ≤ εmax } and θˆ ∈ θ = {θmin ≤ θˆ ≤ θmax }; P2) θ˜ [ −1 Pr oj( sˆ2 T /2) − sˆ2 T /2] ≥ 0 and ε˜ [a−1 Pr oj(a sˆ tanh(ˆs /ω)) − sˆ tanh(ˆs /ω)] ≥ 0. The following result holds. Theorem 4.2: Consider (1) under Assumption 1, the transformed system (15), HOSM observer (16), and control (35) with adaptive laws (36), (37), then: 1) all signals in the closed-loop are semi-globally UUB; 2) the tracking error s converges to a small compact set 

around zero as |s| ≤ 2 ϑ2 /γ2 + V2 (0)e−γ2 t with γ2 and ϑ2 being positive constants defined in (49). Proof: Select the Lyapunov function as V2 =

1 2 g0 ˜ 2 g0 2 s + ε˜ . θ + 2 2 2a

(40)

Consider the following inequalities:

 sW ∗T = sˆ + [T 1]˜z W ∗T g0 θ ∗ sˆ 2 T 1 + + W N c c (43) 2 2g0 ˆ T (ˆs + [T 1]˜z )ˆs −s sˆ βn θˆ T = −βn θ ≤ −g0 θˆ sˆ2 T − βn θˆ sˆ T [T 1]˜z 2 2 c /2g0 (44) ≤ −g0 θˆ sˆ2 T /2 + g12 θmax c     εˆ εˆ ≤ −g0 εˆ sˆ tanh −sβn εˆ tanh ω ω   εˆ [T 1]˜z −βn εˆ tanh ω   εˆ ∗ ≤ −g0 ε sˆ tanh ω   εˆ +g0 ε˜ sˆ tanh (45) + g1εmax c. ω ≤

We have

The derivative of V2 can be deduced as ˙ˆ  − g ε˜ ε˙ˆ /  V˙2 = s s˙ − g0 θ˜ θ/ 0 a   = s [0 T ]e + αn (x¯n ) + βn (x¯n , u θ )u − ydn  2 T   sˆ −1 ˜ −g0 θ  Pr oj 2    sˆ −g0 ε˜ a−1 Pr oj a sˆ tanh ω   = s F2 (x¯n , e, ˆ ydn ) + βn (x¯n , u θ )u + [0 T ]˜z  2 T   sˆ −1 ˜ −g0 θ  Pr oj 2    sˆ −g0 ε˜ a−1 Pr oj a sˆ tanh ω

ˆ ydn ) = [0 T ]T eˆ +αn (x¯n )− ydn is an unknown where F2 (x¯n , e, function of Z = [x¯n , e, ˆ ydn ] ∈ R2n+1 , which can be approximated by HONN (19) as F2 (x¯n , e, ˆ ydn ) = W ∗T (Z ) + ε. 2 From the fact sˆ ≥ 0 and sˆ tanh(ˆs /ω) ≥ 0, and the adaptive laws (36), (37), we know θˆ (t) ≥ 0, t ≥ 0 and εˆ (t) ≥ 0, t ≥ 0 for any nonnegative initial condition θˆ (0) ≥ 0, εˆ (0) ≥ 0. Then based on 0 < g0 ≤ βn (x¯n , u θ ) ≤ g1 , it follows:  V˙2 = s W ∗T + ε − k2 βn sˆ    εˆ + [0 T ]˜z −βn sˆθˆ T − βn εˆ tanh ω  2 T   sˆ −g0 θ˜  −1 Pr oj 2    sˆ −g0 ε˜ a−1 Pr oj a sˆ tanh ω   εˆ 2 ∗T T ˆ ≤ −k2 g0 s + sW − s sˆ βn θ − sβn εˆ tanh ω  2 T   s ˆ + (k2 g1 c + d + ε N ) |s| − g0 θ˜  −1 Pr oj 2    s ˆ −g0 ε˜ a−1 Pr oj a sˆ tanh . (42) ω

 2 T     sˆ V˙2 ≤ −k2 g0 s 2 − g0 θ˜  −1 Pr oj − sˆ 2 T /2 2       s ˆ sˆ −g0 ε˜ a−1 Pr oj a sˆ tanh − sˆ tanh ω ω   s ˆ −g0 ε∗ sˆ tanh ω + (k2 g1 c + d + ε N ) sˆ + (k2 g1 c + d + ε N ) c +

(41)

2 c2 g 2 θmax c 1 + W N c c + 1 + g1 εmax c. 2g0 2g0

(46)

2 c 2 /2g + Denote c22 = c21 c + 1/2g0 + W N c c + g12 θmax c 0 g1 εmax c and c21 = k2 g1 c + d + ε N as bounded positive constants, and consider the properties P1 and P2 of the Pr oj(·)

Then V˙2 can be represented as V˙2 ≤ −γ2 V2 + ϑ2

(49)

where γ2 = 2k2 g0 , ϑ2 = c22 + 0.2785ωc21 + k2 g02 ε˜ 2 / a + k2 g02 θ˜ 2 /  are all bounded positive constants. According to Lyapunov Theorem, we know that s is ultimately bounded, and thus sˆ is also bounded based on (22) and the fact that z˜ is bounded. Consequently, the tracking errors e, eˆ are bounded. This further implies that the control signal u is bounded. Moreover, based on (20), the newly defined states z are also bounded, and thus it follows from (2) and (5), (8), (11), and (14) that z˙ i and x i are all bounded. Similar to the proof of Theorem 4.1, we can also derive the tracking error bound as 

|s| ≤ 2 ϑ2 /γ2 + V2 (0)e−γ2 t (50) which guarantees that the √ tracking error s converges to a compact set lim |s| ≤ 2ϑ2 /γ2 . It is clear from (50) that t →∞ the tracking error can be minimized by select appropriately small ω and large , a (and thus small ϑ2 ). A small observer error z˜ can also improve the tracking performance, i.e., leads to small c21 and c22 . However, the gain k2 should be chosen as a tradeoff between the transient and steady-state performance  since it is included in both γ2 and ϑ2 . Remark 4.3: Compared to control (24), it is shown that control (35) is deduced having only two scalars θˆ and εˆ to be online updated. This imposes significantly reduced computational costs, in particular compared to conventional backstepping control [11], [18], [19], [21]. Furthermore, only one NN is used in the control implementation with the input ˆ ∈ R2n for (24) and Z = [x¯n , e, ˆ ydn ] ∈ R2n+1 for Z = [x¯n , e] (35), where the dimensions are less than other controllers [see following backstepping control (54) for example]. It should also be clarified that the lengthy state transform given in Section II is only used for analysis purpose. The implementation of the proposed control schemes is indeed straightforward [i.e., observer (16) and control (24) with adaptive law (26) for the first scheme and observer (16) and control (35) with adaptive laws (36), (37) for the second scheme]. V. S IMULATION AND E XPERIMENTAL R ESULTS In this section, theoretical developments will be first verified by comparative simulations, and then the practical feasibility and effectiveness of the proposed controllers are experimentally validated in a servo motor system.

Reference

Observer

Output

1 0 -1 -2 0

2

4

6

8

10

12

14

16

18

20

-3 0

2

4

6

8

10 t(s)

12

14

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18

20

3 Control signal

We can define ε∗ = c21 /g0 , and consider the fact [15]   sˆ c21 sˆ − c21 sˆ tanh ≤ 0.2785ωc21. (48) ω

2 1 0 -1 -2

(a) 0.03 Observer Error

(47)

377

2

0.02 0.01 0 -0.01 -0.02 -0.03 0 3

Norm of NN Weights

  sˆ + c22 . V˙2 ≤ −k2 g0 s 2 + c21 sˆ − g0 ε∗ sˆ tanh ω

2

4

6

8

10

12

14

16

18

20

2

4

6

8

10 t(s)

12

14

16

18

20

2.5 2 1.5 1 0.5 0 0

(b) 4

F1 NN

2 NN Approximation

operation in (38) and (39), it follows:

Tracking performance

NA et al.: ADAPTIVE CONTROL FOR NONLINEAR PURE-FEEDBACK SYSTEMS WITH HOSM OBSERVER

0 -2 -4 -6 -8

-10

0

2

4

6

8

10 t(s)

12

14

16

18

20

(c) Fig. 1. Control performance of control (24). (a) Output tracking and control signal. (b) Observer error z˜ 1 and norm of NN weights Wˆ . (c) NN approximation of F1 ( x¯n , e). ˆ

A. Simulation Study Consider a nonlinear pure-feedback system as [18], [21] ⎧ x2 ⎪ ⎨ x˙1 = x 1 + x 2 + 52 3 (51) x˙2 = x 1 x 2 + u + u7 ⎪ ⎩ y = x1. The aim is to make the output y track a given reference yd = 2 sin(t). With the variables defined in (2) and the

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 3, MARCH 2013

Tracking performance

Observer

Output

1 0 -1 -2 0

2

4

6

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10

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20

0

2

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10 t(s)

12

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20

12

14

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18

20

Control signal

3 2 1 0 -1 -2 -3

(a) 0.03 Observer Error

In the simulation, the tracking errors used for the control design are eˆ = zˆ − y¯d and sˆ = [1 1]T eˆ with 1 = 2. Then control (24) with adaptive law (26) and control (35) with adaptive laws (36), (37) are implemented with parameters k1 = k2 = 7,  = 5, a = 1, and σ1 = 0.001, ω = 0.001. The HONN activation function is chosen as σ (x) = 2/(1 + e−10x ) + 1 and the inputs of HONN are the real system variables, i.e., no normalized operation is adopted. Moreover, with respect to a qualified approximation and the reasonable computational costs, the structure of HONN is determined by increasing the number of NN nodes until no further improvement of tracking accuracy can be observed. In this paper, a systematic online tuning approach leads to the final choice of a NN with eight neurons, i.e., L = 8. A further increase in the number of nodes cannot significantly help improving the performance and overlearning effects might be observed. Fig. 1 illustrates the simulation results of control (24) with initial conditions Wˆ (0) = [0, . . . , 0]T and x 1 (0) = 1, x 2 (0) = 0. It is shown that the achieved tracking performance is fairly good with a bounded control signal evaluation as depicted in Fig. 1(a). The HOSM output is also provided in Fig. 1(a). The observer error z˜ 1 and the norm of NN weights Wˆ  are shown in Fig. 1(b), which are all bounded. To validate the HONN approximation performance, Fig. 1(c) depicts profiles ˆ [defined in (27)] and the HONN of the function F1 (x¯n , e) output Wˆ T in (24), which indicates that the approximation is satisfactory (i.e., the covariance of the residual approximation error is 0.0962). Fig. 2 provides the simulation results of control (35) with θˆ (0) = 0 and εˆ (0) = 0. As it can be seen, satisfactory control performance is retained and all signals in the closedloop are bounded. In this case, no information on the function βn (·) is utilized and the online updated parameters are only two scalars, for which the reduced computational costs are occupied. Compared to (35), the introduction of the robust term u r with a signum function in (24) may cause some oscillations in the control signal [Fig. 1(a)] but reduce the tracking error (see Fig. 3). Note that in control (35), the online updated parameter θˆ is the estimation of a combined NN weight parameter θ ∗ = W ∗T W ∗ , and thus it is not straightforward to show the NN approximation. However, the NN approximation performance can be partially justified by tracking control performance [see Fig. 2(a)]. For comparison, the conventional backstepping control is also simulated. For those interested readers, see [18] for the detailed design and analysis of backstepping control for pure-feedback systems. The following control and adaptations

Reference

2

0.02 0.01 0 -0.01 -0.02 -0.03 0 1.5

Adaptive Parameters

corresponding transformed system (15), the HOSM observer for (51) is constructed as ⎧ ⎪ z˙ˆ 1 = v 1 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎨ v 1 = −10 zˆ 1 − y 3 sign(ˆz 1 − y) + zˆ 2 (52) z˙ˆ 2 = v 2 ⎪ 1 ⎪ ⎪ 2 ⎪ v 2 = −15 zˆ 2 − v 1 sign(ˆz 2 − v 1 ) + zˆ 3 ⎪ ⎪ ⎩˙ zˆ 3 = −20sign(ˆz 3 − v 2 ).

2

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ε(t)

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(b) Fig. 2. Control performance of control (35). (a) Output tracking and control signal. (b) Observer error z˜ 1 and adaptive parameters θˆ and εˆ . 1 Backstepping Method 1 Method 2

0.8 Tracking Error

378

0.6 0.4 0.2 0 -0.2 -0.4 0

Fig. 3.

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Comparative tracking errors.

proposed in [18] are utilized for (51) u = −z 1 − c2 z 2 − Wˆ 2T S2 (Z 2 )

(53)

where z 1 = x 1 − yd , z 2 = x 2 − α1 and Z 2 = [x 1 , x 2 , ∂α1 / ∂ x 1 , φ1 ]T with α1 = −c1 z 1 − Wˆ 1T S1 (Z 1 ) Z 1 = [x 1 , y˙d ]T

NA et al.: ADAPTIVE CONTROL FOR NONLINEAR PURE-FEEDBACK SYSTEMS WITH HOSM OBSERVER

∂α1 ∂α1 ∂α1 ˙ˆ y˙d + y¨d + W1 ∂yd ∂ y˙d ∂ Wˆ 1 W˙ˆ i = i (Si (Z i )z i − σi Wˆ i ).

379

φ1 =

(54)

It should be noted that two NNs are utilized in (53) and (54), where the input variable for NNs in (53) and (54) are with high dimensions due to the terms ∂α1 /∂ x 1 , φ1 . Specifically, the precise calculation of α˙ i will be tedious and lengthy for high order systems, which results in the so-called “explosion of complexity” [26]. In this paper, this problem has been circumvented successfully by introducing the state transform (15) and HOSM observer (16). Moreover, Fig. 3 gives comparative tracking errors of control (24) (Method 1), control (35) (Method 2), and (53) (Backstepping). It is shown that backstepping control gives slower tracking error convergence speed compared to (24) and (35). This may be because two NNs are employed in backstepping and thus a longer online learning phase is needed. Among three approaches, control (24) gives the smallest steady-state tracking error, and control (35) takes the least computational costs since only two scalars are online updated. B. Experimental Results To validate the applicability of the proposed control schemes, a turntable servo system with a rotary motor (one degree-of-freedom) is employed as the experimental test-rig (See Fig. 4). This closed-loop control system consists of a XSJ-2 servo motor as the controlled plant, a tachometric dynamo as the position sensor, and an expander board that connects with a personal computer (PC) through an AD/DA converter (PCL-812PG). A compatible PC (Pentium 2.8-GHz, 2-GB RAM) that runs C++ program codes is adopted as the controller. In the online running, the PC receives signals of the motor and then calculates the corresponding control effort. The PC is used to display the results via a graphic interface. A sampling frequency 100 Hz is selected, which is deemed to be faster than the closed-loop demand frequencies. The functional block diagram of the overall system is provided in Fig. 5. It is found that there are unavoidable nonlinearities in this test-rig (e.g., backlash and frictions) due to the mechanical joints, which are difficult to be modeled precisely. The main purpose of the experiments is to show the practical applicability of the proposed controllers (24) and (35) for position tracking. It is noted that the primary dynamics of this test-rig may be captured by a simplified model [41], which can be considered as a specified case of pure-feedback systems. In the experiments, parameters are specified as 1 = 4, k1 = k2 = 0.8,  = a = 0.5, and σ1 = 0.001, ω = 0.001. The size of HONN (i.e., number of neuron) and the sigmoid function are specified as same as those used in simulation. We first choose the position reference as yd = 90 sin(2πt). Fig. 6 depicts the experimental results, where it is shown that both controllers (24) and (35) can achieve satisfactory tracking performance. Since there is a robust term with a signum function in (24), the system evaluations in Fig. 6(a) have more oscillations compared to those of (35) in Fig. 6(b). This fact can be

Fig. 4.

Turntable servo motor system test-rig.

A/D, D/A Converter

Sensor

(PCL812PG) Visual C ++ Complier

Fig. 5.

Amplifiers

XSJ-2 servo motor

Functional diagram of servo system.

observed in the velocity response and control signal in Fig. 6. However, the introduction of this robust term can achieve a smaller tracking error, which will be shown in the following. The aforementioned experimental results illustrate the efficacy of the adaptive element for compensating time-varying dynamics and retaining control performance. To further compare the performance of (24) and (35), extensive experiments have been conducted, and it is also found that control (24) performs slightly better than control (35) in the sense of steady-state tracking error but worse in the cost of oscillated control actions. Figs. 7 and 8 illustrate the comparative results for the references yd = 30 sin(2πt) and yd = 150 sin(πt) with control (24) (Method 1) and control (35) (Method 2), respectively. One can find from these figures that control (24) gives smaller tracking errors in all cases. These observations can also be verified using some performance indices. Here, four indices are considered: the IAE and  ISDE are indices of the tracking error, in which IAE = |e(t)|dt is the integrated absolute error, and ISDE =  (e(t) − e0 )2 dt is the integrated square error between the instant value  and its mean value e0 ; IAU = |u(t)|dt and ISDU = (u(t) − u 0 )2 dt are indices for the control signal, where the former is a measurement of the overall amount of control effort, and the latter measures fluctuations of the control signal around its mean value u 0 . Table I provides comparative results for the time interval t = 0 ∼ 10 s for both controllers (24) and (35) for sinusoidal references with different amplitudes and periods. From Table I, one may conclude that control (24) gives smaller IAE and ISDE in all cases and thus better control

380

IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, VOL. 24, NO. 3, MARCH 2013

TABLE I C OMPARISON OF P ERFORMANCE I NDICES FOR C ONTROL (24) AND (35) W ITH S INUSOID R EFERENCE T =1 60

90

Method

M1

M2

M1

M2

M1

IAE ISDE IAU ISDU

14.86 31.13 1.84 0.41

34.90 144.0 1.78 0.38

19.66 53.52 2.36 0.68

40.90 196.4 2.28 0.62

24.63 45.42 81.85 247.2 2.90 2.75 1.05 0.92

Reference

Position (deg)

Control signal (V) Velocity (deg/s)

100 50 0 -50 -100 0

T =2

45

1

2

3

4

5

6

7

M2

9

M2

M1

M2

M1

M2

M1

M2

32.71 152.0 3.94 1.92

57.05 408.9 3.77 1.75

8.58 12.18 1.85 0.39

35.83 145.5 1.81 0.38

11.78 22.50 2.39 0.66

41.64 198.8 2.31 0.63

13.49 32.19 2.95 1.01

47.86 263.19 2.84 0.95

17.93 54.43 3.79 1.69

54.01 337.8 3.40 1.35

10

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Fig. 7.

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(b) Fig. 6. Experimental results for yd = 90 sin(2π t). (a) Control performance of (24). (b) Control performance of (35). TABLE II C OMPARISON TO PID C ONTROL W ITH yd = 90 sin(2π t/ T ) Period (s)

T = 0.5

T =1

T = 1.5

T =2

T = 2.5

PID M1 M2

135.8 106.1 106.6

78.42 32.71 57.05

27.36 21.04 35.54

25.39 11.78 41.64

25.77 11.67 30.40

performance. However, this is partially at the expense of a large control effort (i.e., IAU) and fluctuations (i.e., ISDU). This is necessary and reasonable since control (24) targets the

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compensation of HONN error with a sliding mode-like term, which requires nonsmooth and fast control action. Finally, conventional PI control is also tested in the same platform. The PID control parameters are adjusted via a heuristic tuning approach for a given position reference, e.g., yd = 90 sin(πt), (i.e., tuned to tradeoff the control actions and the tracking performance). Then comparative experiments for demands yd = 90 sin(2πt/T ) with different periods T

NA et al.: ADAPTIVE CONTROL FOR NONLINEAR PURE-FEEDBACK SYSTEMS WITH HOSM OBSERVER

ACKNOWLEDGMENT

T=0.5

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Fig. 9.

The authors would like to thank the editors and anonymous referees for their constructive suggestions and valuable comments. They would also like to thank Dr. G. Herrmann for his useful suggestion that helped to improve this paper. R EFERENCES

0

-10 0 10

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Comparative tracking errors for yd = 90 sin(2π t/ T ).

are conducted for adaptive controllers (24), (35), and PID control. For sinusoidal references, PID control performs similarly to adaptive control (24) and even better than (35) in the low/middle frequency regime (e.g., T = 2.5 s), whereas it obtains worse error performance in the high frequency regime (e.g., T = 0.5 s). This is also clearly summarized in Table II in terms of IAE. This is, in particular, of interest as linear controllers (e.g., PID) are expected to decrease in control performance at high demand frequencies. As an example, Fig. 9 depicts the comparative tracking errors for T = 0.5, 1.5 and 2.5 s. These experimental results exactly illustrate how the addition of the adaptive element allows for compensation of possible time-varying dynamics to improve the overall performance for various operation scenarios. VI. C ONCLUSION This paper proposed a novel adaptive control framework for nonaffine pure-feedback nonlinear systems. By introducing a coordinate state transform, the state feedback control of a purefeedback system was transformed into the output-feedback control of a canonical system. Consequently, the widely used backstepping control for pure-feedback systems was avoided to remedy the explosion of complexity and the possible circular issue. A high-order sliding mode observer with FT error convergence was employed to estimate unknown states of the transformed system, and two adaptive controllers were developed to achieve tracking control and to guarantee the closed-loop system stability. Specifically, a novel NN structure with only a scalar weight parameter (independent of the number of neurons) was constructed so that the computational costs were reduced. The simplicity in the control implementation renders the developed methods attractive for industrial applications. Simulation and experiments were performed to demonstrate the effectiveness. Future work will be focused on the generalization of this idea for adaptive control of other nonlinear systems.

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Jing Na received the B.S. and Ph.D. degrees from the School of Automation, Beijing Institute of Technology, Beijing, China, in 2004 and 2010, respectively. He was a Visiting Student with the Universitat Politècnica de Catalunya, Barcelona, Spain, in 2008. He was a Research Collaborator with the Department of Mechanical Engineering, University of Bristol, Bristol, U.K., from 2008 to 2009. He has been with the Faculty of Mechanical and Electrical Engineering, Kunming University of Science and Technology, Kunming, China, since 2010. He was a Post-Doctoral Fellow with the Cryogenic System Section, ITER Organization, Cadarache, France, from 2011 to 2012. His current research interests include adaptive control of time-delay systems, neural networks, parameter estimation, adaptive observer design, repetitive control, and nonlinear control and applications.

Xuemei Ren received the B.S. degree from Shandong University, Shandong, China, in 1989, and the M.S. and Ph.D. degrees in control engineering from the Beijing University of Aeronautics and Astronautics, Beijing, China, in 1992 and 1995, respectively. She is currently a Professor with the School of Automation, Beijing Institute of Technology, Beijing. Her current research interests include intelligent systems, neural networks, and adaptive and process control.

Dongdong Zheng was born in Guang’an, Sichuan, China, in 1988. He received the Bachelors and Masters degrees from the School of Automation, Beijing Institute of Technology, Beijing, China, in 2010 and 2012, respectively. He is currently pursuing the Ph.D. degree with Concordia University, Montreal, QC, Canada. His current research interests include neural network control, adaptive identification and control of servo systems, and nonlinear control theory and its applications.

Adaptive control for nonlinear pure-feedback systems with high-order sliding mode observer.

Most of the available control schemes for pure-feedback systems are derived based on the backstepping technique. On the contrary, this paper presents ...
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