Simplified adaptive neural control of strict-feedback nonlinear systems

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Simplified adaptive neural control of strictfeedback nonlinear systems Yongping Pan, Yiqi Liu, Haoyong Yu

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S0925-2312(15)00100-9 http://dx.doi.org/10.1016/j.neucom.2015.01.053 NEUCOM15091

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Neurocomputing

Received date: 26 September 2014 Revised date: 20 December 2014 Accepted date: 28 January 2015 Cite this article as: Yongping Pan, Yiqi Liu, Haoyong Yu, Simplified adaptive neural control of strict-feedback nonlinear systems, Neurocomputing, http://dx. doi.org/10.1016/j.neucom.2015.01.053 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Simplified adaptive neural control of strict-feedback nonlinear systems  Yongping Pan a, Yiqi Liu b , Haoyong Yu a a Department b School

of Biomedical Engineering, National University of Singapore, Singapore 117575

of Automation Science and Engineering, South China University of Technology, Guangzhou 510640, China

Abstract This paper presents a simplified adaptive backstepping neural network control (ABNNC) strategy for a general class of uncertain strict-feedback nonlinear systems. During the backstepping design, all unknown functions at intermediate steps are passed down such that only a single neural network (NN) is needed to approximate a lumped uncertainty at the last step. The closed-loop system achieves practical asymptotic stability in the sense that all involved signals are bounded and the tracking error converges to a small neighborhood of zero. The contribution of this study is that the complexity growing problem of the traditional ABNNC design is substantially eliminated for a general class of uncertain strict-feedback nonlinear systems, where the constraints of control parameters that guarantees closed-loop stability is clearly demonstrated. An illustrative example has been provided to verify effectiveness of the proposed approach. Key words: Adaptive control, backstepping, neural network, function approximation, strict-feedback, nonlinear system.

1

Introduction

Recent years, adaptive approximation-based control using fuzzy systems or neural networks (NNs) has attracted great concern due to its effectiveness of modeling functional uncertainties in nonlinear systems [1]. Some recent results  Corresponding author: H. Yu. Email addresses: [email protected] (Yongping Pan), [email protected] (Yiqi Liu), [email protected] (Haoyong Yu). Preprint submitted to Elsevier Science

17 February 2015

can be referred to [2–12]. The most prominent benefit of applying fuzzy or NN approximation during control synthesis is that the difficulty of system modeling in many practical control problems can be largely alleviated. By the combination of backstepping design and function approximation, adaptive backstepping NN control (ABNNC) has also been developed for some classes of strict-feedback nonlinear systems (SFNSs) with mismatched uncertainties [13–19]. A general class of SFNSs can be expressed as follows: ⎧ ⎪ ⎪ x˙ i = fi (xi ) + gi (xi )xi+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ (i = 1, 2, · · · , n − 1) ⎪ ⎪ ⎪ x˙ n = fn (xn ) + gn (xn )u ⎪ ⎪ ⎪ ⎪ ⎪ ⎩y = x

(1)

1

where xi (t) := [x1 (t), x2 (t), · · · , xi (t)]T ∈ Ri are plant states, u(t) ∈ R and y(t) ∈ R are the control input and the controlled output, respectively, fi (xi ) : Ri → R are nonlinear driving functions, gi (xi ) : Ri → R are control gain functions (i.e. affine terms), and i = 1, 2, · · · , n. In the conventional ABNNC design, the controller complexity grows drastically as the plant order increases owing to two reasons: one is the repeated derivations of virtual control inputs, and the other is the application of multiple approximators. To avoid the repeated derivations of virtual control inputs, a dynamic surface control technique that applies a first-order filter at each backstepping step can be combined to the ABNNC design [20–24]. Yet in this type of approaches, at least n NNs are still needed for a nth-order system. To completely eliminate the complexity growing problem, single NN approximation-based ABNNC was proposed for a special class of uncertain SFNS of (1) with gi (·) = 1 (i = 1, 2, · · · n) in [25]. The key idea in this approach is that all unknown functions at intermediate steps are passed down during backstepping such that only one NN is needed to approximate a lumped unknown function at the last step. This approach leads to a simple control structure which only contains an actual control law with one parameter adaptive law. However, the stability result obtained in [25] is based on a precondition that the optimal NN approximation error is bounded before control, which implies that the plant states are already bounded before control. Therefore, the stability condition in [25] still needs to be further investigated. In this study, a single NN approximation-based ABNNC is presented for a general class of uncertain SFNSs (1), where the constraints of control parameters that guarantees closed-loop stability is clearly demonstrated. The design procedure of the proposed approach is as follows: First, during the backstepping design, all unknown functions of virtual control laws at intermediate steps are passed down to an actual control law; second, an ideal actual control 2

law is proposed to guarantee closed-loop stability; third, only one radial basis function (RBF) NN is applied to approximate a lumped uncertainty in the ideal actual control law; finally, the closed-loop system is proved to be practically asymptotically stable under sufficient constraints of control parameters depended on an initial condition of plant states. The rest of this paper is organized as follows. The considered problem is formulated in Section 2. The procedure of backstepping design is given in Section 3. The proposed approach is provided in Section 4. An illustrative example is given in Section 5. Finally, conclusions are summarized in Section 6. Throughout this paper, R, R+ and Rn denote the spaces of real numbers, positive real numbers and real n-vectors, respectively, | · |,  ·  and  · ∞ denote the absolute value, 2-norm and ∞-norm, respectively, L2 and L∞ denote the spaces of square integrable signals and essentially bounded signals, respectively, min{·}, max{·} and sup{·} represent the functions of minimum, maximum and supremum, respectively, and C k represents the space of functions whose k-order derivatives all exist and are continuous, where k is a positive integer.

2

Problem Formulation

Revisit the system (1) with fi (·) and gi (·) being of C 1 and unknown. Let yd (t) ∈ R be a reference signal, ydi (t) := [yd1 (t), yd2 (t), · · · , ydi (t)]T ∈ Ri , (n) T and yde (t) := [ydn (t), yd (t)]T ∈ Rn+1 , where yd1 (t) = yd (t) and ydi (t) = (i−1) yd (t) with i = 2, 3, · · · , n. Define compact sets Ωx := {xn |xn  ≤ cx }, Ωx0 := {xn |xn  ≤ cx0 }, and Ωde := {yde |yde  ≤ cd }, where cx , cx0 , cd ∈ R+ are constants and cx0 ≤ cx . The following common and reasonable assumptions are exploited in the subsequent development. Assumption 1 : There exist constants gi0 , gi1 ∈ R+ such that 0 < gi0 ≤ |gi (·)| ≤ gi1 with i = 1, 2, · · · , n, ∀x ∈ Ωx . Without loss of generality, let gi (·) > 0. (i)

Assumption 2 : The signal yd (t) satisfies yd (t) ∈ L∞ for i = 0, 1, · · · , n + 1. Define κ0i := 1 and κji := ki ki−1 · · · ki−j+1, j ≤ i for j ≥ 1, where kl ∈ R+ with l = 1, 2, · · · , i are constant control gains, i is a positive integer, and j is a nonnegative integer. The notation κji = ki κj−1 i−1 is frequently used in the subsequent control design. The control objective of this study is to develop a NN-based control strategy for the system (1) under Assumptions 1 and 2 such that the system output y tracks its desired signal yd as accurate as possible. Remark 1 : The approach in [25] considers the system in (1) under an assumption that gi (·) = 1 (i = 1, 2, · · · n). Differing from [25], this study focuses on a general class of SFNSs (1) without this strict assumption. In Assumption 3

1, the controllability condition |gi0 | > 0 can be found in all existing ABNNC approaches, and the locally bounded condition 0 < gi0 ≤ |gi (·)| ≤ gi1 , ∀x ∈ Ωx can be naturally obtained by fi (·) and gi (·) being of C 1 .

3

Backstepping Design

Define the output tracking error z1 (t) := y(t)−yd (t) and virtual tracking errors zi (t) := xi (t) − αi (t) with i = 2, 3, · · · , n, where αi are virtual control inputs T ]T with i = 2, 3, · · · , n. The procedure of defined later. Let xei := [xTi , yd(i+1) backstepping design is as follows. Step 1 : The derivative of z1 is as follows: z˙1 = f1 (x1 ) − y˙ d + g1 (x1 )x2 .

(2)

Choose the virtue control input α2 = −k1 z1 − h1 (xe1 )

(3)

to stabilize the subsystem in (2), where h1 (xe1 ) := (f1 (x1 ) − y˙ d )/g1 (x1 ). Applying x2 = z2 + α2 and (3) to (2) yields z˙1 = g1 (x1 )(−k1 z1 + z2 ).

(4)

Applying (3) to the definition of z2 , one obtains z2 = x2 + k1 (x1 − yd ) − h1 (xe1 ) =

2

j=1

∗ κ2−j 1 (xj − ydj ) + h1 (xe1 )

(5)

where h∗1 (xe1 ) := h1 (xe1 ) + yd2 . Step 2 : The derivative of z2 is as follows: z˙2 = f2 (x2 ) + g2 (x2 )x3 − α˙ 2 .

(6)

From (3) and z1 = y − yd , one get α2 = α2 (xe1 ). Thus, it can be shown that T )y˙ d2 := α2∗ (xe2 ). α˙ 2 = (∂α2 /∂xT1 )x˙ 1 + (∂α2 /∂yd2

Applying the above result to (6) leads to z˙2 = f2 (x2 ) − α2∗ (xe2 ) + g2 (x2 )x3 . 4

(7)

To compensate the interconnected term g1 (x1 )z2 in (4), a correlative interconnected term g1 (x1 )z1 is added and subtracted at the right side of the above equality, which results in z˙2 = f2 (x2 ) + g1 (x1 )z1 − α2∗ (xe2 ) + g2 (x2 )x3 − g1 (x1 )z1 .

(8)

Choose the virtue control law α3 = −k2 z2 − h2 (xe2 )

(9)

to stabilize the subsystem in (8), where 



h2 (xe2 ) := f2 (x2 ) + g1 (x1 )z1 − α2∗ (xe2 ) /g2 (x2 ). Applying x3 = z3 + α3 and (9) to (8) yields z˙2 = g2 (x2 )(−k2 z2 + z3 − g1 (x1 )z1 ).

(10)

Applying (9) with (5) to the definition of z3 , one gets z3 = x3 + k2 z2 + h2 (xe2 ) = (x3 − yd3 ) +

2

+ k2 h∗1 (xe1 ) +

κ3−j (xj j=1 2 h∗2 (xe2 )

− ydj )

where h∗2 (xe2 ) := h2 (xe2 ) + yd3 . Thus, one has z3 =

3

κ3−j (xj − ydj ) + j=1 2

2 j=1

∗ κ2−j 2 hj (xej ).

(11)

Step i (3 ≤ i ≤ n − 1): The derivative of zi is as follows: z˙i = fi (xi ) + gi (xi )xi+1 − α˙ i

(12)

T α˙ i = (∂αi /xTi−1 )x˙ i−1 + (∂αi /∂ydi )y˙ di := αi∗ (xei ).

(13)

where α˙ i is given by

Thus, (12) can be rewritten as z˙i = fi (xi ) − αi∗ (xei ) + gi (xi )xi+1 . Adding and subtracting gi−1 (xi−1 )zi−1 at the right side of the above equality leads to z˙i = fi (xi ) + gi−1 (xi−1 )zi−1 − αi∗ (xei ) + gi (xi )xi+1 − gi−1 (xi−1 )zi−1 . 5

(14)

Choose the virtue control law αi+1 = −ki zi − hi (xei )

(15)

to stabilize the subsystem in (14), where 



hi (xei ) := fi (xi ) − gi−1 (xi−1 )zi−1 − αi∗ (xei ) /gi (xi ). Applying xi = zi + αi and (15) to (14) yields z˙i = gi (xi )(−ki zi + zi+1 − gi−1 (xi−1 )zi−1 ).

(16)

Noting (15) and (11), one obtains zi =

i

κi−j (x − ydj ) + j=1 i−1 j

i−1 j=1

∗ κi−1−j i−1 hj (xej ).

(17)

where h∗j (xej ) := hj (xej ) + yd(j+1) . Step n: The derivative of zn is as follows: z˙n = fn (xn ) + gn (xn )u − α˙ n

(18)

T )y˙ dn := αn∗ (xen ). α˙ n = (∂αn /xTn−1 )x˙ n−1 + (∂αn /∂ydn

(19)

where α˙ n is given by

Then, (18) can be rewritten as z˙n = fn (xn ) + gn (xn )u − αn∗ (xen ). Adding and subtracting gn−1 (xn−1 )zn−1 at the right side of the above equality leads to z˙n = fn (xn ) + gn−1 (xn−1 )zn−1 − αn∗ (xen ) + gn (xn )u − gn−1(xn−1 )zn−1 .

(20)

Design an ideal actual control law u∗ (xen ) = −kn zn − hn (xen )

(21)

to stabilize the last subsystem in (20), where 

hn (xen ) := fn (xn ) + gn−1 (xn−1 )zn−1 

− αn∗ (xen ) /gn (xn ). Applying (15) with (17) to the definition of zn , one gets zn =

n  n−j

n−1 

j=1

j=1

κn−1 (xj − ydj ) +

6

∗ κn−1−j n−1 hj (xej ).

(22)

Now, choose a Lyapunov function candidate Vz (z) =

n

z 2 /2 i=1 i

(23)

with z := [z1 , z2 , · · · , zn ]T for the entire system composed of (4), (10), (16) and (20). The following lemma is established to show the stability result of the closed-loop system under the priori knowledge of plant dynamics. Lemma 1 : For the system (1) satisfying Assumptions 1 and 2, if the actual control law is chosen as u = u∗ in (21), then the closed-loop system achieves exponential stability in the sense of limt→∞ z(t) = 0. Proof: Differentiating Vz in (23) with respect to time t and applying (4), (10), (16), (20) and (21), one gets V˙ z = −k1 g1 (x1 )z12 + z1 z2 g1 (x1 ) − k2 g2 (x2 )z22 + z2 z3 g2 (x2 ) − z1 z2 g1 (x1 ) ··· 2 − kn−1 gn−1 (xn−1 )zn−1 + zn−1 zn gn−1 (xn−1 ) − zn−2 zn−1 gn−2 (xn−2 ) − kn gn (xn )zn2 − zn−1 zn gn−1 (xn−1 ). Thus, it is easy to obtain V˙ z = −

n i=1

ki gi zi2 ≤ −ks Vz ≤ 0

with ks := min{2ki gi0 , i = 1, 2, · · · , n} ∈ R+ , where all the interconnected terms zi−1 zi gi−1 (xi−1 ) with i = 2, 3, · · · , n are completely compensated. The above result implies that the closed-loop system achieves exponential stability in the sense that limt→∞ z(t) = 0.

4

Adaptive Neural Network Control

The ideal actual control law u∗ in (21) is unreliable due to the unknown plant functions fi (·) and gi (·) (i = 1, 2, · · · , n). From the definitions of αi∗ in (5), (13) and (19), αi∗ are of class C 1 with respective to there variables, where i = 2, 3, · · · , n. Since fi (·) and gi (·) (i = 1, 2, · · · , n) and αi∗ (i = 2, 3, · · · , n) are of class C 1 , hi (·) (i = 1, 2, · · · , n) are of class C 1 . Thus, NNs can be applied to approximate certain functions depended on hi (·) (i = 1, 2, · · · , n). To avoid NN approximation at the steps 1 to n − 1 of the backstepping design, the expression of zn in (22) is substituted into (21), which results in u∗ = −

n j=1

κn+1−j (xj − ydj ) − F (xen ) n 7

(24)

where F (·) is a lumped uncertainty defined as F (xen ) :=

n j=1

∗ κn−j n hj (xej )

(25)

with h∗n (xen ) := hn (xen ). Next, a class C 1 linearly parameterized RBF NN [1] ˆ)=W ˆ T Φ(xen ) Fˆ (xen , W

(26)

is applied to approximate F (·) in (25), where Φ(·) : R2n+1 → RM satisfying ˆ ∈ RM is the vector of adjustable Φ ≤ ψ is the vector of basis functions, W weights, ψ ∈ R+ is a certain constant, and M is the number of neurons. Then, a certain actual control law can be determined as follows: u=−

n j=1

ˆ ). κn+1−j (xj − ydj ) − Fˆ (xen , W n

(27)

ˆ |W ˆ  ≤ cw }, Ωxe := {xen |xn ∈ Ωx , yde ∈ Ωde } Define compact sets Ωw := {W where cw ∈ R+ ia a constant. Next, define an optimal NN approximation error ε(xen ) := F (xen ) − Fˆ (xen , W ∗ )

(28)

where W ∗ is a vector of optimal weights given by ∗

W := arg min

ˆ ∈Ωw W





ˆ )| . sup |F (xen ) − Fˆ (xen , W

xen ∈Ωxe

From the universal approximation property of NNs [1], one has the following lemma. Lemma 2 : The optimal approximation error ε in (28) can be bounded by a certain constant ε¯ := sup∀xen ∈Ωxe |ε(xen )| ∈ R+ dominated by the number of neurons. Adding and subtracting gn (·)F (·) at the right side of (20) and applying (24) and (27), one gets ˆ ) − kn zn ) z˙n = gn (xn )(F (xen ) − Fˆ (xen , W − gn−1 (xn−1 )zn−1 . Noting (28), it can be shown that ˜ T Φ(xen ) + ε − kn zn )) z˙n = gn (xn )(W − gn−1 (xn−1 )zn−1 .

(29)

ˆ. ˜ := W ∗ − W in which W ˆ as follows: Choose an adaptive law of W ˆ˙ = γ(z1 Φ(xen ) − σ W ˆ) W 8

(30)

where γ ∈ R+ is a learning rate, and σ ∈ R+ is a small constant. Then, choose a Lyapunov function candidate ˜)= V (z, W

n

z 2 /2 i=1 i

˜ /2γ ˜ TW +W

(31)

for the closed-loop system composed of (4), (10), (16), (29) and (30). The following theorem is established to demonstrate the stability result of this study. Theorem 1 : For the system (1) satisfying Assumptions 1 and 2 driven by the ˆ (0) ∈ Ωw , control law (27) with (26) and (30) and any given xn (0) ∈ Ωx0 and W there exist a suitably large approximation region Ωxe satisfying Ωx ⊇ Ωx0 and suitably large control gains ki satisfying (33) under xen ∈ Ωxe such that the closed-loop system achieves practical asymptotic stability in the sense that all involved signals are uniformly bounded and the tracking error z1 converges to a small neighborhood of zero dominated by λ and ki , where i = 1, 2, · · · , n. Proof: Differentiating V in (31) with respect to time t, applying (4), (10), (16) and (29) and noting the results in the proof of Lemma 1, one obtains V˙ = −

n i=1

ˆ˙ ˜ T Φ(xe ) + ε) − W ˜ T W/γ. ki gi zi2 + gn zn (W

Substituting (30) to the above expression yields n

˜ T Φ(xen ) + gn zn ε ki gi zi2 + gn zn W ˜  + σW ˜ T W ∗. − z1 W Φ(xen ) − σW

V˙ ≤ −

i=1 ˜T

(32)

Noting Assumption 1 and Lemma 2, one obtains ⎧ ⎪ ˜ T Φ ≤ ψ 2 z 2 /(2σ) + σW ˜ 2 /2 ⎪ z1 W ⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ˜ T Φ ≤ (gn1 ψ)2 z 2 /(2σ) + σW ˜ /2 ⎨ gn zn W n

⎪ ⎪ ˜ 2 /2 + σW ∗ 2 /2 ˜ T W ∗ ≤ σW ⎪ σW ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ g z ε ≤ g 2 z 2 /2 + ε¯2 /2 n n

.

n1 n

Applying the above results to (32), one gets V˙ = −(k1 g1 − ψ 2 /2σ)z12 −

n−1 i=2

ki gi zi2

2 − (kn gn − gn1 (ψ 2 /2σ + 1/2))zn2 + ε¯2 /2 + σW ∗ 2 /2.

If the selection of ki (i = 1, 2, · · · , n) satisfies ⎧ ⎪ ⎨ k1

≥ ψ 2 /(2σg10 ), ki > 0(i = 2, · · · , n − 1)

⎪ ⎩k

2 ≥ gn1 (ψ 2 /σ + 1)/(2gn0 )

n

9

(33)

then it can be shown that ε2 + σW ∗ 2 )/2 V˙ = −km Vz + (¯ 2 with km := min{2k1 g1 − ψ 2 /σ, kn gn − gn1 (ψ 2 /2σ + 1/2), 2ki gi0 , i = 2, · · · , n − 1} ∈ R+ . Thus, one obtains

V˙ = −km V + km ϕ

(34)

ˆ ) converges ε2 + σc2w )/2km ∈ R+ , which implies that (xn , W with ϕ := 2c2w /γ + (¯ to the compact set ˆ )|V < ϕ, yde ∈ Ωde }. Ωs := {(xn , W

(35)

Since the bounds ε¯, gi0 and gi1 (i = 1, 2, · · · , n) are obtained on xn ∈ Ωx and yde ∈ Ωde , the selection of ki (i = 1, 2, · · · , n) depends on the sizes of Ωx and ¯ s := Ωx × Ωw − Ωs . Ωde , and (34) is only valid on xen ∈ Ωxe = Ωx × Ωde . Let Ω if ˆ (0)) ∈ Ω01 ⊆ Ωs , (xn (0), W ˆ (t) remain in Ωs , i.e. then xn (t) and W ˆ (t)) ∈ Ωs , ∀t ≥ 0; (xn (t), W Else if

ˆ (0)) ∈ Ω02 ⊂ Ω ¯ s, (xn (0), W ˆ (t) enter and stay inside then V˙ remains negative definite until xn (t) and W Ωs , i.e. ˆ (t)) ∈ Ω02 ∪ Ωs , ∀t ≥ 0. (xn (t), W The above analysis implies that if ki (i = 1, 2, · · · , n) in (33) are chosen on ∀xn ∈ Ωx ⊇ Ωx0 and ∀yde ∈ Ωde so that they are large enough, then xn (t) ∈ Ωx , ∀t ≥ 0 and the result in (34) always holds, ∀xn (0) ∈ Ωx0 and ˆ (0) ∈ Ωw . Solving (34) by the Lemma A.3.2 of [1] yields ∀W V (t) ≤ e−km t V (0) + ϕ, ∀t ≥ 0 which implies V (t) ∈ L∞ and V (t) converges to ϕ. Combining with (31), one ˜ ∈ L∞ . Thus, it is easy to obtain xn , W ˆ , u ∈ L∞ , i.e. all closed-loop gets z, W signals are uniformly bounded. From the above inequality, one also has z12 ≤ e−km t 2V (0) + 2ϕ, ∀t ≥ 0 √ which implies that for any given μ > 2ϕ, there exists a finite time T ≥ 0 such that |z1 | ≤ μ, ∀t ≥ T . From (33) and the definition of ϕ, for a given σ, increasing γ and ki (i = 1, 2, · · · , n) results in the decrease of ϕ. Consequently, the tracking error z1 can be made arbitrary small by the increase of γ and ki (i = 1, 2, · · · , n), i.e. the stability result is practical asymptotic in the sense 10

that z1 converges to a small neighborhood of zero dominated by γ and ki (i = 1, 2, · · · , n) [26]. 2 Remark 2 : In the approach of [25], the stability result similar to (34) is based on a precondition that the optimal NN approximation error ε in (28) is bounded before control, which implies that xen keeps within Ωxe without control. Compared with [25], the differences of the proposed approach include: 1); A wider class of uncertain SFNSs is considered; 2) the initial conditions ˆ (0) are analyzed during stability proof; 3) it is demonstrated of xn (0) and W that the constraint of control gains ki (i = 1, 2, · · · , n) in (33) should be satisfied under the bounds ε¯, gi0 and gi1 (i = 1, 2, · · · , n) on xn ∈ Ωx to ensure closed-loop stability. Remark 3 : To ensure the reliability and safety of modern industrial processes, data-driven methods have been receiving great attention for process monitoring, fault detection and diagnosis [27–29]. The combination of ABNNC with data-driven methods is a very interesting topic. Yet, since this study focuses on ABNNC for uncertain SFNSs, this topic is out of the scope but possible to be further work of this study.

5

An Illustrative Example

Consider a numerical example in [14] with the following strict-feedback nonlinear system: ⎧ ⎪ ⎪ ⎪ x˙ 1 ⎪ ⎪ ⎨

= 0.5x1 + (1 + 0.1x21 )x2

x˙ 2 = x1 x2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ y = x1

+ (2 + cos x1 )u

and a yd generated by the Van der Pol oscillator system: ⎧ ⎪ ⎪ ⎪ x˙ d1 ⎪ ⎪ ⎨

x˙ d2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

= xd2 = −xd1 + β(1 − x2d1 )xd2

yd = xd1

where its phase-plane trajectories approach a limit cycle under β > 0 and nonzero initial states. The procedure of control design is as follows: First, select activation functions μlii (xei ) = exp(−(xei −2(l1 −2))2 /2 /0.82 ) with xe = [xe1 , · · · , xe5 ]T , li = 1, 2, 3 and i = 1, 2, · · · ,5 to construct the vector of basis functions Φ(·) in (26); secˆ (0) = [0, 0, · · · , 0]T ond, choose k1 = 20, k2 = 10, γ = 5000, σ = 0.001 and W 11

for the control law composed of (27) and (30); third, design a signal vector Yd , a filtered version of yd , as Tf (t)Y˙ d +Yd = yd with Yd (0) = xn (0) to avoid initial high gain at the control input [25], where Tf (t) = η exp(−ωt) + τ with η = 1, ω = 2 and τ = 0.01. For simulation, select x2 (0) = [1.2, 1.0]T , xd (0) = [xd1 (0), xd2 (0)]T = [1.5, 0.8]T and β = 0.2 [14]. Control trajectories by the proposed approach are depicted in Fig. 1, where the controlled output y tracks its desired signal yd quickly and accurately under low-gain control input u. It is worth noting that the tracking performance is much better than that of the original literature [14] maybe due to the simpler adjustment of control parameters in the proposed approach. It is also observed that the high-precision tracking is achieved at the cost of oscillation at the beginning of the control input u. To reduce control oscillation, the learning rate γ should not be set to be too high in practice. 5 4

2

y yd

1.5

y tracks yd

3 1 0

2

0.2

0.4

0.6

0.8

1

1 0 −1 −2 0 6

5

10

15

20

25

30

5

10

15 time(s)

20

25

30

Control input u

4 2 0 −2 −4 −6 0

Fig. 1. Control trajectories under the proposed approach.

6

Conclusions

This paper has successfully developed a simplified adaptive NN control strategy for a general class of uncertain SFNSs. The proposed approach not only simplifies the control structure, but also drastically reduces implementation cost. The contributions of this study include: 1) The complexity growing problem of the traditional ABNNC design is substantially eliminated for a wider 12

class of uncertain strict-feedback nonlinear systems; 2) the constraint of control parameters that guarantees closed-loop stability is clearly demonstrated. An illustrative example has been provided to verify effectiveness of the proposed approach.

Acknowledgement This work was supported in part by the Seed Fund of the Engineering Design and Innovation Center, National University of Singapore under Grant no. R261-503-002-133.

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