Adaptive neural control for switched nonlinear systems with unmodeled dynamics and unknown output hysteresis

Neurocomputing 341 (2019) 107–117

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Adaptive neural control for switched nonlinear systems with unmodeled dynamics and unknown output hysteresis Ziliang Lyu a, Zhi Liu a,∗, Yun Zhang a, C.L. Philip Chen b,c a

School of Automation, Guangdong University of Technology, Guangzhou 510006, PR China College of Navigation, Dalian Maritime University, Dalian 116026, PR China c Unmanned System Research Institute, Northwestern Polytechnical University, Xi’an 710072, PR China b

a r t i c l e

i n f o

Article history: Received 20 January 2019 Revised 11 February 2019 Accepted 28 February 2019 Available online 9 March 2019 Communicated by Mou Chen Keywords: Switched nonlinear system Arbitrary switching Adaptive neural control Unmodeled dynamics Output hysteresis

a b s t r a c t This paper aims at addressing the adaptive neural control problem for switched nonlinear systems with output hysteresis and unmodeled dynamics. The switching law in this study is arbitrary. In our model, the unmodeled dynamics are assumed to be of input-to-state practical stability (ISpS). With the help of this assumption, a dynamic normalizing signal is constructed to dominate the unmodeled dynamics. And then, a direct adaptive neural state-feedback control scheme is developed with the help of approximation-based backstepping. The stability analysis shows that the system output is convergent to an adjustable small region of zero asymptotically, and furthermore, all the closed-loop signals are bounded. Finally, we further present two simulation examples to verify the effectiveness of our control scheme.

1. Introduction Adaptive control has attracted numerous attentions over the past years for its ability to provide online estimate of unknown parameters. Backstepping is one of the most promising adaptive control design approaches [1]. In contrast with other approaches, backstepping has the advantage to avoid cancellations of useful nonlinearities flexibly. With the help of adaptive backstepping, some restrictions (e.g., matching conditions and extended matching conditions, see [2–5]) arising in the early stage of adaptive control are removed. By further combing with the universal approximators, such as neural networks (NNs) and fuzzy logic systems (FLSs), the approximation-based backstepping is able to handle the adaptive control design problems when there is no prior knowledge about the structural information of nonlinear functions. More recently, the progress of adaptive backstepping has witnessed many remarkable results reported [6–14]. Among these results, the adaptive compensation for nonsmooth nonlinearities (e.g. saturation [6], dead-zone [9] and hysteresis [10]) is one of the most popular topics. Hysteresis is common in many physical systems and devices, e.g., electro-magnetic fields, piezoelectric actuators and other areas. The hysteresis nonlinearity may degrade the control per∗

Corresponding author. E-mail address: [email protected] (Z. Liu).

https://doi.org/10.1016/j.neucom.2019.02.057 0925-2312/© 2019 Elsevier B.V. All rights reserved.

© 2019 Elsevier B.V. All rights reserved.

formance. In contrast with dead-zone and saturation, hysteresis is more complicated since its output not only depends on the input, but also the change of input. For this reason, the compensation for hysteresis is more difficult as well. With the joint efforts of numerous researchers, many results on the compensation for hysteresis have been reported over the past year. For example, in [15–18], several mathematical models were developed to describe the hysteresis nonlinearity. Among these models, Bouc-Wen hysteresis model [17,18] may be the most widely accepted one. With Bouc-Wen hysteresis model, further investigations on adaptive compensation for hysteresis nonlinearity have been carried out [10,19–21]. To list a few, in [20], an effective hysteresis compensation method was developed with the idea of decomposing the Bouc-Wen hysteresis into a combination of linear term and disturbance-like term. Moreover, any other adaptive inverse-based compensation methods were also reported in [10]. Note that [10,19–21] only investigated the compensation for input hysteresis not the output hysteresis, while the hysteresis phenomenon may also exist in the output channel of control plants. The output hysteresis will lead to the control problem of time-varying unknown control gain (also referred as unknown high-frequency gain in the field of adaptive control). For this reason, the compensation for output hysteresis is more complicated that those for input hysteresis. Moreover, the control scheme designed by [10,19–21] may suffer from poor control performance, since they need to assume the unknown control gain are time-invariant or slow time-varying.

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So far, there are few results reported on the compensation for output hysteresis. In addition, the aforementioned results are limited in non-switched nonlinear systems and they cannot be applied to the switched nonlinear systems directly. Switched nonlinear system is a special type of hybrid system [22]. A single nonlinear switched system may contain multiple subsystems. During the run time of a switched nonlinear system, there exists a switching law orchestrating which subsystem is active. It is well known that the control or stabilizing problem of switched nonlinear systems is more complex than those of nonswitched nonlinear systems, since it needs to face the dual challenges of control design and switching design. In order to deal with these challenges, some useful tools, e.g. common Lyapunov function (CLF) [23], multiple Lyapunov function (MLF) [24] and average dwell time (ADT) [25] have been developed so far. With the help of these tools, many excellent results on control design or stability analysis of switched nonlinear systems were reported [26–31]. In [26], some ADT-based switching conditions to guarantee input-to-state stability (ISS) properties of switched nonlinear systems are presented. As a major approach to establishing the ISS property of switched interconnected nonlinear systems, some MLFbased small-gain conditions were further given in [27,28]. By using backstepping technique, [29–31] successfully addressed the adaptive control problem for switched nonlinear systems with actuator nonlinearity (e.g., dead-zone, saturation and hysteresis). However, these works are restricted since they rely on the assumption that the control plant is free from unmodeled dynamics. Unmodeled dynamics are widespread in many practical devices owing to the existence of measurement noise and modeling errors, etc. The closed-loop system may perform poorly and even become instable if the control scheme without consideration of unmodeled dynamics [32]. So far, there are mainly two approaches to dealing with unmodeled dynamics. The first approach is to dominate the unmodeled dynamics with some designed dynamic signals [33]. And the other approach is to exert small-gain constraints on the nonlinear input-to-output gains (definition see [34]) of the control plant and the unmodeled dynamics, such that, the composition of such nonlinear gains is less than unity [35]. Following these principles, many results have been reported on switched systems [36–41] or non-switched systems [42–47] with unmodeled dynamics. For example, by combining MLF and ADT, the authors in [38] addressed adaptive control problems with the first approach mentioned above. However, the conrol scheme in [38] may be infeasible when there is no priori information about the switching law. More recently, any other novel adaptive control schemes were further proposed in [39,40] by using small-gain and CLF approaches. The advantage of [39,40] is that their approaches allow the switching law is arbitrary. However, the implementation of these small-gain-based approaches may be not an easy task when the switched nonlinear systems simultaneously suffer from unmodeled dynamics and output hysteresis. The main reason is that the nonlinear input-to-output gains are difficult to determine when the system output is coupled with output hysteresis. Therefore, the adaptive control for switched nonlinear systems with unmodeled dynamics and unknown output hysteresis remains an open problem. Motivated by the discussion above, this study investigates the adaptive neural control problem for switched nonlinear systems with unmodeled dynamics and unknown output hysteresis. The contributions of this study are listed as follows: 1. The proposed control scheme can address the adaptive control problem for switched nonlinear systems with unmodeled dynamics and unknown output hysteresis. A dynamics signal is employed in the control scheme to dominate the unmodeled dynamics. In contrast with the existing results in [39,40], the

Controller 1

Subsystem 1

Controller 2

Subsystem 2

Controller M

Subsystem M

Switched Controller

Control Plant

Fig. 1. Switched nonlinear system with output hysteresis.

main difference is that our control scheme can avoid the determination of nonlinear input-to-output gains, which are difficult to determine if the control plant suffers from output hysteresis. 2. Different from the previous results on switched nonlinear system with dead-zone or saturation [29,30], this study consider the more complicated hysteresis nonlinearity. In contrast with [31], where the adaptive control for switched nonlinear systems with input hysteresis has been investigated, this study will consider the output hysteresis. The compensation for output hysteresis is more difficult than the input hysteresis in [31] since the unknown control gain aroused by the output hysteresis is time-varying. To overcome this control difficulty, a Nussbaum function-type gain is employed in the control design. The rest of this paper is organized as follows. The control problem and preliminaries are introduced in Section 2. And then, an adaptive neural state-feedback control scheme is proposed in Section 3 with the help of approximation-based backstepping. Besides, the corresponding stability analysis is also given in this section to prove that the control objective is achieved. Finally, two simulation examples are given in Section 4 to verify the effectiveness of our scheme. Terminologies: A function γ : R+ → R+ is said to be of class K if it is continuous, strictly increasing and is zero at zero. We say that a K-function γ is of class K∞ , if it is unbounded. A KL-function β : R+ × R+ → R+ is a function with the property that for each fixed t, the function β (·, t) is of class K and, for each fixed s, the function β (s, ·) is decreasing and tends to zero at infinity. Notations: Throughout this paper, ◦ denotes the composition operator between two functions; |·| and · denote the absolute value of real numbers and the Euclidean norm of vectors, respectively; R and R+ denote the field of real numbers and the field of nonnegative-real numbers, respectively. 2. Problem formulation and preliminaries 2.1. Problem formulation As shown in Fig. 1, let us consider the switched nonlinear system with unmodeled dynamics and unknown output hysteresis as follows:

z˙ = qσ (t ) (y, z )

x˙ i = xi+1 + fi,σ (t ) (x¯i ) + i,σ (t ) (y, z )

i = 1, 2, . . . , n − 1

x˙ n = uσ (t ) + fn,σ (t ) (x¯n ) + n,σ (t ) (y, z ) y = H ( x1 )

(1)

where σ (t ) : [0, ∞ ) →  = {1, 2, . . . , M} is the right continuous switching law with σ (t ) = k denoting the kth subsystem is active. For i = 1, 2, . . . , n and k = 1, 2, . . . , M, uk , y and x¯i = [x1 , x2 , . . . , xi ]T ∈ Ri are the system inputs, system output and state vectors; fi,k (x¯i ) and qk (y, z) are the smooth nonlinear functions; z˙ = qσ (t ) (y, z ) ∈ R and i,k (y, z) are the so-called unmodeled

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109

dynamics and dynamic disturbances, respectively. In this study, x2 , . . . , xn are measurable, and x1 and z are unmeasurable. In addition, the output y is constrained by the hysteresis nonlinearity formulated as

y = H ( x 1 ) = ν1 x 1 + ν2 ζ

(2)

where ν 1 , ν 2 are nonzero constants with the same signs and ζ is the solution of

ζ˙ = x˙ 1 ϑ (ζ , x˙ 1 ), ζ (t0 ) = 0

Output Layer

(3) Input Layer

where

ϑ (ζ , x˙ 1 ) = 1 − sign(x˙ 1 )β|ζ |m−1 ζ − χ |ζ |m .

(4)

Here, β > |χ | is a positive constant determining the hysteresis’s shape and amplitude, while m ≥ 1 is another positive constant determining the smoothness from initial slope to asymptote’s slope. And furthermore, according to [10, Lemma 1], ζ is bounded:



|ζ (t )| ≤ ζ¯ =

m

1

β +χ

.

(5)

2.2. Control objective and assumptions

Hidden Layer Fig. 2. Structure of RBFNN.

2.3. Preliminaries Lemma 1 [50]. Let θ (s), V(s) ≥ 0 be the smooth functions defined on [0, tf ) and N(s) be the smooth Nussbaum-type function satisfying

 1 s N ( ξ )d ξ = +∞, s→+∞ s 0  s 1 lim inf N (ξ )dξ = −∞. s→+∞ s 0 lim sup

The control objective of this study is to develop an adaptive neural control scheme to guarantee (i) the system output is convergent to an adjustable small region of zero asymptotically; and (ii) all the closed-loop signals are bounded. In order to achieve the control objective, we need to present some standard assumptions as follows: Assumption 1 [33,38,43–47]. The dynamic disturbances satisfy the following inequality

For ∀t ∈ [0, tf ), the smooth functions θ (t), V(t) and 1]θ˙ ds is bounded, if

V (t ) ≤ c0 + e−c1 t



0

t

[g(s )N (θ ) + 1]θ˙ ec1 s ds,

t

0 [ g( s ) N ( θ )

∀t ∈ [0, t f )

+

(9)

(6)

where c0 and c1 are positive constants; g(s ) ∈ [l − , 0 ) ∪ (0, l + ] is a time-varying parameter.

where ψ i,1 (s) and ψ i,2 (s) are unknown nonnegative smooth functions.

Lemma 2 [51]. For any x ∈ R, there exists a positive constant ς such that

|i,k (y, z )| ≤ ψi,1 (|y| ) + ψi,2 (|z| )

Assumption 2 [33,38,43–47]. The unmodeled z-dynamics are ISpS and there exist a common exp-ISpS Lyapunov function Vz (z) for z˙ = qσ (t ) (y, z ) such that

α z (|z| ) ≤ Vz (z ) ≤ α¯ z (|z| ), ∂ Vz q (y, z ) ≤ −λzVz (z ) + γz (|y| ) + dz ∂z k

x ς

+ ς,

 = 0.2785.

(7)

Assumption 3 [7,8]. There exist strictly increasing smooth functions i,k (· ) : R+ → R+ with i,k (0 ) = 0 such that

(8)

Remark 1. Assumption 1 means that the dynamic disturbances i,k are bounded by the sum of ψ i,1 (|y|) and ψ i,2 (|z|). Note that ψ i,1 (s) and ψ i,2 (s) are of class K∞ . It means that i,k is unbounded if y and z are unbounded. Therefore, Assumption 1 is less restricted than [48,49], where the external disturbances are assumed to be bounded. Assumption 2 means that the unmodeled z-dynamics are ISpS. With this assumption, we can construct a dynamic normalizing signal to deal with the unmodeled dynamics. In addition, since x1 is unavailable, Assumption 3 is given to solve this problem. This assumption commonly adopted in the nonlinear systems, for example, see [7,8]. So far, how to remove Assumption 3 remains an challenging question. And our future research will focus on this question.

(10)

Lemma 3 [52]. For any K∞ -function γ and any a, b ∈ R+ , we have

γ ( a + b ) ≤ γ ( 2a ) + γ ( 2b ).

where α z , α¯ z and γ z are K∞ -functions and dz is a known positive constant.

| fi,k (x¯i )| ≤ i,k (x¯i  ).

|x| ≤ x tanh

(11)

2.4. Neural network approximation In control engineering, it is well-known that the radial basis function neural networks (RBFNNs) have excellent approximation performances in modeling nonlinear functions. In this study, we will employ the RBFNNs to approximate the unknown nonlinear system functions. As shown in Fig. 2, the RBFNN generally consists of three layers, i.e., the input layer, the hidden layer and the output layer. In this study, the RBFNN is described as

fnn (X ) = W T φ (X )

(12)

where W = [w1 , w2 , . . . , wL ]T represents the ideal constant weight vector; φ (X ) = [φ1 (X ), φ2 (X ), . . . , φL (X )]T is the basis function vector with φ i (x) being of Gaussian form:



 (X − κ )T (X − κ ) φ (X ) = exp − ,  = 1, . . . , L β2

(13)

where κ ∈ Rn is the center of respective field, β ∈ R is width of the Gaussian function. With the RBFNN above, we have the following lemma.

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Lemma 4 [53]. Let fˆ(X ) : Rn → R be a continuous function defined on a compact set . For any given constant ε > 0 and sufficiently large node number L, there exist a RBFNN WT φ (X) such that

sup | fˆ(X ) − W T φ (X )| ≤ ε . X∈

(14)

3. Adaptive control design and stability analysis 3.1. Adaptive control design In this section, we will design the adaptive control scheme via approximation-based backstepping. Before the the backstepping design procedure, we need to introduce the following coordinate transformations:

ξ1 = y, ξi = xi − πi−1 , i = 2, . . . , n

(15)

where πi−1 are the virtual control laws. In addition, we define xn+1 = uk = πn , zn+1 = xn+1 − πn . Clearly, zn+1 = 0. In order to dominate the unmodeled z-dynamics, we need to introduced a dynamic signal as in [33]:

r˙ = −λ∗z r + y2 γ¯ z (|y| ) + dz

(16)

where λ∗z ∈ (0, λz ) is a designed parameter, dz is defined in (7) and γ¯ z (s ) is a designed nonnegative smooth function with s2 γ¯ z (s ) ≥ γ ( s ). For Step 1 and Step i (i = 2, . . . , n) of the backstepping design procedure, the common virtual laws are designed as

π1 = −N (θ )π¯ 1 ,

(17) Fig. 3. Block diagram for the proposed control scheme.

π¯ 1 = −(k1 + 0.5 )ξ1 − πi = − ( k i + 0 . 5 ) ξ i −

ηˆ 1 ξ1 φ1 (X1 )2 , 2h ¯ 21

ηˆ i ξi φi (Xi )2 2h ¯ 2i

(18)

(19)

where X1 = [y, r]T , Xi = [y, x2 , . . . , xi , ηˆ i , . . . , ηˆ i−1 , θ , r]T . Here, N(θ ) is the Nussbaum-type function specified as N(θ ) := θ 2 cos (θ ) and θ is updated by the adaptive law designed as

θ˙ = −ρ π¯ 1 ξ1 .

(20)

For i = 1, 2, . . . , n, ηˆ i are the estimates of ηi := max k ∈  Wi,k 2 , T φ (X ) specified in next secwhere Wi,k is the weight of RBFNN Wi,k i i tion. We design the adaptive laws for ηˆ i as

ηˆ˙ i =

ri 2 ξi φi (Xi )2 − r¯i ηˆ i . 2h ¯ 2i

In (17)-(21), ki ,

The adaptive control scheme with control law uk = αn , common virtual laws (17), (19) and adaptive laws (20), (21) can guarantee that: i) the system output is convergent to an adjustable small region of zero asymptotically; and ii) all the closed-loop signals are bounded. Proof. Step 1. Consider the following CLF candidate

V1 = r +

1 2 1 2 ξ + η˜ . 2 1 2r1 1

(22)

The time derivative of V1 is given by

V˙ 1 = r˙ + ξ1 ξ˙1 −

1 η˜ 1 ηˆ˙ 1 . r1

(23)

With (2) and (3), we have

(21)

i , ρ , ri and r¯i are designed parameters.



ξ˙1 = μ(t ) ξ2 + π1 + f1,k (x¯1 ) + 1,k (y, z ) where

Remark 2. The configuration of the aforementioned adaptive neural control scheme is shown in Fig. 3. The detailed implementation of the proposed adaptive control scheme can be listed as follows:

μ(t ) = ν1 + ν2 ϑ (ζ , x˙ 1 ).

1. Select appropriate centers of respective fields and widths of T φ ( X ). Gaussian functions, and construct the RBFNNs Wi,k i i 2. Choose appropriate values for the designed parameters λ∗z , ki , i , ρ , ri and r¯i , and determine the common virtual control laws (17), (19) and the adaptive laws (20), (21). 3. Determine the switched controller with uk = πn .

ϑ (ζ , x˙ 1 ) ≥ 1 − (β + χ )|ζ |m

3.2. Stability analysis In this subsection, we summarize the main result of this study with the following theorem. Theorem 1. Consider the switched nonlinear system (1) with unmodeled dynamics and unknown output hysteresis under Assumptions 1-3.

(24)

(25)

With the definition of ϑ (ζ , x˙ 1 ), we obtain that

≥ 1 − (β + χ )|ζ¯ |m = 0

(26)

where ζ¯ is the upper bound of ζ specified in (5). According to the fact that ν 1 and ν 2 are nonzero constants with the same signs, we can examine the following two cases. Case 1 (ν 1 , ν 2 > 0). Since 0 ≤ ϑ (ζ , x˙ 1 ) ≤ 1 + β|ζ |m + |χ| · |ζ |m , we have

ν1 ≤ μ(t ) ≤ ν1 + ν2 (1 + β|ζ |m + |χ | · |ζ |m )  β + |χ | ≤ ν1 + ν2 1 + . β +χ

(27)

Z. Lyu, Z. Liu and Y. Zhang et al. / Neurocomputing 341 (2019) 107–117

Case 2 (ν 1 , ν 2 < 0). Since ν 1 , ν 2 are negative, with the similar analysis of Case 1, it is not difficult to deduce



ν1 + ν2

β + |χ | 1+ β +χ



≤ μ(t ) ≤ ν1 .

Define

(28)

β + |χ | μ = |ν1 |, μ¯ = |ν1 | + |ν2 | 1 + . β +χ

1

ρ

(μN (θ ) + 1 )θ˙ .

(29)

Note that x¯i  ≤ ij=1 |x j |. Together with Young’s inequality and Assumption 3, we can find a positive constant c1 such that

μ¯ 2 ξ12 f12,k (x¯1 )

μξ1 f1,k (x¯1 ) ≤

4 c1

μ¯ 2 ξ12 12,k

≤



+ c1

+ c1 .

4 c1

(30)

Vz (z(t )) ≤ Vz (z0 )e

+e

−λz t



t

(γz (|y(s )| ) + dz )eλz s ds

t0

∗



t

t0

(31)

(r˙ (s ) + λ∗z r (s ) )eλz s ds ∗

= r (t ) + Dr (t0 , t )

(32) ∗ e−λz (t−t0 ) }

)e−λz (t−t0 )

where Dr (t0 , t ) = max{0, Vz (z0 − r0 and r0 is the initial value of r(t). Since 0 < λ∗z < λz , we can find a finite T0 (λ∗z , r0 , z0 ) such that Dr (t0 , t ) = 0, ∀t ≥ t0 + T0 . Here, T0 can be V (z )

chosen as T0 := λ −1 λ∗ ln z r 0 . Therefore, with the help of Young’s z 0 z inequality, Lemma 3 and the fact that α z (|z| ) ≤ Vz (z ) ≤ α¯ z (|z| ), we obtain



μξ1 1,k ≤ μ| ¯ ξ1 | ψ11 (|y| ) + ψ12 (|z| )

≤ μ| ¯ ξ1 | ψ11 (|y| ) + ψ12 ◦ α −1 z ( 2r ) + μ| ¯ ξ1 |ψ12 ◦ α −1 z ( 2 Dr ) ≤



μ¯ ξ ψ11 (|y| ) + ψ12 ◦ α (2r ) 2

+

2 1

μ¯ 2 ξ12 4b1

−1 z

+ b1



4a1

ψ12 ◦ α −1 z ( 2 Dr )

V˙ 1 = −λ∗z r + μξ1 ξ2 + ξ1 (π¯ 1 + fˆ1,k ) +

2

(33)

1

ρ

¯ + νν21ζ )

4 c1

+

μ¯ ξ1 ψi,1 (|y| ) + ψ12 ◦ α (2r ) 2

+



μ¯ ξ1  (

−1 z

4a1

ξ 2 ε2 h ¯ 21 + 1 + 1. 2 2 2

(35)

μN (θ ) + 1 θ˙

1

ρ

ε2 h r¯1 ¯2 2 + η˜ 1 ηˆ 1 + 1 + 1 + dz + a1 + b1 (ψ12 ◦ α −1 z (2Dr )) + c1 . r1 2 2 (36) With Lemma 2, we can obtain



ξ1 ξ 2 μξ1 ξ2 ≤ μξ ¯ 1 ξ2 tanh ς1



+ μ ¯ ς1

(37)

4b1

2

.

r¯1 2 1 η˜ + μN (θ ) + 1 θ˙ 2r1 1 ρ ξ1 ξ 2 ς1



+ d1

(38)



where d1 = r¯1 η12 /(2r1 ) + h ¯ ς1 + ε12 /2 + dz + a1 + b1 ψ12 ◦ ¯ 21 /2 + μ

α −1 z ( 2Dr )

2

+ c1 .

Remark 3. Note that both input and output hysteresis will lead to the unknown control gain problem. In contrast with the input hysteresis in [10,19–21,31], the output hysteresis considered in this study is more complicated since the unknown control gain μ(t) aroused by the output hysteresis is time-varying. The time-varying property of μ(t) makes the approaches designed for input hysteresis compensation [10,19–21,31] infeasible. And there are few results reported on this topic so far. The Nussbaum-type gain technique is a promising method to deal with this problem. However, the condition of Nussbaum-type gain technique is hard to satisfy. In this study, we prove that μ(t ) ∈ [−μ ¯ , − μ ) ∪ ( μ, μ ¯ ], such that, the Nussbaum-type gain technique can be employed to deal with the time-varying unknown control problem aroused by the output hysteresis.

1 2 1 ξ + η˜ i2 . 2 i 2ri

(39)

The time derivative of Vi is given by

≤ −λ∗z r −

μN (θ ) + 1 θ˙

μ¯ 2 ξ1

V˙ 1 ≤ −λ∗z r − k1 ξ12 −

1 V˙ i = V˙ i−1 + ξi ξ˙i − η˜ i ηˆ˙ i ri



where the nonlinear packaged function fˆ1,k is specified as

fˆ1,k = yγ¯ z (|y| ) +

ξ12 φ1 2 +

V˙ 1 ≤ −λ∗z r − k1 ξ12 + μξ1 ξ2 +

Vi = Vi−1 +

+ a1



2 1 − η˜ 1 ηˆ˙ 1 + dz + a1 + b1 ψ12 ◦ α −1 + c1 . z ( 2 Dr ) r1 | y| 2 1,k ν1

2h ¯ 21

Step i (2 ≤ i ≤ n). Consider the following CLF candidate

2

where a1 and b1 are positive constants. Substitute (16), (29)–(33) into (23) and we arrive at

2

η1

+ μξ ¯ 1 ξ2 tanh

where z0 is the initial value of z(t). Substitute (16) into (31), and we have

Vz (z(t )) ≤ Vz (z0 )e−λz (t−t0 ) + e−λz t

With the help of Young’s inequality, it follows that



By using Gronwall’s lemma, we can deduce from (7) that −λz (t−t0 )

|δ1,k (X1 )| ≤ ε1 .

where ς 1 is a positive designed parameter. Together with the inequality η˜ 1 ηˆ 1 ≤ −η˜ 12 /2 + η12 /2, (36) becomes



| y| ν2 ζ¯ |ν1 | + ν1

fˆ1,k = W1T,k φ1 (X1 ) + δ1,k (X1 ),

Substitute (18), (21), (35) into (34), and we have

By combining Case 1 and Case 2 above, we obtain μ(t ) ∈ [ −μ ¯ , − μ ) ∪ ( μ, μ ¯ ], which means that we can invoke Lemma 1 to address the time-varying unknown control gain problem aroused by μ(t). Therefore, with (17), (18), (20), we have

μξ1 π1 = ξ1 π¯ 1 +

With Lemma 4, fˆ1,k can be approximated as

ξ1 fˆ1,k ≤





111

 (34)

+ ξi

j=1

k j ξ j2 −

i−1  r¯ j 2 1 η˜ + (μN (θ ) + 1 )θ˙ 2r j j ρ j=1

¯ i,k − ξi+1 + πi + fi,k (x¯i ) + 

∂πi−1 ∂πi−1 ˙ r˙ − θ ∂r ∂θ

i−1  ∂πi−1 ˙ ∂πi−1 ∂π ∂π ηˆ j − μ i−1 x2 − μ 1 f1,k (x¯1 ) − x ∂ y ∂ y ∂ x j j+1 ∂ η ˆj j=1 j=2  i−1  1 ∂πi−1 − f (x¯ j ) + ξi−1 − η˜ i ηˆ˙ i + di−1 (40) ∂ x j j,k ri j=2

−

i−1 

i−1 

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where

+

i−1  ∂πi−1 ∂πi−1 1,k −  , ∂y ∂ x j j,k j=2  ⎧ ξ1 ξ 2 ⎨μξ ¯ tanh , if i = 2, 1 ς ξi −1 = ⎩ ξi−1 , if i > 2.

¯ i,k = i,k − μ 

ξi fi,k (x¯i ) ≤

4 ci



2 i,k

 i |y| ν2 ζ¯  + + |x j | + ci , | ν1 | ν1 j=2

i−1  ∂πi−1 ∂πi−1 f1,k (x¯1 ) − ξi f (x¯ ) ∂y ∂ x j j,k j j=2    ξi2 ∂πi−1 2 2 |y| ν2 ζ¯ ≤ μ¯ 2 1,k + 4 ci ∂y | ν1 | ν1    j i−1  ∂πi−1 2 2 |y| ν2 ζ¯  +  j,k + + |x p | + ( i − 1 )ci , ∂xj | ν1 | ν1 p=2 j=2

(42)

ξi2

¯ i,k ≤ ξi 

×

μ¯ 2

4ai

∂πi−1 ∂y

2

+

i−1 

ψi1 (|y| ) + ψi2 ◦ α −1 z ( 2r )

ξi2



 μ¯

4bi

+ iai + bi

2

i 

∂πi−1 ∂y

2

+



j=2

i  j=1

+



i−1 



j=2

∂πi−1 ∂xj



2

T fˆi,k = Wi,k φi (Xi ) + δi,k (Xi ),

ξi fˆi,k ≤



+ bi

k j ξ j2 −

j=1





ψ j2 ◦ α −1 z (2Dr ) + ci + (i − 1 )c¯i

i 

where

2



C = min

i=1,...,n n 



i=1

+ bi

+1

ξ 2 ε2 h ¯ 2i + i + i . 2 2 2

(47)

i 

r¯ j 2 1 η˜ + μN (θ ) + 1 θ˙ + ξi ξi+1 + di 2r j j ρ j=1

εi2 /2 +

h ¯ 2i /2 +

i

n 

k j ξ j2 −

j=1

μN (θ ) + 1 θ˙ + D

1

ρ

n 

r¯ j 2 1 η˜ + μN (θ ) + 1 θ˙ + dn , 2r j j ρ

(49)

 λ∗z , 2ki , r¯i ,

ε2 ¯2 r¯i 2 h ηi + i + i + μ ¯ ςi + iai + ici 2ri 2 2

i 



ψ j2 ◦ α −1 z ( 2 Dr ) .

j=1

Integrate (49) over [0, t] and we have

(45) Vn (t ) ≤ Vn (0 )e−Ct +



ψi1 (|y| ) + ψi2 ◦ α −1 z ( 2r )

j=1

k j ξ j2 −

j=1

D=

r¯ j 2 1 η˜ + μN (θ ) + 1 θ˙ 2r j j ρ

2 i,k

2

|δi,k (Xi )| ≤ εi .

r¯i ηi2 /(2ri )

(43)

i |y| ν2 ζ¯  fˆi,k =  + + |x j | 4 ci | ν1 | ν1 j=2     i−1  ξi ∂πi−1 2 2 ∂πi−1 2 + μ¯ + +1 4ai ∂y ∂xj j=2

×

i 

V˙ n ≤ −λ∗z r −

j=1

ξi





to i = n, we finally arrive at

+1

ri



j=2

∂πi−1 ∂xj

di = di−1 + + μ ¯ ςi + iai + bi j=1 ψ j2 ◦ α −1 z (2Dr ) + ici . Note that ξi ξi+1 = 0, and therefore, by iterating (48) form i = 2

1 ξi+1 + πi + fˆi,k − η˜ i ηˆ˙ i + di−1 + iai

i 



Substitute (19), (21), (47) into (45) and we can obtain

where ai , bi , ci and ς i are positive constants. By substituting (41)– (44) into (40), it follows:

+ ξi

ξi2 φi 2 +

≤ −CVn +

i−1 

+

i−1 

(48)

 ∂πi−1 ∂πi−1 ξi ∂πi−1 −μξi x ≤ μξ ¯ i x tanh · x + μ ¯ ςi (44) ∂y 2 ∂y 2 ςi ∂ y 2

j=1

2h ¯ 2i

j=1

j=1

V˙ i ≤ −λ∗z r −

ηi

where

2

2

Similar with (35), we have

V˙ i ≤ −λ∗z r−

2

∂πi−1 ∂xj



where fˆi,k is specified in (46). With Lemma 4, fˆi,k can be approximated as

+1

ψ j2 ◦ α −1 z ( 2 Dr ) ,

i−1 

μ¯

4bi

∂πi−1 ∂y

i−1 i−1 ∂πi−1 ∂πi−1 ˙  ∂πi−1 ˙  ∂πi−1 r˙ − θ− ηˆ j − x ∂r ∂θ ∂ x j j+1 ∂ η ˆj j=1 j=2    i−1   ∂πi−1 ξi ∂πi−1 2 2 |y| ν2 ζ¯ 2 + μ¯ 2 1,k + + 4 ci ∂y | ν1 | ν1 ∂ xj j=2   j |y| ν2 ζ¯  ×  2j,k + + |x p | | ν1 | ν1 p=2  ∂πi−1 ξi ∂πi−1 +μ ¯ x2 tanh · x2 + ξi −1 . (46) ∂y ςi ∂ y

(41)

−μξi



 2

−

With the similar derivation process of (29), (33), we obtain

ξi2



ξi

D e−Ct (1 − e−Ct ) + C D



t 0



μN (θ ) + 1 θ˙ eCs ds. (50)

Clearly, (50) matches the form of (9). Therefore, we can cont clude with Lemma 1 that Vn (t), θ (t) and 0 μN (θ ) + 1 θ˙ ds

t are bounded. And the boundedness of 0 μN (θ ) + 1 θ˙ ds means

t that μN (θ ) + 1 θ˙ e−C (t−s) ds is bounded as well. Similarly, the 0

boundedness of Vn (t) means that r, ξ i and η˜ i are bounded. And note that ηi are constant, which means that ηˆ i are bounded. Retrospect to (17)–(19) and it is not difficult to deduce that π i are bounded.

Z. Lyu, Z. Liu and Y. Zhang et al. / Neurocomputing 341 (2019) 107–117

Define



P = sup t≥0

0

4. Simulation results t



μN (θ ) + 1 θ˙ e−C (t−s) ds

(51)

and we can obtain from (50):

Vn (t ) ≤ Vn (0 )e

−Ct

D P + (1 − e−Ct ) + C D

(52)

which means that

|y| ≤



113

2V (t ) ≤

 



D −Ct 2D 2P 2 Vn (0 ) − e + + . C C D

Therefore, y is convergent to an adjustable small region of zero asymptotically and the proof is completed.  Remark 4. For the stability analysis above, we can see that both the initial condition and the choice of designed parameters will influence the control performance: • Firstly, the transient performance is an explicit function of the initial value of η˜ i (t ). Therefore, we can obtain the better the transient performance if the initial estimates ηˆ i (t0 ) is closer to its true value ηi (t0 ). • Secondly, the convergence rate depends on λ∗z , ki and r¯i . The larger value of λ∗z , ki and r¯i will lead to the faster convergence rate. However, it is worth mentioning that the choice of these designed parameters should be carefully, since an inappropriate choice may lead to poor control performances, large amplitudes of control signal and control energy waste, etc. Therefore, the designed parameters should be chosen according to the available control energy and the control requirement. Remark 5. Although some adaptive control schemes have been developed in [36–38] for switched nonlinear systems with unmodeled dynamics, they all need to exert some ADT constraints on the switching law. With the help of CLF, the switching law in this study is allowed to be arbitrary. Therefore, our approach is still feasible when there is no priori information about the switching law. Remark 6. Unmodeled dynamics may affect the control performance and stability of the closed-loop system. There are mainly two approaches addressing this problem, i.e., the small-gain-based and the dynamic-signal-based approaches. One advantage of the dynamic-signal-based approaches is that they can assist the control performance, although the small-gain-based approaches require less information of the unmodeled dynamics. So far, there are many adaptive control results reported by using the dynamic-signal-based approaches [33,38,41,43–47]. However, the main restriction of these approaches is that they are limited in non-switched system and have not considered the influence of the switchings. And this restriction motivates us to carry out the investigation in this paper. Remark 7. For switched nonlinear systems with arbitrary switchings and unmodeled dynamics, some adaptive control results have been reported in [39,40] more recently. The main idea of these approaches can be divided into three steps: i) determine the nonlinear input-to-output gains of the control plant and the unmodeled dynamics, respectively; ii) exert some constraints on these nonlinear input-to-output gains such that the small-gain condition is satisfied; and iii) establish ISS or ISpS properties of the whole switched nonlinear system with the small-gain theorems. However, the determination of nonlinear input-to-output gains is difficult if the system output is coupled with output hysteresis. In this study, thanks to the constructed dynamic signal in (16), the unmodeled dynamics are successfully dominated, and therefore, we can avoid the determination of nonlinear input-to-output gains.

In this section, two simulation examples will be presented to further verify the effectiveness of our control scheme. Example 1 (Numerical Example). Consider the second-order switched nonlinear system as follows:

z˙ = qσ (t ) (y, z )

x˙ 1 = x2 + f1,σ (t ) (x¯1 ) + 1,σ (t ) (y, z )

x˙ 2 = uσ (t ) + f2,σ (t ) (x¯2 ) + 2,σ (t ) (y, z ) y = H ( x1 )

(53)

where q1 (y, z ) = −z + y2 + 0.5, f1,1 (x1 ) = x21 sin(x1 ), f2,1 (x1 , x2 ) = x21 cos(x2 ) + x21 , 1,1 (y, z ) = z2 , 2,1 (y, z ) = 2z cos(y ); q2 (y, z ) = −z + 0.5y2 + 1, f1,2 (x1 ) = x21 cos(x1 ), f2,2 (x1 , x2 ) = x21 sin(x2 ) + x21 ; 1,2 (y, z ) = z2 sin(y ), 2,2 (y, z ) = z cos(y ). The parameters of H(x1 ) are chosen as: ν1 = ν2 = 1, β = 5, χ = 3.5, m = 2. Choose Vz = z2 and it is not difficult to verify that

∂ Vz q (y, z ) ≤ −0.75z2 + y4 + 1, k = 1, 2. ∂z k Therefore, we can determine α z (s ) = 0.5s2 , α¯ z (s ) = 1.5s2 , λz = 0.75, γz (s ) = s4 and dz = 1, such that, Assumption 2 is satisfied. Select γ¯ z = s2 , λ∗z = 0.5 ∈ (0, λz ) and determine the dynamic signal as

r˙ = −0.5r + y4 + 1. With Theorem 1, the adaptive control scheme can be established as

π1 = −N (θ )π¯ 1 , π¯ 1 = −(k1 + 0.5 )ξ1 −

ηˆ 1 ξ1 φ1 (X1 )2 , 2h ¯ 21

uσ (t ) = −(k2 + 0.5 )ξ2 −

ηˆ 2

2h ¯ 22

ξ2 φ2 (X2 )2 ,

r1 2 ξ1 φ1 2 − r¯1 ηˆ 1 , 2h ¯ 21 r ηˆ˙ 2 = 22 ξ22 φ2 2 − r¯2 ηˆ 2 , 2h ¯2

ηˆ˙ 1 =

θ˙ = −ρ π¯ 1 ξ1 . In order to investigate the influence of different parameters on the control performance, the simulation are divided into two cases. In the first case, k1 = 2, k2 = 1, h ¯ 1 = 10, h ¯ 2 = 1, r1 = 4, r2 = 2, r¯1 = r¯2 = 1 and ρ = 1. In the second case, k1 = 1, k2 = 0. 5, h ¯ 1 = 5, h ¯ 2 = 0.5, r1 = 4, r2 = 2, r¯1 = r¯2 = 1 and ρ = 1. In this numerical example, The RBFNN W1T,k φ1 (X1 ) contains 22 nodes. For  = 1, 2, . . . , 11, the centers κ 1,1, and κ 1,2, are spaced evenly in the interval [−1.0, 1.0] × [0.0, 2.0], and the widths are set as: β1,1, = β1,2, = 0.1. In addition, W2T,k φ2 (X2 ) contains 55 nodes. For  = 1, 2, . . . , 11, the centers κ 2,1, , κ 2,2, , κ 2,3, , κ 2,4, and κ 2,5, are spaced evenly in the interval [−1.0, 1.0] × [−12.0, 6.0] × [−1.0, 1.0] × [−0.5, 4.5] × [0.0, 2.0], and the widths are set as: β2,1, = β2,3, = β2,5, = 0.1, β2,2, = 1.0 and β2,4, = 0.3. Finally, the system is initialized as: x1 (0 ) = 0.2, x2 (0 ) = 0.1, r (0 ) = ηˆ 1 (0 ) = ηˆ 2 (0 ) = 0 and θ (0 ) = 0.1. Select the switching law as in Fig. 4 and the simulation results are shown in Figs. 5–8. Compare the time interval 4.0s-6.0s and 10.0s-12.0s, we can see that the closed-loop signals chatter more violently in the interval 10.0 s–12.0 s. The reason is that the switching in 10.0s-12.0s is faster, and the switched controller needs to change its control input so as to overcome the influence of switching, which leads to the change of the closed-loop signals as well. From Fig. 5 we can see that, y converges to a small region

114

Z. Lyu, Z. Liu and Y. Zhang et al. / Neurocomputing 341 (2019) 107–117

2.2

150

2

100

1.8

Case 1: Case 2:

50

1.6

0

1.4

10

-50

1.2

0 -10

1

-100

0.8 0

-150 0

-20 6

2

4

6

8

10 12 Time(sec)

14

16

18

20

2

7 Time(sec)

6

7.5

8

8

10 12 Time(sec)

14

16

18

20

Fig. 8. Responses of the control law and the virtual control laws.

Fig. 4. Switching law.

0.45

Case 1:

0.4

Case 2:

L

R3

1.5 1

6.5

4

M0, R0

0.35 0.3 6

6.5

7

7.5

8

m, L0

+

0.5

v0 -

0 0.15 0.05 0 -0.05 6

-1 0

2

τ

R1

0.1

-0.5

6.5

4

7

6

7.5

8

R2

8

10 12 Time(sec)

14

16

18

B Vref tri(t,T)

+

20

G

J

Kτ, KB

-

Fig. 5. Responses of x1 (t) and y(t). Fig. 9. Visualization of electromechanical system.

10 Case 1: Case 2:

5 0 -5

0.9

0

0.85

-0.5

0.8

-10

-1

0.75 0.7 2

-15 0

2.5

2

3

4

3.5

6

4

8

-1.5 6

6.5

10 12 Time(sec)

7

7.5

14

8

16

18

20

Fig. 6. Responses of x2 (t) and z(t).

6

Case 1: Case 2:

5

Remark 8. Note that the RBFNNs used in this study are of static structure, that is, only the weights update online, while the centers and the widths of Gaussian do not update. Therefore, the selection of these parameters significantly influences the approximation performance of the RBFNNs and the control performance of the designed control scheme. The Ref. [53] prove that the RBFNNs can uniformly approximate sufficiently smooth functions on closed, bounded subsets, if the centers and widths are chosen on a regular lattice in the respective compact sets. With these constraints, there are mainly two approaches to choosing the parameters of RBFNNs. The first way is to determine the parameters by using the expert experiences and the experiment from practical engineering systems. And the other way is to search the optimal parameters for the RBFNN by using some optimization algorithms, e.g., ant colony optimization [54] and genetic algorithm [55]. Example 2 (Application Example). As shown in Fig. 9, in this example, let us consider the following electromechanical system:

4 3

Mq¨ + Bq˙ + N sin(q ) = I

2

LI˙ = Vε − RI − KB q˙

1

R = σ (t )R1 + 1 − σ (t ) R2 + R3

0 0



2

4

6

8

10 12 Time(sec)

14

16

18

20

Fig. 7. Responses of the adaptive laws and the dynamic signal.

of zero asymptotically, although x1 deviates from zero significantly. It means that the output hysteresis is compensated successfully. In addition, we can see that the output of case 1 converges faster than case 2, which means that the statement in Remark 4 is correct. Furthermore, from Figs. 5–8, we can see that all the closedloop signals are bounded. Therefore, the effectiveness of our control scheme is verified.

where B0 Kτ

N=

mL0 G 2Kτ



+

M0 L0 G Kτ ,

M=

(54) J Kτ

+

mL20 3Kτ

+

M0 L20 Kτ

+

2M0 R20 5Kτ

and

B= ; Vε is the input control voltage, q(t) denotes the angular motor position and I(t) denotes the motor armature. In addition, the switching law σ (t) in (54) is generated by the Pulse Width Modulation (PWM) circuite. Without loss of generality, we specify that σ (t ) = 0 when the switch is connected to R1 , and σ (t ) = 1 when the switch is connected to R2 . The parameters of system (54) are listed in Table 1. Impose the following variable transformations: x1 = q, x2 = q˙ , x3 = I/M and uσ (t ) = Vε /ML. In this examples, we suppose that the system (54) suffers from the unmodeled dynamics z˙ = qσ (t ) (y, z ) and dynamic disturbances 1,σ (t) , 2,σ (t) and 3,σ (t) . And we also suppose that the output of system

Z. Lyu, Z. Liu and Y. Zhang et al. / Neurocomputing 341 (2019) 107–117 Table 1 Parameters of system (54).

115

2.2

Gravity coefficient Rotor inertia Link mass Radius of the load Load mass Link length Viscous friction coefficient at the joint Protective resistance Protective resistance Armature resistance Armature inductance Back-emf coefficient coefficient characterizing the conversion from armature current to torque

G = 9.8N/kg J = 1.625 × 10−1 kg · m2 m = 0.506 kg R0 = 0.023 m M0 = 0.434 kg L0 = 0.305 m B0 = 16.25 × 10−3 N · m · s/rad R1 = 5 R2 = 10 R3 = 5 L = 15H KB = 0.9N · m/A Kτ = 0.9N · m/A

2 1.8 1.6 1.4 1.2 1 0.8 0

2

4

6

8

10 12 Time(sec)

14

16

18

20

14

16

18

20

16

18

20

16

18

20

Fig. 10. Switching law.

1.5

(54) is constrained by the unknown output hysteresis, whose parameters are the same as Example 1. With the variable transformations and the assumptions above, (54) can be transformed to

z˙ = qσ (t ) (y, z )

0.5

x˙ 1 = x2 + f1,σ (t ) (x¯1 ) + 1,σ (t ) (y, z ) x˙ 2 = x3 + f2,σ (t ) (x¯2 ) + 2,σ (t ) (y, z )

0

x˙ 3 = uσ (t ) + f3,σ (t ) (x¯3 ) + 3,σ (t ) (y, z ) y = H ( x1 ) where

f1,0 (x¯1 ) = f1,1 (x¯1 ) = 0, R +R

K

(55)

f2,0 (x¯1 ) = f2,1 (x¯2 ) = K

N −M

-0.5 0

2

4

R +R

∂ Vz q (y, z ) ≤ −0.75z2 + y4 + 1, k = 0, 1. ∂z k And we can also determine α z (s ) = 0.5s2 , α¯ z (s ) = 1.5s2 , λz = 0.75, γz (s ) = s4 and dz = 1. Select γ¯ z = s2 , λ∗z = 0.5 ∈ (0, λz ) and determine the dynamic signal as

r˙ = −0.5r + y4 + 1. With Theorem 1, we can establish adaptive control scheme as

π¯ 1 = −(k1 + 0.5 )ξ1 − π2 = − ( k 2 + 0 . 5 ) ξ 2 −

2h ¯ 21

ηˆ 2

2h ¯ 22

uσ (t ) = −(k3 + 0.5 )ξ3 −

8

10 12 Time(sec)

Fig. 11. Responses of x1 (t) and y(t).

30 20 10 0 -10 -20 -30 0

2

4

π1 = −N (θ )π¯ 1 , ηˆ 1

6

sin(x1 ) −

B B f3,0 (x¯3 ) = − ML x2 − 2ML 3 x3 and f3,1 (x¯3 ) = − ML x2 − 1ML 3 x3 ; q0 (y, z ) = −z + y2 + 0.5, q1 (y, z ) = −z + 0.5y2 + 1; 1,0 (y, z ) = z2 , 1,1 (y, z ) = z2 sin(y ), 2,0 (y, z ) = 2z cos(y ), 2,1 (y, z ) = z cos(y ) and 3,0 = 3,1 = 0. Similar with Example 1, by choosing Vz = z2 , we can obtain

B M x2 ,

1

6

8

10 12 Time(sec)

14

Fig. 12. Responses of x2 (t) and z(t).

ξ1 φ1 (X1 )2 , ξ2 φ2 (X2 )2 ,

ηˆ 3

2h ¯ 23

ξ3 φ3 (X3 )2 ,

r1 2 ξ1 φ1 2 − r¯1 ηˆ 1 , 2h ¯ 21 r ηˆ˙ 2 = 22 ξ22 φ2 2 − r¯2 ηˆ 2 , 2h ¯2 ˙ηˆ 3 = r3 ξ 2 φ3 2 − r¯3 ηˆ 3 , 3 2h ¯ 23

ηˆ˙ 1 =

3.5 3 2.5 2 1.5 1 0.5 0 0

2

4

6

8

10 12 Time(sec)

14

Fig. 13. Responses of the adaptive laws and the dynamic signal.

θ˙ = −ρ π¯ 1 ξ1 , where the designed parameters are select as: k1 = 3, k2 = k3 = 2, h ¯1 = h ¯ 2 = 1, h ¯ 3 = 0.01, r1 = 1, r2 = 0.5, r3 = 0.1, r¯1 = r¯2 = r¯3 = 1 and ρ = 1. In this numerical example, The RBFNN W1T,k φ1 (X1 ) contains 22 nodes. For  = 1, 2, . . . , 11, the centers κ 1,1, and κ 1,2, are spaced evenly in the interval [−1.0, 1.0] × [0.0, 2.0], and the widths are set as: β1,1, = β1,2, = 0.1. W2T,k φ2 (X2 ) contains 55 nodes. For  = 1, 2, . . . , 11, the centers κ 2,1, , κ 2,2, , κ 2,3, ,

κ 2,4, and κ 2,5, are spaced evenly in the interval [−1.0, 1.0] ×

[−12.0, 6.0] × [−1.0, 1.0] × [−0.5, 4.5] × [0.0, 2.0], and the widths are set as: β2,1, = β2,3, = β2,5, = 0.1, β2,2, = 1.0 and β2,4, = 0.3. W3T,k φ3 (X3 ) contains 77 nodes. For  = 1, 2, . . . , 11, the centers κ 3,1, , κ 3,2, , κ 3,3, , κ 3,4, , κ 3,5, , κ 3,6, and κ 3,7, are spaced evenly in the interval [−1.0, 1.0] × [−12.0, 6.0] × [−1.0, 1.0] × [−0.5, 4.5] × [0.0, 2.0] × [−12.0, 6.0] × [−12.0, 6.0], and the widths

116

Z. Lyu, Z. Liu and Y. Zhang et al. / Neurocomputing 341 (2019) 107–117

400

200

0

-200

-400 0

2

4

6

8

10 12 Time(sec)

14

16

18

20

Fig. 14. Responses of the control law and the virtual control laws.

are set as: β3,1, = β3,3, = β3,5, = 0.1, β3,2, = β3,6, = β3,7, = 1.0 and β3,4, = 0.3. Finally, the system is initialized as: x1 (0 ) = 1, x2 (0 ) = 0.5, x3 (0 ) = 0.6, r (0 ) = ηˆ 1 (0 ) = ηˆ 2 (0 ) = ηˆ 3 (0 ) = 0 and θ (0 ) = 0.1. The simulation results are shown in Fig. 10–14. Form the simulation results, we can see that the system output y(t) converges to a small region of zero, and all the closed-loop signals are bounded. Therefore, the effectiveness of our control scheme is verified. 5. Conclusion In this paper, we have investigated the adaptive neural control for a class of switched nonlinear systems with unmodeled dynamics and unknown hysteresis. By using CLF, the switching law is allowed to be arbitrary, and therefore, the approach proposed in this paper is still feasible when there is no priori information of the switchings. The neural-based backstepping technique has been used to design the adaptive control scheme. With the assumption that all subsystems of the unmodeled dynamics are ISpS, a dynamic signal is constructed to dominate the unmodeled dynamics, such that, we do not need to determine any nonlinear input-tooutput gains. In order to deal with the time-varying unknown gain problem aroused by the unknown output hysteresis, a Nussbaum function-type gain is further employed in the adaptive control design. Finally, both stability analysis and simulation examples show that our control scheme is effective and all signals in the closedloop switched nonlinear systems remain bounded. However, this paper only investigates the situation that different subsystems share a common output hysteresis and the adaptive control scheme only uses only one Nussbaum-type function. When the output hysteresis is multiple and there exists huge difference among hysteresis parameters, the control scheme proposed in this study may be of poor performance. Therefore, our future will pay attention to develop another adaptive control scheme based on multiple Nussbaum-type functions. Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grant 61573108, in part by the Natural Science Foundation of Guangdong Province 2016A030313715, and in part by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme. References [1] M. Krstic, P.V. Kokotovic, I. Kanellakopoulos, Nonlinear and Adaptive Control Design, Springer Berlin Heidelberg, 2003. [2] S. Sastry, A. Isidori, Adaptive control of linearizable systems, IEEE Trans. Automat. Control 34 (11) (1989) 1123–1131. [3] A. Teel, R. Kadiyala, P. Kokotovic, S. Sastry, Indirect techniques for adaptive input output linearization of nonlinear systems, Int. J. Control 53 (1) (1991) 193–222.

[4] K. Nam, A. Araposthathis, A model reference adaptive control scheme for pure-feedback nonlinear systems, IEEE Trans. Automat. Control 33 (9) (1988) 803–811. [5] J.B. Pomet, L. Praly, Adaptive nonlinear regulation: estimation from the Lyapunov equation, IEEE Trans. Automat. Control 37 (6) (1992) 729–740. [6] C. Wen, J. Zhou, Z. Liu, H. Su, Robust adaptive control of uncertain nonlinear systems in the presence of input saturation and external disturbance, IEEE Trans. Automat. Control 56 (7) (2011) 1672–1678. [7] B. Chen, C. Lin, X. Liu, K. Liu, Adaptive fuzzy tracking control for a class of mimo nonlinear systems in nonstrict-feedback form, IEEE Trans. Syst., Man, and Cybern. 45 (12) (2015) 2744–2755. [8] H. Wang, B. Chen, K. Liu, X. Liu, C. Lin, Adaptive neural tracking control for a class of nonstrict-feedback stochastic nonlinear systems with unknown backlash-like hysteresis, IEEE Trans. Neural Netw. Learn. Syst. 25 (5) (2014) 947–958. [9] G. Lai, L. Zhi, Z. Yun, C.L.P. Chen, S. Xie, Y.J. Liu, Fuzzy adaptive inverse compensation method to tracking control of uncertain nonlinear systems with generalized actuator dead zone, IEEE Trans. Fuzzy Syst. 25 (1) (2017) 191–204. [10] J. Zhou, C. Wen, T. Li, Adaptive output feedback control of uncertain nonlinear systems with hysteresis nonlinearity, IEEE Trans. Automat. Control 57 (10) (2012) 2627–2633. [11] Y. Shang, B. Chen, C. Lin, Neural adaptive tracking control for a class of high-order non-strict feedback nonlinear multi-agent systems, Neurocomputing 316 (2018) 59–67. [12] H. Wang, B. Chen, C. Lin, Y. Sun, Observer-based neural adaptive control for a class of mimo delayed nonlinear systems with input nonlinearities, Neurocomputing 275 (2018) 1988–1997. [13] Y. Liu, G. Wen, S. Tong, Direct adaptive NN control for a class of discrete-time nonlinear strict-feedback systems, Neurocomputing 73 (13) (2010) 2498–2505. [14] D. Li, D. Li, Adaptive controller design-based neural networks for output constraint continuous stirred tank reactor, Neurocomputing 153 (2015) 159–163. [15] J.W. Macki, P. Nistri, P. Zecca, Mathematical models for hysteresis, Siam Rev. 35 (1) (1993) 94–123. [16] M.A. Krasnoselskii, A.V. Pokrovskii, Systems with Hysteresis, Moscow, Russia: Nauka, 1983. [17] R. Bouc, Forced vibrations of mechanical systems with hysteresis, in: Proceedings of the Fourth Conference on Nonlinear Oscillations, Prague, 1967, 1967. [18] Y.-K. Wen, Method for random vibration of hysteretic systems, J. Eng. Mech. Div. 102 (2) (1976) 249–263. [19] Z. Liu, G. Lai, Y. Zhang, X. Chen, C.L.P. Chen, Adaptive neural control for a class of nonlinear time-varying delay systems with unknown hysteresis, IEEE Trans. Neural Netw. 25 (12) (2014) 2129–2140. [20] J. Zhou, C. Wen, Y. Zhang, Adaptive backstepping control of a class of uncertain nonlinear systems with unknown backlash-like hysteresis, IEEE Trans. Automat. Control 49 (10) (2004) 1751–1759. [21] W. Lv, F. Wang, Y. Li, Finite-time adaptive fuzzy output-feedback control of mimo nonlinear systems with hysteresis, Neurocomputing 296 (2018) 74–81. [22] D. Liberzon, Switching in Systems and Control, Birkhuser Boston, 2003. [23] D. Liberzon, A.S. Morse, Basic problems in stability and design of switched systems, IEEE Control Syst. Mag. 19 (5) (1999) 59–70. [24] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Control 43 (4) (1998) 475–482. [25] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell– time, in: Proceedings of the 38th IEEE Conference on Decision and Control, 3, 1999, pp. 2655–2660. [26] M.A. Muller, D. Liberzon, Input/output-to-state stability and state-norm estimators for switched nonlinear systems, Automatica 48 (9) (2012) 2029–2039. [27] L. Long, J. Zhao, A small-gain theorem for switched interconnected nonlinear systems and its applications, IEEE Trans. Automat. Control 59 (4) (2014) 1082–1088. [28] G. Yang, D. Liberzon, A Lyapunov-based small-gain theorem for interconnected switched systems, Syst. Control Lett. 78 (2015) 47–54. [29] X. Zhao, P. Shi, X. Zheng, L. Zhang, Adaptive tracking control for switched stochastic nonlinear systems with unknown actuator dead-zone, Automatica 60 (60) (2015) 193–200. [30] Y. Dong, Z. Yu, S. Li, F. Li, Adaptive output feedback tracking control for switched non-strict-feedback non-linear systems with unknown control direction and asymmetric saturation actuators, Iet Control Theory Appl. 11 (15) (2017) 2539–2548. [31] X. Wang, H. Li, X. Zhao, Adaptive neural tracking control for a class of uncertain switched nonlinear systems with unknown backlash-like hysteresis control input, Neurocomputing 219 (2017) 50–58. [32] C.E. Rohrs, L.S. Valavani, M. Athans, G. Stein, Robustness of adaptive control algorithms in the presence of unmodeled dynamics, IEEE Trans. Automat. Control 30 (9) (1982) 881–889. [33] Z.P. Jiang, L. Praly, Design of robust adaptive controllers for nonlinear systems with dynamic uncertainties, Automatica 34 (7) (1998) 825–840. [34] Z.P. Jiang, I. Marcels, A small-gain control method for nonlinear cascaded systems with dynamic uncertainties, IEEE Trans. Automat. Control 42 (3) (1997) 292–308. [35] Z.P. Jiang, A combined backstepping and small-gain approach to adaptive output feedback control, Automatica 35 (6) (1999) 1131–1139. [36] G. Shao, X. Zhao, R. Wang, B. Yang, Fuzzy tracking control for switched uncertain nonlinear systems with unstable inverse dynamics, IEEE Trans. Fuzzy Syst. PP (99) (2017) 1.

Z. Lyu, Z. Liu and Y. Zhang et al. / Neurocomputing 341 (2019) 107–117 [37] C. Hua, G. Liu, L. Li, X. Guan, Adaptive fuzzy prescribed performance control for nonlinear switched time-delay systems with unmodeled dynamics, IEEE Trans. Fuzzy Syst. 26 (4) (2018) 1934–1945. [38] Y. Hou, S. Tong, Adaptive fuzzy output-feedback control for a class of nonlinear switched systems with unmodeled dynamics, Neurocomputing 168 (2015) 200–209. [39] S. Tong, Y. Li, Adaptive fuzzy output feedback control for switched nonlinear systems with unmodeled dynamics, IEEE Trans. Cybern. 47 (2) (2017) 295–305. [40] Y. Li, S. Sui, S. Tong, Adaptive fuzzy control design for stochastic nonlinear switched systems with arbitrary switchings and unmodeled dynamics, IEEE Trans. Cybern. 47 (2) (2017) 403–414. [41] W. Zhou, B. Niu, X. Xie, F.E. Alsaadi, Adaptive neural-network-based tracking control strategy of nonlinear switched non-lower triangular systems with unmodeled dynamics, Neurocomputing 322 (2018) 1–12. [42] Y. Li, L. Liu, G. Feng, Robust adaptive output feedback control to a class of non– triangular stochastic nonlinear systems, Automatica 89 (89) (2018) 325–332. [43] T. Zhang, M. Xia, Y. Yi, Adaptive neural dynamic surface control of strict-feedback nonlinear systems with full state constraints and unmodeled dynamics, Automatica 81 (81) (2017) 232–239. [44] X. Xia, T. Zhang, Adaptive output feedback dynamic surface control of nonlinear systems with unmodeled dynamics and unknown high-frequency gain sign, Neurocomputing 143 (2014) 312–321. [45] H. Wang, W. Liu, P.X. Liu, H. Lam, Adaptive fuzzy decentralized control for a class of interconnected nonlinear system with unmodeled dynamics and dead zones, Neurocomputing 214 (2016) 972–980. [46] Z. Li, T. Li, B. Miao, C.L.P. Chen, Adaptive NN control for a class of stochastic nonlinear systems with unmodeled dynamics using dsc technique, Neurocomputing 149 (2015) 142–150. [47] S. Tong, S. Sui, Y. Li, Adaptive fuzzy decentralized control for stochastic large-scale nonlinear systems with unknown dead-zone and unmodeled dynamics, Neurocomputing 135 (2014) 367–377. [48] L. Zhi, W. Fang, Z. Yun, C. Xin, C.L.P. Chen, Adaptive tracking control for a class of nonlinear systems with a fuzzy dead-zone input, IEEE Trans. Fuzzy Syst. 23 (1) (2015) 193–204. [49] Q. Zhou, P. Shi, J. Lu, S. Xu, Adaptive output-feedback fuzzy tracking control for a class of nonlinear systems, IEEE Trans. Fuzzy Syst. 19 (5) (2011) 972–982. [50] S.S. Ge, F. Hong, T.H. Lee, Adaptive neural control of nonlinear time-delay systems with unknown virtual control coefficients, IEEE Trans. Syst., Man, and Cybern., Part B (Cybern.) 34 (1) (2004) 499–516. [51] Polycarpou, M. M., Ioannou, A. P., A robust adaptive nonlinear control design, Automatica 32 (3) (1995) 1365–1369. [52] E.D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control 34 (4) (2002) 435–443. [53] R.M. Sanner, J.E. Slotine, Gaussian networks for direct adaptive control, IEEE Trans. Neural Netw. 3 (6) (1992) 837–863. [54] M. Dorigo, V. Maniezzo, A. Colorni, Ant system: optimization by a colony of cooperating agents, IEEE Trans. Syst., Man, and Cybern., Part B (Cybern.) 26 (1) (1996) 29–41. [55] D.E. Goldberg, Genetic algorithms in search, optimization and machine learning, xiii, Addison-Wesley, 1989, pp. 2104–2116.

Ziliang Lyu received his B.Eng. in the School of Automation, Guangdong University of Technology, Guangzhou, China, in 2017, where he is currently pursuing his M.Eng. degree. His current research interests include bioelectrical signal processing, interferometric signal processing, optical coherence tomography (OCT), machine learning, switched systems and small-gain theory.

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Zhi Liu received the B.S. degree in electrical engineering from the Huazhong University of Science and Technology, Wuhan, China, in 1997, the M.S. degree in electrical engineering from Hunan University, Changsha, China, in 20 0 0, and the Ph.D. degree in electrical engineering from Tsinghua University, Beijing, China, in 2004. He is currently a Professor with the Department of Automation, Guangdong University of Technology, Guangzhou, China. His current research interests include fuzzy logic systems, neural networks, robotics, and robust control.

Yun Zhang received the B.S. and M.S. degrees in automatic engineering from Hunan University, Changsha, China, in 1982 and 1986, respectively, and the Ph.D. degree in automatic engineering from the South China University of Science and Technology, Guangzhou, China, in 1998. He is currently a Professor with the Department of Automation, Guangdong University of Technology, Guangzhou. His current research interests include intelligent control systems, network systems, and signal processing.

C. L. Philip Chen received the M.S. degree in electrical engineering from the University of Michigan, Ann Arbor, MI, USA, in 1985 and the Ph.D. degree in electrical engineering from Purdue University, West Lafayette, IN, USA, in 1988. He was a tenured Professor, the Department Head, and the Associate Dean with two different universities in the U.S. for 23 years. He is currently a Chair Professor with the Department of Computer and Information Science, Faculty of Science and Technology, University of Macau, Macau, China. The University of Macau’s Engineering and Computer Science programs receiving Hong Kong Institute of Engineers’ (HKIE) accreditation and Washington/Seoul Accord is his utmost contribution in engineering/computer science education for Macau as the former Dean of the Faculty. His current research interests include systems, cybernetics, and computational intelligence. Dr. Chen was a recipient of the 2016 Outstanding Electrical and Computer Engineers Award from his alma mater at Purdue University. He was the IEEE SMC Society President from 2012 to 2013 and a Vice President of Chinese Association of Automation (CAA). He has been the Editor-in-Chief of the IEEE TRANSACTION ON SYSTEMS,MAN, AND CYBERNETICS:SYSTEMS, since 2014 and an associate editor of several IEEE TRANSACTIONS. He was the Chair of TC 9.1 Economic and Business Systems of International Federation of Automatic Control from 2015 to 2017, and also a Program Evaluator of the Accreditation Board of Engineering and Technology Education of the U.S. for computer engineering, electrical engineering, and software engineering programs. He is a fellow of AAAS, IAPR, CAA, and HKIE.