Adaptive fuzzy control of switched nonlinear time-varying delay systems with prescribed performance and unmodeled dynamics

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Adaptive fuzzy control of switched nonlinear time-varying delay systems with prescribed performance and unmodeled dynamics Shi Li a , Choon Ki Ahn b , Zhengrong Xiang a,∗ a School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, People’s Republic of China b School of Electrical Engineering, Korea University, Seoul, South Korea

Received 13 February 2018; received in revised form 11 August 2018; accepted 23 October 2018

Abstract This paper considers the problem of adaptive prescribed performance control for a class of switched nonlinear systems with time-varying delays and unmodeled dynamics. Fuzzy logic systems are introduced to deal with the unknown nonlinear terms. Utilizing dynamic surface control and convex combination method, a state-feedback controller is constructed. It is proven that all signals of the closed-loop system are bounded. Finally, simulation results are given to show that the proposed scheme is effective. © 2018 Elsevier B.V. All rights reserved. Keywords: Adaptive fuzzy control; Switched nonlinear systems; State-dependent switching law; Prescribed performance control

1. Introduction In the pasts decades, the fuzzy control has been widely adopted by scholars in the control scheme design procedure of nonlinear systems [1]. In [2], using new nonquadratic Lyapunov functions, the stabilization problem for nonlinear systems which are stood for by a fuzzy model was investigated. The conditions for the stabilization and the observation of T-S nonlinear models in the case, where the premises are partially or not measurable, were provided in [3]. An event-triggered method was introduced to investigate the exponential stabilization problem of T-S fuzzy systems with aperiodic sampling in [4]. In [5], robust stabilization conditions were derived for T-S fuzzy fractional order systems. The considered systems contained actuator saturation and actuator failures. Using small-gain technique, an adaptive fuzzy input saturation control scheme was proposed for nonlinear systems subject to unmodeled dynamics in [6]. Time delay exists widely in many engineering systems. It often degrades system performance. Consequently, the analysis and control of time-delay systems have theoretical and practical importance. L-K functionals [7] and L-R functionals [8] are commonly used to handle time-delay problems. Recently, scholars have obtained many results on the controllers design of nonlinear systems with time delays [9–16]. By dynamic surface control method, Yoo et al. in [9] investigated the strict-feedback nonlinear time-delay systems. The control schemes for large-scale time-delay * Corresponding author.

E-mail address: [email protected] (Z. Xiang). https://doi.org/10.1016/j.fss.2018.10.011 0165-0114/© 2018 Elsevier B.V. All rights reserved.

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systems were developed in [10,11]. Zhou et al. in [12] considered the time-delay adaptive input saturation control of nonlinear systems. In [13], a control method was presented for time-delay MIMO nonlinear systems with dead zones. Zhang and Zhu in [14] employed neural networks to deal with the control problem of strict-feedback time-delay systems. Gao et al. in [15] considered the adaptive control for stochastic nonlinear time-delay systems with unmodeled dynamics. Control methods for time-delay networks was developed in [16]. Choi et al. in [17] presented a novel fuzzy control scheme for T-S fuzzy output constraints time-delay systems. A sampled-data stabilization scheme was developed for T-S fuzzy delays networks in [18]. With the increasing demand for transient performance of real systems, the prescribed performance control has become a hot research topic. The concept of prescribed performance was proposed by Charalampos and George in [19]. In [20–22], some control methods were developed for prescribed performance output-feedback nonlinear systems. Na et al. in [23] proposed a prescribed performance motion control scheme. In [24], the prescribed performance control problem of vehicle active suspensions was investigated. Cui et al. in [25] considered the problem of prescribed performance distributed output consensus for higher-order non-affine nonlinear multi-agent systems. As a type of hybrid nonlinear systems [26], it is important to investigate switched nonlinear systems. Different methods have been proposed to handle the control problem of switched nonlinear systems. Common Lyapunov function methods were used to investigate switched nonlinear systems under arbitrary switching signals in [27–33]. In [34–36], state-dependent switching laws and state feedback controllers of individual subsystems were constructed by introducing the convex combination method. Some tracking control methods were presented in [22,37,38] by using average dwell-time method. Moreover, the multiple Lyapunov function method was widely used in the study of switched nonlinear systems [39–42]. However, finding the proper multiple Lyapunov functions and designing switching laws for some switched systems are difficult. Although several control schemes of time-delay switched nonlinear systems with guaranteed prescribed performance have been proposed, some challenging issues arise: if some subsystems are uncontrollable, can we still guarantee that the tracking error satisfies the prescribed bounds? Moreover, when the systems contain unmodeled dynamics and time-varying delays, how can we design a controller and a switching law? This paper provides positive solutions for these important issues. The following are the main contributions of this paper: (i) All signals of the resulting closed-loop system are bounded even if some subsystems are uncontrollable. This relaxes the strong restrictions on subsystems and the switching laws in some existing literature. (ii) The problem of ‘curse of dimensionality’ existing in some existing control schemes can be overcome by adopting the dynamic surface control method. (iii) The proposed control approach only requires that nonlinear functions are continuous, which make its application wider. The remainder of the paper is organized as follows: The system description is provided in Section 2. Main results are derived in Section 3. Simulation results are given in Section 4. Finally, conclusions are presented in Section 5. Notations: The set of all real numbers is R; The real n-dimensional space is R n ; The real m × n matrix space is m×n ; The two-norm is ·; s¯i denotes [s1 , · · · , si ]T ; y¯i+1 denotes [y2 , · · · , yi+1 ]T . R 2. Problem formulation and preliminaries 2.1. System descriptions and assumptions Consider the following switched nonlinear time-varying delay system: ⎧ ξ˙ = q(ξ, x, t), ⎪ ⎪ ⎨ x˙i = fi,σ (t) (x¯i ) + gi,σ (t) (x¯i )xi+1 + hi,σ (t) (x¯i (t − τi (t))) + i,σ (t) (ξ, x, t), 1 ≤ i ≤ n − 1, x˙ = fn,σ (t) (x) + gn,σ (t) (x)u + hn,σ (t) (x(t − τn (t))) + n,σ (t) (ξ, x, t), ⎪ ⎪ ⎩ n y = x1 ,

(1)

where x¯i = [x1 , · · · , xi ]T ∈ R i ; x = [x1 , · · · , xn ]T ∈ R n are system states; u ∈ R and y ∈ R are system control input and output; ξ ∈ R n0 is the unmodeled dynamics; The switching signal is σ (t) : [0, ∞) → N = {1, 2, · · · , N }. For k ∈ N, i = 1, · · · , n, fi,k (x¯i ), gi,k (x¯i ), and hi,k (x¯i (t − τi (t))) are unknown continuous nonlinear functions. τi (t) is the

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time-varying delay and satisfies 0 ≤ τi (t) ≤ τmax and τ˙i (t) ≤ τ ∗ < 1 (i = 1, 2, · · · , n), τmax and τ ∗ are known positive constants. i,k (ξ, x, t) is the unknown smooth disturbance; q(ξ, x, t) is an unknown Lipschitz function. yd (t) is the desired trajectory. Remark 1. Compared with the system investigated in existing results, both time-varying delays and the unmodeled dynamics are included in system (1), which means that the considered system in our paper is more general. Definition 1 ([43,47]). If there exists a Lyapunov function V (ξ ), and known constants c > 0 and d¯ ≥ 0 satisfying α¯ 1 (ξ ) ≤ V (ξ ) ≤ α¯ 2 (ξ ), ∂V (ξ ) ¯ q(ξ, x) ≤ −cV (ξ ) + γ (x) + d, ∂ξ

(2) (3)

with α¯ 1 (·), α¯ 2 (·), γ (·) being the class K∞ functions, then, unmodeled dynamics ξ in system (1) is exponentially input-state-practically stable (exp-ISpS). Assumption 1 ([47]). The desired trajectory yd (t) is a smooth function and yd , y˙d , and y¨d are bounded. There exists a compact set y ∈ R 3 , such that (yd , y˙d , y¨d )T ∈ y , where y = {(yd , y˙d , y¨d )T : yd + y˙d + y¨d  } with being a known positive constant. Assumption 2 ([13]). The nonlinear function hi,k (x¯i (t − τi (t))) satisfies the following inequality: i    hi,k (x¯i (t − τi (t))) ≤ |hi (x¯i (t − τi (t)))| ≤ μi ϑi,j (xj (t − τj (t))),

(4)

j =1

where μi > 0 is a unknown constant and ϑi,j (xj (t)) is a positive known continuous function. Remark 2. It should be noted out that Assumption 2 is a general assumption in controller design of time-delay systems, and we can find it in existing results, for instance [13–15]. Assumption 3 ([43,47]). There exist unknown nonnegative continuous functions φi1,k (·) and φi2,k (·) such that   i,k (ξ, x, t) ≤ φi1,k (x) + φi2,k (ξ ).

(5)

Assumption 4 ([43,47]). The unmodeled dynamics ξ in system (1) is exp-ISpS. Assumption 5 ([35]). For i = 1, 2, · · · , n, there exist βj such that N 

βj = 1, βj ∈ (0, 1),

(6)

j =1

    N  satisfying ≤ |gi (x¯i )| =  βj gi,j (x¯i ) with gi∗ being constants and the signs of gi (x¯i ) being known. In addij =1  tion, there exist constants g¯ i > 0 satisfying |g˙ i (x¯i )| ≤ g¯ i . In this paper, we assume gi (x¯i ) > 0. 0 < gi∗

Remark 3. Compared with the existing results, it can be seen from Assumption 5 that the signs of gi,k (·) are allowed to be variable in system (1), that is to say, some subsystems of system (1) can be uncontrollable. Lemma 1 ([35,36]). If there exist functions Vn∗ (t), U1,j (t), U2 (t) and scalars βj satisfying (6) and V˙n∗ (t) ≤ U1,σ (t) (t) + U2 (t), it can be obtained V˙n∗ (t) ≤

N  j =1

βj (U1,j (t) + U2 (t)) under the switching law given as follows:

(7)

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σ (t) = arg min{U1,j (t)}.

(8)

j ∈N

Proof. From (8), we have U1,σ (t) (t) =

N 

βj U1,σ (t) (t) ≤

j =1

N 

βj U1,j (t).

j =1

Then, it can be deduced from (7) V˙n∗ (t) ≤

N 

βj U1,j (t) + U2 (t) =

j =1

N 

βj (U1,j (t) + U2 (t)).

2

j =1

Lemma 2 ([43]). If (2) and (3) hold, then, for any constant c¯ ∈ (0, c), t0 > 0, ξ0 = ξ (t0 ) , v0 > 0, and for any continu¯ ≥ 0, a function ous function γ¯ satisfying γ¯ (x) ≥ γ (x), there exists a finite T0 = max {0, ln [V (ξ0 ) /vo ] / (c − c)} D (t0 , t) ≥ 0, ∀t ≥ t0 , and a signal expressed as ¯ v (t0 ) = v0 > 0, v˙ = −cv ¯ + γ¯ (x) + d,

satisfying D (t0 , t) = 0 for t ≥ t0 + T0 and V (ξ ) ≤ v (t) + D (t0 , t) with D(t0 , t) = max 0, e−c(t−t0 ) V (ξ0 )

¯ 0)v − e−c(t−t 0 . Without loss of generality, we assume γ¯ (x) = γ (x). Lemma 3 ([44]). For any real-valued continuous function f (x, y), where x ∈ R m , y ∈ R n , there are smooth scalarvalue functions a(x) ≥ 0 and b(y) ≥ 0 such that |f (x, y)| ≤ a(x) + b(y).

(9)

Lemma 4 ([46]). If F (χ) defined on a compact set χ is a continuous function. Then, for any given constant δ ∗ > 0, there exists a fuzzy logic system Y (χ) = T (χ),

(10)

satisfying

    supχ∈ χ F (χ) − T (χ) ≤ δ ∗ ,

where  = [1 , · · · , M ]T , (χ) = [ϕ1 (χ), · · · , ϕM (χ)]T , and M is the number of rules. 2.2. Prescribed performance The prescribed performance [19] is described as follows: ¯ −δρ(t) < E1 (t) < δρ(t), ∀t ≥ 0,

(11)

where E1 (t) = y(t) − yd (t), ρ(t) = (ρ0 − ρ∞ )e−nt + ρ∞ , and δ, δ, n > 0 and ρ∞ > 0 are design parameters, ¯ ρ0 = ρ(0), and ρ 0 is chosen satisfying ρ0 > ρ∞ > 0 and −δρ(0) < E1 (0) < δρ(0). From (11), we get that E1 (t)

¯ is less than max δρ(0), δρ(0) . 1 (t) Meanwhile, we can deduce that −δ < Eρ(t) < δ¯ should hold for E1 (t) to satisfy the prescribed performance. A conversion function is defined to solve the error constraint control problem, which satisfies: 1 (t) (i) S −1 ( Eρ(t) ) is a smooth function, and it is strictly increasing. ¯ −∞ < S −1 ( E1 (t) ) < +∞. (ii) When −δ < E1 (t) < δ,

ρ(t)

(iii) When

E1 ρ

ρ(t)

→ −δ, lim S −1 ( Eρ1 ) = −∞; When

E1 ρ

¯ lim S −1 ( E1 ) = +∞. → δ, ρ

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1 (t) Choose ζ = S −1 ( Eρ(t) ) = 12 ln S+δ as the conversion function. Then, we can get its reverse function easily: δ−S

S(ζ ) =

δeζ − δe−ζ . eζ + e−ζ

(12)

To ensure that −δ < S(ζ ) < δ¯ holds, ζ ∈ L∞ is required to be provided. Hence, ζ is added into the design procedure. Then, we have E1 ρ˙ ), ζ˙ = r(E˙1 − ρ where r =

1 2ρ(t) [

1 E1 (t) ρ(t) +δ

−

(13) 1 E1 (t) ¯ ρ(t) −δ

] > 0.

We employ the following transformation: 1 (14) ln(δ/δ). 2 Our control objective is to construct an adaptive controller for system (1) such that all states of the corresponding closed-loop system are semi-globally uniformly ultimately bounded (SGUUB). s1 = ζ −

3. Main results 3.1. Adaptive controller design The dynamic surface control method will be employed to design the control scheme. Before the controller design, the change of coordinates is employed as follows: ⎧ ⎨ s1 = ζ − 12 ln(δ/δ), (15) s = xi − ωi , ⎩ i yi = ωi − αi−1 , i = 2, · · · , n, where yi , αi−1 , and ωi will be defined later. Step 1. For Step 1, we select the following Lyapunov function: V1 = Vs1 +

N N−1 T N−1 ˜ T  T ˜ 1  θ˜1,j θ˜1,j 1  g˜ 1,j g˜ 1,j 1 v 1,j 1,j + + + V¯1 + , 2 κ1,j 2 b1,j 2 ς1,j λ0 j =1

j =1

j =1

(16)

t 1 2 ˜ ˜ 1,j will be defined; κ1,j , b1,j , ς1,j , λ0 > 0 where Vs1 = 2g11 s12 , V¯1 = 2(1−τ ˜ 1,j , and  ∗ ) t−τ (t) ϑ1,1 (x1 (z))dz; θ1,j , g 1 are design parameters. The kth subsystem is activated when σ (t) = k. The first virtual control law α1 and the adaptive laws θˆ1,j , gˆ 1,j , and ˆ 1,j are designed as follows: 2 N  s1 βj T1,j (Z1 ) λ∗ (t)s1 λ1 + 1 α1 = − s1 − s1 r − θˆ1,j − 1 , (17) r 2r r j =1 ⎧ 2 ⎨ θ˙ˆ = κ ( s12 − σ1,j θˆ1,j ), j = k, 1,j 1,j 2 T1,j (Z1 ) (18) ⎩ θ˙ˆ = −κ σ θˆ , j = k, 1,j 1,j 1,j 1,j ⎧ ⎪ g˙ˆ = b1,j (s1 x2 Tg1 ,j (Zg1 ) − σg1 ,j gˆ 1,j ), j = k = N, ⎪ ⎨ 1,j g˙ˆ 1,j = −b1,j σg1 ,j gˆ 1,j , j = k = N, (19) ⎪ ⎪ ⎩ g˙ˆ = b (− βj s1 x2 Tg1 ,j (Zg1 ) − σ gˆ ), k = N, 1,j g1 ,j 1,j 1,j βN ⎧ ˙ ˆ ˆ ⎪ ⎪ ⎨ 1,j = ς1,j (s1 T1 ,j (Z1 ) − σ1 ,j 1,j ), j = k = N, ˙ˆ ˆ 1,j , j = k = N, (20)  1,j = −ς1,j σ1 ,j  ⎪ ⎪ βj s1 T1 ,j (Z1 ) ⎩ ˙ˆ ˆ 1,j ), k = N, − σ1 ,j  1,j = ς1,j (− βN

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where λ∗1 (t) =

γ1 2(1−τ ∗ )(s12 +ε12 )

t t−τmax

2 (x (z))dz with γ and ε being positive constants; λ > 0 and σ , σ ϑ11 1 1 1 1 1,j g1 ,j ,

σ1 ,j > 0 are parameters to be determined later and T1,j (Z1 ), Tg1 ,j (Zg1 ), and T1 ,j (Z1 ) are functions which will be ˆ 1,j will be given later. defined; θˆ1,j , gˆ 1,j , and  Then, differentiating V1,k yields N−1 T g˙ˆ  g˜ 1,j f1,k g1,k h1,k 1,k y˙d E1 ρ˙ g˙ 1 1,j + x2 + + − − ] − 2 s12 − V˙1,k = s1 r[ g1 g1 g1 g1 g1 ρg1 b1,j g1 j =1 N θ˜ T θ˙ˆ  1 1,j 1,j 2 2 + [ϑ (x1 (t)) − ϑ1,1 (x1 (t − τ1 (t)))(1 − τ˙ (t))] − 2(1 − τ ∗ ) 1,1 κ1,j j =1

−

N−1  j =1

˙ˆ ˜T   1,j 1,j ς1,j

−

cv ¯ γ¯ (x) d¯ + + . λ0 λ0 λ0

(21)

According to Assumption 2, one has   2 s1 rh1,k  s1 rμ1 ϑ1,1 (x1 (t − τ1 (t)))  r 2 μ21 2 ϑ1,1 (x1 (t − τ1 (t))) ≤ s + ≤ .  2g ∗2 1 g1 g1∗ 2 1

(22)

According to Assumption 3, it can be obtained that      s1 r  s1 r1,k  s1 r    ≤ (x) + φ 11,k  g  φ12,k (ξ ), g1 g1  1   2 2  s1 r    φ11,k (x) ≤ s1 r φ 2 (x) + 1 . 11,k g  4 g1∗2 1

(23) (24)

Based on Definition 1 and Assumption 4, α¯ 1 (·) is a class K∞ function. So α¯ 1−1 (·) is a non-decreasing function. According to Lemma 2 and Lemma 3, we have      s1 r      φ12,k (ξ ) ≤  s1 r  (φ12,k ◦ α¯ −1 (v(t) + D(t, t0 ))) 1 g  g  1    1  s1 r   s1 r  (25) ≤  ∗  ϕ11,k (v(t)) +  ∗  ϕ12,k (D(t, t0 )), g1 g1 where φ12,k ◦ α¯ 1−1 (·) = φ12,k (α¯ 1−1 (·)). Since D(t, t0 ) is bounded, there exists a ϑ1∗ > 0 satisfying ϕ12,j (D(t, t0 )) ≤ ϑ1∗ . Then, we obtain     2 2 2 2 ∗2   s1 r     ϕ11,k (v(t)) +  s1 r  ϕ12,k (D(t, t0 )) ≤ s1 r ϕ 2 (v(t)) + s1 r + 1 + ϑ1 . 11,k  g∗   g∗  4 4 g1∗2 g1∗2 1 1

(26)

Substituting (22), (24), and (26) into (21), we have 2 (x (t)) s1 ϑ1,1 μ21 rf1,k s1 r 2 2 s1 r 2 2 1 1 + ∗2 φ11,k (x) + ∗2 ϕ11,k (v(t)) + + ( + ∗2 )s1 r 2 V˙1,k ≤ s1 ( 2 ∗2 ∗ g1 g1 g1 2(1 − τ )ε1 2g1 g1

s2 g¯ 1 r rE1 ρ˙ 1 ϑ ∗2 s1 rg1,k 1 2 s1 − y˙d − x2 + + 1 + (x1 (t))(1 − 12 ) )+ ϑ1,1 2 ∗ g1 g1 ρ g1 2 4 2(1 − τ ) g1 ε1 ˙ˆ N θ˜ T θ˙ˆ N−1 N−1 T g˙ˆ ˜T    g˜ 1,j  d¯ cv ¯ γ¯ (x) 1,j 1,j 1,j 1,j 1,j − − − + + . − κ1,j b1,j ς1,j λ0 λ0 λ0

+

j =1

Let ∗1,k (Z1 ) =

rf1,k g1

j =1

2

2

r 2 (x) + s1 r ϕ 2 (v(t)) + + s1∗2 φ11,k ∗2 11,k g1 T

(27)

j =1

g1

Z1 = [x T , v, yd , y˙d , ρ, ρ] ˙ . Based on Lemma 4, it yields

2 (x (t)) s1 ϑ1,1 1

2(1−τ ∗ )ε12

+(

μ21 2g1∗2

+

1 )s r 2 + g¯12 s1 − gr1 y˙d g1∗2 1 g1

1 ρ˙ − rE g1 ρ with

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s1 rg1,k x2 1 ϑ1∗2 ∗T T1,k (Z1 ) + s1 D1,k (Z1 ) + + + V˙1,k ≤ s1 W1,k g1 2 4 ˙ N θ˜ T θˆ  s12 1 1,j 1,j 2 (x (t))(1 − ) − ϑ + 1 1,1 ∗ 2 2(1 − τ ) κ1,j ε1 j =1

−

N−1  j =1

T g˙ˆ g˜ 1,j 1,j

b1,j

−

N−1  j =1

˜T  1,j

˙ˆ  1,j

ς1,j

−

cv ¯ γ¯ (x) d¯ + + . λ0 λ0 λ0

(28)

Using Young’s inequality [45], we get 2 s12 T1,k (Z1 ) 1 ∗T s1 W1,k T1,k (Z1 ) ≤ θ1,k + , 2 2 s 2 d ∗2 s1 D1,k (Z1 ) ≤ 1 + 1 , 2 2 2 ∗ where θ1,k = W1,k and θˆ1,k = θ1,k − θ˜1,k is the estimation of θ1,k ; d1∗ is a positive constant. Substituting (18), (29), and (30) into (28), we get 2 s12 T1,k (Z1 ) s 2 s1 rg1,k x2 d ∗2 ϑ ∗2 ˙ +1+ 1 + 1 θˆ1,k + 1 + V1,k ≤ 2 2 g1 2 4

(29) (30)

N  s12 1 2 T ˆ σ1,j θ˜1,j ϑ (x1 (t))(1 − 2 ) + θ1,j + 2(1 − τ ∗ ) 1,1 ε1 j =1

−

N−1  j =1

Let G1,k = N −1 j =1

rg1,k g1 .

T g˙ˆ g˜ 1,j 1,j

b1,j

−

N−1  j =1

˙ˆ ˜T   1,j 1,j ς1,j

−

cv ¯ γ¯ (x) d¯ + + . λ0 λ0 λ0

From Lemma 4, we have G1,k = g T1,k Tg1 ,k (Zg1 ) + Dg1 ,k (Zg1 ), k = N , and G1,N =

(31) 1 βN

(r −

βj g T1,j Tg1 ,j (Zg1 )) + Dg1 ,N (Zg1 ) with Zg1 = [x1 , yd , ρ]T and gˆ 1,k = g 1,k − g˜ 1,k being the estimation of g 1,k .

From (19) and (31), one has 2 s12 T1,k (Z1 ) s2 d ∗2 ϑ ∗2 ˆ 1,k x2 + s1 Dg1 ,k (Zg1 )x2 + 1 + 1 + 1 V˙1,k ≤ θˆ1,k + 1 + s1 G 2 2 2 4 N N−1 2   s 1 2 T ˆ T (x1 (t))(1 − 12 ) + σ1,j θ˜1,j σg1 ,j g˜ 1,j gˆ 1,j ϑ1,1 + θ1,j + ∗ 2(1 − τ ) ε1 j =1 j =1 −

N−1  j =1

˙ˆ ˜T   1,j 1,j ς1,j

−

cv ¯ γ¯ (x) d¯ + + , λ0 λ0 λ0

ˆ 1,k = gˆ T T (Zg1 ), k = N and G ˆ 1,N = where G 1,k

1 βN

(r −

(32) N−1  j =1

T T (Z )). βj gˆ 1,j g1

Similarly, denote 1 ,k = Dg1 ,k (Zg1 )x2 . Then, we have 1 ,k = T1,k T1 ,k (Z1 ) + D1 ,k (Z1 ), k = N , and N−1  βj T1,j T1 ,j (Z1 ) ˆ 1,k = 1,k −  ˜ 1,k being the es+ D1 ,N (Z1 ) with Z1 = [x1 , x2 , yd , ρ]T and  1 ,N = − βN j =1

timation of 1,k . Substituting (20) into (32) yields 2 d2 s12 T1,k (Z1 ) d ∗2 ϑ ∗2 ˙ ˆ 1,k x2 + s1  ˆ 1 ,k + 1 + 1 + 1 + 1 θˆ1,k + s12 + s1 G V1,k ≤ 2 2 2 4

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+

+

N N−1   s12 1 2 T ˆ T ˜ (x (t))(1 − ) + σ + σg1 ,j g˜ 1,j gˆ 1,j ϑ θ θ 1,j 1,j 1,j 1,1 1 ∗ 2 2(1 − τ ) ε1 j =1 j =1 N−1 

˜ T1,j  ˆ 1,j − σ1 ,j 

j =1

cv ¯ γ¯ (x) d¯ + + , λ0 λ0 λ0

ˆ 1 ,k =  ˆ T T1 ,k (Z1 ), k = N , ˆ 1 ,N = − where  1,k

N−1  βj ˆ T1,j T1 ,j (Z1 ) , βN j =1

(33)

and d1 > 0 is a constant.

From Lemma 1, we have V˙1,k ≤ U1,k + U1 ,

(34)

2 2 d s12 T1,k (Z1 ) s12 d1∗2 1 2 1 ˆ 1,k x2 + s1  ˆ 1 ,k − where U1,k = θˆ1,k + s12 + s1 G 2 2(1−τ ∗ ) ϑ1,1 (x1 (t)) ε 2 and U1 = 1 + 2 + 2 1 N N−1 N−1    γ¯ (x) d¯ 1 cv ¯ 2 T θˆ T gˆ ˜T  ˆ σ1,k θ˜1,j σg1 ,j g˜ 1,j σ1 ,j  1,j + 1,j + 1,j 1,j − λ0 + λ0 + λ0 . 2(1−τ ∗ ) ϑ1,1 (x1 (t)) + j =1 j =1 j =1 From (15), we can get x2 = s2 + y2 + α1 . Define V1∗ = V1 . From Lemma 1, it can be obtained that

V˙1∗ ≤

N 

+

ϑ1∗2 4

+

βj (U1,j + U1 )

j =1

≤ −λ1 s12 − λ∗1 (t)s12 + +

N 

s22 y22 s12 1 2 (x (t))(1 − ) + + ϑ 1 2 2 2(1 − τ ∗ ) 1,1 ε12

T ˆ σ1,j θ˜1,j θ1,j +

j =1

N−1 

T σg1 ,j g˜ 1,j gˆ 1,j +

j =1

d ∗2

ϑ ∗2

¯

N−1 

˜ T1,j  ˆ 1,j + C1 − σ1 ,j 

j =1

cv ¯ γ¯ (x) + , λ0 λ0

(35)

2 d

where C1 = 1 + 21 + 41 + λd0 + 21 . To deal with the problem of ‘curse of dimensionality’ in traditional backstepping, a first-order filter is introduced as follows: π2 ω˙ 2 + ω2 = α1 , ω2 (0) = α1 (0),

(36)

where π2 is a positive design parameter. From (36), it can be obtained ω˙ 2 = − πy22 , y˙2 = − πy22 − α˙ 1 . There exists a nonnegative continuous function 2 (s1 , s2 , x, v, y2 , θˆ1,k , yd , y˙d , y¨d , ρ, ρ, ˙ ρ) ¨ such that     y˙2 + y2  ≤ 2 . (37)  π2  Then, we have y2 y˙2 ≤ −

y22 y22 22 + + . π2 2 2

(38)

Step i (2 ≤ i ≤ n − 1). The Lyapunov function for Step i is chosen as follows: Vi = Vsi +

N N−1 T N−1 ˜ T  T ˜ 1  θ˜i,j θ˜i,j 1  g˜ i,j g˜ i,j 1 i,j i,j + + + V¯i , 2 κi,j 2 bi,j 2 ςi,j j =1

j =1

˜ i,j will be defined; V¯i = where κi,j , bi,j , ςi,j > 0 are design parameters; θ˜i,j , g˜ i,j , and  j  i   si2 t 1 2 2 t−τm (t) ϑj,m (xm (z))dz; Vsi = 2gi + 2 yi . j =1 m=1

(39)

j =1

1 2(1−τ ∗ )

×

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ˆ i,j are designed as follows: The virtual control law αi and the adaptive laws θˆi,j , gˆ i,j and  2 N  βj si Ti,j (Zi ) αi = −λi si − θˆi,j − λ∗i (t)si , 2 j =1 ⎧ 2 ⎨ ˙ˆ = κ ( si2 − σi,j θˆi,j ), j = k, θ i,j i,j 2 Ti,j (Zi ) ˙ ⎩ θˆ i,j = −κi,j σi,j θˆi,j , j = k, ⎧ ⎪ g˙ˆ = bi,j (si xi+1 Tgi ,j (x¯i ) − σgi ,j gˆ i,j ), j = k = N, ⎪ ⎨ ˙ i,j gˆ i,j = −bi,j σgi ,j gˆ i,j , j = k = N, ⎪ ⎪ ⎩ g˙ˆ = b (− βj si xi+1 Tgi ,j (x¯i ) − σ gˆ ), k = N, i,j gi ,j i,j i,j βN ⎧ ˆ ˆ˙ ⎪ ⎪ ⎨ i,j = ςi,j (si Ti ,j (x¯i+1 ) − σi ,j i,j ), j = k = N, ˙ ˆ i,j = −ςi,j σi ,j  ˆ i,j , j = k = N,  ⎪ ⎪ βj si Ti ,j (x¯i+1 ) ⎩ ˙ˆ ˆ ), k = N, −σ  = ς (−  i,j

i,j

where λ∗i (t) =

i ,j

βN

γi 2(1−τ ∗ )(si2 +εi2 )

(40)

(41)

(42)

(43)

i,j

j  i   t j =1 m=1

9

t−τmax

2 (x (z))dz with γ and ε being positive constants; λ > 0 and ϑj,m m i i i

Ti,j (Zi ), Tgi ,j (x¯i ), and Ti ,j (x¯i+1 ) will be defined later; σi,j , σgi ,j , σi ,j > 0 are parameters to be determined later; ˆ i,j will be given later. θˆi,j , gˆ i,j , and  Then, taking the derivative of Vi,k yields N θ˜ T θ˙ˆ  fi,k gi,k xi+1 hi,k i,k ω˙ i g˙ i i,j i,j + + + − ] − 2 si2 + V˙¯ i + yi y˙i − V˙i,k = si [ gi gi gi gi gi κi,j gi j =1

−

N−1  j =1

T g˙ˆ g˜ i,j i,j

bi,j

−

N−1  j =1

˙ˆ ˜T   i,j i,j ςi,j

.

From Assumption 2, we have   i i 2 (x (t − τ (t)))  μ2i 2  ϑi,m si hi,k  si μi  m m  ≤ ∗ ϑi,m (xm (t − τm (t))) ≤ ∗2 si + .  gi  2gi gi 2 m=1 m=1 According to Assumption 3, it can be obtained that      si  si i,k  si  ≤  ∗  φi1,k (x) +  ∗  φi2,k (ξ ), gi gi gi   2  si    φi1,k (x) ≤ si φ 2 (x) + 1 . i1,k  g∗  4 g ∗2 i

(44)

(45)

(46) (47)

i

Similar to Step 1, we have      si      φi2,k (ξ ) ≤  si  (φi2,k ◦ α¯ −1 (v(t) + D(t, t0 ))) 1  g∗   g∗  i  i    si   si  ≤  ∗  ϕi1,k (v(t)) +  ∗  ϕi2,k (D(t, t0 )), gi gi

(48)

where φi2,k ◦ α¯ 1−1 (·) = φi2,k (α¯ 1−1 (·)). Since D(t, t0 ) ∈ L∞ , it can be obtained that there exists a ϑi∗ > 0 satisfying ϕi2,j (D(t, t0 )) ≤ ϑi∗ . Then, we obtain        si  s 2 ϕ 2 (v(t)) si2 1 ϑi∗2   ϕi1,k (v(t)) +  si  ϕi2,k (D(t, t0 )) ≤ i i1,k + + + . (49) g  g  4 gi∗2 gi∗2 4 i i

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10

Substituting (45), (47), and (49) into (44), we have

2 2 fi,k si φi1,k (x) si ϕi1,k (v(t)) V˙i,k ≤ si ( + + + gi gi∗2 gi∗2

si

j i   j =1 m=1

2 (x (t)) ϑj,m m

+(

2(1 − τ ∗ )εi2

−

N θ˜ T θ˙ˆ  i,j i,j j =1

Let

∗i,k (Zi )

2gi∗2

+

1 )si gi∗2

si2   2 g¯ i ω˙ i si gi,k xi+1 1 ϑi∗2 1 s − ) + + ) ϑj,m (xm (t)) + yi y˙i + + (1 − i gi gi 2 4 2(1 − τ ∗ ) gi2 εi2 j =1 m=1 j

i

+

μ2i

=

fi,k gi

κi,j

+

−

N−1 

T g˙ˆ g˜ i,j i,j

j =1

2 (x) si φi1,k

gi∗2

T

bi,j

+

−

N−1 

˙ˆ ˜T   i,j i,j ςi,j

j =1

2 (v(t)) si ϕi1,k

gi∗2

si

+

(50)

.

j i   j =1 m=1

2 (x (t)) ϑj,m m

2(1−τ ∗ )εi2

+(

μ2i 2gi∗2

+

1 )s gi∗2 i

+

g¯ i s gi2 i

−

ω˙ i gi

with Zi =

[x T , v, ωi , ω˙ i ] . Based on Lemma 4, it yields ˙ˆ N−1 ˜T   si gi,k xi+1 1 ϑi∗2 i,j i,j ∗T Ti,k (Zi ) + si Di,k (Zi ) + + + + yi y˙i − V˙i,k ≤ si Wi,k gi 2 4 ςi,j j =1

+

j i N θ˜ T θ˙ˆ N−1 T g˙ˆ   g˜ i,j si2   2 1 i,j i,j i,j ) ϑ (x (t)) − − . (1 − m j,m ∗ 2 2(1 − τ ) κi,j bi,j εi j =1 m=1 j =1 j =1

(51)

Then, it can be obtained



2 s 2 Ti,k (Zi ) ∗T si Wi,k Ti,k (Zi ) ≤ i θi,k

2

1 + , 2

(52)

si2 di∗2 + , 2 2 ∗ 2 where θi,k = Wi,k and θˆi,k = θi,k − θ˜i,k is the estimation of θi,k ; di∗ is a positive constant. Substituting (41), (52), and (53) into (51), we get si Di,k (Zi ) ≤

(53)

2 N  si2 Ti,k (Zi ) d ∗2 ϑ ∗2 si gi,k xi+1 si2 T ˆ + σi,j θ˜i,j θˆi,k + + 1 + i + i + yi y˙i + θi,j V˙i,k ≤ 2 gi 2 2 4 j =1

−

N−1  j =1

Let Gi,k = N −1 j =1

gi,k gi .

T g˙ˆ g˜ i,j i,j

bi,j

−

N−1  j =1

˙ˆ ˜T   i,j i,j ςi,j

si2   2 1 ) ϑj,m (xm (t)). (1 − 2(1 − τ ∗ ) εi2 j =1 m=1 i

+

j

From Lemma 4, we get, when k = N , Gi,k = g Ti,k Tgi ,k (x¯i ) + Dgi ,k (x¯i ); and Gi,N =

βj g Ti,j Tgi ,j (x¯i )) + Dgi ,N (x¯i ) and gˆ i,k = g i,k − g˜ i,k being the estimation of g i,k .

From (42) and (54), one has 2 si2 Ti,k (Zi ) s2 d ∗2 ϑ ∗2 ˙ ˆ i,k xi+1 + si Dgi ,k (x¯i )xi+1 + i + 1 + i + i Vi,k ≤ θˆi,k + si G 2 2 2 4 j i  N N−1 2    s 1 2 T ˆ T + ϑj,m (xm (t)) + yi y˙i + σi,j θ˜i,j σgi ,j g˜ i,j gˆ i,j (1 − i2 ) θi,j + 2(1 − τ ∗ ) εi j =1 m=1 j =1 j =1

(54) 1 βN

(1 −

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−

˙ˆ ˜T   i,j i,j ςi,j

j =1

11

(55)

,

ˆ i,k = gˆ T T (x¯i ), k = N , and G ˆ i,N = where G i,k

1 βN

(1 −

N−1  j =1

T T (x¯ )). βj gˆ i,j i

Similarly, denote i ,k = Dgi ,k (x¯i )xi+1 . When k = N , i ,k = Ti,k Ti ,k (x¯i+1 ) + Di ,k (x¯i+1 ); and i,N = N −1 β T T j i,j i ,j (x¯i+1 ) ˆ i,k = i,k −  ˜ i,k being the estimation of i,k . − + Di ,N (x¯i+1 ), and  βN j =1

Substituting (43) into (55) yields 2 d2 si2 Ti,k (Z1 ) d ∗2 ϑ ∗2 ˙ ˆ i,k xi+1 + si  ˆ i ,k + i + 1 + i + i θˆi,k + si2 + si G Vi,k ≤ 2 2 2 4 j i N N−1 2   si   2 1 T ˆ T ˜ ) ϑ (x (t)) + y y ˙ + σ + σgi ,j g˜ i,j gˆ i,j (1 − + θ θ i i i,j i,j i,j j,m m ∗ 2 2(1 − τ ) εi j =1 m=1 j =1 j =1 +

N−1 

˜ Ti,j  ˆ i,j , σi ,j 

(56)

j =1

ˆ i ,k =  ˆ T Ti ,k (x¯i+1 ) and  ˆ i ,N = where  i,k

N−1  j =1

−

ˆ T T ,j (x¯i+1 ) βj  i,j i , βN

and di is a positive constant. Then, we have

V˙i,k ≤ Ui,k + Ui , where Ui,k =

(57)

2 si2 Ti,k (Z1 ) θˆi,k 2

2

ˆ i,k xi+1 + si  ˆ i ,k − 1 ∗ si2 + si2 + si G 2(1−τ ) εi

j i  

2 (x (t)) and U = 1 + ϑj,m m i

j =1 m=1 2 j i  N N−1 N−1 d     1 2 (x (t)) + T gˆ i ˜ T θˆi,j + ˜T  ˆ + ϑ σ σ g ˜ + σi ,j  θ ∗ m i,k g ,j i,j i j,m i,j i,j i,j i,j + yi y˙i . 2 2(1−τ ) j =1 m=1 j =1 j =1 j =1 i  From (15), we can get xi+1 = si+1 + yi+1 + αi . Define Vi∗ = Vl . From Lemma 1, we know l=1

V˙i∗ ≤

i N  

i 

l=1

l=1

+ +

λl sl2 −

1 2(1 − τ ∗ ) i  2 l

l=2 d ∗2

+

ϑi∗2 4

+

βj (Ul,j + Ul )

j =1 l=1 i 

≤−

di∗2 2

2

λ∗l (t)sl2 +

j i  l   l=1 j =1 m=1

2 si+1

−

2

(1 −

i i N−1    y2 1 T ( − 1)yl2 + i+1 + σgl ,j g˜ l,j gˆ l,j πl 2 l=1 j =1

l=2

sl2

i  N 

εl

l=1 j =1

)ϑ 2 (x (t)) + 2 j,m m

γ¯ (x)   cv ¯ ˜ Tl,j  ˆ l,j , + + σgl ,j  λ0 λ0

T ˆ σl,j θ˜l,j θl,j +

i 

Cl

l=1

i N−1

−

(58)

l=1 j =1

ϑ ∗2

2 d

where Cl = 1 + 2l + 4l + 2 l , l = 2, · · · , i. Similar to Step 1, the following filter is employed: πi+1 ω˙ i+1 + ωi+1 = αi , ωi+1 (0) = αi (0),

(59)

where πi+1 is a positive constant. According to (59), we obtain ω˙ i+1 = − πyi+1 , y˙i+1 = − πyi+1 − α˙ i . Similarly, there exists i+1 (x, s¯i+1 , v, y¯i+1 , θˆi,k , i+1 i+1 yd , y˙d , y¨d , ρ, ρ, ˙ ρ) ¨ ≥ 0 such that

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12

    y˙i+1 + yi+1  ≤ i+1 ,  πi+1 

(60)

where s¯i+1 = [s1 , · · · , si+1 ]T and y¯i+1 = [y2 , · · · , yi+1 ]T . Similarly, we have 2 yi+1

yi+1 y˙i+1 ≤ −

πi+1

+

2 yi+1

+

2

2i+1 2

(61)

.

Step n. For Step n, the Lyapunov function is chosen as: Vn = Vsn +

N θ˜ T θ˜  n,j n,j j =1

where V¯n =

+

2κ n,j

N−1 

T g˜ g˜ n,j n,j

2bn,j

j =1

+

j =1 m=1

˜T  ˜  n,j n,j

j =1

j  n   t

1 2(1−τ ∗ )

N−1 

2 t−τm (t) ϑj,m (xm (z))dz,

2ς n,j

Vs n =

sn2 2gn

+ V¯n ,

(62)

˜ n,j will be defined; κn,j > 0, + 12 yn2 ; θ˜n,j , g˜ n,j , and 

bn,j , ςn,j > 0 are design parameters. ˆ n,j , are designed as follows: The controller u and the adaptive laws θˆn,j , gˆ n,j , and  u = −λn sn −

2 N  βj sn Tn,j (Zn ) 2

j =1

θˆn,j − λ∗n (t)sn ,

(63)

2 s2 θ˙ˆ n,j = κn,j ( 2n Tn,j (Zn ) − σn,j θˆn,j ), j = k, θ˙ˆ n,j = −κn,j σn,j θˆn,j , j = k,

(64)

⎧ ⎪ g˙ˆ = bn,j (sn uTgn ,j (x) − σgn ,j gˆ n,j ), j = k = N, ⎪ ⎨ n,j g˙ˆ n,j = −bn,j σgn ,j gˆ n,j , j = k = N, ⎪ gn ,j (x) ⎪ ⎩ gˆ˙ n,j = bn,j (− βj sn uT − σgn ,j gˆ n,j ), k = N, βN

(65)

⎧ ˙ˆ ˆ ⎪ ⎪ ⎨ n,j = ςn,j (sn Tn ,j (Zu ) − σn ,j n,j ), j = k = N, ˙ ˆ n,j = −ςn,j σn ,j  ˆ n,j , j = k = N,  ⎪ ⎪ β s T ˙ˆ j n n ,j (Zu ) ⎩ ˆ n,j ), k = N, − σn ,j  n,j = ςn,j (− βN where λ∗n (t) =

γn 2(1−τ ∗ )(sn2 +εn2 )

j  n   t j =1 m=1

t−τmax

(66)

2 (x (z))dz with γ and ε being positive constants; λ > 2 and the ϑj,m m n n n T

definitions of Tn,j (Zn ), Tgn ,j (x), and Tn ,j (Zu ) with Zu = [x T , u] are similar to Step i; σn,j > 0, σgn ,j , σn ,j > 0 ˆ n,j will be given later. are parameters to be determined; θˆn,j , gˆ n,j , and  Then, we can deduce that N θ˜ T θ˙ˆ  fn,k gn,k u hn,k n,k ω˙ n g˙ n n,j n,j + + + − ] − 2 sn2 + V˙¯ n + yn y˙n − V˙n,k = sn [ gn gn gn gn gn gn κn,j j =1

−

N−1  j =1

T g˙ˆ g˜ n,j n,j

bn,j

−

N−1  j =1

˙ˆ ˜T   n,j n,j ςn,j

.

(67)

Similar to Step i, we have V˙n,k ≤ Un,k + Un , where Un,k and Un are defined as Ui,k and Ui with i = n, respectively.

(68)

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13

Choose the following state-dependent switching law: σ (t) = arg min{ j ∈N

Let Vn∗ =

n 

n 

Ui,j }.

(69)

i=1

Vi . According to Lemma 1, we get

i=1

V˙n∗ ≤

n N  

βj (Ui,j + Ui )

j =1 i=1

≤−

n 

λi si2 −

i=1

+ +

n 

λ∗i (t)si2 −

i=1

n n N−1    1 cv ¯ T ( − 1)yi2 − + σgi ,j g˜ i,j gˆ i,j πi λ0 i=1 j =1

i=2

j i  N n  n  n    si2 2 1 T ˆ ˜ (1 − )ϑ (x (t)) + σ + Ci θ θ m i,j i,j j,m i,j 2(1 − τ ∗ ) εi2 i=1 j =1 m=1 i=1 j =1 i=1 n N−1  

˜ Ti,j  ˆ i,j + σi ,j 

i=1 j =1 d ∗2

n  2 i

i=2

2

+

γ¯ (x) , λ0

(70)

2 d

ϑ ∗2

where Ci = 1 + 2i + 4i + 2 i , i = 2, · · · , n. Let p > 0 be a constant, where Vn∗ (0) ≤ p, and define a compact as = {Vn∗ ≤ p}. Remark 4. The dynamic surface control method is introduced in this section. Compared with the results in [13–15], we do not need to take the derivative of λ∗i (t). The computational complexity of proposed method in this paper is greatly decreased and the design of the controller is much easier. 3.2. Stability analysis Theorem 1. Consider system (1) with Assumptions 1–5 for any bounded initial condition. Controller (63), adaptive laws (18)–(20), (41)–(43), (64)–(66), and state-dependent switching law (69) can guarantee that all signals of the corresponding closed-loop system are SGUUB and the tracking error satisfies the prescribed performance. Proof. Since ∗  y × is a compact set and i+1 (·) and γ¯ (x) are continuous functions, there exist constants ∗i > 0 and γ ∗ > 0 satisfying |i+1 (·)| ≤ ∗i and |γ¯ (x)| ≤ γ ∗ on ∗ , respectively. Then, from (70), we get V˙n∗ ≤ −

n 

λi si2 −

i=1

− where C ∗ =

cv ¯ − λ0

n  i=1

(1 +

n 

λ∗i (t)si2 −

i=1 j =1

i=1 n N −1  

˜T  ˜ σi ,j  i,j i,j 2

i=1 j =1 di∗2 2

+

N n   σi,j

ϑi∗2 4

+

+

2

T ˜ θ˜i,j θi,j −

i

i=1 j =1

1 2(1 − τ ∗ )

N σ θT θ  i,j i,j i,j 2 + 2 j =1

2 d i

n N−1   σg ,j

+

j n  i  

2

(1 −

i=1 j =1 m=1

N−1  σgi ,j g Ti,j gi,j j =1

2

+

T g˜ i,j − g˜ i,j

n  1 ( − 1)yi2 πi i=2

si2 εi2

2 )ϑj,m (xm (t)) + C ∗ ,

N−1  σi ,j Ti,j i,j 2 j =1

)+

n  i=2

(71) ∗2 i 2

+ γ ∗.

To show the boundedness of the system (1), three cases are discussed as follows: Case 1. When |si | ≤ εi , i = 1, · · · , n, we have −λ∗i (t)si2

≤ −γi V¯i +

εi2 γi

t j i  

2(1 − τ ∗ )(si2 + εi2 ) j =1 m=1

t−τmax

2 ϑj,m (xm (z))dz.

(72)

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14

Since ϑj,m (xm (t)) is a continuous function, we define ϑi,max = V˙n∗ ≤ −

n 

λi si2 −

i=1 j =1

i=1

−

N n   σi,j

n N−1   σg ,j i

2

2

T ˜ θ˜i,j θi,j −

i=1 j =1 ≤ −c1 Vn∗ + D1 ,

where D1 = C ∗ +

n  i=1

j =1 m=1

2 (x (t)). Then, we get max ϑj,m m

n n   1 cv ¯ ( − 1)yi2 − γi V¯i − πi λ0 i=2

T g˜ i,j − g˜ i,j

j i  

i=1

−1 σ n N ˜T  ˜   i ,j  i,j i,j

2

i=1 j =1

+ D1 (73)

γi τmax +1 2(1−τ ∗ ) ϑi,max

and c1 = min{2λ1 , · · · , 2λn , σ1,j κ1,j , · · · , σn,j κn,j , σg1 ,j b1,j , · · · , σgn ,j bn,j ,

¯ > 0. σ1 ,j ς1,j , · · · , σn ,j ςn,j , π22 − 2, · · · , π2n − 2, γ1 , · · · , γn , c}

Similar to [47,48], we choose the design parameter as c1 > Dp1 , then V˙n∗ < 0 on Vn∗ = p, which implies that Vn∗ ≤ p is an invariant set. That is to say, if Vn∗ (0) ≤ p, then Vn∗ (t) ≤ p for all t > 0. From (73), we have 0 ≤ Vn∗ (t) ≤ e−c1 t (Vn∗ (0) −

D1 D1 )+ . c1 c1

Hence, all states of the corresponding closed-loop system are SGUUB in Case 1. Case 2. When |si | > εi , i = 1, · · · , n, we have γi −λ∗i (t)si2 ≤ − V¯i . 2 From (71), we obtain V˙n∗ ≤ −

n 

λi si2

−

i=1 j =1

i=1

−

n N−1  

n  N  σi,j

2

T ˜ θ˜i,j θi,j −

σgi ,j T g˜ g˜ i,j − 2 i,j

i=1 j =1 ≤ −c2 Vn∗ + C ∗ ,

n n   γi ¯ 1 cv ¯ 2 ( − 1)yi − Vi − πi 2 λ0 i=2

n N −1  

(74)

i=1

˜T  ˜ σi ,j  i,j i,j

i=1 j =1

2

+ C∗ (75)

where c2 = min{2λ1 , · · · , 2λn , σ1,j κ1,j , · · · , σn,j κn,j , σg1 ,j b1,j , · · · , σgn ,j bn,j , σ1 ,j ς1,j , · · · −2, · · · , γ1 γn 2 ¯ > 0. πn − 2, 2 , · · · , 2 , c} ∗ We choose the design parameter as c2 > Cp , then V˙n∗ < 0 on Vn∗ = p, which implies that Vn∗ ≤ p is an invariant set. That is to say, if Vn∗ (0) ≤ p, then Vn∗ (t) ≤ p for all t > 0. From (75), we get , σn ,j ςn,j , π22

0 ≤ Vn∗ (t) ≤ e−c2 t (Vn∗ (0) −

C∗ C∗ )+ . c2 c2

Hence, all states of the corresponding closed-loop system are SGUUB in Case 2.     Case 3. Define  = { : |s | > ε ,  = 1, · · · , n} and η = {η : sη  ≤ εη , η = 1, · · · , n}(η = ). For  = { : |s | > ε ,  = 1, · · · , n}, define the following Lyapunov function V!∗ =

 ∈!

(Vs +

N N−1 T N−1 ˜ T  T ˜ 1  θ˜,j θ˜,j 1  g˜ ,j g˜ ,j 1 v ,j ,j + + + V¯ + ), 2 κ,j 2 b,j 2 ς,j λ0 j =1

j =1

j =1

we have V˙!∗ ≤ −c V!∗ + C∗ , where c and C∗ are defined similar to c2 and C ∗ given in Case 2.

(76)

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For

 η

15

  = {η : sη  ≤ εη , η = 1, · · · , n}, construct the Lyapunov function as

V!∗η =



(Vsη +

η∈!η

N N−1 T N−1 ˜ T  T ˜ 1  θ˜η,j θ˜η,j 1  g˜ η,j g˜ η,j 1 v η,j η,j + + + V¯η + ), 2 κη,j 2 bη,j 2 ςη,j λ0 j =1

j =1

j =1

we can obtain V˙!∗η ≤ −cη V!∗η + D!η ,

(77)

where cη and D!η are defined similar to c1 and D1 given in Case 1. Follow the proof line, we can obtain that all signals of the resulting closed-loop system are SGUUB in Case 3.

2

˜ i,j , V¯i and v are bounded. According to the aforementioned discussions, si , yi , θ˜i,j , g˜ i,j ,  Remark 5. It can be seen from (73), (75), (76) and (77) that the designed parameters λi , σi,j , σgi ,j , σi ,j , γi , π2 , · · · , πn , κi,j , bi,j , ςi,j , and c¯ are related with the bounds and the convergent rates. Decreasing σi,j , σgi ,j , σi ,j , and γi may lead to reduction of the bounded regions. Increasing λi , σi,j , σgi ,j , σi ,j , κi,j , bi,j , ςi,j , γi , c¯ and decreasing π2 , · · · , πn may accelerate convergence of the system states. Hence, design parameters can be adjusted by considering a tradeoff between the convergence performance and the bounded regions. 4. Demonstrative examples Example 1. Consider the following switched nonlinear system: Subsystem 1: ⎧ ⎪ ξ˙ = q(ξ, x, t), ⎪ ⎪ ⎨ x˙1 = x1 e−0.5x1 + (1 + 0.5 sin x1 )x2 + 2x 2 (t − τ1 (t)) + x1 sin t + ξ 2 sin x1 , 1 ⎪ x˙2 = x1 x22 + 0.15x2 (t − τ2 (t)) sin(x2 (t − τ2 (t))) + (x12 + x22 )sin3 t + x2 ξ sin t sin x1 , ⎪ ⎪ ⎩ y = x1 , Subsystem 2: ⎧ ⎪ ξ˙ = q(ξ, x, t), ⎪ ⎪ ⎨ x˙1 = x12 + (0.5 cos x1 )x2 + 2x12 (t − τ1 (t)) sin x1 + 0.5ξ sin(x1 x23 ), 2 2 ⎪ ⎪ x˙2 = x1 x2 + (2 + cos(x1 x2 ))u + 0.2x2 (t − τ2 (t)) sin(x2 (t − τ2 (t))) + x1 cos t + ξ cos x1 , ⎪ ⎩y =x , 1 where q = −ξ + x12 + 0.5, τ1 (t) = 0.2 + 0.2 sin t, τ2 (t) = 0.5(2 − cos t), ϑ1,1 (x1 ) = x12 , ϑ2,1 (x1 ) = 0, ϑ2,2 (x2 ) = |x2 |, and v˙ = −0.8v + 1.2x12 + 1.5. Since g2,1 (x) = 0, we can obtain that the subsystem 1 is uncontrollable. In addition, from the form of g1,2 (x1 ), we get that the sign of g1,2 (x1 ) is allowed to be variable. Choose the desired trajectory as yd = 0.2sint. Besides, fuzzy logic systems are constructed by the following functions: (xi − 3 + l)2 ), l = 1, 2, 3, 4, 5. i 2 From (18)–(20) and (64)–(66), the following adaptive laws are given: ⎧ ⎨ θ˙ˆ = κ ( s12 r 2 T1,j (Z1 ) 2 − σ1,j θˆ1,j ), j = k, 1,j 1,j 2 ⎩ θ˙ˆ = −κ σ θˆ , j = k, 1,j 1,j 1,j 1,j

˙gˆ = b1,1 (s1 x2 Tg ,1 (Zg ) − σg ,1 gˆ 1,1 ), j = k = 1, 1,1 1 1 1 ˙gˆ = b1,1 (− β1 s1 x2 Tg1 ,1 (Zg1 ) − σg ,1 gˆ 1,1 ), k = 2, μF l = exp(−

1,1

β2

1

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¯ and −δρ (dashed line). Fig. 1. Tracking error E1 (solid line), prescribed performance δρ

Fig. 2. Control signal u.

⎧ ˙ˆ ⎨

ˆ 1,1 ), j = k = 1, 1,1 = ς1,1 (s1 T1 ,1 (Z1 ) − σ1 ,1  β s T (Z )  1 1  ,1 ˙ 1 1 ⎩ ˆ 1,1 ), k = 2, ˆ 1,1 = ς1,1 (− − σ1 ,1  β2 ⎧ 2 ⎨ θ˙ˆ = κ ( s22 − σ2,j θˆ2,j ), j = k, 2,j 2,j 2 T2,j (Z2 ) ⎩ θˆ˙ = −κ σ θˆ , j = k. 2,j

2,j 2,j 2,j

˙ gˆ 2,1 = b2,1 (s2 uTg2 ,1 (x) − σg2 ,1 gˆ 2,1 ), j = k = 1, β1 s2 uTg2 ,1 (x) − σg2 ,1 gˆ 2,1 ), k = 2, g˙ˆ 2,1 = b2,1 (− β2

⎧ ˙ˆ = ς (s T ⎨ 2,1 2,1 2 

ˆ 2 ,1 (Zu ) − σ2 ,1 2,1 ), j = k = 1, β s T (Z ) u 1 2  ,1 ˙ ⎩ 2 ˆ 2,1 = ς2,1 (− ˆ 2,1 ), k = 2, − σ2 ,1  β2 where Zu = [x1 , x2 , u]T . According to (17) and (63), α1 and u are given as 2 2  s1 βj T1,j (Z1 ) λ∗ (t)s1 λ1 + 1 α1 = − s1 − s1 r − θˆ1,j − 1 , r 2r r j =1 2 2  βj s2 T2,j (Z2 ) θˆ2,j − λ∗2 (t)s2 . u = −λ2 s2 − 2 j =1

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Fig. 3. Trajectories y (solid line), yd (dashed line).

Fig. 4. Switching signal σ (t).

Fig. 5. Cascade chemical reactor system.

Choose the parameters as follows: x1 (0) = 0.2, x2 (0) = 0.2, θˆ1,1 (0) = 0.4, θˆ1,2 (0) = 0.2, θˆ2,1 (0) = 0.1, θˆ2,2 (0) = 0.4, ˆ 1,1 (0) = 0,  ˆ 2,1 (0) = 0, ω2 (0) = 0, z = 0.1, v = 0.1, π2 = 0.9, γ1 = γ2 = 0.01, λ1 = 30, gˆ 1,1 (0) = 0, gˆ 2,1 (0) = 0,  λ2 = 150, κ1,1 = 0.02, κ1,2 = 0.03, κ2,1 = 0.02, κ2,2 = 0.015, b1,1 = 30, b2,1 = 15, ς1,1 = 0.25, ς2,1 = 0.2, σ1,1 = 0.1, σ1,2 = 0.15, σ2,1 = 0.15, σ2,2 = 0.1, σg1 ,1 = 0.1, σg2 ,1 = 0.2, σ1 ,1 = 0.2, σ2 ,1 = 0.1, ρ (t) = 0.9e−3t + 0.25, δ = 0.5, δ¯ = 0.8, β1 = 0.6, β2 = 0.4 and ε1 = ε2 = 0.1. From Fig. 1 and Fig. 3, we can obtain that tracking error satisfies the prescribed performance and the system output y can follow the desired trajectory yd under the control signal and switching signal shown in Fig. 2 and Fig. 4.

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¯ and −δρ (dashed line). Fig. 6. Tracking error E1 (solid line), prescribed performance δρ

Fig. 7. Control signal u.

Example 2. Consider a cascade chemical system with two reactors described in [49,50] (see Fig. 5), where the system runs in the environment with some changeable parameters and disturbances. The chemical system can be depicted by the following switched nonlinear time-delay system: ⎧ ξ˙ = q(ξ, x, t), ⎪ ⎪ ⎪ 2 ⎪ x˙1 (t) = −k1,σ (t) x1 (t) − θ1,σ1 (t) x1 (t) − θ1,σ1 (t) x1 (t − τ1 (t)) + 1−R ⎪ V1 x2 (t) + δ1,σ (t) (x1 (t − τ1 (t))) ⎪ ⎪ ⎨ + 1,σ (t) (ξ, x, t), 2 ⎪ x˙2 (t) = −k2,σ (t) x2 (t) − θ2,σ1 (t) x22 (t) + RV21 x1 (t − τ1 (t)) − θ2,σ1 (t) x2 (t) + R ⎪ V2 x2 (t − τ2 (t)) ⎪ ⎪ F ⎪ + V2 u(t) + δ2,σ (t) (x2 (t − τ2 (t))) + 2,σ (t) (ξ, x, t), ⎪ ⎪ ⎩ y = x1 , where N = {1, 2}, θ11 = θ21 = 2, θ1,2 = θ2,2 = 2.5, k1,1 = k2,1 = 0.5, k1,2 = k2,2 = 0.6, R1 = R2 = 0.5, F = 0.5, δ1,1 = 0.5x1 (t − τ1 (t)), δ1,1 = 0.5x1 (t − τ1 (t)), δ1,2 = 0.6x1 (t − τ1 (t)), δ2,1 = 0.5x22 (t − τ2 (t))e0.01x2 (t−τ2 (t)) , δ2,2 = 0.4x22 (t − τ2 (t))e0.01x2 (t−τ2 (t)) , and others are same as Example 1. We choose the parameters as follows: x1 (0) = 0.1, x2 (0) = 0.2, θˆ1,1 (0) = θˆ1,2 (0) = θˆ2,1 (0) = θˆ2,2 (0) = 0, ˆ 1,1 (0) = 0,  ˆ 2,1 (0) = 0, ω2 (0) = 0, z = 0.1, v = 0.1, π2 = 0.9, γ1 = γ2 = 0.01, λ1 = 20, gˆ 1,1 (0) = 0, gˆ 2,1 (0) = 0,  λ2 = 200, κ1,1 = 0.02, κ1,2 = 0.03, κ2,1 = 0.02, κ2,2 = 0.015, b1,1 = 30, b2,1 = 15, ς1,1 = 0.25, ς2,1 = 0.2, σ1,1 = 0.1, σ1,2 = 0.15, σ2,1 = 0.15, σ2,2 = 0.1, σg1 ,1 = 0.1, σg2 ,1 = 0.2, σ1 ,1 = 0.2, σ2 ,1 = 0.1, ρ (t) = 0.9e−3t + 0.25, δ = 0.5, δ¯ = 0.8, β1 = 0.6, β2 = 0.4, and ε1 = ε2 = 0.1.

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Fig. 8. Trajectories y (solid line), yd (dashed line).

Fig. 9. Tracking error E1 with prescribed performance control.

Fig. 10. Tracking error E1 without prescribed performance control.

From Fig. 6 and Fig. 8, we get that our control objective can be achieved under the control signal shown in Fig. 7. In addition, Fig. 9 and Fig. 10 show that superior transient performance and fast convergence rate can be provided by the proposed method in our paper.

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5. Conclusions Using the convex combination technique, we presented a prescribed performance dynamic surface adaptive fuzzy control approach for time-varying delay switched nonlinear systems. The proposed method contained the following advantages: (i) Some subsystems were allowed to be uncontrollable; (ii) The amount of calculation was effectively reduced by utilizing the dynamic surface method; (iii) The system transient performance was also considered. One future topic is sampled-data control for time-delay switched nonlinear systems with Markovian jump parameters. Acknowledgement This work was supported by the National Natural Science Foundation of China (No. 61773211, 61673219). References [1] T.M. Guerra, A. Sala, K. Tanaka, Fuzzy control turns 50: 10 years later, Fuzzy Sets Syst. 281 (2015) 168–182. [2] T.M. Guerra, L. Vermeiren, LMI-based relaxed nonquadratic stabilization conditions for nonlinear systems in the Takagi–Sugeno’s form, Automatica 40 (5) (2004) 823–829. [3] T.M. 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