Asymptotic tracking control for time-delay nonlinear systems with parametric uncertainties and full state constraints

ISA Transactions xxx (xxxx) xxx

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Research article

Asymptotic tracking control for time-delay nonlinear systems with parametric uncertainties and full state constraints ∗

Chun-Xiao Wang a , , Yu-Qiang Wu b , Yan Zhao a , Jia-Li Yu c a

School of Science, Shandong Jianzhu University, Ji’nan, 250101, China Institute of Automation, Qufu Normal University, Rizhao, 276826, China c School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, China b

article

info

Article history: Received 28 May 2018 Received in revised form 21 August 2019 Accepted 23 August 2019 Available online xxxx Keywords: BLF Full state constraints Time-delay Adaptive backstepping control

a b s t r a c t This article concentrates on an adaptive backstepping control design of time-delay strict-feedback uncertain nonlinear systems subject to full state constraints. The tan-type barrier Lyapunov functions (tBLFs) and Lyapunov–Krasovskii function are united together, which successfully get over the difficulties of system design in which the first function is involved to ensure full state constraints satisfaction and the second is established to eliminate the effect of delayed states. By employing a new control scheme, asymptotic tracking performance is arrived, and all the states remain in the desirable regions for all the system running time. Meanwhile, the boundedness of all signals of the closed-loop system is guaranteed. The performance of the control scheme is illustrated through a class of single degree of freedom (1-DOF) time-delay electrostatic microactuator systems. © 2019 Published by Elsevier Ltd on behalf of ISA.

1. Introduction Constraints are very common in physical systems. They may come in different forms, such as, saturation, performance specifications, safety specifications, downtime, etc. Unimaginable consequences will result in if these constraints are violated. For instance, in order to protect the electrode plate of a single degree of freedom (1-DOF) electrostatic microactuator system [1], the movable plate must avoid contacting with the fixed electrodes. Thus, the travel path of the movable plate is constrained. Driven by theoretical challenges and the urgent need of engineering research, control design for the constrained system has received extensive attentions in recent decades [2–20]. Many significant results about constraint-handling have been generated. For example, the set invariance notions [2], the reference governors [3], and the model predictive control [4] are all methods to handle constraint. Additionally, a novel kind of Lyapunov function which was originally proposed by Ngo in [5], called barrier Lyapunov function (BLF), is widely used in the treatment of constraint problems [6–16]. Since the appearance of BLF, it has been successfully applied to the systems subject to output and state constrains, and great achievements have been gained. Originally, BLF-based tracking control was applied to the output constrained nonlinear systems including static and time-varying cases [6,7]. Later, BLFs ∗ Corresponding author. E-mail address: [email protected] (C.-X. Wang).

was gradually extended to address for the partial and full state constrained strict-feedback nonlinear system [8–12]. Moreover, for its practicality, BLF was gradually applied to various types of constrained systems effectively, such as pure-feedback nonlinear systems [13,14], switched nonlinear systems [15], and stochastic systems [16]. BLF was also greatly promoted the development of constrained practical systems, such as electrostatic microactuators [17], uncertain robot [18], spacecraft [19], underactuated crane systems [20] and so on. BLF has proven to be an important tool to deal with system constraints. Moreover, the phenomenon of time-delay commonly exists in various systems inevitably in the field of practical engineering, such as teleoperation systems, robot systems, microwave oscillators, neural network and so on. The existence of time-delay could severely degrade the system’s performance. In addition, it also will make the controller design more complicated. Thus, consideration of time-delay in system design is meaningful and challenging. Over the past decades, there have achieved many fruitful time-delay related results, such as Lyapunov–Krasovskii functional theorem and Razumikhin theorem [21,22]. By constructing suitable Lyapunov–Krasovskii functional, items containing delayed state can be matched, thus ensuring that they are eliminated in subsequent backstepping designs [23–25]. Since then, the research of time-delay systems has undergone a great leap, both theoretically and practically. Based on Lyapunov–Krasovskii functions, all kinds of control methods for nonlinear time-delay systems are developed, such as sliding mode control [26], neural

https://doi.org/10.1016/j.isatra.2019.08.038 0019-0578/© 2019 Published by Elsevier Ltd on behalf of ISA.

Please cite this article as: C.-X. Wang, Y.-Q. Wu, Y. Zhao et al., Asymptotic tracking control for time-delay nonlinear systems with parametric uncertainties and full state constraints. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.038.

2

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network control [27,28], fuzzy control [23,29], adaptive backstepping control [30,31], dynamic surface control [32], L∞ control [33] and so on. Besides constraint and time-delay, system uncertainties are also important factors leading to system instability and dynamic performance degradation. A large number of research results show that adaptive control is a very effective method to deal with parametric uncertain systems [34,35]. In recent years, for many uncertain systems will be subject to output or state constraints, the combination of BLF and adaptive control method has been generated and made great progress [6,8,12,36]. Despite the great progress in this field, it is still unusual to attempt for control design of constrained systems with uncertain parameter and time-delay. It is more difficult and meaningful to control the system when state constraints and dynamic time-delay exist simultaneously, especially for nonlinear systems with parametric uncertainty. Inspired by the above works, this article aims to tackle the state constrained tracking control problem for a class of uncertain time-delay nonlinear systems. Some interesting results obtained in this paper are listed below: (1) With suitable Lyapunov–Krasovskii functionals are constructed, terms including the delayed states can be eliminated in the backstepping design by taking the derivative of the Lyapunov functional. Meanwhile, proper tBLFs are constructed, all states are limited into desirable regions. (2) The tBLF and Lyapunov–Krasovskii function are united together to act as the Lyapunov function, based on which to design the adaptive backstepping procedure. This successfully get over the difficulties of system design in both delayed and constrained states. (3) A new backstepping-based adaptive tuning function design is proposed, which ensures asymptotic tracking performance and the uniform boundedness of the closed-loop system signals. The rest part of this article is organized as follows. Section 2 provides some mathematical preliminaries and statements of the problem. The detailed controller design process is developed in Section 3. Section 4 provides the main results of this paper. Simulation on a 1-DOF electrostatic microactuator is given to demonstrate the effectiveness of the developed control scheme in Section 5. Finally, Section 6 draws the conclusion of this paper. Notation In this article, all the notations are standard. R+ stands for the set of all the nonnegative real numbers, Rn represents the real n-dimensional space. Define x¯ i (t) = (x1 (t), . . . , xi (t))T , and x(t) = (x1 (t), . . . , xn (t))T ∈ Rn . We also denote x¯ i (t − τi ) = (x1 (t − τ1 ), . . . , xi (t − τi ))T , and S¯j (t − τi ) = (S1 (t − τi ), . . . , Sj (t − τi ))T . ∥ · ∥ denotes the Euclidean vector norm in Rm , diag(·) refers to the diagonal matrix. In addition, arguments of the functions will be omitted in the following as long as the context does not cause confusion. 2. Problem statement

hi (·) are uncertain smooth functions with unknown time delays τi , i = 1, . . . , n. The purpose of this article is to design an state-feedback controller to guarantee that the output y tracks the designed trajectory yd (t) asymptotically. Furthermore, for any initial states xi (0), as long as |xi (0)| < kci (kci (i = 1, . . . , n) are positive constants), all the states are limited into the desirable regions: Ωxi := {xi (t) ∈ R, |xi (t)| < kci , ∀t ≥ 0}(i = 1, . . . , n). At the same time, all signals of the closed-loop system remain bounded for all the system running time. For dealing with the state constraints, we do the following coordinates transformation, S1 = x1 − yd , Si = xi − αi−1 , i = 2, . . . , n, where Si (i = 1, . . . , n) denote state tracking errors (or virtual), αi−1 (i = 2, . . . , n) are the stabilizing function to be constructed in Step i of Section 3. All the designed stabilizing functions require bounds: |αi−1 | < α¯ i−1,0 , α¯ i−1,0 are positive constants which will be given in more detail later. Some reasonable assumptions and lemma about the system (1) are necessary for the subsequent development. Assumption 1. gi (·)(i = 1, . . . , n) are known smooth functions which satisfy 0 < g0 ≤ |gi (·)|, where g0 is a positive constant. Without loss of generality, for convenience of discussion, gi (·) is further assumed to be positive in this paper. Assumption 2. The reference signal yd (t) is continuous and differentiable to order n. Specifically, there exist positive ⏐ ⏐ constants (i)

Yi , i = 0, . . . , n, satisfy |yd (t)| ≤ Y0 < kc1 and ⏐yd (t)⏐ ≤ Yi , i = 1, . . . , n, ∀t ≥ 0.

⏐

⏐

Assumption 3 ([32]). The uncertain nonlinear time-delay functions hi (·) : Ri → R (i = 1, . . . , n) satisfy the following inequality.

|hi (x¯ i (t − τi ))| ≤

i ∑

( ) |Sj (t − τi )|qij S¯j (t − τi ) ,

(2)

j=1

where qij (·)(i = 1, . . . , n, j = 1, . . . , n) are known continuous functions. Lemma 1 ([32]). For any real numbers F1 , . . . , Fn , the following inequality holds,

(

n ∑

)2 Fi

≤n

i=1

n ∑

Fi2 .

(3)

i=1

3. Control design via BLF

Consider the following time-delay nonlinear systems x˙ i (t) = fi (x¯ i (t)) + gi (x¯ i (t))xi+1 (t) + hi (x¯ i (t − τi )), i = 1, 2, . . . , n − 1, x˙ n (t) = fn (x(t)) + gn (x(t))u(t) + hn (x¯ n (t − τn )), y(t) = x1 (t),

(1) n

where x(t) ∈ R , y(t) ∈ R and u(t) ∈ R denote the states, output, and control input of the system, respectively. Smooth functions gi (·)(i = 1, . . . , n) are known; fi (·) : Ri → R (i = 1, . . . , n) are uncertain smooth functions satisfy fi (x¯ i ) = θ T ϕi (x¯ i ), where the uncertain vector θ ∈ Ωθ = {θ ∈ Rm , ∥θ∥ ≤ θM , θM ∈ R+ } denotes parametric uncertainties, and ϕ1 , . . . , ϕn are smooth functions;

In this section, the detailed controller design and the corresponding stability analysis for the system (1) are given. The controller design is based on BLF via adaptive backstepping control combined with tuning function. In order to satisfy the state constraints, the following tBLF [11] is introduced: ∗

Vi =

k2b

i

π

( tan

π Si2 2k2b

) , |Si (0)| < kbi , i = 1, . . . , n

(4)

i

where kb1 = kc1 −Y0 > 0 and kbi = kci −α¯ i−1,0 > 0 for i = 2, . . . , n are positive constants denoting the constraints on Si , given by Si ∈ Ωs := {Si ∈ R, |Si | < kbi , i = 1, . . . , n, ∀t ≥ 0}.

Please cite this article as: C.-X. Wang, Y.-Q. Wu, Y. Zhao et al., Asymptotic tracking control for time-delay nonlinear systems with parametric uncertainties and full state constraints. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.038.

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Remark 1. When the state of the system is free, that is kci → ∞, so kbi → ∞, i = 1, · · · , n. We can use L’hopital’s rule to find the limit lim

k2b i

kb →∞ i

π

( tan

π

Si2 2k2b i

)

1

=

2

S˙1 = x˙ 1 − y˙ d = f1 (x1 ) + g1 (x1 )x2 + h1 (¯x1 (t − τ1 )) − y˙ d = θ T ϕ1 (x1 ) + g1 (x1 )(S2 + α1 ) + h1 (¯x1 (t − τ1 )) − y˙ d .

(6)

θ˜ T Γ −1 θ˜ +

n ∑

∫

S12 (τ )q2j1 (S¯1 (τ ))dτ , (7) t −τj

j=1

˙

V˙1 = ϑs1 · θ T ϕ1 + g1 (S2 + α1 ) + h1 (x¯ 1 (t − τ1 )) − y˙d + θ˜ T Γ −1 θˆ

(

)

(n + 1 − j)S12 (t − τj )q2j1 (S1 (t − τj ))

V˙ 1 ≤ −K1 tan

)

α˙ 1 =

+

+ g1 ϑs1 S2 + θ˜ T Γ −1 (θ˙ˆ − Γ Λ1 ) + D1 , (13)

2k2b

(14)

(15)

n ∑

∗

V2 = V1 + V2 +

∫

t

S22 (τ )q2j2 (S¯2 (τ ))dτ .

(n + 1 − j)

(16)

t −τj

j=2

It is obviously V2 > 0, taking the time derivative of (16), then

( V˙ 2 ≤ −K1 tan

π S12

)

( ) + g1 ϑs1 S2 + θ˜ T Γ −1 θ˙ˆ − Γ Λ1 + D1

2k2b 1

(17)

j=2 n ∑

(n + 1 − j)S22 (t − τj )q2j2 (S¯2 (t − τj )).

j=2

Using the famous Young’s inequality, we can infer that 1 4

ϑs21 + h21 (x¯ 1 (t − τ1 )).

(9)

∂α1 1 h1 (x¯ 1 (t − τ1 )) ≤ ϑs22 ∂ x1 4

+ D1 ,

1 2

(

)2

+ h21 (x¯ 1 (t − τ1 )), (18)

1

ϑs22 + h22 (x¯ 2 (t − τ2 )). 2

1 (10)

2

h22 (x¯ 2 (t − τ2 )) + h21 (x¯ 1 (t − τ1 )) =

in which D1 is denoted by n

∑

(n + 1 − j)S12 (t − τj )q2j1 (S1 (t − τj )). (11)

j=1

For convenience and clarity, notions ω1 (x1 ) = ϕ1 as the first regressor vector. Analyze the form of (10), we can design the

2 ∑ 1 j=1

j=1

D1 = h21 (x¯ 1 (t − τ1 )) −

∂α1 ∂ x1

(19)

From the above two inequalities, we can see that the time-delay terms have been isolated, and now we combine them together

4

j)S12 (t)q2j1 (S1 (t))

−ϑs2 ·

ϑs2 · h2 (x¯ 2 (t − τ2 )) ≤

) 1 θ T ϕ1 + g1 (S2 + α1 ) + ϑs1 − y˙d + θ˜ T Γ −1 θ˙ˆ

(n + 1 −

)

π S12

Denote the augmented Lyapunov function candidates V2 as follows

for i = 1, . . . , n. In order to separate the

Substituting (9) into (8), it leads to

n ∑

j=1

) ∂α1 ( T θ ϕ1 + g1 x2 + h1 (x¯ 1 (t − τ1 )) ∂ x1 ( )T 1 ∑ ∂α1 ∂α1 (j+1) ˙ˆ + + y θ. (j) d ˆ ∂θ ∂ yd j=0

(8)

2k2 bi

ϑs1 · h1 (x¯ 1 (t − τ1 )) ≤

(

(n + 1 − j)q2j1 (S1 (t))⎠ ,

In which

time-delay terms, we introduce the Young’s inequality, then

V˙1 ≤ ϑs1 ·

∑

1

− π Si2

⎞

n

in which Λ1 = ϑs1 ω1 is called tuning function. Step 2. For S2 = x2 − α1 , its time derivative is

j=1

(

π S12

)

) ( + ϑs2 θ T ϕ2 + g2 (S3 + α2 ) + h2 (x¯ 2 (t − τ2 )) − α˙ 1 n ∑ + (n + 1 − j)S22 (t)q2j2 (S¯2 (t))

n ∑

cos2

4

2k2b 1

(

(n + 1 − j)S12 (t)q2j1 (S1 (t))

where ϑsi =

1

− θˆ T ω1 + y˙ d − ϑs1

2S1

where K1 is a designed positive constant. By substituting (12) back into (10) and after some arrangements, we have

n ∑

Si

k2b

1

⎜− g1 ⎝

− S1 cos2

j=1

−

)

(12)

t

(n + 1 − j)

where θ˜ = θˆ − θ (θˆ is the estimate of θ ) and Γ = diag(γ1 , . . . , γn ) > 0. Obviously, V1 is continuous and differentiable in Ωs × Ωθ , by taking the time derivative of V1 directly, it yields

+

1 ⎜

π S12

S˙2 = x˙ 2 − α˙ 1 = θ T ϕ2 + g2 (S3 + α2 ) + h2 (x¯ 2 (t − τ2 )) − α˙ 1 .

To deal with the system uncertainty, delayed state, and constrained state simultaneously, the following Lyapunov function is constructed

2

K1 sin

(

Following, the detailed adaptive backstepping design procedure is presented, which is essential to our main results. Step 1. From the first coordinate transformation S1 = x1 − yd , taking the time derivative of it, we have

1

(

⎛

(5)

Hence, if some states are free, we can replace the BLF with a quadratic function. Based on this, the control analysis of the system under free states and constrained states can be unified. In conclusion, the tBLF is more practical than the log-type BLF(lBLF) [14].

V1 = V1∗ +

stabilizing function α1 as

α1 = Si2 .

3

j

h2j (x¯ j (t − τj )).

(20)

Thus the second stabilizing function can be given by

( { K2 sin 1 α2 = − g2

+

π S22

)

k2b

2

2S2

1 ∑ ∂α1 j=0

(j+1)

yd ∂ y(j) d

− θˆ T ω2 +

( +

∂α1 ∂ θˆ

)T

∂α1 g1 x2 ∂ x1

Γ Λ2

Please cite this article as: C.-X. Wang, Y.-Q. Wu, Y. Zhao et al., Asymptotic tracking control for time-delay nonlinear systems with parametric uncertainties and full state constraints. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.038.

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g1 ϑs1 S2

)2

1 1 ∂α1 − ϑs2 − ϑs2 2 4 ∂ x1 ( ) n } 2 ∑ π S2 2 2 ¯ −S2 cos (n + 1 − j)qj2 (S2 (t)) , 2

−

(

Using the same method as before, the time-delay terms are merged

ϑs2

2kb

(21)

k

j=2

2

in which K2 > 0; the second regressor vector ω2 = ϕ2 − ∂α1 ϕ , and the second tuning function Λ2 = Λ1 + ω2 ϑs2 . Then, ∂ x1 1 substituting the stabilizing function (21) into (17), and using the obtained inequalities (18) and (19), the derivative of V2 can be rewritten in the form: V˙ 2 ≤ −

2 ∑

( Kj tan

π Sj2

)

2k2b

j=1

( ) + θ˜ T Γ −1 θ˙ˆ − Γ Λ2

−ϑs2

∂α1 ∂ θˆ

)T (

h2k (x¯ k (t − τk )) +

D2 =

h2j (xj (t

¯

j

j=1

∑k−1

Dk =

k ∑ 1

(22)

V˙ k ≤ −

−τj )) −

n ∑

(n + 1 −

j)S22 (t

2 j )qj2 (S2 (t

¯

−τ

−τj )). (23)

k−1 ∑ ) ∂αk−1 ( T θ ϕj + gj xj+1 + hj (x¯ j (t − τj )) ∂ xj j=1 ( )T k−1 ∑ ∂αk−1 (j+1) ∂αk−1 ˙ˆ + + yd θ. (j) ˆ ∂ θ ∂ y d j=0

n ∑

∫

∑

−

Kj tan

−

)

2k2b

∑

ϑsj

j=2

(

(25)

Sk2 (

τ

¯ τ ))dτ .

)q2jk (Sk (

∂αj−1 ∂ θˆ

+gk−1 ϑsk−1 Sk +

k−1 ∑

θ˙ˆ − Γ Λk−1

+

j=1

+

k−1 ∑ 1 j=1

j

¯

− τj )).

)T

θ˙ˆ

j=1

n ∑

(n + 1 − j)Sk2 (t)q2jk (S¯k (t)).

gk

(27)

(28)

(32)

)

π Sk2 k2b

− θˆ T ωk +

k

2Sk

k−1 ∑ ∂αk−1

k−1 ∑ ∂αk−1 j=1

∂ xj

gj xj+1

(j+1)

yd

∂ y(j) d j=0 ( )T gk−1 ϑsk−1 Sk ∂αk−1 k + Γ Λk − − ϑsk ϑsk 4 ∂ θˆ ( )2 k−1 ∑ j ∂αk−1 −ϑsk 4 ∂ xj j=1 ( ) )T k−1 ( ∑ π Sk2 ∂αj−1 2 + Γ ωk ϑsj − Sk cos 2k2b ∂ θˆ k j=2 } n ∑ × (n + 1 − j)q2jk (S¯k (t)) .

(33)

j=k

Substituting (33) into (32) and simplify the formation of it, we arrive at

j=1

h2j (xj (t

∂αk−1 ∂ θˆ

∑ j ( ∂αk−1 )2 4 ∂ xj

(

Using the Young’s inequality again, it yields: 1

(

k−1

{ Kk sin 1 − αk =

( Dj + ϑsk θ T ϕk + gk (Sk+1 + αk )

ϑs2k + h2k (x¯ k (t − τk )), k ( )2 k−1 k−1 ∑ ∑ ∂αk−1 j ∂αk−1 − ϑsk · hj (x¯ j (t − τj )) ≤ ϑs2k ∂ xj 4 ∂ xj

Dj − ϑsk

Thus, the stabilizing function αk can be given by

j=k

k

gj xj+1

j=k

+

4

∂ xj

⎠

k ∑

+ gk−1 ϑsk−1 Sk +

(26)

j=k

ϑsk · hk (x¯ k (t − τk )) ≤

∂

(j+1) yd (j) yd

4

)

(n + 1 − j)Sk2 (t − τj )q2jk (S¯k (t − τj )).

k−1 ∑ ∂αk−1

⎞

k−1 ∑ ∂αk−1

k

j=1

−

)T ( ) θ˙ˆ − Γ Λk−1

∂αj−1 ∂ θˆ

+ ϑs2k + ϑs2k

+ hk (x¯ k (t − τk )) − α˙ k−1 ) n ∑ + (n + 1 − j)Sk2 (t)q2jk (S¯k (t)) n ∑

( ) + θ˜ T Γ −1 θ˙ˆ − Γ Λk−1

j=1

( ) + θ˜ T Γ −1 θ˙ˆ − Γ Λk−1

)T (

(

ϑsj

)

j=1

j=0

j

k−1

(n + 1 − j)Sk2 (t −τj )q2jk (S¯k (t −τj )). (31)

+ ϑsk · ⎝θ T ωk + gk (Sk+1 + αk ) −

−

Taking the time derivative of Vk leads to

j=1

n ∑

⎛

t −τj

j=k

π Sj2

(30)

∂α

2k2b j

t

(n + 1 − j)

(

Kj tan

j=2

Denote the Lyapunov function Vk as follows for this step,

k−1 ∑

π Sj2

(

j=1

where

V˙ k ≤ −

h2j (x¯ j (t − τj )).

j=k

k−1

S˙k = x˙ k − α˙ k−1 = θ T ϕk + gk (Sk+1 + αk ) + hk (x¯ k (t − τk )) − α˙ k−1 , (24)

Vk = Vk−1 + Vk +

j

h2j (x¯ j (t −τj )) −

k−1 ∑

j=2

∗

j=1

j

With the help of (28)–(31), inequality (27) can be rewritten as

) θ˙ˆ − Γ Λ2 + g2 ϑs2 S3 + D1 + D2 ,

Step k (k = 3, . . . , n − 1). Differentiating Sk leads to

α˙ k−1 =

j

k ∑ 1

h2j (x¯ j (t − τj )) =

k−1 Define ωk = ϕk − ϕj , Λk = Λk−1 + ωk ϑsk as the kth j=1 ∂ xj regressor vector and tuning function, and let

in which D2 is denoted by 2 ∑ 1

k−1 ∑ 1 j=1

j=1

j

(

1

(29)

V˙ k ≤ −

k ∑ j=1

( Kj tan

π Sj2 2k2b

)

( ) + θ˜ T Γ −1 θˆ˙ − Γ Λk

j

Please cite this article as: C.-X. Wang, Y.-Q. Wu, Y. Zhao et al., Asymptotic tracking control for time-delay nonlinear systems with parametric uncertainties and full state constraints. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.038.

C.-X. Wang, Y.-Q. Wu, Y. Zhao et al. / ISA Transactions xxx (xxxx) xxx

−

k ∑

ϑsj

(

j=2

+

k ∑

∂αj−1 ∂ θˆ

)T (

θ˙ˆ − Γ Λk

Then, the following actual controller can be constructed

)

(

Dj + gk ϑsk Sk+1 .

(34)

gn

The coupling term gk ϑsk Sk+1 will be eliminated in the next step. Step n. Differentiating Sn = xn − αn−1 leads to

+

n−1 ∑ ∂αn−1

= θ ϕn + gn u + hn (x¯ n (t − τn )) −

j=0

−

∂

(j+1) yd (j) yd

( −

∂αn−1 ∂ θˆ

n−1 ∑ ( )) ∂αn−1 ( T θ ϕj + gj xj+1 + hj x¯ j (t − τj ) . ∂ xj

)T

− θˆ T ωn +

n

n−1 ∑ ∂αn−1 j=1

∂ xj

gj xj+1

(j+1)

yd

∂ y(j) d )T

j=0

+

n−1 ∑ ∂αn−1

k2b

2Sn

( S˙n

)

π Sn2

[ Kn sin 1 − u =

j=1

T

5

∂αn−1 ∂ θˆ

gn−1 ϑsn−1 Sn

Γ Λn −

n

− ϑsn

ϑsn 4 ( )2 n−1 ∑ j ∂αn−1 −ϑsn 4 ∂ xj j=1 ) ( ] )T n−1 ( ∑ ∂αj−1 π Sn2 2 ¯ 2 q ( S (t)) , (41) + Γ ωn ϑsj − Sn cos n nn 2k2bn ∂ θˆ

θ˙ˆ

(35)

j=1

j=2

Define the following Lyapunov function candidates for the last step: t

∫

∗

Sn2 (

Vn = Vn−1 + Vn +

τ

¯ τ ))dτ .

)q2nn (Sn (

(36)

t −τn

n−1 ∑

( Kj tan

2k2b j

j=1 n−1

−

∑

ϑsj

n−1 ∑

)

−

( ) + θ˜ T Γ −1 θˆ˙ − Γ Λn−1

n ∑

Dn =

n ∑ 1

j

j=1

∂αj−1 ∂ θˆ

)T (

(42)

n ) ∑ θ˙ˆ − Γ Λn + Dj . j=1

h2j (x¯ j (t − τj )) − Sn2 (t − τn )q2nn (S¯n (t − τn )).

(43)

˙

Moreover, the update law can be designed as θˆ = Γ Λn . On the other hand,

Dj + gn−1 ϑsn−1 Sn + ϑsn

n ∑

j n ∑ ∑ 1

Dj =

j=1 n−1 ∑ ∂αn−1

∂ xj

j=1

(θ ϕj + gj xj+1 ) − T

j=1

(

n−1 ∑ ∂αn−1

∂

j=0

n−1 ∑ ∂αn−1

− ϑsn

ϑsj

(

( ) + θ˜ T Γ −1 θ˙ˆ − Γ Λn

2k2b j

In which Dn is denoted by

j=1

−ϑsn

)

Kj tan

j=2

[ · θ T ϕn + gn u + hn (x¯ n (t − τn )) −

π Sj2

j=1

)T ) ∂αj−1 ( ˙ θˆ − Γ Λn−1 ∂ θˆ

(

j=2

+

π Sj2

∂α

(

n ∑

V˙ n ≤ −

We take the time derivative of the (36), and get V˙ n ≤ −

∑n−1

where ωn = ϕn − j=1 ∂nx−1 ϕj , Λn = Λn−1 + ωn ϑsn . Substituting j (38)–(41) into (37), then we have

∂ xj

∂αn−1 ∂ θˆ

(j+1) yd (j) yd

−

h2i (x¯ i (t − τi )) (44)

n n ∑ ∑

(n + 1 −

i)Sj2 (t

−τ

2 i )qij (Sj (t

¯

− τi )).

j=1 i=j

From Assumption 3 and Lemma 1, we have

hj (x¯ j (t − τj ))

1

)T

j=1 i=1

]

i

i

θ˙ˆ + Sn2 (t)q2nn (S¯n (t))

−Sn2 (t − τn )q2nn (S¯n (t − τn )).

h2i (xi (t

¯

− τi )) ≤

≤

(37)

(

1 i

i ∑

)2 |Sk (t − τi )|qik (S¯k (t − τi ))

k=1

i ∑

Sk2 (t − τi )q2ik (S¯k (t − τi )).

(45)

k=1

Using the Young’s inequality, we obtained

ϑsn · hn (x¯ n (t − τn )) ≤

n

Then, substituting (45) into (44), we obtain

1

ϑ 2 + h2n (x¯ n (t − τn )), 4 sn n ( )2 n−1 n−1 ∑ ∑ ∂αn−1 j ∂αn−1 − ϑsn · hj (x¯ j (t − τj )) ≤ ϑs2n ∂ xj 4 ∂ xj j=1

+

n−1 ∑ 1 j=1

j

(38)

n

Dj ≤

j=1

j n i ∑ ∑ ∑

Sk2 (t − τi )q2ik (S¯k (t − τi ))

j=1 i=1 k=1

−

j=1

n n ∑ ∑

(n + 1 − i)Sj2 (t − τi )q2ij (S¯j (t − τi ))

j=1 i=j

h2j

) x¯ j (t − τj ) .

(

(39)

Similarly, time-delay terms in (38) and (39) are conjugated together, yielding 1

n ∑

h2n (x¯ n (t − τn )) +

n−1 ∑ 1 j=1

j

h2j (x¯ j (t − τj )) =

n ∑ 1 j=1

j

h2j (x¯ j (t − τj )).

(40)

= 0.

(46)

Thus the time-delay terms are successfully eliminated. As a result, inequality (42) can be rewritten as V˙ n ≤ −

n ∑ j=1

( Kj tan

π Sj2 2k2b

) ≤ 0.

(47)

j

By now, the controller design procedure is completed .

Please cite this article as: C.-X. Wang, Y.-Q. Wu, Y. Zhao et al., Asymptotic tracking control for time-delay nonlinear systems with parametric uncertainties and full state constraints. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.038.

6

C.-X. Wang, Y.-Q. Wu, Y. Zhao et al. / ISA Transactions xxx (xxxx) xxx

Remark 2. All time-delay terms come from the system and backstepping ∑ design are ingenious combined together, which n expressed as j=1 Dj . With suitable Lyapunov–Krasovskii functionals are constructed, all terms containing the delayed states are eliminated effectively. As a result, inequality (46) is arrived, which is very important for the final asymptotic tracking result. Remark 3. It is worth noting that only one parameter Ki needs to be selected in the controller design which greatly simplifies the design process. State constraints determine the selection of parameters. kbi = kci − α¯ i−1,0 is the constraint for Si , α¯ i−1,0 is a positive constant bound of αi−1 . So in parameter selection for Ki , we just need to make sure kbi = kci − α¯ i−1,0 > 0. Besides, in the design process only one parameter θ needs to be regulated online, which greatly reduces the computational burden. 4. Main result Theorem 1. Consider the system (1) that satisfies Assumptions 1–3. By using the proposed control scheme and selecting the appropriate parameters for it, if S(0) ∈ Ωs , then the following properties hold. (i) The error signals Si , i = 1, 2, . . . , n converge asymptotically to zero; (ii) |xi (t)| < kci , ∀t ≥ 0; (iii) All signals of the closed-loop system remain bounded. Proof. (i) V˙ n ≤ 0 indicates Vn (t) ≤ Vn (0), hence Vn is bounded. From (36), Vn can be rewritten as Vn =

n ∑ k2b

i

π

i=1 n

+

( tan

π Si2

)

2k2b i

n

∑∑

1

+ θ˜ T Γ −1 θ˜ 2

∫

(n + 1 − j)

k=1 j=k

(48)

t

Sk2 (

τ

¯ τ ))dτ .

)q2jk (Sk (

t −τj

So from the result obtained in (47)V˙ n ≤ 0, it follows directly that the tracking errors will converge asymptotically to zero. (ii) Since V˙ n ≤ 0, it is clearly that |Si | < kbi holds. From (i) we know Vn is bounded which indicates θ˜ is bounded. So the boundedness of θˆ = θ˜ + θ can be directly derived from the boundedness of θ and θ˜ . For |yd | ≤ Y0 , |S1 | < kb1 , and in view of S1 = x1 − yd , then we can get |x1 | ≤ |S1 | + yd < kb1 + Y0 . Therefore, as long as kb1 = kc1 − Y0 , then |x1 | < kc1 holds. From (12), we know that the boundedness of α1 can be proved by the boundedness of x1 , yd , y˙ d and θˆ . Suppose |α1 | ≤ α¯ 1,0 , then |x2 | ≤ |α1 | + |S2 | < α¯ 1,0 + kb2 . So as long as kb2 = kc2 − α¯ 1,0 , which leads to |x2 | < kc2 holds directly. Do the same analogy, |xi | < kci (i = 3, . . . , n) hold, as long as kbi = kci − α¯ i−1,0 . Thus it is concluded that the full state constrains are always satisfied during the operation of the system. (iii) From the above proof, we know the state xi , the stabilizing function αi , and the tracking error Si are all bounded. In the same way it can be proved that the actual controller u is bounded. Based on the above proof process, it can be determined that all signals of the closed-loop system remain bounded. 5. Simulation results In this section, simulations will be performed on the model, which is a 1-DOF electrostatic microactuator with bidirectional drive as in Fig. 1, to verify the effectiveness of our control scheme. Cf and Cb are used to represent the capacitances between the movable plate and the upper and lower electrodes, specifically expressed as

ϵA Cf = , l0 − l

ϵA Cb = l0 + l

(49)

where A represents the plate area and ϵ the permittivity of the gap, l ∈ R refers to the air gap between the movable plate and the upper electrode. When both input voltages Vf = Vb = 0, the gap expressed as l0 . The electrostatic forces come from the input voltages Vf and Vb , acting on the movable plate are: Ff = − Fb = −

1 ∂ Cf 2 ∂l 1 ∂ Cb 2 ∂l

Vf2 =

ϵA 2(l0 − l)2 ϵA

Vb2 = −

Vf2 , (50)

2(l0 + l)

V 2. 2 b

Thus, the dynamic equation of the electrostatic microactuator can be expressed as:

(

ϵA m¨l + b(l)˙l + kl =

Vf2 (l0 − l)2

2

−

)

Vb2 (l0 + l)2

=:

ϵA 2

ν

(51)

where k denotes the spring constant, m the mass of the movable electrode, and b(l) represents the nonlinear squeeze film damping. Make the change of variables with large constants σ > 0, β > dl 0: τ = σ t, x1 = ll , x2 = l1 ( dt ), u = βν , and considering the 0 0 system with time-delay, thus yielding the following time-delay strict-feedback form: x˙ 1 (t) = x2 (t) + ∆f¯1 (x1 (t − τ1 )) bc ¯ 1 (t))x2 (t) − k x1 (t) b(x x˙ 2 (t) = − 3 mσ 2 2mσ l0 ϵ Aβ u(t) + ∆f¯2 (x2 (t − τ2 )) + 2mσ 2 l0 y(t) = x1 (t)

(52)

where x¯ 2 (t) = [x1 (t), x2 (t)]T ; y ∈ R denotes the system output; ∆f¯1 (x1 (t − τ1 )), ∆f¯2 (x2 (t − τ2 )) represent the time-delay uncertain ¯ 1 (t)) terms of the system and τ1 , τ2 are time-delay constants; b(x 1 1 ¯ 1 (t)) = is described by b(x + . (1−x1 (t))3 (1+x1 (t))3 Our goal in controlling this electrostatic microactuator is to enable its movable electrode to track a given reference trajectory yd (t) = 0.2 ∗ (sin(2t) + sin(t)) within the air gap. Obviously, the movable electrodes must avoid contact with fixed electrodes at both ends during operation. On the other hand, its speed may also be limited. In our model of the system, these practical constraints can be expressed as constraints on state. Specifically, we assume |x1 | < kc1 = 1, |x2 | < kc2 = 2 in simulation. For simulation, we give the plant parameters values as the reference [1]: bc = 2.659 × 10−21 N · m2 s, m = 1.864 × 10−11 kg, l0 = 1 × 10−6 m, k = 350 N · m−1 , A = 2 × 10−8 m2 , ϵ = 8.859 × 10−12 Fm−1 , σ = 1 × 106 , β = 2 × 1017 , and the time delay uncertain terms ∆f¯1 (x1 (t − τ1 )) = sin(x1 (t − τ1 )), ∆f¯2 (x2 (t − τ2 )) = x1 (t − τ√ 2 )x2 (t − τ2 ), τ1 = τ2 = 0.1. Thus, we can set q11 = 1, q21 = (1 − 1 − 6S12 )/|S1 |, q22 = |S1 |. Then, in this case, the system (52) is a time-delay deterministic system. Based on the design procedure proposed in Section 3, the stabilizing function and the actual controller can be given by

( ) π S2 [ K1 sin 2 1 kb 1 1 1 α1 = − + y˙ d − ϑs1 g1 2S1 4 ( ) 2 ] π S12 ∑ 2 2 − S1 cos (3 − j)q (S (t)) , 1 j1 2 2kb

( u =

1 g2

[ K2 sin −

2S2

(53)

j=1

1

π S22 k2b

2

) 1

+

∑ ∂α1 (j+1) g1 ϑs S2 ∂α1 1 g 1 x2 + yd − ∂ x1 ϑ s2 ∂ y(j) d j=0

Please cite this article as: C.-X. Wang, Y.-Q. Wu, Y. Zhao et al., Asymptotic tracking control for time-delay nonlinear systems with parametric uncertainties and full state constraints. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.038.

C.-X. Wang, Y.-Q. Wu, Y. Zhao et al. / ISA Transactions xxx (xxxx) xxx

7

Fig. 1. 1-DOF electrostatic microactuator with bidirectional [1].

Fig. 2. Trajectories of states and their constraint.

1

ϑs2

2

4

− ϑs2 −

(

∂α1 ∂ x1

)2

( 2

− S2 cos

π S22 2k2b

)

Fig. 3. Trajectory of input signal.

]

q222 (S¯2 (t)) .

(54)

2

Given the initial state value x(0) = [0.1, 0.2]T , and the other design parameters K1 = K2 = 2. With the help of Matlab, substitute the known parameters into α1 to get its maximum absolute value α¯ 1 = 0.6069. As a result, kb1 = kc1 − Y0 = 0.6, kb2 = kc2 − α¯ 1 = 1.3931. With these initial values and parameters, (52)–(54) constitute the closed-loop system. The simulation results are illustrated in Figs. 2–5. Fig. 2 shows the full states constraints are never violated during simulation. We can clearly get the conclusion from Figs. 4 and 5 that the movable plate moves asymptotically along the reference signal in the air gap. On the other hand, if there are some unknown parameters in the system (52), the control goal can also be achieved by using the control scheme proposed in this paper. For instance, the system with uncertain parameters is shown as follows x˙ 1 (t) = x2 (t) + ∆f¯1 (x1 (t − τ1 )) ) ( bc k ¯ x˙ 2 (t) = θ · − b(x1 (t))x2 (t) − x1 (t) mσ 2 2mσ l30 ϵ Aβ + u(t) + ∆f¯2 (x2 (t − τ2 )) 2mσ 2 l0 y(t) = x1 (t)

Fig. 4. Trajectories of y and yd .

(55)

concluded that the unknown parameter has very little influence on the system. Fig. 7 illustrates the trajectories of uncertain parameters. Simulation results demonstrate the effectiveness of the control scheme for time-delay nonlinear systems with state constraints and parametric uncertainties.

in which all parameters are identical to the deterministic case except for the unknown parameter θ . The simulation plots are shown in Figs. 6 and 7. Compare Fig. 6 with Fig. 2, it can be

Remark 4. From Fig. 3, we can see that the control input signal changes sharply for some time. Through analysis, we find that the main reason is that the controller design needs to meet the state constraints. We can solve this problem by setting up

Please cite this article as: C.-X. Wang, Y.-Q. Wu, Y. Zhao et al., Asymptotic tracking control for time-delay nonlinear systems with parametric uncertainties and full state constraints. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.038.

8

C.-X. Wang, Y.-Q. Wu, Y. Zhao et al. / ISA Transactions xxx (xxxx) xxx

Fig. 5. The tracking error.

Fig. 7. trajectories of uncertain parameters.

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work is supported by the National Natural Science Foundation (NNSF) of China under Grant 61803228, 61673243, 6180 3229, 61703232, and 61572112; Taishan Scholar Project of Shandong Province of China under grant 2015162, tsqn201812093; Project of Shandong Province Higher Educational Science and Technology Program, China under grant J18KA361; Doctor Foundation of Shandong Jianzhu University under grant XNBS1950. Fig. 6. States trajectories (with unknown parameter).

some saturation function which will increase the burden of the controller greatly. When both state constraint and input rate saturation are taken into consideration, it is difficult to realize and still recognized as an open problem. 6. Conclusion In this paper, a tBLF based adaptive backstepping control scheme is developed for a class of time-delay nonlinear systems with parametric uncertainties and full state constraints. The tBLF and Lyapunov–Krasovskii function are united together to act as the Lyapunov function, based on which a new adaptive backstepping control scheme is proposed. This successfully get over the difficulties of time delay and full state constraint in system design. It is proved that this control scheme achieves the asymptotic convergence without violating any state constraints. At the same time, all closed-loop signals remain bounded. Finally, we use a class of 1-DOF time-delay electrostatic microactuator system to demonstrate the performance of our control scheme. In our view, the proposed approach can be extended in at least the following three possible areas for better results: (1) controlling time-delay nonlinear systems with both state constraint and input rate saturation; (2) extend this approach to nonlinear time-delay systems with deferred constraints; (3) further consider system with input and output delays.

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Please cite this article as: C.-X. Wang, Y.-Q. Wu, Y. Zhao et al., Asymptotic tracking control for time-delay nonlinear systems with parametric uncertainties and full state constraints. ISA Transactions (2019), https://doi.org/10.1016/j.isatra.2019.08.038.