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Command-filter-based adaptive tracking control for nonlinear systems with unknown input quantization and mismatching disturbances
Applied Mathematics and Computation 377 (2020) 125161
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Command-filter-based adaptive tracking control for nonlinear systems with unknown input quantization and mismatching disturbances Jiali Ma a,c, Ju H. Park b,c,∗, Shengyuan Xu a, Guozeng Cui d, Zhichun Yang b a
School of Automation, Nanjing University of Science and Technology, Nanjing 210094 Jiangsu, PR China School of Mathematical College, Chongqing Normal University, Chongqing 401331 PR China c Department of Electrical Engineering, Yeungnam University, Kyongsan 38541, Republic of Korea d School of Electronic and Information Engineering, Suzhou University of Science and Technology, Suzhou 215009, PR China b
a r t i c l e
i n f o
Article history: Received 6 December 2019 Revised 3 February 2020 Accepted 16 February 2020
Keywords: Uncertain nonlinear systems Input quantization Mismatching disturbances Command filter
a b s t r a c t In this article, the adaptive tracking control problem is investigated for a class of uncertain nonlinear systems with input quantization and mismatching disturbances. The effect of the mismatching disturbances is compensated by introducing smooth functions in the virtual controllers. By combining the novel command filters with backstepping method, an adaptive controller is designed and the problem of ”explosion of complexity” can be solved. It has also been proved that all the signals in the closed-loop systems are bounded and the system output can asymptotically track the reference signal. Finally, simulation examples are provided to verify the effectiveness of the proposed method. © 2020 Elsevier Inc. All rights reserved.
1. Introduction In recent years, the quantized control system has attracted considerable attention because of its application in modern engineering, such as digital systems, network systems and hybrid systems. Due to its practical and theoretical significance, many researchers have paid attention to the control problem of quantized system and great achievements have been obtained, see references [1–10]. Despite these contributions, there still exist some challenging problems in quantized control area. On the one hand, there are few results reporting on the exact tracking of quantized control systems. Only boundedness can be achieved in the existing references [4–9]. Specifically, for strict-feedback nonlinear system, backstepping-based adaptive control scheme was developed in [4] where the boundedness of system states can be ensured. However, the global Lipschitz condition was required for the nonlinear functions and their partial derivatives were assumed to be bounded. Then these assumptions were removed by using the sector bound property of the quantizer in [5]. Besides, global boundedness was also achieved in [7,8] by output feedback control. In order to achieve desired transient performance, asymptotic tracking control and finite-time control were investigated in [11–17]. However, the above references did not take the mismatching disturbances into consideration. When the nonlinear systems have mismatching disturbances, the asymptotic tracking control is not easy to be achieved as shown in [18]. It is difficult to design the virtual controller to effectively compensate the mismatching disturbances. ∗
Corresponding author. E-mail addresses: [email protected] (J. Ma), [email protected] (J.H. Park).
https://doi.org/10.1016/j.amc.2020.125161 0 096-30 03/© 2020 Elsevier Inc. All rights reserved.
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J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
On the other hand, adaptive backstepping scheme has been widely utilized in the existing papers [4–9,11–15,18–21]. As we all know, an inherent problem of the backstepping method is the “explosion of complexity” which is caused by the repeated differentiations of virtual controllers. Recently, great efforts have been devoted to this problem [22–30]. Specifically, dynamic surface control was utilized in [22,23] where the first-order filter was introduced in the backsetpping scheme. However, both papers did not consider the filter errors which may result in poor system performance. Then a modified filter, command filter, was developed in [24,25] where compensation signals were introduced to eliminate the effect of the filter errors. In [28–30], a finite-time command filter was introduced and the tracking error could converge to a desired small neighborhood in finite time under the command-filter-based controller. However, this command filter required the Lipschitz condition of the virtual controllers. Therefore, it is a difficult issue to design a finite-time command filter without requiring the Lipschitz condition. Based on the above discussions, an adaptive asymptotic tracking control problem is studied for a class of uncertain nonlinear systems with input quantization and mismatching disturbances. A priori knowledge of the quantization parameters and the disturbances is not required. By designing the command filter and the compensation signals, a modified adaptive tracking controller can be designed which can ensure the global boundedness of the system states. Besides, the system output can asymptotically track the reference signal. The contributions of our paper can be concluded as follows: (1) Asymptotic tracking control is considered for uncertain nonlinear system with input quantization and a priori knowledge of the quantization parameters is not required; (2) Different from the previous references [4–9], mismatching disturbances are considered in our paper which can be compensated by introducing smooth functions in the virtual controllers; (3) Different from the papers [7,11,12,18] where the reference signal and its nth order derivatives were required, only the information of the reference signal and its first-order derivative is required in our paper; (4) Unlike the results in [24–30], modified finite-time command filters and compensation signals are developed and command-filer based asymptotic tracking controller is designed which can ensure a desired transient performance. The remaining part of this paper is organized as follows: Problem formulation and preliminaries are presented in Section 2. In Section 3, an adaptive controller is designed. And in Section 4, stability analysis is provided. Simulation examples are shown in Section 5 and the conclusion is presented in Section 6. 2. Problem formulation and preliminaries In this article, the following parametric strict-feedback nonlinear system is considered:
⎧ ⎨x˙ i = xi+1 + θ T fi (x¯i ) + di (x, t ), x˙ n = b(x, t )q(u ) + θ T fn (x¯n ) + dn (x, t ), ⎩
(1)
y = x1
where x = [x1 , . . . , xn ]T ∈ Rn is the system state vector. q(u) is the system input where u is the control input. y is the output of the system. fi (x¯i ) ∈ Rr , i = 1, . . . , n, are known nonlinear functions where x¯i = [x1 , . . . , xi ]T . b(x, t) > 0 is an unknown function and θ ∈ Rr are unknown parameters. di (x, t), i = 1, . . . , n, are unknown external disturbances. In this article, the following quantizer is considered [14]
⎧ ui sgn(u ), ⎪ ⎪ ⎨ ui (1 + δ )sgn(u ), q (u ) = ⎪ 0, ⎪ ⎩ q(u(t − )),
ui 1+δ
< |u| ≤ ui , u˙ < 0, or ui < |u| ≤
ui < |u| ≤ 0 ≤ |u| < u˙ = 0
ui , u˙ < 0, 1 −δ u0 , u˙ < 0, 1+δ
or or
ui , u˙ > 0 1 −δ ui (1+δ ) , u˙ > 0 1 −δ
|u| |u| ≤ u0 , u˙ > 0
ui < 1 −δ u0 ≤ 1+δ
(2)
1−ρ where ui = 0 , (i = 1, 2, . . . ), and 0 < ρ < 1 is an unknown parameter. sgn( · ) denotes the sign function. δ = 1+ρ determines quantization density of the quantizer. u0 > 0 determines the size of the dead zone of the quantizer which is also unknown. According to [14], the quantizer can be described as
ρ 1−i u
q(u ) = δ1 (t )u + δ2 (t ) where
δ1 (t ) = and
q (u ) u
,
1,
|u(t )| ≥ u0 |u(t )| < u0
δ2 (t ) =
(3)
0, q(u ) − u,
|u(t )| ≥ u0 |u(t )| < u0
(4)
(5)
Besides, we also have
δ1 ≥ 1 − δ,
|δ2 | ≤ u0 .
(6)
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
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The control objective of this paper is to design a command-filter-based adaptive controller which can ensure that all the signals in the closed-loop systems are bounded and the system output can asymptotically track the reference signal yr . Remark 1. In the existing results, the quantization parameters were always required to be known which were used to design the controller [4–6,13–15,18]. Although the information of quantization parameters was not required in [7–9], only global boundedness can be achieved. Different from the above references, the asymptotic tracking control can be achieved even if the quantization parameters are unknown. In this paper, the following assumptions are provided. Assumption 1. For the unknown function b(x, t), there exist unknown positive constants b and b¯ such that b ≤ b(x, t ) ≤ b¯ . Assumption 2. For disturbance di (x, t), there exists an unknown positive constant d¯i such that |di (x, t )| ≤ d¯i . Assumption 3. The desired trajectory yr and its first order derivative are known and bounded. Remark 2. In the existing results, asymptotic tracking control can only be obtained for nonlinear systems with matching disturbance [11,12]. When the systems have mismatching disturbances, only global boundedness could be achieved [18]. In our paper, the asymptotic tracking control is considered for quantized control systems with mismatching disturbances. By introducing smooth functions in the virtual controllers, it can be guaranteed that the system output can asymptotically track the reference signal. Remark 3. In the existing results about tracking control [5,7–9,11,12,18], the reference signal and its nth order derivatives were always required to be known. In our paper, we just need the information of the reference signal and its first-order derivative. Lemma 1 [31]. Let α be a continuous and piecewise two-order derivable function. Consider the following finite-time-convergent second-order differentiator
˙ ζ1 = ζ2 , ζ˙2 = ε12 (−sat {sgn( fν (ζ1 , α , ζ2 )) × | fν (ζ1 , α , ζ2 )|ν /(2−ν ) } − sat {sgn(ζ2 )|εζ2 |ν } ),
(7)
where ε > 0 is the perturbation parameter, ν ∈ (0, 1).
fν (ζ1 , α , ζ2 ) = ζ1 − α +
sat (s ) =
s,
sgn(ζ2 )|εζ2 |2−ν , 2−ν |s| < ,
sgn(s ),
(8)
|s| ≥ .
Then, there exist τ > 0 and ςτ > 2, ς = ν /(2 − ν ), such that
ζ1 − α = (ε ςτ ),
(9)
can be obtained in finite time, where (ε ς τ ) denotes the approximation of ε ς τ order between ζ 1 and α . Remark 4. In the previous papers [28–30], a finite-time command filter was designed which required the Lipschitz condition of the virtual controller. However, it is difficult to check this condition before the controller design and stability analysis. In our paper, a novel command filter is introduced which removes the Lipschitz condition. Therefore, the proposed command filter is more effective and less conservative. Lemma 2 [31]. Let σ (t) be a positive uniform continuous and bounded function which satisfies
t
lim
t→+∞ t 0
σ (s )ds ≤ σ¯
(10)
where σ¯ is a positive constant. Then for any variable η, the following inequality holds
0 ≤ |η| −
η2 ≤ σ (t ). η + σ 2 (t )
(11)
2
Lemma 3 [31]. For any positive constant σ and variable η, the following inequality holds
0 ≤ |η| − ηtanh
η
σ
≤ ισ
(12)
where ι = 0.2875. Lemma 4 [29]. For ξ i ∈ R, i = 1, . . . , n, 0 < υ ≤ 1, the following inequality holds
n i=1
υ
| ξi |
≤
n i=1
| ξi | υ ≤ n 1 − υ
n i=1
υ
| ξi |
.
(13)
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J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
Lemma 5 [32]. Let V (x ) : Rn → R be a positive definite Lyapunov function. There exist positive constants c1 > 0, c2 > 0 and 0 < υ < 1 such that V˙ + c1V + c2V υ ≤ 0. Then, there exists a positive constant T, such that x(t) ≡ 0 for t ≥ T, where T = c (11−u) ln(1 + 2
c1 1−υ (x(0 ))). c2 V
3. System controller In this section, an adaptive asymptotic controller will be designed based on the command-filter-based backstepping approach. Firstly, define the error signals as follows:
z1 = y − yr ,
(14)
zi = xi − α˜ i−1 , i = 2, . . . , n where α˜ i−1 is the output of the following command filter whose input is αi−1 . And the command filter is designed as
˙ ζi1 = ζi2 , ζ˙i2 = ε12 (−sat {sgn( fν (ζi1 , αi , ζi2 )) × | fν (ζi1 , αi , ζi2 )|ν /(2−ν ) } − sat {sgn(ζi2 )|εζi2 |ν } ),
(15)
And α˜ i = ζi1 is the output of the command filter. Besides, the compensation signals are designed as follows:
⎧˙ ⎨ξ1 = −c1 ξ1 + ξ2 + (α˜ 1 − α1 ) − lsgn(ξ1 ), ξ˙ = −ci ξi − ξi−1 + ξi+1 + (α˜ i − αi ) − lsgn(ξi ), ⎩ i ξ˙n = −cn ξn − ξn−1 − lsgn(ξn ),
(16)
where ci > 0 and l > 0 are design parameters. Based on the above preparation, we define the compensation error signal as
η i = z i − ξi ,
i = 1, . . . , n.
(17)
And the adaptive asymptotical tracking controller can be designed by the following procedures. Step 1: Consider the Lyapunov function candidate
V1 =
1 2 1 η + ϑ˜ 2 , 2 1 2κ 1
(18)
where κ is a positive design parameter. ϑ˜ 1 = ϑ1 − ϑˆ 1 , where ϑˆ 1 is the estimate of ϑ1 and ϑ1 is defined as ϑ1 = d¯1 + l. According to (1), (14) and (16), the derivative of V1 can be expressed as
V˙ 1 = η1 (x2 + θ T f1 + d1 − y˙ r + c1 ξ1 − ξ2 − (α˜ 1 − α1 ) + lsgn(ξ1 )) − ≤ η1 (η2 + α1 + θ T f1 − y˙ r + c1 ξ1 ) + |η1 |ϑ1 −
1 ˜ ˆ˙ ϑ1 ϑ1
κ
1 ˜ ˆ˙ ϑ1 ϑ1
κ
(19)
Then, design the virtual controller α 1 and the adaptive law as
α1 = −c1 z1 − θˆ f1 + y˙ r − ϑˆ1tanh ϑˆ˙ 1 = κη1tanh
η
1 , σ (t )
η
1 , σ (t )
(20) (21)
where θˆ is the estimate of θ ; σ (t) is selected based on Lemma 2. Substituting (20) and (21) into (19) yields
V˙ 1 ≤ −c1 η12 + η1 η2 + η1 θ˜ T f1 + ϑ1 |η1 | − ϑ1 η1 tanh
η
1 , σ (t )
(22)
where θ˜ = θ − θˆ . By using Lemma 3, we have
ϑ1 |η1 | − ϑ1 η1tanh
η
1 ≤ ϑ1 ισ (t ). σ (t )
(23)
Therefore, (22) satisfies
V˙ 1 ≤ −c1 η12 + η1 η2 + η1 θ˜ T f1 + ϑ1 ισ (t ).
(24)
Step i = 2, . . . , n − 1: Consider the Lyapunov function candidate
Vi = Vi−1 +
1 2 1 η + ϑ˜i2 , 2 i 2κ
where ϑ˜ i = ϑi − ϑˆ i , ϑˆ i is the estimate of ϑi and ϑi is defined as ϑi = d¯i + l.
(25)
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
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According to (1), (14) and (16), we have i−1
V˙ i ≤ −
c j η2j + ηi−1 ηi + θ˜ T
j=1
i−1
η j f j + ισ (t )
j=1
i−1
1
ϑ j − ϑ˜i ϑˆ˙ i κ j=1
+ ηi (xi+1 + θ T fi + di − α˜˙ i−1 + ci ξi + ξi−1 − ξi+1 − (α˜ i − αi ) + lsgn(ξi )) i−1
≤−
c j η2j + θ˜ T
j=1
i−1
η j f j + ισ (t )
i−1
1
ϑ j − ϑ˜i ϑˆ˙ i + ηi (ηi+1 + αi + θ T fi − α˜˙ i−1 + ci ξi + zi−1 ) + |ηi |ϑi . κ j=1
j=1
(26)
˙ Then, we design the virtual controller α i and the adaptive law ϑˆ i as
αi = −ci zi − zi−1 − θˆ fi + α˙ i−1 − ϑˆitanh ϑˆ˙ i = κηitanh
η
i . σ (t )
η
i , σ (t )
(27) (28)
Substituting (27) and (28) into (26) yields i
V˙ i ≤ −
c j η2j + θ˜ T
j=1
i
η j f j + ισ (t )
i
j=1
Step n: Define γ = bδ1 = 1
Vn = Vn−1 +
1 r
ϑ j + ηi ηi+1 .
(29)
j=1
and δ 1 = 1 − δ . Consider the Lyapunov function candidate
1 2 1 1 r η + ϑ˜ 2 + θ˜ T θ˜ + γ˜ T γ˜ , 2 n 2κ n 2κ 2κ
(30)
where ϑ˜ n = ϑn − ϑˆ n , ϑˆ n is the estimate of ϑn and ϑn is defined as ϑn = d¯n + l + b¯ u0 ; γ˜ = γ − γˆ , γˆ is the estimate of γ . Then, the derivative of Vn can be expressed as
V˙ n ≤ −
n−1
c j η2j + ηn−1 ηn + θ˜ T
j=1
n−1
η j f j + ισ (t )
j=1
n−1
1
1
ϑ j − ϑ˜n ϑˆ˙ n − θ˜ T θˆ˙ κ κ j=1
+ ηn (bq(u ) + θ fn + dn − α˜˙ n−1 + cn ξn + ξn−1 + lsgn(ξn ) ) − T
=−
n−1
c j η2j + θ˜ T
j=1
n−1
η j f j + ισ (t )
j=1
n−1
1
1
r
κ
γ˜ γˆ˙
r
ϑ j − ϑ˜n ϑˆ˙ n − θ˜ T θˆ˙ − γ˜ γˆ˙ κ κ κ j=1
+ ηn (bδ1 u + bδ2 + θ T fn + dn − α˜˙ n−1 + cn ξn + zn−1 + lsgn(ξn ) ).
(31)
Similarly, the following inequality can be obtained
ηn (bδ2 + dn + lsgn(ξn )) ≤ |ηn |ϑn .
(32)
Therefore, (31) can be rewritten as
V˙ n ≤ −
n−1
c j η2j + θ˜ T
j=1
n−1
η j f j + ισ (t )
j=1
n−1
1
1
ϑ j − ϑ˜n ϑˆ˙ n − θ˜ T θˆ˙ κ κ j=1
+ ηn (bδ1 u + θ fn − α˜˙ n−1 + cn ξn + zn−1 ) + |ηn |ϑn − T
r
κ
γ˜ γˆ˙ .
(33)
˙ ˙ Then, design the system controller u and adaptive laws ϑˆ n , θˆ , γˆ˙ as
u = −
ηn γˆ 2 αn2 , ηn2 γˆ 2 αn2 + σ 2 (t ) η
ϑˆ˙ n = κηntanh θˆ˙ = κ
n
n
σ (t )
,
η j f j − κσ (t )θˆ ,
(34)
(35)
(36)
j=1
γˆ˙ = κηn αn ,
(37)
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J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
where α n is designed as
αn = cn zn + zn−1 + θˆ fn − α˜˙ n−1 + ϑˆntanh
η
n . σ (t )
(38)
Then, substituting (34)–(38) into (33) yields
V˙ n ≤ −
n
c j η2j + ισ (t )
j=1
n
˜ (t )θˆ + bδ1 ηn u + ηn αn − r γ˜ ηn αn . ϑ j + θσ
(39)
j=1
By using Lemma 2, we have
ηn2 γˆ 2 αn2 η γˆ 2 αn2 + σ 2 (t ) ηn2 γˆ 2 αn2 ≤−r ≤ r (σ (t ) − γˆ ηn αn ). ηn2 γˆ 2 αn2 + σ 2 (t )
bδ1 ηn u = − bδ1
2 n
(40)
Then, we have
V˙ n ≤ −
n
c j η2j + ισ (t )
j=1
n
˜ (t )θˆ + r (σ (t ) − γˆ ηn αn ) + ηn αn − r γ˜ ηn αn . ϑ j + θσ
(41)
j=1
According to the definition of r and γ , the following equation can be obtained
−r γˆ ηn αn + ηn αn − r γ˜ ηn αn = 0.
(42)
By combining (41) with (42), we have
V˙ n ≤ −
n
c j η + σ (t ) 2 j
ι
n
j=1
ϑ j + θ˜ θˆ + r .
(43)
j=1
4. Stability analysis Based on the preparation work in Section 3, we have the following results as shown in Theorem 1. Theorem 1. Consider the nonlinear system (1) satisfying Assumptions 1–3. Under the controller (34) and the adaptive laws (21), (28), (35)–(37), global boundedness of the closed-loop system can be achieved and the tracking error y − yr can asymptotically converge to zero. Proof. First, it is easy to prove
1 4
θ˜ θˆ = θ θˆ − θˆ 2 ≤ θ 2 .
(44)
By combining (43) with (44), we have
V˙ n ≤ −
n
c j η2j + σ (t )μ,
(45)
j=1
where μ = ι
n j=1
ϑ j + 14 θ 2 + r. By directly integrating (45) over [t0 , t), we have
Vn (t ) − Vn (t0 ) ≤ −
n
cj
j=1
≤−
n j=1
t0
cj
t
t
t0
η2j ds +
t
t0
μσ (s )ds
η2j ds + μσ¯ .
(46)
From (46), it can be seen that ηi , ϑˆ i , i = 1, . . . , n, θˆ and γˆ are globally bounded. Besides, based on (46), we also have
lim
t→+∞
n j=1
cj
t
t0
η2j ds ≤ Vn (t0 ) + μσ¯
(47)
By using Barbalat lemma, it can be concluded that limt→+∞ η1 (t ) = 0. Next, we will prove the boundedness of ξ . By using Lemma 1, we have that |α˜ i − αi | ≤ (ε ς τ ) can be guaranteed in finite time. Consider the Lyapunov function Vξ = 12 ni=1 ξi2 . According to (16), we have
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
V˙ ξ = −
n
ci ξi2 −
i=1
≤−
n
i=1
ci ξi2 −
i=1
where l =
√ 2 2 l0
V˙ ξ ≤ −
n
n i=1
l | ξi | −
n−1
7
ξi (α¯ i − αi )
i=1
√ n−1 2 l 0 | ξi | − (l1 − (ε ςτ ))|ξi |, 2
(48)
i=1
+ l1 . By choosing appropriate parameter l1 such that l1 − (ε ς τ ) ≥ 0 and using Lemma 4, we have
n i=1
ci ξi2 −
n i=1
√ 2 l0 |ξi | ≤ −c0Vξ − l0Vξυ , 2
(49)
where c0 = mini=1,··· ,n {2ci }, υ = 12 . By using Lemma 5, it can be concluded that ξ i , i = 1, . . . , n, can converge to zero in finite time. According to definition of zi and ηi , we can obtain the boundedeness of xi . Furthermore, since z1 = y − yr and η1 = z1 − ξ1 , we have y − yr = η1 + ξ1 . Considering the asymptotical stability of η1 and the finite-time stability of ξ 1 , we have limt→+∞ (y(t ) − yr (t )) = 0. Thus, the proof is completed. Remark 5. To obtain a better system performance, we can select the parameters ε to be large to accelerate the convergence of the command filter. And we can also choose large value of the parameter cj and l to accelerate the convergence speed of the closed-loop systems. However, large value of the parameters cj may result in large value of the control signal. Therefore, we should select the design parameters according to the demand. Remark 6. In our paper, command-filter based asymptotic tracking control is considered for quantized systems with mismatching disturbances. It should also be pointed out that the proposed method can also be extended to uncertain nonlinear systems with actuator faults. The reason is: As shown in [20], the actuator fault can be modeled as
u = δ1 (t )v + δ2 (t )
(50)
where v is the control input. 0 < δ 1 (t) ≤ 1 denotes the actuation effectiveness. δ 2 (t) denotes uncontrollable additive actuation fault which is bounded. Obviously, it has the same form of system (1). Therefore, the proposed method can be applied to deal with the control problem of nonlinear systems with actuator faults. Remark 7. In future research, we will develop the proposed results to the adaptive tracking control problem with dynamic quantizers, which can guarantee the desired system performance by adaptively updating the dynamic parameter. For system design with dynamic quantizers, references [33] and [34] had given significant design results, where the design conditions for the feedback controller and the quantizer’s parameter are given based on the linear matrix inequality technique. 5. Simulation examples In this section, two simulation examples will be given to illustrate the effectiveness of the proposed command-filterbased backstepping method. Example 1. Considering the following nonlinear system
⎧ ⎨x˙ 1 = x2 + d1 (x, t ), x˙ 2 = b(x, t )q(u ) + x21 + θ sin(x2 ) + d2 (x, t ), ⎩
(51)
y = x1
where θ is an unknown parameter; b(x, t) is an unknown function; di (x, t) are unknown and bounded disturbances. In this part, we will design an adaptive controller such that the output y = x1 can asymptotically track the reference signal yr = 1.5sint + cos1.5t. The actual values for the system parameters are chosen as: b(x, t ) = 0.5, θ = 1, d1 (x, t ) = sint, d2 (x, t ) = cost, and the quantization parameters are chosen as: u0 = 0.02, ρ = 0.2. Based on the results in Section 3, an adaptive controller can be designed. And the control parameters are selected as: c1 = c2 = 5, l = 7, κ = 5, σ (t ) = 0.1e−0.01t . And the initial conditions of the closed-loop system are chosen as: [x1 (0 ), x2 (0 )]T = [0.1, 0]T , [ζ1 (0 ), ζ2 (0 )]T = [0.1, 0]T . And the initial conditions of other signals are chosen as zero. Then the simulation results are shown in Figs. 1–6. The tracking performance can be found in Fig. 1. And the trajectories of the adaptive laws can be found in Figs. 3–6. From Fig. 1, we can see that asymptotic tracking control can be achieved by our proposed method. Example 2. Considering the following uncertain nonlinear system
⎧ x˙ = x2 + θ x21 + d1 (x, t ), ⎪ ⎨ 1 −x2 x˙ 2 = q(u ) + sin(x1 x2 ) + θ 1−e−x22 + d2 (x, t ), 1+e 2 ⎪ ⎩ y = x1 .
(52)
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J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
Fig. 1. The trajectories of y and yr .
Fig. 2. The trajectories of x2 .
Fig. 3. The trajectories of ϑˆ .
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
Fig. 4. The trajectories of θˆ .
Fig. 5. The trajectories of γˆ .
Fig. 6. The trajectories of u.
9
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J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
Fig. 7. The trajectories of y and yr under controller in [11].
Fig. 8. The trajectories of y and yr under controller in [11].
Fig. 9. The trajectories of y and yr under our proposed controller.
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
Fig. 10. The trajectories of ϑˆ under our proposed controller.
Fig. 11. The trajectories of θˆ under our proposed controller.
Fig. 12. The trajectories of γˆ under our proposed controller.
11
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The unknown parameter is chosen as: θ = 0.5. The mismatching disturbances are chosen as: d1 (x, t ) = sin(t ), d2 (x, t ) = cos(t ). The reference signal is chosen as yr = sin(t ). And the initial condition is [x1 (0 ), x2 (0 )]T = [0.1, 0]T , [ζ1 (0 ), ζ2 (0 )]T = [0.1, 0]T , and the initial condition of other signal is zero. The quantization parameters are chosen as: u0 = 0.02, ρ = 0.2. Adaptive asymptotic tracking control has been investigated for system (52) without mismatching disturbances in [11]. When di (x, t ) = 0, by exactly following the results in [11], the simulation result can be found in Fig. 7. As shown in Fig. 7, it can be seen that the system output y can asymptotically track the reference signal yr . However, when di (x, t) = 0, the simulation result by following the results in [11] is shown in Fig. 8. We can see that tracking performance is not desired. Then applying our proposed method to system (52), the system performances are shown in Figs. 9–12. Comparing Fig. 8 with Fig. 9, it can be seen that better tracking performance can be achieved than that in [11]. 6. Conclusion In this article, the command-filter-based adaptive asymptotic tracking control problem has been investigated for uncertain nonlinear systems with input quantization and mismatching disturbances. Different from the previous results, a priori knowledge of the quantization parameters and the disturbances is not required. The mismatching disturbances have been effectively compensated by introducing smooth functions in the virtual controllers. And an adaptive tracking controller has been designed based on the modified command filter and compensation signals. It has been proven that the global boundedness can be achieved and system output can asymptotically track the reference signal. Finally, two simulation examples have been provided to illustrate the effectiveness of the proposed method. Acknowledgements This work of J.H. Park and Z. Yang was supported by the National Natural Science Foundation of China under Grant 11971081, and the Fundamental and Frontier Research Project of Chongqing under Grant cstc2018jcyjAX0144. References [1] J.H. Park, H. Shen, X.H. Chang, T.H. Lee, Recent Advances in Control and Filtering of Dynamic Systems with Constrained Signals, Springer, Cham, Switzerland:, 2018, doi:10.1007/978- 3- 319- 96202- 3. [2] Y. Jiang, J. Zhai, Global practical tracking for a class of switched nonlinear systems with quantized input and output via sampled-data control, Int. J. Contr. Autom. Syst. 17 (2019) 1264–1271. [3] X. Zhang, F. Wang, L. Zhang, Finite time controller design of nonlinear quantized systems with nonstrict feedback form, Int. J. Contr. Autom. Syst. 17 (2019) 225–233. [4] J. Zhou, C. Wen, G. Yang, Adaptive backstepping stabilization of nonlinear uncertain systems with quantized input signal„ IEEE Trans. Autom. Control 59 (2) (2014) 460–464. [5] L. Xing, C. Wen, H. Su, J. Cai, L. Wang, A new adaptive control scheme for uncertain nonlinear systems with quantized input signal, J. Franklin Inst. 352 (12) (2015) 5599–5610. [6] H. Ma, Q. Zhou, L. Bai, H. Liang, Observer-based adaptive fuzzy fault-tolerant control for stochastic nonstrict-feedback nonlinear systems with input quantization, IEEE Trans. Syst. Man Cybern. Syst. 49 (2) (2019) 287–298. [7] L. Xing, C. Wen, Y. Zhu, H. Su, Z. Liu, Output feedback control for uncertain nonlinear systems with input quantization”, Automatica 65 (2016) 191–202. [8] C. Wang, C. Wen, Y. Lin, W. Wang, Decentralized adaptive tracking control for a class of interconnected nonlinear systems with input quantization”, Automatica 81 (2017) 359–368. [9] J. Zhou, C. Wen, W. Wang, Adaptive control of uncertain nonlinear systems with quantized input signal, Automatica 95 (2018) 152–162. [10] Y. Long, J.H. Park, D. Ye, Frequency-dependent fault detection for networked systems under uniform quantization and try-once-discard protocol”, Int. 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[16] Z.M. Li, X.H. Chang, J.H. Park, Quantized output feedback fuzzy tracking control for discrete-time nonlinear networked systems with asynchronous event-triggered constraints, IEEE Trans. Syst. Man Cybern.. To be published doi:10.1109/TSMC.2019.2931530. [17] B.C. Zheng, J.H. Park, Adaptive integral sliding mode control with bounded L2 gain performance of uncertain quantized control systems, IET Control Theory Appl. 9 (2015) 2273–2282. [18] L. Xing, C. Wen, H. Su, Z. Liu, J. Cai, Robust control for a class of uncertain nonlinear systems with input quantization, Int. J. Robust Nonlin. Control 26 (8) (2016) 1585–1596. [19] W. Khan, Y. Lin, S. Khan, N. Ullah, Quantized adaptive decentralized control for interconnected nonlinear systems with actuator faults, Appl. Math. Comput. 320 (2018) 175–189. [20] Y. Zhang, F. Wang, Adaptive neural control of non-strict feedback system with actuator failures and time-varying delays, Appl. Math. Comput. 362 (2019) 124512. [21] Y. Li, D. Ba, S. 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[24] J.A. Farrell, M. Polycarpou, M. Sharma, W. Dong, Command filtered backstepping, IEEE Trans. Autom. Control 54 (6) (2009) 1391–1395. [25] W. Dong, J.A. Farrell, M.M. Polycarpou, V. Djapic, M. Sharma, Command filtered adaptive backstepping„ IEEE Trans. Control Syst. Technol. 20 (3) (2012) 566–580. [26] J. Yu, P. Shi, W. Dong, H. Yu, Observer and command-filter-based adaptive fuzzy output feedback control of uncertain nonlinear systems, IEEE Trans. Ind. Electron 62 (9) (2015) 5962–5970. [27] Y. Hou, S. Tong, Command filter-based adaptive fuzzy backstepping control for a class of switched nonlinear systems, Fuzzy Sets Syst. 314 (2017) 46–60. [28] L. Zhao, J. Yu, C. Lin, Y. Ma, Adaptive neural consensus tracking for nonlinear multiagent systems using finite-time command filtered backstepping, IEEE Trans. Syst. Man Cybern. Syst. 48 (11) (2018) 2003–2012. [29] J. Yu, P. Shi, L. Zhao, Finite-time command filtered backstepping control for a class of nonlinear systems, Automatica 92 (2018) 173–180. [30] J. Yu, L. Zhao, H. Yu, C. Lin, W. Dong, Fuzzy finite-time command filtered control of nonlinear systems with input saturation, IEEE Trans. Cybern. 48 (8) (2018) 2378–2387. [31] X. Wang, Z. Chen, G. Yang, Finite-time-convergent differentiator based on singular perturbation technique, IEEE Trans. Autom. Control 52 (9) (2007) 1731–1737. [32] Z. Sun, M. Yun, T. Li, A new approach to fast global finite-time stabilization of high-order nonlinear system, Automatica 81 (2017) 455–463. [33] X. Chang, R. Huang, H. Wang, L. Liu, Robust design strategy of quantized feedback control, IEEE Trans. Circuits Syst. II Exp. Briefs(a). doi:10.1109/TCSII. 2019.2922311. [34] X. Chang, C. Yang, J. Xiong, Quantized fuzzy output feedback H∞ control for nonlinear systems with adjustment of dynamic parameters„ IEEE Trans. Syst. Man Cybern. Syst.(b). doi:10.1109/TSMC.2018.2867213.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Command-filter-based adaptive tracking control for nonlinear systems with unknown input quantization and mismatching disturbances Jiali Ma a,c, Ju H. Park b,c,∗, Shengyuan Xu a, Guozeng Cui d, Zhichun Yang b a
School of Automation, Nanjing University of Science and Technology, Nanjing 210094 Jiangsu, PR China School of Mathematical College, Chongqing Normal University, Chongqing 401331 PR China c Department of Electrical Engineering, Yeungnam University, Kyongsan 38541, Republic of Korea d School of Electronic and Information Engineering, Suzhou University of Science and Technology, Suzhou 215009, PR China b
a r t i c l e
i n f o
Article history: Received 6 December 2019 Revised 3 February 2020 Accepted 16 February 2020
Keywords: Uncertain nonlinear systems Input quantization Mismatching disturbances Command filter
a b s t r a c t In this article, the adaptive tracking control problem is investigated for a class of uncertain nonlinear systems with input quantization and mismatching disturbances. The effect of the mismatching disturbances is compensated by introducing smooth functions in the virtual controllers. By combining the novel command filters with backstepping method, an adaptive controller is designed and the problem of ”explosion of complexity” can be solved. It has also been proved that all the signals in the closed-loop systems are bounded and the system output can asymptotically track the reference signal. Finally, simulation examples are provided to verify the effectiveness of the proposed method. © 2020 Elsevier Inc. All rights reserved.
1. Introduction In recent years, the quantized control system has attracted considerable attention because of its application in modern engineering, such as digital systems, network systems and hybrid systems. Due to its practical and theoretical significance, many researchers have paid attention to the control problem of quantized system and great achievements have been obtained, see references [1–10]. Despite these contributions, there still exist some challenging problems in quantized control area. On the one hand, there are few results reporting on the exact tracking of quantized control systems. Only boundedness can be achieved in the existing references [4–9]. Specifically, for strict-feedback nonlinear system, backstepping-based adaptive control scheme was developed in [4] where the boundedness of system states can be ensured. However, the global Lipschitz condition was required for the nonlinear functions and their partial derivatives were assumed to be bounded. Then these assumptions were removed by using the sector bound property of the quantizer in [5]. Besides, global boundedness was also achieved in [7,8] by output feedback control. In order to achieve desired transient performance, asymptotic tracking control and finite-time control were investigated in [11–17]. However, the above references did not take the mismatching disturbances into consideration. When the nonlinear systems have mismatching disturbances, the asymptotic tracking control is not easy to be achieved as shown in [18]. It is difficult to design the virtual controller to effectively compensate the mismatching disturbances. ∗
Corresponding author. E-mail addresses: [email protected] (J. Ma), [email protected] (J.H. Park).
https://doi.org/10.1016/j.amc.2020.125161 0 096-30 03/© 2020 Elsevier Inc. All rights reserved.
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J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
On the other hand, adaptive backstepping scheme has been widely utilized in the existing papers [4–9,11–15,18–21]. As we all know, an inherent problem of the backstepping method is the “explosion of complexity” which is caused by the repeated differentiations of virtual controllers. Recently, great efforts have been devoted to this problem [22–30]. Specifically, dynamic surface control was utilized in [22,23] where the first-order filter was introduced in the backsetpping scheme. However, both papers did not consider the filter errors which may result in poor system performance. Then a modified filter, command filter, was developed in [24,25] where compensation signals were introduced to eliminate the effect of the filter errors. In [28–30], a finite-time command filter was introduced and the tracking error could converge to a desired small neighborhood in finite time under the command-filter-based controller. However, this command filter required the Lipschitz condition of the virtual controllers. Therefore, it is a difficult issue to design a finite-time command filter without requiring the Lipschitz condition. Based on the above discussions, an adaptive asymptotic tracking control problem is studied for a class of uncertain nonlinear systems with input quantization and mismatching disturbances. A priori knowledge of the quantization parameters and the disturbances is not required. By designing the command filter and the compensation signals, a modified adaptive tracking controller can be designed which can ensure the global boundedness of the system states. Besides, the system output can asymptotically track the reference signal. The contributions of our paper can be concluded as follows: (1) Asymptotic tracking control is considered for uncertain nonlinear system with input quantization and a priori knowledge of the quantization parameters is not required; (2) Different from the previous references [4–9], mismatching disturbances are considered in our paper which can be compensated by introducing smooth functions in the virtual controllers; (3) Different from the papers [7,11,12,18] where the reference signal and its nth order derivatives were required, only the information of the reference signal and its first-order derivative is required in our paper; (4) Unlike the results in [24–30], modified finite-time command filters and compensation signals are developed and command-filer based asymptotic tracking controller is designed which can ensure a desired transient performance. The remaining part of this paper is organized as follows: Problem formulation and preliminaries are presented in Section 2. In Section 3, an adaptive controller is designed. And in Section 4, stability analysis is provided. Simulation examples are shown in Section 5 and the conclusion is presented in Section 6. 2. Problem formulation and preliminaries In this article, the following parametric strict-feedback nonlinear system is considered:
⎧ ⎨x˙ i = xi+1 + θ T fi (x¯i ) + di (x, t ), x˙ n = b(x, t )q(u ) + θ T fn (x¯n ) + dn (x, t ), ⎩
(1)
y = x1
where x = [x1 , . . . , xn ]T ∈ Rn is the system state vector. q(u) is the system input where u is the control input. y is the output of the system. fi (x¯i ) ∈ Rr , i = 1, . . . , n, are known nonlinear functions where x¯i = [x1 , . . . , xi ]T . b(x, t) > 0 is an unknown function and θ ∈ Rr are unknown parameters. di (x, t), i = 1, . . . , n, are unknown external disturbances. In this article, the following quantizer is considered [14]
⎧ ui sgn(u ), ⎪ ⎪ ⎨ ui (1 + δ )sgn(u ), q (u ) = ⎪ 0, ⎪ ⎩ q(u(t − )),
ui 1+δ
< |u| ≤ ui , u˙ < 0, or ui < |u| ≤
ui < |u| ≤ 0 ≤ |u| < u˙ = 0
ui , u˙ < 0, 1 −δ u0 , u˙ < 0, 1+δ
or or
ui , u˙ > 0 1 −δ ui (1+δ ) , u˙ > 0 1 −δ
|u| |u| ≤ u0 , u˙ > 0
ui < 1 −δ u0 ≤ 1+δ
(2)
1−ρ where ui = 0 , (i = 1, 2, . . . ), and 0 < ρ < 1 is an unknown parameter. sgn( · ) denotes the sign function. δ = 1+ρ determines quantization density of the quantizer. u0 > 0 determines the size of the dead zone of the quantizer which is also unknown. According to [14], the quantizer can be described as
ρ 1−i u
q(u ) = δ1 (t )u + δ2 (t ) where
δ1 (t ) = and
q (u ) u
,
1,
|u(t )| ≥ u0 |u(t )| < u0
δ2 (t ) =
(3)
0, q(u ) − u,
|u(t )| ≥ u0 |u(t )| < u0
(4)
(5)
Besides, we also have
δ1 ≥ 1 − δ,
|δ2 | ≤ u0 .
(6)
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
3
The control objective of this paper is to design a command-filter-based adaptive controller which can ensure that all the signals in the closed-loop systems are bounded and the system output can asymptotically track the reference signal yr . Remark 1. In the existing results, the quantization parameters were always required to be known which were used to design the controller [4–6,13–15,18]. Although the information of quantization parameters was not required in [7–9], only global boundedness can be achieved. Different from the above references, the asymptotic tracking control can be achieved even if the quantization parameters are unknown. In this paper, the following assumptions are provided. Assumption 1. For the unknown function b(x, t), there exist unknown positive constants b and b¯ such that b ≤ b(x, t ) ≤ b¯ . Assumption 2. For disturbance di (x, t), there exists an unknown positive constant d¯i such that |di (x, t )| ≤ d¯i . Assumption 3. The desired trajectory yr and its first order derivative are known and bounded. Remark 2. In the existing results, asymptotic tracking control can only be obtained for nonlinear systems with matching disturbance [11,12]. When the systems have mismatching disturbances, only global boundedness could be achieved [18]. In our paper, the asymptotic tracking control is considered for quantized control systems with mismatching disturbances. By introducing smooth functions in the virtual controllers, it can be guaranteed that the system output can asymptotically track the reference signal. Remark 3. In the existing results about tracking control [5,7–9,11,12,18], the reference signal and its nth order derivatives were always required to be known. In our paper, we just need the information of the reference signal and its first-order derivative. Lemma 1 [31]. Let α be a continuous and piecewise two-order derivable function. Consider the following finite-time-convergent second-order differentiator
˙ ζ1 = ζ2 , ζ˙2 = ε12 (−sat {sgn( fν (ζ1 , α , ζ2 )) × | fν (ζ1 , α , ζ2 )|ν /(2−ν ) } − sat {sgn(ζ2 )|εζ2 |ν } ),
(7)
where ε > 0 is the perturbation parameter, ν ∈ (0, 1).
fν (ζ1 , α , ζ2 ) = ζ1 − α +
sat (s ) =
s,
sgn(ζ2 )|εζ2 |2−ν , 2−ν |s| < ,
sgn(s ),
(8)
|s| ≥ .
Then, there exist τ > 0 and ςτ > 2, ς = ν /(2 − ν ), such that
ζ1 − α = (ε ςτ ),
(9)
can be obtained in finite time, where (ε ς τ ) denotes the approximation of ε ς τ order between ζ 1 and α . Remark 4. In the previous papers [28–30], a finite-time command filter was designed which required the Lipschitz condition of the virtual controller. However, it is difficult to check this condition before the controller design and stability analysis. In our paper, a novel command filter is introduced which removes the Lipschitz condition. Therefore, the proposed command filter is more effective and less conservative. Lemma 2 [31]. Let σ (t) be a positive uniform continuous and bounded function which satisfies
t
lim
t→+∞ t 0
σ (s )ds ≤ σ¯
(10)
where σ¯ is a positive constant. Then for any variable η, the following inequality holds
0 ≤ |η| −
η2 ≤ σ (t ). η + σ 2 (t )
(11)
2
Lemma 3 [31]. For any positive constant σ and variable η, the following inequality holds
0 ≤ |η| − ηtanh
η
σ
≤ ισ
(12)
where ι = 0.2875. Lemma 4 [29]. For ξ i ∈ R, i = 1, . . . , n, 0 < υ ≤ 1, the following inequality holds
n i=1
υ
| ξi |
≤
n i=1
| ξi | υ ≤ n 1 − υ
n i=1
υ
| ξi |
.
(13)
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J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
Lemma 5 [32]. Let V (x ) : Rn → R be a positive definite Lyapunov function. There exist positive constants c1 > 0, c2 > 0 and 0 < υ < 1 such that V˙ + c1V + c2V υ ≤ 0. Then, there exists a positive constant T, such that x(t) ≡ 0 for t ≥ T, where T = c (11−u) ln(1 + 2
c1 1−υ (x(0 ))). c2 V
3. System controller In this section, an adaptive asymptotic controller will be designed based on the command-filter-based backstepping approach. Firstly, define the error signals as follows:
z1 = y − yr ,
(14)
zi = xi − α˜ i−1 , i = 2, . . . , n where α˜ i−1 is the output of the following command filter whose input is αi−1 . And the command filter is designed as
˙ ζi1 = ζi2 , ζ˙i2 = ε12 (−sat {sgn( fν (ζi1 , αi , ζi2 )) × | fν (ζi1 , αi , ζi2 )|ν /(2−ν ) } − sat {sgn(ζi2 )|εζi2 |ν } ),
(15)
And α˜ i = ζi1 is the output of the command filter. Besides, the compensation signals are designed as follows:
⎧˙ ⎨ξ1 = −c1 ξ1 + ξ2 + (α˜ 1 − α1 ) − lsgn(ξ1 ), ξ˙ = −ci ξi − ξi−1 + ξi+1 + (α˜ i − αi ) − lsgn(ξi ), ⎩ i ξ˙n = −cn ξn − ξn−1 − lsgn(ξn ),
(16)
where ci > 0 and l > 0 are design parameters. Based on the above preparation, we define the compensation error signal as
η i = z i − ξi ,
i = 1, . . . , n.
(17)
And the adaptive asymptotical tracking controller can be designed by the following procedures. Step 1: Consider the Lyapunov function candidate
V1 =
1 2 1 η + ϑ˜ 2 , 2 1 2κ 1
(18)
where κ is a positive design parameter. ϑ˜ 1 = ϑ1 − ϑˆ 1 , where ϑˆ 1 is the estimate of ϑ1 and ϑ1 is defined as ϑ1 = d¯1 + l. According to (1), (14) and (16), the derivative of V1 can be expressed as
V˙ 1 = η1 (x2 + θ T f1 + d1 − y˙ r + c1 ξ1 − ξ2 − (α˜ 1 − α1 ) + lsgn(ξ1 )) − ≤ η1 (η2 + α1 + θ T f1 − y˙ r + c1 ξ1 ) + |η1 |ϑ1 −
1 ˜ ˆ˙ ϑ1 ϑ1
κ
1 ˜ ˆ˙ ϑ1 ϑ1
κ
(19)
Then, design the virtual controller α 1 and the adaptive law as
α1 = −c1 z1 − θˆ f1 + y˙ r − ϑˆ1tanh ϑˆ˙ 1 = κη1tanh
η
1 , σ (t )
η
1 , σ (t )
(20) (21)
where θˆ is the estimate of θ ; σ (t) is selected based on Lemma 2. Substituting (20) and (21) into (19) yields
V˙ 1 ≤ −c1 η12 + η1 η2 + η1 θ˜ T f1 + ϑ1 |η1 | − ϑ1 η1 tanh
η
1 , σ (t )
(22)
where θ˜ = θ − θˆ . By using Lemma 3, we have
ϑ1 |η1 | − ϑ1 η1tanh
η
1 ≤ ϑ1 ισ (t ). σ (t )
(23)
Therefore, (22) satisfies
V˙ 1 ≤ −c1 η12 + η1 η2 + η1 θ˜ T f1 + ϑ1 ισ (t ).
(24)
Step i = 2, . . . , n − 1: Consider the Lyapunov function candidate
Vi = Vi−1 +
1 2 1 η + ϑ˜i2 , 2 i 2κ
where ϑ˜ i = ϑi − ϑˆ i , ϑˆ i is the estimate of ϑi and ϑi is defined as ϑi = d¯i + l.
(25)
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
5
According to (1), (14) and (16), we have i−1
V˙ i ≤ −
c j η2j + ηi−1 ηi + θ˜ T
j=1
i−1
η j f j + ισ (t )
j=1
i−1
1
ϑ j − ϑ˜i ϑˆ˙ i κ j=1
+ ηi (xi+1 + θ T fi + di − α˜˙ i−1 + ci ξi + ξi−1 − ξi+1 − (α˜ i − αi ) + lsgn(ξi )) i−1
≤−
c j η2j + θ˜ T
j=1
i−1
η j f j + ισ (t )
i−1
1
ϑ j − ϑ˜i ϑˆ˙ i + ηi (ηi+1 + αi + θ T fi − α˜˙ i−1 + ci ξi + zi−1 ) + |ηi |ϑi . κ j=1
j=1
(26)
˙ Then, we design the virtual controller α i and the adaptive law ϑˆ i as
αi = −ci zi − zi−1 − θˆ fi + α˙ i−1 − ϑˆitanh ϑˆ˙ i = κηitanh
η
i . σ (t )
η
i , σ (t )
(27) (28)
Substituting (27) and (28) into (26) yields i
V˙ i ≤ −
c j η2j + θ˜ T
j=1
i
η j f j + ισ (t )
i
j=1
Step n: Define γ = bδ1 = 1
Vn = Vn−1 +
1 r
ϑ j + ηi ηi+1 .
(29)
j=1
and δ 1 = 1 − δ . Consider the Lyapunov function candidate
1 2 1 1 r η + ϑ˜ 2 + θ˜ T θ˜ + γ˜ T γ˜ , 2 n 2κ n 2κ 2κ
(30)
where ϑ˜ n = ϑn − ϑˆ n , ϑˆ n is the estimate of ϑn and ϑn is defined as ϑn = d¯n + l + b¯ u0 ; γ˜ = γ − γˆ , γˆ is the estimate of γ . Then, the derivative of Vn can be expressed as
V˙ n ≤ −
n−1
c j η2j + ηn−1 ηn + θ˜ T
j=1
n−1
η j f j + ισ (t )
j=1
n−1
1
1
ϑ j − ϑ˜n ϑˆ˙ n − θ˜ T θˆ˙ κ κ j=1
+ ηn (bq(u ) + θ fn + dn − α˜˙ n−1 + cn ξn + ξn−1 + lsgn(ξn ) ) − T
=−
n−1
c j η2j + θ˜ T
j=1
n−1
η j f j + ισ (t )
j=1
n−1
1
1
r
κ
γ˜ γˆ˙
r
ϑ j − ϑ˜n ϑˆ˙ n − θ˜ T θˆ˙ − γ˜ γˆ˙ κ κ κ j=1
+ ηn (bδ1 u + bδ2 + θ T fn + dn − α˜˙ n−1 + cn ξn + zn−1 + lsgn(ξn ) ).
(31)
Similarly, the following inequality can be obtained
ηn (bδ2 + dn + lsgn(ξn )) ≤ |ηn |ϑn .
(32)
Therefore, (31) can be rewritten as
V˙ n ≤ −
n−1
c j η2j + θ˜ T
j=1
n−1
η j f j + ισ (t )
j=1
n−1
1
1
ϑ j − ϑ˜n ϑˆ˙ n − θ˜ T θˆ˙ κ κ j=1
+ ηn (bδ1 u + θ fn − α˜˙ n−1 + cn ξn + zn−1 ) + |ηn |ϑn − T
r
κ
γ˜ γˆ˙ .
(33)
˙ ˙ Then, design the system controller u and adaptive laws ϑˆ n , θˆ , γˆ˙ as
u = −
ηn γˆ 2 αn2 , ηn2 γˆ 2 αn2 + σ 2 (t ) η
ϑˆ˙ n = κηntanh θˆ˙ = κ
n
n
σ (t )
,
η j f j − κσ (t )θˆ ,
(34)
(35)
(36)
j=1
γˆ˙ = κηn αn ,
(37)
6
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
where α n is designed as
αn = cn zn + zn−1 + θˆ fn − α˜˙ n−1 + ϑˆntanh
η
n . σ (t )
(38)
Then, substituting (34)–(38) into (33) yields
V˙ n ≤ −
n
c j η2j + ισ (t )
j=1
n
˜ (t )θˆ + bδ1 ηn u + ηn αn − r γ˜ ηn αn . ϑ j + θσ
(39)
j=1
By using Lemma 2, we have
ηn2 γˆ 2 αn2 η γˆ 2 αn2 + σ 2 (t ) ηn2 γˆ 2 αn2 ≤−r ≤ r (σ (t ) − γˆ ηn αn ). ηn2 γˆ 2 αn2 + σ 2 (t )
bδ1 ηn u = − bδ1
2 n
(40)
Then, we have
V˙ n ≤ −
n
c j η2j + ισ (t )
j=1
n
˜ (t )θˆ + r (σ (t ) − γˆ ηn αn ) + ηn αn − r γ˜ ηn αn . ϑ j + θσ
(41)
j=1
According to the definition of r and γ , the following equation can be obtained
−r γˆ ηn αn + ηn αn − r γ˜ ηn αn = 0.
(42)
By combining (41) with (42), we have
V˙ n ≤ −
n
c j η + σ (t ) 2 j
ι
n
j=1
ϑ j + θ˜ θˆ + r .
(43)
j=1
4. Stability analysis Based on the preparation work in Section 3, we have the following results as shown in Theorem 1. Theorem 1. Consider the nonlinear system (1) satisfying Assumptions 1–3. Under the controller (34) and the adaptive laws (21), (28), (35)–(37), global boundedness of the closed-loop system can be achieved and the tracking error y − yr can asymptotically converge to zero. Proof. First, it is easy to prove
1 4
θ˜ θˆ = θ θˆ − θˆ 2 ≤ θ 2 .
(44)
By combining (43) with (44), we have
V˙ n ≤ −
n
c j η2j + σ (t )μ,
(45)
j=1
where μ = ι
n j=1
ϑ j + 14 θ 2 + r. By directly integrating (45) over [t0 , t), we have
Vn (t ) − Vn (t0 ) ≤ −
n
cj
j=1
≤−
n j=1
t0
cj
t
t
t0
η2j ds +
t
t0
μσ (s )ds
η2j ds + μσ¯ .
(46)
From (46), it can be seen that ηi , ϑˆ i , i = 1, . . . , n, θˆ and γˆ are globally bounded. Besides, based on (46), we also have
lim
t→+∞
n j=1
cj
t
t0
η2j ds ≤ Vn (t0 ) + μσ¯
(47)
By using Barbalat lemma, it can be concluded that limt→+∞ η1 (t ) = 0. Next, we will prove the boundedness of ξ . By using Lemma 1, we have that |α˜ i − αi | ≤ (ε ς τ ) can be guaranteed in finite time. Consider the Lyapunov function Vξ = 12 ni=1 ξi2 . According to (16), we have
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
V˙ ξ = −
n
ci ξi2 −
i=1
≤−
n
i=1
ci ξi2 −
i=1
where l =
√ 2 2 l0
V˙ ξ ≤ −
n
n i=1
l | ξi | −
n−1
7
ξi (α¯ i − αi )
i=1
√ n−1 2 l 0 | ξi | − (l1 − (ε ςτ ))|ξi |, 2
(48)
i=1
+ l1 . By choosing appropriate parameter l1 such that l1 − (ε ς τ ) ≥ 0 and using Lemma 4, we have
n i=1
ci ξi2 −
n i=1
√ 2 l0 |ξi | ≤ −c0Vξ − l0Vξυ , 2
(49)
where c0 = mini=1,··· ,n {2ci }, υ = 12 . By using Lemma 5, it can be concluded that ξ i , i = 1, . . . , n, can converge to zero in finite time. According to definition of zi and ηi , we can obtain the boundedeness of xi . Furthermore, since z1 = y − yr and η1 = z1 − ξ1 , we have y − yr = η1 + ξ1 . Considering the asymptotical stability of η1 and the finite-time stability of ξ 1 , we have limt→+∞ (y(t ) − yr (t )) = 0. Thus, the proof is completed. Remark 5. To obtain a better system performance, we can select the parameters ε to be large to accelerate the convergence of the command filter. And we can also choose large value of the parameter cj and l to accelerate the convergence speed of the closed-loop systems. However, large value of the parameters cj may result in large value of the control signal. Therefore, we should select the design parameters according to the demand. Remark 6. In our paper, command-filter based asymptotic tracking control is considered for quantized systems with mismatching disturbances. It should also be pointed out that the proposed method can also be extended to uncertain nonlinear systems with actuator faults. The reason is: As shown in [20], the actuator fault can be modeled as
u = δ1 (t )v + δ2 (t )
(50)
where v is the control input. 0 < δ 1 (t) ≤ 1 denotes the actuation effectiveness. δ 2 (t) denotes uncontrollable additive actuation fault which is bounded. Obviously, it has the same form of system (1). Therefore, the proposed method can be applied to deal with the control problem of nonlinear systems with actuator faults. Remark 7. In future research, we will develop the proposed results to the adaptive tracking control problem with dynamic quantizers, which can guarantee the desired system performance by adaptively updating the dynamic parameter. For system design with dynamic quantizers, references [33] and [34] had given significant design results, where the design conditions for the feedback controller and the quantizer’s parameter are given based on the linear matrix inequality technique. 5. Simulation examples In this section, two simulation examples will be given to illustrate the effectiveness of the proposed command-filterbased backstepping method. Example 1. Considering the following nonlinear system
⎧ ⎨x˙ 1 = x2 + d1 (x, t ), x˙ 2 = b(x, t )q(u ) + x21 + θ sin(x2 ) + d2 (x, t ), ⎩
(51)
y = x1
where θ is an unknown parameter; b(x, t) is an unknown function; di (x, t) are unknown and bounded disturbances. In this part, we will design an adaptive controller such that the output y = x1 can asymptotically track the reference signal yr = 1.5sint + cos1.5t. The actual values for the system parameters are chosen as: b(x, t ) = 0.5, θ = 1, d1 (x, t ) = sint, d2 (x, t ) = cost, and the quantization parameters are chosen as: u0 = 0.02, ρ = 0.2. Based on the results in Section 3, an adaptive controller can be designed. And the control parameters are selected as: c1 = c2 = 5, l = 7, κ = 5, σ (t ) = 0.1e−0.01t . And the initial conditions of the closed-loop system are chosen as: [x1 (0 ), x2 (0 )]T = [0.1, 0]T , [ζ1 (0 ), ζ2 (0 )]T = [0.1, 0]T . And the initial conditions of other signals are chosen as zero. Then the simulation results are shown in Figs. 1–6. The tracking performance can be found in Fig. 1. And the trajectories of the adaptive laws can be found in Figs. 3–6. From Fig. 1, we can see that asymptotic tracking control can be achieved by our proposed method. Example 2. Considering the following uncertain nonlinear system
⎧ x˙ = x2 + θ x21 + d1 (x, t ), ⎪ ⎨ 1 −x2 x˙ 2 = q(u ) + sin(x1 x2 ) + θ 1−e−x22 + d2 (x, t ), 1+e 2 ⎪ ⎩ y = x1 .
(52)
8
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
Fig. 1. The trajectories of y and yr .
Fig. 2. The trajectories of x2 .
Fig. 3. The trajectories of ϑˆ .
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
Fig. 4. The trajectories of θˆ .
Fig. 5. The trajectories of γˆ .
Fig. 6. The trajectories of u.
9
10
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
Fig. 7. The trajectories of y and yr under controller in [11].
Fig. 8. The trajectories of y and yr under controller in [11].
Fig. 9. The trajectories of y and yr under our proposed controller.
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
Fig. 10. The trajectories of ϑˆ under our proposed controller.
Fig. 11. The trajectories of θˆ under our proposed controller.
Fig. 12. The trajectories of γˆ under our proposed controller.
11
12
J. Ma, J.H. Park and S. Xu et al. / Applied Mathematics and Computation 377 (2020) 125161
The unknown parameter is chosen as: θ = 0.5. The mismatching disturbances are chosen as: d1 (x, t ) = sin(t ), d2 (x, t ) = cos(t ). The reference signal is chosen as yr = sin(t ). And the initial condition is [x1 (0 ), x2 (0 )]T = [0.1, 0]T , [ζ1 (0 ), ζ2 (0 )]T = [0.1, 0]T , and the initial condition of other signal is zero. The quantization parameters are chosen as: u0 = 0.02, ρ = 0.2. Adaptive asymptotic tracking control has been investigated for system (52) without mismatching disturbances in [11]. When di (x, t ) = 0, by exactly following the results in [11], the simulation result can be found in Fig. 7. As shown in Fig. 7, it can be seen that the system output y can asymptotically track the reference signal yr . However, when di (x, t) = 0, the simulation result by following the results in [11] is shown in Fig. 8. We can see that tracking performance is not desired. Then applying our proposed method to system (52), the system performances are shown in Figs. 9–12. Comparing Fig. 8 with Fig. 9, it can be seen that better tracking performance can be achieved than that in [11]. 6. Conclusion In this article, the command-filter-based adaptive asymptotic tracking control problem has been investigated for uncertain nonlinear systems with input quantization and mismatching disturbances. Different from the previous results, a priori knowledge of the quantization parameters and the disturbances is not required. The mismatching disturbances have been effectively compensated by introducing smooth functions in the virtual controllers. And an adaptive tracking controller has been designed based on the modified command filter and compensation signals. It has been proven that the global boundedness can be achieved and system output can asymptotically track the reference signal. Finally, two simulation examples have been provided to illustrate the effectiveness of the proposed method. Acknowledgements This work of J.H. Park and Z. 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