Active disturbance rejection control for uncertain time-delay nonlinear systems

Automatica 112 (2020) 108692

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Brief paper

Active disturbance rejection control for uncertain time-delay nonlinear systems✩ ∗

Maopeng Ran a , , Qing Wang b , Chaoyang Dong c , Lihua Xie a a

School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798, Singapore School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, PR China c School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, PR China b

article

info

Article history: Received 22 November 2017 Received in revised form 25 August 2019 Accepted 23 October 2019 Available online xxxx Keywords: Active disturbance rejection control (ADRC) Extended state observer (ESO) Uncertain nonlinear systems Time-delay

a b s t r a c t This paper addresses the problems of design and analysis of active disturbance rejection control (ADRC) for uncertain nonlinear systems subject to input time-delay. A novel extended state observer (ESO) is first designed as a predictor to realize the predictive state, and the total uncertainty which represents the effects of the unknown nonlinear dynamics and external disturbance. Based on the output of the ESO, a predictive ADRC law is presented. Conditions for the resulting closed-loop system to achieve convergence are obtained. It is shown that the upper bounds of the ESO gains are limited by the time-delay. Numerical simulations are given to illustrate the theoretical results. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Dealing with uncertainty is one of the most important issues in control theory (Petersen & Tempo, 2014). The active disturbance rejection control (ADRC), which was proposed by Han (2009), as an unconventional and efficient design strategy for uncertain systems, has been given a great deal of interest over the past several decades (Huang & Xue, 2014; Sariyildiz, Oboe, & Ohnishi, 2019). In ADRC, the effects of the unknown dynamics and internal/external disturbances are actively estimated and compensated for in real time, using a device called extended state observer (ESO) and ESO-based control laws, respectively. Such control design unburdens the process operator from the tedious task of establishing an accurate system mathematical model and improves anti-interference ability. Up to now, ADRC has been successfully implemented in a wide range of engineering applications (see, Huang & Xue, 2014; Sariyildiz et al., 2019, and references therein). Paralleled with its applications, the progress of the theoretical analysis of ADRC has also been made in motion. In Gao (2003), to make the ADRC more close to practical applications, a parameterized linear ADRC (LADRC) was proposed. Under the assumption ✩ This work was supported by the National Natural Science Foundation of China (61873295, 61833016). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Emilia Fridman under the direction of Editor Ian R. Petersen. ∗ Corresponding author. E-mail addresses: [email protected] (M. Ran), [email protected] (Q. Wang), [email protected] (C. Dong), [email protected] (L. Xie). https://doi.org/10.1016/j.automatica.2019.108692 0005-1098/© 2019 Elsevier Ltd. All rights reserved.

that the differentiation of the defined extended state is bounded, the stability result of LADRC was established in Aguilar-Ibañez, Sira-Ramirez, and Acosta (2017), and Zheng, Gao, and Gao (2012). In Shao and Gao (2017), the exponential stability of LADRC was obtained based on singular perturbation analysis approach. In Guo and Zhao (2013), the convergence of a class of nonlinear ADRC for multi-input multi-output nonlinear systems with large uncertainty was proved. In Jiang, Huang, and Guo (2015), a projected gradient estimator was incorporated into ADRC to lower the requirement of the knowledge of the control coefficient. In Zhao and Guo (2018), the fal-based ESO was considered, which exhibits smaller peaking value. On the other hand, time-delay is an inherent property of various practical control systems (Richard, 2003). The existence of time-delay may cause loss in control performance and even instability of the overall system (Besançon, Georges, & Benayache, 2007). Hence, control design and analysis for time-delay systems have attached a large amount of attention, especially in the fields of robust control (Fridman, 2014; Mazenc & Malisoff, 2017) and adaptive control (Zhang & Lin, 2015). Different from this, only a few works concern the design and analysis of ADRC for timedelay systems. Here, we can cite, for instance, Xia, Shi, Liu, Rees, and Han (2007), Xue, Liu, Chen, and Huang (2016), Zhao and Gao (2014), and Zheng and Gao (2014). However, the approaches in Xia et al. (2007), Zhao and Gao (2014), and Zheng and Gao (2014) mainly concentrated on linear time-invariant systems. In Xue et al. (2016), LADRC was considered for a class of simple time-delay nonlinear systems with globally Lipschitz nonlinearity. Motivated by the facts stated above, in this paper, we consider

2

M. Ran, Q. Wang, C. Dong et al. / Automatica 112 (2020) 108692

ADRC for the following uncertain nonlinear system with input time-varying delay

⎧ ⎨ z˙ (t) = f0 (t , x(t), z(t), ω(t)), x˙ (t) = Ax(t) + B[f (t , x(t), z(t), ω(t)) + u(t − τ (t))], ⎩ y(t) = Cx(t), t ≥ 0,

(1)

where x(t) = [x1 (t), . . . , xn (t)]T ∈ Rn and z(t) ∈ Rp are the states, y(t) ∈ R is the measured output, u(t) ∈ R is the control input with initial condition u(t) = 0 for t < 0, ω(t) ∈ R is the external disturbance, τ (t) is the input time-varying time-delay, f0 (·) : R+ × Rn × Rp × R → Rp and f (·) : R+ × Rn × Rp × R → R are unknown continuously differentiable functions, and the triple (A, B, C ) represents a chain of n integrators, i.e., 0 ⎢0

⎡

⎢.

A=⎢ ⎢ .. ⎣0 0

1 0

0 1

.. .

··· ··· .. .

0 0

0 0

··· ···

.. .

0 0⎥

⎤

0

⎡ ⎤

⎢0⎥ ⎢ ⎥ .. ⎥ ⎥ ∈ Rn×n , B = ⎢ .. ⎥ ∈ Rn×1 , ⎢.⎥ .⎥ ⎦ ⎣0⎦ 1 0

1

C = [1 0 · · · 0] ∈ R1×n . Many practical applications fall into the form of system (1), including the two-link planar manipulator in Fischer, Dani, Sharma, and Dixon (2013), the Euler–Lagrange system in Chakraborty, Obuz, and Dixon (2016), and the underwater vehicle in Ma, Xu, Li, Chu, and Zhang (2018). However, the ADRC technique cannot be directly applied to system (1) since it subjects to time-delay. The objective of this paper is to solve this problem. As far as the authors’ knowledge goes, this paper is the first attempt that considers ADRC for the uncertain time-delay nonlinear system (1). The main contributions of this paper are twofold: (i) The framework of the design of ADRC for time-delay systems is developed, in which a novel ESO acts as a predictor to realize the predictive state and the total uncertainty f (·). (ii) The conditions that guarantee the convergence of the closed-loop system are given by rigorous theoretical analysis. It is found that the upper bounds of the ESO gains are limited by the time-delay. This is a new feature compared with all existing ADRC results. Note that ADRC has been considered for nonlinear systems subject to actuator saturation (Ran, Wang, & Dong, 2016), and nonaffine-in-control problem (Ran, Wang, & Dong, 2017). However, it is stressed that in Ran et al. (2016, 2017), the ESO was designed in the general form, and hence cannot be applied to handle the effects of time-delay. What is more, for uncertain nonlinear systems with input time-delay, the ESO becomes a predictor. The interaction and coupling of time-delay make the analysis of the convergence of the ESO and the closed-loop system more complicated and challenging. These render the contributions of this paper with respect to Ran et al. (2016, 2017) considerable. We should point out that control for input time-delay nonlinear systems has been considered before. Different controllers can be found in the literature, including predictor-based controllers (Bekiaris-Liberis & Krstic, 2017; Krstic, 2010; Sharma, Bhasin, Wang, & Dixon, 2011; Zuo, Lin, & Ding, 2017), high-gain observer based controllers (Ghanes, Leon, & Barbot, 2013; Karafyllis & Krstic, 2013; Lei & Khalil, 2016), and robust controllers (Kamalapurkar, Fischer, Obuz, & Dixon, 2016; Obuz, Klotz, Kamalapurkar, & Dixon, 2017). Note that the approaches in these papers need some prior information of the system nonlinear dynamics (such as the nonlinear dynamics is perfectly known, the second partial derivatives of the nonlinear dynamics are known, the nonlinear dynamics is Lipschitz, and so on). In practice, these information may be unavailable in the general case. What is more, the

approaches in Bekiaris-Liberis and Krstic (2017), Kamalapurkar et al. (2016), Krstic (2010), Obuz et al. (2017), Sharma et al. (2011) and Zuo et al. (2017) are full state feedback. With respect to the aforementioned papers, the approach proposed in this paper provides a novel output feedback solution to control of time-delay nonlinear systems with totally unknown dynamics. Notations. Throughout this paper, ∥ · ∥ represents the Euclidean norm of a vector. ∇ f (·) refers to the gradient of a function f (·). A continuous function α (·) : R+ → R+ is said to belong to class K∞ if it is strictly increasing with α (0) = 0 and α (ν ) → ∞, as ν → ∞. Big O-notation in terms of ν is denoted as O(ν ) and it is assumed that this holds for ν positive and sufficiently small. For any h > 0, C denotes the space of absolutely continuous functions f (·) : [−h, 0] → Rn , which have square integrable first-order derivatives. The space of square integrable functions f (·) : [−h, 0] → Rn is represented by L2 ([−h, 0]; Rn ). Denote e(·) : [−h, ∞) → Rn as an absolutely continuous function; let et (·) ∈ C defined by et (ν ) = e(t + ν ), ν ∈ [−h, 0]; ∥et (ν )∥s = supν∈[−h,0] ∥e(t +ν )∥; e(t ; t0 , e0 ) represents the solution of a given time-delay system with initial time t0 and initial condition e0 . For a Lyapunov functional V (t , et , e˙ t ) : R × C × L2 → R+ , let V˙ (t , et , e˙ t ) = limϑ→0+ sup(1/ϑ )[V (t +ϑ, et +ϑ , e˙ t +ϑ ) − V (t , et , e˙ t )]. 2. ADRC design First of all, for uncertain system (1), it is assumed that: Assumption A1. (Fridman & Shaked, 2003; Obuz et al., 2017) The time-delay τ (t) is continuously differentiable with 0 ≤ τ (t) ≤ h and τ˙ (t) < 1, where h ∈ R+ is the maximum involved delay. Furthermore, the system (1) does not escape to infinity during the time interval [0, h]. Assumption A2. The external disturbance ω(t) and its derivative ω˙ (t) are both bounded. Assumption A3. There exists a (unknown) continuous function ψ (·) : Rn × Rp × R → R+ such that ∀(t , x, z , ω) ∈ R+ × Rn × Rp × R, max {|f (t , x, z , ω)|, ∥∇ f (t , x, z , ω)∥,

∥f0 (t , x, z , ω)∥, ∥∇ f0 (t , x, z , ω)∥} ≤ ψ (x, z , ω). Assumption A4. For all (t , x, ω) ∈ R+ × Rn × R, there exists a (unknown) positive definite function V0 (·) : Rp → R+ such that Lf0 V0 (z) ≤ 0, ∀z : ∥z ∥ ≥ ℓ (∥(x, ω)∥) , where Lf0 V0 (z) denotes the Lie derivative of V0 (z) along the vector field f0 (·), and ℓ(·) is a class K∞ -function. Denote ϱ(t) = t −τ (t). By Assumption A1, ϱ(t) is continuously differentiable and strictly increasing. Let χ (t) = ϱ−1 (t) be the inverse function of ϱ(t). Since ϱ(t) is the delayed time, and χ (t) is the inverse function of ϱ(t), the term χ (t) − t represents the prediction time. Assume χ˙ (t) ∈ [χ1 , χ2 ], where χ1 and χ2 are positive constants. For system (1), one can directly design the control as u(t) = u0 (x(χ (t))) − f (χ (t), x(χ (t)), z(χ (t)), ω(χ (t))),

(2)

where u0 (·) is designed such that the following cascade integrators system is globally asymptotically stable: x˙ (t) = Ax(t) + Bu0 (x(t)).

(3)

Since x(χ (ϱ(t))) = x(t), under control (2), the x(t)-dynamics of system (1) can be transformed into (3). However, the control (2) relies on future information x(χ (t)) and f (χ (t), x(χ (t)), z(χ (t)),

M. Ran, Q. Wang, C. Dong et al. / Automatica 112 (2020) 108692

ω(χ (t))), which are unavailable due to the time-delay and the assumption that the nonlinear function f (·) is unknown. In the following, predictive ESO-based control will be designed to handle this problem. From (1), the dynamics of x(χ (t)) can be obtained as

⎧ x˙ 1 (χ (t)) = χ˙ (t)x2 (χ (t)), ⎪ ⎪ ⎪ ⎪ . ⎪ ⎪ ⎨ .. x˙ n−1 (χ (t)) = χ˙ (t)xn (χ (t)), ⎪ ⎪ ⎪ ⎪ x˙ n (χ (t)) = χ˙ (t)f (χ (t), x(χ (t)), z(χ (t)), ω(χ (t))) ⎪ ⎪ ⎩ + χ˙ (t)u(t).

(4)

Let us define the following extended state xn+1 (χ (t)) = f (χ (t), x(χ (t)), z(χ (t)), ω(χ (t))).

(5)

According to (4) and (5), the ESO is designed as

) ( ⎧ ⎪ ˙ˆ 1 (t) = χ˙ (t)xˆ 2 (t) + χ˙ (t)ε n−1 g1 y(t) − xˆ 1 (ϱ(t)) , ⎪ x ⎪ ⎪ εn ⎪ ⎪ ( ) ⎪ ⎪ ⎪ y(t) − xˆ 1 (ϱ(t)) n−2 ⎪ ˙ ⎪ xˆ 2 (t) = χ˙ (t)xˆ 3 (t) + χ˙ (t)ε g2 , ⎪ ⎪ εn ⎪ ⎪ ⎪ ⎪ ⎨ .. . (6) ) ( ⎪ ⎪˙ y(t) − xˆ 1 (ϱ(t)) ⎪ ⎪ xˆ n (t) = χ˙ (t)xˆ n+1 (t) + χ˙ (t)gn ⎪ ⎪ εn ⎪ ⎪ ⎪ ⎪ ⎪ + χ ˙ (t)u(t) , ⎪ ⎪ ) ( ⎪ ⎪ ⎪˙ y(t) − xˆ 1 (ϱ(t)) ⎪ ⎩ xˆ n+1 (t) = χ˙ (t)ε−1 gn+1 , εn [ ]T where xˆ (t) = xˆ 1 (t), . . . , xˆ n+1 (t) ∈ Rn+1 is the observer state with initial condition xˆ 1 (t) = 0, ∀t ∈ [−τ (0), 0], gi (·) : R → R, i = 1, 2, . . . , n + 1, are continuously differentiable functions to be designed, and ε is a small positive constant. Remark 1. ESO (6) is a predictor for system (1). Compared with the general ESO (Gao, 2003; Guo & Zhao, 2013; Han, 2009; Jiang et al., 2015; Ran et al., 2016, 2017), the main feature of ESO (6) is that the input of the functions gi (·), i = 1, 2, . . . , n + 1, is (y(t) − xˆ 1 (ϱ(t)))/ε n rather than (y(t) − xˆ 1 (t))/ε n . The reason behind this modification is the influence of the input timedelay. In fact, for time-delay system (1), xˆ (t) is the estimation of [xT (χ (t)), xn+1 (χ (t))]T instead of [xT (t), xn+1 (t)]T . We should point out that since the time-delay τ (t) is embedded in the functions gi (·), i = 1, 2, . . . , n + 1, the analysis of ESO (6) becomes very difficult, and the previous ESO results (Guo & Zhao, 2013; Jiang et al., 2015; Ran et al., 2016, 2017) cannot be straightforwardly extended to ESO (6). Based on the output of the ESO (6), the control is given by unom (t) = u0 (xˆ (t)) − xˆ n+1 (t).

(7)

Similar to Freidovich and Khalil (2008), to protect the system from the peaking in the transient period of ESO, the actual control to be injected into the system is modified as

( u(t) = Msat

unom (t)

)

M

,

(8)

where M is the saturation bound selected such that the saturation will not be invoked under state feedback (Lee, Mukherjee, & Khalil, 2015), and sat(·) : R → R is the standard unity saturation function defined by sat(ν ) = sign(ν ) · min{1, |ν|}. Define the scaled estimation error η(t) = [η1 (t), . . ., ηn+1 (t)]T ∈ Rn+1 with

ηi (t) =

xi (χ (t)) − xˆ i (t)

ε n+1−i

, i = 1, 2, . . . , n + 1.

(9)

3

From (4) and (6), the dynamics of η(t) satisfies

⎧ ⎪ ⎪ εη˙ 1 (t) = χ˙ (t) [η2 (t) − g1 (η1 (ϱ(t)))] , ⎪ ⎪ ⎨ .. . ⎪ ⎪ ε η ˙ ˙ (t) [ηn+1 (t) − gn (η1 (ϱ(t)))] , n (t) = χ ⎪ ⎪ ⎩ ε η˙ n+1 (t) = χ˙ (t) [εδ (χ (t)) − gn+1 (η1 (ϱ(t)))] ,

(10)

d(f (χ (t),x(χ (t)),z(χ (t)),ω(χ (t))))

. where δ (χ (t)) = dχ (t) The ESO (6) and the control u0 (·) are designed such that the following assumptions are satisfied. Assumption A5. The functions gi (·), i = 1, 2, . . . , n + 1, are globally Lipschitz with Lipschitz constants Γi , and gi (0) = 0. For all η(t) ∈ Rn+1 , there exist continuous, positive definite, and radially unbounded functions V1 (·), W1 (·) : Rn+1 → R+ and positive constants λ11 , λ12 , λ13 , λ14 , and β1 such that (i) λ11 ∥η(t)∥2 ≤ V1 (η(t)) ≤ λ12 ∥η(t)∥2 ,

λ13 ∥η(t)∥2 ≤ W1 (η(t)) ≤ λ14 ∥η(t)∥2 , n ∑ ∂ V1 (η(t)) (ii) (ηi+1 (t) − gi (η1 (t))) ∂ηi (t) i=1

∂ V1 (η(t)) − gn+1 (η1 (t)) ≤ −W1 (η(t)), ∂ηn+1 (t) ⏐ ⏐ ⏐} {⏐ ⏐ ∂ V1 (η(t)) ⏐ ⏐ ∂ V1 (η(t)) ⏐ ⏐ ⏐ ⏐ ⏐ ≤ β1 ∥η(t)∥. ,...,⏐ (iii) max ⏐ ∂η1 (t) ⏐ ∂ηn+1 (t) ⏐ Assumption A6. The control u0 (·) is globally Lipschitz with u(0) = 0. For all x(t) ∈ Rn , there exist continuous, positive definite, and radially unbounded functions V2 (·), W2 (·) : Rn → R+ and positive constants λ21 , λ22 , λ23 , λ24 , and β2 such that (i) λ21 ∥x(t)∥2 ≤ V2 (x(t)) ≤ λ22 ∥x(t)∥2 ,

λ23 ∥x(t)∥2 ≤ W2 (x(t)) ≤ λ24 ∥x(t)∥2 , n−1 ∑

∂ V2 (x(t)) ∂ V2 (x(t)) + u0 (x(t)) ≤ −W2 (x(t)), ∂ xi (t) ∂ xn (t) i=1 ⏐ ⏐ ⏐ ∂ V2 (x(t)) ⏐ ⏐ ⏐ ≤ β2 ∥x(t)∥. (iii) ⏐ ∂ xn (t) ⏐ (ii)

xi+1 (t)

Remark 2. Though the expressions of Assumptions A5 and A6 may look complicated, they are not restrictive at all. Similar assumptions can be found in Guo and Zhao (2013), and Ran et al. (2016, 2017). The simplest way to satisfy Assumptions A5 and A6 is to design the ESO (6) and the control u0 (·) in the following linear form:

⎧ l1 (y(t) − xˆ 1 (ϱ(t))) ⎪ ⎪ x˙ˆ 1 (t) = χ˙ (t)xˆ 2 (t) + χ˙ (t) , ⎪ ⎪ ε ⎪ ⎪ ⎪ ⎪ ⎪ x˙ˆ (t) = χ˙ (t)xˆ (t) + χ˙ (t) l2 (y(t) − xˆ 1 (ϱ(t))) , ⎪ 2 3 ⎪ ⎪ ε2 ⎪ ⎨ .. . ⎪ ⎪ ⎪ ⎪ ln (y(t) − xˆ 1 (ϱ(t))) ⎪ ˙ ⎪ ⎪ + χ˙ (t)u(t), ⎪ xˆ n (t) = χ˙ (t)xˆ n+1 (t) + χ˙ (t) ⎪ εn ⎪ ⎪ ⎪ ⎪ ⎩ x˙ˆ (t) = χ˙ (t) ln+1 (y(t) − xˆ 1 (ϱ(t))) , n+1 εn+1

(11)

u0 (xˆ (t)) = K xˆ (t),

(12)

4

M. Ran, Q. Wang, C. Dong et al. / Automatica 112 (2020) 108692

where L = [l1 , l2 , . . . , ln+1 ] ∈ Rn+1 and K = [k1 , k2 , . . . , kn ] ∈ R1×n are selected such that the polynomial ν n+1 + l1 ν +· · ·+ ln ν + ln+1 and the matrix A + BK are Hurwitz, respectively. The control based on (11) and (12) can be viewed as a generalization of the LADRC for time-delay systems. Note that in the proposed approach, the system nonlinear dynamics f (·) and the model of the external disturbance ω(t) are totally unknown. In Sanz, Garcia, Fridman, and Albertos (2017, 2018), linear ESO was designed for input time-delay systems. However, the ESO in Sanz et al. (2017) relies on the exact system nonlinear dynamics, and the model of the external disturbance in Sanz et al. (2018) was assumed to be partially known.

Lemma 2 will be proved by contradiction. Suppose that Lemma 2 is false. Then there exist t2 > t1 > t0 such that

⎧ V2 (χ (t1 )) = ζ0 , ⎪ ⎪ ⎨ V (χ (t )) = ζ , 2

2

n−1

δ (χ (t)) =

i=1

dt

(t , ηt , ε ) ≤ −

λ33 ∥ηt ∥2s , ε

+

dω(χ (t))

∂ f (· ) . dχ (t) ∂ω(χ (t))

∂ f (·) ∂ z(χ (t))

(14)

(15)

|Vη (t , ηta , ε) − Vη (t , ηtb , ε)| ≤ λ34 (∥ηta ∥s + ∥ηtb ∥s )∥ηta − ηtb ∥s , (16) where λ31 , λ32 , λ33 , and λ34 are positive constants. Proof. See Appendix. Based on Lemma 1, in the following we are going to show the boundedness of x(χ (t)). Let ζ0 = sup∥x(χ (t))∥≤∥x(χ (0))∥ V2 (x(χ (t))) + 1 and ζ1 ∈ (ζ0 , ∞), where V2 (·) is a positive definite function satisfying Assumption A6. Define the following two compact sets:

Ω0 ={x(χ (t)) ∈ Rn : V2 (χ (t)) ≤ ζ0 }, Ω1 ={x(χ (t)) ∈ Rn : V2 (χ (t)) ≤ ζ1 }. It can be observed that Ω0 ⊆ Ω1 , and x(χ (0)) is an interior point of Ω0 . The next lemma shows that under some conditions, x(χ (t)) remains in the set Ω1 , for any t ∈ [0, ∞). Lemma 2. Consider the closed-loop system formed of (1), (6), and (8). Suppose Assumptions A1 to A6 are satisfied. Then for any x(χ (0)) ∈ Ω0 , there exist ε1∗ and ε -independent positive constant κ such that if h < ε1∗ /κ , then for any ε ∈ (κ h, ε1∗ ) and t ∈ [0, ∞), x(χ (t)) ∈ Ω1 holds. Proof. Note that x(χ (0)) is an interior point of Ω0 . By Assumptions A1 to A4 and the fact that the control is bounded by an ε -independent positive constant, one can conclude that there exists an ε -independent t0 > 0 such that x(χ (t)) ∈ Ω0 , ∀t ∈ [0, t0 ].

(18)

⏐ ⏐ ⏐ dω(χ (t)) ⏐

By Assumptions A2 to A4, and (17), for any t ∈ [0, t2 ], ⏐ dχ (t) ⏐, |f0 (χ (t), x(χ (t)), z(χ (t)), ω(χ (t)))|, and all partial derivatives of f (χ (t), x(χ (t)), z(χ (t)), ω(χ (t))) are all bounded. What is more, |u(t)| ≤ M. Thus there exists an ε -independent positive constant N1 such that

|δ (χ (t))| ≤ N1 , ∀t ∈ [0, t2 ].

Lemma 1. Suppose Assumption A5 is satisfied. Then there exists an ε -independent positive constant κ such that for any ε ≥ κ h, there exists a Lyapunov functional Vη (t , ηt , ε ) for (13) satisfies

d Vη

∂ f (·) ∂ xn (χ (t))

+ f0 (χ (t), x(χ (t)), z(χ (t)), ω(χ (t)))

In this section, the convergence conditions of the closed-loop system formed by plant (1), ESO (6), and control (8) are investigated. For this, we first consider the following system which is simplified from (10):

λ31 ∥ηt ∥2s ≤ Vη (t , ηt , ε) ≤ λ32 ∥ηt ∥2s ,

∑ ∂ f (· ) ∂ f (·) + xi+1 (χ (t)) ∂χ (t) ∂ xi (χ (t)) + (xn+1 (χ (t)) + u(t))

(13)

(17)

The term δ (χ (t)) in (10) can be specified as

3. Convergence analysis

⎧ ε η˙ 1 (t) = χ˙ (t) [η2 (t) − g1 (η1 (ϱ(t)))] , ⎪ ⎪ ⎪ ⎪ ⎨ .. . ⎪ ⎪ ε η ˙ ˙ (t) [ηn+1 (t) − gn (η1 (ϱ(t)))] , n (t) = χ ⎪ ⎪ ⎩ ε η˙ n+1 (t) = −χ˙ (t)gn+1 (η1 (ϱ(t))).

1

⎪ ζ ≤ V2 (χ (t)) ≤ ζ1 , t ∈ [t1 , t2 ], ⎪ ⎩ 0 V2 (χ (t)) ≤ ζ1 , t ∈ [0, t2 ].

(19)

Next, we will prove that there exists an ε1 > 0 such that for any ε ∈ (κ h, ε1 ), ∥η(t)∥ is convergent in the time interval [t0 , t2 ]. Consider a Lyapunov functional Vη (t , ηt , ε ) which satisfies (14) to (16). Let η(10) (t + ϑ ) and η(13) (t + ϑ ) represent the solutions of (10) and (13) at t + ϑ starting from η(t) at t, respectively. Then one has lim sup

1 ( Vη (t + ϑ, η(10) (t + ϑ ), ε )

ϑ ) −Vη (t + ϑ, η(13) (t + ϑ ), ε) ) 1 ( (10) ≤ λ34 lim sup ∥η (t + ϑ )∥s + ∥η(13) (t + ϑ )∥s ϑ ϑ→0+ ( ) × ∥η(10) (t + ϑ ) − η(13) (t + ϑ )∥s 1 ( (10) = 2λ34 ∥ηt ∥s lim sup ∥η (t + ϑ )∥s ϑ ϑ→0+ ) 1 ( (10) ∥η (t + ϑ )∥s − ∥η(13) (t + ϑ )∥s × lim sup ϑ ϑ→0+

ϑ→0+

2

λ34 ∥ηt ∥s ∥(0, . . . , 0, ϑ χ˙ (t)δ (χ (t)))∥ ϑ ≤ 2λ34 χ2 ∥ηt ∥s |δ (χ (t))|. =

(20)

By (14)–(16), (19), and (20), the derivative of Vη (t , ηt , ε ) along (10) in the time interval [0, t2 ] satisfies dVη (t , ηt , ε ) dt

= lim sup

Vη (t + ϑ, η(10) (t + ϑ ), ε ) − Vη (t , ηt , ε )

ϑ

ϑ→0+

= lim sup

Vη (t + ϑ, η(13) (t + ϑ ), ε ) − Vη (t , ηt , ε )

ϑ

ϑ→0+

( + lim sup

Vη (t + ϑ, η(10) (t + ϑ ), ε )

ϑ→0+

−

Vη (t + ϑ, η

(13)

ϑ ) (t + ϑ ), ε )

ϑ λ33 ≤− ∥ηt ∥2s + 2λ34 χ2 ∥ηt ∥s |δ (χ (t))| ε

M. Ran, Q. Wang, C. Dong et al. / Automatica 112 (2020) 108692

λ33 ∥ηt ∥2s + 2λ34 χ2 N1 ∥ηt ∥s ε λ33 2λ34 χ2 N1 √ ≤− Vη (t , ηt , ε ) + √ Vη (t , ηt , ε ). (21) λ32 ε λ31 √ d Vη (t ,ηt ,ε ) dVη (t ,ηt ,ε ) = √ 1 , from (21), one has Since dt dt ≤−

2

d Vη (t , ηt , ε ) dt

≤−

Vη (t ,ηt ,ε )

λ33 √

√

2λ32 ε

Vη (t , ηt , ε ) +

λ34 χ2 N1 . √ λ31

(22)

By (14) and (22), it can be obtained that ∀t ∈ [0, t2 ],

√ ∥ηt ∥s ≤

Vη (t , ηt , ε )

√ λ ( √ 31 ) λ Vη (0, η0 , ε ) 2λ32 λ34 χ2 N1 − 33 t − ≤ ε e 2λ32 ε √ λ31 λ33 λ31 2λ32 λ34 χ2 N1 ε + λ31 λ33 √ λ λ32 2λ32 λ34 χ2 N1 − 33 t ≤ ∥η0 ∥s e 2λ32 ε + ε. (23) λ31 λ31 λ33 It follows from (23) that ∀t ∈ [t0 , t2 ], √ λ λ32 2λ32 λ34 χ2 N1 − 33 t ∥ηt ∥s ≤ ∥η0 ∥s e 2λ32 ε 0 + ε. (24) λ31 λ31 λ33 From (24), one can see that there exists an ε1 > 0 such that for any ε ∈ (κ h, ε1 ), ∥ηt ∥s = O(ε ) holds for t ∈ [t0 , t2 ]. What is more, since xˆ i (t) ≤ |xi (χ (t))| + ε n+1−i |ηi (t)|, i = 1, 2, . . . , n + 1, one can select M > supt ∈[0,t2 ] |u0 (x(χ (t))) − xn+1 (χ (t))| such that the control is out of saturation in the time interval [t0 , t2 ]. By Assumption A6, computing the derivative of V2 (x(χ (t))) along (4) in the time interval [t1 , t2 ] yields dV2 (x(χ (t))) dt

= χ˙ (t)

n−1 ∑

xi+1 (χ (t))

i=1

∂ V2 (x(χ (t))) ∂ xi (χ (t))

) ∂ V2 (x(χ (t))) ( xn+1 (χ (t)) + u0 (xˆ (t)) − xˆ n+1 (t) + χ˙ (t) ∂ xn (χ (t)) n−1 ∑ ∂ V2 (x(χ (t))) = χ˙ (t) xi+1 (χ (t)) ∂ xi (χ (t))

5

such that if h < ε1∗ /κ , then for any ε ∈ (κ h, ε1∗ ) and t ∈ [0, ∞), lim ∥x(t)∥ = O(ε ),

t →∞

sup

t ∈[T (ε ),∞)

|xi (χ (t)) − xˆ i (t)| = O(ε n+2−i ),

(27) (28)

where i = 1, 2, . . . , n + 1, and T (ε ) → 0 as ε → 0. Proof. From Lemma 2, one has that for any ε ∈ (κ h, ε1∗ ) and t ∈ [0, ∞), x(χ (t)) ∈ Ω1 . It follows that (23) holds for all t ∈ [0, ∞). By (23), letting T (ε ) = −ε ln ε n+1 yields (28). What is more, it follows from (25) that (27) holds. This completes the proof of Theorem 1. Remark 3. From Theorem 1, one can see that for time-delay systems, the lower bound of ε is limited by the time-delay. This is a new feature compared with all existing ADRC results. In Freidovich and Khalil (2008), Guo and Zhao (2013), Jiang et al. (2015), Ran et al. (2016, 2017), and Zhao and Guo (2018), it has been shown that, theoretically, the value of ε can be selected to be arbitrarily small to achieve better performance. From this point of view, the results in this paper provide a guideline for the implementation of ADRC for time-delay systems. Remark 4. Not that the uncertainty in system (1) is matched. However, the obtained results can be readily extended to systems with mismatched uncertainty. To accomplish this, a state transformation approach (see, e.g., Meng, Yang, Jagannathan, & Sun, 2014; Zhao & Guo, 2018) can be utilized to transform the systems with mismatched uncertainty into the form of (1). Remark 5. It is worthwhile to mention that the effect of measurement noise on the proposed ESO (6) is comparable to its effect on a standard high-gain observer (Ahrens & Khalil, 2009). Specifically, the estimation error has the order of magnitude O(µ/ε n ), where µ is the bound of the measurement noise. To mitigate the effect of measurement noise, the switch-gain approach proposed by Ahrens and Khalil (2009) can be adopted. 4. Examples In this section, two examples are conducted to verify the obtained theoretical results.

i=1

+ χ˙ (t)u0 (x(χ (t)))

∂ V2 (x(χ (t))) ∂ xn (χ (t))

( ) ∂ V2 (x(χ (t))) + χ˙ (t) ηn+1 (t) + u0 (xˆ (t)) − u0 (x(χ (t))) ∂ xn (χ (t)) ≤ −χ1 W2 (x(χ (t))) + χ2 N2 β2 ∥η(t)∥∥x(χ (t))∥ ( ) χ2 N2 β2 ≤ −χ1 λ23 ∥x(χ (t))∥ ∥x(χ (t))∥ − ∥η(t)∥ , χ1 λ23

Example 1. Consider the following uncertain time-delay nonlinear system

(25)

where N2 is an√ε -independent positive √ constant. In the time

ζ1 ≤ ∥x(χ (t))∥ ≤ , this together with λ21 (24) yields that √one can select ε to be sufficiently small such ζ χ λ that ∥η(t)∥ < √λ 1χ 1N23β , ∀t ∈ [t1 , t2 ]. Namely, there exists an

interval [t1 , t2 ],

ζ1 λ22

21 2 2 2

ε2 ∈ (κ h, ε1 ) such that for any ε ∈ (κ h, ε2 ), dV2 (x(χ (t))) dt

< 0, ∀t ∈ [t1 , t2 ],

(26)

which, contradicts (17). Thus Lemma 2 is verified. Now, we are ready to state our main results. Theorem 1. Consider the closed-loop system formed of (1), (6), and (8). Suppose Assumptions A1 to A6 are satisfied. Then for any x(χ (0)) ∈ Ω0 , there exist ε1∗ and ε -independent positive constant κ

⎧ z˙ (t) = −z(t) + ω(t) + x21 (t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x˙ 1 (t) = x2 (t), x˙ 2 (t) = sin(t) + x21 (t)x2 (t) + z(t)x1 (t)x22 (t) + ω(t) ⎪ ⎪ ⎪ + u(t − τ (t)), ⎪ ⎪ ⎩ y(t) = x1 (t), t ≥ 0,

(29)

where ω(t) = 12 sin(2t), τ (t) = 0.05(1 + sin(t)). For this example, we note that the existing ADRC results cannot be applied directly. However, it can be verified that system (29) satisfies all the conditions in Theorem 1, and consequently the proposed ADRC can be applied. The controller is designed in the linear form (11)–(12), with L = [3, 3, 1]T , ε = 0.05, χ˙ (t) = 1−0.051 cos(t) , K = [−4, −4], and M = 20. Initial conditions are set as x(0) = [1, 1]T , z(0) = 1, and xˆ (0) = [0, 0, 0]T . Figs. 1–2 depict the simulation results, from which one can see that the state x(t) converges to the origin and the ESO achieves satisfactory performance. What is more, Theorem 1 shows that the lower bound of ε is limited by the time-delay. Specifically, for example (29), the closed-loop system becomes unstable as ε reduces to 0.041.

6

M. Ran, Q. Wang, C. Dong et al. / Automatica 112 (2020) 108692

Fig. 1. System response and ESO output (Example 1).

Fig. 3. System response and ESO output (Example 2).

Fig. 2. Control input (Example 1).

Fig. 4. Control input (Example 2).

Example 2. To further verify the proposed approach, a simulation is carried out for an underwater vehicle. The motion of the vehicle with a constant input time-delay can be written as (Ma et al., 2018; Slotine & Li, 1991)

⎧ x˙ 1 (t) = x2 (t), ⎪ ⎪ ⎨ 1 c x˙ 2 (t) = − x2 (t)|x2 (t)| + u(t − τ ) + ω(t), ⎪ m m ⎪ ⎩ y(t) = x1 (t), t ≥ 0.

(30)

The system parameters are set as c = 0.5, m = 1, ω(t) = 0.1 sin(t), and τ = 0.1. The controller is in the linear form (11)– (12), with L = [3, 3, 1]T , ε = 0.4, χ˙ (t) = 1, K = [−1, −2], and M = 10. Initial conditions are set as x(0) = [2, 2]T , and xˆ (0) = [0, 0, 0]T . Fig. 3 shows the convergence of the system state and the ESO estimation error, which illustrates the effectiveness of the proposed approach. Fig. 4 depicts the control input. It can be observed that the saturation is activated in the transient period of the observer. Note that the proposed approach relies on the exact information of the time-delay τ (t). Fig. 5 shows the time response of the system with different τ . One can see that the proposed approach is robust to small perturbations of time-delay. 5. Conclusion In this paper, ADRC has been generalized to uncertain timedelay nonlinear systems. An ESO, which is the essential of ADRC,

Fig. 5. System response with different τ .

was designed to estimate not only the predictive state but also the defined extended state. It has been shown that under the ESO-based predictive ADRC law, the practical convergence of the resulting closed-loop system can be guaranteed. However, the lower bound of the ESO parameter ε depends on the time-delay. The results obtained in this paper provide a solution and guideline for applying ADRC to uncertain time-delay nonlinear systems.

M. Ran, Q. Wang, C. Dong et al. / Automatica 112 (2020) 108692

In further continuing work, along the line, we shall investigate the robustness of ADRC with respect to input time-delay. Acknowledgments The authors appreciate the Associate Editor and anonymous reviewers for their valuable suggestions. Appendix. Proof of Lemma 1

∫t

Since η1 (ϱ(t)) = η1 (t) − t −τ (t) η˙ 1 (ν )dν ≜ η1 (t) − ϖ , system (13) can be further written as

χ1 λ13 ∥η(t)∥2 + h∥η˙ (t)∥2 − 2ε (χ2 Γ β1 )2 2 + ϖ , 2χ1 λ13 ε ∑n+1 where Γ = i=1 Γi . From (A.1), one has

) ( ε 2 η˙ i2 (t) ≤2χ22 ηi2+1 (t) + gi2 (η1 (t) − ϖ ) ) ( ≤2χ22 ηi2+1 (t) + (Γi |η1 (t) − ϖ |)2 ) ( ≤2χ22 ηi2+1 (t) + 2Γi2 η12 (t) + 2Γi2 ϖ 2 ,

dt

=

=

∑

η˙ i (t)

≤χ22 (Γn+1 |η1 (t) − ϖ |)2 ( ) ≤χ22 2Γn2+1 η12 (t) + 2Γn2+1 ϖ 2 .

i=1

χ˙ (t) ε

(

n

∑

(ηi+1 (t) − gi (η1 (t) − ϖ ))

i=1

(A.2)

By Assumption A5 and considering χ˙ (t) ∈ [χ1 , χ2 ], one has dt

∫

2 2

n+1 ∑

( η

t

∥η˙ (ν )∥2 dν

t −h

2 i (t)

+χ

2 2

i=2

+χ

) 4Γi + 2

2Γn2+1

η12 (t)

i=1

( 2 2

n ∑

n ∑

) 4Γi + 2

2Γn2+1

ϖ2

≤c1 ∥η(t)∥2 + c2 ϖ 2 , (A.7) ( ∑ ∑ n n 2 where c1) = χ22 max{2, i=1 4Γi2 + 2Γn2+1 }, c2 = χ22 i=1 4Γi 2 +2Γn+1 . What is more,

t −h

i=1

χ1 W1 (η(t)) + h∥η˙ (t)∥2 − ε n+1 χ2 ∑ Γi β1 |ϖ |∥η(t)∥ + ε

2

i=1

η˙ T (ν )η˙ (ν )dν

∂ V1 (η(t)) ∂ηi (t)

≤−

ε ∥η˙ (t)∥ ≤2χ 2

t

) ∂ V1 (η(t)) −gn+1 (η1 (t) − ϖ ) ∂ηn+1 (t) ∫ t + h∥η˙ (t)∥2 − ∥η˙ (ν )∥2 dν t −h ( n χ˙ (t) ∑ ∂ V1 (η(t)) = (ηi+1 (t) − gi (η1 (t))) ε ∂ηi (t) i=1 ) ∫ t ∂ V1 (η(t)) −gn+1 (η1 (t)) + h∥η˙ (t)∥2 − ∥η˙ (ν )∥2 dν ∂ηn+1 (t) t −h ( n+1 ) χ˙ (t) ∑ ∂ V1 (η(t)) + (gi (η1 (t)) − gi (η1 (t) − ϖ )) . ε ∂ηi (t)

dVη

(A.6)

Combining (A.5) and (A.6) gives

(ν − t + h)η˙ T (ν )η˙ (ν )dν,

∂ V1 (η(t)) + hη˙ T (t)η˙ (t) − ∂ηi (t)

(A.5)

ε 2 η˙ n2+1 (t) ≤χ22 gn2+1 (η1 (t) − ϖ )

where V1 (η(t)) is a positive definite function satisfying Assumption A5. The derivative of Vη with respect to t can be computed as dVη

(A.4)

and

t

∫

t −h

(A.1)

t −h

n+1

∥η˙ (ν )∥2 dν

≤−

Consider the following Lyapunov–Krasovskii functional (Fridman & Shaked, 2003):

∫

t

i = 1, 2, . . . , n,

⎧ ε η˙ 1 (t) = χ˙ (t) [η2 (t) − g1 (η1 (t) − ϖ )] , ⎪ ⎪ ⎪ ⎪ ⎨ .. . ⎪ ⎪ εη˙ n (t) = χ˙ (t) [ηn+1 (t) − gn (η1 (t) − ϖ )] , ⎪ ⎪ ⎩ ε η˙ n+1 (t) = −χ˙ (t)gn+1 (η1 (t) − ϖ ).

Vη = V1 (η(t)) +

7

∫

ϖ ≤h 2

∫

t

|η˙ 1 (ν )| dν ≤ h 2

∫

t

∥η˙ (ν )∥2 dν.

(A.8)

t −h

t −h

Substituting (A.7) and (A.8) into (A.4) yields

χ1 λ13 c1 h c2 h ∥η(t)∥2 + 2 ∥η(t)∥2 + 2 ϖ 2 2 ε ε ε ∫ t (χ 2 Γ β 1 ) 2 2 2 − ∥η˙ (ν )∥ dν + ϖ 2χ λ ε t −h ∫ t 1 13 χ1 λ13 1 ∥η˙ (ν )∥2 dν ≤− ∥η(t)∥2 − 4ε 2 t −h ( ) χ1 λ13 c1 h − − 2 ∥η(t)∥2 4ε ε )∫ t ( 1 c2 h2 (χ2 Γ β1 )2 h − − 2 − ∥η˙ (ν )∥2 dν. (A.9) 2 ε 2χ1 λ13 ε t −h { √ 2(χ Γ β )2 } 4c Let κ = max χ λ1 , 2 c2 , χ2 λ 1 . Then by taking ε ≥ κ h, the 1 13 1 13 ∫t third and fourth terms of (A.9) are nonpositive. Since t −h (ν − t + ∫t h)η˙ T (ν )η˙ (ν )dν ≤ h t −h ∥η˙ (ν )∥2 dν , one can obtain dVη dt

(A.3)

≤−

i=1

≤− + =− −

∫ t χ1 λ13 ∥η(t)∥2 + h∥η˙ (t)∥2 − ∥η˙ (ν )∥2 dν ε t −h χ2 Γ β1 |ϖ |∥η(t)∥ ε ∫ t χ1 λ13 ∥η(t)∥2 + h∥η˙ (t)∥2 − ∥η˙ (ν )∥2 dν 2ε t −h ( )2 χ1 λ13 χ 2 Γ β1 (χ2 Γ β1 )2 2 ∥η(t)∥ − |ϖ | + ϖ 2ε χ1 λ13 2χ1 λ13 ε

∫ χ1 λ13 1 t 2 ≤− ∥η(t)∥ − ∥η˙ (ν )∥2 dν dt 4ε 2 t −h ∫ t 1 χ1 λ13 ≤− V1 (η(t)) − (ν − t + h)η˙ T (ν )η˙ (ν )dν 4λ12 ε 2h t −h κ1 ≤ − Vη , (A.10) ε { } χ λ where κ1 = min 41λ 13 , κ2 . dVη

12

8

M. Ran, Q. Wang, C. Dong et al. / Automatica 112 (2020) 108692

From (A.10), one has Vη (t) ≤

κ

1 Vη (0)e− ε t .

(A.11)

Similar to (A.5)–(A.7), applying Young’s inequality to (13) leads c to ∥η˙ (t)∥2 ≤ ε32 ∥ηt ∥2s , where c3 is an ε -independent positive con2 stant. This ( together)with Assumption A5, one has λ11 ∥η(t)∥ ≤ c3 h2

Vη (t) ≤ λ12 +

√ ∥η(t)∥ ≤

√ ≤

ε2

Vη (t)

λ11

∥ηt ∥2s . It follows from (A.11) that √

≤

Vη (0)e−

κ1 ε t

λ11

κ1 λ12 c3 h2 + ∥η0 ∥s e− 2ε t . 2 λ11 λ11 ε

(A.12)

According to Definition 1.4 in Kharitonov (2012), the origin of system (13) is globally exponentially stable. Let Vη (t , ηt , ε ) =

1

t +T

∫

ε +

∥η(ξ + θ; t , η0 )∥2s dξ t

sup

ξ ∈[t ,t +T ]

where T =

1

α

∥η(ξ + θ; t , η0 )∥2s ,

ln 2β , with α =

κ1 2ε

and β =

√

(A.13) λ12 λ11

+

c3 h2

λ11 ε2

given by

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Chaoyang Dong is a professor in School of Aeronautic Science and Engineering at Beihang University, Beijing, China. He received the Ph.D. degree in Guidance, Navigation and Control in 1996 from Beihang University. He has authored or co-authored more than 100 papers. His research interests include multi-agent systems, distributed control, and modeling and synthesis of electrical systems.

Maopeng Ran received the B.E. and Ph.D. degrees from Beihang University, Beijing, China, in 2012 and 2018, respectively. He is currently a research fellow with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. His research interests include robotics, nonlinear systems control, multi-agent systems, and active disturbance rejection control. Dr. Ran received the Guan Zhao-Zhi Award at the 38th Chinese Control Conference, Guangzhou, China, in 2019.

Lihua Xie received the B.E. and M.E. degrees in electrical engineering from Nanjing University of Science and Technology in 1983 and 1986, respectively, and the Ph.D. degree in electrical engineering from the University of Newcastle, Australia, in 1992. Since 1992, he has been with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a professor and Director, Delta-NTU Corporate Laboratory for Cyber–Physical Systems. He served as the Head of Division of Control and Instrumentation from July 2011 to June 2014. He held teaching appointments in the Department of Automatic Control, Nanjing University of Science and Technology from 1986 to 1989. Dr Xie’s research interests include robust control and estimation, networked control systems, multi-agent networks, localization and unmanned systems. He is an Editor-in-Chief for Unmanned Systems and an Associate Editor for IEEE Transactions on Network Control Systems. He has served as an editor of IET Book Series in Control and an Associate Editor of a number of journals including IEEE Transactions on Automatic Control, Automatica, IEEE Transactions on Control Systems Technology, and IEEE Transactions on Circuits and Systems-II. He was an IEEE Distinguished Lecturer (Jan 2012–Dec 2014) and an elected member of Board of Governors, IEEE Control System Society (Jan 2016–Dec 2018). Dr Xie is a Fellow of IEEE and a Fellow of IFAC.

Qing Wang is a professor in School of Automation Science and Electrical Engineering at Beihang University, Beijing, China. She received the Ph.D. degree in Flight Control, Guidance and Simulation from Northwestern Polytechnical University, Xian, China, in 1996. She has authored or co-authored more than 150 papers. Her current research interests are multi-agent systems, switch control, and fault detection.