Practical tracking control of perturbed uncertain nonaffine systems with full state constraints

Automatica 110 (2019) 108608

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Brief paper

Practical tracking control of perturbed uncertain nonaffine systems with full state constraints✩ ∗

Ye Cao a , Yongduan Song a , , Changyun Wen b a b

Chongqing Key Laboratory of Intelligent Unmanned Systems, School of Automation, Chongqing University, Chongqing, China School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore

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Article history: Received 4 August 2018 Received in revised form 19 May 2019 Accepted 16 August 2019 Available online xxxx Keywords: Full state constraints Practical tracking control Pure-feedback systems Unknown disturbance

a b s t r a c t In this paper, we study the practical tracking control problem for a class of pure-feedback systems subject to full states asymmetric and time-varying constraints, non-vanishing uncertainties and external disturbances. A new robust control scheme is proposed to deal with state constraints. Unlike some existing results for practical tracking control, the proposed method does not involve any switching and is able to deal with asymmetric and time varying state constraints without the need for feasibility conditions. Furthermore, with proper choice of scaling function, three different tracking control results (i.e., ultimately uniformly bounded tracking, practical tracking and asymptotic tracking) can be achieved. Simulation verification further confirms the effectiveness of the proposed approach. © 2019 Published by Elsevier Ltd.

1. Introduction In real word, practical systems usually need to meet constraints in various forms, such as physical stoppages, performance and safety specifications (Krstic & Bement, 2006). For example, in order for an autonomously driving unmanned ground vehicle not to hit the objects (or moving vehicles) on both sides and not to collide with the front vehicle or be hit by the rear vehicle, its position and velocity must be controlled to strictly obey the corresponding constraints. Therefore, how to effectively deal with output and state constraints is of great significance and some methods have been developed based on set invariance (Bürger & Guay, 2010), model predictive control (Mayne, Rawlings, Rao, & Scokaert, 2000) and reference governors (Bemporad, 1998). Recently, the utilization of Barrier Lyapunov Function (BLF) or integral BLF (iBLF) for control of nonlinear systems with output or state constraints has received increasing attention, see for examples in Liu, Gong, Tong, Chen and Li (2018), Liu and Tong (2016), Liu et al. (2018), Tang, Ge, Tee, and He (2016), Tee, Ge, and Tay (2009), Tee, Ren, and Ge (2011), Zhao, Song, and Zhang (2019) ✩ This work was supported in part by the National Natural Science Foundation of China under Grants (No. 61933012, No. 61833013, No. 61860206008, No. 61773081). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Xiaobo Tan under the direction of Editor Miroslav Krstic. ∗ Corresponding author at: Chongqing Key Laboratory of Intelligent Unmanned Systems, School of Automation, Chongqing University, Chongqing, China. E-mail addresses: [email protected] (Y. Cao), [email protected] (Y. Song), [email protected] (C. Wen). https://doi.org/10.1016/j.automatica.2019.108608 0005-1098/© 2019 Published by Elsevier Ltd.

and references therein. In Tee et al. (2009, 2011), BLF-based control schemes are presented for strict-feedback systems with static output constraint and asymmetric time-varying output constraint, respectively. An adaptive control technique is developed for linearly parameterized pure-feedback systems with full state constraints in Liu and Tong (2016). In Tang et al. (2016), a neural network-based control is proposed for strict-feedback systems with time-varying disturbance and novel iBLFs are constructed to handle the unknown affine control gains and state constraints simultaneously. However, note that all the above mentioned results where BLF or iBLF is used to handle state constraints require that the virtual controller αi−1 (i = 2, . . . , n) should satisfy the so-called feasibility conditions −Fi1 < αi−1 < Fi2 , where Fi1 and Fi2 are constraint boundaries. Such requirement is a sufficient condition for the state constraint to be satisfied under the proposed control scheme. Thus offline optimization method is used to select a set of optimal design parameters to meet the feasibility conditions, which increases computational cost and is undesirable in practical application. Moreover, if the states are to be constrained in a small set, no optimal parameters could be identified to satisfy the feasibility condition, rendering the corresponding control scheme inapplicable, as pointed out in Tang et al. (2016). Recently, a non-BLF and non-iBLF based method is proposed to deal with full state constraints for strict-feedback systems without involving feasibility conditions in Zhao and Song (2018). By constructing a nonlinear state-dependent function, the state constraints are guaranteed as long as the boundedness of the proposed state-dependent function is ensured. Although the obstacle caused by state constraints can be handled gracefully, the tracking performance would be affected and it is difficult

2

Y. Cao, Y. Song and C. Wen / Automatica 110 (2019) 108608

to analyze the size of the compact set of tracking error even at steady state phase. In addition, most engineering systems involve non-affine dynamics, non-vanishing uncertainties and unknown disturbances. It is nontrival to achieve zero steady state tracking error asymptotically even for systems with arbitrarily bounded external disturbances, unless additional discontinuous/switching control compensator is included. In this situation, a less ambitious but practical control objective, namely practical tracking is proposed, where the tracking error can converge to any arbitrarily small neighborhood of the origin rather than zero. Some representative results of practical tracking/stabilization for strict-feedback systems can be found in Gong and Qian (2007), Jin, Liu, and Li (2018), Wu et al. (2018), Yan and Liu (2010), Zhang, Yang and Wen (2017), Zhang, Yang, Wen, Wang and Li (2017) and the reference therein. In Gong and Qian (2007), practical tracking is achieved via a dynamic output feedback controller, under the condition that the reference signal belongs to a known constant interval. Inspired by the homogeneous domination technique, nonrecursive practical tracking control schemes are presented in Zhang, Yang, Wen, Wang et al. (2017) for strict-feedback systems with known smooth nonlinear functions. In Jin et al. (2018) and Zhang, Yang, Wen et al. (2017), robust practical stabilization controllers and adaptive practical tracking controllers are proposed for a class of uncertain nonlinear systems satisfying linear growth rate and polynomial growth rate with respect to system nonlinearities, respectively. It is worth mentioning that most of the related literatures impose somewhat serious restrictions on the nonlinearities of systems and the reference signals. Global practical tracking control schemes are investigated for more general class of highorder nonlinear systems by introducing pseudosign functions in Yan and Liu (2010) and Wu et al. (2018). However, undesirable chattering phenomenon inevitably occur due to switching control strategy. Thus new techniques need to be developed to tackle the practical tracking control problem for more general nonlinear systems. In this paper, we will address this problem for a class of purefeedback systems. Firstly, to handle asymmetric time-varying state constraints without involving feasibility condition, we introduce a constrained function, which can impose constraints on the states directly. With certain transformation techniques, a new unconstrained system is obtained by incorporating the constraint boundaries into the original nonlinear system. Then, through scaling the tracking error by a modified exponential function, a robust control scheme is designed to realize practical tracking for state constrained system. In addition, by adjusting the scaling function as required, asymptotically stable and uniformly ultimately bounded results can also be achieved with the same design framework, and the time-varying asymmetric constraints are satisfied for all cases. It is worthy pointing out that the boundedness of control signal is ensured through rigorous analysis even if the scaling function is monotonically increasing with time. Numerical simulations are provided for all cases to demonstrate and verify the effectiveness of the proposed methods. Notations: For function f (x) : R → R, f (x) ∈ L∞ iff its L∞ norm is bounded, where ∥f (x)∥∞ = maxx∈R |f (x)|. A continuous function α : [0, a) → [0, ∞) is said to belong to class K if it is strictly increasing and α (0) = 0. A continuous function β : [0, a) × [0, ∞) → [0, ∞) is said to belong to class K L if, for each fixed s the mapping β (r , s) belongs to class K with respect to r and, for each fixed r, the mapping β (r , s) is decreasing with respect to s and β (r , s) → 0 as s → ∞. Throughout this paper, the argument in a variable or function sometimes is dropped if no confusion is likely to occur. For example, both gi and gi (·) denote gi (x¯ i , x0i+1 ).

2. Problem formulation and preliminaries 2.1. Preliminaries We start by recalling the definition of practically stable for parameterized nonlinear time-varying system of the form x˙ = f (t , x, θ )

(1)

where x ∈ R , t ∈ R≥0 , θ ∈ R is a free control parameter vector and f : R+ × Rn × Rm → Rn is locally Lipschitz in x and piecewise continuous in t for all θ under consideration. n

m

Definition 1 (Chaillet & Loria, 2006; Dasdemir & Zergeroglu, 2015). Let Θ ⊂ Rm be a set of parameters. System (1) is said to be Practically Stable, if given any positive constant ε > 0, there exist θ ∈ Θ and a class K L function h(·) such that the solution of (1) satisfies

∥x(t , θ, t0 , x0 )∥ ≤ ε + h(∥x0 ∥, t − t0 ),

(2)

n

where x0 ∈ R is the initial state and t0 ∈ R≥0 denotes initial time. When (2) is satisfied with ε = 0, system (1) is said to be Asymptotically Stable. This concept has also been similarly used in the stability analysis of nonlinear systems such as Xu and Zhai (2005) and Zhang, Yang, Wen, Wang et al. (2017). In other words, we say that system (1) is practically stable if the size ε of the compact set can be arbitrarily predetermined and diminished by a convenient choice of θ . Note that the parameter θ is typically composed of control gains. Therefore, the aforementioned practically stable constitutes a much stronger property than ultimate boundedness, as it only requires that solutions eventually enter a compact set without leaving it anymore. 2.2. System description and problem formulation Consider a class of unknown pure-feedback systems which is described by x˙ i = fi (x¯ i , xi+1 ) + di (t),

i = 1, . . . , n − 1

x˙ n = fn (x¯ n , u) + dn (t) y = x1

(3)

where xi = [x1 , x2 , . . . , xi ] , i = 1, . . . , n are the state variables, u ∈ R is the input of system, y ∈ R is the output of system, fi (x¯ i , xi+1 ), i = 1, . . . , n − 1 and fn (x¯ n , u) are smooth functions, di (·), i = 1, . . . , n denote unknown time-varying external disturbance. For simplicity of presentation, we define gi (x¯ i , xi+1 ) = ∂ fi (x¯ i ,xi+1 ) ∂ f (x¯ ,u) and gn (x¯ n , u) = n ∂ un . ∂ xi+1 By using the mean value theorem as in Khalil (2002), there must exist x0i+1 and u0 such that T

fi (x¯ i , xi+1 ) = fi (x¯ i , 0) + gi (x¯ i , x0i+1 )xi+1 , i = 1, . . . , n − 1 fn (x¯ n , u) = fn (x¯ n , 0) + gn (x¯ n , u0 )u

(4)

where x0i+1 is a point between zero and xi+1 , and u0 is a point between zero and u. The objective of this paper is to design a control law u(t) such that: (1) the boundedness of all signals is ensured and practical tracking is achieved, i.e. for any ε > 0, there exist corresponding design parameters such that |z1 (t)| = |y(t) − yd (t)| ≤ ε as t → ∞. (2) all states xi (i = 1, . . . , n) preserve certain time-varying asymmetric constraints at all times. More specifically, the states xi (t) are required to satisfy yd (t) − F11 (t)
−Fi1 (t)
i = 2, . . . , n,

(5)

Y. Cao, Y. Song and C. Wen / Automatica 110 (2019) 108608

where Fi1 (t) : R+ → R+ and Fi2 (t) : R+ → R+ are strictly positive time-varying functions, i.e. there exist positive constants εi such that Fi1 (t) > εi and Fi2 (t) > εi , i = 1, . . . , n. Furthermore, their first and second derivatives are continuous and bounded. To achieve the above mentioned goals, some assumptions are presented. Assumption 1. The desired trajectory and its first and second derivatives (namely yd (t), y˙ d (t) and y¨ d (t)) are known, continuous and bounded. Assumption 2. Certain crude structural information on the lumped uncertainties fi (x¯ i , 0)(i = 1, . . . , n) is available to allow unknown constants ai ≥ 0 and known functions ϕi (x¯ i ) ≥ 0 to be extracted, such that

|fi (x¯ i , 0)| ≤ ai ϕi (x¯ i )

(6)

for t ∈ [0, ∞), where ϕ˙ i (x¯ i ) is a continuous function. If x¯ i is bounded, so is ϕi (x¯ i ). Assumption 3. The signs of gi (x¯ i , x0i+1 )(i = 1, . . . , n − 1) and gn (x¯ n , u0 ) are known and there exist unknown constants 0 < g < i g i < ∞ such that g ≤ |gi (·)| ≤ g i for i = 1, . . . , n. i

Assumption 3 implies that the partial derivative gi (·) is strictly either positive or negative. Without losing generality, this paper assumes that 0 < g ≤ gi (·) ≤ g i . In other words, Assumption 3 i

guarantees that gi (·) ̸ = 0 for all time, which is the condition on controllability of system (3). Assumption 4. The unknown time-varying disturbances di (t)(i = 1, . . . , n) are bounded, i.e., |di (t)| < Di , ∀t ≥ 0 with Di as an unknown constant. Remark 1. Assumption 2 allows more general forms of uncertain nonlinear functions than linearly parameterized function fi (x¯ i , 0) = θiT γi (x¯ i , 0) as in Liu and Tong (2016) and Tee et al. (2009, 2011), where θi is an unknown constant vector and γi (x¯ i ) is a known continuous function vector. For instance, consider the uncertain function L(x) = ρ1 cos(ρ2 x) + xe−|ρ3 x| , where ρ1 , ρ2 and ρ3 are unknown constants. Clearly, neither state x nor unknown parameters ρ1 , ρ2 and ρ3 can be factored out from L(x). However, it is effortless to obtain function ϕ (x) = 1 + |x|, such that |L(x)| ≤ 2 2 aϕ (x), with a = max{ρ1 , 1}. Since ϕ (x) ≤ 1 + 12 + x2 = x2 + 23 , we can further choose ϕ (x) =

x2 2

+

3 2

to satisfy the continuously

differentiable condition in Assumption 2. In fact ϕi (x¯ i ) can be easily derived with only crude model information, which can also be seen in Song, Huang, and Wen (2016). In Assumption 3, although gi (·) appears to be similar to the affine terms in a strictfeedback system (Krstic, Kanellakopoulos, & Kokotovic, 1995), a major difference lies in that gi (·) is a function of x¯ i+1 , and thus, it is still a non-affine term in character.

where F1 (t) and F2 (t) are strictly positive and time-varying smooth functions. It is interesting to note that S(x) exhibits the following properties: (i) For −F1 (t) < x(t) < F2 (t), S(x) is smooth and strictly increasing which is thus invertible. (ii) S(x) → ∞ as x(t) → −F1 (t) or x(t) → F2 (t). (iii) S(0) = 0. Based on these properties of S(x), we have the following lemma, which is useful for establishing constraint satisfaction. Lemma 1. For any initial condition satisfying −F1 (0) < x(0) < F2 (0), if S(x) ∈ L∞ , then −F1 (t) < x(t) < F2 (t) holds for all t ∈ [0, +∞). Proof. Refer to the Appendix.

To deal with asymmetric time-varying state constraints, a new system transformation is developed to transform the original constrained system (3) into a new unconstrained system, whose stability can ensure the asymmetric time-varying state constraints given by (5). 3.1. System transformation Firstly, inspired by the work in Zhao and Song (2018), the following function is constructed. x(t) (F1 (t) + x(t))(F2 (t) − x(t))

(7)

□

From z1 = y − yd , the constraint of x1 is equal to the tracking error constraint, i.e. −F11 (t) < z1 (t) < F12 (t). By introducing s1 = z1 and with the aid of Lemma 1, it can be concluded (F +z )(F −z ) 11

1

12

1

that for any initial condition satisfying yd (0) − F11 (0) < x1 (0) < yd (0) + F12 (0), the constraint of z1 is naturally ensured as long as s1 is bounded for t ∈ [0, +∞). It follows that s˙1 = µ1 z˙1 + ν1

(8)

where

µ1 =

F11 F12 + z12 (F11 + z1 )2 (F12 − z1 )2

ν1 = −

[F˙11 F12 + F11 F˙12 + (F˙12 − F˙11 )z1 ]z1 . (F11 + z1 )2 (F12 − z1 )2

(9)

By combining (3) and (4), the tracking problem of full state constrained system is transformed to the stabilization problem of the following system with constrained states xi (i = 2, . . . , n). s˙1 = µ1 (f1 + g1 x2 + d1 − y˙ d ) + ν1 x˙ i = fi + gi xi+1 + di ,

i = 2, . . . , n − 1

x˙ n = fn + gn u + dn .

(10)

To further deal with the state constraints of xi (i = 2, . . . , n), we xi introduce a new transformed state si = (F +x )(F . According −x ) i1

i

i2

i

to Lemma 1, the problem of satisfying pre-specified constraint bound boils down to ensure the boundedness of si for all t > 0. Clearly, the new transformed system dynamic model is given by s˙1 = µ1 f1 + g1 h2 s2 + d1 − y˙ d + ν1

(

)

s˙i = µi fi + gi hi+1 si+1 + di + νi ,

(

)

i = 2, . . . , n − 1

s˙n = µn fn + gn u + dn + νn

(

)

(11)

where hi = (Fi1 + xi )(Fi2 − xi ), µi =

3. Main results

S(x) =

3

Fi1 Fi2 + x2i

(Fi1 + xi )2 (Fi2 − xi )2 ˙ ˙ ˙ [Fi1 Fi2 + Fi1 Fi2 + (Fi2 − F˙i1 )xi ]xi νi = − , i = 2, . . . , n (Fi1 + xi )2 (Fi2 − xi )2

(12)

and all of them are computable and available for control design. Therefore, a new unconstrained system (11) is obtained by incorporating the constraint boundaries into the original nonlinear system (3) and solving the problem satisfying the constrained state condition (5) can be transformed to solving a problem with boundedness signals as the only requirement. Furthermore, from property (iii) of S(x), asymptotic tracking can be achieved if limt →∞ s1 (t) = 0 is followed. Then, we only need to focus on designing controller u to stabilize the transformed system (11).

4

Y. Cao, Y. Song and C. Wen / Automatica 110 (2019) 108608

3.2. Control design and stability analysis

where Ξ1 = βζ1 (µ1 f1 + µ1 g1 h2 w2 + µ1 g1 h2 y2 − µ1 y˙ d + µ1 d1 + ˙ 1 ). By using Assumptions 2–4 and Young’s inequality ν1 + β −1 βw

Before moving on, we introduce a time-varying scaling function β , which plays an important role in our later control design. To gain better insight of the technical development, we first λt consider the case that β = M tanh( eM ), where λ > 0 and M > 1 are design parameters. The function tanh(·) is considered as a smooth approximation of the saturation function and more detailed explanation can be found in Wen, Zhou, Liu, and Su (2011). Based on the properties of function tanh(·), we have the following lemmas which are crucial for the development of practical tracking control.

as in Deng and Krstic (1997), we have

Lemma 2. tanh2 (x) +

For x ∈ R , the following inequality holds tanh(x)

≥ 1.

x

(13)

Proof. Refer to the Appendix.

Lemma 3. For β = M ), it holds that β˙ ≤ λβ and β ≥ where λ > 0 and M > 1 are design parameters. Proof. Refer to the Appendix.

1 , 2

w1 = s1 , wi = si − αif , i = 2, . . . , n. − αif )]3

hi

(14)

(16)

is constructed. This transformation together with other design skills, allows for the aforementioned practical tracking control objective to be achieved, as seen shortly. Furthermore, we define yi = αif −

hi

,

βζ1 µ1 g1 h2 y2 ≤r1 g 1 ζ12 µ21 +

Yi = β yi ,

(2) yd

i = 2, . . . , n.

β

4l1 g

4l1 g

1

1

1 ¯

r1 4g

β

(17)

4g

2

h22 Y22

1

2β ( g¯12 )2

h42 Y24 +

2r1

β g¯12

r1

4g

(22)

1

where r1 > 0 and l1 > 0 are design parameters. By summing both sides of above inequalities, we get

+ g 2 βµ21 ζ12 h22 w22 +

(15)

ζi = βwi

1

β

g12

Ξ1 ≤ g 1 βζ12 (r1 Φ1 + l1 Ψ1 ) + β ∆1

in which ϖi > 0 is a design parameter, hi is given in (12) and αi−1 is the input. Note that αi−1 is an intermediate virtual control which shall be developed at the (i − 1)th step of the design procedure to be presented. Finally, the control signal u is designed at step n. The first-order filter (15) is stable because Yi defined in (17) below is bounded as seen in the sequel. By employing the scaling function, an error transformation

αi−1

βζ1 ν1 ≤ βν1 |ζ1 | ≤ l1 g 1 βζ ν + 2 2 1 1

≤ 2r1 g 1 βζ12 µ21 +

where αif is the output of a first-order filter

αi−1

4r1 g 2 1

2 2 1

1

β D21

βw1 ≤ βλ|ζ1 ||w1 | ≤ l1 g 1 βλ ζ w +

−1 ˙

□

By using the backstepping method, the control design depends on the following transformation of coordinates:

ϖi α˙ if = [β (

βζ1 µ1 d1 ≤ βµ1 |ζ1 ||d1 | ≤ r1 g 1 βµ ζ +

4r1 g

1

βζ1 µ1 g1 h2 w2 ≤βµ1 g1 h2 |ζ1 ||w2 | ≤ g 2 βµ21 ζ12 h22 w22 +

□

λt tanh( eM

−βζ1 µ1 y˙ d ≤ βµ1 |ζ1 ||˙yd | ≤ r1 g 1 βµ ζ ˙ + 2 2 1 1

4r1 g

β

2 2 2 1 1 yd

βζ1 β

+

β a21

βζ1 µ1 f1 ≤ βµ1 a1 |ζ1 |ϕ1 ≤ r1 g 1 βµ21 ζ12 ϕ12 +

β

2r1

h42 Y24

(23)

where Φ1 = µ21 (ϕ12 + y˙ 2d + 3), Ψ1 = λ2 w12 + ν12 and ∆1 = a21

4r1 g

1 4r1 g

+ 1

+ 1

1 2l1 g

+ 1

g¯12

4g

+

2

2 r1

(

g¯12

4g

)2

1

+

D21

4r1 g

. Hence the virtual 1

control law at this step can be constructed as

α1 = −

1

µ1

(k1 + r1 Φ1 + l1 Ψ1 )ζ1

(24)

where k1 > 0 is a design parameter. It is obvious that µ1 ζ1 α1 ≤ 0, thus we have

βζ1 µ1 g1 α1 ≤ −k1 g 1 βζ12 − g 1 βζ12 r1 Φ1 − g 1 βζ12 l1 Ψ1 .

(25)

Now we choose the Lyapunov function candidate as V1 =

1 2

1

ζ12 + Y22 .

(26)

2

Also denote ¯ = [yd , y˙ d , y¨ d ] , ζ¯i = [ζ1 , . . . , ζi ], y¯ i = [y1 , . . . , yi ] and Y¯i = [Y1 , . . . , Yi ]. Now we carry out the control design step

Differentiating (26) and combining (21), (23) and (25), we have

by step.

V˙ 1 ≤ −k1 g βζ12 + β ∆1 + g βµ21 ζ12 h22 w22 +

T

1

Step 1: Differentiating ζ1 and using (11), we have

(18)

According to the definition of si and yi , we have si = wi + yi +

αi−1 hi

, i = 2, . . . , n.

(19)

It follows that

Then, the derivative of

y˙ 2 = α˙ 2f −

2r1

h42 Y24 + Y2 Y˙2 . (27)

h2

=−

Y23

ϖ2

+ ξ2 (·)

(28) α˙

(2)

+

∂α1 ˙ Φ ∂ Φ1 1

+

∂α1 ˙ Ψ ∂ Ψ1 1

2

α1 h˙ 2 h22

with α˙ 1 =

∂α1 ˙ ζ ∂ζ1 1

+

being a continuous function. Then it

follows that (20)

ζ along (20) is

1 2 2 1

ζ1 ζ˙1 = βζ1 µ1 g1 α1 + Ξ1

( α1 )′

where ξ2 (ζ¯3 , x¯ 3 , y¯ 3 , Y¯3 , y¯ d ) = − h1 + ∂α1 µ ˙ ∂µ1 1

ζ˙1 =β[µ1 (f1 + g1 h2 w2 + g1 h2 y2 + g1 α1 ˙ 1 ]. − y˙ d + d1 ) + ν1 + β −1 βw

β

Recalling the definition of Yi in (17) and the first-order filter (15), we have

˙ 1 ζ˙1 = β w ˙ 1 + βw ˙ 1 ]. = β[µ1 (f1 + g1 h2 s2 − y˙ d + d1 ) + ν1 + β −1 βw

2

(21)

β Y24 + βξ2 Y2 + λY22 ϖ2 βY 4 βY 4 βξ 2 β 1 ≤ − 2 + 2 + 2 + + λ2 Y24 + . ϖ2 4 2 4 4

Y2 Y˙2 ≤ Y2 (λβ y2 + β y˙ 2 ) = −

(29)

Y. Cao, Y. Song and C. Wen / Automatica 110 (2019) 108608 1

By choosing

h42

≥

ϖ2

2r1

+

1 4

+ λ2 + ϖ2∗ with ϖ2∗ being a positive

design parameter and combining (29), (27) becomes V˙ 1 ≤ −k1 g βζ12 − βϖ2∗ Y24 + β Γ1 + g βµ21 ζ12 h22 w22 + 1

2

βξ22 2

(30)

where Γ1 = ∆1 + 34 and g βµ21 ζ12 h22 w22 will be handled in the next 2 step. Step i(i = 2, . . . , n − 1): The time derivative of ζi = βwi is

ζ˙i = β[µi (fi + gi hi+1 wi+1 + gi hi+1 yi+1 + gi αi + di ) ˙ i ]. + νi − α˙ if + β −1 βw ζ along (31) is

1 2 2 i

Then the derivative of

ζi ζ˙i = βζi µi gi αi + Ξi

(32)

where Ξi = βζi (µi fi +µi gi hi+1 wi+1 +µi gi hi+1 yi+1 +µi di +νi − α˙ if + ˙ i ). By using Young’s inequality and following the analysis β −1 βw similar to (22), we have

Ξi ≤g i βζi2 (ri Φi + li Ψi ) +

β 2ri

(33)

where Φi = µ2i (ϕi2 + 3), Ψi = α˙ if2 + νi2 + λ2 wi2 and ∆i = 4ri g

3 4li g

+ i

+ i

g¯i2 4g

+

i+1

2 ri

( ¯ 2 )2 gi

4g

+

i

law αi can be constructed as

αi = −

1

µi

D2i 4ri g

. Hence the virtual control i

(ki + ri Φi + li Ψi + µ2i−1 wi2−1 h2i )ζi

(34)

where ki > 0 is a design parameter. Define a positive definite Lyapunov function 1

Vi =

2

1

ζi2 + Yi2+1 .

(35)

2

By combining (32)–(34), it follows that V˙ i ≤ − ki g βζi2 − g βµ2i−1 ζi2−1 h2i wi2 + g

+

β

i

2ri

i

i+1

βµ2i ζi2 h2i+1 wi2+1

Note that y˙ i+1 = α˙ i+1,f −

˙ n ]. ζ˙n = β[µn (fn + gn u + dn ) + νn − α˙ nf + β −1 βw

(36) Yi3+1

)′

(40)

The actual control law u is designed as 1

µn

(kn + rn Φn + ln Ψn + µ2n−1 wn2−1 h2n )ζn

(41)

where kn > 0 is a design parameter, Φn = µ2n (ϕn2 + 1) and 2 Ψn = α˙ nf +νn2 +λ2 wn2 . Define a positive definite Lyapunov function Vn =

1 2

ζn2 .

(42)

Then we have V˙ n ≤ −kn g βζn2 + g βµ2n−1 ζn2−1 h2n wn2 + β Γn n

(43)

n

a2n 4rn g

+ n

1 4r1 g

+ n

1 2ln g

D2n 4rn g

+ n

. Now the results related n

to the closed-loop system under the proposed controller (41) are summarized in the following theorem. Theorem 1. Consider the uncertain nonlinear system (3) subject to asymmetric time-varying full state constraints (5). Suppose Assumptions 1–4 hold, if the control designed in (41) with β = λt M tanh( eM ) is applied, then for any initial condition xi (0)(i = 1, . . . , n) satisfying (5), the following objectives are achieved. (i) The tracking error z1 (t) is bounded by −F11 (t) < z1 (t) < F12 (t), where F11 (t) and F12 (t) represent the transient performance bounds. (ii) Practical tracking is achieved, i.e., for any given ε > 0, there exists a proper scaling gain M ≥ 1 such that the tracking error z1 satisfies |z1 | ≤ ε as t → ∞. (iii) The asymmetric time-varying state constraints (5) are satisfied for all t. (iv) All closed loop signals are bounded. Proof. Now we consider the following Lyapunov function

h4i+1 Yi4+1 + β ∆i + Yi+1 Y˙i+1 . αi

Step n: The time derivative of ζn = βwn is

where Γn =

h4i+1 Yi4+1

+ g i+1 βµ2i ζi2 h2i+1 wi2+1 + β ∆i a2i

where −g βµ2i−1 ζi2−1 h2i wi2 is used to cancel out the corresponding i item in the previous step and g βµ2i ζi2 h2i+1 wi2+1 will be handled i+1 in the next step.

u=− (31)

5

V = V1 + · · · + Vn .

(44)

= −ϖ + ξi+1 (·) where i+1 ∂αi ∂αi ˙ ˙ i + ∂αi Ψ˙ i + ξi+1 (·) = − h + h2 αi with α˙ i = ∂ζ ζi + ∂µ µ ˙ i + ∂∂αΦi Φ ∂ Ψi i i i i+1 i+1 ∂ hi+1 ∂αi 2 2 2 ′ ˙ (µi−1 wi−1 hi ) and hi+1 = ∂ x x˙ i+1 being a continuous ∂ (µ2 w 2 h2 ) i+1

By combining (30), (39) and (43), it follows that

function, then we have

Now we define the following compact sets ΩV = {

h˙ i+1

α˙ i

i−1

(

hi+1

i−1 i

(37)

Therefore i

+(

h4i+1 2ri

+

i

1 4

+

λ2 4

−

1

ϖi+1

)β Yi4+1 +

2

+ β Γi

βϖi∗ Yi4 +

n ∑ βξ 2

i=2

i

i=2

2

+

n ∑

β Γi .

(45)

i=1

∑n

ζi2 + + ˙ + y¨ 2d ≤ i=1

≤ 2p} ⊂ R , Ωd = {[yd , y˙ d , y¨ d ] : B0 } ⊂ R3 , where p and B0 are positive constants specified by the designer which will be addressed later. In such a compact set ΩV × Ωd , there exists a positive constant Πi such that |ξi | ≤ Πi . Thus, n ∑

2n−1

ki g βζi2 −

n ∑

i

i=1

(38)

n ∑

i

i=1

V˙ ≤ −

βµ2i ζi2 h2i+1 wi2+1 i+1

βξi2+1

ki g βζi2 −

2 i=2 Yi

Yi4+1

V˙ i ≤ −ki g βζi2 − g βµ2i−1 ζi2−1 h2i wi2 + g

n ∑

∑n

β + βξi+1 Yi+1 + λYi2+1 ϖi+1 βξ 2 1 1 β 1 ≤ (− + λ2 + )β Yi4+1 + i+1 + + . ϖi+1 4 2 4 4

Yi+1 Y˙i+1 = −

V˙ ≤ −

T

y2d

y2d

βϖi∗ Yi2 + β Θ1

i=2

≤ −β r1 V + β Θ1

(46)

∑n 1

where Γi = ∆i + By choosing ≥ 2r + λ2 + + ϖi∗+1 with i ϖi∗+1 being a positive design parameter, (38) can be rewritten as

ϖi∗ + ≤ y4i + 14 . i=1 Γi and the first inequality holds due to the fact Θ1 ˙ By choosing p > r , we have V < 0 on V = p. Therefore V ≤ p

V˙ i ≤ − ki g βζi2 − βϖi∗+1 Yi4+1 − g βµ2i−1 ζi2−1 h2i wi2 i i

is an invariant set, i.e.∫ if V (0) ≤ p, then V (t) ≤ p for t ≥ 0. t Multiplying (46) by er1 0 β (s)ds , it yields that

3 . 4

+ g i+1 βµ2i ζi2 h2i+1 wi2+1 +

1 ϖi+1

βξi2+1 2

h4i+1

+ β Γi

where r1 = min{2ki g , 2ϖi∗ }, Θ1 =

∑n 1

1 4

(39)

∑n

i

2

2

i=2

Πi2 +

i=2

4

y2i

1

∫t ) d ( r1 ∫ t β (s)ds e 0 V (t) ≤ er1 0 β (s)ds β Θ1 . dt

(47)

6

Y. Cao, Y. Song and C. Wen / Automatica 110 (2019) 108608

Integrating (47) over [0, t ], we have V (t) ≤e−r1

∫t 0

β (s)ds

+ Θ1 e

−r1

V (0)

∫t 0

t

∫

β (s)ds

er1

∫τ

β (s)ds

0

β (τ )dτ .

(48)

0

We compute the second term on the right hand of (48) to get

Θ1 e−r1

∫t

β (s)ds

0

t

∫

er1

∫τ

β (s)ds

0

0

= Θ1 e−r1 =

Θ1 r1

∫t

β (s)ds

0

t

∫

er1

∫τ

β (τ )dτ ∫ τ

β (s)ds

0

d(

0

e−r1

∫t 0

β (s)ds)

0

∫ ⏐ β (s)ds r1 0τ β (s)ds ⏐t

e

0

≤

Θ1 r1

.

(49)

It follows that V (t) ≤ e−r1

∫t 0

β (s)ds

Θ2

V (0) +

r2

≤ V (0) +

Θ1 r1

.

(50)

Therefore, V ∈ L∞ for any initial condition satisfying (5), which indicates that ζi ∈ L∞ and Yi ∈ L∞ . (i) (ii) According to (50), it is obvious that ζi remains in a compact set Ωζi specified by

√ { ⏐ Θ1 } ⏐ Ωζi = ζi ⏐|ζi | ≤ 2(V (0) + ) .

(51)

r1

λt

Based on the fact that limt →∞ β = limt →∞ M tanh( eM ) = M and βw1 = ζ1 , we have

|w1 | ≤

1 M

√ 2(V (0) +

Θ1 r1

)

as t → ∞. Note that w1 (t) =

(52) z1 (t) (F11 (t)+z1 (t))(F12 (t)−z1 (t))

∈ L∞

and −F11 (0) < z1 (0) < F12 (0), then from Lemma 1, it follows that −F11 (t) < z1 (t) < F12 (t). Thus, we further have (F11 (t) + z1 (t))(F12 (t) − z1 (t)) < (F11 (t) + F12 (t))2 . By choosing√F = maxt ≥0 (F11 (t) + F12 (t))2 , it is concluded that |z1 | ≤ F M

2(V (0) +

Θ1 r1

) as t → ∞. Therefore, for any given ε > 0, if

M is selected to satisfy M>

F

ε

√ 2(V (0) +

Θ1 r1

),

(53)

then the tracking error z1 satisfies |z1 | ≤ ε as t → ∞ and practical tracking is achieved. (iii) Since x1 ∈ L∞ and −F11 (t) < z1 (t) < F12 (t), it follows that Φ1 , Ψ1 and α1 are bounded. As y2 and Y2 are bounded, then from (15) and (17), it follows that α2f and α˙ 2f are bounded. According to the definition of w2 in (14), we have s2 is bounded. Together with Lemma 1, we infer that −F21 (t) < x2 (t) < F22 (t) for any initial condition satisfying −F21 (0) < x2 (0) < F22 (0). Similarly, it is also easy to get that −Fi1 (t) < xi (t) < Fi2 (t) for −Fi1 (0) < xi (0) < Fi2 (0)(i = 3, . . . , n). Therefore, the asymmetric time-varying state constraints are satisfied at all times. (iv) By using similar analysis as above, it is not difficult to conclude that αif ∈ L∞ (i = 1, . . . , n) and si ∈ L∞ (i = 1, . . . , n), which further implies that αi−1 (i = 1, . . . , n) and u are bounded. Thus, all signals in the closed loop systems are bounded. This completes the proof. □ Remark 2. It is interesting to note that there are many post ), sible choices for the scaling function β , such as, M tanh( 1+λ M 2

λt

t ), M tanh( eM ) and so on, which impacts decay rate M tanh( 1+λ M of the tracking error during the main control phase. For instance, λt to get a faster convergence rate, we can choose β = M tanh( eM ) with a larger value of λ, or to slow down the convergence rate, we

t can choose β = M tanh( 1+λ ) with a smaller value of λ. On the M other hand, the steady-state tracking accuracy primarily depends on the value of the scaling gain M, which might be conservative to determine it according to (53) due to extensively used mathematical estimations. We can simply set a very large scaling gain M to guarantee the stability first, then tune M to be smaller and smaller while testing the gap between current tracking error accuracy and prescribed error accuracy until satisfactory response curves are achieved.

Remark 3. In Zhao and Song (2018), although the obstacle caused by state constraints can be handled without using the feasibility conditions, the tracking performance would be affected. Accordδ ing to (55) in Zhao and Song (2018), we have z1 = δ1 e, where 2 δ1 = (F11 + x1 )(F12 − x1 )(F11 + yd )(F12 − yd ), δ2 = F11 F12 + x1 yd > 0, and z1 is the tracking error. It should be noted that only z1 ∈ L∞ δ can be derived according to e ∈ L∞ and δ1 ∈ L∞ , and the 2 size of the compact set of tracking error z1 cannot be analyzed even at steady state phase. Besides, when δ2 approaches zero, z1 tends to infinity, rendering a poor tracking effect. To solve this problem, we develop a new system transformation method in this paper and impose the transformation on z1 instead of x1 , which is different from the work in Zhao and Song (2018). Then, through scaling the errors by a modified exponential function, a robust control scheme is designed to realize practical tracking for the state constrained system. As presented in Theorem 1, with the proposed control scheme the tracking error z1 converges to the compact set Ωz1 = {z1 : |z1 | ≤ ε} at steady state phase and the size of the compact set ε can be arbitrarily predetermined and reduced as will by choosing the appropriate parameter M. Thus, both tracking problem and state constraints can be handled gracefully at the same time here. Remark 4. Different from the control schemes in literatures Li, Tong, Liu, and Feng (2017) and Tong, Sui, and Li (2015), which are able to guarantee tracking errors within prescribed performance boundaries, here we focus on ensuring that all states obey the required constraints during the entire process of operation. In addition, in the absence of output constraint, the control scheme in this paper also allows shaping the transient performance of tracking error by choosing the performance boundaries F1 (t) and F2 (t) as desired. In Theorem 1, the scaling function β is bounded by the designed parameter M. It is interesting to note that the scaled variables ζi and Yi can still be proved to be bounded even if β is chosen as some other increasing unbounded functions such as 1+λt , 1+λt 2 or eλt . Furthermore, zero error (asymptotic) tracking results can be achieved by using the same design architecture. To save space, we give the results in the following corollary by only analyzing the case that β = eλt . Corollary 1. Consider the uncertain nonlinear system (3) subject to asymmetric time-varying full state constraints (5). Suppose Assumptions 1–4 hold, if the control scheme (41) with β = eλt is applied, then for any initial condition xi (0)(i = 1, . . . , n) satisfying (5), all the results in Theorem 1 hold except that the practical tracking becomes the perfect Asymptotic Tracking, i.e. limt →∞ z1 (t) = limt →∞ [y(t) − yd (t)] = 0. Proof. From the definition β = eλt , we have β˙ = λβ . Therefore, the following inequality in (22) still holds

˙ 1 ≤ βλ|ζ1 ||w1 | ≤ l1 g βλ2 ζ12 w12 + βζ1 β −1 βw 1

β 4l1 g

. 1

(54)

Y. Cao, Y. Song and C. Wen / Automatica 110 (2019) 108608

∑n 1

2 By employing the Lyapunov function V = 2 i=1 ζi + and using the analysis similar to Theorem 1, we have

∑n 1

i=2

2

V˙ ≤ −β r2 V + β Θ2 where r2 = min{2ki g , 2ϖi } and Θ2 = i 2 i=1 Γi . It follows that

∑n

∫t 0

β (s)ds

Yi2

(55) ∗

V (t) ≤ e−r2

7

V (0) +

Θ2 r2

≤ V (0) +

1 2

∑n

Θ2 r2

2 i=2 Mi

+

.

1 4

∑n

i=2

ϖi∗ + (56)

Therefore, V ∈ L∞ for any initial condition satisfying xi (0) ∈ Ωxi (i = 1, . . . , n), which indicates that ζi ∈ L∞ and Yi ∈ L∞ . Based on the fact that wi = e−λt ζi and yi = e−λt Yi , it is seen that wi and yi are bounded and limt →∞ wi (t) = limt →∞ yi (t) = 0. z1 (t) Note that w1 (t) = (F (t)+z (t))(F , it is ensured that z1 is (t)−z (t)) 11

1

12

1

Fig. 1. Trajectories of q(t) and q˙ (t) subject to the constraint.

bounded and limt →∞ z1 (t) = 0, which further implies that x1 is bounded as desired trajectory yd is bounded. On the other hand, as w1 ∈ L∞ and −F11 (0) < z1 (0) < F12 (0), then from Lemma 1, it follows that −F11 (t) < z1 (t) < F12 (t). The remaining results can be established by using the analysis similar to those in the proof of Theorem 1. The proof is completed. □ Remark 5. Although the controller (41) in Corollary 1 contains time-varying function β = eλt that monotonically increases with time, the control input generated by (41) remains bounded during entire system operation as proven theoretically. One of our key design techniques is to combine β with wi or yi , which enables the scaled variables ζi = βwi and Yi = β yi to remain bounded by (56). As β goes to infinity, wi goes to zero, resulting in the boundedness of ζi and hence of u.

Fig. 2. Control signal u(t).

Remark 6. One of the appealing features of the proposed method is that it provides a unified design framework to achieve different tracking goals by adjusting the scaling function β . Specifically, uniformly ultimately bounded and practically stable results can λt

be realized with β = 1 and β = M tanh( eM ), respectively. In addition, asymptotically stable results are achieved with β = eλt . 4. Simulation results To illustrate our proposed method and evaluate the effectiveness, we apply it to the following robotic manipulator system in Xing, Wen, Liu, Su, and Cai (2017): J q¨ (t) + Bq˙ (t) + MgL sin(q(t)) + d(t) = u(t) where q(t) ∈ R and q˙ (t) ∈ R are the angle and angular velocity of the rigid link, respectively. J ∈ R denotes the rotation inertia of the servo motor, B ∈ R is the damping coefficient, L ∈ R is the length from the axis of joint to the mass center, M ∈ R is the mass of the link, g ∈ R is the gravitational acceleration, d(t) denotes unknown time-varying external disturbance and u(t) ∈ R is the input of the system. Similar to Xing et al. (2017), for simulation the physical parameters are given as J = 1, MgL = 10, B = 2 and the external disturbance d(t) = 5 sin(10t) + 4, while they are all unknown for controller design. The control objective is two-fold: (1) the output q(t) tracks the trajectory qd (t) = sin(t)(rad) with desired tracking accuracy ε = 0.001, i.e. |z(t)| = |q(t) − qd (t)| ≤ 0.001 (rad) when t is sufficiently large; (2) the following full state constraints are achieved:

− 0.5 − 2−0.3t + qd (t)
(57)

In simulation, the initial conditions are set as q(0) = 0.3 (rad) and q˙ (0) = −1 (rad/s) and the control parameters are chosen as

Fig. 3. Tracking error with different scaling functions.

k1 = 0.6, r1 = 0.2, l1 = 2.5, k2 = 1, r2 = 4, l2 = 1.7, λ = 0.7, M = 500 and ϖ2 = 0.1. The simulation results are shown in Figs. 1–2. It is observed from Fig. 1 that the desired tracking goal is achieved and all of the states meet the predefined time-varying asymmetric constraints during the entire operational process. The control signal is also presented in Fig. 2. To qualitatively explore possible reasons of ensuring the states within the given boundary curve, we examine Figs. 1 and 2 at the same time. It is shown that whenever q˙ (t) approaches the boundary curve, our proposed control scheme provides relatively large control action that drive q˙ (t) back from the boundary curve. Now we verify that three different tracking control results can be achieved by respectively choosing different scaling functions λt

β = 1, M tanh( eM ) and eλt . As shown in Fig. 3, asymptotic tracking is achieved in the case that β = eλt , while uniformly ultimately bounded tracking and practical tracking can also be

8

Y. Cao, Y. Song and C. Wen / Automatica 110 (2019) 108608

5. Conclusion In this paper, we consider the control design for stateconstrained pure-feedback systems with non-vanishing uncertainties and unknown time-varying bounded disturbances. Constrained functions are employed to incorporate the constraint boundaries into the original nonlinear system directly. Then a unified design and stability analysis framework is developed to achieve three different tracking results by choosing appropriate scaling function. Compared with existing works, the proposed control schemes are able to deal with asymmetric time-varying state constraints, remove the need of feasibility conditions and achieve desired tracking accuracy. Fig. 4. Control signal with different control schemes.

Appendix A. Proof of Lemma 1

Proof. Now we give the proofs by contradiction. Suppose that x(t) = F3 for t = t1 , where F3 ≤ −F1 (t1 ) or F3 ≥ F2 (t1 ). Since −F1 (0) < x(0) < F2 (0) and x(t) is a continuous function, according to the intermediate value theorem as in Kostrykin and Oleynik (2012), there exists a time instant 0 < ( t2 < ) t1 such that x(t2 ) = −F1 (t2 ) or x(t2 ) = F2 (t2 ). As a result S x(t2 ) = ∞, which leads to a contradiction for the boundedness of S(x). Thus, we know −F1 (t) < x(t) < F2 (t), ∀t ∈ [0, +∞). This completes the proof. □ Appendix B. Proof of Lemma 2

Proof. Let γ (x) = tanh2 (x) + x , then taking the derivative of − γ (x) with respect to x yields dxd γ (x) = 2 tanh(x)sech(x) + sech(x) x tanh(x) . After some calculation, it is easy to get that there exists a 2 x constant 1 < c < 3, such that γ (x) is monotonically increasing within (0, c) and is monotonically decreasing within (c , +∞). Then according to L’Hospital rule as in Taylor (1952), we have 2 (x) tanh(x)′ tanh(x) = limx→0 1−tanh = 1. Therefore, limx→0 x = limx→0 x′ 1 limx→0 γ (x) = limx→+∞ γ (x) = 1 and γ (x) has a lower bound of tanh(x) 1, i.e. tanh2 (x) + x ≥ 1 for x > 0. The proof is completed. □ tanh(x)

Fig. 5. Tracking error with different scaling functions.

Appendix C. Proof of Lemma 3

Proof.

λt eλt 1 tanh2 ( eM λ t Me−λt tanh( eM )

λ

Fig. 6. Control signal with different scaling functions.

λt

realized with β = 1 and β = M tanh( eM ), respectively, which confirms the theoretical results. Besides, the steady-state precision of practical tracking control is high enough to meet most practical applications. The control signals are presented in Fig. 4. It is interesting to observe that no additional control effort is needed as compared to the traditional uniformly ultimately bounded controller. To further illustrate that, under the proposed practical tracking control scheme in Theorem 1, the decay rate of the tracking error is adjustable by the scaling function, we 2 choose different scaling functions M tanh( 1+M0.3t ), M tanh( 1+0M.4t ) 0.5t

and M tanh( e M ). The simulation results are shown in Figs. 5–6. It is observed that a faster convergence rate is achieved by choosing 0.5t β = M tanh( e M ) and a slower convergence rate is achieved by choosing M tanh( 1+M0.3t ), which is consistent with the discussions in Remark 2.

λt

From the definition β = M tanh( eM ), we have β˙ =

[ −

λt

)]. According to Lemma 2, we have tanh2 ( eM ) + λt tanh2 ( eM

λt

− 1 ≥ 0. Thus 1 − ) ≤ Me−λt tanh( eM ). λt By multiplying λe on both sides of the above inequality, it is λt λt concluded that λeλt [1 − tanh2 ( eM )] ≤ λM tanh( eM ), which further implies that β˙ ≤ λβ . On the other hand, we also have β ≥ M tanh( M1 ) ≥ tanh(1) ≥ 21 according the definition of β . □ References Bemporad, A. (1998). Reference governor for constrained nonlinear systems. IEEE Transactions on Automatic Control, 43(3), 415–419. Bürger, M., & Guay, M. (2010). Robust constraint satisfaction for continuous-time nonlinear systems in strict feedback form. IEEE Transactions on Automatic Control, 55(11), 2597–2601. Chaillet, A., & Loria, A. (2006). Uniform global practical asymptotic stability for time-varying cascaded systems. European Journal of Control, 12(6), 595–605. Dasdemir, J., & Zergeroglu, E. (2015). A new continuous high-gain controller scheme for a class of uncertain nonlinear systems. International Journal of Robust and Nonlinear Control, 25(1), 125–141. Deng, H., & Krstic, M. (1997). Stochastic nonlinear stabilization-i: a backstepping design. Systems & Control Letters, 32(3), 143–150, Eq. (312). Gong, Q., & Qian, C. (2007). Global practical tracking of a class of nonlinear systems by output feedback. Automatica, 43(1), 184–189.

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Zhao, K., Song, Y., & Zhang, Z. (2019). Tracking control of MIMO nonlinear systems under full state constraints: A single-parameter adaptation approach free from feasibility conditions. Automatica, (107), 52–60.

Ye Cao received the M.S. degree from Shaanxi normal university, Xi’an, China, in 2016. She is currently pursuing the Ph.D. degree with the School of Automation, Chongqing University, Chongqing, China. Her current research interests include robust adaptive control, neural network, and event-triggered control.

Yongduan Song received the Ph.D. degree in Electrical and Computer Engineering from Tennessee Technological University, Cookeville, TN, USA, in 1992. He held a tenured Full Professor with North Carolina A&T State University, Greensboro, NC, USA, from 1993 to 2008 and a Langley Distinguished Professor with the National Institute of Aerospace, Hampton, VA, USA, from 2005 to 2008. He is currently the Dean of the School of Automation, Chongqing University, Chongqing, China. He was one of the six Langley Distinguished Professors with the National Institute of Aerospace (NIA), Hampton, VA, USA, and the Founding Director of Cooperative Systems with NIA. His current research interests include intelligent systems, guidance navigation and control, bio-inspired adaptive and cooperative systems, rail traffic control and safety, and smart grid. Prof. Song was a recipient of several competitive research awards from the National Science Foundation, the National Aeronautics and Space Administration, the U.S. Air Force Office, the U.S. Army Research Office, and the U.S. Naval Research Office. He is an Associate Editor of a number of journals, including the IEEE TRANSACTIONS ON AUTOMATIC CONTROL, the IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS, and the IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS.

Changyun Wen (F’10) received the B.Eng. degree from Xi’an Jiaotong University, Xi’an in 1983 and the Ph.D. degree from the University of Newcastle, Australia in 1990. From August 1989 to August 1991, he was a Research Associate and then Postdoctoral Fellow at University of Adelaide, Australia. Since August 1991, he has been with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a Full Professor. His main research activities are in the areas of control systems and applications, autonomous robotic system, intelligent power management system, smart grids, cyber–physical systems, complex systems and networks, model based online learning and system identification. Prof. Wen is an Associate Editor of a number of journals including Automatica, IEEE Transactions on Industrial Electronics and IEEE Control Systems Magazine. He is the Executive Editor-in-Chief of Journal of Control and Decision. He served the IEEE Transactions on Automatic Control as an Associate Editor from January 2000 to December 2002. He has been actively involved in organizing international conferences playing the roles of General Chair including IECON 2020, General Co-Chair, Technical Program Committee Chair, Program Committee Member, General Advisor, Publicity Chair and so on. He received the IES Prestigious Engineering Achievement Award 2005 from the Institution of Engineers, Singapore (IES) in 2005. He received the Best Paper Award of IEEE Transactions on Industrial Electronics in 2017. He is a Fellow of IEEE, was a member of IEEE Fellow Committee from January 2011 to December 2013 and a Distinguished Lecturer of IEEE Control Systems Society from February 2010 to February 2013.