Global adaptive output-feedback stabilization for a class of uncertain nonlinear systems with unknown growth rate and unknown output function

Automatica 104 (2019) 173–181

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Global adaptive output-feedback stabilization for a class of uncertain nonlinear systems with unknown growth rate and unknown output function✩ Xuehua Yan a,c , Yungang Liu b , Wei Xing Zheng c ,

∗

a

School of Electrical Engineering, University of Jinan, Jinan 250022, Shandong, PR China School of Control Science and Engineering, Shandong University, Jinan 250061, Shandong, PR China c School of Computing, Engineering and Mathematics, Western Sydney University, Sydney, NSW 2751, Australia b

article

info

Article history: Received 5 January 2017 Received in revised form 9 January 2019 Accepted 9 February 2019 Available online xxxx Keywords: Uncertain nonlinear systems Global adaptive stabilization Unknown output function Unknown growth rate Adaptive control Output feedback

a b s t r a c t This paper addresses the problem of global adaptive stabilization by output-feedback for a class of uncertain nonlinear systems with unknown growth rate and unknown output function. By constructing a suitable Lyapunov function, a new systematic design scheme is proposed to derive an adaptive output-feedback controller with appropriate design parameters based on a dynamic high-gain, under which the system states can be globally regulated to zero. Because the proposed method uses the adaptive technique rather than the time-varying one, it does not require a priori known information about either the growth rate or the upper and lower bounds of the derivative of the unknown output function. This makes the investigated system substantially different from the existing results and also highlights the main contributions of the paper. A practical example is presented to show the validity of the theoretical results. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Global output-feedback stabilization for systems with unmeasured state-dependent growth, as a rather challenging issue in nonlinear control, has received much attention during the last two decades. In handling this issue, the most popular approach to control design is based on the output-feedback domination design by introducing a constant gain or a varying gain/update law (see, e.g., Benabdallah, Ayadi, & Mabrouk, 2014; Benabdallah & Echi, 2016; Choi & Lim, 2005; Koo, Choi, & Lim, 2011; Krishnamurthy & Khorrami, 2004; Lei & Lin, 2006; Liu, 2013; Praly & Jiang, 2004; Qian & Lin, 2002; Qian, Schrader, & Lin, 2003; Yang & Lin, 2005; Zhai, Du, & Fei, 2016; and references therein). The varying gains/update laws can be divided into three categories: adaptive high-gain, time-varying gain, and logic-based switching. Compared to adaptive high-gain, it is sometimes easier to integrate time-varying gain or logic-based switching into the control scheme. When the system uncertainties, arising from unknown ✩ This work was supported in part by the National Natural Science Foundation of China (Nos. 61304013, 61873146), and the Australian Research Council (No. DP120104986). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Martin Guay under the direction of Editor Miroslav Krstic. ∗ Corresponding author. E-mail addresses: [email protected] (X. Yan), [email protected] (Y. Liu), [email protected] (W.X. Zheng). https://doi.org/10.1016/j.automatica.2019.02.040 0005-1098/© 2019 Elsevier Ltd. All rights reserved.

parameters and/or system nonlinearities, are essentially minor, such as known growth rate (see, e.g., Qian and Lin (2002) and Zhai, Ai, and Fei (2013)), a constant gain is usually enough for an output-feedback domination design. Otherwise, it is necessary to introduce a varying gain/update law to counteract large uncertainties. The existing results, except Liu (2013), require that either control coefficients themselves, or at least one of the upper and lower bounds of unknown control coefficients (or generally the derivative of the output function (Zhai et al., 2013)) are known a priori. Recently, global regulation by output-feedback is considered in Liu (2013) for a class of uncertain nonlinear systems, where unknown constant control coefficients are assumed to be of known sign. However, the output function in Liu (2013) is only equivalent to the linear relationship of x1 , and the timevarying gain, rather than the adaptive gain, is employed in the observer and controller. As mentioned in Lei and Lin (2006), it is not easy to implement the time-varying controller from a practical point of view. Therefore, it is still necessary to develop a sound time-invariant adaptive output-feedback control strategy. Unfortunately, when there exist some large uncertainties in control coefficients/output function, it is rather difficult to find an adaptive technique (Liu, 2013). Now, for nonlinear systems with unknown growth rate and unknown output function (see Section 2), an interesting question naturally arises: is it possible to apply adaptive control to achieve global stabilization when the

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X. Yan, Y. Liu and W.X. Zheng / Automatica 104 (2019) 173–181

derivative of the output function has a known sign but unknown upper and lower bounds? To the best of our knowledge, this question still remains unanswered. For the above-mentioned problem, the main difficulty stems from the uncertainties (in the nonlinear terms and unknown output function) which cannot be linearly parameterized. Since the linear parameterization assumption is central in the development of adaptive identification, it is rather difficult (even impossible) to solve the aforementioned problem by the techniques based on parameter identification (Krstic, Kanellakopoulos, & Kokotovic, 1995; Malisoff & Zhang, 2012). With this insight as well as motivated by Lei and Lin (2006, 2007a,b), we use a non-identifier based universal control scheme so that the uncertain nonlinear system can be ‘‘adaptively stabilized". On the other hand, due to larger uncertainties in the output function/control coefficients, the output-feedback controller design becomes much more complicated. By combining the ideas and techniques of universal control and backstepping together and constructing a novel Lyapunov function, this paper resolves the above-mentioned problem and proposes a new adaptive high-gain output-feedback controller for a class of uncertain nonlinear systems with unknown growth rate and unknown output function. More specifically, compared with the closely related works (Lei & Lin, 2006, 2007a,b; Liu, 2008, 2013; Qian & Du, 2012; Shang & Liu, 2015; Shang, Liu, & Zhang, 2009; Yan & Liu, 2013; Zhai et al., 2013, 2016), the main contributions of the paper are threefold. First and most important, the restrictive condition on control coefficients/output function is largely relaxed by the new adaptive technique. In fact, it does not require a priori information (except the sign of the control coefficients/the derivative of the output function) on the growth rate, nor on the upper and lower bounds of the control coefficients/the derivative of the unknown output function. Second, design parameters are introduced in the adaptive control design. With the help of dynamic high-gain instead of constant or time-varying gain, a new coupled observer–controller design scheme is developed with appropriate design parameters, which can effectively deal with the uncertainties in the system nonlinearities and output function. Both the proposed adaptive output-feedback stabilizing controller and the introduced design parameters can be obtained in a recursive way by employing the backstepping-like method. Third, a new Lyapunov function is constructed for the resulting closed-loop system. Different from the common construction of Lyapunov function, a dynamic high-gain with certain parameters is introduced in the Lyapunov function. This gain plays a crucial role in establishing the desired global stability. The organization of this paper is as follows. Section 2 gives the problem formulation. Section 3 presents the design scheme for an adaptive output-feedback stabilizing controller. The main results are stated in Section 4. Section 5 provides an illustrative example. Finally, concluding remarks are made in Section 6. 2. Problem formulation Let us consider the uncertain nonlinear system in the following form:

⎧ ⎨x˙ i = xi+1 + φi (t , x, u), x˙ = u + φn (t , x, u), ⎩ n y = h(x1 ),

i = 1, . . . , n − 1, (1)

where x = [x1 , . . . , xn ]T ∈ Rn is the system state vector with the initial condition x(0); u ∈ R and y ∈ R are the control input and system output, respectively; φi : R+ × Rn × R → R, i = 1, . . . , n are unknown functions, but are continuous in the first argument and locally Lipschitz in the rest arguments; and h : R → R

is an unknown, continuously differentiable function satisfying h(0) = 0. In what follows, we assume that only the output y is measurable. In this paper, we aim to design an adaptive output-feedback controller for system (1), which can guarantee the global regulation of system (1). That is, all signals of the resulting closed-loop system are globally bounded on [0, +∞), while the state of the original system ultimately converges to zero. It is known that for nonlinear system (1) where uncertainties are associated with unmeasurable states, if the power of the nonlinearity growth with respect to unmeasured states is greater than 2, then there are counterexamples for which there exist no output-feedback controls. Hence, a lot of relevant research results are obtained under various structural or growth conditions. One of the common assumptions is that the growth of system nonlinearity is linear with respect to the unmeasurable states. On the other hand, for the system output, there exist many nonlinear output functions with bounded first derivatives, such as h(x1 ) = 2.1x1 + 0.1 sin(x1 ). In practice, as stated in Zhai et al. (2013), there always exists the nonlinear, uncertain relationship between y and x1 , such as between voltage output from the sensor and the real displacement x1 ,1 but the derivative of the nonlinear output function h(x1 ) is bounded. When both the upper and lower bounds of the derivative of the output function are unknown constants, to the best of our knowledge, there are no available results reported in the literature. Furthermore, if uncertainties coexist in the unmeasured state-dependent growth and unknown output function of system (1), then global output-feedback adaptive control design becomes much more formidable due to the lack of an effective observer. Additionally, some practical plants described by (1) are indeed existent, such as an electromechanical system shown in Zhai et al. (2013). We make the following assumptions on system (1). Assumption 1. There exists an unknown positive constant θ , such that

|φi (t , x, u)| ≤ θ (|x1 | + · · · + |xi |),

i = 1 , . . . , n.

Assumption 2. The sign of h′ (s) := unchanged. Moreover, there holds l ≤ ⏐h′ (s)⏐ ≤ ¯l,

⏐

⏐

dh(s) ds

(2)

is known and keeps

∀s ∈ R,

(3)

where l and ¯l are unknown positive constants. Note that one can verify that the system with unknown control coefficients (such as those in Liu (2013), Shang and Liu (2015), Shang et al. (2009) and Yan and Liu (2013)) described by

⎧ ⎨η˙ i = gi ηi+1 + ψi (t , η, u), η˙ = gn u + ψn (t , η, u), ⎩ n y = η1 ,

i = 1, . . . , n − 1, (4)

is an alternative of system (1) with linear output function, where control coefficients gi , i = 1, . . . , n are nonzero, unknown constant parameters. In fact, with the help of the following scaling transformation: 1 xi = ∏ n ηi , i = 1, . . . , n, (5) j=i gj system (4) can be transformed into

⎧ ⎨x˙ i = xi+1 + φi (t , x, u), x˙ = u + φn (t , x, u), ⎩ n y = gx1 ,

i = 1, . . . , n − 1, (6)

1 Application note for an infrared, triangulation-based distance sensor with an analog, nonlinear output. Available at http://zuff.info/SharpGP2D12_E.html.

X. Yan, Y. Liu and W.X. Zheng / Automatica 104 (2019) 173–181

∏n

where g = j=1 gj is a nonzero, unknown constant parameter, and φi = ∏n1 g ψi . So system (6), or system (4), is the special j=i j

case of system (1). It is worth noting that there is no essential difference in handling (4) and (6) under the similar assumptions. Moreover, it is obvious that system (1) also contains those with known control coefficients such as in Lei and Lin (2006) and Qian and Du (2012) where gi = 1, i = 1, . . . , n. It is also worth mentioning that, unless h(x1 ) = x1 , system (1) cannot be transformed into the one in Lei and Lin (2006) (i.e., y = x1 ) with the help of the change of variables such as χ1 = h(x1 ), χi = xi , i = 2, . . . , n. This fact is based on the following discussion. By χ1 = h(x1 ), we have

where φ˜ 1 = (h′ (x1 ) − 1)χ2 + h′ (x1 )χ1 . As long as h′ (x1 ) ̸ = 1, i.e., h(x1 ) ̸ = x1 , according to Assumptions 1 and 2, one can obtain

|φ˜1 | ≤ θ˜ (|x1 | + |x2 |) = θ˜ (|υ (χ1 )| + |χ2 |),

are a powerful tool in nonlinear feedback control (Khalil & Praly, 2014), we employ the following high-gain observer for system (1):

⎧ x˙ˆ = xˆ i+1 − Li ai xˆ 1 , i = 1, . . . , n − 1, ⎪ ⎪ ⎨ i x˙ˆ n = u − Ln an xˆ 1 , ⎪ ⎪ xˆ 2 ⎩˙ y2 L = L2 + L21 + ξn2 , L(0) = 1,

(7)

where υ (·) = h−1 (·). It is not difficult to see that (7) no longer matches Assumption 1 (case i = 1), and hence cannot be applied to the existing theorems such as the one in Lei and Lin (2006). Assumptions 1 and 2, especially Assumption 2, show that the investigated system (1) is essentially different from those in the closely related works (Lei & Lin, 2006, 2007a,b; Liu, 2008; Qian & Du, 2012; Shang & Liu, 2015; Shang et al., 2009; Yan & Liu, 2013; Zhai et al., 2013, 2016). It is known that Assumption 1, as a common condition, is proposed in Lei and Lin (2006) in the context of continuous adaptive output-feedback control. Although Assumption 1 in Liu (2013) is weaker than the one in this paper, the time-varying (nonadaptive) technique is adopted therein to overcome the difficulties caused by unknown constant control coefficients. Assumption 2 highlights the main contributions of this paper, and is weaker than the similar assumptions in the related works. In fact, in Liu (2008), Qian and Du (2012), Zhai et al. (2013) and Zhai et al. (2016) a priori known information is necessary not only on the growth rate, but also on the output function or the upper and lower bounds of unknown control coefficients/the derivative of unknown output function. Although the growth rate in Lei and Lin (2006), Liu (2013), Shang and Liu (2015), Shang et al. (2009) and Yan and Liu (2013) is not required to be known, the output functions are either completely known or equivalent to a linear relationship of x1 , and at least one of the upper and lower bounds of unknown control coefficients in Shang and Liu (2015), Shang et al. (2009) and Yan and Liu (2013) are supposed to be known constants. In addition, the studied systems in Shang et al. (2009) and Shang and Liu (2015) are limited to be the two-dimensional case. Recognizing that the adaptive method is a natural and conventional way to deal with uncertainties, this paper mainly focuses on how to overcome, by adaptive control, the difficulties caused by large uncertainties of control coefficients/output function for nonlinear systems with unmeasurable state-dependent growth. The corresponding results will be unfolded in the next sections. 3. Adaptive output-feedback stabilizing control design 3.1. Observer design In output-feedback control design, the choice of suitable observers is usually very important, especially in the presence of large uncertainties in nonlinear systems. Unfortunately, no unified method has been developed for nonlinear observer design (Krstic et al., 1995). Considering that high-gain observers

(8)

where xˆ = [ˆx1 , . . . , xˆ n ]T , ai > 0, i = 1, . . . , n, are design parameters to be determined, and ξn is defined by the following recursive expressions:

α1 = −sign(h′ )Lµ1 ξ1 ,

{ ξ1 = yL , ξi =

χ˙ 1 = χ2 + φ˜ 1 ,

175

xˆ i Li

− αi−1 , αi = −Lµi ξi ,

(9)

i = 2, . . . , n

with to-be-determined positive constants µi ’s. Remark 1. Note that the introduced observer based on the dynamic high-gain originates from Yan and Liu (2013). The existence of unknown output function/control coefficients makes the observer driven by error ‘‘y − xˆ 1 " and the controller in Lei and Lin (2006) inapplicable. Compared with Lei and Lin (2006), the observer (8) introduced in this paper is driven only by ‘‘u". The difference of observers and controllers renders it necessary to design a suitable dynamic high-gain. Since the controller to be designed will be dependent upon the recursive ξi ’s, it is natural to design a gain update law using the information of y, xˆ 1 , . . . , xˆ n , instead of only y and xˆ 1 as in the original ‘‘universal control" idea proposed in Lei and Lin (2006). For further reasons on the ˙ please refer to inequality (32) and the proof of explicit form of L, Proposition 1. Remark 2. Owing to the existence of unknown output function/control coefficients, the control law in Lei and Lin (2006) cannot be employed directly in this paper. Here, αi ’s in (9) are viewed as the virtual control for the error ξi+1 -subsystem. From (9), it can also be seen that distinct from Lei and Lin (2006), Liu (2008, 2013), Qian and Du (2012), Shang and Liu (2015), Shang et al. (2009), Yan and Liu (2013), Zhai et al. (2013) and Zhai et al. (2016), we replace the constant parameters/time-varying gain (appearing in front of ξi ) by a dynamic high-gain with suitable design parameters in the stabilizing functions αi ’s. This idea successfully overcomes the difficulties caused by unknown growth rate and unknown upper and lower bounds of the derivative of the output function. Motivated by Lei and Lin (2006), consider the re-scaling transformation: xi − xˆ i

xˆ i , zi = i , i = 1, . . . , n. (10) Li L Using (1) and (8), and (10), the whole control system can be expressed in the compact form as

εi =

⎧ L˙ ⎪ ⎨ε˙ = LAε + Φ (t , x, u, L) + ax1 − L Dε, ˙ z˙ = LAz + en Lun − LL Dz , ⎪ ⎩L˙ = z 2 + ξ 2 + ξ 2 , L(0) = 1, n 1 1 where ε = [ε1 , . . . , εn ]T , z = [z1 , . . . , zn ]T , Φ (·) =

]T

φ2 (·), . . . , L1n φn (·) en = [0, . . . , 0, 1]T ⎡ −a1 1 .. .. ⎢ . . A=⎢ ⎣ −an−1 0 −an 0

(11)

[

1 L

φ1 (·), L12

, a = [a1 , . . . , an ]T ∈ Rn , D = diag{1, 2, . . . , n}, ∈ Rn , and

··· .. . ··· ···

0

⎤

.. ⎥ . ⎥ ∈ Rn×n . ⎦

1 0

176

X. Yan, Y. Liu and W.X. Zheng / Automatica 104 (2019) 173–181

In view of Lemma 1 in Praly and Jiang (2004), suitable parameters ai ’s can be chosen such that A is a Hurwitz matrix and there exists a positive definite matrix P = P T > 0 satisfying AT P + PA ≤ −I

and DP + PD ≥ 0.

(12)

Construct V0 (ε ) = ε P ε . Then, in terms of (11) and (12), a simple calculation yields T

V˙ 0 ≤ −L∥ε∥2 + 2ε T P Φ + 2ε T Pax1 ,

∥Φ ∥ ≤ ∥Φ ∥1 ≤ nθ

|zi | + n nθ∥ε∥ +

i=2

nθ l

2ε T P Φ ≤ Θ ∥ε∥2 + 2ε Pax1 ≤

1 L 2

T

1 4

2

∥ε∥ +

|ξ1 |,

+ 12 ξ12 , ∥Pa∥ Lξ12 , 2 i=2 zi 2

√

(15)

where Θ = 4n3 θ 2 ∥P ∥2 + 2n nθ∥P ∥ + 2n 2θ ∥P ∥2 . Combining (13) l and (15) together, one deduces that V˙ 0 ≤ −

(1 2

)

L − Θ ∥ε∥2 +

(1 2

2 2

+

2

) 2 ∥ Pa ∥ L ξ12 2

l

n

+

1∑ 4

zi2 .

(16)

i=2

3.2. Controller design In Zhai et al. (2013), it is proved that when θ and l, ¯l in Assumptions 1 and 2 are known constants, system (1) can be globally stabilized by a linear output-feedback controller. To deal with the uncertainties in the dynamic systems, one of the most effective ways is to apply adaptive control. For a special form of system (1), i.e., system (4), when θ is unknown but the upper and lower bounds of unknown control coefficients are known, an adaptive output-feedback controller is presented in Yan and Liu (2013) to achieve the objective of global regulation. In Shang and Liu (2015) and Shang et al. (2009), global adaptive stabilization by output-feedback is studied for planar nonlinear systems with only known lower bounds of control coefficients, and appropriate update laws are constructed to eliminate the effect in the presence of unknown upper bounds of control coefficients. With a careful investigation, it can be found that in the above-mentioned literature, the common forms of Lyapunov function render that the introduced dynamic high-gain is only used for dealing with unknown growth rate. In this section, in order to overcome the difficulties caused by unknown bounds l and ¯l with adaptive highgain control, we reconstruct a novel Lyapunov function, which is sufficient for the existence of a stabilizing feedback control law. Next, based on the high-gain observer (8), we develop a new adaptive controller and determine the range of the introduced design parameters µi ’s involved in the stabilizing functions and controller. The specific form of the smooth adaptive controller is given as follows: u = Ln+1 αn = −Ln+1+µn ξn

(17)

with the parameters µi ’s satisfying that 1−2σ¯ i−1 2(n−i)+1 1−2σ¯ i−1 +(2(n−i)−1)µi 4(n−i)

2σi−1 − µi−1 < µi <

, i = 1, . . . , n,

µi < σi <

, i = 1, . . . , n − 1,

{

Ln−1−µn−1 xˆ 2

+

+

xˆ n−2 Ln−2−(µn−1 +µn−2 )

+ ···

+ Ln−1+µn +µn−1 +···+µ2 xˆ 2 +Ln+µn +µn−1 +···+µ1 sign(h′ )xˆ 1 ).

∑n 2 l2

xˆ n−1

+

u = −(L1+µn xˆ n + L2+µn +µn−1 xˆ n−1 + · · ·

and moreover,

{

Ln

(14)

it follows from (2) and (3) that

√

xˆ n

L2−(µn−1 +µn−2 +···+µ2 ) y + sign(h′ ) 1−(µ +µ +···+µ +µ ) . n−1 n−2 2 1 L Hence,

Noting that similar to (11) in Zhai et al. (2013),

n ∑

ξn =

(13)

where L˙ ≥ 0 and L(t) ≥ 1 have been used.

|y| |y| ≤ |x1 | ≤ , ¯l l

where σi , i = 1, . . . , n − 1, are parameters, which will be used in the Lyapunov ∑ function, and for notational convenience, σ0 = µ0 := 0, σ¯ i := ij=1 σj . From (9), we obtain

(18)

(19)

From (19), it follows that similar to Lei and Lin (2006), the control law can be directly written without doing a recursive design. Compared with Lei and Lin (2006), this form of expression seems quite complicated, and it may be difficult to choose design parameters. Thus, considering that the current related literature involving unknown output function/control coefficients all use the recursive design idea, we also adopt a backstepping-like procedure, which helps us determine the range of design parameters and make the control design easy to understand. On the other hand, we still adopt the dynamic high-gain to deal with the unknown growth rate θ . However, it can be seen from (8) and (17) that some design parameters µi ’s as powers are introduced in the control law and the dynamic high-gain to overcome the difficulties brought by unknown bounds l and ¯l. As for the auxiliary parameters σi ’s, the introduction of these parameters is only for determining the selection rule of the design parameters µi ’s. In fact, if the Lyapunov function at step k is 1 ξ 2 without introducing parameters chosen as Vk = Vk−1 + 2µ¯ 2L k−1 k σi ’s, then it is not possible to deduce a reasonable choice of design parameters µi ’s. Remark 3. A new adaptive control law (17) based on the dynamic high-gain observer (8) and (9) is constructed in the paper with the suitable design parameters which can be determined according to (18). It is worth pointing out that the design parameters µi ’s introduced in the stabilizing functions αi ’s and control law u indeed exist and can be chosen. Actually, from (18), it is observed that they are coupled with parameters σi ’s, and simultaneously, µ1 is only related to n, µi+1 depends on µi , σ1 , . . . , σi , and σi relies on µi , σ1 , . . . , σi−1 for i = 1, . . . , n − 1. In this way, one can determine the parameters sequentially according to the order µ1 , σ1 , µ2 , σ2 , . . . , µn−1 , σn−1 , µn . Then, we proceed to construct a suitable Lyapunov function, which, together with the designed adaptive control law, guarantees the global stability of the closed-loop system. Step 1: The first Lyapunov function is chosen as V1 = V0 + 1 2 1 2 ξ + 2L z1 . It is clear from (11) and (16) that 2 1 V˙ 1 ≤ −

(1 2

)

L − Θ ∥ε∥2 +

(1 2

+

n ) 1∑ 2 2 2 ∥ Pa ∥ L ξ + zi 1 2

2

4

l

i=2

h L˙ ) + ξ1 Lh′ ε2 + Lh′ z2 + φ1 − ξ1 ′

(

L

3L˙

L

− + z1 z2 . 2L Using the completion of squares, we get −

z2 2 1

a1 z12

⎧ L 2 ′ ¯2 2 ⎪ ⎨ξ1′ Lh ε2 ≤ 4 ∥ε∥ + l Lξ1 , ¯ h lθ 2 ξ φ ≤ l ξ1 , L 1 1 ⎪ ⎩z z ≤ a 1 z 2 + 1 z 2 , 1 2 2 1 2a1 2

(20)

X. Yan, Y. Liu and W.X. Zheng / Automatica 104 (2019) 173–181

By (18), one gets 2σ1 − µ1 < µ2 < substituting (24) into (23) results in

which, together with (20), leads to V˙ 1 ≤ −

(1 4

)

L − Θ ∥ε∥2 +

¯lθ )

+

l 4

2

l

L˙

a1

L

2

ξ12 − ξ12 −

n 1∑

+

(( 2

) 1 ∥Pa∥2 + ¯l2 L +

2

z12 +

(

1

+

2a1

1

)

4

V˙ 2 ≤ −

z22

zi2 + Lh′ ξ1 z2 .

(1 8

)

L − Θ ∥ε∥2 −

(21)

−

z22 ≤ 2L2µ1 ξ12 + 2ξ22 ,

a1 2

n 1∑

z12 +

4

{

Lh′ ξ1 z2 ≤ Lh′ ξ1 ξ2 − lLµ1 +1 ξ12 .

Observing this and 0 < µ1 <

≤ 1, (21) becomes

ν2 =

) V˙ 1 ≤ − L − Θ ∥ε∥2 − lLmax{2µ1 ,1} (Lν1 − θ1 ) ξ12 4 ( ) n L˙ 2 1 a1 2 1∑ 2 1 − ξ1 − z1 + zi + + ξ22 L

2

4

2

i=3

( 22 l

{

∥Pa∥ + ¯l2 + 2

l

+1+

1 − µ1 , if

µ1 ,

1 2

≤ µ1 <

if 0 < µ1 <

/l, and

1 , 2n−1

− 1 2L2σ1

ξ22 . From (9), (11), and (22), ·

ξ2 ξ˙2 − σ1 2σ +1 ξ22 L 1 ) (1 2 L − Θ ∥ε∥ − lLmax{2µ1 ,1} (Lν1 − θ1 ) ξ12 ≤− 4 ) ( n L˙ a1 1∑ 2 1 1 ξ22 + Lh′ ξ1 ξ2 − ξ12 − z12 + zi + + L2σ1

2

4

1

2

i=3

L

L

l 2k−1

k ∑ 1

1

(

(23)

Since L˙ ≥ 0, L ≥ 1, µ1 < σ1 , µ1 > 0, by (2), (3), (9), (10),

Lmax{2µi ,1−2σ¯i−2 −µi−1 } (Lνi − θk ) ξi2

k ( ∑ 1

L2(σi−1 −µi−1 ) − 2k−i

L1+2σ¯i−1 +2(σi−1 −µi−1 ) n 1 ∑

4

zi2 +

i=k+2

1 2

ξi2 −

i−1 ∑

2i−2−j (1 + µj )2

∑k

i=2

νi =

ai )2 (2k +

1 ) 4l2

)

j=1

a1 2

z12

ξk2+1 + L1−2σ¯k−1 ξk ξk+1

(25)

k−3 ¯l

+ 9 · 2k−3¯l2 + 2

⎧ 1 − µi − 2σ¯ i−1 , ⎪ ⎪ ⎪ 1−µi−1 −2σ¯ i−2 ⎨

} 2 l

k−1

l

¯2 θ 2

(l

l2

, and for i = 1, . . . , k,

1−2σ¯

≤ µi < 2(n−i)i−+11 , ⎪ µi + µi−1 − 2σi−1 , ⎪ ⎪ ⎩ 1−µi−1 −2σ¯ i−1 if 2σi−1 − µi−1 < µi < . 2 if

+ 1), 2k +

2

(26)

In fact, suppose that Vk−1 satisfies a similar relationship to (25). Recalling that ξk = zk − αk−1 = zk + Lµk−1 ξk−1 , we can rewrite z˙k as

(14) and using the completion of squares, one has

⎧ ¯2 Lh′ ξ1 ξ2 ≤ 14 lL1+µ1 ξ12 + ll L1−µ1 ξ22 , ⎪ ⎪ ⎪ ⎪ ⎪ − L21σ1 a2 xˆ 1 ξ2 ⎪ ⎪ ( ) ⎪ ⎪ L ⎪ ⎪ ≤ Lξ12 + 4l12 + 4 a22 L1−µ1 ξ22 + 16 ∥ε∥2 , ⎪ ⎪ ⎪ ⎪ 1 ⎪ |h′ |L1+µ1 −2σ1 ε2 ξ2 ≤ 16 L∥ε∥2 + 4¯l2 L1−µ1 ξ22 , ⎪ ⎪ ⎪ ′ 1 +µ − 2 σ ⎪ 1 1z ξ ⎪ ⎨|h |L ( )2 2 ¯l2 ¯ ≤ l + l L1−µ1 ξ22 + 14 lL1+µ1 ξ12 , ⎪ ⎪ ⎪ |h′ | φ ξ ≤ ¯lθ |ξ | · |ξ | ≤ ¯l2 θ 2 ξ 2 + ξ 2 , ⎪ 1 2 1 2 ⎪ 1 2 l ⎪ l2 L1−µ1 +2σ1 ⎪ ⎪ L˙ ′ ⎪ (1 + µ )sign(h ) ξ ξ ⎪ 1 1 2 1 −µ + 2 σ ⎪ 1 1 L ⎪ 2 ⎪ ˙ L˙ ⎪ 2 1) ⎪ ≤ 2LL ξ12 + (1+µ 1−2µ1 +4σ1 ξ2 , ⎪ 2 L ⎪ ⎪ 1 2 1 2µ2 2 1 2 ⎪ ⎪ ⎪ 4 z3 ≤ 2 L ξ2 + 2 ξ3 , ⎩ L1−2σ1 ξ2 z3 = L1−2σ1 ξ2 ξ3 − Lµ2 +1−2σ1 ξ22 .

2k−i

Lmax{2µ1 ,1} (Lν1 − θk ) ξ12

{

L2σ1

1

1−µ1 2

for all k = 3, . . . , n, where θk = max 2θk−1 + 2

a1

ξ2 Lz3 − a2 xˆ 1 − 2 ξ2 + (1 + µ1 )sign(h′ )ξ1 L ) ˙L 1 · 1−µ + 1−µ |h′ |(L2 ε2 + L2 z2 + φ1 ) .

+

a2

¯2

2

L˙

+

L˙

(

if

) − Θ ∥ε∥2 − k+1

i=1

L˙

1

2

( L

i=2

1 . 2

Step 2: Construct V2 = V1 + it follows that

ξ32 + L1−2σ1 ξ2 ξ3 ,

Step k = 3, . . . , n: We proceed to prove the conclusion by 1 ξ 2 satisfies 2L2σ¯k−1 k

V˙ k ≤ − 1 ) a1

1

≤ µ2 < 2n1−3 (1 − 2σ1 ), 1 µ2 + µ1 − 2σ1 , if 2σ1 − µ1 < µ2 < 1−µ . 2

−

{

2

i=4

1 − 2σ1 − µ2 ,

(22) ¯lθ

1

zi2 +

1

induction that Vk = Vk−1 +

a1

+ Lh′ ξ1 ξ2 ,

L

lLmax{2µ1 ,1} (Lν1 − θ2 ) ξ12

L

¯2

1 2n−1

− 2σ1 ). Then,

where θ2 = max{2θ1 + 2l ( l θ2 + 1), 2 + a1 + 2ll + 4¯l2 +¯l + 4a22 + 22 }, l 4l 1 and

(1

V˙ 2 = V˙ 1 +

2

2

Moreover, by (9),

ν1 =

1

1 (1 2n−3

L˙ − Lmax{2µ2 ,1−µ1 } (Lν2 − θ2 ) ξ22 − ξ12 2L ( ) (1 + µ1 )2 L˙ 2(σ1 −µ1 ) − L − ξ22 1−2µ +4σ

i=3

where θ1 =

177

L˙ L˙ ξ˙k = Lzk+1 − ak xˆ 1 − k ξk + (1 + µk−1 ) 1−µ ξk−1 k−1 L L ( L˙ + Lµk−1 Lzk − ak−1 xˆ 1 + (1 + µk−2 )ξk−2 1−µ L

(24)

k−2

L˙ + Lµk−2 Lzk−1 − ak−2 xˆ 1 + (1 + µk−3 )ξk−3 1−µ k−3 L ( L˙ + · · · + Lµ2 Lz3 − a2 xˆ 1 + (1 + µ1 )sign(h′ )ξ1 1−µ 1 L ) ))

(

+

1

L1−µ1

|h′ |(L2 ε2 + L2 z2 + φ1 ) · · ·

.

(27)

178

X. Yan, Y. Liu and W.X. Zheng / Automatica 104 (2019) 173–181

4. Main results

With the help of (25) and (27), we have 1

V˙ k = V˙ k−1 +

≤− −

(L

−

L2σ¯k−1

k−1 ∑

1 2k−1−i

k−1 ( ∑ i=1

+ ξ +L 2

i−1 ∑

L2(σi−1 −µi−1 ) − 2k−1−i

2 k

With the help of (30), we can achieve the control objective of the paper. Before stating the main results, we first present a fundamental proposition to reveal that the boundedness of dynamic high-gain implies that of other variables. So this proposition is vital to the subsequent performance analysis. For the proof of Proposition 1, please simply refer to the proof of Theorem 1 in Lei and Lin (2006).

ξk2

Lmax{2µ1 ,1} (Lν1 − θk−1 ) ξ12

1

L1+2σ¯i−1 +2(σi−1 −µi−1 1

l 2k−2

L˙ L1+2σ¯k−1

Lmax{2µi ,1−2σ¯i−2 −µi−1 } (Lνi − θk−1 ) ξi2

L˙

·

σi

i=1

) − Θ ∥ε∥2 −

2k

i=2

k−1 ∑

ξk ξ˙k −

1−2σ¯ k−2

ξ2 − ) i

ξk−1 ξk +

2i−2−j (1 + µj )2

Proposition 1. For the resulting closed-loop system, if L is bounded on Ω , then z and ε are bounded on Ω as well, and there hold

)

j=1

a1 2

z12 + 1

L2σ¯k−1

n 1 ∑

4

∫ zi2

Ω

ξk ξ˙k .

(28)

following inequalities hold:

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

l L1+µ1 12 2k−1

ξ

(29)

+(θk − 21 )Lmax{2µk , 1−µk−1 −2σ¯k−2 } ξk2

−L1+µk −2σ¯k−1 ξk2 + 21 ξk2+1 + L1−2σ¯k−1 σi ξk ξk+1 +

k−1 ∑ i=1

·

L˙ 2k−i L1+2σ¯i−1

ξi2 +

k−1 ∑

2k−2−i (1 + µi )2

i=1

L˙ L1+2σ¯k−1 +2(σk−1 −µk−1 )

ξk2 .

V˙ n ≤ −

)

2n+1

− Θ ∥ε∥2 −

l 2n−1

Lmax{2µ1 ,1} (Lν1− θn ) ξ12

Theorem 1. Suppose that Assumptions 1 and 2 hold. Under the appropriate choice of design parameters ai ’s and µi ’s, system (1) is globally asymptotically stabilized by the adaptive output-feedback control law (8), (9), and (17). Proof. From (1), (8) and (17), it is not difficult to deduce that the resulting closed-loop system has a unique solution on a small interval [0, ts ) (see Theorem 3.1, page 18 of Hale (1980)). Let Ω := [0, tf ) be the maximal interval on which a unique solution exists, where 0 < tf ≤ +∞ (see Theorem 2.1, page 17 of Hale (1980)). The proof is similar to that in Lei and Lin (2006). For presentation simplicity, here we only give the proof of the boundedness of critical L(t) on [0, tf ). Suppose the contrary, that is, limt →tf L(t) = ˙ it can be seen that L(t) is a contin+∞. With the definition of L, uous and monotonically nondecreasing function, and thus there exists a finite time T ∈ (0, tf ) such that 1

L(t) ≥ max 2n+1 (Θ + 1), (θn + 1) νi ,

(∑ s

2n−2−j (1 + µj )2

−

·

2

Lmax{2µi ,1−2σ¯i−2 −µi−1 } (Lνi − θn ) ξi2 n−i

n ( ∑ 1

i−1 ∑

i=1

j=1

L2(σi−1 −µi−1 ) − 2n−i L˙

L1+2σ¯i−1 +2(σi−1 −µi−1 )

ξi2 −

a1 2

2i−2−j (1 + µj )2

z12 .

) 2(σ 1−µ ) s

s

,

j=1

}

i = 1, . . . , n, s = 1, . . . , n − 1 , ∀t ∈ [T , tf ).

(31)

Using (31), the inequality (30) on [T , tf ) is rewritten as

∑ 1 i=2

∥ε (t)∥2 dt < +∞,

Now, we are in a position to state the main results of this paper.

n

−

Ω

{

With the choice of µk and νk as in (18) and (26), substituting (29) into (28), it is not difficult to verify inequality (25). Therefore, at step n, we can obtain that the smooth adaptive control law (17) renders the Lyapunov function Vn = Vn−1 + 1 ξ 2 to satisfy 2σ¯ n−1 n 2L

( L

∫

where Ω is the maximal interval of solution existence.

i=k+1

Noticing that L˙ ≥ 0, L ≥ 1, σi > µi , i = 1, . . . , k − 1, µk−1 > µk−2 > · · · > µ1 , in terms of (2), (9) and (10), similar to (24), the

⎧1 z 2 ≤ 12 L2µk ξk2 + 12 ξk2+1 , ⎪ ⎪ 4 k+1 ⎪ ⎪ ⎪ ⎪ L1−2σ¯k−2 ξk−1 ξk + 2σ¯1k−1 ξk ξ˙k ⎪ ⎪ L ⎪ ⎪ ⎪ ⎪ ≤ 2k1+1 L∥ε∥2 + (θ 2 + 1)Lξ12 + ⎪ ⎪ ⎪ ⎪ k−1 ⎪ ∑ ⎪ 1 1+µi −2σ¯i−1 2 ⎪ ⎪ + L ξi ⎪ ⎪ k ⎪ 2 −i ⎨ i=2

∥z(t)∥2 dt < +∞,

V˙ n ≤ −∥ε∥2 − min

{

l 2n−1

,

1 2n−2

,

a1

} (∑ n

2

) ξ + 2 i

z12

.

i=1

Consequently,

)

(30)

∫ tf ˙ + ∞ = L(tf ) − L(T ) = L(t)dt T ) ∫ tf ( = ξ12 (t) + ξn2 (t) + z12 (t) dt T

Remark 4. Note that different from Lei and Lin (2006), Liu (2008, 2013), Qian and Du (2012), Shang and Liu (2015), Shang et al. (2009), Yan and Liu (2013), Zhai et al. (2013) and Zhai et al. (2016), the candidate Lyapunov functions at step k = 2, . . . , n involve the dynamic high-gain with appropriate parameters, which play an important role in deducing inequality (30). This is the key ingredient in the proof of the main results below.

1

≤ min

{

l 2n−1

,

1 2n−2

,

a1 2

} Vn (T ) < + ∞,

(32)

which is a contradiction and means that L(t) is bounded on [0, tf ). By the boundedness of L(t) on [0, tf ) and Proposition 1, the remaining part of the proof can be completed in a similar way to that in Lei and Lin (2006).

X. Yan, Y. Liu and W.X. Zheng / Automatica 104 (2019) 173–181

179

Theorem 1 substantially relaxes the assumption on unknown output function/control coefficients in Liu (2008), Shang and Liu (2015), Shang et al. (2009), Yan and Liu (2013), Zhai et al. (2013) and Zhai et al. (2016). The main purpose of this paper is to explore the problem of global stabilization for a class of uncertain nonlinear systems with weaker assumption on the output function/control coefficients by the adaptive method. It is shown that the generalized Theorem 1 implies the theorems of Liu (2008), Shang et al. (2009), Yan and Liu (2013), Zhai et al. (2013) and Zhai et al. (2016). Remark 5. From the above proof, it can be seen that the boundedness of L is a prerequisite for the global boundedness and asymptotic stability of the closed-loop system, and inequality (30) is sufficient to guarantee the boundedness of L. This also confirms the argument stated in Remark 4. In addition, it is shown in (32) that although the original universal control idea L˙ = ε12 in Lei and Lin (2006) is not directly applicable due to the immeasurability of x1 , the appropriately modified form given in the paper can still be used to establish the boundedness of L. 5. An illustrative example In this section, a practical example is given to illustrate the effectiveness of the proposed theoretical results. Example 1. Consider the practical electromechanical system shown in Zhai et al. (2013). The system dynamics is described by M q¨ + Bq˙ + N sin(q) = I ,

{

(33)

LH I˙ = Vε − RI − KB q˙ , 15J +5mL20 +15M0 L20 +6M0 R20

B0 Kτ

mL0 G+2M0 L0 G , 2Kτ

,B = ,N = J is where M = 15Kτ the rotor inertia, m is the link mass, M0 is the load mass, L0 is the link length, R0 is the radius of the load, G is the gravity coefficient, B0 is the coefficient of viscous friction at the joint, q(t) is the angular motor position, I(t) is the motor armature current, and Kτ is the coefficient which characterizes the electromechanical conversion of armature current to torque. LH , R, KB and Vε are the armature inductance, the armature resistance, the back-EMF coefficient and the input control voltage, respectively. Suppose that only q is measurable, and that the system parameters M , LH are unknown constants and may change with temperature and other factors within an unknown finite range. With the same coordinate transformations as in Zhai et al. (2013), i.e., z1 = q, z2 = q˙ , z3 = I and x1 = MLH z1 , x2 = MLH z2 , x3 = LH z3 , system (33) can be rewritten as

K x˙ 3 = u − L BM x2 − ⎪ ⎪ H ⎪ ⎪ ⎪ ⎩y = 1 x . LH M 1

R LH

x3 ,

⎧˙ xˆ 1 ⎪ ⎪ ⎪ ⎨x˙ˆ 2 ˙ˆ 3 x ⎪ ⎪ ⎪ ⎩˙ L

= xˆ 2 − La1 xˆ 1 , = xˆ 3 − L2 a2 xˆ 1 , = u − L3 a3 xˆ 1 , =

(34)

, (35)

y2 L2

+

xˆ 21 L2

(36)

+ ξ32 ,

L(0) = 1,

and

⎧ ⎪ u = − L4+µ3 ξ3 , ⎪ ⎨ ξ3 = xLˆ 33 + Lµ2 ξ2 , ⎪ ⎪ ⎩ξ = xˆ 2 + Lµ1 y , 2 L L2

(37)

where the design parameters can be chosen as a1 = 6, a2 = 1 , µ2 = 19 , µ3 = 31 . It can be verified that 10, a3 = 1, µ1 = 90 such design parameters are suitable. In fact, by solving matrix inequalities (12), we get the positive definite matrix P =

and further becomes B x M 2

According to the statement below Assumption 2, system (34) with unknown control coefficients is an alternative to system (35) with unknown linear output function. For nonlinear systems with unknown control coefficients, such as (34), there are no available results on global adaptive stabilization, without a priori known information on the upper and lower bounds of unknown control coefficients. It is easy to check that (35) meets system (1) with Assumptions 1 and 2, and hence the proposed control scheme can be applied to solve the problem of global stabilization for the practical electromechanical system (33). From (35), it can be seen that h(x1 ) = L 1M x1 . Thus, h′ = L 1M , sign(h′ ) = 1. H H For system (35), according to the design scheme proposed in Section 3, we can construct the observer and controller with the dynamic update law as follows:

[

⎧ z˙1 = z2 , ⎪ ⎪ ⎪ ⎪ ⎨z˙ = 1 z − N sin(z ) − B z , 2 1 M 3 M M 2 ⎪ K 1 R ⎪ z˙3 = L u − L B z2 − L z3 , ⎪ ⎪ H H H ⎩ y = z1 , ⎧ ⎪ x˙ 1 = x2 , ⎪ ⎪ x ⎪ ⎪ ⎨x˙ 2 = x3 − LH N sin( LH1M ) −

Fig. 1. The trajectories of x1 and xˆ 1 in Example 1.

0.1441 0.3644 0.5000

0.3644 3.1271 3.1441

0.5000 3.1441 5.3644

1+3µ

] , 1−2σ +µ

1 31 1 2 and 90 = µ1 < σ1 < 8 1 = 240 , 19 = µ2 < σ2 < = 4 47 1 1 1 14 2 , 0 < µ < , = 2 σ − µ < µ < (1 − 2 σ ) = , = 1 1 1 2 1 180 5 18 3 45 9 1 2σ2 − µ2 < µ3 < 1 − 2σ1 − 2σ2 = 35 with σ1 = 30 , σ2 = 16 . The simulation results displayed in Figs. 1–5 are obtained under the electromechanical system parameters J = 1.625 · 10−3 kg · m2 , m = 0.506 kg, R0 = 0.023 m, M0 = 0.434 kg, L0 = 0.305 m, G = 10 m/s2 , B0 = 16.25 · 10−3 Nms/rad, LH = 25 · 10−3 H , R = 5Ω , Kτ = KB = 0.9 Nm/A, with the initial conditions x1 (0) = 0, x2 (0) = 0, x3 (0) = −2, xˆ 1 (0) = 0.5, xˆ 2 (0) = −1, xˆ 3 (0) = 1. These figures show that the proposed control

180

X. Yan, Y. Liu and W.X. Zheng / Automatica 104 (2019) 173–181

Fig. 5. The trajectory of high-gain L in Example 1.

Fig. 2. The trajectories of x2 and xˆ 2 in Example 1.

Fig. 3. The trajectories of x3 and xˆ 3 in Example 1.

have the unmeasured state-dependent growth with unknown constant rate. Within this context, the problem of nonlinear systems with unknown control direction has not been solved so far. A delicate and yet important case studied here is that the derivative of the unknown output function has a known sign but unknown upper and lower bounds. To this end, we have constructed a new adaptive output-feedback controller by introducing appropriate design parameters. It is worth emphasizing that the proposed control scheme can also be applied to the systems with unknown control coefficients such as Shang et al. (2009) and Yan and Liu (2013), since these classes of systems can be transformed into the studied system with linear output function with the help of coordinate transformation. Besides, the results of this paper can be extended to more general nonlinear systems with the output-polynomial-dependent growth in Lei and Lin (2007a) by introducing two suitable gain update laws, but it may be hard to generalize it to systems under the output-dependent growth condition in Lei and Lin (2007b) due to the inapplicability of the full-order high-gain observer. Future work will be directed at extending the methodology to general nonlinear systems with output-dependent growth condition. Acknowledgments The authors wish to thank the Associate Editor and the four anonymous reviewers for their valuable comments and suggestions which have helped to greatly improve the paper. This work was supported in part by the National Natural Science Foundation of China (Nos. 61304013, 61873146), and the Australian Research Council (No. DP120104986). References

Fig. 4. The trajectory of control law u in Example 1.

design scheme guarantees that all signals of the closed-loop system (35)–(37) are bounded and the states x and xˆ converge to zero. 6. Concluding remarks In this paper, we have studied the global asymptotic stabilization problem for a class of uncertain nonlinear systems which

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181

Xuehua Yan received the Ph.D degree in Control Theory and Control Engineering from Shandong University, Jinan, P. R. China in 2012. She is currently an Associate Professor in the School of Electrical Engineering, University of Jinan, Shandong, P. R. China. She was a Visiting Research Fellow in the School of Computing, Engineering and Mathematics, Western Sydney University, Sydney, Australia from July 2016 to January 2017. Her research interests include nonlinear systems, adaptive control, and practical tracking. Yungang Liu received the Ph.D degree in Control Theory and Control Engineering from Shanghai Jiao Tong University, Shanghai, China in 2000. He is now a Changjiang Scholar Chair Professor with the School of Control Science and Engineering, Shandong University, Jinan, China. His current research interests include stochastic control, nonlinear control design and system analysis, adaptive control and applications in power systems, and big data. He was a recipient of the Guan Zhaozhi Award in the Chinese Control Conference in 2004, the National Outstanding Youth Science Foundation of China in 2013, and the Second Prize of the National Natural Science Award of China in 2015. He currently serves as Associate Editor for seven scientific journals. Wei Xing Zheng received the BSc degree in Applied Mathematics in 1982, the MSc degree in Electrical Engineering in 1984, and the Ph.D degree in Electrical Engineering in 1989, all from Southeast University, Nanjing, China. He is currently a Distinguished Professor at Western Sydney University, Sydney, Australia. Over the years he has also held various faculty/research/visiting positions at Southeast University, China; Imperial College of Science, Technology and Medicine, UK; University of Western Australia; Curtin University of Technology, Australia; Munich University of Technology, Germany; University of Virginia, USA; and University of California-Davis, USA. His research interests are in the areas of systems and controls, signal processing, and communications. He is a Fellow of IEEE. Previously, he served as an Associate Editor for IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, IEEE Transactions on Automatic Control, IEEE Signal Processing Letters, IEEE Transactions on Circuits and Systems-II: Express Briefs, and IEEE Transactions on Fuzzy Systems, and as a Guest Editor for IEEE Transactions on Circuits and Systems-I: Regular Papers. Currently, he is an Associate Editor for Automatica, IEEE Transactions on Automatic Control (the second term), IEEE Transactions on Cybernetics, IEEE Transactions on Neural Networks and Learning Systems, IEEE Transactions on Control of Network Systems, and other scholarly journals. He is also an Associate Editor of IEEE Control Systems Society’s Conference Editorial Board. He has served as Publication Co-Chair of the 3rd IEEE Conference on Control Technology and Applications in Hong Kong, China, August 2019. He is currently the Chair of IEEE Control Systems Society’s Standing Committee on Chapter Activities.