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Disturbance-observer-based nonlinear control for overhead cranes subject to uncertain disturbances
Mechanical Systems and Signal Processing 139 (2020) 106631
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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Disturbance-observer-based nonlinear control for overhead cranes subject to uncertain disturbances Xianqing Wu a,⇑, Kexin Xu a, Xiongxiong He b a b
Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China
a r t i c l e
i n f o
Article history: Received 3 November 2019 Received in revised form 31 December 2019 Accepted 8 January 2020
Keywords: Underactuated system Overhead crane Disturbance observer Feedforward compensation control
a b s t r a c t The regulation and disturbance rejection problems of the underactuated overhead crane system in the presence of uncertain disturbances are considered in this paper. A novel nonlinear control method, along with a finite time disturbance observer, is proposed for the crane system. Different from existing control methods, one important feature of the proposed method is that uncertain disturbances are attenuated and eliminated by the disturbance observer in finite time. More precisely, some transformations are performed with respect to the original dynamic model of the crane system. Then, a finite time disturbance observer is introduced, based on which a nonlinear control method is proposed. Rigorous mathematical analysis is implemented to prove the theoretical derivations. Finally, simulation and experimental tests are carried out to demonstrate the superior performance of the proposed method. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction Underactuated overhead cranes are widely used underactuated mechanical equipments [1–9], which are commonly found and utilized for material transportation in many industrial sites, such as harbors, factories, workshops, and so forth. The control objectives of these systems are to drive the trolley from the initial position to the target position rapidly and precisely as well as to attenuate and eliminate the payload swing at the same time. Though overhead cranes possess many advantages including high transportation efficiency, low energy consumption, simple mechanical structure, etc., the underactuated characteristic of overhead crane systems that only one control input and two degrees of freedom to be controlled makes the control problem of these systems much more challenging. Nevertheless, owing to the widely applications of overhead crane systems, the regulation and disturbance rejection problems of crane systems have attracted considerable attention from the control community during the past few decades. During the past decades, numerous interesting works have been made by a lot of researchers for the control of underactuated crane systems. The existing works about crane control can roughly be divided into two categories: trajectory planning (open-loop control) and feedback control design. For the former, the input shaping technique is one of the most extensively utilized motion planning approaches [10–15]. In particular, in [16], based on the particle swarm optimization technique, an improved input shaping method is designed for three dimensional overhead crane systems, which achieves high payload swing reduction during the control process. Ref. [17] takes several constraints consisting of the permitted swing amplitude
⇑ Corresponding author. E-mail address: [email protected] (X. Wu). https://doi.org/10.1016/j.ymssp.2020.106631 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.
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X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
for the payload, available velocity, acceleration, and jerk for the trolley into consideration, a minimum-time trajectory planning method is proposed on the basis of the quasi-convex optimization approach. To achieve the objectives of safety and energy efficiency, an energy-optimal motion planning method is devised in [18]. In addition, motion planning methods for double pendulum crane systems also have been reported in the literature [19–21]. Compared with the trajectory planning methods, feedback control shows superior performance with respect to uncertain parameters and external disturbances. Plenty of closed-loop control methods have been exploited for crane systems. Specifically, based on the passivity of the crane system, a series of control schemes are given in [22–25]. Assuming that velocity signals are unavailable, some output feedback control methods are designed in [26–28]. Taking the payload hoisting/lowering effect into consideration, in [29], a trajectory tacking control method is designed for the overhead crane systems with varying rope length. Ref. [30] proposes a hybrid control scheme for the overhead cranes having payload hoisting/lowering effect, and it assumes that the viscous damping exists in the system. Moreover, adaptive and sliding model control techniques are employed to deal with system uncertainties or external disturbances [31–43]. In particular, in [43], a trajectory tracking control method is proposed for overhead crane systems, which achieves a finite-time tracking result. Furthermore, many other high-performance control methods are also employed, which mainly include optimal control [44], model predictive control [45,46], intelligent control [47,48], and so forth [49–58]. At present, the control problem of the crane system is still a fairly open topic. On one hand, most existing control methods, especially Lyapunov-based control methods, are designed on the basis of the exact model knowledge of the system, which makes them sensitive to uncertain external disturbances. However, in practice, mechanical systems always suffer from unavoidable factors, such as unmodeled dynamics, external disturbances, and so on. On the other hand, some robust methods, like sliding mode control technique, have been proposed to handle the undesired effects induced by uncertain disturbances. However, the main idea of the existing control methods for the disturbance rejection control of the crane system is to eliminate uncertain disturbances by feedback control, which is a robust way. Furthermore, the robustness of the existing methods is obtained at the price of sacrificing its nominal control performance [59]. In order to overcome the limitations of the existing methods for the crane control, inspired by the development of feedforward compensation control [60–62], the disturbance observer-based control technique is employed to estimate and compensate the effects of uncertain disturbances, and a nonlinear control method is proposed for the regulation of the crane system in this paper. In particular, we use a coordinate transformation to convert the origin dynamic model into a new form, based on which a finite time disturbance observer is presented. Then, an elaborate manifold is introduced and a nonlinear control method is designed to guarantee that the constructed manifold is attractive and invariant. Rigorous theoretical analysis is given to prove the stability of the closed-loop system and the convergence of the state variables. Finally, we demonstrate the superior performance of the proposed method in comparison with exiting nonlinear control methods through simulation and experimental tests. In sum, the suggested control method here exhibits the following remarkable features: 1) To the best of our knowledge, this is the first finite time disturbance observer-based nonlinear control method for the regulation and disturbance rejection of the crane system. 2) Because of the exact compensation of the designed disturbance observer, uncertain disturbances could be completely removed from the system output in finite time. 3) The proposed method belongs to a feedforward-compensation-based method and is continuous without any chattering, which is convenient for practical applications. The remaining parts of this paper are organized as follows. The dynamic model of the crane system is given and converted to a new form through a coordinate transformation in Section 2. Section 3 proposes a disturbance observer and a nonlinear controller with rigorous theoretical stability analysis. In Sections 4 and 5, simulation and hard experimental results are provided to demonstrate the superior control performance and strong robustness of the proposed method. Section 6 summarizes the main work of this paper and discusses our future work.
2. Problem formulation Consider the two-dimensional (2-D) overhead crane systems (as shown in Fig. 1), the dynamic model of which can be described as follows [17,18]:
ðM þ mÞ€x þ ml€h cos h mlh_ 2 sin h ¼ F þ d
ð1Þ
2 ml €h þ ml€x cos h þ mgl sin h ¼ 0
ð2Þ
where xðtÞ and hðtÞ denote the trolley position and the payload angle with respect to the vertical direction, respectively, M and m denote the trolley mass and the payload mass, respectively, l is the cable length, g represents the gravity acceleration, FðtÞ is the control input, and dðtÞ is the lumped term including frictions, uncertain disturbances, unmodeled dynamics, and so on.
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X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
Fig. 1. Structure of the crane system.
For the dynamic model 1,2 of the overhead crane system, after some mathematical arrangements, one can rearrange it as follows:
F ¼ /u u mlh_ 2 sin h ðM þ mÞg tan h €x ¼ g tan h lh€ sec h
ð3Þ ð4Þ
where uðtÞ; /u ðtÞ are auxiliary functions whose expressions are
uðtÞ ¼ €h /d /d ðtÞ ¼
ð5Þ cos h 2
ðM þ m sin hÞl
ð6Þ
d
2
/u ðtÞ ¼ ðM þ m sin hÞl sec h
ð7Þ
For practical mechanical systems, the time derivative of the /d ðtÞ is bounded, that is
j/_ d ðtÞj 6 .d
ð8Þ
þ
where .d 2 R is a prior known constant. In order to facilitate the following control development and stability analysis, the following auxiliary variables are introduced:
v1 ¼ x þ l lnðsec h þ tan hÞ; v2 ¼ v_ 1 ; v3 ¼ g tan h; v4 ¼ v_ 3
ð9Þ
Then, based on (4), one can obtain the following dynamic equations:
8_ v1 ¼ v2 > > > < v_ 2 ¼ v3 1 /ðv3 ; v4 Þ > > v_ 3 ¼ v4 > : v_ 4 ¼ g sec2 hðu þ /d þ 2h_ 2 tan hÞ
ð10Þ
where /ðv3 ; v4 Þ is an auxiliary function with the following expression:
/ðv3 ; v4 Þ ¼
lv24 ðg 2
þ v23 Þ
ð11Þ
3=2
and the auxiliary variable uðtÞ is considered as the control input of the new dynamic model (10). For practical crane systems, the payload swing amplitude usually satisfies jhðtÞj 6 p9 rad and the swing angular velocity usually satisfies that _ jhðtÞj 1 rad=s [10–21], which can be found from the subsequent simulation and experimental tests, thus, v2 ðtÞ 1. In 4
addition, for practical applications, the cable length l is usually shorter than 10 m. Hence, from the practical perspective, we have that
lv24 2 3=2 3Þ
ðg 2 þ v Hence,
6
lv24 0:001lv24 1 g3
ð12Þ
v3 v3 1 /ðv3 ; v4 Þ is valid and (10) can be rewritten as
8_ v1 ¼ v2 > > > < v_ 2 ¼ v3 > > v_ 3 ¼ v4 > : v_ 4 ¼ g sec2 hðu þ /d þ 2h_ 2 tan hÞ
ð13Þ
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For the overhead crane system, the control objective is to drive the trolley to the desired position and eliminate the payload swing at the same time, that is
limxðtÞ ¼ pd ; limhðtÞ ¼ 0
t!1
ð14Þ
t!1
where pd is the desired location of the trolley. From the definitions of the auxiliary variables in (9), one can find that the control objective (14) is equivalent to
lim ðtÞ t!1 1
v
¼ pd ;
lim ðtÞ t!1 3
v
¼0
ð15Þ
Based on this characteristic, the following variables are defined:
n1 ¼ v1 pd ; n2 ¼ v2 ; n3 ¼ v3 ; n4 ¼ v4
ð16Þ
According to the introduced variables (16), the dynamic Eq. (13) can be rewritten as follows:
8_ n1 ¼ n2 > > > < n_ 2 ¼ n3 > > n_ 3 ¼ n4 > : n_ 4 ¼ g sec2 hðu þ /d þ 2h_ 2 tan hÞ
ð17Þ
which will be used for the following control development and stability analysis. Remark 1. Many existing disturbance-observer-based control methods usually assume that the disturbance is a constant or _ ¼ 0. Compared with these existing methods, we have relaxed the the time derivative of the disturbance is limt!1 dðtÞ assumptions on the disturbances from a mathematical viewpoint. Nonetheless, the assumption that the disturbance is differentiable and bounded is still a little strict. In the future, different cases of uncertain disturbances, such as nondifferentiable uncertain disturbances, unbounded disturbances, and so forth, will be taken into consideration when making control design.
3. Main results In this section, a finite time disturbance observer is designed and a nonlinear control method is proposed straightforwardly. Next, the stability of the closed-loop system and the convergence of the state variables are analyzed. The following lemmas will be utilized in the following stability analysis part. Lemma 1 [63]. For the following scalar system:
y_ ¼ c1d sgnðyÞjyja1d c2d sgnðyÞjyja2d
ð18Þ
þ
where c1d ; c2d 2 R are positive constants, a1d P 1 and 0 < a2d < 1, the origin is the globally finite time stable equilibrium point for the system described by (18). Lemma 2 [64]. Consider the following system:
8 x_ 1 ¼ x2 > > > > > x_ ¼ x3 > > < 2 .. . > > > > x_ n1 ¼ xn > > > :_ xn ¼ u
ð19Þ
under the following feedback control law:
u ¼ c1 sgnðx1 Þjx1 ja1 c2 sgnðx2 Þjx2 ja2 cn sgnðxn Þjxn jan
ð20Þ
If c1 ; c2 ; ; cn are chosen to satisfy that the polynomial
pn þ cn pn1 þ þ c2 p þ c1
ð21Þ
is Hurwitz, and a1 ; a2 ; ; an are set to be
ai1 ¼
ai aiþ1 ; i ¼ 2; ; n 2aiþ1 ai
ð22Þ
with anþ1 ¼ 1; an ¼ a; a 2 ð0; 1Þ. Then, the state variables of the system (19) converge to the equilibrium point in finite time.
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Lemma 3 [59]. Consider the following system:
x_ ¼ f ðxÞ þ gðxÞu ¼ hðx; uÞ
ð23Þ
where h : Dx Du ! Rn is locally Lipschitz in x and u, both Dx and Du contain the origin. If x_ ¼ f ðxÞ is asymptotically stable and gðxÞ is continuously differentiable, then the system in (23) is locally input-to-state stable (ISS). 3.1. Disturbance observer design Before proceeding to the disturbance observer design, on the basis of (13), the following auxiliary variable is introduced:
e ¼ f n4
ð24Þ
where the variable fðtÞ is determined by the following equation:
^ d þ 2h_ 2 tan hÞ c1d sgnðeÞjeja1d c2d sgnðeÞjeja2d f_ ¼ g sec2 hðu þ /
ð25Þ
þ
where c1d ; c2d ; a1d ; a2d 2 R are selected in light of the conditions in Lemma 1. Theorem 1. For the following equation:
ed ¼ e_ þ c1d sgnðeÞjeja1d þ c2d sgnðeÞjeja2d
ð26Þ
^ d ðtÞ is generated by the following dynamic equation: If /
2 ^_ d ¼ ðq þ kd Þsgn cos h ed / d g
ð27Þ
where qd ; kd 2 Rþ are positive constants and kd is chosen to satisfy that kd > rd , then eðtÞ will converge to zero in finite time and the disturbances /d ðtÞ can be exactly estimated by the proposed disturbance observer within a finite time. Proof. To prove Theorem 1, consider the auxiliary variable in (24) as follows:
e ¼ f n4
ð28Þ
Differentiating both sides of (28) with respect to time, substituting (25) and the fourth equation of (17) into the resulting _ expression for fðtÞ and n_ 4 ðtÞ, respectively, the following result can be obtained:
e_ ¼ g sec2 hð/^ d /d Þ c1d sgnðeÞjeja1d c2d sgnðeÞjeja2d
ð29Þ
Substituting the expression of e_ ðtÞ in (29) into (26) for e_ ðtÞ, one can obtain that
ed ¼ g sec2 hð/^ d /d Þ ¼ g sec2 h/~ d
ð30Þ
~ d ðtÞ ¼ / ^ d ðtÞ / ðtÞ is defined as the estimation error. According to (30), along with (27), the derivative of the estiwhere / d ~ d ðtÞ can be yielded as follows: mation error /
~_ d ¼ ðq þ kd Þsgnð/ ~ d Þ /_ d / d
ð31Þ
Next, in order to analyze the convergence of the estimation error, the following non-negative function is selected:
V / ðtÞ ¼
1 ~2 / 2 d
ð32Þ
~_ d ðtÞ in (31) into the resulting expression yield that Taking the time derivative of (32) and substituting /
~ d sgnð/ ~dÞ / ~ d /_ d 6 q j/ ~ d j ðkd j/_ d jÞj/ ~ dj V_ / ðtÞ ¼ ðqd þ kd Þ/ d
ð33Þ
Due to that kd > rd P j/_ d ðtÞj, we have
pffiffiffi 1 ~ d j ¼ q 2V 2 ðtÞ V_ / ðtÞ 6 qd j/ d /
ð34Þ
Then, by integrating (34) with respect to time, after performing some mathematical manipulations, one can obtain that 1
1
2V 2/ ðtÞ 6 2V 2/ ð0Þ
pffiffiffi 2qd t
ð35Þ ~
which indicates that there exists a finite time T 1 ¼ j/qd j after which we have d
~d ¼ 0 /
ð36Þ
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X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
which indicates that the estimation error is bounded and converges to zero in finite time. Then, based on (30), after T 1 , (26) becomes
e_ ¼ c1d sgnðeÞjeja1d c2d sgnðeÞjeja2d
ð37Þ 1
By invoking the conclusion of Lemma 1 and the convergent time given in [63], one can conclude that eðtÞ converges to zero in finite time and
eðtÞ; e_ ðtÞ 2 L1
ð38Þ
which, together with (26), further indicates that ed ðtÞ is bounded and convergent. Therefore, it can be concluded from the previous analysis that uncertain disturbances can be exactly estimated in finite time. h 3.2. Control law development In order to achieve the control objective of the crane system, the following manifold is introduced:
Z
t
w ¼ n4 ðtÞ
lðsÞds
0
where
ð39Þ
lðtÞ is defined as
l ¼ k4 sgnðn4 Þjn4 ja4 k3 sgnðn3 Þjn3 ja3 k2 sgnðn2 Þjn2 ja2 k1 sgnðn1 Þjn1 ja1
ð40Þ
where k1 ; k2 ; k3 ; k4 2 Rþ are positive constants and selected to satisfy that
s4 þ k4 s3 þ k3 s2 þ k2 s þ k1 ¼ 0
ð41Þ þ
is a Hurwitz polynomial, and a1 ; a2 ; a3 ; a4 2 R are set to be
ai1 ¼
ai aiþ1
2aiþ1 ai
;
i ¼ 2; 3; 4
ð42Þ
with a5 ¼ 1; a4 ¼ a; a 2 ð0; 1Þ. Next, from the introduced manifold (39), it is interesting to mention that one can construct the relation between the introduced manifold wðtÞ and the control input uðtÞ by taking the time derivative of wðtÞ. Based on this feature, the following Lyapunov function candidate is chosen:
V w ðtÞ ¼
1 2 w 2
ð43Þ
Taking the derivative of V w ðtÞ with respect to time, substituting the time derivative of wðtÞ in (39) and the fourth equation of
_ (17) into the resulting expression for wðtÞ and n_ 4 ðtÞ, respectively, and making some arrangements, the following result can be obtained:
h i V_ w ðtÞ ¼ w g sec2 hðu þ /d þ 2h_ 2 tan hÞ þ l
ð44Þ
Then, to render the manifold wðtÞ attractive and invariant, on the basis of the introduced disturbance observer in Theorem 1, the control input uðtÞ is designed as 2 2 ^ d 2h_ 2 tan h þ j cos h w cos h l uðtÞ ¼ / g g
ð45Þ
^ d ðtÞ can be derived based on the disturbance observer given in Theorem 1. By where j > 1 is a positive control gain and / substituting (45) into (3), the ultimate control input applied to the crane system can be obtained: 2
sin F ¼ ðMþm cos h
hÞl
h
i ^ d þ 2h_ 2 tan h j cos2 h w þ cos2 h l / g g mlh_ 2 sin h ðM þ mÞg tan h
ð46Þ
3.3. Stability analysis In this section, the boundedness and convergence of the closed-loop system’s states will be proven through rigourous theoretical analysis. 1
The proof for this property can be obtained from [63] and hence is omitted for brevity. Interested readers are referred to [63] for more details.
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X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
Theorem 2. The proposed control method, along with the designed disturbance observer, can guarantee that all of the closed-loop system’s states are bounded and convergent, that is
h iT lim x x_ h h_ ¼ ½pd 0 0 0T
t!1
where pd 2 R is the desired position. Proof: In terms of the convergence time of the disturbance observer, the following two cases will be discussed. Suppose ~ d ðtÞ converges to zero at t ¼ T conv . that the estimation error / Case 1. For t < T conv , consider the Lyapunov function candidate V w ðtÞ given in (43) and its time derivative (44), by substituting (45) into (44) for uðtÞ, after making some mathematical arrangements, the following equality can be obtained:
jed j V_ w ðtÞ ¼ ðj #Þw2 #w2 wed 6 ðj #Þw2 ; 8jwj P # where 0 < # < 1. Thus, one can deduce that wðtÞ is bounded for the bounded Next, substituting (45) into (17) for uðtÞ yields the closed-loop system
ð48Þ
ed ðtÞ [65].
3 2 n2 2 3 2_ 3 0 n1 7 6 n 7 6 3 607 6 n_ 7 6 7 7 6 27 6 7þ6 n4 7u 6 7¼6 7 6 405 k 4 n_ 3 5 6 a4 a3 7 4 k4 sgnðn4 Þjn4 j k3 sgnðn3 Þjn3 j 5 1 n_ 4 k2 sgnðn2 Þjn2 ja2 k1 sgnðn1 Þjn1 ja1
ð49Þ
~ d jw. Noting that ðtÞ ¼ / with the bounded input u
3 2 n2 2_ 3 n1 7 6 n3 7 6 n_ 7 6 7 6 27 6 7 n 6 7¼6 4 7 4 n_ 3 5 6 6 k sgnðn Þjn ja4 k sgnðn Þjn ja3 7 5 4 4 3 4 4 3 3 _n4 k2 sgnðn2 Þjn2 ja2 k1 sgnðn1 Þjn1 ja1
ð50Þ
it follows from Lemma 2 that the system (50) is asymptotically stable. Furthermore, according to Lemma 3, it is apparent that ðtÞ. Hence, we can conclude that the states of the closed-loop system (49) are the system (49) is ISS with the bounded input u bounded for t < T conv . ~ d ðtÞ ¼ 0. From Theorem 1, we know that ed ðtÞ ¼ 0 when Case 2. For t P T conv , in this case the estimation error is zero, i.e., / ~ d ðtÞ ¼ 0. Then, (48) becomes /
V_ w ðtÞ ¼ jw2 6 0
ð51Þ 2
By performing similar analysis to Theorem 1, we know that wðtÞ converges to zero in finite time . Furthermore, when wðtÞ ¼ 0, _ we have wðtÞ ¼ 0, that is
n_ 4 ¼ k4 sgnðn4 Þjn4 ja4 k3 sgnðn3 Þjn3 ja3 k2 sgnðn2 Þjn2 ja2 k1 sgnðn1 Þjn1 ja1
ð52Þ
where the time derivative of (39) has been used. It follows from Lemma 2 and the conclusion given in [64] that the state variables of the closed-loop system are convergent in finite time, in the sense that
½n1 n2 n3 n4 T ¼ ½0 0 0 0T
ð53Þ
which, applying (14)–(16), is equivalent to
h
x x_ h h_
iT
¼ ½pd 0 0 0T
ð54Þ
Hereto, the results of this theorem are proven. Remark 2. Although there exist sign functions in the proposed method, the proposed control method is still continuous. To ^ ^ on prove the continuity of the proposed method, one should examine the continuities of /ðtÞ and lðtÞ, respectively. For /ðtÞ, ^_ the basis of the expression of /ðtÞ in (27), by integrating (27) with respect to time, one can obtain that
^ /ðtÞ ¼
Z
0
2
t
cos2 h ðqd þ kd Þsgn ed ds g
The proof for this property is omitted for space limitation.
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^ Then, by employing the e d definition [66], we can obtain that /ðtÞ is continuous. By making similar analysis, we have that ai the functions ki sgnðni Þjni j in lðtÞ are continuous, hence, lðtÞ is also continuous. 4. Simulation results In this section, in order to demonstrate the performance of the proposed control method and assess its efficiency, two groups of simulation tests, included in the following subsections, are carried out by use of Matlab/Simulink. In the first group, existing control methods in [23,25] are chosen for comparison to illustrate the superior performance of the proposed method. In the second group, to further examine the estimation performance of the introduced observer, different from the disturbance in group 1, more complex disturbances are imposed on the crane system during the control process. The expression of the enhanced damping-based control (EDC) method in [23] is
h 2 F ¼ ðM þ m sin hÞ kd ðkp x_ kkp sin h kE h_ cos hÞ þ kE sin h i ð55Þ Rt kp ðx k 0 sin hðsÞds pd Þ m sin hðg cos h þ lh_ 2 Þ wherein kp ; kd ; kE ; k 2 Rþ are positive control gains and turned sufficiently to be chosen as
kp ¼ 0:3;
kd ¼ 3:2;
kE ¼ 1:4;
k ¼ 1:8
The expression of the partially saturated coupled-dissipation control (PSCDC) method in [25] is
Rt 2 F ¼ kp ðM þ m sin hÞ tanh x l 0 sin hðsÞds pd 1l sin h
Mþmkdsin2 h x_ l sin h 1l cos hh_ m sin hðg cos h þ lh_ 2 Þ ð57Þ ðM þ m sin hÞl cos hh_ 2
wherein kp ; kd 2 Rþ are positive control gains. After cautious tuning, the control gains are chosen as
kp ¼ 1:8; kd ¼ 1000 The control gains of the proposed method and the observer parameters are set to be
j ¼ 3; k1 ¼ 1; k2 ¼ 3; k3 ¼ 3:4; k4 ¼ 1:7; a4 ¼ 0:9 qd ¼ 0:01; kd ¼ 1; c1d ¼ 2; c2d ¼ 2; a1d ¼ 30; a2d ¼ 0:5 The system parameters of the crane system are selected as follows:
M ¼ 24 kg;
l ¼ 2 m;
g ¼ 9:8 m=s2
ð59Þ
4.1. Comparison study Consider the payload mass is 12 kg and the desired position is set to be pd ¼ 15 m. During the control process, the following disturbance is imposed on the crane system:
dðtÞ ¼ 5 sinð0:4ptÞ for 0 6 t 6 20 s
ð60Þ
which is generated by use of the simulink block. The simulation results of this group are depicted in Figs. 2–5 and corresponding quantified results are recorded in Table 1, which includes the following specifications: Rise time t r (s): The rise time3 is the time required for the trolley to move from the initial position to the desired position the first time. Maximum payload swing angle hmax (deg): hmax is the maximum of the absolute value of the payload swing angle during the whole control process. Residual swing angle hres (deg): Residual swing angle represents the maximum swing amplitude after 10 s. Positioning error emax (m): Positioning error is the maximum error between the practical position and the desired position after 10 s. Maximum control input F max (N): F max is the maximum control input that required by the controller.
3 Inspired by the transient-response specifications defined in control engineering, the first time that the trolley move from the initial position to the desired position is defined as the rise time, which can demonstrate the transient performance of the crane controllers to some extent.
X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
9
Fig. 2. Simulation results of the EDC method in [23]: trolley position, payload swing, and control input.
Fig. 3. Simulation results of the PSCDC method in [25]: trolley position, payload swing, and control input.
~ d max : Maximum estimation error is defined as the maximum error between the estimated Maximum estimation error / ~ d ðtÞj. ~ d max ¼ max j/ disturbance and the real disturbance, that is / From Figs. 2–4, it can be seen that the trolley and payload can be regulated to the desired positions under the proposed method. Only the trolley is driven to the desired position (with acceptable positioning error) under the existing methods, there exists serious residual payload swing for the comparative methods. From the quantified results in Table 1, one can see that the rise time of the existing EDC method (55) is the shortest among the three methods, and that of the PSCDC method (57) is the longest. Although the amplitude payload swing of the PSCDC method (57) is much smaller. However, other performance specifications of the comparative methods, such as the residual swing angle and positioning error, are much larger than these of the proposed method (46), which indicates that the steady-state performance of the proposed method is the best among the three methods. The obtained simulation results demonstrate that the proposed control method guarantees not only satisfactory transient-response but also smaller steady-state error even in the presence of external disturbances, which is of significant importance for practical applications. Remark 3. From the simulation results obtained in this section, one can find that the transient performance of the three methods is similar. But the steady-state performance of the comparative methods is unsatisfactory owing to the presence of uncertain disturbances. On the contrary, even in the presence of uncertain disturbances, the transient and steady-state performances of the proposed method are satisfactory.
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X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
Fig. 4. Simulation results of the proposed method (46): trolley position, payload swing, and control input.
Fig. 5. Simulation results of the proposed method (46): uncertain disturbance, estimated disturbance, and estimation error.
Table 1 Quantified results for simulation tests of group 1 and group 2. Controllers EDC method (55) Group 1 PSCDC method (57) Group 1 Proposed method (46) Group 1 Proposed method (46) Group 2.1 Proposed method (46) Group 2.2
tr
hmax
hres
emax
F max
~ d max /
8.13 15.86 8.67 10.80 10.90
15.61 7.31 15.05 10.16 10.16
1.31 2.12 0.38 0.11 0.21
0.11 0.17 0.04 0 0
112.5 43.2 70.43 51.44 56.05
NA NA 0.005 0.004 0.222
4.2. Robustness verification In this section, the payload mass is changed to be m ¼ 16 kg and the desired location is chosen as pd ¼ 10 m. To further show the advantage of the proposed disturbance observer, by using simulink block, the following two cases of disturbances are added to the crane system:
sinðtÞ dðtÞ ¼ 10 for 1 þ 0:5t 2 8 0 > > > <5 dðtÞ ¼ > 0:5t þ 5 > > : 5 sinð0:2ptÞ þ 5
0 6 t 6 20 s for for
0 6 t < 2s 2 6 t < 8s
for
8 6 t < 12 s
for 12 6 t 6 20 s
ð61Þ
ð62Þ
X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
11
The results of this simulation are shown in Figs. 6–9, and corresponding quantified results are given in Table 1. Although the payload mass and the final position are different from SubSection 4.1. Furthermore, more complex disturbances including step disturbance, ramp disturbance, periodic disturbance are imposed on the crane system, it can be deserved from the quantified results of Table 1 that the performance of the proposed method is not appreciably affected. In addition, from Figs. 5, 7, 9 and the quantified results in Table 1, one can find that the disturbance observer proposed here can asymptotically estimate different disturbances. The simulation results of this section indicate that uncertain disturbances could be completely removed from the system output and the performance of the closed-loop system is not affected owing to the disturbance observer. 5. Experimental tests In this section, two groups of experimental tests will be implemented on a laboratory prototype (as shown in Fig. 10). Similar to the simulation examination part, the existing EDC and PSCDC methods are chosen for a comparison study. The system parameters of the overhead crane testbed are
M ¼ 9:473 kg;
m ¼ 1 kg;
l ¼ 0:6 m;
g ¼ 9:8 m=s2
ð63Þ
The control gains of the EDC method (55) are
kp ¼ 0:138;
kd ¼ 2:154;
kE ¼ 1:4;
k ¼ 1:8
ð64Þ
and these of the PSCDC method (57) are chosen as
kp ¼ 0:158;
kd ¼ 3:655
Fig. 6. Simulation results of the proposed method (46) for case 1: trolley position, payload swing, and control input.
Fig. 7. Simulation results of the proposed method (46) for case 1: uncertain disturbance, estimated disturbance, and estimation error.
ð65Þ
12
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Fig. 8. Simulation results of the proposed method (46) for case 2: trolley position, payload swing, and control input.
Fig. 9. Simulation results of the proposed method (46) for case 2: uncertain disturbance, estimated disturbance, and estimation error.
The control gains of the proposed method and the observer parameters are selected to be
j ¼ 0:6; k1 ¼ 1; k2 ¼ 3; k3 ¼ 3:4; k4 ¼ 1:7; a4 ¼ 0:2 qd ¼ 0:01; kd ¼ 1; c1d ¼ 2; c2d ¼ 2; a1d ¼ 1:5; a2d ¼ 0:5
5.1. Comparison study In this group, the target position of the trolley is set to be pd ¼ 0:4 m. The experimental results of the three methods are shown in Figs. 11–13, and corresponding quantified results are given in Table 1, some of the specifications are different from the simulation part, which include: Residual swing angle hres (deg): Residual swing angle represents the maximum swing amplitude after 6 s. Positioning error emax (m): Positioning error is the maximum error between the practical position and the desired position after 6 s. From the obtained experimental results, one can find that the cart can be driven to the desired position and the payload swing can be eliminated under different methods. Specifically, the maximum payload swing of the PSCDC method is the smallest among the three methods. But the rise time of the PSCDC method is the longest. By comparing the quantified results of the three methods, one can find that the residual swing angle of the proposed method is much smaller than these of the existing methods. In addition, the positioning error are also smaller than these of the existing EDC and PSCDC methods. These results illustrate that the proposed method shows superior regulation control performance over the existing methods.
X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
Fig. 10. Experimental testbed.
Fig. 11. Experimental results of the EDC method in [23]: trolley position, payload swing, and control input.
Fig. 12. Experimental results of the PSCDC method in [25]: trolley position, payload swing, and control input.
13
14
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Fig. 13. Experimental results of the proposed method (46): trolley position, payload swing, control input, and estimated disturbance.
5.2. Robustness verification In order to illustrate the robustness of the proposed method with respect to uncertain disturbances, external disturbances are manually imposed on the payload during the control process. The target position of the trolley is set to be pd ¼ 0:35 m. In this part, the residual swing angle and position error are measured after 12 s. The corresponding results of this experiment are depicted in Figs. 14–16, respectively, and the detailed quantified results are included in Table 2. It can be noted that, under different control methods, the control objectives can be achieved and the external disturbance can be rejected. However, by comparing the quantified results of the three methods, one can find that the required maximum control input of the EDC method is much larger than that of the other two methods. Although the rise time and the positioning error of the three methods are similar, the residual payload swing of the comparative methods is much larger than that of the proposed method in this paper, which could affect the efficiency of the crane system. In contrast,
Fig. 14. Experimental results of the EDC method in [23]: trolley position, payload swing, and control input.
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Fig. 15. Experimental results of the PSCDC method in [25]: trolley position, payload swing, and control input.
Fig. 16. Experimental results of the proposed method (46): trolley position, payload swing, control input, and estimated disturbance.
Table 2 Quantified results for experimental tests of group 1 and group 2. Controllers
tr
hmax
hres
emax
F max
EDC method (55) Group 1 PSCDC method (57) Group 1 Proposed method (46) Group 1
4.50 5.15 4.75
1.77 1.51 1.66
0.213 0.195 0.079
0.0021 0.0022 0.0018
13.15 12.43 12.78
EDC method (55) Group 2 PSCDC method (57) Group 2 Proposed method (46) Group 2
4.05 3.90 3.75
4.92 4.73 4.65
0.330 0.238 0.110
0.0015 0.0017 0.0013
25.62 13.37 13.99
16
X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
owing to the disturbance observer, disturbances are eliminated quickly, and there is almost no positioning error for the trolley and no residual swing for the payload, which indicate that the proposed method, along with the introduced disturbance observer, shows superior regulation and disturbance rejection performance. Remark 4. From the provided simulation and experimental results, it can be observed that the maximum payload swings of different transportation tasks are different. In addition, one can find that the longer the transportation distance, the larger the maximum payload swing. This problem can be addressed by using ‘‘soft cart start” or trajectory tracking control, which is omitted here because it is beyond the scope of the paper. Remark 5. For nonlinear control systems, it is generally known that there are no guidelines to determine control parameters for nonlinear control methods. After abundant simulation and experimental tests, some rules on how to determine proper control gains for the three methods are obtained. For the comparative methods, the control gains are tuned by use of the experience of a PID tuning procedure. For the proposed method, the observer gains and control gains are all tuned by using the experience of the sliding mode control technique.
6. Conclusions For the regulation and disturbance rejection control of underactuated crane systems, this paper proposes a disturbance observer and a nonlinear feedback control method. The regulation and disturbance rejection properties of the proposed control method are proven through rigorous theoretical analysis. The effectiveness and robustness of the proposed method are illustrated by simulation and experimental tests. The simulation and experimental results have shown that the proposed control method exhibits better regulation and disturbance rejection performance in comparison with the existing control method. In our future work, the payload hoisting/lowering effect will be fully considered when making control design. CRediT authorship contribution statement Xianqing Wu: Resources, Methodology, Writing - original draft, Writing - review & editing, Supervision, Funding acquisition. Kexin Xu: Writing - original draft, Writing - review & editing, Software. Xiongxiong He: Writing - review & editing, Supervision. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors are very thankful for the valuable comments and suggestions from all the reviewers and the associate editor, Dr. Xingjian Jing, which have greatly improved the quality of the paper. They also acknowledge the financial support from the National Natural Science Foundation of China (61803339, 61873239) and the Natural Science Foundation of Zhejiang Province (LQ18F030011, LQ19F030014). References [1] S. Yang, B. Xian, Energy-based nonlinear adaptive control design for the quadrotor UAV system with a suspended payload, IEEE Trans. Industr. Electron. 67 (3) (2020) 2054–2064. [2] X. Liang, Y. Fang, N. Sun, H. Lin, X. Zhao, Adaptive nonlinear hierarchical control for a rotorcraft transporting a cable-suspended payload, IEEE Trans. Syst. Man Cybern.: Syst.https://doi.org/10.1109/TSMC.2019.2931812.. [3] X. Liang, Y. Fang, N. Sun, H. Lin, A novel energy-coupling-based hierarchical control approach for unmanned quadrotor transportation systems, IEEE/ ASME Trans. Mechatron. 24 (1) (2019) 248–259. [4] W. Guo, D. Liu, Nonlinear dynamic surface control for the underactuated translational oscillator with rotating actuator system, IEEE Access 7 (2019) 11844–11853. [5] L.A. Tuan, S.-G. Lee, Modeling and advanced sliding mode controls of crawler cranes considering wire rope elasticity and complicated operations, Mech. Syst. Signal Process. 103 (2018) 250–263. [6] J. Huang, S. Ri, T. Fukuda, Y. Wang, A disturbance observer based sliding mode control for a class of underactuated robotic system with mismatched uncertainties, IEEE Trans. Autom. Control 64 (6) (2019) 2480–2487. [7] J. Huang, S. Ri, L. Liu, Y. Wang, J. Kim, G. Pak, Nonlinear disturbance observer-based dynamic surface control of mobile wheeled inverted pendulum, IEEE Trans. Control Syst. Technol. 23 (6) (2015) 2400–2407. [8] L.A. Tuan, H.M. Cuong, P.V. Trieu, L.C. Nho, V.D. Thuan, L.V. Anh, Adaptive neural network sliding mode control of shipboard container cranes considering actuator backlash, Mech. Syst. Signal Process. 112 (2018) 233–250. [9] N. Uchiyama, H. Ouyang, S. Sano, Simple rotary crane dynamics modeling and open-loop control for residual load sway suppression by only horizontal boom motion, Mechatronics 23 (8) (2013) 1223–1236. [10] K.L. Sorensen, W.E. Singhose, Command-induced vibration analysis using input shaping principles, Automatica 44 (9) (2008) 2392–2397. [11] V.D. La, K.T. Nguyen, Combination of input shaping and radial spring-damper to reduce tridirectional vibration of crane payload, Mech. Syst. Signal Process. 116 (2019) 310–321.
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Contents lists available at ScienceDirect
Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp
Disturbance-observer-based nonlinear control for overhead cranes subject to uncertain disturbances Xianqing Wu a,⇑, Kexin Xu a, Xiongxiong He b a b
Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou 310018, China College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China
a r t i c l e
i n f o
Article history: Received 3 November 2019 Received in revised form 31 December 2019 Accepted 8 January 2020
Keywords: Underactuated system Overhead crane Disturbance observer Feedforward compensation control
a b s t r a c t The regulation and disturbance rejection problems of the underactuated overhead crane system in the presence of uncertain disturbances are considered in this paper. A novel nonlinear control method, along with a finite time disturbance observer, is proposed for the crane system. Different from existing control methods, one important feature of the proposed method is that uncertain disturbances are attenuated and eliminated by the disturbance observer in finite time. More precisely, some transformations are performed with respect to the original dynamic model of the crane system. Then, a finite time disturbance observer is introduced, based on which a nonlinear control method is proposed. Rigorous mathematical analysis is implemented to prove the theoretical derivations. Finally, simulation and experimental tests are carried out to demonstrate the superior performance of the proposed method. Ó 2020 Elsevier Ltd. All rights reserved.
1. Introduction Underactuated overhead cranes are widely used underactuated mechanical equipments [1–9], which are commonly found and utilized for material transportation in many industrial sites, such as harbors, factories, workshops, and so forth. The control objectives of these systems are to drive the trolley from the initial position to the target position rapidly and precisely as well as to attenuate and eliminate the payload swing at the same time. Though overhead cranes possess many advantages including high transportation efficiency, low energy consumption, simple mechanical structure, etc., the underactuated characteristic of overhead crane systems that only one control input and two degrees of freedom to be controlled makes the control problem of these systems much more challenging. Nevertheless, owing to the widely applications of overhead crane systems, the regulation and disturbance rejection problems of crane systems have attracted considerable attention from the control community during the past few decades. During the past decades, numerous interesting works have been made by a lot of researchers for the control of underactuated crane systems. The existing works about crane control can roughly be divided into two categories: trajectory planning (open-loop control) and feedback control design. For the former, the input shaping technique is one of the most extensively utilized motion planning approaches [10–15]. In particular, in [16], based on the particle swarm optimization technique, an improved input shaping method is designed for three dimensional overhead crane systems, which achieves high payload swing reduction during the control process. Ref. [17] takes several constraints consisting of the permitted swing amplitude
⇑ Corresponding author. E-mail address: [email protected] (X. Wu). https://doi.org/10.1016/j.ymssp.2020.106631 0888-3270/Ó 2020 Elsevier Ltd. All rights reserved.
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for the payload, available velocity, acceleration, and jerk for the trolley into consideration, a minimum-time trajectory planning method is proposed on the basis of the quasi-convex optimization approach. To achieve the objectives of safety and energy efficiency, an energy-optimal motion planning method is devised in [18]. In addition, motion planning methods for double pendulum crane systems also have been reported in the literature [19–21]. Compared with the trajectory planning methods, feedback control shows superior performance with respect to uncertain parameters and external disturbances. Plenty of closed-loop control methods have been exploited for crane systems. Specifically, based on the passivity of the crane system, a series of control schemes are given in [22–25]. Assuming that velocity signals are unavailable, some output feedback control methods are designed in [26–28]. Taking the payload hoisting/lowering effect into consideration, in [29], a trajectory tacking control method is designed for the overhead crane systems with varying rope length. Ref. [30] proposes a hybrid control scheme for the overhead cranes having payload hoisting/lowering effect, and it assumes that the viscous damping exists in the system. Moreover, adaptive and sliding model control techniques are employed to deal with system uncertainties or external disturbances [31–43]. In particular, in [43], a trajectory tracking control method is proposed for overhead crane systems, which achieves a finite-time tracking result. Furthermore, many other high-performance control methods are also employed, which mainly include optimal control [44], model predictive control [45,46], intelligent control [47,48], and so forth [49–58]. At present, the control problem of the crane system is still a fairly open topic. On one hand, most existing control methods, especially Lyapunov-based control methods, are designed on the basis of the exact model knowledge of the system, which makes them sensitive to uncertain external disturbances. However, in practice, mechanical systems always suffer from unavoidable factors, such as unmodeled dynamics, external disturbances, and so on. On the other hand, some robust methods, like sliding mode control technique, have been proposed to handle the undesired effects induced by uncertain disturbances. However, the main idea of the existing control methods for the disturbance rejection control of the crane system is to eliminate uncertain disturbances by feedback control, which is a robust way. Furthermore, the robustness of the existing methods is obtained at the price of sacrificing its nominal control performance [59]. In order to overcome the limitations of the existing methods for the crane control, inspired by the development of feedforward compensation control [60–62], the disturbance observer-based control technique is employed to estimate and compensate the effects of uncertain disturbances, and a nonlinear control method is proposed for the regulation of the crane system in this paper. In particular, we use a coordinate transformation to convert the origin dynamic model into a new form, based on which a finite time disturbance observer is presented. Then, an elaborate manifold is introduced and a nonlinear control method is designed to guarantee that the constructed manifold is attractive and invariant. Rigorous theoretical analysis is given to prove the stability of the closed-loop system and the convergence of the state variables. Finally, we demonstrate the superior performance of the proposed method in comparison with exiting nonlinear control methods through simulation and experimental tests. In sum, the suggested control method here exhibits the following remarkable features: 1) To the best of our knowledge, this is the first finite time disturbance observer-based nonlinear control method for the regulation and disturbance rejection of the crane system. 2) Because of the exact compensation of the designed disturbance observer, uncertain disturbances could be completely removed from the system output in finite time. 3) The proposed method belongs to a feedforward-compensation-based method and is continuous without any chattering, which is convenient for practical applications. The remaining parts of this paper are organized as follows. The dynamic model of the crane system is given and converted to a new form through a coordinate transformation in Section 2. Section 3 proposes a disturbance observer and a nonlinear controller with rigorous theoretical stability analysis. In Sections 4 and 5, simulation and hard experimental results are provided to demonstrate the superior control performance and strong robustness of the proposed method. Section 6 summarizes the main work of this paper and discusses our future work.
2. Problem formulation Consider the two-dimensional (2-D) overhead crane systems (as shown in Fig. 1), the dynamic model of which can be described as follows [17,18]:
ðM þ mÞ€x þ ml€h cos h mlh_ 2 sin h ¼ F þ d
ð1Þ
2 ml €h þ ml€x cos h þ mgl sin h ¼ 0
ð2Þ
where xðtÞ and hðtÞ denote the trolley position and the payload angle with respect to the vertical direction, respectively, M and m denote the trolley mass and the payload mass, respectively, l is the cable length, g represents the gravity acceleration, FðtÞ is the control input, and dðtÞ is the lumped term including frictions, uncertain disturbances, unmodeled dynamics, and so on.
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Fig. 1. Structure of the crane system.
For the dynamic model 1,2 of the overhead crane system, after some mathematical arrangements, one can rearrange it as follows:
F ¼ /u u mlh_ 2 sin h ðM þ mÞg tan h €x ¼ g tan h lh€ sec h
ð3Þ ð4Þ
where uðtÞ; /u ðtÞ are auxiliary functions whose expressions are
uðtÞ ¼ €h /d /d ðtÞ ¼
ð5Þ cos h 2
ðM þ m sin hÞl
ð6Þ
d
2
/u ðtÞ ¼ ðM þ m sin hÞl sec h
ð7Þ
For practical mechanical systems, the time derivative of the /d ðtÞ is bounded, that is
j/_ d ðtÞj 6 .d
ð8Þ
þ
where .d 2 R is a prior known constant. In order to facilitate the following control development and stability analysis, the following auxiliary variables are introduced:
v1 ¼ x þ l lnðsec h þ tan hÞ; v2 ¼ v_ 1 ; v3 ¼ g tan h; v4 ¼ v_ 3
ð9Þ
Then, based on (4), one can obtain the following dynamic equations:
8_ v1 ¼ v2 > > > < v_ 2 ¼ v3 1 /ðv3 ; v4 Þ > > v_ 3 ¼ v4 > : v_ 4 ¼ g sec2 hðu þ /d þ 2h_ 2 tan hÞ
ð10Þ
where /ðv3 ; v4 Þ is an auxiliary function with the following expression:
/ðv3 ; v4 Þ ¼
lv24 ðg 2
þ v23 Þ
ð11Þ
3=2
and the auxiliary variable uðtÞ is considered as the control input of the new dynamic model (10). For practical crane systems, the payload swing amplitude usually satisfies jhðtÞj 6 p9 rad and the swing angular velocity usually satisfies that _ jhðtÞj 1 rad=s [10–21], which can be found from the subsequent simulation and experimental tests, thus, v2 ðtÞ 1. In 4
addition, for practical applications, the cable length l is usually shorter than 10 m. Hence, from the practical perspective, we have that
lv24 2 3=2 3Þ
ðg 2 þ v Hence,
6
lv24 0:001lv24 1 g3
ð12Þ
v3 v3 1 /ðv3 ; v4 Þ is valid and (10) can be rewritten as
8_ v1 ¼ v2 > > > < v_ 2 ¼ v3 > > v_ 3 ¼ v4 > : v_ 4 ¼ g sec2 hðu þ /d þ 2h_ 2 tan hÞ
ð13Þ
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For the overhead crane system, the control objective is to drive the trolley to the desired position and eliminate the payload swing at the same time, that is
limxðtÞ ¼ pd ; limhðtÞ ¼ 0
t!1
ð14Þ
t!1
where pd is the desired location of the trolley. From the definitions of the auxiliary variables in (9), one can find that the control objective (14) is equivalent to
lim ðtÞ t!1 1
v
¼ pd ;
lim ðtÞ t!1 3
v
¼0
ð15Þ
Based on this characteristic, the following variables are defined:
n1 ¼ v1 pd ; n2 ¼ v2 ; n3 ¼ v3 ; n4 ¼ v4
ð16Þ
According to the introduced variables (16), the dynamic Eq. (13) can be rewritten as follows:
8_ n1 ¼ n2 > > > < n_ 2 ¼ n3 > > n_ 3 ¼ n4 > : n_ 4 ¼ g sec2 hðu þ /d þ 2h_ 2 tan hÞ
ð17Þ
which will be used for the following control development and stability analysis. Remark 1. Many existing disturbance-observer-based control methods usually assume that the disturbance is a constant or _ ¼ 0. Compared with these existing methods, we have relaxed the the time derivative of the disturbance is limt!1 dðtÞ assumptions on the disturbances from a mathematical viewpoint. Nonetheless, the assumption that the disturbance is differentiable and bounded is still a little strict. In the future, different cases of uncertain disturbances, such as nondifferentiable uncertain disturbances, unbounded disturbances, and so forth, will be taken into consideration when making control design.
3. Main results In this section, a finite time disturbance observer is designed and a nonlinear control method is proposed straightforwardly. Next, the stability of the closed-loop system and the convergence of the state variables are analyzed. The following lemmas will be utilized in the following stability analysis part. Lemma 1 [63]. For the following scalar system:
y_ ¼ c1d sgnðyÞjyja1d c2d sgnðyÞjyja2d
ð18Þ
þ
where c1d ; c2d 2 R are positive constants, a1d P 1 and 0 < a2d < 1, the origin is the globally finite time stable equilibrium point for the system described by (18). Lemma 2 [64]. Consider the following system:
8 x_ 1 ¼ x2 > > > > > x_ ¼ x3 > > < 2 .. . > > > > x_ n1 ¼ xn > > > :_ xn ¼ u
ð19Þ
under the following feedback control law:
u ¼ c1 sgnðx1 Þjx1 ja1 c2 sgnðx2 Þjx2 ja2 cn sgnðxn Þjxn jan
ð20Þ
If c1 ; c2 ; ; cn are chosen to satisfy that the polynomial
pn þ cn pn1 þ þ c2 p þ c1
ð21Þ
is Hurwitz, and a1 ; a2 ; ; an are set to be
ai1 ¼
ai aiþ1 ; i ¼ 2; ; n 2aiþ1 ai
ð22Þ
with anþ1 ¼ 1; an ¼ a; a 2 ð0; 1Þ. Then, the state variables of the system (19) converge to the equilibrium point in finite time.
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Lemma 3 [59]. Consider the following system:
x_ ¼ f ðxÞ þ gðxÞu ¼ hðx; uÞ
ð23Þ
where h : Dx Du ! Rn is locally Lipschitz in x and u, both Dx and Du contain the origin. If x_ ¼ f ðxÞ is asymptotically stable and gðxÞ is continuously differentiable, then the system in (23) is locally input-to-state stable (ISS). 3.1. Disturbance observer design Before proceeding to the disturbance observer design, on the basis of (13), the following auxiliary variable is introduced:
e ¼ f n4
ð24Þ
where the variable fðtÞ is determined by the following equation:
^ d þ 2h_ 2 tan hÞ c1d sgnðeÞjeja1d c2d sgnðeÞjeja2d f_ ¼ g sec2 hðu þ /
ð25Þ
þ
where c1d ; c2d ; a1d ; a2d 2 R are selected in light of the conditions in Lemma 1. Theorem 1. For the following equation:
ed ¼ e_ þ c1d sgnðeÞjeja1d þ c2d sgnðeÞjeja2d
ð26Þ
^ d ðtÞ is generated by the following dynamic equation: If /
2 ^_ d ¼ ðq þ kd Þsgn cos h ed / d g
ð27Þ
where qd ; kd 2 Rþ are positive constants and kd is chosen to satisfy that kd > rd , then eðtÞ will converge to zero in finite time and the disturbances /d ðtÞ can be exactly estimated by the proposed disturbance observer within a finite time. Proof. To prove Theorem 1, consider the auxiliary variable in (24) as follows:
e ¼ f n4
ð28Þ
Differentiating both sides of (28) with respect to time, substituting (25) and the fourth equation of (17) into the resulting _ expression for fðtÞ and n_ 4 ðtÞ, respectively, the following result can be obtained:
e_ ¼ g sec2 hð/^ d /d Þ c1d sgnðeÞjeja1d c2d sgnðeÞjeja2d
ð29Þ
Substituting the expression of e_ ðtÞ in (29) into (26) for e_ ðtÞ, one can obtain that
ed ¼ g sec2 hð/^ d /d Þ ¼ g sec2 h/~ d
ð30Þ
~ d ðtÞ ¼ / ^ d ðtÞ / ðtÞ is defined as the estimation error. According to (30), along with (27), the derivative of the estiwhere / d ~ d ðtÞ can be yielded as follows: mation error /
~_ d ¼ ðq þ kd Þsgnð/ ~ d Þ /_ d / d
ð31Þ
Next, in order to analyze the convergence of the estimation error, the following non-negative function is selected:
V / ðtÞ ¼
1 ~2 / 2 d
ð32Þ
~_ d ðtÞ in (31) into the resulting expression yield that Taking the time derivative of (32) and substituting /
~ d sgnð/ ~dÞ / ~ d /_ d 6 q j/ ~ d j ðkd j/_ d jÞj/ ~ dj V_ / ðtÞ ¼ ðqd þ kd Þ/ d
ð33Þ
Due to that kd > rd P j/_ d ðtÞj, we have
pffiffiffi 1 ~ d j ¼ q 2V 2 ðtÞ V_ / ðtÞ 6 qd j/ d /
ð34Þ
Then, by integrating (34) with respect to time, after performing some mathematical manipulations, one can obtain that 1
1
2V 2/ ðtÞ 6 2V 2/ ð0Þ
pffiffiffi 2qd t
ð35Þ ~
which indicates that there exists a finite time T 1 ¼ j/qd j after which we have d
~d ¼ 0 /
ð36Þ
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X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
which indicates that the estimation error is bounded and converges to zero in finite time. Then, based on (30), after T 1 , (26) becomes
e_ ¼ c1d sgnðeÞjeja1d c2d sgnðeÞjeja2d
ð37Þ 1
By invoking the conclusion of Lemma 1 and the convergent time given in [63], one can conclude that eðtÞ converges to zero in finite time and
eðtÞ; e_ ðtÞ 2 L1
ð38Þ
which, together with (26), further indicates that ed ðtÞ is bounded and convergent. Therefore, it can be concluded from the previous analysis that uncertain disturbances can be exactly estimated in finite time. h 3.2. Control law development In order to achieve the control objective of the crane system, the following manifold is introduced:
Z
t
w ¼ n4 ðtÞ
lðsÞds
0
where
ð39Þ
lðtÞ is defined as
l ¼ k4 sgnðn4 Þjn4 ja4 k3 sgnðn3 Þjn3 ja3 k2 sgnðn2 Þjn2 ja2 k1 sgnðn1 Þjn1 ja1
ð40Þ
where k1 ; k2 ; k3 ; k4 2 Rþ are positive constants and selected to satisfy that
s4 þ k4 s3 þ k3 s2 þ k2 s þ k1 ¼ 0
ð41Þ þ
is a Hurwitz polynomial, and a1 ; a2 ; a3 ; a4 2 R are set to be
ai1 ¼
ai aiþ1
2aiþ1 ai
;
i ¼ 2; 3; 4
ð42Þ
with a5 ¼ 1; a4 ¼ a; a 2 ð0; 1Þ. Next, from the introduced manifold (39), it is interesting to mention that one can construct the relation between the introduced manifold wðtÞ and the control input uðtÞ by taking the time derivative of wðtÞ. Based on this feature, the following Lyapunov function candidate is chosen:
V w ðtÞ ¼
1 2 w 2
ð43Þ
Taking the derivative of V w ðtÞ with respect to time, substituting the time derivative of wðtÞ in (39) and the fourth equation of
_ (17) into the resulting expression for wðtÞ and n_ 4 ðtÞ, respectively, and making some arrangements, the following result can be obtained:
h i V_ w ðtÞ ¼ w g sec2 hðu þ /d þ 2h_ 2 tan hÞ þ l
ð44Þ
Then, to render the manifold wðtÞ attractive and invariant, on the basis of the introduced disturbance observer in Theorem 1, the control input uðtÞ is designed as 2 2 ^ d 2h_ 2 tan h þ j cos h w cos h l uðtÞ ¼ / g g
ð45Þ
^ d ðtÞ can be derived based on the disturbance observer given in Theorem 1. By where j > 1 is a positive control gain and / substituting (45) into (3), the ultimate control input applied to the crane system can be obtained: 2
sin F ¼ ðMþm cos h
hÞl
h
i ^ d þ 2h_ 2 tan h j cos2 h w þ cos2 h l / g g mlh_ 2 sin h ðM þ mÞg tan h
ð46Þ
3.3. Stability analysis In this section, the boundedness and convergence of the closed-loop system’s states will be proven through rigourous theoretical analysis. 1
The proof for this property can be obtained from [63] and hence is omitted for brevity. Interested readers are referred to [63] for more details.
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Theorem 2. The proposed control method, along with the designed disturbance observer, can guarantee that all of the closed-loop system’s states are bounded and convergent, that is
h iT lim x x_ h h_ ¼ ½pd 0 0 0T
t!1
where pd 2 R is the desired position. Proof: In terms of the convergence time of the disturbance observer, the following two cases will be discussed. Suppose ~ d ðtÞ converges to zero at t ¼ T conv . that the estimation error / Case 1. For t < T conv , consider the Lyapunov function candidate V w ðtÞ given in (43) and its time derivative (44), by substituting (45) into (44) for uðtÞ, after making some mathematical arrangements, the following equality can be obtained:
jed j V_ w ðtÞ ¼ ðj #Þw2 #w2 wed 6 ðj #Þw2 ; 8jwj P # where 0 < # < 1. Thus, one can deduce that wðtÞ is bounded for the bounded Next, substituting (45) into (17) for uðtÞ yields the closed-loop system
ð48Þ
ed ðtÞ [65].
3 2 n2 2 3 2_ 3 0 n1 7 6 n 7 6 3 607 6 n_ 7 6 7 7 6 27 6 7þ6 n4 7u 6 7¼6 7 6 405 k 4 n_ 3 5 6 a4 a3 7 4 k4 sgnðn4 Þjn4 j k3 sgnðn3 Þjn3 j 5 1 n_ 4 k2 sgnðn2 Þjn2 ja2 k1 sgnðn1 Þjn1 ja1
ð49Þ
~ d jw. Noting that ðtÞ ¼ / with the bounded input u
3 2 n2 2_ 3 n1 7 6 n3 7 6 n_ 7 6 7 6 27 6 7 n 6 7¼6 4 7 4 n_ 3 5 6 6 k sgnðn Þjn ja4 k sgnðn Þjn ja3 7 5 4 4 3 4 4 3 3 _n4 k2 sgnðn2 Þjn2 ja2 k1 sgnðn1 Þjn1 ja1
ð50Þ
it follows from Lemma 2 that the system (50) is asymptotically stable. Furthermore, according to Lemma 3, it is apparent that ðtÞ. Hence, we can conclude that the states of the closed-loop system (49) are the system (49) is ISS with the bounded input u bounded for t < T conv . ~ d ðtÞ ¼ 0. From Theorem 1, we know that ed ðtÞ ¼ 0 when Case 2. For t P T conv , in this case the estimation error is zero, i.e., / ~ d ðtÞ ¼ 0. Then, (48) becomes /
V_ w ðtÞ ¼ jw2 6 0
ð51Þ 2
By performing similar analysis to Theorem 1, we know that wðtÞ converges to zero in finite time . Furthermore, when wðtÞ ¼ 0, _ we have wðtÞ ¼ 0, that is
n_ 4 ¼ k4 sgnðn4 Þjn4 ja4 k3 sgnðn3 Þjn3 ja3 k2 sgnðn2 Þjn2 ja2 k1 sgnðn1 Þjn1 ja1
ð52Þ
where the time derivative of (39) has been used. It follows from Lemma 2 and the conclusion given in [64] that the state variables of the closed-loop system are convergent in finite time, in the sense that
½n1 n2 n3 n4 T ¼ ½0 0 0 0T
ð53Þ
which, applying (14)–(16), is equivalent to
h
x x_ h h_
iT
¼ ½pd 0 0 0T
ð54Þ
Hereto, the results of this theorem are proven. Remark 2. Although there exist sign functions in the proposed method, the proposed control method is still continuous. To ^ ^ on prove the continuity of the proposed method, one should examine the continuities of /ðtÞ and lðtÞ, respectively. For /ðtÞ, ^_ the basis of the expression of /ðtÞ in (27), by integrating (27) with respect to time, one can obtain that
^ /ðtÞ ¼
Z
0
2
t
cos2 h ðqd þ kd Þsgn ed ds g
The proof for this property is omitted for space limitation.
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^ Then, by employing the e d definition [66], we can obtain that /ðtÞ is continuous. By making similar analysis, we have that ai the functions ki sgnðni Þjni j in lðtÞ are continuous, hence, lðtÞ is also continuous. 4. Simulation results In this section, in order to demonstrate the performance of the proposed control method and assess its efficiency, two groups of simulation tests, included in the following subsections, are carried out by use of Matlab/Simulink. In the first group, existing control methods in [23,25] are chosen for comparison to illustrate the superior performance of the proposed method. In the second group, to further examine the estimation performance of the introduced observer, different from the disturbance in group 1, more complex disturbances are imposed on the crane system during the control process. The expression of the enhanced damping-based control (EDC) method in [23] is
h 2 F ¼ ðM þ m sin hÞ kd ðkp x_ kkp sin h kE h_ cos hÞ þ kE sin h i ð55Þ Rt kp ðx k 0 sin hðsÞds pd Þ m sin hðg cos h þ lh_ 2 Þ wherein kp ; kd ; kE ; k 2 Rþ are positive control gains and turned sufficiently to be chosen as
kp ¼ 0:3;
kd ¼ 3:2;
kE ¼ 1:4;
k ¼ 1:8
The expression of the partially saturated coupled-dissipation control (PSCDC) method in [25] is
Rt 2 F ¼ kp ðM þ m sin hÞ tanh x l 0 sin hðsÞds pd 1l sin h
Mþmkdsin2 h x_ l sin h 1l cos hh_ m sin hðg cos h þ lh_ 2 Þ ð57Þ ðM þ m sin hÞl cos hh_ 2
wherein kp ; kd 2 Rþ are positive control gains. After cautious tuning, the control gains are chosen as
kp ¼ 1:8; kd ¼ 1000 The control gains of the proposed method and the observer parameters are set to be
j ¼ 3; k1 ¼ 1; k2 ¼ 3; k3 ¼ 3:4; k4 ¼ 1:7; a4 ¼ 0:9 qd ¼ 0:01; kd ¼ 1; c1d ¼ 2; c2d ¼ 2; a1d ¼ 30; a2d ¼ 0:5 The system parameters of the crane system are selected as follows:
M ¼ 24 kg;
l ¼ 2 m;
g ¼ 9:8 m=s2
ð59Þ
4.1. Comparison study Consider the payload mass is 12 kg and the desired position is set to be pd ¼ 15 m. During the control process, the following disturbance is imposed on the crane system:
dðtÞ ¼ 5 sinð0:4ptÞ for 0 6 t 6 20 s
ð60Þ
which is generated by use of the simulink block. The simulation results of this group are depicted in Figs. 2–5 and corresponding quantified results are recorded in Table 1, which includes the following specifications: Rise time t r (s): The rise time3 is the time required for the trolley to move from the initial position to the desired position the first time. Maximum payload swing angle hmax (deg): hmax is the maximum of the absolute value of the payload swing angle during the whole control process. Residual swing angle hres (deg): Residual swing angle represents the maximum swing amplitude after 10 s. Positioning error emax (m): Positioning error is the maximum error between the practical position and the desired position after 10 s. Maximum control input F max (N): F max is the maximum control input that required by the controller.
3 Inspired by the transient-response specifications defined in control engineering, the first time that the trolley move from the initial position to the desired position is defined as the rise time, which can demonstrate the transient performance of the crane controllers to some extent.
X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
9
Fig. 2. Simulation results of the EDC method in [23]: trolley position, payload swing, and control input.
Fig. 3. Simulation results of the PSCDC method in [25]: trolley position, payload swing, and control input.
~ d max : Maximum estimation error is defined as the maximum error between the estimated Maximum estimation error / ~ d ðtÞj. ~ d max ¼ max j/ disturbance and the real disturbance, that is / From Figs. 2–4, it can be seen that the trolley and payload can be regulated to the desired positions under the proposed method. Only the trolley is driven to the desired position (with acceptable positioning error) under the existing methods, there exists serious residual payload swing for the comparative methods. From the quantified results in Table 1, one can see that the rise time of the existing EDC method (55) is the shortest among the three methods, and that of the PSCDC method (57) is the longest. Although the amplitude payload swing of the PSCDC method (57) is much smaller. However, other performance specifications of the comparative methods, such as the residual swing angle and positioning error, are much larger than these of the proposed method (46), which indicates that the steady-state performance of the proposed method is the best among the three methods. The obtained simulation results demonstrate that the proposed control method guarantees not only satisfactory transient-response but also smaller steady-state error even in the presence of external disturbances, which is of significant importance for practical applications. Remark 3. From the simulation results obtained in this section, one can find that the transient performance of the three methods is similar. But the steady-state performance of the comparative methods is unsatisfactory owing to the presence of uncertain disturbances. On the contrary, even in the presence of uncertain disturbances, the transient and steady-state performances of the proposed method are satisfactory.
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Fig. 4. Simulation results of the proposed method (46): trolley position, payload swing, and control input.
Fig. 5. Simulation results of the proposed method (46): uncertain disturbance, estimated disturbance, and estimation error.
Table 1 Quantified results for simulation tests of group 1 and group 2. Controllers EDC method (55) Group 1 PSCDC method (57) Group 1 Proposed method (46) Group 1 Proposed method (46) Group 2.1 Proposed method (46) Group 2.2
tr
hmax
hres
emax
F max
~ d max /
8.13 15.86 8.67 10.80 10.90
15.61 7.31 15.05 10.16 10.16
1.31 2.12 0.38 0.11 0.21
0.11 0.17 0.04 0 0
112.5 43.2 70.43 51.44 56.05
NA NA 0.005 0.004 0.222
4.2. Robustness verification In this section, the payload mass is changed to be m ¼ 16 kg and the desired location is chosen as pd ¼ 10 m. To further show the advantage of the proposed disturbance observer, by using simulink block, the following two cases of disturbances are added to the crane system:
sinðtÞ dðtÞ ¼ 10 for 1 þ 0:5t 2 8 0 > > > <5 dðtÞ ¼ > 0:5t þ 5 > > : 5 sinð0:2ptÞ þ 5
0 6 t 6 20 s for for
0 6 t < 2s 2 6 t < 8s
for
8 6 t < 12 s
for 12 6 t 6 20 s
ð61Þ
ð62Þ
X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
11
The results of this simulation are shown in Figs. 6–9, and corresponding quantified results are given in Table 1. Although the payload mass and the final position are different from SubSection 4.1. Furthermore, more complex disturbances including step disturbance, ramp disturbance, periodic disturbance are imposed on the crane system, it can be deserved from the quantified results of Table 1 that the performance of the proposed method is not appreciably affected. In addition, from Figs. 5, 7, 9 and the quantified results in Table 1, one can find that the disturbance observer proposed here can asymptotically estimate different disturbances. The simulation results of this section indicate that uncertain disturbances could be completely removed from the system output and the performance of the closed-loop system is not affected owing to the disturbance observer. 5. Experimental tests In this section, two groups of experimental tests will be implemented on a laboratory prototype (as shown in Fig. 10). Similar to the simulation examination part, the existing EDC and PSCDC methods are chosen for a comparison study. The system parameters of the overhead crane testbed are
M ¼ 9:473 kg;
m ¼ 1 kg;
l ¼ 0:6 m;
g ¼ 9:8 m=s2
ð63Þ
The control gains of the EDC method (55) are
kp ¼ 0:138;
kd ¼ 2:154;
kE ¼ 1:4;
k ¼ 1:8
ð64Þ
and these of the PSCDC method (57) are chosen as
kp ¼ 0:158;
kd ¼ 3:655
Fig. 6. Simulation results of the proposed method (46) for case 1: trolley position, payload swing, and control input.
Fig. 7. Simulation results of the proposed method (46) for case 1: uncertain disturbance, estimated disturbance, and estimation error.
ð65Þ
12
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Fig. 8. Simulation results of the proposed method (46) for case 2: trolley position, payload swing, and control input.
Fig. 9. Simulation results of the proposed method (46) for case 2: uncertain disturbance, estimated disturbance, and estimation error.
The control gains of the proposed method and the observer parameters are selected to be
j ¼ 0:6; k1 ¼ 1; k2 ¼ 3; k3 ¼ 3:4; k4 ¼ 1:7; a4 ¼ 0:2 qd ¼ 0:01; kd ¼ 1; c1d ¼ 2; c2d ¼ 2; a1d ¼ 1:5; a2d ¼ 0:5
5.1. Comparison study In this group, the target position of the trolley is set to be pd ¼ 0:4 m. The experimental results of the three methods are shown in Figs. 11–13, and corresponding quantified results are given in Table 1, some of the specifications are different from the simulation part, which include: Residual swing angle hres (deg): Residual swing angle represents the maximum swing amplitude after 6 s. Positioning error emax (m): Positioning error is the maximum error between the practical position and the desired position after 6 s. From the obtained experimental results, one can find that the cart can be driven to the desired position and the payload swing can be eliminated under different methods. Specifically, the maximum payload swing of the PSCDC method is the smallest among the three methods. But the rise time of the PSCDC method is the longest. By comparing the quantified results of the three methods, one can find that the residual swing angle of the proposed method is much smaller than these of the existing methods. In addition, the positioning error are also smaller than these of the existing EDC and PSCDC methods. These results illustrate that the proposed method shows superior regulation control performance over the existing methods.
X. Wu et al. / Mechanical Systems and Signal Processing 139 (2020) 106631
Fig. 10. Experimental testbed.
Fig. 11. Experimental results of the EDC method in [23]: trolley position, payload swing, and control input.
Fig. 12. Experimental results of the PSCDC method in [25]: trolley position, payload swing, and control input.
13
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Fig. 13. Experimental results of the proposed method (46): trolley position, payload swing, control input, and estimated disturbance.
5.2. Robustness verification In order to illustrate the robustness of the proposed method with respect to uncertain disturbances, external disturbances are manually imposed on the payload during the control process. The target position of the trolley is set to be pd ¼ 0:35 m. In this part, the residual swing angle and position error are measured after 12 s. The corresponding results of this experiment are depicted in Figs. 14–16, respectively, and the detailed quantified results are included in Table 2. It can be noted that, under different control methods, the control objectives can be achieved and the external disturbance can be rejected. However, by comparing the quantified results of the three methods, one can find that the required maximum control input of the EDC method is much larger than that of the other two methods. Although the rise time and the positioning error of the three methods are similar, the residual payload swing of the comparative methods is much larger than that of the proposed method in this paper, which could affect the efficiency of the crane system. In contrast,
Fig. 14. Experimental results of the EDC method in [23]: trolley position, payload swing, and control input.
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Fig. 15. Experimental results of the PSCDC method in [25]: trolley position, payload swing, and control input.
Fig. 16. Experimental results of the proposed method (46): trolley position, payload swing, control input, and estimated disturbance.
Table 2 Quantified results for experimental tests of group 1 and group 2. Controllers
tr
hmax
hres
emax
F max
EDC method (55) Group 1 PSCDC method (57) Group 1 Proposed method (46) Group 1
4.50 5.15 4.75
1.77 1.51 1.66
0.213 0.195 0.079
0.0021 0.0022 0.0018
13.15 12.43 12.78
EDC method (55) Group 2 PSCDC method (57) Group 2 Proposed method (46) Group 2
4.05 3.90 3.75
4.92 4.73 4.65
0.330 0.238 0.110
0.0015 0.0017 0.0013
25.62 13.37 13.99
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owing to the disturbance observer, disturbances are eliminated quickly, and there is almost no positioning error for the trolley and no residual swing for the payload, which indicate that the proposed method, along with the introduced disturbance observer, shows superior regulation and disturbance rejection performance. Remark 4. From the provided simulation and experimental results, it can be observed that the maximum payload swings of different transportation tasks are different. In addition, one can find that the longer the transportation distance, the larger the maximum payload swing. This problem can be addressed by using ‘‘soft cart start” or trajectory tracking control, which is omitted here because it is beyond the scope of the paper. Remark 5. For nonlinear control systems, it is generally known that there are no guidelines to determine control parameters for nonlinear control methods. After abundant simulation and experimental tests, some rules on how to determine proper control gains for the three methods are obtained. For the comparative methods, the control gains are tuned by use of the experience of a PID tuning procedure. For the proposed method, the observer gains and control gains are all tuned by using the experience of the sliding mode control technique.
6. Conclusions For the regulation and disturbance rejection control of underactuated crane systems, this paper proposes a disturbance observer and a nonlinear feedback control method. The regulation and disturbance rejection properties of the proposed control method are proven through rigorous theoretical analysis. The effectiveness and robustness of the proposed method are illustrated by simulation and experimental tests. The simulation and experimental results have shown that the proposed control method exhibits better regulation and disturbance rejection performance in comparison with the existing control method. In our future work, the payload hoisting/lowering effect will be fully considered when making control design. CRediT authorship contribution statement Xianqing Wu: Resources, Methodology, Writing - original draft, Writing - review & editing, Supervision, Funding acquisition. Kexin Xu: Writing - original draft, Writing - review & editing, Software. Xiongxiong He: Writing - review & editing, Supervision. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors are very thankful for the valuable comments and suggestions from all the reviewers and the associate editor, Dr. Xingjian Jing, which have greatly improved the quality of the paper. They also acknowledge the financial support from the National Natural Science Foundation of China (61803339, 61873239) and the Natural Science Foundation of Zhejiang Province (LQ18F030011, LQ19F030014). References [1] S. Yang, B. Xian, Energy-based nonlinear adaptive control design for the quadrotor UAV system with a suspended payload, IEEE Trans. Industr. Electron. 67 (3) (2020) 2054–2064. [2] X. Liang, Y. Fang, N. Sun, H. Lin, X. Zhao, Adaptive nonlinear hierarchical control for a rotorcraft transporting a cable-suspended payload, IEEE Trans. Syst. Man Cybern.: Syst.https://doi.org/10.1109/TSMC.2019.2931812.. [3] X. Liang, Y. Fang, N. Sun, H. Lin, A novel energy-coupling-based hierarchical control approach for unmanned quadrotor transportation systems, IEEE/ ASME Trans. Mechatron. 24 (1) (2019) 248–259. [4] W. Guo, D. Liu, Nonlinear dynamic surface control for the underactuated translational oscillator with rotating actuator system, IEEE Access 7 (2019) 11844–11853. [5] L.A. Tuan, S.-G. Lee, Modeling and advanced sliding mode controls of crawler cranes considering wire rope elasticity and complicated operations, Mech. Syst. Signal Process. 103 (2018) 250–263. [6] J. Huang, S. Ri, T. Fukuda, Y. Wang, A disturbance observer based sliding mode control for a class of underactuated robotic system with mismatched uncertainties, IEEE Trans. Autom. Control 64 (6) (2019) 2480–2487. [7] J. Huang, S. Ri, L. Liu, Y. Wang, J. Kim, G. Pak, Nonlinear disturbance observer-based dynamic surface control of mobile wheeled inverted pendulum, IEEE Trans. Control Syst. Technol. 23 (6) (2015) 2400–2407. [8] L.A. Tuan, H.M. Cuong, P.V. Trieu, L.C. Nho, V.D. Thuan, L.V. Anh, Adaptive neural network sliding mode control of shipboard container cranes considering actuator backlash, Mech. Syst. Signal Process. 112 (2018) 233–250. [9] N. Uchiyama, H. Ouyang, S. Sano, Simple rotary crane dynamics modeling and open-loop control for residual load sway suppression by only horizontal boom motion, Mechatronics 23 (8) (2013) 1223–1236. [10] K.L. Sorensen, W.E. Singhose, Command-induced vibration analysis using input shaping principles, Automatica 44 (9) (2008) 2392–2397. [11] V.D. La, K.T. Nguyen, Combination of input shaping and radial spring-damper to reduce tridirectional vibration of crane payload, Mech. Syst. Signal Process. 116 (2019) 310–321.
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