Research Article
Modeling and energy-based fuzzy controlling for underactuated overhead cranes with load transferring, lowering, and persistent external disturbances
Advances in Mechanical Engineering 2017, Vol. 9(10) 1–13 Ó The Author(s) 2017 DOI: 10.1177/1687814017720086 journals.sagepub.com/home/ade
Menghua Zhang, Xin Ma, Xuewen Rong, Rui Song, Xincheng Tian and Yibin Li
Abstract In practice, vertical load motion is always involved in overhead cranes. In this case, the cable length turns from a constant to a variable, which may induce large amplitude load swing and make it more challenging to develop an appropriate controller. Most existing control methods for varying-cable-length cranes require either linearization or approximation to the original nonlinear dynamics; moreover, the case of external load disturbances is not fully considered. Inspired by these facts, we build the model and suggest an energy-based fuzzy control method for underactuated overhead cranes with load transferring, lowering, and persistent external disturbances. To estimate the persistent external disturbances, we construct a fuzzy disturbance observer. And a strict mathematical analysis of the control method without linearization approximation is presented, providing theoretical support for the superior performance of the proposed controller. Lyapunov techniques and LaSalle’s invariance theorem are used to demonstrate the stability of the closed-loop overhead crane system. Numerical simulation results are included to examine the effectiveness and robustness of the proposed method. Keywords Underactuated overhead crane, anti-swing control, fuzzy disturbance observer, robustness, Lyapunov techniques
Date received: 18 February 2017; accepted: 20 June 2017 Handling Editor: Jiahu Qin
Introduction In recent years, studies on underactuated crane systems have attracted a great deal of interests.1–5 The main purpose of controlling a crane is to drive the trolley to the target position accurately and damp out the unexpected load swing rapidly. As typical underactuated systems, overhead cranes have been widely used in harbors, construction sites, and so on for the transportation of heavy goods. Roughly speaking, the overhead crane operation process can be divided into three steps: (1) load hoisting, (2) horizontal transportation, and (3) load lowering.6–8 Due to the fact that the trolley does
not move in the first step, there is almost no load swing at this stage. In the second step, the load swing is required to keep within a small range, and there should be no or small residual swing as the trolley stops to put
School of Control Science and Engineering, Shandong University, Jinan, China Corresponding author: Xin Ma, School of Control Science and Engineering, Shandong University, Qianfoshan Campus, No. 17923, Jingshi Road, Jinan 250061, Shandong, China. Email:
[email protected]
Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www.creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/ open-access-at-sage).
2 down the load vertically and steadily for step 3. However, due to the presence of inertia and external disturbances, large load swing amplitude is always induced during and at the end of step 2. Taking the above-mentioned facts into account, we attempt to propose an effective control method for the second and third steps to position the trolley accurately while suppressing and eliminating the load swing rapidly. In recent decades, a lot of efforts have been devoted to the anti-swing control problem of underactuated constant-cable-length cranes, and many constructive and useful results have been obtained.9–33 Roughly speaking, the control methods can be sorted as open-loop and closed-loop according to the fact whether state feedback is employed in the anti-swing control laws.25 Open-loop control methods, including input shaping,9–12 offline motion planning,13 and optimal control,14,15 have been proposed for overhead crane systems. Due to the presence of external disturbances, closed-loop control methods are developed to increase the system performance, including sliding-based control,16–19 adaptive fuzzy control,20–22 energy/passivity-based control (PBC),8,23–27 output feedback control,28 nonlinear trajectory planning,29 switching-based emergency control,30 model predictive control,31 nested saturation control,32 and genetic algorithm (GA)-based stable control.33 Nonetheless, the aforementioned control methods are oriented to constant-cable-length overhead crane systems, and their control performance will probably be degraded in the case of cable length variations.34 Moreover, it is more challenging to design appropriate controllers for varying-cable-length cranes as the cable length turns from a constant into a variable. To achieve satisfactory control performance in the presence of cable variation, abundant efforts have been made for varying-cable-length cranes. In the work by Trabia et al.,35 by linearizing the complex crane dynamic around its equilibrium point, a fuzzy logic control method is proposed to suppress and eliminate the unexpected load swing angle. IDA-PBC (interconnection and damping assignment passivity-based control) method is successfully applied to varying-cable-length cranes, which shape the crane system energies to achieve the minimums at the desired equilibrium points.36 By coupling the actuated trolley motion with the unactuated load swing into a sliding surface, sliding-mode-based control methods are developed for crane control subject to varying-cable-length.4,37,38 Neural-network-based frameworks are used to develop intelligent controllers for varying-cable-length cranes and compensate for uncertainties.39 Corriga et al.40 develop a gain-scheduling controller for a linearized varying-cable-length crane system. In the work by Le et al.,41 a nonlinear controller is proposed for an overhead crane system, in which partial feedback linearization technique is used. Garrido et al.42 develop an
Advances in Mechanical Engineering input shaping–based control method with exact load gravity compensation to a three-dimensional (3D) crane. However, the aforementioned control methods for varying-cable-length cranes require either linearization or approximation. Motivated by the fact, Sun et al.43 present a new tracking scheme34 and an adaptive control method for cranes with lowering and horizontal transportation control. In practice, an overhead crane is subject to various continuous disturbances including wind forces and frictions. From a control theoretical viewpoint, fuzzy logic can provide the most general design method which can suppress the external disturbance and cope with system parameter uncertainties in various types of control systems. Therefore, an energy-based fuzzy control method is proposed in this article, achieving accurate load transferring, lowering, and complete disturbances compensating while suppressing and eliminating load swing angle rapidly. Specifically, by introducing some coordinate transformations, the model for underactuated overhead cranes with load transferring, lowering, and persistent external disturbances is first established. Then, we construct a fuzzy disturbance observer to estimate the persistent external disturbances. And then, coupling behavior between the trolley movement and load swing is enhanced by introducing a composite signal, based on which an energy-based fuzzy controller is designed. Next, the origin of the closed-loop system is proven to be asymptotically stable using Lyapunov techniques and LaSalle’s invariance theorem.44 Finally, numerical simulation results are used to verify the control performance of the proposed control method. The main contributions of the proposed control method are summarized as follows: 1.
2.
3.
4.
The proposed control method requires no linearization or approximation operations to the original nonlinear dynamics. The persistent external disturbances are observed precisely and compensated completely, which have significant theoretical importance to analyze the robustness of varying-cable-length cranes. To the best of our knowledge, the mathematical model of overhead cranes with load transferring, lowering, and persistent external disturbances is first built. The new model can guarantee that the process of proving the system stability under the effect of external disturbances becomes more easier. As will be seen from numerical simulation results, the transient performance of the designed controller is improved.
The remaining parts are as follows. In section ‘‘Modeling for underactuated overhead cranes with load transferring, lowering, and persistent external
Zhang et al.
3 seen from Figure 1, the load position in x0 y0 coordinate can be obtained as
x0 m = x0 + l sin u0 y0 m = l cos u0
ð1Þ
where x0 = x cos u0 and y0 = 0 denote the trolley position in x0 y0 coordinate and x0m and y0m represent the load position in x0 y0 coordinate. Using Lagrange’s method, the dynamic equations of underactuated overhead cranes with load transferring, lowering, and persistent external disturbances are provided as follows M + mp cos2 u0 0 €x + mp€l sin u0 + 2mp l_u_ 0 cos u0 cos2 u0 2 + mp lu€0 cos u0 mp l u_ 0 sin u0
Figure 1. Schematic illustration of an overhead crane with load transferring, lowering, and persistent external disturbances.
disturbances,’’ we construct a new model for overhead cranes with load transferring, lowering, and persistent external disturbances in the transformed coordinates. In section ‘‘Main results,’’ the fuzzy disturbance observer and the controller development process, the stability, and convergence analysis are presented. Section ‘‘Numerical simulation results and analysis’’ exhibits numerical simulation results for the proposed energybased fuzzy control method and comparative methods. Section ‘‘Conclusion and future work’’ summarizes the main work of this article.
Modeling for underactuated overhead cranes with load transferring, lowering, and persistent external disturbances Most existing models for overhead cranes take a geodetic coordinate system as the reference coordinate system, for which it is usually difficult to prove the stability of the system when persistent external disturbances exist.45 In this article, we build a mathematical model of overhead cranes with load transferring, lowering, and persistent external disturbances. As can be seen from Figure 1, due to the action of persistent external disturbance d, the load will not be vertically stabilized, but be finally regulated with the cable forming an angle u0 with the vertical direction. To facilitate the controller design and stability analysis, we select x0 y0 coordinate to describe the overhead crane dynamics instead of the standard x y coordinate. In Figure 1, O represents the origin of the x y and x0 y0 coordinates, M and mp represent the trolley mass and the load mass, respectively, l and g are the cable length and the gravitational acceleration, respectively, Fx and Fl denote the control inputs imposed on the trolley and the load, respectively, d represents the persistent external disturbance to the load, and x0 and u0 denote the trolley displacement and the load swing angle in x0 y0 coordinate. As can be
ð2Þ
= ðFx Dx x_ d Þ cos u0 + Mg sin u0 2 mp€l + mp€x0 sin u0 mp l u_ 0 = Fl Dl l_ + mp g cos u0 + d sin u0 cos u0
ð3Þ
2mp ll_u_ 0 + mp l2 €u0 mp€x0 l cos u0 = mp g cos u0 + d sin u0 l sin u0
ð4Þ
To facilitate the subsequent controller development, we rewrite equations (2)–(4) as follows Mðq0 Þq€0 + C q0 , q_0 q_0 + Gðq0 Þ = U
ð5Þ
where q0 2 R3 represents the system state vector, M(q0 ), C(q0 , q_0 ) 2 R3 , G(q0 ), and U 2 R3 are the inertia matrix, the centripetal–Coriolis matrix, the gravity vector, and the control input vector, respectively, which are explicitly defined as 0
M + mp cos2 u0 B cos2 u0 Mðq0 Þ = @ mp sin u0 mp l cos u0
0
0 B C q0 , q_0 = @ 0
0
mp u_ 0 cos u0 0 mp lu_ 0
mp sin u0 mp 0
1 mp l cos u0 C A 0 2 mp l
1 0 mp l_ cos u0 mp lu_ sin u0 C A mp lu_ 0 mp ll_
0
1 u0 ðd + Dx x_ Þ cos u0 Mg sin Gðq0 Þ = @ Dl l_ mp g cos u0 + d sinu0 cos u0 A, mp g cos u0 + d sin u0 l sin u0 0 01 0 1 Fx cos u0 x A, q0 = @ l A U=@ Fl 0 u0 The following assumptions are made for simplification of the model.
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Assumption 1. Subject to the physical constraints, the load swing angle u0 and the swing angle u0 induced by the persistent external disturbance d are kept within the following intervals
Assumption 2. The load can be modeled as a point mass, while the stiffness of the cable can be neglected. The mass and elasticity of the cable have negligible impact on the system dynamics. Assumption 3. The hook mass is very small, which can be ignored. Assumption 4. The wind has little effects on the cable.
Main results Fuzzy disturbance observer In this section, we design a fuzzy disturbance observer to estimate the persistent external disturbance d by extending the results in the work by Kim.46 Then, the load swing angle u0 can be derived according to the relationship between the persistent external disturbance d and the gravity of the load as follows (see Figure 1) ð6Þ
The basic configuration of a fuzzy logic system consists of a fuzzifier, some fuzzy IF-THEN rules, a fuzzy inference engine, and a defuzzifier. The fuzzy inference engine uses the fuzzy IF-THEN rules to perform a mapping from an input linguistic vector x = (x1 , x2 , . . . , xn )T 2 Rn to an output y 2 R. The ith fuzzy rule is written as Ri : If x1 = Ai1 , . . . , xn = Ain , then y = yi
ð7Þ
with Ai1 , . . . , Ain denoting the fuzzy variables and yi a singleton number. Using product inference, center-average, and singleton fuzzifier, the output of the fuzzy system can be expressed as r P
yðxÞ =
yi
i=1
n Q j=1
r P
n Q
i=1
j=1
mAij xj
mAij xj
!
^ T jðxÞ ! =f
n Q
ji =
p p \u0 \ , p\u0 \p 2 2
d u0 = arctan mp g
vector, and jT = (j1 , j2 , . . . , jr )T , with ji denoting the fuzzy basis function, which is explicitly expressed as
j=1
mAij xj
r P
n Q
i=1
j=1
mAij xj
!
ð9Þ
Then, the fuzzy system in equation (8) can be used to approximate the persistent external disturbance d as follows ^ =f ^ T jðxÞ d^ xjf
ð10Þ
where x = (x0 , x_ 0 )T . Consider the following observation dynamic system cos2 u0 ½ðFx Dx x_ Þ m_ = sm + g sin u0 cos2 u0 + M3 cos u0 ^ d + sx_ 0 ð11Þ cos u0 Fl Dl l_ sin u0 M with s.0 being observational parameter. Define the disturbance observation error as z = x_ 0 m
ð12Þ
It can be obtained from equations (2)–(4) that cos2 u0 €x0 = g sin u0 cos2 u0 + M h i cos3 u 0 0 d ð13Þ ðFx Dx x_ Þ cos u0 Fl Dl l_ sin u M It follows from equations (11)–(13) that cos3 u0 ^ z_ + sz = d d^ xjf M
ð14Þ
Let x belong to a compact set Mx . Assuming that there exists an optimal parameter matrix f defined by f = arg
^ min sup d d^ xjf
^ f2M f
ð15Þ
x2Mx
in a convex region Mf , which is defined as
Mf = fj kfk mf
ð16Þ
where mf is a design parameter, and the persistent external disturbance can be exactly written as ð8Þ
where mAij (xj ) denotes the membership function value of the fuzzy variable Aij , r is the number of fuzzy rules, ^ T = (y1 , . . . , yr ) represents the adjustable parameter f
d = d^ðxjf Þ + eðxÞ
ð17Þ
where e(x) denotes a reconstruction error vector satisfying je(x)j e for a positive constant vector e. e(x) will decrease along with an increase in the number of fuzzy rules. Defining the parameter error vector as
Zhang et al.
5 ~ = f f ^ f
ð18Þ
From equations (15), (17), and (18), the dynamics of the disturbance observation error is expressed as 3
3
cos u0 ~ T cos u0 f j ðx Þ eðx Þ z_ + sz = M M
ð19Þ
The Lyapunov function candidate is constructed as V=
1 2 1 cos3 u0 ~ T ~ f f z + 2 2g M
ð20Þ
with g.0 being a control gain. Taking the time derivative of equation (20), inserting for the results of equation (19), yields
cos3 u0 ~ T cos3 u0 f j ðx Þ eðxÞ sz V_ = z M M 3 1 cos u0 ~ T ~_ f f + g M cos3 u0 ~ T cos3 u0 = sz2 f zjðxÞ zeðxÞ M M 1 cos3 u0 ~ T ~_ + f f g M
cos3 u0 ~ T 1 ~_ cos3 u0 2 = sz + f zjðxÞ + f zeðxÞ M M g ð21Þ A fuzzy rule tuning method is selected as ~_ = gzjðxÞ f
ð22Þ
or, equivalently, by definition ^_ = gzjðxÞ f
ð23Þ
one has
Energy-based controller development The energy of the overhead crane system consists of the kinetic and potential energies as E ðt Þ =
1 _0 T q M ðq0 Þq_0 + mp g cos u0 + d sin u0 lð1 cos u0 Þ 2 ð26Þ
whose derivative with respect to time can be calculated as _ ðq0 Þq_0 + M ðq0 Þq€0 _EðtÞ = q_0 T 1 M 2 + mp g cos u0 + d sin u0 lu_ 0 sin u0 + mp g cos u0 + d sin u0 l_ð1 cos u0 Þ T = q_0 M ðq0 Þq€0 + C q0 , q_0 q_0 + mp g cos u0 + d sin u0 lu_ 0 sin u0 + mp g cos u0 + d sin u0 l_ð1 cos u0 Þ T = q_0 ðU Gðq0 ÞÞ + mp g cos u0 + d sin u0 lu_ 0 sin u0 = x_ 0 ½ðFx Dx x_ Þ cos u0 ðd cos u0 Mg sin u0 Þ + l_ Fl Dl l_ + mp g cos u0 + d sin u0 ð27Þ
3
cos u0 V_ = sz2 zeðxÞ M 3 cos u0 s 2 cos6 u0 2 2 e ðxÞ = sz zeðxÞ + z + M 2sM 2 2 s 2 cos6 u0 2 e ðxÞ z + 2sM 2 2 "rffiffiffiffi #2 rffiffiffiffiffiffi s 2 cos6 u0 2 s cos3 u0 1 e ðxÞ = z + z+ eðxÞ 2sM 2 M 2 2 2s
^ is From equation (23), under the assumption that f bounded, the disturbance observation error is uniformly bounded, that is, z 2 L‘ within a region of which size can be kept arbitrarily small. From equations (11)–(14), one can see that z ! 0 (i.e. m ! x_ 0 ) ^ f) ^ approaches the implies that the fuzzy system d(xj actual but unknown disturbance d (i.e. monitor the disturbance well). The persistent external disturbance d is approximately equal to the persistent predictive exter^ Therefore, we assume d = d^ in this nal disturbance d. article.
s 2 cos6 u0 2 e ðxÞ z + 2sM 2 2
ð24Þ Thus, V_ \0 for jzj.
e cos3 u0 sM
ð25Þ
which indicates that the overhead crane system, with Fx and Fl as the inputs and x_ 0 and l_ as the outputs, is typically passive and dissipates.44 The main objective of automatic crane systems is fast and precise positioning and prompt payload swing mitigation. The only way for achieving the aforementioned objective is to fully utilize the dynamic coupling characteristics between the trolley position x0 and the payload swing u0 .28 Different from the control problem of fully actuated systems where decoupling is usually utilized, the coupling behavior should be increased to improve the control performance of underactuated overhead crane systems. Due to the underactuated nature of overhead crane systems, no terms related to the load swing motion (u0 or u_ 0 ). To solve this problem and improve the control performance of the varying-cable-length cranes, a composite signal x is introduced as
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Advances in Mechanical Engineering x = x_ 0 + af ðu0 Þ
where f (u0 ) is a yet-to-construct scalar function and a.0 is a positive control gain. Without loss of generality, the initial position, velocity of the trolley, the initial load swing angle, and angular velocity are considered as zero, that is, x0 (0) = x_ 0 (0) = u0 (0) = u_ 0 (0) = 0. It is straightforward to obtain from equation (28) that x_ = €x0 + au_ 0 f 0 ðu0 Þ
ð29Þ
ðt
0
xdt pdx0 = x pdx0 + a f ðu0 Þdt
0
0
ð30Þ
ðt
ð36Þ
Based on the structure of equation (34), we design the energy-based control laws as follows M + mp cos2 u0 _ 0 u Fx = Dx x_ + d Mg tan u0 + a cos3 u0 0t 1 ð cos u0 kdx x kpx @ xdt pdx0 A
ð37Þ
0
Fl = Dl l_ mp g cos u0 + d sin u0 + amp u_ 0 sin u0 cos u0 kdl l_ kpl el ð38Þ
and ðt
f 0 ðu0 Þ = cos u0 ) f ðu0 Þ = sin u0
ð28Þ
= ex0 + a f ðu0 Þdt
with kpx , kdx , kpl , kdl 2 R+ being positive control gains, el = l pdl representing the error of cable length, and pdl denoting the target cable length.
0
where ex0 denotes the trolley positioning error and pdx0 represents the desired position of the trolley in x0 y0 coordinate. Accordingly, the new state vector K can be expressed as T K = x l_ u_ 0 = x_ 0 + af ðu0 Þ
l_
u_ 0
T
ð31Þ
From equation (5), it is calculated that
0
T lim x0 l u0 x_ 0 l_ u_ 0 = ½pdx0 pdl 0 0 0 0T
t!‘
T lim x l u0 x_ l_ u_ 0 = ½pdx pdl 0 0 0 0T
t!‘
ð32Þ
Inspired by the form of equation (26), a new energylike function is constructed as 0
Theorem 1. The proposed energy-based control laws (37) and (38) guarantee that the system state converges to the equilibrium point in the sense that ð39Þ
or, equivalently
M ðq0 ÞK_ 0 + C q0 , q_0 K0 = U Gðq0 Þ + 2 M + mp cos2 u0 3 a cos2 u0 u_ 0 f 0 ðu0 Þ 6 7 4 amp u_ 0 sin u0 f 0 ðu0 Þ 5 a mp lu_ 0 cos u0 f 0 ðu0 Þ
T
Stability analysis
0
Et ðtÞ = K ½M ðq ÞK + mp g cos u0 +d sin u0 lð1cos u Þ ð33Þ Taking the time derivative of equation (33), inserting the results of equation (32), it is obtained that
ð40Þ
where pdx denotes the desired position of the trolley in x y coordinate, or, equivalently T lim xm l u0 x_ m l_ u_ 0 = ½pdxm pdl 0 0 0 0T
t!‘
ð41Þ with xm and pdxm representing the displacement and desired position of the load in x y coordinate. As can be seen from Figure 1, the following relationships are satisfied for pdx0 , pdx , and pdxm
E_ t ðtÞ = KT M ðq0 ÞK_ 0 + C q0 , q_0 K0 + mp g cos u0 + d sin u0 l_ð1 cos u0 Þ + mp g cos u0 + d sin u0 lu_ 0 sin u0
M + mp cos2 u0 _ 0 0 0 u f ðu Þ = x ðFx Dx x_ d Þ cos u0 + Mg sin u0 + a cos2 u0 2 + l_ Fl Dl l_ + mp g cos u0 + d sin u0 + amp u_ 0 sin u0 f 0 ðu0 Þ + a mp l u_ 0 cos u0 f 0 ðu0 Þ
ð34Þ
To guarantee the last term of E_ t (t) non-positive, the following condition should be satisfied cos u0 f 0 (u0 ) 0 In this article, we choose f (u0 ) as
ð35Þ
pdx0 = pdx cos u0
ð42Þ
pdxm = pdx + l sin u0
ð43Þ
Zhang et al.
7
Proof. To prove Theorem 1, we define the Lyapunov function candidate as 0t 12 ð 1 1 V ðtÞ = Et ðtÞ + cos u0 @ xdt pdx0 A + e2l 2 2
ð44Þ
0
Taking the time derivative of equation (44), along with equations (34), (37), and (38), the following results are derived 2 V_ ðtÞ = kdx cos u0 x 2 kdl l_2 a mp l u_ 0 cos2 u0 0 ð45Þ which clearly indicate that the closed-loop system is Lyapunov stable at the equilibrium point.44 It follows from equations (44) and (45) and (37) and (38) that
V ðtÞ 2 L‘ ) x,
ðt
_ u_ 0 , x_ 0 , Fx , Fy 2 L‘ xdt, el , l, l,
ð46Þ
0
In order to facilitate the stability analysis, let S=
_ u_ 0 V_ ðtÞ = 0 x0 , l, u0 , x_ 0 , l,
ð47Þ
and then define P as the largest invariant set contained in S. Thus, the following conclusions hold in P x = 0, l_ = 0, u_ 0 = 0
ð48Þ
From equation (48), the following results can be obtained x_ = 0, €l = 0, €u0 = 0
0
ð50Þ ð51Þ ð52Þ
sin u = 0
Numerical simulation results and analysis In this section, some simulation tests are performed to validate the performance of the proposed fuzzy disturbance observer and energy-based control method. The overall simulation process is divided into three groups. More precisely, the robustness with respect to various external disturbances is tested in the first group. In the second group, the tolerance of the system to different load masses, different target load positions, and different target cable lengths is further tested. Finally, in the third group, we compare the proposed control method with the partial feedback linearization control law41 and the nonlinear tracking control law.34 For the simulation study, the overhead crane system parameters are set as M = 6:157 kg, mp = 1 kg, g = 9:8 m=s2 The initial position, velocity of the trolley, the initial length, velocity of the cable, and the initial load swing angle and angular velocity are set as x0 ð0Þ = x_ 0 ð0Þ = l_ð0Þ = u0 ð0Þ = u_ 0 ð0Þ = 0, lð0Þ = 0:3 m The target load position and cable length are set as
ð49Þ
It follows from equations (48) and (49) and (2) and (4) that Fx = Dx x_ + d Mg tan u0 Fl = mp g cos u0 + d sin u0 cos u0
Thus, we can conclude that the largest invariant set P contains only the equilibrium point ½x0 l u0 x_ 0 l_ u_ 0 T = ½pdx0 pdl 0 0 0 0T , or ½x l u0 x_ l_ u_ 0 T = ½pdx pdl 0 0 0 0T , or equivalently ½xm l u0 x_ m l_ u_ 0 T = ½pdxm pdl 0 0 0 0T . By employing LaSalle’s invariance theorem,44 the conclusions of equations (47)–(49) are proven.
pdxm = 0:6m, pdl = 0:8m By trial and error, the observational parameter and control gains for the proposed controller are tuned as follows s = 10, g = 50, kpx = 2, kdx = 6:5, kpl = 1:2, kdl = 2 The following membership functions are selected for the premise parts of the fuzzy disturbance observer
Based on Assumption 1 and equation (52), it is derived that
mA1j xj =
u0 = 0
mA2j xj = mA3j xj = mA4j xj = mA5j xj = mA6j xj =
1 1 + exp 5 xj + 0:6 2 exp xj + 0:4 2 exp xj + 0:2 exp x2j 2 exp xj 0:2 2 exp xj 0:4
mA7j xj =
1 1 + exp 5 xj 0:6
ð53Þ
which, together with equations (37) and (50), indicates that ðt
xdt pdx0 = 0 ) ex0 = 0 ) x0 = pdx0
ð54Þ
0
By gathering equations (38) and (51), it is clear that el = 0 ) l = pdl
ð55Þ
It follows from equations (49) and (53) that x_ 0 = 0
ð56Þ
where j = 1, 2, x1 = x0 , and x2 = x_ 0 .
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Advances in Mechanical Engineering
Figure 2. Time history of persistent external disturbance d.
The numerical simulation is implemented in the environment of MATLAB/Simulink.
Simulation group 1: external disturbance robustness verification In this group, the performance of the proposed control method in the presence of various external disturbances is tested. To this end, the persistent external disturbance d is given as shown in Figure 2. The simulation results are provided in Figure 3. As can be seen from Figure 3(a), the persistent external predictive disturbance d^ reaches its desired value after a few seconds. It means that the fuzzy disturbance observer observes the persistent external disturbance well, as expected. It is evident that even in the presence of various persistent external disturbances, the proposed control method can still guarantee satisfactory control performance, indicating strong robustness against different persistent external disturbances.
Simulation group 2: internal disturbance robustness verification To verify the robustness of the proposed control method with respect to different load masses and its effectiveness for different target load positions and different target cable lengths, we examine the following three extreme cases: Case 1. The external disturbance d is 1 N, and the load mass is changed from 1 to 5 kg abruptly at t = 5 s. Case 2. The external disturbance d is 1 N, and the target load position is changed from 0.6 to 1.0 m abruptly at t = 8 s. Case 3. The external disturbance d is 1 N, and the target cable length is changed from 0.8 to 1.5 m abruptly at t = 6 s.
Figure 3. Results of simulation group 1: (a) observed persistent external disturbance and swing angle induced by persistent external disturbance, (b) trolley trajectory, cable trajectory, and load swing angle, and (c) control inputs imposed on the trolley and the load.
The obtained simulation results are shown in Figures 4–6. It is evident that under different situations, the trolley reaches the target position accurately, the cable reaches the target length rapidly, and the load swing is less than 1.1° and converges to zero soon after the trolley stops. We can see from Figures 4–6 that the control performance of the proposed controller is not influenced by the sudden changes in load mass, target load position, and target cable length. This fact directly
Zhang et al.
9
Figure 4. Results of simulation group 2. (Dotted-dashed line) No parameter variations. (Solid line) Case 1: (a) observed persistent external disturbance and swing angle induced by persistent external disturbance, (b) trolley trajectory, cable trajectory, and load swing angle, and (c) control inputs imposed on the trolley and the load.
Figure 5. Results of simulation group 2. (Dotted-dashed line) No parameter variations. (Solid line) Case 2: (a) observed persistent external disturbance and swing angle induced by persistent external disturbance, (b) trolley trajectory, cable trajectory, and load swing angle, and (c) control inputs imposed on the trolley and the load.
demonstrates the strong robustness of the closed-loop system with respect to different load masses, different target positions, and different cable lengths.
method.34 It should be pointed out that the partial feedback linearization control method and the nonlinear tracking control method are designed without considering external disturbances, whose control performance cannot be guaranteed when persistent external disturbances exist. Therefore, we set the persistent external disturbance d as 0 in this case. The expressions of these controllers are as follows:
Simulation group 3: comparative study In this case, to validate the superior performance of the proposed energy-based regulating control method, we compare it with the partial feedback linearization control method41 and the nonlinear tracking control
1. Partial feedback linearization control law41
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Figure 7. Simulation group 3: results of the proposed controller: (a) trolley trajectory, cable trajectory, and load swing angle and (b) control inputs imposed on the trolley and the load.
9 8 > > Kd11 mp sin u0 x_ + Dl Kd12 mp l_ > > > 0 2 > = < 0 _ m sin u ð x p Þ m l u K p11 p dx p m Fl = ð58Þ > pdl Þ + a1 Kd2 mp sin u0 u_ 0 > > > > > + Kp12 mp ðl ; : + a1 Kp2 mp u0 sin u0 mp g cos u0 Figure 6. Results of simulation group 2. (Dotted-dashed line) No parameter variations. (Solid line) Case 3: (a) observed persistent external disturbance and swing angle induced by persistent external disturbance, (b) trolley trajectory, cable trajectory, and load swing angle, and (c) Control inputs imposed on the trolley and the load.
Fx = 9 8 Dx Kd11 M + mp sin2 u0 x_ + Kd12 mp sin u0 l_ > > > > > > 2 0 > > > > = < + Kp11 M + mp sin u ðx pdxm Þ 0 2 0 0 _ Kp12 mp sin u ðl pdl Þ + mp l sin u u > > > > > > a1 Kd2 M + mp sin2 u0 u_ 0 > > > > : 2 0 0 0 0 ; + mp g sin u cos u M + mp sin u a1 Kp2 u ð57Þ
with Kd11 , Kd12 , Kp11 , Kp12 , Kp2 , Kd2 , and a1 representing positive control gains. The control gains for equations (57) and (58) are tuned as Kd11 = 10, Kd12 = 10, Kp11 = 5, Kp12 = 5, Kp2 = 1:8, Kd2 = 2, and a1 = 1. 2. Nonlinear tracking control law34 Fx = kpx ex + mp + M €xd + mp€ld sin u0 + mp l_d u_ 0 2lvx 12x cos u0 + Dx x_ ð59Þ 2 ex kdx e_ x 12x e2x Fl = kpl el + mp€xd sin u0 + mp€ld mp g + 2lvl 12l Dl l_ 2 el kdl e_ l 12l e2l
ð60Þ
where kpx , kdx , kpl , kdl , lvx , and lvl are positive control gains; 1x and 1l denote the maximum allowable tracking
Zhang et al.
Figure 8. Simulation group 3: results of the partial feedback linearization controller: (a) trolley trajectory, cable trajectory, and load swing angle and (b) control inputs imposed on the trolley and the load.
error bounds in directions x and l. After careful tuning, the control gains for equations (59) and (60) are chosen as kpx = 20, kdx = 10, kpl = 45, kdl = 10, lvx = 0:1, and lvl = 0:1. The simulation results are shown in Figures 7–9, and the quantified results are detailed in Table 1. The following seven performance indices are included in Table 1: 1. 2. 3. 4.
5. 6. 7.
Trolley final position pf ; Cable final length lf ; Maximum load swing amplitude u0max ; Load residual swing u0res , which is defined as the maximum load swing angle after the trolley stops; Consumed transportation time ts ; Maximum actuating force imposed on the trolley Fx max ; Maximum actuating force imposed on the load Fl max .
It is concluded from Figures 7–9 and Table 1 that the consumed transportation time of the proposed control method is 7.8 s, 8 s for the partial feedback
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Figure 9. Simulation group 3: results of the nonlinear tracking controller: (a) trolley trajectory, cable trajectory, and load swing angle and (b) control inputs imposed on the trolley and the load.
linearization controller, and 6 s for the nonlinear tracking control method, whereas the positioning and cable length errors are all less than 3 mm. The load swing angle of the proposed control method is within smaller scope (maximum amplitude: 0.99° and residual swing: 0.03°) than that of the partial feedback linearization control method (maximum amplitude: 5.7° and residual swing: 0.8°) and the nonlinear tracking control method (maximum amplitude: 3.6° and residual swing: 1.7°). Although the consumed transportation time of the proposed control method is 1.8 s longer than that of the nonlinear tracking control method, the load swing is much better suppressed and eliminated by the proposed method and there is almost no residual swing when the trolley reaches the desired position.
Conclusion and future work In this article, we build the model and propose an energy-based fuzzy control method for underactuated overhead cranes with load transferring, lowering, and persistent external disturbances. To estimate the persistent external disturbances, a fuzzy disturbance observer is constructed. And a composite signal is introduced to
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Table 1. Quantified results for simulation group 3. Controllers
pf (m)
lf (m)
u0max (8)
u0res (8)
ts (m)
Fxmax (N)
Flmax (N)
Proposed controller Partial feedback linearization controller Nonlinear tracking controller
0.6 0.597 0.601
0.8 0.798 0.8
0.99 5.7 3.6
0.03 0.8 1.7
7.8 8 6
3.1 18.46 3.6
9.8 9.8 9.8
improve the transient performance of the designed controller. Using Lyapunov techniques and LaSalle’s invariance theorem, the equilibrium point of the closedloop system is proven to be asymptotically stable. Numerical simulation results are used to verify the control performance of the proposed control method in terms of accurate trolley positioning and rapid swing elimination. In our future work, we will consider trajectory planning for overhead cranes with load transferring and lowering. Declaration of conflicting interests
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The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
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Funding
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The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was funded by the National High-tech Research and Development (863 Program) of China under Award No. 2015A A042307, Shandong Provincial Scientific and Technological Development Foundation, China, under Award No. 2014 GGX103038, Shandong Province Independent Innovation & Achievement Transformation Special Foundation, China, under Award No. 2015ZDXX0101E01, and the Fundamental Research Funds of Shandong University, China, under Award No. 2015JC027.
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