JST: Smart Systems and Devices
Volume 31, Issue 1, May 2021, 068-075
Fractional-Order Sliding Mode Control of Overhead Cranes
Pham Van Trieu, Pham Duc Toan, Hoang Manh Cuong*
Vietnam Maritime University, Haiphong, Vietnam
*Email: cuonghm@vimaru.edu.vn
Abstract
We constitute a control system for overhead crane with simultaneous motion of trolley and payload hoist to
destinations and suppression of payload swing. Controller core made by sliding mode control (SMC) assures
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the robustness. This control structure is inflexible since using fixed gains. For overcoming this weakness, we
integrate variable fractional-order derivative into SMC that leads to an adaptive system with adjustable
parameters. We use Mittag–Leffler stability, an enhanced version of Lyapunov theory, to analyze the
convergence of closed-loop system. Applying the controller to a practical crane shows the efficiency of
proposed control approach. The controller works well and keeps the output responses consistent despite the
large variation of crane parameters.
Keywords: Fractional-order control, overhead cranes, sliding mode control.
1. Introduction* performance. This study utilizes strong points of both
SMC and FD. We create a flexibly robust controller
Using frequently in industrial transportation, an
by integrating FD into SMC core. The controller
overhead crane is equipped with three mechanisms
tracks trolley and hoist payload to desired positions,
for lifting and transferring material and package in
concurrently keeps the payload swing small at
factories. The modern cranes speedy run or/and
transient state, and suppresses this swing at steady
combine the motions for increasing productivity. This
state. Such the combination applied for overhead
causes imprecise motions and large swing of payload
cranes has not been has not been released until
if cranes do not have suitable control strategies. Many
recently. Proposed control system shows the
control algorithms were proposed for overhead cranes
following advantages:
[1-18]. The articles [1-12] in connection to control of
overhead cranes were published from linear control (i) The controller well immunizes disturbances
[1], feedback linearization [2], pole-placement [3], and is robust with uncertainties due to the action of
linear quadratic optimal control [4], to complicated SMC structures.
nonlinear control [5], model predict control [6],
(ii) Due to using variable FD, control structure
adaptive control approach [7-8], robust control
may vary flexibility to adapt with uncertain
methods [9-10], and modern control techniques such
environment. Tuning FD orders may get the optimal
as fuzzy logic [11], neural networks [12]. Focusing
system responses.
on robust controls, SMC [13] is a control approach
which works effectively for cranes. SMC does not In fact, control formulation of cranes is
require much the modelling precision. It may work classified into low-level control (LLC) and high-level
well with un-modeled dynamics. Additionally, it planning (HLP). LLC deals with precise tracking
shows the robustness when the system faces trolley and crane to destinations while keeping
disturbances and uncertainties. Its advanced versions payload swing small and vanishing at steady-state.
such as terminal SMC [14], fast terminal SMC [15] HLP develops algorithms for motion planning and
make quick finite-time convergence. SMC can trajectories to prevent the obstacles. At modern crane,
combine with fuzzy [16], neural network [17], self- two control options may be combined.
tuning control [18] to achieve both robust and
adaptive features. Accordingly, adaptive mechanisms This study focuses on LLC. We organize the
article as follows. Section 2 provides concept of
are supplemented for estimating uncertainties,
fractional calculus [7] and stability theory for
unknown factors, and disturbances. On the upside,
fractional systems [8-10]. Section 3 describes
fractional calculus has been widely applied to control
engineering in the recent years [22]. Instead of using physical features of an overhead crane through its
fixed first/second-order derivatives for feedback dynamic model. Section 4 designs controller by
combining SMC with FD, analyzing crane stability
signals, fractional-derivative (FD) makes flexible
with Mittag–Leffler sense [9] is also included.
control structure in which it is tunable to get optimal
Section 5 tests the control algorithms on a practical
overhead crane, analyzes and discusses the
ISSN: 2734-9373 application results. Finally, several conclusions and
https://doi.org/10.51316/jst.150.ssad.2021.31.1.9 remarks are represented in Section 6.
Received: 12 January 2021; accepted: 11 March 2021
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JST: Smart Systems and Devices
Volume 31, Issue 1, May 2021, 068-075
Fig. 1. 2D motion of an overhead crane.
2. Fractional Derivative in Control Theorem 2 [22]: Fractional LTI system
We utilize the concept of fractional calculus and α
Dtt x()= Ax () t (5)
related topic to design control system. The following
definition, theorem, and lemma will be applied for with A∈Rn×n is stable if it justifies
analyzing and constituting the control algorithm at
the next sections: arg(eigA )> απ / 2 (6)
Definition 1 [19]: Fractional-order α derivative 3. Dynamical Description
of function f(t) with time defined by Caputo is given
We consider simultaneous motion of trolley mt
as
and payload hoist ml when operating an overhead
1 t crane as in Fig. 1. Three outputs composed of moving
αα= −τm−−1 ττ
Dt ft() ∫ (t ) g ( )d (1) trolley x, lifting (with varying cable length l) payload
Γ−()m α 0
mc, and swinging θ payload are controlled by two
∞ inputs: ut is trolley-pushing force and ul is payload
+ −−tz1
with 0<α<1, m∈Z and Γ=()z∫ et d t being lifting force. bt and br are parameters for frictions at
0 trolley and cable.
gamma function.
Dynamic behavior of cranes is governed by
Lemma 1 [20]: Let x(t)∈Rn be a state vector. actuated model:
The inequality
(mtc+− m ) x m c sinθ l − ml c cos θθ
ααTT≤ (7)
0.5DDtt (xQx ) xQ x (2) 2
+−bxtc 2 m cosθ l θ + ml c sin θθ = u t ( t )
∈ n×n
is held for every positive definite matrix Q R .
−mcsinθ x ++ ( m cl m ) l + bl r
Theorem 1 [21]: If existing a continuously (8)
−−mlθθ2 mgcos = u ( t )
differentiable function V(x,t) satisfying cc l
T
a ab corresponding qa=[x l] , and swinging equation
αα12xx≤≤Vt( , ) x (3)
θ =(1/l )(cos θθ xl −− 2 g sin θ ) (9)
and
describes payload swing θ. Substituting Eq. (9) into
α ≤−α ab
DVt (xx, t) 3 , (4) Eqs. (7), (8) and rearranging leads to a reduced-order
dynamics
then x=0 is global stability in the sense of Mittag-
α α α
Leffler. Here, a, b, 1, 2, and 3 are positive Mqaa++ Bq f(,) q q = U (10)
constants.
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Volume 31, Issue 1, May 2021, 068-075
T equilibrium = is globally stable. The convergence
where, q=[x l θ] are system outputs. M = []mij 22× is s 0
of manifold (11) produces a linear fractional-order
a symmetric matrix with
2 system
m11 = mmtc + sinθ , mm12= 21 = − mc sinθ , and
Dα ()()qq−=−−−λq q βθ (18)
m22 = mmcl + . B=diag(bt,br) denotes damping. t a ad a ad
22T
fqq( , )= [(mlc sinθθ + 0.5 mgc sin 2θ ) −+ ( mlcc θ mgcos θ )] , Physically, θ is always toward 0 due to payload
and U = []uuT is outputs. weight even without control, and gain β supports the
tl fast convergence of θ. Eq. (18) is reduced as
4. Controller Design α
Dt()()qq a−=−− ad λqa q ad (19)
We design a structure for tracking qa to
T
destination qad=[x l] while suppressing payload Applying Theorem 2 to system (19) indicates
swing (θ reaches zero). We consider a sliding that (qa−qad) is locally stabile around 0 for every
manifold containing fractional derivative positive definite matrix λ. Thus, tracking qa to qad is
α achieved.
s=Dt()() qq a −+ ad λqa −+ q ad βθ (11)
Remark 1: Sign function of control law (12) may
where, 0≤α≤1 is fractional-order (FO) of Caputo cause the chattering at system responses. There are
derivative (Definition 1), λ=diag(λ1,λ2) is a positive several ways to reduce this, such as: replacing sign
T
matrix, and β=[β1 0] . A control law is proposed action by situation function or sigmoid function,
compatible with surface (11) by a following using a filter or estimator, higher-order solution,
statement. super-twisting method, and so on. In this article, we
utilize a hyperbolic tangent function
Statement: A structure of fractional sliding mode
−
control eessii−
= =
tanhs [tanhsii ] ss− (20)
(2−α ) ii+
UM=−D {(λq −+ q ) βθ } eei
t a ad (12)
++ −
Bqa f( q , q )ηs sgn replacing for sgns function.
tracks outputs q governed by crane dynamics (7)-(9) Remark 2: Theoretically, sliding surface (11) of
T
to destinations qd=[xd ld 0] asymptotically. Here, controller (12) only assures the infinity convergence.
η=diag(η1,η2) are two positive gains. In fact, the control law (12) is designed based on the
infinity stability of the dynamics
Proof: We begin with a bounded Lyapunov
function Dα s +=ηsgn s0 (21)
T
V =0.5s Ms > 0 (13) An enhanced version [10, 14, 15] of SMC, that
is the so-called Terminal SMC (TSMC), guarantees
Since m=+>( mm sin2 θ ) 0 and
11 tc the finite-time stability of system outputs. By
2
detM = [mt ( m c ++ m l ) mm cl sinθ ] > 0, M is improving the dynamics of sliding surface as
positive definite. Based on Lemma 1, fractional α qp/
D ss++λ ηsgn s0 = (22)
derivative of Lyapunov (13) satisfies
ααT with η=diag(η1,η2) being positive diagonal matrix, q
DVtt≤ sM D s
(14) and p being positive odd integers satisfying q>p, we
T 2(αα− 1) α
=sM[DDt q a +λ t ()] qq a −+ ad β D t θ can obtain the TSMC controller in which the terminal
stability of sliding manifold is held.
Submitting equivalent dynamics (10) and
controller (12) into Eq. (14) yields 5. Results and Analysis
DVαα≤=−sMTT D s sηs D2( α− 1) sgn (15) We check the quality of proposed controller (12)
ttt on crane dynamics (7)-(9) using a laboratory
α −
2( 1) overhead crane whose parameters: mt= 5 kg, mc= 0.85
For α≠1, Caputo derivative Dt sgns = 0
leads inequality (15) to kg, ml= 2 kg, bt= 20 Ns/m, br=Ns/m, and FO-SMC
gains: λ=diag(0.7,0.9), β=[1.2 0]T, η=diag(100,100).
α T
DVt ≤ 0 (16) The initial conditions: q(0)=[0 0.1 0] and q0(0) = .
We use three cases of fractional order α = 0.8, 0.9,
The case α=1 leads inequality (15) to
and 1. The references of trolley and payload hoisting
α
respectively compose of destinations: xd=[0 0.3 0.1
DVt ≤−ηη11 s − 2 s 2 (17)
0.7] m and ld=[0.1 0.4 0.6 0.2] m. The crane
The expressions (13), (16), and (17) fit performances are depicted in Figs. 2-6.
conditions of Theorem 1. This implies that
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1
0.8
0.6
0.4
FO=0.8
FO=0.9
Displacement (m) FO=1
0.2
0
0 5 10 15 20 25
Time (s)
Fig. 2. Trolley motion.
0.7
FO=0.8
FO=0.9
0.6
FO=1
0.5
0.4
0.3
Cable Length (m)
0.2
0.1
0 5 10 15 20 25
Time (s)
Fig. 3. Payload hoist.
50
FO=0.8
40 FO=0.9
FO=1
30
20
10
Angle (degree) 0
-10
-20
0 5 10 15 20 25
Time (s)
Fig. 4. Payload swing.
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Volume 31, Issue 1, May 2021, 068-075
10000
FO=0.8
8000 FO=0.9
FO=1
6000
4000
2000
Force (N)
0
-2000
-4000
0 5 10 15 20 25
Time (s)
Fig. 5. Trolley-moving force.
3000
FO=0.8
2000 FO=0.9
FO=1
1000
0
-1000
Force (N)
-2000
-3000
-4000
0 5 10 15 20 25
Time (s)
Fig. 6. Payload-hoisting force.
Proper selection of FO decides the control As a nature of SMC, the proposed controller
quality. Seen at Fig. 2, FO=0.8 causes much assures the system robustness despite disturbance and
oscillation of trolley motion, both FO=0.9 and 1 uncertain environment. Considering the case FO=1,
assure the destination convergence, FO=0.9 makes we investigate the consistence of output responses
overshoot while FO=1 does not. Varying FO from its when a crane faces parametric uncertainties. In fact,
origin FO=1 can reduce settling time but causes the many crane parameters are variable and thus
overshoots and even steady-state errors. Payload adjustable. A crane can lift and transfer the payload
hoisting (Fig. 3) seems well for cases, however, with various mass mc and volume. The frictions
convergence speed of FO=0.8 is fastest. Payload characterized by bt and br are varied up to
swings are in small boundary (Fig. 4) at transient environment, temperature, and humidity of operation
phase and absolutely suppressed at payload area. We consider the variation of three above-
destinations. Depicted in Figs. 5 and 6, control inputs mentioned parameters with two following cases:
remain keen peaks due to high switched gains of
∆ ∆ ∆ −
controller. Generally, it is hard to say which FO is the Case 1: [ mc bt br]=[100 20 10]%.
best. Finding FO to achieve the optimal responses Case 2: [∆mc ∆bt ∆br]=[−50 −40 30]%.
will be studied in the next article.
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0.7
Origin
0.6 Uncertainties-Case 1
Uncertainties-Case 1
0.5
0.4
0.3
0.2
Displacement (m)
0.1
0
0 5 10 15 20 25
Time (s)
Fig. 7. Robustness of trolley motion.
0.8
0.7
0.6
0.5
0.4
Origin
0.3
Cable Length (m) Uncertainties-Case 1
Uncertainties-Case 2
0.2
0.1
0 5 10 15 20 25
Time (s)
Fig. 8. Robustness of payload-hoist.
10
Origin
8
Uncertainties-Case 1
Uncertainties-Case 2
6
4
2
0
Angle (degree)
-2
-4
-6
0 5 10 15 20 25
Time (s)
Fig. 9. Robustness of payload swing.
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Volume 31, Issue 1, May 2021, 068-075
2500
2000
Origin
Uncertainties-Case 1
1500
Uncertainties-Case 2
1000
500
Force (N)
0
-500
-1000
0 5 10 15 20 25
Time (s)
Fig. 10. Robustness of trolley-pushing force.
1000
Origin
Uncertainties-Case 1
Uncertainties-Case 2
500
0
Force (N)
-500
-1000
0 5 10 15 20 25
Time (s)
Fig. 11. Robustness of payload-hoisting force.
The simulation results when the system is FO that are considered as flexible control gains.
subject to two cases of parametric uncertainties in Trolley and payload lifting responses reach
comparison with original case are depicted in Fig.7 to destinations precisely while well vanishing payload
11. Despite parametric variations, trolley motion swing. Enhancing for 3D motion and integrating
(Fig. 7) and payload swing (Fig. 9) still approach adaptive control approaches will be conducted in the
destination precisely. Parametric uncertainties only future study.
impact on outputs at transient states in which it
Acknowledgments
causes small derivation. Hoisting the payload (Fig. 8)
seems sensitive with the variation of parameters. It This research is funded by Vietnam National
induces much not only transient-state derivations but Foundation for Science and Technology
also steady-state errors. Increasing swished gains Development under grant number 107.01-2019.301.
η=diag(η1,η2) will improve robust feature however
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