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Available online at www.sciencedirect.com
Journal of the Franklin Institute xxx (xxxx) xxx
www.elsevier.com/locate/jfranklin
Nonlinear modeling and robust LMI fuzzy control of
overhead crane systems
Charles Aguiar
a
, Daniel Leite
a , ∗, Daniel Pereira
a
, Goran Andonovski
b
,
Igor Škrjanc
b
a
Department of Automatics, Fed er al University of Lavras, Brazil
b
Faculty of Electrical Engineering, University of Ljubljana, Slovenia
Received 23 April 2019; received in revised form 3 July 2020; accepted 2 December 2020
Available online xxx
Abstract
Overhead cranes are widely used structures for lifting and conveying heavy loads. The development
of feedback control systems for such equipment is important due to the large number of potential ap-
plications and advantages over manual operation concerning stability and robustness. This paper aims
to represent the key nonlinear dynamics of crane systems by means of a state-space fuzzy model with
compact rule-base structure. The fuzzy model is useful to assist the design of a fuzzy controller based
on the concept of parallel compensation. A well-posed conservative linear-matrix-inequality (LMI) fea-
sibility problem is formulated so that a solution guarantees closed-loop Lyapunov stability, bounded
control inputs, quick positioning of the supporting cart, and suppression of load oscillations and colli-
sions. The fuzzy controller is composed by rules with linear control laws derived from local state-space
models. The controller warrants asymptotic convergence of the states. Due to the nonlinear nature of
the fuzzy model and controller, Jacobian linearization is avoided. The proposed fuzzy control approach
for cranes has shown to be more effective and robust than an optimal quadratic controller, and able to
move cargo smoothly and safely to a destination. Particularly, constrained and smoother control inputs
avoid actuator saturation, and tend to increase its lifetime. Laboratory experiments using the LMI fuzzy
controller and actual data validates the approach for cranes in actual scenario.
© 2020 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
∗Corresponding author.
E-mail addresses: charlescaguiar@hotmail.com (C. Aguiar), daniel.leite@ufla.br (D. Leite),
danielpereira@deg.ufla.br (D. Pereira), goran.andonovski@fe.uni-lj.si (G. Andonovski), igor.skrjanc@fe.uni-lj.si
(I. Škrjanc).
https://doi.org/10.1016/j.jfranklin.2020.12.003
0016-0032/© 2020 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
Please cite this article as: C. Aguiar, D. Leite, D. Pereira et al., Nonlinear modeling and robust LMI fuzzy control
of overhead crane systems, Journal of the Franklin Institute, https:// doi.org/ 10.1016/ j.jfranklin.2020.12.003
C. Aguiar, D. Leite, D. Pereira et al. Journal of the Franklin Institute xxx (xxxx) xxx
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1. Introduction
1.1. Contextualization
Overhead and gantry cranes are widely used in industry including the automotive, port, air-
port and civil engineering sectors, to mention a few. This equipment plays a crucial role in the
lifting and transportation of heavy loads such as storage containers, construction blocks, train
wagons, turbines, liquid tanks, among others. Cranes work under the physical principle of
common electrical machines to promote the elevation of loads. They were firstly designed by
the Greek ancient society, where ropes were attached to pulleys so that workers had to apply
smaller forces to raise building blocks. Since then, cranes have evolved into enormous struc-
tures, capable of raising loads of thousands of tons and that should ideally guarantee certain
positioning accuracy, transportation speed and robustness to perturbations and uncertainties
[25,37,47] .
Overhead cranes are characterized by a mobile platform –a cart or trolley – that moves
over elevated rails. A steel cable is usually attached to the inferior side of the platform and is
used to suspend the load. Such cables are equipped with a hook in the lowest end so that the
load can be attached and conveyed by the platform. Fig. 1 shows a typical overhead crane
used in industry.
A challenge related to the operation of cranes concerns the smoothness in which the load
has to be moved toward a reference point [13,22,49] . The starting and stopping movements as
well as disturbances along the path should not cause excessive oscillations for safety reasons
and due to the potential economic losses. Naturally, we are merely allowed to control the
trolley motion while the cargo swing angle is not directly accessible. Hence, an effective
and robust feedback control system is highly recommended to make the platform reaching
a reference point or tracking a specific trajectory while at the same time keeping the load
swing angle close to the vertical axis.
Control design methods for cranes are very often supported by nonlinear or linearized
single-pendulum models assuming sufficiently small swing angles. Nonetheless, several fac-
tors may cause the swing amplitudes to exceed the small angle assumption, namely, (i)
the underactuated nature of crane systems makes its dynamics not fully input-to-output
Fig. 1. Typical overhead crane structure.
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linearizable; (ii) cranes sometimes work in harsh conditions [37] , e.g., subject to strong air
currents or even fixed on vessels, where many external disturbances such as sea waves, ves-
sel movements and winds are present; (iii) during load conveyance, unexpected collisions
may happen; and (iv) unmodeled uncertainties that are difficult to be described with accurate
mathematical expressions, such as those related to friction, are always present. In particular,
cranes are constantly subject to aging, wearing, operation mode variation, faults, seasonality.
Control system performance may degrade because the uncertainty tolerance range tends to be
small [11,13] .
Open-loop trajectory-planning and model-free control methods for cranes are considered
in [16,35,42,47] . In these cases no stability, relative robustness and performance criteria can
be formally analyzed. Optimal feedback and intelligent control of cranes are investigated in
[24,38,39,43] , and [25,44,48,51] , respectively. Deadbeat and robust cooperative control are
discussed in [10,15] . Of particular interest to this study is fuzzy control of overhead cranes.
Since it is not practical to review the state-of-the-art of different research lines on crane
control, the next subsections will focus on fuzzy control methods that are closely related to
the one discussed in this study.
Fuzzy control consists of an intrinsic nonlinear approach to the problem due to overlap-
ping fuzzy membership functions and partial activation of multiple rules containing locally
valid functional models or control laws [3,17] . Besides accurate control inputs, which can
often be provided by model-free controllers, we sometimes want mathematical models that
carry meaningful information about the physical system and, therefore, allow us to analyze
its properties and characteristics as a way to guide the design of controllers capable of at-
taining certain performance requirements. The possibility of analyzing stability, closed-loop
performance and robustness has given rise to a keen interest in model-based fuzzy control
[8,17,30,32] .
When model and fuzzy controller rules own the same antecedent terms, i.e., the same mem-
bership functions, such a control scheme is referred to as parallel distributed compensation
(PDC) [40] . A stable closed-loop system can be achieved by employing the direct Lyapunov
method and using linear matrix inequalities (LMIs). LMIs couple the stability condition given
by a Lyapunov-candidate energy function to the system states and parameters [4,32,40] . If a
feasible solution to a set of LMIs can be determined, then closed-loop stability is theoret-
ically guaranteed [50] . PDC fuzzy control and a smaller number of less conservative LMI
constraints are topics of great importance [1,17] . In the present study we use Sector Nonlin-
earity and Local Approximation in Fuzzy Partition Spaces to achieve a compact fuzzy model
structure and, therefore, a less conservative LMI problem. The LMI fuzzy design problem is
well-posed in the sense of Hadamard.
An advantage of using a fuzzy-PDC-LMI approach to the crane control problem concerns
the possibility to consider multiple control requirements (design specifications and constraints)
by simply appending new LMIs [12,17] . Convex feasibility problems can be solved numer-
ically using interior-point methods [4,17] . Particularly, actuators can only generate limited
control forces. If the control input exceeds the permitted range, then the actuator saturates
and the controller may degenerate into a bangbang controller, which degrades the system
performance and may cause instability. The actuator saturation effect is addressed appending
an additional LMI to restrict the control input. After a literature review in the next section,
Section 1.3 highlights the differences and advantages of the conservative LMI fuzzy control
approach suggested in this paper over related approaches.
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1.2. Literature overview on intelligent control of cranes
This section summarizes recent research related to fuzzy control of cranes. We do not
intend to give an exhaustive review of the literature. The purpose is to overview studies
related to that addressed in this paper.
A fuzzy-tuned PID control method is addressed in [44] . First, transfer function parameter
identification and the conventional root locus method are exploited to find initial gains to
the PID controller. A fuzzy system updates the PID gains aiming robustness to uncertainties
and parameter variations. Traditional PID controllers are argued not to be able to effectively
control crane systems in view of small variations on the true value of the parameters, or
due to unmodeled dynamics. In fact, the results show that the nonlinear fuzzy PID approach
provided a smaller overshoot for the load swing angle considering disturbances and a step on
the trolley position.
An adaptive anti-swing method for uncertain overhead cranes based on sliding mode con-
trol and a fuzzy observer is given in [25] . A robust, almost-critically-damped, closed-loop
system is achieved in the presence of uncertainties such as load mass variation, unmodeled
dynamics, and external disturbances. The fuzzy observer assists the estimation of uncertain-
ties to compensate practical factors. Similarly, in [48] , a fuzzy cerebellar articulation (CMAC)
model is used to estimate uncertainties, namely, gravity and friction, from reference tracking
errors. A PD controller is then used for anti-swing purpose.
A linear quadratic (LQ) fuzzy control scheme for gantry cranes is proposed in [28] . Com-
pared to a non-fuzzy LQ scheme, the fuzzy aspect of the former was able to significantly
reduce steady-state error related to the cart position, which is argued to be caused by friction
effect. In addition, the settling time for both cart position and load swing angle was reduced
in the fuzzy LQ control scheme. In [46] , a Lyapunov-stable H
∞ fuzzy control method for
balancing heavy loads is proposed. The method takes into account modeling uncertainties and
energy loss caused by friction, drag forces and communication delay. The H
∞ fuzzy con-
troller was able to regulate the system states under the influence of disturbances. A smooth
operation of the crane on the way toward a target position is shown.
A fuzzy descriptor that represents the nonlinear crane dynamics exactly using a reduced
number of fuzzy rules –8 rules –is outlined in [6] . A cost functional comprising states and
control inputs is minimized as a typical optimal LQ problem while a fuzzy control law is
derived from the optimization problem. Lyapunov stability is guaranteed. Similarly, a compact
3-rule fuzzy model and a theoretically stable closed-loop system for cranes is discussed in
[51] . Stability is obtained from a Lyapunov–Krasovskii functional. A quarter of a second
delay, and bounded control inputs are considered on the analyses. The controller effectively
drives the crane’s cart to the set point, with small settling time and oscillations.
An anti-swing sliding-mode control scheme subject to unmodeled uncertainties is described
in [38] . Linearization is needless on the approximation of the crane dynamics, while Lyapunov
stability is proved for initial states within a neighborhood of the equilibrium. In [43] , a vector
Lyapunov approach is described to suppress the load swing of a parametrically excited crane
system. In the present paper, unmodeled uncertainties and disturbances are suppressed by the
fuzzy aspect of the model and controller. Lyapunov stability and bounded control inputs are
guaranteed based on the feasibility of LMIs.
The method by Smoczek [33] performs robust anti-swing control of cranes using state
feedback and a fuzzy system. A sensorless architecture, and the use of computer vision,
are proposed. The control approach is based on fuzzy interpolation of the controller and
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accounts for potential parameter variations. The results show small interpolation errors and
small changes of the transient responses (settling and peak times) even for large parameter
variations. Fuzzy control approaches, such as that in [33] and the LMI model-based approach
of the present paper, have shown robustness to uncertainties, disturbances, and parameter
variations.
Advanced crane controllers are found in harbours [24,27] . An energy consumption manage-
ment system for rubber-tyred gantry cranes is suggested in [27] . A fuzzy controller is used to
maximize the benefits of adding energy storage units to the crane system. Actual data from the
Port of Felixstowe validates the fuzzy approach against a standard proportional-integral (PI)
controller. An increase of 32% on energy savings using fuzzy controller is observed, which
outperforms savings yielded by the PI approach by 26%. Predictive fuzzy sliding-mode con-
trol for offshore container cranes is given in [24] . A dynamic model for a simplified offshore
crane is obtained using a Lagrangian method. Load swings are predicted using a Kalman
filter. The sliding surface realizes a target trajectory while suppressing load swings. To avoid
chattering and improve overall performance, the controller parameters, i.e., sliding function,
gains, and rate of change, are tuned from a fuzzy rule scheme. Experiments using a 6-DOF
crane have shown robustness to parameters variations and different occurrences.
Neural control of cranes have been reported. A 3-layer radial-basis-function neural con-
troller that is independent on knowledge of the parameters of cranes is reported in [5] . The
neural network requires system states as inputs only. Its weights update law is derived from
a Lyapunov function so that numerical simulations have shown robust results in position
tracking and anti-sway balancing for loads of 100 and 1000 [Kg]. Aiming at minimizing vi-
bration, a control strategy based on the crane’s inverse dynamics using a feed-forward neural
network is addressed in [29] . Different operation points of the crane system are considered.
Experiments with an industrial crane are conducted and show that the neural method in fact
reduces vibration and steady-state error; and is competitive with other control methods, such as
dual matching control, and shaped reference-based control. Discounted near-optimal feedback
stabilization of a type of crane system is obtained from the solution of the corresponding
Hamilton–Jacobi–Bellman equation combined with neural networks in [45] . A neuro-fuzzy
system able to regulate the load angle and position of a 4-wheel trolley horizontally is ad-
dressed in [34] . Training data for the neuro-fuzzy controller are obtained in closed loop
using a conventional discrete state-space controller designed via pole placement. Different
cargo weights and cable lengths are evaluated. Results show that the neuro-fuzzy controller
provides reduced settling time, maximum overshoot, and steady-state error. By varying the
system parameters, the neuro-fuzzy controller has shown to be more robust than a traditional
state-space controller.
Nevertheless, neural controllers, such as those described in [5] and [29] , are model-free,
i.e., their design depends on the availability of data that reflect all possible behaviors for
multiple training steps. Although neural closed-loop systems are expected to generalize learned
behaviors to many situations, there are a number of factors and occurrences that should be
imposed to the crane system to obtain a consistent set of training data. For instance, never-
seen-before input data, and even input data out of the range of values of the training dataset,
may arise as a result of different wind profiles, load collisions, and types of loads. These may
cause inappropriate transients and instability of neural control systems. As the design of neural
controllers is, in principle, not guided by: parametric model, uncertainty analysis, actuator and
sensor limits, and Lyapunov functions, then, although a nonlinear solution is obtained, its use
in practice tends to be relatively risky, especially compared to the LMI fuzzy control solution
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given in the present paper. Toward autonomous development of general-purpose models and
controllers from online data streams, the area of evolving neuro-fuzzy systems have presented
promising ideas [19,31] .
1.3. Summary and contributions
Given a sequence of set points for the cart position, the present study focuses on the design
of a fuzzy controller that will lead the cart to a destination while regulate velocities and load
swing angle. In other words, given initial states within a neighborhood of the origin, the states
should converge asymptotically to the origin, except the cart position, which should converge
to the reference.
The design of the fuzzy controller is, in principle, based on a fuzzy nonlinear model that
matches the nonlinear dynamic equations of overhead cranes. Original unmodeled uncertainties
(small-scale dissipative forces) tend to be suppressed by the fuzzy aspect of the model and
controller. The fuzzy model is obtained using Sector Nonlinearity and Local Approximation
in Fuzzy Partition Spaces. A compact rule-base structure is obtained, which is important for a
less conservative LMI feasibility problem. Notice that, different from any other paper on the
subject, Local Approximation in Fuzzy Partition Spaces minimizes the number of fuzzy rules
of the model and controller. This avoids further procedures for LMI relaxation. The solution
of the LMI problem directly provides controller gains. Lyapunov stability and bounded control
forces are taken into consideration in the LMI problem. Therefore, fuzzy controller gains are
obtained in this sense.
A variety of computational and laboratory experiments using actual data is performed with
emphasis on closed-loop performance criteria, robustness to different levels of disturbances,
load collisions and parameter variation. A single and double collision of the load during the
transient is a computational experiment being addressed for the first time to evaluate robustness
to abrupt changes of the states of a closed-loop crane system. An optimal Linear Quadratic
(LQ) Controller designed based on a linearized pendulum model is used for comparisons
with the proposed LMI-based fuzzy control approach. The latter is nonlinear and relies on a
compact rule base.
The contributions of this paper are summarized as follows. They are:
• a systematic state-space fuzzy control design method for a type of crane system, which
drives a trolley to set-points smoothly while regulates velocities and load swing angle;
• a compact fuzzy rule-based model – obtained using Sector Nonlinearity and Local Ap-
proximation in Fuzzy Partition Spaces – that matches the nonlinear dynamic equations of
overhead cranes;
• a Lyapunov stable closed-loop system using bounded control inputs (which tend to increase
the actuators lifetime) is guaranteed by solving the proposed LMI feasibility problem;
• a higher level of robustness to disturbances, load collisions, and gradual and abrupt param-
eter variation (uncertainties) is demonstrated in computational and laboratory experiments
using actual data;
• comparative results considering a typical optimal linear quadratic controller for cranes are
also given.
The remainder of this paper is organized as follows. Section 2 describes the nonlinear
and linearized state-space models as well as a 2-step method to construct compact fuzzy
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Fig. 2. Schematic diagram of overhead crane.
rule-based models of overhead crane systems. Section 3 describes the method proposed for
the design of the fuzzy controller. A quadratic Lyapunov function and bounded control inputs
are used on the formulation of the associated LMI problem. An optimal Linear Quadratic
Controller, useful for performance comparisons, is also addressed in this section. Results are
presented in Section 4 . Actual data and laboratory experiments show the usefulness of the
LMI fuzzy control for cranes. Concluding remarks and suggestions for future research are
given in Section 5 .
2. Mathematical model of overhead crane
2.1. Nonlinear state-space model
The dynamic equations of motion of an overhead crane system are given in this section. A
typical diagram of overhead crane is shown in Fig. 2 . Dissipative forces are ignored in view
of their relative small scale. The hook and suspended load are regarded as a single mass point
as the mass of the cargo is usually much larger than that of the hook. As widely assumed in
the literature, the steel cable is modeled as a rigid link, which makes sense for a large range
of swing amplitudes. The Euler–Lagrange method [9] is employed to derive a continuous,
time invariant, nonlinear model of the system.
Cart position in relation to a reference, say the origin O, is given as ( x, y). The nonlinear
equations of motion are
(M + m) ¨x + mL
¨
θcos (θ ) −mL
˙
θ2
sin (θ ) = F,
mL ¨x cos (θ ) + mL
2
¨
θ+ mgL sin (θ ) = 0, (1)
where Mis the cart mass [kg]; mis the load mass [kg]; Lis the cable length [m]. The
force F [N] is the control input produced by any electric motor. Naturally, the unidirectional
movement of the trolley in the xdirection is taken into consideration.
At any time, the system is completely described by four states, namely, cart position and
velocity ( xand ˙ x ) and load swing angle and velocity ( θand
˙
θ), i.e.,
x = [ x
1 x
2 x
3 x
4
]
T
, (2)
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where x
1
= x; x
2
= ˙ x ; x
3
= θ; x
4
=
˙
θ. Notice that velocities are unavailable for measurement
in most cases. In this study, velocity feedback is numerically obtained by differentiating
position and angle signals. However, noise is induced due to many reasons such as sensor
imprecision, electromagnetism, etc. Low-pass filters to further process velocities are important
for a better approximation of the true velocities. Therefore, if not otherwise mentioned, we
consider full state feedback.
After some algebraic manipulations, Eq. (1) may be rewritten in the form ˙ x = f (x, F, t)
as
˙ x
1 = x
2
˙ x
2 =
mL x
2
4
sin (x
3
) + 0. 5 mg sin (2x
3
) + F
a −m cos
2
(x
3
)
˙ x
3 = x
4
˙ x
4 =
ag sin (x
3
) + 0. 5 mL x
2
4
sin (2x
3
) + F cos (x
3
)
L[ m cos
2
(x
3
) −a]
, (3)
where a M + m.
2.2. Linearized state-space model
Linearization of the state-space model (3) is given for the purpose of designing alternative
controllers and completeness of the paper.
The nonlinear state-space model, Eq. (3) , can be linearized at a zero swing angle, x
3
= 0.
Therefore, ˙ x
2
3
≈0, sin (x
3
) ≈x
3 and cos (x
3
) ≈1 are true, and the system dynamics reduces
to
˙ x
1 = x
2
˙ x
2 =
mg
M
x
3
+
1
M
F
˙ x
3 = x
4
˙ x
4 = −(M + m) g
ML
x
3
−1
ML
F. (4)
Let the cart position, x
1
, and the load swing angle, x
3
, be output variables. Thus Eq. (4) is
rewritten as
⎡
⎢
⎢
⎢
⎢
⎣
˙ x
1
˙ x
2
˙ x
3
˙ x
4
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎣
0 1 0 0
0 0
mg
M
0
0 0 0 1
0 0 −(M+ m) g
ML
0
⎤
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎣
x
1
x
2
x
3
x
4
⎤
⎥
⎥
⎥
⎥
⎦
+
⎡
⎢
⎢
⎢
⎢
⎣
0
1
M
0
−1
ML
⎤
⎥
⎥
⎥
⎥
⎦
F
y =
1 0 0 0
0 0 1 0
⎡
⎢
⎢
⎢
⎢
⎣
x
1
x
2
x
3
x
4
⎤
⎥
⎥
⎥
⎥
⎦
, (5)
in which y is the output vector. Eq. (5) is the linear state-space representation of the crane
system. Considering the compact form of expressing Eq. (5) , i.e., ˙
x = A x + B u; y = C x ,
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Fig. 3. Partitions of the range of possible values for θ.
the conditions of complete state controllability (viz. [ B AB A
2
B A
3
B ] has full rank; A is
nonsingular), and observability (viz. [ C
∗A*C* ( A
∗)
2
C ( A
∗)
3
C ] has 4 linearly independent
column vectors) imply the existence of a complete solution to the control design problem.
2.3. Nonlinear fuzzy model by sector nonlinearity
An exact functional fuzzy model for the original crane system (3) can be drawn based on
the Sector Nonlinearity (SNL) method [40] . Let x
3
∈ [ −88
◦, 88
◦] be a range of values for
the load swing angle (local sector), which is imposed by physical limitations, see Fig. 3 .
The SNL method fragments the workspace of x
3 in cregions (sectors). System behavior
in each region is represented by a local linear state-space model. The ith local model, for
i = 1 , . . . , c, is characterized by the pair ( A
i
, B
i
), which stands for the local system and
actuation matrices, respectively. Sectors and state-space models are related by fuzzy rules of
the type
Rule i: IF ( z
i
1
is M
i
1
) AND... AND ( z
i
is M
i
) THEN ˙
x = A
i
x + B
i
F in which
the elements of z
i = [ z
i
1
, . . . , z
i
j
, . . . , z
i
] depend on the states that characterize the ith sector.
Moreover, M
i
j
, i = 1 , . . . , c, are membership functions that granulate (in the sense of cov-
erage and unimodality [18,26] ) the domain of z
j
. Matrices A
i
, i = 1 , . . . , c, are nonsingular.
For practical reasons, z
j should not depend on the control input F . In [40] , a considerable
increase of the model complexity is reported if z
j = f ( x , F ) . The conditions of complete state
controllability and observability of the ith fuzzy local model are analogous to those of the
global linearized model (given immediately after Eq. (5) ). The global functional fuzzy model
is obtained from the smooth combination of its rules. Therefore, fuzzy model and control is
different from switching modeling and switched control [21] since local models are changed
smoothly instead of abruptly. The greater the number of sectors, the greater the number of
rules, and the smaller tends to be the approximation error in relation to the original system.
Nonetheless, a larger amount of sectors increases the complexity of the model and hinders
the design of PDC controllers via LMIs. Additional discussions can be found in [17,40,41] .
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Rewrite Eq. (3) in terms of its nonlinear terms as
˙ x
1
= x
2
˙ x
2
=
1
a −m cos
2
(x
3
)
[ mLx
4
(x
4
sin (x
3
)) + 0. 5 mg sin (2x
3
) + F ]
˙ x
3
= x
4
˙ x
4
=
1
L[ m cos
2
(x
3
) −a]
[ ag sin (x
3
) + 0. 5 mLx
4
(x
4
sin (2x
3
)) + F cos (x
3
)] , (6)
where a = M + m. Thus, if
z
1
(x)
1
a −m cos
2
(x
3
)
z
2
(x) x
4
sin (x
3
)
z
3
(x) sin (2x
3
)
z
4
(x)
1
L[ m cos
2
(x
3
) −a]
z
5
(x) sin (x
3
)
z
6
(x) x
4
sin (2x
3
)
z
7
(x) cos (x
3
) , (7)
then Eq. (6) can be expressed as
˙ x
1 = x
2
˙ x
2 = z
1
[ m Lx
4
z
2
+ 0. 5 m gz
3
+ F ]
˙ x
3 = x
4
˙ x
4 = z
4
[ agz
5
+ 0. 5 mLx
4
z
6
+ F z
7
] . (8)
According to Eq. (7) , the fuzzy model contains seven antecedent terms. If a couple of
membership functions is assumed for each, we have 2
7 fuzzy rules. As stated previously, the
number of rules may have a significant impact on the complexity of the PDC control design.
Nonlinear programming or evolutionary computing algorithms to find feasible values for the
elements of matrices in LMI problems are subject to exponential complexity in relation to
the number of rules; see [7,17,20,36] . The next section describes an approach to reduce the
number of fuzzy rules maintaining model accuracy.
2.4. Reducing the number of fuzzy rules
Local Approximation in Fuzzy Partition Spaces [7,40] is an approach to reduce the number
of fuzzy rules. The method consists in approximating the nonlinear terms in Eq. (7) by a
particular choice of linear terms. In this case, although nonlinear, the fuzzy model becomes
an approximation of the crane system, whereas when 2
c rules were used, the fuzzy model
matched the original nonlinear differential equations. Loss of exactness can, however, be
negligible –as will be shown in the Results Section – while the number of parameters
and hardness of the control design problem are substantially reduced. Frequently, rules can
be removed with no accuracy reduction since the combination of input values of the state
variables that would activate a particular rule is not physically possible.
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Eliminating the majority of the rules is highly beneficial to the design problem because the
number of LMIs to be solved scales exponentially with the linear increasing of the amount of
rules. Instead of applying methods to reduce the conservatism of LMIs by means of relaxation
strategies, we used the Local Approximation in Fuzzy Partition Spaces method. We focus on
a key state variable, the load swing angle, and on the use of a wider pair of membership
functions to cover the workspace of x
3
. A trade-off between model compactness, accuracy,
and easiness of finding a feasible LMI solution is considered.
Aiming at achieving the most compact fuzzy model and controller possible, only three
sectors for the swing angle are taken into consideration. In addition, a single rule is valid for
the sectors that represent extreme (positive and negative) angles. Therefore, only two fuzzy
rules are derived for the model and PDC controller.
For small values of x
3
, Eq. (3) can be rewritten as Eq. (4) . Contrariwise, if x
3 is close to
the extremes, i.e., to ±88
◦, then sin (x
3
) tends to (2/π ) x
3
. Being β cos (88
◦) , we get that
sin (2x
3
) tends to (4/π ) βx
3
1
. The state equations become
˙ x
1 = x
2
˙ x
2 =
2mg
a −mβ2
x
3
+
1
a −mβ2
F
˙ x
3 = x
4
˙ x
4 =
2ag
πL(mβ2
−a)
x
3
+
β
L(mβ2
−a)
F, (9)
Notice that Eqs. (4) and (9) are linear. Hence, x
3 is the unique antecedent term of the ith rule,
i = 1 , 2. The domain of x
3 is granulated in normal (height 1) and complimentary triangular
membership functions, M
i
, i = 1 , 2. Membership functions are associated to sectors. Their
linguistic values are ‘close to 0
◦’ and ‘close to 88
◦’.
Fuzzy membership functions are constrained by
M
1
(x
3
) + M
2
(x
3
) = 1
M
1
(x
3
) p + M
2
(x
3
) q = x
3
, (10)
where
p = min
−88
◦≤x
3
≤88
◦(x
3
)
q = max
−88
◦≤x
3
≤88
◦(x
3
) , (11)
which implies symmetry. pand qare the minimum and maximum values of z
j ( x
3 in this
case).
For a given x
3
, membership degrees are calculated from
M
1
(x
3
) =
1
88
x
3
+ 1 if x
3
< 0
−1
88
x
3
+ 1 if x
3
≥0
(12)
and
M
2
(x
3
) =
−1
88
x
3 if x
3
< 0
1
88
x
3 if x
3
≥0.
(13)
1 Due to the double angle relation for sine, i.e., sin (2α) = 2 sin (α) cos (α ) .
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Fuzzy rules for different sectors are of the form, Rule 1: IF x
3 is M
1
(x
3
≈0) ,
THEN ˙
x = A
1
x + B
1
F Rule 2: IF x
3 is M
2
(x
3
≈±88
◦) ,
THEN ˙
x = A
2
x + B
2
F in which A
i and B
i are the system and control matrices; F is the
control input. From Eqs. (4) and (9) we have
A
1
=
⎡
⎢
⎢
⎢
⎢
⎣
0 1 0 0
0 0
mg
M
0
0 0 0 1
0 0 −ag
ML
0
⎤
⎥
⎥
⎥
⎥
⎦
B
1
=
0
1
M
0 −1
ML
T (14)
and
A
2
=
⎡
⎢
⎢
⎢
⎢
⎣
0 1 0 0
0 0
2mg
a−mβ2 0
0 0 0 1
0 0
2ag
πL(mβ2
−a)
0
⎤
⎥
⎥
⎥
⎥
⎦
B
2
=
0
1
a−mβ2 0
β
L(mβ2
−a)
T
. (15)
The state estimate of the fuzzy model is given as
˙
x =
2
i=1
h
i
(
A
i
x + B
i
F
) (16)
in which
h
i =
M
i
(x
3
)
c
i=1
T (M
i
(x
3
))
(17)
is the activation degree of the ith rule; T is a T -norm, e.g. the algebraic product
T (M
i
(x
3
)) = M
1
(x
3
) M
2
(x
3
) . (18)
3. Control design
3.1. LMI-based fuzzy control
Fuzzy control design based on the PDC concept connects the antecedent part of controller
rules directly with that of the model rules, as described in Eqs. (12) and (13) . Controller
and model have the same membership functions and number of rules, which makes controller
design simpler and faster. A local fuzzy control law may assume several forms [40] . In this
study, local fuzzy controllers are given by state-feedback linear control laws.
Within the framework of state-space fuzzy modeling and PDC control, conditions for
stability and performance can be formulated as an LMI problem. Finding a solution to the
LMI is equivalent to finding a solution to the original problem [4,17,40] . In the following,
Lyapunov stability conditions are written as LMIs. The gains of the controller are determined
from the LMI formulation.
Considering a control algorithm, we assume a sufficiently small time step compared to
the dominant time constant of the system, namely k = 0. 005 , and discrete-time equations.
To obtain feedback stabilizing conditions, the PDC approach is employed so that, in general,
the controller rules are: Rule i: IF z
i
1
is M
i
1
AND... AND z
i
is M
i
THEN
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u
i
(k + 1) = K
i
x (k + 1) where z
i
j
∀ jdepend on the states; M
i
j
∀ jare the same membership
functions as those of the fuzzy model, see Eqs. (12) –(13) ; x (k) is the state vector; and
u
i
(k + 1) denotes the local control input. Additionally, K
i ∈ R
1 ×n
, where nis the system
order, is the gain matrix of the ith local controller to be determined so that the closed-loop
system is asymptotically stable. In particular, according to the approach to give the most
compact fuzzy structure described in Section 2.4 , = 1 , z
1
= x
3 and c = 2.
The overall control input is
u (k + 1) =
c
i=1
h
i
u
i
(k + 1) , (19)
being cthe number of rules and h
i the activation degree as in Eq. (17) . As the time step k
increases, and for a sufficiently small sampling time, Eqs. (16) and (19) can be combined.
The resulting closed-loop system is
x (k + 1) =
c
i=1
c
j=1
h
i
h
j
G
ij
x (k) , (20)
where G
ij A
i
+ B
i
K
j
, or, equivalently,
x (k + 1) =
c
i=1
(h
i
)
2
G
ii
x (k) + 2
c
i<j
h
i
h
j
G
ij
+ G
ji
2 x (k) . (21)
If the unforced system is stable, then K
j can be chosen to improve the transient response.
Unstable systems require primarily the determination of an appropriate K
j to stabilize an
equilibrium.
The stabilization problem based on the Lyapunov criterion [14] can be translated in terms
of finding a feasible solution to a set of LMIs. The following results have been reported in
the literature [7,41] . For completeness, in what follows they are briefly recalled.
Define a candidate Lyapunov function:
V ( x ) = x
T
P x , (22)
where P is a positive definite matrix. Based on Eq. (22) and on the closed loop system (20) ,
the following stabilization theorem was derived and proved in [41] .
Theorem 1 : The closed loop control system (20) is globally exponentially stable if there
exists a positive definite matrix X = P
−1 and a set of matrices Q
i
, i = 1 , . . . , c, such that the
linear matrix inequalities
−X X ( A
i
)
T
+( Q
j
)
T
( B
i
)
T
A
i
X + B
i
Q
j −X
< 0 (23)
are satisfied for all combinations of i, j = 1 , . . . , c.
This is a convex feasibility problem [17] . If a feasible solution ( X , Q
i
) is found, then the
controller gains are
K
i = Q
i
P , i = 1 , . . . , c. (24)
The feasibility of Eq. (23) means that: ( i ) a common P matrix for all rules exists; ( ii )
Eq. (22) is Lyapunov; and ( iii ) the closed loop system (20) is stable using the gains K
i
,
calculated as in Eq. (24) .
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The inequality (23) depends on the terms of the sub-matrices X and Q
i ∀ i. Each Q
i can
be any matrix (not necessarily a square matrix, e.g., any row vector for a single actuator
system), and X can be any positive-definite matrix, that, when combined, make Eq. (23) true.
Therefore, X , the inverse of P , and Q
i ∀ ihave no meaning in the real world. We use the
SeDuMi 1.05 solver [36] to search elements for theses matrices. However, any other search
method can be used, such as evolutionary algorithms and interior-point methods in nonlinear
programming.
3.2. Bounded control input and Hadamard well posedness
If the current state x (k) is known, then the following theorem is useful to handle bounded
control input.
Theorem 2 : Given positive definite matrices X
i = ( P
i
)
−1 and matrices Q
s
= K
s
Z
s
, i, s =
1 , . . . , x . The constraint || u (k + 1) ||
2
≤ζis enforced if the LMIs
1 x (0)
T
x (0) X
i
> 0 (25)
and
X
i ( Q
s
)
T
Q
s ζ2
I
> 0 (26)
i, s = 1 , . . . , x , hold true. x (0) is the initial state.
The proof of Theorem 2 is found in [40] . The value of ζbounds the maximum value
allowed for the 2-norm of the control inputs, that is, ζkeeps the inputs within the limits of
the actuators. The LMIs (25) and (26) can be appended to Eq. (23) in the design of stable
fuzzy controllers satisfying input constraints.
The well posedness of a problem refers to whether or not its formulation meets the
Hadamard criteria. The LMI fuzzy control design problem is well posed if: (i) a solution
exists; (ii) the solution is unique; and (iii) the solution is continuously dependent on the ini-
tial conditions. In our case, we seek a set of controller parameters that guarantees Lyapunov
stability of the closed-loop system as well as bounded inputs, i.e., we seek a region in a
search space that solves the LMI feasibility problem. The solution exists if Theorems 1 and
2 hold simultaneously, as proved in [40] and [41] (condition (i) holds). The region in the
search space is unique since the local state-space models and local controllers are linear and,
therefore, yield a unique convex feasibility region (condition (ii) holds). For any controller
in the loop – namely, for any fuzzy controller derived from a fuzzy model with different
parameters – gradual changes of the initial states provide gradual changes of the transient
responses, which converge to the reference afterwards (condition (iii) holds). The problem is
well-posed.
3.3. Linear quadratic optimal control
Linear quadratic (LQ) optimal control takes into account the linearized model (5) of the
crane system. Different from the conventional pole placement design approach, closed-loop
eigenvalues are not specified directly, but instead, an objective function is optimized. Similar
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to the fuzzy control algorithm, we consider a sufficiently small time step in comparison to
the dominant time constant of the system, namely k = 0. 005 .
Let
J =
1
2
∞
k=0
[ e
T
(k) Q e (k) + u
T
(k) R u (k)] , (27)
where Q and R are positive-definite real symmetric matrices – weight matrices. J is an
objective function to be minimized. The first term in Eq. (27) refers to the error of the states
x in relation to the reference x
re f
. For the regulation problem, i.e. if the reference position of
the cart is assumed to be 0, then e = x . In other words, the reference is the origin of the state
space in this case. The second term refers to the control energy. The LQ design attempts to
minimize J . The elements of Q and R are chosen, respectively, to reflect the preference for
a faster convergence of a specific state to the reference, and for the use of control signals of
smaller amplitude.
Given the control law
u (k) = −Ke (k) . (28)
After simple manipulations and development, see e.g. [23] , the gain matrix is found as
K = ( B
T
PB + R )
−1
( B
T
PA ) , (29)
where P should satisfy the discrete-time Riccati equation,
A
T
PA −P −( A
T
PB )( B
T
PB + R )
−1
( B
T
PA ) + Q = 0 . (30)
If a positive definite P exists, then the closed-loop system matrix, [ A −BK ] , is stable, i.e. its
eigenvalues are within the unit circle (discretized A and B ). The proof can be found in [23] .
Notice that for the crane system under analysis, x
re f = [ x
1 ,re f 0 0 0]
T
, that is, a set point is
given for the cart position while the other states are regulated. Nonetheless, the gains K are
obtained in the same way. The LQ controller is placed in the forward path.
4. Results and discussions
This section compares the behavior of the proposed compact state-space fuzzy model
and of the linearized state-space model in approximating the original crane system. The
performance of the nonlinear closed-loop system using the fuzzy PDC controller is evaluated
and compared to that of the LQ optimal controller designed from the linearized dynamics.
Actuator saturation and system robustness due to different levels of sensor noise, parameter
changes and load collisions are assessed. Additionally, laboratory experiments and real-world
data are considered to validate the fuzzy control approach for cranes in an actual scenario.
Hardware and software details are also given.
Parameters of an actual system are given as M = 400 [Kg] for the trolley mass; m = 3000
[Kg] for the load mass; and L = 2[m] for the length of the steel rope. The gravity is
g = 9 . 81 [ m/ s
2
] and an initial condition is assumed for the states, x (0) = [0 0 0. 1 0] .
Additionally, white noise within the range [ −ϕ , ϕ ] is added to the position and angle mea-
surements, and to the velocities obtained by differentiating positions and angles. The time
step is 0.005.
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4.1. A preliminary open-loop experiment
An early open-loop simulation was done to spotlight the advantage of the nonlinear state-
space fuzzy model over the linearized dynamics. We use a bang-bang input, similar to e.g.
[2] , to evaluate the open loop response using Eq. (3) (original system), Eq. (5) (linearized
model), and Eq. (16) (reduced state-space fuzzy model). In the beginning, k ∈ [0, 1500] , the
input reference is F = 1 [N]; then, for k ∈ [1500, 3000] , F = −1 [N], characterizing the
bang-bang effect. Subsequently, F = 0[N].
The response of the fuzzy system to the bang-bang F reference is essentially similar to
that of the original nonlinear system for all states. At k = 1500, the variation of x
1 becomes
negative due to the negative pulse on the input, which decelerates the trolley. The trolley
drifts to the same position in both cases (original system and fuzzy model), while the load
angle oscillates due to the trolley movement.
While the fuzzy model follows the original angle, x
3
, quite accurately as it withdraws from
the origin, the response of the linearized model for x
3 diverges in some degrees – from 0 to
0.05 [rad] –in relation to the nonlinear system, especially as angles become larger. The local
fuzzy model, whose membership function relative to x
3 has its core in 88
◦, compensates the
deviations from the origin smoothly and proportionally.
4.2. Closed-loop results
State feedback controllers were obtained via the Fuzzy PDC ( Sections 3.1 and 3.2 and LQ
( Section 3.3 ) methods. Gain matrices K are used in the forward path to generate an input F
to the original nonlinear crane system (3) . We consider a reference point to the cart position,
x
1 ,re f = 5 [m] during the first 5000 iterations, and x
1 ,re f = 0[m] after that. The other states
should be regulated, i.e. x
2
, x
3
, x
4
→ 0.
A feasible solution to Eqs. (23), (25) and (26) concatenated was sought. This means that
bounded inputs, ζ= 300 [N], and Lyapunov conditions are included in the fuzzy PDC control
design. The SeDuMi 1.05 solver [36] and the Yalmip toolbox [20] were used in Matlab. We
used an 8 GB RAM, 2.5 GHz microcomputer so that a solution to the LMI problem is found
in less than 1 s.
The symmetric and positive definite matrix X common to all rules is
X =
⎡
⎢
⎢
⎢
⎢
⎣
20. 08 −4. 31 4. 73 −4. 70
−4. 31 33 . 42 4. 84 13 . 47
4. 73 4. 84 3 . 88 −0. 06
−4. 70 13 . 47 −0. 06 11 . 33
⎤
⎥
⎥
⎥
⎥
⎦
. (31)
For each fuzzy rule we have
Q
1 = [262. 05 164. 98 169 . 63 −70. 98]
Q
2 = [512. 77 317 . 95 322. 66 −104. 97] . (32)
The gain matrices of the local fuzzy controllers according to Eq. (24) are
K
1 = [3 . 99 4. 81 32. 69 −10. 15] (33)
and
K
2 = [8 . 86 6 . 83 63 . 60 −13 . 37] . (34)
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The overall control input is calculated at each iteration as in Eq. (19) . Notice that the cor-
responding elements of the gain matrices, Eqs. (33) and (34) , are interpolated in between
depending on the activation degree of the fuzzy rules.
The LQ optimal controller was designed in such a way that the load swing regulation has
priority over the trolley position and their derivatives (speeds). Control errors for the velocities,
x
2 and x
4
, in relation to null references are suppressed as a consequence. A smaller weight is
also admitted to the control energy in Eq. (27) , namely, R = [1] . Care should be taken about
the amplitude of the resulting control input, u, as electrical and mechanical constraints are
inherent to the physical system. The state preference matrix is set to
Q =
⎡
⎢
⎢
⎢
⎢
⎣
2000 0 0 0
0 2000 0 0
0 0 4000 0
0 0 0 2000
⎤
⎥
⎥
⎥
⎥
⎦
. (35)
Several attempts using different values for R and Q were done with focus on the transient
response. The underlying choice provided encouraging results. From matrices A and B of the
linearized model (5) , the LQ design resulted in
K
LQ
= [200 216 . 65 −1199 . 3 −16 . 76] . (36)
These gains place the poles of the closed-loop system in −395 , −1 , −1 . 31 ±j5 . 48 , i.e. the
system is stable; however, system response considering the original nonlinear system (3) in
the loop cannot be previewed with precision due to the small angle assumption and model
linearization.
Figs. 4 and 5 show the results for the states x
1 and x
3 obtained by the fuzzy PDC and
LQ controls, respectively, for the nonlinear crane system. The oscillation aspect supports
clear advantages to the fuzzy controller. For the trolley position and taking into account the
average of the up and down reference steps, the fuzzy control provided a 23.1 [%] overshoot
followed by exponential convergence. The LQ control gave a 43.1 [%] overshoot followed
by exponential convergence. The same analysis for the load swing yielded a maximum angle
of 0.51 [rad] and 0.62 [rad] for the fuzzy and LQ controls. The maximum control input
required in both cases was 295.9 [N] and 792.9 [N], respectively. LMIs restrict the fuzzy
control actions at inferior values, 300 [N], which may increase the lifetime and reliability of
the actuator.
Over the iterations, the fuzzy controller rules are partially activated. Whenever x
3 deviates
from 0
◦by a larger angle, the rule related to angles ‘close to 88
◦’ becomes more active to
suppress this difference more severely. Nonetheless, the rule focused on the sector ‘close to
0
◦’ is more active than the other in all iterations. Stronger actions to larger deviations can also
be noticed by greater absolute values of the gains relative to x
3 and x
4 in Eq. (34) compared
to those in Eq. (33) . In general lines, the performance of the fuzzy system is superior to that
of the LQ system as expected in view of the more realistic design which takes into account
nonlinearities of the physical system, Lyapunov stability and constrained inputs.
4.3. Stability to disturbances and collisions
Experiments to access the robustness of the closed-loop system were performed. First, we
increase the amplitude of white noise in the velocity feedback as velocities are obtained nu-
merically by differentiating positions and angles. Low-pass filters to further process velocities
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Fig. 4. Response of the states x
1 and x
3 and control input for the fuzzy PDC closed-loop system.
Tab le 1
Tolerance to disturbances by the different controls.
Condition Fuzzy PDC LQ optimal
Always stable ϕ ≤0. 010 ϕ ≤0. 004
Stability ≈50% of times ϕ ≈0. 040 ϕ ≈0. 006
Always unstable ϕ ≥0. 050 ϕ ≥0. 011
are important for a better approximation of the true values. Nonetheless, we search for a
limit on the PDC fuzzy and LQ system. Second, we conduct a single and a double collision
experiment during the transient response of the states.
In 10 runs of the control algorithms we seek for boundary values of ϕso that the fuzzy
and LQ closed-loop systems are always stable or unstable. Table 1 shows the results. The
higher values of ϕfor the fuzzy PDC control indicate that it tolerates larger disturbances or
that it possesses a larger relative robustness factor. In the vast majority of the experiments,
instability is triggered by the load swing angle, which is clearly the critical state.
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Fig. 5. Response of the states x
1 and x
3 and control input for the LQ optimal closed-loop system.
A collision of the load with an obstacle during the transient of the cart, specifically at
k = 200, was considered by suddenly forcing the states to
x
1
(k + 1) = x
1
(k) −0. 9
x
2
(k + 1) = 0. 1
x
3
(k + 1) = 0. 15
x
4
(k + 1) = 0. (37)
In addition, a double collision of the load during the step down transient is given as
x
1
(k + 1) = x
1
(k) + 0. 5
x
2
(k + 1) = −0. 2
x
3
(k + 1) = −0. 05
x
4
(k + 1) = −0. 01 (38)
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at k = 5300, and
x
1
(k + 1) = 1 . 5
x
2
(k + 1) = 0. 8
x
3
(k + 1) = 0. 45
x
4
(k + 1) = 0. 75 (39)
at k = 6300. The purpose of this experiment is to evaluate the robustness of the fuzzy control
to an abrupt change. In particular, the shift on the states may not only be caused by collisions,
but also by harsh environmental conditions such as wind bursts or vessel movements. Fig. 6
shows the behavior of the closed-loop fuzzy system. We omit the analogous figures for the
LQ controller as they are essentially the same, except in relation to peaks of the input that
are about 5 times larger after the changes of set points and collisions.
Notice in Fig. 6 that constraints on the amplitude of the control input are still respected
due to the collisions. The maximum absolute amplitude is 300 [N] for the force applied
to the cart, whereas the amplitude in the LQ optimal case reaches values about 5 times
higher. Actuators may saturate for such values and certainly have their lifespan reduced due
to repetitive peaks imposed by the LQ controller. On the other hand, the fuzzy controller
acts relatively smoother in all state variables in all cases. The fuzzy PDC control responded
immediately to the unexpected events and, therefore, to quite irregular transients. It was able
to keep not only the stability of the system, but also acceptable overshoots and settling time.
4.4. Robustness to parametric uncertainty
The robustness of the closed-loop fuzzy and LQ control systems were analyzed considering
a range of values for the load and cable length around the values in which the controllers
were designed, i.e., m = 3000 [Kg] and L = 2[m]. Naturally, if mand Lare known and
are different than such values used in the control design, new gain matrices can be obtained
quickly (in about one second) for both cases, fuzzy and LQ. Therefore, the robustness being
evaluated concerns with an uncertainty on the true parameters of the system. Fig. 7 shows
the instability region for both closed-loop systems.
Interestingly, the stability regions of the controllers are in opposition. The smooth fuzzy
controller is still stable for a considerable range of values around the design conditions,
especially for greater cable lengths and smaller load masses. By contrary, the aggressive LQ
controller, in spite of being also stable for a significant range of values around the design point,
supports smaller cable lengths and greater masses better. Nonetheless, parameter variations
affect the transient performance of the LQ system more severely compared to that of the
fuzzy PDC system.
4.5. Laboratory experiment
Laboratory experiments are conducted to validate the LMI fuzzy control approach using
actual data. Fig. 8 shows the physical crane model, which is supplied by a direct current motor
to drive the cart along the rail, and two incremental encoders with 1024 counts per revolution
to measure the cart’s position and speed, and the load swing angle and angle variation. The
crane system is connected to a laptop through an Arduino I/O board.
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Fig. 6. Evolution of the states due to collisions during the transients for the closed-loop fuzzy PDC system.
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Fig. 7. Stability boundaries of the Fuzzy PDC and LQ optimal control systems. ‘o’: unstable, ‘
∗’: stable.
The Arduino board sends pulse-width modulation signals to the amplifier of the DC motor,
and receives data from the encoders through its analog I/O connections. The real-time oper-
ation of the system is based on the C++ framework, whereas the design of the controller is
done separately, on the MatLab environment. The apparatus is a 2-degree of freedom system
driven by a single DC actuator. Small load swings along the perpendicular axis are neglected.
The parameters of the laboratory crane are m = 2. 0 [kg], M = 7 . 5 [kg], and L = 0. 5
[m], while g = 9 . 81 [ m/s
2
], and the sampling rate is 400 [Hz]. The initial states are x (0) =
[0 0 0 0] . The first target position of the trolley is x
1 ,re f = 0. 6 [m], as exemplified in Fig. 8 .
After 15 s, or k = 12, 000 samples, we change the target position to x
1 ,re f = 0. 2[m]. Then,
after an additional 15-s interval, we set the target position back to x
1 ,re f = 0. 6 [m]. During
the last 15 s, i.e., from k = 24, 000 to k = 36 , 000, a wind perturbation is applied by means
of a floor fan directed along the controllable x
1 axis. We approximate the wind disturbance
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Fig. 8. Laboratory crane system: the LMI fuzzy controller conveys the load smoothly to the set point.
Fig. 9. Experimental vs. simulation results provided by the fuzzy LMI crane controller. Top graph: evolution of the
cart position, and control input; bottom graph: load swing angle, and approximated wind profile using a fan.
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by a linear piecewise function, with three 5-s pieces, related to the acceleration, steady-state,
and deceleration of the fan, as shown in the bottom graph of Fig. 9 . The wind profile affects
the state x
3
; and x
4 as a direct consequence.
Fig. 9 compares simulation and actual closed-loop results provided by the proposed fuzzy
LMI control system. The experimental state data provided by the pair of encoders follow those
obtained in simulation with quite good accuracy, except for a persistent quick small-amplitude
movement of the cart in the actual scenario (see top graph), which does not interfere on the
accomplishment of the tasks. While the cart position tracks and converges to the three different
set points, and the load angle converges to zero in all cases, the control input, whose scale is
given in the right-hand side of the graph, converges to zero for the first two reference steps.
When the wind comes into play ( k = 24, 000), the control force is increased and decreased
gradually, opposed to the wind profile, as a reaction to suppress the wind effect on x
3
. Both
experimental and simulation angles are kept at very low values due to the action of the fuzzy
controller, as expected.
5. Conclusion
This paper presents a state-space fuzzy model of overhead cranes and a fuzzy control
design method based on parallel distributed compensation. We emphasize the diversity of
applications that such equipment may have in the industry and the necessity and advantages
of designing automatic control systems that guarantee its proper operation, performance and
overall safety.
The fuzzy modeling approach considers the Sector Nonlinearity method and reduction of
the size of the rule base by Local Approximation. The latter avoided typical procedures of
linear matrix inequalities relaxation from an initial set of matrix inequalities during control
design. Less conservative LMIs were derived from a quadratic candidate Lyapunov function
and bounded control inputs. It has been demonstrated that the fuzzy model obtained by
reducing the fuzzy rule base approximates the original crane dynamics with negligible errors
in the whole range of operation, and that the LMI design problem is well posed.
Stabilization and performance of the closed-loop system using the PDC fuzzy controller
has been assessed considering different reference inputs for the cart position, different levels
of disturbances (white noise and wind), load collisions, and parameter variations. The fuzzy
approach was compared to a quadratic optimal control approach that assumes linearization
of the original dynamics. While in open-loop the linearized model clearly diverges from the
original nonlinear system for larger swing angles, in closed loop both controllers are able
to conduct the states to the reference, i.e., to move the cart to set points quickly with a
reasonable maximum overshoot, and, simultaneously, to suppress the load swing angle.
Nonetheless, although the quadratic optimal design can provide adequate closed-loop dy-
namics in usual situations, the fuzzy design tolerates larger disturbances, has a larger relative
robustness factor, and gives smaller maximum overshoots using amplitude-constrained inputs,
which tends to increase the actuator lifetime. Laboratory experiments and actual data are
considered to validate the fuzzy control approach for cranes in an actual scenario. The fuzzy
controller has shown to be realistic, nonlinear and smooth.
In the future, scattering fuzzy clustering methods will be considered for the determination
of antecedent membership functions. Second, consecutive missing data from sensors may lead
Lyapunov-stable closed-loop systems to vulnerability and instability. Real-time missing data
imputation methods will be taken into account in the design of fault-tolerant fuzzy controllers.
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Third, we have used the method called Local Approximation in Fuzzy Partition Spaces to
reduce the number of local models. Therefore, with a compact fuzzy model in hands, finding
local fuzzy controllers based on a P -common Lyapunov matrix became easier and faster.
Instead, we may consider a local-model-dependent Lyapunov matrix, P
i ∀ i, for relaxation
of the design problem. Since simultaneous activation of some of the local models may be
impossible, then a smaller LMI is obtained. Fuzzy and neuro-fuzzy modeling of shipboard
cranes, and inclusion of velocity-proportional damping in the model are also future topics.
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
Daniel Leite acknowledges the Serrapilheira Institute (grant number Serra-1812-26777).
Igor Škrjanc is grateful to the Slovenian Research Agency (Program P2-0219: Modeling,
Simulation and Control).
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