HUNGARIAN JOURNAL
OF INDUSTRY AND CHEMISTRY

VESZPRÉM
Vol. 42(2) pp. 71–77 (2014)

THE APPLICATION OF UNKNOWN INPUT ESTIMATORS TO DAMP LOAD
OSCILLATIONS OF OVERHEAD CRANES

BÁLINT PATARTICSB AND BÁLINT KISS

Department of Control Engineering and Information Technology, Budapest University of Technology and Economics,
Magyar Tudósok krt. 2, Budapest, 1111, HUNGARY

BE-mail: patarticsbalint@gmail.com

This paper focuses on the development of state estimation methods for mechanical systems with uncertain frictional pa-
rameters. The goal of the study is to provide reliable angle estimation for state-feedback-based crane control solutions,
designed to reduce load sway. Cranes are underactuated systems, usually unequipped with the sensors necessary to measure
the swinging angle, therefore the damping of their oscillatory behaviour is a challenging task. Two estimators are proposed
for the calculation of the unmeasured states. One is based on an ’unknown input Kalman filter’ (UIKF), the other applies
the ’unscented Kalman filter’ (UKF) with load prediction. Simulation results are provided to demonstrate the accuracy of
the algorithms.

Keywords: overhead crane, state estimation, nonlinear systems, unknown inputs, Kalman filter

Introduction

The sway of the load carried by cranes is an unwanted
phenomenon. Friction may decouple load motion from
the remaining crane mechanism. This causes difficulty
for inexperienced operators, especially when they try to
land the cargo. The friction present in the crane mech-
anism makes the reduction of the swinging particularly
demanding. Therefore the goal of the design of crane con-
trol systems is often to ensure minimal swinging along a
specified trajectory [1–6].

There are numerous algorithms available in the liter-
ature for the solution to this problem. The design of such
a control system can be addressed using soft computing
methods such as fuzzy logic and neural network with
genetic algorithm. These solutions require however the
measurement of the load position, which is often based on
workspace visualization [1]. Hard commuting techniques
are also widely applied for sway reduction. An H2 opti-
mal solution is described in Ref.[2], which also relies on
image processing for the measurement of the load coordi-
nates. There are a variety of algorithms taking advantage
of the flatness property (or exact linearizability) of over-
head cranes [7]. The control systems in Refs.[3] and [4]
consider all the states measurable, however an estimator
is proposed in Ref.[3] for the computation of the uncer-
tain parameters.

In the case of navy cranes, local asymptotic stability
can be achieved without the measurement of the swing-
ing angle, if trajectory planning exploits the flatness prop-
erty of the system [6]. There is also a tracking control
algorithm specifically designed for overhead cranes, that

is capable of effectively reducing the sway despite the
friction [5]. For the calculation of the unmeasured states,
this controller uses a linear observer. In this paper we
propose new methods of state observation for systems
that take into account friction, and are applicable to over-
head cranes. These estimators can replace the linear ob-
server in Ref.[5] or the need to measure the load coor-
dinates in any other state-feedback-based method to en-
hance the precision of the control. Our estimation meth-
ods are based on the assumption that the effect of the fric-
tion on the inputs can be reduced. This is true for most
mechanical systems (including overhead cranes), where
the actuating signals are usually forces and torques.

In the following section, the modelling of Lagrangian
mechanical systems is overviewed, and it is shown how
the friction can be handled as an input disturbance. In the
next section the state estimation techniques are presented
based on the possible approaches of interpreting the fric-
tion. The next section describes the results of the applica-
tion of these methods to the state estimation problem of
overhead cranes and presents the simulation results.

Lagrangian-Based Models of Mechanical Systems

The nonlinear dynamics of the controlled system can be
obtained using the Euler-Lagrange equations. If the gen-
eralized coordinates are q =

(
q1 q2 . . .qn

)T
and the

generalized forces are τ =
(
τ1 τ2 . . .τn

)T
, the Euler-

Lagrange equations read

d

dt

∂L
∂q̇i
−
∂L
∂qi

= τi, (1)



72

where L is the so-called Lagrangian, the difference be-
tween the kinetic and potential energy of the system,
q̇i =

dqi
dt

and i = 1, 2, . . .n.
Using these equations, under some conditions the

controlled system’s model can be written in the form

H(q)q̈ + h(q, q̇) + hs(s) = τ, (2)

where H(q) is the inertia matrix, h(q, q̇) comprises the
centrifugal, Coriolis and gravitational terms, and hs(s)
is the frictional term with s being the vector of the fric-
tional forces. In this form hs(s) can be reorganized to
the right-hand-side of the equation. Introducing the dis-
turbance torques as τ̃ = τ −hs(s), Eq.(2) yields

H(q)q̈ + h(q, q̇) = τ̃. (3)

The state-space representation of the model is

ẋ = f(x, u)

y = g(x),
(4)

where x is the state, u is the input, y is the output of
the system, and f and g are nonlinear functions. This
form can be obtained from Eq.(5) assuming that xT =(
qT q̇T

)
, and the output is not directly dependent on

the torques. The state-space equations read

ẋ =

(
q̇

−H−1(q)h(q, q̇)

)
+

(
0

H−1(q)

)
τ̃

y = g(x).

(5)

The inputs of mechanical systems are usually forces and
torques, consequently they are only dependent on τ̃. This
means, that the effect of the friction can be reduced to the
actuated degrees of freedom of the system.

If it is necessary for the design of the control system,
the nonlinear dynamics can be approximated by a first
order Taylor series expansion around a setpoint (x0, u0).
The linear model then becomes

δẋ = Aδx + Bδu

δy = Cδx,
(6)

where the quantities prefixed with δ mean the distance
from the setpoint, e.g. δx = x−x0, and the matrices are
Jacobians [8].

State Estimation Methods

Since the friction can be interpreted in our setup as an
input disturbance, we consider two ways for its compen-
sation in the state estimation. Either all input variables
can be disregarded, or the additive disturbance can be es-
timated using a load predictor. These ideas form the basis
of the two estimator design techniques described as fol-
lows.

The computations of the controllers implemented in
embedded systems are based on sampled signals. Hence
the design of the estimators is done in discrete-time.

Unknown Input Approach

Consider the discrete-time linear stochastic system model

x[k + 1] = Φx[k] + Γdd[k] + ω[k]

z[k] = Cx[k] + v[k],
(7)

where x is the state, d is the unknown input (disturbance)
of the system and z is the measurement of the output
y[k] = Cx[k]. Φ and Γd are obtained from the matrices
of Eq.(6) in the form of

Φ = eATs, Γd =

∫ Ts
0

eAtB dt, (8)

where Ts is the sample time. The measurement and pro-
cess noise v and ω are assumed to be additive, white, and
Gaussian with mean of zero. Notice, that in Eq.(7), all of
the system inputs are considered to be disturbances.

The state and the unknown input of such a system
can be estimated using an ’unknown input Kalman filter’
(UIKF) [9], if the system satisfies the conditions

dim{d}≤ dim{y} (9)
rank{C} = dim{y} (10)

rank{Γd} = dim{d} (11)
rank{CΓd} = dim{d} . (12)

Additive Disturbance Approach

In the case of input disturbances, load prediction is of-
ten applied. Let us introduce the discrete-time nonlinear
model of a system in the form

x[k + 1] = φ(x[k], ũ[k]) + ω[k]

z[k] = g(x[k]) + v[k],
(13)

where ũ is the disturbed input of the system, and φ and
g are nonlinear functions. We will follow the assumption,
that the disturbance is additive to the input, thus ũ[k] =
u[k] + Λd[k], where u is the vector of actuating signals,
and Λ is a constant matrix.

We also assume a constant disturbance model with
process noise ωd, d[k + 1] = d[k] + ωd[k], resulting in
the model

x̃[k + 1] = φ̃(x̃[k],u[k]) + ω̃[k]

z[k] = g̃(x̃[k]) + v[k].
(14)

In Eq.(14) x̃T =
(
xT dT

)
is the extended state and

ω̃T =
(
ωT ωTd

)
is the extended process noise. The ex-

tended nonlinear mappings of the system are given by
g̃(x̃[k]) = g(x[k]) and

φ̃(x̃[k], u[k]) =

(
φ(x[k], u[k] + Λd[k])

d[k]

)
. (15)

In most cases the continuous-time differential equa-
tion of the observed system is available, in the form



73

M
uF

uC

r
`

θ

J
ρ

cartwinch

m

load

rail

Figure 1: The two-dimensional ovearhead crane

of Eq.(4). The discrete-time state-transition equation can
then be obtained by

φ(x[k], u[k]) = x[k] +

(k+1)Ts∫
kTs

f(x, u[k]) dt, (16)

where the input is piecewise constant and u[k] = u(kTs).
If the integral in Eq.(16) cannot be given in closed form
and Ts is sufficiently small, the left rectangle rule approx-
imation

(k+1)Ts∫
kTs

f(x, u[k]) dt ≈ Tsf(x[k], u[k]). (17)

can be used. Substituting this into Eq.(15), the expression
reads

φ̃(x̃[k],u[k]) =

(
x[k] + Tsf(x[k], u[k] + Λd[k])

d[k]

)
.

(18)
For the observation of the system described

by Eq.(14) and Eq.(18) the ’unscented Kalman fil-
ter’ (UKF) [10] is applicable. We chose one of the
implementation variants of this filter from Ref.[10].

Application of the Methods to Overhead Cranes

System Dynamics

We will now consider the state estimation problem of the
two-dimensional overhead crane (Fig.1). Let us denote
the horizontal displacement of the cart by `, the length
of the rope by r and the angle between the vertical and
the rope by θ. The generalized coordinates of the system
are chosen to be q =

(
` r θ

)T
, and the state is x =(

` r θ ˙̀ ṙ θ̇
)T

. Based on the sensors available,

the output is y =
(
` r ˙̀ ṙ

)T
.

Overhead cranes is a typical example of underactu-
ated systems. It is actuated by two motors, one apply-
ing the force uF on the cart, and another delivering the

estimated system

s

Λ

+
+

u

u

system model

s = η(x,ẋ,u)

ẋ = f(x,ũ) y = g(x)
ũ y∫ẋ x

Figure 2: Dynamics of the estimated system

torque uC to the winch. The system input consists of
these, u =

(
uF uC

)T
.

The parameters of the system are the mass of the cart
M, the mass of the load m, the moment of inertia of the
winch J and the radius of the winch ρ. The acceleration
in the gravitational field is denoted by g.

The following assumptions are made of the system.

• The rope connecting the load to the winch is mass-
less and behaves as a rigid rod during the motion.
• The load is a point mass.
• M, m, J and ρ are known.
• In the initial state of the system the unmeasured

states are zero: θ = 0 rad and θ̇ = 0 rad/s.
• The effect of the aerodynamic resistance is negligi-

ble.

The Lagrangian of overhead cranes reads

L =
1

2
(M + m) ˙̀2 +

1

2

J

ρ2
ṙ2 + +

1

2
m
(
ṙ2 + r2θ̇2

)
+ ml̇ṙ sin θ + ml̇rθ̇ cos θ + mgr cos θ.

(19)

Using Eq.(1), the model can be written in the form of
Eq.(2), where the expressions become

H(q) =



(
M + m sin2 θ

)
m sin θ 0

m sin θ
(
J
ρ2

+ m
)

0

cos θ 0 r


 , (20)

h(q, q̇) = −


m(rθ̇2 + g cos θ) sin θmrθ̇2 + mg cos θ

−2ṙθ̇ −g sin θ


 , (21)

hs(s) = −


sFsC

0


 , τ =


 uF−uC

ρ

0


 , (22)

where sF is the frictional force between the cart and the
rail and sC is the frictional force applied to the winch.

The continuous-time linear model of the crane is com-
puted around the setpoint x0 =

(
0 r0 0 0 0 0

)T
,

u0 =
(
0 mgρ

)T
, and sF = sC = 0 is substituted



74

Table 1: Numeric values of the parameters

Value Unit
M 5 kg
m 0.05 kg
J 3.802 · 10−4 kg m2
ρ 0.02 m
g 9.81 m s−2

Ts 1 ms
r0 0.47 m
`D 0.5 m

into Eq.(5). The resulting Jacobians are

A =




0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
0 0 m

M
g 0 0 0

0 0 0 0 0 0
0 0 −m+M

Mr0
g 0 0 0


 ,

B =




0 0
0 0
0 0
1
M

0
0 − ρ

mρ2+J

− 1
Mr0

0


 ,

C =




1 0 0 0 0 0
0 1 0 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0


 .

(23)

Substituting Eq.(22), the disturbance torque term
in Eq.(3) reads

τ̃ =


 uF + sF−1

ρ
(uC −ρsC)

0


 . (24)

Eq.(24) shows, that the disturbed input of the system can
be introduced in the form ũ = u + Λs, where

Λ =

(
1 0
0 −ρ

)
. (25)

The frictional forces usually depend on the actuating
signals, state variables, and the derivative of the state as
well. The general characteristics of the friction reads as
s = η(x, ẋ, u). The block diagram of the dynamics of
the system is illustrated in Fig.2. Only the system model
block is used for estimator design, which is Eq.(5) if the
effect of the friction is disregarded, hs(s) = 0.

Proposition 1. The discrete-time linear crane model ob-
tained from the matrices in Eq.(23) and using Eq.(8) sat-
isfies the design conditions of the UIKF in Eq.(9-12).

Proof. It is clear that dim{y} = 4, dim{d} = 2 and
rank{C} = 4, thus Eq.(9) and (10) are satisfied. Us-

ing Eq.(8) and the numerical values in Table 1,

Γd = 10
−2




0.0001 0
0 −0.025

−0.0002 0
0.2 0
0 −49.975

−0.4255 0


 ,

CΓd = 10
−2




0.0001 0
0 −0.025

0.2 0
0 −49.975


 .

(26)

We obtain rank{Γd} = 2 and rank{CΓd} = 2, which
are both equal to the number of unknown inputs. This sat-
isfies Eqs.(11) and (12). The continuity argument holds
for the rank of the matrices in Eq.(26), therefore there ex-
ists a neighbourhood around our parameter set where the
design conditions are still satisfied.

Simulation

The estimation results were obtained by simulation of
overhead cranes in a closed-loop scenario using a simple
discrete-time pole-placement-based linear controller. The
simulation also took into account friction. There are var-
ious models of the friction phenomenon, ours included
Coulomb, Stribeck, viscous friction, and stiction [11].
The numerical values of the parameters used in the simu-
lation are given in Table 1.

Because of friction, but mainly stiction, the position
of the crane can not be accurately controlled without in-
tegrators in the controller. Consequently we included the
integral of the measurable positions in the controller de-
sign as proposed on page 309 of Ref. [8].

The outputs to be integrated are yI =
(
` r

)T
. In-

tegrals of these quantities are approximated by the left
rectangle rule as

xI[k + 1] = xI[k] + TsyI[k] = xI[k] + TsCIx[k], (27)

where CI is defined so that yI[k] = CIx[k]. In our case
CI is the first two rows of C in Eq.(23). Using Eq.(27)
and Eq.(7) a new system model is introduced in the form

x̃[k + 1] = Φ̃x̃[k] + Γ̃u[k]

y[k] = C̃x̃[k].
(28)

where the expanded state is x̃T =
(
xT xTI

)
and the

matrices are

Φ̃ =

(
Φ 0

TsCI I

)
, Γ̃ =

(
Γd
0

)
,

and C̃ =
(
C 0

)
.

(29)

For the controller design the process and measurement
noises are omitted.



75

∫
DAC

system
in Fig. 2 ADC

UKF

K

KI

u[k] y y[k]u

u0

+

+-

-

+

yI D -
+

xD

ADC
x[k]

x

yI [k] = (`[k] r[k])
T y[k]

yI [k]

x̂[k]

ŝ[k]

UIKF

x̂[k]

ˆ̃u[k]

u[k]

CKF
x̂[k]

Figure 3: Closed-loop estimation setup used in the simulation

0 1 2 3 4 5 6 7

time [s]

−10
0

10

20

30

40

50

60

u
F

,
ˆ̃ u
F

,
[N

]

uF

UIKF

UKF

0 1 2 3 4 5 6 7

time [s]

−1.0
−0.5

0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5

u
C

,
ˆ̃ u
C

,
[N

m
]

uC

UIKF

UKF

Figure 4: Actuating signals and the estimated inputs of
the system

We want the state-feedback controller to output
u[k] = −K̃x̃[k] = −Kx[k] − KIxI[k]. The gain
K̃ can be computed using the eigenvalue placement
method implemented in Matlab’s place algorithm:
K̃ = place(Φ̃, Γ̃, λ), where λ contains the chosen
eigenvalues of the closed-loop system. In continuous time
they are −1 + 0.5j, −1 − 0.5j, −5.1, −5.2, −5.3 −5.4,
−5.5, and −5.6. The controller gains are

K =




100.594 −0.1273
0.6011 −1.6765
−185.407 0.0458
81.4011 −0.0558
0.0562 −0.3166
−4.2257 −0.0195




T

, and

KI =

(
46.641 1.6026
−0.074 −2.955

)
.

(30)

The closed-loop setup is illustrated in Fig.3. Here the
integration is done using the left rectangle rule, and the

0 1 2 3 4 5 6 7

time [s]

−0.1
0.0

0.1

0.2

0.3

0.4

0.5

m
e
a
su

re
d

st
a
te

s
[m

,
m

/
s]

`

r

˙̀

ṙ

0 1 2 3 4 5 6 7

time [s]

−0.4
−0.3
−0.2
−0.1

0.0

0.1

0.2

u
n

m
e
a
su

re
d

st
a
te

s
[r

a
d

,
ra

d
/
s]

θ

θ̇

Figure 5: State evolution of the controlled crane using
the true states as feedback

ADC and DAC blocks are analogue-digital and digital-
analogue converters respectively. The DAC uses the zero-
order hold method for signal reconstruction. The desired
state of the system is xD =

(
`D r0 0 0 0 0

)T
,

`D is the desired position of the cart and yID = CIxD.
Since we only want to demonstrate the results of

the state estimation, first the real states are fed back
to the controller. To compare the accuracy of our me-
thods to a traditional solution, the classical Kalman filter
(CKF) [12] is also implemented. As a result, the Kalman
filters compute the estimated state of the system x̂[k], and
they also provide information about the input.

The UIKF estimates the friction loaded actuating
signals directly, while the UKF estimates the frictional
forces superposed to the actuating signals. The actuating
signals along with the estimates of the real inputs pro-
vided by the filters can be seen in Fig.4. Here, the esti-
mated input of the UKF is computed as ˆ̃u = u[k] + Λŝ[k].

Because of friction, the real inputs are estimated to be



76

0 1 2 3 4 5 6 7

time [s]

0.00000

0.00005

0.00010

0.00015

0.00020

0.00025

0.00030

0.00035
`
−

ˆ̀
[m

]

UIKF

UKF

0 1 2 3 4 5 6 7

time [s]

−0.001
0.000

0.001

0.002

0.003

0.004

0.005

˙̀
−

ˆ̇ `
[m

/
s]

UIKF

UKF

Figure 6: Error of the position and velocity estimations

much less than the actuating signals. The large offset in
Fig.4 is a consequence of the stiction, the cart will not
move until uF becomes greater than a certain value (in
our case about 28 N). When the cart finally stops mov-
ing, the fluctuation of uF tries to counter the load sway,
during which the measured states do not change. This is
the reason why the estimators find the input to be zero.

Fig.5 shows the transient of the states in a closed-
loop. Results show that the controller can bring the sys-
tem into the desired state, but friction prevents it from
stopping the low amplitude swinging of the load, because
it makes quick, short motions impossible. Error estimates
are illustrated by Figs. 6 and 7. In all cases, the error be-
comes zero when the crane reaches the desired state.

This is not true for the angle and its derivative in Fig.8.
The error decreases at the end of the state transient, but
does not become zero. When the cart stops and the load
swings, the measurements are all constant thus they carry
no information regarding θ and θ̇. In these situations the
cart and winch are usually also motionless due to fric-
tion. Consequently, the estimation error of these quanti-
ties never becomes zero.

It is possible to decrease the estimation error using
two laser slot sensors. With the help of such devices, θ
can be measured accurately in two positions and through
the application of proper sensor fusion techniques the es-
timation of the angle can become more precise in between
the chosen positions as well.

In Fig.9 the estimation error of θ and θ̇ is illustrated
using CKF. The filter was designed for the dynamics
given by Eq.(7) except that it does not consider the inputs
unknown. The results show that the error is substantially
higher than in the case of the UIKF or UKF.

0 1 2 3 4 5 6 7

time [s]

−0.00005

0.00000

0.00005

0.00010

0.00015

r
−
r̂

[m
]

UIKF

UKF

0 1 2 3 4 5 6 7

time [s]

−0.0010
−0.0005

0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035

ṙ
−

ˆ̇ r
[m

/
s]

UIKF

UKF

Figure 7: Error of the rope length and hoisting velocity
estimations

0 1 2 3 4 5 6 7

time [s]

−0.003
−0.002
−0.001

0.000
0.001
0.002
0.003
0.004
0.005

θ
−
θ̂

[r
a
d

]

UIKF

UKF

0 1 2 3 4 5 6 7

time [s]

−0.020
−0.015
−0.010
−0.005

0.000
0.005
0.010
0.015
0.020
0.025

θ̇
−

ˆ̇ θ
[r

a
d

/
s]

UIKF

UKF

Figure 8: Error of the angle and angular velocity
estimations

Fig.8 shows the estimation error of the UKF to be
less. We applied this estimator in a closed-loop scenario
identical to the one presented in Fig.3, but instead of x[k]
the x̂[k] of the UKF was used in the feedback loop. The
state evolution of the crane in this simulation is illustrated
in Fig.10. The system’s behaviour is only slightly differ-
ent from the case when the actual states were used in the
feedback control in Fig.5.



77

0 1 2 3 4 5 6 7

time [s]

−0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4

θ
−
θ̂

[r
a
d
]

0 1 2 3 4 5 6 7

time [s]

−5
−4
−3
−2
−1

0
1
2
3

θ̇
−

ˆ̇ θ
[r

a
d
/
s]

Figure 9: Error of the angle and angular velocity
estimations using CKF

Conclusion

It was shown that in most mechanical systems the dis-
turbing effect of friction can be reduced to the actuating
signals. Based on this statement, two estimation meth-
ods were provided for the computation of the unmeasured
states. One of these techniques considers the inputs com-
pletely unknown and uses UIKF for the estimation. The
other algorithm treats the inputs as distorted by an addi-
tive disturbance, and applies UKF with a load predictor
extension. Simulation results in a closed-loop controlled
scenario were provided to prove the applicability of the
concepts. It was also pointed out, that the precision of
the estimators could be improved using laser slot sensors.
This will be the subject of a forthcoming paper.

REFERENCES

[1] HYLA P.: The crane control systems: A survey,
Proc. 17th Int. Conf. Methods and Models in Au-
tomation and Robotics (MMAR), 2012, 505–509

[2] SCHINDELE D., MENN I., ASCHEMANN H.: Non-
linear optimal control of an overhead travelling
crane, Proc. 18th IEEE Int. Conf. Control Applica-
tions, 2009, 1045–1050

[3] BOUSTANY F., D’ANDREA NOVEL B.: Adap-
tive control of an overhead crane using dynamic
feedback linearization and estimation design, Proc.
IEEE Int. Con. Robotics and Automation, 1992,
1963–1968

0 1 2 3 4 5 6 7

time [s]

−0.1
0.0

0.1

0.2

0.3

0.4

0.5

m
e
a
su

re
d

st
a
te

s
[m

,
m

/
s]

`

r

˙̀

ṙ

0 1 2 3 4 5 6 7

time [s]

−0.4
−0.3
−0.2
−0.1

0.0

0.1

0.2

u
n

m
e
a
su

re
d

st
a
te

s
[r

a
d

,
ra

d
/
s]

θ

θ̇

Figure 10: State evolution of the controlled crane using
the UKF’s estimated states as feedback

[4] NEUPERT J., HILDEBRANDT A., SAWODNY O.,
SCHNEIDER K.: Trajectory tracking for boom
cranes using a flatness based approach, Proc. Int.
Joint Conf. SICE-ICASE, 2006, 1812–1816

[5] RÓZSA T., KISS B.: Tracking control for tow-
dimensional overhead crane, Proc. 8th Int. Conf.
Informatics in Control, Automation, and Robotics,
ICINCO, 2011, 1, 427–432

[6] KISS B., LEVINE J., MULLHAUPT P.: A simple
output feedback PD controller for nonlinear cranes,
Proc. 39th IEEE Conference on Decision and Con-
trol, 2000, 5, 5097–5101

[7] KISS B., LÉVINE J., MÜLLHAUPT P.: Modelling,
flatness and simulation of a class of cranes, Electri-
cal Engineering, 1999, 43, 215–225

[8] LANTOS B.: Theory and design of control systems
I., Akadémia Kiadó, 2001 (in Hungarian)

[9] DAROUACH M., ZASADZINSKI M., ONANA A.B.,
NOWAKOWSKI S.: Kalman filtering with unknown
inputs via optimal state estimation of singular sys-
tems, Int. J. Sys. Sci., 1995, 26(10), 2015–2028

[10] HAYKIN S.S.: Kalman filtering and neural net-
works, John Wiley & Sons, Inc., 2001

[11] ARMSTRONG-HÉLOUVRY B., DUPONT P., DE
WIT C.C.: A survey of models, analysis tools and
compensation methods for the control of machines
with friction, Automatica, 1994, 30, 1083–1138

[12] KALMAN R.E.: A new approach to linear filtering
and prediction problems, Journal of basic Engineer-
ing, 1960, 82, 35–45