11564


FACTA UNIVERSITATIS  
Series: Mechanical Engineering Vol. 21, No 1, 2023, pp. 1 - 20  

https://doi.org/10.22190/FUME230125009B 

© 2023 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND 

Original scientific paper 

DISCRETE-TIME MODEL-BASED SLIDING MODE 

CONTROLLERS FOR TOWER CRANE SYSTEMS 

Anamaria-Ioana Borlea1, Radu-Emil Precup1,2, Raul-Cristian Roman1 

1Politehnica University of Timisoara, Department of Automation and Applied Informatics, 

Timisoara, Romania 
2Romanian Academy – Timisoara Branch, Center for Fundamental and Advanced 

Technical Research, Timisoara, Romania 

Abstract. This paper applies three classical and very popular discrete-time model-

based sliding mode controllers, namely the Furuta controller, the Gao controller, and 

the quasi-relay controller due to Milosavljević, to the position control of tower crane 

systems. Three single input-single output (SISO) control systems are considered, for 

cart position control, arm angular position control and payload position control, and 

separate SISO controllers are designed in each control system. Experimental results 

are included to support the comparison of the three plus three plus three sliding mode 

controllers. 

Key words: Discrete-time model-based sliding mode controllers, Furuta controller, 

Gao controller, Tower crane systems 

1. INTRODUCTION 

Sliding Mode Control (SMC) is a particular kind of variable structure system 

originated in the early 60’s [1]. The main idea of the general Variable Structure Control 

(VSC) laws is to use a high-speed switching control scheme to drive the process state 

trajectory onto a specified hyper-surface, which is commonly called the sliding surface or 

switching surface, and next keep the process state trajectory moving along this surface [2] 

in order to meet the performance specifications imposed to the control system. 

The discontinuous nature of the control signal helps to maintain a high performance 

of SMC and VSC by switching between two distinct control structures [3]. Because of 

 
Received: January 25, 2023 / Accepted March 05, 2023 

Corresponding author: Radu-Emil Precup 
Politehnica University of Timisoara, Department of Automation and Applied Informatics, Bd. V. Parvan 2, 

300223 Timisoara, Romania 

Romanian Academy – Timisoara Branch, Center for Fundamental and Advanced Technical Research, Bd. 
Mihai Viteazu 24, 300223 Timisoara, Romania 

E-mail: radu.precup@aut.upt.ro 



2 A.-I. BORLEA, R.-E. PRECUP, R.-C. ROMAN 

this switching behavior, SMC can have some dead zones for parameters variations, and is 

not sensitive to disturbances [4]. 

This paper is focused on SMC of tower crane systems. Such control approaches are 

outlined as follows. In [5], a continuous-time SMC law is designed based on a nonlinear 

model of a Tower Crane System (TCS). Two variants of discrete-time data-driven SMC 

laws for TCSs are presented in [6]. 

Many papers on different controllers for cranes have been reported in the past years. 

The SMC problem for overhead crane is the subject of [7], which describes a model 

based integral SMC scheme for discrete-time systems. The authors of [8] offer a model-

based combination of SMC. Second-order SMC for controlling the trolley in the XOY 

plane is addressed in [9]. An adaptive fuzzy SMC is developed by [10] for trolley position and 

sway control in the XOY plane, where two linear sliding surfaces are defined for the position 

and sway angle. 

Tower cranes can be found in fewer papers then overhead cranes because of their 

increased complexity. An adaptive control scheme for underactuated tower cranes is 

proposed in [11] to achieve simultaneous slew/translation positioning and swing suppression, 

using this approach is no need for linearize the tower crane dynamical equations around the 

equilibrium point or to neglect nonlinear terms. Controllers which are composed of partial 

feedback linearization and SMC are suggested in [13], guaranteeing the robustness in the 

case of variations of several system parameters. Integral Sliding Mode Control (ISMC) for 

tower cranes is proposed in [13] to ensure precise tracking of the desired position while 

reducing the oscillations of the payload. The controller in [13] is designed using a high 

fidelity nonlinear dynamical model, however, the switching gain must be limited to 

implement the SMC on the real TCS, and as a result a steady-state error will be present in 

the system’s outputs; this shortcoming is overcome using ISMC. 

The purpose of this paper is to apply model-based SMC to a representative nonlinear 

process, namely the TCS. To demonstrate the performance of these approaches, the 

validation through experiments is presented. Three approaches to Discrete Time Sliding 

Mode Control (DTSMC) are described based on the mathematical model of the process: 

the DTSMC law in the approach proposed by K. Furuta [14], the DTSMC law in the 

approach proposed by W.-B. Gao et al. [15] and the DTSMC law of quasi-relay type 

proposed by Č. Milosavljević [16]. 

The remainder of this paper is organized as follows. In Section 2, the DTSMC 

problem formulation and the three control laws are described. The validation study is 

presented in Section 3. Conclusions are presented in Section 4. 

2. MODEL-BASED SLIDING MODE CONTROL PROBLEM FORMULATION 

Let the dynamical discrete-time system be described by the following single input 

linear time-invariant state-space mathematical model: 

 





=

+=
+

,

,
:

1

kk

kkk
u

P
xCy

BxAx
 (1) 

where xk = [xk,1 … xk,n]
T  n is the state vector, T indicates matrix transposition, k 

indicates the discrete time (sampling interval) index, kZ, k0, the (scalar) control input 



 Discrete-Time Model-based Sliding Mode Controllers for Tower Crane Systems 3 

is uk, yk = [yk,1 … yk,p]
T  p is the output vector, and Ann, Bn1 and Cpn 

are the system matrix, the input matrix, the output matrix, respectively [17]. Although the 

model in (1) is linear, the aim of SMC is to control nonlinear processes; however, this 

linear model is considered as it allows a relatively simple analysis and design of the 

control systems according to [14], [15] and [16]. 

Using the notation sk for the switching variable, the sliding hyper-surface 

 n
kk

n

k
sS === }0{ xKx  (2) 

is determined by choosing the gain matrix of the sliding hyper-surface n
1

K  so that 

the system (1) is stable as long as xk remains on S [14]. 

2.1. Furuta DTSMC law 

A type of DTSMC system is proposed by K. Furuta in [14], in which the discrete-time 

control law for the system (1), referred to here as the Furuta DTSMC law, is 

 ,
d

k

eq

kk
uuu +=  (3) 

where d
k

u  is discontinuous control input and the equivalent control input is 
eq

k
u . Imposing 

sk = 0, the control law to keep the state on (2) is given by 

 

,)()(

,

11 n

eq

keq

eq

k
u

−
−−=

=

IAKBKF

xF
 (4) 

with I – the identity matrix of order n. 

The discontinuous control law is described as 

 

.]  ...  [

,

1

1

nT

nd

kd

d

k

ff

u


=

=

F

xF
 (5) 

It is proved in [14] that the system (1) controlled with the control law in (3) is stable if 

the absolute value of the ith element of Fd, i.e. fi, i=1…n, satisfies 

 









−



−

=

,)(if

,)(if0   

,)(if   

,0

,

,0

iikk

iikk

iikk

i

xsf

xs

xsf

f







BK

BK

BK
 (6) 

in which xk,i is the i
th element of xk and i is defined as 

 ,||)(||5.0
1

,

2

,0 
=

=
n

j

jkiki
xxf BK  (7) 

with the amplitude of f0 limited by 

 
,

)(

2
0

1

1

0


=




n

j

j
t

f

BK

 (8) 



4 A.-I. BORLEA, R.-E. PRECUP, R.-C. ROMAN 

where t1 is the first column of the matrix [t1 t2 … tn] in which t1i is the i
th element of t1 

satisfying Kt1=1, Kti=1, i=2…n; furthermore t1 and ti, with i=2…n, are linearly 

independent [14]. 

The gain matrix K shall be designed so that the system [14] 

 
kk

xIAKBKBAx −−=
−

+
)]()([

1

1
 (9) 

is stable. The performance specifications imposed to the control system are zero stationary 

control with respect to constant reference inputs and reasonable settling time. The 

guidelines to design the Furuta DTSMC law such that to meet these performance 

specifications will be given in Section 3.1. 

2.2. Gao DTSMC law 

A type of SMC system is described by W.-B. Gao et al. in [15], where the dynamic 

behavior of the reaching law is as follows: 

 ,)sgn(
1 kskskk

sTsTqss −−=−
+

  (10) 

where Ts is the sampling period, q is a scalar parameter that fulfills qTs(0,1) and >0 is a 
parameter. 

Solving for uk the equation obtained from (1), (2) and (10) leads to the following 

control law, referred to here as the Gao DTSMC law [15]: 

 )}.sgn( ])1([{
)(

1
ksksk

TTqu xKxIAK
BK

+−−


−=   (11) 

Imposing the same performance specifications as in Section 3.1, the guidelines to design 

the Gao DTSMC law will be presented in Section 3.2. 

2.3. Quasi-relay DTSMC law 

Č. Milosavljević describes in detail in [16] the quasi-relay control law for the system 

(1), referred to as the quasi-relay DTSMC law, with the expression 

 ,1),sgn(
1

,
−








−= 

=

nrsxu
k

r

i

ikik
  (12) 

where i>1 are the parameters which must be chosen by the designer. 
The general condition of existence and achievement of the quasi-sliding mode is [16] 

 .0 ,Z ,||||
1


+

kkss
kk

 (13) 

Imposing the same performance specifications as in Section 3.1, the guidelines to design 

the quasi-relay DTSMC law will be specified in Section 3.3. 



 Discrete-Time Model-based Sliding Mode Controllers for Tower Crane Systems 5 

3. VALIDATION CASE STUDY 

The TCS is presented in detail in [6] and [18], as representative process to validate various 

controls algorithms including those discussed in relation with this process. The TCS is a 

nonlinear electromechanical system with a complex dynamic behavior. The system illustrated 

in Fig. 1 has three controlled outputs, namely the cart position y1(m)=x3(m), the arm angular 

position y2(rad)=x4(rad) and the payload position y3(m)=x9(m). 

 

Fig. 1 Block diagram of principle of Tower Crane system [6]. 

The three actuators shown in Fig. 1 are Direct Current (DC) motors, and Pulse Width 

Modulation (PWM) is involved. The variable m1[−1, 1] is the output of the saturation and 

dead zone static nonlinearity specific to the first actuator: 

 

















−−

−

−−−+

−−

=

,)( if,1

,)( if),/())((

,|)(| if,0

,)( if),/())((

,)( if,1

)(

11

1111111

111

1111111

11

1

b

baaba

ac

cbcbc

b

utu

utuuuuutu

utuu

utuuuuutu

utu

tm
 

(14)
 

where t is the continuous time argument, tℜ, t≥0, u1(%)[−100, 100] and u1[−1, 1] is 

the first control input for cart position control, and the values of the parameters in (14) are 

[6] ua1=0.1925, ub1=1 and uc1=0.2. The variable m2[−1, 1] is the output of the saturation 

and dead zone static nonlinearity specific to the second actuator: 

 

















−−

−

−−−+

−−

=

,)( if,1

,)( if),/())((

,|)(| if,0

,)( if),/())((

,)( if,1

)(

22

2222222

222

2222222

22

2

b

baaba

ac

cbcbc

b

utu

utuuuuutu

utuu

utuuuuutu

utu

tm
 

(15)
 

where u2(%)[−100, 100] and next u2[−1, 1] is the second control input for arm angular 

position control, the values of the parameters in (15) are [6] ua2=0.18, ub2=1 and uc2=0.1538. 

The variable m3[−1, 1] is the output of the saturation and dead zone static nonlinearity 

specific to the third actuator: 



6 A.-I. BORLEA, R.-E. PRECUP, R.-C. ROMAN 

 

















−−

−

−−−+

−−

=

,)( if,1

,)( if),/())((

,|)(| if,0

,)( if),/())((

,)( if,1

)(

33

3333333

333

3333333

33

3

b

baaba

ac

cbcbc

b

utu

utuuuuutu

utuu

utuuuuutu

utu

tm
 

(16)
 

where u3(%)[−100, 100] and next u3[−1, 1] is the third control input for payload 

position control, the values of the parameters in (16) are [6] ua3=0.1, ub3=1 and uc3=0.13. 

The nonlinear state-space model of the TCS is [18] 

 
,

,

,

),(

,

,
1

,
1

),(

),(

,

,

,

,

93

42

31

1010

109

2

2

2
8

2

8

1

1

1
7

1

7

66

55

84

73

62

51

xy

xy

xy

fx

xx

m
T

k
x

T
x

m
T

k
x

T
x

fx

fx

xx

xx

xx

xx

P

P

=

=

=

=

=

+−=

+−=

=

=

=

=

=

=

























 

(17)
 

in which the expressions of the nonlinear functions f5, f6 and f10 are 

 

),sin2sin2cos2cos2

sin4sinsin2sinsin2sinsin2

cos4cossin2cos2sin2sin

coscos44(
2

1
),,,...,,,,()(

229

2

2
2

2

98
123

2

2
1

2

83

2108211

1

1
21

1

7
21

2

83

187212

2

19

2

819

2

6

21

2

986105

9

2110532155

xmx
T

k
x

T

xx
xmx

T

k
x

T

xx

xxxxxm
T

k
xx

T

x
xxxx

xxxxxgxxxxxxx

xxxxxxx
x

mmxxxxxff

PP

P





+−−+

+−++

−−+−

+−==

 
(18)

 



 Discrete-Time Model-based Sliding Mode Controllers for Tower Crane Systems 7 

 

),cossin

cossincoscossincoscos

coscossin2sinsin2coscos2

cos2(
cos

1
),,,...,,,,()(

2129

2

2

21

2

98
21

1

1
2

1

7
2219

2

8

2

2

83211082196521985

1106

19

2110532166

xxmx
T

k

xx
T

xx
xm

T

k
x

T

x
xxxxx

xxxxxxxxgxxxxxxxxx

xxx
xx

mmxxxxxff

P

P





−

++−+

−−−+−

−==

 
(19)

 

 

),coscos)(cossin)(cossin)(

sinsin2cossin2sincos2

(
coscos1

1
),,,,...,,,,()(

33
21

2

6

2

5921591269

219652110521106

10

21

3211053211010

L

P

L

L

m

mk
xxxxxxxfxxxfx

xxxxxxxxxxxxx

gx
mxx

mmmxxxxxff

+++++

−++

+


−
+

==



 (20) 

where, as specified in [6], kP1=0.188 m/s and kP2=0.871 rad/s are gains of the first two DC 

motors, TΣ1=0.1 s and TΣ2=0.1 s are time constants for the first two motors in which 

mL=0.33 kg is the payload mass, μL=1600 kg/s is the viscous coefficient associated with 

the payload motion, g=9.81 m/s2 is the gravitational acceleration, kP3=200 kg∙m/s
2 is 

process gain of the third DC motor, and zc(m) is the z coordinate of the payload. 

A part of the model described in Eq. (17) is discretized and next extended by adding 

the discrete-time integral block for zero stationary control error, and introducing a new 

state of this integral block [19]: 

 ,)/1(
,1, kikRkR

eTxx +=
+

 (21) 

where the error is ek=rk−yk, the reference input rk and Ti is the integral time constant (in 

continuous time). This extended model will have the extended state vector: 

 .]  [
T

R

T

e
xxx =  (22) 

Using relations (17) and (22) leads to the state vector for cart position control 

 ,]    [
,,7,3,1

T

kRkkke
xxx=x  (23) 

for arm angular position control 

 ,]    [
,,8,4,2

T

kRkkke
xxx=x  (24) 

and for payload position control 

 .]    [
,,10,9,3

T

kRkkke
xxx=x  (25) 

The gain matrix K is considered to have the following structure, where one subscript 

will be inserted in order to specify one of the three position control systems: 

 ].[
21 R

kkk=K  (26) 



8 A.-I. BORLEA, R.-E. PRECUP, R.-C. ROMAN 

Three separate single input-single output (SISO) control systems are considered, namely 

one for each controlled output y1, y2 and y3. Three discrete-time sliding mode controllers 

discussed in Section 2 are design for each SISO control system. The unified SISO control 

system structure is illustrated in Fig. 2, where DTSMC specifies the sliding mode controller 

(position), DTSMC  {Furuta DTSMC law, Gao DTSMC law, quasi-relay DTSMC law}, 

m  {1, 2, 3} indicates the number of the controlled output, which is also the number of the 

control system, and dm is the disturbance input, not considered in the model (17). 

 

Fig. 2 Unified SISO structure of discrete-time model-based sliding mode control system. 

The subscript k is omitted for the sake of simplicity. 

3.1. Furuta DTSMC law design and implementation 

Summarizing the information given in the previous sections, the guidelines to design 

the Furuta DTSMC law consist of the following steps: 

Step F1. Set the values of Ts to account for the requirements of quasi-continuous digital 

control and Ti, which affects the overshoot and the settling time. 

Step F2. Apply Eq. (9) to obtain the expression of the system matrix. Use Eqs. (21) to (25) 

to express the expression of the extended system matrix (with discrete-time integral block). 

Step F3. Choose K so that the system described by the matrix obtained at step F2 is 

stable. Pole placement can be applied in this regard. 

Step F4. Apply Eq. (8) to get the upper bound of f0, and choose a value which is 

between limits. 

These steps are exemplified as follows. For the cart position control system, the 

sampling period and the integral time constant are chosen as Ts=0.01 s and Ti=0.05 s. The 

expression of the system matrix is obtained using (9) and (23) [20] 

 
.

002000

//1/)2000(/

010

)()(
2212211

1

1 11111

















−

−+=−−=
−

kkkkkkkk
RReeeex

IAKBKBAA
 (27) 

A set of parameters that guarantee the stability of the system with the matrix in Eq. 

(27) is 

 ].005.02.03[
1

−=K  (28) 

For the upper amplitude of f0 described in Eq. (8), the matrix [t1 t2 t3] is set to [K
–1  0  0]; 

this gives 0 < f0 < 15.92 and the value is set to f0=1 [20]. 



 Discrete-Time Model-based Sliding Mode Controllers for Tower Crane Systems 9 

The cart position reference input trajectory is chosen to be the same as in [6]. The 

experimental results obtained for the control system with the controller parameters in Eq. 

(28) are illustrated in Fig. 3. 

 

Fig. 3 The results of cart position control system with Furuta DTSMC law: (a) u1; (b) r1 

(black), y1 (red). 

For the arm angular position control system, the sampling period and the integral time 

constant are chosen as Ts=0.01 s and Ti=0.012 s. The expression of the system matrix is 

obtained using Eqs. (9) and (24) [20] 

 
.

003/25000

//13/)25000(/

010

)()(
2212212

1

2 22222

















−

−+=−−=
−

kkkkkkkk
RReeeex

IAKBKBAA
(29) 

A set of parameters that guarantee the stability of the system with the matrix in (29) is 

 ].0002.02.015.3[
2

−=K  (30) 

For the upper amplitude of f0 described in Eq. (8), the matrix [t1 t2 t3] is set to [K
–1  0  0]; 

this gives 0 < f0 < 3.63 and the value is set to f0=3 [20]. 

The arm angular position reference input trajectory is chosen to be the same as in [6]. 

The experimental results obtained for the control system with the controller parameters in 

Eq. (30) are illustrated in Fig. 4. 



10 A.-I. BORLEA, R.-E. PRECUP, R.-C. ROMAN 

 

Fig. 4 The results of arm angular position control system with Furuta DTSMC law: 

(a) u2; (b) r2 (black), y2 (red). 

For the payload position control system, the sampling period and the integral time 

constant are chosen as Ts=0.01 s and Ti=0.0001 s. The expression of the system matrix is 

obtained using Eqs. (9) and (25) [20] 

 
.

001000000

//1/)1000000(/

010

)()(
2212213

1

3 33333

















−

−+=−−=
−

kkkkkkkk
RReeeex

IAKBKBAA
(31) 

A set of parameters that guarantee the stability of the system with the matrix in (29) is 

 ].0011.00909.59[
3
=K  (32) 

For the upper amplitude of f0 described in (8), the matrix [t1 t2 t3] is set to [K
–1  0  0]; this 

gives 0 < f0 < 107.43 and the value is set to f0=106 [20]. 

The payload position reference input trajectory is chosen to be the same as in [6]. The 

experimental results obtained for the control system with the controller parameters in Eq. 

(32) are illustrated in Fig. 5. 



 Discrete-Time Model-based Sliding Mode Controllers for Tower Crane Systems 11 

 

Fig. 5 The results of payload position control system with Furuta DTSMC law: (a) u3; (b) r3 

(black), y3 (red). 

3.2. Gao DTSMC law design and implementation 

Summarizing the information given in the previous sections, the guidelines to design 

the Gao DTSMC law consist of the following steps: 

Step G1. This step is identical to step F1. 

Step G2. This step is identical to step F2. 

Step G3. This step is identical to step F3. 

Step G4. Set q>0 to fulfill qTs (0,1), and >0. 
These steps are exemplified as follows. For the cart position control system, the 

sampling period and the integral time constant are chosen as Ts=0.01 s and Ti=0.2 s. The 

expression of the system matrix specific to the Furuta DTSMC law results from Eqs. (9) 

and (23) [20] 

 
.

00500

//1/)500(/

010

)()(
2212211

1

1 11111

















−

−+=−−=
−

kkkkkkkk
RReeeex

IAKBKBAA
 (33) 

A set of parameters that guarantee the stability of the system with the matrix in (33) is 

 ].0011.04.0825.0[
1

−=K  (34) 

The other parameters of the control law were set as q=99 such that qTs(0,1) and =200 [20]. 



12 A.-I. BORLEA, R.-E. PRECUP, R.-C. ROMAN 

Using the same reference input trajectory as in Section 3.1 and [6], the experimental 

results obtained for the control system with the controller parameters in Eq. (34) are 

given in Fig. 6. 

For the arm angular position control system, the sampling period and the integral time 

constant are chosen as Ts=0.01 s and Ti=0.2 s. The expression of the system matrix 

specific to the Furuta DTSMC law results from Eqs. (9) and (24) [20] 

 
.

00500

//1/)500(/

010

)()(
2212212

1

2 22222

















−

−+=−−=
−

kkkkkkkk
RReeeex

IAKBKBAA
 (35) 

 

Fig. 6 The results of cart position control system with Gao DTSMC law: (a) u1; (b) r1 

(black), y1 (red). 

A set of parameters that guarantee the stability of the system with the matrix in Eq. 

(35) is 

 ].0011.04.0825.0[
2

−=K  (36) 

The other parameters of the control law were set as q=99 such that qTs(0,1) and =500 [20]. 
Using the same reference input trajectory as in Section 3.2 and [6], the experimental 

results obtained for the control system with the controller parameters in Eq. (36) are 

given in Fig. 7. 



 Discrete-Time Model-based Sliding Mode Controllers for Tower Crane Systems 13 

 

Fig. 7 The results of arm angular position control system with Gao DTSMC law: (a) u2; 

(b) r2 (black), y2 (red). 

For the payload position control system, the sampling period and the integral time 

constant are chosen as Ts=0.01 s and Ti=0.0025 s. The expression of the system matrix 

specific to the Furuta DTSMC law results from Eqs. (9) and (25) [20] 

 

.

0079869184599999/1716871947673-

//141717986918/)5999996871947673(/

010

)()(

221221

3

1

3 33333

















−+=

−−=
−

kkkkkkkk
RR

eeeex
IAKBKBAA

 (37) 

A set of parameters that guarantee the stability of the system with the matrix in Eq. 

(37) is 

 ].00130.092.61[
3
=K  (38) 

The other parameters of the control law were set as q=99 such that qTs(0,1) and =300 [20]. 
Using the same reference input trajectory as in Section 3.3 and [6], the experimental 

results obtained for the control system with the controller parameters in Eq. (38) are 

given in Fig. 8. 



14 A.-I. BORLEA, R.-E. PRECUP, R.-C. ROMAN 

 

Fig. 8 The results of payload position control system with Gao DTSMC law: (a) u3; 

(b) r3 (black), y3 (red). 

3.3. Quasi-relay DTSMC law design and implementation 

Summarizing the information given in the previous sections, the guidelines to design 

the quasi-relay DTSMC law consist of the following steps: 

Step M1. This step is identical to step F1. 

Step M2. This step is identical to step F2. 

Step M3. This step is identical to step F3. 

Step M4. Set the values of the parameters i>1. 
These steps are exemplified as follows. For the cart position control system, the 

sampling period and the integral time constant are chosen as Ts=0.01 s and Ti=0.25 s. The 

expression of the system matrix specific to the Furuta DTSMC law results from Eqs. (9) 

and (23) [20] 

 
.

00500

//1/)500(/

010

)()(
2212211

1

1 11111

















−

−+=−−=
−

kkkkkkkk
RReeeex

IAKBKBAA
 (39) 

A set of parameters that guarantee the stability of the system with the matrix in (39) is 

 ].001.04.09.0[
1

−=K  (40) 

The other parameters of the control law were set as =[3  1  3] [20]. 



 Discrete-Time Model-based Sliding Mode Controllers for Tower Crane Systems 15 

Using the same reference input trajectory as in Section 3.1 and [6], the experimental 

results obtained for the control system with the controller parameters in Eq. (40) are 

illustrated in Fig. 9. 

 

Fig. 9 The results of cart position control system with quasi-relay DTSMC law: (a) u1; 

(b) r1 (black), y1 (red). 

For the arm angular position control system, the sampling period and the integral time 

constant are chosen as Ts=0.01 s and Ti=0.25 s. The expression of the system matrix 

specific to the Furuta DTSMC law results from Eqs. (9) and (24) [20] 

 
.

00400

//1/)400(/

010

)()(
2212212

1

2 22222

















−

−+=−−=
−

kkkkkkkk
RReeeex

IAKBKBAA
 (41) 

A set of parameters that guarantee the stability of the system with the matrix in Eq. 

(41) is 

 ].0011.03.083.0[
2

−=K  (42) 

The other parameters of the control law were set as =[3  1  3] [20]. 

Using the same reference input trajectory as in Section 3.2 and [6], the experimental 

results obtained for the control system with the controller parameters in Eq. (42) are 

illustrated in Fig. 10. 



16 A.-I. BORLEA, R.-E. PRECUP, R.-C. ROMAN 

 

Fig. 10 The results of arm angular position control system with quasi-relay DTSMC law: 

(a) u2; (b) r2 (black), y2 (red). 

For the payload position control system, the sampling period and the integral time 

constant are chosen as Ts=0.01 s and Ti=0.003 s. The expression of the system matrix 

specific to the Furuta DTSMC law results from Eqs. (9) and (25) [20] 

 

.

00100000/3-

//13/)100000(/

010

)()(

221221

3

1

3 33333

















−+=

−−=
−

kkkkkkkk
RR

eeeex
IAKBKBAA

 (43) 

A set of parameters that guarantee the stability of the system with the matrix in (43) is 

 ].0003.01.07[
3

−=K  (44) 

The other parameters of the control law were set as =[50  25  10] [20]. 

Using the same reference input trajectory as in Section 3.3 and [6], the experimental 

results obtained for the control system with the controller parameters in Eq. (44) are 

illustrated in Fig. 11. 

For a fair comparison of the three DTSMC laws, it is accepted that the allowable 

tolerance zone of the controlled output is of  2 % yst, where yst is the steady-state value 

of the controlled output. This is reflected in the measurement of the values of the 

overshoot and the settling time. 

The performance indices overshoot, settling time and stationary control error (or 

steady-state value of control error) are considered in the comparison of the control 



 Discrete-Time Model-based Sliding Mode Controllers for Tower Crane Systems 17 

systems with the three DTSMC laws designed in this paper. The values of these 

performance indices are synthesized in Table 1, Table 2 and Table 3, obtained after three 

test runs for all three control systems with DTSMC laws with extended state vectors 

defined in Eqs. (23), (24) and (25), respectively. The test runs are composed by three step 

signals, three step signals, and four step signals, respectively. 

 

Fig. 11 The results of payload position control system with quasi-relay DTSMC law: 

(a) u3; (b) r3 (black), y3 (red). 

Table 1 Overshoot (%) 

Test run Step Quasi-relay Furuta Gao 

Cart position 1 

2 

3 

4 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

0 

X  0 0 0 

Arm angular 

position  

1 

2 

3 

0 

0 

0 

0 

0 

0 

0 

0 

0 

X  0 0 0 

Payload position  1 

2 

3 

2.7 

0 

5 

3.5 

0 

0 

− 

− 

0 

X  2.6 1.2 − 

X  – average value of the steps which compose the reference signal 



18 A.-I. BORLEA, R.-E. PRECUP, R.-C. ROMAN 

Table 2 Settling time (s) 

Test run Step Quasi-relay Furuta Gao 

Cart position 1 

2 

3 

4 

7.55 

6.2 

8.2 

6 

6.5 

7 

6 

8.6 

4.2 

3.5 

4.4 

4.5 

X  6.987 7.025 4.15 

Arm angular 

position  

1 

2 

3 

7.4 

8.4 

6.6 

5.5 

6 

7.4 

5.4 

5.5 

6.5 

X  7.467 6.3 5.8 

Payload 

position  

1 

2 

3 

4.8 

2.8 

5 

9.6 

7 

9.6 

− 

− 

16 

X  4.2 8.733 − 

X  – average value of the steps which compose the reference signal 

Table 3 Stationary control error (%) 

Cart position Step Quasi-relay Furuta Gao 

 

Arm angular 

position  

1 

2 

3 

4 

1 

1.2 

0.6 

1.5 

0 

0.1 

0 

0 

0 

1.2 

0.1 

0 

X  1 0 0.3 

 

Payload 

position  

1 

2 

3 

1.6 

0 

0.2 

0 

0 

0 

1.6 

0.8 

0 

X  0.6 0 0.8 

 1 

2 

3 

0.4 

0.3 

0 

0.1 

0 

0 

− 

− 

5.2 

X  0.2 0 − 

X  – average value of the steps which compose the reference signal 

Analyzing the values for the overshoot in Table 1 reveals the fact that the smallest 

average value is exhibited by the Furuta DTSMC law. In Table 2, the smallest average 

value of the settling time is obtained by the quasi-relay DTSMC law. Table 3 shows that 

the Furuta DTSMC law exhibits the smallest average value as far as the stationary control 

error is concerned. 

In conclusion, having in mind all three performance indices discussed here, the best 

overall performance is ensured by the Furuta DTSMC law, the second best one is Gao 

DTSMC low and on the third place is Quasi-relay DTSCM low. Nevertheless, the Furuta 

DTSMC law leads to the smallest number of switching of the control inputs, thus the 

efforts at the level of actuators (namely, the power electronics part that produces PWM). 

However, the conclusions of the comparison can be different if other nonlinear processes 

are subjected to sliding mode control, in various fields, including decision-making [21], 

man-computer symbiosis [22], 3D printing objects [23], specific structures of fuzzy 



 Discrete-Time Model-based Sliding Mode Controllers for Tower Crane Systems 19 

systems [24], [25] including those focused on fuzzy control [26], [27], evolving 

controllers [28] and fuzzy cognitive maps [29], VANETs [30], quantum computing [31], 

and telesurgical applications [32]. 

Disturbance inputs were not considered (although the role of a part of possible 

disturbances is highlighted in Fig. 2) because the integral block described in [21] ensures 

the disturbance rejections for certain types of disturbances. It is not guaranteed that the 

ranking of the sliding mode controllers will be kept after rejecting the disturbances. 

4. CONCLUSIONS 

This paper proposed the discrete-time model-based sliding mode control of tower 

crane systems. Three popular control discrete-time control laws were considered and 

compared systematically on the basis of three performance indices obtained after 

measurements in a specific dynamic regime applied to the laboratory equipment. 

The Furuta law was applied to all three control laws to design the gain matrix K in the 

conditions of the application of the equivalent control method [2]. However, the control 

laws are different in terms of different ways to carry out the switching that modifies the 

control system structures making them belong to the general class of variable structure 

systems. The parameters involved in switching are also different. This flexibility is an 

advantage of the control laws designed and implemented in this paper and also confirms 

the degrees of freedom offered by sliding mode control, also showing certain robustness. 

One of the main shortcomings of these control laws is the heuristics in the design of the 

controllers. Another shortcoming is the need for certain initial information on the process 

model. Future research will be focused on mitigating these shortcomings. First, the optimal 

tuning of the free parameters of the sliding mode controllers will be targeted. Second, the 

design of data-driven sliding mode controllers will be carried out starting with [6] and 

compared to model-based sliding mode controllers. Direct relations that involve controller 

tuning parameters and control system performance indices with respect to both reference 

and disturbance inputs will be attempted to be derived in all controller designs. 

Acknowledgement: The research reported in this paper was supported by a grant of the Romanian 

Ministry of Education and Research, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-

2020-0269, within PNCDI III. 

REFERENCES 

1. Emelyanov, S.V., 1967, Variable Structure Control Systems. Nauka, Moscow. 
2. Utkin, V.I., 1977, Variable structure systems with sliding modes, IEEE Transactions on Automatic 

Control, 22(2), pp. 212-222. 

3. Yu, X.-H., Kaynak, O., 2009, Sliding-mode control with soft computing: A survey, IEEE Transactions on 
Industrial Electronics, 56(9), pp. 3275-3285. 

4. Young, K.D., Utkin, V.I., Ozguner, U., 1999, A control engineer’s guide to sliding mode control, IEEE 
Transactions on Control Systems Technology, 7(3), pp. 328-342. 

5. Aboserre, L.T., El-Badawy, A.A., 2020, Robust integral sliding mode control of tower cranes, Journal of 
Vibration and Control, 27(9-10), pp. 1171-1183. 

6. Precup, R.-E., Roman, R.-C., Safaei, A., 2021, Data-Driven Model-Free Controllers, 1st edition. CRC 
Press, Taylor & Francis, Boca Raton, FL. 



20 A.-I. BORLEA, R.-E. PRECUP, R.-C. ROMAN 

7. Xi, Z., Hesketh, T., 2010, Discrete time integral sliding mode control for overhead crane with 
uncertainties, IET Control Theory & Applications, 4(10), pp. 2071-2081. 

8. Qian, D.-W., Yi, J.-Q., 2013, Design of combining sliding mode controller for overhead crane systems, 
International Journal of Control and Automation, 6(1), pp. 132-140. 

9. Bartolini, G., Orani, N., Pisano, A., Usai, E., 2000, Load swing damping in overhead cranes by sliding 
mode technique, Proc. 39th IEEE Conference on Decision and Control, Sydney, NSW, Australia, vol. 2, 

pp. 1697–1702. 
10. Lee, L.-H., Huang, C.-H., Ku, S.-C., Yang, Z.-H., Chang, C.-Y., 2014, Efficient visual feedback method 

to control a three-dimensional overhead crane, IEEE Transactions on Industrial Electronics, 61(8), pp. 

4073-4083. 
11. Sun, N., Fang, Y.-C., Chen, H., Lu, B., Fu, Y.-M., 2016, Slew/translation positioning and swing 

suppression for 4-dof tower cranes with parametric uncertainties: Design and hardware 

experimentation, IEEE Transactions on Industrial Electronics, 63(10), pp. 6407-6418. 
12. Le, T.A., Dang, V.-H, Ko, D.H., An, T.N., Lee, S.-G., 2013, Nonlinear controls of a rotating tower crane 

in conjunction with trolley motion, Proceedings of the Institution of Mechanical Engineers, Part I: Journal 

of Systems and Control Engineering, 227(5), pp. 451–460. 
13. Aboserre, L.T., El-Badawy, A.A., 2021, Robust integral sliding mode control of tower cranes, Journal of 

Vibration and Control, 27(9-10), pp. 1171-1183. 

14. Furuta, K., 1990, Sliding mode control of a discrete system, Systems & Control Letters, 14(2), pp. 145-152. 
15. Gao, W.-B., Wang, Y.-F., Homaifa, A., 1995, Discrete-time variable structure control systems, IEEE 

Transactions on Industrial Electronics, 42(2), pp. 117-122. 

16. Milosavljević, Č., 1985, General conditions for the existence of a quasisliding mode on the switching 
hyperplane in discrete variable structure systems, Automation and Remote Control, 46(3), pp. 307-314. 

17. Spasić, M.D., 2019, Model predictive control based on sliding mode control, Ph.D. thesis, University of 
Niš, Niš, Serbia. 

18. Inteco, 2012, Tower Crane, User’s Manual, Inteco Ltd, Krakow. 
19. Buehler, H., 1986, Reglage par mode de glissement. Presses Polytechniques Romandes, Lausanne. 
20. Borlea, A.-I., 2022, Sliding mode controllers. Validation on a laboratory equipment (in Romanian), 

M.Sc. thesis, Politehnica University of Timisoara, Timisoara, Romania. 

21. Božanić, D., Tešić, D., Marinković, D., Milić, A., 2021, Modeling of neuro-fuzzy system as a support in 
decision-making processes, Reports in Mechanical Engineering, 2(1), pp. 222-234. 

22. Filip, F.G., 2021, Automation and computers and their contribution to human well-being and resilience, Studies 
in Informatics and Control, 30(4), pp. 5-18. 

23. Milićević, I., Popović, M., Dučić, N., Vujičić, V., Stepanić, P., Marinković, D., Ćojbašić, Ž., 2022, Improving 
the mechanical characteristics of the 3D printing objects using hybrid machine learning approach, Facta 

Universitatis-Series Mechanical Engineering, doi: 10.22190/FUME220429036M. 

24. Yapici Pehlivan, N., Turksen, I.B., 2021, A novel multiplicative fuzzy regression function with a multiplicative 
fuzzy clustering algorithm, Romanian Journal of Information Science and Technology, 24(1), pp. 79-98. 

25. Kwak, C.-J., Ri, K.-C., Kwak, S.-I., Kim, K.-J., Ryu, U.-S., Kwon, O.-C., Kim, N.-H., 2021, Fuzzy modus 
ponens and tollens based on moving distance in SISO fuzzy system, Romanian Journal of Information Science 

and Technology, 24(3), pp. 257-283. 

26. Precup, R.-E., Preitl, S., Balas, M., Balas, V., 2004, Fuzzy controllers for tire slip control in anti-lock 
braking systems, Proc. 2004 IEEE International Conference on Fuzzy Systems, Budapest, Hungary, vol. 

3, pp. 1317-1322. 

27. Preitl, Z., Precup, R.-E., Tar, J.K., Takács, M., 2006, Use of multi-parametric quadratic programming in 
fuzzy control systems, Acta Polytechnica Hungarica, 3(3), pp. 29-43. 

28. Škrjanc, I., Blažič, S., Angelov, P., 2014, Robust evolving cloud-based PID control adjusted by gradient 
learning method, Proc. 2014 IEEE Conference on Evolving and Adaptive Intelligent Systems, Linz, 
Austria, pp. 1-6. 

29. Vaščák, J., Hvizdoš, J., Puheim, M., 2016, Agent-based cloud computing systems for traffic management, 
Proc. 2016 International Conference on Intelligent Networking and Collaborative Systems, Ostrava, 
Czech Republic, pp. 73-79. 

30. Boucetta, S.I., Johanyák, Z.C., Pokorádi, L.K., 2017, Survey on software defined VANETs, Gradus, 4(1), 
pp. 272-283. 

31. Osaba, E., Villar-Rodriguez, E., Oregi, I., Moreno-Fernandez-de-Leceta, A., 2021, Hybrid quantum 
computing-tabu search algorithm for partitioning problems: preliminary study on the traveling salesman 

problem, Proc. 2021 IEEE Congress on Evolutionary Computation, Kraków, Poland, pp. 351-358. 
32. Precup, R.-E., Haidegger, T., Preitl, S., Benyó, B., Paul, A.S., Kovács, L., 2012, Fuzzy control solution 

for telesurgical applications, Applied and Computational Mathematics, 11(3), pp. 378-397.