Sliding Mode Control for Inverse Response Systems: A Trajectory
Tracking Study
Gabriel G´omez-Guerra
1,2
a
, Sebasti´an Insuasti
1,2 b
and Oscar Camacho
1 c
1
Colegio de Ciencias e Ingenier´ıas ”El Polit´ecnico”, Universidad San Francisco de Quito USFQ, Quito, Ecuador
2
Instituto de Energ´ıa y Materiales, Universidad San Francisco de Quito USFQ, Quito, Ecuador
Keywords:
Inverse Response, Traj ectory Tracking, Dynamic Sliding Mode Control, Continuous-Stirred Tank Reactor,
TCLab.
Abstract:
This paper introduces and compares three control strategies for systems exhibiting inverse response wi th
variable reference tracking: Dynamic Sliding Mode Control (DSMC), Sliding Mode Control (SMC), and
Proportional-Integral-Derivative (PID) control. These controllers were tested in two cases: using simulations
in a nonlinear i sothermal Continuous-Stirred Tank Reactor (CSTR) and in a modified Temperature Control
Lab (TCLab). The results, both simulations and experimental, show that the DSMC consistently outperforms
both the SMC and the PID controllers, delivering superior tracking performance in controlling the inverse
response behavior when the reference is variable.
1 INTRODUCTION
Inverse response systems, also referred to as non-
minimum phase systems, exhibit an initial reaction
opposite to the desired steady-state outcome when
subjected to a step input. This beh avior po ses chal-
lenges in control system design, often leading to is-
sues like overshoot, extended settling times, and re-
duced stability margins. Inverse response systems
are commonly encounter e d in various industrial ap-
plications, including che mical reactors, distillation
columns, and certain mechanical systems (Liptak
et al., 2018).
Controlling this process is challenging (Ogu n-
naike and Ray, 1994). Various methods have been
employed, including classical control (Liptak et al.,
2018), Smith predicto r-based approaches (Alfaro and
Vilanova, 2012), adaptive control (Chen, 20 01), intel-
ligent control (Estrada, 2021), as well as sliding mode
control (Camacho et al., 1999; Rojas et al., 2005;
Esp´ın et al., 2023).
Implementing classical PID controllers for inverse
response systems re quires careful tuning, as these
processes can cause stability issues. The derivative
control term is often not beneficial for stability. Hig h
a
https://orcid.org/0009-0008-8002-0277
b
https://orcid.org/0009-0001-6958-795X
c
https://orcid.org/0000-0001-8827-5938
controller gain may lead to runaway effects, while low
gain results in sluggish control loops, where increas-
ing speed compromises stability (Ogunnaike and Ray,
1994).
Alternatively, sliding mode control (SMC) is a
non-lin ear app roach widely utilized in both theoret-
ical and practical applications within industrial pro-
cesses (Camacho et al., 1997; Utkin et al., 2020). Its
popularity co mes from its ability to handle uncertain
system conditions and maintain ro bust control perfor-
mance in the face of disturbances (Furat and Eker,
2012; Sardella et al., 2021; Mehta and Bandyopa d-
hyay, 2021; Gambhir e e t al., 2021). In real-world
scenarios, the appeal of the SMC technique lies in its
proficiency in managing highly nonlinear processes,
systems with time delays, changing operating condi-
tions, variations in model parame te rs, disturbances,
and u ncertainties (Utkin et al., 20 20). The SMC strat-
egy defines a surface or manifold tha t guides the pro-
cess output to its desired final value. Specifically, a
sliding surface and its time derivatives are selected
based on performance requirements, with S(t) cho-
sen so that system dynamics are constrained to this
sliding mode surface. Selectin g an a ppropriate slid-
ing mode surface S(t) effectively reduc e s the control
parameters necessary for optimal global control sys-
tem per formance (Utkin et al., 2020). The structure
of an SMC scheme is variable and adapts based on
178
Gómez-Guerra, G., Insuasti, S. and Camacho, O.
Sliding Mode Control for Inverse Response Systems: A Trajectory Tracking Study.
DOI: 10.5220/0013068300003822
Paper published under CC license (CC BY-NC-ND 4.0)
In Proceedings of the 21st International Conference on Informatics in Control, Automation and Robotics (ICINCO 2024) - Volume 1, pages 178-188
ISBN: 978-989-758-717-7; ISSN: 2184-2809
Proceedings Copyright © 2024 by SCITEPRESS – Science and Technology Publications, Lda.

the current system states to the selected sliding sur-
face; however, high-frequency oscillatory responses,
known as chattering, may arise when tracking certain
equilibriu m points (Camacho and Smith , 2000). T he
chattering effect can lead to reduced control perfor-
mance, increased actuator wea r, unnecessary energy
consumption, and system instability.
The dynamic sliding mode control (DSMC) strat-
egy modifies the traditional sliding mode control
(SMC) switching function to create a new fu nction
that diminishes the chattering effect. This function
involves either the first or a higher-order derivative
of the control input, a llowing the shifting of discon-
tinuous components to these derivatives (Sira et al.,
1994; Utkin et al., 2020). Recently, the synthesis of
DSMC designs h as attracted interest (Proa˜no et al.,
2017; G ambhire et al., 2021; Esp´ın et al., 202 3), as
adding extra dynamics to a sliding surface addresses
practical challenges.
Despite the exten sive literature on SMC, it is
worth noting that SMC has not trad itionally been ap-
plied to control non-linear chemical processes that ex-
hibit an inverse response (Castellanos-C´ardenas et al.,
2022; Camacho et al., 1999; Rojas et al., 2005; Esp´ın
et al., 2023; G´omez et al., 2023). In addition, none
of these studies have addressed tracking variable ref-
erences. To the best of our knowledge, th is marks the
first attempt to employ SMC for chemical processes
with inverse response and variable reference tracking.
This paper compares and evaluates sliding mode
control techniques that aim to imp rove the perfor-
mance of variable reference tr a cking. Several stud-
ies have shown that the implementation of variable
temperature profiles (Ahn et al., 2009; Pataro et al.,
2023; Sardella et al., 2021) during the operation of
processes, such as batch reactors, can significantly
enhance their overall efficiency. This optimization
is ac hieved by carefu lly contro lling the thermal envi-
ronment to maximize the release of fermentable sug-
ars. By adjusting the temperature at different stages
of the reaction, the breakdown of complex biomass
components into simpler fermentable sugars is more
effective, leading to improved yields and reduced pro-
cessing times. Therefo re, increasing the efficiency of
the fermentation process also contributes to energy
savings and cost reduction in b iofuel production, a s
demonstra te d in (Vegi and Shastri, 2017). Although
most existing research focuses on regulation o r step
function tracking, this study emphasizes the impor-
tance of dynamic reference tracking for improving
process response and performance.
The paper is divided as follows: in Section
two, some fundamentals are described; Section three
shows the designed controllers; in Section four, the
results by simulations are pre sented; and finally, the
conclusion.
2 FUNDAMENTALS
This Section introduces several concepts that support
the design of the control strategies used in this work.
2.1 Inverse Response System
An inverse response system is characterized by a
transfer function with zeros in the right half of the
plane, causing th e initial slope to oppose the final
steady-state value. An inverse resp onse system can
be modeled by two first-order systems that form the
transfer function described in (1). This model has
been proven to accurately replicate the behavior of th e
real plant (Camacho et al., 2020).
G(s) =
K(−ηs + 1)
(τ
1
s + 1)(τ
2
s + 1)
(1)
2.2 Iinoya-Alpeter Compensator
The controller method aim s to ad dress the challenges
of non-minimum phase behavior, also known as the
inverse response, by employing the Iinoya-Alpeter
compen sator. This compensator uses an Internal
Model Controller to transform the system into a min-
imum phase system. Inverse response systems ar e
non-minimum phase due to a zero in the right-half
plane, w hich introduces instability and lim its practi-
cal closed-loop performance by restricting c ontroller
bandwidth. This leads to delays in plant resp onse and
lower product quality. The Iinoya-Alpeter compen-
sator mitigates th ese adverse effects by introducing
an internal transfer function into the feedback loop,
effectively neutralizing the imp a ct of the zero of the
right half-plane. The block diagram o f the Iinoya-
Alpeter compensator is shown in Figure 1.
Figure 1: Iinoya-Alpeter compensator block diagram.
Where G
p
(s) represents the plant transfer function
without the right half-plane zero, and λ is a parameter,
optimized to th e value 2η, to compen sate for the right-
half-plane zero (Ogunnaike and Ray, 1994). With this
Sliding Mode Control for Inverse Response Systems: A Trajectory Tracking Study
179

control scheme, the controller receives as input a min-
imum phase system given by (2).
G(s) =
K(ηs + 1)
(τ
1
s + 1)(τ
2
s + 1)
(2)
2.3 Inverse Response Systems
Identification
There are several methods for inverse response sys-
tems. In this paper, the Alfaro an d Balaguer identifi-
cation method (Balaguer et al., 2011) is used to model
an inverse response system as a second-order inverse
response mod e l. The procedure can be found in a de-
tailed way in (Balaguer et al., 2011). The resulting
analysis produc es a transfer function as shown in (1).
Another way is to consider a First-Order Plus
Dead Time (FOPDT) system, in which, using a Tay-
lor approximation, the dead tim e term is considered
as the right zero term since it does not account for
the inverse response ter m directly. A FOPDT system
consists of a pole formed by the systems time con-
stant, τ, a proportional gain, K, and a time delay, t
0
,
as represented by the transfer function in (3).
G(s) =
K
τs + 1
e
−t
0
s
(3)
In (Alfaro, 2001) can be found different ap-
proach e s for identification.
3 CONTROL LAWS
This section discusses the control laws applied in this
work: first, a PID controller, followed by a Sliding
Mode Control, and finally a Dynamical Sliding Mode
Controller.
3.1 PID Controller
The first control law proposed is a PID controller. For
this scheme, the system is modeled as an inverse re-
sponse system described in section 2.3.
3.1.1 Controller Design
For this study, a proportional-integral-derivative con-
troller was chosen. The action of the PID controller is
calculated as (4).
u(t) = K
P

e(t) + T
d
de(t)
dt
+
1
T
i
Z
e(t)dt

(4)
Where K
P
, T
d
and T
i
represent the proportional,
derivative and integral parameters of the controller.
The paramete r tuning equations are derived from (Ca-
macho et al., 2020). T he pro portional gain of the con-
troller was calculated as (5).
K
P
=
1
K
τ
1
+ τ
2
+
ηt
0
τ
c
+η+t
0
τ
c
+ η + t
0
!
(5)
Where τ
c
is a tuning parameter selected as (6).
τ
c
t
0
> 0.8 an d τ
c
> 0.1τ (6)
Similarly, the integral gain is calculated as (7).
T
i
= τ
1
+ τ
2
+
ηt
0
τ
c
+ η + t
0
(7)
Finally, the derivative gain is calculated as (8).
T
d
=
ηt
0
τ
c
+ η + t
0
+
τ
1
τ
2
τ
1
+ τ
2
+
ηt
0
τ
c
+η+t
0
(8)
3.2 SMC Controller Design
The SMC algorithm consists of two main stages: the
reaching phase and the sliding mode phase. During
the reaching phase, the state of the system is driven
towards a user-defined surface, while in the sliding
mode phase, the dynamics of the system are con-
strained to follow the dynamics of the sur face, m ak-
ing the system robust to parameter variations and dis-
turbance s (Camacho and Smith, 2000; Salinas et al.,
2018). Known for its robustness and insensitivity to
parameter variations, the conventional SMC theory
applied to inverse response system s can lead to insta-
bility, as noted in (Camacho et al., 1999; Rojas et al.,
2005; Castellanos-C´ardenas et al., 2022).
One way to apply conventional SMC to this kind
of system is to model the system as a First-Order
Plus Dead Time (FOPDT). The controller design fol-
lows the method described by (Camacho et al., 1999),
which uses a proportional-integral-derivative equa-
tion as the surface to ensure reliability a nd robustness.
Applying this procedure to a FOPDT system yields
the result shown in (9).
u
c
(t) =
t
0
τ
K
"

1
t
0
τ
− λ
0

y(t) +
d
2
r(t)
dt
2
+ λ
1
dr(t)
dt
+ λ
0
r(t))
#
+ K
D
S(t)
|S(t)| + δ
(9)
Where, r(t) represents the refere nce trajectory to
be followed, and y(t) is the system output. T he pa-
rameters λ
1
, λ
0
, K
D
, and δ a re chosen to ensure con-
troller stability. Add itionally, S(t) denotes the PID
surface equation, which in this case is g iven by ( 10).
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180

S(t) =
de(t)
dt
+ λ
1
e(t) + λ
0
Z
t
0
e(t)dt (10)
To eliminate the output derivative from the con-
troller action, λ
1
should be set as specified in (11).
λ
1
=
t
0
+ τ
t
0
τ
(11)
Finally, λ
0
was tuned taking into account the con-
straint set in (12).
0 < λ
0
< min

λ
1
η
,
λ
2
1
4

(12)
The reaching controller parame ters K
D
and δ were
manually adjusted.
3.3 Dynamical Sliding Mode Controller
Design
The design begins with the system model, which in-
cludes its inverse response as shown in (1). To miti-
gate the effect of the inverse response, Iinoya’s com-
pensator was applied with λ = 2η, transforming th e
system into a min imum phase system as described in
(2).
Subsequently, the system was expr essed as a dif-
ferential equation in its ”normal form” by applying
the inverse Laplace transform, as illustrate d in (1 3).
d
2
y
∗
(t)
dt
2
= −
(τ
1
+ τ
2
)
(τ
1
τ
2
)
dy ∗ (t)
dt
−
y ∗ (t)
(τ
1
τ
2
)
+
Kη
(τ
1
τ
2
)
du(t)
dt
+
K
(τ
1
τ
2
)
u(t) (13)
From (13) the co ntroller action derivative was iso-
lated as in (14).
Kη
(τ
1
τ
2
)
du(t)
dt
=
d
2
y
∗
(t)
dt
2
+
(τ
1
+ τ
2
)
(τ
1
τ
2
)
dy ∗ (t)
dt
+
y ∗ (t)
(τ
1
τ
2
)
−
K
(τ
1
T τ
2
)
u(t) (14)
A sliding surface need ed to be chosen to define the
system’s dynamics. Given that the proce ss has two
poles and a statio nary error, a proportional-integral-
derivative (PID) surface was selected, as shown in
(15).
S(t) = 2αe(t) + α
2
Z
t
0
e(t)dt +
de(t)
dt
(15)
To ensure that the process remains on the surface,
(15) was derived and set to 0 as presented in (16).
dS(t)
dt
= α
2
e(t) + 2α
de(t)
dt
+
d
2
e(t)
dt
2
= 0 (16)
The error definition, e(t) = r(t) − y(t), was sub -
stituted in (16), and the resulting expression was in-
corporated into the controller algorithm. This substi-
tution simplifies the output derivatives and yield s the
controller action as reported in (17).
Kη
(τ
1
τ
2
)
du(t)
dt
=

(τ
1
+ τ
2
)
(τ
1
τ
2
)
− 2α

dy ∗ (t)
dt
+
y ∗ (t)
(τ
1
τ
2
)
−
K
(τ
1
τ
2
)
u(t) +α
2
e(t) +α
dr(t)
dt
+
d
2
r(t)
dt
2
(17)
Setting α as in (18), the output derivative was
eliminated.
α =
(τ
1
+ τ
2
)
2τ
1
τ
2
(18)
Additionally, a r eaching controller component, in -
cluding parameters K
D
and δ, was designed. As w ith
the SMC, these parameters were manually tuned.
By replacing these considerations with (17) and
isolating the controller action derivative, the DSMC
control law was obtained as (19).
du(t)
dt
=
y ∗ (t)
Kη
+
(τ
1
+ τ
2
)
2
(4τ
1
τ
2
Kη)
e(t) +
(τ
1
+ τ
2
)
Kη
dr(t)
dt
+
τ
1
τ
2
Kη
d
2
r(t)
dt
2
−
u(t)
η
+ K
D
S(t)
|S(t)|+ δ
(19)
4 RESULTS AND DISCUSSION
In this section, the three controller approaches were
tested and evaluated. Two systems are used to eval-
uate the con troller performance: a continuous-stirre d
tank reactor (CSTR) an d a temperature control labo-
ratory (TCLab ). T he CSTR, a non-linear system pre-
viously characterized and tested in (Balaguer et al.,
2011), will be simulated using its physical model as
presented below. The proposed control laws and sys-
tem simulations will b e conducted in Simulin k. The
TCLab, shown in Figure 8, is a laboratory equipme nt
used for real-time temperature control experiments. It
features two controllable heating resistors and is op-
erated via a Simulink interface connected through a
USB serial communication port.
4.1 Continous-Stirred Tank Reactor
(CSTR)
The chemical reactor simulated in this paper is an
isothermal Continuous-Stirred Tank Reactor (CSTR)
that can exhibit inverse response due to a Van der
Sliding Mode Control for Inverse Response Systems: A Trajectory Tracking Study
181

Vusse r eaction. While this system has b e en an alyzed
in previous works (Balaguer et al., 2011), this paper
introdu ces a novel contr ol law for variable reference
tracking, which, to the best o f our knowledge, has
not been proposed before. Figure 2 shows the CSTR
system used to validate the prop osed control laws, as
adapted from (Balaguer et al., 2011).
Figure 2: CST R isothermal reactor (Balaguer et al., 2011).
The variables of interest are the output chemical
concentr ation, C
B
, which is the controlled variable,
and the flow through the reactor, F
r
, which is the ma-
nipulated variable.
4.1.1 Nonlinear Model
According to (Balaguer et al., 2011), the CSTR op-
erates with an in itial conce ntration of the first chemi-
cal, C
A0
, of 2.9175 mol/l and an initial concentration
of the output chemical, C
B0
, of 1.1 mol/l. Due to the
system’s non-linear ity, reference changes should be
within a ±10% range.
For the exam ination of system dynamics, the sub-
sequent assumptions were taken into account:
• He at and density capacities of the reactants are
constant
• The heat loss in coolant jacket is considered neg-
ligible
• The reaction heat and volume rema in constant
• The reaction and reacted material are uniformly
mixed
The mathematical model of the CSTR is obtained
based on exothermic reactions according to (Alfaro
and Vilanova, 2012; Balaguer et al., 2011). A process
reaction in a CSTR can be described as follows:
A
K
1
→ B
K
2
→ C (20)
2A
K
3
→ D (21)
Table 1: Operation values of the continuous stirred reactor
tank.
Model parameters Value
k
1
5
6
min
−1
k
2
5
3
min
−1
k
3
1
6
l · mol
−1
· min
−1
C
Ai
10mol ·l
−1
V 700 l
C
Ao
2.9175mol· l
−1
C
Bo
1.1mol · l
−1
u
o
%
60%
Where A
K
1
→ B stands for an exothermic r e action.
From a mass balance on reactants A and B, the system
can be described as follows:
dC
A
(t)
dt
=
F
r
(t)
V
[C
Ai
−C
A
(t)] − k
1
C
A
(t) − k
3
C
2
A
(t)
(22)
dC
B
(t)
dt
= −
F
r
(t)
V
C
B
(t) + k
1
C
A
(t) = k
2
C
B
(t) (23)
For practical purposes, it should be considered
that the control range of the concentration B is f rom 0
to 1.5714 mol l
−1
; the range of variation of the flow
is from 0 to 634.1719 l min
−1
. In ad dition, the trans-
mitter signal y, process input u, and flow through the
reactor F
r
are presented in percentage.
The sensor-transmitter element takes the form:
y(t)
%
=

100
1.5714

C
B
(t) (24)
The relation for the flow through the reactor as a func-
tion of the process input through the control valve is
linear:
F
r
(t) =

634.1719
100

u(t)
%
(25)
Variables of the mathe matical model for the CSTR
process are detailed as follows:
F
r
: flow through the reactor
V : reactor volume that will remain constant during
operation
C
A
: the concentration of A in the reactor (mol l
−1
)
C
B
: the concentration of B in the reactor (mol l
−1
)
C
Ao
: the concentration of A in steady-state (mol l
−1
)
C
Bo
: the concentration of B in steady- state (mol l
−1
)
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182

k
i
(i = 1, 2, 3): the reaction r ate constants for the three
reactions
y(t)
%
: transmitter signal
u(t)
%
: process input
Table 1 shows the steady-state conditions for each
variable in th e CSTR process.
4.1.2 System Identificatio n as an Inverse
Response Model
The CSTR system was identified as a second-order in-
verse response system (Balaguer et al., 2011) to sim-
plify the con troller design. During the identification
process, a 10% step input change was applied at time
zero to the system. T he transfer function of the iden-
tified CSTR model is given in equation (26).
G
RI
=
0.32(−0.356s + 1)
(0.35s + 1)(0.483s + 1)
(26)
Figure 3 illustrates the behavior of the CSTR sys-
tem along with the model described by equation (26).
The model accurately replicates the system’s b ehav-
ior, although there is a slight discr e pancy in the con-
centration drop. Despite this minor deviation, the
model error is not significant enough to impa ct the
control law.
0 0.5 1 1.5 2 2.5 3 3.5 4
Time [s]
69
69.5
70
70.5
71
71.5
72
72.5
73
73.5
Concentration [%]
CSTR
Inverse Model
FOPDT Model
0 0.5
69.5
70
70.5
Figure 3: CST R system identification.
4.1.3 System Identificatio n as a FOPDT Model
Similarly, the CSTR system was identified as a First-
Order Plus Dead Time ( FOPDT ) model (Seborg et al.,
2016). A step in put with a 10% change was applied to
the CSTR sy stem for this identification. The resulting
transfer function is provided in equation (27).
G
T F
=
0.32
(0.625s + 1)
e
−0.525s
(27)
The model obtained is validated by comparing its
behavior with the actual CSTR system, as shown in
Figure 3. The model accura te ly replicates the behav-
ior of the CSTR system. The detailed graph shows
that the model effectively omits the inverse response
and exhibits only minimal error c ompared to the real
system. This small modeling error is not significant
enoug h to affect the control law.
4.1.4 Tuning Parameters of Controllers
Furthermore, the three control laws were tuned ac -
cording to the procedure outlined in Section 3. The
tuning parameters used in this work are p rovided in
Table 2 for both systems.
Table 2: Tuning parameters for the proposed control laws.
Controller Parameter CSTR TCLab
PID
K
P
3.63 0.63
T
i
0.99 234.81
T
d
0.25 54.67
SMC
λ
0
3.07 2.95E − 5
λ
1
3.50 0.0109
K
D
9.68 2.176
δ 1 1
DSMC
α 2.03 0.0103
K
D
1217.8 5.44
δ 1 1
4.1.5 Performance Indicators
Performance evaluation is performed using the Inte-
gral Squared Error (ISE), a s defined in (28), and the
Integral Squ ared Controller Outp ut (ISCO), as shown
in (29).
ISE =
Z
e(t)
2
dt (28)
ISCO =
Z
u(t)
2
dt (29)
where ISE quantifies performance by integrating
squared error over time and ISCO measures control
signal effort (Seborg et al., 2016; Liptak et al., 2018).
4.1.6 Simulation Results
The CSTR system was tested with three control laws
in two tests: one with a constant frequency variable
referenc e and the other varying the reference oscilla-
tion frequency to evaluate the con trol bandwidth.
Sliding Mode Control for Inverse Response Systems: A Trajectory Tracking Study
183

For the first two seconds of the constant frequency
test, the reference was kept constant at 60% concen-
tration. The rem ainder of the test was designed as
described in Equation (30).
R
C
(t) = 60 + 5 · sin(0.1(t − 2))
+ 2.05(t − 2) · sin(0.4(t − 2)) ( 30)
Figure 4 illustrates the behavior of the CSTR sys-
tem with the three controllers for the re ference de-
scribed in (30).
Figure 4: CSTR system: reference tracking at constant fre-
quency results.
It is observed that the system initially exhibits
strong oscillations around th e reference. However,
once it settles, all thr ee controllers accurately follow
the trajectory. However, during significant upward
variations in the reference, the controllers struggle to
keep up, particularly the SMC controller, as high-
lighted in detail in Figure 4 . I n contrast, Figure 5
presents the controller actions for this experiment.
Figure 5: CSTR system: reference tracking at constant fre-
quency controller action.
Initially, as shown in the first detail of Figure 5, all
three contro llers exhibit strong control action s. How-
ever, in the second detail, the control actions are in-
sufficient to follow the designed reference.
To compare the performanc e of the controllers, an
error threshold was established within a 2
◦
C range
around the reference (Figure 4). This threshold as-
sesses the time each controller takes to reach and stay
within this range, termed the settling time for variable
referenc e .The calculated p erformance indicators are
reported in Table 3
Table 3: CSTR system performance indicators.
PI PID SMC DSMC
ISE 143.4 131.49 19.4
ISCO [·10
4
] 3.74 3.7 3.81
ts [s] 2.8 2.16 1.86
To better visualize the comparison between con-
trollers, Figure 6 illustrates the normalized perfo r-
mance indicators.
Figure 6: CST R system: performance comparison.
The DSMC controller demonstrates superior per-
formance compared to the other two control laws.
However, it is noteworthy that the controller actions
are quite similar across all three laws. The settling
times for the controllers are also comparable . Despite
this, the DSMC controller achieves a smaller error
than the others. On the other hand, between the PID
and SMC schemes, it is observed that the PID outper-
forms the SMC in every indicator except ISCO.
The seco nd test on the CSTR system examined
variable frequency reference tracking to determine the
operating bandwidth. Typically, Bode diagrams ana-
lyze a control system’s frequency response by com-
paring the contro ller’s output with the system’s out-
put for stability and performance. However, this ap-
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proach uses a variable reference as the open-loop in-
put and compares it with the system’s output to assess
the system’s tracking ability.
The primary objective of this analysis is to iden-
tify the reference frequency range in which the con -
troller can o perate optimally. To achieve this, the
magnitude plot will be examined, as it provides a
compariso n of the ratio between the system’s output
and the variable frequency input reference, expressed
in decibels (d B). This analysis is crucial for un der-
standing the system’s dynamic behavior and ensuring
effective control across various operating conditions.
Equation (31) shows how the magnitude plot of the
BODE diagram was calculated. By establishing the
appropriate freq uency range, we can e nhance th e con-
troller’s performance and improve overall system sta-
bility.
|H(s)| = 20 lo g
10




Y (s)
R(s)




(31)
In this context, Y (s) represents the system ou tput,
while R(s) denotes the variable frequency reference.
Consequently, the c loser the magnitude approaches to
0dB, the closer the output w ill a lign with the refer-
ence.
10
-4
10
-2
10
0
10
2
-100
-80
-60
-40
-20
0
20
40
60
80
Magnitude (dB)
PID
SMC
DSMC
Frequency (Hz)
Figure 7: Magnitude bode diagram for variable frequency
reference.
Figure 7 illustrates the re la tionship between the
variable frequency ref erence and the system output.
The Bode diagram reveals that the DSMC con-
troller exhibits a one-to-one correspondence between
the reference an d system output across frequencies
ranging from 0.1 mHz to 0.1 Hz , indicating the high -
est perform ance and bandwidth. The SMC controller
performs well within a bandwidth of 1 mHz to 50
mHz, with a nearly one-to-one correlation. In con-
trast, the PID controller shows the poorest perfor-
mance. The behavior of the PID controller does not
exhibit a distinc t range in which it operates optimally.
4.2 Temperature Control Lab
As previously mentioned, the TCLab is a laboratory
device used for real-time temperature control exper-
iments. It has been extensively characterized in the
literature, demonstrating that the temperature system
can be modeled as a First-Order Plus Dead Time
(FOPDT) sy stem (Rossiter et al., 20 23; Park et al.,
2020; I nsuasti et al., 2022). To model the FOPDT
system as an inverse response system, an inversion
module is added in series with the tempe rature plant.
This module is defined by a transfer function with a
zero in the right-half plane and a pole, as given in (32).
G
I
(s) =
1 − ηs
τ
1
s + 1
(32)
Following the proc ess described in (G´omez et al.,
2023), the inversion module is designed to induce a
temperature dr op of approx imately 5
◦
C. Thus, the
transfer function of the inversion modu le is given by
(33).
G
I
(s) =
1 − 140s
60s + 1
(33)
With the addition of this inversion module, the
temperature p la nt block diagram for the rem ainder of
this paper is as shown in Figure 8 .
Figure 8: Temperature system block diagram.
Figure 9 illustrates the behavior of the temperature
system with the inversion module. As mentioned, the
initial temperature drop is approximately 5
◦
C, for a
step change of 20%.
4.2.1 System Identification as an Inverse
Response Model
The system shown in Figure 8 is mo deled as an in-
verse response system with the transfer function pro-
vided in (1). To identify the system, a step input with
a 20% change was applied, following the method de-
scribed in Section 2.3. The transfer function of the
inverse temperature system is given in (3 4).
G
T I
=
0.86(−163s + 1)
(156.8s + 1)(70.5s + 1)
(34)
Sliding Mode Control for Inverse Response Systems: A Trajectory Tracking Study
185

Figure 9 shows the behavior of both the real sys-
tem and the identified model. The proposed model
accurately r e plicates the beh avior of the real system.
However, since the physical system lacks a cool-
ing me c hanism, the actual tempera ture system takes
longer to achieve the temperature drop compared to
the model.
1000 1200 1400 1600 1800 2000
Time [s]
45
50
55
60
65
70
75
Temperature [
o
C]
TCLab system
Inverse Model
FOPDT Model
Figure 9: Temperature system identification.
4.2.2 System Identificatio n as a FOPDT Model
Following the procedure described in Section 2.3, the
inverse temperature system is identified as a First-
Order Plus Dead Time (FOPDT) model, as given in
(3). For the identification process, a step input with
a 20% change is applied to th e system shown in Fig-
ure 8. The transfer function of the identified system is
provided in e quation (35).
G
T F
=
0.86
(170.9s + 1)
e
−199.96s
(35)
The behavior of the identified FOPDT system is
validated by comparing it with the real system, as
shown in Figure 9. The m odel ac curately captures
the behavior of the system and eliminates the inverse
response, c onfirming that it helps in the design of the
SMC controller by removing the term of inverse re-
sponse from the transfer function.
4.2.3 Experimental Results
To assess the performance of the three control laws
in the temperature system, a variable reference was
designed. For the first 300 seconds of the experiment,
the r eference was kept constant a t 50
◦
C. After this pe-
riod, the reference follows the equation given in (36).
The freq uency of the refer e nce change is kept low to
allow the system to cool naturally, due to the absence
of external cooling in the plant.
R
T
(t) = 50 · sin

1
500
(t − 300)

(36)
Figure 10 illustrates the behavior of the system us-
ing the three proposed control laws for constant fre-
quency r e ference tracking.
Figure 10: Temperature system: reference t racking test re-
sults.
The DSMC controller reaches the reference faster
than the o thers, but all controllers track the reference
effectively. Figur e 11 shows the controller a ctions for
each control law during the experiment.
Figure 11: Temperature system: reference tracking con-
troller action.
It is observed that the DSMC controller exhibits
a stronger initial action to quickly catch up w ith the
referenc e . However, the DSMC controller’s action
then stabilizes within a ran ge of 20% to 32%. Sim-
ilarly, the PID controller’s action also settles within
this range but takes longer to reach the temperature
referenc e . In co ntrast, the SMC controller demon-
strates a higher overall control action but is the slow-
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186

est to respond to significant changes in the reference.
The performance indicators described in Section
4.1.5 were measured to compare the behavior of the
three proposed control laws. Table 4 reports the re-
sults obtained.
The performance indicators measur ed during the
experiment, including the ISE, ISCO, and settling
time, are summarized in Table 4.
Table 4: Temperature system performance indicators.
PI PID SMC DSMC
ISE [·10
5
] 2.66 2.08 0.2873
ISCO [·10
6
] 6.5 9.2 6.95
ts [s] 654 1620 488
Figure 12 presents the normalized performance in-
dicators. The settling time was defined as described in
Section 4.1.6 .
Figure 12: Temperature system: performance comparison.
Figure 12 sh ows the DSMC controller outper-
forms others with smaller error, sm oother action, and
faster tracking, while the SMC controller performs
the worst with slower tracking, higher error, and
higher action.
5 CONCLUSIONS
This study introduced a nd compared three control
laws for systems with inverse response and vari-
able references: a Dynamic Sliding Mode Controller
(DSMC), a Sliding Mode Controller (SMC), an d a
PID contr oller. These control laws were tested on two
different systems: a Continuous- Stirred Tank Reactor
(CSTR) and a software-modified Tempe rature Con-
trol Lab (TCLab). The simulations and experimen-
tal results demonstrate that all three control laws can
track a variable refer ence, but the DSMC controller
outperforms both the SMC and the PID controllers.
Future research could further investigate the ro-
bustness of the Dynamic Sliding Mode Controller
(DSMC) by testing it in more complex environments,
including disturbances and systems with higher time
delays. Additionally, implem enting these control
strategies on hardware systems beyond the TCLab,
such as industrial-scale processes or robotics, would
help validate their practicality and effectiveness in
real world applications. Extending the analysis to
multivariable systems could also offer deeper insights
into the scalability and adaptab ility of the DSMC ap-
proach .
ACKNOWLEDGEMENTS
This work was supported by the Universidad San
Francisco de Quito’s Poli-Grants Program, Grant
24280.
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