Fractional Order-Sliding-Mode Controller for Regulation of a Nonlinear

Chemical Process with Variable Delay

Antonio Di Teodoro

, Marco Herrera

and Oscar Camacho

Colegio de Ciencias e Ingenier

ıas ¨El Polit

ecnico”, Universidad San Francisco de Quito USFQ, Quito, Ecuador

Keywords:

Sliding Mode Control, Fractional Order Calculus, Reduced Order Model, Chemical Processes, Simulation.

Abstract:

The present work shows the application of a new controller based on combining the fractional order calculus

concepts with the sliding mode theory to a non-linear system with variable delay. The power of fractional-order

calculus is used to identify the real process and represent it as a reduced-order model. From this model, the

controller is developed using the sliding-mode control procedure. An SMC based on FOPDT and one based

on fractional calculus are compared using some performance indicators to assess performance quantitatively.

1 INTRODUCTION

The chemical and biochemical engineering ﬁeld has

a lot of control problems, one of which is the pri-

mary focus of current work: regulating variable time

delay processes. In real applications, plant behav-

ior is often affected by unexpected dynamics, out-

side disturbances, etc. (Obando et al., 2023). Dis-

turbances, model errors, unmodeled dynamics, ele-

vated time delay, and poorly deﬁned plant charac-

teristics reduce the efﬁcacy of conventional regula-

tion schemes even for linear time-invariant systems in

chemical processes, especially at the industrial scale.

When designing a process controller, the control

actions become slow if it has a time delay, decreas-

ing the total process performance. Furthermore, in

some cases, it has been seen that delays produce in-

stability in the system, as in (Prado et al., 2022), lead-

ing to the search for more advanced control proposals.

Thus, controlling dynamical systems with delay time

has become a major topic in control theory.

The variable structure control has a method called

the sliding mode control (SMC). SMC is a simple

and reliable method for creating controllers for linear

and non-linear processes, according to (Camacho and

Smith, 2000; Utkin et al., 2020). SMC is one of the

most widely used techniques for managing dynamical

systems because it is relatively insensitive to uncer-

tainties(Camacho and Smith, 2000; Esp

ın et al., 2022;

https://orcid.org/0000-0002-8766-0356

https://orcid.org/0000-0002-3124-2403

https://orcid.org/0000-0001-8827-5938

Utkin et al., 2020). The traditional sliding mode con-

trol always maintains a predesigned sliding variable

S(t), which is generated by a so-called reaching con-

dition, such as S(t)

S(t) < 0, to be zero under high-

frequency switching (Utkin et al., 2020).

Substantial research has been published on the ap-

plication of SMC in industrial engineering process

control; we can name some of them here(Esp

ın et al.,

2022; Xiao and Li, 2016; Dimassi et al., 2019; Ra-

sul and Pathak, 2016; Sardella et al., 2020; Siddiqui

et al., 2020; Herrera et al., 2020; Salinas et al., 2018;

Kadu et al., 2018). Although few studies on SMC de-

sign employ fractional order calculus techniques for

chemical processes, we can highlight some of them

(Di Teodoro et al., 2023; Di Teodoro et al., 2022;

Ullah and Mohammad, 2022; Allahem et al., 2022;

Mehri and Tabatabaei, 2021; Haghighi and Ziaratban,

2020; Ardjal et al., 2021).

The current work illustrates the application of a

novel controller based on fusing notions from sliding

mode theory and fractional order calculus to a non-

linear system with variable delay. First, the actual

process is represented as a reduced-order model us-

ing the strength of fractional-order calculus. Then,

the sliding-mode control approach is used to create

the controller from this model. Finally, some perfor-

mance measures were used to evaluate and compare

an SMC based on FOPDT and an SMC based on frac-

tional calculus.

The paper is divided as follows: in Section

two, some fundamentals are described; Section three

shows the methodology of design; in Section four, the

Di Teodoro, A., Herrera, M. and Camacho, O.

Fractional Order-Sliding-Mode Controller for Regulation of a Nonlinear Chemical Process with Variable Delay.

DOI: 10.5220/0012159400003543

In Proceedings of the 20th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2023) - Volume 1, pages 131-139

ISBN: 978-989-758-670-5; ISSN: 2184-2809

131

results by simulations are presented; and ﬁnally, the

conclusion.

2 BACKGROUND

2.1 Sliding Mode Control

The SMC is a variable structure controller created uti-

lizing nonlinear control techniques. It is a robust con-

trol that reacts appropriately to nonlinear systems un-

der unknown conditions. Furthermore, it is immune

to changes in modeling parameters. The SMC consid-

ers a sliding surface that allows the controlled variable

to transition from an initial state to the desired ﬁnal

state. For this reason, its control law includes a con-

tinuous component to move on the sliding surface and

a discontinuous part for the reachability phase. The

motion of the system on a sliding surface is known as

the sliding mode (Utkin et al., 2020).

The SMC control law U(t), as expressed in (1):

U(t) = U

(t) +U

(t) (1)

(t): It is obtained from the equivalent control

method (Utkin et al., 2020). Keep the controlled vari-

able on the sliding surface σ(t) = 0. The equivalent

control is deduced, considering that the derivative of

the surface is zero.

The sliding condition is given by:

dσ(t)

= 0 (2)

Combined with the previous equation and the

model of the process system, U

(t) is obtained.

(t): It is the discontinuous control part; it al-

lows the system to reach the sliding surface. It incor-

porates a nonlinear element that includes the switch-

ing element of the control law; therefore, U

(t) con-

tains the switching element and is given by:

(t) = K

sign(σ(t)) (3)

is a tuning parameter responsible for the reach-

ing mode.

SMC transitions on the sliding surface cause chat-

tering. Chattering excites system dynamics not mod-

eled, causes vibration of the actuator with wear, and

degrades performance. One way to reduce such im-

pacts is to smooth the non-linear switching function

using soft functions such as the sigmoid function (Ca-

macho and Smith, 2000). Therefore, the discontinu-

ous controller utilizes the sigmoid function:

(t) = K

σ(t)

|σ(t)| + δ

(4)

The convergence condition, often known as at-

tractiveness, ensures that the system dynamics will

always converge on the sliding surface(Utkin et al.,

2020). It is necessary to formulate a Lyapunov func-

tion V (t) > 0 with ﬁnite energy. The candidate Lya-

punov function is deﬁned as follows.

V (t) = 1/2σ

(t) (5)

It is sufﬁcient to make sure that the derivative of

the function V (t) is negative for it to be possible for

the function V (t) to be reduced. Therefore, the condi-

tion of convergence can be written as follows:

dV (t)

dσ(t)

σ(t) < 0 (6)

It states that if the projection of the system trajec-

tories on the sliding surface is stable, then the system

is stable (Li et al., 2010).

2.2 Briefs About Fractional Calculus

2.2.1 Deﬁnitions of Caputo and

Riemann-Liouville Derivative and Their

Connection

Deﬁnition 1. The Riemann − Liouville fractional in-

tegral of order α > 0 is given by (see (Kilbas et al.,

2006; Miller and Ross, 1993; Podlubny, 1994; Kilbas

et al., 1993))





(x) =

Γ(α)

h(t)

(x −t)

1−α

dt, x > a. (7)

We denote by I

) the class of functions h, rep-

resented by the fractional integral (7) of a summable

function, that is, h = I

ϕ, where ϕ ∈ L

(a,b). A de-

scription of this class of functions is given in (Kilbas

et al., 2006; Kilbas et al., 1993; Abbas et al., 2023;

Patel et al., 2023) (L

[a,b] space can be deﬁned as a

space of measurable functions for which the absolute

value is Lebesgue-integrable).

Deﬁnition 2. Let





(x) denote the fractional

Riemann–Liouville derivative of order α > 0, where

h ∈ L

(a,b) (see (Kilbas et al., 2006; Miller and Ross,

1993; Podlubny, 1994; Kilbas et al., 1993; Abbas

et al., 2023))





(x) =





Γ(s−α)

h(t)

(x−t)

α−s+1

dt, (8)

s = [α] + 1,x > a, (9)

where [α] denotes the integer part of α and Γ is the

gamma function.

When 0 < α < 1, then (8) takes the form





(x) =



1−α



(x). (10)

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

132

Example 1.

f (x − a)

(

0, γ = α − 1,

γ+1

γ−α+1

(x − a)

γ−α

, otherwise

With α ∈ (0,1), a > 0, k ∈ N and γ > −1, and an

appropriate f . (See (Ceballos et al., 2020; Ceballos

et al., 2022; Kilbas et al., 1993))

Deﬁnition 3. Let α ≥ 0 and m = [α]. Then, we can

deﬁne the operator

f := I

m−α





f ,

when





f ∈ L

[a,b].

Example 2.

(x − a)

= 0 if β ∈ {0,1,2,. . ., m −

1}.

Lemma 1. Let α ≥ 0 and m = [α] + 1. Suppose that

f is such that

and

exists. Then

f =

f −

m−1

∑

k=0

(x − a)

k−α

Γ(k − α + 1)





f (a).

See (Kilbas et al., 2006; Podlubny, 1994; Kilbas et al.,

1993))

consequently, we have the following lemma:

Lemma 2. Let α ≥ 0 and m = [α] + 1. Suppose that

f is such that

and

exists.

Then

f =

f = D

f . If and only if





f (a) = 0 for all k = 0,..., m − 1.

The semigroup property for the composition of frac-

tional derivatives does not hold in general (see (Pod-

lubny, 1994, Sect. 2.3.6)). In fact, the property:





= D

α+γ

h (11)

holds whenever

( j)

) = 0, j = 0,1,... ,s − 1, (12)

and h ∈ AC

s−1

([a,b]), h

(s)

∈ L

(a,b) and s = [γ] + 1

( AC

([a,b]) denotes the class of functions h, which

are continuously differentiable in the segment [a,b],

up to order s − 1 and h

(s−1)

is absolutely continuous

in [a,b]).

Remark 1. In this paper, we use D

to identify the

fractional operator because we assume that the initial

conditions are equal to zero.

2.2.2 Laplace Transform of Fractional

Derivatives

Deﬁnition 4. Let F(s) := (L f )(s), i.e., say, the

Laplace transform of the function f and α > 0

(See([(Kilbas et al., 1993)])).



L D



(s) = s

F(s) −

l−1

∑

j=0

( j)

α− j−1

where l = [α] + 1 and f

( j)

) =

f (t)|

t−→0

, j =

0,...,l −1. Clearly, if α ∈ (0,1) then



L D



(s) = s

F(s) − f (0

α−1

(13)

2.2.3 The Use of the Mittag-Lefﬂer for G(s)

In order to develop our model, we will make use of the

two-parameter Mittag-Lefﬂer function. This function

is deﬁned classically as

Deﬁnition 5. Let z ∈ C and α > 0. The two-

parameter Mittag-Lefﬂer function is given by

α,β

(z) =

+∞

∑

k=0

Γ(αk + β)

, ∀z ∈ C. (14)

Remark 2. Note that E

α,β

(z) > 0 for all z ∈ C

and

when β = 1, then E

α,β

(z) = E

(z) This function is uni-

formly convergent over C. For α, β ≥ 0 If n = 1 and

r = 1. E

1,1

(z) = E

(z) = exp(z). (see (Kilbas et al.,

2006; Podlubny, 1994; Kilbas et al., 1993))

3 CONTROL DESIGN APPROACH

IN THE CAPUTO SENSE

This section presents the mathematical development

of the sliding mode control based on a FOPID surface.

Since SMC is a model-based controller, let us use an

empirical model of the actual process. Using the reac-

tion curve procedure, the nonlinear process can be ap-

proximated for a ﬁrst-order plus dead time (FOPDT)

model (Smith and Corripio, 2005).

X(s)

U(s)

−t

τs + 1

. (15)

3.1 Time Delay Approximation

A ﬁrst problem encounter in the realization of this

fractional controller is the time delay part (Camacho

and Smith, 2000). In the complex settings, the natu-

ral exponential function e

can be represented as the

well–known Taylor series centered at 0 which is abso-

lutely convergent for z ∈ C. A simple linear approx-

imation is 1 + z where the absolute error

− 1 − z

equals the Taylor remainder R

. On the other hand,

we have that e

−z

thus we can approximate e

−z

1+z

. Indeed, to see this let the approximation

error,



−z

−

1+z



, be bounded by ε

max

> 0. Then,



−z

−

1+z



≤

|z|

|1+z|

max

1,e

−Re

{

}

= ε

max

For example, the time delay e

−t

can be approxi-

mated to

s+1

and, in turn, the FOPDT transfer func-

tion is as follows:

−t

τs + 1

≈

(τs + 1)(t

s + 1)

, (16)

Fractional Order-Sliding-Mode Controller for Regulation of a Nonlinear Chemical Process with Variable Delay

133

that has a second order differential equation form (Ca-

macho and Smith, 2000).

Remark 3. A discussion that seems interesting to us

about the role of the fractional parameter in the expo-

nential function is a future work. A further conjecture

is that we believe that the parameter β is not neces-

sary. Which makes the calculation of the Laplace in-

verse much simpler.

3.2 Fractional Transfer Functions

Let α, β ∈ R, r,n ≥ 0. Then, we deﬁne the fractional

transfer function plus dead time as

X(s)

U(s)



τ(s

) + 1





n,r

))



(17)

where α and β can be interpreted as orders of frac-

tional derivatives and integrals. Note that transfer

function (15) is obtained when α and β approach to

−

Transfer function (17) can be approximated by (see

sec. 3.1 above)

X(s)

U(s)



τ(s

) + 1





n,r

))



≈



τ(s

) + 1



Γ(n)Γ(n + r)

Γ(n + r) +



)



Γ(n)

. (18)

and can be changed to a fractional differential equa-

tion form. Using this approximation in equation (17)

we obtain

Γ(n) s

α+β



Γ(n + r)





Γ(n)



τt

Γ(n + r)

X(s)

τt

[K Γ(n) Γ(n + r)]U (s)

(19)

Using deﬁnition 4 whenever the function X satisﬁes

the conditions of lemma 2 , then



α+β



(t)+



Γ(n + r)

Γ(n)



X)(t)









(t)+



τt

Γ(n + r)

Γ(n)



X(t) =

τt

[K Γ(n + r)]U (t)

(20)

3.3 Fractional Sliding Surface

Let us deﬁne the fractional sliding surface according

to (18) and (20). In the following, we assume the con-

ditions of lemma 2.

Deﬁnition 6. Let α,β ∈ R, m = [α + β], e(t) ∈

m−1

([a,b]) and e

(m)

∈ L

(a,b). Then, the frac-

tional slinding surface, S

α,β

, is deﬁned as



α,β



(t)



α+β

+ λ







(t).

(21)

where λ

,λ

are selected such that S

α,β

= 0, giving

a closed-loop stable response.

Remark 4. In the classical sense (Camacho and

Smith, 2000), our operator converge to (22)

(S e)(t) =



+ λ



e(t)dt. (22)

when n = 2 as α,β approach to 1

−

. The binomial

theorem can be used to generalize n. This is a fasci-

nating topic for further research. Just emphasize that

fractional computation for outside-interval parame-

ters (0,1) is much more difﬁcult to calculate.

Remark 5. Note that the deﬁnitions 2 and 3 allow us

to commute the classical integral operator I

in the

fractional sliding surface operator in deﬁnition 6, i.e.,



α+β

+ λ





α+β

+ λ



. (23)

Classical derivation of operator 6 will be needed in

the following:



α,β



(t) =



α+β

+ λ



[e(t)] .

(24)

where we have used the result in remark 5 followed by

the application of the Fundamental Theorem of Cal-

culus.

The next theorem shows how we can construct the

fractional equivalent control law U

α,β

(t).

Theorem 1. Let α, β ∈ R, m = [α + β], e(t) ∈

m−1

([a,b]) and R

(m)

(t), X

(m)

(t) ∈ L

(a,b). Then,

the fractional control law u

α,β

(t) based in the frac-

tional sliding surface deﬁned in 6 is

α,β

(t) =

τt

K Γ(n + r)









α+β



(t)+ λ





(t)+





(t)+ λ

R(t)+



Γ(n + r)

Γ(n)

− λ







(t)+



− λ







(t)+



τt

Γ(n + r)

Γ(n)

− λ



X(t)







(25)

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

134

Let e(t) = R(t) − X(t). First, we apply operator

(24) to function e(t) to obtain



α,β



(t) =



α,β



(t) −



α,β



(t).

(26)

by the linearity of this fractional operator. Next,



α,β



(t) = 0 guides us, combining equation (20)



α+β



(t)+ λ





(t)+ λ





(t)+ λ

R(t)+



α+β



(t)+



Γ(n + r)

Γ(n)







(t)+









(t)+



τt

Γ(n + r)

Γ(n)



X(t) =



α+β



(t)+





(t)+ λ





(t)+ λ

X(t)+

τt

[K Γ(n + r)]U(t)

(27)

The result follows.

Remark 6. If R : R → R such that R = c, where c ∈

R. Then

R = 0, consequently, D

R(t) = 0 and D

(t) = 0.

The derivatives of the reference value can be ignored

(Camacho and Smith, 2000) without affecting the

control performance, which allows a simpler con-

troller.

Thus, the resulting equivalent controller law is given

as follows:

α,β

(t) =

τt

K Γ(n + r)







R(t)+



Γ(n + r)

Γ(n)

− λ







(t)+



− λ







(t)+



τt

Γ(n + r)

Γ(n)

− λ



X(t)







(28)

The previous equation can be simpliﬁed by letting,

Γ(n + r)

Γ(n)

= λ

(29)

and

= λ

(30)

The sliding (continuous) control law is:

α,β

(t) =

τt

K Γ(n+r)



e(t) +



τt

Γ(n+r)

Γ(n)



X(t)



(31)

The reaching (discontinuous) control law U

α,β

(t) is

deﬁned as follows:

α,β

(t) = K

(α,β)

α,β

(t)



α,β

(t)



+ δ

α,β

(32)

Where K

(α,β)

is the switching gain.

Once the continuous and discontinuous parts of the

SMC, based on the fractional reduced order, have

been obtained; the total fractional order SMC law can

be represented as follows:

α,β

(t) = u

α,β

(t) + K

(α,β)

α,β

(t)



α,β

(t)



+ δ

α,β

. (33)

Until now, to our knowledge, there are no tuning

equations for K

(α,β)

neither δ

α,β

; hence in this paper

they are tuned by trial and error.

3.4 Fractional Results Using r = α and

n = β = 1

M U

α,1

(t) =



α+1

+ λ



R(t)



Γ(α + 1)

− λ







(t)



− λ







(t) +



Γ(α + 1)

τt

− λ



X(t)

(34)

Where M =

τt

Γ(α + 1) and Γ(1) = 0! = 1

3.5 Stability Condition

The dynamics of the system will always converge on

the sliding surface according to the convergence con-

dition; to reach the sliding surface, it is necessary to

create a Lyapunov candidate function V (t) > 0 with

ﬁnite energy. We use the Mittag–Lefﬂer stability the-

orem to do this and get the corresponding systems’

asymptotic stability.

Theorem 2 (See Theorem 5.1 in (Li et al., 2010)). Let

x = 0 be an equilibrium point for the system

x(t) = f (x(t)), t ≥ t

where 0 < α ≤ 1 and I ⊂ R be a domain containing

the origin f ∈ C

(I). Let V (t, x(t)) : [0,∞) × I → R

be a continuously differentiable function and locally

Lipschitz with respect to x such that

i) A||x||

≤ V (t,x(t)) ≤ B||x||

ii) D

V (t, x(t)) ≤ −F||x||

where t ≥ 0, x ∈ I,α ∈ (0, 1), A,B, F,a, and b. are

arbitrary positive constants. Then x = 0 is Mit-

tag–Lefﬂer stable. If the assumptions hold globally

on R, then x = 0 is globally Mittag–Lefﬂer stable.

V (t, x(t)) = x(t)

(2)α−1

α,α

(−λx(t)

(2)α−1

). (35)

Fractional Order-Sliding-Mode Controller for Regulation of a Nonlinear Chemical Process with Variable Delay

135

Setpoint

Control

Valve

Agitator

Figure 1: Mixing Tank process.

Where α ∈ (0,1) , E

α,β

is the Mittag-Lefﬂer function

deﬁned in (14).

This function is locally Lipschitz any interval of R

and satisﬁes

(i 0 < V (t,x(t)) ≤ B||x||

for x ∈ I ⊂ R and any B ∈

R. See (Coloma et al., 2021; Ceballos et al., 2020;

Ceballos et al., 2022)

(ii D

V (t, x(t)) = −λV (t,x(t)) ≤ −F||x||

for any

F,a,b ∈ R, see (Kilbas et al., 2006).

in consequence x = 0 is Mittag–Lefﬂer stable, using

the Remark 4.4 in (Li et al., 2010) (Mittag–Lefﬂer sta-

bility and Generalized Mittag–Lefﬂer stability imply

asymptotic stability) we guarantee asymptotic stabil-

ity of our system.

4 RESULTS AND DISCUSSION

4.1 Mixing Tank Process

The mixing tank shown in Fig. 1 consists of the si-

multaneous entry of the hot W

(t) and cold ﬂow W

(t)

into the process with temperatures T

(t) and T

(t), re-

spectively. The output T

(t) is the temperature of the

mixture measured at a point 125 ft downstream of the

mixing tank. The Fail-Closed (FC) actuator regulates

the cold stream to maintain the desired temperature

within the mixing tank. The control objective con-

sists of maintaining the required mixing temperature

(t) despite disturbances in the hot ﬂow W

(t). The

following determines the time delay between the tank

and the sensor’s position.

(t) = T

(t −t

) (36)

with, transportation lag or delay time:

LAρ

(t) +W

(t)

(37)

where A is the cross section of the pipe, the length

of the pipe L and the density ρ of the contents of the

mixing tank. A complete description of the non-linear

model and parameters in a stationary state and the sys-

tem variables can be found in (Camacho and Smith,

2000).

4.2 Identiﬁcation and Validation of

FOPDT and FO-FOPDT Models

The reaction curve method is used to identify an ap-

proximate model of the non-linear mixing tank pro-

cess. Thus, 10% of the input of the process (m(t))

is made. Figure 2 shows the responses of the tank

mix process, the approximate FOPDT model (Smith

and Corripio, 2005) and the approximate fractional

FO-FOPDT model (Gude and Garc

ıa Bringas, 2022).

Equations (38) and (39) show the approximate mod-

els:

G(s)

FOPDT

−0.8207e

−4.206s

2.227s + 1

(38)

G(s)

FO−FOPDT

−0.8207e

4.142s

2.262s

1.01

+ 1

(39)

As can be seen in the Eqs. (38) and (39). The

controllability relation



> 1



, which makes it dif-

ﬁcult to control (Obando et al., 2023). Also, it can be

considered as a process with a long delay.

0 5 10 15 20 25 30

Time [min]

0.46

0.465

0.47

0.475

0.48

0.485

0.49

0.495

0.5

Transmitter units [pu]

Process

FOPDT

FO-FOPDT

Figure 2: Open Loop Output Responses for step process

input.

The mean square error (MSE) is used to vali-

date the models obtained. Where: MSE

FOPDT

1.4192x10

−7

and MSE

F0−FOPDT

= 9.3819x10

−8

. Al-

though both indices are low, the MSE of the fractional

model was the lowest.

To analyze the performance of the FO-SMC con-

troller, it is compared with a conventional SMC con-

troller presented in (Camacho and Smith, 2000). FO-

SMC controller parameters λ

, λ

and λ

are tuned

according to Section 3. SMC controller parameters

and the FO-FOPDT K

and δ parameters are tuned

according to (Camacho and Smith, 2000). These pa-

rameters are shown in Table 1.

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

136

Table 1: Controller design parameters.

Parameter SMC FO-SMC

0.1179 0.1171

0.6867 0.2425

− 0.4419

0.3833 0.3925

δ 0.7059 0.6894

4.3 Disturbance Rejection Test

In this test, from the operating point T

(t) = 150[

F],

the variations in steps from 250



min



to 150



min



the hot ﬂow W

(t).

Due to the distance of the pipe L = 120[ f t], the

outlet temperature is measured downstream. Further-

more, due to the effect of the disturbance, the varia-

tion of W

(t) causes the time delay of the processes to

increase stepwise from 3.58 to 4.9[min], as shown in

Fig. 3.

0 200 400 600 800 1000 1200 1400

Time [min]

3.5

4.5

Time-Delay [min]

Figure 3: Dead Time variation.

Figure 4 shows the temperature responses for the

disturbance rejection test. It can be seen that for the

last two disturbances, the response of the FO-SMC

controller presents a slight improvement.

0 200 400 600 800 1000 1200 1400

Time [min]

142

144

146

148

150

152

Temperature [

SMC

FO-SMC

Reference

Figure 4: Temperature responses.

The control actions for both controllers have very

similar behavior, as shown in Fig. 5.

0 200 400 600 800 1000 1200 1400

Time [min]

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Controller output [pu]

SMC

F-SMC

Figure 5: Action Controllers.

4.4 Performance Indices Comparison

In Fig. 6 the comparison of the performance indices

of the SMC controller and the FO-SMC is shown. It

can be seen that the performance values obtained for

both controllers are very similar. However, the FO-

SMC controller presents slightly lower ISE and IAE

values.

0 5 10 15 20 25

SMC

FO-SMC

TVu

IAE

23.5

23.51

9.164

9.912

Figure 6: Comparative of performance indexes of the con-

trollers.

5 CONCLUSION

The paper applied a novel SMC based on a fractional-

order model to a variable time-delayed nonlinear pro-

cess. The results showed that the proposed approach

is promising, requiring more exploration of the tun-

ing equations and the effects of λ and β to reach the

sliding surface and reduce chattering.

ACKNOWLEDGEMENTS

The Universidad San Francisco de Quito (USFQ) sup-

ported this work through the Poli-Grants Program un-

der Grant 17965.

Fractional Order-Sliding-Mode Controller for Regulation of a Nonlinear Chemical Process with Variable Delay

137

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