A new fractional-order sliding mode controller via a nonlinear ...

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A new fractional-order sliding mode controller via a nonlinear disturbance observer for a class of dynamical systems with mismatched disturbances Shabnam Pashaei, Mohammadali Badamchizadeh n Faculty of Electrical and Computer Engineering University of Tabriz, Tabriz, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 9 August 2015 Received in revised form 4 April 2016 Accepted 5 April 2016

This paper investigates the stabilization and disturbance rejection for a class of fractional-order nonlinear dynamical systems with mismatched disturbances. To fulfill this purpose a new fractional-order sliding mode control (FOSMC) based on a nonlinear disturbance observer is proposed. In order to design the suitable fractional-order sliding mode controller, a proper switching surface is introduced. Afterward, by using the sliding mode theory and Lyapunov stability theory, a robust fractional-order control law via a nonlinear disturbance observer is proposed to assure the existence of the sliding motion in finite time. The proposed fractional-order sliding mode controller exposes better control performance, ensures fast and robust stability of the closed-loop system, eliminates the disturbances and diminishes the chattering problem. Finally, the effectiveness of the proposed fractional-order controller is depicted via numerical simulation results of practical example and is compared with some other controllers. & 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Sliding mode control Disturbance observer Fractional calculus Lyapunov stability theory Fractional-order mismatched uncertain systems

1. Introduction Non-integer order calculus was introduced by Leibniz in 1695. For about three centuries, the theory of fractional calculus developed as a pure theoretical field of mathematics. The fractional calculus was used only in mathematics and its application in other sciences was not discovered. Since the fractional-order (FO) derivatives and integrals provide a powerful instrument for the description of memory and hereditary properties of different substances, recently many authors pointed out that the fractional calculus is very suitable for modeling and description of properties of many real word processes. This is the most remarkable advantage of the fractional-order models of systems in comparison to the integer-order models. Therefore, it has been known that the FO models of systems are more exact than the integer-order models. This is the main advantage of the fractional-order calculus in comparison to the classical integer-order calculus [1]. Some application of fractional calculus is modeling of speech signal, heat conduction, controlling of autonomous vehicles, electric transmission lines, chemistry and physics [2]. In recent years, the fractional-order controllers have been gained an increasing attention. The idea of utilization the fractional-order n

Corresponding author. Tel.: þ 98 413 3393729; fax: þ 98 413 3300819. E-mail addresses: [email protected] (S. Pashaei), [email protected] (M. Badamchizadeh).

controllers for the control of dynamical systems were presented by Oustaloup. He developed CRONE controller (Commande Robuste d'Ordre Non Entier controller) in 1988 [3,4]. In recent years, the fractional-order PID (FOPID) controller [5], fractional-order TiltIntegral-Derivative (FOTID) compensator [6], fractional-order lead-lag compensators [7], fractional-order optimal controllers [8], fractionalorder adaptive controllers [9] and fractional-order high-gain output feedback control schemes [10] have been studied. Sliding mode control (SMC) techniques have been widely taken into account and developed for the control problems since Utkin's works [11]. SMC is basically a consequence of discontinuous control and is considered as an effective approach for the control of some nonlinear systems that are not stable via continuous state feedback laws. SMC has superiorities such as: simplicity in design, robustness and ability to reject model uncertainties and external disturbances. SMC techniques contain two steps; first, an arbitrary switching surface is considered which provides desirable system performance in the sliding mode (this step is called the existence problem). Second, using the sliding mode theory and Lyapunov stability theory, a control law is designed to ensure the sliding motion (is termed the reachability problem). Both of them are concerned with the asymptotic stability concepts. Consequently, the desired performances are maintained and the system states can reach the equilibrium points [11]. In practice, there are always uncertainties and external disturbances. So, the control and stabilization of the fractional-order dynamical systems in the presence of uncertainties are important topics. There are matched and mismatched uncertainties in dynamical

http://dx.doi.org/10.1016/j.isatra.2016.04.003 0019-0578/& 2016 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Pashaei S, Badamchizadeh M. A new fractional-order sliding mode controller via a nonlinear disturbance observer for a class of dynamical systems with.... ISA Transactions (2016), http://dx.doi.org/10.1016/j.isatra.2016.04.003i

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2

systems. The uncertainties and disturbances which exist in the same channel with the control input are called matched uncertainties. The mismatched uncertainties mean the uncertainties which appear in a different channel from the channel that the control input is presented. In some practical systems, the uncertainties and disturbances are mismatched such as: an aircraft, permanent magnet synchronous motor (PMSM) and MAGnetic LEViation (MAGLEV) suspension system [12]. Most of the existing results on the uncertain systems showed that the conventional form of SMC can guarantee insensitivity to matched uncertainties and external disturbances. But, SMC loses the property of insensitivity in presence of mismatched uncertainties [13]. The stabilization of systems subject to mismatched uncertainties is an interesting topic. For this aim, many researchers proposed several approaches which are combination of different methods with the SMC such as: combination of SMC with the linear matrix inequality (LMI) [13], introduction of a designing method for SMC based on Riccati approach [14], adaptive approach [15], fuzzy [16], neural network [17] and integral sliding mode control (I-SMC) [18]. The chattering is undesirable for the practical systems and actuator since it may direct to the system breakdown and unreasonably large control signal [19]. In order to reduce the chattering problem in SMC and maintain nominal performances, several authors introduced a disturbance observer [20,21]. However, the presented methods in these papers are not available for the mismatched uncertain systems. A sliding mode control via a nonlinear disturbance observer is used to counteract the mismatched uncertainties and reduce chattering in the systems [22]. Researchers have designed a sliding surface based on the disturbance estimation for systems that are subjected to mismatched disturbance to drive the state variables to the desired equilibrium points. The results of this paper have been extended to control of the nth order systems with mismatched uncertainties by proposing an extended disturbance observer and generalization of the sliding surface1 [23]. However, the best knowledge of the prior works shows that there are many papers in control of nonlinear dynamical systems in which the authors have used the fractional calculus for designing the sliding mode controller such as [24–26]. However, to the best of our knowledge, there is no research in which authors have used the fractional calculus for designing of the SMC based on disturbance observer for control of mismatched uncertain systems. While, the utilization of the fractional-order integrals and derivatives in designing of the sliding mode controller can cause robust and finite time stability, reduction the chattering and fast response of the closed loop system in presence of disturbances. Motivated by prior works, in this paper a new fractional-order siding mode control based on disturbance estimation is proposed for a class of fractional-order nonlinear systems in presence of the mismatched disturbances. In addition, for more accurate modeling of the systems, a fractional-order model of MAGLEV system is introduced and the proposed fractional-order control is applied to it because, it has been found that the behavior of many physical systems can be properly described by using the fractional order system theory. Comparison of the simulation results of the proposed method with the integer order SMC, I-SMC, SMC based on the disturbance observer and SMC-EDO shows that the proposed method has better control performance, less overshoot and undershoot, better ability to the mismatched disturbance rejection and alleviating the chattering problem. Moreover, this method is faster and uses less switching gain and observer gains. This paper is organized as follows. Some basic definitions of the fractional-order calculus are introduced in Section 2. In Section 3, 1

SMC-EDO (Sliding Mode Control- Extended Disturbance Observer).

the fractional-order dynamical systems with mismatched disturbances are presented. In Section 4, the design procedure of the proposed fractional-order sliding mode control based on a disturbance observer is exhibited for a class of fractional-order dynamical systems with mismatched disturbances. In Section 5, numerical simulations of the proposed fractional-order sliding mode control based on disturbance observer are presented to show the effectiveness of the proposed control approach. Finally, some conclusions are demonstrated in Section 6.

2. Basic definitions and preliminaries of fractional-order calculus In this section, some basic properties and definitions of the fractional-order integration and differentiation are presented. The main definitions of the fractional derivatives are Riemann–Liouville (RL), Caputo and Grunwald–Letnikov (GL) [1]. Definition 1. The Caputo fractional derivative of order α of a continuous function f ðtÞ is defined as follows: 8 ðmÞ R τÞ < Γ ðm1 αÞ 0t f αðm ; m 1 o α o m; þ 1 dτ ðt τÞ α ð1Þ D f ðtÞ ¼ m : d m f ðtÞ ; α ¼ m; dt where m is the smallest integer number, larger than α. Γ is Gamma function. The definition of Gamma function is given by (2) for all z A R. Z 1 Γ ðzÞ ¼ e u uz 1 du ð2Þ 0

The Gamma function is the generalization of the factorial function and is useful in determining the non-integer values of the factorial function. It should be mentioned that the fractional integral of order α 40 is noted by D α or I α . Property 1. [5]: The following equality holds for the fractional derivative: m h i X ðt t 0 Þα j β βα βj ð3Þ f ðtÞ Dt f ðtÞ Dt α ðDt f ðtÞÞ ¼ Dt t ¼ t 0 Γ ð1 þ α jÞ j¼1 where m 1 r β o m. Property 2. [1]: Similar to integer-order calculus, the fractionalorder integration and differentiation are linear operations Dα ða f ðtÞ þ b gðtÞÞ ¼ aDα f ðtÞ þ bDα gðtÞ

ð4Þ

Property 3. [1]: For α ¼ n, where n is an integer, the operation Dα f ðtÞ gives the same result of the integer order integration and differentiation, and also for α ¼ 0 the following relation is established: D0 f ðtÞ ¼ f ðtÞ

ð5Þ

Lemma 1. [27]: For an integrable function f ðtÞ if there is at least one t 1 A ð0;tÞ such that f ðt 1 Þ a 0, then there is a positive constant L such that I α f ðtÞ ZL. Lemma 2. [28,29]: Let x ¼ 0 be an equilibrium point for the nonautonomous fractional-order system Dα xðtÞ ¼ f ðx; tÞ

ð6Þ

where f ðx; tÞ satisfies the Lipschitz condition with Lipschitz constant l 4 0 and α A ð0; 1Þ. Assume that there is a Lyapunov function Vðt; xðtÞÞ and class-K functions α1 ; α2 and α3 that satisfy

α1 ð‖xkÞ o Vðt; xðtÞÞ o α2 ð‖x‖Þ:

ð7Þ



Dα Vðt; xðtÞÞ o α3 ð‖x‖Þ

ð8Þ

where α A ð0; 1Þ. Then the equilibrium points of the system (6) are asymptotically stable. In this article, the notation Dα is used as the simplified form of Caputo fractional-order derivative.

d

SMC controller

u

Nonlinear system

y

dˆ

3. Problem statement Consider the following class of the fractional-order dynamical systems in the presence of mismatched external disturbances: ( α D x1 ¼ x2 þ dðtÞ; ð9Þ Dα x2 ¼ aðxÞ þ bðxÞu; y ¼ x1

3

ð10Þ

where α A ð0; 1Þ is the order of system, x1 ; x2 are the state variables of the system, dðt Þ is the external disturbance of the system, uðtÞ is the control signal and y is the output of system. It should be mentioned that aðxÞ and bðxÞ are known, smooth nominal functions in term of x and the function bðxÞ a 0 for all t Z 0.

Nonlinear Disturbance Observer Fig. 1. Block diagram of nonlinear disturbance observer with sliding mode control.

controller with the nonlinear disturbance observer is shown in Fig. 1. Lemma 3. [22]: Consider the system (9) and the disturbance observer (12). Suppose that Assumptions 1 and 2 are satisfied for this ^ system and eðtÞ ¼ dðtÞ dðtÞ is the disturbance estimation error. The

^ disturbance estimation dðtÞ of the nonlinear disturbance observer (12) can track the disturbance dðtÞ of the system (9) asymptotically lim eðtÞ ¼ 0 if the observer gain is selected such that lG2 4 0 holds, t-1

Assumption 1. The external disturbance dðtÞ is bounded and this bound is defined by d ¼ supdðtÞ; t 4 0: Assumption 2. The derivative of the external disturbance dðtÞ is _ ¼ 0: bounded and satisfies lim dðtÞ t-1

For this kind of systems subject to mismatched disturbances, the aim of designing controller uðtÞ is making the system output unaffected by the mismatched external disturbance dðtÞ. The nonlinear system (9) is rewritten as follows: Dα X ¼ FðXÞ þ G1 ðXÞu þ G2 ðXÞd: y ¼ x1 :

ð11Þ

where T X ¼ ½x1 ; x2 T , FðXÞ ¼ ½x2 ; aðxÞT ,G1 ðX Þ ¼ 0; bðxÞ ; G2 ðX Þ ¼ ½1; 0T .

which implies that e2 eðtÞ ¼ 0 is globally asymptotically stable. Consider the sliding surface for the fractional-order mismatched uncertain system (9) as ^ sðtÞ ¼ k1 x1 þ k2 ðx2 þ dÞ:

x_ 2 ¼ 4. The proposed fractional-order sliding mode controller based on disturbance observer for a class of fractional-order mismatched uncertain systems

1 _ k1 x_ 1 k2 d^ : k2

D1 α Dα x2 ¼

1 _ k1 x_ 1 k2 d^ : k2

k1 α ^ D x1 Dα d: k2

It is well known that SMC is designed to counteract the disturbances and uncertainties. But in some systems, it is not possible to estimate or obtain the upper bounds of unknown disturbances. In order to stabilize the system, the controller parameters must be increased. In addition, the chattering effect is the biggest disadvantage of SMC. In these cases, the best idea is to use a disturbance observer. This section presents a new fractional-order sliding mode control based on a nonlinear disturbance observer (FOSMC-DO) for a class of FO mismatched uncertain systems. The results of the proposed method are compared with the results of the presented integer-order SMC, I-SMC, SMC-DO2 in [22] and the presented method in [23]. A nonlinear disturbance observer which can estimate the disturbance in (11), is introduced by References [21,22,30] as the following equations:

Dα x2 ¼

p_ ¼ lG2 ðXÞp l½G2 ðXÞlX þ FðXÞ þ G1 ðXÞu: d^ ¼ p þlX:

where ξ1 and ξ2 are constant gains and δ A ð0; 1Þ. Now, the FOSMC law can be written as follows:

ð12Þ

where d^ is the estimation of d,p is the internal state of the nonlinear observer and l is the observer gain. A block diagram of SMC 2

Sliding Mode Control Based on Disturbance Observer.

ð13Þ

where k1 ; k2 are positive constants and d^ is the estimation of disturbance that achieved by the nonlinear disturbance observer (12). To design a fractional-order sliding mode controller based on the disturbance observer, the time derivative of (13) is calculated as presented in the following: _ s_ ðtÞ ¼ k1 x_ 1 þ k2 x_ 2 þ d^ ¼ 0: ð14Þ ð15Þ ð16Þ ð17Þ

According to SMC theory and using (9) and (17) the equivalent control law is obtained as 1 k1 ueq ðtÞ ¼ Dα x1 aðxÞ Dα d^ : ð18Þ bðxÞ k2 This equivalent control can be considered as the mean value of the discontinuous control on the sliding surface. So as to satisfy the sliding condition in the presence of the mismatched disturbances, the discontinuous reaching law is assumed as follows: δ ur ðtÞ ¼ Dα 1 ξ1 sðtÞ þ ξ2 sign ðsðtÞÞsðtÞ : ð19Þ Dα d^ ¼ kc Dα 1 sign ðsðtÞÞ:

uðtÞ ¼ ueq ðtÞ þur ðtÞ 1 k1 Dα x1 aðxÞ kc Dα 1 sign ðsÞ ¼ bðxÞ k2 α1 ξ1 s þ ξ2 sign ðsÞjsjδ : D

ð20Þ

ð21Þ



4

Table 1 The parameters of the MAGLEV suspension [12]

Table 2 The control parameters of the controllers.

Parameters

Definition

Value

Controllers

Parameters

Ms Kb Kf Rc Lc Nc Ap I0 G0

Carriage mass Flux coefficient Force coefficient Coil's resistance Coil's inductance Number of turns Pole face area Nominal current Nominal air gap

1000 kg 0.0015 T m 0.0221 N/T2 10 Ω 0:1 H 2000 0:01 m2 10 A 0:015 m

SMC [21] I-SMC [21] SMC-DO [21] SMC-EDO [22]

k1 ¼ 100; k2 ¼ 20; k3 ¼ 1; kc ¼ 60 k0 ¼ 200; k1 ¼ 100; k2 ¼ 20; k3 ¼ 1; kc ¼ 60 k1 ¼ 100; k2 ¼ 20; k3 ¼ 1; kc ¼ 60; l ¼ ½100; 0; 0 c1 ¼ 10; c2 ¼ 4; kl ¼ 4; ks ¼ 6; l11 ¼ 15 l12 ¼ 50; l13 ¼ 10; l31 ¼ 80; l32 ¼ 50

FOSMC-DO

k1 ¼ 70; k2 ¼ 15; k3 ¼ 1; kc ¼ 0:3; α ¼ 0:99 ξ1 ¼ 30; ξ2 ¼ 30; l ¼ ½80; 0; 0; δ ¼ 0:99

Substituting (9), (20) and (26) into (25) and using sign ðsÞs ¼ jsj and sign2 ðsÞ ¼ 1 one can obtain Dα V r k2 kc jsj k2 ζ 1 bðxÞjsj k2 ζ 2 bðxÞjsjþ β jsj þ k2 kc jsj:

ð27Þ

Dα V r jsjð k2 kc k2 ζ 1 k2 ζ 2 þ β þ k2 kc Þ r jsjð k2 ðζ 1 þ ζ 2 Þ þ β Þ: ð28Þ By holding k2 ðζ 1 þ ζ 2 Þ Z β the system is stable and according to Lemma 2 the system state variables will asymptotically converge to sðtÞ ¼ 0. In order to show that this convergence occurs in finite time the following proof is used. By using Property 1 and taking the fractional integral of both sides of (28) from 0 to t r , we can write Vðt r Þ V α 1 ð0Þ

ðt r Þα 1 r ð k2 ðζ 1 þ ζ 2 Þ þ βÞD α jsj: Γ ðαÞ

ð29Þ

where t r is the reaching time. The reaching time, the time for the system states to reach the sliding surface. So based on Lemma 1 and sðt r Þ ¼ 0 (Vðt r Þ ¼ 0 and I α sðtÞ Z L) it can be written

Fig. 2. Block diagram of MAGLEV suspension system [12].

V α 1 ð0Þ

ðt r Þα 1 rð k2 ðζ 1 þ ζ 2 Þ þ β ÞL: Γ ðα Þ

ð30Þ

Theorem 1 is proposed and then is proved to guarantee the existence of the sliding motion and finite time stability.

ðt r Þα 1 r

Theorem 1. Consider the FO system (9) and the proposed FOSMC (21) then the closed-loop system will be stable in the presence of mismatched disturbances.

tr r

Proof. Select Lyapunov function as: 1 V ¼ s2 : 2

Thus, the state variables of the system will converge to zero in finite time and the proof is complete.

ð22Þ

Motivated by [31], the following assumption could be used X 1 Γ ð1 þ αÞ Dk sDα k s r β jsj: ð23Þ Γ ð1 þ kÞΓ ð1 k þ αÞ k¼1 where β is a positive constant. By taking the fractional-order derivative of (22) and using the mentioned property in [31], it can be written Dα V ¼ sDα s þ

1 X

Γ ð1 þ αÞ Dk sDα k s: Γ ð1 þ kÞΓ ð1 k þ αÞ k¼1

ð24Þ

with considering (23) one can write X 1 Γ ð1 þ αÞ Dα V r sDα s þ Dk sDαk s rsDα s þ βjsj: Γ ð1 þkÞ Γ ð1 k þ α Þ k¼1 ð25Þ The fractional-order derivative of (13) is ^ Dα s ¼ k1 Dα x1 þ k2 Dα x2 þ k2 Dα d:

ð26Þ

Γ ðαÞLðk2 ðζ 1 þ ζ 2 Þ βÞ V α 1 ð0Þ

:

ð31Þ

!1 α

Γ ðαÞLðk2 ðζ 1 þ ζ 2 Þ βÞ V α 1 ð0Þ

:

ð32Þ

5. The numerical example of the proposed FOSMC based on disturbance observer In order to verify the excellent properties of the proposed fractional-order sliding mode control based on disturbance observer for a class of the fractional-order dynamical systems, an example is given in this part. To solve a fractional-order differential equation, Diethelm formulated an approximate numerical approach which is a generalization of the Adams–Bashford– Moulton algorithm. This approach is based on a predictor–corrector technique utilizing the Caputo definition [32]. In the rest of the paper, this approach will be considered to solve the fractionalorder differential equations. Example. MAGLEV suspension system is selected for the simulation. The integer-order model of the MAGLEV suspension system was presented in [12]. Also, the integer-order SMC-DO was designed for this system in [22]. The fractional-order model of the



12

100 SMC I-SMC SMC-DO FOSMC-DO SMC-EDO

40

8 6 S(t)

60

X1

SMC I-SMC SMC-DO FOSMC-DO SMC-EDO

10

80

4

20

2

0

0

-20 -40

-2

0

2

4 6 Time (seconds)

8

-4

10

0

2

1000

0.8

800

0.6 0.4 0.2

10

400 200

0

0

0

2

4 6 Time (seconds)

8

10 -200

0.15

-400 SMC I-SMC SMC-DO FOSMC-DO SMC-EDO

X3

0.1

0.05

0

SMC [21]

2

4 6 Time (seconds)

8

10

Fig. 3. (a)–(c) State responses of the MAGLEV suspension system under the mentioned controllers to an applied step disturbance.

MAGLEV (FO-MAGLEV) suspension system is obtained in the appendix.

ð33Þ

T where x ¼ i; Dα z; ðzt zÞ are the state variables of the system, i is α the current, D z is the vertical electromagnet velocity and ðzt zÞ is the air gap. The input u ¼ V coil is the voltage and d ¼ Dα zt is the rail vertical velocity. A,Bu ,Bd ,C are the state matrix, input matrix,

Asd

Ax1 d

Ax2 d

0.43 96.5 0

Ax3 d

t rsd (s)

Cannot reach to zero I-SMC [21] 0.43 74 0.88 0.112 Cannot reach to zero SMC-DO [21] 11.8 46 0.96 0.072 1.45 SMC-EDO [22] 2.7 91.5 1.05 0.14 2.4 FOSMC-DO 5.1 30 0.81 0.047 1.25

8

10

0.144

t rx1 d (s) Cannot reach to zero 3.4 2 6.5 1.85

Table 4 The definition of the used parameters in Table 3. Parameters Definition Asd

FO-MAGLEV suspension system can be described by ( α D x ¼ Ax þ Bu u þ Bd d;

4 6 Time (seconds)

Table 3 The comparison of the simulation results.

0

0

2

Fig. 5. The time responses of the applied control signals via tanh function.

Controllers

y ¼ Cx;

8


600


U(t)

X2

1

-0.05

4 6 Time (seconds)

Fig. 4. The time responses of the applied sliding surfaces.

1.2

-0.2

5

Ax1 d Ax2 d Ax3 d t rsd t rx1 d t sx1 t sx2

The maximum peak of sliding surface at the time of applying disturbance The maximum peak of x1 at the time of applying disturbance The maximum peak of x2 at the time of applying disturbance The maximum peak of x3 at the time of applying disturbance The time that sliding surface reaches to zero after applying disturbance The time that x1 reaches to zero after applying disturbance The settling time of x1 before applying disturbance The settling time of x2 before applying disturbance



6

100


25 20

I-SMC SMC-DO FOSMC-DO

60 S(t)

X1

15

SMC

80

10

SMC-EDO

40

5 20

0 -5

0

-10 -15

0

5

10

-20

15

0

5

10

15

Time (seconds)

Time (seconds)

Fig. 7. The time responses of the applied sliding surfaces.

0.3 600

0.25


0.2

500


400 300 U(t)

X2

0.15 0.1

200 100 0 -100

0.05

-200 -300

0

-400

-0.05

0

5

10

0

5

15

Time (seconds)

10 Time (seconds)

15

Fig. 8. The time responses of the applied control signals via sign function.

0.05 600

0.04

500


0.03

SMC I-SMC

400

SMC-DO FOSMC-DO

300

0.02

SMC-EDO

U(t)

X3

200

0.01

100 0

0

-100

-0.01

-200 -300

-0.02 0

5

10

15

Time (seconds) Fig. 6. (a)–(c) State responses of the MAGLEV suspension system under the mentioned controllers to an applied impulse disturbance.

-400

0

5

10

15

Time (seconds) Fig. 9. The time responses of the applied control signals via tanh function.



60

0.35

50

0.3

40

0.25

30

0.2

X1

Random Disturbance

0.4

7


20 10

0.15

0 0.1

-10 0.05 0

-20 0

2

4

6

8 10 12 Time (seconds)

14

16

-30

18

0

2

4

6


14

16

18

Fig. 10. The applied random disturbance on the FO-MAGLEV suspension system.

0.6 disturbance matrix and output matrix that are given in below 2 3 Rc K b Nc Ap IoAp

0 Ap 2 L þ K N G L þ K N 6 c b c Go 7 c b c Go o 6 7 2 7 ð34Þ A¼6 0 2K f I o 3 7: 6 2K f M IGo 2 M s Go 5 4 s o 0 0 1 1 A Lc þ K b Nc Gpo

6 Bu ¼ 6 40

3

0.4 0.3 X2

2

7 7: 5

0.2

ð35Þ

0 2 6 G2o 6 Bd ¼ 6 0 4 1

0.1 K b Nc Ap Io Ap

Lc þ K b N c G

o

3 7 7 7: 5

C ¼ ½0; 0; 1:

0 ð36Þ

-0.1

Assume that 2 3 C 6 7 T ¼ 4 CA 5: 2 CA

2

4

6


14

16

18

0.1

ð38Þ


0.08 0.06 0.04 X3

η ¼ Tx:

0

ð37Þ

The system parameters in (34)–(37) are listed in Table 1. A diagram of the MAGLEV suspension system is depicted in Fig. 2. For more information about the MAGLEV suspension system see [12]. For modeling the proposed FOSMC via the disturbance observer on the FO-MAGLEV suspension system, the following equivalent transformation is used [22]:

0.02 0

ð39Þ

-0.02

After substitution of (38) and (39) into (33) the equivalent of the FO-MAGLEV suspension system is written as α

D η ¼ A η þ B u u þ B d d; y ¼ C η;


0.5

ð40Þ

-0.04

0

2

4

6


14

16

18

Fig. 11. (a)–(c) State responses of the MAGLEV suspension system under the mentioned controllers to an applied random disturbance.



in which ;

B u ¼ TBu

;

B d ¼ TBd

;

C ¼ CT

1

:

ð41Þ

Using matrixes A; Bu ; Bd ; C and the parameters in Table 1 give the FO model of the MAGLEV suspension system as 8 α > < D η1 ¼ η2 þ d; Dα η2 ¼ η3 ; > : Dα η ¼ 6233:8η 65:2η 4:8η þ 0:9212u 65:17d; 3

1

2

3

y ¼ η1 :

ð42Þ

After using transformation (38), the nonlinear disturbance observer in (12) is written as ( p_ ¼ lB d ðp þ lηÞ lðA η þ B u uÞ: ð43Þ d^ ¼ p þ lη: The sliding surface (13) is used as follows: ^ sðtÞ ¼ k1 η1 þ k2 η2 þ k3 η3 þk2 d:

ð44Þ

The proposed FOSMC (21) is designed for FO-MAGLEV suspension system (42) as uðtÞ ¼ 6767:04η1 þ 70:77η2 þ 5:21η3 þ 70:74d^

k1 d^ 0:9212k3

k1 k2 kk η η c 2 Dα 1 ðsignðsÞÞ 0:9212k3 2 0:9212k3 3 0:9212k3 Dα 1 ξ1 s þ ξ2 signðsÞjsjδ :


4

ð45Þ

It can be seen from (33) and (42) that the disturbances in the MAGLEV system are mismatched. The results of the proposed FO controller are compared with the results of the integer-order SMC, I-SMC and SMC-DO in [22] and the SMC-EDO in [23]. The initial states of the system are selected as: ½ið0Þ; Dα zð0Þ; ðzt zÞð0Þ ¼ ½0; 0; 0:003:

are the other excellent properties of the proposed method. So this method is cheaper to the practical usages and has better control performance than the other methods when the system is subject to mismatched disturbances. In Fig. 5, so as to demonstrate all of the input signals on a graph, we used tanh function instead of sign function. The comparison results mentioned above are written in Table 3 and the definitions of the used parameters in Table 3 are given in Table 4. Figs. 3–5 and Tables 2 and 3 show that the proposed controller compared with the integer order SMC, I-SMC, SMC-DO and SMC-EDO by using much less switching gain and observer gains has better control performance in the presence of mismatched disturbances. In addition, robust and finite time stability, less overshoot and undershoot, fast response of the closed-loop system and chattering reduction are the other properties of the proposed method. Matched and mismatched disturbances affect the systems and have an undesired effect on the controlled variables. Disturbance rejection controllers are the best suited for mismatched disturbance rejection of the systems. In order to show the better mismatched disturbance rejection of the FO-MAGLEV system by using the proposed controller, after 2 s an external impulse disturbance is applied

3 2 1

S(t)

A ¼ TAT

1

0

ð46Þ

For comparison, the control parameters of the five control methods are written in Table 2. After 1 s a step external disturbance d ¼ 0:7 is applied. The simulation results of the proposed method, SMC, I-SMC, SMC-DO and SMC-EDO are depicted in Figs. 3–5. Based on Figs. 3–5, it is evident that for the time between 0 to 1 s, the time responses of the state variables, control inputs and sliding modes of the five mentioned control methods are the same. But in the presence of external mismatched disturbances, the SMC is sensitive and fails to force the states of the system to reach the favorable equilibrium points. The I-SMC is more practical than SMC because of its robustness. So, the I-SMC can repress the mismatched disturbances better than the SMC. However, the ISMC method has large overshoot, long settling time and poor performance when the system is subject to mismatched disturbances. The SMC-DO has better disturbance rejection property than SMC and I-SMC. But, the used switching gain and observer gains have large values. Also in [22], in order to alleviate the chattering problem the switching gain of this controller was reduced to kc ¼ 15. Then the results of the SMC-DO, SMC and ISMC were compared. However, it was shown that both the SMC and I-SMC controllers were failed to reject the disturbances. Although, the chattering problem is partially reduced, the fine performance of the SMC-DO is destroyed, either. Also, the SMCEDO is used in [23] for mismatched disturbance rejection of the MAGLEV system. According to the simulation results, the disadvantages of the SMC-EDO are large overshoot, undershoot and long settling time. The proposed FOSMC-DO has faster convergence speed and less overshoot than the other methods. Furthermore, the ability to disturbance rejection and chattering reduction by using a much less switching gain and observer gains

-1 -2 -3 -4

0

2

4

6

8

10

12

14

16

18

Time (seconds) Fig. 12. The time responses of the applied sliding surfaces.


500 400 300 U(t)

8

200 100 0 -100 -200 -300

0

2

4

6


14

16

18

Fig. 13. The time responses of the applied control signals via tanh function.



Appendix A

0.15 SMC I-SMC SMC-DO FOSMC-DO SMC-EDO

0.1 0.05

The flux density is given by [12] I B ¼ Kb : G

ðA:1Þ

The force is [12] 2 I F ¼ K f B2 ¼ K f : G

X3

0 -0.05

ðA:2Þ

The fractional-order equation of motion from the fractionalorder Newton's second law [1,12]

-0.1

M s Dβ z ¼ M s g F; 1 o β o 2:

-0.15 -0.2

9

0

1

2

3


7

8

9

10

Fig. 14. The output of the MAGLEV suspension system under the mentioned controllers to an applied white noise.

on the system. In this case, the simulation results of the proposed method, SMC, I-SMC, SMC-DO and SMC-EDO are depicted in Figs. 6–9. It is shown that an impulse disturbance can be rejected when the fractional order system is controlled by the proposed fractional order controller. According to Fig. 8, the proposed FOSMC-DO can repress the chattering reduction better than the integer order SMC, I-SMC, SMC-DO and SMC-EDO. Furthermore, a random disturbance according to Fig. 10 is applied on the FO-MAGLEV system. The simulation results of the five mentioned control methods are depicted in Figs. 11–13. Again, tanh function is used instead of sign function in Figs. 9 and 13. The simulation results show that the random external disturbance is eliminated successfully from the output using the proposed control method. In order to show the possible implementation and practical issues, the white noise with zero mean and 0.1 variance has been applied on the FO-MAGLEV system. The output of the MAGLEV suspension system under the mentioned controllers to an applied white noise is shown by Fig. 14. It can be observed that the proposed method can effectively eliminate the noise effect from the output of the system. Clearly, compared with the traditional SMC, I-SMC, SMC-DO and SMC-EDO controllers, the effect of the external disturbance signals and noise can be significantly reduced by the proposed controller. Therefore, the proposed controller will be useful for the stabilization and mismatched disturbance rejection of a class of nonlinear systems.

ðA:3Þ

Substitution of (A.2) in (A.3) it can be written Kf I 2 ; 1 o β o 2: Dβ z ¼ g Ms G

The electromagnet circuit involved from the electromagnet's coil [12] V coil ¼ IRc þLc

dI dB þ N c Ap : dt dt

ðA:5Þ

By applying (A.1) and considering mentioned property in [1] and after some simplification, (A.6) is obtained ð0 o α o 1Þ Dα I ¼

N c Ap K b I V coil G Rc G Dα G: Iþ GLc þ Nc Ap K b GLc þ Nc Ap K b GðGLc þ N c Ap K b Þ

This paper investigated the stabilization and mismatched disturbance rejection of a class of the fractional-order uncertain dynamical systems. In order to improve the performance and robustness of a class of the fractional-order nonlinear systems subject to mismatched disturbances a new fractional-order sliding mode control by using the nonlinear disturbance observer is proposed. The excellent properties of the proposed FOSMC-DO compared to the SMC, I-SMC, SMC-DO and SMC-EDO include better ability to mismatched disturbance rejection by using less switching gain and observer gains, alleviating the chattering problem. Moreover, the proposed method is faster and has better control performance, less overshoot and undershoot. Finally, the simulation results are exhibited to verify the effectiveness and robustness of the proposed control strategy.

ðA:6Þ

and G ¼ zt Z ) Dα G ¼ Dα zt Dα Z:

ðA:7Þ

From (A.4), (A.6) and (A.7) a nonlinear and FO model of the MAGLEV suspension system is obtained. The MAGLEV suspension system is soft enough to linearization. For linearization, the following definitions are used [12]. B ¼ Bo þ b:

ðA:8Þ

I ¼ I o þ i:

ðA:9Þ

G ¼ Go þ ðzt zÞ:

ðA:10Þ

Z ¼ Z o þ z:

ðA:11Þ β

By assuming at the steady state D Z o ¼ 0, replacing (A.9)–(A.11) in (A.4) and after some simple manipulations (A.12) is obtained Dβ Z ¼

2K f I o M s Go

2

iþ

2K f I o 2 M s Go 3

ðzt zÞ:

ðA:12Þ

The fractional-order V coil is written: V coil ¼ IRc þLc Dα I þ Nc Ap Dα B:

6. Conclusion

ðA:4Þ

ðA:13Þ

Next (A.13) is linearized. Replacement of (A.8)–(A.10) in (A.1) 1 þ Iio 1 ðztGo zÞ Io : ðA:14Þ Bo þ b ¼ K b Go 1 þ ðzt zÞ 1 ðzt zÞ Go Go Note that 1 þ ðztGo zÞ 1 ðztGo zÞ 1. Take the fractional derivative of (A.14) Dα b ¼

K b α K b Io α D i 2 D ðzt zÞ: Go Go

ðA:15Þ

By substituting (A.15) into (A.13) and after some simplification Dα i ¼

Rc Lc þ

þ

K b N c Ap Go

iþ

V coil Lc þ

K b N c Ap Go

G2o

:

K N A I K N A I b c p o Dα zt b c p o Dα z K b N c Ap K N A 2 Lc þ Go Go Lc þ b Goc p ðA:16Þ



10

Dα ðzt zÞ ¼ Dα zt Dα z:

ðA:17Þ

y ¼ zt z:

ðA:18Þ

From (A.12), (A.16)–(A.18) a linear and FO model of the MAGLEV suspension system is obtained. Assume 2 3 2 3 x1 i 6 x 7 6 Dα z 7 α ðA:19Þ 4 25¼4 5 ; u ¼ V coil ; d ¼ D zt : x3 zt z Take the αth-order fractional derivative of (A.19): 2 α 3 2 α 2 3 3 x1 D i D x1 6 Dα x 7 6 Dβ z 6 7 7 4 5 ¼ A4 x2 5 þ Bu u þ Bd d: 25¼4 x3 Dα ðzt zÞ Dα x3 2

3 x1 6x 7 y ¼ C 4 2 5: x3 2

Rc

ðA:20Þ

K b Nc Ap IoA

A

6 Lc þ K b Nc Gpo 6 A¼6 6 2K f MIoG2 s o 4 0 2

0

G2o Lc þ K b N c Gp

o

0 1

2 2K f I o 3 M s Go

3 7 7 7: 7 5

ðA:21Þ

0

3

1 Ap

6 L c þ K b N c Go 7 7: Bu ¼ 6 40 5 0 2

K b Nc Ap Io

ðA:22Þ

6 Go 2 Lc þ K b Nc Gp

6 Bd ¼ 6 0 4 1

C ¼ ½0; 0; 1:

A

o

3 7 7 7: 5

ðA:23Þ

ðA:24Þ

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