A novel fractional-order fuzzy control method based on immersion and invariance approach

Applied Soft Computing Journal 88 (2020) 106043

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Applied Soft Computing Journal journal homepage: www.elsevier.com/locate/asoc

A novel fractional-order fuzzy control method based on immersion and invariance approach Ardashir Mohammadzadeh a , Okyay Kaynak b , a b

∗

Electrical Engineering Department, University of Bonab, Bonab, Iran Bogazici University, Istanbul, Turkey

article

info

Article history: Received 7 January 2019 Received in revised form 4 December 2019 Accepted 17 December 2019 Available online 26 December 2019 Keywords: Immersion and invariance Non-singleton type-2 fuzzy neural network Robust control Fractional-order chaotic systems

a b s t r a c t In most of industrial applications, the dynamics of the system in hand are perturbed by a number of operational conditions. Also the outputs of the sensors always include noise. To alleviate these common problems, this paper presents a novel fuzzy control approach based on the immersion and invariance (I&I) approach under the conditions of unknown dynamics and measurement errors. The adaptation laws for the parameters of the proposed non-singleton type-2 fuzzy neural network (NT2FNN) are derived through a stability analysis based on I&I method. The effectiveness of the proposed membership function (MF) and non-singleton fuzzification is verified by comparison with the conventional Gaussian MF in the presence of measurement errors. The performance of the proposed control method is compared with other techniques and an experimental study is provided to show the capability of the proposed control scheme in real-time applications. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The fractional-order systems have received an increasing attention in recent years. It has been shown that many systems can be described more accurately by using fractional-order calculus such as, financial systems [1], intelligent controllers [2], mechanical systems [3], diffusion waves [4], oscillation of earthquakes [5], electromagnetism [6] and so on. The fractional-order chaotic/hyperchaotic systems are a case of nonlinear systems that have some special characteristics such as, sensitivity to the initial conditions and stochastic/unpredictable behaviors [7]. The chaotic behavior is observed in a wide class of mechanical, electrical, medical and physical systems. The hyperchaotic systems have more than one Lyapunov exponent and display much more complex behavior than regular chaotic systems [2]. Various control methods have been proposed to control and to synchronize chaotic systems. Some of the well-known methods are reviewed in what follows. For instance, in [8,9], the sliding mode technique is applied to synchronize two fractionalorder chaotic systems. The active control method is designed in [10], for a fractional-order financial system and some conditions are obtained to guarantee the stability performance. In [11], the synchronization control of fractional-order chaotic systems is studied and an observer is designed to handle unknown external disturbances. The problem of projective synchronization ∗ Corresponding author. E-mail addresses: [email protected] (A. Mohammadzadeh), [email protected] (O. Kaynak). https://doi.org/10.1016/j.asoc.2019.106043 1568-4946/© 2019 Elsevier B.V. All rights reserved.

is investigated in [12], where some adaptation laws are derived by using Lyapunov stability theorem to update the parameters of controller. In [13], the proportional–integral control method is designed to control the nonlinear chaotic systems and by using linear matrix inequalities the stability conditions are derived. In [14], the backstepping control technique based on MittagLeffler function is presented to stabilize a class of fractional-order systems. The optimal PID controller for chaotic systems is studied in [15]. The nonlinear dynamics of the system are assumed to be known in the most of the cited works. The fuzzy systems are universal approximators and then many systems and process can be modeled based on this property [16,17]. To deal with the uncertainties in the dynamics of the system some fuzzy control methods have also been proposed. For instance the general projective synchronization by using fuzzy system is studied in [18], where the stability of the closed-loop system and boundedness of the tracking errors are proved by Lyapunov approach. In [19], a fuzzy control approach based on Takagi–Sugeno model for a class of nonlinear fractional-order systems is investigated and some stability conditions are derived. The fuzzy sliding mode technique is presented in [20], to control and to synchronize factional-order systems, where a fuzzy system is used in the switching part of the conventional sliding mode technique and it is shown that the states are swiftly reached to the sliding surface. The indirect model reference adaptive fuzzy control based on Lyapunov stability theorem is proposed in [21], to control the fractional order chaotic Duffing oscillator, where the dynamic

2

A. Mohammadzadeh and O. Kaynak / Applied Soft Computing Journal 88 (2020) 106043

system is modeled by a Takagi–Sugeno fuzzy system. The fuzzy backstepping control technique is developed in [22], for a class of nonlinear fractional-order systems. The fuzzy fractional-order systems are studied in [23]. In most of the aforementioned fuzzy control methods, the type-1 fuzzy systems are used. However, it is demonstrated that the type-2 fuzzy controllers have more accuracy with respect to their type-1 counterpart, especially in the noisy environment [24, 25]. The basic fundamentals and designing process of interval and generalized type-2 fuzzy systems have been widely studied [26]. Also the learning of type-2 fuzzy systems in the various approaches such backpropagation, evolutionary algorithms, Kalman filters and Lyapunov methods have been well investigated [27– 30]. However, the type-2 fuzzy control methods are rarely considered in literatures. For instance in [31], a PID fuzzy controller is designed by using the interval type-2 fuzzy systems for a class of fractional-order systems and its performance is compared with the type-1 fuzzy PID controller. In [32], based on the Mittag-Leffler stability theorem a type-2 fuzzy control method is proposed for a class of nonlinear fractional-order chaotic systems. In [33], type-2 fuzzy sliding mode technique is designed to achieve finite-time stability performance. The type-2 fuzzy control based on H∞ technique is studied in [34,35]. In [36], the conventional sliding mode control technique is developed by using type-2 fuzzy systems and it is shown that the type2 fuzzy systems are effective in elimination of the chattering phenomenon. In [37], the input nonlinearities and unknown dynamics are estimated by the type-2 fuzzy systems(T2FSs) and an adaptive fuzzy controller is designed to guarantee the stability of the closed-loop system. The optimality of the T2FSs in control applications is studied [38]. The industrial applications of the fuzzy systems have been frequently investigated. For instance, in [39], a fuzzy based wavelet neural network is used in an adaptive control approach and is applied on the industrial manipulators. In [40], the effectiveness of the fuzzy sliding mode control is examined on the position control of robotic arm. In [41], the performance of the fuzzy PID controller is practically examined on the inverted pendulum system. In [42], the interval type-3 fuzzy systems are practically used for online modeling of the nonlinear systems. In [43], the online adjusting the parameters of the PID controller in industrial applications is studied and the performance of the controller is evaluated on the solid bulk process. In [44], the fuzzy controller is applied on a mobile robot system and in real-time examination the good performance of the fuzzy controller is demonstrated. In [45], a survey on the application of the fuzzy systems and the computational intelligence in industry is given. The performance of type-1 and type-2 fuzzy controllers is experimentally compared by applying on the spherical robot Sphero 2.0 in [46] and it is shown the use of type-2 fuzzy systems improve the performance. Recently the integer-order I&I control method has attracted growing interest and it is studied in some papers. For instance in [47], a backstepping controller is designed based on I&I to attitude control of the space craft. In [48], the I&I controller is applied for a moving-mass flight vehicle. In [49] the adaptive I&I is developed for the quadrotor systems with uncertain parameters. In [50], a controller is designed by I&I theorem for a unmanned aerial vehicle. The procedure of the control design by I&I method and construction of stabilizing laws is explained in [51]. Although the I&I method for integer-order systems are rarely studied in some papers, but as far as the authors know, the I&I method for fractional-order nonlinear systems have not been studied. In most type-2 fuzzy controllers that are proposed in literature, the measurement errors are neglected and also the asymptotic stability performance in the presence of perturbed dynamics

are not fully investigated. Furthermore the computational cost of the proposed type-2 fuzzy controllers is high and the efficiency of these controllers under practical conditions are not investigated. Also the design of stable and robust fuzzy control systems that do not require a Lyapunov functions has been rarely studied. Motivated by the discussion above, a novel non-singleton type2 fuzzy control method is presented in this paper, based on the recently developed immersion and invariance (I&I) technique. It is a method that relies on the notions of system immersion and manifold invariance and, in principle, does not require the knowledge of a Lyapunov function [51]. Also in the proposed scheme the effects of the measurement errors and dynamic uncertainties are compensated and the stability of the closed-loop system is guaranteed. To cope with the measurement errors, a new type-2 fuzzy set with non-singleton fuzzification is proposed. To decrease the computational cost of the type-2 fuzzy system so that it can be applicable in real-word applications, a simple typereduction is used. Furthermore, in the proposed fuzzy system, the number of membership functions for each input is the same and the number of rules is equal to the number of membership functions. This aspect of the proposed controller is verified by experimental implementation. The most important contributions of this paper are summarized in below:

• A non-singleton fuzzification is derived based on the product t-norm for the proposed type-2 membership function.

• A new fuzzy control is presented based on I&I method for a class of uncertain nonlinear fractional-order systems.

• The effectiveness of the proposed method is verified by the simulation and experimental studies. The remaining of this paper is organized as follows. In Section 2, the problem is formulated and some preliminaries are presented. The proposed non-singleton type-2 fuzzy neural network is presented in Section 3. The stability analysis based on I&I theorem is presented in Section 4. The simulation and experimental results are shown in Section 5. Finally conclusions remarks are presented in Section 6. 2. Problem formulation and preliminaries A class of fractional-order chaotic systems are considered as follows: α

Dt 1 y1 = f1 (y1 , . . . , yn ) + d1 (t) + u1 (t)

.. . α Dt n yn = fn (y1 , . . . , yn ) + dn (t) + un (t)

(1)

where, fi , i = 1, 2, . . . , n are unknown but bounded functions, ui , i = 1, 2, . . . , n are control inputs, di (t) , i = 1, 2, . . . , n are external disturbances, y = [y1 , y2 , . . . , yn ]T is the vector of outputs system, and 0 < αi < 1 , i = 1, . . . , n are the fractional derivatives orders. Dαt yi is the fractional derivative of yi . The operation Dαt yi in the sense of Caputo definitions is given as follows: Dαt yi =

1

Γ (m − α )

t

∫ 0

(m)

(τ )

yi

α−m+1

(t − τ )

dτ

(2)

where m is integer so that m − 1 < α < m and Γ (·) is Gamma function. The control objective is to design the control signals ui , i = 1, 2, . . . , n such that the outputs of the system yi , i = 1, . . . , n track the references signals ri , i = 1, . . . , n. The proposed control block diagram for ith subsystem is shown in Fig. 1. As shown in Fig. 1, the unknown function in the dynamics of system (fi (y) + di (t) ) is estimated by the proposed fuzzy system fˆi (θ|y).

A. Mohammadzadeh and O. Kaynak / Applied Soft Computing Journal 88 (2020) 106043

3

Fig. 1. The proposed control block diagram for the ith subsystem.

Assumption 1. Based on the universal approximation property ( ) of the fuzzy systems, there is an optimal fuzzy system fˆi∗ θi∗ |y , with optimal consequent parameters (θi∗ ), such that by using this ( ) ˆ∗ ∗ optimal ( ) fuzzy system, the approximation error εi = fi θi |y − fi θi |y is negligible. It must be noted that, the variation of the vector θi∗ is assumed to be smooth such that the time derivative of θi∗ is considered to be negligible. Considering Assumption 1, the equation of the ith subsystem (1) is rewritten as follows: α

Dt i yi = fˆi∗ θi∗ |y + ui (t)

(

)

(3)

It must be noted that, since the external disturbance di and nonlinear function fi are both considered to be unknown, then it seems logical that the term fi + di is considered as unknown nonlinear function and is estimated by the proposed fuzzy system. Then, the effect of external disturbance is integrated in the approximation error. From (3), the dynamics of the tracking error ei = yi − ri are obtained as follows:

( ) α α Dt i ei = fˆ i θi∗ |y − Dt i ri + ui (t)

(4)

The objective in this paper is to design the control signal ui (t) and to derive the adaptation laws for the consequent parameters of the fuzzy system (θi ) based the immersion and invariance control approach, such that the dynamics of the tracking error (4), to be asymptotically stable. The advantages of the proposed method are:

• The dynamics of the system are considered to be unknown. • The adaptation laws are derived through the robustness and stability analysis.

• The effects of the measurement errors and external disturbances are compensated.

are staying on the manifold:

{ } ϕ = (x, θ) ∈ Rn × Rq |θˆ − θ + β1 (x) = 0

Lemma 2 ([53]). Let f (t) is a continuous function, if m − 1 < α < m ∈ Z + the following relation holds for Caputo derivative: Itα Dαt f (t ) = f (t ) −

Itα f (t ) =

(5) ∗

n

with the equilibrium point x ∈ R and unknown parameter vector θ ∈ Rq , where the functions f (·) and g (·) depend on θ . Assume ∆ that there exist υ (x, θ) such that the system x˙ = f ∗ (x) = f (x) + g (x) υ (x, θ) has a globally asymptotically stable equilibrium at x = x∗ . Then the system (5) is I&I stabilizable such that limt →∞ x (t ) = x∗ , if there exist β1 and β2 such that all trajectories of the following extended system:

(

x˙ = f (x) + g (x) υ x, θˆ + β1 (x)

(

θ˙ˆ = β2 x, θˆ

)

)

∫

1

Γ (α )

f (k) (0)

(8)

t

(t − τ )α−1 f (τ )dτ

(9)

0

Lemma 3 ([54]). Consider x (t ) as a continues function, then for 0 < α < 1 and t ≥ 0, the following inequality holds: 1 2

Dαt x2 (t ) ≤ x (t ) Dαt x (t )

(10)

3. Proposed non-singleton type-2 fuzzy neural network In this section, the proposed NT2FNN is explained. The proposed structure is shown in Fig. 2. The flowchart of the computation process is given in Fig. 3. The output of the NT2FNN is computed step-by-step as follows: Step 1: Get the inputs yi , i = 1, . . . , n. Step 2: Perform the non-singleton fuzzification. Assume that the value of input yi at time t is xi . If the singleton fuzzification is used, it means that the value of input yi is exactly xi . Then the j upper and the lower membership of A˜ i (jth MF for the input yi ), are computed as follows:

(

xi − cA˜ j

⎢ µ ¯ A˜ j = ⎣1 +

i

σ¯ ˜2j

i

)2 ⎤

⎡(

⎥ ⎢ ⎦ exp ⎣

xi − cA˜ j i

xi − cA˜ j i

σ ˜2j ¯ Ai

⎥ ⎦

(11)

Ai

(

⎢ µA˜ j = ⎣1 + i ¯

)2 ⎤

σ¯ ˜2j

Ai

⎡

x˙ = f (x) + g (x) u

k!

where, Itα f (t ) is defined as follows:

⎡

Consider the

m−1 k ∑ t k=0

The following lemmas are used in stability analysis. Lemma 1 (Adaptive I&I Control Technique [52]). following system

(7)

)2 ⎤

⎡(

⎥ ⎢ ⎦ exp ⎣

xi − cA˜ j

)2 ⎤

i

σ ˜2j ¯ Ai

⎥ ⎦

(12)

where cA˜ j , σ¯ j and σ j represent the center, upper width and lower i

A˜ i

¯ A˜ i j

width of the MF A˜ i . By the use of non-singleton fuzzification, the uncertainty of the input yi is modeled by the proposed MF Bi . The membership of Bi is computed as follows:

[

(6)

] [ ] (yi − xi )2 (yi − xi )2 exp − µBi = 1 + σy2i σy2i

(13)

4

A. Mohammadzadeh and O. Kaynak / Applied Soft Computing Journal 88 (2020) 106043

Fig. 2. The structure of NT2FNN.

where, σyi has a constant value and σyi represents the uncertainty of input yi . A large value of the σyi represents a large uncertainty of input yi . If σyi becomes smaller then, the MF Bi approaches to a singleton MF (see Fig. 4). To compute the upper and the j lower membership of A˜ i (jth MF for the input yi ), the value of xi j j is converted to x¯ i and xi (see Fig. 5). By the use of non-singleton j j ¯ fuzzification based on product t-norm, x¯ i and xi are obtained as ¯ follows:

σy2i cA˜ j + σ¯ ˜2j xi

σy2i cA˜ j + σ ˜2j xi i ¯ Ai ¯ = 2 , = 2 2 σyi cA˜ j + σ¯ ˜ j xi ¯ σyi cA˜ j + σ ˜2j xi Ai i i ¯ Ai

j xi

Ai

i

j xi

(14)

The proof of (14) is given in Appendix. Step 3: Compute the memberships. Consider input yi , the upper and the lower memberships j of jth MF for this input (A˜ i ) are computed as follows:

⎡ ⎢ µ ¯ A˜ j = ⎣1 +

(

j

x¯ i − cA˜ j

)2 ⎤

i

⎢ ⎥ ⎦ exp ⎣

σ¯ ˜ j 2

i

⎡(

j

x¯ i − cA˜ j i

σ¯ ˜ j 2

Ai

⎡

(

(15)

⎥ ⎦

Ai

j

xi − cA˜ j

⎢ µA˜ j = ⎣1 + ¯ 2 σ˜j ¯ i ¯ Ai

)2 ⎤

)2 ⎤

⎡(

⎥ ⎢ ⎦ exp ⎣ ¯

i

j

xi − cA˜ j i

σ ˜2j ¯ Ai

)2 ⎤ (16)

⎥ ⎦ j

j

where, xi is the value of input yi at time t, x¯ i and xi are the ¯ fuzzification parameters (see (14)), µ ¯ A˜ j and µA˜ j are the upper and j

i

¯

i

the lower memberships of A˜ i , respectively. Step 4: Find the rule firing. Consider the lth rule as follows: l

l − th Rule : IF yi is A˜ i , i = 1, . . . , n Then fˆ is wl , l = 1, . . . , M

Fig. 3. The flowchart of the computation process of the proposed fuzzy system.

(17)

where M is the number of rules, wl is the consequent parameter corresponding to the lth rule and yi , i = 1, . . . , n are the inputs of NT2FNN. The upper and the lower firing degrees of (17) are obtained as follows: z¯ l =

n ∏

µ ¯ A˜ l , zl = i ¯ i=1

n ∏

µA˜ l i i=1 ¯

(18)

Step 5: Compute the type-reduction parameters. By using Nie–Tan type-reduction method [55], the parameters Zl , l = 1, . . . , M are

computed as follows: Zl =

z¯ l + z l

¯

2 Step 6: Compute the output as follows: fˆ = θ T ζ

(19)

(20)

where,

θ T = [w1 , . . . , wM ] , ζ T = [Z1 , . . . , ZM ] where, θ is the vector of consequent parameters.

(21)

A. Mohammadzadeh and O. Kaynak / Applied Soft Computing Journal 88 (2020) 106043

5

Fig. 4. The non-singleton fuzzification by MF Bi with different σBi .

Fig. 5. The non-singleton fuzzification based on product t-norm.

4. Control design and stability analysis In this section the proposed control signal and the adaptation laws for rule parameters of the proposed fuzzy system are presented. The main results are given in the following theorem. Theorem 1. The ith subsystem in (1), is asymptotically stable if the control signal and the adaptation laws for the free parameters of the proposed fuzzy systems are derived based on the immersion and invariance approach as follows: α

(

)T

ui (t) = −λi ei + Dt i ri − θˆi + δi (ei ) α

Dt i θˆ i = λi Ki ζi ζiT

(

)−1

ζi e i

ζi

(22) (23)

where, ri is the reference signal. θˆi and ζi are defined in (21). Ki > 0 and λi are constant and are chosen such that |arg(Ki )| > αi π2 and |arg(λi )| > αi π2 , are satisfied and δi (ei ) is as follow: Ki ζi ζiT

(

δi (ei ) =

)−1

αi Γ (αi )

ζi

αi

ei

(24)

The main idea behind the control signal (22), is that the uncertain nonlinear functions in the dynamics of the system are estimated by the proposed fuzzy systems as fˆi = θˆiT ζi (see (20)) and then the error feedback controllers are applied. The parameters of the

fuzzy systems θˆi are tuned based on the adaptation laws (23) that are derived through the I&I theorem such that the stability of the closed-loop system to be guaranteed. The details are given in below. Proof. Based on Lemma 1, consider the augmented system as follows:

{

α

T

Dt i ei = θi∗ ζi − Dαi ri + ui (t) α Dt i θˆ i = ψi

(25)

where, θˆi is the estimation of θi∗ , ζi is defined in (21) and ψi is the adaptation law. ( ) In the extended space

ei , θˆi , the following manifold is de-

fined:

ϕi =

{(

ei , θˆi

)

} ∈ Rnθi +1 |θˆi + δi (ei ) − θi∗ = 0

(26)

where, nθi is the number of consequent parameters of the fuzzy system and δi (ei ) is a continuous function must be specified such that the defined manifold (26) is satisfied. From (26), the dynamics of the tracking error is rewritten as follows: α

(

)T

Dt i ei = θˆi + δi (ei )

α

ζi − Dt i ri + ui (t)

(27)

6

A. Mohammadzadeh and O. Kaynak / Applied Soft Computing Journal 88 (2020) 106043

Fig. 6. The trajectories of output and tracking errors in Example 1.

By substituting the control signal ui (22), into dynamics of tracking error (27), we have: αi

Dt ei = −λi ei

(28)

To show that the adaptation law ψi (see (23) and (25)), can render the manifold ϕi (26) invariant, consider the following definition: ∆

υi = θˆi + δi (ei ) − θi∗

(29)

The dynamics of υi are obtained as follows: αi

αi ˆ

αi

Dt υi = Dt θi + Dei δi (ei ) Dt ei

(30)

αi

By substituting Dt θi and Dt ei , from (25) and (27), into (30), we have: α

Dt i υi = ψ(i + ( α

Deii δi (ei )

θˆi + δi (ei ) − υi

)T

ζi − Dαi ri + ui (t)

)

α

α

Dt i υi = ψi − Deii δi (ei ) ζiT υi − λi Deii δi (ei ) ei α

Dt i υi = −Deii δi (ei ) ζiT υi

Dei δi (ei ) = Ki ζ ζ

ζ

(34)

From (33) and (34), one can see that α

Dt i υi = −Ki υi

(35)

By taking fractional integral of both side of (34) one has: α

α

α

( (

Ieii Deii δi (ei ) = Ieii Ki ζi ζiT

)−1 ) ζi

α

α

δi (ei ) =

(39)

Ki ζi ζiT

(

δi (ei ) =

)−1

ζi

α

e i + δi (0) (40) αi Γ (αi ) i By considering δi (0) = 0, Eq. (24) is obtained. Now consider the following Lyapunov function: 1

1 e2 + υi2 (41) 2 i 2 By considering Lemma 3, the fractional order time derivative along the trajectories of V in (41), yields: α

α

α

αi

(42) αi

α

Dt i V ≤ −λi e2i − Ki υi2 < 0

(43)

Then the closed-loop system is asymptotically stable and the stability analysis is completed. 5. Simulations In this section the performance of the proposed method is examined under different conditions. The effectiveness of the proposed control scheme is demonstrated by two simulation examples and one experimental evaluation. Example 1. In this example, the proposed method is applied to control the hyperchaotic Chen system. The system dynamics are given as follows: q

(36)

Dt y1 = 35 (y2 − y1 ) + y4 + d1 + u1 q

Dt y2 = 7y1 − y1 y3 + 12y2 + d2 + u2

From (36) and considering Lemma 2, we obtain: Ieii Deii δi (ei ) = δi (ei ) − δi (0)

(

Substituting Dt ei and Dt υi from (28) and (35), into (42), it is derived that (33)

T −1 i i i

)

)−1 ∫ e i ζi (ei − τ )αi −1 dτ + δi (0) Γ (αi ) 0

Ki ζi ζiT

Dt i V ≤ ei Dt i ei + υi Dt i υi

Considering (33), Dei δi (ei ) is designed such that the dynamics of υi to be asymptotically stable. From (33), it is obvious that υi is α converged to zeros if one has Dt i υi = −Ki υi , in which, Ki > 0 is a α constant and is chosen such that |arg(Ki )| > αi π2 . Then Deii δi (ei ) is considered as follows:

(

(38)

Then from (36)–(38), one has:

(32)

αi

αi

( ( ) Ki (ζ ζ T )−1 ζ ∫ ei ) i i i T −1 Iei Ki ζi ζi ζi = (ei − τ )αi −1 dτ Γ (αi ) 0

V =

From (32) and adaptation law in (23), we have: α

)−1 ) ζi in (36) is written as follows:

αi

(31)

Substituting the control signal ui from (22) into (31), yields: α

( (

Simplification of (39), yields:

αi

αi ˆ

α

From (9), the term Ieii Ki ζi ζiT

q

Dt y3 = y1 y2 − 3y3 + d3 + u3 (37)

q

Dt y4 = y2 y3 + 0.3y4 + d4 + u4

(44)

A. Mohammadzadeh and O. Kaynak / Applied Soft Computing Journal 88 (2020) 106043

7

Fig. 7. Control signal in Example 1.

j

Fig. 8. Comparison of the membership of A˜ i with different fuzzification methods, Example 1. Table 1 Simulation parameters, Example 1. Parameters

Reference

cA˜ 1 = 0, cA˜ 2 = 5, cA˜ 3 = 10, i = 1, . . . , 4

Fig. 2

σ¯ A˜ i = 10, σA˜ i = 5 i = 1, . . . , 4, j = 1, 2, 3 i ¯ i λi = 50, Ki = 10 σyi = 0.1, i = 1, . . . , 4

(34) (14)

i

i

i

Fig. 2

where u1 , u2 , u3 and u4 are the control signals. External disturbances di , i = 1, 2, 3, 4 are Gaussian noise with zero mean and variance 0.1. Initial conditions are yi (0) = 1, i = 1, . . . , 4. The value of fractional-order is q = 0.97 and other parameters are given in Table 1. The trajectories of the outputs and the tracking errors are shown in Fig. 6. The control signals are given in Fig. 7. As it can be seen a good tracking performance is achieved despite of unknown dynamics and perturbed dynamics by the external disturbances. To show the effectiveness of the proposed MF and nonsingleton fuzzification, the simulation is repeated by adding the measurement errors. The measurement errors are considered as

white noise with zero mean and different variance (0.01, 0.05 and 0.1). The values of root mean square errors (RMSEs) for different MFs and different fuzzification methods are shown in Table 2. The performance is also compared with type-1 fuzzy system (T1FS) in Table 2. As it can be seen, the values of RMSEs for the case that type-2 fuzzy system the proposed MF and non-singleton fuzzification are used, are significantly better than other ones. Remark 1. As it can be seen from Table 2, the performance of the control scheme using proposed non-singleton fuzzification is significantly better than Gaussian non-singleton and singleton fuzzification. To illustrate the main reason, consider input yi and assume its MFs in the fuzzification and membership layers are j Bi and A˜ i , respectively. By using proposed fuzzification the memj bership of A˜ i when the value of yi is inaccurate is much closer to its real membership in contrast to Gaussian non-singleton and singleton fuzzification (see Fig. 8). Remark 2. In all simulation and experimental examples the fractional order operators are implemented by using nid Simulink block that has been created by Valerio et al. [56].

8

A. Mohammadzadeh and O. Kaynak / Applied Soft Computing Journal 88 (2020) 106043 Table 2 Comparison of root mean square error (RMSE), Example 1. Noise variance

Fuzzification

MF

e1

e2

e3

e4

Singleton

Gaussian Proposed MF

0.0401 1.2800

0.0271 1.3920

0.0515 1.0687

0.0177 0.3286

Non-singleton

Gaussian Proposed MF

0.0221 0.1424

0.0161 0.0728

0.0243 0.1361

0.0099 0.0476

Singleton

Gaussian Proposed MF

0.0332 1.2334

0.0328 1.5992

0.0350 0.7029

0.0137 0.2166

Non-singleton

Gaussian Proposed MF

0.0244 0.1104

0.0215 0.0638

0.0256 0.1116

0.0103 0.0423

Singleton

Gaussian Proposed MF

9.3441 1.1967

11.7232 1.3296

8.8606 2.2237

11.2892 0.9241

Non-singleton

Gaussian Proposed MF

5.2214 0.0592

6.5864 0.0624

4.8684 0.0483

6.0905 0.0195

Gaussian Proposed MF Gaussian Proposed MF

9.3602 8.3915 8.7253 0.0583

11.7435 10.5256 10.9515 0.0849

8.8759 8.1606 8.2631 0.0553

11.3071 10.1963 10.5104 0.0219

T1FS

9.7613

12.2458

9.2582

11.7943

0

0.01

0.05

Singleton 0.1 Non-singleton 0.1

Singleton

Example 2. In this example, the proposed method is used to control the uncertain fractional-order hyperchaotic Lorenz system. The difference with Example 1 is that the reference systems are not simple, but the reference signals (ri , i = 1, . . . , 4) are the output signals of the Chen system (44). The dynamics of the Lorenz system are given as follows:

Table 3 Comparison of root mean square error (RMSE), Example 2. Synchronization errors

e1

e2

e3

e4

The method of [57] The method of [58] Proposed method

0.3342 0.0291 0.0260

0.2104 0.0680 0.0163

0.1975 0.1533 0.0126

1.7383 0.2707 0.0638

q

Dt y1 = 10 (y2 − y1 ) + y4 + ds1 + u1 q Dt y2 q Dt y3 q Dt y4

= 28y1 − y2 − y1 y3 + ds2 + u2 = y1 y2 − 8/3y3 + ds3 + u3 = −y2 y3 − y4 + ds4 + u4

(45)

where, ds1 = 0.25 cos(6t)y1 − 0.15 sin(t) ds2 = −0.2 cos(2t)y2 + 0.1 sin(3t) ds3 = 0.15 sin(3t)y3 + 0.2 cos(5t) ds4

(46)

= −0.2 cos(2t)y4 − 0.15 cos(t)

Initial conditions are as y1 (0) = 1, y2 (0) = 2, y3 (0) = 3 and y4 (0) = 4. The value of fractional derivative order is q = 0.97. The other control parameters are the same as Example 1. The external disturbances for the reference system (44) are chosen as follows: d1 d2 d3 d4

= −0.25 sin(4t)x1 + 0.1 sin(7t) = 0.1 cos(t)x2 + 0.15 cos(3t) = 0.25 sin(4t)x3 − 0.15 sin(5t) = −0.15 sin(t)x4 + 0.2 cos(2t)

(47)

The tracking performance and tracking errors are shown in Fig. 9. The control signals are given in Fig. 10. It is seen that the tracking errors quickly reach to zero level and there is no chattering phenomena in the control signals. The tracking performance of the proposed method is compared with the sliding mode technique [57] and H∞ based adaptive type-2 fuzzy controller [58]. A comparison of the values of RMSEs is provided in Table 3, which indicate that, the proposed method results in a significantly better performance in comparison to other techniques. It must be noted that in [57], the controller is designed based on the mathematical model of the system and in [58], the measurement errors are neglected, however, in the proposed method the dynamics of the system are unknown and are disturbed by the external perturbations. Example 3. In this example an experimental study is provided to verify the effectiveness of the proposed control scheme in realword problems. The experimental setup of heat transfer system

is depicted in Fig. 11. The dynamics of the system are assumed to be completely unknown. The control objective is to design a controller such that the temperature tracks a reference signal. The control signals are voltages of fan and heater. The value of the fan command is fixed to 5 Volt and the voltages of heater is determined based on the proposed control method. The dynamics of the system is modeled as follows: Dαt y = fˆ θ|y + u(t)

(

)

(48)

( ¯ ) where, fˆ θ|y is NT2FNN. The inputs of the NT2FNN in (48) ¯ are voltage and [temperature at the current and previous ]T samples times (y = u (t − 1) u (t − 2) y (t ) y (t − 1) ). The ¯ controller parameters are the same as in Example 1. The tracking performance by using fractional order and an integer order controller [59], is shown in Fig. 12. As it can be seen the proposed control scheme results in good performance in real-word problems. Remark 3. It can be seen from Fig. 12, that the performance with using fractional order controller is better in contrast to integer order one. This is because the capability of the fractional order calculus for modeling of physical systems is better than integer order one. In other words, if the fractional state space model of the system is represented by the integer order one, to have same performance, more integrals are needed. 6. Conclusion In this paper the control of a class of nonlinear fractionalorder chaotic systems is studied. The dynamics of systems are assumed to be unknown and also the outputs of the system are perturbed by measurement errors. A new membership function and non-singleton fuzzification is proposed to handle the input uncertainties associated with the measurement errors. The consequent parameters of the proposed type-2 fuzzy system are tuned based on the adaptation laws which are derived form immersion and invariance (I&I) control theorem. The I&I control method is

A. Mohammadzadeh and O. Kaynak / Applied Soft Computing Journal 88 (2020) 106043

9

Fig. 9. Tracking performance, Example 2.

Fig. 10. Control signals in Example 2.

Fig. 11. The excremental setup, Example 3.

developed for the fractional-order systems. The proposed control strategy is applied to control two hyperchaotic systems and its performance is compared with sliding mode technique. In addition to simulations, the experimental study shows the capability of the proposed control scheme in practical applications. To verify

the effectiveness of the proposed MF and non-singleton fuzzification, a white noise with zero mean and different variances is added to the outputs as measurement error and then the values of root mean square errors are compared with applying various fuzzification and MFs. The simulation studies presented show

10

A. Mohammadzadeh and O. Kaynak / Applied Soft Computing Journal 88 (2020) 106043

Fig. 12. The tracking performance Example 3.

that the proposed MF and non-singleton fuzzification result in a significantly better performance in the presence of measurement errors. As future studies the proposed control system is intended to be extended to other classes of nonlinear systems such as non-affine nonlinear systems and also the effects of mismatched disturbances are to be considered. In future studies the proposed control system is intended to be extended to other classes of nonlinear systems, such as non-affine nonlinear systems. Additionally, the effects of mismatched disturbances will be considered. Other extensions of the work reported include the development of the proposed fuzzy I&I controller to another class of nonlinear systems and a more thorough investigation of experimental limitations and the computational cost in various practical applications.

j

values of µ ¯ A˜ j µBi and µA˜ j µBi must be obtained. Then to find x¯ i the i

i

¯ be holds: following equation muse d dyi

⎧⎡ ⎪ ⎪ ⎨⎢ ⎢1 + ⎣ ⎪ ⎪ ⎩ [

)2 ⎤

( yi −C j A˜

1+

(yi −xi )2 σy2

]

yi −C j A˜ i

σ¯ 2j A˜

i

⎥ ⎥ ⎦

(A.3)

[ ]} (y −x )2 exp − i 2 i =0 σ yi

i

Consider the following definition

⎡

)2 ⎤

( yi −C j A˜

⎢ s1 = ⎢ ⎣1 +

[ ⎥ ⎥ 1+ ⎦

i

σ¯ 2j A˜ i

Declaration of competing interest

⎡ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.asoc.2019.106043.

i

)2 ⎤

(

⎥ ⎢ ⎥ exp ⎢− ⎦ ⎣

i

σ¯ 2j A˜

⎡

yi −C j A˜

⎢

]

i

(A.4)

)2 ⎤

(

[ ] ⎥ ⎥ exp − (yi −2xi )2 ⎦ σ

i

s2 = exp ⎢ ⎣−

(yi −xi )2 σy2

σ¯ 2j A˜

yi

i

From (A.4), Eq. (A.3), becomes: CRediT authorship contribution statement

)

(

Ardashir Mohammadzadeh: Writing - review & editing. Okyay Kaynak: Writing - review & editing.

2 yi −C j A˜

Assume that the value of input yi at time t is xi . The uncertainty of the input yi is modeled by the proposed MF Bi . The membership of Bi is computed as follows:

[

(yi − xi ) σy2i

2

µBi = 1 +

]

[ exp −

(yi − xi ) σy2i

2

]

)2 ⎤

yi −C j A˜

⎢ µ ¯ A˜ j = ⎢ ⎣1 + i ⎡

i

i

)2 ⎤

( yi −C j A˜

⎢ µA˜ j = ⎢ ⎣1 + ¯ i

⎡

⎥ ⎢ ⎥ exp ⎢− ⎦ ⎣

i

σ 2j ¯ A˜ i

)2 ⎤

( yi −C j A˜

⎥ ⎢ ⎥ exp ⎢− ⎦ ⎣

σ¯ 2j A˜

⎡

i

σ¯ 2j A˜

i

yi −C j A˜ i

σ 2j ¯ A˜ i

( i

A˜

To find ¯ and

j xi

¯

¯

2(yi −xi )

σy2

i

⎥ ⎦ s1 s2 = 0

i

)2

) i

+

σ¯ 2j

(A.2)

i

σ¯ 2j A˜ i )

yi −C j A˜ i

σ¯ 2j A˜ i

⎤

(A.6)

⎥ + (yiσ−2xi ) ⎦ s1 = 0 yi

From (A.6), it is obtained: i

σ¯ ˜2j

in (14) based on product t-norm, the maximum

(yi −xi )2 σy2 i ⎡(

⎢ −⎣

)

yi −C j A˜

i

yi − CA˜ j

i

(yi −xi ) σy2 i

(

⎥ ⎥ ⎦

+

i

A˜ i

A˜

( (yi −xi ) σy2

i

σ¯ 2j

Ai

i

(A.5)

⎤ −

σ¯ 2j

yi −C j A˜

j

i

)

yi −C j A˜

(

where, cAj , σ¯ Aj and σAj are the designable parameters of MF A˜ i . j xi

i

2 yi −C j A˜

(

⎥ ⎥ ⎦ )2 ⎤

(

A˜

The Eq. (A.5) is simplified as follows:

where, σyi is a constant. Now consider ˜ the jth MF in the membership layer for input yi . The upper and the lower membership j of A˜ i is computed as follows: (

s2 +

⎥ 2(y −x ) ⎥ i 2 i s2 + ⎦ σyi

i

σ¯ 2j

(A.1)

j Ai

⎡

yi −C j A˜

⎢ ⎢1 + ⎣

⎢ ⎣−

]

)2 ⎤

(

⎡

(yi −xi )2 σy2

1+

i

⎡ Appendix. Non-singleton fuzzification based on product tnorm

[

i

σ¯ 2j A˜ i

) =−

(yi − xi ) σy2i

(A.7) j

Then from (A.7), Eq. (14) is satisfied. The procedure of finding xi j ¯ in (14), is similar to x¯ i .

A. Mohammadzadeh and O. Kaynak / Applied Soft Computing Journal 88 (2020) 106043

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