J Control Autom Electr Syst DOI 10.1007/s40313-016-0260-4

Robust Rejection of Matched/Unmatched Perturbations from Fractional-Order Nonlinear Systems Sajjad Shoja-Majidabad1

Received: 30 December 2015 / Revised: 1 June 2016 / Accepted: 10 July 2016 © Brazilian Society for Automatics–SBA 2016

Abstract This paper deals with the control of fractionalorder nonlinear systems subjected to matched and unmatched perturbations. A novel control strategy is proposed based on nonlinear block control and sliding mode technique. Based on the block control technique, robust virtual control laws are developed step by step to suppress the mismatched perturbation vector of each block. Afterward, a sliding mode controller is designed to deal with the matched uncertainty vector and stabilize the fractional-order nonlinear system. The tracking errors convergence to zero are proved using fractional-order version of stability theorems. The proposed control strategy is simple and avoids high derivational computations. Eventually, simulation results show that the designed controller attains a satisfied control performance and is robust against matched and unmatched perturbations. Keywords Fractional-order nonlinear system · Nonlinear block control (NBC) · Sliding mode control (SMC) · Unmatched perturbations · Mittag–Leffler stability

1 Introduction Robust control of nonlinear system in the presence of unmatched perturbations (perturbations that appears on different coordinate than the control input) remains one of the most challenging problems in control theory. In other words, it is hardly possible to cancel the unmatched uncertainties in every system by feedback control laws. Therefore, to compensate this type of perturbations it is necessary to apply

B 1

Sajjad Shoja-Majidabad [email protected] University of Bonab, Bonab, East Azerbaijan, Islamic Republic of Iran

some feedback linearization techniques such as: input-output linearization (Isidori 1992), backstepping design Krstic et al. (1995) and nonlinear block control (NBC) (Luk’yanov 1998; Loukianov 2002; Loukianov et al. 2002). However, the main drawback of feedback linearization approaches is that they focus on exact cancelation of nonlinearities in order to get linear behavior of the closed-loop system. Therefore, in the case of modeling error or unmatched uncertainty existence it is useful to combine these techniques with the sliding mode control (SMC) (Loukianov et al. 2011). The SMC technique enables high accuracy tracking and insensitivity to matched perturbations and system modeling errors. Nevertheless, conventional SMC alone is not able to suppress unmatched perturbations (Utkin et al. 1999). The combination of the backstepping design and SMC have been reported in some literatures to compensate unmatched perturbations (Shieh and Shyu 1999; Xia et al. 2013; Wang et al. 2015; Zhang et al. 2015; Chang 2009; Chang and Cheng 2010, 2007; Davila 2013). In Shieh and Shyu (1999), Xia et al. (2013), Wang et al. (2015), Zhang et al. (2015), adaptive backstepping techniques based on SMC are utilized to obtain robustness for induction motor and aerospace systems with mismatched uncertainties. Also some adaptive backstepping-SMC controllers are presented for multi-input (Chang 2009; Chang and Cheng 2010) and multi-input multi-output (Chang and Cheng 2007) nonlinear systems. Refs. Davila (2013), Yang et al. (2013) applied the backstepping strategy which is combined with higher-order and terminal SMCs, respectively. Recently, other combinations of the backstepping technique and sliding mode controllers/observers are reported in different fields of engineering such as: power converters and distributed generation units (Fehr and Gensior 2016; Sun et al. 2015), pneumatic systems (Laghrouche et al. 2014; Taheri et al. 2014), aerospace and flight vehicles (Peng et al. 2015; Chen and Jiang 2014).

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Another method for the problem of unmatched perturbations suppression is based on the Nonlinear Block Controllable form (NBC-form) which is proposed by Luk’yanov (1998), Loukianov (2002), Loukianov et al. (2002). In Loukianov et al. (2002), the SMC is employed to reject the matched uncertainties. Afterward, high gain virtual block control laws are applied to attenuate the unmatched perturbations effect and stabilize the sliding mode dynamics. Similar SMC based on NBC technique is utilized to stabilize various highly nonlinear systems such as: single machine connected to infinite bus (Loukianov et al. 2004; Vazquez et al. 2007), multimachine power systems (Huerta et al. 2009, 2010), induction motor (Astorga et al. 2014), quadrotor system (Vega et al. 2012), etc. In Estrada and Fridman (2010), the NBC and a quasi-continuous higher-order sliding mode technique is proposed to solve mismatched perturbations problem. All of the mentioned studies have been developed for integer-order systems. Therefore, it is valuable to see the matched/unmatched perturbations compensation from fractional-order perspective. Fractional calculus idea was established in the 17th century. This field of mathematic discusses about arbitrary-order derivatives and integrals. For three centuries, this topic was developed as a pure theoretical subject with no applications (Podlubny 1999). But in last three decades, it has been used in different fields of physics and engineering (ShojaMajidabad et al. 2015). From control engineering point of view, fractional calculus has been applied for various dynamical systems modeling and control. In the robust control area, the SMC has been used to stabilize integer-order (Delghavi et al. 2016; Shoja-Majidabad et al. 2015) and mostly fractional-order (Pisano et al. 2010; Jakovljevi et al. 2016; Hosseinnia et al. 2010; Pisano et al. 2012; Aghababa 2015; Shoja-Majidabad et al. 2014; Zhong et al. 2015; ShojaMajidabad et al. 2015; Aghababa 2015; Ullah et al. 2015) dynamical systems. In these literatures, SMC is applied to control fractional-order multi-input multi-output linear system (Pisano et al. 2010), nonlinear chaotic and non-chaotic systems (Jakovljevi et al. 2016; Hosseinnia et al. 2010; Pisano et al. 2012; Aghababa 2015; Shoja-Majidabad et al. 2014), and large-scale systems (Shoja-Majidabad et al. 2014). Moreover, the SMC with chattering-free property (Zhong et al. 2015), terminal sliding surface (Shoja-Majidabad et al. 2015; Aghababa 2015), and fuzzy approximators (Ullah et al. 2015) is proposed for fractional-order nonlinear systems. Nevertheless, there are few studies around unmatched perturbation suppression for fractional-order systems. In Wei et al. (2015), an adaptive backstepping approach is proposed to deal with the unknown mismatched parameters. The SMC with nonlinear observer is designed to reject unmatched uncertainties in Pashaei and Badamchizadeh (2016). However, due to progressions of fractional-order modeling and control, it is necessary to deal more with

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the unmatched perturbations rejection in the fractional-order systems. Inspired by the above discussions, this paper contains the following novelty: (1) Unmatched perturbation compensation for fractionalorder nonlinear systems: For this purpose, the SMC combination with a novel NBC strategy is presented. The SMC is designed to remove the matched perturbations, and the NBC is expressed to reject the unmatched perturbations. (2) The proposed NBC technique is different from the conventional NBC which is reported in Luk’yanov (1998), Loukianov (2002), Loukianov et al. (2002). In other words, the conventional NBC is not easily applicable for systems with complicated fractional derivatives/integrals. However, the proposed NBC tackles this restriction. Moreover, upper bounds of derivatives of perturbations are not necessary to be known. (3) Practical stability of closed-loop system is realized by the fractional-order stability theorems presented in AguilaCamacho et al. (2014), Duarte-Mermoud et al. (2015). This paper proceeds as follows: In Sect. 2, some preliminaries and properties of fractional calculus are expressed. A Caputo derivative-based fractional-order nonlinear system with model uncertainty is presented in Sect. 3. In Sect. 4, a sliding mode block control technique is developed to suppress the matched and unmatched perturbations. In Sect. 5, the simulation results show that the proposed approach attains a satisfactory performance in the presence of perturbations. Finally, a conclusion is given in Sect. 6

2 Fractional Calculus Preliminaries In this section, some definitions, remarks, and lemmas related to fractional calculus are presented. Definition 1 (Li and Deng 2007) The fractional integral (Riemann–Liouville integral) with fractional order α ∈R+ of function f (t) is defined as α I0,t

f (t) =

−α D0,t

1 f (t) = (α)

 0

t

f (τ ) dτ (t − τ )1−α

(1)

where Γ (α) denotes the Gamma function. Definition 2 (Li and Deng 2007) The fractional Caputo derivative of continuous function f (t) ( f (t) ∈ C m [0, t]) is given as α C D0,t

−(m−α) m f (t) = D0,t D f (t)  t 1 f (m) (τ ) = dτ Γ (m − α) 0 (t − τ )1−m+α

(2)

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where m − 1 < α < m, m ∈ N. Definition 3 (Li and Deng 2007) The Riemann–Liouville (RL) fractional derivative of function f (t) is defined as α R L D0,t

−(m−α)

f (t) = D m D0,t

f (t) 

dm

1 Γ (m − α) dt m

0

t

f (τ ) dτ (t − τ )1−m+α

(3)

where m − 1 ≤ α < m, m ∈ N. Lemma 1 (Aguila-Camacho et al. 2014; Duarte-Mermoud et al. 2015) Let x(t) ∈ Rn be a vector of differentiable functions. Then, for any time instant t ≥ t0 , the following relationship holds 1 D α (x T (t)x(t)) ≤ x T (t)C Dtα0 ,t (x(t)), ∀α ∈ (0, 1) 2 C t0 ,t (4) Lemma 2 (Liu et al. 2016) Let x(t) ∈ Rn be a vector of differentiable functions. If a continuous function V : [t0 , ∞) × Rn → R satisfies α C Dt0 ,t V (t, x(t))

≤ −βV (t, x(t))

V (t, x(t)) ≤ V (t0 , x(t0 ))E α (−β(t − t0 )α )

(5)

(6)

where α ∈ (0, 1) and β is a positive constant. Definition 4 (Li et al. 2009) A function frequently used in the solutions of fractional-order equations is the Mittag_Leffler function defined as E α (z) =

k=0

3 Fractional-order Nonlinear System with Matched and Unmatched Perturbations Consider a class of Caputo derivative-based fractional-order nonlinear system with model uncertainty and external disturbance as follows: CD

α

X(t) = F(X, t) + G(X, t)u(t) + ω(X, t)

zk (kα + 1)

(7)

where α > 0 and z ∈ C. The Mittag–Leffler function is corresponding to the exponential function which is useful for integer-order equations solution. Definition 5 (Li et al. 2009) The solution of the system α C D0,t x (t) = f (t, x(t)) is said to be Mittag–Leffler stable if x(t) ≤ {m[x(0)]E α (−λt α )}b

(8)

(9)

where α ∈ (0, 1) is the order of system, X ∈ Rn×1 is the state vector, u ∈ Rr ×1 is the input vector, F(X, t) ∈ Rn×1 and G(X, t) ∈ Rn×r are known nonlinear functions of X and t, and ω ∈ Rs×1 is a vector of bounded perturbation term due to model uncertainties. Now let the fractional-order nonlinear system (9) be rewritten in the following NBC-form with p-blocks. CD

then

∞ 

In this paper, only the Caputo definition is used, since initial conditions of fractional-order differential equation with Caputo derivative are same as integer-order differential equation initial conditions with clear physical interpretations.

CD

α

x 1 (t) = f 1 (x 1 , t) + g 1 (x 1 , t) x 2 (t) + ω1 (x 1 , t)

α

x 2 (t) = f 2 (x 1 , x 2 , t) + g 2 (x 1 , x 2 , t) x 3 (t)

+ ω2 (x 1 , x 2 , t) .. .   α C D x p−1 (t) = f p−1 x 1 , x 2 , . . . , x p−1 , t   + g p−1 x 1 , x 2 , . . . , x p−1 , t x p (t)   + ω p−1 x 1 , x 2 , . . . , x p−1 , t   α C D x p (t) = f p x 1 , x 2 , . . . , x p , t   + g p x 1 , x 2 , . . . , x p , t u (t)   + ω p x1, x2, . . . , x p , t

(10)

where x i is a n i × 1 vector (i = 1, 2, . . . , p). In each block, the vectors x i+1 are regarded as fictitious control vectors and Rank(g i ) = n i where the integers n 1 , n 2 , . . . , n p are the controllability p indices of system (10) with i=1 n i = n. Assumption 1 The whole state vectors x 1 , x 2 , . . . , x p are assumed to be measurable. Assumption 2 The matched and unmatched uncertainty vectors of (10) satisfy

where α ∈ (0, 1), λ > 0, b > 0, E α (·) is the Mittag–Leffler function, m(0) = 0 and m(x) ≥ 0 is locally Lipschitz.

ωi (x 1 , x 2 , . . . , x i , t) ≤ Wi (x 1 , x 2 , . . . , x i , t)

Remark 1 Mittag–Leffler stability implies asymptotical stability.

where Wi (x 1 , x 2 , . . . , x i , t) is a known positive scalar function.

(11)

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−C D α x 2−ref (t) + ω2 (x 1 , x 2 , t)

The above assumption is considered to emphasize that determining the perturbations upper bounds is out of this study scope.

= f 2 (x 1 , x 2 , t) + g 2 (x 1 , x 2 , t)x 3−ref (t) + g 2 (x 1 , x 2 , t)z 3 (t) −C D α x 2−ref (t) + ω2 (x 1 , x 2 , t)

4 Sliding Mode Block Control The control problem is to design a controller such that the state x 1 (t) can track a smooth desired reference x 1−ref (t) in spite of the presence of matched and unmatched perturbations. Now let’s define tracking errors for p-blocks as follows z 1 (t) = x 1 (t) − x 1−ref (t) z 2 (t) = x 2 (t) − x 2−ref (t) .. .

As on the first step, the virtual input vector x 3−ref in (16) is chosen as α x 3−ref (t) = g + 2 (x 1 , x 2 , t)(− f 2 (x 1 , x 2 , t) + C D x 2−ref (t)

− k2 z 2 (t) − ksw2 sgn(z 2 (t)))

α

z p−1 (t) = x p−1 (t) − x ( p−1)−ref (t)

− ksw2 sgn(z 2 (t)) + ω2 (x 1 , x 2 , t)

where X r e f = is the desired state vector. Step 1 By taking derivative C D α from the first block of (12) along (10) and using second block of (12), one can obtain α

z 1 (t) = C D α x 1 (t) − C D α x 1−ref (t)

CD

α

−C D α x 1−ref (t) + ω1 (x 1 , t) + g 1 (x 1 , t) z 2 (t) − C D α x 1−ref (t) + ω1 (x 1 , t)

(13)

To stabilize Eq. (13) dynamics, the virtual input vector x 2−ref is chosen as

− C D α x ( p−1)−ref (t)   + ω p−1 x 1 , x 2 , . . . , x p−1 , t

α x 2−ref (t) = g + 1 (x 1 , t)(− f 1 (x 1 , t) + C D x 1−ref (t)

−k1 z 1 (t) − ksw1 sgn(z 1 (t))) g1 g+ 1

α

  x p−ref (t) = g + p−1 x 1 , x 2 , . . . , x p−1 , t   × (− f p−1 x 1 , x 2 , . . . , x p−1 , t + C D α x ( p−1)−ref (t) − k p−1 z p−1 (t)

z 2 (t) = C D α x 2 (t) − C D α x 2−ref (t) = f 2 (x 1 , x 2 , t) + g 2 (x 1 , x 2 , t)x 3 (t)

123

− ksw( p−1) sgn(z p−1 (t)))

(15)

Step 2 Caputo derivative of second block of (12) along (10) and using third block of (12) yields CD

To stabilize (19) dynamics, the virtual input vector x p−ref can be chosen as

z 1 (t) = −k1 z 1 (t) + g 1 (x 1 , t)z 2 (t) − ksw1 sgn(z 1 (t)) + ω1 (x 1 , t)

(19)

(14)

is right pseudo-inverse of g 1 (i.e., = I n 1 ), where and k1 > 0, ksw1 is the sliding gain correspond to ω1 , and sgn(.) is sign function. Hence, the first block closed-loop dynamics will be as α

z p−1 (t) = C D α x p−1 (t) − C D α x ( p−1)−ref (t)   = f p−1 x 1 , x 2 , . . . , x p−1 , t   + g p−1 x 1 , x 2 , . . . , x p−1 , t x p (t) − C D α x ( p−1)−ref (t)   + ω p−1 x 1 , x 2 , . . . , x p−1 , t   = f p−1 x 1 , x 2 , . . . , x p−1 , t   + g p−1 x 1 , x 2 , . . . , x p−1 , t x p−ref (t)   + g p−1 x 1 , x 2 , . . . , x p−1 , t z p (t)

= f 1 (x 1 , t) + g 1 (x 1 , t) x 2−ref (t)

CD

(18)

This procedure may be performed iteratively obtaining other blocks dynamics. Step p − 1: By taking derivative C D α from ( p − 1)th block of (12) along (10) and the ultimate block of (12), we can get

= f 1 (x 1 , t) + g 1 (x 1 , t) x 2 (t)

g+ 1

z 2 (t) = − k2 z 2 (t) + g 2 (x 1 , x 2 , t) z 3 (t)

(12)

T T T [x 1−ref , x 2−ref , . . . , x (Tp−1)−ref , x Tp−ref ]

CD

(17)

where g + 2 is right pseudo-inverse of g 2 , and k2 > 0, and ksw2 is the sliding gain correspond to ω2 . Then, the second block closed-loop dynamics will be as below CD

z p (t) = x p (t) − x p−ref (t)

(16)

(20)

where g + p−1 is right pseudo-inverse of g p−1 , and k p−1 > 0. Then, the ( p − 1)th block closed-loop dynamics will be as CD

α

z p−1 (t) = −k p−1 z p−1 (t)   + g p−1 x 1 , x 2 , . . . , x p−1 , t z p (t)

J Control Autom Electr Syst

  − ksw( p−1) sgn z p−1 (t)   + ω p−1 x 1 , x 2 , . . . , x p−1 , t

u(t) = g + p (x 1 , x 2 , . . . , x p , t)(− f p (x 1 , x 2 , . . . , x p , t) (21)

+ D α x p−ref (t) − k p σ p (t) − kswp sgn(σ p (t))) (26)

Step p: Caputo derivative of p th block of (12) along (10), yields

with k p > 0 and the sliding gain kswp ≥ W p (x 1 , x 2 , . . . , x p , t) guarantees the sliding surface Mittag–Leffler stability (σ p (t) = 0). Also, the other states will converge near the origin (z 1 (t), . . . , z p−1 (t) → 0) by choosing appropriate control gains k1 , . . . , kp−1 and sliding gains (kswi ≥ Wi (x 1 , x 2 , . . . , x i , t)i = 1, 2, . . . , p − 1).

CD

α

z p (t) = C D α x p (t) − C D α x p−ref (t)   = f p x1, x2, . . . , x p , t   + g p x 1 , x 2 , . . . , x p , t u (t) − C D α x p−ref (t)   + ω p x1, x2, . . . , x p , t

(22)

Finally, the fractional-order nonlinear system (10) can be rewritten in the following transformed NBC form CD

α

z 1 (t) = −k1 z 1 (t) + g 1 (x 1 , t)z 2 (t)

Stability proof Let’s consider the following Lyapunov candidate function which is positive definite V p (t) = 0.5z Tp (t)z p (t)

Taking derivative from (27) and using Lemma 1, we can get CD

− ksw1 sgn(z 1 (t)) + ω1 (x 1 , t)

(27)

α

  V p (t) = 0.5D α σ Tp (t)σ p (t) ≤ σ Tp (t)C D α σ p (t) (28)

α C D z 2 (t) = −k2 z 2 (t) + g 2 (x 1 , x 2 , t)z 3 (t)

−k sw2 sgn(z 2 (t)) + ω2 (x 1 , x 2 , t) .. . CD

α

Inserting (25) in (28) results in CD

z p−1 (t) = −k p−1 z p−1 (t)

α

V p (t) ≤ σ Tp (t)( f p (x 1 , x 2 , . . . , x p , t) +g p (x 1 , x 2 , . . . , x p , t)u(t)

+ g p−1 (x 1 , x 2 , . . . , x p−1 , t)z p (t)

− C D α x p−ref (t) + ω p (x 1 , x 2 , . . . , x p , t))

− ksw( p−1) sgn(z p−1 (t))

(29)

+ ω p−1 (x 1 , x 2 , . . . , x p−1 , t) CD

α

Using the control law (26) yields to

z p (t) = f p (x 1 , x 2 , . . . , x p , t) + g p (x 1 , x 2 , . . . , x p , t)u(t) −

CD

α

x p−ref (t) + ω p (x 1 , x 2 , . . . , x p , t)

CD

α

− kswp sgn(σ p (t)) + ω p (x 1 , x 2 , . . . , x p , t))

(23) Remark 2 To derive the transformed model (23), x (i+1)−ref is considered to be the virtual input vector instead of x i+1 . Because, choosing x i+1 will result complicated equations with barely calculable fractional derivatives.

V p (t) ≤ σ Tp (t)(−k p σ p (t) = − σ Tp (t)k p σ p (t) − kswp σ Tp (t)sgn(σ p (t)) + σ Tp (t)ω p (x 1 , x 2 , . . . , x p , t) 2 ≤ − k p σ p (t) − kswp σ p (t) 1 + σ p (t) ω p (x 1 , x 2 , . . . , x p , t)

(30)

Now let’s define the following sliding manifold in order to stabilize system (23)

√ Since σ p (t) ≤ σ p (t) 1 ≤ n p σ p (t) , then one can get

σ p (t) = z p (t)

CD

(24)

α

Taking derivative C D α from (24) and inserting ultimate block of (23) in it, leads to CD

α

  σ (t) = f p x 1 , x 2 , . . . , x p , t   + g p x 1 , x 2 , . . . , x p , t u (t)   − C D α x p−ref (t) + ω p x 1 , x 2 , . . . , x p , t (25)

Theorem 1 Consider the system (23) with the sliding manifold (24) and Assumptions 1 and 2, then the control law

2 V p (t) ≤ −k p σ p (t) − kswp σ p (t)   + ω p x 1 , x 2 , . . . , x p , t σ p (t) 2 ≤ −k p σ p (t)     − kswp − ω p x 1 , x 2 , . . . , x p , t × σ p (t)

(31)

  Selecting the sliding gain   kswp = W p x 1 , αx 2 , . . . , x p , t ≥ ω p x 1 , x 2 , . . . , x p , t , results in D V p (t) ≤ −k C σ p (t) 2 , which implies CD

α

2 V p (t) ≤ −k p z p (t) = −2k p V p (t)

(32)

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Selecting the first block sliding gain ksw1 = W1 (x 1 , t) ≥ ω1 (x 1 , t), results in

It follows from Lemma 2 that V p (t) ≤ V p (0)E α (−2k p t α )

(33) α

V 1 (t) ≤ −k1 z 1 (t)2 = −2k1 V1 (t)

From Definition 5, we have z p (t) ≤ z p (0)

1 E α (−2k p t α ) 2 , which is said to be Mittag–Leffler stable. Therefore, the final block dynamics will converge to zero in a finite time (σ p (t) = z p (t) = 0).

CD

Now, for the state z p (t) constrained to the sliding surface (σ p (t) = 0), the transformed system (23) reduces to

1

z i (t) ≤ z i (0) E α (−2ki t α ) 2 i = 1, 2, . . . , p − 2 (40)

CD

α

− ksw1 sgn(z 1 (t)) + ω1 (x 1 , t) − ksw2 sgn(z 2 (t)) + ω2 (x 1 , x 2 , t)

CD

.. . α

V (t) =

× sgn(z p−1 (t))

p−1 

+ z Tp−1 (t)ω p−1 (x 1 , x 2 , . . . , x p−1 , t) (41)

(34)

The above inequality can be rewritten as CD

Vi (t) = 0.5z iT (t)z i (t), i = 1, 2, . . . , p − 1

(35)

First, differentiating the candidate function V1 (t) 0.5z 1T (t)z 1 (t) along the first block of (34), results in

=

2 V p−1 (t) ≤ −k p−1 z p−1 (t) + (−ksw( p−1) + ω p−1 (x 1 , x 2 , . . . , x p−1 , t) ) z p−1 (t)

CD

− ksw1 z 1T (t) sgn(z 1 (t)) + z 1T (t)ω1 (x 1 , t) (36)

CD

α

V 1 (t) ≤ −k1 z 1 (t)2 − ksw1 z 1 (t)1 (37)

√ n 1 σ 1 (t), then one can get

V 1 (t) ≤ −k1 z 1 (t)2 +(−ksw1 + ω1 (x 1 , t)) z 1 (t)

123

(43)

CD

α

2 V p−1 (t) ≤ −k p−1 z p−1 (t) = −2k p−1 V p−1 (t) (44)

Which yields Mittag–Leffler stability z p−1 (t) ≤ z p−1 (0)

1 E α (−2k p−1 t α ) 2 .

If the tracking error z 2 (t) converges to zero, we have

+ ω1 (x 1 , t) z 1 (t)

α

Selecting the ( p − 1)th block = sliding gain ksw( p−1) W p−1 (x 1 , x 2 , . . . , x p−1 , t)≥ ω p−1 (x 1 , x 2 , . . . , x p−1 , t) , yields

V 1 (t) = 0.5C D α (z 1T (t)z 1 (t)) ≤ z 1T (t)C D α z 1 (t)

Since σ 1 (t) ≤ σ 1 (t)1 ≤

2 V p−1 (t) ≤ − k p−1 z p−1 (t) − ksw( p−1) z p−1 (t) 1 + z p−1 (t) ω p−1 (x 1 , x 2 , . . . , x p−1 , t) (42)

≤ −k1 z 1T (t)z 1 (t) + z 1T (t)g 1 (x 1 , t)z 2 (t)

α

α

Hence, we have

Vi (t),

i=1

CD

V p−1 (t) = 0.5C D α (z Tp−1 (t)z p−1 (t))

z p−1 (t) = − k p−1 z p−1 (t) − ksw( p−1) sgn(z p−1 (t))

Therefore, the system states convergence rate tuning and unmatched perturbations suppression can be achieved by the control gains k1 , . . . , k p−1 and the sliding gains ksw1 , . . . , ksw( p−1) . Let’s choose a Lyapunov function candidate V (t) as a sum of individual Lyapunov candidates for the each block of (34) as

α

α

≤ −k p−1 z Tp−1 (t)z p−1 (t) − ksw( p−1) z Tp−1 (t) + ω p−1 (x 1 , x 2 , . . . , x p−1 , t)

CD

Hence z 1 (t) will have Mittag–Leffler stability (z 1 (t) ≤ 1 z 1 (0) [E α (−2k1 t α )] 2 ). Proceeding this way for the other blocks of (33) results in

Finally, differentiating the candidate function V p−1 (t) = 0.5z Tp−1 (t)z p−1 (t) along the ( p − 1)th block of (34) results in

z 1 (t) = − k1 z 1 (t) + g 1 (x 1 , t)z 2 (t)

α C D z 2 (t) = − k2 z 2 (t) + g 2 (x 1 , x 2 , t)z 3 (t)

CD

(39)

(38)

Remark 3 The function sgn(z i ) in the proposed main and virtual control laws provokes the chattering phenomena, which can cause vibration and system failure due to excitation of high-frequency unmodeled dynamics. Using the smooth functions like tanh (z i /ρi ) instead of sgn(z i ) is a simple technique to alleviate the mentioned problem (0 < ρi < 1).

J Control Autom Electr Syst

Remark 4 Based on (34), convergence of i th block is related to the next block states variations. In other words, small tracking error (due to tanh (.)) of z i+1 (t) will transfer to z i (t). In this situation, it is reasonable to replace the Mittag–Leffler stability with zero neighborhood convergence for the variables z 1 (t), . . . , z p−1 (t). Remark 5 The perturbation terms ωi (x 1 , x 2 , . . . , x i , t) derivations boundedness is not necessary in the proposed sliding mode block controller.

Example 1 Consider the following perturbed fractionalorder nonlinear system x1 (t) = 2sin(x1 ) + 1.5x2 (t) + ω1 (x1 , t)

CD

0.85

x2 (t) = 0.8x1 (t) x2 (t) + x3 (t) + ω2 (x1 , x2 , t)

CD

0.85

x3 (t) = −x32 + 2u(t) + ω3 (x1 , x2 , x3 , t)

(48)

where z 1 = x1 − x1−ref , z 2 = x2 − x2−ref , and z 3 = x3 − x3−ref . Also, the design parameters are selected as:

k2 = 2, ρ2 = 0.01, ksw2 (x1 , x2 ) = 1 + 0.2 |x1 | + 0.2 |x2 |

In this section, two numerical simulations are performed to show the usefulness and efficiency of the suggested robust control strategy. The first case study is a fractional-order nonlinear system with both matched and unmatched perturbations. The second one is a permanent magnet synchronous motor (PMSM) with unmatched disturbance. It is worthy to notify that the PMSM integer-order derivatives are replaced with fractional-order derivatives only to have a proper case study. Moreover, the fractional-order PMSM model has been presented in some literatures (Yu et al. 2016; Li and Wu 2016) recently. Simulations are carried out using MATLAB toolbox called Ninteger (Valério 2005).

0.85

(47)

k1 = 1, ρ1 = 0.05, ksw1 (x1 ) = 1 + 0.1 |x1 |

5 Simulation Results

CD

x3−ref (t) = −0.8x1 (t)x2 (t) + C D 0.85 x2−ref (t)

z 2 (t) − k2 z 2 (t) − ksw2 (x1 , x2 )tanh ρ2 1 2 u(t) = x + D 0.85 x3−ref (t) − k3 z 3 (t) 2 3 C

z 3 (t) − k sw3 (x1 , x2 , x3 )tanh ρ3

(45)

k3 = 5, ρ3 = 0.01, ksw3 (x1 , x2 , x3 ) = 1 + 0.2 |x1 | + 0.3 |x2 | + 0.2 |x3 | The simulation results are shown in Figs. 1, 2, 3 and 4. Figure 1 shows the trajectory of the first state x1 (t) and its reference x1−ref (t). It is observed that x1 (t) tracks x1−ref (t) properly. The tracking error z 1 (t), control signal u(t), and all states (x1 (t), x2 (t), x3 (t)) are presented in Figs. 2, 3 and 4, respectively. Due to the existence of x32 term in the control law (48), u(t) is more sensitive to negative and positive peaks of x3 . This subject is evident in Fig. 3. Finally, boundedness of the system states is illustrated in Fig. 4. In Figs. 5, 6, 7, and 8, similar simulations are carried ), tanh( z 2ρ(t) ), out by replacing smooth functions tanh( z 1ρ(t) 1 2

) with non-smooth functions (sgn(z 1 (t)), tanh( z 3ρ(t) 3 sgn(z 2 (t)), sgn(z 3 (t))). The switching time is Ts = 0.0001. Remark 6 From the virtual and actual control laws (46– 48) and Figs. 5, 6, 7 and 8, it is evident that applying the non-smooth function makes the proposed controller almost impractical. The main reasons of this problem are fractional

where 2

ω1 (x1 , t) = 0.2 sin(t) + 0.1x1 (t) + 0.12 ω2 (x1 , x2 , t) = 0.3 sin(2t) + 0.2x1 (t) + 0.2x2 (t) − 0.4

0

ω3 (x1 , x2 , x3 , t) = 0.2 sin(2t) + 0.2x1 (t) + 0.3x2 (t)

-2

+ 0.2x3 (t) + 0.3

The purpose is to track x1−ref (t) = 2 sin(0.15t)+4 cos(0.1t)− 4 by x1 .

-4 x1(t)

(x1 (0), x2 (0), x3 (0)) = (0, 0, 0)

-6 1 -8

The system contains three scalar block; hence, the following virtual and main control laws are proposed: 1  −2 sin(x1 ) + C D 0.85 x1−ref (t) − k1 z 1 (t) x2−ref (t) = 1.5

z 1 (t) (46) − ksw1 (x1 )tanh ρ1

0.8 0.6

-10

0.4 -12

0

(t) xx1(t) 1

55

50

(t) xx1-ref(t)

60 100

1-ref

150

200

250

t (sec)

Fig. 1 Signals x1 and x1−ref

123

J Control Autom Electr Syst 1

2

z1(t)

0.8

0

0.6 -2

0.4

-4 x1 (t)

z1(t)

0.2 0

-6

-0.2 -0.4

-8

-0.6 -10

-0.8 -1

x (t) 1

x

0

50

100

150

200

t (sec)

(t)

1-ref

-12 0

250

50

100

150

200

250

t (sec)

Fig. 2 Tracking error z 1 = x1 − x1−ref

Fig. 5 Signals x1 and x1−ref

120

0.5

u(t)

z (t) 1

0.4

100

0.3

80 0.2

60 z1(t)

u(t)

0.1

40

0 -0.1

20

-0.2 -0.3

0

-0.4

-20

0

50

100

150

250

200

-0.5

t (sec)

0

50

100

150

200

250

t (sec)

Fig. 3 Control signal u

Fig. 6 Tracking error z 1 = x1 − x1−ref

20 15

x1(t)

6000

x2(t)

5000

u(t)

x (t) 3

4000

10 5

2000 u(t)

x1,2,3(t)

3000

0

1000 0

-5

-1000 -2000

-10

-3000

-15

0

50

100

150 t (sec)

Fig. 4 States x1 , x2 , x3

123

200

250

-4000

0

50

100

150 t (sec)

Fig. 7 Control signal u

200

250

J Control Autom Electr Syst

x1(t)

10 0 -10 -20

0

50

100

150

200

250

x2(t)

5 0 -5

0

50

100

0

50

100

150

200

250

150

200

250

x3(t)

50 0 -50 -100

t (sec)

Fig. 8 States x1 , x2 , x3

derivatives C D 0.85 x2−ref (t), C D 0.85 x3−ref (t) that can magnify the chattering phenomena. It is important to notify that this problem is dominant for: (1) Systems with fractional-order near 1: In this situation, the fractional derivative C D α xi−ref (t) amplifies the chattering. (2) Systems with more blocks: For these systems, number of virtual control laws are high which result in more fractional derivatives. Therefore, the discontinuous oscillations will be high. As mentioned in Remark 5, replacing the non-smooth functions with smooth ones is a simple way to avoid this problem. Example 2 Here, consider a surface-mounted permanent magnet synchronous motor (PMSM) with fractional-order dynamics (Yu et al. 2016; Li and Wu 2016). For the purpose of control design, ( (t),i d (t), i q (t)) are chosen as state variables. The PMSM perturbed dynamics can be written in the following explicit form:

stator voltages, i d and i q the stator currents, the rotor velocity, n p the number of pole pairs, φv the rotor flux linkage, 3n φ K t = 2p v , TL (t) the load torque, J the moment of inertia, and B the viscous friction coefficient. The parameters of PMSM used in the simulation are given as Liu and Li (2012): stator inductances L d = L q = L = 0.004 H , armature resistance Rs = 1.74 , viscous damping B = 7.403 × 10−5 N, moment of inertia J = 1.74 × 10−4 Kg.m2 , rated speed N = 140 rpm, and rotor flux φv = 0.1167. There are two main purposes in controlling the PMSM system (49): First, the reference speed tracking (z (t) = (t) −

r e f (t)): This goal is realizable by controlling the first and third equations of (49) which are in the NBC-form. Then, the following virtual and main control laws can be proposed as:

B

(t) + C D 0.95 r e f (t) − k z (t) J

z (t) − ksw tanh (50) ρ

Rs u q (t) = L q i q (t) + n p (t) i d (t) Lq n p φv +

(t) + C D 0.95 i q−ref (t) Lq

z q (t) (51) − kq z q (t) − kswq tanh ρq

J i q−ref (t) = Kt

Second, power losses reduction (z d (t) = i d (t) − i d−ref (t) = i d (t)−): here i d (t) should approach to zero to decrease the power losses (P(t) = Rs i d2 (t)). To achieve this goal, the following control law is designed:

Rs i d (t) − n p (t) i q (t) + C D 0.95 i d−ref (t) Ld

z d (t) (52) − kd z d (t) − kswd tanh ρd

u d (t) = L d

Note that the above control laws are suggested by transferring the PMSM system (49) into two subsystems (( (t),i q (t)) and (i d (t))). The control parameters are selected as:

Kt B 0.95 i q (t) − (t) + ω1 (t)

(t) = CD J J R 1 s 0.95 i d (t) = − i d (t) + n p (t)i q (t) + u d (t) CD Ld Ld Rs 0.95 i q (t) = − i q (t) − n p (t)i d (t) CD Lq n p φv 1 −

(t) + u q (t) (49) Lq Lq

ρ = 0.02 k = 2 ksw = 12000

where ω1 (t) = 1J TL (t) is bounded unmatched mechanical disturbance, L d and L q are inductances of d–q axes (L d = L q = L), Rs the stator resistance, u d and u q the

Simulation results are presented in Figs. 9, 10, 11 and 12 under unmatched perturbation (T L (t) = 1.8N .m) which is applied in 1 ≤ t ≤ 4. Figure 9 shows that (t) tracks the reference speed r e f (t) very well in the presence of mechanical

ρd = 0.01 kd = 0 kswd = 500 ρq = 0.01 kq = 0 kswq = 500

123

J Control Autom Electr Syst 150

80

Ω (t) Ω

ref

u (t) d

(t)

70

u (t) q

60

150 50

140

100

ud,q (t)

130 Ω (t)

120 110 1.98

2.02

2

130

50

20

140

10

120 110

0

100 0

0

0.5

1

1.02 1

1.04 1.5

2.98 2 t (sec)

30

2.04

120

100 0.98

40

3

2.5

3.02 3

3.04

3.5

-10

0

0.5

1

4

1.5

2 t (sec)

2.5

3

3.5

4

Fig. 12 Control signals Fig. 9 Mechanical speed 20

10

15

0

10

-10

torque variation. Also, speed tracking error, electrical d–q currents, and d–q control signals are presented in Figs. 10, 11 and 12, respectively. From Fig. 11, it is obvious that the direct current i d (t) is converging to zero in order to reduce the power losses. In Figs. 10 and 11, the reference speed r e f (t) and mismatched perturbation ω1 (t) sudden changes are the reason of sharp variations at times 1, 2 and 3 sec.

z

Ω

-20

5

1.98

2

2.02

2.04

Ω

z (t)

0 -5 -10 -15

6 Conclusion

-20 -25 -30

0

0.5

1

1.5

2 t (sec)

2.5

3

3.5

4

Fig. 10 Speed tracking error 3.5

i (t) d

iq(t)

3 2.5

id,q (t)

2

In this paper, the idea of matched and unmatched perturbations suppression has been successfully developed based on sliding mode block control technique for fractional-order nonlinear systems. The proposed control scheme guarantees stability of the closed-loop system. Simulation results show satisfying tracking performance of the suggested control strategy for two different examples. Moreover, the robustness to various matched and mismatched perturbations are successfully accomplished by SMC and step by step robust NBC, respectively. Finally, it is worthy to notify that the proposed control strategy is applicable for integer-order nonlinear systems by selecting α = 1 and enough smooth functions.

1.5 1

References

0.5 0 -0.5 0

0.5

1

1.5

Fig. 11 Electrical d–q currents

123

2 t (sec)

2.5

3

3.5

4

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