A switching sliding mode control technique for chaos suppression of fractional-order complex systems

Abstract

Switching sliding mode control (SSMC) can be utilized as a robust control technique, which is appropriate for the control of highly non-linear power systems like chaotic systems. The present study proposes a switching sliding mode control technique for control and chaos suppression of non-autonomous fractional-order (FO) nonlinear power systems with uncertainties and external disturbances. In the first step, a novel fractional switching sliding surface is introduced as well as its stability analysis to the origin is demonstrated. In the second step, based on the fractional version of the Lyapunov stability theory, a robust non-singular control law is designed to ensure the convergence of the system trajectories to the proposed sliding surface. Next, the proposed SSMC approach is utilized for designing a single input switching control technique for the stabilization of a class of 3D FO chaotic power systems. In order to evaluate the effectiveness and robustness of the suggested approach in practice, two examples including control and the stabilization of FO chaotic electric motors are illustrated.

Keywords

Switching control non-autonomous fractional-order systems chaotic power systems sliding mode control Lyapunov stability electric motors

Introduction

Control of nonlinear systems, especially chaotic systems, has been studied with growing interest in recent years (Singh and Roy, 2017, 2018; Singh et al., 2018). As stated in Leng et al. (2014), various publications have appeared over the recent years. By using fractional calculus (FC), many complex dynamics can be qualified as the simple system comprising fractional-order (FO) derivatives (Bagley and Torvik, 1983; Haghighi and Roohi, 2012; Petras, 2011; Sun et al., 2018). Moreover, FC plays a useful role in the control theory and system modeling (Petras, 2011; Roohi et al., 2018; Ullah et al., 2017). By using FC in the system model, a control law with the ideal order integration can be designed. In this regard, FO chaotic systems are considered as one of the significant applications in the control engineering, which they have vast applications in the real world. During the last 20 years, a large attempt has been made to explain different possibilities of how to implement the fractional calculus techniques in the modeling and control theory (Caponetto et al., 2010, Concepción et al., 2010). In addition, due to severe nonlinearity, high sensitivity, oscillatory characteristics and fractal motions in FO nonlinear systems, scientists have been attracted to control and stabilize such systems (Bigdeli and Ziazi, 2017b). To suppress undesirable behaviors of FO nonlinear systems, various control methods have been proposed such as: adaptive control (Aghababa et al., 2015; Jay Prakash et al., 2018; Roohi et al., 2015), fuzzy control (Sharma et al., 2017; Wang et al., 2016b), PID control (Zamani et al., 2009), feedback control (Shah et al., 2016), sliding mode control (Binazadeh, 2016; Li et al., 2012; Mobayen, 2015), automatic control (Alomoush, 2010; Debbarma et al., 2014) and optimal control (Nemati and Yousefi, 2016).

Sliding mode control (SMC) is a nonlinear control strategy expressing considerable properties such as robustness, accuracy, simple implementation and immutability to uncertainties (Yin et al., 2012, 2013). As is known, SMC includes two steps as follows:

The first one is to design appropriate sliding surface.

The second one is designing control input for the closed-loop system to change to the desired system specified by the sliding surface.

In recent years, a lot of work have utilized SMC technique for the control and stabilization of FO systems. For instance, in Mohadeszadeh and Delavari (2015), a new robust intelligent nonlinear SMC theory based on a time-varying type-2 fuzzy logic system is studied for the synchronization of FO hyper-chaotic systems. Stabilization of uncertain FO chaotic systems has been investigated in Bigdeli and Ziazi (2017a) by a FO adaptive intelligent SMC. In Mujumdar et al. (2015), by using a stable fractional surface, a sliding mode observer has been proposed for the control of non-commensurate FO systems. In Chen et al. (2016) and Yin et al. (2015), time-varying FO SMC methods have been designed for a class of 3D FO nonlinear systems. Shahri et al. (2016) have introduced a LMI-based robust FO feedback SMC strategy for a general class of uncertain fractional systems subject to saturation element. In Ding and Shen (2016), a SMC method has been proposed for projective synchronization of non-identical FO neural networks. Wang et al. (2016a) have designed a finite-time SMC approach, based on the frequency distributed model, to stabilize FO systems. In Lochan et al. (2018), a global SMC method is designed for the synchronization between the master and the slaves’ trajectories. Also, a second order adaptive time varying SMC is proposed for synchronization of hidden chaotic orbits in a new uncertain 4-D conservative chaotic system (Singh and Roy). An adaptive tuning method has been utilized to design FO non-singular fast terminal SMC for robot manipulator systems in Nojavanzadeh and Badamchizadeh (2016).

In this paper, switching sliding mode control method is designed to control and stabilize non-autonomous FO nonlinear systems with uncertainties and external disturbances. Based on the fractional type of the Lyapunov stability theorem, a novel stable sliding surface is introduced. Afterwards, a suitable non-singular control approach is suggested to guarantee the sliding motion. Moreover, as an application of the suggested non-singular controller, a single input sliding mode control approach is proposed for control and stabilization of a class of 3D FO chaotic systems. To evaluate the usefulness of the designed approaches in particular, numerical simulations are illustrated. Finally, the suggested approach in this study is validated and implemented in hardware-in-the-loop (HIL) based on OPAL-RT to integrate the fidelity of physical simulation and the flexibility of numerical simulation.

To sum up, the main advantages of this study are as follows: (i) designing a novel switching sliding mode surface, which is appropriate for non-integer nonlinear systems; (ii) the proposition of a robust switching control law, to force the system states to reach the proposed switching sliding surface; (iii) the suggested control approach in this study is robust over the system uncertainties and external disturbances; (iv) the suggested non-integer controller can be implemented for non-autonomous FO nonlinear systems; (v) the stability of both proposed sliding surface and the global stability of the closed-loop system are presented by using the fractional version of the Lyapunov stability theorem; (vi) stabilization of a class 3D non-integer non-autonomous chaotic systems via a single SMC input.

The switching logic, and thus the control law, are designed so that the state trajectories reach the surface and remain on it. So, as stated in Nandam and Sen (1995) and Vaidyanathan and Lien (2017), the target audience of this paper is not only intended for students and researchers but will also be valuable for executives, managers, marketing experts and project leaders who would like to apply SMC control to industrial applications such as the control of power systems and electronics, Robotics and so on. The paper presents a new non-integer SMC method to make it possible to improve the efficiency and effectiveness of the proposed control.

This paper is organized as follows. Section 2 is focused on some fundamentals of fractional differential equations (FDEs), a numerical algorithm to solve FDEs and problem statement, respectively. In Section 3, the switching sliding mode controllers are proposed and analytical results are given. Numerical simulations are provided in section 4 to demonstrate the effectiveness of the proposed technique. Finally, in Section 5 the paper is concluded.

Preliminaries and problem statement

Some basic concepts about FC are restated in this section. Also, a numerical approach for solving non-integer differential equations is given. In addition, the problem formulation of non-autonomous FO nonlinear systems is presented in this section.

Fractional calculus

Definition 1: (Podlubny, 1999) The Riemann-Liouville fractional integration of order $α$ of a continuous function $f (t)$ is introduced as follows

{{}_{t_{0}}I}_{t}^{α} = {{}_{t_{0}}D}_{t}^{- α} f (t) = \frac{1}{Γ (α)} \int_{t_{0}}^{t} \frac{f (τ)}{{(t - τ)}^{1 - α}} d τ .

(1)

where $t_{0}$ is initial time and $Γ (.)$ is the Gamma function which is defined by $Γ (z) = \int_{t_{0}}^{\infty} t^{z - 1} e^{- t} d t$ .

Definition 2 (Podlubny, 1999): Suppose that $m - 1 < α \leq m$ and $m \in N$ , then the Caputo fractional derivative of order α of a continuous function $f (t) : R^{+} \to R$ is given by

{{}_{t_{0}}^{c}D}_{t}^{α} f (t) = {{}_{t_{0}}D}_{t}^{- (m - α)} \frac{d^{m}}{d t^{m}} f (t) = \frac{1}{Γ (m - α)} \int_{t_{0}}^{t} \frac{f^{(m)} (τ)}{{(t - τ)}^{α - m + 1}} d τ .

(2)

Since the Caputo fractional derivative definition is more compatible rather than other definitions, in the rest of the paper the Caputo fractional derivative is utilized. Moreover, $I^{α}$ and $D^{α}$ denote the fractional integration and Caputo derivative operators of order $α$ , respectively.

Property 1: (Podlubny, 1999) Let $x (t) \in C^{m} [0, T]$ and $α \in (0, 1)$ . Then, $D^{α} x (t)$ is represented as follows

D^{α} x (t) = \frac{1}{Γ (1 - α)} \int_{t_{0}}^{t} \frac{f' (τ)}{{(t - τ)}^{α}} d τ .

(3)

Property 2 (Li and Deng, 2007): For any constant $φ \in R$ we have $D^{α} φ = 0$ .

Property 3 (Li and Deng, 2007): If $α \in (0, 1)$ and $x (t) \in C^{m} [0, T]$ , then

D^{α} (I^{α} x (t)) = D^{α} (D^{- α} x (t)) = x (t)

(4)

Theorem 1. (Zhang et al., 2015): Let $x (t) \in C^{1} ([0, \infty], R)$ be a continuously differentiable function and $α \in (0, 1)$ . Then, the following inequality holds almost everywhere.

D^{α} | x (t^{+}) | \leq sgn (x (t)) D^{α} x (t)

(5)

Where, $x (t^{+}) = lim_{τ \to t^{+}} x (τ)$ .

Lemma 1. (Hardy et al., 1988): the following inequality will be held if $x_{i} \in R, i = 1, 2, \dots, n$

\begin{matrix} {(| x_{1} (t) | + | x_{2} (t) | + \dots + | x_{n} (t) |)}^{β} \leq {| x_{1} (t) |}^{β} \\ + {| x_{2} (t) |}^{β} + \dots + {| x_{n} (t) |}^{β} \end{matrix}

(6)

where $0 < β \leq 1$ is a real number.

Theorem 2: (Zhang et al., 2015) Let $\bar{x} = 0$ be an equilibrium point for the non-autonomous FO system

D^{α} x (t) = f (t, x (t))

(7)

Then for $t_{0} = 0$ , the FO system (7) is Mittag-Leffler stable at the equilibrium point $\bar{x} = 0$ if there exists a continuous function $V (t, x (t))$ satisfies

α_{1} (‖ x ‖^{a}) \leq V (t, x (t)) \leq α_{2} (‖ x ‖^{ab}),

(8)

D^{β} V (t^{+}, x (t^{+})) \leq - α_{3} (‖ x ‖^{ab})

(9)

where $V (t, x (t)) : [0, \infty) \times D \to R$ satisfies locally Lipschitz condition on $x$ ; $D \subset R^{n}$ is a domain containing the origin; $\overset{\cdot}{V} (t, x (t))$ is piecewise continuous, and $lim_{τ \to t^{+}} \overset{\cdot}{V} (t, x (t))$ exists for any $t \in [0, \infty)$ ; and and $V (t^{+}, x (t^{+})) ≅ lim_{τ \to t^{+}} V (τ, x (τ))$ ; $t \geq 0, β \in (0, 1), α_{1}, α_{2}, α_{3}$ , $a$ and $b$ are arbitrary positive constants. If the assumptions hold globally on $R^{n}$ , then $\bar{x} = 0$ is globally Mittag-Leffler stable.

Remark 1: Mittag–Leffler stability and Generalized Mittag–Leffler stability imply asymptotic stability (Li et al., 2010).

The numerical solution of the non-integer differential equation

Here, the numerical solution of the FO differential equation is done through the modified Adams-Bashforth-Moulton algorithm (Asl and Javidi, 2017; Diethelm et al., 2002). A brief description of the algorithm aforementioned is presented in below.

Let the interested FO differential equation is given by

D^{α} y (t) = f (t, y (t)), 0 \leq t \leq T

(10)

y^{(k)} (0) = y_{0}^{(k)}, k = 0, 1, \dots, m - 1 (m = α)

(11)

where equation (11) can be equivalent to the Volterra integral equation, which is shown below

y (t) = \sum_{k = 0}^{m - 1} y_{0}^{(k)} \frac{t^{k}}{k!} + \frac{1}{Γ (α)} \int_{0}^{t} {(t - τ)}^{α - 1} f (y (τ), τ) d τ .

(12)

Then, it can be written as

\begin{matrix} y_{h} (t_{n + 1}) = & \sum_{k = 0}^{m - 1} c_{k} \frac{t_{n + 1}^{k}}{k!} + \frac{h^{α}}{Γ (α + 2)} f (y_{h}^{p} (t_{n + 1}), t_{n + 1}) \\ + \frac{h^{α}}{Γ (α + 2)} \sum_{j = 0}^{n} a_{j, n + 1} f (y_{h} (t_{j}), t_{j}) \end{matrix}

(13)

where $h = (T / N)$ and $t_{n} = nh$ , $n = 0, 1, \dots, N$ , $N$ denotes the number of step times, $t_{n}$ assumes the $n$ th step time as well as $h$ can be considered as the step time of the integration and remaining factors are also given as follows

\begin{matrix} a_{j, n + 1} = {\begin{matrix} n^{α + 1} - (n - α) {(n + 1)}^{α}, j = 0 \\ {(n - j + 2)}^{α + 1} + {(n - j)}^{α + 1} - 2 {(n - j - 1)}^{α + 1}, 1 \leq j \leq n \\ 1 . j = n + 1 \end{matrix} \\ y_{h}^{p} (t_{n + 1}) = \sum_{k = 0}^{m - 1} c_{k} \frac{t_{n + 1}^{k}}{k!} + \frac{1}{Γ (α)} \sum_{j = 0}^{n} b_{j, n + 1} f (y_{h} (t_{j}), t_{j}) . \\ b_{j, n + 1} = \frac{h^{α}}{α} [{(n + 1 - j)}^{α} - (n - j)^{α}] . \end{matrix}

Problem formulation

Suppose a general class of the non-autonomous uncertain n-dimensional non-integer nonlinear system with unknown uncertainty, external disturbance and control input is given as follows

{\begin{matrix} D^{α} x_{1} (t) = f_{1} (t, x (t)) + Δ f_{1} (t, x (t)) + d_{1} (t) + u_{1} (t) \\ D^{α} x_{2} (t) = f_{2} (t, x (t)) + Δ f_{2} (t, x (t)) + d_{2} (t) + u_{2} (t) \\ ⋮ \\ D^{α} x_{n} (t) = f_{n} (t, x (t)) + Δ f_{n} (t, (t)) + d_{n} (t) + u_{n} (t) \end{matrix}

(14)

where $α \in (0, 1)$ is the power of differentiation of the system. The vector of the system’s states is shown by $x (t) = [x_{1} (t), x_{2} (t), . . ., x_{n} (t)]^{T} \in R^{n}$ while $f_{i} (t, x (t)) \in R, i = 1, 2, \dots, n$ denotes a nonlinear function of $t$ and $x (t)$ . Also, the uncertainty of the system is presented by $Δ f_{i} (t, x (t)), i = 1, 2, \dots, n$ is as well as $d_{i} (t) \in R, i = 1, 2, \dots, n$ shows the term of external disturbance and the control input is expressed as $u_{i} (t) \in R, i = 1, 2, \dots, n$ .

Remark 2: The system (13) is a comprehensive representation of FO dynamical systems, where it consists of prominent class of FO chaotic (hyperchaotic) systems like FO Lorenz system, FO Chen system, FO Lu system, FO gyroscope system, FO Brushless DC (BLDC) Motor system, FO permanent-magnet synchronous motor (PMSM) system, FO Financial System, FO Genesio-Tesi system, FO Chua circuit, FO Duffing system and some other systems.

Assumption 1: In reality, the system’s disturbances and uncertainties are usually unknown, but bounded. Therefore, both of the uncertainty and the disturbance terms are assumed to be bounded by known positive scalars as

| Δ f_{i} (t, x (t)) + d_{i} (t) | < ξ_{i} < \infty, i = 1, 2, \dots, n

(15)

where $ξ_{1}$ , $ξ_{2}$ , …, $ξ_{n}$ expresses unknown positive coefficients.

Assumption 2: It is assumed that the measurability of the system’s states (14) is valid.

Switching sliding mode control technique: Framework and analytical argumentations

Recently, sliding mode control method has been known as a strong procedure for controlling FO chaotic systems. The main superiority of SMC is its robustness in overcoming uncertainties and external disturbances of the system.

Design of nonsingular SSMS

In total, the SMC technique can be expressed in two steps. In the first step, an appropriate sliding surface with some significant properties is designed. Here, a fractional sliding surface is designed as below

\begin{matrix} s_{i} (t) & = x_{i} (t) + I^{α} [η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t))] \\ = x_{i} (t) + D^{- α} [η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t))] \end{matrix}

(16)

where $0 < p < 1$ and $η_{i}, i = 1, 2, \dots, n$ , are positive constants.

Based on the SMC approach, the following equations are met when the system works on the sliding mode (Utkin, 1992)

s_{i} (t) = 0

(17)

So, using equations (16) and (17), we have

\begin{matrix} s_{i} (t) = 0 \Rightarrow x_{i} (t) + I^{α} [η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t))] = 0 \\ \Rightarrow x_{i} (t) = - I^{α} [η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t))] \end{matrix}

(18)

Now, according to the Property 3 and taking $D^{α}$ from both sides of equation (18), one obtains

D^{α} x_{i} (t) = - η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t))

(19)

Theorem 3: By considering the non-integer sliding mode dynamics in (19), the system is stable as well as the state trajectories of the nonlinear plant converge to the equilibrium $x_{i} = 0$ .

Proof: Select the following function as Lyapunov function candidate.

V_{1} (t, x (t)) = ‖ x (t) ‖_{1} = \sum_{i = 1}^{n} | x_{i} |

(20)

Obviously, the Lyapunov candidate (20) satisfies equation (8) in Theorem 2, the following inequality holds almost everywhere

\begin{matrix} D^{α} V_{1} (t^{+}, x (t^{+})) & = D^{α} \sum_{i = 1}^{n} | x_{i} (t^{+}) | = \sum_{i = 1}^{n} D^{α} | x_{i} (t^{+}) | \\ \leq \sum_{i = 1}^{n} [sign (x_{i} (t)) D^{α} x_{i} (t)] \end{matrix}

(21)

Using the FO sliding mode dynamics (19), one obtains

\begin{matrix} D^{α} V_{1} (t^{+}, x (t^{+})) \leq \\ \sum_{i = 1}^{n} [sign (x_{i} (t)) (- η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t)))] \end{matrix}

(22)

Clearly

D^{α} V_{1} (t^{+}, x (t^{+})) \leq - \sum_{i = 1}^{n} [η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p})]

Defining $η = \min {η_{i}, i = 1, 2, \dots, n}$ and using relation (7) in lemma 1 we have

\begin{matrix} D^{α} V_{1} (t^{+}, x (t^{+})) & \leq - η \sum_{i = 1}^{n} [| x_{i} (t) | + {| x_{i} (t) |}^{p}] \\ \leq - η ({‖ x (t) ‖}_{1} + ‖ x (t) ‖_{1}^{p}) \end{matrix}

(23)

Hence, based on the Mittag-Leffler stability condition in Theorem 2, the system (19) is globally Mittag-Leffler stable and state trajectories of the system (19) will converge to zero asymptotically. Therefore, the proof is completed.

Remark 3: Since the sliding surface (16) contains the discontinuous $sign (x_{i})$ function, the undesirable chattering phenomenon might happen. Generally, in many sliding mode implementations, control engineers may experience undesirable phenomenon of oscillations having finite frequency, which is known as ‘chattering’. Chattering is a harmful phenomenon because it leads to low control accuracy, high wear of moving mechanical parts, and high heat losses in electrical power circuits. Thus, it can be one of main obstacles in implementation of sliding mode control. According to Weitzner and Zaslavsky (2003) replacing the $sign (.)$ function by a continuous function such as $\tanh (.)$ is one of the most common methods for reducing the chattering effect. Here, by replacing $\tanh$ function instead of $sign$ function, the method overcomes chattering phenomenon. Thus, to reduce the chattering, the $sign (x_{i})$ function is replaced by the $\tanh (x_{i})$ function in (16), which results the following sliding surface

s_{i} (t) = x_{i} (t) + D^{- α} [η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) \tanh (x_{i} (t))]

(24)

Once a suitable FO sliding surface is chosen, the next step is to design a control law to force the state trajectories of the system (14) onto the sliding surface (19) and remain on it forever. Therefore, to guarantee existence of the sliding motion, a robust switching control law is proposed as follows

\begin{matrix} u_{i} (t) = & - (f_{i} (t, x (t)) + ξ_{i} sign (s_{i} (t)) \\ + η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t)) \\ + ρ_{i} (| s_{i} (t) | + {| s_{i} (t) |}^{q}) sign (s_{i} (t))) \end{matrix}

(25)

where $0 < q < 1$ and $ρ_{i}, i = 1, 2, \dots, n$ , are positive constants as switching gain.

In the following theorem, in order to prove that the sliding motion is accomplished, the fractional stability theorem 1 is utilized.

Theorem 4: Consider the non-integer system (13). Since (13) is controlled by the control law (25); the state trajectories of the system will converge to the sliding surface $s_{i} = 0$ .

Proof: Choose the following Lyapunov function

V_{2} (t, S (t)) = ‖ S (t) ‖_{1} = \sum_{i = 1}^{n} | s_{i} (t) |

(26)

Clearly, the Lyapunov function (26) implies equation (8). Using Theorem 1, the following inequality satisfies almost everywhere

\begin{matrix} D^{α} V_{2} (t^{+}, S (t^{+})) & = D^{α} \sum_{i = 1}^{n} | s_{i} (t^{+}) | = \sum_{i = 1}^{n} D^{α} | s_{i} (t^{+}) | \\ \leq \sum_{i = 1}^{n} [sign (s_{i} (t)) D^{α} s_{i} (t)] \end{matrix}

(27)

Substituting $s_{i}$ from (16) into (27), one has

\begin{matrix} D^{α} V_{2} (t^{+}, S (t^{+})) \\ \leq \sum_{i = 1}^{n} [sign (s_{i} (t)) D^{α} (x_{i} + I^{α} [η_{i} (| x_{i} | + {| x_{i} |}^{p}) sign (x_{i})])] \\ \leq \sum_{i = 1}^{n} [sign (s_{i} (t)) (D^{α} x_{i} (t) + η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t)))] \end{matrix}

(28)

According to $D^{α} x_{i} (t) = f_{i} (t, x (t)) + Δ f_{i} (t, x (t)) + d_{i} (t) + u_{i} (t)$ , we have

\begin{matrix} D^{α} V_{2} (t^{+}, S (t^{+})) & \leq \sum_{i = 1}^{n} [sign (s_{i}) (f_{i} (t, x (t)) + Δ f_{i} (t, x (t)) + d_{i} (t) + u_{i} (t) + η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t)))] \\ \leq \sum_{i = 1}^{n} [sign (s_{i} (t)) (f_{i} (t, x (t)) + u_{i} (t) + η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t))) + sign (s_{i} (t)) (Δ f_{i} (t, x (t)) + d_{i} (t))] \\ \leq \sum_{i = 1}^{n} [sign (s_{i} (t)) (f_{i} (t, x (t)) + u_{i} (t) + η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t))) + | Δ f_{i} (t, x (t)) + d_{i} (t) |] \end{matrix}

(29)

Using Assumption 1 and relation (15), one has

\begin{array}{l} D^{α} V_{2} (t^{+}, S (t^{+})) < \sum_{i = 0}^{n} [s i g n (s_{i} (t)) (f_{i} (t, x (t)) + u_{i} (t) \\ + η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) s i g n (x_{i} (t))) + ξ_{i}] \end{array}

(30)

Inserting $u_{i}$ from (25) into (30), it yields

D^{α} V_{2} (t^{+}, S (t^{+})) < \sum_{i = 0}^{n} [sign (s_{i}) (f_{i} (t, x (t)) + (- (f_{i} (t, x (t)) + ξ_{i} sign (s_{i} (t)) + η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t)) + ρ_{i} (| s_{i} (t) | + {| s_{i} (t) |}^{q}) sign (s_{i} (t)))) + η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) sign (x_{i} (t))) + ξ_{i}]

(31)

Now by some simplifications, and based on lemma 1, we obtain

\begin{matrix} D^{α} V_{2} (t^{+}, S (t^{+})) \\ < \sum_{i = 0}^{n} [sign (s_{i} (t)) (- ρ_{i} (| s_{i} (t) | + {| s_{i} (t) |}^{q}) sign (s_{i} (t)))] \\ < - ρ \sum_{i = 0}^{n} [| s_{i} (t) | + {| s_{i} (t) |}^{p}] < - ρ ({‖ S (t) ‖}_{1} + ‖ S (t) ‖_{1}^{q}) \end{matrix}

(32)

Where $ρ = min {ρ_{i}, i = 1, 2, \dots, n}$ .

Thus, pursuant to Theorem 2, Mittag-Leffler stability condition in equation (9) holds and the system (14) is globally Mittag-Leffler stable and the state trajectories of the FO system (14) will converge to $s_{i} = 0$ asymptotically. Therefore, the proof is done.

Remark 4: Due to using of sign function in the sliding surface and control method, it works like a switch. So, it is called switching sliding mode control.

Remark 5: Chattering phenomenon is the main drawback of SMC approach. One can see that, the proposed switching control law (25) contains the sign function, which can cause the chattering. In order to avoid chattering phenomenon, the switching control input (25) is modified as follows

u_{i} (t) = - (f_{i} (t, x (t)) + ξ_{i} \tanh (s_{i} (t)) + η_{i} (| x_{i} (t) | + {| x_{i} (t) |}^{p}) \tanh (x_{i} (t)) + ρ_{i} (| s_{i} (t) | + {| s_{i} (t) |}^{q}) \tanh (s_{i} (t)))

(33)

Remark 6: As it mentioned in Remark 3, in equation (33), replacing the $\tanh (s_{i})$ function by the $sign (s_{i})$ function in (25) has no influence on the stability and robustness of the switching sliding mode controller.

Design of a single input switching sliding mode controller for a class of three-dimensional FO chaotic systems

Here, applicability of the proposed switching SMC method is shown for a class of three dimensional (3D) FO chaotic systems. Now, a class of 3D FO system with one control input is considered as follows

{\begin{matrix} D^{α} Y = f_{1} (t, Y (t), z (t)) \\ D^{α} z = f_{2} (t, Y (t), z (t)) + Δ f_{2} (t, Y (t), z (t)) + d (t) + u (t) \end{matrix}

(34)

which $α \in (0, 1)$ indicates the power of derivative, $Y (t) = [y_{1} (t), y_{2} (t)]^{T} \in R^{2}$ and $z \in R$ are the system’s states, $f_{1} (t, Y (t), z (t))$ and $f_{2} (t, Y (t), z (t))$ express nonlinear functions of $t, Y (t)$ and $z (t)$ ; $u (t) \in R$ is the control input and $Δ f_{2} (t, Y (t), z (t))$ and $d (t)$ denote the system’s uncertainty and the external disturbance respectively, where they are assumed to be bounded by

| Δ f_{2} (t, Y (t), z (t)) + d (t) | < ξ < \infty,

(35)

where $ξ$ is a positive constants.

Remark 7: The presented non-integer system (34) is very general, which includes an important part of the canonical FO chaotic systems, containing FO Genesio–Tesi system, FO Arneodo system as well as almost all of other 3D FO chaotic systems like FO Lü system, FO Lorenz system, FO Tigan system, FO unified system as well as FO Chen system.

Assumption 4: In FO system (34), the function $f_{1} (t, Y (t), z (t))$ is smooth in the nearby of $z (t) = 0$ . Also, for all Y, the subsystem $D^{α} Y = f_{1} (t, Y (t), z (t))$ can be asymptotically stable around the origin $Y (t) = 0$ . Now, the non-integer sliding surface (16) can be written as below

\begin{matrix} s (t) & = z (t) + I^{α} [η (| z (t) | + {| z (t) |}^{p}) sign (z (t))] \\ = z (t) + D^{- α} [η (| z (t) | + {| z (t) |}^{p}) sign (z (t))] \end{matrix}

(36)

where $0 < p < 1$ . It should be noted that $η$ is considered as a positive constant.

By utilizing the sliding motion property $s (t) = 0$ and applying the $D^{α}$ of it, we have $D^{α} z (t) = - η (| z (t) | + {| z (t) |}^{p}) sign (z (t))$ . Therefore, according to Theorem 4, the stability of the sliding mode dynamics is ensured, meaning $\lim_{t \to \infty} z (t) = 0$ . In addition, based on Assumption 4, we obtain $lim_{t \to \infty} ‖ Y (t) ‖ = 0$ . In the sense that if the convergence of the states of the subsystem $D^{α} z (t) = f_{2} (t, Y (t), z (t)) + Δ f_{2} (t, Y (t), z (t)) + d (t) + u (t)$ to the sliding surface $s (t) = 0$ is guaranteed, then the overall closed-loop FO system (34) will be stable. Thus, the next step is to propose a proper robust SMC approach to confirm the existence of the sliding motion. Consequently, the following single input robust sliding mode law can be designed

u (t) = - (f_{2} (t, Y (t), z (t)) + ξ sign (s (t)) + η (| z (t) | + {| z (t) |}^{p}) sign (z (t)) + ρ (| s (t) | + {| s (t) |}^{q}) sign (s (t)))

(37)

where $0 < q < 1$ and $ρ$ , are positive scalars.

Theorem 5: Consider the 3D FO system (34). While the system (34) can be controlled by the control law (37), then the state trajectories of the system will converge to the sliding surface $s (t) = 0$ .

Proof: choose the following Lyapunov function

V_{3} (t, s (t)) = ‖ s (t) ‖_{1} = | s (t) | .

(38)

Obviously, the Lyapunov function (38) satisfies equation (8) and based on Theorem 1, we have almost everywhere

V_{3} (t^{+}, s (t^{+})) = D^{α} | s (t^{+}) | \leq sign (s (t)) D^{α} s (t) .

(39)

Inserting $D^{α} s (t)$ from (36) into (39), one has

V_{3} (t^{+}, s (t^{+})) \leq sign (s (t)) (D^{α} z (t) + η (| z (t) | + {| z (t) |}^{p}) sign (z (t))) .

(40)

Substituting $D^{α} z (t) = f_{2} (t, Y (t), z (t)) + Δ f_{2} (t, Y (t), z (t)) + d (t) + u (t)$ into the above equation, we have

\begin{matrix} V_{3} (t^{+}, s (t^{+})) & \leq sign (s (t)) (f_{2} (t, Y (t), z (t)) + Δ f_{2} (t, Y (t), z (t)) + d (t) + u (t) + η (| z (t) | + {| z (t) |}^{p}) sign (z (t))) \\ \leq sign (s (t)) (f_{2} (t, Y (t), z (t)) + u (t) + η (| z (t) | + {| z (t) |}^{p}) sign (z (t))) + | Δ f_{2} (t, Y (t), z (t)) + d (t) | \\ < sign (s (t)) (f_{2} (t, Y (t), z (t)) + u (t) + η (| z (t) | + {| z (t) |}^{p}) sign (z (t))) + ξ \end{matrix}

(41)

Inserting $u$ from (37) into (41), one can obtain

\begin{matrix} V_{3} (t^{+}, s (t^{+})) < sign (s (t)) (f_{2} (t, Y (t), z (t)) + (- (f_{2} (t, Y (t), z (t)) + ξ sign (s (t)) + η (| z (t) | + {| z (t) |}^{p}) sign (z (t)) + ρ (| s (t) | + {| s (t) |}^{q}) sign (s (t)))) + η (| z (t) | + {| z (t) |}^{p}) sign (z (t)) + ξ) \end{matrix}

(42)

Now by some manipulations and using lemma 1, we obtain

\begin{matrix} V_{3} (t^{+}, s (t^{+})) & < sign (s (t)) (- ρ (| s (t) | + {| s (t) |}^{q}) sign (s (t))) . \\ < - ρ (| s (t) | + {| s (t) |}^{q}) < - ρ (‖ s (t) ‖_{1} + ‖ s (t) ‖_{1}^{q}) \end{matrix}

(43)

As a result, pursuant to Theorem 2, the Mittag-Leffler stability condition is satisfied and the states of the 3D non-integer system (34) will converge to $s = 0$ asymptotically. Therefore, the proof is completed.

Numerical simulations

In this part, two illustrative examples are shown to indicate the applicability of the suggested switching sliding mode control technique in practice. The first example deals with the control of chaotic FO brushless DC motor (BLDCM) system and the second one is about control and stabilization of non-integer permanent magnet synchronous motor (PMSM) system. MATLAB software is utilized for numerical simulations. The mentioned numerical algorithm in Section 2.2 with $h = 0.01$ as time step of is used to numerical solution the FO differentials.

Control of FO BLDCM system

In this stage the effectiveness and robustness of the suggested non-integer switching sliding mode controller (33) is investigated by stabilizing the FO chaotic BLDCM system. The non-integer BLDCM system has vast applications in the industrial automation and manufacturing engineering, such as: computer systems, electric vehicles and hybrid vehicles, micro-radio-controlled airplane and especially aerospace (Zhou et al., 2015, 2016). The system is described as follows

{\begin{matrix} D^{α} x_{1} (t) = - 0.875 x_{1} (t) + x_{2} (t) x_{3} (t) \\ + \underset{Δ f_{1} (t, x (t)) + d_{1} (t)}{\underset{︸}{0.2 \cos (3 t) x_{1} (t) - 0.15 \sin (2 t)}} + u_{1} (t) \\ D^{α} x_{2} (t) = - x_{2} (t) + 55 x_{3} (t) - x_{1} (t) x_{3} (t) \\ + \underset{Δ f_{2} (t, x (t)) + d_{2} (t)}{\underset{︸}{0.25 \sin (4 t) x_{2} (t) + 0.2 \sin (3 t)}} + u_{2} (t) \\ D^{α} x_{3} (t) = 4 (x_{2} (t) - x_{3} (t)) \\ + \underset{Δ f_{3} (t, x (t)) + d_{3} (t)}{\underset{︸}{0.3 \sin (2 t) x_{3} (t) - 0.25 \cos (4 t)}} + u_{3} (t) \end{matrix}

(44)

As has been mentioned in Zhou et al. (2016), this system has chaotic behavior for $0.96 < α \leq 1$ . Therefore, the FO $α$ is set to 0.97 to ensure the existence of chaos for the FO chaotic BLDCM system. Also, the initial values of the system (44) are randomly selected as $x_{1} (0) = 10, x_{2} (0) = - 5$ and $x_{3} (0) = 5$ , respectively.

Equation (16) is utilized to design the sliding surfaces of the case study

\begin{matrix} s_{i} (t) = x_{i} (t) + 0.02 D^{- 0.97} [(| x_{i} (t) | + {| x_{i} (t) |}^{0.6}) sign (x_{i} (t))], \\ i = 1, 2, 3 . \end{matrix}

(45)

Consequently, due to equation (33), by selecting $ξ_{i} = 0.05, i = 1, 2, 3,$ the appropriate control inputs are achieved using the following equations

\begin{matrix} u_{1} (t) = - (- 0.875 x_{1} (t) + x_{2} (t) x_{3} (t) + ξ_{1} sign (s_{1} (t)) + 0.02 (| x_{1} (t) | + {| x_{1} (t) |}^{0.6}) sign (x_{1} (t)) + 2.5 (| s_{1} (t) | + {| s_{1} |}^{0.6}) \tanh (s_{1} (t))) \\ u_{2} (t) = - (- x_{2} (t) + 55 x_{3} (t) - x_{1} (t) x_{3} (t) + ξ_{2} sign (s_{2} (t)) + 0.02 (| x_{2} (t) | + {| x_{2} (t) |}^{0.6}) sign (x_{2} (t)) + 3.3 (| s_{2} (t) | + {| s_{2} |}^{0.6}) \tanh (s_{2} (t))) \\ u_{3} (t) = - (4 (x_{2} (t) - x_{3} (t)) + ξ_{3} sign (s_{3} (t)) + 0.02 (| x_{3} (t) | + {| x_{3} (t) |}^{0.6}) sign (x_{3} (t)) + 2 (| s_{3} (t) | + {| s_{3} (t) |}^{0.6}) \tanh (s_{3} (t))) \end{matrix}

(46)

In order to show the state trajectories of the controlled uncertain FO BLDCM system (44), Figure 1 is illustrated. Figure 1 shows that the states converge to zero, which indicates that the chaotic motions of the uncertain FO BLDCM system are effectively suppressed. In Figure 2, the time responses of the SSs (45) are revealed. It is seen that the SSs achieve to zero. The time histories of the used control efforts (46) are plotted in Figure 3. Clearly, the control efforts are feasible in practice, without damaging chattering actions.

Now, for comparison of the SSMC method, a SMC designed in Wang et al. (2016a) and an adaptive fuzzy control (AFC) method (Liu et al., 2018) are also applied to stabilize the FO BLDCM system (44). Recently, in Wang et al. (2016a), a finite-time SMC and in Liu et al. (2018) an adaptive fuzzy controller has been proposed for stabilization of the FO systems. Here, the implementation of the control approaches in Wang et al. (2016a) and Liu et al. (2018) are considered as follows, respectively

\begin{matrix} s_{i} (t) = D^{0.97 - 1} x_{i} (t) + D^{- 1} [5 x_{i} (t) + 2 {| x_{i} (t) |}^{0.6} sat (x_{i} (t))], \\ i = 1, 2, 3 . \end{matrix}

(47)

\begin{matrix} u_{i} (t) = - (\begin{matrix} f_{i} (t, x (t)) + 5 x_{i} (t) + 2 {| x_{i} (t) |}^{0.6} sat (x_{i} (t)) + 0.1 s_{i} (t) + \\ (0.1 + 0.1 + {| s_{i} (t) |}^{0.3} sign (s_{i} (t))) + sat (s_{i} (t)) \end{matrix}), \\ i = 1, 2, 3 . \end{matrix}

(48)

\begin{matrix} u_{i} (t) = {\begin{matrix} - (2 + {θ_{i}}^{T} (t) ψ_{i} (x_{i} (t))) sgn (x_{i} (t)) - 5, x_{i} (t) > 0 \\ 0, x_{i} (t) = 0 \\ - (2 + {θ_{i}}^{T} (t) ψ_{i} (x_{i} (t))) sgn (x_{i} (t)) - 5, x_{i} (t) > 0 \end{matrix} \\ D^{0.97} θ_{i} (t) = 10 | x_{i} (t) | ψ_{i} (x_{i} (t)), i = 1, 2, 3 . \end{matrix}

(49)

The state trajectories of the system (44) controlled via suggested SSMC (46) and the SMC method (48), existed in Wang et al. (2016a) and the AFC methodology (49), existed in Liu et al. (2018) are shown in Figure 4. It is seen that even though the system states reach to origin, but there are permanent oscillations in the state trajectories of the system via SMC and AFC methods. Additionally, the time responses of the SS (47) and control efforts (48) and (49), which have been suggested in Wang et al. (2016a) and Liu et al. (2018) are depicted in Figures 5, 6 and 7, respectively. One can see that the chattering phenomenon there exists in sliding surface and control inputs. A discussion of this comparison is summarized in Table 1.

Table 1.

Comparison between the results of this SSMC, SMC and AFC.

Comparison items	The results of this paper	The results in Wang et al. (2016a)	The results in Liu et al. (2018)
Control methodology	Switching sliding mode control (SSMC)	Sliding mode control (SMC)	Adaptive fuzzy control (AFC)
Design parameters	Overall, 3n+2 parameters should be selected	Overall, n+8 parameters should be selected	Overall, n(N+1)+1 parameters should be selected
Chattering phenomenon	There is no chattering phenomenon in both states and controller method.	Chattering phenomenon occurs in both states and controller method.	Chattering phenomenon occurs in both states and controller method.
Magnitude of oscillations	Normal range (based on related reference)	Out of range (based on related reference)	Normal range (based on related reference)
Conclusions	The advantages of the proposed SSMC method: (1) higher robustness; (2) the controller is easier to design and may be more convenient for practical use; (3) chattering free; (4) higher convergence accuracy.	The advantages of the SMC method: (1) the controller is effectively working when the states of the system are known; (2) it uses frequency distributed model; (3) it is applicable for the vast area of systems.	The advantages of AFC method: (1) its robustness is demonstrated; (2) it uses fuzzy function; (3) it is applicable for the vast class of dynamical FO systems.

Control of FO permanent magnet synchronous motor system

Here, the usefulness of the single input switching SMC (36) and (37) are confirmed via the stabilization of the uncertain non-integer permanent magnet synchronous motor (PMSM) chaotic system. Due to simple structure, high speed, great power density, minimal maintenance, and smooth operation of the PMSM system, it is considered one of the most attractive models for researchers in industries (Rahimi et al., 2016).

The FO PMSM system is introduced by the following mathematical equations (Xue et al., 2015)

{\begin{matrix} D^{α} x_{1} (t) = - x_{1} (t) + x_{2} (t) x_{3} (t) \\ D^{α} x_{2} (t) = - x_{2} (t) - x_{1} (t) x_{3} (t) + 100 x_{3} \\ D^{α} x_{3} (t) = 10 (x_{2} (t) - x_{3} (t)) \end{matrix} (t) .

(50)

This system will have undesirable behavior for $α \in [0.94, 1]$ where in this example $α = 0.98$ .

In system (50), if $x_{3} (t) = 0$ , we reach to the following two-dimensional subsystem

{\begin{matrix} D^{α} x_{1} (t) = - x_{1} (t) \\ D^{α} x_{2} (t) = - x_{2} (t) \end{matrix}

(51)

Clearly, the asymptotical stability of equation (51) around the origin $x_{1} (t) = 0$ and $x_{2} (t) = 0$ for all $x_{1}$ and $x_{2}$ can be obtained. Therefore, to satisfy the conditions of Assumption 4, we can consider $Y = [x_{1} (t), x_{2} (t)]^{T}$ and $z = x_{3} (t)$ . Then, the FO system (50) becomes

{\begin{matrix} D^{α} y_{1} (t) = - y_{1} (t) + y_{2} (t) z (t) \\ D^{α} y_{2} (t) = - y_{2} (t) - y_{1} (t) z (t) + 100 z (t) \\ D^{α} z (t) = 10 (y_{2} (t) - z (t)) + \underset{Δ f_{2} (t, Y (t), z (t)) + d (t)}{\underset{︸}{0.15 \sin (2 t) z (t) - 0.1 \cos (3 t)}} + u (t) \end{matrix}

(52)

On the basis of equation (36), the following sliding surface is designed

s (t) = z (t) + 0.2 D^{- 0.98} [(| z (t) | + {| z (t) |}^{0.95}) sign (z (t))]

(53)

Afterward, using the Theorem 4 and equation (37), the following switching control law is proposed

u (t) = - (10 (y_{2} (t) - z (t)) + 0.05 sign (s (t)) + 0.2 (| z (t) | + {| z (t) |}^{0.95}) sign (z (t)) + 3 (| s (t) | + {| s (t) |}^{0.9}) sign (s (t)))

(54)

It should be noted that the initial conditions of the non-integer PMSM system are selected randomly as $y_{1} (0) = 5$ , $y_{2} (0) = 2$ and $z (0) = 3$ . The state trajectories of the non-integer PMSM system, which are controlled by single input SMC (53) and (54), are depicted in Figure 8.

It is pointed that the chaotic movements of the FO PMSM system are resolved. The time history of the used SS (53) and the time response of the single input controller (54) are depicted in Figures 9 and 10, respectively. Clearly, the SS (53) as well as control signal (54) converge to zero. It is shown that the single input sliding mode control technique is feasible in the real-world applications.

Conclusions

In this paper, in order to control and stabilize the FO non-autonomous nonlinear systems with uncertainty and external disturbance, a new switching sliding mode control approach is designed. Besides, as an application of the aforementioned SSMC, a single input SSMC technique is introduced to control a 3D class of FO non-autonomous chaotic systems. Also, the stability of the closed-loop system is analyzed via fractional version of Lyapunov stability theorem. In addition, for validation of the designed SSMC approaches in practice, two important chaotic FO systems in electric motors are investigated. For this purpose, chaos suppression problem of FO BLDCM system and FO PMSM system are employed. It is worth pointing out that the proposed SSMC techniques can be applied for control and stabilization of a prominent class of FO non-autonomous nonlinear systems.

Footnotes

Acknowledgements

The authors gratefully acknowledge the editor and anonymous reviewers for providing constructive comments.

Author Note

Mohammad-Hassan Khooban is currently affiliated with Department of Engineering , Aarhus University , Denmark.

Declaration of conflicting interests

The author(s) declared no potential conflict of interests with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Majid Roohi

Mohammad-Hassan Khooban

Mohammad Pourmahmood Aghababa

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