Adaptive fuzzy fractional-order sliding mode controller for a class of dynamical systems with uncertainty

Abstract

A sliding mode control (SMC) strategy is proposed for a second-order non-linear system using fractional calculus and adaptive fuzzy compensator, named fuzzy fractional-order SMC (FFOSMC). The major contribution of this research includes the formulation of a robust fractional-order controller based on a novel non-integer sliding manifold. The proposed controller utilizes reduced discontinuous switching gain using a fuzzy logic system. A fractional-order adaptation law is derived to update the fuzzy logic system, which has more degrees of freedom to achieve the desired performance. The stability of the closed-loop system is guaranteed using fractional-order Lyapunov theory. The parameters of the proposed control method are optimized using Simulink response optimization. Finally, the performance of the FFOSMC is fully investigated and compared with a conventional SMC controller and an integral SMC controller. From simulation results, it is concluded that the proposed algorithm is effective and more robust.

Keywords

Actuators adaptive control control systems design control theory robust control

Introduction

Fractional calculus (FC) is a research field that has been considered for several centuries (Gutierrez et al., 2010; Petras, 2011). In earlier literature, this topic was assumed purely theoretical with no applications (Tavazoei et al., 2008). In modern times, FC has emerged quickly and found applications in many areas of science and technology due to the higher degree of freedom provided (Chen and Vinagre, 2003; Couceiro et al., 2010; Nagoya, 2012; Oustaloup et al., 1996). The first fractional-order controller was proposed by Oustaloup. He derived a robust fractional-order control scheme called the Command Robuste d’Ordre Non-Entier (CRONE) (Oustaloup, et al., 1996). A fractional-order robust controller was proposed for a non-linear system by Ullah et al. (2015). A frequency-domain approach has been introduced in Vinagre et al. (2000), and a variable fractional-order controller was introduced in Valerio and Costa (2013). Tuning of fractional-order PID controller was introduced by Domek (2013). A high-performance robust control is proposed by Ullah and Wang (2013).

Sliding mode control (SMC) is a robust control scheme with many applications (Perruquetti and Barbot, 2002). A linear manifold is chosen as the error surface and the control law is formulated. To prove the asymptotic stability and desired performance of the closed loop, a Lyapunov theorem is used. Recently, several researchers have shown great interest in formulating a sliding mode robust controller using FC. Most of the literature concentrates on a non-integer proportional-derivative (FOPD) sliding manifold, which has been applied in antilock braking systems (Tang et al., 2013), twin-tank models (Delavari et al., 2010) and robotic flexible joint manipulators (Fayazi and Nabizadeh, 2011). The FOPD controller design mentioned above does not take uncertainties into account.

Most non-linear systems contain non-linearities, parametric uncertainties and external disturbances. An ideal sliding mode controller inevitably has a discontinuous switching function to deal with the above uncertainties. Due to imperfect switching, chattering is introduced in practice. Reducing the gain of the switching function can minimize the chattering phenomenon. Several researchers utilized a fuzzy logic system (FLS) to tune the gain of the switching control in the FOPD controller (Delavari et al., 2010); Fayazi and Nabizadeh, 2011; Zhang et al., 2012). In this paper, an adaptive fuzzy logic controller is used to approximate unknown non-linearities and parametric uncertainties online. Because the gain of the switching function depends on the upper boundary of uncertainties, a rational design of fuzzy logic controller can estimate uncertainties and reduce the gain of the switching function, which can reduce the chattering phenomenon.

The corresponding control law is formulated based on the fractional Lyapunov stability theory to guarantee the sliding condition. Finally, the performance of the proposed fuzzy fractional-order SMC (FFOSMC) is fully investigated and compared with conventional SMC and integral SMC. The results show that the proposed algorithm is effective and more robust.

Definitions of fractional derivatives and integrals

The basic definition of a fractional operator can be denoted by a general fundamental operator ${{}_{a}D}_{t}^{α}$ as a generalization of the differential and integral operators, which is defined as follows (Bai and Feng, 2007).

{{}_{a}D}_{t}^{α} ≅ D^{α} = {\begin{matrix} \frac{d^{α}}{{dt}^{α}} & , R (α) > 0 \\ 1 & , R (α) = 0 \\ \int_{a}^{t} (d τ)^{- α} & , R (α) < 0 \end{matrix}

(1)

Here $α$ represents the order of the fractional operator and $R (α)$ represents the set of real numbers. The three definitions used for the general fractional operator are the Riemann–Liouville (RL) definition, the Caputo definition and the Grunwald–Letnikov (GL) definition (Caputo, 1967; Miller and Ross, 1993; Podlubny, 1999). Now the αth order Riemann–Liouville fractional derivative of function is given by

{{}_{a}D}_{t}^{α} f (t) = \frac{d^{α}}{{dt}^{α}} f (t) = \frac{1}{Γ (m - α)} \frac{d^{m}}{{dt}^{m}} \int_{a}^{t} \frac{f (τ)}{(t - τ)^{α - m + 1}} d τ

(2)

The Riemann–Liouville formula of the αth-order fractional integration can be written as

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

(3)

Here $m$ is the first integer larger than $α$ , such as $m - 1 < α < m$ , $t - a$ is the interval of integration and $Γ$ is Euler’s Gamma function.

The Caputo fractional derivative expression of a continuous function is expressed as

{{}_{a}D}_{t}^{α} f (t) = {\begin{matrix} \frac{1}{Γ (n - α)} \int_{a}^{t} \frac{f^{n} (τ)}{(t - τ)^{α - n + 1}} d τ (n - 1 \leq α < n) \\ \frac{d^{n}}{{dt}^{n}} f (t) (α = n) \end{matrix}

(4)

The GL definition is:

{{}_{a}^{GL}D}_{t}^{α} f (t) = lim_{h \to 0} \frac{1}{h^{α}} \sum_{j = 0}^{[(t - α) / h]} (- 1)^{j} (\begin{matrix} α \\ j \end{matrix}) f (t - jh)

(5)

Here h represents the time step that is increasing and [.] is the integer part,

(\begin{matrix} α \\ j \end{matrix}) = \frac{Γ (α + 1)}{Γ (j + 1) Γ (α - j + 1)}

(6)

Stability theorem for non-integer-order system

The stability of non-integer systems has been discussed in the literature in detail. Consider the following theorem (Matignon, 1998).

Theorem 1. Consider the fractional-order system:

{{}_{0}D}_{t}^{α} x = Ax, x (0) = x_{0}

(7)

Here α is the differential order, $x \in R^{n}$ and $A \in R^{n \times n}$ . The system is asymptotically stable if $| \arg (eig (A)) | > α π / 2$ . In this case, each component of the state decays towards 0 like $t^{- α}$ . Also, this system is stable if $| \arg (eig (A)) | \geq α π / 2$ and those critical eigenvalues that satisfy $| \arg (eig (A)) | = α π / 2$ have geometric multiplicity. Figure 1 shows the stable region for $0 < α < 2$ . Obviously, the stable region of fractional system with $0 < α < 1$ is largest than that of the other two cases.

Figure 1.

Stability region of fractional system.

Non-linear system description and review of control methods

Consider the following second-order non-linear system:

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f (X, t) + g (X, t) u + d (X, t) \\ y = x_{1} \end{matrix}}

(8)

Here $X (t) = {[x_{1} x_{2}]}^{T}$ , $u$ represents the control excitation, $d (X, t)$ represents the unknown disturbance, $f (X, t) = f_{n} (X, t) + Δ f (X, t)$ , $Δ f (X, t)$ is the parametric $g (X, t) = g_{n} (X, t) + Δ g (X, t)$ , $g_{n} (X, t) > 0$ represent nominal system input and $Δ g (X, t)$ is the input discrepancy.

Assumption 1. $L (X, t)$ represents the total uncertainty:

L (X, t) = Δ f (X, t) + Δ g (X, t) u + d (X, t)

(9)

$L (X, t)$ is assumed to satisfy the inequality given as

‖ L (X, t) ‖ \leq δ_{1}

(10)

Here $δ_{1}$ is a known positive constant. The objective of the designed controller is to make $X$ strictly follow its desired signal $X_{d} = {[x_{d} {\overset{\cdot}{x}}_{d}]}^{T}$ . The tracking error is defined as

{\begin{matrix} e_{1} = x_{1} - x_{d} \\ e_{2} = x_{2} - {\overset{\cdot}{x}}_{d} \end{matrix}}

(11)

Fuzzy logic system basic definitions

A fuzzy logic system (FLS) consists of four parts: the knowledge base, the fuzzifier, the fuzzy inference engine working on fuzzy rules and the defuzzifier (Kosko, 1994; Wang, 1997).

The knowledge base for the FLS comprises a collection of fuzzy If–Then rules. The multiple input–multiple output (MIMO) If–Then rule(s) are of the following form:

R = \cup_{l = 1}^{M} R_{l}

(12)

Here

$R_{l}$ : If $x_{1}$ is $A_{1}^{l}$ , $x_{2}$ is $A_{2}^{l}$ ,…, $x_{n}$ is $A_{n}^{l}$ , then $y_{1}$ is $B_{1}^{l}$ ,…, $y_{m}$ is $B_{m}^{l}$

\begin{matrix} U = U_{1} \times U_{2} \times \cdot \cdot \cdot \times U_{n}, & U_{i} \in R \\ V = V_{1} \times V_{2} \times \cdot \cdot \cdot \times V_{m}, & V_{i} \in R \end{matrix}

where M represents the total number of rules; $x = (x_{1}, \cdot \cdot \cdot, x_{n})^{T}$ and $y = (y_{1}, \cdot \cdot \cdot, y_{m})^{T}$ represents the input and output vectors of the fuzzy system, respectively. $A_{i}^{l}$ and $B_{j}^{l}$ are the linguistic variables of the fuzzy sets in the subspace $U_{i}$ and $V_{j}$ , described by their membership functions $μ_{A_{i}^{l}} (x_{i})$ and $μ_{B_{j}^{l}} (y_{j})$ , l = 1, 2, …, M.

Definition 1. The output of a MIMO-FLS with centre-average defuzzifier, product inference and singleton fuzzifier is of the following form:

y_{j} = \frac{\sum_{l = 1}^{M} {\bar{y}}_{j}^{l} (Π_{j = 1}^{n} μ_{A_{i}^{l}} (x_{i}))}{\sum_{l = 1}^{M} (Π_{i = 1}^{n} μ_{A_{i}^{l}} (x_{i}))}, j = 1, 2, . . ., m

(13)

Here ${\bar{y}}_{j}^{l}$ is the point in $V_{j}$ at which $μ_{B_{j}^{l}} (y_{j})$ achieves its maximum value 1.

To maintain a consistent performance of the fuzzy system in situations where there is a large uncertainty or unknown variation in plant parameters and structures, the fuzzy system should be adaptive (Yoo and Ham, 2000). Therefore, Equation (13) can be rewritten as

{\begin{matrix} y_{j} = \sum_{l = 1}^{M} {\bar{y}}_{j}^{l} ξ (x) = Θ_{j}^{T} ξ (x), j = 1, 2, . . ., m \\ ξ (x)_{l} = \frac{Π_{i = 1}^{n} μ_{A_{i}^{l}} (x_{i})}{\sum_{l = 1}^{M} (Π_{i = 1}^{n} μ_{A_{i}^{l}} (x_{i}))}, l = 1, 2, . . ., M \end{matrix}

(14)

where $ξ (x)_{l} = (ξ_{1} (x)_{l}, . . ., ξ_{M} (x)_{l})^{T} \in R^{M}$ is called the fuzzy basis function vector and $Θ_{j} = ({\bar{y}}_{j}^{1}, . . ., {\bar{y}}_{j}^{M})^{T} \in R^{M}$ is called the parameter vector.

It has been proved that the adaptive fuzzy system can approximate any real continuous function over a compact set to arbitrary accuracy.

Lemma 1 (Yoo and Ham, 2000). Let $f (x)$ be a continuous function defined on a compact set ∂. Then for any constant scalar $k > 0$ , there exist a fuzzy logic system and ideal parameter vector $Θ^{*}$ such that

{sup}_{x \in \partial} | f (x) - Θ^{* T} ξ (x)_{l} | \leq k

(15)

Sliding mode control (integer-order)

The concept of SMC is based on variable structure control theory. Classical SMC with a linear error manifold is robust to matched disturbances and uncertainties, and is commonly used for both linear and non-linear systems (Utkin and Chang, 2002). Classical SMC exhibits chattering phenomena due to which its practical implementation is hard to realize. To reduce chattering, several methods have been proposed, such as disturbance observer-based SMC (Kawamura et al., 1992), adaptive fuzzy sliding mode (Ho et al., 2009) and continuous sliding mode (Young et al., 1999). A simple and effective method for chattering reduction is to replace the discontinuous $sgn (.)$ function by a $sat (.)$ function, but the steady-state error does not converge to zero (Seshagiri and Khalil, 2005). To preserve the robustness, Bartolini et al. (2000) proposed a second-order SMC method. Due to the integral action, steady-state error is zero but at the cost of degradation of transient performance. Seshagiri and Khalil (2005) proposed a conditional integral SMC method to recover the transient performance. Integral SMC with a fractional-order reaching law and its application to quadrotor are discussed in Mehmet (2011).

Integer-order sliding mode control (SMC)

A classical sliding surface $s$ is given as:

s = c_{1} e_{1} + e_{2}

(16)

Here $c_{1} > 0$ is the design parameter. Differentiating Equation (16), one can obtain

\overset{\cdot}{s} = c_{1} e_{2} + {\overset{\cdot}{e}}_{2}

(17)

Combining Equations (8), (11) and (17),

\overset{\cdot}{s} = c_{1} e_{2} + f_{n} (X, t) + g_{n} (X, t) u + L (X, t) - {\overset{\cdot\cdot}{x}}_{d}

(18)

To derive the equivalent control set $\overset{\cdot}{s} = 0$ , the desired response can be achieved by choosing the control law as

{\begin{matrix} u = u_{eq} + u_{s} \\ u_{eq} = g_{n} (X, t)^{- 1} [{\overset{\cdot\cdot}{x}}_{d} - f_{n} (X, t) - c_{1} e_{2}] \\ u_{s} = - g_{n} (X, t)^{- 1} k_{s} sgn (s) \end{matrix}

(19)

where $u_{eq}$ is the equivalent control item to drive the nominal part of system, $u_{s}$ is the robust switching control item to handle the lumped uncertainty $L (X, t)$ and a sliding condition will occur if $k_{s} > L (X, t)_{max}$ . Here $L (X, t)_{max}$ represents upper bound of lumped uncertainty and $sgn (•)$ is the signum function defined as

sgn (s) = {\begin{matrix} + 1 & if s > 0 \\ 0 & if s = 0 \\ - 1 & if s < 0 \end{matrix}

(20)

The Lyapunov function is

{\begin{matrix} V = \frac{1}{2} s^{2} \\ \overset{\cdot}{V} = s \overset{\cdot}{s} \end{matrix}}

(21)

Combining Equations (18), (19) and (21), it can be shown that $\overset{\cdot}{V} \leq 0$ provided that switching gain $k_{s} > | L (X, t)_{max} |$

\begin{matrix} \overset{\cdot}{V} & = s \overset{\cdot}{s} = s (L (X, t) - k_{s} sgn (s)) \\ \leq (s | L (X, t) |_{max} - k_{s} | s |) \leq 0 \end{matrix}

(22)

Integral sliding mode control (ISMC)

The integral sliding manifold is defined as

s = c_{0} \int e_{1} + c_{1} e_{1} + e_{2}

(23)

Here $[c_{1}, c_{0}]$ are the design parameters. Differentiating Equation (23), one obtains

\overset{\cdot}{s} = c_{0} e_{1} + c_{1} e_{2} + {\overset{\cdot}{e}}_{2}

(24)

Combining Equations (8), (11) and (24),

\overset{\cdot}{s} = c_{0} e_{1} + c_{1} e_{2} + f_{n} (X, t) + g_{n} (X, t) u + L (X, t) - {\overset{\cdot\cdot}{x}}_{d}

(25)

The desired response can be achieved by choosing control law as

{\begin{matrix} u = u_{eq} + u_{s} \\ u_{eq} = g_{n} (X, t)^{- 1} [{\overset{\cdot\cdot}{x}}_{d} - f_{n} (X, t) - c_{0} e_{1} - c_{1} e_{2}] \\ u_{s} = - g_{n} (X, t)^{- 1} k_{si} sgn (s) \end{matrix}

(26)

Combining Equations (21), (25) and (26), and letting $k_{si} > | L (X, t)_{max} |$ , it can be shown that

\overset{\cdot}{V} = s \overset{\cdot}{s} = s (L (X, t) - k_{si} sgn (s)) \leq s | L (X, t)_{max} | - k_{si} | s | \leq 0

(27)

Fractional-order sliding mode control (FOSMC)

In this section, fuzzy SMC with fractional dynamics is derived for trajectory tracking of a class of second-order uncertain dynamical systems. The fractional-order sliding manifold $s$ is defined as:

{\begin{matrix} s_{p} = c_{1} e_{1} + D^{1 - α} e_{2} \\ s = c_{3} D^{- α} s_{p} + s_{p} = c_{1} c_{3} D^{- α} e_{1} + c_{1} e_{1} + c_{3} D^{- 2 α} {\overset{\cdot}{e}}_{2} + D^{1 - α} e_{2} \end{matrix}

(28)

where $α$ represents the order of the fractional operator $α \in (0, 1)$ and $c_{3}$ is the positive parameter. Apply $D^{α}$ to Equation (28), one obtains

D^{α} s = c_{1} c_{3} e_{1} + c_{1} D^{α} e_{1} + c_{3} D^{1 - α} e_{2} + {\overset{\cdot}{e}}_{2}

(29)

Combining Equations (8), (11) and (29),

\begin{array}{l} D^{α} s = c_{1} c_{3} e_{1} + c_{1} D^{α} e_{1} + c_{3} D^{1 - α} e_{2} + f_{n} (X, t) \\ + g_{n} (X, t) u + L (X, t) - {\overset{\cdot\cdot}{x}}_{d} \end{array}

(30)

The proposed control can be obtained as

{\begin{matrix} u = u_{eq} + u_{s} \\ u_{eq} = g_{n} (X, t)^{- 1} ({\overset{\cdot\cdot}{x}}_{d} - f_{n} (X, t) - c_{1} c_{3} e_{1} - c_{1} D^{α} e_{1} - c_{3} D^{1 - α} e_{2}) \\ u_{s} = - g_{n} (X, t)^{- 1} k_{sa} sgn (s) \end{matrix}

(31)

Here $k_{sa}$ represents gain of discontinuous control $u_{s}$ .

Theorem 2. Considering system (8) satisfying Assumption 1, the proposed fractional-order SMC (31) will stabilize the closed-loop system and the tracking error converges to zero.

The following inequality holds (Aghababa, 2014):

| \sum_{j = 1}^{\infty} \frac{Γ (1 + α)}{Γ (1 - j + α) Γ (1 + j)} D^{j} {sD}^{α - j} s | \leq Ω | s |

(32)

Here $Ω$ is a positive constant.

Proof. Applying $D^{α}$ to Equation (21) (Aghababa, 2014)

D^{α} V = {sD}^{α} s + | \sum_{j = 1}^{\infty} \frac{Γ (1 + α)}{Γ (1 - j + α) Γ (1 + j)} D^{j} {sD}^{α - j} s |

(33)

Apply the concept of Equation (32) to Equation (33):

D^{α} V \leq {sD}^{α} s + Ω | s |

(34)

Combining Equations (30), (31) and (34), we have

\begin{array}{l} D^{α} V \leq s [c_{1} c_{3} e_{1} + c_{1} D^{α} e_{1} + c_{3} D^{1 - α} e_{2} + f_{n} (X, t) + g_{n} (X, t) u + L (X, t) - {\overset{\cdot\cdot}{x}}_{d}] + Ω | s | \\ \leq s [c_{1} c_{3} e_{1} + c_{1} D^{α} e_{1} + c_{3} D^{1 - α} e_{2} + f_{n} (X, t) + g_{n} (X, t) (u_{s} + u_{e q}) + L (X, t) - {\overset{\cdot\cdot}{x}}_{d}] + Ω | s | \\ \leq s [L (X, t) - k_{s a} sgn (s)] + Ω | s | \\ \leq [| L {(X, t)}_{\max} | s - k_{s a} | s |] + Ω | s | < 0 \end{array}

(35)

By letting $k_{sa} > | L (X, t)_{max} |$ , $D^{α} V < 0$ , which means that reaching condition of sliding surface is satisfied and $s = 0$ . At this stage, two proofs are necessary. In first proof, Equation (31) with $s = 0$ is analogous to a fractional-order system of the form $D^{α} x = Ax, x \in R^{n}$ , so its stability is proved using the following theorem.

Theorem 3 (Sabatier et al., 2010). A fractional-order system (28) with $s = 0$ (outcome of Theorem 2) of order $α \to [0 < α < 1]$ is stable if and only if there exists a matrix $P = P^{T} > 0, P \in R^{n * n}$ such that:

(Q)^{\frac{1}{α}})^{T} P + P (Q)^{\frac{1}{α}}) < 0

(36)

Proof. To prove Theorem 3, letting $s = 0$ in Equation (28) yields

D^{α} s_{p} = {Qs}_{p}

(37)

Here $Q \in c_{3}; c_{3} = [nx 1]$ . Now the fractional-order system of Equation (37) with order $α$ is written in the following form (Sabatier et al., 2010):

\overset{\cdot}{h} = {hA}_{f}, A_{f} \in {A_{f}, \cdot \cdot \cdot \cdot \cdot \cdot \cdot A_{f L}}

(38)

h = C_{f} s_{p}

(39)

Here

A_{f} = [\begin{matrix} 0 & \dots & 0 & Q^{\frac{1}{α}} \\ Q^{\frac{1}{α}} & \dots & 0 & 0 \\ \dots & 0 & Q^{\frac{1}{α}} & 0 \end{matrix}]; C_{f} = [\begin{matrix} 0 & \dots & 0 & 1 \end{matrix}]; A_{f} \in A^{nxn}

Alternately, Equation (38) is written as

\overset{\cdot}{h} = A_{f} h, A_{f} = \sum_{i = 1}^{L} λ_{i} Q_{i}; \forall λ_{i} \geq 0; \sum_{i = 1}^{L} λ_{i} = 1

(40)

Therefore setting a positive definite matrix $P$ and based on the LMI method, system (37) with order $α$ is stable if and only

{\begin{matrix} A_{f}^{T} P + A_{f} P < 0 \\ \sum_{i = 1}^{L} λ_{i} (A_{f}^{T} P + A_{f} P) < 0 \\ (A_{f}^{T} P + A_{f} P) < 0 \end{matrix}}

(41)

The third row of Equation (41) is satisfied if

(A^{\frac{1}{α}})^{T} P + A^{\frac{1}{α}} P < 0

(42)

Equation (42) is a sufficient condition with a positive definite matrix $P$ . The necessary condition is represented by Equation (36).

The second proof deals with the convergence property of the fractional-order sliding surface. The following Lemmas are defined as:

Lemma 2. If integral of fractional derivative ${{}_{a}D}_{t}^{α}$ of a function $f (t)$ exits, then (Aghababa, 2014)

{{}_{a}D}_{t}^{- α} ({{}_{a}D}_{t}^{α} f (t)) = f (t) - \sum_{J = 1}^{k} {[{{}_{a}D}_{t}^{α - J} f (t)]}_{t = a} \frac{(t - a)^{α - J}}{Γ (α - J + 1)}

(43)

Here $k - 1 \leq α < k$ .

Lemma 3. The fractional integral operator ${{}_{a}D}_{t}^{- α}$ with $α > 0$ is bounded such that

| | {{}_{a}D}_{t}^{- α} (f) | |_{p} \leq K | | f | |_{p}; 1 \leq p \leq \infty; 1 \leq K \leq \infty

(44)

With $s = 0$ , Equation (37) can be rewritten as

s_{p} = - c_{3} D^{- α} s_{p}

(45)

Now $s_{p} = D^{- 2} D^{2} (s_{p})$ , so Equation (45) is re written as

D^{- 2} D^{2} (s_{p}) = - c_{3} D^{- α} s_{p}

(46)

Using Lemma 2, Equation (46) can be written as

s_{p} (t) - [{{}_{t_{r}}D}_{t}^{2 - 1} s_{p}]_{t = t_{r}} \frac{(t - t_{r})^{2 - 1}}{2} - s_{p} (t_{r}) = - c_{3} D^{- α} s_{p}

(47)

Using Lemma 3, the right-hand side of Equation (47) is written as

- c_{3} D^{- α} s_{p} = - c_{3} k_{\partial} | | s_{p} | |

(48)

Here $k_{\partial}$ is a positive constant. Combining Equations (47) and (48),

| | s_{p} (t) - [{{}_{t_{r}}D}_{t}^{1} s_{p}]_{t = t_{r}} \frac{(t - t_{r})}{2} | | - | | s_{p} (t_{r}) | | \leq - c_{3} k_{\partial} | | s_{p} (t) | |

(49)

Let us suppose that the sliding surface $s_{p} (t = t_{s}) = 0$ ; then it is necessary to prove that $t_{r} \leq t_{s} < \infty$ . Equation (49) is written as

| | [{{}_{t_{r}}D}_{t}^{1} s_{p}]_{t = t_{r}} (t_{s} - t_{r}) | | \leq 2 | | s_{p} (t_{r}) | |

(50)

Simplifying Equation (50) as follows:

t_{r} \leq - \frac{2 | | s_{p} (t_{r}) | |}{| | {\overset{\cdot}{s}}_{p} | |_{t = t_{r}}} + t_{s}

(51)

From Equation (51), it is proved that sliding surface converges to zero in finite time. Theorem 1 can also be used to prove the stability and convergence property as the following.

Fuzzy fractional-order sliding mode control (FFOSMC)

As the control accuracy is influenced by the lumped uncertainty $L (X, t)$ , we propose the adaptive fuzzy compensator $\hat{L} (X, t)$ to handle the lumped uncertainty $L (X, t)$ in the above fractional sliding mode surface. The fuzzy compensator $\hat{L} (X, t)$ is given by (Wang, 1997):

\hat{L} (X | Θ_{g}) = \frac{\sum_{l_{1} = 1}^{k} \sum_{l_{2} = 1}^{k} ({\bar{y}}_{h}^{l_{1} l_{2}} μ_{A_{1}^{l_{1}}} (x_{1}) μ_{A_{2}^{l_{2}}} (x_{2}))}{\sum_{l_{1} = 1}^{k} \sum_{l_{2} = 1}^{k} (μ_{A_{1}^{l_{1}}} (x_{1}) μ_{A_{2}^{l_{2}}} (x_{2}))} = Θ_{g}^{T} ξ_{g} (x_{1}, x_{2})

(52)

where $Θ_{g} = {({\bar{y}}_{g}^{1}, . . ., {\bar{y}}_{g}^{k^{2}})}^{T}$ , $ξ_{g} = {(ξ_{g}^{1}, . . ., ξ_{g}^{k^{2}})}^{T}$ , $ξ_{g}^{i} = \frac{μ_{A_{1}^{l_{1}}} (p_{f}) μ_{A_{2}^{l_{2}}} (x_{1})}{\sum_{l_{1} = 1}^{k} \sum_{l_{2} = 1}^{k} (μ_{A_{1}^{l_{1}}} (p_{f}) μ_{A_{2}^{l_{2}}} (x_{1}))}$ , $i = 1, . . ., k^{2}$ . $A_{1}^{l_{1}}$ and $A_{2}^{l_{2}}$ are fuzzy sets of $x_{1}$ and $x_{2}$ , respectively. $μ_{A_{1}^{l_{1}}} (x_{1})$ and $μ_{A_{2}^{l_{2}}} (x_{2})$ are the membership functions of $x_{1}$ and $x_{2}$ , respectively.

The column vector of the optimal parameter $Θ_{g}^{*}$ can be defined as follows:

Θ_{g}^{*} = \arg min_{Θ_{g} \in Ω_{g}} [sup_{X \in Ω_{X}} | \hat{L} (X_{1} | Θ_{g}) - \hat{L} (X, t) |]

(53)

where $Ω_{g}$ and $Ω_{X}$ are the sets of $Θ_{g}$ and $X$ . With the optimal parameter vector of the FLS, the minimum approximation errors can be defined as

w_{g} = L (X, t) - \hat{L} (X {| Θ}^{*})

(54)

Here, $‖ w_{g} ‖ \leq W_{g}$ , $W_{g}$ is a positive parameter. From Equations (52), (53) and (54), one obtains

L (X, t) - \hat{L} (X | Θ) = Θ_{g}^{* T} ξ_{g} (X) + w_{g} - Θ_{g}^{T} ξ_{g} (X) = {\tilde{Θ}}_{g}^{T} ξ_{g} (X) + w_{g}

(55)

Using similar formulations as given previously, the proposed control is derived as:

{\begin{matrix} u = u_{eq} + u_{s} \\ u_{eq} = g_{n} (X, t)^{- 1} [{\overset{\cdot\cdot}{x}}_{d} - f_{n} (X, t) - \hat{L} (X | Θ) - c_{1} c_{3} e_{1} - c_{1} D^{α} e_{1} - c_{3} D^{1 - α} e_{2}] \\ u_{s} = - g_{n} (X, t)^{- 1} [k_{saf} sgn (s)] \end{matrix}

(56)

where $k_{saf} > w_{g}$ , and there is no doubt that $k_{saf} < k_{sa}; k_{saf} < k_{s}; k_{saf} < k_{si}$ and controller (56) with fuzzy compensator only requires $k_{s} > w_{g}$ , so the application of fuzzy compensator can reduce the gain of the switching function.

Theorem 4. Considering the system of Equation (8) satisfying Assumption 1, the proposed control law (56) guarantees that all system signals are bounded under closed-loop operation, and asymptotic output tracking is also achieved, i.e. e₁→ 0 as t→∞.

Proof. Choose a Lyapunov function candidate to be

V_{s} = s^{2} + \frac{1}{γ_{1}} {\tilde{Θ}}_{g}^{2}

(57)

where $γ_{1} > 0$ and ${\tilde{Θ}}_{g} = Θ_{g}^{*} - Θ_{g}$ . Applying $D^{α}$ to Equation (57):

\begin{array}{l} D^{α} V_{s} = s D^{α} s + | \sum_{j = 1}^{\infty} \frac{Γ (1 + α)}{Γ (1 - j + α) Γ (1 + j)} D^{j} s D^{α - j} s | \\ + \frac{1}{γ_{1}} {\tilde{Θ}}_{g} D^{α} {\tilde{Θ}}_{g} + | \sum_{j = 1}^{\infty} \frac{Γ (1 + α)}{Γ (1 - j + α) Γ (1 + j)} D^{j} {\tilde{Θ}}_{g} D^{α - j} {\tilde{Θ}}_{g} | \end{array}

(58)

Apply the concepts presented in Equation (32) to Equation (58):

D^{α} V_{s} \leq {sD}^{α} s + Ω_{1} s + \frac{1}{γ_{1}} {\tilde{Θ}}_{g} D^{α} {\tilde{Θ}}_{g} + Ω_{2} {\tilde{Θ}}_{g}

(59)

Combining Equations (30), (56) and (59), one obtains:

\begin{matrix} D^{α} V & \leq s [L (X, t) - \hat{L} (X | Θ) - k_{saf} sgn (s)] \\ + Ω_{1} s + \frac{1}{γ_{1}} {\tilde{Θ}}_{g} D^{α} {\tilde{Θ}}_{g} + Ω_{2} {\tilde{Θ}}_{g} \\ \leq s [{\tilde{Θ}}_{g} ξ_{g} (X) + w_{g} - k_{saf} sgn (s)] + Ω_{1} s \\ + \frac{1}{γ_{1}} {\tilde{Θ}}_{g} D^{α} {\tilde{Θ}}_{g} + Ω_{2} {\tilde{Θ}}_{g} \\ \leq - k_{saf} | s | + w_{g} s + {\tilde{Θ}}_{g}^{T} (\frac{1}{γ_{1}} D^{α} {\tilde{Θ}}_{g} + s ξ_{g} (X)) \\ + Ω_{1} s + Ω_{2} {\tilde{Θ}}_{g} \end{matrix}

(60)

From Equation (60), choose the adaptive law as $D^{α} {\tilde{Θ}}_{g} = - γ_{1} ξ_{g} (X) s$ , so that

D^{α} V = - k_{sa f} | s | + w_{g} s + + Ω_{1} s + Ω_{2} {\tilde{Θ}}_{g}

(61)

Therefore, with the necessary condition $k_{saf} > w_{g} + Ω_{1}$ , the sliding surface s will converge to zero and the sliding condition is achieved. The convergence proof is the same as shown in the above section, using Theorem 3. The proposed controller is shown in Figure 2.

Figure 2.

Proposed control scheme.

Numerical results and discussions

In this section, the efficiency of the proposed control method is verified using numerical simulations. A permanent magnet synchronous motor (PMSM) is widely used in industry due to its high power density, high torque to inertia and control in a DC motor (Shyu et al., 2002). Controlling this system has practical importance. Assuming that the stator windings generate a sinusoidal magnetic field, its mathematical model is written as:

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ J {\overset{\cdot}{x}}_{2} = - {Bx}_{2} + P ψ_{f} u + L (X, t) \end{matrix}

(62)

Here the state variables x₁ and x₂ are the angle and velocity of the PMSM system, u represents the control signal and L(X,t) contains the lumped uncertainty and external disturbance. B is the viscous friction coefficient, J is the motor moment inertia constant, P is number of pole pairs and Ψ_f is the flux linkage of the permanent magnet. The nominal valves of system parameters are defined as $B = 1.0 \times 10^{- 4} {N . m . s . rad}^{- 1}$ , $J = 1.02 \times 10^{- 3} {kg . m}^{2}$ , $P = 4$ and $ψ_{f} = 0.214 Wb$ .

To minimize high-frequency chattering, a discontinuous sign function is replaced by a saturation function, which is defined as

sat (s) = {\begin{matrix} + 1 & s > ϕ, \\ s / ϕ & | s | \leq ϕ, \\ - 1 & s < - ϕ . \end{matrix}

(63)

Here $ϕ$ is the boundary layer width. Figure 3 shows controller tuning using Simulink design optimization for the step response. The optimized controller parameters are $f_{n} = B_{P} / J = 0.098$ , $g_{n} = P ψ_{f} / J = 839$ , $c_{1} = 8$ , $c_{3} = 10$ , $k_{s} = 0.05$ , $ϕ = 0.01$ .

Figure 3.

Controller tuning using unit step response optimization.

Practically, it is necessary to solve fractional-order systems numerically; however, it is a tedious job to obtain the exact solution. So approximation is used to realize a fractional differential equation. A popular frequency approximation method is presented by Oustaloup et al. (1996) and is given as:

s^{α} \approx K Π_{i = - N}^{N} \frac{1 + (s / ω_{z, i})}{1 + (s / ω_{p, i})}, α > 0

(64)

K represents gain. For Equation (64) to have a unit gain at 1 rad/s, ω_z,i and ω_p,i are given as

{\begin{matrix} ω_{p, i} = ω_{b} {(\frac{ω_{h}}{ω_{b}})}^{(i + N + (1 / 2) (1 + α) / (2 N + 1))} \\ ω_{z, i} = ω_{b} {(\frac{ω_{h}}{ω_{b}})}^{(i + N + (1 / 2) (1 - α) / (2 N + 1))}, K = ω_{h}^{α} \end{matrix}

(65)

In Equation (65), ω_b and ω_h represent the lower and upper frequency bounds and usually ω_bω_h = 1. For the case α < 0, Equation (65) is inverted. For $| α | > 1$ , the approximation is not satisfactory, so fractional powers of s are split as:

s^{α} = s^{n} s^{δ}, α = n + δ, n \in {mathZ}, δ \in [0, 1]

(66)

Step tracking performance comparison

The reference command is a step angle with θ_ref = 1 rad, and the order of FOSMC and FFOSMC is $α = 0.5$ . The unit step response comparisons of the system for a different fractional-order of FFOSMC controller are shown in Figure 4.

Figure 4.

Step response comparison with different fractional order.

From Figure 4, it is concluded that as the fractional order of the FFOSMC varies from 0.2 to 0.99, the rise and settling times of the system improve significantly. At the same time, the system shows overshoot as the fractional order is increased above 0.6. So for the rest of analysis, the fractional order of the FFOSMC is selected as 0.5.

From Figure 5, it is concluded that the proposed FOSMC and FFOSMC methods track the reference command with the smallest rise time, peak overshoot and settling time, compared with SMC and ISMC. It is concluded that the angle-tracking errors of the proposed methods are the lowest.

Figure 5.

Unit step response comparison.

Control input comparisons are given in Figure 6. From simulation results, it is concluded that the control inputs of both FOSMC and FFOSMC are large in the start of the cycle and then show a sharp decreasing trend, while the control inputs for SMC and ISMC are numerically small for t > 0.1 s. Although the control input of the proposed controller is numerically large in the start of cycle, the large peak exists for a very small time. So the average control effort of FFOSMC is comparable with SMC and ISMC, and far below the saturation limit of the drive.

Figure 6.

Control input comparison.

Robust performance comparison with parametric uncertainties and unmodelled dynamics

In this case, the lumped uncertainty $L (X, t)$ only contains the uncertainty term, which is given by

{\begin{matrix} L (X, t) = Δ f (X) = Δ f_{1} (X) + Δ f_{2} (X) \\ Δ f_{1} (X) = 10 x_{2} \\ Δ f_{2} (X) = 5 x_{1}^{2} \end{matrix}

(67)

Here $Δ f_{1} (X, t)$ are the parametric uncertainties and $Δ f_{2} (X, t)$ represents the unmodelled dynamics. In this case, in order better to compare the robust performance of different controllers, the boundary of $L (X, t)$ is larger than the gain of switching/saturation function. The reference command is a sinusoidal trajectory with $θ_{ref} = \sin 5 t$ , $[x_{10} x_{20}] = [0 5]$ and the order of FOSMC is $α = 0.5$ . Figure 7 is the angle tracking response of the PMSM system with different controllers. The tracking errors of the angle and velocity are given in Figures 8 and 9, respectively.

Figure 7.

The time response of the permanent magnet synchronous motor (PMSM) system for tracking the desired trajectory.

Figure 8.

Tracking errors of angle.

Figure 9.

Tracking errors of velocity.

It is shown that tracking error of FOSMC is smaller compared with conventional SMC and ISMC. Because in this case the gain of switching/saturation function is less than boundary of $L (X, t)$ , the tracking errors of above three controllers do not converge to zero/ $ϕ$ . In comparison, the tracking error of FFOSMC almost converges to zero due to the fuzzy compensator.

The lumped uncertainty $L (X, t)$ and output of fuzzy compensator $\hat{L} (X)$ are shown in Figure 10.

Figure 10.

The lumped uncertainty $L (X, t)$ and the output of fuzzy compensator $\hat{L} (X)$ .

It can be found that the output of fuzzy compensator $\hat{L} (X)$ can approximate the lumped uncertainty $L (X, t)$ .

In the second case, the lumped uncertainty $L (X, t)$ contains the uncertainty term, which is also a function of the control input given as

{\begin{matrix} L (X, t) = Δ f (X, t) + Δ g (X, t) u \\ Δ f_{1} (X) = 10 x_{2} \\ Δ f_{2} (X) = 5 x_{1}^{2} \\ Δ g (X, t) = 100 u \end{matrix}

(68)

The reference command is a sinusoidal trajectory with $θ_{ref} = \sin 5 t$ , $[x_{10} x_{20}] = [0 5]$ and the order of the FOSMC is $α = 0.5$ . Figure 11 is the angle tracking response of the PMSM system with different controllers. The tracking errors of angle and velocity are given in Figures 12 and 13, respectively. From the given results, it is concluded that, compared with SMC, ISMC and FOSMC, the proposed controller is more robust to the lumped uncertainty given in Equation (68) and it ensures the lowest tracking angle and velocity errors.

Figure 11.

Tracking the desired trajectory with lumped uncertainty.

Figure 12.

Tracking errors of angle.

Figure 13.

Tracking errors of velocity.

Performance comparison with adaptive PID

This section presents a performance comparison of FFOSMC with an adaptive PID method. The reference command is a sinusoidal trajectory with $θ_{ref} = \sin 5 t$ , $[x_{10} x_{20}] = [0 5]$ and the order of the FOSMC is $α = 0.5$ . The parameters of the PID controller are selected as $K_{p} = 0.28, K_{i} = 0.15, K_{d} = 0.02$ . An adaptive law is derived as $\hat{L} = - Γ e_{1}$ , where $Γ$ represents the learning rate. The system is subjected to lumped uncertainty of Equation (67). The simulation results are presented for $Γ = 100$ and 350. Figure 14 is the angle tracking response comparison of the PMSM system. For this case, $Γ = 100$ . Compared with the proposed control method, the angle error in the case of adaptive PID is 0.1 rad. Comparative analysis of estimated lumped uncertainty is given in Figure 15.

Figure 14.

Tracking errors of angle.

Figure 15.

Estimated lumped uncertainty.

If the learning rate of the adaptive PID $Γ$ is increased from 100 to 350, then the angle-tracking error is significantly reduced but at the cost of high-frequency oscillations in the estimation loop. The simulation results are given in Figures 16 and 17.

Figure 16.

Tracking error of angle.

Figure 17.

Estimated lumped uncertainty comparisons.

Robust performance comparison with external disturbance

In this simulation, the lumped uncertainty $L (X, t)$ only contains the external load disturbance, which is given by $L (X, t) = d (X, t) = 39.2 \sin (2 t)$ . The position reference is a zero trajectory with $θ_{ref} = 0$ , $[x_{1} x_{2}] = [0 0]$ and the order of the FOSMC is $α = 0.5$ . The tracking errors of the angle and velocity are given in Figures 18 and 19, respectively.

Figure 18.

Tracking errors of angle.

Figure 19.

Tracking errors of velocity.

It is concluded that the tracking error of FOSMC and FFOSMC are smaller compared with conventional SMC and ISMC, but the tracking error of FFOSMC is smallest, due to the fuzzy compensator. The lumped uncertainty $L (X, t)$ and the output of fuzzy compensator $\hat{L} (X)$ are shown in Figure 20.

Figure 20.

The lumped uncertainty $L (X, t)$ and the output of fuzzy compensator $\hat{L} (X)$ .

It can be found that output of fuzzy compensator $\hat{L} (X)$ can approximate the lumped uncertainty $L (X, t)$ .

Conclusions

A new, systematic design of the adaptive fractional-order sliding mode controller (FFOSMC) for PMSM, a position control system, is presented and system stability is guaranteed using a fractional-order Lyapunov theorem. By combining the advantages of the fractional PI^α sliding surface and PD^λ sliding surface, a novel fractional-order sliding surface is proposed. The parameters of the proposed controller are optimized for a step response using Simulink design optimization. The proposed controller has more degrees of freedom to achieve the desired performance. The control performance considering control precision and system robustness of the proposed FFOSMC method is better than that of the conventional SMC method and the integral SMC method. A fractional-order adaptive fuzzy logic controller is used to approximate uncertain non-linearities and parametric uncertainties online. The proposed controller is compared with an adaptive PID controller. From the analysis given in the research article, it is concluded that FFOSMC is more robust and the control signal is less corrupted by high-frequency oscillations. To ensure better robustness, the average control effort required by the FFOSMC is almost the same order compared with all other controllers used in the analysis. The analysis proves that the proposed FFOSMC is superior and feasible for practical implementations.

Footnotes

Conflict of interest

The authors declare that there is no conflict of interest.

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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