Discrete sliding mode control based on a new fractional order power reaching law

Abstract

This paper presents a robust controller design for a discrete linear system, dealing with internal and external disturbances. The main focus is to reduce chattering, minimize undesirable oscillations, and enhance robustness against disturbances using a new discrete sliding mode control. The novel approach involves replacing the classical Gao’s reaching law with a fractional order reaching law based on the Grunwald–Letnikov definition, leading to advantages like a simplified algorithm and improved performance against disturbances. The performances of the suggested method are evaluated through two simulation examples.

Keywords

discrete sliding mode control fractional order power reaching law stability analysis discrete-linear systems disturbances

1. Introduction

During the past years, sliding mode control has received significant attention and relevance due to its attractive merits (Yueheng et al., 2022c; Nan et al., 2021; Yueheng et al., 2022a). These merits include insensitivity to parameter uncertainties, simplicity of implementation, fast convergence, and robustness in the presence of external disturbances (Yueheng et al., 2022b; Znidi et al., 2022a; Mohamed et al., 2019; Martin et al., 2019). Several successful works have been developed in the literature for continuous-time systems. However, with the widespread use of computers in various fields, most of the designed controllers are now implemented on sampled-data systems.

Directly implementing continuous sliding mode control in discrete-time systems has proven to be impossible, due to certain properties that cannot be extended to their discrete-time counterparts. The main reason for these problems is the limited sampling rate in the discrete-time system. To overcome these issues, several discrete sliding mode control schemes have been proposed (Benyazid et al., 2018; Znidi et al., 2022b).

Among the efficient schemes presented, the reaching law method was developed by Gao et al. (1995). The corresponding reaching law consists of two terms: The first one is a proportional term, which ensures convergence to the sliding plane. The second one is a discontinuous term, which consists of a signum function. The “signum” term is generally used to ensure the switching around the sliding function.

The simplicity of implementation and the efficient use of such a method have attracted much attention from researchers. Since then, several switching, no switching, and power types of reaching laws, improving upon the classical methods, have been proposed (Bartoszewicz and Latosinski, 2018; Bartoszewicz and LatosińskiMa, 2016, 2017; Ma et al., 2017; Yazici and Yaylaci, 2017).

Starting with the first type, “Switching reaching laws,” which was first presented by Gao et al. (1995), these proposed types are based on three significant requirements for the sliding phase to be featured. The first initial point of the system should move monotonically around the sliding plane and pass it in finite time (Bartoszewicz and LatosińskiMa, 2016). Once the representative point crosses the switching plane, it will continue to cross it again in each next sampling instant without exceeding an a priori known bandwidth called the “quasi-sliding mode band.”

In contemporary literature, various methodologies have been proposed for sliding mode control. Some authors have replaced the “signum” function with a smoother function, “tangent,” to reduce oscillation amplitudes. Additionally, several switching reaching laws have been published, wherein the constant proportional term is replaced with more sophisticated expressions (Bartoszewicz A et al., 2016b). For instance, Ma et al. (2017) introduced a new switching reaching law using an exponential proportional term.

On the other hand, some authors refute the “switching types” due to the high-frequency oscillations caused by the switching requirement in the control signal (Bartoszewicz A et al., 2016b). To minimize control effort and optimize energy consumption in the control process, various non-switching methodologies have emerged. For example, Bartoszewicz and LatosińskiMa (2014) proposed replacing the constant convergence rate in the classical reaching law with a function of the sliding function. This approach has been successfully applied to control data flow in a communication network, preventing data buffer overflow.

Another modification of the classical reaching law was presented by Kurode S et al. (2011), where an exponential reaching law is applied to control the liquid oscillation damping in a reservoir.

The third type of reaching law is based on adopting a power function used to adjust the control parameters automatically. Authors have proposed replacing the gain of the discontinuous term with a power switching function. Haifeng et al. (2019b) developed a single power reaching law, which has been shown to tolerate the sign function and alleviate chattering phenomena. A similar approach was proposed in by Haifeng M et al. (2019a) focusing on the implementation of a multi-power reaching law. The developed strategy has been applied to control a piezo-motor system. Simulation results demonstrate that the multi-power method ensures better control performance (accuracy and robustness) than the single power reaching law.

It is commonly known that Fractional Order (FO) can be considered the general case of the integer order. Fractional order calculus possesses the property to model and control both integer order and fractional order systems with high accuracy and slight chattering. Therefore, combining the high control accuracy of FO calculus with the powerful advantages of discrete sliding mode control (DSMC) has improved the control performance compared to the integer case. To the best of our knowledge, discrete sliding mode control based on a fractional order power reaching law has been seldom investigated (Haifeng M et al., 2019b, Haifeng and Yangmin, 2020).

Motivated by this emerging research field, a new fractional order DSMC based on the discrete-time Grunwald–Letnikov definition has been proposed.

This innovative approach incorporates a power-rate reaching law into the control model, featuring a key innovation in the use of a hyperbolic tangent function. The hyperbolic tangent function allows real-time adjustment of the system’s convergence rate, offering significant advantages over a fixed-rate approach (Haifeng M et al., 2019b, Haifeng and Yangmin, 2020). This adaptability is particularly valuable when dealing with systems that exhibit changing dynamics over time, leading to smoother transitions, reduced oscillations, and minimized overshoot in system responses. Additionally, the hyperbolic tangent function’s non-linear characteristics make it ideal for handling complex systems with non-linear dynamics, ultimately improving system efficiency and robustness.

The rest of this paper is organized as follows: In Section 2, a new fractional order discrete sliding mode control is applied to a linear discrete system. Two cases are considered: In the first case, no external disturbances or parameter uncertainties are taken into account. The second case is dedicated to dealing with linear systems subject to disturbances. In both cases, a detailed robustness analysis is formulated to ensure the asymptotic convergence of the proposed controller. In Section 3, the stability analysis of the suggested method is demonstrated. The features of the proposed controller are evaluated through two simulation examples and compared to previously developed methods in Section 4. Finally, concluding remarks are provided in Section 5.

2. Discrete sliding mode control based on a new fractional order reaching law

This study focuses on designing a new discrete sliding mode controller for a discrete-time linear system. Therefore, two cases are considered: In the first case, no external disturbances and/or parameter uncertainties are taken into account. Then, the proposed quasi-sliding mode control will be extended to handle systems subject to internal (parameter uncertainties) and external disturbances.

2.1. Case 1: Discrete linear systems without disturbances

The system to be controlled in this case is written as an ideal system’s model:

X (k + 1) = A X (k) + B u (k)

(1)

where X(k) ∈ Rⁿ is the system’s states vector and u(k) ∈ R¹ denotes the control input vector.

The matrix A ∈ R^n×n and B ∈ R^n×1 are supposed to be known.

The sliding function is given by

S (k) = C^{T} X (k)

(2)

where C ∈ Rⁿ is selected in order that (C^TB) is invertible.

The classical Reaching law proposed by Gao et al. (1995) is expressed as

S (k + 1) = (1 - q) S (k) - M s i g n (S (k))

(3)

It is widely recognized that the choice of the controller gains q and M can influence the control performance. To further enhance control efficiency, we introduce an adaptive Fractional-Order (F-O) term Δ^α−1|S(k)|^1−φ(k) in the reaching law expression to adjust the control gain (1 − q). Additionally, a power function |S(k)|^1−φ(k) is adopted to tolerate the “sign” function, which helps alleviate chattering phenomena and reduce undesirable oscillation effects.

The new fractional order reaching law is defined as

\begin{array}{l} S (k + 1) = (1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} \\ - M {| S (k) |}^{1 - φ (k)} Δ^{α - 1} s i g n (S (k)) \end{array}

(4)

with 0 < q < 1; 0 < α < 1

M and q are the controls gains;

α is the fractional order;

Δ^α−1 is the fractional term calculated using Grunwald–Letnikov definition.

φ(k) = tanh (|S(k)|/S (1)) denotes the variable power rate based on a hyperbolic tangent function.

S (1) denotes the initial sliding variable value.

Compared to previous power-rate F-O reaching laws, this method introduces a novel approach by incorporating a “hyperbolic tangent function” into the power term φ(k) instead of using a constant power, as previously employed (Haifeng M et al., 2019b, Haifeng and Yangmin, 2020). This innovation enables faster convergence to the desired trajectory and results in a smaller discontinuous control gain (Katarzyna and Bartoszewicz, 2022). The key feature of this reaching law is its use of the hyperbolic tangent function, which effectively governs the rate of change of the sliding variable during the reaching phase. This approach intentionally maintains a slow convergence rate when the representative point is distant from the sliding surface, effectively reducing control input and conserving energy. However, as the representative point approaches the sliding surface, the convergence rate can be dynamically accelerated.

Remark:

Using the Grunwald–Letnikov definition,

Δ^{α} G (k) = \frac{1}{h^{α}} \sum_{j = 0}^{k} {(- 1)}^{j} (\begin{array}{l} α \\ j \end{array}) G (k - j h)

(5)

The FO control terms are calculated as (Haifeng M et al., 2019b, Haifeng and Yangmin, 2020)

Δ^{α - 1} {| S (k) |}^{1 - φ (k)} = \sum_{j = 0}^{l} {(- 1)}^{j} (\begin{array}{l} α - 1 \\ j \end{array}) {| S (k - j) |}^{1 - φ (k - j)}

(6)

Δ^{α - 1} s i g n (S (k)) = \sum_{j = 0}^{l} {(- 1)}^{j} (\begin{array}{l} α - 1 \\ j \end{array}) s i g n (S (k - j))

(7)

Using the new reaching law, the control law is expressed as

u (k) = {(C^{T} B)}^{- 1} [\begin{array}{l} - C^{T} A X (k) \\ + (1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} \\ - M {| S (k) |}^{1 - φ (k)} Δ^{α - 1} s i g n (S (k)) \end{array}]

(8)

According to Haifeng M et al. (2019b), Haifeng and Yangmin (2020), we denote

ξ = \max {{| S (k - j) |}^{1 - φ (k)} f o r j = 1,2,3, \dots \dots}

(9)

and

β = \sum_{j = 0}^{l} {(- 1)}^{j} (\begin{array}{l} α - 1 \\ j \end{array}) > 0

(10)

In the following, we assume that

δ = \frac{M}{1 - q} > 0

(11)

and

0 < M < \frac{δ}{β ξ}

(12)

Theorem 1

Considering the system described by (1), if the controller is designed as (8), and the assumptions (11) and (12) are respected, then the system satisfies the following conditions (Dehri and Nouri, 2021; Bartoszewicz, 1998):

• ∣S (k + 1)∣ < ∣S(k)∣ if ∣S(k)∣ > δ

• ∣S (k + 1)∣ < δ if ∣S(k)∣ ≤ δ

Which means that the sliding function converges to a quasi-sliding mode band (δ).

Proof

In order to prove the convergence of the sliding function to a quasi-sliding mode band, 4 cases are considered as follows:

• Case 1: S(k) > δ

S (k + 1) = (1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} - M {| S (k) |}^{1 - φ (k)} Δ^{α - 1} s i g n (S (k))

First,

−Mβξ ≤ −M|S(k)|^1−φ(k)Δ^α−1 sign (S(k)) ≤ 0

Second,

0 \leq (1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} \leq (1 - q) S (k) β ξ

Then,

- M β ξ \leq S (k + 1) \leq (1 - q) S (k) β ξ

(13)

- M β ξ - S (k) \leq S (k + 1) - S (k)

and

S (k + 1) - S (k) \leq ((1 - q) β ξ - 1) S (k)

using the assumptions (11) and (12):

M β ξ < δ

which means that

$\frac{M}{δ}$ βξ − 1 < 0

so we can write

(1 - q)βξ - 1 < 0 and S(k) > 0

Thus, S (k + 1) − S(k) < 0

Using equation (13)

- M β ξ + S (k) \leq S (k + 1) + S (k)

and

S (k + 1) + S (k) \leq ((1 - q) β ξ + 1) S (k)

we have

−Mβξ + δ ≤ −Mβξ + S(k)

Respecting the assumption (11) and (12),

−Mβξ + δ > 0 and S(k) > 0

we can write: S (k + 1) + S(k) > 0

Then

∣S(k + 1)∣ < ∣S(k)∣ if S(k) > δ

• Case 2: S(k) < − δ

S (k + 1) = (1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} - M {| S (k) |}^{1 - φ (k)} Δ^{α - 1} s i g n (S (k))

we have

0 ≤ −M|S(k)|^1−φ(k)Δ^α−1 sign(S(k)) ≤ Mβξ

Also

(1 - q) S (k) β ξ \leq (1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} \leq 0

Then

(1 - q) S (k) β ξ \leq S (k + 1) \leq M β ξ

(14)

Thus

((1 - q) β ξ - 1) S (k) \leq S (k + 1) - S (k)

and

S (k + 1) - S (k) \leq M β ξ - S (k)

Using equations (11) and (12)

$\frac{M}{δ}$ βξ − 1 < 0

so we can write

(1 − q)βξ − 1 < 0 and S(k) < 0

Then S (k + 1) − S(k) > 0

Considering (14), we can write

((1 - q) β ξ + 1) S (k) \leq S (k + 1) + S (k)

and

S (k + 1) + S (k) \leq M β ξ + S (k)

we have

M β ξ + S (k) \leq M β ξ - δ < 0

Thus

S (k + 1) + S(k) < 0

Then

∣S (k + 1)∣ < ∣S(k)∣ if S(k) < − δ

• Case 3: 0 ≤ S(k) ≤ δ

We have

−Mβξ ≤ S(k + 1) ≤ (1 − q)S(k)βξ

using equation (12), we can write

−δ ≤ −Mβξ

Thus, −δ < S(k + 1)

In other side, we have

S(k + 1) ≤ (1 − q)S(k)βξ

Replacing (1 − q) by its expression, we obtain

S(k + 1) ≤ (1 − q)S(k)βξ ≤ $\frac{M β ξ}{δ}$ S(k) < δ

Then

∣S (k + 1)∣ < δ if 0 ≤ S(k) ≤ δ

• Case 4: −δ ≤ S(k) ≤ 0

Using equation (14), we have

(1 − q)S(k)βξ ≤ S (k + 1) ≤ Mβξ

we have Mβξ < δ

Thus, S (k + 1) < δ

In other side

−(1 − q)δβξ ≤ (1 − q)S(k)βξ ≤ S(k + 1)we can write

−δ < − (1 − q)δβξ < S(k + 1)

Then

∣S (k + 1)∣ < δ if −δ ≤ S(k) ≤ 0

The verification of the aforementioned conditions proves that a quasi-sliding mode exists. Thus, it can be concluded that the proposed control law, given by equation (8), is stable.

Real and industrial systems are often subjected to numerous undesirable conditions, such as external disturbances and modeling uncertainties, which can significantly impact their performance. The objective of the following section is to mitigate the effects of these undesirable conditions on the system by using a robust controller.

2.2. Case 2: Disturbances consideration

In this part of the study, we consider the non-ideal case, which means that the system to be controlled is subject to disturbances.

The system’s model is described as follows:

X (k + 1) = A X (k) + B (u (k) + d (k))

(15)

with d(k) is the disturbance signal, which is assumed to be bounded.

We denote

D = \max {| C^{T} B d (k) |} > 0

(16)

In this case, the sliding function is expressed as

S (k + 1) = C^{T} A X (k) + C^{T} B u (k) + C^{T} B d (k)

(17)

Using the F-O reaching law given by equation (3), the control law is calculated as follows:

u (k) = {(C^{T} B)}^{- 1} [- C^{T} A X (k) + (1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} - M {| S (k) |}^{1 - φ (k)} Δ^{α - 1} s i g n (S (k))]

(18)

In the following, we assume that

0 < M < \frac{δ - D}{β ξ}

(19)

In order to ensure the efficiency of the proposed method in handling the impact of disturbances on the system and the controller’s performance, a detailed robustness analysis is provided.

Theorem 2

Given the system described by (15), the controller modeled by (18), and the assumption (19), the following conditions are satisfied:

• ∣S (k + 1)∣ < ∣S(k)∣ if ∣S(k)∣ > δ

• ∣S (k + 1)∣ < δ if ∣S(k)∣ ≤ δ

That means that the system’s trajectory converges to a quasi-sliding mode band (δ).

Proof

Replacing the control law by its expression in equation (18), we obtain

S (k + 1) = (1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} - M {| S (k) |}^{1 - φ (k)} Δ^{α - 1} s i g n (S (k)) + C^{T} B d (k)

(20)

In this proof, four cases are considered:

• Case 1: S(k) > δ

With δ is assumed to be equal to $\frac{M}{1 - q}$ .

S (k + 1) = (1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} - M {| S (k) |}^{1 - φ (k)} Δ^{α - 1} s i g n (S (k)) + C^{T} B d (k)

Using equation (13), we have

- M β ξ + D \leq S (k + 1) \leq (1 - q) S (k) β ξ + D

(21)

Thus,

- M β ξ - S (k) + D \leq S (k + 1) - S (k)

and

S (k + 1) - S (k) \leq ((1 - q) β ξ - 1) S (k) + D

Using assumption (19), we can write

M β ξ - δ + D < 0

We have

M β ξ - S (k) + D < M β ξ - δ + D < 0

Then, S(k + 1) − S(k) < 0

Using equations (13) and (21), we have

- M β ξ + S (k) + D \leq S (k + 1) + S (k)

and

S (k + 1) + S (k) \leq (1 - q) S (k) β ξ + S (k) + D

We have

- M β ξ + S (k) + D \geq - M β ξ + δ + D \geq 2 D > 0

That yields to S(k + 1) + S(k) > 0

Then

∣S(k + 1)∣< δ if 0 ≤ S(k) ≤ δ

• Case 2: S(k) < − δ

S (k + 1) = (1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} - M {| S (k) |}^{1 - φ (k)} Δ^{α - 1} s i g n (S (k)) + C^{T} B d (k)

Based on equation (14) and (20):

((1 - q) β ξ - 1) S (k) + D \leq S (k + 1) - S (k)

and

S (k + 1) - S (k) \leq M β ξ - S (k) + D

Respecting the above assumption (19):

M β ξ < δ - D < δ

which means that

\frac{M}{δ β ξ} - 1 < 0 and S (k) < 0

so S(k + 1) − S(k) > 0 Considering equations (14) and (20)

((1 - q) β ξ + 1) S (k) \leq S (k + 1) + S (k)

and

S (k + 1) + S (k) \leq M β ξ + S (k) + D

According to (19), we can write

M β ξ + S (k) + D \leq M β ξ - δ + D < 0

Thus,

S (k + 1) + S (k) < 0

Then

∣S (k + 1)∣ < ∣S(k)∣ if S(k) < − δ

• Case 3: 0 ≤ S(k) ≤ δ

Using equation (21), we can write

−δ < − Mβξ − D < − Mβξ + D

Then −δ < S(k + 1)

Replacing (1 − q) by its expression, we obtain

S(k + 1) ≤ Mβξ $\frac{S (k)}{δ}$ + D < δ

Thus

∣S (k + 1)∣ < δ if 0 ≤ S(k) ≤ δ

• Case 4: −δ ≤ S(k) ≤ 0

Based on equations (19) and (20):

((1 - q) β ξ) S (k) + D \leq S (k + 1) \leq M β ξ + D

we have Mβξ + D < δ

Then, S (k + 1) < δ

Using equation (19), we can write

- δ < - M β ξ - D \leq (1 - q) β ξ S (k) + D

we obtain −δ < S (k + 1) Then

∣S(k + 1)∣ < δ if −δ ≤ S(k) ≤ 0

Subsequently, we demonstrate that by selecting suitable parameters for the control law, our approach guarantees the presence of the quasi-sliding mode within finite number of steps.

Theorem 3

For the system equation (15), if the controller is designed as equation (18), the conditions equations (9) and (10) are respected and Mξβ − D > 0, then there exists a finite number k* = m such that the system trajectory will first get across the sliding surface at most k* + 1 steps, where m is represented as

m = \log_{((1 - q) β ξ)} [\frac{M ξ β - D}{| S_{0} | (1 - ((1 - q) ξ β)) + M ξ β - D}]

Proof

In the following, we prove the theorem by contradiction according to equation (20). Assume that there is an initial state, resulting in the initial point S₀ of the switching function, such that S₀, S₁, …, $S_{k^{*} + 1}$ do not change signs. There are two cases: S₀ > 0 and S₀ < 0.

Case 1:

S₀ > 0: Assume that S₀ > 0 and the system trajectories from the initial state do not cross the switching manifold in k* + 1 number of steps, that is, S_m are nonnegative for all m ≤ k* + 1.

S_{1} \leq (1 - q) ξ β S_{0} - (M ξ β - D)

S_{2} \leq (1 - q) ξ β S_{1} - (M ξ β - D) \leq {((1 - q) ξ β)}^{2} S_{0} - ((1 - q) ξ β) (M ξ β - D) - (M ξ β - D)

\begin{array}{l} . \\ . \\ . \end{array}

S_{m} \leq (1 - q) ξ β S_{m - 1} - (M ξ β - D) \leq {((1 - q) ξ β)}^{m} S_{0} - \sum_{i = 0}^{m - 1} {((1 - q) ξ β)}^{m - 1 - i} (M ξ β - D)

S_{m} \leq {((1 - q) ξ β)}^{m} | S_{0} | - \frac{(M ξ β - D) ({1 - ((1 - q) ξ β)}^{m})}{1 - ((1 - q) ξ β)}

S_{m} \leq \frac{{((1 - q) ξ β)}^{m} [| S_{0} | (1 - ((1 - q) ξ β)) + (M ξ β - D)] - M ξ β - D}{1 - ((1 - q) ξ β)}

We have 0 < q < 1, in this case, (1 − q) is a positive number less than 1. Raising it to a positive exponent m will make it smaller, approaching zero as m becomes larger. (1 − q)^m approaches 0 as m increases. It is straightforward to verify that the real number m satisfies:

\frac{{((1 - q) ξ β)}^{m} [| S_{0} | (1 - ((1 - q) ξ β)) + (M ξ β - D)] - M ξ β - D}{1 - ((1 - q) ξ β)} = 0

Which means that S_m ≤ 0, which contradicts the assumption that S_m > 0 is nonnegative for all m ≤ k* + 1.

Case 2:

S₀ < 0: Assume that S₀ < 0, and S_m is negative for all m ≤ k* + 1; similarly, it follows that:

According to equation (14), we have

S_{1} \geq (1 - q) ξ β S_{0} + (M ξ β + D)

S_{2} \geq (1 - q) ξ β S_{1} + (M ξ β + D) \geq {((1 - q) ξ β)}^{2} S_{0} + ((1 - q) ξ β) (M ξ β + D) + (M ξ β + D)

\begin{array}{l} . \\ . \\ . \end{array}

\begin{array}{l} S_{m} \geq (1 - q) ξ β S_{m - 1} + (M ξ β + D) \\ \geq {((1 - q) ξ β)}^{m} S_{0} + \sum_{i = 0}^{m - 1} {((1 - q) ξ β)}^{m - 1 - i} (M ξ β + D) \\ \geq {((1 - q) ξ β)}^{m} S_{0} + \sum_{i = 0}^{m - 1} {((1 - q) ξ β)}^{m - 1 - i} (M ξ β - D) \end{array}

S_{m} \geq - {((1 - q) ξ β)}^{m} | S_{0} | - \frac{(M ξ β - D) (1 - {((1 - q) ξ β)}^{m})}{1 - ((1 - q) ξ β)}

S_{m} \geq - (\frac{{((1 - q) ξ β)}^{m} [| S_{0} | (1 - ((1 - q) ξ β)) + (M ξ β - D)] - (M ξ β - D)}{1 - ((1 - q) ξ β)})

It is straightforward to verify that the real number m satisfies:

\frac{{((1 - q) ξ β)}^{m} [| S_{0} | (1 - ((1 - q) ξ β)) + (M ξ β - D)] - M ξ β - D}{1 - ((1 - q) ξ β)} = 0

which is again a contradiction to the assumption that S_m < 0 is all negative for all m ≤ k* + 1.

The aforementioned two cases show that k* = m is the smallest integer such that the system trajectories are guaranteed to first cross the switching manifold within k* + 1 steps.

Remark

If D = 0, the expression of the m is reduced to

m = \log_{((1 - q) β ξ)} [M ξ β / | S_{0} | (1 - ((1 - q) ξ β)) + M ξ β]

which means that the system described by equation (1) first cross the switching manifold in m + 1 number of steps.

3. Stability analysis

In order to prove the stability analysis of the proposed method (4), we will adopt the concept of Lyapunov stability.

The Lyapunov function is chosen as follows:

V (S (k)) = {(S (k))}^{2}

(22)

Now, let’s compute the change in the Lyapunov function over one iteration and substitute equation (4):

\begin{array}{l} Δ V (S (k)) = {(S (k + 1))}^{2} - {(S (k))}^{2} \\ = {[(1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} - M {| S (k) |}^{1 - φ (k)} Δ^{α - 1} s i g n (S (k))]}^{2} - {(S (k))}^{2} \\ = {[{(1 - q)}^{2} {(S (k))}^{2} {(Δ^{α - 1} {| S (k) |}^{1 - φ (k)})}^{2} - M {| S (k) |}^{1 - φ (k)} Δ^{α - 1} s i g n (S (k))]}^{2} - {(S (k))}^{2} \\ = M^{2} {({| S (k) |}^{1 - φ (k)})}^{2} {(Δ^{α - 1} s i g n (S (k)))}^{2} \\ + M^{2} {({| S (k) |}^{1 - φ (k)})}^{2} {(Δ^{α - 1} s i g n (S (k)))}^{2} - {(S (k))}^{2} \\ - 2 (1 - q) S (k) Δ^{α - 1} {| S (k) |}^{1 - φ (k)} M {| S (k) |}^{1 - φ (k)} \\ \times Δ^{α - 1} s i g n (S (k)) - {(S (k))}^{2} \end{array}

(23)

Using equations (9) and (10), we have

\begin{array}{l} Δ V (S (k)) \leq {(1 - q)}^{2} {(S (k))}^{2} {(β ξ)}^{2} + M^{2} {(ξ)}^{2} {(β)}^{2} \\ - 2 M (1 - q) {(β ξ)}^{2} ∣ S (k) ∣ - {(S (k))}^{2} \end{array}

(24)

We consider

\begin{array}{l} {(1 - q)}^{2} {(S (k))}^{2} + M^{2} - 2 M (1 - q) ∣ S (k) ∣ - {(S (k))}^{2} \\ = ({(1 - q)}^{2} - 1) {(S (k))}^{2} + M^{2} (1 - 2 \frac{1 - q}{M} ∣ S (k) ∣) \end{array}

(25)

The first term:

({(1 - q)}^{2} - 1) {(S (k))}^{2} < 0 a s 0 < q < 1

Using equation (11), the second term can be expressed as

M^{2} (1 - 2 \frac{∣ S (k) ∣}{δ}) < 0 a s ∣ S (k) ∣ \to δ

Then

{(1 - q)}^{2} {(S (k))}^{2} + M^{2} - 2 M (1 - q) ∣ S (k) ∣ - {(S (k))}^{2} < 0

(26)

Thus, ΔV(S(k)) < 0.

We can conclude that the proposed method is stable.

4. Simulation

The aim of this section is to evaluate the performance of the proposed controller through two numerical examples.

4.1. Example 1

The system to be controlled is a linear, discrete second-order system, described as follows (Gao et al., 1995; Haifeng M et al., 2019b):

{\begin{cases} x (k + 1) = [\begin{array}{l} 1.2 + Δ (k) & 0.1 + Δ (k) \\ Δ (k) & 0.6 + Δ (k) \end{array}] x (k) \\ + [\begin{array}{l} Δ (k) \\ 1 + Δ (k) \end{array}] (u (k) + d (k)) \end{cases}

(27)

where

{\begin{cases} Δ (k) = 0.02 \times \sin (k) \\ d (k) = 0.005 \times \sin (k) i f 5 \leq k \leq 30 \end{cases}

The system’s parameters are supposed to be time varying (Figure 1).

The evolution of the external disturbance is given by Figure 2.

Where the initial values are defined as x (1) = [0.02,0]^T, the switching vector is given by C = [5, 1]. The control’s parameters are chosen as M = 0.02; q = 0.51.

The choice of α and h is taken from the literature. As mentioned in the reference (Haifeng M et al., 2019b), we take the same values α = 0.8 and h = 1.

Our objective is to drive the system’s states to the desire values x_d = [0,0]^T.

The simulation results are presented in Figures 3 –5. Figure 3 shows the evolution of the input signal u(k), and Figure 4 depicts the evolution of the system’s states. The evolution of the sliding function S(k) is shown in Figure 5. The simulation findings clearly demonstrate that the proposed controller is an efficient and powerful method. As a result, the state variables converge asymptotically to the desired values within a finite time. Moreover, the sliding function converges asymptotically to a small quasi-sliding mode band. The presented figures indicate that the proposed method has several advantages, including effectively handling the chattering problem, reducing undesirable oscillation effects, and minimizing the impact of external disturbances and parameter uncertainties.

For comparison, our proposed strategy is compared with the classical reaching law proposed by Gao et al. (1995). The comparative simulation results for the discrete linear system (27) are presented in Figures 6 and 7. Figure 6 shows better convergence of the system’s states using the proposed reaching law (in blue) compared to the system’s states using Gao’s method (in red).

Figure 7 demonstrates that the sliding function using the proposed controller (18) converges to a narrower quasi-sliding mode band compared to Gao’s method (Gao et al., 1995) (using the same control gains).

To further demonstrate the performance superiority of the new method, an additional comparison to another fractional order reaching law proposed by Haifeng M et al. (2019b) is presented in Figures 8 and 9. The power rate of the fractional order reaching law given in Haifeng M et al. (2019b) is chosen to be constant.

The comparison shows that the proposed reaching law provides faster convergence and a smaller quasi-sliding mode band compared to both existing methods (Gao et al., 1995; Haifeng M et al., 2019b).

Besides, to better evaluate the proposed method, a simulation of a real model of a semi-batch reactor is given in the next section.

4.2. Example 2

To further prove the applicability of the proposed method, a SISO model of a real semi-batch reactor is used (Tahri et al., 2021).

{\begin{cases} x (k + 1) = [\begin{array}{l} 0.4468 + Δ (k) & 0.4479 + Δ (k) \\ 0.3503 + Δ (k) & 0.5993 + Δ (k) \end{array}] x (k) \\ + [\begin{array}{l} 0.0482 + Δ (k) \\ 0.1332 + Δ (k) \end{array}] (u (k) + d (k)) \\ y (k) = [\begin{array}{l} 0.0482 + Δ (k) & 0.1332 + Δ (k) \end{array}] x (k) \end{cases}

(28)

where

{\begin{cases} Δ (k) = 0.02 \times \sin (k) \\ d (k) = 0.005 \times \sin (k) i f 5 \leq k \leq 30 \end{cases}

The parameters are assumed to be time varying (Figure 10).

The evolution of the external disturbance is given by Figure 11.

Where x(1) = [0.1,0.2]^T, the switching vector is given by C = [5, 1].

The control’s parameters are chosen as M = 0.02; q = 0.51; h = 1; α = 0.8.

The simulation results are displayed in Figures 12 –17. Figure 12 illustrates the progression of the system’s states, while Figure 13 presents the input signal. Figure 14 depicts the evolution of the sliding surface. A comparison of the three methods applied to the system described by equation (28)—namely, the classical Gao method, the fractional order reaching law proposed in Gao et al. (1995) and Haifeng M et al. (2019b), and the proposed FO reaching law—is shown in Figures 15, 16, and 17.

Notably, Figure 16 demonstrates that the magnitude of the proposed method is smaller in comparison to both existing methods. Moreover, the quasi-sliding mode band is much smaller (Figure 17). The system’s states exhibit a higher accuracy convergence toward the desired values when compared to the state evolution depicted in Gao et al. (1995) and Haifeng M et al. (2019b) (as shown in Figure 15).

The robustness and convergence accuracy of the proposed method are validated through simulation results in two examples. This substantiates that the proposed strategy serves as an effective and potent tool for controlling discrete systems subjected to external disturbances and parameter uncertainties.

5. Conclusions

This paper presents a robust fractional order sliding mode control based on a novel reaching law. Firstly, a DSMC (Discrete Sliding Mode Control) is developed for a discrete linear system under ideal conditions, where disturbances are absent. Next, a comprehensive robustness analysis is provided to demonstrate the convergence of the sliding function to a quasi-sliding mode band within finite time. In the second section, we aim to consider more realistic assumptions. Specifically, we assume that the system to be controlled is subject to external and/or internal disturbances. Through a detailed mathematical analysis, we prove that the proposed controller can guide the system’s states from any initial points to the desired trajectory, effectively mitigating the impact of disturbances with appropriate selection of the controller’s gains. The performance and advantages of the new method are validated using two simulation examples. Moreover, a detailed stability analysis is demonstrated. Furthermore, we conduct comparative simulations to showcase the efficiency of the presented method in handling chattering phenomena, reducing control magnitude, ensuring accurate convergence, and minimizing the effects of disturbances.

For future research, our objective is to implement this approach in practical settings.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Ministry of Higher Education and Scientific Research in Tunisia.

ORCID iD

Aicha Znidi

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