On Adaptive Fractional Dynamic Sliding Mode Control of Suspension System

Abstract

This paper introduces a novel adaptive control method for suspension vehicle systems in response to road disturbances. The considered model is based on an active symmetry quarter car (SQC) fractional order suspension system (FOSS). The word symmetry in SQC refers to the symmetry of the suspension system in the front tires or the rear tires of the car. The active suspension controller is generally driven by an external force like a hydraulic or pneumatic actuator. The external force of the actuator is determined using fractional dynamic sliding mode control (FDSMC) to counteract road disturbances and eliminate the chattering caused by sliding mode control (SMC). In FDSMC, a fractional integral acts as a low-pass filter before the system actuator to remove high-frequency chattering, necessitating an additional state for FDSMC implementation assuming all FOSS state variables are available but the parameters are unknown and uncertain. Hence, an adaptive procedure is proposed to estimate these parameters. To enhance closed-loop system performance, an adaptive proportional-integral (PI) procedure is also employed, resulting in the FDSMC-PI approach. A comparison is made between two SQC suspension system models, the fractional order suspension system (FOSS) and the integer order suspension system (IOSS). The IOSS controller is based on dynamic sliding mode control (DSMC) and a PI procedure (DSMC-PI). The results show that FDSMC outperforms DSMC.

1. Introduction

When designing a suspension system, it is crucial to balance passenger comfort and vehicle handling [1,2]. For passenger comfort, the suspension should be soft to minimize body displacement and acceleration. Conversely, for good vehicle handling, the suspension needs to be stiff to ensure the tires maintain contact with the road in different conditions of the terrain [2,3]. In other words, for good handling of the vehicle, the tire should stick to the road [4]. In passive suspension systems, the mass–spring–damper parameters are fixed, causing the vehicle body to oscillate with the road terrain [5]. This means vibrations are transmitted from the wheels to the vehicle body [6]. However, an active symmetry quarter car (SQC) suspension system can adapt to varying road conditions, reducing both body displacement and acceleration [7]. As we mentioned in the abstract, the word symmetry in SQC refers to the symmetry of the suspension system in the front or rear tires of the car.

Active suspension systems typically include actuators that provide additional forces, such as hydraulic [4] or pneumatic (air) [8], which are determined by feedback control [9]. Numerous studies have explored various control strategies for designing active suspension systems [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30], including H-infinity control [9], optimal control [10], fuzzy control [11,12], neural network (NN) [13], linear quadratic regulator (LQR) or linear quadratic Gaussian (LQG) control [14,15,16], model-free control [17], proportional-integral-derivative (PID) control [18,19], and sliding mode control (SMC) [26,27,28,29,30].

Intelligent approaches such as the fuzzy controller [12] or neural networks [13] can be adaptive but commonly are not robust. In the anthers methods such as PID [19], coefficients can be self-tuned or chosen using the Ziegler–Nichols approach [20]. Algorithms like particle swarm optimization (PSO) [21] and genetic algorithms (GA) [22] have been used to tune PID parameters, improving the dynamic performance and stability of active suspension systems. In [23], fractional order PID controllers were designed using an artificial bee colony algorithm with objective functions such as integral absolute error, integral square error and integral time absolute error. Moreover, [24] analyzed a fuzzy-based PID controller for a half car active suspension system. In this analysis, the suspension’s working space is the criterion under observation. In designing an LQR controller, the selection of weighting matrices is a key issue that directly affects the control action. In [25], the authors presented an approach to the optimal control problem where weighting matrices are not selected by trial and error but are calculated for the time domain.

Despite these advancements, many SMC-based suspension controllers still experience chattering, and in most of the proposed SMC suspension controller approaches, chattering occurs [26,27,28,29,30]. To address this, several methods were proposed [31], such as SMC-based boundary layer (BL) [32], adaptive boundary layer (ABL) [33,34,35], higher-order SMC (HOSMC) [36,37], and dynamic SMC (DSMC) [38,39]. While BL and ABL approaches may lose the invariance property [31,32], HOSMC requires an observer to estimate higher-order derivatives of the system model [40,41,42,43]. DSMC mitigates the discontinuous Signum function effect with a low-pass integrator filter, increasing the states that need to be estimated by an observer [38,39], making it preferable.

Given the unknown and uncertain road terrain, a robust controller is necessary [4]. This paper proposes DSMC for suspension systems to handle model uncertainty, prevent chattering and preserve invariance. The model is based on a fractional order suspension system (FOSS) for a symmetry quarter car (SQC), controlled with FDSMC combined with a PI approach (FDSMC-PI). Both FOSS and PI parameters are adaptively calculated. A comparison with the integer order suspension system (IOSS), controlled by adaptive DSMC and PI (DSMC-PI), shows that FDSMC performs better. Lyapunov’s theory is used to prove the stability of the approaches.

Section 2 provides an overview of fractional derivatives and describes the IOSS and FOSS models. Section 3 presents the background and problem to clarify the proposed FDSMC and FDSMC-PI approaches. In this section, the proposed adaptive structures are also explained. Section 4 and Section 5 cover the simulation results and the conclusions, respectively.

2. Preliminaries of Fractional Model Construction

In this section, we first explain a brief about the fractional derivative and then the suspension system model of SQC is presented.

2.1. Preliminaries of Fractional Calculus

Definition 1.

Caputo q-order differentiation of function $\mathrm{f}(\mathrm{t})$ with respect to time $\mathrm{t}$ denoted by [44,45]:

$${\mathrm{D}}_{{\mathrm{t}}_0}^{\mathrm{q}}\mathrm{f}(\mathrm{t})={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\mathrm{q})}}{\displaystyle {\int }_{{\mathrm{t}}_0}^{\mathrm{t}}\frac{{\mathrm{f}}^{\prime }(\mathrm{\tau })}{{\left(\mathrm{t}-\mathrm{\tau }\right)}^{\mathrm{q}}}\mathrm{d}\mathrm{\tau }}$$

(1)

where $\mathrm{t}>{\mathrm{t}}_0$, $0<\mathrm{q}<1$ and $\mathrm{\Gamma }(\mathrm{q})$ are the Gamma function defined as $\mathrm{\Gamma }(\mathrm{q})={\displaystyle {\int }_0^{\infty }{\mathrm{\tau }}^{\mathrm{q}-1}{\mathrm{e}}^{-\mathrm{\tau }}\mathrm{d}\mathrm{\tau }}$.

Lemma 1.

For any vector $\mathrm{x}(\mathrm{t})={[{\mathrm{x}}_1(\mathrm{t}),{\mathrm{x}}_2(\mathrm{t}),\dots ,{\mathrm{x}}_{\mathrm{n}}(\mathrm{t})]}^{\mathrm{T}}\in {\mathrm{R}}^{\mathrm{n}\times 1}$ and any arbitrary matrix $\mathrm{M}\in {\mathrm{R}}^{\mathrm{n}\times \mathrm{n}}$ with constant elements, we have [44]:

$${\mathrm{D}}_{{\mathrm{t}}_0}^{\mathrm{q}}({\mathrm{x}}^{\mathrm{T}}\mathrm{M}\mathrm{x})\le {({\mathrm{D}}_{{\mathrm{t}}_0}^{\mathrm{q}}\mathrm{x})}^{\mathrm{T}}\mathrm{M}\mathrm{x}+{\mathrm{x}}^{\mathrm{T}}\mathrm{M}({\mathrm{D}}_{{\mathrm{t}}_0}^{\mathrm{q}}\mathrm{x})$$

(2)

A result of this lemma can be written as follows:

$${\mathrm{D}}_{{\mathrm{t}}_0}^{\mathrm{q}}({\mathrm{x}}^{\mathrm{T}}\mathrm{x})\le {({\mathrm{D}}_{{\mathrm{t}}_0}^{\mathrm{q}}\mathrm{x})}^{\mathrm{T}}\mathrm{x}+{\mathrm{x}}^{\mathrm{T}}({\mathrm{D}}_{{\mathrm{t}}_0}^{\mathrm{q}}\mathrm{x})$$

(3)

Remark 1.

From here and in the continuation of this paper, we suppose that the initial time of fractional derivative is zero, i.e., ${\mathrm{t}}_0=0$. Moreover, for simplicity, the subscript ${\mathrm{t}}_0$ is also eliminated and hence we write ${\mathrm{D}}^{\mathrm{q}}$ instead of ${\mathrm{D}}_{{\mathrm{t}}_0}^{\mathrm{q}}$.

2.2. Suspension Model Construction

The dynamic model of an SQC integer order suspension system (IOSS) is as follows [4]:

$${\mathrm{m}}_{\mathrm{s}}{\ddot{\mathrm{z}}}_{\mathrm{s}}=-{\mathrm{b}}_{\mathrm{s}}({\dot{\mathrm{z}}}_{\mathrm{s}}-{\dot{\mathrm{z}}}_{\mathrm{u}})-{\mathrm{k}}_{\mathrm{s}}({\mathrm{z}}_{\mathrm{s}}-{\mathrm{z}}_{\mathrm{u}})+{\mathrm{f}}_{\mathrm{a}}$$

(4)

$${\mathrm{m}}_{\mathrm{u}}{\ddot{\mathrm{z}}}_{\mathrm{u}}=+{\mathrm{b}}_{\mathrm{s}}({\dot{\mathrm{z}}}_{\mathrm{s}}-{\dot{\mathrm{z}}}_{\mathrm{u}})+{\mathrm{k}}_{\mathrm{s}}({\mathrm{z}}_{\mathrm{s}}-{\mathrm{z}}_{\mathrm{u}})-{\mathrm{f}}_{\mathrm{a}}+{\mathrm{b}}_{\mathrm{t}}({\dot{\mathrm{z}}}_{\mathrm{r}}-{\dot{\mathrm{z}}}_{\mathrm{u}})+{\mathrm{k}}_{\mathrm{t}}({\mathrm{z}}_{\mathrm{r}}-{\mathrm{z}}_{\mathrm{u}})$$

(5)

where ${\mathrm{z}}_{\mathrm{s}}$, ${\mathrm{z}}_{\mathrm{u}}$ and ${\mathrm{z}}_{\mathrm{r}}$ are displacement of the body, wheel and road (terrain), respectively. The first and second derivatives of ${\mathrm{z}}_{\mathrm{s}}$ and ${\mathrm{z}}_{\mathrm{u}}$ are velocity and acceleration. Moreover, ${\mathrm{m}}_{\mathrm{s}}$, ${\mathrm{m}}_{\mathrm{u}}$, ${\mathrm{k}}_{\mathrm{s}}$, ${\mathrm{k}}_{\mathrm{t}}$, ${\mathrm{b}}_{\mathrm{s}}$ and ${\mathrm{b}}_{\mathrm{t}}$ are mass, stiffness and damping rate of the body (sprung) and wheel (un-sprung), respectively. Moreover, ${\mathrm{z}}_{\mathrm{s}}-{\mathrm{z}}_{\mathrm{u}}$ is the suspension deflection and ${\mathrm{z}}_{\mathrm{u}}-{\mathrm{z}}_{\mathrm{r}}$ is the tire (wheel) deflection. Variable ${\mathrm{f}}_{\mathrm{a}}$ is the output force of the hydraulic actuator which is placed between the body and the wheel. When ${\mathrm{f}}_{\mathrm{a}}=0$, the suspension system acts passively or we have the passive suspension system (PSS). The aim of an active suspension system (ASS) is to determine a suitable (desired) value for ${\mathrm{f}}_{\mathrm{a}}$ such that the smallest possible values for ${\mathrm{z}}_{\mathrm{s}}$, ${\dot{\mathrm{z}}}_{\mathrm{s}}$ and ${\ddot{\mathrm{z}}}_{\mathrm{s}}$ are achieved. We assumed that all the parameters are unavailable but the variables are were measurable except ${\mathrm{z}}_{\mathrm{r}}$ and its derivative, which are considered as the disturbance. The dynamics of this system are represented in Figure 1.

Now, consider the following SQC fractional order suspension system (FOSS), which is the fractional model of Equations (4) and (5) [46].

$${\mathrm{m}}_{\mathrm{s}}{\mathrm{D}}^{2\mathrm{q}}{\mathrm{z}}_{\mathrm{s}}=-{\mathrm{b}}_{\mathrm{s}}({\mathrm{D}}^{\mathrm{q}}{\mathrm{z}}_{\mathrm{s}}-{\mathrm{D}}^{\mathrm{q}}{\mathrm{z}}_{\mathrm{u}})-{\mathrm{k}}_{\mathrm{s}}({\mathrm{z}}_{\mathrm{s}}-{\mathrm{z}}_{\mathrm{u}})+{\mathrm{f}}_{\mathrm{a}}$$

(6)

$${\mathrm{m}}_{\mathrm{u}}{\mathrm{D}}^{2\mathrm{q}}{\mathrm{z}}_{\mathrm{u}}=+{\mathrm{b}}_{\mathrm{s}}({\mathrm{D}}^{\mathrm{q}}{\mathrm{z}}_{\mathrm{s}}-{\mathrm{D}}^{\mathrm{q}}{\mathrm{z}}_{\mathrm{u}})+{\mathrm{k}}_{\mathrm{s}}({\mathrm{z}}_{\mathrm{s}}-{\mathrm{z}}_{\mathrm{u}})-{\mathrm{f}}_{\mathrm{a}}+{\mathrm{b}}_{\mathrm{t}}({\mathrm{D}}^{\mathrm{q}}{\mathrm{z}}_{\mathrm{r}}-{\mathrm{D}}^{\mathrm{q}}{\mathrm{z}}_{\mathrm{u}})+{\mathrm{k}}_{\mathrm{t}}({\mathrm{z}}_{\mathrm{r}}-{\mathrm{z}}_{\mathrm{u}})$$

(7)

As we mentioned, all variables are measurable except ${\mathrm{z}}_{\mathrm{r}}$. Now, we define the following available state variables:

$${\mathrm{x}}_1={\mathrm{z}}_{\mathrm{s}}, {\mathrm{x}}_2={\mathrm{D}}^{\mathrm{q}}{\mathrm{z}}_{\mathrm{s}}, {\mathrm{y}}_1={\mathrm{z}}_{\mathrm{u}}, {\mathrm{y}}_2={\mathrm{D}}^{\mathrm{q}}{\mathrm{z}}_{\mathrm{u}}$$

(8)

then:

$$\begin{array}{l}{\mathrm{D}}^{\mathrm{q}}{\mathrm{x}}_1={\mathrm{x}}_2\\ {\mathrm{D}}^{\mathrm{q}}{\mathrm{x}}_2=-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{x}}_2-{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{x}}_1+{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{y}}_2+{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{y}}_1+{\displaystyle \frac{1}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{f}}_{\mathrm{a}}\end{array}$$

(9)

$$\begin{array}{l}{\mathrm{D}}^{\mathrm{q}}{\mathrm{y}}_1={\mathrm{y}}_2\\ {\mathrm{D}}^{\mathrm{q}}{\mathrm{y}}_2=+{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{x}}_2+{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{x}}_1-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{y}}_2-{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{y}}_1-{\displaystyle \frac{1}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{f}}_{\mathrm{a}}-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{t}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{y}}_2-{\displaystyle \frac{{\mathrm{k}}_{\mathrm{t}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{y}}_1+{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}\end{array}$$

(10)

and ${\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}}$ is considered an unknown disturbance.

$${\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}=+{\displaystyle \frac{{\mathrm{b}}_{\mathrm{t}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{z}}_{\mathrm{r}}+{\displaystyle \frac{{\mathrm{k}}_{\mathrm{t}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{z}}_{\mathrm{r}}$$

(11)

3. Proposed Adaptive FDSMC Design

Now, we define the following measurable extra variable:

$${\mathrm{x}}_3={\mathrm{D}}^{\mathrm{q}}{\mathrm{x}}_2$$

(12)

Moreover, we define the following sliding variable:

$${\mathrm{s}}_{\mathrm{x}}={\mathrm{L}}_1{\mathrm{x}}_1+{\mathrm{L}}_2{\mathrm{x}}_2+{\mathrm{L}}_3{\mathrm{x}}_3$$

(13)

where the fixed numbers ${\mathrm{L}}_1$, ${\mathrm{L}}_2$ and ${\mathrm{L}}_3$ are the coefficients of sliding variable. The selection method of these parameters is explained in Remark 2.

Now, the sliding variable fractional derivative is as:

$${\mathrm{D}}^{\mathrm{q}}{\mathrm{s}}_{\mathrm{x}}={\mathrm{L}}_1{\mathrm{D}}^{\mathrm{q}}{\mathrm{x}}_1+{\mathrm{L}}_2{\mathrm{D}}^{\mathrm{q}}{\mathrm{x}}_2+{\mathrm{L}}_3{\mathrm{D}}^{\mathrm{q}}{\mathrm{x}}_3={\mathrm{L}}_1{\mathrm{x}}_2+{\mathrm{L}}_2{\mathrm{x}}_3+{\mathrm{L}}_3{\mathrm{D}}^{\mathrm{q}}{\mathrm{x}}_3$$

(14)

then one can write:

$${\mathrm{D}}^{\mathrm{q}}{\mathrm{x}}_3=-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{x}}_3-{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{x}}_2+{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{y}}_2+{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{y}}_2+{\displaystyle \frac{1}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}$$

(15)

Using Equation (10) results in:

$$\begin{array}{l}{\mathrm{D}}^{\mathrm{q}}{\mathrm{x}}_3=-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{x}}_3-{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{x}}_2+{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{y}}_2+{\displaystyle \frac{1}{{\mathrm{m}}_{\mathrm{s}}}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}\left(+{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{x}}_2+{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{x}}_1-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{y}}_2-{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{y}}_1-{\displaystyle \frac{1}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{f}}_{\mathrm{a}}-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{t}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{y}}_2-{\displaystyle \frac{{\mathrm{k}}_{\mathrm{t}}}{{\mathrm{m}}_{\mathrm{u}}}}{\mathrm{y}}_1+{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}\right)\\ =\left(-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}\right){\mathrm{x}}_3+\left(-{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}+{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}\right){\mathrm{x}}_2+\left(+{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}\right){\mathrm{x}}_1+\left(-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\displaystyle \frac{{\mathrm{b}}_{\mathrm{t}}}{{\mathrm{m}}_{\mathrm{u}}}}+{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}\right){\mathrm{y}}_2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\displaystyle \frac{{\mathrm{k}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{u}}}}-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\displaystyle \frac{{\mathrm{k}}_{\mathrm{t}}}{{\mathrm{m}}_{\mathrm{u}}}}\right){\mathrm{y}}_1+\left(-{\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}{\displaystyle \frac{1}{{\mathrm{m}}_{\mathrm{u}}}}\right){\mathrm{f}}_{\mathrm{a}}+\left({\displaystyle \frac{1}{{\mathrm{m}}_{\mathrm{s}}}}\right){\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+\left({\displaystyle \frac{{\mathrm{b}}_{\mathrm{s}}}{{\mathrm{m}}_{\mathrm{s}}}}\right){\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}\end{array}$$

(16)

Equation (16) can be written as follows, with the corresponding coefficients:

$${\mathrm{D}}^{\mathrm{q}}{\mathrm{x}}_3={\mathrm{k}}_3{\mathrm{x}}_3+{\mathrm{k}}_2{\mathrm{x}}_2+{\mathrm{k}}_1{\mathrm{x}}_1+{\mathrm{k}}_5{\mathrm{y}}_2+{\mathrm{k}}_4{\mathrm{y}}_1+{\mathrm{k}}_6{\mathrm{f}}_{\mathrm{a}}+{\mathrm{k}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}$$

(17)

In which $\mathrm{k}=[{\mathrm{k}}_1,{\mathrm{k}}_2,{\mathrm{k}}_3,{\mathrm{k}}_4,{\mathrm{k}}_5,{\mathrm{k}}_6]$ and also ${\mathrm{k}}_{\mathrm{d}}=-{\mathrm{k}}_3$. Moreover, the vector of the state variable is $\mathrm{x}={[{\mathrm{x}}_1,{\mathrm{x}}_2,{\mathrm{x}}_3,{\mathrm{y}}_1,{\mathrm{y}}_2,{\mathrm{f}}_{\mathrm{a}}]}^{\mathrm{T}}$, then we have:

$${\mathrm{D}}^{\mathrm{q}}{\mathrm{x}}_3=\mathrm{k}\mathrm{x}+{\mathrm{k}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}$$

(18)

Then, the Equations (14) and (18) results are:

$${\mathrm{D}}^{\mathrm{q}}{\mathrm{s}}_{\mathrm{x}}={\mathrm{L}}_1{\mathrm{x}}_2+{\mathrm{L}}_2{\mathrm{x}}_3+{\mathrm{L}}_3(\mathrm{k}\mathrm{x}+{\mathrm{k}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}})$$

(19)

Suppose that the coefficient vector $\mathrm{k}$ and coefficient scalars ${\mathrm{k}}_{\mathrm{f}}$ and ${\mathrm{k}}_{\mathrm{d}}$ are unknown and are estimated by $\widehat{\mathrm{k}}$, ${\widehat{\mathrm{k}}}_{\mathrm{f}}$ and ${\widehat{\mathrm{k}}}_{\mathrm{d}}$ such that $\widehat{\mathrm{k}}=[{\widehat{\mathrm{k}}}_1,{\widehat{\mathrm{k}}}_2,{\widehat{\mathrm{k}}}_3,{\widehat{\mathrm{k}}}_4,{\widehat{\mathrm{k}}}_5,{\widehat{\mathrm{k}}}_6]$. Then, the estimation of Equation (20) is as follows:

$${\mathrm{D}}^{\mathrm{q}}{\widehat{\mathrm{s}}}_{\mathrm{x}}={\mathrm{L}}_1{\mathrm{x}}_2+{\mathrm{L}}_2{\mathrm{x}}_3+{\mathrm{L}}_3(\widehat{\mathrm{k}}\mathrm{x}+{\widehat{\mathrm{k}}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\widehat{\mathrm{k}}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}})$$

(20)

Theorem 1.

The following FDSMC control input causes the closed-loop system to be stable.

$${\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}=-{\displaystyle \frac{{\mathrm{L}}_1{\mathrm{x}}_2+{\mathrm{L}}_2{\mathrm{x}}_3+{\mathrm{L}}_3\widehat{\mathrm{k}}\mathrm{x}+{\mathrm{\lambda }}_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})}{{\mathrm{L}}_3{\widehat{\mathrm{k}}}_{\mathrm{f}}}}$$

(21)

If the parameters $\widehat{\mathrm{k}}$ and ${\widehat{\mathrm{k}}}_{\mathrm{f}}$ are calculated from the following adaptive procedure:

$$\begin{array}{l}{\mathrm{D}}^{\mathrm{q}}\widehat{\mathrm{k}}={\mathrm{\gamma }}_1{\mathrm{L}}_3{\mathrm{s}}_{\mathrm{x}}\mathrm{x}\\ {\mathrm{D}}^{\mathrm{q}}{\widehat{\mathrm{k}}}_{\mathrm{f}}={\mathrm{\gamma }}_2{\mathrm{L}}_3{\mathrm{s}}_{\mathrm{x}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}\end{array}$$

(22)

and also:

$${\mathrm{\lambda }}_1={\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}{\mathrm{F}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}+\mathrm{\eta }$$

(23)

in which $\left|{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}\right|\le {\mathrm{F}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}$ and $\mathrm{\eta }$ is a positive number.

Proof. 

Replacing Equation (21) into Equation (20) results in:

$${\mathrm{D}}^{\mathrm{q}}{\widehat{\mathrm{s}}}_{\mathrm{x}}=-{\mathrm{\lambda }}_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})+{\mathrm{L}}_3{\widehat{\mathrm{k}}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}$$

(24)

or:

$${\mathrm{L}}_1{\mathrm{x}}_2+{\mathrm{L}}_2{\mathrm{x}}_3+{\mathrm{L}}_3(\widehat{\mathrm{k}}\mathrm{x}+{\widehat{\mathrm{k}}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\widehat{\mathrm{k}}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}})=-{\mathrm{\lambda }}_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})+{\mathrm{L}}_3{\widehat{\mathrm{k}}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}$$

(25)

or:

$$\begin{array}{l}{\mathrm{L}}_1{\mathrm{x}}_2+{\mathrm{L}}_2{\mathrm{x}}_3+{\mathrm{L}}_3(\mathrm{k}\mathrm{x}+{\mathrm{k}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}})-{\mathrm{L}}_3(\mathrm{k}\mathrm{x}+{\mathrm{k}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}})\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mathrm{L}}_3(\widehat{\mathrm{k}}\mathrm{x}+{\widehat{\mathrm{k}}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}})=-{\mathrm{\lambda }}_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})\end{array}$$

(26)

From Equations (19) and (26), the next equation is obtained:

$${\mathrm{D}}^{\mathrm{q}}{\mathrm{s}}_{\mathrm{x}}={\mathrm{L}}_3(\mathrm{k}-\widehat{\mathrm{k}})\mathrm{x}+{\mathrm{L}}_3({\mathrm{k}}_{\mathrm{f}}-{\widehat{\mathrm{k}}}_{\mathrm{f}}){\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}-{\mathrm{\lambda }}_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})$$

(27)

Defining $\tilde{\mathrm{k}}=\mathrm{k}-\widehat{\mathrm{k}}$ and ${\tilde{\mathrm{k}}}_{\mathrm{f}}={\mathrm{k}}_{\mathrm{f}}-{\widehat{\mathrm{k}}}_{\mathrm{f}}$ results in:

$${\mathrm{D}}^{\mathrm{q}}{\mathrm{s}}_{\mathrm{x}}=-{\mathrm{\lambda }}_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})+{\mathrm{L}}_3\tilde{\mathrm{k}}\mathrm{x}+{\mathrm{L}}_3{\tilde{\mathrm{k}}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}$$

(28)

Now, consider the following Lyapunov function:

$$\mathrm{V}(\mathrm{t})={\displaystyle \frac{1}{2}}{{\mathrm{s}}_{\mathrm{x}}}^2+{\displaystyle \frac{1}{2{\mathrm{\gamma }}_1}}\tilde{\mathrm{k}}{\tilde{\mathrm{k}}}^{\mathrm{T}}+{\displaystyle \frac{1}{2{\mathrm{\gamma }}_2}}{{\tilde{\mathrm{k}}}_{\mathrm{f}}}^2$$

(29)

Based on Lemma 1, one can conclude that:

$${\mathrm{D}}^{\mathrm{q}}\mathrm{V}(\mathrm{t})\le {\mathrm{s}}_{\mathrm{x}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{s}}_{\mathrm{x}}-{\displaystyle \frac{1}{{\mathrm{\gamma }}_1}}\tilde{\mathrm{k}}{\mathrm{D}}^{\mathrm{q}}\widehat{\mathrm{k}}-{\displaystyle \frac{1}{{\mathrm{\gamma }}_2}}{\tilde{\mathrm{k}}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\widehat{\mathrm{k}}}_{\mathrm{f}}$$

(30)

Therefore:

$$\begin{array}{l}{\mathrm{D}}^{\mathrm{q}}\mathrm{V}(\mathrm{t})\le {\mathrm{s}}_{\mathrm{x}}\left(-{\mathrm{\lambda }}_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})+{\mathrm{L}}_3\tilde{\mathrm{k}}\mathrm{x}+{\mathrm{L}}_3{\tilde{\mathrm{k}}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \frac{1}{{\mathrm{\gamma }}_1}}\tilde{\mathrm{k}}\left({\mathrm{\gamma }}_1{\mathrm{L}}_3{\mathrm{s}}_{\mathrm{x}}\mathrm{x}\right)-{\displaystyle \frac{1}{{\mathrm{\gamma }}_2}}{\tilde{\mathrm{k}}}_{\mathrm{f}}\left({\mathrm{\gamma }}_2{\mathrm{L}}_3{\mathrm{s}}_{\mathrm{x}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=-{\mathrm{\lambda }}_1{\mathrm{s}}_{\mathrm{x}}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})+{\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}{\mathrm{s}}_{\mathrm{x}}\le -{\mathrm{\lambda }}_1\left|{\mathrm{s}}_{\mathrm{x}}\right|+{\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}\left|{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}\right|\hspace{0.17em}\left|{\mathrm{s}}_{\mathrm{x}}\right|\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\le \left(-{\mathrm{\lambda }}_1+{\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}{\mathrm{F}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}\right)\hspace{0.17em}\left|{\mathrm{s}}_{\mathrm{x}}\right|\end{array}$$

(31)

Using Equation (23) results in:

$${\mathrm{D}}^{\mathrm{q}}\mathrm{V}(\mathrm{t})\le -\mathrm{\eta }\left|{\mathrm{s}}_{\mathrm{x}}\right|$$

(32)

When using the Lyapunov stability theory, it is concluded that the closed-loop system is stable and proof of the theorem is completed. □

Theorem 2.

The following FDSMC-PI control input causes the closed-loop system to be stable.

$$\begin{array}{l}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}=-{\displaystyle \frac{{\mathrm{L}}_1{\mathrm{x}}_2+{\mathrm{L}}_2{\mathrm{x}}_3+{\mathrm{L}}_3\widehat{\mathrm{k}}\mathrm{x}+{\mathrm{\lambda }}_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})+{\mathrm{u}}_{\mathrm{P}\mathrm{I}}}{{\mathrm{L}}_3{\widehat{\mathrm{k}}}_{\mathrm{f}}}}\\ {\mathrm{u}}_{\mathrm{P}\mathrm{I}}={\widehat{\mathrm{k}}}_{\mathrm{P}}{\mathrm{s}}_{\mathrm{x}}+{\widehat{\mathrm{k}}}_{\mathrm{I}}{\displaystyle {\int }_0^{\mathrm{t}}{\mathrm{s}}_{\mathrm{x}}\mathrm{d}\mathrm{t}}\end{array}$$

(33)

If the adaptive parameters are calculated from the following procedure:

$$\begin{array}{l}{\mathrm{D}}^{\mathrm{q}}\widehat{\mathrm{k}}={\mathrm{\gamma }}_1{\mathrm{L}}_3{\mathrm{s}}_{\mathrm{x}}\mathrm{x}\\ {\mathrm{D}}^{\mathrm{q}}{\widehat{\mathrm{k}}}_{\mathrm{f}}={\mathrm{\gamma }}_2{\mathrm{L}}_3{\mathrm{s}}_{\mathrm{x}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}\\ {\mathrm{D}}^{\mathrm{q}}{\widehat{\mathrm{k}}}_{\mathrm{P}}={\mathrm{\eta }}_1{{\mathrm{s}}_{\mathrm{x}}}^2\\ {\mathrm{D}}^{\mathrm{q}}{\widehat{\mathrm{k}}}_{\mathrm{I}}={\mathrm{\eta }}_2{\mathrm{s}}_{\mathrm{x}}{\displaystyle {\int }_0^{\mathrm{t}}{\mathrm{s}}_{\mathrm{x}}\mathrm{d}\mathrm{t}}\end{array}$$

(34)

Also, as in the previous theorem:

$${\mathrm{\lambda }}_1={\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}{\mathrm{F}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}+\mathrm{\eta }$$

(35)

in which $\left|{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}\right|\le {\mathrm{F}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}$ and $\mathrm{\eta }$ is a positive number.

Proof. 

Replacing Equation (33) into Equation (20) and as with Equation (24) to Equation (28), one can conclude that:

$${\mathrm{D}}^{\mathrm{q}}{\mathrm{s}}_{\mathrm{x}}=-{\mathrm{\lambda }}_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})+{\mathrm{L}}_3\tilde{\mathrm{k}}\mathrm{x}+{\mathrm{L}}_3{\tilde{\mathrm{k}}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}-{\mathrm{u}}_{\mathrm{P}\mathrm{I}}$$

(36)

Now, consider the following Lyapunov function:

$$\mathrm{V}(\mathrm{t})={\displaystyle \frac{1}{2}}{{\mathrm{s}}_{\mathrm{x}}}^2+{\displaystyle \frac{1}{2{\mathrm{\gamma }}_1}}\tilde{\mathrm{k}}{\tilde{\mathrm{k}}}^{\mathrm{T}}+{\displaystyle \frac{1}{2{\mathrm{\gamma }}_2}}{{\tilde{\mathrm{k}}}_{\mathrm{f}}}^2+{\displaystyle \frac{1}{2{\mathrm{\eta }}_1}}{{\widehat{\mathrm{k}}}_{\mathrm{P}}}^2{\displaystyle \frac{1}{2{\mathrm{\eta }}_2}}{{\widehat{\mathrm{k}}}_{\mathrm{I}}}^2$$

(37)

Based on Lemma 1, one can conclude that:

$${\mathrm{D}}^{\mathrm{q}}\mathrm{V}(\mathrm{t})\le {\mathrm{s}}_{\mathrm{x}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{s}}_{\mathrm{x}}-{\displaystyle \frac{1}{{\mathrm{\gamma }}_1}}\tilde{\mathrm{k}}{\mathrm{D}}^{\mathrm{q}}\widehat{\mathrm{k}}-{\displaystyle \frac{1}{{\mathrm{\gamma }}_2}}{\tilde{\mathrm{k}}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\widehat{\mathrm{k}}}_{\mathrm{f}}+{\displaystyle \frac{1}{{\mathrm{\eta }}_1}}{\widehat{\mathrm{k}}}_{\mathrm{P}}{\mathrm{D}}^{\mathrm{q}}{\widehat{\mathrm{k}}}_{\mathrm{P}}+{\displaystyle \frac{1}{{\mathrm{\eta }}_2}}{\widehat{\mathrm{k}}}_{\mathrm{I}}{\mathrm{D}}^{\mathrm{q}}{\widehat{\mathrm{k}}}_{\mathrm{I}}$$

(38)

Therefore:

$$\begin{array}{l}{\mathrm{D}}^{\mathrm{q}}\mathrm{V}(\mathrm{t})\le {\mathrm{s}}_{\mathrm{x}}\left(-{\mathrm{\lambda }}_1\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})+{\mathrm{L}}_3\tilde{\mathrm{k}}\mathrm{x}+{\mathrm{L}}_3{\tilde{\mathrm{k}}}_{\mathrm{f}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}+{\widehat{\mathrm{k}}}_{\mathrm{P}}{\mathrm{s}}_{\mathrm{x}}-{\widehat{\mathrm{k}}}_{\mathrm{I}}{\displaystyle {\int }_0^{\mathrm{t}}{\mathrm{s}}_{\mathrm{x}}\mathrm{d}\mathrm{t}}-{\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\displaystyle \frac{1}{{\mathrm{\gamma }}_1}}\tilde{\mathrm{k}}\left({\mathrm{\gamma }}_1{\mathrm{L}}_3{\mathrm{s}}_{\mathrm{x}}\mathrm{x}\right)-{\displaystyle \frac{1}{{\mathrm{\gamma }}_2}}{\tilde{\mathrm{k}}}_{\mathrm{f}}\left({\mathrm{\gamma }}_2{\mathrm{L}}_3{\mathrm{s}}_{\mathrm{x}}{\mathrm{D}}^{\mathrm{q}}{\mathrm{f}}_{\mathrm{a}}\right)+{\displaystyle \frac{1}{{\mathrm{\eta }}_1}}{\widehat{\mathrm{k}}}_{\mathrm{P}}\left({\mathrm{\eta }}_1{{\mathrm{s}}_{\mathrm{x}}}^2\right)+{\displaystyle \frac{1}{{\mathrm{\eta }}_2}}{\widehat{\mathrm{k}}}_{\mathrm{I}}\left({\mathrm{\eta }}_2{\mathrm{s}}_{\mathrm{x}}{\displaystyle {\int }_0^{\mathrm{t}}{\mathrm{s}}_{\mathrm{x}}\mathrm{d}\mathrm{t}}\right)\\ =-{\mathrm{\lambda }}_1{\mathrm{s}}_{\mathrm{x}}\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}({\mathrm{s}}_{\mathrm{x}})+{\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}{\mathrm{s}}_{\mathrm{x}}\le -{\mathrm{\lambda }}_1\left|{\mathrm{s}}_{\mathrm{x}}\right|+{\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}\left|{\mathrm{f}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}\right|\hspace{0.17em}\left|{\mathrm{s}}_{\mathrm{x}}\right|\le \left(-{\mathrm{\lambda }}_1+{\mathrm{L}}_3{\mathrm{k}}_{\mathrm{d}}{\mathrm{F}}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}}\right)\hspace{0.17em}\left|{\mathrm{s}}_{\mathrm{x}}\right|\end{array}$$

(39)

Using the Equation (35) results:

$${\mathrm{D}}^{\mathrm{q}}\mathrm{V}(\mathrm{t})\le -\mathrm{\eta }\left|{\mathrm{s}}_{\mathrm{x}}\right|$$

(40)

When using the Lyapunov stability theory, it is concluded that the closed-loop system is stable and proof of the theorem is completed. □

4. Simulation Presentation

In this section, two simulations are carried out, DSMC and FDSMC. The parameters of the suspension system have been chosen as in

Table 1. The parameters of suspension system.

Parameter	Value	Unit
${\mathrm{m}}_{\mathrm{s}}$	$243$	$\mathrm{k}\mathrm{g}$
${\mathrm{m}}_{\mathrm{u}}$	$40$	$\mathrm{k}\mathrm{g}$
${\mathrm{b}}_{\mathrm{s}}$	$370$	$\mathrm{N}/(\mathrm{m}/\mathrm{s})$
${\mathrm{b}}_{\mathrm{t}}$	$414$	$\mathrm{N}/(\mathrm{m}/\mathrm{s})$
${\mathrm{k}}_{\mathrm{s}}$	$\mathrm{14,671}$	$\mathrm{N}/\mathrm{m}$
${\mathrm{k}}_{\mathrm{t}}$	$\mathrm{124,660}$	$\mathrm{N}/\mathrm{m}$

[4].

In addition, the parameters of the controllers have been selected as:

$${\mathrm{L}}_1=4, {\mathrm{L}}_2=0.4, {\mathrm{L}}_3=0.01, {\mathrm{\lambda }}_1=2$$

(41)

Remark 2.

The sliding surface coefficients ${\mathrm{L}}_1$, ${\mathrm{L}}_2$ and ${\mathrm{L}}_3$ are selected such that the internal dynamics of the sliding surface (13) are considered to be zero, i.e., in the case of  ${\mathrm{s}}_{\mathrm{x}}=0$, all the state variables ${\mathrm{x}}_1$, ${\mathrm{x}}_2$ and ${\mathrm{x}}_3$ converge to zero. Also, we selected ${\mathrm{\gamma }}_1={\mathrm{\gamma }}_2={\mathrm{\eta }}_1={\mathrm{\eta }}_2=1$ in Equation (34) to have a moderate convergence and to reduce drift of adaptive parameters and moreover, for a good approximate of function $\mathrm{k}\mathrm{x}$ in Equation (18).

Moreover, based on the previous discussion, the structure of the proposed controller is shown and depicted in Figure 2.

As this figure shows, before the input control of the suspension system (FOSS), a fractional integrator is placed to suppress the chattering produced by the SMC in Equation (33). Moreover, the unknown parameters of the system are calculated using the adaptive parameters of Equation (34). To provide these adaptive parameters, a new fractional integrator is needed.

However, the advantages of this approach (like the chattering elimination) are promising for practical applications because it is clear that the low-pass integrator removes all the high frequency produced due to chattering.

The road terrain in Figure 3 is obtained using the following equation, which consists of two triangular-like functions. Considerations such as this terrain are common in the literature [4,6].

$${\mathrm{z}}_{\mathrm{r}}=\left\{\begin{array}{l}0.1\sin (\mathrm{t})\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}:6\le \mathrm{t}\le 7\\ 0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}:\hspace{0.17em}\mathrm{O}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$$

(42)

The simulation was conducted using MATLAB with a fixed step time of 0.0001 for both the fractional dynamic sliding mode control (FDSMC-PI) and the dynamic sliding mode control (DSMC-PI). In FDSMC-PI, the fractional order is chosen as $\mathrm{q}=0.8$, while in DSMC-PI, it is $\mathrm{q}=1$ in all equations. The simulation results are depicted in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. Note that in all figures, the horizontal axis scaling depends on the car’s velocity $\mathrm{V}$ according to the equation $\mathrm{d}=\mathrm{V}\mathrm{t}$, where $\mathrm{d}$ is displacement.

Figure 4 and Figure 5 show that tire displacement and velocity exhibit vibrations in DSMC-PI but are smoother in FDSMC-PI. Additionally, body displacement and velocity are very similar in both cases, as seen in Figure 6 and Figure 7. Figure 8 illustrates the softer suspension between the body and the tire in FDSMC-PI. Passenger comfort, shown in Figure 6, is similar in both controllers, but vehicle handling is better in FDSMC-PI, as indicated in Figure 4 and Figure 9.

Comparing road terrain in Figure 3 with tire displacement in Figure 4, it is evident that the tire adheres to the road in FDSMC-PI, unlike in DSMC-PI. Figure 9 highlights the stiff suspension between the tire and the road in FDSMC-PI. Figure 11 displays the input force for both cases, which can be supplied by a pneumatic or hydraulic system. The sliding surface and function error estimation are shown in Figure 10 and Figure 12, respectively. The absence of chattering in both DSMC-PI and FDSMC-PI is evident from all these figures.

The statistical comparisons of the suspension system for FDSMC-PI and DSMC-PI are provided in

Table 2. The statistical comparison of suspension system.

Methods	FDSMC-PI			DSMC-PI
Parameter	Min	Max	RMS	Min	Max	RMS
${\mathrm{z}}_{\mathrm{r}}$	$-0.0279$	$0.0675$	$0.0089$	$-0.0279$	$0.0657$	$0.0089$
${\mathrm{z}}_{\mathrm{u}}$	$-0.0343$	$0.0643$	$0.0088$	$-0.0437$	$0.0641$	$0.0096$
${\dot{\mathrm{z}}}_{\mathrm{u}}$	$-2.4921$	$0.5105$	$0.0780$	$-2.8705$	$2.1772$	$0.2208$
${\mathrm{z}}_{\mathrm{s}}$	$-3.3888\times {10}^{-4}$	$0.0452$	$0.0038$	$-1.9490\times {10}^{-5}$	$0.0268$	$0.026$
${\dot{\mathrm{z}}}_{\mathrm{s}}$	$-0.2328$	$0.2109$	$0.0227$	$-0.1940$	$0.1908$	$0.0215$
${\mathrm{z}}_{\mathrm{s}}-{\mathrm{z}}_{\mathrm{u}}$	$-0.0459$	$0.0588$	$0.0078$	$-0.0505$	$0.0620$	$0.0092$
${\mathrm{z}}_{\mathrm{u}}-{\mathrm{z}}_{\mathrm{r}}$	$-0.0137$	$0.0643$	$0.0014$	$-0.0352$	$0.0641$	$0.0040$
${\mathrm{s}}_{\mathrm{s}}$	$-0.0143$	$0.2559$	$0.0190$	$-3.2425\times {10}^{-4}$	$0.1661$	$0.0126$
${\mathrm{f}}_{\mathrm{a}}$	$-1.0661\times {10}^3$	$1.2461\times {10}^3$	$115.5866$	$-1.6658\times {10}^3$	$1.3065\times {10}^3$	$152.4935$
$\mathrm{k}\mathrm{x}-\widehat{\mathrm{k}}\mathrm{x}$	$-340.9504$	$169.3054$	$45.1721$	$-365.1951$	$227.6984$	$49.4476$

. These comparisons are based on the minimum (Min), maximum (Max) and root mean square (RMS) of the variables depicted in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. As is clear from this table, all of the statistical variables in FDSMC-PI are better than those in DSMC-PI except for ${\mathrm{z}}_{\mathrm{s}}$, ${\dot{\mathrm{z}}}_{\mathrm{s}}$ and ${\mathrm{s}}_{\mathrm{x}}$. This means that, generally, fractional controllers are better than integer controllers.

5. Conclusions

The symmetry quarter car (SQC) model under consideration utilizes a fractional order suspension system (FOSS) controlled by an external force. To ensure a smooth forced signal and manage road terrain disturbances, fractional dynamic sliding mode control (FDSMC) is employed. In FDSMC, a fractional integral acts as a low-pass filter before the actuator to reduce chattering produced by sliding mode control (SMC). To enhance performance, an adaptive proportional-integral (PI) method is incorporated, resulting in the adaptive FDSMC-PI approach. For comparison, an integer order suspension system (IOSS) with dynamic sliding mode control (DSMC) and a PI method is also implemented (DSMC-PI). Assuming all parameters in both FOSS and IOSS are unknown, an adaptive procedure is used to identify the system parameters. It is demonstrated that chattering is removed in both the DSMC and the FDSMC methods. Additionally, while both approaches offer similar passenger comfort, FDSMC provides better vehicle handling due to improved tire–road adhesion.