Output Feedback Fractional-Order Nonsingular Terminal Sliding Mode Control of Underwater Remotely Operated Vehicles

Abstract

For the 4-DOF (degrees of freedom) trajectory tracking control problem of underwater remotely operated vehicles (ROVs) in the presence of model uncertainties and external disturbances, a novel output feedback fractional-order nonsingular terminal sliding mode control (FO-NTSMC) technique is introduced in light of the equivalent output injection sliding mode observer (SMO) and TSMC principle and fractional calculus technology. The equivalent output injection SMO is applied to reconstruct the full states in finite time. Meanwhile, the FO-NTSMC algorithm, based on a new proposed fractional-order switching manifold, is designed to stabilize the tracking error to equilibrium points in finite time. The corresponding stability analysis of the closed-loop system is presented using the fractional-order version of the Lyapunov stability theory. Comparative numerical simulation results are presented and analyzed to demonstrate the effectiveness of the proposed method. Finally, it is noteworthy that the proposed output feedback FO-NTSMC technique can be used to control a broad range of nonlinear second-order dynamical systems in finite time.

1. Introduction

Fractional calculus, an extension of ordinary integer-order differential and integral calculus, is a 300-year-old mathematical subject. However, its application to engineering and physics has only recently attracted significant attention [1–3]. One active field of the applications is fractional-order controller design.

After the concept of the fractional-order (FO) controller was first proposed [4], many FO control strategies combined with other classical control methods were proposed and investigated for both linear and nonlinear systems. One of these attractive control strategies is the combination of the FO control method and sliding mode control (SMC) technology [5, 6].

As one of the most significant achievements in modern control theory, SMC is a well-known efficient control scheme for handling control problems with model uncertainties and external disturbances [7]. Therefore, SMC has been widely applied in many systems such as underwater vehicles [8–13], mobile manipulator [14], stochastic Markovian jumping systems [15], near space vehicles [16], hysteretic structural systems [17], and chaotic systems [18, 19]. SMC mainly contains two components: a driving part that forces the system states to reach and stay on a stable predescribed sliding surface and a sliding surface that ensures the desired error dynamics of the systems [20]. Usually, the sliding surface is described by arbitrary linear dynamics, and it can only guarantee asymptotic stability, which means the system states need infinite time to converge to the equilibrium point. However, it is widely believed that the finite-time stabilization of dynamical systems may give rise to a high-precision and fast system dynamic response [21]. Thus, terminal sliding mode control (TSMC) and its improved version, nonsingular TSMC (NTSMC), which are variant schemes of traditional SMC and can achieve finite-time stability, were proposed and investigated [21–28]. Inspired by this idea, some control strategies combining fractional calculus and TSMC/NTSMC have been reported for both fractional-order and integer-order systems in the past few years [29–32]. It has been verified that the fractional-order TSMC/NTSMC method can ensure better control performance than the integer-order ones even for the integer-order plants.

Recalling the development of the fractional-order SMC/TSMC/NTSMC methods over the past few years, almost all theoretical studies and practical applications have focused on the full state feedback control strategies, which, however, may be unsuitable in many practical applications due to the immeasurability of the full states. Although a fractional-order dynamic output feedback sliding mode controller has been reported recently [33], it should be mentioned that the method proposed in [33] is designed for a class of fractional-order nonlinear systems and that the traditional linear-hyperplane-based SMC method was adopted instead of TSMC or NTSMC. Meanwhile, to the best of the authors’ knowledge, there has been no study on the development of an output feedback fractional-order TSMC/NTSMC (FO-TSMC/FO-NTSMC) strategy for integer-order systems. Therefore, designing an output feedback fractional-order TSMC/NTSMC (FO-NTSMC/FO-NTSMC) strategy for the integer-order systems still remains an open and challenging problem to be solved.

Thus, in light of the equivalent output injection sliding mode observer (SMO) [34, 35], TSMC technology, and fractional calculus, we propose an output feedback FO-NTSMC scheme for underwater remotely operated vehicles (ROVs), a classical multivariable nonlinear second-order dynamic system, in this paper for the first time. The effects of model uncertainties and external disturbances are also taken into account and the proposed control scheme is able to tackle all of these uncertainties in the system dynamics. The main contributions of this paper are as follows: (1) design an equivalent output injection SMO for ROVs and present the corresponding proof; (2) design a novel fractional-order nonsingular terminal sliding manifold which is applicable for the classical second-order systems of ROVs; (3) design a novel control law to guarantee the reachability of the proposed sliding manifold; (4) prove the finite time stabilization of the closed-loop observer-controller systems with fractional-order dynamics for second-order systems. Finally, the goal of this control scheme is to control ROVs to track the desired trajectory in finite time using only the plants’ output signal in the presence of model uncertainties and external disturbances.

The rest of this paper is organized as follows. In Section 2, some basic definitions and preliminaries of fractional-order calculus are presented. In Section 3, the system description and problem formulation are introduced. In Section 4, the integer-order TSMC and NTSMC are briefly reviewed. In Section 5, the design procedure of the proposed fractional-order output feedback FO-NTSMC is demonstrated. Corresponding stability and reachability analyses are performed. In Section 6, the validation of the proposed method is verified through numerical simulations. Finally, some conclusions are presented in Section 7.

2. Preliminaries

In this section, basic definitions of fractional calculus and a necessary fractional calculus stability lemma are presented. Two of the most commonly adopted definitions are the Riemann-Liouville and Caputo definitions.

Definition 1 (see [36]). The 𝛼th-order Riemann-Liouville fractional derivative of function 𝑓(𝑡) with respect to 𝑡 and the terminal value 𝑡₀ is defined as 𝐷^𝛼𝑓(𝑡)=d^𝛼𝑓(𝑡)d𝑡^𝛼=1Γ()d𝑚−𝛼d𝑡^𝑚𝑚𝑡_𝑡0𝑓(𝜏)(𝑡−𝜏)^{𝛼−𝑚+1}d𝜏 (1) and the 𝛼th-order Riemann-Liouville fractional integration is defined as _𝑡0𝐼^𝛼_𝑡𝑓(𝑡)=1Γ(𝛼)^𝑡_𝑡0𝑓(𝜏)d𝜏(𝑡−𝜏)^1−𝛼, (2) where 𝑚−1<𝛼≤𝑚, 𝑚∈𝑁, and Γ(⋅) is the Gamma function.

Definition 2 (see [36]). The Caputo fractional derivative of order 𝛼 of a continuous function 𝑓(𝑡) is defined as follows: 𝐷^𝛼𝑓(𝑡)=1Γ(𝑚−𝛼)^𝑡₀𝑓^(𝑚)(𝜏)(𝑡−𝜏)^{𝛼−𝑚+1}dd𝜏,𝑚−1<𝛼<𝑚_𝑚d𝑡^𝑚𝑓(𝑡),𝛼=𝑚, (3) where 𝑚 is the first integer larger than 𝛼.

Property 1 (see [36]). If the fractional derivative _𝑡0𝐷^𝛼_𝑡𝑦(𝑡) (𝑘−1≤𝛼<𝑘) of a function 𝑦(𝑡) is integrable, then _𝑡0𝐼^𝛼_𝑡𝑡0𝐷^𝛼_𝑡𝑦(𝑡)=𝑦(𝑡)−^𝑘_𝑗=1𝑡0𝐷_𝑡^𝛼−𝑗𝑦(𝑡)_𝑡=𝑡0𝑡−𝑡₀^𝛼−𝑗Γ.𝛼−𝑗+1 (4)

Lemma 3 (see [37]). The fractional integration operator _𝑡0𝐼^𝛼_𝑡 with ⌊𝛼⌋>0 is bounded as 𝐼^𝛼𝑦_𝑝≤𝐾𝑦_𝑝,1≤𝑝≤∞. (5)

3. System Description and Problem Formulation

The standard form of the kinematics and dynamics equations of ROVs in 4-DOF, described in the earth-fixed coordinate and body-fixed coordinate frames as indicated in Figure 1, can be written as follows [38]: ̇𝜂𝑀̇(𝑣)(𝑣)𝜂𝜂=𝐽𝑣,𝑣+𝐶𝑣+𝐷𝑣+𝑔=𝜏+𝐽^𝑇𝜂𝑑, (6) where 𝜂=[𝑥,𝑦,𝑧,𝜓]^𝑇 denotes the ROV’s location and orientation in the earth-fixed coordinate, whereas 𝑣=[𝑢,𝑣,𝑤,𝑟]^𝑇 denotes the vector of the ROV’s linear and angular velocity expressed in the body-fixed coordinate. 𝑀=𝑀₀+Δ𝑀∈ℝ^4×4 is the inertial matrix including added mass. 𝐶(𝑣)=𝐶₀(𝑣)+Δ𝐶(𝑣)∈ℝ^4×4 represents the Coriolis and centripetal forces. 𝐷(𝑣)=𝐷₀(𝑣)+Δ𝐷(𝑣)∈ℝ^4×4 is the hydrodynamic damping term, and the vector 𝑔(𝜂)=𝑔₀(𝜂)+Δ𝑔(𝜂)∈ℝ^4×1 is a combined force/moment of gravity and buoyancy in the body-fixed coordinate. 𝑀₀, 𝐶₀(𝑣), 𝐷₀(𝑣), and 𝑔₀(𝜂) are the nominal parameter matrices, whereas Δ𝑀, Δ𝐶(𝑣), Δ𝐷(𝑣), and Δ𝑔(𝜂) are the model uncertainties. 𝐽^𝑇(𝜂)𝑑∈𝑅^4×1 is the disturbance force/moment vector expressed in the body-fixed coordinate and 𝜏∈𝑅^4×1 is the system control input.

Figure 1 

Earth-fixed and body-fixed frame.

𝐽(𝜂) is the kinematic transformation matrix which expresses the transformation from the body-fixed frame to earth-fixed frame and can be expressed as follows: 𝐽𝜂=.cos𝜓−sin𝜓00sin𝜓cos𝜓0000100001 (7)

The other simplified parameter matrices can be expressed as follows: 𝑀₀=diag𝑚−𝑋_̇𝑢,𝑚−𝑌_̇𝑣,𝑚−𝑍_̇𝑤,𝐼_𝑧−𝑁_̇𝑟,𝐶₀(𝑣)=000−𝑚_𝑣−𝑌_̇𝑣𝑣000−𝑚_𝑣−𝑋_̇𝑢𝑢𝑚0000_𝑣−𝑌_̇𝑣𝑣−𝑚_𝑣−𝑋_̇𝑢,𝐷𝑢00₀(𝑣)=diag𝑋_𝑢+𝑋_𝑢|𝑢|𝑢,𝑌_𝑣+𝑌_𝑣|𝑣|𝑣,𝑍_𝑤+𝑍_𝑤|𝑤|𝑤,𝑁_𝑟+𝑁_𝑟|𝑟|𝑟,𝑔₀𝜂=[0,0,𝑊−𝐵,0]^𝑇, (8) where 𝑊 and 𝐵 denote the weight and buoyancy of the ROV, respectively.

Before we present the main results, necessary preliminary information is provided [21, 34].

Assumption 4 (see [34]). The MIMO dynamic system given by (6) does not have a finite escape time.

Assumption 5 (see [34]). The control input 𝜏 belongs to the extended 𝐿_𝑝 space, denoted by 𝐿^′_𝑝 in this paper. Any truncation of 𝜏 to a finite time interval is bounded.

Assumption 6 (see [34]). The desired trajectory 𝜂_𝑑 is smooth; that is, 𝜂̇_𝑑 and 𝜂̈_𝑑 are bounded, exist, and are known.

Lemma 7 (see [21]). An extended Lyapunov description of finite-time stability can be given with form of fast TSM as 𝑉̇(𝑥)+𝛼𝑉(𝑥)+𝛽𝑉^𝛾(𝑥)≤0,𝛼>0,𝛽>0,0<𝛾<1, (9) and the settling time can be given by 1𝑇≤𝛼1−𝛾ln𝛼𝑉^1−𝛾𝑥₀+𝛽𝛽. (10)

4. Review of the Integer-Order TSM and NTSM

In this section, definitions of the TSM and NTSM are briefly introduced as a necessary preparation for the output feedback FO-NTSMC design.

Definition 8 (see [21, 24]). The TSM and NTSM are equivalent and can be, respectively, described by the following first-order nonlinear differential equations: 𝑠=𝑒̇+𝛽sig(𝑒)^𝜇𝑠=0,_′=𝑒+𝛽^′sig(𝑒)̇^𝜇′=0, (11) where 𝛽^′=𝛽^−1/𝜇=𝛽^−𝜇′>0,1<𝜇^′=1𝜇<2. (12)

The TSM and NTSM defined in (11) are continuous and differentiable despite the adoption of the absolute value and the signum operator; the first derivatives thereof can be, respectively, expressed as follows [21]: 𝑠=𝑒̈+𝛽𝜇|𝑒|^𝜇−1̇𝑠𝑒,_′=𝑒̇+𝛽^′𝜇^′|𝑒|̇^𝜇′−1𝑒.̈ (13)

5. Main Results

In this section, we will develop an output feedback FO-NTSMC approach for the trajectory tracking control of ROVs in the presence of model uncertainties and external disturbances. First, an equivalent output injection SMO will be established to estimate the ROV’s velocity. Then, a novel fractional-order nonsingular terminal sliding manifold will be proposed to ensure the desired dynamics. Finally, a control law is designed to force the trajectory to reach the designed sliding manifold in finite time and remain on it forever.

5.1. Equivalent Output Injection Sliding Mode Observer Design

In this subsection, the equivalent output injection SMO will be designed and analyzed. The following notation will be used except stated otherwise: 𝑥 represents the estimation of 𝑥 and 𝑥_𝑖 represents the 𝑖th component of the vector 𝑥. The math operations used between two vectors are performed in terms of the corresponding elements. And in this paper, 𝑖=1∼4.

The nonlinear model of ROVs in the earth-fixed coordinate is adopted here to simplify the SMO design procedure [38]: 𝜂̇=𝑣_𝑒=𝐽𝜂𝑣,𝑀^′₀𝜂𝑣̇_𝑒+𝐶^′₀𝑣,𝜂𝑣_𝑒+𝐷^′₀𝑣,𝜂𝑣_𝑒+𝑔^′₀𝜂=𝐽^−𝑇𝜂𝜏+𝑑, (14)(15) where 𝑣_𝑒 is the velocity vector in the earth-fixed frame, and the parameter matrices in (15) can be described as follows: 𝑀^′₀𝜂=𝐽^−𝑇𝜂𝑀₀𝐽⁻¹𝜂,𝐶^′₀𝑣,𝜂=𝐽^−𝑇𝜂𝐶₀(𝑣)−𝑀₀𝐽⁻¹𝜂𝐽̇𝜂𝐽⁻¹𝜂,𝐷^′₀𝑣,𝜂=𝐽^−𝑇𝜂𝐷₀(𝑣)𝐽⁻¹𝜂,𝑔^′₀𝜂=𝐽^−𝑇𝜂𝑔₀𝜂. (16)

Property 2 (see [38]). The parameter matrices have some great properties in earth-fixed frame when 𝑀₀=𝑀^𝑇₀ and 𝑀̇₀=0. Consider 𝑀^′₀𝜂=𝑀₀^′𝑇𝜂>0,∀𝜂∈ℝ^4×1,𝑥^𝑇̇𝑀^′₀𝜂−2𝐶^′₀𝑣,𝜂𝑥=0,∀𝑥∈ℝ^4×1,𝑣∈ℝ^4×1,𝜂∈ℝ^4×1,𝐷^′₀𝑣,𝜂>0,∀𝑣∈ℝ^4×1,𝜂∈ℝ^4×1. (17)

Define 𝑥₁=𝜂 and 𝑥₂=𝑣_𝑒. Then, according to (14) and (15), the following model of ROVs in the earth-fixed coordinate can be obtained: 𝑥̇₁=𝑥₂,𝑀^′₀𝑥₁𝑥̇₂=−𝐶^′₀𝑥₁,𝑥₂𝑥₂−𝐷^′₀𝑥₁,𝑥₂𝑥₂−𝑔^′₀𝑥₁+𝐽^−𝑇𝜂𝜏+𝜏_𝑑, (18) where 𝜏_𝑑=𝑑−Δ𝑀^′𝑥̇₂−Δ𝐶^′𝑥₂−Δ𝐷^′𝑥₂−Δ𝑔^′∈ℝ^4×1 is the lumped uncertainty including model uncertainties and external disturbances.

Assumption 9 (see [39]). The lumped uncertainty 𝜏_𝑑 is local Lipschitz continuous and can be bounded with a constant unknown vector 𝐹(⋅): 𝜏_𝑑<𝐹(⋅)∈ℝ^4×1. (19)

Remark 10. In practical applications, the control inputs of ROVs are obviously bounded which means that if the lumped uncertainty 𝜏_𝑑 is unbounded, we cannot effectively control the trajectory of the ROVs.

Inspired by [34, 39], the equivalent output injection SMO for ROVs is designed as follows: 𝑥̇₁=𝑥₂+𝛾₁sgn𝑥₁,𝑀^′₀𝑥₁𝑥̇₂=−𝐶^′₀𝑥₁,𝑥₂𝑥₂−𝐷^′₀𝑥₁,𝑥₂𝑥₂−𝑔^′₀𝑥₁+𝐽^−𝑇𝜂𝜏+𝛾₂sgn𝑥₂−𝑥₂, (20) where 𝛾₁∈ℝ^4×1 and 𝛾₂∈ℝ^4×1 are positive constant vectors to be designed and 𝑥₁=𝑥₁−𝑥₁ and 𝑥₂=𝑥₂−𝑥₂ are estimation errors. Consider 𝑥₂=𝑥₂+(𝛾₁sgn(𝑥₁))_eq∈ℝ^4×1; (𝛾₁sgn(𝑥₁))_eq is the equivalent output injection, which can be obtained by passing the signal 𝛾₁sgn(𝑥₁)through a low pass filter; more details can be found in [35].

Thus, the observer error dynamics can be obtained in terms of (18) and (20): ̇𝑥₁=𝑥₂−𝛾₁sgn𝑥₁,𝑀^′₀̇𝑥₂=−𝛾₂sgn𝑥₂−𝑥₂+𝜏_𝑑+𝑓(⋅), (21)(22) where 𝑓(⋅)=𝐶^′₀(𝑥₁,𝑥₂)𝑥₂−𝐶^′₀(𝑥₁,𝑥₂)𝑥₂+𝐷^′₀(𝑥₁,𝑥₂)𝑥₂−𝐷^′₀(𝑥₁,𝑥₂)𝑥₂.

Theorem 11. Under Assumptions 4–9, the equivalent output injection SMO (20) for ROVs can guarantee that the estimation errors 𝑥₁ and 𝑥₂ converge to 0 in finite time.

Proof. The proof procedure is similar to those presented in [34, 39] with the exception that a different dynamic model is used here. As demonstrated in [10], the sliding mode technology allows systems to be separately designed and analyzed for each DOF. Hence, the proof procedure will be presented in each separate DOF for simplicity. The proof will be presented in two steps.
Step 1. This step will prove that the estimation error 𝑥₁ will converge to zero in finite time. Choose a Lyapunov candidate for the estimation error dynamic (21) as 𝑉_1𝑖=12𝑥²_1𝑖, (23) where 𝑖=1∼4.
Differentiating 𝑉_1𝑖 with respect to time along (21) yields 𝑉̇_1𝑖=𝑥_1𝑖𝑥̇_1𝑖=−𝑥_1𝑖𝛾_1𝑖+𝑥_1𝑖𝑥_2𝑖≤−𝑥_1𝑖𝛾_1𝑖−𝑥_2𝑖. (24)
According to Assumptions 4, 5, and 9, the estimation error 𝑥_2𝑖 does not have finite escape time. This effectively ensures that the estimation error 𝑥_2𝑖 is in the 𝐿^′_𝑝 space. Thus, if we choose 𝛾_1𝑖>|𝑥_2𝑖|+𝜀_1𝑖, 𝜀_1𝑖>0, then the following inequality can be obtained: 𝑉̇_1𝑖≤−𝜀_1𝑖𝑥_1𝑖. (25) Therefore, the finite time convergence of 𝑥_1𝑖 to 0 will be guaranteed. Choose 𝛾_1𝑖>max_{𝑡∈[0,𝑇′]}|𝑥_2𝑖|, where 𝑇^′ is chosen large enough that 𝛾_1𝑖>|𝑥_2𝑖|+𝜀_1𝑖; thus inequality (25) will always hold. Taking the fact that |𝑥_1𝑖|=√2𝑉^1/2_1𝑖 into consideration, we have 𝑉̇_1𝑖=−√2𝜀_1𝑖𝑉^1/2_1𝑖. (26)
Using the differential inequality principle [40, 41], we can conclude that 𝑉_1𝑖=0 when 𝑡_1𝑖≥𝑡_0𝑖+(|𝑥_1𝑖(𝑡_0𝑖)|/𝜀_1𝑖), where 𝑡_0𝑖 is the initial time. Furthermore, when 𝑡_𝑖>𝑡_1𝑖, 𝑥_1𝑖=0. Hence, on the sliding mode, 𝑥_1𝑖=̇𝑥_1𝑖=0 and 𝑥₂=(𝛾₁sgn(𝑥₁))_eq. Thus, we have sgn(𝑥_2𝑖−𝑥_2𝑖)=sgn((𝛾_1𝑖sgn(𝑥_1𝑖))_eq)=sgn(𝑥_2𝑖). Therefore, the observer error dynamics (21)-(22) of the 𝑖th component can be rearranged as follows: ̇𝑥_1𝑖𝑀=0,_′0𝑖̇𝑥_2𝑖=−𝛾_2𝑖sgn𝑥_2𝑖+𝜏_𝑑𝑖+𝑓(⋅)_𝑖. (27)
Step 2. We will prove that the estimation error 𝑥_2𝑖 will converge to zero in finite time. Choose Lyapunov function candidate 𝑉_2𝑖 as follows: 𝑉_2𝑖=12𝑥²_1𝑖+12𝑀^′_0𝑖𝑥²_2𝑖. (28)
Since 𝑥_1𝑖=0 when 𝑡_𝑖>𝑡_1𝑖, differentiating 𝑉_2𝑖 along (27) yields 𝑉̇_2𝑖=−𝛾₂sgn𝑥₂+𝜏_𝑑+𝑓(⋅)_𝑖𝑥_2𝑖=−𝛾_2𝑖𝑥_2𝑖+𝜏_𝑑+𝑓(⋅)_𝑖𝑥_2𝑖≤−𝑥_2𝑖𝛾_2𝑖−𝜏_𝑑+𝑓(⋅)_𝑖. (29)
If we choose 𝛾_2𝑖≥|(𝜏_𝑑+𝑓(⋅))_𝑖|+𝜀_2𝑖, 𝜀_2𝑖>0 is a positive constant to be designed; then (29) can be rewritten as 𝑉̇_2𝑖≤−𝜀_2𝑖|𝑥_2𝑖|. Applying the same proof procedure indicated in Step 1, we can have that 𝑥_2𝑖 will convergence to 0 as 𝑡_2𝑖≥𝑡_1𝑖+(|𝑥_2𝑖(𝑡_1𝑖)|/𝜀_2𝑖). Therefore, the estimation errors 𝑥_1𝑖 and 𝑥_2𝑖 will converge to 0 in finite time. The proof is completed.

5.2. Output Feedback FO-NTSMC Design

In this subsection, a novel output feedback FO-NTSMC method for the 4-DOF trajectory tracking control of ROVs will be proposed and analyzed using the proposed equivalent output injection SMO. The design procedure mainly involves two steps. First, a novel nonlinear fractional-order nonsingular terminal sliding manifold will be proposed. Then, a control law will be designed to ensure the finite-time reachability of the proposed sliding manifold.

To simplify the application of the equivalent output injection SMO, the dynamic equations (20) will be rewritten in the body-fixed coordinate as follows: 𝜂̇=𝐽𝜂𝑣̂+𝛾₁𝜂,𝑀sgn₀𝑣̇̂=𝐽^𝑇𝜂𝛾₂sgn𝐽𝜂𝑣−𝐽𝜂𝑣̂−𝐶₀(𝑣̂)𝑣̂−𝐷₀(𝑣̂)𝑣̂−𝑔₀𝜂+𝜏, (30) where 𝜂=𝜂−𝜂 is the estimation error and 𝐽(𝜂)𝑣=𝐽(𝜂)𝑣̂+(𝛾₁sgn(𝜂))_eq. (𝛾₁sgn(𝜂))_eq is the equivalent output injection of 𝛾₁sgn(𝜂), which can be acquired by passing signal 𝛾₁sgn(𝜂) through a low pass filter [34, 39].

Define the estimated tracking error and its derivative as 𝑒₁̂=𝜂−𝜂_𝑑,𝑒₂̂=𝐽𝜂𝑣̂−𝜂̇_𝑑. (31)

Then, the estimated tracking error dynamic can be obtained as 𝑒₁̇̂=𝑒₂̂+𝛾₁𝜂,𝑒̇̂sgn₂=𝐽𝜂𝑀₀⁻¹𝐽^𝑇𝜂𝛾₂sgn^→𝑣−𝐻₀̂+𝐽̇𝜂̂𝜂̈𝑣,𝜂+𝜏𝑣−_𝑑, (32) where ^→𝑣=𝐽(𝜂)𝑣−𝐽(𝜂)𝑣̂ and 𝐻₀(𝑣̂,𝜂)=𝐶₀(𝑣̂)𝑣̂+𝐷₀(𝑣̂)𝑣̂+𝑔₀(𝜂).

In light of the TSM defined in (11), the novel fractional-order nonsingular terminal sliding mode (FO-NTSM) is designed as 𝑠̂=𝑒₂̂+𝐷^𝛼−1𝛽sig𝑒̂₁^𝜇, (33) where 0<𝛼<1, 𝛽>0, and 1/2<𝜇<1 are positive parameter matrices to be designed.

A fast-TSM-type reaching law is adopted here [21]: 𝑠̇̂=−𝑘₁𝑠̂−𝑘₂sig(𝑠)̂^𝜌, (34) where 𝑘₁>0, 𝑘₂>0, and 0<𝜌<1 are positive parameter matrices to be designed.

Then, the output feedback FO-NTSMC is designed as follows: 𝜏=𝜏₁+𝜏₂+𝜏₃+𝜏₄,𝜏₁=𝐻₀𝑣̂,𝜂+𝑀₀𝐽⁻¹𝜂𝜂̈_𝑑−𝐽̇𝜂𝑣̂,𝜏₂=−𝑀₀𝐽⁻¹𝜂𝐷^𝛼𝛽sig𝑒̂₁^𝜇,𝜏₃=−𝑀₀𝐽⁻¹𝜂𝑘₁𝑠̂+𝑘₂sig(𝑠)̂^𝜌,𝜏₄=−𝑀₀𝐽⁻¹𝜂𝐾sgn(𝑠)̂, (35) where 𝐾 is a positive constant vector to be designed.

Theorem 12. Consider an estimated tracking error dynamic (32) subjected to the output feedback FO-NTSMC (35). Then, the estimated tracking errors 𝑒₁̂ and 𝑒₂̂ will converge to zero in finite time. Moreover, according to the principle of equivalent output injection SMO, the system trajectory tracking errors 𝑒₁ and 𝑒₂ will converge to 0 in finite time.

Proof. As demonstrated in [42], it is more appropriate to prove the occurrence of the sliding mode via fractional-order Lyapunov stability theorems [2, 43] when the closed-loop systems involve fractional-order dynamics.
Inspired by the proof procedure presented in [31], a Lyapunov function is selected as follows: 𝑉_3𝑖=𝑠_𝑖̂. (36)
Then, differentiating (36) with respect to time yields 𝑉̇_3𝑖=𝑠_𝑖̇̂sgn𝑠_𝑖̂=𝑒₂̇̂+𝐷^𝛼𝛽sig𝑒̂₁^𝜇_𝑖sgn𝑠_𝑖̂. (37)
Substitute the estimated tracking error dynamic (32) and the control law (35) yields 𝑉̇_3𝑖=−𝑘_1𝑖𝑠_𝑖̂+𝑘_2𝑖𝑠̂_𝑖^𝜌𝑖+𝐾_𝑖−𝜉(⋅)𝛾₂sgn^→𝑣_𝑖sgn𝑠_𝑖̂≤−𝑘₁|𝑠|̂+𝑘₂|𝑠|̂^𝜌_𝑖−𝐾_𝑖−𝛾_2𝑖𝜉(⋅), (38) where 𝜉(⋅)=𝐽(𝜂)𝑀₀⁻¹𝐽^𝑇(𝜂).
If we choose 𝐾_𝑖 large enough such that 𝐾_𝑖−𝛾_2𝑖‖𝜉(⋅)‖>0, then inequality (38) can be rewritten as 𝑉̇_3𝑖≤−𝑘₁|𝑠|̂+𝑘₂|𝑠|̂^𝜌_𝑖=−𝑘_1𝑖𝑉_3𝑖−𝑘_2𝑖𝑉^𝜌𝑖_3𝑖. (39)
According to Lemma 7, the finite time occurrence of the sliding mode can be guaranteed. And the settling time can be estimated as 𝑡_3𝑖≤1𝑘_1𝑖1−𝜌_𝑖𝑘ln^1𝑖𝑉^1−𝜌𝑖𝑠̂𝑡₀+𝑘_2𝑖𝑘_2𝑖=1𝑘_1𝑖1−𝜌_𝑖𝑘ln^1𝑖𝑠̂𝑡₀^1−𝜌𝑖+𝑘_2𝑖𝑘_2𝑖, (40) where 𝑠(̂𝑡₀) is the initial value of 𝑠(̂𝑡).
The trajectory on the sliding manifold will be analyzed as follows. On the FO-NTSM, the behavior of the closed-loop system is dominated by the equivalent control law [7]. Differentiating the FO-NTSM with respect to time yields 𝑠_𝑖̇̂=𝑒₂̇̂+𝐷^𝛼𝛽sig𝑒̂₁^𝜇_𝑖=𝐽𝜂𝑀₀⁻¹𝐽^𝑇𝜂𝛾₂sgn^→𝑣−𝐻₀̂+𝐽̇𝜂̂𝜂̈𝑣,𝜂+𝜏𝑣−_𝑑+𝐷^𝛼𝛽sig𝑒̂₁^𝜇_𝑖. (41)
Then, the equivalent control law can be obtained as 𝜏_eq=𝐻₀𝑣̂,𝜂−𝐽^𝑇𝜂𝛾₂sgn^→𝑣+𝑀₀𝐽⁻¹𝜂×𝜂̈_𝑑−𝐽̇𝜂𝑣̂−𝐷^𝛼𝛽sig𝑒̂₁^𝜇. (42)
When the trajectory is on FO-NTSM, we have 𝑠̂=0. Furthermore, it is noteworthy that, according to Assumptions 4–9, the desired trajectory is bounded and smooth, and the estimated tracking errors 𝑒₁̂ and 𝑒₂̂ are in the set 𝐿^′_𝑝. Thus, any finite truncation of the tracking error subject to the equivalent control will be bounded. In addition, according to Theorem 11, if the gains 𝛾₁ and 𝛾₂ are chosen appropriately, the estimated system states will converge to the real ones in finite time 𝑡_2𝑖 regardless of the stability of the closed-loop system. After 𝑡=max{𝑡₂,𝑡₃}, the estimated system states will converge to the real ones and stay on the FO-NTSM. Thus, during the FO-NTSM, substituting the equivalent control law (42) into the estimated tracking error dynamics (32) yields 𝑒₁̇=𝑒₂,𝑒₂̇=−𝐷^𝛼𝛽sig𝑒₁^𝜇, (43) where 𝑒₁=𝜂−𝜂_𝑑 and 𝑒₂=𝐽(𝜂)𝑣−𝜂̇_𝑑 are the real tracking errors of the closed-loop system.
Equation (43) can be rearranged as follows: 𝑒₁̈=−𝐷^𝛼𝛽sig𝑒₁^𝜇. (44)
Now, we will prove that the tracking errors 𝑒₁ and 𝑒₂ will converge to zero in finite time using a proof procedure similar to that of [32]. Define a stopping time as follows: 𝑡_𝑠=inf𝑡≥𝑡∶𝑒(𝑡)=0. (45)
According to Definition 1 and operator _𝑡𝐷_𝑡⁻² and the associativity law, (44) can be rewritten as 𝑒₁(𝑡)−_𝑡𝐷_𝑡𝑒₁(𝑡)_𝑡=𝑡𝑡−𝑡2−𝑒₁𝑡=_𝑡𝐷_𝑡^𝛼−2𝛽sig𝑒₁(𝑡)^𝜇. (46)
According to Lemma 3, we have _𝑡𝐷_𝑡^𝛼−2𝛽sig𝑒₁(𝑡)^𝜇=_𝑡𝐼_𝑡^2−𝛼𝛽sig𝑒₁(𝑡)^𝜇≤𝐾𝛽𝑒^𝜇₁. (47)
Then, substituting (47) into (46) yields 𝑒₁(𝑡)−_𝑡𝐷_𝑡𝑒₁(𝑡)_𝑡=𝑡𝑡−𝑡2−𝑒₁𝑡≤𝐾𝛽𝑒^𝜇₁. (48)
Equation (48) can be rearranged as 𝑒₁(𝑡)−_𝑡𝐷_𝑡𝑒₁(𝑡)_𝑡=𝑡𝑡−𝑡2−𝑒₁𝑡≤𝐾𝛽𝑒^𝜇₁. (49)
Noting that 𝑒₁(𝑡)=0 at 𝑡=𝑡_𝑠, it yields _𝑡𝐷_𝑡𝑒₁(𝑡)_𝑡=𝑡2𝑡_𝑠−𝑡≤𝑒₁𝑡. (50)
If [_𝑡𝐷_𝑡𝑒₁(𝑡)]_𝑡=𝑡=0, then 𝑡_𝑠=𝑡. Otherwise, we have 𝑡_𝑠≤2𝑒₁𝑡𝑒₁̇𝑡+𝑡. (51)
Therefore, ROVs can track the desired trajectory in finite time using only the systems’ output position signal. The proof is completed.

6. Simulation Results

In this section, some numerical simulations are performed to illustrate the effectiveness of the proposed method. The nominal physical parameters of the ROV are listed in Table 1.

Table 1 

Nominal physical parameters of the ROV.

The control parameters are as follows: 𝛾_1𝑖=1.5, 𝛾_2𝑖=80, 𝛽_𝑖=0.1, 𝜇_𝑖=0.7, 𝑘_1𝑖=0.5, 𝑘_2𝑖=0.25, 𝜌_𝑖=0.9, 𝛼_𝑖=0.9, 𝐾=diag{0.5,0.5,2,0.5}, 𝜂(𝑡₀)_𝑖=−0.2, 𝑣(𝑡₀)_𝑖=0, 𝜂(𝑡₀)_𝑖=0, and 𝑣̂(𝑡₀)_𝑖=−0.05. 𝜂(𝑡₀), 𝑣(𝑡₀), 𝜂(𝑡₀), and 𝑣̂(𝑡₀) are the initial values of the real and estimated position and velocity information. The desired trajectory is 𝜂_𝑑𝑖=0.3sin(0.05𝜋𝑡). To illustrate the robustness of the proposed method, time-varying disturbances 𝑑_𝑖=10sin(0.1𝜋𝑡) and a parametric variant of 20%, which indicate that the nominal physical parameters used in the output feedback FO-NTSMC are 20% less than those used in the model, are introduced into the closed-loop system. Furthermore, the dynamics of the propellers are also taken into account. We treat the propellers as one-order initial systems with a time constant of 0.5 seconds.

To compare with the integer-order control method expressed as output feedback IO-NTSMC, we adopt the NTSM manifold defined in the second equation of (11), and then the control law (35) will be changed to 𝜏=𝜏₁+𝜏₂+𝜏₃+𝜏₄,𝜏₁=𝐻₀𝑣̂,𝜂+𝑀₀𝐽⁻¹𝜂𝜂̈_𝑑−𝐽̇𝜂𝑣̂,𝜏₂𝑀=−⁰𝐽⁻¹𝜂sig𝑒̂₂^2−𝜇′𝛽^′𝜇^′,𝜏₃=−𝑀₀𝐽⁻¹𝜂𝑘₁𝑠̂+𝑘₂sig(𝑠)̂^𝜌,𝜏₄=−𝑀₀𝐽⁻¹𝜂𝐾sgn(𝑠)̂, (52) where the new parameter matrices 𝛽^′ and 𝜇^′ can be obtained using (12). The other parameters remain unchanged for the fairness of comparision.

The control performance is listed as in Figures 2, 3, 4, 5, 6, 7, 8, and 9. Performance of the proposed equivalent output injection SMO (20) combined with fractional-order/integer-order dynamics is shown in Figures 2–5, respectively. It is clear that the proposed equivalent output SMO can ensure finite-time convergence to the real system states with both fractional-order and integer-order dynamics in the presence of model uncertainties and external disturbances. Figures 6-7 show the trajectory tracking control performance of the output feedback FO-NTSMC and IO-NTSMC. It is clear that the FO-NTSMC can obtain a faster convergence rate and a better dynamic response at the initial stage than the IO-NTSMC, whereas both of them can achieve great robustness against the lumped uncertainties. Furthermore, Figures 8-9 demonstrate that both of the methods have a very serious chattering problem in the control inputs. This problem is mainly caused by the discontinuous terms in the control laws (35) and (52) referred to as 𝜏₄.

Figure 2 

Estimated position and real position of output feedback FO-NTSMC.

Figure 3 

Estimated position and real position of output feedback IO-NTSMC.

Figure 4 

Estimated velocity and real velocity of output feedback FO-NTSMC.

Figure 5 

Estimated velocity and real velocity of output feedback IO-NTSMC.

Figure 6 

Trajectory tracking performance of output feedback FO-NTSMC and IO-NTSMC.

Figure 7 

Sliding manifold of output feedback FO-NTSMC and IO-NTSMC.

Figure 8 

Control inputs of output feedback FO-NTSMC.

Figure 9 

Control inputs of output feedback IO-NTSMC.

To eliminate the chattering phenomenon, the sign functions in 𝜏₄ of the control laws (35) and (52) are replaced by saturation functions with a boundary layer of 0.005. Corresponding simulation results are shown in Figures 10, 11, 12, 13, 14, 15, 16, and 17. It can be clearly observed that the replacement of the sign function does not have an apparent negative effect on the control performance of either method. In addition, the chattering phenomenon is effectively reduced, as shown in Figures 16-17. It is clear that the FO-NTSMC method can still guarantee a faster convergence rate and a better dynamic response than IO-NTSMC with the boundary layer.

Figure 10 

Estimated position and real position of output feedback FO-NTSMC with boundary layer.

Figure 11 

Estimated position and real position of output feedback IO-NTSMC with boundary layer.

Figure 12 

Estimated velocity and real velocity of output feedback FO-NTSMC with boundary layer.

Figure 13 

Estimated velocity and real velocity of output feedback IO-NTSMC with boundary layer.

Figure 14 

Trajectory tracking performance of output feedback FO-NTSMC and IO-NTSMC with boundary layer.

Figure 15 

Sliding manifold of output feedback FO-NTSMC and IO-NTSMC with boundary layer.

Figure 16 

Control inputs of output feedback FO-NTSMC with boundary layer.

Figure 17 

Control inputs of output feedback IO-NTSMC with boundary layer.

7. Conclusions

In this study, a novel output feedback FO-NTSMC is designed for classical nonlinear second-order systems of ROVs in light of the equivalent injection SMO and TSMC technology and fractional calculus. The model uncertainties and external disturbances are taken into account throughout the design and analysis procedures. The proposed control scheme can effectively ensure the finite-time stabilization of the closed-loop system using only the plant’s output signal. Corresponding stability analysis of the closed-loop system is presented using the fractional-order version of the Lyapunov stability theory. The results of comparative numerical simulation demonstrate the effectiveness and robustness of the proposed control method and its superior performance over that of the output feedback IO-NTSMC.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This study was funded by the Project (no. 51004085) supported by National Science Foundation of China and the Program for Zhejiang Leading Team of S&T Innovation (no. 2010R50036). The authors thank the anonymous reviewers for their detailed and valuable comments which strengthened this paper.