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 [<xref ref-type="bibr" rid="B36">36</xref>]).
The αth-order Riemann-Liouville fractional derivative of function f(t) with respect to t and the terminal value t0 is defined as
(1)Dαf(t)=dαf(t)dtα=1Γ(m-α)dmdtm∫t0tf(τ)(t-τ)α-m+1dτ
and the αth-order Riemann-Liouville fractional integration is defined as
(2)Itαt0f(t)=1Γ(α)∫t0tf(τ)dτ(t-τ)1-α,
where m-1<α≤m, m∈N, and Γ(·) is the Gamma function.
Definition 2 (see [<xref ref-type="bibr" rid="B36">36</xref>]).
The Caputo fractional derivative of order α of a continuous function f(t) is defined as follows:
(3)Dαf(t)={1Γ(m-α)∫0tf(m)(τ)(t-τ)α-m+1dτ,m-1<α<mdmdtmf(t),α=m,
where m is the first integer larger than α.
Property 1 (see [<xref ref-type="bibr" rid="B36">36</xref>]).
If the fractional derivative Dtαt0y(t) (k-1≤α<k) of a function y(t) is integrable, then
(4)Itαt0(Dtαt0y(t))=y(t)-∑j=1k[Dtα-jt0y(t)]t=t0(t-t0)α-jΓ(α-j+1).
Lemma 3 (see [<xref ref-type="bibr" rid="B37">37</xref>]).
The fractional integration operator Itαt0 with ⌊α⌋>0 is bounded as
(5)∥Iαy∥p≤K∥y∥p,1≤p≤∞.
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]:
(6)η˙=J(η)v,Mv˙+C(v)v+D(v)v+g(η)=τ+JT(η)d,
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∈R4×4 is the inertial matrix including added mass. C(v)=C0(v)+ΔC(v)∈R4×4 represents the Coriolis and centripetal forces. D(v)=D0(v)+ΔD(v)∈R4×4 is the hydrodynamic damping term, and the vector g(η)=g0(η)+Δg(η)∈R4×1 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. JT(η)d∈R4×1 is the disturbance force/moment vector expressed in the body-fixed coordinate and τ∈R4×1 is the system control input.
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:
(7)J(η)=[cosψ-sinψ00sinψcosψ0000100001].
The other simplified parameter matrices can be expressed as follows:
(8)M0=diag{m-Xu˙,m-Yv˙,m-Zw˙,Iz-Nr˙},C0(v)=[000-(mv-Yv˙)v000-(mv-Xu˙)u0000(mv-Yv˙)v-(mv-Xu˙)u00],D0(v)=diag{Xu+Xu|u|u,Yv+Yv|v|v,Zwmmm+Zw|w|w,Nr+Nr|r|r},g0(η)=[0,0,W-B,0]T,
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 [<xref ref-type="bibr" rid="B34">34</xref>]).
The MIMO dynamic system given by (6) does not have a finite escape time.
Assumption 5 (see [<xref ref-type="bibr" rid="B34">34</xref>]).
The control input τ belongs to the extended Lp space, denoted by Lp′ in this paper. Any truncation of τ to a finite time interval is bounded.
Assumption 6 (see [<xref ref-type="bibr" rid="B34">34</xref>]).
The desired trajectory ηd is smooth; that is, η˙d and η¨d are bounded, exist, and are known.
Lemma 7 (see [<xref ref-type="bibr" rid="B21">21</xref>]).
An extended Lyapunov description of finite-time stability can be given with form of fast TSM as
(9)V˙(x)+αV(x)+βVγ(x)≤0,α>0,β>0,wwwwwwwwwwwwwwwwwwwi0<γ<1,
and the settling time can be given by
(10)T≤1α(1-γ)lnαV1-γ(x0)+ββ.
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 [<xref ref-type="bibr" rid="B21">21</xref>, <xref ref-type="bibr" rid="B24">24</xref>]).
The TSM and NTSM are equivalent and can be, respectively, described by the following first-order nonlinear differential equations:
(11)s=e˙+βsig(e)μ=0,s′=e+β′sig(e˙)μ′=0,
where
(12)β′=β-1/μ=β-μ′>0,1<μ′=1μ<2.
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]:
(13)s=e¨+βμ|e|μ-1e˙,s′=e˙+β′μ′|e˙|μ′-1e¨.
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.
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 xi 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]:
(14)η˙=ve=J(η)v,(15)M0′(η)v˙e+C0′(v,η)ve+D0′(v,η)ve+g0′(η)=J-T(η)τ+d,
where ve is the velocity vector in the earth-fixed frame, and the parameter matrices in (15) can be described as follows:
(16)M0′(η)=J-T(η)M0J-1(η),C0′(v,η)=J-T(η)[C0(v)-M0J-1(η)J˙(η)]J-1(η),D0′(v,η)=J-T(η)D0(v)J-1(η),g0′(η)=J-T(η)g0(η).
Property 2 (see [38]). The parameter matrices have some great properties in earth-fixed frame when M0=M0T and M˙0=0. Consider
(17)M0′(η)=M0′T(η)>0,∀η∈R4×1,xT[M0′˙(η)-2C0′(v,η)]x=0,∀x∈R4×1,wwwwwwwwwwwwwv∈R4×1,η∈R4×1,D0′(v,η)>0,∀v∈R4×1,η∈R4×1.
Define x1=η and x2=ve. Then, according to (14) and (15), the following model of ROVs in the earth-fixed coordinate can be obtained:
(18)x˙1=x2,M0′(x1)x˙2=-C0′(x1,x2)x2-D0′(x1,x2)x2-g0′(x1)+J-T(η)τ+τd,
where τd=d-ΔM′x˙2-ΔC′x2-ΔD′x2-Δg′∈R4×1 is the lumped uncertainty including model uncertainties and external disturbances.
Assumption 9 (see [<xref ref-type="bibr" rid="B39">39</xref>]).
The lumped uncertainty τd is local Lipschitz continuous and can be bounded with a constant unknown vector F(·):
(19)|τd|<F(·)∈R4×1.
Remark 10.
In practical applications, the control inputs of ROVs are obviously bounded which means that if the lumped uncertainty τd 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:
(20)x^˙1=x^2+γ1sgn(x~1),M0′(x1)x^˙2=-C0′(x1,x^2)x^2-D0′(x1,x^2)x^2-g0′(x1)+J-T(η)τ+γ2sgn(x-2-x^2),
where γ1∈R4×1 and γ2∈R4×1 are positive constant vectors to be designed and x~1=x1-x^1 and x~2=x2-x^2 are estimation errors. Consider x-2=x^2+(γ1sgn(x~1))eq∈R4×1; (γ1sgn(x~1))eq is the equivalent output injection, which can be obtained by passing the signal γ1sgn(x~1)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):
(21)x~˙1=x~2-γ1sgn(x~1),(22)M0′x~˙2=-γ2sgn(x-2-x^2)+τd+f(·),
where f(·)=C0′(x1,x^2)x^2-C0′(x1,x2)x2+D0′(x1,x^2)x^2-D0′(x1,x2)x2.
Theorem 11.
Under Assumptions 4–9, the equivalent output injection SMO (20) for ROVs can guarantee that the estimation errors x~1 and x~2 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 x~1 will converge to zero in finite time. Choose a Lyapunov candidate for the estimation error dynamic (21) as
(23)V1i=12x~21i,
where i=1~4.
Differentiating V1i with respect to time along (21) yields
(24)V˙1i=x~1ix~˙1i=-|x~1i|γ1i+x~1ix~2i≤-|x~1i|[γ1i-|x~2i|].
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 Lp′ space. Thus, if we choose γ1i>|x~2i|+ε1i, ε1i>0, then the following inequality can be obtained:
(25)V˙1i≤-ε1i|x~1i|.
Therefore, the finite time convergence of x~1i to 0 will be guaranteed. Choose γ1i>maxt∈[0,T′]|x~2i|, where T′ is chosen large enough that γ1i>|x~2i|+ε1i; thus inequality (25) will always hold. Taking the fact that |x~1i|=2V1i1/2 into consideration, we have
(26)V˙1i=-2ε1iV1i1/2.
Using the differential inequality principle [40, 41], we can conclude that V1i=0 when t1i≥t0i+(|x~1i(t0i)|/ε1i), where t0i is the initial time. Furthermore, when ti>t1i, x~1i=0. Hence, on the sliding mode, x~1i=x~˙1i=0 and x~2=(γ1sgn(x~1))eq. Thus, we have sgn(x-2i-x^2i)=sgn((γ1isgn(x~1i))eq)=sgn(x~2i). Therefore, the observer error dynamics (21)-(22) of the ith component can be rearranged as follows:
(27)x~˙1i=0,M0i′x~˙2i=-γ2isgn(x~2i)+τdi+f(·)i.
Step 2. We will prove that the estimation error x~2i will converge to zero in finite time. Choose Lyapunov function candidate V2i as follows:
(28)V2i=12x~21i+12M0i′x~22i.
Since x~1i=0 when ti>t1i, differentiating V2i along (27) yields
(29)V˙2i=(-γ2sgn(x~2)+τd+f(·))ix~2i=-γ2i|x~2i|+(τd+f(·))ix~2i≤-|x~2i|(γ2i-|(τd+f(·))i|).
If we choose γ2i≥|(τd+f(·))i|+ε2i, ε2i>0 is a positive constant to be designed; then (29) can be rewritten as V˙2i≤-ε2i|x~2i|. Applying the same proof procedure indicated in Step 1, we can have that x~2i will convergence to 0 as t2i≥t1i+(|x2i(t1i)|/ε2i). 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:
(30)η^˙=J(η)v^+γ1sgn(η~),M0v^˙=JT(η)γ2sgn(J(η)v--J(η)v^)-C0(v^)v^-D0(v^)v^-g0(η)+τ,
where η~=η-η^ is the estimation error and J(η)v-=J(η)v^+(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
(31)e^1=η^-ηd,e^2=J(η)v^-η˙d.
Then, the estimated tracking error dynamic can be obtained as
(32)e^˙1=e^2+γ1sgn(η~),e^˙2=J(η)M0-1(JT(η)γ2sgn(v→)-H0(v^,η)+τ)+J˙(η)v^-η¨d,
where v→=J(η)v--J(η)v^ and H0(v^,η)=C0(v^)v^+D0(v^)v^+g0(η).
In light of the TSM defined in (11), the novel fractional-order nonsingular terminal sliding mode (FO-NTSM) is designed as
(33)s^=e^2+Dα-1[βsig(e^1)μ],
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]:
(34)s^˙=-k1s^-k2sig(s^)ρ,
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:
(35)jτ=τ1+τ2+τ3+τ4,τ1=H0(v^,η)+M0J-1(η)(η¨d-J˙(η)v^),τ2=-M0J-1(η)(Dα[βsig(e^1)μ]),τ3=-M0J-1(η)(k1s^+k2sig(s^)ρ),τ4=-M0J-1(η)Ksgn(s^),
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 e^1 and e^2 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:
(36)V3i=|s^i|.
Then, differentiating (36) with respect to time yields
(37)V˙3i=s^˙isgn(s^i)={e^˙2+Dα[βsig(e^1)μ]}isgn(s^i).
Substitute the estimated tracking error dynamic (32) and the control law (35) yields
(38)V˙3i=-{k1i|s^i|+k2i|s^i|ρi+Ki-(ξ(·)γ2sgn(v→))isgn(s^i)}≤-{k1|s^|+k2|s^|ρ}i-(Ki-γ2i∥ξ(·)∥),
where ξ(·)=J(η)M0-1JT(η).
If we choose Ki large enough such that Ki-γ2i∥ξ(·)∥>0, then inequality (38) can be rewritten as
(39)V˙3i≤-{k1|s^|+k2|s^|ρ}i=-k1iV3i-k2iV3iρi.
According to Lemma 7, the finite time occurrence of the sliding mode can be guaranteed. And the settling time can be estimated as
(40)t3i≤1k1i(1-ρi)lnk1iV1-ρi(s^(t0))+k2ik2i=1k1i(1-ρi)lnk1i|s^(t0)|1-ρi+k2ik2i,
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
(41)s^˙i={e^˙2+Dα[βsig(e^1)μ]}i={11J(η)M0-1(JT(η)γ2sgn(v→)-H0(v^,η)+τ)+J˙(η)v^-η¨d+Dα[βsig(e^1)μ]11}i.
Then, the equivalent control law can be obtained as
(42)τeq=H0(v^,η)-JT(η)γ2sgn(v→)+M0J-1(η)×(η¨d-J˙(η)v^-Dα[βsig(e^1)μ]).
When the trajectory is on FO-NTSM, we have s^=0. Furthermore, it is noteworthy that, according to Assumptions 4–9, the desired trajectory is bounded and smooth, and the estimated tracking errors e^1 and e^2 are in the set Lp′. 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 t2i 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
(43)e˙1=e2,e˙2=-Dα[βsig(e1)μ],
where e1=η-ηd and e2=J(η)v-η˙d are the real tracking errors of the closed-loop system.
Equation (43) can be rearranged as follows:
(44)e¨1=-Dα[βsig(e1)μ].
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:
(45)ts=inf{t≥t-:e(t)=0}.
According to Definition 1 and operator Dt-2t- and the associativity law, (44) can be rewritten as
(46)e1(t)-[Dtt-e1(t)]t=t-(t-t-)2-e1(t-)=Dtα-2t-[βsig(e1(t))μ].
According to Lemma 3, we have
(47)Dtα-2t-[βsig(e1(t))μ]=It2-αt-[βsig(e1(t))μ]≤Kβ∥e1μ∥.
Then, substituting (47) into (46) yields
(48)e1(t)-[Dtt-e1(t)]t=t-(t-t-)2-e1(t-)≤Kβ∥e1μ∥.
Equation (48) can be rearranged as
(49)∥e1(t)-[Dtt-e1(t)]t=t-(t-t-)2∥-∥e1(t-)∥≤Kβ∥e1μ∥.
Noting that e1(t)=0 at t=ts, it yields
(50)∥[Dtt-e1(t)]t=t-2∥(ts-t-)≤∥e1(t-)∥.
If [Dtt-e1(t)]t=t-=0, then ts=t-. Otherwise, we have
(51)ts≤2∥e1(t-)∥∥e˙1(t-)∥+t-.
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.
Nominal physical parameters of the ROV.
Parameters
Value
m/kg
200
W/N
2000
B/N
2000
ZB/m
−0.108
Ix/(kg·m^{2})
25.8
Iy/(kg·m^{2})
30.1
Iz/(kg·m^{2})
37.8
Xu˙/kg
−33.6
Yv˙/kg
−37
Zw˙/kg
−62.9
Nr˙/(kg·m^{2})
−25
Nr/(kg/s)
−170
Yv/(kg/s)
−120
Zw/(kg/s)
−180
Nr/(kg/s)
−170
Xu|u|/(kg/m)
−213
Yv|v|/(kg/m)
−270
Zw|w|/(kg/m)
−410
Nr|r|/(kg/m)
−35
The control parameters are as follows: γ1i=1.5, γ2i=80, βi=0.1, μi=0.7, k1i=0.5, k2i=0.25, ρi=0.9, αi=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 v^(t0)i=-0.05. η(t0), v(t0), η^(t0), and v^(t0) are the initial values of the real and estimated position and velocity information. The desired trajectory is ηdi=0.3sin(0.05πt). To illustrate the robustness of the proposed method, time-varying disturbances di=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
(52)jτ=τ1+τ2+τ3+τ4,τ1=H0(v^,η)+M0J-1(η)(η¨d-J˙(η)v^),τ2=-M0J-1(η)sig(e^2)2-μ′β′μ′,τ3=-M0J-1(η)(k1s^+k2sig(s^)ρ),τ4=-M0J-1(η)Ksgn(s^),
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.
Estimated position and real position of output feedback FO-NTSMC.
Estimated position and real position of output feedback IO-NTSMC.
Estimated velocity and real velocity of output feedback FO-NTSMC.
Estimated velocity and real velocity of output feedback IO-NTSMC.
Trajectory tracking performance of output feedback FO-NTSMC and IO-NTSMC.
Sliding manifold of output feedback FO-NTSMC and IO-NTSMC.
Control inputs of output feedback FO-NTSMC.
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.
Estimated position and real position of output feedback FO-NTSMC with boundary layer.
Estimated position and real position of output feedback IO-NTSMC with boundary layer.
Estimated velocity and real velocity of output feedback FO-NTSMC with boundary layer.
Estimated velocity and real velocity of output feedback IO-NTSMC with boundary layer.
Trajectory tracking performance of output feedback FO-NTSMC and IO-NTSMC with boundary layer.
Sliding manifold of output feedback FO-NTSMC and IO-NTSMC with boundary layer.
Control inputs of output feedback FO-NTSMC with boundary layer.
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.
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