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Information fusion model-free adaptive control algorithm and unmanned surface vehicle heading control ⁎

T

⁎

Quanquan Jiang, Ye Li , Yulei Liao , Yugang Miao, Wen Jiang, Haowei Wu Science and Technology on Underwater Vehicle Laboratory, Harbin Engineering University, Harbin 150001, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Unmanned surface vehicle Heading control Model free adaptive control Input and output information fusion

Based on the model-free adaptive control (MFAC) theory, the heading control problem of unmanned surface vehicles (USVs) with uncertainties is explored. First, as a USV’s heading subsystem does not satisfy the quasilinear assumption of the MFAC theory, a new type of input and output information fusion MFAC, i.e., the IOIF–MFAC algorithm is proposed. The novel algorithm proposed herein renders the MFAC theory applicable to the heading control of USVs. Next, the input and output information of the heading subsystem, namely the rudder angle and heading angle, are combined, and the data model of the heading subsystem is subsequently deduced using a compact format dynamic linearization method. Based on which, the stability of the control system is proved. Finally, the eﬀectiveness and practicability of the IOIF–MFAC algorithm are veriﬁed by simulation and ﬁeld experiments through the “Dolphin IB” test platform developed by our group.

1. Introduction Unmanned surface vehicles (USVs) have gained considerable attention worldwide because of their superiority in military and civilian ﬁelds, including enemy ship tracking, maritime investigation, surface rescue, and water-quality monitoring [1–4]. A favorable heading control performance is the precondition for a USV to perform various tasks smoothly. However, at diﬀerent speeds, the hull wetting area, draft, and other physical parameters will change signiﬁcantly, resulting in hull hydrodynamic coeﬃcient changes. Additionally, a USV exhibits the characteristics of nonlinear, uncertain, and time varying. Furthermore, it is extremely diﬃcult to establish an accurate USV dynamic model owing to the considerable eﬀects of environmental disturbances [5–7]. Therefore, it is crucial to research the heading control of USVs with uncertainties. Currently, many control approaches have been proposed, such as PID control [8,9], fuzzy control [10–12], adaptive control [13,14], sliding mode control [15,16], back-stepping control [17], neural network control [18], and optimization control [19]. However, most of the control methods are still in the stage of theoretical and simulation research. A USV is a complex control system whose motion control challenges [5] are as follows: (1) It is diﬃcult to construct an accurate mathematical model for it, thus rendering it diﬃcult for many control methods based on a mathematical model to ensure its control

⁎

performance in practical applications. (2) Owing to the uncertainties of model perturbation and environmental disturbance, the control algorithm (relying on exact mathematical model for controller design) developed by the conventional "model-oriented" design strategy is poorly adaptive, and it is diﬃcult to guarantee the robustness and stability of the control system. That is to say, it is diﬃcult for the conventional "model-oriented" control method to achieve the desired control eﬀect in practical application, and the physical realization is diﬃcult, thus hindering its engineering application signiﬁcantly. In recent years, the data-driven control method has become a research hotspot. In fact, a PID controller is a classic, typically used "oﬄine" data-driven control method. However, in simulation and ﬁeld experiments, a USV is easily aﬀected by the model perturbation and environmental disturbance force, thus rendering it diﬃcult for the PID controller to maintain a consistent control performance; the parameters need to be readjusted to stabilize the system [20,21]. Therefore, a model-free control method of a USV with good robustness and adaptability is required. The model-free adaptive control (MFAC) algorithm was proposed by Hou in 1994 [22]. Because the design process of the algorithm does not depend on the mathematical model of the controlled system and the pseudo partial derivative is insensitive to the time-varying structure and time-varying phase of the system, the algorithm exhibits strong robustness and adaptability. Wu et al. [23] added tracking diﬀerentiator to model-free adaptive

Corresponding authors. E-mail address: [email protected] (Y. Li).

https://doi.org/10.1016/j.apor.2019.06.008 Received 20 January 2019; Received in revised form 21 May 2019; Accepted 3 June 2019 0141-1187/ © 2019 Elsevier Ltd. All rights reserved.

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generalized Lipschitz condition, that is, (2)|y (k1 + 1) − y (k2 + 1)| ≤ b |um (k1) − um (k2)|

controller and designed a controller suitable for complex systems such as non-linear, time-delay, time-varying and strong coupling. Bu et al. [24], based on model free adaptive control theory, considered the problem of data driven control (DDC) for a class of non-aﬃne nonlinear systems with output saturation. A time varying linear data model for such nonlinear system was ﬁrst established by using the dynamic linearization technique, then a DDC algorithm was constructed only depending on the control input data and the saturated output data. Two simulation examples were exploited to verify the eﬀectiveness of the proposed method. However, the above two control method can not be applied to USV’s heading control because the heading subsystem of a USV does not satisfy the quasi-linear hypothesis of the controlled system based on MFAC theory. In 2018, based on model-free adaptive control theory, Li et al. [25] designed a PID-MFA cascade controller, which realized the heading control of USV through angular velocity guidance. However, due to the complex of the controllers, the control method still remains in the stage of simulation research. Besides, the stability of the USV’s heading subsystem also lacks strict proof. In 2019, aiming at the problem of heading subsystem of USV does not satisfy the quasi-linear assumption, Liao et al [26] redeﬁne the output of USV as the linear sum of heading and angular velocity, based on which the redeﬁned compact format model free adaptive control (RO-CFDL-MFAC) method was proposed. However, the system’s robustness is poor and the control performance is too sensitive to the redeﬁned output gain when the RO-CFDL-MFAC method is applied to the USV’s heading control. In addition, some low-cost USV seldom equipped with angular velocity sensors. Du to environmental disturbance, the angular velocity of USV ﬂuctuates greatly, besides, angular velocity sensors has the problem of measuring noise, thus the stability of the controlled system is poor when using RO-CFDL-MFAC method. The outline of this article is as follows. First, we brieﬂy analyze the challenges encountered in the compact-form dynamic linearization–MFAC (CFDL–MFAC) method in USV motion control applications. Subsequently, the design process of the IOIF–MFAC algorithm is introduced and the stability of the IOIF–MFAC scheme is veriﬁed with rigorous mathematical analysis. Next, the eﬀectiveness of the IOIF–MFAC algorithm is veriﬁed using the “Dolphin IB” small USV to perform simulation studies and ﬁeld contrast experiments. Finally, concludes the work.

|y (k1 + 1) − y (k2 + 1)| ≤ b |um (k1) − um (k2)|

where y (ki + 1) = f (y (ki ), ⋯, y (ki − n y ), um (ki ), ⋯, um (ki − nu )) , i =1, 2; b is a positive constant. From a practical viewpoint, the assumptions above on control objects are reasonable and acceptable. Assumption 1 is a typical constraint on general nonlinear systems in control system design. According to [22], theorem 1 can be obtained. Theorem 1. If a nonlinear system (1) satisﬁes assumptions 1 and 2, when |Δum (k )| ≠ 0 , there must exist a time-varying parameter ϕc (k ) ∈ R named the pseudo partial derivative (PPD) that causes the system of Eq. (1) to transform into the CFDL model of Eq. (3): (3) Δy (k + 1) = ϕc (k ) Δum (k )

Δy (k + 1) = ϕc (k ) Δum (k )

(3)

is bounded at any time. In Eq. where ϕc (k ) Δy (k + 1) = y (k + 1) − y (k ), Δum (k ) = um (k ) − um (k − 1) .

(3),

2.2. Compact-form dynamic linearization based on MFAC scheme According to [22], the CFDL–MFAC scheme is as follows:

um (k ) = um (k − 1) + ϕˆc (k ) = ϕˆc (k − 1) +

ρϕˆc (k ) λ + |ϕˆc (k )|2

(y * (k + 1) − y (k )) (4)

ηΔum (k − 1) × (Δy (k ) − ϕˆc (k − 1) Δum (k − 1)) μ + Δum (k − 1)2 (5)

ϕˆc (k ) = ϕˆc (1), if |ϕˆc (k )| ≤ ε or |Δum (k − 1)| ≤ ε orsign(ϕˆc (k )) ≠ sign(ϕˆc (1))

(6)

ϕc (k ) is the PPD; ϕˆc (k ) is the estimation of the PPD; um (k ) and um (k − 1) denote the controlled system’s input at time k and k − 1, respectively; y * (k + 1) and y (k ) represent the expected and actual output of system (1), respectively; ϕˆc (1) is the initial value of the PPD, where ε is a suﬃciently small positive number; sign(⋅) is the symbolic function; μ > 0 and λ > 0 are weight coeﬃcients to adjust the change rates of the PPD and controller output, respectively; ρ ∈ (0, 1] and η ∈ (0, 1] are the step factors. Eq. (4) is the control law; Eq. (5) is the PPD estimation algorithm; Eq. (6) is the PPD reset algorithm. In the CFDL–MFAC scheme, the dynamic I/O data of the controlled system is used to estimate the time-varying parameter ϕc (k ). The estimated ϕˆc (k ) and the minimized one-step forward forecast error are introduced into the control law; subsequently, the new controller output um (k ) is obtained. Based on the controller output and the new observation y (k ) , the new PPD is estimated as ϕˆc (k + 1) . The whole control process is shown in Fig.1. To ensure the stability of the closed-loop system, the controlled system must satisfy the following assumptions, in addition to assumptions 1 and 2 [22].

2. Introduction of CFDL–MFAC scheme and analysis of USV heading control applications The basic principle of the MFAC method is establishing a dynamic linear data model that is equivalent to a nonlinear system at every operating point, and using the I/O data of the controlled system to estimate the PPD online. Subsequently, the weighted one-step forward controller is designed to realize the data-driven MFAC of nonlinear systems [22]. 2.1. Compact-form dynamic linearization

Assumption 3. For a given bounded expected output signal y * (k + 1) , a bounded control input signal um* (k ) that causes the system output be equal to y * (k + 1) always exists.

Generally, single input and single output discrete-time nonlinear systems can be expressed as follows [22]:

y (k + 1) = f (y (k ), ⋯, y (k − n y ), um (k ), ⋯, um (k − nu ))

(2)

Assumption 4. For any moment k and Δum (k ) ≠ 0 , the PPD sign of the system remains unchanged, i.e., ϕc (k ) > ε > 0 or ϕc (k ) < − ε , where ε is a small positive number.

(1)

where y (k ) ∈ R, um (k ) ∈ R , represent the input and output of the system, respectively, at time k ; n y and nu are the orders of the system. Obviously, the heading system of the USV can be expressed as (1). The following assumptions are made for system (1) [22]:

[22] 2.3. Analysis of CFDL–MFAC scheme in USV heading control applications

Assumption 1. Except for ﬁnite time points, the partial derivatives of f (⋯) with respect to the (n y + 2)th variable is continuous.

The principle of the heading control subsystem of the USV is shown in Fig.2.

Assumption 2. Except for ﬁnite moments, system (1) satisﬁes the 2

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Fig. 1. Principle of CFDL–MFAC scheme. Table 1 Procedure of IOIF–MFAC method for USV heading control. Algorithm IOIF–MFAC method for USV heading control Require Desired heading ψ* (k ) ; Actual heading ψ (k ) ; Initialize λ, μ, η , ρ, γ //Coeﬃcients of IOIF–MFAC method ϕˆ (1) //Initial value of PPD c

u (1), u (2), y (1), y (2) //Initial value of I/O data Repeat e (k ) ⟵y * (k ) − y (k ) //tracking error ϕˆc (k ) ⟵ϕˆc (k − 1) +

ηΔum (k − 1) (Δy (k ) μ + |Δum (k − 1)|2

− ϕˆc (k − 1) Δum (k − 1)) //PPD update

If |Δum (k − 1)| ≤ ε or |ϕˆc (k )| ≤ ε or |sign(ϕˆc (k )) ≠ sign(ϕˆc (1)) /PPD resetting ϕˆc (k ) ⟵ϕˆc (1) End

um (k ) ⟵um (k − 1) +

Fig. 2. Principle of USV heading control subsystem.

heading subsystem of the USV satisﬁes Assumption 4. Prove. Based on formula (7), we can obtain

Δy (k + 1) = Δψ (k + 1) + γ × Δum (k ) Δψ (k + 1) =⎛ + γ ⎞ × Δum (k ) ⎝ Δum (k ) ⎠ ⎜

3. Design and analysis of IOIF–MFAC algorithm

⎟

(9)

Compared with formula (3), we can obtain

Because the USV’s heading subsystem does not satisfy the quasilinear assumption of the MFAC theory, a new type of input-and-outputinformation-fusion MFAC (IOIF–MFAC) algorithm is proposed.

ϕc (k ) =

Δψ (k + 1) +γ Δum (k )

(10)

Because the USV’s heading subsystem satisﬁes assumption 2, formula (11) can be obtained.

3.1. Design of IOIF–MFAC algorithm

|ψ (k1 + 1) − ψ (k2 + 1)| ≤ b |um (k1) − um (k2)|

For the controlled system to satisfy assumption 4 and utilize the input and output information of the USV’s heading subsystem, we integrate the input and output information of the controlled system, based on which the new system output is deﬁned as follows:

(11)

According to formula (11), expression (12) can be obtained.

|Δψ (k + 1)| ≤b |Δum (k + 1)|

(7)

From (12), we know that

The expected system output is as follows:

y * (k + 1) = ψ* (k + 1)

+ 1) − y (k )) // Controller output update

k = k + 1 // Time update Until stop command

where ψd is the expected heading, ψ is the measured heading, δd is the expected rudder angle, Δψ is the heading error. It is noteworthy that the USV’s heading subsystem does not satisfy assumption 4. For example, the rudder angle increases from –20° to –10° at time k , but the USV’s heading angle continues to decrease at time k + 1. Therefore, to ensure the stability of the control system, the CFDL–MFAC scheme should be improved.

y (k + 1) = ψ (k + 1) + γ × um (k )

ρϕˆc (k ) (y * (k λ + |ϕˆc (k )|2

(12) |Δψ (k + 1)| |Δum (k + 1)|

is bounded; thus, −

Δψ (k + 1) Δum (k )

is

Δψ (k + 1) Δum (k )

< b. Without losing generality, we only discuss the case of ϕc (k ) > 0 herein. According to formula (10), for any moment k and Δu (k ) ≠ 0 , Δψ (k + 1) Δψ (k + 1) when Δu (k ) + γ > 0 , i.e., γ > − Δu (k ) , we can obtain ϕc (k ) > 0 . m m Combined with expression (12), when γ > b , assumption 4 holds. In summary, there exists M when γ > M ≥ b , and the USV’s heading subsystem satisﬁes Assumption 4.

bounded and −

(8)

where ψ (k + 1) and ψ* (k + 1) represent the actual and expected heading of the USV, respectively, at time k +1; γ is a positive bounded constant and represents the fusion coeﬃcient; y (k + 1) and y * (k + 1) are the new actual and desired output of the system, respectively. Based on formulas (4)–(8), the IOIF–MFAC scheme can be obtained. The procedure of the IOIF–MFAC method for the USV heading control is shown in Table 1.

Theorem 2. When the fusion coeﬃcient and γ > b, y * (k + 1) = y * =const , we use the IOIF–MFAC scheme. A positive number λ min > 0 exists; when λ > λ min , the following conclusions can be obtained.

3.2. System stability analysis Remark 2. There exists a bounded constant M , when γ > M , and the

the system output tracking error is monotonically convergent, and 3

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lim |y * − y (k + 1)| = 0 .

the closed-loop system is BIBO stable, i.e., the output sequence {y (k )} and input sequence {um (k )} are bounded.

ϕ˜c (k ) = ⎡1 − ⎣

˜

⎤ ϕ (k μ + |Δum (k − 1)|2 ⎦ c

− 1) + ϕc (k − 1) − ϕc (k )

1−

, is

monotonically

=1

(19)

(20)

Based on (18) and (20), we obtain formula (21).

|e (k + 1)| ≤ d2 |e (k )| ≤ d 22 |e (k − 1)| ≤ ⋯≤d 2k |e (1)|

η |Δum (k − × ||ϕ˜c (k − 1)| + |ϕc (k − 1) − ϕc (k )|| μ + |Δum (k − 1)|2

μ + |Δum (k − 1)|2 − 1)|2 ; furthermore,

b+γ 2 λ min

ρϕˆc (k ) ϕc (k ) ρϕˆc (k ) ϕc (k ) =1− 2 ˆ λ + |ϕc (k )| λ + |ϕˆc (k )|2

(13)

(21)

It is proven that conclusion (i) of Theorem 2 is established, i.e., the output tracking error is monotonically convergent. Meanwhile, because conclusion (i) implies that y * (k + 1) = y * =const , y (k + 1) = ψ (k + 1) + γ × um (k ) is bounded. Next, we prove the um (k ) . boundedness of Using the inequality ( λ )2 + |ϕˆc (k )|2 ≥ 2 λ ϕˆc (k ) and the control algorithm (5), we obtain formula (22).

(14) where

<

≤ 1 − ρM1 = d2 <1

1)|2

η | Δum (k − 1)|2

2 λ

According to the equation above, 0 < ρ ≤ 1 and λ > λ min , a constant d2 exists that enables formula (20) to hold.

We take the absolute value on both sides of expression (13) to obtain

|ϕ˜c (k )| ≤ 1 −

(b + γ )

=

Prove. According to remark 2, when γ > b , system (1) satisﬁes assumptions 1, 2, 3 and 4. If condition |ϕˆc (k )| ≤ ε or |Δum (k − 1)| ≤ ε or sign(ϕˆc (k )) ≠ sign(ϕˆc (1)) is met, then ϕˆc (k ) is obviously bounded. We deﬁne ϕ˜c (k ) = ϕˆc (k ) − ϕc (k ) as the estimation error of the PPD and subtract ϕc (k ) from both sides of the PPD estimation algorithm (5); therefore, we obtain η | Δu (k − 1)|2 (13)ϕ˜c (k ) = ⎡1 − μ + |Δum (k − 1)|2 ⎤ ϕ˜c (k − 1) + ϕc (k − 1) − ϕc (k ) m ⎣ ⎦ η | Δum (k − 1)|2

ϕˆc (k ) ϕc (k ) (b + γ ) ϕˆc (k ) (b + γ ) ϕˆc (k ) ≤ ≤ 2 2 λ + |ϕˆc (k )| λ + |ϕˆc (k )| 2 λ ϕˆc (k )

0 < M1 ≤

k →∞

increasing approximately

|Δum (k if |Δum (k − 1)| ≤ ε , a minimum value of ηε 2 exists. Therefore, when |Δum (k − 1)| ≤ ε , there must be a constant 2 μ+ε

d1 that satisﬁes expression (15). 0≤ 1−

η |Δu (k − 1)|2 μ + |Δu (k − 1)|2

c

ρϕˆc (k ) ≤ |e (k )| λ + |ϕˆc (k )|2

ηε 2 μ + ε2 = d1 <1

≤1−

≤

|e (k )|

|e (k )| ≤ M2 |e (k )| (22) ρ 2 λmin

is a bounded positive constant.

According to expressions (21) and (22), we can obtain Eq. (23).

|um (k )| ≤ |um (k ) − um (k − 1)| + |um (k − 1)| ≤ |um (k ) − um (k − 1)| + |um (k − 1) − um (k − 2)| + |um (k − 2)| ≤ |Δum (k )| + |Δum (k − 1)|+⋯+|Δum (2)| + |um (1)| ≤ M2 (|e (k )| + |e (k − 1)| + |e (k − 2)|+⋯+|e (2)|) + |u (1)|

(16)

≤ M2 (d 2k − 1 |e (1)| + d 2k − 2 |e (1)|+⋯+d2 |e (1)|) + |u (1)| d2 |e (1)| + |u (1)| ≤ M2 1 − d2

∼

Formula (13) proves the boundedness of ϕc (k ) . Because ϕc (k ) < γ + b and ϕˆc (k ) = ϕ˜c (k ) + ϕc (k ) , we know that ϕˆc (k ) is bounded. We deﬁne the system tracking error as formula (17).

e (k + 1) = y * − y (k + 1) = ψ* − ψ (k + 1) − γ × um (k )

(23)

The boundedness of um (k ) is proven by Eq. (23); therefore, conclusion (ii) of Theorem 2 holds, i.e., the output sequence y (k + 1) and input sequence um (k ) are BIBO stable. when addressing the tracking problem of a variable expected output signal. First, we create an augmented system as follows:

(17)

Based on formulas (3) and (17), we obtain formula (18).

z (k + 1) = f (y (k ), ⋯, y (k − n y ), um (k ), ⋯, um (k − nu )) − y * (k + 1)

|e (k + 1)| = |y * − y (k + 1)| = |y * − y (k ) − ϕc (k ) Δum (k )|

(24) For this augmented system, the method above can prove its stability and monotonic convergence, i.e., the convergence and stability of the augmented system is equivalent to the convergence and stability of the original nonlinear system (1).

= |e (k ) − ϕc (k ) Δum (k )| ρϕˆc (k ) λ + |ϕˆc (k )|2

ρϕˆc (k ) ϕc (k ) |e (k )| λ + |ϕˆ (k )|2

ρ 2 λ min

where M2 =

|ϕ˜c (k )| ≤ d1 |ϕ˜c (k − 1)| + 2γ

c

2 λ ϕˆc (k )

(15)

≤ d12 |ϕ˜c (k − 2)| + 2d1 γ + 2γ ≤⋯ 2γ (1 − d1k − 1) ≤ d1k − 1 |ϕ˜c (1)| + 1 − d1

≤ 1−

ρϕˆc (k )

≤

According to formulas (10) and (12), ϕc (k ) is bounded and |ϕc (k )| ≤ b + γ . Therefore, it can be deduced that |ϕ (k − 1) − ϕ (k )| ≤ 2(b + γ ) < 2γ . According to expressions (14) and (15), we can obtain expression (16).

= e (k ) − ϕc (k ) e (k )

ρϕˆc (k )(y * − y (k )) λ + |ϕˆ (k )|2

|Δum (k )| =

4. Experimental research and analysis

(18)

We use the “Dolphin IB” USV shown in Fig. 3 as the investigated object to verify the proposed heading control method. The primary parameters of “Dolphin IB” USV are shown in Table 2. According to the theory of ship maneuverability, when the time

From assumption 4 and by resetting algorithm (7), we obtain ϕˆc (k ) ϕc (k ) ≥ 0 [24]. Meanwhile, we let λ min = (b + γ )2 4 . When choosing λ > λ min , a positive constant 0 < M1 < 1 exists that enables formula (19) to hold. 4

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Fig. 3. “Dolphin IB” USV. Fig. 4. Heading and rudder angle response curves without uncertain eﬀect. Table 2 Primary parameters of “Dolphin-IB”.

4.2. Simulation experiments with uncertain eﬀect

Parameters

Parameter values

Length Weight Maximum speed Propulsion method

2.0 m 35.0 kg 1.5 m/s Vector advancement

Two experimental cases were performed. The initial state of the heading control subsystem and the control parameters are consistent those in section 4.1. Case 1. Let model parameter K increase by 250%. The heading angle response curves are shown in Fig. 5. Case 2. Let the USV heading subsystem exhibit a 4 s time delay. The simulation results are shown in Fig. 6.

delay characteristic of a USV’s heading subsystem is considered, the discrete mathematical motion model of the heading subsystem of the USV can be expressed as

ψ (k ) = ψ (k − 1) + ψ˙ (k ) × Ts ⎧ ⎪ ψ˙ (k ) = ψ˙ (k − 1) + ψ¨ (k ) × Ts ⎨ ⎪ ψ¨ (k ) = (Kum (k − τ ) − ψ˙ (k ))/ T ⎩

We used the root mean square (RMS) of the heading track error to measure the control performance of the three controllers. The RMS of the heading tracking errors of 50–100 s, 150–200 s, and 250–300 s of the three simulation experiments above are calculated. The calculation results are shown in Table 3.

(25)

where K and T are the maneuver coeﬃcients; Ts is the sampling time; τ is the time delay constant for the heading subsystem; ψ (k ) , ψ˙ (k ) , and ψ¨ (k ) represent the heading angle, angular velocity, and angular acceleration of the system at time k , respectively. Based on the maneuverability test data and system identiﬁcation method [23], the model parameters of the “Dolphin IB” USV are obtained as K = 0.186, T = 1.068, when the speed of the USV is approximately 1 m/s.

4.3. Field experiments and analysis To further verify the practicability of the IOIF–MFAC method and promote engineering applications, the proposed control method is applied to the heading control of “Dolphin IB.” Field experiments are shown in Fig. 7. Two cases of ﬁeld experiments were performed.

4.1. Simulation experiments without uncertain eﬀect The initial state of the heading control subsystem is ψ (1) = ψ (2) = 00, um (1) = um (2) = 00 . ψ (1) and ψ (2) represent the actual heading angles of the USV at times k = 1 and k = 2 , respectively. um (1) and um (2) represent the heading angular velocities at times k = 1 and k = 2 , respectively. The desired heading angle is set as 120°, 30°, and 120° in the 1 st, 2nd, and 3rd 100 s respectively. The PID controller parameters are Kp = 1.5, KI = 0.12, K d = 0.1. The controller parameters λ= of the IOIF–MFAC algorithm are 0.1, μ = 100, η = 1, ρ = 0.6, ε = 0.1, γ = 2.0 . The controller parameters of the RO–CFDL–MFAC method proposed by Fu [23] are λ = 5, μ = 100, η = 1, ρ = 1, ε = 0.1, K = 10 . The simulation results without external interference are shown in Fig. 4. The simulation results show that when no disturbance occurs in the USV’s heading subsystem, the IOIF–MFAC, PID, and RO-CFDL-MFAC methods exhibit similar control performance.

Fig. 5. Heading angle response curves of case 1. 5

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Fig. 8. Heading response with thruster voltage at 5V. Fig. 6. Heading angle response curves of case 2.

Case 1. The thruster voltage is set at 5 V. At this voltage, the speed of the “Dolphin IB” USV is approximately 0.5 m/s. The controller parameters of the IOIF–MFAC algorithm are The controller λ = 0.6, μ = 100, η = 1, ρ = 0.4, ε = 0.1, γ = 0.5. parameters of the RO–CFDL–MFAC algorithm are PID controller λ = 0.1, μ = 100, η = 1, ρ = 1, ε = 0.1, K = 5.The parameters are Kp = 3.0, KI = 0.01, K d = 1.4 . The expected heading angle is –100°. The experimental results of the heading control under the action of the three controllers are shown in Fig. 8.

Table 3 RMS value of heading tracking error. RMS (°) Simulation condition

PID

RO-CFDLMFAC

IOIF–MFAC

Without disturbance

50–100 s: 0.12 150–200 s: 0.13 250–300 s: 0.12 50–100 s: 2.36 150–200 s: 2.12 250–300 s: 2.16 50–100 s: 16.34 150–200 s: 16.53 250–300 s: 16.70

50–100 s: 0.09 150–200 s: 0.08 250–300 s: 0.08 50–100 s: 0.26 150–200 s: 0.25 250–300 s: 0.23 50–100 s: 15.27 150–200 s: 15.46 250–300 s: 15.23

50–100 s: 0.15 150–200 s: 0.12 250–300 s: 0.13 50–100 s: 0.63 150–200 s: 0.94 250–300 s: 0.43 50–100 s: 1.43 150–200 s: 1.32

Case 1

Case 2

Case 2. Maintain the parameters of the three controllers and set the thruster voltage at 12 V. At this voltage, the speed of the “Dolphin IB” USV is approximately 1.5 m/s. The desired heading angle is –120°. The heading control results are shown in Fig. 9.

250–300 s: 0.95

The RMS heading tracking error of the controlled system for the two cases above are calculated and shown in Table 4.

As shown in Figs. 5 and 6 and Table 3, the PID, RO–CFDL–MFAC, and IOIF–MFAC algorithms exhibit the similar control performance when no disturbance exists. However, when there model perturbation exists in the USV’s heading subsystem, the control performance of the PID algorithm will be deteriorated and the USV’s heading will exhibit obvious oscillations. Additionally, when time delay appears in the controlled system, under the action of the PID and RO–CFDL–MFAC algorithms, the heading subsystem of the USV becomes unstable; however, under the IOIF–MFAC algorithm, the actual heading angle can still converge to the desired heading angle and the controlled system can maintain good stability.

5. Conclusions Aiming at the heading control problem of the USV with uncertain eﬀect, a novel algorithm for USV heading control was proposed. The primary contributions are as follows: (1) By fusing the input and output information of the controlled system, a novel type of IOIF–MFAC algorithm was proposed. The new control algorithm solved the problem of the USV’s heading subsystem not satisfying the quasilinear assumption of the MFAC

Fig. 7. “Dolphin IB” USV ﬁeld experiments. Fig. 9. Heading response with thruster voltage at 12V. 6

Applied Ocean Research 90 (2019) 101851

Q. Jiang, et al.

Table 4 RMS value of heading tracking error. Evaluation criteria

PID

RO–CFDL–MFAC

IOIF–MFAC

RMS (°)

Case 1: 1.04 (10–70 s) Case 2: 3.72 (15–40 s)

Case 1: 2.34 (10–70 s) Case 2: 2.67 (15–40 s)

Case 1: 1.12 (10–70 s) Case 2: 1.42 (15–40 s)

From the RMS of the heading tracking error, it is shown that the tracking performance of the PID algorithm will be signiﬁcantly worse when the USV speed changes. However, the RO–CFDL–MFAC and IOIF–MFAC algorithm can maintain good performances at diﬀerent speeds and better control performances can be achieved when using the IOIF–MFAC method.

theory. (2) Based on the MFAC theory, the compact-format dynamic linearization data model of the proposed algorithm was presented and the stability of the IOIF–MFAC scheme was veriﬁed with rigorous mathematical analysis. (3) Simulation experiments demonstrated that compared with the PID and RO-CFDL-MFAC methods, the IOIF–MFAC algorithm demonstrated stronger robustness., ﬁeld experiments veriﬁed the eﬀectiveness of the proposed method.

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Acknowledgements [12]

We acknowledge the support by the National Key R & D Program of China (No. 2017YFC0305700), the National Natural Science Foundation of China (Grant No. 51779052, U.1806228, 51879057), the Innovation Special Zone of National Defense Science and Technology, Natural Science Foundation of Heilongjiang Province of China (Grant No. QC2016062), the Research Fund from Science and Technology on Underwater Vehicle Laboratory (Grant No. 6142215180102), the Heilongjiang Postdoctoral Funds for Scientiﬁc Research Initiation (Grant LBH-Q17046), the National Natural Science Foundation of China (Grant No. 51579022, 51709214), the Fundamental Research Funds for the Central Universities (Grant HEUCFG201810), and the Qingdao National Laboratory for Marine Science and Technology [Grant No. QNLM2016ORP0406].

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