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Heading MFA control for unmanned surface vehicle with angular velocity guidance

T

⁎

Ye Li, Leifeng Wang, Yulei Liao , Quanquan Jiang, Kaiwen Pan 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 Angular velocity guidance Adaptive Kalman ﬁlter

Based on the model-free adaptive control (MFA) theory, the heading control problem of unmanned surface vehicle (USV) with uncertainties is studied. The compact form dynamic linearization based MFA (CFDL-MFA) control method and its diﬃculties in USV heading control applications are analyzed. Aiming at the time delay of rudder-heading control system and the overshoot characteristic of CFDL-MFA method, the angular velocity guidance algorithm is introduced. The outer loop PID guidance controller calculates the desired angular velocity, and the inner loop MFA controller is used for angular velocity control. The heading control is realized indirectly. Considering sensor noise in applications, a control-oriented adaptive Kalman ﬁlter (AKF) based on dynamic linearized model is proposed, which eﬀectively suppresses the adverse eﬀect of sensor noise on heading control of USV. Numerical simulations of ﬁltering experiments and heading control experiments are completed which demonstrate the validity of the proposed heading MFA control method with angular velocity guidance and the proposed AKF method.

1. Introduction Unmanned surface vehicle (USV) is a kind of surface ship equipped with advanced automatic system [1]. At diﬀerent speed, parameters of wetted area, draft, etc. of the hull change greatly, which causes the hydrodynamic coeﬃcients to change with speed. Besides, environmental disturbances inﬂuence USV deeply in real navigation. In general, USV’s dynamic system is nonlinear, uncertain and time-varying, and it is very diﬃcult to establish an accurate mathematical model of USV [2–4]. Hence, it is meaningful to study the motion control problems of USV with uncertainties. The heading control method is one of the most basic issues in USV control ﬁeld. It is also an earlier applied and fruitful ﬁeld of control theories. Many researchers have made great contributions, mainly including PID control [5–7], optimal control [8–10], adaptive control [11–13], robust control [14,15], sliding mode control [16,17], intelligent control [18–23] etc. Among them, the PID control is the most widely applied method in real applications of USV [5,7,24]. Some controllers based on backstepping and sliding mode control methods are applied in ﬁeld experiments [11,16]. However, most of the control methods still remain in the theoretical study stage.

USV control problems are challenging [2,25]: (1) It is diﬃcult to construct an accurate mathematical model of USV, which makes the control performances of model-based control methods are not satisﬁed in practical applications. (2) Due to the uncertainties of model perturbation and environmental disturbances, the adaptabilities of modeloriented control methods (relying on accurate mathematical model) are poor, and it is hard to guarantee the robustness and stability of the control system. In general, the traditional model-oriented control methods are hard to use in actual applications and the control performances are not guaranteed. It seriously hinders the engineering applications of model-oriented control methods. PID control method is a classic and the most commonly used modelfree control method. The output of the control system changes depending on the actual input and output (I/O) data. But it cannot be treated as the real “data-driven” strategy, since it fails to dig more information from the I/O data to help improve the control performance. It is hard to maintain a consistent control performance by PID control method with uncertainties, including perturbation of model of USV and environmental disturbances. Besides, the parameters of PID control method need adjustment to stabilize the system [6,7,26]. Therefore, a model-free control method of USV with good robustness and

Abbreviations: USV, unmanned surface vehicle; AKF, adaptive Kalman ﬁlter; PPD, pseudo partial derivative; MFA, model-free adaptive; EKF, extended Kalman ﬁlter; BIBO, bounded-input bounded-output; CFDL-MFA, compact form dynamic linearization based MFA; TD, tracking diﬀerentiator; I/O, input and output; SISO, single input and single output; RMS, root mean square ⁎ Corresponding author. E-mail addresses: [email protected], [email protected] (Y. Liao). https://doi.org/10.1016/j.apor.2018.08.015 Received 30 June 2018; Received in revised form 19 August 2018; Accepted 22 August 2018 0141-1187/ © 2018 Elsevier Ltd. All rights reserved.

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where y (k ) ∈ R, u (k ) ∈ R respectively represent the input and output of the system at time k ; n y and nu are the order of the system. The USV heading subsystem is one of the nonlinear SISO systems which can be expressed by Eq. (1). We make the following assumptions about the system of Eq. (1):

adaptability is required. We prefer the heading control method of USV from the perspective of data-driven strategy, i.e., it is designed based on the dynamic I/O data of the controlled system, but not based on the mechanism model of the USV, which is the main purpose of this paper. Minitype USV usually equips low cost sensors. The sensor noise could not be ignored [25]. Many minitype USVs only equip the heading sensor. Some sensors can measure the heading and angular velocity at the same time, but the angular velocity is obtained by the heading diﬀerence. In essence, it is still the heading sensor. The noise of heading sensor inﬂuences the diﬀerential term particularly, which worse the control performance in practice. But the diﬀerential term could not be ignored since it eﬃciently improves the dynamic performance of the control system. The adverse inﬂuence of the sensor noise on angular velocity is the commonly faced challenge in most heading control systems including the commonly used PID method. Usually, the Kalman ﬁlter is used to process the sensor data. It has been theoretically proved that the Kalman ﬁlter provides the optimal recursive estimation under the minimum variance criterion, but it relies on the accurate mathematical model [27]. It cannot guarantee the performance since the inaccuracy of modeling and model parameter perturbation. Han [28] proposed the tracking diﬀerentiator (TD) to pick up the diﬀerential signal from the original signal contaminated by noise. The TD is model free and convenient in discrete system. The output of TD is the smooth approximation of the generalized derivative of the input. However, there is a lag between the output of TD and the actual value. This paper discusses the USV heading control problem with uncertainties. Firstly, we brieﬂy analyze the challenges of the CFDL-MFA method in USV heading control applications; Then, considering the dynamic characteristics of the USV heading control system, the angular velocity guidance algorithm is introduced, and the heading MFA control method with angular velocity guidance is proposed; Next, a control oriented adaptive Kalman ﬁlter (AKF) based on the dynamic linearized model is proposed to suppress the adverse eﬀect of sensor noise; Finally, contrast simulation experiments of ﬁlters including extended Kalman ﬁlter (EKF), TD and the proposed AKF, and simulation experiments of heading control of nominal model and with uncertainties are completed. The simulation results demonstrate the validity of the proposed heading MFA control method with angular velocity guidance and the proposed AKF method.

Assumption 1. The input and output of observable and controllable, i.e., for a y * (k + 1) , there exists a bounded feasible makes the system’s output equal to y * (k +

Assumption 2. Except for ﬁnite moments, the partial derivatives of the (n y + 2)th variables of f (⋯) exist and are continuous. Assumption 3. Except for ﬁnite moments, the system of Eq. (1) satisﬁes the generalized Lipschitz condition, i.e., with any k1 ≠ k2 , k2, k2 ≥ 0 and u (k1) ≠ u (k2) ,

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

(2)

where b > 0 is a constant. The three assumptions are acceptable. Assumption 1 is the basic requirement of the controlled system. If not satisﬁed, the system is uncontrollable. Assumption 2 is a typical constraint for general nonlinear systems in the controller design. Assumption 3 is the upper bound limitation of the change rate of the system’s output, i.e., bounded-input makes bounded-output (BIBO). Most nonlinear systems meet the BIBO assumption. The heading control subsystems of marine vehicles (ships, submarines, USVs, oﬀshore platforms, etc.) meet the above assumptions. Theorem 1 [30]. If a nonlinear system satisﬁes Assumptions 1, 2, and 3, when |Δu (k )| ≠ 0 , there must exist a time-varying parameter ϕ (k ) ∈ R named pseudo partial derivative (PPD) which makes the system of Eq. (1) transform into the CFDL model of Eq. (3):

Δy (k + 1) = ϕ (k ) Δu (k )

(3)

where ϕ (k ) is bounded at any time. In Eq. (3), Δy (k + 1) = y (k + 1) − y (k ) , Δu (k ) = u (k ) − u (k − 1) . Detailed proof of Theorem 1 can be found in the reference [30]. Therefore, the dynamic linearized model of the system of Eq. (1) is:

y (k + 1) = y (k ) + ϕ (k ) Δu (k )

(4)

Eq. (4) is a dynamic linearized representation of a class of discretetime nonlinear systems. It is a control oriented linear time-varying model with incremental form and single parameter. It is essentially diﬀerent from the traditional mechanism models and other linearized models of controlled objects.

2. Analysis of MFA control method and USV heading control applications The MFA control method is a control method for nonlinear system by data-driven strategy, which has been widely applied in industrial control ﬁelds such as transportation, oil reﬁning and chemical industry [29,30]. However, there are few studies in the motion control ﬁeld, such as motion control of aircraft, robot and USV. It is a potential great choice for USV heading control. The basic principle of MFA control method is to establish a dynamic linearized model equivalent to the nonlinear system at every working point. The pseudo partial derivative (PPD) of the dynamic linearized model is estimated online based on the I/O data of the controlled system. The weighted one-step forward controller is designed according to the optimal criterion function [30]. Considering engineering practicality, the compact form dynamic linearization based MFA (CFDL-MFA) control method is the most commonly used form due to fewer amounts of calculation and clear physical meaning of PPD.

2.2. Compact form dynamic linearization based MFA control The CFDL-MFA control method is as follows [30]:

u (k ) = u (k − 1) +

ρϕˆ (k ) (y * (k + 1) − y (k )) λ + |ϕˆ (k )|2

ϕˆ (k ) = ϕˆ (k − 1) +

ηΔu (k − 1) × (Δy (k ) − ϕˆ (k − 1) Δu (k − 1)) μ + Δu (k − 1)2

(5)

(6)

ϕˆ (k ) = ϕˆ (1)

(7)

if |ϕˆ (k )| ≤ ε or |Δu (k − 1)| ≤ ε orsign(ϕˆ (k )) ≠ sign(ϕˆ (1)) where ε is a suﬃciently small positive number; ϕ (k ) is the PPD; ϕˆ (k ) is the estimation of the PPD; ϕˆ (1) is the initial value of the PPD; sign(•) is the symbolic function; ρ ∈ (0, 1] and η ∈ (0, 1] are the step factors; μ > 0 and λ > 0 are weight coeﬃcients to adjust the change rate of PPD and controller output. In the CFDL-MFA control method, the reset algorithm of Eq. (7) improves the tracking performance of PPD for time-varying parameters

2.1. Compact form dynamic linearization In general, a single input and single output (SISO) nonlinear discrete-time system can be expressed as:

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

the system of Eq. (1) are bounded expected output control input u* (k ) , which 1) .

(1) 58

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with inertia, i.e. the angular velocity cannot change suddenly. Hence it is likely to be the large overshoot of heading. When the actual heading is above the desired heading, the rudder angle will change continuously in reverse direction for a long time because of the long overshoot time. So the heading of USV is likely to have overshoot and oscillation. It is hard to improve the control performance only by adjusting the parameters of the controller. Aiming at the dynamic characteristics of USV, the CFDL-MFA control method needs to be improved. For the convenience of expression, the “CFDL” will be omitted in the rest of this paper. Fig. 1. The principle of the CFDL-MFA control system.

3. Heading MFA control method for USV with angular velocity guidance

of the controlled system. In the CFDL-MFA control method, the dynamic I/O data of the controlled system is used to estimate the time-varying parameter ϕ (k ). The estimated ϕˆ (k ) and the minimized one-step forward forecast error is introduced into the control law and then the new controller output u (k ) is obtained. Based on controller output and the new observation y (k ) , the new PPD is estimated as ϕˆ (k + 1) . The whole control process is shown in Fig. 1. The CFDL-MFA control method is designed based on the I/O data of the controlled system which is a data-oriented control method. It is independent of the mathematical model of the controlled system. It is insensitive to the model parameter perturbation and time-varying disturbances. Therefore, it is robust and adaptive, which is superior to model-based control method. However, the CFDL-MFA control method still has disadvantages. The output of the controller is incremental. Besides, the PPD and controller output is unexpected to change violently. As a result, the controller output responses slowly and with large inertia. When the controlled variable does not reach the expected value, the controller output increases continuously. If we apply the CFDL-MFA control method directly into the controlled system with inertia, it is likely to have overshoot and oscillation.

The angular velocity responses more rapidly than heading to rudder, so we introduce the angular velocity guidance algorithm, and the heading MFA control method with angular velocity guidance is proposed. However, it brings a new problem of the sensor noise of angular velocity. In order to suppress the adverse eﬀect of the sensor noise, we propose the control-oriented adaptive Kalman ﬁlter (AKF) based on the dynamic I/O data of the controlled system. The adverse eﬀect of sensor noise on angular velocity should be paid attention to not only in the heading control method in this paper; actually it is a commonly faced challenge in most heading control systems with diﬀerential term. The principle of the heading MFA control system for USV with angular velocity guidance is shown in Fig.3. 3.1. The angular velocity guidance algorithm The controller is the cascade structure consisting of the outer loop angular velocity guidance controller and the inner loop angular velocity controller. The outer loop guidance controller applies PID method to calculate the expected angular velocity. The inner loop MFA angular velocity controller achieves angular velocity control. The heading control is realized indirectly. The outer loop angular velocity guidance controller is as Eq. (8):

2.3. Analysis of CFDL-MFA control in USV heading control applications The USV motion control system consists of the speed control subsystem and heading control subsystem. The principle of motion control system of USV is shown in Fig. 2. In this paper, we mainly discuss the heading control subsystem. For underactuated USV turned by rudder, when rudder angle increases, the angular velocity changes ﬁrstly under turning moment, and then the heading changes. If we take the rudder angle as controller output and take the heading as controlled variable in CFDL-MFA controller directly, when the actual heading is close to but not yet reaches the desired heading, the rudder angle will increase or decrease continuously. When the actual heading is very close to the desired heading, the USV still have large angular velocity. The turning motion of USV is

k

r * (k ) = kp × e (k ) + ki ×

∑ e (j) + kd × j=0

e (k ) − e (k − 1) Ts

(8)

where e (k ) = ψ* (k ) − ψ (k ) is the heading error; kp , ki , kd are proportional, integral and diﬀerential coeﬃcients. The coeﬃcients need to be adjusted according to experience. Ts is the step of the control system. The inner loop angular velocity MFA controller is as Eqs. (9–10):

ϕˆ (k ) = ϕˆ (k − 1) +

ηΔu (k − 1) (Δr (k ) − ϕˆ (k − 1) Δu (k − 1)) μ + Δu (k − 1)2

(9)

where Δu (k ) = u (k ) − u (k − 1) ; Δr (k ) = r (k ) − r (k − 1) ; η is the step

Fig. 2. The principle of a typical USV motion control system. 59

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Fig. 3. Principle of the heading MFA control system for USV with angular velocity guidance.

the observation, i.e. the heading sensor output; k is the time of the discrete control system. Set state variable as X = [ ψ (k ) r (k )]T . The state transition matrix is:

factor; μ is the weight coeﬃcient to adjust the change rate of PPD; ∧

ϕ (k − 1) is the estimated PPD at the last moment. If Δu (k − 1) ≤ ε or ϕˆ (k ) ≤ ε or sign(ϕˆ (k )) ≠ signϕˆ (1) , reset ϕˆ (k ) = ϕˆ (1) . ε is a suﬃciently small positive number. ϕˆ (1) is the initial value of PPD. u* (k + 1) = u (k ) +

ρϕˆ (k ) (r * (k ) − rˆ (k )) λ + |ϕˆ (k )|2

G = ⎡ 1 Δt ⎤ ⎣0 1 ⎦

(10)

The observation matrix is:

where ρ is the step factor; λ is the weight coeﬃcients to adjust the change rate of controller output; u (k ) is the actual controller output at the last moment; u* (k + 1) is the desired controller output. It is not really diﬃcult to adjust the parameters in the heading MFA control method with angular velocity guidance, since the inﬂuences of the parameters on the control performance are clear:

H = [1 0]

(13)

The procedure of the AKF is as follows: (1) One step prediction of state

ˆ (k k − 1) = GXˆ (k ) + [ 0 ϕˆ (k ) × (u (k ) − u (k − 1)) ]T X

(14)

(2) One step prediction of observation

(1) η ∈ (0, 1]. A large η is beneﬁcial to the tracking rate of PPD in the rapid adjustment phase of PPD, while it may cause overshoot and oscillation when the PPD should stabilize. (2) μ > 0 . A large μ is beneﬁcial to the stability of PPD, while it may reduce the tracking rate of PPD to the actual system. (3) ρ ∈ (0, 1]. A large ρ is beneﬁcial to the heading response rate when the actual heading is far from the desired heading, while it may cause overshoot and oscillation of the actual heading when the actual heading is close to the desired heading. (4) λ > 0 . A large λ is beneﬁcial to the stability of the rudder angle, while it may reduce the heading response rate in the rapid adjustment phase of heading control.

ˆ (k ) yˆ (k k − 1) = HX

(15)

(3) One step prediction of covariance matrix

P (k k − 1) = GP (k k − 1) GT

(16)

(4) The ﬁlter gain matrix

K (k ) = P (k k − 1) × (H × P (k k − 1) × HT + R)−1

(17)

where R is the variance of the observation noise, i.e. the variance of the heading sensor noise. (5) State update

ˆ (k ) = X ˆ (k k − 1) + K (k ) × (y (k ) − yˆ (k k − 1)) X

In the heading MFA control method for USV with angular velocity guidance, beneﬁted from the diﬀerential part in outer loop angular velocity guidance controller, when the actual heading is close to the desired heading, the desired angular velocity will gradually reduce to 0. Combining the quick response of angular velocity to rudder and the adaptability of MFA control theory, the inner MFA angular controller can make the actual angular velocity reach the desired angular velocity rapidly. The heading MFA control method for USV with angular velocity guidance is expected to have good control performance and adaptability in theory.

(18)

(6) Covariance matrix update

P (k k ) = (I2 × 2 − K (k )) P (k k − 1)

(19)

where I2 × 2 is two-dimensional unit array. Remark 1. The proposed AKF is control-oriented but not modeloriented, and only applied in control system, which is essentially diﬀerent from other Kalman ﬁltering methods. In the AKF method, the state transition equation is replaced by the dynamic linearized model described by PPD. On the one hand, the dynamic linearized model is established only for control system. It cannot completely play the role of the state transition equation in all application scenarios, like dynamic analysis; on the other hand, the basis of the dynamic linearized model is the I/O data obtained by the control system. It is unable to separate from the control system.

3.2. Adaptive Kalman ﬁlter based on dynamic linearized model The control-oriented adaptive Kalman ﬁlter (AKF) based on the dynamic linearized model is proposed. The dynamic linearized model is real time corrected by the dynamic I/O data of the controlled system. It avoids the adverse eﬀect of the modeling error. Considering the dynamic linearized model expressed in Section 2.1, the state space model of the heading response system of USV is:

⎧ ψ (k ) = ψ (k − 1) + r (k ) × Ts r (k ) = r (k − 1) + ϕ (k ) × (u (k ) − u (k − 1)) ⎨ ⎩ y (k ) = ψ (k )

(12)

Remark 2. In the estimation algorithm of PPD, the parameter μ avoids the sudden change of the PPD caused by sensor noise and the instantaneous singularities (zero denominator), but it restricts the changing rate at the same time. So there could be a lag between the equivalent dynamic linearized model and the actual model. Fortunately, the input and output of the “dynamic linearized model” in this paper are rudder angle and angular velocity of USV respectively. Actually, the “PPD” has a relatively clear physical meaning—the derivative of the angular velocity to the rudder angle. It is usually

(11)

where ψ (k ) is the heading of USV; r (k ) is the angular velocity; y (k ) is 60

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slowly changing. So in the speciﬁc application in this paper, the adverse eﬀect of the lag between the equivalent model and the actual model is weakened, while the beneﬁcial eﬀects of the low pass ﬁlter is strengthened.

3.3. Analysis of the stability of the heading control system The heading MFA control system for USV with angular velocity guidance can be expressed as:

⎧ ψ˙ = r ⎨ ⎩ r˙ = f (u, r )

(20) Fig. 4. The "Dolphin I" minitype USV.

The system of Eq. (20) can be seen as a cascade system of two subsystems. In the ﬁrst line of Eq. (20), the angular velocity is the input and the heading is the output of the subsystem. It is a basic integral element. It can be stable if we apply PID controller into an integral element system with appropriate parameters [31]. In the second line of Eq. (20), the rudder angle is the input and the angular velocity is the output of the subsystem. While the desired angular velocity does not exceed the motion ability of the USV and the rudder angle does not exceed the stall angle, the subsystem meets the three assumptions in Section 2.1. Besides, while the rudder angle increases, the angular velocity will not decrease i.e. it is quasi-linearized. According to MFA control theory, the angular velocity control subsystem is stable [30]. The adaptive Kalman ﬁlter element is for smoothing the sensor noise and reducing the noise serration. It helps the stability of the control system. As a result, the proposed heading MFA control method for USV with angular velocity guidance is stable.

In summary, the procedure of the proposed heading MFA control method for USV with angular velocity guidance is as Table 1.

4. Numerical simulation experiments We take the "Dolphin I" minitype USV as the investigated subject to verify the proposed heading control method and the proposed AKF method by numerical simulation. "Dolphin I" is a minitype catamaran as shown in Fig.4, which is about 2 m long, 1 m wide and 55 kg displacement. According to the theory of ship maneuverability, the discrete mathematical motion model of the heading subsystem of USV can be expressed as:

Table 1 Procedure of the heading MFA control method for USV with angular velocity guidance.

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Table 2 The RMS results between the estimated value and the actual value. Heading(°)

Angular velocity(°/s)

Sensor EKF TD AKF

3.20 62.10 9.98 1.91

Not available 3.44 1.18 1.50

oﬀsets relative to the actual values using the EKF method. Consistently, as shown in Table 2, the RMSs of EKF are larger than of TD, AKF and sensor data (if available), both about the heading and the angular velocity. The estimated headings and angular velocities of TD and AKF track the actual value without unreasonable deviation. As to the angular velocity, the ﬁltering results of TD and AKF are similar, and the corresponding RMSs are very close. As to the heading, the lag of AKF is obviously shorter than of TD. The heading estimation RMS of AKF is much smaller than of TD. Besides, only the heading estimation RMS of AKF is smaller than of original sensor data. In a word, in the USV yawing system, the ﬁltering performance of the proposed AKF is superior to EKF and TD.

Fig. 5. The estimation results of heading.

⎧ ψ (k ) = ψ (k − 1) + r (k ) × Ts r (k ) = r (k − 1) + r˙ (k ) × Ts ⎨ 3 ⎩ r˙ (k ) = (Kδ (k ) − r (k ) − αr (k ))/ T

RMS

4.2. Contrast experiments of nominal model (21)

The rudder angle is set as δ = 10 × sin(0.1 × t ) . A random variable is added into the parameter K of the mathematical motion model in Eq. (21), i.e. K = 0.2782 + 1 × rand(1) , to simulate the stochastic model parameter perturbation. The heading sensor noise w (k ) is set as Gauss white noise with covariance of 10° and mean of 0. Three ﬁlters are tested: the EKF, TD and the proposed AKF method. The state transfer equation in the EKF is based on Eq. (21). The detailed procedure of the EKF is in reference [27]. The detailed description of TD is in reference [28]. The rate coeﬃcient in TD has been adjusted to optimal as 50. The iteration cycle is 0.1 s. The simulation results are shown in Figs. 5 and 6. The root mean square (RMS) results between the estimated values and the actual values are shown in Table 2. From Figs. 5 and 6, with the stochastic model parameter perturbation, both the estimated values of heading and angular velocity exist

The initial state of the heading control subsystem is [ ψ0 r0 ] = [0° 0° s]. The desired heading is 90°. The parameters of the outer loop angular velocity guidance PID controller are Kp = 4, KI = 0.001, K d = 1. The parameters of the inner loop angular velocity MFA controller are λ = 0.1, μ = 10, η = 1, ρ = 0.1, ε = 0.001. Although there is no perturbation of model of USV or environmental disturbances, the proposed AKF method is applied in order to test the accuracy of the prediction of state. As a contrast, the parameters of PID heading controller are Kp = 4, KI = 0.001, K d = 1. For the fairness of the comparison, the parameters of the proposed control method and the contrastive PID control method are manually adjusted to better state, and we make the parameters of the outer loop angular velocity guidance PID controller equal to the parameters of contrastive PID heading controller. The simulation results of the step response experiments of heading control using the two controllers are shown in Figs. 7–9. From Figs. 7–8, for the contrastive PID heading controller, the rise time is about 40 s without overshoot. The rudder angle rises rapidly but decreases early to suppress overshoot. For the MFA controller, the rise time is about 15 s with slight overshoot. The rudder angle rises relatively slowly, and converges rapidly when closing the desired heading.

Fig. 6. The estimation results of angular velocity.

Fig. 7. Heading response comparison of nominal model.

Based on maneuverability test data and system identiﬁcation method [32], the parameters of Eq. (21) is obtained as K = 0.2782, T = 0.4352, α = 0.0094 . 4.1. Contrast experiments of ﬁlters

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Fig. 10. Heading response with uncertainties (PID).

Fig. 8. Rudder angle response comparison of nominal model.

4.3. Contrast experiments with uncertainties The initial states of the heading control subsystem, the controller parameters, and the desired heading are same as in Section 4.2. The heading sensor noise and the stochastic model parameter perturbation are same as in Section 4.1.The proposed AKF is applied in the heading MFA controller with angular velocity guidance. The EKF based on mathematical model of Eq. (21) is applied in the contrastive PID heading controller. The simulation results of the step response experiments of heading control with uncertainties are shown in Figs. 10–14. From Figs.10–11, with the stochastic model parameter perturbation, both the estimated values of heading and angular velocity exist oﬀsets relative to the actual values using the traditional model-based EKF method. From Fig. 12, with stochastic model parameter perturbation, the heading control performance is satisﬁed using the proposed heading MFA control method with angular velocity guidance. The rise time is about 40 s almost without overshoot. Fig. 13 is the partial enlarged drawing of Fig. 12. From Fig. 13, the tendency of estimated heading obtained by the proposed AKF brings into correspondence with the actual heading, and the estimated heading changes more slightly than heading sensor data. The RMS of the estimated heading relative to actual heading is about 1.07, while the RMS of the heading sensor observation relative to actual heading is about 3.16. The proposed AKF

Fig. 9. Angular velocity response of nominal model.

The variation characteristics of rudder angle are consistent with the incremental structure of MFA controller. In comparison, the response time of the contrastive PID heading controller is longer and the overshoot is less. It is because that with the same parameter of the outer loop angular velocity guidance PID controller and the contrastive PID heading controller, the suppressing overshoot eﬀect of the diﬀerential term in the contrastive PID heading controller directly acts on the USV. Fig. 9 shows the estimation result of angular velocity by the proposed AKF and the control result of angular velocity of the inner loop MFA controller. With no perturbation of model or environmental disturbance, the estimated angular velocity is almost equal to the actual angular velocity. When the desired angular velocity changes slightly, the actual angular velocity tracks the desired angular velocity very well. When the desired angular velocity changes with great gradient, there is a little lag between the actual angular velocity and the desired angular velocity. On the one hand, it is because the turning motion of USV is with inertia, i.e. the angular velocity cannot change suddenly. On the other hand, it is because of the incremental structure of MFA controller. The simulation result shows the two controllers both have good control performance without perturbation of model or environmental disturbances.

Fig. 11. Angular velocity response with uncertainties (PID). 63

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Fig. 12. Heading response with uncertainties (MFA).

Fig. 15. Heading response of multi-step signal with uncertainties (MFA).

model parameter perturbation and environmental disturbances, the inner angular velocity MFA controller can still track the desired angular velocity rapidly. The simulation results show that the heading MFA control method for USV with angular velocity guidance has good control performance with stochastic model parameter perturbation and environmental disturbances. The proposed heading MFA control method for USV with angular velocity guidance and the proposed AKF have strong robustness and adaptability.

4.4. Experiments of multi-step signal with uncertainties The desired heading is set as 90°, 180° and 60° in the 1 st, 2nd and 3rd 50 s. The other simulation conditions are same as in Section 4.3. The simulation results are shown in Figs. 15–16: From Figs. 15–16, the actual heading converges to the desired heading of multi-step signal. The estimated values of heading and angular velocity track the tendencies of the actual values with less oscillation than the original sensor data. The proposed heading MFA method with angular velocity guidance and the proposed AKF method show good performance under multi-step signal.

Fig. 13. Partial enlarged drawing of the heading response with uncertainties (MFA).

Fig. 14. Angular velocity response with uncertainties (MFA).

method suppresses the adverse eﬀect of heading sensor noise well. Taking the estimated values of AKF as the controller input is beneﬁcial to the stability of the control system. From Fig. 14, with stochastic

Fig. 16. Angular velocity response of multi-step signal with uncertainties (MFA). 64

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Y. Li et al.

5. Conclusions [4]

Aiming at the USV heading control problem with uncertainties, the heading MFA control method for USV with angular velocity guidance is proposed. Aiming at the adverse eﬀect of sensor noise on control system, the adaptive Kalman ﬁlter is proposed. The main contributes are as the following four points:

[5] [6] [7]

(1) Analysis of the CFDL-MFA control method and the USV motion control applications shows that if we take rudder angle as controller output and take heading as controlled variable in CFDL-MFA controller, the heading of USV is likely to have overshoot, and it is hard to improve the control performance only by adjusting the parameters of the controller. (2) Beneﬁted from the diﬀerential part in outer loop angular velocity guidance controller, the proposed heading MFA control method for USV with angular velocity guidance improves the dynamic characteristic including overshoot and oscillation in USV heading control compared to basic CFDL-MFA control method. (3) The control-oriented adaptive Kalman ﬁlter based on dynamic linearized model is proposed which is independent of the mathematical model of the controlled system. It suppresses the adverse eﬀect of heading sensor noise well with stochastic model parameter perturbation. (4) Contrastive simulation results demonstrate that the heading control performance of the proposed heading control method and the ﬁltering performance of the proposed AKF method are well with stochastic model parameter perturbation and environmental disturbances. The proposed heading MFA control method for USV with angular velocity guidance and the AKF method have strong robustness and adaptability with uncertainties.

[8]

[9] [10]

[11] [12]

[13]

[14]

[15]

[16] [17] [18] [19]

Conﬂicting interests [20]

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

[21]

Acknowledgements [22]

This work was supported by the National Natural Science Foundation of China [grant number 51779052]; the Natural Science Foundation of Heilongjiang Province, China [grant number QC2016062]; the Research Fund from Science and Technology on Underwater Vehicle Laboratory [grant number 614221503091701]; the Heilongjiang Postdoctoral Funds for Scientiﬁc Research Initiation [grant number LBH-Q17046]; the National Key R&D Program of China [grant number 2017YFC0305700]; the Fundamental Research Funds for the Central Universities, China [grant numbers HEUCFP201741, HEUCFG201810]; the National Natural Science Foundation of China [grant numbers 51879057, 51579022, 51709214]; the Qingdao National Laboratory for Marine Science and Technology [grant number QNLM2016ORP0406].

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