Observer-based adaptive neural network control for a class of remotely operated vehicles

0029-8018/ © 2016 Elsevier Ltd. All rights reserved.

Ocean Engineering 127 (2016) 82–89

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Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng

Observer-based adaptive neural network control for a class of remotely operated vehicles

crossmark

⁎

Zhenzhong Chua, , Daqi Zhua, Gene Eu Janb a b

College of Information Engineering, Shanghai Maritime University, Shanghai 201306, China College of Sound and Image Arts, Tainan National University of the Arts, Tainan City 72042, Taiwan, ROC

A R T I C L E I N F O

A BS T RAC T

Keywords: Adaptive control Neural network Observer Remotely operated vehicles

In this paper, a new adaptive neural network control approach is developed for a class of remotely operated vehicles whose velocity state and angular velocity state in the body-fixed frame are unmeasured. Unlike most previous control approaches, it doesn’t need thrust model and the thruster control signal is considered as the input of control system directly. Using local recurrent neural network to approximate the unknown nonlinear functions, an adaptive observer is introduced for state estimation. Under the framework of the backstepping design, adaptive neural network control law is constructed based on the output of local recurrent neural network and state estimation. The stability analysis is given by Lyapunov theorem. The effectiveness of the proposed control scheme is illustrated by simulations.

1. Introduction A remotely operated vehicle (ROV) works in a complex marine environment. In underwater observation, manipulator operation and other tasks, a stable and high precision control system can provide higher working efficiency (Mohan and Kim, 2015; Chen, 2008; Shim et al., 2010). Recently, many control approaches have been proposed for dynamic positioning, trajectory tracking and path following (Souza and Maruyama, 2007; Hoang and Kreuzer, 2007), such as sliding mode control (Zhang and Chu, 2012; Chu et al., 2016b), adaptive control (Miao et al., 2013), neural network control (Chatchanayuenyong and Parnichkun, 2006) and so on. In these control approaches, it is mostly assumed that all the states of ROV system are known. Obviously, this assumption can’t be met by most ROVs. Because of the small ROVs are only equipped with the sensors for position and orientation measurement, but without Doppler Velocity Log (DVL), inertial navigation system and other sensors for velocity and angular velocity measurement (Li et al., 2013; Gao, et al., 2004; Zhang, et al., 2009). Some large ROVs are equipped with DVL, but they sometimes need to perform bottom-following control for some special tasks (Silvestre et al., 2008). If the altitude is very low, DVL may be unable to work. Therefore, it needs to be considered in the controller design of ROVs that the velocity state and angular velocity state in the body-fixed frame cannot be measured directly. Since the position and orientation in the earth-fixed frame can be measured directly, ROV is an observable system. Considering the

⁎

complexity and uncertainty of ROV modeling, some adaptive control methods based on high gain observer have been proposed (Boizot et al., 2010; Hankovic, 1997; Lee and Khalil, 1997; Tong and Li, 2002). One of the advantages of a high gain observer is that the information of the ROV dynamic model is not needed, and a large gain coefficient can be used to guarantee the convergence of state estimation errors, so that the velocity state and angular velocity state in the body-fixed frame can be estimated online. However, the large gain coefficient will be introduced into the control law, which will result in the system output oscillation and affect the tracking quality. Based on the above considerations, the high gain observer is not very suitable for ROV control. In these adaptive control methods, neural network are usually used for adaptively learning of unknown term of ROV dynamic model. Therefore, we consider that if the information from online identification can be used to construct the full order observer. Thus, the large gain coefficient will not be needed and the problem of that how to improve the learning accuracy and speed of neural network is only needed to be taken into account. In addition, the actual input of ROV control system is the thruster control signal (Gan et al., 2004), so ROV is a nonaffine nonlinear system essentially. Thus, it is very difficult to design the control law to obtain the thruster control signal directly. In most previous control approaches, the outputs of controllers are usually thruster thrust, then the thruster control signal is calculated by the thrust model (Yu et al., 2008; Fischer et al., 2014; Lapierre and Jouvencel, 2008). However, the thrust model is related to not only the control signal but also the

Corresponding author. E-mail address: [email protected] (Z. Chu).

http://dx.doi.org/10.1016/j.oceaneng.2016.09.038 Received 1 December 2015; Received in revised form 5 September 2016; Accepted 24 September 2016

Ocean Engineering 127 (2016) 82–89

Z. Chu et al.

γ=diag([γ1, …, γn]). f(x), B(x1), A and b are as shown in (5)–(8), respectively.

advance speed of the propeller (Alessandri et al., 1999), so that it is very difficult to establish the accurate thrust model (Kim and Chung, 2006). In practical control process, the inaccurate thrust model may influence the control performance (Zhang and Chu, 2012). Therefore, the problem that how to obtain the thruster control signal directly without thrust model needs to be considered, and then a high precision tracking control system can be obtained. In this paper, considering the complex nonlinear relationship between thrust and control signal, an affine transformation is carried out for thrust model and a scale factor is introduced, thus the thruster control signal can be seen as the input of the tracking control system directly. Since it is difficult to accurately establish ROV's dynamics model and the velocity state and angular velocity state in the body-fixed frame cannot be measured directly, an adaptive state observer based on local recurrent neural network is proposed to estimate the velocity state and angular velocity state online. Furthermore, the scale factor of thrust model is also estimated by adaptive learning. According to the estimated values of observer and the output of local recurrent neural network, the adaptive control law is designed. The uniformly ultimately bounded of tracking error is analyzed by Lyapunov theory and is verified by simulation results. This paper is organized as follows. ROV's tracking control problem is described in Section 2. In Section 3, the observer-based adaptive neural network tracking controller is given. In Section 4, the effectiveness of the proposed method is verified by simulation results. Finally, we make a brief conclusion of the paper in Section 5.

∂τi (ui ) ∂ui

(7)

⎡0 ⎤ b = ⎢ 6×6 ⎥ ⎣ I6×6 ⎦

(8)

(2)

f (x ) = Wφ (VH ) + ε (3)

ui = ui*

where u=[u1, …, un]

T

(10)

where x=[x1; x2] is the input vector of input layer neurons. H=[x; H1] is the input vector of hidden layer neurons. H1 is the output vector of recurrent layer neurons, which is equal to the output of the hidden layer neurons with recurrent structure. W is the weight matrix between hidden layer neurons and output layer neurons. V is the weight matrix between input layer neurons, recurrent layer neurons and hidden layer

Define x=[x1;x2], x1=η, x2=J(η)v. According to (1) and (2), it can be obtained:

x ̇ = Ax + b ( f (x ) + B (x1) γu )

(9)

In (4), since the nonlinear function item f(x) is usually unknown, neural network is mostly used for online learning. In most neural network based-adaptive controllers, RBF neural network, BP neural network, and recurrent neural network are usually used. However, these neural networks have some disadvantages (Zhang and Chu, 2012). For example, in the RBF neural network-based adaptive controller, if there is a big disturbance or the desired value has an abrupt change, the weights of neural network would take a long time to converge. Although the recurrent neural network can overcome this problem, the learning efficiency of recurrent neural network is very poor. Therefore, the local recurrent neural network were proposed in (Zhang and Chu, 2012; Chu et al., 2016a). Compared with the traditional recurrent neural network and BP neural network, there are only some of the hidden layer neurons regress to recurrent layer in local recurrent neural network. As the training results given in (Chu et al., 2016a), it shows that the local recurrent neural network has the advantages of faster learning speed and good learning performance and it also very suitable for adaptive control for ROVs. In this paper, the local recurrent neural network will be introduced into observer designing, and then the adaptive control law will be constructed. The structure of the local recurrent neural network are shown in Fig. 1. For the local recurrent neural network as shown in Fig. 1, from the nonlinear mapping ability of neural network, there are optimal network weights W, V, such that:

where i=1,…, n, n is the number of thrusters and γi is a scale factor:

γi =

⎡0 I ⎤ A = ⎢ 6×6 6×6 ⎥ ⎣ 06×6 06×6 ⎦

3. Controller design

(1)

ui* + O ((ui − ui*)2) ui = ui*

(6)

where x20, u0 are known positive constants. As can be seen from (4), after the affine transformation, an affine nonlinear system is obtained, which will be convenient for control law design. However, since the ROV's dynamic model is difficult to be established accurately, the nonlinear function item f(x) is unknown. In addition, due to the fact that the ROV is usually not equipped with the sensors to measure the velocity state and angular velocity state in the body-fixed frame, the state x2 is unknown. Therefore, the control objective of this paper is to develop a control law u such that the tracking error is uniformly ultimately bounded under the situation that the ROV dynamic model is unknown and the state x2 is unmeasured.

where η denotes the vector of position and orientation in the earthfixed frame, v is the vector of velocity and angular velocity expressed in the body-fixed frame. M is an inertia matrix including extra mass, the matrix C(v) groups centripetal and Coriolis forces, including the centripetal force and Coriolis force produced by extra mass, D(v) is the hydrodynamic damping term, the vector G(η) is the combined gravitational and buoyancy forces in the body-fixed frame, τd is the external disturbances, J(η) is the kinematic transformation matrix expressing the transformation from the body-fixed frame to earth-fixed frame, τ(u) is the thruster thrust, u is the thruster control signal and B is distribution matrix of thrusters. In Eq. (1), τ(u) is a nonlinear function about the thruster control signal and the advance speed of the propeller, where the latter is a variable that might be difficult to measure in actual. Therefore, the accurate thrust model is very difficult to be established. In this paper, a Taylor expansion will be introduced to convert (1) into an affine nonlinear system, thus the thrust model is not needed and it will also be convenient for control law design. The Taylor expansion of τ(u) about u* is given as:

∂τi (ui ) ∂ui

B (x1) = J (η) M−1B

x2 ≤ x20 , u ≤ u 0

The mathematical model of a ROV in 6 DOF can be described as (Biggs and Holderbaum, 2009):

τi (ui ) = γi ui + τi (ui*) −

(5)

When using (4) to describe the tracking system for a particular ROV system, it has the following property: Property 1. The state x2 and control law u are all bounded, namely:

2. Problem formulation

η ̇ = J (η ) v Mv ̇ + C (v ) v + D (v ) v + G (η) + τd = B τ (u )

f (x ) = J ′(η) v − J (η) M−1 (C (v ) v + D (v ) v + G (η) + τd ) ⎛ ⎞ ∂τ (u ) + B (x1) ⎜⎜τi (ui*) − i i ui* + O ((ui − ui*)2 ) ⎟⎟ ∂ui ui = ui* ⎝ ⎠

(4)

is the vector of thruster control signals. 83

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Z. Chu et al.

…

Output layer

W

…

…

…

Hidden layer

ˆ ˆ )T + lW (Wˆ − W0 )] Wˆ ̇ = −λW [e2 φ (VH

(17)

ˆ ˆ ) Wˆ T e2 Hˆ T + lV (Vˆ − V0 )] Vˆ ̇ = −λV [φ′(VH

(18)

γˆ ̇ = −λ γ [diag(e2T B (xˆ1))diag(u ) + l γ (γˆ − γ0 )]

(19)

⎡ − k11 ⎤ I ⎥ >0 Q = −⎢ 3χ2 ˆ ⎢⎣− k12 l W + 2 + 3χ3 ⎥⎦

(20)

lW ≥

V

…

…

H1 Recurrent layer

Fig. 1. The structure of local recurrent neural networks.

lγ ≥

neurons. ε is the estimation error of the optimal neural network and φ(.) is a sigmoid function. For the local recurrent neural network as shown in Fig. 1, it generally has the following properties: Property 2. For different values of variables, there is:

(13)

2

+ φ (VH ) 2 ) + θ1

1 1 B (xˆ1) 2 u 0 2 + B (xˆ1) 2 u 0 2 + θ3 χ1 χ2

(22)

(23) (24)

3χ1 + θ5 2

(25)

(26)

Consider the Lyapunov function:

(14)

V1 =

1 ∼T∼ x1 x1 2

(27)

Taking the derivative of V1 and substituting (26) into it, we can have:

V1̇ = x∼1T x∼1̇ = x∼1T (x∼2 − k11 x∼1 − k2 sgn(x∼1))

(28)

According to the value of k2, we can get:

(15)

V1̇ ≤ −x∼1T k11 x∼1 − x∼1 θ4 sgn(x∼1)

where Wˆ and Vˆ are the estimated values of W and V, respectively, andHˆ = [xˆ; H1]. Remark 1. In the high gain observer, a larger gain coefficient is usually used to compensate the unknown function item f(x), so that the state estimation errors can converge to zero. Since the large gain coefficient will be introduced into the control law, it would cause the system output oscillation and affect the tracking quality. Compared with the high gain observer, the proposed adaptive state observer (14) uses the information of dynamic model from local recurrent neural network. Through using the local recurrent neural network to compensate the unknown function f(x), a smaller gain coefficient can be chosen. Next, the main conclusion will be presented: Proposition 1. Considering the mathematical model of an ROV expressed as (1), if the adaptive state observer and the control law are chosen as (14) and (16), the updated laws of weights and scale factor are selected as (17)–(19), and the control parameters satisfied (20)– (25), then the tracking error will be uniformly ultimately bounded.

u = (B (xˆ1) γˆ)−1 [−e1 − fˆ (xˆ) + k12 x∼1 + (βe2 )q / p − k3 sgn(e2 ) − k 4 e2]

1 ( φ (VHˆ ) χ3

x∼1̇ = x∼2 − k11 x∼1 − k2 sgn(x∼1)

where xˆ = [xˆ1; xˆ2] is the estimated value of x. x∼ = xˆ − x is state estimation error. K1=[k11; k12], K2=[k2; 06×6]. K12∈ R6×6, k12∈ R6×6, k2∈ R6×6 are all constant positive definite matrix, and the selection method of values will be given later. γˆ is the estimated value of γ. fˆ (xˆ) is the output of local recurrent neural network:

ˆ (VH ˆ ˆ) fˆ (xˆ) = Wφ

+

Proof. In this paper, the backstepping design will be used to analyze the stability of the proposed controller. First, it will be proved that the state estimation error x∼1 is convergence. According to (4) and (14), it can be obtained:

where ϖ0 and υ0 are known positive constants. Using the local recurrent neural network, the designed adaptive state observer in this paper is:

ˆ ) − K1 x∼1 − K2 sgn(x∼1) xˆ ̇ = Axˆ + b ( ˆf (xˆ) + B (xˆ1) γu

2

where e1=xˆ1 − x1d and e2=xˆ2 − xˆ2d are the controller errors, the expression of variable xˆ2d will be given later. W0, V0, γ0, χ(.), θ(.), λW, λV and λγ are all known positive constants, k3 is a positive definite matrix, p and q are all positive odd constants, and p > q, 1 < p/q < 2. β is a positive constant.

where l is a known positive constant, and z1 and z2 are two variables. Property 3. Although the optimal network weights W and V are unknown, for the particular ROV system, the upper bound values can be inferred:

V ≤ υ0

1 ˆ ˆ) φ (VH χ2

k2 ≥ xˆ2 + x20 + θ4

k4 ≥

(11)

(12)

+

(21)

x

W ≤ ϖ0

2

1 ˆ ˆ ˆ ) 2 Hˆ 2 + 1 Wφ ˆ ′(VH ˆ ˆ ) 2 Hˆ 2 lV ≥ Wφ′(VH χ1 χ2 1 ˆ ˆ ) 2 + Wφ ˆ ′(VH ˆ ˆ ) 2 ) Hˆ 2 + θ2 + ( Wφ′(VH χ3

Input layer

φ (z1) − φ (z2 ) ≤ l z1 − z2

1 ˆ ˆ) φ (VH χ1

(29)

By (29), we can know that V1 will converge to zero in a finite time. After x∼1 converging to zero, if the controller error e1=xˆ1 − x1d is bounded convergence, the tracking error δ1=x1−x1d will also be bounded convergence. Therefore, the convergence of controller error e1 will be analyzed. For a given desired value x1d, according to (14), we can get: e ̇ = xˆ − k x∼ − k sgn(x∼ ) − x ̇ (30) 1

2

11 1

2

1

1d

Consider the Lyapunov function:

V2 =

1 T e1 e1 2

(31)

Taking the derivative of V2 and substituting (30) into it, we can have:

(16)

V2̇ = e1T e1̇ = e1T (xˆ2 − k11 x∼1 − k2 sgn(x∼1) − x1̇ d )

(32)

Choose: xˆ2d = k11 x∼1 + k2 sgn(x∼1) + x1̇ d − k5 e1 − k 6 sgn(e1)

(33)

where k5 and k6 are all positive definite matrices. 84

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2 ∼ ˆ) ˆ ˆ ) = ∂φ (VHˆ )/∂VHˆ |VHˆ = VH where φ′(VH is the high order item of ˆ ˆ . o (V H Taylor expansion. Substituting (42) into (41), and then substituting into (40), it can be obtained:

Substituting (33) into (32), then:

V2̇ = −e1T k5 e1 − e1T k 6 sgn(e1)

(34)

From (34), if we select xˆ2 = xˆ2d , then e1 will converge to zero within a limited time. However, xˆ2d is only a virtual control law, and there is controller error e2=xˆ2 − xˆ2d between xˆ2d and xˆ2 . If e2 and x∼2 are all bounded, and x∼1 can converge to zero in a finite time, the tracking error δ2=x2−x2d will also be bounded. Therefore, the real control law (16) will be analyzed and the tracking error will be proved to be uniformly ultimately bounded later. In the process of control law design, in order to avoid the emergence of higher order derivative terms, the first order low pass filter is usually used in backstepping design method (Tong et al., 2011). However, the low pass filter is gradually convergence, so that xˆ2 cannot converge to the true xˆ2d . Based on the above considerations, the filter as shown in (35) is used to ensure that xˆ2 is convergence in a finite time.

xˆ2 = xˆ2d +

1 ̇ p/q xˆ2d β

V3̇ = x∼T Ax∼ − x∼T K1 x∼1 − x∼1Tk2 sgn(x∼1) − e2T k3 sgn(e2 ) − e1T k 6 sgn(e1) ∼ˆ ∼ )+ ˆ ˆ ) + Wφ ˆ ′(VH ˆ ˆ )V H + B (xˆ1) γu + (e2 − δ 2 )T (W͠ φ (VH ˆ (VHˆ ) − Wφ ˆ (VH ) + ω1) − e2T k 4 e2 − e1T k5 e1 + tr[W͠ T λW −1W͠ ̇ ] x∼2T (Wφ ∼T −1 ∼̇ + tr[V λV V ] + tr[γ∼T λ γ −1γ∼]̇ (43) where,

∼ˆ ∼ ˆ ˆ )V ω1 = ε − W͠ φ′(VH H − Wo2 (V Hˆ ) − W͠ φ (VHˆ ) + W͠ φ (VH )

⎤ ⎡ x∼1 ⎤ ⎡ x∼ ⎤T ⎡−k11 I ⎥ ⎢ ⎥ + x∼2T ω1 V3̇ ≤ ⎢ ∼1 ⎥ ⎢ ⎣ x2 ⎦ ⎣−k12 l Wˆ ⎦ ⎣ x∼2 ⎦ ∼ˆ ∼ ) − e Tk e − e Tk e + ˆ ˆ ) + Wφ ˆ ′(VH ˆ ˆ )V − e2T (W͠ φ (VH H + B (xˆ1) γu 2 4 2 1 5 1 ∼ ∼ ∼ T ˆ ˆ ) + Wφ ˆ ′(VH ˆ ˆ ) V Hˆ + B (xˆ1) γu ) − tr[W͠ T lW (Wˆ − W0 )] x2 (W͠ φ (VH ∼T − tr[V lV (Vˆ − V0 )] − tr[γ∼T l γ (γˆ − γ0 )]

(35)

Remark 2. As can been seen from the expression from of the filter (35), the filter is similar to the terminal sliding mode. The main advantage of terminal sliding mode is finite time convergence. We refer to the advantage of the terminal sliding mode, and apply it to the filter in backstepping design method. According to (14) and (35), we can get:

ˆ − k12 x∼1 − (βe2 )q / p e2̇ = fˆ (xˆ) + B (xˆ1) γu

(45) Since,

(36)

According to (4) and (14), the state estimation error can be expressed as:

ˆ − B (x1) γu ) − K1 x∼1 − K2 sgn(x∼1) x∼ ̇ = Ax∼ + b ( ˆf (xˆ) − f (x ) + B (xˆ1) γu

(37)

Consider the Lyapunov function:

1 1

1

2

(46)

∼T ∼T ∼ − tr[V lV (Vˆ − V0 )] ≤ −tr[V lV (V + V − V0 )] lV ∼ 2 lV ≤ − 2 V + 2 V − V0 2

(47)

≤ −

lγ 2

γ∼

2

+

lγ

γ − γ0

2

2

(48)

Then,

⎤ ⎡ x∼1 ⎤ ⎡ x∼ ⎤T ⎡−k11 I ⎥ ⎢ ⎥ + x∼2T ω1 V3̇ ≤ ⎢ ∼1 ⎥ ⎢ ⎣ x2 ⎦ ⎣−k12 l Wˆ ⎦ ⎣ x∼2 ⎦ ∼ˆ ∼ ) + e Tk e − e Tk e − ˆ ˆ ) + Wφ ˆ ′(VH ˆ ˆ )V − e2T (W͠ φ (VH H + B (xˆ1) γu 2 4 2 1 5 1 ∼ l l W ∼ ∼ T ˆ ˆ ) + Wφ ˆ ′(VH ˆ ˆ ) V Hˆ + B (xˆ1) γu ) − x2 (W͠ φ (VH W͠ 2 + W W − W0 2

∼T ∼̇ V3̇ = x∼T x∼ ̇ + e1T e1̇ + e2T e2̇ + tr[W͠ T λW −1W͠ ̇ ] + tr[V λV −1V ] + tr[γ∼T λ γ −1γ∼]̇ = x∼T Ax∼ + x∼ T ( ˆf (xˆ) − f (x ) + B (xˆ )(γˆ − γ ) u ) − x∼T K x∼ − x∼ Tk sgn(x∼ ) 1

− tr[W͠ T lW (Wˆ − W0 )] ≤ −tr[W͠ T lW (W͠ + W − W0 )] l l ≤ − W2 W͠ 2 + W2 W − W0 2

− tr[γ∼T l γ (γˆ − γ0 )] ≤ −tr[γ∼T l γ (γ∼ + γ − γ0 )]

1 ∼T ∼ 1 T 1 1 1 ∼T ∼ x x + e1 e1 + e2T e2 + tr[W͠ T λW −1W͠ ] + tr[V λV −1V ] 2 2 2 2 2 1 + tr[γ∼T λ γ −1γ∼] (38) 2 ∼ ∼ ˆ ˆ ͠ where W = W − W , V = V − V , γ = γˆ − γ . Taking derivative of V3 and substituting (33), (35) to (37) into it, we can have: V3 =

2

(44)

Substituting updated laws (17)–(19) into (43), and since φ (VHˆ ) − φ (VH ) ≤ l||Hˆ − H || = l||x∼2 ||, then:

2

1

−

− e1T k5 e1 − e1T k 6 sgn(e1) +

lV 2

∼ V

2

lV 2

+

2

V − V0

−

lγ 2

γ∼

2

+

lγ 2

2

γ − γ0

2

(49)

∼T ∼ ˆ − k12 x∼1 − (βe2 )q / p ] + tr[W͠ T λW −1W͠ ̇ ] + tr[V λV −1V ̇ ] e2 [e1 + fˆ (xˆ) + B (xˆ1) γu + tr[γ∼T λ γ −1γ∼]̇ T

Since,

∼ˆ ∼ ) ˆ ˆ ) + Wφ ˆ ′(VH ˆ ˆ )V − e2T (W͠ φ (VH H + B (xˆ1) γu

(39)

≤

Substituting control law (16) into (39), then:

1 2χ1

V3̇ = x∼T Ax∼ + x∼2T ( ˆf (xˆ) − f (x ) + B (xˆ1)(γˆ − γ ) u ) − x∼T K1 x∼1 − x∼1Tk2 sgn(x∼1)

3χ1 T e e 2 2 2

B (xˆ1)

2

1 2χ1

+

u

2

γ∼

2

ˆ ˆ) φ (VH

W͠

2

+

1 2χ1

2

ˆ ′(VH ˆ ˆ) Wφ

2

Hˆ

∼ V

2

+

2

− e1T k5 e1 −

(50)

∼T ∼̇ sgn(e1) − e2T k3 sgn(e2 ) − e2T k 4 e2 + tr[W͠ T λW −1W͠ ̇ ] + tr[V λV −1V ] + tr[γ∼T λ γ −1γ∼]̇

e1T k 6

∼ˆ ∼ )≤ ˆ ˆ ) + Wφ ˆ ′(VH ˆ ˆ )V x∼2T (W͠ φ (VH H + B (xˆ1) γu +

(40) 1 2χ2

According to the definition of neural network output (10) and (15), the estimation error of neural network can be expressed as:

∼ f = fˆ (xˆ) − f (x ) ˆ ˆ ) + Wφ (VH ˆ ˆ ) − Wφ (VHˆ ) + Wφ (VHˆ ) − Wφ (VH ) + ε = W͠ φ (VH

u

2

γ∼

ˆ ′(VH ˆ ˆ) Wφ

2

Hˆ

1 2χ2

+ 2

∼ V

2

ˆ ˆ) φ (VH

2

W͠

2

+

2

(51)

x∼2T ω1 ≤ 3χ3 x∼2T x∼2 + (41)

+

ˆ ˆ , we can get: Do the Taylor expansion of φ (VHˆ ) about VH ∼ˆ ∼ ˆ ˆ ) − φ′(VH ˆ ˆ )V φ (VHˆ ) = φ (VH H + o2 (V Hˆ )

B (xˆ1)

2

1 2χ2

3χ2 T x x 2 2 2

1 ( 2χ3

(42) 85

1 ( 2χ3

φ (VHˆ )

2

1 2χ3

ε

ˆ ˆ) Wφ′(VH

2

+ 2

1 2χ3

∼ Wo2 (V Hˆ )

2

ˆ ′(VH ˆ ˆ ) 2 ) Hˆ + Wφ

+ φ (VH ) 2 ) W͠

2

∼ V

2

+

2

(52)

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where V3(0) is the initial value of V3. In (59), since there exist the upper boundary value ϖ0 and υ0 of the ∼ optimal network weights W and V respectively, and ε and o 2 (V Hˆ ) are all bounded by selecting the appropriate local recurrent neural network structure, it can be seen that ρ also bounded. From (61), when t→∞, V3 is bounded by ρ/κ. That is, x∼ , e1 and e2 are all uniformly ultimately bounded. Remark 3. According to (1) and (3), B(x1) and γ are both full rank and greater than zero. However, the updated law (19) can’t ensure γ > 0, that is (B (xˆ1) γˆ)−1 may be singular. For this problem, the regularized inverse (B (xˆ1) γˆ)T [ρ0 I + B (xˆ1) γˆ (B (xˆ1) γˆ)T ]−1 can be used to instead (B (xˆ1) γˆ)−1 in the actual control process, where ρ0 is a positive constant. Since the value of ρ0 can be chosen very small, (B (xˆ1) γˆ)T [ρ0 I + B (xˆ1) γˆ (B (xˆ1) γˆ)T ]−1 is approximated to (B (xˆ1) γˆ)−1. Even if B (xˆ1) γˆ is singular, the control law (16) is well defined (Chu et al., 2016a).

Then,

⎡ x∼ ⎤T ⎡−k11 I V3̇ ≤ ⎢ ∼1 ⎥ ⎢ ⎣ x2 ⎦ ⎢⎣−k12 l Wˆ + ∼ + Δ3 V

2

∼ Wo2 (V Hˆ )

1 2χ3

+ Δ4 γ∼ 2

+

lW 2

⎤⎡ ∼ ⎤ ⎥ ⎢ x1 ⎥ − e1T k5 e1 + e2T Δ1 e2 + Δ2 W͠ + 3χ3 ⎥⎦ ⎣ x∼2 ⎦

3χ2 2

2

1 2χ3

+

W − W0

ε 2

2

+

2

+ lV 2

V − V0

2

+

lγ 2

γ − γ0

2

(53) where,

3χ1 2

Δ1 = −k 4 + l

2

+

+ φ (VH )

2)

Δ2 = − W2 +

1 2χ1

1 ( 2χ3

2

φ (VHˆ )

(54)

ˆ ˆ) φ (VH

1 2χ 2

ˆ ˆ) φ (VH

2

+ (55)

lV 1 ˆ ˆ ˆ ) 2 Hˆ 2 + 1 Wφ ˆ ′(VH ˆ ˆ) + Wφ′(VH 2 2χ1 2χ2 1 ˆ ˆ ) 2 + Wφ ˆ ′(VH ˆ ˆ ) 2 ) Hˆ 2 + ( Wφ′(VH 2χ3

Δ3 = −

Δ4 = −

lγ

+

2

1 B (xˆ1) 2χ1

2

u

2

+

1 B (xˆ1) 2χ2

2

u

2

Hˆ

4. Results

2

For most ROV systems, the distance between the centers of gravity and buoyancy is usually far and the roll angle and pitch angle change slightly. Thus, the tracking control of ROV is usually decomposed into two independent parts: horizontal surface tracking control and vertical surface tracking control. In this paper, only horizontal surface tracking control will be carried out to verify the effectiveness of the proposed method. Namely, only the tracking control of position X, position Y and yaw angle in earth-fixed frame will be considered. The parameters of dynamic models of surge DOF, sway DOF and yaw DOF in body-fixed frame are described as follows (Podder, and Sarkar, 2001; Omerdic and Roberts, 2004): M=diag([312, 312, 4.63]), G(η)=0, τ=diag([1−v1/8, 1−v1/8, 1−v2/8, 1−v2/8])u, C(v)=[0, 0, −250v2; 0, 0, 250v1; 250v2, −250v1, 0], D(v)= diag([148|v1|+100, 148|v2|+100, 280|v3|+230]), B =[1, 1, 0, 0; 0, 0, 1, 1; 0.38, −0.38, 0.38, −0.38], τd=10rand(3,1)v, where v=[v1, v2, v3]T represent the surge speed, sway speed and yaw angular velocity in the body-fixed frame, respectively. rand(3,1)∈ R3×1 represents the random data from –1 to 1. For the given ROV dynamic model, the controller parameters in simulation are chosen as: k11=diag([0.5, 0.5, 4]), k12=diag([1, 1, 10]), χ1=χ2=χ3=0.02, k2= diag([3, 3, 1.2], k3=diag([0.1, 0.1, 0.1]), k4=diag([1, 1, 1]), k5=diag([0.1, 0.1, 5]), k6=diag([0.02, 0.02, 0.2]), W0=0.1, V0=0.1, γ0=1, λW=0.4, λV=0.4, λγ=0.005, p=5, q=3. The number of hidden layer neuron is 12, and the number of recurrent layer neuron is 6. The simulation results are shown in Figs. 2–4. In order to illustrate the effectiveness of the proposed method, high gain observer-based PID controller is used for comparison. For the latter, the observer is used to estimate the state firstly, and then PID controller is designed based on the estimated values. The high gain observer and the PID controller are shown in (62) and (63).

(56)

2

(57)

According to (20)–(25), we can know that:

V3̇ ≤ −x∼T Qx∼ − e1T k5 e1 − e2T θ5 e2 − θ1 W͠ + lV 2

1 2χ3

V − V0

∼ Wo2 (V Hˆ ) 2

+

lγ 2

2

γ − γ0

+

lW 2

W − W0

2

2

∼ − θ2 V

2

− θ3 γ∼

2

+

1 2χ3

ε

2

+

2

≤ − κV3 + ρ (58) where,

ρ=

1 1 ∼ ε 2 + Wo2 (V Hˆ ) 2χ3 2χ3 lγ + γ − γ0 2 2

2

+

lW W − W0 2

κ = 2 min[λ min (Q), λ min (k5), θ1, θ2, θ3, θ5]

2

+

lV V − V0 2

2

(59) (60)

where λmin(Q) and λmin(k5) represent the minimum eigenvalues of the matrix Q and k5, respectively. According to (58), it can be obtained:

0 ≤ V3 ≤

ρ ⎛ ρ⎞ + ⎜V3 (0) − ⎟ exp(−κt ) ⎝ κ κ⎠

(61)

Fig. 2. Simulation results of the proposed method. (a) The actual value and the estimated value of position X, (b) The tracking error and estimation error of position X.

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Fig. 3. Simulation results of the proposed method. (a) The actual value and the estimated value of position Y, (b) The tracking error and estimation error of position Y.

Fig. 4. Simulation results of the proposed method. (a) The actual value and the estimated values of yaw angle, (b) The tracking error and estimation error of yaw angle.

Fig. 5. Simulation results of the PID controller. (a) The actual value and the estimated value of position X, (b) The tracking error and estimation error of position X.

xˆ1̇ = xˆ2 − k11 x∼1 ˆ − k12 x∼1 xˆ2̇ = B (xˆ1) γu ⎛ u = BT (xˆ1)[ρ0 I + B (xˆ1) BT (xˆ1)]−1 ⎜ −kp e1 − kd e1̇ − ki ⎝

feasible by choosing the appropriate controller parameters. But the tracking performances are very different. Based on the proposed method, the state estimation errors can converge to zero in less than 2.0 s. But based on the PID controller, the state estimation errors converge to zero at least 14.0 s. In addition, based on the PID controller, the tracking errors are asymptotic convergence. There also will be an apparent oscillation when the desired value has a step, and the convergence time of the tracking errors is longer. Compared with the PID controller, the proposed method can avoid these problems very well. Simulation results show that the proposed method is effective for

(62)

∫0

t

⎞ e1 dt ⎟ ⎠

(63)

In (62) and (63), the controller parameters are chosen as: k11=diag([2, 2, 40]), k12=diag([2, 2, 40]), kp=diag([5, 5, 40]), kd=diag([0.5, 0.5, 0.5]), ki=diag([0.2, 0.2, 2.0]). The simulation results are shown in Figs. 5–7. As can be seen from the simulation results, two controllers are both 87

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Z. Chu et al.

Fig. 6. Simulation results of the PID controller. (a) The actual value and the estimated value of position Y, (b) The tracking error and estimation error of position Y.

Fig. 7. Simulation results of the PID controller. (a) The actual value and the estimated values of yaw angle, (b) The tracking error and estimation error of yaw angle. Biggs, J., Holderbaum, W., 2009. Optimal kinematic control of an autonomous underwater vehicle. IEEE Trans. Autom. Control 54 (7), 1623–1626. Boizot, N., Busvelle, E., Gauthier, J.P., 2010. An adaptive high-gain observer for nonlinear systems. Automatica 46 (9), 1483–1488. Chatchanayuenyong, T., Parnichkun, M., 2006. Neural network based-time optimal sliding mode control for an autonomous underwater robot. Mechatronics 16 (8), 471–478. Chen, H.H., 2008. Vision-based tracking with projective mapping for parameter identification of remotely operated vehicles. Ocean Eng. 35 (10), 983–994. Chu, Z., Zhu, D., Yang, S.X., 2016a. Observer-based adaptive neural network trajectory tracking control for remotely operated vehicle. IEEE Trans. Neural Netw. Learn. Syst.. http://dx.doi.org/10.1109/TNNLS.2016.2544786. Chu, Z., Zhu, Z., Yang, S.X., Jan, G.E., 2016b. Adaptive sliding mode control for depth trajectory tracking of remotely operated vehicle with thruster nonlinearity. J. Navig.. http://dx.doi.org/10.1017/S0373463316000448. Fischer, N., Hughes, D., Walters, P., Schwartz, E.M., Dixon, W.E., 2014. Nonlinear RISEbased control of an autonomous underwater vehicle. IEEE Trans. Robot. 30 (4), 845–852. Gan, Y., Wang, L.R., Liu, J.C., Xu, Y.R., 2004. The embedded basic motion control system of autonomous underwater vehicle. Robot 26 (3), 246–253. Gao, J.S., Xing, Z.W., Zhang, H.B., 2004. Observer-based neural network adaptive control of underwater vehicles. Robot 26 (6), 515–518. Hankovic, M., 1997. Adaptive nonlinear output feedback tracking with a partial highgain observer and backstepping. IEEE Trans. Autom. Control 42 (1), 106–113. Hoang, N.Q., Kreuzer, E., 2007. Adaptive PD-controller for positioning of a remotely operated vehicle close to an underwater structure: theory and experiments. Control Eng. Pract. 15 (4), 411–419. Kim, J., Chung, W.K., 2006. Accurate and practical thruster modeling for underwater vehicles. Ocean Eng. 33 (5–6), 566–586. Lapierre, L., Jouvencel, B., 2008. Robust nonlinear path-following control of an AUV. IEEE J. Ocean. Eng. 33 (2), 89–102. Lee, K.W., Khalil, H., 1997. Adaptive output feedback control of robot manipulators using high-gain observer. Int. J. Control 67 (6), 869–886. Li, Z., Zheng, C., Sun, D., 2013. Tracking analysis and design for ultra-short baseline installation error calibration. Oceans, 23–27, (San Diego). Miao, B., Li, T., Luo, W., 2013. A DSC and MLP based robust adaptive NN tracking control for underwater vehicle. Neurocomputing 111 (2), 184–189. Mohan, S., Kim, J., 2015. Coordinated motion control in task space of an autonomous underwater vehicle-manipulator system. Ocean Eng. 104, 155–167.

ROV tracking control. 5. Conclusions In this paper, the adaptive neural network tracking control method based on observer is addressed. The proposed method can estimate the state by constructing adaptive state observer and use local recurrent neural network for online learning for unknown item of ROV dynamics model. Based on the output of local recurrent neural network and the estimated values of observer, the adaptive tracking control law is designed to obtain the control signals of thrusters directly. Simulation results show that the proposed method can complete ROV racking control and have a better transition when the desired value has a sudden change and ensure that the state estimation error is convergence and tracking error is uniformly ultimately bounded. Although the proposed method is feasible, the problem of finite time convergence for adaptive neural network estimation error should be paid more attention in the future work. Acknowledgment This project is supported by the National Natural Science Foundation of China under Grants 51509150 and 51575336; Shanghai Municipal Natural Science Foundation under Grant 15ZR1419700. References Alessandri, A., Caccia, M., Veruggio, G., 1999. Fault detection of actuator faults in unmanned underwater vehicles. Control Eng. Pract. 7 (3), 357–368.

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Z. Chu et al.

Tong, S., Li, H.X., 2002. Direct adaptive fuzzy output tracking control of nonlinear systems. Fuzzy Sets Syst. 128 (1), 107–115. Tong, S.C., Li, Y.M., Feng, G., Li, T.S., 2011. Observer- based adaptive fuzzy backstepping dynamic surface control for a class of MIMO nonlinear systems. IEEE Trans. Syst. Man Cybern. – Part B: Cybern. 41 (4), 1124–1134. Yu, J.C., Li, Q., Zhang, A.Q., Wang, X.H., 2008. Neural network adaptive control for underwater vehicles. Control Theory Appl. 25 (1), 9–13. Zhang, L.J., Qi, X., Pang, Y.J., 2009. Adaptive output feedback control based on DRFNN for AUV. Ocean Eng. 36 (3), 716–722. Zhang, M.J., Chu, Z.Z., 2012. Adaptive sliding mode control based on local recurrent neural networks for underwater robot. Ocean Eng. 45, 56–62.

Omerdic, E., Roberts, G., 2004. Thruster fault diagnosis and accommodation for openframe underwater vehicles. Control Eng. Pract. 12 (12), 1575–1598. Podder, T.K., Sarkar, N., 2001. Fault-tolerant control of an autonomous underwater vehicle under thruster redundancy. Robot. Auton. Syst. 34 (1), 39–52. Shim, H., Jun, B.H., Lee, P.M., Baek, H., Lee, J., 2010. Workspace control system of underwater tele-operated manipulators on an ROV. Ocean Eng. 37 (11–12), 1036–1047. Silvestre, C., Cunha, R., Paulino, N., Pascoal, A., 2008. A botton-following preview controller for autonomous underwater vehicles. IEEE Trans. Control Syst. Technol. 17 (2), 257–266. Souza, E.C.D., Maruyama, N., 2007. Intelligent UUVs: some issues on ROV dynamic positioning. IEEE Trans. Aerosp. Electron. Syst. 43 (1), 214–226.

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