Longitudinal motion control of underwater vehicle based on fast smooth second order sliding mode

Accepted Manuscript Title: Longitudinal Motion Control of Underwater Vehicle Based on Fast Smooth Second Order Sliding Mode Author: Jian Yang Jin-fu Feng Duo Qi Yong-li Li PII: DOI: Reference:

S0030-4026(16)30755-0 http://dx.doi.org/doi:10.1016/j.ijleo.2016.06.124 IJLEO 57910

To appear in: Received date: Accepted date:

3-5-2016 29-6-2016

Please cite this article as: Jian Yang, Jin-fu Feng, Duo Qi, Yong-li Li, Longitudinal Motion Control of Underwater Vehicle Based on Fast Smooth Second Order Sliding Mode, Optik - International Journal for Light and Electron Optics http://dx.doi.org/10.1016/j.ijleo.2016.06.124 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Longitudinal Motion Control of Underwater Vehicle Based on Fast Smooth Second Order Sliding Mode *Jian Yang1, Jin-fu Feng1, Duo Qi1, Yong-li Li2 1. School of Aeronautics and Astronautics Engineering, Air Force Engineering University, Xi’an, 710038, China

2. Engineering University of CAPF, Xi’an, 710038, China ABSTRACT

By considering the influence of model uncertainty and external disturbance as well as combining theory of high order sliding mode with the recurrent hermite neural network (RHNN), this paper proposed a nonlinear robust control method for solving the longitudinal motion control problem of underwater vehicles. Based on the characteristics of multiple time scales, the longitudinal motion model was divided into inner and outer loops. The designed fast and smooth second order controller, which applied the discontinuous sign function on the derivative of the control law, was used in the outer loop. Then integration was performed to obtain the continuous sliding mode control law, which was applied to remove chattering and improve convergence rate. Furthermore, based on the controller of outer loop, the composite disturbance of the inner loop was effectively estimated using the combination between the fast smooth second order sliding mode controller and the RHNN disturbance observer. Therefore, the controller was effectively compensated, and the entire system’s stability in finite time was proved. The simulation results confirmed the promising performance and robustness of the proposed control method.

Key word: underwater vehicle; longitudinal motion; second order sliding mode control; recurrent neural network; disturbance observer; finite time stability

0 Introduction The effective development and utilization of oceans are crucial to the survival of mankind and the sustainable development of countries. Underwater vehicles have several advantages such as deep diving depth and low operating and maintenance costs. Therefore it plays an irreplaceable role in marine scientific research and resource exploration. Nowadays, underwater vehicles are also used to fulfill complex tasks, such as marine survey sampling and deep sea platform and underwater measurement. Therefore, aimed at complex task requirements, the basic requirements of these vehicles, namely, maneuverability and safety,

are ensured by their control systems [1–3]. Most underwater vehicles are under-drive system with characteristics such as highly nonlinear, fast time-varying and strong coupling, which lead to a difficulty to obtain its hydrodynamic coefficients accurately, making it difficult to ensure the accuracy of the mathematical model. Meanwhile, in the process of communication with mother ship and base stations, as well as the underwater operations, the waves and underwater currents near the surface of underwater vehicle is the main non-negligible disturbance. Thus it can be seen that, the control of underwater vehicles is facing serious challenges, not only need to get rid of dependence on the accuracy of the mathematical model, but also need strong adaptability and robustness to cope with strong external interference[4-6]. In recent years, many nonlinear control methods, such as feedback linearization[7], gain scheduling[8], adaptive back-stepping control[9], and sliding mode control, have been recently developed to address such challenges. Given its excellent robustness to external disturbances and parameter uncertainties, the sliding mode control method has been widely used in the control system design of underwater vehicles. Yoerger et al[10][11].successfully applied the sliding mode control method to Jason underwater vehicles. Bessa et al.[4]used the same method to control the depth of underwater vehicles, and the effectiveness of the algorithm was proven via simulation. Bagheri et al[12].combined the sliding mode control method with the neural network method to achieve trajectory tracking control for high-performance underwater vehicles, and the usability of the control was further improved. Ming et al[13].designed a nonlinear iterative sliding mode trajectory tracking control based on the engineering decoupling design concept. Apart from accurately tracking the 3D trajectory, the

controller could effectively eliminate the influence of the parameter variations in the vehicle model and the disturbances from the marine environment. The abovementioned studies have mostly based their sliding mode control methods on a linear sliding hyper-plane, thereby requiring the control system to have a relatively high control gain to achieve favorable robustness to external disturbances and parameter uncertainties as well as to accelerate its response time. However, excessive control gain may lead to chattering, which in turn diminishes the performance of the control system. At the same time, the sliding mode controller that is based on a linear sliding surface can only guarantee the asymptotic convergence of the error. However, finite time convergence mostly provides a better dynamic response quality and a more accurate tracking performance [14]. To solve this problem, many experts have recently begun to investigate the higher order sliding mode [15], which obtains the continuous available slide mode control law via integration to reduce chattering and ensure finite time convergence. Tian B [16].proposed a continuous high order sliding mode variable control method that could ensure favorable sliding characteristics and accuracy, eliminate chattering, and address the limitations of the control system. Shtessel et al [17] proposed a second order sliding mode control method based on the super spiral algorithm to ensure finite time convergence, effectively reduce the chattering of the sliding mode variable, and solve the high order derivative problem. Based on the above analysis and the high order sliding mode control method, this paper proposes a fast smooth second order sliding mode control method to solve the chattering and finite-time stability problems that are faced by the complex systems of underwater vehicles with higher parameter uncertainties, stronger external disturbances, and stronger multivariable

coupling. Following the multi-time scale principle, the longitudinal motion model is transformed into a bicyclic ring system with inner and outer loops. The fast smooth second order sliding mode controller is then designed for the outer loop to eliminate chattering, accelerate convergence speed, and obtain the continuous control input by using the second order sliding mode to hide the sign function in the integral term. To address wave disturbance and parameter uncertainty, the fast smooth second order sliding mode controller that is combined with the RHNN disturbance observer is then designed for the inner loop according to the outer loop controller. This controller offers valid estimations of composite disturbance for underwater vehicles and ensures the finite-time stability of the entire closed-loop system. The simulation results validate the effectiveness of the proposed control method.

1 Mathematical Model for the Longitudinal Motion of Underwater Vehicles As shown in Fig 1, the body- and earth-fixed coordinates O-xbybzb and O-xyz are defined, with the earth-fixed coordinate treated as an inertial frame. Table 1 defines the corresponding six-degrees of freedom (DOF) kinematic and dynamic parameters. In the body-fixed coordinate system, the origin of the coordinate system is set as the center of gravity of the underwater vehicle (i.e,(xG, yG, zG) =0). The radius vector from the origin of the body-fixed coordinate to the buoyancy center of the underwater vehicle is (xB, yB, zB). This paper only investigates the longitudinal motion of underwater vehicles. Based on the overall configuration of underwater vehicles, the model test results, and the six-DOF movement mathematical model of underwater vehicles

[18]

, the dynamic model for the

longitudinal motion of underwater vehicles is obtained as follows:

m[ w  uq  xG q  zG q 2 ]  Z q q  Z w w  Z uquq  Z uwuw

(1)

 Z ww w w  Z qq q q  (G  Bo ) cos   u 2 Z uu s  Z wave

I y q  m[ xG (uq  w)  zG wq ]  M q q  M w w  M uquq  M uwuw  M ww w w  M qq q q  ( xGW  xB B0 ) cos 

(2)

( zGW  z B B0 ) sin   M uu u   M wave 2

z  w cos   u sin 

 q

where Zq , Z w , Zuq , M q , M w , M uq et al are the hydrodynamic parameters, G is the mass of vehicle,  s is the elevator angle, I y is the rotational inertia of the vehicle around the yb axis, B0 is the buoyancy of the vehicle when completely sunk, and Zwave and Wwave are the wave

force and moment, respectively. Table 1 defines the other parameters. To facilitate the analysis and synthesis of the control system, the nonlinear equation (1) is linearized using the small perturbation method. We assume that the axial velocity of the underwater vehicle u is constant (i.e. u=u0). Given that the vehicle moves slightly near the equilibrium position, the relationship between fluid damping and the movement parameters is approximately linear. By omitting the higher order terms in the model parameters under slow speed and by letting sinθ≈θ and cosθ≈1, the simplified state equation of the underwater vehicle system is obtained as follows:    A1  Ae1    A2  Au u  Am  Ad

(3)

Where   [ z  ] and    w q  are the state variables, and u   s is the input. The coefficient matrixes are expressed as follows:  u   1 0  Ae1   0  A1     0  0 1 

 H 1 (Z u  Z w ) H 1 (Zuq  Z qq q  mu0 )   H 1u02 M uu  A2   1 uw 0 ww A   u  1 2  1  H u0 Zuu   H (M uwu0  M ww w ) H (M uqu0  M qq q )   H 1Z  m  Z w Z q   H 1 (G  B0 )  Ad   1 wave  Am   1 H  ，  M I  Z   1  w y q  H Wwave   H xB B0  H zB B0 

Given the uncertainties of the hydrodynamic coefficients, the partial coefficient matrices in equation (3) are divided as follows into the known nominal part and the unknown uncertain part:   ( A2*  A2 )  ( Au*  Au )u  Am  Ad

(4)

Where A2* and Au* are the nominal values of the hydrodynamic parameters for the dynamic equations that can be obtained by theory estimation, computational fluid dynamics simulation, and other methods, while△A2 and △Au are the uncertain parts of the system dynamics model coefficients matrix that are caused by the uncertainty of the hydrodynamic coefficients. Where  a* a12*  *  au*1  A  *  A2*   11* *  u  au 2   a21 a22 

From equation (4), we obtain the following:   A1  Ae1  * *   A2  Au u  Am  A2  Au u  Ad

(5)

Let f=A2*η+Au*u+Am, △f=△A2x+△Auu+Ad Equation (5) can be converted as follows:   A1  Ae1    f  f

(6)

The influence of uncertainty part △f on the longitudinal movement of the vehicle cannot be accurately determined. However, the uncertain parts of the physical system are bounded by limiting the system input as follows:

f  0

2 Modeling and Simulation of Wave Disturbance The ocean wave model includes regular and random waves. The regular wave model simply describes the wave as traveling in a cosine plane, but the actual wave is an irregular random wave. In general, the propagation direction of the wave is assumed to be fixed, and the wave can be considered as the superposition of innumerable cosine waves that are independent from each other and have different amplitudes, wavelengths, and random initial phases. Spectral density is introduced to describe the statistical characteristics, frequency components, and corresponding energy relationships of the random wave. The widely used PearsonMoscovici (P-M)wave spectrum is expressed as follows: S ( ) 

 3.11  8.1103 g 2 exp  2 4  5  Hs  

(7)

Where ω is the wave frequency, S(ω) is the spectral density, g is the gravitational acceleration, and Hs is the significant wave height. Based on the wave spectrum, we obtain the wave-up of the random wave at any point in the sea, by which the size of the wave force and moment can be calculated. The wave disturbance near the surface of the underwater vehicle can be decomposed into first and second order wave forces [19]. The first order wave force is a high-frequency periodic force,which amplitude is directly proportional to wave height and frequency is similar to wave frequency. The vehicle performs rolling, pitching, and heaving motions under the influence of this force.

The second order wave force is an average upward force which size is proportional to the squared wave height, amplitude is smaller than that of the first order wave force, and remains constant for a long period. Therefore, this force is usually treated as a constant disturbance. This paper focuses on the control problem of underwater vehicles under the influence of the first order wave force and moment. This paper uses the 3D surface element method to calculate the first order wave force and moment as follows based on the P–M wave spectrum equation: N

N

i

i

Z wave   ai exp[i2 h(t ) / g ]sin(   2V cos  / g ) t   ai exp[i2h(t ) / g ]sin()

(8)

N

M wave (t )   CLV  (1  0.02u cos  )sgn(cos  )aii2 exp[i2 h(t ) / g ]cos(ei )t

(9)

i 1

where ai=[2S(ωi)δω]1/2 and ωe= –ωi–(ωi2u/g)cosβ, h(t) is the distance from the vehicle to the water surface, V is the displacement volume of the vehicle, β is the encounter angle of the vehicle and wave, C is the hydrodynamic coefficient, and N generally ranges between 20 and 40. Therefore, when N = 25 in the 2 sea condition, the underwater vehicle navigates at 2m/s at 2 m depth and β is0.The simulation results of the first order wave force and moment are presented as follows: The dynamic model of underwater vehicle navigation near the water surface can be obtained by integrating the abovementioned wave force and moment algorithm into equation (6).

3 Design of Fast Smooth Second Order Sliding Mode Controller The underwater vehicle faces a typical time-variant and nonlinear multivariable control

problem with strong coupling. The model of the vehicle has high uncertainty and external disturbance, which hinder the conventional design approach from designing a controller for the vehicle. The main feature of the sliding mode variable structure control is that the state of the system switches back and forth between one or more planes in the state space. The structure of the controller changes when the states of the system go through the switching plane. When the system is in sliding mode, the system has invariance for the model uncertainty of the controlled object and the external disturbance. Therefore, this sliding mode is suitable for tracking the control of uncertain nonlinear systems. However, as an intrinsic defect of the traditional sliding mode control, chattering can affect the control accuracy and system performance. The high order sliding mode is an advanced version of the traditional sliding mode that not only offers the same advantages being offered by the sliding mode control but also reduces the chattering and improves the accuracy of the control. Therefore, this paper designs a fast smooth second order sliding mode controller for underwater vehicles. Based on multiple time scales theory [20], this paper divides the vehicle longitudinal control system into inner and outer loops. The picture as follows is the structure of the longitudinal control system. In the designed control system, the longitudinal velocity wc and attitude angular rate qc are used as virtual control inputs to allow the state of the underwater vehicle to track steadily the reference signal of the outer loop in finite time. In the inner loop, the control moment M that is generated by the rudder angle δs and treated as the actual input of the system aims to converge w and q to the desired virtual control wc and qc in finite time. The composite disturbance that contains system uncertainty and wave disturbance is estimated by the

nonlinear disturbance observer. 3.1 Design of the Second Order Sliding Mode Controller for the Outer Loop To suppress control chattering, the second order sliding mode control is used in the outer loop. To improve control accuracy, the outer loop sliding surface that contains the error integral term is defined as follows: t

 1  e1  c1  e1dt

(10)

0

Where e1=ξ– ξc, ξc is the longitudinal control target tracking command, ξc= [zc θc]T, c1=[c11c12]T is the integral parameter, c11, c12>0 , and σ1=[σ12σ22]. Taking the derivation of equation (10), we obtain the following:  1  e1  c1e1

(11)

 A1  Ae1  c  c1e1

To ensure its anti-disturbance performance, convergence precision, and convergence rate, the fast smooth second order sliding mode method for the outer loop that combined with the smooth second order sliding mode control method of Shtessel[17] is proposed in this paper. The sliding mode variable dynamics state is expressed as follows:  1  k1  1

( m 1)/ m

t

sgn( 1 )  k2 1   (k3  1 ( )

( m  2)/ m

sgn( 1 ( )))d

(12)

0

where m>0, k1=[k11k12], k2=[k21k22]，k3=[k31k32]，k11,k12 k21,k22, k31, k32>0. In the above equation, when the initial position is located far from σ1, the convergence rate mainly depends on the linear term in the equation. However, when the initial position approaches σ1=0, the convergence rate mainly depends on the nonlinear terms in the equation. Therefore, the proposed method maintains a fast convergence rate regardless of whether the initial state of

the system is far from the equilibrium point. The fast smooth order sliding mode control law is designed to make the outer loop system asymptotically steady in finite time. ( m 1)/ m 1  sgn  1  k2 1  y1 ) c  A1 (c1e1  c  Ae1  k1  1  ( m  2)/ m y1  k3  1 sgn  1  

(13)

Theorem 1: For the longitudinal control system outer loop of the underwater vehicle, under the action of the control law (13), the system becomes steady and the tracking error converges to 0 in finite time when m≥2 and the appropriate parameters k1, k2, and k3are selected. Proof: Choose the following Lyapunov function: 

V1 

1 2 1 ( m  2)/ m y1   k3 z sgn zdz 2 0

1 m  y12  k3 1(2 m2)/ m 2 2m  2

(14)

Define the tracking error of the longitudinal control system outer loop e2   c

(15)

Substitute the above equation into equation (11) to obtain 1  A1e2  A1c  F1

(16)

Where F1  c1e1  c  Ae1 Take the derivative of the designed Lyapunov function V1 =y1 y1  k3 1

( m 2)/ m

1 sgn 1

Substitute equations (13) and (14) into the above equation to obtain

(17)

V1 =y1 (k3  1  k1  1

( m 1)/ m

=  k3  1 k1k3  1

( m  2)/ m

=k3 A1  1

( m  2)/ m

sgn  1 ( A1e2

sgn  1  k2 1  y1 )

( m  2)/ m

(2 m 3)/ m

sgn  1 )  k3  1

sgn  1 y1  k3 A1  1  k 2 k3  1

( m  2)/ m

(2 m  2)/ m

( m  2)/ m

 k3  1

sgn  1e2  k1k3  1

sgn  1e2  ( m  2)/ m

(2 m 3)/ m

(18)

sgn  1 y1

 k 2 k3  1

(2 m  2)/ m

The inner loop is designed to converge the vertical velocity w and angular attitude rate q to the desired virtual control sign that is produced in the outer loop in finite time. That is, e2is converged to 0 in finite time T2. We then obtain the following: V1 =  k1k3 1

(2 m3)/ m

 k2 k3 1

(2 m 2)/ m

The system is homogeneous(i.e., (σ1, y1) →

0

(19)

(kmσ1, km–1y1)), and the negative

homogeneity degree is –1. Therefore, in finite time, σ1 and its derivative converge to 0. When the tracking error e1 reaches the sliding surface, t

e1  c1  e1dt  0

(20)

0

The system state asymptotically tracks the reference tracking signal. 3.2 Design of the Second Order Sliding Mode Controller based on RHNN Disturbance Observer The inner loop control is designed to construct a controller that enables w and qto track rapidly and accurately the reference signal that is provided by the outer loop. The inner loop sliding surface that contains the error integral term is defined as follows: t

 2  e2  c2  e2 dt

(21)

0

wheree2= η– ηc, ηc is the tracking command that is provided by the outer loop, c2 = [c21

c22]T is the integral parameter, and c21, c22>0. Taking the derivative of equation (21), we obtain the following:  2  A2  Au u  Am c  f  c2e2

(22)

Apart from accurately approximating any nonlinear function, RHNN offers faster convergence speed and less computational burden than the conventional neural network. Therefore, to ensure that the inner loop control system can quickly and steadily achieve the control object of the command tracking on the conditions of uncertainty and external wave disturbance, the RHNN neural network is combined with the fast smooth second order sliding mode method to estimate the parameter uncertainty and external disturbance. Afterward, the inner loop second order sliding mode controller is designed based on the RHNN disturbance observer. The structure of the disturbance observer is presented as follows: Where Δf for the composite disturbance in the loop. An optimal RHNN network weight W2 is introduced to obtain the following: f  W2T 2  

(23)

Where ψ2 is the hidden layer output of the RHNN network,ε2 is the approximation error, and  2   * . W2 is located in the convex set 2   W2

F

*  M 2  , ε , where M2 is the positive

constant. The online estimated value of composite disturbance Δf is computed as follows: fˆ  Wˆ2T 2

(24)

Where Wˆ2T is an estimated value of the optimal weight W2. To improve the convergence rate of the observer, W2 is decomposed into a proportional term W2Pand an integral term W2I [21]. Given thatW2is located in convex setΩ2, we assume that

MP, MI > 0. W2 P P   W2 P  M P  ,

W2 I I   W2 I  M I 

(25)

Therefore, Wˆ2  Wˆ2 P  Wˆ2 I , where Wˆ2 P is the proportional term and Wˆ2 I is the integral term. Based on equations (25) and (26), we obtain the following: f  fˆ  W2T 2    Wˆ2T 2  W2 I T 2  Wˆ2TP 2   2

(26)

Where W2 I T  W2TI  Wˆ2TI and 2  Wˆ2TP 2   is the composite error. We assume that  2  2 , where  2 is the positive constant. The following observer is designed by combining

the RHNN network with the disturbance observer: z2  2 z2  2  A2  Am  Auu  fˆ   sr

(27)

Where z2 is the state variable of the observer, and ιsr is the robust item to be designed. The expression of this item is given in Theorem 2, whereλ2>0. The error dynamics equation of the observer is obtained as follows: eob  2eob  W2 I T 2  Wˆ2TP 2   2   sr

(28)

Where eob is the disturbance observation error variable, and eob = η– z. Combined with the fast smooth order sliding mode method in Section 3.1, the control input for achieving a steady system in finite time can be obtained as follows: ( m 1)/ m 1  sgn  1  k5 1  fˆ   sr  y2 ) u  Au (c2e2  c  A2  Am  k4  2  ( m  2)/ m y2  k6  2 sgn  2  

(29)

Where m>2, k4=[k41k42], k5=[k51k52]，k6=[k61k62]，k41,k42 k51,k52, k61, k62>0. Theorem 2: For the longitudinal control system inner loop of the underwater vehicle, under the action of the control law (29) and the disturbance observer (28),when m≥2 , and the appropriate parametersk4, k5, k6is selected, and the weight adaptive law for RHNN Network is

computed as follows: Wˆ2 P  2 P 2 ( 2  e2 )T

  2 I 2 ( 2  e2 )T Wˆ2 I  M I F      2 I 2 ( 2  e2 )T Wˆ2 I  M I , tr[Wˆ2TI 2 ( 2  e2 )T [ 0 ˆ F W2 I     T ˆ ˆT    (  e )T   2 I tr[W2 I 2 ( 2  e2 ) ]W2 I Wˆ  M I , tr[Wˆ2TI 2 ( 2  e2 )T ]  0 2I 2 2 2 2I  F tr (Wˆ2TIWˆ2 I ) 

(30)

(31)

And ιsr is then expressed as follows:  sr 

ˆ 22 ( 2  e2 )T ˆ 2  2  e2  a2

(32)

Where the parameter adaptive law is computed as follows: ˆ 2  b2  2  e2

(33)

In equations (30) to (33), ˆ 2 is the estimated value of the upper bound for the error, where a2  b2 a2 , a20  0 and  2I ,  2P , b1 , b2 >0. Therefore, the closed-loop system becomes steady in finite time, that is, the tracking error converges to 0 in finite time. Proof: Construct the Lyapunov function as follows. 2 a 1 1 T 1 V2 =  2T  2  eob eob  tr (W2 I TW2 I )  2  2 2 2 2 2 I 2b2 b1

(34)

Take the derivative of equation (34) and substitute equations (22), (28), (29), and (33) to obtain the following:

V2   2T (k4  2

( m 1)/ m

T sgn  1  k5 1  f  fˆ   sr  y2 )  eob (2 eob

 ˆ 1  W2 I T 2  Wˆ2TP 2   2   sr )  tr (W2 I TW2 I )  2 2  a2 2I b2   2T (k4  2

( m 1)/ m

sgn  1  k5 1  W2 I T 2  Wˆ2TP 2   2   sr  y2 ) 

T eob (2 eob  W2 I T 2  Wˆ2TP 2   2   sr ) 

 k5  2  2 eob 2

2



1

2I

2I

tr (W2 I TW2 I ) 

2 ˆ 2 b2

F

2

( 2  eob )T ( 2  eob ) ˆ 2 2 ˆ 2   a2  2  eob  a2 b2

2

 ( 2  eob )T Wˆ2TP 2   2  eob 2 

( 2  eob )T ( 2  eob ) ˆ 2 2 ˆ 2   a2  2  eob  a2 b2 F

(35)

 M I ,we can obtain the following:

V2  k5  2  2 eob

When Wˆ2 I

 a2

T tr (W2 I T   2 I 2 2T   2 I 2 eob  W2 I  )   

( 2  eob )T Wˆ2TP 2   2  eob 2 

According to equation (31), when Wˆ2 I

1

(36)

 M I , we can obtain the following:

V2  k5  2

2

 2 eob

2

 ( 2  eob )T Wˆ2TP 2   2  eob 2 

( 2  eob )T ( 2  eob ) ˆ 2 1  tr (W2 I T 2I RT I  a2   k5  2

2

 2 eob

2

2 ˆ 2 b2

 a2

  2 I [Wˆ2 I 2 ( 2  eob )T ]   ) tr (Wˆ2TIWˆ2 I )  

 ( 2  eob )T Wˆ2TP 2   2  eob  2 

2 ˆ 2 b2

(37)

 a2

( 2  eob ) ( 2  eob ) ˆ 2 tr[Wˆ2 I 2 ( 2  eob ) ]tr (W Wˆ2 I )  )  2  eob  a2 tr (Wˆ2TIWˆ2 I ) T

T

T 2I

Where 1 tr (W2TIWˆ2 I )  [tr (W2TIW2 I )  tr (Wˆ2TIWˆ2 I )  tr (W2TIW2 I )] 2

Given that Wˆ2 I

2 F

(38)

T T  tr (Wˆ2TIWˆ2 I )  M I2  tr (W2TIW2 I ) , we can obtain tr (W2 IW2 I )  0 and tr (W2 IWˆ2 I )  0 .

Moreover, given that tr (Wˆ2 I 2 ( 2  eob )T )  0 , we obtain the following regardless of the value of Wˆ2 I :

V2  k5  2  2 eob 2

2

 ( 2  eob )T Wˆ2TP 2   2  eob 2 

( 2  eob )T ( 2  eob ) ˆ 2 2 ˆ 2   a2  2  eob  a2 b2

By substituting equations (31) and (33), we obtain the following:

(39)

V2  k5  2  2 eob   2 P ( 2  eob )T ( 2  eob ) 2T   2  eob 2  2

2

 2  eob ˆ 22  2  2  eob  a2 ˆ 2  2  eob  a2 2



 k5  2  2 eob 2

2

(40)

0

Therefore, the closed-loop system becomes steady in finite time, that is, the tracking error converges to 0.

4 Simulation Analyses To verify the effectiveness and feasibility of the proposed control method, the Remote Environmental Monitoring Units [22]of the Woods Hole Oceanographic Institution is simulated as an example using Matlab. The hydrodynamic parameters are assumed to have 30% uncertainty. Sea disturbance is computed using equations (8) and (9). The significance wave height Hs is 1m, and the angle of encounter is β=45°. The table below presents the simulation objects and hydrodynamic parameters. The parameters of the designed outer and inner loop control system include the following: k1=[0.1 0.1], k2=[0.2 0.2], k3=[0.1 0.1], k4=[0.8 0.8], k5=[0.2 0.2], k6=[0.1 0.1], c1=[0.1 0.1], m=3, c2=[0.1 0.1], b1=0.01, b2=2, β2I=β2P=2, and λ2=10. The hidden layer of the RHNN disturbance observer takes five nodes. The initial conditions are as follows: initial depth z0=10m, initial pitch angle θ0=0º, initial pitch angle velocity q0= –0.01rad, and initial velocity along the z direction w0=0.15m/s. To test the control preference of the designed controller, let the controller track the sinusoidal reference signal. Figure 5 presents the simulation results. As can be seen from Figure 5, under the influence of model uncertainty and external disturbances, the proposed control method can track the variable reference signal with zero

steady amplitude and phase errorin a very short period following a short delay. The system remains stable, and the output of the control input remains smooth without any chattering. Given that the underwater vehicle has a huge inertia and is slowly time variant, the simulation results indicate that the controller has an excellent capability of tracking the reference signal. To illustrate further the advantages of the designed controller, two control schemes are used and compared in the simulation. Control scheme 1 uses the conventional sliding mode control (CSMC), while control scheme 2 uses the proposed sliding mode control method. The reference signal is –15m. These schemes are used to achieve tracking control. Figure 6 compares the simulation results from these control schemes by examining their curve depth, pitch angle, and rudder bias angle, which is used as the control input. As it can be seen from Figure 6, control scheme 2 has a smaller adjusting time and overshoot than control scheme 1. Specifically, the adjusting times of these schemes are 26.53s and 14.8s, respectively. The overshoot of the control scheme 1 is about 18%, while that of the control scheme 2 is less than 1%. The control input of control scheme 1 demonstrates chattering to some extent. However, such chattering becomes almost negligible in control scheme 2. Therefore, the proposed controller shows a better dynamic quality and tracking performance than the traditional controllers. To test the robustness of the proposed control method, we simulated two situations where the system is in a nominal status or is influenced by the model uncertainty and external disturbances. The reference signal is set to –15m.Figure 7 shows the simulation results. The curve depth and the pitch angle are slightly changed when the proposed control method is used under the influence of model uncertainty and external disturbance. Moreover,

the proposed method does not show any overshoot and can reach the steady state quickly. Therefore, the control method has strong robustness.

5 Conclusion This paper proposes a fast smooth second order sliding mode control method to solve the longitudinal motion control problems of underwater vehicles. This method, which is combined with the second order sliding mode control method and based on the RHNN network disturbance observer, considers both model uncertainty and external disturbances, such as waves. This method not only solves the problems in estimating model uncertainty and composite disturbances but also eliminates chattering. This method also strictly proves the finite-time stabilization of the closed-loop system. Based on the simulation results, the proposed method helps the control system accurately and rapidly tracks the reference command signal even under the influence of model uncertainty and external disturbance. The method also has better dynamic quality and tracking performance than the traditional sliding mode control method.

Acknowledgments The authors would like to express their thanks for the support from the National Natural Science Foundation of China (No. 51541905)

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Figure 1 Reference coordinate and body-fixed coordinates

Figure 2 wine simulation

Reference command

The controller for outer loop

Virtual control

qc , wc

The controller for inner loop

Disturbance observer for inner loop

Control moment

The underwater vehicle model

(2)

(1)

Figure 3 the structure of the longitudinal control system

Second order sliding mode control

c

Underwater vehicle

u



Zd Wine disturbance

Disturbance observer

e2

Figure 4 the structure of the disturbance observer

-4

Depth(m)

simulation trajectory desired trajectory -6

-8

-10

0

10

20

30

40 Time(s)

50

60

70

80

(a) Depth 10

θ

0

-10

-20

0

10

20

30

40 Time(s)

50

60

70

80

50

60

70

80

(b)Pitch angle 40

δs

20 0 -20 -40

0

10

20

30

40 Time(s)

(c)Rudder bias angle Figure 5Sinusoidal reference signal tracking control effect

-9 control scheme 1 control scheme 2

-10

-11

z(m)

-12

-13

-14

-15

-16 0

5

10

15

20

25 Time(s)

30

35

40

45

50

(a)Depth 20 control scheme 1 control scheme 2 15

θ(dgr)

10

5

0

-5

-10

0

5

10

15

20

25 Time(s)

30

35

40

45

50

(b)Pitch angle 30 control scheme 1 control scheme 2 20

δs(dgr)

10

0

-10

-20

-30

0

5

10

15

20

25 Time(s)

30

35

40

45

50

(c)Rudder bias angle Figure 6 Comparison between control schemes 1 and 2 in terms of achieving tracking control in a specific depth

-9 nominal state non-nominal state

-10

-11

z(m)

-12

-13

-14

-15

-16 0

5

10

15

20

25 Time(s)

30

35

40

45

50

(a) Depth 20 nominal state non-nominal state 15

θ(dgr)

10

5

0

-5

-10

0

5

10

15

20

25 Time(s)

30

35

40

45

50

(b)Pitch angle Figure 7 Comparison of the simulation chart in two situations

Tab.1 Kinematic and dynamic parameters

DOF

Motion modes

The components of linear

The components of

velocity/angular velocity in

location/Euler angles in

three axis of body-fixed

three axis of earth-fixed

coordinate

coordinate

1

movement along the xb-axis or x-axis

u

x

2

movement along the yb-axis or y-axis

v

y

3

movement along the zb-axis or z-axis

w

z

4

rotation along the xb-axis or x-axis (Roll)

p

φ

5

rotation along the yb-axis or y-axis (Pitch)

q

θ

6

rotation along the zb-axis or z-axis (Yaw)

r

ψ

Table 2

parameters of simulation objects and hydrodynamic parameters

m

30.5kg

B

308N

Zq

-1393kg.m/rad

G

299m

zb

-0.02m

Zw

-35.5kg

xb

0m

yb

0m

Zuq

-5.22kg/rad

Iy

3.45kg.m2

Zuw

-28.6kg/m

Zuu

-6.15 kg/(mrad)

u0

2m/s

-0.632kg.

m/rad2

Zww

-131.kg/m

Zqq

Mq

-4.88kg.m2/rad

Mw

-1.93kg.m

Muq

-2.00.m/rad

Muw

24.0kg

Mww

3.18kg

Mqq

-188kg.m2/rad2

Muu

-6.15 kg/(rad)