Indian Journal of Geo Marine Sciences Vol. 48 (07), July 2019, pp. 1016-1027

Nonlinear robust integral sliding super-twisting sliding mode control application in autonomous underwater glider

Maziyah Mat-Noh1,2*, M.R. Arshad2, Rosmiwati Mohd-Mokhtar2, Qudrat Khan3, Zainah Md Zain1, & Herdawati Abdul Kadir4

1 Instrumentation and Control Research Cluster (ICE), Faculty of Electrical and Electronics Engineering,

University Malaysia Pahang (UMP), 26600 Pekan, Pahang, Malaysia 2Underwater, Control and Robotics Group (UCRG), School of Electrical and Electronic Engineering,

Engineering Campus,Universiti Sains Malaysia (USM), Nibong Tebal, Pulau Pinang, 14300 Malaysia 3Center for Advanced Studies in Telecommunications, CIIT, Park Road, Chak Shahzad, Islamabad 44000, Pakistan

Faculty of Electrical and Electronic Engineering Universiti Tun Hussein Onn Malaysia

86400 Parit Raja, Batu Pahat Johor, Malaysia

*[E-mail: maziyah@ump.edu.my]

The design of a robust controller is a challenging task due to nonlinear behaviour of the glider and surround environment. This paper presents design and simulation of nonlinear robust integral super-twisting sliding mode control for controlling the longitudinal plane of an autonomous underwater glider (AUG). The controller is designed for trajectory tracking problem in existence of external disturbance and parameter variations for pitching angle and net buoyancy of the longitudinal plane of an AUG. The algorithm is designed based on integral sliding mode control and super-twisting sliding mode control. The performance of the proposed controller is compared to original integral sliding mode and original super-twisting algorithm. The simulation results have shown that the proposed controller demonstrates satisfactory performance and also reduces the chattering effect and control effort.

[Keywords: Autonomous underwater glider (AUG); Integral sliding mode control; Super-twisting sliding mode control; Chattering reduction]

Introduction The autonomous underwater glider (AUG) was

first initiated by Henry Stommel in 19891. The Stommel’s idea inspired many other researchers to involve in this area of research. Finally in 2001, three operational AUGs were developed and tested: Slocum2, Spray3 and Seaglider4, which are currently being used by many agencies and research groups for oceanography data collection5,6. Also many laboratory-scale AUGs were developed for research purposes, such as robotic gliding fish (Michigan State University)7, USM glider8 (University Science Malaysia), FOLAGA9 (Join effort between IMEDEA, ISME and University of Genova, Italy) ALEX10 (Osaka Perfecture University), and ROGUE11 (University of Princetone). The AUG dives under water by actuating its internal sliding mass horizontally, vertically or cylindrically and pumping the ballast back and forth.

The gliders are multi-input-multi-output (MIMO) nonlinear systems. AUG is considered as on under-

actuated system, highly nonlinear, time-varying dynamic behaviour in nature, uncertainties in hydrodynamic coefficients, and also disturbances by ocean currents12. Several control techniques have been proposed to control the motion of the AUG. The survey done by Ullah et al.13 reviewed the control strategies ranging from classical control, proportional-integral-derivative (PID), optimal control linear quadratic regulator (LQR) up to intelligent control such as neural network and fuzzy logic. The PID controller proposed14,15 is the most popular used due to its simple architecture and less tuning parameters. The LQR11,15,16,17,18,19 also offers simple architecture, wherein only two tuning parameters need to be varied to achieve the desired performance. Both PID and LQR provide good performance. However, since the model is linearized about the equilibrium point, the performance of the controller is only effective in a small neighbourhood of the equilibrium.

The model predictive control (MPC)20,21 was designed to control the attitude of Slocum glider. The

MAT-NOH et al.: NONLINEAR ROBUST INTEGRAL SLIDING APPLICATION IN

1017

control architecture was divided into higher-level and lower-level controllers for controlling the internal configuration of the glider and was made the actuator to execute actions for maintaining the imposed internal configurations. The MPC22 used in conjunction with the path-following technique for online tuning of the desired vehicle velocity along with the trajectory and thus validated the 3D motion dynamics of the Slocum glider. Yuan Shan and Zheng Yan21 designed the MPC using one-layer recurrent neural network to improve the computational problem in MPC to control the longitudinal plane of AUG.

The intelligent control23,24,8 do not need a precise mathematical model of the plant; however, it will suffer from high computational time and need high tuning effort to attain a good performance. The sliding mode control (SMC) is another technique used25,26. The boundary layer SMC was proposed25,27 for 1 degree of freedom (DOF) and 2 DOF internal movable sliding mass, respectively. The Taylor’s series expansion method is used in obtaining the linearised model of AUG. Hai Yang and Jie Ma28,29 proposed the SMC for nonlinear system of longitudinal plane of AUG. The reaching law is designed based on rapid-smooth reaching law. The performance is improved using inverse system method where the output equations are differentiated repeatedly until the input appeared in the equations, then the control law are designed based on that equations28,29. Mat-Noh et. al.26 proposed SMC to control the pitching and the net buoyancy of the longitudinal plane system. The control law is designed based on super-twisting sliding mode control (STSMC). The standard STSMC composed only discontinuous part, however the control law consists of equivalent and discontinuous parts26. The intelligent technique had been proposed23, wherein the neural network is used to control the horizontal and vertical plane of the AUG. The controller was designed based on the linearized model.

This paper proposes the combination of two types of SMC control strategies, namely, integral sliding mode and super-twisting sliding mode called integral super-twisting sliding mode control. In the basic study of this algorithm30, the controller was designed for nominal system and system with input disturbance and the performance of ISTSMC was compared to integral SMC. However, in this paper, the proposed controller algorithm is designed for nominal system, system with input disturbance and system with

parameter variations. The performance of the proposed controller is compared to the performance of STSMC and integral SMC (ISMC). With this study, the robustness of the proposed controller is tested and benchmarked with the performance of the original ISMC and STSMC. This paper discusses the mathematical model of the longitudinal plane of the AUG and the detail derivation of control for the proposed controller, ISMC and STSMC (Table 1). Approach and Methods Dynamic model of an AUG

In this paper, the work is based on the work done by Graver, wherein detailed derivation of the motion equation can be found11. Here, only the longitudinal plane is considered. The dynamics of the longitudinal plane is controlled using internal movable sliding mass and the variable ballast mass. The rudder is fixed to stabilize the glider straight motion in longitudinal plane; therefore, the lateral dynamics can be ignored. Hence all the lateral components are equal to zero except for the pitching component. The glider’s reference frame is shown in Figure 1. The notation of the glider is given in Table 2.

A glide path is specified by a desired path angle, ξd and desired speed Vd.

(1)

where θ = pitching angle, α = angle of attack

(2)

The initial coordinates (x’,z’) such that x’ is the

position along the desired path and is defined as

(3)

z’ measures the position of the vehicle in the

direction perpendicular to the desired path. The dynamics of the z’ is given in Eq. (4)

(4)

In most applications, the internal movable mass of

the underwater glider only moves along x-axis5 and together with ballast pumping rate will make the glider dive in water column. Therefore, in this paper

1018

the 1 degremass is consx-axis.

The origrewritten wmotion equaare written in

Control Techniques

PID

LQR

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SMC

Intelligent

Fig

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CI., VOL. 48, NO

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Table 2 —

Motio

Motion in the x

Motion in the y

Motion in the z

Rotation about

Rotation about(pitching)

Rotation about

19

g

D

Requires li Depends on Not robust Based on S

Requires li Needs appr

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due to optimreceding ho

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Chattering signal

Requires upparameters

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linmod funcontroller

The notation us

on Axis

x-direction (surg

y-direction (swa

z-direction (heav

t the x-axis (roll)

t the y-axis

t the z-axis (yaw

Disadvantages

nearised modeln model accuracto disturbances

SISO system

nearised modelropriate choice ffunction to disturbance

iscrete model igher computatiomisation processorizon. s against uncertaonable

effect exists in t

pper bounds of us and disturbanceis not tested with

onally intensivearge effort on pa

del is obtained unction to develop

ed for the glider

Linear and angular velocity

Po

ge) v1 (m/s)

ay) v2 (m/s)

ve) v3 (m/s)

) ω1 (rad/s)

ω2 (rad/s)

w) ω3 (rad/s)

(6)

(7)

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x

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z

ϕ

θ

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MAT-NOH et al.: NONLINEAR ROBUST INTEGRAL SLIDING APPLICATION IN

1019

(8)

(9)

(10)

(11)

where = + + + + + + (12)

(13)

(14)

(15)

16)

(17)

where mem is the net buoyancy), mf1 and mf3 denote

the added masses, D, L, and MDL2 represent the drag, lift and viscous moment of the hydrodynamic force and moment were defined as5

(18)

(19)

(20)

(21)

where , , and are the mass of the hull,

internal movable mass, and displaced fluid, respectivly. , , , , , and are the

hydrodynamic lift, drag and pitching moment

coefficients, respectivly. , and are the linear

and quadratic damping constant coefficients. From Eq. (5-11), it can be noticed that the system

is under-actuated system with two inputs and six outputs. The state and input vectors are written in Eqs. (22) and (23), respectively: = [ ] =[ ] (22)

(23)

However, in this study, only two parameters will be considered, namely, pitching angle, and net buoyancy, as written in Eqs. (24-25). The net

buoyancy is indirectly obtained through the ballast mass, .

(24)

(25)

Controller design

This section discusses the methodology of the controller design. The proposed controller is a combination of two SMC strategies, namely, ISMC and STSMC. Therefore, here three controllers will be designed, namely, the proposed controller, ISMC and STSMC.

Before designing the controller, the motion equations in Eq. (5-11) are rewritten in the general form of nonlinear equation as given in Eq. (26).

(26)

where, ,and are defined as state and input vectors, represent the bounded matched

perturbations, and is bounded with a known norm upper bound,

(27)

where, for .

The Eq. (27) is transformed into the form that is

suitable for controller algorithms as written in Eq. (28)

INDIAN J. MAR. SCI., VOL. 48, NO. 07, JULY 2019

1020

(28)

where .

The equations for the selected output are rewritten

in the form of Eq. (28) as given in Eqs. (29-31).

(29)

(30)

(31)

where and are the external disturbances. The controllers are designed for the tracking problems. The errors of the selected outputs are defined in Eqs. (32) and (33).

(32)

(33)

A. Integral Sliding Mode Control (ISMC): The

integral sliding mode control (ISMC) was proposed by Utkin and Shi in 199631. The advantage of ISMC is that the sliding surface is enforced from the beginning and thus eliminates the reaching phase which is also called ‘no reaching phase’ SMC. The control law is defined as

B. (34)

is the ideal or nominal controller (i.e. the

system without perturbation) which can be designed using any method such as PID, pole-placement, LQR, MPC, etc. and is nonlinear (discontinuous) control design to reject the perturbations, .

The sliding manifold is defined as

(35)

where s0 is the conventional sliding surface, and z is the integral term which can be determined.

The underwater glider is controlled by two control inputs. Thus, two control laws and two sliding surfaces are designed. The ISMC control law and sliding surface are given in Eqs. (36) and (37), respectively.

(36)

(37)

The ideal controls, and are defined based

on a linear control and those are written in Eqs (38) and (39) as

(38)

(39)

The conventional sliding surfaces and are

defined as

(40)

Differentiating Eqs (37) and (40) with respect to

time = + = + + = + ( , , ) −( + ) + ( , ) + (41)

(42)

where and are chosen as

(43)

(44)

Substitute into Eqs. (41) and (42), then and

reduces to Eqs. (45) and (46)

(45)

(46)

The equivalent controls and are

determined as and are written in Eqs. (47) and (48)

(47)

(48)

and the reachability conditions are chosen as

then the n

as =( ) +

The ISMC

and net buoy = +( ) +

C. Supe

(STSMC): Tintroduced alternative tocontrol for ttrajectories otwisting arousliding varia

This algobecause it coand the coparameters.

Fig. 2 — Suphase plane.

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{ + (−( ) Sliding M

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(50

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APPLICATION

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using supentrol law for buoyancy ar

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control laws

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= { +−= − ( , )−

IN

m where the ee discontinuouer-twisting a

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their derivag angle and7), (58), (59)

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are chosen

written in Eqs

+ −( ) − ( ) )

1021

equivalent us control algorithm. e pitching Eqs. (55)

(55)

(56)

atives are d the net ) and (60)

(57)

(58)

(59)

(60)

as

(61)

(62)

as super-

(63)

(64)

s. (65) and

− ) −(65)

(66)

INDIAN J. MAR. SCI., VOL. 48, NO. 07, JULY 2019

1022

D. Integral Super-Twisting SMC (ISTSMC): The proposed control law of ISTSMC is defined based on ISMC control as written in Eqs. (67) and (68).

E.

(67)

(68)

The linear control laws were previously defined in Eqs. (38) and (39); the sliding surfaces, and equivalent controls ( ) are similar to the

one defined in Eqs. (37), (47) and (48). However, the discontinuous controls ( , ) are defined

based on the super-twisting SMC as in Eqs. (63) and (64). Therefore, the nonlinear control laws for ISTSMC are written in Eqs. (69) and (70). = 1 { + (− + 1)( ) + ( , )} −

| | ( ) − ( ) (69)

(70) Finally, the ISTSMC control laws are defined in

Eqs. (71) and (72).

(71)

(72)

The proposed control laws in Eqs. (71) and (72) gains from the advantage of no reaching phase of ISMC algorithm and the sliding surfaces convergence in finite time as the main advantage of STSMC.

F. Stability Analysis: The stability analysis of the proposed controller algorithm ISTSMC is discussed here. To ensure the convergence of the controlled parameters of the plant stabilized at the desired value, the sliding mode must be first ensured. Therefore the analysis of stability is made to ensure sliding mode and the output convergence. This is done using lyapunov stability theorem.

Theorem 1: Consider the nonlinear system in Eq. (28) subjected to bounded uncertainty in Eq. (27)

with assumptions, the system is proper ( ) and

minimum phase where the zero dynamic of the system is asymptotically stable. If the sliding manifolds ( )

as written in Eq. (37) the dynamics of integral terms ( )

as written in Eqs. (43) and (44) and the discontinuous controls ( ) as written in Eqs. (63) and (64), then

the convergence conditions are satisfied. Proof: Let us consider the lyapunov functions and

their time derivatives in Eqs. (73), (74), (75) and (76), respectively.

(73)

(74)

(75)

(76)

Substitute the Eq. (37) along with Eqs. (43) and

(44) into Eqs. (75) and (76), to produce Eqs. (77) and (78).

(77)

(78)

Now substitute the Eq. (69) into Eq. (77), and Eq.

(70) into Eq. (78) which gives

(79)

(80) The following sufficient conditions for finite time

convergence must be satisfied33,34:

(81)

(82)

(83)

Results and Discussion Here, the performance of the proposed controller

ISTSMC is compared to the performance of the ISMC and STSMC. All the designed controllers are

evaluated fodisturbance acontrollers wfrom Graver’Table 3.

The contrswitching frresponses of3 to 8.

All the cothe desired shows the faangle and lethe highest tabout more ISMC proviISTSMC proexhibit in coare in downsmallest chat

Here, thematched ex

Ta

Hull mass,

Internal slidin

Displaced flu

Added mass,

Inertia, , ,Lift coefficien

Drag coefficie

Moment coef

Constant coef

Fig. 3

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were evaluated’s work5. All

rollers were srom 25o dowf the nominal

ontrollers arevalues. From

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than 1 secoides the largeovides the smontrol input, nward trend. ttering in bothe water currxternal distur

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Fig. 4 — Net b

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APPLICATION

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APPLICATION

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k11 = 0.60,

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m k21 =1 x 10

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19

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MAT-NOH et al.: NONLINEAR ROBUST INTEGRAL SLIDING APPLICATION IN

1027

References 1 Stommel, H., The Slocum mission. Oceanography.

1989;2(April 1989) 22-25. 2 Webb, D.C., Simonetti PJ, Jones CP. SLOCUM: an

underwater glider propelled by environmental energy. IEEE J Ocean Eng. 2001;26(4) 447-452.

3 Sherman, J., Davis, R.E., Owens, W.B., Valdes, J., The autonomous underwater glider “Spray.” IEEE J Ocean Eng. 2001;26(4) 437-446.

4 Eriksen, C.C., Osse, T.J., Light, R.D., et al. Seaglider: a long-range autonomous underwater vehicle for oceanographic research. IEEE J Ocean Eng. 2001;26(4) 424-436.

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