Higher-order sliding mode control for trajectory tracking of air cushion vehicle

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IJLEO 56880 1–9

Optik xxx (2015) xxx–xxx

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Higher-order sliding mode control for trajectory tracking of air cushion vehicle

1

2

3

Q1

Yaozhen Han a,b,∗ , Xiangjie Liu a a

4

b

5

State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China School of Information Science and Electrical Engineering, Shandong Jiaotong University, Jinan, China

6

a r t i c l e

7 19

i n f o

a b s t r a c t

8

Article history: Received 26 May 2015 Accepted 17 November 2015 Available online xxx

9 10 11 12 13

18

Keywords: Sliding mode control Air cushion vehicle Trajectory tracking Finite time tracking

20

1. Introduction

14 15 16 17

The air cushion vehicle (ACV) is an amphibian complex system with strong nonlinearity, internal unmodeled dynamics, and external disturbances. This article presents a new multivariable higher-order sliding mode (HOSM) control scheme on trajectory tracking of ACV. With two longitudinal thrusters, the ACV is modeled as an uncertain nonlinear system with less degree of freedom to be actuated. The control approach includes nominal continuous control law and super-twisting second-order sliding mode control part. The former is employed to stabilize nominal systems at origin in finite time and improve transient process. The latter is used to alleviate chattering of controlling force in surge and controlling torque in yaw, and overcome system uncertainties. A Lyapunov approach is used to testify finite time stability. The simulations for straight line and circular trajectories are presented to evaluate the applicability, robustness and superiority of the proposed approach. © 2015 Published by Elsevier GmbH.

The air cushion vehicle (ACV) is a peculiar high-performance amphibious vehicle, which is nowadays being extensively applied in the fields of military and civilian [1]. Adopting a suit of flexible skirt to make the air cushion surround the hull, ACV could sail on riffle, sea surface, meadow, marsh, etc [2]. Moreover, ACV is now becoming a desirable clean form of transportation, due to rising fuel prices and energy shortages. Unlike the regular fully actuated vessels, ACV has three degrees of freedom (sway, surge and yaw) and two independent control inputs (force in surge and torque in yaw), forming an underactuated electromechanical system described by complex nonlinear dynamic [3]. In addition, external disturbances, such as wave, current and wind, always influence the navigation performance, both for terrestrial and maritime operations. For these reasons, motion control of ACV is a hard nut to crack and becomes an active research area. Different advanced control algorithms [2–5] have been presented to show excellent course keeping or course tracking performance. A cooperative controller for circular flocking with

21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

Q2

∗ Corresponding author at: Corresponding author at: State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing 102206, China. Tel.: +86 15966687646. E-mail addresses: [email protected] (Y. Han), [email protected] (X. Liu).

collision avoidance of ACVs is presented [6] as well. However, the main problems that arise in the control of ACV, are path following [1], point stabilization [7,8], and trajectory tracking [9]. For trajectory tracking purpose, it is necessary to develop control methods that impel a vessel to reach and follow a temporal reference. Considering its professional and military background, trajectory tracking control problem for ACV represents a very challenging research topic, being, thus, the problem studied in the present work. Regarding trajectory tracking control problem of ACV, Ariaei and Jonckheere [10] achieve circular trajectory tracking based on linear dynamically varying technique. In [11], a new scheduling approach is proposed and applied for the trajectory tracking for multiple ACVs, yet, the ACV model is linear. Aiming at a nonlinear ACV model, Papers [1,12] propose the trajectory tracking control schemes. Nevertheless, the external disturbances are not considered in these schemes. Paper [13] presents a nonlinear receding horizon controller for an ACV with uncertainties. However, the asymptotic stability of closed-loop system is not guaranteed. In [14], a tracking control law for an ACV with a discrete set of inputs is designed and tested, in [15], a position tracking controller is designed by combining bioinspired neurodynamics model with backstepping algorithm, and a nonlinear Lyapunov-based tracking control scheme is also studied in [16], but for these publications [14–16], the tracking error is turned out to be exponentially convergent to a neighbourhood of the origin.

http://dx.doi.org/10.1016/j.ijleo.2015.11.180 0030-4026/© 2015 Published by Elsevier GmbH.

Please cite this article in press as: Y. Han, X. Liu, Higher-order sliding mode control for trajectory tracking of air cushion vehicle, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.180

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As a consequence, a robust trajectory tracking controller has to be adopted when considering nonlinearity, external disturbances, system stability and traceability for arbitrary trajectory comprehensively. Sliding mode control (SMC) is essentially a nonlinear control means that forces the system to “slide” along a cross-section of the system’s normal behaviour by using a discontinuous control signal. As a variable structure control method, SMC is regarded as a powerful method to design robust controllers, and also find applications in vessel control [17–22]. However, the inherent chattering occurring in the control input could be harmful to the steering engine and drive mechanism [22]. In addition, the constraint of finite time stability and higher relative degree also hinder the application of the traditional SMC technique [23]. Retaining the main advantages of the standard SMC, Higher-order sliding mode (HOSM) is proposed to reduce and (or) remove the chattering effect, and achieve better accuracy [24–26]. In [27], a controller based on second-order sliding mode and differential flatness is proposed to realize trajectory tracking tasks. The robustness with respect to external disturbances is also evaluated, yet, the tracking errors are only exponentially stable. Paper [28] proposes a HOSM control scheme for MIMO uncertain nonlinear system and applies it for trajectory tracking of ACV. The finite time stability is guaranteed, but in fact, the chattering of controlling torque in yaw is not considered and the transient process could not be regulated effectively. Motivated by these considerations, this paper proposes a new HOSM control scheme for trajectory tracking of ACV. The main characteristics of this control strategy could be summarized as: It combines HOSM, dynamic extension algorithm and finite time continuous control, and the finite time stability of closed-loop system is guaranteed. The control objective is to track a predefined temporal trajectory in an environment characterized by the existence of current, wave, etc. The ACV model is firstly converted into input–output form via the differential of sliding variable and dynamic extension algorithm. Then the HOSM control problem is viewed as finite time stabilization of higher-order uncertain integral chain. The robust finite time controller with acceptable chattering and fast transient process is constituted by two parts: a nominal continuous controller which could be regarded as desired trajectories generator, and a robust one which is based on super-twisting second-order SMC and forces the system trajectories to track the desired trajectories. The nominal one is employed to adjust transient process while the latter is used to alleviate chattering in control input and realize robustness. The formal proof of finite time stability is based on quadratic form Lyapunov function. Finally, simulation results are provided to confirm the effectiveness of the proposed approach. The rest of the paper is organized as follows. In Section 2, the ACV model is stated. Section 3 shows the controller design including control problem formulation and HOSM control strategy, and the tracking simulations are carried out in Section 4. Section 5 concludes the work.

with the inertia matrix P = diag{ p11 p22 p33 }, C (v) =   0 0 −p22 v D = diag{ d11 d22 d33 }, J() = 0 0 p11 u , p22 v − p11 u 0   cos − sin 0 cos 0 . sin 0 0 1 The C() is the matrices of Coriolis and centripetal forces. D represents the hydrodynamic damping. J() is the transformational matrix between body-fixed and earth-fixed coordinates. The



vector  = x

y



T

denote the position and the orientation



T

of ACV in the earth-fixed frame.  = u v r are the linear velocities in surge, sway, and angular velocity in yaw. The vec-



T

tor  = a b c represent the controlling forces in surge and sway, and the controlling torque in yaw. In consideration of the simplification of ACV [27] as depicted in Fig. 1, the simplifying conditions could be employed as p11 = p22 , d11 = d33 = 0,

a = p11 u , ˛=

b = 0,

(2)

d22 p22

A model for such symmetric ACV could be directly deduced as

⎧ x˙ = ucos − vsin ⎪ ⎪ ⎪ ⎪ y˙ = usin + vcos ⎪ ⎪ ⎪ ⎨˙ =r

(3)

u˙ = vr + u ⎪ ⎪ ⎪ ⎪ ⎪ v˙ = − ur − ˛v ⎪ ⎪ ⎩

In fact, external disturbances such as wave, current and wind always influence navigation performance of ACV. Some additive perturbations are introduced in the surge velocity equation and the sway dynamics. Eq. (3) becomes

(4)

u˙ = vr + u ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v˙ = − ur − ˛v + v ⎪ ⎩

131 132 133 134 135 136 137 138 139

140

141

142

143 144 145 146

147

r˙ = r + r

3. Controller design

148

3.1. Problem formulation

149

Ex = [x1 , x2 , x3 , x4 , x5 , x6 , x7 , x8 ]T = x, y, , u, v, r, u , ˙ u

˙ =J()

130

⎧ x˙ = ucos − vsin ⎪ ⎪ ⎪ y˙ = usin + vcos ⎪ ⎪ ⎪ ⎪ ⎨ ˙ = r



(1)

129

r˙ = r

In contrast with a general two wheel mobile robot, ACV could move freely sideways in spite of this degree of freedom is not actuated. The general kinematics and dynamic equations of the ACV could be developed based on an earth-fixed coordinate frame and a body-fixed coordinate frame as shown in Fig. 1. The dynamic of a general surface vessel [29] is depicted as P+C(v)+D ˙ =

128

c = p33 r ,

The control objective is to achieve trajectory tracking of the perturbed ACV in finite time. The ACV model (4) is a MIMO nonlinear system, with two control inputs ( u and r ). Since the ACV is underactuated, exact linearization could not be attained directly via a static state feedback. It could be checked that the model does not have a well-defined vector relative degree with respect to the position outputs x and y. Indeed, x¨ and y¨ depend on u rather than on r . It is therefore necessary to constitute a second-order dynamic extension of  u for the exact linearization of model (4). Detailed description about Dynamic Extension Algorithm is referred to [30]. Let Ex denote the extended states and  the new control input, expressed as

2. ACV model

127



 = [1 , 2 ]T = [¨ u , r ]T

T

150 151 152 153 154 155 156 157 158 159 160 161

(5)

162

(6)

163

Please cite this article in press as: Y. Han, X. Liu, Higher-order sliding mode control for trajectory tracking of air cushion vehicle, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.180

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Fig. 1. ACV. Earth fixed x y and body fixed coordinate frames u v.

164 165 166

167

It is assumed that  u (0) = ˙ u (0) = 0, then, a state space representation of the ACV could be written as the following uncertain nonlinear system

⎧ x˙ 1 = x4 cosx3 − x5 sinx3 ⎪ ⎪ ⎪ ⎪ ⎪ x˙ 2 = x4 sinx3 + x5 cosx3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙ 3 = x6 ⎪ ⎪ ⎨ x˙ 4 = x5 x6 + x7



=

169 170 171 172

(7)

⎪ x˙ 5 = − x4 x6 − ˛x5 + v ⎪ ⎪ ⎪ ⎪ ⎪ x˙ 6 = 2 + r ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x˙ 7 = x8 ⎪ ⎪ ⎪ ⎩



173 174 175 176 177

178

defining

x1 − x1ref

x2 − x2ref

the

T



(4)

=

(4) 1 (4)



= (g¯ + g)

2 179

180

g¯ =



 = 1

2

1 2

+ f¯ + f

=

sin x3

sin x3 (˛x5 ) + cos x3 (˛x4 + x7 )

0 −v cosx3 0

−v sinx3

(8)







= I + g g¯ −1 ω + f − g g¯ −1 f¯

(4)

(10)

where I denotes the 2 × 2 identity matrix. The fourth-order SMC of system (7) with respect to the sliding variable  is equivalent to the finite time stabilization of the multivariable uncertain integral chains

⎧⎧ ⎪ q˙ 1,i = q2,i ⎪ ⎪ ⎨ ⎪⎪ ⎪ ⎨ . . ⎪ ⎪. ⎩ ⎪ ⎪ q˙ r −1 ,i = qri ,i ⎪ ⎪ T  ⎩ i

186

187

188 189 190

191



q = qT1



 ,

(4)

⎤

⎢ ⎥ ⎢ + sin x3 (˛3 x5 + x6 (˛2 x4 − 2x8 − ˛x7 ) − 2x62 ˛x5 ) ⎥ ¯ ⎢ ⎥, f =⎢ ⎥ (4) − x6 sin x3 (x6 (2˛x4 + x7 ) + 2˛2 x5 ) ⎣ (−x2ref ⎦

˙ v˛ −  ¨ v − r x7 − v ˛2 − r ˛x4 ) sin x3 ((v x62 + 



T

.

⎢ ⎥ ⎢ + (2v ˛x6 + r ˛x5 − 2˙ v x6 − v r ) cos x3 ) ⎥ ⎢ ⎥, f = ⎢ ¨ v + r ˛x4 ) cos x3 ⎥ ⎣ ((r x7 + v ˛2 − ˙ v ˛ − r x62 −  ⎦ ˙ v x6 − v r ) sin x3 ) + (2v ˛x6 + r ˛x5 − 2

195

196

(j−1)



, qi = q1,i , q2,i , ..., qri ,i

T

and

197

.

⎧ q˙ 1,i = q2,i ⎪ ⎪ ⎪ ⎪ ⎨ .. ⎤

194



1, 2 , 1 ≤ j ≤ ri , qj,i = i

qT2

193

= I + gg¯ −1 ␻ + f − gg¯ −1 f¯

q˙ r1 ,1 , q˙ r2 ,2

with i ∈

192

(11)

198

199

With respect to (11), the design of auxiliary control ω could be scheduled in two aspects. One is to design nominal control law ωnom for nominal integral chain system. The other is to realize system robustness, and chattering reduction of controlling force and torque by adopting the second-order sliding mode algorithm. Thus, ω includes two parts: ωnom and ωr . First, consider the nominal system of (11), (i.e. v = 0 and r = 0), which is represented by two SISO independent integrator chains, construed as

,

(−x1ref − x6 cos x3 (x6 (2˛x4 + x7 ) + 2˛2 x5 )

⎡

(4) 1

185

T

3.2. HOSM control strategy



− cos x3 (˛3 x5 + x6 (˛2 x4 − 2x8 − ˛x7 ) − 2x62 ˛x5 )

183

T



cos x3 (˛x5 ) − sin x3 (˛x4 + x7 )

g =

⎡ 182

vector

cos x3

 181

sliding

where



(9)

184

2

, notice that the vector relative degree of





is the virtual control input. This feedback where ω = ω1 ω2 partially realizes decoupling of the nominal system (i.e. without uncertainty). Hence, system (8) could be indicated as  (4) =

 is [4 4]T . The rth-order SMC approach allows the finite time stabilization of each variable  i and its ri − 1 first time derivatives by defining a advisable control law. Then the fourth-order differential of ␴ is





= g¯ −1 ω − f¯

2



The control objective is to robustly steer the ACV from a given initial position to the reference trajectory x1ref and x2ref in finite time. It is assumed that the feasible reference trajectories x1ref and x2ref are fourth-order time derivable. While

1





x˙ 8 = 1

168

f¯ and g¯ are known items. f and g are unknown uncertainties which depend on disturbances v and r . g¯ is invertible. Apply the following preliminary feedback (9) to system (8)

⎪ q˙ ri −1,i = qri ,i ⎪ ⎪ ⎪ ⎩

(12)

200 201 202 203 204 205 206 207 208

209

q˙ ri ,i = ωnom,i





with i ∈ 1, 2 . Some control algorithms which could stabilize system (12) at the origin in finite time may be adopted [31–33]. A so-called homogeneous control law which is terse in form is proposed in [33], based on the homogeneity theory.

Please cite this article in press as: Y. Han, X. Liu, Higher-order sliding mode control for trajectory tracking of air cushion vehicle, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.180

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218

Lemma 1 ([33]). Choose the positive constants a1,i , . . ., ari ,i which make pri + ari ,i pri −1 + · · · + a2,i p + a1,i be Hurwitz polynomial. Then there is i ∈ (0, 1) such that for any zi ∈ (1 − i , 1), system (12) could be stabilized at the origin in finite time under feedback control

219

nom,i (qi ) = −a1,i sign q1,i

214 215 216 217

  z1,i q1,i  − · · ·    zr ,i − ar ,i sign qr ,i qr ,i  i

For testifying finite time stability formally based on Lyapunov theory, a non-decoupled form of super-twisting algorithm is adopted s

ωr = −k1

s1/2



220

221 222

223 224 225 226 227 228

229 230 231 232 233 234

with z j−1,i = 1, zri ,i = zi .



i

 

i

zj,i zj+1,i / 2zj+1,i − zj,i

238 239 240 241 242 243 244 245 246 247





2, ..., ri , zri +1,i =

  z    z = −a1,i sign q1,i q1,i  1,i − · · · − ari ,i sign qri ,i qri ,i  ri ,i 



237

, j ∈

Theorem 1. Choose the positive constants a1,i , . . ., ari ,i , b1,i , . . ., bri ,i , which make pri + ari ,i pri −1 + · · · + a2,i p + a1,i r r −1 i i + · · · + b2,i p + b1,i be Hurwitz polynomial. Then p + bri ,i p there is i ∈ (0, 1) such that for any zi ∈ (1 − i , 1), system (12) could be stabilized at the origin in finite time under the feedback control

− b1,i q1,i − · · · − bri ,i qri ,i 236



(13)

Lemma 1 is established based on geometry homogeneity theory. When system states are far from the equilibrium point, the rate of convergence is slow, leading to long transient tracking time. For this case, refer to the study on finite time stability of SISO integral chain [34], a revised control law nom,i (qi ) for multivariable integral chains is described as Theorem 1.

ωnom,i (qi ) 235

i

˙ = −m1

248

s (q) = qr1 ,1 , qr2 ,2

249

then

 −

264

(20)

265

Substituting (19) into (17) yields s˙ = −k1

s s1/2

266

+ − k2 s + ˆ

(21)

Theorem 2. Considering uncertain ACV system (7), HOSM with respect to  could be established, and trajectory tracking is achieved in finite time if the control law is designed as



 = g¯ −1 −f¯ + ωnom + ωr



(22)

where ωnom and ωr are defined by (14), (19) and (20).





268 269 270

271

273

1 T 1 ω ωr + T + 2m1 ||s|| + m2 sT s 2 r 2

(23)

274

(14)



2, . . ., ri , zri +1,i = 1,

275

ς = {(s, ) ∈ R2m : s = 0}

(24)

V (s, ) is continuous and differentiable everywhere except on the subspace ς. It is also straightforward that V (.) is positive definite and radially unbounded. Time derivative of (23) is





V˙ (s, ) = k22 + 2m2 sT s˙ +

1 2



k12 + 2m1

+ 2 T ˙ − k2 sT ˙ + s˙ T

 − k1

t

ωnom dt

267

272

Proof. Choosing Lyapunov function V (s, ) =

Remark 1. When system states in (12) are far from the equilibrium point, which means |qj,i | > 1, linear term −b1,i q1,i − · · · − bri ,i qri ,i plays a leading role and the system is asymptotically stable. System states could be satisfied with |qj,i | < 1 after finite time. Then homogeneous control part starts governing and system states converge to equilibrium point in finite time. Similar detailed description for SISO system is referred to [34]. Second, in order to realize system robustness, and guarantee that control objectives are completed, second-order SMC algorithm is employed. To construct sliding mode function

T

s − m2 s s

263

(19)

Defining subspace

where z j−1,i = zj,i zj+1,i / 2zj+1,i − zj,i , j ∈ z ri ,i = zi .



− k2 s +

262

˙ T s + T s˙



||s||1/2

(15)

||s||



 −

 sT s˙

sT s˙



T s

+

276

277 278 279 280

3 sT s˙ k1 k2 2 ||s||1/2



(25)

281

2||s||5/2

0

250

251 252



s˙ (q) = q˙ r1 ,1 , q˙ r2 ,2

T

/ ς, yield Considering (19) and (20), then for all (s, ) ∈ − ωnom

Take ω = ωnom + ωr into account, and substitute (11) into (16), yields





s˙ = Im + g g¯ −1 ωr − g g¯ −1 f¯ + f + g g¯ −1 ωnom 253

254 255 256

257

258 259 260 261

(16)

= ωr + ˆ

(17)

where ˆ = −g g¯ −1 f¯ + f + g g¯ −1 ωnom + g g¯ −1 ωr . In practical applications, control quantities ω are bounded. It is supposed that ˆ satisfies

   ˆ  ≤ 1 s

V˙ (s, )











5 k1 k22 ||s||3/2 2

− k2 m1 + 2k12 k2 ||s|| − k1 m1 +



+

k1 sT

+



T s

2||s||5/2

− k1

(18)

with 1 a positive constant. The super-twisting second-order SMC could be implemented on (17). System robustness and chattering alleviation could be achieved by properly choosing control parameters [35].





= − k2 m2 + k23 ||s||2 − k1 m2 +

T ||s||1/2

 + k12

282



1 3 k ||s||1/2 2 1

sT sT + 3k1 k2 ||s|| ||s||1/2

(26)

+ 2k22 sT − k2 || ||2

1  sT ˆ sT ˆ 3 k1 sT ˆ T s + + k1 k2 k12 + 2m1 2 2 ||s|| ||s||1/2 2||s||5/2

− k1

T ˆ ||s||1/2





2 + k2 + 2m2 sT ˆ − k2 ˆ T

Please cite this article in press as: Y. Han, X. Liu, Higher-order sliding mode control for trajectory tracking of air cushion vehicle, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.180

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Simplifying (26) by taking bounding arguments into account

284

V˙ (s, )



 3

≤ − k2 m2 + k2











+

285

k1 |sT |2 2||s||5/2

+ k12



1 3 k ||s||1/2 2 1

+ +

1

k12 + 2m1

2



2 k2

+ 2m2



+ k1

||s||

|sT

(27)

| T ˆ |

V˙ (s, )

V˙ ≤ −









− k2 m1 + 2k12 k2







+

k1

|| ||2

2||s||1/2



Defining x˜ =

289

˙ ≤− V

290

with   =

(28)



1 ||s||1/2

then from (28)

˜T  x ˜−x ˜T ˘ x ˜ x

11 0 0  22 −k12 /2 −3k1 k2 /2

(29)



k1 /2











,  11 =

k13 /2 + k1 m1 ,

 22 = k1 m2 + 5/2 k1 k22 − 3/2 k1 k2 1 ,

⎡

292

293

294 295 296 297 298 299

300

301

⎤

11 0 − 3/4 k1 1 ⎦, ˘ 11 = k2 m1 + 2k12 k2 − ˘ = ⎣  0 ˘ 22 23 − 3/4 k1 1 ˘ 32 k2 ((1/2)k12 + 2m1 )1 , −k22 − (1/2)k2 1 , 32



˘ 22 = = 23 .

k2 m2 + k23

− (k22

+ 2m2 )1 ,



˘ 23 =



If k1 > 0, k2 > 0, m1 > 0 and m2 > 9k12 k22 /4m1 + 2k22 +



3/2 k2 1 are satisfied, It is easy to verify symmetric matrix  > 0.

Similarly, if k1 > 0, k2 > 21 , m1 > isfied, where



m˘ 1 = m˘ 2



9/16 (k1 1 )

k2 (k2 − 21 )



2

+

m˘ 1

m˘ 2

are sat-



1/2 k12 1 − 2k12 k2 (k2 − 21 )



(30)



2k22 1 + 1/4 k2 21 1 = + 2 (k2 − 21 ) (k2 − 21 ) 2

and m2 >

2

302

 1 = ((9/16)(k1 1 ) (k2 + (1/2)1 ) )/k22 ,

303

(2m1 + (1/2)k12 )1

2

− (9/16)(k1 1 ) )/k2 .

||s||1/2



||s||1/2

min ( )||˜x||2



(33)

(34)

(31) 2 = (k2 (m1 + 2k12 ) −

(35)

On account of V 1/2 >





min (P)||s||1/2 , then for all (s, ) ∈ /ς





min ( )

(36)



min (P) /max (P) .

eq

ωr = I m + g g¯

306

307

  −1 −1

308 309

310

g g¯ −1 f¯ − f − g g¯ −1 ωnom

311 312

313

314 315 316

317

318

319

320

Take notice of (19) and (20), the absolutely continuous trajectories of the Filippov solution could not remain on the set ς\{0}, namely, ς does not include origin when s = = 0. In that ˙ / 0 from (24) are satisfied if way, s(t0 ) = 0 and s(t)| t=t0 = (t0 ) = (s(t0 ), (t0 )) ∈ ς\{0} at the time constant t0 . Thus, at least one si (t) will traverse the origin monotonously during some time interval T 0 ⊂ R including t0 , because i (t) is absolutely continuous and / 0. Consequently, inequality (36) holds almost everywhere (t0 ) = and V (t) is continuous decreasing function with respect to time. Then equilibrium point (s, ) = 0 could be reached in finite time by referring to Lyapunov theorem for differential inclusion [37]. In the end, taking s = = 0 to the right side of (20) and combining ˆ (0) = 0 yields s˙ = 0. Therefore, s = s˙ = 0 holds in finite time. It is easy to verify matrix Im + gg¯ −1 is uniformly invertible. eq Equivalent control of ωr in sliding mode is defined as ωs which could be gotten from (17) when s˙ = 0.



305

T

˜ 2 min ( )||X||

min ( ) 1 V ||s||1/2 max (P)

with  =

3 k1 1 ||s||1/2 || || 2

−k12 /2 −3k1 k2 /2

x˜ T  x˜ ≤ −

V˙ ≤ −V 1/2

k12 + 2m1 1 ||s||

T 1 (||s|| 2 , ||s||, ||␻||) ,



291

2

1

˙ ≤− V

  3 k1 k2 1 ||s||3/2 + k22 + 2m2 1 ||s||2 2

+ k2 1 ||s|||| || +

288

+

1

||s||1/2

1

Using similar arguments to [36], formula (23) could be denoted ˜ T PX ˜ for a suitable symmetric positive definite matrix P ∈ as V = X ˜ 2 . Thus, from (34) R3m×3m and V ≤ max (P)||X||

5 k1 k22 ||s||3/2 2

+ k12 || || + 2k22 ||s|||| || + 3k1 k2 ||s||3/2 || || − k2 || ||2 +

V˙ ≤ −

1 ||s|| − k1 m1 + k13 ||s||1/2 2

287

1

304

2

˜ = ||˜x|| holds Define X˜ = s/||s||1/2 , s, and given that ||X|| for all s, , such that (33) could be rewritten as

ˆ | + k2 | ˆ T |

≤ − k2 m2 + k23 ||s||2 − k1 m2 +

4m1

||s||1/2

Considering (18) and Cauchy–Schwarz inequality yields

286

1 1 ⎪ ⎪  2 2  ⎪ ⎪ 9k1 k2 ⎪ 3 ⎪ + 2k22 + k2 1 , m˘ ⎩ m2 > max 2

Hence, from (29) and in view of min ( )||˜x||2 ≤ x˜ T  x˜ ≤ max ( )||˜x||2

3 k1 |sT ˆ || T s| |sT ˆ | + k1 k2 1/2 2 ||s|| 2||s||5/2

 |sT ˆ |

(32)

m > m˘

|sT | |sT | + 3k1 k2 ||s|| ||s||1/2

+ 2k22 |sT | − k2 || ||2 +

Symmetric matrix  > 0 could be also gotten from undemanding verification. For satisfying  > 0 and > 0 simultaneously, k1 , k2 , m1 and m2 are chosen as

⎧ k1 > 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ k2 > 21

5 ||s||2 − k1 m2 + k1 k22 ||s||3/2 2

− k2 m1 + 2k12 k2 ||s|| − k1 m1 +

5



(37)

eq ωnom + ωr

Substitute ω = into (11), the equivalent closed loop dynamics similar to the pure integral chains (12) is acquired. Since the control law ωnom is constructed based on Theorem 1, an rth order sliding mode versus  is set up in finite time, and trajectory tracking is attained. Remark 2. The proof is constructive, the choices of k1 , k2 m1 , m2 must be satisfied with formula (32). Remark 3. If the time derivatives of the sliding variable could not be detected by suitable measuring equipments, these values are estimated by an exact robust differentiator [23]. 4. Simulation results In this section, computer simulations are carried out to demonstrate effectiveness of the designed controller for an ACV.

Please cite this article in press as: Y. Han, X. Liu, Higher-order sliding mode control for trajectory tracking of air cushion vehicle, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.180

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Particularly, this work is focused on two ordinary trajectory tracking tasks: The first trajectory consists of a straight line traversing the origin of earth-fixed coordinate. The second trajectory is proposed as a circular trajectory, defined in the earth-fixed coordinate frame, centered around the origin. In practical situation, wave and wind always influence navigation performance of the ACV. For verifying the robustness of the proposed controller, system parameter ˛ is arranged to vary from 1.1 to 1.5 randomly. External disturbances v (t) and r (x) are introduced in the system dynamics, where r (t) could influence the yaw rate, and is represented by Gaussian noise with mean value 0 and variance 0.3. v (x) affects the surge and sway acceleration, representing the unmodeled external perturbation forces. It is expressed as

366

v (x) = 0.25 (sin (15x)) + 0.15 cos (15x)

351 352 353 354 355 356 357 358 359 360 361 362 363 364

Fig. 2. The actual and reference trajectory of the straight line.

367 368 369

(38)

The position coordinates x and y are measureable and their first, second and third time derivatives are computed by a third-order exact robust differentiator [23] l0 = , li = i−1 =  (i) ,

i = 1, 2, 3

l˙ 0 = 0 0 = −27.39|l0 − |3/4 sign (l0 − ) + l1 l˙ 1 = 1 370

1 − 28.97|l1 − 0 |2/3 sign (l1 − 0 ) + l2

(39)

l˙ 2 = 2 2 − 45|l2 − 1 |1/2 sign (l2 − 1 ) + l3

Fig. 3. The orientation of ACV in the earth-fixed frame.

l˙ 3 = 3 3 = −900sign (l3 − 2 ) 371

372

with

ត. where l1 , l2 , l3 are the observed values of , ˙ , ¨ 

377

378

xref (t) = 10t,

374 375 376

− 37

sign (¨ 1 ) |¨ 1 |3/5 sign (¨ 2 ) |¨ 2 |3/5

4.1. Tracking of a straight line The developed HOSM controller is particularized for the case of tracking a straight line traversing the origin of the fixed-earth frame. The ACV must follow this line at the constant surge speed while moving away from the origin of coordinates. The desired trajectory is given as

373

ωnom = −23

sign (1 ) |1 |3/7 sign (2 ) |2 |3/7

 s=

− 2.3

s1 s2



=



1 2 ត1  ត2 

! − 4.5

!

˙ 1 ˙ 2

!

!

− 48

sign (˙ 1 ) |˙ 1 |1/2 sign (˙ 2 ) |˙ 2 |1/2

− 12

ត ត 3/4 sign   1  |1 |3/4 ត ត2 | sign 2 |

− 6.4

¨ 1 ¨ 2

!

 

! 

! − 2.8

ត1  ត2 

386

! ,

!



+ z aux , ωr = −0.4 s/s1/2 + v − 0.2s,

379 380

381

382 383 384

Because of this specific selected x and y, the reference ACV orientation angle  ref (t) is calculated as ref (t) = arctan

 10t  10t

=

 4

(41)

The initial positions is set as x(0) = 7, y(0) = 2.9, and the initial ACV orientation angle is taken to be zero. According to Theorems 1 and 2, the control law is designed as

 385

(40)

ω = ωnom + ωr z˙ aux = −ωnom

(42)

388



v˙ = −0.4 s/ s − 0.2s yref (t) = 10t

387

Figs. 2 and 3 shows the actual trajectory and orientation of the ACV. Though there are external disturbances and parameter perturbation, they could both track the set values. Fig. 4 shows linear velocities in surge, sway, and angular velocity in yaw. The controlling force and controlling torque are depicted in Figs. 5 and 6, in which the control chattering are almost eliminated. This could also be observed from the curves of sliding mode surface as shown in Fig. 7. This simulation scenario verified the effectiveness and robustness of the proposed HOSM control algorithm. 4.2. Tracking a circular trajectory For further evaluating the performance of the proposed controller, a circular trajectory is to be tracked in an anti-clockwise direction in the plane (x, y), with a given constant angular velocity

Please cite this article in press as: Y. Han, X. Liu, Higher-order sliding mode control for trajectory tracking of air cushion vehicle, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.180

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Fig. 4. Linear velocities in surge, sway, and angular velocity in yaw.

Fig. 8. Circular trajectory tracking of ACV.

 



 

x ∗ (t) = 10 cos t/6

(43)

y ∗ (t) = 10 sin t/6

For this particular choice of x and y, the nominal orientation angle  ∗ (t) is Q3



 ∗ (t) = arctan

403

404 405

    /6 sin t/6 − ˛ cos t/6     

(/6) cos t/6 + ˛ sin t/6

(44)

= arctan(tan((t/6) − arctan(˛/(/6)))

406

= (t/6) − arctan(˛/(/6))

Fig. 5. Controlling force in surge.

The nominal surge and sway velocities and the nominal yaw angular velocity are calculated according to (3)

⎧       u ∗ (t) = −10 /4 sin arctan ˇ/ /4 ⎪ ⎪ ⎨       ⎪ ⎪ ⎩

v ∗ (t) = 10 /4 cos arctan ˇ/ /4



(45)



407 408

409

r ∗ (t) = /4

The proposed control law (42) is adopted. For highlighting the advantages of the proposed controller, a comparative study of HOSM control algorithm is carried out. The designed controller according to [28] is as ω = ωnom + ωr

(46)



sign (1 ) |1

ωnom = −24

|3/7





sign (2 ) |2 |3/7

Fig. 6. Controlling torque in yaw.



sign (¨ 1 ) |¨ 1 |3/5

− 35 sign (¨ 2 ) |¨ 2

sign (˙ 1 ) |˙ 1

− 50

|1/2



410 411 412 413

414 415

sign (˙ 2 ) |˙ 2 |1/2

⎡



− 10 ⎣

|3/5

 

ត1 | ត1 |3/4 sign 

 

⎤,

416

⎦

ត2 | ត2 |3/4 sign 

ωr = (−0.3||ωnom || + 11) sign(s).

417

418

Fig. 7. Sliding mode surface.

As is shown in Fig. 8, tracking trajectory is faster and more accurate by the proposed HOSM algorithm. To be more clear, Fig. 9 shows the errors between the actual output and the desired trajectory both in x and y directions, indicating the superiority of the proposed HOSM both in accuracy and transient process time comparing with that of Defoort et al. [28]. Figs. 10–12 shows the ACV orientation in the earth-fixed frame, the linear velocities in surge, sway, and angular velocity in yaw under the two algorithms respectively.

Please cite this article in press as: Y. Han, X. Liu, Higher-order sliding mode control for trajectory tracking of air cushion vehicle, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.180

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Fig. 9. Tracking errors in x and y direction.

Fig. 12. The linear velocities in surge, sway, and angular velocity in yaw under (Defoort et al. [28]).

Fig. 10. The orientation of the ACV in the earth-fixed frame. Fig. 13. Controlling torque in yaw.

Fig. 11. The linear velocities in surge, sway, and angular velocity in yaw under the proposed algorithm.

Fig. 14. Controlling force in surge.

5. Conclusion 427 428 429 430 431 432 433 434 435 436

The two controllers efficiently correct the undesirable deviations caused by the persistent perturbations. As is expected, the actual controlling force  u is free of chattering, which is shown in Fig. 13. The chattering of the controlling torque is greatly weakened, as shown in Fig. 14. As a result, the tracking accuracy and manoeuvrability of ACV are enhanced. Due to the smoothness of the actual controlling force and controlling torque, the mechanical wear is reduced and the service life of transmission mechanism could be prolonged. In brief, these results show the robustness and the advantages of the proposed finite time robust controller.

This paper presents a new HOSM robust control method on trajectory tracking control of ACV. The ACV is described by a nonlinear dynamic model and an environment characterized by the presence of external disturbances. This HOSM algorithm is equivalent to finite reaching time control of multivariable uncertain integral chains. A continuous controller that could stabilize the nominal integral chain in finite time is designed. Taking into account the uncertainty items, a non-decoupled super-twisting algorithm is proposed. The finite time stability is proved based on the quadric form Lyapunov function. Straight line and circular tracking

Please cite this article in press as: Y. Han, X. Liu, Higher-order sliding mode control for trajectory tracking of air cushion vehicle, Optik - Int. J. Light Electron Opt. (2015), http://dx.doi.org/10.1016/j.ijleo.2015.11.180

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examples are included to demonstrate the effectiveness of the suggested approach, meaning it is well suited for controlling this kind of high-performance amphibious vehicle. A problem that deserves further study is nonlinear robust control for ACV with input constraint. It also gives rise, for future work, to research on handling this control method for other scenarios where the general problem of MIMO nonlinear uncertain systems applies.

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