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Research ArticleSafety-Guaranteed Trajectory Tracking Control forthe Underactuated Hovercraft with State and Input Constraints
Mingyu Fu Shuang Gao and ChenglongWang
College of Automation Harbin Engineering University Harbin 150001 China
Correspondence should be addressed to Shuang Gao gaussianhrbeueducn
Received 16 May 2017 Revised 14 August 2017 Accepted 24 September 2017 Published 25 October 2017
Academic Editor Rafael Morales
Copyright copy 2017 Mingyu Fu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper develops a safety-guaranteed trajectory tracking controller for hovercraft by using a safety-guaranteed auxiliary dynamicsystem an integral sliding mode control and an adaptive neural network methodThe safety-guaranteed auxiliary dynamic systemis designed to implement system state and input constraints By considering the relationship of velocity and resistance hump thevelocity of hovercraft is constrained to eliminate the effect of resistance hump and obtain better stability And the safety limit ofdrift angle is well performed to guarantee the light safe maneuvers of hovercraft tracking with high velocities In view of the naturalcapabilities of actuators the control input is constrained High nonlinearity andmodel uncertainties of hovercraft are approximatedby employing adaptive radical basis function neural networks The proposed controller guarantees the boundedness of all theclosed-loop signals Specifically the tracking errors are uniformly ultimately bounded Numerical simulations are implemented todemonstrate the efficacy of the designed controller
1 Introduction
A hovercraft (Figure 1) is supported totally by its air cushionwith a flexible skirt system around its periphery to seal thecushion air [1]The hovercraft is able to run at high speed overshallow water rapids ice and swamp where no other craftcan go These ldquospecial abilitiesrdquo have attracted many militaryand civil users with particular mission requirements Thestudy about the safety-guaranteed trajectory tracking controlof underactuated hovercraft moving with high velocitiesis meaningful and challenging to reduce the burden ofpilot
From a detailed review of the available literatures aboutthe trajectory tracking control of hovercraft [2ndash8] only posi-tion errors were considered and the velocities of hovercraftwere not controlled However the velocity is related to theresistance hump of hovercraft From [9] the resistance humpoccurs in the vicinity of Froude number 119865119903 = 1 which canbe calculated by 119865119903 = 119881radic119892119897119888 where 119881 is the velocity ofhovercraft and 119897119888 is the cushion length From [10] tworesistance humps (mainly caused by wave-making drag) are
encountered for hovercraft during the acceleration process Itis shown in [1] that the resistance hump will be crossed as 119865119903increases and the craft will travel with better course stabilityand transverse stability Hence the velocity of hovercraftneeds to be large enough corresponding to large 119865119903 to avoidthe resistance hump and hold the better stability
Moreover drift angle plays a key role in the high-speedmoving process of hovercraft [11 12] If the drift angle exceedsthe angle of drift which corresponds to the maximum ofhydrodynamic forces the behavior of hovercraft will benonstable [13] The dangers caused by the drift angle includestern kickoff plough-in and great heeling [11] Hence safetylimit of drift angle must be strictly obeyed in the high-speed tracking process to ensure safemaneuvers of hovercraft[12] For instance safety limit of a hovercraft shows if speedexceeds 40 knots turning is not allowed if speed is in therange of 25 knotssim35 knots drift angle needs to be within thelimits of 75∘sim2∘
Besides from a practical viewpoint the control input isrestrained to prevent the actuators from going beyond theirnatural capabilities [14 15] And radical basis function neural
HindawiMathematical Problems in EngineeringVolume 2017 Article ID 9452920 12 pageshttpsdoiorg10115520179452920
2 Mathematical Problems in Engineering
Figure 1 A 3Dmodel of hovercraft Photo is from the internationalcooperation project described in Acknowledgments
Propeller
Rudder
Surge force
Yaw moment
w
U
r
u
Figure 2 Diagram of underactuated hovercraft
networks (RBFNNs) are used to stabilize complex nonlineardynamic systems and deal with model uncertainties [16ndash18]
The contributions of this paper are as follows
(i) The velocity of hovercraft is controlled within a rea-sonable range to avoid the effect of resistance humpand keep better stability
(ii) The safety limit of drift angle is obeyed to get light safemaneuvers of hovercraft moving with high velocities
(iii) The control input is constrained to handle inputsaturation
This paper is organized as follows Section 2 establishes a non-linear model of underactuated hovercraft and the transfor-mation of it Section 3 proposes a safety-guaranteed auxiliarydynamic system for state and input constraintsMoreover thecontroller is designed and analyzed in this section Numericalsimulation results are shown in Section 4 and the conclusionis discussed in Section 5
2 Problem Formulation
21 Hovercraft Model Description In general only air pro-pellers and rudders are available for hovercraft as shown inFigure 2 It means only the surge and yaw can be regulateddirectly but without any actuators for their sway motion[19 20]
The nonlinear model of hovercraft is obtained by neglect-ing the roll and pitch motions
[ 119910 ]119879 = 119878 (120595) [119906 V 119908 119903]119879 [[[[[[
V
119903
]]]]]]
= [[[[[[
V119903minus11990611990300
]]]]]]
+119872[[[[[[
119877119906 + 120591119906119877V119877119908 minus 119901119888119878119888119877119903 + 120591119903
]]]]]] (1)
where
119878 (120595) = [[[[[[
cos120595 minus sin120595 0 0sin120595 cos120595 0 00 0 1 00 0 0 1
]]]]]]
119872 =[[[[[[[[[[[
1119898 0 0 00 1119898 0 00 0 1119898 00 0 0 1119869119911
]]]]]]]]]]]
(2)
the signals 119906 V and 119908 represent the surge sway and heavevelocities 119903 is the turn rate 119909 119910 and 119911 denote the position ofhovercraftrsquos mass center in the earth fixed frame 120595 describesyaw angle 119898 and 119869119911 are hovercraftrsquos mass and moments ofinertia 120591119906 and 120591119903 are the control inputs which are providedby the actuators 119901119888 is the average cushion pressure 119878119888 isthe cushion area and [119877119906 119877V 119877119908 119877119903]119879 are the total dragswritten as
119877119898 = 120588119886119881119886119876119877wm = 119862wm1199012119888119861119888120588119908119892 119877sk = 051205881199081198812119886119862sk ( ℎ119897sk)
minus034 119897sk11987805119888+ (28167 (119901119888119897119888 )
minus0259 minus 1)119877wm119877119906 = 119877119906119886 + 119877119906119898 + 119877119906wm + 119877119906sk
= minus051205881198861198812119886119862119906119886119878PP minus 119877119898 cos120573 minus 119877wm cos120573minus 119877sk cos120573
119877V = 119877V119886 + 119877V119898 + 119877Vwm + 119877Vsk
= minus051205881198861198812119886119862V119886119878LP minus 119877119898 sin120573 minus 119877wm sin120573minus 119877sk sin120573
119877119908 = 119877119908119886 + 119877119908wm + 119866= minus051205881198861198812119886119862119908119886119878HP minus 120588119886119908119876 + 119866
Mathematical Problems in Engineering 3
119877119903 = 119877119903119886 + 119877119903119898 + 119877119903wm + 119877119903sk= minus051205881198861198812119886119862119903119886119878HP119867hov minus 119877V119886 times 119909119886 + 119877119906119886 times 119910119886
minus 119877V119898119909119898 + 119877119906119898119910119898 minus 119877Vwm119909wm + 119877119906wm119910wmminus 119877Vsk119909sk + 119877119906sk119910sk
(3)
where the dragrsquos suffix 119886 is the aerodynamic profile drag wmis the wave-making drag 119898 is the air momentum drag sk isthe skirt drag 119862119906119886 119862V119886 119862119901119886 119862119903119886 119862wm and 119862sk are the dragcoefficients 119861119888 is the cushion beam 119897119888 is the cushion length119866 is the weight 119878PP 119878LP and 119878HP are the positive lateral andhorizontal projection areas 120573 is the drift angle 119876 is the fanflow rate of cushion fan ℎ is the distance between baffle andthe bottom of skirtrsquos finger 119897sk is the total peripheral lengthof the skirts 119867hov is the height of hovercraft 120588119886 and 120588119908 areair and water density and (119909119886 119910119886) (119909119898 119910119898) (119909wm 119910wm) and(119909sk 119910sk) are the coordinates of forcersquos acting points119881119886 and 119881119886 in (3) can be obtained by
119881119886= radic[119906 + 119881119908 cos (120573119908 minus 120595)]2 + [V + 119881119908 sin (120573119908 minus 120595)]2
119881119886 = radic1198812119886 + 1199082(4)
in which 119881119908 and 120573119908 are absolute wind speed and directionMore details can be found in [1 9 21]
Remark 1 When a hovercraft is moving on a calm watersurface cushion pressure 119901119888 varies within a narrow range andthe heavemotion is stableThis paper is the research about thehorizontal motion of the hovercraft Hence the heavemotionis not discussed and the cushion pressure 119901119888 is assumed to bea constant
From Figure 2 we have
V = 119906 tan (120573) (5)
In order to make 120573 be the system state and more convenientfor the constraint and control of 120573 an improved model isderived from (1) and (5) that is
[ 119910 ]119879 = 119878 (120595) [119906 119906 tan120573 119908 119903]119879 [[[[[[
120573
119903
]]]]]]
=[[[[[[[
119906119903 tan120573minus sin (2120573)2119906 minus 119903cos2120573
00
]]]]]]]
+119872[[[[[[[[
119877119906 + 120591119906119877V
cos2120573119906119877119908 minus 119901119888119878119888119877119903 + 120591119903
]]]]]]]]
(6)
22 State and Input Constraints Saturation nonlinearities ofactuators can be described by
120591120603 = sat (120591120603119888 120591120603119888max 120591120603119888min) 120603 = 119906 119903 (7)
where 120591120603119888max and 120591120603119888min are the maximum and minimumlimitations of actuators 120591120603119888 are the designed control laws andsat(sdot) is a generalized saturation function with the followingform
sat (120572 120572119872 120572119898) =
120572119872 if 120572 gt 120572119872120572 if 120572119898 le 120572 le 120572119872120572119898 if 120572 lt 120572119898
(8)
Assumption 2 All position orientation velocity and acceler-ation values of hovercraft are available for feedback
Safety limit of 120573 and hump speed of hovercraft need tobe obtained from model and real ship tests [11 12] In thispaper they are assumed to be known and available for thestate constraint Then the safe constraints of system state aredefined as
119906min le 119906 le 119906max120573min le 120573 le 120573max (9)
3 Controller Design
31 Safety-Guaranteed Auxiliary Dynamic System
Proposition 3 A constraint error function is designed as fol-lows
Δ120581 = 1198961199081 (sat (119909 119909119898119886119909 119909119898119894119899) minus 119909)+ 1198961199082 (sat (119906119888 119906119888119898119886119909 119906119888119898119894119899) minus 119906119888) (10)
where 1198961199081 gt 0 1198961199082 gt 0 119909 is the system state and 119906119888 is thedesigned control input
Then an auxiliary dynamic system is designed by
120585 = minus1198961205851120585 minus 120599 (sdot) + 1198961205852Δ1205812100381710038171003817100381712058510038171003817100381710038172 120585 + Δ120581 10038171003817100381710038171205851003817100381710038171003817 ge 120590deadzone (Δ120581 120590) 10038171003817100381710038171205851003817100381710038171003817 lt 120590
(11)
where 120585 is the state of the auxiliary dynamic system 1198961205851 1198961205852 arepositive constants 120590 120590 are positive small design constants 120599(sdot)can be derived from the stability analysis and deadzone(Δ120581 120590)is a dead zone function given by
119889119890119886119889119911119900119899119890 (Δ120581 120590) =
Δ120581 119894119891 Δ120581 gt 1205900 119894119891 minus 120590 lt Δ120581 le 120590Δ120581 119894119891 Δ120581 le 120590
(12)
32 Design of the Desired States Desired reference trajectoryis generated by a virtual ship described in the following form
[[[119889119910119889119889
]]]
= [[[cos120595119889 minus sin120595119889 0sin120595119889 cos120595119889 0
0 0 1]]][[[119906119889setV119889set119903119889set
]]] (13)
4 Mathematical Problems in Engineering
Then the trajectory tracking errors are defined as
(119909119890 119910119890) = (119909 minus 119909119889 119910 minus 119910119889) (14)
For the position tracking the desired states are designed by
119906119889 = (119889 minus 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)) cos120595 + ( 119910119889 minus 119896119910 10038161003816100381610038161199101198901003816100381610038161003816sdot sign (119910119890)) sin120595
120573119889 = arctan(minus119889 minus 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)119906119889 sin120595+ 119910119889 minus 119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890)119906119889 cos120595)
(15)
where 119896119909 gt 0 and 119896119910 gt 0 are control gainsRemark 4 To guarantee the traceability of the referencetrajectory under state constraints the desired reference tra-jectory needs to satisfy the following conditions
(1198621) 120578119889 120578119889 and 120578119889 are all bounded in which 120578119889 = 119909119889119910119889 120595119889(1198622) There exists 119879119903 gt 0 such that for all 119905 gt 119879119903
119906min lt 119906119889 lt 119906max120573min lt 120573119889 lt 120573max (16)
33 Controller Design State tracking errors are defined as
119890119906 = 119906 minus 119906119889119890120573 = 120573 minus 120573119889 (17)
Then two integral sliding mode manifolds are given by
119904119906 = 119890119906 + 1205821 int1199050119890119906 119889120591
119904119903 = 119890120573 + 1205822119890120573 + 1205823 int1199050119890120573 119889120591
(18)
where 1205821 1205822 and 1205823 are positive constantsUsing (6) (17) and (18) the time derivatives of 119904119906 and 119904119903
are expressed as
119904119906 = 119890119906 + 1205821119890119906= 119891119906 (x119906) + 119872119906120591119906 +119872119906119877119906 minus 119889 + 1205821119890119906 (19)
119904119903 = 119890120573 + 1205822 119890120573 + 1205823119890120573= 119891119903 (x119903) + 119872119903120591119903 + 119863119903119908 minus 120573119889 + 1205822 119890120573 + 1205823119890120573 (20)
where
119891119906 (x119906) = 119906119903 tan120573 x119906 = [119906 120573 119903]119879 119891119903 (x119903) = minus119906 minus 221199062 sin (2120573) minus 120573119906 cos (2120573)
+ 120573119903 sin (2120573) x119903 = [119906 119903 120573 120573]119879 119872119906 = 1119898119872119903 = minuscos2120573119869119911 119863119903119908 = minus 120573119877V sin (2120573)119906119898 + 119906V minus 119877V1199062119898 cos2120573
minus 119877119903cos2120573119869119911
(21)
To deal with high nonlinearity and model uncertainties119891119906(x119906) and 119891119903(x119903) are approximated by RBFNNs
119891120603 (x120603) = Wlowast119879120603 H120603 (x120603) + 120576120603 120603 = 119906 119903 (22)
where x120603 isin 119877119868 is the input vector and Wlowast120603 isin 119877119899 is the idealweight vectorH120603(x120603) 119877119868 rarr 119877119899 is the basis function vectorwith element ℎ119894120603(x120603) shown as follows
ℎ119894120603 (x120603) = exp(minus1003817100381710038171003817x120603 minus 120583119894120603100381710038171003817100381721205902119894 ) (23)
where120583119894120603 is the center of the receptive field and120590119894 is thewidthof theGaussian functionThe approximation error 120576120603 satisfies|120576120603| le 120576119873
Using (10) and (11) constraint error functions aredesigned as follows
Δ120581119906 = 1198961199081199061 (sat (119906 119906max 119906min) minus 119906)+ 1198961199081199062 (sat (120591119906119888 120591119906119888max 120591119906119888min) minus 120591119906119888)
Δ120581119903 = 1198961199081199031 (sat (120573 120573max 120573min) minus 120573)+ 1198961199081199032 (sat (120591119903119888 120591119903119888max 120591119903119888min) minus 120591119903119888)
(24)
where 1198961199081199061 gt 0 1198961199081199062 gt 0 1198961199081199031 gt 0 and 1198961199081199032 gt 0Then the auxiliary dynamic system is designed as
120585120603=
(minus1198961205851206031120585120603 minus 120599120603 (sdot) + 1198961205851206032Δ1205811206032100381710038171003817100381712058512060310038171003817100381710038172 120585120603 + Δ120581120603) 10038171003817100381710038171205851206031003817100381710038171003817 ge 120590deadzone (Δ120581120603 120590) 10038171003817100381710038171205851206031003817100381710038171003817 lt 120590
(25)
where 120599120603(sdot) = |119872120603119904120603||120591120603 minus 120591120603119888| and 120603 = 119906 119903
Mathematical Problems in Engineering 5
Finally the control laws are given by
120591119906119888 = 1119872119906 (minus1198961119904119906 minus 1205781 sign (119904119906) + 119889 minus 1205821119890119906 minus119872119906119877119906minus W119879119906H119906 (x119906) minus 120576119873 sign (119904119906) + 119896119904119906120585119906)
(26)
120591119903119888 = 1119872119903 ( 120573119889 minus 1198962119904119903 minus 1205782 sign (119904119903) minus 1205822 119890120573 minus 119863119903119908minus 1205823119890120573 minus W119879119903H119903 (x119903) minus 120576119873 sign (119904119903) + 119896119904119903120585119903)
(27)
where 1198961 1198962 1205781 1205782 119896119904119906 and 119896119904119903 are positive constants W119906 =W119906 +Wlowast119906 W119903 = W119903 +Wlowast119903 and119863119903119908 is defined in (20)
And the adaptive laws areW120603 = 120574120603119904120603H120603 (x120603) 120603 = 119906 119903 (28)
where 120574120603 is the adaptive coefficient
34 Stability Analysis
Theorem 5 If the state tracking errors (17) the desired states(15) the auxiliary dynamic system (25) the surge control law120591119906119888 (26) the yaw control law 120591119903119888 (27) and the adaptive laws(28) are applied to the hovercraft system represented by (6)and for any bounded initial condition the closed-loop controlsystem signals 119904119906 119904119903 120585119906 120585119903 119890119906 119890120573 119909119890 119910119890 and 119908119894119906 and 119908119894119903119894 = 1 2 119899 are uniformly ultimately bounded (UUB)The position and desired state tracking errors can be madearbitrarily small by appropriately selecting design parametersAnd the yaw motion will remain bounded
Proof The following Lyapunov function is defined
119881 = 121199042119906 + 121199042119903 + 12120574119906 W119879119906W119906 +12120574119903 W119879119903 W119903 +
121205852119906+ 121205852119903
(29)
From (19) (20) and (29) can be expressed as
= 119904119906 (119891119906 (x119906) + 119872119906120591119906 +119872119906119877119906 minus 119889 + 1205821119890119906)+ 119904119903 (119891119903 (x119903) + 119872119903120591119903 + 119863119903119908 minus 120573119889 + 1205822 119890120573 + 1205823119890120573)+ W119879119906
W119906120574119906 + W119879119903W119903120574119903 + 120585119906 120585119906 + 120585119903 120585119903
(30)
By defining 120591120603 = Δ120591120603 + 120591120603119888 and 120603 = 119906 119903 and using (22) (26)and (27) we have
= minus11989611199042119906 minus 1205781 10038161003816100381610038161199041199061003816100381610038161003816 + 119904119906 (minusW119879119906H119906 (x119906)) +119872119906119904119906Δ120591119906+ 119896119904119906119904119906120585119906 + (minus120576119873 10038161003816100381610038161199041199061003816100381610038161003816 + 120576119906119904119906) minus 11989621199042119903 minus 1205782 10038161003816100381610038161199041199031003816100381610038161003816+ 119904119903 (minusW119879119903H119903 (x119903)) +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903+ (minus120576119873 10038161003816100381610038161199041199031003816100381610038161003816 + 120576119903119904119903) + 1120574119906 W119879119906 W119906 + 1120574119903 W119879119903 W119903+ 120585119906 120585119906 + 120585119903 120585119903
(31)
By using |120576120603| le 120576119873 and 120603 = 119906 119903 (31) can be rewritten as
le minus11989611199042119906 minus 1205781 10038161003816100381610038161199041199061003816100381610038161003816 + 119904119906 (minusW119879119906H119906 (x119906)) minus 11989621199042119903minus 1205782 10038161003816100381610038161199041199031003816100381610038161003816 + 119904119903 (minusW119879119903H119903 (x119903)) + 119872119906119904119906Δ120591119906+ 119896119904119906119904119906120585119906 +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 1120574119906 W119879119906 W119906+ 1120574119903 W119879119903 W119903 + 120585119906 120585119906 + 120585119903 120585119903
(32)
Substituting adaptive laws (28) into it yields
le minus11989611199042119906 +119872119906119904119906Δ120591119906 + 119896119904119906119904119906120585119906 + 120585119906 120585119906 minus 11989621199042119903+119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 120585119903 120585119903 (33)
The process of stability analysis is respectively discussed inthe following two cases
Case 1When 120585120603 ge 120590 in light of (25) andYoungrsquos inequalitywe have
120585120603 120585120603 = 120585120603Δ120581120603 minus 11989612058512060311205852120603 minus 1198961205851206032Δ1205812120603minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
le 1198961205851206032Δ1205812120603 + 141198961205851206032 1205852120603 minus 11989612058512060311205852120603 minus 1198961205851206032Δ1205812120603minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
= minus(1198961205851206031 minus 141198961205851206032)1205852120603 minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
(34)
Substituting 120591120603 = Δ120591120603 + 120591120603119888 and (34) into (33) and usingYoungrsquos inequality yield
le minus11989611199042119906 + 119896119904119906119904119906120585119906 minus (1198961205851199061 minus 141198961205851199062)1205852119906 minus 11989621199042119903+ 119896119904119903119904119903120585119903 minus (1198961205851199031 minus 141198961205851199032)1205852119903
le minus (1198961 minus 05) 1199042119906 minus (1198961205851199061 minus 141198961205851199062 minus 051198962119904119906)1205852119906minus (1198962 minus 05) 1199042119903 minus (1198961205851199031 minus 141198961205851199032 minus 051198962119904119903)1205852119903minus 1198961W119879119906W119906120574119906 minus 1198962W119879119903 W119903120574119903 + 1198961W119879119906W119906120574119906+ 1198962W119879119903 W119903120574119903 le minus21205831119881 + 1205881
(35)
6 Mathematical Problems in Engineering
where
1205831 = min(1198961 minus 05) (1198961205851199061 minus 141198961205851199062 minus 051198962119904119906) (1198962 minus 05) (1198961205851199031 minus 141198961205851199032 minus 051198962119904119903)
1205881 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903
(36)
Case 2 When 120585120603 lt 120590 in light of (25) and Youngrsquos ine-quality we have
120585120603 120585120603 = 120585120603deadzone (Δ120581120603 120590) le 051205852120603 + 05Δ12058121206030511989621199041206031205852120603 = 11989621199041206031205852120603 minus 0511989621199041206031205852120603 le minus0511989621199041206031205852120603 + 11989621199041206031205902 (37)
Substituting (37) into (33) and using Youngrsquos inequality yield
le minus11989611199042119906 +119872119906119904119906Δ120591119906 + 119896119904119906119904119906120585119906 + 051205852119906 + 05Δ1205812119906minus 11989621199042119903 +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 + 051198722119906Δ1205912119906+ 051198722119903Δ1205912119903 + 0511989621199041199061205852119906 + 0511989621199041199031205852119903 + 051205852119906+ 05Δ1205812119906 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 minus 051205852119906 (1198962119904119906 minus 1)minus 051205852119903 (1198962119904119903 minus 1) + 051198722119906Δ1205912119906 + 051198722119903Δ1205912119903+ 11989621199041199061205902 + 11989621199041199031205902 + 05Δ1205812119906 + 05Δ1205812119903
le minus21205832119881 + 1205882
(38)
where
1205832 = min (1198961 minus 1) (051198962119904119906 minus 05) (1198962 minus 1) (051198962119904119903 minus 05)
1205882 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903 + 051198722119906Δ1205912119906
+ 051198722119903Δ1205912119903 + (1198962119904119906 + 1198962119904119903) 1205902 + (Δ1205812119906 + Δ1205812119903)2
(39)
Synthesizing (35) and (38) we have
le minus2120583119881 + 120588 (40)
where 120583 = min1205831 1205832 and 120588 = max1205881 1205882 with the designparameters 1198961 1198962 119896119904119906 119896119904119903 1198961205851199061 1198961205851199062 1198961205851199031 and 1198961205851199032 satisfying
1198961 gt 11198962 gt 1119896119904119906 gt 1119896119904119903 gt 1
1198961205851199061 minus 141198961205851199062 minus 051198962119904119906 gt 01198961205851199031 minus 141198961205851199032 minus 051198962119904119903 gt 0
(41)
Solving (40) we have
0 le 119881 (119905) le 1205882120583 + [119881 (0) minus 1205882120583] 119890minus2120583119905 (42)
It is obviously seen that 119881(119905) is UUB for all 119881(0) le 1198610 with1198610 being any positive constant Therefore in the light of (29)we know that 119904119906 119904119903 120585119906 120585119903 and 119908119894119906 and 119908119894119903 119894 = 1 2 119899 areUUB for all 119881(0) le 1198610 It can be expressed as
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 + 2 [119881 (0) minus 1205882120583] 119890minus2120583119905 (43)
where 120594 = 119904119906 119904119903 120585119906 120585119903 119908119894119906 119908119894119903It implies that there exists 119879 gt 0 such that for all 119905 gt 119879
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 (44)
where radic120588120583 can be made arbitrarily small by appropriatelyselecting the design parameters
Further the following dynamics are obtained from (18)and (44)
119890119906 + 1205821 int1199050119890119906 119889120591 = 120576119906
119890120573 + 1205822119890120573 + 1205823 int1199050119890120573 119889120591 = 120576120573
(45)
where 120576119906 and 120576120573 are arbitrarily small errorsTo prove that 119890119906 and 119890120573 are UUB the Lyapunov function
is given by
119881119904 = 12 (int1199050119890119906 119889120591)2 + 121198902120573 (46)
The time derivative of this Lyapunov function along thedynamics in (45) is such that
119904 = 119890119906 int1199050119890119906 119889120591 + 119890120573 119890120573
= minus1205821 (int1199050119890119906 119889120591)2 minus 12058221198902120573 + 120576119906 int119905
0119890119906 119889120591
minus 1205823119890120573 int1199050119890120573 119889120591 + 120576120573119890120573
(47)
Mathematical Problems in Engineering 7
From Youngrsquos inequality the following fact is obtained
120576119906 int1199050119890119906 119889120591 le 051205821 (int119905
0119890119906 119889120591)2 + 05 12057621199061205821
minus1205823119890120573 int1199050119890120573 119889120591 le 120582211989021205732 + 120582321205822 (int
119905
0119890120573 119889120591)2
120576120573119890120573 le 02512058221198902120573 + 12057621205731205822
(48)
Then
119904 le minus051205821 (int1199050119890119906 119889120591)2 minus 02512058221198902120573 + 05 12057621199061205821
+ 0512058231205822 (int119905
0119890120573 119889120591)2 + 12057621205731205822 = minus2120583119904119881119904 + 120588119904
(49)
where
120583119904 = min 051205821 0251205822 120588119904 = 05 12057621199061205821 + 0512058231205822 (int
119905
0119890120573 119889120591)2 + 12057621205731205822
(50)
Similar to the analysis in (42)sim(44) there exists 119879119904 gt 0 suchthat for all 119905 gt 119879119904
10038171003817100381710038171205941199041003817100381710038171003817 le radic 120588119904120583119904 (51)
where 120594119904 = int1199050119890119906 119889120591 119890120573
It is obvious that int1199050119890119906 119889120591 and 119890120573 are UUB and will be
arbitrarily small by choosing suitable parametersFrom 119904119906 = 119890119906 + 1205821 int1199050 119890119906 119889120591 we know 119890119906 will be arbitrarily
small From (6) and (15) we have
[ 119906 minus 119906119889119906 tan120573 minus 119906119889 tan120573119889] = 119877[119890 + 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)119910119890 + 119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890)] (52)
where
119877 = [ cos120595 sin120595minus sin120595 cos120595] (53)
It is clear that |119877| = 1 which indicates that it is nonsingularThen we have
119890 = minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887119910119890 = minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887 (54)
where 120576119909119887 120576119910119887 are arbitrarily small errorsFurthermore consider the following Lyapunov function
candidate
119881119901 = 121199092119890 + 121199102119890 (55)
The time derivative of 119881119901 along the dynamics in (54) is suchthat
119901 = 119909119890119890 + 119910119890 119910119890= 119909119890 (minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887)
+ 119910119890 (minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887)= minus119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 119909119890120576119909119887 + 119910119890120576119910119887le minus05119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 05119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 12057621199091198872119896119909 +
12057621199101198872119896119910= minus2120583119901119881119901 + 120588119901
(56)
where
120583119901 = min 05119896119909 05119896119910 120588119901 = 12057621199091198872119896119909 +
12057621199101198872119896119910 (57)
Also similar to the analysis in (42)sim(44) there exists 119879119901 gt 0such that for all 119905 gt 119879119901
1003817100381710038171003817100381712059411990110038171003817100381710038171003817 le radic 120588119901120583119901 (58)
where 120594119901 = 119909119890 119910119890From the reason that 120576119909119887 and 120576119910119887 are arbitrarily small
errors we know that 119909119890 and 119910119890 are UUB andwill be arbitrarilysmall
Also 119906 is continuously differentiable in the movingprocess of hovercraft Hence is bounded From (6) we have
= 119906119903 tan (120573) + 119877119906 + 120591119906119898 (59)
From (15) Remark 4 and the boundedness of 119909119890 and 119910119890we have that 119906119889 and 120573119889 are bounded Then 119906 V and 120573 arebounded from (5) and (17) Furthermore119877119906 is bounded from(3)Therefore it can be concluded that 119903will remain boundedfrom (7) and (59) This concludes the proof
Remark 6 In order to avoid the well-known chatteringproblem the sign function used in the control laws (26) and(27) can be replaced by hyperbolic tangent function whichis continuous such that sign(119904) = tanh(119896119867119904) where 119896119867 isa positive scalar which can be chosen to get a very goodapproximation
4 Simulations
Two different cases are implemented to verify the effective-ness and superiority of the proposed controller In simula-tions the main particulars and constraints of hovercraft areshown in Tables 1 and 2 The water surface is calm withoutwaves
8 Mathematical Problems in Engineering
Table 1 Main particulars of hovercraft
119898 (kg) 40000119869119911 (kgm2) 18 times 106119878PP (m2) 45119878LP (m2) 93119878HP (m2) 260119876 (m3s) 1408119881119908 (knots) 10119897sk (m) 65119861119888 (m) 89119897119888 (m) 236ℎ (m) 1119867hov (m) 59119878119888 (m2) 212120573119908 (deg) 45
Table 2 State and input constraints
Variable Maximum Minimum119906 (knots) 40 25120573 (deg) 8 minus8120591119906 (N) 80000 minus80000120591119903 (Nm) 1 times 105 minus1 times 105
The comparisons of three different methods are carriedout in each case The legend ldquoMethod Ardquo means the methodin [14] the legend ldquoMethod Brdquo means the method withoutstate and input saturation constraints
Saturation coefficients are
119896119904119906 = 121198961199081199061 = 11198961199081199062 = 9030001198961205851199061 = 31119896119904119903 = 12
1198961199081199031 = 11198961199081199032 = 6503001198961205851199031 = 381198961205851199062 = 0081198961205851199032 = 008
120590 = 3120590 = 1
(60)
Case 1 The reference trajectory parameters are
119909119889 (0) = 300m119910119889 (0) = 100m
120595119889 (0) = 45∘119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 0
(61)
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 70∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(62)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 111205781 = 0251205821 = 21198962 = 121205782 = 051205822 = 301205823 = 6120576119873 = 05
(63)
It is observed from Figures 3ndash8 that the proposed trajectorytracking controller is effective Tracking errors of all threemethods converge to arbitrarily small values and headingangle is bounded From the comparisons with Methods Aand B in Figures 9 and 10 the proposed controller canlimit the surge speed and the drift angle into the safe rangeeffectively Figures 11 and 12 show that the input saturation isalso handled by the proposed controller
Case 2 The reference trajectory parameters are
119909119889 (0) = minus200m119910119889 (0) = 50m120595119889 (0) = 45∘
119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 01∘s
(64)
Mathematical Problems in Engineering 9
0 2000 4000 6000 8000 10000 12000
0
5000
10000
15000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
x(m
)
y (m)
Figure 3 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 4 The heading angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
500
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus500
minus1000
Figure 5 The surge position tracking error
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 30∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(65)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 153
0 200 400 600 800 1000 1200 1400
0
500
1000
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
minus500
Figure 6 The sway position tracking error
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus10
minus20
Figure 7 The surge velocity tracking error
0 200 400 600 800 1000 1200 1400
0
5
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus10
minus5
Figure 8 The drift angle tracking error
1205781 = 011205821 = 41198962 = 121205782 = 051205822 = 301205823 = 2120576119873 = 05
(66)
It is obvious from Figures 14ndash19 that only the proposedcontroller can track the trajectory successfully Figures 20 and21 show that the surge speed changes to negative values andthe drift angle exceeds the safety limit in the tracking control
10 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controller
u (k
nots)
Method AMethod B
Figure 9 The surge velocity of hovercraft
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
minus10
minus20
Figure 10 The drift angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
2
4
Time (s)
0 50 1000
5
10
Proposed controllerMethod AMethod B
uc
(N)
times105
times104
Figure 11 The surge control law of hovercraft
process ofMethods A and BThese are absolutely not allowedfor hovercraft In contrast you can see from Figure 20 thatsurge speed can still be positive even large enough to avoidthe influence of the resistance humps under the control of theproposed controller And drift angle is also within the safetylimit from Figure 21 From Figures 22 and 23 we know thatthe input constraint ability of the proposed controller is effec-tive Figures 13 and 24 show the heave position of hovercraft
5 Conclusion
A safety-guaranteed trajectory tracking controller has beenproposed for underactuated hovercraft in this paper Thesafety-guaranteed auxiliary dynamic system is designed todeal with state and input constraints The velocity of hov-ercraft is constrained to eliminate the effect of resistance
0 200 400 600 800 1000 1200 1400minus4
0
Time (s)
0 10 20minus15minus10minus5
05
Proposed controllerMethod AMethod B
rc
(Nm
)
times105
times104
minus2
Figure 12 The yaw control law of hovercraft
0 200 400 600 800 1000 1200 1400minus245
minus24
minus235
Time (s)z
(m)
Proposed controllerMethod AMethod B
Figure 13 The heave position of hovercraft
minus2000 0 2000 4000 6000 8000 10000 12000minus1000
0
1000
2000
3000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
minus200minus100
0100
x(m
)
y (m)
minus20 0 20 40 60 80
Figure 14 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 12000
50
100
150
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 15 The heading angle of hovercraft
hump and obtain better stability The safety limit of driftangle is carried out effectively for safety in the high-speedtrajectory tracking process of hovercraftThe input saturationis handled High nonlinearity and model uncertainties areapproximated by RBFNNs Simulation results have been
Mathematical Problems in Engineering 11
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus100
Figure 16 The surge position tracking error
0 200 400 600 800 1000 1200minus600
minus400
minus200
0
200
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
Figure 17 The sway position tracking error
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus100
Figure 18 The surge velocity tracking error of hovercraft
0 200 400 600 800 1000 1200
0
2
4
6
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus2
Figure 19 The drift angle tracking error of hovercraft
0 200 400 600 800 1000 1200
0
50
Time (s)
0 100 200
0
50
Proposed controllerMethod AMethod B
u(k
nots)
minus50minus50
minus100
Figure 20 The surge velocity of hovercraft
0 200 400 600 800 1000 1200
0
100
200
Time (s)
0 100 2000
20
40
Proposed controllerMethod AMethod B
(d
eg)
minus100
Figure 21 The drift angle of hovercraft
0 200 400 600 800 1000 1200
0
Time (s)
0 50 100
0
Proposed controllerMethod AMethod B
uc
(N)
minus5
minus10
times105
times104
minus5
minus10
Figure 22 The surge control law of hovercraft
0 200 400 600 800 1000 1200
0
10
20
Time (s)
0 10 20
05
1015
Proposed controllerMethod AMethod B
rc
(Nm
)
minus5
times109
times104
Figure 23 The yaw control law of hovercraft
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
Figure 1 A 3Dmodel of hovercraft Photo is from the internationalcooperation project described in Acknowledgments
Propeller
Rudder
Surge force
Yaw moment
w
U
r
u
Figure 2 Diagram of underactuated hovercraft
networks (RBFNNs) are used to stabilize complex nonlineardynamic systems and deal with model uncertainties [16ndash18]
The contributions of this paper are as follows
(i) The velocity of hovercraft is controlled within a rea-sonable range to avoid the effect of resistance humpand keep better stability
(ii) The safety limit of drift angle is obeyed to get light safemaneuvers of hovercraft moving with high velocities
(iii) The control input is constrained to handle inputsaturation
This paper is organized as follows Section 2 establishes a non-linear model of underactuated hovercraft and the transfor-mation of it Section 3 proposes a safety-guaranteed auxiliarydynamic system for state and input constraintsMoreover thecontroller is designed and analyzed in this section Numericalsimulation results are shown in Section 4 and the conclusionis discussed in Section 5
2 Problem Formulation
21 Hovercraft Model Description In general only air pro-pellers and rudders are available for hovercraft as shown inFigure 2 It means only the surge and yaw can be regulateddirectly but without any actuators for their sway motion[19 20]
The nonlinear model of hovercraft is obtained by neglect-ing the roll and pitch motions
[ 119910 ]119879 = 119878 (120595) [119906 V 119908 119903]119879 [[[[[[
V
119903
]]]]]]
= [[[[[[
V119903minus11990611990300
]]]]]]
+119872[[[[[[
119877119906 + 120591119906119877V119877119908 minus 119901119888119878119888119877119903 + 120591119903
]]]]]] (1)
where
119878 (120595) = [[[[[[
cos120595 minus sin120595 0 0sin120595 cos120595 0 00 0 1 00 0 0 1
]]]]]]
119872 =[[[[[[[[[[[
1119898 0 0 00 1119898 0 00 0 1119898 00 0 0 1119869119911
]]]]]]]]]]]
(2)
the signals 119906 V and 119908 represent the surge sway and heavevelocities 119903 is the turn rate 119909 119910 and 119911 denote the position ofhovercraftrsquos mass center in the earth fixed frame 120595 describesyaw angle 119898 and 119869119911 are hovercraftrsquos mass and moments ofinertia 120591119906 and 120591119903 are the control inputs which are providedby the actuators 119901119888 is the average cushion pressure 119878119888 isthe cushion area and [119877119906 119877V 119877119908 119877119903]119879 are the total dragswritten as
119877119898 = 120588119886119881119886119876119877wm = 119862wm1199012119888119861119888120588119908119892 119877sk = 051205881199081198812119886119862sk ( ℎ119897sk)
minus034 119897sk11987805119888+ (28167 (119901119888119897119888 )
minus0259 minus 1)119877wm119877119906 = 119877119906119886 + 119877119906119898 + 119877119906wm + 119877119906sk
= minus051205881198861198812119886119862119906119886119878PP minus 119877119898 cos120573 minus 119877wm cos120573minus 119877sk cos120573
119877V = 119877V119886 + 119877V119898 + 119877Vwm + 119877Vsk
= minus051205881198861198812119886119862V119886119878LP minus 119877119898 sin120573 minus 119877wm sin120573minus 119877sk sin120573
119877119908 = 119877119908119886 + 119877119908wm + 119866= minus051205881198861198812119886119862119908119886119878HP minus 120588119886119908119876 + 119866
Mathematical Problems in Engineering 3
119877119903 = 119877119903119886 + 119877119903119898 + 119877119903wm + 119877119903sk= minus051205881198861198812119886119862119903119886119878HP119867hov minus 119877V119886 times 119909119886 + 119877119906119886 times 119910119886
minus 119877V119898119909119898 + 119877119906119898119910119898 minus 119877Vwm119909wm + 119877119906wm119910wmminus 119877Vsk119909sk + 119877119906sk119910sk
(3)
where the dragrsquos suffix 119886 is the aerodynamic profile drag wmis the wave-making drag 119898 is the air momentum drag sk isthe skirt drag 119862119906119886 119862V119886 119862119901119886 119862119903119886 119862wm and 119862sk are the dragcoefficients 119861119888 is the cushion beam 119897119888 is the cushion length119866 is the weight 119878PP 119878LP and 119878HP are the positive lateral andhorizontal projection areas 120573 is the drift angle 119876 is the fanflow rate of cushion fan ℎ is the distance between baffle andthe bottom of skirtrsquos finger 119897sk is the total peripheral lengthof the skirts 119867hov is the height of hovercraft 120588119886 and 120588119908 areair and water density and (119909119886 119910119886) (119909119898 119910119898) (119909wm 119910wm) and(119909sk 119910sk) are the coordinates of forcersquos acting points119881119886 and 119881119886 in (3) can be obtained by
119881119886= radic[119906 + 119881119908 cos (120573119908 minus 120595)]2 + [V + 119881119908 sin (120573119908 minus 120595)]2
119881119886 = radic1198812119886 + 1199082(4)
in which 119881119908 and 120573119908 are absolute wind speed and directionMore details can be found in [1 9 21]
Remark 1 When a hovercraft is moving on a calm watersurface cushion pressure 119901119888 varies within a narrow range andthe heavemotion is stableThis paper is the research about thehorizontal motion of the hovercraft Hence the heavemotionis not discussed and the cushion pressure 119901119888 is assumed to bea constant
From Figure 2 we have
V = 119906 tan (120573) (5)
In order to make 120573 be the system state and more convenientfor the constraint and control of 120573 an improved model isderived from (1) and (5) that is
[ 119910 ]119879 = 119878 (120595) [119906 119906 tan120573 119908 119903]119879 [[[[[[
120573
119903
]]]]]]
=[[[[[[[
119906119903 tan120573minus sin (2120573)2119906 minus 119903cos2120573
00
]]]]]]]
+119872[[[[[[[[
119877119906 + 120591119906119877V
cos2120573119906119877119908 minus 119901119888119878119888119877119903 + 120591119903
]]]]]]]]
(6)
22 State and Input Constraints Saturation nonlinearities ofactuators can be described by
120591120603 = sat (120591120603119888 120591120603119888max 120591120603119888min) 120603 = 119906 119903 (7)
where 120591120603119888max and 120591120603119888min are the maximum and minimumlimitations of actuators 120591120603119888 are the designed control laws andsat(sdot) is a generalized saturation function with the followingform
sat (120572 120572119872 120572119898) =
120572119872 if 120572 gt 120572119872120572 if 120572119898 le 120572 le 120572119872120572119898 if 120572 lt 120572119898
(8)
Assumption 2 All position orientation velocity and acceler-ation values of hovercraft are available for feedback
Safety limit of 120573 and hump speed of hovercraft need tobe obtained from model and real ship tests [11 12] In thispaper they are assumed to be known and available for thestate constraint Then the safe constraints of system state aredefined as
119906min le 119906 le 119906max120573min le 120573 le 120573max (9)
3 Controller Design
31 Safety-Guaranteed Auxiliary Dynamic System
Proposition 3 A constraint error function is designed as fol-lows
Δ120581 = 1198961199081 (sat (119909 119909119898119886119909 119909119898119894119899) minus 119909)+ 1198961199082 (sat (119906119888 119906119888119898119886119909 119906119888119898119894119899) minus 119906119888) (10)
where 1198961199081 gt 0 1198961199082 gt 0 119909 is the system state and 119906119888 is thedesigned control input
Then an auxiliary dynamic system is designed by
120585 = minus1198961205851120585 minus 120599 (sdot) + 1198961205852Δ1205812100381710038171003817100381712058510038171003817100381710038172 120585 + Δ120581 10038171003817100381710038171205851003817100381710038171003817 ge 120590deadzone (Δ120581 120590) 10038171003817100381710038171205851003817100381710038171003817 lt 120590
(11)
where 120585 is the state of the auxiliary dynamic system 1198961205851 1198961205852 arepositive constants 120590 120590 are positive small design constants 120599(sdot)can be derived from the stability analysis and deadzone(Δ120581 120590)is a dead zone function given by
119889119890119886119889119911119900119899119890 (Δ120581 120590) =
Δ120581 119894119891 Δ120581 gt 1205900 119894119891 minus 120590 lt Δ120581 le 120590Δ120581 119894119891 Δ120581 le 120590
(12)
32 Design of the Desired States Desired reference trajectoryis generated by a virtual ship described in the following form
[[[119889119910119889119889
]]]
= [[[cos120595119889 minus sin120595119889 0sin120595119889 cos120595119889 0
0 0 1]]][[[119906119889setV119889set119903119889set
]]] (13)
4 Mathematical Problems in Engineering
Then the trajectory tracking errors are defined as
(119909119890 119910119890) = (119909 minus 119909119889 119910 minus 119910119889) (14)
For the position tracking the desired states are designed by
119906119889 = (119889 minus 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)) cos120595 + ( 119910119889 minus 119896119910 10038161003816100381610038161199101198901003816100381610038161003816sdot sign (119910119890)) sin120595
120573119889 = arctan(minus119889 minus 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)119906119889 sin120595+ 119910119889 minus 119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890)119906119889 cos120595)
(15)
where 119896119909 gt 0 and 119896119910 gt 0 are control gainsRemark 4 To guarantee the traceability of the referencetrajectory under state constraints the desired reference tra-jectory needs to satisfy the following conditions
(1198621) 120578119889 120578119889 and 120578119889 are all bounded in which 120578119889 = 119909119889119910119889 120595119889(1198622) There exists 119879119903 gt 0 such that for all 119905 gt 119879119903
119906min lt 119906119889 lt 119906max120573min lt 120573119889 lt 120573max (16)
33 Controller Design State tracking errors are defined as
119890119906 = 119906 minus 119906119889119890120573 = 120573 minus 120573119889 (17)
Then two integral sliding mode manifolds are given by
119904119906 = 119890119906 + 1205821 int1199050119890119906 119889120591
119904119903 = 119890120573 + 1205822119890120573 + 1205823 int1199050119890120573 119889120591
(18)
where 1205821 1205822 and 1205823 are positive constantsUsing (6) (17) and (18) the time derivatives of 119904119906 and 119904119903
are expressed as
119904119906 = 119890119906 + 1205821119890119906= 119891119906 (x119906) + 119872119906120591119906 +119872119906119877119906 minus 119889 + 1205821119890119906 (19)
119904119903 = 119890120573 + 1205822 119890120573 + 1205823119890120573= 119891119903 (x119903) + 119872119903120591119903 + 119863119903119908 minus 120573119889 + 1205822 119890120573 + 1205823119890120573 (20)
where
119891119906 (x119906) = 119906119903 tan120573 x119906 = [119906 120573 119903]119879 119891119903 (x119903) = minus119906 minus 221199062 sin (2120573) minus 120573119906 cos (2120573)
+ 120573119903 sin (2120573) x119903 = [119906 119903 120573 120573]119879 119872119906 = 1119898119872119903 = minuscos2120573119869119911 119863119903119908 = minus 120573119877V sin (2120573)119906119898 + 119906V minus 119877V1199062119898 cos2120573
minus 119877119903cos2120573119869119911
(21)
To deal with high nonlinearity and model uncertainties119891119906(x119906) and 119891119903(x119903) are approximated by RBFNNs
119891120603 (x120603) = Wlowast119879120603 H120603 (x120603) + 120576120603 120603 = 119906 119903 (22)
where x120603 isin 119877119868 is the input vector and Wlowast120603 isin 119877119899 is the idealweight vectorH120603(x120603) 119877119868 rarr 119877119899 is the basis function vectorwith element ℎ119894120603(x120603) shown as follows
ℎ119894120603 (x120603) = exp(minus1003817100381710038171003817x120603 minus 120583119894120603100381710038171003817100381721205902119894 ) (23)
where120583119894120603 is the center of the receptive field and120590119894 is thewidthof theGaussian functionThe approximation error 120576120603 satisfies|120576120603| le 120576119873
Using (10) and (11) constraint error functions aredesigned as follows
Δ120581119906 = 1198961199081199061 (sat (119906 119906max 119906min) minus 119906)+ 1198961199081199062 (sat (120591119906119888 120591119906119888max 120591119906119888min) minus 120591119906119888)
Δ120581119903 = 1198961199081199031 (sat (120573 120573max 120573min) minus 120573)+ 1198961199081199032 (sat (120591119903119888 120591119903119888max 120591119903119888min) minus 120591119903119888)
(24)
where 1198961199081199061 gt 0 1198961199081199062 gt 0 1198961199081199031 gt 0 and 1198961199081199032 gt 0Then the auxiliary dynamic system is designed as
120585120603=
(minus1198961205851206031120585120603 minus 120599120603 (sdot) + 1198961205851206032Δ1205811206032100381710038171003817100381712058512060310038171003817100381710038172 120585120603 + Δ120581120603) 10038171003817100381710038171205851206031003817100381710038171003817 ge 120590deadzone (Δ120581120603 120590) 10038171003817100381710038171205851206031003817100381710038171003817 lt 120590
(25)
where 120599120603(sdot) = |119872120603119904120603||120591120603 minus 120591120603119888| and 120603 = 119906 119903
Mathematical Problems in Engineering 5
Finally the control laws are given by
120591119906119888 = 1119872119906 (minus1198961119904119906 minus 1205781 sign (119904119906) + 119889 minus 1205821119890119906 minus119872119906119877119906minus W119879119906H119906 (x119906) minus 120576119873 sign (119904119906) + 119896119904119906120585119906)
(26)
120591119903119888 = 1119872119903 ( 120573119889 minus 1198962119904119903 minus 1205782 sign (119904119903) minus 1205822 119890120573 minus 119863119903119908minus 1205823119890120573 minus W119879119903H119903 (x119903) minus 120576119873 sign (119904119903) + 119896119904119903120585119903)
(27)
where 1198961 1198962 1205781 1205782 119896119904119906 and 119896119904119903 are positive constants W119906 =W119906 +Wlowast119906 W119903 = W119903 +Wlowast119903 and119863119903119908 is defined in (20)
And the adaptive laws areW120603 = 120574120603119904120603H120603 (x120603) 120603 = 119906 119903 (28)
where 120574120603 is the adaptive coefficient
34 Stability Analysis
Theorem 5 If the state tracking errors (17) the desired states(15) the auxiliary dynamic system (25) the surge control law120591119906119888 (26) the yaw control law 120591119903119888 (27) and the adaptive laws(28) are applied to the hovercraft system represented by (6)and for any bounded initial condition the closed-loop controlsystem signals 119904119906 119904119903 120585119906 120585119903 119890119906 119890120573 119909119890 119910119890 and 119908119894119906 and 119908119894119903119894 = 1 2 119899 are uniformly ultimately bounded (UUB)The position and desired state tracking errors can be madearbitrarily small by appropriately selecting design parametersAnd the yaw motion will remain bounded
Proof The following Lyapunov function is defined
119881 = 121199042119906 + 121199042119903 + 12120574119906 W119879119906W119906 +12120574119903 W119879119903 W119903 +
121205852119906+ 121205852119903
(29)
From (19) (20) and (29) can be expressed as
= 119904119906 (119891119906 (x119906) + 119872119906120591119906 +119872119906119877119906 minus 119889 + 1205821119890119906)+ 119904119903 (119891119903 (x119903) + 119872119903120591119903 + 119863119903119908 minus 120573119889 + 1205822 119890120573 + 1205823119890120573)+ W119879119906
W119906120574119906 + W119879119903W119903120574119903 + 120585119906 120585119906 + 120585119903 120585119903
(30)
By defining 120591120603 = Δ120591120603 + 120591120603119888 and 120603 = 119906 119903 and using (22) (26)and (27) we have
= minus11989611199042119906 minus 1205781 10038161003816100381610038161199041199061003816100381610038161003816 + 119904119906 (minusW119879119906H119906 (x119906)) +119872119906119904119906Δ120591119906+ 119896119904119906119904119906120585119906 + (minus120576119873 10038161003816100381610038161199041199061003816100381610038161003816 + 120576119906119904119906) minus 11989621199042119903 minus 1205782 10038161003816100381610038161199041199031003816100381610038161003816+ 119904119903 (minusW119879119903H119903 (x119903)) +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903+ (minus120576119873 10038161003816100381610038161199041199031003816100381610038161003816 + 120576119903119904119903) + 1120574119906 W119879119906 W119906 + 1120574119903 W119879119903 W119903+ 120585119906 120585119906 + 120585119903 120585119903
(31)
By using |120576120603| le 120576119873 and 120603 = 119906 119903 (31) can be rewritten as
le minus11989611199042119906 minus 1205781 10038161003816100381610038161199041199061003816100381610038161003816 + 119904119906 (minusW119879119906H119906 (x119906)) minus 11989621199042119903minus 1205782 10038161003816100381610038161199041199031003816100381610038161003816 + 119904119903 (minusW119879119903H119903 (x119903)) + 119872119906119904119906Δ120591119906+ 119896119904119906119904119906120585119906 +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 1120574119906 W119879119906 W119906+ 1120574119903 W119879119903 W119903 + 120585119906 120585119906 + 120585119903 120585119903
(32)
Substituting adaptive laws (28) into it yields
le minus11989611199042119906 +119872119906119904119906Δ120591119906 + 119896119904119906119904119906120585119906 + 120585119906 120585119906 minus 11989621199042119903+119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 120585119903 120585119903 (33)
The process of stability analysis is respectively discussed inthe following two cases
Case 1When 120585120603 ge 120590 in light of (25) andYoungrsquos inequalitywe have
120585120603 120585120603 = 120585120603Δ120581120603 minus 11989612058512060311205852120603 minus 1198961205851206032Δ1205812120603minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
le 1198961205851206032Δ1205812120603 + 141198961205851206032 1205852120603 minus 11989612058512060311205852120603 minus 1198961205851206032Δ1205812120603minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
= minus(1198961205851206031 minus 141198961205851206032)1205852120603 minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
(34)
Substituting 120591120603 = Δ120591120603 + 120591120603119888 and (34) into (33) and usingYoungrsquos inequality yield
le minus11989611199042119906 + 119896119904119906119904119906120585119906 minus (1198961205851199061 minus 141198961205851199062)1205852119906 minus 11989621199042119903+ 119896119904119903119904119903120585119903 minus (1198961205851199031 minus 141198961205851199032)1205852119903
le minus (1198961 minus 05) 1199042119906 minus (1198961205851199061 minus 141198961205851199062 minus 051198962119904119906)1205852119906minus (1198962 minus 05) 1199042119903 minus (1198961205851199031 minus 141198961205851199032 minus 051198962119904119903)1205852119903minus 1198961W119879119906W119906120574119906 minus 1198962W119879119903 W119903120574119903 + 1198961W119879119906W119906120574119906+ 1198962W119879119903 W119903120574119903 le minus21205831119881 + 1205881
(35)
6 Mathematical Problems in Engineering
where
1205831 = min(1198961 minus 05) (1198961205851199061 minus 141198961205851199062 minus 051198962119904119906) (1198962 minus 05) (1198961205851199031 minus 141198961205851199032 minus 051198962119904119903)
1205881 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903
(36)
Case 2 When 120585120603 lt 120590 in light of (25) and Youngrsquos ine-quality we have
120585120603 120585120603 = 120585120603deadzone (Δ120581120603 120590) le 051205852120603 + 05Δ12058121206030511989621199041206031205852120603 = 11989621199041206031205852120603 minus 0511989621199041206031205852120603 le minus0511989621199041206031205852120603 + 11989621199041206031205902 (37)
Substituting (37) into (33) and using Youngrsquos inequality yield
le minus11989611199042119906 +119872119906119904119906Δ120591119906 + 119896119904119906119904119906120585119906 + 051205852119906 + 05Δ1205812119906minus 11989621199042119903 +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 + 051198722119906Δ1205912119906+ 051198722119903Δ1205912119903 + 0511989621199041199061205852119906 + 0511989621199041199031205852119903 + 051205852119906+ 05Δ1205812119906 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 minus 051205852119906 (1198962119904119906 minus 1)minus 051205852119903 (1198962119904119903 minus 1) + 051198722119906Δ1205912119906 + 051198722119903Δ1205912119903+ 11989621199041199061205902 + 11989621199041199031205902 + 05Δ1205812119906 + 05Δ1205812119903
le minus21205832119881 + 1205882
(38)
where
1205832 = min (1198961 minus 1) (051198962119904119906 minus 05) (1198962 minus 1) (051198962119904119903 minus 05)
1205882 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903 + 051198722119906Δ1205912119906
+ 051198722119903Δ1205912119903 + (1198962119904119906 + 1198962119904119903) 1205902 + (Δ1205812119906 + Δ1205812119903)2
(39)
Synthesizing (35) and (38) we have
le minus2120583119881 + 120588 (40)
where 120583 = min1205831 1205832 and 120588 = max1205881 1205882 with the designparameters 1198961 1198962 119896119904119906 119896119904119903 1198961205851199061 1198961205851199062 1198961205851199031 and 1198961205851199032 satisfying
1198961 gt 11198962 gt 1119896119904119906 gt 1119896119904119903 gt 1
1198961205851199061 minus 141198961205851199062 minus 051198962119904119906 gt 01198961205851199031 minus 141198961205851199032 minus 051198962119904119903 gt 0
(41)
Solving (40) we have
0 le 119881 (119905) le 1205882120583 + [119881 (0) minus 1205882120583] 119890minus2120583119905 (42)
It is obviously seen that 119881(119905) is UUB for all 119881(0) le 1198610 with1198610 being any positive constant Therefore in the light of (29)we know that 119904119906 119904119903 120585119906 120585119903 and 119908119894119906 and 119908119894119903 119894 = 1 2 119899 areUUB for all 119881(0) le 1198610 It can be expressed as
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 + 2 [119881 (0) minus 1205882120583] 119890minus2120583119905 (43)
where 120594 = 119904119906 119904119903 120585119906 120585119903 119908119894119906 119908119894119903It implies that there exists 119879 gt 0 such that for all 119905 gt 119879
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 (44)
where radic120588120583 can be made arbitrarily small by appropriatelyselecting the design parameters
Further the following dynamics are obtained from (18)and (44)
119890119906 + 1205821 int1199050119890119906 119889120591 = 120576119906
119890120573 + 1205822119890120573 + 1205823 int1199050119890120573 119889120591 = 120576120573
(45)
where 120576119906 and 120576120573 are arbitrarily small errorsTo prove that 119890119906 and 119890120573 are UUB the Lyapunov function
is given by
119881119904 = 12 (int1199050119890119906 119889120591)2 + 121198902120573 (46)
The time derivative of this Lyapunov function along thedynamics in (45) is such that
119904 = 119890119906 int1199050119890119906 119889120591 + 119890120573 119890120573
= minus1205821 (int1199050119890119906 119889120591)2 minus 12058221198902120573 + 120576119906 int119905
0119890119906 119889120591
minus 1205823119890120573 int1199050119890120573 119889120591 + 120576120573119890120573
(47)
Mathematical Problems in Engineering 7
From Youngrsquos inequality the following fact is obtained
120576119906 int1199050119890119906 119889120591 le 051205821 (int119905
0119890119906 119889120591)2 + 05 12057621199061205821
minus1205823119890120573 int1199050119890120573 119889120591 le 120582211989021205732 + 120582321205822 (int
119905
0119890120573 119889120591)2
120576120573119890120573 le 02512058221198902120573 + 12057621205731205822
(48)
Then
119904 le minus051205821 (int1199050119890119906 119889120591)2 minus 02512058221198902120573 + 05 12057621199061205821
+ 0512058231205822 (int119905
0119890120573 119889120591)2 + 12057621205731205822 = minus2120583119904119881119904 + 120588119904
(49)
where
120583119904 = min 051205821 0251205822 120588119904 = 05 12057621199061205821 + 0512058231205822 (int
119905
0119890120573 119889120591)2 + 12057621205731205822
(50)
Similar to the analysis in (42)sim(44) there exists 119879119904 gt 0 suchthat for all 119905 gt 119879119904
10038171003817100381710038171205941199041003817100381710038171003817 le radic 120588119904120583119904 (51)
where 120594119904 = int1199050119890119906 119889120591 119890120573
It is obvious that int1199050119890119906 119889120591 and 119890120573 are UUB and will be
arbitrarily small by choosing suitable parametersFrom 119904119906 = 119890119906 + 1205821 int1199050 119890119906 119889120591 we know 119890119906 will be arbitrarily
small From (6) and (15) we have
[ 119906 minus 119906119889119906 tan120573 minus 119906119889 tan120573119889] = 119877[119890 + 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)119910119890 + 119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890)] (52)
where
119877 = [ cos120595 sin120595minus sin120595 cos120595] (53)
It is clear that |119877| = 1 which indicates that it is nonsingularThen we have
119890 = minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887119910119890 = minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887 (54)
where 120576119909119887 120576119910119887 are arbitrarily small errorsFurthermore consider the following Lyapunov function
candidate
119881119901 = 121199092119890 + 121199102119890 (55)
The time derivative of 119881119901 along the dynamics in (54) is suchthat
119901 = 119909119890119890 + 119910119890 119910119890= 119909119890 (minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887)
+ 119910119890 (minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887)= minus119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 119909119890120576119909119887 + 119910119890120576119910119887le minus05119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 05119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 12057621199091198872119896119909 +
12057621199101198872119896119910= minus2120583119901119881119901 + 120588119901
(56)
where
120583119901 = min 05119896119909 05119896119910 120588119901 = 12057621199091198872119896119909 +
12057621199101198872119896119910 (57)
Also similar to the analysis in (42)sim(44) there exists 119879119901 gt 0such that for all 119905 gt 119879119901
1003817100381710038171003817100381712059411990110038171003817100381710038171003817 le radic 120588119901120583119901 (58)
where 120594119901 = 119909119890 119910119890From the reason that 120576119909119887 and 120576119910119887 are arbitrarily small
errors we know that 119909119890 and 119910119890 are UUB andwill be arbitrarilysmall
Also 119906 is continuously differentiable in the movingprocess of hovercraft Hence is bounded From (6) we have
= 119906119903 tan (120573) + 119877119906 + 120591119906119898 (59)
From (15) Remark 4 and the boundedness of 119909119890 and 119910119890we have that 119906119889 and 120573119889 are bounded Then 119906 V and 120573 arebounded from (5) and (17) Furthermore119877119906 is bounded from(3)Therefore it can be concluded that 119903will remain boundedfrom (7) and (59) This concludes the proof
Remark 6 In order to avoid the well-known chatteringproblem the sign function used in the control laws (26) and(27) can be replaced by hyperbolic tangent function whichis continuous such that sign(119904) = tanh(119896119867119904) where 119896119867 isa positive scalar which can be chosen to get a very goodapproximation
4 Simulations
Two different cases are implemented to verify the effective-ness and superiority of the proposed controller In simula-tions the main particulars and constraints of hovercraft areshown in Tables 1 and 2 The water surface is calm withoutwaves
8 Mathematical Problems in Engineering
Table 1 Main particulars of hovercraft
119898 (kg) 40000119869119911 (kgm2) 18 times 106119878PP (m2) 45119878LP (m2) 93119878HP (m2) 260119876 (m3s) 1408119881119908 (knots) 10119897sk (m) 65119861119888 (m) 89119897119888 (m) 236ℎ (m) 1119867hov (m) 59119878119888 (m2) 212120573119908 (deg) 45
Table 2 State and input constraints
Variable Maximum Minimum119906 (knots) 40 25120573 (deg) 8 minus8120591119906 (N) 80000 minus80000120591119903 (Nm) 1 times 105 minus1 times 105
The comparisons of three different methods are carriedout in each case The legend ldquoMethod Ardquo means the methodin [14] the legend ldquoMethod Brdquo means the method withoutstate and input saturation constraints
Saturation coefficients are
119896119904119906 = 121198961199081199061 = 11198961199081199062 = 9030001198961205851199061 = 31119896119904119903 = 12
1198961199081199031 = 11198961199081199032 = 6503001198961205851199031 = 381198961205851199062 = 0081198961205851199032 = 008
120590 = 3120590 = 1
(60)
Case 1 The reference trajectory parameters are
119909119889 (0) = 300m119910119889 (0) = 100m
120595119889 (0) = 45∘119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 0
(61)
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 70∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(62)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 111205781 = 0251205821 = 21198962 = 121205782 = 051205822 = 301205823 = 6120576119873 = 05
(63)
It is observed from Figures 3ndash8 that the proposed trajectorytracking controller is effective Tracking errors of all threemethods converge to arbitrarily small values and headingangle is bounded From the comparisons with Methods Aand B in Figures 9 and 10 the proposed controller canlimit the surge speed and the drift angle into the safe rangeeffectively Figures 11 and 12 show that the input saturation isalso handled by the proposed controller
Case 2 The reference trajectory parameters are
119909119889 (0) = minus200m119910119889 (0) = 50m120595119889 (0) = 45∘
119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 01∘s
(64)
Mathematical Problems in Engineering 9
0 2000 4000 6000 8000 10000 12000
0
5000
10000
15000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
x(m
)
y (m)
Figure 3 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 4 The heading angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
500
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus500
minus1000
Figure 5 The surge position tracking error
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 30∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(65)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 153
0 200 400 600 800 1000 1200 1400
0
500
1000
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
minus500
Figure 6 The sway position tracking error
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus10
minus20
Figure 7 The surge velocity tracking error
0 200 400 600 800 1000 1200 1400
0
5
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus10
minus5
Figure 8 The drift angle tracking error
1205781 = 011205821 = 41198962 = 121205782 = 051205822 = 301205823 = 2120576119873 = 05
(66)
It is obvious from Figures 14ndash19 that only the proposedcontroller can track the trajectory successfully Figures 20 and21 show that the surge speed changes to negative values andthe drift angle exceeds the safety limit in the tracking control
10 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controller
u (k
nots)
Method AMethod B
Figure 9 The surge velocity of hovercraft
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
minus10
minus20
Figure 10 The drift angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
2
4
Time (s)
0 50 1000
5
10
Proposed controllerMethod AMethod B
uc
(N)
times105
times104
Figure 11 The surge control law of hovercraft
process ofMethods A and BThese are absolutely not allowedfor hovercraft In contrast you can see from Figure 20 thatsurge speed can still be positive even large enough to avoidthe influence of the resistance humps under the control of theproposed controller And drift angle is also within the safetylimit from Figure 21 From Figures 22 and 23 we know thatthe input constraint ability of the proposed controller is effec-tive Figures 13 and 24 show the heave position of hovercraft
5 Conclusion
A safety-guaranteed trajectory tracking controller has beenproposed for underactuated hovercraft in this paper Thesafety-guaranteed auxiliary dynamic system is designed todeal with state and input constraints The velocity of hov-ercraft is constrained to eliminate the effect of resistance
0 200 400 600 800 1000 1200 1400minus4
0
Time (s)
0 10 20minus15minus10minus5
05
Proposed controllerMethod AMethod B
rc
(Nm
)
times105
times104
minus2
Figure 12 The yaw control law of hovercraft
0 200 400 600 800 1000 1200 1400minus245
minus24
minus235
Time (s)z
(m)
Proposed controllerMethod AMethod B
Figure 13 The heave position of hovercraft
minus2000 0 2000 4000 6000 8000 10000 12000minus1000
0
1000
2000
3000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
minus200minus100
0100
x(m
)
y (m)
minus20 0 20 40 60 80
Figure 14 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 12000
50
100
150
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 15 The heading angle of hovercraft
hump and obtain better stability The safety limit of driftangle is carried out effectively for safety in the high-speedtrajectory tracking process of hovercraftThe input saturationis handled High nonlinearity and model uncertainties areapproximated by RBFNNs Simulation results have been
Mathematical Problems in Engineering 11
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus100
Figure 16 The surge position tracking error
0 200 400 600 800 1000 1200minus600
minus400
minus200
0
200
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
Figure 17 The sway position tracking error
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus100
Figure 18 The surge velocity tracking error of hovercraft
0 200 400 600 800 1000 1200
0
2
4
6
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus2
Figure 19 The drift angle tracking error of hovercraft
0 200 400 600 800 1000 1200
0
50
Time (s)
0 100 200
0
50
Proposed controllerMethod AMethod B
u(k
nots)
minus50minus50
minus100
Figure 20 The surge velocity of hovercraft
0 200 400 600 800 1000 1200
0
100
200
Time (s)
0 100 2000
20
40
Proposed controllerMethod AMethod B
(d
eg)
minus100
Figure 21 The drift angle of hovercraft
0 200 400 600 800 1000 1200
0
Time (s)
0 50 100
0
Proposed controllerMethod AMethod B
uc
(N)
minus5
minus10
times105
times104
minus5
minus10
Figure 22 The surge control law of hovercraft
0 200 400 600 800 1000 1200
0
10
20
Time (s)
0 10 20
05
1015
Proposed controllerMethod AMethod B
rc
(Nm
)
minus5
times109
times104
Figure 23 The yaw control law of hovercraft
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
119877119903 = 119877119903119886 + 119877119903119898 + 119877119903wm + 119877119903sk= minus051205881198861198812119886119862119903119886119878HP119867hov minus 119877V119886 times 119909119886 + 119877119906119886 times 119910119886
minus 119877V119898119909119898 + 119877119906119898119910119898 minus 119877Vwm119909wm + 119877119906wm119910wmminus 119877Vsk119909sk + 119877119906sk119910sk
(3)
where the dragrsquos suffix 119886 is the aerodynamic profile drag wmis the wave-making drag 119898 is the air momentum drag sk isthe skirt drag 119862119906119886 119862V119886 119862119901119886 119862119903119886 119862wm and 119862sk are the dragcoefficients 119861119888 is the cushion beam 119897119888 is the cushion length119866 is the weight 119878PP 119878LP and 119878HP are the positive lateral andhorizontal projection areas 120573 is the drift angle 119876 is the fanflow rate of cushion fan ℎ is the distance between baffle andthe bottom of skirtrsquos finger 119897sk is the total peripheral lengthof the skirts 119867hov is the height of hovercraft 120588119886 and 120588119908 areair and water density and (119909119886 119910119886) (119909119898 119910119898) (119909wm 119910wm) and(119909sk 119910sk) are the coordinates of forcersquos acting points119881119886 and 119881119886 in (3) can be obtained by
119881119886= radic[119906 + 119881119908 cos (120573119908 minus 120595)]2 + [V + 119881119908 sin (120573119908 minus 120595)]2
119881119886 = radic1198812119886 + 1199082(4)
in which 119881119908 and 120573119908 are absolute wind speed and directionMore details can be found in [1 9 21]
Remark 1 When a hovercraft is moving on a calm watersurface cushion pressure 119901119888 varies within a narrow range andthe heavemotion is stableThis paper is the research about thehorizontal motion of the hovercraft Hence the heavemotionis not discussed and the cushion pressure 119901119888 is assumed to bea constant
From Figure 2 we have
V = 119906 tan (120573) (5)
In order to make 120573 be the system state and more convenientfor the constraint and control of 120573 an improved model isderived from (1) and (5) that is
[ 119910 ]119879 = 119878 (120595) [119906 119906 tan120573 119908 119903]119879 [[[[[[
120573
119903
]]]]]]
=[[[[[[[
119906119903 tan120573minus sin (2120573)2119906 minus 119903cos2120573
00
]]]]]]]
+119872[[[[[[[[
119877119906 + 120591119906119877V
cos2120573119906119877119908 minus 119901119888119878119888119877119903 + 120591119903
]]]]]]]]
(6)
22 State and Input Constraints Saturation nonlinearities ofactuators can be described by
120591120603 = sat (120591120603119888 120591120603119888max 120591120603119888min) 120603 = 119906 119903 (7)
where 120591120603119888max and 120591120603119888min are the maximum and minimumlimitations of actuators 120591120603119888 are the designed control laws andsat(sdot) is a generalized saturation function with the followingform
sat (120572 120572119872 120572119898) =
120572119872 if 120572 gt 120572119872120572 if 120572119898 le 120572 le 120572119872120572119898 if 120572 lt 120572119898
(8)
Assumption 2 All position orientation velocity and acceler-ation values of hovercraft are available for feedback
Safety limit of 120573 and hump speed of hovercraft need tobe obtained from model and real ship tests [11 12] In thispaper they are assumed to be known and available for thestate constraint Then the safe constraints of system state aredefined as
119906min le 119906 le 119906max120573min le 120573 le 120573max (9)
3 Controller Design
31 Safety-Guaranteed Auxiliary Dynamic System
Proposition 3 A constraint error function is designed as fol-lows
Δ120581 = 1198961199081 (sat (119909 119909119898119886119909 119909119898119894119899) minus 119909)+ 1198961199082 (sat (119906119888 119906119888119898119886119909 119906119888119898119894119899) minus 119906119888) (10)
where 1198961199081 gt 0 1198961199082 gt 0 119909 is the system state and 119906119888 is thedesigned control input
Then an auxiliary dynamic system is designed by
120585 = minus1198961205851120585 minus 120599 (sdot) + 1198961205852Δ1205812100381710038171003817100381712058510038171003817100381710038172 120585 + Δ120581 10038171003817100381710038171205851003817100381710038171003817 ge 120590deadzone (Δ120581 120590) 10038171003817100381710038171205851003817100381710038171003817 lt 120590
(11)
where 120585 is the state of the auxiliary dynamic system 1198961205851 1198961205852 arepositive constants 120590 120590 are positive small design constants 120599(sdot)can be derived from the stability analysis and deadzone(Δ120581 120590)is a dead zone function given by
119889119890119886119889119911119900119899119890 (Δ120581 120590) =
Δ120581 119894119891 Δ120581 gt 1205900 119894119891 minus 120590 lt Δ120581 le 120590Δ120581 119894119891 Δ120581 le 120590
(12)
32 Design of the Desired States Desired reference trajectoryis generated by a virtual ship described in the following form
[[[119889119910119889119889
]]]
= [[[cos120595119889 minus sin120595119889 0sin120595119889 cos120595119889 0
0 0 1]]][[[119906119889setV119889set119903119889set
]]] (13)
4 Mathematical Problems in Engineering
Then the trajectory tracking errors are defined as
(119909119890 119910119890) = (119909 minus 119909119889 119910 minus 119910119889) (14)
For the position tracking the desired states are designed by
119906119889 = (119889 minus 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)) cos120595 + ( 119910119889 minus 119896119910 10038161003816100381610038161199101198901003816100381610038161003816sdot sign (119910119890)) sin120595
120573119889 = arctan(minus119889 minus 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)119906119889 sin120595+ 119910119889 minus 119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890)119906119889 cos120595)
(15)
where 119896119909 gt 0 and 119896119910 gt 0 are control gainsRemark 4 To guarantee the traceability of the referencetrajectory under state constraints the desired reference tra-jectory needs to satisfy the following conditions
(1198621) 120578119889 120578119889 and 120578119889 are all bounded in which 120578119889 = 119909119889119910119889 120595119889(1198622) There exists 119879119903 gt 0 such that for all 119905 gt 119879119903
119906min lt 119906119889 lt 119906max120573min lt 120573119889 lt 120573max (16)
33 Controller Design State tracking errors are defined as
119890119906 = 119906 minus 119906119889119890120573 = 120573 minus 120573119889 (17)
Then two integral sliding mode manifolds are given by
119904119906 = 119890119906 + 1205821 int1199050119890119906 119889120591
119904119903 = 119890120573 + 1205822119890120573 + 1205823 int1199050119890120573 119889120591
(18)
where 1205821 1205822 and 1205823 are positive constantsUsing (6) (17) and (18) the time derivatives of 119904119906 and 119904119903
are expressed as
119904119906 = 119890119906 + 1205821119890119906= 119891119906 (x119906) + 119872119906120591119906 +119872119906119877119906 minus 119889 + 1205821119890119906 (19)
119904119903 = 119890120573 + 1205822 119890120573 + 1205823119890120573= 119891119903 (x119903) + 119872119903120591119903 + 119863119903119908 minus 120573119889 + 1205822 119890120573 + 1205823119890120573 (20)
where
119891119906 (x119906) = 119906119903 tan120573 x119906 = [119906 120573 119903]119879 119891119903 (x119903) = minus119906 minus 221199062 sin (2120573) minus 120573119906 cos (2120573)
+ 120573119903 sin (2120573) x119903 = [119906 119903 120573 120573]119879 119872119906 = 1119898119872119903 = minuscos2120573119869119911 119863119903119908 = minus 120573119877V sin (2120573)119906119898 + 119906V minus 119877V1199062119898 cos2120573
minus 119877119903cos2120573119869119911
(21)
To deal with high nonlinearity and model uncertainties119891119906(x119906) and 119891119903(x119903) are approximated by RBFNNs
119891120603 (x120603) = Wlowast119879120603 H120603 (x120603) + 120576120603 120603 = 119906 119903 (22)
where x120603 isin 119877119868 is the input vector and Wlowast120603 isin 119877119899 is the idealweight vectorH120603(x120603) 119877119868 rarr 119877119899 is the basis function vectorwith element ℎ119894120603(x120603) shown as follows
ℎ119894120603 (x120603) = exp(minus1003817100381710038171003817x120603 minus 120583119894120603100381710038171003817100381721205902119894 ) (23)
where120583119894120603 is the center of the receptive field and120590119894 is thewidthof theGaussian functionThe approximation error 120576120603 satisfies|120576120603| le 120576119873
Using (10) and (11) constraint error functions aredesigned as follows
Δ120581119906 = 1198961199081199061 (sat (119906 119906max 119906min) minus 119906)+ 1198961199081199062 (sat (120591119906119888 120591119906119888max 120591119906119888min) minus 120591119906119888)
Δ120581119903 = 1198961199081199031 (sat (120573 120573max 120573min) minus 120573)+ 1198961199081199032 (sat (120591119903119888 120591119903119888max 120591119903119888min) minus 120591119903119888)
(24)
where 1198961199081199061 gt 0 1198961199081199062 gt 0 1198961199081199031 gt 0 and 1198961199081199032 gt 0Then the auxiliary dynamic system is designed as
120585120603=
(minus1198961205851206031120585120603 minus 120599120603 (sdot) + 1198961205851206032Δ1205811206032100381710038171003817100381712058512060310038171003817100381710038172 120585120603 + Δ120581120603) 10038171003817100381710038171205851206031003817100381710038171003817 ge 120590deadzone (Δ120581120603 120590) 10038171003817100381710038171205851206031003817100381710038171003817 lt 120590
(25)
where 120599120603(sdot) = |119872120603119904120603||120591120603 minus 120591120603119888| and 120603 = 119906 119903
Mathematical Problems in Engineering 5
Finally the control laws are given by
120591119906119888 = 1119872119906 (minus1198961119904119906 minus 1205781 sign (119904119906) + 119889 minus 1205821119890119906 minus119872119906119877119906minus W119879119906H119906 (x119906) minus 120576119873 sign (119904119906) + 119896119904119906120585119906)
(26)
120591119903119888 = 1119872119903 ( 120573119889 minus 1198962119904119903 minus 1205782 sign (119904119903) minus 1205822 119890120573 minus 119863119903119908minus 1205823119890120573 minus W119879119903H119903 (x119903) minus 120576119873 sign (119904119903) + 119896119904119903120585119903)
(27)
where 1198961 1198962 1205781 1205782 119896119904119906 and 119896119904119903 are positive constants W119906 =W119906 +Wlowast119906 W119903 = W119903 +Wlowast119903 and119863119903119908 is defined in (20)
And the adaptive laws areW120603 = 120574120603119904120603H120603 (x120603) 120603 = 119906 119903 (28)
where 120574120603 is the adaptive coefficient
34 Stability Analysis
Theorem 5 If the state tracking errors (17) the desired states(15) the auxiliary dynamic system (25) the surge control law120591119906119888 (26) the yaw control law 120591119903119888 (27) and the adaptive laws(28) are applied to the hovercraft system represented by (6)and for any bounded initial condition the closed-loop controlsystem signals 119904119906 119904119903 120585119906 120585119903 119890119906 119890120573 119909119890 119910119890 and 119908119894119906 and 119908119894119903119894 = 1 2 119899 are uniformly ultimately bounded (UUB)The position and desired state tracking errors can be madearbitrarily small by appropriately selecting design parametersAnd the yaw motion will remain bounded
Proof The following Lyapunov function is defined
119881 = 121199042119906 + 121199042119903 + 12120574119906 W119879119906W119906 +12120574119903 W119879119903 W119903 +
121205852119906+ 121205852119903
(29)
From (19) (20) and (29) can be expressed as
= 119904119906 (119891119906 (x119906) + 119872119906120591119906 +119872119906119877119906 minus 119889 + 1205821119890119906)+ 119904119903 (119891119903 (x119903) + 119872119903120591119903 + 119863119903119908 minus 120573119889 + 1205822 119890120573 + 1205823119890120573)+ W119879119906
W119906120574119906 + W119879119903W119903120574119903 + 120585119906 120585119906 + 120585119903 120585119903
(30)
By defining 120591120603 = Δ120591120603 + 120591120603119888 and 120603 = 119906 119903 and using (22) (26)and (27) we have
= minus11989611199042119906 minus 1205781 10038161003816100381610038161199041199061003816100381610038161003816 + 119904119906 (minusW119879119906H119906 (x119906)) +119872119906119904119906Δ120591119906+ 119896119904119906119904119906120585119906 + (minus120576119873 10038161003816100381610038161199041199061003816100381610038161003816 + 120576119906119904119906) minus 11989621199042119903 minus 1205782 10038161003816100381610038161199041199031003816100381610038161003816+ 119904119903 (minusW119879119903H119903 (x119903)) +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903+ (minus120576119873 10038161003816100381610038161199041199031003816100381610038161003816 + 120576119903119904119903) + 1120574119906 W119879119906 W119906 + 1120574119903 W119879119903 W119903+ 120585119906 120585119906 + 120585119903 120585119903
(31)
By using |120576120603| le 120576119873 and 120603 = 119906 119903 (31) can be rewritten as
le minus11989611199042119906 minus 1205781 10038161003816100381610038161199041199061003816100381610038161003816 + 119904119906 (minusW119879119906H119906 (x119906)) minus 11989621199042119903minus 1205782 10038161003816100381610038161199041199031003816100381610038161003816 + 119904119903 (minusW119879119903H119903 (x119903)) + 119872119906119904119906Δ120591119906+ 119896119904119906119904119906120585119906 +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 1120574119906 W119879119906 W119906+ 1120574119903 W119879119903 W119903 + 120585119906 120585119906 + 120585119903 120585119903
(32)
Substituting adaptive laws (28) into it yields
le minus11989611199042119906 +119872119906119904119906Δ120591119906 + 119896119904119906119904119906120585119906 + 120585119906 120585119906 minus 11989621199042119903+119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 120585119903 120585119903 (33)
The process of stability analysis is respectively discussed inthe following two cases
Case 1When 120585120603 ge 120590 in light of (25) andYoungrsquos inequalitywe have
120585120603 120585120603 = 120585120603Δ120581120603 minus 11989612058512060311205852120603 minus 1198961205851206032Δ1205812120603minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
le 1198961205851206032Δ1205812120603 + 141198961205851206032 1205852120603 minus 11989612058512060311205852120603 minus 1198961205851206032Δ1205812120603minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
= minus(1198961205851206031 minus 141198961205851206032)1205852120603 minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
(34)
Substituting 120591120603 = Δ120591120603 + 120591120603119888 and (34) into (33) and usingYoungrsquos inequality yield
le minus11989611199042119906 + 119896119904119906119904119906120585119906 minus (1198961205851199061 minus 141198961205851199062)1205852119906 minus 11989621199042119903+ 119896119904119903119904119903120585119903 minus (1198961205851199031 minus 141198961205851199032)1205852119903
le minus (1198961 minus 05) 1199042119906 minus (1198961205851199061 minus 141198961205851199062 minus 051198962119904119906)1205852119906minus (1198962 minus 05) 1199042119903 minus (1198961205851199031 minus 141198961205851199032 minus 051198962119904119903)1205852119903minus 1198961W119879119906W119906120574119906 minus 1198962W119879119903 W119903120574119903 + 1198961W119879119906W119906120574119906+ 1198962W119879119903 W119903120574119903 le minus21205831119881 + 1205881
(35)
6 Mathematical Problems in Engineering
where
1205831 = min(1198961 minus 05) (1198961205851199061 minus 141198961205851199062 minus 051198962119904119906) (1198962 minus 05) (1198961205851199031 minus 141198961205851199032 minus 051198962119904119903)
1205881 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903
(36)
Case 2 When 120585120603 lt 120590 in light of (25) and Youngrsquos ine-quality we have
120585120603 120585120603 = 120585120603deadzone (Δ120581120603 120590) le 051205852120603 + 05Δ12058121206030511989621199041206031205852120603 = 11989621199041206031205852120603 minus 0511989621199041206031205852120603 le minus0511989621199041206031205852120603 + 11989621199041206031205902 (37)
Substituting (37) into (33) and using Youngrsquos inequality yield
le minus11989611199042119906 +119872119906119904119906Δ120591119906 + 119896119904119906119904119906120585119906 + 051205852119906 + 05Δ1205812119906minus 11989621199042119903 +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 + 051198722119906Δ1205912119906+ 051198722119903Δ1205912119903 + 0511989621199041199061205852119906 + 0511989621199041199031205852119903 + 051205852119906+ 05Δ1205812119906 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 minus 051205852119906 (1198962119904119906 minus 1)minus 051205852119903 (1198962119904119903 minus 1) + 051198722119906Δ1205912119906 + 051198722119903Δ1205912119903+ 11989621199041199061205902 + 11989621199041199031205902 + 05Δ1205812119906 + 05Δ1205812119903
le minus21205832119881 + 1205882
(38)
where
1205832 = min (1198961 minus 1) (051198962119904119906 minus 05) (1198962 minus 1) (051198962119904119903 minus 05)
1205882 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903 + 051198722119906Δ1205912119906
+ 051198722119903Δ1205912119903 + (1198962119904119906 + 1198962119904119903) 1205902 + (Δ1205812119906 + Δ1205812119903)2
(39)
Synthesizing (35) and (38) we have
le minus2120583119881 + 120588 (40)
where 120583 = min1205831 1205832 and 120588 = max1205881 1205882 with the designparameters 1198961 1198962 119896119904119906 119896119904119903 1198961205851199061 1198961205851199062 1198961205851199031 and 1198961205851199032 satisfying
1198961 gt 11198962 gt 1119896119904119906 gt 1119896119904119903 gt 1
1198961205851199061 minus 141198961205851199062 minus 051198962119904119906 gt 01198961205851199031 minus 141198961205851199032 minus 051198962119904119903 gt 0
(41)
Solving (40) we have
0 le 119881 (119905) le 1205882120583 + [119881 (0) minus 1205882120583] 119890minus2120583119905 (42)
It is obviously seen that 119881(119905) is UUB for all 119881(0) le 1198610 with1198610 being any positive constant Therefore in the light of (29)we know that 119904119906 119904119903 120585119906 120585119903 and 119908119894119906 and 119908119894119903 119894 = 1 2 119899 areUUB for all 119881(0) le 1198610 It can be expressed as
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 + 2 [119881 (0) minus 1205882120583] 119890minus2120583119905 (43)
where 120594 = 119904119906 119904119903 120585119906 120585119903 119908119894119906 119908119894119903It implies that there exists 119879 gt 0 such that for all 119905 gt 119879
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 (44)
where radic120588120583 can be made arbitrarily small by appropriatelyselecting the design parameters
Further the following dynamics are obtained from (18)and (44)
119890119906 + 1205821 int1199050119890119906 119889120591 = 120576119906
119890120573 + 1205822119890120573 + 1205823 int1199050119890120573 119889120591 = 120576120573
(45)
where 120576119906 and 120576120573 are arbitrarily small errorsTo prove that 119890119906 and 119890120573 are UUB the Lyapunov function
is given by
119881119904 = 12 (int1199050119890119906 119889120591)2 + 121198902120573 (46)
The time derivative of this Lyapunov function along thedynamics in (45) is such that
119904 = 119890119906 int1199050119890119906 119889120591 + 119890120573 119890120573
= minus1205821 (int1199050119890119906 119889120591)2 minus 12058221198902120573 + 120576119906 int119905
0119890119906 119889120591
minus 1205823119890120573 int1199050119890120573 119889120591 + 120576120573119890120573
(47)
Mathematical Problems in Engineering 7
From Youngrsquos inequality the following fact is obtained
120576119906 int1199050119890119906 119889120591 le 051205821 (int119905
0119890119906 119889120591)2 + 05 12057621199061205821
minus1205823119890120573 int1199050119890120573 119889120591 le 120582211989021205732 + 120582321205822 (int
119905
0119890120573 119889120591)2
120576120573119890120573 le 02512058221198902120573 + 12057621205731205822
(48)
Then
119904 le minus051205821 (int1199050119890119906 119889120591)2 minus 02512058221198902120573 + 05 12057621199061205821
+ 0512058231205822 (int119905
0119890120573 119889120591)2 + 12057621205731205822 = minus2120583119904119881119904 + 120588119904
(49)
where
120583119904 = min 051205821 0251205822 120588119904 = 05 12057621199061205821 + 0512058231205822 (int
119905
0119890120573 119889120591)2 + 12057621205731205822
(50)
Similar to the analysis in (42)sim(44) there exists 119879119904 gt 0 suchthat for all 119905 gt 119879119904
10038171003817100381710038171205941199041003817100381710038171003817 le radic 120588119904120583119904 (51)
where 120594119904 = int1199050119890119906 119889120591 119890120573
It is obvious that int1199050119890119906 119889120591 and 119890120573 are UUB and will be
arbitrarily small by choosing suitable parametersFrom 119904119906 = 119890119906 + 1205821 int1199050 119890119906 119889120591 we know 119890119906 will be arbitrarily
small From (6) and (15) we have
[ 119906 minus 119906119889119906 tan120573 minus 119906119889 tan120573119889] = 119877[119890 + 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)119910119890 + 119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890)] (52)
where
119877 = [ cos120595 sin120595minus sin120595 cos120595] (53)
It is clear that |119877| = 1 which indicates that it is nonsingularThen we have
119890 = minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887119910119890 = minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887 (54)
where 120576119909119887 120576119910119887 are arbitrarily small errorsFurthermore consider the following Lyapunov function
candidate
119881119901 = 121199092119890 + 121199102119890 (55)
The time derivative of 119881119901 along the dynamics in (54) is suchthat
119901 = 119909119890119890 + 119910119890 119910119890= 119909119890 (minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887)
+ 119910119890 (minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887)= minus119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 119909119890120576119909119887 + 119910119890120576119910119887le minus05119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 05119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 12057621199091198872119896119909 +
12057621199101198872119896119910= minus2120583119901119881119901 + 120588119901
(56)
where
120583119901 = min 05119896119909 05119896119910 120588119901 = 12057621199091198872119896119909 +
12057621199101198872119896119910 (57)
Also similar to the analysis in (42)sim(44) there exists 119879119901 gt 0such that for all 119905 gt 119879119901
1003817100381710038171003817100381712059411990110038171003817100381710038171003817 le radic 120588119901120583119901 (58)
where 120594119901 = 119909119890 119910119890From the reason that 120576119909119887 and 120576119910119887 are arbitrarily small
errors we know that 119909119890 and 119910119890 are UUB andwill be arbitrarilysmall
Also 119906 is continuously differentiable in the movingprocess of hovercraft Hence is bounded From (6) we have
= 119906119903 tan (120573) + 119877119906 + 120591119906119898 (59)
From (15) Remark 4 and the boundedness of 119909119890 and 119910119890we have that 119906119889 and 120573119889 are bounded Then 119906 V and 120573 arebounded from (5) and (17) Furthermore119877119906 is bounded from(3)Therefore it can be concluded that 119903will remain boundedfrom (7) and (59) This concludes the proof
Remark 6 In order to avoid the well-known chatteringproblem the sign function used in the control laws (26) and(27) can be replaced by hyperbolic tangent function whichis continuous such that sign(119904) = tanh(119896119867119904) where 119896119867 isa positive scalar which can be chosen to get a very goodapproximation
4 Simulations
Two different cases are implemented to verify the effective-ness and superiority of the proposed controller In simula-tions the main particulars and constraints of hovercraft areshown in Tables 1 and 2 The water surface is calm withoutwaves
8 Mathematical Problems in Engineering
Table 1 Main particulars of hovercraft
119898 (kg) 40000119869119911 (kgm2) 18 times 106119878PP (m2) 45119878LP (m2) 93119878HP (m2) 260119876 (m3s) 1408119881119908 (knots) 10119897sk (m) 65119861119888 (m) 89119897119888 (m) 236ℎ (m) 1119867hov (m) 59119878119888 (m2) 212120573119908 (deg) 45
Table 2 State and input constraints
Variable Maximum Minimum119906 (knots) 40 25120573 (deg) 8 minus8120591119906 (N) 80000 minus80000120591119903 (Nm) 1 times 105 minus1 times 105
The comparisons of three different methods are carriedout in each case The legend ldquoMethod Ardquo means the methodin [14] the legend ldquoMethod Brdquo means the method withoutstate and input saturation constraints
Saturation coefficients are
119896119904119906 = 121198961199081199061 = 11198961199081199062 = 9030001198961205851199061 = 31119896119904119903 = 12
1198961199081199031 = 11198961199081199032 = 6503001198961205851199031 = 381198961205851199062 = 0081198961205851199032 = 008
120590 = 3120590 = 1
(60)
Case 1 The reference trajectory parameters are
119909119889 (0) = 300m119910119889 (0) = 100m
120595119889 (0) = 45∘119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 0
(61)
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 70∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(62)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 111205781 = 0251205821 = 21198962 = 121205782 = 051205822 = 301205823 = 6120576119873 = 05
(63)
It is observed from Figures 3ndash8 that the proposed trajectorytracking controller is effective Tracking errors of all threemethods converge to arbitrarily small values and headingangle is bounded From the comparisons with Methods Aand B in Figures 9 and 10 the proposed controller canlimit the surge speed and the drift angle into the safe rangeeffectively Figures 11 and 12 show that the input saturation isalso handled by the proposed controller
Case 2 The reference trajectory parameters are
119909119889 (0) = minus200m119910119889 (0) = 50m120595119889 (0) = 45∘
119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 01∘s
(64)
Mathematical Problems in Engineering 9
0 2000 4000 6000 8000 10000 12000
0
5000
10000
15000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
x(m
)
y (m)
Figure 3 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 4 The heading angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
500
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus500
minus1000
Figure 5 The surge position tracking error
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 30∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(65)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 153
0 200 400 600 800 1000 1200 1400
0
500
1000
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
minus500
Figure 6 The sway position tracking error
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus10
minus20
Figure 7 The surge velocity tracking error
0 200 400 600 800 1000 1200 1400
0
5
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus10
minus5
Figure 8 The drift angle tracking error
1205781 = 011205821 = 41198962 = 121205782 = 051205822 = 301205823 = 2120576119873 = 05
(66)
It is obvious from Figures 14ndash19 that only the proposedcontroller can track the trajectory successfully Figures 20 and21 show that the surge speed changes to negative values andthe drift angle exceeds the safety limit in the tracking control
10 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controller
u (k
nots)
Method AMethod B
Figure 9 The surge velocity of hovercraft
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
minus10
minus20
Figure 10 The drift angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
2
4
Time (s)
0 50 1000
5
10
Proposed controllerMethod AMethod B
uc
(N)
times105
times104
Figure 11 The surge control law of hovercraft
process ofMethods A and BThese are absolutely not allowedfor hovercraft In contrast you can see from Figure 20 thatsurge speed can still be positive even large enough to avoidthe influence of the resistance humps under the control of theproposed controller And drift angle is also within the safetylimit from Figure 21 From Figures 22 and 23 we know thatthe input constraint ability of the proposed controller is effec-tive Figures 13 and 24 show the heave position of hovercraft
5 Conclusion
A safety-guaranteed trajectory tracking controller has beenproposed for underactuated hovercraft in this paper Thesafety-guaranteed auxiliary dynamic system is designed todeal with state and input constraints The velocity of hov-ercraft is constrained to eliminate the effect of resistance
0 200 400 600 800 1000 1200 1400minus4
0
Time (s)
0 10 20minus15minus10minus5
05
Proposed controllerMethod AMethod B
rc
(Nm
)
times105
times104
minus2
Figure 12 The yaw control law of hovercraft
0 200 400 600 800 1000 1200 1400minus245
minus24
minus235
Time (s)z
(m)
Proposed controllerMethod AMethod B
Figure 13 The heave position of hovercraft
minus2000 0 2000 4000 6000 8000 10000 12000minus1000
0
1000
2000
3000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
minus200minus100
0100
x(m
)
y (m)
minus20 0 20 40 60 80
Figure 14 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 12000
50
100
150
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 15 The heading angle of hovercraft
hump and obtain better stability The safety limit of driftangle is carried out effectively for safety in the high-speedtrajectory tracking process of hovercraftThe input saturationis handled High nonlinearity and model uncertainties areapproximated by RBFNNs Simulation results have been
Mathematical Problems in Engineering 11
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus100
Figure 16 The surge position tracking error
0 200 400 600 800 1000 1200minus600
minus400
minus200
0
200
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
Figure 17 The sway position tracking error
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus100
Figure 18 The surge velocity tracking error of hovercraft
0 200 400 600 800 1000 1200
0
2
4
6
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus2
Figure 19 The drift angle tracking error of hovercraft
0 200 400 600 800 1000 1200
0
50
Time (s)
0 100 200
0
50
Proposed controllerMethod AMethod B
u(k
nots)
minus50minus50
minus100
Figure 20 The surge velocity of hovercraft
0 200 400 600 800 1000 1200
0
100
200
Time (s)
0 100 2000
20
40
Proposed controllerMethod AMethod B
(d
eg)
minus100
Figure 21 The drift angle of hovercraft
0 200 400 600 800 1000 1200
0
Time (s)
0 50 100
0
Proposed controllerMethod AMethod B
uc
(N)
minus5
minus10
times105
times104
minus5
minus10
Figure 22 The surge control law of hovercraft
0 200 400 600 800 1000 1200
0
10
20
Time (s)
0 10 20
05
1015
Proposed controllerMethod AMethod B
rc
(Nm
)
minus5
times109
times104
Figure 23 The yaw control law of hovercraft
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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4 Mathematical Problems in Engineering
Then the trajectory tracking errors are defined as
(119909119890 119910119890) = (119909 minus 119909119889 119910 minus 119910119889) (14)
For the position tracking the desired states are designed by
119906119889 = (119889 minus 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)) cos120595 + ( 119910119889 minus 119896119910 10038161003816100381610038161199101198901003816100381610038161003816sdot sign (119910119890)) sin120595
120573119889 = arctan(minus119889 minus 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)119906119889 sin120595+ 119910119889 minus 119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890)119906119889 cos120595)
(15)
where 119896119909 gt 0 and 119896119910 gt 0 are control gainsRemark 4 To guarantee the traceability of the referencetrajectory under state constraints the desired reference tra-jectory needs to satisfy the following conditions
(1198621) 120578119889 120578119889 and 120578119889 are all bounded in which 120578119889 = 119909119889119910119889 120595119889(1198622) There exists 119879119903 gt 0 such that for all 119905 gt 119879119903
119906min lt 119906119889 lt 119906max120573min lt 120573119889 lt 120573max (16)
33 Controller Design State tracking errors are defined as
119890119906 = 119906 minus 119906119889119890120573 = 120573 minus 120573119889 (17)
Then two integral sliding mode manifolds are given by
119904119906 = 119890119906 + 1205821 int1199050119890119906 119889120591
119904119903 = 119890120573 + 1205822119890120573 + 1205823 int1199050119890120573 119889120591
(18)
where 1205821 1205822 and 1205823 are positive constantsUsing (6) (17) and (18) the time derivatives of 119904119906 and 119904119903
are expressed as
119904119906 = 119890119906 + 1205821119890119906= 119891119906 (x119906) + 119872119906120591119906 +119872119906119877119906 minus 119889 + 1205821119890119906 (19)
119904119903 = 119890120573 + 1205822 119890120573 + 1205823119890120573= 119891119903 (x119903) + 119872119903120591119903 + 119863119903119908 minus 120573119889 + 1205822 119890120573 + 1205823119890120573 (20)
where
119891119906 (x119906) = 119906119903 tan120573 x119906 = [119906 120573 119903]119879 119891119903 (x119903) = minus119906 minus 221199062 sin (2120573) minus 120573119906 cos (2120573)
+ 120573119903 sin (2120573) x119903 = [119906 119903 120573 120573]119879 119872119906 = 1119898119872119903 = minuscos2120573119869119911 119863119903119908 = minus 120573119877V sin (2120573)119906119898 + 119906V minus 119877V1199062119898 cos2120573
minus 119877119903cos2120573119869119911
(21)
To deal with high nonlinearity and model uncertainties119891119906(x119906) and 119891119903(x119903) are approximated by RBFNNs
119891120603 (x120603) = Wlowast119879120603 H120603 (x120603) + 120576120603 120603 = 119906 119903 (22)
where x120603 isin 119877119868 is the input vector and Wlowast120603 isin 119877119899 is the idealweight vectorH120603(x120603) 119877119868 rarr 119877119899 is the basis function vectorwith element ℎ119894120603(x120603) shown as follows
ℎ119894120603 (x120603) = exp(minus1003817100381710038171003817x120603 minus 120583119894120603100381710038171003817100381721205902119894 ) (23)
where120583119894120603 is the center of the receptive field and120590119894 is thewidthof theGaussian functionThe approximation error 120576120603 satisfies|120576120603| le 120576119873
Using (10) and (11) constraint error functions aredesigned as follows
Δ120581119906 = 1198961199081199061 (sat (119906 119906max 119906min) minus 119906)+ 1198961199081199062 (sat (120591119906119888 120591119906119888max 120591119906119888min) minus 120591119906119888)
Δ120581119903 = 1198961199081199031 (sat (120573 120573max 120573min) minus 120573)+ 1198961199081199032 (sat (120591119903119888 120591119903119888max 120591119903119888min) minus 120591119903119888)
(24)
where 1198961199081199061 gt 0 1198961199081199062 gt 0 1198961199081199031 gt 0 and 1198961199081199032 gt 0Then the auxiliary dynamic system is designed as
120585120603=
(minus1198961205851206031120585120603 minus 120599120603 (sdot) + 1198961205851206032Δ1205811206032100381710038171003817100381712058512060310038171003817100381710038172 120585120603 + Δ120581120603) 10038171003817100381710038171205851206031003817100381710038171003817 ge 120590deadzone (Δ120581120603 120590) 10038171003817100381710038171205851206031003817100381710038171003817 lt 120590
(25)
where 120599120603(sdot) = |119872120603119904120603||120591120603 minus 120591120603119888| and 120603 = 119906 119903
Mathematical Problems in Engineering 5
Finally the control laws are given by
120591119906119888 = 1119872119906 (minus1198961119904119906 minus 1205781 sign (119904119906) + 119889 minus 1205821119890119906 minus119872119906119877119906minus W119879119906H119906 (x119906) minus 120576119873 sign (119904119906) + 119896119904119906120585119906)
(26)
120591119903119888 = 1119872119903 ( 120573119889 minus 1198962119904119903 minus 1205782 sign (119904119903) minus 1205822 119890120573 minus 119863119903119908minus 1205823119890120573 minus W119879119903H119903 (x119903) minus 120576119873 sign (119904119903) + 119896119904119903120585119903)
(27)
where 1198961 1198962 1205781 1205782 119896119904119906 and 119896119904119903 are positive constants W119906 =W119906 +Wlowast119906 W119903 = W119903 +Wlowast119903 and119863119903119908 is defined in (20)
And the adaptive laws areW120603 = 120574120603119904120603H120603 (x120603) 120603 = 119906 119903 (28)
where 120574120603 is the adaptive coefficient
34 Stability Analysis
Theorem 5 If the state tracking errors (17) the desired states(15) the auxiliary dynamic system (25) the surge control law120591119906119888 (26) the yaw control law 120591119903119888 (27) and the adaptive laws(28) are applied to the hovercraft system represented by (6)and for any bounded initial condition the closed-loop controlsystem signals 119904119906 119904119903 120585119906 120585119903 119890119906 119890120573 119909119890 119910119890 and 119908119894119906 and 119908119894119903119894 = 1 2 119899 are uniformly ultimately bounded (UUB)The position and desired state tracking errors can be madearbitrarily small by appropriately selecting design parametersAnd the yaw motion will remain bounded
Proof The following Lyapunov function is defined
119881 = 121199042119906 + 121199042119903 + 12120574119906 W119879119906W119906 +12120574119903 W119879119903 W119903 +
121205852119906+ 121205852119903
(29)
From (19) (20) and (29) can be expressed as
= 119904119906 (119891119906 (x119906) + 119872119906120591119906 +119872119906119877119906 minus 119889 + 1205821119890119906)+ 119904119903 (119891119903 (x119903) + 119872119903120591119903 + 119863119903119908 minus 120573119889 + 1205822 119890120573 + 1205823119890120573)+ W119879119906
W119906120574119906 + W119879119903W119903120574119903 + 120585119906 120585119906 + 120585119903 120585119903
(30)
By defining 120591120603 = Δ120591120603 + 120591120603119888 and 120603 = 119906 119903 and using (22) (26)and (27) we have
= minus11989611199042119906 minus 1205781 10038161003816100381610038161199041199061003816100381610038161003816 + 119904119906 (minusW119879119906H119906 (x119906)) +119872119906119904119906Δ120591119906+ 119896119904119906119904119906120585119906 + (minus120576119873 10038161003816100381610038161199041199061003816100381610038161003816 + 120576119906119904119906) minus 11989621199042119903 minus 1205782 10038161003816100381610038161199041199031003816100381610038161003816+ 119904119903 (minusW119879119903H119903 (x119903)) +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903+ (minus120576119873 10038161003816100381610038161199041199031003816100381610038161003816 + 120576119903119904119903) + 1120574119906 W119879119906 W119906 + 1120574119903 W119879119903 W119903+ 120585119906 120585119906 + 120585119903 120585119903
(31)
By using |120576120603| le 120576119873 and 120603 = 119906 119903 (31) can be rewritten as
le minus11989611199042119906 minus 1205781 10038161003816100381610038161199041199061003816100381610038161003816 + 119904119906 (minusW119879119906H119906 (x119906)) minus 11989621199042119903minus 1205782 10038161003816100381610038161199041199031003816100381610038161003816 + 119904119903 (minusW119879119903H119903 (x119903)) + 119872119906119904119906Δ120591119906+ 119896119904119906119904119906120585119906 +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 1120574119906 W119879119906 W119906+ 1120574119903 W119879119903 W119903 + 120585119906 120585119906 + 120585119903 120585119903
(32)
Substituting adaptive laws (28) into it yields
le minus11989611199042119906 +119872119906119904119906Δ120591119906 + 119896119904119906119904119906120585119906 + 120585119906 120585119906 minus 11989621199042119903+119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 120585119903 120585119903 (33)
The process of stability analysis is respectively discussed inthe following two cases
Case 1When 120585120603 ge 120590 in light of (25) andYoungrsquos inequalitywe have
120585120603 120585120603 = 120585120603Δ120581120603 minus 11989612058512060311205852120603 minus 1198961205851206032Δ1205812120603minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
le 1198961205851206032Δ1205812120603 + 141198961205851206032 1205852120603 minus 11989612058512060311205852120603 minus 1198961205851206032Δ1205812120603minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
= minus(1198961205851206031 minus 141198961205851206032)1205852120603 minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
(34)
Substituting 120591120603 = Δ120591120603 + 120591120603119888 and (34) into (33) and usingYoungrsquos inequality yield
le minus11989611199042119906 + 119896119904119906119904119906120585119906 minus (1198961205851199061 minus 141198961205851199062)1205852119906 minus 11989621199042119903+ 119896119904119903119904119903120585119903 minus (1198961205851199031 minus 141198961205851199032)1205852119903
le minus (1198961 minus 05) 1199042119906 minus (1198961205851199061 minus 141198961205851199062 minus 051198962119904119906)1205852119906minus (1198962 minus 05) 1199042119903 minus (1198961205851199031 minus 141198961205851199032 minus 051198962119904119903)1205852119903minus 1198961W119879119906W119906120574119906 minus 1198962W119879119903 W119903120574119903 + 1198961W119879119906W119906120574119906+ 1198962W119879119903 W119903120574119903 le minus21205831119881 + 1205881
(35)
6 Mathematical Problems in Engineering
where
1205831 = min(1198961 minus 05) (1198961205851199061 minus 141198961205851199062 minus 051198962119904119906) (1198962 minus 05) (1198961205851199031 minus 141198961205851199032 minus 051198962119904119903)
1205881 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903
(36)
Case 2 When 120585120603 lt 120590 in light of (25) and Youngrsquos ine-quality we have
120585120603 120585120603 = 120585120603deadzone (Δ120581120603 120590) le 051205852120603 + 05Δ12058121206030511989621199041206031205852120603 = 11989621199041206031205852120603 minus 0511989621199041206031205852120603 le minus0511989621199041206031205852120603 + 11989621199041206031205902 (37)
Substituting (37) into (33) and using Youngrsquos inequality yield
le minus11989611199042119906 +119872119906119904119906Δ120591119906 + 119896119904119906119904119906120585119906 + 051205852119906 + 05Δ1205812119906minus 11989621199042119903 +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 + 051198722119906Δ1205912119906+ 051198722119903Δ1205912119903 + 0511989621199041199061205852119906 + 0511989621199041199031205852119903 + 051205852119906+ 05Δ1205812119906 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 minus 051205852119906 (1198962119904119906 minus 1)minus 051205852119903 (1198962119904119903 minus 1) + 051198722119906Δ1205912119906 + 051198722119903Δ1205912119903+ 11989621199041199061205902 + 11989621199041199031205902 + 05Δ1205812119906 + 05Δ1205812119903
le minus21205832119881 + 1205882
(38)
where
1205832 = min (1198961 minus 1) (051198962119904119906 minus 05) (1198962 minus 1) (051198962119904119903 minus 05)
1205882 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903 + 051198722119906Δ1205912119906
+ 051198722119903Δ1205912119903 + (1198962119904119906 + 1198962119904119903) 1205902 + (Δ1205812119906 + Δ1205812119903)2
(39)
Synthesizing (35) and (38) we have
le minus2120583119881 + 120588 (40)
where 120583 = min1205831 1205832 and 120588 = max1205881 1205882 with the designparameters 1198961 1198962 119896119904119906 119896119904119903 1198961205851199061 1198961205851199062 1198961205851199031 and 1198961205851199032 satisfying
1198961 gt 11198962 gt 1119896119904119906 gt 1119896119904119903 gt 1
1198961205851199061 minus 141198961205851199062 minus 051198962119904119906 gt 01198961205851199031 minus 141198961205851199032 minus 051198962119904119903 gt 0
(41)
Solving (40) we have
0 le 119881 (119905) le 1205882120583 + [119881 (0) minus 1205882120583] 119890minus2120583119905 (42)
It is obviously seen that 119881(119905) is UUB for all 119881(0) le 1198610 with1198610 being any positive constant Therefore in the light of (29)we know that 119904119906 119904119903 120585119906 120585119903 and 119908119894119906 and 119908119894119903 119894 = 1 2 119899 areUUB for all 119881(0) le 1198610 It can be expressed as
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 + 2 [119881 (0) minus 1205882120583] 119890minus2120583119905 (43)
where 120594 = 119904119906 119904119903 120585119906 120585119903 119908119894119906 119908119894119903It implies that there exists 119879 gt 0 such that for all 119905 gt 119879
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 (44)
where radic120588120583 can be made arbitrarily small by appropriatelyselecting the design parameters
Further the following dynamics are obtained from (18)and (44)
119890119906 + 1205821 int1199050119890119906 119889120591 = 120576119906
119890120573 + 1205822119890120573 + 1205823 int1199050119890120573 119889120591 = 120576120573
(45)
where 120576119906 and 120576120573 are arbitrarily small errorsTo prove that 119890119906 and 119890120573 are UUB the Lyapunov function
is given by
119881119904 = 12 (int1199050119890119906 119889120591)2 + 121198902120573 (46)
The time derivative of this Lyapunov function along thedynamics in (45) is such that
119904 = 119890119906 int1199050119890119906 119889120591 + 119890120573 119890120573
= minus1205821 (int1199050119890119906 119889120591)2 minus 12058221198902120573 + 120576119906 int119905
0119890119906 119889120591
minus 1205823119890120573 int1199050119890120573 119889120591 + 120576120573119890120573
(47)
Mathematical Problems in Engineering 7
From Youngrsquos inequality the following fact is obtained
120576119906 int1199050119890119906 119889120591 le 051205821 (int119905
0119890119906 119889120591)2 + 05 12057621199061205821
minus1205823119890120573 int1199050119890120573 119889120591 le 120582211989021205732 + 120582321205822 (int
119905
0119890120573 119889120591)2
120576120573119890120573 le 02512058221198902120573 + 12057621205731205822
(48)
Then
119904 le minus051205821 (int1199050119890119906 119889120591)2 minus 02512058221198902120573 + 05 12057621199061205821
+ 0512058231205822 (int119905
0119890120573 119889120591)2 + 12057621205731205822 = minus2120583119904119881119904 + 120588119904
(49)
where
120583119904 = min 051205821 0251205822 120588119904 = 05 12057621199061205821 + 0512058231205822 (int
119905
0119890120573 119889120591)2 + 12057621205731205822
(50)
Similar to the analysis in (42)sim(44) there exists 119879119904 gt 0 suchthat for all 119905 gt 119879119904
10038171003817100381710038171205941199041003817100381710038171003817 le radic 120588119904120583119904 (51)
where 120594119904 = int1199050119890119906 119889120591 119890120573
It is obvious that int1199050119890119906 119889120591 and 119890120573 are UUB and will be
arbitrarily small by choosing suitable parametersFrom 119904119906 = 119890119906 + 1205821 int1199050 119890119906 119889120591 we know 119890119906 will be arbitrarily
small From (6) and (15) we have
[ 119906 minus 119906119889119906 tan120573 minus 119906119889 tan120573119889] = 119877[119890 + 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)119910119890 + 119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890)] (52)
where
119877 = [ cos120595 sin120595minus sin120595 cos120595] (53)
It is clear that |119877| = 1 which indicates that it is nonsingularThen we have
119890 = minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887119910119890 = minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887 (54)
where 120576119909119887 120576119910119887 are arbitrarily small errorsFurthermore consider the following Lyapunov function
candidate
119881119901 = 121199092119890 + 121199102119890 (55)
The time derivative of 119881119901 along the dynamics in (54) is suchthat
119901 = 119909119890119890 + 119910119890 119910119890= 119909119890 (minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887)
+ 119910119890 (minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887)= minus119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 119909119890120576119909119887 + 119910119890120576119910119887le minus05119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 05119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 12057621199091198872119896119909 +
12057621199101198872119896119910= minus2120583119901119881119901 + 120588119901
(56)
where
120583119901 = min 05119896119909 05119896119910 120588119901 = 12057621199091198872119896119909 +
12057621199101198872119896119910 (57)
Also similar to the analysis in (42)sim(44) there exists 119879119901 gt 0such that for all 119905 gt 119879119901
1003817100381710038171003817100381712059411990110038171003817100381710038171003817 le radic 120588119901120583119901 (58)
where 120594119901 = 119909119890 119910119890From the reason that 120576119909119887 and 120576119910119887 are arbitrarily small
errors we know that 119909119890 and 119910119890 are UUB andwill be arbitrarilysmall
Also 119906 is continuously differentiable in the movingprocess of hovercraft Hence is bounded From (6) we have
= 119906119903 tan (120573) + 119877119906 + 120591119906119898 (59)
From (15) Remark 4 and the boundedness of 119909119890 and 119910119890we have that 119906119889 and 120573119889 are bounded Then 119906 V and 120573 arebounded from (5) and (17) Furthermore119877119906 is bounded from(3)Therefore it can be concluded that 119903will remain boundedfrom (7) and (59) This concludes the proof
Remark 6 In order to avoid the well-known chatteringproblem the sign function used in the control laws (26) and(27) can be replaced by hyperbolic tangent function whichis continuous such that sign(119904) = tanh(119896119867119904) where 119896119867 isa positive scalar which can be chosen to get a very goodapproximation
4 Simulations
Two different cases are implemented to verify the effective-ness and superiority of the proposed controller In simula-tions the main particulars and constraints of hovercraft areshown in Tables 1 and 2 The water surface is calm withoutwaves
8 Mathematical Problems in Engineering
Table 1 Main particulars of hovercraft
119898 (kg) 40000119869119911 (kgm2) 18 times 106119878PP (m2) 45119878LP (m2) 93119878HP (m2) 260119876 (m3s) 1408119881119908 (knots) 10119897sk (m) 65119861119888 (m) 89119897119888 (m) 236ℎ (m) 1119867hov (m) 59119878119888 (m2) 212120573119908 (deg) 45
Table 2 State and input constraints
Variable Maximum Minimum119906 (knots) 40 25120573 (deg) 8 minus8120591119906 (N) 80000 minus80000120591119903 (Nm) 1 times 105 minus1 times 105
The comparisons of three different methods are carriedout in each case The legend ldquoMethod Ardquo means the methodin [14] the legend ldquoMethod Brdquo means the method withoutstate and input saturation constraints
Saturation coefficients are
119896119904119906 = 121198961199081199061 = 11198961199081199062 = 9030001198961205851199061 = 31119896119904119903 = 12
1198961199081199031 = 11198961199081199032 = 6503001198961205851199031 = 381198961205851199062 = 0081198961205851199032 = 008
120590 = 3120590 = 1
(60)
Case 1 The reference trajectory parameters are
119909119889 (0) = 300m119910119889 (0) = 100m
120595119889 (0) = 45∘119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 0
(61)
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 70∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(62)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 111205781 = 0251205821 = 21198962 = 121205782 = 051205822 = 301205823 = 6120576119873 = 05
(63)
It is observed from Figures 3ndash8 that the proposed trajectorytracking controller is effective Tracking errors of all threemethods converge to arbitrarily small values and headingangle is bounded From the comparisons with Methods Aand B in Figures 9 and 10 the proposed controller canlimit the surge speed and the drift angle into the safe rangeeffectively Figures 11 and 12 show that the input saturation isalso handled by the proposed controller
Case 2 The reference trajectory parameters are
119909119889 (0) = minus200m119910119889 (0) = 50m120595119889 (0) = 45∘
119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 01∘s
(64)
Mathematical Problems in Engineering 9
0 2000 4000 6000 8000 10000 12000
0
5000
10000
15000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
x(m
)
y (m)
Figure 3 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 4 The heading angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
500
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus500
minus1000
Figure 5 The surge position tracking error
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 30∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(65)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 153
0 200 400 600 800 1000 1200 1400
0
500
1000
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
minus500
Figure 6 The sway position tracking error
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus10
minus20
Figure 7 The surge velocity tracking error
0 200 400 600 800 1000 1200 1400
0
5
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus10
minus5
Figure 8 The drift angle tracking error
1205781 = 011205821 = 41198962 = 121205782 = 051205822 = 301205823 = 2120576119873 = 05
(66)
It is obvious from Figures 14ndash19 that only the proposedcontroller can track the trajectory successfully Figures 20 and21 show that the surge speed changes to negative values andthe drift angle exceeds the safety limit in the tracking control
10 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controller
u (k
nots)
Method AMethod B
Figure 9 The surge velocity of hovercraft
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
minus10
minus20
Figure 10 The drift angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
2
4
Time (s)
0 50 1000
5
10
Proposed controllerMethod AMethod B
uc
(N)
times105
times104
Figure 11 The surge control law of hovercraft
process ofMethods A and BThese are absolutely not allowedfor hovercraft In contrast you can see from Figure 20 thatsurge speed can still be positive even large enough to avoidthe influence of the resistance humps under the control of theproposed controller And drift angle is also within the safetylimit from Figure 21 From Figures 22 and 23 we know thatthe input constraint ability of the proposed controller is effec-tive Figures 13 and 24 show the heave position of hovercraft
5 Conclusion
A safety-guaranteed trajectory tracking controller has beenproposed for underactuated hovercraft in this paper Thesafety-guaranteed auxiliary dynamic system is designed todeal with state and input constraints The velocity of hov-ercraft is constrained to eliminate the effect of resistance
0 200 400 600 800 1000 1200 1400minus4
0
Time (s)
0 10 20minus15minus10minus5
05
Proposed controllerMethod AMethod B
rc
(Nm
)
times105
times104
minus2
Figure 12 The yaw control law of hovercraft
0 200 400 600 800 1000 1200 1400minus245
minus24
minus235
Time (s)z
(m)
Proposed controllerMethod AMethod B
Figure 13 The heave position of hovercraft
minus2000 0 2000 4000 6000 8000 10000 12000minus1000
0
1000
2000
3000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
minus200minus100
0100
x(m
)
y (m)
minus20 0 20 40 60 80
Figure 14 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 12000
50
100
150
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 15 The heading angle of hovercraft
hump and obtain better stability The safety limit of driftangle is carried out effectively for safety in the high-speedtrajectory tracking process of hovercraftThe input saturationis handled High nonlinearity and model uncertainties areapproximated by RBFNNs Simulation results have been
Mathematical Problems in Engineering 11
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus100
Figure 16 The surge position tracking error
0 200 400 600 800 1000 1200minus600
minus400
minus200
0
200
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
Figure 17 The sway position tracking error
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus100
Figure 18 The surge velocity tracking error of hovercraft
0 200 400 600 800 1000 1200
0
2
4
6
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus2
Figure 19 The drift angle tracking error of hovercraft
0 200 400 600 800 1000 1200
0
50
Time (s)
0 100 200
0
50
Proposed controllerMethod AMethod B
u(k
nots)
minus50minus50
minus100
Figure 20 The surge velocity of hovercraft
0 200 400 600 800 1000 1200
0
100
200
Time (s)
0 100 2000
20
40
Proposed controllerMethod AMethod B
(d
eg)
minus100
Figure 21 The drift angle of hovercraft
0 200 400 600 800 1000 1200
0
Time (s)
0 50 100
0
Proposed controllerMethod AMethod B
uc
(N)
minus5
minus10
times105
times104
minus5
minus10
Figure 22 The surge control law of hovercraft
0 200 400 600 800 1000 1200
0
10
20
Time (s)
0 10 20
05
1015
Proposed controllerMethod AMethod B
rc
(Nm
)
minus5
times109
times104
Figure 23 The yaw control law of hovercraft
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Finally the control laws are given by
120591119906119888 = 1119872119906 (minus1198961119904119906 minus 1205781 sign (119904119906) + 119889 minus 1205821119890119906 minus119872119906119877119906minus W119879119906H119906 (x119906) minus 120576119873 sign (119904119906) + 119896119904119906120585119906)
(26)
120591119903119888 = 1119872119903 ( 120573119889 minus 1198962119904119903 minus 1205782 sign (119904119903) minus 1205822 119890120573 minus 119863119903119908minus 1205823119890120573 minus W119879119903H119903 (x119903) minus 120576119873 sign (119904119903) + 119896119904119903120585119903)
(27)
where 1198961 1198962 1205781 1205782 119896119904119906 and 119896119904119903 are positive constants W119906 =W119906 +Wlowast119906 W119903 = W119903 +Wlowast119903 and119863119903119908 is defined in (20)
And the adaptive laws areW120603 = 120574120603119904120603H120603 (x120603) 120603 = 119906 119903 (28)
where 120574120603 is the adaptive coefficient
34 Stability Analysis
Theorem 5 If the state tracking errors (17) the desired states(15) the auxiliary dynamic system (25) the surge control law120591119906119888 (26) the yaw control law 120591119903119888 (27) and the adaptive laws(28) are applied to the hovercraft system represented by (6)and for any bounded initial condition the closed-loop controlsystem signals 119904119906 119904119903 120585119906 120585119903 119890119906 119890120573 119909119890 119910119890 and 119908119894119906 and 119908119894119903119894 = 1 2 119899 are uniformly ultimately bounded (UUB)The position and desired state tracking errors can be madearbitrarily small by appropriately selecting design parametersAnd the yaw motion will remain bounded
Proof The following Lyapunov function is defined
119881 = 121199042119906 + 121199042119903 + 12120574119906 W119879119906W119906 +12120574119903 W119879119903 W119903 +
121205852119906+ 121205852119903
(29)
From (19) (20) and (29) can be expressed as
= 119904119906 (119891119906 (x119906) + 119872119906120591119906 +119872119906119877119906 minus 119889 + 1205821119890119906)+ 119904119903 (119891119903 (x119903) + 119872119903120591119903 + 119863119903119908 minus 120573119889 + 1205822 119890120573 + 1205823119890120573)+ W119879119906
W119906120574119906 + W119879119903W119903120574119903 + 120585119906 120585119906 + 120585119903 120585119903
(30)
By defining 120591120603 = Δ120591120603 + 120591120603119888 and 120603 = 119906 119903 and using (22) (26)and (27) we have
= minus11989611199042119906 minus 1205781 10038161003816100381610038161199041199061003816100381610038161003816 + 119904119906 (minusW119879119906H119906 (x119906)) +119872119906119904119906Δ120591119906+ 119896119904119906119904119906120585119906 + (minus120576119873 10038161003816100381610038161199041199061003816100381610038161003816 + 120576119906119904119906) minus 11989621199042119903 minus 1205782 10038161003816100381610038161199041199031003816100381610038161003816+ 119904119903 (minusW119879119903H119903 (x119903)) +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903+ (minus120576119873 10038161003816100381610038161199041199031003816100381610038161003816 + 120576119903119904119903) + 1120574119906 W119879119906 W119906 + 1120574119903 W119879119903 W119903+ 120585119906 120585119906 + 120585119903 120585119903
(31)
By using |120576120603| le 120576119873 and 120603 = 119906 119903 (31) can be rewritten as
le minus11989611199042119906 minus 1205781 10038161003816100381610038161199041199061003816100381610038161003816 + 119904119906 (minusW119879119906H119906 (x119906)) minus 11989621199042119903minus 1205782 10038161003816100381610038161199041199031003816100381610038161003816 + 119904119903 (minusW119879119903H119903 (x119903)) + 119872119906119904119906Δ120591119906+ 119896119904119906119904119906120585119906 +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 1120574119906 W119879119906 W119906+ 1120574119903 W119879119903 W119903 + 120585119906 120585119906 + 120585119903 120585119903
(32)
Substituting adaptive laws (28) into it yields
le minus11989611199042119906 +119872119906119904119906Δ120591119906 + 119896119904119906119904119906120585119906 + 120585119906 120585119906 minus 11989621199042119903+119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 120585119903 120585119903 (33)
The process of stability analysis is respectively discussed inthe following two cases
Case 1When 120585120603 ge 120590 in light of (25) andYoungrsquos inequalitywe have
120585120603 120585120603 = 120585120603Δ120581120603 minus 11989612058512060311205852120603 minus 1198961205851206032Δ1205812120603minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
le 1198961205851206032Δ1205812120603 + 141198961205851206032 1205852120603 minus 11989612058512060311205852120603 minus 1198961205851206032Δ1205812120603minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
= minus(1198961205851206031 minus 141198961205851206032)1205852120603 minus 10038161003816100381610038161198721206031199041206031003816100381610038161003816 1003816100381610038161003816120591120603 minus 1205911206031198881003816100381610038161003816
(34)
Substituting 120591120603 = Δ120591120603 + 120591120603119888 and (34) into (33) and usingYoungrsquos inequality yield
le minus11989611199042119906 + 119896119904119906119904119906120585119906 minus (1198961205851199061 minus 141198961205851199062)1205852119906 minus 11989621199042119903+ 119896119904119903119904119903120585119903 minus (1198961205851199031 minus 141198961205851199032)1205852119903
le minus (1198961 minus 05) 1199042119906 minus (1198961205851199061 minus 141198961205851199062 minus 051198962119904119906)1205852119906minus (1198962 minus 05) 1199042119903 minus (1198961205851199031 minus 141198961205851199032 minus 051198962119904119903)1205852119903minus 1198961W119879119906W119906120574119906 minus 1198962W119879119903 W119903120574119903 + 1198961W119879119906W119906120574119906+ 1198962W119879119903 W119903120574119903 le minus21205831119881 + 1205881
(35)
6 Mathematical Problems in Engineering
where
1205831 = min(1198961 minus 05) (1198961205851199061 minus 141198961205851199062 minus 051198962119904119906) (1198962 minus 05) (1198961205851199031 minus 141198961205851199032 minus 051198962119904119903)
1205881 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903
(36)
Case 2 When 120585120603 lt 120590 in light of (25) and Youngrsquos ine-quality we have
120585120603 120585120603 = 120585120603deadzone (Δ120581120603 120590) le 051205852120603 + 05Δ12058121206030511989621199041206031205852120603 = 11989621199041206031205852120603 minus 0511989621199041206031205852120603 le minus0511989621199041206031205852120603 + 11989621199041206031205902 (37)
Substituting (37) into (33) and using Youngrsquos inequality yield
le minus11989611199042119906 +119872119906119904119906Δ120591119906 + 119896119904119906119904119906120585119906 + 051205852119906 + 05Δ1205812119906minus 11989621199042119903 +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 + 051198722119906Δ1205912119906+ 051198722119903Δ1205912119903 + 0511989621199041199061205852119906 + 0511989621199041199031205852119903 + 051205852119906+ 05Δ1205812119906 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 minus 051205852119906 (1198962119904119906 minus 1)minus 051205852119903 (1198962119904119903 minus 1) + 051198722119906Δ1205912119906 + 051198722119903Δ1205912119903+ 11989621199041199061205902 + 11989621199041199031205902 + 05Δ1205812119906 + 05Δ1205812119903
le minus21205832119881 + 1205882
(38)
where
1205832 = min (1198961 minus 1) (051198962119904119906 minus 05) (1198962 minus 1) (051198962119904119903 minus 05)
1205882 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903 + 051198722119906Δ1205912119906
+ 051198722119903Δ1205912119903 + (1198962119904119906 + 1198962119904119903) 1205902 + (Δ1205812119906 + Δ1205812119903)2
(39)
Synthesizing (35) and (38) we have
le minus2120583119881 + 120588 (40)
where 120583 = min1205831 1205832 and 120588 = max1205881 1205882 with the designparameters 1198961 1198962 119896119904119906 119896119904119903 1198961205851199061 1198961205851199062 1198961205851199031 and 1198961205851199032 satisfying
1198961 gt 11198962 gt 1119896119904119906 gt 1119896119904119903 gt 1
1198961205851199061 minus 141198961205851199062 minus 051198962119904119906 gt 01198961205851199031 minus 141198961205851199032 minus 051198962119904119903 gt 0
(41)
Solving (40) we have
0 le 119881 (119905) le 1205882120583 + [119881 (0) minus 1205882120583] 119890minus2120583119905 (42)
It is obviously seen that 119881(119905) is UUB for all 119881(0) le 1198610 with1198610 being any positive constant Therefore in the light of (29)we know that 119904119906 119904119903 120585119906 120585119903 and 119908119894119906 and 119908119894119903 119894 = 1 2 119899 areUUB for all 119881(0) le 1198610 It can be expressed as
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 + 2 [119881 (0) minus 1205882120583] 119890minus2120583119905 (43)
where 120594 = 119904119906 119904119903 120585119906 120585119903 119908119894119906 119908119894119903It implies that there exists 119879 gt 0 such that for all 119905 gt 119879
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 (44)
where radic120588120583 can be made arbitrarily small by appropriatelyselecting the design parameters
Further the following dynamics are obtained from (18)and (44)
119890119906 + 1205821 int1199050119890119906 119889120591 = 120576119906
119890120573 + 1205822119890120573 + 1205823 int1199050119890120573 119889120591 = 120576120573
(45)
where 120576119906 and 120576120573 are arbitrarily small errorsTo prove that 119890119906 and 119890120573 are UUB the Lyapunov function
is given by
119881119904 = 12 (int1199050119890119906 119889120591)2 + 121198902120573 (46)
The time derivative of this Lyapunov function along thedynamics in (45) is such that
119904 = 119890119906 int1199050119890119906 119889120591 + 119890120573 119890120573
= minus1205821 (int1199050119890119906 119889120591)2 minus 12058221198902120573 + 120576119906 int119905
0119890119906 119889120591
minus 1205823119890120573 int1199050119890120573 119889120591 + 120576120573119890120573
(47)
Mathematical Problems in Engineering 7
From Youngrsquos inequality the following fact is obtained
120576119906 int1199050119890119906 119889120591 le 051205821 (int119905
0119890119906 119889120591)2 + 05 12057621199061205821
minus1205823119890120573 int1199050119890120573 119889120591 le 120582211989021205732 + 120582321205822 (int
119905
0119890120573 119889120591)2
120576120573119890120573 le 02512058221198902120573 + 12057621205731205822
(48)
Then
119904 le minus051205821 (int1199050119890119906 119889120591)2 minus 02512058221198902120573 + 05 12057621199061205821
+ 0512058231205822 (int119905
0119890120573 119889120591)2 + 12057621205731205822 = minus2120583119904119881119904 + 120588119904
(49)
where
120583119904 = min 051205821 0251205822 120588119904 = 05 12057621199061205821 + 0512058231205822 (int
119905
0119890120573 119889120591)2 + 12057621205731205822
(50)
Similar to the analysis in (42)sim(44) there exists 119879119904 gt 0 suchthat for all 119905 gt 119879119904
10038171003817100381710038171205941199041003817100381710038171003817 le radic 120588119904120583119904 (51)
where 120594119904 = int1199050119890119906 119889120591 119890120573
It is obvious that int1199050119890119906 119889120591 and 119890120573 are UUB and will be
arbitrarily small by choosing suitable parametersFrom 119904119906 = 119890119906 + 1205821 int1199050 119890119906 119889120591 we know 119890119906 will be arbitrarily
small From (6) and (15) we have
[ 119906 minus 119906119889119906 tan120573 minus 119906119889 tan120573119889] = 119877[119890 + 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)119910119890 + 119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890)] (52)
where
119877 = [ cos120595 sin120595minus sin120595 cos120595] (53)
It is clear that |119877| = 1 which indicates that it is nonsingularThen we have
119890 = minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887119910119890 = minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887 (54)
where 120576119909119887 120576119910119887 are arbitrarily small errorsFurthermore consider the following Lyapunov function
candidate
119881119901 = 121199092119890 + 121199102119890 (55)
The time derivative of 119881119901 along the dynamics in (54) is suchthat
119901 = 119909119890119890 + 119910119890 119910119890= 119909119890 (minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887)
+ 119910119890 (minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887)= minus119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 119909119890120576119909119887 + 119910119890120576119910119887le minus05119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 05119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 12057621199091198872119896119909 +
12057621199101198872119896119910= minus2120583119901119881119901 + 120588119901
(56)
where
120583119901 = min 05119896119909 05119896119910 120588119901 = 12057621199091198872119896119909 +
12057621199101198872119896119910 (57)
Also similar to the analysis in (42)sim(44) there exists 119879119901 gt 0such that for all 119905 gt 119879119901
1003817100381710038171003817100381712059411990110038171003817100381710038171003817 le radic 120588119901120583119901 (58)
where 120594119901 = 119909119890 119910119890From the reason that 120576119909119887 and 120576119910119887 are arbitrarily small
errors we know that 119909119890 and 119910119890 are UUB andwill be arbitrarilysmall
Also 119906 is continuously differentiable in the movingprocess of hovercraft Hence is bounded From (6) we have
= 119906119903 tan (120573) + 119877119906 + 120591119906119898 (59)
From (15) Remark 4 and the boundedness of 119909119890 and 119910119890we have that 119906119889 and 120573119889 are bounded Then 119906 V and 120573 arebounded from (5) and (17) Furthermore119877119906 is bounded from(3)Therefore it can be concluded that 119903will remain boundedfrom (7) and (59) This concludes the proof
Remark 6 In order to avoid the well-known chatteringproblem the sign function used in the control laws (26) and(27) can be replaced by hyperbolic tangent function whichis continuous such that sign(119904) = tanh(119896119867119904) where 119896119867 isa positive scalar which can be chosen to get a very goodapproximation
4 Simulations
Two different cases are implemented to verify the effective-ness and superiority of the proposed controller In simula-tions the main particulars and constraints of hovercraft areshown in Tables 1 and 2 The water surface is calm withoutwaves
8 Mathematical Problems in Engineering
Table 1 Main particulars of hovercraft
119898 (kg) 40000119869119911 (kgm2) 18 times 106119878PP (m2) 45119878LP (m2) 93119878HP (m2) 260119876 (m3s) 1408119881119908 (knots) 10119897sk (m) 65119861119888 (m) 89119897119888 (m) 236ℎ (m) 1119867hov (m) 59119878119888 (m2) 212120573119908 (deg) 45
Table 2 State and input constraints
Variable Maximum Minimum119906 (knots) 40 25120573 (deg) 8 minus8120591119906 (N) 80000 minus80000120591119903 (Nm) 1 times 105 minus1 times 105
The comparisons of three different methods are carriedout in each case The legend ldquoMethod Ardquo means the methodin [14] the legend ldquoMethod Brdquo means the method withoutstate and input saturation constraints
Saturation coefficients are
119896119904119906 = 121198961199081199061 = 11198961199081199062 = 9030001198961205851199061 = 31119896119904119903 = 12
1198961199081199031 = 11198961199081199032 = 6503001198961205851199031 = 381198961205851199062 = 0081198961205851199032 = 008
120590 = 3120590 = 1
(60)
Case 1 The reference trajectory parameters are
119909119889 (0) = 300m119910119889 (0) = 100m
120595119889 (0) = 45∘119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 0
(61)
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 70∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(62)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 111205781 = 0251205821 = 21198962 = 121205782 = 051205822 = 301205823 = 6120576119873 = 05
(63)
It is observed from Figures 3ndash8 that the proposed trajectorytracking controller is effective Tracking errors of all threemethods converge to arbitrarily small values and headingangle is bounded From the comparisons with Methods Aand B in Figures 9 and 10 the proposed controller canlimit the surge speed and the drift angle into the safe rangeeffectively Figures 11 and 12 show that the input saturation isalso handled by the proposed controller
Case 2 The reference trajectory parameters are
119909119889 (0) = minus200m119910119889 (0) = 50m120595119889 (0) = 45∘
119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 01∘s
(64)
Mathematical Problems in Engineering 9
0 2000 4000 6000 8000 10000 12000
0
5000
10000
15000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
x(m
)
y (m)
Figure 3 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 4 The heading angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
500
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus500
minus1000
Figure 5 The surge position tracking error
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 30∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(65)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 153
0 200 400 600 800 1000 1200 1400
0
500
1000
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
minus500
Figure 6 The sway position tracking error
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus10
minus20
Figure 7 The surge velocity tracking error
0 200 400 600 800 1000 1200 1400
0
5
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus10
minus5
Figure 8 The drift angle tracking error
1205781 = 011205821 = 41198962 = 121205782 = 051205822 = 301205823 = 2120576119873 = 05
(66)
It is obvious from Figures 14ndash19 that only the proposedcontroller can track the trajectory successfully Figures 20 and21 show that the surge speed changes to negative values andthe drift angle exceeds the safety limit in the tracking control
10 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controller
u (k
nots)
Method AMethod B
Figure 9 The surge velocity of hovercraft
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
minus10
minus20
Figure 10 The drift angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
2
4
Time (s)
0 50 1000
5
10
Proposed controllerMethod AMethod B
uc
(N)
times105
times104
Figure 11 The surge control law of hovercraft
process ofMethods A and BThese are absolutely not allowedfor hovercraft In contrast you can see from Figure 20 thatsurge speed can still be positive even large enough to avoidthe influence of the resistance humps under the control of theproposed controller And drift angle is also within the safetylimit from Figure 21 From Figures 22 and 23 we know thatthe input constraint ability of the proposed controller is effec-tive Figures 13 and 24 show the heave position of hovercraft
5 Conclusion
A safety-guaranteed trajectory tracking controller has beenproposed for underactuated hovercraft in this paper Thesafety-guaranteed auxiliary dynamic system is designed todeal with state and input constraints The velocity of hov-ercraft is constrained to eliminate the effect of resistance
0 200 400 600 800 1000 1200 1400minus4
0
Time (s)
0 10 20minus15minus10minus5
05
Proposed controllerMethod AMethod B
rc
(Nm
)
times105
times104
minus2
Figure 12 The yaw control law of hovercraft
0 200 400 600 800 1000 1200 1400minus245
minus24
minus235
Time (s)z
(m)
Proposed controllerMethod AMethod B
Figure 13 The heave position of hovercraft
minus2000 0 2000 4000 6000 8000 10000 12000minus1000
0
1000
2000
3000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
minus200minus100
0100
x(m
)
y (m)
minus20 0 20 40 60 80
Figure 14 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 12000
50
100
150
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 15 The heading angle of hovercraft
hump and obtain better stability The safety limit of driftangle is carried out effectively for safety in the high-speedtrajectory tracking process of hovercraftThe input saturationis handled High nonlinearity and model uncertainties areapproximated by RBFNNs Simulation results have been
Mathematical Problems in Engineering 11
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus100
Figure 16 The surge position tracking error
0 200 400 600 800 1000 1200minus600
minus400
minus200
0
200
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
Figure 17 The sway position tracking error
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus100
Figure 18 The surge velocity tracking error of hovercraft
0 200 400 600 800 1000 1200
0
2
4
6
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus2
Figure 19 The drift angle tracking error of hovercraft
0 200 400 600 800 1000 1200
0
50
Time (s)
0 100 200
0
50
Proposed controllerMethod AMethod B
u(k
nots)
minus50minus50
minus100
Figure 20 The surge velocity of hovercraft
0 200 400 600 800 1000 1200
0
100
200
Time (s)
0 100 2000
20
40
Proposed controllerMethod AMethod B
(d
eg)
minus100
Figure 21 The drift angle of hovercraft
0 200 400 600 800 1000 1200
0
Time (s)
0 50 100
0
Proposed controllerMethod AMethod B
uc
(N)
minus5
minus10
times105
times104
minus5
minus10
Figure 22 The surge control law of hovercraft
0 200 400 600 800 1000 1200
0
10
20
Time (s)
0 10 20
05
1015
Proposed controllerMethod AMethod B
rc
(Nm
)
minus5
times109
times104
Figure 23 The yaw control law of hovercraft
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
where
1205831 = min(1198961 minus 05) (1198961205851199061 minus 141198961205851199062 minus 051198962119904119906) (1198962 minus 05) (1198961205851199031 minus 141198961205851199032 minus 051198962119904119903)
1205881 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903
(36)
Case 2 When 120585120603 lt 120590 in light of (25) and Youngrsquos ine-quality we have
120585120603 120585120603 = 120585120603deadzone (Δ120581120603 120590) le 051205852120603 + 05Δ12058121206030511989621199041206031205852120603 = 11989621199041206031205852120603 minus 0511989621199041206031205852120603 le minus0511989621199041206031205852120603 + 11989621199041206031205902 (37)
Substituting (37) into (33) and using Youngrsquos inequality yield
le minus11989611199042119906 +119872119906119904119906Δ120591119906 + 119896119904119906119904119906120585119906 + 051205852119906 + 05Δ1205812119906minus 11989621199042119903 +119872119903119904119903Δ120591119903 + 119896119904119903119904119903120585119903 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 + 051198722119906Δ1205912119906+ 051198722119903Δ1205912119903 + 0511989621199041199061205852119906 + 0511989621199041199031205852119903 + 051205852119906+ 05Δ1205812119906 + 051205852119903 + 05Δ1205812119903
le minus (1198961 minus 1) 1199042119906 minus (1198962 minus 1) 1199042119903 minus 051205852119906 (1198962119904119906 minus 1)minus 051205852119903 (1198962119904119903 minus 1) + 051198722119906Δ1205912119906 + 051198722119903Δ1205912119903+ 11989621199041199061205902 + 11989621199041199031205902 + 05Δ1205812119906 + 05Δ1205812119903
le minus21205832119881 + 1205882
(38)
where
1205832 = min (1198961 minus 1) (051198962119904119906 minus 05) (1198962 minus 1) (051198962119904119903 minus 05)
1205882 = 1198961120574119906 W119879119906W119906 +1198962120574119903 W119879119903 W119903 + 051198722119906Δ1205912119906
+ 051198722119903Δ1205912119903 + (1198962119904119906 + 1198962119904119903) 1205902 + (Δ1205812119906 + Δ1205812119903)2
(39)
Synthesizing (35) and (38) we have
le minus2120583119881 + 120588 (40)
where 120583 = min1205831 1205832 and 120588 = max1205881 1205882 with the designparameters 1198961 1198962 119896119904119906 119896119904119903 1198961205851199061 1198961205851199062 1198961205851199031 and 1198961205851199032 satisfying
1198961 gt 11198962 gt 1119896119904119906 gt 1119896119904119903 gt 1
1198961205851199061 minus 141198961205851199062 minus 051198962119904119906 gt 01198961205851199031 minus 141198961205851199032 minus 051198962119904119903 gt 0
(41)
Solving (40) we have
0 le 119881 (119905) le 1205882120583 + [119881 (0) minus 1205882120583] 119890minus2120583119905 (42)
It is obviously seen that 119881(119905) is UUB for all 119881(0) le 1198610 with1198610 being any positive constant Therefore in the light of (29)we know that 119904119906 119904119903 120585119906 120585119903 and 119908119894119906 and 119908119894119903 119894 = 1 2 119899 areUUB for all 119881(0) le 1198610 It can be expressed as
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 + 2 [119881 (0) minus 1205882120583] 119890minus2120583119905 (43)
where 120594 = 119904119906 119904119903 120585119906 120585119903 119908119894119906 119908119894119903It implies that there exists 119879 gt 0 such that for all 119905 gt 119879
10038171003817100381710038171205941003817100381710038171003817 le radic 120588120583 (44)
where radic120588120583 can be made arbitrarily small by appropriatelyselecting the design parameters
Further the following dynamics are obtained from (18)and (44)
119890119906 + 1205821 int1199050119890119906 119889120591 = 120576119906
119890120573 + 1205822119890120573 + 1205823 int1199050119890120573 119889120591 = 120576120573
(45)
where 120576119906 and 120576120573 are arbitrarily small errorsTo prove that 119890119906 and 119890120573 are UUB the Lyapunov function
is given by
119881119904 = 12 (int1199050119890119906 119889120591)2 + 121198902120573 (46)
The time derivative of this Lyapunov function along thedynamics in (45) is such that
119904 = 119890119906 int1199050119890119906 119889120591 + 119890120573 119890120573
= minus1205821 (int1199050119890119906 119889120591)2 minus 12058221198902120573 + 120576119906 int119905
0119890119906 119889120591
minus 1205823119890120573 int1199050119890120573 119889120591 + 120576120573119890120573
(47)
Mathematical Problems in Engineering 7
From Youngrsquos inequality the following fact is obtained
120576119906 int1199050119890119906 119889120591 le 051205821 (int119905
0119890119906 119889120591)2 + 05 12057621199061205821
minus1205823119890120573 int1199050119890120573 119889120591 le 120582211989021205732 + 120582321205822 (int
119905
0119890120573 119889120591)2
120576120573119890120573 le 02512058221198902120573 + 12057621205731205822
(48)
Then
119904 le minus051205821 (int1199050119890119906 119889120591)2 minus 02512058221198902120573 + 05 12057621199061205821
+ 0512058231205822 (int119905
0119890120573 119889120591)2 + 12057621205731205822 = minus2120583119904119881119904 + 120588119904
(49)
where
120583119904 = min 051205821 0251205822 120588119904 = 05 12057621199061205821 + 0512058231205822 (int
119905
0119890120573 119889120591)2 + 12057621205731205822
(50)
Similar to the analysis in (42)sim(44) there exists 119879119904 gt 0 suchthat for all 119905 gt 119879119904
10038171003817100381710038171205941199041003817100381710038171003817 le radic 120588119904120583119904 (51)
where 120594119904 = int1199050119890119906 119889120591 119890120573
It is obvious that int1199050119890119906 119889120591 and 119890120573 are UUB and will be
arbitrarily small by choosing suitable parametersFrom 119904119906 = 119890119906 + 1205821 int1199050 119890119906 119889120591 we know 119890119906 will be arbitrarily
small From (6) and (15) we have
[ 119906 minus 119906119889119906 tan120573 minus 119906119889 tan120573119889] = 119877[119890 + 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)119910119890 + 119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890)] (52)
where
119877 = [ cos120595 sin120595minus sin120595 cos120595] (53)
It is clear that |119877| = 1 which indicates that it is nonsingularThen we have
119890 = minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887119910119890 = minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887 (54)
where 120576119909119887 120576119910119887 are arbitrarily small errorsFurthermore consider the following Lyapunov function
candidate
119881119901 = 121199092119890 + 121199102119890 (55)
The time derivative of 119881119901 along the dynamics in (54) is suchthat
119901 = 119909119890119890 + 119910119890 119910119890= 119909119890 (minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887)
+ 119910119890 (minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887)= minus119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 119909119890120576119909119887 + 119910119890120576119910119887le minus05119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 05119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 12057621199091198872119896119909 +
12057621199101198872119896119910= minus2120583119901119881119901 + 120588119901
(56)
where
120583119901 = min 05119896119909 05119896119910 120588119901 = 12057621199091198872119896119909 +
12057621199101198872119896119910 (57)
Also similar to the analysis in (42)sim(44) there exists 119879119901 gt 0such that for all 119905 gt 119879119901
1003817100381710038171003817100381712059411990110038171003817100381710038171003817 le radic 120588119901120583119901 (58)
where 120594119901 = 119909119890 119910119890From the reason that 120576119909119887 and 120576119910119887 are arbitrarily small
errors we know that 119909119890 and 119910119890 are UUB andwill be arbitrarilysmall
Also 119906 is continuously differentiable in the movingprocess of hovercraft Hence is bounded From (6) we have
= 119906119903 tan (120573) + 119877119906 + 120591119906119898 (59)
From (15) Remark 4 and the boundedness of 119909119890 and 119910119890we have that 119906119889 and 120573119889 are bounded Then 119906 V and 120573 arebounded from (5) and (17) Furthermore119877119906 is bounded from(3)Therefore it can be concluded that 119903will remain boundedfrom (7) and (59) This concludes the proof
Remark 6 In order to avoid the well-known chatteringproblem the sign function used in the control laws (26) and(27) can be replaced by hyperbolic tangent function whichis continuous such that sign(119904) = tanh(119896119867119904) where 119896119867 isa positive scalar which can be chosen to get a very goodapproximation
4 Simulations
Two different cases are implemented to verify the effective-ness and superiority of the proposed controller In simula-tions the main particulars and constraints of hovercraft areshown in Tables 1 and 2 The water surface is calm withoutwaves
8 Mathematical Problems in Engineering
Table 1 Main particulars of hovercraft
119898 (kg) 40000119869119911 (kgm2) 18 times 106119878PP (m2) 45119878LP (m2) 93119878HP (m2) 260119876 (m3s) 1408119881119908 (knots) 10119897sk (m) 65119861119888 (m) 89119897119888 (m) 236ℎ (m) 1119867hov (m) 59119878119888 (m2) 212120573119908 (deg) 45
Table 2 State and input constraints
Variable Maximum Minimum119906 (knots) 40 25120573 (deg) 8 minus8120591119906 (N) 80000 minus80000120591119903 (Nm) 1 times 105 minus1 times 105
The comparisons of three different methods are carriedout in each case The legend ldquoMethod Ardquo means the methodin [14] the legend ldquoMethod Brdquo means the method withoutstate and input saturation constraints
Saturation coefficients are
119896119904119906 = 121198961199081199061 = 11198961199081199062 = 9030001198961205851199061 = 31119896119904119903 = 12
1198961199081199031 = 11198961199081199032 = 6503001198961205851199031 = 381198961205851199062 = 0081198961205851199032 = 008
120590 = 3120590 = 1
(60)
Case 1 The reference trajectory parameters are
119909119889 (0) = 300m119910119889 (0) = 100m
120595119889 (0) = 45∘119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 0
(61)
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 70∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(62)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 111205781 = 0251205821 = 21198962 = 121205782 = 051205822 = 301205823 = 6120576119873 = 05
(63)
It is observed from Figures 3ndash8 that the proposed trajectorytracking controller is effective Tracking errors of all threemethods converge to arbitrarily small values and headingangle is bounded From the comparisons with Methods Aand B in Figures 9 and 10 the proposed controller canlimit the surge speed and the drift angle into the safe rangeeffectively Figures 11 and 12 show that the input saturation isalso handled by the proposed controller
Case 2 The reference trajectory parameters are
119909119889 (0) = minus200m119910119889 (0) = 50m120595119889 (0) = 45∘
119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 01∘s
(64)
Mathematical Problems in Engineering 9
0 2000 4000 6000 8000 10000 12000
0
5000
10000
15000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
x(m
)
y (m)
Figure 3 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 4 The heading angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
500
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus500
minus1000
Figure 5 The surge position tracking error
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 30∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(65)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 153
0 200 400 600 800 1000 1200 1400
0
500
1000
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
minus500
Figure 6 The sway position tracking error
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus10
minus20
Figure 7 The surge velocity tracking error
0 200 400 600 800 1000 1200 1400
0
5
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus10
minus5
Figure 8 The drift angle tracking error
1205781 = 011205821 = 41198962 = 121205782 = 051205822 = 301205823 = 2120576119873 = 05
(66)
It is obvious from Figures 14ndash19 that only the proposedcontroller can track the trajectory successfully Figures 20 and21 show that the surge speed changes to negative values andthe drift angle exceeds the safety limit in the tracking control
10 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controller
u (k
nots)
Method AMethod B
Figure 9 The surge velocity of hovercraft
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
minus10
minus20
Figure 10 The drift angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
2
4
Time (s)
0 50 1000
5
10
Proposed controllerMethod AMethod B
uc
(N)
times105
times104
Figure 11 The surge control law of hovercraft
process ofMethods A and BThese are absolutely not allowedfor hovercraft In contrast you can see from Figure 20 thatsurge speed can still be positive even large enough to avoidthe influence of the resistance humps under the control of theproposed controller And drift angle is also within the safetylimit from Figure 21 From Figures 22 and 23 we know thatthe input constraint ability of the proposed controller is effec-tive Figures 13 and 24 show the heave position of hovercraft
5 Conclusion
A safety-guaranteed trajectory tracking controller has beenproposed for underactuated hovercraft in this paper Thesafety-guaranteed auxiliary dynamic system is designed todeal with state and input constraints The velocity of hov-ercraft is constrained to eliminate the effect of resistance
0 200 400 600 800 1000 1200 1400minus4
0
Time (s)
0 10 20minus15minus10minus5
05
Proposed controllerMethod AMethod B
rc
(Nm
)
times105
times104
minus2
Figure 12 The yaw control law of hovercraft
0 200 400 600 800 1000 1200 1400minus245
minus24
minus235
Time (s)z
(m)
Proposed controllerMethod AMethod B
Figure 13 The heave position of hovercraft
minus2000 0 2000 4000 6000 8000 10000 12000minus1000
0
1000
2000
3000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
minus200minus100
0100
x(m
)
y (m)
minus20 0 20 40 60 80
Figure 14 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 12000
50
100
150
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 15 The heading angle of hovercraft
hump and obtain better stability The safety limit of driftangle is carried out effectively for safety in the high-speedtrajectory tracking process of hovercraftThe input saturationis handled High nonlinearity and model uncertainties areapproximated by RBFNNs Simulation results have been
Mathematical Problems in Engineering 11
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus100
Figure 16 The surge position tracking error
0 200 400 600 800 1000 1200minus600
minus400
minus200
0
200
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
Figure 17 The sway position tracking error
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus100
Figure 18 The surge velocity tracking error of hovercraft
0 200 400 600 800 1000 1200
0
2
4
6
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus2
Figure 19 The drift angle tracking error of hovercraft
0 200 400 600 800 1000 1200
0
50
Time (s)
0 100 200
0
50
Proposed controllerMethod AMethod B
u(k
nots)
minus50minus50
minus100
Figure 20 The surge velocity of hovercraft
0 200 400 600 800 1000 1200
0
100
200
Time (s)
0 100 2000
20
40
Proposed controllerMethod AMethod B
(d
eg)
minus100
Figure 21 The drift angle of hovercraft
0 200 400 600 800 1000 1200
0
Time (s)
0 50 100
0
Proposed controllerMethod AMethod B
uc
(N)
minus5
minus10
times105
times104
minus5
minus10
Figure 22 The surge control law of hovercraft
0 200 400 600 800 1000 1200
0
10
20
Time (s)
0 10 20
05
1015
Proposed controllerMethod AMethod B
rc
(Nm
)
minus5
times109
times104
Figure 23 The yaw control law of hovercraft
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
From Youngrsquos inequality the following fact is obtained
120576119906 int1199050119890119906 119889120591 le 051205821 (int119905
0119890119906 119889120591)2 + 05 12057621199061205821
minus1205823119890120573 int1199050119890120573 119889120591 le 120582211989021205732 + 120582321205822 (int
119905
0119890120573 119889120591)2
120576120573119890120573 le 02512058221198902120573 + 12057621205731205822
(48)
Then
119904 le minus051205821 (int1199050119890119906 119889120591)2 minus 02512058221198902120573 + 05 12057621199061205821
+ 0512058231205822 (int119905
0119890120573 119889120591)2 + 12057621205731205822 = minus2120583119904119881119904 + 120588119904
(49)
where
120583119904 = min 051205821 0251205822 120588119904 = 05 12057621199061205821 + 0512058231205822 (int
119905
0119890120573 119889120591)2 + 12057621205731205822
(50)
Similar to the analysis in (42)sim(44) there exists 119879119904 gt 0 suchthat for all 119905 gt 119879119904
10038171003817100381710038171205941199041003817100381710038171003817 le radic 120588119904120583119904 (51)
where 120594119904 = int1199050119890119906 119889120591 119890120573
It is obvious that int1199050119890119906 119889120591 and 119890120573 are UUB and will be
arbitrarily small by choosing suitable parametersFrom 119904119906 = 119890119906 + 1205821 int1199050 119890119906 119889120591 we know 119890119906 will be arbitrarily
small From (6) and (15) we have
[ 119906 minus 119906119889119906 tan120573 minus 119906119889 tan120573119889] = 119877[119890 + 119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890)119910119890 + 119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890)] (52)
where
119877 = [ cos120595 sin120595minus sin120595 cos120595] (53)
It is clear that |119877| = 1 which indicates that it is nonsingularThen we have
119890 = minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887119910119890 = minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887 (54)
where 120576119909119887 120576119910119887 are arbitrarily small errorsFurthermore consider the following Lyapunov function
candidate
119881119901 = 121199092119890 + 121199102119890 (55)
The time derivative of 119881119901 along the dynamics in (54) is suchthat
119901 = 119909119890119890 + 119910119890 119910119890= 119909119890 (minus119896119909 10038161003816100381610038161199091198901003816100381610038161003816 sign (119909119890) + 120576119909119887)
+ 119910119890 (minus119896119910 10038161003816100381610038161199101198901003816100381610038161003816 sign (119910119890) + 120576119910119887)= minus119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 119909119890120576119909119887 + 119910119890120576119910119887le minus05119896119909 100381610038161003816100381611990911989010038161003816100381610038162 minus 05119896119910 100381610038161003816100381611991011989010038161003816100381610038162 + 12057621199091198872119896119909 +
12057621199101198872119896119910= minus2120583119901119881119901 + 120588119901
(56)
where
120583119901 = min 05119896119909 05119896119910 120588119901 = 12057621199091198872119896119909 +
12057621199101198872119896119910 (57)
Also similar to the analysis in (42)sim(44) there exists 119879119901 gt 0such that for all 119905 gt 119879119901
1003817100381710038171003817100381712059411990110038171003817100381710038171003817 le radic 120588119901120583119901 (58)
where 120594119901 = 119909119890 119910119890From the reason that 120576119909119887 and 120576119910119887 are arbitrarily small
errors we know that 119909119890 and 119910119890 are UUB andwill be arbitrarilysmall
Also 119906 is continuously differentiable in the movingprocess of hovercraft Hence is bounded From (6) we have
= 119906119903 tan (120573) + 119877119906 + 120591119906119898 (59)
From (15) Remark 4 and the boundedness of 119909119890 and 119910119890we have that 119906119889 and 120573119889 are bounded Then 119906 V and 120573 arebounded from (5) and (17) Furthermore119877119906 is bounded from(3)Therefore it can be concluded that 119903will remain boundedfrom (7) and (59) This concludes the proof
Remark 6 In order to avoid the well-known chatteringproblem the sign function used in the control laws (26) and(27) can be replaced by hyperbolic tangent function whichis continuous such that sign(119904) = tanh(119896119867119904) where 119896119867 isa positive scalar which can be chosen to get a very goodapproximation
4 Simulations
Two different cases are implemented to verify the effective-ness and superiority of the proposed controller In simula-tions the main particulars and constraints of hovercraft areshown in Tables 1 and 2 The water surface is calm withoutwaves
8 Mathematical Problems in Engineering
Table 1 Main particulars of hovercraft
119898 (kg) 40000119869119911 (kgm2) 18 times 106119878PP (m2) 45119878LP (m2) 93119878HP (m2) 260119876 (m3s) 1408119881119908 (knots) 10119897sk (m) 65119861119888 (m) 89119897119888 (m) 236ℎ (m) 1119867hov (m) 59119878119888 (m2) 212120573119908 (deg) 45
Table 2 State and input constraints
Variable Maximum Minimum119906 (knots) 40 25120573 (deg) 8 minus8120591119906 (N) 80000 minus80000120591119903 (Nm) 1 times 105 minus1 times 105
The comparisons of three different methods are carriedout in each case The legend ldquoMethod Ardquo means the methodin [14] the legend ldquoMethod Brdquo means the method withoutstate and input saturation constraints
Saturation coefficients are
119896119904119906 = 121198961199081199061 = 11198961199081199062 = 9030001198961205851199061 = 31119896119904119903 = 12
1198961199081199031 = 11198961199081199032 = 6503001198961205851199031 = 381198961205851199062 = 0081198961205851199032 = 008
120590 = 3120590 = 1
(60)
Case 1 The reference trajectory parameters are
119909119889 (0) = 300m119910119889 (0) = 100m
120595119889 (0) = 45∘119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 0
(61)
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 70∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(62)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 111205781 = 0251205821 = 21198962 = 121205782 = 051205822 = 301205823 = 6120576119873 = 05
(63)
It is observed from Figures 3ndash8 that the proposed trajectorytracking controller is effective Tracking errors of all threemethods converge to arbitrarily small values and headingangle is bounded From the comparisons with Methods Aand B in Figures 9 and 10 the proposed controller canlimit the surge speed and the drift angle into the safe rangeeffectively Figures 11 and 12 show that the input saturation isalso handled by the proposed controller
Case 2 The reference trajectory parameters are
119909119889 (0) = minus200m119910119889 (0) = 50m120595119889 (0) = 45∘
119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 01∘s
(64)
Mathematical Problems in Engineering 9
0 2000 4000 6000 8000 10000 12000
0
5000
10000
15000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
x(m
)
y (m)
Figure 3 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 4 The heading angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
500
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus500
minus1000
Figure 5 The surge position tracking error
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 30∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(65)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 153
0 200 400 600 800 1000 1200 1400
0
500
1000
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
minus500
Figure 6 The sway position tracking error
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus10
minus20
Figure 7 The surge velocity tracking error
0 200 400 600 800 1000 1200 1400
0
5
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus10
minus5
Figure 8 The drift angle tracking error
1205781 = 011205821 = 41198962 = 121205782 = 051205822 = 301205823 = 2120576119873 = 05
(66)
It is obvious from Figures 14ndash19 that only the proposedcontroller can track the trajectory successfully Figures 20 and21 show that the surge speed changes to negative values andthe drift angle exceeds the safety limit in the tracking control
10 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controller
u (k
nots)
Method AMethod B
Figure 9 The surge velocity of hovercraft
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
minus10
minus20
Figure 10 The drift angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
2
4
Time (s)
0 50 1000
5
10
Proposed controllerMethod AMethod B
uc
(N)
times105
times104
Figure 11 The surge control law of hovercraft
process ofMethods A and BThese are absolutely not allowedfor hovercraft In contrast you can see from Figure 20 thatsurge speed can still be positive even large enough to avoidthe influence of the resistance humps under the control of theproposed controller And drift angle is also within the safetylimit from Figure 21 From Figures 22 and 23 we know thatthe input constraint ability of the proposed controller is effec-tive Figures 13 and 24 show the heave position of hovercraft
5 Conclusion
A safety-guaranteed trajectory tracking controller has beenproposed for underactuated hovercraft in this paper Thesafety-guaranteed auxiliary dynamic system is designed todeal with state and input constraints The velocity of hov-ercraft is constrained to eliminate the effect of resistance
0 200 400 600 800 1000 1200 1400minus4
0
Time (s)
0 10 20minus15minus10minus5
05
Proposed controllerMethod AMethod B
rc
(Nm
)
times105
times104
minus2
Figure 12 The yaw control law of hovercraft
0 200 400 600 800 1000 1200 1400minus245
minus24
minus235
Time (s)z
(m)
Proposed controllerMethod AMethod B
Figure 13 The heave position of hovercraft
minus2000 0 2000 4000 6000 8000 10000 12000minus1000
0
1000
2000
3000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
minus200minus100
0100
x(m
)
y (m)
minus20 0 20 40 60 80
Figure 14 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 12000
50
100
150
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 15 The heading angle of hovercraft
hump and obtain better stability The safety limit of driftangle is carried out effectively for safety in the high-speedtrajectory tracking process of hovercraftThe input saturationis handled High nonlinearity and model uncertainties areapproximated by RBFNNs Simulation results have been
Mathematical Problems in Engineering 11
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus100
Figure 16 The surge position tracking error
0 200 400 600 800 1000 1200minus600
minus400
minus200
0
200
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
Figure 17 The sway position tracking error
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus100
Figure 18 The surge velocity tracking error of hovercraft
0 200 400 600 800 1000 1200
0
2
4
6
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus2
Figure 19 The drift angle tracking error of hovercraft
0 200 400 600 800 1000 1200
0
50
Time (s)
0 100 200
0
50
Proposed controllerMethod AMethod B
u(k
nots)
minus50minus50
minus100
Figure 20 The surge velocity of hovercraft
0 200 400 600 800 1000 1200
0
100
200
Time (s)
0 100 2000
20
40
Proposed controllerMethod AMethod B
(d
eg)
minus100
Figure 21 The drift angle of hovercraft
0 200 400 600 800 1000 1200
0
Time (s)
0 50 100
0
Proposed controllerMethod AMethod B
uc
(N)
minus5
minus10
times105
times104
minus5
minus10
Figure 22 The surge control law of hovercraft
0 200 400 600 800 1000 1200
0
10
20
Time (s)
0 10 20
05
1015
Proposed controllerMethod AMethod B
rc
(Nm
)
minus5
times109
times104
Figure 23 The yaw control law of hovercraft
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
Table 1 Main particulars of hovercraft
119898 (kg) 40000119869119911 (kgm2) 18 times 106119878PP (m2) 45119878LP (m2) 93119878HP (m2) 260119876 (m3s) 1408119881119908 (knots) 10119897sk (m) 65119861119888 (m) 89119897119888 (m) 236ℎ (m) 1119867hov (m) 59119878119888 (m2) 212120573119908 (deg) 45
Table 2 State and input constraints
Variable Maximum Minimum119906 (knots) 40 25120573 (deg) 8 minus8120591119906 (N) 80000 minus80000120591119903 (Nm) 1 times 105 minus1 times 105
The comparisons of three different methods are carriedout in each case The legend ldquoMethod Ardquo means the methodin [14] the legend ldquoMethod Brdquo means the method withoutstate and input saturation constraints
Saturation coefficients are
119896119904119906 = 121198961199081199061 = 11198961199081199062 = 9030001198961205851199061 = 31119896119904119903 = 12
1198961199081199031 = 11198961199081199032 = 6503001198961205851199031 = 381198961205851199062 = 0081198961205851199032 = 008
120590 = 3120590 = 1
(60)
Case 1 The reference trajectory parameters are
119909119889 (0) = 300m119910119889 (0) = 100m
120595119889 (0) = 45∘119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 0
(61)
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 70∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(62)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 111205781 = 0251205821 = 21198962 = 121205782 = 051205822 = 301205823 = 6120576119873 = 05
(63)
It is observed from Figures 3ndash8 that the proposed trajectorytracking controller is effective Tracking errors of all threemethods converge to arbitrarily small values and headingangle is bounded From the comparisons with Methods Aand B in Figures 9 and 10 the proposed controller canlimit the surge speed and the drift angle into the safe rangeeffectively Figures 11 and 12 show that the input saturation isalso handled by the proposed controller
Case 2 The reference trajectory parameters are
119909119889 (0) = minus200m119910119889 (0) = 50m120595119889 (0) = 45∘
119906119889set (119905) = 35 knotsV119889set (119905) = 0119903119889set (119905) = 01∘s
(64)
Mathematical Problems in Engineering 9
0 2000 4000 6000 8000 10000 12000
0
5000
10000
15000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
x(m
)
y (m)
Figure 3 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 4 The heading angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
500
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus500
minus1000
Figure 5 The surge position tracking error
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 30∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(65)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 153
0 200 400 600 800 1000 1200 1400
0
500
1000
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
minus500
Figure 6 The sway position tracking error
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus10
minus20
Figure 7 The surge velocity tracking error
0 200 400 600 800 1000 1200 1400
0
5
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus10
minus5
Figure 8 The drift angle tracking error
1205781 = 011205821 = 41198962 = 121205782 = 051205822 = 301205823 = 2120576119873 = 05
(66)
It is obvious from Figures 14ndash19 that only the proposedcontroller can track the trajectory successfully Figures 20 and21 show that the surge speed changes to negative values andthe drift angle exceeds the safety limit in the tracking control
10 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controller
u (k
nots)
Method AMethod B
Figure 9 The surge velocity of hovercraft
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
minus10
minus20
Figure 10 The drift angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
2
4
Time (s)
0 50 1000
5
10
Proposed controllerMethod AMethod B
uc
(N)
times105
times104
Figure 11 The surge control law of hovercraft
process ofMethods A and BThese are absolutely not allowedfor hovercraft In contrast you can see from Figure 20 thatsurge speed can still be positive even large enough to avoidthe influence of the resistance humps under the control of theproposed controller And drift angle is also within the safetylimit from Figure 21 From Figures 22 and 23 we know thatthe input constraint ability of the proposed controller is effec-tive Figures 13 and 24 show the heave position of hovercraft
5 Conclusion
A safety-guaranteed trajectory tracking controller has beenproposed for underactuated hovercraft in this paper Thesafety-guaranteed auxiliary dynamic system is designed todeal with state and input constraints The velocity of hov-ercraft is constrained to eliminate the effect of resistance
0 200 400 600 800 1000 1200 1400minus4
0
Time (s)
0 10 20minus15minus10minus5
05
Proposed controllerMethod AMethod B
rc
(Nm
)
times105
times104
minus2
Figure 12 The yaw control law of hovercraft
0 200 400 600 800 1000 1200 1400minus245
minus24
minus235
Time (s)z
(m)
Proposed controllerMethod AMethod B
Figure 13 The heave position of hovercraft
minus2000 0 2000 4000 6000 8000 10000 12000minus1000
0
1000
2000
3000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
minus200minus100
0100
x(m
)
y (m)
minus20 0 20 40 60 80
Figure 14 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 12000
50
100
150
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 15 The heading angle of hovercraft
hump and obtain better stability The safety limit of driftangle is carried out effectively for safety in the high-speedtrajectory tracking process of hovercraftThe input saturationis handled High nonlinearity and model uncertainties areapproximated by RBFNNs Simulation results have been
Mathematical Problems in Engineering 11
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus100
Figure 16 The surge position tracking error
0 200 400 600 800 1000 1200minus600
minus400
minus200
0
200
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
Figure 17 The sway position tracking error
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus100
Figure 18 The surge velocity tracking error of hovercraft
0 200 400 600 800 1000 1200
0
2
4
6
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus2
Figure 19 The drift angle tracking error of hovercraft
0 200 400 600 800 1000 1200
0
50
Time (s)
0 100 200
0
50
Proposed controllerMethod AMethod B
u(k
nots)
minus50minus50
minus100
Figure 20 The surge velocity of hovercraft
0 200 400 600 800 1000 1200
0
100
200
Time (s)
0 100 2000
20
40
Proposed controllerMethod AMethod B
(d
eg)
minus100
Figure 21 The drift angle of hovercraft
0 200 400 600 800 1000 1200
0
Time (s)
0 50 100
0
Proposed controllerMethod AMethod B
uc
(N)
minus5
minus10
times105
times104
minus5
minus10
Figure 22 The surge control law of hovercraft
0 200 400 600 800 1000 1200
0
10
20
Time (s)
0 10 20
05
1015
Proposed controllerMethod AMethod B
rc
(Nm
)
minus5
times109
times104
Figure 23 The yaw control law of hovercraft
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
0 2000 4000 6000 8000 10000 12000
0
5000
10000
15000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
x(m
)
y (m)
Figure 3 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 4 The heading angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
500
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus500
minus1000
Figure 5 The surge position tracking error
The initial values of hovercraft model are119909 (0) = 0119910 (0) = 0120595 (0) = 30∘119906 (0) = 35 knots120573 (0) = 0119903 (0) = 0
(65)
The controller parameters are
119896119909 = 06119896119910 = 061198961 = 153
0 200 400 600 800 1000 1200 1400
0
500
1000
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
minus500
Figure 6 The sway position tracking error
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus10
minus20
Figure 7 The surge velocity tracking error
0 200 400 600 800 1000 1200 1400
0
5
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus10
minus5
Figure 8 The drift angle tracking error
1205781 = 011205821 = 41198962 = 121205782 = 051205822 = 301205823 = 2120576119873 = 05
(66)
It is obvious from Figures 14ndash19 that only the proposedcontroller can track the trajectory successfully Figures 20 and21 show that the surge speed changes to negative values andthe drift angle exceeds the safety limit in the tracking control
10 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controller
u (k
nots)
Method AMethod B
Figure 9 The surge velocity of hovercraft
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
minus10
minus20
Figure 10 The drift angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
2
4
Time (s)
0 50 1000
5
10
Proposed controllerMethod AMethod B
uc
(N)
times105
times104
Figure 11 The surge control law of hovercraft
process ofMethods A and BThese are absolutely not allowedfor hovercraft In contrast you can see from Figure 20 thatsurge speed can still be positive even large enough to avoidthe influence of the resistance humps under the control of theproposed controller And drift angle is also within the safetylimit from Figure 21 From Figures 22 and 23 we know thatthe input constraint ability of the proposed controller is effec-tive Figures 13 and 24 show the heave position of hovercraft
5 Conclusion
A safety-guaranteed trajectory tracking controller has beenproposed for underactuated hovercraft in this paper Thesafety-guaranteed auxiliary dynamic system is designed todeal with state and input constraints The velocity of hov-ercraft is constrained to eliminate the effect of resistance
0 200 400 600 800 1000 1200 1400minus4
0
Time (s)
0 10 20minus15minus10minus5
05
Proposed controllerMethod AMethod B
rc
(Nm
)
times105
times104
minus2
Figure 12 The yaw control law of hovercraft
0 200 400 600 800 1000 1200 1400minus245
minus24
minus235
Time (s)z
(m)
Proposed controllerMethod AMethod B
Figure 13 The heave position of hovercraft
minus2000 0 2000 4000 6000 8000 10000 12000minus1000
0
1000
2000
3000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
minus200minus100
0100
x(m
)
y (m)
minus20 0 20 40 60 80
Figure 14 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 12000
50
100
150
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 15 The heading angle of hovercraft
hump and obtain better stability The safety limit of driftangle is carried out effectively for safety in the high-speedtrajectory tracking process of hovercraftThe input saturationis handled High nonlinearity and model uncertainties areapproximated by RBFNNs Simulation results have been
Mathematical Problems in Engineering 11
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus100
Figure 16 The surge position tracking error
0 200 400 600 800 1000 1200minus600
minus400
minus200
0
200
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
Figure 17 The sway position tracking error
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus100
Figure 18 The surge velocity tracking error of hovercraft
0 200 400 600 800 1000 1200
0
2
4
6
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus2
Figure 19 The drift angle tracking error of hovercraft
0 200 400 600 800 1000 1200
0
50
Time (s)
0 100 200
0
50
Proposed controllerMethod AMethod B
u(k
nots)
minus50minus50
minus100
Figure 20 The surge velocity of hovercraft
0 200 400 600 800 1000 1200
0
100
200
Time (s)
0 100 2000
20
40
Proposed controllerMethod AMethod B
(d
eg)
minus100
Figure 21 The drift angle of hovercraft
0 200 400 600 800 1000 1200
0
Time (s)
0 50 100
0
Proposed controllerMethod AMethod B
uc
(N)
minus5
minus10
times105
times104
minus5
minus10
Figure 22 The surge control law of hovercraft
0 200 400 600 800 1000 1200
0
10
20
Time (s)
0 10 20
05
1015
Proposed controllerMethod AMethod B
rc
(Nm
)
minus5
times109
times104
Figure 23 The yaw control law of hovercraft
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200 140020
40
60
80
Time (s)
Proposed controller
u (k
nots)
Method AMethod B
Figure 9 The surge velocity of hovercraft
0 200 400 600 800 1000 1200 1400
0
10
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
minus10
minus20
Figure 10 The drift angle of hovercraft
0 200 400 600 800 1000 1200 1400
0
2
4
Time (s)
0 50 1000
5
10
Proposed controllerMethod AMethod B
uc
(N)
times105
times104
Figure 11 The surge control law of hovercraft
process ofMethods A and BThese are absolutely not allowedfor hovercraft In contrast you can see from Figure 20 thatsurge speed can still be positive even large enough to avoidthe influence of the resistance humps under the control of theproposed controller And drift angle is also within the safetylimit from Figure 21 From Figures 22 and 23 we know thatthe input constraint ability of the proposed controller is effec-tive Figures 13 and 24 show the heave position of hovercraft
5 Conclusion
A safety-guaranteed trajectory tracking controller has beenproposed for underactuated hovercraft in this paper Thesafety-guaranteed auxiliary dynamic system is designed todeal with state and input constraints The velocity of hov-ercraft is constrained to eliminate the effect of resistance
0 200 400 600 800 1000 1200 1400minus4
0
Time (s)
0 10 20minus15minus10minus5
05
Proposed controllerMethod AMethod B
rc
(Nm
)
times105
times104
minus2
Figure 12 The yaw control law of hovercraft
0 200 400 600 800 1000 1200 1400minus245
minus24
minus235
Time (s)z
(m)
Proposed controllerMethod AMethod B
Figure 13 The heave position of hovercraft
minus2000 0 2000 4000 6000 8000 10000 12000minus1000
0
1000
2000
3000
Starting point
Proposed controllerMethod A
Method BDesired trajectory
minus200minus100
0100
x(m
)
y (m)
minus20 0 20 40 60 80
Figure 14 The actual and desired trajectory of hovercraft
0 200 400 600 800 1000 12000
50
100
150
Time (s)
Proposed controllerMethod AMethod B
(d
eg)
Figure 15 The heading angle of hovercraft
hump and obtain better stability The safety limit of driftangle is carried out effectively for safety in the high-speedtrajectory tracking process of hovercraftThe input saturationis handled High nonlinearity and model uncertainties areapproximated by RBFNNs Simulation results have been
Mathematical Problems in Engineering 11
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus100
Figure 16 The surge position tracking error
0 200 400 600 800 1000 1200minus600
minus400
minus200
0
200
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
Figure 17 The sway position tracking error
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus100
Figure 18 The surge velocity tracking error of hovercraft
0 200 400 600 800 1000 1200
0
2
4
6
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus2
Figure 19 The drift angle tracking error of hovercraft
0 200 400 600 800 1000 1200
0
50
Time (s)
0 100 200
0
50
Proposed controllerMethod AMethod B
u(k
nots)
minus50minus50
minus100
Figure 20 The surge velocity of hovercraft
0 200 400 600 800 1000 1200
0
100
200
Time (s)
0 100 2000
20
40
Proposed controllerMethod AMethod B
(d
eg)
minus100
Figure 21 The drift angle of hovercraft
0 200 400 600 800 1000 1200
0
Time (s)
0 50 100
0
Proposed controllerMethod AMethod B
uc
(N)
minus5
minus10
times105
times104
minus5
minus10
Figure 22 The surge control law of hovercraft
0 200 400 600 800 1000 1200
0
10
20
Time (s)
0 10 20
05
1015
Proposed controllerMethod AMethod B
rc
(Nm
)
minus5
times109
times104
Figure 23 The yaw control law of hovercraft
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
xe
(m)
minus100
Figure 16 The surge position tracking error
0 200 400 600 800 1000 1200minus600
minus400
minus200
0
200
Time (s)
Proposed controllerMethod AMethod B
ye
(m)
Figure 17 The sway position tracking error
0 200 400 600 800 1000 1200
0
100
200
300
Time (s)
Proposed controllerMethod AMethod B
e u(k
nots)
minus100
Figure 18 The surge velocity tracking error of hovercraft
0 200 400 600 800 1000 1200
0
2
4
6
Time (s)
Proposed controllerMethod AMethod B
e (d
eg)
minus2
Figure 19 The drift angle tracking error of hovercraft
0 200 400 600 800 1000 1200
0
50
Time (s)
0 100 200
0
50
Proposed controllerMethod AMethod B
u(k
nots)
minus50minus50
minus100
Figure 20 The surge velocity of hovercraft
0 200 400 600 800 1000 1200
0
100
200
Time (s)
0 100 2000
20
40
Proposed controllerMethod AMethod B
(d
eg)
minus100
Figure 21 The drift angle of hovercraft
0 200 400 600 800 1000 1200
0
Time (s)
0 50 100
0
Proposed controllerMethod AMethod B
uc
(N)
minus5
minus10
times105
times104
minus5
minus10
Figure 22 The surge control law of hovercraft
0 200 400 600 800 1000 1200
0
10
20
Time (s)
0 10 20
05
1015
Proposed controllerMethod AMethod B
rc
(Nm
)
minus5
times109
times104
Figure 23 The yaw control law of hovercraft
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
0 200 400 600 800 1000 1200Time (s)
Proposed controllerMethod AMethod B
z(m
)
minus15
minus2
minus25
minus3
Figure 24 The heave position of hovercraft
presented to illustrate the effectiveness of the proposed con-troller
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
The supports of the National Natural Science Foundationof China (Grant no 51309062) and the project ldquoResearchon Maneuverability of High Speed Hovercraftrdquo (Project no2007DFR80320) are gratefully acknowledged
References
[1] Y Liang and A Bliault Theory amp Design of Air Cushion CraftArnold 2000
[2] H Sira-Ramırez ldquoDynamic second-order sliding mode controlof the hovercraft vesselrdquo IEEE Transactions on Control SystemsTechnology vol 10 no 6 pp 860ndash865 2002
[3] J Aranda ldquoA Control for tracki-ng and stabilization of anunder-actuated nonlinear RC hovercraftrdquo International Feder-ation of Auto-matic Control 2006
[4] J Zhao and J Pang ldquoTrajectory control of underactuated hover-craftrdquo inProceedings of the 2010 8thWorldCongress on IntelligentControl and Automation (WCICA rsquo10) pp 3904ndash3907 July 2010
[5] RMorales H Sira-Ramırez and J A Somolinos ldquoLinear activedisturbance rejection control of the hovercraft vessel modelrdquoOcean Engineering vol 96 pp 100ndash108 2015
[6] G G Rigatos and G V Raffo ldquoInputndashoutput linearizing con-trol of the underactuated hovercraft using the derivative-freenonlinear kalman filterrdquo Unmanned Systems vol 03 no 02 pp127ndash142 2015
[7] K Shojaei ldquoTrajectory tracking control of autonomous under-actuated hovercraft vehicles with limited torquerdquo in Proceedingsof the 2014 2nd RSIISM International Conference on Roboticsand Mechatronics (ICRoM rsquo14) pp 468ndash473 October 2014
[8] K Shojaei ldquoNeural adaptive robust control of underactuatedmarine surface vehicles with input saturationrdquo Applied OceanResearch vol 53 pp 267ndash278 2015
[9] M Cohen T Miloh and G Zilman ldquoWave resistance of ahovercraft moving in water with nonrigid bottomrdquoOcean Engi-neering vol 28 no 11 pp 1461ndash1478 2001
[10] Y Yang Z K Zhang and M D Amp ldquoiscussion and practiceabout hump transition problem of high-density air medium-low speed ACVrdquo Ship amp Boat 2014
[11] H Fu ldquoAnalysis and consider- ation on safety of all-lift hover-craftrdquo Ship amp Boat 2008
[12] M Tao and W Chengjie Hovercraft performance and skirt-cushion system dynamics design National Defence IndustryPress 2012
[13] G Zilman ldquoMathematical simula- tion of hovercraft maneu-verin- grdquo Proceedings of the Twenty-Third American Towing tankConference pp 373ndash382 1993
[14] J Du X Hu M Krstic and Y Sun ldquoRobust dynamic position-ing of ships with disturbances under input saturationrdquo Auto-matica vol 73 pp 207ndash214 2016
[15] M Chen S S Ge and B Ren ldquoAdaptive tracking control ofuncertain MIMO nonlinear systems with input constraintsrdquoAutomatica vol 47 no 3 pp 452ndash465 2011
[16] S Li YWang J Tan and Y Zheng ldquoAdaptive RBFNNsintegralsliding mode control for a quadrotor aircraftrdquoNeurocomputingvol 216 pp 126ndash134 2016
[17] K Xia and W Huo ldquoRobust adaptive backstepping neuralnetworks control for spacecraft rendezvous and docking withinput saturationrdquo ISA Transactions vol 62 pp 249ndash257 2016
[18] J Feng and G-X Wen ldquoAdaptive NN consensus tracking con-trol of a class of nonlinear multi-agent systemsrdquo Neurocomput-ing vol 151 no 1 pp 288ndash295 2015
[19] T Elmokadem M Zribi and K Youcef-Toumi ldquoTrajectorytracking sliding mode control of underactuated AUVsrdquo Non-linear Dynamics vol 84 no 2 pp 1079ndash1091 2016
[20] T Elmokadem M Zribi and K Youcef-Toumi ldquoTerminal slid-ing mode control for the trajectory tracking of underactuatedAutonomousUnderwaterVehiclesrdquoOceanEngineering vol 129pp 613ndash625 2017
[21] V K Dyachenko Resistance of Air Cushion Vehicle CentralResearch Institute of ACAD Krylov 1999
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpswwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 201
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
