A Trajectory Tracking Robust Controller of Surface ... - Semantic Scholar
IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. **, NO. **, AUGUST 2013
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A Trajectory Tracking Robust Controller of Surface Vessels with Disturbance Uncertainties Yang Yang, Jialu Du, Hongbo Liu, Chen Guo, Ajith Abraham Senior Member, IEEE
Abstract—This paper considers the problem of a trajectory tracking control for marine surface vessels with unknown time-variant environmental disturbances. The proposed mathematical model of the surface ship movement includes Coriolis and centripetal matrix and nonlinear damp terms. The observer is constructed to provide an estimation of unknown disturbances, which is applied to design a novel trajectory tracking robust controller through a vectorial backstepping technique. It is proved that the controller can force the ship to track the arbitrary reference trajectory and guarantees that the signals of the closed-loop trajectory tracking control system are globally uniformly ultimately bounded. The simulation results and comparisons illustrate that the proposed controller is effective and robust to external disturbances. Index Terms—trajectory tracking control of vessels, disturbance observer, vectorial backstepping, nonlinear, robust
I. I NTRODUCTION Rajectory tracking control of surface vessels is an important control problem. It is of great significance for navigation in safety, energy-saving and emission-reduction [1]. It has attracted a great deal of attention from the control systems community both in theory and in practice. In [2], the simplified linear model was used to develop an adaptive high precision track controller for ships through a combination of feedforward and linear-quadratic-Gaussian (LQG) feedback control. In fact, the tracking control for a ship has an inherently nonlinear character. Taking advantage of the model free intelligent control techniques, the fuzzy logic control and proportional-integral-derivative (PID) track autopilot of ships was presented in [3] and a ship neural network trajectory tracking controller was developed in [4]. In recent years, several significant results have been presented using nonlinear control techniques through ship maneuvering mathematical models. Jiang [5] proposed two global tracking control laws for under actuated vessels
T
This work was supported partly by the National Natural Science Foundation of China (51079013, 61074053 and 61173035), the Applied Basic Research Program of Ministry of Transport of China (2012-329225-070 and 2011-329-225-390), the Higher Education Research Fund of Education Department of Liaoning (LT2010013), China, and the Program for New Century Excellent Talents in University (NCET-11-0861). Y. Yang, J. Du, and C. Guo are with the School of Information Science and Technology at Dalian Maritime University, Dalian 116026, China. email: [email protected], [email protected], [email protected]. H. Liu is with the Institute for Neural Computation, University of California San Diego, La Jolla, CA 92093, with an affiliate appointment in the School of Information Science and Technology at Dalian Maritime University, Dalian, 116026 China (9Corresponding author. e-mail: [email protected]). A. Abraham works at Machine Intelligence Research Labs, Seattle, Washington 98071, USA. e-mail: [email protected].
using Lyapunov’s direct method. Petterson and Nijmeijer [6] illustrated a semi-global exponential stabilization of the tracking error for any desired trajectory using an integrator backstepping approach. Furthermore, they developed an exponential tracking control law of the reference trajectory for the ship using tracking control of chained form systems through a coordinate transformation, which was validated by the experimental results on a scale 1:70 model of an offshore supply vessel in the laboratory [7]. Yu et al. [8] introduced the second-level sliding mode surface approach to design a trajectory tracking control law for an under actuated ship with parameter uncertainties. Wondergem et al. [9] considered the problem of trajectory tracking for fully-actuated surface ships in the presence of the Coriolis and centripetal induced forces and moments and a nonlinear damping. They presented a controller-observer output feedback control scheme with semi-global exponential stability by its ship motion mathematical model. On the other hand, the ships in the sea are always exposed to the environmental disturbances induced by wind, waves, and ocean currents. It is necessary to develop robust controllers for external disturbances. Under constant disturbances, a nonlinear trajectory tracking control law was designed for fully-actuated ship simultaneously considering Coriolis and centripetal matrix and nonlinear damp in [10]. Aschemann and Rauh [11] presented two alternative nonlinear control approaches to track the trajectories through the extended linearisation technique. The tracking accuracy was improved significantly by introducing a compensating control action provided by a disturbance observer for constant disturbances. Using a backstepping technique, a discontinuous feedback control law [12] and a new family of smooth time-varying dynamic feedback laws [13] have been derived respectively. Mostly the mathematical model of ships doesn’t simultaneously consider Coriolis and centripetal matrix and nonlinear damp terms, or the uncertain time-variant environmental disturbances aren’t dealt with during the control design procedures. However, the sea state is constantly changing during the navigation of ships. For under-actuated ships, Do [14] provided a solution for the practical stabilization through several nonlinear coordinate changes, the transverse function approach, the backstepping technique, the Lyapunov’s direct method, and utilization of the ship dynamics. For fully-actuated surface vessels, this paper presents a novel approach to solve the trajectory tracking control problem. The mathematical model of the ship movement
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simultaneously contains Coriolis and centripetal matrix and nonlinear damp terms. The disturbances induced by wind, waves and currents are considered. Our proposed approach is featured such that a disturbance observer is introduced to estimate time-variant uncertain environmental disturbances.
Xo X
xg
r V
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x
G
O
because environmental energy applied to the ship is limited. The matrix R(ψ) is rotation matrix defined as cos ψ − sin ψ 0 cos ψ 0 R(ψ) = sin ψ (3) 0 0 1 with the property R−1 (ψ) = RT (ψ). Here M is nonsingular, symmetric and positive definite inertia matrix, C(ν) is the matrix of Coriolis and centripetal terms, and D(ν) is the damping matrix. They are respectively m11 0 0 m22 m23 , M = 0 (4) 0 m32 m33
II. P ROBLEM F ORMULATION
u
2
v
0 0 C(ν) = m22 v + m23 r
Y
y
Yo
Fig. 1. Definition of the earth-fixed OXo Yo and the body-fixed AXY coordinate frames.
Definition of the reference coordinate frames of ship motion is illustrated in Figure 1, where OXo Yo is the earth-fixed frame and AXY is the body-fixed frame. The coordinate origin O of the earth-fixed reference frame OXo Yo is the original position of the desired trajectory. The axis OXo is directed to the North and OYo is directed to the East. The coordinate origin A of the body-fixed frame is taken as the geometric center point of the ship structure. The axis AX is directed from aft to fore, the axis AY is directed to starboard, and the normal axis AZ is directed from top to bottom. Under the assumption that the ship is port-starboard symmetric, the gravity center G is located a distance xg between the gravity center of the ship and the origin of the body-fixed frame along axis AX. The vector η = [x, y, ψ]T is the actual track of the ship in the earth-fixed frame, consisting of the ship position (x, y) and orientation ψ ∈ [0, 2π]. The vector ν = [u, v, r]T is the velocity vector of ships in the body-fixed frame. The variables u, v and r are respectively the forward velocity (surge), the transverse velocity (sway) and the angular velocity in yaw of the ship. Surge is decoupled from sway and yaw. Neglecting the motions in heave, pitch and roll, the 3-DOF nonlinear motion equations of a surface ship can be expressed as [15]: η˙ = R(ψ)ν
(1)
M ν˙ + C(ν)ν + D(ν)ν = τ + b
(2)
where τ = [τ1 , τ2 , τ3 ]T is the control input vector; b(t) = [b1 (t), b2 (t), b3 (t)]T is the vector representing unknown and time-variant external environmental disturbances due to wind, waves and ocean currents in the body-fixed frame. Here, it is assumed that the changing rate of disturbances ˙ is bounded, i.e. ∥ b(t) ∥6 Cd < ∞, where Cd is a nonnegative constant. The above assumption is reasonable
d11 (u) D(ν) = 0 0
0 −m22 v − m23 r , 0 m11 u −m11 u 0 (5) 0 0 d22 (v, r) d23 (v, r) . d32 (v, r) d33 (v, r)
(6)
In above equations m11 = m − Xu˙ , m22 = m − Yv˙ , m23 = mxg − Yr˙ , m32 = mxg − Nv˙ , m33 = Iz − Nr˙ , d11 (u) = −Xu − Xu|u| |u|, d22 (v, r) = −Yv − Yv|v| |v| − Yr|r| |r|, d23 (v, r) = −Yr − Y|v|r |v| − Y|r|r |r|, d32 (v, r) = −Nν − N|v|v |v| − N|r|v |r|, d33 (v, r) = −Nr − N|v|r |v| − N|r|r |r| where m is the mass of the ship, Iz is the moment of inertia about the yaw rotation, and the other symbols, for example Yu˙ = ∂Y /∂ u, ˙ are referred to as hydrodynamic derivatives. A reader may consult [16] for more details. The control objective in this paper is to design a feedback control law τ for the system (1)–(2) such that the position and orientation η(t) of ships tracks arbitrary smooth reference trajectory ηd (t), while it is guaranteed that all signals of the resulting closed-loop trajectory tracking system of a ship are globally uniformly ultimately bounded. Assumption 1: The desired smooth reference signal ηd is bounded and has the bounded first and second time derivatives η˙ d and η¨d . III. C ONTROLLER D ESIGN In this section, a disturbance observer is designed to estimate the unknown time-variant external environmental disturbances of the system (1)–(2). Then, we present the robust trajectory tracking controller for ships that solves the control objective as stated in Section II. The closed-loop
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trajectory tracking control system of a ship mainly consists of two parts: the ship subjected to external disturbances, and the trajectory tracking controller with the disturbance observer. The schematic diagram is depicted in Figure 2. Observer reference trajectory
bˆ
error
Controller
Wind, waves and ocean currents b actual position and orientation of the ship
d
Fig. 2.
Diagram of the trajectory tracking control system of a ship.
A. Disturbance Observer Design By means of the exponential convergent observer for a general nonlinear system from the reference [14], we constructed the disturbance observer for the disturbance vector b of the system (1)–(2) as follows: ˆb(t) = β + K0 M ν (7) β˙ = −K0 β − K0 [−C(ν)ν − D(ν)ν + τ + K0 M ν] (8) where ˆb is a disturbance estimation, K0 is a 3-by-3 positive definite symmetric observer gain matrix, and β is a three dimension intermediate auxiliary vector generated by (8). Define the estimation error vector ˜b(t) = [˜b1 (t), ˜b2 (t), ˜b3 (t)]T of disturbance vector b as: ˜b(t) = b(t) − ˆb(t) (9) From (7), (8) and (2), we have ˆb˙ = β˙ + K0 M ν˙ = K0 [b − (β + K0 M ν)] = K0 (b − ˆb) Then the derivative of (9) along (10) is ˜b˙ = b˙ − K0 (b − ˆb) = b˙ − K0˜b.
3
where c=
Cd2 , 4ε
(16)
α = 2[λmin (K0 ) − ε],
(17)
λmin (K0 ) − ε > 0,
(18)
and λmin (·) represents the smallest eigenvalue of a matrix. Therefore, we have the following theorem. Theorem 1: The disturbance observer (7)–(8) guarantees that the disturbance estimation error ˜b of disturbances exponentially converges to a√ball Ωb centered at the origin with the radius Rd = Cd /[2 ε(λmin (K0 ) − ε)]. The estimation error ˜b of disturbances can be made arbitrarily small by appropriately adjusting the design matrix K0 and parameter ε satisfying the condition (18). Proof: Solving (15), we have c [ c ] −αt 0 ≤ Ve (t) ≤ + Ve (0) − e (19) α α It is known from (19) that Ve is ultimately bounded and exponentially converges to a ball centered at the origin with the radius RV = Cd2 /[8ε(λmin (K0 )−ε)]. Furthermore, it is known from the definition of Ve that the disturbance estimation error ˜b exponentially converges to √ a ball Ωb centered at the origin with the radius Rd = Cd /[2 ε(λmin (K0 ) − ε)]. Therefore, the theorem is proved. Remark 1: In the case Cd = 0, i.e., the disturbance vector is unknown constant vector, the disturbance observer is exponentially stable. The disturbance estimation error ˜b exponentially converges to zero. B. Control Law Design
(10)
Let the desired position and orientation of ships be ηd = [xd , yd , ψd ]T . First define the error vectors as follows:
(11)
Consider the following Lyapunov function candidate: 1 Ve = ˜bT ˜b. (12) 2 The time derivative of Ve along the solution of (11) is ˙ V˙ e = ˜bT (−K0˜b + b) (13) T T ˙ = −˜b K0˜b + ˜b b. According to the following complete square inequality ˜bT b˙ ≤ ε˜bT ˜b + 1 b˙ T b, ˙ (14) 4ε where ε is a small positive constant, (13) can be rewritten as 1 V˙ e ≤ − λmin (K0 )˜bT ˜b + ε˜bT ˜b + b˙ T b˙ 4ε C2 ≤ − 2[λmin (K0 ) − ε]Ve + d 4ε (15) ≤ − αVe + c
ηe = η − ηd
(20)
X e = ν − X1
(21)
where X1 is the stabilization function vector of subsystem (1), ν is taken as the virtual control input vector. The control law design consists of two steps. Step 1: Consider the following Lyapunov function candidate: 1 V1 = ηeT ηe . (22) 2 The derivative of ηe is given by η˙ e = η˙ − η˙ d = R(ψ)Xe + R(ψ)X1 − η˙ d .
(23)
Then the time derivative of V1 along the solution of (23) is V˙ 1 = ηeT η˙ e = ηeT [R(ψ)X1 − η˙ d ] + ηeT R(ψ)Xe .
(24)
We choose the stabilization function vector X1 = R−1 (ψ)(−C1 ηe + η˙ d )
(25)
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where C1 is a positive definite symmetric design parameter matrix. Substituting (25) into (24) yields V˙ 1 = ηeT [R(ψ)R−1 (ψ)(−C1 ηe + η˙ d ) − η˙ d ] + ηeT R(ψ)Xe = −ηeT C1 ηe + ηeT R(ψ)Xe .
Substituting (30) into (29) results in V˙ 2 = − ηeT C1 ηe + XeT [RT (ψ)ηe − C(ν)ν − D(ν)ν + C(ν)ν + D(ν)ν + M X˙1 − R(ψ)ηe − C2 Xe − ˆb + b − M X˙1 ] − ˜bT K0˜b + ˜bT b˙ ˙ = − ηeT C1 ηe − XeT C2 Xe − XeT ˜b − ˜bT K0˜b + ˜bT b. (35)
(26)
The coupling term ηeT R(ψ)Xe will be cancelled in the next step. Step 2: From (2) and (21), we have
Considering (14) and the following complete square inequalities
X˙e = ν˙ − X˙1 =M
−1
[−C(ν)ν − D(ν)ν + τ + b − M X˙1 ].
− XeT ˜b ≤ ε1 XeT Xe +
(27)
1 ˜T ˜ b b. 2
(28)
In terms of (11), (26) and (27), the time derivative of V2 is ˙ V˙ 2 = V˙ 1 + XeT M X˙e + ˜bT ˜b = −ηeT C1 ηe + XeT [RT (ψ)ηe − C(ν)ν − D(ν)ν ˙ (29) + τ + b − M X˙1 ] − ˜bT K0˜b + ˜bT b. We choose the control input vector as τ = C(ν)ν + D(ν)ν + M X˙1 − RT (ψ)ηe − C2 Xe − ˆb (30) where C2 is a positive definite symmetric design parameter matrix. According to (20) and the property R−1 (ψ) = RT (ψ), we calculate the derivative of X1 as follows: X˙1 =R˙ T (ψ)[−C1 (η − ηd ) + η˙ d ]
(31)
+ R−1 (ψ)[−C1 (η˙ − η˙ d ) + η¨d ]. In addition, we have from (3) −r sin ψ −r cos ψ 0 ˙ R(ψ) = r cos ψ −r sin ψ 0 0 0 0 cos ψ − sin ψ 0 0 −r cos ψ 0 r 0 = sin ψ 0 0 1 0 0 =R(ψ)S(r) 0 −r 0 where S(r) = r 0 0. Then, we obtain 0 0 0
0 0 0 (32)
(33)
By substituting (7), (20), (21) and (33) into (30), (30) can be written as τ =− (M S R C1 + R + C2 R C1 )(η − ηd ) + M R η¨d T
T
T
T
T
+ (M S T RT + M RT C1 + C2 RT )η˙ d + [C(ν) + D(ν) − M RT C1 R − C2 − K0 M ]ν − β.
(36) (37)
where ε1 is a small positive constant, then, (35) can be rewritten as V˙ 2 ≤ − λmin (C1 )ηeT ηe − λmin (C2 M −1 )XeT M Xe 1 ˜T ˜ + ε1 XeT Xe + b b − λmin (K0 )˜bT ˜b + ε˜bT ˜b 4ε1 1 + b˙ T b˙ 4ε ≤ − 2 min[λmin (C1 ), λmin (C2 M −1 ) − ε1 λmax (M −1 ), 1 1 (38) λmin (K0 ) − − ε]V2 + Cd2 4ε1 4ε where λmin (C2 M −1 ) − ε1 λmax (M −1 ) > 0, λmin (K0 ) −
1 − ε > 0, 4ε1
(39) (40)
and λmax (·) represents the largest eigenvalue of a matrix. Therefore, there is the following theorem. Theorem 2: Under Assumption 1, for the 3-DOF nonlinear motion mathematical model of ships with unknown time-variant disturbances given by (1) and (2), the control input vector τ described by (34) together with (8) guarantees that the actual trajectory of ships tracks the arbitrary reference trajectory with the desired accuracy and all the signals of the closed-loop trajectory tracking system of ships are globally uniformly ultimately bounded by appropriately choosing the design parameter matrix C1 , C2 and K0 satisfying the conditions (39) – (40). Proof: Notate µ = min[λmin (C1 ), λmin (C2 M −1 ) − ε1 λmax (M −1 ), 1 − ε], λmin (K0 ) − (41) 4ε1
X˙1 =[R(ψ)S(r)]T [−C1 (η − ηd ) + η˙ d ] + RT (ψ)[−C1 (η˙ − η˙ d ) + η¨d ].
1 ˜T ˜ b b 4ε1
− XeT C2 Xe ≤ −λmin (C2 M −1 )XeT M Xe
Consider the augmented Lyapunov function candidate: 1 V2 = V1 + XeT M Xe + 2
4
(34)
Cd2 . 4ε Then (38) can be rewritten as σ=
V˙ 2 (t) ≤ −2µV2 (t) + σ. Solving the above inequility, we have [ σ σ ] −2µt 0 ≤ V2 (t) ≤ + V2 (0) − e . 2µ 2µ
(42)
(43)
(44)
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It is seen from (44) that V2 (t) is globally uniformly ultimately bounded. Hence, ηe , Xe and ˜b are globally uniformly ultimately bounded according to (28), then X1 and ν are globally uniformly ultimately bounded. From the boundedness of ηd and b, we know that η and ˆb are bounded. From (28) and (44), we can obtain √ [ ] σ σ −2µt ∥z1 ∥ ≤ + 2 V2 (0) − e . (45) µ 2µ √ It follows that, for any µz1 > σ/µ , there exits a constant Tz1 > 0, such that ∥z1 ∥ ≤ µz1 for all t > Tz1 . Therefore, the trajectory tracking error z1 of the ship can converge √ to the √ compact set Ωze := {z1 ∈ R3 | ∥z1 ∥ ≤ σ/µ}. Since σ/µ can be made arbitrarily small if the design parameters C1 , C2 , and K0 are appropriately chosen, the actual trajectory of the ship can track the arbitrary reference trajectory. Theorem 2 is thus proved. IV. S IMULATIONS AND C OMPARISONS In this section, the simulation studies are carried out on CyberShip II, which is a 1:70 scale replica of a supply ship of the Marine Cybernetics Laboratory in Norwegian University of Science and Technology. The ship has the length of 1.255m, mass of 23.8kg and other parameters of the ship are given in detail in [17]. We carry out the simulations with two different disturbances. In the simulations, the reference trajectory is chosen as bellow xd = 4 sin(0.02t) yd = 2.5(1 − cos(0.02t)) ψd = 0.02t
(46)
Fig. 3. Constant external disturbances b1 , b2 , b3 and their estimations ˆb1 , ˆb2 , ˆb3 .
which is an ellipse. A. Trajectory tracking under constant disturbances
reference trajectory actual path
5 4 3 y/m
In this section, the disturbance vector is set as b = [2N, 2N, 2N · m]T which corresponds to the environmental disturbances due to slowly varying wind, waves and currents. Assume the initial conditions of the system are [x(0), y(0), ψ(0), u(0), v(0), r(0)]T = [1m, 1m, π/4rad, 0m/s, 0m/s, 0rad/sT and the initial state of the disturbance observer is ˆb(0) = [0, 0, 0]T . The design parameter matrices are taken as C1 = diag([0.05, 0.05, 0.05]), C2 = diag([120, 120, 120]), K0 = diag([2, 2, 2]) such that the conditions (39) and (40) are satisfied for 0.125 < ε1 < 9.6509 and 0 < ε < 1.9741. The results are depicted in Figures 3–7. The external disturbances b and its estimate value ˆb are depicted in Figure 3 from which it is clearly seen that the disturbance observer provides the rapidly exponentially convergent estimation of unknown disturbances within about 1.5s as proved in Theorem 1. From Figure 4, it is observed that the proposed controller is able to force the ship to track the reference trajectory. Furthermore, the curves of the desired and actual positions and yaw angles are presented in Figure 5, which shows
2 1 0 −5
0 x/m
5
Fig. 4. Actual trajectory and reference trajectory in xy-plane under constant disturbances.
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0
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Fig. 5. Desired and actual position (x, y) and yaw angle ψ under constant disturbances.
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Fig. 7. Surge control force τ1 , sway control force τ2 and yaw control torque τ3 under constant disturbances.
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Fig. 6. Surge velocity u, sway velocity v and yaw rate r under constant disturbances.
that the actual ship position (x, y) and yaw angle ψ can track the desired trajectory ηd = [xd , yd , ψd ]T at a good precision in around 40s. The curves of the surge velocity u, sway velocity v and yaw rate r versus time are shown in Figure 6. The corresponding control inputs are presented in Figure 7, which shows that the control force and torque are smooth and reasonable. These results reveal that all the signals of the closed-loop trajectory tracking system of ships are globally uniformly ultimately bounded as proved in Theorem 2. Therefore, the proposed trajectory tracking controller is effective for the ship with uncertain constant disturbances. Fig. 8. Time-variant external disturbances b1 , b2 , b3 and their estimations ˆb1 , ˆb2 , ˆb3 .
B. Trajectory tracking under time-variant disturbances In this section, the disturbance vector is set as b(t) =[b1 (t), b2 (t), b3 (t)]T in Figures 8–12, which exhibit almost the same control 1.3+ 2.0 sin(0.02t) + 1.5 sin(0.1t)N performance as under constant disturbances despite the = −0.9 + 2.0 sin(0.02t − π/6) + 1.5 sin(0.3t)N . time-variant disturbances. It is obvious that the designed − sin(0.09t + π/3) − 4 sin(0.01t)N · m controller is effective when the ship is exposed to both The initial conditions of the system and the design pa- unknown constant and time-variant disturbances, which rameters of controller are same as the counterparts in the demonstrates that the proposed controller is robust against first case of subsection III-A. The results are depicted unknown environmental disturbances.
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7
τ1 /N
5 reference trajectory actual path
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τ2 /N
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y/m
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0 −10 0 10 0 −10 0
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Fig. 9. Actual trajectory and reference trajectory in xy-plane under timevariant disturbances.
Fig. 12. Surge control force τ1 , sway control force τ2 and yaw control torque τ3 under time-variant disturbances.
C. Performance comparisons
m
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In this section, we compare the tracking performance of the designed controller (34) in this paper with the controller without disturbance observer:
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r/rad · s−1
v/m · s−1
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Fig. 10. Desired and actual position (x, y) and yaw angle ψ under time-variant disturbances.
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+ [M (S T RT + RT Ccm1 ) + Ccm2 RT ]η˙ d + M RT η¨d + [C(ν) + D(ν) − M RT Ccm1 R − Ccm2 ]ν ∫ t − Kcm [ν + RT Ccm1 (η − ηd ) − RT η˙ d ]dð 0
(47)
ψd ψ
5
τcm = − (M S T RT Ccm1 + RT + Ccm2 RT Ccm1 )(η − ηd )
Fig. 11. Surge velocity u, sway velocity v and yaw rate r under under time-variant disturbances.
which is designed using the backstepping approach for the ship with constant disturbances in [10] with gains Ccm1 = diag([0.05, 0.05, 0.05]), Ccm2 = diag([120, 120, 120]) and Kcm = diag([2, 2, 2]). Figures 13 and 14 illustrate the comparison of tracking performance between the two different controllers under constant disturbances and time-variant disturbances respectively. It can be seen from Figure 13 that both the controller exhibit similarly good transient and steady-state performances under the constant disturbances. However, under time-variant disturbances, it is seen from Figure 14 that the controller τ with disturbance observer in this paper performs better than the backstepping controller τcm with faster decay of tracking error and lower steadystate error value, since our observer provides an estimation of unknown disturbances. In contrast, τcm does not have disturbance compensation, it results in a larger tracking error norm. To quantitatively compare the two controller performance, the performance under both constant and timevariant disturbances is summarized in Table I where xe = xd − x and ye = yd − y representing the error between the desired and actual positions, ψe = ψd − ψ representing the error between the desired and actual yaw angles, and tf inal = 300s. Table I clearly shows that the controller τ has better steady-state and transient performances than the backstepping controller τcm .
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resulting closed-loop trajectory tracking system of the ship are globally uniformly ultimately bounded. Furthermore, the simulation results on an offshore supply ship model has illustrated that our controller is effective and robust to external disturbances. Our proposed trajectory tracking control scheme can provide good transient and steady-state performance for the considered ship system. Future research work would extend the proposed method to address the robust adaptive output feedback tracking of ships subjected to external disturbances and model uncertainties only depending on the position information η = [x, y, ψ]T . From a practical viewpoint, it is convenient since it doesn’t have to measure directly the velocities ν = [u, v, r]T .
Controller τ Controller τcm
||z1 ||
1.5
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0 0
Fig. 13. bances.
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300
Comparison of tracking performance under constant distur-
R EFERENCES 2
Controller τ Controller τcm
||z1 ||
1.5
1
0.5
0 0
Fig. 14. bances.
8
50
100
150 t/s
200
250
300
Comparison of tracking performance under time-variant distur-
V. C ONCLUSIONS In this paper, a trajectory tracking robust control law has been designed for fully-actuated surface vessels in the presence of uncertain time-variant disturbances due to wind waves, and ocean currents. Both Coriolis and centripetal matrix and nonlinear damp term have been considered in the nonlinear ship surface movement mathematical model. The control strategy is introduced by the vectorial backstepping technique with our disturbance observer. The disturbance observer is employed to compensate disturbance uncertainties. It has been proved that the signals of the TABLE I P ERFORMANCE INDEX COMPARISON OF CONTROLLERS τ UNDER DIFFERENT DISTURBANCES .
Disturbances
Constant
AND τcm
Time-variant
Controller
τ
τcm
τ
τcm
settling time ts (s) ∫ tf inal |xe |dt(m · s) 0 ∫ tf inal |ye |dt(m · s) 0 ∫ tf inal |ψ e |dt(rad · s) 0
39 19.4765 17.8455 13.6354
56 25.6207 40.7659 35.1542
41 19.8225 18.2010 13.5816
64 52.0868 55.9768 44.8413
[1] Z. Li, J. Sun, and S. Oh, “Path following for marine surface vessels with rudder and roll constraints: an MPC approach,” in Proceedings of American Control Conference, St. Louis, MO, USA, June 2009, pp. 3611–3616. [2] T. Holzhuter, “LQG approach for the high-precision track control of ships,” Control Theory and Applications, IEE Proceedings-, vol. 144, no. 2, pp. 121–127, 1997. [3] Z. Ma, D. Wan, and L. Huang, “A novel fuzzy PID controller for tracking autopilot,” Ship Electronic Engineering, vol. 19, no. 6, pp. 21–25, 1999. [4] G. Zhang and G. Ren, “Self-tuning adaptive control for ship’s track based on neural network,” Journal of Central South University of Technology, vol. 38, no. 1, pp. 62–68, 2007. [5] Z. Jiang, “Global tracking control of underactuated ships by Lyapunov’s direct method,” Automatica, vol. 38, no. 2, pp. 301–309, 2002. [6] K. Pettersen and H. Nijmeijer, “Tracking control of an underactuated surface vessel,” in Proceedings of the 37th IEEE Conference on Decision and Control, Tampa, Florida, USA, December 1998, pp. 4561–4566. [7] ——, “Underactuated ship control: theory and experiments,” International Journal of Control, vol. 74, no. 14, pp. 1435–1446, 2001. [8] R. Yu, Q. Zhu, G. Xia, and Z. Liu, “Sliding mode tracking control of an underactuated surface vessel,” IET Control Theory & Applications, vol. 6, no. 3, pp. 461–466, 2012. [9] M. Wondergem, E. Lefeber, K. Y. Pettersen, and H. Nijmeijer, “Output feedback tracking of ships,” Control Systems Technology, IEEE Transactions on, vol. 19, no. 2, pp. 442–448, 2011. [10] J. Du, L. Wang, and C. Jiang, “Nonliear ship trajectory tracking control based on backstepping algotirhm,” Ship Engineering, vol. 32, no. 1, pp. 41–44, 2010. [11] H. Aschemann and A. Rauh, “Nonlinear control and disturbance compensation for underactuated ships using extended linearisation techniques,” in Proceedings of 8th IFAC Conference on Control Applications in Marine Systems, Rostock, Germany, Sep. 2010, pp. 167–172. [12] J. Ghommam, F. Mnif, A. Benali, and N. Derbel, “Asymptotic backstepping stabilization of an underactuated surface vessel,” Control Systems Technology, IEEE Transactions on, vol. 14, no. 6, pp. 1150– 1157, 2006. [13] J. Ghommam, F. Mnif, and N. Derbel, “Global stabilisation and tracking control of underactuated surface vessels,” IET Control Theory & Applications, vol. 4, no. 1, pp. 71–88, 2010. [14] K. Do, “Practial control of underactuated ships,” Ocean Engineering, vol. 37, no. 13, pp. 1111–1119, 2010. [15] T. Fossen, Marine Control Systems Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Trondheim, Norway: Marine Cyernetics, 2002. [16] SNAME, “The society of naval architects and marine engineers. nomenclature for treating the motion of a submerged body through a fluid,” Technical and Research Bulletin, no. 1-5, 1950. [17] R. Skjetne, V. Smogeli, and T. I. Fossen, “Modeling, identification, and adaptive maneuvering of Cybership II: A complete design with experiments,” in Proceedings of IFAC Conference on Control Applications in Marine System, Ancona, Italy, Jul. 2004, pp. 203– 208.
1
A Trajectory Tracking Robust Controller of Surface Vessels with Disturbance Uncertainties Yang Yang, Jialu Du, Hongbo Liu, Chen Guo, Ajith Abraham Senior Member, IEEE
Abstract—This paper considers the problem of a trajectory tracking control for marine surface vessels with unknown time-variant environmental disturbances. The proposed mathematical model of the surface ship movement includes Coriolis and centripetal matrix and nonlinear damp terms. The observer is constructed to provide an estimation of unknown disturbances, which is applied to design a novel trajectory tracking robust controller through a vectorial backstepping technique. It is proved that the controller can force the ship to track the arbitrary reference trajectory and guarantees that the signals of the closed-loop trajectory tracking control system are globally uniformly ultimately bounded. The simulation results and comparisons illustrate that the proposed controller is effective and robust to external disturbances. Index Terms—trajectory tracking control of vessels, disturbance observer, vectorial backstepping, nonlinear, robust
I. I NTRODUCTION Rajectory tracking control of surface vessels is an important control problem. It is of great significance for navigation in safety, energy-saving and emission-reduction [1]. It has attracted a great deal of attention from the control systems community both in theory and in practice. In [2], the simplified linear model was used to develop an adaptive high precision track controller for ships through a combination of feedforward and linear-quadratic-Gaussian (LQG) feedback control. In fact, the tracking control for a ship has an inherently nonlinear character. Taking advantage of the model free intelligent control techniques, the fuzzy logic control and proportional-integral-derivative (PID) track autopilot of ships was presented in [3] and a ship neural network trajectory tracking controller was developed in [4]. In recent years, several significant results have been presented using nonlinear control techniques through ship maneuvering mathematical models. Jiang [5] proposed two global tracking control laws for under actuated vessels
T
This work was supported partly by the National Natural Science Foundation of China (51079013, 61074053 and 61173035), the Applied Basic Research Program of Ministry of Transport of China (2012-329225-070 and 2011-329-225-390), the Higher Education Research Fund of Education Department of Liaoning (LT2010013), China, and the Program for New Century Excellent Talents in University (NCET-11-0861). Y. Yang, J. Du, and C. Guo are with the School of Information Science and Technology at Dalian Maritime University, Dalian 116026, China. email: [email protected], [email protected], [email protected]. H. Liu is with the Institute for Neural Computation, University of California San Diego, La Jolla, CA 92093, with an affiliate appointment in the School of Information Science and Technology at Dalian Maritime University, Dalian, 116026 China (9Corresponding author. e-mail: [email protected]). A. Abraham works at Machine Intelligence Research Labs, Seattle, Washington 98071, USA. e-mail: [email protected].
using Lyapunov’s direct method. Petterson and Nijmeijer [6] illustrated a semi-global exponential stabilization of the tracking error for any desired trajectory using an integrator backstepping approach. Furthermore, they developed an exponential tracking control law of the reference trajectory for the ship using tracking control of chained form systems through a coordinate transformation, which was validated by the experimental results on a scale 1:70 model of an offshore supply vessel in the laboratory [7]. Yu et al. [8] introduced the second-level sliding mode surface approach to design a trajectory tracking control law for an under actuated ship with parameter uncertainties. Wondergem et al. [9] considered the problem of trajectory tracking for fully-actuated surface ships in the presence of the Coriolis and centripetal induced forces and moments and a nonlinear damping. They presented a controller-observer output feedback control scheme with semi-global exponential stability by its ship motion mathematical model. On the other hand, the ships in the sea are always exposed to the environmental disturbances induced by wind, waves, and ocean currents. It is necessary to develop robust controllers for external disturbances. Under constant disturbances, a nonlinear trajectory tracking control law was designed for fully-actuated ship simultaneously considering Coriolis and centripetal matrix and nonlinear damp in [10]. Aschemann and Rauh [11] presented two alternative nonlinear control approaches to track the trajectories through the extended linearisation technique. The tracking accuracy was improved significantly by introducing a compensating control action provided by a disturbance observer for constant disturbances. Using a backstepping technique, a discontinuous feedback control law [12] and a new family of smooth time-varying dynamic feedback laws [13] have been derived respectively. Mostly the mathematical model of ships doesn’t simultaneously consider Coriolis and centripetal matrix and nonlinear damp terms, or the uncertain time-variant environmental disturbances aren’t dealt with during the control design procedures. However, the sea state is constantly changing during the navigation of ships. For under-actuated ships, Do [14] provided a solution for the practical stabilization through several nonlinear coordinate changes, the transverse function approach, the backstepping technique, the Lyapunov’s direct method, and utilization of the ship dynamics. For fully-actuated surface vessels, this paper presents a novel approach to solve the trajectory tracking control problem. The mathematical model of the ship movement
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simultaneously contains Coriolis and centripetal matrix and nonlinear damp terms. The disturbances induced by wind, waves and currents are considered. Our proposed approach is featured such that a disturbance observer is introduced to estimate time-variant uncertain environmental disturbances.
Xo X
xg
r V
A
x
G
O
because environmental energy applied to the ship is limited. The matrix R(ψ) is rotation matrix defined as cos ψ − sin ψ 0 cos ψ 0 R(ψ) = sin ψ (3) 0 0 1 with the property R−1 (ψ) = RT (ψ). Here M is nonsingular, symmetric and positive definite inertia matrix, C(ν) is the matrix of Coriolis and centripetal terms, and D(ν) is the damping matrix. They are respectively m11 0 0 m22 m23 , M = 0 (4) 0 m32 m33
II. P ROBLEM F ORMULATION
u
2
v
0 0 C(ν) = m22 v + m23 r
Y
y
Yo
Fig. 1. Definition of the earth-fixed OXo Yo and the body-fixed AXY coordinate frames.
Definition of the reference coordinate frames of ship motion is illustrated in Figure 1, where OXo Yo is the earth-fixed frame and AXY is the body-fixed frame. The coordinate origin O of the earth-fixed reference frame OXo Yo is the original position of the desired trajectory. The axis OXo is directed to the North and OYo is directed to the East. The coordinate origin A of the body-fixed frame is taken as the geometric center point of the ship structure. The axis AX is directed from aft to fore, the axis AY is directed to starboard, and the normal axis AZ is directed from top to bottom. Under the assumption that the ship is port-starboard symmetric, the gravity center G is located a distance xg between the gravity center of the ship and the origin of the body-fixed frame along axis AX. The vector η = [x, y, ψ]T is the actual track of the ship in the earth-fixed frame, consisting of the ship position (x, y) and orientation ψ ∈ [0, 2π]. The vector ν = [u, v, r]T is the velocity vector of ships in the body-fixed frame. The variables u, v and r are respectively the forward velocity (surge), the transverse velocity (sway) and the angular velocity in yaw of the ship. Surge is decoupled from sway and yaw. Neglecting the motions in heave, pitch and roll, the 3-DOF nonlinear motion equations of a surface ship can be expressed as [15]: η˙ = R(ψ)ν
(1)
M ν˙ + C(ν)ν + D(ν)ν = τ + b
(2)
where τ = [τ1 , τ2 , τ3 ]T is the control input vector; b(t) = [b1 (t), b2 (t), b3 (t)]T is the vector representing unknown and time-variant external environmental disturbances due to wind, waves and ocean currents in the body-fixed frame. Here, it is assumed that the changing rate of disturbances ˙ is bounded, i.e. ∥ b(t) ∥6 Cd < ∞, where Cd is a nonnegative constant. The above assumption is reasonable
d11 (u) D(ν) = 0 0
0 −m22 v − m23 r , 0 m11 u −m11 u 0 (5) 0 0 d22 (v, r) d23 (v, r) . d32 (v, r) d33 (v, r)
(6)
In above equations m11 = m − Xu˙ , m22 = m − Yv˙ , m23 = mxg − Yr˙ , m32 = mxg − Nv˙ , m33 = Iz − Nr˙ , d11 (u) = −Xu − Xu|u| |u|, d22 (v, r) = −Yv − Yv|v| |v| − Yr|r| |r|, d23 (v, r) = −Yr − Y|v|r |v| − Y|r|r |r|, d32 (v, r) = −Nν − N|v|v |v| − N|r|v |r|, d33 (v, r) = −Nr − N|v|r |v| − N|r|r |r| where m is the mass of the ship, Iz is the moment of inertia about the yaw rotation, and the other symbols, for example Yu˙ = ∂Y /∂ u, ˙ are referred to as hydrodynamic derivatives. A reader may consult [16] for more details. The control objective in this paper is to design a feedback control law τ for the system (1)–(2) such that the position and orientation η(t) of ships tracks arbitrary smooth reference trajectory ηd (t), while it is guaranteed that all signals of the resulting closed-loop trajectory tracking system of a ship are globally uniformly ultimately bounded. Assumption 1: The desired smooth reference signal ηd is bounded and has the bounded first and second time derivatives η˙ d and η¨d . III. C ONTROLLER D ESIGN In this section, a disturbance observer is designed to estimate the unknown time-variant external environmental disturbances of the system (1)–(2). Then, we present the robust trajectory tracking controller for ships that solves the control objective as stated in Section II. The closed-loop
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trajectory tracking control system of a ship mainly consists of two parts: the ship subjected to external disturbances, and the trajectory tracking controller with the disturbance observer. The schematic diagram is depicted in Figure 2. Observer reference trajectory
bˆ
error
Controller
Wind, waves and ocean currents b actual position and orientation of the ship
d
Fig. 2.
Diagram of the trajectory tracking control system of a ship.
A. Disturbance Observer Design By means of the exponential convergent observer for a general nonlinear system from the reference [14], we constructed the disturbance observer for the disturbance vector b of the system (1)–(2) as follows: ˆb(t) = β + K0 M ν (7) β˙ = −K0 β − K0 [−C(ν)ν − D(ν)ν + τ + K0 M ν] (8) where ˆb is a disturbance estimation, K0 is a 3-by-3 positive definite symmetric observer gain matrix, and β is a three dimension intermediate auxiliary vector generated by (8). Define the estimation error vector ˜b(t) = [˜b1 (t), ˜b2 (t), ˜b3 (t)]T of disturbance vector b as: ˜b(t) = b(t) − ˆb(t) (9) From (7), (8) and (2), we have ˆb˙ = β˙ + K0 M ν˙ = K0 [b − (β + K0 M ν)] = K0 (b − ˆb) Then the derivative of (9) along (10) is ˜b˙ = b˙ − K0 (b − ˆb) = b˙ − K0˜b.
3
where c=
Cd2 , 4ε
(16)
α = 2[λmin (K0 ) − ε],
(17)
λmin (K0 ) − ε > 0,
(18)
and λmin (·) represents the smallest eigenvalue of a matrix. Therefore, we have the following theorem. Theorem 1: The disturbance observer (7)–(8) guarantees that the disturbance estimation error ˜b of disturbances exponentially converges to a√ball Ωb centered at the origin with the radius Rd = Cd /[2 ε(λmin (K0 ) − ε)]. The estimation error ˜b of disturbances can be made arbitrarily small by appropriately adjusting the design matrix K0 and parameter ε satisfying the condition (18). Proof: Solving (15), we have c [ c ] −αt 0 ≤ Ve (t) ≤ + Ve (0) − e (19) α α It is known from (19) that Ve is ultimately bounded and exponentially converges to a ball centered at the origin with the radius RV = Cd2 /[8ε(λmin (K0 )−ε)]. Furthermore, it is known from the definition of Ve that the disturbance estimation error ˜b exponentially converges to √ a ball Ωb centered at the origin with the radius Rd = Cd /[2 ε(λmin (K0 ) − ε)]. Therefore, the theorem is proved. Remark 1: In the case Cd = 0, i.e., the disturbance vector is unknown constant vector, the disturbance observer is exponentially stable. The disturbance estimation error ˜b exponentially converges to zero. B. Control Law Design
(10)
Let the desired position and orientation of ships be ηd = [xd , yd , ψd ]T . First define the error vectors as follows:
(11)
Consider the following Lyapunov function candidate: 1 Ve = ˜bT ˜b. (12) 2 The time derivative of Ve along the solution of (11) is ˙ V˙ e = ˜bT (−K0˜b + b) (13) T T ˙ = −˜b K0˜b + ˜b b. According to the following complete square inequality ˜bT b˙ ≤ ε˜bT ˜b + 1 b˙ T b, ˙ (14) 4ε where ε is a small positive constant, (13) can be rewritten as 1 V˙ e ≤ − λmin (K0 )˜bT ˜b + ε˜bT ˜b + b˙ T b˙ 4ε C2 ≤ − 2[λmin (K0 ) − ε]Ve + d 4ε (15) ≤ − αVe + c
ηe = η − ηd
(20)
X e = ν − X1
(21)
where X1 is the stabilization function vector of subsystem (1), ν is taken as the virtual control input vector. The control law design consists of two steps. Step 1: Consider the following Lyapunov function candidate: 1 V1 = ηeT ηe . (22) 2 The derivative of ηe is given by η˙ e = η˙ − η˙ d = R(ψ)Xe + R(ψ)X1 − η˙ d .
(23)
Then the time derivative of V1 along the solution of (23) is V˙ 1 = ηeT η˙ e = ηeT [R(ψ)X1 − η˙ d ] + ηeT R(ψ)Xe .
(24)
We choose the stabilization function vector X1 = R−1 (ψ)(−C1 ηe + η˙ d )
(25)
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where C1 is a positive definite symmetric design parameter matrix. Substituting (25) into (24) yields V˙ 1 = ηeT [R(ψ)R−1 (ψ)(−C1 ηe + η˙ d ) − η˙ d ] + ηeT R(ψ)Xe = −ηeT C1 ηe + ηeT R(ψ)Xe .
Substituting (30) into (29) results in V˙ 2 = − ηeT C1 ηe + XeT [RT (ψ)ηe − C(ν)ν − D(ν)ν + C(ν)ν + D(ν)ν + M X˙1 − R(ψ)ηe − C2 Xe − ˆb + b − M X˙1 ] − ˜bT K0˜b + ˜bT b˙ ˙ = − ηeT C1 ηe − XeT C2 Xe − XeT ˜b − ˜bT K0˜b + ˜bT b. (35)
(26)
The coupling term ηeT R(ψ)Xe will be cancelled in the next step. Step 2: From (2) and (21), we have
Considering (14) and the following complete square inequalities
X˙e = ν˙ − X˙1 =M
−1
[−C(ν)ν − D(ν)ν + τ + b − M X˙1 ].
− XeT ˜b ≤ ε1 XeT Xe +
(27)
1 ˜T ˜ b b. 2
(28)
In terms of (11), (26) and (27), the time derivative of V2 is ˙ V˙ 2 = V˙ 1 + XeT M X˙e + ˜bT ˜b = −ηeT C1 ηe + XeT [RT (ψ)ηe − C(ν)ν − D(ν)ν ˙ (29) + τ + b − M X˙1 ] − ˜bT K0˜b + ˜bT b. We choose the control input vector as τ = C(ν)ν + D(ν)ν + M X˙1 − RT (ψ)ηe − C2 Xe − ˆb (30) where C2 is a positive definite symmetric design parameter matrix. According to (20) and the property R−1 (ψ) = RT (ψ), we calculate the derivative of X1 as follows: X˙1 =R˙ T (ψ)[−C1 (η − ηd ) + η˙ d ]
(31)
+ R−1 (ψ)[−C1 (η˙ − η˙ d ) + η¨d ]. In addition, we have from (3) −r sin ψ −r cos ψ 0 ˙ R(ψ) = r cos ψ −r sin ψ 0 0 0 0 cos ψ − sin ψ 0 0 −r cos ψ 0 r 0 = sin ψ 0 0 1 0 0 =R(ψ)S(r) 0 −r 0 where S(r) = r 0 0. Then, we obtain 0 0 0
0 0 0 (32)
(33)
By substituting (7), (20), (21) and (33) into (30), (30) can be written as τ =− (M S R C1 + R + C2 R C1 )(η − ηd ) + M R η¨d T
T
T
T
T
+ (M S T RT + M RT C1 + C2 RT )η˙ d + [C(ν) + D(ν) − M RT C1 R − C2 − K0 M ]ν − β.
(36) (37)
where ε1 is a small positive constant, then, (35) can be rewritten as V˙ 2 ≤ − λmin (C1 )ηeT ηe − λmin (C2 M −1 )XeT M Xe 1 ˜T ˜ + ε1 XeT Xe + b b − λmin (K0 )˜bT ˜b + ε˜bT ˜b 4ε1 1 + b˙ T b˙ 4ε ≤ − 2 min[λmin (C1 ), λmin (C2 M −1 ) − ε1 λmax (M −1 ), 1 1 (38) λmin (K0 ) − − ε]V2 + Cd2 4ε1 4ε where λmin (C2 M −1 ) − ε1 λmax (M −1 ) > 0, λmin (K0 ) −
1 − ε > 0, 4ε1
(39) (40)
and λmax (·) represents the largest eigenvalue of a matrix. Therefore, there is the following theorem. Theorem 2: Under Assumption 1, for the 3-DOF nonlinear motion mathematical model of ships with unknown time-variant disturbances given by (1) and (2), the control input vector τ described by (34) together with (8) guarantees that the actual trajectory of ships tracks the arbitrary reference trajectory with the desired accuracy and all the signals of the closed-loop trajectory tracking system of ships are globally uniformly ultimately bounded by appropriately choosing the design parameter matrix C1 , C2 and K0 satisfying the conditions (39) – (40). Proof: Notate µ = min[λmin (C1 ), λmin (C2 M −1 ) − ε1 λmax (M −1 ), 1 − ε], λmin (K0 ) − (41) 4ε1
X˙1 =[R(ψ)S(r)]T [−C1 (η − ηd ) + η˙ d ] + RT (ψ)[−C1 (η˙ − η˙ d ) + η¨d ].
1 ˜T ˜ b b 4ε1
− XeT C2 Xe ≤ −λmin (C2 M −1 )XeT M Xe
Consider the augmented Lyapunov function candidate: 1 V2 = V1 + XeT M Xe + 2
4
(34)
Cd2 . 4ε Then (38) can be rewritten as σ=
V˙ 2 (t) ≤ −2µV2 (t) + σ. Solving the above inequility, we have [ σ σ ] −2µt 0 ≤ V2 (t) ≤ + V2 (0) − e . 2µ 2µ
(42)
(43)
(44)
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5
It is seen from (44) that V2 (t) is globally uniformly ultimately bounded. Hence, ηe , Xe and ˜b are globally uniformly ultimately bounded according to (28), then X1 and ν are globally uniformly ultimately bounded. From the boundedness of ηd and b, we know that η and ˆb are bounded. From (28) and (44), we can obtain √ [ ] σ σ −2µt ∥z1 ∥ ≤ + 2 V2 (0) − e . (45) µ 2µ √ It follows that, for any µz1 > σ/µ , there exits a constant Tz1 > 0, such that ∥z1 ∥ ≤ µz1 for all t > Tz1 . Therefore, the trajectory tracking error z1 of the ship can converge √ to the √ compact set Ωze := {z1 ∈ R3 | ∥z1 ∥ ≤ σ/µ}. Since σ/µ can be made arbitrarily small if the design parameters C1 , C2 , and K0 are appropriately chosen, the actual trajectory of the ship can track the arbitrary reference trajectory. Theorem 2 is thus proved. IV. S IMULATIONS AND C OMPARISONS In this section, the simulation studies are carried out on CyberShip II, which is a 1:70 scale replica of a supply ship of the Marine Cybernetics Laboratory in Norwegian University of Science and Technology. The ship has the length of 1.255m, mass of 23.8kg and other parameters of the ship are given in detail in [17]. We carry out the simulations with two different disturbances. In the simulations, the reference trajectory is chosen as bellow xd = 4 sin(0.02t) yd = 2.5(1 − cos(0.02t)) ψd = 0.02t
(46)
Fig. 3. Constant external disturbances b1 , b2 , b3 and their estimations ˆb1 , ˆb2 , ˆb3 .
which is an ellipse. A. Trajectory tracking under constant disturbances
reference trajectory actual path
5 4 3 y/m
In this section, the disturbance vector is set as b = [2N, 2N, 2N · m]T which corresponds to the environmental disturbances due to slowly varying wind, waves and currents. Assume the initial conditions of the system are [x(0), y(0), ψ(0), u(0), v(0), r(0)]T = [1m, 1m, π/4rad, 0m/s, 0m/s, 0rad/sT and the initial state of the disturbance observer is ˆb(0) = [0, 0, 0]T . The design parameter matrices are taken as C1 = diag([0.05, 0.05, 0.05]), C2 = diag([120, 120, 120]), K0 = diag([2, 2, 2]) such that the conditions (39) and (40) are satisfied for 0.125 < ε1 < 9.6509 and 0 < ε < 1.9741. The results are depicted in Figures 3–7. The external disturbances b and its estimate value ˆb are depicted in Figure 3 from which it is clearly seen that the disturbance observer provides the rapidly exponentially convergent estimation of unknown disturbances within about 1.5s as proved in Theorem 1. From Figure 4, it is observed that the proposed controller is able to force the ship to track the reference trajectory. Furthermore, the curves of the desired and actual positions and yaw angles are presented in Figure 5, which shows
2 1 0 −5
0 x/m
5
Fig. 4. Actual trajectory and reference trajectory in xy-plane under constant disturbances.
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0
xd x
τ1 /N
m
5 0 50
100
150
200
250 yd y
5 50
100
150
200
250
300 ψd ψ
5 0 0
50
100
150 t/s
200
250
300
r/rad · s−1
v/m · s−1
u/m · s−1
Fig. 5. Desired and actual position (x, y) and yaw angle ψ under constant disturbances.
50
100
150
200
250
300
50
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300
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100
150 t/s
200
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300
−5 −10 0 0
τ3 /N · m
rad
0 0 10
−2 −4 0 0
300 τ2 /N
m
−5 0 10
6
−2 −4 0
Fig. 7. Surge control force τ1 , sway control force τ2 and yaw control torque τ3 under constant disturbances.
0.1 0 −0.1 0 0.1
50
100
150
200
250
300
50
100
150
200
250
300
50
100
150 t/s
200
250
300
0 −0.1 0 0.05 0
−0.05 0
Fig. 6. Surge velocity u, sway velocity v and yaw rate r under constant disturbances.
that the actual ship position (x, y) and yaw angle ψ can track the desired trajectory ηd = [xd , yd , ψd ]T at a good precision in around 40s. The curves of the surge velocity u, sway velocity v and yaw rate r versus time are shown in Figure 6. The corresponding control inputs are presented in Figure 7, which shows that the control force and torque are smooth and reasonable. These results reveal that all the signals of the closed-loop trajectory tracking system of ships are globally uniformly ultimately bounded as proved in Theorem 2. Therefore, the proposed trajectory tracking controller is effective for the ship with uncertain constant disturbances. Fig. 8. Time-variant external disturbances b1 , b2 , b3 and their estimations ˆb1 , ˆb2 , ˆb3 .
B. Trajectory tracking under time-variant disturbances In this section, the disturbance vector is set as b(t) =[b1 (t), b2 (t), b3 (t)]T in Figures 8–12, which exhibit almost the same control 1.3+ 2.0 sin(0.02t) + 1.5 sin(0.1t)N performance as under constant disturbances despite the = −0.9 + 2.0 sin(0.02t − π/6) + 1.5 sin(0.3t)N . time-variant disturbances. It is obvious that the designed − sin(0.09t + π/3) − 4 sin(0.01t)N · m controller is effective when the ship is exposed to both The initial conditions of the system and the design pa- unknown constant and time-variant disturbances, which rameters of controller are same as the counterparts in the demonstrates that the proposed controller is robust against first case of subsection III-A. The results are depicted unknown environmental disturbances.
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τ1 /N
5 reference trajectory actual path
5
0 −5 0 10
τ2 /N
4
y/m
3
τ3 /N · m
2 1
−5
0 x/m
100
150
200
250
300
50
100
150
200
250
300
50
100
150 t/s
200
250
300
0 −10 0 10 0 −10 0
0
50
5
Fig. 9. Actual trajectory and reference trajectory in xy-plane under timevariant disturbances.
Fig. 12. Surge control force τ1 , sway control force τ2 and yaw control torque τ3 under time-variant disturbances.
C. Performance comparisons
m
5
In this section, we compare the tracking performance of the designed controller (34) in this paper with the controller without disturbance observer:
xd x
0
m
−5 0 10
50
100
150
200
250
300 yd y
5
rad
0 0 10
50
100
150
200
250
300
0 0
50
100
150 t/s
200
250
300
r/rad · s−1
v/m · s−1
u/m · s−1
Fig. 10. Desired and actual position (x, y) and yaw angle ψ under time-variant disturbances.
0.1 0 −0.1 0 0.1
50
100
150
200
250
300
50
100
150
200
250
300
50
100
150 t/s
200
250
300
0 −0.1 0 0.05 0
−0.05 0
+ [M (S T RT + RT Ccm1 ) + Ccm2 RT ]η˙ d + M RT η¨d + [C(ν) + D(ν) − M RT Ccm1 R − Ccm2 ]ν ∫ t − Kcm [ν + RT Ccm1 (η − ηd ) − RT η˙ d ]dð 0
(47)
ψd ψ
5
τcm = − (M S T RT Ccm1 + RT + Ccm2 RT Ccm1 )(η − ηd )
Fig. 11. Surge velocity u, sway velocity v and yaw rate r under under time-variant disturbances.
which is designed using the backstepping approach for the ship with constant disturbances in [10] with gains Ccm1 = diag([0.05, 0.05, 0.05]), Ccm2 = diag([120, 120, 120]) and Kcm = diag([2, 2, 2]). Figures 13 and 14 illustrate the comparison of tracking performance between the two different controllers under constant disturbances and time-variant disturbances respectively. It can be seen from Figure 13 that both the controller exhibit similarly good transient and steady-state performances under the constant disturbances. However, under time-variant disturbances, it is seen from Figure 14 that the controller τ with disturbance observer in this paper performs better than the backstepping controller τcm with faster decay of tracking error and lower steadystate error value, since our observer provides an estimation of unknown disturbances. In contrast, τcm does not have disturbance compensation, it results in a larger tracking error norm. To quantitatively compare the two controller performance, the performance under both constant and timevariant disturbances is summarized in Table I where xe = xd − x and ye = yd − y representing the error between the desired and actual positions, ψe = ψd − ψ representing the error between the desired and actual yaw angles, and tf inal = 300s. Table I clearly shows that the controller τ has better steady-state and transient performances than the backstepping controller τcm .
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resulting closed-loop trajectory tracking system of the ship are globally uniformly ultimately bounded. Furthermore, the simulation results on an offshore supply ship model has illustrated that our controller is effective and robust to external disturbances. Our proposed trajectory tracking control scheme can provide good transient and steady-state performance for the considered ship system. Future research work would extend the proposed method to address the robust adaptive output feedback tracking of ships subjected to external disturbances and model uncertainties only depending on the position information η = [x, y, ψ]T . From a practical viewpoint, it is convenient since it doesn’t have to measure directly the velocities ν = [u, v, r]T .
Controller τ Controller τcm
||z1 ||
1.5
1
0.5
0 0
Fig. 13. bances.
50
100
150 t/s
200
250
300
Comparison of tracking performance under constant distur-
R EFERENCES 2
Controller τ Controller τcm
||z1 ||
1.5
1
0.5
0 0
Fig. 14. bances.
8
50
100
150 t/s
200
250
300
Comparison of tracking performance under time-variant distur-
V. C ONCLUSIONS In this paper, a trajectory tracking robust control law has been designed for fully-actuated surface vessels in the presence of uncertain time-variant disturbances due to wind waves, and ocean currents. Both Coriolis and centripetal matrix and nonlinear damp term have been considered in the nonlinear ship surface movement mathematical model. The control strategy is introduced by the vectorial backstepping technique with our disturbance observer. The disturbance observer is employed to compensate disturbance uncertainties. It has been proved that the signals of the TABLE I P ERFORMANCE INDEX COMPARISON OF CONTROLLERS τ UNDER DIFFERENT DISTURBANCES .
Disturbances
Constant
AND τcm
Time-variant
Controller
τ
τcm
τ
τcm
settling time ts (s) ∫ tf inal |xe |dt(m · s) 0 ∫ tf inal |ye |dt(m · s) 0 ∫ tf inal |ψ e |dt(rad · s) 0
39 19.4765 17.8455 13.6354
56 25.6207 40.7659 35.1542
41 19.8225 18.2010 13.5816
64 52.0868 55.9768 44.8413
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