Proceedings of the 34th Chinese Control Conference July 28-30, 2015, Hangzhou, China
Dynamic Matrix Control for Dynamic Positioning System of Ships HU Xin, WANG Rui, LIU Yongchao, DU Jialu* School of Information Science and Technology, Dalian Maritime University, Dalian 116026 E-mail:
[email protected] Abstract: This paper presents dynamic matrix control (DMC) for dynamic positioning (DP) system of ships with uncertain ship dynamics and unknown time-varying environmental disturbances. The presented controller is able to make the ship’s position and heading converge to the desired values. Any a priori knowledge of the ship dynamics and disturbances isn’t required in our control design. Simulation studies on a dynamically positioned ship are carried out and the simulation results illustrate the effectiveness of our proposed control scheme. Key Words: Dynamic Positioning Systems, Dynamic Matrix Control, Uncertainties
1
Introduction
Offshore exploration and exploitation of ocean resources at an unprecedented scale have led to the increasing interest in the development of the dynamic positioning (DP) system. The DP system is an automatic control system that acts to maintain the vessel position and heading at a reference point by means of the vessel’s own propulsion systems[1]. The DP system is used in a wide range of vessel types and in different marine operations such as marine construction, wreck investigation and pipe-laying[2]. As such, the DP technology plays an important part in the offshore industry aimed at improving the efficiency and safety of ocean exploitation techniques. In the 1960s, the first DP system based on the PID control technique came to existence[3]. Subsequently, a new model-based control concept, which is based on multi-variable optimal control and Kalman filtering techniques, was employed into the DP control design[4]. With the development of control theories, many nonlinear control schemes have been developed for the DP control design so as to handle the inherent nonlinear characteristics of ships. Fossen and Grøvlen derived a globally exponentially stable nonlinear control law for DP by employing the vectorial backstepping method, where the environmental disturbances are ignored[5]. In the presence of the unknown time varying environmental disturbances, Du et al. proposed a robust nonlinear control design by combining the adaptive techniques with the vectorial backstepping method, where the unknown bounds of the disturbances were estimated through the adaptive laws[6]. All the aforementioned control designs for the DP system require a priori knowledge of the ship dynamics. In practice, the ship dynamics are difficult to be determined since they are related with ship’s operational and environmental conditions which are constantly changing. Considering the uncertain ship dynamics, Wang et al.[7] designed a nonlinear controller for the DP system based on the nonlinear model *Corresponding author This work is supported partly by National Natural Science Foundation (NNSF) of China under Grant 51079013, partly by the Applied Basic Research Program of Ministry of Transport of China under Grant 2012-329-225-070, partly by the Fundamental Research Funds for the Central Universities under Grant 3132014332.
predictive control (MPC) algorithm where the environmental disturbances are neglected. In fact, both the ship dynamics and environmental disturbances are difficult to be determined. Simultaneously dealing with both the uncertain ship dynamics and unknown disturbances are more practical. Under the assumption that both the ship dynamics and the disturbances are unknown constants, Do developed a global robust adaptive control design for the DP system by using the projection algorithm and vectorial backstepping method, where the projection operators were utilized to design the adaptive laws for the unknown constant parameters and disturbances[8]. However, the environmental disturbances acting on ships are unknown and time-varying. Since the late of 1970s, the dynamic matrix control (DMC) has been developed and applied into many control areas[9-11]. The DMC belongs to the model predictive control. Based on the DMC, the control law can be calculated online. Furthermore, the DMC doesn’t require any a priori knowledge of the model parameters and has the good robustness against the environmental disturbances. Motivated by the above works, this paper presents DMC for the DP system. Both the uncertain ship dynamics and unknown disturbances are considered in our proposed control design. Any a priori knowledge of the ship dynamics and disturbances isn’t required. In addition, the simulation studies are carried out on a dynamically positioned ship, and the simulation results illustrate the effectiveness of our proposed control design. The remaining parts of this paper are organized as follows. Section 2 presents the mathematical model of dynamically positioned ships. Section 3 investigates the DMC for the DP system. Section 4 provides the simulation studies to illustrate the effectiveness of the proposed control design. And the conclusions are drawn in the end.
2
Mathematical Model of Dynamically Positioned Ships
Fig. 1 shows the reference coordinate frames of ship motion. OX 0Y0 denotes the earth-fixed frame, where the axis OX 0 is directed to the north, OY0 is directed to the east, and OZ 0 points downwards normal to the earth’s surface. The coordinate origin O of the earth-fixed frame is taken as any point on the earth’s surface. ox0 y0 denotes the
1008
body-fixed frame, where the axis ox0 is directed from aft to
where X
force, the transverse axis oy0 is directed to starboard, and the normal axis oz0 is directed from top to bottom. The coordinate origin A is taken as the gravity center of the ship. The X 0Y0 and x0 y0 planes in these two frames coincide with the still water surface.
A
C 3
[K T ,X T ]T R 6 , Y
K R3 , U
I 3u3 º ª0 3u3 6u6 , B «0 »R 1 M D ¬ 3u3 ¼ >I 3u3 03u3 @ R 3u6 .
E
W R3 ,
ª 03u3 º 6u3 « M 1 » R , ¬ ¼
DMC for the Dynamic Positioning Systems
The DMC mainly consists of three parts: model prediction, rolling optimization and feedback revising, which are repeated online. For a given controlled object, the step responses of the object are ai a (iT c) , i 1, , p ,
0
where T c is the sampling period and p is the prediction horizon. The object step response vector [a1 , a2 , , a p ]
0
Fig. 1: The earth-fixed and body-fixed coordinate frames.
The nonlinear mathematical model of dynamically positioned ships is described as follows[12]. K R (\ )X (1) MX DX W Z (2) T where the vector K [ x, y,\ ] is the position vector consisting of ship position ( x, y ) and heading \ in the earth-fixed frame. The vector X [u , v, r ]T is the velocity vector consisting of surge velocity u , sway velocity v and yaw rate r in the body-fixed frame. The rotation matrix R (\ ) is defined as
R (\ )
ªcos(\ ) sin(\ ) 0º « sin(\ ) cos(\ ) 0» « » «¬ 0 0 1»¼
will be used to describe the system predictive model[13]. At the sampling point t k , the control input is u (k ) . The responses of the system at the sampling points t k 1, , k p are notated by
y0 ( k 1 | k ), y0 (k 2 | k ), , y0 (k p | k ) (7) which are called the predictive initial values. At the sampling points t k , , k m 1 , the control input is added by 'u ( k ), 'u ( k 1), , 'u ( k m 1) , respectively. Thus, the responses of the object are represented by yˆ ( k 1 | k ), yˆ (k 2 | k ), , yˆ (k p | k ) . (8) where m is the control horizon. According to the superposition principle, we have yˆ ( k 1 | k ) y0 (k 1 | k ) a1'u (k ) yˆ (k 2 | k ) y0 (k 2 | k ) a1'u (k 1) a2 'u (k ) yˆ (k p | k ) y0 (k p | k ) a pm1'u (k m 1)
(3)
M is the inertia matrix including added mass forces and moments. D is a linear damp matrix. W [W 1 ,W 2 ,W 3 ] is the control input vector consisting of the surge force W 1 , sway force W 2 and yaw moment W 3 . Z [Z1 , Z 2 , Z3 ] is the disturbance vector consisting of the disturbance forces in surge Z1 and in sway Z 2 , and disturbance moment Z3 in yaw. Here, we assume that M and D in (2) are unknown and the environmental disturbance Z in (2) is time-varying and unknown. The dynamic parameters M and D are related with the ship’s own operating conditions which are constantly changing. The changing environment is difficult to be predicted. Therefore, the assumptions are reasonable and practical. Under the condition that the heading \ is very small, we obtain (4) R (\ ) # I 3u3 In the light of (4), we can transfer (1)-(2) into (5) X AX BU EZ Y CX (6)
a p 'u (k )
(9)
Thus, we have ª yˆ(k 1| k ) º « yˆ(k 2 | k ) » » « » « » « ˆ ( | ) y k p k ¼ ¬
T
ª a1 0 «a « 2 a1 « « a a ¬« p p1
0 ºª 'u(k) º ª y0 (k 1| k) º 0 »»« 'u(k 1) » « y0 (k 2 | k) » » »« « » » « »« »« » » « a pm1 ¼»¬'u(k m 1)¼ ¬ y0 (k p | k)¼
(10) which is called as the predictive model. 3.1
Predictive Model for Dynamically Positioned Ship
In this section, we firstly establish the predictive model for the dynamically positioned ship (5)-(6). When the surge force W 1 , sway force W 2 and yaw moment W 3 are taken as step signals respectively, the step response vectors in surge, sway and yaw are a11 , a12 , a13 under W 1 , a21 , a22 , a23 under W 2 , and a31 , a32 , a33 under W 3 . All
aij
[aij ,1 (T ),, aij , N ( NT )] ( i 1,2,3 , j 1,2,3 ) are
1 u N vectors with T as the sampling period and N as the prediction horizon. Notate the matrix Aij as
1009
Aij
ª aij ,1 «a « ij , 2 « « «¬aij , N
0
º 0 »» » » aij , N M 1 »¼ N uM
aij ,1 aij , N 1
0
Here, yai and y di are the actual and desired values of the (11)
c
where M c is the control horizon. According to (10), we can obtain the dynamic matrix for the dynamically positioned ship (5)-(6) as
A
ª A11 « « A21 « A31 ¬
A12 A22 A32
A13 º » A23 » A33 »¼ 3 N u3 M c
(12)
At the sampling point t k , the control input is U (k ) . The step responses of the dynamically positioned ship model (5)-(6) at the sampling points t k 1, , k N are represented by the following vector (13) Y0 ( k ) [Y01 (k ); Y02 (k ); Y03 (k )]3 N u1 with Y0i ( k ) [ y0i (k 1 | k ), , y0i ( k N | k )]TN u1 (14) where i 1,2,3 . At the sampling points t inputs in surge, sway
k ,, k M c 1 , the control
and
are added by 'ui (k ), 'ui (k 1),, 'ui (k M c 1) , i 1,2,3 , respectively. Notate 'U (k ) ['U 1 (k ); 'U 2 (k ) ; 'U 3 (k )]3 M c u1 (15)
ship position and heading, respectively, and D ir are the smoothing factors. In order to compute the optimal control increment, the online optimization is needed at every sampling point. The objective function of the optimization is chosen as
min J (k ) [W (k ) Y0 (k )]T Q[W (k ) Y0 (k )] 'U T (k ) R'U (k ) where ªQ11 Q «« Q22 «¬
Thus, the predictive outputs of the object are represented by Y (k ) [Y1 (k ); Y2 (k ); Y3 (k )]3 N u1 (17) with Yi (k ) [ yi ( k 1 | k ), , yi (k N | k )]TN u1 (18) According to (10), we can obtain the predictive model of the dynamically positioned ship as
Y0 (k ) A 'U ( k )
Y (k ) 3.2
(19)
Rolling Optimization
Let the desired target vector of the ship position and heading be K d [ y d 1 , y d 2, , y d 3 ]T . Let the reference outputs of the dynamically positioned ship at the sampling points t k 1, , k N be
W (k ) [W1 (k );W2 (k );W3 (k )]T3 N u1
with
wi (k r ) D y ai (k ) (1 D ) y di , r 1, , N , i 1,2,3 r i
and
diag (ri1 ,, riM c ) R are the weight matrices. Here, diag () denotes a dialog matrix. Let
wJ (k ) w'U (k )
0
(24)
we have
'U * (k ) [ 'U 1* (k ); 'U 2* (k ) ; 'U 3* (k )]3 M c u1 ( A T QA R) 1 A T Q[W (k ) Y0 (k )]
(25)
where
'U i* (k ) ['ui* (k ), 'ui* (k 1), , 'ui* (k M c 1)]TM c u1 (26) Once the optimal input 'U i* ( k ) is computed, only the first input 'ui* (k ) is implemented and the rest is discarded. Notate 'U ( k ) [ 'u1* (k ), 'u 2* ( k ), 'u 3* (k )]T3u1 (27) Thus, we have U (k ) U (k 1) 'U (k ) (28) 3.3
Feedback Revising
Consider the uncertain ship dynamics and unknown time-varying disturbances, there exist some errors between the predictive output vector Y (k ) and the actual output vector Ya (k ) . Under the control input (28), the actual outputs of the dynamically positioned ship in surge, sway and yaw at the sampling time t k 1, , k N are notated as
yai ( k 1 | k ),, yai (k N | k ) , i 1,2,3 . Then we have the actual output vector Ya (k ) of the dynamically
positioned ship
(20)
Ya (k ) [Ya1 (k ); Ya 2 (k ); Ya 3 ( k )]T3 N u1
where
Wi (k ) [ wi ( k 1), , wi (k N )]TN u1 .
R22
º » with » R33 »¼ 3 M u3M c c
M c uM c
Rii
with
(16)
ª R11 « « «¬
diag ( qi1 , , qiN ) R N u N
Qii
yaw
'U i (k ) ['ui (k ), 'ui (k 1), , 'ui ( k M c 1)]TM c u1
º » ,R » Q33 »¼ 3 N u3 N
(23)
(29)
with
Yai ( k ) [ y ai ( k 1 | k ),, y ai (k N | k )]TN u1 (30)
(21)
r i
(22)
Define the error vector between the predictive outputs and the actual outputs e(k ) Ya (k ) Y (k ) (31) Define the corrected vector as
1010
Yˆ (k ) Y (k ) He(k ) ª H 11 « where the design matrix H H 22 « «¬
(32)
r3,1
r3,5
º » » H 33 »¼ 3 N u3 N with H ii diag ( hi1 ,, hiN ) R N u N , i 1,2,3 . Yˆ (k ) [Yˆ1 (k ); Yˆ2 (k ); Yˆ3 (k )] R 3 N with Yˆi (k ) R N , i 1,2,3 . Here, Yˆ (k ) is used to update the predictive initial vector at the next sampling point k 1 as Y0 (k 1) S 0Yˆ (k ) (33)
h2,1
h2, 49
D
D
S
D
0
º 0 7 » 1.492 u 10 » 6.521 u 1010 »¼
1.7746 u 10 6
º » 1.7746 u 10 6 » 7.1506 u 108 »¼
r1,1
r1, 5
0.017
,
r2,1
q3, 49 r2,5
2606.1 , 0.0089
30
xy -plane.
20
DMC PID
0 -20
0
50
100
150
200
40 20 -20
\/q
DMC PID
0 0
50
100
150
200
0 DMC PID
-0.5 -1
0
50
100 t/s
Fig. 3: Ship’s actual position
150
200
( x, y ) and heading \ .
2 0 150
DMC PID 200
150
DMC PID 200
150
DMC PID 200
-2 50
100
0 -2 0
The design parameters are chosen as T 0.4 s , N 49 , M c 5 , q1,1 q1, 49 449.32 ,
179.73 , q3,1
DMC PID
50
100
0.02
K (0) [0m, 0m, 0 0 ]T and X (0) [0m / s, 0m / s, 0 0 / s]T .
q 2, 49
0.2 .
10
0
Let the desired position and heading of the ship be K d [ 20m, 20m, 0 0 ]T , and the initial states as
q2,1
,
2
0
2.2204 u 105
1 .6
49 3
40
y/m
ª2.2204 u 10 4 « 0 « « 0 ¬
1.492 u 10 7
D
10 20 x/m Fig. 2: Trajectory of the ship in
v/ms -1
M
D
h3, 49 1 3
0
u/m s -1
To validate the proposed DP controller based on the DMC in Section 3, the simulations are carried out on a dynamically positioned ship in two different disturbance cases in this section. Furthermore, performance comparisons between the PID controller in [14] and the proposed DP controller are presented to show the effectiveness and robustness of the proposed DP controller. 4.1 Performance of Proposed DP Controller Based on DMC In this subsection, the proposed DP controller based on DMC is tested on a dynamically positioned ship which is 175m in length. The M and D in the motion model (2) are[14]
0 3.346 u 10 7
D
49 2
,
0
Simulations
ª2.642 u 10 7 « 0 « « 0 ¬
h3,1
,
1
with
ª0 1 0 0 º «0 0 1 0 » « » S «0 0 0 » . « » « 1» «¬0 0 0 1»¼ Nu N Based on the above processes, the control input (28) can be generated iteratively. Under this control input, the dynamically positioned ship can converge to the desired target values.
4
D
1 2
20 y/m
º » » S »¼ 3 N u3 N
1 .3
h1, 49
30
x/m
S0
ªS « « «¬
49 1
h1,1
,
Case 1: The simulations are carried out in moderate sea. The significant wave height Hs=2m. The simulation results are depicted using solid line in Figs. 2-6.
,
1011
r/ qs -1
where
1 1
0.000225
0 -0.02 0
50
100 t/s
Fig. 4: Ship’s surge velocity u , sway velocity rate r .
v and yaw
6
0
20
50
100
150
x/m
W1/N
5 0 -5 -10
x 10
DMC PID 200
-20
6
x 10
0
50
100
150
-20
\/q
W3/Nm
x 10
0
50
Fig. 5: Ship’s surge force
DMC PID 150 200
100 t/s
W 1 , sway force W 2 ,
yaw moment
u/ms-1
4
Z1/N
5 0
50
100
150
200
v/ms-1
Z2/N
x 10
-2.5 0
50
100
150
Z3/Nm
r/qs-1 50
100 t/s
150
50
0
50
DMC PID 150 200
100 t/s
( x, y ) and heading \ .
DMC PID 0
50
100
150
200
0 -2 50
100
150
DMC PID
200
0 50
Fig. 9: Ship’s surge velocity
Z3
DMC PID
100 t/s
150
u , sway velocity v
DMC PID
200
and yaw rate
r.
6
30
10 5 0 -5
x 10
0
DMC PID 50
100
150
200
W2/N
6
10 5 0 -5
10
x 10
0
DMC PID 50
100
150
200
6
W3/Nm
y/m
0
0
Case 2: The simulations are carried out in High Sea in this case. The significant wave height Hs=9m. The simulation results are depicted using solid line in Figs. 7-11.
20
2 0 -2 -4
200
Fig. 6: Disturbance forces Z1 , Z 2 and disturbance moment in moderate sea when Hs=2m.
100
DMC PID 150 200
-0.05
W1/N
0
200
0.05
x 10
3 2
1 0 -1 -2
0
200
5
4
150
2
4
-2
100
Fig. 8: Ship’s actual position
W3 .
x 10
0
50
0
DMC PID 200
6
5 0 -5 -10
0
20
y/m
W2/N
5 0 -5 -10
DMC PID
0
0 0
10 x/m
20
Fig. 7: Trajectory of the ship in
30
0
x 10
-5 -10
0
50
100 t/s
Fig. 10: Ship’s surge force W 1 , sway force
xy -plane.
1012
DMC PID 150 200
W2 ,
yaw moment
W3 .
5
4
Z1/N
10
x 10
In this paper, based on the DMC, the controller has been designed for the DP system with uncertain ship dynamics and unknown time-varying environmental disturbances. The control design doesn’t require any a priori knowledge both of the ship dynamics and disturbances. The effectiveness of our proposed control design has been illustrated by simulation studies on a dynamically positioned ship. In the future, we will further do the experiments to validate the proposed control scheme based on the model ship which is being built in the laboratory.
5 0
0
50
100
150
200
50
100
150
200
5
Z2/N
x 10 -1.8 -1.85
0 6
References
Z3/Nm
x 10 3.6
[1]
3.5
[2]
0
50
100 t/s
150
200
Fig. 11: Disturbance forces Z1 , Z 2 and disturbance moment in high sea when Hs=9m.
Z3
[3]
4.2 Performance Comparisons with the PID controller In this subsection, the performance comparisons between the PID controller and the proposed DP controller are conducted. The PID controller is given as
W PID
Conclusion
³
t
K p z1 (t ) K i z1 (W t )dW t K d z1 (t ) 0
(34)
where z1 (t ) K (t ) K d , K p , K i , and K d are the control gains. K p diag(3 u104 ,3 u104 ,1.5 u108 ) , K i
[4]
[5]
diag (10,10,10) ,
and Kd diag(1.1 u 106 ,1.1 u 106 ,7.5 u 109 ) . The simulation studies for the PID controller are carried out in two cases in subsection 4.1. The simulation results for case 1 are depicted using dash line in Figs. 2-5 and the simulation results for case 2 are depicted using dash line in Figs. 7-10. From Fig. 2, it is observed that both the proposed controller and the PID controller can keep the ship at the desired target position in the xy -plane with satisfactory performance in two cases. Further, Fig. 3 shows that the ship position ( x, y ) and heading \ arrive and keep at the desired target value in around 100s. The ship velocities in surge, sway and yaw are shown in Fig. 4. The corresponding control forces and moment are plotted in Fig. 5, which shows that they are smooth and reasonable. In addition, Fig. 6 presents the environmental disturbance forces and moment acting on ships. Furthermore, it is observed from Figs. 7-10 that the proposed DP controller exhibits the similar positioning performance in each of the two cases. However, it is observed from Figs. 7 and 8 that the PID controller can’t position the ship at the desired target values in the simulation time 200s in the case 2. The above results illustrate that our proposed DP controller for the dynamically positioned ship is effective and robust in the presence of unknown time-varying disturbances.
[6]
[7]
[8] [9]
[10]
[11] [12] [13] [14]
1013
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