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Ship stabilization control using an adaptive input disturbancepredictor

Jingyang. Liu, Robert. Allen, Hong. Yi, Yufang. Zhang

Abstract When ships travel on the oceans, changes in the seastates and the sailing conditions will induce significant uncertainhydrodynamics, leading to a deterioration in the performance oftraditional stabilization systems. To overcome this problem, acombination of Model Predictive Control (MPC) and anadaptive input disturbance predictor is proposed. Thiscombination predicts the wave disturbance force by using apredictive model of the input disturbance and thencompensating for the predicted disturbance within the MPCframework. This has the advantages of MPC and the adaptivemodel, and avoids the complicated robust tuning of a state

observer which is commonly used within the MPC frameworkto reject output disturbances. Model Predictive Control is betterthan classical control at dealing with constraints, and theadaptive disturbance model enhances the ship adaptabilitywhen traveling in varying sea conditions. Very good predictionsof the ship motion are obtained with less degradation underchanges of sailing conditions, thus achieving very good closedloop performance in the MPC framework. An adaptive inputdisturbance predictor based on the time series Auto Regressive(AR) model is used in a numerical simulation which shows thatthis combination works very well.

I. I NTRODUCTION HIP stabilization control has been studied for many years.

The book of Perez [1] gives us a comprehensive pictureof the development of roll stabilization control since the1960s. Good roll stabilization performance has been achieved

by various control strategies through active fin and rudderstabilizers [2]-[4], and several commercial systems can now

be found on the market.Despite this development, much work remains to be done.

For ships traveling on the oceans, changes in the sea statesand sailing conditions will induce significant uncertainhydrodynamics which cannot be described accurately by thenominal model used in control system design [5]. To dealwith such a problem, some conservative designs, such as PID

and H-infinity can produce a less sensitive performance atspecified frequencies [6]. They are, however, based on theassumption that the wave power spectral density will notchange, which is often not true. Another morestraightforward approach is to measure the approachingwaves and use a feedforward channel to compensate.However, this is restricted by the lack of measurementmethods, which are still not ready for practical applications.Oda, Ohstu and Hotta [7], [8] proposed an adaptive rudderroll stabilizer based on an Auto Regressive Moving Average(ARMA) model which actually falls into the framework ofGeneral Predictive Control(GPC). Perez and Goodwin [1], [9]

proposed a new framework based on Model PredictiveControl (MPC). Although his main purpose was to use theadvantages of MPC to deal with the constraints, it did give asolution to address the changes of sea states and sailingconditions. His main idea was to identify a wave disturbancemodel to predict the wave induced motion and then embed itwithin the MPC framework to compensate. Whenever thesailing condition changes, or every 20 minutes, it updates thestate-space disturbance model using the Kalman filter, duringwhich time the control action is switched off. This is good

enough for large ships, such as oil tankers, which havemoderate speed and a relatively constant sailing course. Butfor high speed ships such as fast ferries or naval vessels,sailing conditions can be regarded as constantly changing.

Manuscript received October 6, 2009.J. Liu is with the School of Naval Architecture, Ocean and Civil

Engineering, Shanghai Jiao Tong University, Shanghai, 200240 China(phone: +86(0)2134207163; fax: +86(0)2134207163; e-mail:

jy_liu@sjtu.edu.cn).R. Allen is with Institute of Sound and Vibration Research, University of

Southampton, Southampton, SO17 1BJ, UK. (e-mail: ra@soton.ac.uk).H. Yi is with the School of Naval Architecture, Ocean and Civil

Engineering, Shanghai Jiao Tong University, Shanghai, 200240 China(e-mail: yihong@sjtu.edu.cn).

Y. Zhang is with the School of Naval Architecture, Ocean and CivilEngineering, Shanghai Jiao Tong University, Shanghai, 200240 China(e-mail: zhangyf@sjtu.edu.cn).

Moreover, as Maciejowski [10 pp. 56-61] points out, thesystem model augmented with a disturbance model can beused to reject the immeasurable disturbance in MPC, but thisrequires a well designed state observer to estimate thedisturbance states. If it is not, uncertainties in both the plantmodel and the disturbance model will induce large errors inthe estimated states which are likely to degrade the

performance of the MPC. Efforts to create an effectiveadaptive disturbance predictor could also be renderedineffective by these errors. To enhance the robustness, robustobservers can be chosen [11], or we can use the methods

proposed by Lee and Yu [12]. However, such methods willincrease the complexity of the design.

This paper will give an alternative method to avoid the useof the augmented model. It simply calculates the waveinduced force or moment instead of the wave induced motion.Section 2 will first extend the basic MPC algorithm to dealwith the input disturbance in time series form. Section 3derives the input disturbance predictor based on an AR model

S

2010 8th IEEE International Conference onControl and AutomationXiamen, Chi na, Jun e 9-11, 2010

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and section 4 presents the control algorithm. Section 5shows the numerical results of the closed loop performanceand the discussed. Conclusions are drawn in section 6.

I. COMBINATION OF MPC AND INPUT DISTURBANCE Assume that a dynamic system has this discrete state space

form:

( 1) ( ) ( )( ) ( )

x k Ax k Bu y k Cx k

+ = +=k (1)

The MPC controller should also respect constraints on controlvariables as well as the constrained outputs,

min max( )u u k u min max( )u u k u (2)min max( ) y y k y

In which are the control increments.( ) ( ) ( 1)u k u k u k = The core of the MPC algorithm is to minimize the

following cost function [10 Ch. 4]:1 1

2 2

0 0

( ) ( | ) ( ) ( | ) p u H H

Qi i

R

J k y k i k r k i u k i k

= == + + + +

(3)

In the cost function, are the predicted outputs attime k based upon and are the predictedcontrol inputs, are the input references at the timestep k ; Here we use rather than which is common inLinear Quadratic control. Matrices are the weightingmatrices which are assumed to be constant over the predictionhorizon.

( | y k i k + ))

)i+ k ( |u k i k +

(r k i+i+ u u

,Q P

p H and u H are the prediction and control horizons.

The cost function (3) can be written as2( ) ( ) ( )Q

2

R J k Y k k U = + (4)

Where ( | )( )

( 1 | p

y k k

Y k

y k H k

= + )

)

)

(5)

( | )( )

( 1 | p

r k k

k

r k H k

= + (6)

( | )( )

( 1 | p

u k k

U k

u k H k

= +

(7)(Q, ,Q)Q diag= , are the weightingmatrices. By deriving the prediction expressions, we canwrite

(R, ,R) R diag=

( ) ( ) ( 1) ( )Y k x k u k U k = + + (8)Where

2

1 p H

C

CA

CA

CA

=

(9)

2

0

0

p H i

i

CB

CAB CB

C A

= B

+ =

(10)

2 1

0 0

0 0 00 0

0

p p H H i i

i i

CB

CAB CB

C A B C A

= =u H B

+ =

(11)

Set the error as( ) ( ) ( ) ( 1) E k k x k u k = (12)Then change it to

( ) ( ) ( ) ( 1)k E k x k u k = + + (13)Substitute it and (8) into equation (4) and use these predictionexpressions (5)-(7), we can get

T T T J U H U U G E QE = + (14)Where

2 (T G QE = )k (15)T H Q R= + (16)

This is a standard Quadratic Programming (QP) problem.Thus, the optimal control signals can be obtain by minimize

( ) J k under certain constrains.arg minopt T T T

U U U H U U G

= + E QE (17)

. .s t L U M L , M are the constraints matrices.

The real ship motion in seaways suffers a continuous wavedisturbance in a force-superposition form, take the rollmotion as an example, can be expressed as:

b b fins wave I p Dp G

p

+ + = +

= (18)

In which, denotes the roll angle, and p is the roll rate.

The roll moment generated by the fins and waves are denotedas b fins and

bwave on the right hand side of (18).

The relation from the fin angle to b fins can be expressed as:

1 2b fins K p K = + (19)

Details of obtaining the coefficients of (18) and (19) (theinertia I , damping D , static restoring matrix G and

1K , 2K ) be find in [1].Substitute (19) into (18) and rearrange it, we can get the

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standard continues state space form as:

11 1 1

1 2

0 1 0( )

bwave I I G I D K I K p p

= + + (20)

Using the zero-order hold method and instead withand

u1 b

wave I with , we obtain the standard discrete form as:

( 1) ( ) ( ) (

( ) ( )

) x k Ax k Bu k

y k Cx k

k + = + +

= (21)

where, [ ]T x p = , .2 2C I =If the future prediction of can be gotten from some

kinds of predictive methods. Then we can embed them withinthe MPC algorithm. Apply the same route above; we get theMPC prediction expression as:

( ) ( ) ( 1) ( ) ( 1)Y k x k u k U k k = + + + (22)Here, is the prediction of input disturbance whichcan be expressed as:

( 1)k

2

0

( 1)

( 1) ( | 1)

( 1)

( 1 | p H i

pi

C k

CA k C k k

T k

C A k H i k

=

+ =

+

1)

T

(23)

Set the error as( ) ( ) ( ) ( 1) ( 1) E k k x k u k k c

= (24)The input disturbance is embedded within the MPCframework directly, and we can still write the cost function asa quadratic programming form

T T J U H U U G E QE = + The only differences from the basic MPC algorithm are thatwe need to calculate the past time series of from the sensormeasurements and use a predictive algorithm to get the

prediction .( 1)k

II. ADAPTIVE INPUT DISTURBANCE PREDICTOR In the system (20) , means the state vector can be

measured directly. From this we can easily obtain:2 2C I =

( 1) ( ) ( 1) ( )k x k Ax k Bu k = (25)This expression reveals that the time series of the

disturbance can be calculated from the attitudemeasurements. Although the uncertainties of A and alsoinduce errors, they will not accumulate to the next time step.Consequently the estimation will never get too far away fromthe real disturbance, which makes using the adaptive methodmore practical.

B

In the derivation of section 2, the prediction of the waveinduced motion is not involved. Hence the MPC algorithmdoes not require a specific form of the predictor but just thefuture time series of the input disturbance. Thus, any

predictor can be included into the above framework whennecessary. From this point, the AR model, which is verysuitable for ship motion prediction [13] is selected here.

An AR model has the form:

1 1( ) ( 1) ( 2) ( ) ( )n n y k y k y k y k n e k = + + + + (26)

n is the order of the model which can be obtained by theminimum AIC Estimate procedure [14]. is Gaussiannoise. The model coefficients can be updated at everysample using the least mean square method. By doing thisrecursively, an adaptive predictive model is produced.

( )e k

i

III. COMBINATION OF MPC AND ADAPTIVE INPUTDISTURBANCE PREDICTOR

As mentioned previously, to combine the MPC with inputdisturbances, the only modification of the MPC algorithm isto append the matrix (23) in which all of the unknown areobtained from a predictive model. Here, we use the AR modeldescribed in section 3.

So the full control system design routine is, at everysample step:

1) Get the ship motion measurements and currentcontrol input signal .

( ) y k ( )u k

2) Estimate the past sample ( 1)k by using (25).3) Update the AR predictive model with the new ( 1)k .4) Predict sequence ( | 1) ( 1| 1) pk k k H k + using

the updated AR predictive model (26).5) Solve the MPC problem (17) using the algorithm

described in section 2 to obtain the optimal control sequence.6) Adopt the first control variable of the optimal control

sequence to update the control command.

IV. NUMERICAL SIMULATION For comparison, we adopted the same nonlinear vessel

model used by Perez and Goodwin [1], [9]. The nonlineardynamic feature of fins is discarded here for simplification.Since though, constraints are considered in the MPC designto include the mechanical characteristics of fins.

The parameters of the simulation are: Wave spectrum:ITTC; 1/3 4 H m= ; 7.5T s= ; Ship speed: 15kt; sample time:0.25s; MPC predicted horizon: 10 samples. To change thesailing conditions, the wave direction begins to vary at 50sfrom 60 degrees to 45 degrees.

Fig. 1 shows the simulation results. The open loop performance is the ship response to waves without activecontrol. The combination of MPC with an adaptive

disturbance model achieves a very good closed loop performance even the sailing conditions are changing. Thecase of MPC with a state space disturbance model which was

pre-identified with a wave angle as 60 degrees degradessignificantly after the sailing conditions depart from thedesign conditions at which the model parameters wereestimated. This demonstrates the advantages of thecombination of MPC and an adaptive input disturbance

predictor in dealing with changing sailing conditions.

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The following simulations investigate the tolerance tomodel uncertainties of these two methods. Both methods arecarried out with one accurate model and one inaccurate modelwhich was derived by increasing the diagonal elements of thestate matrix A by 20%. The simulation parameters are thesame as in Fig. 1, except that the wave angle is now keptconstant as 90 degrees.

Fig. 2 shows simulation results of the combination of MPCand an adaptive input disturbance model. As section 3discussed, the use of an input disturbance model avoids thestate estimation. It will not induce large errors due to themodel uncertainties, and thus achieves a good robustness.The use of an output disturbance model needs a well designedobserver to decrease the state estimation errors due to themodel uncertainties. Fig. 3 shows that the inaccurate modelcaused an unacceptable close loop performance. In this case aKalman filter observer was used to estimate the states butwithout robustness consideration.

V. CONCLUSION In this paper, a general framework combining MPC with an

arbitrary adaptive input disturbance predictor is proposed.Instead of the output disturbance, the input disturbance can be

calculated directly from the measurements. This avoids theobserver design and will not introduce accumulated errors.By this simple extension, the MPC algorithm is then modifiedto be able to deal with the input disturbance.

The whole study is based on the linear MPC framework,which is usually a good approximation in ship motion control.In some cases, if the ship motion faces large amplitudeoscillation, nonlinear MPC may be used with this kind of

predictor and this will be investigated in future studies.Simulation results show that this method works better than

the state space output disturbance model case when thesailing condition varies, and the avoidance of the observercan more easily achieve a better robustness to modeluncertainties.

Although the AR model based predictor is the mainmethod adopted here, other predictive methods based on the

past time series can also be used within this framework. Statespace expression of the predictive model is not essential.

0 50 100 150 200 250 300-20

-15

-10

-5

0

5

10

15

20

Time (sec)

R o

l l a n g

l e ( d e g

Open loopMPC with AR disturbance modelMPC with State space dis turbance model

Fig. 1. Comparison of control performance with different disturbancemodels with changing sailing conditions (Wave direction varies from 60 to45 degrees).

The simulations in this paper only used fin roll stabilizersfor demonstration. Indeed, this approach can be applied toother stabilization problems such as longitudinal motionstabilization of catamarans, and rudder roll stabilizations.

0 50 100 150 200 250 300-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

Time (sec)

R o

l l a n g

l e ( d e g

)

Accurate modelInaccurate model

Fig. 2. Comparison of a combination of MPC and an adaptive inputdisturbance model with both accurate and inaccurate models

R EFERENCES [1] T. Perez, Ship Motion Control , Springer, 2005.[2] J.E. Conolly. Rolling and its stabilization by fins, Transactions of the

Royal Institution of Naval Architects , 1969.[3] M. Blanke, P. Haals, and K.K. Andreasen. Rudder roll damping

autopilot experience in Denmark, in Proc. Of IFAC work shopCAMS89 , Lyngby, Denmark, 1989.

[4] J. van Amerongen, P. van der Klugt, and H. van Nauta Lemke, Rudderroll stabilization for ships, Automatica , vol. 26, pp. 679-690, 1990.

0 50 100 150 200 250 300-15

-10

-5

0

5

10

Time (sec)

R o

l l a n g

l e ( d e g

)

Accurate modelInaccurate model

Fig. 3. Comparison of a combination of MPC and a state space outputdisturbance model with both accurate and inaccurate models.

[5] A.R.M.J. Lloyd. Roll stabilization by rudder, 4th Ship ControlSystem Symposium-SCSS , The Netherlands, 1975.

[6] M.Blanke, J. Adrian, K. Larsen, and J. Bentsen. Rudder roll dampingin coastal region sea conditions, in Proc. Of 5th IFAC Conference on

Manoeuvring and Control of Marine Craft , MCMC2000 , 2000.[7] H. Oda, K. Ohtsu, and T. hotta. A study on roll stabilization by rudder

control, Journal of Japan Institute of Navigation , 1995.[8] H. Oda, K. Ohtsu, and T. hotta. Statistical analysis and design of a

rudder roll stabilization system, Control Engineering Practice , vol. 4, pp. 351-358, 1996.

[9] T. Perez, G. C. Goodwin, Constrained predictive control of ship finstabilizers to prevent dynamic stall, Control Engineering Practice, vol.16, pp.482-494, 2008.

[10] J. M. Maciejowski. Predictive control with constrains. Prentice Hall,2002,

[11] Z. Gao, S. Hu and F. Jiang. A novel motion control design approach based on active disturbance rejection, Proceedings of the 40th IEEEconference on Decision and Control , Orlando, Florida USA, December2001, pp. 4877-4882.

[12] J. H. Lee and Z. H. Yu. Tuning of model predictive controllers forrobust performance, Computers in Chemical Engineering , vol. 18,

pp.15-37, 1994.[13] I. Yumori, Real Time Prediction of Ship Response to Ocean Waves

Using Time Series Analysis, OCEANS 1984, vol.13, pp. 1082-1089.[14] H. Akaike, T. Nakagawa, Statistical analysis and control of dynamic

systems , Kluwer Academic Publishers, 1994.

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