Nonlinear Output Feedback and Locally Optimal Control of Dynamically Positioned Ships: Experimental Results

Copyright © IFAC Control Applications in Marine Systems, Fukuoka, Japan, 1998

NONLINEAR OUTPUT FEEDBACK AND LOCALLY OPTIMAL CONTROL OF DYNAMICALLY POSITIONED SHIPS: EXPERIMENTAL RESULTS

J ann Peter Strand * Thor 1. Fossen *

* Department of Engineering Cybernetics, Norwegian University

of Science and Technology, N-7034 Trondheim, Norway. e-mail: [email protected]@itk.ntnu.no

Abstract: In this paper an output feedback controller for dynamically positioned ships is presented. The control law is designed in the framework of locally optimal backstepping design, where locally an 1ioo -optimal controller is found. Globally the system is rendered inverse optimal by construction of the cost function. A nonlinear observer is used in the output feedback backstepping design. Results from experimental tests with a model ship are presented. Copyright @ 1998 IFAC Keywords: Dynamic positioning, nonlinear output feedback control, locally optimal backstepping, experimental tests.

ships was derived in Strand et al. (1998). The proposed control system has been implemented on a model ship, scale 1:70, and the results from the experiments are reported.

1. INTRODUCTION

Traditional control systems for dynamic positioning (DP) of ships are based on different design methodologies within linear optimal control theory, often a Kalman filter in conjunction with a linear quadratic controller, see S0rensen et al. (1996) and the references therein for an overview. A first attempt towards nonlinear output feedback control of dynamically positioned ships was made in Fossen and Gr0vlen (1998) by using backstepping (Krstic et al., 1995). This work has been further extended by Aarset et al. (1998) to include wave filtering and integral action, where a passive nonlinear observer proposed in Fossen and Strand (1999) is used in the observer backstepping.

2. SHIP MODEL Consider the model of Fossen and Strand (1999):

~ = Aw~

+ Ewww

iJ=J('If;)v .

1

b= -T- b + Ebwb MiJ=-Dv+JT('If;)b+Tthr y = "l

In this paper an output feedback DP controller is constructed by backstepping in order to meet one local and one global design objective. The first objective is to find an optimal controller for the linearized ship model. The second objective is to achieve global inverse optimality for the nonlinear system. In the construction of the control law, a locally optimal backstepping methodology is used, based on the results of Ezal et al. (1997). A locally optimal state feedback controller for

+ "lw = "l + Cw~ '

(1)

(2) (3) (4)

(5)

It is common to separate the motion of a ship into

a low-frequency (LF) model (2)-(4) and a wavefrequency (WF) model (1). Here "l = [x, y, 'If;JT represents the position (x , y) of the ship and the orientation (heading) 'If; of the ship in the horizontal plane relative to an earth-fixed frame, see Fig. 1. The body-fixed velocities are represented by the vector v = tu, v,rJT. M is the inertia

83

x

A3 Stable ship. This implies that the linear damping matrix D = DT > O. A4 The stochastic white noise terms Ww and Wb are treated as zero-mean bounded signals. A5 Based on physical considerations the vessel velocities are bounded. y

3. OBSERVER An observer is needed to recover the low-frequency part of the position and heading measurements, the so-called wave filtering. Moreover, estimates of non-measured velocities must be found in addition to bias estimates, which accounts for slowly varying environmental loads and unmodelled effects. The observer used in this paper was proposed by Fossen and Strand (1999):

Fig. 1. Reference frames. matrix (including hydrodynamic added inertia) and D is a linear damping matrix. All the environmental loads, such as 2nd-order wave-induced forces, ocean currents and wind, are lumped into a slowly-varying bias term b E ~, where T and Eb are constant diagonal matrices and Wb E ~3 is a white noise vector. The control inputs are the forces developed by the propulsion system, decomposed in surge, sway and yaw, and represented by the vector 7thr. The WF motion is modeled as a linear model of order p, where ~ E ~3 .p and TJ w = [xw, Yw, 'lj;w]T is the output. In this paper it is assumed that only position and heading are measured. The WF motion is added to the LF response (5). From (5) this is the measurement vector y E ~3. When designing tracking control systems, the reference vector is a smooth time varying signal TJd E er with corresponding reference velocities Vd = [Ud, Vd, rdjT. Hence:

ry = J('lj;)v,

ryd = J('lj;d)Vd,

~

[

cosa -sina sin a cos a

o

0

0]° .

where

S

-1 = [0 1 0

0]

000 0

-DD + JT('lj;y)b JT ('lj;y)K4Y

r, + ew~

(13) (14)

where Y = y - fj and K1 E ~2,pX3,K2,K3,K4 E ~3x3 are observer gain matrices. The passivity and stability properties of the observer are fully treated in Fossen and Strand (1999), where it is shown that the observer error dynamics is input-to-state stable (ISS) with respect to the disturbances Ww and Wb. If we disregard these zero-mean disturbances in the analysis, the observer error dynamics is proved to be GES.

(6)

(7)

(8)

4. CONTROLLER DESIGN

Time differentiation of (8) yields the vessel parallel kinematic equation (Strand et al., 1998):

J('lj; - 'lj;d)v - rdSe - Vd

(12)

fj =

Note that J-1(a) = JT(a) . Deviations from the desired position and heading relative to this vesselparallel frame are given as:

e=

-T- 1b+ K 3 y

+7thr +

1

e = JT('lj;d)(TJ - TJd)'

ry = J('lj;y)D + K 2y

(10) (11)

b= Mf/ =

where the rotation matrix J(a) is: J ( a) =

= Aw~ + K 1 y

Since we do not want the WF part of the vessel motion to enter the feedback loop, we use the LF observer equations in the controller design. This design is done in the framework of locally optimal backstepping (LOB), a new design formalism derived by Ezal et al. (1997). A state feedback controller for ships using LOB was derived in Strand et al. (1998). In this paper regulation is considered such that TJd == constant and Vd == O. The bias estimate b produced by the observer is treated as a feedforward term, and a new control input is defined as:

(9)

= _sT.

The following assumptions are made for the stability considerations in the forthcoming sections:

Al Low speed assumption. In this case. the inertia matrix M = MT > 0 and M = 0 (Newman, 1977). A2 J('lj;)::::: J('lj; + 'lj;w) or J('lj;) ::::: J('lj;y), where 'lj; denotes the measured heading. This is a g~od assumption since the magnitude of the wave-induced yaw motion 'lj;w will be less than 5 degrees in extreme weather situations and less than 1 degree during normal operation.

(15) For notational simplicity, let 'lj;ey ~ 'lj;y - 'lj;d' By using (8), (9) and (15) the low frequency observer equations (11) and (13) can be formulated in vessel parallel variables as: 84

where the sub-matrices are of dimension 3 x 3. By using (22) the GARE (21) transforms into:

(16) I fj = _M- Dv + M-lu + H 2('l/Jey, 'l/Jd)iJ. (17) where HI and H2 are bounded matrices containing the observer gain matrices:

,F6.+6..A+ fIofI'[ - BR- I BT) 6. + Q = 0

6. ( ~2

(23)

.A = LAL-l, fIo = LHo and Q = L-TQL- l . The system will be mapped into new

where

HI ('l/Jd) = J('l/Jd)K2 H2('l/Jey, 'l/Jd) = M-I f[('l/Jey)JT('l/Jd)K4

coordinates by the diffeomorfism

z = (x) = Lx + (x)

Note that this system is in block strict-feedback form, which makes it suitable for backstepping controller design.

where z = [zf, z:IY. Here, (24) consists of a linear term Lx and (x) which contains higher order terms. By application of (24) the linear part of the system (18) becomes in new coordinates:

4.1 Locally Optimal Backstepping

(25) where the 3 x 3 sub-matrices of .A and fIo are

The LF observer equations (16)-(17) can be linearized about a working point where J(O) = I, and written compactly as: :i; =

Ax + Bu + H oiJ

.All A2l

(18)

where x = [eT , vT]T. Here, iJ is treated as a disturbance term. The control law is designed to meet one local and one global objective. The system (18) with the linear control law Ul must satisfy the cost functional:

J1 =

100 [xTQx + uT RUl - ,.liJTiJ]dt

J = loo[q(x) + uTR*(x)u - ,2iJTiJ]dt.

0

=

.A12 = I, 2 I -!vl- DLo - Lo, A22 = -M- D - Lo, I

+ H 2(0 , 'l/Jd)'

By using the Lyapunov function

V = zT6.z

(26)

for the transformed system (25) we get by completion of the squares and inserting the GARE (23) :

(19)

if:s -zTQz-u7Ru+,2iJT y where the 'Hoo-optimal controller for (25) is: (27)

(20)

The backstepping of the nonlinear system has to be performed step-wise. At each step i the stabilizing function is chosen to contain the terms from the linear backstepping. The" bad" nonlinearities at each step are cancelled, whereas "helpful" nonlinearities can be kept. At the final step the control law is selected. By using the Lyapunov function (26) a positive definite function q(z) and a positive definite matrix R*(z) are constructed to satisfy:

This is an inverse approach, since q(x) and R*(x) are not specified beforehand. It is required that q(x) is positive definite and R*(x) = R;(x) > O\fx where R*(O) = Rand qxx(O) = Q. The two design objectives can be met by applying LOB. The linear control law for (18) is Ul = - R- l BT Px where P = pT > 0 is found by solving the generalized algebraic Riccati equation (GARE):

+Q =

= Lo,

fIoI = HI ('l/Jd) , fIo2 = -LoHI('l/Jd)

where Q and R are positive definite cost matrices. The second objective is to achieve global inverse optimality with respect to the cost functional

PA+ATp+ P (..1.. H 0 HT - BR- l BT) P ')'2 0

(24)

if

:s -q(z) -

uT R*(z)u + ,2iJTiJ

(28)

which also must satisfy the locally optimal requirements in z coordinates.

(21)

For systems in block strict feedback form, P can be factored into I 4.2 Nonlinear Backstepping

(22)

Step 1: Define the first error variable as Zl = Hence:

where L is a lower triangular matrix with identity matrices along the diagonal and 6. = 6.T > 0 is a block diagonal matrix. In this particular case we have:

6.

1

=

6. 1

[ 0

0] 6. 2 '

Zl = al

[ I 0] L = - Lo I

+ (Jeyv - ad + HliJ·

e.

(29)

Here Jeyv is the virtual control and al is the stabilizing function for the first step. In order to match the linear backstepping al = LoZl where the second error variable is:

This is obtained by thc uniquc Cholesky factor of p-1 =

(30)

UTU , where 1:::.1 / 2L = U- T .

85

4.3 Control Law

Step 2: Time differentiation of (30) yields: Z2 = JeyD

+ jeyv - al

(31)

Starting with the Lyapunov function (26) we get:

where it can be shown that:

jey = (rS

11 =

+ ;PwS - rdS) Jey .

zT(.6.A + AT .6.)z + 2zT .6.Byu + 2zT HoY + zT(.6.N + N T .6.)z + 2zT HyY + (;zT(.6.Y + yT .6.)z. (40)

(32)

Here r is the LF yaw velocity and ;Pw is the WF part of the m~tion. In regulation r d = O. By substitution of v with the observer equation (17), using (32) and substituting for Jeyv = Z2 + LoZl the expression for Z2 becomes:

We do completion of the squares of:

2zT .6.Byu- = Ilu + R;l(-)BT .6.zII~R. T y -z .6.ByR;l(-)B;.6.z + _uT R*(-)u,

Hr

2zT .6.HoY = -~ IIY +,2 .6.z11 T .6.H fIT.6.z + 2y-Ty+...L z ")'2 0 0 I ,

l

Z2 = -JeyM- DJ~z2 - JeyM- l DJ~LoZl 1

+JeyM- u + (r -L~Zl

.

(41)

2 (42)

"V

.

+ 'l/J w)SZ2 + (r + 'l/Jw)SLoZl - LoZ2 - LoHdj + Jey H2Y (33) + ;Pw) as r + 'l/J w = f + {;

(43)

We propose to rewrite (r

-

Here {; ~ i' + ;Pw is a bounded signal as stated in the following proposition.

- zT .6.ByR;l(-)B;.6.z + zT(.6.N + iF .6.)z 1 T - -T T T + "2z2 .6.2H2yH2y.6.2z2 + {;z (.6.Y + Y .6.)z

and the following bound can be found: (35)

Cl

+ IIu + R;l(-)B~ .6.zll~. -

The bound on i' is established by the ISS property of the observer. The signal ;Pw is the time derivative of the WF motion in yaw and must be bounded, based on physical considerations.

-

-

-

-

-

:::; alllzll1

Az + By('l/Jey)u + HoY

2

+ {;zT(.6.Y + yT .6.)z 2 + a21hl1

NI

TTZ Qz = zl QUZl

+ 2z2T-Q12Zl + z2T-Q22Z2 (46)

The cross-terms involving Zl and Z2 in 11 can be removed by completion of the squares of:

+ 2z'{ (.6. 2N l - (12) Zl QIl (.6.2Nl - (12) z211~1I

-ZiQllZl

+ 1'S (38)

=-

IIZl -

T-T-lT-T -1+Z2 Q 12 Q Il Q12 Z2 + Z2 (NI .6. 2QIl .6.2N l - T - 1 -T - 1 -NI .6. l2 Qll Q12 - Q 12 Q I1 .6.2N1)Z2. (47)

+ (;Yz

+N('l/Jey, 1')z + H('l/Jey)Y

(44)

where al = {;maxc~ and a2 = (4 + c5ma~i3'2). Here, Et £2 102, f3l and f32 are positive scalars, where the first can be chosen freely. Now, rewrite:

where NI (0,0) = N2 (0 , 0) = O. The backstepping creates a mapping into new coordinates:

z=

2

(45)

(37)

N2('l/Jey, f) = M-I D - JeyM- l DJ~

2

I

1

NI ('l/Jey,f) = N 2('l/Jey,f)Lo

IIY +,2 H; .6.z11

+ 10 2H~ .6.z11 + (,2 + ci)iJT Y

~z'{ .6.2H2yH~.6.2z2

+ A22Z2 + Nlz l + N2 Z2 + JeyM- u +{;SZ2 + (;SLoz l - LoHlY + Jey H2Y (36)

where the nonlinear terms are collected into and N2 which depends on l' and 'l/Jey:

12 IIY

10

~2

Since H2y is a bounded matrix and Y is a constant matrix the following bound can be found by completion of the squares and by using (35):

Note that the yaw velocity estimate f is given in terms of Zl and Z2 by l' = ZJ3} + ai 3 } where (-) {;} denotes the i-th vector element. By adding and subtracting terms to obtain A 2l Zl and A 22 Z2 and using (34) we have: Z2 = A 2l Zl

-zTQz - uT R*(-)u - zT .6.BT R- l B.6.z

11 =

Proposition 1. The signals i' and ;Pw are bounded

I{;I = Ii' + ;Pw I :::; {;max

-T

where R* (-) R* (-) > 0 is the nonlinear cost matrix to be determined later and " Cl are positive scalars. By using the above expressions and inserting the GARE (23) into (40) we have:

(34)

The following nonlinear terms are collected into a new matrix

(39)

-

where Hy(O) = 0 and

,Do

-T

--1

-

-T

--1-

"Eo('l/Jey,r) = NI .6. 2Q1l .6. 2N1 - NI .6. 2 Qll Q12 -T -T - 1 +.6. 2N 2 + N2 .6. 2 - Q12 QIl .6.2N 1 (48)

Hy( 'l/Jey) = [H2}'l/Jey) ] where to(O, 0) = O. Thus,

with H2y('l/Jey) = Jey H2('l/J ey ) - H2(0). 86

11 can be written as:

V:S -//Zl - 0 1/ (~2Nl - (12) Z21/~11

GNC laboratory

-Zf(022 - Of20111( 12)Z2 - UT R*(.)u 1 1 A -zf(~2J.ey M-I R* (.)M- JT ey~2 - ~2M-l R- 1M-I ~ 2 - t 0 (.,. 'l-'ey' r')) Z2 +(·l + ci)fjT y + I/u + k;l( .)B[ ~z,,2_

Basin

x

R.

2 +a111 z111 + fr211z2112.

y

(49)

The nonlinear control law is chosen as:

u = _R-;I(·)M-l J?;,.~2Z2

Fig. 2. Experimental setup (left) and Cybership I (right).

(50)

where R* (-) remains to be decided. Define


If Conjecture 2 is not satisfied, nonlinear damping must be added to the control law.

(51)

Control Law. One possible choice for R.(·) that makes sure that the matrix is invertible and continuous for all z is:

(52)

t(1/Jey, r) ~ M J?;,.~21to(1/Jey , f)~21 Jey M(53)

R-;IC) = 11-1 (1/1ey, f)J + 11-2 (1/1 ey ,f)R;; 1(1/1 ey ) (58)

where R;;l(O) .= R and t(O,O) = O. Thus, the expression for V can be written compactly as:

where J is the 3 x 3 identity matrix and

11-1 (1/1ey, f) = max {O, >'d ,

V:S -q(.) - uT R.(·)u + Cl + d)yTy +al IIzdl2 + fr211z2112

(54)

11-2 (1/1ey , f) = { (1 + 1>'1

where

k

)-1 : otherwise

>'1 = >'max(t(1/Jey' f)), >'2 = >'max(R;;I(1/1ey))

qC) = I/ Z1 -

OIl (~2Nl(1/Jey,f) -

for any k 2: 0.:. Here, >'max(') denotes the largest eigenvalue of R;;I(1/Jey ) and t(1/J ey ,f) for the current value of 1/Jey and f. By this choice of 11-2(' ) the control effort will be reduced in cases where the nonlinearities are helpful for the stability. In this case a certainty equivalence controller has been found, since the control is equal to the state feedback version (Strand et al., 1998), only that the actual states are replaced by the estimates and the measured heading 1/1ey .

(12) z211Q2_

I I

+
-

-T -

1-

+Z2 (Q22 - Q1 2Q11 QI2)Z2.

(55)

In the further design we must make sure that q(.) is positive definite and that V is an ISS Lyapunov function with respect to y. The obstacle are the last two terms in (54).

Conjecture 2. A lower quadratic bound on q(.) can be found that satisfies

5. EXPERIMENTAL RESULTS

(56) where Cl > al and C2 > fr2. In the general case, the quadratic lower bound on q(.) cannot be established since the growth of q(.) depends on the nonlinearities in NI. However, for this particular system such a bound can be found by using Assumption A5 in the analysis. !:::, Under Conjecture 2, the expression for

V:S

~ >'2I

The proposed output feedback controller has been implemented and tested at the Guidance, Navigation and Control (GNC) Laboratory, NTNU. We have used Cybership I, a model ship of scale 1:70, see Fig. 2. The length of the model ship is Lm = 1.19 m and the mass is mm = 17.6 kg. The experimental results are scaled to full scale by requiring that the Proude number is constant (Bis scaling, see Fossen (1994)).

V becomes:

-(1- 8)q(·) - uTR*(·)u + ;;;?yTy

(57)

where 1'2 = "12 +d and 0 < 8 < 1. This shows that the system is ISS with respect to y. If we disregard the zero-mean disturbance terms Ww and Wb in the vessel model (1)-(5), the observer error y is exponentially decaying and the origin z = 0 of the system (39) is GAS. Then it remains to design the cost matrix R. (.) such that q(.) is positive definite. This is obtained by requiring that:

The length and mass of a typical DP operated supply ship is Ls = 70 · Lm meters and ms = 4590 tones. A drawing of the experimental setup is presented in Fig. 2. The experiments are performed as follows: Initially there are practically no environmental loads acting on the ship. After 50 seconds a wave generator is switched on and after 400 seconds wind loads are generated using a fan directed approximately 30 degrees at the port bow of the ship. The wind generator is switched

R:;l (1/Jey' f) 2: R;;l (1/J ey ) + t( 1/Jey ,f) 87

l0r-~---r--~--~--r-~---r--~--~~

I

5

.g

0 _.,,-..,.....=-J-

.~ Io'!

~L:E::SJ

-s

o

200

400

(.00

"00

1000

1200

1«10

IGOO

11100

2000

'lE\; ,d o

Cij

f~!

-150

.lJ

o

-: -155 .S

~ -160r-.......~'If

:I:

.165

time (scc]

Fig. 3. Experimental DP results: x-position (upper plot), y-position (middle) and heading (lower).

1100

1000

1100

10400

1(.00

11100

2000

r5J 400

(.00

1100

1000

1200

1400

If\OO

I¥OO

2000

time (sec]

Aarset, M. F., J. P. Strand and T. L Fossen (1998). Nonlinear vectorial observer backstepping with integral action and wave filtering for ships. In: Proc. IFAC Con! On Control Appl. In Marine Syst. (CAMS'98). Fukuoka, Japan. Ezal, K, Z. Pan and P. V. Kokotovic (1997). Locally optimal backstepping design. In: Proc. 36th IEEE Conf. On Decision and Control. San Diego. pp. 1767-1773. Fossen, T. L (1994). Guidance and Control of Ocean Vehicles. John Wiley & Sons Ltd. Fossen, T. L and A. Gr~vlen (1998). Nonlinear output feedback control of dynamically positioned ships using vectorial observer backstepping. IEEE Trans. on Cont. Syst. Techn. 6(1), 121-128. Fossen, T. Land J. P. Strand (1999). Passive nonlinear observer design for ships using lyapunov methods: full-scale experiments with a supply vessel. Automatica. AUT-35(1). Krstic, M., L Kanellakopoulos and P. V. Kokotovic (1995). Nonlinear and Adaptive Control Design. John Wiley & Sons Inc. Newman, J. N. (1977). Marine Hydrodynamics. MIT Press. Cambridge, Massachusetts. S~rensen, A. J., S. L Sagatun and T. L Fossen (1996) . Design of a dynamic positioning system using model-based control. Journal of Control Engineering Practice 4(3), 359-368. Strand, J. P., K Ezal, T. L Fossen and P. V. Kokotovic (1998). Nonlinear control of ships: a locally optimal design. In: Proc. IFAC Nonlinear Cont. Syst. Design Symp. (NOLCOS'98). Enschede, The Netherlands,. pp. 732-737.

·1(,0

-1(,(;

(000

REFERENCES

Zoom-in of vessel heading [dcg] '-r:-~----~-----r----~":':"'--~----,

-1(,4

200

400

Fig. 5. Experimental DP results: Plot of the bias estimates in surge (upper plot), sway (middle) and yaw (lower).

.1700:----=2oo:--~---,'----'---IOOO.L--1-'200---14....00--1-'600---18-'-OO--2.JOOO

·1"

:wo

I~'~

1(l~"----7.:;-----~--__:7:_--__:'t====i"'=,=:::!JIO'" lime [sec]

Fig. 4. Experimental DP results: Zoom-in of the heading measurement (dotted) and it's lowfrequency estimate (solid). off after 1500 seconds. In Fig. 3 the measured position and heading are plotted together with the LF estimate. To illustrate the wave filtering properties of the observer a zoom-in of the heading measurement and the corresponding LF estimate is shown in Fig. 4. The effect of the wind loads are clearly illustrated by the bias estimates in Fig. 5.

6. CONCLUSIONS An output feedback controller for dynamically positioned ships has been derived, where a separation principle is obtained. The controller is developed in the framework of locally optimal backstepping design, where a nonlinear observer is used in the backstepping. If the zero mean disturbances are disregarded in the analysis, the total system is rendered globally asymptotically stable. With disturbances the system is input-to-state stable. The control system has been implemented on a model ship and plots from the experiments illustrate the performance. 88