On the pang-bang principle and its application to a problem in ship manoeuvring

MATHEMATISCH INSTITUUT

UNIVERSITEIT

GRONINGEN

REPORT TW-64

On the pang-bang principle

and its application to a problem ¡n ship manoeuvring

by

G. J. Olsder

ARCHIEF

Lab.

y. Scheepsbouwkunde

Technische Hogeschoòl

Deift

Report TW-6

On the pang-bang principle

and its application to a problem in ship manoeuvring

by

G.J., Olsder

Summary. _{A linear time optimal control process is considered under the}

assumption that both the control and its derivative with respect to time

are bounded. The theory is applied to a problem in ship control.

Acknowledgement. I am indebted to prof.dr. J.A. Sparenberg for the many valuable discussions on the subject of this report.

Contents.

2

page

1. Introduction ₃

2. Statement of the problem

3. Normal systems 6

. The theorem of Chang ₇

5. The existence of optimal control 9

6. Number of switches

7. _{Relation between the switch points and the initial value of (t) 18}

8. Application to ship control 20

9. Numerical solution and results 23

10. The Algol program 30

1. Introduction

-3

In [ 5 J _{the following was stated: 'A ship follows a certain rectilinear} course and wants to change to another one. The initial course and the final

one are prescribed. It is asked to determine the rudder angle u as a function

of time to realize the prescribed course change as quickly as possible".

The solution of this problem was obtained with the aid of the maximum

principle. The result was that the behaviour of the rudder as bang-bang,

i.e. during the manoeuvre the rudder angle assumed only its _{two extremal}

values, and jumped instantaneously from the one to the other one and vice

versa. In practice this is difficult to realize. Therefore in this paper

the velocity of the rudder angle is assumed to be bounded in addition to the bounds on the rudder angle itself.

it is known that in this type of problems the control variable is either

situated at the boundary of its admissible region ("bang"-regicns), or it is

not at its boundary and moves with extremal velocity (tpangU_regions). It is

not easy to decide which time intervals during the manoeuvre are bang and

which are pang.

In this paper a theorem is proved which gives, under rather general

assumptions, the least upperbound for the number of switches of the control

variable. A switch is defined to be a transition, either of bang to pang or of pang to pang. By the latter, viz, pang to pang, is meant that the

rudder velocity jumps from its upper- to its lowerbound or vice versa.

2. Statement of the problem.

We consider a system described by the following linear differential

equation

dx

(2.1) Ax + au,

where t denotes the time, A is an n x n matrix with constant elements, _a

is a vector consisting of n constant elements and u is a scalar.

The vector x (x1,...,x)T, _{where T denotes the transpose, represents the}

state of the system. The quantity u, which will be _{a function of time,} represents the control. So, if a control function u(t), t0ttf and _an initial value x0 at to are given, the behaviour of x(t) has been _determined

as a function of time during to t _{t. The eigenvalues of A are} supposed to be real.

We will speai'z of an admissible control if u(t) is continuous and satisfies the following conditions:

(2.2) (2.3)

at all points t to t tf where

ii(t)

exists. A dot above a symbol denotes its derivative with respect to time. The _{constants M and K are} positive.

Because two restrictions on u(t) exist, it is convenient _{to consider} u itself as a state variable, which will be denoted by

_{y, and}

_{i as the} new control, which will be called y.

System (2.1) becomes (2.z+) where (2.5) A o... ..o u(t) <N,

h(t)

<K

By by, a o u(t ) u(tf) O o 'k

/

'y

\fl n+1! \\U

In the new formulation y will be called admissible _{if v(t) is a piecewise}

continuous function of time and

(2.6)

_!v(t)I

K, t

< t < t

0=

=

f

-5

Inequality (2.2) yields

(2.7)

y1(t)< M,

which means that the state space in which y(t) is allowed to vary, is

consüained. Let Y denote the admissible region for y(t), so Y consists of all those states (

)T

for which y<M.

It is asked to steer the system from a given initial value

o T f T

y (y.1,...,y,O) to y (O,...,O,O) by means of an admissible control

under the condition y(t) E Y, in the least possible time.

The manoeuvre will start at t = O, which is no restriction, because C-,)

(2.1) is autonomous, and will end at tf* So y(0) y° and Y(tf)

yf

We also assume that system (2.1) is normal, which will be defined in

3. Normal Systems.

To ensure the uniqueness of the solution of the problem we shall

assume that system (2.1) is normal. That is, if there are two controllers

u1(t) and u2(t), both measurable and only restricted to

u1(t)LK, 1

1,2

and satisfying the usual maximum principle, which steer x° to being

the initial and final value of system (2.1) respectively, then u1(t)

u(t)

almost everywhere.

A necessary and sufficient condition for system (2.1) to be normal

is that

(3.1) rank _[ a, Aa,

A2a,..., A1a

_] n.

This is a usual condition for control systems, because it states that the

system is controllable.

For a more complete treatment of the concept of normal and controllable

systems vid. [LI. J. It is easy to show with the aid of (2.8) that when (2.1) is normal then also (2.LI.) possesses this property.

A result [- J which will be used later on is the following.

Theorem. Let system (2.1) be normal, then for each nonzero vector p, consisting of n elements,

(3.2)

(tA)

The notation (p,q) denotes the innerproduct of the vectors p and q. This

theorem can easily be translated in terms of system (2.L1).

Proof. Suppose the contrary. Then there exists a vector p such that

(3.3) (p,etAa) o.

Let t0 to get (p,a) O. Next differentiate (3.3) with respect to t and

again set t0 to obtain (p,Ab) O.

By repeated differentiation of (3.3) with respect to t, we get

(3.) (p,a) (p,Aa) .. (p,A 1a) O.

But this means that [a, Aa,...,Anhla] has dependent rows, contrary to

the assumption that (2.1) is normal.

L. Theorem of Chang

Because we deal with a control problem in a bounded state space,

the solution rests heavily on the following theorem, which is due

to Chang [2,3 J

Theorem: Let Y, the region in which y(t) is allowed to stay, be closed

and convex. Further assume that system (2.L) is normal.

Let (t) be the trajectory, corresponding to t), where '(t) and

(t) satisfy (2.6) and (2.7) respectively, and joining the two given

points y° and yf, in a minimal time. In case of such an optimal control a bounded covariant vectorfunction (t), with (n+1) elements

,

can be found in such a way that (t), (t) and (t) satisfy

By -t- by dt -(.2) dt -- -B + (t). n((t)) , r',

(',3) (,By + by) max (ip, By-t- by)

!vL

where the scalar function (t) 0 if (t) is an interior point of Y

and (t) >0 if (t) is a boundary point of Y;

is the outward normal unit vector from Y at

The maximum in (L.3) is taken over all possible control functions v(t),

which satisfy v(t) jK.

Conversely, if a (t) can be found in such a way that (t), (t) and (t) satisfy (.1), (.2) and (.3), then (t) is the unique minimal time path and (t) the unique control, transferring y° to yf

In the original theorem the following assumptions are made on the

boundary of Y, denoted by Y;

a a unique normai n(y) exists at every point y 3Y. the partial derivatives are uniformly bounded.

y1

every point y within a certain distance d1 from Y is on one and

only one of the normals n(y).

there exists a distance d2 and a constant e> O such that at every point y within distance d2 from BY there is at least some y,

with _Iv K, satisfying

(n(y), By -t- by) < -e.

-8

It is easily seen that all these conditions are satisfied in our case,

Moreover, a penetrating hypothesis is made in the paper by Chang. This hypothesis

demands that it will be possible to keep y(t) at yf during O t t with the aid of an admissible control. Because here yf = (O,... ,O,O) such an admissible control is v(t) O and the hypothesis is satisfied.

Remark that this theorem is identical with the 'classic' maximum principle

if y(t) does not touch 3Y during the whole manoeuvre, because in that case

t) O.

The theorem states that if an optimal control exists, then it is unique.

(5.2) v(t)

Iy

_flL

(t)

<s

1

,

with an arbitrary, but fixed s> O.

Every point y° cao be steered to the origin in a finite time by a control which satisfies (51) if and only if the following two conditions

hold

system (2.4) is normal,

every cigenvalue A of A satisfies ReA s O.

Taking s = .minI1,K) it is clear that an admissible control always exists,

steering an arbitrarily y° to the origin

With this control y(t) remains inside Y durìng the manoeuvre.

5b.At least one of the eigenvalues of A is positive.

Let us consider the control problem described by (2.1)-(2.3) in this

case. A priori nothing can be said about the existence of an admissible

control, which satisfies (2.2) and (2.3). The existence depends on the

values of x0, K and M. For prescribed x° and M it is easy to give a bound

-9

5. Existence of optimal control.

Before trying to construct the time optimal control, we must first prove

that such a control exists. A kncwn property in the theory of control is

that, if there exists an admissible control v(t), i.e. v(t) satisfies (2.6)

and the corresponding path satisfies (2.7) and transfers y° to f, then an

optimal control v(-t) exists.

So if we are able to construct an admissible control steering y° to yt, within Y, then the existence of the optimal control is ensured.

.1

In the following we distinguish two cases; a every eigenvalue Aof A

satisfies A <O; b. at least one eigenvalue of A is positive.

5a.Every eigenvalue of A satisfies A O.

The following theorem, proved in [] and here reformulated with regard to the problem we are dealing with, will be used.

Theorem: Consider system (2.4) where the control function now satisfies

t

(5.1) v(t) I

f

v(t) dt < s

K > O for K in such a way that for each K < K an admissible control,

satisring (2.2) and (2.3) and steering x° to does not exist. Conversely,

the existence is not assured for every K > K. Properly speaking, here only

a bound for the nonexistence will be given.

Let us write A in the Jordan canonical form by means of a

transformation. It is known that the optimal control (t) is

under such a transformation. After this every symbol used in

o

space is provided with an asterix. The initial value is X , fX

is x and is again the origin.

Equation (2.1) becomes

X

(5.3)

_{CA x +bu.}

dx x x x

Let us denote an eigenvalue of A, which is positive, by a, and assume

that _{er appears} in a Jordan block which is an 1x1 matrix, because otherwise

the calculations and results are more complicated. Taking the rthi row of

(5.3), belonging to the eigenvalue a, we get

dt dx dt X r - x X

-

-ax

_rr

i-b u. r

We consider (5.L1.) to be the differential equation for a new, independent

oX fX

control problem to transfer to _{Xr under the restrictions (2.2)}

and (2.3) as quickly as possible. If an admissible control for this new problem does not exist, an admissible control for the problem _{as defined in}

(2.1) - (2.3) does not exist either.

Let us assume that bX > O and x < O, the other cases can be tated analogously. Then it follows from

(5,L)

that

b M x

(55)

r

<X°

(<0)

is a necessary condition for the existence of _{a solution, because otherwise}

xr0X _{cannot be steered to the origin whatever the value of K may be. From}

now on it is assumed that (5.5) is satisfied.

Now we try to get x0 to the origin as quickly as possible, so by means

of the optimal control, if it exists. It is known from the control theory

that if an admissible control exists which steers

X°

to x then also

the optimal control exists and if an optimal control does not exist, then

an admissible control does not exist either, So for the existence of an

OX X

admissible control steering Xr to it is suficient to consider the existence of the optimal control.

10

-suitable invariant

the transformed the endpoint

It is intuitively clear and it also follows from section 6 that,when an

optimal control concerning the problem of (5,L) exists it has the form of

fig.5.la (type I) or of fig. 5.lb (type II).

-t-M

o

(5.8)

Fig.5.la and b. Possible forms of the optimal control. In 5.la u(t)

reaches its boundary uM, in 5.lb u(t) does not reach that boundary.

If the control u(t) reaches the boundary u +M during the manoeuvre then it

is of type I and otherwise it is of type II. This difference can also be

stated in another way: if tf >2M/K then the control is of type I and if

tf < 2M/K it is of type II. So in the (IK,tf) plane the curve

separates the two regions where the control is of type I respecting type II.

Pn elementary calculation shows that when the control is of type I then

2 x

x a b

(5.6) tf ---ln(1 r r r

i K a

_{K(e_rM/K}

1)

and when it is of type II then

2 tf = - - ln(1 X 2 0 _X

-a

x b r r r K

We now consider the dependence of the control and its type on the

parameter K. For K sufficiently large, the optimal control exists and is of

type I. If K decreases continuously there are two possibilities for the

optimal control, depending on the value of Xr Either the control remains

of type I for each 1K for which the control exists, or it changes over to

type II for certain K. These two possibilities, case (b1) and case (b2)

respectively, will be investigated now. It is emphasized that these

possibilities depend only on the value of xr0;

0r and

bX

_{remain the}

same in both cases.

u u A II tf t arctgk tf t O

(5.6)

tf = 2M/K

tf tf for K case (b ) 1 case (b2)

KK

₀₂

K I -

12-Case (b1). For each K for which the control exists, it is of type I.

So we must deal with formula (5.7). If we let K decrease, tf will increase.

For a certain K, the argument of the logarithm in (5,7) will be zero, and hence tf _{So, this special K, which will be denoted by K0 is uniquely}

determìnd by

aM/K

x

o x 2

(5.9) K (e r o - 1) = X b a

o r r r

and is a lowerbound for the existence of an optimal cortrol. For each

K < K an optimal control and hence any control steering Xr0 to the origin does not exist.

Case (b2). The parameter K is decreased continuously and for certain

K _{K1 the control changes over from type I to type II. This is drawn} in figure 5.2 in the (K,tf) plane. Point P in this figure is such a point

where type I changes over to type II. So for K> K1 formula (5.7) is valid.

Let us investigate (5.7) on the interval K< « . For K _{K1, tf} has

I

an intersection with the curve tf 2M/K. For K + K, tf approaches

I

infinity, see figure 5,2, and hence the curves (5.7) and (5.6) must have

another point of intersection Q (fig.5.2).

Fig.5.2. tf as a function of K for two different values of x°. The dashed _curve

tf 2M/K separates the regions of type I and type II.

Let such an intersection ] at K K2. A calculation shows that (5.6) and

(5.7) have precisely two points of intersection, K1 and

1(2.

At this moment we know that for K < K K2 and for K K1 formula (5.7) is valid.

Let us now deal with the region 1(2 < K < K1. For 1K a little bit"

smaller than K1 we must deal with formula (5.8), which has _{an intersection} with (5.6) at K = K1. Investigating formula (5.8) it appears that tf also

II

-

_-

_:-

2M/K II

has two points of intersection with curve (5.6), precisely at K = K1 and K K2! So by decreasing K, the control changes over from type II to type I

at K _1<2

The conclusion is that for the lowerbound K we have the relation (5.9)

independent of the fact that case (b1) or case (b2) occurs. Hence for

determining the underbound for K only case (b1) is relevant.

The somewhat surprising result that a control of type II changes over to

type I again for decreasing K can also be macb clear by considering the

trajectory xrX(t) and control u(t) belonging to K K0 (5.9). This trajectory

is drawn in figure 5.3. For t M/K the right hand side of (5.4) is zero,

X X

so x_r (t) remains at point - b N/a

r r _bX r r X o X r X X r

Fig.5.3. The trajectory xrX(t) in the limit case K

_K.

The corresponding control is of type I, however with an interval of value M which is infinitely long. By a continuity reasoning it is claer _{that for a}

1K with K < K < K0± s , s _{sufficiently small, the optimal control}

corresponding to the problem with that K, will be of type I too, independent of what will occur for an increasing K.

In this way we have a lowerbound for K for each positive eigenvalue _of the matrix A. The largest of these bounis _{a lowerbound, not necessarily}

the largest lowerbound, of 1< for the original problem.

t

-6. Number of switches.

The optimal trajectory

j(t)

can

be divided in parts which belong to

Y and which belong to the interior of Y. The 'classic' _{maximum principle}

holds for parts of the optimal trajectory _{(t), which are interior to Y,}

i.e.

ßi-1(t)

< M. If _{(t) is at the boundary of Y for a time interval}

of positive measure, then by (2.4)

(6.1)

So the control function is composed of parts of the curves

(6.2) y -i-K, y -K and y =

Now it follows from above that

i-1(t)

_{qualitatively shows the behaviour}

of

figuo 6.1, 11% fl+1 -t-M --M B1 A

dy

n-t-1 v(t) O.

dt

Fig6.i. Possible form of y (t).

n-t-1

The points A, where _{(t) leaves the boundary of Y, viz. A1, A4 and A6,}

and

where '(t)

changes

sign,

_{viz. A2, A3, A5, completely characterize} and _{as a function of time. The points A., i 1,2,3,... are called}

switch

points.

Writing (4.3)

as

(6.3) (ip, by) max by)

and

using (2.5) this formula becomes

(6.4)

(1(t))

max (1(t).v).

y <K

So,if

_(t)O,then

ni-1 (6.5) (t) K sgn( _1(t)), n 'L

and if i_n-1(t) O, v(t) is not determined, Conversely, if _x (t) ± M on n+1

A4A5B3

t

-to this end (L.2); (6.7) z

K

AT T a r', 11) (t) / +

where the i-sign is valid when y z i-M and the -sign when y z - M.

ni-i ni-1

It follows that (t) satisfies

ni-1 (6.8) d n+1 h(t) ± c(t), dt -where r'.

(6.9)

h(t) z a1 i1(t) -t- . . . + a n

We flow have to examino the behaviour of h(t). By virtue of a well known

theorem on linear equations with constont coefficients each function

.(t), iz1...,n has the form

Alt _Amt

(6.10) f1(t) e -t- + f (t)e

m

where A1, A are all the distinct eigenvalues of the matrix _AT and f 1(t), f(t) are polynomials. The degree of f.(t) is less than the multiplicity of A Consequently, each linear combination of the

functions .(t), izi,..., n has the form of (6.10).

All the numbers A1,..., Am are real, since all the eigenvalues of A,

and hence of _AT, are real. í lemma proved in [7] states that a function of the form (6.10) does not have more than (n-i) real roots. Hence we find

that h(t), defined in (6.9) does not have more than (n-1) real roots. Now we prove the following.

Lemma: The function (t) is continuous in t.

Proof. It follows directly from (6.7) that the first n elements of (t)

are

continuous

in t, because 1(t),..., (t) is the solution of a linear

homogeneous differential equation with constant coefficients. So by (6.9)

(t)

15

-a time interv-al of positive me-asure, then O on that interval and hence

(6.6) _+1(t) 0.

16

-h(t) is also continuous in t.

As long as (t) Y, (6.6) holds and as long as (t) is an interior point of Y, (6.8) holds with (t) 0. So for these cases 1(t) is continuous too. The only points where +1(t) can be discontinuous are

the points where a transition of a pang - to a bang-region takes place

or vice versa. These transition-poinare called junction-points.

The proof of Changes theorem gives the following information; at the

junction-points

p, p,...

the jump condition holds, i.e. (6.11) -i-0)

-

'k°

'k

or in our case

(6.12) _-

+i(0) -

± y

for some constants >0. The +sign at the right hand side of (6.12) is

valid when (t) i-M, the -sign when (t) -M. We know that

QyO) and

exist and are bounded, because

n+1 is already known to be

piecewise continuous and is bounded by Changes theorem.

Consider an interval where ''(t) is bang, and where

flT

(O,...,O,+1), This for instance corresponds with interval B1A1 in

figure 5.1. The other case, T (O,...,O,-1) can be treated analogously. It is kncv (6.5), (6.6) that during (6.13) < t < :i4i _(t) O i-1 n+1 (6.1L) < t _< p : (t) O , k ni-1

-(6.15) < t _< + : (t) < O +1 2 n-i-1

where _{and 62 are positive and sufficiently small. From (6.12) and (6.13)} it follows that

(6.16)

_{n+i(T0) =}

and

(6.17)

with and >0. In order to be not in conflict with (6.lLi-) and (6.15) it is necessary that O _{and herewith the continuity of}

- 17

-We are now ready to state and to prove the main result of this paper.

Theorem. Consider the control process (2.1), where A has only real

eigen-values. The control variable satisfies (2.2) and (2.3). If an optimal

o f

-control exists, which steers x to x , then it does not have more than n switch-points.

Proof. We will prove that between two successive switch-points A. and A. i ii-1

there is at least one real root of h(t), see (6.9).

Because h(t) has at most (n-1) real roots, at most n switch-points A1 can

exist.

Let the switches A. and A. occur at the time -r. and T. respectively.

i il-1 i

There exist a t. and a t. with T. < t. < t. < r. such that,

1,1 i,r i i,l i,r

=

for t1 < t

_< 1(t) ' Y and that, (6,6), 1(t) z O for

-r. < t < t. and t. < t < -r. . By this construction of t. and

i

=

i,l i,r

r1

1,1

r' +1(t) is at a switch-point

and

br at the boundary of Y for

z .

and

z , so in both cases

1,1 i,r

(6.18) (t. ) z i (t. ) z

n+1 i,l n-i-1

i,r

as follows from (6.5) and (6.6). It follows from the construction of

t.

and

t. that (t) O on t. < t < t. and hence on this interval

i,l i,r i,l

lPn+1 satisfies

(6.19)

d1

z h(t).

follows that is a continuously differentiable function

< t

and

from the lemma above it follows that (t) is

r,l n+1

O <t <te. Therefore, using the mean value theorem, a

5 exists in such a way that

d

()

(6.20) n-t-1

z

_h(S)

z _Q

dt

The point 6 _{is a root of h(t)}

which

_{lies between r. and T. and therefore}

the proof is complete. From (6.8) it

ont.

<t

1,1 continuous on

t.

i,l

<o

< t. i,r

7. Relation between the switch-points

andt'

initial value of i(t),

For the construction of the optimal control it is sufficient to know

the switch-points. Suppose that n switch points T1,..., T, with

O < T1 < T <.<t< t are given in such a way that the control, completely

2 --n

f

f

characterized by these points, transfers the system from

y

to y

It

follows from

Chang's theorem that these n switch points aro uniquely

determined.

The function (t) is characterized by its differential equation (6.7).

We will show that we can find (ni-1) initial values for

_1(t)

at

t0 in such a way that the control corresponding to those values by

(L.1)-(L.3) possesses the prescribed switch-points

_T,...,T.

The initial value of Xt) at t0 will he given by

D (oO)T

The vector consisting of the first n components of (t) will be denoted

by

(t)

and the vector consisting of the first n components of ij will

he denoted by

By (6.7)

T

(7.1)

(t)

etA

o

and hence h(t), defined in (6.9) can be written as

T

_-tA

(7.2)

h(t)

= (a, e 0) O

,e

-tA

a).

For each interval (T.,T.+i), i 1,2,...n-1 there exist a

t. and a t. , as defined in section 6, in such a way that

i,r

t.

_i,r

_T

t.

_i,r

-tA

(7.3)

_f

h(t) dt ( ,

f

e a dt) z o i 1,..,,n-1.

ti1

_ti1

These are (n-1) conditions for Ç0 to be determined. We also demand a

normalization condition for

Ç,

because (7.3) is homogeneous in

Theorem: Within a normalization condition Ç is uniquely determined by

(7.3).

Proof: Note that h(t) O for each Ç O, see section 3, so the conditicns

in (7.3) are not trivial fore O.

For (7.3) to be solvable in Ç°, it is necessary and sufficient that the

colurruis

(7.4)

are

linearly independent.

t.

i,r

-tA

fe a dt

i 1,..,,n-1,

ti,1

-Suppose the contrary, so the colurnrs (7 .Li) are linearly dependent and

defining an arbitrarily t and t with t < t < t , the n,l n,r n-1,r n,l n,r columns (7.5) t. i ,r tA

f

e

adt,

i1,..,,n

t. i,l

are also linearly dependent. The columns of (7.5) constitute a square

matrix and the rows of this matrix are linearly dependent too. So a

nonzero vector p of n elements exists in such a way that

t.

T -tA

(7.6) (p , e a) dt o, i 1,...,

t.

1,1

These conditions state that in the interior of each of the intervals

-tA

(t lj

) at least one real root of (p,e a) exists. Because the

intervals (t.

r are disjoint in pairs by their construction, the function (DT, etAa) possesses at least n real roots.

By the theorem in section 3 we know that (pTe_tAa) is not identically

zero, and behaves like the function given by (6.10). By the same reasoning as

atfollowing formula (6.1O)it is known that (DT, e tAa)does not have more

than (n-1) real roots. Hence we have a contradiction. So the columns (7»--)

are linearly independent and the proof is complete.

To find the initial value of the remaining component

+1' we know that (7.7) So satisfies n-i-1

tr

(7.8) _{1n+1 +} h(t) dt = 0, to'1

where t is defined as in the same way as t. i1,...(n-í) and i,r

toi

0,

In this way we have found a (t) in such a way that the solution of (L.1) - (L.,3), with the above determined initial values , gives the control with switch-points T1 ,.. as was required.

o

o

if T1 M/K and if T1 < M/K

-(8.1)

where 8 _{represents the angular velocity of the ship in the horizontal plane,}

n represents the lateral velocity of the ship, r represents the angular velocity and u the rudder angle. The quantities b1, c, i3,Ll,5 are constants,

which characterize the type of ship. We suppose that, as generally being the

case _{in practice ,the eigenvalues of the matrix in the right hand side of} (8.1) are real and distinct.

The control variable u, which is a function of time, satisfies the

following conditions (8.2) ¡u(t) - M,

(8.3) ü(t) I K.

The problem will be how to choose u(t) as a function of time, subject to

(8.2) and (8.3) in order to steer the ship from an initial course (e,n,

r)

(',0,0) to

_{the final course (O,n,r) (0,0,0) in the least}

possible time. In addition it is required that u(t) = u(tf) = 0, where

t denotes the initial time and t the final time.

o f

As in the previous sections, it is convenient to consider u(t) _{as a} state variable and ù(t) will be the new control variable, which will be

denoted by v(t). In the new formulation the equations of motion become

dy By -t- by,

(o

fo

,

b-

and B u

10

0 1 0 O b3 b4 b5 o c C C. O 1.1.

-

20

-8. Application to the time optimal course changing of a ship.

A ship follows a certain rectilinear course and wants to change

to another one. The initial and final course are prescribed. The problem

is how to change the rudder angle with time to realize the prescribed

change of course as quickly as possible.

In order to define the mechanical behaviour of the ship we need its

equations of motions[1,53, which contain the rudder angle u as a control

variable. We assume that the ship moves with constant velocity through

undisturbed water and air and consider only its motion in the horizontal

plane. Then the following linearized equations of motion are relavant;

d

00

1

O b3 h4

O C3 C1.1.

r'

The control variable v(t) satisfies

(8.6) !v(t) <K.

The problem of changing course as quickly as possible with two restrictions on

the control variable is reduced to a time optimal control problem with a

bounded state variable u(t). The initial state at t0zO is denoted by

y0 z

(1000)T

and the final condition f z (0,0,0,0) , which will be achieved

at time tf

The general solution of (8.4) is

(8.7) y(t) etE { y°

+ f

b y(s) ds }

From section 6 it is known that always y(s) z t K, or y(s) z O and that there exist at most three switch points

_'t1,

-r2 and 'r3.

See figure 8.1.

u

tfB (8.8) Y(tf) = Q z e

-M

Fig.8.1. Fossible forms of the optimal u(t).

For the construction of the solution we first ignore the boundary

cons train on U and then the contributions which belong to intervals, where u(t) does not satisfy (8.2) are subtracted. It is assumed that the manoeuvre starts with y z -i-K.

21 -T3+W y° +

K{f - f

_+f

-j -sE e b ds O T1 't2 w -t1 ± K

-5

₊ T1 2T1-w e_CB _{b ds} . H(w-'t1) -sE + K{ -t-f

_-

_f

} e b ds. H(w-2-1-T1) -w+(3 T2- T1 )/2 w-t-( T2+ T3)/2 -sE

K{ -j

₊

_f

} e b do. H(W-T3+T2) T3 -w+(2T3-2)/2 r / 2 3 t u -i-M

22

-where w M/K and H denotes the Heaviside function;

(8.9) H(p) El if p > O, H(p) E O if p < O.

The vector equation (8.8) yIelds four scalar equations, which contain three

unknown variables r1, T2 and T3. This seems to be a contradiction. However the scalar equation in (8.8) which belongs to u(t), which is the fourth

component of y(t), forms an Identity, because the way of construction of

(6.8) sees to it that u(tf) O, whatever the values of T1,T2 and

t3

are.

Hence three equations with three unknown variables are left, from which

9. Numerical solution and results.

By means of the generalized Newton Raphson technique, (N.R.T.),E6],

and T3 can be solved numerically from (8.8). A disadvantage of

the

N.R,T

is that convergence to

T1,T2

andT3 depends on

the choice of

the starting values. We can avoid this difficulty in the following way.

Take different values for the maximal angular velocity of the rudder,

K, say K1, i=1,2,3,... and let

XIK. with 0< À<1 and

K1

sufficiently

largeU. More precisely,

K1 must be chosen so large that we can use the

values of the switches which belong to the optimal control with K

as initial values for the control problem with K K1. The control problem with K has been treated extensively in [ 5]

Now we continue as follows: as starting values for the problem with

K K.+1, take the T1,-r2 and T3, which are the solution of the problem with K K1. We can improve this method if we extrapolate quadratically the -c, j1,2,3, found for the probms with K K., K K.1 and K

to find the new starting value for K K. . If we denote the t, which

i+1 _J

belongs to the process with K K by t'., the starting value

T. for the

process

with

K 1<11 is given by

(9.1) T = ()?±X+1)T1. - X(À2+À1)

T1

+À A.'2,

j

= 1,2,3, as

can

easily be verified.

In this way it turned out to be possible to obtain the solution for

each K

>

0, whenever it exists.

To write the three relevant scalar equations of (8.8)

explicitly,

we

transform the matrix B to its Jordan canonical structure. B is defined

by (8.5) and its canonical form is

1 0

(9.2)

Bx=

_(o

(o

o o o

O O

o o

where a

and

are

the elgenvalues of (b3 b (9.3) _{c3 c} X Let us substitute y Qy in (8.+); X dy = Ql BQyX +

Q1

b y.

The matrix Q

is nonsingular and is determined by Q1BQ

BX.

In the new

coordinate system the initial value becomes Q_1y0 _{and the final situation}

Qlyf

2'4 -X

The expression etB , which now appears in (8.8), has the form

/1 t o o

(9.5)

f

o i o o

O e O

O O e

It is clear by considering (9,5) and (8.8) that the second equation of

(8.8), with B replaced by BX, will be the identity, because the second row of (9.5) contains only a constant.

The calculations have been carried out for two types of ships, a stable

one,a and are negative, and an unstable one, not both a and 13 are negative.

The characteristic coefficients of these ships are, see [5

The eigenvaiues a and 13 of the stable ship are -0,3623 and -3,2515 and of the

unstable ship +0,01L.7 and -2,715.

The results are shown in figures 9.1 -9.. The ship changes its course angle

with -1 radiais. The maximum rudder deviation, M, is 0,15 radiais. The

numerical solutions with the N.R.T. were started at K 10 radiais por unit

of time. The multiplicator À was chosen to be 0,8.

In figures 9.1 and 9.2

T.,

j1,2,3 as well as the points p,q,r, which are roughly defined to be the points on the time-axis where the rudder angle

is zero, are given as a function of K. The general definition of _{p,q and r}

is given in the following section.

In figures 9.3 and 9»+ u is given as a function of t for several values

of K. For the two ships under consideration the maximum of K valuo will he about

i radial per unit of time.

The difference between the stable and unstable ship is clear. The quantity

(T3-T1)

remains small in comparison with T1 for stable ships; for unstable

ships the 7correction timefl

(T3-T1)

is about the same as T1, for every K. It is seen that the length

_(T3-T2)

is small with respect to T1 and

(T2-T1)

for stable

coefficients stable unstable

b3 -0,89L6 -0,5 b -0,2856 -0,3 b5 +0,1L.53 +0,1795 c3 -L4,392 -3,8 c -2,719 -2,2 c5 -1,225 -1,307

25

-as well -as unstable ships and for every K. This means that in essence we

have two important steering periods, which is in agreement with Nomo[5].

The difference also appears from figures 9.3 and 9I, Let us compare for

instance the behaviour of the rudder for KzO,1 at the second steering period.

The rudder of the unstable ship reaches the boundary the rudder of the stable one does not reach the boundary.

Fig. 9.1. The

characteristic times T1, T2, T3 (dashed lines)

and p, q, r (drawn lines) plotted

against K for the stable ship. M

= .15 radiais. Course change

lo

8 6 h 2 10 2 's 's .5.

--ti

meaning of the symbols

Fig. 9.2. The characteristic times

r, T2

T3 (dashed lines) and p, q, r (drawn lines) plotted

against K for the unstable ship. M = .15 radiais. Course change = -1 radiais.

101

i

Fig. 9.3. The rudder angle u as a function of time for several values of K for the stable ship. M = .15 radiais. Course _{change -1 radiais.}

28

-I I I I I I I

4 6 8 10 12 14 16

o 2

29

-/

K1.5

K1

KO.7

\

KO.3

K O

vv

0 2 L 6 8 10 12

iL

16

Fig. 9.. The rudder angle u as a function of time for second values at _K

for the unstable ship.

M = .5

-

30

-10.

The Algol program.

In stead of T1 ,T2 and T3 the points p,q and r, already mentioned at the end of section 9, are used in the calculations, see figure 8.1.

Point

p is the intersection of the line with tangent 1< throui the point

(T. _{uL1))with the t-axis. The i-sign is valid if u(T1)> O, otherwise} the -sign

is. In an

analogous way the points q and r are defined with

respect to T2 and T3, The points i, , - determine p,q,r uniquely and.

conversely. Formula (8,8) becomes

tfB

_p!2

(p+q)/2 (q-s-r)/2 r (10.1) 7(tf) O e y + K{f

-f

1

- f

} e bds O p12 (p-i-q)/2 (q-t-r)/2 p-w

p/2

-sB

-f

}e

bds.H(p-2w)

p12

w q-w

(p-t-q)/2

-sB

-K{f

-f

}e

hds.H(q-p-2w)

(pi-q)/2

pi-w

r-w

(q+r)/2

_-'B

-f

}ebds,H(r-q-2w)

(q+r)/2 q+w

The program given here, calculates the values of

_{p,q and r for several}

K?s. The program is written in Algol 60, with duc observance of the

Alcu'contion, fn' the Telefunken TRL

computer of the

_{Rekencentrurn}

_of

the University of Groningen.

The program serves to print the values of

p,q, and r for several

values of K. For each K a Newton Raphson technique is needed to solve

the values of p,q

and

r.

The convergence to the correct

p,q and r is also

printed.

The road instruction at line 123 requires the values of the variables

h3,

b5, b5(0),

C3,

c, e5, e6 (co), DM

(zó) and

_{THETA (}

required

change

of course).

NORFIX is an output procedure: NORFIX (

('i.... ?)

) prints the words

placed between the string quotes;

NORFIX (FIXPRT,105,Z) prints the value

of Z with one digit before

and 5 digits after the decimal point.

The output procedure NLCR

means: new line and carriage return.

- 3_ -Description of the program.

The procedure JOINVERSION, line 5, is a standard procedure and

calculates the elements of the inverted matrix of A, when A is given.

JOINVERSION is called on by JOINVERSION (A,B,C,E), where the _{array A is the}

matrix to be inverted, the real variable C is the value of the _determinant

of A. _{Then this value is smaller then the real variable B, the program jumps} to label E.

The procedures INT and F, lines 6-43, calculated the _{value cf the three}

relevant scalar functions of the left hand side of (8.8).

The procedures DIVIN and DER, lines 44-88, calculate the _derivatives of the three relevant equations with respect to the three variables _{n, q}

and r.

The procedure GEHN (A,B,C,D), lines 89-122, carries out the generalized Newton Raphson techniqie. The array A represents the starting values.

After having passed this procedure, A represents the _{array to which Newton} Raphson converged. B is the required _{accuracy of the solution. The arrays} Cand D are a lower- and upperbound for the values of A _{to be found. In}

particular cases it is possible that Newton Raphson does _{not converge. when}

this occurs the interval between C and D is halved. The new estimate will give us a new C or D _{and so on. This has been built in in the procedure.}

The lines 124-136 carry out the transformation described _{by formula (7.4).} The lines 137-167 calculate the values of _{,q and r for the case K} co

For t metJod to bton these values see [s].

The lines 168-196 calculate the values of p,q and r for several values of

O BEGIN'IREALtB3,B4,B5,B6,C3,C4,C5,C6,R,RT,D,A1 ,A2,A,M1,M2,U,THETA,XNUL,

1

K,T1,T2,T3,P1,P2,P3,P4,L,W,KBG,.FF.,DM,AD,LAB,LALB,LC,LL,N1...

2

tARRAY'S(/1..4,l..4/),BB,X(/1..4/),TS,TT,TU,TR(/1..3/>.,

3 'INTEGER'G,N,I,J.,... 4

'BOOLEAN'KA,BOEL.,

5 'PROCEDURE' JO.I.NVERSION..,'CODE'...., 32 -6

'REAL''PROCEDURE'INT(Z,T).,

7

'INTEGEP'Z.,

8

_'REAL'T.,

9 BEGIN''IFIZ'EQUAL.'j'THEN' INT.=P1*T_P2*T*T.*...5., lo

.,IFZ.EQUALt2,THEN,INT.=_P3*(EXP(_A1*T))/A1.,

11 'IF'Z'EQUAL'3'THEN' INT.=P4*(.EXP(A.2*T...)../.A2..,

12 'END'.,'REAL''PROCEDURE'F(A,B,C,Z).,

13

'INTEGER'Z.,

14

'REAL'A,B,C.,

15 'BEG IN' 16

IIFtAINOTLESStLL'AND'(BA)INOTLESS'LL'AND'(CB)'NQTLESS'LL'THEN$

17

F.=INT(Z,O)±INT(Z,L)+INT(Z,AL)INT(Z,A+L)INT(Z,BL)+INT(Z,B+L)

18 +INT Z CL ) TNT (Z, C). 19

IF'A'NOTLESS'LL'AND'(BA)'NOTLESS'LL'AND'(CB)

..'LESS'L.L'THEN' 20

F.=INT(Z,O)+INT(Z,L)+INT(Z,AL)TNT(2,A+L)INT(Z,BL)

21

2*INT(Z,(B+C)/2)INT.(Z,C),,

.

22

IF'A'NOTLESS'LL'AND' (BA)

'LESS'LL'AND'(CB) 'NOTLESS'LL'THEN'

23 ...F..=_INT(Z,0)+TNT(.Z,L..)+INT.,(.Z.,A_L.)_2*INT(Z,(A+B4t2)+INT(Z,B+L}

24

+INT(Z,CL)INT(Z,C).,

25 'IF'A'NOTLESS'LL'AND'.(BA) ...'LESS'LL'AND'(CB)

.

.'LESS'LL'.THEN.'

26

F.=_INT(Z,0)+INT(Z,L)+INT(Z,A_L)_2*INT(Z,(A+B)/2)

27

+2*INT(Z,(B+C)/2)INT(Z,C).,

28 'IF'A

'LESS'LL'AND'(BA)'NOTLESS'LL'AND'(CB)'NOTLESS'LL'JHEN'

29

F.=INT(Z,0)+2*INT(Z,A/2)INT(Z,A+L)INT(Z,BL.)+.iN.T(Z,B+L.)

30

+INT(Z,CL)INT(Z,C).,

31 'IF'A

.'LESS'LL'AND' (BA)'NOTLESS'LL'AND'(CB)

.

'LESS'LL'THEN'

32

F.=_INT(Z,0)+2*INT(Z,Af2)INT(Z,A+L)INT(Z,BL)

33 +2*INT(L.,(B+C)/2)_INT(Z,C.).,.

34 'IF'A

'LESS'LL'AND'(BA)

'LESS'LL'AND'(CB)'NOTLESS'LL'THEN'

35

F._INT(Z,O)+2*INT(Z,A/2)-2*INT(Z,(A+B)t2)+INT(Z,B+L)

36

+INT( Z,CL)INT(Z,C).,

37 '.IF'A

'LESS'LL'AND'(BA)

'LESS'LL.'AND'(C.ß) 'LESS'LL'THE.N'

38

F.=INT(Z,0)±2*INT(Z,A/2)-2*INT(Z,(A+B)/2)

39 ....+2*INT(Z, (B+C)/2)INT.(Z,C) 40

' IF'Z'EQUAL'1'THEN'F.=F+XNUL/K.,

41

_{IFzEQUAL2sTHEN.F.=F*EXP(A1*C,),.,}

42

' IFZIEQUAL'3ITHENF.=F*EXp(A2*C).,

43 'END.'..,

44

'REAL' 'PROCEDURE' DIVIN(Z,T).,

45

'INTEGER' Z.,

46

'REAL' T.,

47 'BEGIN' 'IF' Z 'EQUAL'.. 1

'THEN' DIVIN.(P1P2*T).,.

. .

48

'IF' Z 'EQUAL' 2 'THEN' DIVIN.=P3*EXP(_A1*T).,

49

'IF' Z 'EQUAL' 3

'THEN' DIVIN.=P4*EXP(._A2*T).,,

50 'END'., 'REAL' 'PROCEDURE' DER(Z,ZZ,A,B,C).,

51

'INTEGER' Z,ZZ.,

52

'REAL' A,B,C.,

53 'BEGIN' 'IF' ZZ 'EQUAL'

i 'THEN'

54 'BEGIN'

55 'IF'.. A 'NOTLESS'LL'AND' (BA) 'NOTLESS'LL!.THEN' 56

DER.DIVIN(Z,A+L)DIVIN(Z,AL).,

57 ' IF'A ...'NOTLESS'LL'AND' (BA) ... 'LESS'LL'.THEN.' 58

DER.=DIVIN(Z,(A+B)/2)+DIVIN(Z,AL).,

59 'IF'A

60

DER.+DIvrN(z,A/2)DIvtN(z,A+L).,

61 'IF'A . 'LESS'LL'.AND'.(BA) LESS$LL..THENI

62

63 'END'.,

64

'IF' ZZ 'EQUAL' 2

'THEN'

65 'BEGIN'

66

'IF' (BA)'NOTLESS'LL'AND' (CB) 'NOTLESS'LL'THEN'

67

_{DER.DIVIN(Z,BLJ+DIVIN(Z,B+L).,}

69

DER.=-DIVIN(Z,B-L)+DIVIN(Z,(B+C) /2).

70 ' IF'(B-A)

LESSILL'AND'(C-B)tNOTLESS'LL'THEN'

71

DER.=-DIVIN(Z,(A+B)/2)+DIVIN(Z,B+L).,

72 'IF' (B-A) 'LESS'LL'AND' (C-B) 'LESS'LL' THEN' 73

DER.=-DIVIN(Z,(A+B)/2)+DIVIN(Z,(B+C)/2).,

74 'END'.,

75

'IF' ZZ 'EQUAL' 3 'THEN'

76 'BEGIN'

77 'IF' (C-B) 'NOTLESS'LL'THEN' 78

DER.=-DIVIN(Z,C)+DIVIN(Z,C-L).,

79 'IF' (C-B)

'LESS'LL'THEN'

80

DER.=-DIVIN(Z,C)+DIVIN(Z,(B+C)/2).,

81 'END'.,

82

,TF,Z,EQUALI2tTHENIDER.=DER*EXP(A1*C).,

83

,IF,Z,EQUAL,3ITHENIDER.DER*EXP(A2*C).,

84

'IF'ZZ'EQUAL'2'THEN'

85

'BEGIN''IF'Z' EQUALI2.THENSDER.=DER+A1*F(A,B,C,2).,

86

SIFIZ,EQUAL,3,THEN$DER.DER+A2*E(A,B,C,3).,

87 'END'.,

88 'END'.,

89

'PROCEDURE' GENN(TT,V,K0,KB).,

90

'REAL'KO,KB,V.,

91

'ARRAY'TT.,

92'BEGIN' 'INTEGER' I,J,L.,

93

IARRAYSTD,TE,KBG,KOG,FIE(/1..3/),FG(/1..301..3/).,

94

'REAL'A,B,C,D,U.,

95

'FOR'I.=1'STEP'1'UNTIL'3.1'D0'

96,BEGINtKOG(/i/).=KO.,KBG(/I/).XKB.,

97 'END'.L.=0.

---98 NOP,..A.=TT(!1/).,B.=TT(/2/-). ,C.=TT (//).,L.=L#1

99

'FOR'I.=1'STEP'1'UNTIL'3.1'DO'

100 'BEGIN'TD(/I/) .=TT (/It).,FIE(/T/).=F(A,B,C,T).,

101

'FOR'J.=1'STEP'1'UNTIL'3.1'DO'

102

FG (/1 ,J/).DER (I,J ,A , B,C).,

103 'END'

.,JOTNVERSION(FG,.f-15,U,SING).,

104

'FOR'I .=1'STEP' 1'UNTIL'3.i'DO'

105 'BEGIN' TE( / I / ) .=0.,

106

107 108 109 110

111

112

FOR 'J. =1' STEP '1' UNI IL' 3. 1' 00

TEC III) .TE(

II /)+FG(

/1 ,J/ )*FIE( /J/) .

IF'TE(/I/)

'GREATER'O'THEN'KBG(/I/),TD(/I/)'ELSE'

KOG(/I/).=TD(/I/).,

TT(/I/).=TT(/I/)-TE( III).,

IF'TT(/I/)'GREATER'KBG(/I/)'OR' TT(/I/)'LESS'KOG(/I/) 'THEN'

113 'END'., NORFIX(NLCR,'('ITERATIE')',FIXPRT,200,L).,

114 NORFIX('('SWITCHES')',TT)., ...

115

_{NORFIX(NLCR,'('FIEWAARDEt'4')',FIE).,}

116

D.=O. ,

117

FOR' I.1' STEP'l' UNI IL' 3 .1'DO'

118

D.=D+ABS(TT(/I/)-TD(/I/H.,

119

'IF'D'GREATER'V

'THEN''GOTO'NOP.,'GOTO'FIN.,

120 SING..NORFIX('I'MATRIX SINGULIER')').,

121 FIN..

122 'END'.,

123

READ(83,B4,B5,86,C3,C4,C5,C6,THETA,DM).,

124

R.=B3*C4-C3*B4. ,RT.=63*C5-C3*B5. ,A.=B3+C4. ,D.=A*A-4*R.,

125

410=(A+SQRT(D))*.5.,A2.=(A_SQRT(D))*.5.,M1.=C3/(B3-A1).,

126 M2.=C3/ (B3-A2)., 127

128

5(/2,4/) .=5(/3,3/) .S(/4,3/).=1. S(/1,2/.-C3/RT., ...

129

_{5(/1,1/).=-R/RT.'S(/13/).=R3/RT.,S(/3,2/).=-M1.,S(/4,2/).=-M2.,}

130

S(/3,4/).=(B5*M1+C5)/A1.,S(/4,4/).=(-B5*M2C5)/A2.,

131

XNUL.=S(/1,1/)*THETA.,BB(/1/).=0.,BB(/2/).=86.,BB(/3/).=C6.,

132

XNUL.=-XNUL.,

. ..

133

BB( /4/) .=1., ' FOR ' I.1' STEP'l'UNT IL

'4.1'D0'

134 'BEGIN'FF.=O. ,'FOR'J.=l'STEP' I'UNTIL'4.1'D0$

135

FF,=FF+5(/I,J/*BB(/J/).,X(/I/).=FF.,

136 'END' .,P1.=X(/1/).,P2.=X(/2/)..,P3.=X(/3/).,P4.=X(/4/).,

-

-3I-138

_{'BOOLEAN'BB.,}

139

_{GAMMA.=THETA.,S.LN(.2.)/A2.,ST.=LN(2.i../A1.,M1.(A183)/B4.,}

140

M2.=(A2-B3)/B4.

141

P.=(C4*B5_B4*C5)*DM/R.,Q.RT*DM/R..,

142

P.=-P.,Q.=-Q.,

143 AAP..TOG.=o.,TBG.=l.,T1.TBG.,BB.'TRUE'..,

144 ELKE..'IF'Al'GREATER'O'AND'Tl'NOTLESS'ST'THEN'

145 'BEGIN' .T1.=T1-.1.,'GOTO'ELKE.,

146 'END'.,

147 U1.=.t1.2*EXP(A1*.T.1.)4..,U2.=(1-2*EXP(-A2*T1.)..)., 148

tIFIBB'THENITB,=S+f-8'ELSE'TB.=TBB.,TA.TB.,

149 R00T...IF$EXP(A2*TB)N0TLESS12'TI-fEN'

150 'BEGIN'TB.=(TA+S) /2.,TA.T3., 'GOTO'ROOT.,

151 'END'.,

152

_{T2.=TBNl/A1)*LN((EXP(A1*TB)_2)/U1)_(1/A2)*LN((EXP(A2*TB)-2)/U2))}

153 ....f(i./.(i...2*EXp(_A1*TB))_1/(1_2*EXp(_A2*TB))).., 154

'IF'ABS(T2-TB)'LESS'f-9'THEN''GOTO'CONTIN 'ELSE'

155 'BEGIN' TB.=T2.,'GOTO'ROOT.,

156 'END'.,

157 CONTIN..T3.=T2.,

158

T2.=(1/A1)*LNNEXP(A1*T2)_2)/U1).,

159

_{THETA.=Q*(2*T1_T3T2).,T3.T2-T3.,}

160

'IF'ABS(THETA-GAMMA)'LESS'.f--4'THEN''GOT0'TOX.,

161

' IF'THETA'LESS'GAMMA'THEN'

162 ,BEGINTBG.=T1.,T1.=.5*(TBGT0G).,BB.$FALSE, .,TBB.TB.,

163 'END'. 'ELSE'

164 'BEGIN'' IFBBSTHENTBG.2*TBG.,TOG.T].,T1.a.5*(TBG+TOG).,

165 'END'.,'GOTQ'ELKE.,

166 'END'.,

167 lOX..

...

168

TR(/1/).=TS(/1/).=TT(/1/).T1.,

169 -

TR(/2/).=T5(/2/).TTC/2/).=T2.,

170

_{TR(/3/).=TS(/3/).=TT(/3/).=T3.,}

171

172

LA.=LAB*LAB+LAB+1.,LB.=LAB*LA.,LC.=LAB*LAB*LAB.,

173

K.=10.'

174

_{NORFIX(NLCR,'('ONEINDIG')',FIXPRT,104,TT).,}

175

AD.=f-6.,

176 PAK..NORFIX(NLCR,FIXPRT,104,TT).,

177

NORFIX('('DIT ..rS...NIEUWE STARTWAARDE'..).t..)...,

178

L.DM/K.,

179

LL.2*L.,

180

GENN(TT,AD,0,30).,

181 'IF'( .

TT(/1/).2*L)LESS*QTHEN

182

NORFIX('(' FIRST PART DOES NOT REACH THE BOUNDARY')').,

183

_{IF(TT(/2/)_TT(/1Ì)-2*L)'LESS01THEN1}

184

NORFIX('(' SECOND PART DOES NOT REACH THE BOUNDARY')').,

185... ,I.F,(TT(/3/)_TT(/2/)_2*L)LESS1O,THENI...

186

NORFIX('(' THIRD PART

DOES NOT REACH THE BOUNDARY')').,

187

NORFIX('('HELLING BOOGTANGENS')',K).,,

188

NEWPAGE., NLCR.,

189 .'FOR.'I.=.1.'STEP'1'.UNTIL.'.3..Î'DO'

190 'BEGIN'TU(/I/)

.=TT(fI/)*LA-TS(/1/)*LB+TR(/I/)*L.C.,

191

TR(/I/).=TS.L/I./.)....,T.S(/1/)..=TT(.t.I.t)..,TT(/.I./.).=TU(./ij..).,

192 'END'.,

19

K.=LAB*K.,

194

'IF' K 'GREATER' .01 'THEN' 'GOTO' PAK,,,

References.

1. Abkowitz, M.A., Lectures on ship hydrodynamics-steering and manoeuvrability. Hydro og Aero dynamisk Lab. Lyngby Denmark.

Report no. Hy-5 (i96Lf).

2, Chang, S.S.L., Optimal control in bounded phase space.

Automatica, Vol.1, Pergamon Press, New York, 1962, pp.55-67.

3. Chang, S.S.L., An extension of Ascoli's theorem and its application to the

theory of optimal control.

Transactions of the Am.Math.Soc. vol.115, 1965.

Lee, E,B. and Markus, L., Foundations of optimal control theory.

John Wiley, New York, 1967.

Olsder, G.J., On the time optimal course changing of ships.

Journl of Eng. Math. vol.3-2 (1969).

Ostrowski, A.M., Solutions of equations and systems of equations.

Academic Press 1966.

Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidse, R.V. and Mishchenko,E.F.,

The mathematical theory of optimal processes. John Wiley, New York, 1962.