On the time optimal bang bang control of linear multi-variable systems with small initial perturbations

ON THE TIME OPTIMAL BANG BANG CONTROL

OF LINEAR MULTIVARIABLE SYSTEMS

WITH SMALL INITIAL PERTURBATIONS

8m

Mskulwsg Z 228 co osiit

7L 01e 7117$- Fw 015 781811

ON THE TIME OPTIMAL BANG BANG CONTROL

OF LINEAR MULTIVARIABLE SYSTEMS

WITH SMALL INITIAL PERTURBATIONS

ON THE TIME OPTIMAL BANG BANG CONTROL

OF LINEAR MULTIVARI.ABLE SYSTEMS

WITH SMALL INITIAL PERTURBATIONS

PROEFSCHRIFT

TER VERKRIJGING VAN HET DOCTORAAT IN DE

WISKUNDE EN NÄTUURWETENSCHAPPEN

AAN DE RIJKSUNIVERSITEIT TE GRONINGEN

OP GEZAG VAN DE RECTOR MAGNIFICUS DR. A. WATTEL

IN, HET OPENBAAR TE VERDEDIGEN OP

DONDERDAG 9 SEPTEMBER 1911

DES NAMIDDAGS TE.4 UUR

DOOR

GEERT JAN OLSDER

1.. Introduction.

SOme function theoretical results I. 8

Some function theoretical results II. 17

Definitions and known results of control theory. 27

*

L -systems. 31

Formülation of the main theorem. ₃₅

Elucidation of the main theorem by means of (2,1)- and (2,2)- systems. ₄₀

Some nonlinear equations. ₄₈

First term in the expansions of the switching-times and final time. 58 IO. Higher order terms in the expansions of the switching-times and final time. 63

The formal power series are convergent power series. ₆₉

The solution found is time optimal. 74

The main theorem is applicable to almost all x0. ₈₀

Some additional flotes on the main theorem. ₈₇

Numerical experience with (4,1)- and (4,2)- systems. ₉₁

References. ₉₈

The last fifteen years control theory has found its feet and has already reached a certain degree of completeness. Two schools in particular, the one of Pontragin in the U.S.S.R. and the one of Bellman in the U.S.A., have greatly stimulated the development of control theory. A well-known branch within the field of control theory on which a great deal of work has been done, is linear time optimal control theory. In the latter a system is considered that can be steered by means of one or more control components. The linear system has to be steered in such a way that a certain aim is achieved as quickly as possible. The control components may only take on values which are situated in a region previously given.

In this speçial field a fair amount of publications have appeared, both of a theoretical [4], [5] and of a numerical [12] nature (numbers between square brackets refer to

publi-cations to be found in the list of references). Much attention has been paid to control problems in which one control component occurs and also to the numerical methods of

/

solution with regard to this. Relatively little, however, has been published on problems with more than one control component. The theory treated in this thesis deals with such control problems. This theory, which is origitial, shows how the control components act together to make the system achieve the ultimate aim - which must be near the initial position - as quickly as possible. We will first try to make this clear by meansof a simple example, preceded by an introductory one.

Consider a system, such as a trolley, which moves along a horizontal track without friction. The position z7 of the trolley, which has mass one, is described by Newton's law of motion

2 d z dt2

where t is the time and Suppose that the initial

-c (c > O) and that of bringing the trolley mal time. This has to be u7(t) subject to the con

ro

ij r

i

r0i L I 7] + I O O [z2 [i]

u7(t) is the external controlling force applied to the trolley. position at time t = O of e trolley along thé track is the initial velocity x2 is zero. We consider the problem to z7 = O where it has to arrive with velocity z2 = O in

mini-done by means of a (possibly discontinuous) controlling force

s train t

!u1t't)I r

The solution of the time optimal control problem introduced above is intuitively clear. The optimal control u(t) is a maximal accelerating force followed by a maximal

deceler-or

[x

ätion until the mêchanism stòps exactly at the required position x O. The critical time fo the switcki from ácceleration tò deceleration is precisely ha1way the process. A simple calculation shows that the optimal trajectory in the (s7, plane consists of two parts of parabo]as of which the fotmulas arè

These parabolas have beeñ sketched in fig. (1.1).

û(t

Êig. (1.1). 'The, optimal trajectory in th (x', x2) plane.

First the controlling force u7 = #1 is applied to the trolley and the trolley moves ¿long the parabola s7 = + -, c. At' the moment that s - íc (tee figuré), i.e. after /e units of time, the controlling force, u1 jtps horn #1 to -1 and the ttòlley continues along the parabola X

-

until it reaches the origin after 2/c units of time. Hence the. optimal control is

f#l,

0<t</c

-

L-i

,

Vc < t 2/c the minimal time of the process is 2/c units of time.

between the two systems is that in the latter example a control component u2(t) has been &c1

added, whiáh acts directly on the ve1ocity.- The problem will be the same as above:

the initi4l position and velocity are s1

-c (c

> O)

and

2 = O

respectively and the

system must be steered to x.7

= z2 = O

as quickly as possible by means of

the

two control

components

u7(t) and u2(t),

which are constrained to

Ju1t'tfl 1 ju2(t)j 1.

With the aid of control theoretical methods this problem can be solved. The again intuitively clear: during the whole manoeuvre is u2(t) = +1 and u7(t) erates (u7 = #1) and then decelarates (u1

= -1).

The switch from u7 #1 to again haif-waythe process. A simple calculation shows that the minimal time

units of time

and

that the optimal control u(t),

u(t) is..

r

L-i, -1

vt-2#2/Y,

u

can

direc-u2(t) #1 O t

-22/Tn.

Let us consider both examples from one point of view. A difference between these

examples is that in the first one the. time necessary for the manoeuvre (if the manoeuvre starts at t = O this time will be called the final time) is proportional to /c heréas in the second one the final

time

is about proportional tO c for c sufficiently small

(then -2#2/7 e). This difference appears to be verr characteristic of the two systems.

Also if instead of (z -c, z2 = O) .the initial position and velocity aré

Xl = cl.c, x2

where and 02 are constants which satisfYJ c

z

7, then thi differén behàviour remains valid. In the case of the f.irst sytem one finds that for almost all values of and 02 (restricted to I c # 7) the final timé is about proportional to le for e

sufficiently small. In the case of the sécond system oãe finds that fOr all values of e and 02, again satisfying the restriction, the final time is about proportional to L fOr

e sufficiently small.

A rough explanation of this differénce is that in the first example the control influence directly only the velocity in the (x7 z2) plane of the system in the tion

[fl

of this plane, whereas in the second exañple the control (u1, u2) can in-fluence directly the velocity in the directions

[ ]

and [o]

. The last twò vectors

span the (z7, z2) plane.

We will now generalize the examples mentioned. A system is described by the differential

equation .

solution is first ccel_

u1 -1

is

- Ax # Bu,

dt

where

the n-vector -s denotes the state of the system and the.r'vector u is the control; t is the time. The matrices A aùd B (of size nxn and nxr

respectively)

have constant

elements.

The components u(t), i'l,...,±', of the control vector u(t) are restricted

to

i , i = 1,..o,r

It is

required

to choose th control u(t) in such a way that the system is steered from the initial point s cx0 (0 is an n-vector with

length one and

is

a

positive

scalar) at t O to the origin as quickly as possible.

For a large class of systems the time optimal control (the control we are looking for) shows a

bang-bang

behaviour, i.e. the control-components are piecewise constant and take on only the values-+1 and -1. Points on the time-axis at which a control

component

jumps from -i to -i--1 or vice versa are called switches. The time at which such a switch occurs is called a

switching-time.

Under

certain conditions concerning the system and, the direction of the initial value, the following facts about the time optimal control, which is

bang-bang,

will be proved in this thesis,

provided

thàt

t is

sufficiently small.,

10. _{All control components together have precisely (n-1) switòhes. For the validity of}

this property it ia not necessary that all the eigenvalues

of

the matrix A are real. A known fact is E5 that if these eigenvalues are real then each control component

separately has at most (iz-1) switches.

-Simple criteria exist which give the nz.anber

of

switches per control component. These nznbera are p--1 or p (and the sun of these nunbers is n-1). The uantity p is -the smallest natural nunber in euch a way that p ? - . If one knows the switching-times of a control component, the starting-sign

of

this component (i.e. #1) has to be known in order to determine- this control component completely as a fwiction of time. There are also simple criteria which give the starting-signs of the different

cOntrol components.

3° The final time and the switching-times are analytic fwtctions of where p has been defined above. (We assune that sC is fixed and that

It is

real). These fwic.tions have the value zero if their argwnent is zero, and if they are developed in a Taylor 8eries with respect to their argu'nent, the coefficients of the linear terms are not equal to zero. Hence for

t

sufficiently small the final time of the manoeuvre is about proportional to

Consider for instance (4, 1)-, (4, 2)-, (4, 3)- and (4, 4)-systems, which means that of these four systems n = 4 and r - 1, 2., 3 and 4 respectively. The approximate

It follows that for p > i (ie. r < n) we must know the switching-times wid the final

time very exactly in order to steer the system from cx0 to the origin. An error of

order e in one of the switching-times or the final time - such an error is small with

respect to the duration of the whole process if c is sufficiently small - 00.74858 that

the final situation of the ey8tsm has again a distance of order

from the origin (the

d-tstance from a poi.nt x wi.th components Xp

to the orig-n z.s deftned to be

{

¿}). In thi8 way we can conclude that for p > i and e sufficinetly small the

.7_=1

-criterion of time optimality is not a fortw'iate one2 if we want to steer the system to

the origin exactly. This is a mathematical formulation of the generally accepted

engi-neering principie that in order to reduce small initial perturbations one should not

use bang-bwg control whiôh ià moreover often difficult to realize for tf sufficiently

emallJ. Instead of this a (continuous) linear. feecack control is frequently used.

It also follows that: for e sufficiently small the systems. move with an approximately

constant velocity in the x-plane during thé time interväl betwéen t = O and the first switching-time along a mòre or less straight line (i.e. -in the direction of Bu). Hence the system reaches a distance of order from the origin which is large (if p > 1) with respect to the initial perturbation which had the distance

e.

The three properties mentioned above are formulate4 mpre precisely in the "main theorem't in chapter 6. The whole thesis is concerned with this main theorem and its proof. The chapters 9 up to and including 13 are the heart of the proof and ari outline of these chapters and also of chapters 7 and 8 is given at the end of chapter 6.

The main theorem - formulated under the condition that, e is sufficiently small - will probably also be valid for many systems which are governed by nonlinear differential equations of the form

-- =

_{g (x, t) # B'(x3 t) u}

dt

provided that these nonlinear equations can be linearized to the form (1.1).

In the. proof of the main theorem use is made of the concept of formal power series which

is introduced 'in chapter 2. In that chapter some definitions and (known) results con-cerning the theory of functions are given. In chapter 3 a new function theoretical result is derived. The important aim of these two chapters is a generalization of the implicit function theorem, which is crucial for the proof of the main theorem.

Suppose we are given the functions

f(zi3...,zk.; w), i

1,...3k,

of the complex vari-ables zlJ...JZk and w which satisfy O) = O and which are analytic in a neighbourhood of the origin z7 _...

_zk

w O. The, implicit function theorem states

that the equations

w) ₂

1,...,k

the Jacobian of the functions

f

with respect to the variables

z.

is not

equal

to zero at z

= ... =

_zk

= w = O. That the Jacobian is not equal to zero is a sufficient, but not a necessary condition for the equationS

f ± O

to have an analytic

solution. In chapter 3 this condition will be weakened. Note that if the Jacobian is

Ò then the solution mentioned is unique; this is in general not true if only the

weaker conditións are satisfied. It is possible in that case that more

than

one analytic solution exists.

In chapter 4 a short review is given of known òntrol theoretical facts. Forj a more extensive introduction one

can

consult [4).

In chapter 5 the class of systems is introduced for which the main theárem is válid. The relation to generally known classes of systems, such as controllable systems, is shown.

In chapter 14 something is said about possibleextensions of the main theorem. The

main

theorem has been formulated under sufficient conditions but simple examples indicate that these conditions

can

bè weakened.

The main theorem offers é numerical method to calculate the time optimal control; one only has to construct thé power series mentioned in

30

Some numerical reSults obtained by this method, especially for (4,2) systemE, have been given in chapter 15.

Such a

(4,2)

system

stems from

aproblèm in ship

manoeuvrirg, which

is as follows. A ship

follows a rectilinear coürse

and

must be steered to é parallel course as quickly as possible. There are twó control components: one rudder at the

stern and

one at the bow

of the ship.

This way of determining the time optimal control seems to be applicable for systems with

n and r

rather large. However, the condition "c must be

sufficiently small" may

not be satisfied if one deals with practical time optimal control problems.

Some notations and, conventions.

The n-dimensionäl real vector space is denoted by R.

Let

z

be

a (column) vector in

then the transpose of

z

is denoted by

z'.

The equality

s =

O means that all components

of the vector

s

are zero. The norm of

s,

denoted by

_{il I)} ,

is defined by

if

we suppose

that are the components of z.

-To distinguish between vectors we sometimes write

1, 2,

;. the

numbers 1, 2,...

are indices

and

not exponents. The innerproduct of two vectors

ad

2 is

written

as

12

(s

,s ).

I-f we are given

a square matrix A,

then

det[A]

denotes the determinant of A. If this

determinant is

zero,A is

called singulär

and otherwise it is called

regular or

If a is a scalar, then, if a O, Bgn(a) ₌ a/Ial, where

_Ial

is the absolute value of a. In case a = O, the sgn-relation is nàt defined.

We will not distinguish between the notations a1

k and a7 k In botb cases these

3..., notations denote the sane quantity.

The real intervals a < t < b, a t b, a < t b and a < t < b will be denoted by

(a,b), [a,bl, (a,b] and [a,b) respectively.

In this thesis the following abbrèviations will be used: r.h.s. right-hand side,

l.h.s. left-hand side, f.p.s. formal power seriés, c.p.s. convergent power series, t.o.c. time optimal control,

a.e. almost everywhere.

In this thesis'some paragraphs -are headed by "Remark" or "Observation". Remarks are essential for understanding the investigation; Observations are not strictly necessary and can be omitted.

2. SOME FUNCTION THEORETICAL RESULTS I.

For the sake of completeness we start with some "known" definitions and related results. The definitions agree with those used in [i] and [2]. For a more complete treatment of the concepts to be introduced these references can be consulted. In this chapter all variables are supposed to be complex, unless stated differently.

Definition (2.1). ([1], page 30). A function f(zl,...,zk), k 1, is analytic in a domain of its variables if in some neighbourhood of every point (z',... ,z) of the domain it is the sum of an (absolutely) convergent power series

71

(2.2)

E a. . (z z°) ... (z

z0)

i i k k

Theorem (2.3). ([1], page 31). Ari analytic function is continuous and has (mixed) par-tial derivatives of all orders which are likewise analytic.

Theorem (2.4).([1], page 33). An analytic function of analytic functions is again ana-lytic, which is specified as follows. Suppose that

l

_k

(2.5)

E a. . Z7 ...2

Oi 'k° l"k

converges for ¡z. < r., j

= 7,..

.,k ; the quantities r are positive. For these z the

function represented by the summation (2.5) is denoted by f(z73.. .,z7). Furthermore,

suppose that the functions

(2.6) z. 4.(w7,...,wj), .(O,...,O) = O, j

= 1,...,k

are power series convergent for IwI <

p, i

= 1,. ..,l ; the quantities p are again

positive. If for these w the values z given by (2.6) satisfy z1

r, j

= 1,...,k

then the function

(2.7) f(47(w7,. _{. .,w l),...,dk(w i,..}.,w))

is analytic in the domain IwI <

p, i

i,...,l , and its power series can be obtained

by formal substitution.

Theorem (2.8) ([1], page 39; the implicit function theorem). If the functions

(2.9)

_{fi'27'.2k1; zk7,...,zkk), f,.(O;O)}

O,

= 7,...,k7

are analytic functions of the k7 # k2 variables in a neighbourhood of the origin and if the Jacobian

(2. 14)

with any (complex) coefficients {a}. The series is said to be zero if all coefficients

are zero.

-It is not necessary for a f.p.s to h4vè a positive radius of convergence.

l.

_tic

If . = E

z ,

j

1,2,

are two f.p.s.1 then the linear

corn-J

_17JJ17(O

bination (a, B constants)

and the product

are defined by and

il

ic

Z.

a. . z ...z

hl3,h1C0

iic

+

l

_k = E

a7

z1

i

k

a. .

i

k

+ß.a.

'

2

a'!

. =

Z...E

_a

a

t7*

p1+v_i1 uk+Vkik

1"k

2

respectively. It is easily verified that the total of the series (2.14) is a commutative ring without null-divisors.

In the foliowing we will use

f.p.s.

as well as convergent power series (c.p.s.). For the sake of clearness we will use the symbol

=

for the

equality

of two f.p.s., i.e. all the

(2. 10)

O

for

Z1 -

_{= Zk}

_=.Zk

+1 = =

zk k

= 0

,

then

the equations

.1 7

72

(2.11)

fJ(zl,...32k; Zk#1,...,Zkk)

=

0,

j =

1,...,k1 ,

have a Imique solution

(2.12)

=

zj(zk13...szkk)3

1,...,

J

vanishing for Zk

+1 = =

zk k

O and

analytic in a neighbourhood of the origin.

1

12

Definition (2.13). ([1], page 3). If zl,...,zk

are given

variables, then a formal power series (f.p.s.) is an expression of the

fòrm

corresponding coefficients cóincide; for the equality of two

C.p.S.

the usual

symbol

= will be used. For example, if

_j1

az is

a f.p.s.,,

then

(2. 15)

j1J

.a .z O a. O, j ±

1, 2,...

J

The symbols

is also use4 if a f.p.s. is represented by a

livariablete

So C E

j=1

means that the fp.s. is represented by .

If

E a.z is convergent, then twill be

equalto thesum-(being a function of z) of this convergent series.

The following theorem, thé proof of which can be foundi'h [i], pagés 6, 7 and 8, will be needed in this chapter.

Theorem (2.16). Let (2. 77) '-,

"1

'1

"1

_

dl

'2k1k2

l

"k1k2

i

be f.p.s. with _a0 = 0., i = 7.,....,k7. f the

(k1

x k1) matrix of whith the (i,j)h

element is given by

_ä0

.,,

.. ..,

,

the ntber 1 being the j-th

index, is regular,

then the equations

(2.18)

fj(z.l,...Jzk.

Zk13..i3Zk.#k) =0, i

=

,..

have one and. only one solution of thé kind

il

(2.21)

E ..a. i-

J1..

.b. (2.19) z. = =0

01"k2

2

1"k7k2

=

i,.,k1

In this solution is .b00 = O, i 1.,...,k7

,

and the other coefficients are uniquely determinded by recurrence relations (The f p s (2 19) are said to be a solution of

(2 18)

if

the f p s obtained by substitition of (2 19) in (2 17) are all zero)

Lemma (2.20). If in the c.p.s.

i

Ea.

z ...-..

1 k

.1

convérgent within domain containing 27

= .o.

= 0 , evey term has at least one factor z., j fixed, then f can be dividéd by z. and thé quotient is again an analytic function (in the same domain as f is).

Proof The absolute value of each term in the power series of the quotient is majorized by the absolute valué

of

thé corresponding term in the power series of the analytic

(2. 23)

Also a fp.s. is given:

(2. 24)

f(z,w) Z

a. zw

i,jO

' Z

bw

Nòw we

can form f(,w), which is a f.p.s. in w. If

(2.25)

f(,w)

=

O ,

(2.26)

(,w)

O

where. the f.p.s. (,w) is obtained by substitution of in the expansion of

) z

i-1

-

(z,w)

Z ia..z a Z

i1,j=O

then the f.p.s. (2.24) has a positive radius of convergence and hence the function (w) =

bw

, analytic in a neighbourhood of w = O, satisfies f((w).,w) _O

i= i

Remark (2.27).. From (2.24) and (2.25) it is easily proved that f(030) O, or O ., and from (2.26) that

f(z,w)

. O . In the corollary, directly after the proof of theorém

(2.22), it appears that condition (2.26) can be weakened.

I/I

Proof of theorem (2.22). We assume that infinitely many b. are O, because othèrwise a polynomial would appear in (2.24) and then the. convergénceis evident.

If

-(z,w) O for z = w =

O ,

or, which is the same,

o O, the implicit function

theorem can be applied. This theorem states that f(z,w) E O uniquely defines a function

z

= z'w),

analytic in a neighbourhood of w = O, which satisfies f(z(w),w)

r O

in this neighbourhood. Hence, according to theorems (2.4) and (2.16) the f.p.s.. (2.24) must represent this analytic function and 'then theorem (2.22) has been ptoved.

In the remainder of thé proof

te

assune that

(2. 28)

=0.

z-mO

Suppose thàt

(2. 29)

C

bw1

#

b7

+

function

and hence the quotient is analytic.. I/I

az.

Theorem (2.22). (due to Hautus [3]). The function f('z,w) is analytic in a neighbourhood

with b 0, 7 < < , and define

(2.30) h - (,w) _h\)w'#hVlw'1

# ...

withh 0, 1v<

Now we perform a transformation z

î and

in the following way:

(2.31)

z

=

(b#zJw

;

(b#)w1

The variable is, a r, a f.p.s. in w .;

(2.32) b

w.

i-=1

The function

f(z*,w) is 4efined in

a neighbourhood Q2 of

z _{W =} O

as

(2.33)

f(z*w)

=f((b#z*)w ,w) =

=

a..

(b

*)1

7#1p i,j=0

The neighbourhood Q2 has been defined in such a way that if (z*,W) E Q2 then ((b #z*)whi,w) Q1. The neighbourhood Q, has been chosen in this way, because now

li

theorem (2.4) can be applied which states that

f(z*,W)

is an analytiô function in Q2. Hence a neighbourhood Q of w .0 exists, Q contained in Q2, in such a way that

f(z*,w)

is 'represented by the c.p.s.

(2.34)

?(z*,w)

=

..

_ii

z*iwi

l,ê=0

in Each coefficient in this expansion is a

function

of finitely

many a.and

b.

It directly follows that, if is substituted in (2.34),

(2.35)

?(,w)

f((b*)w,w)

f(,w)

0.

The power appeared in the r.h.S. of (2.33). Define

(2.36') m =

min{j#iu

0)

which means that ni is the minimal value of j#ii under the condition # 0. Supposé that this minimum ni is achieved for i0 and some j, which is equal to ni. The power

series in the r.h.s. of (2.33) has the term aomwm, with a0 O

We now remark that if ni is ahieved for i0, ni is also achieved for an i i

and

some

*

j, because f( ,w) = O would otherwise be impossible. Suppose the contrary; the mini-mum ni is only achieved for i0,

j-in.

In this case

*

,', where a.. =

a.

l.,7

17#m

From

(2.40)

it follows that

(2.41)

f*(*w)

O Moreover,

(2.42)

z*w)

=

(z*w)

= 3z

=

WflI#A

((*),w)

and

so

(2.43)

(*w)

-rn#J

1f

((b,1*)w,w)

w

.Ç (z,w)

_hw'

hiw'

'

....

0.

To summarize, originally we dealt with the function f(z,w) and the variables z, w and.

; after the transformations (2.31) a new function f*(z*,w) was obtained and the

corresponding variables became z*, w and The function f*(z*w) has the same prop-erties as the function f(z,w); it is analytic in a neighbourhood Q of z = w = 0, where it has the c.p.s. (2.39). If the f.'p.s. given in (2.32), is substituted in the c.p.s. (2.39), a new f.p.s. f*(*,w) is obtained, which satisfies, as shown in

(2.41) and (2.43),

Wmf*(Ç*,W)

(*)

_Q

*

Because

f(*,w)

= O , the coefficient of must be zero in the f.p.s. f(

,w)

. How-ever, if we substitute the f.p.s. in (2.37), the coefficient of m in the f.p.s.

is aom , which is O.

So we know that the minimum m is achieved for an i , which means that n ? ii.

Sup-pose that m .i _; this can,. apart 'from possibly (i=O,, jm), only be realized for (i1 j0), which implies czW O. However, according to (2.28) this is impossible and hence

(2.38)

.

m>

j'.

The function

(z*w)

can be divided by m. The quotient, to be denoted by

f*(z*w)

, is again an analytic function in Q according to

lemma (2.20).

In the neighbourhood the

*

*

function f

(z

w) satisfies

(2. 39).

f*(*w)

= E z

(2.44)

f*(*w)

(w)

In addition, we know that

(2.45) h* (,z') hvWV_rfl+P #

...

₃

where v-m#jj

< y,

according to ineq. (2.38).

lÉ v-rn+i.i > 1, then .*(z*,w) O ± w O, and we repeat the whole process of

z

_*

_*

performing new transformations from z and to new variables, until we have

* *

(2.46) h = h # h -w +

V v+1

which is obtained after a finite nuber of transfòrmations as dèscribed above.

Suppose that during that last run, i.e. after the last transformation, we deal with the function

f*_*

and the variables w

and

Then we know that

f** (*..*)

is an analytic function in a neighbourhood 2 of z = w O ,

and

(2.47)

where

(2.48)

for sme k 0. If the contrary is supposed, it is easily shown with the aid of (2.47) and (2.48) that

(2.49)

Moreover,we know that

(2.50)

f

*

(*w)

O

*..*

1-b<.w

, -z.= 1 £

(.*)

_{hy #}

h7

+ ..

with

h.

O , which implies that.

(2.51)

(z",w)I

O .

ea-.

-

z

=w=0

NOw the. implicit function theorem

can

be applied to

f**

(z**w)

O , which states that is an analytic

Èuncton

of w in the neighbourhood ò

_{w ± O.}

This function is unique,

and

hence, according to theorems (2.4) and (2.16), the f.p.s. represents

this analytic function. So the f.p.s. has a positive radius of convergence. Performing the inverse transformatIons, we find that is an añalytic function of w in

a neighbourhood ofw O. I/I -

-As already said in remark (2.27), the conditions under which theorem (2.22) has been formulated can be weakened, which is shown in the following corollary.

f

*. *

(z

*. .*

,w)

*.*

=0_.

(2.60)

then

(2.67)

j*=fl7jfl{j

¡

a.

-l-m (z,w) a # # 0)

Corollary (2.52). The function f(z,w), not identical to zero, is analytic in a neighbour-hood of z w = O and has in the c.p.s.

(2.53) f(z,w) =

_a.

ziw

i,j-O Also a f.p.s. is given:

(2.54)

. E b.w

i1

If f(,w) O, where f(w) is obtained by substitution o- (2.54) in (2.53), then the f.p.s. in (2.54) has a psitive radius of convergence and hence the function

.

b.wi _{, defined and analytic in a neighbourhood of. w = O, satisfies f(tw) s O} in this neighbourhood.

Proof. Define.m

_i

to be the minimum of,j for which an a.. O exists. Because f(z,w) 0,

1-a

In7 is finite. From lemma (2.20) it follows that the function f(z,w)., defined by -m

(2.55)

f(z,w) = w f(z,w)

is again analytic in , where it has the powér series expansion

(2.56)

f(z3w)

Z

..

zI

i,j-O 2O with

..

= a. . . We also know that

'Z-7

(2.57) w f(iw) = f(r,w) = O ,

from which it follows that

(2.58)

f(t,w) = O

Define

(2.59) in nnn{j (t,w) O) ,

2

aza

where (,w) is the f.p..s. obtained by substitution of the f.p.s. _{in the power} series expansion (at z w 0) of the k-th derivative (with respect to a). of f(a,w). That in2 is finite can be shown as follows. If

(2.62) and hence m2 <

i.

Define (2.63)

(w)

O az1-m2-1. ?(z,w) - (z,w) - m2-1 3z

then according to theorem (2.3)

7

is analytic in Moreover, from the definition of

7

it follows that r

(2.64)

7(w)

O

(,w)

O

Now theorem (2.22) can be applied to the funtion

7

and the f.p.s and hence is

analytic in W in a neighbourhood of W O. This analytic function w) satisfies

3. SOME FUNCTION THEORETICAL RESULTS II.

The result given in theorem (2.22) can be generalized to a system of functions. As in chapter two, all variables are supposed to be complex, unless stated differently.

Lemma (3.1). We are given a function f(z7,. ..zk, w), f(O,. ..,O. O) = O, represented by the (absolutely) convergent power series

(3.2)

f

E a . z

7"k#7

in a neighbourhood 2 of z7

= ... =

_zk = W = O. Hence

f

is analytic in . We suppose that

(3.3)

=

(z71.. .Jzk

w)I

= O

- O

According to the implicit fùnction theorem, the equation

f(z13.

'2k

w) O can be

solved with respect to z7 in terms of the remaining variables:

(3.4) = w), g(O,...,O; O) O

where g is an analytic function in some domain containing the point z2 . w = O. Hence,, near this point, g can be represented by its power series expansion

2 k

ki

(3.5) z = Z c. . z ...z w i . . 1...i. 2 k 1l7<47O 2

ki

If f.p.s. are- given, (3.6) z w , i 7,. .,k

which satisfy, if substituted in (3.2),

(3.7)

_{'i'"'k w)}

O

then, if 211Ck are substituted in (3.5),

(3.8)

= g(,.

k

w)

Proof. Suppose that w) is given by the

f.p.s.

(3.9) w) E

_{d w}

then (3.8) states that b7 d -for all i > 7.

Suppose (3.8) is not true and hence _d for some i. Define

(3.70)

N-min{j

h7.d. i>1},

(3.13) 7(w) =

(2w),

_'k'

w) It is easily seen 1 i N, then a

blNdN_ biN. H

It remains to be (3. 15) (3. 17)

that ,. d. for 1< i < N. If we can

1i t.

-contradiction has been obtained, because from ence (3.8) is true.

proved that . b . for 1 < i < N . For thät

1i li;

=

def

(3. 14) f7(z1;w) - f(z1,

2'"'k

f(z,

. It is clear that eW which satisfy 0 ,j N. The f.p.s. b1 w

= . w is uniquely determined by ( ;w) 0. Hençe

1

?1

7 1 1

functions of the coefficients ei., and are functions of ë...

13 l_3

Let us consider the dependence of b . on the coefficients e.. :j

13 -. 2.3

of b . w in the f.p.s. (3.14) we get a new f.p.s. in w i=1 7l

cients are zero. The coefficient of w gives

0 1,0,. ..,0

j

is uniquely determined bSrf7(7;w)

from which it appears that b77 is uniquely determined

càéfficiènt 'ow gives

e10 .. b72 e20

)=

E

e..z

w,

2.3

..,<;w)E

jzw''

l,3O

0 and that e.. = for all i and those j

+ . b11 e02 'O

from which it appears that b72 is uniquely determined as a function of b77 and hence of e70, e71, e02 and e07. In this way we continue. It is b is a fünction of b , i _< < m-7 and e.., O < i < m, O _< j < m.

7m 13

=

=

2.3

=.=

cients b77, b72,...,biN are all uniquely determined by the quantities

0jN.

The coefficients l2,...,lN.depend on.., O <

prove that

1i1

=

(3.10) it follows that

purpose consider

= O and the c.p.s. the coefficients b7. are

detail. By substitution of which all

coeffi-as a function of e70 and e07. The

iN, 0jN, in

e10, e17, e02 and easily seen that Hence the coeff

i-0<iN,

exactly the same way as b71, b12,...,biN depend on e... Because e. = for all i and N, we have proved that b7

=

i = 1,. ..,N.

'/'

N

(3. 17) =

E b. w3 , i

= 2, . . ., k

j=1

The functïoñsE., 2 i <k are analytic in win each bounded part of the w-plane. By the implicit function theorem the function 7'w) is uniquely defined by

(3.12) f(7(w), 2(w),.. .,k(w); w) O

and is an analytic functiOn of win a neighbourhood of w O. Nòw (w satisfies

(3.20)

Df7 w) Dfk ¡ .

,

(3.22) det E b.. wa., j=i ?-7 are given, which satisfy

(3.27)

i'k

w) 0 ,

j = 1,....,k

where the f.p.s. w) has been obtained by substitution of (3.20)

in

the

c.p.s. at the r.h.s. of (3.19). If

Df7

j

w)

k

with

-

0. Then it is easily verified that the Jacobian

(3.24)

D(zl,...,zk)

z1

= ... =

2k =

and the implicit function theorem (2.8) can &e applied which states that

f.(zl,...,zk; w) = 0,

j = 1,...,k,

uhiquely determine

zi,...,zk as analytic functions

Theorem (3.18). The functions fj(zla...Jzki. w),

j

1,...,k,aré analytic ma

neigh-bourhood of w = O and have in the following c.p.s.

oe l k

ki

(379)

fd(zlJ...izk; w) =

1

E _{z7 ....zk} w , a

= 1,...,k

k1°

i k#7 Suppose f.p.s., Df.

where an arbitrary element in this matrix,

j-- (7,...Jck; w) has been obtained by sub-a

stitution of (3.20) in the power series expansion of at z7 ... z w

0,

then o

the f.p.s. (3.20) have positive radii of convergence-and hence the fuñctions

E _{b.. w0, i i,...,k, analytic in a neighbourhood of w}

_{= Ò,}

_satisfy

f.(cl(w),...,k(w) w) 0, i =

2-Remark, (3.23). From (3.20) and.(3.21) it follows that 0) 0, i

r O, and from (3.22). it follows that fi(zl3...,zk; w) 0, i

= i,...,k. I/I

Proof of theorem (3.18). We

assume that for at least one

i

infinitely many

1,2,...,are 0, because otherwise the f.p.s. (3.20) are all polynomials in w

_and

then the convergence is evident.

Suppose that the f.p.s. represented by the. l.h.s. of (3.22) has the form E _{d. w0,}

of w in à neighbourhood ofw O. Hence, according to theotem (2.16),

repre-sent these functions. NOw theorem (3. 18) has been proved. For: theremaindèr of the proof we assume d = O. Because this remainder is rather lengthy, it will be split up into fOur parts, to be denoted by I

II,

III and IV.

I. We distinguish two cases:

(z,.. .,Zkj w) =

O atzl_...=zk=w_OfOralli,i_1,...,k,

f.

for at least one i

and j:

-- (z7,..

_'2k

w) O at z7

... =zk

w O.

a

t-,,

If case

(11)

occurs, all symbols except w are provided with a

(i

e -i. i - )

and

the proof is continued at II, where,, for notationa. cpnvenience,

af7

j

w) O at z7

... =

w = O.

For the time being we assume that

7

case (i) appears. Note thàt now the f.p.s. at the l.h.s. of (3.22) tarts with apart from a nonzero constant - w > k.

Suppose that

-

u.

ij.#7

(3.25)

ç.=b.

_t

w'#b.

wt

#...,i=

tl_li#7

with

b.

O añd

i < .. ,

and define

w)

_hj

w +

v..#7

ta

h..

_tuv.

w.

#7 i-a

with h.. O. Possibly ç. O for somè i; in this case we take

u.

= . HOwever, for

t

Z.

at least one

i

we have

_u.

< . An analogous ituàtioripreaï1s for h.. Posibly for

some h.. no finite number v. exists which satisfies

h..

0; in that case.h.. ± 0.

i-a i-a

tav

ta

Howeve, from (3. 22) it follows that for each

i

at least one

j

exists for which

'a

is

finite.

Taking u

_i'

we mnt±odue new variables zi- and ç in the following way:

(3.27) z =

(q.+ z)wu;

_i

=('q1 + ç)w",

i

= 7,..,k

where q.

₌ biu if

u.

= u and else q. = O . The variable ç is again a f.p.s. in w;

= b. u w u

if

(3. 28)

if u.= u

3 (3. 26) h..

ta

The functions

w),

i

= 7,..

.,k,are defined in a neighbourhood 2 of = = w O as

...-(3.29)

f(z

. . .,z; w)

fÍ

'f((q1#z)wU,.

..,

(q#z)wU; w)

₌

j

#.f(j

#_

.-#i

J k

i

k#1

i

k

*

in w . 11

.(q#z)

}, _y = --i" k-i-1

m1

The neighbourhood

2 has been defined in such a way that, if (z,..-.,z, w)

-,

then

((q1#)w,..., (q#z)w', w)

E Hence the functions _{w), j}

are analytic functions in 22. Then a neighbourhood of

= ..

_{= =} w = O exists in such a manner that

(3.30)

f.(z,...,z; w).

= E .. k+1

_,

i

ê

i''kr°

Each coefficient in this expansion (j fixed) is a function of finitely many an4 finitely many

It directly follows.that, if ,

i

_{= i,.. .,k, are substituted in (3.30),}

(3. 37)

w)

ji''k

w) O, i

= 1,...,k

ikf1i(i7#..

The power w appeared at the r.h.s. of (3.29). Define

(3.32)

n? k

=

ki

+ u E

(7 17=7

.a .

1. k+i

A reasoning quite analogous to the one given in the proof of theorem (2.22) shows that

(3.33)

m

>

The functions f.('z,...,4; w) can be divided by w. The quotient-functions, _{to be de} noted by fi(zlJ...zk; w), are again analytic functions in Q7 according _{to lemma (2.20).}

In Q the functions w) have the expansions

*

*

*

*

*1

*k

ki

(3.34)

fi7,.,2k; w)

= Z Zi

Z)<

w

_,

= i,...,k

-

i''ki0

7 k-i-7

with .a .

= ..

. . From (3.31) it is clear that 1

k#i 1_i..

(3.35) _{w) 0, j =}

Moreover, it is easily proved that (just as in the proof of theorem (2.22)):

*

def. 3f

*

*

y.

.-m#u

(3.36)

h..

_{=== --}

w) =

h.

w

...

_,

_i

= 1,...,k;

i

= 7,..,k.

Because -(3.37)

_*

-w

O(Zl,...,Zk

'tik

a(z73.. .,Zk)

a(zl.zk)

it follows from (3.22) that

(3.38) det where

*

(3.39) -;;-:

(zl,...,zk; w)

J a(f1, . .J1

_*

_*

J.1

_*

*.

;

(r;1, . . ., k-w)

_{. ... -T-;}

l3 ' az7 .

*

* afk

*

7'"'k

w) ...

Ti''k-

" az k (i-m)

To summarize: òriginally e dealt with the functions f and the variables W and after the trànsformations (3.27) new functiOns were obtained and the corresponding variables became ô arid r;. The functions havé the same properties as the

func-tions f.; the.y are analytic in a neighbourhood Q of z7

= ..., =

= w =

O , hete they

have the c.p.s. (3.34). If the f.'p.s. r;, i

= 1,...,k,

given in (3..28), are substituted in the ô.p.s. (3.34), then the relations (3.35) are valid. The analogon of (3.22) is

(3.38).

-If vçm+

1 for all

and

j, then

for all i and j, and we repeat the whole process of performing new transformations of thé kind (3.27), until we have, for at least one i and one

j

(3.40)

hY

h.. # h.. w ...

i7V..

zJ\..#1

with h.. O Relation (3.40) is obtained after a finite number of transformations. V

j

Contrary tö the proof of theoreti (2.22) and for, the sake of economy, we will not ue

- rL?? _*

*

here the repeated index but instead of this the sytñbol . o instead of

we now write ... Suppose that after the last transformation we deal with the functions and the variables

.,

w and

..

Then we know that w),

i = 1,...,k,

are analytic functions in a neighbourhood of 0 and

(3.41)

w) 0,

i

1,..

.,k

=0

0

(3.42)

=

.

ii

¿4

,

i

=

¿i=1

In addition we know that

(3.43)

det

and that (3.40) is valid for at least one i and one j. In order to simplify the notation we take i ₌ j

=

1..

II. We perform new coordinate transformations of the following form:

(3.44)

zi_fl(zl,.,zk; w);

¡=

z

,

i

= 2,...3k

and

-77'

'k

w)

O;

=

,

i

= 23...,k

From (3.40), with

i = j = 1,

it directly follows that w) o at

i

=

_{= 2k = W}

O and hence the implicit function theorem can be applied to the first equation of (3.44). This theorem states that is an analytic function of

Z2,...,2k.andw

in a neighbourhood of

w = O;

(3. 46)

=

¡.,,

.. .,;

w)

¡.. .,Z,; w)

From lemma (3.!) it follows that

(3.47)

w) w)

because

_w),

w) E ¡,

_{and heñce}

af7

_f7

.g

(3.48)

=.

=1,

z it appears that

(3.49)

_-*

_w)I

_o.

i k

-*

3z

This will be needed later on.

w)

w)

w)

w) z7

We now introduce the functions

(3.50)

w)

w),

w), j

1,...,k ,

_*

_*

* ...

which are analytic in some neighbourhood of z

= ... = z

0. For

j

1 defi-nition (3.50) can be simplified to

(3.52)

?.*(...,Çj

w) 0,

j =

With theaid of (3.41), (3.45) and (3.47) it is easily verified that

(3.54)

* *

We will now prove that if are substituted in

. -. .

, the, resulting

f.p.s., to be denoted by p, is not

equal

to zero. Because s

(3.53)

'''<

'i''

(,

,z)

(213 . . .,

z)

a(z

r,,.

32k)

(z.t, ... .,

Zk) (z

1' . .

.32k)

where

;

=det

.0 1

0...0

O 0

it follows that the f.p.s. p is the product of two other f.p.s. in w

1

1''k

w)

...

1''k

w)

(3.55)

det

-

_-

'k

-(7, .

. w)

. ..

(73 .

_k w) 3?

.3Ç:; w)

Both these f.p.s. are . O ; the first f.p.s. at the r.h.s. of. (3.55) according to

(3.43); the second f.p.s. starts with a nonzero constant according to (3.49). All f.p.s. in,w constitute a ring without null-divisors andhencé p 0.

III. Let us now consider the functions

(3.56)

w)

d

?'0,

z2,...,zk; w), j = 2,...,k

These functions are analytic in a neighbourhood of ... = W = 0. Moreover,

(3.57)

j''1

w)

O, j = 2,...,k

Because

iO ...

O

3?;

3?;

(3.58)

it follows that (3.59) (3. 61)

-det

af2

w)

3f2

3(f23.. ''k

,-*

-*

3Z2, .

_. Z7)

3k

*

-*

w) 3Zk

To resume: we are given now

k-1

functions,

(3.60)

_..,;

w), j = 2,...,k

,

analytic in a neighbourhood of ...

=i

= w = O and we are also given

k-1 f.p.s.;

If these f.p.s. are substituted in (3.60), then we obtain (3.57), and if these f.p.s. are substituted in the Jacobian

.1

a f.p.s. results which is 0.

Hence our original problem

(k

analytic functions and k f.p.s.) has been reduced, after a finite number of steps, to an analogous problem with

_k-1

analytic functions and

k-1

f.p.s..

IV. In this way we continue: the whole process, described in this proof, is repeated until one of the'three following situations arises.

(i) Suppose i functions and i f.p.s. with

i < i < k

are left and these i f.p.s. are all

det

polynomials in

w (i.e.

of each of the i f.p.s. only fihitely many coefficieÎts are # O). The convergence of these i f p s is evident and a straightforward calculation, in which we perform thé i'ñversè transføìma'tions, ShOws that the f.p.s.. have positive radii of convergence, which had to be proved

(ii) Suppose i functioii, .

6

. (s1,.. w), j 13

.

,Z

and if.p.s.; i 1,...,i, are left, which satisfy O., j=1,..,,l.

If the JacObian.

(3.63)

'à (s

is O at s w O, then the implicit. functiöh theorem can be applied, which states, in connction with théòrém (2.16), that are analytic. functions

o w in the neighbourhood of w = 0. Again performing the inverse transformations9, we obtain the result that the f.ps have positive radii of convergence.

(jj j) Otre function, given by

(3.64) g(s; w)

and one f.p.s., o , are left, This f.p.s. satisfies

(3.65)

. g(ci; w) (oj w) 0

Now theorem (2.22) can be applied, which states that o is an analytic function of w in a neighbourhood of W 0, and hence, if we perform the inverse transformations, the

4. DEFINITIONS AID KNOWN RESULTS OF CONTROL THEORY.

The definitions used in this chapter are in agreement with the definitions given in [4];this reference

can

also be consulted for a more complete treatment of concepts to be introduced. The quantities which are considered in this chapter are real.

A system is considered of which the motion is mathematically characterized by the lin-ear vector differential equation

(4.1)

where

the vector x = (x73...,x)' cálled the state variable, represents the state of the system in the n-diinensial (real) vectorspace R, (' detiores the transpose);

t denotes the time;

A is an n x n matrix with constant elements, this matrix is called the _system matrix;

B, called the control matrix, is an n. x r matrix with constant elements. The columns of B, none of which is identical to the zero vectOr, are denoted by

ba,j1,...,r

(y) the vector u =

(73,),

_{called the control, is a measürable vector-valued} function of time, with values uct) constrained to

(4.2) _{u.(tfl <} 7.

j =

At time t ₌ O the system has been situated at _{x0, where the vector x0 lies in}

_R

_and satisfies Hx°JI = i, L being the Euclidian norm, andwherê c -is a positive scalar. Let

x(t;u) denote the absolutely continuous solution of _{(4.I),satisfying almost everywhere}

(a.e)

(43) d.x(tu) _{- Ax(t;u) Bu(t)}

with initial data x(O;u) . This solution is sometimes called the trajectory of

the system (with respect to u(t) and _{z,°). If there is no risk of confusion we simply} write x(t) instead of x(t;u).

The problem is to determine a control _{u (if it exists), subject to its constraints,} in such a way that the solution x(t;u*) _{reaches the origin in R}

in mininum

_time

T, with O T < Such a control will be termed time optimal.

Some results related to the time optimal control (t.o.c) are more easily described in y-coordinates, to be defined, as in x-coordinates. _{The transforniation of x - to}

y -coordinates is defined as

=tA o

y=e

x-ex

It is well known that

etA

is regular for all, finite values of

t.

In the new coordi-nates the system must be steered from the oi.gin at

t = O

to the poitl

-x0 as

quickly as possible The differential equation of the motion of the system tn y-coordi-nates becomes

(4.5)

dt

= eBU

and

y(t;u)

will be the solution Of this equation satisfying y(O;u) = O. Define

(4.6)

R*(t)

{y(tu)

I u

measurable,

Iu.(t)l

i, j = 1,...,r}

* .

The set

R (t)

is called the reachable set at time

t

and consists of all possible values that solutions of (4.5) can asürne at time

t,

if all admissible controls are

used. The reacháble set is symmetric with respect to the origin, convex and compact

f or all

t > O

and is continuous in

t

with respect to the Hausdorff metric [4] In the

Hausdor,ff metric the distance of two nonempty, compact subsets A arid A2 in the n-dimensional Euclidian vector space is defined as

(4.7)

dde.

p(A73A2)

j{ max

w1 A7

d(w7,A2) #

max

d(w2,A7)

w2 A2

where

(4.8) d(w71A2)

min

{IJw-w7Ii}

j

w2A2

d(w2,A7)

min {11w2-w7 w7

t

A7

With the aid of the above mentioned properties of reachable sets it can be proved that if there is an admissible control u (i.e. it satisfies its constraints) and a t1 O

for which

y(t7;u)

-

tx0,

then there exists a t.o.c.. If this t.o.c. reaches the point - _{at time T, then this point lies on the boutidar'y of}

R*(T).

Definition (4.9). The linear system (4.1) is conttôllale if fo each pair of points x7.and s2 ih ¡, not restricted to

_¡IsII

= 1, i =

7,2,

and eàch t2 > O a bounded

meas-urable conrol

u(t)

exists, which steers the system from point xl at time

t

t7 = O

to 2 at time t =

t2.

Note that in this definition it is not necessary for the control to satisfT the con-straints (4.2). The definition 'of cont:rollabiiity given here, is équivalent to the definition of full controllability in [4]. Because in this referencè the elements of' the matrices/I and B are functions of time, several definitions of controllability have been introdüced in [4]. In cse A and B are constant, all these definitions

coiñ-I

An equivalent definition of controllability is.: system (4.1) is controllable ifthe origin is an interior point of

R*(t)

for each t > O A property of controllable systems is that the corresponding reachable set

R*(t)

is expanding, i. e

R*(t)

is contained in the interior of

R*(t*)

for all O

t < t

A computable criterion for controllability is that system (4.1) is controllable ifand only if the

n x nr

matrix

(4.1b)

, .

'

[B;AB,...,An-1B]

has rank

n.

Def-inition (4.11). For controllable systems the., controllability-indexvt1O] is defined

as (v

isintger,

1.) .

(4. 12)

. = min{ I rik [B,

AB, . . .,A1B ]

n

Theorem (4.13). Suppose (4.1) is controllable. A. necessary and sufficient condition for the control u (again satisfying (4.2)), which steers the point cx0 to the origin in the x-coordiñates, to be time optimal is that it has the form

*

o-tAj

(4.14)

u.(t)

' agn(14, , e

b ) , ,y = 1,...,r ,

for some n-vector P0 which' satisfies _IJ'P°IJ = 1..

The vector 11,0 appearing in (4.14) has a geometrical meaning. It is the normal of a support hyperplaneof the convex set R*(T) at- the point cx0, pointing away from this set. From this it is clear that different vectors may exist, which satisfy (4.14). It is not necessary that for a given ip, the t.oc. u*(t)

is

uniquely determinéd a.e. by (4.14.). .Itis possible that (P°,

eb

O for some j. However, in the main theo-rem of this thesis, theotheo-rem (6.9), we will be only interested in time optimal controls

for which a vector P0 exists in süch a way that these. controls are uniquely determined a.e. by (4.14).

Definition (4.15). The. point -cx0, which belongs to the boundary of R*(T), is an exposed point of

R*(T)

if

R(T)

has a support hyperplane at -cr0 which has only -cx0

as its intersection, with R*(T). . . .

-Theorem (4.16). The t.o.c.

u*(t)

is uniquely determined a.e. oñ [O,T] by

(4.17)

u,(t) = sgn[(ip°

e_t4b)]

_{, j =}

o o .. .

o.

. *

for some vector _4' with 4' = 7 if and only if cx is an exposed point of

R (T).

Moreover, the control u*(t) is unique in this case., though different vectors P0 may exist to satisfy (4.17). (The control u*(t) is said to be unique, if no other control

u**(t),

with u**(t)

u*(t)

_{on a set 'of positive measure on [O,T] , exists, which, in}

the y-space, steers the system from the origin to the point ex or in the x-space from the point cr' to the origin, in the same time as u*(t) does).

If (4'°,

e_tAbl)

0,

then it has à finite nunber of zeros on the interval E0,Tj, be-cause

CtAba)

is an analytic function of

ta

Then u?t) is defined nd Iu(t) = .1

on[O,T ]with the éxception of at most a finite number of t-values, and we say that the j-th component of he control u*(t) is bang-bang. 1f al;l components are bang-bang we simply say that the control is bang-bang. Points on the open interval (0,T), where one or more of the control components Jumps from -1 to 4-1 or vice versa, are called switches. Timès at which such switches occur are called switching-times.

In theorem (4.16) the condition for unique determination of the t.o.c. by (4.17) has been expressed in a condition on the point -cxc. Suppose all boundary points of R*(T)

are exposed points. Then to each boundary point at least one vector corresponds in such a way that the t.o.c. is unique and is uniquely determined a.e. by (4.17). I-f all boundary points ae exposed points añd if R*(T) contàinS more than one point, we say that the convex set R*(T) is strictly convex.

Definition (4.18).. System (4.1) is called normal ifR*(t) is strictly convex for each

t > 0.

Thus, if the system is normal, the t.o.c., if. it exists, is always unique and uniquely determined a.e. by (4.17) for some vector

Theorem (4.19). System (4.1) is normal if and only if the square matrices

(4.20)

[b0, Ab0,...,Alb0]

,

=

have rank

n.

Note that system (4.1) is normal if and only if it is controllable with respect to each of its control components sepárately. A normal system is always controllable; the cOnverse is not true.

Let us now consider normal systems. The t.-o.c. is a piecewise constant function .of time. Since system (4.1) is time-invariant, we are able to determine the t.o.c. as a function of the state x only. Hnce we find sets or regions of the state space over which the control is constant (on such a set for instance u7 +1,

u2 =

U

-1).

We feel intuitively that it is plausible that thesé sets will be separated by curves in R2, by surfaces in R3 and by hypersurfaces in tile n-dimensional space. As f ar as the author knows, no general proof of this "intuition" exists. The separating sets are called switching-curses, switching-surfaces or switchinghypersurfaces [5]. The concept of switching-(hyper)surfaces is not only restricted to normal systems as

5. SYSTEMS

In the previous chapter controllable systems have been defined and, _{as a subclass,} nor-mal systems. Before formulating the maiñ theorem, we will define another subclass _of controllable systems and show how it is related to normai systems.

*

Systems which belong to the subclass to be introduced, are called _{L -systems. To define} L-systems, it is necessary for us to restrict oursèlvès to controllable _{systems of}

which the cali ns i.e.

r < n.

V

-For systems of which 'rank

[B] = r,

_{let p denote the smallest natural nùber which is} greater than or equal tO . _{and let us define the integer i as}

r

(5.1)

i

_{n - l - (p-1)r}

The integers p and i will appear to be important in the Vrest of this thesis. It: is

easi-ly seen that

(5.2)

O<l<r.-7

Definition (5.3). Systems of which rank.[B]

r

_{are called ile-systems if the following} square matrices are regular:

(5.4) L

_-

C

p-2 p-1 i p-1 £ q i

-

B, AB,...,A

_{B, A}

b 13.,A

_b

_{A b}

The nuinbêrs(i11...3i1) are number i17 is equal to one numbers [i7,...,i1), then

_q

in (5.4) is defined as

(5.5)

' L

..

₌

l_li.

ing matrix

L:

a polynomial in

det[L'*

.*

l-1,...,7-ilf.

spanned by the ces exjsts, the

rank

[B] = r.

In this. case

Obviously all L*systems are controllable

and have the smallest possible controllabiLi-ty-index, i.e. v _p

Observation (5.6). In this remark it will be shown that almost all systems are

L

-sys-tenis. Choose an admissible combination of thé _nuthbers

arid fix it;

these numbers wil-l. be denoted by i7,...,j7. _{Consider the dependence of the}

correspond-.* _{on the elements of 4 and}

_{B. The}

determinant of this matriÑ is

.,_/-l-I-1

these elements. If this polynomial j _{not identical to zeto, then} ]is zero only on a hypersurface in the (real) vectorspace

i

* elements of A and B. Because only _{a finite number of different L -matri}

-

.flXfl/-flxr

re is only a finite number of such hypersurfaces _{in R to which} an arbitrary choice of different numbers of [1,...,r

J. The

of the ni.mthers

(1,...,r).

If i17 is equal to one of the is defined as _q = p and èlse

q = p-l.

If pl, the 1*_matrix

i

q i

Systems which are not L-systes correspond.

It remains to be shOwn thàt the above mentiOned polynomial is, not identical to zero. that end we will give matrices.A'.4P4 for which det[L* .*

)

0.

*

The coluthri vectors of

L .* * can be arranged as

i j

k-i 1

2 2 k.. -1 2. k -7 r

(5.7) [ b , Ab b , b, Ab ,...,A

2

b ,..,A

r b ],

where the numbers k. are nonnegative integers (k. p-7, p or p#l) which satisfy

t t

r

-S-.

/

_{5. 8i}

-

,

- .- t

-The matrix (5 7), formed by the rearranged columnvectors of L* * is denoted by

-L.

In L only those b' appear as a coluninvector for which i is such that k 1. Hence it is sufficient to give only those bt_vectors. If k. 1., then ht is defined as

(Ö,...,0,7,O,...,0)', with the number 7 at the (n -

tl

k.)-th place. If A is defined

j=1 -as (5.9) A

t

hen (5.10)

det[L*

.

.*

-]=

-

# det.[t]det

1 0 ...O

In this remark we

assumed

ftat each point in the veòtor space has an equal chance of representing a (control) systeth. Frm an engineering point of view this is of coursé questionable. III

By means of four examples:we will show that thé interrelátion of the introduced sub-classes of controllable systems, i.e. normal - and L -systems, is as sketched in fig.

(5.1). Before start-ing with the examples we introduce the notation:(n,r) system,

whih

means that the state vector of the system under consideration has n omponents, and that

the control vector ha r components.

-The examples have been given in the table on page 34. -The numbers in the first col of

this table correspond to the same numbers in f1ig. (5.1)-. Thüs for instance ,núber 7 is a controllable

sytêm

which is not normal and not an L*_systern.; the matrix L is

singu-lär.

Note that, given a fixed n and r,

it

is not always possible to give four examples

i

O...0

.1

Fig. (5.1) The interrêlation of normal - and L*_systems.

corresponding to the four numbers in fig. (5.1). If for instance r1, the subclasses introduced coincide with the claàs of controllable systems, and hence an example of a system which is controllable, but not normal, does not exist.

Remark (5.11). It is easily seen that the-introduced-subclasses of systems are invari-ant with respect to coordinate transformations of the form Px, where is the state vector in the new coordinates and P is a regular thatrix with constant lèments.

I/I

"x"

means that thé system matrix belongs tó the (sub)class concerned. u_ . example n r .

pl

E 1J. o 0 u E Z (O s.

0*

E ai

i

O0øC

ww

,0Ua

1Ja)iI

-system matrix

A

- -control matrix B

.10

Ó 0 -1 0

0.00

-1 o]. Q 1]

01

3 2 2. 0 x L 2

-J

O O O

O-2

0 0 O

O-3 O

Ó,Ö 0-4

I I 1 2 i 3

14

4 2 2 I

x

-:

x

_.1,2..

L

*

.3 -1 0 O

Ö-2 O

O O-3

rl

-Ï

Iii

Li

o

3220

x x

-4

fool

i

Loi

i

Iii

ii

L'].

.

2120 xxx

-6. FORMULATION OF THE MAIÑ THEOREM.

At time t = C the system, given by (4.1), has been situated at x(0)

tx,

_{where the} vector z0 satisfies IJx°IJ i and where

t

is a positive scalar. The system must be steered to the origin in R as quickly as possible. It is known that the.t.o.c. exists if the system is controllable and if

t

is sufficiently small. The time at which the system reaches the origin by means of the t.o.c. is denoted by _T. If -cx° is an exposed point of R*(T), the. t.o.c. is unique

_and

shows a bang-bang behaviour.

The solution of the differential equation (4.1) can be written as t

tA o -sA

(6.1)

z(t)

_{e {cx}

_e

_Bu(s)ds

- o

We suppose that the t.o.c. which in the x-space transfers the system from cx0 to the origin, exists and that

-tx0

is an exposed point of R*(T). If the t.o.c. is

súbsti-tuted,. the result is.

t t

11 12 n T

(6.2) _{0 =}

e1x'i'

=

EX0 # (

-

+

... #

)eb'da

o t t 11 _7n t21 t22

-I

o t21 t t

rl

r2

nT

1

nT

+ (-1)

2

j

)ib2ds

t 2n 2 o t

_rl

t

rn

_r

where t.. is the time at which the j-th switch of _{the i-th control component u. occurs}

1_1

2-and

ni

is the number of switches of the i-th _{control component. It wiLl appear that for} many systems, many values of z0 and _{sufficient]y small, the total number of switches} equals

n-l.

Let us restrict ourselves to this _{case for a moment. Then the vector eq.} (6.2) consists of

_n

equations with _{n unknowns, viz, the switching-times and the final} time. Also the unknown signs with which _{the u.(t) start, the ttstarting_signs!t}

havé to be determined.

Observation (6.3). One might t-hink that these _{n equations determine the signs and the} unknowns uniquely; this is in general _{not true. Even if we suppose the natural}

restric-tion C < t.1 < t.

< ...

< t. < T, i

= l,...,r,

more solutions may exist. However, the t.o.c. is unique and has to be extracted from all the solutions obtained from eq. (6.2). Of course the solution with the _{Smallest value of T is the right}

one. More will be said about this later on. I/I

Iñ the main theorem an explicit criterion will _{be given about the number of switches} and about the starting-signs of the different _{control components. To this end we need}

the concept of characteristic matrices.

*

For L -systems the (square) characteristic

matrix

M. . (which is called an M-matrix) is defined as

p-1j7

-p-li1

(6.4) M. .

= c,

(-A)

b

(-A)b

(-A)

_b'

4.

#-b?r)]

p! P! where

r

-AB

(-A)2B

(6.5) J 1s

,.,

L

2!

'p-l)!

J

and where the numbers

j1...,j

are añ- arbitrary permutation of the numbers {l,2,..,r} with the two separate restrictions

(6.6) i-1

_<2

_{< <}

¿il, a4.1

<a12 <-...

If p = 1, the matrix J is not defined and does not appear in the Pd-matrix (6.4).

We will now discuss the choice of the + signs in the last column of the M-matrix (6.4). To this end cOnsider the vector _{J°J =} 1, defined to be perpendicular to the first n - i columns of M. . , which are linearly independent. Of the two remaining

possi-a.

bilities for 'J , - opposite directions - , one is chosen and fixed. Later on a. criterion

will be formulated which decides on the two possibilities, but at this moment this is

irrelevant. The signs of , i l#l,.. .,r, in the läst column of this

charac-teristic matrix are nw determined by

p-1 i.

ágn(°,

A b )

Because we restricted ourselves to L*_systems, these sgn- relations are well defined. Another interpretation of the choice of the + signs can be given. Th signs are chosen in such a way that the innerproduct of d A (i- ± ... + b') has its maximal value. From this interpretation it follows that the characteristic matri M,rji is regular..

Obviously ( different M-matrices exist for a syste.; different in the sense f differ-ent sequences of indices.. In case i O, or.when there :is no risk of confusion, the

indices of the M-matrices will be omit-ted.

-wé introduce the flotation M{j; x°} , which denotes a a trix equal to the matrix M, except in the j-th column, which is- replaced by the column vector x0. -This notation will álso be used with respect to other matrices and other coli.mn vectors.

In the proof of the main theorem it will appear that to: almost all unit vectors z0 a-unique M-mätrix corresponds in such a way that

(6.7)

(6.8) dt[M{(p-i)r

i

x°}] <

Idet[MCn;

x°}] I

i

=

ll