Almost disturbance decoupling with bounded peaking

Almost disturbance decoupling with bounded peaking

Citation for published version (APA):

Trentelman, H. L. (1986). Almost disturbance decoupling with bounded peaking. SIAM Journal on Control and Optimization, 24(6), 1150-1176. https://doi.org/10.1137/0324070

DOI:

10.1137/0324070

Document status and date: Published: 01/01/1986

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ALMOST DISTURBANCE DECOUPLING WITH BOUNDED

PEAKING*

HARRY L. TRENTELMAN’

Abstract. Thispaperisconcerned with a generalization of the almost disturbance decouplingproblem by state feedback. Apart from approximate decoupling from the external disturbancesto a first to-be-controlledoutput, we require asecond output tobeuniformly bounded with respect to the accuracyof

decoupling. The problemis studiedusingthegeometric approachtolinear systems. Weintroducesome newalmost controlledinvariant subspacesand studytheirgeometricstructure. Necessary and sufficient conditionsforthesolvabilityof the aboveproblemare formulatedintermsofthesecontrolledinvariant

subspaces.Aconceptual algorithmis introducedtocalculatethe feedbacklaws needed to achieve thedesign purpose.

Key words, almost disturbance decoupling, almost invariant subspaces, linear systems, geometric approach, high gain feedback,output stabilization

AMS(MOS)subject classifications. G3-B28,G3-BS0, G3-C05, G3-C15, G3-C35, G3-C45,G3-C60

1. Introduction. In this paper we are concerned with the problem of almost disturbancedecoupling bystatefeedback asintroducedbyWillems

[20].

Thisproblem dealswiththesituation in which we cannot achieve exactdecoupling from the external disturbances to an exogenous output channel as, for example, in

[22],

but only approximate decouplingwithanydesireddegreeof accuracy.In general,thefeedback gain necessaryto achieve this will increase asthedesireddegree of accuracyincreases. Itmay then happenhoweverthatsome of thestate variablestendto peakexcessively.

Itisofconsiderablepracticalinterest toknow whenit ispossibleto achieve disturbance decoupling within any desired degree of accuracy, while this peaking phenomenon will not occur.

The systemthatwe will be considering in this paperis given by the equations Yc Ax

+

Bu

+

Gd,

(1.1)

_z

Hx,

z

H2x,

where the control

u(t),

thestate

x(t),

the disturbance

d(t)

and the outputs

z(t)

and

z2(t)

are real vectors offinite dimensions. Wewill assume that the vector

z2(t)

is an enlargement of

z(t),

i.e., thereis amatrix M such that

_H

MH2.

Iffor any positive real number e afeedbackmatrix

F,

canbe chosen such thatintheclosed loop system with zeroinitial condition, forall disturbancesd

(.)

in the unitballof

Lp[O, )

wehave

(1.2)

_{IIz ll}

<-thenwesay that for the system under consideration the

Lp-almost

disturbance decoup-ling problem from d to

_z

is solvable. After choosing

_F

to achievethis approximate decoupling, the output

z2(t)

ofcoursedependsoneand,forcertain disturbancesd

(.),

itmay then happen that

_z=ll

_,.o

-

ase 0,i.e.,asthe accuracy of decouplingincreases.

* Receivedbythe editors April28, 1983,and in revisedformJune1, 1985.

"

Departmentof Mathematics andComputing Science,EindhovenUniversityofTechnology,5600MB Eindhoven, the Netherlands.Thisresearch wassupported bytheNetherlands Organization forthe Advance-mentofPureScienceResearch(Z.W.O.).

As

an example, considerthe system

(1.1)

with

A= 0 B= G=

0

H1-"

(1

0

0),

HE

1

0 Define afeedbackmatrix

_F

by

Itcanthen be verified that the impulse response from thedisturbance d to zl isgiven

by

+3

9

W

e(t):’-"

nle(’--JF)tG

e-3t/e(1-t+--

t2

k e

282

/

and that

_Wl,

e.

Hence,

for any 1-< p-<

_o,

theabove feedbackmatrix

_F

achieves

Lp-amost

disturbancedecouplingfrom d toZl.Onthe otherhand, however,the impulse responsefrom d to _z2 iscalculatedto be

1+3

9

t2

t e

+2e

2

W2

(t):=

HEeA+a)tG

e-3/

2._7.

t2

2e 27 81

t2

---

+

--andit canbeverifiedthat

o(1 / e)

->3_ase->0, i.e.,wehaveobtained almost disturbancedecoupling from dto

_z

atthecostof highlyundesiredpeaking behaviour of the output

z(t).

The question which we ask in this paper is this: When is itpossible to choose

_F

such that simultaneously

(1.2)

holds and there exists a constant C (independent

of

e)

such that

_for

all disturbances d

(.)

inthe unit ball

_of

Lp[0,

)

wehave

1.3)

_z=

<--

c

for

all e? That is, the output

z2(t)

is bounded uniformly as e tends to zero. Ifthis behaviour isachieved,wesay thatwehave

Lp-bounded

peaking from dto z2. Problems

ofthis kindhavebeenconsidered before.Francisand Glover

[3]

considered abounded peaking problemin the context ofcheap control. More recently, Kimura

[9]

found conditions that guarantee bounded peaking in the context of perfect regulation. We willstudy the above problem using the bynowwell known concepts of almost controlled invariantand almost controllabilitysubspace

[19],

[20].

Wewillalsousetheapproach offrequency domain description of geometric concepts asinitiated in Hautus

[5].

The outline ofthis paperis as follows. In 2we will introduce somenotational conventions usedin this paper andstate somepreliminary results andbackground.

Section 3 contains a description of the mainproblemwe willbe concernedwith in thispaper. In 4 we will introducethedisturbancedecouplingproblemwithoutput stability. Thisproblem is anextensionofthe

(exact)

disturbance decoupling problem

as treated in

[22].

Its solution willbe needed to solve our mainproblem, but isalso importantin its ownright.In 5we willderive anecessaryconditionforthe solvability of

(ADDPBP)p.

This condition willbe in the form ofa subspace inclusioninvolving an almost controlled invariant subspace. Section 6 contains an investigation ofthe geometric structure of the almost controlled invariant subspace that was introduced in 5. In 7 these structural results will be used to prove that for certain classes of systems the subspace inclusion derived in 5 in fact constitutes a necessary and sufficient condition for solvability of

(ADDPBP)p.

The sequences of state feedback maps thatachievethe design purposewillbeconstructed explicitly.Section 8 contains somecorollaries ofour main resultand some extensions. In 9 a numerical example is worked out to illustrate the computational feasibility of our theory. Finally, the paper closeswith someconcluding remarksin 10.Several technicaldetails ofproofs in this paperare deferred to Appendices

A,

B andC.

2. Preliminaries andbackground. Inthis section we will introduce some notation used in this paper and review somerelevant facts oncontrolled invariant and almost controlledinvariantsubspaces. Alsosomebasicfacts onthe convergence of subspaces willbe given.

2.1. In this paper the following notation willbe used: If is a normed vector space, we will write

_I1"

for the norm on

.

If l"[0, c), is a measurable function, thenwe will denote

[[/(t)ll

p_dt ?-|esssup k. t>o lip if 1

=<

p

<

c,

ifp

=.

If

_II/11

<

o,

we will say that l

Lp[O, oo).

If M is a square matrix then

tr(M)

will denoteitsspectrum. If

A1

and

_A:

aresetsof complexnumbers then

A1

u)

A:

willdenote

their disjointunion. Foranypositive integer n, we will denote _n :=

{1,

2,

,

n}.

Considerthe system

(1.1).

Let

u(t)

:=

_",

x(t)

:=

",

d(t)

:=q,

z(t)

Lrl

:=Rp, _and

_{z(t) Lr}

2:=Rp2. Let

A, B, G,

H1

and

H2

be real matrices of appropriate

dimensions. We will write ’[i := ker

_Hi

(i 1,

2),

:=imB and

_AF

:=A+ BF. The reachable subspace will be denoted by

(A[

):=

d

+A

+...+

A"-.

A

collection of subspaces

,

2," ",

k

will be called a chainin if

=

:

=.

=

k.

If0#b we will denote :=span b.

If Vc is AF-invariant, the restriction of

AF

to Vwill be denoted by

AF[

Wewill write

_AF

I./V

or/F

for the quotient mapinducedby

_AF

onthefactor space

/V (see [22]).

If

V

and are both

AF-invariant

and

V,

then

AF[

V/

will

denote the map inducedby

AF[V

onthe factor

space

V/ogr. Wedefinethe canonical projectionP"

/V

byPx:=x

+

V:If B :=

PB,

then

(AF, B)

will be called the system induced in

/V.

If H"

Lr

is a linear map and Vcker

H,

then H"

/V Lr

is defined byHP H. Adistribution

_f

9’/ (i.e., thespace offinite-dimensionalvalued distributions with support on

[0,

oo))

will be called a Bohl distribution ifthere exist

N0f/t(i)

+f-1

Here

f-l(t):=

KeLtM,

vectors

_f

and matrices

K, L,

M such that

_f=

Y,i-(o) denotes Dirac’s delta and (i) its ith distributional derivative,

_f

will be called regularif

_f

0 (i 0,.

,

N)

and impulsive iff_ 0.

2.2. Wewill now reviewsomebasicfacts from geometriccontroltheory. If cg

is a subspace, then

V*(’[)

will denote the largest

(A, B)-

or controlled invariant subspace in Y{and

*(Y0

will denote the largest controllability subspacein Yf

[22].

IfCg

is asymmetric subset ofthe complex plane C(i.e.,h

Cg:>

Cg

and

Cg

contains atleast one point of the real axis),then

g*(t

)

willdenote the largest stabilizability

subspace in Y(

_([5]

or 11

]).

Asubspace //’ac willbe calledalmost controlled invariant ifforall_Xo l/’a and for all e>0 there is a state trajectory

x(.

such that

x(0)=

Xo and d(,, x,(t))<-e for all t. Asubspace

a

c _{willbecalled an almost controllabilitysubspace}ifforall

Xo,

x

Y

there is a T

>

0 suchthat forall e

>

0there is astatetrajectory

_x

(.)

such that

x(O)=xo, x(T)=Xl

and

d(Ya,x(t))<-e

for all t. Basic facts on these classes ofsubspacescanbefoundin

[19]

or

[20]

(see

also

[17]). A

subspace

a

c isalmost controlled invariant if and only if

+

,,

where is controlled invariant and

a

is an almost controllability subspace. Asubspace

a

is an almost controllability

k

in such that subspace ifand onlyifthere is a map F"

T

07/ and a chain

{

i}i=l

=I+AF:+’"

"+AkF-k.

We will say that is a singly generated almost controllabilitysubspaceifthereis amap F"

%

a vectorb andaninteger k

>

0 such that

,

d03

AFd"

"03

AkF-L

Again, for Y’c

,

T’*(YQ

will denote the largest almost controlled invariant subspace in

Y"

and

*(Y{)

the largest almost controllability subspace in

’’.

We will denote

Rb*(Y’):=

+

A*(Y’)

and

T’b*(Y():= T’*(Yc)

+

b*(Y{).

The subspace

//’b*(Y’)

plays an important role inthe problem of almost disturbancedecoupling. In fact, in [20] the following resultwas obtained:

PROPOSITION2.1. Considerthesystem Ax

+

Bu,

z Hx. Then

_for

alle

>

0 there

exists a map F’gT-ll such that

_{IIHexp[t(A+BF,)]GII,<-_e if}

and only

_if

imGc

T’b*(ker

H).

LetY{:=kerH.The space

Vb*(Y{)

willbe called the space of distributionallyweakly

unobservable states with respect to the output z.

b*’(Y’)

will be called the space of strongly controllablestates with respect tothe output z. Forthisterminology see

[6].

Aproof of the following result canbe foundin [1, Lemma 1]. LEMMA2.2. LetY{

.

Then the following equalities hold"

(i)

b*(Y’)

f’l Y’-"

*(Y{),

(ii)

*(:]’)

f’)

V*(Y’)

*(Y0,

(iii)

*(Y()

f)

*(Y{)

*(Y0.

[3

This paperwill sometimesdeal with a newsystem

(A, BW),

obtainedby

_del.eting

the part of the inputmatrixBlyingin

*(Y{).

Thissystemisobtained

_b.y

taking

=

suchthat

0]

q (

f3 o//.,(y/))= and by letting Wbeamap suchthat =imBW

(see

also

[1]).

The supremal almost controllability subspace contained in Y{with respect tothisnewsystem

(A,

B

W)

willbedenoted by

*(Y{).

Wewillcorrespondingly denote

o

+

A*,(Y[)

by

*(Y().

The following result follows from[1, Lemma

2]"

LMMA2.3.

Assume

nowthat

T"

is

(A,

B)-invariant. Let Fbe such that

(A

+

BF)

Let

(AF, B)

be the systeminduced in

/T"

and P"

-

/V

thecanonicalprojection.

Wethen havethe followingresult:

LEMMA2.4.

_If

_t

isan almostcontrollabilitysubspacewith respectto

(A, B),

then

Pt

isan almost controllability subspace with respect to

(AF, B).

Proof

Let

_Pxo

and

_Px

be in

Pta,

with Xo,Xl

a.

There is a T>0 and, forall

e>0, atrajectory

x(.)

such that

x(0)=

Xo,

x(T)=x

and

d(a,X(t))<=e

for all t.

It can be seen immediately that

z

(t):= Px,(t)

is a trajectory of the system

(AF, B).

Moreover,

z(0)=Pxo,

z(T)=Px

and

d(Pa,z,(t))=infRllPr--Px(t)ll

Wewill also needthe following proposition, which is provenin

[17,

Thm.

2.39]

(see

also

[15]

or

[16]).

PROPOSITION 2.5. Considerthe system

Ax

+

Bu.Let

a

beanalmost controllabil-ity subspace.

Suppose

A is a symmetric set

_of

dim

(A

g)-dim

a

complex numbers. Then thereis an

(A,

B)-invariant subspace

V

anda map F"

-

such that VO)

(A[)

and

r(AF]

)

A.

To concludethis section, we shall recall some factson left-invertibility oflinear systems. Again consider the system

Ax

+

Bu,

z Hx.

Assume

that the map B is injective.

We

willsay that the system

(A,

B, H)

is left-invertible ifthetransfermatrix

H(Is-A)-IB

is an injective rational matrix. The following resultwas provenin

[22,

Ex.

4.4] (see

also [6, Thm.

3.26]).

LEMMA2.6.

(A,

B, H)

is a

_{left-invertible}

system

_if

andonly

_if

*(ker

H)=

0. [3

2.3. In the following, we will review some basic facts on the frequency domain approachtothe geometric concepts ofthis paper. Wewilldenote

[s]

(respectively,

(s), +(s))

for the set ofall n-vectorswhose components are polynomials

(respec-tively, rational functions, strictly proper rational functions) with coefficients in R. If

c

=R",

then

5’’[s]

(respectively,

5’[(s), r+(s))

willdenote thesetofall

(s) [s]

(respectively,

(s), +(s))

withthe property that

sO(s)

5’[for alls.Slightly generalizing adefinitionbyHautus

[5],

ifforagiven x thereare rationalfunctions

:(s)

(s)

and

,

to(s) (s)

such that

x=(Is-A)(s)+Bto(s)

forall s, we willsay that x has a to)-representation.

For a description of

(almost)

controlled invariant subspaces in terms of

(,

to)-representations,wereferto

[5], [12], [13]

and

[17].

Weshall need the following fact:

LEMMA 2.7. Let

.

Then we have" x

_{*b(7[) if}

and only

_if

x has a

_(,

to)-representation with

(s)

’’[s]

and

to(s)

a//[s].

U

2.4. Finally, we will recall some facts on the convergence of subspaces. In this paper we will use the common notion ofconvergence ofsubspaces in the sense of Grassmanian topology. Let {%; e

> 0)

be a sequence of subspaces of of fixed dimension.Itcanbe proven that

%

-

(e

0)

ifand onlyifthereis abasis

{v,

,

va}

for Vand there are bases

{vl(e),’’

’,

Vq(e)}

of

%

such that, for all i, vi(e)vi as e

-

0

(convergence

in

).

Wewillneedthe followinglemma,which canbe provenby

standard means:

LEMMA

2.8.

Suppose

Vl,

,

Vq areindependentvectorsand

v,(e)

-

_vfor

alli. Then

for

esufficientlysmall,vl

(e),"

.,

Vq

e)

arelinearlyindependent.Consequently,

ifV

-

V

and

_t’

tV,

where

V

f’)/4/"

_{0},

then

_for

e sufficiently small

%

f’)

_/V

_{0}

and

_%

O)

3. Mathematicalproblemformulation. Considerthe system

(1.1).

Wewill assume that

z2(t)

is an enlargement of

z(t),

that is, thereis a matrix M suchthat

_H1

_MH2

or, equivalently,

(3.1)

ker

_H2

=:

_X2

c

’’

:=ker

_H.

Fromnow on,

(3.1)

willbeastandingassumption.Throughoutthispaperwewill also assume thatB is injective.

Consider the following synthesis problem. Fix 1-<__p<_-. We will say that the

Lp-almost

disturbancedecouplingproblemwithbounded peaking

(ADDPBP)p

issolvable ifthereis aconstant C andfor all e

>

0thereis afeedback map

F"

suchthat,

with the feedback lawu

_Fx

inthe closedloopsystem for

x(0)

0for all d

Lp[O, ),

the following inequalities hold:

(3.3)

_IIz=ll

_CIIdll,

Notethatif wetake

_H

H,

we obtainthe original

_L-almost

disturbancedecoupling problem,

(ADDP)p,

without the requirement of bounded peaking

(see [20]

or

[17]).

Anotherinteresting specialcaseis totake

_H

=/,whichcorrespondstothe requirement of bounded peaking

of

theentirestate vector.

Inthe present paper, necessaryand sufficientgeometricconditionsfor the solvabil-ity of the aboveproblemwillbe derived for the casesp 1, p 2 and p oo. Wewill firstshow how the solvability of

(ADDPBP)

can

be

expressedin terms of the closed loop impulse response matrices from the disturbance d to the outputs

_z

and z,

respectively. If

_F"

q/isa statefeedback map, then denote the closed loop transition matrixby

(3.4)

T( t)

:=e(’+)’

and let

(3.5)

’(s)

:=

(Is-A-

BF)-’

denote its Laplace transform.Wethen have the following:

LEMMA3.1. Fixp

{1, oO}.

Then

(ADDPBP)p

is solvable

_if

andonly

_if

there is a

constant Cand

_for

all e

>

0 a

_feedback

map

F"

all such that

IIn

_LOlls,-<-e

and

IIH=

TOII,

<=

C.

(ADDPBP)

is solvable and only

_if

there is a constant C and

_for

all e>0 a

feedback

map

_F

such that

_{H T}

(s)O

and

_H

’

(s)O

areasymptotically stable and such that sup

IIH,

L(i,o)Oll

<--

and sup

IIU=L(i)OII-<-

C.

Proof.

Theprooffollows immediately from the fact that for p 1 and for p oo the

_L-included

norm of the closed loop convolution operator from d to

_z

equals exactly the

_L-norm

ofits kernel,i.e.,

IIHTGIIL,.

The secondstatementfollows from the fact that the L.-inducednormof theconvolutionoperator from d to

z

equals the

H-norm

supa

IlU,(i,o)Oll

(see,

for example,

[23).

[3

4. Disturbancefleeouplingwithstabilityconstraints. Prior toconsiderations involv-ing the peakinvolv-ing behaviour of the enlarged output z2, we should make sure that the

outputz2is in

Lv[O,

oo)

atall.

Hence,

animportant part of thesolutionof theproblem

posedin 3is to constructthe required feedback maps

F

insuch awaythat,for any d

_L[0,

),

inthe closedloopsystemwith

x(0)=

0 wehavez2

Lv[0,

co).

Therefore,

in this sectionthe followingvariation onthe well known

(exact)

disturbancedecoupling problem

[22]

will beconsidered. Again, consider the system given by

(1.1).

The usual disturbance decoupling problem is concerned with the determination ofa feedback map F" 0// such that in the closed loop system the external disturbance d does not influence a specified output Zl. We will consider the more general situation in

which simultaneously we demand stability of thesecond, larger, output z2.

Inthis section, C_g, the stability set, willbe agiven subset of the complex plane C which is symmetric. Asymptotic stability is thus obtained by taking C g=

{h

cC"Reh

<

0}. A

rational matrix or rational vector is called stable if all its poles arein

Cg.

Wewillconsider the followingproblem"

(DDPOS)

the disturbance decoupling

problem with outputstabilization is said to be solvable ifthere is a feedback map F

such that

H(Is-AF)-G=O

and

H2(Is-A)-IG

is stable.

Inordertobe abletoformulateconditionsfor the solvability of the above problem, introduce the following subspace:

DEFINITION 4.1.

*Vg(ff’l,

’2)

will denote the subspace of all points xc

₁

for whichthereis a

(:,

to)-representationwith

(s)c

l,+(s),

to(s)c

q/+(s) and

H2:(s)

is stable.

Thus,interpretedinthetimedomain,

Vg(Y(,

’{2)

isthesubspace consisting ofall points in which a regular Bohl state trajectory starts that lies entirely in Y{. The components ofthistrajectory modulo

’2

are stable. It follows immediately from the definition that

Vg(Y{1, ’{2)

is contained in

V*(Y{).

By the assumption

(3.1),

if a

trajectoryliesin ’{2,the same is truefor’{1.Consequently we alsohavethe inclusion

ff’*({2

(22

c/’g(’{l {2)

We notethat Definition 4.1 is ageneralizationof a definitionbyHautus

[5].

His space

_S

(see

[5, p.

706])

coincides with

Vg(Y{1,

’{2)

if ’r is taken to be

.

The following theorem canbe provento be completely analogousto [5, Thm.

4.3]:

THEOREM 4.2.

(,,

)=

*(c,)+

*().

D

Notethatitfollowsfrom the above theoremthat

Vg

({1,

ff2)iscontrolledinvariant. The nexttheorem provides the key step in thesolution ofDDPOS. The result states thatwhat can be done in Definition 4.1 by open loop control can in fact be done by state feedback"

THEOREM 4.3.

ere

existsa mapF" such that

(4.1)

AFg(Y, Y2)

C

g(Y,

(4.2)

AF*(Y2)

*(Y2),

(4.3)

(AF

g(Y[,

Y2)/*(Y2))

c

_Cg.

Proof

Duringthisproof, denote

g

:=

_g(Y,

_Y2).

Since

*(Y2)

g

andsince both spaces are controlled invariant, they are compatible

(see

[22, Ex.

9.1]). Hence,

there is a map

Fo’

such that

_Ao*(Y2)

*(Y2)

and

AFog

c

g.

_{Let @:=}

@

g

and let Vbe anymatrix such that @ imBE

Consider the controllability subspace

(AFo[).

By

the

facts that

g

and

AFog

g,

this controllability subspaceis contained in

_Y.

Since any controllability subspaceisalsoastabilizability subspace,it mustbecontained inthe largest stabilizabil-ity subspace

(Y)

in

_Y1.

Itthen follows that

_(Y),

so

(4.4)

(Y{)=

.

Next,

we claim that

_(E{1)

is

_{AFo-invariant.}

First, since it is

(A,

B)-invariant, we have

_{Ao(C,) (C,)+.}

On the other hand,

Ao(C,)Ao,

Hence,

again using

(K1)g,

we obtain

AFo(E{,)((E{1)+)g

(E{1)

+

(

g)=

(E{1).

The last equality follows from

(4.4).

Using

(4.4)

and [5, Prop. 2.16], we deduce that the pair

_(AFol

(E{1),

BV)

is stabilizable.

Let

_PI"

g

g/*(E{2)

be.the canonicalprojection. Let

(AFo,

BV)

be thesystem induced in

g/*(E{2).

It can easilybe seen, for example, by using arank test

(see

[4]

or[5, Thm.

2.13]),

that the latter systemis stabilizable.

Hence,

there is a map

_F1

onthefactor space such that

(AFo +

BVF1)

Cg.

Now,

let

_F1

beany mapon

g

such that

_F

_FP

anddefine

_F1

arbitrary on a complement of

g.

Define F :=

_Fo

+

VF1.

Since

FI

*(C:)=

_Fol

*(C=) ("

I"

denotes"restriction

to"),

wethen have

AF*(E{=)

*(E{)

and itcan beverifiedthat Fig. 1 commutes.

Weare now in apositionto prove themain resultofthis section. THEOREM 4.4. DDPOS issolvable

_iff

imG

g(E{1, E{2).

Proo

()

Choose Fasin

eorem

4.3. Then

_(AFlim

G) E{1,whichyields the decouplingfrom d to

_z.

FIG.

Let

/2

be as in the Fig. 1 and let

tF’.’-AFlt/’g(ff{1,{2)/ff’*(ff{2).

Then

HE(IS-AF)-IG

I2(Is--,F)-IP1G,

which is stablesince

tr(AF)

c

_Cg.

(==>)

_IfF

is such that

HI(IS-AF)-IG-O

and

H2(Is-AF)-IG

is stable then for

d let

:(s)

:=

_(Is-

_AF)-iGd

and

to(s)

:=

_F(s).

Then clearly Gd

(Is-A)(s)+Bto(s)

and H_(s)is stable. [3

Remark 4.5. Ifin the above problem we take

_HI

_HE

H,

DDPOS reduces to the ordinary disturbance decoupling problem DDP

(see [22]).

Inthis case we have,

denoting if{:=_ker

_H,

_Vg(7{1,

_’{2)

_Vg*(20

+

_V*(20

_V*(2{)

_If we take

_HI

=0 and

HE--H,

we arrive at OSDP as studied in Hautus

[5].

The solvability ofthis problem requires the existence of a state feedback F such that

H(Is-AF)-IG

is stable.

Necessary and sufficient conditions can be found by noting that

Vg(2{,ff’2)=

Vg*()+

V*(20.

As

also noted in [5], if we take

_HI

=0,

HE

H and im G=

,

the above theorem provides necessaryand sufficient conditions forthe solvability of the output stabilizationproblem, OSP.

5. Anecessarygeometric condition for thesolvabilityof

(ADDPBP)p.

Inthis section we shall establish a necessary condition for the solvability of

(ADDPBP)p.

This condition will be inthe form ofa subspace inclusion. The proofis rathertechnical and some ofthe details are deferred to Appendix A. In the rest ofthis paper, the stability setwillbe takento be Cg

{A

C

IRe

h

< 0}.

Consider the system

=

Ax+

Bu,

_z

HlX, z2

H2x

and assume that

(3.1)

is

satisfied. The following subspace willplay an important role inthe sequel:

DEFINITION 5.1.

//’b(ff{1, ’{2)

will denote the subspace of all x X that have a

(s

c,

to)-representationwith

so(to) ’{(s), to(s) (s)

and

H2(s)

isproper and stable. Interpreted inthetime domain,

Vb(ff{,

ffQ) consists exactly of those points in that can serve as an initial conditionforsome Bohl distributional trajectory that

lies

entirelyin :7{1,while thevectorof components of the trajectory modulo

2

isthesum ofa stable regular Bohl functionand a Diracdelta.

It follows immediately from the definitionand [12,Thm.

4.1]

that

//’b(’{, ’tr2)

is contained in

Vb*(ff{),

thesubspace of distributionally weakly unobservablestates with respecttotheoutputZl.Itisalso immediatethat

Vg

(ff{,

:7{2)

is contained in

Vb

(’{,

TQ). Weare now in apositionto state the mainresultof this section:

THEOREM 5.2. Fixp

{1,

2,

oo}.

Then thefollowing holds:

{(ADDPBP)p

issolvable}:=>{im Oc T’b(’{1,

’{2)}.

Inthe remainder ofthis section we will establish aproof of the abovetheorem. Again,considerthe system Ax

+

Bu,

Zl HlX,z2

H2x.

Assume

that fore

>

0,

u(t)

is aregularBohl input. LetXoX. Let

zl.(t)

and

z2.(t)

be the outputs corresponding totheabove input andinitial condition

x(0)

Xo. Denote

i.(s)

forthe corresponding Laplace transforms of

zi,(t).

Wethen have the followinglemma:

LEMMA5.3. Suppose thateither

_of

the followingconditions is

_satisfied:

(ii)

l,e(S)

and

J2,e(s)

arestable

for

all e, sup,oa

II ,.(fo)ll- 0

-0 adthere

exists a constantCsuch that supo,en

z,e

(ito)]]

<-C

for

alle.

Then Xoe

Vb

(Y{1, Y{2).

71

A

detailed proof of Lemma 5.3 can be found in Appendix A. The idea of the proofisthe following. First we notethat the initial condition Xo abovehas for each e>0 a

(,

to)-rcpresentation

Xo=(Is-A)e(s)+Btoe(s).

Here

toe(s)

isthe (rational)

Laplace transform of

ue(t).

Using the asymptotic behaviour as described by the condition (i) or (ii) above, we then analyse the limiting behaviour for e->0 of the sequences ofrational vectors

e(s)

and

toe(s).

This willleadto a

(:,

to)-represcntation

for Xo with the properties described in Definition 5.1. To conclude this section we apply Lemma 5.3 to obtainthe following:

Proof of

Theorem 5.2.

Assume

that

(ADDPBP)p

issolvable.

Let

Xoeim G. Let

_Fe

be as in Lemma 3.1 and define

ue(t):= FeTe(t)Xo.

Then, depending onp, one of the conditions (i) or(ii) in Lemma 5.3 is satisfied. Itfollows that Xoe Vb(X1,

Y{2).

D

6. Thegeometric structure ofSra(f’l,;/’2). Inthe sequel,it will turn outthat under certain assumptionsonthe system

(1.1)

the subspaceinclusioninTheorem 5.2isalso asufficientconditionfor the solvability of

(ADDPBP)p.

Inordertoprovethis andto be abletoconstruct the required feedback maps,we need more detailed information on the geometric structure ofthe subspace ]’/’b(Y{1,

Y{2)

as introduced in the previous section. In the present section, we will first show that the subspace //’b(Y{,

Y{2)

can always be written as the sum of the subspace

g(Y{,

X2) (see

4)

together with an almost controllability subspace depending on

Y{

and 3’{2. Using this result, we will show thatif either

*(Y{1)-

{0}

orY{2

{0},

then T’b(Y{1,

X2)

admits a decomposition into the direct sum of

Wg(X1,

Y{2)

togetherwith anumberof singly generatedalmost controllability subspaces,with aparticular position with respectto the subspaces Y{

andY{2. The main resultofthis section willbe the following theorem:

THEOREM 6.1.

Assume

that

]*(Y{1)

{0}

orthat3’{2

{0}.

Then thereisan integer

m’

,

thereare integersrl,

,

r,,,,M

andvectorsbl,

,

b,,, andthereis a map

F"

->

all such that

(6.1)

//’b(Y{1,

Y{E)

o//.g(y{,

Y{2)@

()

’i,

i=1 where with j=l

(6.2)

and --2

(6.3)

_@

AJF-1/i

c_Y{2. j=l

If in thestatementof the above theoremoneof the integers riissuch thatri-1

<

1 or_ri-2

<

1, then the correspondingsumsin

(6.2)

or

(6.3)

are understoodto be equal to

{0}.

Itwill turn out in theproofof Theorem6.1 that in the case that

*(Y{1)= {0}

the integer

m’

may be chosentobe equalto m (=dim

).

Inthe case that Y{2

{0}

it will appear that

m’

may be chosentobe equalto m dim

V*(Y{1)

and alsothat

inthis casethe integers rimay be takentobe either1 or 2. Since

]/’g(ffl,

{0})--(see

Theorem

4.2)

thetheorem thus states that

T’b(Y[I,

{0})

is equaltothe direct sum of

_T’*(Y[I)

together withanumber of singlygeneratedalmostcontrollabilitysubspaces which are equal to eitherspan{hi} (with0#

_bi

)

orspan{hi,

AFb},

with{b,

Av-b}

linearly independentand

_b

/’1

Q

The resultof Theorem6.1 willbeinstrumental inthenextsection, wherewe will considerthe sufficiency of thesubspace inclusion imGc

_{b(Y(l, Y{2)}

_{for solvability of}

(ADDPBP)p

andpropose a"scheme" for calculationof the required feedback maps.

Intheremainderof the present section we willestablish aproofof Theorem 6.1.

We

introducethe following subspace:

DEFINITION 6.2.

?b(Yfl,

if{2) will denote the subspace of all x that have a

(:,

to)-representation with

(S)fffl[S], (.O(S) 0[S]

and

H2(S)

is constant (i.e., if

,(s)=Y’,oX,S’

then

HExi=O

for i->1).

Interpretedinthe timedomain, b(Y’l,

Y’2)

consistsexactly of those points in that canbe driven to 0 along apurely distributional Bohl trajectory that liesentirely in

Y(,

while thevector of components ofthis trajectory modulo

X2

is a Dirac delta.

It follows immediately from the definition and Lemma 2.7 that every point in

b(X, Y2)

is strongly controllable withrespectto the output

_z.

Moreover,

it is also immediate that every point x thatis strongly controllable with respectto the output

Z2, is an element of

b(fffl,

if{E).

Hence,

theinclusion /

b*(fff2)c

b(fffl, Yf)C

b*(Y()

holds. In fact, we havethe following nice result: THEOREM 6.3.

(i)

b(ff/,, Y[2)

+

A(b*(Y/2)

f-)

_Yfl),

(ii)

T’b([1,

if/2)

T’g([,

Y[2)+b(Y[, Y/2).

Proof.

(i) Suppose that

x=(Is-A)s(s)+ Bto(s),

with

(s)

[l[S],

w(s) ll[s]

N

and

H2:(s)

is constant. Let

_s(s)=Y.,=oX,

S and

w(s)=Y

N+_{--o uis.} Obviously,

:(s)=

Xo+S(s)

and w(s)=uo+stoi(s), where

sl(s)X2[s]

and

Wl(S)

q/Is].

Hence,

x=

Buo-Axo+

SXo+

s2(s)-As(s)+

BSWl(S)

and by equating powers itfollows that

(6.4)

x

_-Axo

+

Buo,

(6.5)

-Xo

(Is-A)l(S)+ Btol(S).

Therefore,

Xob*(X2)

(see

Lemma

2.7).

Since also XoY{, we obtain x

3+A(b*(’{2)f’iY/’).

Conversely, if

_x=Buo-Axo

with Xob*(ff/’2)fqY{1, there is

s(s)X2[s]

and

tol(S)e ?/Is]

such

that-Xo=(Is-A)s(s)+Bto(s).

Defining then

:(s)

:=

_Xo+

_s(s)

and

to(s):=

Uo+

sto(s),

we obtaina

(:,

to)-representation of x with

s(s)

Y{[s],

to(s)

//Is] and

H2(s)

H2xo

isconstant.

(ii) Assumethatx T’b(Y{,

Y{2).

Thereis a

(:,

to)-representationforx with

s(s)

Y((s),

to(s) //(s)

and

H2:(s)

properandstable.

Decompose s(s)

:l(s)+ :2(s)

and

to(s)

tol(s)

+

to2(s),

where

:l(s)

and

to(s)

are polynomialvectorsand

s2(s)

and

to2(s)

arestrictlyproper. Obviously,

s(s)

fffl[s],

s2(s)

ff/’.+(s),

toi(s)

a//[s]

and

to2(s)

+(s). Moreover, H2s(s)

is constant and

H2:2(s)

is strictly proper and stable.

Now,

sincetheleft-handsideof this equationisproper and the right-hand sideis a poly-nomial vector, both sides must, in fact, be constant. Thus, there is a vector

_x

such that

x=(Is-A)l(S)+Bto(s)=x-(Is-A)2(s)-Bto2(s).

It follows that

XG

?b(Ytrl, ’f2)

and x _xl

7/’g(Yfl,

r2).

Since x

_{xa +}

(x

x),

we obtain that x

Fg(Yf,

Yf2)+b(Yf, Yt’2).

The converse inclusion follows immediately from the definitions.

The importance of the abovetheoremisthatitshows,togetherwithTheorem4.2,

that

Thus,the space

Vb(’(a, ’(2)

can,in principle, be calculated usingexisting algorithms. The stabilizabilitysubspace and the controlled invariantsubspace appearing in

(6.6)

can be calculated using the invariant subspace algorithm ISA [22, p.

91]

and a construction as in [22, p. 114]. The almost controllability subspace

b*(Y’2)

can be calculated using the almost controllability subspace algorithm

(ACSA)’ [20].

Forany fixed subspace ’’c

,

this algorithmis defined by

(6.7)

’+1(’/)

+ A(’

(’)

’/);

ff(’r)

{0}.

Itis well known, see [20], that

(6.7)

defines a nondecreasing sequence of subspaces which reaches a limit after a finite numberofiterations.

Moreover,

this limitequals

ff"(ff’)

b*(ff{).

Inthe sequel, denote

(6.8)

i(ffCl,

ffLr2):-’-

ffi(ffLr2)

f’)

_ffLrl

Using the properties of the sequence

-i(ff.)

statedabove, togetherwithTheorem

6.3,

the following resultisimmediate:

.LEMMA 6.4.

i(’’1,

’’2)is a nondecreasingsequence which reaches a limit

_after

a

finite

number

_of

iterations. This limit equals

"(Y{, Y{2)=

Rb*(Y{2)

Y{1. Consequently,

(6.9)

b(27{, Yff2)

?

_""

A"(?7{, YLr2).

[

Other properties of the sequence

ff(3’{,

’2)

areproveninLemmaB.1, Appendix

B. Using these properties, we obtainthe following lemma:

LEMMA 6.5. Assumethat

*(71)=

{0}.

Then there is a chain {i}’= in anda map F allsuch that

(6.10)

b(’{, 72)

AF’"

"AF,,

(6.11)

-

AF-’iC

Xl, i=l

(6.12)

)

i-AF

,

i=2

(6.13)

dim

_i-dim

A-

dim

(7[, 7’)/-1(’c,

3’c2)

].

Proof.

See Appendix B.

We are now in a position to establish a proof of Theorem 6.1 for the case

*(C,)

{0}"

Proof of

Theorem 6.1

(Case

1"

*(’/’)=

{0}).

Duringthis proofwe will denote

(7’, [:)

by

,

_V

(J’, J2)

by

_V

and

Vg

(7’1, ’’)

by

Vg.

According to Theorem 6.5 we have that

V

Vg +

.

We claimthatthe latter sum is a direct sum. Indeed, this follows immediately from the facts that

_V

c

_V*(’[)

and c

_*(J),

while

V*(’’)

*(’)=

*(’[1)= {0) (see

Lemma

2.2).

By Lemma 6.5 there is a chain

{@)’=1

in @ andamap Fsuchthat

(6.10)

to

(6.13)

hold. Let

_@

be the firstsubspace in thechain which is not zero, i.e.,

_@

{0)

and

_@

{0}

forj-14-1,-..,n. Choose a basis for @ as follows. First choose a basis {b,...,

b,}

for

@.

Extend this to a basis{b,

,

b,, b,

,

b,_)

for@_

(here,

d

:=dim @i).Continue thisprocedure until wehave abasis for @.

AFJi,

V,

the followingvectorsformabasisfor

_b

By

the fact thatdim dim

AFbl,

Arba,

AF

-1

bd_,

AtF

-

b,

AF

-1

bd,

AF

-

bdrl,

AFbl,’",

_AFbd,

AFbdr+l,’’’,

AFbd,_,’’’,’’’,AFbal,

It may immediately be verified that the above list of vectors can be rearranged to

At’-1

hi},with obtain m subspaces

_i

:=span{b, Afl,

r,-2b,}

c

span

{b,

AFbi,

_AF

and

Ar-3h

span

{b, A,.

,--

,

This completes the proofofTheorem6.1 for the case that

*(ff{1)=

{0}.

Inthe remainder ofthissection, we will set up aproofof Theorem 6.1, the case that if{2

.

{0}.

Inthe following, let beasubspaceof such that

[

*(ff{1)]

Let W be a map such that =imBW and let

(Y{1):

+A({),

where

(Y{)

denotes the supremal almost controllability subspace contained in ff{ with respectto the system

(A,

B

W) (see

also Lemma

2.3).

Define

(6.14)

(Y{)

:=

+

A(

if{l).

We will show that if {2

{0},

then b(Y{,

Y{2)

has a decomposition into the direct sumof

g(Y{1,

Y{2)

(which,inthatcase, isequalto

_(Y{))

andthesubspace

(Y{)"

LA 6.6. Let { bea subspace

_of

.

en

(6.15)

V(C,, {0})=

(C,

{0)

(C).

Proo

Inthis proof, denote

g

:=

g(Y{,

{0}).

Also, let :=

_*({).

Since

({0})

,

itfollows from Theorem 6.3 that

(c, (0)

+

+

A[

++A[()

C]

++A[+(

C)].

Now,

notethat

₁

*(Y{)

(see

[22,Thm.

5.5]).

Consequently,

_A

*({)+

g

+

.

Hencewe.find

(c, {0)=

+

+

A(

g

+ 1

+

+

A(

{).

Again, by the fact that

*(Y{1)

c

g,

we have

(c,, {0})=

+

+

A(

C).

Finally,since

g

*(Y{)

and

(Y{1) ({),

itfollows from Lemma2.3 that the sumin

(6.15)

is direct.

Using the abovelemma we may now obtainthe followingproof of Theorem 6.1, the case that Y{2

{0};

oofofeorem

6.1

(Case

2:{2

{0}).

Weclaimthat

(Y{)=

A(

{).

To

_rove

this,assumethat thereis a vector 0 x such that x

A,

with avector in Y{. Define := span

{}.

Since

A

+

,

is controlled invariant. Since also {,we findthat

*({).

Itfollows that

*({)

{0}

and hence that x=0. This yields a contradiction.

Next,

we claim that dim

dim

A(

{1). Assume

the contrary. Then wemayfind a vector 0 x

Y{

such

that Ax=0. It follows that span

{x}

is a controlled invariant subspace contained in { and hence that x

*(ff{)

{0}.

Again, this is a contradiction.

Now,

choose a basis for

(ff{1)

as follows" first choose abasis b,...,

b

of {1. Extendthis to a basis {b,..., b, b+,...,

bm,}

of

.

By

the above, the vectors {b,..., b, Abe,..., Abe, b+,..., b,}formabasisfor

(Y{).

Thesevectors can berearranged

intoone- and two-dimensional singlygeneratedalmost controllability subspaces with the properties

(6.2)

and

(6.3).

This completesthe proof of Theorem6.1. 1

7. The main result. In the present section we will combine our results of the previous sections to show thatif the system

(1.1)

is such thatit satisfies at least one of the followingtwoproperties:

(7.1)

the system

(A,

B, HI)

isleft-invertible,

(7.2)

the mapping

_H2

isinjective,

then the subspace inclusion imGc

_{b(’, X2)}

is both a necessary and sufficient condition for solvability of the

Lp-almost

disturbance decoupling problem with bounded peaking

(ADDPBP)p

for the values p 1, p 2 andp

Recallfrom 5 that forthesevaluesof p thelattersubspaceinclusion wasalready

showntobeanecessary conditionwithouttheextraassumptions

(7.1), (7.2).

Herewe shall, in fact, prove that if either

(7.1)

or

(7.2)

holds then imGc

//’b(/’, ffLr2)

is a sufficientconditionfor solvability of

(ADDPBP)p

for

all 1

The following resultisthemainresult ofthispaper:

THEOREM7.1. Assumethatatleastone

_of

thetwoconditions

(7.1), (7.2)

is

_satisfied.

Let p{1, 2,

oo}.

Then

(ADDPBP)p

issolvable

_if

andonly

_if

im

In orderto obtain a proofof the latter statement, we will prove the following"

LEMMA

7.2.

Assume

thatat leastone

_of

the two conditions

(7.1), (7.2)

is

_satisfied.

Let

T(t)

and

"(s)

be

_defined

by

(3.4)

and

(3.5).

Then the following statements are equivalent"

(i) Thereexigtsa constantCandasequence

{F;

e

> 0}

such that

IIH

TII,-

0

(e

O)

and

_H2

_T

G L,

--<

C

V

e.

(ii) ThereexistsaconstantCand asequence

{F;

e

_>0}

suchthat,

for

all e,

H

G

and are stable and

supllnZ(i,o)ll-O

(-0)

and sup

II/-/=L(i,o)ll-<-

C,

.

(iii) imG //’b(ff,

2).

Note that the implications (i)(iii) and

(ii)(iii)

follow immediately from

Lemma5.3. Alsonotethat oncewehave proven the abovelemma,aproof ofour main result Theorem 7.1 may be obtainedby combining Theorem 5.2 and Lemma3.1. We stress that the implications (iii)=>(i) in the above, in fact, yields sufficiency of the subspaceinclusionim G

b(’’l, ffLr2)

for solvability of

(ADDPBP)pfor

all 1

<-_p

<__oo. The idea of the proofof the implication (iii)

:=>

(i) of

Lemma

7.2 is as follows. First we notethat left-invertibility of the system

(A, B, HI)

isequivalenttothecondition

*(:)

{0} (Lemma 2.6),

while injectivity of the map

HE

is equivalentto

_’2

{0}.

Thus,under the assumptions ofLemma 7.2, //’b

(’’1, 2)

may bedecomposed according to

(6.1),

(6.2)

and

(6.3).

Each of the singlygeneratedalmost controllabilitysubspaces

Li

appearing in this decomposition will then be approximated by sequences of con-trolled invariant subspaces

{;

e>0}.

If we then define

:=

_{//’g0))’l L,}

the sequence

{o//.;

e

>0}

will converge to

//’b(’l, ’2).

In this sense, imG is "almost contained" inthecontrolledinvariantsubspace

.

Thesubspace

o//.

in turn is"almost contained"in ffr

(where

thelatter "almost" should be interpretedintheLl-sense,see also

[20]),

while its distancefrom

’/’2

is uniformly bounded with respectto e. Using the structure of the above, we will construct a particular sequence of feedback maps

{F;

e

> 0}

suchthat

(A

/

_BF)

F

F.

Finally itwillbe shown that this sequence has the properties required by (i) and (ii) in Lemma7.2. To startwith, we will show howasinglygeneratedalmostcontrollability subspacecanbe approximatedby control-led invariant subspaces. Let be and let :=g0)’.

"0)AkF-g.

For i_k and e>0,

definevectorsin by

(7.3)

xl(e):=(I+eav)-lb,

x,(e):=(I+eav)-Iavx,_(e).

Notethat thematrixinversions in the above expressionsare definedfor e sufficiently small.

Moreover,

it canbe seenimmediately that

x,(e)

A

-

b(e

0).

Thus itfollows from Lemma 2.8 that

for

e sufficiently small, the vectors {xi(e), _k} are linearly independent. Foreach e, define a subspace

e

by

(7.4)

e

:=_span

_{x(e),

.,

_Xk(e)}.

Assume

u q/ _{is suchthat b}

_Bu

_{and define amap}

_Fe

_"*e

_Uby

(7.5)

Fex,(e)

:=

e-’

u, (i_k).

The main properties of the sequences

{e;

e

> 0}

and

{F

e;e

> 0}

are summarized in the following lemma:

LEMMA7.3. For _kwe have

x(

e -->

Ai-

as e-->O. Consequently,

e

->

.

Each

e

is controlledinvariantand, with

_Fe

_defined

by

(7.5), (AI

+

BFe)&e

e.

Moreover,

a matrix

_of

(A

+

BEe)le

isgiven by

(7.6)

Me:

Finally,

_for

each e,

e

c

(A]),

the reachablesubspace

_of

(A,

B).

Proof.

The claim

x(e)

A-Ib

is immediate. Sincethevectors

_A-b

are a basis for

,

itfollows from 2 and 4that

e

.

Using

(7.3)

and

(7.5),

itmay be verified by straightforwardcalculationthat

(Av

+

_BFe)x(e)=-=1

eJ-’-xj(e).

Itfollows that

e

is indeed

_AF

+

BFe-invariant

and thatamatrixof the maprestricted to

we

isgiven by

(7.6).

Finally, to prove that

e

is contained in the reachable subspace, make a Taylor expansionto findthat

(I

+

eAF)

-1

m=0

(-e)mArfl

Itthenfollows immediately that

xl(e)E(AFI)

for all e. The samefollows for

X2(e), X3(e)

etc.

Wenotethataslightly ditterentconstructionleadingtoanapproximating sequence forasinglygenerated controllabilitysubspacewasdescribed in

[13].

Theconstruction describedby us howeverexhibits animportant propertywhich willbe formulated in the following lemma. The proof ofthis result is straightforward but rathertechnical andwill be deferred toAppendixC.

LEMMA7.4. Let :=

_)/k=

_A-

_it

be such that

₎

_k=2

_A

i--2

F

rl

and

)/k=

A-3/

772.

Let

_x(e)

and

_Fe

beas

_defined

above. Then the following holds" there is constant C

such that

_for

all _k:

(7.7)

]]H

eA+")’x,(e)llL,O

ase

O,

(7.8)

IIH_

e’+)’x,(e)ll,

<-C

_for

alle. [3

Now,

in orderto completeaproof ofLemma 7.2,weneedonemorepreliminary result.

Up

to nowwe have constructed a sequence of controlled invariant subspaces convergingtoasingly generated almost controllability subspace anddefined afeedback mapon eachof these controlled invariant subspaces.

By

applying the decomposition

theorem, Theorem 6.1, and applying the aboveconstruction to each appearing in

(6.1),

we can find a sequence of controlled invariant subspaces

e

converging to

)__m’

.

Intheobviouswaywe can define amap

_Fe

on

e.

Nowthe question is,can we define

_Fe

appropriatelyon asubspace complementaryto

e

?Thenextconstruction

theorem states that, indeed, we can. It is here that we will use the results on exact disturbance decoupling with stability constraints from 4. In the following,

Vb

:=

Vb(:7/.1,

:7/.2),

Vg

:=

_{Vg(:7/.1, :7/.2)}

and

b

:=

_{b(:7/.1, :7/’2)}

are denoted:

THEOREM 7.5. Considerthesystem

(1.1).

LetAbeasymmetricset

_of

dim

[((A

)

+

Vg)/F

b

]

complex numbers. Then there is a map

FI"

-->

71 and a subspace c such

that the followingconditions are

_satisfied"

(7.9)

(7.10)

(7.11)

(7.12)

(7.13)

(7.14)

(A+

BF1)

T’g

c

_T’g,

A

+

B

F1)

V*(:7/.2)

V*(:7/.2),

tr(A

+

_BFI

/

*(’C2))

C,

(A+

BF)(T’g

)

)

tr(A

+

BF,

O )/

)

A.

Proof.

Let

_Fo"

->

q/ _{be amap that satisfies the conditions}

_(4.1),

_(4.2)

and

(4.3)

of Theorem 4.3. Let P"

->

/Vg

be the canonical projection. Let

_(AFo,

B)

be the system inducedby

_(AFo,

B)

inthefactor space

/Vg.

Since

Vb

Vg

+

b

and kerP

Vg,

wehave

_PVb

Pb.

By

Lemma2.4andthe factthat

b

is an almostcontrollability subspace,itfollows that

P’

b is analmostcontrollability subspacewith respecttothe

system

(/Fo,

/)"

By [22, Prop. 1.2],

P(A[)=(.Folim

).

LetA be as above. It can easily be verifiedthat #A= dim

[(,Folim

B)/tb].

Thus,we may apply Proposition 2.5 to find an

_{(gFo,/)-invariant}

subspace c

_/Vg

anda

map/"

/Vg

suchthat

(7.15)

P@

(olim

(7.16)

(AFo +

BF)

Z,

(7.17)

tr(fiFo+//

)

A.

Now let c _{be any subspace such that} po _and

Lr

fq

_Vg

_{0}.

Define a map

FI"

->

0-// by

_F1

:=

_Fo

/FP. We contend that the subspace and the map

_F1

satisfy theclaimsof thetheorem.To prove

(7.9)

to

(7.11),

notethat

F11

Vg

_Fol

Vg.

Theclaim

(7.12)

can be proven as follows: From

(7.15)

we have

P(Vb

+)=

P(AI3).

Hence,

since

Vg

c_Vb,

_t/’b

W

--

_t/’g

_W(Al

tJ ). Assume

xe

_Vb

c

.

Then

PxePVbf’={0}.

Thus, x kerPf)

Vg

[-1

_{0}.

Itfollows that

Vb

+

Vb

.

To prove

(7.13),

note by using

(7.16)

that

P(A+BF)(Vg)=

P(AFo +

BFP)(

Vg

)

Z)

(AFo +

BF1)

c

Lr

P(

Vg

09

).

Finally,

(7.14)

follows immediately from

(7.17).

Weare now in a positionto complete the proof ofLemma7.2:

Proof of

Lemma 7.2. (i)

=:>

(ii). This follows immediately from the fact that the

L2-induced

norm of a convolution operatoris bounded from aboveby the

Ll-norm

ofits kernel

(see,

for example,

[2]).

(iii)

=>

(i). Inthis partwe will construct asequence of feedback maps

{F;

e

> 0}

suchthat,foreachx Vb,

Lx

0and

H= Lx

-<-

c

forall e,forsome constant

C. The construction is divided into five steps"

1. Decomposition.Apply Theorem6.1 tofind adecomposition

i--1

with

_{=)5__}

_AF

,

such that

(6.2)

and

(6.3)

hold.

2. Approximation

_of

singly generatedcontrollabilitysubspaces. Foreach

,

apply the construction

(7.3)

to

(7.6).

Thus we find vectors

x(e)

(i

),

subspaces := span

{x(e);

}

and maps

_F"

&

0// suchthat

Moreover,

by applying Lemma 7.4, there are constants Cj such that

(7.19)

[[H1

e(av+BFS)txJ)(e)iiL,’-’>O(e

--->

0),

(7.20)

[IH2

e(Av+BFs’)txJ)(e)IIL,

<=

C foralle.

3. Composition. Sincethe

_Z#s

areindependent,itfollows from Lemma2.8that for e sufficiently small the

_Z#s

(J

_m’) are independent. Define :=

Z#l""

"m’-It

follows that

-

.

Define now

F

-

by defining

FI

:=

_(F

₊

_Fs

(j

_m’).

4. Construction

_offeedback

outside

_t.

To define a mapon a complement of

,

letAc

Cg

be asymmetric setofdim

[((A[

)+

o//,,)/o//,]

complex numbers and apply the constructiontheorem Theorem 7.5to find asubspace c and amap

FI"

such that

(7.9)

to

(7.14)

are satisfied. Inthe remainderofthis proof, denote by

.

Wemay then prove the following:

LEMMA7.6. Foralle sufficiently small the following holds"

(7.21)

T’g

b

Proof

By Lemma 6.6,

T’b3=T’gY.

Since, for each e,

(Lemma 7.3),

itfollows from

(7.12)

that c

T’gY.

Since

-->

,

we obtain from Lemma 2.8that

VI (T’g

Y)=

{0}

for e sufficiently small. The

equality,(7.21)

now follows immediately by noting that for e sufficiently smalldim dim

.

5.

Definition

of

the sequence

{F;

e

> 0}.

Let

W

be an arbitrary subspace of such that

=

T’gY

/4/’. In this (e-dependent) decomposition of define

F"

-->

by

F]T’gY:=FIlT’gY,

_FI=FI

and

F

arbitrary on

Wecontendthat the sequence

{ F;

e

> 0}

defined in thiswaysatisfiesthe condition

(i) ofLemma7.2.Toprove this,firstlet x

T’g.

Since

_F

T’g

F1

T’g,

wehave by

(7.9)

and the fact that

T’g

c_Y(1that

x(AvlT’g)=

Y(1 forall e.Thus, forall e,

H1T(t)x=O

for all t. Let

_Av,

and

H2

be defined by the following commutative diagram (Fig.

2),

in which

_P

isthe canonicalprojection:

AF

P

P

,/*(yc)

AF

’,/

(yc)

FIG. 2

Wethen have

H2T(t)x=

fflzeAF,

tXfor all e. Itfollows from

(7.11)

that

H2Tx

is in LI[0,

oo)

with, obviously,

Ll-norm

independent ofe. Tocomplete the proofit now suffices toshow that forallxe

,,

_N

rx

,-,

0and

N.

rx

,

isuniformly bounded withrespectto e.Since

b

isspanned by thevectors

_A

-1

b,itsuffices totgke_x

A

--1

_b.

By

(7.21),

we have

b

c

_@N.

_{Thus, there} are vectors

_v(e)e

and

coecients

z0(e)

e such that

(7.22)

A-lb,

v(e)+

Y,

_E

_{’o(e)xS)(e)}

j=l i=1

(i, j)#

(s, l)

andthat

’st(e)

1

(e

0). Now,

for a 1, 2wehave

7.23)

m’

j=li=1

By

(7.19),

note that for a 1 the last term converges to 0 as e0. Using

(7.20),

it followsthat for a 2thelast term is bounded fromabove, independent of e.

Finally, we will show that for both a 1, 2 the first term on the right in

(7.23)

tends to 0 as e 0. For this, let

A,

and

_H

be defined by the commutative diagram

(Fig.

3)

(P

isthecanonical projection):

AFt

FIG. 3

By (7.14),

note that

o’(AF,)=

Ac

_Cg.

_Moreover,

Since

_v(e)

-->0,

_this^

expression tends to 0as

(e

-->

0).

Let

A,

and

_H:

be definedby thecommutativediagram(Fig.

4)

(P3

isthecanonical

AFt

projection)"

FIG. 4

It can be verified that

tr(.F,)=tr(.F,)Wcr(AF,

lt/’g/*(?7[2)),

which, by

(7.11)

and

(7.14)

iscontained in

Cg.

Itfollows that

whichagain converges to 0as e 0. This completes the proof ofLemma 7.2.

Remark 7.7. Itis worthwhile to pointout which freedominthe spectrum assign-mentwe havein

A

+

BF

whenweusetheconstructionofthe sequence

{F;

e>

0}

as in theproofofLemma 7.2.

The lattice diagram (Fig.

5)

shows the hierarchy of the relevant subspaces in combinationwith the freedominthe spectrum of

A

+

BF.

Denote

o//.

:=

g

_{(fill, fir2)})

NotebyLemma7.3that the spectrum of the map

A

+

_BFI

consistsofaneigenvalue in -e-1 with multiplicityequalto dim

[l/’b/t/’g].

fixed, independent ofe assignable, independent ofe tendsto -oo as eO stabilizable, independent ofe fixed, independent ofe FIG. 5

8. Somespecial cases and extensions. Inthis section we will consider somespecial cases of the maintheorem, Theorem 7.1,and spendafew words on someextensions ofthis result.Oneinteresting special caseof

(ADDPBP)p

isthesituationthatwetake

H1

H and

_H2

I. This corresponds to the almost disturbance decoupling problem withbounded peaking of theentire state vector. Denote

r

:=ker/-/.Since,byTheorem

4.2,

g(:][,

{0})=

o//g,([)

+

*({0})=

o//.g,(y[)

and since, by Theorem 6.3,

b(Y[,

{0})=

3

+

A(

b*({0})

f3

)

3

+

A(

fq

ff),

wehavethe following corollary of Theorem7.1" COROLLARY8.1. Fixp

{

1,2,

}.

Then the

Lp-almost

disturbancedecoupling

prob-lem with bounded peaking

of

the entire state vector is solvable

_if

and only

_if

imG

*(

C

+

+

A

n

c).

Next,

wewillspendsomewordsonpossibleextensionsof the results ofthispaper. First we wouldlike to pointoutthat,while

(ADDPBP)p

is a nontrivial extension of

(ADDP)p,

we might also consider an extension of the

Lp-Lq

almost disturbance decoupling problem

(ADDP)’,

see

[20]

or

[17].

This would lead to the following synthesis problem"

We will saythat the

Lp- Lq

almost disturbance decouplingproblem with bounded peaking

(ADDPBP)’

is solvableifthere is a constant C and, for all e

>

0, afeedback map

_F

#/suchthatwiththefeedbacklaw u

Fx,

inthe closed loop system for

x(0)

0there holds,

for

all 1

<=

p<-_q<-

_oo,

forall d

Lq[0, )

Itmay beshownthat the solvabilityofthe above problemisequivalenttotheexistence ofasequence of feedback maps

{F,;

e

>

0}

andaconstant C such that for both p 1 and

p=oo

_{IIHTGII,-O(-O)}

and

_{IIH=LGII=,-<-}

C for all e.

Atheory analogoustothe oneabove may be developed aroundthis problem. It canbe shownthat,again under the assumption that either

(A, B, H)

isleft-invertible orthat

_H2

isinjective, anecessary and sufficient condition for the solvability ofthis problem isthat

imGc

V(’tc,,

ff[2)

+

(9*(fftc2)

f"l

’[1).

Formore details, the readerisreferred to

17].

Afinal extension of the results of the present paperisthesituation in which we requireinternal stability of the closed loop system. Thiswould lead to the following synthesis problem: We willsaythat the

Lp-almost

disturbancedecouplingproblem with

bounded peaking andstrong stabilization

(ADDPBPSS)p

issolvable ifthe followingis true.There is a constant C and forall e

>

0 and all real numbers S a feedback map

F,s:-

suchthat, with the feedback law u

_F,sX,

in the closed loop system for

x(0)

=0for all d

Lp[0, c)

theinequalities

(3.2)

and

(3.3)

holdandsuchthatRe

tr(A+

BF,s)

<=

S.

Thus, we require that the spectrum of the closed loop matrix can be located to the leftof any vertical line Res S in thecomplex plane. Itmaybeproventhat if at least one oftheconditions

(7.1),

(7.2)

hold,then for p {1,2,

}

the latterproblemis solvable if andonlyif

(A,

B)

is controllable and

(8.1)

Wenote that if

(A,

B, H1)

is aleft-invertible system then the inclusion

(8.1)

becomes imGc

+

A[

b*(9’/2)

f3

(see

Theorem

6.3).

If

_H2

is injective then

(8.1)

becomes imGc

*(3’g’,)+

+

A[

CI

Again, for detailsthe readeris referredto

17].

9. A worked example. To illustrate the theory developed in this paper and to demonstrate its computational feasibility, in this section we will present a worked example. We will consider a linear system with two outputs and check whether

(ADDPBP)p

issolvable forthis system.

Next,

we willactually computeasequence of feedback mappings thatachieves ourdesignpurpose. The system thatwillbeconsidered isgiven by

(t)=Ax(t)+ Bu(t)+Gd(t), Zl(t)= H1x(t), z2(t)

H2x(t),

with

0 -1 0 0 0 A= 0 0 1 0., B= 0 0

00

0 0 0

10

0

01

0 0 0 0 and

H1

(0

0 0 1 0),

_H2

I5x5

0

Thus,

5

andq/=

_2.

_Denote

_9’ci

_ker

_Hi.

_{Theroute that we willtakeis asfollows.} First,we will check whether the subspaceinclusion imGc

_{Vb(9’’l, 72)}

holdsto see if

(ADDPBP)p

issolvable. Itturns outthatthis is indeed true.After this,we willfollow closely the linesof the developmentin 7 and construct arequired sequence

{Fn}.

As before,

Cg={h

C[ReA

<0}

and the subspaces

//’g*(ffrl)

and

t/’g(Yffl,

ffff2)

are taken with respecttothis stabilityset. Letthe standard basis vectors in

5

bedenoted by

Using the algorithm ISA

(see

[22, p.

91])

and a construction as in [22, p. 114], we calculate that

g(Yf,

Yf2)

g*(Yf)=

span{el,

e2}

(since

*(Yf2)=

{0}).

Thus, by Theorem 4.4, DDPOSis notsolvable forthe above system. Since

_Yf2

{0},

by Theorem 6.3 weshould checkifthesubspace inclusion imG

Vg*(Yf)+

+

A(

Yf)

holds.