The Generalised Discrete Algebraic Riccati Equation in LQ optimal control

0.2 MB·English

by  Augusto Ferrante

The Generalised Discrete Algebraic Riccati Equation
in LQ optimal control ∗
2
1 Augusto Ferrante‡ Lorenzo Ntogramatzidis⋆
0
2
‡Dipartimento diIngegneria dell’Informazione

n
a Universita` diPadova,viaGradenigo, 6/B–35131Padova,Italy
J
[email protected]
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1 ⋆DepartmentofMathematics andStatistics

] CurtinUniversity, PerthWA,Australia.
C
O [email protected]
.
h
January 19, 2012
t
a
m
[

1
Abstract
v
4
0
This paper investigates the properties of the solutions of the generalised discrete
7
3 algebraic Riccati equation arising from the solution of the classic infinite-horizon
1. linear quadratic control problem. In particular, a geometric analysis is used to study
0 the relationship existing between the solutions of the generalised Riccati equation
2
and the output-nulling subspaces of the underlying system and the corresponding
1
: reachability subspaces. This analysis reveals the presence of a subspace that plays
v
i an important role in the solution of the related optimal control problem, which is
X
reflected in the generalised eigenstructureofthecorresponding extendedsymplectic
r
a pencil.
Inestablishingthemainresultsofthispaper,severalancillayproblemsonthediscrete
Lyapunovequationandspectral factorisationare alsoaddressed and solved.
Keywords: Generalised discrete algebraic Riccati equation, LQ optimal control,
extendedsymplecticpencil, output-nullingsubspaces,reachabilitysubspaces.
∗PartiallysupportedbytheItalianMinistryforEducationandResearch(MIUR)underPRINgrantn. 20085FFJ2Z).
1 Introduction
Ever since in the early sixties Kalman described in his pioneering papers [11, 12] the crucial role of
Riccati equations in the solution of the linear quadratic (LQ) optimal control and filtering problems,
the range of control and estimation problems where Riccati equations have been discovered to play
a fundamental role has been increasing dramatically. Indeed, in the last fifty years Riccati equations
have been found to arise also in linear dynamic games with quadratic cost criteria, spectral factorisation
problems, singular perturbation theory, stochastic realization theory and identification, boundary value
problems for ordinary differential equations, invariant embedding and scattering theory. For this reason,
Riccatiequationsareuniversallyregardedasacornerstoneofmoderncontroltheory. Severalmonographs
have been entirely devoted to providing a general and systematic framework for the study of Riccati
equations,seee.g. [20,13, 9,1].
In the continuous time, the structure of the solution of a linear-quadratic problem strongly depends
on the rank of the matrix penalising the control in the performance index, which is traditionallydenoted
by R. When R is non-singular the optimal control can be found by solving a Riccati equation (which is
differential or algebraic depending on the horizon of the performance index). Indeed, in this case, such
Riccati equation – which explicitly involves the inverse of R – is well-defined. But when R is singular,
a solution of the problem is guaranteed to exist for all initial conditions only if the class of allowable
controlsisextended toincludedistributions[7, 21, 15], and theRiccati equationisnotdefined.
In the discrete time, the classic solution of the infinite-horizon LQ problem is traditionally expressed
in termsofthesolutionX oftheRiccati equation
X=ATXA−(ATXB+S)(R+BTXB)−1(BTXA+ST)+Q, (1)
where A∈Rn×n, B∈Rn×m,Q∈Rn×n, S∈Rn×m and R∈Rm×m aresuch that
Q S
P =def =P T ≥0. (2)
ST R
(cid:20) (cid:21)
MatrixP isusuallyreferredtoasPopovmatrix. ThesetofmatricesS =(A,B;Q,R,S)isoftenreferredto
asPopovtriple,seee.g. [9]. Equation(1)istheso-calledDiscreteRiccatiAlgebraicEquationDARE(S ).
Noticethat now it is not theinverseofR that explicitlyappears in theRiccati equation but theinverseof
the term R+BTXB, which can be non-singular even when R is singular. Nevertheless, even though the
distinctionbetweenthecasesinwhichRisinvertibleorsingularneedsnotbeconsidered,veryofteneven
in the discrete time it is assumed that R is non-singular because this assumption considerably simplifies
severalunderlyingmathematicalderivations.
However,eventhesolutionto theinfinite-horizonLQ problemexpressedin termsofmatricessatisfy-
ing this equation is somehow restrictive. Indeed, an LQ problem may have solutions even if DARE has
no solutions, and the optimal control can be written in this case as a state feedback written in terms of a
matrixX such thatR+BTXB issingularand satisfiesthemoregeneral Riccati equation
X = ATXA−(ATXB+S)(R+BTXB)†(BTXA+ST)+Q, (3)
ker(R+BTXB)⊆ker(ATXB+S), (4)
where the matrix inverse in DARE(S ) has been replaced by the Moore-Penrose pseudo-inverse, see
[16]. Equation (3) is known in the literature as the generalised discrete-time algebraic Riccati equa-
tion GDARE(S ). The GDARE(S ) with the additional constraint on its solutions given by (4) is some-
times referred to as constrainedgeneralised discrete-time algebraicRiccati equation, herein denoted by
1
CGDARE(S ). It is obvious that (3) constitutes a generalisation of the classic DARE(S ), in the sense
that any solution of DARE(S ) is also a solution of GDARE(S ) – and therefore also of CGDARE(S ) be-
cause(4)isautomaticallysatisfiedsinceker(R+BTXB)=0 –butthevice-versa isnottrueingeneral.
m
Despite its generality, this type of Riccati equation has not yet received a great deal of attention in the
literature. It has only been marginally studied in the monographs [17, 9, 1] and in the paper [3]. The
only comprehensive contributions entirely devoted to the study of the solutions of this equation are [8]
and [18]. The former investigates conditions under which the GDARE(S ) admits a stabilising solution
in terms of the deflating subspaces of the extended symplectic pencil. The latter studies the connection
between the solutions of this equation and the rank-minimising solutions of the so-called Riccati linear
matrix inequality. In pursuing this task, the authors of [18] derived a series of results that shed some
light into the structural properties of the solutions of the generalised Riccati equation, and in particular
in the fundamental role played by the term R+BTXB. An example is the important observation ac-
cording to which the inertia of this matrix R+BTXB – that from now on we will denote by R for the
X
sake of conciseness – is independent of the particular solution X satisfying CGDARE(S ), [18, Theorem
2.4]. This implies that a given CGDARE(S ) cannot have one solution X = XT such that R+BTXB is
non-singular and another solutionY =YT for which R+BTYB is singular. As such, i) if X is a solution
of DARE(S ), then all solutions of CGDARE(S ) will also satisfy DARE(S ) and, ii) if X is a solution of
CGDARE(S ) such that R+BTXB is singular, then DARE(S ) does not admit solutions. The results pre-
sented in [18] are established in the very general setting in which the Popov matrix P is not necessarily
positivesemidefiniteas in(2).
In this paper we are interested in the connection of the use of the CGDARE(S ) in the solution of
optimal control or filtering problems – the so-called H -DARE in the terminology of [18]. It is often
2
taken for granted that the generalised discrete-time Riccati equation generalises the standard DARE(S )
in the solution of the infinite LQ optimal control problem in the same way in which [16] established
that the generalised Riccati difference equation generalises the standard Riccati difference equation in
the solution of the finite-horizon LQ problem. However, to the best of the authors’ knowledge, this fact
has neverbeen presented in a direct, self-contained and rigorous way. Thus, the first aim of this paper is
to fill this gap, by showing in an elementary, yet rigorous, way, the connection of the CGDARE(S ) and
the solution of the standard infinite-horizon LQ optimal control problem. The second aim of this paper
is to provide a geometric picture describing the structure of the solutions of the CGDARE(S ) in terms
of the output nulling subspaces of the original system S and the corresponding reachability subspaces.
Indeed, under the usual assumption of positive semidefiniteness of the Popov matrix, the null-space of
R is independent of the solution X of CGDARE(S ). Even more importantly, this null-space is linked
X
to the presence of a subspace – that will be identified and characterised in this paper – which plays
an important role in the characterisation of the solutions of CGDARE(S ), and also in the solution of
the related optimal control problem. This subspace does not depend on the particular solution X, nor
does the closed-loop matrix restricted to this subspace. This new geometric analysis will reveal that the
spectrum of the closed-loop system is divided into two parts: the first depends on the solution X of the
CGDARE(S ), whilethesecond– coincidingexactlywiththeeigenvaluesoftheclosed-looprestrictedto
thissubspace–isindependentofitand doesnotappearinthegeneralisedeigenstructureoftheextended
symplectic pencil. At first sight, this fact seems to constitute a limitation in the design of the optimal
feedback, because it means that regardless of the solution of the generalised Riccati equation chosen for
the implementation of the optimal feedback, the closed-loop matrix will always present a certain fixed
eigenstructure as part of its spectrum. However, when R+BTXB is singular, the set of optimal controls
presentsafurtherdegreeoffreedom–whichisalsoidentifiedin[17,Remark4.2.3]–thatallowstoplace
all thepoles oftheclosed-loopsystematthedesired locationswithoutchangingthecost.
Several other important ancillary results of independent theoretical interest are derived in this paper.
2
These include interesting considerations on the solutions of Hermitian Stein equations and spectral
factorisationresultsthat generalisetheclassicones inmorethanonedirection.
2 Linear Quadratic optimal control and CGDARE
InthissectionweanalysetheconnectionsbetweenLinearQuadratic(LQ)optimalcontrolandCGDARE.
Most of the results presented in this section are considered “common wisdom”. However, we have not
been able to find a place where they have been derived in detail, so we believe that this section may
be useful. Consider the classic LQ optimal control problem. In particular, consider the discrete linear
time-invariantsystemgovernedby
x =Ax +Bu , (5)
t+1 t t
whereA∈Rn×nandB∈Rn×m,andlettheinitialstatex ∈Rnbegiven. Theproblemistofindasequence
0
ofinputsu ,witht =0,1,...,¥ , minimisingthecostfunction
t
¥
J(x ,u)=def (cid:229) xT uT Q S xt . (6)
0 t t ST R u
t
t=0 (cid:20) (cid:21)(cid:20) (cid:21)
(cid:2) (cid:3)
Before we introduce the solution of the optimal control problem, we recall some classic linear al-
gebra results which will be useful in the sequel. We also give a proof of these results for the sake of
completeness.
P P
11 12
Lemma 2.1 ConsiderthesymmetricpositivesemidefinitematrixP= . Then,
PT P
12 22
h i
(i) kerP ⊇kerP ;
12 22
†
(ii) P P P =P ;
12 22 22 12
†
(iii) P (I−P P )=0;
12 22 22
(iv) P −P P† PT ≥0.
11 12 22 12
Proof: (i)SinceP=PT≥0,twomatricesCandDexistsuchthatP=[C D]T[C D]sothatP =CTD
12
and P =DTD. Let x∈kerP . Then, 0=xTDTDx=||Dx||2, which givesDx=0. This in turn implies
22 22
that x∈kerP . (ii) The inclusion kerP ⊇kerP can be rewritten as imPT ⊆imP . Thus, a matrix
12 12 22 12 22
K∈Rn×m exists such that P =KP . On post-multiplying both sides of this identity by P† P we
12 22 22 22
† †
obtain P P P =KP P P =KP =P . (iii) Since as already proved kerP ⊆kerP , a matrix
12 22 22 22 22 22 22 12 22 12
† † †
K existssuch thatP =KP . Therefore, P (I−P P )=KP (I−P P )=K(P −P P P )=
12 22 12 22 22 22 22 22 22 22 22 22
K(P22−P22)=0. (iv)It followsdirectly fromP11−P12P2†2P1T2 =[I −P12P2†2] PP1T1 PP12 −P†IPT ≥0.
12 22 22 12
Wenowintroducesomenotationthatwillbeusedthroughoutthepaper. Fhirst,toainhymatrixiX =XT∈
Rn×n weassociatethefollowingmatrices:
Q =def Q+ATXA−X, S =def ATXB+S, R =def R+BTXB, (7)
X X X
G =def I −(R+BTXB)†(R+BTXB)=I −R†R , (8)
X m m X X
K =def (R+BTXB)†(BTXA+ST)=R†ST, (9)
X X X
A =def A−B(R+BTXB)†(BTXA+ST)=A−BK , (10)
X X
Q S
P =def X X . (11)
X ST R
X X
(cid:20) (cid:21)
3
† †
ThetermR R istheorthogonalprojectorthatprojectsontoimR =imR sothatG istheorthogonal
X X X X X
projector that projects onto kerR . Hence, kerR = imG . When X is a solution of CGDARE(S ),
X X X
then K is the corresponding gain matrix, A the associated closed-loop matrix, and P is the so-called
X X X
dissipationmatrix. It is easy to see that all symmetricand positivesemidefinitesolutions of GDARE(S )
satisfy(4), and are thereforesolutionsofCGDARE(S ). In fact, ifX is positivesemidefinite,wefind
Q +X S AT
X X = X A B +P ≥0.
ST R BT
X X
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:2) (cid:3)
Therefore, applyingLemma2.1wefind(4),thatcanberewrittenaskerR ⊆kerS andalsoasS G =
X X X X
0.
Thefollowingfundamentalresultholds.
Theorem 2.1 Suppose that for every x there exists an input u ∈ Rm, with t ∈ N, such that J(x ,u) is
0 t 0
finite. Then we have:
1. CGDARE(S ) admits symmetric solutions: A solution X¯ =X¯T ≥0 may be obtained as the limit of
the sequence of matrices generated by iterating the generalised Riccati difference equation (see
(14)below)with zeroinitialcondition.
2. Thevalueof theoptimalcostisxTX¯x .
0 0
3. X¯ is theminimumpositivesemidefinitesolutionofCGDARE(S ).
4. Theset ofall optimalcontrolsminimising(6)can beparameterisedas
u =−K x +G v , (12)
t X¯ t X¯ t
with arbitraryv .
t
Proof: (1). Consider the finite horizon LQ problem consisting in the minimisation of the performance
indexwithzero terminalcost
J =def T(cid:229) −1 xT uT P xt (13)
T t t u
t
t=0 (cid:20) (cid:21)
(cid:2) (cid:3)
subject to (5) with assigned initial state x ∈ Rn. The optimal control is obtained (see e.g. [16]) by
0
iterating, backward in time starting from the terminal condition P (T) = 0, the generalised Riccati
T
differenceequationP (t)=R[P (t+1)],whereR[·]istheRiccati operatordefined as
T T
R[P]=def ATPA−(ATPB+S)(R+BTPB)†(BTPA+ST)+Q (14)
andtheoptimalvalueofthecostisJ∗(x )=xTP (0)x . Letusnowconsiderthe“reversetime”sequence
T 0 0 T 0
ofmatricesdefinedasXt =def Pt(0). SincePt (t)=Pt −t(0)forallt ≤t ,thesequence{Xt}t∈N isobtainedby
iteratingthegeneralised Riccati differenceequationforward withinitialconditionX =0. Thesequence
0
{Jt∗(x0) =def x0TXtx0}t∈N is obviously monotonically non-decreasing (it is the sequence of optimal costs
overintervalsofincreasinglengthst). Hence,thesequences{Xt}t∈N,and{R+BTXtB}t∈N aremonoton-
icallynon-decreasingsequencesofpositivesemidefinitematrices. Wenowshowthatthesesequencesare
bounded. Assume,bycontradiction,thatlimt→+¥ kXtk=+¥ .Thesequence{Xt1= kXXttk}t∈N isbounded.
Thus, there exists a converging sub-sequence {X1}. Let X¯1 be its limit. Clearly kX¯1k=1: let x1 ∈Rn
ti 0
be such that kx1k=1 and (x1)TX¯1x1 =1. Since we assumed that for any x there exists a trajectory that
0 0 0 0
4
renders J defined in (6) finite, we have that there exist a constant m and an input trajectory u1 such that
0
J∗(x1)≤J(x1,u1)≤m , where thefirst inequalityfollowsfrom theoptimalityofthecost J∗(x1) and the
ti 0 0 0 ti 0
factthat,foragivenu1,theindex(6)isasumofinfinitenon-negativetermswhichisgreaterthanorequal
to the sum of the first t terms of the sum. On the other hand we have J∗(x1)=kX k(x1)TX1x1 →+¥ ,
i ti 0 ti 0 ti 0
which isacontradiction.
Since {Xt}t∈N is non-decreasing and bounded, it admits limit X¯ for t → ¥ . Then, limt→¥ Xt =
limt→¥ Xt+1 = limt→¥ R[Xt] = X¯. Thus, if limt→¥ R[Xt] = R[X¯], then R[X¯] = X¯, i.e. X¯ is a positive
semidefinite solution of CGDARE(S ). To prove that this is indeed the case, it is sufficient to show that
† †
limt→¥ RXt = RX¯. In fact, the pseudo-inverse is the only possible source of discontinuity in the Riccati
iteration. Toprovethelatterequality,considerthesequence{R+BTXtB}t∈N. Sinceitisamonotonically
non-decreasing sequenceofpositivesemidefinitematrices,thechain ofinclusions
ker(R+BTX B)⊇ker(R+BTX B)⊇ker(R+BTX B)⊇...
0 1 2
holds. Clearly,thereexistat¯suchthatforanyt ≥t¯thischainbecomesstationary,i.e.,foranyt ≥t¯there
holds ker(R+BTX B) = ker(R+BTX B). This implies that a change of coordinates independent of
t t+1
t exists such that in the new basis R =R+BTX B=diag{R0,O}, where {R0} , is a non-decreasing
Xt t t t t≥t¯
sequence of positive definite matrices. Clearly, limt→¥ RXt =RX¯, so that, in this basis, RX¯ has the form
RX¯ = R+BTX¯B = diag{R0,O}, where R0 =def limt→¥ Rt0. Moreover, since the sequence {Rt0} is non-
decreasing, R0 is alsononsingular,so that(R0)−1 →(R0)−1.Thus,in thechosen basiswehaveindeed
t
(R0)−1 O (R0)−1 O
R† =(R+BTX B)† = t −→ =R†.
Xt t O O O O X¯
(cid:20) (cid:21) (cid:20) (cid:21)
(2). Let
J◦(x )=def infJ(x ,u). (15)
0 0
u
Clearly, J◦(x ) ≥ J∗(x ) = xTX x . Then, by taking the limit, we get J◦(x ) ≥ xTX¯x . We now show
0 t 0 0 t 0 0 0 0
thatthetime-invariantfeedbackcontrolu∗=def −K x yieldsthecostxTX¯x ,whichisthereforetheoptimal
t X¯ t 0 0
valueofthecost. ConsiderthecostindexJ =def J +xTX¯x ,whereJ isdefinedin(13). Itfollowsfrom
T,X¯ T T T T
[16,SectionII],seealso[10],thatanoptimalcontrolforthisindexisgivenbythetime-invariantfeedback
u∗ = −K x and the optimal cost does not depend on the length T of the time interval and is given by
t X¯ t
J∗ = xTX¯x . Notice that for this conclusion we only need the fact that X¯ is a positive semi-definite
T,X¯ 0 0
solutionofCGDARE(S ). Nowwehave
¥
xTX¯x ≤ J◦(x )≤J(x ,u∗)= (cid:229) xT (u∗)T P xt
0 0 0 0 t t u∗
t=0 (cid:20) t (cid:21)
(cid:2) (cid:3)
T
= lim (cid:229) xT (u∗)T P xt
T→¥ t=0 t t (cid:20) ut∗ (cid:21)
= lim J∗ (cid:2)−xTX¯x ≤(cid:3)lim xTX¯x =xTX¯x . (16)
T→¥ T,X¯ T T T→¥ 0 0 0 0
Comparing the first and last term of the latter expression we see that all the inequalities are indeed
equalities,sothat theinfimumin(15)is aminimumand itsvalueis indeedxTX¯x .
0 0
(3). SupposebycontradictionthatthereexistanotherpositivesemidefinitesolutionX˜ ofCGDARE(S )
andavectorx ∈Rn suchthatxTX˜x <xTX¯x . Takethetime-invariantfeedbacku˜ =−K x . Thesame
0 0 0 0 0 t X˜ t
argumentthatledto(16)nowgivesJ(x ,u˜)≤xTX˜x <xTX¯x ,whichisacontradictionbecausewehave
0 0 0 0 0
shownthatxTX¯x is theoptimalvalueofthecost functionJ.
0 0
5
(4). Let U betheset ofoptimalcontrolinputsat timet =0. Let u ∈Rm and x =Ax +Bu bethe
0 0 1 0 0
correspondingstateatt =1. Clearly theoptimalcostcan bewritten as
X¯ O x
J∗ =xTX¯x = xT uT 0 .
0 0 0 0 O O u
0
(cid:20) (cid:21)(cid:20) (cid:21)
(cid:2) (cid:3)
Moreover,u ∈U ifandonlyiftheoptimalcost can bewritteninthefollowingalternativeform:
0 0
x
J∗ = xTX¯x + xT uT P 0
1 1 0 0 u
0
(cid:20) (cid:21)
(cid:2) AT (cid:3) x
= xT uT X¯ A B +P 0 .
0 0 BT u
0
(cid:18)(cid:20) (cid:21) (cid:19)(cid:20) (cid:21)
(cid:2) (cid:3) (cid:2) (cid:3)
By subtracting the first expression from the second, we get that u ∈ U if and only if
0 0
xT uT QX¯ SX¯ x0 =0.SinceX¯,andhenceP = QX¯ SX¯ ,arepositivesemidefinite,thisisequiva-
0 0 (cid:20)SXT¯ RX¯(cid:21) u0 X¯ (cid:20)SXT¯ RX¯(cid:21)
(cid:2)lentto QX¯(cid:3)SX¯ x0 (cid:2)=0(cid:3). Finally,thisisequivalenttou =−R† STx +G v ,wherev ∈Rm isarbitrary,
(cid:20)SXT¯ RX¯(cid:21) u0 0 X¯ X¯ 0 X¯ 0 0
because thecolu(cid:2)mn(cid:3)sofGX¯ forma basisforkerRX¯. By iteratingthisargumentforallt =1,2,..., weget
(12).
3 Preliminary technical results
In this section, we present several technical results that will be used in the sequel. Most of these are
ancillary resultsontheStein equationandon spectral factorisationofindependentinterest.
3.1 The Hermitian Stein equation
In this section, we give some important results on the solutions X of the so-called Hermitian Stein
equation(knownalsoas thediscrete-timeLyapunovequation):
X =ATXA+Q, (17)
where A,Q∈Rn×n and Q=QT ≥0.
Lemma 3.1 Let X be a solution of the Hermitian Stein equation (17). Then, kerX is A-invariant and is
containedinthenull-spaceofQ.
Proof: Letl ∈C beontheunitcircleand suchthat (A+l I ) isinvertible. Wecan re-write(17)as
n
X =ATX(A+l I )−l ATX+Q, (18)
n
so that
(l AT+I )X = ATX(A+l I )+Q=l ATX(l ∗A+I )+Q,
n n n
sincel ison theunitcircle(which meansin particularthat l ∗ =l −1). Thisis equivalentto
X(l ∗A+I )−1
n
=l (l AT+I )−1ATX+(l AT+I )−1Q(l ∗A+I )−1. (19)
n n n
6
Let x ∈ kerX. On pre-multiplying (19) by x ∗ and post-multiplying it by x , we obtain x ∗(l AT+
I )−1Q(l ∗A+I )−1x =0, and since (l AT+I )−1Q(l ∗A+I )−1 is Hermitian and positive semidefi-
n n n n
nite,we get
Q(l ∗A+I )−1x =0. (20)
n
Let us now post-multiply (19) by the same vector x . We get X(l ∗A+I )−1x = 0, which means that
n
kerX is (l ∗A+I )−1-invariant. Hence, it isalso (l ∗A+I )-invariantand therefore A-invariant. In view
n n
of(20), kerX =(l ∗A+I )−1kerX is alsocontainedin thenull-spaceofQ.
n
We recall that equation (17) has a unique solution if and only if A is unmixed, i.e. for all pairs
l ,l ∈s (A)wehavel l 6=1. In thiscasewehavethefollowingresult.
1 2 1 2
Lemma 3.2 Let A be unmixed and X be the unique solution of (17) where Q=QT ≥0. Then, kerX is
theunobservablesubspaceofthepair(A,Q).
A O
Proof: Letthepair(A,Q)bewrittenintheKalmanobservabilitycanonicalform, i.e.,A= 11 and
A A
21 22
Q= Q1 O , wherethepair(A ,Q )is completelyobservable. Let uswrite(17)inthisbashis. We fiind
11 1
O O
h i
X X AT AT X X A O Q O
11 12 = 11 21 11 12 11 + 1
XT X O AT XT X A A O O
(cid:20) 12 22 (cid:21) (cid:20) 22 (cid:21)(cid:20) 12 22 (cid:21)(cid:20) 21 22 (cid:21) (cid:20) (cid:21)
⋆ ⋆ Q O
1
= + .
⋆ AT X A O O
(cid:20) 22 22 22 (cid:21) (cid:20) (cid:21)
Therefore,X satisfiesthehomogeneousHermitianSteinequationX =AT X A . SinceAisunmixed,
22 22 22 22 22
thesubmatrixA isunmixed,andthereforeX =0istheuniquesolutionofX =AT X A . Assuch,
22 22 22 22 22 22
theHermitianSteinequationin thisbasiscan besimplifiedas
X X ⋆ AT X A Q O
11 12 = 11 12 22 + 1 .
XT O AT XT A O O O
(cid:20) 12 (cid:21) (cid:20) 22 12 11 (cid:21) (cid:20) (cid:21)
Again,sinceAisunmixed,foralll ∈s (A )andl ∈s (A )wehavethatl l 6=1,andthetop-right
1 11 2 22 1 2
block of thelatterequation yields theuniquesolutionX =0. Therefore, weget thefollowingequation
12
for X : X = AT X A +Q . In view of Lemma 3.1 the unique solution X of the latter equation
11 11 11 11 11 1 11
has trivial kernel because the pair (A ,Q ) is observable. This implies that kerX = im O where the
11 1
I
partition is consistent with the block structure of X. On the other hand, this subspacehisiexactly the
unobservablesubspaceofthepair(A,Q).
Remark 3.1 When thesolutionX oftheHermitianStein equation(17)isnotunique,byLemma3.1the
null-spaceofX isstillA-invariantandiscontainedinthenull-spaceofQ,butitcouldbestrictlycontained
into the unobservable subspace of the pair (A,Q) – which we recall is the largest A-invariant subspace
contained in the null-space of Q – without necessarily being equal to it. Moreover, it is possible that
noneofthesolutionsoftheHermitianSteinequationaresuchthatkerX coincideswiththeunobservable
subspaceof thepair (A,Q). Considerfor exampletheHermitianStein equation(17) withA= 1 0 and
1 1
Q= 1 0 . In this case, it is easy to see that the set of all solutionsof the Hermitian Stein equahtioni(17)
0 0
h i a −1
is X = 2 : a ∈R . The null-space of any solution of the Hermitian Stein equation is zero.
−1 0
(cid:26)(cid:20) 2 (cid:21) (cid:27)
Hence, noneofthesolutionsoftheHermitianStein equationissuchthat itskernel isequal toim 0 .
1
h i
7
Thelastresult weneed isthefollowing.
Lemma 3.3 Let A∈Rn×n,F ∈Rn×n, B∈Rn×m andassumeX ∈Rn×n issuchthat
AT X
XF = . (21)
BT O
(cid:20) (cid:21) (cid:20) (cid:21)
Then, BT(AT)kX = 0 for all k ≥ 0, i.e., imX is contained in the unobservable subspace of the pair
(AT,BT).
Proof: We first prove that BTX = 0. Let us choose a basis in which F is written as F = diag{N,F},
I
whereN isnilpotentandF isinvertible. LetusdecomposeX accordingly,i.e.,X = X X . Itisvery
I 1 2
easytoseethatATX N=X impliesX =0. Infact,bymultiplyingsuchequationbyAT andN totheleft
1 1 1
(cid:2) (cid:3)
and to the right, respectively,we obtain X =(AT)kX Nk for all k≥0. By choosing k to be greater than
1 1
the nilpotency index of N, we get X =(AT)kX Nk =0. From (21) we also obtain BTX F =0, which
1 1 2 I
impliesBTX =0sinceF isinvertible. Therefore, BTX =0.
2 I
ThesameargumentcanbeiteratedtoprovethatBT(AT)kX =0forallk≥0. Indeed,bypre-multiplying
the first of (21) by AT we get AT(ATX)F =ATX. By pre-multiplying the same equation by BT, we get
BT(ATX)F = 0 since we already proved that BTX is zero. Hence, we can write these two equations as
AT (ATX)F = ATX andre-applythesameargumentusedabovetoshowthatBTATX =0,andsoon.
BT O
h i h i
3.2 Spectral Factorisation
Since as aforementioned the Popov matrix P is assumed symmetric and positive semidefinite, we can
considerafactorisationoftheform
Q S CT
P = = C D , (22)
ST R DT
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:2) (cid:3)
where Q=CTC, S=CTD and R=DTD. Let us define the rational matrixW(z)=def C(zI −A)−1B+D.
n
The spectrum F (z)=def W∼(z)W(z) – whereW∼(z)=def WT(z−1) – associated with the Popov triple S can
bewrittenas
Q S (zI −A)−1B
F (z)= BT(z−1I −AT)−1 I n ,
n n ST R I
n
(cid:20) (cid:21)(cid:20) (cid:21)
(cid:2) (cid:3)
whichisalsoreferred toasPopovfunctionassociatedwithGDARE(S ), [9]. Thematrixinequalityforan
unknown matrix X =XT of the form P ≥0 is called the discrete Riccati linear matrix inequality, and
X
is hereindenoted byDRLMI(S ). Let us alsodefine
AT XA−X ATXB
L(X)=def P −P = .
X BTXA BTXB
(cid:20) (cid:21)
NoticethatL(X)is alinearfunctionofX.
Lemma 3.4 ([18, p.322],see e.g. [2]foradetailed proof). ForanyX =XT ∈Rn×n, thereholds
(zI −A)−1B
F (z)= BT(z−1I −AT)−1 I P n . (23)
n n X
I
n
(cid:20) (cid:21)
(cid:2) (cid:3)
8
Theorem 3.1 Let r denotethenormalrankofthespectrumF (z).1 If X isasolutionofCGDARE(S ),the
rankof R is equaltor. If X isa solutionof DRLMI(S ), therankofR isat mostequaltor.
X X
Proof: Let us consider X =XT such that P ≥0. By Lemma 2.1, this means in particular that (i) R
X X
is positivesemidefinite, (ii) kerS ⊇kerR , and (iii)Q −S R†ST is positivesemidefinite. Notice that
X X X X X X
(iii)meansthatX satisfies thediscreteRiccati inequality
D(X)=def ATXA
−(ATXB+S)(R+BTXB)†(BTXA+ST)+Q−X ≥0.
Therefore, wecan writeD(X)=HTH forsomematrixH ,which leadsto theexpression
X X X
S R† HT
P = X X R R†ST I + X H O . (24)
X I X X X O X
(cid:20) (cid:21) (cid:20) (cid:21)
(cid:2) (cid:3) (cid:2) (cid:3)
By plugging(24)into(23)weseethat thespectrum F (z)=W ∼(z)W(z)can bewrittenas
∼ ∼
F (z) = W (z)W (z)+W (z)W (z)
1 1 2 2
∼ ∼ W1(z)
= W (z) W (z) ,
1 2 W (z)
2
(cid:20) (cid:21)
(cid:2) (cid:3)
whereW (z)isgivenby
1
1 (zI −A)−1B
W (z) = R2 R†ST I n
1 X X X I
n
(cid:20) (cid:21)
1 (cid:2) (cid:3)
= R2(R† ST(zI −A)−1B+I ),
X X X n m
andW (z)is givenby
2
(zI −A)−1B
W (z)= H O n =H (zI −A)−1B.
2 X I X n
n
(cid:20) (cid:21)
(cid:2) (cid:3)
1
Notice thatW (z)=R2T (z), where T (z)=def R† ST(zI −A)−1B+I is square and invertiblefor all but
1 X X X X X n m
finitely many z∈C. Its inverse can be written as T−1(z)=I −R†ST(zI −A )−1B. Thus, the normal
X m X X n X
rank of
1
TX−∼(z)F (z)TX−1(z)= RX21 TX−∼(z)W2∼(z) "W2(z)RTX2X−1(z)#
h i
1
is equal to the normal rank r of F (z). Then, the rank of R2, which equals that of R , is not greater than
X X
r. Now consider the case where X =XT is a solution of CGDARE(S ). In this case, the term H in (24)
X
iszero,andthereforesoistherationalfunctionW (z). Assuch,W (z)isasquarespectralfactorofF (z),
2 1
i.e.,W∼(z)W(z)=W∼(z)W (z).Moreover,T −∼(z)F (z)T−1(z)=R , whichimpliesthatwhenX =XT
1 1 X X X
is asolutionofCGDARE(S ), therank ofR is exactlyr.
X
1ThenormalrankofarationalmatrixM(z)isdefinedasnormrankM(z)d=efmaxz∈CrankM(z). TherankofM(z)isequal
toitsnormalrankforallbutfinitelymanyz∈C.
9