Find a reasonable necessary and sufficient frequency domain condition, i.e, a condition in terms of the rational matrix ∂W, or possibly in terms of the two-variable rational matrix W, for the existence of a real symmetric negative semi-definite solution of the algebraic Riccati equation

B.D.O

BDO Anderson

HL Trentelman

JC Willems

JW Polderman

VM Popov

Trentelman, Harry L.

ARTS repository - University of Groningen

   University of GroningenWhen does the algebraic Riccati equation have a negative semi-definite solution?Trentelman, Harry L.Published in:EPRINTS-BOOK-TITLEIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Trentelman, H. L. (1999). When does the algebraic Riccati equation have a negative semi-definite solution?In EPRINTS-BOOK-TITLE University of Groningen, Johann Bernoulli Institute for Mathematics andComputer Science.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 16-07-2023When does the algebraic Riccati equation have anegative semi-denite solution?Harry L. TrentelmanNovember 24, 19991 IntroductionIn this contribution we want to draw the readers's attention to an open problemthat concerns the existence of certain solutions to the algebraic Riccati equa-tion. Since its introduction in control theory by Kalman in the beginning ofthe sixties, the algebraic Riccati equation has known an impressive range of ap-plications, such as linear quadratic optimal control, stability theory, stochasticltering and stochastic control, stochastic realization theory, the synthesis oflinear passive networks, dierential games, and H1 optimal control and robuststabilization. For an overview of the existing literature on the algebraic Riccatiequation, we refer to [3].In this note, we deal with the existence of real symmetric solutions to thealgebraic Riccati equation, in particular with the existence of negative semi-denite solutions. It is well-known (see [9]) that the existence of a real symmet-ric solution to the algebraic Riccati equation is equivalent to a given frequencydomain inequality along the imaginary axis. In [9], it was also stated that theexistence of a negative semi-denite solution is equivalent to this frequency do-main inequality holding for all complex numbers in the closed right half of thecomplex plane. Soon after the appearance of [9], a correction [10] appeared inwhich it was outlined that this statement is not correct, and that the frequencydomain inequality in the closed right half plane is a necessary, but not sucientcondition for the existence of a negative semi-denite solution.Since then, several attempts have been made to obtain a convenient neces-sary and sucient frequency domain condition for the existence of a negativesemi-denite solution. In this note we discuss some of these conditions. We alsoexplain that we consider these conditions not to be satisfactory yet, and there-fore we claim the problem of formulating a sensible frequency domain conditionto be still an open problem.Research Institute for Mathematics and Computing Science, P.O. Box 800, 9700 AVGroningen, The Netherlands, e-mail: H.L.Trentelman@math.rug.nl12 The algebraic Riccati equationLet A 2 Rnn and B 2 Rnm be such that (A;B) is a controllable pair. Also,let Q 2 Rnn be such that Q = QT , let S 2 Rmn , and let R 2 Rmm be suchthat R > 0. We deal with the following algebraic Riccati equation:ATK +KA+Q  (KB + ST )R 1(BTK + S) = 0 (2.1)In order to give a frequency domain condition for the existence of a real sym-metric solution K, we dene the matrix of two-variable rational functions Win the indeterminates  and  byW (; ) := R+BT (I  AT ) 1ST + S(I  A) 1B+BT (I  AT ) 1Q(I  A) 1B (2.2)which is often called the Popov function associated with (2.1). (see [7]). Thecoecients of this matrix are quotients of real polynomials in the indeterminates and . With this two-variable rational matrixW we associate a (one-variable)rational matrix @W by dening @W () := W ( ; ), i.e., obtained by taking =   and  = . It was shown in [9] that the algebraic Riccati equation (2.1)has a real symmetric solution if and only if@W (i!)  0 for all ! 2 R, i! 62 (A) (2.3)In [9] it was claimed that the two-variable rational matrix W also provides theclue to the existence of a negative semi-denite solution. Indeed, [9], Theorem4 states that (2.1) has a real symmetric solution K  0 if and only ifW (; )  0 for <e()  0,  62 (A) (2.4)Unfortunately, as was noted in [10], this statement is not correct: in generalthe condition (2.4) is only a necessary conditition, but not a sucient one. It iscertainly interesting to note that for certain important special cases (2.4) doesprovide a necessary and sucient condition. For one important special casethis result is known as the bounded real lemma. This special case in concernedwith the situation that Q =  CTC, S = 0, and R = I. In that case, after achange of variable from K to  K, the Riccati equation becomesATK +KA+CTC +KBBTK = 0 (2.5)while we have W (; ) = I  GT ()G(). The condition (2.4) then becomesGT ()G()  I for <e()  0,  62 (A)which indeed is well-known to be equivalent to the existence of a solutionK  0to the algebraic Riccati equation (2.5) (see, for example [2]). In [5], a numberof additional special cases in which the frequency domain inequality (2.4) is anecessary and sucient condition were established. We will not go into thesespecial cases here. Instead, we want to discuss an alternative condition on thetwo-variable rational matrix W that was proven in [4] to be equivalent to theexistence of a negative semi-denite solution to (2.1). In order to rederive thiscondition here, we make use of recent results in [12] on quadratic dierentialforms and dissipativity of linear dierential systems23 Dissipativity of linear dierential systemsAs was also noted in [9], the study of the algebraic Riccati equation can beput into the more general framework of studying quadratic storage functionsfor linear systems with quadratic supply rates. In that context, an importantrole is played by the linear matrix inequality L(K)  0, whereL(K) :=  ATK +KA+Q KB + STBTK + S R  (3.1)It can be shown that L(K)  0 if and only if the quadratic function V (x) := xTKx satises the inequalityddtV (x(t))  xT (t)Qx(t) + 2xT (t)STu(t) + uT (t)Ru(t) (3.2)for all t 2 R, for all x and u satisfying the dierential equation _x = Ax+Bu. Theinequality (3.2) is called the dissipation inequality for the system _x = Ax+Buwith supply rate xTQx + 2xTSTu + uTRu. If a function V (x) satises thisinequality, it is called a storage function. The dissipation inequality expressesthe property that along trajectories x and u of the system the increase ininternal storage cannot exceed the rate at which storage is supplied to thesystem. If such a function V (x) exists, we call the system dissipative withrespect to the given supply rate. It can be shown that the system _x = Ax +Bu with supply rate xTQx + 2xTSTu + uTRu is dissipative if and only if thefrequency domain condition (2.3) (so along the imaginary axis) holds. Thus,(2.3) is equivalent to the existence of a real symmetric solution K to the linearmatrix inequality L(K)  0. Moreover, if this condition holds, there existreal symmetric solutions K  and K+ such that any real symmetric solutionK satises K   K  K+. The function V1(x) :=  xTK+x is then thesmallest, and the function V2(x) :=  xTK x is the largest storage function.Also, K  and K+ are solutions of the algebraic Riccati equation (2.1). Theseconsiderations show that the existence of a negative semi-denite solution to(2.1) is equivalent to the existence of a negative semi-denite solution to thelinear matrix inequality (3.1), equivalently, to the existence of a positive semi-denite storage function of the system _x = Ax+Bu with supply rate xTQx+2xTSTu+ uTRu.The general problem of the existence of storage functions was recently put inthe framework of quadratic dierential forms for linear dierential systems [12],[8]). An important role in these references is played by two-variable polynomialmatrices, i.e., matrices whose coecients are real polynomials in two indeter-minates, say  and . A two-variable polynomial matrix  can be representedas (; ) =Xk;j k;jkjwhere the k;j are matrices with real coecients, k; j 2 N, and the sum is anite one.  is called symmetric if (; )T = (; ). Each symmetric q  q3two-variable poynomial matrix  induces a quadratic dierential form (QDF),i.e., a map Q : C1(R;Rq )! C1(R;R), dened byQ(`) :=Xk;j (dk`dtk )Tk;j dj`dtjAssociated with , we dene the (one-variable) polynomial matrix @ by@() := ( ; ). Note that this polynomial matrix is para-hermitian, i.e.,(@( ))T = @(). A QDF Q is called average non-negative if for all` 2 C1(R;Rq ) of compact support we haveZ 1 1Q(`)dt  0In [12] this property was shown to be equivalent with the existence of a sym-metric two-variable polynomial matrix 	 such thatddtQ	(`)  Q(`) for all ` 2 C1(R;Rq ) (3.3)It was also shown that Q is average non-negative if and only if @ is non-negative along the imaginary axis, i.e., @(i!)  0 for all ! 2 R. We nowexplain how this result can be used to rederive the condition 2.3 for the ex-istence of a real symmetric solution to the algebraic Riccati equation. Con-sider the controllable system _x = Ax + Bu, more precisely, the system  =(R;Rn+m ;B), with time axis R, signal space Rn+m and behaviorB := f(x; u) 2C1(R;Rn+m) j _x = Ax + Bug: Any controllable system also admits an imagerepresentation (see ([6, 11]). Consider the image representation xu  =  C( ddt)Bp( ddt )I  ` (3.4)Here, p() is the characteristic polynomial of A, i.e., p() := det(I   A), andC() is the polynomial matrix dened by C() := p()(I   A) 1, that is, theclassical adjoint of I A, appearing when one applies Cramer's rule to computethe inverse (I  A) 1. We haveB = f(x; u) 2 C1(R;Rnm) j there exists ` 2 C1(R;Rm) such that 3.4gso (3.4) indeed denes an image representation of our system . Now denethe symmetric two-variable polynomial matrix  by(; ) :=  C()Bp()I T  Q STS R  C()Bp()I  (3.5)It is immediate that if x, u and ` are related by (3.4), then for the QDF Qassociated with  we have Q(`) = xTQx+2xTSTu+uTRu: Assume now thatQ is average non-negative, equivalently, there exists 	 such that ddtQ	  Q.It was proven in ([8]) that any suchQ	 can be represented as a (static) quadraticfunction of any state of the underlying system. In our case, a (minimal) state isgiven by x = C( ddt )B`, so there exists a real symmetric matrix K 2 Rnn suchthat if x and ` are related by x = C( ddt )B`, then Q	(`) = xTKx. Collectingthese facts we nd that the following three statements are equivalent:41. R1 1(xTQx + 2xTSTu + uTRu)dt  0 for all x 2 C1(R;Rn) and u 2C1(R;Rm) of compact support, satisfying _x = Ax+Bu2. there exists a real symmetric matrix K 2 Rnn such that ddtxTKx xTQx + 2xTSTu + uTRu for all x 2 C1(R;Rn) and u 2 C1(R;Rm )satisfying _x = Ax+Bu3. @(i!)  0 for all ! 2 RAs noted before, condition (2) is equivalent to the existence of a real symmetricsolution of the algebraic Riccati equation. Note that for  given by (3.5) wehave (; ) = p()p()W (; ) with W the two-variable rational matrix givenby (2.2). Hence, along the imaginary axis we have @(i!) = jp(i!)j2@W (i!).Therefore, condition (3) is equivalent to the frequency domain inequality (2.3).Thus we have re-established the fact that the existence of a real symmetricsolution to the algebraic Riccati equation is equivalent to the frequency domaininequality (2.3).4 The existence of negative semi-denite solutionsLet us now study what the set-up of QDF's for linear dierential systems can tellus on the existence of negative semi-denite solutions to the algebraic Riccatiequation. To start with, let  be an arbitrary q  q symmetric two-variablepolynomial matrix. The associated QDF Q is called half-line non-negative iffor all ` 2 C1(R;Rq ) of compact support we haveZ 0 1Q(`)dt  0It was proven in [12] that a QDF Q is half-line non-negative if and only ifthere exists a symmetric two-variable polynomial matric 	 such that Q	  0and such that (3.3) holds. Applying this fact to our system , again usingthat it has an image representation given by (3.4), we this time nd that thefollowing two statements are equivalent:1. R 0 1(xTQx + 2xTSTu + uTRu)dt  0 for all x 2 C1(R;Rn) and u 2C1(R;Rm) of compact support, satisfying _x = Ax+Bu2. there exists a real symmetric matrix K 2 Rnn , K  0, such thatddt ( xTKx)  xTQx + 2xTSTu + uTRu for all x 2 C1(R;Rn) andu 2 C1(R;Rm ) satisfying _x = Ax+BuAs before, condition (2) is equivalent to the existence of a negative semi-denitereal symmetric solution to the algebraic Riccati equation. In the following, weestablish a condition on the two-variable polynomial matrix  that is equivalentto condition (1). It turns out that in this way we reobtain the frequency domaincondition for the existence of a negative semi-denite solution to the algebraicRiccati equation that was obtained before in [4]. First note that condition (1)5is equivalent to the condition that for all ` 2 C1(R;Rm) of compact supportwe haveZ 0 1 A( ddt )B`p( ddt )` T  Q STS R  A( ddt )B`p( ddt )`  dt  0 (4.1)By an approximation argument, this condition is equivalent to the same in-equality holding for all ` 2 C1(R;Rm) such that R 0 1 k`k2dt is nite. Now, letN 2 N, and let 1; 2; : : : ; N be N distinct complex numbers in <e() > 0.Also, let v1; v2; : : : vN be arbitrary vectors in Cm . Consider the function`(t) := NXi=1 eitvi (4.2)Applying (4.1) to this function ` (silently moving from real valued functions tocomplex valued functions), we obtainNXi=1 NXj=1 Z 0 1(e( i+j)tvi  A( i)Bp( i)I T  Q STS R  A(j)Bp(j)I  vj)dt  0which, after integration, yieldsNXi=1 NXj=1 vi ( i; j)i + j vj  0 (4.3)Since, for xed complex numbers 1; 2; : : : ; N , this holds for all v1; v2; : : : vN ,this implies that the hermitian mN mN matrix whose (i; j)th block is equalto the mm matrix ( i;j)i+j is positive semi-denite. If, conversely, (4.3) holdsfor any choice of N , any choice of distinct complex numbers 1; 2; : : : ; N in<e() > 0, and any choice of vectors v1; v2; : : : vN in Cm , then clearly (4.1) holdsfor any function ` of the form (4.2). Again by an approximation argument, theinequality (4.1) must then hold for all ` 2 C1(R;Rm) of compact support. Thisproves that any of the statements (1) and (2) is equivalent with the followingcondition (3):3. for all N 2 N, and for any choice 1; 2; : : : ; N of distinct complex num-bers in <e() > 0, the hermitian mN mN matrix ( i;j)i+j i;j=1;2;:::Nis positive semi-denite.We now express condition (3) in terms of the two-variable rational matrix W .It is easily veried that( i; j)i + j i;j=1;2;:::;N = DW ( i; j)i + j i;j=1;2;:::;N DHere D is the blockdiagonal matrix whose ith diagonal block is equal to p(i)I(with I the m  m identity matrix). Thus we claim that condition (3) isequivalent to the following condition (4):64. for all N 2 N, and for any choice 1; 2; : : : ; N of distinct complex num-bers in <e() > 0 such that i 62 (A), i = 1; 2; : : : ; N , the hermitianmN mN matrix W ( i;j)i+j i;j=1;2;:::N is positive semi-denite.Indeed, if i 62 (A), i = 1; 2; : : : ; N , then D is nonsingular. From this the proofof the implication (3) ) (4) is obvious. The proof of the converse implicationthen follows by using a continuity argument.We have now shown that the existence of a negative semi-denite real sym-metric solution of the algebraic Riccati equation is equivalent to condition (4)on the two-variable rational matrix W . Thus we have re-established the condi-tion that was proposed in [4] as a correct alternative for the errorous frequencydomain condition (2.4). A similar condition was obtained in [1] in related workon spectral factorization.We remind the reader that the title of this note is: \When does the algebraicRiccati equation have a negative semi-denite solution?" We have re-derivedthe necessary and sucient frequency domain condition (4) for this to hold.Yet, in our opinion, this condition is not a satisfactory one, since not only itrequires one to check non-negativity of an innite number of hermitian matrices,but also there is no upper bound to the dimension of these matrices. In viewof this, we formulate the following open problem:Find a reasonable necessary and sucient frequency do-main condition, i.e., a condition in terms of the rationalmatrix @W , or possibly in terms of the two-variable ratio-nal matrix W , for the existence of a real symmetric nega-tive semi-denite solution of the algebraic Riccati equation(2.1).References[1] B.D.O. Anderson, \Corrections to: Algebraic properties of minimal de-gree spectral factor". Automatica, Vol. 11, pp. 321 - 322, 1975.[2] B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis- A Modern Systems Theory Approach, Prentice-Hall, Englewood Clis,N.J., 1973.[3] S. Bittanti, A.J. Laub, J.C. Willems (Eds.), The Riccati Equation,Springer Verlag, Berlin, 1991.[4] B.P. Molinari, \Conditions for non-positive solutions of the linear matrixinequality". IEEE Trans. Automat. Contr., Vol. AC-29, pp. 804 - 806,1975.[5] P.J. Moylan, \On a frequency condition in linear optimal control theory".IEEE Trans. Automat. Contr., Vol. AC-29, pp. 806, 1975.[6] J.W. Polderman and J.C. Willems, Introduction to Mathematical SystemsTheory, Springer Verlag, 1997. 7[7] V.M. Popov, L'Hyperstabilite des systemes automatiques, Dunod, Paris,1973.[8] H.L. Trentelman and J.C. Willems, \Every storage function is a statefunction", Systems and Control Letters, 32, pp. 249 - 259, 1997.[9] J.C. Willems, \Least squares stationary optimal control and the algebraicRiccati equation", IEEE Trans. Automat. Contr., Vol. AC-16, pp. 621 -634, 1971.[10] J.C. Willems, \On the existence of a nonpositive solution to the Riccatiequation", IEEE Trans. Automat. Contr., Vol. AC-19, pp. 592 - 593, 1974[11] J.C. Willems, \Paradigms and puzzles in the theory of dynamical sys-tems", IEEE Trans. Automat. Contr., Vol. 36, pp. 259-294, 1991.[12] J.C. Willems and H.L. Trentelman, \On quadratic dierential forms". Toappear in SIAM J. Contr. Optim..8

English

When does the algebraic Riccati equation have a negative semi-definite solution?

University of Groningen Research Database

  
 University of Groningen
When does the algebraic Riccati equation have a negative semi-definite solution?
Trentelman, Harry L.
Published in:
EPRINTS-BOOK-TITLE
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Publication date:
1999
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Trentelman, H. L. (1999). When does the algebraic Riccati equation have a negative semi-definite solution?
In EPRINTS-BOOK-TITLE University of Groningen, Johann Bernoulli Institute for Mathematics and
Computer Science.
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Download date: 10-02-2018
When does the algebraic Riccati equation have a
negative semi-denite solution?
Harry L. Trentelman

November 24, 1999
1 Introduction
In this contribution we want to draw the readers's attention to an open problem
that concerns the existence of certain solutions to the algebraic Riccati equa-
tion. Since its introduction in control theory by Kalman in the beginning of
the sixties, the algebraic Riccati equation has known an impressive range of ap-
plications, such as linear quadratic optimal control, stability theory, stochastic
ltering and stochastic control, stochastic realization theory, the synthesis of
linear passive networks, dierential games, and H
1
optimal control and robust
stabilization. For an overview of the existing literature on the algebraic Riccati
equation, we refer to [3].
In this note, we deal with the existence of real symmetric solutions to the
algebraic Riccati equation, in particular with the existence of negative semi-
denite solutions. It is well-known (see [9]) that the existence of a real symmet-
ric solution to the algebraic Riccati equation is equivalent to a given frequency
domain inequality along the imaginary axis. In [9], it was also stated that the
existence of a negative semi-denite solution is equivalent to this frequency do-
main inequality holding for all complex numbers in the closed right half of the
complex plane. Soon after the appearance of [9], a correction [10] appeared in
which it was outlined that this statement is not correct, and that the frequency
domain inequality in the closed right half plane is a necessary, but not sucient
condition for the existence of a negative semi-denite solution.
Since then, several attempts have been made to obtain a convenient neces-
sary and sucient frequency domain condition for the existence of a negative
semi-denite solution. In this note we discuss some of these conditions. We also
explain that we consider these conditions not to be satisfactory yet, and there-
fore we claim the problem of formulating a sensible frequency domain condition
to be still an open problem.

Research Institute for Mathematics and Computing Science, P.O. Box 800, 9700 AV
Groningen, The Netherlands, e-mail: H.L.Trentelman@math.rug.nl
1
2 The algebraic Riccati equation
Let A 2 R
nn
and B 2 R
nm
be such that (A;B) is a controllable pair. Also,
let Q 2 R
nn
be such that Q = Q
T
, let S 2 R
mn
, and let R 2 R
mm
be such
that R > 0. We deal with the following algebraic Riccati equation:
A
T
K +KA+Q  (KB + S
T
)R
 1
(B
T
K + S) = 0 (2.1)
In order to give a frequency domain condition for the existence of a real sym-
metric solution K, we dene the matrix of two-variable rational functions W
in the indeterminates  and  by
W (; ) := R+B
T
(I  A
T
)
 1
S
T
+ S(I  A)
 1
B
+B
T
(I  A
T
)
 1
Q(I  A)
 1
B (2.2)
which is often called the Popov function associated with (2.1). (see [7]). The
coecients of this matrix are quotients of real polynomials in the indeterminates
 and . With this two-variable rational matrixW we associate a (one-variable)
rational matrix @W by dening @W () := W ( ; ), i.e., obtained by taking
 =   and  = . It was shown in [9] that the algebraic Riccati equation (2.1)
has a real symmetric solution if and only if
@W (i!)  0 for all ! 2 R, i! 62 (A) (2.3)
In [9] it was claimed that the two-variable rational matrix W also provides the
clue to the existence of a negative semi-denite solution. Indeed, [9], Theorem
4 states that (2.1) has a real symmetric solution K  0 if and only if
W (

; )  0 for <e()  0,  62 (A) (2.4)
Unfortunately, as was noted in [10], this statement is not correct: in general
the condition (2.4) is only a necessary conditition, but not a sucient one. It is
certainly interesting to note that for certain important special cases (2.4) does
provide a necessary and sucient condition. For one important special case
this result is known as the bounded real lemma. This special case in concerned
with the situation that Q =  C
T
C, S = 0, and R = I. In that case, after a
change of variable from K to  K, the Riccati equation becomes
A
T
K +KA+C
T
C +KBB
T
K = 0 (2.5)
while we have W (; ) = I  G
T
()G(). The condition (2.4) then becomes
G
T
(

)G()  I for <e()  0,  62 (A)
which indeed is well-known to be equivalent to the existence of a solutionK  0
to the algebraic Riccati equation (2.5) (see, for example [2]). In [5], a number
of additional special cases in which the frequency domain inequality (2.4) is a
necessary and sucient condition were established. We will not go into these
special cases here. Instead, we want to discuss an alternative condition on the
two-variable rational matrix W that was proven in [4] to be equivalent to the
existence of a negative semi-denite solution to (2.1). In order to rederive this
condition here, we make use of recent results in [12] on quadratic dierential
forms and dissipativity of linear dierential systems
2
3 Dissipativity of linear dierential systems
As was also noted in [9], the study of the algebraic Riccati equation can be
put into the more general framework of studying quadratic storage functions
for linear systems with quadratic supply rates. In that context, an important
role is played by the linear matrix inequality L(K)  0, where
L(K) :=

A
T
K +KA+Q KB + S
T
B
T
K + S R

(3.1)
It can be shown that L(K)  0 if and only if the quadratic function V (x) :=
 x
T
Kx satises the inequality
d
dt
V (x(t))  x
T
(t)Qx(t) + 2x
T
(t)S
T
u(t) + u
T
(t)Ru(t) (3.2)
for all t 2 R, for all x and u satisfying the dierential equation _x = Ax+Bu. The
inequality (3.2) is called the dissipation inequality for the system _x = Ax+Bu
with supply rate x
T
Qx + 2x
T
S
T
u + u
T
Ru. If a function V (x) satises this
inequality, it is called a storage function. The dissipation inequality expresses
the property that along trajectories x and u of the system the increase in
internal storage cannot exceed the rate at which storage is supplied to the
system. If such a function V (x) exists, we call the system dissipative with
respect to the given supply rate. It can be shown that the system _x = Ax +
Bu with supply rate x
T
Qx + 2x
T
S
T
u + u
T
Ru is dissipative if and only if the
frequency domain condition (2.3) (so along the imaginary axis) holds. Thus,
(2.3) is equivalent to the existence of a real symmetric solution K to the linear
matrix inequality L(K)  0. Moreover, if this condition holds, there exist
real symmetric solutions K
 
and K
+
such that any real symmetric solution
K satises K
 
 K  K
+
. The function V
1
(x) :=  x
T
K
+
x is then the
smallest, and the function V
2
(x) :=  x
T
K
 
x is the largest storage function.
Also, K
 
and K
+
are solutions of the algebraic Riccati equation (2.1). These
considerations show that the existence of a negative semi-denite solution to
(2.1) is equivalent to the existence of a negative semi-denite solution to the
linear matrix inequality (3.1), equivalently, to the existence of a positive semi-
denite storage function of the system _x = Ax+Bu with supply rate x
T
Qx+
2x
T
S
T
u+ u
T
Ru.
The general problem of the existence of storage functions was recently put in
the framework of quadratic dierential forms for linear dierential systems [12],
[8]). An important role in these references is played by two-variable polynomial
matrices, i.e., matrices whose coecients are real polynomials in two indeter-
minates, say  and . A two-variable polynomial matrix  can be represented
as
(; ) =
X
k;j

k;j

k

j
where the 
k;j
are matrices with real coecients, k; j 2 N, and the sum is a
nite one.  is called symmetric if (; )
T
= (; ). Each symmetric q  q
3
two-variable poynomial matrix  induces a quadratic dierential form (QDF),
i.e., a map Q

: C
1
(R;R
q
)! C
1
(R;R), dened by
Q

(`) :=
X
k;j
(
d
k
`
dt
k
)
T

k;j
d
j
`
dt
j
Associated with , we dene the (one-variable) polynomial matrix @ by
@() := ( ; ). Note that this polynomial matrix is para-hermitian, i.e.,
(@( ))
T
= @(). A QDF Q

is called average non-negative if for all
` 2 C
1
(R;R
q
) of compact support we have
Z
1
 1
Q

(`)dt  0
In [12] this property was shown to be equivalent with the existence of a sym-
metric two-variable polynomial matrix 	 such that
d
dt
Q
	
(`)  Q

(`) for all ` 2 C
1
(R;R
q
) (3.3)
It was also shown that Q

is average non-negative if and only if @ is non-
negative along the imaginary axis, i.e., @(i!)  0 for all ! 2 R. We now
explain how this result can be used to rederive the condition 2.3 for the ex-
istence of a real symmetric solution to the algebraic Riccati equation. Con-
sider the controllable system _x = Ax + Bu, more precisely, the system  =
(R;R
n+m
;B), with time axis R, signal space R
n+m
and behaviorB := f(x; u) 2
C
1
(R;R
n+m
) j _x = Ax + Bug: Any controllable system also admits an image
representation (see ([6, 11]). Consider the image representation

x
u

=

C(
d
dt
)B
p(
d
dt
)I

` (3.4)
Here, p() is the characteristic polynomial of A, i.e., p() := det(I   A), and
C() is the polynomial matrix dened by C() := p()(I   A)
 1
, that is, the
classical adjoint of I A, appearing when one applies Cramer's rule to compute
the inverse (I  A)
 1
. We have
B = f(x; u) 2 C
1
(R;R
nm
) j there exists ` 2 C
1
(R;R
m
) such that 3.4g
so (3.4) indeed denes an image representation of our system . Now dene
the symmetric two-variable polynomial matrix  by
(; ) :=

C()B
p()I

T

Q S
T
S R

C()B
p()I

(3.5)
It is immediate that if x, u and ` are related by (3.4), then for the QDF Q

associated with  we have Q

(`) = x
T
Qx+2x
T
S
T
u+u
T
Ru: Assume now that
Q

is average non-negative, equivalently, there exists 	 such that
d
dt
Q
	
 Q

.
It was proven in ([8]) that any suchQ
	
can be represented as a (static) quadratic
function of any state of the underlying system. In our case, a (minimal) state is
given by x = C(
d
dt
)B`, so there exists a real symmetric matrix K 2 R
nn
such
that if x and ` are related by x = C(
d
dt
)B`, then Q
	
(`) = x
T
Kx. Collecting
these facts we nd that the following three statements are equivalent:
4
1.
R
1
 1
(x
T
Qx + 2x
T
S
T
u + u
T
Ru)dt  0 for all x 2 C
1
(R;R
n
) and u 2
C
1
(R;R
m
) of compact support, satisfying _x = Ax+Bu
2. there exists a real symmetric matrix K 2 R
nn
such that
d
dt
x
T
Kx 
x
T
Qx + 2x
T
S
T
u + u
T
Ru for all x 2 C
1
(R;R
n
) and u 2 C
1
(R;R
m
)
satisfying _x = Ax+Bu
3. @(i!)  0 for all ! 2 R
As noted before, condition (2) is equivalent to the existence of a real symmetric
solution of the algebraic Riccati equation. Note that for  given by (3.5) we
have (; ) = p()p()W (; ) with W the two-variable rational matrix given
by (2.2). Hence, along the imaginary axis we have @(i!) = jp(i!)j
2
@W (i!).
Therefore, condition (3) is equivalent to the frequency domain inequality (2.3).
Thus we have re-established the fact that the existence of a real symmetric
solution to the algebraic Riccati equation is equivalent to the frequency domain
inequality (2.3).
4 The existence of negative semi-denite solutions
Let us now study what the set-up of QDF's for linear dierential systems can tell
us on the existence of negative semi-denite solutions to the algebraic Riccati
equation. To start with, let  be an arbitrary q  q symmetric two-variable
polynomial matrix. The associated QDF Q

is called half-line non-negative if
for all ` 2 C
1
(R;R
q
) of compact support we have
Z
0
 1
Q

(`)dt  0
It was proven in [12] that a QDF Q

is half-line non-negative if and only if
there exists a symmetric two-variable polynomial matric 	 such that Q
	
 0
and such that (3.3) holds. Applying this fact to our system , again using
that it has an image representation given by (3.4), we this time nd that the
following two statements are equivalent:
1.
R
0
 1
(x
T
Qx + 2x
T
S
T
u + u
T
Ru)dt  0 for all x 2 C
1
(R;R
n
) and u 2
C
1
(R;R
m
) of compact support, satisfying _x = Ax+Bu
2. there exists a real symmetric matrix K 2 R
nn
, K  0, such that
d
dt
( x
T
Kx)  x
T
Qx + 2x
T
S
T
u + u
T
Ru for all x 2 C
1
(R;R
n
) and
u 2 C
1
(R;R
m
) satisfying _x = Ax+Bu
As before, condition (2) is equivalent to the existence of a negative semi-denite
real symmetric solution to the algebraic Riccati equation. In the following, we
establish a condition on the two-variable polynomial matrix  that is equivalent
to condition (1). It turns out that in this way we reobtain the frequency domain
condition for the existence of a negative semi-denite solution to the algebraic
Riccati equation that was obtained before in [4]. First note that condition (1)
5
is equivalent to the condition that for all ` 2 C
1
(R;R
m
) of compact support
we have
Z
0
 1

A(
d
dt
)B`
p(
d
dt
)`

T

Q S
T
S R

A(
d
dt
)B`
p(
d
dt
)`

dt  0 (4.1)
By an approximation argument, this condition is equivalent to the same in-
equality holding for all ` 2 C
1
(R;R
m
) such that
R
0
 1
k`k
2
dt is nite. Now, let
N 2 N, and let 
1
; 
2
; : : : ; 
N
be N distinct complex numbers in <e() > 0.
Also, let v
1
; v
2
; : : : v
N
be arbitrary vectors in C
m
. Consider the function
`(t) :=
N
X
i=1
e

i
t
v
i
(4.2)
Applying (4.1) to this function ` (silently moving from real valued functions to
complex valued functions), we obtain
N
X
i=1
N
X
j=1
Z
0
 1
(e
(


i
+
j
)t
v

i

A(


i
)B
p(


i
)I

T

Q S
T
S R

A(
j
)B
p(
j
)I

v
j
)dt  0
which, after integration, yields
N
X
i=1
N
X
j=1
v

i
(


i
; 
j
)


i
+ 
j
v
j
 0 (4.3)
Since, for xed complex numbers 
1
; 
2
; : : : ; 
N
, this holds for all v
1
; v
2
; : : : v
N
,
this implies that the hermitian mN mN matrix whose (i; j)th block is equal
to the mm matrix
(


i
;
j
)


i
+
j
is positive semi-denite. If, conversely, (4.3) holds
for any choice of N , any choice of distinct complex numbers 
1
; 
2
; : : : ; 
N
in
<e() > 0, and any choice of vectors v
1
; v
2
; : : : v
N
in C
m
, then clearly (4.1) holds
for any function ` of the form (4.2). Again by an approximation argument, the
inequality (4.1) must then hold for all ` 2 C
1
(R;R
m
) of compact support. This
proves that any of the statements (1) and (2) is equivalent with the following
condition (3):
3. for all N 2 N, and for any choice 
1
; 
2
; : : : ; 
N
of distinct complex num-
bers in <e() > 0, the hermitian mN mN matrix

(


i
;
j
)


i
+
j

i;j=1;2;:::N
is positive semi-denite.
We now express condition (3) in terms of the two-variable rational matrix W .
It is easily veried that

(


i
; 
j
)


i
+ 
j

i;j=1;2;:::;N
= D


W (


i
; 
j
)


i
+ 
j

i;j=1;2;:::;N
D
Here D is the blockdiagonal matrix whose ith diagonal block is equal to p(
i
)I
(with I the m  m identity matrix). Thus we claim that condition (3) is
equivalent to the following condition (4):
6
4. for all N 2 N, and for any choice 
1
; 
2
; : : : ; 
N
of distinct complex num-
bers in <e() > 0 such that 
i
62 (A), i = 1; 2; : : : ; N , the hermitian
mN mN matrix

W (


i
;
j
)


i
+
j

i;j=1;2;:::N
is positive semi-denite.
Indeed, if 
i
62 (A), i = 1; 2; : : : ; N , then D is nonsingular. From this the proof
of the implication (3) ) (4) is obvious. The proof of the converse implication
then follows by using a continuity argument.
We have now shown that the existence of a negative semi-denite real sym-
metric solution of the algebraic Riccati equation is equivalent to condition (4)
on the two-variable rational matrix W . Thus we have re-established the condi-
tion that was proposed in [4] as a correct alternative for the errorous frequency
domain condition (2.4). A similar condition was obtained in [1] in related work
on spectral factorization.
We remind the reader that the title of this note is: \When does the algebraic
Riccati equation have a negative semi-denite solution?" We have re-derived
the necessary and sucient frequency domain condition (4) for this to hold.
Yet, in our opinion, this condition is not a satisfactory one, since not only it
requires one to check non-negativity of an innite number of hermitian matrices,
but also there is no upper bound to the dimension of these matrices. In view
of this, we formulate the following open problem:
Find a reasonable necessary and sucient frequency do-
main condition, i.e., a condition in terms of the rational
matrix @W , or possibly in terms of the two-variable ratio-
nal matrix W , for the existence of a real symmetric nega-
tive semi-denite solution of the algebraic Riccati equation
(2.1).
References
[1] B.D.O. Anderson, \Corrections to: Algebraic properties of minimal de-
gree spectral factor". Automatica, Vol. 11, pp. 321 - 322, 1975.
[2] B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis
- A Modern Systems Theory Approach, Prentice-Hall, Englewood Clis,
N.J., 1973.
[3] S. Bittanti, A.J. Laub, J.C. Willems (Eds.), The Riccati Equation,
Springer Verlag, Berlin, 1991.
[4] B.P. Molinari, \Conditions for non-positive solutions of the linear matrix
inequality". IEEE Trans. Automat. Contr., Vol. AC-29, pp. 804 - 806,
1975.
[5] P.J. Moylan, \On a frequency condition in linear optimal control theory".
IEEE Trans. Automat. Contr., Vol. AC-29, pp. 806, 1975.
[6] J.W. Polderman and J.C. Willems, Introduction to Mathematical Systems
Theory, Springer Verlag, 1997.
7
[7] V.M. Popov, L'Hyperstabilite des systemes automatiques, Dunod, Paris,
1973.
[8] H.L. Trentelman and J.C. Willems, \Every storage function is a state
function", Systems and Control Letters, 32, pp. 249 - 259, 1997.
[9] J.C. Willems, \Least squares stationary optimal control and the algebraic
Riccati equation", IEEE Trans. Automat. Contr., Vol. AC-16, pp. 621 -
634, 1971.
[10] J.C. Willems, \On the existence of a nonpositive solution to the Riccati
equation", IEEE Trans. Automat. Contr., Vol. AC-19, pp. 592 - 593, 1974
[11] J.C. Willems, \Paradigms and puzzles in the theory of dynamical sys-
tems", IEEE Trans. Automat. Contr., Vol. 36, pp. 259-294, 1991.
[12] J.C. Willems and H.L. Trentelman, \On quadratic dierential forms". To
appear in SIAM J. Contr. Optim..
8


   University of GroningenWhen does the algebraic Riccati equation have a negative semi-definite solution?Trentelman, Harry L.Published in:EPRINTS-BOOK-TITLEIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Trentelman, H. L. (1999). When does the algebraic Riccati equation have a negative semi-definite solution?In EPRINTS-BOOK-TITLE University of Groningen, Johann Bernoulli Institute for Mathematics andComputer Science.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 06-06-2022When does the algebraic Riccati equation have anegative semi-denite solution?Harry L. TrentelmanNovember 24, 19991 IntroductionIn this contribution we want to draw the readers's attention to an open problemthat concerns the existence of certain solutions to the algebraic Riccati equa-tion. Since its introduction in control theory by Kalman in the beginning ofthe sixties, the algebraic Riccati equation has known an impressive range of ap-plications, such as linear quadratic optimal control, stability theory, stochasticltering and stochastic control, stochastic realization theory, the synthesis oflinear passive networks, dierential games, and H1 optimal control and robuststabilization. For an overview of the existing literature on the algebraic Riccatiequation, we refer to [3].In this note, we deal with the existence of real symmetric solutions to thealgebraic Riccati equation, in particular with the existence of negative semi-denite solutions. It is well-known (see [9]) that the existence of a real symmet-ric solution to the algebraic Riccati equation is equivalent to a given frequencydomain inequality along the imaginary axis. In [9], it was also stated that theexistence of a negative semi-denite solution is equivalent to this frequency do-main inequality holding for all complex numbers in the closed right half of thecomplex plane. Soon after the appearance of [9], a correction [10] appeared inwhich it was outlined that this statement is not correct, and that the frequencydomain inequality in the closed right half plane is a necessary, but not sucientcondition for the existence of a negative semi-denite solution.Since then, several attempts have been made to obtain a convenient neces-sary and sucient frequency domain condition for the existence of a negativesemi-denite solution. In this note we discuss some of these conditions. We alsoexplain that we consider these conditions not to be satisfactory yet, and there-fore we claim the problem of formulating a sensible frequency domain conditionto be still an open problem.Research Institute for Mathematics and Computing Science, P.O. Box 800, 9700 AVGroningen, The Netherlands, e-mail: H.L.Trentelman@math.rug.nl12 The algebraic Riccati equationLet A 2 Rnn and B 2 Rnm be such that (A;B) is a controllable pair. Also,let Q 2 Rnn be such that Q = QT , let S 2 Rmn , and let R 2 Rmm be suchthat R > 0. We deal with the following algebraic Riccati equation:ATK +KA+Q  (KB + ST )R 1(BTK + S) = 0 (2.1)In order to give a frequency domain condition for the existence of a real sym-metric solution K, we dene the matrix of two-variable rational functions Win the indeterminates  and  byW (; ) := R+BT (I  AT ) 1ST + S(I  A) 1B+BT (I  AT ) 1Q(I  A) 1B (2.2)which is often called the Popov function associated with (2.1). (see [7]). Thecoecients of this matrix are quotients of real polynomials in the indeterminates and . With this two-variable rational matrixW we associate a (one-variable)rational matrix @W by dening @W () := W ( ; ), i.e., obtained by taking =   and  = . It was shown in [9] that the algebraic Riccati equation (2.1)has a real symmetric solution if and only if@W (i!)  0 for all ! 2 R, i! 62 (A) (2.3)In [9] it was claimed that the two-variable rational matrix W also provides theclue to the existence of a negative semi-denite solution. Indeed, [9], Theorem4 states that (2.1) has a real symmetric solution K  0 if and only ifW (; )  0 for <e()  0,  62 (A) (2.4)Unfortunately, as was noted in [10], this statement is not correct: in generalthe condition (2.4) is only a necessary conditition, but not a sucient one. It iscertainly interesting to note that for certain important special cases (2.4) doesprovide a necessary and sucient condition. For one important special casethis result is known as the bounded real lemma. This special case in concernedwith the situation that Q =  CTC, S = 0, and R = I. In that case, after achange of variable from K to  K, the Riccati equation becomesATK +KA+CTC +KBBTK = 0 (2.5)while we have W (; ) = I  GT ()G(). The condition (2.4) then becomesGT ()G()  I for <e()  0,  62 (A)which indeed is well-known to be equivalent to the existence of a solutionK  0to the algebraic Riccati equation (2.5) (see, for example [2]). In [5], a numberof additional special cases in which the frequency domain inequality (2.4) is anecessary and sucient condition were established. We will not go into thesespecial cases here. Instead, we want to discuss an alternative condition on thetwo-variable rational matrix W that was proven in [4] to be equivalent to theexistence of a negative semi-denite solution to (2.1). In order to rederive thiscondition here, we make use of recent results in [12] on quadratic dierentialforms and dissipativity of linear dierential systems23 Dissipativity of linear dierential systemsAs was also noted in [9], the study of the algebraic Riccati equation can beput into the more general framework of studying quadratic storage functionsfor linear systems with quadratic supply rates. In that context, an importantrole is played by the linear matrix inequality L(K)  0, whereL(K) :=  ATK +KA+Q KB + STBTK + S R  (3.1)It can be shown that L(K)  0 if and only if the quadratic function V (x) := xTKx satises the inequalityddtV (x(t))  xT (t)Qx(t) + 2xT (t)STu(t) + uT (t)Ru(t) (3.2)for all t 2 R, for all x and u satisfying the dierential equation _x = Ax+Bu. Theinequality (3.2) is called the dissipation inequality for the system _x = Ax+Buwith supply rate xTQx + 2xTSTu + uTRu. If a function V (x) satises thisinequality, it is called a storage function. The dissipation inequality expressesthe property that along trajectories x and u of the system the increase ininternal storage cannot exceed the rate at which storage is supplied to thesystem. If such a function V (x) exists, we call the system dissipative withrespect to the given supply rate. It can be shown that the system _x = Ax +Bu with supply rate xTQx + 2xTSTu + uTRu is dissipative if and only if thefrequency domain condition (2.3) (so along the imaginary axis) holds. Thus,(2.3) is equivalent to the existence of a real symmetric solution K to the linearmatrix inequality L(K)  0. Moreover, if this condition holds, there existreal symmetric solutions K  and K+ such that any real symmetric solutionK satises K   K  K+. The function V1(x) :=  xTK+x is then thesmallest, and the function V2(x) :=  xTK x is the largest storage function.Also, K  and K+ are solutions of the algebraic Riccati equation (2.1). Theseconsiderations show that the existence of a negative semi-denite solution to(2.1) is equivalent to the existence of a negative semi-denite solution to thelinear matrix inequality (3.1), equivalently, to the existence of a positive semi-denite storage function of the system _x = Ax+Bu with supply rate xTQx+2xTSTu+ uTRu.The general problem of the existence of storage functions was recently put inthe framework of quadratic dierential forms for linear dierential systems [12],[8]). An important role in these references is played by two-variable polynomialmatrices, i.e., matrices whose coecients are real polynomials in two indeter-minates, say  and . A two-variable polynomial matrix  can be representedas (; ) =Xk;j k;jkjwhere the k;j are matrices with real coecients, k; j 2 N, and the sum is anite one.  is called symmetric if (; )T = (; ). Each symmetric q  q3two-variable poynomial matrix  induces a quadratic dierential form (QDF),i.e., a map Q : C1(R;Rq )! C1(R;R), dened byQ(`) :=Xk;j (dk`dtk )Tk;j dj`dtjAssociated with , we dene the (one-variable) polynomial matrix @ by@() := ( ; ). Note that this polynomial matrix is para-hermitian, i.e.,(@( ))T = @(). A QDF Q is called average non-negative if for all` 2 C1(R;Rq ) of compact support we haveZ 1 1Q(`)dt  0In [12] this property was shown to be equivalent with the existence of a sym-metric two-variable polynomial matrix 	 such thatddtQ	(`)  Q(`) for all ` 2 C1(R;Rq ) (3.3)It was also shown that Q is average non-negative if and only if @ is non-negative along the imaginary axis, i.e., @(i!)  0 for all ! 2 R. We nowexplain how this result can be used to rederive the condition 2.3 for the ex-istence of a real symmetric solution to the algebraic Riccati equation. Con-sider the controllable system _x = Ax + Bu, more precisely, the system  =(R;Rn+m ;B), with time axis R, signal space Rn+m and behaviorB := f(x; u) 2C1(R;Rn+m) j _x = Ax + Bug: Any controllable system also admits an imagerepresentation (see ([6, 11]). Consider the image representation xu  =  C( ddt)Bp( ddt )I  ` (3.4)Here, p() is the characteristic polynomial of A, i.e., p() := det(I   A), andC() is the polynomial matrix dened by C() := p()(I   A) 1, that is, theclassical adjoint of I A, appearing when one applies Cramer's rule to computethe inverse (I  A) 1. We haveB = f(x; u) 2 C1(R;Rnm) j there exists ` 2 C1(R;Rm) such that 3.4gso (3.4) indeed denes an image representation of our system . Now denethe symmetric two-variable polynomial matrix  by(; ) :=  C()Bp()I T  Q STS R  C()Bp()I  (3.5)It is immediate that if x, u and ` are related by (3.4), then for the QDF Qassociated with  we have Q(`) = xTQx+2xTSTu+uTRu: Assume now thatQ is average non-negative, equivalently, there exists 	 such that ddtQ	  Q.It was proven in ([8]) that any suchQ	 can be represented as a (static) quadraticfunction of any state of the underlying system. In our case, a (minimal) state isgiven by x = C( ddt )B`, so there exists a real symmetric matrix K 2 Rnn suchthat if x and ` are related by x = C( ddt )B`, then Q	(`) = xTKx. Collectingthese facts we nd that the following three statements are equivalent:41. R1 1(xTQx + 2xTSTu + uTRu)dt  0 for all x 2 C1(R;Rn) and u 2C1(R;Rm) of compact support, satisfying _x = Ax+Bu2. there exists a real symmetric matrix K 2 Rnn such that ddtxTKx xTQx + 2xTSTu + uTRu for all x 2 C1(R;Rn) and u 2 C1(R;Rm )satisfying _x = Ax+Bu3. @(i!)  0 for all ! 2 RAs noted before, condition (2) is equivalent to the existence of a real symmetricsolution of the algebraic Riccati equation. Note that for  given by (3.5) wehave (; ) = p()p()W (; ) with W the two-variable rational matrix givenby (2.2). Hence, along the imaginary axis we have @(i!) = jp(i!)j2@W (i!).Therefore, condition (3) is equivalent to the frequency domain inequality (2.3).Thus we have re-established the fact that the existence of a real symmetricsolution to the algebraic Riccati equation is equivalent to the frequency domaininequality (2.3).4 The existence of negative semi-denite solutionsLet us now study what the set-up of QDF's for linear dierential systems can tellus on the existence of negative semi-denite solutions to the algebraic Riccatiequation. To start with, let  be an arbitrary q  q symmetric two-variablepolynomial matrix. The associated QDF Q is called half-line non-negative iffor all ` 2 C1(R;Rq ) of compact support we haveZ 0 1Q(`)dt  0It was proven in [12] that a QDF Q is half-line non-negative if and only ifthere exists a symmetric two-variable polynomial matric 	 such that Q	  0and such that (3.3) holds. Applying this fact to our system , again usingthat it has an image representation given by (3.4), we this time nd that thefollowing two statements are equivalent:1. R 0 1(xTQx + 2xTSTu + uTRu)dt  0 for all x 2 C1(R;Rn) and u 2C1(R;Rm) of compact support, satisfying _x = Ax+Bu2. there exists a real symmetric matrix K 2 Rnn , K  0, such thatddt ( xTKx)  xTQx + 2xTSTu + uTRu for all x 2 C1(R;Rn) andu 2 C1(R;Rm ) satisfying _x = Ax+BuAs before, condition (2) is equivalent to the existence of a negative semi-denitereal symmetric solution to the algebraic Riccati equation. In the following, weestablish a condition on the two-variable polynomial matrix  that is equivalentto condition (1). It turns out that in this way we reobtain the frequency domaincondition for the existence of a negative semi-denite solution to the algebraicRiccati equation that was obtained before in [4]. First note that condition (1)5is equivalent to the condition that for all ` 2 C1(R;Rm) of compact supportwe haveZ 0 1 A( ddt )B`p( ddt )` T  Q STS R  A( ddt )B`p( ddt )`  dt  0 (4.1)By an approximation argument, this condition is equivalent to the same in-equality holding for all ` 2 C1(R;Rm) such that R 0 1 k`k2dt is nite. Now, letN 2 N, and let 1; 2; : : : ; N be N distinct complex numbers in <e() > 0.Also, let v1; v2; : : : vN be arbitrary vectors in Cm . Consider the function`(t) := NXi=1 eitvi (4.2)Applying (4.1) to this function ` (silently moving from real valued functions tocomplex valued functions), we obtainNXi=1 NXj=1 Z 0 1(e( i+j)tvi  A( i)Bp( i)I T  Q STS R  A(j)Bp(j)I  vj)dt  0which, after integration, yieldsNXi=1 NXj=1 vi ( i; j)i + j vj  0 (4.3)Since, for xed complex numbers 1; 2; : : : ; N , this holds for all v1; v2; : : : vN ,this implies that the hermitian mN mN matrix whose (i; j)th block is equalto the mm matrix ( i;j)i+j is positive semi-denite. If, conversely, (4.3) holdsfor any choice of N , any choice of distinct complex numbers 1; 2; : : : ; N in<e() > 0, and any choice of vectors v1; v2; : : : vN in Cm , then clearly (4.1) holdsfor any function ` of the form (4.2). Again by an approximation argument, theinequality (4.1) must then hold for all ` 2 C1(R;Rm) of compact support. Thisproves that any of the statements (1) and (2) is equivalent with the followingcondition (3):3. for all N 2 N, and for any choice 1; 2; : : : ; N of distinct complex num-bers in <e() > 0, the hermitian mN mN matrix ( i;j)i+j i;j=1;2;:::Nis positive semi-denite.We now express condition (3) in terms of the two-variable rational matrix W .It is easily veried that( i; j)i + j i;j=1;2;:::;N = DW ( i; j)i + j i;j=1;2;:::;N DHere D is the blockdiagonal matrix whose ith diagonal block is equal to p(i)I(with I the m  m identity matrix). Thus we claim that condition (3) isequivalent to the following condition (4):64. for all N 2 N, and for any choice 1; 2; : : : ; N of distinct complex num-bers in <e() > 0 such that i 62 (A), i = 1; 2; : : : ; N , the hermitianmN mN matrix W ( i;j)i+j i;j=1;2;:::N is positive semi-denite.Indeed, if i 62 (A), i = 1; 2; : : : ; N , then D is nonsingular. From this the proofof the implication (3) ) (4) is obvious. The proof of the converse implicationthen follows by using a continuity argument.We have now shown that the existence of a negative semi-denite real sym-metric solution of the algebraic Riccati equation is equivalent to condition (4)on the two-variable rational matrix W . Thus we have re-established the condi-tion that was proposed in [4] as a correct alternative for the errorous frequencydomain condition (2.4). A similar condition was obtained in [1] in related workon spectral factorization.We remind the reader that the title of this note is: \When does the algebraicRiccati equation have a negative semi-denite solution?" We have re-derivedthe necessary and sucient frequency domain condition (4) for this to hold.Yet, in our opinion, this condition is not a satisfactory one, since not only itrequires one to check non-negativity of an innite number of hermitian matrices,but also there is no upper bound to the dimension of these matrices. In viewof this, we formulate the following open problem:Find a reasonable necessary and sucient frequency do-main condition, i.e., a condition in terms of the rationalmatrix @W , or possibly in terms of the two-variable ratio-nal matrix W , for the existence of a real symmetric nega-tive semi-denite solution of the algebraic Riccati equation(2.1).References[1] B.D.O. Anderson, \Corrections to: Algebraic properties of minimal de-gree spectral factor". Automatica, Vol. 11, pp. 321 - 322, 1975.[2] B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis- A Modern Systems Theory Approach, Prentice-Hall, Englewood Clis,N.J., 1973.[3] S. Bittanti, A.J. Laub, J.C. Willems (Eds.), The Riccati Equation,Springer Verlag, Berlin, 1991.[4] B.P. Molinari, \Conditions for non-positive solutions of the linear matrixinequality". IEEE Trans. Automat. Contr., Vol. AC-29, pp. 804 - 806,1975.[5] P.J. Moylan, \On a frequency condition in linear optimal control theory".IEEE Trans. Automat. Contr., Vol. AC-29, pp. 806, 1975.[6] J.W. Polderman and J.C. Willems, Introduction to Mathematical SystemsTheory, Springer Verlag, 1997. 7[7] V.M. Popov, L'Hyperstabilite des systemes automatiques, Dunod, Paris,1973.[8] H.L. Trentelman and J.C. Willems, \Every storage function is a statefunction", Systems and Control Letters, 32, pp. 249 - 259, 1997.[9] J.C. Willems, \Least squares stationary optimal control and the algebraicRiccati equation", IEEE Trans. Automat. Contr., Vol. AC-16, pp. 621 -634, 1971.[10] J.C. Willems, \On the existence of a nonpositive solution to the Riccatiequation", IEEE Trans. Automat. Contr., Vol. AC-19, pp. 592 - 593, 1974[11] J.C. Willems, \Paradigms and puzzles in the theory of dynamical sys-tems", IEEE Trans. Automat. Contr., Vol. 36, pp. 259-294, 1991.[12] J.C. Willems and H.L. Trentelman, \On quadratic dierential forms". Toappear in SIAM J. Contr. Optim..8

Harry L. Trentelman

Crossref

University of Groningen

   University of GroningenWhen does the algebraic Riccati equation have a negative semi-definite solution?Trentelman, Harry L.Published in:EPRINTS-BOOK-TITLEIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:1999Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Trentelman, H. L. (1999). When does the algebraic Riccati equation have a negative semi-definite solution?In EPRINTS-BOOK-TITLE University of Groningen, Johann Bernoulli Institute for Mathematics andComputer Science.CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019When does the algebraic Riccati equation have anegative semi-denite solution?Harry L. TrentelmanNovember 24, 19991 IntroductionIn this contribution we want to draw the readers's attention to an open problemthat concerns the existence of certain solutions to the algebraic Riccati equa-tion. Since its introduction in control theory by Kalman in the beginning ofthe sixties, the algebraic Riccati equation has known an impressive range of ap-plications, such as linear quadratic optimal control, stability theory, stochasticltering and stochastic control, stochastic realization theory, the synthesis oflinear passive networks, dierential games, and H1optimal control and robuststabilization. For an overview of the existing literature on the algebraic Riccatiequation, we refer to [3].In this note, we deal with the existence of real symmetric solutions to thealgebraic Riccati equation, in particular with the existence of negative semi-denite solutions. It is well-known (see [9]) that the existence of a real symmet-ric solution to the algebraic Riccati equation is equivalent to a given frequencydomain inequality along the imaginary axis. In [9], it was also stated that theexistence of a negative semi-denite solution is equivalent to this frequency do-main inequality holding for all complex numbers in the closed right half of thecomplex plane. Soon after the appearance of [9], a correction [10] appeared inwhich it was outlined that this statement is not correct, and that the frequencydomain inequality in the closed right half plane is a necessary, but not sucientcondition for the existence of a negative semi-denite solution.Since then, several attempts have been made to obtain a convenient neces-sary and sucient frequency domain condition for the existence of a negativesemi-denite solution. In this note we discuss some of these conditions. We alsoexplain that we consider these conditions not to be satisfactory yet, and there-fore we claim the problem of formulating a sensible frequency domain conditionto be still an open problem.Research Institute for Mathematics and Computing Science, P.O. Box 800, 9700 AVGroningen, The Netherlands, e-mail: H.L.Trentelman@math.rug.nl12 The algebraic Riccati equationLet A 2 Rnnand B 2 Rnmbe such that (A;B) is a controllable pair. Also,let Q 2 Rnnbe such that Q = QT, let S 2 Rmn, and let R 2 Rmmbe suchthat R > 0. We deal with the following algebraic Riccati equation:ATK +KA+Q  (KB + ST)R 1(BTK + S) = 0 (2.1)In order to give a frequency domain condition for the existence of a real sym-metric solution K, we dene the matrix of two-variable rational functions Win the indeterminates  and  byW (; ) := R+BT(I  AT) 1ST+ S(I  A) 1B+BT(I  AT) 1Q(I  A) 1B (2.2)which is often called the Popov function associated with (2.1). (see [7]). Thecoecients of this matrix are quotients of real polynomials in the indeterminates and . With this two-variable rational matrixW we associate a (one-variable)rational matrix @W by dening @W () := W ( ; ), i.e., obtained by taking =   and  = . It was shown in [9] that the algebraic Riccati equation (2.1)has a real symmetric solution if and only if@W (i!)  0 for all ! 2 R, i! 62 (A) (2.3)In [9] it was claimed that the two-variable rational matrix W also provides theclue to the existence of a negative semi-denite solution. Indeed, [9], Theorem4 states that (2.1) has a real symmetric solution K  0 if and only ifW (; )  0 for <e()  0,  62 (A) (2.4)Unfortunately, as was noted in [10], this statement is not correct: in generalthe condition (2.4) is only a necessary conditition, but not a sucient one. It iscertainly interesting to note that for certain important special cases (2.4) doesprovide a necessary and sucient condition. For one important special casethis result is known as the bounded real lemma. This special case in concernedwith the situation that Q =  CTC, S = 0, and R = I. In that case, after achange of variable from K to  K, the Riccati equation becomesATK +KA+CTC +KBBTK = 0 (2.5)while we have W (; ) = I  GT()G(). The condition (2.4) then becomesGT()G()  I for <e()  0,  62 (A)which indeed is well-known to be equivalent to the existence of a solutionK  0to the algebraic Riccati equation (2.5) (see, for example [2]). In [5], a numberof additional special cases in which the frequency domain inequality (2.4) is anecessary and sucient condition were established. We will not go into thesespecial cases here. Instead, we want to discuss an alternative condition on thetwo-variable rational matrix W that was proven in [4] to be equivalent to theexistence of a negative semi-denite solution to (2.1). In order to rederive thiscondition here, we make use of recent results in [12] on quadratic dierentialforms and dissipativity of linear dierential systems23 Dissipativity of linear dierential systemsAs was also noted in [9], the study of the algebraic Riccati equation can beput into the more general framework of studying quadratic storage functionsfor linear systems with quadratic supply rates. In that context, an importantrole is played by the linear matrix inequality L(K)  0, whereL(K) :=ATK +KA+Q KB + STBTK + S R(3.1)It can be shown that L(K)  0 if and only if the quadratic function V (x) := xTKx satises the inequalityddtV (x(t))  xT(t)Qx(t) + 2xT(t)STu(t) + uT(t)Ru(t) (3.2)for all t 2 R, for all x and u satisfying the dierential equation _x = Ax+Bu. Theinequality (3.2) is called the dissipation inequality for the system _x = Ax+Buwith supply rate xTQx + 2xTSTu + uTRu. If a function V (x) satises thisinequality, it is called a storage function. The dissipation inequality expressesthe property that along trajectories x and u of the system the increase ininternal storage cannot exceed the rate at which storage is supplied to thesystem. If such a function V (x) exists, we call the system dissipative withrespect to the given supply rate. It can be shown that the system _x = Ax +Bu with supply rate xTQx + 2xTSTu + uTRu is dissipative if and only if thefrequency domain condition (2.3) (so along the imaginary axis) holds. Thus,(2.3) is equivalent to the existence of a real symmetric solution K to the linearmatrix inequality L(K)  0. Moreover, if this condition holds, there existreal symmetric solutions K and K+such that any real symmetric solutionK satises K  K  K+. The function V1(x) :=  xTK+x is then thesmallest, and the function V2(x) :=  xTK x is the largest storage function.Also, K and K+are solutions of the algebraic Riccati equation (2.1). Theseconsiderations show that the existence of a negative semi-denite solution to(2.1) is equivalent to the existence of a negative semi-denite solution to thelinear matrix inequality (3.1), equivalently, to the existence of a positive semi-denite storage function of the system _x = Ax+Bu with supply rate xTQx+2xTSTu+ uTRu.The general problem of the existence of storage functions was recently put inthe framework of quadratic dierential forms for linear dierential systems [12],[8]). An important role in these references is played by two-variable polynomialmatrices, i.e., matrices whose coecients are real polynomials in two indeter-minates, say  and . A two-variable polynomial matrix  can be representedas(; ) =Xk;jk;jkjwhere the k;jare matrices with real coecients, k; j 2 N, and the sum is anite one.  is called symmetric if (; )T= (; ). Each symmetric q  q3two-variable poynomial matrix  induces a quadratic dierential form (QDF),i.e., a map Q: C1(R;Rq)! C1(R;R), dened byQ(`) :=Xk;j(dk`dtk)Tk;jdj`dtjAssociated with , we dene the (one-variable) polynomial matrix @ by@() := ( ; ). Note that this polynomial matrix is para-hermitian, i.e.,(@( ))T= @(). A QDF Qis called average non-negative if for all` 2 C1(R;Rq) of compact support we haveZ1 1Q(`)dt  0In [12] this property was shown to be equivalent with the existence of a sym-metric two-variable polynomial matrix 	 such thatddtQ	(`)  Q(`) for all ` 2 C1(R;Rq) (3.3)It was also shown that Qis average non-negative if and only if @ is non-negative along the imaginary axis, i.e., @(i!)  0 for all ! 2 R. We nowexplain how this result can be used to rederive the condition 2.3 for the ex-istence of a real symmetric solution to the algebraic Riccati equation. Con-sider the controllable system _x = Ax + Bu, more precisely, the system  =(R;Rn+m;B), with time axis R, signal space Rn+mand behaviorB := f(x; u) 2C1(R;Rn+m) j _x = Ax + Bug: Any controllable system also admits an imagerepresentation (see ([6, 11]). Consider the image representationxu=C(ddt)Bp(ddt)I` (3.4)Here, p() is the characteristic polynomial of A, i.e., p() := det(I   A), andC() is the polynomial matrix dened by C() := p()(I   A) 1, that is, theclassical adjoint of I A, appearing when one applies Cramer's rule to computethe inverse (I  A) 1. We haveB = f(x; u) 2 C1(R;Rnm) j there exists ` 2 C1(R;Rm) such that 3.4gso (3.4) indeed denes an image representation of our system . Now denethe symmetric two-variable polynomial matrix  by(; ) :=C()Bp()ITQ STS RC()Bp()I(3.5)It is immediate that if x, u and ` are related by (3.4), then for the QDF Qassociated with  we have Q(`) = xTQx+2xTSTu+uTRu: Assume now thatQis average non-negative, equivalently, there exists 	 such thatddtQ	 Q.It was proven in ([8]) that any suchQ	can be represented as a (static) quadraticfunction of any state of the underlying system. In our case, a (minimal) state isgiven by x = C(ddt)B`, so there exists a real symmetric matrix K 2 Rnnsuchthat if x and ` are related by x = C(ddt)B`, then Q	(`) = xTKx. Collectingthese facts we nd that the following three statements are equivalent:41.R1 1(xTQx + 2xTSTu + uTRu)dt  0 for all x 2 C1(R;Rn) and u 2C1(R;Rm) of compact support, satisfying _x = Ax+Bu2. there exists a real symmetric matrix K 2 Rnnsuch thatddtxTKx xTQx + 2xTSTu + uTRu for all x 2 C1(R;Rn) and u 2 C1(R;Rm)satisfying _x = Ax+Bu3. @(i!)  0 for all ! 2 RAs noted before, condition (2) is equivalent to the existence of a real symmetricsolution of the algebraic Riccati equation. Note that for  given by (3.5) wehave (; ) = p()p()W (; ) with W the two-variable rational matrix givenby (2.2). Hence, along the imaginary axis we have @(i!) = jp(i!)j2@W (i!).Therefore, condition (3) is equivalent to the frequency domain inequality (2.3).Thus we have re-established the fact that the existence of a real symmetricsolution to the algebraic Riccati equation is equivalent to the frequency domaininequality (2.3).4 The existence of negative semi-denite solutionsLet us now study what the set-up of QDF's for linear dierential systems can tellus on the existence of negative semi-denite solutions to the algebraic Riccatiequation. To start with, let  be an arbitrary q  q symmetric two-variablepolynomial matrix. The associated QDF Qis called half-line non-negative iffor all ` 2 C1(R;Rq) of compact support we haveZ0 1Q(`)dt  0It was proven in [12] that a QDF Qis half-line non-negative if and only ifthere exists a symmetric two-variable polynomial matric 	 such that Q	 0and such that (3.3) holds. Applying this fact to our system , again usingthat it has an image representation given by (3.4), we this time nd that thefollowing two statements are equivalent:1.R0 1(xTQx + 2xTSTu + uTRu)dt  0 for all x 2 C1(R;Rn) and u 2C1(R;Rm) of compact support, satisfying _x = Ax+Bu2. there exists a real symmetric matrix K 2 Rnn, K  0, such thatddt( xTKx)  xTQx + 2xTSTu + uTRu for all x 2 C1(R;Rn) andu 2 C1(R;Rm) satisfying _x = Ax+BuAs before, condition (2) is equivalent to the existence of a negative semi-denitereal symmetric solution to the algebraic Riccati equation. In the following, weestablish a condition on the two-variable polynomial matrix  that is equivalentto condition (1). It turns out that in this way we reobtain the frequency domaincondition for the existence of a negative semi-denite solution to the algebraicRiccati equation that was obtained before in [4]. First note that condition (1)5is equivalent to the condition that for all ` 2 C1(R;Rm) of compact supportwe haveZ0 1A(ddt)B`p(ddt)`TQ STS RA(ddt)B`p(ddt)`dt  0 (4.1)By an approximation argument, this condition is equivalent to the same in-equality holding for all ` 2 C1(R;Rm) such thatR0 1k`k2dt is nite. Now, letN 2 N, and let 1; 2; : : : ; Nbe N distinct complex numbers in <e() > 0.Also, let v1; v2; : : : vNbe arbitrary vectors in Cm. Consider the function`(t) :=NXi=1eitvi(4.2)Applying (4.1) to this function ` (silently moving from real valued functions tocomplex valued functions), we obtainNXi=1NXj=1Z0 1(e(i+j)tviA(i)Bp(i)ITQ STS RA(j)Bp(j)Ivj)dt  0which, after integration, yieldsNXi=1NXj=1vi(i; j)i+ jvj 0 (4.3)Since, for xed complex numbers 1; 2; : : : ; N, this holds for all v1; v2; : : : vN,this implies that the hermitian mN mN matrix whose (i; j)th block is equalto the mm matrix(i;j)i+jis positive semi-denite. If, conversely, (4.3) holdsfor any choice of N , any choice of distinct complex numbers 1; 2; : : : ; Nin<e() > 0, and any choice of vectors v1; v2; : : : vNin Cm, then clearly (4.1) holdsfor any function ` of the form (4.2). Again by an approximation argument, theinequality (4.1) must then hold for all ` 2 C1(R;Rm) of compact support. Thisproves that any of the statements (1) and (2) is equivalent with the followingcondition (3):3. for all N 2 N, and for any choice 1; 2; : : : ; Nof distinct complex num-bers in <e() > 0, the hermitian mN mN matrix(i;j)i+ji;j=1;2;:::Nis positive semi-denite.We now express condition (3) in terms of the two-variable rational matrix W .It is easily veried that(i; j)i+ ji;j=1;2;:::;N= DW (i; j)i+ ji;j=1;2;:::;NDHere D is the blockdiagonal matrix whose ith diagonal block is equal to p(i)I(with I the m  m identity matrix). Thus we claim that condition (3) isequivalent to the following condition (4):64. for all N 2 N, and for any choice 1; 2; : : : ; Nof distinct complex num-bers in <e() > 0 such that i62 (A), i = 1; 2; : : : ; N , the hermitianmN mN matrixW (i;j)i+ji;j=1;2;:::Nis positive semi-denite.Indeed, if i62 (A), i = 1; 2; : : : ; N , then D is nonsingular. From this the proofof the implication (3) ) (4) is obvious. The proof of the converse implicationthen follows by using a continuity argument.We have now shown that the existence of a negative semi-denite real sym-metric solution of the algebraic Riccati equation is equivalent to condition (4)on the two-variable rational matrix W . Thus we have re-established the condi-tion that was proposed in [4] as a correct alternative for the errorous frequencydomain condition (2.4). A similar condition was obtained in [1] in related workon spectral factorization.We remind the reader that the title of this note is: \When does the algebraicRiccati equation have a negative semi-denite solution?" We have re-derivedthe necessary and sucient frequency domain condition (4) for this to hold.Yet, in our opinion, this condition is not a satisfactory one, since not only itrequires one to check non-negativity of an innite number of hermitian matrices,but also there is no upper bound to the dimension of these matrices. In viewof this, we formulate the following open problem:Find a reasonable necessary and sucient frequency do-main condition, i.e., a condition in terms of the rationalmatrix @W , or possibly in terms of the two-variable ratio-nal matrix W , for the existence of a real symmetric nega-tive semi-denite solution of the algebraic Riccati equation(2.1).References[1] B.D.O. Anderson, \Corrections to: Algebraic properties of minimal de-gree spectral factor". Automatica, Vol. 11, pp. 321 - 322, 1975.[2] B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis- A Modern Systems Theory Approach, Prentice-Hall, Englewood Clis,N.J., 1973.[3] S. Bittanti, A.J. Laub, J.C. Willems (Eds.), The Riccati Equation,Springer Verlag, Berlin, 1991.[4] B.P. Molinari, \Conditions for non-positive solutions of the linear matrixinequality". IEEE Trans. Automat. Contr., Vol. AC-29, pp. 804 - 806,1975.[5] P.J. Moylan, \On a frequency condition in linear optimal control theory".IEEE Trans. Automat. Contr., Vol. AC-29, pp. 806, 1975.[6] J.W. Polderman and J.C. Willems, Introduction to Mathematical SystemsTheory, Springer Verlag, 1997.7[7] V.M. Popov, L'Hyperstabilite des systemes automatiques, Dunod, Paris,1973.[8] H.L. Trentelman and J.C. Willems, \Every storage function is a statefunction", Systems and Control Letters, 32, pp. 249 - 259, 1997.[9] J.C. Willems, \Least squares stationary optimal control and the algebraicRiccati equation", IEEE Trans. Automat. Contr., Vol. AC-16, pp. 621 -634, 1971.[10] J.C. Willems, \On the existence of a nonpositive solution to the Riccatiequation", IEEE Trans. Automat. Contr., Vol. AC-19, pp. 592 - 593, 1974[11] J.C. Willems, \Paradigms and puzzles in the theory of dynamical sys-tems", IEEE Trans. Automat. Contr., Vol. 36, pp. 259-294, 1991.[12] J.C. Willems and H.L. Trentelman, \On quadratic dierential forms". Toappear in SIAM J. Contr. Optim..8

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When does the algebraic Riccati equation have a negative semi-definite solution?

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