- Email: [email protected]
On the bounded real lemma
Systems & Control Letters 45 (2002) 339 – 346
www.elsevier.com/locate/sysconle
On the bounded real lemma $ Akos L$aszl$o Department of Mathematics, TTK, University of Pecs, Pecs H-7624, Ifjusag u tja 6, Hungary Received 20 June 2000; received in revised form 20 August 2001
Abstract In this paper we deal with special generalizations of the well-known bounded real lemma. We develop general statements for comparison of rational transfer functions in the sense of quadratic forms on the imaginary axis. The main results can c 2001 Elsevier Science B.V. All rights reserved. be expressed by means of Hamiltonian matrices. Keywords: Moore–Penrose pseudo inverse; Frobenius norm; Hamiltonian matrix; The imaginary axis; Contractions; Transfer functions
1. Introduction In recent years, H ∞ control problems in various guises have become the subjects of intensive research, mainly by electrical engineers. Anderson’s ?rst version of the bounded real lemma presented simple conditions under which a transfer function is contractive on the imaginary axis. So it rendered possible to determine the H ∞ norm of a transfer function, and the lemma became a signi?cant element of proofs of hundreds of papers (and some books). See, e.g. [2,3]. An H ∞ feedback controller is regarded as optimal one if the resulting closed loop transfer function is stable and has minimal H ∞ norm. Roughly speaking a closed loop transfer function T1 (s) is “better” than T2 (s) if T1 ∞ 6 T2 ∞ ; or equivalently sup T1 w2 6 sup T2 w2
w2 ¡1
w2 ¡1
holds, where the indicated norms are the well known H2 and H∞ norms, i.e. ∞ 1=2 1 ∗ w (j!)w(j!) and T ∞ := sup max [T (j!)]: w2 := 2 −∞ ! However the above conditions do not preclude the possibility of the existence of an input w for which T2 w2 ¡ T1 w2 holds, or in other words w may provide a larger output energy for T2 . $ L$aszl$o). E-mail address: [email protected] (A. c 2001 Elsevier Science B.V. All rights reserved. 0167-6911/01/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 1 ) 0 0 1 9 0 - 6
Laszlo / Systems & Control Letters 45 (2002) 339 – 346 A.
340
Here arises the question in a natural way: what is the necessary and suIcient condition of the validity of the relation T2 w2 ¿ T1 w2 for all w ∈ L2 [ − ∞; ∞), or equivalently of T2 (j!)T2∗ (j!) − T1 (j!)T1∗ (j!) ¿ 0 or
T2∗ (j!)T2 (j!) T1∗ (j!) T1 (j!)
I
for all ! ∈ R;
(1)
¿0
for all ! ∈ R:
(2)
The latter gives also an equivalent formulation of (1), since the left-hand side of (1) is exactly the Schur complement of the 22 block of (2). Note that the above sense of “majorization” provides only a partial ordering among transfer matrices of the same number of columns. We can treat the question in a more general context without making any additional eKorts. Our solution involves a third transfer function and a special factorization property.
2. Preliminaries and notations Throughout the paper, we use the Doyle convention, i.e., a transfer matrix in terms of state-space data is denoted by A B (W (s) :=) (3) : = C(Is − A)−1 B + D: C D We shall use the following simple considerations. If the matrix D in (3) is square and invertible, then we obtain a realization for W (s)−1 : W (s)−1 = D−1 − D−1 C(Is − A× )−1 BD−1 ;
(4)
where A× : = A − BD−1 C (See, e.g. [3]). The Kalman decomposition of the pair (A; B) shows us that the set of uncontrollable eigenvalues of A and A + BK coincides, where K is any feedback matrix of suitable size. Similarly, the unobservable eigenvalues of (C; A) and (C; A + FC) also coincide, and we conclude that the realization (4) has exactly the same uncontrollable or unobservable modes as (3). In this paper 0 and I means the zero and the identity matrix, respectively. If the sizes are not clear from the context then it will be indicated by index, e.g., I3 denotes the identity matrix of the size 3 × 3; while 02×3 is the zero matrix of the size 2 × 3: X ∗ stands for the complex conjugate transpose of a matrix X: Let X ∈ Cn×m have a singular value decomposition given by X = UV; where U ∈ C n×n and V ∈ C m×m are unitary matrices, ˆ 0r×(m−r) = ; 0(n−r)×r 0(n−r)×(m−r) where ˆ = diag { 1 ; 2 ; : : : ; r }; 1 ¿ 2 ¿ · · · ¿ r ¿ 0 are the singular values of X; ?nally r is the rank of X: De?ne by −1 0r×(n−r) ˆ † ∗ X := V U ∗; 0(m−r)×r 0(m−r)×(n−r)
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the Moore–Penrose pseudo inverse of X . It is easy to see that XX † and X † X are hermitian projectors, furthermore X † = lim X ∗ ( I + XX ∗ )−1 = lim ( I + X ∗ X )−1 X ∗ : →0
(5)
→0
The usual 2-norm of the complex matrix X will be denoted by X . The Frobenius norm of the matrix X can be de?ned as r √ √ ∗ ∗ X F : = trace X X = trace XX =
( k )2 : k=0
Furthermore introduce the notations C+ and C0 ; for the right half-plane and the imaginary axis, respectively, and (X ) is the number of eigenvalues of the matrix X in C+ : 3. Main results In this paper we will be concerned with transfer functions of the form Tk (s) = Ck (Is − Ak )−1 Bk + Dk ;
k = 1; 2; 3:
(6)
We need some notations to formulate our main results. Suppose that the matrices Ak ; Bk ; Ck ; Dk ; k = 1; 2; 3 in realizations (6) are of the sizes nk × nk ; nk × mk ; pk × nk ; pk × mk ; k = 1; 2; 3; respectively. Suppose that m1 = m2 and p1 = p3 . Introduce the following blockmatrices: 0 B1 0 A1 0 0 0 ; B˜ 3 = 0 ; A = 0 A2 0 ; B D = B 2 0 CD =
0
A3
0
D2∗ C2
0
C1
0
C3
0
;
B3 D3∗
C˜ 2 = [0 C2
0];
R=
B3 D2∗ D2
D1∗
D1
D3 D3∗
:
The following lemma collects some natural facts on the relation of the Moore–Penrose pseudoinverse and the ?nite dimensional version of the Douglas theorem. Lemma. Assume that R ¿ 0: (a) Then there exists a m3 × p2 contractive matrix X (i.e.; X 6 1) such that D1 = D3 XD2 : (b) Let Dk have the singular value decompositions Dk = Uk k Vk ; where Uk ∈ C matrices and ˆk 0 ; k = 0 0
(7) pk ×pk
V ∈C
ˆk are rk × rk matrices (k = 2; 3:) Then (i) F11 0 V2 D 1 = U3 0 0 −1 −1 where F11 is an r3 × r2 matrix; and Xˆ F : = ˆ3 F11 ˆ2 is a contractive matrix.
mk ×mk
are unitary
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(ii) The matrix X is a contractive solution of (7) if and only if Xˆ F Xˆ 12 ∗ U2∗ ; X = V3 Xˆ 21 Xˆ 22 where Xˆ 12 ; Xˆ 21 and Xˆ 22 are some matrices of the sizes r3 ×(p2 −r2 ); (m3 −r3 )×r2 ; (m3 −r3 )×(p2 −r2 ); respectively; such that Xˆ F Xˆ 12 Xˆ : = Xˆ 21 Xˆ 22 is still contractive. (iii) The solution XF : =
D3† D1 D2†
= V3∗
Xˆ F 0 0
0
U3∗
has the minimal Frobenius norm among all contractive solutions of (7). (c) If; in addition; we assume; that rank R = rank D2 + rank D3 ;
(8)
then XF strictly contractive; i.e.; XF ¡ 1: We omit the proof of the above lemma because (a) and (b) are well-known (see e.g. [1]) and can be obtained by using the singular value decompositions for D2 and D3 , whereas (c) is immediate because (8) implies the positive de?niteness of the restriction of R to the sum of the ranges of D2∗ D2 and of D3 D3∗ . Remark 1. It is easy to see that the strict inequality R ¿ 0 implies relation (8). Introduce the Hamiltonian matrix ∗ A − BD R−1 CD BD R−1 BD∗ − B˜ 3 B˜ 3 H= : ∗ C˜ 2 C˜ 2 − CD∗ R−1 CD −A∗ + CD∗ R−1 BD∗ Theorem. Assume that A has no eigenvalues on C0 ; and R ¿ 0: Then the following statements are equivalent: (i) ∗ T2 (s)T2 (s) T1∗ (s) ¿ 0 for all s ∈ C0 : T1 (s) T3 (s)T3∗ (s) (ii) H has no eigenvalues on C0 . (iii) The equation T1 (s) = T3 (s)Q(s)T2 (s)
for all s ∈ C;
has a rational solution Q such that Q(s) ¡ 1 for all s ∈ C0 ; furthermore rank T2 (s) ≡ m2 ; rank T3 (s) ≡ p3 for all s ∈ C0 :
(9)
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343
Proof. We have
A
∗ −B˜ 3 B˜ 3
CD
−BD∗
BD
∗ G(s) := C˜ 2 C˜ 2 −A∗
= [CD
−
BD∗ ]
CD∗ R Is − A
∗ B˜ 3 B˜ 3
−1
∗ −C˜ 2 C˜ 2 Is + A∗
BD
CD∗
+ R:
and according to the block matrix inversion lemma this gives −
G(s) = R + [CD
∗
(Is − A)−1
BD∗ ]
∗ (Is + A∗ )−1 C˜ 2 C˜ 2 (Is − A)−1 (T2 (−s)) R ∗ T2 (s) (T1 (−s)) R ∗ = : T1 (s) T3 (s)(T3 (−s)) R ∗
−(Is − A)−1 B˜ 3 B˜ 3 (Is + A∗ )−1 (Is + A∗ )−1
BD
CD∗
Thus at a point s = j! (! ∈ R) of the imaginary axis G(j!) =
T2∗ (j!)T2 (j!)
T1∗ (j!)
T1 (j!)
T3 (j!)T3∗ (j!)
;
which is obviously hermitian on C0 . Furthermore from (4) we see that
∗ BD R−1 BD∗ − B˜ 3 B˜ 3 A − BD R−1 CD ∗ ∗ −1 ∗ ∗ −1 ∗ ˜ ˜ G −1 (s) = C 2 C 2 − CD R CD −A + CD R BD −R−1 CD R−1 BD∗
BD R−1
CD∗ R−1 ; R−1
so H is the “A-matrix” of G −1 : The above realization of G −1 has no uncontrollable or unobservable modes on C0 . Indeed, the uncontrollable or unobservable modes of the realization of G −1 coincide with those of the realization of G; but
A ∗ C˜ 2 C˜ 2
∗
−B˜ 3 B˜ 3 −A∗
and
A
have the same eigenvalues on the imaginary axis, and we have assumed that A has no eigenvalues on C0 . Thus G −1 has no poles on C0 if and only if H has no eigenvalues on C0 . (i) ⇒ (ii): We have G(j!) ¿ 0 for all ! ∈ R; and hence G(j!)−1 is a continuous rational function on R: Thus G −1 has no poles on the imaginary axis. (ii) ⇒ (i): The rational matrix G −1 has no poles and zeros on C0 ; thus, all the eigenvalues of G(j!) are nonzero, continuous, real-valued functions in !: This implies that (G(j!)) is a constant function in !; and by lim!→0 G(j!) = R ¿ 0 we ?nd that (G(j!)) ≡ m.
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(i) ⇒ (iii): By the identity (T (−s)) R ∗ = (T (s))∗ for all s ∈ C0 and (5) we know that the rational functions R ∗ T2 (s))−1 (T2 (−s)) R ∗ V2 (s) := lim ( I + (T2 (−s)) →0
V3 (s) := lim ( I + (T3 (−s)) R ∗ T3 (s))−1 (T3 (−s)) R ∗ →0
give the pseudoinverses on the imaginary axis of T2 and T3 , respectively. Easy to see that the elements of the resulting matrix are of the form polynomial + rational function. Thus according to (b) (iii) of our Lemma and by the Remark 1 we ?nd that the matrix Q(s) := V2 (s)T1 (s)V3 (s); is a strictly contractive solution of (9) for each s ∈ C0 , and since the function T1 (s) − T3 (s)Q(s)T2 (s) is analytic, leaving out of consideration the poles, we can state that Q is a solution of (9) for almost all point of C. (But, in general, it is no longer contractive on the whole C.) (iii) ⇒ (i): For any ! ∈ R the inequality Q(j!) ¡ I is equivalent with Q∗ (j!) Im2 ¿ 0: M := Q(j!) Ip3 De?ne
N (j!) :=
T2 (j!)
0
0
T3 (j!)∗
:
The conditions of (iii) imply rank N ≡ m2 + p3 which gives G(j!) = (N (j!))∗ MN (j!) ¿ 0; concluding the proof. Remark 2. The above theorem remains valid if any of the triples (Ak ; Bk ; Ck ); k = 1; 2; 3; is absent; or in other words n1 = 0; n2 = 0 or n3 = 0.
4. Applications to the “majorization” inequalities Namely, for T3 (s) ≡ I , our theorem gives the necessary and suIcient conditions of (1) and (2). In this case our matrices become B1 0 A1 0 ; BD = ; B˜ 3 = 0; A= 0 A2 B2 0 CD = H=
0
D2∗ C2
C1
0
;
A − BD R−1 CD
C˜ 2 = [0 C2 ];
R=
BD R−1 BD∗
∗ C˜ 2 C˜ 2 − CD∗ R−1 CD −A∗ + CD∗ R−1 BD∗
D2∗ D2 D1∗ D1
I
:
:
(10)
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According to the block matrix inversion lemma ˆ −1 ˆ −1 D1∗ R − R ; R−1 = −1 −1 −D1 Rˆ I + D1 Rˆ D1∗ where Rˆ := D2∗ D2 − D1∗ D1 , thus the expressions in (10) can be written as B1 −1 −1 ∗ Rˆ [B1∗ B2∗ ]; BD R BD = B2 ∗ ˆ −1 B1∗ −C1∗ D1∗ Rˆ −1 B2∗ D R −C 1 1 ; CD∗ R−1 BD∗ = −1 −1 C2∗ D2 Rˆ B1∗ C2∗ D2 Rˆ B2∗ ∗ ∗ ˆ −1 D1∗ C1 −C1∗ D1 Rˆ −1 D2∗ C2 C + C D R C 1 1 1 1 ; CD∗ R−1 CD = −1 −1 −C2∗ D2 Rˆ D1∗ C1 C2∗ D2 Rˆ D2∗ C2 0 0 ∗ ˜ ˜ C2 C2 = : 0 C2∗ C2
(11)
In order to formulate the conditions of the “majorization” (1) introduce the new matrices C1 0 A1 0 ; Cˆ := ; Aˆ := 0 A2 0 C2 B1 D1 ˆ ˆ B := ; D := ; B2 D2 −Ip1 0 ∗ ˆ J := ; Rˆ := Dˆ J D: 0 Ip2 Hence in this case our Hamiltonian H is of the form −1 ∗ ˆ −1 Dˆ ∗ J Cˆ Bˆ Rˆ Bˆ Aˆ − BR ; Hm := ∗ ∗ ∗ ∗ −1 ∗ −1 ∗ Cˆ J Cˆ − Cˆ J Dˆ Rˆ Dˆ J Cˆ −Aˆ + Cˆ J Dˆ Rˆ Bˆ
(12)
and the theorem becomes Corollary. Assume that Aˆ has no eigenvalues on C0 ; and Rˆ ¿ 0. Then the following statements are equivalent: (i) T2∗ (s)T2 (s) ¿ T1∗ (s)T1 (s) for all s ∈ C0 : (See inequality (1).) (ii) Hm has no eigenvalues on C0 . (iii) The equation T1 (s) = Q(s)T2 (s) for all s ∈ C; has a rational solution Q such that Q(s) ¡ 1
for all s ∈ C0 ;
furthermore rank T2 (s) ≡ m2 ; for all s ∈ C0 :
346
Laszlo / Systems & Control Letters 45 (2002) 339 – 346 A.
Remark 3. For T2 (s) ≡ I the matrix Hm gives the well known Hamiltonian of the “original” bounded real lemma. 5. Conclusion General tools have been derived for comparison of rational transfer functions. We have no assumptions connected with controllability or observability made, but we have supposed that the “A” matrices of the realizations have no eigenvalues on the imaginary axis. Under further assumptions on the matrices our results can be formulated also with algebraic Riccati equations or dissipation inequalities. Acknowledgements I wish to thank GyUorgy Michaletzky for the helpful discussions. References [1] T. Ando, Concavity of certain maps on positive de?nite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (1979) 203–241. [2] J.C. Doyle, K. Glover, P.P. Khargonekar, F. Francis, Statespace solutions to standard H2 and H∞ control problems, IEEE Trans. Automat. Control 34 (1989) 831–847. [3] P. Lancaster, L. Rodman, Algebraic Riccati Equations, Clarendon Press, Oxford, 1995.
www.elsevier.com/locate/sysconle
On the bounded real lemma $ Akos L$aszl$o Department of Mathematics, TTK, University of Pecs, Pecs H-7624, Ifjusag u tja 6, Hungary Received 20 June 2000; received in revised form 20 August 2001
Abstract In this paper we deal with special generalizations of the well-known bounded real lemma. We develop general statements for comparison of rational transfer functions in the sense of quadratic forms on the imaginary axis. The main results can c 2001 Elsevier Science B.V. All rights reserved. be expressed by means of Hamiltonian matrices. Keywords: Moore–Penrose pseudo inverse; Frobenius norm; Hamiltonian matrix; The imaginary axis; Contractions; Transfer functions
1. Introduction In recent years, H ∞ control problems in various guises have become the subjects of intensive research, mainly by electrical engineers. Anderson’s ?rst version of the bounded real lemma presented simple conditions under which a transfer function is contractive on the imaginary axis. So it rendered possible to determine the H ∞ norm of a transfer function, and the lemma became a signi?cant element of proofs of hundreds of papers (and some books). See, e.g. [2,3]. An H ∞ feedback controller is regarded as optimal one if the resulting closed loop transfer function is stable and has minimal H ∞ norm. Roughly speaking a closed loop transfer function T1 (s) is “better” than T2 (s) if T1 ∞ 6 T2 ∞ ; or equivalently sup T1 w2 6 sup T2 w2
w2 ¡1
w2 ¡1
holds, where the indicated norms are the well known H2 and H∞ norms, i.e. ∞ 1=2 1 ∗ w (j!)w(j!) and T ∞ := sup max [T (j!)]: w2 := 2 −∞ ! However the above conditions do not preclude the possibility of the existence of an input w for which T2 w2 ¡ T1 w2 holds, or in other words w may provide a larger output energy for T2 . $ L$aszl$o). E-mail address: [email protected] (A. c 2001 Elsevier Science B.V. All rights reserved. 0167-6911/01/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 1 ) 0 0 1 9 0 - 6
Laszlo / Systems & Control Letters 45 (2002) 339 – 346 A.
340
Here arises the question in a natural way: what is the necessary and suIcient condition of the validity of the relation T2 w2 ¿ T1 w2 for all w ∈ L2 [ − ∞; ∞), or equivalently of T2 (j!)T2∗ (j!) − T1 (j!)T1∗ (j!) ¿ 0 or
T2∗ (j!)T2 (j!) T1∗ (j!) T1 (j!)
I
for all ! ∈ R;
(1)
¿0
for all ! ∈ R:
(2)
The latter gives also an equivalent formulation of (1), since the left-hand side of (1) is exactly the Schur complement of the 22 block of (2). Note that the above sense of “majorization” provides only a partial ordering among transfer matrices of the same number of columns. We can treat the question in a more general context without making any additional eKorts. Our solution involves a third transfer function and a special factorization property.
2. Preliminaries and notations Throughout the paper, we use the Doyle convention, i.e., a transfer matrix in terms of state-space data is denoted by A B (W (s) :=) (3) : = C(Is − A)−1 B + D: C D We shall use the following simple considerations. If the matrix D in (3) is square and invertible, then we obtain a realization for W (s)−1 : W (s)−1 = D−1 − D−1 C(Is − A× )−1 BD−1 ;
(4)
where A× : = A − BD−1 C (See, e.g. [3]). The Kalman decomposition of the pair (A; B) shows us that the set of uncontrollable eigenvalues of A and A + BK coincides, where K is any feedback matrix of suitable size. Similarly, the unobservable eigenvalues of (C; A) and (C; A + FC) also coincide, and we conclude that the realization (4) has exactly the same uncontrollable or unobservable modes as (3). In this paper 0 and I means the zero and the identity matrix, respectively. If the sizes are not clear from the context then it will be indicated by index, e.g., I3 denotes the identity matrix of the size 3 × 3; while 02×3 is the zero matrix of the size 2 × 3: X ∗ stands for the complex conjugate transpose of a matrix X: Let X ∈ Cn×m have a singular value decomposition given by X = UV; where U ∈ C n×n and V ∈ C m×m are unitary matrices, ˆ 0r×(m−r) = ; 0(n−r)×r 0(n−r)×(m−r) where ˆ = diag { 1 ; 2 ; : : : ; r }; 1 ¿ 2 ¿ · · · ¿ r ¿ 0 are the singular values of X; ?nally r is the rank of X: De?ne by −1 0r×(n−r) ˆ † ∗ X := V U ∗; 0(m−r)×r 0(m−r)×(n−r)
Laszlo / Systems & Control Letters 45 (2002) 339 – 346 A.
341
the Moore–Penrose pseudo inverse of X . It is easy to see that XX † and X † X are hermitian projectors, furthermore X † = lim X ∗ ( I + XX ∗ )−1 = lim ( I + X ∗ X )−1 X ∗ : →0
(5)
→0
The usual 2-norm of the complex matrix X will be denoted by X . The Frobenius norm of the matrix X can be de?ned as r √ √ ∗ ∗ X F : = trace X X = trace XX =
( k )2 : k=0
Furthermore introduce the notations C+ and C0 ; for the right half-plane and the imaginary axis, respectively, and (X ) is the number of eigenvalues of the matrix X in C+ : 3. Main results In this paper we will be concerned with transfer functions of the form Tk (s) = Ck (Is − Ak )−1 Bk + Dk ;
k = 1; 2; 3:
(6)
We need some notations to formulate our main results. Suppose that the matrices Ak ; Bk ; Ck ; Dk ; k = 1; 2; 3 in realizations (6) are of the sizes nk × nk ; nk × mk ; pk × nk ; pk × mk ; k = 1; 2; 3; respectively. Suppose that m1 = m2 and p1 = p3 . Introduce the following blockmatrices: 0 B1 0 A1 0 0 0 ; B˜ 3 = 0 ; A = 0 A2 0 ; B D = B 2 0 CD =
0
A3
0
D2∗ C2
0
C1
0
C3
0
;
B3 D3∗
C˜ 2 = [0 C2
0];
R=
B3 D2∗ D2
D1∗
D1
D3 D3∗
:
The following lemma collects some natural facts on the relation of the Moore–Penrose pseudoinverse and the ?nite dimensional version of the Douglas theorem. Lemma. Assume that R ¿ 0: (a) Then there exists a m3 × p2 contractive matrix X (i.e.; X 6 1) such that D1 = D3 XD2 : (b) Let Dk have the singular value decompositions Dk = Uk k Vk ; where Uk ∈ C matrices and ˆk 0 ; k = 0 0
(7) pk ×pk
V ∈C
ˆk are rk × rk matrices (k = 2; 3:) Then (i) F11 0 V2 D 1 = U3 0 0 −1 −1 where F11 is an r3 × r2 matrix; and Xˆ F : = ˆ3 F11 ˆ2 is a contractive matrix.
mk ×mk
are unitary
Laszlo / Systems & Control Letters 45 (2002) 339 – 346 A.
342
(ii) The matrix X is a contractive solution of (7) if and only if Xˆ F Xˆ 12 ∗ U2∗ ; X = V3 Xˆ 21 Xˆ 22 where Xˆ 12 ; Xˆ 21 and Xˆ 22 are some matrices of the sizes r3 ×(p2 −r2 ); (m3 −r3 )×r2 ; (m3 −r3 )×(p2 −r2 ); respectively; such that Xˆ F Xˆ 12 Xˆ : = Xˆ 21 Xˆ 22 is still contractive. (iii) The solution XF : =
D3† D1 D2†
= V3∗
Xˆ F 0 0
0
U3∗
has the minimal Frobenius norm among all contractive solutions of (7). (c) If; in addition; we assume; that rank R = rank D2 + rank D3 ;
(8)
then XF strictly contractive; i.e.; XF ¡ 1: We omit the proof of the above lemma because (a) and (b) are well-known (see e.g. [1]) and can be obtained by using the singular value decompositions for D2 and D3 , whereas (c) is immediate because (8) implies the positive de?niteness of the restriction of R to the sum of the ranges of D2∗ D2 and of D3 D3∗ . Remark 1. It is easy to see that the strict inequality R ¿ 0 implies relation (8). Introduce the Hamiltonian matrix ∗ A − BD R−1 CD BD R−1 BD∗ − B˜ 3 B˜ 3 H= : ∗ C˜ 2 C˜ 2 − CD∗ R−1 CD −A∗ + CD∗ R−1 BD∗ Theorem. Assume that A has no eigenvalues on C0 ; and R ¿ 0: Then the following statements are equivalent: (i) ∗ T2 (s)T2 (s) T1∗ (s) ¿ 0 for all s ∈ C0 : T1 (s) T3 (s)T3∗ (s) (ii) H has no eigenvalues on C0 . (iii) The equation T1 (s) = T3 (s)Q(s)T2 (s)
for all s ∈ C;
has a rational solution Q such that Q(s) ¡ 1 for all s ∈ C0 ; furthermore rank T2 (s) ≡ m2 ; rank T3 (s) ≡ p3 for all s ∈ C0 :
(9)
Laszlo / Systems & Control Letters 45 (2002) 339 – 346 A.
343
Proof. We have
A
∗ −B˜ 3 B˜ 3
CD
−BD∗
BD
∗ G(s) := C˜ 2 C˜ 2 −A∗
= [CD
−
BD∗ ]
CD∗ R Is − A
∗ B˜ 3 B˜ 3
−1
∗ −C˜ 2 C˜ 2 Is + A∗
BD
CD∗
+ R:
and according to the block matrix inversion lemma this gives −
G(s) = R + [CD
∗
(Is − A)−1
BD∗ ]
∗ (Is + A∗ )−1 C˜ 2 C˜ 2 (Is − A)−1 (T2 (−s)) R ∗ T2 (s) (T1 (−s)) R ∗ = : T1 (s) T3 (s)(T3 (−s)) R ∗
−(Is − A)−1 B˜ 3 B˜ 3 (Is + A∗ )−1 (Is + A∗ )−1
BD
CD∗
Thus at a point s = j! (! ∈ R) of the imaginary axis G(j!) =
T2∗ (j!)T2 (j!)
T1∗ (j!)
T1 (j!)
T3 (j!)T3∗ (j!)
;
which is obviously hermitian on C0 . Furthermore from (4) we see that
∗ BD R−1 BD∗ − B˜ 3 B˜ 3 A − BD R−1 CD ∗ ∗ −1 ∗ ∗ −1 ∗ ˜ ˜ G −1 (s) = C 2 C 2 − CD R CD −A + CD R BD −R−1 CD R−1 BD∗
BD R−1
CD∗ R−1 ; R−1
so H is the “A-matrix” of G −1 : The above realization of G −1 has no uncontrollable or unobservable modes on C0 . Indeed, the uncontrollable or unobservable modes of the realization of G −1 coincide with those of the realization of G; but
A ∗ C˜ 2 C˜ 2
∗
−B˜ 3 B˜ 3 −A∗
and
A
have the same eigenvalues on the imaginary axis, and we have assumed that A has no eigenvalues on C0 . Thus G −1 has no poles on C0 if and only if H has no eigenvalues on C0 . (i) ⇒ (ii): We have G(j!) ¿ 0 for all ! ∈ R; and hence G(j!)−1 is a continuous rational function on R: Thus G −1 has no poles on the imaginary axis. (ii) ⇒ (i): The rational matrix G −1 has no poles and zeros on C0 ; thus, all the eigenvalues of G(j!) are nonzero, continuous, real-valued functions in !: This implies that (G(j!)) is a constant function in !; and by lim!→0 G(j!) = R ¿ 0 we ?nd that (G(j!)) ≡ m.
Laszlo / Systems & Control Letters 45 (2002) 339 – 346 A.
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(i) ⇒ (iii): By the identity (T (−s)) R ∗ = (T (s))∗ for all s ∈ C0 and (5) we know that the rational functions R ∗ T2 (s))−1 (T2 (−s)) R ∗ V2 (s) := lim ( I + (T2 (−s)) →0
V3 (s) := lim ( I + (T3 (−s)) R ∗ T3 (s))−1 (T3 (−s)) R ∗ →0
give the pseudoinverses on the imaginary axis of T2 and T3 , respectively. Easy to see that the elements of the resulting matrix are of the form polynomial + rational function. Thus according to (b) (iii) of our Lemma and by the Remark 1 we ?nd that the matrix Q(s) := V2 (s)T1 (s)V3 (s); is a strictly contractive solution of (9) for each s ∈ C0 , and since the function T1 (s) − T3 (s)Q(s)T2 (s) is analytic, leaving out of consideration the poles, we can state that Q is a solution of (9) for almost all point of C. (But, in general, it is no longer contractive on the whole C.) (iii) ⇒ (i): For any ! ∈ R the inequality Q(j!) ¡ I is equivalent with Q∗ (j!) Im2 ¿ 0: M := Q(j!) Ip3 De?ne
N (j!) :=
T2 (j!)
0
0
T3 (j!)∗
:
The conditions of (iii) imply rank N ≡ m2 + p3 which gives G(j!) = (N (j!))∗ MN (j!) ¿ 0; concluding the proof. Remark 2. The above theorem remains valid if any of the triples (Ak ; Bk ; Ck ); k = 1; 2; 3; is absent; or in other words n1 = 0; n2 = 0 or n3 = 0.
4. Applications to the “majorization” inequalities Namely, for T3 (s) ≡ I , our theorem gives the necessary and suIcient conditions of (1) and (2). In this case our matrices become B1 0 A1 0 ; BD = ; B˜ 3 = 0; A= 0 A2 B2 0 CD = H=
0
D2∗ C2
C1
0
;
A − BD R−1 CD
C˜ 2 = [0 C2 ];
R=
BD R−1 BD∗
∗ C˜ 2 C˜ 2 − CD∗ R−1 CD −A∗ + CD∗ R−1 BD∗
D2∗ D2 D1∗ D1
I
:
:
(10)
Laszlo / Systems & Control Letters 45 (2002) 339 – 346 A.
345
According to the block matrix inversion lemma ˆ −1 ˆ −1 D1∗ R − R ; R−1 = −1 −1 −D1 Rˆ I + D1 Rˆ D1∗ where Rˆ := D2∗ D2 − D1∗ D1 , thus the expressions in (10) can be written as B1 −1 −1 ∗ Rˆ [B1∗ B2∗ ]; BD R BD = B2 ∗ ˆ −1 B1∗ −C1∗ D1∗ Rˆ −1 B2∗ D R −C 1 1 ; CD∗ R−1 BD∗ = −1 −1 C2∗ D2 Rˆ B1∗ C2∗ D2 Rˆ B2∗ ∗ ∗ ˆ −1 D1∗ C1 −C1∗ D1 Rˆ −1 D2∗ C2 C + C D R C 1 1 1 1 ; CD∗ R−1 CD = −1 −1 −C2∗ D2 Rˆ D1∗ C1 C2∗ D2 Rˆ D2∗ C2 0 0 ∗ ˜ ˜ C2 C2 = : 0 C2∗ C2
(11)
In order to formulate the conditions of the “majorization” (1) introduce the new matrices C1 0 A1 0 ; Cˆ := ; Aˆ := 0 A2 0 C2 B1 D1 ˆ ˆ B := ; D := ; B2 D2 −Ip1 0 ∗ ˆ J := ; Rˆ := Dˆ J D: 0 Ip2 Hence in this case our Hamiltonian H is of the form −1 ∗ ˆ −1 Dˆ ∗ J Cˆ Bˆ Rˆ Bˆ Aˆ − BR ; Hm := ∗ ∗ ∗ ∗ −1 ∗ −1 ∗ Cˆ J Cˆ − Cˆ J Dˆ Rˆ Dˆ J Cˆ −Aˆ + Cˆ J Dˆ Rˆ Bˆ
(12)
and the theorem becomes Corollary. Assume that Aˆ has no eigenvalues on C0 ; and Rˆ ¿ 0. Then the following statements are equivalent: (i) T2∗ (s)T2 (s) ¿ T1∗ (s)T1 (s) for all s ∈ C0 : (See inequality (1).) (ii) Hm has no eigenvalues on C0 . (iii) The equation T1 (s) = Q(s)T2 (s) for all s ∈ C; has a rational solution Q such that Q(s) ¡ 1
for all s ∈ C0 ;
furthermore rank T2 (s) ≡ m2 ; for all s ∈ C0 :
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Laszlo / Systems & Control Letters 45 (2002) 339 – 346 A.
Remark 3. For T2 (s) ≡ I the matrix Hm gives the well known Hamiltonian of the “original” bounded real lemma. 5. Conclusion General tools have been derived for comparison of rational transfer functions. We have no assumptions connected with controllability or observability made, but we have supposed that the “A” matrices of the realizations have no eigenvalues on the imaginary axis. Under further assumptions on the matrices our results can be formulated also with algebraic Riccati equations or dissipation inequalities. Acknowledgements I wish to thank GyUorgy Michaletzky for the helpful discussions. References [1] T. Ando, Concavity of certain maps on positive de?nite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (1979) 203–241. [2] J.C. Doyle, K. Glover, P.P. Khargonekar, F. Francis, Statespace solutions to standard H2 and H∞ control problems, IEEE Trans. Automat. Control 34 (1989) 831–847. [3] P. Lancaster, L. Rodman, Algebraic Riccati Equations, Clarendon Press, Oxford, 1995.