Ralf L M Peeters

Martine Olivi

Bernard Hanzon

Continuous-time lossless systems, boundary interpolation and pivot structures

Keywords: Lossless systems, balanced realization, interpolation theory, canonical form, pivot structure, Schur algorithm, linear fractional transformations

Balanced realizations of lossless systems can be generated from the tangential Schur algorithm using linear fractional transformations. In discrete-time, canonical forms with pivot structures are obtained by interpolation at infinity. In continuous-time case interpolation at infinity is on the stability boundary, leading to an angular derivative interpolation problem. In the scalar case, the balanced canonical form of Ober can be recovered in this way. Here we generalize to the multivariable case. It is shown that boundary interpolation can be regarded as a limit of classical interpolation with interpolation points tending to the imaginary axis. Some pivot structures can be generated, but no complete atlas is obtained. However, for input normal pairs an atlas of admissible pivot structures can be generated in a closely related way.

INTRODUCTION

Lossless systems have several applications in linear systems and control theory. In system identification [START_REF] Bruls | Linear and non-linear system identification using separable least-squares[END_REF], in H 2 -model order reduction [START_REF] Leblond | Weighted H 2approximation of transfer functions[END_REF]; [START_REF] Fulcheri | Matrix rational H 2approximation: A gradient algorithm based on Schur analysis[END_REF]), and in digital signal processing [START_REF] Vaidyanathan | The role of lossless systems in modern digital signal processing: a tutorial[END_REF]; [START_REF] Strang | Wavelets from Filter Banks[END_REF]). They are also central in the construction of parameterizations of various classes of linear systems, see [START_REF] Ober | Balanced realizations: canonical form, parametrization, model reduction[END_REF]; [START_REF] Hanzon | Overlapping blockbalanced canonical forms for various classes of linear systems[END_REF]; [START_REF] Alpay | On the differential structure of matrix-valued rational inner functions[END_REF]. In that context balanced realizations of lossless systems have been pursued. See [START_REF] Peeters | Balanced realization of lossless systems: Schur parameters, canonical forms and applications[END_REF] for a survey on this, mainly in discrete-time, where the tangential Schur algorithm and related results from interpolation theory [START_REF] Ball | Interpolation of rational matrix functions[END_REF]) are employed.

In [START_REF] Hanzon | Balanced parametrizations of stable SISO all-pass systems in discrete-time[END_REF] balanced realizations (A, B, C, D) for the discrete-time scalar case were constructed with a triangular structure in the matrix [B, A] and also in the controllability matrix. Then in [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF] the discrete-time multivariable case was presented within the framework of interpolation theory, with interpolation points away from the stability boundary. In [START_REF] Peeters | Canonical lossless state-space systems: Staircase forms and the Schur algorithm[END_REF] pivot structures in [B, A] as well as in the controllability matrix were obtained, for interpolation at infinity and with direction vectors subject to certain admissibility conditions. If the admissibility conditions are dropped, then subdiagonal pivot structures in [B, A] arise, but pivot structures in the controllability matrix may be lost, see [START_REF] Hanzon | Subdiagonal pivot structures and associated canonical forms under state isometries[END_REF]. In [START_REF] Peeters | Parametrization of matrix-valued lossless functions based on boundary interpolation[END_REF] the discrete-time multivariable case is studied with interpolation at the stability boundary. This construction can be viewed as a limit of classical interpolation.

To obtain parameterizations for lossless systems in continuous-time, the bilinear transformation can be used, but it typically destroys the pivot structures. This is particularly inconvenient for balance-and-truncate type approximation methods, and also for numerical procedures to put a system in canonical form and extract the parameters. A balanced canonical form for continuous-time scalar lossless systems was first presented in Ober (1987a,b). This canonical form has an upper triangular [B, A] and an upper triangular controllability matrix; the matrix A is a Schwarz matrix. In [START_REF] Hanzon | Lossless scalar functions: Boundary interpolation, Schur algorithm and Ober's canonical form[END_REF] it was shown how it relates to a Schur algorithm with interpolation points on the stability boundary at infinity. We now complete the picture by addressing the continuoustime multivariable case. In Section 2 we give a background on continuous-time (J-)lossless matrices and (J-)balanced realizations. In Section 3 we discuss a tangential Schur algorithm in continuous-time, with interpolation points not on the imaginary axis, and associated balanced realizations. In Section 4 the case with interpolation points on the imaginary axis is addressed. In Section 5 Ober's scalar balanced canonical form is discussed and in Section 6 its construction is generalized to the multivariable case. It is found that pivot structures can only be generated for input normal pairs, taking interpolation points at infinity and with direction vectors chosen from the standard basis vectors subject to admissibility conditions. Section 7 concludes the paper. Proofs are largely omitted because of space limitations.

CONTINUOUS-TIME LOSSLESS MATRICES AND BALANCED REALIZATIONS

We denote by Π + := {s ∈ C; Re s > 0} and Π -:= {s ∈ C; Re s < 0} the open right and left half-planes, respectively. For any matrix function R(s), we define the matrix functions R * (s) and R (s) by R * (s) := R(s) * , and R (s) := R * (-s).

(1) (Here * denotes Hermitian transposition and s denotes the complex conjugate of s.) Note that if s lies on the imaginary axis, then R (s) = R(s) * .

A p × p rational function G(s) is called (continuous-time) lossless if at all points of analyticity s ∈ C it holds that

G(s)G(s) * ≤ I p for all s ∈ Π + , (2)
G(s)G(s) * = I p for all s ∈ iR, (3)
G(s)G(s) * ≥ I p for all s ∈ Π -.

(4) The function G(s) is lossless if and only if G * (s) is lossless. If G(s) is lossless of (generalized) McMillan degree n, then its n poles (including multiplicities) are all in Π -. The inverse of G(s) exists and is given by G(s) -1 = G (s).

If (A, B, C, D) is a balanced realization of such a lossless function G(s), then it is minimal (i.e., controllable and observable) and it holds that

DD * = I p (D is p × p unitary), (5)
C = -DB * (of size p × n), (6)
A + A * = -BB * , (7) so that both Gramians are equal to I n . Conversely, if (A, B, C, D) satisfies these three conditions, then the associated transfer function

G(s) = D + C(sI n -A) -1 B is lossless of McMillan degree ≤ n.
In that case, the McMillan degree of G(s) equals n if and only if A is (continuous-time) asymptotically stable (i.e., all its eigenvalues are in Π -), in which case (A, B, C, D) is minimal and balanced. Thus, starting from a given (minimal) balanced realization, the set of all (minimal) balanced realizations of G(s) is generated by the set of linear state-space isometries, i.e., by state-space basis transformations from the unitary group U (n).

The Grasse product

Let (A, B, C, D) be a state-space realization of a transfer function G(s), then the associated block-partitioned matrix

R = D C B A (8) is called the realization matrix. For two (p + n) × (p + n) realization matrices R 1 = D 1 C 1 B 1 A 1 and R 2 = D 2 C 2 B 2 A 2 (not
necessarily minimal and not necessarily corresponding to lossless transfer functions), we define their Grasse product by:

R 1 R 2 = D 1 D 2 D 1 C 2 + C 1 B 1 D 2 + B 2 B 1 C 2 + A 1 + A 2 . ( 9
)
Note that the Grasse product can equivalently be defined in terms of conventional matrix multiplication as:

R 1 R 2 = R 1 I p 0 0 0 R 2 + R 1 0 0 0 I n + 0 0 0 I n R 2 . ( 10
)
Proposition 1. The Grasse product has the following properties:

(a) (R 1 R 2 ) R 3 = R 1 (R 2 R 3 ) (associativity). (b) The matrix E = I p 0 0 0 is the (unique) neutral element: for all R it holds that E R = R E = R. (c) (R 1 R 2 ) * = R * 2 R * 1 . (d) R = D C B A has a 'Grasse inverse' S if and only if D -1 exists. It satisfies R S = S R = E
and is given by

S = D -1 -D -1 C -BD -1 -(A -BD -1 C) . ( 11
) If R corresponds to G(s) then S corresponds to G(-s) -1 . (e) R = R 1 R 2 has a
T Θ (G)(s) = [Θ 1 (s)G(s) + Θ 2 (s)][Θ 3 (s)G(s) + Θ 4 (s)] -1 . (12) If G(s) is rational of McMillan degree n and Θ(s) is rational of McMillan degree m, then T Θ (G) is rational of McMillan degree ≤ n + m. If Θ(s) and Φ(s) are two 2p × 2p matrix functions, then the LFT T ΘΦ = T Θ • T Φ
is a composition of LFTs (on the intersection of domains where both expressions are well defined). In particular:

T -1 Θ = T Θ -1 .
When dealing with a domain of lossless functions G(s) it is useful to consider J-lossless matrices Θ(s) and their realizations, where J = I p 0 0 -I p . Here, a 2p × 2p matrix

Θ(s) is called (continuous-time) J-lossless if it holds that Θ(s)JΘ(s) * ≤ J for all s ∈ Π + , (13)
Θ(s)JΘ(s) * = J for all s ∈ iR,

Θ(s)JΘ(s) * ≥ J for all s ∈ Π -. (14)
A J-lossless matrix Θ(s) is invertible; its inverse is given by Θ(s) -1 = JΘ (s)J. If a J-lossless matrix is proper rational, then it has a (minimal) J-balanced realization (A, B, C, D), i.e., which satisfies

DJD * = J (D is constant J-unitary), (16)
C = -DJB * (of size 2p × m), (17)
A + A * = -BJB * . ( 18
)
We now have the following result.

Θ w,u,v (s) = I 2p + (w + w) (s -w) v u v u * J ( v 2 -u 2 ) . (19)
This matrix function is J-lossless of McMillan degree 1 having its pole at w. Theorem 5. Let F (s) be a p×p lossless function of McMillan degree n ≥ 1. Let w ∈ Π + be an arbitrary interpolation point. Then there exists a normalized direction vector u ∈ C p with u = 1, for which the Schur vector v := F (w)u satisfies v < 1.

For such choices of w, u, and v, there exists a unique p × p lossless function G(s) of McMillan degree n -1 for which

F = T Θw,u,v (G). ( 20
)
The tangential Schur algorithm yields, from a lossless function F (s) of McMillan degree n, a finite sequence of Schur vectors v = (v n , v n-1 , . . . , v 1 ) and a unitary p × p matrix F 0 , associated with -a sequence w = (w n , w n-1 , . . . , w 1 ) of interpolation points in Π + , -a sequence u = (u n , u n-1 , . . . , u 1 ) of normalized direction vectors with u k = 1 (k = 1, . . . , n).

It works as follows. Let F n = F , then for k = n, n-1, . . . , 1:

put v k = F k (w k )u k . • If v k < 1, let F k-1 = T -1 Θw k ,u k ,v k F k , which has degree k -1. • If v k = 1 then stop.
A local coordinate map ϕ(w, u) can be associated with the sequences w and u and with F 0 by ϕ(w, u) :

F (s) ∈ V(w, u) → (v 1 , v 2 , . . . , v k , F 0 ), ( 21
)
where the domain V(w, u) of the map consists of functions F for which v k < 1 for all k = n, n -1 . . . , 1 and F 0 ∈ U (n).

Computing balanced realizations

In [START_REF] Peeters | Linear fractional transformations and balanced realization of discrete-time stable all-pass systems[END_REF]

(A, B, C, D) = A, [B 1 B 2 ] , C 1 C 2 , D 11 D 12 D 21 D 22 . ( 23
)
Then a state-space realization ( Ã, B, C, D) of

F = T Θ (G) is obtained by D C B Ã = T ∆ (R e ), with R e = D 0 C 0 I m 0 B 0 A (24) and ∆ =        D 11 C 1 0 D 12 0 0 B 1 A 0 B 2 0 0 0 0 I n 0 0 0 D 21 C 2 0 D 22 0 0 0 0 0 0 I m 0 0 0 0 0 0 I n        ]. ( 25
)
This theorem directly applies to the situation at hand and takes a simple form which mimics what we have in discretetime (Hanzon et al., 2006, Thm. I.6.4): the Schur recursion can be computed on the level of state-space realizations by a Grasse product.

Theorem 7. A balanced realization ( Ã, B, C, D) of F = T Θw,u,v (G) is computed from a balanced realization (A, B, C, D) of G(s) by the formula D C B A = V 0 0 0 D 0 C 0 0 0 B 0 A U 0 0 0 * , ( 26
)
in which U and V are two (p + 1) × (p + 1) Grasse unitary matrices, defined in terms of w, u and v by

U =     Ip (w + w) 1 -v 2 u - (w + w) 1 -v 2 u * - 1 2 (w + w) (1 -v 2 )     (27) and V =     Ip (w + w) 1 -v 2 v - (w + w) 1 -v 2 v * - 1 2 v 2 (w + w) (1 -v 2 ) + w -w     (28)
Proof. Note that (w+w) 1-v 2 is a positive real number and let λ be its positive square root. A J-balanced realization of Θ w,u,v is given by

D = I 2p , C = λ v u , B = -λ v u * J, A = w. (29)
Applying Thm. 6 yields

D C B A =   D λ(v -Du) C λ(u * -v * D) w -λ 2 (1 -v * Du) -λv * C B -λBu A   .
(30) The same expression is obtained when working out the Grasse product (26) using definitions ( 27) and (28).

INTERPOLATION OF CONTINUOUS-TIME LOSSLESS FUNCTIONS ON THE IMAGINARY AXIS

Boundary interpolation

Lemma 8. Let F (s) be a p × p rational lossless function, σ ∈ iR a point on the imaginary axis, and u ∈ C p a given normalized direction vector. Then the scalar expression

ρ = -u * F (σ) * F (σ)u, (31) is nonnegative real: ρ ≥ 0.
A well-posed boundary interpolation problem not only involves a Schur vector v, but it also includes the value ρ of the so-called angular derivative. This is due to the fact that v = 1, because F (σ) is unitary on the boundary iR, causing one real degree of freedom to occur (now captured by ρ). It can be stated as follows: Problem 9. (GADI) Characterize all the rational lossless functions F (s) of McMillan degree n ≥ 1 satisfying the interpolation conditions

F (σ)u = v, v * F (σ)u = -ρ (32
) where σ ∈ iR, u and v are unit vectors, and ρ > 0 is the angular derivative.

Any solution of this generalized angular derivative interpolation problem can be represented as a linear fractional transformation of a lossless function G(s) of McMillan degree n -1 such that G(σ)u = v, see [START_REF] Ball | Interpolation of rational matrix functions[END_REF].

Description of the solutions

The J-lossless matrix function associated with the LFT involved in the solution of the GADI Problem is, up to a constant J-unitary right multiplier, given by

Φ σ,u,v,ρ (s) = I 2p - 1 ρ(s -σ) v u v u * J. ( 33
D C B A = V 0 0 0 D 0 C 0 0 0 B 0 A U 0 0 0 * , (35)
in which U and V are the Grasse unitary matrices defined in terms of w, u and v by

U =     I p 1 √ ρ u - 1 √ ρ u * - 1 2ρ     , V =     I p 1 √ ρ v - 1 √ ρ v * σ - 1 2ρ     . ( 36
)
Note that the recursion step ( 35) can be rewritten as

D C B A =       D v -Du √ ρ C u * -v * D √ ρ σ - 1 -v * Du ρ - v * C √ ρ B - Bu √ ρ A       , (37)
and that boundary interpolation can be viewed as a limit of classical interpolation by observing that lim w→σ 1 -F (w)u 2 w + w = ρ.

(38)

THE SCALAR CASE: BOUNDARY INTERPOLATION AND OBER'S CANONICAL FORM

Ober's canonical form

In the scalar case, a real continuous-time lossless scalar function G(s) of McMillan degree n has a unique realization (A, b, c, d) with A in Schwarz form, see Ober (1987a); see also [START_REF] Ober | Balanced realizations: canonical form, parametrization, model reduction[END_REF][START_REF] Ober | Balanced parametrization of classes of linear systems[END_REF]. The realization matrix R is then given by

R =         - β 0 • • • 0 β - β 2 2 α 1 0 -α 1 0 . . . . . . 0 α n-1 0 -α n-1 0         (39)
with = ±1, and β, α 1 , α 2 , . . . , α n-1 all positive real numbers. Let A n-k be defined as the tridiagonal matrix

A n-k =      0 α k+1 0 -α k+1 0 . . . . . . . . . 0 α n-1 0 -α n-1 0      , (40)
of which the characteristic polynomial is denoted by ∆ n-k (s) = det(sI n-k -A n-k ). Then ∆ j is a monic polynomial, which is even (odd) for j even (odd), and it satisfies the 2nd order recursion ∆

n-k (s) = s∆ n-k-1 (s) + α 2 k+1 ∆ n-k-2 (s). (41) It then holds that G(s) equals G(s) = - ∆ n (s) -β 2 2 ∆ n-1 (s) ∆ n (s) + β 2 2 ∆ n-1 (s) . (42)
G(s) satisfies interpolation conditions at ∞ on the stability boundary:

G(∞) = -, lim s→∞ s 2 G (s) = β 2 . ( 43
)
It can be generated from the Schur algorithm for boundary interpolation at infinity, as explained below; see also [START_REF] Hanzon | Lossless scalar functions: Boundary interpolation, Schur algorithm and Ober's canonical form[END_REF].

Boundary interpolation on the imaginary axis and a Schur algorithm

The corresponding well-posed angular derivative interpolation (ADI) problem, see [START_REF] Ball | Interpolation of rational matrix functions[END_REF], describes the scalar version of the GADI Problem 9. For a scalar lossless function f (s), the conditions (32) reduce to: A Schur algorithm for real scalar lossless functions with the interpolation points all located at the origin (σ = 0) is now given by the recursion for j = n, n -1, . . . , 1:

f (σ) = ξ, f (σ) = -ξρ (
f j (0) = ξ, f j (0) = -ξρ j , (47)
f j-1 = T -1 Θ σ j ,ρ j ,ξ (f j ). ( 48
)
It takes f = f n to the set of parameters (ξ, ρ 1 , ρ 2 , . . . , ρ n ) with ξ = f (0) = ±1.

Connection with Ober's canonical form

To move the interpolation points from σ = 0 to ∞, let F j (s) = f j (1/s) (which is also lossless) and compute F (s) = F n (s) using the reversed Schur algorithm from F 0 = ξ = 1. Let a j = ρ j /2. Separating the odd and even parts of the numerators and the denominators, we get F j (s) = Pj -aj Pj-1 Pj +aj Pj-1 , where P j is a monic polynomial, which is even (odd) for j even (odd), satisfying the 2nd order recursion P j+1 (s) = sP j (s) + a j a j+1 P j-1 (s).

(49)

  Note that every lossless G(s) of McMillan degree n admits such a balanced realization (A, B, C, D). Then ( A, B, C, D) is another balanced realization of G(s) if and only if D = D and there exists a (unique) n × n unitary matrix Q for which A = QAQ * , B = QB and C = CQ * .

  Theorem 3. If G(s) is p × p lossless of McMillan degree n and the 2p × 2p matrix Θ(s) is J-lossless of McMillan degree m, then F (s) = T Θ(s) (G(s)) is p × p lossless of McMillan degree ≤ n + m.3. PARAMETRIZATION OF LOSSLESS MATRICES USING THE TANGENTIAL SCHUR ALGORITHMThe following theorem describes one iteration step in the continuous-time version of the tangential Schur algorithm, where the purpose is to characterize a lossless function F (s) of McMillan degree n ≥ 1 satisfying a given interpolation condition, in terms of an LFT on a lossless function G(s) of McMillan degree n -1. Definition 4. Let w ∈ Π + be an arbitrary interpolation point, u ∈ C p with u = 1 a normalized direction vector, and v ∈ C p with v < 1 a Schur vector. Then the 2p × 2p matrix function Θ w,u,v (s) is defined as

)

  Then the following result holds. Theorem 10. (a) Let σ ∈ iR, u ∈ C p with u = 1, v ∈ C p with v = 1, and ρ ∈ R + . Let Φ σ,u,v,ρ be the elementary J-lossless factor given by (33). Let G(s) be a p × p rational lossless function of McMillan degree n such that G(σ)u = v. Let F (s) be defined byF = T Φσ,u,v,ρ (G). Then F (s) is a p × p rational lossless function of McMillan degree n + 1 such that F (σ)u = v, v * F (σ)u = -ρ. (34) (b) Let F (s) be a p × p rational lossless function of McMillan degree n + 1. Let σ ∈ iR, and let u ∈ C p with u = 1 be such that ρ := -u * F (σ) * F (σ)u is strictly positive. Let v := F (σ)u (which satisfies v = 1), and let Φ σ,u,v,ρ be the elementary J-lossless factor (33). Let G(s) be defined by G = T -1 Φσ,u,v,ρ (F ). Then G(s) is a p × p rational lossless function of McMillan degree n such that G(σ)u = v. 4.3 Balanced state-space realizations Applying the approach developed in Section 3.1 we have the following result. Theorem 11. A balanced realization ( Ã, B, C, D) of F = T Φσ,u,v,ρ (G) is computed from a balanced realization (A, B, C, D) of G(s) by the formula

  44) with σ ∈ iR and ξ = v/u satisfying |ξ| = 1. The angular derivative ρ satisfies ρ > 0 if the McMillan degree of f (s) is ≥ 1. All the solutions are then given by f = T Θ σ,ρ,ξ (g) (45) where Θ σ,ρ,ξ (s) is the J-lossless matrix function Θ σ,ρ,ξ (s) = Jg(s) ranges over the lossless functions of McMillan degree deg(g) = deg(f ) -1 which satisfy g(σ) = -ξ. (Compared to the expression (33) we have included the right multiplier J, which accounts for the changed sign in the condition g(σ) = -ξ.)

  Grasse inverse S if and only if R 1 has a Grasse inverse S 1 and R 2 has a Grasse inverse S 2 ; it satisfies S = S 2 S 1 .

A (p + n) × (p + n) realization matrix R will be called Grasse unitary if it holds that R R * = E. Note that the set of Grasse unitary matrices forms a group: the Grasse product of two Grasse unitary matrices yields a Grasse unitary matrix. In connection with realization matrices corresponding to (balanced) realizations of (continuoustime) lossless functions, the following properties hold. Proposition 2. (a) If R corresponds to a (minimal) balanced realization of a lossless transfer function, then R is Grasse unitary. (b) Conversely, if R is Grasse unitary then R corresponds to a lossless transfer function. The associated realization (A, B, C, D) is balanced (hence minimal) if and only if in addition A is asymptotically stable.

2.2 LFTs and J-lossless matrices

Let Θ(s) = Θ 1 (s) Θ 2 (s) Θ 3 (s) Θ 4 (s)

be a 2p × 2p block-partitioned matrix function with blocks of size p × p. Then the linear fractional transformation (LFT) T Θ associated with Θ(s) is defined to take any p × p matrix G(s) for which Θ 3 (s)G(s) + Θ 4 (s) is invertible to the p × p matrix given by

Comparing with Eqn. (41) it follows that

For j = 1, . . . , n a balanced realization of F j (s) is obtained by the following recursion from Thm. 11:

which recovers (up to a change of signs in A n ) the balanced canonical form of Ober, with = -1.

THE MULTIVARIABLE CASE: INTERPOLATION AT INFINITY AND PIVOT STRUCTURES

In the multivariable case we can also map s to 1/s, to move the interpolation condition from σ = 0 to ∞. Theorem 12. Let F (s) be a p × p rational lossless function

Then

Now, the J-lossless function Ψ u,v,ρ (s) fails to be proper, but Ψ u,v,ρ (1/s) = Φ 0,u,v,ρ (s) is proper. Thm. 8 of [START_REF] Peeters | Linear fractional transformations and balanced realization of discrete-time stable all-pass systems[END_REF] also allows one to deal with this situation and to compute a balanced realization ( Ã, B, C, D) of F (s) from a balanced realization (A, B, C, D) of G(s) using a J-balanced realization of Φ 0,u,v,ρ (s):

Proof. By a direct application of (Peeters et al., 2001, Thm. 8) we get that the realization matrix of T Θ (G) is given by

If u is chosen to be a standard basis vector e k , then the kth column of B is a pivot-1 column. However, if [B, A] has a pivot structure, then [ B, A] will only have a pivot structure for every choice of v if Bu is a pivot-1 column too. This confirms that pivot structures are indeed obtained in the scalar case (when k = 1 is the only option). But in the multivariable case a pivot structure is only generated in this way when the same choice u = e k is made in every iteration of the (reversed) tangential Schur algorithm. (Then only column k of B (and B) will be a pivot column, while A (and A) is upper-Hessenberg.) This does not admit the generation of an atlas of pivot structures.

To construct an atlas of pivot structures for input-normal pairs (A, B), the following modifications to the above procedure can be made. ( 1 Theorem 14. Let H be the constant J-unitary matrix

with h such that h * u = 0, µ = 1 2 h * h + γ, and γ ∈ iR. Then the realization matrix of a balanced realization

If in addition [B, A] has an admissible pivot structure (i.e., with A having a staircase structure) and u = e k is an admissible choice for the direction vector, then [ B, A] again has an admissible pivot structure.

The vectors h and the scalars γ of the iteration steps together serve to parameterize all possible admissible pivot structures for [B, A], and to generate them by applying a sequence of LFTs. But to revert the procedure it is not possible to apply a sequence of LFTs, because the associated matrices happen to be singular. Instead, one uses the explicit formula given in the theorem.

Note that an admissible pivot structure for [B, A] implies the controllability matrix to have a pivot structure too. If one drops the admissibility condition, then pivot structures in [B, A] in general are destroyed -which is different from the discrete-time case where subdiagonal pivot structures are obtained.

CONCLUSIONS

We have presented a tangential Schur algorithm with boundary interpolation and generalized angular derivative conditions for multivariable lossless systems in continuoustime. It is established that it can be regarded as a limit of classical interpolation with interpolation points tending to the imaginary axis. Balanced realizations in terms of the parameters accompany the algorithm. The procedure underlying Ober's canonical form (with interpolation at infinity and triangular pivot structures) is generalized to the multivariable case. It is found that no complete atlas of pivot structures is obtained. However, for input normal pairs an atlas of admissible pivot structures can still be generated with LFTs in a closely related way.