Ralf Peeters
email: ralf.peeters@maastrichtuniversity.nl

Martine Olivi

Bernard Hanzon

Balanced realization of lossless systems: Schur parameters, canonical forms and applications

Keywords: Lossless systems, balanced realization, interpolation theory, Schur parameters, canonical forms

Lossless systems have many applications in systems and control theory, including signal processing, filter design, system identification, system approximation, and the parameterization of classes of linear systems. In this survey paper we address the issue of parameterization of the space of rational lossless matrix functions by successfully combining two approaches. The first approach proceeds in state-space from balanced realizations and triangular pivot structures of reachability matrices. The second approach concerns interpolation theory with linear fractional transformations and the tangential Schur algorithm. We construct balanced realizations (and canonical forms) in terms of the Schur parameters encountered in the tangential Schur algorithm, and conversely, we interpret balanced realizations in discrete-time and in continuous-time in terms of Schur parameters. A number of application areas are discussed to illustrate the importance of this theory for a variety of topics.

INTRODUCTION

Lossless systems of finite order have applications in various areas of linear systems and control theory. In the area of system identification, approximation and model reduction, they have for instance been used in connection with the Douglas-Shapiro-Shields factorization, see e.g., [START_REF] Douglas | Cyclic vectors and invariant subspaces for the backward shift operator[END_REF]; [START_REF] Baratchart | On the topological structure of inner functions and its use in identification[END_REF]; [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], to perform H 2 -model order reduction of stable linear systems. They feature in the separable least squares method for system identification, see [START_REF] Bruls | Linear and non-linear system identification using separable least-squares[END_REF], where the identification problem is reduced to an optimization problem over the set of input normal pairs. In the design of orthogonal wavelets (and multiwavelets) from filter banks, see [START_REF] Strang | Wavelets from Filter Banks[END_REF], they play a central role as the lossless polyphase matrices, used to filter the phases of an input signal to produce a decompostion into approximation and detail coefficients. See [START_REF] Vaidyanathan | The role of lossless systems in modern digital signal processing: a tutorial[END_REF] for a tutorial on applications of lossless systems in digital signal processing.

Lossless systems also play an important role in the construction of parameterizations of various classes of linear systems, with special properties of these classes, such as stability, built in. See, e.g., [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]. In this context, balanced realizations of lossless systems have been pursued. Balanced realizations are well known to have numerical advantages and to be useful for model reduction, especially in conjunction with balance-and-truncate type procedures.

A balanced canonical form for SISO lossless systems in continuous-time was presented in Ober (1987a); see also [START_REF] Ober | Balanced realizations: canonical form, parametrization, model reduction[END_REF]. In the constructions of Hanzon andOber (1997, 1998) for other classes of systems, the case of continuous-time lossless systems also plays a central role. In the SISO case, the resulting canonical form for lossless systems is balanced with a positive upper triangular reachability matrix. In the MIMO case, Kronecker indices and nice selections are used to arrive at balanced overlapping canonical forms.

For discrete-time lossless systems, canonical forms can be obtained indirectly by transfering results from the continuous-time domain to the discrete-time domain by application of a bilinear transformation. However, this approach destroys certain nice properties of the continuoustime canonical form: truncation of state components no longer leads to reduced order systems that are balanced and in canonical form, and upper triangularity of the reachability matrix is lost. In [START_REF] Hanzon | Balanced parametrizations of stable SISO all-pass systems in discrete-time[END_REF], similar ideas are applied to the SISO discrete-time lossless case directly. This produces a balanced canonical form with the desired properties, which can in fact be parametrized using Schur parameters.

The class of lossless transfer functions of finite order is bijectively related to the class of rational inner functions, Preprints of the 15th IFAC Symposium on System Identification Saint- Malo, France, July 6-8, 2009 of which the differential structure is studied in [START_REF] Alpay | On the differential structure of matrix-valued rational inner functions[END_REF]. There, a parametrization for the MIMO case is obtained by means of a recursive procedure, the tangential Schur algorithm, which involves Schur parameter vectors, interpolation points and normalized direction vectors.

In [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF] this interpolation theoretic approach was successfully combined with the balanced statespace approach, to deal with the MIMO discrete-time lossless case directly. In that paper, the atlases developed in [START_REF] Alpay | On the differential structure of matrix-valued rational inner functions[END_REF] are supplied with balanced statespace realizations in terms of the parameters used in the tangential Schur algorithm. It involves linear fractional transformations which replace the unitary matrix multiplications introduced in [START_REF] Hanzon | Balanced parametrizations of stable SISO all-pass systems in discrete-time[END_REF].

In [START_REF] Peeters | Canonical lossless state-space systems: Staircase forms and the Schur algorithm[END_REF] it was investigated which choices for the parameters in the tangential Schur algorithm enable one to deal with specified Kronecker structures, thereby generalizing positive upper triangularity of the reachability matrices to canonical forms with pivot structures.

The interpretation of continuous-time balanced realizations of lossless functions exhibiting an upper triangular reachbility matrix in terms of interpolation theory and Schur parameters, turns out to involve interpolation conditions at the stability boundary, more specifically at infinity. In [START_REF] Hanzon | Lossless scalar functions: Boundary interpolation, Schur algorithm and Ober's canonical form[END_REF] the original continuous-time canonical form of Ober (1987a) was analyzed in detail in this way, using the results of [START_REF] Ball | Interpolation of rational matrix functions[END_REF] on boundary interpolation problems. A generalization of this approach to the MIMO case completes the picture and is currently under preparation.

In this paper we present a survey of the above results in a unified way. In Section 2 we present a background on realization theory and balanced realizations of stable systems. Section 3 deals with lossless function as well as with J-inner functions, which also play an important role in the interpolation theoretic approach to the parameterization of lossless systems. In Section 4 the tangential Schur algorithm is developed, and the various results connecting balanced realizations and Schur parameters are presented. Section 5 provides details on application areas and Section 6 concludes the paper.

BALANCED STATE-SPACE REALIZATIONS OF LINEAR SYSTEMS

Realization theory and Lyapunov balancing in discrete time

Consider a linear time-invariant state-space system in discrete time, with m inputs and p outputs:

x t+1 = Ax t + Bu t , (1)
y t = Cx t + Du t . (2)
Here we have: t ∈ Z (discrete time); x t ∈ C n (the state at time t) for some nonnegative integer n (the state space dimension); u t ∈ C m (the input at time t) and y t ∈ C p (the output at time t). The sizes of the complex-valued matrices A, B, C and D are n × n, n × m, p × n and p × m, respectively. The p × m rational transfer matrix of such a system is given by G

(z) = D + C(zI n -A) -1 B.
G(z) = D + C(zI n -A) -1 B
(3) for some quadruple of matrices (A, B, C, D), then called a state-space realization of G(z). To such a realization we associate the block-partitioned realization matrix R:

R = D C B A . (4) A realization (A, B, C, D) of G(z)
T such that (A ′ , B ′ , C ′ , D ′ ) = (T AT -1 , T B, CT -1 , D); equivalently: D ′ C ′ B ′ A ′ = I p 0 0 T D C B A I m 0 0 T -1 . (5)
As is well-known, an input pair (A, B) is reachable if the associated n × nm reachability matrix K = B AB . . . A n-1 B has full row rank n. An output pair (C, A) is observable if the pn × n observability matrix

O =     C CA . . . CA n-1     has full column rank n. A state-space realization (A, B, C, D) is minimal if and only if (A, B
) is reachable and (C, A) is observable; cf., e.g., [START_REF] Kailath | Linear Systems[END_REF].

In this case, the poles of G(z) are the eigenvalues of A, and its McMillan degree is equal to the sum of the degrees (multiplicities) of all the poles of G(z) (see (Kailath, 1980, Sect. 6.5)). This definition for the McMillan degree generalizes to the larger class of proper and non-proper rational functions by including a possible pole at infinity with its associated degree.

Let (A, B, C, D) be some realization of a transfer function.

If the eigenvalues of A all belong to the open unit disk, then the matrix A is called (discrete-time) asymptotically stable, and (A, B, C, D) an asymptotically stable realization. If (A, B, C, D) is an asymptotically stable realization, then the controllability Gramian W c and the observability Gramian W o are well defined as the exponentially convergent series

W c = ∞ k=0 A k BB * (A * ) k , (6)
W o = ∞ k=0 (A * ) k C * CA k . ( 7
)
These Gramians are the unique (and positive semidefinite) solutions of the respective Lyapunov-Stein equations This concept of (Lyapunov) balanced realizations was first introduced in [START_REF] Moore | Principal component analysis in linear systems: Controllability, observability, and model reduction[END_REF] in the continuous-time case and used for model reduction. In [START_REF] Pernebo | Model reduction via balanced state space representations[END_REF] the same was done for the discrete-time case. The concept of balancing has since been extended to several other classes of linear systems as well as to nonlinear systems. Balanced realizations are now a well-established tool which often exhibit good numerical properties.

W c -AW c A * = BB * , (8)
W o -A * W o A = C * C. (9)
Any realization (A, B, C, D) which is minimal and asymptotically stable, is known to be similar to a balanced realization. Such a balanced realization is in general not unique. Similarity transformation matrices T which do not affect balancing are permutation matrices (allowing one to order the squared Hankel singular values on the main diagonal of W c = W o ) and diagonal scaling matrices with unimodular entries along the main diagonal (including sign matrices). If some singular value has a multiplicity larger than one, other transformation matrices exist which do not affect balancing too.

The continuous-time case and the bilinear transform

A continuous-time state-space system (A, B, C, D) refers to the set of equations

ẋt = Ax t + Bu t , (10)
y t = Cx t + Du t , (11)
with t ∈ R. Here, the transfer matrix is similarly given by G(s) = D+C(sI n -A) -1 B and the concepts of minimality, reachability, observability and input-output equivalence are introduced entirely analogous to the discrete-time case.

A continuous-time system (A, B, C, D) and the matrix A are called asymptotically stable if the eigenvalues of A are all located in the open left half of the complex plane. If (A, B, C, D) is an asymptotically stable realization, then the controllability Gramian W c and the observability Gramian W o are well defined as the exponentially convergent integrals

W c = ∞ -∞ e At BB * e A * t dt, (12)
W o = ∞ -∞ e A * t C * Ce At dt. ( 13
)
These Gramians are the unique (and positive semidefinite) solutions of the Lyapunov equations A well-known way to transfer results from discrete-time to continuous-time and vice versa, is by means of the bilinear transform m : C → C, given by:

AW c + W c A * = -BB * , (14)
A * W o + W o A = -C * C. (
m : z → s = z -1 z + 1 , (16)
of which the inverse is given by m -1 : s → z = 1+s 1-s . If (A, B, C, D) is a discrete-time stable realization of a transfer matrix G(z), then the transfer matrix G(s) = G( z-1 z+1 ) has a continuous-time stable state-space realization ( A, B, C, D) given by

D C B A = D -C(A + I n ) -1 B √ 2C(A + I n ) -1 √ 2(A + I n ) -1 B (A -I n )(A + I n ) -1 (17) Conversely it holds that: D C B A = D + C(I n -A) -1 B √ 2 C(I n -A) -1 √ 2(I n -A) -1 B (I n + A)(I n -A) -1 (

Model order reduction: balance-and-truncate

A popular and relatively easy technique for model order reduction within the class of stable linear systems, is offered by the balance-and-truncate approach. First a balanced realization (A, B, C, D) is computed for which the Gramians are monotonically decreasing along their main diagonal. Then the original model order n is reduced to a desired lower order k < n by mere truncation of the state vector, retaining only its first k entries. The matrices A, B and C are likewise replaced by their corresponding left upper submatrices A 11 , B 1 and C 1 , of sizes k×k, k×m and p × k, respectively. The popularity of this method is largely due to the computational simplicity of the method and the possibility to choose a reduced model order by monitoring and thresholding the values on the main diagonal of the Gramians, in combination with the experience that the approximations are often quite good in practice. It is shown in [START_REF] Glover | All optimal Hankel-norm approximations of linear multivariable systems and their L ∞ -error bounds[END_REF] that an L ∞ -error bound is available for the approximations. A drawback of the approach is the lack of an explicit optimization criterion which is minimized by this procedure. When a preferred approximation criterion is available, approximations of the balance-and-truncate type can also be used as a starting point for a numerical local search technique.

Both in discrete-time and in continuous-time, the balanceand-truncate procedure is easily seen to yield a stable approximation of order ≤ k. It may happen, though, due to loss of reachability or observability that the reduced model order is actually less than k. For certain subclasses of systems, however, it can be guaranteed that the reduced model order does in fact equal k by putting additional constraints on the parameterized form of the realization (A, B, C, D). We shall encounter such a situation for some parameterizations of the class of lossless systems, discussed below. Finally, note that model order reduction by the balance-and-truncate procedure in general does not commute with the bilinear transformation between discrete-time and continuous-time.

LOSSLESS SYSTEMS AND J-INNER FUNCTIONS

Discrete-time lossless, inner, J-lossless and J-inner functions

When working in discrete-time, for any rational matrix function R(z) we shall define the associated matrix functions R * (z) and R ♯ (z) by

R * (z) := R(z) * ,
and R ♯ (z) := R * (z -1 ). ( 19) Discrete-time lossless matrix functions are introduced according to the following definition. Definition 3.1. In discrete-time, a square p × p rational matrix function G(z) is called lossless (or stable all-pass) if at all points of analyticity it holds:

G(z) * G(z) ≥ I p , for |z| < 1, (20)
G(z) * G(z) = I p , for |z| = 1, (21)
G(z) * G(z) ≤ I p , for |z| > 1. ( 22
)
Of these three conditions, each pair actually implies the remaining third condition and the inequalities are all to be understood in the sense of the partial ordering of positive semi-definite Hermitian matrices. Note that the Bode amplitude (gain) diagram for a SISO lossless system is constant 1, for all z = e iω , from which this class of systems derives its name.

The poles of a rational lossless function are all located inside the open unit disk. A lossless function is invertible: Lossless functions have many applications in systems and control theory, including system identification and digital signal processing. See [START_REF] Vaidyanathan | The role of lossless systems in modern digital signal processing: a tutorial[END_REF] for a tutorial and many further properties of lossless systems. See also [START_REF] Genin | On Σ-lossless transfer functions and related questions[END_REF].

it holds that G(z) -1 = G ♯ (z). When G(z) is lossless, also G * (z)
When dealing with lossless (matrix) functions, we shall make use of interpolation theory. To this end it shall be convenient to generalize the concept of lossless functions in the following way, see also [START_REF] Potapov | The multiplicative structure of Jcontractive matrix functions[END_REF]; [START_REF] Dym | J-contractive matrix functions, reproducing kernel spaces and interpolation[END_REF]; [START_REF] Genin | On Σ-lossless transfer functions and related questions[END_REF]. Let Σ be a k × k signature matrix, i.e., a diagonal matrix having only the entries ±1 along its main diagonal. Definition 3.2. Let Σ be a k × k signature matrix. Then a k × k rational matrix function Θ(z) is called discrete-time Σ-lossless if at all points of analyticity it holds:

Θ(z) * ΣΘ(z) ≥ Σ, for |z| < 1, (23)
Θ(z) * ΣΘ(z) = Σ, for |z| = 1, (24)
Θ(z) * ΣΘ(z) ≤ Σ, for |z| > 1. ( 25
)
As before, this definition contains redundancy in the sense that each one of these three defining properties is implied by the other two.

Any rational matrix function Θ(z) which satisfies the property Θ(z) * ΣΘ(z) = Σ for |z| = 1 is called Σ-unitary.

For such functions the inverse is given by Θ(z

) -1 = ΣΘ ♯ (z)Σ. A function is called Σ-inner if it is (-Σ)-lossless. If Θ(z) is Σ-lossless, then Θ * (z) is also Σ-lossless whereas both Θ ♯ (z)
and Θ(z) -1 are Σ-inner. The class of Σ-lossless functions is closed under multiplication.

To be consistent with the earlier definition of lossless functions, note that an I k -lossless function is just called lossless. Likewise, an I k -inner function is simply called inner. Throughout this paper we will be much concerned with lossless and inner functions of size p × p, and with J-unitary and J-inner functions of size 2p × 2p, where J denotes the following signature matrix:

J = I p 0 0 -I p . ( 26
)
Note that J-inner and J-lossless functions in general may have poles everywhere in the complex plane, but inner functions are analytic inside the unit disk and lossless functions are analytic outside the unit disk (including at the point at infinity). J-Lossless functions play a central role in the theory of chain scattering systems, see [START_REF] Kimura | Chain-Scattering Approach to H ∞ -Control[END_REF].

Σ-Balanced realizations of Σ-lossless matrix functions in discrete time

In state-space, balanced realizations of discrete-time lossless systems are surprisingly easy to characterize. It is wellknown (cf., e.g., [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF] and the references given there) that R = D C B A is a balanced realization matrix of a lossless system if and only if R is a unitary matrix and A is asymptotically stable. It then holds that the Gramians are given by W c = W o = I n . If R is unitary and A has one or more eigenvalues on the unit circle, then the associated system is still lossless, but minimality does not hold.

Note that all balanced realizations of a lossless system are obtained from a given one, by applying unitary state-space transformations. The fact that all the Hankel singular values are identical to 1 makes that there is maximal freedom in choosing a balanced realization. This freedom can be used to build additional nice properties into the realization.

The following result applies to the more general class of discrete-time Σ-lossless systems. A proper Σ-lossless transfer function G(z) of McMillan degree n always admits a minimal Σ-balanced state-space realization (A, B, C, D), i.e., one for which the realization matrix satisfies 3.3 J-unitary and J-inner functions Some particular J-inner functions that are intensively used in interpolation theory are the constant J-unitary matrices. We have the following result, for which details and proofs can be found in [START_REF] Dym | J-contractive matrix functions, reproducing kernel spaces and interpolation[END_REF]. Theorem 1. Every constant J-unitary matrix can be represented in a unique way as follows:

D C B A Σ 0 0 I n D * B * C * A * = Σ 0 0 I n . ( 27
M = H(E) P 0 0 Q , ( 28
)
where P and Q are p × p unitary matrices and H(E) denotes the Halmos extension of a strictly contractive p×p matrix E (i.e., such that I -E * E > 0). This Halmos extension H(E) is defined by

H(E) = (I -EE * ) -1/2 E(I -E * E) -1/2 E * (I -EE * ) -1/2 (I -E * E) -1/2 . ( 29
)
It holds that H(E) is Hermitian, J-unitary and invertible with inverse

H(E) -1 = H(-E).
The important class of elementary J-inner factors (i.e., Jinner functions of McMillan degree one) is characterized by the following theorem. Theorem 2. Let Θ(z) be an elementary J-inner factor with a pole at z = 1/w. Then, apart from a constant J-unitary multiplier on the right, Θ(z) must be in one of the following three forms:

(

1) If w ∈ C, |w| = 1, Θ(z) = I 2p + z -w 1 -wz -1 xx * J x * Jx , (30)
for some vector

x ∈ C 2p such that x * Jx 1-|w| 2 > 0. (2) If |w| = 1, Θ(z) = I 2p + δ (z + w) (z -w) xx * J, (31)
for some nonzero x ∈ C 2p such that x * Jx = 0, and for some real δ > 0.

(

3) If w = ∞, then Θ(z) = I 2p + 1 z -1 xx * J x * Jx , (32)
for some x ∈ C 2p such that x * Jx < 0.

The continuous-time case

In the continuous-time case, the definition of a lossless matrix function is as follows. Definition 3.3. In continuous-time, a square p × p rational matrix function G(s) is called lossless (or stable all-pass) if at all points of analyticity it holds:

G(s) * G(s) ≥ I p , for Re(s) < 0, (33)
G(s) * G(s) = I p , for Re(s) = 0, (34)
G(s) * G(s) ≤ I p , for Re(s) > 0. ( 35
)
The analogous extension to Σ-lossless functions Θ(s) now reads: Definition 3.4. Let Σ be a k × k signature matrix. Then a k × k rational matrix function Θ(z) is called continuoustime Σ-lossless if at at all points of analyticity it holds:

Θ(s) * ΣΘ(s) ≥ Σ, for Re(s) < 0, (36)
Θ(s) * ΣΘ(s) = Σ, for Re(s) = 0, (37)

Θ(s) * ΣΘ(s) ≤ Σ, for Re(s) > 0. ( 38
)
A balanced realization (A, B, C, D) of a continuous-time Σ-lossless system is characterized by the following identities:

A + A * = -BΣB * , (39)
C = -DΣB * , (40)

Kronecker structures, triangular forms and truncation of the state vector

When choosing a balanced realization of a lossless system, one has the freedom of the unitary group to build in additional properties by means of state-space transformation. One property of interest is upper triangularity of the reachability matrix K = B AB . . . A n-1 B , as upper triangularity is preserved under state vector truncation in a balance-and-truncate approach to model order reduction.

Generically, by applying QR-decomposition to K, a unitary state-space transformation matrix can be computed which makes the reachability matrix K upper triangular while preserving balancedness. It depends on the Kronecker structure of the lossless system whether this is indeed possible. In such a case, the block-partitioned n × (m + n) matrix (B, A) is also upper triangular and state truncation preserves this property too.

As a result, state truncation for such a balanced realization of a continuous-time lossless system is easily seen to produce a balanced realization of a reduced order approximation system. In the discrete-time lossless case, balancedness is generally lost. It may be restored by reorthonormalization of the last m columns of the truncated realization matrix, as balancedness requires a unitary realization matrix, but this affects the dynamical matrix and results in a definitely less attractive procedure than in the continuous-time case.

When dealing with any given Kronecker structure, a generalized procedure can be developed in which an associated suitable selection of n columns of the reachability matrix K is made upper triangular. This has been worked out in [START_REF] Peeters | Canonical lossless state-space systems: Staircase forms and the Schur algorithm[END_REF] using so-called nice selections, yielding a procedure which can be applied both in continuoustime and in discrete-time. This produces atlases of coordinate charts, exhibiting convenient triangularity properties which are preserved under state truncation, together covering the manifold of p × p lossless systems.

INTERPOLATION THEORY FOR LOSSLESS SYSTEMS

Linear fractional transformations

Linear fractional transformations occur extensively in representation formulas for the solution of various interpolation problems, see [START_REF] Ball | Interpolation of rational matrix functions[END_REF]. They are at the heart of the parametrization of lossless functions through the tangential Schur algorithm. The properties of a linear fractional transformation of matrices were studied in a form adapted to the needs of J-theory in [START_REF] Potapov | Linear fractional transformations of matrices[END_REF] and in an algebraic setting in [START_REF] Young | Linear fractional transformations in rings and modules[END_REF].

Let Θ ∈ C 2p×2p (z) be an invertible rational matrix in z, block partitioned into four p×p blocks as Θ = Θ 1 Θ 2 Θ 3 Θ 4 .

Associated with Θ, let the linear fractional transformation T Θ be defined to act on rational matrices G ∈ C p×p (z) in the following way:

T Θ : G → (Θ 4 G + Θ 3 )(Θ 2 G + Θ 1 ) -1 .
(42) Then the following group property holds for the LFTs associated with two 2p × 2p invertible matrices Φ and Ψ:

T Φ • T Ψ = T ΦΨ . (43) Consequently, the inverse of T Θ is given by T -1 Θ = T Θ -1 .
It also holds that T Φ = T Ψ if and only if there exists a scalar function λ ∈ C(z) for which Φ = λΨ. The latter still holds true, if the set of rational functions G on which the LFTs T Θ act, is restricted to the real lossless functions; see [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF].

We have the following result, which explains the importance of LFTs with J-inner matrix functions for the construction of lossless functions. For a proof, see for instance (Ball et al., 1990, Sect. 18.2.).

If the J-inner matrix function Θ(z) happens to be constant (i.e., of McMillan degree 0), then the associated LFT T Θ (G) does not affect the McMillan degree of the lossless function G.

The tangential Schur algorithm

In [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] balanced statespace canonical forms have been constructed for various classes of linear systems, with special properties of these classes (such as stability) built in. A balanced canonical form for continuous-time SISO lossless systems was presented in Ober (1987a). This continuous-time lossless case is central to the constructions of [START_REF] Hanzon | Overlapping blockbalanced canonical forms and parametrizations: the SISO case[END_REF], [START_REF] Hanzon | Overlapping blockbalanced canonical forms for various classes of linear systems[END_REF]. In the scalar case, it is balanced with a positive upper triangular reachability matrix, while Kronecker indices and nice selections are used in the multivariable case to arrive at balanced overlapping canonical forms.

For discrete-time lossless systems, application of the bilinear transform preserves balancedness, but it destroys positive upper triangularity of the reachability matrix, which is unattractive for balance-and-truncate type procedures. Therefore, in [START_REF] Hanzon | Balanced parametrizations of stable SISO all-pass systems in discrete-time[END_REF] a balanced canonical form for discrete-time lossless SISO systems was developed directly, which is parameterized by Schur parameters and which yields an upper triangular reachability matrix K. The unitary realization matrix R is factored into a product of elementary unitary matrices, thus facilitating a recursive construction of the balanced canonical form. This SISO balanced canonical form was generalized to the MIMO case in [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF], using linear fractional transformations and the tangential Schur algorithm.

The tangential Schur algorithm yields a flexible approach to the recursive construction of a parametrization of the space of lossless systems of a given order n. It is derived from the method of [START_REF] Alpay | On the differential structure of matrix-valued rational inner functions[END_REF]; [START_REF] Fulcheri | Matrix rational H 2approximation: A gradient algorithm based on Schur analysis[END_REF], where the tangential Schur algorithm is used to construct an infinite atlas of generic overlapping parameterizations for the space of p × p inner functions of McMillan degree n. The relationship between these two situations is constituted by the map R(z) → R ♯ (z), which relates p × p inner functions of McMillan degree n to p × p lossless functions of McMillan degree n.

In the context of inner functions, the tangential Schur algorithm consists of an iterative procedure by which a given p × p inner function of McMillan degree n is reduced in n iteration steps to a p × p inner function of McMillan degree 0 (i.e., to a constant unitary matrix). In each iteration step the McMillan degree of the inner function at hand is reduced by 1, by application of a suitable linear fractional transformation which is chosen to meet a particular interpolation condition, and which involves an associated J-inner matrix function of McMillan degree 1. The actual parametrization procedure consists of the reverse process, by which a chart of inner functions of McMillan degree n is constructed in n iteration steps, starting from an initial unitary matrix. The choice of interpolation points and (normalized) direction vectors may serve to index such a chart, while the local coordinates correspond to the n Schur vectors.

In the general form of the tangential Schur algorithm discussed in [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF], each elementary J-inner factor has its pole outside the closed unit disk at z = 1/w. It can be represented as:

Θ u,v,w,ξ,H (z) = (44) =     I 2p + (z -w)(1 -wξ) (1 -wz)(ξ -w) -1 u v u v * J (1 -v 2 )     H,
where (1) w ∈ C, |w| < 1, (the interpolation point),

(2) u ∈ C p×1 , u = 1, (the normalized direction vector),

(3) v ∈ C p×1 , v < 1, (the Schur parameter vector), (4) ξ ∈ C, |ξ| = 1, (a normalization point on the circle), ( 5) and H is a 2p × 2p constant J-unitary matrix. This representation is of the form (1) in Thm. 2, as seen by imposing: x = u v , with u = 1 and thus v < 1; the right J-unitary factor H is chosen as H = Θ(ξ).

The following proposition further clarifies the structure of Θ u,v,w,ξ,H (z) and of the action of the associated LFT: Proposition 4. The J-inner matrix function Θ u,v,w,ξ,H (z) can be factored as: 47) is lossless of McMillan degree n + 1 and satisfies the interpolation condition

Θ u,v,w,ξ,H (z) = H(uv * )S u,w (z)S u,w (ξ) -1 H(uv * ) -1 H, (45)
G(z) = T Θ u,v,w,ξ,H (z) (G(z)) (
G(1/w)u = v. ( 48
)
The reverse process is addressed by the following proposition. It describes the construction of a lossless matrix function G(z) of reduced degree n from a given lossless matrix function G(z) of McMillan degree n + 1 proceeding from an interpolation condition of the form (48). See also, e.g., [START_REF] Ball | Interpolation of rational matrix functions[END_REF]; [START_REF] Alpay | On the differential structure of matrix-valued rational inner functions[END_REF]; [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF].

Proposition 6. Let G be a p×p lossless function of McMillan degree n + 1 which satisfies an interpolation condition of the form G(1/w)u = v, (49) in which w ∈ C is an interpolation point with |w| < 1, u ∈ C p×1 is a normalized direction vector with u = 1 and v ∈ C p×1 is a Schur vector satisfying v < 1. Let ξ ∈ C be an arbitrary number of modulus 1, and let H be an arbitrary constant J-unitary matrix. Then G admits the representation

G = T Θ u,v,w,ξ,H (G),
(50) for some p × p lossless function G of McMillan degree n.

Note that in the standard case with w = 0 the value of G(∞) corresponds to the direct feedthrough term D of any state-space realization ( A, B, C, D) of G(z), so that the interpolation condition then takes the form Du = v. This case is often encountered when constructing canonical forms with additional triangularity properties.

The tangential Schur algorithm can now be described as follows. Theorem 7. (Tangential Schur algorithm). Let G (n) be a p × p lossless transfer matrix of McMillan degree n. For k = n, . . . , 1, let interpolation points w k ∈ C be given with

|w k | < 1, let constants ξ k ∈ C be given with |ξ k | = 1, and let mappings H k : (u, v, w, ξ) → H k (u, v, w, ξ) be given which assign a constant 2p × 2p J-unitary matrix H k (u, v, w, ξ) to each quadruple (u, v, w, ξ) ∈ C p × C p × C × C with u = 1, v < 1, |w| < 1 and |ξ| = 1.
Then for k = n, . . . , 1, there exist vectors u k ∈ C p with u k = 1, such that the vectors v k ∈ C p constructed recursively by the following formulas, each have length v k < 1: 52)

v k := G (k) (1/w k )u k , (51)
Θ k := Θ(u k , v k , w k , ξ k , H k (u k , v k , w k , ξ k )), (
G (k-1) := T -1 Θ k (G (k) ). ( 53
)
With such a choice of the unit vectors u k (k = n, . . . , 1) each of the functions G (k) is lossless of McMillan degree k and one can write

G (n) = T Θn (T Θn-1 (. . . (T Θ1 (G (0) )) . . .)) = = T ΘnΘn-1•••Θ1 (G (0) ), ( 54
)
where G (0) is a constant unitary matrix.

The tangential Schur algorithm offers a recursive way to reduce a given p × p lossless matrix function of McMillan degree n to a unitary constant matrix G (0) . In each of the n steps the McMillan degree is decreased by one. Each step is associated with an interpolation condition. The reversed tangential Schur algorithm can be used to generate a parameterized chart of lossless systems, in which the Schur vectors v 1 , v 2 , . . . , v n and the unitary matrix G (0) serve as the parameters. The set of values for w k , u k , ξ k and the set of mappings H k at the n recursion steps can serve to index a generic chart for the differentiable manifold of lossless functions of degree n, if the mappings H k are sufficiently smooth. Such a chart involves np + 1 2 p(p -1) real degrees of freedom in the real case, and 2np + p 2 real degrees of freedom in the complex case.

State-space realizations

In [START_REF] Hanzon | Balanced realizations of discrete-time stable all-pass systems and the tangential Schur algorithm[END_REF] it is established under which conditions a recursion step of the (reversed) tangential Schur algorithm can be implemented on the state-space realization level by means of straightforward unitary matrix multiplication of the realization matrix R. More precisely, we first introduce the following definition of a mapping F U,V . Definition 4.1. For any two matrices U and V of size (p + 1) × (p + 1), the associated mapping F U,V is defined to act on a proper rational p × p matrix function G(z) as follows:

F U,V : G(z) → F 1 (z) + F 2 (z)F 3 (z) z -F 4 (z) , (55)
with F 1 (z) of size p × p, F 2 (z) of size p × 1, F 3 (z) of size 1 × p and F 4 (z) scalar, defined by:

F (z) = F 1 (z) F 2 (z) F 3 (z) F 4 (z) = V 1 0 0 G(z) U * . ( 56
)
Then the next result establishes that a state-space realization of G(z) = F U,V (G(z)) can in fact be obtained by operating directly on an available realization matrix R of G(z) through pre-and post-multiplication by the matrices V and U * . Proposition 8. Let U and V be two matrices of size (p + 1) × (p + 1). Let G(z) be a p × p proper rational transfer function admitting a state-space realization (A, B, C, D).

Then a state-space realization ( A, B, C, D) of G(z) = F U,V (G(z)) with (n + 1)-dimensional state-space is given by:

D C B A = V 0 0 I n 1 0 0 0 D C 0 B A U * 0 0 I n . (57)
Clearly, if we are dealing with a balanced realization of a lossless function G(z), then the realization matrix R is unitary and pre-and post-multiplication with unitary matrices V and U * preserves this property, so that G(z) is again lossless and the state-space realization ( A, B, C, D) is balanced if and only if A is asymptotically stable.

When employing a linear fractional transformation involving an elementary J-inner factor Θ u,v,w,ξ,H (z), we have the following major result. Theorem 9. If T Θ u,v,w,ξ,H (z) coincides with a mapping F U,V with U and V unitary, then

Θ u,v,w,ξ,H (z) = H(uv * )S u,w (z)H(wuv * ) P 0 0 Q , (58)
for some u, v ∈ C p with u = 1, v < 1, some w ∈ C with |w| < 1, and some p × p unitary matrices P and Q.

Equivalently it holds that

H = H(uv * )S u,w (ξ)H(wuv * ) P 0 0 Q . ( 59
)
In that case one can take U = U 1 0 0

P and V = V 1 0 0 Q , where U =    √ 1-|w| 2 √ 1-|w| 2 v 2 u I p -(1 + w √ 1-v 2 √ 1-|w| 2 v 2 )uu * w √ 1-v 2 √ 1-|w| 2 v 2 √ 1-|w| 2 √ 1-|w| 2 v 2 u *   (60) V =    √ 1-|w| 2 √ 1-|w| 2 v 2 v I p -(1 - √ 1-v 2 √ 1-|w| 2 v 2 ) vv * v 2 √ 1-v 2 √ 1-|w| 2 v 2 - √ 1-|w| 2 √ 1-|w| 2 v 2 v *    (61)
For the special case w = 0 it follows that

U = u I p -uu * 0 u * , (62)
V = v I p -(1 -1 -v 2 ) vv * v 2 1 -v 2 -v * . (63)
In the SISO case (i.e.: p = 1) it is natural to choose u = 1

and U = I 2 . Then V = v 1 -v 2 1 -v 2 -v * . It
follows that the realization matrix R attains a positive upper-Hessenberg form, for which the reachability matrix is positive upper triangular; this yields the canonical form presented in [START_REF] Hanzon | Balanced parametrizations of stable SISO all-pass systems in discrete-time[END_REF]. In the MIMO case, when u is chosen to be a standard basis vector, the matrix U becomes a permutation matrix. This situation is further investigated in [START_REF] Peeters | Canonical lossless state-space systems: Staircase forms and the Schur algorithm[END_REF], where it is established which choices of standard basis vectors in the consecutive recursion steps are required to yield reachability matrices with particular pivot structures, corresponding to prescribed Kronecker structures. In [START_REF] Hanzon | Subdiagonal pivot structures and associated canonical forms under state isometries[END_REF] it is shown that all choices of standard basis vectors produce subdiagonal canonical forms.

If the interpolation point w is chosen to be a pole of the lossless function G(z) and if u is chosen to be a normalized vector in the kernel of G( w-1 ), it follows that the Schur vector attains the value v = 0. Repeating this procedure for each step of the tangential Schur algorithm produces the Potapov factorization of G(z), i.e.: G(z) is factored into a product of lossless functions of degree 1. This is a useful property, which is used in the software package RARL2, see [START_REF] Olivi | RARL2, Matlab based software package developed by Team APICS at INRIA[END_REF], to change the local parameterization whenever the current parameterization is considered to become less well conditioned.

If G(z) is real, it is straightforward to arrive at a real parameterization by requiring the interpolation points, direction vectors, and so on, all to be real.

In another line of research, presented in [START_REF] Peeters | Linear fractional transformations and balanced realization of discrete-time stable all-pass systems[END_REF], it was investigated how an available state-space realization of a proper J-lossless matrix Θ(z -1 ) and a statespace realization of a lossless function G(z) can jointly be employed to directly yield a state-space realization of the lossless function G(z) = T Θ(z) in the form of an LFT acting on an extended realization matrix. When the realization for Θ(z -1 ) is J-balanced and the realization for G(z) is balanced, then the realization thus obtained for G(z) is again balanced.

Theorem 10. Let Θ(z) = Θ 1 (z) Θ 2 (z) Θ 3 (z) Θ 4 (z) be J-inner of
McMillan degree m. Suppose that the J-lossless function Θ(z -1 ) is proper, for which a J-balanced state-space realization is given by (A,

B 1 B 2 , ( C 1 C 2 ) , D 1 D 2 D 3 D 4 ).
Let G(z) be lossless of McMillan degree n, and suppose that a balanced realization is given by (A, B, C, D), which gives rise to an extended realization matrix R e given by

R e = D 0 C 0 I m 0 B 0 A . ( 64
)
Let ∆ be defined in terms of the partitioned J-balanced realization of Θ(z -1 ) by:

∆ =        D 1 C 1 0 D 2 0 0 B 1 A 0 B 2 0 0 0 0 I n 0 0 0 D 3 C 2 0 D 4 0 0 0 0 0 0 I m 0 0 0 0 0 0 I n        . ( 65
)
Then a balanced realization ( A, B, C, D) for the lossless function G(z) = T Θ(z) is given by the LFT

R = D C B A = T ∆ (R e ). (66)
When restricting to elementary J-inner factors Θ(z), however, note that this procedure generates alternative balanced state-space realizations, not previously produced by the realization procedures presented earlier.

Boundary interpolation

Above it was shown that upper triangular structures in the reachability matrix are conveniently obtained in discretetime upon choosing the interpolation points w at zero and the direction vectors u from the set of standard basis vectors in a suitable way. Application of the bilinear transform takes these realizations into balanced realizations for continuous-time lossless systems. However, this approach does not allow one to build positive upper triangularity into the continuous-time reachability matrix.

In [START_REF] Hanzon | Lossless scalar functions: Boundary interpolation, Schur algorithm and Ober's canonical form[END_REF] it was established for the SISO case that positive upper triangularity of the reachability matrix in continuous-time (as exhibited by Ober's canonical form, see Ober (1987b,a)) does in fact correspond to choosing the interpolation points w at infinity. For the associated discrete-time case (related by the bilinear transform) the elementary J-inner factor Θ(z) is of the form (2), with w chosen on the unit circle at w = -1.

In discrete-time the corresponding boundary interpolation problem not only involves a condition of the form G(w -1 )u = v with u = 1, but also a second condition of the form v * G ′ (w -1 )u = -w -1 ρ, with a real positive value for ρ. At w = -1 these conditions take the form

G(-1)u = v, (67)
v * G ′ (-1)u = ρ. ( 68
)
The fact that G(z) is unitary for each point on the unit circle makes that v = 1, thus leaving one (real) degree of freedom. The property that ρ is real and positive, stems from the fact that for a lossless function G(z), the matrix -zG(z) * G ′ (z) is positive semi-definite for all z on the unit circle (and u can always be chosen such that positivity of ρ holds, provided that the degree of G(z) is at least 1). See for instance (Ball et al., 1990, Ch. 21) for a treatment of such boundary interpolation problems.

To apply this theory to the case at hand, note that it is inconvenient to address the derivative of a continuous-time lossless function G(s) at s = ∞ directly. Instead, a useful way to proceed is by studying the related function G(s -1 ) and its derivatives at zero. This approach has been worked out in [START_REF] Hanzon | Lossless scalar functions: Boundary interpolation, Schur algorithm and Ober's canonical form[END_REF].

For the continuous-time lossless MIMO case, positive upper triangularity can also be built into the reachability matrix of a balanced realization, by generalizing the results of [START_REF] Hanzon | Lossless scalar functions: Boundary interpolation, Schur algorithm and Ober's canonical form[END_REF]. These recent results will appear elsewhere.

APPLICATIONS

Parameterization of stable systems: input normal forms

The parameterizations presented above have been developed primarily for lossless systems in discrete-time or in continuous-time. However, they are easily employed for parameterizing the class of stable systems of a given order n too; see also [START_REF] Hanzon | Overlapping blockbalanced canonical forms for various classes of linear systems[END_REF]. To this end, observe that a stable system can generically be brought into canonical form by first making the system input normal (i.e., the reachability Gramian W c equals the identity matrix I n ) and then using orthogonal state-space transformations (which leave input normality unaffected) to make the reachability matrix K positive upper triangular. Obviously, instead of input normality and a positive upper triangular reachability matrix, one may also concentrate on output normality and a positive lower triangular observability matrix. In the MIMO case, Kronecker indices and nice selections can be used to develop a canonical form along similar lines (see [START_REF] Peeters | Canonical lossless state-space systems: Staircase forms and the Schur algorithm[END_REF]).

In terms of realization matrices, it suffices to parameterize the part involving the input normal pair (A, B) and then to supply arbitrary matrices C and D, with C such that observability holds. Here it is helpful to notice that without loss of generality the reversed tangential Schur algorithm can be started from G (0) = I p . Thus, the parameterization of all input normal pairs (A, B) for which K is positive lower triangular only involves the sequence of Schur vectors.

System identification: separable least squares

In linear system identification, the method of separable least squares (SLS) was first introduced in [START_REF] Golub | The differentiation of pseudoinverse and nonlinear least squares problems whose variables separate[END_REF]. In [START_REF] Bruls | Linear and non-linear system identification using separable least-squares[END_REF] one proceeds by reducing the nonlinear least squares optimization problem at hand, to a new optimization problem over the space of output normal pairs (C, A). In this approach, the matrices B and D are analytically optimized for the original criterion function in terms of A and C, and subsequently eliminated from the problem. As we have seen in the previous subsection, output normal pairs (C, A) are intimately related to lossless systems satisfying a normalization condition, and they can be conveniently parameterized with Schur vectors. The state-space approach discussed in the present paper allows one to proceed in the discrete-time case with parameterizations which are constructed as a product of unitary matrices.

L 2 -System approximation and RARL2

In L 2 -system approximation, the Douglas-Shapiro-Shields factorization, see [START_REF] Douglas | Cyclic vectors and invariant subspaces for the backward shift operator[END_REF], can be used to factor a given strictly proper stable rational transfer matrix function H into a product GP in which G is lossless of the same McMillan degree as H and P is unstable. The factor G is unique up to a right unitary factor. The L 2approximation problem can subsequently be reformulated, analytically, entirely in terms of G. Parameterization of the class of lossless systems according to the state-space methods described here, allows one to handle the stability constraint conveniently, by optimizing over the class of lossless systems.

In [START_REF] Marmorat | Matrix rational H 2 -approximation: a statespace approach using Schur parameters[END_REF] this approach was worked out both in discrete-time and in continuous-time, using the balanced state-space parameterizations discussed above. Later, the procedure was implemented in the Matlab software package RARL2, see [START_REF] Olivi | RARL2, Matlab based software package developed by Team APICS at INRIA[END_REF], which employs a chart switching strategy to move to the center of a new parameter chart when the current approximation moves too far away from the center of the current chart. This employs the fact that each lossless system can be put at the center of an adapted chart, where all the Schur vectors v k are zero, by means of the Potapov factorization. The distance to the center of a chart is intuitively measured by the lengths of the Schur vectors used to represent the lossless system in that chart: when the length of a Schur vector becomes equal to 1, the boundary of a chart is reached and degeneration phenomena may occur.

We find it interesting to note that the numerical conditioning of the implemented computational procedure is considerably better for the balanced state-space realizations in discrete-time with their associated unitary matrix factorizations, when compared to an earlier implementation which performed analogous computations directly on the level of transfer functions (see [START_REF] Fulcheri | Matrix rational H 2approximation: A gradient algorithm based on Schur analysis[END_REF]).

Orthogonal wavelets from filter banks

In modern digital signal processing, an important class of wavelets is constituted by the orthogonal wavelets that are generated by power complementary filter banks. In [START_REF] Strang | Wavelets from Filter Banks[END_REF] a detailed exposition of this topic is given. In its classical form, the procedure is as follows.

First, two copies of the same discrete-time input signal are passed through a lowpass filter C(z) and a highpass filter D(z), to produce approximation coefficients and detail coefficients, respectively. These output sequences are then downsampled by a factor 2, i.e., only the even-indexed values (the even phases of the signals) are stored. When the filters C(z) and D(z) are power complementary (see also [START_REF] Vaidyanathan | The role of lossless systems in modern digital signal processing: a tutorial[END_REF]), no information loss occurs and the output sequences admit an interpretation in terms of associated orthonormal bases, spanned by the socalled scaling function (used for approximation purposes) and the wavelet function (used to analyze the details). One then proceeds repeatedly, in a likewise fashion, by processing the approximation signal (the downsampled output of the filter C(z)) to compute approximation coefficients and detail coefficients on coarser scales.

For each level of the analysis, an equivalent procedure can also be considered, in which the input signal is first split into its two phases (the even and odd phases), which are then jointly processed by a 2 × 2 polyphase filter H(z) to produce the same (downsampled) sequences of approximation and detail coefficients as before. It then follows that orthonormality of the wavelet bases, and power complementarity of the filters C(z) and D(z), are in fact equivalent to H(z) being (discrete-time) lossless. The theory presented here for the construction of parameterizations of lossless systems can thus be applied to facilitate parametric optimization over the class of lossless 2 × 2 matrix functions of a given degree n. This is particularly useful for wavelet design, especially when a wavelet is designed for a specific application. Initial investigations using this approach are presented in [START_REF] Karel | Optimal discrete wavelet design for cardiac signal processing[END_REF].

Extensions to more complicated situations are currently under investigation. One issue of practical interest concerns the number of vanishing moments that a wavelet has. As it turns out, the problem of building in vanishing moments can be reformulated as an interpolation problem for 2 × 2 lossless systems on the stability boundary (at z = 1) involving higher order derivatives. Another obvious extension concerns multiwavelets, where the polyphase matrix is generalized to become a lossless matrix of size 2r × 2r with r > 1; see also [START_REF] Peeters | Multiwavelet design for cardiac signal processing[END_REF].

CONCLUSIONS

In this paper we have presented an overview of the main results which link balanced parameterizations of lossless systems to interpolation theory and Schur parameters in various settings: discrete-time and continuous-time, SISO and MIMO. It is indicated which choices in the tangential Schur algorithm (in particular which choice of interpolation points) supports the construction of triangular pivot structures in reachability matrix. It is possible to deal with special Kronecker structures and restrictions to important subclasses (such real lossless systems) are conveniently handled. Certain extensions of the presented theory, not discussed here, are also available in the literature. Most notably these involve Nudelman interpolation problems and symmetric inner functions. Further lines of research concern the application domains indicated above, which often present specific additional constraints that require special attention.

  18) In fact, if W c and W o denote the Gramians for the discrete-time system (A, B, C, D), then they also constitute the Gramians for the associated continuous-time system ( A, B, C, D), showing that (A, B, C, D) is discretetime balanced if and only if ( A, B, C, D) is continuous-time balanced.

  and G(z) * are lossless. Note that a lossless function G(z) is unitary at each point z on the unit circle. The determinant of a lossless matrix is a scalar lossless function of the same McMillan degree. As we shall see below, a lossless matrix of McMillan degree n can always be factored into a product of n lossless matrices of McMillan degree 1, its so-called Potapov factorization.

  ) Conversely, if this property holds, then G(z) is proper Σlossless of McMillan degree ≤ n. For a Σ-balanced statespace realization (A, B, C, D) it holds that the solutions to the associated Lyapunov-Stein equations with modified right-hand sides BΣB * and C * ΣC, respectively, again satisfy W c = W o = I n .

  proper Σ-lossless rational matrix function admits such a Σ-balanced realization. When the right-hand sides of the Lyapunov equations are modified to -BΣB * and -C * ΣC, respectively, it holds again that W c = W o = I n . The relations for balanced realizations of lossless functions are simply obtained by setting Σ to the identity matrix. If the bilinear transform is applied to the z-domain, taking a Σ-lossless discrete-time function Θ(z) into a Σlossless continuous-time function Θ(s) = Θ( z-1 z+1 ), then any associated Σ-balanced realization (A, B, C, D) for Θ(z) is transformed into a Σ-balanced realization ( A, B, C, D) for Θ(s) according to Eqns. (17)-(18).

  Theorem 3. Let Θ be J-inner of size 2p × 2p and of McMillan degree m. If G is p × p lossless and of McMillan degree n, then the matrix function G = T Θ (G) is welldefined, lossless and of McMillan degree ≤ n + m.

  where H(uv * ) denotes the Halmos extension of the strictly contractive matrix uv * , and where S u,w (z) := I p + (b w (z) -1) uu * 0 0 I p , (46) with b w (z) = z-w 1-wz . Recall that an LFT involving a product of matrices comes down to a composition of LFTs involving each of the individual factors. Also, an LFT involving a constant Junitary matrix does not change the McMillan degree. Therefore, only the LFT involving the factor S u,w (z) can affect the McMillan degree. The block-diagonal structure of S u,w (z) implies that for this LFT postmultiplication takes place by (I p + (b w (z) -1) uu * ) -1 = I p + 1-wz z-w -1 uu * . The next proposition (see Hanzon et al. (2006)) establishes the effect of the LFT associated with Θ u,v,w,ξ,H (z) on a lossless function G(z) of McMillan degree n. Proposition 5. Let G(z) be a lossless function of McMillan degree n. Then

  Two state-space systems are called input-output equivalent if they have the same transfer matrix. Here, two rational (matrix) functions are identified if they agree almost everywhere, i.e.: common factors are always canceled. Note that a transfer matrix in this causal set-up is always proper (i.e., analytic at infinity), the direct feedthrough (gain) D being the value at infinity.

	Realization theory establishes that, conversely, any p × m rational matrix function G(z) analytic at infinity can be
	written in the form


Fanben Meng

Yves-Marie Hervault

Lucie Norel

Karine Costuas

Colin Van Dyck

Victor Geskin

Jérôme Cornil

Huey Hoon Hng

Stéphane Rigaut

Xiaodong Chen

Photo-modulable Molecular Transport Junctions based on Organometallic Molecular Wires †

Photo-modulable molecular transport junctions are developed via on-wire lithography-fabricated nanogap functionalized with a dithienylethene unit bearing two ruthenium fragments. A reversible and repeatable bi-state conductive switching upon alternate irradiation of UV and visible light can be distinctly observed. Theoretical calculations further suggest that bi-directional isomerization is due to the ruthenium moieties that modulate judiciously the electronic coupling between the photochromic part and the metal electrodes, and that the differences in electronic structure between the two isomers (open and closed states) are responsible for conductivity switching.

Introduction

Molecular transport junctions (MTJs) are essential structures for developing molecule-based electronics. 1 Building molecular function into MTJs to exploit advanced nanodevices with promising applications is both a formidable challenge and an opportunity that, if realized, could have a revolutionary impact on molecular electronics. 2 Several functional MTJs, such as diodes, 3 transistors, 4 switches, 5 and memristors, 6 have been made to show promising application in fields from information technology to nano-bio interface. Integrating photo-switchable molecules, 7 such as dithienylethylene (DTE), stilbenes, and azobenzenes, into nanodevices is receiving increasing attention. 8 For example, for DTE molecules in solution, 9 the photo-switching is induced by structure cyclization from a cross-conjugated open state to a π-conjugated closed state under ultraviolet (UV) light, while the reverse process is triggered by visible region irradiation. 9 Nevertheless, in the solid state devices, the covalent connection between metal electrodes and DTE molecules may cause just one-way switching from closed to open state, due to the strong electronic coupling formed in such a configuration in MTJs, 10 thus affecting molecular functionality in photochromic isomerization.

In order to solve this problem, we designed and synthesized a novel and longer photochromic molecule, compound 1, by covalently bonding two ruthenium fragments [HS-C 6 H 4 -C≡C(dppe) 2 Ru] + (dppe=1,2-bis(diphenylphosphino)ethane) on either ends of a DTE unit.

The ruthenium complexes are expected to achieve an efficient coupling along the entire conjugated path, 11 while decreasing the strong electronic coupling between the gold electrodes and the DTE groups observed with direct connections. Besides, such fragments are known to maintain the photoisomerization properties of DTEs, characterized by two open (o) and closed (c) states, in order to ensure electronic switching between the π-conjugated and non-conjugated states. 12 In addition, their intrinsic low oxidation potential close to the Fermi level of gold should favour highest occupied molecular orbital (HOMO) mediated-tunneling. 13 On-wire lithography (OWL), 14 a chemistry-based nanofabrication technique, is a high-throughput method used to prepare nanogaps, affording precise control of the gap feature sizes down to 1 nm. This method relies on the electrochemical deposition of multi-segmented metal nanowires within anodized aluminium oxide membranes and selective chemical etching of a sacrificial metal layer, which determines the gap size. The obtained nanogap structures have been proven to ensure effective electrical connection to target molecules 15,16 and to provide a promising potential platform to fabricate functional MTJs, such as rectifiers and memristors. 6,17 In this contribution, we combine the advantages of photo-isomerization of DTE molecules and MTJs fabrication based on OWLproduced nanogaps to explore a photo-controllable reversible switching nanodevice with two-state conductance. Importantly, we found that the organometallic strategy with the DTE molecule 1 leads to a proper modulation of the strong electronic coupling between the gold electrodes and the DTE groups, solving the problem of one-way switching from closed to open state, 10 without the use of gold nanoparticles for example. 18 In addition, theoretical calculations were performed in order to assess the geometries of both closed and open forms of 1 and understand the geometric reorganization upon irradiation, as well as the transmittance process through the molecule when incorporated between gold electrodes.

Results and discussion

The targeted binuclear compound 1 was readily obtained from the reaction between the diethynyl-substituted dithienylethene and the vinylidene complex [Cl(dppe) 2 Ru=C=CH-p-C 6 H 4 -SAc](OTf) in the presence of a base and of a non-coordinating salt, as previously described for analogous complexes (Figure 1a). 19 The vinylidene precursor was obtained beforehand by the coordination of the alkyne AcS-p-C 6 H 4 -CCH to the 16-electron complex [Cl(dppe) 2 Ru](OTf). Both complexes were fully characterized (ESI †) and 1 shows very similar characteristics to those of the previously reported analogue complex, [Cl-(dppe) It is to be noted that dithienylethenes can adopt two conformations, with the two thiophene rings in mirror symmetry (parallel conformation) and in the pseudo-C 2 symmetry (anti-parallel conformation), and that the photocyclization reaction can only proceed from the anti-parallel conformation. Importantly, as expected on the basis of 1 H and 31 P NMR characterizations and previously demonstrated by theoretical calculations for the analogue complex, 12 only the antiparallel isomer is formed due to steric hindrance. Therefore, as highlighted in Figure 1b, isomerization studies in toluene unambiguously show that the photochromic conversion process upon irradiation of 1o to the very stable 1c complex is fully reversible in UV-vis cells ([c] ≈ 10 -5 mol•L -1 ). More specifically, the colourless complex 1o displays an absorption at λ max = 359 nm (ε = 84000 L•mol -1 •cm -1 ). The transition associated with this excitation occurs between the HOMO, π (DTE) and π/d [AcS-p-C 6 H 4 -C 2 -Ru-C 2 ] in character, and the lowest unoccupied molecular orbital (LUMO) which is the corresponding π* of the DTE, where the LUMO, in particular, possesses two lobes on the methylated carbon atoms that can overlap upon rotation to lead to a ring closure (see below). 12 Upon irradiation with UV light (~350 nm), this band vanishes, while a broad absorption assigned to the deep green closed isomer 1c appears at λ max = 713 nm (ε = 45000 L•mol -1 •cm -1 ). This absorption corresponds mainly to a HOMO (Ru d/π -C 2 DTE π ) → LUMO (Ru d/π -C 2 DTE π* ) transition which removes bonding electrons between the two methylated carbon atoms. It was previously shown that the broadness of the band is due the overlapping absorptions of different rotamers of 1c in solution with various orientations of the metallic units with respect to the DTE plane. 12 This 1c-containing solution can be further bleached back to a colourless solution of 1o under visible light (~700 nm) via ring opening, with quantitative recovering of the initial spectra. Complete conversions in both directions were confirmed with 31 P and 1 H NMR studies, with characteristic resonance shifts of the eight equivalent phosphorus atoms, and of the thiophene and methyl group protons (ESI †). Then, in order to better predict the behaviour of the device, theoretical studies of molecular conformations and electronic structures were performed (computational details, ESI †). The geometry optimization of simplified structures ‡ of 1o and 1c in vacuum reveal different three-dimensional geometrical arrangements (Figure 2a), due to the modification of the DTE structure  The 1o skeleton presents an angle of roughly 135° between the two acetylide branches (projected straight lines) with a distance of 33.1 Å between the terminal sulfur atoms. The 1c conformation is slightly more spread out with a distance of 34.2 Å between the two anchor atoms and an angle of 158° between the two conjugated chains. Considering the S-S axis as reference, the two molecules can be included in cylinders of same circumference (ca. 15.2 Å). When comparing the change in geometry between 1c and 1o, a shortening of 1 Å of the sulfursulfur distance is observed in 1o. It is attributed to a change in direction of the acetylide chain axis which is almost collinear in the closed form. The adaptability of 1o is remarkable since a constraint of only 0.03 eV is needed to keep the same molecular length as that of 1c (S-S distance of 34.2 Å). This suggests that the structural changes between 1c and 1o within nanogaps can occur without detachment of the molecule from the electrode.

The electronic structures of the simplified 1o and 1c adducts are given in Figure 2b. The closed form 1c is less stable than 1o by 0.26 eV. The electronic structure of 1c (that has one more double bond than 1o) displays an additional MO in the HOMO-LUMO gap (i.e., the new HOMO) which has bonding character between the two carbon atoms of the newly formed bond. Important for conduction, this orbital is also delocalized over the conjugated carbon skeleton, including the ruthenium centres. From the above calculated results of both molecular conformations and electronic structures, it is shown that the conformational isomerization between 1o and 1c can be achieved even when covalently immobilized within the nanogaps. Meanwhile, the different molecular orbital arrangement of 1c and 1o

anticipates a conductive switching of 1-based MTJs, suggesting that such a DTE molecule would be a promising candidate for fabrication of photo-modulable MTJs. Experimentally, ~5 μm Au wire structures with ~3 nm gaps, (which are consistent with the length of 1) were synthesized and cast onto a silicon substrate (containing a 600 nm SiO 2 layer) with patterned gold microelectrodes. Then, the wires were connected to the microelectrodes by e-beam lithography and subsequent thermal deposition of chromium and gold. Scanning electron microscopy (SEM) revealed that the two ends of an Au nanowire were connected to two separate microelectrodes (Figure 3a), with the gap on the wire around 3 nm (Figure 3b) as expected. The unmodified nanogap device was immersed in a 5 ml tetrahydrofuran (THF) solution of dithiol terminated 1c (after deprotection of thiol, by eliminating the acetyl group with 5 μL of 28% ammonia) for 24 h. Through the formation of Au-S bonds between the two sides of the nanogap and 1c, the molecules can be immobilized into the nanogap, as is depicted in Figure 3c. These covalent bonds will ensure a strong electronic coupling between 1c and the electrodes 15,16 to achieve the photo-controllable conductance switching due to the photochromic properties of 1c (Figure 3d). 11 Two-terminal I-V characterization of the ~3 nm gap devices was conducted under a vacuum of 5 × 10 -5 Torr at room temperature in the dark both before and after 1c assembly. The blank nanogap devices showed no conductance within the noise limit of the measurement (<1 pA, Figure S4, ESI †). However, when the ~3 nm gap was loaded with a 1c monolayer, the devices exhibited an obvious I-V response (black curve in Figure 3e) at low bias (≤ 1 V). The observed symmetry of the curves indicates that the 1c molecules were covalently bound to each of the gold electrodes on opposite sides of the gap. If the molecules were bound only to one contact, the I-V curves would display asymmetry due to different injection barriers. 15 This I-V characteristic explicitly suggests a successful fabrication of 1c-based transport junctions. The yield for working devices is 22% (21 out of 94 devices with I > 1 nA on a single chip with an area of 1 cm × 1cm). In addition, it should be mentioned that the magnitude of the current under 1 V bias obtained from different 1c-based devices varied in the range from 300 µA to 10 nA (Figure S5, ESI †). This disparity is presumably mainly due to the different number of 1c molecules that are actually incorporated into the nanogap via chemical bonds with the gold surface at each side. The roughness and morphology of the gold surface, and the ± 0.2 nm gap size variation, may be among the reasons that cause these variations. More interestingly, the I-V curves of 1c-functionalized nanogap devices exhibited light-triggered switching. (Figure 3e and Figure S5, ESI †) After the device was irradiated with visible light at a wavelength of 700 nm for 150 min, the current decreased by about an order of magnitude, as shown in Figure 3e (red curve), thus indicating that the device is switched from high conductance (ON state) to low conductance (OFF state). Light at 365 nm was then employed to irradiate the device for 30 min. It can be observed that the current surged back (blue curve in Figure 3e), suggesting the reverse switching of the device to the low-resistive ON state. Though the conductivity of the device after switching was a somewhat lower than in the initial state, the 1c-based devices have a high potential as light-triggered switches under different excitations. This bi-directional transformation is consistent with the photochromic properties of molecule 1 in solution. Note that the temperature-dependent I-V measurement was also performed in a large temperature range (20K-300K) (Figure S6, ESI †) and just less than twofold variation was observed, eliminating the influence of thermal effects on the device conductivity, and confirming the stability of the high conductance state in the dark (vide infra). Therefore, the photo-controllable structural rearrangement of 1 definitely affects the conductivity of the device, since the π-conjugation pattern in the closed form facilitates the electron transport along the molecule wire. 15 It is worth noting that we also employed 1o to substitute 1c for device fabrications with same gap size and that these 1o-based devices showed very low conductivity and no photochromic behaviour as 1c-based devices (<10 pA, 1 V bias, Figure S7, ESI †), suggesting that the MTJs cannot be achieved with 1o molecules. The failure could be ascribed to the conformation of 1o, as discussed in the theoretical studies, which is less linear and rigid than 1c, and hence less favourable to form a connection on the two opposite nanoelectrodes.

Although these mono-grafted 1o molecules may experience the photo-isomerization under UV light, we still cannot measure the conductivity change of the devices that is dominated by the disconnection at one side of nanogap. Note that, based on the calculated 1o conformation, double bonding onto the same side of the gap could be proposed but the steric hindrance due to the phenyl groups would disfavour it (Figure S8, ESI †) if molecules lie down on the surface. In addition, for 1o molecules connected to both nanoelectrodes, there is no energy level suitable to support high conductivity of the device (see theoretical calculations).

Furthermore, in order to analyse these experimental current characteristics of the devices, we have calculated the transmission spectra of both the closed and open structures (as obtained in the gas-phase geometry with the constraint of matching the interelectrode gap size) when incorporating them in a gold nanojunction. Current flowing through the junction is obtained by integration of this spectrum in the bias transmission window, according to the Landauer formula. 20,21 Transmissions at both equilibrium (zero bias) and under a bias of 0.6 V were calculated in order to take into account the polarization effects arising from the electric field. 22 Transmissions for each isomer are separately reported in Figure 4a. The closed isomer 1c spectrum exhibits an intense peak of transmission near the Fermi level thus allowing for high conductivity. The applied bias induces a shift of this peak, keeping it at the limit of the transmission window and hence restricting the possibility of this isomer to theoretically achieve high conductivity. 23 On the other hand, the open isomer 1o exhibits a small peak collapsing under bias. The theoretical closed/open current ratio at 0.6 V is around 40 which is consistent with the one order of magnitude observed experimentally (Figure 3e), taking into account the high number of molecules in experimental nanojunctions as against our single-molecule calculations. By computing the molecular projected self-consistent Hamiltonian (MPSH) spectrum, we can associate each transmission peak of the spectra with a molecular state. 21,22 Formally, the MPSH states can be described as an electronic reorganization of the isolated isomers upon substitution of the terminal hydrogen atoms by the gold electrodes. Relevant MPSH states are reported in Fig 4b . In comparison to the gas-phase calculations, we clearly identify the HOMO as the orbital aligned near the Fermi level for both isomers. The intense peak observed for the closed isomer 1c is driven by the good delocalization of the MPSH HOMO on the entire photochromic core between the ruthenium linkers, which are crucial to ensure a smooth transmission pathway. However, this orbital shows a moderate weight on the linkers and the gold surface, leading to a valuable reduction of hybridization with the metal, which helps to avoid the quenching of the photochromic switch 24 in order to maintain photoreversibility. Thus, it is the ruthenium moieties that judiciously adjust the electronic coupling between the photochromic part and the metal electrodes. Under bias, the photochromic part remains unchanged while a polarization of the linker towards the positive side takes place. The final shape rationalizes the conserved intensity of the peak under bias and points to its pinning to the lower (positive) Fermi level. In contrast, the HOMO of the open isomer 1o in the junction has a small weight on the central photochromic part; this breaks the delocalization and gives rise to a weak transmission peak near the Fermi level. Under bias, the orbital is completely changed and gets strongly polarized leading to the collapse of the weak transmission peak.

Finally, we have demonstrated that such a 1c-functionalized nanodevice can be used as a photo-modulable switch, for which reversibility, repeatability, and stability upon irradiation during ON/OFF cycles are all key characteristics. The ON/OFF cycle test conducted under 1 V bias is shown in Figure 5a. The repeatable wave-shape current response is generated by alternate irradiation of light with wavelengths at 700 nm and 365 nm, reflecting the reversible conductive switching of the device between ON and OFF. It is clearly observed that upon 700 nm irradiation, the conductance of the device gradually decreases following the isomerization from 1c to 1o, while upon 365 nm irradiation, the conductance is progressively enhanced due to the recovery of the 1c arrangement. The conductivity of nanodevice is dominated by the amount ratio between 1c and 1o bridging the nanogap and therefore the dynamics of 1 isomerization can be studied through the conductance change of the devices. The switching dynamics can be reflected by the conductance change of nanodevices, as shown in Figure 5. The reaction rate of the UV-triggered isomerized process of 1 immobilized between two nanoelectrodes is apparently higher than that of the visible light-triggered process and the ratio of the reaction constants (slopes of ln(G) vs time in Figure S9, ESI †) k 365 to k 700 is about 3. Although, it is consistent with the behaviours of the parent complex 12 and 1 (Figure S10, ESI †) in solution showing a much faster isomerization under UV irradiation than under visible light, the kinetics of molecular isomerization is uncomparable due to the distinctly different condition and microenvironment of the photochromic molecules in these two system, such as the medium (vacuum, organic solvent), molecular state (up to thousands of immobilized in a nanogap, 1 μM free in solution), and even the light path (penetration into a nanogap, direct irradiation). In addition, in order to test whether the conductive switching is really photo-related, we also measured the device in dark for 1 h, and found no notable resistive change. The reversible light-triggered conductive switching can be repeatedly operated for several cycles, maintaining a stable bi-state of system. Nevertheless, a progressive loss of reversibility is observed in the course of such a cycling experiment, as often observed. 8 This can be the result of the stochastic behaviour of thiols on gold leading to the desorption of molecules from the substrates, 25 that is increased by the slight additional constraint upon opening. One has also to consider the effect of UV irradiation, causing the decomposition of the molecules, 26 and the ductility of gold in the reorganization process occurring upon opening. 27 The reorganizations that follow bond breaking with the surface or the gold wire pulling § might be unfavourable for re-anchoring and/or for reclosure respectively. We could also observe that, in comparison to single-molecule studies on a related molecule, 8b the irradiation time required to reach the photostationary state is longer. This is reasonable, since the nanogap can contain thousands of molecules, and the number of those isomerised is a function of exposure time. 28 Finally, it is also worth noting that the observed closed/open ratio about one order of magnitude is consistent with the theoretical calculation of not only molecule 1 in gold nanogap here, but also relative molecules in previous reports, 29 which is also comparable with conductance change observed in single-molecule switches on other DTE molecules. 8c,8e,30

Conclusions

In summary, we have demonstrated a photo-controllable MTJ nanodevice based on the association of a ruthenium-based organometallic molecular wire including a DTE unit and an OWL-generated nanogap. The obtained devices perform bi-directional conductive switching upon irradiation with UV and visible light. The theoretical calculations performed on 1 reveal (i) that the ruthenium acetylide moieties decrease judiciously the electronic coupling of the DTE fragment with the metal electrodes in its open form, allowing the system to close upon irradiation, and (ii) that the delocalization of the HOMO of the closed isomer lying near the Fermi level of the junction is high enough to lead to a good conductivity. The switching in conductivity is explained by a weak delocalization in the photochromic part in addition to the polarization of the HOMO level of the open form lying near the Fermi level. Therefore, such a photo-controllable device not only shows a potential application in functional nanocircuits, but also provides a promising platform to clarify the nature of charge transport across single molecules.

Fig. 1

 1 Fig. 1 (a) Scheme of synthesis of the molecular switch 1; (b) UV-Vis absorption spectra of 1o and 1c in toluene.

Fig. 2

 2 Fig. 2 (a) Calculated molecular conformations of 1o and 1c viewed in three different orientations; (b) molecular orbital diagrams of 1o (left) and 1c (right). The contour values used for the MO representation are ±0.03 (e/borh) ½ .

15

 15

Fig. 3

 3 Fig. 3 (a) SEM image of a device fabricated with an OWL-generated nanowire; (b) SEM image of ~3nm gap in the nanowire; (c) illustration of a prepared nanowire device with 1c molecules spanning the ~3 nm gap; (d) the scheme of isomerization of 1 covalently bonded into the nanogap under light at 365 nm and 700 nm light; (e) Representative I-V characteristics of ~3 nm OWL-generated gap devices loaded with 1c molecules, as measured in the dark under vacuum (black curve: initial device; red curve: device after 700 nm irradiation for 2.5 h; blue curve: device after subsequent 365 nm irradiation for 30 min);

Fig. 4

 4 Fig. 4 (a) Transmission spectra (in log scale) of the closed and open isomers around the Fermi level of the electrodes which are set to zero for convenience. The black line corresponds to the calculation at equilibrium and the red line in non-equilibrium to observe the bias effect. The red dashed line delimits the transmission window at 0K showing the part of the transmission spectra which must be integrated to obtain the current at 0.6V. The HOMO transmission peaks discussed in the text are highlighted in blue; (b) MPSH state identified as the HOMO orbital for the closed and open isomers. The first row corresponds to the calculations at equilibrium and the second to the non-equilibrium calculations showing the polarization effects.

Fig. 5

 5 Fig. 5 (a) Current response of an 1c-based nanogap device at -1 V under alternate irradiation of light at 700 nm and 365 nm for ON/OFF cycle test (the black range represents the state of the device in the dark); (b) conductance (G) of an 1c-based nanogap device under alternate irradiation of light at 365 nm (blue) and 700 nm (red) for one ON/OFF cycle.

  2 Ru-C≡C-(C 15 S 2 F 6 H 6 )-C≡C-Ru(dppe) 2 -Cl].12

Acknowledgements

The authors thank the support from the Singapore Ministry of Education Tier2 (MOE2010-T2-1-017), Merlion program, Université de Rennes 1, the CNRS, the ANR (ANR-09-JCJC-0025), PHC Tournesol FR 2012. This work was performed using HPC resources from GENCI-CINES and GENCI-IDRIS (Grant 2010-80649). C.V.D. and J.C. are FNRS research fellows.

Notes and references

a School of Materials Science and Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore. E-mail: chenxd@ntu.edu.sg; Web: http † Electronic Supplementary Information (ESI) available: Experimental processes of 1 synthesis, and nanodevice fabrication and measurement; Computational Details; Experimental I-V curves of blank device, different 1c-based devices, and 1o-based device; Geometries of 1o arrangements. See DOI: 10.1039/b000000x/ ‡ Phenyl groups of the dppe were replaced by hydrogen atoms to reduce computational cost § Gold wire pulling also modifies the conductivity.