Balanced Realization of Lossless Systems: Schur Parameters ...

15th IFAC SYSID (SYSID 2009)Saint-Malo, France, July 6-8, 2009These Gramians are the unique (and positive semidefinite)solutions of the respective Lyapunov-Stein equationsW c − AW c A ∗ = BB ∗ , (8)W o − A ∗ W o A = C ∗ C. (9)Moreover, under asymptotic stability of A it holds thatW c is positive definite if and only if the pair (A, B) isreachable, and W o is positive definite if and only if the pair(C, A) is observable. A minimal and asymptotically stablerealization (A, B, C, D) of a transfer function is called abalanced realization if its observability and controllabilityGramians are both diagonal and equal. Then W c = W o =diag{σ 1 , σ 2 , . . . , σ n } in which the positive real quantitiesσ 1 , σ 2 , . . . , σ n denote the (Hankel) singular values of thesystem.This concept of (Lyapunov) balanced realizations was firstintroduced in Moore (1981) in the continuous-time caseand used for model reduction. In Pernebo and Silverman(1982) the same was done for the discrete-time case. Theconcept of balancing has since been extended to severalother classes of linear systems as well as to nonlinearsystems. Balanced realizations are now a well-establishedtool which often exhibit good numerical properties.Any realization (A, B, C, D) which is minimal and asymptoticallystable, is known to be similar to a balancedrealization. Such a balanced realization is in general notunique. Similarity transformation matrices T which do notaffect balancing are permutation matrices (allowing oneto order the squared Hankel singular values on the maindiagonal of W c = W o ) and diagonal scaling matrices withunimodular entries along the main diagonal (including signmatrices). If some singular value has a multiplicity largerthan one, other transformation matrices exist which do notaffect balancing too.2.2 The continuous-time case and the bilinear transformA continuous-time state-space system (A, B, C, D) refersto the set of equationsẋ t = Ax t + Bu t , (10)y t = Cx t + Du t , (11)with t ∈ R. Here, the transfer matrix is similarly given byG(s) = D+C(sI n −A) −1 B and the concepts of minimality,reachability, observability and input-output equivalenceare introduced entirely analogous to the discrete-time case.A continuous-time system (A, B, C, D) and the matrixA are called asymptotically stable if the eigenvalues ofA are all located in the open left half of the complexplane. If (A, B, C, D) is an asymptotically stable realization,then the controllability Gramian W c and the observabilityGramian W o are well defined as the exponentiallyconvergent integralsW c =W o =∫ ∞−∞∫ ∞−∞e At BB ∗ e A∗t dt, (12)e A∗t C ∗ Ce At dt. (13)These Gramians are the unique (and positive semidefinite)solutions of the Lyapunov equationsAW c + W c A ∗ = −BB ∗ , (14)A ∗ W o + W o A = −C ∗ C. (15)Under continuous-time asymptotic stability of A it holdstoo that W c is positive definite if and only if the pair(A, B) is reachable, and W o is positive definite if and onlyif the pair (C, A) is observable. Once again, a minimaland asymptotically stable realization (A, B, C, D) of atransfer function G(s) is called a balanced realization ifits observability and controllability Gramians are bothdiagonal and equal.A well-known way to transfer results from discrete-time tocontinuous-time and vice versa, is by means of the bilineartransform m : C → C, given by:m : z ↦→ s = z − 1z + 1 , (16)of which the inverse is given by m −1 : s ↦→ z = 1+s1−s .If (A, B, C, D) is a discrete-time stable realization of aG( z−1z+1transfer matrix G(z), then the transfer matrix Ĝ(s) =) has a continuous-time stable state-space realization(Â, ̂B, Ĉ, ̂D) given by( ) ̂D Ĉ̂B Â =(D − C(A + In ) −1 B √ )√2C(A + I n ) −12(A + In ) −1 B (A − I n )(A + I n ) −1(17)Conversely it holds that:( ) ( √ )D C ̂D + Ĉ(I= √ n − Â)−1 ̂B 2 Ĉ(I n − Â)−1B A 2(In − Â)−1 ̂B (In + Â)(I n − Â)−1 (18)In fact, if W c and W o denote the Gramians for thediscrete-time system (A, B, C, D), then they also constitutethe Gramians for the associated continuous-time system(Â, ̂B, Ĉ, ̂D), showing that (A, B, C, D) is discretetimebalanced if and only if (Â, ̂B, Ĉ, ̂D) is continuous-timebalanced.2.3 Model order reduction: balance-and-truncateA popular and relatively easy technique for model orderreduction within the class of stable linear systems,is offered by the balance-and-truncate approach. First abalanced realization (A, B, C, D) is computed for whichthe Gramians are monotonically decreasing along theirmain diagonal. Then the original model order n is reducedto a desired lower order k < n by mere truncation of thestate vector, retaining only its first k entries. The matricesA, B and C are likewise replaced by their correspondingleft upper submatrices A 11 , B 1 and C 1 , of sizes k×k, k×mand p × k, respectively. The popularity of this method islargely due to the computational simplicity of the methodand the possibility to choose a reduced model order bymonitoring and thresholding the values on the main diagonalof the Gramians, in combination with the experiencethat the approximations are often quite good in practice.It is shown in Glover (1984) that an L ∞ -error boundis available for the approximations. A drawback of theapproach is the lack of an explicit optimization criterionwhich is minimized by this procedure. When a preferred275

15th IFAC SYSID (SYSID 2009)Saint-Malo, France, July 6-8, 2009approximation criterion is available, approximations of thebalance-and-truncate type can also be used as a startingpoint for a numerical local search technique.Both in discrete-time and in continuous-time, the balanceand-truncateprocedure is easily seen to yield a stableapproximation of order ≤ k. It may happen, though, dueto loss of reachability or observability that the reducedmodel order is actually less than k. For certain subclassesof systems, however, it can be guaranteed that the reducedmodel order does in fact equal k by putting additionalconstraints on the parameterized form of the realization(A, B, C, D). We shall encounter such a situation forsome parameterizations of the class of lossless systems,discussed below. Finally, note that model order reductionby the balance-and-truncate procedure in general doesnot commute with the bilinear transformation betweendiscrete-time and continuous-time.3. LOSSLESS SYSTEMS AND J-INNER FUNCTIONS3.1 Discrete-time lossless, inner, J-lossless and J-innerfunctionsWhen working in discrete-time, for any rational matrixfunction R(z) we shall define the associated matrix functionsR ∗ (z) and R ♯ (z) byR ∗ (z) := R(z) ∗ , and R ♯ (z) := R ∗ (z −1 ). (19)Discrete-time lossless matrix functions are introduced accordingto the following definition.Definition 3.1. In discrete-time, a square p × p rationalmatrix 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 theremaining third condition and the inequalities are all tobe understood in the sense of the partial ordering ofpositive semi-definite Hermitian matrices. Note that theBode amplitude (gain) diagram for a SISO lossless systemis constant 1, for all z = e iω , from which this class ofsystems derives its name.The poles of a rational lossless function are all locatedinside the open unit disk. A lossless function is invertible:it holds that G(z) −1 = G ♯ (z). When G(z) is lossless, alsoG ∗ (z) and G(z) ∗ are lossless. Note that a lossless functionG(z) is unitary at each point z on the unit circle. Thedeterminant of a lossless matrix is a scalar lossless functionof the same McMillan degree. As we shall see below,a lossless matrix of McMillan degree n can always befactored into a product of n lossless matrices of McMillandegree 1, its so-called Potapov factorization.Lossless functions have many applications in systems andcontrol theory, including system identification and digitalsignal processing. See Vaidyanathan and Doǧanata (1989)for a tutorial and many further properties of losslesssystems. See also Genin et al. (1983).When dealing with lossless (matrix) functions, we shallmake use of interpolation theory. To this end it shall beconvenient to generalize the concept of lossless functions inthe following way, see also Potapov (1960); Dym (1989);Genin et al. (1983). Let Σ be a k × k signature matrix,i.e., a diagonal matrix having only the entries ±1 along itsmain diagonal.Definition 3.2. Let Σ be a k × k signature matrix. Then ak × k rational matrix function Θ(z) is called discrete-timeΣ-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 sensethat each one of these three defining properties is impliedby the other two.Any rational matrix function Θ(z) which satisfies theproperty 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) isΣ-lossless, then Θ ∗ (z) is also Σ-lossless whereas both Θ ♯ (z)and Θ(z) −1 are Σ-inner. The class of Σ-lossless functionsis closed under multiplication.To be consistent with the earlier definition of losslessfunctions, note that an I k -lossless function is just calledlossless. Likewise, an I k -inner function is simply calledinner. Throughout this paper we will be much concernedwith lossless and inner functions of size p × p, and withJ-unitary and J-inner functions of size 2p × 2p, where Jdenotes the following signature matrix:( )Ip 0J =. (26)0 −I pNote that J-inner and J-lossless functions in general mayhave poles everywhere in the complex plane, but innerfunctions are analytic inside the unit disk and losslessfunctions are analytic outside the unit disk (including atthe point at infinity). J-Lossless functions play a centralrole in the theory of chain scattering systems, see Kimura(1997).3.2 Σ-Balanced realizations of Σ-lossless matrix functionsin discrete timeIn state-space, balanced realizations of discrete-time losslesssystems are surprisingly easy to characterize. It is wellknown(cf., e.g., Hanzon ( et al. )(2006) and the referencesD Cgiven there) that R = is a balanced realizationB Amatrix of a lossless system if and only if R is a unitarymatrix and A is asymptotically stable. It then holds thatthe Gramians are given by W c = W o = I n . If R is unitaryand A has one or more eigenvalues on the unit circle, thenthe associated system is still lossless, but minimality doesnot hold.Note that all balanced realizations of a lossless system areobtained from a given one, by applying unitary state-space276

15th IFAC SYSID (SYSID 2009)Saint-Malo, France, July 6-8, 2009transformations. The fact that all the Hankel singularvalues are identical to 1 makes that there is maximalfreedom in choosing a balanced realization. This freedomcan be used to build additional nice properties into therealization.The following result applies to the more general classof discrete-time Σ-lossless systems. A proper Σ-losslesstransfer function G(z) of McMillan degree n always admitsa minimal Σ-balanced state-space realization (A, B, C, D),i.e., one for which the realization matrix satisfies(D CB A) ( ) ( )Σ 0 D ∗ B ∗0 I n C ∗ A ∗ =( )Σ 0. (27)0 I nConversely, if this property holds, then G(z) is proper Σ-lossless of McMillan degree ≤ n. For a Σ-balanced statespacerealization (A, B, C, D) it holds that the solutionsto the associated Lyapunov-Stein equations with modifiedright-hand sides BΣB ∗ and C ∗ ΣC, respectively, againsatisfy W c = W o = I n .3.3 J-unitary and J-inner functionsSome particular J-inner functions that are intensivelyused in interpolation theory are the constant J-unitarymatrices. We have the following result, for which detailsand proofs can be found in Dym (1989).Theorem 1. Every constant J-unitary matrix can be representedin a unique way as follows:M = H(E)(P 00 Q), (28)where P and Q are p × p unitary matrices and H(E)denotes the Halmos extension of a strictly contractive p×pmatrix E (i.e., such that I − E ∗ E > 0). This Halmosextension H(E) is defined by( )(I − EEH(E) =∗ ) −1/2 E(I − E ∗ E) −1/2E ∗ (I − EE ∗ ) −1/2 (I − E ∗ E) −1/2 . (29)It holds that H(E) is Hermitian, J-unitary and invertiblewith inverse H(E) −1 = H(−E).The important class of elementary J-inner factors (i.e., J-inner functions of McMillan degree one) is characterizedby the following theorem.Theorem 2. Let Θ(z) be an elementary J-inner factor witha pole at z = 1/w. Then, apart from a constant J-unitarymultiplier on the right, Θ(z) must be in one of the followingthree forms:(1) If w ∈ C, |w| ≠ 1,( ) z − w xx ∗Θ(z) = I 2p +1 − wz − 1 Jx ∗ Jx , (30)for some vector x ∈ C 2p xsuch that∗ Jx1−|w|> 0.2(2) If |w| = 1,(z + w)Θ(z) = I 2p + δ(z − w) xx∗ J, (31)for some nonzero x ∈ C 2p such that x ∗ Jx = 0, and forsome real δ > 0.(3) If w = ∞, then( ) 1 xx ∗Θ(z) = I 2p +z − 1 Jx ∗ Jx , (32)for some x ∈ C 2p such that x ∗ Jx < 0.3.4 The continuous-time caseIn the continuous-time case, the definition of a losslessmatrix function is as follows.Definition 3.3. In continuous-time, a square p × p rationalmatrix 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) nowreads:Definition 3.4. Let Σ be a k × k signature matrix. Then ak × k rational matrix function Θ(z) is called continuoustimeΣ-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-timeΣ-lossless system is characterized by the following identities:A + A ∗ = −BΣB ∗ , (39)C = −DΣB ∗ , (40)DΣD ∗ = Σ. (41)Any proper Σ-lossless rational matrix function admits sucha Σ-balanced realization. When the right-hand sides of theLyapunov equations are modified to −BΣB ∗ and −C ∗ ΣC,respectively, it holds again that W c = W o = I n . Therelations for balanced realizations of lossless functions aresimply 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 Σ-lossless continuous-time function ̂Θ(s) = Θ( z−1z+1), then anyassociated Σ-balanced realization (A, B, C, D) for Θ(z) istransformed into a Σ-balanced realization (Â, ̂B, Ĉ, ̂D) for̂Θ(s) according to Eqns. (17)-(18).3.5 Kronecker structures, triangular forms and truncationof the state vectorWhen choosing a balanced realization of a lossless system,one has the freedom of the unitary group to build inadditional properties by means of state-space transformation.One property of interest is upper triangularity of thereachability matrix K = ( B AB . . . A n−1 B ) , as uppertriangularity is preserved under state vector truncation ina balance-and-truncate approach to model order reduction.Generically, by applying QR-decomposition to K, a unitarystate-space transformation matrix can be computedwhich makes the reachability matrix K upper triangularwhile preserving balancedness. It depends on the Kroneckerstructure of the lossless system whether this is277

15th IFAC SYSID (SYSID 2009)Saint-Malo, France, July 6-8, 2009indeed possible. In such a case, the block-partitioned n ×(m + n) matrix (B, A) is also upper triangular and statetruncation preserves this property too.As a result, state truncation for such a balanced realizationof a continuous-time lossless system is easilyseen to produce a balanced realization of a reduced orderapproximation system. In the discrete-time lossless case,balancedness is generally lost. It may be restored by reorthonormalizationof the last m columns of the truncatedrealization matrix, as balancedness requires a unitary realizationmatrix, but this affects the dynamical matrix andresults in a definitely less attractive procedure than in thecontinuous-time case.When dealing with any given Kronecker structure, a generalizedprocedure can be developed in which an associatedsuitable selection of n columns of the reachability matrixK is made upper triangular. This has been worked out inPeeters et al. (2007) using so-called nice selections, yieldinga procedure which can be applied both in continuoustimeand in discrete-time. This produces atlases of coordinatecharts, exhibiting convenient triangularity propertieswhich are preserved under state truncation, together coveringthe manifold of p × p lossless systems.4. INTERPOLATION THEORY FOR LOSSLESSSYSTEMS4.1 Linear fractional transformationsLinear fractional transformations occur extensively in representationformulas for the solution of various interpolationproblems, see Ball et al. (1990). They are at theheart of the parametrization of lossless functions throughthe tangential Schur algorithm. The properties of a linearfractional transformation of matrices were studied in aform adapted to the needs of J-theory in Potapov (1988)and in an algebraic setting in Young (1984).Let Θ ∈ C 2p×2p (z) be an invertible rational matrix ( in ) z,Θ1 Θblock partitioned into four p×p blocks as Θ =2.Θ 3 Θ 4Associated with Θ, let the linear fractional transformationT Θ be defined to act on rational matrices G ∈ C p×p (z) inthe 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 LFTsassociated 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 −1Θ = T Θ −1.It also holds that T Φ = T Ψ if and only if there exists ascalar function λ ∈ C(z) for which Φ = λΨ. The latterstill holds true, if the set of rational functions G on whichthe LFTs T Θ act, is restricted to the real lossless functions;see Hanzon et al. (2006).We have the following result, which explains the importanceof LFTs with J-inner matrix functions for the constructionof lossless functions.Theorem 3. Let Θ be J-inner of size 2p × 2p and ofMcMillan degree m. If G is p × p lossless and of McMillandegree n, then the matrix function Ĝ = T Θ(G) is welldefined,lossless and of McMillan degree ≤ n + m.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 LFTT Θ (G) does not affect the McMillan degree of the losslessfunction G.4.2 The tangential Schur algorithmIn Ober (1987b); Hanzon and Ober (1998) balanced statespacecanonical forms have been constructed for variousclasses of linear systems, with special properties of theseclasses (such as stability) built in. A balanced canonicalform for continuous-time SISO lossless systems was presentedin Ober (1987a). This continuous-time lossless caseis central to the constructions of Hanzon and Ober (1997),Hanzon and Ober (1998). In the scalar case, it is balancedwith a positive upper triangular reachability matrix, whileKronecker indices and nice selections are used in the multivariablecase to arrive at balanced overlapping canonicalforms.For discrete-time lossless systems, application of the bilineartransform preserves balancedness, but it destroyspositive upper triangularity of the reachability matrix,which is unattractive for balance-and-truncate type procedures.Therefore, in Hanzon and Peeters (2000) a balancedcanonical form for discrete-time lossless SISO systems wasdeveloped directly, which is parameterized by Schur parametersand which yields an upper triangular reachabilitymatrix K. The unitary realization matrix R is factored intoa product of elementary unitary matrices, thus facilitatinga recursive construction of the balanced canonical form.This SISO balanced canonical form was generalized to theMIMO case in Hanzon et al. (2006), using linear fractionaltransformations and the tangential Schur algorithm.The tangential Schur algorithm yields a flexible approachto the recursive construction of a parametrization of thespace of lossless systems of a given order n. It is derivedfrom the method of Alpay et al. (1994); Fulcheri andOlivi (1998), where the tangential Schur algorithm isused to construct an infinite atlas of generic overlappingparameterizations for the space of p × p inner functionsof McMillan degree n. The relationship between these twosituations is constituted by the map R(z) ↦→ R ♯ (z), whichrelates p × p inner functions of McMillan degree n to p × plossless functions of McMillan degree n.In the context of inner functions, the tangential Schuralgorithm consists of an iterative procedure by which agiven p × p inner function of McMillan degree n is reducedin n iteration steps to a p × p inner function of McMillandegree 0 (i.e., to a constant unitary matrix). In eachiteration step the McMillan degree of the inner functionat hand is reduced by 1, by application of a suitablelinear fractional transformation which is chosen to meeta particular interpolation condition, and which involvesan associated J-inner matrix function of McMillan degree1. The actual parametrization procedure consists of thereverse process, by which a chart of inner functions ofMcMillan degree n is constructed in n iteration steps,starting from an initial unitary matrix. The choice ofinterpolation points and (normalized) direction vectors278

15th IFAC SYSID (SYSID 2009)Saint-Malo, France, July 6-8, 2009function in terms of A and C, and subsequently eliminatedfrom the problem. As we have seen in the previous subsection,output normal pairs (C, A) are intimately relatedto lossless systems satisfying a normalization condition,and they can be conveniently parameterized with Schurvectors. The state-space approach discussed in the presentpaper allows one to proceed in the discrete-time case withparameterizations which are constructed as a product ofunitary matrices.5.3 L 2 -System approximation and RARL2In L 2 -system approximation, the Douglas-Shapiro-Shieldsfactorization, see Douglas et al. (1970), can be used to factora given strictly proper stable rational transfer matrixfunction H into a product GP in which G is lossless ofthe same McMillan degree as H and P is unstable. Thefactor G is unique up to a right unitary factor. The L 2 -approximation problem can subsequently be reformulated,analytically, entirely in terms of G. Parameterization ofthe class of lossless systems according to the state-spacemethods described here, allows one to handle the stabilityconstraint conveniently, by optimizing over the class oflossless systems.In Marmorat et al. (2002) this approach was worked outboth in discrete-time and in continuous-time, using thebalanced state-space parameterizations discussed above.Later, the procedure was implemented in the Matlabsoftware package RARL2, see Olivi and Marmorat (2003),which employs a chart switching strategy to move tothe center of a new parameter chart when the currentapproximation moves too far away from the center of thecurrent chart. This employs the fact that each losslesssystem can be put at the center of an adapted chart,where all the Schur vectors v k are zero, by means ofthe Potapov factorization. The distance to the centerof a chart is intuitively measured by the lengths of theSchur vectors used to represent the lossless system in thatchart: when the length of a Schur vector becomes equalto 1, the boundary of a chart is reached and degenerationphenomena may occur.We find it interesting to note that the numerical conditioningof the implemented computational procedure is considerablybetter for the balanced state-space realizationsin discrete-time with their associated unitary matrix factorizations,when compared to an earlier implementationwhich performed analogous computations directly on thelevel of transfer functions (see Fulcheri and Olivi (1998)).5.4 Orthogonal wavelets from filter banksIn modern digital signal processing, an important class ofwavelets is constituted by the orthogonal wavelets that aregenerated by power complementary filter banks. In Strangand Nguyen (1996) a detailed exposition of this topic isgiven. In its classical form, the procedure is as follows.First, two copies of the same discrete-time input signal arepassed through a lowpass filter C(z) and a highpass filterD(z), to produce approximation coefficients and detailcoefficients, respectively. These output sequences are thendownsampled by a factor 2, i.e., only the even-indexedvalues (the even phases of the signals) are stored. When thefilters C(z) and D(z) are power complementary (see alsoVaidyanathan and Doǧanata (1989)), no information lossoccurs and the output sequences admit an interpretation interms of associated orthonormal bases, spanned by the socalledscaling function (used for approximation purposes)and the wavelet function (used to analyze the details). Onethen proceeds repeatedly, in a likewise fashion, by processingthe approximation signal (the downsampled output ofthe filter C(z)) to compute approximation coefficients anddetail coefficients on coarser scales.For each level of the analysis, an equivalent procedurecan also be considered, in which the input signal is firstsplit into its two phases (the even and odd phases), whichare then jointly processed by a 2 × 2 polyphase filterH(z) to produce the same (downsampled) sequences ofapproximation and detail coefficients as before. It thenfollows that orthonormality of the wavelet bases, andpower complementarity of the filters C(z) and D(z), are infact equivalent to H(z) being (discrete-time) lossless. Thetheory presented here for the construction of parameterizationsof lossless systems can thus be applied to facilitateparametric optimization over the class of lossless 2 × 2matrix functions of a given degree n. This is particularlyuseful for wavelet design, especially when a wavelet isdesigned for a specific application. Initial investigationsusing this approach are presented in Karel et al. (2005).Extensions to more complicated situations are currentlyunder investigation. One issue of practical interest concernsthe number of vanishing moments that a wavelethas. As it turns out, the problem of building in vanishingmoments can be reformulated as an interpolation problemfor 2 × 2 lossless systems on the stability boundary (atz = 1) involving higher order derivatives. Another obviousextension concerns multiwavelets, where the polyphasematrix is generalized to become a lossless matrix of size2r × 2r with r > 1; see also Peeters et al. (2006).6. CONCLUSIONSIn this paper we have presented an overview of the mainresults which link balanced parameterizations of losslesssystems to interpolation theory and Schur parameters invarious settings: discrete-time and continuous-time, SISOand MIMO. It is indicated which choices in the tangentialSchur algorithm (in particular which choice of interpolationpoints) supports the construction of triangular pivotstructures in reachability matrix. It is possible to deal withspecial Kronecker structures and restrictions to importantsubclasses (such real lossless systems) are convenientlyhandled. Certain extensions of the presented theory, notdiscussed here, are also available in the literature. Mostnotably these involve Nudelman interpolation problemsand symmetric inner functions. Further lines of researchconcern the application domains indicated above, whichoften present specific additional constraints that requirespecial attention.REFERENCESAlpay, D., Baratchart, L., and Gombani, A. (1994). Onthe differential structure of matrix-valued rational innerfunctions. Operator Theory: Advances and Applications,73, 30–66.282

15th IFAC SYSID (SYSID 2009)Saint-Malo, France, July 6-8, 2009Ball, J., Gohberg, I., and Rodman, L. (1990). Interpolationof rational matrix functions, volume 45 of OperatorTheory: Advances and Applications. Birkhäuser.Baratchart, L. (1990). On the topological structure of innerfunctions and its use in identification, volume 8, 51–59.Birkhäuser.Bruls, J., Chou, C., Haverkamp, B., and Verhaegen, M.(1999). Linear and non-linear system identificationusing separable least-squares. European Journal ofControl, 5(1), 116–128.Douglas, R., Shapiro, H., and Shields, A. (1970). Cyclicvectors and invariant subspaces for the backward shiftoperator. Annales de l’Institut Fourier (Grenoble), 20,37–76.Dym, H. (1989). J-contractive matrix functions, reproducingkernel spaces and interpolation, volume 71 of CBMSlecture notes. American Mathematical Society, RhodesIsland.Fulcheri, P. and Olivi, M. (1998). Matrix rational H 2 -approximation: A gradient algorithm based on Schuranalysis. SIAM Journal on Control and Optimization,36(6), 2103–2127.Genin, Y., Van Dooren, P., Kailath, T., Delosme, J., andMorf, M. (1983). On Σ-lossless transfer functions andrelated questions. Linear Algebra and its Applications,50, 251–275.Glover, K. (1984). All optimal Hankel-norm approximationsof linear multivariable systems and their L ∞ -errorbounds. International Journal of Control, 39(6), 1115–1193.Golub, G. and Pereyra, V. (1973). The differentiationof pseudoinverse and nonlinear least squares problemswhose variables separate. SIAM Journal of NumericalAnalysis, 10(2), 413–432.Hanzon, B. and Ober, R. (1997). Overlapping blockbalancedcanonical forms and parametrizations: theSISO case. SIAM Journal of Control and Optimization,35, 228–242.Hanzon, B. and Ober, R. (1998). Overlapping blockbalancedcanonical forms for various classes of linearsystems. Linear Algebra and its Applications, 281, 171–225.Hanzon, B., Olivi, M., and Peeters, R. (2006). Balancedrealizations of discrete-time stable all-pass systems andthe tangential Schur algorithm. Linear Algebra andApplications, 418, 793–820.Hanzon, B., Olivi, M., and Peeters, R. (2008). Losslessscalar functions: Boundary interpolation, Schur algorithmand Ober’s canonical form. In Proceedings ofthe 47th Conference on Decision and Control, Cancun,1845–1850.Hanzon, B., Olivi, M., and Peeters, R. (2009). Subdiagonalpivot structures and associated canonical forms understate isometries. In Proceedings of SYSID 2009, St.Malo.Hanzon, B. and Peeters, R. (2000). Balanced parametrizationsof stable SISO all-pass systems in discrete-time.Mathematics of Control, Signals, and Systems, 13, 240–276.Kailath, T. (1980). Linear Systems. Prentice-Hall.Karel, J., Peeters, R., Westra, R., Moermans, K., Haddad,S., and Serdijn, W. (2005). Optimal discrete waveletdesign for cardiac signal processing. In Proceedings ofthe 27th IEEE EMBS Annual International Conference(EMBC 2005), Shanghai, 2769–2772.Kimura, H. (1997). Chain-Scattering Approach to H ∞ -Control. Birkhäuser.Leblond, J. and Olivi, M. (1998). Weighted H 2 -approximation of transfer functions. Mathematics ofControl, Signals, and Systems, 11, 28–39.Marmorat, J., Olivi, M., Hanzon, B., and Peeters, R.(2002). Matrix rational H 2 -approximation: a statespaceapproach using Schur parameters. In Proceedingsof the 41st Conference on Decision and Control, LasVegas, 4244–4249.Moore, B. (1981). Principal component analysis in linearsystems: Controllability, observability, and model reduction.IEEE Transactions on Automatic Control, AC-26,17–32.Ober, R. (1987a). Asymptotically stable allpass transferfunctions: canonical form, parametrization and realization.In Proceedings of the IFAC World Congress, Munich.Ober, R. (1987b). Balanced realizations: canonical form,parametrization, model reduction. International Journalof Control, 46, 643–670.Olivi, M. and Marmorat, J. (2003). RARL2, Matlabbased software package developed by Team APICSat INRIA, Sophia-Antipolis, France. Available at:http://www-sop.inria.fr/apics/RARL2/rarl2.html.Peeters, R., Hanzon, B., and Olivi, M. (2001). Linearfractional transformations and balanced realization ofdiscrete-time stable all-pass systems. In Proceedingsof the 1st IFAC Symposium on System Structure andControl, Prague.Peeters, R., Hanzon, B., and Olivi, M. (2007). Canonicallossless state-space systems: Staircase forms and theSchur algorithm. Linear Algebra and Applications, 425,404–433.Peeters, R., Karel, J., Westra, R., Haddad, S., and Serdijn,W. (2006). Multiwavelet design for cardiac signalprocessing. In Proceedings of the 28th IEEE EMBSAnnual International Conference (EMBC 2006), NewYork City, 1682–1685.Pernebo, L. and Silverman, L. (1982). Model reductionvia balanced state space representations. IEEE Transactionson Automatic Control, AC-27, 382–387.Potapov, V. (1960). The multiplicative structure of J-contractive matrix functions. Amer. Math. Soc. Transl.(2), 15, 131–243.Potapov, V. (1988). Linear fractional transformations ofmatrices. Amer. Math. Soc. Transl. (2), 138, 21–35.Strang, G. and Nguyen, T. (1996). Wavelets from FilterBanks. Cambridge-Wellesley.Vaidyanathan, P. and Doǧanata, Z. (1989). The role oflossless systems in modern digital signal processing: atutorial. IEEE Transactions on Education, 32(3), 181–197.Young, N. (1984). Linear fractional transformations inrings and modules. Linear Algebra and its Applications,56, 251–290.283