Distance between behaviors and rational representations

DISTANCE BETWEEN BEHAVIORS AND RATIONALREPRESENTATIONSH.L. TRENTELMAN AND S.V. GOTTIMUKKALA ∗Abstract. In this paper we study notions of distance between behaviors of linear differentialsystems. We introduce four metrics on the space of all controllable behaviors which generalize existingmetrics on the space of input-output systems represented by transfer matrices. Three of these aredefined in terms of gaps between closed subspaces of the Hilbert space L 2 (R). In particular wegeneralize the ‘classical’ gap metric. We express these metrics in terms of rational representationsof behaviors. In order to do so, we establish a precise relation between rational representationsof behaviors and multiplication operators on L 2 (R). We introduce a fourth behavioral metric asa generalization of the well-known ν-metric. As in the input-output framework, this definition isgiven in terms of rational representations. For this metric we however establish a representation free,behavioral characterization as well. We make a comparison between the four metrics, and comparethe values they take, and the topologies they induce. Finally, for all metrics we make a detailedstudy of necessary and sufficient conditions under which the distance between two behaviors is lessthan one. For this, both behavioral as well as state space conditions are derived in terms of drivingvariable representations of the behaviors.Key words. behaviors, linear differential systems, gap metric, ν-metric, rational representations,multiplication operatorsAMS subject classifications. 93B20, 93B28, 93B36, 93C051. Introduction. This paper deals with notions of distance between systems.In the context of linear systems with inputs and outputs, several concepts of distancehave been studied in the past. Perhaps the most well-known distance concept is thatof gap metric introduced by Zames and El-Sakkary in [28], and extensively used byGeorgiou and Smith in the context of robust stability in [7]. The distance betweentwo systems in the gap metric can be calculated, but the calculation is by no meanseasy and requires the solution of an H ∞ optimization problem, see [6]. A distanceconcept which is equally relevant in the context of robust stability is the so-calledν-gap, introduced by Vinnicombe in [22], [21]. Computation of the ν-gap betweentwo systems is much easier than that of the ordinary gap, and basically requirescomputation of the winding number of a certain proper rational function, followedby computation of the L ∞ -norm of a given proper rational matrix. A third distanceconcept is that of L 2 -gap, which is the most easy to compute, but which is not at alluseful in the context of robust stability, as shown in[21]. More recently an alternativenotion of gap for linear input-output systems was introduced by Ball and Sasane in[13], allowing also non-zero initial conditions of the system.In this paper we will put the above four distance concepts into a more general,behavioral context, extending them to a framework in which the systems are notnecessarily identified with their representations (e.g. transfer matrices), but in which,instead, their behaviors, i.e. the spaces of all possible trajectories of the systems,form the core of the theory. This idea was put forward for the first time in [12].Indeed, we will introduce four metrics on the set of all (controllable) behaviors witha fixed number of variables that we will call the L 2 -metric, the Zames (Z) metric,the Sasane-Ball (SB) metric, and the Vinnicombe (V) metric. The first three will∗ Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen,P. O Box 800, 9700 AV Groningen, The Netherlands Phone:+31-50-3633998, Fax:+31-50-3633800h.l.trentelman@rug.nl.1

2be defined in terms of the concept of ’gap’ between closed subspaces of the Hilbertspace L 2 (R, C q ) of square integrable functions, the fourth one, the V-metric, will bedefined in terms of representations of the behaviors. Of course, no a priori inputoutputpartition of the system variables needs to be given. Our set-up will howeverbe applicable also to the ‘classical’ input-output framework. We will establish severalbehavioral, representation free characterizations of properties of the metrics we haveintroduced. We will also study the interrelation between the metrics and compare thetopologies they induce.We want to mention that the idea of distance between behaviors was also studiedin a more general framework in [3]. The latter paper deals with behaviors as generalsubsets of the set of all functions from time axis to signal space (not necessarilyrepresentable by higher order linear differential equations), and introduces a notionof distance between such behaviors.A key ingredient in our paper will be the notion of rational representation ofbehaviors, recently introduced in [27]. Whereas, originally, behaviors of linear differentialsystems were defined as kernels and images of polynomial differential operators,in [27] it was explained how they also allow representations as ‘kernels’ and ‘images’of ‘rational differential operators’ in a mathematically consistent, natural way. Infact, for a given behavior, there is freedom in the choice of the rational matrices usedfor its representation, and they can for example be chosen to be proper, bounded onthe imaginary axis, stable, prime and inner, all at the same time. In this paper wewill use these properties of the rational representations to describe the relationshipbetween ‘kernels’ and ‘images’ of the ‘rational differential operators’ on the one hand,and kernels and images of the (operator theoretic) multiplication operators associatedwith the rational matrices on the other.Using the relation between rational representations of behaviors and multiplicationoperators, we will on the one hand express the L 2 -metric, Z-metric and SB-metricin terms of rational representations, and on the other hand give a representation freecharacterization of the V-metric. As a special case, this will provide a representationfree characterization of the ‘classical’ ν gap in the input-output framework.For each of the four metrics we will also characterize under which conditionsthe distance between two behaviors is strictly less than one. For the L 2 - metricand the V-metric this will turn out to be relatively easy, and we obtain behavioralcharacterizations for this. However, for the Z-metric and the SB-metric this is moreinvolved, and we will make a detailed study of this problem using driving variablestate representations of the behaviors involved. This will also involve the problem ofstate represention of the kernel of a Toeplitz operator with an invertible symbol.The outline of this paper is as follows. In Section 2 we review behaviors oflinear differential systems, and introduce the L 2 -metric, Z-metric and SB-metric. InSection 3 we briefly review rational kernel and image representations of behaviors. Wealso show that behaviors admit rational image representations in which the rationalmatrices are proper and stable, right prime, inner, and have no zeros. An analogousresult is proven for rational kernel representations. In Section 4 we establish in detailthe relation between rational image and kernel representations of behaviors on the onehand, and the images and kernels of the ‘classical’ multiplication operators associatedwith these representations on the other. Using this relation, in Section 5 we expressall three behavioral metrics that were introduced in Section 2 in terms of rationalrepresentations of the behaviors. We also show that our definitions of Z-metric andSB-metric generalize ‘classical’ gap metrics for input-output systems represented by

transfer matrices. In Section 6 we introduce the fourth metric, the V-metric. Unlikethe other three metrics, the definition of this metric is in terms of representations ofthe behaviors, involving the notion of winding number. We will in this section derivea new, representation-free, behavioral characterization of this metric. Section 7 dealswith a comparison of the four metrics. We will both compare the values they take, andalso the topologies they induce. In Section 8, for each of the metrics we find conditionsunder which the distance between two behaviors is strictly less than the value one.For the L 2 -metric and the V-metric this issue is readily dealt with, and we obtainbehavioral characterizations. For the Z-metric and the SB-metric this is a harderproblem, and in Sections 9 and 10 we study this problem using driving variable staterepresentations of the behaviors. This also involves the study of Toeplitz operatorswith an invertible symbol. The paper closes with conclusions in Section 11.1.1. Basic concepts and notation. We now introduce the basic concepts andnotation used in this paper. We will denote the ring of polynomials with real coefficientsby R[ξ]. The field of real rational functions is denoted by R(ξ), the ring ofproper real rational functions by R(ξ) P . As usual, a proper real rational function willbe called stable if its poles are in C − := {λ ∈ C | Re(λ) < 0}. It is called anti-stableif its poles are in C + := {λ ∈ C | Re(λ) > 0}. RL ∞ will denote the ring of all properreal rational functions without poles on the imaginary axis, and RH ∞ denotes thering of all proper and stable real rational matrices. RH∞ − will denote the ring ofproper antistable real rational functions.For a given ring R, a matrix G with coefficients in R is called right prime overR (left prime over R) if there exists a matrix G + with coefficients in R such thatG + G = I (GG + = I). In this paper the condition of primeness will occur withrespect to the rings R[ξ], R(ξ) P , RL ∞ , RH ∞ and RH∞. − We will be using matriceswith coefficients from the above rings. In order to streamline notation we will supressthe dimensions. For example, the spaces of all real rational matrices with coefficientsin RL ∞ or RH ∞ will again be denoted by RL ∞ or RH ∞ .For a given real rational matrix G we denote G ∗ (ξ) := G ⊤ (−ξ). A proper realrational matrix is called inner if G ∗ G = I and co-inner if GG ∗ = I. Note that ifG ∈ RL ∞ is inner (co-inner), then it is right prime (left prime) over RL ∞ . Theanalogous statement is not true for left and right primeness over RH ∞ . G is calledunitary if G ∗ G = GG ∗ = I. We denote the usual infinity norm of G ∈ RL ∞ by ‖G‖ ∞ .For a given complex matrix M, σ max M and σ min M denote the largest and smallestsingular value, respectively. Note that ‖G‖ ∞ = sup ω∈R σ max G(iω). For a given realrational matrix G, its zeros are the roots of the nonzero numerator polynomials inthe Smith-McMillan form of G (see [27]).For a given real rational function g without poles or zeros on the imaginary axis,the winding number of g is defined as the net number of counterclockwise encirclementsof the origin by the (closed) contour g(λ) as λ traverses in counterclockwise directiona standard D-contour enclosing all poles and zeros of g in C + . The winding numberof g is denoted by wno(g), and is equal to the difference Z −P , where Z is the numberof zeros and P is the number of poles of g in C +In this paper, we will only consider real-valued signals. For a given integer q,we denote by L 2 (R, R q ) the space of all Lebesgue measurable functions w : R → R qsuch that ∫ ∞−∞ ‖w(t)‖2 dt < ∞. This is a Hilbert space with inner product given by< w 1 , w 2 >= ∫ ∞−∞ w 1(t) ⊤ w 2 (t)dt. The usual L 2 -norm is denoted by ‖ · ‖ 2 . In thesequel, in the notation we will mostly suppress the dimension q, and simply denotethis Hilbert space by L 2 (R). The subset of all signals w such that w(t) = 0 for3

4almost all t < 0 is a closed subspace of L 2 (R) and is denoted by L 2 (R + ). Likewise,L 2 (R − ) will denote the closed subspace consisting of all signals w such that w(t) = 0for almost all t > 0. Obviously, L 2 (R − ) ⊥ = L 2 (R + ). The orthogonal projections ofL 2 (R) onto L 2 (R + ) and L 2 (R − ) are denoted by Π + and Π − , respectively.In addition to their time-domain descriptions, signals allow descriptions in the frequencydomain. For given integer q, we denote by∫ L 2 (iR, C q ) the space of all Lebesguemeasurable functions W : iR → C q such that 1 ∞2π −∞ ‖W (iω)‖2 dω < ∞. This is again∫a Hilbert space, with inner product given by < W 1 , W 2 >= 1 ∞2π −∞ W 1(iω) ∗ W 2 (iω)dω.Again, we suppress the dimension q in the notation and denote this space by L 2 (iR).We will denote the usual Hardy space of of ∫ all complex valued functions W that areanalytic in C + 1 ∞and that satisfy sup σ>0 2π −∞ W 1(σ +iω) ∗ W 2 (σ +iω)dω < ∞ by H 2 .This space can be identified with a closed subspace of L 2 (iR) (see [5]).The usual Fourier transformation is denoted by F. The Fourier transform W (iω)of a (real valued) signal w ∈ L 2 (R) satisfies the property W (−iω) = W (iω), where vdenotes the componentwise complex conjugate of v ∈ C q . Define(1.1) S := {W ∈ L 2 (iR, C q ) | W (−iω) = W (iω) for all ω ∈ R}.It is well-known that F is a linear transformation and that it defines a bijectionbetween L 2 (R) and the subpace S. Furthermore, FL 2 (R + ) = H 2 ∩ S and FL 2 (R − ) =H2 ⊥ ∩ S. The inverse of F will be denoted by F −1 .We will denote by L loc (R, R q ) the space of al measurable functions w from Rto R q that are locally integrable, i.e. for all t 0 , t 1 the integral ∫ t 1t 0‖w(t)‖dt is finite.For systems of linear differential equations R( d dt)w = 0, solutions w are understoodto be in this space, and the differential equation is understood to be satisfied indistributional sense. If the dimensions are clear from the context we use the notationL loc .2. Distance between behaviors. In the behavioral context, a linear differentialsystem is defined as a triple Σ = (R, R q , B), with R the time axis, R q the signalspace, and B ⊂ L loc (R, R q ) the behavior, which is equal to the space of solutions of a finitenumber of higher order, linear, constant coefficient differential equations. For anysuch system there exists a real polynomial matrix R such that B is equal to the spaceof solutions of the system of differential equations R( d dt)w = 0. This is then called apolynomial kernel representation of the behavior B and we write B = ker R( d dt). Theset of all linear differential systems with q variables is denoted by L q . The subset ofall controllable ones is denoted by L q cont. We denote by m(B) (the input cardinality)the number of inputs of B. For an overview of the basic material on behaviors, werefer to [11], [26].In this section we will introduce three metrics on the space L q cont of behaviors ofcontrollable linear differential systems, inspired by the several notions of ‘gap’ in thecontext of input-output systems represented by transfer matrices. The general ideais to associate with every controllable behavior B a suitable subspace of the Hilbertspace L 2 (R), and in this way define a metric on L q cont in terms of the usual metric onthe set of closed subspaces of L 2 (R). This can be done in several ways, and each ofthese choices will lead to a particular metric on L q cont. In later sections, we will studythese metrics, and compare them.In order to set the scene, we now first review some standard material on the gapbetween closed subspaces of a Hilbert space (see e.g. [1] or [23]). For a given Hilbert

space H, the directed gap between two closed subspaces V 1 and V 2 of H is defined asgap(V ⃗ 1 , V 2 ) :=supv 1∈V 1‖v 1‖=1The gap between V 1 and V 2 is then defined asinf ‖v 1 − v 2 ‖.v 2∈V 2gap(V 1 , V 2 ) := max( gap(V ⃗ 1 , V 2 ), gap(V ⃗ 2 , V 1 )).The gap between two subspaces always lies between zero and one, i.e. 0 ≤ gap(V 1 , V 2 ) ≤1 for all V 1 , V 2 . It is also well-known that the gap between two subspaces can beexpressed in terms of the norms of the orthogonal projection operators onto thesesubspaces. More specific, gap(V ⃗ 1 , V 2 ) = ‖Π V ⊥2Π V1 ‖. Here, Π V is the orthogonal projectionof H onto V. Also, gap(V 1 , V 2 ) = ‖Π V1 − Π V2 ‖. Another relevant fact isthat the gap does not change after taking orthogonal complements, in other words,gap(V 1 , V 2 ) = gap(V ⊥ 1 , V ⊥ 2 ).In this paper, the relevant Hilbert space will always be H = L 2 (R). The directedgap and gap between two closed linear subspaces of the Hilbert space L 2 (R) will bedenoted by gap ⃗ L2and gap L2, respectively.We now introduce the following metrics on the space L q cont of controllable lineardifferential systems.2.1. L 2 -metric. The first metric that we consider is the one that is directlyinduced by the gap on the Hilbert space L 2 (R). We will call it the L 2 -metric:Definition 2.1. Let B 1 , B 2 ∈ L q cont. The L 2 -metric, denoted by d L2 (B 1 , B 2 ),is defined as the gap between B 1 ∩ L 2 (R) and B 2 ∩ L 2 (R) in L 2 (R):d L2 (B 1 , B 2 ) := gap L2(B 1 ∩ L 2 (R), B 2 ∩ L 2 (R)).The L 2 -metric measures the distance between two behaviors as the gap between theirL 2 -behaviors over the whole real line.2.2. Zames metric. The second metric that we introduce is obtained by intersectingthe behaviors with the subspace L 2 (R + ) of all signals that are zero in thepast. We will call it the Zames metric because in the input-output transfer matrixcontext it will turn out to coincide with the ‘classical’ gap metric.Definition 2.2. Let B 1 , B 2 ∈ L q cont. The Zames metric, denoted by d Z (B 1 , B 2 ),is defined as the gap between B 1 ∩ L 2 (R + ) and B 2 ∩ L 2 (R + ) in the Hilbert spaceL 2 (R):d Z (B 1 , B 2 ) := gap L2(B 1 ∩ L 2 (R + ), B 2 ∩ L 2 (R + )).In the sequel we will often use the shorthand terminology Z-metric.2.3. Sasane-Ball metric. A third metric that we will consider is obtained byprojecting the L 2 -behaviors onto the future, and subsequently taking their gap in theHilbert space L 2 (R). It will be called the Sasane-Ball metric since it will turn outto coincide with the behavioral distance introduced in the input-output framework in[13]. Recall that Π + is the orthogonal projection of L 2 (R) onto L 2 (R + ).Definition 2.3. Let B 1 , B 2 ∈ L q cont. The Sasane-Ball metric, denoted byd SB (B 1 , B 2 ), is defined as the gap between Π + (B 1 ∩ L 2 (R)) and Π + (B 2 ∩ L 2 (R)) inL 2 (R):d SB (B 1 , B 2 ) := gap L2(Π + (B 1 ∩ L 2 (R)), Π + (B 2 ∩ L 2 (R))).5

6The Sasane-Ball metric measures the distance over the future time axis, with arbitrarypast. The Hilbert space is again taken as L 2 (R). We will often use the shorthandterminology SB-metric.In the sequel we will make a detailed study of the above three metrics, andexpress their properties in terms of rational representations of the behaviors and theirassociated multiplication operators. Later, we will also introduce a fourth metric,the Vinnicombe metric. As in the input-output case, the definition of the latter canonly be given in terms of representations, since it does not seem to allow a naturalinterpretation in terms of gap in Hilbert space.3. Rational representations of behaviors. In addition to polynomial representations,behaviors admit rational representations (see [27]). In particular, for areal rational matrix G a meaning can be given to the equation G( d dt)w = 0 and to theexpression ker G( d dt). For this we need the concept of left coprime factorization of arational matrix G over R[ξ]. A factorization of such G as G = P −1 Q with P and Qreal polynomial matrices is called a left coprime factorization if (P Q) is left primeover R[ξ] and det(P ) ≠ 0. Following [27], if G = P −1 Q is a left coprime factorizationover R[ξ] then we define G( d dt )w = 0 if Q( d dt)w = 0 and(3.1) ker G( d dt ) := ker Q( d dt ).In this way every linear differential system also admits representations as the ‘kernelof a rational matrix’. If G is a rational matrix, we call a representation of B asB = {w ∈ L loc (R, R q ) | G( d dt)w = 0} a rational kernel representation of B and oftenwrite B = ker G( d dt). For a more detailed exposition on rational representations ofbehaviors the reader is referred to [27], [15].Therein, it can also be found that a linear differential system is controllable ifand only if its behavior B admits a representation(3.2) B = {w ∈ L loc (R, R q ) | ∃v ∈ L loc1 (R, R m ) s.t. w = G( d dt )v }for some integer m and some real rational matrix G with m columns. The equationw = G( d dt )v should be interpreted as ( ( )I − G( d dt )) wv= 0, whose meaning wasdefined above. The representation (3.2) is called a rational image representation, andwe often write B = im G( d dt). The minimal m required can be shown to be equalto m(B), the input cardinality of B, and is achieved if and only if G has full columnrank.If G is a real rational matrix then for the behavior B = im G( d dt) we can obtaina polynomial image representation as follows, using right coprime factorization overR[ξ]. A factorization G = MN −1 with M and N real polynomial matrices is called aright coprime factorization over R[ξ] if „ «MN is right prime over R[ξ] and det(N) ≠ 0.Lemma 3.1. Let G be a real rational matrix and let G = MN −1 be a rightcoprime factorization over R[ξ]. Then im G( d dt ) = im M( d dt ).Proof: Let G = P( −1 Q ) be a left coprime factorization over R[ξ]. Then we obviouslyMhave ( P −Q ) = 0. Using this, it can be shown thatNker ( P ( d ) −Q( d ) ) ( )M(ddt dt= im) dtN( d ) dt(see [19], Prop. 3.2). Since (I − G) = P −1 (P − Q) is a left coprime factorization,we have w = G( d dt )v ⇔ (I − G( d dt ))col(w, v) = 0 ⇔ (P ( d dt ) −

Q( d dt ))col(w, v) = 0. Thus we obtain w ∈ im G( d dt ) ⇔ ∃v such that w = G( d dt )v.In turn this is(equivalent) (with: ∃v)such that (P ( d dt ) − Q( d dt))col(w, v) = 0 ⇔w M(d∃v, l such that =) dtv N( d ) l ⇔ ∃l such that w = M( d )l ⇔ w ∈ im M( d ).dt dtdt□A given behavior allows rational image and kernel representations in which the rationalmatrices satisfy certain desired properties. In particular, they can be chosen to beproper, stable, prime and (co-)inner at the same time. The precise statement is asfollows:Theorem 3.2. Let B ∈ L q cont. There exists a real rational matrix G such thatB = im G( d dt) where G satisfies the following four properties1. G ∈ RH ∞ ,2. G is right prime over RH ∞ ,3. G is inner,4. G has no zeros.If B ∈ L q then there exists a real rational matrix ˜G such that B = ker ˜G( d dt), where˜G satisfies the following three properties1. ˜G ∈ RH∞ ,2. ˜G is left prime over RH∞ ,3. ˜G is co-inner.Proof: We first prove the existence of G satisfying the properties 1, 2, 3 and 4 suchthat B = im G( d dt ). Since B ∈ Lq cont, by [27], Theorem 9, there exists G 1 ∈ RH ∞ ,right prime over RH ∞ and having no zeros, such that B = im G 1 ( d dt). We now adaptG 1 in such a way that also property 3 is satisfied.Define Z := G ∗ 1G 1 . By right primeness of G 1 it is easily verified that Z is biproper.Further we have Z ∗ = Z and Z has no poles and zeros on the imaginary axis. LetG 1 = MN −1 be a right coprime factorization over R[ξ]. Now let L be a squarepolynomial matrix such that M ∗ M = L ∗ L, and L is Hurwitz. Indeed such L existsand is obtained by polynomial spectral factorization of M ∗ M. Define W := LN −1 .Since Z is biproper, we havedeg det(N ∗ N) = deg det(M ∗ M) = deg det(L ∗ L).Therefore W is biproper and since N and L are both Hurwitz, we have W, W −1 ∈RH ∞ . Define G := G 1 W −1 . Clearly G ∈ RH ∞ and G ∗ G = I. Further, since G 1is right prime over RH ∞ , there exists G + 1 ∈ RH ∞ such that G + 1 G 1 = I. DefineG + := W G + 1 . Clearly G+ ∈ RH ∞ and G + G = I, so G is right prime over RH ∞ .Clearly G = ML −1 is a right coprime factorization over R[ξ]. Therefore im G( d dt ) =im M( d dt ) = im G 1( d dt ) = B.If B ∈ L q , from Theorem 5, [27], it admits a rational kernel representationB = ker ˜G 1 ( d dt ) such that ˜G 1 is right prime over RH ∞ . Using polynomial spectralfactorization of ˜G1 ˜G∗ 1 a rational matrix ˜G can then be obtained such that B =ker G( d dt) and the conditions are satisfied.□Obviously, the above theorem also holds with RH ∞ replaced by RH − ∞, the space ofproper and anti-stable real rational matrices. In this paper we will often use rationalimage and kernel representations of B that satisfy some, or all, of the properties statedin Theorem 3.2. In general, if B = im G( d dt ) = ker ˜G( d dt ), then obviously ˜GG = 0.7

8If, in addition, G is inner and ˜G is co-inner, then it is immediate that the rationalmatrix (G ˜G ∗ ) is unitary and therefore also GG ∗ + ˜G ∗ ˜G = I.To conclude this section, we review the notion of dual behavior, see [18], Section10, and [20]. For a given behavior B ∈ L q cont we define its dual behavior B ∗ byB ∗ := {w ∈ L loc (R, R q ) |∫ ∞−∞w(t) ⊤ w ′ (t)dt = 0 for all w ′ ∈ B with compact support}.It can be shown that B ∗ ∈ L q cont and that m(B ∗ ) = q − m(B). Moreover, in thepolynomial context B = im M( d dt ) if and only if B∗ = ker M ∗ ( d dt). Using Lemma 3.1,this carries over to rational representations: for real rational G we have B = im G( d dt )if and only if B ∗ = ker G ∗ ( d dt ).4. Rational representations and multiplication operators. In this sectionwe will study the relation between rational representations of behaviors and classicalmultiplication operators on L 2 (R). In particular we will clarify the connection betweenrational kernel and image representations, and the kernels and images of the associatedmultiplication operators.With any real rational matrix G ∈ RL ∞ we can associate a unique linear operatorG : L 2 (iR) → L 2 (iR) whose action is defined by the multiplication W ↦→ GW . IfG ∈ RH ∞ , then the subspace H 2 is invariant under the multiplication operator, i.e.GH 2 ⊂ H 2 . In this paper we will focus on system descriptions in the time-domain.Let F denote the Fourier transformation. Then, with any G ∈ RL ∞ we associate atime-domain ‘multiplication operator’ in the usual way as follows:Definition 4.1. Let G ∈ RL ∞ . The operator M G : L 2 (R) → L 2 (R) is definedby M G := F −1 G F.We will call M G the multiplication operator with symbol G. Of course, M G can beinterpreted as a convolution operator, but we will not use this fact here. Obviously,if G 1 , G 2 ∈ RL ∞ , then M G1G 2= M G1 M G2 . It is a well-known fact that the operatorM G is an isometry, i.e. ‖M G w‖ 2 = ‖w‖ 2 for all w ∈ L 2 (R) if and only if G isinner, i.e. G ∗ G = I. For any G ∈ RL ∞ , the operator norm ‖M G ‖ is equal tothe L ∞ -norm ‖G‖ ∞ . Also, if G 1 , G 2 , G 3 ∈ RL ∞ and G 1 is inner and G 3 is coinner,then ‖M G1 M G2 M G3 ‖ = ‖M G2 ‖. The restriction of M G to L 2 (R + ) is denotedby M G | L2(R + ). The composition Π + M G | L2(R + ): L 2 (R + ) → L 2 (R + ) is called theToeplitz operator with symbol G. It will be denoted by T G . If G ∈ RH ∞ , then L 2 (R + )is invariant under M G . In this case, M G is called causal. Also, then M G | L2(R + )= T G .We will now study the connection between rational representations of behaviorsand multiplication operators. In particular, with any p × q real rational matrix G ∈RL ∞ we can associate the linear differential behaviors ker G( d dt ) ⊂ L loc(R, R q ) andim G( d dt ) ⊂ L loc(R, R p ). On the other hand G defines a multiplication operator M Gwith ker M G ⊂ L 2 (R) and im M G ⊂ L 2 (R). We will now study the relation betweenthese different ‘kernels’ and ‘images’. We first prove the following useful lemma:Lemma 4.2. Let G, ˜G ∈ RL ∞ be such that im G( d dt ) = ker ˜G( d dt). If G is rightprimeand ˜G is left-prime (over RL ∞ ) then im M G = ker M ˜G.If G, ˜G ∈ RH ∞ andG is right-prime and ˜G is left-prime (over RH ∞ ) then im T G = ker T ˜G.Proof: Since G ˜G = 0 we have M ˜GM G = 0, whence im M G ⊂ ker M ˜G.Conversely, letG + , ˜G + ∈ RL ∞ such that G + G = I and ˜G ˜G + = I. Then clearly( G+˜G)((G ˜G + I 0) =0 I),

whence GG + + ˜G + ˜G = I. Then M ˜Gw= 0 implies w = M G M G +w. Since M G +w ∈L 2 (R), the result follows. The second statement follows in a similar manner □The above lemma is instrumental in proving the following basic relation betweenrational representations and multiplication operators:Theorem 4.3. Let G ∈ RL ∞ . Then the following hold:1. ker G( d dt ) ∩ L 2(R) = ker M G .2. If G is right-prime (over RL ∞ ) then im G( d dt ) ∩ L 2(R) = im M G .3. If G ∈ RH ∞ then ker G( d dt ) ∩ L 2(R + ) = ker T G .4. If G ∈ RH ∞ is right-prime (over RH ∞ ) then im G( d dt ) ∩ L 2(R + ) = im T G .Proof: (1.) Let G = P −1 Q be a right coprime factorization over R[ξ]. Then w ∈ker G( d dt ) ∩ L 2(R) if and only if Q( d dt )w = 0 and w ∈ L 2(R). This holds if and only ifQ(iω)W (iω) = 0 and W ∈ S, where W = Fw and S is the subspace of L 2 (iR) givenby (1.1). Since P has no roots on the imaginary axis, the latter is equivalent withP −1 (iω)Q(iω)W (iω) = 0 and W ∈ S, equivalently, w ∈ L 2 (R) and M G w = 0.(2.) Let ˜G ∈ RL ∞ be left prime and such that im G( d dt ) = ker ˜G( d dt). Then, byLemma 4.2, im M G = ker M ˜G,and the result follows from (1.).Finally, proofs of (3.) and (4.) can be given in a similar manner, using a leftprime ˜G ∈ RH ∞ and with L 2 (R) replaced by L 2 (R + ) and S replaced by H 2 ∩ S. □In general, for a given behavior B, its intersection with L 2 (R) is called an L 2 -behavior.L 2 -behaviors have been studied before, see e.g. [24], or more recently [9].Suitable rational image and kernel representations of a given controllable behaviorimmediately yield explicit expressions for the orthogonal projection of L 2 (R) onto theassociated L 2 -behavior and its orthogonal complement:Lemma 4.4. Let B ∈ L q cont and let B = im G( d dt ) = ker ˜G( d dt ), with G, ˜G ∈RL ∞ inner and co-inner, respectively. Then the orthogonal projection of L 2 (R) ontoB ∩ L 2 (R) is given by the multiplication operator M GG ∗. The orthogonal projectionof L 2 (R) onto (B ∩ L 2 (R)) ⊥ is given by the multiplication operator M ˜G∗ ˜G.Proof: In order to prove the first statement note that that M GG ∗ is a projector:(M GG ∗) 2 = M GG ∗ GG ∗ = M GG ∗, it is self-adjoint: (M GG ∗)∗ = M (GG ∗ ) ∗ = M GG ∗ and,by Theorem 4.3, its image im M GG ∗ is equal to im M G = B ∩ L 2 (R). The secondstatement follows from the fact that M ˜G∗ ˜G= I − M GG ∗.□A related issue arises if one wants to put the notion of dual behavior in the Hilbertspace context and, in particular, relate duality and orthogonality. The following resultholds:Lemma 4.5. Let B ∈ L q cont. Then (B ∩ L 2 (R)) ⊥ = B ∗ ∩ L 2 (R).Proof: (⊂) If w ∈ L 2 (R) satisfies ∫ ∞−∞ w⊤ (t)w ′ (t)dt = 0 for all w ′ ∈ B ∩ L 2 (R),then also for all w ′ ∈ B of compact support. Hence w ∈ B ∗ ∩ L 2 (R). (⊃) Ifw ∈ B ∗ ∩ L 2 (R) then ∫ ∞−∞ w⊤ (t)w ′ (t)dt = 0 for all w ′ ∈ B of compact support. By adensity argument (using controllability of B) the integral can then be shown to be 0for all w ∈ B ∩ L 2 (R).□5. Distance between behaviors and rational representations. Using therelation between rational representations and multiplication operators established inSection 4, in the present section we will for each of the three metrics introduced inSection 2 study how to compute their values in terms of rational representations of thebehaviors. We will also show that these behavioral metrics are in fact generalizationsof ‘classical’ gaps studied previously in the input-output transfer matrix context.9

105.1. L 2 -metric. Obviously, by Theorem 4.3, if for i = 1, 2, G i ∈ RL ∞ is rightprimeand B i = im G i ( d dt ), and if ˜G i ∈ RL ∞ is such that B i = ker ˜G i ( d dt), thend L2 (B 1 , B 2 ) = gap L2(im M G1 , im M G2 ) = gap L2(ker M ˜G1, ker M ˜G2).The following result is well known in the context of input-output systems, see [21],[22]. Here, we state it in the context of rational representations of behaviors, and forcompleteness include a proof:Theorem 5.1. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Let G 1 , G 2 ∈ RL ∞ suchthat B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt ) with G 1, G 2 inner. Also, let ˜G 1 , ˜G 2 ∈ RL ∞such that B 1 = ker ˜G 1 ( d dt ) and B 2 = ker ˜G 2 ( d dt ), with ˜G 1 , ˜G 2 co-inner. Thend L2 (B 1 , B 2 ) = ‖ ˜G 2 G 1 ‖ ∞ = ‖ ˜G 1 G 2 ‖ ∞ .Proof: According to Lemma 4.4 we have gap ⃗ L2(B 1 ∩L 2 (R), B 2 ∩L 2 (R)) = ‖M ˜G∗M2 ˜G2 G1G ∗‖ = 1‖M G ∗2M ˜G2G 1M G ∗1‖ = ‖M ˜G2G 1‖ = ‖ ˜G 2 G 1 ‖ ∞ . Thus, d L2 (B 1 , B 2 ) = max{‖ ˜G 2 G 1 ‖ ∞ , ‖ ˜G 1 G 2 ‖ ∞ }.We prove that ‖ ˜G 2 G 1 ‖ ∞ = ‖ ˜G 1 G 2 ‖ ∞ . Indeed, using the fact that G 2 G ∗ 2 + ˜G ∗ ˜G 2 2 =I, and pre- and postmultiplying this expression by G ∗ 1 and G 1 , respectively, wesee that ( ˜G 2 G 1 ) ∗ ˜G2 G 1 + (G ∗ 2G 1 ) ∗ G ∗ 2G 1 = I. This yields σmax( 2 ˜G 2 G 1 )(iω) = 1 −σmin 2 (G∗ 2G 1 )(iω) for all ω ∈ R. Since the singular values of (G ∗ 2G 1 )(iω) and (G ∗ 1G 2 )(iω)coincide, this implies σ max ( ˜G 2 G 1 )(iω) = σ max ( ˜G 1 G 2 )(iω) for all ω, whence ‖ ˜G 2 G 1 ‖ ∞ =‖ ˜G 1 G 2 ‖ ∞ .□Since the gap does note change by taking orthogonal complements in Hilbert space,by applying Lemma 4.5 we immediately obtain that the L 2 -metric is invariant underdualization of behaviors:Lemma 5.2. d L2 (B 1 , B 2 ) = d L2 (B ∗ 1, B ∗ 2).5.2. Zames metric. We will first show that Definition 2.2 generalizes the ‘classical’definition of gap metric in the input-output framework. Indeed, suppose we havetwo systems, with identical number of inputs and outputs, given by their transfer matricesG 1 and G 2 . In [7] the gap δ(G 1 , G 2 ) is defined as follows. Let G i = M i N −1ibe normalized right coprime factorizations with N i , M i ∈ RH ∞ . Then, following[7], the gap between G 1 and G 2 is defined as the L 2 -gap between the images of thecorresponding Toeplitz operators (the ‘graphs’):δ(G 1 , G 2 ) = gap L2(im T “ N 1M 1 ” , im T “ N 2M 2 ” ).This can be interpreted in the behavioral set-up as follows. The system with transfermatrix G i has in fact (input-output) behavior B i given by the rational imagerepresentation( ) (ui=y i)IG i ( d dt ) v i .Moreover, by [15], Theorem 7.4, an alternative rational image representation of B i isgiven by( ) (ui Ni ( d= dt ) )y i M i ( d dt ) v i ′ .

11By Theorem 4.3 we therefore obtainδ(G 1 , G 2 ) = gap L2(B 1 ∩ L 2 (R + ), B 2 ∩ L 2 (R + )),which indeed equals d Z (B 1 , B 2 ) as defined by Definition 2.2. This shows our claim.In terms of rational representations, the metric defined in Definition 2.2 can becomputed in terms of solutions of two H ∞ optimization problems. The followingproposition is a generalization of a well-known result by T. Georgiou (see [6]) on thecomputation of gap metric in the input-output framework using normalized coprimefactorizations of transfer matrices. We formulate the result here in a general frameworkusing rational representations of behaviors. A proof can be obtained by simplyadapting the proof given in [17] in the input-output framework.Proposition 5.3. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Let G 1 , G 2 ∈ RH ∞ besuch that B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt ), with G 1 and G 2 inner and rightprimeover RH ∞ . Then we have(5.1) gap ⃗ L2(B 1 ∩ L 2 (R + ), B 2 ∩ L 2 (R + )) = inf ‖G 1 − G 2 Q‖ ∞Q∈RH ∞and hence(5.2) d Z (B 1 , B 2 ) = max{ infQ∈RH ∞‖G 1 − G 2 Q‖ ∞ ,inf ‖G 2 − G 1 Q‖ ∞ }.Q∈RH ∞We conclude this subsection with the following result that was obtained in an inputoutputframework in [21] (see also [17], Theorem 4.7). The result expresses computationof the distance in the Z-metric as a single optimization problem. The proof from[21] immediately carries over to our framework, and will be omitted.Proposition 5.4. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Let G 1 , G 2 ∈ RH ∞ besuch that B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt ), with G 1 and G 2 inner and rightprimeover RH ∞ . Thend Z (B 1 , B 2 ) =inf ‖G 1 − G 2 Q‖ ∞ .Q,Q −1 ∈RH ∞Remark 5.5. As was already mentioned in the introduction, in [3] a notionof gap between behaviors was introduced in a more general context, with behaviorsas arbitrary subsets of the set of all functions from time axis to signal space. Thisnotion of distance was inspired by the gap metric for nonlinear input-output systemsintroduced in [8]. It can be shown that for the special case of controllable lineardifferential systems (as is being considered in the present paper) the behavioral gapin [3] specializes to our Zames metric.5.3. Sasane-Ball metric. In this subsection we show that our definition, Definition2.3, generalizes the gap as defined by Sasane in [12] and Ball and Sasane in [13].In [13], for a given minimal input-state-output system ẋ = Ax + Bu, y = Cx + Duwith state space R n and stable p × m transfer matrix G, the ‘extended graph’ isdefined as the subspace((5.3) G(G) :=0Ce At 1I(t)) ( )R n I+ LT 2 (R + , R m )Gof the Hilbert space L 2 (R + , R m+p ). Here, 1I(t) denotes the indicator function of R +and T G is the Toeplitz operator with symbol G. For stable G the ‘ordinary’ graph in

12the gap context is given by(IT G)L 2 (R + , R m ),so the difference lies in the first term in (5.3), which takes into account arbitraryinitial conditions on the system. In [13] the following metric is then defined on thespace of stable p × m transfer matrices:δ ′ (G 1 , G 2 ) = gap L2(G(G 1 ), G(G 2 )).We will now show that for any given transfer matrix G the extended graph is in factequal to the image of the intersection of the input-output behavior with L 2 (R) underthe orthogonal projection onto L 2 (R + ):( ) I(5.4) G(G) = Π + ( imG( d dt ) ∩ L 2 (R) ).Indeed, from Theorem 4.3 the righthand side of (5.4) equals( ) IΠ + LM 2 (R) =G( )( )IΠ + LM 2 (R − I) + Π + LG M 2 (R + ) =G( ) ( )IIim + im .T GH + GHere H + G denotes the Hankel operator Π +M G | L2(R − ): L 2 (R − ) → L 2 (R + ). Since(A, B) is reachable, im H + G = CeAt 1I(t)R n . This proves((5.4).)From this we concludeIthat for the two input-output behaviors B i = imG i( d dt ) we have δ ′ (G 1 , G 2 ) =d SB (B 1 , B 2 ) as defined by Definition 2.3.We now turn to the problem of computing for two given behaviors their distancein the SB-metric. It turns out that not much work needs to be done for this, sincethe SB-metric is in a sense dual to the Z-metric. We first prove the following lemma.Lemma 5.6. Let B ∈ L q cont. The orthogonal projection of B ∩ L 2 (R) ontoL 2 (R + ) is equal to the orthogonal complement in L 2 (R + ) of B ∗ ∩ L 2 (R + ):Π + (B ∩ L 2 (R)) = ( B ∗ ∩ L 2 (R + ) ) ⊥∩ L2 (R + ).Proof: (⊂) Let w + ∈ Π + (B ∩ L 2 (R)), and let w + = Π + w with w ∈ B ∩ L 2 (R).Take any v ∈ B ∗ ∩ L 2 (R + ). Since L 2 (R + ) ⊂ L 2 (R), by Lemma 4.5 we have v ∈( B ∩ L 2 (R) ) ⊥ . Thus ∫ ∞−∞ v⊤ w + dt = ∫ ∞v ⊤ w0 + dt = ∫ ∞v ⊤ wdt = ∫ ∞0 −∞ v⊤ wdt = 0.Thus w + ∈ ( B ∗ ∩ L 2 (R + ) ) ⊥ .(⊃) First note that Π + (B ∩ L 2 (R)) = Π + ( B ∗ ∩ L 2 (R) ) ⊥ . Since Π + = Π ∗ +,the latter equals ( Π −1+ (B ∗ ∩ L 2 (R)) ) ⊥, the orthogonal complement of the inverseimage under Π + of B ∗ ∩ L 2 (R). Now let w ∈ ( B ∗ ∩ L 2 (R + ) ) ⊥ ∩ L 2 (R + ). Take anyv ∈ Π −1+ (B ∗ ∩ L 2 (R)). Then v + := Π + v ∈ B ∗ ∩ L 2 (R + ). Thus ∫ ∞−∞ w⊤ v + dt = 0.Therefore, ∫ ∞−∞ w⊤ vdt = ∫ ∞w ⊤ vdt = ∫ ∞w ⊤ v0 0 + dt = ∫ ∞−∞ w⊤ v + dt = 0. We conclude

that w ∈ ( Π −1+ (B ∗ ∩ L 2 (R)) ) ⊥= Π+ (B ∩ L 2 (R)). This completes the proof of thelemma.□By applying this lemma, we obtain the following theorem that expresses the distancebetween two behaviors in the SB-metric in terms of the distance of the dual behaviorsin the Z-metric:Theorem 5.7. Let B 1 , B 2 ∈ L q cont. Then d SB (B 1 , B 2 ) = d Z (B ∗ 1, B ∗ 2)Proof: By Lemma 5.6 we have(5.5) d SB (B 1 , B 2 ) = gap L2( ( B ∗ 1 ∩ L 2 (R + ) ) ⊥∩ L2 (R + ), ( B ∗ 2 ∩ L 2 (R +) ⊥∩ L2 (R + )).For i = 1, 2, let Π i be the orthogonal projection of L 2 (R) onto B ∗ i ∩ L 2(R + ). ThenI − Π i is the orthogonal projection onto (B ∗ i ∩ L 2(R + )) ⊥ . As before let Π + be theorthogonal projection onto L 2 (R + ). Clearly Π i Π + = Π + Π i = Π i . It is easily verifiedthat ((I − Π i )Π + ) 2 = (I − Π i )Π + , that(5.6) im (I − Π i )Π + = ( B ∗ i ∩ L 2 (R + ) ) ⊥∩ L2 (R + ),and that (I − Π i )Π + is self-adjoint. Hence (I − Π i )Π + is in fact the orthogonalprojection onto the subspace (5.6). As a consequence we find that (5.5) is equal to‖(I − Π 1 )Π + − (I − Π 2 )Π + ‖ = ‖Π 1 − Π 2 ‖ = gap L2(B ∗ 1 ∩ L 2 (R + ), B ∗ 2 ∩ L 2 (R + ))13which by Definition 10.2 equals d Z (B ∗ 1, B ∗ 2). This completes the proof.□As a consequence, for given controllable behaviors the distance in the SB-metric canbe computed by computing the distance between the dual behaviors in the Z-metric.Again, this involves the solutions of two H ∞ optimization problems:Theorem 5.8. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Let ˜G 1 , ˜G 2 ∈ RH − ∞ be suchthat B 1 = ker ˜G 1 ( d dt ) and B 2 = ker ˜G 2 ( d dt ), with ˜G 1 and ˜G 2 co-inner and left-primeover RH − ∞. Then we have(5.7) d SB (B 1 , B 2 ) = max{ infQ∈RH ∞‖ ˜G ∗ 1 − ˜G ∗ 2Q‖ ∞ ,inf ‖ ˜G ∗ 2 − ˜G ∗ 1Q‖ ∞ }.Q∈RH ∞Proof: Note that B ∗ 1 = im ˜G ∗ 1( d dt ) and B∗ 2 = im ˜G ∗ 2( d dt ), that m(B∗ 1) = m(B ∗ 2), andthat ˜G ∗ 1, ˜G ∗ 2 ∈ RH ∞ are inner and right-prime over RH ∞ . The result then follows byapplying Proposition 5.3.□Remark 5.9. According to Theorem 7 in [12], for the special case of stableinput-state-output systems the concept of distance between behaviors that was introducedin [12] coincides with the SB-metric defined in our paper. Most likely, thedistance concept from [12] in fact coincides with the SB-metric for general controllablebehaviors. This issue is left for future research.6. Vinnicombe metric. In [21], [22], G. Vinnicombe proposed a notion of distancebetween transfer matrices in the input-output framework often referred to asthe ν-gap (see also [2], [14]). The main difference between the ν-gap and both theL 2 -gap and the usual gap metric studied in [7] is that the ν-gap does not have an apparent,direct interpretation in terms of ‘gap between subspaces’ of the Hilbert spaceL 2 (R). Instead, in computing the value of the ν-gap between two transfer matrices,

14an important role is played by the winding number of a rational matrix associatedwith the given transfer matrices.In the present section we will generalize the notion of ν-gap, and introduce ametric on the set of controllable behaviors with the same input cardinality. This willyield a representation free characterization of the ν-gap between two systems.Definition 6.1. Let B 1 , B 2 ∈ L q cont, and m(B 1 ) = m(B 2 ). Let G 1 , G 2 ∈ RH ∞be inner and right-prime (over RH ∞ ) such that B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt ).We define the Vinnicombe metric d V (B 1 , B 2 ) by⎧⎨ gap L2(im M G1 , im M G2 ) if det(G ∗ 2G 1 )(iω) ≠ 0 ∀ω ∈ R(6.1) d V (B 1 , B 2 ) :=and wno det(G ∗⎩2G 1 ) = 0,1 otherwiseIt should of course be checked whether this definition is correct, in the sense that thedefinition of d V (B 1 , B 2 ) is independent of the rational matrices G 1 , G 2 . For this, weprove the following lemma:Lemma 6.2. Let G, G ′ ∈ RH ∞ be inner and right-prime. Then im G( d dt ) =im G ′ ( d dt ) if and only if there exists a constant unitary matrix U such that G′ = GU.Proof: From [15], im G( d dt ) = im G′ ( d dt) if and only if there exists a nonsingularrational matrix U such that G ′ = GU. Let G + , G ′+ ∈ RH ∞ be left inverses of G andG ′ , respectively. Then U = G + G ′ so U ∈ RH ∞ . Also, U −1 = G ′+ G, so U −1 ∈ RH ∞ .Finally, note that I = G ′∗ G ′ = U ∗ G ∗ GU = U ∗ U, so U −1 = U ∗ . Since U ∗ ∈ RH∞ − weconclude that U is constant.□To prove that Definition 6.1 is correct, let G ′ 1, G ′ 2 ∈ RH ∞ be alternative rationalmatrices, both inner and right-prime, such that B 1 = im G ′ 1( d dt ) and B 2 = im G ′ 2( d dt ).Obviously, im M Gi = im M G ′i= B i ∩ L 2 (R). Also, from the previous lemma wehave that G ′ i = G iU i for constant unitary matrices U i . Thus, G ′∗2 G ′ 1 = U2 ∗ G ∗ 2G 1 U 1so det(G ∗ 2G 1 )(iω) ≠ 0 ∀ω ∈ R if and only if det(G ′∗2 G ′ 1)(iω) ≠ 0 ∀ω ∈ R. Also,wno det(G ∗ 2G 1 ) = 0 if and only if wno det(G ′∗2 G ′ 1) = 0.A proof of the fact that d V (B 1 , B 2 ) as defined above indeed defines a metric (onthe subset of L q cont of all controllable behaviors with the same input cardinality), canbe given by adapting the corresponding proof in the input-output setting. For thiswe refer to [21]. As short hand terminology, in the sequel we will refer to this metricas the V-metric.Of course, computing the gap between two controllable behaviors in the Vinnicombemetric only involves checking an appropriate winding number, possibly followedby a computation of the gap in the L 2 -metric. The following result followsimmediately from Theorem 5.1:Theorem 6.3. Let B 1 , B 2 ∈ L q cont, and m(B 1 ) = m(B 2 ). Let G 1 , G 2 ∈ RH ∞ beinner and right-prime (over RH ∞ ) such that B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt ).Also, let ˜G 1 , ˜G 2 ∈ RL ∞ such that B 1 = ker ˜G 1 ( d dt ) and B 2 = ker ˜G 2 ( d dt ), with ˜G 1 , ˜G 2co-inner. Then⎧⎨d V (B 1 , B 2 ) :=⎩‖ ˜G 2 G 1 ‖ ∞ (= ‖ ˜G 1 G 2 ‖ ∞ )1 otherwiseif det(G ∗ 2G 1 )(iω) ≠ 0 ∀ω ∈ Rand wno det(G ∗ 2G 1 ) = 0,The original definition of V-metric as given in [21], as well as its generalizationgiven above, are not entirely satisfactory, since they are given in terms of the rational

matrices representing the systems. In the remainder of this section, we will instead establisha representation-free characterization of the distance between two controllablebehaviors in the V-metric, no longer using the matrices appearing in their rationalrepresentations. Before doing this, we first introduce some additional material onlinear differential systems.6.1. More about behaviors. In Section 2 we have introduced linear differentialsystems as those systems whose behavior can he represented as the kernel of a polynomialdifferential operator, B = ker R( d dt), whith R a real polynomial matrix. Anotherrepresentation is a latent variable representation, defined through polynomial matricesR and M by R( d dt )w = M( d dt )v, with B = {w ∈ L loc | ∃v ∈ L loc such that R( d dt )w =M( d dt)v}. The variable v is called a latent variable. If the latent variable has theproperty of state (see [11], [18]) then the latent variable is called a state variable,and the latent variable representation is called a state representation of B. For anyB ∈ L q many state representations exist, but the minimal number of components ofthe state variable in any state representation of B is an invariant for B. This nonnegativeinteger is called the McMillan degree of B, denoted by n(B). It is a basic factthat the McMillan degree of B and its dual B ∗ are the same: n(B) = n(B ∗ ). Thefollowing lemma expresses the McMillan degree in terms of rational representations.Lemma 6.4. Let G be a proper real rational matrix. Then the following holds:1. If G is left prime over R(ξ) P and G = P −1 Q is a left coprime factorizationover R[ξ], then the McMillan degree of the behavior ker G( d dt) is equal todeg det(P ).2. If G has no zeros, is right prime over R(ξ) P and G = MN −1 is a rightcoprime factorization over R[ξ], then the McMillan degree of the behaviorim G( d dt) is equal to deg det(N).Proof: (1.) The crux is that if Q is a full row rank polynomial matrix with prows, then the McMillan degree of ker Q( d dt) is equal to the maximum of the degreesof the determinants over all p × p minors of Q (see [11]). Now let G have p rowsand be left prime over R(ξ) P . Then by [27], page 240, it has a biproper p × pminor, say ˜G. The corresponding p × p minor of Q, say ˜Q, satisfies ˜G = P −1 ˜Q.In addition, for every p × p minor ˜Q ′ , P −1 ˜Q′ is proper. Thus for every minor ˜Q ′ wehave deg det( ˜Q ′ ) ≤ deg det(P ), while deg det( ˜Q) = deg det(P ). This proves the claim.(2.) This is proven along the same lines, using the fact that if M is a full column rankpolynomial matrix with m columns, having no zeros, then the McMillan degree ofim M( d dt) is equal to the maximum of the degrees of the determinants over all m × mminors of M.□Next we will briefly discuss autonomous behaviors, see [11], page 66. Let B ∈ L q .We call the behavior autonomous if it has no input variables, i.e. if m(B) = 0.Being autonomous is reflected in kernel representation as follows: B is autonomousif and only if there exists a square, nonsingular polynomial matrix R such that B =ker R( d dt ). Also, a given B ∈ Lq is autonomous if and only if it is a finite dimensionalsubspace of L loc (R, R q ). In fact, its dimension is then equal to the degree of thepolynomial det(R). Also, this number is equal to the McMillan degree of B, i.e.dim B = n(B).The roots of the polynomial det(R) are called the frequencies of B. They onlydepend on B, since if B = ker R ′ ( d dt) is a second kernel representation of B withR ′ square and nonsingular we must have R ′ = UR for some unimodular polynomialmatrix U. A frequency λ with Re(λ) = 0, i.e. λ lies on the imaginary axis, is called15

16an imaginary frequency. The following is easily seen, and we omit the proof:Lemma 6.5. Let B ∈ L q . let M be a polynomial matrix with q columns. Then Bis autonomous if and only if M( d dt)B is autonomous. Let λ ∈ C be such that M(λ)has full column rank. Then λ is a frequency of B if and only if λ is a frequency ofM( d dt )B.If B is autonomous and has no imaginary frequencies then B = B stab ⊕ B antiuniquely, with B stab stable (i.e. lim t→∞ w(t) = 0 for all w ∈ B stab ) and B anti antistable(i.e. lim t→−∞ w(t) = 0 for all w ∈ B anti ).6.2. A representation free approach to the Vinnicombe metric. In thissubsection we present a representation free approach to the Vinnicombe metric. Wefirst prove a lemma that expresses the winding number appearing in the definition ofthe Vinnicombe metric in terms of McMillan degrees associated with the underlyingbehaviors:Lemma 6.6. Let B 1 , B 2 ∈ L q cont with m(B 1 ) = m(B 2 ). Let G 1 , G 2 ∈ RL ∞ suchthat B i = im G i ( d dt ) Then1. G ∗ 2G 1 is a nonsingular rational matrix if and only if B 1 ∩ B ∗ 2 is autonomous.2. (G ∗ 2G 1 )(iω) is nonsingular for all ω ∈ R if and only if B 1 ∩B ∗ 2 is autonomousand has no imaginary frequencies.Furthermore, if G 1 , G 2 ∈ RH ∞ , G 2 is left-prime over RL ∞ and (2.) above holds thenwno det(G ∗ 2G 1 ) = n(B 2 ) − dim(B 1 ∩ B ∗ 2) antiwhere (B 1 ∩ B ∗ 2) anti denotes the anti-stable part of B 1 ∩ B ∗ 2.Proof: Let G 1 = MN −1 be a right coprime factorization over R[ξ], and G ∗ 2 = P −1 Qa left coprime factorization over R[ξ]. Then B 1 = im M( d dt). Furthermore, sinceB ∗ 2 = ker G ∗ 2( d dt ), we have B∗ 2 = ker Q( d dt ). It is then easily verified that B 1 ∩ B ∗ 2 =M( d dt ) ker(QM)( d dt ).(1.) Now G ∗ 2G 1 = P −1 QMN −1 , hence G ∗ 2G 1 is nonsingular if and only if QM isnonsingular, equivalently, B 1 ∩ B ∗ 2 is autonomous. Statement (2.) then follows fromLemma 6.5. Assume now that G 1 , G 2 ∈ RH ∞ , G 2 is left-prime over RL ∞ , and thecondition (2.) holds. We havedet(G ∗ 2G 1 ) =det(QM)det(P ) det(N)The winding number wno det(G ∗ 2G 1 ) is equal to number of roots of det(QM) in C +minus the number of roots of the product det(P ) det(N) in C + . The number ofroots of det(QM) in C + is equal to dim(B 1 ∩ B ∗ 2) anti , while the number of roots ofdet(P ) det(N) in C + is equal to the degree of det(P ) (note that N is Hurwitz, P isanti-Hurwitz). Finally, from left-primeness of G 2 , by Lemma 6.4 the degree of det(P )is equal to the McMillan degree of B ∗ 2, which is equal to the McMillan degree of B 2 .This completes the proof.□This immediately leads to the following result which expresses the distance in theV-metric between two given behaviors completely in terms of the behaviors, and nolonger in terms of their rational representations:Theorem 6.7. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Then{dL2 (B 1, B 2) if B 1 ∩ B ∗ 2 autonomous, has no imaginaryd V (B 1 , B 2 ) =frequencies and dim(B 1 ∩ B ∗ 2) anti = n(B 2),1 otherwise

17We conclude this subsection with establishing some basic properties of the V-metric. In contrast to the L 2 -metric, the V-metric is not invariant under dualization.The following result result gives conditions under which invariance does hold.Theorem 6.8. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Then the following holds:1. If n(B 1 ) = n(B 2 ) then d V (B 1 , B 2 ) = d V (B ∗ 1, B ∗ 2).2. If d V (B 1 , B 2 ) < 1 and d V (B ∗ 1, B ∗ 2) < 1 then d V (B 1 , B 2 ) = d V (B ∗ 1, B ∗ 2) ifand only if n(B 1 ) = n(B 2 ).Proof: (1.) Consider the conditions: B 1 ∩ B ∗ 2 is autonomous, has no imaginary frequenciesand dim(B 1 ∩B ∗ 2) anti = n(B 2 ). Obviously, since (B ∗ 1) ∗ = B 1 , and by the assumptionthat n(B 2 ) = n(B 1 ) = n(B ∗ 1), this set of conditions is equivalent with: B ∗ 2 ∩(B ∗ 1) ∗ is autonomous, has no imaginary frequencies and dim(B ∗ 2 ∩(B ∗ 1) ∗ ) anti = n(B ∗ 1).Now we distinguish between two cases: (a.) the above equivalent sets of conditionshold. Then d V (B 1 , B 2 ) = d L2 (B 1 , B 2 ) and d V (B ∗ 2, B ∗ 1) = d L2 (B ∗ 2, B ∗ 1). By symmetrywe then obtain d V (B 1 , B 2 ) = d L2 (B 1 , B 2 ) = d L2 (B 2 , B 1 ) = d L2 (B ∗ 2, B ∗ 1) =d V (B ∗ 2, B ∗ 1) = d V (B ∗ 1, B ∗ 2). (b.) the conditions do not hold. In that case bothd V (B 1 , B 2 ) = 1 and d V (B ∗ 2, B ∗ 1) = 1, so again d V (B 1 , B 2 ) = d V (B ∗ 1, B ∗ 2).(2.) From d V (B 1 , B 2 ) < 1 it follows that dim(B 1 ∩B ∗ 2) anti = n(B 2 ). On the otherhand, d V (B ∗ 2, B ∗ 1) < 1 implies that dim(B ∗ 2 ∩ (B ∗ 1) ∗ ) anti = n(B ∗ 1). Thus, n(B 2 ) =n(B ∗ 1) = n(B 1 ).□Example 6.9. We give an example in which dualization changes the values ofthe V-metric if the McMillan degrees of the behaviors are not equal. Define B i :=im G i ( d dt ), with G 1 (ξ) := √ 1 ( )( )1 1 1, G 2 1 2 (ξ) :=ξ + √ .5 ξ − 2Note that n(B 1 ) = 0 while n(B 2 ) = 1. We compute (G ∗ 2G 1 )(ξ) = − ξ+3 √2(ξ−√5).Clearly its winding number is unequal to 0, so we have d V (B 1 , B 2 ) = 1. Now,B ∗ 1 = ker G ∗ 1( d dt ) = im H 1( d dt ) and B∗ 2 = ker G ∗ 2( d dt ) = im ˜H 2 ( d dt ) = im H 2( d dt), whereH 1 (ξ) := √ 1 ( ) −1, ˜H2 (ξ) := − 1( )2 1ξ + √ ( ) 1 ξ + 21 −ξ−2 , H2 (ξ) :=5ξ + √ .5 1We compute (H ∗ 2 H 1 )(ξ) =‖ ˜H 2 H 1 ‖ ∞ . Now, ( ˜H 2 H 1 )(ξ) =ξ−1√2(ξ−√5). Its winding number is equal to 0, so d V (B ∗ 1, B ∗ 2) =ξ+3√2(ξ−√5), which indeed yields ‖ ˜H 2 H 1 ‖ ∞ = 3 √10< 1.We conclude this subsection by stating a result that was proven in [21] in aninput-output setting, and that expresses the computation of the distance betweentwo controllable behaviors in the V-metric as an optimization problemProposition 6.10. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Let G 1 , G 2 ∈ RH ∞ beinner and right-prime (over RH ∞ ) such that B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt ).Thend V (B 1 , B 2 ) =inf ‖G 1 − G 2 Q‖ ∞ .Q,Q −1 ∈RL ∞,wno det(Q)=0Recall that computation of the Z-metric was formulated in an analogous way inProposition 5.4. Again, the proof given in [21] carries over to our framework, and isomitted here.

18Remark 6.11. Although in Theorem 6.7 we established a representation freecharacterization of the V-metric, there still remains the question whether this metriccan be given a ‘gap in the Hilbert space’ interpretation like the Z-metric and theSB-metric, for example by intersecting the behaviors with some ‘natural’ subspace ofL 2 (R), or by applying a suitable projection. This question remains unanswered, andis left for future research.7. Comparison of the metrics. In this section we will compare the metricsthat we introduced in Sections 2 and 6. It will turn out that the L 2 -gap is dominatedby the V-gap, which in turn is dominated by the Z-gap. The L 2 -gap is also dominatedby the SB-gap. However, the SB-gap will in general turn out to be incomparable withboth the V-gap as well as the Z-gap. We will also compare the topologies induced bythe metrics. Generalizing a result from [21], we will find that the topologies inducedby the V-metric and the Z-metric coincide. We will also show that if we restrict theV-metric and the SB-metric to the subset L q cont(n) of all controllable behaviors offixed McMillan degree n, then they induce the same topology on that subset. Thisnew result will generalize a result from [13] on stable input-output systems.Our first proposition is a simple generalization of results from [21]:Proposition 7.1. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Thend L2 (B 1 , B 2 ) ≤ d V (B 1 , B 2 ) ≤ d Z (B 1 , B 2 ).Proof: The inequality between d L2 and d V follows immediately from Theorem 6.7.The one between d V and d Z follows by combining Proposition 5.4 and Proposition6.10. □Next, we study the question how the SB-metric relates to the other metrics. We firstcompare with the L 2 -metric and the V -metric:Theorem 7.2. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Then the following hold:1. d L2 (B 1 , B 2 ) ≤ d SB (B 1 , B 2 ).2. If n(B 1 ) = n(B 2 ) then d V (B 1 , B 2 ) ≤ d SB (B 1 , B 2 ).Proof: (1.) d L2 (B 1 , B 2 ) = d L2 (B ∗ 1, B ∗ 2) ≤ d Z (B ∗ 1, B ∗ 2) = d SB (B 1 , B 2 ). The firstequality follows from the fact that the L 2 -metric is invariant under dualization, thesecond follows from Proposition 7.1, the third follows from Theorem 5.7. (2.) ByTheorem 6.8, If n(B 1 ) = n(B 2 ) then d V (B 1 , B 2 ) = d V (B ∗ 1, B ∗ 2). Next, by Proposition7.1 and Theorem 5.7, respectively, d V (B ∗ 1, B ∗ 2) ≤ d Z (B ∗ 1, B ∗ 2) = d SB (B 1 , B 2 ). □Remark 7.3. According to the previous theorem, on every set L q cont(n)consisting of all controllable behaviors with fixed McMillan degree n, the V-metricis dominated by the SB-metric. In general these two metrics turn out to be incomparable.If for two given behaviors we have d V (B 1 , B 2 ) < 1 then of coursed V (B 1 , B 2 ) = d L2 (B 1 , B 2 ) ≤ d SB (B 1 , B 2 ). However, in the following example wepresent two behaviors such that d SB (B 1 , B 2 ) < d V (B 1 , B 2 ).Example 7.4. Consider the behaviors B 1 and B 2 from Example 6.9. Wecomputed that d V (B 1 , B 2 ) = 1 and d V (B ∗ 1, B ∗ 2) = √ 310. It was shown in [21], page241, that for B 1 and B 2 with input cardinality equal to 1, their distance in the V-metric coincides with that in the Z-metric. Thus d Z (B ∗ 1, B ∗ 2) = √ 310. This impliesthat d SB (B 1 , B 2 ) = 3 √10< 1. Finally, we compare the SB-metric with the Z-metric. By Theorem 5.7 d Z (B ∗ 1, B ∗ 2) = d SB (B 1 , B 2 ). Therefore these two metricsare again incomparable in the sense that d Z ≰ d SB nor d Z ≱ d SB . In fact, for every

pair of behaviors B 1 , B 2 ∈ L q cont we have d Z (B 1 , B 2 ) < d SB (B 1 , B 2 ) if and only ifd Z (B ∗ 1, B ∗ 2) > d SB (B ∗ 1, B ∗ 2).Next, we will turn to a comparison of the topologies induced by the metrics. Itfollows from the inequalities given in this section that the topology induced by L 2 -metric is coarser than those induced by the others. In fact, this topology can beshown to be strictly coarser than the other topologies, and it was argued in [21] that,due to this fact, it is in general not useful in robust control.By generalizing Theorem IV.4 in [21], it can be shown that for any pair B 1 , B 2 ∈L q cont there exists a constant 0 < c ≤ 1 (depending on B 1 ) such that(7.1) c d Z (B 1 , B 2 ) ≤ d V (B 1 , B 2 ) ≤ d Z (B 1 , B 2 ).Obviously, this inequality implies that the topologies induced by the Z-metric andthe V-metric coincide. We will now compare the topologies of the SB-metric and theV-metric.Theorem 7.5. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Assume that n(B 1 ) =n(B 2 ). Then there exists 0 < c ≤ 1 such thatc d SB (B 1 , B 2 ) ≤ d V (B 1 , B 2 ) ≤ d SB (B 1 , B 2 ).Proof: By Theorems 6.8 and 7.2, under the assumption n(B 1 ) = n(B 2 ) we haved V (B 1 , B 2 ) = d V (B ∗ 1, B ∗ 2) and d V (B 1 , B 1 ) ≤ d SB (B 1 , B 1 ). Using (7.1) there exists0 < c ≤ 1 such that c d Z (B ∗ 1, B ∗ 2) ≤ d V (B ∗ 1, B ∗ 2). This then implies c d SB (B 1 , B 2 ) ≤d V (B 1 , B 2 ).□As a consequence, for any integer n the V-metric and the SB-metric considered asmetrics on the subset L q cont(n) of all controllable behaviors of fixed McMillan degreen, induce the same topology. Obviously this topology then also coincides with theone induced by the Z-metric on L q cont(n). The issue whether the Z-metric and theSB-metric define the same topology was posed as an open problem in [12], page 1222.Our result gives an answers to this question in full generality. The result was obtainedbefore in [13] for input-output systems with stable transfer matrices.8. Properties of the metrics. In this section we will take a closer look at themetrics introduced in Sections 2 and 6 and establish several properties. Our mainfocus will be on expressing these properties in behavioral terms.It is well-known that for subspaces V 1 , V 2 of the Euclidean space R n with thestandard inner product we havegap(V 1 , V 2 ) < 1 if and only if V 1 ∩ V ⊥ 2 = {0}.In the sequel we will study the question how this generalizes to the metrics that wedefined on the space of controllable behaviors. In particular, for each of the metrics wehave defined we will study the question: what are necessary and sufficient conditionsunder which the distance between two behaviors is strictly less than one?8.1. Properties of the L 2 -metric and the V-metric. We start off withanswering the question posed in the introduction for the L 2 -metric. The followinglemma gives necessary and sufficient conditions in terms of the rational matricesappearing in image representations of the behaviors.Lemma 8.1. Let B 1 , B 2 ∈ L q cont with m(B 1 ) = m(B 2 ). Let G 1 , G 2 ∈ RL ∞ suchthat B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt ) with G 1, G 2 inner. Then the followingthree statements are equivalent:19

201. d L2 (B 1 , B 2 ) < 1,2. det(G ∗ 2G 1 )(iω) ≠ 0 for all ω ∈ R and G ∗ 2G 1 is biproper,3. G ∗ 2G 1 is nonsingular and (G ∗ 2G 1 ) −1 ∈ RL ∞ .Proof: Let ˜G 2 ∈ RL ∞ be co-inner and such that B 2 = ker ˜G 2 ( d dt). By Theorem 5.1,d L2 (B 1 , B 2 ) = ‖ ˜G 2 G 1 ‖ ∞ . From the proof of Theorem 5.1, recall that‖ ˜G 2 G 1 ‖ ∞ = sup σ max ( ˜G 2 G 1 )(iω) = 1 − inf σ min(G ∗ 2G 1 )(iω).ω∈Rω∈RObviously, inf ω∈R σ min ((G ∗ 2G 1 )(iω)) > 0 if and only if (G ∗ 2G 1 )(iω) is nonsingular forall ω ∈ R and lim ω→∞ σ min ((G ∗ 2G 1 )(iω)) > 0, equivalently, (2.) holds. Clearly, (2.)and (3.) are equivalent.□The property of biproperness in the above can be characterized equivalently in termsof the associated behaviors as follows:Lemma 8.2. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Let G 1 , G 2 ∈ RL ∞ be rightprime, such that B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt ). Then G∗ 2G 1 is biproper if andonly if B 1 ∩ B ∗ 2 is autonomous and dim(B 1 ∩ B ∗ 2) = n(B 1 ) + n(B 2 ).Proof: (⇒). By Lemma 6.6, G ∗ 2G 1 biproper implies that B 1 ∩ B ∗ 2 is autonomous.Let G 1 = MN −1 be a right coprime factorization over R[ξ], and G ∗ 2 = P −1 Q a leftcoprime factorization over R[ξ]. Now, G ∗ 2G 1 = P −1 QMN −1 . By biproperness we havedeg det(QM) = deg det(P )+deg det(N). Also, as in the proof of Lemma 6.6, we haveB 1 ∩B ∗ 2 = M( d dt ) ker(QM)( d dt ). This implies that dim(B 1 ∩B ∗ 2) = deg det(QM). Onthe other hand, by Lemma 6.4, deg det(P ) = n(B ∗ 2) = n(B 2 ) and deg det(N) = n(B 1 ).(⇐). The converse implication is proven by reversing the above argument. □This immediately yields the following behavioral characterization:Theorem 8.3. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Then the following areequivalent:1. δ L2 (B 1 , B 2 ) < 1,2. B 1 ∩ B ∗ 2 is autonomous, has no imaginary frequencies anddim(B 1 ∩ B ∗ 2) = n(B 1 ) + n(B 2 ),Proof: This follows by combining Lemmas 6.6, 8.1 and 8.2.□Remark 8.4. Note that this theorem gives necessary and sufficient conditionsunder which the gap between the subspaces B 1 ∩ L 2 (R) and B 2 ∩ L 2 (R)) of theHilbert space L 2 (R) is smaller than 1. Interestingly, these conditions involve specificsystem theoretic properties of the underlying behaviors, in particular autonomy of anintersection, absence of nonzero periodic signals, and an equality involving McMillandegrees. In this section we want to extend the above results to the other three metrics.For the V-metric this extension is straightforward:Lemma 8.5. Let B 1 , B 2 ∈ L q cont and m(B 1 ) = m(B 2 ). Let G 1 , G 2 ∈ RH ∞ beinner and right-prime (over RH ∞ ) such that B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt ).Then the following are equivalent:1. d V (B 1 , B 2 ) < 1,2. d L2 (B 1 , B 2 ) < 1 and wno det(G ∗ 2G 1 ) = 0.If any of these conditions hold then d V (B 1 , B 2 ) = δ L2 (B 1 , B 2 ).Proof: Assume d V (B 1 , B 2 ) < 1. Then obviously d L2 (B 1 , B 2 ) < 1. By Definition 6.1we also have wno det(G ∗ 2G 1 ) = 0. Conversely, if d L2 (B 1 , B 2 ) < 1 then by Lemma8.1 det(G ∗ 2G 1 )(iω) ≠ 0 for all ω ∈ R. Together with wno det(G ∗ 2G 1 ) = 0 this yieldsd V (B 1 , B 2 ) = d L2 (B 1 , B 2 ) < 1.□

This immediately yields the following behavioral characterization for the distancebetween two controllable behaviors in the V-metric to be smaller than 1:Theorem 8.6. Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Then the following areequivalent:1. d V (B 1 , B 2 ) < 1,2. d L2 (B 1 , B 2 ) < 1 and dim(B 1 ∩ B ∗ 2) anti = n(B 2 ),3. d L2 (B 1 , B 2 ) < 1 and dim(B 1 ∩ B ∗ 2) stab = n(B 1 ).Proof: The equivalence of (1.) and (2.) follows by combining Lemmas 6.6 and 8.5.To prove the equivalence of (2.) and (3.) note that, by Theorem 8.3, d L2 (B 1 , B 2 ) < 1implies dim(B 1 ∩ B ∗ 2) = n(B 1 ) + n(B 2 ). Then obviously dim(B 1 ∩ B ∗ 2) stab = n(B 1 )if and only if dim(B 1 ∩ B ∗ 2) stab = n(B 1 ).□218.2. Properties of the Z-metric and the SB-metric. We now study thequestion posed in in the introduction to this section for the Z-metric and the SBmetric.In particular we will establish behavioral characterizations for the properiesd Z (B 1 , B 2 ) < 1 and d SB (B 1 , B 2 ) < 1. For these two metrics this issue is moreinvolved than for the L 2 -metric and the V-metric. Our route will be to first derivegeneral Hilbert space characterizations, and next translate these into behavioral terms.We first recall the notion of Fredholm operator (see [10], [21]):Definition 8.7. Let H be a Hilbert space, and F : H → H a boundedlinear operator. F is called a Fredholm operator if im F is closed and dim(ker F ) andcodim(im F ) are finite.Now, for an arbitrary Hilbert space H and closed subspaces V 1 and V 2 of H, thefollowing conditions under which the gap between V 1 and V 2 is smaller than 1 arewell-known (see [21], [14]:Proposition 8.8. Let H be a Hilbert space and let V 1 and V 2 be closed subspaces.Let Π V2 | V1 be the orthogonal projection onto V 2 restricted to V 1 . Then the followingstatements are equivalent:1. gap(V 1 , V 2 ) < 1,2. Π V2 | V1 is Fredholm, V 1 ∩ V ⊥ 2 = {0} and V 2 ∩ V ⊥ 1 = {0}.Note that obviously ker Π V2 | V1 = V 1 ∩ V ⊥ 2 . This general result is immediately applicablein our context, with Hilbert space L 2 (R + ). Denote the gap in this Hilbertspace by gap L2(R ). Let B + 1 , B 2 ∈ L q cont. For convenience, use the short hand notationB 2 1 := B 1 ∩ L 2 (R + ) and B 2 2 := B 2 ∩ L 2 (R + ). By Definition 2.2, d Z (B 1 , B 2 ) =gap L2(B 2 1, B 2 2) = gap L2(R )(B 2 1, B 2 2) and hence we immediately conclude the following:+ Proposition 8.9. Let B 1 , B 2 ∈ L q cont. Then d Z (B 1 , B 2 ) < 1 if and only if1. Π B 22| B 21is Fredholm,2. B 2 1 ∩ (B 2 2) ⊥ = {0} and B 2 2 ∩ (B 2 1) ⊥ = {0}.Our aim in the sequel is to reformulate the conditions (1.) and (2.), obtaining moretransparent, behavioral, system theoretic ones, in line with the conditions that weobtained for the L 2 -metric and the V-metric in the previous subsection. We willnow first deal with the Fredholm condition (1.). Surprisingly, it turns out that thiscondition is equivalent to the condition that the distance in the L 2 -metric is less thanone:Theorem 8.10. Let B 1 , B 2 ∈ L q cont. Let G 1 , G 2 ∈ RH ∞ be inner and rightprime over RH ∞ , such that B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt ). Then Π B 2| 2 B 2 is1Fredholm if and only if T G ∗2 G 1is Fredholm. If, in addition, m(B 1 ) = m(B 2 ), then thefollowing are equivalent:

221. T G ∗2 G 1is Fredholm,2. G ∗ 2G 1 is nonsingular and (G ∗ 2G 1 ) −1 ∈ RL ∞ ,3. d L2 (B 1 , B 2 ) < 1.Proof: Recall that B 2 1 = im T G1 and B 2 2 = im T G2 . It was shown in [21] and [14]that Π B 22| B 21is Fredholm if and only if T G ∗2 G 1is Fredholm. The condition that T G ∗2 G 1is Fredholm can be expressed equivalently as the invertibility condition (2.) on therational matrix G ∗ 2G 1 . Indeed, by [21], page 39, (see also [4]), for a given square matrixG ∈ RL ∞ the Toeplitz operator T G is Fredholm if and only if det G(iω) ≠ 0 for allω ∈ R and G has a proper inverse, equivalently G is nonsingular and G −1 ∈ RL ∞ .By Lemma 8.1 this is equivalent to the condition that the distance between B 1 andB 2 in the L 2 -metric is less than 1.□In the next section we will make a detailed study of condition (2.) in Proposition 8.9,by, in fact, explicitly computing the subspace intersections B 2 1∩(B 2 2) ⊥ and B 2 2∩(B 2 1) ⊥in terms of driving variable state space representations of the behaviors B 1 and B 2 .This will then yield behavioral, as well as state space characterizations of the propertyd Z (B 1 , B 2 ) < 1.To conlude this subsection, we take a brief look at the SB-metric. By Theorem5.6, d SB (B 1 , B 2 ) = d Z (B ∗ 1, B ∗ 2), so a characterization of the property d SB (B 1 , B 2 )

9.1. Computing the kernel of Toeplitz operators with invertible symbol.In this subsection we will, for a given invertible real rational matrix, computethe kernel of the associated Toeplitz operator in terms of the constant real matricesobtained from a state space realization of the rational matrix.Let G ∈ RL ∞ be nonsingular such that G −1 ∈ RL ∞ . Consider the Toeplitzoperator T G : L 2 (R + ) → L 2 (R + ) with symbol G. Let G(s) = C(sI − A) −1 B + D be arealization, possibly non-minimal, where A ∈ R n×n has no imaginary axis eigenvalues.Since G is biproper, D is nonsingular. We denote by X − (A) the stable subspace ofA, i.e.X − (A) := {x 0 ∈ R n | limt→∞e At x 0 = 0}.Likewise, X + (A) denotes the antistable subspace of A. Clearly R n = X − (A)⊕X + (A).Let 1I(t) be the indicator function of R + . Denote the unobservable subspace of thepair (−D −1 C, A − BD −1 C) by N. We now compute the kernel ker T G of the Toeplitzoperator T G :Theorem 9.2. Let X 0 := [X − (A − BD −1 C) + N] ∩ X + (A). Then the kernel ofT G is equal toker T G = {v ∈ L 2 (R + ) | ∃x 0 ∈ X 0 : v(t) = −1I(t)D −1 Ce (A−BD−1 C)t x 0 }.Consequently ker(T G ) is a finite dimensional subspace of L 2 (R + ) with dimension equalto dim(X 0 ) − dim(X + (A) ∩ N).Proof: Let v ∈ L 2 (R + ) be in ker T G . Let w = M G v. This w is the unique w ∈ L 2 (R)given by ẋ = Ax + Bv, w = Cx + Dv. Since Π + w = 0, we must have that w(t) = 0for t ≥ 0. Using the fact that D is nonsingular, this implies that for t ≥ 0 we havev(t) = −D −1 Ce (A−BD−1C)t x(0). Since v ∈ L 2 (R + ) we must have v(t) → 0 as t → ∞.This implies that x(0) must be contained in X − (A − BD −1 C) + N, the sum of thestable subspace and the unobservable subspace. Next we prove that x(0) ∈ X + (A).Let S be a coordinate transformation in R n such thatS −1 AS =( )A− 0, S −1 B =0 A +(B−B +),with A − Hurwitz and A + anti-Hurwitz. Partition S = (S − S + ). Then im(S + ) isequal to the X + (A). For t ≥ 0 the state trajectory x(t) is explicitly given byx(t) =∫ t0S − e A−(t−s) B − v(s)ds +∫ ∞tS + e A+(t−s) B + v(s)ds,where the integration starts at 0 due to the fact that v(t) = 0 for t ≤ 0. By evaluatingx(t) at t = 0 this yieldsx(0) =∫ ∞0S + e A+(t−s) B + v(s)ds,which obviously is contained in im(S + ) = X + (A).Conversely, let v(t) = −1I(t)D −1 Ce (A−BD−1C)t x 0 with x 0 ∈ X 0 . Let w = M G v.Again, this w is the unique w ∈ L 2 (R) given by ẋ = Ax + Bv, w = Cx + Dv. Weclaim that, in fact, w is given by{Cew(t) =At x 0 t < 0,0 t ≥ 0.23

24First note that, since x 0 ∈ X + (A), this w is in L 2 (R). We will now to prove that wsatisfies the equations ẋ = Ax+Bv, w = Cx+Dv with x(0) = x 0 . Indeed, for t < 0 itis given that v(t) = 0. Thus the equations become ẋ(t) = Ax(t), w(t) = Cx(t), whichis indeed satisfied for t < 0 by the given w. For t ≥ 0, define x(t) := e (A−BD−1 C)t x 0 .Then v(t) = −D −1 Cx(t) and hence ẋ(t) = Ax(t) + Bv(t) for t ≥ 0. Finally, 0 =Cx(t) + Dv(t) for t ≥ 0.We have now shown that w(t) = 0 for t ≥ 0 so Π + w = 0. This implies Π + M G v =0, so v ∈ ker(T G ).Let {x i , i = 1, 2, . . . , r} be a basis for the subspace X 0 ∩N and extend it to a basis{x i , i = 1, 2, . . . , k} of X 0 . Then a basis for ker(T G ) is given by{−1I(t)D −1 Ce (A−BD−1 C)t x i , i = r + 1, 2, . . . , k}.Thus dim(ker(T G )) = dim(X 0 ) − dim(X 0 ∩ N).observation that X 0 ∩ N = X + (A) ∩ N.The result then follows from the□9.2. Driving variable and output nulling representations of behaviors.We will now review some basic facts on driving variable and output nulling representationsof behaviors. For details we refer to [25], [20], [16]. We first consider drivingvariable (DV) representations.Let A ∈ R n×n , B ∈ R n×m , C ∈ R q×n , D ∈ R q×m , and consider the equations(9.1) ẋ = Ax + Bv, w = Cx + Dv.These equations represent the so-called full behavior(9.2)B DV (A, B, C, D) := {(w, x, v) ∈ L loc (R, R q )×L loc (R, R n )×L loc (R, R m ) | (9.1) holds }.In (9.1), we interpret w as manifest variable and (x, v) as latent variables. Thus, B DVis a latent variable representation of its external behavior given by(9.3)B DV (A, B, C, D) ext = {w | ∃(x, v) such that (w, x, v) ∈ B DV (A, B, C, D)}.In fact, in (9.1), x is a state variable and v is an auxiliary variable, called the drivingvariable. Further, if B = B DV (A, B, C, D) ext then we call B DV a driving variablerepresentation of B. A driving variable representation of B is called minimal if thestate dimension n and the driving variable dimension m are minimal over all drivingvariable representations. It can be shown that this holds if and only if n = n(B) andm = m(B).Next we review output nulling (ON) representations. Let A ∈ R n×n , B ∈ R n×q ,C ∈ R p×n , D ∈ R p×q and consider the equations(9.4) ẋ = Ax + Bw, 0 = Cx + Dw.The full behavior represented by these equations is given by(9.5) B ON (A, B, C, D) := {(w, x) ∈ L loc (R, R q ) × L loc1 (R, R n ) | (9.4) holds }.In (9.4), we interpret w as manifest variable and x as a latent variable. Thus, B ONis a latent variable representation of its external behavior given by(9.6)B ON (A, B, C, D) ext = {w | ∃x such that (w, x) ∈ B ON (A, B, C, D)}.

Also in (9.4), x is a state variable. If B = B ON (A, B, C, D) ext then we call B ON anoutput nulling representation of B. An output nulling representation of B is calledminimal if the state dimension n and the dimension of the equation space p areminimal over all output nulling representations. This holds if and only if n = n(B)and p = q − m(B).There is the following duality between DV and ON representations: if B ∈L q cont then B DV (A, B, C, D) is a minimal DV representation of B if and only ifB ON (−A ⊤ , C ⊤ , −B ⊤ , D ⊤ ) is a minimal ON representation of the dual behavior B ∗ .The following result from [16], Theorem 5.8, will be used in the sequel:Lemma 9.3. Let G ∈ R(ξ) P . Let G(ξ) = C(ξI − A) −1 B + D be a realization with(A, B) a controllable pair and (C, A) an observable pair. Then B DV (A, B, C, D) isa minimal DV-representation of im G( d dt ) if and only if G is right prime over R(ξ) Pand has no zeros.9.3. State representation of B 1 ∩ B ∗ 2 and B 2 ∩ B ∗ 1. In this subsection wewill establish state representations of B 1 ∩ B ∗ 2 and B 2 ∩ B ∗ 1 in terms of realizationsof rational image representations of B 1 and B 2 .Let B 1 , B 2 ∈ L q cont with m(B 1 ) = m(B 2 ). A standing assumption throughoutthis section will be that G 1 , G 2 ∈ RH ∞ are right prime over RH ∞ , B 1 = im G 1 ( d dt )and B 2 = im G 2 ( d dt ), and G 1 and G 2 have no zeros. According to Theorem 3.2, suchG 1 and G 2 exist. Now realize G i (ξ) = C i (ξI − A i ) −1 B i + D i , i = 1, 2 with (A i , B i )controllable and (C i , A i ) observable. We then have G ∗ 2(ξ) = −B ⊤ 2 (ξI + A ⊤ ) −1 C ⊤ 2 +D ⊤ 2 . According to Lemma 9.3, this yields the following minimal DV representation ofB 1 :(9.7) ẋ 1 = A 1 x 1 + B 1 v, w 1 = C 1 x 1 + D 1 v.Furthermore, a minimal ON representation of B ∗ 2 is given by(9.8) ẋ 2 = −A ⊤ 2 x 1 + C ⊤ 2 w 2 , 0 = −B ⊤ 2 x 2 + D ⊤ 2 w 2 .25Now define(9.9) (A1 0A :=C2 ⊤ C 1 −A ⊤ 2) ( )B1, B :=C2 ⊤ , C := (DD2 ⊤ C 1 − B2 ⊤ ), D := D2 ⊤ D 1 .1Clearly, then (G ∗ 2G 1 )(ξ) = C(ξI − A) −1 B + D. Also, it is easily verified that a staterepresentation of the intersection B 1 ∩ B ∗ 2 is given by(9.10)( ) (ẋ1 A1 0=ẋ 2 C2 ⊤ C 1 −A ⊤ 2) ( ) ( )x1 B1+x 2 C2 ⊤ v,D 1( )(9.11) 0 = (D2 ⊤ C 1 − B2 ⊤ x1) + Dx2 ⊤ D 1 v,2(9.12) w = C 1 x 1 + D 1 vi.e., w ∈ B 1 ∩ B ∗ 2 if and only if there exists x 1 , x 2 and v such that (9.10), (9.11) and(9.12) hold.

26If we assume that (G ∗ 2G 1 ) −1 ∈ RL ∞ , then D = D ⊤ 2 D 1 is nonsingular. Eliminatingthe variable v from (9.10), (9.11) and (9.12), and, writing x = col(x 1 , x 2 ), analternative state representation of B 1 ∩ B ∗ 2 is then given by(9.13) ẋ = (A − BD −1 C)x, w = ( (C 1 0) − D 1 D −1 C ) x.We now address the issue of minimality of the state representation (9.13). Let A i bean n i × n i matrix (i = 1, 2). By minimality of the DV and ON representations (9.7)and (9.8) above, we have n(B 1 ) = n 1 and n(B ∗ 2) = n 2 . Thus we obtain:Lemma 9.4. Assume that (G ∗ 2G 1 ) −1 ∈ RL ∞ . Then (9.13) is a minimal staterepresentation of B 1 ∩ B ∗ 2.Proof: By Lemma 8.2, B 1 ∩B ∗ 2 is autonomous, and we have dim(B 1 ∩B ∗ 2) = n(B 1 )+n(B 2 ). As a consequence, since n(B ∗ 2) = n(B 2 ), the state space dimension of thestate representation (9.13), being equal to n 1 + n 2 , is equal to the McMillan degreedim(B 1 ∩ B ∗ 2) of B 1 ∩ B ∗ 2, and hence the state representation is minimal. □In particular this implies that the state is observable from the manifest variable, so forany w there exists exactly one x = col(x 1 , x 2 ) such that (9.13) holds. Observabilityis equivalent to observability of the pair ((C 1 0) − D 1 D −1 C, A − BD −1 C).Clearly, using observability, the stable part (B 1 ∩ B ∗ 2) stab of the autonomousbehavior B 1 ∩ B ∗ 2 consists of those external trajectories w whose corresponding statetrajectory x passes through the stable subspace of A − BD −1 C, in other words,(9.14) (B 1 ∩ B ∗ 2) stab = { w ∈ B 1 ∩ B ∗ 2 | x(0) ∈ X − (A − BD −1 C) }.Of course, a similar representation holds for the antistable part (B 1 ∩ B ∗ 2) anti .The following lemma states that, although (−D −1 C, A − BD −1 C) does not needto be observable, we do have that it is detectable:Lemma 9.5. Assume that (G ∗ 2G 1 ) −1 ∈ RL ∞ . Let N be the unobservable subspaceof the pair (−D −1 C, A − BD −1 C). Then N ⊂ X − (A − BD −1 C).Proof: Under the assumption, the state representation (9.13) is minimal. Let x 0 ∈ N,x 0 = col(x 10 , x 20 ), and let w be the corresponding external trajectory. In (9.13)we then have −D −1 Cx = 0, so w = C 1 x 1 , with ẋ 1 = A 1 x 1 . Since A 1 is Hurwitz,w(t) → 0 as t → ∞. By observability of (9.13) this implies x 0 ∈ X − (A − BD −1 C).□Note that x 0 is of the form col(0, x 20 ) if and only if x 0 ∈ X + (A). The following willbe very useful:Lemma 9.6. Assume that (G ∗ 2G 1 ) −1 ∈ RL ∞ . Then X + (A) ∩ N = {0}.Proof: Let x 0 ∈ X + (A) and in (9.13) assume that the corresponding state trajectoryx satisfies D −1 Cx = 0. Since then ẋ 1 = A 1 x 1 and since x 0 = (0, x 20 ) we find thatx 1 = 0, so x 2 satisfies ẋ 2 = −A ⊤ 2 x 2 , 0 = −B2 ⊤ x 2 . As (A 2 , B 2 ) was chosen to becontrollable, this implies that x 2 = 0, which yields x 0 = 0.□As a consequence of the representation (9.14) we can also compute the following:Lemma 9.7. dim(B 1 ∩ B ∗ 2) stab = dim X − (A − BD −1 C).We now turn to establishing a state representation of B 2 ∩ B ∗ 1. As before, letG i (ξ) = C i (ξI − A i ) −1 B i + D i , i = 1, 2 with (A i , B i ) controllable and (C i , A i ) observable.Now define(9.15) ( ) ( )A ′ A2 0:=C1 ⊤ C 2 −A ⊤ , B ′ B2:=1C1 ⊤ , C ′ := (DD1 ⊤ C 2 − B1 ⊤ ), D ′ := D1 ⊤ D 2 .2

Under the assumption (G ∗ 1G 2 ) −1 ∈ RL ∞ (equivalently (G ∗ 2G 1 ) −1 ∈ RL ∞ ), a minimalstate representation of the autonomous behavior B 2 ∩ B ∗ 1 is then given by(9.16)ddt x′ = (A ′ − B ′ D ′−1 C ′ )x ′ , w ′ = ( (C 2 0) − D 2 D ′−1 C ′) x ′ .where x ′ = col(x 2 , x 1 ). Our aim is to relate this explicitly to the state representation(9.13) of B 1 ∩ B ∗ 2. Indeed, the quadruple (9.15) is similar to the dual of (A, B, C, D):if we defineS :=( 0 −II 0then SA ′ S −1 = −A ⊤ , SB ′ = −C ⊤ , C ′ S −1 = B ⊤ and D ′ = D ⊤ . Thus, introducingthe new state variable z := Sx ′ , we get the following minimal state representation forB 2 ∩ B ∗ 1(9.17) ż = −(A − BD −1 C) ⊤ z, w ′ = ( (C 2 0) − D 2 D −⊤ B ⊤) z.The advantage of this representation is that it explicitly displays its relation with thestate representation (9.13) of B 1 ∩ B ∗ 2. This will be useful in the sequel. Note thatthis representation is again observable. Therefore(9.18) (B 2 ∩ B ∗ 1) stab = { w ′ ∈ B 2 ∩ B ∗ 1 | z(0) ∈ X − (−(A − BD −1 C) ⊤ ) }.Since X − (−(A−BD −1 C) ⊤ ) = X − (A−BD −1 C) ⊥ , in addition to the result of Lemma9.7 we have(9.19) dim(B 2 ∩ B ∗ 1) stab = n 1 + n 2 − dim X − (A − BD −1 C)9.4. Representation of B 2 1 ∩ (B 2 2) ⊥ and B 2 2 ∩ (B 2 1) ⊥ . In this final subsectionwe wrap up things, and establish explicit representations of the subspace intersectionsB 2 1 ∩ (B 2 2) ⊥ and B 2 2 ∩ (B 2 1) ⊥ in terms of realizations of rational image representationsof B 1 and B 2 .Let B 1 , B 2 ∈ L q cont, m(B 1 ) = m(B 2 ). Recall the short hand notation B 2 1 :=B 1 ∩ L 2 (R + ) and B 2 2 := B 2 ∩ L 2 (R + ). Again, let G 1 , G 2 ∈ RH ∞ be right primeover RH ∞ , such that B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt ), and such that G 1 andG 2 have no zeros. Recall from Lemma 9.1 that B 2 1 ∩ (B 2 2) ⊥ = T G1 ker T G ∗2 G 1. Underthe additional condition that (G ∗ 2G 1 ) −1 ∈ RL ∞ (equivalently, d L2 (B 1 , B 2 ) < 1, seeTheorem 8.10), a minimal state representation of B 1 ∩ B ∗ 2 is given by (9.13).In finding a representation of the intersection B 2 1 ∩ (B 2 2) ⊥ , the subbehavior ofB 1 ∩ B ∗ 2 of all external trajectories w whose corresponding state trajectory x passesboth through the intersection of the stable subspace of A − BD −1 C, and the antistablesubspace of A turns out to be crucial. Define(9.20) (B 1 ∩ B ∗ 2) + := { w ∈ B 1 ∩ B ∗ 2 | x(0) ∈ X + (A) }.Note that dim(B 1 ∩ B ∗ 2) + = n 2 , the MacMillan degree of B 2 .In the sequel, let P + : (R q ) R → (R q ) R denote the map that projects functionsfrom R to R q onto their future: (P + w)(t) := w(t)1I(t). Then, by applying Theorem9.2 we find that the subspace B 2 1 ∩ (B 2 2) ⊥ is the image of (B 1 ∩ B ∗ 2) stab ∩ (B 1 ∩ B ∗ 2) +under this projection:),27

28Theorem 9.8. Assume (G ∗ 2G 1 ) −1 ∈ RL ∞ . Let A, B, C and D be given by (9.9).Then we haveB 2 1 ∩ (B 2 2) ⊥ = P +((B1 ∩ B ∗ 2) stab ∩ (B 1 ∩ B ∗ 2) + )Furthermore, the dimension of B 2 1∩(B 2 2) ⊥ is equal to dim X − (A−BD −1 C)∩X + (A).Proof: If w 1 ∈ B 2 1 ∩ (B 2 2) ⊥ then it is of the form w 1 = T G1 v 1 with v 1 ∈ ker T G ∗2 G 1.By Theorem 9.2 and Lemma 9.5 we have v 1 (t) = −1I(t)v(t) withv(t) = D −1 Ce (A−BD−1 C)t x 0for some x 0 ∈ X − (A − BD −1 C) ∩ X + (A). Since G 1 ∈ RH ∞ we have w 1 = T G1 v 1 =M G1 v 1 , which is the unique solution in L 2 (R) of ẋ 1 = A 1 x 1 + B 1 v 1 , w 1 = C 1 x 1 +D 1 v 1 . For t ≥ 0 we have v 1 (t) = v(t), so for t ≥ 0 we must have w 1 (t) = w(t),where ( w(t) is determined by the equations ẋ = (A − BD −1 C)x, x(0) = x 0 , w =(C1 0) − D 1 D −1 C ) x. Clearly, w ∈ (B 1 ∩ B ∗ 2) stab ∩ (B 1 ∩ B ∗ 2) + and w 1 = P + w.Conversely, let w ∈ (B 1 ∩ B ∗ 2) stab ∩ (B 1 ∩ B ∗ 2) + and consider P + w. By definition,( w is determined by the equations ẋ = (A − BD −1 C)x, x(0) = x 0 , w =(C1 0) − D 1 D −1 C ) x, with x 0 ∈ X − (A − BD −1 C) ∩ X + (A). Definev(t) := −D −1 Cx(t).Then by Theorem 9.2 and Lemma 9.5 we have v 1 (t) = −1I(t)v(t) ∈ ker T G ∗2 G 1. Definew 1 := T G1 v 1 . Then, again, w 1 = M G1 v 1 is the unique trajectory in L 2 (R) thatsatisfies the equations ẋ 1 = A 1 x 1 + B 1 v 1 , w 1 = C 1 x 1 + D 1 v 1 . Since for t ≥ 0 wehave v 1 (t) = v(t), we must also have w 1 (t) = w(t) for t ≥ 0, so P + w = w 1 ∈T G1 ker T G ∗2 G 1= w 1 ∈ B 2 1 ∩ (B 2 2) ⊥ .Finally, from observability of the state representation (9.13) the dimension of(B 1 ∩ B ∗ 2) stab ∩ (B 1 ∩ B ∗ 2) + is equal to dim(X − (A − BD −1 C) ∩ X + (A)). Obviouslythis must also be the dimension of B 2 1 ∩ (B 2 2) ⊥ .□Next we turn to representing the dual intersection B 2 2 ∩ (B 2 1) ⊥ . Thus, as before,we introduce the subbehavior of B 2 ∩B ∗ 1 of all external trajectories w ′ such that theircorresponding state trajectory x ′ passes through X + (A ′ ) (with respect to the staterepresentation (9.16)):(9.21) (B 2 ∩ B ∗ 1) + := { w ′ ∈ B 2 ∩ B ∗ 1 | x ′ (0) ∈ X + (A ′ ) }.It is easily verified that in terms of the alternative state representation (9.17) we have(9.22) (B 2 ∩ B ∗ 1) + = { w ′ ∈ B 2 ∩ B ∗ 1 | z(0) ∈ X + (−A ⊤ ) }so analogously to Theorem 9.8 we find that if (G ∗ 2G 1 ) −1 ∈ RL ∞ thenMoreover, the dimension ofX + (−A ⊤ ).B 2 2 ∩ (B 2 1) ⊥ = P +((B2 ∩ B ∗ 1) stab ∩ (B 2 ∩ B ∗ 1) + ) .B 2 2 ∩ (B 2 1) ⊥ is equal to dim X − (−(A − BD −1 C) ⊤ ) ∩10. Properties of the metrics, continued. Using the detailed analysis inthe previous section, we will now continue our study of the Z-metric. We will alsoreturn to the L 2 -metric, the V-metric and the SB-metric. Let B 1 , B 2 ∈ L q cont withm(B 1 ) = m(B 2 ). Again, a standing assumption throughout this section will be thatG 1 , G 2 ∈ RH ∞ are right prime over RH ∞ , B 1 = im G 1 ( d dt ) and B 2 = im G 2 ( d dt), andG 1 and G 2 have no zeros. In addition we now assume that both G 1 and G 2 are inner.Throughout this section, the constant real matrices A, B, C and D are obtained fromminimal realizations of G 1 and G 2 , and are given by (9.9).

10.1. Behavioral characterizations for Z-metric and V-metric. From Theorem8.6 recall that for controllable behaviors B 1 , B 2 with the same number of inputs,d V (B 1 , B 2 ) < 1 if and only if d L2 (B 1 , B 2 ) < 1 and the dimension of (B 1 ∩ B ∗ 2) stabis equal to the MacMillan degree of B 2 . Also for the Z-metric conditions can now beformulated in terms of the stable part of B 1 ∩ B ∗ 2. First note the following immediateconsequence of Theorem 9.8:Lemma 10.1. Assume that (G ∗ 2G 1 ) −1 ∈ RL ∞ . Let (B 1 ∩ B ∗ 2) + be defined by(9.20). Then B 2 1 ∩ (B 2 2) ⊥ = 0 if and only if (B 1 ∩ B ∗ 2) stab ∩ (B 1 ∩ B ∗ 2) + = {0}.By applying Proposition 8.9 and Theorem 9.8, this immediately leads to thefollowing behavioral characterization. Again, let (B 1 ∩B ∗ 2) + be defined by (9.20) andlet (B 2 ∩ B ∗ 1) + be defined by (9.21).Theorem 10.2. Let B 1 , B 2 ∈ L q cont with m(B 1 ) = m(B 2 ). Then d Z (B 1 , B 2 ) < 1if and only if the following three conditions hold:1. d L2 (B 1 , B 2 ) < 1,2. (B 1 ∩ B ∗ 2) stab ∩ (B 1 ∩ B ∗ 2) + = {0},3. (B 2 ∩ B ∗ 1) stab ∩ (B 2 ∩ B ∗ 1) + = {0}.Also in case of the V-metric, the behavior intersections appearing in conditions(2.) and (3.) of Theorem 10.2 turn out to crucial in order to characterize distance lessthan one. Indeed, the following theorem gives an alternative for the characterizationof Theorem 8.6 (see also [21], page 41):Theorem 10.3. Let B 1 , B 2 ∈ L q cont with m(B 1 ) = m(B 2 ). Then d V (B 1 , B 2 ) < 1if and only if the following two conditions hold:1. d L2 (B 1 , B 2 ) < 1,2. dim(B 1 ∩ B ∗ 2) stab ∩ (B 1 ∩ B ∗ 2) + = dim(B 2 ∩ B ∗ 1) stab ∩ (B 2 ∩ B ∗ 1) + .The condition (2.) above is equivalent todim B 2 1 ∩ (B 2 2) ⊥ = dim B 2 2 ∩ (B 2 1) ⊥ .Proof: We will show that under the assumption that condition (1.) holds (equivalently,(G ∗ 2G 1 ) −1 ∈ RL ∞ ), condition (2.) is equivalent with dim(B 1 ∩ B ∗ 2) stab =n(B 1 ). Indeed, it was shown in the previous subsection that(10.1) dim(B 1 ∩ B ∗ 2) stab ∩ (B 1 ∩ B ∗ 2) + = dim X − (A − BD −1 C) ∩ X + (A)anddim(B 2 ∩ B ∗ 1) stab ∩ (B 2 ∩ B ∗ 1) + = dim X − (−(A − BD −1 C) ⊤ ) ∩ X + (−A ⊤ ).Now note thatX − (−(A − BD −1 C) ⊤ ) ∩ X + (−A ⊤ ) =X − (A − BD −1 C) ⊥ ∩ X + (A) ⊥ = (X − (A − BD −1 C) + X + (A)) ⊥ .As a consequence we obtain:(10.2) dim(B 2 ∩ B ∗ 1) stab ∩ (B 2 ∩ B ∗ 1) + = n 1 + n 2 − dim X − (A − BD −1 C) + X + (A).Next,n 1 + n 2 − dim X − (A − BD −1 C) + X + (A)= n 1 + n 2 − ( dim X − (A − BD −1 C) + dim X + (A) − dim X − (A − BD −1 C) ∩ X + (A) )= n 1 − dim X − (A − BD −1 C) + dim X − (A − BD −1 C) ∩ X + (A).29

30This is equal to dim X − (A − BD −1 C) ∩ X + (A) if and only if the condition n 1 =dim X − (A − BD −1 C) holds, equivalently dim(B 1 ∩ B ∗ 2) stab = n(B 1 ).□Thus, as indicated before in [21], page 41, under the assumption that the distance inthe L 2 -metric is less than 1, the distance in the V-metric is less than 1 if and only ifthe dimensions of the intersections (B 1 ∩ B ∗ 2) stab ∩ (B 1 ∩ B ∗ 2) + and (B 2 ∩ B ∗ 1) stab ∩(B 2 ∩ B ∗ 1) + are equal, whereas the distance in the Z-metric is less than 1 if and onlyif these intersections have dimension zero.10.2. State space characterizations. In this final subsection we will collectthe relevant material from Section 9 and formulate state space conditions for thedistance in our metrics to be less than 1, in terms of the minimal driving variablerepresentations of B 1 and B 2 obtained by realization of G 1 and G 2 . We will firstestablish such conditions for the L 2 -metric:Theorem 10.4. Let B 1 , B 2 ∈ L q cont with m(B 1 ) = m(B 2 ). Then d L2 (B 1 , B 2 ) < 1if and only if the following two conditions hold:1. D = D2 ⊤ D 1 is nonsingular,2. A − BD −1 C has no imaginary axis eigenvalues.Proof: (⇒) By Lemma 8.1, G ∗ 2G 1 is biproper. This implies D = D2 ⊤ D 1 is nonsingular.By Theorem 8.3, B 1 ∩B ∗ 2 is autonomous and its dimension is equal to n(B 1 )+n(B 2 ).Thus (9.13) is a minimal state representation, so it is observable. Finally, again byTheorem 8.3, B 1 ∩ B ∗ 2 has no imaginary frequencies, so none of the eigenvalues ofA − BD −1 C can lie on the imaginary axis.(⇐) D = D2 ⊤ D 1 nonsingular implies that G ∗ 2G 1 is biproper. By Lemma 8.2,B 1 ∩ B ∗ 2 is autonomous and its dimension is equal to n(B 1 ) + n(B 2 ). Also, (9.13)is a state representation of B 1 ∩ B ∗ 2. Since A − BD −1 C has no eigenvalues on theimaginary axis, B 1 ∩ B ∗ 2 has no imaginary frequencies. Theorem 8.3 then yieldsd L2 (B 1 , B 2 ) < 1.□Next, we turn to the V-metric again:Theorem 10.5. Let B 1 , B 2 ∈ L q cont with m(B 1 ) = m(B 2 ). Denote n 1 := n(B 1 ).Then d V (B 1 , B 2 ) < 1 if and only if the following conditions hold:1. d L2 (B 1 , B 2 ) < 12. dim X − (A − BD −1 C) = n 1 .Proof: If d V (B 1 , B 2 ) < 1 then d L2 (B 1 , B 2 ) < 1. Under this condition B 1 ∩ B ∗ 2is autonomous and by Lemma 9.7, dim(B 1 ∩ B ∗ 2) stab = dim X − (A − BD −1 C). ByTheorem 8.6 the latter equals n 1 . The converse is proven in a similar way. □Finally, we give a characterization for the Z-metric:Theorem 10.6. Let B 1 , B 2 ∈ L q cont and m(B 1 ) = m(B 2 ). Then d Z (B 1 , B 2 ) < 1if and only if the following four conditions hold:1. d L2 (B 1 , B 2 ) < 12. X − (A − BD −1 C) ⊕ X + (A) = R n1+n2 .Proof: d Z (B 1 , B 2 ) < 1 if and only if d L2 (B 1 , B 2 ) < 1 and conditions (2.) and (3.)of Theorem 10.2 hold. By (10.1), condition (2.) is equivalent with X − (A−BD −1 C)∩X + (A) = {0} and by (10.2) condition (3.) is equivalent with X − (A − BD −1 C) ∩X + (A) = R n1+n2 .□Note that, since dim X + (A) = n 2 , the condition X − (A − BD −1 C) ⊕ X + (A) = R n1+n2implies that dim X − (A − BD −1 C) = n 1 . This indeed confirms that d Z (B 1 , B 2 ) < 1implies d V (B 1 , B 2 ) < 1, as we already knew.

11. Conclusions. In this paper we have studied notions of distance betweenlinear differential systems. We have introduced four metrics on the space of all controllablebehaviors. Three of these have been defined in terms of gaps between closedsubspaces of the Hilbert space L 2 (R). After having established the relation betweenrational representations of behaviors and classical multiplication operators, we haveexpressed these metrics in terms of the proper rational matrices appearing in therational representations. We have introduced a fourth metric on the space of controllablebehaviors as a generalization of the ν-metric. As in the input-output framework,this definition has been given in terms of rational representations. For this metric, wehave established a representation free, behavioral characterization as well. We havealso made a comparison between the four metrics, and have compared the values theytake, and the topologies they induce. Finally, for all metrics we have made a detailedstudy of necessary and sufficient conditions under which the distance between twobehaviors is less than one. For this, both behavioral as well as state space conditionshave been derived in terms of driving variable representations of the behaviors.31REFERENCES[1] N.I. Akhiezer and I.M. Glazman, em Theory of Linear Operators in Hilbert Space, Vol. I,Frederick Ungar, New York, 1961.[2] J.A. Ball and A.J. Sasane, ‘Extension of the ν-metric, the H ∞ case’,http://arxiv.org/abs/1010.1843, 2010.[3] Wenming Bian, Mark French and Harish Pillai, ’An intrinsic behavioural approach to the gapmetric. SIAM Journal on Control and Optimization, 47, (4), pp. 1939 - 1960, 2008.[4] R.G. Douglas, Banach algebra techniques in operator theory, second edition, Graduate Textsin Mathematics, 179, Springer-Verlag, New York, 1998.[5] B.A. Francis, A Course in H ∞ Control Theory, Lecture Notes in Control and InformationSciences, Springer-Verlag, 1987.[6] T.T. Georgiou, ‘On the computation of the gap metric’, Systems and Control Letters, 11, pp.253-257, 1988.[7] T.T. Georgiou and M.C. Smith, ‘Optimal robustness in the gap metric’, IEEE Transactionson Automatic Control, Vol. 35, pp. 673-686, 1990.[8] T.T. Georgiou and M.C. Smith, ’Robustness analysis of nonlinear feedback systems: an inputoutputapproach’, IEEE Transactions on Automatic Control, Vol. 42, No. 9, pp. 1200 -1221, 1997.[9] M.E.C. Mutsaers and S. Weiland, ‘Rational representations and controller synthesis of L 2behaviors’, Automatica, 48, pp. 1-14, 2012.[10] N.K. Nikolski, Treatise On the Shift Operator, Springer verlag, 1980.[11] J.W. Polderman and J.C. Willems, Introduction to Mathematical Systems Theory: a BehavioralApproach, Springer-Verlag, Berlin, 1997.[12] A.J. Sasane, ’Distance between behaviours’, Internat. J. Control, 76, pp. 1214 - 1223, 2003.[13] J.A. Ball and A.J. Sasane, ‘Equivalence of behavioral distance and the gap metric’, Systemsand Control Letters, 55, pp. 214-222, 2006.[14] A. J. Sasane, ‘The new ν-metric induces the classical gap topology’,http://arxiv.org/abs/1012.0427, 2010.[15] S. V. Gottimukkala, S. Fiaz and H. L. Trentelman ‘Equivalence of rational representations ofbehaviors’, Systems and Control Letters, Vol. 60, pp. 119-127, 2011.[16] S.V. Gottimukkala, H.L. Trentelman and S. Fiaz, ‘Realization and elimination in rational representationsof behaviors’, Systems and Control Letters, Vol. 62, pp. 708-714, , 2013.[17] J.A. Sefton and R.J. Ober, ‘On the gap metric and coprime factor perturbations’, Automatica,Vol. 29, pp. 723-734, 1993.[18] J.C. Willems and H.L. Trentelman, ‘On quadratic differential forms’, SIAM Journal of Controland Optimization, Vol. 36, No. 5, pp. 1703 - 1749, 1998.[19] H.L. Trentelman and J.C. Willems, ‘H ∞-control in a behavioral context: the full informationcase’, IEEE Transactions on Automatic Control, Vol. 44, No. 3, pp. 521 - 536, 1999.[20] J.C. Willems and H.L. Trentelman, ‘Synthesis of dissipative systems using quadratic differentialforms - part I, IEEE Transactions on Automatic Control, Vol. 47, nr. 1, pp. 53 - 69, 2002.

32[21] G. Vinnicombe, Uncertainty and Feedback: H ∞ loop-shaping and the ν-gap metric, ImperialCollege Press, 2001.[22] G. Vinnicombe, ‘Frequency domain uncertainty and the graph topology’, IEEE Transactionson Automatic Control, Vol. 38, pp. 1371-1383, 1993.[23] J. Weidman, Linear Operators in Hilbert Space, Springer Verlag, Berlin, 1980.[24] S. Weiland and A.A. Stoorvogel, ‘Rational representations of behaviors: interconnectabilityand stabilizability’, Math. Contr. Sign. & Systems, Vol.10, pp. 125-164, 1997.[25] J.C. Willems, ‘Input-output and state-space representations of finite-dimensional linear timeinvariantsystems’, Linear Algebra and its Applications, Vol. 50, pp. 581- 608, 1983.[26] J.C. Willems, ‘The behavioral approach to open and interconnected systems’, IEEE ControlSystems Magazine, pp. 46-99, December 2007.[27] J.C. Willems and Y. Yamamoto, “Behaviors defined by rational functions,” Linear algebra andits applications, Vol. 425, pp. 226-241, 2007.[28] G. Zames and A.K. El-Sakkary, ‘Unstable systems and feedback, The gap metric’, Proceeedingsof the Allerton Conf., pp. 380-385, 1980.