Extension of the $\nu$-metric for stabilizable plants over $H^\infty$

An abstract $\nu$-metric was introduced by Ball and Sasane, with a view towards extending the classical $\nu$-metric of Vinnicombe from the case of rational transfer functions to more general nonrational transfer function classes of infinite-dimensional linear control systems. In this short note, we give an important concrete special instance of the abstract $\nu$-metric, by verifying that all the assumptions demanded in the abstract set-up are satisfied when the ring of stable transfer functions is the Hardy algebra $H^\infty$. This settles the open question implicit in \cite{BalSas2}.


Introduction
We recall the general stabilization problem in control theory. Suppose that R is a commutative integral domain with identity (thought of as the class of stable transfer functions) and let F(R) denote the field of fractions of R. The stabilization problem is: Given P ∈ (F(R)) p×m (an unstable plant transfer function), find C ∈ (F(R)) m×p (a stabilizing controller transfer function), such that H(P, C) := P I (I − CP ) −1 −C I ∈ R (p+m)×(p+m) (is stable).
In the robust stabilization problem, one goes a step further. One knows that the plant is just an approximation of reality, and so one would really like the controller C to not only stabilize the nominal plant P 0 , but also all sufficiently close plants P to P 0 . The question of what one means by "closeness" of plants thus arises naturally. So one needs a function d defined on pairs of stabilizable plants such that (1) d is a metric on the set of all stabilizable plants, (2) d is amenable to computation, and (3) stabilizability is a robust property of the plant with respect to this metric (that is, whenever a plant P 0 is stabilized by a controller C, then there is a small enough neighbourhood of the plant P 0 consisting of plants which are stabilized by the same controller C). Such a desirable metric, was introduced by Glenn Vinnicombe in [13] and is called the ν-metric. In that paper, essentially R was taken to be the rational functions without poles in the closed unit disk or, more generally, the disk algebra, and the most important results were that the ν-metric is indeed a metric on the set of stabilizable plants, and moreover, one has the inequality that if P 0 , P ∈ S(R, p, m), then µ P,C ≥ µ P 0 ,C − d ν (P 0 , P ), where µ P,C denotes the stability margin of the pair (P, C), defined by µ P,C := H(P, C) −1 ∞ . This implies in particular that stabilizability is a robust property of the plant.
The problem of what happens when R is some other ring of stable transfer functions of infinite-dimensional systems was left open in [13]. This problem of extending the ν-metric from the rational case to transfer function classes of infinite-dimensional systems was addressed in [1]. There the starting point in the approach was abstract. It was assumed that R is any commutative integral domain with identity which is a subset of a Banach algebra S satisfying certain assumptions, labeled (A1)-(A4), which are recalled in Section 2. Then an "abstract" ν-metric was defined in this setup, and it was shown in [1] that it does define a metric on the class of all stabilizable plants. It was also shown there that stabilizability is a robust property of the plant.
In [13], it was suggested that the ν-metric in the case when R = H ∞ might be defined as follows. (Here H ∞ denotes the algebra of bounded and holomorphic functions in the unit disk {z ∈ C : |z| < 1}.) Let P 1 , P 2 be unstable plants with the normalized left/right coprime factorizations Here G k , G k arise from P k (k = 1, 2) according to the notational conventions given in Subsection 2.5 below, and · * has the usual meaning, namely: G * 1 (ζ) is the transpose of the matrix whose entries are complex conjugates of the entries of the matrix G 1 (ζ), for ζ ∈ T. Also in the above, for a matrix M ∈ (L ∞ ) p×m , T M denotes the Toeplitz operator from (H 2 ) m to (H 2 ) p , given by where M ϕ is considered as an element of (L 2 ) p and P (H 2 ) p denotes the canonical orthogonal projection from (L 2 ) p onto (H 2 ) p .
In [2], we showed that the above does work for the case when R is the smaller class QA of quasianalytic functions in the unit disk. We proved this by showing that this case is just a special instance of the abstract ν-metric introduced in [1]. A perusal of the extensive literature on Fredholm theory of Toeplitz operators from the 1970s lead to this choice of R = QA and S = QC (the class of quasicontinuous functions) as conceivably the most general subalgebras of H ∞ and L ∞ which fit the setup of [1].
In this article, we use a different idea to tackle the problem of defining a new metric in the case when R = H ∞ . We first notice that when R is the disk algebra A(D), then there is no problem in defining the ν-metric; see [1, §5.1]. We then handle the H ∞ case by using the observation that the restrictions of a function f ∈ H ∞ to the smaller disks with radii r < 1 give rise to elements in the disk algebra by dilating these restrictions to bigger disks of radius 1. In other words, f r defined via are all elements of A(D). We then use these restrictions in a suitable manner to define the ν-metric.
The paper is organized as follows: (1) In Section 2, we recall the general setup with the assumptions and the abstract metric d ν from [1]. (2) In Section 3, we specialize R to a concrete ring of stable transfer functions, namely R = H ∞ , and show that our abstract assumptions hold in this particular case. Moreover in the Subsection 3.2, we will show that when our extended ν-metric is restricted to rational plants, we obtain the classical ν-metric, hence showing that we have obtained a genuine extension.

Recap of the abstract ν-metric
We recall the setup from [1]: (A1) R is commutative integral domain with identity. (A2) S is a unital commutative complex semisimple Banach algebra with an involution · * , such that R ⊂ S. We use inv S to denote the invertible elements of S. (A3) There exists a map ι : inv S → G, where (G, +) is an Abelian group with identity denoted by •, and ι satisfies (I3) ι is locally constant, that is, ι is continuous when G is equipped with the discrete topology. (A4) x ∈ R∩(inv S) is invertible as an element of R if and only if ι(x) = •.
We recall the following standard definitions from the factorization approach to control theory.
2.1. The notation F(R): F(R) denotes the field of fractions of R.
2.2. The notation F * : If F ∈ R p×m , then F * ∈ S m×p is the matrix with the entry in the ith row and jth column given by F * ji , for all 1 ≤ i ≤ p, and all 1 ≤ j ≤ m.

2.3.
Right coprime/normalized coprime factorization: For a matrix P ∈ (F(R)) p×m , a factorization P = N D −1 , where N, D are matrices with entries from R, is called a right coprime factorization of P if there exist matrices X, Y with entries from R such that XN + Y D = I m . If moreover N * N + D * D = I m , then the right coprime factorization is referred to as a normalized right coprime factorization of P .

2.4.
Left coprime/normalized coprime factorization: For a matrix P ∈ (F(R)) p×m , a factorization P = D −1 N , where N , D are matrices with entries from R, is called a left coprime factorization of P if there exist matrices X, Y with entries from R such that N X + D Y = I p . If moreover N N * + D D * = I p , then the left coprime factorization is referred to as a normalized left coprime factorization of P .
2.5. The notation G, G, K, K: Given P ∈ (F(R)) p×m with normalized right and left factorizations P = N D −1 and P = D −1 N , respectively, we introduce the following matrices with entries from R: Similarly, given a C ∈ (F(R)) m×p with normalized right and left factorizations C = N C D −1 C and C = D −1 C N C , respectively, we introduce the following matrices with entries from R: 2.6. The notation S(R, p, m): S(R, p, m) denotes the set of all elements P ∈ (F(R)) p×m that possess a normalized right coprime factorization and a normalized left coprime factorization.
We now recall the definition of the metric d ν on S(R, p, m). But first we specify the norm we use for matrices with entries from S. and refer to it as the Gelfand norm. Here M denotes the entry-wise Gelfand transform of M , and · denotes the induced operator norm from C m to C p . For the sake of concreteness, we fix the standard Euclidean norms on the vector spaces C m to C p .
The maximum in (2.1) exists since M is a compact space when it is equipped with Gelfand topology, that is, the weak- * topology induced from L(S; C). Since we have assumed S to be semisimple, the Gelfand transform is an injective map. If M ∈ S 1×1 = S, then we note that there are two norms available for M : the one as we have defined above, namely M S,∞ , and the norm M S of M as an element of the Banach algebra S. But throughout this article, we will use the norm given by (2.1).
The following was proved in [1]: Definition 2.4. Given P ∈ (F(R)) p×m and C ∈ (F(R)) m×p , the stability margin of the pair (P, C) is defined by The number µ P,C can be interpreted as a measure of the performance of the closed loop system comprising P and C: larger values of µ P,C correspond to better performance, with µ P,C > 0 if and only if C stabilizes P .
The following was proved in [1]: The above result says that stabilizability is a robust property of the plant, since if C stabilizes P 0 with a stability margin µ P,C > m, and P is another plant which is close to P 0 in the sense that d ν (P, P 0 ) ≤ m, then C is also guaranteed to stabilize P . We will now introduce a Banach algebra, C b (A ρ ), which will serve as the Banach algebra S in our abstract set up.
Notation 3.1. Given ρ ∈ (0, 1), let A ρ be the open annulus Proposition 3.2. Let ρ ∈ (0, 1). With the norm defined by Proof. The verification of the claims is straightforward. We just give the proof of the semisimplicity. Recall that a commutative complex Banach algebra is called semisimple if its radical ideal, namely the intersection of all the maximal ideals of the Banach algebra is 0. We also know that kernels of complex homomorphisms are maximal ideals. For z ∈ A ρ , the map Since the radical ideal is contained in the intersection of the kernels of the complex homomorphisms ϕ z , z ∈ A ρ , it must be zero.
Then I is an injective map.
Proof. The map I is a linear transformation. Suppose that I(f ) = 0 for some f ∈ H ∞ . This means that the restriction of f to the annulus A ρ is identically 0, and as f is holomorphic in D, f must be zero in the whole disk D. Hence f = 0.
Henceforth we will identify H ∞ as a subset of C b (A ρ ) via this map I.
We will now define an index on invertible elements of S = C b (A ρ ).
Notation 3.4. We use the notation C(T) for the Banach algebra of complexvalued continuous functions defined on the unit circle T := {z ∈ C : |z| = 1}, with all operations defined pointwise, with the supremum norm: and the involution · * defined pointwise: , then for each r ∈ (ρ, 1), the map F r : T → C, given by belongs to inv C(T), and so each F r has a well-defined (integral) winding number w(F r ) ∈ Z with respect to 0.
Moreover, we now show by the local constancy of the winding number We use the fact that the winding numbers w(ϕ), w(ψ) with respect to 0 of ϕ, ψ : T → C \ {0}, are the same if ϕ, ψ are homotopic; see for example [3, §2.7.10, p.50].
As the annulus K := {z ∈ C : r ≤ |z| ≤ r ′ } is compact, it follows that there is a m > 0 such that F (z) lies in C \ mD for all z ∈ K. Also, F is uniformly continuous on K, and so we can choose N large enough so that with we have that Fix an n ∈ {0, 1, 2, . . . , N − 1}. Set ϕ = F rn and ψ = F r n+1 . Then ϕ, ψ belong to inv C(T). Consider the map H : H is a homotopy from ϕ to ψ. In particular it follows from the above that ψ = H(·, 1) ∈ inv C(T), and that the winding numbers of ϕ and ψ are identical. So it follows that . This completes the proof.
By the preceding discussion, we see that W is well-defined. We will now prove a sequence of results aimed towards verifying the assumptions (A3) and (A4) in our abstract setup.
Proof. For f, g ∈ inv C(T), we have w(f g) = w(f ) + w(g), and so it follows that for F, G ∈ inv C b (A ρ ), and r ∈ (ρ, 1), This completes the proof.
This completes the proof. Proposition 3.9. Let ρ ∈ (0, 1). Then W : inv C b (A ρ ) → Z is locally constant, that is, it is continuous when Z is equipped with the discrete topology.
Proof. Let F ∈ inv C b (A ρ ). Let r ∈ (ρ, 1). By the local constancy of the map w : inv C(T) → Z, it follows that there is a δ > 0 such that for all This proves the desired local constancy of W .
Finally we have the following analogue of the classical Nyquist criterion. ("Only if" part) Let G ∈ C b (A ρ ) be the inverse of F := I(f ). If r ∈ (ρ, 1), then f r := f (r·) ∈ A(D) and f r ∈ inv C(T). Since W (F ) = w(f r ) = 0, it follows again by the Nyquist criterion for the disk algebra recalled above, that f r is invertible in A(D). In other words, f (rz) = 0 for all z ∈ D. It follows from here, as the choice of r ∈ (ρ, 1) was arbitrary, that f (z) = 0 for all z ∈ D, that is, f has a pointwise inverse g : D → C. Moreover, g is holomorphic in D. We have f (z)g(z) = f (z)G(z) = 1 (ρ < |z| < 1), and so it follows that G(z) = g(z) (ρ < |z| < 1). Hence by the maximum modulus principle, sup showing that g ∈ H ∞ . Consequently, f ∈ inv H ∞ . This completes the proof of the "only if" part.
Proof. Since H ∞ is a commutative integral domain with identity, (A1) holds.
(A2) follows from the results in Propositions 3.2 and 3.3. Indeed, the set C b (A ρ ) is a unital, commutative, complex, semisimple Banach algebra with the involution · * defined earlier in Proposition 3.2. Moreover, the map The definition of the abstract ν-metric given in Definition 2.2, now takes the following concrete form. For P 1 , P 2 ∈ S(H ∞ , p, m), with the normalized left/right coprime factorizations where the notation is as in Subsections 2.1-2.6.

It now follows from (3.2) and (3.3) that µ
Consequently, F C b (Aρ),∞ = F ∞ := sup z∈Aρ F (z) . Now suppose that A ∈ (H ∞ ) p×m . Then we have we we have used the vector valued version of the Maximum Modulus Principle (see for example [9, p.50]) to obtain the second equality. This completes the proof.
In light of the above result, the abstract ν-metric now takes the following form.
For P 1 , P 2 ∈ S(H ∞ , p, m), with the normalized left/right coprime factorizations where the notation is as in Subsections 2.1-2.6.
Remark 3.13. We also remark that the set S(H ∞ , p, m) coincides with the set of stabilizable plants, using the following two facts: Summarizing, our main result is the following, where the stability margin of a pair (P, C) ∈ S(H ∞ , p, m) × S(H ∞ , m, p) is Corollary 3.14. d ν given by (3.4) is a metric on the set of stabilizable plants over H ∞ . Moreover, if P 0 , P belong to S(H ∞ , p, m) and C ∈ S(H ∞ , m, p), then µ P,C ≥ µ P 0 ,C − d ν (P 0 , P ).

3.1.
Irrelevance of ρ ∈ (0, 1) in the definition of the ν-metric for S(H ∞ , p, m). Consider the condition Clearly only the tail end of the winding numbers are relevant, and so the noninvertibility in C b (A ρ ) owing to the noninveribility of det((G 1 | rT ) * G 2 | rT ) for small r's in (ρ, 1) should not really matter. We remedy this problem by taking the pointwise limit as ρ ր 1 of the ν-metrics corresponding to the ρ's in (0, 1).

d ∞
ν is an extension of the "classical" ν-metric. In [13], the νmetric for rational plants (and more generally elements of S(A(D), p, m)) was defined as follows. For P 1 , P 2 ∈ S(A(D), p, m), with the normalized left/right coprime factorizations where the notation is as in Subsections 2.1-2.6. In this subsection we will show that our ν-metric, defined by (3.5), coincides exactly with the above metric defined by (3.6), when the data P 1 , P 2 belong to S(A(D), p, m) (instead of the bigger set S(H ∞ , p, m)).
Hence ϕ ∈ inv C(T). Since the winding number map w : inv C(T) → Z is locally constant, we can also conclude that w(ϕ) = 0. Hence . This completes the proof.

d ∞
ν is an extension of the ν-metric defined for R = QA in [2]. In [2], a ν-metric was defined when R = QA, and we recall the definition below.
First of all, we use the notation QC for the C * -subalgebra of L ∞ (T) of quasicontinuous functions: QC := (H ∞ + C(T)) (H ∞ + C(T)). The Banach algebra QA of analytic quasicontinuous functions is QA := H ∞ QC. For P 1 , P 2 ∈ S(QA, p, m), with the normalized left/right coprime factorizations where the notation is as in Subsections 2.1-2.6. In this subsection we will show that our ν-metric, defined by (3.5), coincides exactly with the above metric defined by (3.7), when the data P 1 , P 2 belong to S(QA, p, m) (instead of the bigger set S(H ∞ , p, m)).
Moreover, then the Fredholm index of T f (namely, dim(ker T f )−dim(ker T * f )) is the negative of the winding number with respect to the origin of the curves f (r) for 1 − δ < r < 1.
Using Sarason's result ([11, Lemma 6]) recalled above, and the local constancy of winding numbers, we will now show that for all r's close enough to 1 the maps ζ ϕr → det((G 1 (rζ)) * G 2 (rζ)) (ζ ∈ T) are invertible as elements of C(T), and moreover their winding numbers are all 0. Indeed, we have for suitable scalar g 1i , g 2i ∈ QA and indices i. But by [9, Part A, Section 3.4], g 1i | rT = g 1i,(r) and g 2i | rT = g 2i,(r) . Also, by Sarason's result, for all i's Since for all r's close enough to 1, the ϕ (r) are uniformly bounded away from 0, it follows that also the ϕ r are uniformly bounded away from 0. In particular, they are all elements of inv C(T) for r's sufficiently near 1. Finally, by the local constancy of winding numbers, it follows that also the winding numbers of ϕ r are all 0 for all r's close enough to 1.
Hence when det(G * 1 G 2 ) is considered as an element F of C b (A ρ ), we have that F r are invertible in C(T) for all r's close enough to 1, and their winding numbers are 0. Thus the condition (C) is satisfied for all ρ's close enough to 1. Hence d ρ ν (P 1 , P 2 ) = G 2 G 1 ∞ = d ν,QA (P 1 , P 2 ) for all ρ's close enough to 1. Consequently, d ∞ ν (P 1 , P 2 ) = d ν,QA (P 1 , P 2 ) (< 1). Now suppose that d ∞ ν (P 1 , P 2 ) < 1. Then d ρ ν (P 1 , P 2 ) is a constant < 1 for all ρ's sufficiently close to 1. This implies that the condition (C) is satisfied for all ρ's close enough to 1. Hence the maps ζ ϕr → det((G 1 (rζ)) * G 2 (rζ)) (ζ ∈ T) are all elements of inv C(T) for all r's close enough to 1, and moreover, their winding numbers are all equal to 0. Owing to the invertibility in C b (A ρ ), it follows that these maps ϕ r are uniformly bounded away from 0 for all r's close enough to 1. Set ϕ to be the map From the above observations, the maps ϕ (r) are uniformly bounded away from 0 for all r's sufficiently near 1 and moreover their winding numbers are all 0. But then by Douglas's result recalled above (or see [9, Corollary 4.5.11]), the operator T ϕ is invertible. In particular, it is Fredholm with Fredholm index 0. Hence d ν,QA (P 1 , P 2 ) = G 2 G 1 ∞ = d ∞ ν (P 1 , P 2 ) (< 1). This completes the proof.

A computational example
Consider the transfer function P given by we can then transplant the plant to the unit disk. In this manner, we can endow S(H ∞ (C >0 , p, m) also with the ν-metric. As an illustration, we will calculate the ν-metric between a pair of plants arising from this P when there is uncertainty in the parameter a or T . A normalized (left=right) coprime factorization of P above is given by P = N/D, where where T, a 1 , a 2 > 0. Set s := ϕ(z) for z ∈ A ρ , ρ ∈ (0, 1). Then It is clear that z → |f (ϕ(z))| is bounded on D. It can be shown that for |a 1 − a 2 | small enough, the real part of f (s) is nonnegative and bounded away from zero for all s ∈ C such that Re(s) ≥ 0, as shown below. Then there is a δ 0 small enough such that for all 0 ≤ δ < δ 0 , there is a m > 0 such that Re(f (s)) > m > 0 (s ∈ C ≥0 ). · δ. Choose δ 0 so that 1 + √ 2 2R · δ < ǫ 2 for all 0 ≤ δ < δ 0 . Thus whenever |s| > R, we have for all such δ that and so Re s − a √ 2s + a · s − a − δ √ 2s + a + δ > 1 4 . But clearly for s ∈ C ≥0 , Re |s| 2 e −2Re(s)T ( √ 2s + a)( √ 2s + a + δ) ≥ 0.
We have