A Generalized Chordal Metric in Control Theory Making Strong Stabilizability a Robust Property

An abstract chordal metric is defined on linear control systems described by their transfer functions. Analogous to a previous result due to Partington (Linear Operators and Linear Systems. An Analytical Approach to Control Theory. London Mathematical Society Student Texts, vol. 60, Cambridge University Press, Cambridge, 2004) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^\infty $$\end{document}, it is shown that strong stabilizability is a robust property in this metric.


Introduction
The aim of this note is to give an extension of a result due to Jonathan Partington (recalled below in Proposition 1.1) saying that strong stabilizability is a robust property of the plant in the chordal metric. The new point in this article is that we prove that the analogous result holds in an abstract setting, hence expanding the domain of applicability from the original setting of unstable plants over H ∞ to ones over arbitrary rings of stable transfer functions satisfying mild assumptions. (Here, as is usual in the control engineering literature, H ∞ denotes the Hardy algebra of bounded holomorphic functions defined in the complex open right half plane {s ∈ C : Re(s) > 0}.) We recall the general stabilization problem in control theory. Suppose that R is an integral domain with identity (thought of as the class of stable transfer functions) and let F(R) denote the field of fractions of R. Then the stabilization problem is: Given p ∈ F(R) (an unstable plant transfer function), find c ∈ F(R) (a stabilizing controller transfer function), such that (the closed loop transfer function) H (p, c) := p 1−pc pc 1−pc pc 1−pc c 1−pc belongs to R 2×2 (that is, it is stable).
The demand above that H (p, c) ∈ R 2×2 guarantees that the "closed loop" transfer function of the signal map in the interconnection of p and c as shown in Fig. 1, is stable (so after the interconnection, "nice" signals are indeed mapped to nice signals). A stronger version of the problem is when we require a stable controller c ∈ R which stabilizes p. If such a c exists, then we say that p is strongly stabilizable.
In the robust stabilization problem, one goes a step further than the stabilization problem. 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 d.
There are various known metrics which do the job, notably the gap metric [12], the graph metric [10] and the Vinnicombe ν-metric (see [11] for the rational transfer function case and [1,9] for its recent extension for nonrational transfer functions). This last metric is in some sense the "best" one, as it is comparatively easy to compute and admits some sharp robustness results. The Vinnicombe metric itself arose from a very natural idea of defining a metric between meromorphic functions in the complex right half plane, namely the pointwise chordal metric, defined below. This metric has been studied by function theorists (see for example [4]), since it is a natural analogue of the H ∞ distance between bounded analytic functions, and it can be used for functions with poles in a disk. The use of the chordal metric to study robustness of stabilizability was made by El-Sakkary [8].
If p 1 , p 2 are two meromorphic functions in the open right half plane, then the chordal distance κ between p 1 , p 2 is This metric has the interpretation that it is the supremum of the pointwise Euclidean distance between the points p 1 (s) and p 2 (s) on the Riemann sphere. Recall that the stereographic projection allows the identification of the extended complex plane C ∪ {∞} with the unit sphere S of diameter 1 in R 3 , where the point z = 0 in the complex plane corresponds to the south pole S of the sphere S and the point z = ∞ corresponds to the north pole N of S. Points P C in the complex plane can be identified with a corresponding point P S on the sphere S, namely the one in S which lies on the straight line joining P C and N . See Fig. 2 , and let c ∈ H ∞ be such that Set k := c ∞ and g = g 0 ∞ . If then p is also stabilized by c.
This follows from the following key estimate, which gives a lower bound on the chordal distance; see [

Abstract set-up and main result
Our main result is given in Theorem 1.4 below. We will assume throughout the following: (A1) R is a commutative ring without zero divisors and with identity.
(A3) allows identification of elements of R with elements of S. So in the sequel, if x is an element of R, we will simply write x (an element of S!) instead of ι(x). We will denote by F(R) the field of fractions over R. An element p ∈ F(R) is said to have a coprime factorization over R if where n, d ∈ R, d = 0 and there exist x, y ∈ R such that nx + dy = 1.
We define the subset of coprime factorizable plants over R to be the set The maximal ideal space of S is denoted by M(S). If x ∈ S, then we denote by x the Gelfand transform of x. Also, we set If p 1 , p 2 ∈ S(R), then the chordal distance κ between p 1 , p 2 , which have coprime factorizations The function κ given by the above expression is well-defined. Indeed, if then n 1 d 1 = n 1 d 1 , and so, for each ϕ ∈ M(S), we have n 1 (ϕ) d 1 (ϕ) = n 1 (ϕ) d 1 (ϕ).Using this one can see that and so it follows that the expression in the definition of κ is independent of any particular choice of a coprime factorization of either plant.
We have the following result.

Proposition 1.3 κ is a metric on S(R).
Proof The proof is straightforward, but we give the details as they elucidate the use of the basic assumptions in our abstract setting.
Our main result is the following, which we will prove in the next section.
Proof of Theorem 1.4 Let p 0 = n 0 d 0 and p = n d be coprime factorizations of p 0 and p. Since c stabilizes p 0 , it follows in particular that Moreover, since (n 0 , d 0 ) are coprime in R, there exist elements x, y ∈ R such that n 0 · x + d 0 · y = 1. Hence it follows that So d 0 − n 0 c is invertible as an element of R. In particular, it is also invertible as an element of S, and so Suppose that d − nc is invertible as an element of R, then and so H (p, c) ∈ R 2×2 , showing that p is also stabilized by c, and we are done. So suppose that d − nc is not invertible as an element of R. Then d − nc is not invertible in S too, since by assumption (A4), R is a full subring of S. Thus there is a ϕ 0 ∈ M(S) such that We consider the following cases.
Suppose first that d 0 (ϕ 0 ) = 0. By the coprimeness of (d 0 , n 0 ), we have n 0 (ϕ 0 ) = 0. Then we have where we have used (2.2) to obtain the last equality. But Thus if a is any number such that 0 < a < 1, we have Finally, suppose that d 0 (ϕ 0 ) = 0. If n 0 (ϕ 0 ) = 0, then , ∞ , using (2.2), and proceeding in the same manner as above, we obtain (2.4) once again. Suppose now that n 0 (ϕ 0 ) = 0. We have Using (2.2) we have that . , we obtain which contradicts the hypothesis. Hence d − nc is invertible as an element of R, and hence p is stabilized by c.