Convergence and Compactness of Families of Proper Plants in the ...
Convergence and Compactness of Families of Proper Plants in the Graph Topology Mathukumalli Vidyasagar∗ Abstract— The graph topology plays a central role in characterizing the robustness of feedback systems. In particular, it provides necessary and sufficient conditions for the transfer matrix of a stabilized closed-loop system to be continuous with respect to the controller. If in addition we confine our attention to a compact set of controllers, we can draw much stronger conclusions, for example, the uniform continuity of the closed-loop transfer matrix. This motivates a detailed study of convergence in the graph topology, and a characterization of compactness in this topology. Readily verifiable necessary and sufficient conditions for a set (of controllers) to be compact the graph topology are not available at present. Following our preliminary results in an earlier paper, the present paper gives a simple characterization of convergence in the graph topology in the field of fractions associated with the disc algebra. The result is then applied to characterize compact sets in the graph topology in the field of fractions associated with the disc algebra. An application to the problem of approximate system design is also discussed.
I. I NTRODUCTION Many control system design problems inherently involve system approximation. Indeed, the system model itself is an approximation since no model can be exact. One then encounters the following question: Let a model M be an approximation of system Σ in some sense. Let K be a controller that is designed for M. How do we guarantee the performance of the closed-loop system when we connect K to Σ? It is possible to discuss the performance of the closedloop system once a controller K is chosen, but we should note that the controller K is yet to be chosen at the time of design. One can keep track of the performance by checking its performance at each step, but this can be troublesome if the approximation demension is large. Instead, one may discuss the convergence of the closed-loop performance with respect to an appropriate topolgy (e.g., the graph topology discussed discuss below), but even then we need to have an a priori estimate of such a performance before a controller is selected. It is well appreciated that the graph topology (or the gap metric) is the appropriate topology for discussion of closed-loop performance [14], [15], [25]. Suppose {Pλ } is a family of plant models parametrized by some parameter ∗ [email protected], Erik Jonsson School of Engineering & Computer Science, Room 3.908A, ECSS, The University of Texas at Dallas, Richardson, TX 75080-3021, U. S. A. Author to whom all correspondence should be addressed. ∗∗ [email protected], Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, JAPAN.
Yutaka Yamamoto∗∗ λ assuming values in some topological space Λ, and let Pλ0 denote the ‘nominal’ plant model. Then there exists a controller that stabilizes Pλ for all λ in some neighborhood of λ0 , and in addition the closed-loop transfer function is continuous at λ0 , if and only if the open-loop transfer function Pλ is continuous in the graph topology at λ0 . If the topology were too strong, the continuity requirement on Pλ would become too stringent, whereas if it were too weak the above property would fail. That is, the graph topology is the weakest topology that makes the closed-loop stability a robust property. This topology is metrizable and there are several metrics that generate the same topology, one of which is the gap metric. To be more specific, consider the following problem: Suppose that we are given a sequence of plant approximants Pn , n = 1, 2, . . ., that converges to the “true” plant P, and we have a prescribed set K from which a controller K may be chosen. Can we guarantee the performance of the closedloop in the limit? We do not know a priori the controller K until it is chosen; on the other hand, we do wish to guarantee the closed-loop performance for the whole class of problems where K is chosen from K . In other words, we aim at guaranteeing the convergence of the design problem. Such an approximation problem occurs frequently for sampled-data control system design [21], [23] and for distributed parameter systems [1], and others. Precisely stated, the question is the following: Can we ensure that the closed-loop transfer matrix H(Pn , K) converges uniformly to H(P, K) for every controller K ∈ K ? If the set of controllers K is ‘too large’ then uniform convergence will not hold. Thus the set K has to be ‘sufficiently small’ in order for uniform convergence to hold. It is easy to see that a simple sufficient condition for such uniform convergence is that the set K should be compact in the graph topology. That is the motivation for studying the question of compactness in the graph topology. We have studied this problem in [23] and given some results for the case where the set of stable systems is H∞ . These consisted of some necessary conditions and some sufficient conditions, with a fairly significant ‘gap’ between them. In this paper we proceed further to give a more complete characterization of convergence of families of proper plants in the field of fractions associated with the disc algebra Ad on the unit disc (and its continuous-time counterpart Ac ); see subsection II-A below. This characterization then yields a natural characterization of compactness in the graph topology, and answers the question raised above.
The paper is organized as follows: Section II gives preliminary materials on various function algebras (subsection II-A) and coprimenss (subsection II-B). In Section III, we proceed to give an in-depth study on a metric that gives the graph topology. This metric leads to characterizations of compactness in the graph topology in Section IV. We will also discuss an application to approximate system design in Section V.
below, H∞ is “too big”, and many of the arguments below do not hold in this larger set. Now let us turn to the continuous-time case. Let Sc denote the set of rational functions that do not have any poles in the closed right half-plane nor at infinity. If f ∈ Sc , then its inverse Laplace transform, call it h, is of the form
II. P RELIMINARIES
where δ (·) denotes the unit impulse distribution, h0 is a real constant, and ha (·) is a sum of decaying exponentials. For a function f ∈ Sc , we define
A. Various Algebras In this section we introduce various function algebras that correspond to stable systems in both continuous-time as well as discrete-time. We begin with the discrete-time case. As is customary, let D denote the open unit disc of the complex plane, and T the unit circle. Let Sd denote the set of rational functions that do not have any poles in the closed unit disc. For purely technical reasons, namely the ability to invoke the Gel’fand theory of commutative Banach algebras, we allow all rational functions to have complex numbers as coefficients. Notwithstanding this assumption, it is easy to modify all of the results below to make them applicable to functions with real coefficients. Clearly Sd consists of transfer functions of lumped stable discrete-time systems (i.e., those whose minimal state space realization is finitedimensional and asymptotically stable). Suppose f (·) ∈ Sd . Then f (z) has a Taylor series expansion of the form ∞
f (z) = ∑ fi zi , i=0
where the sequence { fi } is absolutely summable. On Sd we can define two distinct norms, namely ∞
k f k1 := ∑ | fi |, k f k∞ := max | f (e jθ )|. i=0
θ ∈[0,2π)
(1)
It is well known that k f k1 and k f k∞ correspond to the gain of the system with transfer function f (·), when the input and output spaces are taken as `1 , and as `2 , respectively. Under each of these norms, the set Sd is a normed linear algebra, in the sense that it is a ring under pointwise addition and pointwise multiplication, and the norm is submultiplicative. But Sd is not complete under either norm. Depending on which norm we use to complete Sd , we get two different Banach algebras as the completion. If we take the completion of Sd in the norm k · k1 , then the resulting Banach algebra is `˜1 , the set of z-transforms of sequences in `1 . If we take the completion of Sd in the norm k · k∞ , then the resulting Banach algebra is the so-called ‘disc algebra’, consisting of functions that are analytic on D, and continuous (and therefore bounded) on the closed unit disk, and denoted here by Ad . It is easy to see that k f k∞ ≤ k f k1 for all f ; hence it follows that Sd ⊆ `˜1 ⊆ Ad . At this point the reader may wonder why we do not introduce the Banach algebra H∞ , the set of functions that are analytic on the open unit disc and essentially bounded on the closed unit circle. As we shall see
h(t) = h0 δ (t) + ha (t),
k f k1 := |h0 | +
Z ∞ 0
(2)
|ha (t)|dt, k f k∞ := sup | f ( jω)|.
(3)
ω
It is easy to see that k f k∞ ≤ k f k1 for all f . The set Sc is a normed algebra under each of these norms. Again, as in the discrete-time case, depending on which norm we use to complete Sc , we get different completions. It is obvious from comparing (2) and (3) that the completion of Sc under k · k1 must be a subset of the direct sum R ⊕ L˜ 1 , where L˜ 1 denotes the set of Laplace transforms of functions in L1 [0, ∞). For later reference, we introduce the symbol L˜ 1+ to denote R⊕ L˜ 1 . Actually, L˜ 1+ is the completion of Sc under k · k1 , as shown in [17]; in turn the proof in [17] is based on [10], where it is shown that every function in L1 [0, ∞) can be approximated arbitrarily closely by a sum of decaying exponentials. Let us denote the completion of Sc under k · k∞ by Ac . Then Ac consists of functions that are analytic in the open right half-plane, and are continuous (and therefore bounded) on the one-point compactification of the closed right half-plane plus the point at infinity. It is easy to see that there is a oneto-one correspondence between the sets Ac and Ad by the familiar bilinear mapping z = (s−1)/(s+1). Thus a function f belongs to Ad if and only if the associated function s−1 g : s 7→ f s+1 belongs to Ac . Until now we have discussed the set of ‘proper stable rational functions’ in the continuous-time case, its discretetime analog, and the completions of each of these sets in several appropriate norms. Now we discuss the set of ‘strictly proper stable rational functions’ in the continuous-time case and its discrete-time analog. Consider first the continuoustime case, and suppose that either f ∈ L˜ 1+ := R ⊕ L˜ 1 or f ∈ Ac . In either case, the symbol f (∞) is unambiguously defined. If f is a rational function, then it is common to think of f as being ‘strictly proper’ if f (∞) = 0. So long as we restrict ourselves to the Banach algebras L˜ 1+ or Ac , we can continue to use the same definition. That is, we define a function f to be ‘strictly proper’ if f (∞) = 0. Let us denote the set of all such functions by I∞ . Then it is easy to see that I∞ is a closed ideal in both Banach algebras L˜ 1+ and Ac . It is a ready consequence of the Riemann-Lebesgue lemma [8, p. 22] that, in L˜ 1+ , I∞ is precisely equal to L˜ 1 , the set of Laplace transforms of functions in L1 [0, ∞). On the other
hand, in Ac , I∞ is strictly larger than L˜ 1 . Note that if we try to define a ‘strictly proper’ function in H∞ , we run into the technical difficulty that there is no unambiguous value f (∞) for elements in this larger Banach algebra. This is the main motivation for restricting attention to the smaller Banach algebras L˜ 1+ and Ac . When we move to the discrete-time case, the notion of a strictly proper function is not so natural. Instead, we can just choose a point e jα ∈ T, and define Iα as Iα := { f ∈ Ad : f (e jα ) = 0}. Of special interest is the set I0 , consisting of all functions f (·) in Ad such that f (1) = 0. B. Coprimeness In order to avoid a proliferation of symbols, thoughout the rest of the paper, we use the symbol B for a Banach algebra, which can represent anyone of `˜1 , L˜ 1+ , Ac , Ad , and let F denote the associated field of fractions. We shall also require an ideal I in B. In the continuous-time case, the symbol I denotes I∞ , the ideal of strictly proper functions in B, whereas in the discrete-time case, the symbol I denotes Iα , the ideal of functions that vanish at e jα . Definition 1: Suppose n, d ∈ B. Then we say that n, d are coprime1 if there exist other elements x, y ∈ B such that xn + yd = 1. More generally, suppose N, D are matrices with elements from B with the same number of columns. Then we say that N, D are right-coprime if there exist matrices X,Y with elements with elements from B such that XN +Y D = I, where I denotes the identity matrix over B with the appropriate dimensions. Since the above relationship can be written as N [ X Y ] = I, D it is clear that N, D are right-coprime if and only if the matrix A = [ N T DT ]T has a left inverse in the set of matrices with elements in B, which we express in the interests of brevity as ‘A has a left inverse over B’. The notion of leftcoprimeness is defined in an entirely analogous fashion. Definition 2: Suppose P is a matrix over F . Then a pair (N, D) is said to be a right-coprime factorization (rcf) of P if (i) Det D 6= 0 in the algebra B (so that D−1 is well-defined as a matrix over F ), (ii) P = ND−1 , and (iii) (N, D) are rightcoprime. A left-coprime factorization is defined analogously. It is possible to give an abstract necessary and sufficient condition for right-coprimeness in terms of the set of maximal ideals of B. Recall that an ideal I in B is said to be ‘maximal’ if it is a proper subset of B and is not contained in any other ideal of B (other than B itself). The situation 1 also
referred to as B´ezout.
becomes particularly elegant if B is a commutative Banach algebra over the field of complex numbers C. In this case, the well-known Gel’fand-Mazur theorem [6], [2] states that the quotient field B/I ∼ = C for every maximal ideal I ∈ M . As a result, for each I ∈ M , the coset [a]I := a + I is (or more accurately, can be uniquely identified with) a complex number, which is denoted by aˆI . Note that aˆI and [a]I are two different ways of denoting the same object. The association a 7→ aˆI as I varies over I ∈ M maps the set B into the set of complex-valued functions on M . This mapping is called the Gel’fand transform of a. The so-called ‘carrier space topology’ on M is the weakest topology on M in which the mapping aˆ : M → C is continuous for every a ∈ B. It is customary to denote by Ω the set M with the carrier space topology, and individual elements of Ω (which are actually maximal ideals of B) by ω. One of the most useful results of Gel’fand theory is that, in the carrier-space topology, the set Ω is compact [2]. Moreover, if the Banach algebra B is ‘semisimple’, meaning that the intersection of all maximal ideals of B consists of the singleton set {0}, then the Gel’fand transform is a one-to-one mapping from B into C(Ω), the set of continuous functions on Ω. The carrier spaces of the various Banach algebras under consideration here are well-known in the literature. The relevant theorem is found on [8, p. 161] for the discretetime case, and the continuous-time case follows by the familiar bilinear transformation z = (s − 1)/(s + 1). Both L˜ 1+ and Ac have the same carrier space, namely the one-point compactification of the closed right half-plane. Thus, for each point s in the set C+e := {s : Re s ≥ 0} ∪ {∞}, the set of functions Is = { f ∈ B : f (s) = 0} forms a maximal ideal of L˜ 1+ and of Ac . Conversely, if I is any maximal ideal of B, then there is a unique point s ∈ C+e such that I is precisely Is . In particular, the set of strictly proper functions I∞ forms a maximal ideal of both L˜ 1+ and Ac . Similarly, in the discrete-time case, both `˜1 and the disc algebra Ad have the same carrier space, namely the closed unit disk D. For every z ∈ D, the corresponding set Iz := { f ∈ B : f (z) = 0} is a maximal ideal of B; conversely every maximal ideal of B is of the above form for some z ∈ D. In particular, I1 is a maximal ideal of both `˜1 and Ad . We note in passing that the famous ‘corona theorem’ of Carleson [4] states that the open unit disk is dense in the maximal ideal space of H∞ ; but to date there is no characterization of the complete maximal ideal space of H∞ . Yet another very important result in Gel’fand theory is that the spectrum of an element a ∈ B (i.e., the set of λ for which λ 1 − a is not invertible) consists precisely of the set {a(ω), ˆ ω ∈ Ω}. Hence λ 1 − a has an inverse in B if and only if a(ω) ˆ 6= λ for every ω ∈ Ω. In particular a has an inverse
in B if and only if a(ω) ˆ 6= 0 ∀ω. In the specific situations being studied here, the Gel’fand transform is effectively the identity map, i.e., the injection of B into the set C(Ω) of continuous functions on Ω; so we can drop the ‘hat’ for the transform. This leads to the following easy consequence [16, Lemma 8.1.9]: Lemma 1: Let a1 , . . . , an ∈ B. Then there exist x1 , . . . , xn ∈ B such that n
∑ xi ai = 1
i=1
if and only if Rank[a1 (ω) . . . an (ω)] = 1, ∀ ω ∈ Ω. The result extends readily to matrices as well. Observe now that a pair (N, D) over B is right-coprime if and only if it is left-invertible over B. Hence we now give the following general characterization of left invertibility [16, Thm. 8.1.12]: Theorem 1: Suppose A ∈ Bn×m with n ≥ m. Then A admits a left inverse in Bm×n if and only if Rank A(ω) = m, ∀ ω ∈ Ω. (4) As an application of the above theorem, suppose N ∈ Bn×m , D ∈ Bm×m . Then (N, D) are right-coprime if and only if N(s) Rank = m, ∀ s ∈ C+e , D(s) in the continuous-time case, and if and only if N(z) = m, ∀ z ∈ D, Rank D(z) in the discrete-time case. Until now we have defined the notion of coprimeness for matrices over B, and the notion of a coprime factorization (left or right) for matrices over F . Next we cite some results to establish when such factorizations exist. It is already known [18] that not every fraction a/b in F has a coprime factorization. Basically it is a matter of choosing functions a, b ∈ B such that they do not have any common zeros in the carrier space Ω, but their zeros cluster at a common point on the boundary of Ω. In fact, [16, Corollary 8.1.6] states that every fraction associated with a commutative ring has a coprime factorization if and only if the underlying ring is a Bezout domain, and none of the four Banach algebras `˜1 , L˜ 1+ , Ac , Ad is a Bezout domain. Therefore, if we fix integers n, m and examine all matrices in F n×m , then some matrices will have not have a coprime factorization of any kind, either right or left. This therefore raises the following question: Suppose a matrix P over F has a right-coprime factorization; does it also have a left-coprime factorization? The answer is given by [16], Theorems 8.1.66 and 8.1.68. Note that, for each of the four choices of B, the associated carrier space Ω is a contractible set, since it is the closed unit disk in the discrete-time case, and is homeomorphic to the closed unit disk in the continuous-time case. Therefore we can assert that if a matrix P over F can an rcf, it also has an lcf, and vice versa.
III. C ONVERGENCE IN THE G RAPH T OPOLOGY A. The Graph Topology Suppose for the moment that B is a normed algebra, which need not even be complete, with two properties: (i) The set of units of B is an open set, and (ii) the map u 7→ u−1 on the set of units is continuous. It goes without saying that both properties hold if B is actually a Banach algebra, i.e., is complete; see for example [2]. However, even if one were to restrict attention to the sets Sc and Sd which are not complete, these assumptions still hold; see [16, Lemma 2.2.8]. Given a normed algebra B satisfying the above conditions, let F denote the associated field of fractions. So we can think of B as the set of ‘stable’ systems, while F is the set of ‘unstable’ systems. The graph topology, introduced in [15] and amplified in [16, Chapter 7], extends the topology on B to a topology on F . Specifically, suppose n, d ∈ B are coprime with d 6= 0, and let p = n/d. Then the set of all fractions n0 /d 0 , where n0 , d 0 belong to some open balls around n, d respectively, and in addition the ball around d 0 does not contain zero, is defined as a neighborhood of p. In the case of multi-input, multi-output systems, suppose N, D are matrices over B, with D being square and nonsingular (meaning that its determinant is not the zero element of B). Then P = ND−1 is well-defined as a matrix over F . Now suppose that in addition N, D are right-coprime. Then the set of all ratios N 0 (D0 )−1 , where N 0 , D0 belong to open balls around N, D respectively, and in addition, all matrices in the ball around D are nonsingular, constitutes a neighborhood of P. The significance of the graph topology is that it is the weakest topology in which feedback stability is a continuous property. Specifically we have the following result [15, Theorem 7.2.29]:2 Theorem 2: Suppose {Pλ } is a family of plant models parametrized by some parameter λ assuming values in some topological space Λ, and let Pλ0 denote the ‘nominal’ plant model. Then there exists a controller that stabilizes Pλ for all λ in some neighborhood of λ0 , and in addition the closedloop transfer function is continuous at λ0 , if and only if the open-loop transfer function Pλ is continuous in the graph topology at λ0 . B. Metrizing the Graph Topology In view of the importance of the graph topology, several attempts have been made to define a metric that induces this topology. Chief among these are the gap metric [25], the graph metric [15], the ν-metric [19] and the pointwise gap metric [13]. While all of these have been studied fairly widely in the literature, there is another metric due to Praagman [12] that has not attracted much attention. However, as shown below, by building on this metric, it is possible to arrive at a very simply defined and easily computable metric that induces the graph topology. For this purpose, we introduce a new notion, with the somewhat 2 Note
that the topology of the set Λ is not essential for the theorem.
unmanageable name and acronym ‘special normalized rightcoprime factorization (snrcf)’; perhaps it can be pronounced as ‘snerf’ by treating the ‘c’ as silent. We also need to define notions of a plant being ‘proper’ and ‘strictly proper’ in an abstract setting. Definition 3: Suppose B is either L˜ 1+ or Ac and let I denote the ideal I∞ . Alternatively, suppose B is either `˜1 or Ad and let I denote the ideal Iα for some α ∈ [0, 2π). In either case, let F denote the field of fractions associated with B, and suppose P is a matrix over F with an rcf (N, D). Then we say that P is strictly proper if N is a matrix over I , and proper if Det D 6∈ I . Given any rcf (N, D) of P, it is well-known that all rcfs of P are of the form (NU, DU) where U is a unit matrix, i.e., has an inverse over B. One consequence of this fact is that whether P is strictly proper or not is independent of which rcf is used in the definition above. Moreover, it is easy to see that, in case B equals L˜ 1+ or Ac , the above definitions correspond to the common usage of these phrases. Finally, note that if P is proper, then one can define [P]I := [N]I ([D]I )−1 to be its ‘value at infinity’ (or some other chosen point in the complex plane). Observe that [P]I is unambiguously defined, in that the right side of this equation is the same no matter which rcf (N, D) of P is used, and that [P]I is just a matrix of complex numbers. Finally, observe that if P is proper, then the modified plant Ps := P − [P]I is strictly proper. Next we define a ‘normalized’ rcf, which was first introduced in [15], and then extend the definition to identify one ‘special normalized’ rcf amongst many, in the case where the plant is proper. We need one more symbol for that purpose. For a M is a matrix over B, the symbol M ∗ denotes the conjugate transpose of M. Definition 4: Suppose P is a matrix over F . Then a normalized right-coprime factorization (nrcf) of P is a pair (N, D) with the property that N [ N ∗ D∗ ](ω) (ω) = I, ∀ ω ∈ ∂ Ω, (5) D where ∂ Ω denotes the boundary of the carrier space Ω. Now suppose that an ideal I in B is specified, and that P is strictly proper. Then the special nrcf (snrcf) of P is the nrcf of P with the additional property that N 0 = . (6) D I I Note that (5) embraces both continuous-time and discretetime systems. In the continuous-time case, ∂ Ω is the imaginary axis plus the point at infinity, whereas in the discretetime case ∂ Ω is the unit circle T. At this juncture it may not be obvious why there exist any nrcfs at all, and why there exists a unique snrcf. The explanation has to do with spectral factorization. Lemma 2: Suppose a plant P has an rcf (N1 , D1 ) over B, which equals anyone of `˜1 , L˜ 1+ , Ac , Ad . Then there exists an
nrcf of P that is unique up to right multiplication by a unitary matrix. Proof: Consider first the discrete-time case, with B equal to either `˜1 or Ad . Suppose P is a plant and that (N1 , D1 ) is a right-coprime factorization of P. Then, as we have seen earlier, it follows that the matrix A1 (z) = [N1T (z) DT1 (z)]T has full column rank at all z ∈ D. Now let us examine the matrix F(e jα ) = A∗1 (e jα ) A1 (e jα ). Clearly F(e jα ) is square and Hermitian for all α. Since both N1 and D1 belong to B, the matrix F(e jα ) is bounded above uniformly with respect to α. Finally, since N1 and D1 are also right-coprime, it follows that the matrix F(e jα ) is also bounded below uniformly with respect to α. In other words, there exist constants ε > 0 and µ < ∞ such that εI ≤ F(e jα ) ≤ µI, ∀ α ∈ [0, 2π). Hence, by the spectral factorization theorem there exists a unit matrix U over B such that F(e jα ) = U ∗ (e jα ) U(e jα ), ∀ α ∈ [0, 2π). So if we now define A(z) = A1 (z) [U(z)]−1 =
N1 (z) D1 (z)
· [U(z)]−1 ,
then A∗ A = [U ∗ ]−1 A∗1 A1U −1 = [U ∗ ]−1 FU −1 = [U ∗ ]−1U ∗UU −1 = I. Therefore A(·) is an nrcf of P. The continuous-time case follows by bilinear transformation. There are various versions of the spectral factorization theorem that one can invoke to deduce the existence of a suitable U. For rational matrices, the result was introduced to the engineering community by Youla [24]. However, a more general result, which requires only that F(·) ∈ L1 (T) and that log Det F(·) ∈ L1 (T), was proved by Wiener and Masani [20]. Note that the spectral factor U is unique up to left multiplication by a unitary matrix, whence the nrcf is unique up to right multiplication by a unitary matrix. Now we move on to the existence of a srncf for strictly proper plants P. Let (N1 , D1 ) now denote any nrcf of P (we already know that an nrcf exists). Since P is strictly proper, it follows that [N1 ]I = 0, and since (N1 , D1 ) is an nrcf, this in turn implies that [D1 ]I is some unitary matrix, call it O. Then N1 · O∗ A= D1 is the unique snrcf. Now we come to the main new result of this section. Theorem 3: Suppose P1 , P2 are strictly proper plants of the same dimension, and denote their snrcfs by N2 N1 . A1 = , A2 = D1 D2 Then the quantity ds (P1 , P2 ) := kA1 − A2 k
induces the graph topology on the set of all strictly proper plants. Proof: The proof is based on the result of Praagman [12], which can be stated as follows: Let U denote the set of unitary matrices of the appropriate dimension. Given two plants P1 , P2 , not necessarily strictly proper, define dPr := min kA1 − A2Uk. U∈U
Then dPr induces the graph topology on B. It might be mentioned here that the paper of Praagman specifically refers to the factorization of rational matrices in its title, whereas here we wish to apply to irrational matrices as well. Therefore in an appendix we restate the results of Praagman for the more general case (and in the process also, we hope, improve the clarity). Now suppose P1 , P2 are strictly proper, and let A1 , A2 denote snrcfs of P1 , P2 , not just nrcfs. Since an snrcf is obtained by right-multiplying an nrcf by a unitary matrix, we can, without loss of generality, choose A1 , A2 to be snrcfs in the definition of dPr . Therefore the task before us now is to prove that dPr and ds induce the same topology on the set of strictly proper plants. For this purpose, suppose {Pi } is a sequence of strictly proper plants, and P0 is another strictly proper plant, all of them having the same dimension. The objective is to show that dPr (Pi , P0 ) → 0 ⇐⇒ ds (Pi , P0 ) → 0. Now the definition makes it clear that dPr (P1 , P2 ) ≤ ds (P1 , P2 ). So the “if” part is obvious and we need to prove just the “only if” part. For this purpose, suppose dPr (Pi , P0 ) → 0 as i → ∞, and choose a sequence of unitary matrices {Ui } such that kAiUi − A0 k → 0 as i → ∞. Now evaluate both sides modulo the ideal I . This leads to (Ai )I Ui → (A0 )I as i → ∞. Notice however that, since Ai , A0 are snrcfs, we have that 0 (Ai )I = (A0 )I = . I Therefore it follows that Ui → I as i → ∞. Since matrix inversion is continuous, this implies that Ui−1 → I as i → ∞. Finally, we conclude that lim Ai = lim AiUi · lim Ui−1 = A0 · I = A0 .
i→∞
i→∞
i→∞
Therefore ds (Pi , P0 ) → 0 as i → ∞. Theorem 3 gives a metric on the set of strictly proper plants. Now, by adapting the proof of [16, Proposition 7.2.41], it is possible to extend the definition to proper plants. A couple of preliminary results are needed for that purpose. Lemma 3: Suppose B, I are as above, that P0 is a proper plant, and that {Pi } is a sequence that converges to P0 in the graph topology. Then Pi is proper for all sufficiently large i.
Proof: Since {Pi } converges to P0 , it follows that there is a sequence of rcfs (Ni , Di ) of Pi and an rcf (N0 , D0 ) of P such that Ni → N0 , Di → D0 . Moreover, since P0 is proper, Det D0 6∈ I . Since Det Di → Det D0 , and since I is a closed set (so that its complement is an open set), it follows that Det Di 6∈ I for all sufficiently large i; that is, Pi is proper for all sufficiently large i. Lemma 4: Suppose {Pi } is a sequence of proper plants, and that P0 is also proper. Define complex matrices [Pi ]I and [P0 ]I as before, and define the associated strictly proper plants Ps,i , Ps,0 by Ps,i := Pi − [Pi ]I , Ps,0 := P0 − [P0 ]I . Then Pi → P0 in the graph topology if and only if (i) the sequence of complex matrices {[Pi ]I converges to the complex matrix P0 ]I , and (ii) the sequence of strictly proper plants {Ps,i } converges to Ps,0 in the graph topology. Proof: Let (Ns,i , Ds,i ) be any rcf of the strictly proper plant Ps,i . Then it is easy to see that (Ns,i + [Pi ]I Ds,i , Ds,i ) is an rcf of Pi . The proof is now obvious. It is evident that the set of proper n × m plants can be viewed as a direct product of the set Cn×m of complex (constant) matrices and the set of n×m strictly proper plants. Therefore the importance of Lemma 4 lies in showing that the graph topology on the set of proper n×m plants is just the product of the topology on Cn×m and the graph topology on the set of n × m strictly proper plants. As shown in Examples 7.2.42 and 7.2.44 of [16], in general the graph topology is not a product topology. Hence Lemma 4 is a useful result. Finally we come to the metric for proper plants. Theorem 4: Given two proper plants P1 , P2 , define their ‘values at infinity’ to be the complex matrices [P1 ]I , [P2 ]I , and associated strictly proper plants Ps,i := Pi − [Pi ]I . Then the metric ds (P1 , P2 ) := k[P1 ]I − [P2 ]I k + ds (Ps,1 , Ps,2 ) induces the graph topology on the set of proper plants. The proof is omitted as it is obvious. Moreover, we are justified in using the same symbol ds for both proper as well as strictly proper plants, because it is consistent to do so. IV. C OMPACTNESS IN THE G RAPH T OPOLOGY In the previous section, we have presented a metric that induces the graph topology on proper plants. Using this result, in this section we present several results on the compactnes of families of proper plants. There are three results in all: One abstract necessary and sufficient condition, which in turn leads one readily verifiable necessary condition and another readily verifiable sufficient condition. Throughout the section, the emphasis is on a set P of proper plants, all of equal dimension. Associated with P is a set of complex numbers PI , corresponding to the ‘values at infinity’ of all plants in P, and a set of strictly proper plants Ps , consisting of the strictly proper part of every plant in P.
Theorem 5: For each P ∈ P, let AP denote the unique snrcf of Ps , the strictly proper part of P. Then P is compact in the graph topology if and only if the following two conditions hold: 1) The set of complex matrices PI is compact, and 2) The set {AP , P ∈ P} is compact in the set of matrices over B of appropriate dimension. Proof: As shown in Theorem 4, a sequence {Pi } in P converges to a limit P0 in P in the graph topology, if and only if (i) the associated sequence of ‘values at infinity’ {Pi,I } converges to P0,I , and (ii) the associated sequence of snrcfs {APi } converges to AP0 in the set of matrices over B. The theorem then follows. Though the proof of the above theorem is simple, its importance should not be underestimated. It converts the problem of checking whether a family of potentially unstable plants is compact in the graph topology, to one of checking the compactness of a set of constant matrices, and checking the compactness of a set of stable matrices. The first step is easy: Since the set of matrices (the ‘values at infinity’) is finite-dimensional, the set PI is compact if and only it is closed and bounded. The second step is not trivial, as B is an infinite-dimensional space. Nevertheless, as we shall see below, it is possible to derive some useful sufficient conditions for compactness. Before that, we state an obvious necessary condition. Theorem 6: With all notation as above, the set P is compact in the graph topology only if the set of matrices PI is closed and bounded. So in particular a family of proper plants whose ‘values at infinity’ are not uniformly bounded cannot be compact in the graph topology. The sufficient condition for compactness is based on the well-known Montel’s theorem in complex analysis; see e.g., [5]. Let S be an open subset of C. A family of analytic functions F is said to be normal with respect to S if every sequence in F contains a subsequence that converges uniformly (to an analytic function) over every compact subset of S. Montel’s theorem states, quite simply, that every family F that is locally bounded is normal. In other words, if for every compact subset X of S, there exists an upper bound M for | f (z)|, z ∈ X, f ∈ F , then F is normal. We begin with the discrete-time case. Theorem 7: With the notation above, suppose there exists a α ∈ [0, 2π) such that 1) The set of matrices PIα is compact, and 2) There is an open set U containing the closed unit disk D such that the family of snrcfs {AP , P ∈ P} is analytic and uniformly bounded over U. Then the family P is compact in the graph topology. Proof: Since U is open and {AP , P ∈ P} is uniformly bounded over U, it follows from Montel’s theorem that the family {AP , P ∈ P} is normal. Thus every sequence of the form {APi , P ∈ P} contains a subsequence that converges uniformly over D to an analytic function. By Theorem 5,
this plus the first condition are enough to imply that the family P is compact. Observe that since every AP is an snrcf, its norm is bounded by one over T and hence on D. However, once z gets outside D, in principle kA(z)k could grow very fast. This is why we need to impose the condition that the family {AP , P ∈ P} is uniformly bounded over U. The continuous-time case is entirely similar. Theorem 8: With the notation above, suppose there exists a σ > 0 such that 1) The set of matrices PI∞ is compact, and 2) The family of snrcfs {AP , P ∈ P} is analytic and uniformly bounded over the half-plane {s : Re s ≥ −σ }. Then the family P is compact in the graph topology. The proof is omitted as it is entirely similar to that in the discrete-time case. V. A PPLICATION TO A PPROXIMATE S YSTEM D ESIGN As briefly mentioned in the Introduction, sampled-data systems and distributed parameter systems are often discretized in the spatial domain and this yields a sequence of finite-dimensional systems (with increasing dimensions) that is shown to “converge” to the original system. For example, the so-called averaging approximation for delay-differential systems is such an example [1]; see also [17] for a more general setting on finite-dimensional approximations. We start by noting the following lemma on the continuity of the closed-loop operator with respect to K. This is nothing but a consequence of the graph topology [15]: Lemma 5: Consider the closed-loop operator H(P, K) where K is assumed to be stabilizing. Then H(P, K) is jointly continuous in P and K with respect to the graph topology. Let us now state the following theorem: Theorem 9: Let Pn be a sequence of plants that converges to a plant P0 in the graph topology, and let K be a set of controllers K such that i) every K ∈ K is stabilizing for all approximating plants, and ii) K is compact with respect to the graph topology. Then H(Pn , K) converges uniformly to H(P0 , K) for K ∈ K . Before giving a proof, let us note the following: 1) If Pn does not converge to P0 in the graph topology, then H(Pn , K) does not converge to H(P0 , K); 2) If K were not compact, the uniformity of the convergence of H(Pn , K) is not guaranteed. The first claim is obvious from the nature of the graph topology. More specifically, consider the plants P0 =
1 , s+1
Pn =
s − εn (s + 1)(s + εn )
(7)
where εn is a positive sequence that converges to 0. Any constant 0 < K < 1 stabilizes Pn and P0 for sufficiently large n, but H(Pn , K) does not converge to H(P0 , K).
For the second claim, consider 1 Pn = (8) s + 1 + εn and let εn be as above. Let K = (0, ∞). Then any K ∈ K stabilizes Pn , but the convergence of H(Pn , K) is not uniform in K since the convergence of K(1+Pn K)−1 to K(1+P0 K)−1 cannot be uniform in K ∈ K . Sketch of Proof Take ε > 0 and take any K ∈ K . There exists N such that d(H(Pn , K), H(P0 , K)) < ε for all n ≥ N. By Lemma 5, there exists a neighborhood B(K, δ ) := {K 0 : d(K 0 , K) < δ } of K such that d(H(Pn , K 0 ), H(P0 , K 0 )) < ε for all n ≥ N(K, ε) and K 0 ∈ B(K, δ ). This yields a covering of the controller set: K =
[
B(K, δ ).
K∈K By the compactness of K , there exists a finite subcovering
K = B(K1 , δ1 ) ∪ · · · ∪ B(Km , δm ). Taking Nmax := {N(K1 , ε), . . . , N(Km , ε)}, we readily have that n ≥ Nmax implies d(H(Pn , K), H(P0 , K)) < ε
(9)
for all K ∈ K . Remark 1: If we take the gain of the frequency response operator (see, e.g., [22]), then we can prove that the frequency response operator gain of fast-sample/fast-hold approximation converges uniformly to the original frequency response operator gain on such a compact set K . This is indeed proven in [21] and in [23]. Indeed, in [21] the characterization of compactness was only preliminary, and we gave an improvement in [23]. However, the characterization of compactness given here is completely new, and gives a clearer understanding of these results. Indeed, the early result given in [21] gave a motivation for the present work. VI. C ONCLUDING R EMARKS We have given a complete characterization of convergence in the graph topology based on the notion of special normalized right-coprime factorizations. This leads further to characterizations of compact sets in the graph topology for proper plants within the field of fractions associated with the disc algebras Ad and Ac . A result on the approximate system design has also been given.
R EFERENCES [1] H. T. Banks and J. A. Burns, “Hereditary control problems: numerical methods based on averaging approximations,” SIAM J. Control & Optimiz., 16-2: 169–208, 1979. [2] S. K. Berberian, Lectures in Functional Analysis and Operator Theory, Springer, 1973. [3] M. Cantoni and G. Vinnicombe, “Linear feedback systems and the graph topology”, IEEE Trans. Autom. Control, 47(5), 710-719, May 2002. [4] L. Carleson, “Interpolations by bounded analytic functions and the corona problem,” Annals of Mathematics, 76, No. 3: 547–559, 1962. [5] J. B. Conway, Functions of One Complex Variable, I, Springer, 1986. [6] I. M. Gel’fand, S. Raikov and G. Shilov, Commutative Normed Rings, Nauka, Moscow, 1949. [7] T. T. Georgiou, “On the computation of the gap metric”, Systems & Control Letters, 11, 253-257, 1988. [8] K. Hoffman, Banach Spaces of Analytic Functions, Dover, Mineola, NY, 2007 (Reprint of 1962 book). [9] T. T. Georgiou and M. C. Smith, “Optimal robustness in the gap metric”, IEEE Trans. Autom. Control, 35(6), June 1990. [10] D. W. Kammler, “Approximation with sums of exponentials in L p [0, ∞)”, J. Approx. Thy., 16, 384-408, 1976. [11] J. W. Polderman, “A note on the structure of two subsets of the parameter space in adaptive control”, Systems & Control Letters, 7, 25-34, 1986. [12] C. Praagman, “On the factorization of rational matrices depending on a parameter”, Systems & Control Letters, 9, 387-392, 1987. [13] Li Qiu and E. J. Davison, “Feedback stability under simultaneous gap metric uncertainties in the plant and controller”, Systems & Control Letters, 18, 9-22, 1992. [14] M. Vidyasagar, “The graph metric for unstable plants and robustness estimates for feedback stability,” Proc. 21st IEEE CDC, 148–151, 1982. [15] M. Vidyasagar, “The graph metric for unstable plants and robustness estimates for feedback stability,” IEEE Trans. Autom. Control, AC-29, No. 5: 403–418, 1984. [16] M. Vidyasagar, Control System Synthesis, MIT Press, 1985. [17] M. Vidyasagar and B. D. O. Anderson, ”Approximation and stabilization of distributed systems by lumped systems,” Systems & Control Lett., 12-2: 95–101, 1989. [18] M. Vidyasagar, H. Schneider and B. A. Francis, “Algebraic and topological aspects of feedback stabilization”. IEEE Trans. Autom. Control, 27(4), 880-894, August 1982. [19] G. Vinnicombe, “Frequency domain uncertainty and the graph topology”, IEEE Trans. Autom. Control, 38(9), 1371-1383, Sept. 1993. [20] N. Wiener and P. Masani, “The prediction theory of multivariate stochastic processes”, Acta Mathematica, 98(1), 111-150, 1957. [21] Y. Yamamoto, B. D. O. Anderson and M. Nagahara, “Approximating sampled-data systems with applications to digital redesign,” Proc. 41st IEEE CDC, : 3724–3729, 2002. [22] Y. Yamamoto, A. G. Madievski and B. D. O. Anderson, “Approximation of frequency response for sampled-data control systems,” Automatica, 35: 729–734, 1999. [23] Y. Yamamoto, M. Vidyasagar, “Compact Sets in the graph topology and applications to approximation of system design,” Proc. 50th IEEE CDC, pp. 621–626, 2011. [24] D. C. Youla, “On the factorization of rational matrices”, IRE Trans. Inform. Thy., 17, 172-189, 1961. [25] G. Zames and A. El-Sakkary, “Unstable systems and feedback: The gap metric,” Proc. Allerton Conf., 380–385, 1980.
I. I NTRODUCTION Many control system design problems inherently involve system approximation. Indeed, the system model itself is an approximation since no model can be exact. One then encounters the following question: Let a model M be an approximation of system Σ in some sense. Let K be a controller that is designed for M. How do we guarantee the performance of the closed-loop system when we connect K to Σ? It is possible to discuss the performance of the closedloop system once a controller K is chosen, but we should note that the controller K is yet to be chosen at the time of design. One can keep track of the performance by checking its performance at each step, but this can be troublesome if the approximation demension is large. Instead, one may discuss the convergence of the closed-loop performance with respect to an appropriate topolgy (e.g., the graph topology discussed discuss below), but even then we need to have an a priori estimate of such a performance before a controller is selected. It is well appreciated that the graph topology (or the gap metric) is the appropriate topology for discussion of closed-loop performance [14], [15], [25]. Suppose {Pλ } is a family of plant models parametrized by some parameter ∗ [email protected], Erik Jonsson School of Engineering & Computer Science, Room 3.908A, ECSS, The University of Texas at Dallas, Richardson, TX 75080-3021, U. S. A. Author to whom all correspondence should be addressed. ∗∗ [email protected], Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, JAPAN.
Yutaka Yamamoto∗∗ λ assuming values in some topological space Λ, and let Pλ0 denote the ‘nominal’ plant model. Then there exists a controller that stabilizes Pλ for all λ in some neighborhood of λ0 , and in addition the closed-loop transfer function is continuous at λ0 , if and only if the open-loop transfer function Pλ is continuous in the graph topology at λ0 . If the topology were too strong, the continuity requirement on Pλ would become too stringent, whereas if it were too weak the above property would fail. That is, the graph topology is the weakest topology that makes the closed-loop stability a robust property. This topology is metrizable and there are several metrics that generate the same topology, one of which is the gap metric. To be more specific, consider the following problem: Suppose that we are given a sequence of plant approximants Pn , n = 1, 2, . . ., that converges to the “true” plant P, and we have a prescribed set K from which a controller K may be chosen. Can we guarantee the performance of the closedloop in the limit? We do not know a priori the controller K until it is chosen; on the other hand, we do wish to guarantee the closed-loop performance for the whole class of problems where K is chosen from K . In other words, we aim at guaranteeing the convergence of the design problem. Such an approximation problem occurs frequently for sampled-data control system design [21], [23] and for distributed parameter systems [1], and others. Precisely stated, the question is the following: Can we ensure that the closed-loop transfer matrix H(Pn , K) converges uniformly to H(P, K) for every controller K ∈ K ? If the set of controllers K is ‘too large’ then uniform convergence will not hold. Thus the set K has to be ‘sufficiently small’ in order for uniform convergence to hold. It is easy to see that a simple sufficient condition for such uniform convergence is that the set K should be compact in the graph topology. That is the motivation for studying the question of compactness in the graph topology. We have studied this problem in [23] and given some results for the case where the set of stable systems is H∞ . These consisted of some necessary conditions and some sufficient conditions, with a fairly significant ‘gap’ between them. In this paper we proceed further to give a more complete characterization of convergence of families of proper plants in the field of fractions associated with the disc algebra Ad on the unit disc (and its continuous-time counterpart Ac ); see subsection II-A below. This characterization then yields a natural characterization of compactness in the graph topology, and answers the question raised above.
The paper is organized as follows: Section II gives preliminary materials on various function algebras (subsection II-A) and coprimenss (subsection II-B). In Section III, we proceed to give an in-depth study on a metric that gives the graph topology. This metric leads to characterizations of compactness in the graph topology in Section IV. We will also discuss an application to approximate system design in Section V.
below, H∞ is “too big”, and many of the arguments below do not hold in this larger set. Now let us turn to the continuous-time case. Let Sc denote the set of rational functions that do not have any poles in the closed right half-plane nor at infinity. If f ∈ Sc , then its inverse Laplace transform, call it h, is of the form
II. P RELIMINARIES
where δ (·) denotes the unit impulse distribution, h0 is a real constant, and ha (·) is a sum of decaying exponentials. For a function f ∈ Sc , we define
A. Various Algebras In this section we introduce various function algebras that correspond to stable systems in both continuous-time as well as discrete-time. We begin with the discrete-time case. As is customary, let D denote the open unit disc of the complex plane, and T the unit circle. Let Sd denote the set of rational functions that do not have any poles in the closed unit disc. For purely technical reasons, namely the ability to invoke the Gel’fand theory of commutative Banach algebras, we allow all rational functions to have complex numbers as coefficients. Notwithstanding this assumption, it is easy to modify all of the results below to make them applicable to functions with real coefficients. Clearly Sd consists of transfer functions of lumped stable discrete-time systems (i.e., those whose minimal state space realization is finitedimensional and asymptotically stable). Suppose f (·) ∈ Sd . Then f (z) has a Taylor series expansion of the form ∞
f (z) = ∑ fi zi , i=0
where the sequence { fi } is absolutely summable. On Sd we can define two distinct norms, namely ∞
k f k1 := ∑ | fi |, k f k∞ := max | f (e jθ )|. i=0
θ ∈[0,2π)
(1)
It is well known that k f k1 and k f k∞ correspond to the gain of the system with transfer function f (·), when the input and output spaces are taken as `1 , and as `2 , respectively. Under each of these norms, the set Sd is a normed linear algebra, in the sense that it is a ring under pointwise addition and pointwise multiplication, and the norm is submultiplicative. But Sd is not complete under either norm. Depending on which norm we use to complete Sd , we get two different Banach algebras as the completion. If we take the completion of Sd in the norm k · k1 , then the resulting Banach algebra is `˜1 , the set of z-transforms of sequences in `1 . If we take the completion of Sd in the norm k · k∞ , then the resulting Banach algebra is the so-called ‘disc algebra’, consisting of functions that are analytic on D, and continuous (and therefore bounded) on the closed unit disk, and denoted here by Ad . It is easy to see that k f k∞ ≤ k f k1 for all f ; hence it follows that Sd ⊆ `˜1 ⊆ Ad . At this point the reader may wonder why we do not introduce the Banach algebra H∞ , the set of functions that are analytic on the open unit disc and essentially bounded on the closed unit circle. As we shall see
h(t) = h0 δ (t) + ha (t),
k f k1 := |h0 | +
Z ∞ 0
(2)
|ha (t)|dt, k f k∞ := sup | f ( jω)|.
(3)
ω
It is easy to see that k f k∞ ≤ k f k1 for all f . The set Sc is a normed algebra under each of these norms. Again, as in the discrete-time case, depending on which norm we use to complete Sc , we get different completions. It is obvious from comparing (2) and (3) that the completion of Sc under k · k1 must be a subset of the direct sum R ⊕ L˜ 1 , where L˜ 1 denotes the set of Laplace transforms of functions in L1 [0, ∞). For later reference, we introduce the symbol L˜ 1+ to denote R⊕ L˜ 1 . Actually, L˜ 1+ is the completion of Sc under k · k1 , as shown in [17]; in turn the proof in [17] is based on [10], where it is shown that every function in L1 [0, ∞) can be approximated arbitrarily closely by a sum of decaying exponentials. Let us denote the completion of Sc under k · k∞ by Ac . Then Ac consists of functions that are analytic in the open right half-plane, and are continuous (and therefore bounded) on the one-point compactification of the closed right half-plane plus the point at infinity. It is easy to see that there is a oneto-one correspondence between the sets Ac and Ad by the familiar bilinear mapping z = (s−1)/(s+1). Thus a function f belongs to Ad if and only if the associated function s−1 g : s 7→ f s+1 belongs to Ac . Until now we have discussed the set of ‘proper stable rational functions’ in the continuous-time case, its discretetime analog, and the completions of each of these sets in several appropriate norms. Now we discuss the set of ‘strictly proper stable rational functions’ in the continuous-time case and its discrete-time analog. Consider first the continuoustime case, and suppose that either f ∈ L˜ 1+ := R ⊕ L˜ 1 or f ∈ Ac . In either case, the symbol f (∞) is unambiguously defined. If f is a rational function, then it is common to think of f as being ‘strictly proper’ if f (∞) = 0. So long as we restrict ourselves to the Banach algebras L˜ 1+ or Ac , we can continue to use the same definition. That is, we define a function f to be ‘strictly proper’ if f (∞) = 0. Let us denote the set of all such functions by I∞ . Then it is easy to see that I∞ is a closed ideal in both Banach algebras L˜ 1+ and Ac . It is a ready consequence of the Riemann-Lebesgue lemma [8, p. 22] that, in L˜ 1+ , I∞ is precisely equal to L˜ 1 , the set of Laplace transforms of functions in L1 [0, ∞). On the other
hand, in Ac , I∞ is strictly larger than L˜ 1 . Note that if we try to define a ‘strictly proper’ function in H∞ , we run into the technical difficulty that there is no unambiguous value f (∞) for elements in this larger Banach algebra. This is the main motivation for restricting attention to the smaller Banach algebras L˜ 1+ and Ac . When we move to the discrete-time case, the notion of a strictly proper function is not so natural. Instead, we can just choose a point e jα ∈ T, and define Iα as Iα := { f ∈ Ad : f (e jα ) = 0}. Of special interest is the set I0 , consisting of all functions f (·) in Ad such that f (1) = 0. B. Coprimeness In order to avoid a proliferation of symbols, thoughout the rest of the paper, we use the symbol B for a Banach algebra, which can represent anyone of `˜1 , L˜ 1+ , Ac , Ad , and let F denote the associated field of fractions. We shall also require an ideal I in B. In the continuous-time case, the symbol I denotes I∞ , the ideal of strictly proper functions in B, whereas in the discrete-time case, the symbol I denotes Iα , the ideal of functions that vanish at e jα . Definition 1: Suppose n, d ∈ B. Then we say that n, d are coprime1 if there exist other elements x, y ∈ B such that xn + yd = 1. More generally, suppose N, D are matrices with elements from B with the same number of columns. Then we say that N, D are right-coprime if there exist matrices X,Y with elements with elements from B such that XN +Y D = I, where I denotes the identity matrix over B with the appropriate dimensions. Since the above relationship can be written as N [ X Y ] = I, D it is clear that N, D are right-coprime if and only if the matrix A = [ N T DT ]T has a left inverse in the set of matrices with elements in B, which we express in the interests of brevity as ‘A has a left inverse over B’. The notion of leftcoprimeness is defined in an entirely analogous fashion. Definition 2: Suppose P is a matrix over F . Then a pair (N, D) is said to be a right-coprime factorization (rcf) of P if (i) Det D 6= 0 in the algebra B (so that D−1 is well-defined as a matrix over F ), (ii) P = ND−1 , and (iii) (N, D) are rightcoprime. A left-coprime factorization is defined analogously. It is possible to give an abstract necessary and sufficient condition for right-coprimeness in terms of the set of maximal ideals of B. Recall that an ideal I in B is said to be ‘maximal’ if it is a proper subset of B and is not contained in any other ideal of B (other than B itself). The situation 1 also
referred to as B´ezout.
becomes particularly elegant if B is a commutative Banach algebra over the field of complex numbers C. In this case, the well-known Gel’fand-Mazur theorem [6], [2] states that the quotient field B/I ∼ = C for every maximal ideal I ∈ M . As a result, for each I ∈ M , the coset [a]I := a + I is (or more accurately, can be uniquely identified with) a complex number, which is denoted by aˆI . Note that aˆI and [a]I are two different ways of denoting the same object. The association a 7→ aˆI as I varies over I ∈ M maps the set B into the set of complex-valued functions on M . This mapping is called the Gel’fand transform of a. The so-called ‘carrier space topology’ on M is the weakest topology on M in which the mapping aˆ : M → C is continuous for every a ∈ B. It is customary to denote by Ω the set M with the carrier space topology, and individual elements of Ω (which are actually maximal ideals of B) by ω. One of the most useful results of Gel’fand theory is that, in the carrier-space topology, the set Ω is compact [2]. Moreover, if the Banach algebra B is ‘semisimple’, meaning that the intersection of all maximal ideals of B consists of the singleton set {0}, then the Gel’fand transform is a one-to-one mapping from B into C(Ω), the set of continuous functions on Ω. The carrier spaces of the various Banach algebras under consideration here are well-known in the literature. The relevant theorem is found on [8, p. 161] for the discretetime case, and the continuous-time case follows by the familiar bilinear transformation z = (s − 1)/(s + 1). Both L˜ 1+ and Ac have the same carrier space, namely the one-point compactification of the closed right half-plane. Thus, for each point s in the set C+e := {s : Re s ≥ 0} ∪ {∞}, the set of functions Is = { f ∈ B : f (s) = 0} forms a maximal ideal of L˜ 1+ and of Ac . Conversely, if I is any maximal ideal of B, then there is a unique point s ∈ C+e such that I is precisely Is . In particular, the set of strictly proper functions I∞ forms a maximal ideal of both L˜ 1+ and Ac . Similarly, in the discrete-time case, both `˜1 and the disc algebra Ad have the same carrier space, namely the closed unit disk D. For every z ∈ D, the corresponding set Iz := { f ∈ B : f (z) = 0} is a maximal ideal of B; conversely every maximal ideal of B is of the above form for some z ∈ D. In particular, I1 is a maximal ideal of both `˜1 and Ad . We note in passing that the famous ‘corona theorem’ of Carleson [4] states that the open unit disk is dense in the maximal ideal space of H∞ ; but to date there is no characterization of the complete maximal ideal space of H∞ . Yet another very important result in Gel’fand theory is that the spectrum of an element a ∈ B (i.e., the set of λ for which λ 1 − a is not invertible) consists precisely of the set {a(ω), ˆ ω ∈ Ω}. Hence λ 1 − a has an inverse in B if and only if a(ω) ˆ 6= λ for every ω ∈ Ω. In particular a has an inverse
in B if and only if a(ω) ˆ 6= 0 ∀ω. In the specific situations being studied here, the Gel’fand transform is effectively the identity map, i.e., the injection of B into the set C(Ω) of continuous functions on Ω; so we can drop the ‘hat’ for the transform. This leads to the following easy consequence [16, Lemma 8.1.9]: Lemma 1: Let a1 , . . . , an ∈ B. Then there exist x1 , . . . , xn ∈ B such that n
∑ xi ai = 1
i=1
if and only if Rank[a1 (ω) . . . an (ω)] = 1, ∀ ω ∈ Ω. The result extends readily to matrices as well. Observe now that a pair (N, D) over B is right-coprime if and only if it is left-invertible over B. Hence we now give the following general characterization of left invertibility [16, Thm. 8.1.12]: Theorem 1: Suppose A ∈ Bn×m with n ≥ m. Then A admits a left inverse in Bm×n if and only if Rank A(ω) = m, ∀ ω ∈ Ω. (4) As an application of the above theorem, suppose N ∈ Bn×m , D ∈ Bm×m . Then (N, D) are right-coprime if and only if N(s) Rank = m, ∀ s ∈ C+e , D(s) in the continuous-time case, and if and only if N(z) = m, ∀ z ∈ D, Rank D(z) in the discrete-time case. Until now we have defined the notion of coprimeness for matrices over B, and the notion of a coprime factorization (left or right) for matrices over F . Next we cite some results to establish when such factorizations exist. It is already known [18] that not every fraction a/b in F has a coprime factorization. Basically it is a matter of choosing functions a, b ∈ B such that they do not have any common zeros in the carrier space Ω, but their zeros cluster at a common point on the boundary of Ω. In fact, [16, Corollary 8.1.6] states that every fraction associated with a commutative ring has a coprime factorization if and only if the underlying ring is a Bezout domain, and none of the four Banach algebras `˜1 , L˜ 1+ , Ac , Ad is a Bezout domain. Therefore, if we fix integers n, m and examine all matrices in F n×m , then some matrices will have not have a coprime factorization of any kind, either right or left. This therefore raises the following question: Suppose a matrix P over F has a right-coprime factorization; does it also have a left-coprime factorization? The answer is given by [16], Theorems 8.1.66 and 8.1.68. Note that, for each of the four choices of B, the associated carrier space Ω is a contractible set, since it is the closed unit disk in the discrete-time case, and is homeomorphic to the closed unit disk in the continuous-time case. Therefore we can assert that if a matrix P over F can an rcf, it also has an lcf, and vice versa.
III. C ONVERGENCE IN THE G RAPH T OPOLOGY A. The Graph Topology Suppose for the moment that B is a normed algebra, which need not even be complete, with two properties: (i) The set of units of B is an open set, and (ii) the map u 7→ u−1 on the set of units is continuous. It goes without saying that both properties hold if B is actually a Banach algebra, i.e., is complete; see for example [2]. However, even if one were to restrict attention to the sets Sc and Sd which are not complete, these assumptions still hold; see [16, Lemma 2.2.8]. Given a normed algebra B satisfying the above conditions, let F denote the associated field of fractions. So we can think of B as the set of ‘stable’ systems, while F is the set of ‘unstable’ systems. The graph topology, introduced in [15] and amplified in [16, Chapter 7], extends the topology on B to a topology on F . Specifically, suppose n, d ∈ B are coprime with d 6= 0, and let p = n/d. Then the set of all fractions n0 /d 0 , where n0 , d 0 belong to some open balls around n, d respectively, and in addition the ball around d 0 does not contain zero, is defined as a neighborhood of p. In the case of multi-input, multi-output systems, suppose N, D are matrices over B, with D being square and nonsingular (meaning that its determinant is not the zero element of B). Then P = ND−1 is well-defined as a matrix over F . Now suppose that in addition N, D are right-coprime. Then the set of all ratios N 0 (D0 )−1 , where N 0 , D0 belong to open balls around N, D respectively, and in addition, all matrices in the ball around D are nonsingular, constitutes a neighborhood of P. The significance of the graph topology is that it is the weakest topology in which feedback stability is a continuous property. Specifically we have the following result [15, Theorem 7.2.29]:2 Theorem 2: Suppose {Pλ } is a family of plant models parametrized by some parameter λ assuming values in some topological space Λ, and let Pλ0 denote the ‘nominal’ plant model. Then there exists a controller that stabilizes Pλ for all λ in some neighborhood of λ0 , and in addition the closedloop transfer function is continuous at λ0 , if and only if the open-loop transfer function Pλ is continuous in the graph topology at λ0 . B. Metrizing the Graph Topology In view of the importance of the graph topology, several attempts have been made to define a metric that induces this topology. Chief among these are the gap metric [25], the graph metric [15], the ν-metric [19] and the pointwise gap metric [13]. While all of these have been studied fairly widely in the literature, there is another metric due to Praagman [12] that has not attracted much attention. However, as shown below, by building on this metric, it is possible to arrive at a very simply defined and easily computable metric that induces the graph topology. For this purpose, we introduce a new notion, with the somewhat 2 Note
that the topology of the set Λ is not essential for the theorem.
unmanageable name and acronym ‘special normalized rightcoprime factorization (snrcf)’; perhaps it can be pronounced as ‘snerf’ by treating the ‘c’ as silent. We also need to define notions of a plant being ‘proper’ and ‘strictly proper’ in an abstract setting. Definition 3: Suppose B is either L˜ 1+ or Ac and let I denote the ideal I∞ . Alternatively, suppose B is either `˜1 or Ad and let I denote the ideal Iα for some α ∈ [0, 2π). In either case, let F denote the field of fractions associated with B, and suppose P is a matrix over F with an rcf (N, D). Then we say that P is strictly proper if N is a matrix over I , and proper if Det D 6∈ I . Given any rcf (N, D) of P, it is well-known that all rcfs of P are of the form (NU, DU) where U is a unit matrix, i.e., has an inverse over B. One consequence of this fact is that whether P is strictly proper or not is independent of which rcf is used in the definition above. Moreover, it is easy to see that, in case B equals L˜ 1+ or Ac , the above definitions correspond to the common usage of these phrases. Finally, note that if P is proper, then one can define [P]I := [N]I ([D]I )−1 to be its ‘value at infinity’ (or some other chosen point in the complex plane). Observe that [P]I is unambiguously defined, in that the right side of this equation is the same no matter which rcf (N, D) of P is used, and that [P]I is just a matrix of complex numbers. Finally, observe that if P is proper, then the modified plant Ps := P − [P]I is strictly proper. Next we define a ‘normalized’ rcf, which was first introduced in [15], and then extend the definition to identify one ‘special normalized’ rcf amongst many, in the case where the plant is proper. We need one more symbol for that purpose. For a M is a matrix over B, the symbol M ∗ denotes the conjugate transpose of M. Definition 4: Suppose P is a matrix over F . Then a normalized right-coprime factorization (nrcf) of P is a pair (N, D) with the property that N [ N ∗ D∗ ](ω) (ω) = I, ∀ ω ∈ ∂ Ω, (5) D where ∂ Ω denotes the boundary of the carrier space Ω. Now suppose that an ideal I in B is specified, and that P is strictly proper. Then the special nrcf (snrcf) of P is the nrcf of P with the additional property that N 0 = . (6) D I I Note that (5) embraces both continuous-time and discretetime systems. In the continuous-time case, ∂ Ω is the imaginary axis plus the point at infinity, whereas in the discretetime case ∂ Ω is the unit circle T. At this juncture it may not be obvious why there exist any nrcfs at all, and why there exists a unique snrcf. The explanation has to do with spectral factorization. Lemma 2: Suppose a plant P has an rcf (N1 , D1 ) over B, which equals anyone of `˜1 , L˜ 1+ , Ac , Ad . Then there exists an
nrcf of P that is unique up to right multiplication by a unitary matrix. Proof: Consider first the discrete-time case, with B equal to either `˜1 or Ad . Suppose P is a plant and that (N1 , D1 ) is a right-coprime factorization of P. Then, as we have seen earlier, it follows that the matrix A1 (z) = [N1T (z) DT1 (z)]T has full column rank at all z ∈ D. Now let us examine the matrix F(e jα ) = A∗1 (e jα ) A1 (e jα ). Clearly F(e jα ) is square and Hermitian for all α. Since both N1 and D1 belong to B, the matrix F(e jα ) is bounded above uniformly with respect to α. Finally, since N1 and D1 are also right-coprime, it follows that the matrix F(e jα ) is also bounded below uniformly with respect to α. In other words, there exist constants ε > 0 and µ < ∞ such that εI ≤ F(e jα ) ≤ µI, ∀ α ∈ [0, 2π). Hence, by the spectral factorization theorem there exists a unit matrix U over B such that F(e jα ) = U ∗ (e jα ) U(e jα ), ∀ α ∈ [0, 2π). So if we now define A(z) = A1 (z) [U(z)]−1 =
N1 (z) D1 (z)
· [U(z)]−1 ,
then A∗ A = [U ∗ ]−1 A∗1 A1U −1 = [U ∗ ]−1 FU −1 = [U ∗ ]−1U ∗UU −1 = I. Therefore A(·) is an nrcf of P. The continuous-time case follows by bilinear transformation. There are various versions of the spectral factorization theorem that one can invoke to deduce the existence of a suitable U. For rational matrices, the result was introduced to the engineering community by Youla [24]. However, a more general result, which requires only that F(·) ∈ L1 (T) and that log Det F(·) ∈ L1 (T), was proved by Wiener and Masani [20]. Note that the spectral factor U is unique up to left multiplication by a unitary matrix, whence the nrcf is unique up to right multiplication by a unitary matrix. Now we move on to the existence of a srncf for strictly proper plants P. Let (N1 , D1 ) now denote any nrcf of P (we already know that an nrcf exists). Since P is strictly proper, it follows that [N1 ]I = 0, and since (N1 , D1 ) is an nrcf, this in turn implies that [D1 ]I is some unitary matrix, call it O. Then N1 · O∗ A= D1 is the unique snrcf. Now we come to the main new result of this section. Theorem 3: Suppose P1 , P2 are strictly proper plants of the same dimension, and denote their snrcfs by N2 N1 . A1 = , A2 = D1 D2 Then the quantity ds (P1 , P2 ) := kA1 − A2 k
induces the graph topology on the set of all strictly proper plants. Proof: The proof is based on the result of Praagman [12], which can be stated as follows: Let U denote the set of unitary matrices of the appropriate dimension. Given two plants P1 , P2 , not necessarily strictly proper, define dPr := min kA1 − A2Uk. U∈U
Then dPr induces the graph topology on B. It might be mentioned here that the paper of Praagman specifically refers to the factorization of rational matrices in its title, whereas here we wish to apply to irrational matrices as well. Therefore in an appendix we restate the results of Praagman for the more general case (and in the process also, we hope, improve the clarity). Now suppose P1 , P2 are strictly proper, and let A1 , A2 denote snrcfs of P1 , P2 , not just nrcfs. Since an snrcf is obtained by right-multiplying an nrcf by a unitary matrix, we can, without loss of generality, choose A1 , A2 to be snrcfs in the definition of dPr . Therefore the task before us now is to prove that dPr and ds induce the same topology on the set of strictly proper plants. For this purpose, suppose {Pi } is a sequence of strictly proper plants, and P0 is another strictly proper plant, all of them having the same dimension. The objective is to show that dPr (Pi , P0 ) → 0 ⇐⇒ ds (Pi , P0 ) → 0. Now the definition makes it clear that dPr (P1 , P2 ) ≤ ds (P1 , P2 ). So the “if” part is obvious and we need to prove just the “only if” part. For this purpose, suppose dPr (Pi , P0 ) → 0 as i → ∞, and choose a sequence of unitary matrices {Ui } such that kAiUi − A0 k → 0 as i → ∞. Now evaluate both sides modulo the ideal I . This leads to (Ai )I Ui → (A0 )I as i → ∞. Notice however that, since Ai , A0 are snrcfs, we have that 0 (Ai )I = (A0 )I = . I Therefore it follows that Ui → I as i → ∞. Since matrix inversion is continuous, this implies that Ui−1 → I as i → ∞. Finally, we conclude that lim Ai = lim AiUi · lim Ui−1 = A0 · I = A0 .
i→∞
i→∞
i→∞
Therefore ds (Pi , P0 ) → 0 as i → ∞. Theorem 3 gives a metric on the set of strictly proper plants. Now, by adapting the proof of [16, Proposition 7.2.41], it is possible to extend the definition to proper plants. A couple of preliminary results are needed for that purpose. Lemma 3: Suppose B, I are as above, that P0 is a proper plant, and that {Pi } is a sequence that converges to P0 in the graph topology. Then Pi is proper for all sufficiently large i.
Proof: Since {Pi } converges to P0 , it follows that there is a sequence of rcfs (Ni , Di ) of Pi and an rcf (N0 , D0 ) of P such that Ni → N0 , Di → D0 . Moreover, since P0 is proper, Det D0 6∈ I . Since Det Di → Det D0 , and since I is a closed set (so that its complement is an open set), it follows that Det Di 6∈ I for all sufficiently large i; that is, Pi is proper for all sufficiently large i. Lemma 4: Suppose {Pi } is a sequence of proper plants, and that P0 is also proper. Define complex matrices [Pi ]I and [P0 ]I as before, and define the associated strictly proper plants Ps,i , Ps,0 by Ps,i := Pi − [Pi ]I , Ps,0 := P0 − [P0 ]I . Then Pi → P0 in the graph topology if and only if (i) the sequence of complex matrices {[Pi ]I converges to the complex matrix P0 ]I , and (ii) the sequence of strictly proper plants {Ps,i } converges to Ps,0 in the graph topology. Proof: Let (Ns,i , Ds,i ) be any rcf of the strictly proper plant Ps,i . Then it is easy to see that (Ns,i + [Pi ]I Ds,i , Ds,i ) is an rcf of Pi . The proof is now obvious. It is evident that the set of proper n × m plants can be viewed as a direct product of the set Cn×m of complex (constant) matrices and the set of n×m strictly proper plants. Therefore the importance of Lemma 4 lies in showing that the graph topology on the set of proper n×m plants is just the product of the topology on Cn×m and the graph topology on the set of n × m strictly proper plants. As shown in Examples 7.2.42 and 7.2.44 of [16], in general the graph topology is not a product topology. Hence Lemma 4 is a useful result. Finally we come to the metric for proper plants. Theorem 4: Given two proper plants P1 , P2 , define their ‘values at infinity’ to be the complex matrices [P1 ]I , [P2 ]I , and associated strictly proper plants Ps,i := Pi − [Pi ]I . Then the metric ds (P1 , P2 ) := k[P1 ]I − [P2 ]I k + ds (Ps,1 , Ps,2 ) induces the graph topology on the set of proper plants. The proof is omitted as it is obvious. Moreover, we are justified in using the same symbol ds for both proper as well as strictly proper plants, because it is consistent to do so. IV. C OMPACTNESS IN THE G RAPH T OPOLOGY In the previous section, we have presented a metric that induces the graph topology on proper plants. Using this result, in this section we present several results on the compactnes of families of proper plants. There are three results in all: One abstract necessary and sufficient condition, which in turn leads one readily verifiable necessary condition and another readily verifiable sufficient condition. Throughout the section, the emphasis is on a set P of proper plants, all of equal dimension. Associated with P is a set of complex numbers PI , corresponding to the ‘values at infinity’ of all plants in P, and a set of strictly proper plants Ps , consisting of the strictly proper part of every plant in P.
Theorem 5: For each P ∈ P, let AP denote the unique snrcf of Ps , the strictly proper part of P. Then P is compact in the graph topology if and only if the following two conditions hold: 1) The set of complex matrices PI is compact, and 2) The set {AP , P ∈ P} is compact in the set of matrices over B of appropriate dimension. Proof: As shown in Theorem 4, a sequence {Pi } in P converges to a limit P0 in P in the graph topology, if and only if (i) the associated sequence of ‘values at infinity’ {Pi,I } converges to P0,I , and (ii) the associated sequence of snrcfs {APi } converges to AP0 in the set of matrices over B. The theorem then follows. Though the proof of the above theorem is simple, its importance should not be underestimated. It converts the problem of checking whether a family of potentially unstable plants is compact in the graph topology, to one of checking the compactness of a set of constant matrices, and checking the compactness of a set of stable matrices. The first step is easy: Since the set of matrices (the ‘values at infinity’) is finite-dimensional, the set PI is compact if and only it is closed and bounded. The second step is not trivial, as B is an infinite-dimensional space. Nevertheless, as we shall see below, it is possible to derive some useful sufficient conditions for compactness. Before that, we state an obvious necessary condition. Theorem 6: With all notation as above, the set P is compact in the graph topology only if the set of matrices PI is closed and bounded. So in particular a family of proper plants whose ‘values at infinity’ are not uniformly bounded cannot be compact in the graph topology. The sufficient condition for compactness is based on the well-known Montel’s theorem in complex analysis; see e.g., [5]. Let S be an open subset of C. A family of analytic functions F is said to be normal with respect to S if every sequence in F contains a subsequence that converges uniformly (to an analytic function) over every compact subset of S. Montel’s theorem states, quite simply, that every family F that is locally bounded is normal. In other words, if for every compact subset X of S, there exists an upper bound M for | f (z)|, z ∈ X, f ∈ F , then F is normal. We begin with the discrete-time case. Theorem 7: With the notation above, suppose there exists a α ∈ [0, 2π) such that 1) The set of matrices PIα is compact, and 2) There is an open set U containing the closed unit disk D such that the family of snrcfs {AP , P ∈ P} is analytic and uniformly bounded over U. Then the family P is compact in the graph topology. Proof: Since U is open and {AP , P ∈ P} is uniformly bounded over U, it follows from Montel’s theorem that the family {AP , P ∈ P} is normal. Thus every sequence of the form {APi , P ∈ P} contains a subsequence that converges uniformly over D to an analytic function. By Theorem 5,
this plus the first condition are enough to imply that the family P is compact. Observe that since every AP is an snrcf, its norm is bounded by one over T and hence on D. However, once z gets outside D, in principle kA(z)k could grow very fast. This is why we need to impose the condition that the family {AP , P ∈ P} is uniformly bounded over U. The continuous-time case is entirely similar. Theorem 8: With the notation above, suppose there exists a σ > 0 such that 1) The set of matrices PI∞ is compact, and 2) The family of snrcfs {AP , P ∈ P} is analytic and uniformly bounded over the half-plane {s : Re s ≥ −σ }. Then the family P is compact in the graph topology. The proof is omitted as it is entirely similar to that in the discrete-time case. V. A PPLICATION TO A PPROXIMATE S YSTEM D ESIGN As briefly mentioned in the Introduction, sampled-data systems and distributed parameter systems are often discretized in the spatial domain and this yields a sequence of finite-dimensional systems (with increasing dimensions) that is shown to “converge” to the original system. For example, the so-called averaging approximation for delay-differential systems is such an example [1]; see also [17] for a more general setting on finite-dimensional approximations. We start by noting the following lemma on the continuity of the closed-loop operator with respect to K. This is nothing but a consequence of the graph topology [15]: Lemma 5: Consider the closed-loop operator H(P, K) where K is assumed to be stabilizing. Then H(P, K) is jointly continuous in P and K with respect to the graph topology. Let us now state the following theorem: Theorem 9: Let Pn be a sequence of plants that converges to a plant P0 in the graph topology, and let K be a set of controllers K such that i) every K ∈ K is stabilizing for all approximating plants, and ii) K is compact with respect to the graph topology. Then H(Pn , K) converges uniformly to H(P0 , K) for K ∈ K . Before giving a proof, let us note the following: 1) If Pn does not converge to P0 in the graph topology, then H(Pn , K) does not converge to H(P0 , K); 2) If K were not compact, the uniformity of the convergence of H(Pn , K) is not guaranteed. The first claim is obvious from the nature of the graph topology. More specifically, consider the plants P0 =
1 , s+1
Pn =
s − εn (s + 1)(s + εn )
(7)
where εn is a positive sequence that converges to 0. Any constant 0 < K < 1 stabilizes Pn and P0 for sufficiently large n, but H(Pn , K) does not converge to H(P0 , K).
For the second claim, consider 1 Pn = (8) s + 1 + εn and let εn be as above. Let K = (0, ∞). Then any K ∈ K stabilizes Pn , but the convergence of H(Pn , K) is not uniform in K since the convergence of K(1+Pn K)−1 to K(1+P0 K)−1 cannot be uniform in K ∈ K . Sketch of Proof Take ε > 0 and take any K ∈ K . There exists N such that d(H(Pn , K), H(P0 , K)) < ε for all n ≥ N. By Lemma 5, there exists a neighborhood B(K, δ ) := {K 0 : d(K 0 , K) < δ } of K such that d(H(Pn , K 0 ), H(P0 , K 0 )) < ε for all n ≥ N(K, ε) and K 0 ∈ B(K, δ ). This yields a covering of the controller set: K =
[
B(K, δ ).
K∈K By the compactness of K , there exists a finite subcovering
K = B(K1 , δ1 ) ∪ · · · ∪ B(Km , δm ). Taking Nmax := {N(K1 , ε), . . . , N(Km , ε)}, we readily have that n ≥ Nmax implies d(H(Pn , K), H(P0 , K)) < ε
(9)
for all K ∈ K . Remark 1: If we take the gain of the frequency response operator (see, e.g., [22]), then we can prove that the frequency response operator gain of fast-sample/fast-hold approximation converges uniformly to the original frequency response operator gain on such a compact set K . This is indeed proven in [21] and in [23]. Indeed, in [21] the characterization of compactness was only preliminary, and we gave an improvement in [23]. However, the characterization of compactness given here is completely new, and gives a clearer understanding of these results. Indeed, the early result given in [21] gave a motivation for the present work. VI. C ONCLUDING R EMARKS We have given a complete characterization of convergence in the graph topology based on the notion of special normalized right-coprime factorizations. This leads further to characterizations of compact sets in the graph topology for proper plants within the field of fractions associated with the disc algebras Ad and Ac . A result on the approximate system design has also been given.
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