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A new characterization of invariant subspaces of H2 and applications to the optimal sensitivity problem
Systems & Control Letters 54 (2005) 539 – 545 www.elsevier.com/locate/sysconle
A new characterization of invariant subspaces of H 2 and applications to the optimal sensitivity problem Kenji Kashima∗ , Yutaka Yamamoto Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan Received 3 February 2004; received in revised form 23 September 2004; accepted 3 October 2004 Available online 21 December 2004
Abstract This paper gives a new equivalent characterization for invariant subspaces of H 2 , when the underlying structure is specified by the so-called pseudorational transfer functions. This plays a fundamental role in computing the optimal sensitivity for a certain important class of infinite-dimensional systems, including delay systems. A closed formula, easier to compute than the well-known Zhou–Khargonekar formula, is given for optimal sensitivity for such systems. An example is given to illustrate the result. © 2004 Elsevier B.V. All rights reserved. Keywords: Pseudorational transfer function; Infinite-dimensional systems; Optimal sensitivity; H ∞ control; Beurling–Lax theorem
1. Introduction Let P (s) be the transfer function of the delaydifferential system: x(t) ˙ = x(t − 1) + u(t), y(t) = x(t) − x(t − 1),
−1 < < 0, > 1.
By −1 < < 0, P (s) = (es − )/(ses − ) belongs to H ∞ . Consider the sensitivity minimization problem for P with a given stable weighting function W (s): opt := inf W − P ∞ . (1) ∈H ∞
∗ Corresponding author.
E-mail addresses: [email protected] (K. Kashima), [email protected] (Y. Yamamoto). 0167-6911/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2004.10.004
Invoking the inner-outer factorization P (s)=m(s) P0 (s), we can reduce this to one of finding opt := inf W − m∞ . (2) ∈H ∞
When W is proper and rational [3,4], which covers all practically interesting cases, a beautiful state space formula is known [9,12,17]. In this formula, the optimal performance level is computed from the singularity of a matrix, in the form det m(−H )|22 , where A|22 denotes the (2, 2) block of the matrix A, and H denotes the Hamiltonian arising from W and (see Theorem 8 below). This result was first proved by Zhou and Khargonekar [18] for the pure delay m(s) = e−Ls , and later extended to general inner functions by [9,12,17] and others. This result has played a key role in obtaining the solution to the H ∞ problem for sampled-data
540
K. Kashima, Y. Yamamoto / Systems & Control Letters 54 (2005) 539 – 545
systems [1,7]. It also has a natural extension to the two-block case [5,8]. It should however be noted that for distributed plants as above it is not at all trivial to obtain the inner–outer factorization P = mP 0 . This in general requires the computation of unstable zeros of P, and there can be infinitely many of them, for example, in the case of neutral delay-differential systems. For such cases, applying the (generalized) Zhou–Khargonekar formula as described above is at best approximate by computing zeros. This motivates the present study. We attempt to derive a more direct formula for the computation of the optimal sensitivity. To this end, we confine our consideration to the class of pseudorational transfer functions introduced by the latter author [13]. This class of transfer functions admits a type of fractional representation, which in turn leads to a state space representation. This is also very effective in studying delay systems [13], repetitive control [16], etc. Such a representation is in close connection with left shiftinvariant subspaces of H 2 , and it is this relationship that leads us to a simple characterization of the inner part of the transfer function. It should also be noted that such a representation yields yet another form of the Beurling–Lax theorem [6,10]. We then obtain a simplified form of the generalized Zhou–Khargonekar formula (Theorem 9). The most attractive feature of this result is that we do not need explicit numerical computations of poles and zeros of the transfer function. An example is given to illustrate the results.
Lemma 1 (Yamamoto [14]). For , ∈ D + (R), ( ∗ ) = () + ().
(4)
Dually, for distributions , having support bounded on the right, r( ∗ ) = r() + r().
(5)
For a distribution , −1 denotes its inverse, when it exists, with respect to convolution. We have the following corollary. Corollary 2. For ∈ D + (R) with −1 also in D + (R), (−1 ) = −().
(6)
Similarly, if both and −1 have support bounded on the right, then r(−1 ) = −r().
(7)
Proof. According to (4), () + (−1 ) = ( ∗ −1 ) = () = 0, when both and −1 belong to D + (R). The second equality is obtained similarly.
2. Preliminaries 2.1. Distributions and supports We need some notions from distribution theory [11]. For a given distribution , supp denotes its support. The following quantities are of importance; we allow them to be ±∞, but when supp is bounded, they are both finite: () := inf {t : t ∈ supp }, r() := sup {t : t ∈ supp }.
elements of E (R− ). D + (R) is the space of distributions having support bounded on the left. Clearly E (R− ) ⊂ D + (R). Both E (R− ) and D + (R) constitute an algebra with respect to convolution ∗. The action of a distribution on a test function ∈ D, D being the space of infinitely differentiable functions on (−∞, ∞) with compact support, is denoted by , . The following lemma is a consequence of Titchmarsh’s theorem on convolution [2]:
(3)
Let E (R− ) denote the space of distributions having compact support in (−∞, 0]; the Dirac delta distribution a at point a (a 0), its derivative a , etc., are
We now give the definition of pseudorational impulse responses. This class plays a crucial role in realization, modeling, and control of infinitedimensional systems, particularly delay-differential systems [13,14]. Definition 3. Let f be a distribution having support in [0, ∞). It is said to be pseudorational if there exist q, p ∈ E (R− ) such that 1. q −1 exists over D + (R), 2. ord q −1 = −ord q,
K. Kashima, Y. Yamamoto / Systems & Control Letters 54 (2005) 539 – 545
3. f can be written as
quite effective in computing concrete realizations; for details, see [13,14]. What is important to note here is that
f = q −1 ∗ p, where ord q denotes the order of a distribution q [11]. For a distribution , its Laplace transform is denoted by ˆ (s). Similarly, for a set X of distributions, Xˆ := {fˆ : f ∈ X} provided that every f ∈ X is Laplace transformable. If f is pseudorational, its associated transfer function fˆ is also said to be pseudorational. p As usual, H p and H− denote the Hardy spaces on the open right- and left-half complex plane, respectively. The spaces H 2 and H−2 are the spaces of the Laplace transform of functions in L2 [0, ∞) and L2 (−∞, 0], respectively [6]. The Hilbert space of functions square integrable on the imaginary axis will be denoted by L2 (j R). Clearly L2 (j R) = H 2 ⊕ H−2 . Let ˜be the mirror image of the distribution defined by ˜, (t) := , (−t),
∈ D.
(˜) = −r().
(9)
2.2. Pseudorational impulse responses and associated state space Let := L2loc [0, ∞) be the space of all locally Lebesgue square integrable functions. is equipped with the following natural left-shift semigroup t [13]:
∈ , t 0, s 0.
Let f = q −1 ∗ p be pseudorational. We associate to it the following state space:
X := {x ∈ : q ∗ x ∈ E (R− )}.
Thus when 1/qˆ is stable, it is the orthogonal complement of a right-shift invariant subspace of L2 [0, ∞), and hence gives another representation of the Beurling–Lax theorem [6]. This is the theme of Section 3, but before proceeding further, we conclude this section by giving the following characterization for exponential stability. Lemma 4 (Yamamoto [15]). The left-shift semigroup t in Xq is exponentially stable, i.e., there exist positive constants C, such that t x Ce−t x, if and only if there exists c > 0 such that Re − c
f or all such that q( ˆ ) = 0.
3. Characterization of the inner part
For the Laplace transforms, ˆ ˜(s) := ˆ (−s). Then ˆ ˜ ∈ p H− if and only if ˆ ∈ H p . For a closed subspace Y of a Hilbert space X, its orthogonal complement will be denoted by X Y .
q
• X q is a left-shift invariant subspace of , and • when 1/qˆ is stable, it is indeed a closed subspace of L2 [0, ∞).
(8)
By definition, we clearly have
(t )(s) := (s + t),
541
(10)
It is easy to see that X q is a t -invariant closed subspace of . This space gives a standard state space for pseudorational impulse responses, and is indeed
The objective of Section 3 is to derive a formula for the inner function m arising from the representation Xq for a pseudorational impulse response q −1 ∗p. The precise meaning is as follows: Let q −1 ∗ p be pseudorational, and suppose that q satisfies the condition of Lemma 4. Then by a result in [16] that Xq ⊂ L2 [0, ∞). This implies that Xˆ q is a left shift-invariant closed subspace of H 2 . Consequently, its orthogonal complement H 2 Xˆ q is a right shiftinvariant closed subspace of H 2 . According to the Beurling–Lax theorem [6], there exists a unique inner function m, ˆ modulo constants, such that H 2 Xˆ q = mH ˆ 2 . In other words, Xˆ q = H (m) ˆ := H 2 mH ˆ 2. It is also known that [6] ˆ xˆ ∈ H−2 }. H (m) ˆ = {xˆ ∈ H 2 : m˜
(11)
When 1/qˆ were rational, this inner m ˆ is simply the Blaschke product consisting of zeros of q. ˆ But when it is irrational, it is not easy to find m. ˆ The following theorem gives a formula for this.
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K. Kashima, Y. Yamamoto / Systems & Control Letters 54 (2005) 539 – 545
Theorem 5. Let 1/q(s) ˆ be stable and pseudorational. Then Xˆ q = H (m) ˆ for m ˆ given by
This yields q ∗ x = (q) ∗ q˜∗ . Therefore, by (5) and (9), we obtain
(12)
r(q ∗ x) = (q) + r(q˜) + r() = (q) − (q) + r() = r() 0,
We first prove that (12) indeed gives an inner function.
because ∈ L2 (−∞, 0]. Furthermore, since (q ∗ x) = (q) + (x) is finite, q ∗ x belongs to E (R− ). This implies H (m) ˆ ⊂ Xˆ q .
m ˆ = e−(q)s
q˜(s) ˆ . q(s) ˆ
Lemma 6. For ∈ E (R− ) such that 1/ˆ(s) is stable,
ˆ ˜(s) m ˆ = e−()s ˆ (s) is an inner function. Proof. Since clearly |m| ˆ = 1 on the imaginary axis, it suffices to prove that m ˆ belongs to H ∞ . Recall that m ˆ is in H ∞ if and only if convolution with m defines a bounded linear operator on L2 [0, ∞). Take an arbitrary x ∈ L2 [0, ∞), i.e., xˆ ∈ H 2 , and we show ˆ on the imaginary axis, m∗x ∈ L2 [0, ∞). Since |m|=1 m ˆ xˆ ∈ L2 (j R) and this implies m ∗ x ∈ L2 (−∞, ∞). Furthermore, we have
We are now ready to proof Theorem 5.
∞
1−
n=1
s n
exp
s n
,
n=1
| n |2
< ∞.
Hence ∞ |Re n | < ∞, 1 + | n |2 n=1
Proof of Theorem 5. Let us first show Xˆ q ⊂ H (m). ˆ ˆ ∈ Xˆ q ⊂ H 2 . By (10) this means q ∗ ∈ Take any
ˆ ∈ L2 (j R). It follows E (R− ). Since m ˆ is inner, m˜ ˆ
−1 from (7) and (9) that r((q˜) ) = −r(q˜) = (q) and r(m˜∗ ) = r(−(q) ) + r((q˜)−1 ) + r(q ∗ ) = r(q ∗ ) 0. i.e., m˜ˆ ˆ ∈ and This yields m˜∗ ∈ ˆ hence we have Xˆ q ⊂ H (m). Conversely, take xˆ ∈ H (m) ˆ ⊂ H 2 . Then by (11), we have qˆ xˆ ∈ H−2 . e−(q)s q˜ ˆ
q(s) ˆ = eas
∞ |Re n |
by (4), (6) and (9) (note also that r() 0). Thus m ∗ x ∈ L2 [0, ∞), and hence m ˆ is an inner function.
ˆ := m˜ ˆ xˆ =
Remark 7. The obtained inner m ˆ is expressible as the product of e−Ls and a Blaschke product. To see this, recall the infinite product representation of q(s) ˆ [15]:
where
(m ∗ x) = () + (˜) + (−1 ) + (x) = − r() + (x) 0
L2 (−∞, 0],
For example, let W (s) = 1/(ses − c). This is clearly pseudorational, where q = ( −1 − c) and (q) = −1. Applying Theorem 5 to (12), we have s −se−s − c ce + s = H . Xˆ q = H es ses − c ses − c
H−2 ,
and this is a sufficient condition for the convergence of the Blaschke product consisting of zeros of q(s) ˆ ˆ q(s) ˆ is of the form ebs mb (s) [6]. Hence e−(q)s q˜(s)/ where mb (s) is the Blaschke product consisting of the zeros of q. ˆ 4. Optimal sensitivity problem Consider the optimal weighted sensitivity problem: opt := inf W (s) − Pˆ (s)(s) . (13) ∈H ∞
∞
Here W (s) is a stable, rational and strictly proper weighting function, and we assume that the given plant
K. Kashima, Y. Yamamoto / Systems & Control Letters 54 (2005) 539 – 545
543
Pˆ (s) is pseudorational and belongs to H ∞ . This problem is reducible to that of finding opt := inf W (s) − m(s) ˆ (s)∞ ,
For the proof, let us begin by decomposing Pˆ as ˆ P = Pˆi · Pˆ0 , where
where m ˆ is the inner part of Pˆ . For this reduced problem, we can apply the method given in [18,9,12,17] as follows: Let (A, B, C) be a minimal realization of W. Define its Hamiltonian H by A BB T / . (14) H := −C T C/ −AT
(19)
∈H ∞
The following result holds: Theorem 8 (Zhou–Khargonekar, Lypchuk, Smith and Tannenbaum Smith, Yamamoto [18,9,12,17]). Let m, ˆ H be as above. Suppose that m ˆ is analytic on the set of eigenvalues of H . Then the optimal sensitivity opt is given by max{ : det m˜(H ˆ )|22 = 0},
(15)
where A|22 denotes the (2, 2) block of matrix A when partitioned accordingly as (14). As we noted in Section 1, however, m ˆ is not necessarily easy to find for general infinite-dimensional systems. Even if it is given in the form of a Blaschke product, it is still not trivial to compute the optimal sensitivity according to (15). We here give a more direct representation that is much more suitable for computation. To this end, we assume that the numerator of Pˆ is factorized into the product of stable and anti-stable parts. Then we state our main result as follows: Theorem 9. Suppose that the plant Pˆ is factorized as pˆ 1 (s)pˆ 2 (s) Pˆ (s) = ∈ H ∞, q(s) ˆ
(16)
ˆ −1 , where p1 , p2 , q ∈ E (R− ) such that q(s) −1 −1 r(p )s ∞ ∈ H . Assume also that pˆ 1 (s) , e 2 pˆ 2˜(s) 1/pˆ 2 is analytic on the set of eigenvalues of H . Define L := −(q) + (p1 ) − r(p2 ). Then the optimal sensitivity opt is given by (17) max{ : det eLH pˆ 2˜(H )pˆ 2 (H )−1 |22 = 0}.
pˆ 2 (s) Pˆi (s) := e−Ls · , pˆ 2˜(s) pˆ 1 (s)pˆ 2˜(s) Pˆ0 (s) := eLs · . q(s) ˆ
(18)
We start with the following lemma. Lemma 10. The functions Pˆi and Pˆ0 belong to H ∞ , and Pˆi is inner. Proof. By the assumption above, both functions have no unstable poles. Notice that, on the imaginary axis, ˆ ˆ ˆ ˆ |Pi | = 1 and |P0 | = |P | P < ∞. It is thus suffi∞ cient to show that Pi and P0 have support in [0, ∞), that is, (Pi ) 0 and (P0 ) 0. The rest is similar to the proof of Lemma 6. By (4), (6) and (9), we have (Pi ) = L + (p2 ) − (p2˜) = − (q) + (p1 ) − r(p2 ) + (p2 ) + r(p2 ) = (P ) 0, (P0 ) = − L + (p1 ) + (p2˜) − (q) = (q) − (p1 ) + r(p2 ) + (p1 ) + (p2˜) − (q) = 0.
(20)
Proof of Theorem 9. Let opt be defined by opt = inf W − Pˆi .
(21)
∈H ∞
∞
We first show that the optimal sensitivity opt defined by (13) is equal to opt . Then Theorem 8 combined with formula (18) would yield the desired result. Note first that := Pˆ0 belongs to H ∞ for arbitrary ∈ H ∞ by Lemma 10. Hence we easily see opt opt . But showing opt opt is less trivial because Pˆ0−1 does not necessarily belong to H ∞ . The function Pˆ0−1 is analytic on the right half plane and satisfies that (P0−1 )=−(P0 )=0 by (6) and (20). Notice that we can decompose Pˆ0−1 as e−(p1 )s er(p2 )s (q) · · e q(s), ˆ Pˆ0 (s)−1 = pˆ 1 (s) pˆ 2˜
(22)
by the definition of L := −(q) + (p1 ) − r(p2 ). Since pˆ 1 (s)−1 ∈ H ∞ and ((p1 ) ∗ p1−1 ) = 0 by
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K. Kashima, Y. Yamamoto / Systems & Control Letters 54 (2005) 539 – 545
(4) and (6), we can show that e−(p1 )s /pˆ 1 (s) belongs to H ∞ as in Lemma 6. Also by the assumption on p2 , er(p2 )s /pˆ 2˜(s) ∈ H ∞ . Next −(q) ∗ q has compact support in [0, ∞). Therefore, from the Paley–Wiener–Schwartz theorem [11], there exist C > 0 and a positive integer k such that |e(q)s q(s)| ˆ C(1 + |s|)k ,
Re s 0.
Pˆ0−1 (T s + 1)
1.4 1.2 1 0.8 0.6
> 0, e(q)s q(s)/(T ˆ s+
0.2
0.4
0 0.2
0.4
γ
0.6
0.8
Fig. 1. 0 < < 1(dash) and = 2 (solid).
∈ H ∞, k
and we have lim W − Pˆ T T →0
1.6
(23)
This implies that, for arbitrary T 1)k is in H ∞ and so is Pˆ0−1 /(T s + 1)k . Now for any ∈ H ∞,
T :=
2 1.8
= lim W − Pˆi k T →0 ∞ (T s + 1) ∞ = W − Pˆi . ∞
This clearly yields opt opt . Recall that Pˆi is an inner function satisfying Pˆi˜(s)= Ls e pˆ 2 (s)pˆ 2˜(s) from (18) and Lemma 10. Therefore, the desired result follows by applying Theorem 8 to (21). Remark 11. If the given plant Pˆ has no unstable zeros, the associated inner function is a simple delay Pˆi = e−(P )s . 4.1. Example Let us compute the optimal sensitivity by using Theorem 9. Consider the weighting function W (s) and the plant Pˆ (s) as follows: 1 pˆ es − , Pˆ (s) = = 2s , s+1 qˆ 2e − 1 ( > 0, = 1),
W (s) =
with the Hamiltonian of W given by −1 −1 H = . −−1 1 Notice that the optimal sensitivity opt W ∞ = 1. Observe that Pˆ (s) has infinitely many zeros. If 0 < < 1, Pˆ (s) has no unstable zeros. Hence the
optimal sensitivity
is given as the maximum satisfying det eH |22 = 0, independent of , since L = −(q) + (p) = 1. On the contrary, in the case of > 1, Pˆ (s) has infinitely many unstable zeros, i.e., s = ln + 2n j (n ∈ Z). Then L = −(q) − r(p) = 2 and from (17) we have
opt = max{ : det(eH (I − eH )(eH − I )−1 |22 ) = 0}. Fig. 1 shows the absolute values of the determinant of the corresponding matrices above. When the plant has no unstable zeros, the optimal sensitivity is about 0.44. On the other hand in the case of =2 the optimal value is approximately 0.80. 5. Conclusion We have given an equivalent characterization for invariant subspaces of H 2 , when the underlying structure is specified by the so-called pseudorational transfer functions. We have derived a simple closed formula for the optimal sensitivity for such systems. This formula does not require explicit numerical computations of poles and zeros of the associated transfer functions. References [1] B. Bamieh, J.B. Pearson, A general framework for linear periodic systems with application to H∞ sampled-data control, IEEE Trans. Automat. Control 37 (1992) 418–435. [2] W.F. Donoghue, Distributions and Fourier Transforms, Academic Press, New York, 1969. [3] C. Foias, H. Özbay, A. Tannenbaum, Robust Control of Infinite Dimensional Systems, Springer, Berlin, 1996.
K. Kashima, Y. Yamamoto / Systems & Control Letters 54 (2005) 539 – 545 [4] C. Foias, A. Tannenbaum, G. Zames, Weighted sensitivity minimization for delay systems, IEEE Trans. Automat. Control 31 (1986) 763–766. [5] K. Hirata, Y. Yamamoto, A. Tannenbaum, A Hamiltonianbased solution to the two block H ∞ problem for general plants in H ∞ and rational weights, Systems Control Lett. 40 (2000) 83–95. [6] K. Hoffman, Banach Spaces of Analytic Functions, PrenticeHall, New York, 1962. [7] P.T. Kabamba, S. Hara, Worst case analysis and design of sampled data control systems, IEEE Trans. Automat. Control 38 (1993) 1337–1357. [8] K. Kashima, H. Özbay, Y. Yamamoto, On the mixed sensitivity optimization problem for stable pseudorational plants, Proc. Fourth IFAC Workshop on Time Delay Systems, Rocquencourt, France, 2003. [9] T.A. Lypchuk, M.C. Smith, A. Tannenbaum, Weighted sensitivity minimization: General plants in H ∞ and rational weights, Linear Algebra Appl. 8 (1988) 71–90. [10] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. [11] L. Schwartz, Théorie des Distributions, Herman, Paris, 1966.
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[12] M.C. Smith, Singular values and vectors of a class of Hankel operators, Systems Control Lett. 12 (1989) 301–308. [13] Y. Yamamoto, Pseudorational input/output maps and their realizations: a fractional representation approach to infinitedimensional systems, SIAM J. Control Optimization 26 (1988) 1415–1430. [14] Y. Yamamoto, Reachability of a class of infinite-dimensional linear systems: an external approach with applications to general neutral systems, SIAM J. Control Optimization 27 (1989) 217–234. [15] Y. Yamamoto, Equivalence of internal and external stability for a class of distributed systems, Math. Control Signals Systems 4 (1991) 391–409. [16] Y. Yamamoto, S. Hara, Relationships between internal and external stability for infinite-dimensional systems with applications to a servo problem, IEEE Trans. Automat. Control 33 (1988) 1044–1052. [17] Y. Yamamoto, K. Hirata, A. Tannenbaum, Some remarks on Hamiltonians and the infinite-dimensional one block H ∞ problem, Systems Control Lett. 29 (1996) 111–117. [18] K. Zhou, P.P. Khargonekar, On the weighted sensitivity minimization problem for delay systems, Systems Control Lett. 8 (1987) 307–312.
A new characterization of invariant subspaces of H 2 and applications to the optimal sensitivity problem Kenji Kashima∗ , Yutaka Yamamoto Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan Received 3 February 2004; received in revised form 23 September 2004; accepted 3 October 2004 Available online 21 December 2004
Abstract This paper gives a new equivalent characterization for invariant subspaces of H 2 , when the underlying structure is specified by the so-called pseudorational transfer functions. This plays a fundamental role in computing the optimal sensitivity for a certain important class of infinite-dimensional systems, including delay systems. A closed formula, easier to compute than the well-known Zhou–Khargonekar formula, is given for optimal sensitivity for such systems. An example is given to illustrate the result. © 2004 Elsevier B.V. All rights reserved. Keywords: Pseudorational transfer function; Infinite-dimensional systems; Optimal sensitivity; H ∞ control; Beurling–Lax theorem
1. Introduction Let P (s) be the transfer function of the delaydifferential system: x(t) ˙ = x(t − 1) + u(t), y(t) = x(t) − x(t − 1),
−1 < < 0, > 1.
By −1 < < 0, P (s) = (es − )/(ses − ) belongs to H ∞ . Consider the sensitivity minimization problem for P with a given stable weighting function W (s): opt := inf W − P ∞ . (1) ∈H ∞
∗ Corresponding author.
E-mail addresses: [email protected] (K. Kashima), [email protected] (Y. Yamamoto). 0167-6911/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2004.10.004
Invoking the inner-outer factorization P (s)=m(s) P0 (s), we can reduce this to one of finding opt := inf W − m∞ . (2) ∈H ∞
When W is proper and rational [3,4], which covers all practically interesting cases, a beautiful state space formula is known [9,12,17]. In this formula, the optimal performance level is computed from the singularity of a matrix, in the form det m(−H )|22 , where A|22 denotes the (2, 2) block of the matrix A, and H denotes the Hamiltonian arising from W and (see Theorem 8 below). This result was first proved by Zhou and Khargonekar [18] for the pure delay m(s) = e−Ls , and later extended to general inner functions by [9,12,17] and others. This result has played a key role in obtaining the solution to the H ∞ problem for sampled-data
540
K. Kashima, Y. Yamamoto / Systems & Control Letters 54 (2005) 539 – 545
systems [1,7]. It also has a natural extension to the two-block case [5,8]. It should however be noted that for distributed plants as above it is not at all trivial to obtain the inner–outer factorization P = mP 0 . This in general requires the computation of unstable zeros of P, and there can be infinitely many of them, for example, in the case of neutral delay-differential systems. For such cases, applying the (generalized) Zhou–Khargonekar formula as described above is at best approximate by computing zeros. This motivates the present study. We attempt to derive a more direct formula for the computation of the optimal sensitivity. To this end, we confine our consideration to the class of pseudorational transfer functions introduced by the latter author [13]. This class of transfer functions admits a type of fractional representation, which in turn leads to a state space representation. This is also very effective in studying delay systems [13], repetitive control [16], etc. Such a representation is in close connection with left shiftinvariant subspaces of H 2 , and it is this relationship that leads us to a simple characterization of the inner part of the transfer function. It should also be noted that such a representation yields yet another form of the Beurling–Lax theorem [6,10]. We then obtain a simplified form of the generalized Zhou–Khargonekar formula (Theorem 9). The most attractive feature of this result is that we do not need explicit numerical computations of poles and zeros of the transfer function. An example is given to illustrate the results.
Lemma 1 (Yamamoto [14]). For , ∈ D + (R), ( ∗ ) = () + ().
(4)
Dually, for distributions , having support bounded on the right, r( ∗ ) = r() + r().
(5)
For a distribution , −1 denotes its inverse, when it exists, with respect to convolution. We have the following corollary. Corollary 2. For ∈ D + (R) with −1 also in D + (R), (−1 ) = −().
(6)
Similarly, if both and −1 have support bounded on the right, then r(−1 ) = −r().
(7)
Proof. According to (4), () + (−1 ) = ( ∗ −1 ) = () = 0, when both and −1 belong to D + (R). The second equality is obtained similarly.
2. Preliminaries 2.1. Distributions and supports We need some notions from distribution theory [11]. For a given distribution , supp denotes its support. The following quantities are of importance; we allow them to be ±∞, but when supp is bounded, they are both finite: () := inf {t : t ∈ supp }, r() := sup {t : t ∈ supp }.
elements of E (R− ). D + (R) is the space of distributions having support bounded on the left. Clearly E (R− ) ⊂ D + (R). Both E (R− ) and D + (R) constitute an algebra with respect to convolution ∗. The action of a distribution on a test function ∈ D, D being the space of infinitely differentiable functions on (−∞, ∞) with compact support, is denoted by , . The following lemma is a consequence of Titchmarsh’s theorem on convolution [2]:
(3)
Let E (R− ) denote the space of distributions having compact support in (−∞, 0]; the Dirac delta distribution a at point a (a 0), its derivative a , etc., are
We now give the definition of pseudorational impulse responses. This class plays a crucial role in realization, modeling, and control of infinitedimensional systems, particularly delay-differential systems [13,14]. Definition 3. Let f be a distribution having support in [0, ∞). It is said to be pseudorational if there exist q, p ∈ E (R− ) such that 1. q −1 exists over D + (R), 2. ord q −1 = −ord q,
K. Kashima, Y. Yamamoto / Systems & Control Letters 54 (2005) 539 – 545
3. f can be written as
quite effective in computing concrete realizations; for details, see [13,14]. What is important to note here is that
f = q −1 ∗ p, where ord q denotes the order of a distribution q [11]. For a distribution , its Laplace transform is denoted by ˆ (s). Similarly, for a set X of distributions, Xˆ := {fˆ : f ∈ X} provided that every f ∈ X is Laplace transformable. If f is pseudorational, its associated transfer function fˆ is also said to be pseudorational. p As usual, H p and H− denote the Hardy spaces on the open right- and left-half complex plane, respectively. The spaces H 2 and H−2 are the spaces of the Laplace transform of functions in L2 [0, ∞) and L2 (−∞, 0], respectively [6]. The Hilbert space of functions square integrable on the imaginary axis will be denoted by L2 (j R). Clearly L2 (j R) = H 2 ⊕ H−2 . Let ˜be the mirror image of the distribution defined by ˜, (t) := , (−t),
∈ D.
(˜) = −r().
(9)
2.2. Pseudorational impulse responses and associated state space Let := L2loc [0, ∞) be the space of all locally Lebesgue square integrable functions. is equipped with the following natural left-shift semigroup t [13]:
∈ , t 0, s 0.
Let f = q −1 ∗ p be pseudorational. We associate to it the following state space:
X := {x ∈ : q ∗ x ∈ E (R− )}.
Thus when 1/qˆ is stable, it is the orthogonal complement of a right-shift invariant subspace of L2 [0, ∞), and hence gives another representation of the Beurling–Lax theorem [6]. This is the theme of Section 3, but before proceeding further, we conclude this section by giving the following characterization for exponential stability. Lemma 4 (Yamamoto [15]). The left-shift semigroup t in Xq is exponentially stable, i.e., there exist positive constants C, such that t x Ce−t x, if and only if there exists c > 0 such that Re − c
f or all such that q( ˆ ) = 0.
3. Characterization of the inner part
For the Laplace transforms, ˆ ˜(s) := ˆ (−s). Then ˆ ˜ ∈ p H− if and only if ˆ ∈ H p . For a closed subspace Y of a Hilbert space X, its orthogonal complement will be denoted by X Y .
q
• X q is a left-shift invariant subspace of , and • when 1/qˆ is stable, it is indeed a closed subspace of L2 [0, ∞).
(8)
By definition, we clearly have
(t )(s) := (s + t),
541
(10)
It is easy to see that X q is a t -invariant closed subspace of . This space gives a standard state space for pseudorational impulse responses, and is indeed
The objective of Section 3 is to derive a formula for the inner function m arising from the representation Xq for a pseudorational impulse response q −1 ∗p. The precise meaning is as follows: Let q −1 ∗ p be pseudorational, and suppose that q satisfies the condition of Lemma 4. Then by a result in [16] that Xq ⊂ L2 [0, ∞). This implies that Xˆ q is a left shift-invariant closed subspace of H 2 . Consequently, its orthogonal complement H 2 Xˆ q is a right shiftinvariant closed subspace of H 2 . According to the Beurling–Lax theorem [6], there exists a unique inner function m, ˆ modulo constants, such that H 2 Xˆ q = mH ˆ 2 . In other words, Xˆ q = H (m) ˆ := H 2 mH ˆ 2. It is also known that [6] ˆ xˆ ∈ H−2 }. H (m) ˆ = {xˆ ∈ H 2 : m˜
(11)
When 1/qˆ were rational, this inner m ˆ is simply the Blaschke product consisting of zeros of q. ˆ But when it is irrational, it is not easy to find m. ˆ The following theorem gives a formula for this.
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K. Kashima, Y. Yamamoto / Systems & Control Letters 54 (2005) 539 – 545
Theorem 5. Let 1/q(s) ˆ be stable and pseudorational. Then Xˆ q = H (m) ˆ for m ˆ given by
This yields q ∗ x = (q) ∗ q˜∗ . Therefore, by (5) and (9), we obtain
(12)
r(q ∗ x) = (q) + r(q˜) + r() = (q) − (q) + r() = r() 0,
We first prove that (12) indeed gives an inner function.
because ∈ L2 (−∞, 0]. Furthermore, since (q ∗ x) = (q) + (x) is finite, q ∗ x belongs to E (R− ). This implies H (m) ˆ ⊂ Xˆ q .
m ˆ = e−(q)s
q˜(s) ˆ . q(s) ˆ
Lemma 6. For ∈ E (R− ) such that 1/ˆ(s) is stable,
ˆ ˜(s) m ˆ = e−()s ˆ (s) is an inner function. Proof. Since clearly |m| ˆ = 1 on the imaginary axis, it suffices to prove that m ˆ belongs to H ∞ . Recall that m ˆ is in H ∞ if and only if convolution with m defines a bounded linear operator on L2 [0, ∞). Take an arbitrary x ∈ L2 [0, ∞), i.e., xˆ ∈ H 2 , and we show ˆ on the imaginary axis, m∗x ∈ L2 [0, ∞). Since |m|=1 m ˆ xˆ ∈ L2 (j R) and this implies m ∗ x ∈ L2 (−∞, ∞). Furthermore, we have
We are now ready to proof Theorem 5.
∞
1−
n=1
s n
exp
s n
,
n=1
| n |2
< ∞.
Hence ∞ |Re n | < ∞, 1 + | n |2 n=1
Proof of Theorem 5. Let us first show Xˆ q ⊂ H (m). ˆ ˆ ∈ Xˆ q ⊂ H 2 . By (10) this means q ∗ ∈ Take any
ˆ ∈ L2 (j R). It follows E (R− ). Since m ˆ is inner, m˜ ˆ
−1 from (7) and (9) that r((q˜) ) = −r(q˜) = (q) and r(m˜∗ ) = r(−(q) ) + r((q˜)−1 ) + r(q ∗ ) = r(q ∗ ) 0. i.e., m˜ˆ ˆ ∈ and This yields m˜∗ ∈ ˆ hence we have Xˆ q ⊂ H (m). Conversely, take xˆ ∈ H (m) ˆ ⊂ H 2 . Then by (11), we have qˆ xˆ ∈ H−2 . e−(q)s q˜ ˆ
q(s) ˆ = eas
∞ |Re n |
by (4), (6) and (9) (note also that r() 0). Thus m ∗ x ∈ L2 [0, ∞), and hence m ˆ is an inner function.
ˆ := m˜ ˆ xˆ =
Remark 7. The obtained inner m ˆ is expressible as the product of e−Ls and a Blaschke product. To see this, recall the infinite product representation of q(s) ˆ [15]:
where
(m ∗ x) = () + (˜) + (−1 ) + (x) = − r() + (x) 0
L2 (−∞, 0],
For example, let W (s) = 1/(ses − c). This is clearly pseudorational, where q = ( −1 − c) and (q) = −1. Applying Theorem 5 to (12), we have s −se−s − c ce + s = H . Xˆ q = H es ses − c ses − c
H−2 ,
and this is a sufficient condition for the convergence of the Blaschke product consisting of zeros of q(s) ˆ ˆ q(s) ˆ is of the form ebs mb (s) [6]. Hence e−(q)s q˜(s)/ where mb (s) is the Blaschke product consisting of the zeros of q. ˆ 4. Optimal sensitivity problem Consider the optimal weighted sensitivity problem: opt := inf W (s) − Pˆ (s)(s) . (13) ∈H ∞
∞
Here W (s) is a stable, rational and strictly proper weighting function, and we assume that the given plant
K. Kashima, Y. Yamamoto / Systems & Control Letters 54 (2005) 539 – 545
543
Pˆ (s) is pseudorational and belongs to H ∞ . This problem is reducible to that of finding opt := inf W (s) − m(s) ˆ (s)∞ ,
For the proof, let us begin by decomposing Pˆ as ˆ P = Pˆi · Pˆ0 , where
where m ˆ is the inner part of Pˆ . For this reduced problem, we can apply the method given in [18,9,12,17] as follows: Let (A, B, C) be a minimal realization of W. Define its Hamiltonian H by A BB T / . (14) H := −C T C/ −AT
(19)
∈H ∞
The following result holds: Theorem 8 (Zhou–Khargonekar, Lypchuk, Smith and Tannenbaum Smith, Yamamoto [18,9,12,17]). Let m, ˆ H be as above. Suppose that m ˆ is analytic on the set of eigenvalues of H . Then the optimal sensitivity opt is given by max{ : det m˜(H ˆ )|22 = 0},
(15)
where A|22 denotes the (2, 2) block of matrix A when partitioned accordingly as (14). As we noted in Section 1, however, m ˆ is not necessarily easy to find for general infinite-dimensional systems. Even if it is given in the form of a Blaschke product, it is still not trivial to compute the optimal sensitivity according to (15). We here give a more direct representation that is much more suitable for computation. To this end, we assume that the numerator of Pˆ is factorized into the product of stable and anti-stable parts. Then we state our main result as follows: Theorem 9. Suppose that the plant Pˆ is factorized as pˆ 1 (s)pˆ 2 (s) Pˆ (s) = ∈ H ∞, q(s) ˆ
(16)
ˆ −1 , where p1 , p2 , q ∈ E (R− ) such that q(s) −1 −1 r(p )s ∞ ∈ H . Assume also that pˆ 1 (s) , e 2 pˆ 2˜(s) 1/pˆ 2 is analytic on the set of eigenvalues of H . Define L := −(q) + (p1 ) − r(p2 ). Then the optimal sensitivity opt is given by (17) max{ : det eLH pˆ 2˜(H )pˆ 2 (H )−1 |22 = 0}.
pˆ 2 (s) Pˆi (s) := e−Ls · , pˆ 2˜(s) pˆ 1 (s)pˆ 2˜(s) Pˆ0 (s) := eLs · . q(s) ˆ
(18)
We start with the following lemma. Lemma 10. The functions Pˆi and Pˆ0 belong to H ∞ , and Pˆi is inner. Proof. By the assumption above, both functions have no unstable poles. Notice that, on the imaginary axis, ˆ ˆ ˆ ˆ |Pi | = 1 and |P0 | = |P | P < ∞. It is thus suffi∞ cient to show that Pi and P0 have support in [0, ∞), that is, (Pi ) 0 and (P0 ) 0. The rest is similar to the proof of Lemma 6. By (4), (6) and (9), we have (Pi ) = L + (p2 ) − (p2˜) = − (q) + (p1 ) − r(p2 ) + (p2 ) + r(p2 ) = (P ) 0, (P0 ) = − L + (p1 ) + (p2˜) − (q) = (q) − (p1 ) + r(p2 ) + (p1 ) + (p2˜) − (q) = 0.
(20)
Proof of Theorem 9. Let opt be defined by opt = inf W − Pˆi .
(21)
∈H ∞
∞
We first show that the optimal sensitivity opt defined by (13) is equal to opt . Then Theorem 8 combined with formula (18) would yield the desired result. Note first that := Pˆ0 belongs to H ∞ for arbitrary ∈ H ∞ by Lemma 10. Hence we easily see opt opt . But showing opt opt is less trivial because Pˆ0−1 does not necessarily belong to H ∞ . The function Pˆ0−1 is analytic on the right half plane and satisfies that (P0−1 )=−(P0 )=0 by (6) and (20). Notice that we can decompose Pˆ0−1 as e−(p1 )s er(p2 )s (q) · · e q(s), ˆ Pˆ0 (s)−1 = pˆ 1 (s) pˆ 2˜
(22)
by the definition of L := −(q) + (p1 ) − r(p2 ). Since pˆ 1 (s)−1 ∈ H ∞ and ((p1 ) ∗ p1−1 ) = 0 by
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K. Kashima, Y. Yamamoto / Systems & Control Letters 54 (2005) 539 – 545
(4) and (6), we can show that e−(p1 )s /pˆ 1 (s) belongs to H ∞ as in Lemma 6. Also by the assumption on p2 , er(p2 )s /pˆ 2˜(s) ∈ H ∞ . Next −(q) ∗ q has compact support in [0, ∞). Therefore, from the Paley–Wiener–Schwartz theorem [11], there exist C > 0 and a positive integer k such that |e(q)s q(s)| ˆ C(1 + |s|)k ,
Re s 0.
Pˆ0−1 (T s + 1)
1.4 1.2 1 0.8 0.6
> 0, e(q)s q(s)/(T ˆ s+
0.2
0.4
0 0.2
0.4
γ
0.6
0.8
Fig. 1. 0 < < 1(dash) and = 2 (solid).
∈ H ∞, k
and we have lim W − Pˆ T T →0
1.6
(23)
This implies that, for arbitrary T 1)k is in H ∞ and so is Pˆ0−1 /(T s + 1)k . Now for any ∈ H ∞,
T :=
2 1.8
= lim W − Pˆi k T →0 ∞ (T s + 1) ∞ = W − Pˆi . ∞
This clearly yields opt opt . Recall that Pˆi is an inner function satisfying Pˆi˜(s)= Ls e pˆ 2 (s)pˆ 2˜(s) from (18) and Lemma 10. Therefore, the desired result follows by applying Theorem 8 to (21). Remark 11. If the given plant Pˆ has no unstable zeros, the associated inner function is a simple delay Pˆi = e−(P )s . 4.1. Example Let us compute the optimal sensitivity by using Theorem 9. Consider the weighting function W (s) and the plant Pˆ (s) as follows: 1 pˆ es − , Pˆ (s) = = 2s , s+1 qˆ 2e − 1 ( > 0, = 1),
W (s) =
with the Hamiltonian of W given by −1 −1 H = . −−1 1 Notice that the optimal sensitivity opt W ∞ = 1. Observe that Pˆ (s) has infinitely many zeros. If 0 < < 1, Pˆ (s) has no unstable zeros. Hence the
optimal sensitivity
is given as the maximum satisfying det eH |22 = 0, independent of , since L = −(q) + (p) = 1. On the contrary, in the case of > 1, Pˆ (s) has infinitely many unstable zeros, i.e., s = ln + 2n j (n ∈ Z). Then L = −(q) − r(p) = 2 and from (17) we have
opt = max{ : det(eH (I − eH )(eH − I )−1 |22 ) = 0}. Fig. 1 shows the absolute values of the determinant of the corresponding matrices above. When the plant has no unstable zeros, the optimal sensitivity is about 0.44. On the other hand in the case of =2 the optimal value is approximately 0.80. 5. Conclusion We have given an equivalent characterization for invariant subspaces of H 2 , when the underlying structure is specified by the so-called pseudorational transfer functions. We have derived a simple closed formula for the optimal sensitivity for such systems. This formula does not require explicit numerical computations of poles and zeros of the associated transfer functions. References [1] B. Bamieh, J.B. Pearson, A general framework for linear periodic systems with application to H∞ sampled-data control, IEEE Trans. Automat. Control 37 (1992) 418–435. [2] W.F. Donoghue, Distributions and Fourier Transforms, Academic Press, New York, 1969. [3] C. Foias, H. Özbay, A. Tannenbaum, Robust Control of Infinite Dimensional Systems, Springer, Berlin, 1996.
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