Bounded real and positive real balanced truncation for infinite-dimensional systems

Bounded real balanced truncation for infinite-dimensional systems is considered. This provides reduced order finite-dimensional systems that retain bounded realness. We obtain an error bound analogous to the finite-dimensional case in terms of the bounded real singular values. By using the Cayley transform a gap metric error bound for positive real balanced truncation is subsequently obtained. For a class of systems with an analytic semigroup, we show rapid decay of the bounded real and positive real singular values. Together with the established error bounds, this proves rapid convergence of the bounded real and positive real balanced truncations.


Introduction
In model reduction the aim is to approximate a system with many degrees of freedom by a system with few degrees of freedom. In this article we are interested in the case where the original system has infinitely many degrees of freedom. Examples of such systems are systems described by partial differential equations or delay differential equations.
Approximation of controlled partial differential equations by standard numerical methods such as finite elements often gives results that are far from optimal [20]. A rigorous verification of this observation depends on two things: 1) an error analysis of these standard numerical methods and 2) determining what the optimal approximation results (approximately) are. Lyapunov balanced truncations are close to optimal approximations and are therefore important in rigorously verifying the above fundamental observation.
Lyapunov balanced truncation was introduced for finite-dimensional systems by Moore [18] and a crucial aspect is the error bound which was independently derived by Enns [7] and Glover [8]. In (1), σ k are the singular values of the Hankel operator of the system and N and n are the orders of the original and truncated systems respectively. The theory of Lyapunov balanced truncation has been extended to the infinite-dimensional case by Glover, Curtain and Partington [9]. Some assumptions made there were proven to be redundant in [12]. The upshot is that the balanced truncation error-bound (1) (now with N = ∞) continues to hold in the infinite-dimensional case. There is also the trivial lower bound which holds for any reduced order system of dimension n. Combined these bounds show that Lyapunov balanced truncation is indeed close to optimal. An analysis of the singular values of Hankel operators shows that in many applications these singular values converge to zero at a rate faster than any polynomial rate (whether the rate is in fact exponential is -for partial differential equation examples-an open problem) [20,21]. This implies that Lyapunov balanced truncations in these applications converge at a rate faster than any polynomial rate. Standard numerical methods such as finite elements do not converge as fast in these applications (so-called higher order methods are in fact not higher order for these applications because of lack of smoothness). See [20] and also the example in Section 8.
A downside of Lyapunov balanced truncation is that in general any energy relation in the original system is not necessarily retained in the reduced order system. In the finite-dimensional case the alternative methods called bounded real [19] and positive real balanced truncation [6] do retain such an energy relation. In this article we generalise these methods to the infinite-dimensional case. In particular, we prove the corresponding error bounds. For a special class of systems we also provide an analysis of the singular values involved. We conclude that for a large class of systems there exist approximations that preserve the relevant energy relation and that converge much faster than those provided by the standard numerical methods. This is illustrated by the numerical example of a boundary controlled heat equation in Section 8.

Statements of main results
There are two classical notions of dissipativity in control theory: on the one hand there are the systems called impedance passive, passive or positive real and on the other hand there are the systems called scattering passive, contractive or bounded real.
Our first main result considers the bounded real case. Precise definitions of the notions involved are given later in this article. Theorem 1.1. Let G ∈ H ∞ (C + 0 ; B(U , Y )) be a strictly bounded real function with summable bounded real singular values and with U and Y finitedimensional. Then for each nonnegative integer n there exists a bounded real rational function of McMillan degree ≤ n, denoted G n and called the reduced order transfer function obtained by bounded real balanced truncation, such that where σ k are the bounded real singular values.
Our second main result considers the positive real case. Again, precise definitions of the notions involved are given later in this article. Theorem 1.2. Let J ∈ H ∞ (C + 0 ; B(U )) be a strictly positive real function with summable positive real singular values and with U finite-dimensional. Then for each nonnegative integer n there exists a positive real rational function of McMillan degree ≤ n, denoted J n and called the reduced order transfer function obtained by positive real balanced truncation, such that whereδ is the gap metric [16, p.197, p.201] and σ k are the positive real singular values.

Organisation of the article
In Section 2 we review bounded real and positive real balanced truncation in the finite-dimensional case. We do this so that we can highlight some features that are essential in the infinite-dimensional case, but typically are not given much prominence in the finite-dimensional case. Section 3 summarises the aspects of well-posed linear systems, optimal control and spectral factor systems that are needed to prove the main results of this article. Section 4 is the technical heart: here we construct the bounded real balanced truncation and prove Theorem 1.1. These results are converted via the Cayley transform, discussed in Section 5, to the positive real case in Section 6, which contains the proof of Theorem 1.2. In Section 7 we discuss the asymptotic behavior of the bounded real and positive singular values for a class of systems. Finally, Section 8 contains the already mentioned specific example of a boundary controlled heat equation.

Review of the finite-dimensional case
Model reduction by bounded real balanced truncation and positive real balanced truncation for rational functions, introduced in Opdenacker & Jonckheere [19] and Desai & Pal [6] respectively, is reviewed. The survey article by Gugercin & Antoulas [10], as well as Antoulas [2] include summaries of some of the material in this section. In our review we emphasise the aspects that are important in the generalisation to non-rational functions. Let U and Y denote finite-dimensional Hilbert spaces, which are the input and output spaces respectively. We recall that a rational function G : C + 0 → B(U , Y ) belongs to H ∞ if and only if G is proper and every pole of G is in the open left-half complex plane. Given such a G it is possible to write for some finite-dimensional space X and operators with A Hurwitz. The quadruple of operators (4) (and implicitly the space X ) is called a realisation of G and is denoted by [ A B C D ]. Moreover, we can always choose [ A B C D ] such that the associated input-state-output systeṁ y(t) = Cx(t) + Du(t), is minimal (i.e. controllable and observable).

Bounded real balanced truncation
Definition 2.1. A function G ∈ H ∞ (C + 0 ; B(U , Y )), where U and Y are Banach spaces, is bounded real if and such a G ∈ H ∞ (C + 0 ; B(U , Y )) is strictly bounded real if the above inequality is strict.
1. Synonymously with the term 'bounded real' the terms Schur, contractive and scattering passive are used. In the model reduction literature [2] the term 'bounded real balanced truncation' seems to have become standard and therefore we use this terminology.
2. Note that, in spite of the terminology, there is no realness assumption in Definition 2.1. However, if such an assumption is made about the original system, then realness of the reduced order system can be concluded.
Bounded real balanced truncation makes use of the well-known Bounded Real Lemma, see Anderson & Vongpanitlerd [1], which gives a state space characterisation of bounded real functions.
C D ], the following are equivalent. (i) G is bounded real.
(ii) There exists a triple of operators (P, K, W ) with and P positive and self-adjoint satisfying the bounded real Lur'e equations Moreover, if either of the above hold then there are positive self-adjoint solutions P m , P M to (7) such that for any self-adjoint solution P of (7) we have The extremal operators P m , P M are the optimal cost operators of the bounded real optimal control problems, namely: The minimisation problems (9) are subject to the minimal input-state-output realisation (5).
Proof. See [1]. There it is assumed that dim U = dim Y , but the result is true in general.
If P = P * > 0 is a solution of (7), for some K, W then an elementary calculation shows that P −1 > 0 solves the dual bounded real Lur'e equations for some operators L : Y → X , X : Y → Y . By the Bounded Real Lemma, there are extremal self-adjoint solutions Q m , Q M to (10) such that for any selfadjoint solution Q to (10); 0 < Q m ≤ Q ≤ Q M . In particular, it is not difficult to see that Remark 2.4. Solutions of the bounded real Lur'e equations are generally not unique. We expand on this further, as it will be important later in this article. Given solutions (P, K, W ) and (Q, L, X) of the bounded real Lur'e equations (7) and dual bounded real Lur'e equations (10) respectively, the first components P and Q do not in general uniquely determine the other two respective components. If we fix (P, K, W ) and (Q, L, X) as above then the operators K , W and L , X defined by unitary, are such that (P, K , W ) and (Q, L , X ) are also solutions of (7) and (10) respectively.

Definition 2.5. The minimal realisation [ A B
C D ] of a bounded real rational transfer function is bounded real balanced, or in bounded real balanced co-ordinates, if The nonnegative square roots of the eigenvalues of the product P m P −1 M are called the bounded real singular values, which we denote by (σ k ) m k=1 , each with (geometric) multiplicity r k (so that m k=1 r k = dim X ). The bounded real singular values are ordered such that σ k > σ k+1 > 0 for each k.
To define the bounded real balanced truncation, for n < m let X n and Z n denote the sum of the first n and last m − n eigenspaces of Π respectively. Then with respect to the orthogonal decomposition X = X n ⊕ Z n , the operators A, B, C and Π split as The truncated system with realisation A11 B1 C1 D is called the bounded real balanced truncation and its transfer function is called the reduced order transfer function obtained by bounded real balanced truncation. We note that this reduced order transfer function is uniquely determined by the original transfer function, i.e., it does not depend on the particular bounded real balanced realisation that is chosen for truncation.
Given a bounded real balanced realisation [ A B C D ], observe that from (7a), the optimal cost operator Π = P m satisfies the Lyapunov equation Similarly from (10a), Π = Q m satisfies the Lyapunov equation Since A is stable it follows that Π is both the controllability and observability Gramian of the extended system  (13) and (14) we see that Σ E is Lyapunov balanced [18], [22] and that the bounded real singular values of [ A B C D ] are the Lyapunov singular values of Σ E (i.e. the singular values of the Hankel operator of Σ E ). The Lyapunov balanced truncation of (15) is C D ] is recovered by omitting the blocks L 1 , K 1 , X, W and zero. This corresponds to restricting to and projecting onto the original input and output spaces U and Y respectively. Therefore bounded real balanced truncation of [ A B C D ] can be seen as Lyapunov balanced truncation of Σ E . This relation is used in [19] in proving the following theorem, which is the main result for bounded real balanced truncation in the finite-dimensional case.
Theorem 2.6. Let G ∈ H ∞ (C + 0 ; B(U , Y )) denote a rational bounded real function and let (σ j ) m j=1 denote its bounded real singular values, each with multiplicity r j . For r < m let G r denote the reduced order transfer function obtained by bounded real balanced truncation. Then G r is bounded real and the following error bound holds If [ A B C D ] is a minimal, bounded real balanced realisation of G, then the bounded real balanced truncation A11 B1 C1 D is stable. If additionally G is strictly bounded real, then G r has MacMillan degree r = r j=1 r j and A11 B1 C1 D is minimal and bounded real balanced.
Proof. See Theorem 2 and Section IV of [19]. The assumption there that G is strictly bounded real is not needed to prove that G r is bounded real and that A 11 is stable. The authors also assume throughout that U = Y , but this isn't needed and the proof for the general case is essentially the same.
Our approach to the infinite-dimensional case makes extensive use of the above connection with Lyapunov balanced truncation (more so than the finitedimensional case does). This approach necessitates careful consideration of the non-uniqueness of the extended system.

Positive real balanced truncation
Definition 2.7. An operator valued analytic function J : We say that the analytic function J : Remark 2.8. 1. The term strictly positive real is used for various slightly different concepts in the literature, as described in, for example, Wen [38]. The condition (18)  2. We do not assume that a positive real function is real on the real axis as is sometimes done in the literature.
3. Synonymously with the term 'positive real function' the terms impedance passive function, Weyl function, Titchmarsh-Weyl function and Caratheodory-Nevanlinna function are used. In the model reduction literature the term 'positive real balanced truncation' seems to have become standard and therefore we use this terminology.
Positive real balanced truncation is identical in spirit to bounded real balanced truncation and was proposed in the finite-dimensional case in [6]. The key ingredient is the Positive Real Lemma, which analogously to the Bounded Real Lemma provides a state-space characterisation of positive real functions. (ii) There exists a triple of operators (P, K, W ) with P : X → X , K : X → U , W : U → U , and P positive and self-adjoint satisfying the positive real Lur'e equations If either of the above hold then there exist positive, self-adjoint solutionsP m ,P M to (19) such that any self-adjoint solution P to (19) satisfies The extremal operatorsP m ,P M are the optimal cost operators of the positive real optimal control problems, namely: The minimisation problems (21) are subject to the minimal input-state-output realisation (5) of J.
The nonnegative square roots of the eigenvalues ofP mP  Proof. See Harshavardhana et al. [15], [14] and the references therein.
Remark 2.11. The analogous H ∞ error bound does not hold for positive real balanced truncation; a counter-example can be found in Guiver & Opmeer [13]. This is because there is not the same connection to Lyapunov balanced truncation [18] as in the bounded real balanced truncation case. In fact in the positive real case an H ∞ error bound seems less natural as positive real functions need not belong to H ∞ . Instead a gap metric error bound holds, whereδ is the gap metric and σ k are the positive real singular values. The bound (22) is also proven in [13] (and was at the same time independently established by Timo Reis). Note that our second main result of this paper, Theorem 1.2, is the expected generalisation of (22).

Preliminaries
In this section we collect the infinite-dimensional results from the literature that we shall require to prove our main results. Recall that we are seeking to approximate H ∞ (C + 0 ; B(U , Y )) functions, which in contrast to Section 2 need not be rational, and thus state-space representations will generally be infinite-dimensional. In this work U and Y denote the input and output spaces respectively, which are always assumed to be finite-dimensional Hilbert spaces.
Transfer functions belonging to H ∞ (C + 0 ; B(U , Y )) can be realised by wellposed linear systems and Section 3.1 contains the corresponding notation and key required material. Sections 3.2 and 3.3 describe the optimal control problems and spectral factor systems respectively that we will require for bounded real and positive real balanced truncation.

Well-posed linear systems
Well-posed linear systems on L 2 go back to the work of Salamon [25], [26]. The monograph of Staffans [31] is dedicated to the study of general well-posed linear systems, and we shall make use of many results from this text. We remark that there are several different but equivalent formulations in the literature of a well-posed linear system. Although we use many results from [37], we have chosen to use the formulation of [31] so as to more readily apply results from that book. The equivalence between the formulations in [37] and [31] is shown in [31,Section 2.8].
For precise definitions of the following objects we refer the reader to [31, Section 2.2]. We denote by Σ = (A, B, C, D) on (Y , X , U ) (respectively, the output, state and input spaces) an L p well-posed linear system with state x and output y given by for input u ∈ L p loc (R + ; U ). We shall mostly be using L 2 well-posed systems, though we shall also need L 1 well-posed systems. In the above (A t ) t≥0 is a strongly continuous semigroup on the state-space X , B t 0 is the input map (with initial time 0 and final time t), C 0 the output map and D 0 the inputoutput map (both with initial time 0). We remark that the finite-dimensional input-state-output system (5) has operators A t , B t 0 , C 0 and D 0 given by , the operators B t 0 , C 0 and D 0 can be expressed in terms of the master operators B, C and D and vice versa. There is no issue, therefore, with using the master operators A, B, C and D. For example, for the finite-dimensional system (5), B, C and D are given by Remark 3.2. We collect some notation we shall need. Let π + and π − denote the projections from L 2 (R) onto L 2 (R + ) and L 2 (R − ) respectively. We let τ t denote the bilateral shift by t on L 2 (R), i.e. for t, s ∈ R, (τ t u)(s) = u(t + s).

Remark 3.3.
We assume that the reader is familiar with the generators of a well-posed linear system. The control operator and observation operator of well-posed linear systems date back to Weiss, [33] and [34] respectively. We shall also require the notion of a regular transfer function, as introduced by Weiss [36], and an operator node, system node, a compatible operator node and an admissible feedback operator. All of these concepts can be found in [31] and the latter are only drawn upon in some of the proofs of our later results, and are not needed for understanding the statements of those results.
The term realisation of an input-output (linear, time-invariant, causal) map D on L p refers to an L p well-posed linear system with input-output map D, see [31, Definition 2.6.3] for more details. The transfer function G of an L p well-posed system is defined as (see [31,Definition 4.6 defined for Re s > ω A (the growth bound of A). The transfer function G is usually understood, however, as the "Laplace transform of the input-output map", which by [31, Corollary 4.6.10] is equivalent to the above definition. We refer the reader to [31,Corollary 4.6] or Weiss [35] for more information.
The transfer function in (25) determines D uniquely and hence by a realisation of a transfer function we mean a realisation of the input-output map D related to G by (25).
The following result is well-known and simply states that every H ∞ function has a (stable) L 2 well-posed realisation, with Hilbert space state space.
with X a Hilbert space such that the following stability assumptions hold: We call such a system (in particular satisfying (26)-(29)) a stable L 2 well-posed linear system.
Proof. This is well-known and follows from, for example, [37,Theorem 4.2].
We need the following notion of a dual transfer function and a dual realisation.
Given an L 2 well-posed linear system Σ = (A, B, C, D) on (Y , X , U ) (Hilbert spaces) we call the L 2 well-posed linear system Σ d given by The reflection R acts on L 2 (R), and we view elements of L 2 (R + ) or L 2 (R − ) as belonging to L 2 (R) by extending by zero. Given an input Remark 3.6. It is proven in [31, Theorem 6.2.3] that Σ d is an L 2 well-posed linear system. Furthermore, it is easy to see that a transfer function is (strictly) bounded real if and only if its dual is (dual in the sense of the above definition). Similarly, for (strictly) positive real transfer functions.
The following result describes some properties of dual systems, notably that the dual system realises the dual transfer function, and is again taken from [31].

Optimal control problems
Bounded real (positive real) balanced truncation makes use of the unique optimal cost operators of the scattering supply rate (impedance supply rate) optimal control problems described below. The following results are special cases of [37, Proposition 7.2] that can also be found in Staffans [28], and are the first instances of why we restrict attention to the strictly bounded real (positive real) case.
Lemma 3.8. Let Σ = (A, B, C, D) denote a stable L 2 well-posed linear system with strictly bounded real transfer function. Then the optimal control problem: over all u ∈ L 2 (R + ; U ) subject to (23), has a solution in the sense that for any where the optimal control is unique and is given by and P m : X → X is bounded and satisfies P m = P * m ≥ 0 and Proof. See [37,Proposition 7.2]. The assumption that the transfer function G is strictly bounded real is equivalent to see [37,Section 7] and hence I − π + D * Dπ + is boundedly invertible. Therefore the optimal control u opt and optimal cost operator P m in (34) and (35) respectively are well-defined. Furthermore, in [37] it is assumed that G is weakly regular (with zero feedthrough), but that is not needed for this proof.
Corollary 3.9. Using the assumptions and notation of Lemma 3.8 let Σ d denote the dual system from Definition 3.5. The dual optimal control problem: for Proof. The result follows immediately from Definition 3.5, Remark 3.6 and Lemma 3.8.
Lemma 3.10. Let Σ = (A, B, C, D) denote a stable L 2 well-posed linear system with strictly positive real transfer function. Then the optimal control problem: over all u ∈ L 2 (R + ; U ) subject to (23), has a solution in the sense that for any where the optimal control is unique and is given bỹ andP m : X → X is bounded and satisfiesP m =P * m ≥ 0 and The assumption that the transfer function is strictly positive real is equivalent to see [37,Section 7] and hence (Dπ + + π + D * ) is boundedly invertible. Therefore the optimal controlũ opt and optimal cost operatorP m in (39) and (40) respectively are well-defined. Furthermore, in [37] it is assumed that the transfer function is weakly regular (with zero feedthrough), but that is not needed for this proof.
Corollary 3.11. Using the assumptions and notation of Lemma 3.10 let Σ d denote the dual system from Definition 3.5. The dual optimal control problem: Proof. The result follows immediately from Definition 3.5, Remark 3.6, and Lemma 3.10.

Extended systems
Here we recall some results on spectral factorisations and particularly spectral factor systems developed in [37]. This is the second instance where we require strict bounded realness of G. The material here is used in the next section to construct the bounded real balanced truncation. and The functions θ and ξ are uniquely determined up to multiplication by a unitary operator in B(U ) and B(Y ) respectively. Specifically, if θ 0 satisfies (42) and ξ 0 satisfies (43) then the sets of all spectral factors satisfying (42) and (43) are given by respectively. Proof.
The assumption that G is strictly bounded real implies that The existence of the spectral factor θ satisfying θ, θ −1 ∈ H ∞ (C + 0 ; B(U )), the equality (42) and uniqueness up to a unitary transformation follows from Rosenblum & Rovnyak [24,Theorem 3.7]. The claims regarding ξ follow from the above and duality.
) denote a spectral factor from Lemma 3.12 satisfying (42) with input-output map D θ . Define where In the above and observability Gramian P m given by (33), i.e. the optimal cost operator of the optimal control problem (32).
denote a spectral factor from Lemma 3.12 satisfying (43) with input-output map D ξ . Define where Then B E is bounded and Σ E2 : and controllability Gramian Q m given by (36), which is the optimal cost operator of the dual optimal control problem.
Proof. The claims follow immediately once we note that Σ E2 is the dual of the system constructed in Lemma 3.13 applied to the dual transfer function G d instead of G (and therefore now using the spectral factor ξ instead of θ from Lemma 3.12).
Remark 3.15. For a fixed strictly bounded real transfer function G and stable L 2 well-posed realisation Σ of G there are many extended output systems Σ E1 and many extended input systems Σ E2 owing to the non-uniqueness of the spectral factors θ and ξ from Lemma 3.12. However, given any Σ E1 , every other extended output system is determined by Σ E1 and a unitary operator U ∈ B(U ). As such we say that from G and Σ we obtain a family of extended output systems, parameterised by U . Similarly for Σ E2 , now parameterised by unitary V ∈ B(Y ).

Bounded real balanced truncation
In this section we construct the bounded real balanced truncation of a strictly bounded real function and prove Theorem 1.1. We note that existence of bounded real balanced realisations in the infinite-dimensional case is shown in [31, Theorem 11.8.14], however bounded real balanced truncation is not addressed there. We construct the bounded real balanced truncation by relating it to the Lyapunov balanced truncation of a certain extended system, as outlined (for the finite-dimensional case) in Section 2.1. The details of the construction of this extended system are given in Section 4.1. Subsequently in Section 4.2 we define the bounded real balanced singular values and the bounded real balanced truncation and prove Theorem 1.1.

Extended Hankel operators and transfer functions
Given a stable L 2 well-posed realisation of the strictly bounded real function G we seek to combine an extended output system Σ E1 and an extended input system Σ E2 from Lemmas 3.13 and Lemma 3.14 respectively into one (jointly) extended system with transfer function of the form where χ is yet to be determined. Towards this end, we first consider a (extended) Hankel operator constructed from an (extended) output map C E and (extended) input map B E from Lemmas 3.13 and 3.14 above respectively. The extended transfer function G E will subsequently be defined in terms of this extended Hankel operator.
Since the extended systems from Lemmas 3.13 and Lemma 3.14 are not unique, we in fact obtain a family of extended systems, parameterised by two unitary operators. Compare this construction with that in the finite-dimensional case, described in Remark 2.4.
Hankel operators are well-studied objects, with unfortunately many different conventions being used in the literature. We say that an operator where τ t i is the usual left shift by t ≥ 0 on L 2 (R + ; Z i ) with adjoint (τ t i ) * the corresponding right shift.
Remark 4.1. We adopt the convention of Hankel operators mapping forwards time to forwards time. Therefore it is necessary to include a reflection operator R (as in Definition 3.5) in our definition of Hankel operator of a well-posed linear system when compared to [31].
) denote a strictly bounded real function with stable L 2 well-posed realisation Σ. Let θ, ξ denote spectral factors as in Lemma 3.12 and let C E and B E denote the output map and input map from Lemma 3.13 and 3.14 respectively. Define the bounded operator H E by where R is the reflection from Definition 3.5. Then H E is a Hankel operator.
The operator H E is independent of the choice of realisation Σ of G and depends on the spectral factors chosen as follows. If H E (θ 0 , ξ 0 ) is the Hankel operator for the choice of spectral factors θ 0 , ξ 0 and H E (θ, ξ) is the Hankel operator for spectral factors θ, ξ related to θ 0 , ξ 0 by (44), then the Hankel operators are related by acting by (pointwise) multiplication. The same is true for [ I 0 0 U ], only now acting on L 2 (R + ; Y U ). Proof of Lemma 4.2: A proof that that the operator H E is Hankel in the sense of (54) can be found in [11, Lemma 6.3.1.]. The result follows readily from the intertwining properties of input maps and output maps of well-posed linear systems.
To see that H E is independent of the choice of Σ we calculate and using the formulae (45) for C θ and (52) for B ξ gives that this equals By inspection of (57), for given spectral factors θ and ξ, H E depends only on the terms D, D θ , D ξ and their adjoints and inverses where applicable. We recall that an input-output map is completely determined by its transfer function (and vice versa). Therefore, (57) depends only on G and the spectral factors θ and ξ. By their construction in Lemma 3.12 the spectral factors are certainly independent of the stable L 2 well-posed realisation of G and hence so is H E . Equation (56) follows from (57) and the (easily established) relations Again U and V are here understood as operators respectively acting on L 2 (R + ; U ) and L 2 (R + ; Y ) by pointwise multiplication, and certainly commute with π + , π − and R.
is called the family of extended Hankel operators of G. We recall the definition of singular values, nuclear operators and what we mean by the transfer function of a Hankel operator.
Definition 4.6. For a bounded linear operator T : Z 1 → Z 2 between Banach spaces, the k th singular value s k is defined as The operator T is nuclear if its singular values (s k ) k∈N are summable, i.e. k∈N s k < ∞.
Remark 4.7. In this work we use the term singular value in a non-standard manner. Given the above definition, we call σ k the k th singular value of T , but counted with multiplicities, so that if s 1 = s 2 = · · · = s p1 and s p1 > s p1+1 , for some p 1 ∈ N then we set σ 1 = s 1 = s 2 = · · · = s p1 , σ 2 = s p1+1 = . . . , and so on. As such, our k th singular value σ k has multiplicity p k ∈ N and satisfies σ k > σ k+1 , however note that σ k need not necessarily be the distance of T to rank k − 1 operators. Using this convention the operator T is nuclear if We remark that if all the singular values are simple, then our convention and the usual convention coincide.
The following facts about Hankel operators are well-known, and proofs of these assertions are included in, for example, [11]. Given a bounded Hankel operator H : L 2 (R + ; Z 1 ) → L 2 (R + ; Z 2 ) a function ψ satisfying , is called a symbol for H, where L is the unilateral Laplace transform, R C is the reflection (R C φ)(s) = φ(−s) and M ψ is multiplication by ψ. In general H may have many symbols but every bounded Hankel operator has a symbol φ ∈ where φ 1 can be extended analytically to the right-half complex plane and φ 2 can be extended analytically to the left-half complex plane. The components φ 1 and φ 2 are unique up to an additive constant. We call φ 1 the analytic part of φ in C + 0 . We define a transfer function corresponding to H as the analytic part of a symbol in C + 0 of H (which up to an additive constant is uniquely determined by H), plus an arbitrary constant operator.
Proof. This is a condensed version of [12,Proposition 3.4].
The operatorsŨ andṼ are unitary and from this an easy calculation shows that for bounded T : It is also easy to see that for any n ∈ N the map T →Ũ TṼ is a bijection of rank n operators to rank n operators. Therefore for n ∈ N s n (H 0 E ) = inf H 0 E − T : rank T < n = inf H E −Ũ TṼ : rank T < n = s n (H E ).
By counting with multiplicities it follows that σ k (H 0 E ) = σ k (H E ) for every k ∈ N, which completes the proof. We are now able to construct our desired extended transfer function.
) denote a strictly bounded real function and assume that G has a nuclear family of extended Hankel operators. Let H E denote a member of this family corresponding to the spectral factors θ, ξ. Then there exists χ ∈ H ∞ (C + 0 ; B(Y , U )) such that is regular and is a transfer function of H E . The feedthrough of χ can without loss of generality be taken equal to zero. Therefore we let denote the bounded operator such that The components D, D θ and D ξ of D E are the feedthroughs of G, θ and ξ respectively. By always fixing the feedthrough of χ as zero, for each G, θ and ξ there is a one-to-one correspondence between Hankel operators H E and transfer functions G E given by (59).
Proof. The existence of χ and the regularity of G E (and hence G and the spectral factors θ and ξ) follows from Proposition 4.8. By that result the Hankel operator H E determines G E uniquely up to an additive constant, which is determined by the feedthroughs of G, θ and ξ and the choice of χ having feedthrough zero. From its construction, we see that the original transfer function G and the spectral factors θ and ξ are components of the extended transfer function G E . The next lemma describes how we can obtain L p well-posed realisations of G and θ from L p well-posed realisations of G E . We shall need those later in this work.
are L p well-posed realisations of G and θ respectively. Here P U denotes the orthogonal projection of Y U onto U and P Y denotes the orthogonal projection of Y U onto Y . If A, B, C and D denote the generators of (A, B, C, D), then A, B| U , P Y C and P Y D| U and A, B| U , P U C and P U D| U are the generators of the above realisations of G and θ respectively. Furthermore, if (A, B, C, D) is a stable L 2 well-posed realisation of G E then the realisations in (61) are stable L 2 well-posed realisations of G and θ respectively.
Proof. It is routine to verify that the two systems in (61) satisfy the conditions of [31, Definition 2.2.1], and hence are L p well-posed. Since the generators are unique, a short calculation demonstrates that the formulae given are indeed the generators. The final claim is immediate from the definition of a stable L 2 wellposed realisation since restriction and projection are bounded operations.

Bounded real balanced truncation
We now define the bounded real singular values and bounded real balanced truncation for strictly bounded real functions. These in principle depend on the choice of extended system, but Remark 4.15 and Lemma 4.20 will demonstrate that this dependence is trivial.  2. Our next result shows that the above definition is consistent with the finite-dimensional version in Section 2.1. There the bounded real singular values were defined as the square roots of the eigenvalues of the product of the bounded real optimal cost operators. An analogous approach in the infinite-dimensional case is trickier because although the product of the optimal cost operators is bounded, it is not a priori clear why it should have (nonnegative, real) eigenvalues. However, we prove that when the bounded real singular values are summable (which is always true in the finite-dimensional case) then the definitions coincide.
Lemma 4.16. Let G ∈ H ∞ (C + 0 ; B(U , Y )) denote a strictly bounded real function and let P m and Q m denote the optimal cost operators of the optimal control problems from Lemma 3.8 and Corollary 3.9 respectively corresponding to a given stable L 2 well-posed realisation of G. Then the bounded real singular values are summable if and only if P m Q m is compact and the square roots of its eigenvalues are summable. If these conditions hold then apart from possibly zero the bounded real singular values are precisely the square roots of the eigenvalues of P m Q m (which therefore depend only on G).
Proof. For brevity we give an outline of the proof. For all the details see [11,Lemma 6.3.12]. Choose a stable L 2 well-posed realisation of G so that P m and Q m are given by (35) and (36) respectively. For some choice of spectral factors as in Lemma 3.12, let C E and B E denote the extended output operator and extended input operator from Lemmas 3.13 and 3.14 respectively. By those results it follows that P m = C * E C E and Q m = B E B * E . Let H E denote the Hankel operator given by (55), which is a member of the family of extended Hankel operators of G. By Definition 4.14 the bounded real singular values are the singular values of H E .
Combining the following two facts for bounded operators T, S : X → X on a Hilbert space • if T is compact then T S and ST are compact, • T is compact if and only is T * is , it is not difficult to prove that H E is compact if and only if P m Q m (equivalently Q m P m ) is. Using the fact that for λ = 0 λ ∈ σ(T S) ⇐⇒ λ ∈ σ(ST ), it follows that when H E is compact, H * E H E and Q m P m have the same nonzero eigenvalues and, arguing carefully with eigenvectors, we see that these eigenvalues have the same multiplicities. We recall that if H E is compact then its singular values are precisely the (nonnegative) square roots of the eigenvalues of H * E H E . These observations combined prove the first assertion. From Lemma 4.2 it follows that H E is independent of the stable L 2 wellposed realisation of G chosen, hence so are its singular values and thus when the bounded real singular values are summable, we see that the non-zero eigenvalues of P m Q m also depend only on G.
We very briefly recap some of the results from [12] on Lyapunov balanced truncation in the infinite-dimensional case.
A nuclear Hankel operator H : L 2 (R + ; Z 1 ) → L 2 (R + ; Z 2 ) (with Z 1 , Z 2 finite-dimensional) is necessarily given by an integral operator of the form with h ∈ L 1 (R + ; B(Z 1 , Z 2 )) satisfying h(t) := n∈N λ n (Re a n )e ant , t > 0, for sequences (λ n ) n∈N ∈ 1 (B(Z 1 , Z 2 )) and (a n ) n∈N ⊂ C − 0 . The series in (63) converges absolutely and uniformly on t > 0. This result follows from the decompositions of [4], and a proof can be found in [12, where W m,p denotes the usual Sobolev space. Recall that the Schmidt pairs of a (compact) operator T : L 2 (R + ; Z 1 ) → L 2 (R + ; Z 2 ) are the eigenvectors of T * T and T T * respectively, with respect to the eigenvalue σ 2 i . Here (σ i ) i∈N are the singular values of T . The Schmidt pairs of the operator H given by We recall the well-posed realisations and their generators that are a crucial ingredient for the Lyapunov balanced truncation of [12].   Z 2 )) and G ∈ H ∞ (C + 0 ; B(Z 1 , Z 2 )) denote a Hankel operator given by (62), its kernel and transfer function respectively. Then for 1 ≤ p < ∞ the shift realisation Σ sr p of G from Lemma 4.17 is an L p well-posed linear realisation of G and has generators A, B and C given by Here W −1,p (R + ; Z 2 ) is the dual space of W 1,p 0 (R + ; Z 2 ). When p = 1 the control operator B is bounded and is defined by Define the truncation space which by (64) is a subspace of L 1 , L 2 and W 1,1 (R + ; Z 2 ), and also the projection The operator P n is clearly linear and well-defined (bounded even) by the Hölder inequality E of G E and for n ∈ N, let X n and P n denote the space and projection from (69) and (70) respectively. Define the operators Here P Y is the orthogonal projection of Y U onto Y . We call the finitedimensional system on (Y , X n , U ) generated by n D the reduced order system obtained by bounded real balanced truncation (determined by is called the reduced order transfer function obtained from G by bounded real balanced truncation. Note that the bounded real balanced truncation depends on the choice of extended transfer function G E . The next lemma shows that different choices of G E give rise to bounded real balanced truncations of G that are unitarily equivalent to one another. In particular, they all give rise to the same reduced order transfer function in (72). Lemma 4.20. Let G ∈ H ∞ (C + 0 ; B(U , Y )) denote a strictly bounded real function with summable bounded real singular values. Then the bounded real balanced truncation is unique up to a unitary transformation, determined by the choice of extended transfer function G E . Every bounded real balanced truncation gives rise to the same reduced order transfer function obtained by bounded real balanced truncation, which is therefore independent of the above choice.
Proof. For the choice of spectral factors θ 0 and ξ 0 , let G 0 E denote the resulting member of the family of extended transfer functions of G. If the spectral factors θ and ξ are related to θ 0 and ξ 0 by (44) and G E is the corresponding extended transfer function, then G 0 E and G E are related by where U ∈ B(U ), V ∈ B(Y ) are unitary. The relation (73) readily follows from the version for the corresponding extended Hankel operators (56) and our definition of the feedthrough D E of G E in (60).
denote the generators of the exactly observable shift realisation on L 1 of G 0 E . It is readily seen that generate the exactly observable shift realisation on are Schmidt pairs for H E . Therefore for n ∈ N and in fact [ I 0 0 U ] : X 0 n → X n is an isomorphism. Furthermore if P 0 n denotes the projection of L 1 (R + ; Y U ) onto X 0 n , defined analogously to P n in (70) then With these observations we are able to describe how the bounded real balanced truncations of G 0 E and G E are related. By definition of the projections P n , P 0 n and the operators A E and A 0 E we see that for 1 ≤ i ≤ n and 1 ≤ k ≤ p i and The Schmidt pair relations (74) and the fact that [ I 0 0 U ] is unitary imply that the expressions in (76) and (77) are equal. Since this equality holds on a basis for X n we infer that Similarly, using the projection relation (75) yields which implies that As with A 0 E and A E , the operators C 0 E and C E are the same and we see that Finally, We see from (78) Having defined the bounded real balanced truncation, we now seek to prove Theorem 1.1. The next lemma describes some of the properties of the reduced order system obtained by bounded real balanced truncation, and is the key ingredient in proving Theorem 1.1.
) denote a strictly bounded real function with summable bounded real singular values and for n ∈ N let G n denote the reduced order transfer function obtained by bounded real balanced truncation. Then G n is rational, bounded real and for every choice of extended transfer function the resulting bounded real balanced truncation from Definition 4.19 is a stable realisation of G n .
Proof. By Lemma 4.20 the bounded real balanced truncations are all unitarily equivalent to one another. Since the stability of A of the realisation [ A B C D ] is invariant under unitary transformation, it does not matter which bounded real balanced truncation we pick. Therefore, for this proof we pick a member G E of the family of extended transfer functions arbitrarily and for notational convenience we denote the bounded real balanced truncation by That G n is rational is clear, as is a realisation on a finite-dimensional state-space. It was proven in [12, Proposition 6.12] that A n is stable (see also [12,Definition 5.5]).
It remains to see that G n is bounded real and for this we use the Bounded Real Lemma, Lemma 2.3. We seek a solution (P, K, W ), with P : X n → X n self-adjoint and positive, of the bounded real Lur'e equations (7) (subject to the realisation we claim that P := I : X n → X n , K := C U n : X n → U , W := D θ : U → U , solve (7) and we proceed to verify equations (7a), (7b) and (7c).
In [11,Lemma 5.3.16] (see also [12,Proposition 6.12]), it is proven that the Lyapunov equation holds as an operator equation on X n equipped with the L 2 inner product. Using (82) we can rewrite (83) as which is (7a). We now verify the second equation (7b), i.e. we demonstrate that Firstly, by applying Lemma 4.13 we obtain L 2 well-posed realisations of G and θ from the output normal realisation (Lemma 4.17) of G E . We denote the generators of these realisations of G and θ by (A, B, C, D) and (A, B, C θ , D θ ) respectively. Applying [37,Theorem 12.4] to these realisations (noting that and also shows that with domain consisting of the x ∈ X such that the above limit exists. If for the truncation space X n we have X n ⊂ W 1,2 (R + ; Y U ), then we can restrict the equality (86) to X n . Noting that the generators of the output normal realisation on L 2 and those of the exactly observable shift realisation on L 1 agree on the intersection of their domains, we thus obtain as an operator equation from X n to Y (both finite-dimensional), which when adjointed gives (85), as required.
In general we however do not have X n ⊂ W 1,2 (R + ; Y U ) so that a somewhat more involved argument is needed. The essential argument is as follows (a more detailed argument can be found in [11,Lemma 6.3.16.]). We first restrict (86) to the intersection D(A) ∩ W 1,2 (R + ; Y U ). We then use that the generators of the output normal realisation on L 2 and those of the exactly observable shift realisation on L 1 agree on the intersection of their domains to replace the operators in (86) by their L 1 well-posed equivalents. By continuity and density, that version of (86) in fact holds on W 1,1 (R + ; Y U ). We can then restrict to X n , which is always a subset of W 1,1 (R + ; Y U ), to obtain (87), which as above implies (85).
It remains to prove that the third equation (7c) of the bounded real Lur'e equations holds, i.e.
In Staffans [29, Corollary 7.2] (see also [37,Remark 12.9]) the following formula is given relating the feedthroughs of the original transfer function G and the spectral factor θ (both of which are regular by Lemma 4.11): That the limit on the right hand side of (89) is zero for strongly stable realisations of transfer functions with an impulse response in L 1 has been proven in the PhD thesis of Mikkola [17,Theorem 9.1.15]. Our realisation of G derived from Σ sr 2 E satisfies these hypotheses, thus establishing (88). An alternative proof that the limit on the right hand side of (89) is zero, using the Coifman-Rochberg decomposition, is also given in [11,Lemma 6.3.16.].
Therefore, we have proven that which states that (I Xn , C U n , D θ ) is a (self-adjoint, positive) solution of the bounded real Lur'e equations and hence G n is bounded real.
We now have all the ingredients to prove Theorem 1.1.
Proof of Theorem 1.1: From Lemma 4.21 we have that the reduced order transfer function obtained by bounded real balanced truncation G n is rational and bounded real. It remains to prove the error bound. By Lemma 4.9, every Hankel operator H E of an extended system Σ E with transfer function G E , is nuclear. So [12, Theorem 2.3] applied to G E yields where (G E ) n is the Lyapunov balanced truncation of G E (not the bounded real balanced truncation), and σ k are the Lyapunov singular values of G E and so are also the bounded real singular values of G, by Definition 4.14. By construction of G E in (59) we have that Moreover, by construction of the bounded real balanced truncation and Lyapunov balanced truncation (see (72) and [12, Definition 5.5] respectively) Together these yield Combining (90) and (91) gives the result.

The Cayley transform
As is well-known, bounded real and positive real systems are related by the Cayley transform (also known as the diagonal transform and as the Möbius transform). Here we collect the material we shall need in order to be able to convert bounded real balanced truncation to positive real balanced truncation.    (v)S : BR ∩ D(S) → P R is a bijection.
Proof. The proofs of these assertions are elementary and are not given here. See [11,Lemma 7.1.8.] for detailed proofs. The arguments used are very similar to those in, for example, Belevitch [3, p.160, p.189].
Corollary 5.6. Let G ∈ H ∞ (C + 0 ; B(U )) be strictly bounded real, where U is finite-dimensional. Then G is regular if and only ifS(G) is.
Proof. The proof is elementary, and can be found in [11,Corollary 7.1.10.].
The next result is contained within [30,Theorem 5.2], although the formulae (92) are not given there, and demonstrates that given a well-posed realisation of a strictly bounded real function G, we can obtain a well-posed realisation of (the strictly positive real function)S(G) with the same state.
is an L p well-posed realisation ofS(G) on (U , X , U ). Moreover the state trajectories of Σ G with input u and output y and ΣS (G) with input v and output w given by are the same.
Proof. See [30,Theorem 5.2]. As mentioned in the proof of that result, the relationship can be seen as (negative identity) static output feedback with external control v. The relationship corresponds to adding an extra feedthrough term. From these observations and the formulae for the closed loop (well-posed) linear system from [31, Theorem 7.1.2] the formulae in (92) follow.

Positive real balanced truncation
In this section we define the positive real balanced truncation of a strictly positive real function J ∈ H ∞ (C + 0 ; B(U )) with summable positive real singular values, and prove the gap metric error bound Theorem 1.2. To do so we make use of the material gathered in Section 5. The next result is crucial for linking positive real balanced truncation to bounded real balanced truncation. Lemma 6.1. Let Σ J denote a stable L 2 well-posed linear system with strictly positive real transfer function J ∈ H ∞ (C + 0 ; B(U )). LetP m andQ m denote the optimal cost operators of the positive real optimal control problems from Lemma 3.10 and Corollary 3.11 subject to Σ J respectively. Let ΣS (J) denote the L 2 well-posed realisation given by (92). Then the optimal cost operators of the bounded real optimal control problems from Lemma 3.8 and Corollary 3.
respectively. Let P m denote the optimal cost operator of the bounded real optimal control problem (32) subject to the realisation ΣS (J) . A long, but elementary, calculation using (94) shows that Remark 6.6. The result of Lemma 6.5 is an infinite-dimensional version of [19,Lemma 3]. We remark, however, that the transformation (15) in [19] is not the same transformation as (5.3). As such the formulae in (95) are slightly different to those in [19,Lemma 3]; namely there is a difference in signs. Definition 6.7. Let J ∈ H ∞ (C + 0 ; B(U )) denote a strictly positive real function with summable positive real singular values, and let G E denote a member of the family of extended transfer functions of G :=S(J). Let (Ã,B,C,D) denote the generators of the L 1 well-posed realisation of J from Lemma 6.5. We define the operatorsÃ n ,B n andC n bỹ A n := P XnÃ | Xn : X n → X n ,B n := P XnB : U → X n , where X n is the truncation space (69). The input-state-output system generated by Ã nBñ CnD is called the reduced order system obtained by positive real balanced truncation (determined by G E ). We call J n given by J n (s) :=C n (sI −Ã n ) −1B n +D, the reduced order transfer function obtained by positive real balanced truncation.
The next lemma demonstrates that the positive real balanced truncation is determined by J up to a unitary transformation and thus that the reduced order transfer function J n is uniquely determined by J. From the Positive Real Lemma it follows that J n is positive real and it is clearly rational since it has a realisation with finite-dimensional state-space. Therefore by Lemma 5.5 (ii), J n ∈ D(S) and another elementary calculation using (98) shows thatS(J n ) = G n . Therefore by Lemma 5.5 (v), G n ∈ D(S) andS(J n ) = G n . We note that the commutative diagram is well defined in the sense that it is independent of G E . Furthermore, the above observations have demonstrated that it does indeed commute.
We now gather the ingredients required to prove the gap metric error bound for positive real balanced truncation, which we formulated as Theorem 1.2. Lemma 6.9. The map F given by F : is an isometric isomorphism. WithŠ the Cayley transform of Remark 5.2 and D ∈ D(Š) we have where G(D) denotes the graph of D.
Proof. The simple proof is left to the reader.
We remind the reader of the definition of the gap metric, for closed subspaces of a Hilbert space and for closed operators, see also Kato [16,p. 197,p.201]. Definition 6.10. For M , N non-empty closed subspaces of a Hilbert space X , the gap is defined asδ where P M , P N are the orthogonal projections of X onto M and N respectively. For closed linear operators S, T : X → Z , where Z is a Hilbert space, the gap between S and T is defined aŝ where G(S) and G(T ) denote the graphs of S and T respectively.
The following elementary lemma shows that the gap metric is invariant under isometries. Proof. This is elementary and a proof can be found in [11,Lemma 7.2.11.], for example.
We now have all of the ingredients to prove the gap metric error bound Theorem 1.2.
Proof of Theorem 1.2: Since J is strictly positive real with summable bounded real singular values, the hypotheses of Lemma 6.5 are satisfied and thus the positive real balanced truncation J n of Definition 6.7 is well-defined. From Lemmas 5.5 and 6.3 the transfer function G :=S(J) is strictly bounded real with summable bounded real singular values. Therefore all the assumptions of Theorem 1.1 are satisfied and so the error bound (2) holds for G and its bounded real balanced truncation G n .

Asymptotic behavior of bounded real and positive real singular values
Our main results, Theorems 1.1 and 1.2, each have two key assumptions. We require that the transfer function is strictly bounded real (respectively strictly positive real) and has summable bounded real singular values (respectively summable positive real singular values). Here we provide a large class of examples where the latter condition is satisfied.
As we have seen in Lemma 4.16, the bounded real singular values are summable precisely when the Hankel singular values of a (equivalently every) member of the family of extended Hankel operators of G are summable. Therefore, we seek conditions which ensure that a Hankel operator is nuclear. The next result is taken from [21]. In what follows X α denote interpolation spaces, see for example, [31, Section 3.9] and S p is the Schatten class; the linear operators whose singular values form a sequence in p . In particular, S 1 is the class of nuclear operators.
Theorem 7.1. Assume that A generates an exponentially stable analytic semigroup, B ∈ B(U , X β ), C ∈ B(X α , Y ) and D ∈ B(U , Y ), with α − β < 1 and that at least one of U and Y is finite-dimensional. Then the Hankel operator of this system is in S p for all p > 0.
Given a stable L 2 well-posed realisation of the strictly bounded real function G with generators (A, B, C, D), and choice of spectral factors θ and ξ as in Lemma 3.12 it follows from Lemma 4.11 that (A, B E , C E , D E ) generate a stable L 2 well-posed realisation of the extended transfer function G E . Here are the generators of B E and C E from (50) and (45) respectively and D E is as in (60). It is not a priori clear how unbounded C E and B E are because it is not presently clear how unbounded the components C θ and B ξ are. However, under the assumption of strict bounded realness, we are able to formulate the next result which provides checkable conditions for the summability of the bounded real singular values.
Proposition 7.2. Assume that A generates an exponentially stable analytic semigroup on X , B ∈ B(U , X β ), C ∈ B(X α , Y ) and D ∈ B(U , Y ), with α − β < 1 and that both U and Y are finite-dimensional. Then (A, B, C, D) are the generators of a stable L 2 well-posed linear system. If the transfer function of this system is strictly bounded real, then the bounded real singular values belong to p for every p > 0. In particular, they are summable and moreover decay faster than any polynomial rate.
Proof. That (A, B, C, D) are the generators of a stable L 2 well-posed linear system follows from [31,Theorem 5.7.3]. In Staffans [27,Theorem 1] it is proven that under our assumptions the operator C θ from (86) is bounded X α → U .
Hence C E is bounded X α → Y U . Arguing analogously in the dual case we deduce that B ξ satisfies where we have identified (X * ) γ with X −γ as X is a Hilbert space.
We conclude that B E is bounded U Y → X β . Therefore from Theorem 7.1, the Hankel operator of the extended system belongs to S p and hence the bounded real singular values belong to p .
The next result is a corresponding version of the above for positive real systems. Corollary 7.3. Assume that A generates an exponentially stable analytic semigroup on X , B ∈ B(U , X β ), C ∈ B(X α , Y ) and D ∈ B(U , Y ), with α ∈ [0, 1] and α − β < 1 and that U is finite-dimensional. Then (A, B, C, D) are the generators of a stable L 2 well-posed linear system. If the transfer function of this system is strictly positive real, then the positive real singular values belong to p for every p > 0. In particular, they are summable and moreover decay faster than any polynomial rate.
Proof. Denote the transfer function associated to (A, B, C, D) by J. From Lemma 5.5, the function G :=S(J) is strictly bounded real and from Lemma 6.3 the bounded real singular values of G are the positive real singular values of J. We seek therefore to apply Proposition 7.2, and in order to do so we require a state-space realisation of G. As argued in the proof of Lemma 6.5, the Cayley transform of operators the operatorÃ (where −B(I + D) −1 C = ∆ in the notation of [5]) generates an analytic semigroup on X and the interpolation spaces X δ andX δ corresponding to A andÃ respectively are equal for all δ ∈ [α − 1, β + 1]. Thus B ∈ B(U , X β ) = B(U ,X β ), andC ∈ B(X α , U ) = B(X α , U ), since trivially α, β ∈ [α − 1, β + 1]. It remains to see thatÃ generates an exponentially stable semigroup. By the same results from [31] above we can "go back again", and recover the realisation for J from that of G, namely
The PDE (105) can be written in the form (5) (e.g. as in [21]), with A generating an analytic, exponentially stable contraction semigroup on X = L 2 (0, 1). Here C is the trace operator, which is bounded X α → C for all α > 1 4 . Furthermore, B = C * , and hence B is bounded C → X β for all β < − 1 4 . Finally, D = 1. Therefore, using Remark 7.4, we see that the conditions on the operators in Corollary 7.3 are satisfied and hence (105) has summable positive real singular values (belonging to p for all p > 0, in fact).
We have approximated the heat equation (105) using several standard numerical discretisation methods. Unfortunately, computing the distance in the gap metric between these discretisations and the infinite-dimensional system is intractable. Therefore we have used a piecewise linear finite element (FE) approximation with N = 50 degrees of freedom as a substitute for the infinitedimensional system. The relevant gap metric distances can then be computed using the gapmetric function in MATLAB. The log of the gap metric error versus the number of degrees of freedom in the numerical discretisation is plotted in Figure 1.
Computing the positive real balanced truncation of the infinite-dimensional system is also intractable. Therefore we again take the piecewise linear FE approximation with N = 50 degrees of freedom as a substitute for the infinitedimensional system and compute the positive real balanced truncation of this system. We note that this is the usual procedure for approximating balanced truncations of PDEs. Again, Figure 1 contains the log of the gap metric error between the positive real balanced truncation and the piecewise linear FE approximation with N = 50 versus the number of degrees of freedom in the positive real balanced truncation. It can be observed that positive real balanced truncation is vastly superior to the other numerical discretisation methods. Figure 1 also contains the gap metric error bound for the positive real balanced truncation based on the positive real singular values of the piecewise linear FE approximation with N = 50 degrees of freedom. It can be seen that for n ≥ 8 this error bound is in fact smaller than the error as computed by the gapmetric function in MATLAB. This is due to the inaccuracy of the gapmetric function in MATLAB which has a maximal tolerance of 10 −5 , which for n ≥ 8 is larger than the actual error. With this in mind, it is clear that for this example our gap metric error bound is tight and for n ≥ 8 it is in fact a better approximation of the actual error than the error computed by the gapmetric function in MATLAB.