THE GENERALISED SINGULAR PERTURBATION 1 APPROXIMATION FOR BOUNDED REAL AND POSITIVE 2 REAL CONTROL SYSTEMS

. The generalised singular perturbation approximation (GSPA) is considered as a model reduction scheme for bounded real and positive real lin- ear control systems. The GSPA is a state-space approach to truncation with the deﬁning property that the transfer function of the approximation interpo- lates the original transfer function at a prescribed point in the closed right half complex plane. Both familiar balanced truncation and singular perturbation approximation are known to be special cases of the GSPA, interpolating at inﬁnity and at zero, respectively. Suitably modiﬁed, we show that the GSPA preserves classical dissipativity properties of the truncations, and existing a priori error bounds for these balanced truncation schemes are satisﬁed as well.

between the transfer function G and its reduced order approximation G r . Here bound (1.2) is known to be achieved (that is, equality holds in (1.2)) for certain 15 single-input single-output (SISO) systems, see [29], and a lower bound in the multi-16 input multi-output (MIMO) case has recently been derived in [38]. For more infor- 17 mation on balanced truncation, the reader is referred to the survey paper [18] or the cation has led to numerous further developments, some of which we discuss further 20 below, as well as, for example, to infinite-dimensional systems: [8,14,16,24,35,49]. 21 In the frequency domain, balanced truncation for rational functions is a model 22 reduction scheme which yields a rational approximation with the property that it 23 interpolates the original function at infinity. Roughly, by applying the same method 24 to a rational function now with argument 1/s instead of s, another reduced order 25 rational transfer function is obtained, which now interpolates the original at zero. 26 Interpolating at zero is a frequency domain property of the so-called singular per-27 turbation approximation (SPA), in particular meaning that the steady-state gains 28 are equal. From a dynamical systems perspective, singular perturbation approxi- 29 mation decomposes the state variables into those with "fast" and "slow" dynamics, 30 and assumes that the "fast" variables are at equilibrium, meaning that differen-31 tial equations simplify to algebraic equations. For linear systems these algebraic 32 equations are easily solvable, which leads to a model with fewer differential equa- 33 tions, and hence fewer states. The mapping s to 1/s mentioned above is called the 34 reciprocal transformation and provides a relationship between SPA and balanced 35 truncation. This relationship was exploited in [30] to show that the singular per-36 turbation approximation of a balanced, minimal, linear system admits the same 37 H ∞ error bound (1.2), as well as retaining minimality and stability of the original. 38 To the best of our knowledge, the provenance of the reciprocal transformation in 39 systems and control theory is unclear, and it now forms part of the subjects' "folk- for ξ ∈ C, Re (ξ), Im (ξ), ξ and |ξ| denote its real part, imaginary part, complex 4 conjugate and modulus, respectively. We let C 0 denote the set of all complex num-5 bers with positive real part. For n ∈ N, R n and C n denote the familiar real and 6 complex n-dimensional Hilbert spaces, respectively, both equipped with the inner 7 product ·, · which induces the usual 2-norm · 2 . For m ∈ N, let R n×m and C n×m 8 denote the normed linear spaces of n × m matrices with real and complex entries, 9 respectively, both equipped with the operator norm, also denoted · 2 , induced 10 by the · 2 norm on R n or C n . The superscript * denotes the complex-conjugate 11 transpose (and, importantly, the adjoint with respect to the above inner product). For m, p ∈ N, the space of analytic functions C 0 → C p×m is denoted by H(C 0 , C p×m ). 17 The subset of functions which are additionally bounded with respect to the norm is denoted by H ∞ (C 0 , C p×m ).

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2. The generalised singular perturbation approximation. We gather ele-20 mentary and notational preliminaries before recalling the generalised singular per- 21 turbation approximation and describing some properties. 22 We consider the linear control system (1.1) where, as usual, u, x and y denote the 23 input, state and output and 24 (A, B, C, D) ∈ C n×n × C n×m × C p×n × C p×m , for some m, n, p ∈ N 1 . In practice, the quadruple (A, B, C, D) is real-valued and 25 in many situations, the matrix D does not play a role. As such, we use the triple 26 (A, B, C) when the choice of D, which need not be zero, is unimportant. 27 The triple (A, B, C) is said to be stable if A is Hurwitz, that is, every eigenvalue  38 1 The material which follows holds if we assume that A : X → X , B : U → X , C : X → Y and D : U → Y are bounded linear operators between finite-dimensional complex Hilbert spaces U , X and Y which, of course, is equivalent to our formulation once bases are chosen for U , X and Y. transfer function is the dimension of a minimal state-space realisation, see [43, 1 Remark 6.7.4, p.299].

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Recall that the stable triple (A, B, C) is called (internally or Lyapunov) balanced 3 if there exists a Σ such that 4 AΣ + ΣA * + BB * = 0 and A * Σ + ΣA + C * C = 0 . (2.2) If Σ satisfies (2.2), then necessarily Σ equals both the controllability and observ-5 ability Gramians of the linear system specified by (A, B, C), that is, (hence the terminology balanced) and is consequently self-adjoint and positive semi- of (A, B, C), which we shall assume throughout the paper are simple (that is, each 13 has algebraic and geometric multiplicity equal to one). As singular values, the σ j 14 are ordered so that In practical applications, a basis of the state-space is chosen so that Σ is a diagonal 16 matrix, with the terms σ j on the diagonal. 17 Singular perturbation approximations are defined in terms of conformal partitions 18 of (A, B, C), denoted by where A 11 ∈ R r×r , B 1 ∈ R r×m , C 1 ∈ R p×r and so on, for some r ∈ n − 1. Of course, for some r ∈ n − 1 and ξ ∈ C, Re(ξ) ≥ 0 assume that ξ ∈ σ(A 22 ). The quadruple is called the generalised singular perturbation approximation of (1.1).

CHRIS GUIVER
In light of the ordering (2.3), Σ 1 and Σ 2 contain the larger and smaller eigenvalues 1 of Σ, respectively. perturbation approximation is that the generalised singular perturbation approximation is well-defined. Furthermore,

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Under the assumption that A is Hurwitz, we would of course expect x 2 (t) → 0 25 as t → ∞ in the absence of control, that is, when u = 0. ♦

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We recall two results which shall play a key role in constructing the dissipativity 27 preserving GSPA in Sections 3 and 4. no proof is given. Again for completeness, a proof is provided in the Appendix. references the reader is referred to, for example, [45,46]. The term 'real' in bounded 20 real refers to the sometimes-made assumption that G is real on the real axis. It is 21 true that many physically motivated systems enjoy such a property, but we do not (with variable Z), for some K ∈ C m×n and W ∈ C m×m , which are extremal in the 38 sense that any other positive semi-definite solution P of (3.1) satisfies P m ≤ P ≤ (also with variable Z) for some L ∈ C n×p and X ∈ C p×p . We say that the realisation If G H ∞ < 1, then G ξ r may be chosen with the above properties and, additionally, 33 to have McMillan degree r and G ξ r H ∞ < 1.

THE GENERALISED SINGULAR PERTURBATION APPROXIMATION 9
The next result pertains to existence and approximation of so-called spectral factors, 1 and spectral "sub"-factors, particularly of reduced order transfer functions obtained 2 by bounded real GSPA. Here H * denotes s → (H(s)) * for matrix-valued rational 3 functions H of a complex variable.

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Proposition 3.4. Imposing the notation and assumptions of Theorem 3.3, the 5 following statements hold.
The spectral factors R ξ r and S ξ r have state-space realisations with the same 11 dimension as those for G ξ r and may be chosen with the interpolation property  "square" systems, meaning the input and output spaces have the same dimension, 20 m = p, and that a rational, C m×m -valued function G is said to be positive real if 21 Re where ∆ is the set of poles of G. The assumption that G is rational implies that G Lur'e equations (with variable Z), for some K ∈ C m×n and W ∈ C m×m , which are extremal in the 24 sense that any other positive semi-definite solution P of (4.2) satisfies P m ≤ P ≤ (also with variable Z) for some L ∈ C n×m and X ∈ C m×m . We say that (A, B, C, D) Adopting the nomenclature convention used in [20], we say that the rational, approximation of order r ∈ n − 1, is well-defined and the following statements hold.
holds. Finally, if G is strongly positive real, then G ξ r as above may be chosen to 19 have McMillan degree r and be strongly positive real as well. 20 In certain cases, the error bound (4.5) may be used to derive a more conservative 21 (that is, worse), but a priori, bound. The reader is referred to [19, Remark 3.6.11] 22 for more details. (i) There exists a proper, rational, C m×m -valued function R such that The functions R and R ξ r may be chosen with the property that R(ξ) = R ξ r (ξ) 3 and, further, R ξ r and G ξ r have state-space realisations with the same dimen-4 sion.

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If G ∈ H ∞ , then R and R ξ r may be chosen to belong to H ∞ as well. In this case it follows that 5. Examples.
against real s > 0 for several ξ j > 0, for the cases r = 1 and r = 2, respectively. Here 13 R is a spectral factor for I −G * G and R ξ r is a sub-spectral factor for I −(G  Example 5.2. The paper [38, Section V] considers model reduction of RC ladder 1 circuit arrangements. The first circuit in that paper, which we consider here, has 2 two current sources which gives rise to MIMO control system with the state-space    In order to prove Theorems 3.2 and 3.3, we draw on the material presented in 12 Section 2, and also require three technical lemmas, stated and proven first.
hold for some K, P ∈ C m×n and Q, W ∈ C m×m , then is a realisation of a spectral factor S.

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Proof of Lemma 6.1: To prove statement (i), let x 0 ∈ C n , u be a continuous control and x = x(·; u, x 0 ) the corresponding differentiable state. From (6.1) we have that Integrating both sides of (6.3) between 0 and t ≥ 0 gives By a continuity and density argument, the inequality (6.4) holds for all u ∈ L 2 with 8 corresponding continuous state x. With zero initial state x 0 = 0, it follows that 9 the input u and output y satisfy y L 2 ≤ u L 2 , and hence (A, B, C, D) is bounded 10 real.

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Statement (ii) follows from an elementary calculation using the equalities in (6.1). Indeed, let s ∈ iR and consider Statement (iii) is proven similarly, only instead using the equalities in (6.2). The 1 details are omitted. property.

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Evidently, if ξ ∈ iR, then the resulting simplification of (6.14) and (6.15) im- 11 plies that (A ξ , B ξ , C ξ , D ξ ) is bounded real balanced, completing the proof of state-12 ment (i). 13 We proceed to prove statements (ii) and (iii), treating the cases ξ ∈ C 0 and ξ ∈ iR 14 separately. Assume that ξ ∈ C 0 . The first equation in (6.14) implies that every 15 eigenvalue of A ξ has non-positive real part. Suppose that A ξ v = ηi v for some η ∈ R 16 and v ∈ C r . Forming the inner product 17 (A * ξ Σ 1 + Σ 1 A ξ + C * ξ C ξ )v, v , and using (6.14), it follows that and, as A is Hurwitz, we deduce that v = 0. Recalling our supposition that A ξ v = ηi v, we conclude that A ξ is Hurwitz as well.

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For ξ ∈ C 0 , and for statement (iii), we shall require (A, B, C, D) defined in (6.5), The first equations in (6.17) and (6.18) may respectively be rewritten as If ξ ∈ iR, then a consequence of the simplification of (6.21) and (6.20) is that Hurwitz, again invoking the assumption that the singular values are simple implies 8 that the spectra of Σ 1 and Σ 2 are disjoint. Statement (2) of Lemma 6.2 implies 9 that ξ ∈ σ(A ξ ) and that (6.6) holds, from which it is routine to verify that A ξ is 10 Hurwitz, since A 11 is, and ξ ∈ iR. The proof of statement (ii) is complete.
Invoking (6.10), we now see that implying that G is not strictly bounded real. The above proof is easily altered by 6 taking p 0 = 0 in the case that as G ξ r is continuous at infinity. 8 It remains to consider ξ ∈ iR. We first establish that (A 11 , B 1 , −C 1 , D) is strictly 9 bounded real. For which purpose, the inequality (6.23) implies that D 2 < 1, and follows from the Bounded Real Lemma and by construction that Σ 1 and Σ −1 1 are 12 solutions of the bounded real algebraic Riccati equation . In light of [50, Theorem 13.19], it suffices to prove that A E is Hurwitz, that is, that 16 Σ 1 is a stabilizing solution of (6.30). Elementary manipulation of (6.30) for both 17 Z = Σ 1 and Z = Σ −1 1 shows that Subtracting (6.31) from (6.32) gives 2 A * E Π + ΠA E + ΠB 1 R −1 B * 1 Π = 0 , from which we see that every eigenvalue of A E has non-positive real part. Now 3 suppose that v ∈ C r and ω ∈ R are such that A E v = iωv. Forming the inner Finally, noting that (A E , B 1 ) is controllable, as (A 11 , B 1 ) is, we conclude from (6.33) is strictly bounded real. Finally, invoking (6.11) and that ξ ∈ iR, we estimate that whence (A ξ , B ξ , C ξ , D ξ ) is strictly bounded real. for all ξ ∈ C 0 ∪ iR by Theorem 3.2. By construction, the realisation is the GSPA of that in (6.35), where K ξ , L ξ , W ξ and X ξ are given by (6.16).

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Letting J ξ r denote the transfer function of (6.36) and invoking Theorem 2.4 yields For statement (ii), let ξ ∈ iR, and let R ξ r ∈ H ∞ (C 0 , C m×m ) and S ξ r ∈ H ∞ (C 0 , C p×p ) 7 be defined by the realisations respectively, where K ξ , L ξ , W ξ and X ξ are given by (6.16). Appealing to (6.14), 9 (6.15), and invoking statements (ii) and (iii) of Lemma 6.1, it follows that R ξ r and 10 S ξ r are spectral factors of G ξ r in the sense of (3.4), as required. By their definitions 11 in.

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The error bound (3.5) follows by combining (6.37) with the identity (which follows by construction) where ♯ denotes an entry we are not concerned with.

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A straightforward calculation shows that establishing the first inequality in (3.8). The dual case is proven similarly, us-24 ing (6.15), and invoking statement (iii) of Lemma 6.1 with V ξ r ∈ H ∞ (C 0 , C p×2p ) 25 defined by the realisation 6.2. The positive real generalised singular perturbation approximation.

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The proof of the next lemma is very similar to that of Lemma 6.1, and is thus 29 omitted. We have also omitted the corresponding statements pertaining to the dual 30 positive real equations as, although they do hold, we shall not require them.

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Lemma 6.4. If (A, B, C, D) with transfer function G and Σ ≥ 0 are such that 1 A * Σ + ΣA = −K * K − P * P , for some appropriately sized K, P , Q and W , then the following statements hold.
where ∆ denotes the set of poles of G. 6 We shall employ the so-called Cayley Transform S : Here D(S) contains all G ∈ H(C 0 , C m×m ) where the above formula makes sense (at where φ = (ξI − A 22 ) −1 and K ξ , W ξ , L ξ , X ξ are given by (6.16).

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In light of (6.41), an application of statement (i) of Lemma 6.4 yields that (A ξ , B ξ , C ξ , D ξ ) 5 is positive real. Evidently, if ξ ∈ iR, then the resulting simplification of (6.41) 6 and (6.42) implies that (A ξ , B ξ , C ξ , D ξ ) is positive real balanced, completing the 7 proof of statement (i). 8 The proof that A ξ is Hurwitz when ξ ∈ C 0 is the same as that in the proof of 9 Theorem 3.2, only using the first equation in (6.41), instead of (6.14). The details 10 are therefore omitted. where K, W, L and X are given by (6.19).

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When ξ ∈ iR, then a consequence of the first equations in (6.43) and (6.44) is implies that the spectra of Σ 1 and Σ 2 are disjoint. Statement (2) of Lemma 6.2 20 yields that ξ ∈ σ(A ξ ). Consequently, A ξ − ξI is invertible, and thus from (6.6) we 21 see that A 11 = (A ξ − ξI) −1 . It is now routine to verify that A ξ is Hurwitz, since 22 A 11 is, and ξ ∈ iR. We have proven statement (ii).
which are well-defined by assumption (A.3). 7 Block wise inspection of the two inequalities (A.1) and (A.2) yields the relationships: , and ) . An elementary sequence of calculations, using the definitions of A s , B s and C s and 8 the above inequalities, gives Since s ∈ σ(A) (indeed, σ(A) ⊆ E ξ ), it follows from (A.10) that z = 0 and thus 11 v = 0, proving that s ∈ σ(A s ).

CHRIS GUIVER
Moreover, since C s v 2 ≥ 0 for all v ∈ C n−r , by considering any eigenvalue λ of A s 1 with corresponding eigenvector v and the inequality where we have used that s ∈ σ(A s ), and so  For notational convenience in the following arguments set ζ = Re(ξ) > 0. Rearranging (A.8) yields that where p := 1 + 2ζs and we have used (6.9). Similarly, from (A.9), we see that where again we have used (6.9). Combining (A.13) and (A.14) gives = λ m (−2ζ∆Σ 2 ∆ * + pΣ 2 ∆ * + p∆Σ 2 )(−2ζΣ 2 + p∆ − * Σ 2 + pΣ 2 ∆ −1 ) . Now assume that just one singular value is omitted in the reduced order system, so that Σ 2 = σ n I. Invoking the assumption that the singular values are simple, it follows that the reduced order system has a scalar state. Then where we have used that |p| = |p| = 1 and that ∆ and ∆ * = ∆ are scalar quantities. 1 We investigate the second term in (A.15) and estimate that by geometric considerations and in light of (A.11). Thus the second term in (A.15) is non-positive, and so , which are of the form (A.1) and (A.2), respectively. 12 We now use a telescoping series and the triangle inequality to show that which is (A.4), as required. 1 The proof of (A.5) in the case that ξ ∈ iR follows via the same argument used is well-defined for all ξ ∈ C 0 ∪ iR. Suppose first that ξ ∈ C 0 . Straightforward 7 algebraic manipulation using the definition of (A ξ , B ξ , C ξ , D ξ ) in (2.5), the decom-8 position (2.6) and the equations (2.2) shows that the following Lyapunov inequalities If ξ ∈ iR, then it follows immediately from inspection of (A.16) and (A.17) 12 that (A ξ , B ξ , C ξ ) is balanced, proving statement (ii). 13 We prove statement (i) first assuming that ξ ∈ C 0 . Inequality (A.17) implies that 14 every eigenvalue of A ξ has non-positive real part. Suppose that A ξ v = ηi v for some 15 η ∈ R and v ∈ C r . Forming the inner product , v , and using (A.17), it follows that as Re (ξ) > 0. Since Σ 2 > 0, we infer that and, as A is Hurwitz, we deduce that v = 0. Recalling our supposition that A ξ v = 21 ηi v, we conclude that A ξ is Hurwitz as well.

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For observability, let λ ∈ C and v ∈ C n be such that A ξ v = λv and C ξ v = 0. Note that The error bound (2.9) now follows from subtracting (6.11) from (6.10) in Lemma 6.3 from (6.6), as every partition in (2.6) gives rise to a Hurwitz A ξ , by Theorem 2.3.

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If ξ ∈ iR, then the result follows from the error bound (A.5), also in Lemma A.1.

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Here we have applied statement (3) of Lemma 6.2 to the first equality in (6.6) to 5 infer that A 11 is Hurwitz.