L 2 -optimal model reduction for unstable systems using iterative rational

Transcription

1 L 2 -optimal model reduction for unstable systems using iterative rational Krylov algorithm Caleb agruder, Christopher Beattie, Serkan Gugercin Abstract Unstable dynamical systems can be viewed from a variety of perspectives We discuss the potential of an inputoutput map associated with an unstable system to represent a bounded map from L 2(R) to itself then develop criteria for optimal reduced order approximations to the original (unstable) system with respect to an L 2-induced Hilbert-Schmidt norm Our optimality criteria extend the eier-luenberger interpolation conditions for optimal H 2 approximation of stable dynamical systems Based on this interpolation framework, we describe an iteratively corrected rational Krylov algorithm for L 2 model reduction A numerical example involving a hardto-approximate full-order model illustrates the effectiveness of the proposed approach I INTRODUCTION We consider single input/single output (SISO) linear dynamical systems described via a state space representation, ẋ(t) = Ax(t) bu(t) H : y(t) = c T () x(t), where A R n n, b, c R n We consider input functions u(t) defined on the entire real line with u(t) 2 dt < To avoid trivialities, we assume, without loss of generality to our results, that the representation in () is a minimal realization of H We have particular interest in cases where the system order, n, is very large that the system may be unstable, ie, A may have some eigenvalues with positive real parts Our goal will be to find, for any order r n, a reduced-order model represented as ẋr (t) = A r x r (t) b r u(t) H r : y r (t) = c T (2) r x r (t), that optimally approximates the full order system () with respect to an error measure described in Section II-C that exten in a natural way the usual H 2 system norm ost model reduction methodologies, such as balanced truncation [], [], Hankel norm approximation [7], or optimal H 2 approximation [4], were developed originally for asymptotically stable dynamical systems systems having all their poles in the left half-plane However, there exist prominent applications where model reduction of unstable systems becomes a vital tool One such case is controller reduction Controllers are designed to drive a plant into desirable robust performance settings, see [5] However, many controller design techniques, such as LQG H This work was supported in part by NSF under DS C agruder, C Beattie, S Gugercin are with the Department of athematics, Virginia Tech, Blacksburg, VA, , USA calebm, beattie, gugercin}@vtedu metho, lead to controllers that have the same order as the plant to be controlled; see, for example, [3], [5] references therein for more details Large order plants lead to large order controllers High-order controllers are problematic for real-time applications due to the potential for degraded numerical accuracy, computational lags that may be difficult to compensate for, the complex supporting hardware that becomes necessary Hence, one would like to replace the original high-order controller with a low order but high-fidelity approximation Since controllers are usually unstable systems [2], the controller reduction problem lea directly to a model reduction problem involving unstable systems We refer the reader to [4], [5], [2], [5], [7] for some recent works on model reduction of unstable dynamical systems We note that most of these works are based on balanced truncation, unlike the framework we pursue here which uses rational Krylov metho an interpolatory framework for model reduction A L 2 Systems II BACKGROUND Denote by L 2 (R) the set of functions with finite energy : L 2 (R) = f } f(t) 2 dt < by L n 2 (R) the corresponding set of vector-valued functions on R: } L n 2 (R) = x(t) R n x(t) 2 dt < Define an operator A : L n 2 (R) L n 2 (R) as A x = ẋ Ax defined on all vector-valued functions x(t) L n 2 (R) having components that are absolutely continuous, with derivative ẋ L n 2 (R) as well A is densely defined in L n 2 (R) is a - map of its domain onto L n 2 (R) as long as the matrix A has no imaginary eigenvalues If A has imaginary eigenvalues then there will be f L n 2 (R) such that Ax = f has no solution in L n 2 (R) Indeed, if A has purely imaginary eigenvalues at ±ıω Az = ıω z then f(t) = cos(ω t) t 2 Re(z ) sin(ω t) t 2 Im(z ) L n 2 (R) but every solution to Ax = f grows asymptotically like x(t) arcsinh(t) which grows logarithmically at ± ; thus, there can be no solution in L n 2 (R) Conversely, if A has no imaginary eigenvalues, then for every f L n 2 (R) we have an explicit representation of the unique x L n 2 (R) that solves Ax = f In the context of the dynamical system given in (), the resulting input-output

2 map H : u y then appears as a convolution operator y(t) = [c T A b u](t) = h(t τ) u(τ) dτ : if A is stable, ie, all eigenvalues of A have strictly negative real part, then variation of parameters gives immediately, [A f](t) = x(t) = t ea(t τ) f(τ) dτ, y(t) =[c T A b u](t) = h(t) = c T e At b for t for t < h(t τ) u(τ) dτ if A is antistable, ie, eigenvalues have positive real parts, then [A f](t) = x(t) = e A(t τ) f(τ) dτ, t y(t) =[c T A b u](t) = h(t) = h(t τ) u(τ) dτ for t c T e At b for t < Note that in (3) the value of y(t) is independent of future values of u; the corresponding h(t) is causal For (4), the value of y(t) is independent of past values of u; the corresponding h(t) is anticausal The case where A is a general unstable matrix without purely imaginary eigenvalues, ie, some eigenvalues have strictly positive real parts the remaining have strictly negative parts, is only slightly more involved Let U U be maximal invariant subspaces of A associated with stable antistable eigenvalues, respectively This means that if X X are matrices having columns that constitute bases for U U : U = Ran(X ) U = Ran(X ), then dim(u ) dim(u ) = n, the n n matrix X = [X X ] is invertible, there are square matrices such that [ ] A[X X ] = [X X ] (5) with stable antistable (meaning that is stable) Let Y = (X ) T be partitioned as Y = [Y Y ] to conform with the partitioning of X Define the stable antistable spectral projectors for A, respectively, as Π = X (Y ) T Π = X (Y ) T To determine L n 2 (R) solutions to Ax = f, we use stable antistable spectral projectors to separate () into stable antistable subsystems with x = Π x x = Π x: ẋ (t) = Ax (t) Π bu(t) H : ẋ (t) = Ax (t) Π bu(t) y(t) = c T ( x (t) x (t) ) (6), An explicit representation of L n 2 (R) solutions to Ax = f can be found via variation of parameters as before for each of the subsystems So, finally (3) (4) if A is unstable, then for any f L n 2 (R), calculate [A f](t) = x(t) = Π x(t) Π x(t) = t e A(t τ) Π f(τ) dτ t e A(t τ) Π f(τ) dτ The system map H : u y appears explicitly as y(t) = [c T A b u](t) = with h(t) = h(t τ) u(τ) dτ c T e At Π b for t c T e At Π b for t < H maps functions u L 2 (R) to y L 2 (R), as before However, values of y(t) now depend on both past future values of u; h(t) is noncausal Notice that (7) reduces to (3) in the stable (causal) case to (4) in the antistable (anticausal) case B Frequency Domain Representation Laplace transforms are the usual approach to obtaining a frequency domain representation of systems having the form of () Although the usual (unilateral) Laplace transform exten in a natural way to the bilateral Laplace transform: L[u](s) = u(t) e st dt, L[u](s) will not exist for Re(s) unless u(t) has a sufficiently rapid decay at either ± ; u L 2 (R) does not by itself imply that L[u](s) exists Instead, let ŷ(ω) û(ω) denote Fourier transforms of y(t) u(t), respectively Applying a Fourier transform to (7) yiel after some manipulation: ŷ(ω) = c T (ıωi A) Π b û(ω) c T (ıωi A) Π b û(ω) = c T (ıωi A) b û(ω) = H (ıω) û(ω) H (ıω) û(ω) = H(ıω) û(ω), introducing transfer functions H (s) = c T (si A) Π b, H (s) = c T (si A) Π b, H(s) = c T (si A) b Notice that the total transfer function, H(s), splits naturally into the sum of a stable transfer function, H (s), an antistable transfer function, H (s) corresponding to the splitting of A into stable antistable components Notice also [ that the evident ] realizations [ for H ] H, A Π ie, H := b A Π c T H := b c T, are nonminimal both systems have the same apparent order as H itself inimal realizations are immediate from the block decomposition in (5):» H (s) = c T X (si ) Y T b b := c T» (8) H (s) = c T X (si ) Y T b b := c T where we use c ±T = c T X ± b ± = (Y ± ) T b (7)

3 C The L 2 (ir) norm Definition 2: Let L 2 (ir) denote the set of meromorphic functions, G(s) such that G(ıω) 2 dω < L 2 (ir) is a Hilbert space is relevant here because transfer functions of the systems we consider are elements of L 2 (ir) If G(s) H(s) are elements of L 2 (ir) that are real-valued on R (ie, if they represent real dynamical systems), then their inner product is defined as G, H L2 = = G(ıω)H(ıω)dω G( ıω)h(ıω)dω (9) the L 2 (ir)-norm of H is ( /2 H L2 = H(ıω) dω) 2 () Definition 22: Let H 2 (C ) denote the set of functions, G(s) that are analytic for s in the open right half plane, C = s Re(s) > }, such that sup x> G(x ıy) 2 dy < () Similarly, let H 2 (C ) denote the set of functions analytic in the open left half plane, C = s Re(s) < }, such that sup x< G(x ıy) 2 dy < (2) The following well-known result describes an orthogonal direct sum decomposition of L 2 (ir) Theorem 23: L 2 (ır) is an orthogonal direct sum of H 2 (C ) H 2 (C ): L 2 (ır) = H 2 (C ) H 2 (C ) Proof: Let H L 2 (ir) suppose that H has poles in each of the left right half-planes of C There exist H H 2 (C ) H H 2 (C ) so that H = H H Notice that the product H ( s)h (s) is analytic in the left half plane For any R >, define a semicircular contour in the left half-plane: Γ R = z z = iω with ω [ R, R]} [ π z z = Re iθ with θ 2, 3π ]} 2 Using stard arguments the Cauchy-Goursat theorem, H, H L2 = H ( iω)h (iω)dω = lim H ( s)h (s) = R 2πı Γ R Corollary 24: Given L 2 systems H, H r as in () (2) H H r 2 L 2 = H H r 2 H 2(C ) H H r 2 H 2(C ) Thus the quality of the approximation, H r H is determined by how well the stable antistable components of H r approximate the corresponding components of H Using the residue theorem, we obtain a form of the L 2 inner product that can evaluated as a finite sum Denote by res[f (s), µ] the residue of F at µ Theorem 25: Let G(s), H(s) be in L 2 (R) Suppose H(s) has poles at µ, µ 2,, µ m labeled so that the first k poles are stable, µ,, µ k } C, the last m k poles are antistable µ k,, µ m } C Decompose G H into stable antistable parts G = G G H = H H Then, G, H L2 = Proof: k G ( µ i ) res[h (s), µ i ] (3) i= m l=k G ( µ l ) res[h (s), µ l ] G, H L2 = G G, H H L2 = G, H L2 G, H L2 k = res[g ( s)h (s), µ i ] i= m l=k res[g ( s)h (s), µ l ] The last equality is an application of the residue theorem The conclusion then follows from the observation that G ( s) is analytic in the left half-plane that G ( s) is analytic in the right half-plane See also Lemma 24 of [8] III OPTIAL L 2 ODEL REDUCTION Given an L 2 system, H as in (), we consider reduced order systems, H r of order r as in (2), which are best approximations to H with respect to the L 2 norm: H H r L2 = min H r L2 (4) dim( r)=r For stable systems, the L 2 minimization problem (4) reduces to H 2 minimization For the analysis of that special case, see [8], [4], [9] the references therein Note that the model reduction problem we consider here is different from the finite-horizon model reduction approaches [4], [7] used for unstable dynamical systems where the fullorder model H remains causal but as a consequence is not a mapping from L 2 (R) to L 2 (R) Since the output y(t) can grow without bound, only a finite-horizon time window can be considered The set of transfer functions associated with rth-order dynamical systems is not convex, so the optimal approximation problem (4) allows for multiple minimizers Indeed there may be local minimizers that do not solve (4) Definition 3: A reduced order system, H r, is a local minimizer for (4) if, for all ɛ > sufficiently small, H H r L2 H r for all dynamical systems with dim( H r r L2 Cɛ for some constant C r L2 (5) r ) = r

4 Theorem 32: Suppose H L 2 H r is a local L 2 - minimizer to H in the sense of (5) Suppose further that H r has simple poles: λ,, λ r } Then H H r, s λ L2 = l H H r, (6) (s λ l ) 2 L 2 = The proof follows closely the pattern of proof described in Theorem 5 of [3] (for stable systems), is only summarized here Proof: The definition (5) lea to H H r 2 L 2 H r 2 L 2 H H r H r So, H H r, H r By choosing H r r H H r 2 L 2 H r 2 H H r, H r r 2 L 2 r L2 2 H r r 2 L 2 r L2 r 2 L 2 appropriately one can obtain (6): pick r = ±ɛ s λ ±ɛ res[h r, H r r = λ l ] l (s λ l )(s λ l ± ɛ) ; then let ɛ A Interpolation-based optimality conditions We now offer new interpolatory L 2 optimality conditions that extend the interpolatory optimal H 2 conditions [9], [8] from stable dynamical system settings to unstable ones Theorem 33: Given an L 2 -system, H(s), as described in (), let H r (s) be a local minimizer of dimension r for the optimal L 2 model reduction problem (4) Suppose further that H r (s) has simple poles, λ i } r, ordered in such a way that the first k poles are stable the last r k poles are antistable: λ,, λ k } C λ k,, λ r } C Then H r ( λ i) = H ( λ i) H r ( λ j) = H ( λ j) for i =,, k d H r d H r = d H s= λi s= λj = d H s= λi (7) s= λj for j = k,, r Proof: Evidently, res[, λ if j = l s λ j ] = l otherwise Pick an index l k From (3) (6), we find = H H r, s λ L2 l kx = H ( λ j) H r ( λ j) j= rx j=k H ( λ j) Hr ( λ j) = H ( λ l ) H r ( λ l ) res[ (s λ l ), λ j] res[ (s λ l ), λ j] Next, note that H ( s) H r ( s) is analytic at s = λ l : H ( s) H r ( s) (s λ l ) 2 = H ( λ l ) H r ( λ l ) (s λ l ) 2 so we also have H ( λ l ) H r ( λ l ) s λ l = H H r, (s λ l ) 2 L 2 k = res[ H ( s) H r ( s) (s λ, λ j ] l ) 2 j= = H ( λ l ) H r ( λ l ) Similar arguments hold for k l r IV ITERATED INTERPOLATION A The Interpolation Problem Consider the system H described by A, b, c as in (), with associated stable antistable quantities ±, b ±, c ± as described in (8) The interpolatory model reduction problem involves finding a system (2) so that H r (s) interpolates H(s) (perhaps also derivative values), at selected interpolation points that are designated by shifts, σ i } r i= The conditions for optimal L 2 approximation described in (7) also involve an additional feature: Hermite interpolation is necessary for both stable antistable subsystems Toward this end, suppose two sets of distinct shifts are given: σ i } k i= C σ i } r i=k C, that are each closed under conjugation (ie so that shifts within each set are either real or occur in conjugate pairs) We wish to find a reduced order system H r (s) with stable antistable components, H r (s) H r (s), respectively, so that H r (σ i ) = H (σ i ) H r (σ j ) = H (σ j ) Notice that for i =,, k for j = k,, r d H r d H r = d H s=σi s=σj = d H H ± (s) = c T (si A) Π ± b = c ±T (si ± ) b ± d H ± s=σi (8) s=σj = c T (si A) 2 Π ± b = c ±T (si ± ) 2 b ±, so we may form reduced order interpolants to the stable antistable components H ± (s) independently based on the stable/antistable components of A

5 Define matrices V k, W k, V r k W r k as V k = ˆ(σ I ) b,, (σ k I ) b R 2 c T (σ I ) 3 W T k = S T 6 4 c T (σ k I ) 7 5 V r k = ˆ(σ k I ) b,, (σ ri ) b R 2 c T (σ k I ) 3 W T r k = S T 6 4 c T (σ ri ) 7 5 (9) S ± R ± represent (invertible) change-of-bases matrices Since the shifts are distinct closed under conjugation, W T k V k W T r k V r k are invertible S ± R ± can be chosen so that V k, W k, V r k W r k are real matrices, W T k V k = I k, W T r k V r k = I r k Corollary 4: Suppose distinct shifts σ i } r i= are given as described above suppose real matrices V k, W k, V r k W r k are computed as described in (9) Define H r (s) = H r (s) Hr (s) with H r (s) =c T V k (si k ) W k b H r (s) = c T V r k (si r k ) W r k b where k = WT k V k r k = W T r k V r k Then H r satisfies the interpolation conditions in (8) The proof is omitted but follows directly from the related interpolation properties true for rational Krylov subspaces used to reduce stable systems B Proposed Algorithm (L2-IRKA) The L 2 optimality conditions (7) reveal that H r Hr are Hermite interpolants to H H at mirror images of the poles of H r Hr, respectively Hence, as in the case of the H 2 problem, the optimal interpolation points depend on a reduced system yet to be computed are not known a priori The strategy we propose iteratively corrects the interpolation points until the necessary conditions are met The resulting Algorithm 42 outlined here is inspired by the Iterative Rational Krylov Algorithm (IRKA) of [8] Algorithm 42: ITERATIVE RATIONAL KRYLOV ALGORITH FOR L 2-OPTIAL ODEL REDUCTION (L2-IRKA) [ ] A b ) Decompose the full order system H := c T into minimal [ stable ] antistable subsystems: [ ] H := b c T H := b c T 2) ake an initial selection of σ i for i =,, r that is closed under conjugation ordered in such a way that σ,, σ k } C σ k,, σ r } C Fix a convergence tolerance 3) Compute V k, W k, V r k W r k according to (9) 4) while (relative change in σ i > tol) a) k = WT k V k, r k = W T r k V r k b) Update the shifts: σ,, σ r } = λ( k )} λ( r k )}; update k to be the total number of shifts in C ; relabel the shifts so that σ,, σ k } C σ k,, σ r } C c) Compute update V k, W k, V r k W r k according to (9) 5) k = WT k V k, r k = W T b k c T k = WT k b, b r k = W T = c T V k, c T r k b, r k V r k, r k = c T V r k Upon convergence H r = H r Hr, will satisfy the L 2 optimality conditions (7) We observe convergence behavior similar to that of IRKA; alternative stopping criteria continue to be studied For example, careful use of system error norms as opposed to relative shift change may be advantageous The final reduced order model is given as ẋ k (t) = k x k (t) b k u(t) H r : ẋ r k (t) = r k x r k (t) b r k u(t) (2) y r (t) = c T k x k (t) c T r k x r k (t), Unlike the Iterative Rational Krylov Algorithm of [8], our proposed method iterates on two sets of interpolation points, originating at each step from stable antistable reduced order poles A key feature of Algorithm 42 is that it adjusts the number of stable poles (k) unstable poles (r k) during the iteration so that the user does not need to determine this beforeh This is similar to the balanced truncation method of [5] where, for a given r, dimensions of H r Hr are chosen according to the Hankel singular values of H H One need not specify the orders of H r Hr ; they are chosen automatically by the algorithm In the form we have presented, Algorithm 42 will carry a practical restriction on system size due to the computation of the block decomposition (5) in Step The balanced truncation method of [5] carries a similar limitation Even so, system orders of a few thous will present little difficulty circumstances are even more favorable if either the stable or antistable invariant subspace is of modest dimension does not grow with overall system order note that (5) does not require a full eigendecomposition for A odifications to Algorithm 42 that can take advantage of these circumstances will be evaluated in future work V A NUERICAL EXAPLE We illustrate the performance of our proposed method on an unstable model having 8 stable 2 antistable poles The pole distribution, for both the stable antistable poles, is chosen to reflect a condenser distribution, making the system very hard to reduce (see [6] for more details) The normalized Hankel singular values, σ k /σ, defined in [6] for unstable systems, are depicted in Figure The slow decay of the singular values confirms that the system is hard to approximate Indeed, only near k 5 does the normalized Hankel singular value, σ k /σ, pass the 3 level (recall the full system order is ) We reduce the order of the system for r = 2 up through r = 3, in increments of 2, using our Algorithm 42 compare with the balanced truncation method of [6] The

6 resulting L 2 error for every r is plotted in Figure 2 As the figure illustrates, our proposed method consistently yiel better L 2 performance than balanced truncation Even though our proposed method is geared towar L 2 model reduction, we also investigate its performance in terms of the the L norm, which is defined as H L = sup H(ıω) ω R Figure 3 depicts the resulting relative L errors for both balanced truncation our proposed method as r varies from 2 to 3 Even though the balanced truncation method of [5] is precisely intended for L -based approximation, our proposed L 2 -based method performs as well as balanced truncation even in terms of the L norm Indeed, Algorithm 42 outperformed balanced truncation with respect to the L error measure in out of 5 cases Balanced truncation was better only in four cases: r = 4, r =, r = 4 r = 8 This further supports the effectiveness of our proposed approach for model reduction of unstable systems The Bode plots of the full-order model H, the two reduced-order models for r = 3 are plotted in Figure 4 Both reduced models show a good match with the original model, especially for lower frequencies The Bode plots of the two error systems are shown in Figure 5 illustrate that our proposed method can yield better L error performance notice that balanced truncation produces a higher peak VI CONCLUSIONS By representing an unstable dynamical system as a noncausal bounded input-output map from L 2 (R) to itself, we are able to derive necessary conditions for optimal model reduction of the original system with respect to an L 2 -induced Hilbert-Schmidt norm The optimality conditions reveal that stable antistable components of the optimal reducedorder model must be Hermite interpolants to the corresponding components of the original model at the mirror images of the stable antistable reduced-order poles, respectively )**))!) ( )**," )-)**))**," &!! &!!& &!!" )! " # $ % &! &" &# &$ &% "! "" "# "$ "% '! ( Fig 2 /234/), " )5((6( Relative L 2 error as r varies 78)9(:;<8,"!=>? This naturally exten the eier-luenberger interpolation conditions for optimal H 2 approximation Based on these interpolatory L 2 optimality conditions, we developed an iteratively corrected rational Krylov algorithm that successively adjusts the interpolation points until the necessary optimality conditions are reached REFERENCES [] P K Aghaee, A Zilouchian, S Nike-Ravesh, A H Zadegan, Principle of frequency-domain balanced structure in linear systems model reduction, Computers Electrical Engineering, vol 29, no 3, pp , ay 23 [2] A C Antoulas, Approximation of Large-Scale Dynamical Systems, Philadelphia: Society for Industrial Applied athematics, 25 [3] A C Antoulas, C Beattie, S Gugercin, Interpolatory model reduction for large-scale linear dynamical systems, Efficient odeling Control of Large-Scale Systems, J ohammadpour K Grigoriadis E, Springer-Verlag, ISBN , in-press, 2 ) "!! / ,:;-<269-=,6>?927-@29?= &!! /234/),! )5((6( 78)9(:;<8,"!=>? ) "!!" "!!#!, --! " "!!$ "!!% "!!& )**))!) ( )**,! )-)**))**,! &!!& &!!" "!!' "!!( "!!)! "! #! $! %! &! '! (! )! *! "!! &!!' )! " # $ % &! &" &# &$ &% "! "" "# "$ "% '! ( Fig Decay of Hankel singular values Fig 3 Relative L error as r varies

7 34,5!23!" #!" %!" &!"!!" "!"!! 67)897:7':;)'<99!7()(-=()<)!7()(>7)9!"!&!"!$!"!#!"!%!"!&!"!!!" "!"! '()*,(-/)2 4,2 A&!BCDE [] C T ullis R A Roberts, Synthesis of minimum roundoff noise fixed point digital filters, IEEE Trans on Circuits Systems, CAS- 23:, pp: , 976 [2] A Varga, odel reduction software in the SLICOT library, in Applied Computational Control, Signals, Circuits, ser The Kluwer International Series in Engineering Computer Science, B Datta, Ed Boston, A: Kluwer Academic Publishers, vol 629, pp [3] A Varga BDO Anderson, Accuracy-enhancing metho for balancing-related frequency-weighted model controller reduction, Automatica, vol 39, pp , 23 [4] DA Wilson, Optimum solution of model reduction problem, in Proc Inst Elec Eng, pp 6-65, 97 [5] K Zhou with JC Doyle K Glover, Robust optimal control, Prentice Hall, 996 [6] K Zhou, G Salomon E Wu, Balanced realization model reduction for unstable systems, Int J Robust Nonlinear Control, vol 9, no 3, pp 83-98, arch 999 [7] A Zilouchian, Balanced structures model reduction of unstable systems, Proceedings of the IEEE Southeastcon 9, pp 98 2, 99 Fig 4 Bode plots of H the reduced models for r = 3 [4] Barahona, AC Doherty, Sznaier, H abuchi, JC Doyle, Finite horizon model reduction the appearance of dissipation in Hamiltonian systems, Proceedings of the 4st IEEE Conference on Decision Control, pp , 22 [5] S Barrachina, P Benner, ES Quintana-Ortí, G Quintana-Ortí Parallel Algorithms for Balanced Truncation of Large-Scale Unstable Systems, Proceedings of 44th IEEE Conference on Decision European Control Conference ECC 25, pp , 25 [6] CA Beattie S Gugercin, Krylov-based model reduction of second-order systems with proportional damping, Proceedings of the 44th IEEE Conference on Decision Control / European Control Conference, pp , 25 [7] K Glover, All Optimal Hankel-norm Approximations of Linear utilvariable Systems their L -error Boun, Int J Control, 39: 5-93, 984 [8] S Gugercin, A C Antoulas, C Beattie, H 2 model reduction for large-scale linear dynamical systems, SIA J atrix Anal Appl, vol 3, no 2, pp , June 28 [9] L eier DG Luenberger, Approximation of Linear Constant Systems, IEE Trans Automat Contr, Vol 2, pp , 967 [] B C oore, Principal Component Analysis in Linear System: Controllability, Observability odel Reduction, IEEE Transactions on Automatic Control, AC-26:7-32, 98 67)897:7':;))((7(<7)9 6-9=>(?@= A%!BCDE!"! 34,5!2!4 (,5!23!" "!"!!!"!%!"!&!"!#!"!$!"!%!"!!!" "!"! '()*,(-/)2 Fig 5 Bode plots of the error models for r = 3