Krylov-based minimization for optimal H 2 model reduction

Transcription

1 Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec , 27 Krylov-based iniization for optial H 2 odel reduction Christopher A. Beattie and Serkan Gugercin Abstract We present an approach to odel reduction for linear dynaical systes that is nuerically stable, coputationally tractable even for very large order systes, produces a sequence of onotone decreasing H 2 error nors, and (under odest hypotheses is globally convergent to a reduced order odel that is guaranteed to satisfy first-order optiality conditions with respect to H 2 error. I. INTRODUCTION Suppose we are given a stable single-input/single-output linear dynaical syste described by the transfer function H(s := c(si A 1 b, (1 with A R n n, b,c T R n. We have particular interest in cases where the syste order, n, is very large and our goal is to find, for any given order r n, a reduced-order odel described by H r (s := c r (si r A r 1 b r (2 with A r R r r, and b r,c T r Rr, such that H r (s is the optial H 2 approxiation to H(s, i.e where H H r H2 = G H2 := in H G r H2. (3 G r: stable ( 1 + 1/2 G(jω dω 2 (4 2π A reduced-order odel that is a global iniizer for H 2 error criterion is guaranteed to exist in the single-input/singleoutput case. However, existence of global iniizers in the ulti-input/ulti-output case is still an open question. Finding global iniizers can be very difficult so the ore odest goal of finding local iniizers is ore usual. Optial H 2 approxiation in this sense has been investigated extensively; see for instance [1], [3], [28], [9], [21], [5], [3], [29], [4], [2], [2] and references therein. Most existing optial H 2 ethods require dense atrix operations, e.g., solving a series of Lyapunov equations. This rapidly becoes intractable as diension increases, so such ethods are rarely usable even for ediu scale probles. We propose here a coputationally effective Krylov-based iniization algorith for optial H 2 approxiation, that is effective even for systes having on the order of any thousands of state variables. This work was supported in parts by the NSF Grants DMS and DMS and AFOSR Grant FA Christopher A. Beattie and Serkan Gugercin are with Departent of Matheatics, Virginia Tech,, 46 McBryde Hall, Blacksburg, VA, , USA {beattie,gugercin}@ath.vt.edu We will construct reduced order odels H r (s by a Galerkin projection process. Let V R n r and Z R n r be given so that Z T V = I r. Then with x r (t R r, Vx r (t R n will approxiate x(t by forcing Z T (Vx r (t AVx r (t b u(t = The reduced order odel in (2 is then obtained as follows: A r = Z T AV, b r = Z T b, c T r = c T V. (5 The reduced order transfer function, H r (s, is deterined principally by the range spaces of V and Z. We will chose the coluns of V and Z to span rational Krylov subspaces designed to iniize the H 2 error. A. Moent atching and rational Krylov ethods Given H(s as in (1, odel reduction by oent atching aounts to finding a reduced-order syste H r (s that interpolates both the values of H(s and its derivatives, at selected points σ k in the coplex plane. For reasons we discuss later, we consider only Herite interpolation; so the oent atching proble requires finding A r, b r, and c r so that H r (σ k = H(σ k and H r(σ k = H (σ k for k = 1,...,r or equivalently, c T (σ k I A 1 b = c T r (σ k I r A r 1 b r c T (σ k I A 2 b = c T r (σ ki r A r 2 b r and for k = 1,...,r. (The quantity c T (σ k I A (j+1 b is called the j th oent of H(s at σ k. Rational interpolation by projection was first proposed by Skelton et. al. in [1], [31], [32]. Grie [16] showed how one can obtain the required projection using the rational Krylov ethod of Ruhe [26]. We state a particular version of Grie s result that suffices to effect Herite interpolation: Proposition 1.1: [16] Consider H(s = c(si A 1 b, distinct shifts given as σ 1, σ 2,..., σ r, and atrices V and Z with Z T V = I, and Ran(V = span { (σ 1 I A 1 b,, (σ r I A 1 b } (6 Ran(Z = span { (σ 1 I A T 1 c,, (σ r I A T 1 c }. (7 The reduced order syste H r defined by A r = Z T AV, b r = Z T b, c T r = ct V, atches two oents of H(s at each of the interpolation points σ k, k = 1,, r. Proposition 1.1 shows that Krylov-based ethods are able to atch oents without ever coputing the explicitly. This is iportant since the coputation of oents is, in /7/$ IEEE. 4385

2 46th IEEE CDC, New Orleans, USA, Dec , 27 general, ill-conditioned. This is one of the ain advantages of the Krylov-based ethods [12]. Unlike graian based odel reduction ethods such as balanced truncation (see [6], Krylov-based odel reduction constructs the odeling subspaces V and Z using only atrix-vector ultiplications and soe sparse linear solvers. These approaches can be ipleented iteratively, which can broaden its range of coputational effectiveness even further; for details, see also [14], [15]. B. First-order optiality conditions for H 2 odel reduction Here, we present first-order necessary conditions for optial H 2 approxiation that are due to Meier and Luenberger [5]. These conditions aount to requiring the optial reduced order odel to be a Herite interpolant to the fullorder syste at particular points. This otivates our usage of Krylov-based ethods. Assue that H(s has siple poles λ 1, λ 2,... λ n. Let φ i denote the associated residues of H(s at the poles λ i, so that φ i = li s λi H(s(s λ i, i = 1,, n. Siilarly, let H r (s has siple poles ˆλ 1, ˆλ 2,... ˆλ r with residues ˆφ i = li s ˆλi H(s(s ˆλ i, i = 1,, r. Then Theore 1.1: [5] Given the syste H(s = n let H r (s = r φ k s λ k, ˆφ k s ˆλ k be an optial solution to the H 2 odel reduction proble (3. Then, H r (s interpolates H(s and its first derivative at ˆλ i, i = 1,...,r, i.e., H r ( ˆλ k = H( ˆλ k and H r ( ˆλ k = H( ˆλ k, for i = 1,...,r. (8 Recent work by Gugercin et al. [2] provides a new and siple proof of these necessary conditions for H 2 -optiality, also showing equivalence with a variety of other (apparently distinct necessary conditions for H 2 -optiality. Theore 1.1 also has straightforward extensions to systes with ultiple poles; see, for exaple, [6], [22], [5]. Theore 1.1 illustrates why the rational Krylov fraework is copelling for the optial H 2 proble; the firstorder conditions (8 lead to Herite interpolation at specific interpolation points ˆλ i, for i = 1,...,r. Based on this observation, a Krylov-based algorith was developed in [2] that generates a reduced odel satisfying the first-order necessary conditions (8. We build on this Krylov/interpolation fraework and incorporate strategies that assure a step-wise decrease in the H 2 error H H r H2 and guarantee a superlinear rate of convergence to a (local iniizer. The key to such an error descent ethod will be gradient and Hessian expressions for H H r H2 viewed ultiately as a function of the interpolation points {σ i } r i=1. II. H 2 ERROR GRADIENTS AND HESSIANS The reduced odels that we consider are constructed via a Galerkin projection: A r = Z T AV, b r = Z T b, c T r = c T V, where (9 using rational Krylov subspaces as in Proposition 1.1 Ran(V = span { (σ i I A 1 b } r i=1 and (1 Ran(Z = span { (σ i I A T 1 c } r i=1. where σ k are distinct coplex nubers. H r (s, and consequently, the H 2 error J = H H r 2 H 2 both depend on the r free paraeters {σ i } r i=1, so we seek a forulation that treats {σ i } r i=1 as the optiization paraeters. We begin with an expression for the H 2 error recently proved by Gugercin and Antoulas [19], [18], [6]: Lea 2.1: Given the full-order syste H(s = n φ k s λ k and any reduced-order odel H r (s = r ˆφ k, the H s ˆλ 2 nor of the error syste, denoted by k J := H(s H r (s 2 H 2, is given by J = n i=1 φ i (H( λ i H r ( λ i + r j=1 ˆφ j (H r ( ˆλ j H( ˆλ j. (11 Both H r (s and J present theselves here as functions of the 2r paraeters {ˆλ i } r i=1 (reduced-order poles and {ˆφ i } r i=1 (reduced-order residues. So it is straightforward to first derive the gradient and Hessian of J with respect to these variables. Although we can derive an H 2 -descent optiization algorith directly with respect to these 2r paraeters, we have found it far ore effective to transfor the proble further so that the H 2 -descent can be organized with respect to the r paraeters, {σ i } r i=1. We can deterine the two Jacobian atrices representing ˆλ i and ˆφ i, and use the chain rule to introduce this further change of variables to obtain the gradient and Hessian of J with respect to the interpolation shifts, {σ i } r i=1. This fors the core of a Krylov/interpolation-based H 2 -descent optiization algorith. A. Gradient of J with respect to poles and residues Gugercin et al. [2] derived the derivatives of the cost function J with respect to ˆφ k and ˆλ k : Theore 2.1: [2] Let H(s = n φ k s λ k and H r (s = r ˆφ k where both H(s and H s ˆλ r (s have distinct poles k as before. Then, J ˆφ = 2H( ˆλ + 2H r ( ˆλ, = 1...r, (12 J ( = 2ˆφ H ( ˆλ H r ˆλ ( ˆλ, = 1...r. (13 If we define the 2r-diensional paraeter vector q = [ˆφ1,..., ˆφr, ˆλ ] T 1,..., ˆλr, (14 Theore 2.1 defines the gradient of J with respect to q: [ ] T J J J J q J = ˆφ,..., 1 ˆφ,,...,. (15 r ˆλ 1 ˆλ r 4386

3 46th IEEE CDC, New Orleans, USA, Dec , 27 B. Hessian of J with respect to poles and residues By differentiating (12 and (13 with respect to ˆφ p and ˆλ p (followed by soe tedious anipulations, we obtain secondorder partial derivatives of J with respect to the pole-residue paraeters ˆφ and ˆλ: Theore 2.2: Let H(s = n φ k s λ k and H r (s = ˆφ k as above. Then, r s ˆλ k 2 J 2 ˆφ p ˆφ = ˆλ + ˆλ, (16 p 2 J ˆλ p ˆλ = 4ˆφ ˆφp (ˆλ + ˆλ p 3 for p, (17 2 J 2ˆλ = 2ˆφ (H ( ˆλ H r ( ˆλ ˆφ 2, (18 2 J 2ˆφ p ˆλ p ˆφ = (ˆλ + ˆλ for p (19 p 2 2 J ( ˆλ ˆφ = 2 H ( ˆλ H r( ˆλ + ˆφ (2 2ˆλ 2 Theore 2.2 in effect, gives the Hessian of J with respect to the 2r-diensional paraeter vector q defined in (14, : 2 qj = ˆλ 3 { 2 J ˆφi, i = 1,...,r,, q i = q i q j ˆλ i r, i = r + 1,...,2r. (21 Reark 2.1: One can develop a descent ethod (either line search or trust-region for optial H 2 odel reduction cobining Theores 2.1 and 2.2. However, for an r th order reduced odel, the optiization variable q will have a diension 2r. In the next section, we will develop a Krylovbased Newton ethod where the only variables will be the r interpolations points {σ i } r i=1 ; hence the nuber of variables will be reduced to half without any loss of accuracy in the underlying optiization proble. Our nuerical experience suggests that the Krylov-based fraework converges faster than the pole-residue forulation. C. The Jacobian atrix J λ := ˆλ i Let λ(σ = [ˆλ 1, ˆλ 2,..., ˆλ r ] T denote the r-tuple of the reduced order poles ephasizing the fact that we now view the reduced order poles as a function of interpolation points only. Recall that the reduced odels will be obtained via rational Krylov projection as shown in (9 and (1. Since, the gradient and Hessian of the H 2 cost with respect to {ˆφ i } and {ˆλ i } are already obtained in the previous section, what reains to obtain for a Krylov-based optiization ethod are the Jacobians representing ˆφ i and ˆλ i. We will derive these two quantities using the underlying Krylov-based reduction fraework in (9 and (1. Define the coplex r-tuple σ = [σ 1, σ 2,..., σ r ] T C r together with related atrices V and W: V = [ (σ 1 I A 1 b,..., (σ r I A 1 b ] (22 c(σ 1 I A 1 W T =. c(σ r I A 1 By this notation, the reduced-order atrix A r in (9 becoes A r = Z T AV where Z T = (W T V 1 W T ; hence satisfying Z T V = I r. As before, ˆλ i and ˆφ i will denote, respectively, the poles and residues of the reduced order syste. We ay calculate the entries of the Jacobian atrix of λ(σ by differentiating the eigenvalue relation W T AVˆx i = ˆλ i W T Vˆx i and using the definitions of V and W in (22. This yields Lea 2.2: Let ˆx be a unit eigenvector of A r = (W T V 1 W T AV associated with ˆλ i, so W T AVˆx = ˆλ i W T Vˆx. Then, where ˆλ i = ˆx T j W (AVˆx T ˆλ i Vˆx ˆx T W T + Vˆx (ˆx T W T A ˆλ iˆx T W T j Vˆx ˆx T W T Vˆx j W T = W T = e j c(σ j I A 2 and (23 (24 j V = V = (σ j I A 2 be T j σ. j ( The entries of the Jacobian atrix J λ = b λi provide a easure of the sensitivity of the reduced order poles, λ(σ, to perturbations of σ. Notice that Vˆx and ˆx T W T are Galerkin approxiations to right and left eigenvectors, respectively, of A. (26 shows how to copute J λ. This requires solving a sall r r generalized eigenvalue proble to copute ˆλ i and ˆx, and 2r additional linear solves to copute (σ i I n A 2 b and (σ i I n A T 2 c T. However, since constructing V and W T already requires coputing (σ i I n A 1 b and (σ i I n A T 1 c T, J λ does not require additional factorizations, only soe additional triangular solves are needed. D. The Jacobian atrix J φ = ˆφ i Define the r-tuple φ(σ = [ˆφ 1, ˆφ 2,..., ˆφ r ] T. As above, notation φ(σ ephasizes the fact that the reduced order residues is a function of the interpolation ( points only. Also, siilar to J λ, the entries of J φ = ˆφi will provide a easure of the sensitivity of the reduced order residues, φ(σ, to perturbations of σ. Lea 2.3: Given H(s = c(si A 1 b, let the reduced odel H r (s = c r (si r A r 1 b r be obtained as in (9 with Z = W(W T V 1 where V and W are as in (22. Define M = W T AV, N = W T V, and the generalized 4387

4 46th IEEE CDC, New Orleans, USA, Dec , 27 eigendecoposition MX = NXΛ with a diagonal atrix of reduced syste poles (eigenvalues Λ = diag(ˆλ i and an invertible atrix of (right eigenvectors X. Define Y = (NX 1, an associated atrix of left eigenvectors so that Y M = ΛY N. Then, the residues of H r (s is given by Moreover, φ i = e T i ˆφ i = e T i Y W T bcvxe i. (25 [ ( j Y W T bcvx + Y ( j W T bcvx +Y W T bc( j VX + Y W T bcv( j X ] e i (26 where j X = X solves the Sylvester equation N 1 M( j X ( j XΛ + Q =. (27 with Q = N 1 ( j MX N 1 ( j NXΛ X( j Λ and j Y = Y satisfies j Y = Y [( j NX + N( j X]Y. (28 where j W T and j V are as in (24, and j M = j N = M = ( j W T AV + W T A( j V N = ( j W T V + W T ( j V Proof: (25 is obtained by transferring the reduced odel into odal for. On the other hand, (26 follows fro differentiating (25 with respect to σ j. Lea 2.3 explains how to copute J φ. Note that J φ requires solving r sall r r Sylvester equations as in (27. In addition to having sall diension r, (27 can be solved easily due to diagonal coefficient ter Λ. Once ore, the additional coputational cost due to Jacobian coputation is sall. Moreover, even though the Sylvester equation (27 is singular, it is guaranteed to be consistent by construction. E. The gradient and Hessian of J with respect to shifts σ Having established the two Jacobians J λ and J φ, we are now ready to state the ain result of the paper stating the gradient and Hessian of the H 2 error with respect to shifts in a Krylov-based odel reduction setting: Lea 2.4: Given the full-order odel H(s, let H r (s = r ˆφ k be obtained via Krylov-based odel s ˆλ k reduction as in (9-(1 with shifts σ = [σ 1, σ 2,..., σ r ] T. Define [ ] Jφ J = (29 J λ where J λ and J φ as defined in (23 and (26 Then, σj, the gradient of the H 2 error J = H H r 2 H 2 with respect to σ is given by σj = J T ( q J (3 where q J is given by (15. Moreover, σj 2, the Hessian of J with respect to σ, is σ 2 J = ( JT q 2 J r ( J J + i=1 ˆφ σ 2 ˆφ i + J σˆλ 2 i i ˆλ i (31 where qj 2 is as in (21, and σ 2 ˆφ i and σˆλ 2 i represent, respectively, the Hessian of ˆφ i and ˆλ i with respect to σ. Proof: (3 follows fro cobining Theore 2.1 and Leas 2.2 and 2.3. (31 follows fro Theore 2.2, and Leas 2.2 and 2.3. We can establish Lipschitz continuity of J and σj. To ephasize dependence on the interpolation points σ = [σ 1, σ 2,..., σ r ] T, we use a superscript [σ] in the reduced order odel. Theore 2.3: Let H(s be defined as in (1 and σ = [σ 1, σ 2,..., σ r ] T. Denote the reduced-order odel associated with σ as H r [σ] (s calculated fro (9-(1 with shifts σ = [σ 1, σ 2,..., σ r ] T. The H 2 error viewed as a function of the shifts, J (σ : C r R is defined as J (σ = H H r [σ] 2 H 2. For soe finite J >, let N be the level set N = {σ : J (σ J }. Then, both J (σ and σj (σ are Lipschitz continuous functions on the level set N. Proof: The result follows fro observing the continuity, differentiability and boundedness of J (σ and σj (σ on the level set N. III. A KRYLOV-BASED MINIMIZATION ALGORITHM Our iniization algorith is structured overall as a line search/odified Newton ethod. We present only a generic description without details (such as terination and step acceptance criteria associated with standard optiization ipleentation. For such details, we refer to [24]. We have also developed a trust region variant of this approach that we do not discuss here but will include in the full paper. Note that the Hessian expression (31 suggests that the calculation of σ 2 J will require the copution of the individual Hessians σ 2 ˆφ i and σˆλ 2 i. However in (31 these ters are ultiplied by eleents of q J which will becoe negligible in the vicinity of a iniizer. In our nuerical ipleentations, we approxiate σ 2 J either using a Gauss- Newton approxiation or a partial secant update B = J T ( 2 q J J (32 B = J T ( 2 qj J + S (33 where S is a least-change secant approxiation to ( r J i=1 ˆφ 2 i σ ˆφ i + J 2 ˆλ i σˆλ i as described, for exaple, of Dennis et al. [11] for nonlinear least squares probles. Further odifications to B are also perfored to assure that it is safely positive definite (thus insuring a descent direction; see [24]. 4388

5 46th IEEE CDC, New Orleans, USA, Dec , 27 Algorith 3.1: Krylov-based H 2 iniization 1 Choose initial shifts σ ( = [σ ( 1,..., σ( r ] T 2 for k =, 1, 2,... until convergence a Copute σj ( σ (k using (3 b Copute the Hessian approxiation B k using (32 or (33. If necessary, odify B k to be positive definite. c Copute the odified Newton direction: p (k = B 1 k σj ( σ (k d Copute a step length α (k that satisfies the Wolfe conditions, i.e., with σ (k + = σ(k + α k p (k σ (k + c 1α k J J σj T σj σ (k + p (k c 2 σj with < c 1 < c 2 < 1. e Update σ (k+1 = σ (k + = σ (k + α (k p (k σ (k + σ (k T p (k, σ (k T p (k f Copute the reduced order odel H (k+1 r for new shifts σ (k+1 3 end The following theore is a direct consequence of Algorith 3.1 and Theore 2.3. Theore 3.1: Given H(s = c(si A 1 b and the r th order reduced odel, H r (k, obtained via Algorith 3.1 with the corresponding H 2 error J (k = H H (k 2. Assue H 2 that the initial reduced odel H r ( is a stable dynaical syste for each k with r (s is stable. Then H r (k (s ( J (k+1 J (k, k =, 1, 2,.... and li J σ (k = k Moreover, if, upon convergence, the Hessian 2 σj is positive definite, then Algorith 3.1 converges to a local iniizer. Reark 3.1: In coputing the step length α (k, we only ention that the final step length ust satisfy Wolfe conditions, even though different ways of accoplishing this are available. The details of how the line search is perfored is oitted. For details, we refer the reader to [24]. Reark 3.2: This is the first Krylov-based descent algorith for optial H 2 odel reduction. Previous descent approaches require solving two large-scale Lyapunov equations to copute the search direction. In our case, the doinant cost to calculate search directions are sparse linear solves. To our knowledge, this is the first approach where the Hessian (hence a Newton direction is used in optial H 2 approxiation. A. A low order exaple IV. NUMERICAL EXAMPLES In this exaple, we illustrate the application of the proposed for a siple fourth order syste fro [3] with the state-space representation A = , b = 4 1 ct = 1. We reduce the order to r = 3, 2, 1 using the proposed Krylov-based descent algorith with line search, i.e. Algorith 3.1. The resulting relative H 2 error nors, H Hr H 2 H H2, are tabulated in Table I. In each case, the ethod yields the inial H 2 error nors as listed in [3], [2] staying in nuerically effective Krylov-fraework and guaranteeing a descent in each step. Convergence behavior of H Hr H 2 H H2 and σj for r = 1, 2 and 3 are shown, respectively, in Figures 1, 2 and 3. In each case, convergence is very fast and quadratic behavior of the Newton step is clearly reflected. TABLE I RELATIVE H 2 ERRORS r = 1 r = 2 r = 3 H H r H2 H H / H / H Nor of the gradient k: Iteration index Fig. 1. Evolution of Algorith 3.1 for r = Nor of the gradient k: Iteration Index B. A rando odel Fig. 2. Evolution of Algorith 3.1 for r = 2 The full-order odel is a stable rando odel of order n = 5. We reduce the order to r = 4 using Algorith 4389

6 46th IEEE CDC, New Orleans, USA, Dec , 27 / H Nor of the gradient k: Iteration Index Fig. 3. Evolution of Algorith 3.1 for r = The convergence behavior for this exaple is shown in Figure 4. As in the previous, the proposed ethod quickly converges to the optial odel while staying in Krylovfraework. / H Nor of the gradient k: Iteration Index Fig. 4. Evolution of Algorith 3.1 for the rando odel for r = 3 REFERENCES [1] L. Baratchart, M. Cardelli and M. Olivi, Identification and rational l 2 approxiation: a gradient algorith, Autoatica, 27: (1991. [2] P. Fulcheri and M. Olivi, Matrix rational H 2 approxiation: a gradient algorith based on Schur analysis, SIAM Journal on Control and Optiization, 36: (1998. [3] D.C. Hyland and D.S. Bernstein, The optial projection equations for odel reduction and the relationships aong the ethods of Wilson, Skelton, and Moore, IEE. Trans. Autoat. Contr., Vol. 3, No. 12, pp , [4] A. Lepschy, G.A. Mian, G. Pinato and U. 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