Rational Krylov Methods for Model Reduction of Large-scale Dynamical Systems

Transcription

1 Rational Krylov Methods for Model Reduction of Large-scale Dynamical Systems Serkan Güǧercin Department of Mathematics, Virginia Tech, USA Aerospace Computational Design Laboratory, Dept. of Aeronautics & Astronautics, MIT, October 21, 2005, Cambridge, MA 1

2 Outline 1. Introduction and Problem Statement 2. Motivating Examples 3. Rational Krylov-Interpolation Framework 4. Current Trends in Rational Krylov Projection 5. Conclusions 2

3 Introduction Consider an n th order single-input/single-output system G(s): ẋ(t) = Ax(t) + bu(t) G(s) : G(s) = c(si n A) 1 b y(t) = cx(t) = n(s) d(s) u(t) R: input, x(t) R n : state, y(t) R: output A R n n, b,c T R n. Will assume R(λ i (A)) < 0 Need for improved accuracy = Include more details in the modeling stage In many applications, n is quite large, n O(10 6, 10 7 ), Untenable demands on computational resources = 3

4 Model Reduction Problem: Find ẋ r (t) = A r x r (t) + b r u(t) y r (t) = c r x r (t) G r (s) = c r (si r A r ) 1 b r where A r R r r, b r,c T r R r, with r n such that 1. y y r is small. 2. The procedure is computationally efficient. A B A r B r C D C r D r G r (s): used for simulation or designing a reduced-order controller 4

5 Model reduction through projection: a unifying framework. Construct Π = VZ T, where V,Z R n r with Z T V = I r : ẋ r = Z T AV }{{} :=A r x r (t) + Z T b }{{} u(t), y r (t) = }{{} cv x r (t) :=b r :=c r What is the approximation error e(t) := y(t) y r (t)? G(s): Associate a convolution operator S: S : u(t) y(t) = (Su)(t) = (g u)(t) = t g(t τ)u(τ)dτ. g(t) = ce At b for t 0: Impulse response. Transfer function: G(s) = (Lg)(s) = c(si A) 1 b. 5

6 The H Norm : 2-2 induced norm of S: G(s) H = sup u 0 y 2 u 2 = sup u 0 Su 2 = sup G(jw) u 2 2 w R G G r = Worst output error y(t) y r (t) 2 u(t) 2 = 1. The H 2 Norm : L 2 norm of g(t) in time domain: G(s) 2 H 2 = 0 trace[g T (t)g(t)]dt = 1 2π + trace[g (jw)g(jw)]dw G(s) G(jw) (db) G(s) freq (rad/sec) 6

7 Motivating Example: Simulation A Tunable Optical Filter: (Data: D. Hohlfeld, T. Bechtold, and H. Zappe) An optical filter, tunable by thermal means. Silicon-based fabrication. The thin-film filter: membrane to improve thermal isolation Wavelength tuning by thermal modulation of resonator optical thickness The device features low power consumption, high tuning speed and excellent optical performance. 7

8 Modeling: A simplified thermal model to analyze/simulate important thermal issues: 2D model and 3D model Meshed and discretized in ANSYS 6.1 by the finite element methods The Dirichlet boundary conditions at the bottom of the chip. A constant load vector corresponding to the constant input power of of 1 mw for 2D model and 10 mw for 3D model The output nodes located in the membrane Eẋ(t) = Ax(t) + bu(t), y(t) = cx(t) 2D: n = 1668, nnz(a) = 6209, nnz(e) = D: n = , nnz(a) = , nnz(e) =

9 Motivating Example: Control International Space Station (ISS) Modules: ISS: a complex structure of many modules. More than 40 Shuttle flights to complete the assembly. Each module modeled by 1000 state variables For ISS, controllers will be needed for many reasons: 1. to reduce the oscillatory motions; 2. to fix the orientation of the space station with respect to some desired direction Controllers to be implemented on-board Controllers of low complexity needed due to: hardware, radiation, throughput and testing issues 9

10 Flex Structure Variation During Assembly Stage 1R - Dec 99 Stage 4A - Mar 00 Stage 13A - May 02 Figure 1: ISS: Evolution of the frequency response as more modules are added (Data: Draper Labs) 10

11 SVD based approximation methods Singular Value Decomposition (SVD) provides low-rank approximation of matrices. What are the singular values of G(s)? Two Lyapunov Equations: AP + PA T + bb T = 0, A T Q + QA + c T c = 0, 0 < P : Reachability Gramian 0 < Q : Observability Gramian Hankel Singular Values: σ i (Σ) := λ i (PQ) for i = 1,...,n. States difficult to reach correspond to small eigenvalues of P. States difficult to observe correspond to small eigenvalues of Q. 11

12 Balanced Truncation Method States difficult to reach States difficult to observe? Make P = Q by a basis change: x = Tx, dett 0 Ā = TAT 1, b = Tb, c = ct 1. P = UU T, U T QU = RΣ 2 R T = T b := Σ 1/2 R T U 1 In the balanced basis: P = Q = Σ = diag(σ1,...,σ n ) Projection Framework: P = UU T, Q = LL T, U T L = WΣY T, Σ = diag(σ 1,...,σ n ). Z := LY 1 Σ 1/2 1, V := UW 1 Σ 1/2 1. VZ T : oblique projector G r (s) = A r b r c r 0 = ZT AV Z T b cv 0. 12

13 Properties of the reduced model : 1. G r (s) is asymptotically stable. 2. G(s) G r (s) 2 (σ r σ n ). Drawbacks in large-scale settings : Requires U and L 1. O(n 3 ) arithmetic operations 2. O(n 2 ) memory requirement Remedy: σ i (Σ), λ i (P) and λ i (Q) decay rapidly. = Find Ũ, L R n k s.t. P := ŨŨ P and Q := L L T Q. (Approximate) balancing using Ũ and L. O(n 2 k) arithmetic operations, O(nk) memory requirement. Penzl [2000], Li/White [2000]: LR-Smith, G/Sorensen/Antoulas [2002]: Modified LR-Smith, Willcox/Peraire [2002]: Balanced POD 13

14 Model Reduction via Interpolation Rational Interpolation: Given G(s), find G r (s) so that G r (s) interpolates G(s) and certain number of its derivatives at selected frequencies σ k in the complex plane ( 1) j j! d j G(s) ds j = ( 1)j s=σk j! d j G r (s) ds j, s=σk for k = 1,...,K, and j = 1,...,J ( 1)j j! d j G(s) ds j = c(σ k I A) (j+1) b: s=σk = j th moment of G(s) at σ k =: η σk (j). 14

15 For σ 1 = σ 2 =... = σ K = σ = η (j) σ = ca j 1 b : j th Markov Parameter Moment matching problem: Partial realization Solution: Lanczos and Arnoldi procedures For arbitrary σ η (j) σ = c(σi A) (j+1) b Moment matching problem: Rational Interpolation Solution: Rational Lanczos/Arnoldi procedures For multiple points = rational Krylov method 15

16 Moments are very ill-conditioned to compute Markov Parameters, σ = 10 2 Moments at σ = Ball et al. [1990], Antoulas et al. [1990], Pillage et al. [1993]: Moment matching problem with explicit moment computation. = Total loss of accuracy in large-scale settings Feldman and Freund [1995]: Padé via Lanczos ( Krylov setting) Moment matching without moment computation Iterative implementation 16

17 The Arnoldi Procedure for σ = 1. Given A R n n, b R n, and c T R n, 2. v 1 := b b, w := Av 1; α 1 := v T 1 w f 1 := w v 1 α 1 ; V 1 := (v 1 ); H 1 := (α 1 ) 3. For j = 1, 2,, r 1 β j := f j, v j+1 := f j β j V j+1 := (V j v j+1 ), Ĥ j = H j β j e T j w := Av j+1, h := Vj+1w, T f j+1 = w V j+1 h ) H j+1 := (Ĥj h AV = VH + fe T r, V T V = I r, V T f = 0, V R n r A r = H, b r = b e 1, c r = cv 17

18 G r (s) matches the first r Markov parameters of G(s) without computing them Only matrix-vector multiplications. No SVD, or Schur decomposition required Range(V) = Span ( [ b, Ab,, A r 1 b ] ) Lanczos Procedure: Two-sided Arnoldi. Range(V) = Span ( [ b, Ab,, A r 1 b ] ) Range(Z) = Span ( [ c T, A T c T,, (A T ) r 1 c T ] ) G r (s) matches the first 2r Markov parameters of G(s) without computing them For arbitrary σ, A (σi A) 1, b (σi A) 1 b, c c(σi A T ) 1 18

19 Approximation of a Building Model Los Angeles University Hospital: n = 48 to r = Bal. Red. 40 Σ H(jw) (db) σ = Σ σ= Σ Freq (rad/sec) 19

20 Rational Krylov Method Lanczos/Arnoldi: Moment matching at a single point. Large error around other frequencies Match the moments around various frequencies Better approximation over a broad range Solution by projection: First by Skelton et. al. [1985] Grimme [1997]: Efficient computation by Krylov projection 20

21 Given 2r interpolation points: {σ i } 2r i=1 Set V = Span [ (σ 1 I A) 1 b,, (σ r I A) 1 b ], and Z = Span [ (σ r+1 I A T ) 1 c T,, (σ 2r I A T ) 1 c T ], Make Z T V = I r. A r = Z T AV, b r = Z T b, c r = cv = G(σ i ) = G r (σ i ) for i = 1,...,2r Moment matching without explicit moment computation 21

22 Given r interpolation points: {σ i } r i=1 Set V = Span [ (σ 1 I A) 1 b,, (σ r I A) 1 b ], and Z = Span [ (σ 1 I A T ) 1 c T,, (σ r I A T ) 1 c T ], Make Z T V = I r. A r = Z T AV, b r = Z T b, c r = cv = G(σ i ) = G r (σ i ), and d ds G(s) = d s=σi ds G r(s) Moment matching without explicit moment computation s=σi Efficient computations of V and Z: Dual Rational Arnoldi, Rational Lanczos, Rational Power Krylov.(Grimme [1997]) 22

23 Why to choose model reduction via rational interpolation? van Dooren et al. [2004], Halevi [2005]: Generically, any reduced model G r (s) can be obtained via interpolation. Interpolation points = Zeroes of G(s) G r (s). Example: G(s) = G(s) G r (s) = 1 s 2 + 3s + 2 V = (σ 1 A) 1 b = A 1 b, BUT: and G r (s) = 0.2 s s(s 2) s s s σ 1 = 0, σ 2 = 2 Z = (σ 2 A) T c T = (2I A) T c T What is a good selection of interpolation points? Similar to polynomial approximation of complex functions. 23

24 Advantages of Krylov based methods: Iterative implementation Only matrix-vector multiplications and sparse LU decompositions O(r 2 n) arithmetic operation and O(nr) storage requirements Drawbacks: Stability is not guaranteed No global error bound Selection of σ i is an ad-hoc process. 24

25 Stability: Implicit Restart (Sorensen et al.[1995]) Let A r,b r,c r be obtained via r steps of Arnoldi algorithm. Let A r have one unstable eigenvalue µ, i.e., R(µ) > 0. Define b := (µi n A)b Run r 1 steps of Arnoldi process on the pair (A, b) The new reduced system will be asymptotically stable Matching the moments of Ĝ(s) = c(si A) 1 b Trade-off between guaranteed stability and exact moment matching. 25

26 Least-squares model reduction (G./Antoulas [2003]) Bilinear transformation and reduction in the discrete-time Combines the SVD and Krylov reduction V : Krylov-side, Z : SVD-side Stability and moment matching Fourier model reduction (Willcox/Megretski [2005]) Bilinear transformation and reduction in the discrete-time Stability and moment matching 26

27 An H 2 Error Expression (G./Antoulas [2002]) G(s) 2 H 2 = cpc T = b T Qb. φ i := G(s)(s λ i ) s=λi and φ j := G r (s)(s λ j ) s= λj. G(s) G r (s) 2 H 2 = n φ i (G( λ i ) G r ( λ i )) + i=1 r j=1 ) φ j (G r ( λ j ) G( λ j ). Error due to mismatch at λ i and λ j. Try to kill the first term σ i = λ i (A). Select λ i (A) with large φ i. 27

28 CD Player and Selection of σ i Dynamics of a CD Player with 120 states, m = p = 1. Reduce the order to r = 14 by Rational Krylov G r: Choosing σ i = λ i (A). G 1,G 2,G 3 : Arbitrary σ i. Best 3 among 2000 models. G r G 1 G 2 G 3 H H Table 1: H 2 and H errors for the CD Player Example Does there exist an optimal shift selection? 28

29 Optimal H 2 approximation Problem: Given a stable dynamical system G(s), find a reduced model G r (s) that satisfies G r (s) = arg G(s) Ĝ(s) H2. min deg(ĝ) = r Ĝ : stable First-order conditions: (Meier and Luenberger [1967]) d G( ˆλ i ) = G r ( ˆλ i ), and ds G(s) = d s= ˆλi ds G r(s) s= ˆλi Match the first two moments at the mirror images of the Ritz values. First-order conditions as interpolation. Rational Krylov Framework 29

30 For the H 2 problem, simply set σ i = ˆλ i ˆλ i NOT known a priori = Needs iterative rational steps An Iterative Rational Krylov Algorithm (IRKA): (G..Beattie/Antoulas [2004]) 1. Choose σ i for i = 1,..., r. 2. V = Span [ (σ 1 I A) 1 b,, (σ r I A) 1 b ], 3. Z = Span [ (σ 1 I A T ) 1 c T,, (σ r I A T ) 1 c T ], Z T V = I r. 4. while [relative change in σ j ] > ǫ (a) A r = Z T AV, (b) σ i λ i (A r ) for i = 1,..., r (c) V = Span [ (σ 1 I A) 1 b,, (σ r I A) 1 b ]. (d) Z = Span [ (σ 1 I A T ) 1 c T,, (σ r I A T ) 1 c T ], Z T V = I r. 5. A r = Z T AV, b r = Z T b, c r = cv Upon convergence, first-order conditions satisfied via Krylov projection framework, no Lyapunov solvers 30

31 ISS 12a Module n = Reduce to r = 2 : 2 : 60 Compare with balanced truncation Comparision between IRKA and BT 1 Evolution of IRKA for r=2 Real(σ 1 ) Relative H 2 error 10 1 BT Imag(σ 1 ) IRKA r: reduced order Relative H 2 error Number of iterations 31

32 Example: Optimal Cooling of Steel Profiles ( P. Benner) G(s) = c(se A) 1 b, n = 79, 841 r = 6 via IRKA 10 1 Bode Plots of G(s) and G r (s) (IRKA) G(s) G r (s) 10 2 H(jw) frequency (rad/sec) G(s) G 1 (s) =

33 van Dooren et al.[2005]: Model reduction of interconnected systems via rational Krylov Reduce each subsystem via interpolation Reduced-interconnection interpolates the full-model Bai [2003]: Krylov-reduction of second-order systems: Mẍ(t) + Gẋ(t) + Kx(t) = bu(t), y(t) = cx(t) A = G(s) : 0 I M 1 K M 1 G,B = q(t) = Aq(t) + B u(t) y(t) = Cq(t) 0 M 1 b,c T = ct Apply model reduction in the first-order state-space Transfer back to second-order form Structure is lost 0 T 33

34 find a matrix W R n r such that W T W = I r M r = W T MW, G r = W T GW, K r = W T KW, b r = W T b, and c r = cw M r ẍ r (t) + G r ẋ r (t) + K r x r (t) = b r u(t), y r (t) = c r x r (t) Second-order Krylov subspaces (Second-order Arnoldi Iteration): W spans the required Krylov subspace but constructed directly in the second-order framework Interpolation conditions hold. Antoulas/Sorensen [2003]: Passivity preserving Krylov-based model reduction G(jw) + (G(jw)) > 0. G(s) + G (s) = H(s)H (s) Interpolation points: Zeros of H(s) Passivity is preserved. 34

35 Conclusions and Future Work: Rational Krylov framework for model reduction In the SISO case, any G r (s) can be obtained via interpolation Match moments without moment computation Cost is reduced from O(n 3 ) to O(r 2 n) To guarantee stability Implicit restart Least-squares reduction (G./Antoulas) Fourier reduction (Willcox/Megretski) Optimal selection of interpolation points Interpolation condition for optimal H 2 approximation Iterative Rational Krylov Algorithm On-going and future work - Interconnected systems - Second-order systems - Controller reduction - Optimal approximation - Guaranteed error estimates - Balancing via Krylov 35

36 Controller reduction for large-scale systems Consider an n th order plant G(s) = c(si A) 1 b n th κ order stabilizing controller: K(s) = c K (si A K ) 1 b K + d K LQG, H control designs n κ = n (i) Complex hardware (ii) Degraded accuracy (iii) Degraded computational speed Obtain K r (s) of order r n κ to replace K(s) in the closed loop. 36

37 Controller reduction via frequency weighting Small open loop error K(s) K r (s) not enough. Minimize the weighted error: W o (s)(k(s) K r (s))w i (s). How to obtain the weights W o (s) and W i (s)? If K(s) and K r (s) have the same number of unstable poles and if [K(s) Kr (s)]]g(s)[i + G(s)K(s)] 1 < 1, or [I + G(s)K(s)] 1 G(s)[K(s) K r (s)] < 1, = K r (s) stabilizes G(s). = 37

38 For stability considerations: W i (s) = I and W o (s) = [I + G(s)K(s)] 1 G(s) or W o (s) = I and W i (s) = G(s)[I + G(s)K(s)] 1. To preserve closed-loop performance: W i (s) = [I + G(s)K(s)] 1 and W o (s) = [I + G(s)K(s)] 1 G(s). Solved by frequency-weighted balancing (Anderson and Liu [1989], Schelfhout and De Moor [1996], Varga and Anderson [2002] ). Requires solving two Lyapunov equations of order n + n κ. A i P + PA T i + b i b T i = 0, A T o Q + QA o + c T o c o = 0, A i,b i : K(s)W i (s), A o,c o : W o (s)k(s) Balance P and Q. 38

39 Controller-reduction via Krylov Projection How to modify IRKA for the controller reduction problem? Let W i (s) = I and W o (s) = [I + G(s)K(s)] 1 G(s) A K P + PA T K + b K b T K = 0 }{{} unweighted Lyapunov eq. A T wq + QA w + c T wc w = 0. }{{} weighted Lyapunov eq. Z = K(A T,C T, σ i ), and V = K(A,B, µ j ) Z and σ i : Reflect W o (s): the closed-loop information. σ i = jw i over the region where W o (jw) is dominant V and µ j : Obtained in an (optimal) open loop sense. µ j : From an iterative rational Krylov iteration 39

40 An Iterative Rational Krylov Iteration for Controller Reduction: 1. Choose σ i = jw i, for i = 1,..., r where w i is chosen to reflect W o (jw). 2. Z = Span [ (σ 1 I A T K ) 1 c T K (σ ri A T K ) 1 c T K ] with Z T Z = I r. 3. V = Z 4. while [relative change in µ j ] > ǫ (a) A r = Z T A K V, (b) µ j λ i (A r ) for j = 1,..., r (c) V = Span [ (µ 1 I A K ) 1 b K (µ r I A K ) 1 ] b K with Z T V = I r. 5. A r = Z T A K V, b r = Z T b K, c r = c K V Z K r (s) includes the closed loop information V K r (s) is optimal in a restricted H 2 sense Π = ZVT 40

41 International Space Station Module 1R: n = 270. G(s) is lightly damped Long-lasting oscillations. K(s) is designed to remove these oscillations. n κ = x 10 3 Impulse response of the open loop system 6 x 10 3 Impulse response of the closed loop system Amplitude 2 1 Amplitude time (sec) time (sec) Reduce the order to r = 19 using iterative Rational Krylov and to r = 23 using one-sided frequency weighted balancing 41

42 FWBR: Frequency-weighted balancing with W i (s) = I and W o (s) = [I + G(s)K(s)] 1 G(s). IRK-CL: Iterative Rational Krylov - Closed Loop version: σ i reflect the weight W o (s). 30 Bode Plot of W o (s) = T(s) = [ I + G(s)K(s)] 1 G(s) Singular Values (db) Frequency (rad/sec) σ i = j logspace( 1, 2, 10) rad/sec 42

43 Singular Values (db) ISS Example: Bode Plots of reduced closed loop systems T T IRK CL T FW20 T FW23 Relative Errors H error T T FW T T FW23 T T IRK CL Frequency (rad/sec) ISS Model: Error in closed loop systems T T FW20 Relative Errors H 2 error T T FW T T FW23 T T IRK CL Singular Values (db) T T FW23 Weighted Errors H 2 error W i (K K FW20 ) < T T IRK CL W i (K K FW23 ) < 1 W i (K K IRK CL ) < Frequency (rad/sec) 43

44 An Unstable Model: n=2000. K(s) of order n κ = 2000 stabilizes the model. 70 Impulse response of G(s) Amplitude Impulse Response of T(s) Amplitude time (sec) K(s) has four unstable poles. 44

45 Reduce the order to r = 14: Stabilizing controller K r (s) has 4 unstable poles as desired Bode Plots of T(s) and T r (s) T(s) T r (s) 0.5 Impulse Response T(s) T r (s) Singular Values Amplitude Singular Values 10 0 Bode Plots of K(s) and K r (s) 10 1 K(s) K r (s) Amplitude Response to u(t)= sin (4t) T(s) T r (s) Frequency (rad/sec) time (sec) 45

46 Part II: Conclusions and Future Work n >> 10 6 : Forces usage of Inexact Solves in Krylov-based reduction Perturbation effects: Backward and forward error analysis framework Good/Optimal shift selection robust with respect to inexact solves I-IRKA (Locally) optimal reduced models for n > 10 6 without user intervention Acceleration strategies Open issues: Global H 2 and/or H perturbation effects Modifications to GMRES, effective preconditioning strategies Scalable parallel versions A large-scale easy-to-use model reduction toolbox Modify the algorithms to fit into the framework of, e.g., Trilinos Implementation on Virginia Tech.-System X 46

47 Effect of Initialization: CD Player Model n = 120 to r = 8 via IRKA (Logarithmic grid) Logarithmic Grid Sweep 8 th Order Reduced Model Relative H 2 Error Imaginary Axis Real Axis

48 Starting point: H 2 error expression (Gugercin and Antoulas [2003]) G(s) with poles λ i and G r (s) with poles λ j φ i := G(s)(s λ i ) s=λi and φ j := G r (s)(s λ j ) s= λj. G(s) G r (s) 2 H 2 = n φ i (G( λ i ) G r ( λ i )) + i=1 r j=1 ) φ j (G r ( λ j ) G( λ j ). Open loop error due to mismatch at λ i and λ j 48

49 Given Σ = The Lanczos Procedure: SISO Case [ A C B D ] = v 1 := 1 B and z 1 := sgn(cb) C T CB CB AV = V T + fet r A T Z = ZT T + ge T r, V T Z = I r, V T g = Z T f = 0 Σ r = [ Z T AV CV Z T B D ] = [ T sgn(cb) CB e T 1 Σ r is unique and matches 2r Markov parameters CB e1 D ] Im(V ) = Im ( [ B AB A r 1 B ] ) =: Im (R r (A, B)) Im(Z) = Im ( [ C T A T C T (A T ) r 1 C T ] ) =: Im ( O r (A, C) T) 49

50 Define H r := η 1 η 2 η r η 2 η 3 η r η r η r+1 η 2r 1, where η i = CA i 1 B. Lanczos requires deth i 0 for i = 1,, r. Violated for many systems Look-Ahead Lanczos deth i 0 can be avoided if tridiagonal structure not required. Only requirement: deth r 0. Arnoldi Procedure: The r th order reduced model Σ r matches only r Markov parameters and is not unique. For arbitrary σ, A (σi A) 1, B (σi A) 1, C C(σI A T ) 1 50