Model reduction of large-scale systems

Transcription

1 Model reduction of large-scale systems An overview Thanos Antoulas joint work with Dan Sorensen URL: www-ece.rice.edu/ aca CITI Lecture, Rice, February 2005 Model reduction of large-scale systems p.1

2 Support: Collaborators: NSF + NSF ITR Grants Dan Sorensen (CAAM), TA (ECE) Paul van Dooren (Belgium) Ahmed Sameh (Purdue) Mete Sozen (Purdue) Ananth Grama (Purdue) Chris Hoffmann (Purdue) Kyle Gallivan (Florida State) Serkan Gugercin (Virginia Tech) Model reduction of large-scale systems p.2

3 Contents 1. Problem statement 2. Motivating examples 3. Approximation methods (very briefly) SVD Krylov Krylov/SVD 4. Open problems/concluding remarks Model reduction of large-scale systems p.3

4 The big picture Physical System + Data discretization Modeling ODEs Model reduction reduced # of ODEs PDEs Simulation Control Model reduction of large-scale systems p.4

5 Dynamical systems y 1 ( ) y 2 ( ). y p ( ) x 1 ( ) x 2 ( ). x n ( ) u 1 ( ) u 2 ( ). u m ( ) We consider explicit state equations Σ : ẋ(t) = f(x(t),u(t)), y(t) = h(x(t),u(t)) with state x( ) of dimension n m,p. Model reduction of large-scale systems p.5

6 Problem statement Given: dynamical system Σ = (f,h) with: u(t) R m, x(t) R n, y(t) R p. Problem: Approximate Σ with: ˆΣ = ( ˆf,ĥ), u(t) Rm, ˆx(t) R k, y(t) R p, k n : (1) Approximation error small - global error bound (2) Preservation of stability, passivity (3) Procedure computationally efficient Model reduction of large-scale systems p.6

7 Approximation by projection Unifying feature of approximation methods: projections. Let V, W R n k, such that W V = I k Π = VW is a projection. Define ˆx = W x. Then ˆΣ : d dt ˆx(t) = W f(vˆx(t), u(t)) y(t) = h(vˆx(t), u(t)) Thus ˆΣ is "good" approximation of Σ, if x Πx is "small". Model reduction of large-scale systems p.7

8 Special case: linear dynamical systems Σ: ẋ(t) = Ax(t) + Bu(t), y(t) = Cx(t) + Du(t) Σ = A B R (n+p) (n+m) C D Problem : Approximate Σ by projection: Π = V W, Π 2 = Π, ˆΣ = Â Ĉ ˆB ˆD = W AV CV W B D R (k+p) (k+m), k n Norms : H norm: worst output error y(t) ŷ(t) for u(t) = 1. H 2 -norm: h(t) ĥ(t) Model reduction of large-scale systems p.8

9 Motivating Examples: Simulation/Control 1. Passive devices VLSI circuits Thermal issues 2. Data assimilation North sea forecast Air quality forecast Sensor placement 3. Biological/Molecular systems Honeycomb vibrations MD simulations Heat capacity 4. CVD reactor Bifurcations 5. Mechanical systems: Windscreen vibrations Buildings 6. Optimal cooling Steel profile 7. MEMS: Micro Electro-Mechanical Systems Elf sensor Model reduction of large-scale systems p.9

10 Passive devices : VLSI circuits 1960 s: IC 1971: Intel : Intel Pentium IV 10µ details 2300 components 64KHz speed 0.18µ details 42M components 2GHz speed 2km interconnect 7 layers Model reduction of large-scale systems p.10

11 Passive devices : VLSI circuits Conclusion: Simulations are required to verify that internal electromagnetic fields do not significantly delay or distort circuit signals. Therefore interconnections must be modeled. Electromagnetic modeling of packages and interconnects resulting models very complex: using PEEC methods (discretization of Maxwell s equations): n SPICE: inadequate Source: van der Meijs (Delft) Model reduction of large-scale systems p.11

12 Passive devices : Thermal management of semiconductor device Semiconductor package ANSYS grid Temperature distribution Semiconductor manufacturers provide to customers models of heat dissipation. Procedure: from finite element models generated using ANSYS, a reduced order model is derived, which is provided to the customer. Model reduction of large-scale systems p.12

13 Weather: Storm surge forecast depth Wick n North Shields Huibertgat Harlingen Delfzijl Den Helder Lowestoft IJmuiden-II HoekvanHolland Sheerness Vlissingen Dover m Wave propagation in shallow sea described by shallow water equations. Computational complexity: 60, 000 equations more than 2 days computational time. Allowed: 6 hours! Actually, there are 8 measurement stations Data assimilation Kalman Filter Propagate low-rank factors of data covariance matrix. Source: Verlaan, Heemink (Delft University) 0 Model reduction of large-scale systems p.13

14 Weather: Air quality simulations Data assimilation for atmospheric chemistry models: model the formation and transport of air pollutants (CO). Chemistry transport model (CTM): governed by PDEs + boundary conditions. After spatial discretization ODEs. Grid: x, y direction 1km = 100 points; z direction 10km = 30 points. Terra satellite pictures of CO concentration at the pacific rim Source: Glowinski, He (University of Houston) Model reduction of large-scale systems p.14

15 CVD reactors Stability analysis of steady flows as a function of Grashof, Rayleigh, Reynolds numbers. 16 million variable FEM model of CVD reactor (full 3D Navier Stokes) Used P_ARPACK on Sandia-Intel Tflop, 1024 processors. Computed 6 eigenvalues of largest real part Source: Lehoucq, Salinger (Sandia) Model reduction of large-scale systems p.15

16 Mechanical systems: cars Car windscreen simulation subject to acceleration load. Problem: compute noise at points away from the window. PDE: describes deformation of a structure of a specific material; FE discretization: 7564 nodes (3 layers of 60 by 30 elements). Material: glass with Young modulus N/m 2 ; density 2490kg/m 3 ; Poisson ratio 0.23 coefficients of FE model determined experimentally. The discretized problem has dimension: 22,692. Notice: this problem yields 2nd order equations: Mẍ(t) + Cẋ(t) + Kx(t) = f(t). Source: Meerbergen (Free Field Technologies) Model reduction of large-scale systems p.16

17 Mechanical Systems: Buildings Earthquake prevention Yokohama Landmark Tower: two active tuned mass dampers damp the dominant vibration mode of 0.185Hz. Source: S. Williams Model reduction of large-scale systems p.17

18 Mechanical Systems: Buildings Building Height Control mechanism Damping frequency Damping mass CN Tower, Toronto 533 m Passive tuned mass damper Hancock building, Boston 244 m Two passive tuned dampers 0.14Hz, 2x300t Sydney tower 305 m Passive tuned pendulum 0.1,0.5z, 220t Rokko Island P&G, Kobe 117 m Passive tuned pendulum Hz, 270t Yokohama Landmark tower 296 m Active tuned mass dampers (2) 0.185Hz, 340t Shinjuku Park Tower 296 m Active tuned mass dampers (3) 330t TYG Building, Atsugi 159 m Tuned liquid dampers (720) 0.53Hz, 18.2t Source: S. Williams Model reduction of large-scale systems p.18

19 MEMS: Elk sensor Mercedes A Class semiconductor rotation sensor Source: Laur (Bremen) Model reduction of large-scale systems p.19

20 Approximation methods: Overview Model reduction of large-scale systems p.20

21 Approximation methods: Overview SVD Nonlinear systems POD methods Empirical Gramians Linear systems Balanced truncation Hankel approximation Model reduction of large-scale systems p.20

22 Approximation methods: Overview Krylov SVD Realization Interpolation Lanczos Arnoldi Nonlinear systems POD methods Empirical Gramians Linear systems Balanced truncation Hankel approximation Model reduction of large-scale systems p.20

23 Approximation methods: Overview Krylov SVD Realization Interpolation Lanczos Arnoldi Nonlinear systems POD methods Empirical Gramians Linear systems Balanced truncation Hankel approximation Krylov/SVD Methods Model reduction of large-scale systems p.20

24 SVD Approximation methods SVD: A = UΣV. A prototype problem. Supernova Clown Singular values of Clown and Supernova Supernova: original picture Clown: original picture Clown: rank 6 approximation green: clown red: supernova (log log scale) Supernova: rank 6 approximation Supernova: rank 20 approximation Clown: rank 12 approximation Clown: rank 20 approximation Singular values provide trade-off between accuracy and complexity Model reduction of large-scale systems p.21

25 POD: Proper Orthogonal Decomposition Consider: ẋ(t) = f(x(t), u(t)), y(t) = h(x(t), u(t)). Snapshots of state: X = [x(t 1 ) x(t 2 ) x(t N )] R n N Model reduction of large-scale systems p.22

26 POD: Proper Orthogonal Decomposition Consider: ẋ(t) = f(x(t), u(t)), y(t) = h(x(t), u(t)). Snapshots of state: X = [x(t 1 ) x(t 2 ) x(t N )] R n N SVD: X = UΣV U k Σ k Vk, k n. Approximate the state: ξ(t) = U kx(t) x(t) U k ξ(t), ξ(t) R k Model reduction of large-scale systems p.22

27 POD: Proper Orthogonal Decomposition Consider: ẋ(t) = f(x(t), u(t)), y(t) = h(x(t), u(t)). Snapshots of state: X = [x(t 1 ) x(t 2 ) x(t N )] R n N SVD: X = UΣV U k Σ k Vk, k n. Approximate the state: ξ(t) = U kx(t) x(t) U k ξ(t), ξ(t) R k Project state and output equations. Reduced order system: ξ(t) = U kf(u k ξ(t), u(t)), y(t) = h(u k ξ(t), u(t)) ξ(t) evolves in a low-dimensional space. Model reduction of large-scale systems p.22

28 POD: Proper Orthogonal Decomposition Consider: ẋ(t) = f(x(t), u(t)), y(t) = h(x(t), u(t)). Snapshots of state: X = [x(t 1 ) x(t 2 ) x(t N )] R n N SVD: X = UΣV U k Σ k Vk, k n. Approximate the state: ξ(t) = U kx(t) x(t) U k ξ(t), ξ(t) R k Project state and output equations. Reduced order system: ξ(t) = Ukf(U k ξ(t), u(t)), y(t) = h(u k ξ(t), u(t)) ξ(t) evolves in a low-dimensional space. Issues with POD: (a) Choice of snapshots, (b) singular values not I/O invariants. Model reduction of large-scale systems p.22

29 The Hankel singular values Trade-off between accuracy and complexity for linear dynamical systems is provided by the Hankel Singular Values, which are computed as follows: Gramians P = 0 e At BB e A t dt, Q = 0 e A t C Ce At dt To compute gramians solve 2 Lyapunov equations : AP + PA + BB = 0, P > 0 A Q + QA + C C = 0, Q > 0 σ i = λ i (PQ) σ i : Hankel singular values of system Σ Model reduction of large-scale systems p.23

30 Approximation by balanced truncation There exists basis where P = Q = S = diag (σ 1,, σ n ). This is the balanced basis of the system. In this basis partition: A = A 11 A 12 A 21 A 22, B = B 1 B 2, C = (C 1 C 2 ), S = Σ Σ 2 ROM obtained by balanced truncation ˆΣ = A 11 B 1 C 1 Σ 2 contains the small Hankel singular values. ( Projector: Π = V W I k where V = W = 0 ). Model reduction of large-scale systems p.24

31 Properties of balanced reduction 1. Stability is preserved 2. Global error bound : σ k+1 Σ ˆΣ 2(σ k σ n ) Model reduction of large-scale systems p.25

32 Properties of balanced reduction 1. Stability is preserved 2. Global error bound : σ k+1 Σ ˆΣ 2(σ k σ n ) Drawbacks 1. Dense computations, matrix factorizations and inversions may be ill-conditioned 2. Need whole transformed system in order to truncate number of operations O(n 3 ) Model reduction of large-scale systems p.25

33 Approximation methods: Krylov Krylov Realization Interpolation Lanczos Arnoldi Nonlinear systems POD methods Empirical Gramians Krylov/SVD Methods SVD Linear systems Balanced truncation Hankel approximation Model reduction of large-scale systems p.26

34 Krylov approximation methods Given Σ = ( A C B D ), expand transfer function around s 0 : G(s) = η 0 + η 1 (s s 0 ) + η 2 (s s 0 ) 2 + η 3 (s s 0 ) 3 + Moments at s 0 : η j. Model reduction of large-scale systems p.27

35 Krylov approximation methods Given Σ = ( A C B D ), expand transfer function around s 0 : G(s) = η 0 + η 1 (s s 0 ) + η 2 (s s 0 ) 2 + η 3 (s s 0 ) 3 + Moments at s 0 : η j. Find ˆΣ = ( Â ˆB Ĉ ˆD ), with Ĝ(s) = ˆη 0 + ˆη 1 (s s 0 ) + ˆη 2 (s s 0 ) 2 + ˆη 3 (s s 0 ) 3 + such that for appropriate l: η j = ˆη j, j = 1, 2,,l Model reduction of large-scale systems p.27

36 Key fact for numerical reliability But: computation of moments is numerically problematic ( ) A B Solution: given Σ = moment matching methods can be implemented in a numerically efficient way: C moments can be matched without moment computation iterative implementation : ARNOLDI LANCZOS Special case: Expansion around infinity: η j : Markov parameters. Problem: partial realization. Expansion around zero: η j : moments. Problem: Padé interpolation. In general: arbitrary s 0 C: Problem: rational interpolation. Model reduction of large-scale systems p.28

37 Properties of Krylov methods (a) Number of operations: O(kn 2 ) or O(k 2 n) vs. O(n 3 ) efficiency (b) Only matrix-vector multiplications are required. No matrix factorizations and/or inversions. No need to compute transformed model and then truncate. Model reduction of large-scale systems p.29

38 Properties of Krylov methods (a) Number of operations: O(kn 2 ) or O(k 2 n) vs. O(n 3 ) efficiency (b) Only matrix-vector multiplications are required. No matrix factorizations and/or inversions. No need to compute transformed model and then truncate. (c) Drawbacks global error bound? ˆΣ may not be stable. Remedy: implicit restart. Lanczos breaks down if det H i = 0. Remedy: look-ahead. ˆΣ tends to approximate the high frequency poles of Σ. Remedy: match expansions around other frequencies (rational Lanczos). Model reduction of large-scale systems p.29

39 Approximation methods: Krylov/SVD Krylov Realization Interpolation Lanczos Arnoldi Nonlinear systems POD methods Empirical Gramians SVD Krylov/SVD Methods Linear systems Balanced truncation Hankel approximation Model reduction of large-scale systems p.30

40 Application: LA Administration Building 2nd order model: Mẍ + Dẋ + Kx = Bu, y = Cx, x R n, n = Proportional damping: D = αm + βk Model reduction of large-scale systems p.31

41 300 Damped Evals alpha = 0.67, beta = Freq. Response alpha = 0.67, beta = reduced original Response G( jω) Frequency ω Projection matrices: W = V, spanv = spanned by eigenvectors of 200 dominant eigenvalues ˆM = V MV, ˆD = V DV, ˆK = V KV, ˆB = V B, Ĉ = CV. Model reduction of large-scale systems p.32

42 Bowen Large-Scale Laboratory (Purdue) Test facility: Model reduction of large-scale systems p.33

43 Next application: WTC more than 1,000,000 degrees of freedom Model reduction of large-scale systems p.34

44 Projectors and complexity Projectors are a unifying feature of model reduction methods Project with Π = V W, Π 2 = Π, to get reduced model: ˆx = (W AV )ˆx + (W B)u, y = (CV )ˆx + Du Quality is measured in terms of the frequency response G(jω) = C(jωI A) 1 B + D and in particular its peak, called the H -norm, or its 2-norm (energy) called the H 2 -norm. Computational complexity: n: dimension of original system; k: dimension of reduced system; α: average number of non-zero elements per row of A Dense operations SVD: solve for gramians n 3 αkn SVD: perform balancing n 3 αkn Krylov kn 2 αkn Approximate/ sparse operations Model reduction of large-scale systems p.35

45 Software Goal: reliable solution of large dimensional numerical problems encountered in complex control applications. Turning numerically reliable algorithms into high performance, portable and robust numerical software relies on numerical linear algebra, computer science, computer hardware, etc. General purpose software including model reduction: MATLAB, SCILAB Model reduction software: SLICOT Parallel model reduction software: PSLICOT For large-scale eigenvalue problems: ARPACK SUGAR: MEMS simulation MOZART: Ozone propagation Model reduction of large-scale systems p.36

46 Approximation methods: Properties Krylov Realization Interpolation Lanczos Arnoldi Nonlinear systems POD methods Empirical Gramians SVD Krylov/SVD Methods Linear systems Balanced truncation Hankel approximation Model reduction of large-scale systems p.37

47 Approximation methods: Properties Properties Krylov Realization Interpolation Lanczos Arnoldi Nonlinear systems POD methods Empirical Gramians SVD Linear systems Balanced truncation Hankel approximation numerical efficiency n 500 Krylov/SVD Methods Model reduction of large-scale systems p.37

48 Approximation methods: Properties Properties numerical efficiency n 500 Krylov Realization Interpolation Lanczos Arnoldi Nonlinear systems POD methods Empirical Gramians Krylov/SVD Methods SVD Linear systems Balanced truncation Hankel approximation Properties Stability Error bound n 500 Model reduction of large-scale systems p.37

49 Open problems Decay rate of the Hankel singular values Choice of expansion points for rational Krylov Choice of spectral zeros in passive model reduction Iterative SVD model reduction methods Model reduction of uncertain systems Model reduction of second-order systems Model reduction of descriptor (DAE) systems Model reduction for e.g. parabolic PDE systems Parallel algorithms for sparse computations in model reduction Development/validation of control algorithms based on reduced models In general: Model reduction of structured systems Model reduction of large-scale systems p.38

50 References A.C. Antoulas, Approximation of large-scale dynamical systems, Advances in Design and Control, SIAM, Philadelphia (2005). Dan Sorensen s web site: sorensen Thanos Antoulas web site: aca Model reduction of large-scale systems p.39