Stochastic dynamical modeling:

Stochastic dynamical modeling: Structured matrix completion of partially available statistics Armin Zare www-bcf.usc.edu/ arminzar Joint work with: Yongxin Chen Mihailo R. Jovanovic Tryphon T. Georgiou Beer Talk Seminar Series, USC, May 18th, 2017 1 / 43

Stochastic dynamical modeling: Structured matrix completion of partially available statistics Armin Zare www-bcf.usc.edu/ arminzar Joint work with: Yongxin Chen Mihailo R. Jovanović Tryphon T. Georgiou Beer Talk Seminar Series, USC, May 18th, 2017 2 / 43

3 / 43 Control-oriented modeling ẋ = A x + B u y = C x stochastic input linearized dynamics stochastic output Objective account for second-order statistics using stochastically forced linear models combine physics-based with data-driven modeling

4 / 43 Motivating application: flow control Image credit: technology: application: challenge: shear-stress sensors/surface-deformation actuators surface modification turbulence suppression analysis, optimization and control for complex flow dynamics

5 / 43 stochastic input linearized dynamics stochastic output Proposed approach view second-order statistics as data for an inverse problem identify forcing statistics to account for available statistics

5 / 43 stochastic input linearized dynamics stochastic output Proposed approach view second-order statistics as data for an inverse problem identify forcing statistics to account for available statistics Key questions Can we identify forcing statistics to reproduce available statistics? Can this be done by white-in-time stochastic process?

Outline Structured covariance completion problem embed available statistical features in low-complexity models complete unavailable data (via convex optimization) Algorithms Alternating Direction Method of Multipliers (ADMM) Alternating Minimization Algorithm (AMA) Turbulence modeling case study: turbulent channel flow Summary and outlook 6 / 43

STRUCTURED COVARIANCE COMPLETION 7 / 43

8 / 43 Response to stochastic input stochastic input u ẋ = Ax + Bu stochastic output white-in-time u state covariance X Lyapunov equation A X + X A = B ΩB

8 / 43 Response to stochastic input stochastic input u ẋ = Ax + Bu stochastic output white-in-time u state covariance X Lyapunov equation A X + X A = B ΩB colored-in-time u A X + X A = B H H B H = lim t E (x(t) u (t)) + 1 2 B Ω Georgiou, IEEE TAC 02

9 / 43 Theorem X = X 0 is the steady-state covariance of (A, B) rank there is a solution H to B H + H B = (A X + XA ) [ AX + XA ] B B 0 = rank [ 0 ] B B 0 Georgiou, IEEE TAC 02

Problem setup known elements of X A X + X A = (B H + H B ) }{{} Z Knowns system matrix A partially available entries in X Unknowns missing entries of X disturbance statistics Z { input matrix B input power spectrum H 10 / 43

11 / 43 Number of input channels A X + X A = (B H + H B ) }{{} Z number of input channels: limited by the rank of Z Chen, Jovanović, Georgiou, CDC 13

12 / 43 Inverse problem Convex optimization problem minimize X, Z log det (X) + γ Z subject to A X + X A + Z = 0 physics X ij = G ij for given i, j available data

12 / 43 Inverse problem Convex optimization problem minimize X, Z log det (X) + γ Z subject to A X + X A + Z = 0 physics X ij = G ij for given i, j available data nuclear norm: proxy for rank minimization Z := σ i (Z) Fazel, Boyd, Hindi, Recht, Parrilo, Candès, Chandrasekaran,...

13 / 43 Filter design filter linear system d u y ξ = A f ξ + B d ẋ = A x + B u u = K ξ + d y = C x white-in-time input filter dynamics A f E (d(t 1 ) d (t 2 )) = δ(t 1 t 2 ) = A B K ( ) 1 K = 2 B H X 1

14 / 43 Equivalent representation Linear system with filter [ ] [ ] [ ] [ ẋ A BK x B = + ξ 0 A BK ξ B y = [ C 0 ] [ ] x ξ ] d

14 / 43 Equivalent representation Linear system with filter [ ] [ ] [ ] [ ẋ A BK x B = + ξ 0 A BK ξ B y = [ C 0 ] [ ] x ξ ] d coordinate transformation [ ] x = φ [ I 0 I I ] [ x ξ ] reduced-order representation [ ] [ ẋ A BK BK = φ 0 A y = [ C 0 ] [ ] x φ ] [ x φ ] + [ B 0 ] d

15 / 43 Low-rank modification white noise d filter colored noise u linear dynamics x colored input: ẋ = A x + B u white noise d modified dynamics x low-rank modification: ẋ = (A B K) x + B d

State-feedback interpretation 16 / 43 white noise d filter colored noise u linear dynamics x colored input: ẋ = A x + B u white noise d + colored noise u linear dynamics x K ẋ = (A B K) x + B d

17 / 43 Restricting the number of input channels stochastic input ẋ = Ax + Bu stochastic output

17 / 43 Restricting the number of input channels stochastic input ẋ = Ax + Bu stochastic output Full rank B B = I (excite all degrees of freedom) A X + XA = H H H = A X

17 / 43 Restricting the number of input channels stochastic input ẋ = Ax + Bu stochastic output Full rank B B = I (excite all degrees of freedom) A X + XA = H H H = A X Complete cancellation of A! ẋ = 1 2 X 1 x + d

ALGORITHMS 18 / 43

19 / 43 Covariance completion problem minimize X, Z log det (X) + γ Z subject to A X + X A + Z = 0 X ij = G ij for given i, j

20 / 43 Primal and dual problems Primal minimize X, Z log det (X) + γ Z subject to A X + B Z C = 0

20 / 43 Primal and dual problems Primal minimize X, Z log det (X) + γ Z subject to A X + B Z C = 0 Dual maximize Y 1, Y 2 log det (A Y ) G, Y 2 subject to Y 1 2 γ A adjoint of A; Y := [ Y1 Y 2 ]

21 / 43 SDP characterization Z = Z + Z, Z + 0, Z 0 minimize X, Z +, Z log det (X) + γ trace (Z + + Z ) subject to A X + B Z C = 0 Z + 0, Z 0

22 / 43 Customized algorithms Alternating Direction Method of Multipliers (ADMM) Boyd et al., Found. Trends Mach. Learn. 11 Alternating Minimization Algorithm (AMA) Tseng, SIAM J. Control Optim. 91

23 / 43 Augmented Lagrangian L ρ (X, Z; Y ) = log det (X) + γ Z + Y, A X + B Z C + ρ 2 A X + B Z C 2 F

23 / 43 Augmented Lagrangian L ρ (X, Z; Y ) = log det (X) + γ Z + Y, A X + B Z C + ρ 2 A X + B Z C 2 F Method of multipliers minimizes L ρ jointly over X and Z ( X k+1, Z k+1) := argmin X, Z L ρ (X, Z; Y k ) Y k+1 := Y k + ρ ( A X k+1 + B Z k+1 C )

24 / 43 Iterative ADMM steps X k+1 Z k+1 ADMM vs AMA := argmin L ρ (X, Z k ; Y k ) X := argmin L ρ (X k+1, Z; Y k ) Z Y k+1 := Y k + ρ ( A X k+1 + B Z k+1 C )

24 / 43 Iterative ADMM steps X k+1 Z k+1 ADMM vs AMA := argmin L ρ (X, Z k ; Y k ) X := argmin L ρ (X k+1, Z; Y k ) Z Y k+1 := Y k + ρ ( A X k+1 + B Z k+1 C ) Iterative AMA steps X k+1 Z k+1 := argmin L 0 (X, Z k ; Y k ) X := argmin L ρ (X k+1, Z; Y k ) Z Y k+1 := Y k ( + ρ k A X k+1 + B Z k+1 C )

25 / 43 Iterative ADMM steps ADMM vs AMA X k+1 := argmin L ρ (X, Z k ; Y k ) X := argmin L ρ (X k+1, Z; Y k ) Z Y k+1 := Y k + ρ ( A X k+1 + B Z k+1 C ) Z k+1 Iterative AMA steps X k+1 := argmin L 0 (X, Z k ; Y k ) X := argmin L ρ (X k+1, Z; Y k ) Z Y k+1 := Y k ( + ρ k A X k+1 + B Z k+1 C ) Z k+1

26 / 43 Z-update minimize Z V k := ( A 1 X k+1 + (1/ρ) Y1 k ) = U Σ U svd Z + ρ 2 Z V k 2 F soft thresholding singular value thresholding Z k+1 = U S γ/ρ (Σ) U complexity: O(n 3 )

27 / 43 X-update in AMA minimize X log det (X) + Y k, A X explicit solution: X k+1 = ( A Y k) 1 A adjoint of A complexity: O(n 3 )

28 / 43 X-update in ADMM minimize X log det (X) + ρ 2 A X U k 2 F optimality condition: X 1 + ρ A ( A X U k) = 0 challenge: solution: non-unitary A proximal gradient algorithm

29 / 43 Proximal gradient method Proximal algorithm linearize ρ 2 A X U k 2 F around X i add proximal term µ 2 X X i 2 F optimality condition: µ X X 1 = ( µ I ρ A A ) X i + ρ A ( U k) = V Λ V

29 / 43 Proximal gradient method Proximal algorithm linearize ρ 2 A X U k 2 F around X i add proximal term µ 2 X X i 2 F optimality condition: µ X X 1 = ( µ I ρ A A ) X i + ρ A ( U k) = V Λ V explicit solution: X i+1 = V diag (g) V g j = λ j 2µ + ( ) 2 λj + 1 2µ µ complexity per iteration: O(n 3 )

30 / 43 Y -update in AMA ( Y1 k+1 = sat γ Y k 1 + ρ k A 1 X k+1) Y 1 2 γ ( Y2 k+1 = Y2 k + ρ k A2 X k+1 G ) saturation operator sat γ (M) = M S γ (M) saturation of singular values

31 / 43 Y -update in AMA ( Y1 k+1 = sat γ Y k 1 + ρ k A 1 X k+1) Y 1 2 γ ( Y2 k+1 = Y2 k + ρ k A2 X k+1 G ) Dual problem minimize Y 1, Y 2 subject to ) log det (A 1 Y 1 + A 2 Y 2 Y 1 2 γ G, Y 2

32 / 43 Properties of AMA Covariance completion via AMA proximal gradient on the dual problem sub-linear convergence with constant step-size Step-size selection Barzilla-Borwein initialization followed by backtracking positive definiteness of X k+1 sufficient dual ascent Software Dalal & Rajaratnam, arxiv:1405.3034 Zare, Chen, Jovanović, Georgiou, IEEE TAC 17 http://www.ece.umn.edu/ mihailo/software/ccama/

TURBULENCE MODELING 33 / 43

34 / 43 output covariance: Φ(k) := lim t E (v(t, k) v (t, k)) v = [ u v w ] T Turbulent channel flow k horizontal wavenumbers known elements of Φ(k) A = [ ] A11 0 A 12 A 22

35 / 43 Key observation Lyapunov equation A X + X A = B ΩB λ i (A X DNS + X DNS A ) i White-in-time stochastic excitation too restrictive! Jovanović & Georgiou, APS 10

normal stresses One-point correlations shear stress y y Nonlinear simulations Solution to inverse problem 36 / 43

37 / 43 γ dependence Φ(k) Φdns(k) F Φdns(k) F 100 γ

Two-point correlations; γ = 300 38 / 43 nonlinear simulations covariance completion Φ 11 Φ 12

Verification in stochastic simulations R τ = 180; k x = 2.5, k z = 7 uu uv y y Direct Numerical Simulations Linear Stochastic Simulations E t Zare, Jovanović, Georgiou, J. Fluid Mech. 17 39 / 43

SUMMARY AND OUTLOOK 40 / 43

Summary Framework for explaining second-order statistics using stochastically forced linear models Complete statistical signatures to be consistent with the known dynamics Filter design Low-rank dynamical modification Covariance completion problem Convex optimization AMA vs ADMM Application to turbulent channel flow Reasonable completion of two-point correlations Linear stochastic simulations 41 / 43

Challenges Turbulence modeling modeling higher-order moments development of turbulence closure models design of flow estimators/controllers Algorithmic alternative rank approximations (e.g., iterative re-weighting, matrix factorization, r norm) Grussler, Zare, Jovanović, Rantzer, CDC 16 improving scalability Theoretical conditions for exact recovery 42 / 43

Publications 43 / 43 Theoretical and algorithmic developments Zare, Chen, Jovanović, Georgiou, IEEE TAC 17 Application to turbulent flows Zare, Jovanović, Georgiou, J. Fluid Mech. 17 Stochastic modeling of spatially evolving flows Ran, Zare, Hack, Jovanović, ACC 17 Use of the r norm in covariance completion problems Grussler, Zare, Jovanović, Rantzer, CDC 16 Reformulation as a perturbation to system dynamics Zare, Jovanović, Georgiou, CDC 16 Zare, Dhingra, Jovanović, Georgiou, CDC 17 (submitted)