1 / 21 Perturbation of system dynamics and the covariance completion problem Armin Zare Joint work with: Mihailo R. Jovanović Tryphon T. Georgiou 55th

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1 1 / 21 Perturbation of system dynamics and the covariance completion problem Armin Zare Joint work with: Mihailo R. Jovanović Tryphon T. Georgiou 55th IEEE Conference on Decision and Control, Las Vegas, Dec , 2016

2 2 / 21 Control-oriented modeling ẋ = A x + Bf y = Cx stochastic input linearized dynamics stochastic output Objective combine physics-based with data-driven modeling account for statistical signatures of dynamical systems using stochastically forced linear models

3 3 / 21 Response to stochastic input stochastic input f ẋ = Ax + Bf stochastic output white-in-time f state covariance X Lyapunov equation! A X + X A = B B

4 3 / 21 Response to stochastic input stochastic input f ẋ = Ax + Bf stochastic output white-in-time f state covariance X Lyapunov equation! A X + X A = B B colored-in-time f A X + X A = B H H B Georgiou, IEEE TAC 02 H = lim E (x(t) t!1 d (t)) B

5 4 / 21 Filter design filter linear system d f y = A f + Bd ẋ = Ax + Bf f = K + d y = Cx white-in-time input filter dynamics E (d(t 1 ) d (t 2 )) = (t 1 t 2 ) A f = A BK 1 K = 2 B H X 1

6 5 / 21 Equivalent representation Linear system with filter apple apple ẋ A BK = 0 A BK y = C 0 apple x apple x + apple B B d

7 5 / 21 Equivalent representation Linear system with filter apple apple ẋ A BK = 0 A BK y = C 0 apple x apple x + apple B B d? coordinate transformation apple x = apple I 0 I I apple x? reduced-order representation apple apple ẋ A BK BK = 0 A apple x + apple B 0 d y = C 0 apple x

8 6 / 21 Reduced-order representation white noise d filter colored noise f linear system x colored input: ẋ = Ax + B f white noise d modified dynamics x modified dynamics: ẋ = (A B K) x + B d

9 State-feedback interpretation 7 / 21 white noise d filter colored noise f linear system x colored input: ẋ = Ax + B f white noise d + colored noise f linear system x K ẋ = (A B K) x + B d

10 8 / 21 Linear constraint on X white noise d filter colored noise f linear system x colored input: A X + X A = BH HB white noise d + colored noise f linear system x K state-feedback design: (A BK) X + X (A BK) = B B

11 9 / 21 Restricting the number of input channels stochastic input f ẋ = Ax + Bf stochastic output

12 9 / 21 Restricting the number of input channels stochastic input f ẋ = Ax + Bf stochastic output Full rank B e.g. B = I (excite all degrees of freedom) AX + XA = H H ) H = AX

13 9 / 21 Restricting the number of input channels stochastic input f ẋ = Ax + Bf stochastic output Full rank B e.g. B = I (excite all degrees of freedom) AX + XA = H H ) H = AX Reduced-order representation ẋ = 1 2 X 1 x + d complete cancellation of A! Zare, Chen, Jovanović, Georgiou, IEEE TAC 16 Zare, Jovanović, Georgiou, J. Fluid Mech 17 (to appear); arxiv:

14 10 / 21 Objectives dynamics: ẋ = Ax + Bu + Bd control: u = Kx modified dynamics: ẋ = (A BK) x + Bd X = (A b i ki T ) x + Bd i

15 10 / 21 Objectives dynamics: ẋ = Ax + Bu + Bd control: u = Kx modified dynamics: ẋ = (A BK) x + Bd X = (A b i ki T ) x + Bd i objective 1: row-sparsity of K

16 10 / 21 Objectives dynamics: ẋ = Ax + Bu + Bd control: u = Kx modified dynamics: ẋ = (A BK) x + Bd X = (A b i ki T ) x + Bd i objective 1: row-sparsity of K objective 2: minimize J = lim t!1 E (u (t) u(t)) = trace (K X K )

17 11 / 21 Minimum energy covariance completion minimize J(K, X) + X ke i Kk 2 i control energy row-sparsity-promoting penalty function? >0 performance vs sparsity tradeoff

18 12 / 21 Minimum energy covariance completion minimize K, X, trace (KXK ) + nx w i ke i Kk 2 subject to (A B K) X + X (A B K) + B B = 0 i=1 X ij = G ij (i, j) 2I X 0, 0

19 12 / 21 Minimum energy covariance completion minimize K, X, trace (KXK ) + nx w i ke i Kk 2 subject to (A B K) X + X (A B K) + B B = 0 i=1 X ij = G ij (i, j) 2I X 0, 0 Change of variables Y := KX Schur complement yields! SDP characterization

20 13 / 21 Change of variables Y := KX row-sparse structure preserved Polyak, Khlebnikov, Shcherbakov, ECC 13 Dhingra, Jovanović, Luo, CDC 14

21 14 / 21 Minimum energy covariance completion Step 1: identification minimize X, Y, trace YX 1 Y + nx w i ke i Y k 2 subject to A X + X A B Y Y B + B B = 0 Iterative reweighting: X ij = G ij (i, j) 2I X 0, 0 i=1 Use Y from previous iteration to form weights w + i = 1 ke i Y k 2 + Candès, Wakin, Boyd, J. Fourier Anal. Appl. 08

22 15 / 21 Minimum energy covariance completion Step 2: polishing B 2 eliminate columns of B corresponding to row-sparsity structure of Y from Step 1 minimize X, Y, trace YX 1 Y subject to A X + X A B 2 Y Y B 2 + B B = 0 X ij = G ij (i, j) 2I X 0, 0

23 16 / 21 ẋ(k,t) = A(k) x(k,t) + (k,t) y(k,t) = C(k) x(k,t) x = k apple v2, y = 2 4 v 1 v 2 v 3 Example 3 5 horizontal wavenumbers colored-in-time input

24 16 / 21 ẋ(k,t) = A(k) x(k,t) + (k,t) y(k,t) = C(k) x(k,t) x = k apple v2, y = 2 4 v 1 v 2 v 3 Example 3 5 horizontal wavenumbers colored-in-time input output covariance: (k) := lim t!1 E (y(k,t) y (k,t)) known elements of (k)

25 Example 17 / 21 ẋ(k,t) = A(k) x(k,t) + (k,t) y(k,t) = C(k) x(k,t) x = k apple v2, y = 2 4 v 1 v 2 v horizontal wavenumbers colored-in-time input known elements of (k) output covariance: (k) := lim t!1 E (y(k,t) y (k,t)) ẋ = Ax + Bu + Bd y = Cx u = K x

26 Completion of two-point covariances = 10 4, N = 11 true covariances covariance completion recovery before polishing: 90% recovery after polishing: 96% 18 / 21

27 19 / 21 Iterative reweighting number of inputs w/o iter. reweighting 4 w/ iter. reweighting

28 Retained inputs 20 / 21 N = 11, x = apple v2, u = apple u1 u 2 =0 =0.05 =0.5 = 50 u 1 u 2

29 Concluding remarks Covariance completion as a static feedback design? minimum control energy? limited number of input channels Minimum energy covariance completion problem? change of variables? SDP characterization Future work development of customized optimization algorithms application to turbulence modeling Acknowledgments University of Minnesota Doctoral Dissertation Fellowship NSF Award CMMI AFOSR Award FA / 21