An ADMM algorithm for optimal sensor and actuator selection

Transcription

1 An ADMM algorithm for optimal sensor and actuator selection Neil K. Dhingra, Mihailo R. Jovanović, and Zhi-Quan Luo 53rd Conference on Decision and Control, Los Angeles, California, / 25

2 2 / 25 Motivation Systematic approach to selecting sensors and actuators Phasor Measurement Units in power networks Autonomous formations of vehicles Flexible wing aircraft

3 3 / 25 Recent work Particular measurement models Linear measurements (Joshi, Boyd 09) Cramer-Rao lower bound (Roy, Chepuri, Leus 13) Optimal experiment design (Kekatos, Giannakis, Wollenberg 12) Related approaches Maximize effect of actuators (Summers, Lygeros 14) ADMM for nonconvex formulation (Masazade, Fardad, Varshney 12) Convex characterization as Semidefinite Program SDP formulation (Polyak, Khlebnikov, Shcherbakov 13) Discrete time (Munz, Pfister, Wolfrum 14)

4 4 / 25 Problem formulation LTI system with many potential actuators ẋ = Ax + B 1 d + B 2 u [ ] [ Q 1/2 0 z = x + 0 R 1/2 ] u Objective: Minimize closed loop H 2 norm using a few actuators

5 Actuator selection via regularization 5 / 25 minimize J(K) + γ i e T i K 2 variance amplification (closed loop H 2 norm) row-sparsity-promoting penalty function

6 Actuator selection via regularization minimize J(K) + γ i e T i K 2 variance amplification (closed loop H 2 norm) row-sparsity-promoting penalty function γ specifies relative importance of sparsity γ = 0 uses all actuators Lin, Fardad, Jovanovic, ACC 11, TAC 13 5 / 25

7 6 / 25 State feedback H 2 problem minimize X,K trace ( Q X) + trace(r K X K T) subject to (A B 2 K) X + X (A B 2 K) T + B 1 B T 1 = 0 X 0

8 7 / 25 LMI formulation Standard change of variables Y := KX yields convex form minimize X,Y trace ( Q X) + trace(r Y X 1 Y T) subject to A X B 2 Y + X A T Y T B T 2 + B 1 B T 1 = 0 X 0

9 8 / 25 Challenge: impose structure on K in (X, Y) coordinates Linear constraints on K become nonlinear on Y and X

10 Row-sparsity is preserved 9 / 25

11 10 / 25 Actuator selection - Semidefinite Program minimize X,Y trace(q X) + trace(r Y X 1 Y T ) + γ e T i Y 2 }{{} f (X,Y) subject to A X B 2 Y + X A T Y T B T 2 + B 1 B T 1 = 0 X 0 Promote row sparsity of Y instead of K Polyak, Khlebnikov, Shcherbakov, ECC 13 Munz, Pfister, Wolfrum, TAC 14

12 11 / 25 Computational complexity trace( R Y X 1 Y T ) = trace(r Θ) Worst case computational complexity: O ( (n + m) 6)

13 12 / 25 Alternating Direction Method of Multipliers (ADMM) Splitting method for convex functions with linear constraints Form augmented Lagrangian L ρ (X, Y, Λ) := f (X, Y) + Λ, h(x, Y) + ρ 2 h(x, Y) 2 F h(x, Y) := A X B 2 Y + X A T Y T B T 2 + B 1 B T 1

14 13 / 25 Update variables separately X k+1 = arg min X L ρ (X, Y k, Λ k ) Y k+1 = arg min Y L ρ (X k+1, Y, Λ k ) Λ k+1 = Λ k + ρ h(x k+1, Y k+1 ) Boyd, Parikh, Chu, Peleato, Eckstein 11

15 14 / 25 minimize Y Y-minimization γ e T i Y 2 + ρ 2 B 2 Y + Y T B T 2 + V k 2 F Group LASSO

16 14 / 25 minimize Y Y-minimization γ e T i Y 2 + ρ 2 B 2 Y + Y T B T 2 + V k 2 F Group LASSO Proximal method: sequentially approximate with minimize Y γ e T i Y Y V k 2 F Computational complexity of O(n 2 m)

17 15 / 25 X-minimization minimize X trace ( X Q + X 1 Yk T R Y ) + ρ A X + X AT + 2 U k 2 F Can formulate as SDP (worst case computational complexity O(n 6 ))

18 15 / 25 X-minimization minimize X trace ( X Q + X 1 Yk T R Y ) + ρ A X + X AT + 2 U k 2 F Can formulate as SDP (worst case computational complexity O(n 6 )) Projected Newton s method Find search directions with conjugate gradient (n 5 ) Project onto {X : X 0}

19 16 / 25 Sensor Selection: SDP Formulation Change of coordinates Z := X o L

20 17 / 25 Example - Flexible Wing Aircraft Select subset of sensors to minimize estimation error

21 Using less than half the sensors degrades performance by only 20% 18 / 25

22 19 / 25 Example - Mass Spring System Dynamics p i = (p i+1 p i ) + (p i 1 p i ) + d i Sensors Position of each mass Velocity of each mass

23 20 / 25 Computation time for γ = 100 n states and n outputs 50 states and m outputs

24 21 / 25 Conclusions Convex characterization of sensor or actuator selection Splitting algorithm Solve simpler sub-problems Scaling dependent on number of states Future work More efficient X-minimization Customized Interior Point Method Alternative splitting methods

25 22 / 25 Acknowledgements Support: NASA JPFP Fellowship MnDRIVE Graduate Scholars Program NSF ECCS Collaboration on aircraft example Marty Brenner, NASA Claudia Moreno and Harald Pfifer

26 23 / 25 Dual Problem Lagrangian g(y) = maximize e T i U γ U, Y L(X, Y, Λ, M) = trace(qx + X 1 Y T RY) + U, Y + Λ, AX + XA T BY Y T B T + V M, X where M 0 [ X L Y L ] = [ Q X 1 Y T RYX 1 + A T Λ + ΛA 2RYX 1 + U 2B T 2 Λ ] Dual Problem maximize V, Λ subject to Q + ΛA + A T Λ (B T 2 Λ 1 2 U)T R 1 (B T 2 Λ 1 2 U) 0 e T i U 2 γ

27 24 / 25 Sensor Selection ẋ = A x + B 1 d y = C x + η Observer estimates state from output ˆx = A ˆx + L (y ŷ)

28 24 / 25 Sensor Selection ẋ = A x + B 1 d y = C x + η Observer estimates state from output ˆx = A ˆx + L (y ŷ) Steady-state variance of the estimation error x ˆx J(L) = trace (X o B 1 B T 1 + X o L L T ) Observability gramian X o 0 (A L C) T X o + X o (A L C) + I = 0.

29 25 / 25 Sensor Selection: SDP Formulation Change of coordinates Z := X o L minimize X o, Z trace ( X o B 1 B T 1 + X 1 o Z Z T) + γ r Z e i 2 i = 1 subject to A T X o + X o A C T Z ZC + I = 0 X o 0