Structured Stochastic Uncertainty

Transcription

1 Structured Stochastic Uncertaint Bassam Bamieh Mechanical Engineering, UNIVERSITY OF CALIFORNIA AT SANTA BARBARA () Allerton, Oct 12 1 / 13

2 Stochastic Perturbations u M Perturbations are iid gains: u(k) = (k) (k) M is LTI nec & suff condition is the H 2 norm kmk 2 2 < 1 () Allerton, Oct 12 2 / 13

3 Stochastic Perturbations u M Perturbations are iid gains: u(k) = (k) (k) M is LTI nec & suff condition is the H 2 norm kmk 2 2 < 1 LMI proof (S. Bod, 84) x(k + 1) = (A + BC (k)) x(k) Linear Sstem w Mult. Noise P(k) :=E x(k)x(k) T ) P(k + 1) = A P(k) A T + BC P(k) C T B T LMI cond. for P(K) k!1! 0 = LMI cond. for kmk 2 2 < 1 () Allerton, Oct 12 2 / 13

4 Structured Stochastic Perturbations Perturbations 1 (t),..., n(t) are iid G n 3 5 nec & suff cond. for Mean Square Stabilit given b matrix of H 2 norms! 02 kg 11 k 2 2 kg 1n k C. 5A < 1 kg n1 k 2 2 kg nn k 2 2 (Hinrichsen&Pritchard 95, Lu&Skelton 02, Elia 04) () Allerton, Oct 12 3 / 13

5 Structured Stochastic Perturbations Perturbations 1 (t),..., n(t) are iid G n 3 5 nec & suff cond. for Mean Square Stabilit given b matrix of H 2 norms! 02 kg 11 k 2 2 kg 1n k C. 5A < 1 kg n1 k 2 2 kg nn k 2 2 (Hinrichsen&Pritchard 95, Lu&Skelton 02, Elia 04) Proof involves LMIs and scalings cf. time-varing L 2 and L 1 -norm bounded perturbations Not clear how to generalize to correlated s () Allerton, Oct 12 3 / 13

6 Applications of Structured Stochastic Perturbations Network dnamics with link/node failures, etc. (Elia, Patterson & Bamieh) x(t + 1) =Ax(t) +! MX µ i (t) b i b i i=1 x(t) Linear sstem w. multiplicative noise () Allerton, Oct 12 4 / 13

7 Applications of Structured Stochastic Perturbations Network dnamics with link/node failures, etc. PDEs with random coefficients (Elia, Patterson & Bamieh) ( discrete space ) ( continuous (x, t) = (A + B (x, t) C) (x, t) I e.g. problems from (x, t) = ( apple + apple(x, (x, apple x (x, t) x G () Allerton, Oct 12 4 / 13

8 Stochastic Hdrodnamic Stabilit and Turbulence x Uncertain base flow U(x,, t) = Ū(, t) + (x,, t) spatiotemporal correlations of should be dialed into the model () Allerton, Oct 12 5 / 13

9 AN INPUT-OUTPUT APPROACH TO STRUCTURED STOCHASTIC PERTURBATIONS () Allerton, Oct 12 6 / 13

10 An IO Approach to Stochastic Stabilit Look at the dnamics of the correlation matrix sequences uk, etc. stochastic vector signals u d z G 1 deterministic, matrix-valued signals d u e n 3 Gk l ul G k l=0 l w z n 82 <6 := E 6 4 : k X 7 7[ 1 5 n] 9 = ek e w " monotone sstem ; The loop gain operator plas a central role! 1 X L(X) := Gl X Gl l=0 complexit of L scales w # of perturbations, not state space dimension B. B AMIEH, Structured stochastic uncertaint, th Annual Allerton Conference. AFOSR, Aug / 28

11 IO notion of Mean Square Stabilit Def: G is Mean-Square Stable (MSS) if for white input process u, output process has uniforml bounded variance k = X l g k l u l k = X l g 2 k l l G u u G k := E{ k k } apple X k g 2 k! {z } sup k u k k 2 Z + = kgk 2 2 k u k 1 () Allerton, Oct 12 7 / 13

12 IO notion of Mean Square Stabilit Def: G is Mean-Square Stable (MSS) if for white input process u, output process has uniforml bounded variance k = X gk l ul k = l u G X g2k l l l G u MSS of feedback sstems (insert disturbances, d1, d2 white) u1 d1 2 G1 G2 d1 1 u2 d2 u1 1 2 u2 d2 MSS Feedback Stabilit, All internal signals have uniforml bounded variance sequences () Allerton, Oct 12 7 / 13

13 nec & suff Small Gain Condition (SISO) d u G z e w iid ) z is white ( whiten s e) ) zk = e k u is also white ) k apple kgk 2 2 u k is colored, but uncorrelated with w ) ek = k + w go around the loop with the variance sequences 1 kgk 2 2 k u k 1 apple w + d suff () Allerton, Oct 12 8 / 13

14 nec & suff Small Gain Condition (SISO) d u G z e w iid ) z is white ( whiten s e) ) zk = e k u is also white ) k apple kgk 2 2 u k is colored, but uncorrelated with w ) ek = k + w go around the loop with the variance sequences 1 kgk 2 2 k u k 1 apple w + d suff kgk ) uk unbounded as k! 1 nec don t need to construct destabilizing s! () Allerton, Oct 12 8 / 13

15 nec & suff Structured Small Gain Condition Look at the dnamics of the correlation matrix sequences uk, etc. d u Ʃd G Ʃu k X Gk l ul G k Ʃ l l=0 z n < 6 7 := E : n () 3 e 5 w $ 1 n 9 = ; Ʃz ek Ʃe Ʃw " monotone sstem Allerton, Oct 12 9 / 13

16 nec & suff Structured Small Gain Condition Look at the dnamics of the correlation matrix sequences uk, etc. d u Ʃd G Ʃu k X Gk l ul G k Ʃ l l=0 z n < 6 7 := E : n 3 e 5 w $ 1 n 9 = ; Ʃz going around the loop I L ( uk ) d +! 1 X L(X) := Gl X G l ek Ʃe Ʃw " monotone sstem w loop gain l=0 () Allerton, Oct 12 9 / 13

17 Properties of the Loop Operator L L(X) :=! 1X G l X G l l=0 L maps pos. s. def. matrices to pos. s. def. matrices It is thus cone invariant for the cone of pos. s. def. matrices & 9 a Perron eigenvalue and corresponding pos. s def. eigenmatrix (Parrilo & Khatri 00) () Allerton, Oct / 13

18 Properties of the Loop Operator L L(X) :=! 1X G l X G l l=0 L maps pos. s. def. matrices to pos. s. def. matrices It is thus cone invariant for the cone of pos. s. def. matrices & 9 a Perron eigenvalue and corresponding pos. s def. eigenmatrix these properties impl (1 (L)) uk apple I MSS stabilit condition (nec & suff) (Parrilo & Khatri 00) L ( uk ) apple d + w (L) < 1 () Allerton, Oct / 13

19 Special Case of iid s s iid! = I!! 1X 1X L(X) := I G l X G l = diag G l X G l l=0 l=0 The eigen-matrices of L must be diagonal matrices! L(X) = X How does L act on diagonal matrices? " w1... w n Therefore # = L " v1... v n #!, " w1 #.. w n apple eigs(l) = eigs kg ij k kg 11k 2 2 kg 1nk 2 " 2 v1 # 6 = kg n1k 2 2 kg nnk 2 v n 2 () Allerton, Oct / 13

20 Mutuall Correlated s (but temporall white)! 1X L(X) := G l X G l l=0 L : R n n! R n n In general, it involves terms like 1X g ij (k) g lm (k) k=0 inner products between subsstems impulse responses At worst: L represented as an n 2 n 2 matrix () Allerton, Oct / 13

21 Further Work Robust Performance and Correlations Spatial correlations in s and G have special structure e.g. spatial invariance ) simpler conditions useful for applications to large-scale sstems Partial Differential Equations L is a map on pos. s. spatial operators Temporal correlations in s??? probabl involves other aggregates of the impulse response sequence {g k } () Allerton, Oct / 13