Distributed Robustness Analysis Anders Hansson Division of - PowerPoint PPT Presentation

Distributed Robustness Analysis Anders Hansson Division of Automatic Control Link¨ oping University August 22–23, 2016

Outline Robustness Analysis Chordal Sparsity in Semidefinite Programming Domain- and Range-Space Decomposition Proximal Splitting Methods Domain-Space Decomposition Revisited Interior-Point Methods Summary

Robustness Analysis Consider the following uncertain system, p = Gq , q = ∆( p ) , (1) where G ∈ RH p × m is a transfer function matrix, and ∞ ∆ : L p 2 → L m 2 is a bounded and causal operator. The uncertain system in (1) is said to be robustly stable if the interconnection between G and ∆ remains stable for all ∆ in some class.

Integral Quadratic Constraints Let ∆ : L p 2 → L m 2 be a bounded and causal operator. This operator is said to satisfy the IQC defined by Π, i.e., ∆ ∈ IQC(Π), if � � T � � � ∞ v v ∀ v ∈ L p Π dt ≥ 0 , 2 , (2) ∆( v ) ∆( v ) 0 where Π is a bounded and self-adjoint operator. Assuming that Π is linear time-invariant and has a transfer function matrix representation, the IQC in (2) can be written in the frequency domain as � � ∗ � � � ∞ v ( j ω ) � v ( j ω ) � Π( j ω ) d ω ≥ 0 , (3) � � ∆( v )( j ω ) ∆( v )( j ω ) −∞ v and � where ˆ ∆( v ) are the Fourier transforms of the signals

Stability Theorem Theorem (IQC analysis) The uncertain system in (1) is robustly stable, if 1. for all τ ∈ [0 , 1] the interconnection described in (1) , with τ ∆ , is well-posed; 2. for all τ ∈ [0 , 1] , τ ∆ ∈ IQC(Π) ; 3. there exists ǫ > 0 such that � G ( j ω ) � ∗ � G ( j ω ) � Π( j ω ) � − ǫ I , ∀ ω ∈ [0 , ∞ ] . (4) I I Proof. See Megretski and Rantzer, 1997.

Example If ∆ is a linear operator, i.e. q = ∆ p , where ∆ = δ I , δ ∈ [ − 1 , 1], then � X ( j ω ) � Y ( j ω ) Π( j ω ) = Y ( j ω ) ∗ − X ( j ω ) where X ( j ω ) = X ( j ω ) ∗ � 0 and Y ( j ω ) = − Y ( j ω ) ∗ . Typically Π is parameterized with basis functions.

Collection of Uncertain Systems Consider a collection of uncertain systems: p i = G i pq q i + G i pw w i z i = G i zq q i + G i zw w i (5) q i = ∆ i ( p i ) , and let p = ( p 1 , . . . , p N ), q = ( q 1 , . . . , q N ), w = ( w 1 , . . . , w N ) and z = ( z 1 , . . . , z N ).

Interconnection of Uncertain Systems       w 1 z 1 Γ 11 Γ 12 · · · Γ 1 N       w 2 z 2 Γ 21 Γ 22 · · · Γ 2 N       = (6)       . . . . . ... . . . . .       . . . . . w N z N Γ N 1 Γ N 2 · · · Γ NN � �� � � �� � � �� � w z Γ Each of the blocks Γ ij are 0-1 matrices. Interconnected uncertain system: p = G pq q + G pw w z = G zq q + G zw w (7) q = ∆( p ) w = Γ z , where G ⋆ • = diag( G 1 ⋆ • , . . . , G N ⋆ • ) and ∆ = diag(∆ 1 , . . . , ∆ N ).

Lumped Formulation Eliminate w : p = ¯ Gq , q = ∆( p ) , (8) where ¯ G = G pq + G pw ( I − Γ G zw ) − 1 Γ G zq . The interconnected uncertain system is robustly stable if there exists a matrix ¯ Π such that � ¯ � ∗ � ¯ � G ( j ω ) G ( j ω ) ¯ Π( j ω ) � − ǫ I , ∀ ω ∈ [0 , ∞ ] , (9) I I for some ǫ > 0. LMI is dense .

Sparse Formulation Theorem Let ∆ ∈ IQC(¯ Π) . If there exist ¯ Π and X = xI ≻ 0 such that   ∗     ¯ ¯ G pq G pw Π 11 0 Π 12 0 G pq G pw       − Γ T X Γ Γ T X 0 0 G zq G zw G zq G zw        � − ǫ I , (10)    ¯ ¯   I 0 Π 21 0 Π 22 0 I 0 0 0 X Γ 0 − X 0 I I for ǫ > 0 and for all ω ∈ [0 , ∞ ] , then the interconnected uncertain system in (7) is robustly stable.

Sparsity in SDPs General SDP (new definition of x ): c T x minimize (11a) S , x m � F 0 + x i F i + S = 0 , subject to S � 0 . (11b) i =1 with S ∈ S n , x ∈ R m , c ∈ R m and F i ∈ S n for i = 0 , . . . , m . Slack variable S inherits sparsity pattern from problem data. Solvers like DSDP (Benson and Ye, 2005) and and SMCP (Andersen, Dahl and Vandenberghe, 2010) make use of this structure.

Sparsity Graph A sparsity pattern is a set E ⊆ {{ i , j } | i , j ∈ { 1 , 2 , . . . , n }} . A matrix A ∈ S n is said to have a sparsity pattern E if A i , j = A j , i = 0, whenever i � = j and { i , j } / ∈ E , or equivalently A ∈ S n E . The graph G = ( V , E ) with V = { 1 , 2 , . . . , n } is called the sparsity graph associated with the sparsity pattern.

Chordal Graphs and Sparsity Patterns   0 0 0 0 0 x x x   x x x 0 0 0 0 0     x x x x x 0 0 0     0 0 x x x 0 0 0   A =   0 0 x x x x ∗ x     0 0 0 0 0 x x x     0 0 0 0 ∗ x x x 0 0 0 0 0 x x x

Cliques and Clique Trees A maximal clique C i is a maximal subset of V such that its induced subgraph is complete. A tree of maximal cliques for which C i ∩ C j for i � = j is contained in all the cliques on the path connecting C i and C j is said to have the clique intersection property . (Always exists.)

Sparse Cholesky Factorization A sparsity pattern E is chordal if and only if any positive definite E has a Cholesky factorization PAP T = LDL T with matrix A ∈ S n P T ( L + L T ) P ∈ S n E for some permutation matrix P , which is related to the clique intersection property. After permutation sparse postive definite matrices with chordal sparsity pattern have sparse Cholesky factorizations with no fill-in.

Chain of Uncertain Systems δ 1 δ 2 δ N p 1 q 1 p 2 q 2 p N q N z 1 z 2 z N − 1 2 2 G 1 ( s ) G 2 ( s ) G N ( s ) · · · z 2 z 3 z N 1 1

Average CPU Time 10 3 SeDuMi (lumped) SDPT3 (lumped) 10 2 DSDP (sparse) SMCP (sparse) CPU time (seconds) 10 1 10 0 10 − 1 10 − 2 0 20 40 60 80 100 120 140 160 180 200 N

Test for Positive Semidefiniteness (Grone et al., 1984) A partially specified matrix A ∈ S n can be completed to a positive semidefinite matrix if and only if A C i � 0 where C i are the maximal cliques of the graph for the specified entries. ( A C i denotes the sub-matrices obtained by picking out the columns and rows indexed by C i ) Example:   � 1 � � 1 � 1 1 / 2 ? 1 / 2 1 / 3   � 0 ⇔ 1 / 2 1 1 / 3 � 0 & � 0 1 / 2 1 1 / 3 1 ? 1 / 3 1

Dual SDP Primal problem again: c T x minimize (12a) S , x m � F 0 + x i F i + S = 0 , subject to S � 0 . (12b) i =1 with chordal S with cliques C j , j = 1 , . . . , p . Dual SDP: tr ZF 0 minimize (13a) Z subject to tr ZF i = c i , i = 1 , . . . , m (13b) Z � 0 (13c)

Domain-Space Decomposition (Fukuda et al., 2000) Write F i = � p j =1 E j F i j E T with E j containing columns of identity j matrix indexed by clique C j . (Not unique) Since tr ZF i = � p j =1 tr E T j ZE j F i j , equivalent dual problem is: p � tr Z C j F 0 minimize (14a) j Z j =1 p � tr Z C j F i subject to j = c i , i = 1 , . . . , m (14b) j =1 Z C j � 0 i = 1 , . . . , p (14c)

Consensus Constraints Equivalantly in decoupeled form: p � tr Z j F 0 minimize (15a) j Z j =1 p � tr Z j F i subject to j = c i , i = 1 , . . . , m (15b) j =1 Z j � 0 , j = 1 , . . . , p (15c) � � E T E i Z i E T − E j Z j E T E i , j = 0 , (15d) i , j i j ∀ i , j , where i are children of j in a clique tree with the clique intersection property, and where j are all non-leaf nodes of the tree. E i , j contains the columns of the identity matrix indexed by C i ∩ C j .

Range-Space Decomposition (Fukuda et al., 2000) The dual of the previous problem is c T x minimize (16a) x , U m � subject to F 0 x i F i j + j + G j ( U ) � 0 j = 1 , . . . , p (16b) i =1 where x ∈ R m , with � G j ( U ) = E T k E k , j U k , j E T E T j E i , j U i , j E T k , j E k − i , j E j i ∈ ch ( j ) where U i , j ∈ S | C i ∩ C j | , and where k is the parent of j in the clique tree. (For the root and for the leafs some of the terms are not there) Often the above LMIs are loosely coupled, i.e. many F i j are zero.

Example Find x = ( x 1 , . . . , x 4 ) such that   0 x 1 x 2   � 0 x 2 x 1 x 3 0 x 3 x 4 is equivalent to find ( x , u ) such that � x 1 � � − u � x 2 x 3 � 0 & � 0 x 1 + u x 2 x 3 x 4

Decomposition and Product Space Formulation Feasibility problem from range-space decomposition can with v = ( x , U ) be phrased as find (17a) v subject to v ∈ C j , j = 1 , . . . , p , (17b) where � � m � � � F 0 x i F i C j = v j + j + G j ( U ) � 0 i =1 Let C j = { s j ∈ R |J j | | E T J j s j ∈ C i } , ¯ j = 1 , . . . , p , (18) such that s j ∈ ¯ J j s j ∈ C j , where E J j are composed of C j implies E T rows of the identity matrix indexed by the set J j , which is the set of i such that v i is constrained by C j . Let I i = { k | i ∈ J k } , i.e. the set of indeces of constraints, which depends on v i .

Example revisited Find ( x , u ) = ( x 1 , x 2 , x 3 , x 4 , u ) such that � x 1 � � − u � x 2 x 3 � 0 & � 0 x 2 x 1 + u x 3 x 4 Hence J 1 = { 1 , 2 , 5 } ; J 2 = { 3 , 4 , 5 } and I 1 = { 1 } ; I 2 = { 1 } ; I 3 = { 2 } ; I 4 = { 2 } ; I 5 = { 1 , 2 }