Rational Functions and Optimal Decentralized Control Sanjay Lall - PowerPoint PPT Presentation

Rational Functions and Optimal Decentralized Control Sanjay Lall Stanford University Control, Optimization, and Functional Analysis: Synergies and Perspectives Workshop in honor of Professor Bill Helton October 3, 2010

2 Outline • Quadratic invariance • Convexity of the image of a rational function • Representations of rationals • State-space solutions

3 Quadratic Invariance with M. Rotkowitz

4 Decentralized Controller Synthesis � A + BX ( I − DX ) − 1 C � minimize subject to X ∈ S • S is a subspace, e.g., � � �� x y + 3 z � � S = � x, y ∈ R � 0 y • A, B, C, D are given matrices • In control, S corresponds to the set of decentralized controllers • General problem is computationally intractable

5 Rational Functions − 16 x + 12 y + 3 z + 32 x y + 40 x z + 16 x 2   6 x + 8 y + 16 x z − 1   A + BX ( I − DX ) − 1 C =   8 x + 4 y − 3 z + 16 x y + 28 x z + 8 x 2 − 2     6 x + 8 y + 16 x z − 1 Coefficient matrices Subspace of variables � 0 � � 3 � � x � 1 0 0 A = B = X = 2 0 1 x + 2 y y z     4 4 4 3 4 0 C = D =     4 2 2 Coeffs. and vars. may be functions, e.g., in H 2 or H ∞ .

6 Youla Parameterization � A + BX ( I − DX ) − 1 C � minimize subject to X ∈ S Let h ( X ) = − X ( I − DX ) − 1 , and use the change of variables Q = h ( X ) gives equivalent problem minimize � A − BQC � subject to h ( Q ) ∈ S Works when there is no constraint that X ∈ S

7 Quadratic Invariance The subspace S is called quadratically invariant under D if XDX ∈ S for all X ∈ S S is quadratically invariant under D if and only if ⇒ X ( I − DX ) − 1 ∈ S X ∈ S ⇐ Equivalently: QI ⇐ ⇒ h ( S ) = S .

8 Convex Optimization minimize � A − BQC � subject to Q ∈ S • Convex program • Given optimal Q , let X = − Q ( I − DQ ) − 1 • Centralized problem (i.e., without X ∈ S constraint) reduces to 4-block problem: has state-space solution for H 2 and H ∞ • No general state-space solution

9 Theorem • U and Y are Banach spaces • D : U → Y is compact • S ⊂ L ( Y , U ) is a closed subspace � � • Let M = X ∈ L ( Y , U ) ; ( I − DX ) is invertible • Let h ( X ) = − X ( I − DX ) − 1 Then the subspace S is quadratically invariant under D if and only if h ( S ∩ M ) = S ∩ M

10 Example Suppose S and D are given by       ∗ 0 0 0 0 0 0 0 0 0     ∗ ∗ 0 0 0 0 ∗ 0 0 0               D = ∗ ∗ ∗ 0 0 S = X | X = 0 ∗ 0 0 0         ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ 0 0             ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗   S is quadratically invariant, since for general X ∈ S   0 0 0 0 0 0 ∗ 0 0 0     XDX ∼ 0 ∗ 0 0 0     ∗ ∗ ∗ 0 0   ∗ ∗ ∗ 0 ∗

11 Example p p G 1 G 2 G 3 p p c c c c c c K 1 K 2 K 3 • p is the propagation delay • c is the communication delay • Quadratic invariance if c ≤ p

12 Delay Structure In the above problem D and X are structured according to λ p ∗ λ 2 p ∗ λ 3 p ∗ λ c ∗ λ 2 c ∗ λ 3 c ∗     ∗ ∗ λ 2 p ∗ λ 2 c ∗ λ p ∗ λ p ∗ λ c ∗ λ c ∗ ∗ ∗         D = X =     λ 2 p ∗ λ 2 c ∗ λ p ∗ λ p ∗ λ c ∗ λ c ∗ ∗ ∗         λ 3 p ∗ λ 2 p ∗ λ 3 c ∗ λ 2 c ∗ λ p ∗ λ c ∗ ∗ ∗ Quadratically invariant if c ≤ p , since XDX has the structure λ 2 r ∗ λ 3 r ∗ λ r ∗   ∗ λ 2 r ∗ λ r ∗ λ r ∗ ∗     XDX = where r = min { c, p }   λ 2 r ∗ λ r ∗ λ r ∗ ∗     λ 3 r ∗ λ 2 r ∗ λ r ∗ ∗

13 Convexity with Laurent Lessard

14 Convexity Reduced rational optimization to minimize � A − BQC � Q ∈ h ( S ) subject to Are there any cases when h ( S ) is convex, but h ( S ) � = S ?

15 Convexity S is quadratically invariant under D iff h ( S ) is convex. • Hence if h ( S ) is convex, then h ( S ) = S • S is quadratically invariant iff { X ( I − DX ) − 1 | X ∈ S } is convex

16 Theorem Suppose • U and Y are Banach spaces, D ∈ L ( U , Y ) � � • Let M = X ∈ L ( Y , U ) ; ( I − DX ) is invertible • Let h ( X ) = − X ( I − DX ) − 1 If • S ⊂ L ( Y , U ) is a closed double-cone • T ⊂ L ( Y , U ) is convex • h ( S ∩ M ) = T ∩ M Then T ∩ M = S ∩ M .

17 Algebraic Framework with Laurent Lessard

18 Algebraic Version • R is a commutative ring with 2 a unit • D ∈ R m × n • S is an R -module X ∈ R m × n | ( I − DX ) is invertible � � • Let M = If S is quadratically invariant, then h ( S ∩ M ) = S ∩ M • Removes compactness requirements on D • Necessary for particular rings, e.g. proper rational functions

19 2 a unit If 2 �∈ U ( R ) then the above can fail. On the integers, let  �      2 x y z 0 0 0 �   � S = y z 0 x, y, z ∈ Z D = 0 0 1  , �     � z 0 0 0 1 0  � S is QI, since 2 yz z 2 0   z 2 XDX = 0 0   0 0 0 But X ∈ S does not imply h ( X ) ∈ S , e.g.,     0 0 1 1 1 1 X = 0 1 0 ⇒ h ( X ) = 1 1 0 =     1 0 0 1 0 0

20 General Rationals • QI tells us about convexity of { X ( I − DX ) − 1 | K ∈ S } • What about convexity of C = { A + BX ( I − DX ) − 1 C | X ∈ S } • QI depends on the representation used for C . • How to test convexity independent of representation?

21 General Rationals Define � � A + BX ( I − DX ) − 1 C | X ∈ S C = We’d like to solve minimize � X � subject to X ∈ C If S is quadratically invariant under D , then C is linear � � C = A − BQC | Q ∈ S

22 Example   a b 1 b 2 b 2   x 1 � A B � c 1 d 1 0 0   = X = x 2     c 1 d 1 0 0 C D   x 3 c 2 d 2 d 3 d 3   a b 1 b 2 � ′ � A B � x 1 � X ′ = = c 1 d 1 0   C D x 2 x 3 c 2 d 2 d 3 • Both examples have the same set C • The first is not QI, but the second is.

23 Internal Quadratic Invariance � A B � S is called internally quadratically invariant under if C D W 2 SW 1 is QI with respect to D • W 1 and W 2 are projectors satisfying � B � � � range W 1 = range null W 2 = null C D D • Independent of choice of W 1 and W 2 when S is a module • Projectors can always be chosen to be proper

24 Feedback Transformations W 1 and W 2 are projectors satisfying � B � � � range W 1 = range null W 2 = null C D D Add W i into the feedback loop without changing the closed-loop map. z w z w A B A B C D C D W 1 W 2 X W 2 XW 1

25 Theorem • P is rational and proper • D is strictly proper • S is a module in the set of proper rationals • Let h ( K ) = K ( I − DK ) − 1 If S is internally quadratically invariant, then Bh ( S ) C = BSC

26 Delay Example   H 11 z − 1 H 12 w 1 r 2 z − 1 G 1 G 2 z − 2 H 21 z − 1 H 22 r 1 w 2 z − 1   D =   z − 1 z − 1 H 21 H 22   z − 1   z − 1 H 11 z − 2 H 12 y 1 y 2 u 1 u 2 � K 11 K 12 � 0 0 X = K 1 K 2 0 0 K 22 K 21 S is not quadratically invariant with respect to D , so use   z − 1 1 0 0 z − 2 z − 1 1 � 1 0 � 0 0   W 1 = W 2 =   z − 1 1 + z − 2 0 1 0 1 0     z − 1 z − 2 0 0

27 Delay Example w 1 r 2 w 1 r 2 z − 1 z − 1 G 1 G 2 G 1 G 2 r 1 w 2 r 1 w 2 z − 1 z − 1 z − 1 u 1 y 1 y 2 u 2 z − 1 z − 1 y 1 y 2 u 1 u 2 z − 1 K 1 K 2 K 1 K 2 quadratically invariant internally quadratically invariant

28 State-Space with John Swigart

29 State-Space Solutions Is there a state-space method for solving minimize � A − BQC � subject to Q ∈ H 2 Q ∈ S • Riccati equations • Separation structure • Order of optimal controller

30 Two-Player LQR � x 1 ( t + 1) � � A 11 � � x 1 ( t ) � � B 11 � � u 1 ( t ) � 0 0 = + + v ( t ) x 2 ( t + 1) x 2 ( t ) u 2 ( t ) A 21 A 22 B 21 B 22 Minimize S 1 S 2 N 1 � Cx ( t ) + Du ( t ) � 2 � lim N + 1 E N →∞ t =0 K 1 K 2 Objective: pick controller of the form � � u 1 ( t ) = γ 1 t, x 1 (0) , . . . , x 1 ( t ) � � u 2 ( t ) = γ 2 t, x 1 (0) , . . . , x 1 ( t ) , x 2 (0) , . . . , x 2 ( t )

31 LFT Formulation � P 11 + P 12 K ( I − P 22 K ) − 1 P 21 � minimize subject to K is stabilizing K is block lower z w P 11 P 12 P 21 P 22 y u � C � � 0 D � ( zI − A ) − 1 � � • P = + I B 0 0 I K

32 A Standard Heuristic Optimal centralized solution � u 1 � � F 11 F 12 � � x 1 � = u 2 F 21 F 22 x 2 where F = − ( D T D + B T XB ) − 1 B T XA Common heuristic can be unstable: u 1 = F 11 x 1 + F 12 x est Player 1 replaces x 2 with x est 2 2 u 2 = F 21 x 1 + F 22 x 2