Parameterization of Stabilizing Controllers for Interconnected Systems Michael Rotkowitz G D p D p D p D 1 G 2 G 3 G 4 p G 5 K 1 K 2 K 3 K 4 Contributors: Sanjay Lall, Randy Cogill K 5 Department of Signals, Sensors, and Systems Royal Institute of Technology (KTH) Control, Estimation, and Optimization of Interconnected Systems: From Theory to Industrial Applications CDC-ECC 05 Workshop, 11 December 2005
2 M. Rotkowitz, KTH Overview Motivation: Optimal Constrained Control Linear Time-Invariant Quadratic invariance Optimal Control over Networks Summation Nonlinear Time-Varying Iteration New condition
3 M. Rotkowitz, KTH Block Diagrams Two-Input Two-Output z y P 11 P 12 P 21 G u w z P 12 u K y P 11 P 21 w K G Classical K G
4 M. Rotkowitz, KTH Standard Formulation z y P 11 P 12 P 21 G K u w minimize subject to P 11 + P 12 K(I GK) 1 P 21 K stabilizes P
5 M. Rotkowitz, KTH Communicating Controllers G 1 G 2 G 3 G 4 G 5 D D D D D D D D K 1 K 2 K 3 K 4 K 5 Control design problem is to find K which is block tri-diagonal. u 1 K 11 DK 12 0 0 0 y 1 u 2 u 3 u 4 = DK 21 K 22 DK 23 0 0 y 2 0 DK 32 K 33 DK 34 0 y 3 0 0 DK 43 K 44 DK 45 y 4 u 5 0 0 0 DK 54 K 55 y 5
6 M. Rotkowitz, KTH General Formulation The set of K with a given decentralization constraint is a subspace S, called the information constraint. z P 11 P 12 P 21 G w y u We would like to solve K minimize subject to P 11 + P 12 K(I GK) 1 P 21 K stabilizes P K S For general P and S, there is no known tractable solution.
7 M. Rotkowitz, KTH Change of Variables - Stable Plant minimize subject to P 11 + P 12 K(I GK) 1 P 21 K stabilizes P Using the change of variables R = K(I GK) 1 we obtain the following equivalent problem minimize subject to P 11 + P 12 RP 21 R stable This is a convex optimization problem.
8 M. Rotkowitz, KTH Change of Variables - Stable Plant z P 11 w P 12 u K y R P 21 G Using the change of variables R = K(I GK) 1 we obtain the following equivalent problem z P 11 w P 12 R P 21 This is a convex optimization problem.
9 M. Rotkowitz, KTH Breakdown of Convexity - Stable Plant minimize subject to P 11 + P 12 K(I GK) 1 P 21 K stabilizes P K S Using the change of variables R = K(I GK) 1 we obtain the following equivalent problem minimize subject to P 11 + P 12 RP 21 R stable R(I + GR) 1 S
10 M. Rotkowitz, KTH Breakdown of Convexity - Stable Plant z P 11 w P 12 u K y R P 21 G Using the change of variables R = K(I GK) 1 we obtain the following equivalent problem minimize subject to P 11 + P 12 RP 21 R stable R(I + GR) 1 S
11 M. Rotkowitz, KTH Quadratic Invariance The set S is called quadratically invariant with respect to G if KGK S for all K S
12 M. Rotkowitz, KTH Quadratic Invariance The set S is called quadratically invariant with respect to G if KGK S for all K S Main Result S is quadratically invariant with respect to G if and only if K S K(I GK) 1 S
13 M. Rotkowitz, KTH Quadratic Invariance The set S is called quadratically invariant with respect to G if KGK S for all K S Main Result S is quadratically invariant with respect to G if and only if K S K(I GK) 1 S Parameterization {K K stabilizes P, K S} = { } R(I + GR) 1 R stable, R S
14 M. Rotkowitz, KTH Optimal Stabilizing Controller Suppose G R sp and S R p. We would like to solve minimize subject to P 11 + P 12 K(I GK) 1 P 21 K stabilizes P K S
15 M. Rotkowitz, KTH Optimal Stabilizing Controller Suppose G R sp and S R p. We would like to solve minimize subject to P 11 + P 12 K(I GK) 1 P 21 K stabilizes P K S If S is quadratically invariant with respect to G, we may solve which is convex. minimize subject to P 11 + P 12 RP 21 R stable R S
16 M. Rotkowitz, KTH Arbitrary Networks............ i t ij p ij j............ Assuming transmission delays satisfy the triangle inequality (i.e. transmissions take the quickest path) S is quadratically invariant with respect to G if t ij p ij for all i, j
17 M. Rotkowitz, KTH Distributed Control with Delays G D p D p D p D 1 G 2 G 3 G 4 p G 5 K 1 K 3 K 2 K 4 K 5 K 3 sees information from G 3 immediately G 2, G 4 after a delay of t G 1, G 5 after a delay of 2t G 3 is affected by inputs from K 3 immediately K 2, K 4 after a delay of p K 1, K 5 after a delay of 2p
18 M. Rotkowitz, KTH Distributed Control with Delays G D p D p D p D 1 G 2 G 3 G 4 p G 5 K 1 K 3 K 2 K 4 K 5 S is quadratically invariant with respect to G if t p
19 M. Rotkowitz, KTH Distributed Control with Delays G D p D p D p D 1 G 2 G 3 G 4 p G 5 D c D c D c D c D c D c D c K 1 K 3 K 2 K 4 K 5 K 3 sees information from G 3 after a delay of c G 2, G 4 after a delay of c + t G 1, G 5 after a delay of c + 2t G 3 is affected by inputs from K 3 immediately K 2, K 4 after a delay of p K 1, K 5 after a delay of 2p
20 M. Rotkowitz, KTH Distributed Control with Delays G D p D p D p D 1 G 2 G 3 G 4 p G 5 D c D c D c D c D c D c D c K 1 K 3 K 2 K 4 K 5 Without computational delay, S is quadratically invariant with respect to G if t p
21 M. Rotkowitz, KTH Distributed Control with Delays G D p D p D p D 1 G 2 G 3 G 4 p G 5 D c D c D c D c D c D c D c K 1 K 3 K 2 K 4 K 5 Without computational delay, S is quadratically invariant with respect to G if t p If computational delay is also present, then S is quadratically invariant with respect to G if t p + c n 1
22 M. Rotkowitz, KTH Two-Dimensional Lattice K 11 K 12 K 13 K 21 K 22 K 23 K 31 K 32 K 33 Assuming controllers communicate along edges
23 M. Rotkowitz, KTH Two-Dimensional Lattice K 11 K 12 K 13 G 11 D p G 12 D p G 13 D p D p D p K 21 G 21 D p K 22 G 22 D p K G 23 23 D p D p D p K 31 K 32 K 33 G 31 D p G D 32 p G 33 Assuming controllers communicate along edges Assuming dynamics propagate along edges
24 M. Rotkowitz, KTH Two-Dimensional Lattice K 11 K 12 K 13 G 11 D p G 12 D p G 13 D p D p D p K 21 G 21 D p K 22 G 22 D p K G 23 23 D p D p D p K 31 K 32 K 33 G 31 D p G D 32 p G 33 S is quadratically invariant with respect to G if t p
25 M. Rotkowitz, KTH Two-Dimensional Lattice K 11 K 12 K 13 K 21 K 22 K 23 K 31 K 32 K 33 Assuming controllers communicate along edges
26 M. Rotkowitz, KTH Two-Dimensional Lattice K 11 K 12 K 13 G 11 G 12 G 13 K 21 G 21 K 22 G 22 K G 23 23 K 31 K 32 K 33 G 31 G 32 G 33 Assuming controllers communicate along edges Assuming dynamics propagate outward so that delay is proportional to geometric distance
27 M. Rotkowitz, KTH Two-Dimensional Lattice K 11 K 12 K 13 G 11 G 12 G 13 K 21 G 21 K 22 G 22 K G 23 23 K 31 K 32 K 33 G 31 G 32 G 33 S is quadratically invariant with respect to G if t p 2
28 M. Rotkowitz, KTH Sparsity Example Suppose G S = K K For arbitrary K S KGK Hence S is quadratically invariant with respect to G.
29 M. Rotkowitz, KTH Same Structure Synthesis Suppose G S = K K Then KGK so S is not quadratically invariant with respect to G. In fact, K(I GK) 1
30 M. Rotkowitz, KTH Small Gain If a < 1 then (1 a) 1 = 1 + a + a 2 + a 3 +... for example ( 1 1 ) 1 = 1 + 1 2 2 + 1 4 + 1 8 +... more generally, if A < 1 then (1 A) 1 = I + A + A 2 + A 3 +...
31 M. Rotkowitz, KTH Not So Small Gain Let Q = I + A + A 2 + A 3 +... then AQ = A + A 2 + A 3 +... subtracting we get (I A)Q = I and so Q = (I A) 1
32 M. Rotkowitz, KTH Convergence to Unstable Operator 25 Impulse Response 20 Amplitude 15 10 5 Let W (s) = 2 s+1 0 0 0.5 1 1.5 2 2.5 Time (sec) Plot shows impulse response of N k=0 W k for N = 1,..., 7 Converges to that of I I W = s 1 s+1 In what topology do the associated operators converge?
33 M. Rotkowitz, KTH Conditions for Convergence: Inert Very broad, reasonable class of plants and controllers Basically, impulse response must be finite at any time T Arbitrarily large Includes the case G R sp, K R p
34 M. Rotkowitz, KTH Proof Sketch K r y u G KGK S for all K S Suppose K S. Then K(I GK) 1 = K + KGK + K(GK) 2 +... S
35 M. Rotkowitz, KTH Consider Nonlinear Let Q = I + A + A 2 + A 3 +... then AQ = A + A 2 + A 3 +... subtracting we get (I A)Q = I and so Q = (I A) 1
36 M. Rotkowitz, KTH Block Diagram Algebra K r y u G For a given r, we seek y, u such that y = r + Gu u = Ky and then define Y, R such that y = Y r = (I GK) 1 r u = Rr = K(I GK) 1 r
37 M. Rotkowitz, KTH Iteration of Signals K r y u G We can define the following iteration, commensurate with the diagram y (0) = r u (n) = Ky (n) y (n+1) = r + Gu (n)
38 M. Rotkowitz, KTH Iteration of Signals and Operators K r y u G We can define the following iterations, commensurate with the diagram y (0) = r u (n) = Ky (n) y (n+1) = r + Gu (n) Y (0) = I R (n) = KY (n) Y (n+1) = I + GR (n)
39 M. Rotkowitz, KTH Iteration of Signals K r y u G We then get the following recursions y (0) = r u (0) = Kr y (n+1) = r + GKy (n) u (n+1) = K(r + Gu (n) )
40 M. Rotkowitz, KTH Iteration of Signals K r y u G We then get the following recursions y (0) = r u (0) = Kr y (n+1) = r + GKy (n) u (n+1) = K(r + Gu (n) ).. y = lim n y(n) u = lim n u(n)
41 M. Rotkowitz, KTH Iteration of Operators K r y u G We then get the following recursions Y (0) = I R (0) = K Y (n+1) = I + GKY (n) R (n+1) = K(I + GR (n) ).. Y = lim n Y (n) R = lim n R(n)
42 M. Rotkowitz, KTH Conditions for Convergence Very broad, reasonable class of plants and controllers Arbitrarily large Includes the case G strictly causal, K causal Includes the case G R sp, K R p
43 M. Rotkowitz, KTH Parameterization - Nonlinear {K K stabilizes G} = { } R(I + GR) 1 R stable C.A. Desoer and R.W. Liu (1982) V. Anantharam and C.A. Desoer (1984)
44 M. Rotkowitz, KTH New Invariance Condition K 1 (I ± GK 2 ) S for all K 1, K 2 S
45 M. Rotkowitz, KTH New Invariance Condition K 1 (I ± GK 2 ) S for all K 1, K 2 S Invariance under Feedback If this condition is satisfied, then K S K(I GK) 1 S
46 M. Rotkowitz, KTH New Invariance Condition K 1 (I ± GK 2 ) S for all K 1, K 2 S Invariance under Feedback If this condition is satisfied, then K S K(I GK) 1 S Parameterization {K K stabilizes G, K S} = { } R(I + GR) 1 R stable, R S
47 M. Rotkowitz, KTH Proof Sketch K 1 (I ± GK 2 ) S K 1, K 2 S Suppose that K S. Then R (0) = K S. If we assume that R (n) S, then R (n+1) = K(I + GR (n) ) S Thus R (n) S for all n Z +. R = lim n R(n) S
48 M. Rotkowitz, KTH Conclusions Quadratic invariance allows parameterization of all (LTI) stablizing decentralized controllers. Similar condition allows parameterization of all (NLTV) stabilizing decentralized controllers These condition are satisfied when communications are faster than the propagation of dynamics.
49 M. Rotkowitz, KTH Quadratic Invariance The set S is called quadratically invariant with respect to G if KGK S for all K S????????? K 1 (I ± GK 2 ) S for all K 1, K 2 S
50 M. Rotkowitz, KTH Open Questions / Future Work Unstable plant Is there a weaker condition which achieves the same results? Perhaps K(I ± GK) S K S? When is the optimal (possibly nonlinear) controller linear? When (else) does it hold? What should we call it?