Parameterization of Stabilizing Controllers for...
Transcript of Parameterization of Stabilizing Controllers for...
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Parameterization of StabilizingControllers for Interconnected Systems
Michael Rotkowitz
Dt Dt Dt Dt
Dp Dp Dp DpG1 G2 G3 G4 G5
K5K4K3K2K1
Contributors: Sanjay Lall, Randy Cogill
Department of Signals, Sensors, and SystemsRoyal Institute of Technology (KTH)
Control, Estimation, and Optimization of Interconnected Systems:From Theory to Industrial ApplicationsCDC-ECC’05 Workshop, 11 December 2005
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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
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Block Diagrams
Two-Input Two-Output
P11 P12P21 G
K
w
uy
z
K
G
P11
P21P12
w
u y
z
Classical
K G
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Standard Formulation
P11 P12P21 G
K
w
uy
z
minimize ‖P11 + P12K(I − GK)−1P21‖subject to K stabilizes P
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Communicating Controllers
D DD D
D DD D
K5
G5G4
K4
G1 G2 G3
K3K2K1
Control design problem is to find K which is block tri-diagonal.
u1u2u3u4u5
=
K11 DK12 0 0 0DK21 K22 DK23 0 0
0 DK32 K33 DK34 00 0 DK43 K44 DK450 0 0 DK54 K55
y1y2y3y4y5
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P11 P12P21 G
K
w
uy
z
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General FormulationThe set of K with a given decentralizationconstraint is a subspace S, called theinformation constraint.
We would like to solve
minimize ‖P11 + P12K(I − GK)−1P21‖subject to K stabilizes P
K ∈ S
• For general P and S, there is no known tractable solution.
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Change of Variables - Stable Plant
minimize ‖P11 + P12K(I − GK)−1P21‖subject to K stabilizes P
Using the change of variables
R = K(I − GK)−1
we obtain the following equivalent problem
minimize ‖P11 + P12RP21‖subject to R stable
This is a convex optimization problem.
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Change of Variables - Stable Plant
K
G
P11
P21P12
w
u y
z
R
Using the change of variables
R = K(I − GK)−1
we obtain the following equivalent problem
P11
P21P12
wz
R
This is a convex optimization problem.
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Breakdown of Convexity - Stable Plant
minimize ‖P11 + P12K(I − GK)−1P21‖subject to K stabilizes P
K ∈ S
Using the change of variables
R = K(I − GK)−1
we obtain the following equivalent problem
minimize ‖P11 + P12RP21‖subject to R stable
R(I + GR)−1 ∈ S
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Breakdown of Convexity - Stable Plant
K
G
P11
P21P12
w
u y
z
R
Using the change of variables
R = K(I − GK)−1
we obtain the following equivalent problem
minimize ‖P11 + P12RP21‖subject to R stable
R(I + GR)−1 ∈ S
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Quadratic InvarianceThe set S is called quadratically invariant with respect to G if
KGK ∈ S for all K ∈ S
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Quadratic InvarianceThe set S is called quadratically invariant with respect to G if
KGK ∈ S for all K ∈ S
Main ResultS is quadratically invariant with respect to G if and only if
K ∈ S ⇐⇒ K(I − GK)−1 ∈ S
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Quadratic InvarianceThe set S is called quadratically invariant with respect to G if
KGK ∈ S for all K ∈ S
Main ResultS 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}
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Optimal Stabilizing ControllerSuppose G ∈ Rsp and S ⊆ Rp.
We would like to solve
minimize ‖P11 + P12K(I − GK)−1P21‖subject to K stabilizes P
K ∈ S
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Optimal Stabilizing ControllerSuppose G ∈ Rsp and S ⊆ Rp.
We would like to solve
minimize ‖P11 + P12K(I − GK)−1P21‖subject to K stabilizes P
K ∈ S
If S is quadratically invariant with respect to G, we may solve
minimize ‖P11 + P12RP21‖subject to R stable
R ∈ Swhich is convex.
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Arbitrary Networks
i jtij
pij
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Assuming transmission delays satisfy the triangle inequality(i.e. transmissions take the quickest path)
S is quadratically invariant with respect to G if
tij ≤ pij for all i, j
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Distributed Control with Delays
Dt Dt Dt Dt
Dp Dp Dp DpG1 G2 G3 G4 G5
K5K4K3K2K1
• K3 sees information from G3 immediatelyG2, G4 after a delay of tG1, G5 after a delay of 2t
• G3 is affected by inputs from K3 immediatelyK2, K4 after a delay of pK1, K5 after a delay of 2p
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Distributed Control with Delays
Dt Dt Dt Dt
Dp Dp Dp DpG1 G2 G3 G4 G5
K5K4K3K2K1
S is quadratically invariant with respect to G if
t ≤ p
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Distributed Control with Delays
Dc DcDc Dc Dc
Dt Dt Dt Dt
Dp Dp Dp DpG1 G2 G3 G4 G5
K5K4K3K2K1
Dc Dc Dc Dc Dc
• K3 sees information from G3 after a delay of cG2, G4 after a delay of c + tG1, G5 after a delay of c + 2t
• G3 is affected by inputs from K3 immediatelyK2, K4 after a delay of pK1, K5 after a delay of 2p
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Distributed Control with Delays
Dc DcDc Dc Dc
Dt Dt Dt Dt
Dp Dp Dp DpG1 G2 G3 G4 G5
K5K4K3K2K1
Dc Dc Dc Dc Dc
Without computational delay,S is quadratically invariant with respect to G if
t ≤ p
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Distributed Control with Delays
Dc DcDc Dc Dc
Dt Dt Dt Dt
Dp Dp Dp DpG1 G2 G3 G4 G5
K5K4K3K2K1
Dc Dc Dc Dc Dc
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 + cn − 1
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Two-Dimensional Lattice
Dt Dt Dt
Dt
Dt Dt
Dt
Dt
Dt
Dt
Dt
DtK11 K12 K13
K23K22K21
K31 K32 K33
Assuming controllers communicate along edges
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Two-Dimensional Lattice
Dt Dt Dt
Dt
Dt Dt
Dt
Dt
Dt
Dt
Dt
DtK11 K12 K13
K23K22K21
K31 K32 K33
G11 G12 G13
G23G22G21
G31 G32 G33
Dp
Dp
Dp
Dp
Dp
Dp
Dp
Dp Dp
Dp Dp
Dp
Assuming controllers communicate along edges
Assuming dynamics propagate along edges
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Two-Dimensional Lattice
Dt Dt Dt
Dt
Dt Dt
Dt
Dt
Dt
Dt
Dt
DtK11 K12 K13
K23K22K21
K31 K32 K33
G11 G12 G13
G23G22G21
G31 G32 G33
Dp
Dp
Dp
Dp
Dp
Dp
Dp
Dp Dp
Dp Dp
Dp
S is quadratically invariant with respect to G if
t ≤ p
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Two-Dimensional Lattice
Dt Dt Dt
Dt
Dt Dt
Dt
Dt
Dt
Dt
Dt
DtK11 K12 K13
K23K22K21
K31 K32 K33
Assuming controllers communicate along edges
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Two-Dimensional Lattice
Dt Dt Dt
Dt
Dt Dt
Dt
Dt
Dt
Dt
Dt
DtK11 K12 K13
K23K22K21
K31 K32 K33
G12 G13
G23G22
G32 G33
G11
G21
G31
Assuming controllers communicate along edges
Assuming dynamics propagate outwardso that delay is proportional to geometric distance
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Two-Dimensional Lattice
Dt Dt Dt
Dt
Dt Dt
Dt
Dt
Dt
Dt
Dt
DtK11 K12 K13
K23K22K21
K31 K32 K33
G12 G13
G23G22
G32 G33
G11
G21
G31
S is quadratically invariant with respect to G if
t ≤ p√2
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Sparsity ExampleSuppose
G ∼
• ◦ ◦ ◦ ◦• • ◦ ◦ ◦• • • ◦ ◦• • • • ◦• • • • •
S =
K∣
∣ K ∼
◦ ◦ ◦ ◦ ◦◦ • ◦ ◦ ◦◦ • ◦ ◦ ◦• • • ◦ ◦• • • ◦ •
For arbitrary K ∈ S
KGK ∼
◦ ◦ ◦ ◦ ◦◦ • ◦ ◦ ◦◦ • ◦ ◦ ◦• • • ◦ ◦• • • ◦ •
Hence S is quadratically invariant with respect to G.
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Same Structure SynthesisSuppose
G ∼
◦ ◦ ◦• ◦ ◦◦ • •
S =
K∣
∣ K ∼
◦ ◦ ◦• ◦ ◦◦ • •
Then
KGK ∼
◦ ◦ ◦◦ ◦ ◦• • •
so S is not quadratically invariant with respect to G.
In fact,
K(I − GK)−1 ∼
◦ ◦ ◦• ◦ ◦• • •
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Small Gain
If |a| < 1 then
(1 − a)−1 = 1 + a + a2 + a3 + . . .
for example
(
1 − 12
)−1= 1 +
1
2+
1
4+
1
8+ . . .
more generally, if ‖A‖ < 1 then
(1 − A)−1 = I + A + A2 + A3 + . . .
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Not So Small Gain
Let
Q = I + A + A2 + A3 + . . .
then
AQ = A + A2 + A3 + . . .
subtracting we get
(I − A)Q = I
and so
Q = (I − A)−1
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Convergence to Unstable Operator
0 0.5 1 1.5 2 2.50
5
10
15
20
25
Impulse Response
Time (sec)
Am
plitu
de
• Let W (s) = 2s+1• Plot shows impulse response of
∑Nk=0 W
k for N = 1, . . . , 7
• Converges to that of II−W =s+1s−1
• In what topology do the associated operators converge?
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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 ∈ Rsp, K ∈ Rp
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Proof Sketch
K
G
r y u
KGK ∈ S for all K ∈ S
Suppose K ∈ S. Then
K(I − GK)−1 = K + KGK + K(GK)2 + . . . ∈ S
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Consider Nonlinear
Let
Q = I + A + A2 + A3 + . . .
then
AQ = A + A2 + A3 + . . .
subtracting we get
(I − A)Q = I
and so
Q = (I − A)−1
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Block Diagram Algebra
K
G
r y u
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)−1ru = Rr = K(I − GK)−1r
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Iteration of Signals
K
G
r y u
We can define the following iteration, commensurate with the diagram
y(0) = r
u(n) = Ky(n)
y(n+1) = r + Gu(n)
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Iteration of Signals and Operators
K
G
r y u
We can define the following iterations, commensurate with the diagram
y(0) = r Y (0) = I
u(n) = Ky(n) R(n) = KY (n)
y(n+1) = r + Gu(n) Y (n+1) = I + GR(n)
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Iteration of Signals
K
G
r y u
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))
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Iteration of Signals
K
G
r y u
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 = limn→∞ y
(n) u = limn→∞ u
(n)
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Iteration of Operators
K
G
r y u
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 = limn→∞Y
(n) R = limn→∞R
(n)
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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 ∈ Rsp, K ∈ Rp
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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)
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New Invariance Condition
K1(I ± GK2) ∈ S for all K1, K2 ∈ S
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New Invariance Condition
K1(I ± GK2) ∈ S for all K1, K2 ∈ S
Invariance under FeedbackIf this condition is satisfied, then
K ∈ S ⇐⇒ K(I − GK)−1 ∈ S
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New Invariance Condition
K1(I ± GK2) ∈ S for all K1, K2 ∈ S
Invariance under FeedbackIf 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}
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Proof Sketch
K1(I ± GK2) ∈ S ∀ K1, K2 ∈ 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 = limn→∞R
(n) ∈ S
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Conclusions
• Quadratic invariance allows parameterization of all (LTI) stablizingdecentralized controllers.
• Similar condition allows parameterization of all (NLTV) stabilizing de-centralized controllers
• These condition are satisfied when communications are faster than thepropagation of dynamics.
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Quadratic InvarianceThe set S is called quadratically invariant with respect to G if
KGK ∈ S for all K ∈ S
?????????
K1(I ± GK2) ∈ S for all K1, K2 ∈ S
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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?