Analysis of Control Systems on Symmetric Conesivanpapusha.com/pdf/jordan_cdc2015_slides.pdf ·...
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Analysis of Control Systemson Symmetric Cones
Ivan Papusha Richard M. Murray
California Institute of Technology
Control and Dynamical Systems
IEEE Conference on Decision and Control, Osaka, Japan
December 17, 2015
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Stability of a linear system
Is the system x = Ax asymptotically stable?
A =
− 10 1 5 12 − 9 2 71 0 − 41 04 1 3 − 9
• structurally dense
• no easy way to escape calculating eigenvalues, Lyapunov matrix. . .
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Stability of a linear system
Is the system x = Ax asymptotically stable?
A =
− 10 1 5 12 − 9 2 71 0 − 41 04 1 3 − 9
• structurally dense
• no easy way to escape calculating eigenvalues, Lyapunov matrix. . .
(eigenvalues: −41.157,−4.7465,−11.548± 1.4405i)
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Stability of a linear system
Is the system x = Ax asymptotically stable?
A =
− 10 1 5 12 − 9 2 71 0 − 41 04 1 3 − 9
• structurally dense
• no easy way to escape calculating eigenvalues, Lyapunov matrix. . .
(eigenvalues: −41.157,−4.7465,−11.548± 1.4405i)
can we do better?
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Stability of a linear system
Is the system x = Ax asymptotically stable?
A =
− 10 1 5 12 − 9 2 71 0 − 41 04 1 3 − 9
• structurally dense
• no easy way to escape calculating eigenvalues, Lyapunov matrix. . .
(eigenvalues: −41.157,−4.7465,−11.548± 1.4405i)
can we do better?
yes if we exploit cone structure of A
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Age-old problem
question: When is the linear dynamical system
x(t) = Ax(t), A ∈ Rn×n, x(t) ∈ Rn
globally asymptotically stable? (x(t) → 0 as t → ∞ for all initialconditions)
answer: solved!
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Age-old problem
answer: The linear system is stable if and only if
1. all eigenvalues of the matrix A have negative real part,
Re(λi (A)) < 0, i = 1, . . . , n.
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Age-old problem
answer: The linear system is stable if and only if
1. all eigenvalues of the matrix A have negative real part,
Re(λi (A)) < 0, i = 1, . . . , n.
2. given a positive definite matrix Q = QT ≻ 0, there exists a uniquematrix P ≻ 0 satisfying the Lyapunov equation
ATP + PA+ Q = 0.
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Age-old problem
answer: The linear system is stable if and only if
1. all eigenvalues of the matrix A have negative real part,
Re(λi (A)) < 0, i = 1, . . . , n.
2. given a positive definite matrix Q = QT ≻ 0, there exists a uniquematrix P ≻ 0 satisfying the Lyapunov equation
ATP + PA+ Q = 0.
3. the following linear matrix inequality holds,
P = PT ≻ 0, ATP + PA ≺ 0.
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Age-old problem
answer: The linear system is stable if and only if
1. all eigenvalues of the matrix A have negative real part,
Re(λi (A)) < 0, i = 1, . . . , n.
2. given a positive definite matrix Q = QT ≻ 0, there exists a uniquematrix P ≻ 0 satisfying the Lyapunov equation
ATP + PA+ Q = 0.
3. the following linear matrix inequality holds,
P = PT ≻ 0, ATP + PA ≺ 0.
4. there exists a quadratic Lyapunov function V : Rn → R,
V (x) = 〈x ,Px〉which is positive definite (V (x) > 0 for all x 6= 0) and decreasing(V < 0 along system trajectories).
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Age-old problem
answer: The linear system is stable if and only if
3. the following linear matrix inequality holds,
P = PT ≻ 0, ATP + PA ≺ 0.
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Quadratic Lyapunov function
∃P = PT ≻ 0, ATP + PA ≺ 0
Lyapunov’s theorem ⇓x = Ax is stable
V (x)
x1 x2
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Quadratic Lyapunov function
∃P = PT ≻ 0, ATP + PA ≺ 0
Lyapunov’s theorem ⇓ ⇑ for all linear systems
x = Ax is stable
V (x)
x1 x2
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Three cones
A proper cone is closed, convex, pointed, has nonempty interior, andclosed under nonnegative scalar multiplication.
• nonnegative orthant
Rn+ = {x ∈ Rn | xi ≥ 0, for all i = 1, . . . , n}
• second order (Lorentz) cone
Ln+ = {(x0, x1) ∈ R× Rn−1 | ‖x1‖2 ≤ x0}
• positive semidefinite cone
Sn+ = {X ∈ Rn×n | X = XT � 0}
these cones are self-dual and symmetric (cone of squares, Jordan algebra)
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Cone invariance
DefinitionThe system x = Ax is invariant with respect to the cone K if eA(K ) ⊆ K .
• once the state enters K , it never leaves
x(0) ∈ K ⇒ x(t) ∈ K for all t ≥ 0
• equivalently, A is cross-positive
x ∈ K , y ∈ K∗, and 〈x , y〉 = 0
⇒ 〈Ax , y〉 ≥ 0
x(t)
KK∗
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Linear Lyapunov function
∃p ∈ Rn, pi > 0, (Ap)i < 0, i = 1, . . . , n
Lyapunov’s theorem ⇓x = Ax is stable
V (x)
x1 x2
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Linear Lyapunov function
∃p ∈ Rn, pi > 0, (Ap)i < 0, i = 1, . . . , n
Lyapunov’s theorem ⇓ ⇑ for Rn+-invariant linear systems
x = Ax is stable
V (x)
x1 x2
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Lorentz Lyapunov function
∃p ∈ intLn+, Ap ∈ − intLn
+
Lyapunov’s theorem ⇓x = Ax is stable
V (x)
x1 x2
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Lorentz Lyapunov function
∃p ∈ intLn+, Ap ∈ − intLn
+
Lyapunov’s theorem ⇓ ⇑ for Ln+-invariant linear systems
x = Ax is stable
V (x)
x1 x2
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General theorem
Let L : V → V be a linear operator on a Jordan algebra V withcorresponding symmetric cone of squares K , and assume thateL(K ) ⊆ K . The following statements are equivalent:
(a) There exists p ≻K 0 such that −L(p) ≻K 0
(b) There exists z ≻K 0 such that LPz + PzLT is negative definite on V .
(c) The system x(t) = L(x) with initial condition x0 ∈ K isasymptotically stable.
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General theorem
Let L : V → V be a linear operator on a Jordan algebra V withcorresponding symmetric cone of squares K , and assume thateL(K ) ⊆ K . The following statements are equivalent:
(a) There exists p ≻K 0 such that −L(p) ≻K 0
(b) There exists z ≻K 0 such that LPz + PzLT is negative definite on V .
(c) The system x(t) = L(x) with initial condition x0 ∈ K isasymptotically stable.
x = Ax is K -invariant
⇓Lyapunov function obtained by conic programming over K
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Simple example
A is cross-positive (Metzler) with respect to nonnegative orthant K = Rn+
A =
−10 1 5 12 −9 2 71 0 −41 04 1 3 −9
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Simple example
A is cross-positive (Metzler) with respect to nonnegative orthant K = Rn+
A =
−10 1 5 12 −9 2 71 0 −41 04 1 3 −9
• linear Lyapunov function V = 〈p, x〉 suffices:
p =
1.43922.77880.220791.9704
≻Rn+0 Ap =
−8.5383−7.8967−7.6133−8.536
≺Rn+0
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Simple example
A is cross-positive (Metzler) with respect to nonnegative orthant K = Rn+
A =
−10 1 5 12 −9 2 71 0 −41 04 1 3 −9
• linear Lyapunov function V = 〈p, x〉 suffices:
p =
1.43922.77880.220791.9704
≻Rn+0 Ap =
−8.5383−7.8967−7.6133−8.536
≺Rn+0
• quadratic representation: there exists z ∈ Rn+ such that p = z ◦ z
V (x) = 〈x ,Pzx〉, Pz = diag(z2), z =
√1.4392√2.7788√0.22079√1.9704
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Transportation network example
x0
x1
x2
x3
x4
• Directed transportation networkx1, . . . , x4 (Rantzer, 2012), augmentedwith a catch-all buffer x0.
x0x1x2x3x4
=
ℓ00 ℓ01 ℓ02 ℓ03 ℓ040 −1− ℓ31 ℓ12 0 00 0 −ℓ12 − ℓ32 ℓ23 00 ℓ31 ℓ32 −ℓ23 − ℓ43 ℓ340 0 0 ℓ43 −4− ℓ34
x0x1x2x3x4
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Transportation network example
x0
x1
x2
x3
x4
• Directed transportation networkx1, . . . , x4 (Rantzer, 2012), augmentedwith a catch-all buffer x0.
• Metzler substructure
x0x1x2x3x4
=
ℓ00 ℓ01 ℓ02 ℓ03 ℓ040 −1− ℓ31 ℓ12 0 00 0 −ℓ12 − ℓ32 ℓ23 00 ℓ31 ℓ32 −ℓ23 − ℓ43 ℓ340 0 0 ℓ43 −4− ℓ34
x0x1x2x3x4
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Transportation network example
x0
x1
x2
x3
x4
• Directed transportation networkx1, . . . , x4 (Rantzer, 2012), augmentedwith a catch-all buffer x0.
• Metzler substructure
• ℓ00, . . . , ℓ04 have no definite sign
x0x1x2x3x4
=
ℓ00 ℓ01 ℓ02 ℓ03 ℓ040 −1− ℓ31 ℓ12 0 00 0 −ℓ12 − ℓ32 ℓ23 00 ℓ31 ℓ32 −ℓ23 − ℓ43 ℓ340 0 0 ℓ43 −4− ℓ34
x0x1x2x3x4
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Lorentz cone-invariant dynamics
x0
x1 x2
x0
x1 x2
Figure 1: Embedded focus along x0-axis
A dynamics matrix A is Ln+-invariant if and only if there exists
ξ ∈ R, AT Jn + JnA− ξJn � 0. (∗)
Provided this condition holds, A is (Hurwitz) stable if and only if thereexists p ≻Ln
+0 with Ap ≺Ln
+0.
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Technical summary
Algebra: Real Lorentz SymmetricV Rn Rn Sn
K Rn+ Ln
+ Sn+
〈x , y〉 xT y xT y Tr(XY T )
x ◦ y xiyi (xT y , x0y1 + y0x1)12(XY + YX )
Pz , z ∈ intK diag(z)2 zzT − zT Jnz2
Jn X 7→ ZXZ
V (x) = 〈x ,Pz (x)〉 xT diag(z)2x xT(
zzT − zT Jnz2
Jn
)
x ‖XZ‖2F
Free variables in V (x) n n n(n + 1)/2dynamics L x 7→ Ax x 7→ Ax X 7→ AX + XAT
L is cross-positive A is Metzler A satisfies (∗) by construction−L(p) ≻K 0 (Ap)i < 0 ‖(Ap)1‖2 < (−Ap)0 AP + PAT ≺ 0
Stability verification LP SOCP SDP
Table 1: Summary of dynamics preserving a cone
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Questions
Is (say) H∞ control synthesis possible via
• LP?
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Questions
Is (say) H∞ control synthesis possible via
• LP? (yes, for Rn+-invariant systems, Rantzer 2012)
• SDP?
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Questions
Is (say) H∞ control synthesis possible via
• LP? (yes, for Rn+-invariant systems, Rantzer 2012)
• SDP? (yes, in general, Yakubovich 1960s. . . )
• SOCP?
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Questions
Is (say) H∞ control synthesis possible via
• LP? (yes, for Rn+-invariant systems, Rantzer 2012)
• SDP? (yes, in general, Yakubovich 1960s. . . )
• SOCP? (conjecture: yes)
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Questions
Is (say) H∞ control synthesis possible via
• LP? (yes, for Rn+-invariant systems, Rantzer 2012)
• SDP? (yes, in general, Yakubovich 1960s. . . )
• SOCP? (conjecture: yes)
Are these three cones the end of the story?
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Questions
Is (say) H∞ control synthesis possible via
• LP? (yes, for Rn+-invariant systems, Rantzer 2012)
• SDP? (yes, in general, Yakubovich 1960s. . . )
• SOCP? (conjecture: yes)
Are these three cones the end of the story?
• yes
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Questions
Is (say) H∞ control synthesis possible via
• LP? (yes, for Rn+-invariant systems, Rantzer 2012)
• SDP? (yes, in general, Yakubovich 1960s. . . )
• SOCP? (conjecture: yes)
Are these three cones the end of the story?
• yes (kind of)
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Symmetric cone categorization
If K is a (finite dimensional) symmetric cone, then it is a cartesianproduct
K = K1 × K2 × · · · × KN ,
where each Ki is one of (e.g., Faraut 1994)
• n × n self-adjoint positive semidefinite matrices with real, complex,or quaternion entries
• 3× 3 self-adjoint positive semidefinite matrices with octonion entries(Albert algebra), and
• Lorentz cone
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Contributions
• analysis idea comes from the cone inclusion
nonnegative orthant ⊆ second-order cone ⊆ semidefinite cone
LP ⊆ SOCP ⊆ SDP
easy → harder → hardest
• characterized new class of linear systems that admit SOCP-basedanalysis without any loss
• unified existing analysis frameworks
• algebraic connections with a mature theory (Jordan algebras)
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Thanks!
more information: Ivan Papusha, Richard M. Murray. Analysis ofControl Systems on Symmetric Cones, IEEE CDC, 2015.
www.cds.caltech.edu/~ipapusha
funding: NDSEG, Powell Foundation, STARnet/TerraSwarm, Boeing
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