Control Systems on Three-Dimensional Lie Groups Equivalence and Controllability

Rory Biggs∗ and Claudiu C. Remsing

Geometry and Geometric Control (GGC) Research Group Department of (Pure and Applied) Rhodes University, Grahamstown 6140, South Africa

13th European Control Conference June 24–27, 2014, Strasbourg, France

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 1 / 23 Outline

1 control systems

2 Classification of systems Classification of 3D Lie groups Solvable case Semisimple case

3 Controllability characterizations

4 Conclusion

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 2 / 23 Outline

1 Invariant control systems

2 Classification of systems Classification of 3D Lie groups Solvable case Semisimple case

3 Controllability characterizations

4 Conclusion

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 3 / 23 Invariant control affine systems

Left-invariant control affine system

` (Σ)g ˙ = Ξ(g, u) = g (A + u1B1 + ··· + u`B`), g ∈ G, u ∈ R

state space: G is a connected (matrix) with g ` input set: R dynamics: family of left-invariant vector fields Ξ(·, u) 0 trace: Γ = A + Γ = A + hB1,..., B`i is an affine subspace of g A ∈ Γ0 ←→ homogeneous, A ∈/ Γ0 ←→ inhomogeneous.

Σ: A + u1B1 + ··· + u`B`.

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 4 / 23 Trajectories, controllability, and full rank

` admissible controls: piecewise continuous curves u(·) : [0, T ] → R trajectory: absolutely continuous curve s.t.g ˙ (t) = Ξ(g(t), u(t)) controllable: exists trajectory from any point to any other full rank: Lie(Γ) = g; necessary condition for controllability.

Characterization of full-rank systems on 3D Lie groups 1-input homogeneous: none

1-input inhomogeneous: A, B1,[A, B1] linearly independent

2-input homogeneous: B1, B2,[B1, B2] linearly independent 2-input inhomogeneous: all 3-input: all.

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 5 / 23 Equivalence

Detached feedback equivalence Σ and Σ0 are detached feedback equivalent if 0 ` ` 0 ∃φ :G → G , ϕ : R → R such that Tg φ · Ξ(g, u) = Ξ (φ(g), ϕ(u))

specialization of feedback equivalence diffeomorphism φ preserves left-invariant vector fields

Proposition Full-rank systems Σ and Σ0 equivalent if and only if there exists a 0 0 Lie group isomorphism φ :G → G such that T1φ · Γ = Γ .

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 6 / 23 Outline

1 Invariant control systems

2 Classification of systems Classification of 3D Lie groups Solvable case Semisimple case

3 Controllability characterizations

4 Conclusion

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 7 / 23 Classification of 3D Lie algebras

Eleven types of real 3D Lie algebras (e.g., [Mub63]) 3 3g — R Abelian g2.1 ⊕ g1 — aff (R) ⊕ R cmpl. solvable g3.1 — Heisenberg h3 nilpotent

g3.2 cmpl. solvable

g3.3 — book Lie algebra cmpl. solvable 0 g3.4 — semi-Euclidean se (1, 1) cmpl. solvable a g3.4, a > 0, a 6= 1 cmpl. solvable 0 g3.5 — Euclidean se (2) solvable a g3.5, a > 0 exponential 0 g3.6 — pseudo-orthogonal so (2, 1), sl (2, R) simple 0 g3.7 — orthogonal so (3), su (2) simple

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 8 / 23 Classification of 3D Lie groups

3D Lie groups (e.g., [GOV94]) 3 2 3 3g1 —4— R , R × T, R × T, T g2.1 ⊕ g1 —2 — Aff ( R)0 × R, Aff (R)0 × T ∗ g3.1 —2—H 3,H3 = H3/ Z(H3(Z)) g3.2 —1—G 3.2

g3.3 —1—G 3.3 0 g3.4 —1 — SE (1 , 1) a a g3.4 —1—G 3.4 0 g3.5 — N — SE (2), n-fold cov. SEn(2), univ. cov. SEf (2) a a g3.5 —1—G 3.5 g3.6 — N — SO (2, 1)0, n-fold cov. A(n), univ. cov. Ae g3.7 —2 — SO (3), SU (2).

∗ Only H3,An, n ≥ 3, and Ae are not matrix Lie groups. R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 9 / 23 Case study: systems on the H3

1 y x 0 y x H3 : 0 1 z h3 : 0 0 z = xE1 + yE2 + zE3 0 0 1 0 0 0

Theorem

On H3, any full-rank system is equivalent to exactly one of the following systems

(1,1) Σ : E2 + uE3 (2,0) Σ : u1E2 + u2E3 (2,1) Σ1 : E1 + u1E2 + u2E3 (2,1) Σ2 : E3 + u1E1 + u2E2 (3,0) Σ : u1E1 + u2E2 + u3E3.

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 10 / 23 Case study: systems on the Heisenberg group H3

Proof sketch (1/2)

yw − vz x u    x, y, z, u, v, w ∈  d Aut(H ) = Aut(h ) = 0 y v : R 3 3   yw − vz 6= 0  0 z w 

P3 DP3 E Single-input system Σ with trace Γ = i=1 ai Ei + i=1 bi Ei .

  a2b3 − a3b2 a1 b1 ψ =  0 a2 b2 ∈ Aut(h3), ψ · (E2 + hE3i) = Γ. 0 a3 b3

So Σ is equivalent to Σ(1,1). Two-input homogeneous system with trace Γ = hA, Bi; similar argument holds.

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 11 / 23 Case study: systems on the Heisenberg group H3

Proof sketch (2/2)

Two-input inhomogeneous system Σ with trace Γ = A + hB1, B2i. 0 If E1 ∈ hB1, B2i, then Γ = A + hE1, B2i; like single-input case there exists automorphism ψ such that ψ · Γ = E3 + hE1, E2i.

If E1 ∈/ hB1, B2i, construct automorphism ψ such that ψ · Γ = E1 + hE2, E3i. (2,1) (2,1) Σ1 and Σ2 are distinct as E1 is eigenvector of every automorphism. Three-input system: trivial.

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 12 / 23 Case study: systems on the SO (3)

3×3 > SO (3) = {g ∈ R : gg = 1, det g = 1}

 0 −z y  so (3) :  z 0 −x = xE1 + yE2 + zE3 −y x 0

Theorem On SO (3), any full-rank system is equivalent to exactly one of the following systems

(1,1) Σα : αE1 + uE2, α > 0 (2,0) Σ : u1E1 + u2E2 (2,1) Σα : αE1 + u1E2 + u2E3, α > 0 (3,0) Σ : u1E1 + u2E2 + u3E3.

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 13 / 23 Case study: systems on the orthogonal group SO (3)

Proof sketch d Aut(SO (3)) = Aut(so (3)) = SO (3) Classification procedure similar, though more involved.

Product A • B = a1b1 + a2b2 + a3b3 is preserved by automorphisms. Critical point C•(Γ) at which an inhomogeneous affine subspace is tangent to a sphere Sα = {A ∈ so (3) : A • A = α} is given by A • B C•(Γ) = A − B B • B  −1   •   B1 • B1 B1 • B2 A • B1 C (Γ) = A − B1 B2 . B1 • B2 B2 • B2 A • B2

ψ · C•(Γ) = C•(ψ · Γ) for any automorphism ψ ∈ SO (3). Scalar α2 = C•(Γ) • C•(Γ) invariant under automorphisms.

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 14 / 23 Outline

1 Invariant control systems

2 Classification of systems Classification of 3D Lie groups Solvable case Semisimple case

3 Controllability characterizations

4 Conclusion

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 15 / 23 Some controllability criteria for invariant systems

Sufficient conditions for full-rank system to be controllable system is homogeneous state space is compact the direction space Γ0 generates g, i.e., Lie (Γ0) = g there exists C ∈ Γ such that t 7→ exp(tC) is periodic the identity element 1 is in the interior of the attainable set A = {g(t1): g(·) is a trajectory such that g(0) = 1, t1 ≥ 0}. ([JS72])

Systems on simply connected completely solvable groups Condition Lie (Γ0) = g is also necessary. ([Sac09])

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 16 / 23 Characterization for systems on 3D groups

Theorem

1 a On Aff (R)0 × R, H3, G3.2, G3.3, SE (1, 1), and G3.4, a full-rank system is controllable if and only if Lie (Γ0) = g.

2 On SEn(2), SO (3), and SU (2), all full-rank systems are controllable.

3 On Aff (R) × T, SL (2, R), and SO (2, 1)0, a full-rank system is controllable if and only if it is homogeneous or there exists C ∈ Γ such that t 7→ exp(tC) is periodic.

4 a On SEf (2) and G3.5, ∗ 0 a full-rank system is controllable if and only if E3 (Γ ) 6= {0}.

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 17 / 23 Characterization for 3D groups

Proof sketch (1/2)

1 Completely solvable simply connected groups; characterization known ([Sac09]).

2 The groups SO (3) and SU (2) are compact, hence all full-rank systems are controllable.

SEn(2) decomposes as of vector space and compact subgroup; hence result follows from [BJKS82].

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 18 / 23 Characterization for 3D groups

Proof sketch (2/2)

3 Study normal forms of these systems obtained in classification. Full-rank homogeneous systems are controllable. For each full-rank inhomogeneous system we either explicitly find A ∈ Γ such that t 7→ exp(tA) is periodic or prove that some states are not attainable by inspection of coordinates ofg ˙ = Ξ(g, u). As properties are invariant under equivalence, characterization holds.

4 Study normal forms of these systems obtained in classification. Condition invariant under equivalence. Similar techniques with extensions ([JS72]); however for one system on a G3.5 we could only prove controllability by showing 1 ∈ int A.

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 19 / 23 Outline

1 Invariant control systems

2 Classification of systems Classification of 3D Lie groups Solvable case Semisimple case

3 Controllability characterizations

4 Conclusion

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 20 / 23 Conclusion

Summary Characterization of controllability for systems on 3D Lie groups. Normal forms for controllable systems on 3D Lie groups.

Outlook Cost-extended systems associated to optimal control problems.

R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 21 / 23 ReferencesI

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R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 22 / 23 ReferencesII

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R. Biggs, C.C. Remsing (Rhodes) Control Systems on 3D Lie Groups ECC 2014 23 / 23