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Control Systems on Three-Dimensional Lie Groups Equivalence and Controllability 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 Mathematics (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 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 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 2 G; u 2 R state space: G is a connected (matrix) Lie group with Lie algebra 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 2 Γ0 ! homogeneous, A 2= 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 9φ :G ! G , ' : R ! R such that Tg φ · Ξ(g; u) = Ξ (φ(g);'(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 Heisenberg group H3 21 y x3 20 y x3 H3 : 40 1 z5 h3 : 40 0 z5 = 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) 82yw − vz x u 3 9 < x; y; z; u; v; w 2 = d Aut(H ) = Aut(h ) = 0 y v : R 3 3 4 5 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 . 2 3 a2b3 − a3b2 a1 b1 = 4 0 a2 b25 2 Aut(h3); · (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 2 hB1; B2i, then Γ = A + hE1; B2i; like single-input case there exists automorphism such that · Γ = E3 + hE1; E2i. If E1 2= hB1; B2i, construct automorphism such that · 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 orthogonal group SO (3) 3×3 > SO (3) = fg 2 R : gg = 1; det g = 1g 2 0 −z y 3 so (3) : 4 z 0 −x5 = 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 = fA 2 so (3) : A • A = αg 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 · C•(Γ) = C•( · Γ) for any automorphism 2 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 2 Γ such that t 7! exp(tC) is periodic the identity element 1 is in the interior of the attainable set A = fg(t1): g(·) is a trajectory such that g(0) = 1; t1 ≥ 0g. ([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 2 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= f0g. 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 semidirect product 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.
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