Cost-extended control systems on SO(3)

Transcription

1 Cost-extended control systems on SO(3) Ross M. Adams Mathematics Seminar April 16, 2014 R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 1 / 34

2 Outline 1 Introduction 2 Control systems on SO(3) 3 Cost-extended systems 4 Hamilton-Poisson systems 5 Conclusion R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 2 / 34

3 Outline 1 Introduction 2 Control systems on SO(3) 3 Cost-extended systems 4 Hamilton-Poisson systems 5 Conclusion R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 3 / 34

4 Problem statement SO(3) = { g R 3 3 g g = 1, det g = 1 } Group of rotations Three-dimensional, connected, compact Lie group Optimal control problems on SO(3) Detached feedback equivalence of control systems Classify cost-equivalent systems Solve optimal control problems Hamilton-Poisson systems R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 4 / 34

5 The Lie algebra so (3) = { A R 3 3 A + A = 0 } Basis: E 1 = E 2 = E 3 = Commutator relations: [E 2, E 3 ] = E 1 [E 3, E 1 ] = E 2 [E 1, E 2 ] = E 3 Lie algebra automorphisms: Aut(so(3)) = SO(3) R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 5 / 34

6 Outline 1 Introduction 2 Control systems on SO(3) 3 Cost-extended systems 4 Hamilton-Poisson systems 5 Conclusion R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 6 / 34

7 LiCA systems Left-invariant control affine system Σ = (SO(3), Ξ) The dynamics Ξ : SO(3) R l T SO(3), 1 l 3 are left invariant (g, u) Ξ(g, u) = g Ξ(1, u) The parametrisation map Ξ(1, ) : R l T 1 SO(3) = so(3) is affine u A + u 1 B u l B l so(3) We assume B 1,..., B l are linearly independent R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 7 / 34

8 Terminology The trace Γ of the system Σ is Γ = im(ξ(1, )) so(3) = A + Γ 0 = A + B 1,..., B l Σ is called homogeneous if A Γ 0 inhomogeneous if A Γ 0 Σ has full rank provided the Lie algebra generated by Γ equals the whole Lie algebra Lie(Γ) = so(3) R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 8 / 34

9 Controllability Trajectory Absolutely continuous curve g( ) : [0, T ] SO(3) satisfying a.e. Controllability ġ(t) = Ξ(g(t), u(t)) Σ = (SO(3), Ξ) is called controllable if for any g 0, g 1 SO(3) exists a trajectory taking g 0 to g 1 Necessary conditions for controllability SO(3) is connected The trace Γ has full-rank SO(3) is compact Σ has full-rank Σ is controllable R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 9 / 34

10 Detached feedback equivalence Let Σ = (SO(3), Ξ) and Σ = (SO(3), Ξ ) Σ and Σ are (locally) DF-equivalent if there exist N, N 1, and a (local) diffeomorphism Φ = φ ϕ : N R l N R l, φ(1) = 1, such that for all g N and u R l T g φ Ξ (g, u) = Ξ (φ(g), ϕ(u)) Characterization Two full rank systems are DF-equivalent iff ψ Aut(so(3)) such that ψ Γ = Γ R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 10 / 34

11 DF-equivalence on SO(3) Proposition Any full rank system on SO(3) is DF-equivalent to exactly one of the systems Ξ (1,1) α (1, u) = αe 1 + u 1 E 2 Ξ (2,0) (1, u) = u 1 E 1 + u 2 E 2 Ξ (2,1) α (1, u) = αe 1 + u 1 E 2 + u 3 E 3 Ξ (3,0) (1, u) = u 1 E 1 + u 2 E 2 + u 3 E 3 Here α > 0 parametrizes families of non-equivalent class representatives R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 11 / 34

12 Outline 1 Introduction 2 Control systems on SO(3) 3 Cost-extended systems 4 Hamilton-Poisson systems 5 Conclusion R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 12 / 34

13 Optimal control problems Let Σ = (SO(3), Ξ) An (invariant) optimal control problem is specified by ġ(t) = g(t) Ξ(1, u(t)) g(0) = g 0, g(t ) = g 1, g 0, g 1 SO(3), T > 0 fixed J (u( )) = T 0 (u(t) µ)q(u(t) µ) dt min, µ R l. Here Q is a positive definite l l matrix. Cost-extended system (Σ, χ) Σ = (SO(3), Ξ) χ : R l R, u (u µ)q(u µ) Boundary data (g 0, g 1, T ) specifies a unique problem R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 13 / 34

14 Cost-extended systems Cost-equivalence (Σ, χ) and (Σ, χ ) are C-equivalent affine isomorphism ϕ : R l R l such that φ Aut(SO(3)) and an for some r > 0. The diagrams T 1 φ Ξ(1, u) = Ξ (1, ϕ(u)) and χ ϕ = rχ R l ϕ R l R l ϕ R l Ξ(1, ) g T 1 φ g Ξ (1, ) χ R δ r R χ commute R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 14 / 34

15 Classification T Σ Given Σ = (SO(3), Ξ), let T Σ denote the group of feedback transformations leaving Σ invariant { } T Σ = ϕ Aff(R l ) : ψ d Aut(SO(3)), ψ Ξ(1, u) = Ξ(1, ϕ(u)). Proposition (Σ, χ) and (Σ, χ ) are C-equivalent iff ϕ T Σ such that χ = rχ ϕ for some r > 0. For χ : u (u µ) Q (u µ) and ϕ : u Ru + x we have where µ = R 1 (x µ) R l. (χ ϕ)(u) = (u µ ) R Q R(u µ ) R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 15 / 34

16 Two-input homogeneous systems Proposition Every (2, 0) system is C-equivalent to (Σ (2,0), χ 1 αβ ) or (Σ(2,0), χ 2 α), where Σ (2,0) = (SO(3), Ξ (2,0) ) and χ 1 αβ = (u 1 α 1 ) 2 + β(u 2 α 2 ) 2, α 1, α 2 0, 0 < β < 1, χ 2 α = (u 1 α) 2 + u2 2, α 0. Proof sketch Every (2, 0) system is DF-equivalent to Σ (2,0) = (SO(3), Ξ (2,0) ) where Ξ (2,0) (1, u) = u 1 E 1 + u 2 E 2 Calculate feedback transformations T Σ (2,0) R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 16 / 34

17 Proof sketch cont. 1 0 In matrix form Ξ (2,0) (1, u) = Checking the condition ψ Ξ (2,0) (1, u) = Ξ (2,0) (1, ϕ(u)) gives Thus c 1 = c 2 = 0 a 1 a 2 a [ ] b 1 b 2 b = 0 1 ϕ11 ϕ 12 ϕ c 1 c 2 c ϕ 22 As ψ SO(3) = a 3 = b 3 = 0 and c 3 = ±1 Therefore T Σ (2,0) = {ϕ ϕ O(2)} R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 17 / 34

18 Proof sketch cont. 1 0 In matrix form Ξ (2,0) (1, u) = Checking the condition ψ Ξ (2,0) (1, u) = Ξ (2,0) (1, ϕ(u)) gives Thus c 1 = c 2 = 0 a 1 a 2 a [ ] b 1 b 2 b = 0 1 ϕ11 ϕ 12 ϕ 0 0 c ϕ 22 As ψ SO(3) = a 3 = b 3 = 0 and c 3 = ±1 Therefore T Σ (2,0) = {ϕ ϕ O(2)} R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 18 / 34

19 Proof sketch cont. 1 0 In matrix form Ξ (2,0) (1, u) = Checking the condition ψ Ξ (2,0) (1, u) = Ξ (2,0) (1, ϕ(u)) gives Thus c 1 = c 2 = 0 a 1 a [ ] b 1 b = 0 1 ϕ11 ϕ 12 ϕ 0 0 ± ϕ 22 As ψ SO(3) = a 3 = b 3 = 0 and c 3 = ±1 Therefore T Σ (2,0) = {ϕ ϕ O(2)} R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 19 / 34

20 Proof sketch cont. (Σ, χ) is C-equivalent to (Σ (2,0), χ 0 ), for some [ ] χ 0 : u (u µ) a1 b Q (u µ), Q = b a 2 ϕ 1 O(2) such that ϕ 1 Q ϕ 1 = diag (γ 1, γ 2 ), γ 1 γ 2 > 0 Therefore χ 1 (u) = 1 (χ 0 ϕ 1 )(u) = (u µ ) diag (1, β) (u µ ) α 1 where 0 < β 1, µ R 2. If β 1, then diag (σ 1, σ 2 ) O(2), σ 1, σ 2 { 1, 1}, are the only transformations left preserving the quadratic form diag (1, β). Thus χ 1 αβ = (u 1 α 1 ) 2 + β(u 2 α 2 ) 2, α 1, α 2 0, 0 < β < 1 R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 20 / 34

21 Proof sketch cont. If β = 1, then any ϕ O(2) preserves diag (1, 1) α 0 and θ R such that µ 1 = α cos θ and µ 2 = α sin θ [ ] cos θ sin θ Thus for ϕ 2 = we have sin θ cos θ χ 3 (u) = (χ 2 ϕ 2 )(u) = ( u [ ]) ( α u 0 Therefore, every such system is C-equivalent to [ ]) α 0 (Σ (2,0), χ 2 α) : { Ξ (2,0) (1, u) = u 1 E 1 + u 2 E 2 χ 2 α(u) = (u 1 α) 2 + u 2 2, α 0. Each value of α defines a distinct equivalence class R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 21 / 34

22 Outline 1 Introduction 2 Control systems on SO(3) 3 Cost-extended systems 4 Hamilton-Poisson systems 5 Conclusion R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 22 / 34

23 Optimal control problems Question How do we solve a given optimal control problem? We have A family (Ξ(, u)) u R l of dynamical systems A cost function χ : u (u µ) Q(u µ) we want to minimize Consider (Σ, χ) Construct a family of Hamiltonian functions on T SO(3) H λ u (ξ) = λχ(u) + ξ(ξ(g, u)), λ { 1 2, 0} Left-trivialize cotangent bundle, i.e., T SO(3) = SO(3) so(3) Then H λ u (g, p) = λχ(u) + p(ξ(1, u)) which is an element of C (so(3) ). R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 23 / 34

24 Neccessary conditions Pontryagin maximum principle (PMP) Let (ḡ( ), ū( )) be a solution of an optimal control problem on [0, T ]. Then ξ( ) : [0, T ] T SO(3), with ξ(t) Tḡ(t) SO(3) and λ 0 such that H λ ū(t) (λ, ξ(t)) (0, 0) (1) ξ(t) = H ū(t) λ (ξ(t)) (2) (ξ(t)) = max Hu λ (ξ(t)) = constant. (3) u Trajectory ḡ( ) is a projection of the integral curve ξ( ) of H λ ū(t) Any pair (g( ), u( )) satisfying the PMP is called a trajectory-control pair We only consider the case when λ = 1 2 (normal extremals) R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 24 / 34

25 Poisson structure Structure on so(3) p = p 1 E1 + p 2E2 + p 3E3 = [ p 1 p 2 ] p 3 so(3) Lie-Poisson bracket of F, G C (so(3)): {F, G} (p) = p ([df (p), dg(p)]) Hamiltonian vector field: H[F] = {F, H} Quadratic Hamilton-Poisson system (so(3), H) H : p pa + pqp, A so(3) Equations of motion: ṗ i = p ([E i, dh(p)]) R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 25 / 34

26 Example Consider the family of cost-extended systems (Σ (2,0), χ 2 α) : { Ξ (2,0) (1, u) = u 1 E 1 + u 2 E 2 χ 2 α(u) = (u 1 α) 2 + u 2 2, α 0. We have H u (p) = 1 2 Then ( (u1 α) 2 + u2 2 ) + p(u1 E 1 + u 2 E 2 ) H u 1 = (u 1 α) + p 1 = 0 = u 1 = p 1 + α H u 2 = u 2 + p 2 = 0 = u 2 = p 2 The optimal Hamiltonian is given by H(p) = αp (p2 1 + p2 2 ) R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 26 / 34

27 Relation to Hamilton-Poisson systems Recall that Ξ(1, u) = A + l i=1 u ib i Let B be the 3 l matrix where the i th column of B is the coordinate vector of B i in the basis {E 1, E 2, E 3 }. Then Ξ(1, u) = A + Bu Proposition Any ECT (g( ), u( )) of (Σ, χ) is given by ġ = Ξ(g(t), u(t)) and u(t) = Q 1 B p(t) + µ Here p( ) : [0, T ] so(3) is an integral curve of the Hamilton-Poisson system on so(3) specified by H(p) = p(a + Bµ) pbq 1 B p. R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 27 / 34

28 Affine equivalence Definition Systems G and H are A-equivalent if affine automorphism ψ such that ψ G = H Proposition The following systems are equivalent to H: H ψ : where ψ - linear Poisson automorphism H (p) = pa + p(rq)p : where r 0 H + C : where C - Casimir function R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 28 / 34

29 Classification on so (3) H(p) = pqp 1 2 p2 1 p p2 2 Conditions α 1, α 2 > 0 α 1 α 3 > 0 α 1 > α 4 > 0 or α 1 = α 4 > 0 H(p) = pa + pqp α 1 p p2 1 p p2 1 p 1 + α 1 p p2 1 α 1 p 1 + p p2 2 α 1 p 2 + p p2 2 α 1 p 1 + α 2 p 2 + p p2 2 α 1 p 1 + α 3 p 3 + p p2 2 α 1 p 1 + α 2 p 2 + α 4 p 3 + p p2 2 R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 29 / 34

30 Example Consider (Σ (2,0), χ 2 α) for α > 0 The optimal Hamiltonian H(p) = αp (p2 1 + p2 2 ) is equivalent to H 1 1 (p) = p p2 1 Indeed, let ψ : p p α α 2 0 α 0 0 Then we have ψ H = H 1 1 ; or more specifically, (T ψ H)(p) αp 2 = αp 2 p 3 = ( H 1 1 ψ)(p) α (α + p 1 ) p 3 R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 30 / 34

31 An important relationship Proposition (Σ, Ξ) and (Σ, Ξ ) are C-equivalent Remark The converse is not true associated H and H are A-equivalent However, we can still solve for the optimal controls of any given cost-extended system R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 31 / 34

32 Counter example (Σ (2,0), χ 2 0 ) Ξ (2,0) (1, u) = u 1 E 1 + u 2 E 2 χ 2 0 : T 0 (u 1(t) 2 + u 2 (t) 2 )dt min (Σ (3,0), χ 1 β ) H 2 (p) = 1 2 (p2 1 + p2 2 ) Ξ (3,0) (1, u) = u 1 E 1 + u 2 E 2 + u 3 E 3 χ 1 β : T 0 (u 1(t) 2 + u 2 (t) 2 + βu 3 (t) 2 )dt min, 0 < β < 1 H 3 (p) = 1 2 (p2 1 + p β p2 3 ) H 2 and H 3 are both equivalent to H(p) = 1 2 p2 1 R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 32 / 34

33 Outline 1 Introduction 2 Control systems on SO(3) 3 Cost-extended systems 4 Hamilton-Poisson systems 5 Conclusion R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 33 / 34

34 Conclusion Summary Related the notions of DF-equivalence, C-equivalence, and A-equivalence Obtained each type of classification for SO(3) Related work Final integration procedure on SO(3) Control systems on SO(4) Note that so(4) = so(3) so(3) R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 34 / 34