Optimal Control of Drift-Free Invariant Control Systems on the Group of Motions of the Minkowski Plane

Transcription

1 Optimal Control of Drift-Free Invariant Control Systems on the Group of Motions of the Minkowski Plane Dennis I. Barrett*, 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 4 7, 014, Strasbourg, France Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

2 Outline 1 Invariant optimal control The semi-euclidean group SE(1, 1) 3 Classification of cost-extended systems 4 Extremal controls Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC 014 / 15

3 Introduction Context invariant control systems on 3D Lie groups invariant optimal control problems (with quadratic cost) Problem classify cost-extended systems determine extremal controls calculate extremal trajectories Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

4 Invariant control systems Drift-free left-invariant control system Σ = (G, Ξ) state space G matrix Lie group with Lie algebra g dynamics Ξ : G R l T G left-invariant: Ξ(g, u) = g Ξ(1, u) parametrization map: Ξ(1, ) : R l g, u u 1 B u l B l trace Γ = B 1,..., B l g is l dimensional Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

5 Trajectories and controllability Admissible controls and trajectories admissible control: piecewise cont. curve u( ) : [0, T ] R l trajectory corresponding to u( ): abs. cont. curve g( ) : [0, T ] G such that ġ(t) = Ξ(g(t), u(t)) for a.e. t [0, T ] (g( ), u( )) is a trajectory-control pair Controllability Σ is controllable if any two points can be joined by a trajectory necessary and sufficient: trace Γ generates g Restrict to controllable systems Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

6 Equivalence of control systems Detached feedback equivalence Σ is DF -equivalent to Σ if φ Diff(G), ϕ GL(R l ) such that T g φ Ξ(g, u) = Ξ (φ(g), ϕ(u)), for every g G, u R l trajectories of DF -equivalent systems in 1-to-1 correspondence Algebraic characterization Σ and Σ DF -equivalent φ Aut(G) such that T 1 φ Γ = Γ Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

7 Optimal control Invariant optimal control problem left-invariant control system Σ = (G, Ξ) boundary data (initial state g 0, final state g 1, terminal time T > 0) quadratic cost: χ(u) = u Q u, u R l, Q R l l is PD ġ = g(u 1 B u l B l ) g(0) = g 0, g(t ) = g 1, T > 0 fixed J [u( )] = T 0 χ(u(t)) dt min (OCP) Minimize cost functional J over trajectory-control pairs (g( ), u( )) of Σ subject to boundary data Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

8 Pontryagin lift Lifting to the cotangent bundle lift the problem to the cotangent bundle T G = G g using Pontryagin Maximum Principle: G-invariant Hamiltonian function H : T G R induced Hamilton-Poisson system (g, H) Hamilton-Poisson systems on g = (g, {, }) Lie-Poisson bracket: {F, G}(p) = p, [df (p), dg(p)] Hamiltonian vector field H = {, H} Proposition The (normal) extremal controls of an optimal control problem (OCP) are linearly related to the integral curves of (g, H) Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

9 Cost-extended systems Associate to each optimal control problem a cost-extended system (Σ, χ) Cost equivalence (Σ, χ) is C-equivalent to (Σ, χ ) if φ Aut(G), ϕ GL(R l ) such that T g φ Ξ(g, u) = Ξ (φ(g), ϕ(u)) and χ ϕ = rχ for some r > 0 optimal (resp. extremal) trajectory-control pairs of C-equivalent systems in 1-to-1 correspondence Characterization (same underlying control system) (Σ, χ) and (Σ, χ ) are C-equivalent ϕ GL(R l ) such that ϕ preserves Σ: φ Aut(G) such that T 1 φ Ξ(1, u) = Ξ(1, ϕ(u)) χ = rχ ϕ for some r > 0 Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

10 The semi-euclidean group SE(1, 1) Lie group and Lie algebra SE(1, 1) : x cosh θ sinh θ se(1, 1) : x 0 θ y sinh θ cosh θ y θ 0 group of isometries of (R, ), where x y = x 1 y 1 + x y connected and simply connected Standard basis E 1 = E = E 3 = [E, E 3 ] = E 1 [E 3, E 1 ] = E [E 1, E ] = 0 Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

11 Classification of cost-extended systems Cost-extended systems on SE(1, 1) Let (Σ, χ) be a controllable drift-free cost-extended system. If Σ is two-input, then (Σ, χ) is C-equivalent to { Ξ (,0) (1, u) = u 1 E 1 + u E 3 χ (,0) (u) = u 1 + u If Σ is three-input, then (Σ, χ) is C-equivalent to a system { Ξ (3,0) (1, u) = u 1 E 1 + u E + u 3 E 3 χ (3,0) λ (u) = u 1 + λu + u 3 (0 < λ 1) Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

12 Proof sketch (two-input) DF -equivalence Every two-input system Σ = (SE(1, 1), Ξ) is DF -equivalent to the system C-equivalence Σ (,0) = (SE(1, 1), Ξ (,0) ), Ξ (,0) (1, u) = u 1 E 1 + u E 3 every cost-extended system is C-equivalent to a system (Σ (,0), χ), [ ] χ(u) = u α1 β u β α [ 1 β ] [ ] α1 α β then ϕ 1 = α 1, ϕ = α 1 0 preserve Σ (,0) and (χ ϕ 1 ϕ )(u) = u [ r 0 0 r ] u = rχ (,0) (u), r = α 1α β α 1 > 0 Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

13 Extremal controls of two-input system (normal) extremal controls: u 1 = p 1, u = p 3 p( ) int. curve of H (,0) (p) = 1 (p 1 + p3) E E 0 E 1 E E 0 E 1 (a) C(p) > 0 Equations of motion ṗ 1 = p p 3 ṗ = p 1 p 3 ṗ 3 = p 1 p C(p) = p1 p is a Casimir E E 0 E 1 E 3 0 (b) C(p) = 0 0 E 0 E 1 Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

14 Integration Integral curves p( ) of H (,0) Let H (,0) (p(0)) = h 0 > 0 and C(p(0)) = c 0 0. If c 0 > 0, then there exist t 0 R and σ { 1, 1} such that p(t) = p(t + t 0 ) for every t If c 0 = 0, then there exist t 0 R and σ 1, σ { 1, 1} such that p(t) = q(t + t 0 ) for every t p 1 (t) = σω dn(ωt, k) p (t) = σkω cn(ωt, k) p 3 (t) = kω sn(ωt, k) q 1 (t) = σ 1 Ω sech(ωt) q (t) = σ 1 σ Ω sech(ωt) q 3 (t) = σ Ω tanh(ωt) Ω = h 0 k = 1 c 0 /Ω Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15

15 Conclusion Sub-Riemannian structures invariant SR structure associated to every cost-extended system thus, on SE(1, 1) (up to isometric group automorphisms): [ ] 1 0 sub-riemannian structures: D 1 = E 1, E 3, g 1 = Riemannian structures: g1 λ = 0 λ Outlook determine optimal trajectories classification of cost-extended systems with drift Barrett, Biggs, Remsing (RU) Control Systems on SE(1, 1) ECC / 15