Station keeping problem

Transcription

1 Station keeping problem Eric Goubault Benjamin Martin Sylvie Putot 8 June 016 Benjamin Martin Station keeping problem 8 June / 18

2 Introduction Hybrid autonomous systems Consider the system: 9x f px, uq where x P R n states, u P R m control (discrete number of possible controls). System is autonomous as u only depends on states. V -stability A function V pxq : R n Ñ R is V -stable if: V ě 0 Ñ 9 V ă 0 Let V : tx : V pxq ď 0u. If the system is V -stable, then from any initial states, after a given time, the trajectory enters the set V and never exits it. Benjamin Martin Station keeping problem 8 June 016 / 18

3 Introduction Illustration (Luc Jaulin) Benjamin Martin Station keeping problem 8 June / 18

4 Introduction Problem example: station keeping of a planar robot (1) Cartesian coordinates $ & 9x cospθq 9y sinpθq % 9θ u $ & % Polar coordinates 9φ sin φ d ` u d 9 cos φ 9α sin φ d with φ θ ` α π. Benjamin Martin Station keeping problem 8 June / 18

5 Introduction Problem example: station keeping of a planar robot () " 9φ sin φ d ` u d 9 cos φ with control law, e.g: # 1 if cos φ ď u sin φ otherwise? Question Is it certain that from any initial state, the robot eventually stays around a beacon centred at the origin? Benjamin Martin Station keeping problem 8 June / 18

6 Introduction Problem example: station keeping of a planar robot (3) Π 4 Π Π 4 0 Π 4 Π 4 Π Benjamin Martin Station keeping problem 8 June / 18

7 Introduction Transformation into a polynomial system Idea Try to prove the existence of an (algebraic) invariant for the system which allows to show what we want. Only possible with algebraic systems, thus adding extra variables: h sin φ, g cos φ and e 1 d. We obtain a polynomial system: $ 9h phe ` uqg & 9g phe ` uqh 9φ he ` u d 9 g % 9e ge with h ` g 1 and de 1. Benjamin Martin Station keeping problem 8 June / 18

8 Introduction Darboux polynomials Let L f ppq be the Lie derivative of polynomial p P Rrxs, (with respect to the flow f ). Definition A polynomial p is Darboux if L f ppq qp with q P Rrxs. In our case, is Darboux: e with cofactor ge, g ` h with cofactor 0. If u constant, pu ` ehq with cofactor ge If u h, dh with cofactor g. If u h, he with cofactor gpe 1q Benjamin Martin Station keeping problem 8 June / 18

9 Introduction Darboux polynomials Let L f ppq be the Lie derivative of polynomial p P Rrxs, (with respect to the flow f ). Definition A polynomial p is Darboux if L f ppq qp with q P Rrxs. In our case, is Darboux: e with cofactor ge g ` h with cofactor 0. If u constant, pu ` ehq with cofactor ge If u h, dh with cofactor g. If u h, he with cofactor gpe 1q Darboux polynomials can be used to derive invariant/variant as rational or logarithmic functions, e.g. [Goubault et al., ACC 014]. Benjamin Martin Station keeping problem 8 June / 18

10 Constant Control Invariant Consider u constant, then since u ` eh and e are Darboux with same cofactor, then ˆu ` eh L f e L f pu ` ehqe pu ` ehql f pe q e 4 0. This implies u`eh e condition e 0, h 0 : is constant (invariant). We have then for any initial u ` eh e u ` e 0h 0 e0 e0pu ` ehq ` e pu ` e 0 h 0 q 0, and as e, e 0 ą 0 that the sign of u ` eh is maintained. Benjamin Martin Station keeping problem 8 June / 18

11 Constant Control Invariant Consider u constant, then since u ` eh and e are Darboux with same cofactor, then ˆu ` eh L f e L f pu ` ehqe pu ` ehql f pe q e 4 0. This implies u`eh e condition e 0, h 0 : is constant (invariant). We have then for any initial u ` eh e u ` e 0h 0 e0 e0pu ` ehq ` e pu ` e 0 h 0 q 0, and as e, e 0 ą 0 that the sign of u ` eh is maintained. Ñ Can be obtained e.g. with [Goubault et al., ACC 014] or [Ghorbal and Platzer, TACAS 014] Benjamin Martin Station keeping problem 8 June / 18

12 Constant Control Invariant regions 8 V cst : V cst : V cst : " pφ, dq : u ` sinpφq * ă 0 d " pφ, dq : u ` sinpφq * 0 d " pφ, dq : u ` sinpφq * ą 0 d Benjamin Martin Station keeping problem 8 June / 18

13 Proportional control Invariant: proportional control Recall: dh is Darboux with cofactor g, and 9 d g. We can deduce: L f plogp dh q dq Lp dh q dh ` g 0. (since d ą 0, dh can be replaced by dh if h ă 0, dh otherwise) Benjamin Martin Station keeping problem 8 June / 18

14 Proportional control Invariant: proportional control Recall: dh is Darboux with cofactor g, and 9 d g. We can deduce: L f plogp dh q dq Lp dh q dh ` g 0. (since d ą 0, dh can be replaced by dh if h ă 0, dh otherwise) This implies logp dh q d is constant (invariant). For any initial condition d 0, h 0 : logp dh q d logp d 0 h 0 q d 0 logp dh q d logp d 0 h 0 q ` d 0 0. Benjamin Martin Station keeping problem 8 June / 18

15 Proportional control Invariant regions 10 V pro : tpφ, dq : logp sinpφqd q d ă u 8 V pro : tpφ, dq : logp sinpφqd q d u 6 4 V pro : tpφ, dq : logp sinpφqd q d ą u Benjamin Martin Station keeping problem 8 June / 18

16 Switched System Switched system Recall: u #? 1 if g ď h otherwise. Benjamin Martin Station keeping problem 8 June / 18

17 Switched System Switched system Recall: u #? 1 if g ď h otherwise. Consequence: all the proportional invariant regions are cut (not existing), but an invariant region in the constant case exists, maximal when V cst 1{ (i.e., the region tangent at the vertical line at φ π{4, at point p π{4,? {q). Benjamin Martin Station keeping problem 8 June / 18

18 Switched System Switched system Recall: u #? 1 if g ď h otherwise. Consequence: all the proportional invariant regions are cut (not existing), but an invariant region in the constant case exists, maximal when V cst 1{ (i.e., the region tangent at the vertical line at φ π{4, at point p π{4,? {q). We observe when proportional control is applied: 9 V cst ă 0; on the frontier of the region φ π{4 and φ π{4: flow enter when d ą 1, flow exit when d ă 1. Benjamin Martin Station keeping problem 8 June / 18

19 Switched System Switched system Recall: u #? 1 if g ď h otherwise. Consequence: all the proportional invariant regions are cut (not existing), but an invariant region in the constant case exists, maximal when V cst 1{ (i.e., the region tangent at the vertical line at φ π{4, at point p π{4,? {q). We observe when proportional control is applied: 9 V cst ă 0; on the frontier of the region φ π{4 and φ π{4: flow enter when d ą 1, flow exit when d ă 1. ñ Can the region defined by V cst ď 1{ be reached from anywhere? Benjamin Martin Station keeping problem 8 June / 18

20 Switched System Decomposition of the state space d φ Benjamin Martin Station keeping problem 8 June / 18

21 Switched System Decomposition of the state space d Regions 5,6,7 contains the invariant. φ Benjamin Martin Station keeping problem 8 June / 18

22 Switched System Analysis From region 5 to 6: flow enters at d 0 P r1,? s. By construction, enters 7 and then 5 at d 1 satisfying: logp d 0 h 0 q d 0 logp d 1 h 0 q d 1 d 1 e d 1 d 0 e d 0, with h 0 sinp π{4q. Benjamin Martin Station keeping problem 8 June / 18

23 Switched System Analysis From region 5 to 6: flow enters at d 0 P r1,? s. By construction, enters 7 and then 5 at d 1 satisfying: logp d 0 h 0 q d 0 logp d 1 h 0 q d 1 d 1 e d 1 d 0 e d 0, with h 0 sinp π{4q. Lambert-W function W pzq is defined as the solution to z xe x. Here, d 1 W p d 0 e d 0 q. From [Stewart, 009], it satisfies: d 0 ď W p d 0 e d 0 q ď 1{d 0. Therefore, d 1 P r d 0, 1{d 0 s Ñ d 1 P r?, 1s. Benjamin Martin Station keeping problem 8 June / 18

24 Switched System Analysis From region 5 to 6: flow enters at d 0 P r1,? s. By construction, enters 7 and then 5 at d 1 satisfying: logp d 0 h 0 q d 0 logp d 1 h 0 q d 1 d 1 e d 1 d 0 e d 0, with h 0 sinp π{4q. Lambert-W function W pzq is defined as the solution to z xe x. Here, d 1 W p d 0 e d 0 q. From [Stewart, 009], it satisfies: d 0 ď W p d 0 e d 0 q ď 1{d 0. Therefore, d 1 P r d 0, 1{d 0 s Ñ d 1 P r?, 1s. Recall d 1 a pq{ belongs to the maximal invariant. Hence, two cases: d 1 P r?,? {s or d 1 P r? {, 1s Benjamin Martin Station keeping problem 8 June / 18

25 Switched System First case: sliding mode Sliding mode on surface S tpφ, dq : φ π{4, d P r? {, 1su: flows in 5 are oriented towards 7 and vice versa. Benjamin Martin Station keeping problem 8 June / 18

26 Switched System First case: sliding mode Sliding mode on surface S tpφ, dq : φ π{4, d P r? {, 1su: flows in 5 are oriented towards 7 and vice versa. Note: f pφ, dq ˆ h d ` 1 g f`pφ, dq ˆ h d h g. From [Fillipov, 1988; Liberzon, 003], the system behaves as the unique convex combination f λ λf ` p1 λqf` normal to the normal of the S (vector p1, 0q). I.e. f λ with λ satisfying xf λ, p1, 0qy 0. Benjamin Martin Station keeping problem 8 June / 18

27 Switched System First case: sliding mode Sliding mode on surface S tpφ, dq : φ π{4, d P r? {, 1su: flows in 5 are oriented towards 7 and vice versa. Note: f pφ, dq ˆ h d ` 1 g f`pφ, dq ˆ h d h g. From [Fillipov, 1988; Liberzon, 003], the system behaves as the unique convex combination f λ λf ` p1 λqf` normal to the normal of the S (vector p1, 0q). I.e. f λ with λ satisfying xf λ, p1, 0qy 0. ñ f λ p0, gq with g?{. d is strictly decreasing with constant speed: d eventually reaches? { Benjamin Martin Station keeping problem 8 June / 18

28 Switched System Second case Exit surface is S tpφ, dq : φ π{4, d P r?,? {su. Benjamin Martin Station keeping problem 8 June / 18

29 Switched System Second case Exit surface is S tpφ, dq : φ π{4, d P r?,? {su. Flow enters region 5, with trajectory verifying V cst constant. In addition, V cst is monotonically decreasing on S: the outermost trajectory starts at d 0?. Benjamin Martin Station keeping problem 8 June / 18

30 Switched System Second case Exit surface is S tpφ, dq : φ π{4, d P r?,? {su. Flow enters region 5, with trajectory verifying V cst constant. In addition, V cst is monotonically decreasing on S: the outermost trajectory starts at d 0?. We deduce, from the def. of V cst, that the trajectory encounters φ π{4 at d 1 P r? {, 1s: entering into the sliding mode. Benjamin Martin Station keeping problem 8 June / 18

31 Switched System Second case Exit surface is S tpφ, dq : φ π{4, d P r?,? {su. Flow enters region 5, with trajectory verifying V cst constant. In addition, V cst is monotonically decreasing on S: the outermost trajectory starts at d 0?. We deduce, from the def. of V cst, that the trajectory encounters φ π{4 at d 1 P r? {, 1s: entering into the sliding mode. Proof of V -stability of V cst ` 1{ Benjamin Martin Station keeping problem 8 June / 18

32 Conclusion Conclusion A simple problem involves a non-trivial algebraic proof of stability. Benjamin Martin Station keeping problem 8 June / 18

33 Conclusion Conclusion A simple problem involves a non-trivial algebraic proof of stability. algebraic invariants can be automatically generated... Benjamin Martin Station keeping problem 8 June / 18

34 Conclusion Conclusion A simple problem involves a non-trivial algebraic proof of stability. algebraic invariants can be automatically generated but not the proof itself: only assisting the proof; Benjamin Martin Station keeping problem 8 June / 18

35 Conclusion Conclusion A simple problem involves a non-trivial algebraic proof of stability. algebraic invariants can be automatically generated but not the proof itself: only assisting the proof; in particular, specific functions (e.g. Lambert W function) can be involved; Benjamin Martin Station keeping problem 8 June / 18

36 Conclusion Conclusion A simple problem involves a non-trivial algebraic proof of stability. algebraic invariants can be automatically generated but not the proof itself: only assisting the proof; in particular, specific functions (e.g. Lambert W function) can be involved; can the proof be more direct? Benjamin Martin Station keeping problem 8 June / 18