Some results on energy shaping feedback

Transcription

1 Some results on energy shaping feedback Andrew D. Lewis Bahman Gharesifard 13/08/2007 Department of Mathematics and Statistics, Queen s University andrew@mast.queensu.ca URL: PhD student, Department of Mathematics and Statistics, Queen s University The problem Given an open-loop mechanical control system Σ ol = (Q, G ol,v ol ): feedback Σ ol = (Q, G ol,v ol,f) Σ = (Q, G,V,F ) The osed-loop system Σ = (Q, G,V ) should have some desired properties, e.g., an equilibrium point q 0 should be stable.

2 In equations: Given: G ol γ (t)γ (t) = G ol dv ol (γ(t)) + m u a (t)g ol F a (γ(t)). a=1 Find: Feedback controls u shp : TQ R m such that the osed-loop system has governing equations G γ (t)γ (t) = G dv (γ(t)) + G F (γ (t)). Form of F : F = F,diss + F,gyr where 1. F,diss is a dissipative force and 2. F,gyr is a quadratic gyroscopic force. Recall: A quadratic gyroscopic force is of the form F gyr (v q );w q = B gyr (w q,v q,v q ), where B gyr is a (0,3)-tensor field satisfying (denote F gyr (v q ) = B gyr(v q )). Punchline: Preserves energy. B gyr (u q,v q,w q ) = B gyr (v q,u q,w q )

3 Assumed procedure: 1. Find osed-loop (kinetic energy)/(quadratic gyroscopic force): G ol F kin (γ (t)) = G γ (t)γ (t) + G B,gyr(γ (t)) G ol γ (t)γ (t) 2. Find osed-loop potential energy: 3. Find osed-loop control: F pot (γ(t)) = G ol G dv (γ(t)) dv ol (γ(t)). }{{} Λ m u a shp(v q )G ol F a (q) = F kin (v q ) F pot (q). a=1 Today: Ignore dissipative forces. Objectives of approach What are the possible osed loop energies, E (v q ) = 1 2 G (v q ) + V (q)? For stabilisation: Want Hess V (q 0 ) > 0? Main limitation for stabilisation: only works for systems that are linearly stabilizable, i.e., doesn t work for hard systems (i.e., requiring discontinuous feedback) if there are benefits, they are global in nature.

4 Potential shaping: Partial literature review and comments 1. M. Takegaki and S. Arimoto, A new feedback method for dynamic control of manipulators, Trans. ASME Ser. G J. Dynamic Systems Measurement Control, vol. 103, no. 2, pp , A. J. van der Schaft, Stabilization of Hamiltonian systems, Nonlinear Anal. TMA, vol. 10, no. 10, pp , A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden, and G. Sánchez de Alvarez, Stabilization of rigid body dynamics by internal and external torques, Automatica J. IFAC, vol. 28, no. 4, pp , Hamiltonian approach (IDA-PBC): 4. R. Ortega, M. W. Spong, F. Gómez-Estern, and G. Blankenstein, Stabilization of a ass of underactuated mechanical systems via interconnection and damping assignment, IEEE Trans. Automat. Control, vol. 47, no. 8, pp , Lagrangian approach with symmetry: 5. A. M. Bloch, N. E. Leonard, and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem, IEEE Trans. Automat. Control, vol. 45, no. 12, pp , A. M. Bloch, D. E. Chang, N. E. Leonard, and J. E. Marsden, Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping, IEEE Trans. Automat. Control, vol. 46, no. 10, pp , 2001.

5 Equivalence of Lagrangian and Hamiltonian setting: 7. D. E. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden, and C. A. Woolsey, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM Contrôle Optim. Calc. Var., vol. 8, pp , G. Blankenstein, R. Ortega, and A. J. van der Schaft, The matching conditions of controlled Lagrangians and IDA-passivity based control, Internat. J. Control, vol. 75, no. 9, pp , Geometric formulation and weak integrability results: 9. D. R. Auckly and L. V. Kapitanski, On the λ-equations for matching control laws, SIAM J. Control Optim., vol. 41, no. 5, pp , D. R. Auckly, L. V. Kapitanski, and W. White, Control of nonlinear underactuated systems, Comm. Pure Appl. Math., vol. 53, no. 3, pp , Extension to general Lagrangians: 11. J. Hamberg, General matching conditions in the theory of controlled Lagrangians, in Proceedings of the 38th IEEE CDC. Phoenix, AZ: IEEE, Dec. 1999, pp J. Hamberg, Simplified conditions for matching and for generalized matching in the theory of controlled Lagrangians, in Proceedings of the 2000 American Control Conference, Chicago, IL, June 2000, pp Extension to nonholonomic systems: 13. D. V. Zenkov, Matching and stabilization of the unicye with rider, in Proc. IFAC Workshop on Lagrangian and Hamiltonian Methods in Nonlinear Control, Princeton, NJ, March 2000, pp Linear systems: 14. D. V. Zenkov, Matching and stabilization of linear mechanical systems, in Proceedings of MTNS 2002, South Bend, IN, Aug

6 A theorem on potential energy shaping The setup We have an open-loop system (Q, G ol,v ol, {F 1,...,F m }) and have applied a control to shape the kinetic energy to G. Define Λ = G ol G. Let F be the codistribution generated by {F 1,...,F m } (control forces are thus F-valued). Assume constant rank. Let F be the codistribution Λ 1 (F). Lemma 1 Given: G ol, V ol, G, and F. A force F taking values in F gives a osed-loop potential V if and only if F(q) = Λ dv (q) dv ol (q), q Q. The potential energy shaping partial differential equation Define Q R = Q R and π: Q R Q by π(q,v ) = q. A potential function V defines a section of Q R : q (q,v (q)). We have the map Φ d : J 1 Q R T Q satisfying Φ d (j 1 V (q)) = dv (q). Abbreviate α = Λ 1 dv ol. Let π F : T Q T Q/F be the canonical projection. Define R pot = {j 1 V (q) J 1 Q R π F Φ d (j 1 V (q)) = π F α (q)}.

7 Proposition 1 A section F of F is a potential energy shaping feedback if and only if F = Λ dv dv ol for a solution V to R pot. What s the point of all the fanciness? You get a partial differential equation in the framework for applying the Goldschmidt 1 theory for integrability of partial differential equations. 1 J. Differential Geom., 1, , 1967 The statement Let I 2 (F ) be the two-forms in the algebraic ideal generated by F. Theorem 1 Let (Q, G ol,v ol,f) be an analytic simple mechanical control system and let G be an analytic Riemannian metric. Let j 1 V (q 0 ) R pot. Assume that q 0 is a regular point for F and that F is integrable in a neighbourhood of q 0. Then the following statements are equivalent: (i) there exists a neighbourhood U of q 0 and an analytic potential energy shaping feedback F defined on U which satisfies Φ d (j 1 V (q 0 )) = F (q 0 ) + α (q 0 ); (ii) there exists a neighbourhood U of q 0 such that dα (q) I(F,q ) for each q U. Moreover, if V,1 and V,2 are two osed-loop potential functions, then d(v,1 V,2 )(q) F,q for each q Q.

8 The working version If F is not integrable, replace it with the largest integrable codistribution contained in it. Since F is integrable choose coordinates (q 1,...,q n ) for Q such that Write α = G,ij G jk V ol ol q }{{ k dq i. } α,i F,q = span R {dq 1 (q),...,dq r (q)}. The potential shaping partial differential equation has a solution if α,a q b = α,b q a, a,b {r + 1,...,n}. If V is some solution to the potential shaping partial differential equation, then any other solution has the form V (q 1,...,q n ) = V (q 1,...,q n ) + F(q 1,...,q r ). Discussion The proof of the existence part of the theorem is not constructive. It merely tells you that there are no obstructions to constructing a Taylor series solution order-by-order. Note that the integrability condition for potential shaping is a condition on α = Λ 1 dv ol. This is dependent on G. The point: A bad design for G can make it impossible to do any potential energy shaping. Note that we require integrability of F = Λ 1 (F). This is dependent on G. The point: A bad design for G can make it impossible to achieve any flexibility in the character of the possible osed-loop potential functions.

9 An affine connection formulation of kinetic energy shaping For a Riemannian metric G, define KE G : TQ R by KE G (v q ) = 1 2 G(v q,v q ). An affine connection is G-energy-preserving if L γ (t)ke G (γ (t)) = 0 for every geodesic γ of. Lemma 2 is G-energy preserving if and only if Sym( G) = 0. Theorem 2 Given: G ol and F. The solutions to the following problems are in 1 1 correspondence: (i) when does there exist G and a gyroscopic tensor B such that G γ (t)γ (t) + G B (γ (t)) G ol γ (t)γ (t) G ol (F); (ii) when does there exist G and a G -energy preserving connection such that γ (t)γ (t) G ol γ (t)γ (t) G ol (F).

10 The kinetic energy shaping partial differential equation Geometric formulation of partial differential equation Recall that the set of torsion-free affine connections on Q is an affine subbundle Aff 0 (Q) = {Γ T Q J 1 TQ Γ π 1 0 = id TQ, modelled on S 2 (T Q) TQ. (j 1 Y Γ(Y ))(X) (j 1 X Γ(X))(Y ) = [X,Y ]} Let Y KE = { (Γ, G) Aff 0 (Q) S 2 +(T Q) Γ is G-energy preserving }. Define Φ LC : J 1 S 2 +(T Q) Aff 0 (Q) by Φ LC (j 1 G) = G. Define the quasilinear partial differential equation R kin = { (j 1 Γ(q),j 1 G(q)) J 1 Y π F (Γ(q) Φ KE (j 1 G(q))) = 0 }, where π F : S(T Q) TQ S(T Q) TQ/G ol (F) is the canonical projection. An observation Define two subsets of Aff 0 (Q): Aff 0 (Q,F, G ol ) = G ol + S 2 (T M) coann(f), EP(Q) = { Aff 0 (Q) is G-energy preserving for some G}. The solutions (, G ) to R kin are then described by asking that Aff 0 (Q,F, G ol ) EP(Q). Aff 0 (Q,F, G ol ) is easy to understand. What about EP(Q)? And when EP(Q) what does { G Sym( G) = 0} look like?

11 Relationship to an inverse problem in calculus of variations Consider the following subset of EP(Q): LC(Q) = { Aff 0 (Q) is the Levi-Civita connection for some G}. The problem was initially investigated by Eisenhart and Veblen 1 who give necessary conditions and a sufficient condition with strong hypotheses. Comparison of problems: LC(Q) G = 0 has solution EP(Q) Sym( G) = 0 has solution The Eisenhart and Veblen problem is nice: it has an involutive symbol. The symbol for our generalisation is not involutive work to do here. 1 Proceedings of the National Academy of Sciences of the United States of America, 8, 19 23, 1922 Summary The method of energy shaping has been applied in certain cases, sometimes with some generality. However... The question, If I give you a system, can you determine whether it can be stabilised using energy shaping remains unresolved.