Numerical Computational Techniques for Nonlinear Optimal Control

Transcription

1 Seminar at LAAS Numerical Computational echniques for Nonlinear Optimal Control Yasuaki Oishi (Nanzan University) July 2, 2018 * Joint work with N. Sakamoto and. Nakamura

2 1. Introduction Optimal control of a nonlinear system Hamilton--Jacobi--Bellman eq. Power series method [Al'brekht 61][Lukes 69]... Stable-manifold method [Sakamoto--van der Schaft 08] cf. [Yamashita--Shima 98] swing-up of a pendulum, control under input saturation, nonlinear optimal servo,... 2

3 his talk: Numerical computational techniques for improvement of the stable-manifold method ime evolution of a Hamiltonian system numerically unstable to compute numerical methods that preserve the structure of the Hamiltonian system Choice of initial points based on trials and errors shooting method 3

4 2. Stable-manifold method Problem Given a plant xt ( ) = fxt ( ( )) + Gxt ( ( )) ut ( ), obtain an input ut () that minimizes the objective fn. ò 0 xt Qxt + ut Rut t () () () () d. where f(0) = 0, Q O, R O Hamiltonian 1-1 Hxp (, ) = pfx ()- pgxr () gx () p+ xqx 4 Hamiltonian system H H xt () = ( xt (), pt ()), pt () =- ( xt (), pt ()) p x 4

5 Hamiltonian sys. H H xt xp pt xp p x () = (, ), () =- (, ) (, xp) origin stable manifold {( xp, ) p= px ( )} Riccati stab. sol. P {( xp, ) p= 2 Px} Optimal input: 1 ut =- R gx px 2-1 ( ) ( ) ( ) 5

6 Stable-manifold method simulate the Hamilton. sys. in backward time Issues precisely compute it around the origin Choose px ( ) so that the set {(, xpx ())} contains the trajs. Simulation of the Hamilton. sys. is numeric. unstable numerical methods that preserve the structure of the Hamiltonian system Choice of initial points is based on trials and errors shooting method 6

7 3. Structure preservation H H xt = xt pt pt =- xt pt p x initial point: ( x(0), p(0)) = ( x, p ) 0 0 () ( (), ()), () ( (), ()) Symplecticity ((), xt pt ()) ( x, p ) 0 0 ( Dx, Dp ) 0 0 time evolution of the perturbation ( Dx, Dp ) 0 0 ( Dxt ( ), Dpt ( )) æo I öædx() t ö ( xt () pt ()) D D = -I O Dp() t çè øèç ( Dxt ( ), Dpt ( )) ø Preservation of the Hamiltonian Hxt ((), pt ()) = const. const. 7

8 Numerical method Given a step size h > 0, obtain ( x, p ) that -k -k approximates (( x -kh), p( - kh)) for k = 1,2,, K ( x, p ) ( x, p ) ( x, p ) -K -K Standard classic Runge--Kutta method no symplecticity; no preserv. of the Hamiltonian Structure-preserving numerical methods numerically stable [Hairer--Lubich--Wanner 02] 8

9 Gauss method (of order 4) Symplectic numerical method For k = 0, 1,, K -1 g g 1 2 æ H ö p = ç H çè- x ø æ H ö p = ç H çè- x ø æ ö æ ö çè ø è ø æxö æx ö k h æ 1 3 ö - g hg p = p ç -k 4 è ø çè ø ç 4 6 è ø x x -k-1 -k h g 1 g 2 p = + + p -- k 1 ç -k 2 ( ) 1 2 æxö æx ö æ k 1 3 ö - h hg g p = p èç ø èç -k ø ç 4 6 è ø 4 Need to solve nonlinear eqs. for g and g

10 q() t Example: swing-up of a pendulum [Sakamoto 13] x ut () æqö = ç q çè ø Q = 0.01 I, R = 2 P = Riccati stabilizing solution ( x, p ) = ( x,2 Px ), x = ( ) h = p 1 x 1 p 2 x 2 time RK method time Gauss method 10

11 Values of the Hamiltonian Hxp (, ) is constant theoretically It can be used to see the quality of computation Hamiltonian H RK method Gauss method time Gauss method is a little better? 11

12 Averaged-vector-field method [Quispel--McLaren 08] Preserving the Hamiltonian value For k = 0, 1,, K -1 æ 1 H x ö æx ö æ ( xr ( ), pr ( )) ö -k-1 - k p : h dr p = - p ò0 H çè - ( xr ( ), pr ( )) -- k 1 ø çè - k ø ç è x ø where xr () = (1- rx ) + rx, pr () = (1- rp ) + rp -k -k-1 -k -k-1 Comput. of the integral Gaussian quadrature 1æ H H ( xr ( ), pr ( )) ö æ ( xr ( ), pr ( )) ö p p i i ò dr w 0 H» å i H ç - ( xr ( ), pr ( )) i 1 ( xr ( ), pr ( )) x = - è ø çè x i i ø Need to solve a nonlinear eq. for ( x, p ) k -- k 12

13 Example Hamiltonian H RK method AVF method time AVF method keeps the Hamiltonian constant It does not necessarily mean its superiority 13

14 4. Shooting method ( x, p ) ( x, p ) ( x, p ) -K -K ( x, p( x )) * * How to choose an initial point that leads to a given x? * Idea ( x, p ) -K -K ( Dx, Dp ) ( x, p ) ( x, p ) ( Dx,2 PDx ) -K -K ( Dx, Dp ) Evolution of the perturb. is linear possible to choose Dx0 so that D x = x -x * -K -K a kind of Newton method 14

15 ime evolution of the perturbation d dt æ 2 2 æ H H xö ( xt ( ), pt ( )) ( xt ( ), pt ( )) æ 2 xö D D p x p 2 2 p = H H - ( xt ( ), pt ( )) - ( xt ( ), pt ( )) p èçd ø ç D 2 çè ø è x x p ö ø æ x ö æ I ö D 0 x 2P x = D 2P ç D ç è 0ø è ø 0 numerically æ x ö D -1 ç p çèd -1 ø æu ö -1 = ç x V D çè -1 ø 0 æ x ö æu ö D -K -K = Dx p V çd ç è -Kø è -Kø 0 For choose D x = U D x = x - x -K -K 0 * -K Dx For a new initial pt. ( x +D x, p + 2 PDx ), compute a sequence Repeat until convergence 0 15

16 x 2 Example: swing-up of a pendulum x h x 0 * = ( ) = 0.002, - Kh =-0.45 = (4-10) x * computational time: 52 s ( p,0) 4th 3rd 2nd 1st x 1 rajectory for the swing up can be computed 16

17 Swing-up with multiple swings 1. Compute trajectories for 1 swing 17

18 Swing-up with multiple swings 1. Compute trajectories for 1 swing 2. Extend the simulation time for each trajectory Find a trajectory close to 2 swing 18

19 Swing-up with multiple swings 1. Compute trajectories for 1 swing 2. Extend the simulation time for each trajectory 3. Run the shooting method from the trajectory 19

20 Swing-up with 1 to 7 swings cf. [Horibe--Sakamoto 17] 20

21 Swing-up with 1 to 7 swings cf. [Horibe--Sakamoto 17] Obj. func swing 2-swing 3-swing 4-swing 5-swing 6-swing 7-swing

22 Swing-up with 7 swings 22

23 Application to the Furuta pendulum rajectory with 3 swings is successfully computed 23

24 5. Summary Application of numerical computational techniques to the stable-manifold method Structure-preserving numerical methods for stable computation of a Hamiltonian system Shooting method for systematic choice of an initial pt. Reduction of know-how factors of the stable-manifold method 24