Two-Point Boundary Value Problem and Optimal Feedback Control based on Differential Algebra

Transcription

1 Two-Point Boundary Value Problem and Optimal Feedback Control based on Differential Algebra Politecnico di Milano Department of Aerospace Engineering Milan, Italy Taylor Methods and Computer Assisted Proofs Barcelona, 2-7 June 28

2 Motivation Space trajectory and space system design is always affected by uncertainties Uncertainties due to navigation systems Uncertainties in modeling the dynamical environment Operating conditions will generally differ from the nominal design Suitable algorithms must be developed to estimate the effects of the previous uncertainties design control corrections to compensate possible errors Differential algebra is applied to get: HO expansions of the solution of TPBVPs HO expansions of the solution of optimal control problems

3 Outline Notes on Differential Algebra (DA) High Order expansion of the flow of ODEs High Order Two-Point Boundary Value Problem (TPBVP) Solver Lambert problem Station keeping (SK) around Halo orbits Optimal control problem Indirect methods Optimal feedback control based on Differential Algebra Lunar Landing

4 Notes on Differential Algebra DA is an algebra of Taylor polynomials, which can be readily implemented in a computer environment DA enables the automatic computation of the Taylor expansion of any function of variables up to the arbitrary order f v n Unlike standard automatic differentiation tool, the analytic operations of differentiation and antiderivation are introduced A DA number can be seen as a Taylor Model without the interval remainder bound: TM DA (P α,f,i α,f ) P α,f n D v nd v is the DA framework for Taylor polynomials of variables and order n v

5 Expansion of ODEs Flow (1/2) Consider the ODE initial value problem: ẋ = f(x), x() = x Any integration scheme is based on algebraic operations, involving the evaluation of f at several integration points x Replacing with and carrying out all the operations in enable the evaluation of the Taylor nd v [x ] n D v expansion of the ODE flow Example: explicit Eulerʼs scheme [x] k+1 = [x] k + f([x] k ) h [x] k+1 n At each step, is the -th Taylor expansion of the flow of the ODE

6 Expansion of ODEs Flow (2/2) Example: 2-Body Problem Eccentricity:.5 Starting point: pericenter Integration scheme: Runge-Kutta (variable step, order 8) Order of the flow expansion: 5 Sensitivity analysis with respect to the initial conditions y [AU] !.5!1!1.5!3!2.5!2!1.5!1! x [AU] An uncertainty box of size.1 AU on the initial position is propagated by means of the 5th order expansion of the flow

7 Expansion of ODEs Flow (2/2) Example: 2-Body Problem Eccentricity:.5 Starting point: pericenter Integration scheme: Runge-Kutta (variable step, order 8) Order of the flow expansion: 5 Sensitivity analysis with respect to the initial conditions y [AU] !.5!1!1.5!3!2.5!2!1.5!1! x [AU] An uncertainty box of size.1 AU on the initial position is propagated by means of the 5th order expansion of the flow

8 Expansion of ODEs Flow (2/2) Example: 2-Body Problem Eccentricity:.5 Starting point: pericenter Integration scheme: Runge-Kutta (variable step, order 8) Order of the flow expansion: 5 Sensitivity analysis with respect to the initial conditions y [AU]!1!1.2!1.4!1.6!1.8 exact n = 1 n = 2 n = 5!2!1! x [AU] An uncertainty box of size.1 AU on the initial position is propagated by means of the 5th order expansion of the flow

9 High Order TPBVP solver (1/3) Example: Lambert problem Given initial position: final position: time of flight: tof Find the initial velocity,, the spacecraft must have to reach in tof Various algorithms exist to identify a nominal solution of this TPBVP, based on iterative procedures Suppose an error occurs on r i r i r f δr i What is the new v i? v f r f r i v i r i v i

10 High Order TPBVP solver (2/3) Given a nominal solution, Use DA to expand the flow of the ODE w.r.t. and r i v i v i Build the following map: ( ) ( [ ] )( ) δrf Mrf δri = δr i [I ri ] δv i Invert it: ( ) ( [ ] ) 1 ( ) δri Mrf δrf = δv i [I ri ] δr i, to the Lambert problem: ( ) ( [ ] )( δrf Mrf δri = [ ] δv f Mvf δv i By imposing δr f =, the previous map delivers a Taylor series expansion of the solution of the TPBVP in δr i )

11 High Order TPBVP solver (3/3) Given a displacement from the nominal initial position, δr i, the evaluation of the previous map delivers the corrections to the nominal initial velocity, δv i, to reach the final desired nominal position, Test case: Earth-Mars transfer (Mars Express) y [AU] nominal r f 1.5 Mars 1.5 Sun!.5!1 Earth!1.5!1! x [AU]

12 High Order TPBVP solver (3/3) Given a displacement from the nominal initial position, δr i, the evaluation of the previous map delivers the corrections to the nominal initial velocity, δv i, to reach the final desired nominal position, Test case: Earth-Mars transfer (Mars Express) y [AU] !.5!1 r f error box of size.1 AU Earth Sun Mars!1.5!2!1.5!1! x [AU]

13 High Order TPBVP solver (3/3) Given a displacement from the nominal initial position, δr i, the evaluation of the previous map delivers the corrections to the nominal initial velocity, δv i, to reach the final desired nominal position, Test case: Earth-Mars transfer (Mars Express) y [AU] !.5!1 no corrections Earth Sun Mars!1.5!2!1.5!1! x [AU] y [AU] !.5!1 r f 5-th order corrections Earth Sun Mars!1.5!2!1.5!1! x [AU]

14 SK around Halo Orbits (2/2) Station Keeping on a Halo orbit around L1: Given a nominal halo orbit, design the correction maneuvers to compensate dynamical perturbations TPBVP formulation: given a displacement from the current nominal state, cancel the error after a given time Example: Reference halo orbit (Az = 8 km) Uncertainty on spacecraft position at intersection with x-z Error cancellation after.5 period Y.6.4.2!.2!.4 No corrections L X

15 SK around Halo Orbits (2/2) Station Keeping on a Halo orbit around L1: Given a nominal halo orbit, design the correction maneuvers to compensate dynamical perturbations TPBVP formulation: given a displacement from the current nominal state, cancel the error after a given time Example: Reference halo orbit (Az = 8 km) Uncertainty on spacecraft position at intersection with x-z Error cancellation after.5 period Y.6.4.2!.2!.4 1st order corrections L X

16 SK around Halo Orbits (2/2) Station Keeping on a Halo orbit around L1: Given a nominal halo orbit, design the correction maneuvers to compensate dynamical perturbations TPBVP formulation: given a displacement from the current nominal state, cancel the error after a given time Example: Reference halo orbit (Az = 8 km) Uncertainty on spacecraft position at intersection with x-z Error cancellation after.5 period Y.6.4.2!.2!.4 3rd order corrections L X

17 SK around Halo Orbits (2/2) Station Keeping on a Halo orbit around L1: Given a nominal halo orbit, design the correction maneuvers to compensate dynamical perturbations TPBVP formulation: given a displacement from the current nominal state, cancel the error after a given time Example: Reference halo orbit (Az = 8 km) Uncertainty on spacecraft position at intersection with x-z Error cancellation after.5 period Y.6.4.2!.2!.4 1th order corrections L X

18 Optimal Control Problem Consider the dynamical system: Determine u that minimizes the performance index: The following constraints are usually imposed Boundary conditions at final time : Path constraints on the control variables: Consider the particular constraints: Direct methods: based on reducing the optimal control problem to a nonlinear programming problem Indirect methods: based on reducing the optimal control problem to a Boundary Value Problem (BVP) t f

19 Indirect Methods (1/3) Constraints are added to the performance index introducing two kinds of Lagrange multipliers: for the final constraints and (costate variables) for dynamics and path constraints The problem is then reduced to identifying a stationary point of J by

20 Indirect Methods (2/3) Dynamics Constraints

21 Indirect Methods (3/3) The problem consists on finding the functions, and by solving the differential-algebraic system: differential algebraic Euler-Lagrange (EL) equations The previous differential-algebraic system must be coupled to additional boundary and path constraints The optimal control problem is reduced to a boundary value problem on a differential-algebraic system of equations (DAE)

22 Lunar Landing (1/4) Nominal initial conditions: pericenter of a 2x1 km descent polar orbit Final conditions: Lunar south pole (altitude: 2 m; velocity: 3 m/s downward) Dynamics: ẍ = µ r 3 x + u 1 ÿ = µ r 3 y + u 2 z = µ r 3 z + u 3 and x 2 km y 1 km

23 Lunar Landing (2/4) Using order reduction, the dynamics reads: Performance index: minimize the quadratic functional Constraints: fixed and, fixed final time Euler-Lagrange equations Differential part:

24 Lunar Landing (3/4) Processing the last algebraic equation leads to: which can be inserted in the differential equations: ẋ = f(x, u) u = λ 1,2,3 λ 1,2,3 = M T u where All constraints simply reduce to: x(t )=x x(t f )=x f f x = [ I M The original problem is reduced to a simple Two Point Boundary Value Problem (TPBVP) ]

25 Lunar Landing (4/4) Nominal solution: altitude [km] altitude profile time [min] u i [m/s 2 ] ! time [min] Optimal feedback control: Suppose an error δx occurs on the initial state; i.e., x(t )=x + δx Find the new optimal control history u(t) control history u x u y u z

26 Optimal Feedback Control using DA (1/2) Given the nominal solution to the previous problem, ( x, ū, ū ), consider an error in the initial state [x] = x + δx Using a DA-based Runge-Kutta integrator, expand the flow of the dynamics w.r.t. x, u, and u [ ] δx f Mxf δx [ ] δu f = Muf δu [ ] δ u f M uf δ u Build the following map: ( δxf δx ) = ( [ ] ) δx Mxf δu [I x ] δ u

27 Optimal Feedback Control using DA (2/2) Invert the previous map: By imposing δx f =, the previous map delivers a Taylor series expansion of the solution of the optimal control problem in δx x δx δu Given an error on,, the corrections and are computed from the Taylor expansions δ u δx δu δ u = ( [ ] ) 1 ( ) Mxf δxf [I x ] δx The new optimal control history is obtained by integrating the dynamics starting from the new initial conditions

28 Lunar Landing: Feedback Control Dynamics: r = µ r 3 r + u Maximum errors: position: 1 km velocity: 5 m/s altitude [km] nominal without corrections time [min] position error [km] x 2 km without corrections y 1 km time [min]

29 Lunar Landing: Feedback Control Dynamics: altitude [km] r = µ r 3 r + u Maximum errors: position: 1 km velocity: 5 m/s nominal without corrections with corrections time [min] position error [km] x 2 km y without corrections with corrections 1 km time [min]

30 Lunar Landing: Feedback Control Dynamics: altitude [km] r = µ r 3 r + u Maximum errors: position: 1 km velocity: 5 m/s nominal without corrections with corrections time [min]! u i [m/s 2 ] x 2 km y 1 km! u x! u y! u z time [min]

31 References P. Di Lizia, R. Armellin, E. A. Finzi, and M. Berz High Order Robust Guidance of Interplanetary Trajectories based on Differential Algebra Journal of Aerospace Engineering, Sciences and Applications, Vol. I, n. 1, 28 (available at P. Di Lizia Robust Space Trajectory and Space System Design using Differential Algebra Ph.D. dissertation, Politecnico di Milano, 28 (soon available at

32 Two-Point Boundary Value Problem and Optimal Control of Spacecraft Dynamics based on Differential Algebra Politecnico di Milano Department of Aerospace Engineering Milan, Italy Taylor Methods and Computer Assisted Proofs Barcelona, 2-7 June 28