Determination of optimal feedback terminal controllers for general boundary conditions using generating functions

Park, Chandeok; Scheeres, Daniel J.

doi:10.1016/j.automatica.2006.01.015

Cited by 63 publications

(61 citation statements)

References 8 publications

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“…The termination tolerance in (12) is 10 −9 . The optimal solutions found replicate those already known in the literature [10,11], indicating the effectiveness of the developed method.…”

Section: Low-thrustsupporting

confidence: 76%

“…This problem is taken from the literature where a solution is available, for comparison's sake [10,11]. Consider the planar, relative motion of two particles in a central gravity field expressed in a rotating frame with normalized units: the length unit is equal to the orbital radius, the time unit is such that the orbital period is 2 , and the gravitational parameter is equal to 1.…”

Section: Low-thrustmentioning

confidence: 99%

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Approximate Solutions to Nonlinear Optimal Control Problems in Astrodynamics

Topputo

Bernelli‐Zazzera

2013

ISRN Aerospace Engineering

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A method to solve nonlinear optimal control problems is proposed in this work. The method implements an approximating sequence of time-varying linear quadratic regulators that converge to the solution of the original, nonlinear problem. Each subproblem is solved by manipulating the state transition matrix of the state-costate dynamics. Hard, soft, and mixed boundary conditions are handled. The presented method is a modified version of an algorithm known as "approximating sequence of Riccati equations. " Sample problems in astrodynamics are treated to show the effectiveness of the method, whose limitations are also discussed.

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“…The termination tolerance in (12) is 10 −9 . The optimal solutions found replicate those already known in the literature [10,11], indicating the effectiveness of the developed method.…”

Section: Low-thrustsupporting

confidence: 76%

Section: Low-thrustmentioning

confidence: 99%

Approximate Solutions to Nonlinear Optimal Control Problems in Astrodynamics

Topputo

Bernelli‐Zazzera

2013

ISRN Aerospace Engineering

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“…The optimal feedback control for linear systems with quadratic objective functions is addressed through the matrix Riccati equation: this is a matrix differential equation that can be integrated backward in time to yield the initial value of the Lagrange multipliers [2]. The same problem has been tackled in an elegant fashion using the Hamiltonian dynamics and exploiting the properties of the generating functions [5]. With this approach it is possible to devise suitable canonical transformations, satisfying the Hamilton-Jacobi equation, that also verify Hamilton-JacobiBellman equation of the optimal feedback control problem.…”

Section: Introductionmentioning

confidence: 99%

Feedback Optimal Control of Low-thrust Orbit Transfer in Central Gravity Field

Owis¹

2013

IJACSA

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I. ABSTRACTLow-thrust trajectories with variable radial thrust is studied in this paper. The problem is tackled by solving the HamiltonJacobi-Bellman equation via State Dependent Riccati Equation(STDE) technique devised for nonlinear systems. Instead of solving the two-point boundary value problem in which the classical optimal control is stated, this technique allows us to derive closed-loop solutions. The idea of the work consists in factorizing the original nonlinear dynamical system into a quasi-linear state dependent system of ordinary differential equations. The generating function technique is then applied to this new dynamical system, the feedback optimal control is solved. We circumvent in this way the problem of expanding the vector field and truncating higher-order terms because no remainders are lost in the undertaken approach. This technique can be applied to any planet-to-planet transfer; it has been applied here to the Earth-Mars low-thrust transfer. II. INTRODUCTIONHistorically, the optimal low-thrust transfers have been tackled first with indirect and then with direct methods. The former stem from the Pontryagin's maximum principle that uses the calculus of variations [1,2]; the latter aim at solving the problem via a standard nonlinear programming procedure [3]. Even if it can be demonstrated that both approaches lead to the same result [4], the direct and indirect methods have different advantages and drawbacks, but they require in any case the solution of a complex set of equations: the Euler-Lagrange equations (indirect methods) and the Karush-Kuhn-Tucker equations (direct methods). The guidance designed with these methods is obtained in an open-loop context. In other words, the optimal path, even if minimizing the prescribed performance index, is not able to respond to any perturbation that could alter the state of the spacecraft. Furthermore, if the initial conditions are slightly varied (e.g. the launch date changes), the optimal solution needs to be recomputed again. The outcome of the classical problem is in fact a guidance law. expressed as a function of the time,u = u(t), t ∈ [t0, tf ], being t0 and tf the initial and final time, and u the control vector, respectively. This paper deals with the optimal feedback control problem applied to the low thrust interplanetary trajectory design. With this approach the solutions that minimize the performance index are also functions of the generic initial state x0; the outcome is in fact a guidance law written as u = u(x0, t0, t), t t ∈ [t0, tf ]. This represents a closed-loop solution: given the initial conditions (t0,x0) it is possible to extract the optimal control law that solves the optimal control problem. Moreover, if for any reason the state is perturbed and assumes the new value (t0 , x0 ) = (x0 + δx, t0 + δt), we are able to compute the new optimal solution by simply evaluating so avoiding the solution of another optimal control problem. This property holds by virtue of the closed loop characteristics of the control law that can be vie...

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“…An alternative approach was recently proposed by Park and Scheeres [12], which relies on the theory of canonical transformations and their generating functions for Hamiltonian systems. More specifically, canonical transformations are able to solve boundary value problems between Hamiltonian coordinates and momenta for a single flow field.…”

Section: Introductionmentioning

confidence: 99%

“…The main difficulty of this approach is finding the generating functions via the solution of the Hamilton-Jacobi equation. This problem was solved in [12] by expanding the generating function in power series of its arguments.…”

Section: Introductionmentioning

confidence: 99%