An SQP method for the optimal control of large-scale dynamical systems

Gill, Philip E.; Jay, Laurent O.; Leonard, Michael W.; Petzold, Linda R.; Sharma, Vivek

doi:10.1016/s0377-0427(00)00310-1

Cited by 52 publications

(31 citation statements)

References 5 publications

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“…Some of these are general purpose nlp solvers such as snopt (Gill et al, 2000(Gill et al, , 2002, which is based on an sqp framework, or ipopt (Wächter and Biegler, 2006) which is based on an ip framework. Others are developed primarily for optimal control problems, such as acado (Houska et al, 2011) which provides a framework for using a variety of algorithms for direct optimal control.…”

Section: Background and Motivationmentioning

confidence: 99%

Structure-Exploiting Numerical Algorithms for Optimal Control

Nielsen¹

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Cover illustration: An illustration of the structure of the Karush-Kuhn-Tucker matrix for the unconstrained finite-time optimal control problems considered in this thesis. Thank you Jonas Linder for helping me with the design.Linköping studies in science and technology. Dissertations.No. 1848 Tillägnas hela min underbara familj Structure-Exploiting Numerical Algorithms for Optimal Control AbstractNumerical algorithms for efficiently solving optimal control problems are important for commonly used advanced control strategies, such as model predictive control (mpc), but can also be useful for advanced estimation techniques, such as moving horizon estimation (mhe). In mpc, the control input is computed by solving a constrained finite-time optimal control (cftoc) problem on-line, and in mhe the estimated states are obtained by solving an optimization problem that often can be formulated as a cftoc problem. Common types of optimization methods for solving cftoc problems are interior-point (ip) methods, sequential quadratic programming (sqp) methods and active-set (as) methods. In these types of methods, the main computational effort is often the computation of the second-order search directions. This boils down to solving a sequence of systems of equations that correspond to unconstrained finite-time optimal control (uftoc) problems. Hence, high-performing second-order methods for cftoc problems rely on efficient numerical algorithms for solving uftoc problems. Developing such algorithms is one of the main focuses in this thesis. When the solution to a cftoc problem is computed using an as type method, the aforementioned system of equations is only changed by a low-rank modification between two as iterations. In this thesis, it is shown how to exploit these structured modifications while still exploiting structure in the uftoc problem using the Riccati recursion. Furthermore, direct (non-iterative) parallel algorithms for computing the search directions in ip, sqp and as methods are proposed in the thesis. These algorithms exploit, and retain, the sparse structure of the uftoc problem such that no dense system of equations needs to be solved serially as in many other algorithms. The proposed algorithms can be applied recursively to obtain logarithmic computational complexity growth in the prediction horizon length. For the case with linear mpc problems, an alternative approach to solving the cftoc problem on-line is to use multiparametric quadratic programming (mp-qp), where the corresponding cftoc problem can be solved explicitly off-line. This is referred to as explicit mpc. One of the main limitations with mp-qp is the amount of memory that is required to store the parametric solution. In this thesis, an algorithm for decreasing the required amount of memory is proposed. The aim is to make mp-qp and explicit mpc more useful in practical applications, such as embedded systems with limited memory resources. The proposed algorithm exploits the structure from the qp problem in the parametric solution in order to reduc...

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Section: Background and Motivationmentioning

confidence: 99%

Structure-Exploiting Numerical Algorithms for Optimal Control

Nielsen¹

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“…The discretization of the Lagrange-d'Alembert principle [30] leads to structure-preserving time-stepping equations, which serve as equality constraints for the resulting finite dimensional nonlinear optimization problem [31]. This problem can be solved by standard nonlinear optimization techniques such as sequential quadratic programming (see, e.g., Gill et al [32,33], Powell [34], and Han [35]). …”

Section: B Local Optimal Controlmentioning

confidence: 99%

Trajectory Design Combining Invariant Manifolds with Discrete Mechanics and Optimal Control

Moore

Ober‐Blöbaum

Marsden

2012

Journal of Guidance, Control, and Dynamics

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A mission design technique that combines invariant manifold techniques, discrete mechanics, and optimal control produces locally optimal low-energy trajectories. Previously, invariant manifolds of the planar circular restricted three-body problem have been used to design trajectories with relatively small midcourse change in velocity V. A different method of using invariant manifolds is explored to design trajectories directly in the four-body problem. Then, using the local optimal control method DMOC (Discrete Mechanics and Optimal Control), it is possible to reduce the midcourse V to zero. The influence of different boundary conditions on the optimal trajectory is also demonstrated. These methods are tested on a trajectory that begins in Earth orbit and ends in ballistic capture at the moon. Impulsive DMOC trajectories require up to 19% less V than trajectories using a Hohmann transfer. When applied to low-thrust trajectories, DMOC produces an improvement of up to 59% in the mass fraction and 22% in travel time when compared with results from shooting methods. = mass consumption, kg P = spacecraft Q = configuration space q = configuration variable q k = discrete configuration variable r a , r p = radius of elliptical orbit at apoapsis and perigee, km r M H , a H = perigee radius, semimajor axis of Hohmann transfer ellipse, km T = low-thrust magnitude, N T k = discrete thrust magnitude applied at node k, N T max = maximum allowable thrust, N TQ = state space T t = normalized time denoting transition from low thrust to no thrust T x;k = discrete thrust at node k in x direction, N T y;k = discrete thrust at node k in y direction, N t k = discrete time variable u = continuous control parameter u k = discrete control variable u x = control force in x direction u y = control force in y direction v r = normalized radial velocity v x;k = normalized discrete velocity at node k in x direction v y;k = normalized discrete velocity at node k in y direction x i , y i = x, y position of body i for i E; S; M V = scales nondimensional velocity to meters per second V = magnitude of change in velocity, m=s or km=s V C = V necessary to enter elliptical capture orbit at moon, km=s V H = V necessary to leave Earth orbit using a Hohmann transfer, km=s V i = V for circular orbit insertion at body i, i E; M, m=s or km=s V traj = V applied throughout trajectory, m=s t = discrete time grid t = refined discrete time grid = variational operator q k = discrete configuration variation i = angle of body i with respect to x axis of coordinate system for i M; S, rad = mass parameter of planar circular restricted three-body problem k = discrete thrust optimization variable, N = angle of semimajor axis of elliptical orbit at the moon with respect to the Earth-moon x axis, deg = phase of Earth-moon frame with respect to sun-Earth frame, deg = effective potential in rotating frame ! i = normalized rotation rate of body i for i M; S

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“…A considerable amount of work has been conducted for calculating sensitivities for transient simulations for ordinary differentiable equations (ODEs) and differentiable algebraic systems (DAEs) [155] [145] [146] [50] [51] [91]. [196].…”

Section: Transient Simulationsmentioning

confidence: 99%