120 research outputs found
Successive Convexification of Non-Convex Optimal Control Problems and Its Convergence Properties
This paper presents an algorithm to solve non-convex optimal control
problems, where non-convexity can arise from nonlinear dynamics, and non-convex
state and control constraints. This paper assumes that the state and control
constraints are already convex or convexified, the proposed algorithm
convexifies the nonlinear dynamics, via a linearization, in a successive
manner. Thus at each succession, a convex optimal control subproblem is solved.
Since the dynamics are linearized and other constraints are convex, after a
discretization, the subproblem can be expressed as a finite dimensional convex
programming subproblem. Since convex optimization problems can be solved very
efficiently, especially with custom solvers, this subproblem can be solved in
time-critical applications, such as real-time path planning for autonomous
vehicles. Several safe-guarding techniques are incorporated into the algorithm,
namely virtual control and trust regions, which add another layer of
algorithmic robustness. A convergence analysis is presented in continuous- time
setting. By doing so, our convergence results will be independent from any
numerical schemes used for discretization. Numerical simulations are performed
for an illustrative trajectory optimization example.Comment: Updates: corrected wordings for LICQ. This is the full version. A
brief version of this paper is published in 2016 IEEE 55th Conference on
Decision and Control (CDC). http://ieeexplore.ieee.org/document/7798816
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Verification of successive convexification algorithm
In this report, I describe a technique which allows a non-convex optimal control problem to be expressed and solved in a convex manner. I then verify the resulting solution to ensure its physical feasibility and its optimality. The original, non-convex problem is the fuel-optimal powered landing problem with aerodynamic drag. The non-convexities present in this problem include mass depletion dynamics, aerodynamic drag, and free final time. Through the use of lossless convexification and successive convexification, this problem can be formulated as a series of iteratively solved convex problems that requires only a guess of a final time of flight. The solution’s physical feasibility is verified through a nonlinear simulation built in Simulink, while its optimality is verified through the general nonlinear optimal control software GPOPS-II.Aerospace Engineerin
A recursively feasible and convergent Sequential Convex Programming procedure to solve non-convex problems with linear equality constraints
A computationally efficient method to solve non-convex programming problems
with linear equality constraints is presented. The proposed method is based on
a recursively feasible and descending sequential convex programming procedure
proven to converge to a locally optimal solution. Assuming that the first
convex problem in the sequence is feasible, these properties are obtained by
convexifying the non-convex cost and inequality constraints with inner-convex
approximations. Additionally, a computationally efficient method is introduced
to obtain inner-convex approximations based on Taylor series expansions. These
Taylor-based inner-convex approximations provide the overall algorithm with a
quadratic rate of convergence. The proposed method is capable of solving
problems of practical interest in real-time. This is illustrated with a
numerical simulation of an aerial vehicle trajectory optimization problem on
commercial-of-the-shelf embedded computers
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Lossless convexification of optimal control problems
textThis dissertation begins with an introduction to finite-dimensional optimization and optimal control theory. It then proves lossless convexification for three problems: 1) a minimum time rendezvous using differential drag, 2) a maximum divert and landing, and 3) a general optimal control problem with linear state constraints and mixed convex and non-convex control constraints. Each is a unique contribution to the theory of lossless convexification. The first proves lossless convexification in the presence of singular controls and specifies a procedure for converting singular controls to the bang-bang type. The second is the first example of lossless convexification with state constraints. The third is the most general result to date. It says that lossless convexification holds when the state space is a strongly controllable subspace. This extends the controllability concepts used previously, and it recovers earlier results as a special case. Lastly, a few of the remaining research challenges are discussed.Aerospace Engineerin
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