A sequential quadratic programming method is proposed for solving nonlinear optimal control problems subject to general path constraints including mixed state-control and state only constraints. The proposed algorithm further develops on the approach proposed in [1] with objective to eliminate the use of a high number of time intervals for arriving at an optimal solution. This is done by introducing an adaptive time discretization to allow formation of a desirable control profile without utilizing a lot of intervals. The use of fewer time intervals reduces the computation time considerably. This algorithm is further used in this thesis to solve a trajectory planning problem for higher elevation Mars landing.
We introduce a new approach for global multiobjective optimization of trajectories in continuous nonlinear dynamical systems that can provide rigorous, arbitrarily tight bounds on the objective values and state paths realized by (Pareto-)optimal trajectories. By controlling all sources of error, our resulting method is the first global trajectory optimization method that can reliably handle nonconvex nonlinear dynamical systems with substantial instabilities, such as the notoriously ill-behaved multi-body gravitational systems governing interplanetary space trajectories. Rigorous finite-dimensional global optimization methods based on space partitioning (branch and bound) do not directly extend to infinite-dimensional problems of trajectory optimization, lacking a way to exhaustively partition an infinite-dimensional space. Thus existing generic methods for deterministic global trajectory optimization rely on direct discretization of the control variables, if not also the state variables. While the resulting errors may prove inconsequential for relatively stable (conservative/dissipative) systems, they severely influence results in unstable systems that arise in many aerospace applications, and whose chaotic sensitivities offer great potential for inexpensive trajectory control. In order to achieve higher accuracy, current programs for interplanetary trajectory optimization typically use problem-specific control parameterizations with local optimization methods (commonly, multiple shooting with sequential quadratic programming), combined with stochastic or expert-guided sampling to seek global optimality. This approach substantially relies on pre-existing intuition about the character of optimal solutions, and provides no guarantees on the global optimality of solutions obtained. The requirements for expert guidance and judgment of uncertainties tend to drive up costs and restrict innovation for the trajectory solutions that play a crucial role in early conceptual design for deep space missions. The thesis takes a new approach to avoid unaccountable discretization errors. Using a specially designed exhaustive partition of the (finite-dimensional) state space into subregions, we construct a finite transition graph between these subregions, such that each trajectory of interest maps to a finite path (transition sequence) in the graph, where each transition trajectory lies in a local state space neighborhood of its corresponding subregions. For any such path, the cost of any corresponding trajectory can be bounded below by the sum of lower bounds on the cost of each stepwise transition. Provided that the transition bounds converge to exact bounds with increasing refinement of the state space partition, an adaptive refinement can produce asymptotically convergent bounds on optimal trajectories. We compute a lower bound on each stepwise transition between state subregions by a novel "interval linearization" technique that simultaneously considers all possible trajectories between two subregions that lie within a local neighborhood. This technique first linearizes the dynamics on the local neighborhood, and replaces the remaining nonlinear terms by interval enclosures of their values over the neighborhood. We then derive a nonlinear system-of-equations solution to a corresponding pointwise generalized linear optimal control problem with time-varying coefficients. Finally, using interval methods, we compute enclosures to the solutions of these equations as the coefficients for the nonlinear terms range over the previously computed enclosures on the neighborhood. This technique effectively confines the difficulties of the infinite-dimensional trajectory space to a local neighborhood, where they can be contained by rigorous approximation. While our approach can in principle be applied to compute a complete optimal control policy over the entire state space for a given target, practical efficiency in most cases demands adaptive restriction of the state space to trajectories between particular start and goal subregions. We introduce a bidirectional "bounded path" algorithm, generalizing efficient graph shortest path algorithms, which permits simultaneously identifying the shortest path(s) in the transition graph-to direct adaptive refinement-and identifying state space subregions whose intersecting path bounds exceed a threshold-to prune subregions that cannot intersect optimal trajectories. By expanding a generally nonconvex dynamical flow to a finite graph admitting this Dijkstra-like search procedure, the transition graph may be seen as "unfolding" the state space to leverage some of the same efficiencies as level-set methods for convex dynamical systems. The structure of our method yields additional practical advantages. It is the first global trajectory optimization method indifferent to the forms of the optimal controls, requiring no prior knowledge and dealing naturally with unbounded controls, singular arcs, and certain types of control constraints. By augmenting the state space to represent additional objective functions, it can provide adaptive sampling enclosures of a bounded Pareto front, directly according to the refinement of the state space and independent of further user input. Finally, its persistent data structures built on state space decomposition can provide reusable "maps" indicating regions of interest, that can jump-start refinement for related trajectory optimization problems with small variations in their defining parameters, as may readily arise in engineering design. We demonstrate the behavior of our method first on two simple trajectory optimization problems (single- and multiobjective) for illustrative purposes, and then on two more complex problems (single- and multiobjective) related to current problems of interest in astrodynamics and robotics (respectively). In each case, our results prove consistent with known or strongly conjectured solutions for these problems obtained from highly problem-specific analysis, and overcome the apparent limitations of a benchmark direct multiple shooting method. We also discuss the potential for our method to address important open problems in spaceflight trajectory optimization, given future work to improve the scalability of our implementation.
Specialists working in the areas of optimization, mathematical programming, or control theory will find this book invaluable for studying interior-point methods for linear and quadratic programming, polynomial-time methods for nonlinear convex programming, and efficient computational methods for control problems and variational inequalities. A background in linear algebra and mathematical programming is necessary to understand the book. The detailed proofs and lack of "numerical examples" might suggest that the book is of limited value to the reader interested in the practical aspects of convex optimization, but nothing could be further from the truth. An entire chapter is devoted to potential reduction methods precisely because of their great efficiency in practice.
This book focuses on Augmented Lagrangian techniques for solving practical constrained optimization problems. The authors: rigorously delineate mathematical convergence theory based on sequential optimality conditions and novel constraint qualifications; orient the book to practitioners by giving priority to results that provide insight on the practical behavior of algorithms and by providing geometrical and algorithmic interpretations of every mathematical result; and fully describe a freely available computational package for constrained optimization and illustrate its usefulness with applications.
Sequential quadratic programming methods as developed by Wilson, Han, and Powell have gained considerable attention in the last few years mainly because of their outstanding numerical performance. Although the theoretical convergence aspects of this method and its various modifications have been investigated in the literature, there still remain some open questions which will be treated in this paper. The convergence theory to be presented, takes into account the additional variable introduced in the quadratic programming subproblem to avoid inconsistency, the one-dimensional minimization procedure, and, in particular, and 'active set' strategy to avoid the recalculation of unnecessary gradients. This paper also contains a detailed mathematical description of a nonlinear programming algorithm which has been implemented by the author.
Timely and well-based decision-making plays a crucial role in improving the performance, success rate, and safety of autonomous aerospace systems, especially for those involving multi-agent networks, hybrid dynamical systems, and operations under uncertain environments. Many space missions benefit from the performance and resilience improvements that employing optimal decision-making strategies significantly increases autonomy level, maneuverability, and multi-task capability. However, in real-world applications, a wide range of aerospace systems involve decisions at different autonomy levels and they are usually coupled with each other, which makes it challenging to find the optimal mixed-variable decisions. Optimization and optimal control are the main tools in decision-making. This dissertation aims to develop a systemic rank-constrained optimization methodology for the decision-making of autonomous systems in space missions where the system states are represented by mixed discrete and continuous variables. Rank constrained optimization is to optimize a convex function subject to a convex set of constraints and a rank constraint on the unknown matrix. It has received increasing attention in the areas of matrix completion, signal processing, and model reduction, just to name a few. However, the connection between rank-constrained optimization, especially for rank one-constrained optimization, and mixed-variable decision-making problems has not been well established. In fact, any discrete variable can be regarded as a continuous variable with a polynomial equality constraint. Meanwhile, many system dynamics can be converted into polynomial constraints through discretization and conversion of expressions. Thus, a mixed-variable decision-making problem could be cast as a polynomial optimization problem, which can be expressed as an equivalent quadratically constrained quadratic programming (QCQP) problem by introducing extra variables and quadratic equalities. Furthermore, a general QCQP can be equivalently transformed into a linear matrix programming problem by introducing a to-be-determined rank-one matrix. This dissertation focuses on establishing computationally efficient programs to solve the resulting rank constrained optimization problems and to evaluate the effectivity, efficiency, and performance of the proposed methodologies in decision-making for autonomous systems stemming from applications closely related to space missions. The products contain (1) development of a unified modeling route to formulate a decision-making problem to a general QCQP or rank-constrained optimization problems (RCOP), (2) proposition of four different sequential algorithms, named alternating minimization algorithm (AMA) combined with penalty function, Alternating Projection Approach (APA), Alternating Rank Minimization Approach (ARMA), and Customized Alternating Direction Method of Multipliers (ADMM), to solve the resulting QCQP or RCOP, (3) applications in space missions including a mission planning for spacecraft rendezvous and docking mission, and a fuel optimal guidance for Mars entry, powered descent, and landing mission, (4) applications in other areas including a sensor localization mission of a multi-agent system, and a UAV path-planning problem. Work in this dissertation removes a computational bottleneck in solving a broad class of challenging mixed-variable optimization problems using a uniform formulation associated with a standard routine. The research products, composed of theoretical analysis, algorithm developments, and practical applications, collectively contribute to the full autonomy of a wide class of autonomous systems in space missions.
A review of the computational method of conjugate gradients for linear and nonlinear operator equations is given with emphasis in applying this technique to state variable constraint control problems. The first and second Frechet derivatives of the performance functional are derived. The search directions generated in the iteration process for the optimal control are locally conjugate with respect to the second Frechet derivative. The convergence is along the expanding sequence of sets, the itersection of the linear spaces spanned by the search directions and the set of admissible controls. The computational technique is applied to two state variable constraint problems, in one of which a penalty function is employed to convert the constraint problem to an unconstrained one in addition to the approach considering the constraints directly. For this same problem the method of steepest descent also is studied, and comparison of the results obtained is made and discussed. (author).
This is a long-overdue volume dedicated to space trajectory optimization. Interest in the subject has grown, as space missions of increasing levels of sophistication, complexity, and scientific return - hardly imaginable in the 1960s - have been designed and flown. Although the basic tools of optimization theory remain an accepted canon, there has been a revolution in the manner in which they are applied and in the development of numerical optimization. This volume purposely includes a variety of both analytical and numerical approaches to trajectory optimization. The choice of authors has been guided by the editor's intention to assemble the most expert and active researchers in the various specialities presented. The authors were given considerable freedom to choose their subjects, and although this may yield a somewhat eclectic volume, it also yields chapters written with palpable enthusiasm and relevance to contemporary problems.
