Matthew W. Harris

Minimum Time Rendezvous of Multiple Spacecraft Using Differential Drag

This paper presents a method to solve the minimum time rendezvous problem of multiple spacecraft using differential drag. The problem is to rendezvous any number of vehicles using only the relative aerodynamic drag between the vehicles. Each vehicle is equipped with drag plates that can be opened or closed, thus modulating the drag force acting on the vehicle. By actuating the drag plates, the aerodynamic drag becomes the control used to accomplish the rendezvous. The optimal control problem is relaxed to a convex problem, and it is proved that a solution of the relaxed problem exists that is also a solution of the original problem. This process is called lossless convexification, and it leads to a solution method based on solving a finite number of linear programming problems. This method offers guaranteed convergence to the global minimum in polynomial time and does not require an initial guess. Two examples are solved with two and five vehicles and the results are compared with existing technologies. B...

Customized Real-Time Interior-Point Methods for Onboard Powered-Descent Guidance

This paper presents a new onboard-implementable, real-time convex optimization-based powered-descent guidance algorithm for planetary pinpoint landing. Earlier work provided the theoretical basis of convexification, the equivalent representation of the fuel-optimal pinpoint landing trajectory optimization problem with nonconvex control constraints as a convex optimization problem. Once the trajectory optimization problem is convexified, interior-point method algorithms can be used to solve the problem to global optimality. Though having this guarantee of convergence motivated earlier convexification results, there were no real-time interior point method algorithms available for the computation of optimal trajectories on flight computers. This paper presents the first such algorithm developed for onboard use and flight-tested on a terrestrial rocket with the NASA Jet Propulsion Laboratory and the NASA Flight Opportunities Program in 2013. First, earlier convexification results are summarized and the result...

Lossless convexification of non-convex optimal control problems for state constrained linear systems

This paper analyzes a class of finite horizon optimal control problems with mixed non-convex and convex control constraints and linear state constraints. A convex relaxation of the problem is proposed, and it is proved that a solution of the relaxed problem is also a solution of the original problem. This process is called lossless convexification, and its generalization for problems with state constraints is the primary contribution of the paper. Doing so enables the use of interior point methods of convex optimization to obtain global optimal solutions of the original non-convex problem. The approach is also demonstrated on example problems.

Optimal Solutions for Quasiplanar Ascent over a Spherical Moon

also assumed that the transfer is quasiplanar and that the thrust pitch and yaw angles are small. The solution of the resulting two-point boundary value problem involves modified thrust integrals and requires three quadratures and one iteration. A result of this solution is that a variable multiplier can be assumed constant, leading to an analytical solution (without quadrature and without iteration) in terms of the well known thrust integrals. These are new solutionsfortheconstantthrust,minimumtimetransferproblemoverasphericalmoon.Bothsolutionsaretestedin a sample and hold guidance scheme. First, the out-of-plane initial conditions are set to zero. The planar results show that both solutions satisfy the final conditions, consume the same mass, and use approximately the same thrust pitch angle history. Second, to test out-of-plane performance, the out-of-plane initial conditions are made nonzero. The results are similar. Because the second solution is analytical, it merits further consideration as an onboard guidance algorithm.

Convex Necessary and Sufficient Conditions for Density Safety Constraints in Markov Chain Synthesis

This paper introduces a new approach for the synthesis of Markov chains with density safety constraints. Specific safety constraints considered are: (i) Density upper bound constraints; (ii) Bound on the rate of change of density. The proposed approach is based on a new mathematical result that formulates these constraints as linear, and hence convex, inequality constraints. The main contribution of the paper is that the new convex constraints are proved to be equivalent, necessary and sufficient, to the original constraints. This result enables the formulation of the Markov chain synthesis problem as a Linear Matrix Inequality (LMI) optimization problem with additional constraints on the steady state probability distribution, ergodicity, and feasible state transitions. The LMI formulation presents a necessary and sufficient design condition for reversible Markov chains, that is, it is not conservative. When the reversibility is not required, the LMI condition is only sufficient due to the ergodicity constraint. Since LMI problems can be solved to global optimality in polynomial time by using Interior Point Method algorithms, the proposed LMI-based design approach is numerically tractable.

Convex optimization formulation of density upper bound constraints in Markov chain synthesis

This paper introduces a new approach for the synthesis of Markov chains with density upper bound constraints. The proposed approach is based on a new mathematical result that formulates the density upper bound constraints, known also as safety constraints, as linear, and hence convex, inequality constraints. It is proved that the new convex constraints are equivalent, necessary and sufficient, to the density upper bound constraints, which is the main contribution. Next, this result enabled the formulation of the Markov chain synthesis problem as an Linear Matrix Inequality (LMI) optimization problem with additional constraints on the steady state probability distribution, ergodicity, and state transitions. The LMI formulation presents an equivalent design formulation in the case of reversible Markov chains, that is, it is not conservative. When reversibility assumption is relaxed, the LMI condition is only sufficient due to the ergodicity constraint, i.e., it is conservative. Since LMI problems can be solved to global optimality in polynomial time by using interior point method (IPM) algorithms of convex optimization, the proposed LMI-based design approach is numerically tractable.

Lossless convexification for a class of optimal control problems with quadratic state constraints

This paper presents lossless convexification for a class of finite horizon optimal control problems with non-convex control constraints and quadratic state constraints. Some special cases where the state at most touches the state constraint have been addressed previously in the literature. In this paper, the convexification results are generalized to allow optimal trajectories with boundary arcs. There are a number of practical examples that belong to the class of problems studied here. The optimal control problems considered have convex cost, specialized linear dynamics, quadratic state constraints, and non-convex control constraints. Hence, the control constraints are the single source of non-convexity. The control set is relaxed to a convex set by introducing a scalar slack variable. It is shown that optimal solutions of the relaxed problem are also optimal solutions of the original problem, hence the term lossless convexification. The main contribution of this paper is to extend the lossless convexification to the problem with quadratic state constraints. The proof uses a maximum principle with state variable inequality constraints and requires an assumption on the bounds of an external disturbance. A numerical example is presented to illustrate the approach. Because the numerical problem is a second order cone problem, convergence to the global minimum is guaranteed in a deterministic, finite number of steps.

Lmi-based control of stick-slip oscillations in drilling

A control system for controlling stick-slip oscillations in a drilling system is disclosed. The control system includes a processor and a memory communicatively coupled to the processor with computer program instructions stored therein. The instructions are configured to, when executed by the processor, cause the processor to receive a communication including a value of a drilling system parameter, determine based on the value of the drilling system parameter, a value for a torque to be applied to a drill string to reduce the effect of stick-slip oscillations, and transmit to a drive system the value for the torque to be applied by the drive system. The value for the torque to be applied by the drive system is determined by solving a set of linear matrix inequalities including a disturbance minimization linear matrix inequality, a first magnitude constraint linear matrix inequality, and a second magnitude constraint linear matrix inequality.

Rendezvous Using Differential Drag and Feedback Control

This paper presents a control scheme to achieve minimum time rendezvous of two spacecraft using only drag panels and a reaction wheel. Both vehicles are equipped with drag plates that can be deployed or not thus modulating the drag forces. The minimum time problem is solved by relaxing the control set to a convex set and proving that a solution of the relaxed problem solves the original problem. This process is called lossless convexification, and it leads to a solution method based on solving a finite number of linear programming problems. This method offers guaranteed convergence to the global minimum in polynomial time and does not require an initial guess. Because of nonlinearities, disturbances, and unmodeled dynamics, feedback is required to track the optimal trajectory. The feedback scheme uses the spacecraft attitude thus respecting the original constraints and avoiding the need for propellant and thrusters. Consequently, rendezvous using differential drag is achieved with high precision and without thrusters.

Lossless convexification for a class of optimal control problems with linear state constraints

This paper presents lossless convexification for a class of finite horizon optimal control problems with convex cost, linear dynamics, linear state constraints, and non-convex control constraints. There are a number of examples that belong to this class of problems since many practical problems are constrained to evolve in a bounded state space. The control set is relaxed to a convex set by introducing a scalar slack variable, and it is proved that optimal solutions of the relaxed problem are optimal solutions of the original problem, hence the term lossless convexification. Extending the proof to problems with state boundary arcs is the main theoretical contribution of this paper. The practical implication is that the non-convex optimization problem can be solved as a convex problem with guaranteed convergence properties. A numerical example is presented to illustrate the approach.