Global solution of multi-objective optimal control problems with multi agent collaborative search and direct finite elements transcription

This paper addresses the solution of optimal control problems with multiple and possibly conflicting objective functions. The solution strategy is based on the integration of Direct Finite Elements in Time (DFET) transcription into the Multi Agent Collaborative Search (MACS) framework. Multi Agent Collaborative Search is a memetic algorithm in which a population of agents performs a set of individual and social actions looking for the Pareto front. Direct Finite Elements in Time transcribe an optimal control problem into a constrained Non-linear Programming Problem (NLP) by collocating states and controls on spectral bases. MACS operates directly on the NLP problem and generates nearly-feasible trial solutions which are then submitted to a NLP solver. If the NLP solver converges to a feasible solution, an updated solution for the control parameters is returned to MACS, along with the corresponding value of the objective functions. Both the updated guess and the objective function values will be used by MACS to generate new trial solutions and converge, as uniformly as possible, to the Pareto front. To demonstrate the applicability of this strategy, the paper presents the solution of the multi-objective extensions of two well-known space related optimal control problems: the Goddard Rocket problem, and the maximum energy orbit rise problem

Multi-objective optimal control of ascent trajectories for launch vehicles

This paper presents a novel approach to the solution of multi-objective optimal control problems. The proposed solution strategy is based on the integration of the Direct Finite Elements Transcription method, to transcribe dynamics and objectives, with a memetic strategy called Multi Agent Collaborative Search (MACS). The original multi-objective optimal control problem is reformulated as a bi-level nonlinear programming problem. In the outer level, handled by MACS, trial control vectors are generated and passed to the inner level, which enforces the solution feasibility. Solutions are then returned to the outer level to evaluate the feasibility of the corresponding objective functions, adding a penalty value in the case of infeasibility. An optional single level refinement is added to improve the ability of the scheme to converge to the Pareto front. The capabilities of the proposed approach will be demonstrated on the multi-objective optimisation of ascent trajectories of launch vehicles

Solving multi-objective dynamic travelling salesman problems by relaxation

This paper describes a method to solve Multi-objective Dynamic Travelling Salesman Problems. The problems are formulated as multi-objective hybrid optimal control problems, where the choice of the target destination for each phase is an integer variable. The resulting problem has thus a combinatorial nature in addition to being a multi-objective optimal control problem. The overall solution approach is based on a combination of the Multi Agent Collaborative Search, a population based memetic multi-objective optimisation algorithm, and the Direct Finite Elements Transcription, a direct method for optimal control problems. A relaxation approach is employed to transform the mixed integer problem into a purely continuous problem, and a set of smooth constraints is added in order to ensure that the relaxed variables of the final solution assume an integer value. A special set of smooth constraints is introduced in order to treat the mutually exclusive choices of the targets for each phase. The method is tested on two problems: the first is a motorised Travelling salesman problem described in the literature, the second is a space application where a satellite has to de-orbit multiple debris. For the first problem, the approach is generating better solutions than those reported in the literature

Direct transcription of optimal control problems with finite elements on Bernstein basis

The paper introduces the use of Bernstein polynomials as a basis for the direct transcription of optimal control problems with Finite Elements in Time. Two key properties of this new transcription approach are demonstrated in this paper: Bernstein bases return smooth control profiles with no oscillations near discontinuities or abrupt changes of the control law, and for convex feasible sets, the polynomial representation of both states and controls remains within the feasible set for all times. The latter property is demonstrated theoretically and experimentally. A simple but representative example is used to illustrate these properties and compare the new scheme against a more common way to transcribe optimal control problems with Finite Elements in Time

A relaxation approach for hybrid multi-objective optimal control : application to multiple debris removal missions

This paper presents a novel approach to the solution of multi-phase multi-objective hybrid optimal control problems. The proposed solution strategy extends previous work which integrated the Direct Finite Elements Transcription (DFET) method to transcribe dynamics and objectives, with a memetic strategy called Multi Agent Collaborative Search (MACS). The problem is reformulated as two non-linear programming problems: a bi-level and a single level one. In the bi-level formulation the outer level, handled by MACS, generates trial control vectors that are then passed to the inner level, which enforces the feasibility of the solution. Feasible control vectors are then returned to the outer level to evaluate the corresponding objective functions. The single level formulation is also run periodically to ensure local convergence to the Pareto front. In order to treat mixed integer problems, the heuristics of MACS have been modified in order to preserve the discrete nature of integer variables. For the single level refinement and the inner level of the bi-level approach, discrete variables are relaxed and treated as continuous. Once a solution to the relaxed problem has been found, a smooth constraint is added to systematically force the relaxed variables to assume integer values. The approach is first tested on a simple motorised travelling salesmen problem and then applied to the mission design of a multiple debris removal mission

Multi-objective optimal control of re-entry and abort scenarios

This paper presents a novel approach to the solution of multi-phase multi-objective optimal control problems. The proposed solution strategy is based on the integration of the Direct Finite Elements Transcription (DFET) method, to transcribe dynamics and objectives, with a memetic strategy called Multi Agent Collaborative Search (MACS). The original multi-objective optimal control problem is reformulated as two non-linear programming problems: a bi-level and a single level one. In the bi-level problem the outer level, handled byMACS, generates trial control vectors that are then passed to the inner level, which enforces the feasibility of the solution. Feasible control vectors are then returned to the outer level to evaluate the corresponding objective functions. A single level refinement is then run to improve local convergence to the Pareto front. The paper introduces also a novel parameterisation of the controls, using Bernstein polynomials, in the context of the DFET transcription method. The approach is first tested on a known atmospheric re-entry problem and then applied to the analysis of ascent and abort trajectories for a space plane

Multi-objective optimisation under uncertainty with unscented temporal finite elements

This paper presents a novel method for multi-objective optimisation under uncertainty developed to study a range of mission trade-offs, and the impact of uncertainties on the evaluation of launch system mission designs. A memetic multi-objective optimisation algorithm, named MODHOC, which combines the Direct Finite Elements in Time transcription method with Multi Agent Collaborative Search, is extended to account for model uncertainties. An Unscented Transformation is used to capture the first two statistical moments of the quantities of interest. A quantification model of the uncertainty was developed for the atmospheric model parameters. An optimisation under uncertainty was run for the design of descent trajectories for a spaceplane-based two-stage launch system

Direct solution of multi-objective optimal control problems applied to spaceplane mission design

This paper presents a novel approach to the solution of multi-phase multi-objective optimal control problems. The proposed solution strategy is based on the transcription of the optimal control problem with Finite Elements in Time and the solution of the resulting Multi-Objective Non-Linear Programming (MONLP) problem with a memetic strategy that extends the Multi Agent Collaborative Search algorithm. The MONLP problem is reformulated as two non-linear programming problems: a bi-level and a single level problem. The bi-level formulation is used to globally explore the search space and generate a well spread set of non-dominated decision vectors while the single level formulation is used to locally converge to Pareto efficient solutions. Within the bi-level formulation, the outer level selects trial decision vectors that satisfy an improvement condition based on Chebyshev weighted norm, while the inner level restores the feasibility of the trial vectors generated by the outer level. The single level refinement implements a Pascoletti-Serafini scalarisation of the MONLP problem to optimise the objectives while satisfying the constraints. The approach is applied to the solution of three test cases of increasing complexity: an atmospheric re-entry problem, an ascent and abort trajectory scenario and a three-objective system and trajectory optimisation problem for spaceplanes

GTOC 9 : Results from University of Strathclyde (team Strath++)

The design and planning of space trajectories is a challenging problem in mission analysis. In the last years global optimisation techniques have proven to be a valuable tool for automating the design process that otherwise would mostly rely on engineers’ expertise. The paper presents the optimisation approach and problem formulation proposed by the team Strathclyde++ to address the problem of the 9th edition of the Global Trajectory Optimisation Competition. While the solution approach is introduced for the design of a set of multiple debris removal missions, the solution idea can be generalised to a wider set of trajectory design problems that have a similar structure

Robust multi-objective optimisation of a descent guidance strategy for a TSTO spaceplane

This paper presents a novel method for multi-objective optimisation under uncertainty developed to study a range of mission trade-offs, and the impact of uncertainties on the evaluation of launch system mission designs. A memetic multi-objective optimisation algorithm, MODHOC, which combines the Direct Finite Elements transcription method with Multi Agent Collaborative Search, is extended to account for model uncertainties. An Unscented Transformation is used to capture the first two statistical moments of the quantities of interest. A quantification model of the uncertainty was developed for the atmospheric model parameters. An optimisation under uncertainty was run for the design of descent trajectories for the Orbital-500R, a commercial semi-reusable, two-stage launch system under development by Orbital Access Lt