Dharmashankar Subramanian

Dynamic optimization strategies for three-dimensional conflict resolution of multiple aircraft

Free flight is an emerging paradigm in air traffic management. Conflict detection and resolution is the heart of any free-flight concept. The problem of optimal cooperative three-dimensional conflict resolution involving multiple aircraft is addressed by the rigorous numerical trajectory optimization methods. The conflict problem is posed as an optimal control problem of finding trajectories that minimize a certain objective function while the safe separation between each aircraft pair is maintained. The initial and final positions of the aircraft are known and aircraft models with detailed nonlinear point-mass dynamics are considered. The protection zone around the aircraft is modeled to be cylindrical in shape. A novel formulation of the cylindrical protection zone is proposed by the use of continuous variables. The optimal control problem is converted to a finite dimensional nonlinear program (NLP) by the use of collocation on finite elements. The NLP is solved by the use of an interior point algorithm that incorporates a novel line search method. A reliable initialization strategy that yields a feasible solution on simple models is also proposed and adapted to detailed models. Several resolution scenarios are illustrated. The practical issue of flyability of the generated trajectories is addressed by the ability of our mathematical programming framework to accommodate detailed dynamic models.

MILP and NLP Techniques for centralized trajectory planning of multiple unmanned air vehicles

We consider the problem of optimal cooperative three-dimensional conflict resolution involving multiple unmanned air vehicles (UAVs) using numerical trajectory optimization methods. The conflict problem is posed as an optimal control problem of finding trajectories that minimize a certain objective function while maintaining the safe separation between each UAV pair. We assume the origin and destination of the UAV are known and consider UAV models with simplified linear kinematics. The main objective of this report is to present two different approaches to the solution of the problem. In the first approach, the optimal control is converted to a finite dimensional nonlinear program (NLP) by using collocation on finite elements and by reformulating the disjunctions involved in modeling the protected zones by using continuous variables. In the second approach the optimal control is converted to a finite dimensional mixed integer linear program (MILP) using Euler discretization and reformulating the disjunctions involved with the protected zones by using binary variables and Big-M techniques. Based on results of extensive random simulations, we compare time complexity and optimality of the solutions obtained with the MILP approach and the NLP approach. NLPs are essential to enforce flyability constraints on more detailed UAV models. Moreover, any nonlinear extensions to the problem cannot be dealt with by MILP solvers. The main objective of this paper is to open the route to the use of MILP solutions (based on simple linear UAV models) in order to initialize NLP solvers which allow the use of dynamic UAV models at any desired level of detail

3D Conflict Resolution of Multiple Aircraft via Dynamic Optimization

Free flight is an emerging paradigm in Air Trac Management (ATM). In this paper, we focus on the problem of cooperative 3D conflict resolution among multiple aircraft by posing it as an optimal control problem of finding trajectories that minimize a certain objective function while maintaining safe separation between each aircraft pair. We assume the origin and destination of the aircraft are known and consider aircraft models with detailed nonlinear point-mass dynamics. The protection zone around the aircraft is modeled to be cylindrical in shape. We also extend the modeling framework to accommodate no-fly zones of finite height or otherwise. A novel formulation of the cylindrical protection zone using continuous variables. We address the solution of this problem using rigorous numerical trajectory optimization methods. The optimal control problem is converted to a finite dimensional NonLinear Program (NLP) using collocation on finite elements. We solve the NLP using an Interior Point algorithm that incorporates a novel line search method. We also propose a reliable initialization strategy that yields a feasible solution on simple models and is also adapted to detailed models. Resolution scenarios including cases with no-fly zones are illustrated.

A Nonlinear Hybrid Life Support System: Dynamic Modeling, Control Design, and Safety Verification

We present control design for a variable configuration CO2 removal (VCCR) system, which exhibits a hybrid dynamical character due to the various modes in which one needs to operate the system. The VCCR is part of an overall NASA Air Recovery System of an intended human life support system for space exploration. The objective of the control system is to maintain CO2 and O concentrations in the crew cabin within safe bounds. We present a novel adaptation of the model predictive control technique to a nonlinear hybrid dynamic system. We exploit the problem structure and map the hybrid optimization problem into a continuous nonlinear program (NLP) with the aid of an appropriate representation of time and set definitions. We present a systematic approach for designing the objective function for the nonlinear model predictive control (NMPC) regulation problem that achieves a long-term, cyclic steady state. We also present a simple switching feedback controller and compare the performance of the two controllers during off-nominal and failure conditions to highlight the benefits of a systematically designed NMP controller. We then perform safety verification of both control designs-the model predictive control with techniques from statistical learning theory and the switching feedback controller with Barrier certificates computed using sum of squares programming. The two approaches yield consistent results.

On optimizing the selection of business transformation projects

To compete and thrive in a changing business environment, a business can adapt by initiating and successfully carrying out business transformation projects. In this paper we propose a methodology for the optimal selection of such transformational projects. We propose a two-stage methodology based on (1) correlation analytics for identifying key drivers of business performance and (2) advanced portfolio-optimization techniques for selecting optimal business-transformation portfolios in the face of resource constraints, budget constraints, and a rich variety of business rules. We illustrate our methodology through a case study from the electronics industry.

Trajectory control of multiple aircraft : An NMPC approach

A multi-stage nonlinear model predictive controller is derived for the real-time coordination of multiple aircraft. In order to couple the versatility of hybrid systems theory with the power of NMPC, a finite state machine is coupled to a real time optimal control formulation. This methodology aims to integrate real-time optimal control with higher level logic rules, in order to assist mission design for flight operations like collision avoidance, conflict resolution, and reacting to changes in the environment. Specifically, the controller is able to consider new information as it becomes available. Stability properties for nonlinear model predictive control are described briefly along the lines of a dual-mode controller. Finally, a small case study is presented that considers the coordination of two aircraft, where the aircraft are able to avoid obstacles and each other, reach their targets and minimize a cost function over time.

An XML-based language for the Research & Development pipeline management problem

Abstract Process management frameworks, such as Sim-Opt [AIChE J. 10 (2001) 2226], which addresses the Research & Development (R&D) pipeline management problem with mathematical programming and discrete-event simulation give rise to formulations that are extremely data-intensive and have complex hierarchical data-requirements. This necessitates a data model that can be used to model any given problem instance in the form of a structured input language. Further, the language requires a parser that reads and interprets any input instance in order to capture the input data in memory and allow the formulation and solution of the corresponding optimization and simulation models. In the past, structured documentation languages have been designed for this purpose. However, such customized languages often lead to a strong coupling between the language definition and the parser implementation. Any redefinition or extension of the language to accommodate changes in the problem scope and/or optimization/simulation formulations would imply a customized extension of the parser, thus leading to software engineering difficulties. One solution to the above difficulties is provided by the Extensible Markup Language (XML) technology, a recent advance in software technology that enables extensibility and data abstraction and provides efficient data structuring parsers and object orientation. XML imposes the requirement of specifying data in an inherently hierarchical structure and provides generic parsers that do not require any re-design upon language extensions or redefinitions. This paper describes an XML-based language that has been developed for the R&D pipeline management problem, with the keywords, structural syntax, and data content models for representing all aspects of the problem. It also discusses the practical issue of effectively accessing data that gets stored in the document object model (DOM) upon parsing, by designing a set of problem definition classes (PDC), which organize the data stored in the generic DOM structure into an effective set of data structures that facilitate formulation generation. Efforts to integrate the language, the DOM parser, and the PDC in a discrete event simulation application for the R&D pipeline problem are also discussed.

Prescriptive Analytics for Allocating Sales Teams to Opportunities

For companies with large sales forces whose sellers approach business clients in teams, the problem of allocating sales teams to sales opportunities is a critical management task for maximizing the revenue and profit of the company. We approach this problem via predictive and prescriptive analytics, where the former involves data mining to learn the relationship between sales team composition and the revenue earned for different types of clients and opportunities, and the latter involves optimization to find the allocation of sales resources to opportunities that maximizes expected revenue subject to business constraints. In looking at the overall sales force problem, we focus on the interplay between the data mining and optimization components, making sure to formulate the two aspects in a jointly tractable and effective manner. We perform a sensitivity analysis of the optimization component to provide further insight into the interaction between prediction and prescription. Finally, we provide an empirical study using real-world data from a large technology companys sales force. Our results demonstrate that by using these analytics, we can increase revenue by 15%.

Iterative estimation maximization for stochastic linear programs with conditional value-at-risk constraints

We present a new algorithm, iterative estimation maximization (IEM), for stochastic linear programs with conditional value-at-risk constraints. IEM iteratively constructs a sequence of linear optimization problems, and solves them sequentially to find the optimal solution. The size of the problem that IEM solves in each iteration is unaffected by the size of random sample points, which makes it extremely efficient for real-world, large-scale problems. We prove the convergence of IEM, and give a lower bound on the number of sample points required to probabilistically bound the solution error. We also present computational performance on large problem instances and a financial portfolio optimization example using an S&P 500 data set.

An importance sampling method for portfolio cvar estimation with Gaussian copula models

We developed an importance sampling method to estimate Conditional Value-at-Risk for portfolios in which inter-dependent asset losses are modeled via a Gaussian copula model. Our method constructs an importance sampling distribution by shifting the latent variables of the Gaussian copula and thus can handle arbitrary marginal asset distributions. It admits an intuitive geometric explanation and is easy to implement. We also present numerical experiments that confirm its superior performance compared to the naive approach.