Vipin Gopal

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.

Quadratic programming algorithms for large-scale model predictive control

Abstract Quadratic programming (QP) methods are an important element in the application of model predictive control (MPC). As larger and more challenging MPC applications are considered, more attention needs to be focused on the construction and tailoring of efficient QP algorithms. In this study, we tailor and apply a new QP method, called QPSchur, to large MPC applications, such as cross directional control problems in paper machines. Written in C++, QPSchur is an object oriented implementation of a novel dual space, Schur complement algorithm. We compare this approach to three widely applied QP algorithms and show that QPSchur is significantly more efficient (up to two orders of magnitude) than the other algorithms. In addition, detailed simulations are considered that demonstrate the importance of the flexible, object oriented construction of QPSchur, along with additional features for constraint handling, warm starts and partial solution.

Interior point SQP strategies for large-scale, structured process optimization problems

Successive quadratic programming (SQP) has been the method of choice for the solution of many nonlinear programming problems in process engineering. However, for the solution of large problems with SQP based codes, the combinatorial complexity associated with active set quadratic programming (QP) methods can be a bottleneck in exploiting the problem structure. In this paper, we examine the merits of incorporating an interior point QP method within an SQP framework. This provides a novel interpretation of popularly used predictor-corrector interior point (IP) methods. The resulting large-scale SQP algorithm, with an interior point QP, also allows us to demonstrate significant computational savings on problems drawn from optimal control and nonlinear model predictive control.

Large scale inequality constrained optimization and control

Since Karmarkars work (1984), interior point methods in linear programming have triggered a tremendous amount of activity. The applicability of interior-point methods for the efficient solution of nonlinear programming problems has also been of interest, and has shown huge potential benefits. This has tremendous impact in process control, especially since optimal control and model predictive control problems, hitherto considered unsolvable, could be solved in a realistic time. In this article, we outline some recent developments in interior point methods for the solution of linear and nonlinear programming problems followed by a summary of the recent work for applying these concepts in control. We conclude with a review of current status and a discussion of future directions.

Addressing Multiobjective Control: Safety and Performance through Constrained Optimization

We address systems which have multiple objectives: broadly speaking, these objectives can be thought of as safety and performance goals. Guaranteeing safety is our first priority, satisfying performance criteria our second. In this paper, we compute the systems safe operating space and represent it in closed form, and then, within this space, we compute solutions which optimize a given performance criterion. We describe the methodology and illustrate it with two examples of systems in which safety is paramount: a two-aircraft collision avoidance scenario and the flight management system of a VSTOL aircraft. In these examples, performance criteria are met using mixed-integer nonlinear programming (MINLP) and nonlinear programming (NLP), respectively. Optimized trajectories for both systems demonstrate the effectiveness of this methodology on systems whose safety is critical.

Nonsmooth dynamic simulation with linear programming based methods

Abstract Process simulation has emerged as a valuable tool for process design, analysis and operation. In this work, we extend the capabilities of iterated linear programming (LP) for dealing with problems encountered in dynamic nonsmooth process simulation. A previously developed LP method is refined with the addition of a new descent strategy which combines line search with a trust region approach. This adds more stability and efficiency to the method. The LP method has the advantage of naturally dealing with profile bounds as well. This is demonstrated to avoid the computational difficulties which arise from the iterates going into physically unrealistic regions. A new method for the treatment of discontinuities occurring in dynamic simulation problems is also presented in this paper. The method ensures that any event which has occurred within the time interval in consideration is detected and if more than one event occurs, the detected one is indeed the earliest one. A specific class of implicitly discontinous process simulation problems, phase equilibrium calculations, is also examined. A new formulation is introduced to solve multiphase problems.

A Successive Linear Programming Approach for Initialization and Reinitialization after Discontinuities of Differential-Algebraic Equations

Determination of consistent initial conditions is an important aspect of the solution of differential-algebraic equations (DAEs). Specification of inconsistent initial conditions, even if they are only slightly inconsistent, often leads to a failure in the initialization problem. In this paper, we present a successive linear programming (SLP) approach for the solution of the DAE derivative array equations for the initialization problem. The SLP formulation handles roundoff errors and inconsistent user specifications, among other things, and allows for reliable convergence strategies that incorporate variable bounds and trust region concepts. A new consistent set of initial conditions is obtained by minimizing the deviation of the variable values from the specified ones. For problems with discontinuities caused by a step change in the input functions, a new criterion is presented for identifying the subset of variables which are continuous across the discontinuity. The SLP formulation is then applied to determine a consistent set of initial conditions for further solution of the problem in the domain after the discontinuity. Numerous example problems are solved to illustrate these concepts.

Active multi-model control for dynamic maneuver optimization of unmanned air vehicles

We present a wavelet-based multi-resolution dynamic maneuver optimization method for UAV route planning. We use a combination of evolutionary computing and interior point based dynamic optimization algorithm to satisfy constraints imposed by vehicle dynamics as well as obstacle avoidance.

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.