Arvind U. Raghunathan

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.

An Interior Point Method for Mathematical Programs with Complementarity Constraints (MPCCs)

Interior point methods for nonlinear programs (NLPs) are adapted for solution of mathematical programs with complementarity constraints (MPCCs). The constraints of the MPCC are suitably relaxed so as to guarantee a strictly feasible interior for the inequality constraints. The standard primal-dual algorithm has been adapted with a modified step calculation. The algorithm is shown to be superlinearly convergent in the neighborhood of the solution set under assumptions of MPCC-LICQ, strong stationarity and upper level strict complementarity. The modification can be easily accommodated within most nonlinear programming interior point algorithms with identical local behavior. Numerical experience is also presented and holds promise for the proposed method.

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

Mathematical programs with equilibrium constraints (MPECs) in process engineering

Mathematical programs with equilibrium constraints (MPECs) form a relatively new and interesting subclass of nonlinear programming problems. In this paper we propose a novel method of solving MPECs by appropriate reformulation of the equilibrium conditions. The reformulation can be easily incorporated in a certain class of interior point algorithms for nonlinear optimization. The algorithm used in the study follows a primal-dual interior point approach and shows encouraging results on a test suite of MPECs. The algorithm is also able to perform optimization of distillation columns with phase changes and tray optimization using only continuous variables. We also consider a number of topics to improve performance of the algorithm and to identify classes of process engineering problems that can be handled as MPECs.

An MPEC formulation for dynamic optimization of distillation operations

We consider the dynamic optimization of chemical processes with changes in the number of equilibrium phases. Recent work has shown that transitions in the number of phases can be modeled as a mathematical program with equilibrium constraints (MPEC). This study generalizes the MPEC to consider dynamic characteristics. In particular, we describe a simultaneous discretization and solution strategy for dynamic optimization problems with complementarity constraints. These discretized problems are then solved with IPOPT-C, a recently developed barrier method for MPEC problems. Our approach is applied to two distillation examples. In the first, we consider the optimal startup of a binary batch distillation problem. In the second, we consider the dynamic operation of a cryogenic column for the separation of natural gas liquids. Both cases demonstrate the effectiveness of the approach on large scale MPEC problems.

Global optimization of multi-period optimal power flow

In this work, we extend the algorithm proposed in [1] to solve multi-period optimal power flow (MOPF) problems to global optimality. The multi-period version of the OPF is time coupled due to the integration of storage systems into the network, and ramp constraints on the generators. The global optimization algorithm is based on the spatial branch and bound framework with lower bounds on the optimal objective function value calculated by solving a semidefinite programming (SDP) relaxation of the MOPF. The proposed approach does not assume convexity and is more general than the ones presented previously for the solution of MOPF. We present a case study of the IEEE 57 bus instance with a time varying demand profile. The integration of storage in the network helps to satisfy loads during high demands and the ramp constraints ensure smooth generation profiles. The SDP relaxation does not satisfy the rank condition, and our optimization algorithm is able to guarantee global optimality within reasonable computational time.

Infeasibility detection in alternating direction method of multipliers for convex quadratic programs

We investigate the infeasibility detection in the alternating direction method of multipliers (ADMM) when minimizing a convex quadratic objective subject to linear equalities and simple bounds. The ADMM formulation consists of alternating between an equality constrained quadratic program (QP) and a projection onto the bounds. We show that: (i) the sequence of iterates generated by ADMM diverges, (ii) the divergence is restricted to the component of the multipliers along the range space of the constraints and (iii) the primal iterates converge to a minimizer of the Euclidean distance between the subspace defined by equality constraints and the convex set defined by bounds. In addition, we derive the optimal value for the step size parameter in the ADMM algorithm that maximizes the rate of convergence of the primal iterates and dual iterates along the null space. In fact, the optimal step size parameter for the infeasible instances is identical to that for the feasible instances. The theoretical results allow us to specify a practical termination condition for infeasibility and the performance of such criterion is demonstrated in a model predictive control application.

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.

Learning to Rank 3D Features

Representation of three dimensional objects using a set of oriented point pair features has been shown to be effective for object recognition and pose estimation. Combined with an efficient voting scheme on a generalized Hough space, existing approaches achieve good recognition accuracy and fast operation. However, the performance of these approaches degrades when the objects are (self-)similar or exhibit degeneracies, such as large planar surfaces which are very common in both man made and natural shapes, or due to heavy object and background clutter. We propose a max-margin learning framework to identify discriminative features on the surface of three dimensional objects. Our algorithm selects and ranks features according to their importance for the specified task, which leads to improved accuracy and reduced computational cost. In addition, we analyze various grouping and optimization strategies to learn the discriminative pair features. We present extensive synthetic and real experiments demonstrating the improved results.

Optimal Stabilization Using Lyapunov Measures

Numerical solutions for the optimal feedback stabilization of discrete time dynamical systems is the focus of this technical note. Set-theoretic notion of almost everywhere stability introduced by the Lyapunov measure, weaker than conventional Lyapunov function-based stabilization methods, is used for optimal stabilization. The linear Perron-Frobenius transfer operator is used to pose the optimal stabilization problem as an infinite dimensional linear program. Set-oriented numerical methods are used to obtain the finite dimensional approximation of the linear program. We provide conditions for the existence of stabilizing feedback controls and show the optimal stabilizing feedback control can be obtained as a solution of a finite dimensional linear program. The approach is demonstrated on stabilization of period two orbit in a controlled standard map.