Robin L. Raffard

Distributed optimization for cooperative agents: application to formation flight

We present a simple decentralized algorithm to solve optimization problems involving cooperative agents. Cooperative agents share a common objective and simultaneously pursue private goals. Furthermore, agents are constrained by limited communication capabilities. The algorithm is based on dual decomposition techniques and appears to be very intuitive. It solves the dual problem of an artificially decomposed version of the primal problem, replacing one large computationally intractable problem with many smaller tractable problems. It returns a feasible solution to the primal problem as well as an upper bound on the distance between this solution and the global optimum. Both convex and nonconvex examples are presented, the complexity of the convex case is analyzed, and the savings in complexity are demonstrated for both examples. Finally, by showing that there is no duality gap in these examples, optimality is certified.

Adjoint-based control of a new eulerian network model of air traffic flow

An Eulerian network model for air traffic flow in the National Airspace System is developed and used to design flow control schemes which could be used by Air Traffic Controllers to optimize traffic flow. The model relies on a modified version of the Lighthill-Whitham-Richards (LWR) partial differential equation (PDE), which contains a velocity control term inside the divergence operator. This PDE can be related to aircraft count, which is a key metric in air traffic control. An analytical solution to the LWR PDE is constructed for a benchmark problem, to assess the gridsize required to compute a numerical solution at a prescribed accuracy. The Jameson-Schmidt-Turkel (JST) scheme is selected among other numerical schemes to perform simulations, and evidence of numerical convergence is assessed against this analytical solution. Linear numerical schemes are discarded because of their poor performance. The model is validated against actual air traffic data (ETMS data), by showing that the Eulerian description enables good aircraft count predictions, provided a good choice of numerical parameters is made. This model is then embedded as the key constraint in an optimization problem, that of maximizing the throughput at a destination airport while maintaining aircraft density below a legal threshold in a set of sectors of the airspace. The optimization problem is solved by constructing the adjoint problem of the linearized network control problem, which provides an explicit formula for the gradient. Constraints are enforced using a logarithmic barrier. Simulations of actual air traffic data and control scenarios involving several airports between Chicago and the U.S. East Coast demonstrate the feasibility of the method

Cell packing influences planar cell polarity signaling

Some epithelial cells display asymmetry along an axis orthogonal to the apical-basal axis, referred to as planar cell polarity (PCP). A Frizzled-mediated feedback loop coordinates PCP between neighboring cells, and the cadherin Fat transduces a global directional cue that orients PCP with respect to the tissue axes. The feedback loop can propagate polarity across clones of cells that lack the global directional signal, although this polarity propagation is error prone. Here, we show that, in the Drosophila wing, a combination of cell geometry and nonautonomous signaling at clone boundaries determines the correct or incorrect polarity propagation in clones that lack Fat mediated global directional information. Pattern elements, such as veins, and sporadic occurrences of irregular geometry are obstacles to polarity propagation. Hence, in the wild type, broad distribution of the global directional cue combines with a local feedback mechanism to overcome irregularities in cell packing geometry during PCP signaling.

Network Congestion Alleviation Using Adjoint Hybrid Control: Application to Highways

This paper derives an optimization-based control methodology for networks of switched and hybrid systems in which each mode is governed by a partial differential equation (PDE). We pose the continuous controller synthesis problem as an optimization program with PDEs in the constraints. The proposed algorithm relies on an explicit formulation of the gradient of the cost function, obtained via the adjoint of the PDE operator. First, we show how to use the result of the optimization to synthesize on/off control strategies. Then, we generalize the method to optimal switching control of hybrid systems over PDEs: the system is allowed to switch from one mode (or PDE) to another at times which we synthesize to minimize a given cost. We derive an explicit expression of the gradient of the cost with respect to the switching times. We implement our techniques on a highway congestion control problem using Performance Measurement System (PeMS) data for the California I210 for a 9 mile long strip with 26 on-ramps (controllable with red/green metering lights) and off-ramps (uncontrollable).

Market-Based Air Traffic Flow Control with Competing Airlines

To advance efficient and equitable use of the U.S. National Airspace System during weather disruptions, air traffic management is modeled as a network flow optimization program that explicitly incorporates airline preference information. Market mechanisms are proposed to perform distributed computation of efficient solutions based on novel cost metrics that accurately reflect the cost of delays to the airlines. Two distinct types of network flow models are presented to demonstrate the tradeoff between computational complexity and control input flexibility. The discrete path flow model describes a simplified approach by fixing flow velocity and limiting rerouting options, thereby satisfying the primary assumptions of general equilibrium theory to ensure efficiency of the market outcome. The continuous link flow model allows additional control inputs to better align with current methods used in dealing with adverse weather conditions; namely, ground-delay programs, miles-in-trail restrictions, and flight reroutes. The market-mechanism outcome for both models is shown to be preferred by all airlines over a solution determined without incorporating preference information. Simulation results are presented for feasible problem sizes of both flow models and demonstrate the gains that can be achieved by implementing market mechanisms for air traffic management.

A Cooperative Distributed Approach to Multi-Agent Eulerian Network Control: Application to Air Traffic Management ∗

Based on an Eulerian network model, air traffic flow in the National Airspace System is optimized using second order adjoint methods for hyperbolic PDEs. Multiple airlines with independent cost structures are added to previous work on air traffic flow modeling and optimization. A cooperative, distributed solution methodology is proposed that allows airlines to keep cost information private by distributing the computational workload between the airlines while ensuring efficient outcomes. The Nash Bargaining Solution is presented as a possible solution concept, and a distributed method for its calculation is presented. I. Introduction Management of today’s National Airspace System (NAS) has become an extremely complicated task. Pilots and controllers must continuously manage numerous complex subsystems and adapt to frequent changes in traffic flow, spacing and routing caused by variable weather conditions and other network delays. Specifically, the national Air Traffic Control System Command Center (ATCSCC) is responsible for continuously updating decisions concerning traffic flow to maximize network throughput. The current methodology is based on a sophisticated set of rules described in playbooks that have been developed through years of controller experience. This procedure is neither automated nor optimal, and requires constant human intervention to allow the system to operate. Airlines are an integral element of the NAS and are developing more and more advanced methods of aircraft management, as supervised by Airline Operations Centers (AOCs). Their stake in aircraft flow decisions is quite high, as their profitability is directly tied to on-time performance, and significant costs are incurred by flight delays, cancellations and rerouting. Furthermore, their preferences change dynamically, based on highly variable parameters such as aircraft load factor and percentage of connecting passengers per flight. Unfortunately, airlines are currently only able to affect decisions over a small part of the network flow problem, by swapping slots with their own aircraft and deciding which flights to delay when required. For smooth flow of their aircraft over the NAS network, the airlines must rely on the savvy and equity of the ATCSCC controllers who attempt to minimize schedule disruption while not necessarily minimizing incurred costs. This need not be the case. Consider a futuristic scenario for the NAS, where airlines cooperate with each other in a distributed manner, supervised by the FAA, to dynamically reroute and reschedule flights based on restrictions imposed by weather conditions and network delays in such a manner that their individual profits are maximized, while all safety requirements are met and the airspace is used efficiently. In sucha scenario, real-time collaboration amongst the airlines will drive the efficient use of resources by incorporating airline specific preferences between routes into the flow optimization and ultimately, the passenger will be better served.

Second order adjoint-based optimization of ordinary and partial differential equations with application to air traffic flow

We present an algorithm to implement the second order Newton method on ordinary differential equation (ODE) and partial differential equation (PDE) optimization programs. The algorithm is based on the direct computation of the Newton step without explicitly calculating the second derivative (Hessian) of the objective function. The method poses the search for the Newton step as a convex quadratic optimization program. We apply our method to (a) dynamical systems driven by ODEs and to (b) constrained PDE optimization programs in the context of air traffic flow. In both cases, our implementation of the Newton method shows much faster convergence than first order algorithms, while not significantly increasing computational time.

Adjoint-based optimal control of the expected exit time for stochastic hybrid systems

In this paper, we study the problem of controlling the expected exit time from a region for a class of stochastic hybrid systems. That is, we find the least costly feedback control for a stochastic hybrid system that can keep its state inside a prescribed region for at least an expected amount of time. The stochastic hybrid systems considered are quite general: the continuous dynamics are governed by stochastic differential equations, and the discrete mode evolves according to a continuous time Markov chain. Instead of adopting the usual Hamilton-Jacobi viewpoint, we study the problem directly by formulating it as a PDE constrained optimization problem, and propose a solution using adjoint-based gradient descent methods. Numerical results of the proposed approach are presented for several representative examples, and, for the simple case, compared with analytical results.

Toward efficient and equitable distributed air traffic flow control

In order to ensure efficient and equitable use of the National Airspace, the problem of air traffic flow control is posed as a PDE constrained optimization program which explicitly incorporates distinct preferences for independent airlines. Novel cost metrics are introduced to capture the real-world cost of delays to the airlines, and control parameters are aligned with the actual inputs the FAA can manipulate in dealing with adverse weather conditions, which include ground-delay programs, miles-in-trail restrictions and flight reroutes. A market mechanism is proposed that allows for distributed computation of efficient and equitable solutions. The resulting solution preferentially reduces delays for expensive flights, but compensates less expensive flights via transfer payments

Automatic Parameter Identification via the Adjoint Method, with Application to Understanding Planar Cell Polarity

A key focus of systems biology has been the development of models, at the appropriate level of abstraction, to help understand different biological processes. This development usually proceeds in iterative fashion, in which the structure of the model is chosen to represent certain hypotheses about how the system operates and parameters for this structured model are chosen. Often, the first experiment is to ask if a robust set of parameters exists so that the model reproduces all or most of the observed biological data. The model is tested against this actual data and for its predictive capabilities. As new data and/or new understanding arises, the structure of the model may be altered, and new parameters selected. In protein regulatory networks, the number of states to model is typically large and depends on the number of proteins of interest, the parameter spaces are large, and the most appropriate models are nonlinear functions of the states. Thus it is becoming increasingly important to develop fast, efficient, scalable methods for large scale parameter identification. This paper presents an adjoint-based algorithm for performing automatic parameter identification on differential equation based models of biological systems. The algorithm solves an optimization problem, in which the cost reflects the deviation between the observed data and the output of the parameterized mathematical model, and the constraints reflect the governing parameterized equations themselves. Preliminary results of the application of this algorithm to a previously presented mathematical model of planar cell polarity signaling in the wings of Drosophila melanogaster are presented