Ameer K. Mulla

Computation of feedback control for time optimal state transfer using Groebner basis

Abstract Computation of time optimal feedback control law for a controllable linear time invariant system with bounded inputs is considered. Unlike a recent paper by the authors, the target final state is not limited to the origin of state-space, but is allowed to be in the set of constrained controllable states. Switching surfaces are formulated as semi-algebraic sets using Groebner basis based elimination theory. Using these semi-algebraic sets, a nested switching logic is synthesized to generate the time optimal feedback control. However, the optimal control law enforces an unavoidable limit cycle in the time-optimal trajectory for most non-origin target points. The time-period of this limit-cycle is dependent on the target position. This dependence is algebraically characterized and a method to compute the time-period of the limit-cycle is provided. As a natural extension, the set of constrained controllable states is also computed.

Distributed computation of minimum time consensus for multi-agent systems

The problem of computing the minimum time to consensus of multiple identical double-integrator agents is considered. A distributed algorithm for computing the final consensus target state is proposed. Local feedback time optimal control laws are synthesized to drive each agent to the computed final consensus target state. Each agent is assumed to know the states of every other agent. As a part of the algorithm, every possible triplet of agents compute their mutual minimum time to consensus. The maximum value among these triplet minimum times is shown to be the required minimum time to consensus for the entire group. Since the required computation can be performed for each triple separately, it can be distributed evenly among the agents.

Gröbner basis computation of feedback control for time optimal state transfer

The synthesis of time-optimal feedback control of a single input, continuous time, linear time invariant system with bounded inputs is considered. Unlike a recent paper by the authors, the target final state is not necessarily the origin of the state space. Semi-algebraic representations of switching surfaces corresponding to the bang-bang time-optimal control are computed using a Gröbner basis based elimination algorithm. These surfaces are then used to synthesize a nested switching logic for time-optimal feedback control. Non-origin target points introduce unavoidable limit cycles in time optimal trajectories, whose period depends on the target position. This dependence is algebraically characterized and a method to compute the periods of the limit cycles is proposed. As a natural extension, we also provide a semi-algebraic characterization of the set of all points reachable with constrained inputs.

Efficient Task Allocation in Services Delivery Organizations

Service delivery organizations are constantly under pressure to meet service level agreements (SLAs), operate under tight costs, utilize their resources well and to improve the operational performance. A well-planned task allocation plays an important role in meeting these objectives. This work considers the problem of efficient task allocation to employees in such organizations with the aim of meeting the SLAs by minimizing the number of tasks missing their deadlines and the magnitude of these deadline misses taking into account employees skills, productivity, utilization and fairness. For this problem, it proposes an integer linear programming (ILP) based solution which is SLA-, resource skill-, productivity-, utilization-and fairness-aware. It also proves an approximation guarantee for the problem using the primal-dual technique. Further, in empirical evaluations using real-world transaction processing data from a large services delivery organization, the proposed ILP-based solution outperforms the currently practiced manual allocation in the organization. It is able to reduce the number of deadline violations by 90% and the magnitude of violations by 95% and increases the productivity of resources by 23%-38%. Further, the ILP-based solution never took more than 2 mins to allocate transactions in our evaluations making it a promising solution to deploy in real-time.

Communication preserving min-max time consensus tracking

A collection of double integrator agents is considered, where each agent has bounded input and a limited range for receiving information from other agents. One autonomous node (the Root) generates an unknown reference trajectory to which other agents are required to converge in minimum possible time, using only locally available information. A distributed algorithm is proposed to identify a directed spanning tree rooted at the Root agent. The child agent of each edge of this selected tree uses min-max time control strategies to converge to the states of the parent node. Necessary and sufficient conditions are derived for limiting the inter-agent distance so as to preserve the tree communication structure chosen. Under these conditions, the proposed control strategy turns out to be the global communication preserving min-max strategy for the entire collection.

A model for lane-less traffic with local control laws

In this paper, a new model for traffic on roads with multiple lanes is developed, where the vehicles do not adhere to lane discipline. Assuming identical vehicles, the dynamics is split along two independent directions - the Y-axis representing the direction of the traffic and the X-axis representing the lateral or the direction perpendicular to the traffic direction. Different influence graphs are used to model the interaction between the vehicles in these two directions. The instantaneous accelerations of each vehicle, in both X and Y directions, are functions of the measurements from the neighbouring vehicles according to these influence graphs. Under time invariant influence structure, expected for example, in dense traffic, the collection converges to a layered formation with fixed inter-vehicle distances. In general, the formation is BIBO stable with the velocity and inter vehicle separations oscillating between a finite number of equilibrium points.

Computation of the target state and feedback controls for time optimal consensus in multi-agent systems

ABSTRACT N identical agents with bounded inputs aim to reach a common target state (consensus) in the minimum possible time. Algorithms for computing this time-optimal consensus point, the control law to be used by each agent and the time taken for the consensus to occur, are proposed. Two types of multi-agent systems are considered, namely (1) coupled single-integrator agents on a plane and, (2) double-integrator agents on a line. At the initial time instant, each agent is assumed to have access to the state information of all the other agents. An algorithm, using convexity of attainable sets and Hellys theorem, is proposed, to compute the final consensus target state and the minimum time to achieve this consensus. Further, parts of the computation are parallelised amongst the agents such that each agent has to perform computations of O(N2) run time complexity. Finally, local feedback time-optimal control laws are synthesised to drive each agent to the target point in minimum time. During this part of the operation, the controller for each agent uses measurements of only its own states and does not need to communicate with any neighbouring agents.

Min–Max Time Consensus Tracking With Communication Guarantee

In this paper, a collection of double integrator agents with bounded inputs is considered. Communication is possible between any two agents only if the inter-agent distance is less than a fixed threshold. A special node, referred to as the “leader,” generates an unknown reference trajectory to which all the other agents are required to converge in the shortest possible time. Assuming the initial communication graph to be connected, a directed spanning tree rooted at the leader is identified using a local algorithm. The dynamics of any two agents connected by an edge in the selected tree are modeled as a time-optimal pursuit–evasion game, while maintaining the inter-agent communication link. Using the corresponding feedback saddle-point strategies, local min–max time control laws for each pair of agents are formulated. For the selected tree, the proposed collection of local min–max time control strategies is shown to be the communication preserving global min–max time strategy.

A dynamic model for lane-less traffic

The problem of modeling road traffic, when the drivers do not obey lane discipline, in considered. Identical vehicles are assumed to influence each other based on proximity and visual feedback. This percolation of influence across suitably defined layers of vehicles, in modeled using weighted directed graphs. Drivers accelerate, decelerate or maneuver sideways with the objective of maintaining safe inter-vehicle distances between the cars. This target inter-vehicle distance varies with the absolute velocity of the cars. When the influence structure is time invariant, e.g. in dense traffic, the collection converges to a layered formation with inter-vehicle distances based on vehicle velocities. In general, for a changing influence topology the inter-car distance can oscillate, while remaining uniformly bounded.

Implementation of distributed consensus on an outdoor testbed

In this paper, a distributed multi-agent consensus algorithm with theoretically provable convergence properties is implemented outdoors with three quadrotors. These quadrotors communicate with each other over an incomplete communication graph. Each quadrotor computes its position using only GPS data and transmits this information to the neighbouring quadrotors using Xbee wireless modules. The relative position information from all the neighbours is used by each quadrotor to compute the required pitch and roll angles, according to the consensus algorithm. It is shown that this distributed control law successfully navigates all the quadrotors to an autonomously decided consensus point. Details of the implementation and the consensus experiment are presented.