arXiv subjects

Ketan Savla

Publications and source records attributed to Ketan Savla.

34 records · Page 2Linked to original sources

arXiv2017

A Principled Approximation Framework for Optimal Control of Semi-Markov Jump Linear Systems

We consider continuous-time, finite-horizon, optimal quadratic control of semi-Markov jump linear systems (S-MJLS), and develop principled approximations through Markov-like representations for the holding-time distributions. We adopt a phase-type approximation for holding times, which is known to be consistent, and translates a S-MJLS into a specific MJLS with partially observable modes (MJLSPOM), where the modes in a cluster have the same dynamic, the same cost weighting matrices and the same control policy. For a general MJLSPOM, we give necessary and sufficient conditions for optimal (switched) linear controllers. When specialized to our particular MJLSPOM, we additionally establish the existence of optimal linear controller, as well as its optimality within the class of general controllers satisfying standard smoothness conditions. The known equivalence between phase-type distributions and positive linear systems allows to leverage existing modeling tools, but possibly with large computational costs. Motivated by this, we propose matrix exponential approximation of holding times, resulting in pseudo-MJLSPOM representation, i.e., where the transition rates could be negative. Such a representation is of relatively low order, and maintains the same optimality conditions as for the MJLSPOM representation, but could violate non-negativity of holding-time density functions. A two-step procedure consisting of a local pulling-up modification and a filtering technique is constructed to enforce non-negativity.

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arXiv2017

Queue Length Simulation for Signalized Arterial Networks and Steady State Computation under Fixed Time Control

We consider traffic flow dynamics for a network of signalized intersections, where the outflow from every link is constrained to be equal to a given capacity function if the queue length is positive, and equal to the minimum of cumulative inflow and capacity function otherwise. In spite of the resulting dynamics being discontinuous, recent work has proved existence and uniqueness of the resulting queue length trajectory if the inter-link travel times are strictly bounded away from zero. The proof, which also suggests a constructive procedure, relies on showing desired properties on contiguous time intervals of length equal to the minimum among all link travel times. We provide an alternate framework to obtain queue length trajectories by direct simulation of delay differential equations, where link outflows are obtained from the provably unique solution to a linear program. Existence and uniqueness of the solution to the proposed model for traffic flow dynamics is established for piecewise constant external inflow and capacity functions, and the proposed method does not require travel times to be bounded away from zero. Additionally, if the external inflow and capacity functions are periodic and satisfy a stability condition, then there exists a globally attractive periodic orbit. We provide an iterative procedure to compute this periodic orbit. A periodic trajectory is iteratively updated for every link based on updates to a specific time instant when its queue length transitions from being zero to being positive. The update for a given link is based on the periodic trajectories computed in the previous iteration for its upstream links. The resulting iterates are shown to converge uniformly monotonically to the desired periodic orbit.

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arXiv2016

Robustness of DC Power Networks under Weight Control

We study, possibly distributed, robust weight control policies for DC power networks that change link susceptances, or weights in response to balanced disturbances to the supply-demand vector. The margin of robustness for a given control policy is defined as the radius of the largest $l_1$ ball in the space of balanced disturbances under which the link flows can be asymptotically contained within their specified limits. For centralized control policies, the control design as well as margin of robustness are obtained from solution to an non-convex weight control problem. We establish relationship between feasible sets for DC power flow and associated network flow, which is used to establish an upper bound on the margin of robustness in terms of the min cut capacity. This bound is proven to be tight if the network is tree-like, or if the lower bound of the operation range of weight control is zero. An explicit expression for the flow-weight Jacobian is derived and is used to devise a projected sub-gradient algorithm to solve the relaxed weight control problem. An exact multi-level programming approach to solve the weight control problem for reducible networks, based on recursive application of equivalent bilevel formulation for relevant class of non-convex network optimization problems, is also proposed. The lower level problem in each recursion corresponds to replacing a sub-network by a (virtual) link with equivalent weight and capacities. The equivalent capacity function for tree-reducible networks is shown to possess a strong quasi-concavity property, facilitating easy solution to the weight control problem. Robustness analysis for natural decentralized control policies that decrease weights on overloaded links, and increase weights on underloaded links with increasing flows is provided for parallel networks. Illustrative simulation results for a benchmark IEEE network are included.

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arXiv2016

Service Rate, Busy Period & Throughput Analysis of a Horizontal Traffic Queue

We consider a horizontal traffic queue (HTQ) on a periodic road segment, where vehicles arrive according to a spatio-temporal Poisson process, and depart after traveling a distance that is sampled independently and identically from a spatial distribution. When inside the queue, the speed of a vehicle is proportional to a power $m >0$ of the distance to the vehicle in front. The service rate of HTQ is equal to the sum of the speeds of the vehicles, and has a complex dependency on the state (vehicle locations) of the system. We show that the service-rate increases (resp., decreases) in between arrivals and departures for $m<1$ (resp., $m>1$) case. For a given initial condition, we define the throughput of such a queue as the largest arrival rate under which the queue length remains bounded. We extend the busy period calculations for M/G/1 queue to our setting, including for non-empty initial condition. These calculations are used to prove that the throughput for $m=1$ case is equal to the inverse of the time required to travel average total distance by a solitary vehicle in the system, and also to derive a probabilistic upper bound on the queue length over a finite time horizon for the $m>1$ case. Finally, we study throughput under a release control policy, where the additional expected waiting time caused by the control policy is interpreted as the magnitude of the perturbation to the arrival process. We derive a lower bound on throughput for a given combination of maximum allowable perturbation, for $m<1$ and $m>1$ cases. In particular, if the allowable perturbation is sufficiently large, then this lower bound grows unbounded as $m \to 0^+$. Illustrative simulation results are also presented.

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arXiv2016

Convexity and Robustness of Dynamic Traffic Assignment and Freeway Network Control

We study the use of the System Optimum (SO) Dynamic Traffic Assignment (DTA) problem to design optimal traffic flow controls for freeway networks as modeled by the Cell Transmission Model, using variable speed limit, ramp metering, and routing. We consider two optimal control problems: the DTA problem, where turning ratios are part of the control inputs, and the Freeway Network Control (FNC), where turning ratios are instead assigned exogenous parameters. It is known that relaxation of the supply and demand constraints in the cell-based formulations of the DTA problem results in a linear program. However, solutions to the relaxed problem can be infeasible with respect to traffic dynamics. Previous work has shown that such solutions can be made feasible by proper choice of ramp metering and variable speed limit control for specific traffic networks. We extend this procedure to arbitrary networks and provide insight into the structure and robustness of the proposed optimal controllers. For a network consisting only of ordinary, merge, and diverge junctions, where the cells have linear demand functions and affine supply functions with identical slopes, and the cost is the total traffic volume, we show, using the maximum principle, that variable speed limits are not needed in order to achieve optimality in the FNC problem, and ramp metering is sufficient. We also prove bounds on perturbation of the controlled system trajectory in terms of perturbations in initial traffic volume and exogenous inflows. These bounds, which leverage monotonicity properties of the controlled trajectory, are shown to be in close agreement with numerical simulation results.

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arXiv2014

Stability Analysis and Control Synthesis for Dynamical Transportation Networks

We study dynamical transportation networks in a framework that includes extensions of the classical Cell Transmission Model to arbitrary network topologies. The dynamics are modeled as systems of ordinary differential equations describing the traffic flow among a finite number of cells interpreted as links of a directed network. Flows between contiguous cells, in particular at junctions, are determined by merging and splitting rules within constraints imposed by the cells' demand and supply functions as well as by the drivers' turning preferences, while inflows at on-ramps are modeled as exogenous and possibly time-varying. First, we analyze stability properties of dynamical transportation networks. We associate to the dynamics a state-dependent dual graph whose connectivity depends on the signs of the derivatives of the inter-cell flows with respect to the densities. Sufficient conditions for the stability of equilibria and periodic solutions are then provided in terms of the connectivity of such dual graph. Then, we consider synthesis of control policies that use a combination of turning preferences, speed limits, and ramp metering, in order to optimize convex objectives. We first show that, in the general case, the optimal control synthesis problem can be cast as a convex optimization problem, and that the equilibrium of the controlled network is in free-flow. If the control policies are restricted to speed limits and ramp metering, then the resulting synthesis problem is still convex for networks where every node is either a merge or a diverge junction, and where the dynamics is monotone. These results apply both to the optimal selection of equilibria and periodic solutions, as well as to finite-horizon network trajectory optimization. Finally, we illustrate our findings through simulations on a road network inspired by the freeway system in southern Los Angeles.

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arXiv2014

Robust Network Routing under Cascading Failures

We propose a dynamical model for cascading failures in single-commodity network flows. In the proposed model, the network state consists of flows and activation status of the links. Network dynamics is determined by a, possibly state-dependent and adversarial, disturbance process that reduces flow capacity on the links, and routing policies at the nodes that have access to the network state, but are oblivious to the presence of disturbance. Under the proposed dynamics, a link becomes irreversibly inactive either due to overload condition on itself or on all of its immediate downstream links. The coupling between link activation and flow dynamics implies that links to become inactive successively are not necessarily adjacent to each other, and hence the pattern of cascading failure under our model is qualitatively different than standard cascade models. The magnitude of a disturbance process is defined as the sum of cumulative capacity reductions across time and links of the network, and the margin of resilience of the network is defined as the infimum over the magnitude of all disturbance processes under which the links at the origin node become inactive. We propose an algorithm to compute an upper bound on the margin of resilience for the setting where the routing policy only has access to information about the local state of the network. For the limiting case when the routing policies update their action as fast as network dynamics, we identify sufficient conditions on network parameters under which the upper bound is tight under an appropriate routing policy. Our analysis relies on making connections between network parameters and monotonicity in network state evolution under proposed dynamics.

eess.SY↗

arXiv2014

Throughput Optimality and Overload Behavior of Dynamical Flow Networks under Monotone Distributed Routing

The paper investigates the throughput behavior of single-commodity dynamical flow networks governed by monotone distributed routing policies. The networks are modeled as systems of ODEs based on mass conversation laws on directed graphs with limited flow capacities on the links and constant external inflows at certain origin nodes. Under monotonicity assumptions on the routing policies, it is proven that a globally asymptotically stable equilibrium exists so that the network achieves maximal throughput, provided that no cut capacity constraint is violated by the external inflows. On the contrary, should such a constraint be violated, the network overload behavior is characterized. In particular, it is established that there exists a cut with respect to which the flow densities on every link grow linearly over time (resp. reach their respective limits simultaneously) in the case where the buffer capacities are infinite (resp. finite). The results employ an $l_1$-contraction principle for monotone dynamical systems.

math.OC↗

arXiv2012

Robust Distributed Routing in Dynamical Networks with Cascading Failures

Robustness of routing policies for networks is a central problem which is gaining increased attention with a growing awareness to safeguard critical infrastructure networks against natural and man-induced disruptions. Routing under limited information and the possibility of cascades through the network adds serious challenges to this problem. This abstract considers the framework of dynamical networks introduced in our earlier work [1,2], where the network is modeled by a system of ordinary differential equations derived from mass conservation laws on directed acyclic graphs with a single origin-destination pair and a constant inflow at the origin. The rate of change of the particle density on each link of the network equals the difference between the inflow and the outflow on that link. The latter is modeled to depend on the current particle density on that link through a flow function. The novel modeling element in this paper is that every link is assumed to have finite capacity for particle density and that the flow function is modeled to be strictly increasing as density increases from zero up to the maximum density capacity, and is discontinuous at the maximum density capacity, with the flow function value being zero at that point. This feature, in particular, allows for the possibility of spill-backs in our model. In this paper, we present our results on resilience of such networks under distributed routing, towards perturbations that reduce link-wise flow functions.

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arXiv2011

Robust Distributed Routing in Dynamical Flow Networks - Part II: Strong Resilience, Equilibrium Selection and Cascaded Failures

Strong resilience properties of dynamical flow networks are analyzed for distributed routing policies. The latter are characterized by the property that the way the inflow at a non-destination node gets split among its outgoing links is allowed to depend only on local information about the current particle densities on the outgoing links. The strong resilience of the network is defined as the infimum sum of link-wise flow capacity reductions under which the network cannot maintain the asymptotic total inflow to the destination node to be equal to the inflow at the origin. A class of distributed routing policies that are locally responsive to local information is shown to yield the maximum possible strong resilience under such local information constraints for an acyclic dynamical flow network with a single origin-destination pair. The maximal strong resilience achievable is shown to be equal to the minimum node residual capacity of the network. The latter depends on the limit flow of the unperturbed network and is defined as the minimum, among all the non-destination nodes, of the sum, over all the links outgoing from the node, of the differences between the maximum flow capacity and the limit flow of the unperturbed network. We propose a simple convex optimization problem to solve for equilibrium limit flows of the unperturbed network that minimize average delay subject to strong resilience guarantees, and discuss the use of tolls to induce such an equilibrium limit flow in transportation networks. Finally, we present illustrative simulations to discuss the connection between cascaded failures and the resilience properties of the network.

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arXiv2011

Robust Distributed Routing in Dynamical Flow Networks - Part I: Locally Responsive Policies and Weak Resilience

Robustness of distributed routing policies is studied for dynamical flow networks, with respect to adversarial disturbances that reduce the link flow capacities. A dynamical flow network is modeled as a system of ordinary differential equations derived from mass conservation laws on a directed acyclic graph with a single origin-destination pair and a constant inflow at the origin. Routing policies regulate the way the inflow at a non-destination node gets split among its outgoing links as a function of the current particle density, while the outflow of a link is modeled to depend on the current particle density on that link through a flow function. The dynamical flow network is called partially transferring if the total inflow at the destination node is asymptotically bounded away from zero, and its weak resilience is measured as the minimum sum of the link-wise magnitude of all disturbances that make it not partially transferring. The weak resilience of a dynamical flow network with arbitrary routing policy is shown to be upper-bounded by the network's min-cut capacity, independently of the initial flow conditions. Moreover, a class of distributed routing policies that rely exclusively on local information on the particle densities, and are locally responsive to that, is shown to yield such maximal weak resilience. These results imply that locality constraints on the information available to the routing policies do not cause loss of weak resilience. Some fundamental properties of dynamical flow networks driven by locally responsive distributed policies are analyzed in detail, including global convergence to a unique limit flow.

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arXiv2011

Stability Analysis of Transportation Networks with Multiscale Driver Decisions

Stability of Wardrop equilibria is analyzed for dynamical transportation networks in which the drivers' route choices are influenced by information at multiple temporal and spatial scales. The considered model involves a continuum of indistinguishable drivers commuting between a common origin/destination pair in an acyclic transportation network. The drivers' route choices are affected by their, relatively infrequent, perturbed best responses to global information about the current network congestion levels, as well as their instantaneous local observation of the immediate surroundings as they transit through the network. A novel model is proposed for the drivers' route choice behavior, exhibiting local consistency with their preference toward globally less congested paths as well as myopic decisions in favor of locally less congested paths. The simultaneous evolution of the traffic congestion on the network and of the aggregate path preference is modeled by a system of coupled ordinary differential equations. The main result shows that, if the frequency of updates of path preferences is sufficiently small as compared to the frequency of the traffic flow dynamics, then the state of the transportation network ultimately approaches a neighborhood of the Wardrop equilibrium. The presented results may be read as a further evidence in support of Wardrop's postulate of equilibrium, showing robustness of it with respect to non-persistent perturbations. The proposed analysis combines techniques from singular perturbation theory, evolutionary game theory, and cooperative dynamical systems.

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arXiv2010

Maximally Stabilizing Task Release Control Policy for a Dynamical Queue

In this paper, we introduce a model of dynamical queue, in which the service time depends on the server utilization history. The proposed queueing model is motivated by widely accepted empirical laws describing human performance as a function of mental arousal. The objective of this paper is to design task release control policies that can stabilize the queue for the maximum possible arrival rate, assuming deterministic arrivals. First, we prove an upper bound on the maximum possible stabilizable arrival rate for any task release control policy. Then, we propose a simple threshold policy that releases a task to the server only if its state is below a certain fixed value. Finally, we prove that this task release control policy ensures stability of the queue for the maximum possible arrival rate.

math.OC↗

arXiv2008

On Endogenous Reconfiguration in Mobile Robotic Networks

In this paper, our focus is on certain applications for mobile robotic networks, where reconfiguration is driven by factors intrinsic to the network rather than changes in the external environment. In particular, we study a version of the coverage problem useful for surveillance applications, where the objective is to position the robots in order to minimize the average distance from a random point in a given environment to the closest robot. This problem has been well-studied for omni-directional robots and it is shown that optimal configuration for the network is a centroidal Voronoi configuration and that the coverage cost belongs to $Θ(m^{-1/2})$, where $m$ is the number of robots in the network. In this paper, we study this problem for more realistic models of robots, namely the double integrator (DI) model and the differential drive (DD) model. We observe that the introduction of these motion constraints in the algorithm design problem gives rise to an interesting behavior. For a \emph{sparser} network, the optimal algorithm for these models of robots mimics that for omni-directional robots. We propose novel algorithms whose performances are within a constant factor of the optimal asymptotically (i.e., as $m \to +\infty$). In particular, we prove that the coverage cost for the DI and DD models of robots is of order $m^{-1/3}$. Additionally, we show that, as the network grows, these novel algorithms outperform the conventional algorithm; hence necessitating a reconfiguration in the network in order to maintain optimal quality of service.

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arXiv2006

Traveing Salesperson Problems for a double integrator

In this paper we propose some novel path planning strategies for a double integrator with bounded velocity and bounded control inputs. First, we study the following version of the Traveling Salesperson Problem (TSP): given a set of points in $\real^d$, find the fastest tour over the point set for a double integrator. We first give asymptotic bounds on the time taken to complete such a tour in the worst-case. Then, we study a stochastic version of the TSP for double integrator where the points are randomly sampled from a uniform distribution in a compact environment in $\real^2$ and $\real^3$. We propose novel algorithms that perform within a constant factor of the optimal strategy with high probability. Lastly, we study a dynamic TSP: given a stochastic process that generates targets, is there a policy which guarantees that the number of unvisited targets does not diverge over time? If such stable policies exist, what is the minimum wait for a target? We propose novel stabilizing receding-horizon algorithms whose performances are within a constant factor from the optimum with high probability, in $\real^2$ as well as $\real^3$. We also argue that these algorithms give identical performances for a particular nonholonomic vehicle, Dubins vehicle.

cs.RO↗

arXiv2006

Asymptotic constant-factor approximation algorithm for the Traveling Salesperson Problem for Dubins' vehicle

This article proposes the first known algorithm that achieves a constant-factor approximation of the minimum length tour for a Dubins' vehicle through $n$ points on the plane. By Dubins' vehicle, we mean a vehicle constrained to move at constant speed along paths with bounded curvature without reversing direction. For this version of the classic Traveling Salesperson Problem, our algorithm closes the gap between previously established lower and upper bounds; the achievable performance is of order $n^{2/3}$.

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