Solmaz S. Kia

Distributed convex optimization via continuous-time coordination algorithms with discrete-time communication

This paper proposes a novel class of distributed continuous-time coordination algorithms to solve network optimization problems whose cost function is a sum of local cost functions associated to the individual agents. We establish the exponential convergence of the proposed algorithm under (i) strongly connected and weight-balanced digraph topologies when the local costs are strongly convex with globally Lipschitz gradients, and (ii) connected graph topologies when the local costs are strongly convex with locally Lipschitz gradients. When the local cost functions are convex and the global cost function is strictly convex, we establish asymptotic convergence under connected graph topologies. We also characterize the algorithms correctness under time-varying interaction topologies and study its privacy preservation properties. Motivated by practical considerations, we analyze the algorithm implementation with discrete-time communication. We provide an upper bound on the stepsize that guarantees exponential convergence over connected graphs for implementations with periodic communication. Building on this result, we design a provably-correct centralized event-triggered communication scheme that is free of Zeno behavior. Finally, we develop a distributed, asynchronous event-triggered communication scheme that is also free of Zeno with asymptotic convergence guarantees. Several simulations illustrate our results.

A centralized-equivalent decentralized implementation of Extended Kalman Filters for cooperative localization

We present a novel decentralized cooperative localization algorithm for mobile robots. The proposed algorithm is a decentralized implementation of a centralized Extended Kalman Filter for cooperative localization. In this algorithm, instead of propagating cross-covariance terms, each robot propagates new intermediate local variables that can be used in an update stage to create the required propagated cross-covariance terms. Whenever there is a relative measurement in the network, the algorithm declares the robot making this measurement as the interim master. By acquiring information from the interim landmark, the robot the relative measurement is taken from, the interim master can calculate and broadcast a set of intermediate variables which each robot can then use to update its estimates to match that of a centralized Extended Kalman Filter for cooperative localization. Once an update is done, no further communication is needed until the next relative measurement. The communication graph can be a time-varying directed graph with the only requirement that it should have a spanning tree rooted at the interim master.

Periodic and event-triggered communication for distributed continuous-time convex optimization

We propose a distributed continuous-time algorithm to solve a network optimization problem where the global cost function is a strictly convex function composed of the sum of the local cost functions of the agents. We establish that our algorithm, when implemented over strongly connected and weight-balanced directed graph topologies, converges exponentially fast when the local cost functions are strongly convex and their gradients are globally Lipschitz. We also characterize the privacy preservation properties of our algorithm and extend the convergence guarantees to the case of time-varying, strongly connected, weight-balanced digraphs. When the network topology is a connected undirected graph, we show that exponential convergence is still preserved if the gradients of the strongly convex local cost functions are locally Lipschitz, while it is asymptotic if the local cost functions are convex. We also study discrete-time communication implementations. Specifically, we provide an upper bound on the stepsize of a synchronous periodic communication scheme that guarantees convergence over connected undirected graph topologies and, building on this result, design a centralized event-triggered implementation that is free of Zeno behavior. Simulations illustrate our results.

A distributed dynamical solver for an optimal resource allocation problem over networked systems

In this paper, we consider an optimal resource allocation problem over networked systems where the global cost function is a sum of local convex cost functions of the agents. This optimization problem is subject to an affine equality constraint which represents the demand that the agents should meet through weighted contribution of their cost variables. We propose a novel distributed continuous-time algorithm that solves the problem over networks with connected graph communication topology. We also present an extension of our algorithm to solve allocation problems subject to multiple affine constraints. We demonstrate our results through a numerical example.

Cooperative Localization for Mobile Agents: A Recursive Decentralized Algorithm Based on Kalman-Filter Decoupling

Technological advances in ad hoc networking and the miniaturization of electromechanical systems are making possible the use of large numbers of mobile agents (for example, mobile robots, human agents, and unmanned vehicles) to perform surveillance, search and rescue, transport, and delivery tasks in aerial, underwater, space, and land environments. However, the successful execution of such tasks often hinges upon accurate position information, which is needed in lower-level locomotion and path-planning algorithms. Common techniques for the localization of mobile robots are the classical preinstalled beacon-based localization algorithms, fixed feature-based simultaneous localization and mapping (SLAM) algorithms, and Global Positioning System (GPS) navigation. However, these localization techniques work based on assumptions such as the existence of distinct and static features that can be revisited often or line of sight to GPS satellites, which may not be feasible for operations such as search and rescue, environment monitoring, and oceanic exploration. In the case of GPS navigation, there is also a current concern about signal jamming for outdoor navigation, especially for unmanned aerial vehicle coordination and control. Instead, cooperative localization is emerging as an alternative localization technique that can be employed in such scenarios.

Cooperative localization under message dropouts via a partially decentralized EKF scheme

For a team of mobile robots with limited onboard resources, we propose a partially decentralized implementation of an extended Kalman filter for cooperative localization. In the proposed algorithm, unlike a fully centralized scheme that requires, at each timestep, information from the entire team to be gathered together and be processed by a single device, we only require that the robots communicate with a central command unit at the time of a measurement update. In addition, the computational and storage cost per robot in terms of the size of the team is reduced to O(1). Moreover, the algorithm is robust to occasional in-network communication link failures while the estimation update of the robots receiving the update message is of minimum variance. We demonstrate the performance of the algorithm in simulations.

Multi-stage anti-windup for LTI systems with actuator magnitude and rate saturation

We consider systems with actuator magnitude and rate limitations, and assume there are high performance compensators with highly desirable properties as long as the actuator limitations are not violated. A multi-stage anti-windup (AW) scheme is presented that provides stability and performance measures, for commands with known bound. More critically, it uses different gains for different levels of saturation, allowing for more aggressive anti-windup gains when the command signals are moderately above the actuator limitations. The benefits are shown through an illustrative example.

A server-client based distributed processing for an Unscented Kalman filter for cooperative localization

For a group of mobile robots with communication and computation capabilities, we consider a cooperative localization algorithm based on Unscented Kalman filtering. We present a server-client paradigm to distribute the computational cost of this algorithm among team members. The highest computational cost of the Unscented Kalamn filter comes from calculating the collective covariance matrix of the team and its square root, normally obtained by Cholesky decomposition. Our server-client based computationally distributed algorithm is centered on identifying an appropriate Cholesky decomposition algorithm which allows a coordinated computational task allocation among team members.

New anti-windup structure for magnitude and rate limited inputs and peak-bounded disturbances

A new structure for the anti-windup (AW) compensation for rate limited actuation is proposed which is less conservative than structures currently used. For peak bounded disturbances or reference inputs, we develop AW augmentation loops for both magnitude and rate actuator saturation. To reduce conservatism further, the proposed technique is combined with a multi-stage AW loops to obtain different gains for different levels of saturation.

An Augmented Lagrangian distributed algorithm for an in-network optimal resource allocation problem

This paper studies distributed solutions for an optimal resource allocation problem over networked systems with connected graph communication topologies. The problem setting consists of a group of agents in a network cooperatively meeting a demand by supplying a resource whose commitment incurs a cost on them. The objective in the optimal resource allocation problem is to obtain a commitment value for each agent such that the total cost, which is the sum of the costs of the agents, is minimized. In this paper we discuss how using ideas from Augmented Lagrangian method for convex optimization problems with affine constrains, we can arrive at a distributed solutions whose convergence guarantees holds for networks where the local costs are convex. We also show that if the local costs are all strongly convex and their gradients are globally Lipschitz then the convergence guarantees are exponential and the results can be extended to a special class of time-varying network interaction topologies. Simulations illustrate our results.