Maben Rabi

A Randomized Incremental Subgradient Method for Distributed Optimization in Networked Systems

We present an algorithm that generalizes the randomized incremental subgradient method with fixed stepsize due to Nedic and Bertsekas [SIAM J. Optim., 12 (2001), pp. 109-138]. Our novel algorithm is particularly suitable for distributed implementation and execution, and possible applications include distributed optimization, e.g., parameter estimation in networks of tiny wireless sensors. The stochastic component in the algorithm is described by a Markov chain, which can be constructed in a distributed fashion using only local information. We provide a detailed convergence analysis of the proposed algorithm and compare it with existing, both deterministic and randomized, incremental subgradient methods.

Optimal stopping for event-triggered sensing and actuation

Novel event-triggered sensing and actuation strategies are presented for networked control systems with limited communication resources. Two architectures are considered: one with the controller co-located with the sensor and one with the control co-located with the actuator. A stochastic control problem with an optimal stopping rule is shown to capture two interesting instances of these architectures. The solution of the problem leads to a parametrization of the control alphabet as piecewise constant commands. The execution of the control commands is triggered by stopping rules for the sensor. In simple situations, it is possible to analytically derive the optimal controller. Examples illustrate how the new event-based control and sensing strategies outperform conventional time-triggered schemes.

A simple peer-to-peer algorithm for distributed optimization in sensor networks

We propose a distributed algorithm that solves a special class of optimization problems using only peer-to- peer communication. One application is parameter estimation problems in sensor networks. Current decentralized algorithms for solving this class of optimization problems typically rely on passing around a parameter estimate in a ring consisting of all network nodes. In our algorithm, which extends the randomized incremental subgradient method with fixed stepsize due to Nedic and Bertsekas, nodes maintain individual estimates and need to exchange information only with their neighbors. We establish approach of the solution to an interval around the optimum value. We illustrate the algorithms performance, in terms of convergence rate and communication cost relative to alternative schemes, through several numerical examples.

Event-triggered strategies for industrial control over wireless networks

New event-based sampling strategies can support the efficient use of radio resources in wireless control systems. Motivated by the recent introduction of wireless network nodes in process control industry, we consider the particular demands these closed-loop systems set on the wireless communication and the influence the communication has on the control performance. In the paper, it is pointed out that by letting sensor nodes transmit only when needed, it is possible to minimize the communication bandwidth utilization in these systems. We show how classical control strategies commonly based on periodic sampling, such as proportional-integral-derivative control and minimum variance control, can be cast in an event-based setting in which decentralized communication decisions are taken suitable for commonly used contention-based medium access control protocols. Event-triggered sampling for estimation is also reviewed. Simulated examples illustrate the results.

Adaptive Sampling for Linear State Estimation

When a sensor has continuous measurements but sends limited messages over a data network to a supervisor which estimates the state, the available packet rate fixes the achievable quality of state estimation. When such rate limits turn stringent, the sensors messaging policy should be designed anew. What are the good causal messaging policies ? What should message packets contain ? What is the lowest possible distortion in a causal estimate at the supervisor ? Is Delta sampling better than periodic sampling ? We answer these questions under an idealized model of the network and the assumption of perfect measurements at the sensor. For a scalar, linear diffusion process, we study the problem of choosing the causal sampling times that will give the lowest aggregate squared error distortion. We stick to finite-horizons and impose a hard upper bound on the number of allowed samples. We cast the design as a problem of choosing an optimal sequence of stopping times. We reduce this to a nested sequence of problems each asking for a single optimal stopping time. Under an unproven but natural assumption about the least-square estimate at the supervisor, each of these single stopping problems are of standard form. The optimal stopping times are random times when the estimation error exceeds designed envelopes. For the case where the state is a Brownian motion, we give analytically: the shape of the optimal sampling envelopes, the shape of the envelopes under optimal Delta sampling, and their performances. Surprisingly, we find that Delta sampling performs badly. Hence, when the rate constraint is a hard limit on the number of samples over a finite horizon, we should should not use Delta sampling.

Multiple Sampling for Estimation on a Finite Horizon

We discuss some multiple sampling problems that arise in finite horizon real-time estimation when there is an upper limit on the number of allowable samples. Measuring estimation quality by the aggregate squared error, we compare the performances of the best deterministic, level-triggered and the optimal sampling schemes. We restrict the signal to be either a Wiener or an Ornstein-Uhlenbeck process. For the Wiener process, we provide closed form expressions and series expansions, whereas for the Ornstein Uhlenbeck process, procedures for numerical computation. Our results indicate that the best level-triggered sampling is almost optimal when the signal is stable

Estimation over heterogeneous sensor networks

Design trade-offs between estimation performance, processing delay and communication cost for a sensor scheduling problem is discussed. We consider a heterogeneous sensor network with two types of sensors: the first type has low-quality measurements, small processing delay and a light communication cost, while the second type is of high quality, but imposes a large processing delay and a high communication cost. Such a heterogeneous sensor network is common in applications, where for instance in a localization system the poor sensor can be an ultrasound sensor while the more powerful sensor can be a camera. Using a time-periodic Kalman filter, we show how one can find an optimal schedule of the sensor communication. One can significantly improve estimation quality by only using the expensive sensor rarely. We also demonstrate how simple sensor switching rules based on the Riccati equation drives the filter into a stable time-periodic Kalman filter.

Level-triggered control of a scalar linear system

In networked control systems, because of limits on communication rates, control waveforms are frequently piece-wise constant with limits on rates at which they can switch levels. It is, therefore, natural to expect event-triggered control switchings to perform more efficiently than time-triggered ones. This article verifies the validity of this supposition. A method for designing good level-triggered control schemes is obtained by reducing the continuous time problem to one in discrete time. Then, by numerical procedures, the performance of the level-triggered scheme is computed for comparison with that of the periodically switched control scheme.

Sampling of diffusion processes for real-time estimation

This paper addresses the causal sampling of observations of a diffusion process that results in a good quality continuous estimator based upon these samples. The optimal sampling scheme with a fixed number of samples is found by solving an optimal (multiple) stopping problem. This is solved explicitly in a special case. A class of threshold approximations is also described.

Separated Design of Encoder and Controller for Networked Linear Quadratic Optimal Control

For a networked control system, we consider the problem of encoder and controller design. We study a discrete-time linear plant with a finite horizon performance cost, comprising of a quadratic function of the states and controls, and an additive communication cost. We study separation in design of the encoder and controller, along with related closed-loop properties such as the dual effect and certainty equivalence. We consider three basic formats for encoder outputs: quantized samples, real-valued samples at event-triggered times, and real-valued samples over additive noise channels. If the controller and encoder are dynamic, then we show that the performance cost is minimized by a separated design: the controls are updated at each time instant as per a certainty equivalence law, and the encoder is chosen to minimize an aggregate quadratic distortion of the estimation error. This separation is shown to hold even though a dual effect is present in the closed-loop system. We also show that this separated design need not be optimal when the controller or encoder are to be chosen from within restricted classes.