Abhishek Tiwari

Estimation with Information Loss: Asymptotic Analysis and Error Bounds

In this paper, we consider a discrete time state estimation problem over a packet-based network. In each discrete time step, the measurement is sent to a Kalman filter with some probability that it is received or dropped. Previous pioneering work on Kalman filtering with intermittent observation losses shows that there exists a certain threshold of the packet dropping rate below which the estimator is stable in the expected sense. That work assumes that packets are dropped independently between all time steps. However we give a completely different point of view. On the one hand, it is not required that the packets are dropped independently but just that the information gain πg, defined to be the limit of the ratio of the number of received packets n during N time steps as N goes to infinity, exists. On the other hand, we show that for any given πg, as long as πg> 0, the estimator is stable almost surely, i.e. for any given Ε > 0, the error covariance matrix Pkis bounded by a finite matrix M, with probability 1 -Ε.We also give explicit formula for the relationship between M and Ε. We consider the case where the observation matrix is invertible.

Analysis of dynamic sensor coverage problem using Kalman filters for estimation

Abstract We introduce a theoretical framework for the dynamic sensor coverage problem for the case with multiple discrete time linear stochastic systems placed at spacially separate locations. The objective is to keep an appreciable estimate of the states of the systems at all times by deploying a few limited range mobile sensors. The sensors implement a Kalman filter to estimate the states of all the systems. In this paper we present results for a single sensor executing two different random motion strategies. Under the first strategy the sensor motion is an independent and identically distributed random process and a discrete time discrete state ergodic Markov chain under the second strategy. For both these strategies we give conditions under which a single sensor fails or succeeds to solve the dynamic coverage problem. We also demonstrate that the conditions for the first strategy are a special case of the main result for the second strategy.

Polyhedral cone invariance applied to rendezvous of multiple agents

In this paper, we pose the N-scalar agent rendezvous as a polyhedral cone invariance problem in the N dimensional phase space. The underlying dynamics of the agents are assumed to be linear. We derive a condition for positive invariance for polyhedral cones. Based on this condition, we demonstrate that the problem of determining a certificate for rendezvous can be stated as a convex feasibility problem. Under certain rendezvous requirements, we show that there are no robust closed-loop linear solutions that satisfy the invariance conditions. We show that the treatment of the rendezvous problem on the phase plane can be extended to the case where agent dynamics are non-scalar.

Cone invariance and rendezvous of multiple agents

Abstract In this article is presented a dynamical systems framework for analysing multi-agent rendezvous problems and characterize the dynamical behaviour of the collective system. Recently, the problem of rendezvous has been addressed considerably in the graph theoretic framework, which is strongly based on the communication aspects of the problem. The proposed approach is based on the set invariance theory and focusses on how to generate feedback between the vehicles, a key part of the rendezvous problem. The rendezvous problem is defined on the positions of the agents and the dynamics is modelled as linear first-order systems. These algorithms have also been applied to non-linear first-order systems. The rendezvous problem in the framework of cooperative and competitive dynamical systems is analysed that has had some remarkable applications to biological sciences. Cooperative and competitive dynamical systems are shown to generate monotone flows by the classical Muller—Kamke theorem, which is analysed using the set invariance theory. In this article, equivalence between the rendezvous problem and invariance of an appropriately defined cone is established. The problem of rendezvous is cast as a stabilization problem, with a the set of constraints on the trajectories of the agents defined on the phase plane. The n-agent rendezvous problem is formulated as an ellipsoidal cone invariance problem in the n-dimensional phase space. Theoretical results based on set invariance theory and monotone dynamical systems are developed. The necessary and sufficient conditions for rendezvous of linear systems are presented in the form of linear matrix inequalities. These conditions are also interpreted in the Lyapunov framework using multiple Lyapunov functions. Numerical examples that demonstrate application are also presented.

A Platform for Cooperative and Coordinated Control of Multiple Vehicles

The Caltech Multi-Vehicle Wireless Testbed (MVWT) is an experimental plat-form for investigating the increasingly important intersecting frontiers of reliable distributed computation, communication and control. The testbed consists of eight autonomous vehicles equipped with onboard sensing, communication and computation. The vehicles are underactuated and exhibit nonlinear second-order dynamics, key properties that capture the essence of similar real-world applications at the forefront of cooperative control.

Ellipsoidal cones and rendezvous of multiple agents

In this paper we use ellipsoidal cones to achieve rendezvous of multiple agents. Rendezvous of multiple agents is shown to be equivalent to ellipsoidal cone invariance and a controller synthesis framework is presented. We first demonstrate the methodology on first order LTI systems and then extend it to rendezvous of systems that are force driven.

Estimation of linear stochastic systems over a queueing network

In this paper, we consider the standard state estimation problem over a congested packet-based network. The network is modeled as a queue with a single server processing the packets. This provides a framework to consider the effect of packet drops, packet delays and bursty losses on state estimation. We use a modified Kalman Filter with buffer to cope with delayed packets. We analyze the stability of the estimates with varying buffer length and queue size. We use high order Markov chains for our analysis. Simulation examples are presented to illustrate the theory.

Dynamic Sensor Coverage with Uncertainty Feedback : Analysis Using Iterated Maps.

This paper presents an analysis of the dynamic sensor coverage problem with uncertainty feedback. We consider a simple case of two spatially separate uncertain systems 1 and 2. In an earlier paper we introduced the dynamic sensor coverage problem and gave two stochastic sensor motion algorithms to solve the problem. We take a deterministic approach in this paper, the sensor decides to measure system 1 or 2 based on the relative uncertainty of its estimates of the states of the two systems. Error covariance is used as a metric for uncertainty of estimates. Based on the sensor measurements the error covariance evolves according to the Lyapunov or the Riccati map. The uncertainty space is partitioned and each partition has a different sensor motion decision associated with it. For a certain class of partitions we prove the existence and local stability of a unique periodic steady state orbit. We prove global stability for a scalar special case. We also show by way of an example that by changing certain parameters in these partitions stable orbits of higher periods can be obtained. Implications of this work and comparisons with existing work in the sensor scheduling and sensor coverage literature are also presented. In the end we present a discussion on future extensions of this work. Simulation examples are provided to illustrate the main concepts