Sanjay P. Bhat

Bio: Sanjay P. Bhat is an academic researcher from Tata Consultancy Services. The author has contributed to research in topics: Lyapunov function & Lyapunov equation. The author has an hindex of 26, co-authored 89 publications receiving 8327 citations. Previous affiliations of Sanjay P. Bhat include Harvard University & Indian Institute of Technology Bombay.

Finite-Time Semistability Theory with Applications to Consensus Protocols in Dynamical Networks

Abstract: This paper focuses on semistability and finite-time stability analysis and synthesis of systems having a continuum of equilibria. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. In this paper, we merge the theories of semistability and finite-time stability to develop a rigorous framework for finite-time semistability. In particular, finite-time semistability for a continuum of equilibria of continuous autonomous systems is established. Continuity of the settling-time function as well as Lyapunov and converse Lyapunov theorems for semistability are also developed. In addition, necessary and sufficient conditions for finite-time semistability of homogeneous systems are addressed by exploiting the fact that a homogeneous system is finite-time semistable if and only if it is semistable and has a negative degree of homogeneity. Finally, we use these results to develop a general framework for designing semistable protocols in dynamical networks for achieving coordination tasks in finite time.

Semistability theory for differential inclusions with applications to consensus problems in dynamical networks with switching topology

Abstract: This paper focuses on semistability and finite-time semistability for discontinuous dynamical systems. Semistability is the property whereby the solutions of a dynamical system converge to Lyapunov stable equilibrium points determined by the system initial conditions. Using these results we develop a framework for designing semistable protocols in dynamical networks with switching topologies. Specifically, we present distributed nonlinear static and dynamic output feedback controller architectures for multiagent network consensus with dynamic communication topologies.

Energy equipartition and the emergence of damping in lossless systems

Abstract: The principle of equipartition of energy is usually viewed as a statistical result formulated in terms of the probability distribution of configurations, that is, entropy. In this paper we use deterministic linear systems techniques to analyze the vibrational energy of systems of undamped coupled oscillators with identical coupling. Our approach is based on time averaging of squared outputs of the system and thus avoids both recurrence and statistical arguments. We first consider a single undamped oscillator and show that the time averaged potential energy and the time-averaged kinetic energy converge to the same value. Next, we consider a collection of n identical undamped oscillators with lossless coupling. As in the case of a single oscillator, equipartition of energy holds for the total system kinetic and potential energies. We focus on the equipartition of oscillator energy, that is, the equal distribution of energy among oscillators, regardless of the form of energy. We derive expressions for the transient and steady-state behavior of each oscillator.

Continuous, bounded, finite-time stabilization of the translational and rotational double integrators

Abstract: A class of bounded, continuous, time-invariant, finite-time stabilizing feedback laws is given for the double integrator. These controllers are modified to obtain finite-time stabilizing feedbacks for the rotational double integrator that do not exhibit "unwinding".

Second-Order Systems with Singular Mass Matrix and an Extension of Guyan Reduction

Abstract: The set of consistent initial conditions for a second-order system with singular mass matrix is obtained. In general, such a system can be decomposed (i.e., partitioned) into three coupled subsystems of which the first is algebraic, the second is a regular system of first-order differential equations, and the third is a regular system of second-order differential equations. Under specialized conditions, these subsystems are decoupled. This result provides an extension of Guyan reduction to include viscous damping.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Finite-Time Stability of Continuous Autonomous Systems

Abstract: Finite-time stability is defined for equilibria of continuous but non-Lipschitzian autonomous systems. Continuity, Lipschitz continuity, and Holder continuity of the settling-time function are studied and illustrated with several examples. Lyapunov and converse Lyapunov results involving scalar differential inequalities are given for finite-time stability. It is shown that the regularity properties of the Lyapunov function and those of the settling-time function are related. Consequently, converse Lyapunov results can only assure the existence of continuous Lyapunov functions. Finally, the sensitivity of finite-time-stable systems to perturbations is investigated.

Nonlinear Feedback Design for Fixed-Time Stabilization of Linear Control Systems

Abstract: Two types of nonlinear control algorithms are presented for uncertain linear plants. Controllers of the first type are stabilizing polynomial feedbacks that allow to adjust a guaranteed convergence time of system trajectories into a prespecified neighborhood of the origin independently on initial conditions. The control design procedure uses block control principles and finite-time attractivity properties of polynomial feedbacks. Controllers of the second type are modifications of the second order sliding mode control algorithms. They provide global finite-time stability of the closed-loop system and allow to adjust a guaranteed settling time independently on initial conditions. Control algorithms are presented for both single-input and multi-input systems. Theoretical results are supported by numerical simulations.

Coverage control for mobile sensing networks

Abstract: This paper presents control and coordination algorithms for groups of vehicles. The focus is on autonomous vehicle networks performing distributed sensing tasks where each vehicle plays the role of a mobile tunable sensor. The paper proposes gradient descent algorithms for a class of utility functions which encode optimal coverage and sensing policies. The resulting closed-loop behavior is adaptive, distributed, asynchronous, and verifiably correct.