Shiva Shankar

A Behavioral Approach to Control of Distributed Systems

This paper develops a theory of control for distributed systems (i.e., those defined by systems of constant coefficient partial differential operators) via the behavioral approach of Willems. The study here is algebraic in the sense that it relates behaviors of distributed systems to submodules of free modules over the polynomial ring in several indeterminates. As in the lumped case, behaviors of distributed ARMA systems can be reduced to AR behaviors. This paper first studies the notion of AR controllable distributed systems following the corresponding definition for lumped systems due to Willems. It shows that, as in the lumped case, the class of controllable AR systems is precisely the class of MA systems. It then shows that controllable 2-D distributed systems are necessarily given by free submodules, whereas this is not the case for n-D distributed systems,

Algebraic geometric aspects of feedback stabilization

n \ge 3

Geometric Completeness of Distribution Spaces

. This therefore points out an important difference between these two cases. This paper then defines two notions of autonomous distributed systems which mimic different properties of lumped autonomous systems. Control is the process of restricting a behavior to a specific desirable autonomous subbehavior. A notion of stability generalizing bounded input--bounded output stability of lumped systems is proposed and the pole placement problem is defined for distributed systems. This paper then solves this problem for a class of distributed behaviors.

The Lattice Structure of Behaviors

This paper develops a theory of feedback stabilization for SISO transfer functions over a general integral domain which extends the well-known coprime factorization approach to stabilization. Necessary and sufficient conditions for stabilizability of a transfer function in this general setting are obtained. These conditions are then refined in the special cases of unique factorization domains (UFDs), Noetherian rings, and rings of fractions. It is shown that these conditions can be naturally interpreted geometrically in terms of the prime spectrum of the ring. This interpretation provides a natural generalization to the classical notions of the poles and zeros of a plant.The set of transfer functions is topologized so as to restrict to the graph topology of Vidyasagar [IEEE Trans. Automatic Control, AC-29 (1984), pp. 403–418], when the ring is a Bezout domain. It is shown that stability of a feedback system is robust in this topology when the ring is a UFD.This theory is then applied to the problem of sta...

An obstruction to the simultaneous stabilization of two n-D plants

This paper introduces a notion of geometric completeness for spaces of distributions, modelled after the notion of a complete variety in algebraic geometry. It is related to the following elimination problem for systems of PDE: consider the set of homogeneous solutions of a system of PDE in some space of distributions. When is the projection of this set onto some of its coordinates also the set of homogeneous solutions of a system of PDE?

Can One Control the Vibrations of a Drum

If a linear, continuous, shift invariant distributed system is considered as a (dynamical) system converting input signals to output signals, then this information is encapsulated in the impulse response or the transfer function of the system. The set of all transfer functions has the structure of a ring, corresponding to the operations of parallel and cascade connections of two systems. However, in the behavioral theory of Willems, a system is not described in terms of its input-output transformation property. Indeed, the concept of a behavior does not even need the notions of inputs and outputs and is therefore more fundamental than the classical concept of a system given by its transfer function. The question then arises as to what is the structure of the set of all behaviors. This paper argues that the relevant structure here is that of a modular lattice.

Singularities in mechanisms: the local trajectory tracking problem

This paper calculates an obstruction to the simultaneous stabilization of twon-D plants. This obstruction is topological and lies in the (singular) cohomology of a subset of ℂn that arises naturally in the problem. Given any family ofn-D plants, the vanishing of the corresponding cohomology classes of every pair of plants in the family is thus a necessary condition for the simultaneous stabilization of the family.

Regular implementation in the space of compactly supported functions

AbstractThis paper considers the problem of constructing feedback stabilizing controllers for the wave operator on

Singularities in mechanisms II—trajectory tracking in the 3-axis wrist

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