Harish K. Pillai

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,

Lossless and Dissipative Distributed Systems

n \ge 3

Decomposable subspaces, linear sections of Grassmann varieties, and higher weights of Grassmann codes

. 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.

On the Number of Linear Feedback Shift Registers With a Special Structure

This paper deals with linear shift-invariant distributed systems. By this we mean systems described by constant coefficient linear partial differential equations. We define dissipativity with respect to a quadratic differential form, i.e., a quadratic functional in the system variables and their partial derivatives. The main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate. The proof of this result involves the construction of the dissipation rate. We show that this problem can be reduced to Hilberts 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions.

A parametrisation for dissipative behaviours ― the matrix case

We consider the question of determining the maximum number of points on sections of Grassmannians over finite fields by linear subvarieties of the Plucker projective space of a fixed codimension. This corresponds to a known open problem of determining the complete weight hierarchy of linear error correcting codes associated to Grassmann varieties. We recover most of the known results as well as prove some new results. A basic tool used is a characterization of decomposable subspaces of exterior powers, that is, subspaces in which every nonzero element is decomposable. Also, we use a generalization of the Griesmer-Wei bound that is proved here for arbitrary linear codes.

On homomorphisms of n-D behaviors

Given a primitive polynomial , of degree , we deal with the problem of finding the number of possible linear feedback shift register realizations, with m-input m-output delay elements, such that the corresponding characteristic polynomial is . We show the equivalence between these realizations and a set of specially structured matrices. Furthermore, the number of realizations is computed for some special cases.

A parametrization for dissipative behaviors

We study linear, time-invariant dynamical systems that are dissipative with respect to a generalised power defined by a quadratic differential form. We address several cases, in an increasing order of complexity, and show how dissipative systems can be parametrised. In this process, we also establish a number of results for polynomial matrices that are of independent interest. The present article is a generalisation of our earlier work that dealt with single-input single-output dissipative systems (Pendharkar, I., and Pillai, H.K. (2004), ‘A Parametrization for Dissipative Behaviours’, Systems and Control Letters, 51, 123–132).

Voltage Regulation of a Boost Converter in Discontinuous Conduction Mode: A Simple Robust Adaptive Feedback Controller

Different modeling procedures applied to a physical system may result in behaviors which are distinct but nevertheless share many structural properties. Such behaviors are isomorphic in a sense which we formalize and characterize in this paper. More generally, we introduce a natural notion of homomorphisms between behaviors of multidimensional systems, generalizing recent work of Fuhrmann. A generalization of strict system equivalence (in the sense of Fuhrmann) is shown to describe the relationship between generalized state-space descriptions in the nD case.

Representation Formulae for Discrete 2D Autonomous Systems

The behavioral equivalent of single input single output (SISO) systems are behaviors with two manifest variables. Passive SISO systems can, therefore, be viewed as J-dissipative behaviors with two manifest variables. Here the special matrix J defines a QDF that captures the passivity property of SISO systems. In this paper, we investigate more general QDFs QΦs induced by some operator Φ. These QDFs define some relation between the input, the output and their derivatives of a SISO system. We characterize all behaviors that are dissipative with respect to the prescribed QDF QΦ. In fact, we parametrize all the behaviors dissipative with respect to QΦ in terms of those dissipative with respect to the special QDF QJ induced by the matrix J. Similar results can also be given for lossless systems.

Systems with sector bound nonlinearities: A behavioral approach

Ideal switches in power converters are typically implemented using unidirectional semiconductor devices that may lead to a new operation mode generically called discontinuous conduction mode (DCM). The DCM arises when the ripple, that is, sustained oscillations of small amplitude, is large enough to cause the polarity of the signal (current or voltage) applied to the switch to reverse. Due to the presence of diodes, switches are assumed to operate unidirectionally, but in DCM this unidirectionality assumption is violated. In classicalconverter topologies, DCM appears very frequently in low load operating modes. More interestingly, to achieve high performance some new converters are purposely designed to operate all the time in DCM [1].