Virendra Sule

Feedback Stabilization Over Commutative Rings: The Matrix Case

This paper provides a solution of the feedback stabilization problem over commutative rings for matrix transfer functions. Stabilizability of a transfer matrix is realised as local stabilizability over the entire spectrum of the ring. For stabilizable plants, certain modules generated by its fractions and that of the stabilizing controller are shown to be projective compliments of each other. In the case of rings with irreducible spectrum, this geometric relationship shows that the plant is stabilizable if and only if the above modules of the plant are projective of ranks equal to the number of inputs and the outputs. If the maxspectrum of the ring is Noetherian and of zero (Krull) dimension, then this result shows that the stabilizable plants have doubly coprime fractions. Over unique factorization domains the above stabilizability condition is interpreted in terms of matrices formed by the fractions of the plant. Certain invariants of these matrices known as elementary factors, are defined and it is shown that the plant is stabilizable if and only if these elementary factors generate the whole ring. This condition thus provides a generalization of the coprime factorizability as a condition for stabilizabilty. A formula for the class of all stabilizing controllers is then developed that generalizes the previous well-known formula in factorization theory. For multidimensional transfer functions these results provide concrete conditions for stabilizabilty. Finally, it is shown that the class of polynomial rings over principal ideal domains is an additional class of rings over which stabilizable plants always have doubly coprime fractions.

Algebraic geometric aspects of feedback stabilization

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...

Novel modeling and solution approach for repeated finite-element analysis of eddy-current systems

In this paper, we present an efficient modeling and computational scheme for a repeated solution of an eddy-current system with different values of the supply frequency as well as of the permeability and conductivity of the eddy-current region. The scheme is based on a general parametric expression obtained for the finite-element (FE) solution with the supply frequency, permeability, and conductivity as parameters. The algorithm allows for numerically efficient updating of the solution for different values of the parameters through the solution of a much smaller sparse linear system, instead of a repeated solution of the entire FE model. Moreover, if required, the solution can be computed only over a small region of interest, making the scheme ideally suited to many coupled-field problems. As an application, the scheme is applied to a typical bar-plate eddy-current system, excited by nonsinusoidal currents. The time variations of the magnetic field are computed as a superposition of responses computed for a number of harmonics. An a priori estimate for the difference between responses to two harmonics has been obtained, which can be used as a frequency-sensitivity measure to avoid computation of responses to all individual harmonics. The applicability of the approach to general transient excitations and further possible developments are identified.

AN EFFICIENT COMPUTATIONAL MODELING AND SOLUTION SCHEME FOR REPEATED FINITE ELEMENT ANALYSIS OF COMPOSITE DOMAINS WITH VARYING MATERIAL PROPERTY

Many engineering applications require efficient computation of the solution of the Poisson equation defined over one, two or three dimensions, representing a wide variety of physical systems. The most popular method used for this purpose is the finite element method (FEM). An important class of applications in modeling and simulation require repeated computation of the solution over a given geometry, for different values of the relevant material property in a region within the domain. They can arise in the areas of nondestructive testing and evaluation, field-control in electromagnetic applications, as well as computer-aided design of devices and systems. In this work, a linear algebraic study of the dependence of the coefficient (stiffness) matrix on the varying material property is used to develop a general expression of the finite element solution. Based on this, a novel modeling and simulation scheme is suggested which significantly reduces the computational cost of the repetitive solution stage, as well as the size of the numerical model to be stored. Numerical simulations are presented for a simple illustrative case and a few possible applications are suggested in different engineering fields.

A Behavioral Approach to the Control of Discrete Linear Repetitive Processes

This paper formulates the theory of linear discrete time repetitive processes in the setting of behavioral systems theory. A behavioral, latent variable model for repetitive processes is developed and for the physically defined inputs and outputs as manifest variables, a kernel representation of their behavior is determined. Conditions for external stability and controllability of the behavior are then obtained. A sufficient condition for stabilizability is also developed for the behavior and it is shown under a mild restriction that, whenever the repetitive system is stabilizable, a regular constant output feedback stabilizing controller exists. Next, a notion of eigenvalues is defined for the repetitive process under an action of a closed-loop controller. It is then shown how under controllability of the original repetitive process, an arbitrary assignment of eigenvalues for the closed-loop response can be achieved by a constant gain output feedback controller under the above restriction. These results on the existence of constant gain output feedback controllers are among the most striking properties enjoyed by repetitive systems, discovered in this paper. Results of this paper utilize the behavioral model of the repetitive process which is an analogue of the 1D equivalent model of the dynamics studied in earlier work on these processes.

State space approach to behavioral systems theory: the Dirac–Bergmann algorithm

This paper develops an approach to behavioral systems theory in which a state space representation of behaviors is utilised. This representation is a first order hybrid representation of behaviors called pencil representation. An algorithm well known after Dirac and Bergmann (DB) is shown to be central in obtaining a constraint free and observable (CFO) state space representation of a behavior. Results and criteria for asymptotic stability, controllability, inclusions and Markovianity of behaviors are derived in terms of the matrices of this representation which involve linear algebraic processes in their computation.

Steady-state frequency response for periodic systems

This paper extends the concept of steady-state frequency response, well known in the theory of linear time-invariant (LTI) systems, to linear time-varying systems with periodic coeAcients, called periodic systems. It is shown that for an internally stable periodic system there exist complete orthogonal systems of real periodic functions ff n g and fc n g called eigenfunctions, such that for the inputs fn every output of the system converges in steady state to sncn, where sn are non-negative real numbers. The set of all such numbers is called the singular frequency response of the system. In the case of LTI systems, the singular frequency response turns out to be consisting of the magnitudes of the sinusoidal frequency responses of the system. The singular frequency responsefsng is shown to be the singular spectrum of a compact operator associated with the system and has all the characteristics of the magnitude frequency response of LTI systems. A state-space realization of this operator and its adjoint leads to an alternative formulation of inverse of the singular frequency response as eigenvalues arising from a boundary value problem with periodic boundary values. # 2001 The Franklin Institute. Published by Elsevier Science Ltd. All rights reserved.

On control laws for discrete linear repetitive processes with dynamic boundary conditions

Repetitive processes are characterized by a series of sweeps, termed passes, through a set of dynamics defined over a finite duration known as the pass length. On each pass an output, termed the pass profile, is produced which acts as a forcing function on, and hence contributes to, the dynamics of the next pass profile. This can lead to oscillations in the sequence of pass profiles produced which increase in amplitude in the pass-to-pass direction and cannot be controlled by application of standard control laws. Here we give new results on the design of physically based control laws for so-called discrete linear repetitive processes which arise in applications areas such as iterative learning control.

Rigorous discrete time linearization of periodically switched circuits with respect to duty cycle perturbations

ABSTRACT This paper considers the well-known problem of deriving a linear model of dynamics of periodically switched circuits w.r.t. small perturbations in duty cycle (or switching instances) as external control inputs. A rigorous approach to this problem is developed and is shown that the linearized model is shift invariant and discrete time in nature. This is at variance with the well-known model, which is linear time invariant continuous time referred as state space averaging (SSA) model. SSA model ignores commutativity conditions in matrices of state space model due to varying parameters over intervals as well as the discrete nature of control input. The proposed method of linearization considers the problem of linearization in a neighbourhood of a periodic solution. The monodromy matrix for state transition over all phases of switching is considered to account for non-commuting matrices of parameters. Similarly discrete nature of the input changing once in every period of switching leads to the discrete model. This methodology is applicable for multiple independently switched circuits and takes into account orders of switching once the nominal periodic solution over which linearization is sought is fixed. This paper gives the detailed theory as well as illustrative examples to prove the usefulness of the proposed methodology.

One variable polynomial division approach for elliptic curve arithmetic over prime fields

This paper presents an approach for point addition and doubling on elliptic curves over finite fields Fp which is based on one variable polynomial division. This is achieved by identifying the plane Fp × Fp with the extension field Fp2 and transforming the elliptic curve equation as well as line equations arising in point addition or doubling into polynomials in one variable. Hence the intersection of the line with the curve is analogous to roots of the gcd between these polynomials. We show that this approach to arithmetic involves considerable scope for decomposition and parallel computation.