Ishan Pendharkar

A parametrisation for dissipative behaviours ― the matrix case

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).

A parametrization for dissipative behaviors

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

The class of nonlinear dynamical systems obtained by interconnection of a linear, time-invariant dynamical system and a memoryless, time-invariant, sector-bound nonlinearity has been widely studied in literature. In this paper it is shown that many important results for this class of systems can be elegantly unified using behavioral theory of dynamical systems. Systems with slope-restricted nonlinearities are also considered.

Kalman–Yakubovich lemma in the behavioural setting

In this paper, the Kalman–Yakubovich lemma is formulated and proved in the behavioural theoretic setting. The behavioural framework is used to generalize the lemma, for behaviours with two manifest variables, with respect to supply rates that are quadratic differential forms (QDFs).

Vector-exponential time-series modeling for polynomial J-spectral factorization

An iterative algorithm to perform the J-spectral factorization of a para-Hermitian matrix is presented. The algorithm proceeds by computing a special kernel representation of an interpolant for a sequence of points and associated directions determined from the spectral zeroes of the to-be-factored matrix.

Kalman-Yakubovich Lemma in the Behavioral Setting

Abstract In this paper, the Kalman-Yakubovich lemma is formulated and proved in the behavioral-theoretic setting. It is shown that such a formulation yields a physically appealing interpretation of the lemma. This formulation can be used to generalize the lemma with respect to supply rates defined by Quadratic Differential Forms (QDFs).

On the construction of quadratically stable switched linear systems with multiple component systems

We consider the stability of switched linear systems with multiple component systems of the n-th order. Using ideas from dissipative dynamical systems we construct a set of LTI systems sharing a quadratic Lyapunov function. Switched systems composed of subsystems from this set are exponentially stable for arbitrary switching sequences.

Interpolation with bilinear differential forms

We present a recursive algorithm for modeling with bilinear differential forms. We discuss applications of this algorithm for interpolation with symmetric bivariate polynominals, and for computing storage functions for autonomous systems.