A complete account is given of the theory of so-called dissipative dynamical systems. The concept of dissipativeness is defined as a general input-output property which includes, as notable special cases, passivity and other properties related to finite-gain. The aim is to treat input-output and state properties side-by-side with emphasis on exploring connections between them. The key connection is that a dissipative system in general possesses a set of energy-like functions of the state. The properties of these functions are studied in some detail. It is demonstrated that this connection represents a direct generalization of the well-known Kalman-Yakubovich lemma to arbitrary dynamical systems. Applications to stability theory and passive system synthesis are briefly discussed for non-linear systems.

Dissipative Dynamical Systems: Basic Input-Output and State Properties

Adaptive nonlinear filters equipped with polynomial models of nonlinearity are explained. The polynomial systems considered are those nonlinear systems whose output signals can be related to the input signals through a truncated Volterra series expansion or a recursive nonlinear difference equation. The Volterra series expansion can model a large class of nonlinear systems and is attractive in adaptive filtering applications because the expansion is a linear combination of nonlinear functions of the input signal. The basic ideas behind the development of gradient and recursive least-squares adaptive Volterra filters are first discussed. Adaptive algorithms using system models involving recursive nonlinear difference equations are then treated. Such systems may be able to approximate many nonlinear systems with great parsimony in the use of coefficients. Also discussed are current research trends and new results and problem areas associated with these nonlinear filters. A lattice structure for polynomial models is described. >

Adaptive polynomial filters

Most magnetic resonance imaging sequences employ field gradients and amplitude modulated RF pulses to excite only those spins lying in a specific plane. The fidelity of the resulting magnetization distribution is crucial to overall image resolution. Conventional RF-pulse design techniques rely on the small tip-angle approximation to Bloch's equation, which is inadequate for the design of 90° and 180° pulses. This paper demonstrates the existence of a selective pulse, and provides a sound mathematical and computational basis for pulse design. It is shown that the pulses are optimal in the class of piecewise continuous functions of duration T. An optimal pulse is defined as the pulse on the interval that achieves a magnetization profile "closest" to the desired distribution. Optimal control theory provides the mathematical basis for the new pulse design technique. Computer simulations have verified the efficacy of the 90° and the 180° inversion and "pancake-flip" optimal pulses.

Optimal Control Solutions to the Magnetic Resonance Selective Excitation Problem

In this paper, we investigate the stochastic stabilization problem for a class of bilinear continuous time-delay uncertain systems with Markovian jumping parameters. Specifically, the stochastic bilinear jump system under study involves unknown state time-delay, parameter uncertainties, and unknown nonlinear deterministic disturbances. The jumping parameters considered here form a continuous-time discrete-state homogeneous Markov process. The whole system may be regarded as a stochastic bilinear hybrid system that includes both time-evolving and event-driven mechanisms. Our attention is focused on the design of a robust state-feedback controller such that, for all admissible uncertainties as well as nonlinear disturbances, the closed-loop system is stochastically exponentially stable in the mean square, independent of the time delay. Sufficient conditions are established to guarantee the existence of desired robust controllers, which are given in terms of the solutions to a set of either linear matrix inequalities (LMIs), or coupled quadratic matrix inequalities. The developed theory is illustrated by numerical simulation.

/pdf/on-stabilization-of-bilinear-uncertain-time-delay-stochastic-3mo3tpajeu.pdf

On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters

https://www.dtc.umn.edu/s/resources/gg-se.pdf

A bibliography on nonlinear system identification

Recently, attention has been focused on the class of bilinear systems, both for its applicative interest and intrinsic simplicity. In fact, it appears that many important processes, not only in engineering, but also in biology, socio-economics, and ecology, may be modeled by bilinear systems. Moreover, since their nonlinearity is due to products between input and state variables, this class frequently may be studied by techniques similar to those employed for linear systems. This work is intended to motivate the interest of bilinear systems and to present the current state of research in its various aspects. After an introductory section, in which theoretical and applicative aspects of bilinear systems are enlightened, four other sections follow, respectively, devoted to structural properties, mathematical models, identification and optimization. In a final section, some concluding remarks are made on still open problems and possible trends for future research.

G. DiPillo

Papers

Bilinear systems: An appealing class of "nearly linear" systems in theory and applications