Classical Linear System Theory Research Articles

Coherent observers for linear quantum stochastic systems

The theory of observers is a basic part of classical linear system theory. The purpose of this paper is to develop a theory of coherent observers for linear quantum systems. We provide a class of coherent quantum observers, which track the observables of a linear quantum stochastic system in the sense of mean values, independent of any additional quantum noise in the observer. We prove that there always exists such a coherent quantum observer described by quantum stochastic differential equations in the Heisenberg picture, and show how it can be designed to be consistent with the laws of quantum mechanics. We also find a lower bound for the mean squared estimation error due to the uncertainty principle. In addition, we explore the quantum correlations between a linear quantum plant and the corresponding coherent observer. It is shown that considering a joint plant–observer Gaussian quantum system, entanglement can be generated under the condition that appropriate coefficients of the coherent quantum observer are chosen, and this issue is illustrated in an example. These results pave the way towards observer-based quantum control.

State-space Approach to Pricing Design in OSNR Nash Game

The static nature of the noncooperative power control game model in optical networks makes it difficult to study and design an appropriate pricing scheme. In this paper, we derive a first-order best response dynamics from the game-theoretical model and formulate a general multi-input and multi-output (MIMO) state-space model. We use classical linear system theory to explain the controllability of the pricing and the observability of the power states. We use the output regulator theory to design a pricing policy for the network for a given optical signal-to-noise ratio (OSNR) target.

On analysis of limit cycles in positioning systems near Striebeck velocities

The paper deals with the analysis of friction induced oscillatory behavior in mechanical control systems at non-zero velocities. It is considered that in the positioning system under investigation the mechanical plant is controlled by linear control algorithm and the reference position trajectory is time varying. To perform the limit cycle analysis near the Striebeck velocity, a novel linearly parameterized friction model is introduced for the separation of high and low velocity regimes (under and over the Striebeck velocity). Based on this model, the limit cycles that occurs in mechatronic systems near Striebeck velocities can be predicted and studied using techniques from classical linear system theory.

A Multi-dimensional Systems Theory Framework for Binary Mathematical Morphology

Binary mathematical morphology is a set theoretical approach to multidimensional signal processing. It enables extraction of shape features and is thus a well known and successfully applied kind of operation for image processing and recognition tasks. Nevertheless, a system theoretical treatment of these operations seems to be difficult, due to its mathematical origin of integral geometry. A system theoretical description of binary mathematical morphology is given in the present paper that allows to design realizations of binary morphological operations following methods from classical linear systems theory in a very favorable way. The advantage of the proposed correspondence description is exemplary shown in the development of a matching technique for shape varying object clusters.

Phase-type distributions and representations: Some results and open problems for system theory

In this paper we consider phase-type distributions. These distributions correspond to the random hitting time of an absorbing Markov chain. They are used for modelling various random times, in particular, those which appear in manufacturing systems as processing times, times to failure, repair times, etc. The Markovian nature of these distributions allows the use of very efficient matrix based computer methods for performance evaluation. In this paper we give a system theory oriented introduction to phase-type distributions. We concentrate mainly on the representation problem which consists of finding a Markov chain associated with some phase-type distribution. This is a realization problem in the sense of system theory with a lot of links with the classical linear system theory but also with a number of constraints which make the problem harder but more interesting. Indeed this problem has strong connections with the positive realization problem in control theory. The paper recalls known results, gives some new results, and points out the main remaining problems.

Theoretical properties of recursive neural networks with linear neurons

Recursive neural networks are a powerful tool for processing structured data, thus filling the gap between connectionism, which is usually related to poorly organized data, and a great variety of real-world problems, where the information is naturally encoded in the relationships among the basic entities. In this paper, some theoretical results about linear recursive neural networks are presented that allow one to establish conditions on their dynamical properties and their capability to encode and classify structured information. A lot of the limitations of the linear model, intrinsically related to recursive processing, are inherited by the general model, thus establishing their computational capabilities and range of applicability. As a byproduct of our study some connections with the classical linear system theory are given where the processing is extended from sequences to graphs.

Max-plus algebra and system theory: Where we are and where to go now

More than sixteen years after the beginning of a linear theory for certain discrete event systems in which max-plus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison with classical linear system theory, there are areas which are practically untouched, mostly because the corresponding mathematical tools are yet to be fabricated. This is the case of the geometric approach of systems which is known, in the classical theory, to provide another important insight to system-theoretic and control-synthesis problems, beside the algebraic machinery. A preliminary discussion of geometric aspects in the max-plus algebra and their use for system theory is proposed in the last part of the paper.

Max-Plus Algebra and System Theory: Where We Are and Where to Go Now

More than sixteen years after the beginning of a linear theory for certain discrete event systems in which max-plus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison with classical linear system theory, there are areas which are practically untouched, mostly because the corresponding mathematical tools are yet to be fabricated. This is the case of the geometric approach of systems which is known, in the classical theory. to provide another important insight to system-theoretic and control-synthesis problems. beside the algebraic machinery. A preliminary discussion of geometric aspects in the max-plus algebra and their use for system theory is proposed in the last part of the paper.

Synchronization and Linearity: An Algebra for Discrete Event Systems

This book proposes a unified mathematical treatment of a class of 'linear' discrete event systems, which contains important subclasses of Petri nets and queuing networks with synchronization constraints. The linearity has to be understood with respect to nonstandard algebraic structures, e.g. the 'max-plus algebra'. A calculus is developed based on such structures, which is followed by tools for computing the time behaviour to such systems. This algebraic vision lays the foundation of a bona fide 'discrete event system theory', which is shown to parallel the classical linear system theory in several ways.

The internal resonance relations in the pause states of a nonlinear LCR circuit

Electrical current i( t) ( t is time) responses, having pauses, and named pause states, of an LCR circuit with a nonlinear (ferro- electric) capacitor C, whose voltage/charge characteristic v c( q) is not strictly prescribed in the analysis, to a periodic rectangular- wave input voltage wave v( t) are considered. The pauses of the current function are the intervals where the function oscillates, but are relatively small. There is a resonance relation between the amplitudes of the pause and nonpause oscillations in the pause states which are, in the usual sense, nonresonant states, since the extreme value of the nonpause oscillations is nonresonant with respect to the input. The pause oscillations may be considered as a local time, resonantly suppressed response which is an extension of the resonance situations which are traditionally considered. There is a specific distribution of the “potential” and “kinetic” energies of the oscillations, between the pause and nonpause intervals. We have here both a simple feature of a nonlinear response, which is rare, and a qualitatively interesting oscillatory state which is relevant, in many details, to both the nonlinear and linear versions of the system. (The relevant analysis is missing in the classical linear system theory which is almost completely devoted to the topics of spectrum and harmonic superposition or state-space equations which are not the most suitable tools for the analysis of complicated waveforms.) The system-theoretic interest in the study of the states is not limited, however, by the topic of the resonance relation, as is finally discussed. The purpose of the work is to introduce the concept of the pause state as a concept in system theory.

Approximation procedures for the optimal control of bilinear and nonlinear systems

Optimal control problems for bilinear systems are studied and solved with a view to approximating analogous problems for general nonlinear systems. For a given bilinear optimal control problem, a sequence of linear problems is constructed, and their solutions are shown to converge to the desired solution. Also, the direct solution to the Hamilton-Jacobi equation is analyzed. A power-series approach is presented which requires offline calculations as in the linear case (Riccati equation). The methods are compared and illustrated. Relations to classical linear systems theory are discussed.