It is well known that a convolutional code can be viewed as a linear system over a finite field. In this paper we develop this viewpoint for convolutional codes using several recent innovations from the systems theory literature. In particular we define codes as behaviors of a set of compact support time trajectories over a vector space. We also consider several different representations of codes, in particular generalized first order representations. As an application of these ideas, we present a BCH construction technique for convolutional codes that yields optimal high rate codes.

The article reviews dierent denitions for a convolutional code which can be found in the literature. The algebraic dierences between the denitions are worked out in detail. It is shown that bi-innite support systems are dual to nite-support systems under Pontryagin duality. In this duality the dual of a controllable system is observable and vice versa. Uncontrollability can occur only if there are bi-innite support trajectories in the behavior, so nite and half-innite-support systems must be controllable. Unobservability can occur only if there are nite support trajectories in the behavior, so bi-innite and half-innite-support systems must be observable. It is shown that the dierent denitions for convolutional codes are equivalent if one restricts attention to controllable and observable codes. Keywords: Convolutional codes, linear time-invariant systems, behavioral system theory. 1 Introduction It is common knowledge that there is a close connection between linear syst...

This contribution deals with the synthesis of supervisory control for hybrid systems \Sigma with discrete external signals. Such systems are in general neither l- complete nor representable by finite state machines. We find the strongest l-complete approximation (abstraction) \Sigma l for \Sigma, represent it by a finite state machine, and investigate the control problem for the approximation. If a solution exists, we synthesize the maximally permissive supervisor for \Sigma l . We show that it also solves the control problem for the hybrid system \Sigma. If no solution exists, approximation accuracy can be increased by computing the strongest k-complete abstraction \Sigma k , k ? l. Most of this paper is set within the framework of Willems' behavioural systems theory. 1 Introduction The topic of this paper is supervisory control of time invariant hybrid systems with discrete external (input and output) signals. Roughly speaking, the external behaviour (the set of external signals) o...

The problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative. Our formulation and solution of this problem is completely representation-free. The system dynamics are specified by a differential behavior, and the performance is specified through a quadratic differential form. We view control as interconnection: a controller constrains a distinguished set of system variables, the control variables. The resulting behavior of the to-be-controlled variables is called the controlled behavior. The constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior, obtained by setting the control variables equal to zero, and the plant behavior, obtained by leaving the control variables unconstrained. The main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications. The essential requirement is a coupling condition, an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior.

Switched linear systems exhibit a continuous state evolving along the continuous flow of time according to linear time invariant differential equations. Furthermore, a discrete interface to the environment is provided, acting on input signals by switching between a finite number of differential equations and generating output signals when the continuous state crosses certain boundaries. We suggest a conservative approximation scheme based on sampling, state partitioning and-completion realized by a finite past induced state machine. The control problem is investigated on the approximation level. If a solution exists, it also solves the problem for the switched linear system.

In this thesis we take a detailed look at the algebraic structure of convolutional and quasi-cyclic codes using the tools and methods of linear systems theory. Let F q be a finite field with q elements. In particular, we define convolutional codes as linear, right shift invariant, compact support behaviors in (F n ) Z+ . We then examine the concepts of observability, controllability, and minimality for convolutional codes as defined above. We show how convolutional codes are dual to the class of autoregressive behaviors. We compare compact support convolutional codes to non-compact support convolutional codes. In addition, we derive first order representations of convolutional codes on a purely module theoretic. We also examine the properties of these representations and give conditions for observability and minimality. Using the systems theoretic structure of convolutional codes we present two code constructions. For the first one we choose n; k; q and ffi 2 Z+ , such that q ffi ...

Recently a smooth compactification of the space of linear systems with n states, m inputs and p outputs has been discovered. In this paper we obtain a concrete interpretation of this compactification as a space of discrete-time behaviors. We use both homogeneous polynomial representations and generalized first-order representations, and provide a realization theory to link these to each other.

In this dissertation some of the theory of multidimensional convolutional codes is developed. Given a finite field, F, consider the polynomial ring R = F[z 1 ; :::; z m ] in m indeterminates over F. An m-dimensional convolutional code of length n is an R-submodule of R n . With this definition, many results and techniques from commutative algebra and algebraic geometry are available for the study of m-dimensional convolutional codes. These connections are explored in this dissertation (especially in Chapter 3). Much attention in the literature has been given to 1-dimensional convolutional codes, a slight amount to 2-dimensional convolutional codes, and almost none to higher-dimensional convolutional codes. It is worth noting that m-dimensional convolutional codes are a nontrivial generalization of the 1-dimensional case. There are interesting differences between the 1- and 2-dimensional cases, and again between the 2- and 3-dimensional cases. Many of these differences are con...

this article, we give an introduction to symbolic dynamics and then discuss some common themes in coding theory, automata theory and system theory. Although these subjects have grown up somewhat independently, there is a strong connection among them. In particular, they all study systems, representations of systems, and transformations from one system to another. For additional reading on the connections among these subjects, we refer the reader to the Mutilingual Dictionary, hereafter referred to as the Dictionary, which appears in this volume. For ease of reading, most of the terms that we use are defined within this article.

This contribution treats the estimation of reachable states for time invariant hybrid systems. Using the framework provided by Willems' behavioural systems theory, we suggest a method based on l-complete approximations, which can be realized by finite state machines. The approximating behaviour is a superset of the original behaviour. Hence, the estimate of reachable states based on an l-complete approximation can be shown to be conservative, i. e. the exact set of reachable states is guaranteed to be contained in the estimate. Because of this property our method is adequate for verification tasks where the state variable has to remain within a certain specification. KEYWORDS Hybrid systems, reachable states, verification, behavioural approach, l-complete approximations. INTRODUCTION Within the scope of this paper, a hybrid state space is the product of a finite set and an R vector space. The considered hybrid systems are discrete time state systems with hybrid state space. It is ...