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25
The Behavior of Convolutional Codes
, 1995
"... 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 ..."
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Cited by 47 (16 self)
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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.
State maps for linear systems
 SIAM J. Control Opt
, 1997
"... Abstract. Modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differentialalgebraic equations. As such, statespace models are not a natural starting point for modeling, while they have utmost importance in the simulation and contro ..."
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Cited by 20 (10 self)
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Abstract. Modeling of physical systems consists of writing the equations describing a phenomenon and yields as a result a set of differentialalgebraic equations. As such, statespace models are not a natural starting point for modeling, while they have utmost importance in the simulation and control phase. The paper addresses the problem of computing state variables for systems of linear differentialalgebraic equations of various forms. The point of view from which the problem is considered is the behavioral one, as put forward in [J. C. Willems, Automatica J. IFAC, 22 (1986),
Connections between Linear Systems and Convolutional Codes
 Codes, Systems, and Graphical Models
, 2000
"... 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 biinnite support systems are dual to nitesupport systems under Pontryagin duality. In this duality the dual ..."
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Cited by 19 (4 self)
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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 biinnite support systems are dual to nitesupport 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 biinnite support trajectories in the behavior, so nite and halfinnitesupport systems must be controllable. Unobservability can occur only if there are nite support trajectories in the behavior, so biinnite and halfinnitesupport 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 timeinvariant systems, behavioral system theory. 1 Introduction It is common knowledge that there is a close connection between linear syst...
Supervisory Control Of Hybrid Systems Via lComplete Approximations
, 1998
"... 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 lcomplete approximation (abstraction) \Sigma l for \Sigma, re ..."
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Cited by 19 (13 self)
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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 lcomplete 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 kcomplete 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...
Synthesis of Dissipative Systems Using Quadratic Differential Forms: Part I
 IEEE TRANSACTIONS ON AUTOMATIC CONTROL
, 2002
"... 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 representationfree. The system dynamics are specified by a differential behavior, and the performance is specified ..."
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Cited by 17 (5 self)
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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 representationfree. 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 tobecontrolled 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.
Discrete Control Of Switched Linear Systems
 Proceedings of the European Control Conference
, 1999
"... 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 equa ..."
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Cited by 14 (5 self)
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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 andcompletion 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.
Algebraic Description And Construction Of Error Correcting Codes: A Linear Systems Point Of View
, 1997
"... In this thesis we take a detailed look at the algebraic structure of convolutional and quasicyclic 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 be ..."
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Cited by 10 (0 self)
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In this thesis we take a detailed look at the algebraic structure of convolutional and quasicyclic 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 noncompact 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 ...
Homogeneous Behaviors
, 1996
"... 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 discretetime behaviors. We use both homogeneous polynomial representations and gen ..."
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Cited by 9 (6 self)
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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 discretetime behaviors. We use both homogeneous polynomial representations and generalized firstorder representations, and provide a realization theory to link these to each other.
Multidimensional Convolutional Codes
, 1998
"... 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 mdimensional convolutional code of length n is an Rsubmodule of R n . With this definit ..."
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Cited by 8 (1 self)
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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 mdimensional convolutional code of length n is an Rsubmodule of R n . With this definition, many results and techniques from commutative algebra and algebraic geometry are available for the study of mdimensional convolutional codes. These connections are explored in this dissertation (especially in Chapter 3). Much attention in the literature has been given to 1dimensional convolutional codes, a slight amount to 2dimensional convolutional codes, and almost none to higherdimensional convolutional codes. It is worth noting that mdimensional convolutional codes are a nontrivial generalization of the 1dimensional case. There are interesting differences between the 1 and 2dimensional cases, and again between the 2 and 3dimensional cases. Many of these differences are con...
The behavioral approach to open and interconnected systems
 Control Systems Magazine
, 2007
"... During the opening lecture of the 16th IFAC ..."