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System Identification, Approximation and Complexity
 International Journal of General Systems
, 1977
"... This paper is concerned with establishing broadlybased systemtheoretic foundations and practical techniques for the problem of system identification that are rigorous, intuitively clear and conceptually powerful. A general formulation is first given in which two order relations are postulated on a ..."
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Cited by 34 (23 self)
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This paper is concerned with establishing broadlybased systemtheoretic foundations and practical techniques for the problem of system identification that are rigorous, intuitively clear and conceptually powerful. A general formulation is first given in which two order relations are postulated on a class of models: a constant one of complexity; and a variable one of approximation induced by an observed behaviour. An admissible model is such that any less complex model is a worse approximation. The general problem of identification is that of finding the admissible subspace of models induced by a given behaviour. It is proved under very general assumptions that, if deterministic models are required then nearly all behaviours require models of nearly maximum complexity. A general theory of approximation between models and behaviour is then developed based on subjective probability concepts and semantic information theory The role of structural constraints such as causality, locality, finite memory, etc., are then discussed as rules of the game. These concepts and results are applied to the specific problem or stochastic automaton, or grammar, inference. Computational results are given to demonstrate that the theory is complete and fully operational. Finally the formulation of identification proposed in this paper is analysed in terms of Klirâ€™s epistemological hierarchy and both are discussed in terms of the rich philosophical literature on the acquisition of knowledge. 1
Interconnection of Object Specifications
 Formal Methods and Object Technology
, 1996
"... ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, and ap ..."
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Cited by 8 (2 self)
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ing yet further from reality, we might proscribe the simultaneous effect of two or more methods on an object's state; doing so, we impose a monoid structure on the fixed set of methods proper to an object class. Applying methods one after the other corresponds to multiplication in the monoid, and applying no methods corresponds to the identity of the monoid. A monoid is a set M with an associative binary operation ffl M : M \ThetaM ! M , usually referred to as `multiplication', which has an identity element e M 2 M . If M = (M; ffl M ; e M ) is a monoid, we often write just M for M, and e for e M ; moreover for m;m 0 2 M , we usually write mm 0 instead of m ffl M m 0 . For example, A , the set of lists containing elements of A, together with concatenation ++ : A \ThetaA ! A and the empty list [ ] 2 A , is a monoid. This example is especially important for the material in later sections. A monoid homomorphism is a structure preserving map between the carriers of ...
ObjectOriented Hybrid Systems of Coalgebras plus Monoid Actions
 Algebraic Methodology and Software Technology (AMAST
, 1996
"... . Hybrid systems combine discrete and continuous dynamics. We introduce a semantics for such systems consisting of a coalgebra together with a monoid action. The coalgebra captures the (discrete) operations on a state space that can be used by a client (like in the semantics of ordinary (nontempora ..."
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Cited by 5 (1 self)
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. Hybrid systems combine discrete and continuous dynamics. We introduce a semantics for such systems consisting of a coalgebra together with a monoid action. The coalgebra captures the (discrete) operations on a state space that can be used by a client (like in the semantics of ordinary (nontemporal) objectoriented systems). The monoid action captures the influence of time on the state space, where the monoids that we consider are the natural numbers monoid (N; 0; +) of discrete time, and the positive reals monoid (R0 ; 0; +) of real time. Based on this semantics we develop a hybrid specification formalism with timed method applications: it involves expressions like s:meth@ff, with the following meaning: in state s let the state evolve for ff units of time (according to the monoid action), and then apply the (coalgebraic) method meth. In this formalism we specify various (elementary) hybrid systems, investigate their correctness, and display their behaviour in simulations. We furthe...
Algebraic representation of dynamics and behavior for continuoustime linear systems
 Math. Systems Theory
, 1991
"... continuoustime systems An algebraic approach to continuoustime linear systems is presented which closely parallels the discretetime decomposable systems approach of Arbib and Manes, as well as the older k[z]module theory of linear systems of Kalman. The focal point of the presentation is a class ..."
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Cited by 1 (1 self)
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continuoustime systems An algebraic approach to continuoustime linear systems is presented which closely parallels the discretetime decomposable systems approach of Arbib and Manes, as well as the older k[z]module theory of linear systems of Kalman. The focal point of the presentation is a class of topological rings, termed R+rings, which play the same role for continuous time that k[z] does for discretetime. Each such ring R defines a class of toplogical modules, termed the (R)modules, which may be naturally identified with a class of locally equicontinuous semigroups, called the (R)semigroups. Thus, just as discretetime linear dynamics are coextensive with k[z]modules, so too are continuoustime linear dynamics coextensive with (R)modules. This identification underlies the development of a purely algebraic theory of behavior and realization for continuoustime linear systems. The specific choice of R determines the type of dynamics allowed. For example, taking R to be the ring of all measures on the nonnegative reals yields dynamics described by the class of all semigroups, while choosing R to be the ring of all L 1 measures yields dynamics whose responses vanish at infinity.