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Rational semimodules over the max-plus semiring and geometric approach to discrete event systems

Stéphane Gaubert, Ricardo Katz (2004)

Kybernetika

We introduce rational semimodules over semirings whose addition is idempotent, like the max-plus semiring, in order to extend the geometric approach of linear control to discrete event systems. We say that a subsemimodule of the free semimodule 𝒮 n over a semiring 𝒮 is rational if it has a generating family that is a rational subset of 𝒮 n , 𝒮 n being thought of as a monoid under the entrywise product. We show that for various semirings of max-plus type whose elements are integers, rational semimodules...

Reachability and observability of linear systems over max-plus

Michael J. Gazarik, Edward W. Kamen (1999)

Kybernetika

This paper discusses the properties of reachability and observability for linear systems over the max-plus algebra. Working in the event-domain, the concept of asticity is used to develop conditions for weak reachability and weak observability. In the reachability problem, residuation is used to determine if a state is reachable and to generate the required control sequence to reach it. In the observability problem, residuation is used to estimate the state. Finally, as in the continuous-variable...

Realization of nonlinear input-output equations in controller canonical form

Arvo Kaldmäe, Ülle Kotta (2018)

Kybernetika

In this paper necessary and sufficient conditions are given which guarantee that there exists a realization of a set of nonlinear higher order differential input-output equations in the controller canonical form. Two cases are studied, corresponding respectively to linear and nonlinear output functions. The conditions are formulated in terms of certain sequence of vector spaces of differential 1-forms. The proofs suggest how to construct the transformations, necessary to obtain the specific state...

Realization theory for linear and bilinear switched systems: A formal power series approach

Mihály Petreczky (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The paper represents the first part of a series of papers on realization theory of switched systems. Part I presents realization theory of linear switched systems, Part II presents realization theory of bilinear switched systems. More precisely, in Part I necessary and sufficient conditions are formulated for a family of input-output maps to be realizable by a linear switched system and a characterization of minimal realizations is presented. The paper treats two types of switched systems. The...

Realization theory for linear and bilinear switched systems: A formal power series approach

Mihály Petreczky (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is the second part of a series of papers dealing with realization theory of switched systems. The current Part II addresses realization theory of bilinear switched systems. In Part I [Petreczky, ESAIM: COCV, DOI: 10.1051/cocv/2010014] we presented realization theory of linear switched systems. More precisely, in Part II we present necessary and sufficient conditions for a family of input-output maps to be realizable by a bilinear switched system, together with a characterization of minimal...

Realization theory for linear and bilinear switched systems: A formal power series approach

Mihály Petreczky (2011)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is the second part of a series of papers dealing with realization theory of switched systems. The current Part II addresses realization theory of bilinear switched systems. In Part I [Petreczky, ESAIM: COCV, DOI: 10.1051/cocv/2010014] we presented realization theory of linear switched systems. More precisely, in Part II we present necessary and sufficient conditions for a family of input-output maps to be realizable by a bilinear switched system, together with a characterization of minimal...

Realization theory for linear and bilinear switched systems: A formal power series approach

Mihály Petreczky (2011)

ESAIM: Control, Optimisation and Calculus of Variations

The paper represents the first part of a series of papers on realization theory of switched systems. Part I presents realization theory of linear switched systems, Part II presents realization theory of bilinear switched systems. More precisely, in Part I necessary and sufficient conditions are formulated for a family of input-output maps to be realizable by a linear switched system and a characterization of minimal realizations is presented. The paper treats two types of switched systems. The...

Reduction and transfer equivalence of nonlinear control systems: Unification and extension via pseudo-linear algebra

Ülle Kotta, Palle Kotta, Miroslav Halás (2010)

Kybernetika

The paper applies the pseudo-linear algebra to unify the results on reducibility, reduction and transfer equivalence for continuous- and discrete-time nonlinear control systems. The necessary and sufficient condition for reducibility of nonlinear input-output equation is presented in terms of the greatest common left factor of two polynomials describing the behaviour of the ‘tangent linearized system’ equation. The procedure is given to find the reduced (irreducible) system equation that is transfer...

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