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Nonlinear

Analysis.

Theory,

Methods

&Applications, Vol. Proc. 2nd World

Pergamon

Rimed

PII:

PSEUDO EDGAR

SO362-546X(96)00143-5

DYNAMIC

CHACONt,

GISELA Universidad

tDepartamento $Departament.o t&part

amento

HYBRID

de Sistemas

SYSTEMS

DE SARRAZINSand de loa Andes,

M&la,

de Computacidn. Facultad de Iqpierk de Matembticas. Facultad de Cienciaa. de Control.

Facukad

Key words and phrases. Hybrid Dynamic Systems, time Dynamic Systems, Realization of systems.

30, No. 4. pp. 2533-2537. 1997 Congress of Nonlinear Analysrs 0 1997 Else&x Science Ltd in Great Britain. All rights reserved 0362-546X/!97 $17.00 + 0.00

FERENC

Venezuela. ornail:

SZIGETIS

ecbaconQii.ula.ve

e-mail: gcoviQciens.ula.ve de Ingenie& e-mail: szigetiQing.ula.ve

Discrete

Event

Dynamic

Systems,

Continuous

1JNTRODUCTION

Dynamic Systems (DS) whose behavior results from the interaction of continuous time processes with discrete-event processes, are called Hybrid Dynamic Systems (HYDS). These systems arise in a wide range of applications [l], which explains why a generally accepted notion of HYDS has not yet appeared. In the search for a general mathematical concept of such systems, we have previously presented a linguistic description of some continuous time processes [2]. This approach can be justified by means of the non linear realization theory of the systems [3]. In this way, we obtain a more accurate mathematical description of the dynamic interaction between the continuous device and the discrete event subsystems of the HYDS. We modeled a DS by means of the triplet (X, S, Cp), where X is the state space, S a transformation semigroup, and ip : X x S -+ X the state transition function. In a classical Continuous Time Dynamic System (C-T DS) S is the set of real numbers R, and states evolve with time, according to a set of differential equations. Following the framework developed by Ramadge and Wonham in [4] a Discrete Event Dynamic System (DEDS) can be modeled as a DS over an alphabet U (or event set), where the change of the states takes place in response to the events. In both C-T DS and DEDS theories, it is very natural to use semigroups acting over state spaces which are not everywhere defined; hence we introduce the concept of Pseudo Dynamic Systems (Ps-DS), where we suppose the existence of a partially defined semigroup action. These results are developed in Section 2. Following these preliminaries we present our main result in Section 3, concerning the concept of Pseudo Dynamic Hybrid Systems and some applications. 2.PSEUDO

DYNAMIC

SYSTEMS

At this point we give an outline of the basic results required in this paper. For a more detailed discussion we refer the reader to [2]. Let (X,S,@) be a DS as in the Introduction, and considerthe set S, = {u E S/@(z,u)!}, where the symbol “!” denotes “is defined”. Then the partial mapping Cp: X x S + X is given by aI : S, + X, O,(u) = @(z,u), u E S,. If S is a monoid with neutral element 8, and we supposethat 6 E S, for al’\ z E X, then @,q= Id, is defined everywhere. Thus we have the following definition: DEFINITION 2.1. A Pseudo Dynamic System is a DS (X, S, @), where @ is a partially defined semigroupaction, over X x S. The domain of @is the subset U {z} x S, C X x S. ZEX

2533

2534

Second World

Congress

of Nonlinear

Analysts

EXAMPLE 2.2. Let U’ denote the set of all finite strings of elements in an alphabet U, including the empty string 6. Let L be an arbitrary prefix-closed language in U’. For u E L define the sub-language L, c U’, such that v E L, iff the concatenation uv E L. Let Q : L x U’ + L, a,,(u) = uv, v E L,. Then (L, u’, a) is a Ps-DS. EXAMPLE 2.3. Let D C 9V’ be a domain, f : D + 8” a smooth vector field. Let t I-+ @t(t) = a({, t) be the complete solution of the initial value problem i(t) = f (z(t)), z(O) = .$. Hence t C) @t(t) is the solution defined over the maximal interval [O,Z’

Lihat lebih banyak...
Analysis.

Theory,

Methods

&Applications, Vol. Proc. 2nd World

Pergamon

Rimed

PII:

PSEUDO EDGAR

SO362-546X(96)00143-5

DYNAMIC

CHACONt,

GISELA Universidad

tDepartamento $Departament.o t&part

amento

HYBRID

de Sistemas

SYSTEMS

DE SARRAZINSand de loa Andes,

M&la,

de Computacidn. Facultad de Iqpierk de Matembticas. Facultad de Cienciaa. de Control.

Facukad

Key words and phrases. Hybrid Dynamic Systems, time Dynamic Systems, Realization of systems.

30, No. 4. pp. 2533-2537. 1997 Congress of Nonlinear Analysrs 0 1997 Else&x Science Ltd in Great Britain. All rights reserved 0362-546X/!97 $17.00 + 0.00

FERENC

Venezuela. ornail:

SZIGETIS

ecbaconQii.ula.ve

e-mail: gcoviQciens.ula.ve de Ingenie& e-mail: szigetiQing.ula.ve

Discrete

Event

Dynamic

Systems,

Continuous

1JNTRODUCTION

Dynamic Systems (DS) whose behavior results from the interaction of continuous time processes with discrete-event processes, are called Hybrid Dynamic Systems (HYDS). These systems arise in a wide range of applications [l], which explains why a generally accepted notion of HYDS has not yet appeared. In the search for a general mathematical concept of such systems, we have previously presented a linguistic description of some continuous time processes [2]. This approach can be justified by means of the non linear realization theory of the systems [3]. In this way, we obtain a more accurate mathematical description of the dynamic interaction between the continuous device and the discrete event subsystems of the HYDS. We modeled a DS by means of the triplet (X, S, Cp), where X is the state space, S a transformation semigroup, and ip : X x S -+ X the state transition function. In a classical Continuous Time Dynamic System (C-T DS) S is the set of real numbers R, and states evolve with time, according to a set of differential equations. Following the framework developed by Ramadge and Wonham in [4] a Discrete Event Dynamic System (DEDS) can be modeled as a DS over an alphabet U (or event set), where the change of the states takes place in response to the events. In both C-T DS and DEDS theories, it is very natural to use semigroups acting over state spaces which are not everywhere defined; hence we introduce the concept of Pseudo Dynamic Systems (Ps-DS), where we suppose the existence of a partially defined semigroup action. These results are developed in Section 2. Following these preliminaries we present our main result in Section 3, concerning the concept of Pseudo Dynamic Hybrid Systems and some applications. 2.PSEUDO

DYNAMIC

SYSTEMS

At this point we give an outline of the basic results required in this paper. For a more detailed discussion we refer the reader to [2]. Let (X,S,@) be a DS as in the Introduction, and considerthe set S, = {u E S/@(z,u)!}, where the symbol “!” denotes “is defined”. Then the partial mapping Cp: X x S + X is given by aI : S, + X, O,(u) = @(z,u), u E S,. If S is a monoid with neutral element 8, and we supposethat 6 E S, for al’\ z E X, then @,q= Id, is defined everywhere. Thus we have the following definition: DEFINITION 2.1. A Pseudo Dynamic System is a DS (X, S, @), where @ is a partially defined semigroupaction, over X x S. The domain of @is the subset U {z} x S, C X x S. ZEX

2533

2534

Second World

Congress

of Nonlinear

Analysts

EXAMPLE 2.2. Let U’ denote the set of all finite strings of elements in an alphabet U, including the empty string 6. Let L be an arbitrary prefix-closed language in U’. For u E L define the sub-language L, c U’, such that v E L, iff the concatenation uv E L. Let Q : L x U’ + L, a,,(u) = uv, v E L,. Then (L, u’, a) is a Ps-DS. EXAMPLE 2.3. Let D C 9V’ be a domain, f : D + 8” a smooth vector field. Let t I-+ @t(t) = a({, t) be the complete solution of the initial value problem i(t) = f (z(t)), z(O) = .$. Hence t C) @t(t) is the solution defined over the maximal interval [O,Z’

Somos uma comunidade de intercâmbio. Por favor, ajude-nos com a subida ** 1 ** um novo documento ou um que queremos baixar:

OU DOWNLOAD IMEDIATAMENTE