Pseudo dynamic hybrid systems

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

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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’<). Then (D, 8+, a) is a Ps-DS. EXAMPLE

2.4. Let U be an arbitrary

u : [O,T,)

-+ U, with

set, and let S(U) denote the set of the U-valued

step functions

T, = ,& ti, t; 2 0, u(t) = Ui E Uf t E [tl + ” ‘+ ti-17 tl + * *. +ti-1

+ ti),

i= 1,2,..., k. We will use the notation u = S(&d = S((tl, . . . , tk), (~1,. . . , uk)). The concatenation of step functions u, v in S(U) is defined by u(t) 21.

v(t)

O
= 1

- T,)

v(t

T, 5

t

< Tt4 + TV

S(U) is a semigroup respect to the concatenation of step functions.The empty function B : [0, 0) + U is the neutral element of S(U). Let D c %” be a domain; then a,, : D x %+ + D is the pseudodynamic corresponding to f,, : D + 3?“, as in Example 2.3. Fix the state ze E D, and say that a step function u E S(U), is admissible for zo if 1.

t

I-+ Q,, (20,

t)

can be defined over [0,

2.

t

I+ @‘ua(21,

t)

can be defined over

k. t where2k-1

[tl,

be 2:;lj:kt!,,

tl

where 21 = a,,, (20, +

ts];

tl).

where 22 = a,,, (21,

defined

Over

t2).

[tl+.“+tk-l,tl+“‘+tk-l+tk);

t;e;).

In other words the control system the domain [0, T,) of the control u. Let SzO = {U E S(U)

tl];

i(t)

: u is admissible

= f(z(t),

u(t)) = f+l(z(t)),

for ze} c S (U).

Then

z(O) = 20, has solution

the dynamic

@ (% u) = @ur (@tlksl (* * * (@tq (20, h) , tz) * - ‘) , tk) .

over

@ given by

(2.1)

is well defined over S,, . Hence (D, S(U), @) is a Ps-DS. PSEUDO-DYNAMIC SYSTEMS WITH OUTPUT MAPPINGS 2.5. Suppose that a Ps-DS (X,S,@) is equipped with an Output Mapping (OM) q : X + Y and an output set Y. Then we say that the quintuple (X, Y, S, 8, cp) is a PseudoDynamic System with Output Mapping (Ps-DS&OM).If (Xr,Yr,Sr, @r,(~r) is another Ps-DS&OM, then the triplet (f,g,h) of the mappings f : X + Xr, g : Y + Yl, h : S -+ .!& is a dynamic preserving mapping if:

Second World

a. h is a semigroup b. The diagram

(monoid)

Congress

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2535

Analysts

homomorphism.

(2.2) commutes.

Here commutativity

implies

the condition

that h(S,)

c &I(,).

THE REALIZATION OF INPUT-OUTPUT MAPPINGS 2.6. Let S be a semigroup, Y an arbitrary set. A partial mapping p : S + Y with domain L C S, B E L, is called an input-output mapping. If (X, Y, S, a, ‘p) is a Ps-DS&OM and ze E X is a fixed point, then the sextuple (X, zu, Y, S, @, cp) is an initialized Ps-DS&OM. If an initialized PsDS&OM is given, then the partial mapping defined on the domain S,, by the formula pzo (u) = cp (@( ze, u)) is an input-output mapping. Using Nerode’s equivalence relation [5], we give a realization of the partially defined input-output mapping p : S + Y. Let SO c S be the domain of p. If u, u E Se, then u N v if 1. s, = s,, 2. p(uw)

= p(ww),

for all w E S, = S,.

The equivalence class of an element I will be denoted by [u]. Let X be the set of all classes. The initial state is zu = [@I. Let Sl,l = S,,. Obviously, Sl,,l depends on the class of u. The partially defined dynamic @ is given by

@[u]: S[“]+ x, The output Moreover

mapping

cp : X + Y is given by cp ([u]) = p(u), which cp (%0(u))

Hence, the initialized CLASSICAL

q&4 = @w, 4 = b4.

Ps-DS&OM

= cp (@[O](4)

= cp (bl)

= f (4%

U c 92”. Consider

u(t)) I Y(t) = h W)

only on the clsss [u].

= P(U)

(X, ze, Y, S, @, ‘p) realizes the input-output

CONTROL SYSTEMS 2.7. Let D c R”, w

= cpWI)

depends

mapping

the control 7

p : S -+ Y.

system (2.3)

If we admit integrable controls where f : D x U + R”, and h : D + Rk are smooth functions. u : [O, Z’,,) + U c V’, then it is known that we can approximate the control u by a sequence (ul) of controls such that a. ul : [0, !.I’,) + U are step functions,

~11--t u almost everywhere,

b. the trajectories z~ corresponding to ul with the same initial condition ~(0) uniformly to the trajectory z, corresponding to the original control u.

= z(O), converge

Hence the output sequence is yl = h o ~1. Therefore we can consider the step controls instead of the integrable ones. Following the Example 2.4, (D, S(U),@) is a Ps-DS, with @ defined by (2.1). Then by 2.5 we can define an initialized Ps-DS&OM (0, <, IR”, S(U), @, h) and an input-output system pt (4 = h (W,

4).

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The following question now arises: When can the input-output mapping P be obtained from a classical control system? We have seen in 2.6, that any input-output mappings p can be realized by an abstract Ps-DS&OM. The crucial point is to equip the abstractly constructed realization with topological and differential structures. B. Jakubczyk [3], has defined the smoothness of the input-output mapping. Roughly speaking its smoothness means that the functions t= @I,...,

tl)+p(S(t,1L))~R~

aresmoothforallqlE&U,

l=l,2,...

He also has defined time invertibility, which permits the extension of the monoid S(U) to a group. Subsequently, he defined a rank, which is the dimension of a minimal smooth realization over an smooth differential manifold X, where the state transitions @(z, u) are defined by a smooth control system over X, as in Example 2.3 . This fact is a very important theoretical support of our linguistic point of view. The abstract linguistic description of the continuous time smooth systems preserves the richness of the differential structures which can be reconstructed by the Jakuboczky’s theorem. 3.

PSEUDO

DYNAMIC

HYBRID

SYSTEMS

Hybrid Dynamic Systems (HYDS) contain basicly two distinct types of components; subsystems with continuous dynamics (6) and subsystems with discrete event dynamics (k), that interact with each other. In earlier work we modelled the dynamic behavior of the continuous time and the discrete event system as a Ps-DS&OM as defined in 2.5. For the details of this framework we refer the reader to PIThen we have c = (C, zu,Y’, SC, @‘, hc) and E = (E, 8, YE, SE, @El hE) as Ps-DS&OM. In this context we present the following definition. DEFINITION 3.1. A Pseudo Dynamic Hybrid System (PsD-HYS) is a triplet (6, E, 1) where 6 and 2 are Ps-DS&OM and f = (fe, fr, fz) is a dynamic preserving mapping such that fe(zo) = 6.

We note that the commutativity

of the following diagram, implies that jr($) cxsc

%

(ro,rl)r

ExSE

c

5

Sfo

q

E

C S&l.

YC lh

3

YE

3.2. We elaborate further on the concepts discussed above by considering two specific exmaples of classical Hybrid Dynamic Systems.

APPLICATIONS

3.2.1 Switching Systems with hysteresis. Consider a Switching System with hysteresis, with is a finite number of modes, given by the diferential equations i = Ii(z), z E Xi, i = 1, .. .. k, where Xi are overlapping operation domains in %” equipped with a given switching rule. Thii system could be considered as a pair of dynamic subsystems (6, @ with a dynamic preserving mapping f = (fe, fr, fz). 6 is the classical DS given by the modes of the switching system, considering the state space as

EXAMPLE

X = fi X;, and fi is constructed positi&‘of

as in Example 2.1 where a word u = ur, . . . , uk is the possible

the switches, or given rule. Then fe(z) = Ui, if z E Xi. If z(t) at time moment t belongs

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to severals Xj, j # i, then we consider that ui is occuring until the trajectory does not leave the region Xi. The consideration of controlled switching systems, where-switching rules admit some ambiguity in the dynamics, could be presented in a similar way, with f a multivalued defined function. This idea will be developed in another paper. 3.2.2. Classical Hybrid Controlled System. Consider a plant given by a classical control system as in 2.7. As we have seen, it can be described by a Ps-DS&OM (d, <, 9?“, S(U), @, h), w h ere the controls are the reference signals. If we optimize the reference signals with respect to a cost function, we obtain an operation mode. The resulting system is a switching system where we can select a mode among the modes u E R,. Here R is the set of events asociated to the operation modes of the given plant. When an event u occours, a new operation mode appears in the plant. In the processing unit described by these classical control models, faults or non tolerable changes can also occur. The different events of this type are drawn from the set R,,. The subfixes c and RC of R denote the controlled and non controlled adjectives. We suppose that if a fault (a non controlled event u) occurs, then the plant functions in an uncontrolled default mode, also described by a differential equation. Then the event description of the behavior of the working modes of the plant can be given by a (DEDS), which is a PsDS&OM (e, 0, U, (a, U n,,)*, @, hE). In this case our PsD-HYS consists of the pair (6, E) Ps-DS&OM which describe the event dynamic preserving mapping j = (fu, fr, fs): EXAMPLE

Ex

and fu(Q

D x S(U) 1 (fo, fl)

8

(Q,UR,,)’

s

D

1

R,,

1 fo

1 f2

E

3

,

u

= 0. We can say that the plant image is contained

in the discrete event dynamical

model.

ACKNOWLEDGEMENT

The authors gratefully acknowledge the following sources of financil support: The first author recived partial support by the Venezuelan organism CONICIT under grant I-22, and was also supported by CDCHT-IJLA under grant I-405-92-A. The second author was supported by CDCHT-ULA under grant C-716-9502-C. REFERENCES 1. ANTSAKLIS Mediterranean

(1994). 2. SZIGETI of complex

P., LEMMON Spposium

M. & STIVER

J., Modeling

and Des@

of Hybrid

in Control

k Automation

pp.

on New Directions

F., CHACON E.k DE SABRAZIN G., Hybrid dinamic system. Submitted to IEEE transaction on automatic

3. JAKUBCZYK B., Existence and uniqueuess of nalizatious tion, 18, pp. 455-471. (1980). 4. RAMADGE P.J.k WONHAM W. M., Supervisory coutml and Optimization. 25 (l), pp. 206230 (1981). 5. NERODE

A., Liuear

automaton

trausformatious.

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