Notes on control and observation in cellular automata ... - CiteSeerX

Notes on control and observation in cellular automata models S. EL YACOUBI and A. EL JAI Systems Theory Laboratory. University of Perpignan 52, P. Alduy Avenue. 66860 Perpignan Cedex FRANCE

Abstract: This paper deals with Cellular automata in the context of systems theory which concerns the study of abstract properties of systems and some techniques for analysing their behaviour taking into account input-outputs. We investigate the possibility to introduce in an appropriate way control and observation variables in cellular automata models. Some results related to additive cellular automata are given. Keywords : Cellular automata, distributed parameter systems, control, observation.

1

Introduction

research. The purpose of this paper is to introduce control and observation in cellular automata models to make them more useful in systems theory. We examine the particular case of additive cellular automata for which some important properties are exploited. It turns out that the presented paper is only introductory and much work can still be done in this context. A further study is going to be conducted towards elaboration of appropriate optimization tools and efficient computational techniques for the implementation of control and observation problems in terms of cellular automata models.

Cellular automata are a class of discrete dynamical systems introduced by John von Neumann as models of self-reproduction and then widely studied in a variety of disciplines (e.g., computer science, physics, mathematics, biology, ecology) with different aims particularly, as modelling and simulation of spatio-temporal phenomena. For systems studied in terms of inputs (controls) and outputs (observation), called distributed parameter systems (DPS), the description was mainly based on partial differential equations (PDE). However the complexity of real world systems leads to difficulties both in control theory and in the model implementation. An alternative of such approach is proposed considering cellular automata models. They constitute a very promising tool for describing complex natural systems in terms of local interactions between a large number of identical components. Despite their simplicity, they offer a good candidate for avoiding numerical and implementation problems.

2 2.1

Basic definitions Distributed Parameter Systems

Definition 2.1 Distributed parameter systems (DPS) is generally defined by a triple of operators (A, B, C) where the operator A defines the dynamics of the system and generate a strongly continuous semi-group of operators (St )t≥0 . B and C are the control and observation operators.

The classical approach in systems theory is based on modelling, analysis and control. In this context, all the literature on cellular automata was until now, mainly focused on modelling. Analysis and control of such systems remain open fields of

The mathematical exploration of systems theory in the case of DPS needs the following data to be precised : 1

1. The space domain Ω, which usually is an open bounded and regular subset of IRd . 2. The state, control and observation spaces denoted by Z, U and O which relay information on the regularity of the state of the system, the controls and the observation.

where the cells ci for i = 1, · · · , n are connected to c by a closeness or influence relationship and n is the size of N (c). Let f be a local transition function defined as follows : f : S N (c) −→ S

In real applications the control space U and the observation space O include information on the number of actuators and sensors. For example if U = L2 (0, T ; IRp ) the system is excited by p actuators. The case where O = L2 (0, T ; IRq ) is related to measurements via q sensors. For x ∈ Z, u ∈ U and y ∈ O, the linear state equation is given by : (

x˙ = Ax + Bu y = Cx

Cellular automata models

CA are discrete mathematical models consisting of a regular lattice L, a set that can be finite or infinite, composed of identical elements called cells and located in a space domain Ω. Each cell is characterized by its state which takes values in a finite set of states S and is updated according to a local transition function f . In general, the state set is a finite commutative ring of integers modulo k ≥ 2 given by S = {0, 1, · · · , k − 1}

which associate a new state to each neighbourhood configuration considered as a map from N (c) to S. We recall the elementary definition of a cellular automaton [5].

Definition 2.2 A cellular automaton A is given by the quadruple (L, S, N, f ).

(1)

A very wide literature is devoted to DPS approach via the operators A, B and C. Conversely, given a triple (A, B, C), under some convenient conditions, there exists an abstract linear system (S) such that A is the generator of the semi-group of (S), B the control operator and C the observation operator on (S) (Realization theory). That is why we will extend cellular automata modelling approach to fit as closely as possible with the usual approach of DPS.

2.2

(4)

There is a difference between the cell state which corresponds to a certain value in S and the state of the system which corresponds to the configuration of the cellular automaton. This configuration is defined by the map s : L −→ S c −→ s(c)

(5)

The cellular automaton evolves through a succession of global states or configurations which define its trajectory, by the iteration of its global rule F : SL → SL s → F (s)

(6)

The time evolution of the CA starting from an initial configuration s0 defines the orbit of s under F , i.e. the sequence of configurations s, F (s), F 2 (s), F 3 (s), · · · , F n (s)

(7)

(2) where F n (s) is the nth iteration of s under F .

in which the usual operations are considered using modular arithmetics. For c ∈ L, we denote by N (c), the set of cells that interact with the cell c called the neighbourhood of cell c. It can be defined by the following mapping N N : L −→ Ln c −→ N (c) = (c1 , c2 , · · · , cn )

(3)

3

Cellular automata as distributed parameter systems

We first focus on the classical autonomous CA. The following definition is based on the previous statements with some connections that make CA as good models in DPS study.

3.1

Autonomous case

Definition 3.1 The Local evolution of an autonomous cellular automaton A is defined by the couple A = (f, N )

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where f (4) defines the local dynamics and N (3) describes the local interactions between the cells of L. The state set is the set of all the configurations and will be denoted by X with X = SL

(9)

This set plays a similar role to what the state space is for DPS. The local CA function f induces a global map, defined by (6), which describe the global dynamics of the CA. The relation between the local dynamics, given by the transition function f , and the global dynamics in the whole domain L is the following : F (s)(c) = f (s|N (c)) ; ∀c ∈ L

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where s|N (c) designates the restriction of configuration s to the subset N (c). Definition 3.2 Given an initial configuration s0 , an autonomous CA can be defined in terms of the global dynamics by the state equation : (

st+1 = F st s0 ∈X

(11)

and then completely determined by the coupe (s0 , F ) (12) This presentation is similar to that of discrete DPS where the state equation is often stated by xk+1 = Axk

; x0 ∈ X

(13)

and A is the transition operator defined on the state space. We restrict ourselves to CA defined on L = Z d , (d ≥ 1) where Z is the set of integers and introd duce a metric over the state space X = S Z . Let δ : S × S → {0, 1} defined by : (

δ(i, j) =

0 if 1 if

i=j i 6= j

For x, y ∈ X we consider the distance dδ : X × X → IR+ given by dδ (x, y) =

X δ(x(c), y(c)) c∈Z d

2||v||∞

where ||v||∞ designates the maximum of the absolute value of the components of cell c. It is easy to check that δ and dδ are two distances respectively over S and X. Furthermore, the topology associated to dδ coincides with the product topology induced by the discrete topology of S. We have then the following result Proposition 3.3 The state set X = S L equipped with the distance dδ is a compact metric space and the global dynamics mapping F is continuous according to the topology induced by δ. The proof of this result can be found in [8] We can then establish a similarity with the usual distributed parameter semi-group approach by noting that the sequence {F i }i≥0 defined in (7) plays the same role than the semi-group, usually denoted by (St ). Furthermore we have the result. Proposition 3.4 The global dynamics operator F of the cellular automaton defined in (12) satisfies the following semi-group properties : 1. F 0 = Id , where Id is the identity operator in X, 2. F t+s = F t oF s , for all s, t {0, 1, 2, · · ·},

∈

I

=

3. dδ (F t (x), x) −→ 0 when t → 0+ , for all x ∈ X. The proof of this result is immediate It is easy to see that the solution of system (6) given by st = F t (s0 ) has the same form than the solution of DPS in the autonomous case (13), given by x(t) = St (x0 ) where (St )t≥0 is the semi-group generated by the operator A. In what follow,we consider cellular automata models from systems theory point of view taking into account the interactions between the system

and its environment, via inputs and outputs. The model must so be completed by control and measurement functions. Let us consider the following hypothesis. • L = Z d is a cellular domain, the elements of which are denoted by ci = c(i1 ,...,id ) . • I = {0, 1, · · · , T } is a discrete time horizon. • Lp (resp. Lq ), is a sub-domain which defines the region of the lattice L where the cellular automaton is excited (resp. observed). It contains p (resp. q) cells which may be connected or not. Without loss of generality, we can consider that the controls defined on Lp and the measurements considered via the output function in Lq are real numbers.

3.2

Control in cellular automata

Let us consider the control space U = C(I × Lp , IR)

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which consists of all the bounded controls defined as maps u given by : u : I × Lp −→ IR (t, ci ) −→ ut (ci ) Thus, we can consider the control operator G given by d G : U −→ S Z (15) u −→ Gu which defines the way how the control is applied to the cellular automaton through the cells of Lp . The map G is defined for all control variable ut , by a local control function g : IR −→ S such that : (

Gut (c) =

g(ut (c) for c ∈ Lp 0 elsewhere

(16)

Introducing the characteristic function χLp = 1 for c ∈ Lp 0 elsewhere we can write, (

Gut = g(ut (.))χLp

(17)

and then consider the cellular automata as a controlled system. Definition 3.5 A controlled CA can be locally defined by the triple Ac = (f, g, N )

(18)

where f and g are respectively the local transition and control functions and N the neighbourhood given in (3). The global evolution of Ac is described by the following state equation : (

st+1 = F (st + Gut ) s0 ∈X

(19)

where F is the global dynamics of the autonomous CA and G is the global control function. Note that the equations (11) and (19) are identical when G = 0. The system is excited in the sub-region Lp . Elsewhere it evolves according to the global dynamics function F . The result of this combination is a new global dynamics denoted by F, which defines completely the controlled cellular automaton. We have the following result. Proposition 3.6 For all t ∈ I and s a given CA configuration, the global dynamics F is connected to the couple of local functions (f, g) by the relation

F(s) = f (s|N + g(u|N ∩Lp )χN ∩Lp )

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where u ∈ U and s|N is the restriction of the map s to the neighbourhood elements. Proof: For all c ∈ L, F(s)(c) = f ((s + Gu)(N (c))) = f (s(N (c)) + (Gu)(N (c))). Using (17), we obtain : F(s)(c) = f (s(N (c)) + (g(u(.))χLp )(N (c))). For c0 ∈ N (c), if c0 ∈ Lp then (g(u(.))χLp )(c0 )) = g(u(c0 )). Elsewhere, the result is zero. therefore, (g(u(.))χLp )(N (c)) = (g(u(N (c) ∩ Lp ))χN (c)∩Lp ). The solution of the equation (19) has a very complicated form in the general case, given by :

st = F (· · · (F (s0 + Gu0 ) + · · ·) + Gut−1 )

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for a control policy u = (u0 , u1 , · · · , ut−1 ) which determines all the controls ui exciting the system at time 0 ≤ i < t.

3.3

Observation in cellular automata models

In DPS, the output function provides information (measurements) on the system via sensors. Thus an observation operator denoted C is considered. This operator gives a relation between the state space and the observation space. With Lq the subregion where the measurements are considered, the observation space is defined by O = C(Im × Lq , IR)

(22)

where Im = {0, 1, · · · , Tm } defines the measurements time horizon (Tm can be equal to T ). The observation space (22) consists of all the bounded measurements made in the cells of the subregion Lq and given by a measurement variable (output) denoted by θ:

augmented with the output function θt = Hst ; t ∈ Im

It may be called distributed cellular automaton (DCA) and completely defined by (s0 , F, G, H). The state equation (25) describes the controlled dynamical system and (26) is the usual output function which gives information on the system state, via the subregion Lq . The observation operator H given in (23) depends on the nature of the measurements; it may describe the cases of a pointwise sensors (observation in many isolated cells), zone sensor, and also boundary sensors. The case of impulse measurements may also be considered when Im is reduced to {tm1 , tm2 , · · · , tms }. The DCA given in (25)-(26) is very close to the usual discrete DPS

θ : Im × Lq −→ IR (t, ci ) −→ θt (ci )

(

We define the global observation operator H by H : SZ s

d

−→ O −→ Hs

(23)

which associate a measurement to each real state s. This global operator induces a local observation function h : S −→ IR. such that (Hs)(t, ci ) = h(st (ci ) for all ci ∈ Lq . In the above definition, it is assumed implicitly that the output have q components, each one is associated to one cell of Lq . Definition 3.7 An observed cellular automata Ao is locally defined by the triple Ao = (f, h, N )

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The global observation is given via the output equation : θ = Hs Now, we can give the complete description of CA in terms of inputs and outputs by the following result

4 4.1

xk+1 = Axk + Buk ; x0 ∈ X yk = Cxk ; k = 1, 2, 3, · · ·

(

st+1 = F (st + Gut ) ; t ∈ I s0 ∈ S

(25)

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Case of additive cellular automata Additive cellular automata

Additive cellular automata (linear and homogeneous) constitute a very important class of models which capture the most interesting behaviours of complex systems despite their simplicity. The basic results on CA deal with additive global dynamics owing to their algebraic structure. Considering addition in S modulo its cardinality k, addition in the configurations space S L is defined by for all s1 , s2 ∈ S L : ∀c ∈ L, (s1 + s2 )(c) = s1 (c) + s2 (c)

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Definition 4.1 A global dynamics F is additive if for every pair of configurations s1 , s2 ∈ S L , F (s1 + s2 ) = F (s1 ) + F (s2 )

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This definition is equivalent to the local condition of additive CA f (st (N (c))) =

Proposition 3.8 The cellular automaton in the sense of DPS is given by the state equation

(26)

X

ai st (ci )

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1≤i≤n

for some scalar a0 , a1 , · · · , an which are called the weights or coefficients of cells in the neighbourhood N of size n.

4.2

Additive distributed cellular automata

Let us consider the DCA given by the equation (25) where F is an additive global map. We have the following result. Proposition 4.2 The solution of the state equation (25) for an additive CA is given by : st = F t (s0 ) +

t−1 X

F t−τ (Guτ )

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τ =0

Proof : Starting from s0 , we consider the intermediate state s00 = s0 + Gu0 which add to s0 the effect of action u. The next state is calculated by applying F and then

allow the resolution of the control problem. The control problem may be stated in the following form :

(P )

 min J(u)   u∈U ad

under the functional constraint   (f, g, N )

In DPS case, there are general existence and uniqueness results for problem (P ) with a quadratic cost functional [9]. In order to use similar approaches for CA, additional structures need to be defined for the considered state and control spaces. The task is under consideration and simulations of simple problems are in progress.

s1 = F (s00 ) = F (s0 ) + F (Gu0 ) s01 = F (s0 ) + F (Gu0 ) + Gu1 and then

References

s2 = F 2 (s0 ) + F 2 (Gu0 ) + F (Gu1 ) Successive applications of this two operations give the result The solution given below, has a similar form than in DPS case (1) : Z

[2] B. Chopard and M. Droz, Cellular Automata Modeling of Physical Systems, Cambridge University Press, 1998. [3] J. Conway, The game of Life, Scientific American, 1970.

t

St−τ Buτ dτ

x(t) = St x0 +

[1] A. Adamatzky, Identification of cellular automata, Taylor & Francis Ed., 1994.

0

where (St )t≥0 is the semi-group generated by the linear operator A. It allows us to make analogy between additive CA and linear DPS.

4.3

Control problem

Various control problems may be considered for additive DCA which are generally stated as optimization problems. Assume that we want to achieve a given objective on the system state at a given time, we can formulate the following problem : for u = (u1 , · · · , up ) a control variable in a set of admissible controls Uad ⊂ U, as a vector where ui : I −→ IR for 1 ≤ i ≤ p, we consider a realvalued cost functional J which may depend on the state s, the control u (and then on Lp ), the objectif to be achieved and the time horizon I. J : Uad −→ IR+ u 7−→ J(u)

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The possible choices of the cost functional J will lead to different control problems. The properties and the regularity of cost functional J will

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