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Distributed systems analysis via sensors and actuators
Sensors and Acruafors A, 29 (1991)
Distributed
1
l-11
systems analysis via sensors and actuators
A. El Jai IMPIUniversity (Received
of Pepignan,
Avenue
de Wlleneuve,
66025
May 22, 1990; in revised form December
Perpignan
Cddtx
11, 1990; accepted
(France) January
18, 1991)
Abstract The purpose of this paper is to give some results structures and systems analysis. Abstract concepts of to the controllability and observability of systems developed results are illustrated by many examples Keyworak
sensors,
actuators,
system
analysis,
related to actuators described of specific
distributed
1. Introduction Nowadays, the technology and theory of sensors and actuators from their own research area, combining both physical and chemical aspects. Thermal, chemical and mechanical transducers are necessarily linked with electronic circuits, which are very often designed to interface the transducers with the model manipulation implemented in computers. In any case, many scientists work on the conceptual aspects of sensors and actuators, design aspects, sensitivity problems, etc. Sensors and actuators can play a fundamental role in the understanding of any real system. Fundamental aspects of the knowledge of a system are developed by theoreticians from a mathematical point of view, and there is very often a wide gap between this and a concrete comprehension of the system. It is now time to link some of this theoretical work with concrete considerations of input-output problems. This is the purpose of this paper. Sensors and actuators form an important link between a system and its environment. Sensors have a passive role and allow the system evolution to be measured. Actuators have an active role and are used to excite the system. This means that the actuators will appear in a right hand side member of a model and the sensors will define an output
the link between actuator and sensor and sensors are introduced and applied by partial differential equations. The geometries and systems.
systems.
function. In any case sensors and actuators are considered via a space variable. Mathematically speaking, the space variable is present in all systems described by partial difequations, integral ferential equations, integro-differential equations, etc. Let us focus our interest on systems described by partial differential equations. Hundreds of researchers throughout the world work on such systems. Very interesting results related to systems theory have been developed since the 197Os, but most of these results are still inaccessible to engineers or automaticians. This gap can be bridged by the introduction of sensor and actuator concepts. Many papers have been devoted to these aspects. The book by El Jai and Pritchard [l] is focused on this approach and this research area produces more and more results every day. In this paper we shall show that many theoretical aspects of systems theory can be seen from the viewpoint of sensors and actuators and will therefore become more easily accessible to physicists and other scientists. 2. Systems analysis Systems analysis consists of a set of certain notions which enables a better knowledge and understanding of the system to be obtained_ In this Section we recall and study the most important of these notions. In the next
1991
-
Elsevier
Sequoia,
Lausanne
2
Sections we show how they can be linked to sensor and actuators structures. Systems analysis can be done from a purely theoretical viewpoint as for abstract differential equations. But the study may be also become concrete, in some sense, by using the structure of sensors and actuators. Thus one can study the concept of controllability via actuator structures; that of observability via sensor structures; that of stabilizability via actuator structures; the concept of detectability via sensor structures; compensators via actuators and sensor structures; the concept of observer via sensor structures, and so on. More details of these concepts and different theoretical approaches are found in the literature. For an approach through sensors and actuators, see refs. 1-3. In this paper our interest is focused on controllability and observability linked with sensor and actuator structures, their locations and numbers. It is awkward to try to control a system before analysing it and before looking whether or not it is possible to steer ,its state to a prescribed position. It is also important to know if the measurement output has some consistency with respect to the evolution of the system. One has to know if the system is stable or can be stabilized, etc. These notions can be studied only if a model has been established. This model will simulate the system evolution and then the analysis can be done on the model. In this paper we shall try to give some results as rigorously as possible while avoiding too much mathematics. For more details, the reader should consult the references. The model we consider is given by the following differential equation: i(t) =/&z(f) +Bu(t)
0 < t< T
z(0) =zo
(1)
with the output function y(t)=Cz(t)
O
(2)
We shall go into further details of our hypothesis later. The interest of such a model is that it generalizes naturally the finite-dimensional models and concerns many types of partial differential equations, as will be shown later. Moreover, the results are pre-
sented in a homogeneous form for all these systems. The hypothesis is as follows: * In the finite-dimensional case A, B and C are matrices of order (n Xn), (n Xp) and (q X n) such that with some regularity on the input function, the solution of eqn. (1) is given by z(t) = exp(At)zO +
f F
J
exp[A(t - r))&(r)
d7
0
(3) The state space is I??‘, the control space is UP and the output space is Rq. * In the case of partial differential equations one must be more careful with the hypothesis allowing the same solution as (3). We need the different functions to be at least squareintegrable on the time interval (0, 7). The state space 2 is infinite dimensional. If the system (l)-(2) is controlled by p actuators and observed via q sensors we have u
EL2(0, T, W) and y EL*(O, T, UP)
(4)
Moreover, we need some regularity on the dynamics operator-A such that exp(At) is well defined: A is supposed to be the generator of a semi-group (exp(At))l,o such that for any state z ~2, the map t +exp(At)z is continuous. Then the solution (3) is always valid. * In the case of partial differential equations, the state space 2 is a function space. In any case one needs some mathematical hypothesis on 2 such that it becomes suitable for analytical manipulations. Usually one has to consider the space 2 as a separable Hilbert space. Different cases of partial differential equations will be developed below. Let us recall some of the fundamental notions that occur in systems analysis. 2. I. Controllability In the case of partial differential equations one has to consider the fact that it is never possible to steer the system exactly to a given state. Many different definitions (exact, weak controllability) are given and developed in the literature [4-71. We consider the following definition.
3
For this purpose we can suppose that the system (1) is autonomous (U = 0) because of the linearity. In this case the output (2) gives
Definition of controllability The system (1) is said to be controllable on the time interval (0, 7’) if for any desired state zd EZ and E > 0, there exists a control u EL*(O, T; W) such that: 112(T) -zd 11< E
~0) = C exp(-W(O) (8) Let us consider the operator K defined by
(3
y(.)CLZ(O,
K(.):z(O) EZ -
wherez(7J is the state reached by the system (1) when excited by the control u at time T.
T; W)
(9)
such that
This definition means that, given a desired state zd in Z, one can find a control that steers the system to a state which is very close to rd. It is possible to show that the control which steers the system to the desired state zd can be expressed by
~0) = WHO) (10) Obviously the system will be observable if, in some sense, the operator K leads from an initial state z(0) to a unique observation y(.). Mathematically this is equivalent to saying that K is an injection. So we use the following definition.
u(t)=B* exp[A*(T-t)] -1
X
exp[A(T-t)JBB*
exp[A*(T-t)]
x C-G - exp@ T)z,)
elf
(6)
The above formula gives an open-loop control structure. A feedback version is obtained by replacing the initial state by the current one: u(t) = B* exp[A*( T- t)] -1
X
exp[A(T-t)pB*
exp[A*(T-t)]
do
Definition of observability The system (1) with the output (2) is said to be observable if Qr(K)
function
= (01 .
(Ker
(11)
(K) is the kernel of the operator
K={z~Zlfi=O}.)
This definition is equivalent to our previous explanation. In any case it is possible to show that the adjoint of the operator K is given by T
K*z=
exp(A *t)C*z(r) dt
s
(12)
0
X
(zd- exp[A(T--N4~))
(7)
For the proof of these results, one has just to put the control (6) in the system (1) and look at the solution at time T. One must be careful when applying formula (7) because of the singularity problem that occurs in the neighbourhood of t = T (this does not happen in the open-loop case) [S]. 2.2. Observability
When a feedback control law has to be implemented one needs knowledge of the current state; the observability problem occurs when one has to reconstruct the state of the system. The problem is whether knowledge of the dynamics of system (l), together with the output function (2), is sufficient to reconstruct the initial state t(O).
such that the state to be reconstructed given by T
is
-1
exp(A*t)C*C
z(O)=
U0
exp(At) & >
T x
ap(A*t)C*y(t)
(13)
dt
s 0 I.e.,
the
reconstruction
operator
is
(K*K)-‘K*.
Formulae (6) and (13) are the same as in the case of systems described by differential equations (finite-dimensional case). The problem which occurs in the case of partial differential equations (infinite-dimensional case) is that these formulae are not regular. That is, the operator to be inversed is not bounded,
4
for instance, an error on the measurement can lead to an initial state which is not acceptable.
3. Mathematical sensors
concept of actuators and
Actuators (sensors) are the intermediaries between a system and its environment. Their structure will depend on the geometry of the support, the location of the support and the spatial distribution of the action (measurement) on the support. Let Cl be the spatial domain in which the system (1) is modelled (usually 0 is bounded CR” with n < 3). We can have many geometries for the support. Among the current situations, we may have: (i) pointwise or zone actuators (sensors) inside the physical domain a; (ii) pointsvise or zone actuators (sensors) on the boundary of the physical domain Q (iii) filament actuators (sensors) in the domain CR; iv mobile pointwise actuator (sensor) in th; Jomain a. Various situations will be represented in the following Sections. Definitions (1) Suppose that Sziis a closed set included in the geometrical domain R in which the system (1) is modelled and giELZ(Q)+ A zone actuator (sensor) is the couple (Q, gi) where: * C& defines the geometrical support of the actuator (sensor), j, gi defines the spatial distribution of the action (measurement) on the support sLi. For the definition one needs some smoothness for the function gi so that the mathematical techniques are rigorous. (2) A pointwise actuator (sensor) is the couple (b, &b) where: * b E fi is the location of the control (measurement), * &, is the Dirac mass concentrated in b. In the case of a boundary actuator (sensor) the definitions are the same with QcX2 (the boundary of a), or biEan. In the case where the system is excited (observed) byp actuators
(sensors) we shall consider (Q, gi)rciGP with sZicCI and giEL2(Q), Vi = 1, 2, . . . , p_ (3) The actuators (C&n,, gi),4iGP are said to be strategic if the system excited by these actuators is controllable on (0, r) for any T>O. The previous definition means that strategic actuators are those which allow the system to be steered to chosen desired a priori states. (4) The sensors (Di, jJIGiCq are said to be strategic if the system (1) with the output function (2) is observable on (0, r) for any T> 0. The output function (2) depends on the sensor structures. When the sensors are strategic, the knowledge of the output function and the dynamics of the system (1) are sufficient to reconstruct the state of the system, and the formula (13) is well defined. In the following Sections we shall see that these definitions have some consistency with systems analysis. Let us consider some examples of sensors and actuators for classical systems. 3.1. Examples of actuators Zone case in a diffusion system Consider the system excited by p zone actuators (Q, gj)14i
y(x, O)=O Y(&
t)=O
(14) (1%
xecl
(5, t)Eafixlo,
q
The system given by eqns. (14) and (15) models, for instance, a diffusion system excited by p zone burners located inside the domain 0. Such a system is a particular case of system (1). We have to consider Rz = AZ whose
domain
is HA(a)
nH’(O)
and BU zigIgiUi with B:(L*(O, T)y +
9(A)
In this case the hypothesis considered at the beginning of the paper is true. With such
the control
Fig. 4. Pointwise
actuators.
actuators one can consider any other boundary conditions (see Figs. l-3). Cl
Fig. 1. Actuator
Distribution
=I
with circular
Pointwise case in a di$usion system
support.
Consider the system (see Fig. 4) excited by p pointwise actuators (bi, &)rqi
of the ~ontrd
% =Ay+ 56(x-b&,(t) i-
y(x, 0) I I :
’
, =,-‘I
Fig. 2. Actuator
’ ’ ’
a1
al+‘1
with rectangular
Support
of
t)=O
Y(5,
the control
=o
(16)
XEn
(5,~)~~~XlO9 IT
S on the right-hand side is the Dirac mass. This model corresponds to a system excited only at the points bi E Sz (the case of pointwise burners in a diffusion system). System (16) is a particular case of system (1) with
*
and
=1
support.
Bu= 2 S(X-bi)ui with B:RP i-l
Supports
n
AZ = AZ whose domain is HA(n) nN2(fi)
I’ ,
, 0
4 L ,
(x, t)~Rx]0,
1
-%4*
Other boundary conditions can be considered.
of
the controls
Boundary case in a di@sion
system
Consider the system excited by p zone boundary actuators (rl,, gi)r Gi
$
=Ay
y(x, O)=O
Qx]O,
XEa-2
Y(!ff t, =i~l~i(04(t)
Fig. 3. Actuators
in circular
domain.
q (17) (6
t, E X2x
lo, q
where &CC&I and gjEL2(Q Vi, 1
6
on its boundary. This situation, shown in Fig. 5, occurs with many physical systems. We can show that such a system is once more a particular case of system (1). In the case shown in Fig. 6 of p boundary pointwise actuators (b, 6,,), pimp one has to consider the system ;
=Ay
y(x, O)=O
fix]O,
q
XECI
(18)
where bi E Kk ‘di, 1
Fig. 6. Pointwise
boundary
actuators.
3.2. Example of output and sensors Let us give the expression for output function with respect to the sensor structures. We suppose that the system (1) with the output function (2) is observed via q sensors (Di,jJ1 cicg where the (D& Gi G4 are the sensor supports and g)l
If the ith sensor measurement
is of zone
type it gives the
(20) If the jth sensor
is of pointwise
type we have
(21) Finally output Fig. 5. Boundary
zone actuators.
Y (0
the structure
= C-e)
of the operator
C in the
7
desired state. The second result is related to the possibility of an adequate location for a given type of action (burner, etc.) such that the system is controllable. The proof of these results is given elsewhere [l]. Let us now give a result of characterization of the actuator structures such that they are strategic. Let (~~j) be an eigenfunction system of the operator A associated with the eigenvalues (A,), with r,, the multiplicity of A,. Fig. 7. Different types of sensors.
is given Cij =
(23)
by
(cpi, f;.)LYD.) %(bi) (
Then sult:
in the zone case in the pointwise case (22)
where 1
4. Existence and characterization strategic actuators and sensors
of
In this section we shall give some fundamental results related to strategic actuators (the results for sensors are similar). The results are related to the link between the controllability concept and the structure and number of actuators. Result 1 For a given system modelled by (l), if the suPPorts (Q)I 6i -sp of the actuators are given, there always exist distributions (gi)l that the actuators (C!i,gi)l
we have the following
fundamental
re-
Result 3 The actuators (SL,, gj)l~icp are strategic if and only if: (a) p*swrn (b) rank G, =r, Vn where G, is the matrix of orderp X r,, defined by (G,)ij
= (gi, qnj)Lz(n) =
f n,
g&)cPni(x)
* (24)
l
and l,ci,cr,.
This rank condition links the number, the location and the structure of the actuators to their possibility to steer the system to the desired prescribed states. In the case of pointwise actuators located in bjEn, condition (b) is valid with (Gn)i,j = qnj(bi)
(25)
We shall see later how this result can lead to concrete choices of strategic actuators. These results are the same for sensor structures linked with the observability problem.
5. Examples of practical cases Let us apply the previous results to the case of diffusion systems in one or two dimensions. Similar results for hyperbolic systems are given in refs. 9 and 10. In one space dimension we consider a diffusion system defined in Cnl= 10, 1[ and in
8 the two-dimensional case fi* = IO, a[ x 10, 6[. The time interval is [0, 7J with T>O. Let
(26) and the boundaries
n
~=ail,x]O,
be
and 7
=aGXlO,
77
5.1. Pointwtie actuators Let us consider the system pointwise actuators (b, &Jl~j&
Y@,
fY
0) = 0
Y(E,o=o
by p
(28)
L
In the one-dimensional of the system (28) are AR =-n2rr2
excited
(27)
case the eigenvalues
, n=l , 2 , . . .
(29)
r,, = 1, Vn, associated
with multiplicity the eigenfunctions cp,(x)=&sinnnx,
n=l,
2, . . .
with (30)
So part (a) of Result 3 in Section 4 shows that thep=l actuator may be strategic. Part (b) of Result 3 implies that the actuator (b, 6,) is strategic if and only if: Vn > 1, sin(nd)#O=nd#k~ob4.Q
(31)
Remark The result (31) may be surprising, but for implementation problems one has to consider a finite number of modes so that only a finite number of non-strategic locations for the actuator are to be considered. In the two-dimensional case the system (28) is considered by replacing Q, by Q, and Ci by CZ. In this case the eigenvalues are A,,=-
(
1 rr2
;+g
corresponding
G&~, x2)=
(32)
to the eigenfunctions
&
the multiplicity one except if
nml
sin7
of which
(a2/b2) # a rational
number
.
sin
mm2 b
may be more ( E Q)
(33) than (34)
In this case one actuator may be strategic. Let b= (b,, b2) be the actuator location. The same approach leads to a strategic actuator if and only if @,/a)
and (&lb)
If the actuators the system is
P Q
(35)
are located
on the boundary,
?Y - -Ay=O
at
Y@,
0) = 0
Y(E,
‘)
sz’
=i$lW5--bi)ui(t)
(36)
Ci
Mathematically speaking the Result 3 is more difficult, but the (a) and (b) characterization still works. If the system (36) is two dimensional, for example, then with condition (34) the actuator (b, 6,) with b = (b,, b2) is strategic if and only if (35) is confirmed. Obviously the previous remark still holds. Many other situations with standard geometry which can be obtained by symmetry considerations are developed in ref. 1. For the observability problem, the same characterization results are given elsewhere [l-3, 111. 5.2. Zone actuators With p zone actuators system (25) becomes % - AY =
(C$, gi)lsi_,
the
j$lg&)Ui(t) Qi
Y(X,0) = 0
fY
YG t)=O
ci
(37)
In the one- or two-dimensional cases, the eigenvalues and the eigenfunctions are given as in the previous Section. Let us give details of the case of a2 with condition (34). One actuator (a,, gl) may be sufficient to steer the system to any desired state. The support of the actuator may have any arbitrary shape. We shall consider the following two situations: A rectangular support (see Fig. 2) 0, = fi [q -Ii, ai + ZJccl* i-l
(38)
9
From Result 3, we can show that the actuator (SL,, gi) is not strategic if a,/a (or a,/b) EQ and the function gi is symmetric about the line x=a, (y=a,).
A natural cost function to be considered is the control norm, which in this case means the cost of transferring the system, to the desired state:
s 1
A circular support (see Fig. 1) Let c = (c,, c2) be the centre and C!, =D(c,
r) ca2
of the support (39)
In this case the actuator (&, g,) is not strategic if cl/a (or c2/b) EQ and gi is symmetric with respect to the line x =cl (or y =cl). If the actuators are located on the boundary of s1*, the system is described by ?Y - -Ay=O at Y(X,
0) = 0
i-l2
Y(&
r,=i$l&5(#ui(fl
L
(40)
and the results are similar to those of the distributed zone case. Other practical cases are those where the support may be adjacent to one or more sides of the boundary C2. Consider, for example, the case where one actuator is located on the side (see Fig. 5):
{(x,y)/OGxGa,
y=O}
then the actuator support is [a -II, a 1 + I,] X (0) and it is not strategic if a,fa E Q.
6. Optimization location
of actuator structure and
When one has to conceive a system and then to ‘imagine’ the choice of actuators, all the parameters of the actuator may be in some sense optimized. One has to select the parameters to be optimized, and a cost function related to the objective of the problem, which depends on these parameters. This can lead to very complicated problems. If we consider, for instance, the case of one actuator (C&, gl) which excites the system (37) (with p = l), then we can suppose the following parameters to be unknown: (i) the geometry and location of fir; (ii) the spatial distribution g, on I(z,.
J= llUtl12=
ul(t)’ dt
(41)
0
One can also consider more complicated quadratic functions. We shall now give some of the known methods for solving each of these steps.
6.1. Choice of the geomeny fl,
and location
of
There are two approaches for finding the optimal geometry of the actuator support IR,. The first is parametric and based on the introduction of some parameters (LY,.) which define the geometry and the location a, of the actuator and then to minimize the cost J with respect to the (ai). The second is an ‘optimum design’ technique which needs more mathematics. A deformation field V and the Eulerian derivative of J with respect to !C& in the direction of V have to be considered. Both techniques have been used for some specific problems and implemented for numerical simulations [12-141. Example
of the parametric approach
Let us consider the two-dimensional case and assume that technological constraints prescribe the actuator support to be circular (Figs. 1 and 3) and defined by D(a, R) where a = (a,, a,) E IR and the radius R is such that D(a, R) cfk The optimization problem is then f
min
1(D(a,
(a, a2, R)
J
(42)
R) cfl
where the cost J= llul II2 depends implicitly on the actuator support parameters a,, a2 and R. 6.2. Choice of the distribution g, As in Section 6.1., two approaches are possible [3, 151. The first is once more parametric and based on the identification of some convenient parameters which define the function gl. For example, one can consider, in the one-dimensional case, the function
of the numerical data of the dimensions Sz). For more details see ref. 16.
of
References
Fig. 8. Graph
of g, for different values of the curvature
exp~_a,z2)
g1cQ
{exp[~~(~-x~)2-~l~21-1} 1 -cr,P
=
0 elsewhere if x1 - I ,
(43)
if the zone support is 0, = [x1 -I, x1 + I] where x1 is the centre of the action zone, I is the half length of the action zone; a, is the curvature of the function g1 (see Fig. 8). If the location of the actuator is fixed, only the parameter a, in (43) has to be optimized. This parameter can be chosen in ] - 00, + Q)[. For LY,--+ - 00 the control distribution is of Dirac type and for cyl --f + 00 the control distribution is uniform on [x1 -1, -7t1+I].
7. Conclusions From the engineer’s point of view, a condition that may be surprising is Result 3 of Section 4, which concerns the lowest number of actuators (sensors) necessary to ensure controllability (observability). In practice it is always possible to excite (observe) a system even by one actuator (sensor) or by a finite number p < supr, of actuators (sensors). The problem is lhether the conditions (a) and (b) of Result 3 are ‘reasonable’. In fact it can be proved that it is always possible to ensure controllability (observability) in the sense defined in Section 2 by one actuator (sensor) if the domain CR is replaced by LX* such that the distance d(C.k, a*) between R and OR*is arbitrarily small (up to the precision
1 A. El Jai and A. J. Pritchard, Sensors and controls in the analysis of distributed systems, Ellis Horwood Series in Mathematics and its Applications, Wiley, New York, 1988. 2 A. El Jai and M. Amouroux, Automatique des SystPmes Distribzk, Hem-&s, Paris, 1990. 3 A. El Jai and A. J. Pritchard, Actuators and sensors in distributed systems, Int. J. ConrroI, 46 (1987) 1139-1153. 4 M. Amouroux and A. El Jai (eds.), Proc. Fifih JFAC Symp. Control of Distributed Parameter Systems, ZFAC Symposia Series, No. 3, Pergamon, Oxford, 1990. 5 R. F. Curtain and A. J. Pritchard, Infinite dimensional linear systems theory, Lecture Notes in Control and Information Science, Vol. 8, Springer, Berlin, 1978. 6 Y. Sakawa, Controllability for partial differential equations of parabolic type, SUM J. Control, 12 (1974) 389-400. 7 R. Triggiani, Controllability and observability in Banach spaces with bounded operators, SIAM J. Control, I3 (1975) 462491. On the singularity which 8 A. El Jai and M. Amourow, arises in an optimal feedback problem, MECO Congress, Tunir, Sept. 1982. 9 A. El Jai, Controllability and observability via actuators and sensors for a class of hyperbolic systems, Proc. Fouah IFAC Symp. Control of Distributed Parameter Systems, Los Angeles, CA, U.S_4., June 1986. 10 A. El Jai and J. Bouyaghroumni, Numerical approach for exact pointwise controllability of hyperbolic systems, Proc. Fifih IFAC Symp. Control of DistributedParameter Systems, Perpignan, France, June 1989, IFAC Symposia Series, No. 3, Pergamon, Oxford, pp. 465-471. 11 A. El Jai and M. Amouroux (eds.), Pmt. First Int. IFAC Workshop Sensors and Actuators in Distributed Systems, PeqGgnan, France, Dec. 1987. 12 M. Amouroux and J. P. Babary, On optimization of zones of action for an optimal control problem for distributed parameter systems, ht. .I. Control, 29 (1979) 861-869. 13 A. El Jai and A. Gonzalez, Actionneurs et contralabilitt des systtmes hyperboliques,AP’l; 23 (4) (1989) 369-384. 14 A. El Jai and A. Najem, Optimal actuator Iocation in a diffusion process, Lecture Notes in Control and Information Science, Vol. 62, Springer, Berlin, 1984, pp. 407417. 15 A. El Jai and A. Belfekih, Sur le choix de la rtpartition spatiale du contrble dans les systemes distribues, RAIROIAPII, 21 (5) (1987) 493-505. 16 A. El Jai and S. El Yacoubi, On actuator number in distributed systems, Math. Control Syst. SignalsMCSS, submitted for publication.
I1
Biography AbdeZhaq El Jai is a professor at the University of Perpignan in France. He was a professor at the Faculty of Sciences of Rabat in Morocco until 1985. He obtained his Doc-
torat d’Etat in Automatics (1978) at LAAS/ UPS in Toulouse, France. Professor El Jai has written two books on distributed systems, has been the chairman of various international meetings and is on the IFAC committee on theory.
Distributed
1
l-11
systems analysis via sensors and actuators
A. El Jai IMPIUniversity (Received
of Pepignan,
Avenue
de Wlleneuve,
66025
May 22, 1990; in revised form December
Perpignan
Cddtx
11, 1990; accepted
(France) January
18, 1991)
Abstract The purpose of this paper is to give some results structures and systems analysis. Abstract concepts of to the controllability and observability of systems developed results are illustrated by many examples Keyworak
sensors,
actuators,
system
analysis,
related to actuators described of specific
distributed
1. Introduction Nowadays, the technology and theory of sensors and actuators from their own research area, combining both physical and chemical aspects. Thermal, chemical and mechanical transducers are necessarily linked with electronic circuits, which are very often designed to interface the transducers with the model manipulation implemented in computers. In any case, many scientists work on the conceptual aspects of sensors and actuators, design aspects, sensitivity problems, etc. Sensors and actuators can play a fundamental role in the understanding of any real system. Fundamental aspects of the knowledge of a system are developed by theoreticians from a mathematical point of view, and there is very often a wide gap between this and a concrete comprehension of the system. It is now time to link some of this theoretical work with concrete considerations of input-output problems. This is the purpose of this paper. Sensors and actuators form an important link between a system and its environment. Sensors have a passive role and allow the system evolution to be measured. Actuators have an active role and are used to excite the system. This means that the actuators will appear in a right hand side member of a model and the sensors will define an output
the link between actuator and sensor and sensors are introduced and applied by partial differential equations. The geometries and systems.
systems.
function. In any case sensors and actuators are considered via a space variable. Mathematically speaking, the space variable is present in all systems described by partial difequations, integral ferential equations, integro-differential equations, etc. Let us focus our interest on systems described by partial differential equations. Hundreds of researchers throughout the world work on such systems. Very interesting results related to systems theory have been developed since the 197Os, but most of these results are still inaccessible to engineers or automaticians. This gap can be bridged by the introduction of sensor and actuator concepts. Many papers have been devoted to these aspects. The book by El Jai and Pritchard [l] is focused on this approach and this research area produces more and more results every day. In this paper we shall show that many theoretical aspects of systems theory can be seen from the viewpoint of sensors and actuators and will therefore become more easily accessible to physicists and other scientists. 2. Systems analysis Systems analysis consists of a set of certain notions which enables a better knowledge and understanding of the system to be obtained_ In this Section we recall and study the most important of these notions. In the next
1991
-
Elsevier
Sequoia,
Lausanne
2
Sections we show how they can be linked to sensor and actuators structures. Systems analysis can be done from a purely theoretical viewpoint as for abstract differential equations. But the study may be also become concrete, in some sense, by using the structure of sensors and actuators. Thus one can study the concept of controllability via actuator structures; that of observability via sensor structures; that of stabilizability via actuator structures; the concept of detectability via sensor structures; compensators via actuators and sensor structures; the concept of observer via sensor structures, and so on. More details of these concepts and different theoretical approaches are found in the literature. For an approach through sensors and actuators, see refs. 1-3. In this paper our interest is focused on controllability and observability linked with sensor and actuator structures, their locations and numbers. It is awkward to try to control a system before analysing it and before looking whether or not it is possible to steer ,its state to a prescribed position. It is also important to know if the measurement output has some consistency with respect to the evolution of the system. One has to know if the system is stable or can be stabilized, etc. These notions can be studied only if a model has been established. This model will simulate the system evolution and then the analysis can be done on the model. In this paper we shall try to give some results as rigorously as possible while avoiding too much mathematics. For more details, the reader should consult the references. The model we consider is given by the following differential equation: i(t) =/&z(f) +Bu(t)
0 < t< T
z(0) =zo
(1)
with the output function y(t)=Cz(t)
O
(2)
We shall go into further details of our hypothesis later. The interest of such a model is that it generalizes naturally the finite-dimensional models and concerns many types of partial differential equations, as will be shown later. Moreover, the results are pre-
sented in a homogeneous form for all these systems. The hypothesis is as follows: * In the finite-dimensional case A, B and C are matrices of order (n Xn), (n Xp) and (q X n) such that with some regularity on the input function, the solution of eqn. (1) is given by z(t) = exp(At)zO +
f F
J
exp[A(t - r))&(r)
d7
0
(3) The state space is I??‘, the control space is UP and the output space is Rq. * In the case of partial differential equations one must be more careful with the hypothesis allowing the same solution as (3). We need the different functions to be at least squareintegrable on the time interval (0, 7). The state space 2 is infinite dimensional. If the system (l)-(2) is controlled by p actuators and observed via q sensors we have u
EL2(0, T, W) and y EL*(O, T, UP)
(4)
Moreover, we need some regularity on the dynamics operator-A such that exp(At) is well defined: A is supposed to be the generator of a semi-group (exp(At))l,o such that for any state z ~2, the map t +exp(At)z is continuous. Then the solution (3) is always valid. * In the case of partial differential equations, the state space 2 is a function space. In any case one needs some mathematical hypothesis on 2 such that it becomes suitable for analytical manipulations. Usually one has to consider the space 2 as a separable Hilbert space. Different cases of partial differential equations will be developed below. Let us recall some of the fundamental notions that occur in systems analysis. 2. I. Controllability In the case of partial differential equations one has to consider the fact that it is never possible to steer the system exactly to a given state. Many different definitions (exact, weak controllability) are given and developed in the literature [4-71. We consider the following definition.
3
For this purpose we can suppose that the system (1) is autonomous (U = 0) because of the linearity. In this case the output (2) gives
Definition of controllability The system (1) is said to be controllable on the time interval (0, 7’) if for any desired state zd EZ and E > 0, there exists a control u EL*(O, T; W) such that: 112(T) -zd 11< E
~0) = C exp(-W(O) (8) Let us consider the operator K defined by
(3
y(.)CLZ(O,
K(.):z(O) EZ -
wherez(7J is the state reached by the system (1) when excited by the control u at time T.
T; W)
(9)
such that
This definition means that, given a desired state zd in Z, one can find a control that steers the system to a state which is very close to rd. It is possible to show that the control which steers the system to the desired state zd can be expressed by
~0) = WHO) (10) Obviously the system will be observable if, in some sense, the operator K leads from an initial state z(0) to a unique observation y(.). Mathematically this is equivalent to saying that K is an injection. So we use the following definition.
u(t)=B* exp[A*(T-t)] -1
X
exp[A(T-t)JBB*
exp[A*(T-t)]
x C-G - exp@ T)z,)
elf
(6)
The above formula gives an open-loop control structure. A feedback version is obtained by replacing the initial state by the current one: u(t) = B* exp[A*( T- t)] -1
X
exp[A(T-t)pB*
exp[A*(T-t)]
do
Definition of observability The system (1) with the output (2) is said to be observable if Qr(K)
function
= (01 .
(Ker
(11)
(K) is the kernel of the operator
K={z~Zlfi=O}.)
This definition is equivalent to our previous explanation. In any case it is possible to show that the adjoint of the operator K is given by T
K*z=
exp(A *t)C*z(r) dt
s
(12)
0
X
(zd- exp[A(T--N4~))
(7)
For the proof of these results, one has just to put the control (6) in the system (1) and look at the solution at time T. One must be careful when applying formula (7) because of the singularity problem that occurs in the neighbourhood of t = T (this does not happen in the open-loop case) [S]. 2.2. Observability
When a feedback control law has to be implemented one needs knowledge of the current state; the observability problem occurs when one has to reconstruct the state of the system. The problem is whether knowledge of the dynamics of system (l), together with the output function (2), is sufficient to reconstruct the initial state t(O).
such that the state to be reconstructed given by T
is
-1
exp(A*t)C*C
z(O)=
U0
exp(At) & >
T x
ap(A*t)C*y(t)
(13)
dt
s 0 I.e.,
the
reconstruction
operator
is
(K*K)-‘K*.
Formulae (6) and (13) are the same as in the case of systems described by differential equations (finite-dimensional case). The problem which occurs in the case of partial differential equations (infinite-dimensional case) is that these formulae are not regular. That is, the operator to be inversed is not bounded,
4
for instance, an error on the measurement can lead to an initial state which is not acceptable.
3. Mathematical sensors
concept of actuators and
Actuators (sensors) are the intermediaries between a system and its environment. Their structure will depend on the geometry of the support, the location of the support and the spatial distribution of the action (measurement) on the support. Let Cl be the spatial domain in which the system (1) is modelled (usually 0 is bounded CR” with n < 3). We can have many geometries for the support. Among the current situations, we may have: (i) pointwise or zone actuators (sensors) inside the physical domain a; (ii) pointsvise or zone actuators (sensors) on the boundary of the physical domain Q (iii) filament actuators (sensors) in the domain CR; iv mobile pointwise actuator (sensor) in th; Jomain a. Various situations will be represented in the following Sections. Definitions (1) Suppose that Sziis a closed set included in the geometrical domain R in which the system (1) is modelled and giELZ(Q)+ A zone actuator (sensor) is the couple (Q, gi) where: * C& defines the geometrical support of the actuator (sensor), j, gi defines the spatial distribution of the action (measurement) on the support sLi. For the definition one needs some smoothness for the function gi so that the mathematical techniques are rigorous. (2) A pointwise actuator (sensor) is the couple (b, &b) where: * b E fi is the location of the control (measurement), * &, is the Dirac mass concentrated in b. In the case of a boundary actuator (sensor) the definitions are the same with QcX2 (the boundary of a), or biEan. In the case where the system is excited (observed) byp actuators
(sensors) we shall consider (Q, gi)rciGP with sZicCI and giEL2(Q), Vi = 1, 2, . . . , p_ (3) The actuators (C&n,, gi),4iGP are said to be strategic if the system excited by these actuators is controllable on (0, r) for any T>O. The previous definition means that strategic actuators are those which allow the system to be steered to chosen desired a priori states. (4) The sensors (Di, jJIGiCq are said to be strategic if the system (1) with the output function (2) is observable on (0, r) for any T> 0. The output function (2) depends on the sensor structures. When the sensors are strategic, the knowledge of the output function and the dynamics of the system (1) are sufficient to reconstruct the state of the system, and the formula (13) is well defined. In the following Sections we shall see that these definitions have some consistency with systems analysis. Let us consider some examples of sensors and actuators for classical systems. 3.1. Examples of actuators Zone case in a diffusion system Consider the system excited by p zone actuators (Q, gj)14i
y(x, O)=O Y(&
t)=O
(14) (1%
xecl
(5, t)Eafixlo,
q
The system given by eqns. (14) and (15) models, for instance, a diffusion system excited by p zone burners located inside the domain 0. Such a system is a particular case of system (1). We have to consider Rz = AZ whose
domain
is HA(a)
nH’(O)
and BU zigIgiUi with B:(L*(O, T)y +
9(A)
In this case the hypothesis considered at the beginning of the paper is true. With such
the control
Fig. 4. Pointwise
actuators.
actuators one can consider any other boundary conditions (see Figs. l-3). Cl
Fig. 1. Actuator
Distribution
=I
with circular
Pointwise case in a di$usion system
support.
Consider the system (see Fig. 4) excited by p pointwise actuators (bi, &)rqi
of the ~ontrd
% =Ay+ 56(x-b&,(t) i-
y(x, 0) I I :
’
, =,-‘I
Fig. 2. Actuator
’ ’ ’
a1
al+‘1
with rectangular
Support
of
t)=O
Y(5,
the control
=o
(16)
XEn
(5,~)~~~XlO9 IT
S on the right-hand side is the Dirac mass. This model corresponds to a system excited only at the points bi E Sz (the case of pointwise burners in a diffusion system). System (16) is a particular case of system (1) with
*
and
=1
support.
Bu= 2 S(X-bi)ui with B:RP i-l
Supports
n
AZ = AZ whose domain is HA(n) nN2(fi)
I’ ,
, 0
4 L ,
(x, t)~Rx]0,
1
-%4*
Other boundary conditions can be considered.
of
the controls
Boundary case in a di@sion
system
Consider the system excited by p zone boundary actuators (rl,, gi)r Gi
$
=Ay
y(x, O)=O
Qx]O,
XEa-2
Y(!ff t, =i~l~i(04(t)
Fig. 3. Actuators
in circular
domain.
q (17) (6
t, E X2x
lo, q
where &CC&I and gjEL2(Q Vi, 1
6
on its boundary. This situation, shown in Fig. 5, occurs with many physical systems. We can show that such a system is once more a particular case of system (1). In the case shown in Fig. 6 of p boundary pointwise actuators (b, 6,,), pimp one has to consider the system ;
=Ay
y(x, O)=O
fix]O,
q
XECI
(18)
where bi E Kk ‘di, 1
Fig. 6. Pointwise
boundary
actuators.
3.2. Example of output and sensors Let us give the expression for output function with respect to the sensor structures. We suppose that the system (1) with the output function (2) is observed via q sensors (Di,jJ1 cicg where the (D& Gi G4 are the sensor supports and g)l
If the ith sensor measurement
is of zone
type it gives the
(20) If the jth sensor
is of pointwise
type we have
(21) Finally output Fig. 5. Boundary
zone actuators.
Y (0
the structure
= C-e)
of the operator
C in the
7
desired state. The second result is related to the possibility of an adequate location for a given type of action (burner, etc.) such that the system is controllable. The proof of these results is given elsewhere [l]. Let us now give a result of characterization of the actuator structures such that they are strategic. Let (~~j) be an eigenfunction system of the operator A associated with the eigenvalues (A,), with r,, the multiplicity of A,. Fig. 7. Different types of sensors.
is given Cij =
(23)
by
(cpi, f;.)LYD.) %(bi) (
Then sult:
in the zone case in the pointwise case (22)
where 1
4. Existence and characterization strategic actuators and sensors
of
In this section we shall give some fundamental results related to strategic actuators (the results for sensors are similar). The results are related to the link between the controllability concept and the structure and number of actuators. Result 1 For a given system modelled by (l), if the suPPorts (Q)I 6i -sp of the actuators are given, there always exist distributions (gi)l
we have the following
fundamental
re-
Result 3 The actuators (SL,, gj)l~icp are strategic if and only if: (a) p*swrn (b) rank G, =r, Vn where G, is the matrix of orderp X r,, defined by (G,)ij
= (gi, qnj)Lz(n) =
f n,
g&)cPni(x)
* (24)
l
and l,ci,cr,.
This rank condition links the number, the location and the structure of the actuators to their possibility to steer the system to the desired prescribed states. In the case of pointwise actuators located in bjEn, condition (b) is valid with (Gn)i,j = qnj(bi)
(25)
We shall see later how this result can lead to concrete choices of strategic actuators. These results are the same for sensor structures linked with the observability problem.
5. Examples of practical cases Let us apply the previous results to the case of diffusion systems in one or two dimensions. Similar results for hyperbolic systems are given in refs. 9 and 10. In one space dimension we consider a diffusion system defined in Cnl= 10, 1[ and in
8 the two-dimensional case fi* = IO, a[ x 10, 6[. The time interval is [0, 7J with T>O. Let
(26) and the boundaries
n
~=ail,x]O,
be
and 7
=aGXlO,
77
5.1. Pointwtie actuators Let us consider the system pointwise actuators (b, &Jl~j&
Y@,
fY
0) = 0
Y(E,o=o
by p
(28)
L
In the one-dimensional of the system (28) are AR =-n2rr2
excited
(27)
case the eigenvalues
, n=l , 2 , . . .
(29)
r,, = 1, Vn, associated
with multiplicity the eigenfunctions cp,(x)=&sinnnx,
n=l,
2, . . .
with (30)
So part (a) of Result 3 in Section 4 shows that thep=l actuator may be strategic. Part (b) of Result 3 implies that the actuator (b, 6,) is strategic if and only if: Vn > 1, sin(nd)#O=nd#k~ob4.Q
(31)
Remark The result (31) may be surprising, but for implementation problems one has to consider a finite number of modes so that only a finite number of non-strategic locations for the actuator are to be considered. In the two-dimensional case the system (28) is considered by replacing Q, by Q, and Ci by CZ. In this case the eigenvalues are A,,=-
(
1 rr2
;+g
corresponding
G&~, x2)=
(32)
to the eigenfunctions
&
the multiplicity one except if
nml
sin7
of which
(a2/b2) # a rational
number
.
sin
mm2 b
may be more ( E Q)
(33) than (34)
In this case one actuator may be strategic. Let b= (b,, b2) be the actuator location. The same approach leads to a strategic actuator if and only if @,/a)
and (&lb)
If the actuators the system is
P Q
(35)
are located
on the boundary,
?Y - -Ay=O
at
Y@,
0) = 0
Y(E,
‘)
sz’
=i$lW5--bi)ui(t)
(36)
Ci
Mathematically speaking the Result 3 is more difficult, but the (a) and (b) characterization still works. If the system (36) is two dimensional, for example, then with condition (34) the actuator (b, 6,) with b = (b,, b2) is strategic if and only if (35) is confirmed. Obviously the previous remark still holds. Many other situations with standard geometry which can be obtained by symmetry considerations are developed in ref. 1. For the observability problem, the same characterization results are given elsewhere [l-3, 111. 5.2. Zone actuators With p zone actuators system (25) becomes % - AY =
(C$, gi)lsi_,
the
j$lg&)Ui(t) Qi
Y(X,0) = 0
fY
YG t)=O
ci
(37)
In the one- or two-dimensional cases, the eigenvalues and the eigenfunctions are given as in the previous Section. Let us give details of the case of a2 with condition (34). One actuator (a,, gl) may be sufficient to steer the system to any desired state. The support of the actuator may have any arbitrary shape. We shall consider the following two situations: A rectangular support (see Fig. 2) 0, = fi [q -Ii, ai + ZJccl* i-l
(38)
9
From Result 3, we can show that the actuator (SL,, gi) is not strategic if a,/a (or a,/b) EQ and the function gi is symmetric about the line x=a, (y=a,).
A natural cost function to be considered is the control norm, which in this case means the cost of transferring the system, to the desired state:
s 1
A circular support (see Fig. 1) Let c = (c,, c2) be the centre and C!, =D(c,
r) ca2
of the support (39)
In this case the actuator (&, g,) is not strategic if cl/a (or c2/b) EQ and gi is symmetric with respect to the line x =cl (or y =cl). If the actuators are located on the boundary of s1*, the system is described by ?Y - -Ay=O at Y(X,
0) = 0
i-l2
Y(&
r,=i$l&5(#ui(fl
L
(40)
and the results are similar to those of the distributed zone case. Other practical cases are those where the support may be adjacent to one or more sides of the boundary C2. Consider, for example, the case where one actuator is located on the side (see Fig. 5):
{(x,y)/OGxGa,
y=O}
then the actuator support is [a -II, a 1 + I,] X (0) and it is not strategic if a,fa E Q.
6. Optimization location
of actuator structure and
When one has to conceive a system and then to ‘imagine’ the choice of actuators, all the parameters of the actuator may be in some sense optimized. One has to select the parameters to be optimized, and a cost function related to the objective of the problem, which depends on these parameters. This can lead to very complicated problems. If we consider, for instance, the case of one actuator (C&, gl) which excites the system (37) (with p = l), then we can suppose the following parameters to be unknown: (i) the geometry and location of fir; (ii) the spatial distribution g, on I(z,.
J= llUtl12=
ul(t)’ dt
(41)
0
One can also consider more complicated quadratic functions. We shall now give some of the known methods for solving each of these steps.
6.1. Choice of the geomeny fl,
and location
of
There are two approaches for finding the optimal geometry of the actuator support IR,. The first is parametric and based on the introduction of some parameters (LY,.) which define the geometry and the location a, of the actuator and then to minimize the cost J with respect to the (ai). The second is an ‘optimum design’ technique which needs more mathematics. A deformation field V and the Eulerian derivative of J with respect to !C& in the direction of V have to be considered. Both techniques have been used for some specific problems and implemented for numerical simulations [12-141. Example
of the parametric approach
Let us consider the two-dimensional case and assume that technological constraints prescribe the actuator support to be circular (Figs. 1 and 3) and defined by D(a, R) where a = (a,, a,) E IR and the radius R is such that D(a, R) cfk The optimization problem is then f
min
1(D(a,
(a, a2, R)
J
(42)
R) cfl
where the cost J= llul II2 depends implicitly on the actuator support parameters a,, a2 and R. 6.2. Choice of the distribution g, As in Section 6.1., two approaches are possible [3, 151. The first is once more parametric and based on the identification of some convenient parameters which define the function gl. For example, one can consider, in the one-dimensional case, the function
of the numerical data of the dimensions Sz). For more details see ref. 16.
of
References
Fig. 8. Graph
of g, for different values of the curvature
exp~_a,z2)
g1cQ
{exp[~~(~-x~)2-~l~21-1} 1 -cr,P
=
0 elsewhere if x1 - I ,
(43)
if the zone support is 0, = [x1 -I, x1 + I] where x1 is the centre of the action zone, I is the half length of the action zone; a, is the curvature of the function g1 (see Fig. 8). If the location of the actuator is fixed, only the parameter a, in (43) has to be optimized. This parameter can be chosen in ] - 00, + Q)[. For LY,--+ - 00 the control distribution is of Dirac type and for cyl --f + 00 the control distribution is uniform on [x1 -1, -7t1+I].
7. Conclusions From the engineer’s point of view, a condition that may be surprising is Result 3 of Section 4, which concerns the lowest number of actuators (sensors) necessary to ensure controllability (observability). In practice it is always possible to excite (observe) a system even by one actuator (sensor) or by a finite number p < supr, of actuators (sensors). The problem is lhether the conditions (a) and (b) of Result 3 are ‘reasonable’. In fact it can be proved that it is always possible to ensure controllability (observability) in the sense defined in Section 2 by one actuator (sensor) if the domain CR is replaced by LX* such that the distance d(C.k, a*) between R and OR*is arbitrarily small (up to the precision
1 A. El Jai and A. J. Pritchard, Sensors and controls in the analysis of distributed systems, Ellis Horwood Series in Mathematics and its Applications, Wiley, New York, 1988. 2 A. El Jai and M. Amouroux, Automatique des SystPmes Distribzk, Hem-&s, Paris, 1990. 3 A. El Jai and A. J. Pritchard, Actuators and sensors in distributed systems, Int. J. ConrroI, 46 (1987) 1139-1153. 4 M. Amouroux and A. El Jai (eds.), Proc. Fifih JFAC Symp. Control of Distributed Parameter Systems, ZFAC Symposia Series, No. 3, Pergamon, Oxford, 1990. 5 R. F. Curtain and A. J. Pritchard, Infinite dimensional linear systems theory, Lecture Notes in Control and Information Science, Vol. 8, Springer, Berlin, 1978. 6 Y. Sakawa, Controllability for partial differential equations of parabolic type, SUM J. Control, 12 (1974) 389-400. 7 R. Triggiani, Controllability and observability in Banach spaces with bounded operators, SIAM J. Control, I3 (1975) 462491. On the singularity which 8 A. El Jai and M. Amourow, arises in an optimal feedback problem, MECO Congress, Tunir, Sept. 1982. 9 A. El Jai, Controllability and observability via actuators and sensors for a class of hyperbolic systems, Proc. Fouah IFAC Symp. Control of Distributed Parameter Systems, Los Angeles, CA, U.S_4., June 1986. 10 A. El Jai and J. Bouyaghroumni, Numerical approach for exact pointwise controllability of hyperbolic systems, Proc. Fifih IFAC Symp. Control of DistributedParameter Systems, Perpignan, France, June 1989, IFAC Symposia Series, No. 3, Pergamon, Oxford, pp. 465-471. 11 A. El Jai and M. Amouroux (eds.), Pmt. First Int. IFAC Workshop Sensors and Actuators in Distributed Systems, PeqGgnan, France, Dec. 1987. 12 M. Amouroux and J. P. Babary, On optimization of zones of action for an optimal control problem for distributed parameter systems, ht. .I. Control, 29 (1979) 861-869. 13 A. El Jai and A. Gonzalez, Actionneurs et contralabilitt des systtmes hyperboliques,AP’l; 23 (4) (1989) 369-384. 14 A. El Jai and A. Najem, Optimal actuator Iocation in a diffusion process, Lecture Notes in Control and Information Science, Vol. 62, Springer, Berlin, 1984, pp. 407417. 15 A. El Jai and A. Belfekih, Sur le choix de la rtpartition spatiale du contrble dans les systemes distribues, RAIROIAPII, 21 (5) (1987) 493-505. 16 A. El Jai and S. El Yacoubi, On actuator number in distributed systems, Math. Control Syst. SignalsMCSS, submitted for publication.
I1
Biography AbdeZhaq El Jai is a professor at the University of Perpignan in France. He was a professor at the Faculty of Sciences of Rabat in Morocco until 1985. He obtained his Doc-
torat d’Etat in Automatics (1978) at LAAS/ UPS in Toulouse, France. Professor El Jai has written two books on distributed systems, has been the chairman of various international meetings and is on the IFAC committee on theory.