On the Notion of Duality for Linear Delay Systems

Copyright e> IFAC Linear Time Delay Systems, Grenoble, France, 1998

ON THE NOTION OF DUALITY FOR LINEAR DELAY SYSTEMS H. Mounier· J. Rudolph··

• Departement Automatique-Productique, Ecole des Mines de Nantes, 4, rue Alfred Kastler, La Chantrerie, 44307 Nantes, France, E-mail: [email protected] •• Institut fUr Regelungs- und Steuerungstheorie, TU Dresden, Mommsenstr. 13, D-01062 Dresden, Germany, E-mail: [email protected]

Resume: La notion du dual (ou de l'adjoint) d'un systeme est etendue au cas des systemes lineaires (stationnaires ou non) a retards ponctuels. n en resulte que la representation d'etat duale d'une representation retardee n'est pas necessairement retardee. La dualite entre diverses notions de commandabilite et d'observabilite est exhibee.

Abstract: The notion of the dual (or adjoint) of a system is extended to the case of linear (constant or time-varying) systems with constant pointwise delays. It results that the dual of a retarded state representation needs not be retarded. The duality between several notions of controllability and observability is exhibited. Copyright e> 1998 IFAC Keywords: Adjoint, controllability, duality, equivalence, observability, time delay systems.

1. INTRODUCTION

:i:

= Ax + Bu,

A nice discussion of this subject can be found in Chapter 9 of Kailath' book (Kailath, 1980), and classical work on this is e.g. (Kalman, 1962; Luenberger, 1969; nchmann et al., 1984). Although the notion of adjoint spaces is involved rather than duals (Luenberger, 1969), it seems not to be uncommon to continue to talk about duality.

y=Cx,

where the dual system can be simply defined as

.X.:. . = A T -+ C T -u, X

-y

y = C(t)x,

the adjoint system is obtained by transposition and time reversion:

The duality concept is nowadays folklore in the theory of linear time invariant systems. The probably most frequently used definition is based on the state representations :i:

= A(t)x + B(t)u,

= B T -x.

It is also well-known that this simple approach, based on matrix transposition, is insufficient for a useful definition in the time-varying case. Instead, the notion of adjoint system has to be introduced, as follows. Given the time-varying system

As well-known, based on the duality concept one may establish such useful dualities as the one between controllability and observability, or be25

with F( ft) = (hj ( ft)) a matrix the entries of which are polynomials in the derivation operator ft with coefficients in a differential field k (Le., roughly, a field the elements of which are differentiable and the derivatives are again in the field), and thus linear differential operators. When performing computations on this type of representation one has to multiply the coefficients, and such a multiplication has to be properly defined, namely as

tween (static) state feedback and (static) output injection. It is thus very natural to ask: Is there a reasonable and useful generalization of the duality concept to linear systems with delays? Surprisingly, this question seems to be open. Therefore, an answer is attempted here. The point of departure of the present study has been the module theory of linear systems, as proposed by M. Fliess (Fliess, 1990) and, more. particularly, its use for the study of linear delay systems (Mounier, 1995) and of the duality of delayfree linear time-varying systems (Rudolph, 1996). However, this algebraic framework will not be used here (this will be published elsewhere); instead the results are rel?0rted using only standard linear algebra.

'r/a E k,

(1)

With this one has d () -d az

t

d . = az + az. dt

Observe that the multiplication is commutative if and only if all the coefficients are constants, Le., iff 'r/a E k, il. = O. This is easily seen on the following example:

The structure of the paper is as follows. In the next section the delay-free time-varying case. is briefly reconsidered. In Section 3 a definition of a reverse-time adjoint system for a retarded state representation is proposed. The observation that advances are involved in the corresponding representation leads to the consideration of a notion of equivalence, which is called IS-equivalence, and which - on the basis of invariance w.r.t. to time shift - allows to introduce an equivalent system which again is a delay-system. This is the subject of Section 4. The following section, Section 5, concerns the generalization of the duality concept to general (weak) state representations. In Section 6 the context of state representations is left, and instead the duality of delay systems in any general representation is considered. Section 7 is devoted to the discussion of the resulting duality between several concepts of controllability and related observability notions.

d(d) dt a dt

cP = a.ddt + a dt2

d(d) cP :/; a dt dt = a dt2 .

(With this multiplication the coefficients of the system representation are elements of the noncommutative ring k[ftl.) The duality concept for these systems has been discussed in detail in (Rudolph, 1996), and there the dual (or adjoint) system has been defined by considering the equation

(_yT,uT,O)+WTp(ft) =0.

(2)

Here the differential operators, namely the entries of p (ft) are applied from the right. The dual (or adjoint) system is then defined by going back to a system "with operators on the left":

(?)

+ pT (-1,)

to

~ O.

(3)

This passage "from right to left" requires the of factors and, therefore, the use of transposition: (M N)T = NT MT. Moreover, it leads to the requirement of a proper definition for the commuted multiplication of the polynomial coefficients. One wants to have x . y = y 0 x, with o a new multiplication, which yields commut~tion

2. DUALITY OF DELAY-FREE TIME-VARYING SYSTEMS The basic algebraic properties used hereafter may be found in a book such as the one by P. M. Cohn (Cohn, 1985).

a

As already mentioned in the introduction, the duality concept for (delay-free) linear time-varying systems requires some kind of time reversion. The algebraic reason for this can be understood for example by considering a system with a distinguished input u = (Ul,"" um) and a distinguished output y = (Yl,"" Yp) in a general representation:

0

d dt

d

.

d. a

= dt . a = a + a· dt

d

+ dt 0 a.

(4)

(Algebraically speaking, one has the opposite ring of k[ftl.) One may now rewrite the relations by replacing the multiplication 0 by . and the derivation ft by - ft· Indeed, then a 0 ft is rewritten as a·( -ft) and with a·( -ft) = -fta+il. one recovers the original multiplication rule (of k[ftj). These considerations provide an algebraic explanation for the introduction of reverse time. More details on this subject can be found in (Rudolph, 1996). 26

For the present discussion of the delay systems one may retain the fact that besides matrix transposition also time reversion is required. Furthermore, a formula generalizing (3) will be derived below.

the delay operators in 6 by their inverses, namely the advances in 6- 1 = {c511, ... , <5;1} - where, clearly, <5;1 maps f(t) on f(t + hr), . .. , f(t + h r ) so that <5 i c5;l = 1. With this, together with matrix transposition and the change between B (resp. C) and CT (resp. B T ), one obtains

3. DUALITY OF RETARDED STATE REPRESENTATIONS

_!!-.x = A T ( r 1 )x + C T (6- 1 )u dt

y = B T ( r 1 )x + D T (6- 1 )u.

Guided by the above discussion of the timevarying case, one may now define the dual of a delay-system by taking into account transpositions of matrices and time reversion. A retarded linear state representation (time invariant or not) is given by r

Consider the very simple example: :i;

(5a)

-x

j=OI=O y(t)

L Ci,k(t)X(t - kh i ) +

i=O k=O r L

ND

L Dj,l(t)u(t -lhj )

(5b)

j=OI=O

Using the set 6 = {c51 , ..• , c5r } of the localized delay operators of constant amplitude, Le., operators which map a function f(t) on f(t - hr), .. . , f(th r ), this may be rewritten as: r

:i;

r

NA

i=O k=O r

Y

=L

r

L Ci,kO;X

i=O k=O

(6a)

ND

+ L L Dj,IO~u j=O 1=0

(6b)

In these representations the matrices Ai,k, Bj,l> Ci,k, Dj,l, of appropriate sizes, have entries in the (difference-differential) field of coefficients k, which may for example be taken as the reals in the time invariant case, and as rational functions on some appropriate time interval in the time-varying case. Set A(6) = 2:;=0 2:~':0 Ai,kof, B(6)

The entries of those matrices are elements of the (noncommutative) polynomial ring k[ft,6) and the multiplication satisfies 'Va E k

d

d

dt c5i = Oi dt'

8

(7)

LFj,i(ft, 6)

i=l

The preceding system may then be rewritten as

= A(6)x + B(t5)u, Y = C(6)x + D(6)u

:i;

Zi

= 0,

j

= 1, ... ,N

where the linear difference-differential operators are polynomials in the di~erentiationoperator dt and the delay operators m 6 = {<51, .. · , <5r }, with coefficients in an appropriate differencedifferential field k, as discussed above. This general description includes state representations or

Ij,i

(8a) (8b)

Now observe that here time reversion consists not only in replacing by but also in replacing

ft

y = x.

In order to get a general definition, consider the delay systems in the general representation:

= oi(a)c5i ,

i = 1, ... , r.

= il,

In the preceding example, the multiplication by <5 was use<;! in order to obtain a delay system from a system involving an advance. But of course one could as well have used 152 , 03 , etc. Therefore, one must define an appropriate notion of equivalence, which allows to relate systems obtained by applying (positive and negative) powers of the delay operators on the defining equations.

2:;=0 2:~~ Bj,lo~, C(6) = 2:~=0 2::;0 Ci,kc5f, D(6) = 2:;=0 2:~ Dj,/c5~.

c5ia

Y = x.

4. t5-EQUIVALENCE

=

~ a = a ~ + a,

= t5x.

This is called an adjoint (or dual) of the original delay system. Notice that while the original system has been defined by a retarded state representation, the dual delay system obtained is not of the same form: it is of the advanced type. Actually, it does not even admit a retarded state representation (Mounier, 1995). The generalization of this approach is the subject of the next section.

j=O 1=0

Ne

= <5- 1il,

-<5x

NB

= L L Ai,kOfx + L L Bj.lo~u,

y

One may now wonder, whether there is a way to get an appropriate definition of the dual as a delay system, instead of an "advance system". In the preceding example for instance, one may observe that the advance may be eliminated by shifting the state equation of the reverse time adjoint by applying <5. This yields the delay system

Ne

=L

= u,

(No index is used in the case of a single delay.) Time reversion together with the change between B (resp. C) and CT (resp. B T ) yields

:i;(t) = L

r

(9b)

The latter will be called the reverse time adjoint of the system (8).

NA

L Ai,k(t)X(t - khd + i=O k=O r NB L L Bj,l(t)u(t -lhj ),

(9a)

ft

27

implicit representations of any type. The system variables z = (Zl, ... , z.) may contain states, inputs, etc. Of course, this last representation can be rewritten in a matrix form as F(1I,6)z = 0,

and the definition of F( 11,6) is then clear. Let be S(6) a square (N x N) matrix with entries that are all products of (arbitrary, finite, negative or positive) powers of the elements of 6 and which has a non-zero determinant det S(6) '10.

K(t5), L.:(6) (Mounier, 1995). Notice, that here the representations considered before have been further generalized by introducing time derivatives of the inputs in the output equation - one may see e.g. (Mounier, 1995) for a discussion of this subject.

The reverse time adjoint of the state representation (10) may now be defined as: IJ

_K(0-1)

X = A T (o-l)X + LCT(6- 1)11(i) i=O

Consider now any system S(6)F(1I,6)z

(l1a) 11

= 0,

i.e., related with the one defined by F( ft, 6) via left multiplication by S(6), and call it S-related to the one defined by F. Then, clearly, every system is S-related to itself (with S the identity matrix). Moreover, if G(ft,6) = S(6)F(ft,6), then S(6)-IG(ft,6) = F(ft,6) and S(6)-1 exists and has also entries that are all products of powers of the elements of 6 - this is easy to prove. Finally, if G(ft,6) = SI (6)F(ft, 6) and H(ft,6) = S2(6)G(ft,6), then H(ft,6) = S2(6)Sl(6)F(ft,6), and with SI(6) and S2(6) also their product has entries that are powers of the elements of 6. The relation is thus symmetric, reflexitive, and transitive and one may state the definition: Two systems are called 6 -equivalent if they are S -related in the above sense. The term "6-equivalence" is used because of the intrinsic (representation independent) definition that may be given in the module theoretic approach. It also indicates that this equivalence is based on the fact that the systems are invariant under time shifts, Le., applications of delays and advances - cf. (Willems, 1991).

-6'\Q6- 1 )

x=

T

6>' [A (o-l)X

+ I:f:oCT(o-l)l1(i)]

11 = [B T (o-l)X + I:J=o(-1)iVi(6-1)11(i)]

6>' L.:( 06>'

1

(12a)

)

(12b)

where the multi-index notation ~>.

Cl

_ (/1 1"~1 1"~2 1"~r • (/2 ••• (/r

-

with ..x E N'" is used for the sake of better readability. This latter system is obviously 6-equivalent to (11), by construction. For any sufficiently large .Ai, i = 1, ... , r, one obtains a delayed representation. It is called a dual of the (weak) state representation (10). The dual is therefore defined up to 6-equivalence.

Let be given a linear delay system with a distinguished input u = (u}, ... ,urn) and a distinguished output y = (Y1, . .. ,Yp). This system can be represented by

Not all delay systems admit retarded state representations as discussed in Section 3, but in general also the time derivatives of the state may be delayed (A simple example was given above.). The general (weak) state representations are thus defined as:

x = A(6)x + I:f=oBi (6)u(i) L.:(6) y = C(6)x + I: J=oVi (6)u(i)

Now, with sufficiently large non-negative integers .Ai, i = 1,..., r, the preceding equations can be transformed into a representation of a delay system:

6. DUALITY OF SYSTEMS IN A GENERAL REPRESENTATION

5. DUALITY OF GENERAL (WEAK) STATE REPRESENTATIONS

K(6)

L.:(o-l) 11 = B T (6- 1)x + L( -l)iVJ(o-l)l1 W i=O (lIb)

P (f.,6)

G)

= o.

(lOa)

The dual of this system representation may be defined as

(lOb)

with A(6),Bi (6),C(6), Vi (6), K(6), L.:(6) over k[6] (i.e., polynomials in the delays contained in 6 with coefficients in an appropriate field k), with appropriate sizes. Here, K( 6) and L.:(6) are full rank diagonal matrices. If one of the entries of K( 6) has a degree larger than 1 these state representations are called neutral or advanced depending on the degrees of the entries of the matrices

(~) +6"pT (-f.,6-') '" = 0

(13)

with v = {VI, ... , vr }, the input u = (U1, ... , up), called the dual input, and the output fi = (Y1'··· ,Ym), called the dual output. Here, again, the integers VI,··· ,Vr are to be chosen

28

Again this can be related to the classical notion of observability over k[«5] (see (Lee and Olbrot, 1981)) and be nicely formalized in the module theoretic approach (Mounier, 1995; Fliess and Mounier, 1998). One should also consider the strong relations to the classical concepts in the delay-free case - see e.g. (Fliess, 1990; Willems, 1991; Rudolph, 1996).

large enough in order to get a delay system representation. Moreover, the dual is defined up to «5-equivalence again. One may verify that the previous definition of the dual (weak) state representation fits into the thus-obtained more general definition, as follows. Equation (10) can be rewritten as

-~(~,«5) 0 (K(<<5)ft-A(<<5))) (U) =0. ( -'D(cu,«5) .c(<<5) -C(<<5) ~

It has become apparent in the study of several technological examples of linear delay systems (Mounier et al., 1997; Petit et al., 1997) and underlined in the module theoretic approach (Mounier, 1995; Fliess and Mounier, 1998), that a useful generalization of the previous notions can be obtained by admitting the use of advances in the preceding notions of controllability and observability. This yields the following: A linear delay system with input u = (U1,' .. ,urn), output Y = (Y1,"" Yp), and additional (latent) variables z = (Z1,'" ,Zq) is called 15 -free controllable (Mounier, 1995; Fliess and Mounier, 1998) if the following type of equations is satisfied by the system variables

Now (13) reads

(1)

0=

+

6

V

»)_ (,.,1)

-iJT(_f,.r 1

-IF(-f..rl) 0

( (_.qr1)f._.A(r1»)T

£T(r 1 ) _cT(r ) '

"'2

=

and after introducing x -Wl as the dual state and A = v, it is easy to recover (12). Remark: In an abstract context it is in fact possible to define the dual systems wjthout even distinguishing the inputs and the outputs: it is sufficient to distinguish the observable subsystem (Rudolph, 1996). 7. DUALITY OF CONTROLLABILITY AND OBSERVABILITY

and

For all the notions of controllability and observability considered in this section the references (Mounier, 1995; Fliess and Mounier, 1998) may be considered for further details, examples, and criteria.

where now the entries of all the matrices are polynomials in ft, <51 , ... , <5r and <511 , ... ,15; 1. It is called «5-observable if the following type of equations is satisfied by the system variables

= A linear delay system with input u (U1,'" ,urn), output y = (Y1, ... ,Yp), and additional (latent) variables Z = (Z1,"" Zq) is called free controllable (Mounier, 1995; Fliess and Mounier, 1998) if it admits a representation of the form

Clearly, this can be considered as a natural extension. Its usefulness comes from the fact that, firstly, (as opposed to the free controllability) it is rather frequently encountered in practice, and, secondly, it allows for an elegant approach to path planning and open loop trajectory tracking (Mounier et al., 1997) (which by the way is very much related to flatness of nonlinear systems (Fliess et al., 1995), and also forms a point of departure for the generalization of this notion to systems with delays in (Mounier and Rudolph, 1997) and to boundary controlled linear distributed parameter syste~s (Fliess et al., 1997).).

u = D(ft,«5)w,y = N(ft,«5)w,z = Q(ft,«5)w, with W

= Mdft, «5)u + M2 (ft, «5)y + M3 (ft, «5)z,

where the entries of all the matrices are polynomials in ft, <5 1 , ... ,<5r with coefficients in an appropriate (difference-differential) field k. For retarded state representations, as discussed in Section 3 above, this notion can be shown to correspond to reachability in the sense of linear systems over commutative rings (see, e.g., (Sontag, 1976)). It can be formalized in the module theoretic approach as the freeness of the system module, wherefrom the term (Mounier, 1995; Fliess and Mounier, 1998). Analogously, the system is called observable (Mounier, 1995; Fliess and Mounier, 1998) if it admits a representation of the form

z

= P(ft, «5)u+R(ft, «5)y

A(ft,«5)y

In some cases yet another generalization is helpful, which is called 7r-free controllability, and its counterpart a-determination (Mounier, 1995; Fliess and Mounier, 1998). Here, instead of inverses of the delays, inverses of (more general) polynomials of these delays are allowed. One then has:

= B(ft,«5)u 29

U

• d = D(di' a, 1r -1 )w, •

d

z=Q(di,a,1r

-1

Fliess, M. (1990). Some basic structural properties of generalized linear systems. Systems Control Lett. 15, 391-396. Fliess, M. and H. Mounier (1998). Controllability and observability of linear delay systems: an algebraic approach. COCV (Control, Optimization and Calculus of Variations) . To appear. (URL: http://www.emath.fr/COCV/). Fliess, M., H. Mounier, P. Rouchon and J. Rudolph (1997). Systemes lineaires sur les operateurs de Mikusinski et commande d'une poutre flexible. In: ESAIM Proc. "Elasticite, viscoelasticite et controle optimal", Huitiemes entretiens du centre Jacques Cartier. Vol. 2. Lyon. pp. 183-193. (URL: http://www.emath.fr/proc/VoI.2/). Fliess, M., J. Levine, P. Martin and P. Rouchon (1995). Flatness and defect of nonlinear systerns: introductory theory and examples. Internat. J. Control 61, 1327-1361. nchmann, A., I. Niirnberger and W. Schmale (1984). Tim~varying polynomial matrix systems. Int. J. Control 40, 329-362. Kailath, T. (1980). Linear Systems. Prentice-Hall. Englewood Cliffs, New Jersey. Kalman, R.E. (1962). Canonical structure of linear dynamical systems. Proc. Nat. Acad. Sci. 48, 59OO. Lee, E.B. and A. Olbrot (1981). Observability and related structural results for linear hereditary systems. Internat. J. Control 34, 1061-1078. Luenberger, D.G. (1969). Optimization by Vector Space Methods. WHey. New York. Mounier, H. (1995). Proprietes structurelles des systemes lineaires a retards : aspects theoriques et pratiques. These de doctorat, Universite Paris Sud, Orsay. Mounier, H. and J. Rudolph (1997). First steps towards flatness based control of a class of nonlinear chemical reactors with delays. In: Proc. 4th European Control Conference. Brussels, Belgium. nO 508. Mounier, H., P. Rouchon and J. Rudolph (1997). Some examples of linear delay systems. RAIRO-JESA-APII 31, 911-925. Petit, N., Y. Creff and P. Rouchon (1997). afreeness of a class of linear delay systems. In: Proc. 4th European Control Conference. Brussels, Belgium. nO 520. Rudolph, J. (1996). Duality in time-varying linear systems: A module theoretic approach. Linear Algebra Appl. 245, 83-106. Sontag, E. D. (1976). Linear systems over commutative !jngs: a survey. Richerche di A utomatica 1, 1-34. Willems, J.C. (1991). Paradigms and puzzles in the theory of dynamical systems. IEEE 7rans. Automat. Control 36, 259-294.

)w

and analogous relations for the observability: •

d

z=P(di,a,a •

d

A(di,a,a

-1

-1

•

d

)u+R(di,a,a •

d

)y=B(di,a,a

-1

-1

)y

)u

A first result is the observation: For every afree controllable (resp. a-obsenJable) linear delay system there exists a free controllable (resp. obsenJable) a-equivalent one. This can be very easily seen on the representations used for the characterization of these properties here. Now the following duality result can be stated: A linear delay system is a-determined if and only if it has a 1r-free controllable dual. Conversely, a linear delay system is 1r-free controllable if and only if it has a a-determined dual. Together with the above side result on the equivalence of free controllability and a-free controllability up to 0equivalence one immediately gets the corollary: The dual of a a-free controllable system is obsenJable, the dual of a a-obsenJable one is free controllable. Brief sketch of the proof: Consider P(ft,a)

(~) = 0,

with

v

= (~)

(14)

and set P(ft,a) = (Pv(ft,a) I Pz(ft,a». With w T = (w;,w;) and :vT(_yT, uT) one gets wTP(ft,a)

+ (:vT,O) = 0

Now the system (14) is a-determined if there exists a representation over k[ft, 0, a-I] which does not require latent variables z. Then one has w;Pv(ft,a,a- 1 ) +:vT = 0 and, as Pv is left regular, after "passing to the left" and time shift one has the representation of a 1r-free controllable system.

8. CONCLUSION It seems that with the duality concept proposed here most duality results from the delay-free case can be generalized rather directly to the systems with delays (time-varying or not). The present considerations may be treated in the module theoretic approach. Acknowledgement: This work was partially supported by the G.D.R.-P.R.C. Automatique.

9. REFERENCES Cohn, P.M. (1985). Free Rings and their Relations. Academic Press. Londres.

30