On the Definition of Decoupling for Implicit and Generalized Systems†

Copyright © IFAC System Structure and Control, Nantes, France, 1995

ON THE DEFINITION OF DECOUPLING FOR IMPLICIT AND GENERALIZED SYSTEMSt

E. DELALEAU· and P. S. PEREIRA da SILVA"" • Laboratoire des signaux et systemes, CNRS - Supelec, Plateau de Moulon, F-91192 Gif-sur- Yvette, France. e-mail: delaleau~lss.supelec.fr •• Dep. Eng. Eletronica - Escola Politecnica - Universidade de Sao Paulo, Av. ProJ. Luciano Gualberto, travessa 3, no. 158, Cidade Universiltiria, Sao Paulo - SP Cep 05508-900, Brazil. e-mail: paulo~lac . poli.usp.br

Abstract. A definition of decoupling valid for implicit and generalized systems is presented and compared with many existing ones. It is shown that all these definitions are equivalent for Kalman dynamics, i.e., dynamics with state equations not depending on the derivatives of the inputs.

Resume. Une definition de decouplage applicable aux systemes implicites ou generalises est proposee et comparee la plupart des definitions existantes de decouplage. Toutes ces definitons coincident pour les dynamiques kalmaniennes, c 'est-a-dire sans derivees de l'entree. Keywords. Disturbance decoupling, generalized systems, implicit systems, inputoutput decoupling, time-varying linear systems.

1 INTRODUCTION The problems of input-output decoupling and disturbance decoupling have been studied in great detail in linear and nonlinear system theory. See Wonham (1985), Isidori (1989) or Nijmeijer and van der Schaft (1990) and the references therein for a survey.

1989, Nijmeijer and van der Schaft 1990) can be summarized as follows:

The output y is decoupled from the disturbance w (or y is not affected by w) if for each fixed initial condition x(to) = Xo , and each fixed control function v(·), the output function y(.) is the same for every disturbance function w(·).

This work deals with the definition of decoupling for implicit and generalized systems using the differential algebraic approach of Fliess (1989) . Some antinomies with the standard notion of decoupling are exhibited via a simple example. Consider a control system given by

Y

f(x,v,w), h(x),

The disturbance decoupling problem consists in finding a control law, e.g., a dynamic state feedback in such a way that the output of the closed loop system is decoupled from the disturbance (see Respondek (1991), Cao and Zheng (1992), Huijberts et al. (1992), Delaleau and Fliess (1994), Perdon et al. (1993), Pereira da Silva (1993)).

(1 )

where x = (Xl , . .. , xn) is the state, v (Vl, ... , vm ) the control input , w = (wl, . .. , wq) the disturbance input, and Y = (Yl , . .. , Yp) the output. Essentially, the definition of a disturbance decoupled system (Wonham 1985, Isidori

This notion is perhaps too strong or not adapted for an implicit system or a system whose state variable representation depends on input derivatives (generalized systems) as can be seen on the following example.

tThe work of P . S . Pereira da Silva has been financed by FAPESP , grants 92/4826-2 and 91/0508-3.

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Example. l Consider the linear dynamics

x y

=

v - ~

(2a)

+~

(2b)

x

Based on the last statement, the output of system (2a-2b) is declared to be decoupled from the disturbance. The paper establishes that the two notions of decoupling coincide for explicit systems without derivatives of the input like (1). However, the second notion is in general weaker than the first one, and is also valid for systems described by (algebro-)differential equations and depends neither on initial conditions nor on a particular state variable representation.

where v is the control input, ~ is the disturbance input and x is the state. Let Xo = 0, v(t) == 0, ~l(t) == 0 and ~2(t) == 1. The corresponding output functions will be ydt) == 0 and Y2(t) == 1. Hence, the output is not decoupled from the disturbance. Computing the derivative of the output one gets iJ = v . It is clear that , apart from an influence in the initial condition y(to) , the disturbance ~ cannot affect the evolution of y . Taking the Laplace transform of iJ = v leads to Y(s) = ~ V(s) + OTI(s) , where Y(s), V(s) and TI(s) denotes the Laplace transforms of y , v and ~. So, from an input-output point of view, the output of this dynamics is decoupled from the disturbance ~ . Notice that system (2a-2b) may appear in the context of implicit systems: Setting 6 = x and 6 = x - ~ yields the implicit linear system E~ A~ + B v + H~ , y C ~ + J ~ where

The paper is organized as follows: Section 2 gives a brief summary of the differential algebraic approach to linear (Fliess 1990) and nonlinear (Fliess 1989) systems. The notion of decoupling is presented in the Section 3 and comparisons with most of existing definitions are presented in the Section 4. 2

PRELIMINARlES A. Linear systems (see Fliess (1990) for details)

= = ~ = (~l ,6) , E = (~ ~), A = (~1 ~), B = ( ~ ), H = ( ~1 ) , C = (0 1) , and

2.1 A linear system is a finitely gererated k[ft]module, where k is the field of coefficients. An output of a linear system A is a finite subset y = (Yl, ... , yp) of A. An input of a linear system A is a set e = (el , . .. , e s ) of A such that the quotient k-module A/[e] is torsion. A linear dynamics is a linear system A where an input e is distinguished; it is denoted as A/[e].

J = ( 1 ).

Generally speaking, a system (linear or not) can be described by a set of differential equations of the form F.(y , ... , y (a ;) , v, .. . , v(tJ. ), ::v , ... , t::)'h;), z , ... , z (6;») = 0

i = 1, ... , s, where y , v, and ~ are the external variables, namely the output , the control input, and the disturbance input , where Z = (Zl , . . . , zv) are the latent variables (state variables, internal variables, ...), and where Cli, f3i, /i , 6i E IN , i = 1, ... , s .

From an input-output point of view, one can give the following definition 2 , already considered in (Delaleau 1993, Delaleau and Fliess 1994, Delaleau and Pereira da Silva 1994) :

2.2 Any input can be decomposed into two parts: the control input (or control for short) v = (VI, ... , v m ) and the disturbance input (or disturbance for short) ~ = (t:vl,"" t:vq ) . The disturbance represents the effects, generally undesirable, that influences the system. The control u represents the input variables that can be assigned freely and hence are supposed to be differentially independent. The control and disturbance inputs are supposed not to interact , i.e., [v] n [t:v] = {O} . For the sake of simplicity the disturbance is considered to be differentially independent3 .

2.3 A state of a linear dynamics A/[v, tv] is a set x = (Xl, .. . , xn) of A such that its canonical image in A/[v , tv] is a basis of A/[v , t:v] considered as a k-vector space. A state variables representation of the dynamics A/[v, ~] reads

The output y is decoupled of the disturbance ~ if the system variables do not satisfy differential equations of the form

x= A which depend explicitly on y and their derivatives.

~

X

+

L Bi finite

v{i)

+

L Hi

t:v{;)

finite

or

=

The state is said to be K alman if B; 0 and Hi = 0 for i > 0 and generalized otherwise. For

1 Delaleau

(1993) 2This definition will be stated later in precise mathematical terms .

3This hypothesis is implicit in most of the works.

26

an output y of A the output equation reads y= Cx+

L

D;v(i)

finite

+L

J;c:7(i)

fin ite

B . Nonlinear systems (see Fliess (1989) for de-

tails)

Fig. 1. Canonical form for decoupled system .

2.4 A nonlinear system is a finitely generated differential field extension K/k, where k is the field of coefficients. An output of a linear system K is a finite set y = (YI, . . . , yp) of K. An input e of a nonlinear system K is a set e = (el, . .. , e3 ) such that the extension K/k(e) is differentially algebraic. A nonlinear dynamics is a nonlinear system K where an input e is distinguished . A non linear dynamics with input e is denoted by K/k(e).

and possibly from the choice of the state and other latent variables. It is also valid for implicit or/and time-varying linear systems.

3.3 When the Condition of Definition 3.1 is satisfied, it is possible to exhibit a "canonical" state variables representation showing a decomposition for which the output is clearly seen to be decoupled from the disturbance. As dim A/[v, c:7] = dim A/[y, v, c:7] +dim [y, v, c:7]/[v, c:7], and because [y, v, c:7]/[v, c:7] == [y, v]/[v], it is always possible to construct a state z = (z, z) of A/[v, c:7] such that z = (Zl,".' znJ is a state of [y, v]/[v], and Z (Zl, .. . , Zn,) is a state of A/[y, v, c:7], with nl + n2 = dim A/[v, c:7]. And in that case a state variables representation reads

2.5 As in the linear case any input can be decomposed into two parts: the control input v (VI' ... ' v m ) and the disturbance input c:7 (c:7l, ... , c:7 q ) and the same hypothesis are considered (see Paragraph 2.2) .

=

=

=

2.6 A state of a nonlinear system K/k(v, c:7) is a (non differential) transcendence basis x = (Xl , . .. , x n ) of the extension K/k(v, c:7) . A state variables represention of the dynamics K/k(v , c:7) reads 1:' ( . I.'j Xi,X

, V, . . .

0

, v (Cl;) ,t4i, . .. , ti:7 (13;)) --

(5a) finite

A21 Z + A22 Z +

Z

(4)

+ .L: B21 u(1) + .L: HI c:7(I)

i = 1, . .. , n

finite

where Pi , i = 1, . . . , n are polynomials over k . The state x is said to be Kalman if Qi Pi 0, i = 1, . .. , n , and generalized otherwise. For an output y of K the outputs equations read

y

= =

H J·(y·J' x , v , .

."

vb .) , ~ 1ooV ,

. . . , ~(,,;)) 1000V

=

L

CZ+

(5b)

finite

DI u(l)

(5c)

finite

hence A is decomposed into two subsystems (see [y, v, Z], which is observable Figure 1), El (Fliess 1990) with output y decoupled from c:7, and E2 = [v, c:7, z], which in influenced by the disturbance and is unobservable because Zj ~ [y, v, w], j = 1, . . . , n2 (Fliess 1990).

=

- 0 -

j = 1, .. . , p

3 DIST . DECOUPLING This Section gives a precise mathematical statement to what is an output decoupled from a disturbance in the linear and nonlinear cases. Consequences and equivalent characterizations of these definitions are developped .

B. Nonlinear systems -

The nonlinear counter-

part of Definition 3.1 is

Definition 3.4 Let K/k(v , w) be a nonlinear dynamics with control v and disturbance w . The output y of K /k is decoupled from the disturbance c:7 if the fields k(w) and k(y, v) arek-algebraically

A . Linear systems - Recall the definition already considered in Delaleau and Fliess (1994) .

disjoint.

Definition 3.1 Let A/[v, c:7] be a linear dynamics with control v and disturbance c:7. The output y of A is decoupled from the disturbance c:7 if [y, v, c:7] = [y , v] EI7 [c:7]

3.5 As in the linear case it is an input-output concept, completely independent from the choice of the state and other latent variables. It is equivalent to say that there is no polynomial differential equation of the form (3), with coefficients over k, relating the output, the control , and the disturbance components.

3.2 This is an input-output definition completely independent of the representation of the system

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3.6 For a dynamics K.jk(v, tv) with output y, decoupled from the disturbance, the system K. jk can be decomposed into two subsystems (see Figure 1): the first one is decoupled from the disturbance and its output is y; the second one is influenced by the disturbance and is unobservable. Indeed, because k(tv) and k(y, v) are k-algebraically disjoints and because tr.d o K.jk(v, tv) = n is finite, tr.d o k(y, v)jk(v) = tr.d o k(y, v, Q)jk(v, Q) = nl is finite and nl $ n. Consequently it is always possible to construct a state z = (z, i) of K.jk(v,tv) where z = (=I, ... ,Zn,) is a transcendence basis of the extension k(y, v)jk(v) and i = (ZI,"" zn2) is a transcendence basis of the extension K.jk(y, v, tv), with nl + n2 = n. Notice that the components of i are transcendent over k(y, v, Q) and hence unobservable (Diop and Fliess 1991). With that choice, a state variables representation K.jk(v, Q) with output y always reads !. F 1i ( Zi,Z,V,

...

i

,v (a») = 0

over k(v)(x}. Proof. Using Kahler differentials the statement "k(y, v) and k(Q) are k-algebraically disjoint" is easily seen to be equivalent to [dy, dV] n [dQ] = 0 (see Johnson (1969». In the same manner, the statement "k(y, v)(x) is algebraic over k(v)(z)" is equivalent to [dy] C span,c {dx, [dvn. Assume first that [dy] C span,c {dx, [dvn. As z is a state of the dynamics K./k(v, tv) and because v and tv are differentially independent, it follows easily that [dy, dv] n [dtv] = O. Assume now that [dy, dv] n [dQ] = O. This implies that the K.[~]-modules [dy, dv, dtv]j[dv, dtv] and [dy, dv]j[dv] are isomorphic and hence that [dy, dv]/[dv] is torsion. Consider that [dy] is not contained in spanx:; {dx, [vn. In particular there exists some 7] E [dy] of the

= 2:~=1 aidxi + 2:~1 2:j=o bijdvp) + 2:r=IEj=OCijdQ~j) where ai,bij,cij E K. and

form

(6a)

= 1, ... , nl

F2j(Ei, Z, Z, v, . .. ,v(.8) , t:7, .•. ,t:7 b ») = 0

(Cl,B, ... , cq,B) i= O. Differentiating 7] and using spanx:; {dx} c spanx:; {dz, [dv], dQ} , it is easy to show that any linear combination of the form E7=0 ei7](i), where ei E K., will contain a nonzero term (of maximal order on tv) given by Er=o e",ci,Btv~"'+,B). So the canonical projection of TJ on [dy, dv]j[dv] cannot be a torsion element and there is a contradiction.

(6b)

j = 1, ... ,n2

HIc (Ylc , Z, v, . .. , v(.8») = 0

7]

(6c)

k = 1, ... ,p

where FIi, F2j, and Hk are polynomials over k. Observe that it clear on those equations that y is decoupled from tv.

-

C. Comments and other characterization - This part applies for both linear and nonlinear systems, but for the sake of brevity it is written in the nonlinear language only.

Proposition 3.10 Let K./k(v, tv) be a nonlin-

ear dynamics with state x such that the components of x are algebraic over k(v)(z, tv, ... , tv(a). The output y is decoupled from the disturbance, according to Definition 3.4, if and only if k(y, v)(x, Q, . .. , tv(a-l) is algebraic over k(v)(x, tv, ... , Q(a-I)} .

3.7 In the decomposition of figure 1 the decoupling of the output from the disturbance appears undoubtly. If a system is already given under a state variables model it is would be necessary to perform generalized transformations, involving input (both control and disturbance input) and its derivatives, to obtain the state z = (z, i) from the original state x. That is the source of the antinomy presented in the Introduction. For instance, in Example 1, Z = x + Q and i = 0, and the decomposition reads i = v, y = z.

Proof. It is quite analogous to the proof of the Proposition 3.9 and it is left to the reader. _

3.11 This last Proposition allows to explain the phenomenon of dependence on the initial conditions of the output components on the disturbance. Definition 3.4 implies that k(y, v)/k(v) is differentially algebraic and hence each output component satisfies a differential equation of the form

3.8 For dynamics given under state variable models, the dependence of the output on the disturbance can be checked on the state variables equations . Two cases arise whether the state is classicalor generalized with respect to the disturbance input ({3i = 0 or (3i i= 0 in (4).

(7)

=

where Qk, k 1, . .. ,p, are polynomials over k. This ensures that the evolution of the output is really decoupled from the disturbance, at least for nonsingular trajectories for which the implicit relation (7) can be made explicit with respect to

Proposition 3.9 Let K.jk(v , tv) be a nonlinear

dynamics with state x such that the components of x are algebraic over k( v) (x, tv). The output y is decoupled from the disturbance, according to Definition 3.4, if and only if k(y, v)(x) is algebraic

(It.)

Yk

28

.

to Definition 3.1. The output of a proper linear time-invariant system is decoupled from the disturbance according to Definition 4.2 if and only if this output is decoupled from the disturbance according to Definition 3.1.

However, if the system is given under the form of a generalized state variables representation, according to Proposition 3.10 the output components also satify R le ( Yk(I) ,X,t:7 , .. "tv (0-1) ,U, . ..

,U

(Aktl)

=0

(8)

Proof. As the disturbance input is differentially indepent, [tv] is torsion-free and hence [y, v, tv] = [y, v] EB [tv] is equivalent to [y, V, ri7] = [y, v] EB [w], where - : A ---* A denotes the formal Laplace transform (see Fliess (1994)). As [y, v]j[v] is torsion y = Tu and therefore Definition 3.1 is equivalent 6 to 4.l. The second part of the Proposition follows from the fact that for proper time-invariant linear system described by (10), the matrix T2 in (9) is the Laplace transform of C hto e[(t-T)A] Htv(r)dr in (11) . •

where Ri<, k = 1, .. . , p, are polynomials over k. Setting t = 0 in (8) shows the dependence on the initial conditions of the output and its derivatives on a finite number (given by a-I) of derivatives of the components of the disturbance and on the state and control input and its derivatives. That arises only on state variable representation and that is the reason for which a state dependent definition cannot work for generalized of implicit systems.

4

COMPARISON B. Nonlinear case - In the nonlinear case there exists only state dependent definitions

A. Linear case - In the linear case there exist two definitions of decoupling , one given in the transfer matrix approach, and the other on state equations .

Definition 4.4 7 The output y is decoupled from the disturbance tv if for every initial state xo, for every control function v(·) and for all t ~ 0

Consider a linear time-invariant system with transfer equation

y(t,xo, v , tvl) = y(t, Xo, v, tv2) for

(9)

every

pazr

of

disturbance

functions

tvl( ·), an(-).

where Y(s), U(s), and TI(s), denote respectively the Laplace transforms of the output, the control input and the disturbance input .

In (Perdon et al. 1993, Pereira da Silva and Pinto Leite 1993), the notion of decoupling corresponds to the fact that the output y and its derivatives of any order can be determined as functions of the state and the control input v and its derivatives. This definition may be stated in the differential algebraic language as follows:

Definition 4.1 4 The output is decoupled from the disturbance if T2 == 0 in (9) . Consider a linear time-invariant system with state variable equations Ax+Bv+Htv

Cx

y

Definition 4.5 Let K,jk(v, tv) be a nonlinear dynamics with state x and output y that is classic. Then the output y is decoupled from the disturbance tv if [dy] C span~ {dx, [v]}.

(10)

Definition 4.2 5 The output of (10) is decoupled from the disturbance if

cft

e[(t-T)A]Htv(r)dr

=0

Proposition 4.6 If the output y of a nonlinear system is decoupled from the disturbance tv according to Definition 4.4 then y is decoupled from tv according to Definition 3.4. For a system given under a Kalman form (1), if y is decoupled from tv according to Definition 3.4 then y is decoupled tv according to Definition 4.4.

(11)

to

Proposition 4.3 The output of a linear system is decoupled from the disturbance according to Definition 4.1 if and only if it is decoupled according

6This proof show that Definition 4.1, originaly stated for time-invariant proper linear system extends to timevarying and/ or non proper linear systems. 7Isidori (1989) , Nijmeijer and van der Schaft (1990)

4 Bhattacharyya (1974) sWonham (1985)

29

Proof. (Sketch) Assume that y is not decoupled from f;;;J according to Definition 3.4. If x is a state of the dynamics K/k(v, f;;;J), according to Propositions 3.9 and 3.10 the output components depend effectively on f;;;J and hence y is not decoupled according to Definition 4.4

Diop, S. and M. Fliess (1991) . On nonlinear observability. In C. Commault, D. Normand-Cyrot, l.-M. Dion, L. Dugard, M. Fliess, A. Titli, G. Cohen, A. Benveniste and I. D. Landau (Eds.). 'Proc. of the 1st European Control Conference'. Hermes. Paris. pp. 152-157. Fliess, M. (1989). 'Automatique et corps differentiels'. Forum Math. 1, 227-238. Fliess, M. (1990). 'Some basic structural properties of generalized linear systems'. Systems Control £ett. 15, 391-396. Fliess, M. (1994). 'Une intrepretation algebrique de la transformation de Laplace et des matrices de transfert'. Linear Algebra Appl. 203-204, 429442. Fliess, M., l. Levine, P. Martin and P. Rouchon (1993). 'Linearisation par bouclage dynamique et transformations de Lie-Ba.cklund'. C. R. Acad. Sd. Paris Ser. I Math. t. 317,981-986. Fliess, M., J. Levine, P. Martin and P. Rouchon (1994). Nonlinear control and Lie-Ba.cklund transformations: Towards a new differential geometric standpoint. In 'Proc. 33rd IEEE Conference on Decision and Control'. Lake Buena Vista, FL. pp. 981-986. Huijberts, H. l. C., H. Nijmeijer and L. L. M. van der Wegen (1992). 'Dynamic disturbance decoupling for nonlinear systems'. SIAM J. Control Optim. 30, 336-349. Isidori, A. (1989). Nonlinear Control Systems. SpringerVerlag. Berlin. lohnson, J. (1969). 'Kahler differentials and differential algebra'. Ann. of. Math 89, 92-98. Nijmeijer, H. and A. J. van der Schaft (1990). Nonlin. ear Dynamical Control Systems. Springer-Verlag. New York. Perdon, A. M., Y.-F. Zheng, C. H. Moog and G. Conte (1993). 'Disturbance decoupling for nonlinear systems: a unified approach'. Kyber. netika (Prague) 29, 479-484. Pereira da Silva, P. S. (1993). 'On the nonlinear dynamic disturbance decoupling problem'. J. Math. Systems Estim. Control. to appear. Pereira da Silva, P. S. and V. M. Pinto Leite (1993). Disturbance decoupling by regular dynamic feedback for affine nonlinear systems: a linear algebraic approach. In 'Proc. IFAC-World Congress'. Vol. 6. Sidney. pp. 387-390. Respondek, W. (1991). Disturbance decoupling via dynamic feedback. In B. Bonnard, B. Bride, J.-P. Gauthier and I. Kupka (Eds.). 'Controlled Dynamical Systems'. Vol. 8 of Progr. Systems Control Theory. Birkhiiuser. pp. 347-357. Wonham, W. M. (1985). Linear Muitivariable Control: a Geometric Approach. 3rd edn. Springer-Verlag. New York.

Consider a system, given under Kalman form (1). Assume now that y is decoupled from f;;;J according Definition 3.4. Because of Proposition 3.9 a similar reasoning than the one of paragraph 3.11 can be used to write a differential equation of the form (7). Since y~j) = hjj(x,v,v, ... ,v U- 1 )), all the initial conditions of (7) do not depend on the disturbance and so y is decoupled from f;;;J according to Definition 4.4. •

Proposition 4.7 The notions of decoupling from the disturbance of Definitions 4.5 and 3.4 are equivalent for systems given under Kalman form (1) .

Proof. Apply Kalher differentials to the Condition of Proposition 3.9. •

5 CONCLUSION All the ideas developed here can be adapted to deal with the input-output decoupling problem. The definition of decoupling can be restated in the differential geometric approach of infinite jets and prolongations of Fliess et al. (1993), Fliess et al. (1994)). This allows to consider systems which are not only described by polynomial equations.

REFERENCES Bhattacharyya, S. P. (1974). 'Disturbance rejection in linear systems'. Internat. J. Control 5, 633-637. Cao, L. and Y.-F. Zheng (1992). 'Disturbance decoupling via dynamic feedback'. Internat. J. Systems Sci. 23, 683-694. Delaleau, E. (1993) . Sur les derivees de I'entree en representation et commande des systemes non lineaires. These de doctorat, Universite Paris-XI. Orsay. Delaleau, E. and M. Fliess (1994) . Nonlinear disturbance rejection by quasi-static state feedback. In U. Helmke, R. Mennicken and l. Saurer (Eds.). 'Systems and Networks: Mathematical Theory and Applications (MTNS'93, Volume Il, Invited and Contributed Papers)'. Akademie Verlag. Berlin. pp. 109-112. Delaleau, E. and P. S. Pereira da Silva (1994). 'Conditions de rang pour le rejet dynamique de perturbations'. C. R . Acad. Sci. Paris Ser. I Math. 319, 1121-1126.

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