Some Remarks on Quasi-Static Feedback of Generalized States

Copyright @ IFAC New Trends iD Design of Control Systems, Smoleoice, Slovak Republic, 1994

SOME REMARKS ON QUASI-STATIC FEEDBACK OF GENERALIZED STATES J. RUDOLPH- and E. DELALEAU·· • Inltitut fur S"stemd"namil: und Regelungltechnil:, UnitJersitat Stuttgart, Poltfach 801140, D-70S11 Stuttgart, GERMANY. •• Laborotoire des Signaux et SlIstemes, C.N.R.S. - Supelec, Plateau de Moulon, F-91192 Gif-sur- YtJette Cede:r, FRANCE.

Abstract. Three simple technical examples show the usefulness of, and the need for, generalized state representations of nonlinear systems which depend on time derivatives of the input. Quasi-static feedback of the generalized states is shown to be a powerful means for input-output decoupling, disturbance rejection, and linearization. Keywords. Decoupling, disturbance rejection, generalized states, linearization, nonlinear systems, quasi-static feedback .

sharing some properties of the classical states, w.r.t . state feedback for instance. In this light, the so-called well formed systems have been introduced recently (Rudolph 1993). They generalize the usual dynamics of the type Z = !(x, u) with full rank jacobian matrix ~ to which they are equivalent by quasi-static feedback of x. This class of feedback, which can be seen as in between static and dynamic feedback, has been introduced by Delaleau and Fliess (1992) together with an algebraic interpretation of the well-known structure algorithm. They have shown that any classical right invertible system is decouplable via a feedback of this type . This quasi-static feedback is characterized by the fact that the input u is calculated from the state and a finite number of derivatives of the new input v , and vice versa:

During the last years, some interest has been accorded to so-called generalized state representations of non linear systems. These are state representations involving a finite number of time derivatives of the system input:

°.

. . ... ,u (Q» = , A i (z;,z,u,u, t

%=

1, ... ,n.

They appear very naturally in the differential algebraic approach to linear and non linear systems (see e.g. (Fliess 1989)). The states leading to this type of representations are easily characterized in that framework. Some earlier references mainly concern problems related to observability and observers «WiJliamson 1977, Birk and Zeitz 1988, Zeitz 1984) e.g.). The standard technical example is by now an overhead crane (Fliess et al. 1993). As a consequence, the question has been considered under which conditions there exists a change of state such that the new state is of the classical type, i. e. the corresponding state representation does not involve any time derivatives of the input: Zi = h(x, u), i = 1, ... , n . This problem has been solved recently by Delaleau (1993a) and by Delaleau and Respondek (1993) (see their references for prior results.) . By the way, these results can be interpreted as a solution of the so-called realization problem .

~"Ui , X,V , V , .. . ,v(ro»

= 0,

lPi(Vi, x, U, u,

= 0, i = 1, .. . , m.

... ,

u(r o»

The use of quasi-static feedback for several other classical synthesis problems has been considered: so, the disturbance rejection (Delaleau and Fliess 1993, Delaleau and Pereira da Silva 1994), the feedback linearization (Rudolph 1994), and the model matching (Rudolph and Delaleau 1993). Most of these results concern the feedback of classical states. The question of equivalence under quasi-static feedback of generalized states has been considered in (Rudolph 1994) .

Nonetheless, and as a consequence of these results, not all systems admit classical states. This leads to the problem of systems classification w.r.t. the existence of peculiar types of generalized states,

In the present paper, we discuss several aspects of static and quasi-static feedback of generalized states. The problems we are interested in are 51

input-output decoupling, feedback linearization, and disturbance rejection. We only give rather vague formulations, without entering into mathematical details. Everything can be made precise using the differential algebraic approach.

abIes which are related with the state and the input by relations of the form

Bi(Yi,x,e,e, .. . ,e(P;»=O, i= 1, ... ,p, with coefficients in k .

On three examples, we show how generalized state representations may occur in technical systems. The first one is the above-mentionned overhead crane model from (Fliess et al. 1993) . For this model, we illustrate the usefulness of static feedback of a generalized state for input-output decoupling. Furthermore, we introduce a quasi-static feedback of a generalized state which transforms the system into a linear controllable dynamics in Brunovsky form . The second example is a simple electrical network which does not admit a classical state, either. We design a disturbance rejecting and linearizing state feedback for this network. As a third example, we consider a cascade of two chemical reactors. By interpreting the second reactor as being driven by the first one, a generalized state representation results. We show how several decoupling and linearizing quasi-static feedback strategies can be derived.

=

An unperturbed dynamics, i.e., one with e u, is called fiat if there exists all output '!I. called a fiat (or linearizing) output, with the followiJlg two properties. 1. The components of y are not related by any differential equation P(y, iI, . .. , yb» = 0 with coefficients in k. 2. Any system variable z, i. e., the state x, the input u, and all their derivatives, can be calculated from y and its derivatives, i.e., it satisfies a relation of the type Q(z, y, iI, ... , yb» = 0 with coefficients in k. Such a flat (unperturbed) dynamics is equivalent (by endogenous feedback) to a linear controllable one (cf. (Fliess et al. 1992» . Analogously, we call a perturbed dynamics, i.e., one with W non-void, perturbed fiat if there exists an output y, also called a fiat (or linearizing) output, sharing the two preceding properties with coefficients in the differential field generated by W over k. This can be understood as admitting wand any of its derivatives as parameters in the respective relations.

1 SYSTEMS In this section, we introduce some basic system theoretic notions all of which can be properly defined in the differential algebraic framework (see, e.g., (Fliess and Glad 1993)) or in the infinite dimensional differential geometric framework recently proposed by Fliess et al. (1993) .

2 QUASI-STATIC STATE FEEDBACK Let be given a dynamics with a state representation (1). A new (control) input v = (Vl, ... , vrn) can be introduced via relations of the form

In the present paper, we consider dynamics defined by (generalized) state representations of the form

Ai(Xi,x,e,e, ... ,e(a.» = 0, i = 1, ... ,n

(2)


(1)

.. .,u(r o» = 0, i = 1, ... , m.

(3)

This is done in such a way that 1. x is a state of the dynamics thus-obtained, too, and 2. one can get u from relations of the form

with coefficients, or parameters, in some differential field k (one may think of JR) . The set e = (e 1, . .. , e rn + q ) is called the mput, X=(Xl, ... ,X n ) the (generalized) state corresponding to (1) . A state representation (1) and the corresponding state x are called classical if the orders Qi, i = 1, ... , n are all zero. Note that classical states do not necessarily exist for all dynamics (Delaleau 1993a, Delaleau and Respondek 1993).

Ji(Ui,X,V,V, ... ,v(ro »

= O, i= 1, ... ,m.

We may also introduce a new state

=

(4)

x by

=

Here, the
We distinguish two kinds of inputs, viz. control inputs u = (Ul, ... , urn) and perturbation inputs W == (Wl,"" w q ). These are assumed not to interact, which means that there are no relations between the components of e (u, w) and their derivatives.

=

We may also associate outputs with a dynamics. Such an output is a set Y (Yl, ' .. , Yp) of vari-

=

52

Pereira da Silva 1994, Fliess et al. 1994, Rudolph 1994) . For dynamics with classical state representations, differential algebraic rank conditions have been given for the solvability by quasi-static state feedback of the decoupling problem (Delaleau and Fliess 1993) and of the disturbance rejection problem (Delaleau and Fliess 1993, Delaleau and Pereira da Silva 1994). It has also been shown there that for those systems these problems both can always be solved by a quasi-sI at.ic feedback of a classical state. The solvability of the feedback linearization problem by so-called endogenous feedback has been shown in (Fliess et al. 1992) to be equivalent to the flatness of the dynamics. Results concerning the solvability of this problem by quasi-static state feedback are given in (Rudolph 1994) . These references use the differential algebraic framework . For other results, also in the differential geometric frame, see their references.

u can be calculated without including additional dynamics in the feedback. The fact that, unless Q is void, the representation (1) depends on the perturbation input Q leads to the definition of quasi-static feedback of z with perturbation measurement (in a broader sense). In this case, the relations (3), (4) read
= 0,

for i = 1, ... , m. Once more, one can interpret this as just allowing the perturbation Q and its derivatives to appear as additional parameters in the relations of definition. In the differential algebraic approach this can be done by simply including Q in the differential field of coefficients. More precise differential algebraic definitions, based on the mathematical concept of filtrations, can be found in (Delaleau and Fliess 1992, Delaleau and Fliess 1993, Rudolph 1994) . 3 FEEDBACK SYNTHESIS PROBLEMS In this section we briefly recall the definitions of the three classical control synthesis problems input-output decoupling, disturbance rejection, and feedback linearization.

4

EX. 1: AN OVERHEAD CRANE

Let us derive a state representation of the crane schematically shown in figure 1. The cartesian

A dynamics given by (1) with an output y is said to be (input-output) decoupled if each of the output components depends on only one control input component and if it does effectively depend on this control input component . The decoupling problem now consists in finding a state feedback in such a way that the dynamics is decoupled w.r.t. the new control input. We may say here that in the decoupling problem one is looking for a feedback of z in order that y~6;) = v; with some 6; ~ 0, for i = 1, ... , p.

Y m

Z

Fig. 1. Crane in IR3 coordinates (X, Y, Z) of the load are given by

Consider a dynamics (1) with output y, control input u, and perturbation input Q. The disturbance rejection (or disturbance decoupling) problem consists in finding, if possible, a feedback law and a new control input v, such that the output y is no longer influenced by the perturbation input, but depends on the new control input.

X = Rsin {} cos t/J + Dz Y = R sin t/J + Dy Z = Rcos{} cos t/J Here R is the length of the rope, Dz and D'6 are the horizontal positions of the trolley. From Newtons law we get

The feedback linearization problem consists in finding a state feedback such that the dynamics admits a state representation of a controllable linear dynamics with the new state and the new control input. Without loss of generality, this representation can be taken in the Brunovsky canonical form.

where m is the mass of the load and T is the tension in the rope .

Precise differential algebraic definitions of these problems, as well as several results on their solvability, can be found in e.g. (Delaleau and Fliess 1992, Delaleau and Fliess 1993, Delaleau and

Eliminating T from these equations and choosing Ul = D z , u? = D~ ,and U3 = R as the input, z = ({), RO, t/J, Rt/J) as state variables, and y = (X, Y, Z) as output yields the (generalized)

mX =

-Tsin {} cos t/J

mY = -Tsint/J mi = -Tcos{} cos t/J

53

+ mg

e. =

state representation:

x - RCOS%3 %2 • %2%4 tan %3 %2 = R I

·

%3 •

X.

-

=~ R = - %~.ta.n R

%3

. 9 sm

%1 -

Rjp

=

=

=

e,

DII cos X3

-

x = Rsin %1 COSX3 + D" Y = Rsin %3 + D" Z = RcosxI COSX3 It has been shown in (Delaleau 19936) that there

does not exist a classical state for this system 1 . Nonetheless, it is possible to eliminate either R by in~roducing a particular x = ~l (x, R) or D", and DII via a peculiar i = ~2(x,D""D""DII,iJlI) (see (Delaleau 19936». In order to achieve inputoutput decoupling for this system, we set Vl = X, V2 Y, and V3 Z . This defines a new input which is related to u by a static feedback of the ~eneralized state x . For, one has v = ~(x, u), u = ~(x, v) and, moreover, x is a state for the dynamics with input v, too.

5 EX. 2: NONLINEAR CIRCUIT Consider the circuit of figure 2.

=

The flatness of the crane has been proven by Fliess et al. (1994). In (Fliess et al. 1993) also a linearizing endogenous feedback strategy is discussed in detail. The flatness can be easily seen on the (equivalent) algebraic model equations:

Fig. 2. Nonlinear circuit

(6) (7) (8)

(Z-g)(X-D",)=XZ (Z - g)(Y - Dy) = YZ (X_D",)2+(Y_D II )2+Z2=R2

In order to get a model of this circuit, one considers the idealized model of an operational amplifier and the well established equation of the diode: iD = I,(exp(VD/Vr)-I), where I, and Vr are characteristic parameters of the diode (Hasler and Neirynck 1985) . Moreover, suppose that the two operational amplifiers operate in their linear mode and that the two diodes are identical. Writing down equations for the components and KirTZl/ RI, is Cl ci7, i8 choff laws yields: il -I, (exp(vt/Vr ) - 1), i7 = I, (exp( -vt/Vr) - 1), i3 (Vl - U)/R3, i2 Vt/R2 , i4 -v,/R
Obviously, the cartesian coordinates y = (X , Y, Z) of the load form a flat output. For, one can get D", from (6), DII from (7), and then R from (8) . As we have seen above, the state dimension of the system is four . We introduce a (generalized) state e as 6 X, 6 X, 6 Y, and e4 1'. D!!rivi~g (8) once, we ge.t (X - . D",)(X - D",) + (Y - DII )(Y - D II ) + Z Z = RR and deriving it ~nce mo~e (~ - D",)2 + (X -:- D",)(-?C - D",) + (1' D II )2+(Y-DII )(Y-DII )+Z2+ZZ = R2+RR. A simple calculation now shows that, unless Z 0 or R = 0, we can express Z , Z, and Z as Z = oo(e, u), Z = 01(e, u, u), and Z 02(C u, u, u) .

=

QO({, u)

Now introduce Vl X, V2 Y, and V3 Z as a new input . Of course e is a state of the new dynamics and v can be calculated from u, U and u. Moreover, we get D", from (6), DII from (7), and then R from (8) in terms of {, v, V, and ii . We have thus defined a qua.C;i-stiltic feedback of { which transforms the system into Brullon:kv form . Observe that, as the static feedback of ~ derived above, this feedback also decouples the output y. We thus conclude that, by using generalized states, firstly, the crane is linearizable also by a quasi-static feedback, and, secondly, depending on the choice of such states, one may get a different dimension of the "observable part" of the decoupled system.

D-

- " COS XI

. -gCOS%ISID%3

R% - Eft + D" sin XI sin X3

=

(Q2({, u, u, ii) - g)({3 - U2)

=

=

=

= =

=

~

=

=

=

=

One may choose e = (u , TZl) as input and the voltage v, as output . The voltage u is a control input and the voltage TZl is a perturbation input. One can check that this circuit defines a one dimensional dynamics and one can choose x = v, as the state variable. The corresponding state representation reads

=

The state representation corresponding to reads

=

= =

thus

6=6

x = (R,+R,)Vr R,R,C,

e2 = (Q2({, u, u, ii) -

argsinh (~

__,,_+_u_

g)(6 - uI) Qo({, u)

R.C,

2R, I,

+ C'''') + 21,

R,C,

This state representation depends on the first derivative of TZl and, according to (Glad 1988, Delaleau and Respondek 1993), no classical state does exist for this system .

I The non-existence of a classical state for the planar crane considered by Fliess et al. (1993) has been treated in (Dela.leau and Respondek 1993).

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Setting v = v, = 4>(%, U, Q, w) defines a new control input which achieves the disturbance rejection . This control input is related to u by a static state feedback of the generalized state % with perturbation measurement . For, % remains a state for the input (v, Q), and u = t/;(%, V , =-,0) . As a practical consideration, observe that the disturbance and its first derivative can be measured via the currents i l and is (Q = Rli l and tb = is/Cd. One also easily verifies that the system is perturbed fiat, with v, the fiat output, and that the feedback we have derived does also linearize the system.

= CAI can be used as a desired trajectory for the first one. This trajectory can be imposed (at least asymptotically), via state feedback linearization for instance (see below).

U3

We choose y = (CA2' T2) as the output and design a decoupling feedback in order to be able to design desired trajectories for c~2(t) and T;(t) independently. Introducing new input variables as VI := CA2 = (CAI - CA2)q2

+ rA(cA2, T2)

V2 := T2 = h(CA2' T2) + (4)(CAI , CAd - T2)q2

+ U2

defines a decoupling and linearizing quasi-static feedback of the state (CA2' T 2) as U3 = CAI = (VI +CA2q2 -rA(cA2 , T 2 »/q2

EX. 3: CASCADE OF CHEMICAL REACTORS Consider the model of a cascade of two chemical stirred tank reactors. In both reactors the elementary chemical reaction A - B takes place . If in both reactors the volume is constant one may obtain the following equations (cr. (Aris 1965)): 6

= (CAD - CAI)ql + rA(cAI,Td TI = h(CAI' Td + (To - Tdql + UI

CAI

U2

= -h(CA2,T2)-(4>(U3 , U3)-T2)Q2+V2

(13) (14)

Here we may replace U3 and U3 by using (13) . This yields U2 as a function of the variables VI,VI,V2 , CA2 ,T 2. Alternatively, we might use (9) . Outer loops of the f?rm VI = c~2(t) - AI(CA2 C~2(t)) and V2 = T;(t) - A2(T2 - T;(t)) with AI , A2 > 0 yield linear error dynamics allowing to track the desired trajectories C~2(t) and T;(t) asymptotically. Note that VI = C12(t) - AI(VI c12(t)) can be used then.

(9) (10)

CA2 = (CAI - CA2)q2 + r A(CA2, T2) (12) T2 = h(CA2' T 2) + (TI - T2)q2 + U2 Here CAI, CA2 are the respective concentrations of the reactant A and T I , T2 are the respective temperatures in the tanks 1 and 2. The quantities ql, q2, CAD, and To are constant parameters. The functions r A and h depend on the type of the reaction. Their partial derivatives w.r.t . TI (resp . T 2) are different from zero in the domain of interest . The inputs are the (normalized) heat exchange rates UI and U2 . (11)

In a quite similar manner we may calculate a linearizing feedback law in order to get the control U3 = CAI as required for the feedback loop (13) . One may also use other techniques. Introducing z := CAI as a new state variable allows to calculate TI =: 4>(CAI, z) from (9) and

z=

(:::1 -Ql) z + :~ +(To - 4>(CAI , z))ql

With back

By the series structure of the system , one may regard the second reactor as being driven by the first one (and U2) , as follows . Suppose a controller can be found that achieves tracking of a desired trajectory C~I (t) by using the input UI . We will derive such a control strategy in the sequel. Then , for the second reactor U3 := CAI can be considered as a ( "virtual") control input . Because of (9) , the temperature TI cannot be considered as a second such ("virtual") control input , but one obtains an expression for TI as TI = : 4>(CAI,(:,U) from (9), at least locally. Using this expression, with U3 := CAI, one gets a generalized state representation for the second subsystem as

UI

V3

=

:=

[V3

[h(CAlt 4>(CAI , z»+

+ UI]

.

i, we may calculate a linearizing feed-

+

A (ql -~) z] jar _ aCAI aT!

-h(CAI,4>(CAJ,Z» - (To - 4>(CAI,Z»ql .

(15)

Imposing an asymptotically stable linear error dynamics , analogously to what we did above for the first reactor, we achieve U3 = C~ 1 (t) (VI + CA2q2 - r A( CA2, T 2))/ q2= C~2(t) - Al (CA2 C~2(t)) + CA2Q2 - r A(CA2 , T2))/Q2 asymptotically. Note that, because the input U3 = CAI from (13) is realized only asymptotically, while CA2 does not influence T2 the converse does not hold true. Note, moreover , that these are "purely local" results. Therefore, the linear closed loop for the subsystem with state CAI , CAI , corresponding to the first tank , should be designed with a sufficiently fast dynamics and the desired trajectories must be chosen with care.

=

= (U3 - C,t2)q2 + rA(cA2 , T2) T2 = h(CA2, T 2 ) + (4)(U3, U3) - T2)q2 + U2

CA2

where now U2 and U3 are considered as the inputs. Only if 4> is affine in its second argument (This is usually not the case.), the derivative CAI can be eliminated by a change of state coordinates (Glad 1988, Delaleau and Respondek 1993).

We come now back to the four dimensional original dynamics (9)-( 12) . A classical decoupling strategy would be as follows . The control U2 is calculated using equation (14) which yields T2 V2 .

If now, by using the two inputs U2 and U3, a control strategy is found for the second reactor , this

=

55

In order to calculate "lone has to calculate the third derivative of C..n . Introducing v" := c~J allows to solve for "1 as

Delaleau, E. and M. Fliess (1992) . 'Algorithme de structure , filtrations et decouplage' . C. R. Acad. Sci. Pari" 315, Serie I , 101-106. Delaleau, E . and M. Fliess (1993) . Nonlinear disturbance rejection by quasi-static state feedback . In 'Proc. MTNS'93, Regensburg' . To appear. Delaleau, E. and P. S. Pereira da Silva (1994) . Rank conditions for the dynamic disturbance decoupling problem. In 'Proc. 33rd IEEE CDC'. To appear . Delaleau, E. and W . Respondek (1993). 'Lowering the orders of derivatives of controls in generalized state space systems' . J. Math . S""tem" Edim . Control. To appear . Fliess, M. (1989) . 'Automatique et corps differentiels'. Forum Math. 1, 227-238 . Fliess, M. and S .. T . Glad (1993). An algebraic approach to linear and nonlinear contro!' In H. 1. Trentelman and J . C. Willems (Eds.). 'Essays on Control: Perspectives in the Theory and its Applications '. Vo!. 14 of Progre"" in S"". tem" and Control Theory. Birkhauser. Boston. pp. 223-267. Fliess, M., J. Levine and P. Rouchon (1993) . 'Generalized state variable representation for a simplified crane description' . Int . J . Control 85, 277283. Fliess, M., J. Levine and P. Rouchon (1994) . 'Flatness and defect of nonlinear systems: Introductory theory and examples' . Int . J . Control. To appear. Fliess, M., J. Levine, P. Martin and P. Rouchon (1992) . 'Sur les systemes non lineaires differentiellement plats' . C. R . Acad. Sei . Pari" t. 315, Serie I, 619-624. Fliess, M., J. Levine, P. Martin and P. Rouchon (1993). Towards a new differential geometric setting in nonlinear contro!' In ' Internat. Geometrical Coil.'. World Scientific. Singapore. To appear . Glad , S. T . (1988). Nonlinear state space and input output descriptions using differential polynomials. In J . Descusse, M. Fliess, A . Isidori and D. Leborgne (Eds.). 'New Trends in Nonlinear Control Theory' . Springer-Verlag. pp. 182-189. Hasler, M. and J. Neirynck (1985) . Circuit.! non line· aire". Presses Poly techniques Romandes, Lausanne. Rudolph, J. (1993) . ' Une forme canonique en bouclage quasi statique'. C. R . Aead. Sei . Pari" t . 316, Serie I, 1323-1328. Rudolph, J. (1994) . 'Well-formed dynamics under quasi-static state feedback' . Submitted. Rudolph , J. and E . Delaleau (1993) . Nonlinear right-model matching by proper compensation or quasi-static feedback. In J. W . Nieuwenhuis, C . Praagman and H. 1. Trentelman (Eds.). 'Proc . 2nd European Control Conference' . pp. 1528-1533. Williamson, D. (1977) . 'Observation of bilinear systems with application to biological control'. Au· tomatica 13 , 243- 254 . Zeitz , M. (1984) . 'Observability ciUlonical (phasevariable) form for non-linear time-variable systems' . Int. J. Control 15, 949-958 .

(16) We also observe that y = (CA2 , T 2 ) is a flat output. The variable CA1 can be calculated with (13), then T1 = t1J( CA 1, CA 1) allows to get also T1 as a function of CA2, T2 and their derivatives. The inputs are obtained from (14) and (15) . The feedback (14),(16) is a quasi-static feedback of the classical state (CA1,T1 , CA2,T2 ) . Observe that the state representation corresponding to the state (T2 , CA2, CA2, CA2) is linear (in Brunovsky form). But this state is related to the original one by a generalized state transformation . One may also express (14),(16) as a quasi-static feedback of this generalized state. This corresponds to first introducing the generalized state and then using the feedback of that state. From a stabilization point of view, the feedback linearizing the whole system may in general be preferable to (14),(15) . Nonetheless, this type of feedback may be much more difficult, or impossible, to achieve in examples with higher dimension. One easily verifies that fi = (CA1, CA2) is a flat output, too. Analogous to what we did for the first subsystem, we may introduce CA2 as a new state component. Introducing Vs := CA2 then allows to calculate a linearizing static state feedback "2 = CP(CA1,T1 , CA2,T2 , V3 , vs) . But, of course, this feedback does not decouple CA2 and T2 . Finally, one may also consider To as a perturbation input instead of a parameter . One easily verifies that then the system is perturbed flat , and the feedbacks defined above are feed backs with measurement of the (easily measurable) perturbation To . From a practical point of view, the situation is more difficult for the other parameters, because their derivatives will be required for the feedback. REFERENCES Aris, R. (1965) . Introduction to the Anal""i" 01 Chem. ical Reactor". Prentice-Hall, Englewood Cliffs. Birk, J. and M. Zeitz (1988) . 'Extended Luenberger observers for nonlinear multivariable systems'. Int . J. Contro/47 , 1823- 1836 . Delaleau, E. (1993a) . Lowering orders of input deriva· tives in generalized state representations of nonlinear systems. In M. Fliess (Ed .). ' Nonlinear Control Systems Design 1992'. Pergamon Press. Oxford . pp. 347-351. Delaleau, E . (1993b). Sur le" derivee" de I'entree en repre"entation et eommande de" """teme" non linea ire". These de Doctorat , Univ . Paris XI.

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