* Received 6 July 1995; revised 14 August 1997; received in final form 16 February 1998. An earlier version was presented at the 1st IFAC Workshop on New Trends in Design of Control Systems (Slovenice, Slovak Republic, October 1994. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor Tamer Bas ,ar. Corresponding author Dr E. Delaleau. Tel. #33 1 69851715; Fax #33 1 69413060; E-mail de- laleau@lss.supelec.fr. - Institut fu¨r Regelungs- und Steuerungstheorie, Technische Universita¨ t Dresden, Mommsenstr. 13, D—01062 Dresden, Germany. ‡Universite´ Paris-sud, Centre scientifique d’Orsay, Labora- toire des signaux et syste`mes, C.N.R.S.-Supe´lec, Plateau de Moulon, F-91192 Gif-sur-Yvette, France. PII: S0005 – 1098(98)00047 – 8 Automatica, Vol. 34, No. 8, pp. 993—999, 1998  1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0005-1098/98 $19.00#0.00 Brief Paper Some Examples and Remarks on Quasi-Static Feedback of Generalized States* J. RUDOLPH- and E. DELALEAU‡ Key Words—Decoupling; disturbance rejection; generalized states; linearization; nonlinear control systems; quasi-static state feedback; trajectory tracking. Abstract—Three simple nonlinear examples with physical back- ground are considered. They illustrate how generalized state representations, which depend on time derivatives of the input, occur. It is shown with these examples how quasi-static feedback of such generalized states may be used for input—output decoup- ling, disturbance rejection, and linearization—and thus for trajectory tracking.  1998 Elsevier Science Ltd. All rights reserved. 1. Introduction Generalized state representations of nonlinear systems are characterized by a dependence on time derivatives of the input variables: x R  "F  (x, u, u R , 2 , u  ), i"1, 2 , n. Such representations have been considered at several oc- casions in the literature, e.g. Zadeh and Desoer (1963), pp. 40—41, Williamson (1977), Freedman and Willems (1978), Zeitz (1984) and Birk and Zeitz (1988). A formal treatment and justification has been given in the differential algebraic approach to nonlin- ear systems (Fliess 1990; Fliess and Glad, 1993). Unlike the linear ones, nonlinear systems do not necessarily admit a repres- entation of the form x R "f (x, u) (Freedman and Willems, 1978; van der Schaft, 1987; Glad, 1988; Delaleau and Respondek, 1995). The tracking control of an overhead crane seems to be the only technological example considered in the literature in the context of feedback control (Fliess et al., 1993). As in the state representation, time derivatives of the input may also be introduced in the feedback laws: u  "  (x, v, v R , 2 , v  ), i"1, 2 , m. Such equations have been considered in Perdon et al. (1990) for the study of a canonical form for classical state representations x R "f (x, u). Invertible feedback laws defined by these equations are sufficient for the decoupling of classical right-invertible sys- tems (Delaleau and Fliess, 1992). Such a feedback can be seen as being in-between static and dynamic feedback and, therefore, it is called ‘‘quasi-static state feedback’’ (Delaleau and Fliess, 1992). Quasi-static state feedback has been used in several other classical synthesis problems: the disturbance rejection (Delaleau and Fliess, 1994; Delaleau and Pereira da Silva, 1994), the feedback linearization (Rudolph, 1995; Delaleau and Rudolph, 1995; Rudolph et al., 1995; Rothfu{ et al. 1996), and the model matching (Rudolph and Delaleau, 1993). The notion of control- led invariance under quasi-static state feedback has been ad- dressed in Huijberts and Moog (1994) and Andiarti (1995). Most of these results concern the feedback of classical states. The question of equivalence under quasi-static feedback of generaliz- ed states has been considered in Rudolph (1995) and Delaleau and Rudolph (1995). The objective of the present paper is twofold: three tutorial examples serve to show, firstly, how generalized state repres- entations may occur as models of physical systems, and, second- ly, how the quasi-static feedback of such generalized states allows solving several control problems. The advantage of these tools is that neither the state of the plant nor the one of the controller is extended. This allows obtaining simpler control designs and faster loops. The first example is the extension to the three-dimensional space of the above-mentioned overhead crane model from (Fliess et al., 1993). A quasi-static feedback of a generalized state allows transforming the system into a linear controllable system in a Brunovsky´ form. On this basis, asymptotic trajectory track- ing can easily be achieved. The second example is a simple electrical circuit which, as the crane model, does not admit any classical state. A disturbance rejecting and linearizing state feed- back is defined for this circuit. As a third example, a cascade of two chemical reactors is considered. The analysis of the system structure is simplified by interpreting the second reactor as being driven by the first one. A generalized state representation results for the second reactor. It is shown how several decoupling and linearizing quasi-static state feedback strategies can be derived. The control design on the three examples is simplified by their flatness (Fliess et al., 1995a)—see also Rothfu{ et al. (1997) for an elementary introduction—and, thus, their dynamic feedback linearizability (Isidori et al., 1986; Charlet et al., 1989). For the investigation of the circuit example the flatness concept is ex- tended to systems with disturbances. The paper is written in an intuitive language without entering into mathematical details. Everything can be formalized using the differential algebraic approach (see the references cited above) or the differential geometric approach of infinite jets and prolongations (Fliess et al., 1994). The main interest being in generalized state representations and quasi-static state feedback, especially references in which these topics have been studied are given. For the many interest- ing results on the control of classical nonlinear state representa- tions, in particular, in the differential geometric approach, one may see the textbooks (Isidori, 1995; Nijmeijer and van der Schaft, 1990), or (Slotine and Li, 1991). 2. Systems, generalized states, and flatness A(control ) system is defined by its differential equations E  (w, 2 , w)"0, i"1, 2 , N, (1) 993