2418 Copyright @ 1996IFAC 2b-144 13rh Triennial World Congress. San Francisco. ESSENTIAL ORDERS AND NONLINEAR DECOUPLING C. Califano' , S. Monaco' and D. Normand - Cyrot ·· * DipoHJ'tilJlE'ut() di Jnfol7lwtim fl Si. ... r,flm.i..'itic8, U.niveJ'sita di Rnmll: "La Sapi wlZI L". \111 tm/os.'ii8.UI/ 18. (/(}Jej4 Rmllll, ITALY Phone: +J9 ' 6 -4458[j3S0, E-mail PJJul), ': +39-6-44585873, [-nail lDonacoGitcas pur. c aspur . it *. dt -s SiglJlll1x S},stimws, CNRS- ESE. 1-'lntellu de ],.fo U/(J II . 91192 C if s ur Yvdt!' Crx/f!x. FRANCE Plu'IH': +33-J6 ·· 9851748. E- mail cyroU . lss.supele c. tr Abst ract. Tbe concept of es .. 'ientiai orders, recently introduced in is her{! ()xtended to the nonlillear discrete-time context . On this basis it is shuWIJ bow to compute a minimum order compensator for soh 'ing the restricted noni.cteracting control problem for right-invertible submersive systems. Keywords: Noninteracting control. differential algebra, structure at infinity, discrete-tiloe systems. INTRODUCTIO N During the last two decades nonlinear systems have been widely investigated, either in a geometric or in an alge- braic framework. leading to many interesting results. For nonlinear continuous-time systelns, the assumption of dynamics affine with to th e control variables, nat urally suggests the use of differe nti al geometric tools a nd algebraic concepts to sol ve mall Y synthesis problems of gre at interest: disturb ance and noninter- acting control, adaptive control, output regulation, Ho::; control, between the others (Isidori. 1989; Nijmeijer and V"dIl der Shaft , 1990; Battilotti. 1994 ; Marino and Tomei, 1995). Unf ortunately, in a discrete· time c. ontext, once the lin- e ar structure is lost , the analysis requires an heavy te<:h- nical apparatus . Th;' aspect d earl)' appears from the first works (Monaco and Normand- Cyrot , 1983; Griz- zle, 1986a). It follows th at although the conditions for th e existence of the solution of many important (;ont.rol This work hR.CI betm partia ll y s upport· ed by itaJian fund .. M.U. R.S.T.40 % problems can always be forniUlated in t erms of differ- ential geometric c;on cepts it i:) a lmost a.lways compu ta- tionally ha rd t.o work out. f" ..xplicit solutions. Differenti al algebra allows t(') uvercome, to some ex- tent , some of the!)e difficulties; as alread y underlined in (Fliess. 1990; Grizzle, 199:3; Aranda-Bricaire et a l. , 1994:), where computations based 011 the differentials of the varia.bles lead to a linear algebraic: frA..mework. Along these lUl e!:i we hereafta in the treate- ment of disc rete-time sys tems, the conce pt of essential orders. The main goal is to ::- how how such integers I in analogy with the co ntinuous-time case (Glumineau and I VIoog , 1!)89; et al. , 1992), gi ve rise to easier comput a tion of a minimum order compensa tor for t he restric te d noninter act ing con1 ro1 problem. The procedure, which from the structure a l- gorithm as proposed in GrizzlE \ (1993) improves pr evi- ous result .s (Monaco and \"ormand-Cy rot , 1984; Gri;6- zle, 1986b; Monaco and Nonlland 1987; Lee and Marcus, 1987; l\.·ionaco et al. : 1989) and can moreover be applied to solve r r. lated The paper lli organised as foll ows. In section 1 the al· gebraic framework in which the work is developed is