Memorandum COSOR Controlled invariance of nonlinear systems: nonexact forms speak louder than exact forms H.J.C. Huijberts C.H.
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1 EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of 11athematics and Computing Science Memorandum COSOR Controlled invariance of nonlinear systems: nonexact forms speak louder than exact forms H.J.C. Huijberts C.H. Moog Eindhoven, March 1993 The Netherlands
2 Eindhoven University of Technology Department of Mathematics and Computing Science Probability theory, statistics, operations research and systems theory P.O. Box MB Eindhoven - The Netherlands Secretariat: Telephone: Dommel building ISSN
3 Controlled invariance of nonlinear systems: nonexact forms speak louder than exact forms* H.J.C. Huijberts t C.H. Moogt Abstract A general setting is developed which describes controlled invariance for nonlinear control systems and which incorporates the previous approaches dealing with controlled invariant distributions. The main feature of the theory developed is that it is able to clarify the controlled invariance of distributions which are not necessarily integrable. The latter are of major importance for the geometric description of e.g. dynamic feedback problems. 1 Introduction During the last two decades, nonlinear control theory,vas developed thanks to the increasing number of researchers involved in this area. A main stream of the research in the 80's was the generalization, at least partially, of the so called geometric approach which proved to be particularly efficient for linear time-invariant systems (see [17] for an overview). In this linear theory, controlled invariance is a fundamental notion. The study of controlled invariance for nonlinear systems of the form :i; = f(x) +g(x)u (1) where x E lr n, U E lr m, was initiated in [2]. In this paper invariants were sought under feedback transformations U = a(x) +v (2) Later on, controlled invariance was tackled by various authors ([11],[7],[13],[14]). The group of feedback transformations acting on (1) was enlarged to transformations of the form u = a(x) + j3(x)v (3) where j3( x) is square and locally invertible. These works yielded the definition of a controlled invariant distribution. The key was found for the solution of synthesis problems, such as the disturbance decoupling problem and the noninteracting control problem, via regular (or invertible) static state feedback (see the textbooks [9],[15] for an overview). "The authors thank HP, Harm van Essen and Eduardo Aranda for suggesting the title of the paper. tdepartment of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. ILaboratoire d'automatique de Nantes, Ecole Centrale de Nantes/Universite de Nantes, Unite Associee au C.N.R.S. 823, 1 rue de la Noe, Nantes Cedex 03, France. Research was partly performed while the second author was a Visiting Professor at Eindhoven University of Technology, supported by the Dutch Systems and Control Theory Network. 1
4 Some limits of this by now well established theory appeared at the end of the 80's in the characterization of left- or right-invertibility for nonlinear systems or for synthesis problems involving dynamic feedback. A nice understanding of these problems is provided by a differential algebraic theory ([6]). The goal of this paper is to introduce a generalized notion of controlled invariance by allowing an enlarged class of feedback transformations acting on (1). The motivation is to clarify a number of pathological situations. Relations exist with both the differential geometric and the differential algebraic approach, but these will not be outlined in this paper. In the sequel we consider a nonlinear control system (1), where the entries of j(x) and g(x) are meromorphic functions of x. It is assumed that rank g(x) = m and that n ~ 1. In Section 2 we define the generalized notion of invariance with respect to the dynamics (1). Section 3 is devoted to controlled invariance and related properties. 2 Invariant subspaces vve follow the notations and setting of [5]. Let K denote the field of meromorphic functions of x, u, u," " u(n-l). [; is the formal vector space spanned by {dx, du, du,' ",du(n-l)} over K. The notation dx stands for {dxi,''', dxn} and duck) for {du~k),..., d~)}. Let X := span,,:{dx} and U := span,lc{du,du,'",du(n-l)}. Consider a subspace n c X. Define n= span,lc{w IwEn} (4) n where W = 'L Wi(X, u, u,' ", u(n-l))dxi and time-derivation is defined by W = 'Li=l (Widxi + i=l Widxi)' Thus W E span,lc{dx,du}. Definition 2.1 A subspace n C,:t' is said to be invariant with respect to (1) if nc n + span,lc{du} (5) Remark 2.2 Let K k be the field of meromorphic functions of x, U,' ",u(k) and define K' = U Kk ken Then (5) is equivalent to the statement that (n + span,lc,{du(k) I k ~ vector space, with the derivation defined above. O}) is a differential Example 2.3 Let ~ be an involutive invariant distribution for (1) and let (XI, X2) be a local system of coordinates such that ~ = span{8~)' Then in the coordinates (XI,X2), (1) takes the form (d. [9],[15]) Xl =!l(xl) +9l(Xl)U (6) X2 = hex!, X2) +92(Xl' X2)U Let n := span,lc{dxd. Then n= span,lc{dxd = span,lc{d(jl(xd + gl(xl)u)} c n + span,lc{du} (7) Hence n is invariant in the sense of Definition 2.1. By means of n, the invariance of ~ is described in a dual form. \Vhen a given subspace is not invariant, it is interesting whether or not there exists a feedback transformation that renders it invariant. This is the topic of the next section. 2
5 3 Controlled invariant subspaces In this section we define and characterize the controlled invariance of subspaces 0 C X under quasi-static state feedback. In Subsection 3.1 we first define quasi-static state feedback, based on [3],[4]. In Subsection 3.2 we give a definition of controlled invariance under quasi-static state feedback. We then investigate when subspaces are controlled invariant under a regular static state feedback (3). These feedbacks are a special case of quasi-static state feedback. It will turn out that the conditions for controlled invariance under regular static state feedback allow us to formulate a condition for controlled invariance of (not necessarily involutive) distributions via regular static state feedback. Subsection 3.2 is concluded with a condition for controlled invariance of subspaces 0 C X under quasi-static state feedback. In Subsection 3.3 we make some remarks about the smallest controlled invariant subspace containing some given subspace. Throughout this section we employ the following terminology. A vector w C [; is called exact if there exists a 4> E JC such that w = d4>. A subspace 0 C E of dimension r is called exact if there exist functions 4>1"",4>1' E JC such that 0 = spanjc {d4>1,..., d4>1' }. Given subspaces 0 1 C O2 C [;, (02/01) is said to be exact if there exist functions 4>1,"., 4>d E [;, with d = dim(02) - dim(ot), such that O2 = 0 1 EB spanjc{d4>l,'",d4>d}, or, in other words, (02/01) is isomorphic to an exact subspace of E. Consider a subspace 0 C E. Then clearly {O} C 0 is exact. Furthermore, if 0 1 C 0, O2 C 0 are exact, then also 0 1+O2 C 0 is exact. Hence there exists a unique maximal exact subspace in O. 3.1 Quasi-static state feedback Consider the nonlinear system (1). A generalized static state feedback for (1) is a feedback of the form u= (x,v,...,v(1')) where v E JRm denotes the new controls. Let JCv denote the field of meromorphic functions of {x, {v(k) I k ~ On and define the formal vector space [;v := spanjcjd~ I ~ E JC v }. As in [3],[4], we define the following filtrations ([1]) of [;v: V-I Vk.- spanjcjdx}.- spanjcjdx,dv,...,dv(k)} (k~o) (8) (9) U-1.- spanx::jdx} Uk.- spanx::jdx,d,...,d (k)}(k~o) (10) The filtrations Uk and Vk are said to have bounded difference ([1]) if there exists an s E IN such that for all k ~ -1 Uk C Vk+s Vk C Uk+s (11) Definition 3.4 ([3],[4]) u given by (8) is said to be a quasi-static state feedback for (1) if the filtrations Uk and Vk have bounded difference. Remark 3.5 It is easily verified that a regular static state feedback (3) is a quasi-static state feedback. 3
6 The following result is also easily proved. Proposition 3.6 u given by (8) is a quasi-static state feedback if and only if locally there exist an s E IN and a function 7/;(x, u,..., u(s)) such that locally (8) is equivalent to v=7/;(x,u,...,u(s)) (12) 3.2 Controlled invariance Definition of controlled invariance Consider the control system (1) together with a quasi-static state feedback (8) and define V := spankjdv(k) I k ~ o}. Definition 3.7 A subspace n c X is said to be controlled invariant for (1) if there exists a quasi-static state feedback (8) such that for (1,8) one has The definition of controlled invariance given in Definition 3.7 is in accordance with the well known definition of a controlled invariant distribution. Recall from e.g. [9],[15] that a distribution,6 is controlled invariant if there exists a regular static state feedback (3) such that (13) [f+go:,,6] c,6 (14) where (g{3)*i denotes the i-th column of (g{3). Define n =,61. = {w I (w,r) == 0, 'ir E~} Then for n, (14) translates into Lf+gan c n 1:-(9{3).in c n(i=l,..,m) (15) (16) Let wen. Then for (1,3) we have m W= I:- f+gaw+i)vil(g{3).iw + (w, (g{3)*i)dvi) i=l From this equality it immediately follows that ~ is controlled invariant if and only if (13) is satisfied for n =,61.. (17) 4
7 3.2.2 Controlled invariance via regular static state feedback In this subsection we investigate under what conditions a subspace n C X is controlled invariant under regular static state feedback. Recall from Subsection that a regular static state feedback is a special sort of quasi-static state feedback. A first result is the following. Proposition 3.8 Consider a d-dimensional subspace n ex. Assume that n is controlled invariant under a quasi-static state feedback of the form u = <p(x, v). Then n admits a basis WI,'",Wd with n Wi = EWij(x)dxj j=1 Proof Assume that n = span}c{wl,'",wd}, with n Wi = EWij(X,u)dXj j=1 Let A(x, u) be the matrix with entries Wij (i = 1" subspace (over IC) of X 6? span}c{du}, it may be characterized by (18) (19)..,di j = 1",., n). Viewing n as a linear n = rowspan}c (A(x, u) 0) Similarly, nis characterized by (20) where n= rowspan}c(b(x,u,u) (Ag)(x,u)) n 8A m 8A ( n 8 ) B(x, u, u) =E "1j":(x, U)Xi(X, u) +E ~Uj + A(x, u) fx(x) +E ag. u (22) i=1 X t j=1 uj i=1 X t with Ix the Jacobian of f. Since n is rendered invariant via u = (x,v) there exist matrices P(x,v,v) and Q(x,v) such that or B(x,, ~)dx + (Ag)(x, <p)d = P(x, v, v)a(x, )dx + Q(x, v)dv B(x, <p,~) = P(x, v, v)a(x, ) - (Ag)(x, <p) x(x, v) (Ag)(x, <p)<pv(x, v) = Q(x, v) Since <Pv(x, v) is invertible, this yields B(x,,~) = P(x, v,v)a(x, <p) - Q(x, v)<pv(x, v)-i<px(x, v) Furthermore, by the Inverse Function Theorem there exists a function 1/J(x, u) such that u = (x,v) is equivalent to v = 1/J(x,u) and 1/Jx(x,u) = -<Pv(X,1/J(x, U))-I<px(X, 1/J(x, u)). Hence (25) yields (21) (23) (24) (25) B(x, u, u) = P(x, u, u)a(x, u) +Q(x, u)1/jx(x, u) (26) 5
8 where P(x, u, it) = P(x, 'I/;(x, u),, (x, u, it) and Q(x, u) = Q(x, 'I/;(x, u)). Taking partial derivatives with respect to iti, we obtain Obviously, 2-8A 8P -;:;- = ~A(x, u) (i = 1"", m) uui UUi =0 8~:' (i,j=l,..,m) Ui Uj Hence there exist matrices RiC x, u) (i = 1" 8A -;:;- = Ri(x, u)a(x, u) uui.., m) such that Using arguments from the theory of linear time-varying ordinary differential equations this yields that A(x, u) is of the form A(x,u) = ii>(x,u)\lj(x) with ii>(x, u) square invertible. Hence n = rowspanx; (A(x, u) 0) = rowspanx; (\lj(x) 0) (29) which establishes our claim. If n = rowspanx; (A(x, U,"', u(l)) 0) with.e > 1, the claim is established by using the same arguments together with an induction argument. For (1), let 9 denote the distribution spanned by the input vector fields. Define the subspace g.1 C X by 9.1 = {x E X I (w, g) == 0, Yg E 9} (30) Theorem 3.9 Let n c X be a subspace which admits a basis satisfying (i8). Then n is controlled invariant if and only if -- (i) (nn9.1)cn (ii) nnn is exact. Moreover, ifn satisfies (i) and (ii), then it can be rendered invariant via a regular static state feedback. (27) (28) Proof (sufficiency) Assume that (i) and (ii) hold. N?te that n +nc spanx;{dx, du}. Let n c X be such that n = (n n g.1) EEl n. Assume that n n X # {O}. This implies that there is an wen, w# 0, such that ~ E X and hence we (n n g.1), which gives a contradiction. Thus nn X = {O} (31) By (ii), there exists a VI (x, u) such that n +n=n EEl spanx;{dvd (32) 6
9 Since (i) and (31) hold, we must have that (8vd8u) has full row rank. Then there exists a V2(U) such that (8vj8u) is square and invertible, where v = (v[ vn T. By (32) we now have that Moreover, since (8v j 8u) is invertible, there exists a 'Ij;( X, v) such that u = 'Ij;( X, v). Hence 'Ij; defines a quasi-static state feedback and thus n can be rendered invariant via quasi-static state feedback. Since we are dealing with a control system (1) that is affine in u, it is easily seen that v can be taken affine in u and thus 'Ij; can be taken affine in v. This implies that n can be rendered invariant via a static state feedback (3). (necessity) Assume that n is controlled invariant and that it is rendered invariant via a regular quasi-static state feedback u = (x, v). It is easily verified that (33) (34) Hence condition (i) must hold. Consider a basis {WI,...,w r } and n Wi = 2: uij(x)dxj j=i of n where (35) (36) Let Al (X) be the matrix with entries Uij(X) (i = 1". " d; j = 1". " n) and A2(x) the matrix with entries Uij(x) (i = d + 1,..., r; j = 1,"',n). Viewing n as a linear subspace (over IC) of X EEl span,ddu}, it may be characterized by (37) Similarly, nis characterized by n= rowspanlc ( BI(x, u) 0 ) B 2 (x,u) (A 2 g)(x) (38) where n [) Bi(X,U) = Ai(X,1L) + Ai(X)(Jx(x) +L a:,u) (i= 1,2) i=l X t with fx the Jacobian of f. Since n is rendered invariant via u = (x,v), there exist matrices PI(x,v), P 2(x,v), Q(x,v) such that or B2(X, (x, v))dx + (A 2 g)(x)d = P 1 (x, v)a 1 (x) + P 2 (x, v)a 2 (x) + Q(x, v)dv (40) (39) (41) 7
10 Hence B2(X, <jj(x, v» (A2g)(x)<jJv(X, v) Q(x,v) Since v is invertible, there exists a function 1/J(x, u) with the property that u = <jj(x, v) is equivalent to v = 1/J(x, u). Moreover, by the Inverse Function Theorem, we have <jjx(x, v) = -1/Ju(x, (x, V»-l1/Jx(x, (x, v» v(x,v) = 1/Ju(x,<jJ(x,v»-l Then equations (42) and (43) yield B2(X, u) (A2g)(x) = Q(x, u)v'u(x, u) where Fi(x,u) = Pi(X,1/J(X,u» (i = 1,2) and Q(x,u) = Q(x,1/J(x,u». Using a similar argument as in the sufficiency-part, it can be shown that (A 2 g)(x) has full row rank. Hence there exists a regular static state feedback u = (3(x)u such that (A 2 g{3)(x) = (R(x) 0), with R(x) invertible. This means that without loss of generality we may assume that (A 2 9)(x) itself is invertible. By (44) we then have that Q(x,u) is also invertible, and hence by (44), rowspanjc(b 2 (x,u) (A2g)(x» = rowspanjcq(x, U)-l (B 2 (x, u) (A 2 g)(x» = (42) (43) (44) (45) Noting that rowspanjc (Q(x, u)-l(.i\(x, u)al(x) + F 2 (x, u» rowspanjc (Bl(x, u) 0) C rowspanjc (~~~:~ ~) + 1/Jx(x, u) 1/Ju(x, u») the equality (45) yields which implies that (n +n)/n is exact. Al(X) 0 ) rowspanjc A 2 (x) 0 ( 1/Jx(x,u) 1/Ju(x,u) (46) Theorem 3.9 has an interesting consequence. Recall from e.g. [9],[15] that necessary and sufficient conditions for (local) controlled invariance of distributions are only known in case the distribution under consideration is involutive. Using Theorem 3.9, one can also give conditions for controlled invariance of noninvolutive distributions. 8
11 Define the distribu Corollary 3.10 Consider a (not necessarily involutive) distribution ~. tion ~o by ~o = {r I [f,r] E~, [9,r] C~} n ~ Then ~ is controlled invariant if and only if [f,~] c.6.+9 (i) [9,~] c ~ + g (ii) ~o = ~ n ~o where ~o denotes the involutive closure of Proof Define 11 := ~.1.. By Theorem 3.9, 11 is controlled invariant if and only if (11n 9.1.) C 11 (47) and there exists a function <p such that 11 +n= 11 EEl span}(;{d } Note that (47) is equivalent to (48) (49) Now ~n{t I [f,r] E ~+9, [g,r] C ~+9} (49) and (50) yield ~ C {T I [f,t] E ~ +g, [g,t] C ~ + g} (51) 9
12 which is equivalent to (i). Furthermore, we have (n + <2)1. = {r 1< Wl +412, r >= 0, VWl,W2 E n} = Equalities (48) and (52) imply that Llo = Ll n Ker d {r 1< w,r >= 0, Vw E n} n {r 1< 41,r >= 0, Vw En} = Ll n {r 1< Lf+guW,r >= 0, Vw E n} = Ll n {r I Lf+gu < w,r > - < W,Lf+gur >= 0, Vw E n} = (52) Ll n {r 1< W,Lf+gur >= 0, Vw E n} = Lln{r I [1+gu,r] E Ll} = Lln{r I [1,r] E Ll,[9,r] C Ll} = Llo Now define on IR n + m the distributions Ll e := Ll x {OJ, Ll3 := Llo x {OJ. Then according to (53) there exists an involutive distribution A on IRn+m such that Let.6.3 denote the involutive closure of Llo. By (54) we have that Ll3 C A and hence, by involutivity of A,.6.3 C A. This implies that Since we clearly have Ll3 C Lle n.6.3, this means that Llo= Lle n.6.0' Condition (ii) now follows by noting that Ao=.&0 x {OJ. Remark 3.11 If Ll is involutive, then it can be shown by using the Jacobi-identity that Llo is also involutive. Hence for involutive Ll condition (ii) in Theorem 3.9 is automatically satisfied and thus the result of Theorem 3.9 is in accordance with existing results (see e.g. [9],[15]) The general case Let us again consider a general subspace n C X. Define by induction: no.- {OJ no.- n (53) (54) (55) nk+l := nk +nk+l Furthermore, define 10
13 Theorem 3.12 Let n ex. If - (i) (nn9.l)cn (ii) nko-l + fho-l is exact. nko-1 then n is controlled invariant for (1). Proof From the definition of k*, there exist vector valued dvl,"" dvko in, where each dvi is non-empty, such that {h = span{dvd C nostno (56) {d. (k"-i) d (ko-2) d} C n ko- 1 span VI, v + nko-l 2,, VkO n ko-l Note that from (ii) the last inclusion in (56) is in fact an equality. We now have. (k* 1) n ko- 1 + n ko- 1 = n ko- 1 + span{dv 1 -,.., dvp} C (57) n+span{dv(k) I k ~ O} It remains to be shown that v defines a quasi-static state feedback. From the above construction, one has VI <PI (x, u) V2 = <P2(x, vi, VI, u) (58) Vk o <Pko(x,{v111 ~ i ~ k* -1,0 ~ j ~ k* - i},u) From (i), (8(<PI'..., <Pk* )/8u) has full row rank. Thus there exists VkO+l = <Pk*+l(u) such that (8(<Pb''',<Pko+l)/8u) is square invertible. From the Inverse Function Theorem, there exists a function 'lj; such that u = 'lj;(x, v, v,"', v k O). Remark 3.13 The above theorem only gives sufficient conditions for the controlled invariance of a subspace n C X. It is easily proved that condition (i) is also a necessary condition. It is the authors' conjecture that condition (ii) is also necessary. For a proof of this conjecture a better understanding of quasi-static state feedback seems to be needed. This remains a topic for further research. 11
14 3.2.4 The smallest controlled-invariant subspace containing a subspace Given a subspace II C X, it is unclear whether (or under what conditions) there exists a smallest controlled invariant subspace containing II. This is due to the fact that for two controlled invariant subspaces 0 1, O2 ex, we do not necessarily have that 0 1 n02 is controlled invariant, so that we cannot use the "standard" arguments (as in e.g. [17],[9],[15]). In this subsection we will give some comments on this question. A first result is the following: Proposition 3.14 Consider a subspace II C X and define II* := X n (II +11(1) li(n)). Let 0 C X be a controlled invariant subspace containing II. Then II* C O. Proof By definition, there exists a quasi-static state feedback such that nco + V. It can easily be shown by induction that this implies that O(k) C 0 + V (k E IN). Hence also II + II(1) II(n) C 0 + V. Thus, 11* = X n (II II(n)) C X n (0 +V) = 0 The subspace II* defined in Proposition 3.14 is a candidate for being the smallest controlled invariant subspace containing II. If II is exact, it can be shown that indeed it is. This may be shown in the following way. Let r = dim II and choose meromorphic functions hi (x),..., h r (x) such that II = spanx:::{dh 1,, dh r }. Next consider the system x = f(x)+g(x)u y = h(x) Then for this system, II* = X n Y; where Y = spanx:::{dy,., dy(n)}. (The subspace X n y was introduced in [18] for the study of the minimal order input-output decoupling problem.) If the system (59) is right-invertible, one can construct a quasi-static state feedback which renders II* invariant by using the construction in [16]. If (59) is not right-invertible, the same construction, together with Lemma 1 from [12] may be used to show that II* is controlled invariant. Summarizing, we have the following result: Theorem 3.15 Consider a subspace II C X which is exact. Then IT* := X n (IT II(n)) is the smallest controlled invariant subspace containing IT. 4 Conclusions A generalized notion of controlled invariance under quasi-static state feedback for nonlinear systems was introduced. It was shown that this notion coincides with the"classical" notion of a controlled invariant distribution under regular static state feedback. Using the generalized notion of controlled invariance, a condition for the controlled invariance of not necessarily involutive distributions was derived. For a subspace 0 ex, sufficient conditions for controlled invariance under quasi-static state feedback were given. This paper leaves some interesting open questions, which are a topic for further research. A first question is whether the sufficient conditions for controlled invari;llll'(' under quasi-static state feedback that were derived are also necessary. A second questioll is whether (or under (59) 12
15 what conditions) there exists a smallest controlled invariant subspace containing some given subspace. It seems that for the answer to both questions a better understanding of quasi-static state feedback is needed. Finally, let us remark that throughout we have restricted ourselves to "Kalmanian" systems and to subspaces n c X. However, the definition of controlled invariance and the characterizations of controlled invariance in this paper can, mutatis mutandis, be translated to non-kalmanian systems and subspaces n C X xu. References [1] Atiyah, M.F., and LG. MacDonald, Introduction to commutative algebra, Addison Wesley, Reading Mass., [2] Brockett, R.W., Feedback invariants for nonlinear systems, Proceedings IFAC Congress, Helsinki, 1978, pp [3] Delaleau, E., and ~1. Fliess, An algebraic interpretation of the structure algorithm with an application to feedback decoupling, Proceedings NOLCOS 1992, Bordeaux, France, pp [4] Delaleau, E., and M. Fliess, Algorithme de structure, filtrations et decouplage, C.R. Acad. Sci. Paris, t.315, Serie I, (1992), pp [5] Di Benedetto, M.D., J.W. Grizzle and C.H. Moog, Rank invariants of nonlinear systems, SIAM J. Control Optimiz., 27, (1989), pp [6] Fliess, M., Automatique et corps differentiels, Forum Math., 1, (1989), pp [7] Hirschorn, R.M., (A, B)-invariant distributions and disturbance decoupling of nonlinear systems, SIAM J. Control Optimiz., 19, (1981), pp [8] Huijberts, H.J.C., H. Nijmeijer and L.L.M. van del' Wegen, Dynamic disturbance decoupling for nonlinear systems, SIAM J. Control Optimiz., 30, (1992), pp [9] Isidori, A., Nonlinear control systems, Springer Verlag, Berlin, [10] Isidori, A., Control of nonlinear systems via dynamic state feedback, in Algebraic and geometric methods in nonlinear control theory, M. Fliess and M. Hazewinkel, eds., Reidel, Dordrecht, [11] Isidori, A., A.J. Krener, C. Gori-Giorgi and S. Monaco, Nonlinear decoupling via feedback: a differential geometric approach, IEEE Trans. Automat. Control, AC-26, 1981, pp [12] Moog, C.H., Nonlinear decoupling and structure at infinity, MCSS, 1, (1988), pp [13] Nijmeijer, H., Controlled invariancefor affine control systems, Int. J. Control, 34, (1981), pp
16 [14] Nijmeijer, H., and A.J. van der Schaft, Controlled invariance for nonlinear systems, IEEE Trans. Automat. Control, AC-27, (1982), pp [15] Nijmeijer, H., and A.J. van der Schaft, Nonlinear dynamical control systems, Springer Verlag, New York, [16] Perdon, A.11., Y.F. Zheng, C.H. Moog and G. Conte, Disturbance decoupling for nonlinear systems: a unified approach, Proceedings 2nd IFAC Workshop on System Structure and Control, Prague, 1992, pp [17] Wonham, W.M., Linear multivariable control: a geometric approach (3rd Edition), Springer Verlag, Berlin, [18] L. Cao and Y.F. Zheng, On minimal compensators for decoupling control, Syst. Contr. Lett., pp
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