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Output Rank and Flatness of Regular Implicit Systems
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OUTPUT RANK AND FLATNESS OF REGULAR IMPLICIT SYSTEMS Paulo Sergio Pereira da Silva _,1 Nadeia Aparecida Veloso Pazzoto-
- Escola Politecnica da USP - PTC Cep 05508-900 - Sao Paulo - SP - BRAZIL Email: paulo~lac.usp.br, Fax: +(55)(11)38185718
Abstract: A regular implicit system is a non linear explicit control system S and a finite set of constraints under a geometric assumption of regularity. We show that the output rank of such a class of systems may be deduced from the dimensions of some codistributions defined on S. We develop necessary and sufficient conditions for testing if a given output z of a regular implicit system is a flat output. The relative dynamic extension algorithm is presented and a geometric interpretation is also given. This algorithm may be used for constructing dynamic linearizing feedback laws and/or the solution of the dynamic input-output decoupling problem for implicit systems. Copyright \i:) 20011FAC Keywords: Nonlinear systems, implicit systems, differential flatness, decoupling, feedback linearization.
1. INTRODUCTION
Let
r
(Fliess et al., 1999). We may construct the codistributions Yk = span {dx, dy , ... , dy(k)}, Yk = span {dy, . . . , dy(k)} and Zk = span {dz, ... , dz(k)} . We may consider that r is a subset of S defined by r = {~ E S I y(k) = 0, k E 1N} (Prop. 1 shows that r is a immersed submanifold of S) .
be an implicit system of the form
i:(t) = f(x(t)) z(t) y(t)
+ g(x(t))u(t)
= 4>(x(t)) = h(x(t)) = 0
(la) (lb) (lc)
Definition 1. The implicit system r is said to be regular if (i) The codistributions Yk and Yk are nonsingular for k = 0, .. . , n for every point ~ E r. (ii) Let v = n . The codistributions Lk = Yv + k + Zk and .c k = Yv+k + Zk are nonsingular for k = 0, . . . , n for every point ~ E r.
where x(t) E lRn is the pseudo-state of the system, u(t) E lRm is the pseudo-input 2, z(t) E lRP is the output and Yi(t), i = 1, ... , r are the constraints. The explicit system given by (la) will be denoted by S. Consider the system S with output 3 y = h(x), in the framework of
In this paper we show that the output rank (see (Fliess, 1989)) of a regular implicit system can be deduced from the dimensions of the codistributions .c k and L k . This result may be considered as a generalization of the ones of (Di Benedetto et al., 1989; Delaleau and Pereira da Silva, 1998) for this class of implicit systems.
1 The first author was partially supported by CNPq and by FAPESP under grants 300492/95-2 and 97 / 04668-1. The second author is partially supported by CAPES. 2 Note that u is not a differentially independent input for r, since the constraints y == 0 induce differential relations linking the components of u. By the same reasons , x is not a state of r. 3 We regard y = h{x) as an output instead of being a constraint.
225
literature (Campbell, 1982; Liyi, 1989; Campbell, 1990). Solvability of nonlinear implicit differential equations is considered in (Brenan et al. , 1995; Rheinboldt, 1991). Other problems like controllability (Lin and Ahmed, 1991), stabilization (McClamroch, 1990; Chen and Shayman, 1992), and feedback control (Christodoulou and I§ik, 1990; Liu and Celikovsky, 1997), have already been considered. Feedback linearization and flatness of implicit systems has been studied for instance by (Liu, 1993; Kawaji and Taha, 1994; Pereira da Silva and Correa Filho, 2(00). The approach considered here for considering implicit systems is strongly related to the ideas of (Fliess et al., 1995).
Let p(y) be the rank of explicit system S with output yand let p(z) be the rank of implicit system r with output z . We show that the sequence of integers (1k = dim Lk - dim Lk-l - p(y) plays the role of the structure ate infinity of the implicit system (1). In particular, the output rank of the implicit system is shown to be equal to p(z) = dimL n -dimLn _ l - p(y) (see Thm. 1). Let m = dim u . We show that z is a flat output of the implicit system r if and only if the output rank of the implicit system is equal m - p(y) and furthermore span {dx , du} c Ln . These conditions can be deduced directly from the dimensions of the co distributions Lk and Lk (see Thm. 2) . Our approach will follows the infinite dimensional geometric setting recently introduced in control theory (Fliess et al., 1993; Fliess et al., 1999) in combination with the ideas presented in (Pereira da Silva and Cor rea Filho, 2000) .
It can be shown that an implicit system r defined by (la)-(lc) that obeys the condition (i) of Def. 1 can be regarded as an immersed system in the explicit system S defined by (la). More precisely
It is important to point out that our results shows effective ways for computing the output rank and control laws for dynamic feedback linearization and/or decoupling of an implicit system r, without the need of transforming r into an explicit system. In fact , note that the relative dynamic extension algorithm for affine systems relies only on multiplications and matrix inversions.
Proposition 1. (Pereira da Silva and Correa Filho, 2000) Let S be the system associated to (la) in the sense of (Fliess et al., 1999). Let r = {~ E S I y(k)(~) = 0, k E IN} . Assume that r is nonempty. Then the subset reS has a structure of immersed (embedded) manifold in S. Let L : r -+ S be the canonical insertion. We can define a Cartan field 8r on r by the equation L. 8r (-y) = 1; 0 L(-y)" Er. Equipped with this Cart an field , r is a system such that L is a LieBiicklund immersion.
2. PRELIMINARlES AND NOTATION 2.1 Notation
The idea 4 of the proof of this fact is to execute the dynamic extension algorithm considering the explicit system S given by equation (la) with output y = h(x) . The condition (i) of Def. 1 assures that the dynamic extension algorithm may be executed without any local singularities. In the step n of this algorithm we have computed a new state representation (x , u) where 5 x = _in)) . U- _- (w, J.L ) , were h - (1 ) ( X , Yl , ... , Yn w -__Yk(n+l ) and J.L = un) . Note that the new state equations are affine, i. e. , they are of the form
The field of real numbers will be denoted by JR. The set of real matrices of n rows and m columns is denoted by JRn xm. The matrix MT stands for the transpose of M . The set of natural numbers {I, ... , k} will be denoted by Lk1- We will use the standard notations of differential geometry in the finite and infinite dimensional case (Zharinov, 1992) . A brief overview of the infinite dimensional approach of (Fliess et al., 1999) is presented in Appendix A. Some notations and definitions of Appendix A are used along the paper (e . g. the definition of system as a diffiety, the definition of state representation and differential flatness) .
i = z
For simplicity, we abuse notation, letting (Zl ' Z2) stand for the column vector (zf, zI) T, where Zl and Z2 are also column vectors. Let x = (Xl , . .. ,xn ) be a vector of functions (or a collection of functions). Then {dx} stands for the set {dxl> ... , dx n } .
j(x)
+ g(x)w + g(x)J.L
= ~(x) = (j>(x)
(2)
By the properties of the dynamic extension algorithm (see (Pereira da Silva, 2000)), we have span {dX} = span {dx,dy , ... ,dy(n)} and span {dx , du} = span {dx , du , dy , ... ,dy(n+l)}. After that we may choose (z, v) of the form z = (za, Zb) , v = (Va , Vb) by a convenient staticstate feedback such that Vb = J.L, span {dza }
2.2 Implicit systems 4 An important detail that is overlooked in the present sketch of the proof of Prop. 1 is the construction of the smooth atlas of r. 5 Using the same notation of (Di Benedetto et al., 1989; Pereira da Silva, 2000).
Linear singular (or implicit) systems are an important class of control systems and many papers and books on this subject are found in the
226
= span {dy(O) , . . . ,dy(n)} , and span {dza , dvik ) :
of the components of z , we may assume that the first al rows of Cl (x) are locally independent. Then there exist a partition Z = (Zl , Zl), where dim Zl = al and a regular static-state feedback with new input (VI, w), such that
k = O, .. . , r} = span {dy(k) : k = O, . . . ,r+n} for all rEIN . Note that, by construction we have span {di} = span {dz} and span {di,du} = span {dz, dv} .
J.L
The proof of Prop. 1 also shows that (Zb, Vb) is a state representation of r and in the coordinates {t , Za , Va} for rand {t , Za, Zb, Va , Vb} for S, the immersion L is given by L(t, Zb, Vb) = (t, 0, Zb, 0, Vb) . Furthermore, the Cart an field of S is of the form
where
=
VI
= Ol(X,W) + .81 (X)Vl
(VI, vd, such that
1t
Zl
.h
=
VI
= £l(x , w, vd
Add the following dynamic extension
and let fi,l = VI. We stress that .8l(X) is locally nonsingular.
Step k. In the step k - 1 we have constructed a state representation Xk-l
(4)
= ik-l(Xk-d + 9k-I(Xk-l)Wk-l+ (5)
9k-l (xk-dJ.Lk-l z(k-l )
= c,6k-l(Xk-d
- (-X,W,Wl ,·· · ,Wk_2 , zl -(1) _(k-l») ' , .. . , zk_l whereXk_lWk-l = Wk-2 and J.Lk-l = (Vk-l, fi,k-d ·
Note that Zb is chosen in a way that {dz b, dz a } is a basis of span {dx , dy , . . . , dy(n)} . In particular one may choose Zb as a convenient subset of x . Note that Vb = J.L is an input for r . Notice also that the dynamic extension algorithm may be computed in a way that J.L is a subvector of u. In other words, J.L is the differentially independent part of u and the constraints y == induce differential relations linking the other components of u . This explains why we call x by "pseudo-state" and u by "pseudo-input" of r .
Compute Z(k)
°
=
ak(xk-l) + bk(Xk-l)Wk-l+ Ck(Xk-dJ.Lk-l
Let ak = rank Ck(Xk-d and assume that this rank is locally constant around some Xk-l . Up to a reordering of the components of z, we may assume that the first ak rows of Ck(Xk-d are locally independent. Then there exist a partition Z = (Zk, Zk), where dim Zk = ak and a regular static-state feedback with new input (Vk , Wk-l) defined by
3. RELATIVE DYNAMIC EXTENSION ALGORITHM
J.Lk-l = Ok(Xk-l ,Wk-l)
The following algorithm is instrumental for studying flatness and decoupling of implicit systems:
where Vk
+ .8k(Xk-d v k
= (Vk, Vk) , such that _(k)
_
zk = Vk , (k) ,(k)(_ ) zk = zk Xk ,Wk-l , Vk
Algorithm 1. (Relative Dynamic Extension Algorithm) Execute n steps of the dynamic extension algorithm for the explicit system S with output y = h(x) obtaining the state representation with state x and input u given by (2) . Then execute the following steps: Step 1. Compute
Add the following dynamic extension Wk-l = Wk Vk = {.Lk
and let fi,k = h . We stress that .8k(Xk-d is locally nonsingular .
Let al = rank Cl (x) and assume that this rank is locally constant around xo. Up to a reordering
The following result summarizes the main geometric properties of the Relative Dynamic Extension
227
implicit system. In particular, the relative output rank coincides with the rank of the implicit system. More precisely
Algorithm for time-invariant nonlinear systems. We stress that the list of integers {eTl," " eTn}, where n = dimx, is strongly related to the algebmic structure at infinity (see (Di Benedetto et al. , 1989)) and is called relative structure at infinity. The integer p(z) = eTn is called relative output mnk at a point ~ E S.
Theorem 1. Consider the relative structure at infinity {eTl ' ... , eTn } as defined in Lemma 1. Let r be the regular implicit system (1) and consider the state representation (Xb , Vb) of r with state equations defined by (4) . Then {eTl' ... ,eTn } is the algebraic structure at infinity of r (with respect to this state representation) . In particular, the output rank of the implicit system r coincides with p(z) = eTn .
Lemma 1. Let S be the time-invariant system given by (la) with classical state representation (x, u) and classical time-invariant output y = h(x). Assume that the output rank of the explicit system S is given by p(y) . Let Vk be the open and dense set of regular points of the codistributions Y" Yi (see Def. 1), for i = 0, . . . , n and of .cj , L j for j E {O, .. . , k} . In the kth step of the relative dynamic extension algorithm, one may construct around ~ E Vk , a new local classical state representation (:h, Uk) of the system S with state fh = (i, w, . .. ,w{k-l), zP) , ... , zkk»), input
Sketch of Proof. The idea of the proof is to apply Prop. 1 and to show that the Lie-Biicklund immersion L : r --+ S is such that L*.c k = span {dx b, d(z 0 L), . .. ,d(z{k) 0 L)} and dim.c k = dimL*£k + p(y) (see (4)) . 0
Uk = (w{k), J.Lk), where J.Lk = (~~k) , P-k) , and output z{k) =
5. FLAT OUTPUTS FOR REGULAR IMPLICIT SYSTEMS The following result may be regarded as a generalization for implicit systems of a result of (Martin, 1992) (see also (Pereira da Silva, 2000)).
that z~k) C zk:i ) and to choose P-k+l C P-k . (iii) Let € E Vk. The sequence eTk = dim(Lkle)) dim(£k-de) - p(y) is nondecreasing, the sequence Pk = dim(Lkle) - dim(L k- 1 Id - p(y) is non increasing, and both sequences converge to the same integer p( z), called the relative output rank at ~, for some k* ~ n = dimx . (iv) Let Sk C Vk be the open neighborhood of a given ~ E Vk , such that the dimensions of £j,Lj j E {O, . .. ,k} are constant inside Sk . We have Sk = Sk- for k ~ k*. Furthermore Lknspan{dx}lv = Lk-_lnspan{dx}lv for every v E Sk- and k ~ k*. (v) For k ~ k*, one may choose Zk = Zk- in Uk- . Furthermore, Lk+l = Lk + span {w{k-l), zkk+l)} 1
for k ~ k* . (vi) Let Y dim
Theorem 2. Let r be the regular implicit system (1) . Let p(y) be the output rank of the explicit system S defined by (la) with output y = h(x) . Then z =(x) is a flat output of r if and only if p(z) = dimz = m - p(y) , where m = dimu, and Ln :J span {dx,du} . This is equivalent to have p(z) = dim z and n+ ~~=1 eTi = p(z)(n+ 1), where n = dim£n- dimL n . Sketch of Proof. Applying Prop. lone may show that L*Lk = span {d(ZOL), . . . ,d(z{k)OL)} and L*.c k = span {dXb, d(z 0 L) , ... , d(z{k) 0 L)} . In particular, to say that Lk :J span {dx} is equivalent to say that span {d(z 0 L), . .. ,d(z{k) 0 L)} :J span {dXb, dVb}, i.e. , the state and the input of r is a function of z and its derivatives (see (Martin, 1992; Pereira da Silva, 2000)). The second affirmation is a consequence of (Pereira da Silva, 2000, Prop. 7.4) . 0
= span {dy{k) Ik E DV} . Then eTk =
c::;"!y and Pk
=
dim L:~;!Y' In particular
we have p(z) = dim .c:~;!Y'
Proof. The proof of this lemma is based on the same techniques of (Pereira da Silva, 2000) and it is omitted by reasons of space. 0
6. CONCLUSIONS The results of this paper may be useful for studying flatness and the dynamic decoupling problem for implicit systems. It is not difficult to show that the relative dynamic extension algorithm constructs a regular dynamic state feedback for the implicit system that solves the dynamic linearization problem and/ or the dynamic inputoutput decoupling problem.
4. RELATIVE OUTPUT RANK AND OUTPUT RANK OF IMPLICIT SYSTEMS In this section we show that , for regular implicit systems, the relative structure at infinity coincides with the structure at infinity of the corresponding
228
7. REFERENCES
McClamroch, N. H. (1990) . Feedback stabilization of control systems described by a class of nonlinear differential-algebraic equations. Systems fj Control Letters 15, 53-60. Pereira da Silva, P. S. (2000) . Some geometric properties of the dynamic extension algorithm. Technical Report BT / PTC / 0008. Escola Politecnica da Universidade de Siio Paulo. available in http://www.lac.usp.br/ ... pauloj. Pereira da Silva, P. S. and C. Cor rea Filho (2000) . Relative flatness and flatness of implicit systems. Technical Report BT /PTC /xxxx. Escola Politecnica da Universidade de Sao Paulo. To appear in SIAM J . Contr. Optimiz, available in http://www.lac.usp.br/..-.pauloj. Pomet, J .-B. (1995). A differential geometric setting for dynamic equivalence and dynamic linearization . In: Geometry in Nonlinear Control and Differential Inclusions (B. Jackubczyk, W . respondek and T . Rzezuchowski, Eds.). Banach Center Publications. Warsaw. pp. 319-339. Rheinboldt (1991) . On the existence and uniqueness of solutions of non linear semi-implicit differential-algebraic equations. Nonlinear Analysis, Theory, Methods fj Appl. 16, 64266 I. Zharinov, V. V. (1992) . Geometrical A spects of Partial Differentials Equations. World Scientific. Singapore.
Brenan, K. E., S. L. Campbell and L. R. Petzold (1995) . Numerical Solution of Initial- Value Problems in Differential-Algebraic Equations. Springer-Verlag. New York. Campbell, S. L. (1982) . Singular Systems of Differential Equations. Pitman. London. Campbell, S. L. (1990) . Descriptor systems in the 90's. In: Proc. 29th IEEE Conf. Dec. Control. pp. 442-447. Chen, X. and M. A. Shayman (1992) . Dynamics and control of constrained non linear systems with application to robotics. In: Proc. Am. Control Conference. pp. 2962- 2966. Christodoulou, M. A. and C. I~ik (1990) . Feedback control for nonlinear singular systems. Internat. J. Control 51(2), 487-494. Delaleau, E. and P. S. Pereira da Silva (1998) . Filtrations in feedback synthesis: Part I - systems and feedbacks. Forum Math. 10(2) , 147174. Di Benedetto, M. D., J . W . Grizzle and C. H. Moog (1989) . Rank invariants of non linear systems. SIAM J. Control Optim. 21, 658672. Fliess, M. (1989) . Automatique et corps differentiels. Forum Math. 1, 227-238. Fliess, M., J. Levine, P. Martin and P. Rouchon (1993). Linearisation par bouclage dynamique et transformations de Lie-Biicklund. C. R. Acad. Sci. Paris Sir. I Math. 311,981986. Fliess, M., J . Levine, P. Martin and P. Rouchon (1995). Index and decomposition of non linear implicit differential equations. In: IFAC Conf. System Structure and Control. Fliess, M., J . Levine, P. Martin and P. Rouchon (1999) . A Lie-Biicklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44(5) , 922-937. Kawaji, S. and E . Z. Taha (1994) . Feedback linearization of a class of nonlinear descriptor systems. In : Proc. 33rd IEEE Conf. Dec. Control. Vol. 4. pp. 4035-4037. Lin, J . Y. and N. U. Ahmed (1991) . Approach to controllability problems for singular systems. Int. J. Control 22, 675-690. Liu , X. P. (1993) . On linearization of nonlinear singular control systems. In: Proc. American Control Conference. pp. 2284-2287. Liu, Xiaoping and Sergej Celikovsky (1997) . Feedback control of affine nonlinear singular control systems. Internat. J. Control 68(4) , 753774 . Liyi, Dai (1989) . Singular Control Systems. Springer-Verlag. Martin , P . (1992) . Contribuition a l'etude des systemes differentiellement plats. PhD thesis. Ecole des Mines. Paris.
Appendix A. DIFFIETIES AND SYSTEMS In this appendix we recall briefly the main concepts of the infinite dimensional geometric setting of (Fliess et al., 1993; Pomet, 1995; Fliess et al., 1999) . ./RA-Manifolds. Let A be a countable set. Denote by ./RA the set of functions from A to ./R. One may define the coordinate function Xi : ./RA -> ./R by Xi(() = ((i), i E A . This set can be endowed with the Frechet topology (i. e. , an inverse limit topology (Zharinov, 1992)). A basis of this topology is given by the subsets of the form B = {( E ./RA I IXi(O - Oil < fi, i E F} , where F is a finite subset of A , Oi E ./R and fi is a positive real number for i E F . A function
Copyright Cl IFAC Nonlinear Control Systems, St. Petersburg, Russia, 200 I
c:
0
C>
Publications www.elsevier.comllocatelifac
OUTPUT RANK AND FLATNESS OF REGULAR IMPLICIT SYSTEMS Paulo Sergio Pereira da Silva _,1 Nadeia Aparecida Veloso Pazzoto-
- Escola Politecnica da USP - PTC Cep 05508-900 - Sao Paulo - SP - BRAZIL Email: paulo~lac.usp.br, Fax: +(55)(11)38185718
Abstract: A regular implicit system is a non linear explicit control system S and a finite set of constraints under a geometric assumption of regularity. We show that the output rank of such a class of systems may be deduced from the dimensions of some codistributions defined on S. We develop necessary and sufficient conditions for testing if a given output z of a regular implicit system is a flat output. The relative dynamic extension algorithm is presented and a geometric interpretation is also given. This algorithm may be used for constructing dynamic linearizing feedback laws and/or the solution of the dynamic input-output decoupling problem for implicit systems. Copyright \i:) 20011FAC Keywords: Nonlinear systems, implicit systems, differential flatness, decoupling, feedback linearization.
1. INTRODUCTION
Let
r
(Fliess et al., 1999). We may construct the codistributions Yk = span {dx, dy , ... , dy(k)}, Yk = span {dy, . . . , dy(k)} and Zk = span {dz, ... , dz(k)} . We may consider that r is a subset of S defined by r = {~ E S I y(k) = 0, k E 1N} (Prop. 1 shows that r is a immersed submanifold of S) .
be an implicit system of the form
i:(t) = f(x(t)) z(t) y(t)
+ g(x(t))u(t)
= 4>(x(t)) = h(x(t)) = 0
(la) (lb) (lc)
Definition 1. The implicit system r is said to be regular if (i) The codistributions Yk and Yk are nonsingular for k = 0, .. . , n for every point ~ E r. (ii) Let v = n . The codistributions Lk = Yv + k + Zk and .c k = Yv+k + Zk are nonsingular for k = 0, . . . , n for every point ~ E r.
where x(t) E lRn is the pseudo-state of the system, u(t) E lRm is the pseudo-input 2, z(t) E lRP is the output and Yi(t), i = 1, ... , r are the constraints. The explicit system given by (la) will be denoted by S. Consider the system S with output 3 y = h(x), in the framework of
In this paper we show that the output rank (see (Fliess, 1989)) of a regular implicit system can be deduced from the dimensions of the codistributions .c k and L k . This result may be considered as a generalization of the ones of (Di Benedetto et al., 1989; Delaleau and Pereira da Silva, 1998) for this class of implicit systems.
1 The first author was partially supported by CNPq and by FAPESP under grants 300492/95-2 and 97 / 04668-1. The second author is partially supported by CAPES. 2 Note that u is not a differentially independent input for r, since the constraints y == 0 induce differential relations linking the components of u. By the same reasons , x is not a state of r. 3 We regard y = h{x) as an output instead of being a constraint.
225
literature (Campbell, 1982; Liyi, 1989; Campbell, 1990). Solvability of nonlinear implicit differential equations is considered in (Brenan et al. , 1995; Rheinboldt, 1991). Other problems like controllability (Lin and Ahmed, 1991), stabilization (McClamroch, 1990; Chen and Shayman, 1992), and feedback control (Christodoulou and I§ik, 1990; Liu and Celikovsky, 1997), have already been considered. Feedback linearization and flatness of implicit systems has been studied for instance by (Liu, 1993; Kawaji and Taha, 1994; Pereira da Silva and Correa Filho, 2(00). The approach considered here for considering implicit systems is strongly related to the ideas of (Fliess et al., 1995).
Let p(y) be the rank of explicit system S with output yand let p(z) be the rank of implicit system r with output z . We show that the sequence of integers (1k = dim Lk - dim Lk-l - p(y) plays the role of the structure ate infinity of the implicit system (1). In particular, the output rank of the implicit system is shown to be equal to p(z) = dimL n -dimLn _ l - p(y) (see Thm. 1). Let m = dim u . We show that z is a flat output of the implicit system r if and only if the output rank of the implicit system is equal m - p(y) and furthermore span {dx , du} c Ln . These conditions can be deduced directly from the dimensions of the co distributions Lk and Lk (see Thm. 2) . Our approach will follows the infinite dimensional geometric setting recently introduced in control theory (Fliess et al., 1993; Fliess et al., 1999) in combination with the ideas presented in (Pereira da Silva and Cor rea Filho, 2000) .
It can be shown that an implicit system r defined by (la)-(lc) that obeys the condition (i) of Def. 1 can be regarded as an immersed system in the explicit system S defined by (la). More precisely
It is important to point out that our results shows effective ways for computing the output rank and control laws for dynamic feedback linearization and/or decoupling of an implicit system r, without the need of transforming r into an explicit system. In fact , note that the relative dynamic extension algorithm for affine systems relies only on multiplications and matrix inversions.
Proposition 1. (Pereira da Silva and Correa Filho, 2000) Let S be the system associated to (la) in the sense of (Fliess et al., 1999). Let r = {~ E S I y(k)(~) = 0, k E IN} . Assume that r is nonempty. Then the subset reS has a structure of immersed (embedded) manifold in S. Let L : r -+ S be the canonical insertion. We can define a Cartan field 8r on r by the equation L. 8r (-y) = 1; 0 L(-y)" Er. Equipped with this Cart an field , r is a system such that L is a LieBiicklund immersion.
2. PRELIMINARlES AND NOTATION 2.1 Notation
The idea 4 of the proof of this fact is to execute the dynamic extension algorithm considering the explicit system S given by equation (la) with output y = h(x) . The condition (i) of Def. 1 assures that the dynamic extension algorithm may be executed without any local singularities. In the step n of this algorithm we have computed a new state representation (x , u) where 5 x = _in)) . U- _- (w, J.L ) , were h - (1 ) ( X , Yl , ... , Yn w -__Yk(n+l ) and J.L = un) . Note that the new state equations are affine, i. e. , they are of the form
The field of real numbers will be denoted by JR. The set of real matrices of n rows and m columns is denoted by JRn xm. The matrix MT stands for the transpose of M . The set of natural numbers {I, ... , k} will be denoted by Lk1- We will use the standard notations of differential geometry in the finite and infinite dimensional case (Zharinov, 1992) . A brief overview of the infinite dimensional approach of (Fliess et al., 1999) is presented in Appendix A. Some notations and definitions of Appendix A are used along the paper (e . g. the definition of system as a diffiety, the definition of state representation and differential flatness) .
i = z
For simplicity, we abuse notation, letting (Zl ' Z2) stand for the column vector (zf, zI) T, where Zl and Z2 are also column vectors. Let x = (Xl , . .. ,xn ) be a vector of functions (or a collection of functions). Then {dx} stands for the set {dxl> ... , dx n } .
j(x)
+ g(x)w + g(x)J.L
= ~(x) = (j>(x)
(2)
By the properties of the dynamic extension algorithm (see (Pereira da Silva, 2000)), we have span {dX} = span {dx,dy , ... ,dy(n)} and span {dx , du} = span {dx , du , dy , ... ,dy(n+l)}. After that we may choose (z, v) of the form z = (za, Zb) , v = (Va , Vb) by a convenient staticstate feedback such that Vb = J.L, span {dza }
2.2 Implicit systems 4 An important detail that is overlooked in the present sketch of the proof of Prop. 1 is the construction of the smooth atlas of r. 5 Using the same notation of (Di Benedetto et al., 1989; Pereira da Silva, 2000).
Linear singular (or implicit) systems are an important class of control systems and many papers and books on this subject are found in the
226
= span {dy(O) , . . . ,dy(n)} , and span {dza , dvik ) :
of the components of z , we may assume that the first al rows of Cl (x) are locally independent. Then there exist a partition Z = (Zl , Zl), where dim Zl = al and a regular static-state feedback with new input (VI, w), such that
k = O, .. . , r} = span {dy(k) : k = O, . . . ,r+n} for all rEIN . Note that, by construction we have span {di} = span {dz} and span {di,du} = span {dz, dv} .
J.L
The proof of Prop. 1 also shows that (Zb, Vb) is a state representation of r and in the coordinates {t , Za , Va} for rand {t , Za, Zb, Va , Vb} for S, the immersion L is given by L(t, Zb, Vb) = (t, 0, Zb, 0, Vb) . Furthermore, the Cart an field of S is of the form
where
=
VI
= Ol(X,W) + .81 (X)Vl
(VI, vd, such that
1t
Zl
.h
=
VI
= £l(x , w, vd
Add the following dynamic extension
and let fi,l = VI. We stress that .8l(X) is locally nonsingular.
Step k. In the step k - 1 we have constructed a state representation Xk-l
(4)
= ik-l(Xk-d + 9k-I(Xk-l)Wk-l+ (5)
9k-l (xk-dJ.Lk-l z(k-l )
= c,6k-l(Xk-d
- (-X,W,Wl ,·· · ,Wk_2 , zl -(1) _(k-l») ' , .. . , zk_l whereXk_lWk-l = Wk-2 and J.Lk-l = (Vk-l, fi,k-d ·
Note that Zb is chosen in a way that {dz b, dz a } is a basis of span {dx , dy , . . . , dy(n)} . In particular one may choose Zb as a convenient subset of x . Note that Vb = J.L is an input for r . Notice also that the dynamic extension algorithm may be computed in a way that J.L is a subvector of u. In other words, J.L is the differentially independent part of u and the constraints y == induce differential relations linking the other components of u . This explains why we call x by "pseudo-state" and u by "pseudo-input" of r .
Compute Z(k)
°
=
ak(xk-l) + bk(Xk-l)Wk-l+ Ck(Xk-dJ.Lk-l
Let ak = rank Ck(Xk-d and assume that this rank is locally constant around some Xk-l . Up to a reordering of the components of z, we may assume that the first ak rows of Ck(Xk-d are locally independent. Then there exist a partition Z = (Zk, Zk), where dim Zk = ak and a regular static-state feedback with new input (Vk , Wk-l) defined by
3. RELATIVE DYNAMIC EXTENSION ALGORITHM
J.Lk-l = Ok(Xk-l ,Wk-l)
The following algorithm is instrumental for studying flatness and decoupling of implicit systems:
where Vk
+ .8k(Xk-d v k
= (Vk, Vk) , such that _(k)
_
zk = Vk , (k) ,(k)(_ ) zk = zk Xk ,Wk-l , Vk
Algorithm 1. (Relative Dynamic Extension Algorithm) Execute n steps of the dynamic extension algorithm for the explicit system S with output y = h(x) obtaining the state representation with state x and input u given by (2) . Then execute the following steps: Step 1. Compute
Add the following dynamic extension Wk-l = Wk Vk = {.Lk
and let fi,k = h . We stress that .8k(Xk-d is locally nonsingular .
Let al = rank Cl (x) and assume that this rank is locally constant around xo. Up to a reordering
The following result summarizes the main geometric properties of the Relative Dynamic Extension
227
implicit system. In particular, the relative output rank coincides with the rank of the implicit system. More precisely
Algorithm for time-invariant nonlinear systems. We stress that the list of integers {eTl," " eTn}, where n = dimx, is strongly related to the algebmic structure at infinity (see (Di Benedetto et al. , 1989)) and is called relative structure at infinity. The integer p(z) = eTn is called relative output mnk at a point ~ E S.
Theorem 1. Consider the relative structure at infinity {eTl ' ... , eTn } as defined in Lemma 1. Let r be the regular implicit system (1) and consider the state representation (Xb , Vb) of r with state equations defined by (4) . Then {eTl' ... ,eTn } is the algebraic structure at infinity of r (with respect to this state representation) . In particular, the output rank of the implicit system r coincides with p(z) = eTn .
Lemma 1. Let S be the time-invariant system given by (la) with classical state representation (x, u) and classical time-invariant output y = h(x). Assume that the output rank of the explicit system S is given by p(y) . Let Vk be the open and dense set of regular points of the codistributions Y" Yi (see Def. 1), for i = 0, . . . , n and of .cj , L j for j E {O, .. . , k} . In the kth step of the relative dynamic extension algorithm, one may construct around ~ E Vk , a new local classical state representation (:h, Uk) of the system S with state fh = (i, w, . .. ,w{k-l), zP) , ... , zkk»), input
Sketch of Proof. The idea of the proof is to apply Prop. 1 and to show that the Lie-Biicklund immersion L : r --+ S is such that L*.c k = span {dx b, d(z 0 L), . .. ,d(z{k) 0 L)} and dim.c k = dimL*£k + p(y) (see (4)) . 0
Uk = (w{k), J.Lk), where J.Lk = (~~k) , P-k) , and output z{k) =
5. FLAT OUTPUTS FOR REGULAR IMPLICIT SYSTEMS The following result may be regarded as a generalization for implicit systems of a result of (Martin, 1992) (see also (Pereira da Silva, 2000)).
that z~k) C zk:i ) and to choose P-k+l C P-k . (iii) Let € E Vk. The sequence eTk = dim(Lkle)) dim(£k-de) - p(y) is nondecreasing, the sequence Pk = dim(Lkle) - dim(L k- 1 Id - p(y) is non increasing, and both sequences converge to the same integer p( z), called the relative output rank at ~, for some k* ~ n = dimx . (iv) Let Sk C Vk be the open neighborhood of a given ~ E Vk , such that the dimensions of £j,Lj j E {O, . .. ,k} are constant inside Sk . We have Sk = Sk- for k ~ k*. Furthermore Lknspan{dx}lv = Lk-_lnspan{dx}lv for every v E Sk- and k ~ k*. (v) For k ~ k*, one may choose Zk = Zk- in Uk- . Furthermore, Lk+l = Lk + span {w{k-l), zkk+l)} 1
for k ~ k* . (vi) Let Y dim
Theorem 2. Let r be the regular implicit system (1) . Let p(y) be the output rank of the explicit system S defined by (la) with output y = h(x) . Then z =
= span {dy{k) Ik E DV} . Then eTk =
c::;"!y and Pk
=
dim L:~;!Y' In particular
we have p(z) = dim .c:~;!Y'
Proof. The proof of this lemma is based on the same techniques of (Pereira da Silva, 2000) and it is omitted by reasons of space. 0
6. CONCLUSIONS The results of this paper may be useful for studying flatness and the dynamic decoupling problem for implicit systems. It is not difficult to show that the relative dynamic extension algorithm constructs a regular dynamic state feedback for the implicit system that solves the dynamic linearization problem and/ or the dynamic inputoutput decoupling problem.
4. RELATIVE OUTPUT RANK AND OUTPUT RANK OF IMPLICIT SYSTEMS In this section we show that , for regular implicit systems, the relative structure at infinity coincides with the structure at infinity of the corresponding
228
7. REFERENCES
McClamroch, N. H. (1990) . Feedback stabilization of control systems described by a class of nonlinear differential-algebraic equations. Systems fj Control Letters 15, 53-60. Pereira da Silva, P. S. (2000) . Some geometric properties of the dynamic extension algorithm. Technical Report BT / PTC / 0008. Escola Politecnica da Universidade de Siio Paulo. available in http://www.lac.usp.br/ ... pauloj. Pereira da Silva, P. S. and C. Cor rea Filho (2000) . Relative flatness and flatness of implicit systems. Technical Report BT /PTC /xxxx. Escola Politecnica da Universidade de Sao Paulo. To appear in SIAM J . Contr. Optimiz, available in http://www.lac.usp.br/..-.pauloj. Pomet, J .-B. (1995). A differential geometric setting for dynamic equivalence and dynamic linearization . In: Geometry in Nonlinear Control and Differential Inclusions (B. Jackubczyk, W . respondek and T . Rzezuchowski, Eds.). Banach Center Publications. Warsaw. pp. 319-339. Rheinboldt (1991) . On the existence and uniqueness of solutions of non linear semi-implicit differential-algebraic equations. Nonlinear Analysis, Theory, Methods fj Appl. 16, 64266 I. Zharinov, V. V. (1992) . Geometrical A spects of Partial Differentials Equations. World Scientific. Singapore.
Brenan, K. E., S. L. Campbell and L. R. Petzold (1995) . Numerical Solution of Initial- Value Problems in Differential-Algebraic Equations. Springer-Verlag. New York. Campbell, S. L. (1982) . Singular Systems of Differential Equations. Pitman. London. Campbell, S. L. (1990) . Descriptor systems in the 90's. In: Proc. 29th IEEE Conf. Dec. Control. pp. 442-447. Chen, X. and M. A. Shayman (1992) . Dynamics and control of constrained non linear systems with application to robotics. In: Proc. Am. Control Conference. pp. 2962- 2966. Christodoulou, M. A. and C. I~ik (1990) . Feedback control for nonlinear singular systems. Internat. J. Control 51(2), 487-494. Delaleau, E. and P. S. Pereira da Silva (1998) . Filtrations in feedback synthesis: Part I - systems and feedbacks. Forum Math. 10(2) , 147174. Di Benedetto, M. D., J . W . Grizzle and C. H. Moog (1989) . Rank invariants of non linear systems. SIAM J. Control Optim. 21, 658672. Fliess, M. (1989) . Automatique et corps differentiels. Forum Math. 1, 227-238. Fliess, M., J. Levine, P. Martin and P. Rouchon (1993). Linearisation par bouclage dynamique et transformations de Lie-Biicklund. C. R. Acad. Sci. Paris Sir. I Math. 311,981986. Fliess, M., J . Levine, P. Martin and P. Rouchon (1995). Index and decomposition of non linear implicit differential equations. In: IFAC Conf. System Structure and Control. Fliess, M., J . Levine, P. Martin and P. Rouchon (1999) . A Lie-Biicklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44(5) , 922-937. Kawaji, S. and E . Z. Taha (1994) . Feedback linearization of a class of nonlinear descriptor systems. In : Proc. 33rd IEEE Conf. Dec. Control. Vol. 4. pp. 4035-4037. Lin, J . Y. and N. U. Ahmed (1991) . Approach to controllability problems for singular systems. Int. J. Control 22, 675-690. Liu , X. P. (1993) . On linearization of nonlinear singular control systems. In: Proc. American Control Conference. pp. 2284-2287. Liu, Xiaoping and Sergej Celikovsky (1997) . Feedback control of affine nonlinear singular control systems. Internat. J. Control 68(4) , 753774 . Liyi, Dai (1989) . Singular Control Systems. Springer-Verlag. Martin , P . (1992) . Contribuition a l'etude des systemes differentiellement plats. PhD thesis. Ecole des Mines. Paris.
Appendix A. DIFFIETIES AND SYSTEMS In this appendix we recall briefly the main concepts of the infinite dimensional geometric setting of (Fliess et al., 1993; Pomet, 1995; Fliess et al., 1999) . ./RA-Manifolds. Let A be a countable set. Denote by ./RA the set of functions from A to ./R. One may define the coordinate function Xi : ./RA -> ./R by Xi(() = ((i), i E A . This set can be endowed with the Frechet topology (i. e. , an inverse limit topology (Zharinov, 1992)). A basis of this topology is given by the subsets of the form B = {( E ./RA I IXi(O - Oil < fi, i E F} , where F is a finite subset of A , Oi E ./R and fi is a positive real number for i E F . A function
./R is smooth if
./R is a smooth function . Only the dependence on a finite number of coordinates is allowed . From this notion of smoothness, one can easily state the notions of vector fields and differential forms 6 on ./RA and smooth mappings from ./RA 6 We stress that the forms are finite combina tions of the form aI, dxl" where Ii is the multi index (ji ,l , . .. , j i ,r,J , the aI i are s mooth functions, dx1i =
Li
229
x = {xi , i E lnl} U = {u~k)1,j E lml,k E IN} where TO"!j;-l(t , x, U) = t . The set of functions x = (Xl, ... ,xn ) is called state and u = (Ul' . . . ,urn) is called input. In these coordinates the Cartan field is locally written by
to IRB. The notion of IRA-manifold can be also established easily as in the finitely dimensional case (Zharinov, 1992). Given an IRA-manifold P , COO(P) denotes the set of smooth maps from P to IR. Let Q be an IRB-manifold and let tjJ : P --+ Q be a smooth mapping. The corresponding tangent and cotangent mapping will be denoted respectively by tjJ.: TpP --+ Tq,(p)Q and tjJ' : T;(p)Q --+ T;P. The map tjJ : P --+ Q is called an immersion if, around every ~ E P and tjJ(~) E Q, there exist local charts of P and Q such that, in these coordinates
=
Note that J; may depend on t, x and a finite number of elements of U . In this sense, the state representation defined here is said to be generalized , since one accepts that fi may depend on the derivatives of the input . If the fi depend only on (x, u) for i E ln 1, then the state representation is said to be classical. A state representation of a system 5 is completely determined by the choice of the state x and the input u and will be denoted by (x, u). An output y of a system 5 is a set of functions defined on S.
x.
Diffieties. A diffiety M is a IRA manifold equipped with a distribution L\ of finite dimension r, called Gartan distribution. A section of the Cart an distribution is called a Gartan field. An ordinary diffiety is a diffiety for which dim L\ = 1 and a Cartan field aM is distinguished and called the Cartan field. In this paper we will only consider ordinary Diffieties that will be called simply by Diffieties.
System associated to differential equations. Now assume that a control system is given by a set of equations
i=
(A.2)
One can always associate to these equations a diffiety 5 of global coordinates 1j; = {t, x , U} and Cart an field given by (A.I). Flatness. We present now a simple definition of flatness in terms of coordinates 7. A system 5 equipped with Cartan field and time function t = T is locally flat around ~ E 5 if there exists a set of smooth functions y = (Yl , .. . , Ym) , called li E lm 1, j E flat output , such that the set {t, IN} is a (local) coordinate system of 5 around ~ E 5 , where y;j) = -ft jYi Note that the Cartan field is locally given by :
-ft
-ft.
Tt
1
Xi = fi(t , x,U, .. . , u(Qi), i E lnl Yj = T/j(X , U , . . . , u(Q}»), j E lpl
A Lie-Biicklund mapping tjJ : M >--> N between Diffieties is a smooth mapping that is compatible with the Cartan fields , i.e. , tjJ.aM = aN0tjJ. A LieBiicklund immersion (respectively, submersion) is a Lie-Backlund mapping that is an immersion (resp., submersion) . A Lie-Backlund isomorphism between two diffieties is a diffeomorphism that is a Lie-Backlund mapping. Context permitting, we will denote the Cart an field of an ordinary diffiety M simply by Given a smooth object tjJ defined on M (a smooth function , field or form) , then L d tjJ will be denoted by ;p and L nd tjJ = di(n), n E
y;j)
Tt
IN.
Systems. The set of real numbers IR have a trivial structure of diffiety with the Cartan field given by the operation of derivation of smooth functions. A system is a triple (5, IR. T) where 5 is a diffiety equipped with Cartan field and T : 5 f---> IR is a Lie-Backlund submersion . The global coordinate function t of IR represents tzme, that is chosen for once and for all. A LieBacklund mapping between two systems (5. JR , T) and (5' , JR , T' ) is a time-respecting Lie-Backlund mapping di : 5 >--> 5' , i.e. , T' = TO di . Context permitting. the system (5. JR. T) is denoted simply by 5 .
-ft
d a dt = at
as
"" ""
+ L
jElN
L
' Elml
(j +l )
Yi
a
ay,(j)
State Space Representation and Outputs. A local state representation of a system (5, JR , T ) is a local coordinate system. l.-' = {t. x. U} where
dX; L! /\ .
/\ dX jL r,
7 For more intrinsic definitions and some yariations. see ( Fliess e t al. . 1993: Fliess et al. 1999).
On the other hand . the field s are
(poss ibly' ) infinite s ums of the form'\' , ai -aaXI . ~l E.""I.
230