Nonlinear control and combinatorics of words 0 ... - Semantic Scholar

Nonlinear control and combinatorics of words Matthias Kawski Department of Mathematics Arizona State University Tempe, Arizona 85287 1

Keywords: Nonlinear control, controllability, free Lie algebra, combinatorics of words. AMS Classi cation: Abstract. This article attempts to bring together some problems in nonlinear control

systems and in algebraic combinatorics of words. One the control side recent results in controllability, nilpotent approximating systems, and applications to path planning are surveyed. On the other side, after a brief survey of various bases for free Lie algebras recent developments in the algebraic combinatorics of words are discussed. These two areas are linked together by the applications to and of the Chen series, Fliess series, and product expansions of these series, culminating in the introduction of chronological products.

0 Introduction This is a survey article about an area of overlap of nonlinear control theory and algebraic combinatorics. The unifying theme is the increasingly sophisticated treatment, analysis and usage of the Chen-Fliess series. On the control side we restrict our attention to the problems of local controllability, nilpotent approximations and very brie y, the path planning problem. On the combinatorial side we survey the development of the theory of bases for free Lie algebras, and then concentrate on the problem of resolving the identity map on some free algebra, which leads to the problem of nding explicit formulas for the dual bases of the Poincare-Birkho-Witt basis built on bases of the free Lie algebra. In trying to be accessible to audiences of both control theorists and the combinatorists, most topics are motivated at an elementary level, and complete proofs are given only where they are particularly illuminating (mostly in the last sections which unite 1 This work was partially supported by NSF-grants 90-07547 and 93-08289.

1

the control and the combinatorial points of view). The article gives an extensive list of references where original proofs can be found; while this list is far from complete, an eort has been made to include many dierent points of view. The organization of the article is as follows: Starting with an applied mechanical problem, the controllability property is de ned and several recently obtained conditions for controllability are reviewed, emphasizing the importance of the internal structure of the iterated Lie barckets involved. In the second section nilpotent control systems are investigated: After presenting coordinate realizations of nilpotent systems, algorithms for the construction of nilpotent approximating systems are given. Both of these rely on the graded and ltered structures of the Lie algebras of vector elds. The section mentions stability results of such approximations regarding controllability and stabilization. The third section reviews the history of adaptations of the Chen path integrals to control, then called the Chen-Fliess series, presents Sussmann's explicit product expansion of the series, and concludes with some remarks about applications to the path planning problem. The second part of the article takes a more combinatorial point of view and begins with elementary properties of various algebras associated with words, with particular emphasis on the shue product, its role in the characteraization of Lie elements and of exponential Lie series. The next section reviews the developemnt of various bases for free Lie algebras: Starting with P. Hall's work in group theory, various bases have been discovered, and eventually all have been shown to come from one underlying principle. Nonetheless, there are subtile dierences in the suitability of these to applications in control. The last section tries to put the combinatorial and the control theoretic prespective even closer together. One focus is on the relation of Sussmann's product expansion of the Fliess series in entirely control theoretic terms and a combinatorial resolution of the identity map in an algebra of words. The other focal point is the chronological product, which combinatorially essentially is no more than a pre-shue product, yet, which from a geometric-control point of view appears to play a much more fundamental role.

2

1 Controllability This section gives the basic de nitions relating to controllability and brie y reviews the main criteria in use for deciding controllability. Throughout the section we refer to the following example for motivation and illustration. Consider a penny rolling without slipping on a plane, but able to rotate about its vertical axis (MClamroch and Bloch [1]). Denote by (y; z) the point of contact with the plane R , by  the steering angle (e.g. with respect to the y-axis, and by  the rolling angle (i.e. about the horizontal symmetry axis, relative to a xed orientation). Further let T and T be the torques about the vertical and a suitable horizontal axis. The equations of motion are: 2

1

2

J  J  y_ z_ 1

2

T T r_ cos  r_ sin 

= = = =

1

(1)

2

where Ji denote moments of inertia and r is the radius of the penny. One typical controllability question to ask is: By appropriately choosing the torques Ti (as functions of time), is it possible to get from any given initial state (position, orientation, and angular velocity) to any desired terminal state? One may consider unbounded torques, or more realistically torques bounded by some constants. A local controllability question to ask is whether it is possible to reach any nearby state (from a given state) without having to go through long excursions. If it is possible to reach a certain state, one may further ask which is the best (optimal) way to reach that state. Optimality criteria could be to minimize the integral of the energy, to minimize time, or many others. Once it is known that one can reach a state, one may ask for an algorithm that produces a control (in this case, torques as functions of time) that steers the system to that state (a special case of the path planning problem). To phrase such questions in a generalized setting introduce the state variable x = _ 2 M = R  T  R that takes values in the six dimensional (y=r; z=r; ; ; _ ; ) manifold M (here T denotes the torus [0; 2]  [0; 2] with appropriately identi ed edges). As controls ui take the normalized torques Ji? Ti . Also introduce the vector 2

2

2

2

1

3

elds f; g ; g 1

2

f (x) = x cos x g (x) = @x@ 5 g (x) = @x@ 6 5

4

@ @x1

+ x sin x 5

4

@ @x2

+x

5

@ @x3

+x

6

@ @x4

(2)

1

2

(or write the vector elds as column vectors f (x) = (x cos x ; x sin x ; x ; x ; 0; 0; )T , g (x) = (0; 0; 0; 0; 1; 0)T and g = (0; 0; 0; 0; 0; 1)T ). With this, the controlled equations of motion are in the following general form (3) of a nonlinear control system (that is ane in the controls). Before coming back to the example, we review the some standard terminology and a few main criteria for controllability. In the following consider controlled dynamical systems of the form 5

1

4

5

4

5

6

2

x_ = f (x) +

m X j =1

uj gj (x)

(3)

where the state x takes values in an analytic manifold 2 M n, f and gj are analytic vector elds on M , and the control u(
) takes values in a nonempty compact subset U  Rm containing 0 2 Rm . (If 0 62 U, pick any constant u 2 U, de ne new vector elds f~(x) = f (x) + P uj gj (x), g~j (x) = gj (x), and new controls u~(t) = u(t) ? u taking values in U~ = fw 2 Rm : w + u 2 Ug which now contains zero.) The reachable set Rp (T ) from p 2 M at time T is the set of all endpoints x(T; u) of trajectories x(
; u) of (3) that start at p at time t = 0 and that correspond to admissible controls u de ned on [0; T ]. Here take the set of all measurable U-valued mappings that are de ned on [0; 1) as the set of admissible controls U . When appropriate also consider controls de ned on nite intervals [0; T ]. (In other places, various authors require admissible controls to have various other regularity properties, e.g. to be of class C r , or piecewise constant, piecewise linear, piecewise C r , and the like. For a detailed study of how controllability properties depend on the regularity properties of the controls and the regularity properties of the vector elds see Grasse [2]). Of the many de nitions for various special kinds of controllability that have been given in the literature we only mention the following: The system (3) is globally controllable if every terminal state q 2 M can be reached from every initial point p 2 M in some time T < 1 by some admissible control; i.e., Tp2M ST 0. The 0

0

0

0

4

system (3) is small-time locally controllable (STLC) about p if p 2 intRp(T ) for every T >0 The next task is to nd simple criteria that allow one to algorithmically decide whether a system is controllable or not. In the case of linear systems x_ = Ax + Bu with x 2 Rn and u 2 Rm with matrices A and B of the appropriate dimensions such a criterion is given by the Kalman-rank-condition: The (linear) system is controllable if and only if the compound matrix formed from all the columns of the matrices fB; AB; A B; : : : An? B g has full rank. Here controllable may be taken to mean e.g. globally controllable with unbounded piecewise constant controls, or STLC about zero with controls taking values in the cube [?1; 1]m . The key observation towards similarily elegant criteria for general nonlinear systems on a manifold, is that the matrix products in the Kalman-rank-condition are to be replaced by Lie products of the vector elds f and gj (compare e.g. Sussmann, [3, 4]). The Lie product (or Lie bracket) of two smooth vector elds v and w is de ned as the unique vector eld [v; w] such that [v; w]' = w(v') ? v(w') for every smooth function ' 2 C 1(M ). In local coordinates write [v; w] = (Dv)w ? (Dw)v where Dv is the Jacobian matrix of the (column) vector eld v. For longer iterated brackets it is convenient to also use the notation (ad v; w) = w and inductively (adj v; w) = [v; (adj v; w)]. Going back to the example of a rolling penny one easily nds that [g ; g ] = 0, [f; g ] = @x@ 4 whereas e.g. [f; g ](x) = cos x @x@ 1 + sin x @x@ 2 + @x@ 3 , and [f; [f; g ]](x) = ?x sin x @x@ 1 + x cos x @x@ 2 . Lie brackets not only are de ned coordinate-free (or alternatively, they transform as desired under coordinate transformations), but moreover, in the analytic case that we consider here, they contain all the geometric information of the system. This is made precise in the theorem (for detailed de nitions see the original reference): 2

1

0

+1

1

2

5

1

4

5

4

4

2

2

4

Theorem 1.1 (Sussmann [3]) Let M; M~ be simply connected real analytic manifolds

and let L; L~ be complete transitive Lie subalgebras of the Lie algebras of all analytic vector elds on M and M~ , respectively. Suppose that  is a Lie algebra isomorphism from L onto L~ , and p 2 M and q 2 M~ are such that image of the isotropy algebra of L at p under  is the isotropy algebra of L~ at q. Then there exists a unique dieomorphism F : M ! M~ such that F (p) = q and such that F (X ) = (X ) for every X 2 L.

5

Here, and in the following, L(f; g) is the Lie algebra spanned by all iterated Lie brackets of the vector elds f and g ; : : : ; gm, and W (p) = fv(p) : v 2 Wg for a set W of vector elds. If m = 1 also use the notation S j for the linear span of all iterated Lie brackets containing j times the vector eld g = g and an arbitrary number the eld f . One of the best studied cases of controllability is that of small-time local controllability about a rest point p of the uncontrolled vector eld f (i.e. f (p) = 0). For a detailed survey of the known conditions see Kawski [5]. 1

1

x_ = f (x) +

m X j =1

x 2 M n; x(0) = p; f (p) = 0; u 2 [?1; 1]m

uj gj (x)

(4)

with analytic vector elds f and gj . We list the main results for such systems.

Theorem 1.2 System (4) is accessible from p if and only if dimL(f; g)(p) = n. Theorem 1.3 (Linear condition) System (4) is STLC about p if dimS (f; g)(p) = n. 1

The following most simple example on M = R illustrates the need to consider brackets with several factors g. 2

x_ = u ju(
)j  1 (5) k x_ = x x =0 Here g(x) = @x@ 1 and ((adk g; f )(0) = (?)k k! @x@ 1 , while every iterated bracket containing any nonzero number of f 's and fewer than k factors g vanishes at x = 0. Clearly this system is accessible from x , but if k is an even number it is not STLC from x (or controllable in practically any sense from x ) because no points x 2 R with x < 0 can be reached from x . This motivates the Hermes condition, proved for planar systems by Hermes [6], and in the general case by Sussmann [7]: 1

2

0

1

0

0

0

0

2

2

0

Theorem 1.4 (Hermes condition) System (4) (with m = 1) is STLC about p if it is accessible and S k (f; g)(p)  S k? (f; g)(p) for every integer k > 0. 2

2

1

A corresponding necessary condition for STLC was given by Stefani [8] in 1985:

Theorem 1.5 (Stefani) If (ad k g; f )(p) 62 S k? (f; g)(p) then the system (4) (with 2

2

m = 1) is not STLC about p.

6

1

In 1985 (Stefani [9]) showed that the system x_ = u; y_ = x; z_ = x y is STLC. In this case the @z@ direction is obtained only from the bracket [[f; [f; g]]; g]; g]; g], containing the controlled vector eld g an even number of times. The following theorem generalizes the Hermes condition to the multi-input case, allows for more exible weight assignments, and puts this example into a general framework: 3

Theorem 1.6 (Sussmann [10]) Suppose that there exists a weight  2 [0; 1] such that whenever k is odd and `1 ; `2; : : : `m are all even then

X

L k0;`0 (f; g)(0) (6) where the sum extends over all (k0 ; `0) such that k0 + Pj `0j < k + Pj `j . Then system (4) is STLC about x = 0. L k;` (f; g)(0)  (

)

(

)

0

Going back to the rolling penny, the following iterated brackets span the tangent space at x = 0 (and hence the system is accessible from x ): (ad [f; g ]; [f; g ])(0) = @x@ 1 , [[f; g ]; [f; g ]](0) = @x@ 2 , [f; g ](0) = @x@ 1 + @x@ 3 , [f; g ](0) = @x@ 4 , g (0) = @x@ 5 , and g (0)) = @ @x6 . Upon closer investigation one sees that the system is also STLC about x by virtue Sussmann's general condition for controllability. Practically this means that it is possible e.g. to reach in arbitrarily short time (and thus without large excursions) states of the form x = (0; 0; 0; "; 0; 0) (with small " 6= 0) from x = 0 using suitable torques. The following two examples (Kawski [5]) do not satisfy the above condition, yet by other means were shown to be STLC: In the system 0

1

2

0

2

1

2

1

2

1

2

0

0

x_ x_ x_ x_

1 1 1 1

u x x xx

= = = =

(7)

1

3 1

2

3

the only bracket giving the @x@ 4 -direction, [[f; [f; g]]; [[[f; g]; g]; g]] contains an even number of factors g and an odd number of factors f . In the system

x_ x_ x_ x_

1 1 1 1

= = = =

u x x x +x

(8)

1

3 1

2 3

7

7 2

the only brackets giving the @x@ 4 -direction, are f = (ad (ad g; f )f ) and f = (ad [f; g]; f ). Here the rst with six gs and three f s clearly corresponds to the uncontrollable term x , but there is no admissible weighting that will make the other bracket f with seven gs and eight f s be of overall lower weight. Nonetheless, it is shown in [5] that the term x may dominate the term x even for arbitrarily small times and control bounds. A most recent more general controllability result by Agrachev and Gamkrelidze [11] tries to put these examples into a more general framework. These examples and theorems clearly exhibit how conditions for controllability rely on the analysis of the iterated Lie brackets of the vector elds that de ne the system. The more general the conditions are, the more they require a detailed analysis of the internal structure of the brackets involved. 2

3

7

2 3

7 2

3 2

2 Nilpotent control systems If L is any Lie algebra de ne the central descending series as the sequence of ideals L i  given by L = L and inductively L i = [L; L i ] = f[V; W ] : V 2 L; W 2 L i g. The Lie algebra L is called nilpotent if its central descending series terminates, i.e. if there is an integer k such that L k = f0g. A control system (4) is called a nilpotent system if the Lie algebra generated by the vector elds f; g ; : : : gm is nilpotent. The importance of nilpotent systems is that one one side they are very convenient to work with, e.g. solution curves x(
; u) may be found by simple quadratures (after a suitable decomposition), and on the other side the class of nilpotent systems is suciently rich that, when used as approximating systems, they preserve many geometric properties of the original system. We introduce some terminology (compare Goodman [12]): For a xed choice of coordinates (x ; : : : xn) on Rn and a sequence of positive integers r ; : : : rn de ne a one-parameter family of dilations  = (t )t> by t (x) = (tr1 x ; : : : ; trn xn). A polynomial p = p(x) is homogeneous of degree k  0 w.r.t. the dilation  if p  t = tmp for all t > 0. Denote by Pk the set of all polynomials that are homogeneous of degree k. A vector eld V on Rn with polynomial coecents is called homogeneous of degree ` if X Pk  Pk ` for all k  0; ?`. Write n` for the set of all vector elds that are homogeneous of degree ` with respect to . For example if r = (1; 2; 5) then p(x) = x + x + x x 2 H whereas the vector eld ( )

(1)

( +1)

( )

( +1)

( )

1

1

1

0

1

+

6 1

8

3 2

1

3

6

V (x) = @x@ 1 + x @x@ 3 2 n? . Note that a dilation induces gradations and ltrations on the spaces of polynomials P = Lk Pk and vector elds with polynomial coecients V = L` n`. Speci cally, one easily veri es that Pk P`  Pk ` for all k; `  0, and similarily [nk ; n`]  nk l . To see the latter, if V 2 nk , W 2 n` and p 2 Ps with s  ?k ? ` and s  0 then [V; W ]p = W (V p) ? V (Wp)  V (Ps k ) + W (Ps `)  Ps k `. With N` = [j`nj , also consider the ltration V = S` N`. Again [Nk ; N`]  Nk `. If % > rj for all dilation exponents rj , then N?% = ;. Consequently, if V ; : : : V m 2 N? are vector elds (with polynomial coecients) of negative homogeneous degrees, then they generate a nilpotent Lie algebra. The following theorem essentially arms that the converse is true also. 2 2

1

0

+

+

+

+

+ +

+

1

1

Theorem 2.1 (Kawski [13]) Let V ; : : : V m be real analytic vector elds on a real 1

analytic manifold M n , which generate the nilpotent Lie algebra L = L(V 1 ; : : : V m ). If p 2 M is such that dimL(p) = n then there are local coordinates (x1 ; : : : xn) about p and a family of dilations  = (t )t>0 such that relative to these coordinates the vector elds V 1 ; : : : V m have polynomial coecients and are of degree (?1) with respect to the dilation .

A slightly stronger theorem with weakened regularity hypotheses was proved by Grabowski [14]. The proof of the theorem is constructive: Key steps in the construction are the selection of the dilation exponents

ri = maxfj  1 : dimL j (p) = (n + 1) ? j g ( )

(9)

a choice of vector elds W i 2 L ri such that fW (p); : : : ; W n(p)g are linearly independent and nally the de nition of the preferred coordinates (x ; : : : xn) as the inverse of the map (10) (x) = (exp xn W n)  : : : (exp x W )(p) (

)

1

1

1

1

that is de ned on a neighbourhood of 0 2 Rn. Here the map (t; q) ! (exp tW )(q) denotes the ow of the vector eld W . If a control system is not nilpotent then is may be possible to nilpotentize it by using a suitable feedback u = (x) + (x)v where  and are analytic Rm and Rmm -valued functions, respectively, (x) is invertible for every x 2 M , and v is considered a new 9

control. The new vector elds f~(x) = f (x) + Pj j (x)gj (x) and g~i(x) = Pj ji (x)gj (x) may sometimes be chosen in such a way that the Lie algebra they generate is nilpotent. Presently little is known when such a nilpotentizing feedback is possible. A necessary condition was given in Hermes, Lundell, Sullivan [15]. Recently, nilpotentizing fedback was used by Murray [16, 17] in the path planning problem. When such exact nilpotentization cannot be achieved one may try to approximate the system x_ = f (x)+Pj uj gj (x) by a nilpotent system x_ = f~(x)+Pj uj g~j (x). The following construction is due to Hermes [18], see also Stefani [19]: Suppose that the original system is accessible at x = 0. Find iterated Lie brackets f1 ; : : : ; fn of lengths r : : : rn of the vector elds f and gj that are linearly independent at x = 0 and that are such that the lengths ri are minimal. If necessary perform a linear change of coordinates to achieve that fi (0) = @x@ i . Consider the family of dilations de ned by these coordinates and the exponents ri. Expand the vector elds f and gi in homogeneous polynomials and truncate the series at terms of degree 0 for f and ?1 for gi. E.g. if f (x) = P1 j ?% fj (x), P then set f~(x) = j ?% fj (x). Then f~ 2 N and g~i 2 N? , and consequently the Lie algebra generated by the vector elds f(adj gi) : 1  i  n; j  0g is nilpotent. (To obtain L(f;~ g~) nilpotent simply truncate the series for f also at terms of degree ?1.) This construction assures that the approximating system is also accessible. Moreover, if this approximating system is STLC, then the original system is also STLC [18]. Going back to the example of the rolling penny from the preceding section, choose e.g. V = g , V = g , V = [f; g ], V = [f; g ], V = [V ; V ] and V = [V ; V ]. The corresponding dilation exponents are r = (1; 1; 2; 2; 4; 6). Introduce the new coordinates z = (x ; x ; x ; x ; x ; x ? x ) and obtain the new system with vector elds f~(z) = z @z@3 + z @z@4 + z z @z@5 + z z @z@6 2 n g~ (z) = @z@1 2 n? , and g~ (z) = @z@2 2 n? . Another important application of systems whose vector elds are homogeneous with respect to some dilation is in the problem of feedback stabilization, which is to nd a feedback law u = (x) (i.e. the control is a suciently regular function of the state), such that the closed loop system x_ = f (x) + Pi i(x)gi(x) is (asymptotically) stable about the origin x = 0. In this case the main results are of the type: If a feedback law that is homogeneous w.r.t. a dilation asymptotically stabilizes a system that also is homogeneous then the same feedback law will locally symptotically stabilize a system that is perturbed by vector elds that are of higher degrees w.r.t. the dilation. For 0

1

0

=

0 =

1

1

2

5

1

6

2

2

4

3

3 4

0

3

2

2

1

4

1

1

5

3

4

6

5

6

3

2 3 4

0

1

1

10

2

1

details see e.g. Hermes [20] and Rosier [21]. With the results mentioned above it makes sense to consider nilpotent systems as model systems that one may want to consider rst when studying e.g. controllability and stabilizability properties. For a more optimal control perspective compare the investigations of the structure of the small-time reachable sets of free nilpotent systems, and for higher order normal forms by Krener and Kang [22] and Krener and Schattler [23]. The problems arise how to generate all possible nilpotent systems, and moreover, how to pick a suitable coordinate realization of these systems. We will return to the rst question when considering realizations of free nilpotent systems in the following sections. Regarding the second question consider the following two nilpotent systems on R : 4

8 > x_ > > < x_ > x_ > > : x_

1

2

3

4

= = = =

8 > y_ > > < y_ > y_ > > : y_

u x x xx

1 2

1

3

2

1

4

3

u y y y

= = = =

1

(11)

2

2 2

In both cases the Lie algebras generated by the vector elds are four dimensional, spanned by the elds fg; [f; g]; [f; [f; g]]; [[f; [f; g]; [f; g]]g and by theorem 1.1 the systems are equivalent up to a dieomorphism. In this case this dieomorphism takes the form of the coordinate change y = (x ; x ; x ; x x ? x ). Clearly the second system is not STLC, and consequently the rst one is not STLC either. While this lack of controllability is apparent in the second realization, it is by no means obvious in the rst. This exempli es that not all coordinate realizations (in polynomial cascade form) are equal, and the question arises how to pick a coordinate representation that is particularly suitable for e.g. studying controllability. To conclude this section consider the following example of a nilpotent dilation homogeneous system which shows the need to not only carefully choose the coordinates, but 1

2

11

3

3

2

4

also the iterated Lie brackets one calculates (compare [5]).

8 > x_ > > > x_ > > > < x_ > x_ > > > x_ > > > : x_

1 2 3 4 5 6

= u = x = x = x = x = x

x_ x_ x_ x_ x_ x_

7

8

1

1 2

2 1

1 6

3 1

1 24

9

10

4 1

11

5

12

= xx = x = x = xx = x = xx 1 6

3 1

2

6

1 2

2 3

1 4

2 1

(12)

2 2

1 24

7

2

4

While this system as a whole is clearly not STLC, consider the projection R ; (t) = f(x ; x ; x ; x ; x )(t; u)g. of the reachable sets onto a ve dimensional subspace. Calculate the following iterated Lie brackets all containing three times the factor f and four times the factor g (i.e. all of the type of possible obstructions to controllability according to Sussmann's general theorem): (3 4) 0

8

9

10

11

12

V = (ad f; (ad g; f ))(0) = @x@ 8 V = (ad (ad g; f ); f ))(0) = @x@ 9 (13) V = (ad [g; f ]; (ad g; f ))(0) = @x@10 V = [f; [[f; g]; (ad g; f )]](0) = @x@11 V = [(ad f; g); (ad g; f )](0) = @x@12 The rst three brackets correspond to obviously uncontrollable directions, exempli ed by three supporting hyperplanes of R ; (t), i.e. xj (t; u)  0, j = 8; 9; 10 for all (t; u). On the other hand it has been shown [5, 11] that the other two directions are independently controllable, i.e. there are no further supporting hyperplanes. While there are many more iterated Lie brackets containing three f s and four gs, all others can be expressed as linear combinations of the above only using anticommutativity and the Jacobi identity (the above form a basis for the homogeneous component L ; (f; g) of the free Lie algebra generated by f and g considered as indeterminants.) The above choice nicely splits into bases for the two and the three dimensional subspaces of controllable and uncontrollable directions; clearly not every basis is expected to do this. Very loosely stated this leads to the problem of nding a basis of formal brackets for the free Lie algebra on (m + 1) generators that splits into bases for the controllable and the uncontrollable directions. A nal remark in this section concerns the use of nilpotent systems for the path planning problem. Considering systems of the form (4), initialized at a point p 2 M , the problem 8

2

9

2

10

2

4

2

2

3

11

12

2

3

(3 4) 0

(3 4)

12

is to give an algorithm that given a desired terminal state q will automatically generate an admissible control u = uq that will steer the system from p to q. In the special case of nilpotent systems (or nilpotentizable systems) (assumed to be globally controllable) several solutions have been given, compare e.g. Laariere and Sussmann [24], Sussmann [25], Jacob [26, 27], Murray and Sastry [16]. First one brings the systems into cascade form by transforming to preferred coordinates (e.g. using theorem 2.1). Next one picks a suitable family of controls u that are parametrized by a nite number of real parameters  = ( ; : : : ; s), e.g. linear combinations of sinusoids with suitably related frequencies, piecewise constant or polynomial controls etc. Using the polynomial cascade form of the system explicitly compute the endpoints of the corresponding trajectories x(
; u), and nally symbolically or numerically invert this map to obtain  as a function of the endpoint. To obtain one general all-purpose algorithm one may consider the largest possible nilpotent systems, i.e. nilpotent systems that are as free as possible. In the next sections it is shown how to obtain such systems, but it remains open which of the in general many equivalent systems one should use, a question which again is closely related to picking a suitable basis for the free nilpotent Lie algebra. 1

3 Lie series in control This section surveys the key role played by the Chen-Fliess series in the study of ane nonlinear systems. The following sections will return to this series from a more combinatorial and algebraic perspective, here the emphasis is on the control theoretic point of view. In 1957 K. T. Chen [28] associated the formal power series () = 1 +



1 X Z X

dxi1 : : : dxip Xi1 : : : Xip



p=1

(14)

to a (smooth) path  : [a; b] ! Rn. The Xi are noncommutative indeterminants and R starting from the line integral  dxi, the integrals for p  2 are inductively de ned by

Z



dxi1 : : : dxip =

Z b Z a



 : : : dx dx ip?1 dip (t) t i1

13

(15)

where t denotes the portion of  ranging from a to t. It is shown that the logarithm of this series is a Lie element. The formal series is employed to derive certain Euclidean invariants of the path . In the early seventies M. Fliess (e.g. [29, 30]) realized the importance of this formal series for control theoretic purposes. Rewritten in various forms this formal series is still at the heart of many studies of properties of nonlinear control systems. It plays a particularly central role in the problem of nding state space realizations of inputoutput systems, typical are bilinear and nonlinear realizations studied by Krener [31], Jacubczyk [32], Crouch and Lamnabhi-Lagrarigue [33, 34] and Sontag and Wang. We follow the presentation of the Chen-Fliess series as in Sussmann [7]: Consider the free associative algebra A = A(X ; : : : ; Xm) generated by m indeterminants X ; : : : Xm . For a multiindex I = (i ; : : : is) with integers 0  ij  m set XI = Xi1 Xi2 : : : Xis . Also consider the algebra A^ = A^(X ; : : : ; Xm) of formal power series in X ; : : : ; Xm with real coecients. Let U be the set of Lebesgue integrable functions de ned on some interval [0; T ] and taking values in Rm. For a point P 2 A^ and a u(
) 2 U consider the dierential equation in A^ (initial value problem) 1

1

1

1

1

m dS = S (X + X uiXi); S (0) = P: dt j

(16)

0

=1

A solution to (16) is a function S : [0; T ] ! A^ such that S (0) = P and (16) holds for each coecient. Hence, if P = P pI XI and S (t) = P sI (t)XI then for each I = (i ; : : : is ) 1

sI (0) = pI and s_I (t) = sJ (t)uis (t)

R

(17)

where J = (i ; : : : is? ). In the particular case that P = 1, the solution is sI (t) = t uI , where the integral stands for 1

Zt 0

uI =

1

Z t Z ts Z ts?1 0

0

0

:::

Z t2 0

uis (ts)uis?1 (ts? ) : : : ui2 (t )ui1 (t ) dt dt : : : dts 1

2

1

1

0

(18)

2

If u 2 U de ne Ser(T; u) { the formal series of u { to be S (T ) where S (
) is the solution of (16) with initial condition P = 1. The set U is a semi-group under the concatenation product, de ned as follows: If u; v 2 U have domains [0; T ] and [0; T ], respectively, de ne the product u]v 2 U with domain [0; T + T ] by u]v(t) = u(t) if t  T and u]v(t) = v(t ? T ) otherwise. A key property of Ser is given in: 1

1

1

14

2

2

1

Theorem 3.1 (Sussmann [7]) The map Ser : U ! A^ is a semi-group monomorphism, i.e. Ser is injective and satis es Ser(u]v) =Ser(u)Ser(v).

Returning to the control system, we now use fj in place of gj for better notation

x_ = f (x) +

m X

j =1

uj fj (x)

(19)

with x 2 M n, and here we consider C 1 vector elds fj on M . Each vector eld fj is a rst-order dierential operator on the space C 1(M ) of smooth functions on M . For a multi-index I = (i ; : : : is ) with 0  ij  m the product fI = fij : : : fis is an s-order partial dierential operator on C 1(M ). Upon substituting the vector elds fi for the indeterminants Xi one obtains the formal series of partial dierential operators on M 1

X ZT ! Serf (T; u) = u I fI I

(20)

0

Applying the operators fI to a smooth function  2 C 1(M ) one obtains a formal series of smooth functions on M

X ZT ! Serf (T; u) = uI (fI ) I

(21)

0

In Sussmann [7] it is shown that this series Serf (T; u) is an asymptotic series for the propagation of  along trajectories of (19). In the case that the vector elds fj and the function  are analytic the series Serf (T; u) converges to (x(T; u; x )): 0

Theorem 3.2 (Sussmann [7]) Consider system (19) with fj analytic vector elds, K  M compact,  2 C ! (M ) and A > 0. Then there exists T > 0 such that for every x 2 K , for every u 2 U with ju(
)j  A that is de ned on an interval [0; Tu]  [0; T ] 0

the solution curve x(t; u; x0 ) is de ned on [0; Tu], the series Serf (t; u) converges to (x(t; u; x0 )) and the convergence is uniform as long as x 2 K , u 2 U with ju(
)j  A, and Tu  T .

One of the many uses of this series is in the proof of necessary conditions for controllability, e.g. of Theorem 1.5. Assuming that (ad k g; f )(p) is linearly independent of all brackets in S k? (f; g) for some k > 0 there is a smooth function  : Rn ! R with (p) = 0, (ad k g; f )(p) = 1 and f (p) = 0 for all f 2 S k? (f; g). After suitable manipulations of the series and integrations by parts etc. it is shown that essentially 2

2

1

2

2

15

1

R

(x(t; u; p)) = ((2k)!)? T (u(t)) k dt + o(jT j k ) form which one concludes that for small T the reachable set from p cannot contain a neighbourhood of p, i.e. the system is not STLC from p. In Kawski [35] a similar necessary condition for STLC, dealing with brackets that have four factors g and three factors f was established. The proof required a partial factorization of the Fliess series in order to be able to get useful estimates of the iterated integrals. More speci cally, all terms in the series involving three or fewer gs were rewritten in terms of a Poincare-Birkho-Witt basis with corresponding iterated integrals simpli ed after integrations by parts. Then an argument similar to the one used by Stefani can be used to prove this necessary condition. Already this proof made it very apparent that the way in which the Fliess series is written is not the most convenient one for applications to control. Also, the Lie series character of the series, already shown by Chen [28], is not at all apparent from the formula. We will come back to the question when a formal power seris is an exponential Lie series in the next sections. In 1985 Sussmann [36] gave a product expansion of the Fliess series as a directed in nite product of exponentials: 1

2

0

2 +1

Ser(T; u) =

Y

H 2H

exp(CH (T; u)H )

(22)

Here, the product is taken over a Hall basis of the free Lie algebra generated by the indeterminants X ; X ; : : : ; Xm, see below, and explicit formulas for the coecients CH (T; u) as iterated integrals of the controls u are given. Here, a Hall set is de ned (compare [37]) as a totally ordered subset H of the set M = M(X ; : : : Xm) of formal brackets such that 0

1

0

1. Each generator Xi is in H 2. If H; H 0 2 H, H  H 0 then length(H )  length(H 0) 3. If H 2 M is not a generator, so that H = [H ; H ] with Hi 2 M, then H 2 H if and only if H ; H 2 H with H < H and either (i) H is a generator, or (ii) H = [H ; H ] with H  H . 1

1

2

21

22

2

1

21

2

2

2

1

Note that as formal brackets e.g. [X ; [X ; [X ; X ]]] and [X ; [X ; [X ; X ]]] are dierent, whereas the Jacobi identity implies that as elements of a Lie algebra they are equal. 0

1

0

16

1

1

0

0

1

Under the canonical map from the space M of formal brackets into the free Lie algebra L = L(X ; : : : Xm ) the image of a Hall set is a basis for L, compare M. Hall [38]. To every formal bracket H 2 H and every control u 2 U de ned on [0; T ] associate two functions cH (u) and CH (u) also de ned on [0; T ]. cH (u) will be integrable and R CH (u)(t) = t cH (u)(s) ds. If H = Xi then let CXi = ui. If H = (adk H ; H ) (as usual (ad v;
) denotes the mapping w ! [v; w]) with H ; H 2 H (and either H is a generator or its left factor is dierent from H ) then de ne cH (u) = k1! (CH1 (u))k cH2 (u) (23) The proof of the product expansion given in [36] uses standard dierential equations techniques (variation of parameters). That article also gives bounds for the iterated integrals, thus providing for convergence results for the in nite product. Section 6 will return to such in nite product expansions, that time from a more combinatorial point of view, showing that the formuals for the coecients in this product expansion in some sense date back as far as 1958 Schutzenberger [39], however, there clearly without any integrals or any convergence results. 0

1

0

1

2

2

2

1

4 Words and the Shue Product We rather closely follow the terminology and notation of Lothaire [40]. Let Z be a nite set that we call an alphabet. Its elements are called letters. A word over Z is any nite sequence w = (a ; a ; : : : ; as) with aj 2 Z . The length jwj of w is the length s of the sequence. Write e for the empty word (empty sequence). The set of all words of length s is denoted by Z s, and the set of all words is Z  . It is equipped with a binary operation obtained by concatenating two sequences (a ; a ; : : : ; ar )(b ; b ; : : : ; as) = (a ; a ; : : : ; ar ; b ; b ; : : : ; as). As this product is clearly associative write the word (noncommutative monomial) as w = a a : : : as. The empty word e satis es we = w = ew for any word w 2 Z . The set Z  together with the concatenation product has a monoid structure. The closely related free semigroup is Z = Z  n feg. Note that Z  has the graded structure n r s whenever w 2 Z r and z 2 Z s. Z  = L1 n Z , where wz 2 Z The free associative algebra over Z , that is the linear space of nite linear combinations of words w 2 Z  with real coecients, is denoted by A. Also refer to A as the algebra of 1

2

1

1

2

1

2

1

2

+

=0

+

17

2

1

2

noncommutative polynomials in m = jZ j indeterminants. This space also has a natural n n graded structure A = L1 n A where each homogeneous component A is the subspace spanned by all words of length n. Write A for the algebra of Z . Finally, A is the linear space of all formal power series Pw2Z  cw w (with real coecients cw ). To construct the free Lie algebra L = L(Z ) generated by Z rst consider the free magma M = M(Z ), the set of all parenthesized words over Z (compare Bourbaki [37]): By induction de ne the sets Z i by setting Z = Z and Z s = Srs? Z r  Z s?r . E.g. if Z = fa; b; cg then (((a; c); b); (b; c)) 2 Z . The free magma is M(Z ) = S1 s Z s and may also be identi ed with the set of all formal brackets over Z , compare the preceding section. The algebra of the magma with real coecients is denoted Lib(Z ), e.g. ((a; b); c) ? 3(b; c) 2 Lib(Z ). The free algebra L over Z is the quotient algebra L(Z ) = Lib(Z )=J where J is the two sided ideal generated by the set f(a; a); ((a; b); c) + ((b; c); a) + ((c; a); b) : a; b; c 2 Lib(Z )g. E.g. as elements of Lib(fa; bg the formal brackets (a; (b; (a; b))) and (b; (a; (a; b))) are distinct elements, however as elements of L they are identical. Note that each element in M that is not in Z has uniquely determined left and right factors, and thus a unique factorisation, whereas elements of the free Lie algebra do not have right or left factors. Consider the map h~ : Lib(Z ) ! A de ned on basis elements (!; ) 2 M n Z by h~ ((!; )) = h~ (!)h~ () ? h~ ()h~ (!), and h~ (a) = a for a 2 Z . It is clear that J lies in the kernel of h~ , and thus there is a map h : L ! A such that h~ = h   where  : Lib ! L is the projection map. Elements x 2 A that lie in the image of h are called Lie elements. For example a = h(a) 2 Z  A and aab ? 2aba + baa = h((a; (a; b))) 2 A are Lie elements. A simple criterion to decide when an element x 2 A is Lie uses the diagonal mapping, that is the algebra morphism
: A ! A A that is de ned on letters a 2 Z as =0

+

( )

+

(1)

(5)


(a) = e a + a e

( )

1 =1

( )

(

=1

)

( )

(24)

Thus e.g.
(ab) = ab e + b a + a b + e ab. Note that, however,
(ab ? ba) = (ab ? ba) e + e (ab ? ba). Let G  A be the set of elements w 2 A such that

18


(w) = e w + w e. Clearly Z  G . Moreover, if w; z 2 G then
(wz ? zw) = (e w + w e)(e z + z e) ? (e z + z e)(e w + w e) = (e wz + z w + w z + wz e) ? (e zw + w z + z w + zw e) = (e (wz ? zw) + (wz ? zw) e (25) Thus the set G is a Lie subalgebra of A (with the commutator product) and it contains all Lie elements. The following theorem asserts that the converse is true, also.

Theorem 4.1 (Friederich's criterion) An element w 2 A is Lie if and only if
(w) = w e + e w. One may identify A with the space of all real valued linear mappings de ned on A. Speci cally, for  = Pw2Z  w w 2 A and x = Pw2Z  xw w 2 A de ne the linear map ~

X ~(x) == w xw w2Z 

(26)

This sum is nite since for every x 2 A all except a nite number of the coecients xw vanish. The shue product is de ned as the transpose X : A A ! A of the diagonal mapping
: A ! A A. Speci cally, for words v; w 2 Z  de ne vX w as the unique element in A satisfying

=

for all z 2 Z 

(27)

For practical calculation the following recursive formula, in terms of the concatenation product, which may also serve as an alternative de nition, is more convenient. For any word w 2 Z  set eX w = w = wX e. For (possibly empty) words w; z 2 Z  and letters a; b 2 Z de ne eX a = aX e = a and or equivalently

(aw)X (bz) = a(wX (bz)) + b((aw)X z)

(28)

(wa)X (zb) = (wX (zb))a + ((wa)X z)b

(29)

19

Then extend linearly to A . The equivalence of these two de nitions is straightforward: Let a; b; c 2 Z and v; w; z 2 Z . Using the Kronecker-delta

< (aw) (bz);
(cv) > < (aw) (bz); (e c)
(v) > + < (aw) (bz); (c e)
(v) > b;c < (aw) z;
(v) > +a;c < w (bz);
(v) > b;c < (aw)X z; v > +a;c < wX (bz); v > < b((aw)X z) + a(wX (bz)); cv > (30) The shue product is obviously commutative. Via a somewhat lengthy, but straightforward calculation one may also directly verify that the shue product is also associative. Every element x 2 A can be written as x = P1 n xn where each xn is a nite linear combination of words of length n. The formal power series x 2 A is called a Lie element if each xn is a Lie element. For every x 2 A with zero constant term (i.e. the coecient of the empty word is zero) de ne exp(x) = Pnj (xn=n!) and log(e + x) = P1 (?)j xj =j where as usual x = e etc. One easily veri es that log(exp(x)) = x and j exp(log(1 + x) = (1 + x). In [28] Chen showed that the logarithm of the Chen series is a Lie element, or in other words that the series is an exponential Lie series. In [41] Ree gave the following criterion for exponential Lie series, which is closely related to the Friedrich's criterion: Theorem 4.2 (Ree 1958 [41]) A series x = P  x w 2 A with x = 1 is an < (aw)X (bz); cv > = = = = =

=0

=0

=1

+1

0

w2Z

w

e

exponential Lie series, i.e. log(x) is a Lie element, if and only if the coecients xw satisfy the shue relations, that is xw xz =  (wx z), where  (Pv2Z  cv v) = Pv2Z  cv xv .

Using this criterion, it is easy to verify:

Proposition 4.3 The Chen-Fliess series is an exponential Lie series To every word w 2 Z  associate an iterated integral. Speci cally, let U be the space of integrable functions de ned on intervals [0; T ] or [0; 1) with values in Rm. (If necessary, extend a control u de ned on a nite interval [0; T ] by setting u(t) = 0 for all t > T .) Inductively de ne the map ~ : Z   R  U . If T  0, and u 2 U then +

~(e; T; u) = 1 R ~(aw; T; u) = T ua(t)~(w; t; u)dt 0

20

and if a 2 Z; and w 2 Z 

(31)

(Here the letters a 2 Z index the component functions ua of the Rm-valued function u. This amounts to choosing an ordering of Z .) Extend ~ linearly to all of A. If x = Pw2Z  xw w 2 A , T  0 and u 2 U and if the series Pw2Z  xw ~ (w; T; u) is absolutely convergent then also use the symbol ~(x; t; u) for this series. A slightly dierent view is to consider the associated map  mapping A into the space of maps from U into the space of locally absolutely continuous functions on R , i.e.  : A ! Mappings(U ; AC ([0; 1); R)) de ned on Z  by (w)(u)(T ) = ~(w; u; T ). We refer to the image of A under S as the space of iterated integrals I , and use the convenient notation w = (w) for w 2 Z , and more generally x = (x) = Pw2Z  xw w for x = Pw2Z  xw w 2 A. While not hard, it still is instructive to once directly verify the proposition, using Ree's theorem: Since the linearity properties are clear we only need to verify that wx z = w z for all w; z 2 Z , i.e. wx z (T; u) = w (T; u)z (T; u) for all u and all T . Proceeding by induction on the sum k = jwj + jzj of the lengths of the words w and z, the start is trivial: If k = 0, i.e. w = e = z then wX z = eX e = e and also ee = 1 = e. The case k = 1 is similarily simple. The pattern already emerges in the case of jwj = jzj = 1, i.e. both w = a and z = b being letters. ax b(T; u) = Sab ba (T; u) = ab (T; u)+ ba (T; u) R  R R R (32) =  T ua(t) t ub( d dt + T ub(t) t ua ( ) d dt R R = T ua (t) dt T ub(t) dt = a(T; u) b(T; u) after one simple integration by parts. For the induction step use the recursive characterization of the shue product in terms of the concatenation product. For letters a; b 2 Z and words w; z 2 Z  +

+

0

0

0

0

0

0

 aw X bz (T; u) = a wX bz b aw X z (T; u) = a wx bz (T; u) + b aw x z (T; u) ZT ZT = ua(t)wx bz (t; u) dt + ub(t) aw x z (t; u) dt (de nition of ) ZT ZT = ua(t)w (t; u)bz (t; u) dt + ub(t)aw (t; u)z (t; u) dt (induction hypo.) ! ! ZT d ZT d = dt aw (t; u) bz (t; u) dt + aw (t; u) dt bz (t; u) dt (defn. of ) (

)

(

)

(

0

0

0

(

(

))

(

))+ ((

((

(

)

)

)

)

)

(

0

0

0

21

)

= aw (T; u)bz (T; u)

(integration by parts)

(33)

This establishes that  is an algebra homomorphism from the algebra of noncommuting polynomials (power series) with the shue product to the algebra of iterated integrals with pointwise multiplication. As a consequence there exists a Lie elements  such that Ser(T; u) = exp( (T; u)). Sussmann gave an explicit formula for  = log(Ser(T; u)) in [36]. However, even after symmetrizing manipulations the formula is still too unwieldly for many control theoretic applications. Alternatively, one may try to express the series Ser(T; u) as an (in nite) product of exponentials of simple Lie elements. This immediately leads to the problem of nding a suitable basis for L and then obtaining explicit formula for the coecients, that again are iterated integrals.

5 Hall bases The rst bases for free Lie algebras have their root in the collecting process of P. Hall [43] who was investigating higher commutators in free groups. Witt [44] demonstrated that there is an isomorphism between free Lie algebras and the higher commutator groups of free groups. For further group theoretical investigations see e.g. Meier-Wunderli [45]. The rst applications to Lie algebras are by M. Hall [47] and Magnus [46]. Compare section 3 for the standard de nition of Hall bases (also: Bourbaki [37]). Since then various bases for Free Lie algebras have been introduced. The bases by Sirsov [48] and Chen-Fox-Lyndon [49], originally thought to be dierent, turned out to be essentially the same. Using the notation from the last section, a Lyndon word is any word w 2 Z  that in lexicographical order is strictly less than all its cyclic permutations, i.e. if w = uv with u; v 2 Z  then uv < vu. For any Lyndon word w 2 Z  n Z let %(w) 2 Z  be the longest proper right factor of w that is a Lyndon word. One may show that the corresponding left factor (w) is also a Lyndon word. The factorisation (w) 2 M of a Lyndon word w is (w) = w if w 2 Z and (((w)); (%(w))) if w 2 Z  n Z . A basis for the free Lie algebra is obtained when these formal brackets in M are mapped into the free Lie algebra L (compare the discussion of Hall bases in the preceding sections). In the seventies Viennot demonstrated that all the known bases for free Lie algebras arise 22

from the underlying principle of unique factorisations of the free monoid Z . Speci cally, a complete factorisation of Z  is an ordered subset B = fui : i  1g  Z  such that every word w 2 Z  can be written in a unique way as w = ui1 ui2 ui3 : : : uir with each uij 2 B and ui1  ui2  : : : uir . Closely related to Viennot's factorisations are Lazard factorisations, compare Bourbaki (the elimination theorem x2.9 in [37]) or Lothaire (problem 5.4.5 in [40]). In a special case this asserts that if a 2 Z then L = L(Z ) is the direct sum of the one-dimensional subspace fta : t 2 Rg and the Lie subalgebra that admits as a basic family the set f(adk a; b) : k  0; b 2 Z n fagg. We shall refer to the single large family of bases for free Lie algebras singled out by Viennot as general Hall bases. The dierence to the de nition of Hall bases in the narrow sense as given in section 3 is that the compatibility of the ordering of the bases with the length of the basis elements is replaced by the weaker requirement that the any Hall element is larger than its left factor. A generalized Hall set over Z is de ned to be a totally ordered subset H  M(Z ) of formal brackets that satis es the following (compare Viennot [50] Theorem 1.2): 1. Each generator a 2 Z is in H 2. If H 2 M is not a generator, so that H = [H ; H ] with Hi 2 M, then H < H , 1

2

1

3. If H = [H ; H ] 2 M with Hi 2 M, then H 2 H if and only if H ; H 2 H with H < H and either (i) H is a generator, or (ii) H = [H ; H ] with H  H . 1

1

2

2

1

2

2

21

22

2

21

1

Melancon/Reutenauer [52, 53] give essentially the same de nition, however they replace each bracket (a; b) 2 M by R(a; b) = (R(b); R(a)) and R(a) = a for letters a 2 Z . Also their ordering is reversed: \ 0 and i  i  : : : is form a basis for U , referred to as the Poincare-Birkho-Witt basis for U . In the case that L = L(Z ) is the free algebra over the alphabet Z , we may identify the universal enveloping algebra U of L with the free associative algebra A = A(Z ), (for technical details compare Bourbaki [37] corollary II.3.1.1). Of main interest to us are explicit formulas for the dual basis to the PBW-basis obtained from Hall sets as described above. For illustration, if a; [a[ab]]; [ab]; [[ab]b]; b are the rst ve basis elements of an basis for L(fa; bg), then the following are elements of the corresponding PBW basis: 1

1

1

a;

1

2

3

1

1

2

b; aa; [ab] = ab ? ba; ba; bb; aaa; [a[ab]] = aab ? 2aba + baa; [ab]a = aba ? baa; baa; [[ab]b] = abb ? 2bab + bba; b[ab] = abb ? bab; bba; bbb;

For w 2 Z , let w~ denote the linear map w~ : A ! R de ned by w~(z) = w;z for any word z 2 Z , i.e. w~(z) = 0 if z 6= w and w~(w) = 1. We want to express the dual basis to the PBW-basis as linear combinations (that turn out to be nite linear combinations) of the maps w~ for w 2 Z . If we write fSh : h 2 PBW ? basisg for the dual basis to the PBW basis, then one easily calculates the following relations for the above example (all 26

missing matrix entries are zero):

10 1 aC CC BB af 0 1 0 10 1 f CC B C ab S 1 0 a ~ a B C CC @ A=@ A@ A B C f ~ B C Sb 0 1 b A @ ba CA 1 bbe 10 1 0 1 0 g CC BB aaa CC BB Saaa CC BB 1 g CC CC BB aab BB S a ab CC BB 1 2 1 CC BB aba C BB S CC BB 1 1 CC BB g CCC BB ab a CC BB gC CC BB baa BB Sbaa CC BB 1 CC BB g CCC CC = BB BB 1 2 1 C BB abb CC BB S ab b CC BB C BB Sb ab CC BB g CC 1 1 C CC BBB bab CC BB BB g CCC 1 C CA BB@ bba B@ Sbba CA B@ gA 1 Sbbb bbb 0 BB Saa BB S ab BB S @ ba Sbb [

[ [

(34)

]]

[

]

[

] ]

[

]

1 0 CC BB 1 CC BB 1 ?1 CC = BB 1 A @

(35)

]

In this example the basis is formed from Lyndon words, and the resulting transformation matrix has a remarkable structure, being triangular with ones on the diagonal and nonnegative integer values above. (The block structure is due to the multigrading structure of A(Z ).) This is not a coincidence, and general results have been proved compare e.g. Melancon and Reutenauer [51] and Garsia [57]. The rst general formula for the dual basis to the PBW basis resulting from Hall sets (in the narow sense) was obtained by Schutzenberger in 1958, see the notes of the Seminaire Dubreil [39]. In 1989 Melancon and Reutenauer obtained an explicit expression for the dual basis corresponding to the Lyndon basis for L. They remarked that it is an surprising fact that exactly the same formula holds for the Hall-Sirsov basis as shown in [39]. Further work by Melancon and Reutenauer [52] clari ed the situation, compare also the next section. The formula that was derived in both cases makes use of the shue product in A . Speci cally, in the case of Lyndon words [51] Se = e~, if z = aw is Lyndon with a 2 Z , w 2 Z  then Saw = aSw , and if w = wi1 wi2 : : : wsis 2 Z  is any word decomposed into Lyndon words w > w > : : : ws then Sw = i !i !1: : : i ! Swi11 X Swi22 X : : : X Swiss (36) s 1

1

2

2

1

2

27

where the exponents denote shue exponentiation w = w and wk = wX wk . The main surprise was that the formula Saw = aSw for Lyndon words is essentially exactly the same in the case of Hall words studied by Schutzenberger. We will come back to this in the next section. Note that Sussmann's formula [36] from 1985 for the iterated integrals contains in a dierent setting essentially the same combinatorial formula for the dual basis of the PBW basis constructed from a Hall basis. 1

+1

6 Chronological products The left and right translations by a letter a 2 Z are the linear maps a ; %a : A ! A, de ned on words w 2 Z  by a = aw and %a (w) = wa. The transposes a.; /a : A ! A are de ned on words by

< a.w; z >=< w; a(z) > and < w/a; z >=< w; %a(z) >

(37)

for all z 2 Z . In particular, if w = vb with b 2 Z and v 2 Z  then vb/a = a;b v. Observe that a.; /a are derivations on A when considered with the shue product, compare Ree [41, 42]. Speci cally, for v; w; z 2 Z  and a 2 Z

< (vX w)/a; z > = = = = =

< vX w; za > < v w;
(z)
(a) > < v w;
(z)(a e) > + < v w;
(z)(e a) > < (v/a) w;
(z) > + < v (w/a);
(z) > < (v/a)X w + vX (w/a); z >

(38)

Alternatively, using the recursive characterization of the shue product, for letters a; b; c 2 Z and words v; w 2 Z  ((vb)X (wc))/a = (((vX (wc))b + ((vb)X w)c) /a = (vX (wc))b;a + ((vb)X w)c;a = ((vb)/a)X (wc) + (vb)X ((wc)/a)

(39)

The calculations for the left translation and derivation are analogous. In general the composition of two D and D is not a derivation, but their commutator [D ; D ] = D  D ? D  D always is a derivation. The set of derivations forms a Lie algebra under composition. Here identify the elements of the Lie algebra L = L(z) with 1

1

2

1

2

2

2

1

28

derivations on A (with the shue product). In particular, for u; v 2 L and W 2 A inductively (on the length of u and v) de ne

w/[u; v] = (w/v)/u ? (w/u)/v

(40)

The (right) chronological product is the linear map  : A  A ! A de ned on words as follows: If a 2 Z , and w; z 2 Z 

w  e = 0 and w  (za) = (w  z + z  w)a

(41)

In general a (right) chronological product is a linear map de ned on a linear space satisfying the following identity:

r  (s  t) = (r  s + s  r)  t

(42)

As a consequence the commutative map (r; s) ! r s = r  s + s  r is associative:

r (s t) = = = =

r  (s  t + t  s) + (s  t + t  s)  r (r  s + s  r)  t + (r  t + t  r)  s + (s  t + t  s)  r (r  s + s  r)  t + t  (r  s + s  r) (r s) t

(43)

The chronological products of words are closely related to the shue product:

w  z + z  w = wX z

(44)

On the space of absolutely continuous functions de ned on intervals [0; T ] de ne a (right) chronological product by (using g_ to denote the derivative of g) (f  g)(t) =

Zt 0

f (s)g_ (s) ds

(45)

One easily veri es that this product satis es the identity (42):

R

(f  (g  h))(t) = t f (s)g(s)h_ (s) ds  R R R = t s f ()g_ () d + s g()f_() d h_ (s) ds = ((f  g + g  f )  h)(t) 0

0

0

0

29

(46)

Another chronological algebra considered by Liu and Sussmann (private communication) de nes a chronological product on the space of polynomial functions in one variable by setting xn  xm = n xn m. Agrachev and Gamkrelidze [58] have used chronological products extensively in the study of products of one-parameter families of dieomorphisms. However, while our chronological product is a pre-associative product, theirs is a pre-Lie product and is required to satsify the identity: a?(b?c)?b?(a?c) = (a?b)?c?(b?a)?c. A geometrical application is in terms of left invariant connections: (X; Y ) ! rX Y . This map is a chronological product if and only if the connection is at, i.e. [rX ; rY ] = r X;Y . While a. (for a 2 Z ) still is a derivation on A now considered as a chronological algebra 1

[

+

]

a.(w  z) = (a.w)  z + w  (a.z)

(47)

/a no longer is a derivation: (w  (za))/b = (wX z)a/b = a;b(wX z)

(48)

Here, w; z 2 Z  and a; b 2 Z . This property of chronological products complements the following property of Hall words (see Viennot [50], also Melancon [52])

Theorem 6.1 Let H  M be a Hall set (in the general sense). Suppose z = wa is a deparenthesized Hall word with w 2 Z , and a 2 Z , and w has the unique factorisation into a nonincreasing product of Hall words w = hs : : : h h with hs  : : :  h  h then 2

1

the parenthesization of z = wa is (wa) = ((hs ); (: : : ((h2 ); ((h1 ); a)))

2

1

Very closely related to this property is the following most simple recursive formula for a free control system. Using the alphabet Z to label the controls ua; a 2 Z , and a Hall set H  A(Z ) to label the coordinate directions xH ; H 2 H a free system is explicitly given by 8 < x_ a = ua if a 2 Z (49) : x_ H;K = xH x_ K if (H; K ) 2 H Recall that due to the unique factorsiation properties one may as well use the deparenthesized word HK as an index. In practical applications, e.g. investigations of controllability properties, reachable sets (e.g. Krener and Schattler [22]) or in path planning (

)

30

algorithms (see e.g. Murray [17]), one usually considers a nite dimensional nilpotent subsystem. One only needs to take care that if x H1 ;H2 is one coordinate direction in the system, then also xH1 and xH2 are coordinate directions. Using the convenient notation of xw for xhs


xh2 xh1 if w factors as w = hs : : : h h into a nonincreasing product of Hall elements, this system may be written in the usual form x_ = Pa2Z uafa (x). Without worrying about the free ini nite dimensional case, the vector elds take formally the form X fa(x) = xH
a @x@ (50) H H 2H Using the multi-grading of the Lie algebra generated by the vector elds one easily shows that in particular fH (0) = cH @x@H for some nonzero constant cH , and consequently the Lie algebra L(f ) generated by these vector elds is isomorphic to the free Lie algebra L = L(Z ). All of this makes perfect sense if one only considers suitable subsets of H corresponding to nilpotent Lie algebras. These realizations of free nilpotent Lie algebras are closely related to the general procedure in theorem 2.1. For alternative constructions see Grayson and Grossman In terms of chronological products the same system is written as (

)

2

1

8 < x_ a = ua : x H;K = xH  xK

if a 2 Z (51) if (H; K ) 2 H If the last letter of the (deparenthesized) Hall word HK is a 2 Z , i.e. HK = wa for some w 2 Z  and w factors w = hs : : : h h with hs  : : :  h  h then a complete expansion of the formula in (49) is (

)

2

1

x_ wa = xhs
xh2 xh1 ua

2

1

(52)

or in terms of chronological products

xwa = xhs  (: : : (xh3  (xh2  xh1 )) : : :)

(53)

This formula makes the analogy to the iterated inetgrals in Sussmann's product expansion equation (22) very clear. The only dierence is the omission of the multi-factorials (they reappaear as the nonzero constant cH above). In a vector space V with basis feigi and dual basis feigi the identity map id : V ! V may be written as id = Pi ei ei with the usual identi cation Hom(V; V )  = V V 31

where V  denotes the dual of V . In our case the words w 2 Z  (including the empty word e) form a basis for the free associative algebra A of (noncommuting) polynomials, and we write w~ for the elements of the dual basis, now considered as elements of A , the associative algebra of noncommuting power series with the letters in Z considered as indeterminants. Thus the identity map id : A ! A may be written as X id = w w~ (54) w2Z 

If H  M is a Hall set, write H^  L for the corresponding Hall-basis for the free Lie algebra L = L(Z ), and let P  A be the corresponding PBW basis. Then X id = b S~b (55) b2P

and this latter expression can be rewritten as a directed in nite product (compare e.g. Sussmann [36], Agrachev and Gamkrelidze [58, 59] and Melancon/Reutenauer [51]) X ~ Y (56) b Sb = exp(Sh h^ ) b2P

h2H

^

where h^ stands for the the image of h 2 M in L under the natural projection map. (Usually we do not distinguish between h and h^ .) Finally, we come back to the product expansion of the Chen-Fliess series in control. For brevity let us consider the control system (19) de ned in terms of the vector elds fa , a 2 Z ; and consider the series Serf obtained from the series Ser (by substituting the vector elds fa for the indeterminants Xa). The shue algebra homomorphism  or ~ (section 4 equation (31)) mapping words R w 2 A (with the shue product) to iterated integrals ~(w)(u)(T ) = T uw (with pointwise multiplication) combined with the homomorphism mapping words w to the partial dierential operators f w may then be combined (in the sense of Hopf algebras) to map the formula (54) to the Chen Fliess series (20). Recall that given a xed initial point x 2 M , disregrading any complications arising form lack of convergence (compare section 4 when there are no problems), the series Serf maps the pair (; u) 2 C ! (M )  U into (x(Tu ; u)), the value of  along the the trajectory of (19). The correspondence between the in nite products (56) and (22) is given by the same shue algebra homomorphism  combined with the Lie algebra homomorphism mapping parenthesized words h 2 L(Z ) to Lie brackets f h of vector elds. 0

0

32

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36