Time varying systems can be viewed as input affine nonlinear systems, by augmenting the state with the time variable as WDŒt; xtT and rewriting the system equations as:
P Dg0./C Xm
iD1
gi./ui;
withg0./D 1
A.t /x
,gi./D 0
Bi.t /
, andBi is theith column ofB.
A distribution will be invariant on an open set U under the vector fields gi if and only if Œgi; j D @@jgi @gi
@ j 2 ./;for allj 2 and 2 U;wherej; j D1; d i m./are vector fields locally spanning;see Remark 6.1 on pp. 44 of Isidori (1989).
Controllability depends on the rank of the smallest distribution that containsg and is invariant under the vector fieldf, given by the following algorithm : 0 Dg; iC1 Di CŒf; ias the limiting distribution ofDlimi!1i.
For the linear affine system the distributioni is spanned exactly by the vectorsBi.t /given by the Silverman–Meadows algorithm.
If is involutive, by the Frobenius theorem, one can determine the transformation that de-composes system equations in the controllability form. To do this, it is necessary to solve partial differential equations of the form.@x/ıj D 0, wherefıjg span the distribution, for details see Isidori (1989).
B Vector Fields
f W Rn ! Rn is a smooth vector fieldif all of its coordinate functions are real valued functions of xT D
x1 x2 : : : xn
T
with continuous partial derivatives of any order. These mappings may be represented in the form ofn-dimensional column vectors of real valued functions. The dual object is called acovector field, which is a smooth mapping assigning to each pointx an element of the dual space.Rn/. A special covector field is the so-calleddifferentialof a real-valued function defined on an open subsetU ofRn:
d.x/WD @
@x WDh@
@x1
@
@x2 : : : @x@
n
i: The derivative ofalongf is defined as
Lf.x/WDdf D Xn
iD1
@
@xi
fi.x/;
which is a real-valued function. TheLie productŒf; gof two vector fieldsf andgis a vector field of the form
Œf; g.x/ D @g
@xf .x/ @f
@xg.x/:
The last operation of frequent use involves a covector field! and a vector fieldf: Lf!.x/ DfT.x/
@!T
@x T
C!.x/@f
@x and the result is a covector field, the derivative of!alongf.
The differential operations introduced above can be related to each other in the following way:
if˛; ˇare real-valued functions andf; gare vector fields then
Œ˛f; ˇg.x/ D˛.x/ˇ.x/Œf; g.x/C.Lfˇ.x//˛.x/g.x/ .Lg˛.x//ˇ.x/f .x/;
if˛; ˇare real-valued functionsf a vector field and!a covector field then L˛fˇ!.x/D˛.x/ˇ.x/.Lf!.x//Cˇ.x/h!.x/; f .x/id˛.x/
C.Lfˇ.x//˛.x/!.x/:
B Vector Fields
Suppose we have d smooth vector fields f1; : : : fd, all defined on the same open setU. The vectorsf1.x/; : : : ; fd.x/span a subspace ofRn:
.x/WDspanff1.x/; : : : ; fd.x/g
which is called asmooth distribution. Starting from the dual objects, if we have!1; : : : ; !d smooth covector fields we can define a subspace of.Rn/:
˝.x/WDspanf!1.x/; : : : ; !d.x/g and this mapping is called asmooth codistribution.
A distributionis said to beinvolutiveif1; 2 2implies thatŒ1; 22 . Letbe a distribution, theannihilatorofat pointxis
?.x/WD fw2.Rn/Ww.v/D0for allv 2.x/g: Analogously, for a codistribution˝one has
˝?.x/WD fv2 RnWw.v/D0for allw 2˝.x/g:
It is well-known that the dual space ofRnis isomorph with itself, i.e., for everyw2.Rn/there exists one and only onew 2 Rnsuch that
w.v/ D hw; vi WDwTv:
The effect of a linear mapA W Rn ! Rn on covectors, i.e., on the elements of .Rn/ can be expressed as:
AQW.Rn/!.Rn/ A.wQ /DwA:
Hence the notionWAmakes sense for any "cosubspace" of covectors W. This cosubspace, gener-ated by the covectorsw1T; : : : ; wkT, is:
W Dspanfw1T; : : : ; wTkg
and it is often identified with the row image of the matrixW, with rowswT1; : : : ; wkT.
A control system on a smooth n-dimensional manifold M is a collection F of smooth vector fields depending on independent parametersw DŒw1; ; wm2 Rmcalled control inputs such thatw.t /belongs to a suitable class of real valued functionsW, called admissible controls.
A dynamical system can be considered as a nonlinear polysystem of the form P
xDf .x.t /; w.t //; x.0/D0; (B.1) where in general, it is assumed thatx 2 M andf .:; w/; w 2 is an analytic (smooth) vector field onM: It is supposed that M is ann-dimensional real analytic manifold (para-compact and connected).
B Vector Fields
Associated with the system (B.1), denote byAF.x; t /the set of all elements attainable fromx at timet:For eachx 2M; AF.x/D [t0AF.x; t /:
Under the Lie bracket, and the pointwise addition, the space of all analytic vector fields on M becomes a Lie algebra; Li e.F/denotes the subalgebra generated by F. For each q 2 M, Li eq.F/ is a subspace of TqM, the tangent space of M at q. A set of vector fields F on a connected smooth manifoldM is called bracket-generating (full-rank) if Li eqF D TqM for all q 2M.
Families of vector fieldsF andG are said to be (strongly)equivalentifLi e.F/DLi e.G/and AF.q; T / D AG.q; T /for allq 2 M and for allT > 0, where the overbar denotes the closure of the sets. The Lie SaturateLS.F/of a family of vector fieldsF is the union of families strongly equivalent toF.
In general it is difficult to construct the Lie saturate explicitly, however one can construct a completely ascending family of compatible vector fields –Lie extension– starting from a given set F of vector fields. A vector fieldf is called compatible with the systemF ifAF[f.q/AF.q/
for allq 2 M. SinceLS.F/is a closed convex positive cone in Li e.F/, a possibility to obtain compatible vector fields is extension by convexification, see Jurdjevic (1997): forf1; f2 2 F and any nonnegative functions˛1; ˛2 2C1.M /the vector fields˛1f1C˛2f2is compatible withF. IfLS.F/contains a vector spaceV , thenLi e.V/LS.F/.
B.1 Normal controllability
Let us denote byefwtx0 the solution of the equationP D fw./; .0/ D x0. Then for a given vector fieldF one can consider the (positive) orbits of the vector field, i.e.,
˚;xq 0.!/.T /Defwqtqefwq 1tq 1 efw2t2efw1t1x0
where D.t1; t2; tq/; ti 0withT DPq
jD1tj and! D.w1; w2 wq/2q; fwi 2F. We will use˚q for˚;0q .!/with fixed!.
A pointy 2M is callednormally reachablefrom anx2 M if there exist a finite sequence of vector fields ffi; i D 1; qg and N 2 Rq
C such that ˚ ;xqN D y and the mapping 2 Rq
C ! ˚;xq , which is defined in an open neighborhood ofN, has rankn DdimM atN.
As a consequence of the surjective mapping theorem, Bartle (1976) Theorem41:6, one has that there is a neighborhoodV ofy such that the pointsz 2 V are normally reachable points fromx.
Let us denote byN.x/the set of normally reachable points. It follows that ifN.x/is not empty, then it has a nonempty interior. A fundamental result is Theorem4:3in Sussmann (1976):
Theorem 1: Let F be a system of Cr vector fields on the CrC1 manifold M, 1 r 1. Then the following conditions are equivalent:
i. F is controllable
ii. F is normally controllable
iii. M is connected and, for everyx 2M,x is normally accessible fromx.
B Vector Fields
Remark 16: Further details concerning the relation between controllability and normal controllability can be found in Grasse (1985), too. In Grasse and Sussmann (1990) it is proved that globally controllable smooth systems are controllable by using piecewise constant controls. The key point here is that for a globally controllable system every point has the normal accessibility property. Actually the interior points of the reachability set are reachable by piecewise constant controls, for details see Sussmann (1987).