IEEE TRANSACTIONS OX AUTOWTIC CONTROL, VOL. AC-28, NO.
8, AUGUST 86 1983
[ 161 K. D. Glazebrook and D. ,M. Jon-. “Some best possible results for a discounted one armed bandit.” Metrrku. to be published. [I71 P. R. Kumar and T. I. Seidman, “On the optimal solutlon of the one-armed bandit adaptive control problem.” I E E E Tram. .4uromur. Contr.. vol. AC-26. pp. 1176- 1184.
19x1. [ 181 I. M e i h p n and G. W a s . “Multiple feedback at a single sen’er station.“ St%. Proc. ;?pp/.. vol. 5. pp. 195-205. 1977. [I91 P. h-ash. “Optimal allocation of resources between research projects.’‘ Pb.D. disserta-
tion. Cambridge University. Cambridge. England. 1973. [20] P. Nash and I. C. Gittins, “A Hamiltonian approach to optimal stochastic r ~ o u r c c allocation,”
[email protected]. Proh.. vol. 9. pp. 55-68. 1977. [?I] K. C Sevcik. “The use of service time distributions in scheduling,” Univ. Toronto. Toronto, Canada. Tech. R e p CSRG-14. 1972. [22] P. Whittle. “Multi-armed bandits and the Gittins index,” J . Rqv. Stur. Soc. Ser. B . vol. 42, pp. 143-149. 1980.
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present here, weuse as key tools the so-called (regular) controllability distributions introduced in [8]. In this way, our approach completely fits in the systematic work on the generalization of Wonham’s geometric approach to linear systems (see, e.g., [3]-[IO]).We note that a parallel decomposition as in (1.3a) has been studied in [ I I].We also note that similar results are derived in [4]and in a different style in [ 11. The main purpose of this note is to show that the solution of the nonlinear RDP can also be derived by directly generalizing the theory of [ 131. 11. PROBLEM FORMULATION Recall the following definitions (see [3]-[9]). Definition 2.1: An involutive distribution D of fixed dimension, on M , is controlled inoariant for the system ( I . 1a) if there_exists a feegback of the form ( 1.2) such that the modified dynamics i = A ( x ) +:X B, ( x ) L’,leave D invariant; Le.,
Feedback Decomposition of Nonlinear Control Systems HENK NIJMEIJER
[k,D]cD Abstruct -By using the recently developed (differential) geometric proach to nonlinear systems, afeedbackdecompositionfornonlinear control systems is derived:
ap
[,,D]cD,
i=l;..,m.
Definition 2.2: An involutive distribution D of fixed dimension. on M , is a regular controllability distribution of the system ( I . 1a) if it is controlled invariant for the system, and moreover,
I. INTRODUCTION Consider a control system of the form
D = involutive closure of { ad:-B,I k E N ,i E I}
n7
i=A(x)+
(1.la)
Bl(X)U1 1 = I
z,=H,(x),
i=l
,
... m
(1.lb)
I
where x are local coordinates of a smooth n-dimensional manifold M , A , B , ; . . . B , are smooth vector fields on M , and H,: M N . is a smooth output map from ,M to a smooth p , - ( p , > 1) dimensional manifold Nt for i = l ; . . . m . We assume that each Hi, i = l ; . . . m , is a surjective submersion. Furthermore, we will assume that the system (1.la) is strongly accessible (see [12]). In this note we will study the static state feedback noninteracting control problem. That is (see [4]), we seek a control law of the form --f
U=(U(X)+B(X)C
( 14
for a certain subset I c (I;. .,m}. Instead of the above notion of controlled invariance, it is sufficient to use a somewhat weaker concept. Definition 2.3: An involutive distribution D of fixed dimension, on M , is locally controlled irzuariant for the system (1.1a) if locally around each point x. E M there exisJtS a feedback of the form (1.2) such thatthe modified dynamics i = A ( .x ) +:1 B, (x)o, leave D invariant. Similarly, one defines a local version of Definition 2.2: the regular local controllability dlstributions. In considering the static state feedback noninteracting controlproblem. we seek regular local controllability distributions R . .,R , defined by R , : = involutive closure of { adi-i, I k E N}
(2.1)
where A and B, are as in (1.3a), i = 1.; .,m. where a: M - . R m , /3: M + R m X mare smooth maps, / 3 ( x ) = - ( / 3 , , ( ~ )is ) Remark: In the local coordinates of (1.3a) we se_e that R, = nonsingularforallxin~~,andu=(u,.~~~,~,)~~IW~.LetA(~)=A(x) sp_an{a/dz,}, and clearly each distribution R, satisfies [ A , R , ] c R, and + X y L I B , ( x ) a , ( x )and B , ( x ) = X . , ” = I B , ( ~ ) P , i ( x %en, ). in suitable local [B,,R,]cR,,j=l;.~,m,i=l..~.,m. coordinates the modified dynamics 4= A (x)+ X?=,B, (x) u , should read Assuming (2. l), we see that R,ckerH,.=:K,
j z i , i,j=l;..,m
(2.2)
which means exactly that uj(.) does not affect the output z,( .). f o r j == i. Second, we have the nonlinear version of output controllabilig; that is, H,*(R,)=TN,
..
where x = ( . x I ; . .xrn) with each x, and I, possibly being a vector. For linear systems. the above problem- the restricted decoupling problem (RDP)-has been solved under the additional assumption that the set of outputs is “complete”: i.e.. f’ I ker H, = 0 (see [13]). In the solution we Manuscript received June 24. 1982. The author a’as with Stichting Mathematisch Centrum. Kruislaan 413, 1098 SJ Amsterdam. The Netherlands. He is now with the Department of Applied Mathematics. Ta,ente University of Technol%y, hschede, The h‘etherlands.
i=l,..-,m.
(2.3)
This follows from the fact that the system ( I . la) is strongly accessible, so that (1.3a) is-also strongly accessible. But then each of the systems i ,= A , ( x , ) + B,( x,) u, is strongly accessible, and by the fact that the map H, is a sqective submersion we see that the set of reachable output values has nonempty interior in N, for all i = 1,. .. ,m. Thus, the static state feedback noninteracting control problem can be stated as follows. Given the system (I.la) (l.lb), find($possible) a local feedback taw of the form (1.2) such that (2.2) and (2.3) hold for the distributions R, defined bv (2.1). Now, as in the linear case, there is a compatibility problem (see [13]). Clearly, ifwe have controlled invariant distributions D l , . . . ,Dnl. then by no means does it follow that there exists a local feedback (1.2) which leaves each of them invariant. Therefore, we make the follorving assumption:
00l8-9286/83/0800-0861$01.00 Q1983 IEEE
862
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. nt
AC-28,NO. 8, AUGUST 1983
B,(x)=(O,-’~.O.Bi(x,),O,.‘.,O)~.
n kerH,. = 0 ,=I
Next, from (3.2b) we see that
which means that the map H : M + N 1 @ N 2 @ ... (3A’,,,,
[ A . R * , + - - . + R * , , ] ~ R ~ + . . . + R C , , + S ~ ~ ~(3.4) {E~) H(x)=(HI(x):-.,H,,,(x))
is locally injective.
and therefore we can construct a local feedback u-= B ( x ) a , ( x )such that = A(x)+ ~ , ( x ) a l ( xsatisfies ) (cf. [3]) [ A , Rf + . . . + R’,] c Rf + . . . + R*,,. Similarly, for thedistribution R ; + ... + R f - l + R:+l + . . . RZt. we construct a feedback u = x)&,( x) such that the modified dynamics leave this distribution invariant. Finally, by applying the total feedback u = E l ( x ) a , ( x ) +. . . + E n , ( x ) a , , ( x ) , we obtain A(x) = ( A l ( ? c ) , ) , A 2 ( x , ) . . . . , A , ( x , ) ) . So we have established a local feedback ( I .2) such that the modified dynamics are as in (1.3a). and also from (3.1) (l.3b) is satisfied. Furthermore, we note that each system x, = A i ( x i ) + B,( x , ) ~ ‘ ,is strongly accessible, and we have that A(r)
e(
111. W I N THEOREhf
Define R: : = supremal regular local controllability distribution in
n kerH,.,
i=l:...m
.
,si
Remark: R: is well defined (see [6], [SI), but the dimension probably is not fixed. Theorem 3.1: Under the assumption (2.4) and the assumption that each R: has fixed dimension, i = I; . ., m , the static state feedback noninteracting control problem is solvable in a local fashion if and onb if
Rf
+ Ki = TM.
(3.1)
R:
nt
( n ker Hr.)nkerH,. n kerff,. =
r=
T==l
= 0.
I
Since R: C K,. i = 1:. . , m , it follows that the R: are independent. In the next step, wewill show that the R: are compatible; i.e.. there is a local feedback (1.2) which leaves each of the distributions R: invariant. From (3.1) wesee that for each i = l ; . . , m . R:*O. For if R:=O for an i E (1,- . .,m}, then K , = TM. which means that z, = H,( x) is constant. Therefore. we know by the independence of the R: that locally there exist with E R: nspan( B , ; . . ,&}. i = independent vector fields 0 = I: . I , M . So. span(B,: . .,B,} = span(EI: . . , E , } . We also have that dim R: 2 p , (by assumption R: has fixed dimension). and thus from the independency of the R:. we have RT = + . . . + R*, = TM. Thus, the distributions R;: . ’.RZ, are simultaneously integrable (see [ 1 I. Definition 3.1 and Lemma 3.11). So locally around each point x,, E M there exist coordinates such that R: = span(d/ax,}. i = 1.; ..m. with each x, possibly being a vector. Now, from the fact that the distributions R: are locally controlled invariant we have
6
6
Iv.
6
e}.
E
N}.
i = I ,. . ’ ,m .
REhtARKS
I ) In [ I I . Lemma 3.11 the distributions D , ; . . . D , should be independent; Le.. for each disjoint subset I, and I, of (I; . . , L ) , one has that D“ n 0 ~ 2= 0. 2) [ a d ~ - B , . a d ~ - B ~ ] = O f o r a l l k , ( ~ N a n d i = i ( s e e a l s o , [ l l ] ) . 3) If the number of output channels is smaller than the number of inputs, the above procedure still works in a slightly modified way. l Namely, there are more thanoneindependent vector fields in Rf f span(B,; . ..B,,}. and/or there exist some additional input vector fields B I which do not belong toone of the -distributions R:,-but-after B : ( x 2 ) ; . ., applqing feedback-also have the form Bk(x)= (Bi(xl), By(.xn,))‘. These vector fields are superfluous for the whole control synthesis of the system. 4) Each of the systems xi=A , ( + , ) + B , ( x l ) v , .z, = H,(x,) is strongly invertible (see [2]). This has also been clarified in a geometric way in [9], and follows directly from the condition that R: K , = TM! so R: is not contained in ker H,.. We also note that thesituation described ic Theorem 3.1 iseven more special. Namely. the system . i , = A , ( x , ) + B , ( x , ) u , is strongly invertible nith respect to each of the components of the output
-
REFERENCES D. Claude. “Decouphng of nonlinear systems.” S w r . Conrr. L e r r . . vol. 1. pp. 242-248, 1982. R. M. Hirschom. “InvertibiliIy of nonlinear control systems.“ STAM J . Conrr.. vol. 17. pp. 289-297. 1979. “ ( A . B)-mvariant distributions and disturbance decoupling of nonlinear s p terns.” S I A M J . Conrr. Optimir.. vol. 19. pp. 1-19. 1981. A. Isidori, A. J Krener. C Con-Giorgi. and S. Monaco. “Nonlinear decoupling via feedback: A differential geometric approach.” IEEE Tram. Automar. Conrr.. vol. 26, pp. 331-345. 1981. -. “Locally (f.g)-invariant distributions.” S w . Conrr. Len.. vol. 1. pp. 12-15, 1981. A J. Krener and A. Isidori. ( A d / . C )“Invariant and controllability distributions.” in Feedbuck Conrrol of Lineur and Nonhnwr Sxsrenrr (Lecture Notes in Conuol and Information Sciences 39). pp. 157-164. H. Nilmeijer. “Controlled inbariance for affine control systems.’* Inr. J . Conrr., vol. 34. pp R25-833. 1981. -. “Controllability distributions for nonhnear systems.” Svsr. Conrr. Lerr.. vol. 2. pp. 122-129. 1982. - “lnvertlbiht> of aifine nonlinear control systems: A geometric approach.” Sysr. Conrr. Lpn.. vol. 2. pp. 163-168. 1982. H. NiJmeijer and A. I. Van Der Schaft. “Controlled invariance for nonlinear systems.” I E E E Tram. .4uromar. Conrr.. vol. 27. pp 904-914. 1982 W Remondek. r . “On decomvosition oi nonlinear control svstems.’*Svsr. Conrr. Lerr.. vol I. pp 301-308. 19x2 H. J. Sussmann and V. Jurdjebic. “Controllability of nonlinear systems.”J. Dtflerenrru/ ~ouurro~rr. bo1 12. DD .. 95-1 16. 1972. W M Wonham. &near Mulr~ranubleConrrol. A Geometric Approach, 2nd ed New York: Springer-Verlag. 1979.
-.
E R:, i = I:. .,m. where the last equality follows from the fact that Note also that the distribution Rf + . . . + RC,, is involutive. cf. [ I 11. Noy, from (3.3) and [5]. 171 it follows that there locally exists a vector field Bl such that ~ p a n ( d , ) = s p a n ( ~and ~ ) [ B , , R*;+ . . . + R * , , ] c R I + . . . + RC,,. Therefore, in the coordinate system constructed above we have that Bl(.~)=(Bl(.~l),o.~~’,o)~. Similarly. we construct vector fields B,. i = 2: . ..m . such that [ B , , R; + . . . + R IR?- :1, , + . . . + R ’ , ] C R ; + . . . + R : - 1 + R;+l+..’ + R*,, and span{ d,) = span( Thus.
involutive closure of { adi-B,lk
Conversely. from the fact thatthe R: are supremal relative to the condition (2.2) and from (2.3)-which is equivalent to R , + K,= TM-it 0 follows that (3.1) is necessary.
Proof: Assume (?.I) holds, then (2.2) and (2.3) are true for R f . We show next that the K , : = nI *,ker H,.. i = I; . . , m , are independent. Indeed,
c
=
.
