controllability, nonlinear systems, geometric approach, involutive distribution, nonlinear control systems, Department of Applied Mathematics, decoupling, linear systems, analytic systems, Pergamon Press, attitude control, system, output controllability, local controllability, output function, rigid body, control system
Content:
.Von:maar Annlysp. Theory. .Uerhods & Applicnnorrr. Primed in Great Bntaln.
Vol. 8. No. 3. pp. ?73279. 1984
0362546X44 13.M) + 40 @ 1984 Pergamon Press Ltd.
THE TRIANGULAR DECOUPLING PROBLEM FOR NONLINEAR
control systems HENK NIJMEIJER Twente
University of Technology, Department of
Applied Mathematics, P.O. Box 217, 7500 AE Enschede,
THE NETHERLANDS (Receioed 3 May 1983; received for publicadon 20 June 1983) Key words and phrases: Nonlinear
control systems. differential geometric methods. controllability distributions, triangular decoupling.
1. INTRODUCTION CONSIDER a control system of the form
(l.la)
zi = Hi(X), i = 1,. . . ,p
(l.lb)
u here x are local coordinates of a smooth ndimensional manifold M, A, BI , . . , B, are smooth
vector fields on M and H; : M += Ni is a smooth output map from M to a smooth pIdimensional manifold A'i for i = 1, . . , p. We assume that each Hi is a surjective submersion. In this note we will study the (staticstatefeedback) Triangular Decoupling Problem (T.D.P.). That is, we seek a control law of the form
u = 4x) + P(x)0
(1.2)
where CYM: + W", /3: M+ KY"""' are smooth maps, /3(x) = (&(x)) is nonsingular for all x in
A4 and u = (~1, . . . , u,)`E W". Let A(X) = A(x) +IZEl B,(x)cq(x) and B,(x) =
Z:;"=,B,(x)&(x). Then the modified dynamicsk = A(X) + XF I Bi(x)ui should control the output
z,,i=l,...
, p sequentially, i.e. Cl controls zI, possibly changing the values ~2, . . , zp.
then Liz controls ZZ, possibly changing the values of ~3, . . , zp, with the requirement that
z1 be left unaffected and so forth, with d, controlling zp without influencing zl, . . , zp_,
(`here the Li, are vectors such that ul, . . . , u,) = (cl, . . . 6,)). For
linear systems the
triangular Decoupling Problem has been solved completely, sed [3, 11, 12,211. In the solution
we present here we use as key tools the so called regular controllability distributions, introduced
in [14]. In this way our approach completely fits in the systematic work on the generalization
of the geometric approach to linear systems, see e.g. [610, 13181. Note that in the T.D.P.
the partial decoupling of the outputs is weaker than achieving complete dynamic interacting,
which for a special casethe Restricted Decoupling Problemhas been solved in [16].
Recall the following
2. PROBLEM FORMULATION definitions, see [610, 141. 273
Definirion 2.1. An involutive distribution D on M is conwolfed inoariant for the system (1. la) if there exists a feedback of the form (1.2) such that the modified dynamics k = A(x) + Xi &(x)u, leaves D invariant. i.e. [ii,D]CD [B,. D] CD, i = 1,. , m. Definition 2.2. An involutive distribution on M is a regular controllability distribution of the system (l.la) if it is controlled invariant for the system and moreover D = involutive closure of {adf Bi/ k E h, i E I} for a certain subset I C (1, . . , m}. Instead of the above notion of controlled invariance we will use a slightly weaker concept, which is much easier to handle. Definition 2.3. An involutive distribution D on M is locally controlled invariant for the system (l.la) if locally around each point x0 E M there exists a feedback of the form (1.2) such that the modified dynamics i = A(X) + XEi B,(x)u, leaves D invariant. A locally controlled invariant distribution of fixed dimension can easily be characterized, see [S, 131.
THEOREM 2.4.Let D be an involutive distribution of fixed dimension on M and suppose that
the distribution D fl Span{B,,
, B,} has fixed dimension. Then D is locally controlled
invariant if and only if
[A, D] C D + Span{Bi,. . . , B,}
[B,, D] C D + Span{Bi,. . , B,}, i = 1,. . . , m.
Remark 1. In theorem 2.4 the assumption that D has fixed dimension is essential. Therefore
one usually requires this already in definition 2.3, see e.g. [S, 131. Similarly one defines a local
version of definition 2.2: the regular local controllability distributions.
Finally we need a definition of output controllability, see also [16]. Consider the system
(l.la) together with an output function H:M + N. Assume that His a surjective submersion.
Let D be the controllability distribution of (l.la), see [14, 201, i.e. D = involutive closure of
{ad~BJkEFV,i=l,.
. . ,m}. Thenwehave
Definition 2.5. The system (l.la) with output function H: M+ N is output controllable if H.(D) = TN, where D is the controllability distribution of (1. la).
Remark 2. This notion of output'controllability
is similar to the notion of strong accessibility
for a system, [20]. Namely if we denote by &(x0) the reachable set of (l.la) at time f from
x0, then the system is output controllable if H(R,(xo)) has nonempty interior in IV.
It is now easy to see that the local version of the Triangular Decoupling Problem can be formalized, as for linear systems, in the following way: given the system (l.la, b) find (if possible) a local feedback law of the form (1.2) and regular local controllability distributions
Ths rriangular decoupling problem for nonlinear conirol systems
275
R,..
, R, such that we have
1l R, C ,cI Ker H,. i = 1, . . , p
and R, i Ker H,. = TM.
(2.1) (2.2)
In (2.1) the vacuous condition at i = 1 just says RI C TM. Define R; = supremal regular local controllability distribution in ni:i Ker H,..
Remark 3. R,: is well defined, see [lo, 141 and may be computed via the Controllability Subdistribution Algorithm of [lo]. but the dimension of R;(s) may change if x varies in M.
3.MAINTHEOREM
THEOREM 3.1. Under the assumption that for each i = 1. . . p, Rr as well as R; fl Span{Br. . . . B,} have fixed dimension, T.D.P. is solvable in a local fashion if and only if
R; + Ker Hi. = TM, i = 1. . . . , p.
(3.1)
Proof. The necessity of (2.2) follows from the maximality of the R:. For sufficiency we have to show that the RI are comparibfe; although each R,: is locally controlled invariant. it by no
means follows that there exists a local feedback law (1.2) which leaves each of them invariant.
From (2.1) it is clear that
R; 3 R; 3 . . > R;.
(3.2)
According to [19] we can choose local coordinates (xr. . . . xp_ ,) on M such that
eachxjpossibly being a vector.
Ri is locally controlled invariant, so
[A, Ri] C Rp + SpanIB,, . . . , B,}
[Bi, Ri] C RL + Span{Br, . . . , B,}. i = 1.. . . , m. I
(3.3)
By theorem such that
2.4 this is equivalent
to the fact that there exists a local feedback [A, R,] CR; [ Bi, Ri] C RP, i = 1, . , m
~1= a(x)  p(x) u (3.4 1
(here A and I?; are defined as in Section 1). In our local coordinates this means that
A'(x,, . . .
A(x) = _
YxI)r)]) B,(x) = [;;;;;:: : : ;;:+;;I)
(3.5)
[ A'@?, . . . ,x,J
i=l,.
. . , m, where A', respectively Bj, represents the first xldimensional
(=dimRi)
component of the vector field A, respectively Bi and A', respectively 812, the remaining
276
H. NIJMEIJER
components of A respectively B,. Also Rimiis locally controlled invariant. so
[A, Ri_L]CRj_I Span{Bi, ....B.}
(3.6)
[&,R;_,]CR,l7 Span{Bi,.
, B,}, i = 1,. , m.
By using the second component of the vector fields A and fii as in (3.5) and the fact that the dimension of Ri_IIISpan{ I!?,, . . . , f?,} modulo Rj equals the dimension of RjIIISpan{ Br. 8,) minus the dimension of Ri IISpan{ fir, . . , B,}, i.e. is constant by assumption. we `deduce, according to [6, 8, 131, that we can find a local feedback u = L?(X) + &~)ti such that the new vector fields A and B, satisfy (3.4) as well as [A, R;_I]cR,, (3.7) [B,.R,,]CR,,,i=l,....m.
Or, in our local coordinates
(3.8)
i=l,.
. , m, where A' (Bt) is the first x,dimensional (=dim R,)component of A( B,). A'
(II?`) is the second x2dimensional (=dim Rj_Idim Ri) component of A( B,), A'( @) rep
resents the remaining component of A( I?,). Notice that this second local feedback law u =
C?(X) + p(x)(i is independent of xl, i.e. u = &(x2, . . , xp_ *) + &x2, . . . xp ~)a. Repetition
of the above argument yields

i=l,.
( m, where Ai( gi) represents the jth x,dimensional component of A( B,). That is.
we have shown that the distributions R,:are compatible. Next we will use the fact that the
R,T'arse regular local controllability distributions. Usin, 0 this we see that (eventually after a
permutation on the new input functions (fir,
, tim)) there exists a partitioning of the set
{l, . . , m} into p subsets Zk, k = 1, . . , p such that 1, = (1, . . . ml}, II = (1. . . . ml.
. . . , mz}, . . , /,={l,.
. . , m} with the propertyjEZi:GRik1
for k=l..
p.
Therefore our system after applying feedback has the form
B,i(x.,,. ,X,l)
Ly(x:,
. ,XpI)
B,qxp.xpl)
I:
0
"/ I
Furthermore partitioning
The triangular decoupling problem for nonlinear control s);stsms
qx,, . . ,xp_J B,?(Xl.. . ,Xp_l)
B,l(x,.. .Xr_[) 0
,&I, _O 0, + Bp`( x/l 13xp.xpI )
ic I E b lp  I
;
I
0 0
0 0
277 6,. (3.10)
we obtain from R,' cn iit Ker H, for the output functions the following
Zl = H&4,,)
zpel = HpeL(xz,. . ,xp_J zp = Hp(Xl, . xpl).
(3.11)
Finally we note that the condition (3.1). R; + Ker H,. = TM. automatically leads to the notion of output controllability. For example the matrix (dHl/dx,(xp, ,Y~_~)) has full rank and so forth. q Remarks. (i) The system (1. la) is strongly accessible. see [20], ifR;, the supremal controllability distribution. equals TM. If R; = TM we can skip the x~ 1 component in (3.10) and (3.11). (ii) The decomposition given here is different from the cascade decomposition given in [19] (see also [9]). (iii) In some cases one can derive conditions for invertibility for the "subsystems" with Up_, as input function and z, as output function; see [15] for a
Geometric Interpretation of invertibility.
4. AN EXAMPLE:
THE RIGID BODY
We will illustrate the Triangular Decoupling Problem by a simple example of controlling the rigid body. For a mathematical description of a control system on the rigid body together with various results on controllability of the system we refer to [l, 2. 1. 51. The setting used here is similar as in [18]. Consider the system on SO(3) X R'
I? = S(o)R
where R E SO (3) represents the position of a rigid body with respect to an inertial set of axes in Wj, 0 = (wi , (oh, m)' E W3 is the
angular velocity of the rigid body, (CL,. LIZ, 1~;)' are the
275 controls of the system and
As output functions we consider zI = Hi(r. CL)=) last row of the matrix R (4.2) z: = H?(r, w) = second row of R.
i.e. Hi:SO(3) x R'+ 5' and Hl:S0(3) x W'  S'. Similar as in [lS] we will first solve a simpler T.D.P.. namely let r = (r,. r2. rj)' be the first column of R. Then (4.1) reduce to
cyr2_  wr3
0
0
0
w3rl + wlr3
0
0
0
mr1  wlrz
0
0
bl*m
+ a;' Ul + 0
0
U? +
0
bwlm
0
021
0
bm(c)2
0
0_
a;'
(4.3)
where b, = a;`(az
 a3), bl = a;`(a3  a,) and bj = a;`(al zl = Hl(r, (0) = rj z? = H?(r, w) = r?
 a?). Instead
of (4.2) we obtain: (4.4)
According distribution distribution
to theorem 3.1 we only have to compute the supremal regular controllability Ri in Ker Hi. For this we first compute the supremal controlled invariant D in Ker Hi..
Then, see [18], D = Span Xi, X2} where
r2 
Xl(r, 4 =
rl
0
X2(r,
4=
w
 u1
(4.5)
O_
Now it is straightforward to show that D is also a regular controllability distribution and therefore we obtain Ri = D (see also [lo]). Note that the dimension of Ri is not fixed on SO(3) x Z', but on the open submanifold of S_O(3) x 2' where rlrlwlti_ # 0 we certainly have that R; has fixed dimension and R; + Ker Hp = T(SO(3) X i&?`). Finally we note that the sy.stem (4.3) is strongly accessible, i.e. R ; = T(SO(3) x X3), see e.g. [4, 51, and thus RI i Ker fi,* = T( SO(3) x Raj). Therefore by theorem 3.1 the T.D.P. is solvable. The decou
The triangular decoupling problem for nonlinear control systems pling feedback law is given by, see [18],
[;;j=
~::I':0E"l+
["
;I:
[Zj.
279 (1.6)
Finally we see that by the same coupe de grcice as in [lS] this feedback law (5.6) also solves the Triangular Decoupling Problem for the system (1.1. 1) on the open and dense submanifold of SO(3) x R3 where T~T?W~Q + 0. 5. CONCLUSION By generalizing the geometric approach to linear
SYSTEMS THEORY, we were able to solve the Triangular Decoupling Problem for
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H Nijmeijer