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Further results on noninteracting control with internal stability of nonlinear systems
FURTHER RESULTS ON NONINTERACTING CONTROL WITH INT...
14th World Congress ofIFAC
F..2c-23-2
Copyright © 1999 IFAC 14th Triennial World Congress Beijing) P.R. China j
FURTHER RESULTS ON NONINTERACTING CONTROL WITH INTERNAL STABILITY OF NONLINEAR SYSTEMS X .. Xia*
'" Department of Electrical and Electronic Engineering? University of Pretoria, PretoT'ia 0002, South Africa. Fax: +27 (12) 362 5000" Email: [email protected].
Abstract: The parameterization approach is taken further to deal with the noninteracting control problem v.. .ith internal stability of nonlinear systems. First a general conclusion and some sufficient conditions for the solvability of the problem are given. A cOJTlplete solution is then provided to the noninteracting control problem with internal stability for a restricted cJass of nonlinear systems. C;opyright Cc) 1999 If~A(---; Keywords: nonlil1car systems, non-interacting control, stability, controllability distribution
1. INTR,ODUCTION
bility. Exa.mples a.nd counterexarnples are demonstrated in (Xia, 1991). In (BattiJotti 1992), a sufficient condition was given for the nonlinear noninteracting control with stability via a dynamic state feedback. 1
rrherc has been considerable interest in the problem of noninteracting COIltrol \vith internal stabili~y of nonlinear systems (Xia and Gao, 1993; Battilotti~ 1994). According to the partition of the outputs~ the nonlinear noninteracting control problem can be divided into two cases: the Morgan's problem (the one-one decoupling problem), and the block de coupling problem..At present, there are rela.tively complete solutions to the nonlinear stable Morgan '8 problem under both (nondegeneratc) static state fecdbacks and (nondegenerate) dynaruic state feedbaeks, under some mild regularity conditions (Battilotti and Dayawansa, 1991). }or the general case of the stable block decollpling probleTI1, the research results are limited (Grizzle and Isidori, 1989; Battilotti, 1992; Xia, 1993; Xia, 1994). In (Grizzle and Isidori, 1989), Grizzle and Isidori extended their results in (Isidori and Grizzle, 1988), and obtained a necessary condition for the stable block decoupling via nondegenerate sta.tic state feed backs, tha,t is~ the so-called p* -dynamics is asymptotically stable. A parameteriza.tion approach was taken in (Xia, 1993) to deal with the static block decoupling problem with exponential internal sta-
The parameterization approach (Xia, 1993) has been ShO\i\rn to be a useful method to deal with problems of structural design with internal stabjlity (Xi a, 1994; Xia, 1996). This approach is further explored in this paper to continue studying the noninteracting control problem with internal stability of nonlinear systems ...~ general result on the solva.bility of the problem is first given. COllditions are then found for the uniqueness of the set of compatible controlla.bility distributions that can arise as solutions to the noninteracting control problem. Sufficient conditions for the solvability of the noninteracting control problem with stability of this class of systems are thus given. Final1y~ a complete solution is provided to the problem for a restricted class of nonlinear systems.
2. A GENBR,.l\L ANALYSIS Consider a nonlinea.r control system of the follo\ving form~
2748
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
FURTHER RESULTS ON NONINTERACTING CONTROL WITH INT...
rn
X
=
f(x)
+
L
gi(X)Ui~
(1 )
i=l
Yj
hj(x)~j:::::: 1, ... ,fJ-,
the compatible friend (Xia-, 1993) of {Pl , ... 'PJ..'}, then the composite system writes in the ne\v coordinates as, with obvious abuse of notations,
Xi ~
where x E 1V C IR.n, Yj E JRVj, LJ=l Vj p~ [,91,' .. ' g'm are ~rrlooth vector fields, and hI, ... , hJ.1. are smooth mappings.
XIi'+l
J.L
Pt
. -. ~
(5)
X M , Xp,-+-l )Vj
p.+l
lex) + L
XJL +2 ==
g~+2,j(X)Vj
j=1
;::=
G(x) ::= span{g(x)}~ 7n.. For i == ], ... ~ J-l, denote as
§p,+l,j (Xl,
j=l
(gl(X), ... , gm(x», a.nd assume that dimG =
=== kerdh i , g(x)
h (Xi) + 9ii(Xi)Vi, i ;::;: 1, ... ,/1, 1,.,,+1 (Xl ~ · .. ,X{l.~ X'p,+l) +L
v\tithout loss of generality, it is assumed tha.t Xo =::: 0 is an equilibrium of I(x), i.e., f(O) = 0, and only the local case is considered.
Let K i
.=
Yi
== hi ( Xi) ,
for
i:::::: 1 ~ ... ~ JL.
the ma""timal controUabiIity distribution of the system (1) contained in K i , and P* ::= Pt. Denote as 0* the maximal controllability distribution of the system (1) contained in P* ~ and as Po ~< j, giG> the strong accessibility distribution of the system (1). Under some regularity conditions, all these distributions exist, and are computable through standard a.lgorithms (Sussmann and Jurdjevic~ 1972).
ii) Denote as F(Pl , . . . , PJ.1) the compatible friend set of {PI, ... ~ P?l}' then (Q~ fJ) E FCPl, ... 'PIJ-) if and only jf (a, (3) takes the following fornl
Itis "\Jlell-kno\vn (Grizzle and Isidori~ 1989) that a necessary and ~uffieient condition for the solvabili ty of the I10ninterac Ung control problem is the existence of a set of controllability dist.ributions {Pl~ . · . 'P/-L} such that
Proof: Equation (4) is proved in (Xia, 1993), Leuuna 6. i) can be proved along similar lines as in (Grizzle and Isidori ~ 1989). For a proof of ii), refer to (Xia, 1993)~ Theorem 3.
n::l
Pi ~Ki G ~Pi + n(G 11 Pj
)
== G,
(2)
j-=J:.i
for 'i == 1, ... ~ I).. For simpli city, assume that any set of controllability distributions satisfying (2) also satisfy the aforenlentioned regularity condition, and is referred to as a solution to the noninteracting control problem.
For the convenience of the discussion, assume that the set of controllability distribution {Pi, ... ,P~} are given in the following form: Pi == ker dTi(x). (3) This amounts to solving jt partial diffe.rential equations whose solvability is guaranteed by the
Frobenius theorem. Theorem 1 .. The set of cont.rollability distributions given by (3) is a solution to the nonlnteract.ing control problem if and only if p
rankLgT
== L,rankLgTi ,
(4)
i=l
""There T represents a rnatrix with
TOVltS
Ti -
Under condition (4), the follo\ving conclusions hold:
i) Let P :=: nr=l Pi, 0 be the maximal controllability distribution contained in P. There exist smooth functions Tfl.+l (x) and TjL+2(X), together with 7; (x), ... , T.u (x), constituting a local coordinate transformation, such that 0 ker{dT1 , ..• ,dT.u-~l}' Let u ~ o:o(x) + 13o(x)v be
Q
= Qo
+80
[
6(~1)] :
e~(x~)
[1]1(~1)] ,
i3 === 130
~(x)
:
~~(x~)
.(6)
~(x)
In (Grizzl€ and Isidori, 1989), it "",~as proved (in Theorem 3 ...1) that the dynamics i~;.t+l === fJ.1.+1 (0, ... ,0, x J1 + l) does not depend on the choice of (a, fJ) E F(P1 , . . . , Pg), \vhose asymptotic stability thus constitutes a necessity for the solvability of the stable noninteracting control problem. Necessary and sufficient conditions are given in the following theorem.
Theorem. 2. Suppose {PI,"" P~} is a solution to the noninteracting control problem of the system Cl). Without loss of generality! th€ system (1) is assuTned to be in the form of (.5). Then there exist a pair (0, (3) E F(P1 , ••• ,Pi-J.) such tha.t the stable noninteracting control probJenl is solvable if and only if i) Subsystem (7) is stabilizable by a static state i = 1, ... ,J..t;
feedback~
for each
ii) Subsystem
X.u+ 1
:=
j(O, ... ~ 0, XJL+l)
is asymptotically stable; jii) Subsystem
x.u+2 ~ if..L+2(O, ... , O,XJ.l+2)
+9J-l+2 J.l'+1 (0, ... ~ O~ X,u+2)V~+1 is stabilizable by a static state feedback. t
(8)
2749
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
FURTHER RESULTS ON NONINTERACTING CONTROL WITH INT...
Proof: (sufficiency) For each i ::::: 1, ... ,p.~ let Vi == Lti(Xi) be the stabilizing feedbacks in eondition i), and let v,u.+l :-:::: G,u +1 (X JL +2) be the stabilizing feedback in condition iii). From (6), the pair defined by the following feedback v =
Clt~Xl) 1+ 1.J.}~
[
Ctp.(xJ.J J (Xp;+2)
Pt
== ker{dT,u-l'.3' dTi } ,
0* = ker{dTI , .. ·, dTtl" dT/-L+l, dTJ-L.t-3}. The corresponding decomposition can be written as
Xi == h(Xi' Xj.t+3)
+ gii(Xi, X,u+3) Vi , i
X/L+l ::::::: ffL+l (Xl, ... , X tt , X.u+11
.
=== 1, ... , I-'--
Xp:+3)
/L
Q~---t
--T-
belong to F(P1,."~P~). The drift term of the lllorlified dynamics corresponding to this feedback
is
L 9,J,+J
,j (XJ , . . . 'XJ-o X p ,+)
,Xt-t-~·3)Vj
j=l
J.l+l
1;11-+2::::::: j(x) --t-
L
gJl+2,j(X)V.i
j=1
li ~ h(Xi) + 9ii(xi)ai(xi) 1;L+ 1
= iJ-1,'I-l (Xl
~
X,u+3
'Xl-l~ X.u+1)
...
f-l
+ L g~+Ci(.1.~l ~ ... ,Xp., Xj.t+l)a:i(Xi) (9) i.:-o::l M
flJ-+ 2 =-=
fp,+2(X)
+ LgJL+2.i(x)ai(xi) i=]
+9t-t+ 2 ,.u+ 1 (x)ap;+l (X;.t+2) Since O:i.(O) == O~ for i == 1, ... ,f£~ one has
j I l l l (O~
,0, X.u+l) === fp,+l (O~
J1-f.+2(O~
,0, X,u+2)
:=
!p+2 (O~
,0, X Ii,+l), ,0, X.u+2)
... l 0, X tt +2)a: fL +l (xf.t+ 2 ), so (9) is in a. ]o\\r triangular form. From (Vidyasagar, 1980) ~ the systerIl (9) is asymptotically stable. +gj.t+2,j1,+1 (0,
(necessity) From (6), the free term of the modifIed dynanlics of the syst.em (5) corresponding to any feedback pair belonging to F(Pr , . .. , PIl.) takes the form of (9), with only {l;.u+l (X~~+2) being replaced by Dp.+l (x). If (9) is asymptotically stable? then directly from the stability concepts, the conditions in the theorem hold. Even if the stability or stabilizability of the subsystems in Theorem 2 are assumed to be known, the conditions are still not checkable, since there is usually not a unique solution to the noninteracting control problem.
=
jtJ.+3(Xp+3)~
X.u+3), for i = 1, . , . ,!-t. then 0* nG #- O. 'There js an elernent (a column) gt+2~ 1.+1 (x) ~ 0 in glJ--12,~-j'1 (x). Suppose, with9ut loss of generality, that the first component of g~+2,M+l (x) is different from zero. Denote the first component of X~+2 as :1:/-0'.+2 ~ i.e., deCOITlpOSe X.u+2 == (X.u+2, XJ.t+2) , where X,u+2 E JR. Define the distri bution PI =:: ker{ dXl, dXp.+2, dX.u+3}, Yi If 0*
;;=
hi(Xi;
1= 0,
then, by (Xia, 1993), Lemma 2.], it is easy to verify that {p] i P';, ... , P;} satisfies (2), that iS l it is a solution to the noninteracting control problem. But by the construction of P 1 ~ Pt n G i:- PI n G. "Thus PI =j:. Pt, and the solution is not unique. In summarizing, one has the follov..ring theorem. Theorem 3. Suppose the noninteracting control problem of the system (1) is locally solvable at xo, and the maximal solution {Pt, ... , P;} is regular.
If • the fixed dynamics of {Pi, ... ~ F;} is asymptotically stabIe, • the subsystems (7) and (8) corresponding to {Pt ~ ... , P;} are asymptotically stabilizable by sta.tic state feed backs ~ then the problem of noninteracting control with internal stability is locally solva.ble at Xo for the
system (1). Lemma 1. A solution {Pl , ... ~P~} to the noninteracting control probleJJl is unique if and only if Pt == Pt, for i = 1, ... ~ j..L, and 0* == o.
Proof: Sufficiency is proved in (Grizzle and 1989). For
necessity~
Isidori~
note first,
0* C rlf~l Pt.
(10)
Take (et,;3) E F(?; , ... ,PI:)) apply Theorem 1 to obtain a decoupled decomposition for the closed loop system corresponding to {P1· ~ ... 'Pt:}' But due to (10), in characteriz.ing the coordinate transforJuation X == T(x), one can further deeompose T i S11 ch tha.t
Po
If 0* == 0, the a.bove two conditions are also necessary. For linear controllable systems, ii) is automatically satisfied, and i) is also necessary for the solv~bi1ity of the problem. The above assertions about linear systems are proved in (Grizzle and lsidori, 1989). In the linear case, these results also say that a stabilizing and decoupling feedback can be chosen from F(Pt: ... ~ Pt:) when the problem is solvable. In (Xia) 1994), an example was given to show that it is not the case for nonlinear systems. In the next sec.tioo, a nontrivial class of systems will be identified for which a complete solution will be
=== ker dT:U+3 ,
2750
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FURTHER RESULTS ON NONINTERACTING CONTROL WITH INT.. ,
found for the solvability of noninteracting control problem with internal stability.
CO::VfPT~ETE
3. A
SOLUTJ01\'" FOR. A CLASS OF SYSTEMS
Consider a nonlinear system with three inputs and two outputs l
± ::= f (x) + gl (x )111 + g2(.L:)11.2 + g3 (.T) 1J3, Yl==h 1 (x)) yz \V here
h 2 (x), E lR n ,
:=
:L~
PI n G + P2 n G ::;::; G, (12) also satisfy the regularity condition~ i.e., the distributions G,Pl,P2,Pl n G, P2 n G and O~ the maxirnal controllability distribution contained in PI n P2, are nonsingular in a neighborhood of Xo ~
o.
First of all, from (H3), one can easily verify that c Li.j, i. e., Pt is properly c.ontained in ~~ 8 onlv if ~8a == 0 and .E.L a $2 := 0 and that P2* C ~~ X4
pr
~
only if Xo ==
o.
-i!;
=== 0 and if-; ::=::: 0, in a neighborhood of So one has the follov\I"ing four situations:
== A 2; and P'; == .60 2;
A) Pt ::::: L1i and P;'
f ~ gi, and h j are all smooth.
A fe""r assulnption will be made to specify the class of systems. (X),g2(X),g:~(x)}
(H1) {gl
14th World Congress of IFAC
is involutive;
(H2) the relative degrees rl and r2 of the outputs Yl and Y2, respectively, exist and such that the decoup]jng rnatrix defined by £91 hi (x) £9'2 hI (x) £93 hI (x) ] [ L g1 L>fl h 2 (x) L g2 Ljl h 2 (x) L g3 Ljl h 2 (x)
Ltl
Lt!
Lt!
is of full row rank for all x in the neighborhood of the equilibrium x == 0 of the system;
It is \vell-kno\vn that, under the above t~vo assumptions~ the system is decouplable by a static state feedback (Xia and Gao, 1993), and the closed-loop system can be written as~ with abuse of notations,
B) Pi
c
C)
Pi
D)
Pt c
~-ote
~i
Pi
== ~i a.nd
C
D.r and P; C
that X
c
}J'"
.6. 2; ~2;
means X is properly contained
in Y. Each of the above four situations is to be discussed. D) is the simplest case, since in this case one has 81 8f &f 8Xl == 0, (JX2 == 0, 8X4 == 0, the system is actually not strongly accessible (Sussmann and Jurdjevic, 1972). Stable decoupling is possible if and only if the inaccessible dynamics defined by is asymptotically stable.
Now, consider the case A). A set of two controllability distributions PI a.nd P z contained iD ker dh 1 and ker dh2 , respectively, can arise as a solution to the nonint.eracting control problem if and only if (12) holds. Since PI ~ Pt, P2 ~ P2*' ~ dim G ~ 3 and by the property of (11) controllability distributions, there can only be the -- ( X2,1,··· ~ X2,T2 .,following three cases: J.
X3 ::::::: f(Xl ~ X2~ xs, X4) X4 ~ 'l/,3
.
rh'leh
111 '\\
- ( Xl,l,·."
Xl -
Xl~rl
)
an
cl
X2
Denote as ~i and d:i the maximal controlled invariant distributions contained in ker dh 1 and kerdh z , respectively. Then by the above assumptions,
D.i
:=:
kcr{ dXl1,
.... ~ dXl r1
}
A; == ker{ dX21 , ..• , dX2r2} And denote as Pt and P:; the maximal controllability distributions contained in ker dh 1 and ker dh 2 : respectively. Then one has, for i === 1, 2,
Pt
~
6t·
The third assumption is the regularity condition assumed in the previous section for the general case. (H3) Any set of t\VO controlla.bility distributions (PI ~ P2) satisfying
AI) Pl = Pt and P2 :::;:;: p,;; A2) P 1 :;,= Pt and P2 n G = span{gl with k(x) being a. smooth function;
A3) P z === Pi and Pi n G == span{g2 with k(x) being a. smooth function;
+
k(X)g3}
+ k(X)g3}
For AI), since P == PI n P2 == span{ 0: ':X4}~ the maximal controllability dist.ribution B contained in P, which by assumption (H3) is nonsingular, can be either 2 dimensional or .1 dimensional. dim 0 == 2, then 0 == P. By Theorem 2, the system is stabJy decouplabJe if and only jf
~rhen
X3 === f(O, 0,
X4
is feedback
:=
'U3 ~
X3, X4),
(13)
stabilizable~
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Copyright 1999 IFAC
ISBN: 0 08 043248 4
FURTHER RESULTS ON NONINTERACTING CONTROL WITH INT...
v\Then dim 0 is
==
14th World Congress of IFAC
X4 ===
1, then the fixed mode of (P1 , P2)
X3
=
1(0, 0, X3~ 0)
whose stability then constitutes, by Theorem 2 a necessary a.nd sufficie.nt condition for stable decoupling of the system. 1
For A2), since P2 C P; ~ one can assume P2 = ker{ dX211 ... , dX2r:l ~ dT1 ~ .. · ~ dTs }
for some smooth functions of x, T], ... , T s , such that {dX21 ~ ... , dX2rz , dT1 , •.• ~ (1'T:-,;} is an independe.nt set~ One can prove that either s
== ]
or s
..v ]
If s ~ 1, then performing a coordinate transformation Zi == X2.-' i == 1,2,3, Z4 = Tl (x), it is then easy to see that the fixed rnodc of (Pi, P2) is X3 = f(O,O,X3~O), (17) whose stability constitutes the necessary and sufficient condition for stable decoupling of the system. For case A3), one can sirnilarly obtain that stable decoupling is possible if and only if (17) is asymptotically stable or .f.L = 0 and the subsystem vXz ~
2.
==:
In fact, due to Pt = ker{ dXll ~ . . . ~ dX]r1}~ the equation (12) and the assunlption of P2 n G = span{g] + k(X)g3}' one knoV\Ts that there is at least onc Ti such that ~;~ f= O. Without loss of generality, assume that ~aaT -/:: O. 'T'hen by X4 rI'heorem 4w2 (Xia, 1993), a compatible friend of (Pl~ P2) can be computed by =Ul,
=
LITl which, because £93 T l .=:= V3
as
112
Ha
By definj tioIl, P z =<
+ LgT1
re
Xl. T1
(J 4)
>,
dX3~
dX4}.
f
O.
Bl) PI
Since PlnP2 :::= 0, by Theorem 2, stable decoupling is possible if and only jf the subsystem
X2,T2
=:;:
= Pt
1, ...
and P2
=
X2
~ 8°X4 },
8
, a 'aa }. X3
X4
P2*;
+
k(x)g~}
B3) P2 == P'; and Pl n G === span{g2 \vith k(x) being a smooth function;
+
k(X).93}
By Theorem 2, (P1,PZ ) in Bl) is a solution to the noninteracting control problem \vith internal stability if and only if ~
x]
Xl.T] -===
'11]
Xl, i
2:;3 =
,1:+ 1 ,'i
f(Xt,
~
1, ... , 1"J
,T2
-
1
(] 9) x:~)
stabilizable~
In the case of 82), onc can prove along similar lines as in the proof of A2), that P2 takes the form P2 = ker{ dx], dT}
x
OXl
~
V
J
By regularity assumptions on P2 this can be the
i
dx 3} == span{:
0,
B2) P 1 ::= P; and P2 n G :::= span{gl ",rith k(x) being a smooth function;
is feedback
case only if
X2,i:::;'::: x2,i+l,
l,
P{ == ker{dx2} ::::;:: span{ ea
pi- :::;
, dX2r2'
if;;
Next, consider case B). In this case, ELaa = O~ and X4
Analogous to case A) ~ there are only three cases for a Ret of two controllability distributions (PI, P2) to be a solution to the noninteracting control problem:
~ (0 ... 0 1 0 ... 0 * *)T (15) for k == 1 ~ ... , r l ) in which the 1 is at the k-th position: and * could be nonzero. Thus, dim P2 ~ 7'1, implying dim n - r1 == rz + 2. So, s :::; 2. s > 0, because P2 is properly contained in P2'.
=:
(18)
X3, X4)
Xl
+ k(X)g3)
8f
1, ... ,rl - 1,
X4 =::U3 is feedback stabilizable.
-I=- 0, can be denoted
1, 9 I span{gl + k(X)g3}
2, then actually P z ~ ker{ dX21, . . .
== f(Xl~
P; :=ker{ dx
in \vhich (f~ g) is the lllOdified dynanlics corresponding to the feedback (l4)~ It can be calculated that
==
Xl,i+l,i;:::.:
:;:::::: Ul
X3
'U~
== V2, == a(x) + trJ(X)1J,
adjl- k (gl
===
Xl,i
Vz:-= Uz,
- If s
U3
is feedback stabilizable;
-1,
U2
(16)
gr
for a smooth function T of such that t=- O. Note that the reasoning in the proof of A2) for the case of s =::;;; 2 is not valid because otherwise one would have == 0, contradicting to the fact that P2 === ker dX2 ~lS0 there is only one T ~
P-
Since Pt I1 P z =::;;; 0, there is no fixed. mode for (Pl~ P2), again by Theorem 2, (PI, P2) is a solution to the noninteracting control problem with
2752
Copyright 1999 IFAC
ISBN: 0 08 043248 4
FURTHER RESULTS ON NONINTERACTING CONTROL WITH INT...
stability if and only if the subsystem (19) is stabilizable.
For case B3), one can prove, along similar lines as in the proof of A2)~ that Pi takes the form Pi ;::= ker{ dx 2, dX3, dT} 1\.~hjch aT -i- O.
in
OX4
I
T js a smooth function of x such that
'I
Sinc.e .PI P2 == 0) there is no fixed nlode for (PI, P2), once aga.in by Theorem 2, (Pl~ P2) is a solution to the noninteracting control problem with stability if and only if the subsystem Xl,i = Xl,i+l, Xl!rl
==:
-i
==
l~
... ,rl - 1,
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of nonlinear systems has been given. Conditions have been found for the uniqueness of the set of compatible controllability distributions that can arise as solutions to the noninteracting control problem. Sufficient conditions have thus been pre~ scnted for the solvability of the problem of this class of systems. Finally, a complete solution has been provided to the problem for a restricted class of nonlinear systems. It c.an be noted that an the examples a.nd counterexamples in (Xia, 1994) fit into SOTIle c.ategories in Theorem 4.
Acknowledgment The author would like to thank Professor L K. Craig for his valuable comments during the preparation of this paper.
Ul
(20) 5:4
== ·U.3
is feedback stabilizable. The analysis of situation C) is symmetri.c to situation B) ~ a necessary and sufficient condition for the existence of a solution to the nonjnteracting c.ontrol problem with stability is that (remernbering that P== 0 and -#!== 0) VXl V X4 XZ,i
= X2,i+l, i
;;:::=
1, ... , r2
-
1
X2,r2 =:: 'U,2
(21 )
X3 = f(X2' X3) is feedback stabilizable, or :1;2,i
X2,r2
X3
== X2.i-t-l, i ==
1, ... , r2
-
1
== 'i.!2 =:
f(X2' X3)
X4 == V,3 is feedback stabilizabJe.
(22)
Tu sunlInary~ for the sysiem (11)~ a complete solution to the noninteracting control problem with stability is provided by the follovling theorem.
Theorem 4. lJnder aRsumption
(H1)~
(H2) and
(H3), the noninteracting control problem Vw~ith stability is solvable for the system (11) if and only if one of the follo\ving conditions holds: 1) The subsystem (13) is feedback stabilizable;
Ft
2) == 0 and the subsystem (16) is feedback stab] 11zab1e; :i) ~ == 0 and the subsystem (18) is feedback stablhzable; 4. CONCLUSIONS t'ollo1ving previous \vork (Xia, ] 993; Xia, 1994; Xia, 1996), the parameterization approach ha.s been taken further to deal with the non intera.cting control problem \vith internal stability of nonlinear systems. A general result on the noninteracting c.ontrol problem v.lith internal stability
5. REFERENCES Battilott.i, S. (1992). A sufficient condition for nonlinear noninteracting control with stability via dynamic state feedback: blockpart.itioned outputs. International Journal of Control 55, 1141 ~ 1160. Battilotti, S. (1994) . .lVoninteracting Control with Stability for Nonlinear Systems. SpringerVerlag. Berlin. Ba.ttilotti~ S. and W. P. Dayawansa (1991). Nonintera.cting contro.l '\vith stability for a class of nonIinear systems. f3ystems and Control Letters 19~ 327-338. Grizzle, J. ,rv. and A. Isidori (1989). Block noninteracting control with stability via static state feedback. Mathematics of Control: Signals and Systerns 2, 315-914. Isidori~ A. and J. ,\\T, Grizzle (1988). Fixed modes and nonlinear noninteracting control with stability. IEE'E Transactions on Automatic Control AC-33, 907--914. Sussmann, H. J. and \T. Jurdjevic (1972). Controllability of non linear systems. Journal of Differential Equations 12, 95-116. \ridyasagar, M. (1980). Decomposition techniques for large-scale systems with nonadditive interactions, sta.bility and stabilizability. IEEE Transactions on Automatic Control AC25, 773 - 779. Xia, X. (1993). Parameterization of decoupling control laws for affine nonlincar systems. IEEE 1mnsactions on Automatic Control AC-38, 916 - 928~ Xia, X. (1994). E·xamples and counterexamples in stable noninteracting control of nonlinear systems. Acta ..411.tomatica Sinica 20, 186 190. Xia, X. (1996). A parameterization approach to disturbance decoupling problem with stability of nonlinear systems. Automatica 32, 607 - 610. Xia~ X. and \\'. Gao (1993). ]\lonlinear Systems Control and Decoupling. .A.cademic Press. Beijing,{in Chinese).
2753
Copyright 1999 IFAC
ISBN: 0 08 043248 4
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F..2c-23-2
Copyright © 1999 IFAC 14th Triennial World Congress Beijing) P.R. China j
FURTHER RESULTS ON NONINTERACTING CONTROL WITH INTERNAL STABILITY OF NONLINEAR SYSTEMS X .. Xia*
'" Department of Electrical and Electronic Engineering? University of Pretoria, PretoT'ia 0002, South Africa. Fax: +27 (12) 362 5000" Email: [email protected].
Abstract: The parameterization approach is taken further to deal with the noninteracting control problem v.. .ith internal stability of nonlinear systems. First a general conclusion and some sufficient conditions for the solvability of the problem are given. A cOJTlplete solution is then provided to the noninteracting control problem with internal stability for a restricted cJass of nonlinear systems. C;opyright Cc) 1999 If~A(---; Keywords: nonlil1car systems, non-interacting control, stability, controllability distribution
1. INTR,ODUCTION
bility. Exa.mples a.nd counterexarnples are demonstrated in (Xia, 1991). In (BattiJotti 1992), a sufficient condition was given for the nonlinear noninteracting control with stability via a dynamic state feedback. 1
rrherc has been considerable interest in the problem of noninteracting COIltrol \vith internal stabili~y of nonlinear systems (Xia and Gao, 1993; Battilotti~ 1994). According to the partition of the outputs~ the nonlinear noninteracting control problem can be divided into two cases: the Morgan's problem (the one-one decoupling problem), and the block de coupling problem..At present, there are rela.tively complete solutions to the nonlinear stable Morgan '8 problem under both (nondegeneratc) static state fecdbacks and (nondegenerate) dynaruic state feedbaeks, under some mild regularity conditions (Battilotti and Dayawansa, 1991). }or the general case of the stable block decollpling probleTI1, the research results are limited (Grizzle and Isidori, 1989; Battilotti, 1992; Xia, 1993; Xia, 1994). In (Grizzle and Isidori, 1989), Grizzle and Isidori extended their results in (Isidori and Grizzle, 1988), and obtained a necessary condition for the stable block decoupling via nondegenerate sta.tic state feed backs, tha,t is~ the so-called p* -dynamics is asymptotically stable. A parameteriza.tion approach was taken in (Xia, 1993) to deal with the static block decoupling problem with exponential internal sta-
The parameterization approach (Xia, 1993) has been ShO\i\rn to be a useful method to deal with problems of structural design with internal stabjlity (Xi a, 1994; Xia, 1996). This approach is further explored in this paper to continue studying the noninteracting control problem with internal stability of nonlinear systems ...~ general result on the solva.bility of the problem is first given. COllditions are then found for the uniqueness of the set of compatible controlla.bility distributions that can arise as solutions to the noninteracting control problem. Sufficient conditions for the solvability of the noninteracting control problem with stability of this class of systems are thus given. Final1y~ a complete solution is provided to the problem for a restricted class of nonlinear systems.
2. A GENBR,.l\L ANALYSIS Consider a nonlinea.r control system of the follo\ving form~
2748
Copyright 1999 IFAC
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14th World Congress ofIFAC
FURTHER RESULTS ON NONINTERACTING CONTROL WITH INT...
rn
X
=
f(x)
+
L
gi(X)Ui~
(1 )
i=l
Yj
hj(x)~j:::::: 1, ... ,fJ-,
the compatible friend (Xia-, 1993) of {Pl , ... 'PJ..'}, then the composite system writes in the ne\v coordinates as, with obvious abuse of notations,
Xi ~
where x E 1V C IR.n, Yj E JRVj, LJ=l Vj p~ [,91,' .. ' g'm are ~rrlooth vector fields, and hI, ... , hJ.1. are smooth mappings.
XIi'+l
J.L
Pt
. -. ~
(5)
X M , Xp,-+-l )Vj
p.+l
lex) + L
XJL +2 ==
g~+2,j(X)Vj
j=1
;::=
G(x) ::= span{g(x)}~ 7n.. For i == ], ... ~ J-l, denote as
§p,+l,j (Xl,
j=l
(gl(X), ... , gm(x», a.nd assume that dimG =
=== kerdh i , g(x)
h (Xi) + 9ii(Xi)Vi, i ;::;: 1, ... ,/1, 1,.,,+1 (Xl ~ · .. ,X{l.~ X'p,+l) +L
v\tithout loss of generality, it is assumed tha.t Xo =::: 0 is an equilibrium of I(x), i.e., f(O) = 0, and only the local case is considered.
Let K i
.=
Yi
== hi ( Xi) ,
for
i:::::: 1 ~ ... ~ JL.
the ma""timal controUabiIity distribution of the system (1) contained in K i , and P* ::= Pt. Denote as 0* the maximal controllability distribution of the system (1) contained in P* ~ and as Po ~< j, giG> the strong accessibility distribution of the system (1). Under some regularity conditions, all these distributions exist, and are computable through standard a.lgorithms (Sussmann and Jurdjevic~ 1972).
ii) Denote as F(Pl , . . . , PJ.1) the compatible friend set of {PI, ... ~ P?l}' then (Q~ fJ) E FCPl, ... 'PIJ-) if and only jf (a, (3) takes the following fornl
Itis "\Jlell-kno\vn (Grizzle and Isidori~ 1989) that a necessary and ~uffieient condition for the solvabili ty of the I10ninterac Ung control problem is the existence of a set of controllability dist.ributions {Pl~ . · . 'P/-L} such that
Proof: Equation (4) is proved in (Xia, 1993), Leuuna 6. i) can be proved along similar lines as in (Grizzle and Isidori ~ 1989). For a proof of ii), refer to (Xia, 1993)~ Theorem 3.
n::l
Pi ~Ki G ~Pi + n(G 11 Pj
)
== G,
(2)
j-=J:.i
for 'i == 1, ... ~ I).. For simpli city, assume that any set of controllability distributions satisfying (2) also satisfy the aforenlentioned regularity condition, and is referred to as a solution to the noninteracting control problem.
For the convenience of the discussion, assume that the set of controllability distribution {Pi, ... ,P~} are given in the following form: Pi == ker dTi(x). (3) This amounts to solving jt partial diffe.rential equations whose solvability is guaranteed by the
Frobenius theorem. Theorem 1 .. The set of cont.rollability distributions given by (3) is a solution to the nonlnteract.ing control problem if and only if p
rankLgT
== L,rankLgTi ,
(4)
i=l
""There T represents a rnatrix with
TOVltS
Ti -
Under condition (4), the follo\ving conclusions hold:
i) Let P :=: nr=l Pi, 0 be the maximal controllability distribution contained in P. There exist smooth functions Tfl.+l (x) and TjL+2(X), together with 7; (x), ... , T.u (x), constituting a local coordinate transformation, such that 0 ker{dT1 , ..• ,dT.u-~l}' Let u ~ o:o(x) + 13o(x)v be
Q
= Qo
+80
[
6(~1)] :
e~(x~)
[1]1(~1)] ,
i3 === 130
~(x)
:
~~(x~)
.(6)
~(x)
In (Grizzl€ and Isidori, 1989), it "",~as proved (in Theorem 3 ...1) that the dynamics i~;.t+l === fJ.1.+1 (0, ... ,0, x J1 + l) does not depend on the choice of (a, fJ) E F(P1 , . . . , Pg), \vhose asymptotic stability thus constitutes a necessity for the solvability of the stable noninteracting control problem. Necessary and sufficient conditions are given in the following theorem.
Theorem. 2. Suppose {PI,"" P~} is a solution to the noninteracting control problem of the system Cl). Without loss of generality! th€ system (1) is assuTned to be in the form of (.5). Then there exist a pair (0, (3) E F(P1 , ••• ,Pi-J.) such tha.t the stable noninteracting control probJenl is solvable if and only if i) Subsystem (7) is stabilizable by a static state i = 1, ... ,J..t;
feedback~
for each
ii) Subsystem
X.u+ 1
:=
j(O, ... ~ 0, XJL+l)
is asymptotically stable; jii) Subsystem
x.u+2 ~ if..L+2(O, ... , O,XJ.l+2)
+9J-l+2 J.l'+1 (0, ... ~ O~ X,u+2)V~+1 is stabilizable by a static state feedback. t
(8)
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FURTHER RESULTS ON NONINTERACTING CONTROL WITH INT...
Proof: (sufficiency) For each i ::::: 1, ... ,p.~ let Vi == Lti(Xi) be the stabilizing feedbacks in eondition i), and let v,u.+l :-:::: G,u +1 (X JL +2) be the stabilizing feedback in condition iii). From (6), the pair defined by the following feedback v =
Clt~Xl) 1+ 1.J.}~
[
Ctp.(xJ.J J (Xp;+2)
Pt
== ker{dT,u-l'.3' dTi } ,
0* = ker{dTI , .. ·, dTtl" dT/-L+l, dTJ-L.t-3}. The corresponding decomposition can be written as
Xi == h(Xi' Xj.t+3)
+ gii(Xi, X,u+3) Vi , i
X/L+l ::::::: ffL+l (Xl, ... , X tt , X.u+11
.
=== 1, ... , I-'--
Xp:+3)
/L
Q~---t
--T-
belong to F(P1,."~P~). The drift term of the lllorlified dynamics corresponding to this feedback
is
L 9,J,+J
,j (XJ , . . . 'XJ-o X p ,+)
,Xt-t-~·3)Vj
j=l
J.l+l
1;11-+2::::::: j(x) --t-
L
gJl+2,j(X)V.i
j=1
li ~ h(Xi) + 9ii(xi)ai(xi) 1;L+ 1
= iJ-1,'I-l (Xl
~
X,u+3
'Xl-l~ X.u+1)
...
f-l
+ L g~+Ci(.1.~l ~ ... ,Xp., Xj.t+l)a:i(Xi) (9) i.:-o::l M
flJ-+ 2 =-=
fp,+2(X)
+ LgJL+2.i(x)ai(xi) i=]
+9t-t+ 2 ,.u+ 1 (x)ap;+l (X;.t+2) Since O:i.(O) == O~ for i == 1, ... ,f£~ one has
j I l l l (O~
,0, X.u+l) === fp,+l (O~
J1-f.+2(O~
,0, X,u+2)
:=
!p+2 (O~
,0, X Ii,+l), ,0, X.u+2)
... l 0, X tt +2)a: fL +l (xf.t+ 2 ), so (9) is in a. ]o\\r triangular form. From (Vidyasagar, 1980) ~ the systerIl (9) is asymptotically stable. +gj.t+2,j1,+1 (0,
(necessity) From (6), the free term of the modifIed dynanlics of the syst.em (5) corresponding to any feedback pair belonging to F(Pr , . .. , PIl.) takes the form of (9), with only {l;.u+l (X~~+2) being replaced by Dp.+l (x). If (9) is asymptotically stable? then directly from the stability concepts, the conditions in the theorem hold. Even if the stability or stabilizability of the subsystems in Theorem 2 are assumed to be known, the conditions are still not checkable, since there is usually not a unique solution to the noninteracting control problem.
=
jtJ.+3(Xp+3)~
X.u+3), for i = 1, . , . ,!-t. then 0* nG #- O. 'There js an elernent (a column) gt+2~ 1.+1 (x) ~ 0 in glJ--12,~-j'1 (x). Suppose, with9ut loss of generality, that the first component of g~+2,M+l (x) is different from zero. Denote the first component of X~+2 as :1:/-0'.+2 ~ i.e., deCOITlpOSe X.u+2 == (X.u+2, XJ.t+2) , where X,u+2 E JR. Define the distri bution PI =:: ker{ dXl, dXp.+2, dX.u+3}, Yi If 0*
;;=
hi(Xi;
1= 0,
then, by (Xia, 1993), Lemma 2.], it is easy to verify that {p] i P';, ... , P;} satisfies (2), that iS l it is a solution to the noninteracting control problem. But by the construction of P 1 ~ Pt n G i:- PI n G. "Thus PI =j:. Pt, and the solution is not unique. In summarizing, one has the follov..ring theorem. Theorem 3. Suppose the noninteracting control problem of the system (1) is locally solvable at xo, and the maximal solution {Pt, ... , P;} is regular.
If • the fixed dynamics of {Pi, ... ~ F;} is asymptotically stabIe, • the subsystems (7) and (8) corresponding to {Pt ~ ... , P;} are asymptotically stabilizable by sta.tic state feed backs ~ then the problem of noninteracting control with internal stability is locally solva.ble at Xo for the
system (1). Lemma 1. A solution {Pl , ... ~P~} to the noninteracting control probleJJl is unique if and only if Pt == Pt, for i = 1, ... ~ j..L, and 0* == o.
Proof: Sufficiency is proved in (Grizzle and 1989). For
necessity~
Isidori~
note first,
0* C rlf~l Pt.
(10)
Take (et,;3) E F(?; , ... ,PI:)) apply Theorem 1 to obtain a decoupled decomposition for the closed loop system corresponding to {P1· ~ ... 'Pt:}' But due to (10), in characteriz.ing the coordinate transforJuation X == T(x), one can further deeompose T i S11 ch tha.t
Po
If 0* == 0, the a.bove two conditions are also necessary. For linear controllable systems, ii) is automatically satisfied, and i) is also necessary for the solv~bi1ity of the problem. The above assertions about linear systems are proved in (Grizzle and lsidori, 1989). In the linear case, these results also say that a stabilizing and decoupling feedback can be chosen from F(Pt: ... ~ Pt:) when the problem is solvable. In (Xia) 1994), an example was given to show that it is not the case for nonlinear systems. In the next sec.tioo, a nontrivial class of systems will be identified for which a complete solution will be
=== ker dT:U+3 ,
2750
Copyright 1999 IFAC
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FURTHER RESULTS ON NONINTERACTING CONTROL WITH INT.. ,
found for the solvability of noninteracting control problem with internal stability.
CO::VfPT~ETE
3. A
SOLUTJ01\'" FOR. A CLASS OF SYSTEMS
Consider a nonlinear system with three inputs and two outputs l
± ::= f (x) + gl (x )111 + g2(.L:)11.2 + g3 (.T) 1J3, Yl==h 1 (x)) yz \V here
h 2 (x), E lR n ,
:=
:L~
PI n G + P2 n G ::;::; G, (12) also satisfy the regularity condition~ i.e., the distributions G,Pl,P2,Pl n G, P2 n G and O~ the maxirnal controllability distribution contained in PI n P2, are nonsingular in a neighborhood of Xo ~
o.
First of all, from (H3), one can easily verify that c Li.j, i. e., Pt is properly c.ontained in ~~ 8 onlv if ~8a == 0 and .E.L a $2 := 0 and that P2* C ~~ X4
pr
~
only if Xo ==
o.
-i!;
=== 0 and if-; ::=::: 0, in a neighborhood of So one has the follov\I"ing four situations:
== A 2; and P'; == .60 2;
A) Pt ::::: L1i and P;'
f ~ gi, and h j are all smooth.
A fe""r assulnption will be made to specify the class of systems. (X),g2(X),g:~(x)}
(H1) {gl
14th World Congress of IFAC
is involutive;
(H2) the relative degrees rl and r2 of the outputs Yl and Y2, respectively, exist and such that the decoup]jng rnatrix defined by £91 hi (x) £9'2 hI (x) £93 hI (x) ] [ L g1 L>fl h 2 (x) L g2 Ljl h 2 (x) L g3 Ljl h 2 (x)
Ltl
Lt!
Lt!
is of full row rank for all x in the neighborhood of the equilibrium x == 0 of the system;
It is \vell-kno\vn that, under the above t~vo assumptions~ the system is decouplable by a static state feedback (Xia and Gao, 1993), and the closed-loop system can be written as~ with abuse of notations,
B) Pi
c
C)
Pi
D)
Pt c
~-ote
~i
Pi
== ~i a.nd
C
D.r and P; C
that X
c
}J'"
.6. 2; ~2;
means X is properly contained
in Y. Each of the above four situations is to be discussed. D) is the simplest case, since in this case one has 81 8f &f 8Xl == 0, (JX2 == 0, 8X4 == 0, the system is actually not strongly accessible (Sussmann and Jurdjevic, 1972). Stable decoupling is possible if and only if the inaccessible dynamics defined by is asymptotically stable.
Now, consider the case A). A set of two controllability distributions PI a.nd P z contained iD ker dh 1 and ker dh2 , respectively, can arise as a solution to the nonint.eracting control problem if and only if (12) holds. Since PI ~ Pt, P2 ~ P2*' ~ dim G ~ 3 and by the property of (11) controllability distributions, there can only be the -- ( X2,1,··· ~ X2,T2 .,following three cases: J.
X3 ::::::: f(Xl ~ X2~ xs, X4) X4 ~ 'l/,3
.
rh'leh
111 '\\
- ( Xl,l,·."
Xl -
Xl~rl
)
an
cl
X2
Denote as ~i and d:i the maximal controlled invariant distributions contained in ker dh 1 and kerdh z , respectively. Then by the above assumptions,
D.i
:=:
kcr{ dXl1,
.... ~ dXl r1
}
A; == ker{ dX21 , ..• , dX2r2} And denote as Pt and P:; the maximal controllability distributions contained in ker dh 1 and ker dh 2 : respectively. Then one has, for i === 1, 2,
Pt
~
6t·
The third assumption is the regularity condition assumed in the previous section for the general case. (H3) Any set of t\VO controlla.bility distributions (PI ~ P2) satisfying
AI) Pl = Pt and P2 :::;:;: p,;; A2) P 1 :;,= Pt and P2 n G = span{gl with k(x) being a. smooth function;
A3) P z === Pi and Pi n G == span{g2 with k(x) being a. smooth function;
+
k(X)g3}
+ k(X)g3}
For AI), since P == PI n P2 == span{ 0: ':X4}~ the maximal controllability dist.ribution B contained in P, which by assumption (H3) is nonsingular, can be either 2 dimensional or .1 dimensional. dim 0 == 2, then 0 == P. By Theorem 2, the system is stabJy decouplabJe if and only jf
~rhen
X3 === f(O, 0,
X4
is feedback
:=
'U3 ~
X3, X4),
(13)
stabilizable~
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FURTHER RESULTS ON NONINTERACTING CONTROL WITH INT...
v\Then dim 0 is
==
14th World Congress of IFAC
X4 ===
1, then the fixed mode of (P1 , P2)
X3
=
1(0, 0, X3~ 0)
whose stability then constitutes, by Theorem 2 a necessary a.nd sufficie.nt condition for stable decoupling of the system. 1
For A2), since P2 C P; ~ one can assume P2 = ker{ dX211 ... , dX2r:l ~ dT1 ~ .. · ~ dTs }
for some smooth functions of x, T], ... , T s , such that {dX21 ~ ... , dX2rz , dT1 , •.• ~ (1'T:-,;} is an independe.nt set~ One can prove that either s
== ]
or s
..v ]
If s ~ 1, then performing a coordinate transformation Zi == X2.-' i == 1,2,3, Z4 = Tl (x), it is then easy to see that the fixed rnodc of (Pi, P2) is X3 = f(O,O,X3~O), (17) whose stability constitutes the necessary and sufficient condition for stable decoupling of the system. For case A3), one can sirnilarly obtain that stable decoupling is possible if and only if (17) is asymptotically stable or .f.L = 0 and the subsystem vXz ~
2.
==:
In fact, due to Pt = ker{ dXll ~ . . . ~ dX]r1}~ the equation (12) and the assunlption of P2 n G = span{g] + k(X)g3}' one knoV\Ts that there is at least onc Ti such that ~;~ f= O. Without loss of generality, assume that ~aaT -/:: O. 'T'hen by X4 rI'heorem 4w2 (Xia, 1993), a compatible friend of (Pl~ P2) can be computed by =Ul,
=
LITl which, because £93 T l .=:= V3
as
112
Ha
By definj tioIl, P z =<
+ LgT1
re
Xl. T1
(J 4)
>,
dX3~
dX4}.
f
O.
Bl) PI
Since PlnP2 :::= 0, by Theorem 2, stable decoupling is possible if and only jf the subsystem
X2,T2
=:;:
= Pt
1, ...
and P2
=
X2
~ 8°X4 },
8
, a 'aa }. X3
X4
P2*;
+
k(x)g~}
B3) P2 == P'; and Pl n G === span{g2 \vith k(x) being a smooth function;
+
k(X).93}
By Theorem 2, (P1,PZ ) in Bl) is a solution to the noninteracting control problem \vith internal stability if and only if ~
x]
Xl.T] -===
'11]
Xl, i
2:;3 =
,1:+ 1 ,'i
f(Xt,
~
1, ... , 1"J
,T2
-
1
(] 9) x:~)
stabilizable~
In the case of 82), onc can prove along similar lines as in the proof of A2), that P2 takes the form P2 = ker{ dx], dT}
x
OXl
~
V
J
By regularity assumptions on P2 this can be the
i
dx 3} == span{:
0,
B2) P 1 ::= P; and P2 n G :::= span{gl ",rith k(x) being a smooth function;
is feedback
case only if
X2,i:::;'::: x2,i+l,
l,
P{ == ker{dx2} ::::;:: span{ ea
pi- :::;
, dX2r2'
if;;
Next, consider case B). In this case, ELaa = O~ and X4
Analogous to case A) ~ there are only three cases for a Ret of two controllability distributions (PI, P2) to be a solution to the noninteracting control problem:
~ (0 ... 0 1 0 ... 0 * *)T (15) for k == 1 ~ ... , r l ) in which the 1 is at the k-th position: and * could be nonzero. Thus, dim P2 ~ 7'1, implying dim n - r1 == rz + 2. So, s :::; 2. s > 0, because P2 is properly contained in P2'.
=:
(18)
X3, X4)
Xl
+ k(X)g3)
8f
1, ... ,rl - 1,
X4 =::U3 is feedback stabilizable.
-I=- 0, can be denoted
1, 9 I span{gl + k(X)g3}
2, then actually P z ~ ker{ dX21, . . .
== f(Xl~
P; :=ker{ dx
in \vhich (f~ g) is the lllOdified dynanlics corresponding to the feedback (l4)~ It can be calculated that
==
Xl,i+l,i;:::.:
:;:::::: Ul
X3
'U~
== V2, == a(x) + trJ(X)1J,
adjl- k (gl
===
Xl,i
Vz:-= Uz,
- If s
U3
is feedback stabilizable;
-1,
U2
(16)
gr
for a smooth function T of such that t=- O. Note that the reasoning in the proof of A2) for the case of s =::;;; 2 is not valid because otherwise one would have == 0, contradicting to the fact that P2 === ker dX2 ~lS0 there is only one T ~
P-
Since Pt I1 P z =::;;; 0, there is no fixed. mode for (Pl~ P2), again by Theorem 2, (PI, P2) is a solution to the noninteracting control problem with
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FURTHER RESULTS ON NONINTERACTING CONTROL WITH INT...
stability if and only if the subsystem (19) is stabilizable.
For case B3), one can prove, along similar lines as in the proof of A2)~ that Pi takes the form Pi ;::= ker{ dx 2, dX3, dT} 1\.~hjch aT -i- O.
in
OX4
I
T js a smooth function of x such that
'I
Sinc.e .PI P2 == 0) there is no fixed nlode for (PI, P2), once aga.in by Theorem 2, (Pl~ P2) is a solution to the noninteracting control problem with stability if and only if the subsystem Xl,i = Xl,i+l, Xl!rl
==:
-i
==
l~
... ,rl - 1,
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of nonlinear systems has been given. Conditions have been found for the uniqueness of the set of compatible controllability distributions that can arise as solutions to the noninteracting control problem. Sufficient conditions have thus been pre~ scnted for the solvability of the problem of this class of systems. Finally, a complete solution has been provided to the problem for a restricted class of nonlinear systems. It c.an be noted that an the examples a.nd counterexamples in (Xia, 1994) fit into SOTIle c.ategories in Theorem 4.
Acknowledgment The author would like to thank Professor L K. Craig for his valuable comments during the preparation of this paper.
Ul
(20) 5:4
== ·U.3
is feedback stabilizable. The analysis of situation C) is symmetri.c to situation B) ~ a necessary and sufficient condition for the existence of a solution to the nonjnteracting c.ontrol problem with stability is that (remernbering that P== 0 and -#!== 0) VXl V X4 XZ,i
= X2,i+l, i
;;:::=
1, ... , r2
-
1
X2,r2 =:: 'U,2
(21 )
X3 = f(X2' X3) is feedback stabilizable, or :1;2,i
X2,r2
X3
== X2.i-t-l, i ==
1, ... , r2
-
1
== 'i.!2 =:
f(X2' X3)
X4 == V,3 is feedback stabilizabJe.
(22)
Tu sunlInary~ for the sysiem (11)~ a complete solution to the noninteracting control problem with stability is provided by the follovling theorem.
Theorem 4. lJnder aRsumption
(H1)~
(H2) and
(H3), the noninteracting control problem Vw~ith stability is solvable for the system (11) if and only if one of the follo\ving conditions holds: 1) The subsystem (13) is feedback stabilizable;
Ft
2) == 0 and the subsystem (16) is feedback stab] 11zab1e; :i) ~ == 0 and the subsystem (18) is feedback stablhzable; 4. CONCLUSIONS t'ollo1ving previous \vork (Xia, ] 993; Xia, 1994; Xia, 1996), the parameterization approach ha.s been taken further to deal with the non intera.cting control problem \vith internal stability of nonlinear systems. A general result on the noninteracting c.ontrol problem v.lith internal stability
5. REFERENCES Battilott.i, S. (1992). A sufficient condition for nonlinear noninteracting control with stability via dynamic state feedback: blockpart.itioned outputs. International Journal of Control 55, 1141 ~ 1160. Battilotti, S. (1994) . .lVoninteracting Control with Stability for Nonlinear Systems. SpringerVerlag. Berlin. Ba.ttilotti~ S. and W. P. Dayawansa (1991). Nonintera.cting contro.l '\vith stability for a class of nonIinear systems. f3ystems and Control Letters 19~ 327-338. Grizzle, J. ,rv. and A. Isidori (1989). Block noninteracting control with stability via static state feedback. Mathematics of Control: Signals and Systerns 2, 315-914. Isidori~ A. and J. ,\\T, Grizzle (1988). Fixed modes and nonlinear noninteracting control with stability. IEE'E Transactions on Automatic Control AC-33, 907--914. Sussmann, H. J. and \T. Jurdjevic (1972). Controllability of non linear systems. Journal of Differential Equations 12, 95-116. \ridyasagar, M. (1980). Decomposition techniques for large-scale systems with nonadditive interactions, sta.bility and stabilizability. IEEE Transactions on Automatic Control AC25, 773 - 779. Xia, X. (1993). Parameterization of decoupling control laws for affine nonlincar systems. IEEE 1mnsactions on Automatic Control AC-38, 916 - 928~ Xia, X. (1994). E·xamples and counterexamples in stable noninteracting control of nonlinear systems. Acta ..411.tomatica Sinica 20, 186 190. Xia, X. (1996). A parameterization approach to disturbance decoupling problem with stability of nonlinear systems. Automatica 32, 607 - 610. Xia~ X. and \\'. Gao (1993). ]\lonlinear Systems Control and Decoupling. .A.cademic Press. Beijing,{in Chinese).
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