State-Dependent Scaling Design for
Robust Backstepping via Output Feedback 12
Hiroshi Ito y3
y
Department of ControlEngineering and Science, Kyushu Instituteof Technology
680-4 kawazu,Iizuka,Fukuoka 820-8502,Japan
Phone: (+81)948-29-7717, Fax: (+81)948-29-7709
E-mail: [email protected]
Extended Abstract: Thispaperconsidersglobalrobuststabilizationofaclassofnonlinearsys-
temsviaoutputfeedback. Anewapproachtooutput-feedbackbacksteppingisprop osed. The
approach provides us with a systematic design pro cedure which can handle output-feedback
stabilization problems of strict-feedback nonlinear systems in a unied way. More imp or-
tantly, the approach by itself has a mechanism of achieving robust stabilization against a
generalclass of structured uncertaintiesin the pro cedure. Compared with the state-feedback
global stabilization, the the class of uncertainties which has b een treated by the literature
of global robust stabilizationproblems via output feedback is quite restricted in spite of the
practical importanceof consideringvarious lo cationsand structure of uncertainties. The ap-
proach presented in this pap er can b e considered as an successful extension of the author's
state-dependentdesignforstate-feedbackbacksteppingtothe outputfeedbackcase. Thereby,
this paper showsthe p ower of the general conceptof state-dependentscaling design for non-
linear systems control by lo oking at output-feedback stabilization problems, esp ecially in a
backsteppingmanner. The scaling approachallows usto treats b oth staticand dynamic un-
certainty in an unied way and , in addition, b e able to clarify the dierence b etween their
1
Technical Rep ort in Computer Science and Systems Engineering, Log Number CSSE-3, ISSN 1344-8803. 1999c
KyushuInstituteofTechnology
2
Thecurrentversionofthepap erwascompletedbyJanuary20,1999whentheauthorvisitedtoDept. ofAppl. Mech.
&Eng. Sci.,UniversityofCaliforniaat SanDiego,LaJolla,CA92093-0411,USA.
3
Authorforcorresp ondence
heritsadvantages of SDscaling design suchasautomatic computationof backstepping based
on optimization. Controllers in this pap er are dynamic feedback which consists of observer
and feedback gain(or controller). The essential dierence b etween nominal stabilization and
robust stabilization is described. It is shownthat observerdesign cannot b e separated glob-
ally from controller design. The observer should b e designed strong enough to comp ensate
\nonlinear size" of the uncertainty on the entire state-space. The coupling is natural and
inevitableinrobuststabilizationasitisforlinearsystems. Inaddition,fornonlinearsystems,
nonlinearity of the coupling is crucial for global stabilization which cannot b e comp ensated
globallybyeitherfeedback-gainorobserver-gainindep endently. Thisfactcontrastswithnom-
inal stabilization in which it is possible to stabilize the whole system globally by designing
controllerstrongenoughwhenevertheobserverdynamicsbyitselfdesigntob eonlystable(or,
viceversa). Strongobserversrequiredforrobuststabilizationmaynot existunlessthe output
havethe full information ofthe state. Ifthe nonlinear size of uncertainties are small enough,
the global robust stabilization can b e certainly achieved. This paper shows the condition of
allowablesize and nonlinearityof uncertainties for whichrobuststabilization can b edone via
backstepping. Theconditionisconsideredastheindex whichdescrib esthe largestallowable
size of uncertainty in robust stabilization via linear H 1
control. Indeed, for linear systems,
theconditionof hascouplingb etweenfeedbackgainand observerdesign(orRiccatiinequal-
ities). In addition tothe coupling,the conditionof the uncertainty sizein this pap erexhibits
arecursiveformb ecause of backstepping. Anotherfeature of the outputbacksteppingpro ce-
duresin thispaperis that itdo esnot requireYoung's inequality. Instead, the paperuses the
Schurcomplementsformula whichgivesanecessary and sucient condition for negativityof
a quadratic form. This paper also prop oses a novel recursive pro cedure of robust observer
design,which resembles backsteppingor forwardingfor controllerdesign.
KeyWords: robust backstepping; state-dep endent scaling; global robust stability; output
feedback;observerdesign; input-to-state stability; matrix inequality; convex optimization.
1 Intro duction
Forglobalstabilizationofuncertainnonlinearsystemsintheso-calledstrict-feedbackform,backstepping
requires domination of uncertain nonlinearities at each step of its recursive pro cedure(Krstic et al.,
1995; Freeman and Kokotovic, 1996). Such domination is achieved through the choice of appropriate
dominating functions which satisfy certain inequalities in the Lyapunov derivative corresp onding to
the lo cationsand characteristics of uncertain comp onentsin the system. Ito and Freeman(1998a) has
shown that state-dep endentscaling providesus with asystematic and unied metho dfor constructing
suitabledominatingfunctions in robustbackstepping designfor state-feedback.
The idea of state-dependent(SD) scaling design was prop osed in Ito (1998a), which was motivated
by the fact that scaling factors of small-gain conditions are allowed to be functions of state variables
conditions(Ito, 1996). The drawback of nonlinear H
1
controlas a nonlinear design tool can be over-
comein the sense that SD scaling has the ability to enlargestability regions insmall-gain type robust
stabilization(Ito,1998b). Themetho dologyofSDscalingdesignisapplicablenotonlytostrictfeedback
systems, but also toother general classes of nonlinear systems(Ito, 1998a; Ito, 1998b). The concept of
SD scaling design is general enough to formulate a wide variety of robust nonlinear control problems
and it is amenable to computational optimizationtechniques. When it comes to uncertain systems in
which encompasses a large class of uncertain nonlinear systems with structured, memoryless and dy-
namicuncertainties. Ithasb een shownthatrobustbacksteppingcanb edescribedasrecursiveselection
ofappropriatescalingfactors(Ito and Freeman,1998a; Itoand Freeman, 1998b). The backsteppingcan
be p erformedby computational optimizationas well.
Ifthestatevariablesarenotavailableforfeedback,onemaysimplygiveupseekingglobalstabilization
and settle for semi-global stabilization. On this standp oint, there are a lot of pap er dealing with
semi-global stabilizationby output feedback. The idea of input saturation and high-gain observer has
been successful for such semi-global stabilization (Esfandiari and Khalil, 1992; Khalil and Esfandiari,
1993; Lin and Sab eri, 1995). Teel and Praly (1995) and Teel and Praly (1994) proposed a useful
semi-global backstepping lemma and high-gain observers with saturating control for dynamic output
feedback. By using these semi-global techniques, a robust stabilization problem was also considered
intensively for a certain type and lo cation of unstructured uncertainty, namely, robustness against
unknown stable zero dynamics. It is possible to deal with unknown parameters in such a semi-global
stabilization as well, e.g.(Lin and Qian, 1998). However, from anther view p oint, given an uncertain
system, semi-global stabilizationusing high-gain and saturation maybemeaningful onlyif the system
cannotbe globally stabilized.
Therearealsoglobalresultsforoutput-feedbackstabilizationofnonlinearsystemsinaformofstrict-
feedback or chain of integrators. However, the typical results, e.g.(Krstic et al., 1995) are applicable
onlytononlinear systemswhose nonlinearitiesinthe systemequationdonot dep endonthe statesthat
are not measured. It is, however, not clear what is the essential ingredient of this assumption, apart
fromtheirtechniqueofconstructingobserversand controllers. Asidefrominverseoptimality,discussion
about robust global stabilization via this type of output feedback is absent in spite of their practical
importance.
The rst objective of this pap er is to prop ose a unied procedure to achieve robust and global
stability via output feedback for the class of uncertainty which is as large as the uncertainty tackled
in the state-feedback control literature, namely, uncertain systems in the strict-feedback form with
nonlinearly bounded uncertainties. In other words, this objective it to enlarging the class of nominal
modelsandesp eciallythestructureofuncertainty,thatcanb egloballystabilizedunderoutputfeedback,
Inorderto accomplishthis rst objective,this pap er successfully extendsthe author'sstate-dependent
design for state-feedback backstepping to the output feedback case. By doing that, the p ower of the
general concept of state-dependent scaling design for nonlinear systems control is shown as well. The
author'sp ositionisseekingglobalstabilizationinsteadofsettlingforsemi-globalstabilizationfromthe
beginning. Thereby,the pap er claries essentialpoints required to make global stabilizationrobust as
desired. Thus, the second objective is to characterize the essential dierence b etween nominal global
stabilization and robust global stabilization in output feedback control. The robustness in this paper
ismoredesirable inthat the size and lo cationofuncertaintyis prescrib ed a priori, whichis completely
dierent from the inverse optimal type of robustness. The backstepping is develop ed without the
assumptionthatrequiresthenominal systemtohavenonlinearitiesdep endingonlyonmeasuredstates,
i.e.,(y)whereyistheoutput. Thispaperclariesthatthefeedback-gainpartofoutputfeedbackdesign
byitselfdoesnotneed toexcludenonlinearitiessuchas(y)x,wherexisunnecessarilymeasured. This
paper describ es what kindof task isessentially requiredfor observer design in sucha case. The pap er
also shows a condition on which robust stabilization can b e achievedglobally for aprescrib ed class of
uncertainties. Itwill b eshownthat, exclusivelyfornonlinearsystems,\nonlinear size"ofuncertainties,
appeared ascoupling, is crucial for global robust stabilization, which cannot comp ensated globally by
eitherfeedback-gain orobserver-gain independently.
The idea of SD scaling approachto backsteppingin this pap er is asfollows:
introduce co ordinate transformation tothe entire closed-lo op system in order tocreate a freedom
incho osingLyapunovfunctions
use the Schurcomplementsformulato extract arecursive structure of design
solvethe design problem byselecting SD scalingand co ordinate transformationrecursively.
Furthermore,itis completed by
show that the design problem isrecursively linear inthe parameters of SD scalingand coordinate
transformation
show the existence of solutions
provide computational formulas and analyticalsolutions
One feature of the backstepping prop osed in this pap er is that the procedures are amenable to auto-
matednumerical computation based on convex optimization. Since the backstepping is p erformed by
domination, it is unnecessary to use precise parameters of systems in the control law, which prevents
thecontrollerfromhavinglong andcomplicatedterms. Anotherimp ortantfeatureofthis pap er isthat
the output backstepping is shown to b e feasible without using Young's inequality. Instead, the paper
uses the Schur complementsformula which gives a necessary and sucient condition for negativity of
aquadraticform.
The author needs to explain the standp oint of this pap er since it is quite dierent from those of
nonlinear adaptivecontrol and many of backstepping papers. The author's p oint of view is similar to
thatof linear robust stabilizationvia H 1
control. The roles of H 1
types robust controlare
provide a metho dof solving(more precisely, trying tosolve) the problem
characterize acondition under whichthe robust stabilization is solvable
provide information ab out how large size of uncertainty is allowable.
The latter two roles are necessary since the problem by itself does not always have the solution. The
reasonwhythis situation o ccurs isthatwe specify the nominal systemand structure andsize ofuncer-
tainties a priori. A robust stabilization problem is solvable obviously if the uncertainty is suciently
small. This typ e of robust controlis attractive in the sense that itprovides us with a way to obtain a
control law evenif it is not as go od as we originally desired. It theoretically p ersuades us to give up
seeking unreasonably large uncertainty, inparticular, since the constantly scaled H 1
control is neces-
saryand sucientforachievingrobust stabilizationagainst time-varyinguncertainty. Thispap er looks
atglobal robuststabilizationof nonlinear systemsfromthe same p ointof view. In addition,this paper
demonstratesa classof uncertainsystems whichisalwaysrobustlystabilizable forarbitrarilylargesize
andarbitrarilyfastgrowthorderuncertainties. The latterstandp oint ismorecommon inthe nonlinear
literature. In this way,this pap er takespractically goo d positions fromb oth the sides.
6
1
6
0
w z
u y
Figure1: Uncertainnonlinear plant 6
P
2 Uncertain nonlinear systems
Considerthe uncertain nonlinear system6
P
shown inFig.1. Here, 6
0
denotesanominal plantand 6
1
represents the uncertainty and modeling error of the plant. We assume that the nominal part 6
0 is
described by
6
0 :
8
<
: _
x=A(y)x+B(y)w+G(y)u
z =C(y )x
y=C
y (y)x
;
x(t)2R n
;u(t)2R 1
w (t)2R p
;z(t)2R p
y (t)2R r
(1)
The matrix-valued functionsA, B, C, Gand C
y
are assumed tobeC 0
functions. The vectors wand z
are denedas
w= 2
6
6
6
4 w
1
w
2
.
.
.
w
n 3
7
7
7
5
; z = 2
6
6
6
4 z
1
z
2
.
.
.
z
n 3
7
7
7
5
;
w
i
(t)2R p
i
z
i
(t)2R p
i
p
i
0; p= P
n
i=1 p
i
i=1;2;111;n
(2)
Suppose thatthe uncertain system 6
1
has the following structure of nonlinear mappings 1:z 7!w.
1=block-diag[1
1
;1
2
;111;1
n
]; (3)
where some of the mappings 1
i :z
i 7! w
i
, i= 1;2;:::;n can b e zero in vector size. Each uncertainty
1
i
is allowedto havethree typ es of components:
1
i :z
i
= 2
4 z
id
z
is
z
ir 3
5
7!w
i
= 2
4 w
id
w
is
w
ir 3
5
; w
i
= 2
4 1
id
0 0
0 1
is 0
0 0 1
ir 3
5
z
i
: (4)
Here,1
id
representsadynamicsystem. 1
is
and1
ir
denotefullstaticandrepeatedstaticscalarsystems,
respectively. It is unnecessary for 1
i
tohave alltypes of uncertainty. The dynamicuncertainty 1
id is
denedby
1
id :
_ x
1
i
=f
1
id (x
1
i
;z
id
;t)
w
id
=h
1
id (x
1
i
;z
id
;t)
; (5)
wheref
1
id
(0;0;t)=0andh
1
id
(0;0;t) =0aresatisedforallt0and f
1
id and h
1
id
are vector-valued
C 0
functions. The full staticpart 1
is
isdescribedby
1
is :w
is
=h
1
is (z
is
;t); (6)
whereh
1
is
is avector-valued C 0
function and h
1
is
(0;t)=0 forall t0. The rep eated staticpart 1
ir
isdenedwith r
i
>1 copies of astatic scalar system
ir :
1
ir
= ri
diag
j=1
ir
=
ir I
r
i
;
ir : w
ir
=h
ir (t)z
ir
(7)
whereh
ir
isa scalar-valuedC 0
function. Weconsider the followingclass of uncertainty6
1 .
1
(i) 1
id
has L
2
0gain less than or equal to 1 with a radially unbounded C 1
storage function V
1i (x
1
i )
satisfyingV
1i
(0) =0. (ii) 1
is
satiseskz
1
is k
2
kw
1
is k
2
for all t2[0;1). (iii)
ir
satises kz
ir k
2
kw
ir k
2
for all t2[0;1).
The uncertain system 6
P
has an equilibrium p oint at x
cl
= 0 when u 0. Roughly sp eaking, the
gain of eachuncertainty is assumed to b e lessthan or equal to unity. Uncertainty havingsuper-linear
growth (and thusunbounded gain) can still beincluded bya judicious choiceof the nonlinear weights
B (x)and C(x). Indeed,6
0
not onlydescrib es anominal plant,but also can includeinformation ab out
input-outputnonlinearitiesofuncertainty. The manipulationtochooseanappropriatepair of (6
0
;6
1 )
takingnonlinearityintoaccountisessentiallysimilartotheideaofintro ducingfunctionsofsignalnorms
(Sontag and Wang, 1996; Sontag,1998; Mareels and Hill, 1992). Note that 6
0
also describes how the
uncertainty aects the nominal plant such as geometrical lo cations, structures of uncertainties where
uncertainparametersarepresent. RememberthatB(x)andC(x)sp ecifythe\nonlinearsize"(including
size,nonlinearity,lo cation and structure) of uncertainties.
3 SD scaling analysis for observer-feedback control
With denitions of the uncertainty 6
1
in mind, several sets of real-valued scaling matrices will b e
dened. For notational simplicity, we assume that 1
id
and 1
is
are square insize of input and output
vectors for alli=1;2;:::;n. Forthe dynamic uncertainty 1
id
, wedene
L
id :=fL
id
=
id I
id :
id
>0g: (8)
Here, I
id
denotesanidentity matrix whichis compatible in size with the vector z
id
. For the full static
uncertainty1
is
, aset of scalingis denedby
L
is :=fL
is
=
is (y;x)I^
is :
is
(y;^x)>08(y;^x)2R r
2R n
g: (9)
Inthe case of the rep eatedstatic uncertainty1
ir
, we dene twosets of scaling matrices.
L
ir
:=fL
ir :L
T
ir
(y;x)^ =L
ir
(y;^x); L
ir
(y ;x)^ >0 8(y;x)^ 2R r
2R n
g: (10)
R
ir
:=fR
ir :R
T
ir
(y;x)^ =0R
ir
(y;^x)8(y;x)^ 2R r
2R n
g: (11)
Here,b othL
ir
and R
ir
are squarematriceswhosesize isthesameasthedimensionof z
is
. Thesescaling
matrices are used to estimate the worst case value of the time-derivative of Lyapunov functions(Ito,
1998b;Ito and Freeman, 1998a). LetL
i
(y;x)^ and R
i
(y;x)^ b e dened by
L
i :=
8
<
: L
i
(y;^x)=
2
4 L
id
0 0
0 L
is
(y ;x)^ 0
0 0 L
ir (y;x)^
3
5
: L
id 2L
id
L
is 2L
is
L
ir 2L
ir 9
=
;
(12)
R
i :=
8
<
: R
i
(y;x)^ = 2
4
0 0 0
0 0 0
0 0 R
ir (y;x)^
3
5
:R
ir 2R
ir 9
=
;
; (13)
for i = 1;2;:::;n. Note that a constant > 0 satises I 2 L
i
and 0 2 R
i
. Now dene two sets of
scalingmatrices for the whole 6
1
asfollows:
L:=
L= n
blo ck-diag
i=1 L
i
(y ;x);^ L
i 2L
i
(14)
R:=
R= n
block-diag
i=1 R
i
(y;x);^ R
i 2R
i
: (15)
case where constant scalings are used for time-varying uncertainty. Scaling matrices for static uncer-
tainty are chosenas functions of outputand state estimate, while static uncertainty arising ina linear
system isusually not distinguishedfrom dynamicuncertainty(Ito,1996).
We next consider robust stabilization of the uncertain nonlinear system 6
P
by dynamic output
feedback. Weemploy the full order observer
(
_
^
x=A(y )^x+Y(y;x)(y^ 0y )^ +G(y)u
^ y=C
y (y)^x
(16)
Byusing the signal ^xestimated bythe observer,we choosea dynamic outputfeedback law as
u=K(y;^x)^x: (17)
Then,the closed-loop system is writtenas
d
dt
x
^ x
=
A GK
YC
y
A0YC
y
+GK
x
^ x
+
B
0
w (18)
Wenowcharacterizerobuststabilizationof6
P
usingSDscaling,quadraticLyapunovfunctionsanda
dieomorphicco ordinate change. Consider the dieomorphismbetweenx^2R n
and ^2R n
asfollows:
^
=S(y;x)^^ x: (19)
The time-derivativeof ^ is obtained as
_
^ =
"
@S
@y
1
^ x;
@S
@y
2
^ x;111;
@S
@y
n
^ x
#
C
y _ x+
"
@S
@x^
1
^ x;
@S
@x^
2
^ x;111;
@S
@x^
n
^ x
#
_
^
x+S(y ;x)^ _
^
x=V(y;x)^ x_ +T(y;x)^ _
^ x:(20)
Denex~=^x0x. Then,
d
dt
^
~ x
=
V(y;^x) T(y;x)^
0I I
d
dt
x
^ x
;
x
^ x
=
"
S(y ;x)^ 01
0I
S(y ;x)^ 01
0
#
^
~ x
(21)
The closed-loopsystem b ecomes
d
dt
^
~ x
=
(V +T)(A+GK)S 01
0(VA+TYC
y )
0 A0YC
y
^
~ x
+
VB
0B
w (22)
We also introduce anotherco ordinate transformationto x:~
=W~x (23)
whereW is aconstantnon-singular matrix. The closed-lo op system on the coordinate (;^ ) is
d
dt
^
=
"
(V +T)(A+GK)S 01
0(VA+TYC
y )W
01
0 W(A0YC
y )W
01
#
^
+
VB
0WB
w (24)
z =C h
S 01
0W 01
i
^
(25)
The followingdescrib es the main idea of the SD scalingapproach to the outputfeedback problem.
Theorem 1 (i) Suppose that there existconstant symmetric matrices P and P such that
N(y;^x)=
"
S 0T
(A+GK) T
(V +T) T
P+P(V +T)(A+GK)S 01
0P(VA+TYC
y )W
01
0W 0T
(VA+TYC
y )
T
P W
0T
(A0YC
y )
T
W T
~
P +
~
PW(A0YC
y )W
01
#
<0 (26)
P >0;
~
P >0 (27)
are satised for all (y;x)^ in R r
2 R n
, then the nominal nonlinear system 6
0
is globally uniformly
asymptotically stabilized by the dynamic output feedback (16-17). Furthermore, a Lyapunov function is
given by V(x;x)^ = ^ T
P^+ T
~
P.
(ii)Supposethat thereexistconstant symmetricmatricesP,
~
P and scalingfunctions L2L andR 2R
suchthat
M(y ;x)^ =
2
6
6
6
6
6
6
6
6
4 (
S 0T
(A+GK) T
(V +T) T
P+
P(V +T)(A+GK)S 01
)
PVB+S 0T
C T
R T
S 0T
C T
L 0P(VA+TYC
y )W
01
B T
V T
P+RCS 01
0L 0 0B
T
W T
~
P0R CW 01
LCS 01
0 0L 0LCW
01
0W 0T
(VA+TYC
y )
T
P 0
~
PWB0W 0T
C T
R T
0W 0T
C T
L (
W 0T
(A0YC
y )
T
W T
~
P+
~
PW(A0YC
y )W
01 )
3
7
7
7
7
7
7
7
7
5
<0(28)
P >0;
~
P >0 (29)
are satised for all (y;^x) in R r
2R n
, then the uncertain nonlinear system 6
P
is global ly uniformly
asymptotically stabilized by the dynamic output feedback (16-17) for any admissible uncertainty 6
1 .
Furthermore, a Lyapunov function is given by V(x;x)^ =^ T
P^+ T
~
P+ P
n
i=1
id V
1i (x
1
i ).
The analysis problem of robust stability is reduced into the existence of scaling matrices which make
M negative. This is actually considered as the denition of the state-dependent scaling approach to
outputfeedback control with full-orderobservers.
Although the representation (22) may seem to allow us to use a sort of separation between state-
feedback stabilization and observer design somehow at a glance, it is certainly not true for nonlinear
systemsstabilization. To explain this point,we need the followinglemma.
Lemma 1 Consider a symmetric matrix
M =
M
11 M
12
M T
12 M
22
(30)
(i) Schur complements formula : M <0 isequivalent to
M
22
<0; M
11 0M
12 M
01
22 M
T
12
<0 (31)
(ii)Young's inequality : M <0is satised if
M
22 +0
01
<0; M
11 +M
12 0M
T
12
<0 (32)
0= 2
6
6
6
6
4
1
0 111 0
0
2 .
.
. .
.
.
.
.
. .
.
. .
.
.
0
0 111 0
n 3
7
7
7
7
5
>0 (33)
0>M
11 +M
12 0M
T
12
>M
11 +M
12 0M
T
12 0M
12
(0+M 01
22 )M
T
12
=M
11 0M
12 M
01
22 M
T
12
(34)
Instead, the purpose is to show that the pair of inequalities in (32) is an alternative expression of
Young'sinequality:
2y T
z y T
0y+z T
0 01
z (35)
wherey and z are vectors. It is easilyveriedthat
x
z
T
M
11 M
12
M T
12 M
22
x
z
= x T
M
11
x+2x T
M
12 z+z
T
M
22
z (36)
x T
M
11 x+x
T
M
12 0M
T
12 x+z
T
0 01
z+z T
M
22
z (37)
=
x
z
T
"
M
11 +M
12 0M
T
12
0
0 M
22 +0
01
#
x
z
(38)
Theinequalities(32)ofmatricesarenottheformulawhichisusuallycalledYoung'sinequality. However,
thispap er referstothat asYoung'sinequalityinorder todistinguishthat fromthe Schurcomplements
formula. It is also true that the inequalities (32) has app eared as an ordinary Young's inequality of
vectors or scalars in nonlinear systems control. A common role of Young's inequality is to get rid of
productsoftwovectorsintheLyapunovderivativeandtogetadecoupledquadraticexpression. Thisis
explainedintheproof oftheabovetheorem. TheSchurcomplementsformulalooksatthe negativityin
termsofmatrices insteadofthe scalarvalueofquadraticforms. The Schurcomplementsformulagives
a necessary and sucient condition while Young's inequality is only sucient. The idea of Young's
inequality is toreplace the full information of the matrix 0M 01
22
with simple scalar parameters
i ata
price of lo osing necessity. Actually, an alternative statement of the Schur complements formula is as
follows: M <0 holds if and only if there existsa diagonal matrix 0>0such that
M
22 +0
01
<0; M
11 0M
12 M
01
22 M
T
12
<0 (39)
are satised. Compared with the Schur complements, Young's inequality is conservative. The Schur
complementsissuperiortoYoung'sinequalityinthissensealthoughYoung'sinequalityisacommonto ol
innonlinear systems design(Krstic et al., 1995; Freeman and Kokotovic, 1996; Sepulchre et al., 1997).
From this standpoint, this pap er replaces the task of Young'sinequality with the Schurcomplements.
Theoutputfeedbackdesignwill beshowntohaverecursivestructures forbacksteppingevenifYoung's
inequalityisnotused. Inotherwords,thispap erproposesbacksteppingpro cedureswithoutintroducing
any conservatism in solving problems recursivelyexcept that Theorem 1 is a sucient condition (note
that a recursive structure of solution by itself may have unnecessary conservatism). This may not
only allows the design to tolerate large size of uncertainties, but also prevent controllers from having
unnecessary high gain and harmfully rst or slowgrowth order.
Corollary 1 Assume that there exists a constant matrix
~
P >0 suchthat
H(y;^x):=W 0T
(A0YC
y )
T
W T
~
P +
~
PW(A0YC
y )W
01
<0 (40)
holdsfor all (y ;^x)2R r
2R n
(i)Suppose that there exists a constant matrix P >0 such that the inequality
N(y;x)^ :=N
11
(y ;x)^ 0N
12
(y;^x)H 01
(y ;x)N^ T
12
(y;^x)<0 (41)
N
11
(y ;x)^ :=S 0T
(A+GK) T
(V +T) T
P+P(T +V)(A+GK)S 01
; N
12
(y;^x):=P(VA+TYC
y )W
01
(42)
issatisedfor all (y;^x)in R 2R , then thenominal nonlinear system 6
0
isglobal lyuniformly asymp-
totically stabilized by the dynamic output feedback (16-17). Moreover, if 6
0
is a linear system and if
S is constant, the set of conditions (41) and (40) is equivalent to the existence of P > 0 and
~
P > 0
satisfying
N
11 :=S
0T
(A+GK) T
(V +T) T
P+P(V +T)(A+GK)S 01
<0 (43)
H :=W 0T
(A0YC
y )
T
W T
~
P +
~
PW(A0YC
y )W
01
<0 (44)
(ii)Suppose thatthereexist a constant matrix P >0and scalingfunctions L2L andR 2R suchthat
theinequality
M(y;x)^ :=M
11
(y;x)^ 0M
12
(y;x)H^ 01
(y;^x)M T
12
(y;x)^ <0 (45)
M
11
(y ;x)^ :=
2
6
6
6
4 (
S 0T
(A+GK) T
(V +T) T
P+
P(T +V)(A+GK)S 01
)
PVB+S 0T
C T
R T
S 0T
C T
L
B T
V T
P+R CS 01
0L 0
LCS 01
0 0L
3
7
7
7
5
(46)
M
12
(y ;x)^ :=
2
6
4
P(VA+TYC
y )W
01
B T
W T
~
P+R CW 01
LCW 01
3
7
5
(47)
is satised for all (y ;x)^ in R r
2R n
, then the uncertain nonlinear system 6
P
is global ly uniformly
asymptotically stabilized by the dynamic output feedback (16-17) for any admissible uncertainty6
1 .
Proof : (i)The conditions(41) and (40)are straightforwardfrom(26) byusing theSchurcomplements
formula. Obviously,(41) and (40) imply (43) and (44). Now,supp ose that P >0 is asolution to (43)
witha constant S for a linear system 6
0 . Let
~
P b e a solution to(44). If
~
P in (41) is replaced by
~
P,
the inequality (41) is satisedfor a sucientlarge constant >0.
(ii) Itis straightforward fromthe Schurcomplementsformula.
Thetwoinequalities (43) and (44) inthis corollary merely represent the separation principle for linear
systems. The conditions (43) and (44), however,do not guarantee global stability for nonlinear 6
0 . If
6
0
isnonlinear, inthe ab ove proof may be required to b e unbounded as y or x^ goes to 61. If is
afunction of (y ;x),^ there isno guarantee that there exists a Lyapunov function V which is consistent
with
@V
@[^ T
; T
] T
=2 h
^ T
P ; T
(y;^x)
~
P i
for (43) and (44). It is, however, true that (41) can b e satised semi-globally by a sucient large
constant . We may achieve semi-global stabilization by using the separation (43-44) and taking into
accountthelevelsetoftheLyapunovfunctionV(x;^x)= T
P + T
~
Pdeformedby. Thispap erdo es
not pursue this obvious direction of semi-global stabilization since it doesnot capture essentialp oints
required for global and nonlinear stabilization. This pap er, instead, is fo cused on global stabilization
andcharacterizesrequirements for globalstabilization. As for robuststabilization, we cannotseparate
observerdesigncompletelyfromrobuststabilizationinaglobalsense. Theseparationargumentin(i)of
Corollary1isnotapplicableto(ii) eitherevenforlinear6
0
b ecauseofthecouplingtermM
12
(esp ecially
thetermB T
W T
~
P)b etweenfeedbackandobserverinM <0. Infact,linearrobustcontroltheorytellsus
thatobserverdesignmustb ecoupledwithrobusticationofstabilizationagainstuncertainties. Inother
words, the observer should b e designed strong enough by taking intoaccountthe eect of uncertainty
androbustness objectives.
This section denes the class of uncertain nonlinear systems to which output backstepping design via
SDscaling will apply. The output equationof the system issupp osed to given by
y =x
1
(48)
orequivalently
C
y
=[1 0 111 0] (49)
This case is sometimes called output feedback in the nonlinear control literature(Krstic et al., 1995).
This pap er deals with the uncertain nonlinear system 6
P
under the following structural assumptions.
First,we assume that A and Gcan b e writtenin the form
A(x
1 )=
2
6
6
6
6
6
6
4 a
11 a
12
0 111 111 0
a
21 a
22 a
23
0 0
.
.
. .
.
. .
.
. .
.
. .
.
. .
.
.
.
.
. .
.
. .
.
. .
.
.
0
a
n01;1 a
n01;2
111 111 a
n01;n
a
n;1 a
n;2
111 111 a
n;n 3
7
7
7
7
7
7
5
;G(x
1 )=
2
6
6
6
4 0
.
.
.
0
a
n;n+1 3
7
7
7
5
: (50)
withC 0
scalar functions a
ij
of the measured state x
1
. The function a
ij (x
1
)is required tosatisfy
a
i;i+1 (x
1
)6=0; 1in; 8x
1
2R (51)
Asfor functions B and C, weassume
B(x
1 )=
2
6
6
6
6
4 B
11
0 111 0
B
21 B
22 .
.
. .
.
.
.
.
. .
.
. .
.
.
0
B
n;1
111 B
n;n01 B
n;n 3
7
7
7
7
5
; C(x
1 )=
2
6
6
6
6
4 C
11
0 111 0
C
21 C
22 .
.
. .
.
.
.
.
. .
.
. .
.
.
0
C
n;1
111 C
n;n01 C
n;n 3
7
7
7
7
5
(52)
whereB
ij (x
1 )2R
12pi
and C
ij (x
1 )2R
pi21
. Then, the uncertainty aects the system as
B(x)w = 2
6
6
4
B
11 1
1 C
11
0 0 111
B
21 1
1 C
11 +B
22 1
2 C
21 B
22 1
2 C
22 0
.
.
.
.
.
.
.
.
. .
.
. 3
7
7
5 2
6
6
4 x
1
x
2
x
3
.
.
. 3
7
7
5
(53)
This expression is only an aid for illustrating the structure of the uncertainty and is mathematically
ambiguous. The op erator 1
i
in the ab ove equationdo es not represent matrix multiplication but non-
linear mappings which can have dynamics with initial conditions. For simplicity, this paper assumes
that the system do es not haveany uncertaintiesin the virtual controlcoecientswhich app ear in the
backsteppingprocedure. Itiscertainly p ossibletoextend theidea of SDscalingeasilytothe uncertain
system which has 1 blocks in a more general manner as in Ito and Freeman (1998a). Because each
entryB
ii 1
j C
ji
aboveis scalar,a rep eatedstatic uncertaintycan always be representedby ascalar full
staticuncertainty. However,weincludethe rep eatedrepresentationherebecauseitallowsmoredegrees
offreedom in the scaling designand italso prepares the wayfor amultivariableversion of our results.
Twotypes of prop ertiesof observers will b e used inthis pap er.
Ordinary observer: The observer-gainY(x
1
) ischosen as aC 0
functionmatrix suchthatthere exist
a constant matrix
~
P and a C 0
function matrix Q
y (x
1
) satisfying
H(x
1
):=(A0YC
y )
T
~
P +
~
P(A0YC
y
)<0Q
y
(54)
~
P >0; Q
y
>0 (55)
1
Robust observer: Given a matrix-valued function 0(x
1
) >0. The C 0
observer-gain function Y(x
1 )
andthe constant matrix W are chosen such that there exists a constant diagonal matrix
~
P satisfying
H(x
1
):=W 0T
(A0YC
y )
T
W T
~
P +
~
PW(A0YC
y )W
01
<00 01
(56)
~
P >0 (57)
hold for all x
1
2R. Note that H <00 01
<0 isequivalentto 0<0H 01
<0.
The requirement of robust observer is stronger than that of ordinary observer. A robust observer
is an ordinary observer. The converse is not true. Suppose that
~
P > 0 is a solution to (54). We can
decompose the matrix into
~
P = W T
3W with a lower triangular W and a diagonal matrix 3. This
means that (56) is satised by replacing 0 01
with W 0T
Q
y W
01
. However, 0 01
W
0T
Q
y W
01
is not
guaranteedatall. Thersttwotermsonthelefthandsideof(56)corresp ondtotheLyapunovderivative
of the observer error system. The robust observer requires that the observer error system is stable to
a degree prescrib ed by 0. That is why the robust observer can b e used for making a control system
robustagainstuncertainties. The function0isconsidered asanindexof robustness. The smaller0>0
is, the more robust the resulting observer is. This will be explained later on. Note that for a certain
class of systems, it is always p ossible to construct observer gains required for ordinary observers and
robustobservers. The observerdesign will b e explained inSection 8.
5 Backstepping design for output feedback
WenowdirectourattentiontoadieomorphismS(y;x)^ inasp ecialform. Thedieomorphismwilllead
us to a recursive structure with which output feedback backstepping is prop osed. We thereby extend
the robust backstepping procedure presented in Ito and Freeman(1998a) for output feedback design.
Thebackstepping iscarried out successfully by selecting SD scalingmatrices recursively.
Let ^x
[k]
denotethe state ofthe observerx^
1
through ^x
k :
^ x
[k ]
=[^x
1
;x^
2
;111;^x
k ]
T
: (58)
Consider smooth scalar-valued functions s
1 (x
1 ), s
2 (x
1
;x^
1
), 111, s
n01 (x
1
;^x
[n02]
) which are to b e deter-
minedina recursivemanner froms
1
through s
n01
. We denea dieomorphism S(x
1
;x)^ asfollows:
S 01
(x
1
;x^
[n02]
)= 2
6
6
6
6
4
1 0 0 111 0
s
1
1 0 111 0
0 s
2 1
.
.
.
0
.
.
. .
.
. .
.
. .
.
. .
.
.
0 111 0 s
n01 1
3
7
7
7
7
5
(59)
S(x
1
;x^
[n02]
)= 2
6
6
6
6
4
1 0 0 111 0
0s
1
1 0 111 0
s
1 s
2
0s
2 1
.
.
.
0
.
.
.
.
.
. .
.
. .
.
. .
.
.
(01) n01
s
1 111s
n01
111 s
n02 s
n01 0s
n01 1
3
7
7
7
7
5
: (60)
Thesmo oth function V(x
1
;x^
[n01]
)and T(x
1
;x^
[n01]
)in (20) are obtained as
V(x
1
;x^
[n01]
)= 2
6
6
6
6
4
0 0 0 111 0
?
1;1;1
0 0 111 0
?
1;2;2 0 0
.
.
.
0
.
.
. .
.
. .
.
. .
.
. .
.
.
?
1;n01;n01
0 0 111 0 3
7
7
7
7
5
(61)
T(x
1
;x^
[n01]
)= 6
6
6
6
6
6
4
1 0 0 0 111 0
?
1;0;1
1 0 0 111 0
?
1;2;2
?
1;1;2
1 0
.
.
.
0
?
1;3;3
?
1;3;3
?
1;2;3 1
.
.
.
0
.
.
.
.
.
. .
.
.
.
.
.
.
.
. .
.
.
?
1;n01;n01
111 111 ?
1;n01;n01
?
1;n02;n01 1
7
7
7
7
7
7
5
; (62)
where?
1;i;j
denotes anyfunction dep ending onlyon(x
1
;^x
[i]
) and the functions s
1
through s
j
and their
partialderivatives. We choosea feedbackgain (17) inthe form of
K = h
(01) n01
s
1 1 11s
n
111 0s
n01 s
n s
n i
(63)
where s
n (x
1
;x^
[n01]
) is another smo oth function yet to b e determined. Then, the matrices in (26) and
(28)for the closed-lo op system b ecome
N:=
"
^
S T
^
A T
(V +T) T
P+P(V +T)
^
A
^
S 0P(VA+TYC
y )W
01
0W 0T
(VA+TYC
y )
T
P W
0T
(A0YC
y )
T
W T
~
P +
~
PW(A0YC
y )W
01
#
(64)
M:=
2
6
6
6
6
6
6
4
^
S T
^
A T
(V +T) T
P+P(V +T)
^
A
^
S PVB+S 0T
C T
R T
S 0T
C T
L 0P(VA+TYC
y )W
01
B T
V T
P+RCS 01
0L 0 0B
T
W T
~
P0R CW 01
LCS 01
0 0L 0LCW
01
0W 0T
(VA+TYC
y )
T
P 0
~
PWB0W 0T
C T
R T
0W 0T
C T
L (
W 0T
(A0YC
y )
T
W T
~
P+
~
PW(A0YC
y )W
01 )
3
7
7
7
7
7
7
5
(65)
^
A:=[A G];
^
S :=
S 01
01110 s
n
We nowrestrict P to diagonal:
P = n
diag
i=1 P
i
; P
i
>0; P
[k ]
= k
diag
i=1 P
i
(66)
We also considerthe co ordinate transformation W of ~xin alowertriangular form:
W = 2
6
6
6
6
4 W
11
0 0 111 0
W
21 W
22
0 111 0
W
31 W
32 W
33 .
.
.
0
.
.
. .
.
. .
.
. .
.
. .
.
.
W
n1
111 W
n;n02 W
n;n01 W
n;n 3
7
7
7
7
5
(67)
Denesystem matrices forthe rst k state and input comp onents by
^
A
[k ] (x
1 )=
2
6
6
6
6
6
4 a
11 a
12
0 111 111 0
a
21 a
22 a
23
0 111 0
.
.
. .
.
. .
.
. .
.
. .
.
. .
.
.
a
k 01;1 a
k 01;2
111 111 a
k 01;k 0
a
k 1 a
k 2
111 111 a
kk a
k ;k +1 3
7
7
7
7
7
5
(68)
B
[k]
(x
1 )=
2
6
6
4 B
11
0 111 0
B
21 B
22 .
.
. .
.
.
.
.
. .
.
. .
.
.
0
B
k 1
111 B
k ;k 01 B
k k 3
7
7
5
; C
[k ] (x
1 )=
2
6
6
4 C
11
0 111 0
C
21 C
22 .
.
. .
.
.
.
.
. .
.
. .
.
.
0
C
k1
111 C
k ;k 01 C
k k 3
7
7
5
(69)
In a similar manner, S
[k ] (x
1
;x^
[k 02]
), S
[k ] (x
1
;^x
[k 02]
), V
[k ] (x
1
;^x
[k 01]
), T
[k ] (x
1
;^x
[k 01]
), W
[k ]
and W
[k ] are
denedask2k upp er left partsof S, S 01
, V,T, W and W 01
, resp ectively. Let
^
S
[k]
(x
1
;^x
[k 01]
)=
"
S 01
[k ]
01110 s
k
#
;
~
P
[k ]
=
"
~
P
[k ]
?
0;0;0
?
0;0;0
~
P
k k
#
(70)
A
[k ]
= 2
6
6
6
6
6
4 A
[k 01]
0
.
.
.
0
a
k 01;k
a
k ;3 a
k k 3
7
7
7
7
7
5
; Y
[k]
=
Y
[k 01]
Y
k
(71)
C
y[k ]
=[
1 0 111 0 0
]=[ C
y [k 01]
0
]; C
y [1]
=1 (72)
Scalingmatrices are also denedrecursivelyas
L
[k]
:=
(
L
[k ]
= k
blo ck-diag
i=1 L
i (x
1
;^x
[i02]
); L
i 2L
i )
(73)
R
[k ] :=
(
R
[k ]
= k
blo ck-diag
i=1 R
i (x
1
;^x
[i02]
); R
i 2R
i )
: (74)
Now,wedeneN
[k ] (x
1
;x^
[k 01]
) byaddingsubscript [k ]toeverymatrix inthe righthandside of (64).
Byusing
[H]
[k ] (x
1 )=
[H]
[k 01]
?
1;0;0
?
1;0;0 [H]
k k
; [H]
[n]
=H (75)
the matrixN
[k ]
can b e representedby
N
[k]
(x
1
;x^
[k 01]
)=
"
N
[k ]11 (x
1
;x^
[k 01]
) N
[k ]12 (x
1
;x^
[k 01]
)
N T
[k ]12 (x
1
;x^
[k 01]
) H
[k ] (x
1 )
#
; N
[n]
=N (76)
N
[k]11 :=
^
S T
[k ]
^
A T
[k ] (V
[k ] +T
[k]
) T
P
[k ] +P
[k ] (V
[k ] +T
[k ] )
^
A
[k ]
^
S
[k]
; N
[k ]12 :=P
[k]
(V
[k ] A
[k ] +T
[k ] Y
[k ] C
y [k ] )W
01
[k]
(77)
We also dene
~
M
[k ] as
~
M
[k]
(x
1
;x^
[k 01]
)=
"
M
[k ]11 (x
1
;x^
[k 01]
) Q T
k M
12 (x
1
;x^
[n01]
)
M T
12 (x
1
;^x
[n01]
)Q
k
H(x
1 )
#
;
~
M
[n]
=M (78)
M
[k]11 :=
2
6
6
4
^
S T
[k ]
^
A T
[k ] (V
[k ] +T
[k ] )
T
P
[k ] +P
[k ] (V
[k ] +T
[k ] )
^
A
[k ]
^
S
[k ] P
[k ] V
[k ] B
[k ] +S
0T
[k ] C
T
[k ] R
T
[k ] S
0T
[k ] C
T
[k ] L
[k]
B T
[k ] V
T
[k ] P
[k ] +R
[k]
C
[k ] S
01
[k ]
0L
[k ]
0
L
[k ] C
[k ] S
01
[k ]
0 0L
[k ] 3
7
7
5 (79)
M
12 :=
2
6
4
P(VA+TYC
y )W
01
B T
W T
~
P+RCW 01
LCW 01
3
7
5;
Q
k
= 2
6
6
6
6
4 I
k 0 0
0 0 0
0 I
q
0
0 0 0
0 0 I
q
3
7
7
7
7
5
;
Q
n
=I
n+2P
(80)
whereI
k
denotesak2k identity matrixand q:=
P
k
i=1 p
i
. NotethatM
[k ]11
=
Q T
k M
11
Q
k
holds. Wecan
verify the following.
Theorem 2 Suppose 1kn.
(i-a) N
[k]
does not include fs
k +1
;s
k +2
;111;s
n g.
[k ] k
(i-c) Every entry of N
[k ]
is simultaneously ane in all the entries of P
[k ] .
(i-d) N
[k ]
<0 implies N
[k 01]
<0 unless k=1.
(ii-a)
~
M
[k ]
does notinclude either fs
k +1
;s
k +2
;111;s
n g, fL
k +1
;L
k +2
;111;L
n
g or fR
k +1
;R
k+2
;111;R
n g.
(ii-b) Every entryof
~
M
[k ]
is simultaneously ane in L
k , R
k and s
k .
(ii-c) Every entry of
~
M
[k]
is simultaneouslyane in all the entries of L
[k ] , R
[k ]
and P
[k ] .
(ii-d)
~
M
[k ]
<0 implies
~
M
[k 01]
<0 unless k =1.
Although the system is nonlinear in state variables, the ab ove theorem shows that the problem of
SD scaling is recursively linear in decision variables(or design parameters). This suggests that the
nonlinearity of thesystem essentiallydo notmake the problem seriously dicult. The characterof the
problem still remains the same asthat of robust linear design inthis sense.
OnthebasisofTheorem2,thispap erprop osesthefollowingproceduresofbacksteppingforfeedback
gaindesign.
Nominal backstepping: Solve
N
[k ] (x
1
;x^
[k 01]
)<0; 8(x
1
;^x
[k 01]
)2R2R k 01
(81)
fors
k
fromk =1 through k =n.
Robust backstepping : Solve
~
M
[k ] (x
1
;x^
[k 01]
)<0; 8(x
1
;x^
[k 01]
)2R2R k01
(82)
forfs
k
;L
k
;R
k
g fromk =1 through k =n.
Both the procedures supp ose that P,
~
P and Y are given. The ab ove pro cedures can be carried
out recursively since the pro cess ofnding decisionparameters atStep k do es not requireany decision
parameters to b e found at Step k +1;k+2;:::;n. The recursive pro cedures can b e also justied in
thatStep k is anecessary step for accomplishing Step k+1;k+2;:::;n.
Theorem 3 (i) If the whole procedure of nominal backstepping iscompleted from k =1 through k =n
properly, the parameters fs
1
;s
2
;:::;s
n
g solve
N(x
1
;^x
[n01]
)<0; 8(x
1
;x^
[n01]
)2R2R n01
(83)
(i) If the whole procedure of robust backstepping is completed from k = 1 through k = n properly, the
parameters fs
1
;s
2
;:::;s
n g , fL
1
;L
2
;:::;L
n
g and fR
1
;R
2
;:::;R
n
g solve
M(x
1
;x^
[n01]
)<0; 8(x
1
;x^
[n01]
)2R2R n01
(84)
Forinstance, the problem of ndingfL
k
;R
k
;s
k
g satisfying
~
M
[k ]
<0 is a convexoptimization problem.
The nominal backstepping and the robust backstepping for output feedback design via SD scaling are
amenable to computation based on optimizationas ithas been shown for state-feedback design in Ito
and Freeman (1998a). It is ready for automated numerical calculation by computer. The recursive
design prop osedin this section do esnot require precise knowledge of each system parameter since the
designisbasedondominationinsteadofcancelation. Anexactlycancelingformulaisconsideredasone
special solution to the domination. Moreover, the domination approach can b e exploited to get rid of
the propagationof complicated and long terms inthe controllawK.
Thesubsequentsectionsinvestigatewhether thesolutionsexistornotintherecursivepro cedures. In
otherwords,acondition ofthe allowablesize and nonlinearityof uncertaintywill b ederived. Existence
conditions and analytical solutions will be develop ed. Furthermore, a class of systems which can b e
always robustly stabilizable against arbitrarily large uncertainties by output feedback via the robust
backsteppingwill b e shown.
This section transforms the nominal backstepping and robust backstepping into problems which are
suitablefor nding analytical solutions. No conservatism will b e intro ducedin this section. Solving a
transformedproblem isequivalent top erforming the backstepping inSection5.
Denethe followingtwo functions.
N
[k ] (x
1
;x^
[k 01]
):=N
[k ]11 (x
1
;x^
[k 01]
)0N
[k ]12 (x
1
;^x
[k 01]
)H 01
[k]
(x
1 )N
T
[k ]12 (x
1
;x^
[k 01]
) (85)
M
[k ] (x
1
;x^
[k 01]
):=M
[k ]11 (x
1
;^x
[k 01]
)0
Q T
k M
12 (x
1
;x^
[n01]
)H 01
M T
12 (x
1
;x^
[n01]
)
Q
k
(86)
Fromthe Schurcomplements,the equivalence
N
[k ]
<0 , N
[k ]
<0 (87)
M
[k ]
<0 ,
~
M
[k ]
<0 (88)
are obviously true on the assumption that H < 0 holds. Due to structures of S, W and the strict-
feedback formof 6
0
, we can provethe following for
N
[k ] and
M
[k ] .
Theorem 4 Suppose 2kn.
(i)The symmetric matrix
N
[1]
(x
1 )=
~
9
1 (x
1
) (89)
depends only on s
1 .
N
[k ] (x
1
;x^
[k 01]
)<0 is equivalent to
"
N
[k 01]
(x
1
;x^
[k 02]
)
~
8
k (x
1
;^x
[k01]
)
~
8 T
k (x
1
;^x
[k 01]
)
~
9
k (x
1
;x^
[k 01]
)
#
<0 ; (90)
where
~
8
k
dependsonlyon(s
1
;111;s
k 01
)andtheirpartialderivatives. Thesymmetricmatrix
~
9
k
depends
ons
k .
(ii) Assume that
~
P is diagonal:
~
P = n
diag
i=1
~
P
i
;
~
P
i
>0;
~
P
[k ]
= k
diag
i=1
~
P
i
(91)
Then,the symmetric matrix
M
[1]
(x
1 )=9
1 (x
1
) (92)
depends only on (L
1
;R
1
) and s
1 .
M
[k ] (x
1
;x^
[k 01]
)<0 is equivalent to
"
M
[k 01]
(x
1
;x^
[k 02]
) 8
k (x
1
;x^
[k 01]
)
8 T
k (x
1
;^x
[k 01]
) 9
k (x
1
;x^
[k 01]
)
#
<0 ; (93)
where 8
k
depends only on (L
[k ]
;R
[k ]
) and (s
1
;111;s
k 01
) and their partial derivatives. The symmetric
matrix9
k
depends on (L
k
;R
k
) and s
k .
Proof : Recall that
^
S
[k ] (x
1
;^x
[k01]
)= 2
6
6
6
6
6
4
^
S
[k 01]
(x
1
;x^
[k 02]
) 0
.
.
.
0
1
0 s
k 3
7
7
7
7
7
5
; S 01
[k ] (x
1
;x^
[k 02]
)=
"
S 01
[k 01]
(x
1
;x^
[k 03]
) 0
?
1;k 02;k 01 1
#
^
A
[k]
(x
1 )=
^
A
[k 01]
(x
1 ) 0
?
1;0;0 a
k ;k +1
; B
[k ] (x
1 )=
B
[k 01]
(x
1 ) 0
?
1;0;0 B
k k
C
[k ] (x
1 )=
C
[k01]
(x
1 ) 0
?
1;0;0 C
k k
=
C
[k 01]
(x
1 ) 0
C
k ;3 C
kk
=
C
[k 01]
(x
1 ) 0
C
k ;0
V
[k ] (x
1
;^x
[k 01]
)=
V
[k 01]
(x
1
;x^
[k 02]
) 0
?
1;k 01;k01 0
; T
[k ] (x
1
;x^
[k 01]
)=
T
[k 01]
(x
1
;^x
[k 02]
) 0
?
1;k 01;k 01 1
L
[k ] (x
1
;x^
[k 02]
)=
L
[k 01]
(x
1
;x^
[k 03]
) 0
0 L
k
; R
[k ] (x
1
;x^
[k 02]
)=
R
[k 01]
(x
1
;x^
[k 03]
) 0
0 R
k
Then,we have the following.
P
[k ] (V
[k ] +T
[k ] )
^
A
[k ]
^
S
[k ]
=
2
6
4 (
P
[k 01]
(V
[k 01]
(x
1
;x^
[k 02]
)+T
[k 01]
(x
1
;x^
[k 02]
))
^
A
[k 01]
(x
1 )
^
S
[k 01]
(x
1
;x^
[k 02]
) )
0
P
k01 a
k 01;k
P
k
?
1;k 01;k 01
P
k (a
kk +a
k ;k +1 s
k +?
1;k 01;k 01 )
3
7
5
P
[k ] V
[k ] B
[k ]
=
P
[k 01]
V
[k 01]
(x
1
;^x
[k 02]
)B
[k 01]
(x
1 ) 0
?
1;k01;k 01
0
P
[k ] V
[k ] A
[k ] W
01
[k ]
=
P
[k 01]
V
[k 01]
(x
1
;x^
[k 02]
)A
[k 01]
(x
1
) 0
P
k
?
1;k 01;k 01
P
k
?
1;k 01;k 01
P
[k ] T
[k]
Y
[k ] C
y ;[k ] W
01
[k ]
=
"
P
[k 01]
T
[k 01]
(x
1
;x^
[k 02]
)Y
[k 01]
(x
1 )W
01
k k
0 111 0 0
P
k
?
1;k01;k 01
0 111 0 0
#
:
Thus,we obtain
N
[k ]12 H
01
[k]
N T
[k]12
=
"
N
[k01]12 H
01
[k 01]
N T
[k 01]12
?
1;k01;k 01 P
k
P
k
?
1;k 01;k01
P 2
k
?
1;k 01;k 01
#
(94)
N
[k ]
=
N
[k 01]
?
1;k 01;k 01
?
1;k 01;k01 2P
k (a
k k +a
k;k +1 s
k +?
1;k 01;k 01 )
=
"
N
[k 01]
~
8
k
~
8 T
k
~
9
k
#
(95)
Toprove the claim for
M
[k ]
, we need
~
P
[k ] W
[k ] B
[k ]
=
"
~
P
[k 01]
W
[k 01]
B
[k 01]
(x
1
) 0
?
1;0;0
~
P
k k W
kk B
k k
#
(96)
L
[k ] C
[k]
S 01
[k]
=
"
L
[k01]
(x
1
;x^
[k03]
)C
[k 01]
(x
1 )S
01
[k 01]
(x
1
;x^
[k 03]
) 0
L
k (C
k ;3
?
1;k03;k 02 +C
k ;k
?
1;k 02;k 01
) L
k C
k k
#
(97)
R
[k ] C
[k]
S 01
[k ]
=
"
R
[k 01]
(x
1
;x^
[k 03]
)C
[k 01]
(x
1 )S
01
[k 01]
(x
1
;^x
[k 03]
) 0
R
k (C
k ;3
?
1;k 03;k 02 +C
k ;k
?
1;k 02;k 01
) R
k C
k k
#
(98)
L
[k ] C
[k]
W 01
[k]
=
"
L
[k 01]
(x
1
;x^
[k 03]
)C
[k 01]
(x
1 )W
01
[k 01]
0
L
k C
k ;0
?
0;0;0
L
k C
k k W
01
k k
#
(99)
R
[k ] C
[k]
W 01
[k]
=
"
R
[k 01]
(x
1
;x^
[k 03]
)C
[k 01]
(x
1 )W
01
[k 01]
0
R
k C
k ;0
?
0;0;0
R
k C
k k W
01
kk
#
(100)
Now,let U denote
U(x
1
;x^
[n02]
)=0(B T
W T
~
P +R CW 01
)H 01
(
~
PWB+W 0T
C T
R T
) (101)
The followingrecursivenotation is used.
U
[k ] (x
1
;x^
[k 02]
)=
"
U
[k 01]
(x
1
;x^
[k03]
) U
3;k (x
1
;^x
[k 02]
)
U T
3;k (x
1
;x^
[k 02]
) U
k k (x
1
;x^
[k 02]
)
#
; U
kk (x
1
;x^
[k 02]
)2R p
k 2p
k
(102)
