Canonical forms for linear-quadratic optimal control problems

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Title:
Canonical forms for linear-quadratic optimal control problems
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v, 91 leaves : ill. ; 28 cm.
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Creator:
Khargonekar, Pramod Prabhakar, 1956-
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Subjects / Keywords:
Control theory   ( lcsh )
Riccati equation   ( lcsh )
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bibliography   ( marcgt )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1981.
Bibliography:
Includes bibliographical references (leaves 86-90).
Statement of Responsibility:
by Pramod Prabhakar Khargonekar.
General Note:
Typescript.
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Vita.

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Full Text












CANONICAL FORMS FOR LINEAR-QUADRATIC
OPTIMAL CONTROL PROBLEMS











By


PRAMOD


PRABHAKAR KHARGONEKAR


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY


UNIVERSITY


OF FLORIDA


1981









ACKNOWLEDGEMENTS


rish


express


sincere


gratitude


to all


those who


contributed


I w
towards


making


this work


possible.


been


Professor R
a constant


KALMAN,


source


the chairman


encouragement


of my


super


during my


entire


committee,


graduate work.


The main motivation


several


control


invaluable


problems.


for this work was


discu


am also


ssions


provided


his research


with him on linear-quadratic


grateful


for his early


proposal
optimal


intuition


the relevance


of the


Riccati transformation


group


ctral


theory


of linear-quadratic


optimal


control


problems.


Without


the financial


support


which


he arranged


for me for the


last


three


years,


this


work


would


not exist


toda


would like


my supervisory


express


committee:


sincere


Professor


apprec
s T. E


nation


to the other members


BULLOCK


LONG,


M. POPOV,


V. SHAFFER.


Discu


ssions


with


Professor


BULLOCK


on the linear-quadr


atic


optimal


control


and algebraic


problem


geometry


and with


helped


Profe


a great


ssor
deal


LONG


gaining


on invariant


an under


ending


problems


considered


in this


sser


station.


A doctorate


more


than a


dissertation--it


a slow


educational


process.


KAMEN,


grat
A.


fully


acknowledge


TMTJENBAUI4,


Profe


BA SIL;E,


ssors


and others


EMRE


SONTAG,


for their guidance


help.


would also


like


express


my deep appreciation


the untiring


help and


unending pati


ence


of my fellow


students


and fri


ends


Tryphon,


Bulent,


Ja ime,


Satis


am also


teful


to Ms. Eleanor


Onoda


for doing an


excellent


typing


job.


course,


no research


would


reported


here were


it not for the


constant


love


encouragement


of my


parents.


To them,


am eternally








This


research


was supported


in part


US Army


Research


Grant


DAAG29-80-C0050 and


US Air


Force Grant AFOSR76-3034D


through


the Center


for Mathematical System


Theory,


University


of Florida,


Gainesville,


32611,


USA.










TABLE


OF CONTENTS


ACKNOWLEDGEMENTS


ABSTRACT


CHAPTER


INTRODUCTION


* S S S S S S 1


RICCATI


TRANSFORMATION


GROUP


The Linear-Quadratic Optimizati
Riccati Transformation Group .
Invariants, Group Actions, and


Problem


Canonical F


S S S


TII.


STATE


SPACE


AND FEEDBACK GROUPS


Invariants
Action of


for the State Space
the Feedback Group .


Group


Action


ACTION


OF THE RICCATI


GROUP--I


S S S S S S S S S S S S S S


Finite Time Problems--I .
The Orthogonal Group .
Finite Time Problems--I
Infinite Time Zero Terminal


*~~~ S S S S


Case


ACTION


OF THE


RICCATI


GROUP- II


S S S S 5 5 S S S S S 5 5


The Algebraic


Riccati


Equation Approach


S S 5


SYSTEM


THEORETIC


APPLICATIONS


Spectral Theory of Linear
Other Applications .


dratic


Optimal


ntrol.


REFERENCES


BIOGRAPHICAL


S S S S S S S S *


SKETCH









Abstract


of Dissertation


Presented


to the Graduate


Council


of the


University


of Florida


in Partial Fulfillment


of the Requirements


for the Degree






CANONICAL FORMS


of Doctor






FOR LINEAR


of Philosophy


'- QUADRATIC


OPTIMAL CONTROL PROBLEMS








By


PRAMOD


PRABHAKAR


KHARGONEKAR


August,


1981


Chairman:
Cochairman:


Dr. R.


Kalman


Bullock


Major


Department


Electrical Engineering


This work


acting
with q


on spaces


uadratic


concerned
of various


with


types


cost criteria.


the Riccati


of linear


Canonical


group


optimal


forms


of transformations


control
complete


problems
systems


of invariants


are


obtained.


are studio


ed for


These canonical


this


group action and


forms


the Riccati


several results
transformation


group are


shown


In particular,


problems,


to have
spectral


sever


theory


Riccati differential


al important


of linear
equation,


theoretic


-quadratic
positive


optimal


applications.


control


real transfer


functions


are


inve


solution
"infinite


stigated


via the Riccati


problem of


time


free


transformation group.


existence


terminal state


solution


problems


complete


to the so called


is given


terms


non-


negativity


a symmetric


matrix.


This


solution


is also related









CHAPTER


INTRODUCTION


Linear


-quadratic


optimal


control


problem


one of the most


fundamental


problems


modern


system


theor


Optimal


control


1 and


filtering


problems


to the


in system


control


concepts


theory


eory was


reach


today.


done


ability


e pione
KALMAN


obser


ering work


[1960].


vabilit


on linear


Some


other


-the


concepts


-quadratic


papers


which


opt imal


made


substantial


include


There are


contributions


POPOV


seven


1:19624)


ral t


WILLEMS


books


the theory


[1971]


of linear


-quadratic


J ONCKHEERE


on the subject


which


optimal


SILVERMAN


we mention


control

[1978].


BROCKETT


1970]


ANDERSON


and MOORE


[1971],


and WILLEMS


[1981].


Despite


voluminous


research


on this


problem,


some


aspec


the th


eory


have not


been


fully


Invest


tigated.


This di


ssertation


explores


one such


direc


tion.


Very


which
Action


corre


mathematical


spends


GL(n)


some


(the


objec


notion


set of


admit


a group


equivalence


nx n


nonsingular


among


transformations
these objects.


matri


on the


space


n X n matrices


underlying


notion


y conjugation
equivalence i


a very g


this


case


ood example


arises


from


this.


change


of bas


The main mathematics


problem


such


a situation


to find


quantities


which


remain


invariant


under


transformation.


In the example


above,


invariant

The basic


factors


idea


that


eigenvalues


are


these invariant


some


the invariant


quantities


reveal


quantities.


the fundamental


features


of the mathematical


objects


under


consideration.


Mathematical


example,


tem th


GL(n)


eory


provides


acts on the set


examples
reachable


this


systems


type


in a


of problems.


natural


way.


This


Again a


action


detailed


corresponds
investigation


a change


of this


of basis


problem from an


in the state


invariant


space.


theoretic


point

POPOV
some


of view


1:197


assoc


iated.


has revealed


KAL;MAN


1:1974),


problems.


basic


TANNENBAUM


section 4


structure


1980]


we bri


of reachable


an account


efly


scuss


syst


ems.


of this


some


and
the


results


The main


problem


+ln a


O C a


-I-


station


arises from analogous


considera-


,1 irl 13 U .1. ~3~G










research


proposal.


This


group


of transformations


soon


became


known as


the Riccati


four


group.


components:


Roughly


state


speaking,


space


the Riccati


transformations,


group
input


is built


space


transforma-


tions,
Riccati


is to find


linear


state variable


differential


invariants


feedback,


equation.


canonical


change


main
forms


problem of


of variables


dissertation


for linear-quadratic


optimal


control


problems


of Chapters


IV and


under


the action


V show that


of the Riccati


group.


we are substantially


succe


Our re
ssful


suits


in resolving


this problem.


The results


are very


satisfactory for


case


equal


Kronecker


indices.


We also


treat


case


of strictly unequal


Kronecker


indices
easy to


explicitly.
formulate.


The results


However,


in the


in any


general


concre


case


situation


do not seem to


techniques


developed in


this


dissertation


can be used


obtain


invariants


canonical


forms.


We will


now briefly


describe


the contents


of the following


chapters.


Chapter


transformation


II is devoted


group.


a precise


We analyze


each


formulation

"component"


of the Riccati


Riccati


group


separate


group.
KALMAN


The bas


combine


ideas


four


this


chapter


components


are


esse


into a
ntially


single


transformation


contained


purpose


Chapter


III is


to review


some well


known material


on the action


of the state


space


group


feedback


group


on the


space


reachable


pairs.


We define a


new


canonical


form for


the action


of the feedback group


called


"dual


Kronecker


canonical


form"


. This


canonical form


There


is based


are two main


on the duality


contributions


of the


in Chapter


associated


ITT.


Young


We show that


s diagrams.
the set


of reachable


systems


with


iven


set of unorder


Krone


cker


indices


an affine


variety;


a geometric


quotient


for the action


of the general


linear


group


on this


space


exists


the geometric


quotient


turns


out to


a Euclidean


invariants


space,


due to POPOV


the quotient


r19723.


This


re sult


given
should


by the


well known


be viewed as









DAVISON [1976]
reachable pair


in the


Kronecker


the stabilizer


canonical


in the feedback group


form.


Chapter


IV contains


the first results


on the invariants


canonical


forms
satis


for the
factory


action
for the


of the Riccati


case


equal


group.


Kronecker


The r


results


are very


We also


treat


case


strictly un


equal


Kronecker


indices


expli


city.


Some


the results


this


chapter


turn


out t


useful


in Chapter


system theoretic


applications.


In Chapter
canonical


we devel


forms.


Here


a real


another


we assume


symmetric


approach


that


assoc


solution.


invariants


iated algebraic


Again,


the results


very


satisfact


case


equal


Kronecker


indices.


difficulties


in the


neral


case


are analogous


the difficulties


Chapter


Chapter


contains


some


theoret


applications


the results


of Chapters


IV and


The main


application


spec


tral


theory


linear-quadratic


JONCKHEERE


optimal


and SILVERMANT


control


[197


problems.
for the di


This theory


score


-time


was developed


case.


ey analyze


the so called


"infinite


time


free terminal


state linear


-quadratic


optimal


control
problem
certain
a direct


problem"


from a


boundedness


self-adjoint
calculation


Hilbert


of infima


Hilbert
of the


spec


space


operators


reduced


rator.


trum of


point


view.


nonnegativity


However,


it turns


out that


We consider


JONCKKEERE a
We show that


the continuous-


SILVERMAN


the existence


1978]


time


analog


as formula


solution


the problem
originally


to the optimization


[1971].


problem remains


invariant under


the action


of the


Riccati


group.


However,


the Hilbert


space


operator


defined


JONCKHEERE


and SILVERMAN


[1978]


does not remain


invariant


under


the Riccati


group.


We first


compute


a Hilb


space


operator which


is the continuous-time


anal


the operator defined


JONCKHEERE


and SILVERMAN


[1978].


Riccati


group


leads


to a


major


simplific


ation


in the


derivation


operator.


then


show


formulated


indices.


Riccati


equation has


obtaining


are


space


this operator


is not tractable.


considered


WILLEMS










of boundedness


of infima


can be reduced


to checking;


nonnegativity


symmetric


matrix.


We rederive


a generalized


version


of this


result


using


certain


developed


simple results


KALMAN,


from


the theory


and NARENDRA


[196


"minimal


This


give s


energy
a very


control"


simple


system


theoretic


interpretation


of the


spec


tral


theoretic


results.


We also


examine


fast algorithms


Popov's
for the


equency


Riccati


function,


differe


ntial


itive


equa


tion


real matrices


due to


KAILATH


[1975]
Under


and CASTI [
certain mild


1974]


in the light


conditions,


we complete


Riccati


transformation


y parametrize


scalar


group.
itive


real


functions.


Further work


on the


fast


algorithms


from


cati


group
scalar
intere


point


positive


view


real


seems


promising.


functions


General


the matrix


nation


case


our results


also


considerable


We close


assumptions.


this


introduction


Throughout


with


a few words


this dissertation,


on notation


we will


work with


d standing
the field


real


numb


ers.


denotes


m-dime


nsional real


vector


space


consisting


m-length


column


vectors with


real entries.


denotes


set of


mx


n matrices


over


the field


real


numbers


, b]


denotes


the Hilbert


space


A polynomial


square
called


integrable
a Hurwitz


functions
polynomial


on the interval


iff the


roots


are in the


open


left


half


plane.


undefined


terminolo


from


invariant


is referred


theory and algebraic


to FOGARTY


[1965],


geometry
MIUMFORD


is standard


[1965],


for which


TANNENBAUM


the reader


[1980].


rrXn
R









CHAPTER


RICCATI


TRANSFORMATION


GROUP


In this chapter,


we shall


introduce


the Riccati


transformation


group,


quadratic


KALMAN


which arises


optimi

[1975]


naturally as


zation


in a


problems


research


a group


This


propose


of transformations


group


was originally


Roughly


of linear-
formulated


speaking,


Riccati


transformation


group


consists


state


space


transformations,


input


space


of variables


transformations
in the Riccati


linear


state


equation.


variable


shall


feedback,


scuss


each


change


these


transformations

group action.


separately


then


combine


ese


into


a single


The Linear


Quadratic


Optimization


Problem.


The "dynamic


for a


continuous-


optimization


time,


finite


problem with

dimensional,


integral


linear


quadratic


stem


criteria


is defined as


follows


minimize


the functional


(T)Qx(T)


+ 2x'(T)Su(T)


u' (T)Ru()


+ x'(0)Pox(0),


over all


continuous


valued


input


functions


Ut r)


with


respect


to the solutions


n-th


order


continuous


-time


linear


(1.2)


it(c


= Fx( )


+ Gu(T),


= c.


Here


are respect


tively


n x n real matrices


assumed t
definite.


symmetric


We also assume


matrices


that


is always


assumed


is reachable.


itive


It should


noted


that


1n slTTi A+.nt


the system


as well as


tn lTiT


the quadratic


rpE3 F,1 tsz


cost


It In PP4\ 1.11


~n~nsnl nrnss


criteria are


niir T" r e1l ts


time-
wh i ph


n X


n X


are


(1. 1)


J(u(


I. f


.










(1.3)


c, N)


1
k=-N


(k)Qx(k)


+ 2x'(k)Su(k)


+ u'(k)Ru(k)]


+ x'(0)Pox(0)


over all


valued


input


sequences


u(k),


with


respect


to the


n-th


order


discret


e-time


linear


system.


(1.4)


= Fx(k)


+ Gu(k),


x(-N)


= c.


, x(k)


Here
that


is not assumed


are as before.


positive


The only


difference


definite.


linear


quadrati


optimi


nation


problems


defin


ed above


have


been


studied


extensively


over the


last


two decades.


The literature


surrounding


these problems


grown


enormously.


reader


referred


seminal
WILLEMS


paper
[1971]


of KALMAN [1960].
. JONCKHEERE and


also KALMAN


[1964]


SILVERMAN


POPOV
refer


[1964],
ences


cited


there.


It is


well


known


that


the solution


the above


continuous-time


problem
optimal


base


control


on the (matrix)


input


Riccati differential


equation.


iven


u~&r)


x ( r),


= L(r)


= Fx (r)
-


+ Gu(r),


X,( t


where


LtC'r)


-l(G'P(r)


is the solution


of the Riccati


differential


equation


(1.6)


-JP( r)


= F'P(-r)


+ P(t)F


- (P(P')G


Here we


implicitly assume


that


the solution


to (1.6)


exists.


+ S'),


-l(P(7)G


C,


J(u( )


U~ (


dx+( 7)
dx*
d2










Chapter


VI for a discussion


of the general


case.


A similar


feedback


solution


base


on the Riccati


differe nce


equation


can be given


for the


discret


e-time


problem.


thus


see that


the optimi


zation


problem


as well as


its solution


can be fully


Since


features


described
the initial


of the optimi


in terms


state


zation


the "problem data"


plays


blem,


no spec


1 role


we do not consider


in various
it in the


"problem data"
be identified


Thus


with


every


linear


quadratic


optimization


a six tuple


Conve


problem


rsely,


six tuple


of matrices


of appropriate


sizes,


reachable,


are symmetric


positive


definite


then


we can define


a linear


quadratic


optimization


problem


ociat


ed with


In view


these


ementary


serva


tions,


we define


(finite


time)


optim


nation


data


space


(1.7)


:= ((F,


>( _


_mxm
X R


X R


reachable,


symmetric,


positive


definite).


Similar


considerations


also


apply


discrete-time


linear


quadratic


optimization


probl


ems.


In particular


we have


discrete-time


optimization


data


space


RInn


:= (-F,


R~

nxn
x R


reachable,


symmetric).


An important


problem


closely related


to the linear-quadratic


opt imiza-


tion


problem is


so-called


infinite


time


zero


terminal


state


linear-


quadratic


optimal


control


problem defined a


follows


Minimize


(1.8)


=io[x'(T)Qx(T)


+ 2x'(T)Su(T)


u'(T)Ru(-) ]d'r


ass


can


>(~


><~


)<~


t &~


: (F


nXn
R


F ,


J(u(.)










over


possible


functions


m
R -valued,
Necessary a


continuous,

d sufficient


square


integrable


conditions


input


for the existence


solution
the above


this


infinite


problem are well known.


time


zero


terminal


In particular,


state problem


exists


a solution


if and


only


if the algebraic


Riccati


equation


(1.9)


+PF


- (PG +


+ S)


admits


a rea


1 symmetric


solution.


case


(1.9)


a real symmetric


solution,


it has a unique


real


symmetric


solution


such that


eigenvalues


- GR-(G'P
aGR (C PP


are in


the (clo


s e&)


left


half


plane;


Re(A(F


GRGP +
-GR (G'P+


WILLEMS


1971]


an exce


llent


reference


for a


detailed analysis


of the


algebraic


Riccati equation.


It is


clear


from


above


discussion


that


the infinite


time


zero


terminal

matrices


data


space


state problems


are completely


In view


for the


infinit


time


zero


specific
this we


terminal


ed b


define

state


quintuple


the optimi


zation


problems


DS := ((F,


X R

_~


_~


Nt tXm


reachable,


symmetric,


positive


definite.


Analogous


considerations


apply


to the discrete-time


case.


Riccati


Transformation


Group.


purpose


the above


dynamic


this


section


optimization


introduce


problems


some


which


transformations o
naturally from the


problem


statement and


the associate


feedback


solution


based


on the


Pic reati


di fferen. al.


erniaBti nfl


-l~pc


st)


nXn
R









:- T


the n


the linear


quadratic


optimization


problem


(1.1)


transforms


minimize


-1i
c,


-= t [z'(T)T'QTz(T)
o


+ 2z'(T)T'Su(T)


u'( )Ru(T)]dz


z' (O)T'P


Tz(O),


subject


to the linear


dcz(T)
dcT


system


--TFTz(T)


Gu(Tr),


Thus,


the problem


data


transform


, T'QT,


Riccati diff


and only i

equation a


erential

T'P(T-)T


ssoc


iated


T'PoT).
equation


Furthermore
associated


is the solution


with


VT,


1,


the solution


with


to the Riccati


T'QT,


differ


T' PoT).


ential


It is


easy


eck that


optimal


control


inputs


for the original


problem


transformed


problem are


same.


Conse


quently,


state


space


basis


chan


do not alter


basic


features


linear quadratic


optimization


problems.


Summarizing,


GL(n)


acts


on the data


space


(2.1)


x GL(n)


1TT,


T-G,
T ,


T'QT,


T'S,


T' PoT).


Finally,


reachable


we note


pairs


that


has b


the action


een


considered


GL(n)
from


on the


seven


space


ral points


view


in the literature.


TANNNBAIJM


[1980]


and the


See,


for example


reference


cited


POPOV
there.


1972 ],


KALMMAI


summarize


[1974],


some


~~~~c' ~ ~ ~ ~ ~ i-i- ---I-rrf ~


x(t)


J(U()


D :


H(T


n L~.


-


I, \ rm


on_.~


---- f--_ *>n -*IK- -1^~









Change


of basis


in the input


space.


This


transformation


similar


to the state


space


transformation


scusse


d above.


mx m


nonsingular


matrix.


We introduce


new input


variables


v(.)


v( r)


Proceeding as


this


leads


to the transformation


(2.2)


D X GL(m)


'-

V'RV,


Linear


state variable


feedback.


This


perhaps


the most


nontrivial


transformation of


the original


problem.


Suppose


we choose


N; t


:= u(t)


where


n matrix and


u(.)


continuous


-valued


function


The linear


system


then


transforms


= (F


+ GL)x(-r)


+ Gu(T),


= c.


We also


have


X'Q)


+ 2x' (T)Su(T)


+ u'(T)Ru(T) =


X'Qra)


+ SL +


L' S'


+ L'RL]x(r)


+ 2x'(r)(


+ L'R)x(T)


+ v'(C)RV(t).


Thus


the orig


:= (F


inal


+ GL,


problem data


+ L'RL


tran


+ SL + L'S


+ L'R,


sform to


It is


routine


verification


that


a solution


the Riccati


differential


equation


asso


ciated


with


if and


only


P(T)


is the


solution
(whenever


the Riccati diffe


they


exist).


rental


The optimal


equation


associated


feedback matrices


with


L(d)


L(dL)


-1
V-


-x r


dx(


D :


Po)


'. R


Ut ~t)









Summarizing,


linear


state variable


feedback


induces


transformation


(2.4)


nRlxn
x R


-

+ GL,


+ SL + L'S


+ L'RL,


+ L'R,


It has


een recog


nized.


prev


iously


in liter


nature


that


edba


transformation


does


not alter


most


the basic


features


the dynamic


optimi

[1975]


zation


blem.


JONCKHEERE and


See,


for example,


SILVERMAN


[197


SILVERMAN


etc.


[197


Feedback


MOLINARI


tran


sformation


stabilize


system


JONCKHEERE


and SILVERMAN


we have


problem does
differential


shown


above


not alter
equation


the form


under
does n


linear


eedbac


.ot alter


solution


and the


quadratic


the Ri


optimal


optimi
ccati


zation


feedback matrices


related


a simple


trans


lati


as in (2.5).


of variables


the Riccati


differential


equation.


This


tra ns


linear
differ


formation


quadratic


ntial


arises


optimi


from


zation


fact


blem


that


given


solution


erms


to the


the Riccati


equation


(1.6)


-dP(T)
dir


= F'P(T)


+ P(t )F


P(T)


+ S)R


1(P(r)G


+ S)


Suppose


we perform


the change


of variables


P(rT)


:= P(r)


where
obtain


a N: U


the differential


CO ns


tant


symmetric


matrix.


equation


A
5) )


= F'P
=FP


+ P(r)F


+ F'M +


a ('P(T)G


+ MG)


use


are


(Q


H (F


pt ~)









As is


clear


from


this trans


formed


equation,


the Riccati


differential


equation


assoc


iated with


the data


+ F'M + MF,


+ MG,


t rari


formed


b MG).

problem


Furthermore


cost


assoc


iated


with


+ F'M +


+ 2x'(T)(


+ MG)u(r)


7)Ru(


m) 3dm


+ x'(0)(


- M)x(O).


Recalling


that


x(r)


is governed


= Fx(-r)


+ Gu(t),


we have


+ 2x'(T)Su.(T)


+ u' (T)Ru(t)


( 'r)MX~rC))3d


+ x'(o)P


- x' (O)I x(O)


- c'Mc.


Thus


if and


only


every


fixe


initial


an infima


state
exists


an infima


Further


exists


an input


function


u(s)


fixed


minimizes


initial


if and


state


only


Finally,


P(T)


minimizes
a solution


each


and only
optimal


also


A
P(r)


feedback


preserves


a solution


in matrices


most


the b


are the


asic


(1.6),


same.


atur


Thus,


the associated


this


tran


the problem.


sformation
conclude,


Sym(n)


be the additive


group


nx n


symmetric


matrices


then have


(2.6)


the transformation


X Sym(n)


dT


+ I(


=~f


D:


t S,


7:)(Q


J(u( )


J,(u( )









This


transformation


occurs


seven


ral different


contexts


in system


theory


literature.


In the study


stability


and optimality


this


transformation arises


POPOV


[197


In the study


stochastic


reali
CANDY


this


[1975]


optimal


zation


[197


transformation


consider
control.


problem,


this transformation has


the closely related


occurs


a similar


in ANDERSON


ransformat ion


been


problem


[1967]


for the


cons


idered


positive
BERNHARD


inverse


real matrices,


COHEN


problem


We will


now combine


the four


different


types


transformation


scuss


ed above


. This will


cons


titute


the action


of the Riccati


group


on the data


space


Consider


the set


(2.7)


GL(n),


GL(m),


ym(n) ).


We define


a binary


operation


between any


two elements


given


o (T2,


2' L2'


:= (TTT2,


V'V2,
1 2


VL
1 2


+ LT 2,
1 2s


+ (T) -


T').


It is


easy


check that


with


the binary


operation


a group.


We just
inverse
define


note that


, O,0)
(T-. V


serves


as the identity and


, V-ILT,


- T'MT).


Po)'


^ A'


where


now


mXn
R









+ F'M +


+ T'(S


+ MG)L +


L' (S


+ MG)'T +


L'RL,


:= V'RV,


+ MG)


+ L'R)V,


+ M)T.


In an


Riccati


entire


group


analogous


on the


manner


space


we can define
corresponding


the action


of the


to the infinit


e-time


zero


terminal


stat


problems


where


are as above.


It is


easy


check


that


the function


satisfies


the following


rules:


(2.9)


and


, 72


cp(d.,


(2.10)


P (cl,


=c.


In fact


the binary


operation


was defined


precisely


that


(2.9)


hold.


Invariant s,


Group


Actions,


Canonical


Forms.


an abstract


that


acts


denoted multiple


the right)


catively

iff there


a set.


a map


such


that


have


w1 )


=i(~z


Wj,


ii(z,


2),


= z.


(Here


algebraic


is the


identity


transformation


The triple


space


is called


the (right)


w): w in W} i


call


ed an


action
called


=- i'(


*-= T'


H(i,


Y2


MF)T


-t DS:


cl(z,


Ccl(zl










Now let


other


said to be invariant


(under


a function.


the action


Then


iff for


An invariant


is called


a complex


te invariant


iff for


implies


ri~i(


orbit


In this


of finding


set theor


etic


a set


some


The set


is a complete


setup,


that


there


invariant

a trivial


parametrizes


orbit


said


solution


space.


parametrize
surjective.

the problem
e can choose


to be the set


general,


very


orbits


useful.


It is


mor e


However
desirabi


such


a soluti


to have


with


some


math


ematical


structure.


group


called


an algebra


group


an algebraic


ety,
->W:


W i-)


the
-1


two

are


maps C: W x W
morphisms. Now


suppose


r-W W2

algebr


and

aic


group


an algebra


vari


such


that


Z W


a morphism.


We then


say that


acts


morphically.


In such


a situati

variety.
invariant

excellent

theoretic


on, we may require


This problem


theory


pioneer


erence


aspec


the parametri


one of


central


MEJMFOBID


an expository account


of linear


system


to be an algebraic


problems


TANNENBAUM

f geometric


the modern


1980]


geometric


is an


invariant


theory.


action


is a right


action


D follows


directly


from


(2.10).

machine


group
seem


owev


,there
metric


main


have


are certain


invariant


difficulty


algebraic


lies


ometric


difficulties


theory


in the


struc


to the


fact


ture.


in directly


action


that


applying


Riccati
does not


principal r


eason


for this
symmetric


seems


matrix


be the
is well


condition


known


> 0.


The positivity


equivalent


certain


polynomial


inequalities


the entries


Thus,


it does not


seem


to be


space


vari


-*


f(


f(i


,W))


-IW:


CL (










However,


whether


is linearly reductive


seems


to be


a difficult


question as well.


The next


three chapters


are concerned


with answering


questions


invariants


canonical forms


for the action


We do


some


aspects


of modern


invariant


theory,


but only


after


certain


preliminary reductions


of the original


problem to


somewhat


more


mannageable


situations.


use









CHAPTER


III.


STATE


SPACE AND


FEEDBACK


GROUPS


purpose


this


chapter


is to


review


some relevant


results


about


the action

reachable


of the


systems.


state


space


The results


eedback groups


described here will


on the


space


be the main


preliminary

the action


material


of the Riccati


obtaining


group.


invariants


This


chapter


canonical


is divided


forms


into


sections:


first


we di


scuss


invariants


canonical


forms


for the


action


of the state


feedback group.

several years.


Most


space


group,


and next


of these results


We present


a new result


we di


have


on the


scuss


n known

action


e action

for the


of the


past


state


space


group which


be viewed


as a g


geometric


counterpart


some


previously


known results


on invariants


due to POPOV


[197


Let us define


space


of reachable


systems


:= ((F,


Rnxn


reachable);


denotes


of all reachable


teams with


states


inputs


basis


- As


changes


GL(n)


prec
acts


ending


chapter,


group


state


space


the following way:


x GL(n)


i- (T-1FT,


T-1G) .


It is


easy


to verify


that


a right


action


GL(n)


feedback group


defined


:= ((T,


GL(n),


GL(m),


Rrtn)n3


a group


under


the multiplication


given


L1) o(T2,


S(1T2'


according


r. ,


acts


21


-.1


-,1


-t C:


nXm
R


*


,,









right


action


identified with


reason


that


subgroups


leads us


groups


of the Riccati


study actions


GL(n)
group;
these


can be


in fact,


groups


this


on the


naturally


is the main


space


reachable


systems.


Invariants


for the State


Space


Group


Action.


The action


several


points


of view.


GL(n)


complete


has been


studied.


set of invariants


extensively
canonical


from


forms,


and a


geome


tric quotient


have


been


discovered.


We shall describe


some


of these results


the interested


HAZEWINKEL


reader
KALMAN


very


briefly.


referred


[197


detailed


POPOV


HAZEWINKEL


[1972],


1977],


proofs
KALMAN


TANNENBAUM


discussion


[1972,


1974],


1980],


BYRNES


GAUGER


C J977]


and others.


Define


the reachabilit


y matrix map


(4.1)


p: Z R


t (G,


F1G


=. p(F,


Number


the columns


p(F,


lexicographically


(from left


to right)


- 1)1,


- l)m.


nice


selection


defin


a subset


v of these indices


size


such


that


belongs


then


belongs


< i.


For a


nice


selection


v define


:nXn


PV(F,


where


p (F,


denotes


nX n


submatrix


p(F,


consisting


the columns


results


of KALMAN


indexed


[1974]


by the


nice selection


is that


reachable


of the
if and


only


p (F,


is nonsingular


some


nice


selection


(4.2)


REMARK.


each


nice


selection


define


open affine


subsets


X Rnxm


(F,


tlXn
R









It is easy to


see that


GL(n)


acts


in such a


that


: i.


is stable under


the action


GL(n).


Further,


: v a nice


selection)


constitutes


an affine


open


cover


TATINENBAUM


tJ-98O2


shows


that


GL(n)


acts


and hence


on each


with


closed


MUMFORD


orbits.


TANITENBAUM


Using
then


metric


proves


that


invariant


a geometric


theory


quotient


the action


Finally


shown


, using


that


the action

out to be a


GL(n)


the techniques


"patch up"
v


GL(n)


smooth,


irreducible,


exists


of modern


such a

exists.


quas


for each

algebraic


way that


This


nice selection


geometry,


geometric


geometric


iprojective


quoti


variety


it is


quotient

ent turns


of dimension


However,


we will


not work directly with


this


ometric


quoti


ent.


Following


section,


POPOV


called


1972],


we will


the Kronecker


now describe


nice selection,


how a

can be


special


nice


naturally


associated


with


every


reachable


a reachable


pair.


Consider the


ordered


set of vectors


' ~(g1,


Fgl',


" .,


Fn-lg


where


is the


column


the matrix


In other


words,


cons


ists of


columns


p(F,


list


ed from


the left


to the


right


A vector


said to


an antecedent


another vector


im +


the list


< pm +


can now


occurs


define


some well known


the left


invariants


FPg
F g in
qof the pa
of the pair


DEFINITIONS.


be in


an integer,


Then


j-th


unordered


Kronecker


index


is the


smallest


eger


such


that


is linearly


dependent


upon


antecedents.


unordered


Kronecker


S..,t


Integers


indices

obtained


K2, ..., K


of the pair (

by reordering


are called t
The integers


in a


* a a -


n-l
F


II


..










condition


=1,


for the sake of notational and


We will


computational


often make


simplicity.


this assumption

Let us define


E: rank


:= ((F,


= m}.


be ih


with


unordered Kronecker


indices


There


Kronecker


a ni


i ndic e


selection

s, defined


naturally


as follows:


ass


ociated with


a (double)


the unorder


index


longs


called


chas


the definitions


the Kronecker


it is


easily


section


seen


of the pair
that


(t'4)


det PYK(F,


0.


We will


now


describe


some


of the main r


results


POPOV


the invariants


of the action


GL(n)


reachable


pair.


For the


sake of


notational


simplic


rank


belo


that


there


exists


Then


a unique


Corollary


of (indexed)


real


of POPOV
numbers


tJ1972J


states


= 1,


= 1,


- 1,


= 0,


=1,


min(K.,


+ 1,


- 1);


= 0,


= 1,


min (K.,
1


such


(4,6)


that


Ki
F -g.


Sa. Fk
k ijk


where


for each


the summation


with


respect


runs


over


index set

F gi on


above


this


list


index


defined above.


its ante


depends
set by


cedents.


Nijk]
Note


only upon


where


determines


that


the ind


the unordered


= (K1,


the linear

ex set of


Kronecker


dependence


S,
tJjk


indices;


s in the
we denote


K).
M


cardinality


denote


the lexicographicallyy)


~~~~.~h L.I --.4- ,


-)


m;


~aijk:


t '"')


cr(F,


I,, \


_L L









orbit


under


the action


GL(n)


if and


only


the unordered


Kronecker


indices

= t(F2'


, G)


, G2)


are the


same


Further,


= (K ,


is an


m-tuple


positive


integers


satisfying


-tuple


such


that

a(F,


real


number


then


are unorder


there


exists


Kronecker


a pair
indices


= Cy.


Roughly


a complete,


will


speaking,


unordered


independent


now describe


Kronecker
invariants


algebro-


indices
for the


geometric


cons


action


structure


titute


GL(n)


set of


reachable


pairs


S(K1,


havin


..., Km)
* *4m


same uno
a fixed s


rder


Kronecker


integers


indices


satisfyin


= n.


4(K)


subset


defined


4(K)


the unordered


Kronec


mndi


Km.
m


now have


(4.8)


THEOREM.


= K ,1


S.., K )


as above.


Then


4(K)


has the structure


an affine


vari


The function


a: 2(K)


is a regular


surj


active


map.


pair


N
R K)


constitute


a geo


metric


quot i


4(K)


under


action


GL(n).


PROOF.


first


some


notation.


x Rnxm


each int
Anti(F, G


eger


denote


the submatrix


consi


Rnxn

sting


the columns


inde


the nice


Ant.(F,
Define


selec


tion


consists


satisfying


+ mj


of antecedent


+ mK.
which


In other words,


are


the (polynomial)


... + K


.. + K


are


ces


f K~


n,


: (F


a(F1'


Ha(F,










Here


denotes


the (fixed)


number


of columns


Ant.(F,
1


denote


the set of integers


We will


now give




equations


such


which give


that


as a subset


x Rnxm


We start


noting


that


(4.4)


det PK (F,


is a necessary


condition


Ant.(F,


condition


that


F .g s
For any


Kig
gi
should


linearly
linearly


matrix


F


to be in
dependent
dependent


m.(B)
J i


2(K).
upon
upon


denote


e additional
antecedents


the columns


j-th lar


minor


- i,


...


Then we


must


have


(4.9)


m.(Anti(F,


FKig.)


= 0,


for all


suffic


sible


ient


It is


order


easily


seen


that


*...


that


are


be the unordered


necessary
Kronecker


tnti


Following the


standard


proce


dure of


algebraic


geometry,


we now


introduce
projective


a new variable


variety)


Then


the (real)


2( K)


solutions


isomorphic


a quasi-


the equations


(4.10)


det PVk


the Equations


(4.9).


This


proves


our assertion


that


4(K)


has the


structure


of an affine


variety


We will


now show that


a: Z(K)


a surje


results


ctive


of POPOV


regular

[1972]


map.


Surjectivity


as described in


follows


Theorem


directly


(4.7).


from


prove


that


I. nran r1 r


mmii nlm


ces


~Xn
R


~h nw


fh s~ f


j r ^









is regular


for each


index


in the index


Now,


~(PvF,


a regular


map on


2(K).


It follows


that


22(K)


( (i,


j) -th


entry


(pvK(FI


G))-1


is also

ijk


(4.12)


a regular


(implicitly)


FKigi
i


for each


given


ijFk


denote


ition


of

G))


F gk


n PvK(F,


Multiplying


(4.12)


q-th


row


(pVKQE',


we ge


= a.
ijk


As p..' s
a regular


and
map.


FKigi
Hence


are regular


a sure


2(K),


active


it follows


regular


that


map.


Finally,


, G )


S(K)


belong


to the


same


orbit


under


the action


GL(n)


if and only


G2).


It follows


directly


from


the definition


metric


quotient,


(see


MUMFORD


[1965],) that


RNK)


a ge


ometric


quotient.


The above


reformulation


result


should


results


viewed as


of POPOV


1972)].


geometric


invariant


We see that


theoretic
a disjoint


union


of affine


varieties


corresponding


various


choices


of unordered


Kronecker


indic


quotient with


on each of these affine


respect


the action


varieties


GL(n)


ex:


a ge


ists.


metric


In our work,


2( K),


Kig


R:


G))-l


de t


(Pij :


(cp,,,


'Sqn)F


a,
1Jk


(F1'


Cr(F1'


~~jF2~


w


v m







geometric


quotient


for the


action


GL(n)


Z(K)


can be


deduced


Am a


much


Section


with


osed


FOGARTY


it then
quotient
quotient


easier way.
Corollary


orbits.

[1965].)


follows
exists
can be


from


It is known


1.6]


Recall


Having
MUMFORD


The point


shown


that


from TANNENBAUM


GL(n)


that


GL(n)


proved


that


pages


here


course


acts


[1980,


Part


and hence


a linearly r


Z(K)


-350]


is


eductive


is an affine


that


a geo


that


Z(K),
group.


variety,


metric


geometric


N)
BK).


(4.114)


REMARK.


Results


the above


theorem


should


compared


with


certain results


of HAZEWINKEL


[1980].


It is shown


there


that


construct
general,


tions


the well


"discontinuous


known


"control


However,


canonical


one restricts


forms"I


are


attention


these


constru


continuous.


actions
It is


2(K),


well known


then


that


these


cons


tructions


these constructions


are indeed
are closely


related

regular


to the invariant


maps


4(K).


s OX.


Thus


We have


the continuity


shown


that


a,


s are


of the constructions


canonical


forms


2(K)


is natural.


Action


of the Feedback


Group.


now turn


our atte


nation


to the


action


the feedback


group


This problem has


een


complete


ely r


resolved


due to


the deep and


striking results


of BRUNOVSKY


[1970],


ROSENBROCK


[1970],


KALMAN


[1972 ],


POPOV


[1972],


WOIIIAM


1974],


others.


The main


purpose


this


section


describe


start


with


the stabilizer
a theorem that


in the feedback group
describes a complete


in a


pair


invariants


under


the action


of the feedback group.


THEOREM.


The ordered


Kronecker


indices


constitute


a complete


set of invariant


of the action


of the feedback group


various


proofs


of this


result,


the reader


is referred


to the


papers


mentioned above


be in


with


ordered Kronecker


indices









:= block


diag


... Fm]


:= block diag


... gm ;
m^_ f^


where


are


K. X1
1


matrices


of the form


F






The pair
In fact,


called


a triple


the Kronecker


that


tran


canoni


1
iform

form


sfhrins


can be


cons


truct


using


the invaria


ijk


In other words,


there


a regular


S(K)


such


that


transforms


to the Kronecker


canonical


form.


our purposes,


however,


it is


more


useful


choose


a slightly


different


canonical


form


under the


feedback group.


In order


to describe


this


canonical


r := K


*


form,


we shall


first set


a given


some


of ordered


notation.
Kronecker


indices
indices


Define


the Young'


matrix


mx n matrix


assoc


iated with


this set of


give


Y (i,
K


i,
O,


otherwise.


Then


the dual


Kronecker


indices


are


defined


21+


m
j.'Ji


= 1i,


map


-t rf


X Ki,










be the unique


integers


such


that


"r(1)


r(l)+l


* -


* =


r(q-1)


r q-1)+l


r(q)


- m.


To each


eger


we can uniquely


associate another


eger


PB(i)


defi


(i))


Finally,


define


r(2)


r(q-l)'


We note


that


case


the Kronecker


indices


are all


equal;


= n/m,


then


- m,


=1,


r(1)


=11.


As we will


see i


the following,


case


of equal


Kronecker


indices


is relatively uncomplicated


and leads


very


nt results.


be in


Then


with

it is


order


easy


Kronecker

see that


indices


feedback


(5.2)


equivalent


Here


"0" in


Aj it.


entry


matrix consisting


zeroes,


- IT


is the
1


x +l
1+1


matrix


the form


I -.


- 2


.. f = K


=...


... >


K >
m -


, G )


K2


"2


m,


r(l)


(1)










consisting


zeroes.


case


the Kronecker


indices


are equal,


we have


SF2


We call
see that


, G )
* G
, G)


the dual


Kronecker


can be obtained


canonical


from


form.


ff15s


easy to


a permutation


state variables


Here again,


there


exists


a regular


map,


("built"


out of


d: Z(K)


Sd(F,


such


that


d(F,


dissertation,


some


transforms


, G )


choice of Krone


the stabilizer


in the


will
cker


eedback


denote the


indices


group


. We


dual


now


Kronecker


turn


a pair


Throughout
canonical


this
form


our attention


, G ).


Stabilizer


in the


canonical


feedback group


form)


has been


the pair
obtained


the Kronecker


WANG


and DAVISON


[197


BROCKETT


[1977 ],
however


F1I[MRMA'NN


turns


and WILLEMS


out that


[1979].


Ricca


group


the stabilizer


problem,

the dual


Kronecker


canonical


form)


is more


conve


nient.


be ih


the feedback


group.


We partition


SF


iT'r


a,
rJk


























where
matrix.


T.h
The


stabilizer


x2.,'


X cr


subgroup


,G)
SG


is character


following


THEOREM.


, G)


be i


the dual


Kronecker


canonical


form


as described


above.


Then


a triple


, L )


belongs


to the stabilizer


, G)


and only


, L )


satisfies:


- 0,


dat V.i


S0,


- 1,


S..,


= 0,


>1,


where


B(i)
















where


where


is a


Xc"k


matrix


satisfying


- 1 + r(k


- 1)),


r>i


where


B( i)


finally,


= 0,


= 2,


r(i-l) (I.
1


S..,


The proof


of this result


omitted


since it is


a very


tedious


cumb


ersome


a very


but straightforward


simple


corollary


of the


direct
above


verifi


cation.


theorem which


We will
describes


now state


> j,


B( i)


~(j),










(5.4)


COROLLARY.


*
,G


in the dual Kronecker


canonical


form with


Kronecker


ii)


belongs


indices


to the


... = K


stabilizer


= n/m.

,G*)


Then a


triple


if and only


= O,


det V


S0.


The above corollary


is much


easier


to verify


directly.


In the


following
Corollary


chapters,


(5.4)


will


we shall make


extensive


be especially useful


use of these


since


in this


results.


case


results


are easy


to formulate


'n an


explicit


form.


K2


= V,









CHAPTER


ACTION


OF THE RICCATI


GROUP--I


In this


chapter we


shall develop an approach


which


leads


to a


complete
Riccati


invariants


group


on certain


canonical forms


subsets


for the action


of the "optimization data


of the


space"


The re


suits


this chapter are


most


satisfactory for the


case


of equal


Kronecker


indi


ces.


In other


cases


the results


are


similar;


however


seems


to be difficult


formulate


these


results


in closed form due


notational and


which also


runs


computational


into


somewhat


complexity.
analogous


An alternative approach


difficulties


is developed


the next


in Chapter


chapter.


VI where


Some


of the results


we discuss


stem


of this chapter will


theoretic


be used


applications


interpretations


of the action


the Riccati


transformation


group.


This


chapter


is divided into


four


parts.


First we


scuss


finite-time


invariant


problems,


theory


next we
q-tuples


give


an expos


squar e


itory account


matrices


under


the orthogonal


group,


we then


the infinite


reconsider


h on


zon-zero


finite
final


time
state


problems,
problems.


and finally we


consider


Finite


Time


Problems-- I.


We recall


that


the Riccati


group


acts


on the data


space


((F,


where


= T-FT +


T-1GL,


= T-IGV,


+ F'M+


+ T'(S


+ MG)L +


i2 (S


+ MG) T


+ L'RL,


-- V'RV,


= (T'(S


+ MG)


+ L'R)V,


-'(


h
H(F,


MF) T









We start with


the following


(6.1)


PROPOSITION.


be ih


and let


ordered


Kronecker


indices


Then there exists a quadruple


in the Riccati


group


such


that


some


Further,


any two


six-tuples


belong


to the


same


orbit


if and


0)
only


if the ordered


(and hence


Kronecker


=F2


indices


there


= G


exists


are equal,

a triple


in the stabili


zer


in the feedback group


, G


such


that


(6.2)


'QIT
1-


'S'T
1


= (T )


1s


'RV
1


(6.4)


= (V


C) 'RIV


[Here


*G
,G )


is the dual


Kronecker


canonical


form


explained


Chapter


PROOF.


By the


techniques


of Chapter


there


exists


a triple


such


that


= T -T +


T-GL,


= T V.-1


Choose


M = P


Then


cP((F,


Po)


A
r S,


= (F


a A.


0,


... >


S)'R L


cp((F


+ (L









Suppose


there


exists


a quadruple


such


that


(P((*;




Then


-1 *
= TT y
i r1


-1 *
+ T GIL,
1'


-1 *
= T GIV.
1


Hence,
be the


dual

= G*


feedb


by feedback i
same as those


Kronecker
*


nvariance,


the Kronecker


Gj).


canonical form,


Hence


ack group


(T,
G
,G )


(Fi,


it follows


belongs


indices


Gl)
that


f (F
S*
(F2,

= F2


the stabili


2' G2
G2)
=: F


zer


must


in the


Further,


0 = T'(0


Since

where


+ M)T


nonsingular


, L )


= T'MT.


Hence


ongs


the stabili


, G )


Now the


Equations


6.4)


JI ow


immediately


from the


definition


Converse


Kronecker

Kronecker

Further,

feedback


hold,


then


indi


(F1,
since


ces,


canoni


if there


group


it fo


form


have


( F,


it follows


exists


such


flows


dire


ctly


that


from


that


in the stabili


Equations
definite


.2),


the
are
*


zer


same


ordered


in the dual


= G2


in the


(6.3),


(6.10


that


*k


, 0))


Hence,

to the


same


(F2'


belong


orbit.


This


proposition


shows


that


the problem of the


action


Riccati


S~~~~~~~~~~ jj*. *.II.f V~~AA- --


=X*
_


L,


=: P


---- -----


1


t 0),


L i,,


1


n ir










Thus,


it is


Kronecker

Kronecker


natural


indices.

indices.


to study this

The simplest


problem for various


among


these


cases


choices


is the


case


of equal


Let us define


a subset


of the optimization


problem data


space


:= ((F,


: the ordered


Kronecker


indices


are equal).


It is obvious


that


is stable


under


the action


of the


Riccati


group.


be in


such


that


Kronecker


indices


=1%


S= K


= n/m.


Propos


ition


there exists


such


that


cp((F,


-(F


Further,
feedback


we need to


group


consider


, G )


the action
on matrices


of the stabilizer


of the


nx n


matrix


and any


nx m


matrix


be the


mxm


matrices


such that


With


this


notation,


we now have


(6.5)


PROPOSITION.


such that


a -- --


are


/


t &,


K2


Po)


&,


1


II









Further,


two such


six tuples


with


equal Kronecker


indices


belong to


same


orbit


if and


only


if there


exist


orthogonal matrix


such


that


(6.6)


V'Q. i
QijV


2
= Qij ,


t~,


s2
~ k'


= 1,


PROOF.


Proposition


(6.1),


there


exists


a quadruple


such


that


cp((F,


mXin


nonsingular


matrix


such


that


V'RV


=1.


Such


a V


exists


since


a pos


itive


definite


symmetric


matrix.


Define


* CV,


Then


S..,i


it is


easy


see that


,o)


-(F


Suppose


there


exists


such


that


cp((F


, QI


= (F


, 2


, 0).


Then


Propos


ition


belongs


stabilizer


the feedba


group


G
, G)


Hence


L = O.


Further,


implies


that


ortho


onal.


Corollar


y (5.4),


=4'i


ag (V,


..., V)


Equations


(6.6)


follow


directly


from Equations


and (6.5).


Conversely


(6.6)


holds,


then


is any


it follows


mx m


directly


from


orthogonal


matrix


such


the definition


that
that


cp((F


I,


= (F


, Q


, 0).


where T


:= diag


This completes


the proof.


(F


L,


w m r


cp((F


0)


(6.1)


0))


0) ,


R_









reduced


to the action


of the orthogonal group


on matrices


given
well,
group
digre
under


(6.6).


the problem


on a


bunch


ssion and


It will


seen later


is essentially that


of matrices.


describe


action


In view


the invariant


that


in some


the action


of this


theory


other


cases


of the orthogonal


we make


q-tuples


a brief


matrices


orthogonal group.


The Orthogonal


Group


purpose


of this


section


to describe


certain


somewhat


recent


results


on the invariants


the action


the ortho


onal group


on a


space


SIBIRSKII


matrices

[1967],


Most


which


these


were


later


result
redis


are originally


covered


due to


an extended


form


PROCESS


relevant
involving


We will


our problems.


other


state


only those


For detail


classical groups,


proofs


see SIBIRISKII


results


which


and relat


[1967]


are


ed results


PROCESS


[1976].


be the


space


q-tuples


m X


matrices.


group


O(m)


mX m


(real)


ortho


nal matrices


acts


as g


iven


X: W


x 0(m)


A ),
q


i- (V'A V,


V'A2V,


The problem


is to find


invariants


the action


P1W)


denote


the ring
see that


real


R[W]


valued


polynomial functions


is isomorphic


to the


ring


It is


easy


of polynomials


indeterminate


with real


coeffi


clients.


be the subring


R[W]


consisting


of those


functions


which


remain


invariant


under


action


0(m)


14W)


belongs


and only


A ))
9.


= f((V'A V,
1


V'A V,
2


for all


0(m).


Fv rv rv ii] r:3C,1.---aa-4 n-. -a-a.


*


(A1'


f((n,,


1.,,,,,,,,.L?


n









function


tr f:


A')).
q


now have


the main


THEOREM.
The ring


SIBIRSKII


is ge


1967,


nerated


Lemma


as an


PROCESS


-algebra


Theorem


the elements


tr f


where


a monomial


in noncommutative


indeterminate


q-tuples


..., A)


with
and


deg

(B ,


- 1.


Further,


belong


same


orbit


if and


only


tr f ((A1,


= tr f


((BI,


for all monomials


in the above


mentioned


generating


set.


be all


the distinct


monomials


in the


noncommutative


variables


1' 2'


egree


- 1.


iven


q- tuple


= (A1,


..., A )


Tr A


:= (tr


Then,


light


above


theorem,


Tr A


a complete


invariant


for the action


O(m)


is known


group.


that


Thus,


affine vari
immediately


Theorem


ety.


that


gives


the orthog
the action


Theorem
ring o:


explicit


onal


group


a linearl

of FOGARTY


f invariant


formulae


a line


y re


early re


ductive


[196


duct ive


group


on an


it follows


is finitely


for a set


generated.


nerators.


now consider


O(m)


the problem of


a private


canonical


forms


communication,


for the action


Process


indicated


I' I SI -I.


tr f2(A) ,


tr fN(A)).


R:


t B,,


) i B


~~ tr(f(A1'


t fN


f,(n)


(7.1)


Ir I


.m


i I


r.


I I









subsets


the Euclidean


topology)


For any


integer


define


- (A


= (Al,


the (real)


eigenvalues


+ A.)
1


are distinct).


Now,


be in


an orthogonal matrix


such


that


V'(A.


+ A!)


= diag


*..,


Further


V'A V
1i


=


+ A.)


= diag


+ V'(A


Al)


+ a skew-symmetric


matrix.


We call


a matrix


diagonal matrix


quasi-skew-


(with


decrease


symmetric
entries)


is the


and a


sum


skew-symmetric


matrix.
matrix


Thus,


any
such


can be


that


quas


transformed


i-skew-


by an


orthogonal


symmetric.


be such


that


are quas


i-skew-s


ymmetric.


belong to


same


orbit


under


O(m),


then


there


exists


orthogonal matrix


such


that


particular,


V'A.V


= B.,


= 1,


V'A.V
2l


= B.


Since
parts


are quasi-


and
1


are


skew-symmetric,


equal.


Since


it follows


the diagonal


that


the diagonal


entries


distinct,


it then follows


from elementary


linear


algebra


that


= diag


- 1.


m


...>


n ,


are


~ t 1










Finally,


we note


that


each


a dense


open sub


set of


For,


belongs


if and


only if


the characteristic


polynomial


+ A.)
1


has no repeated root


Using resultants


it is


easy


show that


the complement


an algebraic


set of strictly


lower


pension.


Finite


Time


Problems--II


now resume


our discussion


of Section


was shown


Proposition


(6.5)


that


for the


case


equal


Kronecker


indices


, every


orbit


contains


a six-tuple


of the form


denote


((n/m)


+ (n/m))


-.=: q-tuple


consi


sting


m x m matrices


Qij'


ordered


in some


immediately from Proposition


fixed


Theorem


way.

(7.1)


It
that


now follows


Tr(Q,


forms


a complete


invariant.


More


preci


sely


we have


(8.1)
exist


THEOREM.


such


be ih


Let (
that


under

denot


acti


Tr (Q,


Then


is in the

the Riccati


same


there


orbit


group


Then


, S,


-* TR[(F,


a well
S(F1,
- 1' -.


defined function.


Further,


1'P
1> 01


belong


same


orbit


if and


two six-tuples


- (F2,


only


TR[dl]


2' $2'
= TR[d


PROOF.


The theorem follows


immediately


from


Propositions


(7.1).


We not


that


using


techniques


outlined at


the end


of the


previous


section,


we can obtain


canonical


forms


for the


action


of the


Riccati group.


T7. --!1-I-r -- 1 1I -1 _1


TR: D1


P o)


&t


N
R


- -


0)


(F,


-_


-


1-


r


II


.. -, L








the ordered Kronecker


:= ((F,


indices


K >
m -


In this


case


we shall


outline


a proce


dure that


leads


a canonical


form


(and hence


group


a complete


In orde:


set of invaria

r to motivate


for the


this procedure,


action


of the Riccati


we start


with


example.


= 1.


The dual


Kronecker


canonical form


, L)


belo


to the stabilizer


of the feedback group


, G )


and only


vii
V11
0
t


V11


1


#0,


v21,


arbitrary.


of the
lower


sition


Propo
form


triangular


we can


restrict


A
,Q,


matrix


our attention


given


such


to six-tuples


there


exists


that


) 'RV


Then,


= I.


transforms


are


... >


A
, Q,


8.3)















Then


under


action


transforms


1. I


where


(Ii)


'ST +


Let us consider


the transformation


that


undergoes:


(s.


denotes


ij-th


entry


31' s


-4 ~J~2


+ ts


-4 s2J~.,


-s


Choose

can be


Then


transformed


This shows


that any


such


element


that


Further


(with

there


=0)


, Q2


, I,
bel


same


orbit


if and


2,

only


exist


Tn other words.


)I with


a canonical


45


I,


+ (L


(T


+ (T


+ (L


+ t,


-S2


+1


,


1 i










Cons


equently,


modulo


the action


of such


signature


matric


the entries


in the above


form


constitute


a complete


of invariants.


In
brought


general,


into


we start


the form


noting that any

, Q, R, S, 0)


element


a suitable


can


element


the Riccati


group.


Here


SG)


denotes


the dual


Kronecker


canonical


form as


sociated


with


Kronecker


indices


K > Kg
1 2


***


K >
m -


Spec


ializing
v* *.
V L )


Theorem


belo


(5.5)


this


stabili


case


zer


it is


in th


easy


feedback


see that


group


,G)


if and


only


Lr],


0O,


= 1,


T. .,


itive


satisfy


definite,


their


the formulae
e exists an


in Theorem


upper


(5.3


triangular


Since


matrix


such


that


=1.


Then, with


in the stabilizer


e appropriate
*(T ,
, G ), T ,


such


that


transforms


A
, Q,


, 0)


Furthermore,


some


elements


, Q2


if there


, 0)


exists


belong

, L)


to the


same


orbit


in the stabilizer


under


if and


only


in the feedback group


~~iif nh th


) 'RV


*^
Ct'


erlFh









(6.5)


hold.


is lower


triangular,


this


implies


that


= dia


...,


- 1.


Let us consider


the action


such


elements


the feedback group


as given


Partition


- [S'
1'


si].
r"


Choose

action


It is


then


easy


to verify


that


Under


-Os.
1


+ Til) i


... + T'.S
ri


+ L!'
i'


=1,


AL
free.


be the set of indices


Then


we can recurs


ively


such that

solve for


the

L


ij-th entry

such that the


(ij)-th


entry


zero


start


with


It is


clear


that


can be chosen so that


the entries


correspond-


indices


are zero.


can


easily verified


that


this


choice


fixes


r(r-l)'


T(r-1)


T21.


Next we


consider


+ r(r-l)


Since


is fixed


we can


find


trivialize


certain


entries


which


corres


pond


indi


ces


Clearly,


this


procedure


Hence


that


can be recursively


we may assume


entries


complete


to obtain


that


corresponding


the desired


in such a


indices


form


are zero.


us denote


the set of such matrices


It is


straightforward,


though


tedious,


verify


that


p


I'


r-1


+ 1


r-2)


&,


r


IQi


/V


r


A










= diag


= diag


- 0,


we have


(8.1*)


=Q2


In other words,


with


a canonical


form modulo


the action


signature


matrices


as described


in (8.4).


now define


an equivalence


relation


pair s


matrices


two pairs


n x n symmetric


of matrices


, sl)
S1. S


matrices


with


2 in
in


are said


be sign


equivalent


if there


exist


such


that


Equations


(8.4)


hold.


Clearly,


sign


equivalence


an equivalence


relation.


denote


the equivalence


class of the pair


denote


the collection of


these equivalence


sses.


It is


obvious


that


each


consists


a finite


collection


of pairs


matrices.


can


now


state the main


(8.5)
Then


THEOREM.


there


exist matri


ces


with


same


orbit


SC

(F,


element


such


that


under


acti


of the Riccati


group.


Further,


the assignment


A [Q


qs: D2


a well defined


qs(dl)


function


= qs(


from


onto


Finally,


if and only


for any
belong


same


orbit


under


the Riccati Group.


The above


theorem


gives


a complete


set of invariants


in terms


- 11 fl .1 -I- 2--


C ),
m


f j,


=3-1


&r


Eai) '


(E1'


-t Qs:


" "


-- "


rr 11


17


*


L-










Thus


we see that


Theorems


(8.1)


(8.5)


solve


the problem of


invariant s


distinct


of these two
the action o


canonical


Kronecker


cases.


forms


for the


indices.


Analy


the orthogonal


group and


cases
neral
neral


of equal


case


case


the "affine


Kronecker


indices


a "combination"


requires
action"


combining


However,


seems


very


difficult


to formulate


any general


results


along


this direction.


entirely differ


In the next


ent approach


which


chapter,


also leads


we shall


consider


invariants


canonical


forms,


runs


into


analo


gous


diffi


culties.


Infinite


Time-Zero


Terminal


State


Case--


discus


the action


in Chapter


the Riccati


this problem arises


group


when


on quintuples


we consider


In this


section,


we restrict


our attention


the subset


data


space


which


corres


ponds


to optimization


problems


with


equa


Kronecker


luciA


ces.


have


not been able


to extend


techniques


this section


neral


case


obtain


invariant


ca nonica


forms.


the next


chapter,


we shall reexamine


this


blem


om an


entire


different


point


view which


leads


invari


ants


and canonical


forms.


Let us define


be the subset


.*- ((F,


: the


ordered


Kronecker


indices


are equal).


We start


choosing


a Hurwitz


X(s) = sr


polynomial


- als


- t2s


example


oose


-(S


+ 1)r


Given


a reachable


pair


with


Kronecker


indices


- n/m


there


st a


triple


Pc i i


group


such


that


T-1T


-+ TGL
+ T GL =


T-GV
T GV


-G


where


is given


-IX


&,





















r m


r-1 m


CaI'
1 i


=: F


+ GL.


It is


easy


verify


that


the chara


cteri


stic


polynomial


and hence


the eigenvalues


are in the left


half


plane


now have


a preliminary


(9.E1


LEMMA.


A triple


belongs


to the stabilizer


of the


feedback group


if and


= a nonsingular


mXm


only


matrix,


= diag


L = 0.


PROOF.


971


+ T-1GL =
+ TGL =


T-'GV


then


T-1T


T+ TG


*LT +


T-1G


L= F


+GL.


It follows


that


T~F


+ T-1
+ T G


+L


- VL ]


, L)
+LT
a


Corollary


SG )
G)


(5.4),
if and


belongs
-VL )


(T,
only


to the stabili


belongs


+LT
a


to the
-VL)


zer


if and


only


stabilizer


belongs


to the stabilizer


= an


mx m


nonsingular matrix,


F









However,
belongs


= diag


* ri


then


the stabilizer


LT =
cc


if and


VL .
a;


Thus,


only


= an


nonsingular


matrix,


= diag


The above


lemma


shows


that


abili


zer


the feedback


group


dual


Krone


cker


canonic


al form


, G )


same


as the stabilizer


of the feedback group


Returning


to the consideration


of the Riccati


group action


DS1,


we first


have


the following


(9.2)


PROPOSITION.


transformed


to a


quintuple


quintuple (]
of the form


can be


Further,


. belong
belong


to the


same


orbit


under


the Riccati


group


if and


only


if there


exists


orthogonal


matrix


such


that


V') sV


diag


PROOF.


Choose


a triple


such


that


-ITT


+ T-1GL =


-'cv


*C
=0


Sbe such


that


' Q,


Sl )


Since
matrix


1 is positive


such


definite,


that


there


= 1.


exists


mym


nonsingular


With


= diag


we have


_ S2


t &t


V~









some


Since


the eigenvalues


are in the left


half
there


plane,
exists


it is


well known


in the theory


n x n symmetric


matrix


of Liapunov


such


equation


that


that


i )'M +


= Qn.
- -


Then


4lr((F


= (F


,Q2,


, 0,


- Mi).


Combining


all the transformations


above,


there


exists


the Riccati


group


such


that


- (F


where


- MG.


This


proves


the first


part


of the


propo


sition.


Now,


, I,


belong


same


orbit


under


if and.


only


if there


exists


such


that


(9.3)


PT 1


T + T-1G


L=F


(9.4)


-1 *
T GV


(9.5)


T'((F


)'M +
S


MF )T +


T'(S1


+ MG)L


+ L'(S1


+ MG)'T


+ L'L=


(9.6)


+ MG)


+ L')


(9.7)


= I.


Equations


(9.5)


(9.4)


are equivalent


= diag (V,


*


,M))


JI( (F,









= 0.


Again,


since


it follows


the eigenvalues


from


the theory


are in the


apunov


equation


open
that


left


half


M= O.


plane,


Finally,


equivalent


north


nal.


Thus,


Equations


(9.3)


through


(9.7)


T'S


are equivalent


= S2


some


ortho


onal


matrix


= diag


..., V)


.1


ence,


belong


same


orbit


under


if and


only


if there


exists


an ortho


onal matrix


such


that


diag


V')sV


see that


the problem


of Ric


cati


group ac


tion on


i esse


ntially


reduces


to the action


the ortho


gonal group


on a


space


matrices.


can now apply


res


ults of S


section


t a complete


set of


invariants


canonical


forms


for the action


Given


matrix


we partition


sl].
r


denote


r- tuple


mx m


matrices


can now state


the following


(9.8)


THEOREM.


be in


Then


there


exists


nx m


matrix


such that


, R,
(F


, O,


same


orbit


under


action


of the


Riccati


group.


TRS[(F,


denote


Then


TRS:


= S2


-- [S,


I,


'M+


t "')


H TR










PROOF.


The theorem follows


immediately from Proposition


(9.2)


and (7.1).


We note


that


using


the techniques


outlined at


the end


Section


we can obtain
certain dense


canoni


open


forms


under the


ortho


onal


group


subs


The idea


of using


Liapunov


equation


generalizes


to finite


time


problems


as well.


We state


the result


A
=(F,


(9.9)


THEOREM.


be in


and let


an arbitrary


= (F,


Hurwitz


polynomial


egree


orbit


Then


under


there

such


exists


that


charact


eristic


polynomial


PROOF.


mXm


nonsingular


matrix


such


that


A
V'RV


=1.


By well known results


on state-feedback,


there


exists


m xn matrix


such


that


-GL)


= X(s).


ROSENBROCK


[1970]


for a proof


of this.)


It now follows


from the


definition


that


A


- (F,


It follows


from


the theory


of Liapunov


equation


that


there


exists


n symmetric


matrix


such


that


cp(d,


-(F,


some


For the


forms


case


of equal


invariants


Kronecker


following


the t


indict


technique s


we can


obtain


developed


canonical


in this


chapter.


Tn nava-tno1 4 a i-~ n4 rt~ ~' A r~ e n4 lnA 4 r OOnnn' flll-tol


=: d


fn nnnonal


kn~.tMtIIYI


I-<


Gin; I


Fnnmn


Cn nk n: nlrn


nnC fnn~ C








CHAPTER


ACTION


OF THE RICCATI


GROUP--II


In this


This


invariants
approach


chapter


, we shall

canonical


is based.


on the


develop an alternative


forms


asso


for the action


ciat


ed algebraic


approach


of the Riccati


Riccati


obtain-


group.


equation.


Under


symmetric

closed fo


cases
in the
feedba


assumption


solution,

rm results

equal and


neral


ck group


case


plays


that


this algebraic


we derive


Riccati


invariants


for invariants

strictly unequal


arise


a very


due to


equation has


canonical


canonical


Kronecker


the fact


critical role


that

- jus


indi


forms


ces.


forms.

only f


a real

We get


or the


The difficulties


the stabilizer

t as in Chapter


in the


This


stabilizer

it difficult


subgroup,


in general,


formulate


appears


neral results


very


complicated making


on invariants.


Some


results


which


are di


of this chapter


scusse


have


d in the next


direct

chapter


syst


theoretic


appli


cations


The Algebraic


Riccati


Equation Approach


start


y defining


certain


subsets


of the optimization


data


spaces


quintuple


recall


that


algebraic


(10.1)


Riccati


+ PF + Q


equation


(ARE)


- (PG


= 0.


-l~pG+


define


the ARE


with


has a real


(10. 1)


symmetric


associated


solution).


In an analogous


way,


we have


* ((F


the ARE


(10.1)


has a real


symmetric


solution).


-t Ir II 0


a* -


t S)


n 11


rm 1 ir


I I









(10.2)
belongs
Riccati


PROPOSITION.


if and


group


only


:= (F,


if there


exists


be ih


-- (T,


Then


in the


such that


cp(d,


-= (F,


, 0, Po
' o


some


such


that


the eigenvalues


are in the


closed left


half


plane.


PROOF.


symmetric


exists


solution.


a unique


belongs


Furth


solution


ermore


such


then


it i
that


known


the ARE


that


(10.1)
in this


a real


case


there


with


- 1(MG +
-RI(MG+


+ GL)


has eigenvalues


in the closed


left


half


plane.


(See


WILLEMS


[1971,


Theorem


Then


F' M+ ME'


+ MG)L


+ MG)'


+ L'RL,


+ F'M + MF


- (MG +


$146. +


and also


+ MG)


+ L'R


= 0.


Therefore,


with


T = I,


=1,


we have


cp(d,(T,


M)) =(F,


where


+ GL =


- GR-1MG +
-O R (MU+-


- M.


Finally,


was chosen


such


that


the roots


are in the closed


&,


L,


+ L'(


r Lt


R,


G,


& +









some


A
r, R,


Then


definition


we have


(10.3)


= T'(Q


+ F'M +


MF)T+


+ MG)L +


L'(S


+MG)


T + L'RL = 0,


(10.4)


Since


= (T'(S +


is nonsingular


+ L'R)V


, (10.4)


= 0.


implies


(10.5)


= -T'(


+ MG)R


Substituting


= T'(Q


into


(10.3),


+ F'M +


we have


+ MG)R


+ MG)'T


- T'(S


+ MG)R


+ MG)


T+T


+ MG)R


+ F'M +


- (S


+ MG)R


+ MG) ')T.


nonsingular,


it follows


that


a solution


to the ARE


(10.1).


Hence,


belongs


The above


group


Riccati


propos


interpretation


equation


ition may


regarded as


of the condition


be solvable.


We shall


that


the Riccati


assoc


consider


transformation


iated algebraic
consequences o


this


result


problems


on spec


tral


as developed


rece


eory
ntly


of linear-quadratic


JONCKHEERE


optimal


S ILVERMAN


control


[197


Thus


for each


in the orbit


such


there
that


exists
the ei


a sixtuple


values


are in


the closed left


of the


Riccati


half


plane


group on


Therefore
we may re


for the analysis


strict


our attention


the action
to elements


of the form


such


that


the roots


are in


the closed left


half


plane.


The conditions


under which any


two such


s ixtuDles


belong to


same


orbit


under


are


Given


in the following


=T'(


R,


Po)


h
~ (F,


(P (d,


(lo. 5)


MF)T


T'


+MG)









= (F2,


and
1 ----


same


nonsingular


be ih


are


orbit
matri


in the closed


under


x T


left


if and


and an


hand.


only


mx m


be such
plane.
if there


nonsingular


that
Then


the eigenvalues


belong


exists
matrix


such


that


T-'FT


(10.7)


-1
T G V


V'R V


= G2'

= R2


PROOF


(10.7) holds.


Suppose


there


It then follows


exist
from


nonsingular


matrices


the definition


such


that


that


cp((F1,


ol~'


= (F2,


02)


Conversely,


suppose


there


exists


= (T,


such


that


cP(d.1,


Then we


have


V'R V


Furthermore,


T' (FM +


MF1) T


+ T'MG L
1


+ L' GMT
1


+ L'R L =
1


+ L'R )V
1


= 0.


is nonsingular,


it follows


that


- 2'MG B


Consequently


T'(F{M +


- MGIR


'GtIYT =


is nonsingular,


satisfies


F'M +
1


- MG R
1


-l GM =


F~3'


d2'


(T'MG1









As the


e nvalue s


are the left


half


plane,


it follows


that


eige


value s


- GR


1
SG;M


are also in the closed left


half


plane.


Thus,


a real


symmetric


solution


to the ARE (10.1)


associate


with


such


the do


sed left


in the closed left


half
half


that


plane.
plane,


the ei


Since


values
the eig


G1


-1
G'M
1 "1


envalues


n x n matrix consi


are


are also


sting


zeroes


also


one such


solution.


However


, by


uniqueness


such


a solution


WILLEMS


[1971,


Theorem


we must


have


= 0.


Consequently,


Thus,


T- F1T
1


- F2,


-1
T G V
1


V'R V
1


= R2.


thus


see that


every


element


can be brought


into


the form


a suitable


such


a form is


unique up


state


element


space


the Riccati


transformations


group.
(GL(n))


Further,


the input


space


transformati


ons


(GL(m)).


We will


now show that


these


actions


are


intimately related


with


stabilizer


of the feedback group.


We state the result


in the following


(10.8)


PROPOSITION.


be a reachable


pair.


Then


there


nonsingular


matrices


such


that


T-1FT


- G L,


some


n matrix


Furthermore,


m x n matrices


there


exist


nonsingular


matrices


such


that


(10.9)


*-1F
V (F


*1) T
- G L1)T


- G L2,


) -1 GV
2 ) G V


if and


only


if there


exists


such


that


, L )


belongs


the stabili


zer


in the feedback


group


*G
,G)




*-l1 *
) (L1T


+ L*).


see


= G


*E


G2'


t R1'


T
ol


-1GV









T-FT


Defining


+ -1GL
+ T GL5


:= V-1L ,


j*


T1TT


= F*


T-1GV


we have


- G L,


T-1V
T GV


Equations


hold


(T ) )


if and.


*(V
G (V


only if


- LT )


*)- 1 *v*
') G V


Now,


(10.10) holds


stabilizer


holds


in the


if and only if


if and o
feedback


there


nly


group
exists


SV* L


V L )


LT )*


belongs


Equivalently,


to the


(10.9)


in the stabilizer


the feedback


group


such


that


*
- LLT


V L2


= L




* -1 *
) (TTT


*+ ).
+ L).


us now return


to the consideration of


the action of


the Riccati


group


Using the


above


propos


ition,


we now have


following


(10.11)
exist n
are in


THEOREM.


Let

, R,


natrices


the closed left


A
= (F,
such


o -
half


G, Q,
that


plane


A A


4-


the eige


and there


be in


Then


value s
exists
exists 7


such


there


- GL
that


S(FL,


, 0,


PO).
0


Further,


,' P


*
= (F
) in


-GL,
11'


such


0,
that


= (F2


eigenvalues


C- ~I' 3


- G2
j 2L


- GL1


*


(10.10)


= F


=*


cA


+ (T


..


r


"


L


L









- F2


1= G2


= G ),


there


exists


in the


stabilizer


the fe


edback group


, G )


such


that


(10.12)


(10.13)


)'RV
R V


) p


(10.14)


PROOF.
= (F,


that


position (10.2),


belongs


there


exist


P
under
under


to the orbit


such


Further


, by


Propo


sition(10.8) there


exist


nonsingular


matrices


such


that


T-TT


- G L,


T GVv


Hence,


-= (FL'


, 0,


where


= V'RV


= T'P T.
O


This


proves


the first


part


of the


theorem.


under


Propos
if and


ition(l0.6)


only


if there


exist


belong


nonsingular


to the


same


matrices


orbit


such


that


-1 *
T (F1


- G L)T
1 1


(10.15)


-1 *
T G V
1


T' PolT


V'R V
1


R2


Hence
(F -
Z


the order


2L:2


Kronecker


are


indices


same.


Hence


X GL
1 121
= F2 = F


Further,


Proposition


"I


n(10.8)


Equation (10.15) holds


if and


only


*c -X-


+ *),


-(LT


T
01


= G


= 2


L=: F


(V


(T


= G2


(P(d,









*) -1( *
= (v ) (L1T


S'RV


o2


T )'P


The above


theorem shows


that


problem of


Riccati


group action


reduced to


the action of


the stabilizer


of the feedback


group


on matrices


scuss


ed in Chapter


stabilizer


depends


only


on the Kronecker


indices.


This


leads


us to study the


problem for various


equal


Kronecker


indi


choices


ces


of Kronecker
the simplest


indices.


to analyze


Again,


case


as in Sections


Following


case


the techniques


strictly unequal


developed
Kronecker


in Section
indices.


we can also


In fact


treat


the results


here


closely parallel


results


of Sections


Before


analyzing


special


cases,


we have


following


COROLLARY.


(lo.i6
there


exist


h ..
-Ci, a


matrices


such


A


that


be in


Then
- F*
:= F


the eigenvalues


-GL


in the closed left


half


plane and


:= (FL,


belongs


to the orbit


PROOF.


Note


that


ordered Kronecker


indi


lower


belongs


triangular


to the


choose Q = S
L- such that


matrix


stabilizer


there

, G ).


- G L ,


the eigenvalues


are


exists

By Th


such


eorem


that


(10.11),


we may


for an appropriate


in the left


half plane.


choose


a lower


triangular


matrix


such


that


V ARV*
V 'RV


=1.


Then


, 0,


- (FL'


+ L ),


can


are


Po) ,


cp((~








L= (V


*) LT


Noting that


-1^ T
TT


= FL,


the result


follows.


We start with


case


of equal


Kronecker


indices.


:= ((F,


the Kronecker


indices


are equal).


Given any


m xn matrix


n xn matrix


be partitioned


L = [L


where


:= n/m.


Now,


Theorem (10.11) for


= (F1


DlR'


there


exist


matrices


such


that


= (F


- G L2,


, O,


belongs
matrix


to the orbit


such


that


V' R2V


under


be an


mXm


nonsingular


= I.


:= diag


Then


it is


easy


see that


p(d2,


= (F


, 0,


R;1


Le],









mRRd.2


:= Tr(P


where


denote


are following the


the collection


notation


developed


of matrices
in Chapter I


Sections


now state


main


(10.17)


THEOREM.


TRR:


~~N


Then


a well


defined function.


Further,


D1R'
same


TRR[ d]


orbit


= TR [d2]


if and


only


belong


under


PROOF.


the argument


just


before the theorem


there


exists


an element


under


= (F


under


of the form


- G L,
= (F
)' Po2)


Further,


G 0,
- G L ,


I,


if and only


if there


'It
0, I,
long


exists


every


the
the


orbit


same


orbit


stabilizer


the feedback


group


such


that


(10.18)


and (10.12) and (10.14) hold.


Since


the Kronecker


indices


*G
, G)


equal,


it follows


from Corollary


(10.18)


that


mx m


orthogonal matrix,


= 0,


= diag


thus,
only


see that


belong


to the


same


orbit


under


= (V )'L1T


w


can


*) 'V


are


v")


G Le,









where


mxm


orthogonal matrix and


= diag


V*).


p ,I


now follows


from


eorem


(7.1)


that


a well defined function


inR[a)


~ThR


if and


only


belong to


same


orbit


under


This


Esse


theorem


ntially,


very


similar


the problem


to Theorem


same:


(8.1)


analyze


of the previous


action


chapter.


of the ortho


onal


group


on a


space


of matrices.


case


strictly unequal


Kronecker


indices


can be treat


ed using


the techniques


developed


in Section


Intuitively


speaking,


plays


the role


plays


the role


Here


we state


the results


without


proofs


since


the proofs


follow the


pattern


Section


We start


defining


:= ((F,


the Kronecker


indices


... K


now have


(10.19)


THEOREM.


-(Ps


AQ A
, Q, R


-0


element


D2R.


Then


there


matri


ces


with


such


that


eigenvalues


-GL


are


in the closed left


half plane


= (F


-GL


, 0,


belo


the orbit


under


Riccati


group.


Further


assignment


A P
d we [P


a well


Ap(dl)
orbit


defined


Sp(d2)
under the


function.


if and
Riccati


only


Finally,


belong


to the


D2R~


same


group.


Here we


are


following


the notation


of Section


eore


m (10.19)


should


compared with


Theorem


(8.5)


Roughly


speaking,


in the approach


are


Qs :










Theorems


(8.5)


(10.19)


give


a complete


set of invariants


in terms


finite


collection


of matrices.


We conclude this


section with


case


of infinite


time-zero


terminal state


problems.


We start


with


(10.20)


there


THEOREM.


exists


mx n


the eigenvalue s


= (F1,


matrix
+ G L)


be in
matrix


and an


are in the closed left


DSR.


Then


such


that


half plane


d= (F

d = (
DS s
R -


+ G L,


belongs


to the


F1


uch


-that t
that t


;he eigenvalues


= (F
S2
- GL )
1


same


orbit


- G2L2'
and


Further,


)p 2,


- G L )
2 2


0)
are


in the closed left


half


plane,


belong


same


orbit


under


only


are


if th
same,


ordered


(hence,


exists


Kronecker


indices


, G


in the stabilizer


= G2
of the


there


feedback


group


SG )


such


that


)'R1V
1


= R2.
-" 2/


= (V


*1 -IT
) (LIT


*+
+ L).


PROOF.


Recall


as it does


observation,


proof


that


except
follows


the Riccati


that


from


group


the last


the proof


term


acts


is omitted.


With


same
this


of Theorem (10.11)


Again,


unequal


one can analyze


Kronecker


indices.


the special
The results


cases


are very


equal and strictly


similar


to those


obtained in


Chapter


Section


We omit


the details.


=: F









CHAPTER VI.


SYSTEM THEORETIC


APPLICATIONS


In this chapter,


we will


examine


some


of the system


theoretic


applications


interpretations


of the Riccati


transformation group


the techniques


developed


in the last


two chapters.


The main


application
optimization


of our results
problems, dev


is to the spectral
eloped recently by


theory of linear-quadratic
JONCKHEERE and SILVERMAN


[1978].


We will


show that


their results


can be considerably


extended


simplified


we will

infima


show


using


by using

infinite


the Riccati


the Riccati


time


transformation


group


free terminal


that


group.


the bounded


state


case


In particular,


ness


is equivalent


the positivity


Hilbert


space


nx n


operators which


mmetric


are


matrix.
central


We will


objects


show that


of study


JONCKHEERE


SILVERMAN


[197


are not invariant


under


Riccati


transformation group.


We will


examine


the behavior


of Popov's


frequency


function


under


the Riccati


transformation group.


We will also


show


that


sitive


real


transfer


functions


the Riccati


transformation


group are


closely related.


11. Spectral


Theory


of Linear


Quadratic


Optimal


Control


spec


tral


theory


linear


quadratic


optimal


control


problems


ently


been


developed


JONCKHEERE and


SILVERMAN


[1978]


relate the problems


of boundedness


of the


infima


to the spectra


of certain Hilbert


space


operators.


Here


we will


examine


these problems


the light


of Riccati


continuous-time versions


transformation
of the discret


group.
e-time


will analyze


problems


inve


stigated


JONCKHEERE and


SILVERMAN


1978]


Our techniques


and results


apply


equally well


to the discrete-time


problems.


We start
that we will


precisely


consider.


defining


the dynamic


Following WILLEMS


[1971


optimization
], we define


problems
the infinite


time
asso


free


ciated


terminal


with


state
= (F,


linear


quadratic


optimization


problem


as follows.


Infimize


rec









J (u(.))


-f o10


x' (t)


+ 2x'(t)Su(t)


+ u'(t)Ru(t) ]dt


+ x'(0) Pox(0),


subject


to the linear


dx(t)/dt


= Fx(


system

t) + Gu(t),


lim x(t)
t->--o


over


= 0,


u(.)


oo, O]


:= (f:


o, 0]


: f(t) I


square


integrable


o, 0]}.


Here
that


the infimum


the boundedness


taken


the infima


as the infimum of


of infinite


time


limits


It turns


free terminal


state problems


of critical


importance


in the exist


ence


solutions


the Riccati


differe


ntial


equations.


We state the rele


vant


results


the following


THEOREM.


= (F,


be in


Then


following


statement s


are


equivalent.


For the
associate


infinite
ed with


time


free


terminal


state


problem


Jr(u(.))


where e


the infimum is


taken


over


The Riccati


differential


equation associated


with


(11.5)


- dP(t)/dt


= F'P(t)


+ P(t)F


- (P(t)G


+ S)R


-l(p(t)G +


P(0)


Lm(.
in(


m
2-


S,


(11.


,R"










(ii i)


For all


the finite


time


linear


quadratic


optimization


problem


associate


with


has a solution.


For all


near


quadratic


the (


optimi


zero


initial


nation


state)


blem


assoc


finite


iated


time

with


a solution.


Sketch


follows


ANDERSON

between


esse


of {i 1.)


ntially


and MOORE


from

1971,


(iii)


"'>r UOiF'


V ertn


the class

Section

follows


cal r


results


(iii)


KALMAN


details.


rom


the reach


1960


Equivalence


ability


implication


-4(1)


follows


from a


rivial modifi


cation


result


of WILLEMS


1971,


eore


m 1]


now prove


finite


time


linear


- (iv) .


quadratic


optimi


Suppose

nation


that for


blem


some


associate


with


does


not have


a solution.


Then


there


exists


a sequence


Cu1 11j=~,


~i


such


that


lim J(u.
* ar 1n


- 0


Define


m


O, fo

(uit),


: u^(t)


Then


belongs


cc, o]


for all


Now with


x(t)


we have


Hence,


inf J (u(-))


- .CO


This


contradicts


The result,


thus,


follows.


.~~~-.~I .IA f I


-t


Jf(u('))


o0


F'


1 I


Lmt _
2


7(u(


._ J_ ^


mr r


n n


*


I


-










of this


section


the general


show that


it is


case


both
order


natural and.


necessary to


t a satisfactory


consider


theory.


In fact,


it turns


out that


contains


most


critical


information


for the boundedness


of infima.


The next


theorem shows


that


the existence


of solutions


to the various


dynamic
Riccati


optimization


problems


remains


invariant


under


the action


group.


(11.4)


TIIEOBEM.


= (F,


be in


= (F,


be in


in the Riccati


group


< 0.


Then a


solution


to the infinite


time


free terminal


state respectc


tively,


finite


time,


infinite


time


zero


terminal


state)


linear-quadratic


optimization


problem as


social


with


spec


tively,


exists


if and


only


a solution


to the


same


problem a


associated


with


in a,


(respectively,


P(d,


, 7))


exists.


terminal


PROOF.
state


It follows


linear


from


quadratic


Theorem (11.2) that


optimization


the infinite


problem a


associated


time
with


free


,P0)


has a solution


if and


only


if the associated Ri


ccati


differ


ntial


equation (11.3) has


a solution


for all


<0.


However,


P(t)
with


T' (P(t)


with


is the solution


then
- M)T


cp(cl,


differential


it is


to the Riccati


easy


check (by


a solution


Converse


equation


differential


direct


Riccati


P(t)


associate


equation associate


substitution)


differential


is the solution


ed with


the n


that


equation associated


the Riccati

) -l(t)T-


is the


solution


It now follows


the Riccati


from


differential


Theorem (11.2) that


equation as


the infinite


time


sociated


free


with


terminal


state problem as
problem associate


sociated
ed with


with

cp(d,


has a solution
has a solution.


if and


only


if the


same


The proofs


for the finite


time


problems


the infinite


time


zero


terminal


state


problems


are


similar.


I-nv 1- A.1..- -I... S <1 -1-.- ---A A--a --C


O' a


R,


O a(d,


CP d,


I- --


I


1.1. -.1 ~..


_ -









orbit


This


s imple


observation allows


us to


extend and


simplify


the results


of JONCKHEERE


SILVERMAN


1978]


as we will


demonstrate


in the following results.


any point


Then,


Theorem


(9.9)
such


of
that


:= F


Section


- (F,


+GL


it follows


that


'I,


is asymptotically


there
belong


stable.


exist
s to


Hence


matrices
the orbit


where


in the light


Theorems (11.2) and (ll.4), we


, I,


need.


where


only


asympt


consider


otically


points of
stable.


the form


(Note


that


we have


dropped


from


as the actual


canonical


form of


does
start


not play


with


optimization


an important


the infinite


problem


time


associate


role


in the following


free terminal


with


state


a six tuple


development.)


linear
in the


quadr
form


atic


asymptotically


stable.


Then


the associate


cost


criterion


(11.5)


= [012x
-m


(t)Su(t)


+ u' (t)Ru(t)]dt


subject


to the linear


x(t)


= Fx(t)


system

+ Gu(t),


lim x(t)
t-4--


= 0.


It is well


known


that


the solution


to the above


equation


is given


x(t)


F(t-to) (t
== e 'ux(t


t
+ fte
o


F(t-)Gu) i.
Gut ) dr.


is asymptotically


stable,


taking limit


-4 0i


we get


(11.6)


x(t)


Substituting (11.6)


into (11.5)


we get


=
=(F,


A
, Q,


S, Po)


+ x'(0)P x(0),


J,(u())


=ItF(t-I) Gu(r)dr.









where


(11.7)


Em(-


c, 0],


(11.8)


(11.9)


: L(-2


-emL
2


=1,


(11.10)


(pu) (t)


(1. 11)


(P2u)(t)


:= f w(t
-CO


- T)u(T)dT,


(11.12)


(p u) (t)


-F'tp eo
Po~e


Gu(T) dT,


w(t)


SI

* 0.1

0


FtG

-F' tS


> 0

<0

=- 0.


0, 0],


let us define


:= fu'(t)v(t)dt.


Then


(11.14)


in this


Jf(u)


notation,


=

the Hilbert


space


operator


defined above.


We note


that


+ P2


whereas


a Wiener-Hopf


Sis


type


a "Hankel like"


operator


(with


operator.


the "kernel"


Thus,


we have


R6(t)


a decompo


+ w(t)
sition


the sum of


a Wiener-Hopf


operator and a


"Hankel


like"


operator.


This


decomposition


corresponds


to the decomposition


obtained


-Lp(-


:= u(t),


m
2


'- P1


: = G'


~, ol,


(11. 13)









theoretic


significance


of these


decompositions


remains unclear.


case,
Hopf


it is clear


operator)


from


corresponds


the definitions


that


to the integral


the Wiener-


quadratic


criteria


whereas


Hankel like


matrix
natural


case,


tool


to "account"


operator)
the final


corresponds


to final


state weighing matrix


for the so called


"Hankel


state weighing


is the


perturbation"


(see


JONCIGIEERE:


and SILVERMAN


[197


now have


the following


simple


(11.15)


LEMMA.


a bounded linear


self


adjoint


operator.


PROOF.


self


adjoint.


Clearly,


Since


is a linear


asymptotic


ally


operator
stable,


a bounded


it follows


from


definition


that


there


exist


such


that


I w(t) U


It then follows


from GOHBERG and


KRE flN


1960,


page


that


bounded


operator.


asymptotic


ally


stable


we can


define


.- IIf~e
* -O


It is


easy


to check


that


Hence


is also
5


a bounded


operator.


Since


='


it follows


from


YOSIDA


[1964,


page


197]


that


self


adjoint.


Finally,


p u> =
5~


v'(t)G'


-F't t o
Po


'(T)G'e


-F' pofO
P^f .


our


, 0],


o


-FtGG'


-F' tdtJ(


t)


Ifl
L (-


-P 'G
Gut t) d~dt


-Ft
Gu(t)dtd.e









Hence


is self


adjoint.


Thus,


is the sum of bounded linear


self


adjoint


operators


Hence,


a bounded


linear


self


adjoint


operator.


now analyze


the boundedness


the infima


for infinite


time


free terminal


state linear


quadratic


optimization


problems


in terms


the Hilber t


space


operator


In particular,


we have


(11.16)


THEOREM.


= (F,


be in


such


that


is asymptotically


stable.


Then


the following


statement


are


equivalent.


The Riccati


differential


equation (11.5) associated


with


has a solution for


< 0.


The operator


associated


with


is positive


semi-


definite.


PROOF.


Note that


Jf(u)
f


, pu>


where


is a bounded


linear


self


adjoint


operator.


Hence,


standard re


suits


on th


spectral


eory


self adjoint


operators,


if and


only


is positive


semidefinite


The conclusion


follows


from


Theorem (11.2).


This


sets


along the


the sta


lines


ge for a detailed
of JONCKHEERE and


investigation


S ILVERMAN


on the spectrum


[1978,


1980].


will not


pursue


this


avenue


here


as that


our main


purpose.


fact,


a direct analysis


JONCKHEERE and


defining another


SILVERMAN


Hilbert


of the spectrum of


[1978,
space


1980]


overcome


operator whose


seems
these


spectrum


be difficult.


difficulties


differs


from


that


is not


Pf only
necessary


zero.
compute
itself.


However,


in the light


the spectrum of


It may


be easier to


of Theorem


ass


oc elated


(11.4),
with


check the positivity


somS,
some


other


point


on the


orbit


a matter


of fact,


turns


out to


be the


case.


be in


It follows


directly from


the results


of WTTIJLEM.


r1IQ7 .


Theorems


I. C *t


I that


it is


necessary


can


S,


h
(F,


h
P o)










to admit


a real


symmetric


solution


in order


that


the Riccati


differential


equation(11.5) associated


with


have


a solution for


< 0.


However,
solution,


if the algebraic


then


Riccati equation (11.17) has


we are in the setup


of Chapter


a real symmetric


In particular,


Corollary (10.16)


stable,


, O,
(but


implies that
) belongs


not be


there


exist


the orbit


asymptotically


matrices


such


stable).


that


such
F :


However,


that


+ GL


order


us e


stable.


which


give
this


the operator


We will


we need


first


the resulting


operator
section.


theoretic


restrict


turns


assume


our attention


out to


interpretations


that
n to


asymptoti
of these


asymptotically


those
cally


points
stable.


consideration


We will


later


= (F,


be in


such


that


asymptotically


stable.


It follows


directly


from our


previous


results


that


the operator


in this


case


(11.18)


=PL


where


are


as in(l.


and (11.12).


We will


now show that


itive


semidefinite


(resp


ectively,


definite),


if and


only


a certain
definite).


n symmetric
e main result


matrix


based


itive


on the


semidefinite


following


respectively,


simple


(11.19)


LEMMW.


be any two


Hilbert


spaces


a: H1


two bounded linear


operators.


be the identity


complex


number


1,


elongs


operators.
nectrum of


Then,


only


belon


to the


spec


trum of


+c43.


PROOF


- 1)


a complex


-1 Now,
Now,


number


suppose


in the spectrum of
does not belong t


the spectrum of


+c43a.


Then


-1)1


an invertible


operator.


It follows


that


is also


-A


invertible.


is easy


F


i H2


-~ HL


H1


+ ga


-, H1


" H2


-t Bcx.


- ag


GcrB









But this


implies


that


- 1) I2


is invertible.


This


contradicts


fact


is in


spec


trum


+ g2


Hence,


must


belong


ctrum of


czf3.


converse


follows


a symmetrical argument.


We now return


the operator


to the analysis


given


in (11.12)


of the operator
can be written


Notice


that


(11.20)


= P4P5 '


where


(11.21)


: u -Po0 e
Ic-0


-Ft
Gu(t)dt,


: Rn


(11.22)


i(
2(-


03 0


x p x,


x)(t)


:= G'e


-F't


We now apply


the result


of Lemma (11.19) to


check


the positivity


(11.25)


THEOREM.


= (F,


, 0,


be in


such


that


is asymptotically


stable.


Define


the matrix


-FtGGf


(11.21*)


Then,


the operator


PfK


associate


with


positive


semidefinit e


(resp


actively,


definite


if and


only


if the eil


values


+ P
0


are


nonnegative


(resp


actively,


itive).


PROOF.


Note


that


is the identity


operator


where


are


given


by (11.21),


(11.2


asymptotically


now apply


stable,


the result


are


of Lemma (11.19)


bounded linear
. It follows


operators.


that


(real)


can
number


belongs


the spectrum of


if and


only


if it belongs


spectrum of


-F't
dt.


n := 10


03


Ba


PqP









However,


-FtGG
eGG'e


(p5%) (x)


-.F't


= PoiX.


Consequently,


the spectrum


consi


of the eige


values


of the


matrix


+ P


modulo


Hence


sitive


semidefinite


(respectively,
are nonnegative


definite


respect


if and


tively,


only


if the


eigenvalues


positive).


Thus,
assoc


neces


we see that


ated.


sary and


with
suffi


in order


a six tuple


ent to


check
= (F,


check the


the positivity


the operator


it is


positivity


the eigenvalues


+ P T.
0


Even


though


are


symmetric


matrices


may not


a symmetric


matrix.


Therefore,


a prior


it is


not obvious


that


+ P7


real
itive


eige


values


definite.


. Note
Further,


that


reachabilit


we may write


+Po
0


+ Po).


Thus,
a pos


+ P7 1
0


itive


can be written as


definite


matrix.


a product


can now use


a symmetric


the classic


matrix


cal results


regular


pencils


GANTMACHER


[1959,


quadratic


Chapter


forms.


The reader


Section


referred


In particular,


we have


(11.25)


PROPOSITION.


n Xn real


symmetric


matrices


itive


nonnegati
symmetric


(resp


matrix


definite.


actively


+ Po)


Then,
itive'


eige


if and


positive


values


only


+ P T)


if the


semidefinite


n real


efinit


PROOF.


We can write


+ Po7)
+Pw)


+ Po7T)
+pw)*


o)


(n


(TT










it follows


that


from GANTMACHER


the eigenvalues


[1959,
+ Pw 7)


Chapter


Section


are nonnegative


Theorem 10]


(respectively,


positive)


if and


only


positive


semidefinite


(definite).


Note


that


+P~)


a real


symmetric


matrix.


now


summarize


some


of the main results


section


the following


(11.26)
Riccati
for all
a real


THEOREM.


differential


only


symmetric


equation(11.5) a


if the


solution.


assoc


Now,


ssoc


lated


suppose


-o)
lated.


algebraic


that


be in


with


Riccati


the algebraic


Then


as a solution

equation has


Riccati


equation


exist

orbit

then


associated


such


the Riccati


for all


with


that


is stable.


differ
if and

positive


ential


only


a rea


symmetric


= (F,


case


equation


if the


semidefinite


solution.
) belongs


asymptoti


associated with d
n real symmetric


cally


Then there


to the


stable,


has a solution


matrix


where


7T *=


-F'tt
r dt.


Some
Ao)


remarks


are in order


here


The trans


mainly r


formation


requires


A
, Q,


from


a real symmetric


solution


a pos
proce

there


itive

dure


algebraic


definite


find.


are several


Riccati


matrix.


solutions
efficient


equation


course,


to the


there


algebraic


algorithms


finding a

s no known


Riccati


actually


square
finite


equation.


compute


root


However,


the stabilizing


solution t

completely


the algebraic


clarify and


Riccati


equation.


extend the invest


case,


tion


started


our results

JONCKHEERE


S ILVERMAN


[1978].


We note


that


our condition


that


be asymptotically


stable


is sati


sfied


those points


generically
D for which


the data


the algebraic


space


Riccati


corresponding

equation has a


real


symmetric


solution.


Aq -


'I l~ a % -f .I.-- --- !-. -a r a -i4


(~, hG


'FtGG'


?,C ,,,, hLnt: nn


f\


-


,,, L,,1


L1,,,,, L,'


_.









control


KALMAN,


NARENDRA~


[1963].


We now proceed


to describe


these


considerations.


= (F,


be in


consider the


zero


initial


state)


finite


time


linear


quadratic


optimization


problem a


associated


with


= 0,


(for


< 0).


Minimize


u'(t)u(t)dt


subject


to the linear


system


dx(t)/dt


= Fx(t)


+ Gu(t),


= 0.


Clearly,
energy"'
state x


the first
whereas th


term


in the


e second t


erm


Now,


cost


criterion


corres


then


ponds


the solut


corresponds
a weighing
ion to this


to "control
on the final
problem,


obviously,
using the


then


results


on minimal


energy


control


above problem
as developed


can be


KALMAN


solved


, HO,


NAREITDRA


tJ196


Let us define,


for each


7(t)


-F-rGG


-F' d
drt.


Since


reachable


it follows


that


w(t)


itive


definite


for all
Consider


an arbitrary


the set


U(t,


of input


but fixed


functions


vector


Ll[t,


such


that
tran
time


with
sfers


Then


the state at time


system from
Corollary


the state


Section


zero


to the state


KALMAN,


and NARENDRA


implies


that


u' ()u(T)dT


can now state


the following


(11.27)


THEOREM.


be a reachable


pair.


Then


+ x'(0)P x(0),


= x wT(t)


J(u(


:(0)


w









lim r(t) -
t- -co


exists.


is asymptotically


stable


-1
= 7v


then


PROOF.


be any
define


vector


arbitrary


but fixed


Lm[t


->Rm


A
u(t)


evident


that


for each


A
and u

U(tl,


L(t)


xo)


= u(t)


U(t U( t2,


elongs


It

Xo)


follows


that


mrn
U(tin X
U(t2,xo) ^2


Now,


from the result


of KALMAN,


and NARENDRA


[1965,


Corollary


Section


quoted above,


we have


x'o(t
0


< xot(t )


above


inequality


holds


for all


Since


7(t)


positive


definite


for all


it follows


that


X'o(t)
0


-1
X
0


Thus,


X'Tn(t)-
0


a monotonically


increasing


function


bounded


below


Therefore


lim x'(Trt)
t-a--o0


exists


each


Consequently,


- I_ \ -l


now


U(tl,xo


U(t,,


u (( 't)U( T) dr


u'( Z)U( T)dT.









is asymptotically


stable,


then


by the


definition


lim w(t)
t.-*-oo


= 7> 0.


elementary results


of analysis,


we may


conclude


that


lim r( t) -
t->-co


=77


now ready to


state


our main


(11.28)
solution


"a


THEOREM.


to the Riccati


differential


equation


be in


exists


Then,


for all


only


if the algebraic


Riccati


equation


assoc


lated


with


solution.


Suppose


a solution.


belongs


that


algebraic


Then there
o the orbit


Riccati equation a


exist


such


Furth


ermore,


ssoc


that


the Ri


iated

= (F,


with


ccati


differential
if and only


equation


nxn


ass


ociated


real


with


mmetric


a solution


matrix


for all


itive


semi-


definite


PROOF.


(11.


. The exist


The first


part


ence


the theorem

such that


follows


from


Theorem


, I,


longs


the orbit


follows


from


Corollar


.= (F, G,
y (10.16)


Chapter


It now follows


from


Theorem (11.4) that


the Riccati


differential


equation


associated


with


has a solution


for all


if and


only


if the


Riccati


differential


equation


associate


ed with


has a solution for


Now,


asso


ciated


with


Theor


em (11.2),


has a solution


Riccati


differe


for all


ntial


only


equation
if for all


<0 ,


the (


zero


initial


state)


finite


time


linear


quadratic


optimi


zation


problem a


ssoc


iated with


a solution.


The cost


criteria


this


case


= fou


1('r)U()d'r


x'(0)Pox(0).


are


(F


t G,


t,


J(u()









inf
ueL*[t,0]


J(u(


jnfb
xc:Rn


x' (7(t)


Hence,


this


finite


time


optimization


problem has


a solution


if and


only


(T(t) -1


itive


semidefinite.


Theorem


(11.27)


implies


that


<)-1
or -- 1`


Thus


, again


y Theorem(11.27),


(r( t)


positive


semidefinite


for all


if and.


only


itive


semidefinite.


Conse


quently,


the Riccati


diff


erential


equation


assoc


lated


with


a solution
particular,


if and


only


asymptotic


ally


itive


stable


semidefinite


then


Note


that


Theorems (11.27)


and (11.28)


their


imply


the results


Theor


em(ll. 26).


Roughly


speaking,


we conclude


that


Riccati differential


equation
Riccati


decomposes
equation (o


into


parts


equivalently,


one part


Wie ner-Hopf


consi


operator


algebraic


the other


part


consists


minimum


energy


control


with


final


state


weighing.


The conditions


well
it is


understood.


quite


easy


solvability of
e, for example,
check positive


algebraic


WILIJEMS~


Riccati


[1971].


equation are


On the


semidefiniteness


other


hand,


nx n


symmetric


matrix.


This


can


done


for example,


y computing all


principal minors.


(11.29)
before


REMARK.


that


has a solution


the Riccati


= (F,
differ


for all


be in


ential


only


equation (11. )
if the algebraic


It was


noted


associated with
Riccati equation


associated
natural op


with


erator


has a real


theoretic


symmetric


interpretation.


solution.


This result


Here we will


describe


relevant
the lines


results without


of ,TJONCKHEREP


proofs.


The proofs


andf RTT.VEPRMAN


r o7


( I I


be constructed along


'.iiin.a the


standard


techniques


)x.
o-









where
is the


is asymptotically


sum of


stable.


Then,


iven


associated


by (11.10)


- (11.12).


a Wiener-Hopf


can be
class.


that


Therefore


that
Hopf


shown


order


rator.


that


Using


is a complete
5


prov


techniques
is contain


that


be positive


operators,


(for whi


itive


from


stability
operator


perturbation


ctrum
it is


semidefinite


semidefinite.


see GOHBERG


the theory


KBETIN


1960]


the trace


theory


necessary
of Wiener-


ROSENBLUM


the spectrum


+ p2)


given


values


"frequ


ency


function"


(taME


- F)-1G


- F')


-Is,


for all


Hence,


is positive


semidefinite


if and


only


for all


well known
equation as


equival


sociated


nonnega
ent to


with


tivit


the solvability


thus


see that


algebraic


order


that


itive


Riccati


semidefinite
equation have


it is


necess


that


assoc


a solution.


We close this


section


examining


the behavior


of Popov'


frequency


domain


function


+ S'(sI


-F)-1G


+ G'(-


- F')-S


- F)


associated


with


= (F,


under


the action


of the


Riccati


group.


be ih


:= (d,


operator


can then


spectrum of


the asymptotic


continuous


in the


for all


Riccati


iated algebraic


+ G'(-


- F') Q(


:= R


+ Pe)


+ Pe)


i P2)


~( W)


+ G'


~(W)


is (LU)


T,








b









(11.50)


=v(


-'


sl- F')-1L') (


s)(I


- L(sI


- F)-1G)V.


case,


the algebraic


Riccati


equation associated


with


has a real


symmetric


solution,


then


there


exist


matrices


such


that


-(F,


=1.


Equation (11.30)


which


shows


that


is well known


this corresponds


to a


spec


be closely related


tral


factor


to algebraic


zation
Riccati


equation.


12. Other


Applications


This section


relationship


mainly


between


intended


Riccati


tran


o show that
formation


there


a close


group and


itive


real matrices.


show that


so called


"positive


real


lemma"


rederiv


obtain a
Finally,
Riccati


ed from


complete
we make


differential


the Riccati


transformation


parametrization


some remarks


on the


equation


scalar


fast
CASTI


gr oup


point
itive


algorithms


tJ-97h23


view.


real


We also


functions.


for solvin


KAILA2JI


[19731


z(s)


Z(s)


said to


mx m
sitive


matrix


real rational


functions.


Then


real


elements


z(s)


are analytic


in the region


Re(s)


of the complex


plane,


the matrix


itive


semidefinite


for all


complex


numbers


such


that


where


the complex


conjugate


transpose


Z(s).


The property


of positive re


alness


turns


out to be


critical


importance


network


sive network


theoretic aspects


synthesis.


positive


NEWCOMB


real


functions.)


1966]
Using


for the
certain


preliminary


synthesis


steps,


(see


ANDERSON


VONGPANILTERD


+ z(


"i1(9)


t 0)


"acs,










z(oo)


:= lim Z(s)
S~-- O


transfer


function.
analytic


exists.


Further


In other words,


z(cm)


+ z(co)


a proper


is nonsingular.


Since


assumptions


rational,


are easily


using


seen


elementary r
e equivalent


ealization


theory


existence


the above
of matrices


of appropriate


sizes


such


that


+ H(sI


where


ca non


nonsingular


asymptomati


call


stable.


In the


light of these


iderations


we may


restate the problem


as f


follows.


nx


nx m


mx m


real


matrices


respect


vely


canonical,


asympt


otically


stable


nons


ingular.


Find


conditions


such that


(12.1)


:= J


+ J'


+ H(sl


- +
G +


c'(s


I-F')


>0,


for all


the complex


plane


such


that


It is well


known


that
half


in thi
plane


case


itive


semidefiniteness


the cl


osed right


repla


(12.2)


>0,


for all


(12.1)


hold


Then
defi"


t ha t


a necessary


ujte


However


condition


notice


that


the P


. Hence,


opoy


frequency


a resu


It diue


function


assoc


to WTLLEM ;


ated.


1971,


with


eorem


we see


that


is pos


itive


real


only


itive


definite


the algebraic


c. ati equation


associated


with


a real


symnie


trick


solution.


now we


are


in the


up of


Chapter


In particular,


proo


Proposition(10.2)


implies


that


algebraic


a real


Riccati


symmetric


equation


solution


associate


if and


with


if there


= (F,


exists


1,


, H')


, H')


=I


R,


iw)









for
half


some


plane.


such


Rewriting


that

(12.


the eigenvalues


explicitly,


are in the left


we get


CJ2A4)


(12.5)


(12.6)


V'RV


F'M +


+ J')V


MF = L'RL,


iG = -L'R,


(12.7)


+ GL,


Equations


(12.4)


- (12.7)


are


easily


seen


to be equivalent


to the


called


"P05


itive


real


and ANDERSON


lemma"

1967].


equations


This


shows


due to YAKUBOVIC


that


the positive


[1962],


real


KALMAN
lemma


is closely


related


the Riccati


transformation


group.


now consider


spec


case


of scalar


positive


real functions.


In this


case


be normal


1/2.


Thus,


(12.8)


= 1/2


+ h(sI


- F)-1


Further,

control


without


canonical


loss


of generality


be taken


to be in


form:


(12.9)


Then


the characteristic


polynomial


(12.10)


~s)


:= det (sI


-F)


=3


- a'


now consider


the following: problem.


. ea


I,


+ M


a,, ,,,, a


U~ U


V'(J


* I









such


matrice
Choose


that


n vector


are in the (close


left


given


such


half


plane


(12.8)


that


is positive


values


be the unique


real.


+ gt)


solution


to the Liapunov


equation


(12.11)


F'M +


t vi.


stence


uniqueness or
M F'T
M = f e e


follows


from asymptotic


stability


FTt.)


fact,


define


(12.12)


It is


easy


to verify


that


= 1/2


+ h/(sI


a positive


real


function.


can now state


the following


(12.15)


THEOREM.
defined


fixed


by (12.9),


be asymptotically


real


numbers,


stable.


Define


the set


* -


*(


= 1/2


+ h(sI


is positive


real).


be the set


the eigenvalues


are


in the


(cbos


left


half


plane).


Then


-4:


a-H*1>


a bijection.


-t gR


Ixn
R










PROOF.


From


the arguments


iven above,


it is clear


that


well


defined.


Suppose


belongs


Then


there


exist


matrices


such


that


F'M +


IvF = 2%,


the eigenvalues


are in the closed left


half


plane.


Hence


belongs


This


proves


that


surjective.


suppose


that


there


exist


such


that


h (29


hA
-- h(


= h.


The n


there


exist


symmetric


matrices


such


that


- 'Mi


=- i,


F'M.


+ M.F
1


= V2
iei,


= i,


Hence,


are solutions


to the algebraic


Riccati


equation


F'P +


+ h')


= 0,


such that


the eigenvalues


both


are in the


closed right


half


plane.


However


, by WILLEMS


1971,


eorem


such


a solution


is unique,


and,


ence,


. This


proves


that


ective.


This


note


that


theorem


completely answers


the main step


in obtaining a


the question rais


formula


ed above.
involves


solution


of the


(linear)


Liapunov


equation (12.11).


Using


Kronecker


e2


he'












closed


Combining this
form expression


formula


with (12.12), we


can


as a subset


obtain a
is the set


such


that


the eigenvalues


+ gi


are in the closed


left


half


plane.


Using the


classical


stability theory and Hurwitz


determinants,


can be described


by a


system of


polynomial


inequalities.


(See,
such


also


that


KAL~MAN


(h2,


[1979].)


We note


that


there


is not observable


exist


one such


choice.


Define


is observable).


:= (2


Then,


- g'M,


belongs


F)
some


if and


observable.


suitable


belongs


only
Thus,


polynomial


be describe


ed as


a subset


inequalities.


We close this section with


a remark


on the fast


algorithms


solving

[1974].


differential


Riccati


differential


equation


= (F,


equation (11.3) associated


due to KAILATH


be in
with


[197


Consider
The main


CASTI


the Riccati
observation


KAILATH


[1973 ]


CASTI


[1974]


is that


r := rank


= rank


+ P F + Q
o


- (PoG +
0


then


the Riccati


nonlinear


differe


differ
ntial


ential


equations.


equation may


be replaced


In particular,


by a


set of


much


smaller
It turns
Riccati
further


than


out that


tran


then
this


formation


reduction


this
rank


group.


Riccati


eads


to a


considerable


invariant


This


shows


equation along


that


under


saving


of computation.


action


one cannot


the lines


of the


achieve


of KAILATH


t 197 3)


CASTI


[1974]


using


Riccati


transformation


group.


However,


completely


satisfactory


explanation


of this


reduction


still remains


elusive.









REFEENCES


B. D.


O. ANDERSON


1967]


"A system


theory


criterion for


positive


real matrices"


SIAM J.


Control,


171-182.


B. D


. O. ANDERSON


B. MOORE


[1971]


Linear


Optimal


Control,


Prentice-Hall,


Englewood


Cliffs,


B. D


0. ANDERSON


S. VONGPANITLERD


[1975]


Network Analysis


Synthesis


, Prentice-Hall,


Englewood


Cliffs,


P. BERNHARD and G.


COHEN


[197


"Etude


d'une


problem
reduction


function
command


de la t


quentielle


optimal


avec


int
une


ce problem"


ervenant dans
application a
Revue RAIRO,


July


R. W.


pages


BROCKETT


CJ1970)


Finite


Dimensional Linear


Systems,


Wiley,


York.


[1977]


"The


geometry


Research


reports


the set


contr


of the Automatic


ollable
Control


linear


systems"


Laboratory,


Nagoya


Universe it


1-7.


1970]


"A classification


of linear


controllable


systems"


Kybernatika,


187.


BYRNES


and M.


GAUGER


1977]


"Decidabilit
applications
Advances in


criteria


the moduli


Mathematics,


for the similarity problem,


of linear


with


dynamical systems"


9-90.


CANDY


1975]


"Reali


zation


of invariant


system descriptions


from Markov


SI~~~~~ SS


P. BRUNOVSKY


.. i Il


n clI


n~


II









CASTI


[1974]


"Matrix


Riccati


generalized


equations,
functions


dimensionality reductions,


Utilitas


Math.,


95-110.


FOGARTY


1965]


Invaria nt


Theory,


Benjamin,


New York.


P. A.


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BIOGRAPHICAL


SKETCH


Pramod


Prabhakar


KHARGONEKAR was


born


on August


,19


in Indore,


INDIA,
Bachej&:


to Prabhakar KH

r of Technology


ARGONEKAR


degree


and Leela K
electrical


HARGONEKAR.

engineering


He obtained


from


the Indian


Institute


of Technology,


Bombay


in 1977.


He obtained


his Master


Science


degree


in mathematics


in 1980


from


the University


of Florida.











certify that


conforms
adequate,


I have


to acceptable


in scope


read


standards
quality,


this


study and
scholarly


that


prese


as a dissertation


in my


opinion


nation and


fully


the degree


Doctor


of Philosophy.


Dr. R.


Graduate
Electri


E. Kalman,


Rese


arch


Chairman
Professor


cal Engineering


certify that


conforms
adequate,


to acceptable


scope


have


read


standards
quality,


study an
scholarly


as a di


sser


that


pres


station


in my


entation
for the


opinion


and


fully


gree


Doctor


of Philosophy.


Dr. T. E


Bullock,


Co-Chairman


Prof


essor


of Electrical Engineering


certify that


have


read


this


study


that


in my


opinion


conforms
adequate,


to acceptable


in scope


standards
quality,


of scholarly pre
as a dissertation


sentation


fully


for the degree


Doctor


of Philosophy.


Dr. R; L. Long
Assistant Professor


of Math


tics


certify that


conforms
adequate,


to acceptable
in scope and


have read
standards


quality,


this study an
of scholarly


.that
prese


as a dissertation


my opinion


nation and


fully


for the degree


Doctor


of Philosophy.


' 4


4- Arl


me. v~


R&


~La/~L










certify that


conforms
adequate,


to acceptable


scope


have read
standards
quality,


this


study


that


of scholarly pre
as a dissertation


se


in my
nation


opinion


for the degree


fully
of


Doctor


of Philosophy.


C/ tfL2


V. Shaffer


Prof


essor


of Electrical Engineering


This


dissertation


of Engineering and


was submitted


was


accepted


to the Graduate


as partial


Faculty


fulfillment


of the College
of the requirements


for the degree


of Doctor


of Philosophy


August,


1981.


Dean,


Dean,


College


Graduate


of Engineering


School


I
,~kk/


e~c-~






















































UNIVERSITY OF FLORIDA


1262 08394 207 7