Continuous Time and Discrete Time Lyapunov Equations: Review and New Directions

CONTINUOUS

TIME

EQUATIONS

AND :

DISCRETE

REVIEW

TIME

AND

NEW

Ahmad J.A. De

Department

I .

LYAPUNOV DIRECTIONS

A. Mohammad Abreu-Garcia

of Electrical Engineering The University of Akron Akron, OH 44325-3904

INTRODUCTION

Since theory

its

reintroduction

continues

and design

in the

to play a vital

of control

systems.

1950's,

Lyapunov's

role

in the analysis

This

role

is evident

through a large number of direct and indirect applications in controls. Originally

Lyapunov

to test the stability Lyapunov functions tant tools tems, vide

the

eling.

Lyapunov for

functions

systems.

However,

and analysis

equations.

stability,

system's

Lyapunov

have lead to one of the most impor-

Not

but

properties

In particular,

robustness,

of dynamical

in the design

a test

dynamical

introduced

they

of control

only

sys-

do they pro-

also

define

most

relevant

to system mod-

controllability,

observability,

and optimality can be extracted from these

equations. CONTROL AND DYNAMIC SYSTEMS, VOL. 74 Copyright 9 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

253

254

AHMAD A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

This

chapter

Lyapunov's on

some

system

presents

theory

new

and

and

its

novel

modeling.

is n o t e d

In

nique

has

that

senting

this

example

of the

Section

II

Lyapunov's

theory,

theorems

tion

also

sents

This

both

into

brief

to

these

intuitive

major

analytical

interpretation

of

jor a p p l i c a t i o n s

of the L y a p u n o v equations.

equations

and

an example.

presents The

a continuization

in Sections VI and VIII

II .

HISTORICAL

were

first

amount

1950's,

for

solv-

Section V of these

technique

references

ma-

as

are

given

Lyapunov

func-

respectively.

from their name,

introduced

work on the s t a b i l i t y late

these

BACKGROUND

As can be inferred tions

and

sec-

IV summarizes

for the application

conclusion

of

III which pre-

methods

Section

to

impor-

This

equations.

out new d i r e c t i o n s

and

equations.

and n u m e r i c a l

good

review

ing the L y a p u n o v

points

a

sections.

equations,

is. f o l l o w e d by Section

pre-

followed

historical

Lyapunov

tech-

that

gives

can be

four

the

an

that

divided a

technique.

of this

chapter

theory.

related

gives

equations.

this

in

equations

It is felt

of Lyapunov's

gives

tant

in

of

emphasis theory

Lyapunov

version

[1,2].

new d i r e c t i o n s

is

this

a new c o n t i n u i z a t i o n

technique

chapter

review

with

of

particular,

in

take full advantage This

applications

a preliminary

appeared

historical

applications

are u t i l i z e d to devise It

a brief

they

by Lyapunov

of dynamical

have

of investigation.

been

in 1892

systems.

under

a

in his

Since the

considerable

It should be noted that much

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

of

the

results

the 1960's

in

[3-14,

Developments ers,

optimal

Riccati

Equation

[19,20],

control

dynamical

control

by

Moore

problem

control

important Lyapunov

results

in this

area

systems,

this

and

work

only the

addressed

THE

ORIGIN

OF

EQUATIONS

Consider =

Ax,

and define v(x(t))

:

THE

x(0)

=

respect

first

H-infinity

to

the

robust

of

case

The

the most Although

linear

is relevant

interested

Then

to time yields

to

reader

for more details.

FUNCTIONS

(zero input)

system

[23,24]

(I)

v(x(t))

as (2)

x T(t)Px(t),

dimensions.

and

LYAPUNOV

function

where P is any symmetric proper

linear

others

x0,

a Lyapunov =

the

for both

LYAPUNOV

the autonomous

and

[22] .

references

THE

geometrical

summary

theory

oth-

Algebraic

is presented.

here.

is referred to the above

to

problems

the

in

direct

reduction

extensions

a brief

developed

of

order

follows

has

the

by W o n h a m

which

nonlinear

A.

study

solution

its

among

Lyapunov's

through

in model

and model m a t c h i n g

In that

include,

systems

with

developed

therein] .

the

[21],

been

through

design

[15-18],

applications

introduced

have

applications

optimization

method,

of

field

and references and

parameter

properties

this

255

positive

definite

differentiating

matrix Eq.

with

(2) with

256

AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA

v(x(t))

=

xT(t)P~(t)

and substituting Eq. (x(t))

=

(I) into Eq.

x T(t) [PA

According

to

+ iT(t)Px(t) (3) gives

+ ATp]x(t) .

Lyapunov's

17,22,23,25,

and references

be

definite

negative

(3)

for

(4)

stability therein],

system

theory

~}(x(t))

(I)

to

[15has

be

to

stable.

This implies that PA

where

+ ATp

Q is

trix. or the

=

-Q

some

Lyapunov's

of the

in the analysis be

servability

positive

semi-definite

first

of the d i f f e r e n t i a l

the dynamics

will

(5)

is referred to as Lyapunov's

method.

solution

0,

symmetric

This m e t h o d

second

As

<

method

equations

system which makes

and design of control seen

later,

Eq.

(5)

LYAPUNOV

EQUATIONS

was

independence

Lj (t) .

requires

describing

systems.

reduces

to the

ob-

Lyapunov equation upon the choice Q = cTc,

The Grammian ear

direct

it impractical

where C is the output matrix of the dynamic B.

ma-

AS

originally of

Essentially,

linearly independent

any L i (t)

two

system.

GRAMMIANS

used to study the functions

and Lj (t)

if their Grammian,

are

lin-

L i(~) said

and to be

defined as

tf

Go (Li, Lj)

=

is non-singular.

i

Li (t-~) Lj (~) d~, 0

(6)

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

Similarly, Grammians =

the

controllability

of the d y n a m i c

Ax

and

257

observability

system

+ Bu

(7a)

y = Cx + Du,

(7b)

can be w r i t t e n

as

[26]

t

Wc (t, to)

i

=

e A(t-~) B B T e AT(t-~) d~,

(8)

e AT(t-~) C T C e A (t-~) d~.

(9)

0

Wo (t0, t ) = 0

It is e a s y to show that then as to Lyapunov AWc

and

0 and

=

+ WoA--

are

equations

test

(8) - (9) s a t i s f y

the

BBT ,

(i0)

_

cTc,

(11)

termed

respectively

play

Lyapunov

an

of c o n t r o l

systems.

for stability,

they

as

the

controllability

equations.

important

and c o n t r o l l a b i l i t y

C.

--) oo, Eqs.

q

-

observability

design

t

matrix

equations

+ WcAT

ATWo

which

-~

if A is a s t a b i l i t y

role

in

The the

In a d d i t i o n

also d e f i n e

latter

two

analysis

and

to p r o v i d i n g

a

the o b s e r v a b i l i t y

as can be seen next.

CONTROLLABILITY,

OBS ERVAB

I L I TY,

AND

MINIMALITY

In

this

section,

controllability

and

basic

definitions

observability,

and

and hence

tests

for

minimal-

258

AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA

ity,

via L y a p u n o v

play

a central

equations

role

are given.

in the

modeling

These and

concepts

design

prob-

lem.

Definitionat

t=to,

which

if

will

origin.

System one

can

pletely

state

the

is true

states,

then

it

then

if

controllability

A W c + Wc AT

t=t o

-

-BBT

tial

System

where

times

can

the

Test-

said

to

be

com-

If

system

controllable the

(7)

if and

unique

is only

solution

(12)

be

is

and

all

said

determined and

u(t)

t0
to

states

be

from

over

If this

initial

Test"

it is c o m p l e t e l y Grammian,

observable the

output

a period

is true then

state o b s e r v a b l e

Grammian

observability

Lyapunov

is

and all

[23,27].

Grammian,

(7)

y(t)

Observability then

(7)

times

the

q

said to be c o m p l e t e l y

stable

x(t o ) to

[23,24] .

functions

[t0,tl],

state

for all initial

u(t)

equation

if x(t o)

input

controllable

initial

is c o m p l e t e l y

is n o n - s i n g u l a r

Definition-

be

function

system

stable

to

input

Grammian

to the L y a p u n o v

said an

controllable

Controllability

the

is

construct

transfer

If this

initial

(7)

If

observable the unique

of

at and

time

for all ini-

system

(7)

is

[23]. system

(7)

if and only solution

is if

to the

equation

ATw O + WoA

is n o n - s i n g u l a r

=

-cTc,

[23,24].

(13)

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

System

Definition-

reducible number

if

the

needed

(7)

is

number

to

said

of

realize

to be m i n i m a l

states

the

259

n

is

the

corresponding

or

ir-

minimum transfer

function. It s h o u l d be n o t e d that, output zero not

(SISO)

case,

(MIMO) given

clear

in

case.

input

a s y s t e m will be reducible

cancellations as

in the single

occur.

the

However

multiple

A test

the

input

for m i n i m a l i t y

single if pole

situation

multiple for both

in the f o l l o w i n g t h e o r e m which

is

output

cases

is

is s t a t e d without

a proof. Theorem-

System

ducible

if

it

(7)

is

is

both

said

to

be

minimal

controllable

and

or

irre-

observable

[23,24] . Methods a given fer

for

reducible

function

rely

obtaining

on

are

available the

of the system.

Lyapunov

Stability

and

if

trix Q there

Test:

each

exists

+

PA

D .

INTERPRETATION

unobservable

(7)

is

stable

a b i l i t y concepts

if

symmetric

ma-

a positive

definite

symmetric

ma-

[23,24]

=-Q.

importance

all

definite

(14)

AND

CONTROLLABILITY

The

and

They

positive

trix P which satisfies ATp

System

for

from the trans-

[20,23,24,28].

controllable

subspaces

for

realizations

one or even d i r e c t l y

isolating

only

irreducible

of

IMPORTANCE AND

the

OF

OBSERVABILITY

controllability

introduced earlier

in this

and

observ-

chapter

is

260

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

evident

in state

a powerful given tion

feedback.

technique

system.

to

In this

available

on the

the

system.

system

is m o d i f i e d

tions. the

In

If

the

states

If the

about

all

strategy,

is

the

system

the

the

is m a n i p u input of

the

specifica-

observable,

then

could be d e d u c e d

available

is

informa-

behavior

some d e s i r e d

states

of

the control

completely

and m a d e

feedback

response

of the

achieve

system

the

manner,

is

observer

[24].

this to

state

that m o d i f i e s

system

information

a state

reshape

control

lated via a c o n t r o l l e r to

Briefly,

via

to the c o n t r o l l e r

completely

controllable,

one

is able to design

a controller

that a r b i t r a r i l y places

the poles

system,

shaping

sponse

of

ability

peated

thus

the

system

re-

[24] .

Theorems

tions

the

on stability,

using

the

have been here.

Lyapunov

addressed

However,

the p h y s i c a l

controllability,

this,

suppose

that

state

x(0)=a,

the

functions/equations

earlier

and will

it is w o r t h w h i l e

interpretation system time

and observ-

not

is

response

SISO of

be

re-

investigating

of these equations. (I)

equa-

with

this

To do initial

system

is

given by h(tf)

=

eA(tf-to)a.

This means

(~5)

that the c o n t r o l l a b i l i t y G r a m m i a n

integral of hTh,

i.e

it

is a m e a s u r e

is the

of the e n e r g y

of

the time response. Another the impulse

interpretation

is

obtained

response of the system

by

considering

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

=

Ax

+

Bu,

x(0)

The impulse

response

h(tf)

eA(tf)B.

Thus

=

the

=

(16)

0.

of this

s y s t e m is given by (17)

controllability

Grammian

sents the e n e r g y of the impulse A

more

261

interesting

in this

repre-

response.

interpretation

c o n s i d e r i n g the f o l l o w i n g

case

is

obtained

by

o p t i m i z a t i o n problem:

Given the s y s t e m =

Ax

+

Bu,

it is d e s i r e d

x(0)

to d e s i g n

e n e r g y that d r i v e s final time t f > It has b e e n and o n l y (8)

=

if the

the

(~8)

x0,

a control states

input u with m i n i m u m

to the o r i g i n

at a g i v e n

to. shown

that

the

solution

controllability

is n o n - s i n g u l a r

[26] .

The

is p o s s i b l e

Grammian optimal

given

control

if

by Eq. law

is

given by

U.

where

-- B T e A T (t-t~

=

star d e n o t e s

demonstrates Grammian

The

the optimal

the n e c e s s i t y

to be invertible,

condition

(19)

(to, tf) x0,

control.

(19)

for the c o n t r o l l a b i l i t y i.e of full rank as a

for c o n t r o l l a b i l i t y .

observability

considering

the

Grammian

same a u t o n o m o u s

can

be

= Cx(t).

interpreted

s y s t e m with

taken as y(t)

Equation

(20)

by

the output

262

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

Now,

with

knowledge

interval state input Eq.

[t0, tf],

x 0.

of it

Without

the

is loss

of

find

Solving

into Eq.

over

both

the

the

initial

the

control

for x(t)

from

(20) gives

C e A(t-t0) x0,

multiplying sults

to

output

generality,

zero.

(18) and s u b s t i t u t i n g =

and

desired

u can be c o n s i d e r e d

y(t)

input

(21)

sides

of

Eq.

(21)

by

eAT(t-t~

T

re-

in __ e A T (t-to)c T C e A (t-to) X0,

e AT(t-to) CTy (t)

integrating [to,tf],

Eq.

both

sides

of

Eq.

(22)

(22)

over

the

interval

(22) becomes

itf 0

eAT(z-t~ CTy (~) dZ

=

itf 0

eAT(z-to)cTce A(~-t~ d~xo,

(23)

or x0 =

which

indicates

Grammian solution It

to

can

given by

be

the

eAT(~-t~

(~) d~,

necessity

for

the

order

to

invertible

to the p r o b l e m

should

Grammian This

Wo I (to, tf)

also

be

represents be

seen

in

being noted

a measure by

(24)

observability have

a unique

considered. that of

considering

the the the

observability output output

energy. energy

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

i

h

263

(25)

tf yT (~) y (I;) d~, 0

e A T (~-t0)c T c e A (~-t0) dZx0,

=

(26)

d to

=

x~W0 (to, tf) x0.

E.

THE

DISCRETE

Consider

x(k+l)

The

=

x(0)

Lyapunov

time

EQUATIONS system

[15]

(27)

= x 0,

function

x (k) T p x (k) .

change

=

V(k+l)

AV

=

x(k)T(ATpA

means

stable,

ATpA

can

as

(29)

-V(k),

that

the

or if one

(28)

in V is d e f i n e d

AV

This

one

the

LYAPUNOV

discrete

= Ax(k),

and define V

the

(27)

_p)x(k)

for

change

.

system

(27)

in V m u s t

be

(30) to

be

asymptotically

non-positive

definite,

writes

-

p

=

derive

tinuous

time

meaning

of

-

(31)

Q,

similar

case.

stability

The

arguments only

in b o t h

to

those

difference cases

[24].

of

the

is t h a t

con-

of the

264

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

Except and

for

theorems

equations case.

are

similar

_

Wc

ATWo A

-

W o =_

time

-- -

of the

to

equations

AWcAT

the

meaning

concerning

These

where

stability, discrete

those

of

definitions

time

the

Lyapunov

continuous

time

are

BBT,

(32)

cTc,

(33)

matrices

[A,B,C]

are

those

of

the

discrete

system.

This

concludes

retical

III.

aspects

the

Lyapunov

the

OF

THE

early

equation

LYAPUNOV

1960 's,

This

importance

and

review

of

the

theo-

functions/equations.

has b e e n

investigation.

extreme

historical

of L y a p u n o v

SOLUTIONS

Since

of

the

the

under

is

EQUATIONS

solution

a considerable

particularly

vast

to

due

the

amount

to

their

number

of

applications

took

on

three

in

the area of controls. The

research

in

this

area

directions.

Namely,

tion

implications,

and

tions,

its

and n u m e r i c a l

In the solution, [29],

one

finds

Barnett

Barnett [16].

in this

Section

II.

et

aspects

closed

the

solu-

form e x p l i c i t

solu-

the et

key

al

al

theoretical papers

[30] . [15],

direction the

have

by

aspects

Gantmacker that most

already

interested

of

Taussky

Other m a i n

It s h o u l d be n o t e d

However

of

solutions.

direction,

and O s t r o w s k i

include

sults

first

theoretical

different

been reader

et

the al

references [31],

and

of the

re-

addressed

in

is r e f e r r e d

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

to

references

above

and

references

therein

265

for

more

details. In the the

second

research

when

the

Jordan

gives

is

focused

system

form

in this

direction,

is

or any

direction

a summary

continuous Other [34] .

It

require

in

other

be

special

form

Power

time

are

that

transformation

of

This

is

a

solution such

[32,33].

to

as

A key p a p e r Power

for both

Lyapunov

due

noted

form.

form.

form s o l u t i o n s

discrete

solutions,

the

to H.M.

contributions

the

a

form

finding

canonical

is due

should

canonical

given

and

closed

around

of s p e c i a l

time

major

the

equations.

Peter

Lancaster

explicit the

the

solutions

system

numerically

to

a

demanding

procedure. In

the

third

finds

two

main

where

the

error

direction, approachesin

imized.

For

referred

to

al

Bartels

[35],

based This

on

details the

widely

A.

work et

reducing

approach

Bartels

and

the

is

methods

this

of

K.

HIGHLIGHTS

this for

presented.

first

Zietak

A matrix most

is

one

iterative

successively

min-

the

reader

is

[11-14],

Peters

et

approach

is

The

second

to

a lower

This

here

THE

one

reliable

[36].

OF

is

solutions,

approach,

[36] .

u s e d and is g i v e n

LYAPUNOV

In

of

the

Stewart

The

solution

al

the

numerical

Schur

and

approach

is is

form.

due the

to

most

in detail.

SOLUTION

OF

THE

EQUATIONS

section

a

detailed

the

solution

The

first

of the

method,

due

discussion Lyapunov

of

three

equations

to B a r n e t t

is

and Story

266

AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA

[15],

is

method

based

is due

on

the

to B a r t e l s

m e t h o d is due to S.J.

1.

Solution via

As

was

Barnett

This

using

the L y a p u n o v

+ ATp

=

The

[36].

second

The

third

[3] .

Equations

Products

earlier,

to s t u d y the t h e o r e t i c a l

PA

Stewart

Lyapunov

Kronecker

Story.

Consider

and

product.

Hammarling

of

mentioned

and

Kronecker

this

method

is

aspects

method

is

normally

due

utilized

of the solution.

equation

-Q,

Kronecker

to

(34)

products,

this

equation

can be w r i t t e n

as

[(AT|

where

+

vec(A)

transpose

(I|

T) ] v e c ( P )

defines

of the

A|

=

the v e c t o r

rows of A,

a11B a12B

--"

-

vec(Q),

f o r m e d by

(35)

stacking

alnB ,

......

the

and

. amlB

=

(36)

amnB

a s s u m i n g A is mxn. Equation A has

no

solution

vec(P)

(35)

has

eigenvalues

a unique on

the

solution imaginary

if and axis

only

[15],

if the

is g i v e n by

=

-

[ (AT|

+

(I|

T) ] - i v e c ( Q ) .

(37)

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

It

should

inversion

be

noted

that

of an n2xn 2 matrix.

is not u s e d for actual

2.

This

Bartels

is

one

and

of

popular MATLAB

AP

Stewart

the

most

is why this m e t h o d

Algorithm

numerically

For e x a m p l e

the

more

=

reliable

it is u s e d

and

in the

general

case

of

the

Lyapunov

-Q,

(38)

Q is symmetric.

lution

the

lets c o n s i d e r the e q u a t i o n

+ PB

where

This

involves

s o f t w a r e package.

consider

equation,

solution

calculations 9

widely used algorithms.

To

the

267

if and

only

Equation

(38) has

if any e i g e n v a l u e s

a unique

[~i of

A

so-

and

~j

of B s a t i s f y

(Xi +

To

~j ~

solve

0

Eq.

for

(38)

g e s t e d the f o l l o w i n g

i)

Reduce

the

all

for

=

uTAu

=

and

P,

j.

(39)

Bartels

and

Stewart

sug-

steps 9

matrix

unitary transformation

A

i

to

a

lower

Schur

form

using

a

U as follows

Azl

0

.--

0

A21

A22

".

.

_ Apl

Ap2

---

App

where each m a t r i x Aii is at most

,

_

2x2.

(40)

268

2)

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

In

Schur

B

same

form

as

=

where 3)

the

V

The

vTBv

is

fashion,

B

B11

BI2

.--

Blq

0

B22

.--

B2q

0

-..

0

Bqq

=

also

reduce

a unitary

transformed

Q and

to

upper

triangular

(41)

_

matrix 9 P are

given

by

N

Q11 Q

=

uTQv

......

Q1q (42)

=

_

QpI

......

Qpq

P11

......

Plq

_

and

P

=

uTpv

(43)

=

Ppl

4)

Recursively,

(k=l,2,...,p;

It

should

be

solve

......

for

the

Ppq _

blocks

of P

as

k-i

L-I

j=l

i=l

follows

1= 1,2,...,q).

noted

that

the

(44)

solution

of

Eq.

(44)

is

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

easy to carry mension

out

of 2x2.

since One

ber of c o m p u t a t i o n s B=A T.

This

Schur

should

a m a x i m u m di-

also notice

that

is r e d u c e d by almost

the num-

one half when

is needed.

of

the

computational

decomposition

(2+4g) n 3 flops,

and

where

tions

for

The actual

the

is one

due to

addition number

form

reduction

of Eq.

to

be

average

Schur

solution

is

estimated

a flop

and g is the

required

burden

is

multiplication,

verge.

have

is due to the fact that only one Schur de-

composition Most

all blocks

269

the

about

and

of

one

iterato

(44) requires

conabout

7n 3 flops.

3.

S.J.

Hammarling

Hammarling

Algorithm

1982,

modified

the

Bartels/Stewart

a l g o r i t h m such that the solution to the L y a p u n o v equations

is

Hammarling directly

more was

numerically

able

for the

to

upper

solve

well the

conditioned.

Lyapunov

triangular

Cholesky

equations factors

of

the Grammians.

B.

SOLUTION

OF

THE

DISCRETE

LYAPUNOV

EQUATIONS

As for

in the

the

possible system. reader

continuous

discrete

time

for specific For

details

time

Lyapunov

canonical of

is r e f e r r e d to N.J.

and references

case,

this

equations

solutions are

representations type

Young

t h e r e i n such as

explicit

of

of the

solutions,

[9], K e q q i a n WU

[38].

only

the [37],

270

AHMAD A. MOHAMMAD AND J. A. DE A B R E U - G A R C I A

It s h o u l d be n o t e d that discrete ment

time

The

discrete

common

time

continuous rithms

procedure

efficient

practice

Lyapunov

time

can

nature

Lyapunov equation prevented

of n u m e r i c a l l y

tion.

the n o n l i n e a r

be

to

the d e v e l o p for the solu-

in n u m e r i c a l l y

equation

one w h e r e used

algorithms

solving

is to convert

efficient obtain

of the

the

it to a

and r e l i a b l e

algo-

solution.

This

the

is o u t l i n e d next.

Consider

the

discrete

Lyapunov

equation

given

by

[32,33] ATLA

-

L

=

- Q,

(45)

u s i n g the t r a n s f o r m a t i o n

A

=

(B+I) (B-I)

converts

the

continuous

BLb

+

discrete

LbB

=

(A-I)

it is

conditioned. for

the

tance.

=

Lb 2 (B-I) ,

(B_I)T

time

Lyapunov

(46)

equation

to

the

(47)

(47)

- Q. can

be

or

S.J.

Bartels/Stewart

of

L

time L y a p u n o v e q u a t i o n

Equation

However,

-I ,

should

be

solution

of

for

out

demanding

the d e v e l o p m e n t this

Lb

using

H a m m a r l i n g 's

pointed

numerically Thus,

solved

problem

that and

either

algorithm.

the

inversion

might

be

ill

of a new t e c h n i q u e is

of

great

impor-

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

C.

A

NOTE

ON

THE

DESCRIPTOR

SOLUTION

OF

Ex

= Ax

y =

EQUATIONS

system

+ Bu

(48a)

Cx,

where

(48b) -I

(sE-A)

condition

The

THE

LYAPUNOV-LIKE

Consider the regular d e s c r i p t o r

exist

as

a

sufficient

and

necessary

for regularity.

expected

Grammians

controllability

for this

and

system satisfy

observability

[39]

A W c ET

+ E W c AT =

- BB T

(49)

ATWo E

+ ETWo A

- C TC.

(50)

Under

the

=

assumption

consistent

initial

eliminate

any

that

the

conditions

system

is

(initial

impulsive

behavior),

given theorems

concerning

reachability,

and

similar

stability

However,

the

unique.

In addition,

ness

with

of the

Grammians to

the

corresponding

divided

to

these 1987

about

part the

of

with

conditions

that

Lewis

regular

systems.

equations [40],

the

is

not

has shown that

in general.

existence he has

two parts, the

noncausal

system part

has

observability,

He also

for these Grammians

Briefly, into

regular

Frank

for

satisfied

solution.

causal

of

definition

theorems

are

those

Bender

are not

gave a more general gether

to

solution

these equations

ing

271

and

shown one

uniquethat

the

correspond-

and of

to-

the

the

other

system.

272

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

According

to

Bender,

the

teachability

Grammians

are

given by

r (AWc~ E T + ~.wo~A ~) r r

= - CoBB Tr T

(~.wo2~. ~ - Aw~2A =) r

while the o b s e r v a b i l i t y

= r162

are e x t r e m e l y applications tions.

izations, technique solution

and

the

one

hence,

to obtain to this

(54)

the coefficients

series expansion solutions This

require

For e x a m p l e

(53)

= CT-~CTCr

difficult. that

Tcr

respectively,

s o ,s -I in the Laurent

that

are given by

= - r

CT_~(~.TWo2~. - ;JWo2A) r

It is clear

(52)

Grammians

r (;JWo~. + ~.TWo~A)r

where ~-I and ~0 are,

(51)

the

can

specific

for these

solution

not

reduced

of H(s).

is a m a j o r

can not

of

obtain utilize

equations

drawback

in

of these

equa-

balanced

real-

the

order models.

balancing However,

p r o b l e m has been p r o p o s e d

a in

[2].

This

concludes

the

review

of

the

solutions

to

the

Lyapunov equations.

IV.

IMPORTANT

APPLICATIONS

LYAPUNOV

Different have

already

OF

THE

the

Lyapunov

EQUATIONS

applications been

of

introduced

However,

it is

felt

that

of these

applications

equations

in the p r e v i o u s

a more

is n e e d e d

comprehensive for the

sake

section. summary of clar-

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

ity and completeness

of this

of these applications

A.

LYAPUNOV

These

Consider

=

theory

represent

and

a summary

here.

+ Bu,

the

in control stable

x(0)

=

first

application

system design

[41].

system given by

(55 .a)

x0,

= Cx.

(55.b)

It is desired to design uTu

Thus,

CONTROLLERS

the minimal

Ax

y(t)

is p r e s e n t e d

controllers

of Lyapunov's

chapter.

273

a control

input u such that

___ 1

such

(56)

that

the

initial

state

returns

problem

starts

to the

origin

as rapidly as possible. The

solution

Lyapunov V

xTpx

=

x T[PA

that

symmetric unique PA

+ ATp]x

A

is

symmetric

Substituting

The

by

assuming

a

(57)

=

control

2uTBTpx.

which

(58)

implies

semi-definite

positive

definite

Q,

that

there

for

any

exists

(59)

(59) into Eq. +

input

a

P such that

-Q.

Eq.

- xTQx

+

stable

positive

+ ATp

=

this

function

=

Notice

to

(58) yields

2uTBTpx. u

should

(60) be

chosen

.such

that

V

is

274

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

negative

with

the

largest

magnitude

clear that u must be parallel two

vectors

must

have

ond term negative.

to BTpx.

opposite

signs

It

However, to make

It

is

the

sec-

Thus u should be taken as

known

(61)

important

optimal LQR

is

these

BTpx

u

the

possible.

to

notice

controller

design

that

among

gives

only

this

all the

controller

controllers. linear

is The

optimal

con-

troller.

B.

LYAPUNOV

EQUATIONS

ANALYS

This

has already

beginning

of

time

descriptor

earlier. suffer

It

especially

responding transfer

chapter.

was

some

the

that

to

at the

discrete

also

addressed

these

extensions

problems.

descriptor

case

into two

This

is

where

the

subsystems

and p o l y n o m i a l

parts

cor-

of the

function.

LYAPUNOV

EQUATIONS OF

Controllability, ity analysis

and

the

proper

MINIMALITY

have

out

to be s e p a r a t e d

to

addressed

were

serious in

been

Extensions

systems

pointed

evident

system needs

C.

this

from

STABILITY

IS

application

and

IN

already

implications

THE

systems

addressed of

these

THE

SYSTEM

observability,

of dynamic been

DEFINE

and

hence

via Lyapunov

earlier.

concepts

The in

minimalequations

importance

control

system

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

design the

were also

considered

applications

sign,

observer

lems

have

noted

design,

been

that

of these

concepts

and

discussed

the

in details

of

controller

However,

these

systems

suffers

problems.

mainly

arise

tion

and

latter

the

two

problems

solution

cases.

of

Lyapunov

These

problems

de-

probit

concepts

crete time and d e s c r i p t o r These

Among

realization

in detail.

application

therein.

the

minimal

275

was

in dis-

from serious

in the

defini-

equations have

in

been

the

briefly

introduced earlier.

D .

LYAPUNOV

EQUATIONS

IN

MODEL

ORDER

REDUCTION

B.C. the

Moore

[21]

demonstrated

controllability

'balanced

was

used

der m o d e l s Lyapunov

constitute

and o b s e r v a b i l i t y in the

same paper

of d y n a m i c a l

equations

and is essential tion

observability

coordinates,

controllability fact

and

algorithm

how the eigenvalues

a measure

to obtain This

introduced

later;

the This

reduced

application

the most

in the d e v e l o p m e n t

in

of

of each state.

systems.

is p r o b a b l y

Grammians,

of

important

orof one

of the continuizahence

it

is

intro-

duced here in detail.

1.

Since

The

Technique

its i n t r o d u c t i o n

technique

has

[42-53].

By

niques

Balancing

triggered now,

(BT)

by B.C. Moore, an

there

to obtain b a l a n c e d

intensive

are

several

realizations

the b a l a n c i n g

wave

of

research

numerical

tech-

[see references

276

AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA

above] . rather

However,

than the

they

only

concept.

differ

An o u t l i n e

in

the

of this

details technique

is g i v e n next. Consider

y

the

system

[50,52,53]

=

Ax

+

Bu

(62a)

=

Cx

+

Du,

(62b)

and the c o r r e s p o n d i n g AWc

+ Wc AT

ATw o +

Next,

consider

Wc =

where

WoA

Lyapunov

equations

=

-BBT

(63)

=

-cTc.

(64)

the s i n g u l a r

value decomposition

of Wc

U c ~ c U T,

Uc

is

(65)

unitary,

~c

is

diagonal

with

entries

(;i~(Yi+ 1 - Let _

TI

and

=

~i/2

UcLc

apply

a

,

(66)

similarity

transformation

to

system

(62)

to get

-1/2

A1

=

Zc

CI

=

CUcLc

,.,1/2

.T..

Uc~UcZ~c

, B1

=

-1/2

Zc

uTB, (67)

1/2 T..... .., 1/2 Wcl

=

The next

Wol

I,

Wol

=

UcWoUc~c

step is to p e r f o r m

= UoiZoiUoT1-

Choosing

~c

T 2 as

.

(68)

an SVD on Wol as (69)

277

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

= Uo Zo /' and

applying

system

(67)

a

(70)

similarity

gives

transformation

the b a l a n c e d

using

T 2 to

system

_i14 T

Ab

-

/~oi U o l A I U o l ~ o I14 1 ,

Bb

=

~I/4UolB~ ir 2~oi

Cb

(71a)

_-I14

CiUol2~ol

=

,

(71b)

with the new G r a m m i a n s

Wcb

=

_I14 T i14 Lol U o l l U o l ~ c

Wob

=

Lol

_-I14

It mal

should

and

-iI~ uTIWoIUoIEol

be

output

tained

from

_I12 ~oI =

=

noted normal

Eq.

i/2 Eol =

=

that,

the

by

E,

well

realizations

(71)

(72)

~,

(~i>_(~i+l .

known

can be

applying

the

(73)

input

nor-

readily

ob-

similarity

transformations

Ti

T 2 T IE ~/2

=

for input

To =

normal,

nique

and

T 2 T I ~ -I/2

for output The

(74)

normal.

final is

(75)

to

step

in this

look

for

model

a break

order in

the

reduction singular

techvalues

such that

(~r>>(~r + I ' and t r u n c a t e

the b a l a n c e d

system

after the

r th state.

278

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

To

illustrate

the

last

step,

consider

the

balanced

system

:[A11 A12][xl x2

A21

A22

bl

u (t)

(76a)

b2

x2

~: [c,c.][x.]

(76b)

X2 with

the G r a m m i a n s

Wc

=

o]

Wo =

'

~2

then

the

Xr

=

AllXr

Yr

=

CllXr,

with

r th o r d e r m o d e l

can be t a k e n

as

+ bllU

(78a)

(78b)

Granunians

Wcr

For

reduced

(77)

I (Z~)>>k (Z2), min max

-

Wor

=

simplicity,

renamed

as

2.

~i

-- ~'r

the

(79)

reduced

model

matrices

are

[A r,B r,C r] .

Properties

of

the

Technique a)

The

entries

of

the

system.

between

order

them

t r i x H of the

of ~

are

There

a n d the system-

Balancing

[44, 4 5 , 5 2 ]

called is

an

singular

the

second

interesting values

Specifically,

order

modes

relationship

of the

Hankel

ma-

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

279

2 k i ( H H T) = ~ib)

In general,

and the

the

eigenvalues

controllability

of the

level

of

of

the

Grammians

controllability

are

and

observability a quantization

observability

of

each state. c) In balanced coordinates, as it is observable d) The subsystems

a state is as controllable

(Wo=Wc=~).

[Aii,Bi,C i] in Eq.

(76) are balanced

with Woi = Wci = ~ie) T r a c e ( W o o ) = ]~ D C 2 Wco = C r o s s f)

There

is

an

gain

of

the

system,

Grammian. upper

bound

on

the

frequency

error

given by ]E ( s ) ~ C g)

The

(s I-A) -IB-c I (s I-All )-IB iI~-<2 T r a c e (~2).

sub-matrices

Aii

are

asymptotically

stable

if

~i, ~2 have no common entries. Again, time

extensions

and

o f this

descriptor

drawbacks

that

application

systems

prevent

or

suffer

limit

its

to discrete

from

the

same

application

to

these systems. E.

LYAPUNOV AND

Briefly, use norms

the

ROBUST

basic

going

system

analysis into

IN

H-INFINITY

CONTROL

the H-infinity

in the

Without

EQUATIONS

and robust control problems

no rms

IHL,

and design

details,

the

I~2,

I~F,

of control solution

and

~L~2

systems. to

these

280

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

p r o b l e m s boils down to the solution of the known algebraic For

Riccati

more

equation

details,

paper by G l o v e r Simply,

tem

finity hand, a

internally

norm

of

Usually, inal

the

the robust

controller

Although chapter,

stable

control

the

key

a norm

consists

of

the c l o s e d loop sysminimizing

[22,54,55].

On

p r o b l e m consists an

the

the

in-

other

of d e s i g n i n g

uncertain

system.

system is given by a known nombound

on

the

fluctuations

from

system. these

it

is

topics

observed

are that

systems

and d i f f i c u l t i e s

in

LYAPUNOV

not

considered

extensions

to

in

this

discrete

suffer from the same drawthe

lution of L y a p u n o v equations

F.

to

problem

while

stabilizes

time and d e s c r i p t o r backs

equations.

referred

control

system

that

and

the nominal

is

Lyapunov

that makes

the u n c e r t a i n

plant

reader

H-infinity

a controller

to be

the

[22] .

the

designing

the

and/or

EQUATIONS

definition

and

the

so-

for these systems.

IN

SYSTEM

TRANSFORMATIONS

Two ture.

important

These a p p l i c a t i o n s

1.

The bility tain

applications

Similarity

singular and/or

input

izations

are given next.

Transformations

value

decomposition

observability

normal,

introduced

are cited in the litera-

output

of the

Grammians normal,

earlier.

The

and

was

controllaused

balanced

importance

to

ob-

real-

of these

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

realizations

is evident

through

their

281

crucial

role

in

model order reduction.

2.

Continuization

Although directly of

a

the

suggests

that

investigated technique

on

the

its

This

resulting

is p r e s e n t e d

been

the

Grammians

discrete

used

time

transformation,

direction

has

been

in a new c o n t i n u i z a t i o n

Lyapunov

next

not

equations.

This

details

serve

in more

to

for new Lyapunov based applications.

DIRECTIONS

LYAPUNOV

A.

[2]

and

of

bilinear

possibility.

as an example

NEW

the

have

equality

system

using

in

discretization

equations the

time

based

technique

IN

THE

APPLICATION

novel

application

OF

THEORY

INTRODUCTION

In

this

section

equations

is

directions

in

out.

area,

continuous when

V.

Lyapunov

in this

version

and

More

a

developed. utilizing

Furthermore, these

specifically,

of

Lyapunov some

equations

are

a new c o n t i n u i z a t i o n

new

pointed

algorithm

is presented. The p r o p o s e d tinuization tions

and,

tinuization

continuization

scheme hence,

based is

a numerically

reliable

to

used

the

the

widely

LBCT

eliminates

on the use

termed

technique'

technique

'the

(LBCT). and

bilinear

less

of L y a p u n o v

Lyapunov

This

expensive

equa-

based

technique

transform.

the b i l i n e a r

is a new con-

con-

offers

substitute

Furthermore,

transform

drawbacks,

282

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

especially, tial

the

numerical

transient

error

Moreover,

this

the

widely

other

It is also the

zero

cial of

this

tween

Lyapunov

In this

and

that new

viewed

this

it

section,

approach

continuization

system

(CTS).

(LBCT)

time

that

system

(DTS)

The p r o p o s e d

guarantees

provided

advantage be-

modeling. may

lead

and

dis-

one

into a continuous

continuization

tech-

the s t a b i l i t y of the resulting starts

with

a stable

DTS.

The

is b a s e d on the assumption that both the CTS

B and output

dinates,

knowledge

Lyapunov

spe-

are u t i l i z e d to

It is shown that with a p p r o p r i a t e

the

and

PROBLEM

L y a p u n o v equations

input m a t r i x

of

as

provides

system

to

hold.

transform

and the DTS should share the same Hankel ues.

order

Another main

theory

method.

Preliminary

a discrete

technique

can be

ini-

superior

zero

link

is h o p e d of

the

direct

CONTINUIZATION

1.

nique

the

this

the b i l i n e a r

technique. is

of

high

schemes b a s e d on Lyapunov theory.

THE

convert

technique, both

the

numerically

techniques

development

B.

is

stability

it

cretization

CTS

hold

approach

the

time

used

of this

In addition, to

technique

order

and

characteristic

shown that

cases

problems

and DTS,

one

equations

A.

Different

CTS

models

can

of

the

solve

for the

choices

matrix

choices

in b a l a n c e d

Hankel

the CT

C,

singular val-

two

singular continuous

system's

dynamic

of B and C result

corresponding

to

different

of the coorvalues time matrix

in d i f f e r e n t

discretization

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

techniques.

For

Euler's

method,

inverse

of

tion

of

while

Euler

be

that

the

the

of

ately.

As

B

in

all

this

tion

original

comes

the

more

CTS

resulting the

accurate

will

as

are

methods), trans-

normally been

dynamics DTS

chosen

and

in

to the

bilinear

CTS

as

a combina-

method

ob-

used. matrix

provided appropri-

discretization

give

This

the

is

model

only

system.

it

original

matrices

method

be close

in the

transform

of

chosen

transform

continuization

techniques, of

C as

in the

those C

will

bilinear

the

and

CTS

(actually

bilinear

precisely CTS

the

result

eigenvectors

will

method

of B and

will

had

if B and C are

resulting

and

choice

form m e t h o d

The

the

Euler's

the

tained

example,

283

an

approxima-

approximation

sampling

time

T

be-

becomes

smaller.

to

In c o n t r o l

theory,

it is e s s e n t i a l

that

one be able

transform

a

CTS

and

vice

For

example,

CTS

model

given

in

and

simulation

is

asked

computer.

To

discretized

using

techniques In

the

asked

to

achieve

to

construct

for CTS

techniques

simulate

are

and

the

state

one

While

is

not

first

usually and

is

discretization understood,

[57,58].

a unique of

is

system

and well

nonlinearity space

a

discretization

model.

of

given

a digital

model

of the

lack

versa.

[56] .

implemented

the

CTS

problems,

are

is

it u s i n g

available

method

techniques is

one

behavior

a CTS

easily

this

in

time

the

the

identification

continuization

DTS

of

a DTS

problems,

this,

one

discrete

techniques

reason

to

such as E u l e r ' s

system

given

into

The

main

mapping

from

identification

representations

[59] .

In

284

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

the p r o p o s e d technique, taining

a

CTS

lined.

This

method

servability among of

other

the

model

from

CTS

stable.

given

the

both

provided

that

Theoretically,

However,

of

equations

for

matrix

and

C,

emphasize tion of

a given

time

that this m e t h o d The

solution

of

to

out-

and

ob-

guarantees,

and m i n i m a l i t y

the

given

there

satisfy input

T.

will

both

matrix

It

DTS be

is in-

is

Lyapunov B,

output

important

to

can be used for discretiza-

only p r o b l e m

due

is

to a given DTS model.

will

set

sampling

as well.

the

them

model

This

stability

corresponding

one

DTS

for ob-

controllability

finitely m a n y CTSs only

procedure

equations.

properties,

and

a

uses

Lyapunov

resulting

minimal

a systematic

the

is the n o n - u n i q u e n e s s

nonlinear

nature

of

the

introducing

the

discrete time L y a p u n o v equations. The

following

continuization retical

section,

algorithm

aspects.

starts without

Theoretical

by

discussing

aspects

its

theo-

follow the algo-

rithm outline.

2.

The

Continuization

Algorithm

Let the DTS be given by Xk+ 1 = A d X k + Yk = where

CdXk

+

subscript

It is d e s i r e d the

behavior

singular

of

values

tain degree

BdU k

(80a)

DdUk

(80b)

d denotes

discrete

time.

to find a CTS model the

given

the

Hankel

of the CTS be those of the DTS,

a cer-

of accuracy,

DTS.

that approximates

in terms

Letting

of norms,

is guaran-

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

teed.

More

specifically,

either

of the

system

can be p r e c i s e l y

the

tems.

If the

original

stable

DTS

is

[HI2 or same

285

the

DC

gain

for b o t h

sys-

then

so will

be

its CTS c o u n t e r p a r t . It can

should

use

the

lined b e l o w The with

be

for stable

solution

to

transform

in the worst technique nique.

More

without

the

available

DT

recent

need

ATbZAdb

where

~

subscript

-- Z

arise

that

=

-

out-

balanced The pro-

the

DTS

the

Therefore,

balance

a

DTS

transform

are

alleviate to the

the

use

developed

[2] .

DTS

be

will

on the

t r a n s f o r m tech-

bilinear to

be

f r o m the p r o p o s e d

to

via

the

DT

solution of

Thus,

assumed

of

Euler's without in

bal-

(Adb, Bdb, Cdbr Dd, T) .

s a t i s f y the DTS L y a p u n o v e q u a t i o n s

BdbB~b T

=

-- CdbC db

db stands the

use

the

into

method.

equations

is d i a g o n a l

Denoting

will

and g i v e n by

=

one

starts

of a DTS m a y be b a s e d

approach

s y s t e m must

Z

DTS

requires

techniques

recently

anced coordinates

-

problem

is due here.

a novel

generality,

AdNZ,A~N

given

Moreover,

problem,

m e t h o d has been

This

to

DTS's

procedure

and f r o m the b i l i n e a r

Lyapunov

of

scheme

errors

[60] .

balancing

the

continuization

case,

itself

the

of c a u t i o n

The b a l a n c i n g

bilinear

in

continuization of

continuization

balanced.

anti-stable

system

the

A note

for

systems.

a transformation

posed

loss

that

reciprocal

coordinates.

the

noted

with

-

=

p

(81)

Q,

(82)

entries

for d i s c r e t e

sampling

time

T,

(~i_>(~j i f

i>j.

and b a l a n c e d the

input

The

system.

and output

286

AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA

matrices

of the CTS Bcb,

Bcb =

Bdbl~,

This

choice

Euler's

method,

trices CTS

except

Ccb can be taken as

Ccb = is

(83)

Cdb/~.

justified

for example,

if

one

does

not c h a n g e

for the m u l t i p l i c a t i o n

dynamics

matrix

Acb

is then

notices

these ma-

factor

chosen

to

that

~.

The

satisfy

the

CTS L y a p u n o v e q u a t i o n s

so

Acb%

+

~ATb

=

-- P / T - -

ATb ~

+ ~Acb

=

-

that

the

DTS.

method

and the

and

output

while

in

bilinear

the

noticing

or Eq.

transform one

same

of

the

combination

of

t r a n s f o r m method-

the

is

are

Grammians a

taken

as

values

method.

It

is

Acb

separately,

the

solution

if

one

uniqueness

of

there are two s o l u t i o n s Then

solves

the

from

for

the s o l u t i o n

Euler's taken

as

worthwhile either will

Acb

Eq.

not

from

be

Eqs.

is unique.

solution,

Acb I and Acb 2 that

it follows

in are

for

simultaneously,

(84)-(85).

the

solves

(84)-(85)

Eqs.

(85)

singular

However,

show

Qc,

bilinear

unique.

To

(84)

strategy

Hankel

if

(85)

-

matrices

the

that

=

have

this

method

(84)

will

Clearly,

Euler's input

CTS

Q/T

Pc

suppose

that

satisfy both

that

Acbl%

+

ZAcTbl =

-- Pc

(86)

Acb2~

+

T ~Acb2

- Pc-

(87)

Subtracting

Eq.

=

(87) f r o m Eq.

(86) and l e t t i n g

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

Acb I - Acb 2 =

287

N

yields N]~

+

]~N T =

This e q u a t i o n

reduces

(~jnij + (Yinji

Similarly, must

(88)

O.

-

f r o m the

which

+

NT~

=

observability

Lyapunov

equation,

reduces

0,

N

(90)

to

(~inij + nji(~j =

Comparing

Eqs.

(~i~(~j i f

i~j,

nij

=

nji

and

is

seen

it

that

(91) , that

only

that

solution

to

to

the

Acb 2.

solution

solution

of

of Eqs. these

simple p a r a m e t e r i z a t i o n

[61] 9

Provided

i~j,

acbii =

the

provided

is

that Acb I =

the

that

and

-- 0,

implies

Returning

(91)

0.

(89)

these two e q u a t i o n s

seen

(89)

0.

satisfy

~N

which

to

(~i~(~j i f

- Pcii 2(Yi

two

(84)-(85),

it

equations

Acb can be w r i t t e n

is

is

a

as (92)

288

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

for the d i a g o n a l

elements

of Acb,

and

Pcijf~j -- qcij(~i acbij =

for the off d i a g o n a l the

fact

choice

that

the

of b a l a n c e d

Clearly, (Acb,

elements

system

Ccb,

as

This

is due

and e x p l a i n s

to the

coordinates.

Dc=Dd)

it

of Acb.

is b a l a n c e d

from L y a p u n o v

Bcb,

balanced

(93)

2 2 ((~i -- (~j)

theory,

is

satisfies

both both

the

resulting

stable, Lyapunov

system

minimal Eqs.

and

(84)

and

(86). 3.

This

Theoretical

is best

summarized

For

Theorem-

Aspects

any

given

(Adb, Bdb, Cdb, Dd, ~, T) , modes,

there

balanced

exists

CTS model

Bob = B and

Acb

/fT,

is

a

of

the

LBCT

in the following

stable, with

minimal

distinct

unique,

theorem-

balanced second

minimal,

order

stable,

(Acb, Bcb, Ccb, Dc=Dd, ~),

DTS

and

where

Ccb = C b/f{,

the

unique

solution

of

the

CT

Lyapunov

equations

AcbZ

+

~ATb

=

T - BcbBcb

=

- TBdbBTb

T Acb~

+ ~Acb

=

-

T CcbCcb

=

- TC~Cdb.

the

CTS

Moreover,

the f o l l o w i n g

approximation

properties-

of

the

DTS

model

has

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

i)

The

model 2)

CTS

model

is m i n i m a l

There

minimal

a

solution

if

the

DTS

for

Eqs.

(84)-(85),

for

a

of 2) is unique.

4) The I~2 n o r m of the DTS CTS

stable

of T, B, and C.

3) The s o l u t i o n

the

and

and stable.

exists

fixed choice

is

289

(notice

that

we

error

in

is equal only

to the ~HI2 n o r m

deal

with

the

of

strictly

p r o p e r part) . 5)

The

initial

the

step

response

is

identi-

the

Hankel

cally zero.

Proof

:

(i) G u a r a n t e e d by L y a p u n o v equations. (2,3) (4)

Proved A

in s e c t i o n

direct

consequence

S i n g u l a r values (5)

First,

same value

it

p r o p e r part

4.

is

noted term

the

that

D.

is i d e n t i c a l l y

Comparison

with

matching

Euler's

both

systems

Moreover,

initial

of

share the

the

the

initial strictly

zero.

Other

Techniques

the p r o p o s e d

and

from

response

with

In order to v a l i d a t e parison

of

of the DT and CT systems.

feedthrough theorem,

2.

the

technique,

Bilinear

a com-

transform meth-

ods is p r e s e n t e d next. To model

set

the

stage,

suppose

and the c o n t i n u o u s

discretization In this

case,

technique the

first

model using

that

the

are r e l a t e d a

relation between

sampling the DTS

discrete

via Euler's period

T.

and the CTS

290

A H M A D A. M O H A M M A D AND J. A. DE AB R E U - G A R C I A

state

space m a t r i c e s

Adb =

I

+ TAcb,

Substituting (I

+

Eq.

Bdb =

(94)

TAcb)~(I

where

Wcb

will

be

anced' Thus, to

is

(95a) (95b)

BobBc b method,

must

satisfy

T Bcb Bcb,

-

that

Hankel

t e r m in Eqw. <<

(96)

necessarily

diagonal.

this having

for the CTS

same

TIAcb~AT~

T - TBcbBcb

=

from Euler's

its G r a m m i a n s

the

(94)

(81) gives

AcTb = -

later

in order

nonlinear

=

not

shown

with

have

+

- ~

Ccb~-

equation

T WcbAcb

AcbWcb+

, Crib =

into Eq.

as o b t a i n e d

the CT L y a p u n o v

Bcb~

+TATb)

TAcb AcTb + Acb The CTS,

is given by

system the

obtained

norm

as

However is

it

'semi-bal-

same eigenvalues. via Euler's

that

of

the

(95b) must be close

method

DTS,

the

to zero or

IBcbBT~

(97a)

or

TIAc << ~SdbS~/(~max. By comparison, LBCT This

has

the

the

same

s y s t e m must

system

Hankel

satisfy

(97b)

obtained norm

as

via the p r o p o s e d that

the CT Lyapunov

of

the

DTS.

equations

Acb~

+

~cb

=-

B cbBTb,

(98)

~cb~

+

~ALb

T -- _ CcbCcb"

(99)

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

Thus

the

have

a

system

resulting

different

values

from

However

from the p r o p o s e d

dynamic

the

system

both

models

contrast

to

matrix

and

obtained

share

the

LBCT

Hankel

via

same

291

singular

inverse

input

will

Euler.

and

output

matrices.

In

proposed put,

algorithm

bilinear

will

have

and output matrices

singular the

the

values.

proposed

system

algorithm via

different

while

Moreover,

obtained

transform

the

the

dynamics,

in-

sharing the

the

is

method,

system

strictly

bilinear

same Hankel

resulting

proper

from

while

transform

is

the

simply

proper. In

addition

to

for the previous important

providing

theorem,

(97b)

to

Of course

stability

Equation to

Euler's

be

gives

algorithm

tions

this

different

proof

following

the

in

Euler's

method.

by the theorem. use

and

the

This

for the p r o p o s e d

inverse

of

Lyapunov

accuracy

eigen

less expensive

equa-

tests

analysis

with

stability/accuracy

line

of

investigation

for the test has

[2] .

the

to notice

has

approach.

that Eq.

generalized

in the delta

equation

the

stability

(97).

used to define

to

suggests

It is i n t e r e s t i n g

and DTS

novel

have the

the condition

replacing

Eq.

been p u r s u e d 3)

the above

is g u a r a n t e e d

provide

method

on

close

(97b)

computationally based

alternate

consequences-

I) Equation

2)

an

operator

been

Grammians approach

reached

In the

(95b)

for both

[62] .

independently

delta

has been

operator

CTS

However, using

a

approach,

292

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

this e q u a t i o n

is d e r i v e d on the basis

ergy control origin there

input that takes the initial

in a f i x e d time. is no

addition,

ones.

equations

However,

tions

are

used

similarity surprise

assumption

to

between

since

operator

are

here

the

state to the

In the previous

optimality

in the delta

Lyapunov

of a m i n i m u m en-

the

from

original

derive

the

two

delta

start

approach,

obtained

the

to

development,

generalized

Lyapunov

generalized

approaches operator

In

the original

the

DTS

with.

equa-

ones.

The

s h o u l d be of no

approach

is

essen-

tially a v a r i a t i o n of Euler's method. 4)

Equation

assumption its

that

a

continuous

proaches to

(95b)

zero

Lyapunov

This

Eq.

when

a

to

system

the

the

logical

converges

sampling

seen by

This

into

proof

time

c o u l d be

(95b) .

equation

a solid

discrete

version

zero. in

gives

time

letting

transforms

continuous

the

one

to ap-

T equal discrete

giving

the

a f o r e m e n t i o n e d proof. 5)

Inspection

of Eq.

o b t a i n e d via Eqs. that

obtained

(95b)

(92)-(93)

via

the

reveals

that

the CTS model

is always more

inverse

Euler's

stable than

method.

can be seen by n o t i c i n g that the error term, t e r m in Eq. 6)

The

(95b),

previous

between

tion/ i d e n t i f i c a t i o n

5.

It the

Further

should

CTS

is

be

establishes

equations

and

a solid

system

link

simula-

concepts.

Devel opment s

noted

equal

the first

is g r e a t e r than or equal to zero.

development

Lyapunov

This

to

that twice

the the

the

LCBT

steady trace

state of

gain

the

of

cross

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

Grammian

Wco

[21].

of

a matrix

is

it

is n o t e d

that

related

to

values

of

the

the

coordinates,

First of

is n o t e d

its

eigenvalues

Hankel

singular

following

and

the

that

the

eigenvalues.

the

the

Wco

sum

it

of W c o

293

trace

Secondly

are

directly

In

balanced

values.

r e l a t i o n b e t w e e n the e i g e n -

Hankel

singular

values

of

the

CTS holds : 2

This

can

be

exploited

to m a t c h

the

steady

state

gain

in the p r e c e d i n g

sec-

as follows:

I) Solve

for the CTS as o u t l i n e d

tion. 2)

Find

DTS,

the

steady

state

gains

for

both

the

CTS

and

say sc and sd r e s p e c t i v e l y .

3) C o m p u t e the s c a l i n g 4)

Find

the

scaled

5)

Solve

for

the

s i n g u l a r values,

6.

factor set

CTS

of

using

f~=sd/sc. Hankel

the

singular

scaled

set

of

values

Hankel

as o u t l i n e d above 9

Numerical

Example

and

Further

Discussion

It is w o r t h w h i l e tion t e c h n i q u e

comparing

with the well

U s i n g a M A T L A B program, of A p p e n d i x

the p r o p o s e d

continuiza-

known b i l i n e a r

transform.

a continuous

model

1 was f o u n d in three d i f f e r e n t

(i) The s t a n d a r d b i l i n e a r

t r a n s f o r m method.

for the DTS ways"

294

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

(ii)

Our

Hankel DTS,

proposed

singular

continuization

values

as

technique

obtained

by

using

the

balancing

the

and

(iii)

Our

scaled

proposed

set

of

continuization

the

in the p r e v i o u s

Hankel

technique

singular

values

using

as

a

obtained

section.

(iv) The Zero O r d e r H o l d method. The

results

i.

Different

are

compared

shown

of

responses with

in Figs.

As

for

the

step gave

is b a s i c a l l y

due

maximum

from

Hankel

is

est

maximum

ear

gave

state

where

arguments The tem

as

order

the

hold

In

is

for the

is

terms

of

the

impulse

obtained

via

the

the

proposed

the

uses

the

from

(iii)

as the

of

fact

error

the

via

relative

to that

to

system

bilinear

both methods

relatively

responses

and

the

as well

frequency

hold,

method

via

This

while

obtained

errors

due

The

obtained

its m a x i m u m

signal

error.

error

this

that

are

transform,

term.

that

performance

has

initial

system

final

This

transform

to

results

bilinear

initial

that

it is c l e a r

superior

The

models

chapter.

obtained

The

error.

transform.

linear

a zero

(notice

and

the

in A p p e n d i x

continuous

feed-through

values

initial

(error/signal), (iii)

shown

DTS.

high

comparable

method) .

zero

above

relatively to the

singular

gave

of the

responses,

transform

transform

are

original

(ii) g a v e

error

bilinear

the

process

1-8 at the end of the

as expected,

obtained

this

error (ii)

the

that

in the

small.

low-

and

bilinthe

bi-

initial Similar

response.

of the c o n t i n u o u s

bilinear

transform,

algorithm

are

time

sys-

the

zero

shown

in

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

Figs.

5-8.

proposed all

low

This

It

is

clear

from

algorithm,

matches

frequencies

within

is

specially

true

these the

zero

the

for

figures, order

system's

the

system

295

that

the

hold

for

band-width. obtained

via

scaling. In tion

summary,

this

of L y a p u n o v

section

presented

equations.

This

a link b e t w e e n

continuization

niques

one

on

Lyapunov method els. same

the

on

the

for

CTS

model

or a s c a l e d given

the

will

two

the

for

DTS.

is

provided

theory

The

technique

provides

equations

steady hoped

a

have

mod-

either

the

singular

values

a scaling

factor

responses

that

this

technique

new

continuization

of the

of this

minimal the

input

and

technique

CTS

version

previous

parameterization balanced

a

step

choices

feature

DTS

of

state

developing

of

given

to

that

a stable

in

for

Hankel

shown

converse

tech-

stability

chosen

is

A main

that

provides

the

on d i f f e r e n t

The

application

and d i s c r e t i z a t i o n

of the

the

to p r o d u c e

a given

be

It

way

based

ability

Lyapunov

can

applica-

versions

It was

matrices.

true

CTS

to m a t c h

the

techniques output

hand.

systems.

pave

and

version

DTS.

can be c h o s e n of

other

obtaining

The

of the

hand

a new

is of

statement of

coordinates

the

DTS

can

be

of

the

found. The

choice

CTS p l a y s ity

of

equated

the

input

an i m p o r t a n t

this

technique.

to t h e i r DTS

justification method.

of

for

However,

role In

and

output

in the the

counterparts

this

choice

matrices

accuracy

foregoing

and v a l i d these

s c a l e d by i / ~ . was

it s h o u l d be b o r n e

based

on

were The

Euler's

in m i n d that

this

296

A H M A D A. M O H A M M A D AND J. A. DE A B R E U - G A R C I A

is not the only choice.

An alternate

choice c o u l d use

the b i l i n e a r t r a n s f o r m or the zero order hold method.

VI

.

CONCLUSION

In

this

chapter,

a

background

material

for

presented. solution brief

historical

This

was

the

Lyapunov

of

the

Finally,

a new

system modeling

established. links

were

development equations between

Possible pointed of this

were

out.

equations

were

of

this

review,

theory

Lyapunov

a

pos-

were p o i n t e d out. based

on

in full details.

the New

and Lyapunov t h e o r y were

In

to

the

exploit course

the G e n e r a l i z e d

these of

and

(97) which equations

system

simulation

lays the foundation in

the

the

Lyapunov

i n d e p e n d e n t l y d e r i v e d and a direct

Lyapunov

the

of the L y a p u n o v

technique

directions

technique

e s t a b l i s h e d via Eq. utilizing

Throughout

was p r e s e n t e d

a brief

Furthermore,

applications

continuization

and

a discussion

of the a p p l i c a t i o n s

Lyapunov equations links b e t w e e n

by

equations.

was presented.

sible e x t e n s i o n s

Lyapunov

followed

review of important

equations

review

analysis

link was for of

d i s c r e t i z a t i o n techniques.

VII

.

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P.T. ,

AP P END IX

CONT INUI ZAT ION

Given

1

EXAMPLE

Discrete

Time

System

(all matrices

are given

in

MATLAB notation) ad=[0.9000

0.I000;-0.i000

0.9000];bd=[l

I] ',cd=[l

i]; Balanced Discrete

time System

adb =[0.8500

0.1118;-0.1118

bdb =[1.4554

0.3438] ';cdb =[1.4554

Continuized

System via Bilinear

at =[-15.9543 bt =[1.5468

0.9500]; -0.3438];

Transform

12.3560;-12.3560

-4. 4153];

0.4413] ';ct =[154.6776

-44.1340];

dt =[-1.0497] ; Continuized Approach

Balanced

System

via

Lyapunov

Equations

~

".

~

I 9

~,

o 9 0 0 b -~ ~,

II

o

"" I

9

-

~:)

o

9

".

~)

(.,)

o

o

!'0 0

~ ~ 0 ~

o o

II

~

I1

~,

~

I

9

~'::::1 ~1

r---,

,-,

I- L

II ,--,

,--,-

~

~

3:: o

c0

9

-(.~

II ~ ~

() 0

(~

.~

;,.,,

I-L

II ,---,

~

I bO

o tn --.] O~

I ~:)

o

9

r,,)

I

II ,---,

II

o

r~

~

,--',

r.~

0

~

~1

,--]

~,

C)

304

AHMAD A. MOHAMMAD AND J. A. DE ABREU-GARCIA

S~p R e s p o n s e

15,

? -5 0 Fig.

50 100 Time=kT, T=.01

I. Step response, C T S v i a

~14

.

.

:

~ t..A.':, :-i...............

150 sec

200

Bilinear Transform

:

:

!. ..............

!..............

"~" 1D

,

[1

U

50

IDD

15D

Time=kT, T=. 01 sec Fig. 2. Step response,

CTS via LBCT.

20D

CONTINUOUS

AND DISCRETE

9 15

TIME LYAPUNOV EQUATIONS

305

S~ep Response

~~~ f,o '

Fig.

0

50 I00 150 T J m e = k T , T = . 01 sec

3.

Step

2 .........

CTS

y

"~

.

|

.

~ 9

o o

~ o

:

:

:

9

, m"

I

... : ......... :.., . . . . .

% ~

,--2 ~

via Modified

LBCT

Error in Step Response

I

~0

response,

200

.~ ("",

'~

/ o

;~ 9

.

~:

.

.. : _ -

:

-_ ~.~-: .--- . ~ . : - ,

:

.~-: ---

...!. .............. i. .............

9

~

:

:

.

: .

!

o

9

,

i

i

i

5o

1oo

15o

TJme=kT,

Fig.

4.

Step

response

T=.

errors

01

sec

2oo

306

A H M A D A. M O H A M M A D A N D J. A. D E A B R E U - G A R C I A

Frequency :=: 102

9

,

,

Response, 9

9

9

, , ,

9

9

Magnitude 9

9

9

9

, , ,

VS

,

,

9

9

W ,

,

,,

0

N

I .

.

6,

o

\ I !

-rt r-t "r't " ' ~ Oo |

,"':";'t'ev

100

1O, Frequency

Fig.

5.

Frequency

Frequency o pa

0

"o oo

!

9 10o

9 .....

9

w

102 in Radians

response,

Response.

103

Magnitude

Phase

VS

%r

-v-

i

-50

o ~

~.,

-100

~ -o o .

I

.o %=

I

"O00OOOO0 O

N -1so

I

.rl "r-I

:

-200 10o

.

.

.

.

.

.

.

,,

101

,

,

,

. . . . _ _

102

Frequency w in Radians Fig. 6. Frequency response, Phase

9

9

9 = .

.

103

CONTINUOUS AND DISCRETE TIME LYAPUNOV EQUATIONS

Frequency Response, 102

9

,

.

,

.

o

, .

,

,

307

~agnitude VS

.

.

.

,

,

,

,

. . . .

,

. .

.

,

.

,

,

,

.

.

.

I

i

i

I

i

0 Dq I

9 6~

I0, -

,~

-

~i,

-

o~

-

~

10o

-

0 ~

" -

!

-

I

.

10-1

.

.

.

.

, , ,

lOo

. . ,

Frequency Fig.

7. Frequency Frequeny

0 L~

. .

i

102

1 o1

~

in

radians

response,

Response,

Magnitude Phase

9

,

,

,

,

.

I

I

I

I

I

I/

,

VS

W

, ,

,

.

.

.

.

II

I

!

|

|

i

.

.

.

211...'_- .......

I

9

\'\

-20

i--I =L

-40

-

-60

-

o I,.4 o X:::

'

I

-80 i

-I00

m

100

i

I

I

I

I

Ig

I

101

Frequency Fig.

8. Frequency

102

w

in

radians

response,

Phase

!

i

1_

103