multivariable feedback design

8/11/2019 Multivariable Feedback Design

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IEEE

TRANSACTIONS

ON AUTOMATIC

CONTROL, VOL.

AC-26. NO. I . FEBRUARY

I98I

Multivariable

eedback

esign:

oncepts

for

a

ClassicallModern

ynthesis

JOHN C. DOYLE

eNo

GUNTER STEIN.

MEMBER.

EEE

Abtt

l4ct-Tbb

poper

pnesetrts

prrcdcd

OeOg

perspecOve

n muld_

vrrlabh

feedbrdr

cootol

problemc.

It

rcvlews

tte

b.slc kore-foe&oct

deflgr ir

tbe

lrce

of ure.rtalndes-and

genenllze

ltrwr

sirgbltrDnt,

sltrgt€.otrQut

SEO)

statencnts

and

consfalnts

of the

d6lg!

probl€m

to

muldlrput,

EuldouQut (MIMO)

cas€.

Tt?o

mJor MIMO

deslg!

ap

prooch€s

srt tten

evahrrted In

the

context

of th* re$lts.

I.

Ir+rnooucrroN

T

HE last

two

decades

have

brought

major develop-

r

ments

in

the

mathematical

theory

of

multivariable

linear

iime

invariant

feedback

systems.

hese nclude

the

celebrated

state

space

concept

or

system

description

and

the notions

of mathematical

optimization

for

controller

synthesis

U,

[2].

Various

time-domain-based

analytical

and

computational

tools

have

been

made

possible

by

these

deas.

The

developments

also

include

certain

gener-

alizations

of frequency-domain

concepts

which

offer

anal-

ysis

and synttresis

ools

n

the

classical

ingle-inpu!

5ingle-

output (SISO)

tradition

[3],

[4].

Unfortunately,

however,

the two decades have also brought a growing schism

between

practitioners

of feedback

control

design

and

its

theoreticians.

The

theory

has

increasingly

concentrated

on

analytical

issues

and

has

placed

little

emphasis

on issues

which

are important

and interesting

from

the

perspective

of

design.

This

paper

is

an attempt

to

express

he latter

persp€c-

tive

and to

examine

the

extent

to

which

modern

results

.

u1smsaningful

o

it. The

paper

begins

with

a review

of the

fundamental

practical

issue

n

feedback

design-namely,

how

to

achieve

the

benefits

of feedback

in

the

face

of

uncertninties.

Various

types

of

uncertainties

which

arise n

physical

systems

are

briefly

described

and so-called

..un-

structured

uncertainties"

are

singled

out

as

generic

errors

which

are

associated

with

all

design

models.

The

paper

then

shows

how

classical

SISO

statements

f

the feedback

design

problem

in

the face

of unstructured

uncertainties

can

be reliably generalized

to multiinput,

multioutput

(MIMO)

systems,

and it

develops

MIMO generalizations

Manuscript

rcccived

lrdarch !],

1980;

revised

Octobcr

6, 19E0.

This

rcscarch

was

supported

by the

ONR

uader

Contract

N0001+75-COl.l4.

!y

the_

_QpE

undcr

Contract

ET-78-C-01-3391,

nd

by NASA

undei

Grant

NGL22ffi-124.

. .J.

C.

Do.yle-isrit:q

th-:e

ys-tems

nd Rescarch

Ccntcr,

Honcywc[

Inc,,

Mim-^eapolis,

MN

55413

and

thc University

of

California,

Bdkebi

Ci

94720.

. -9.

St€iq.is

-yt! -99- 9yrr"--!

aqd

Rcscarch

C.catcr,

Honcywell,

Inc.,

Minn,capolis, MN 55413 and thc Departmcnt of Elcctrical Enchl,crini

?nd.

Com-puter

Scienccs,

Massachusctts

Institutc

of Tcchnolofr,

CamI

bridgc,

MA

02139.

of

the classical

Bode gain/phase

constraints

[5],

[6]

which

limit

ultimate

performance

of feedback

n

the

face

of

such

uncertainties.

Severalproposed

MIMO

design

procedures

4re 6;4mined

next

in

the

context

of

the fundamental

feedback

design ssue.

These

nclude

the recent

requency

domain

inverse

Nyquist

anay (INA)

and

characteristic

loci (CL)

methods

and

the

well-known

linear-quadratic

Gaussian (LQG) procedure. The INA and CL metlods

are

found

to

be effective,

but

only in

special

cases,while

LQG

methods,

if

used

properly,

have

desirable

general

features.

The

latter

are

fortunate

consequences

of

quadratic

optimiiation,

not

explicitly

sought

after

or tested.

for

by tle tleoretical

developers

f the

procedure.

practi-

tioners

should find

them

valuable

or

design.

II. Frrosecr

Fur.rpeurvrers

We will

deal with

the

standard

eedback

configuration

illustrated

in Fig.

l. It

consists

of the

interconnected

lant

(G)

and controller (iK)

forced

by commands r),

mea-

surement

noise

(4),

and

disturbances d).

The

dashed

precompensator

P)

is

an

optional

element

used o

achieve

deliberate

command

shaping

or to

repres"ol

4 lelrrnity

feedback

system

n

equivalent

unity

feedback

orm.

All

disturbances

are

assumed

o

be reflected

o

the measured

outputs

(y),

all

signals

are multivariable,

n

general,

and

both

nominal

matlematical

models

or

G and

K are finite

dimensional

inear

time invariant

(FDLTI)

systems

with

transfer

function

matrices

G(s)

and

K(s).

Then

it is

well

known

that the

configuration,

if

it

is

stable,

has

the

following

major

properties

I

)

Input-Output

Behaoior:

y : G K ( I + G K ) - t ( r - r r ) + ( r + G K ) - t d

( l )

A

e : r - y

:

(/+

GK)-

(r

-

d) +

GK(I

+

GK)-

n.

2)

System

Sensitioity

[7]:

AH"r

-

( I+G,K)- tAH"t .

(2)

(3)

In (3),

A.F/.,

and

Aflo,

denote

changes

n

the

closed-loop

system

and

changes

n

a nominally

equivalent

openJoop

system,

respectively,

caused

by changes

n

the

plant

G,

i .e . ,

Q ' :G+AG.

Equations(l)-(3) summarize he fundamentalbenefits

and

design

objectives

nherent

n feedback

oops.

Specifi_

cally,

(2)

shows

trat

the

loop's

errors

in

the

presence

of

0018-9286

/81 0200-0004$00.7s

r98

rEEE



DoYI-E

AND

STEIN:

MULTIVARIABLE

FEEDBACK

DESIGN

d

)

which

show

that

rpturn difference

maenitudes

approxi-

mate

the

loop

gains,

o-[GK),

whenever

these

are

large

compared

with

unity.

Evidently,

good

multivariable

feedback

oop

design

boils down

to achieving

high

loop

gains n the necessaryrequency ange.

Despite

the

simplicity

of this

last statement,

t

is clear

from

years of

research

and desigrr

activity

that

feedback

design

s

not

trivial.

This

is true because

oop

gainscannot

be

made

arbitrarily

high over

arbitrarily

large

frequency

ranges.

Rather,

they

must satisfy

certain

performance

tradeoffs

and

design

limitations.

A major

performance

tradeoff,

for example,

concerns

command

and disturbance

error

reduction

versus sensor

noise

error

reduction

[10].

The conflict

between

hese

wo objectives

s evident

n

(2).

Large

o_[GK(i<^r)]

alues over a

large

frequency

range

make

errors

due

to

r and

d small.

However,

they

also

make errors due to 4 large because

his

noise

s

"passed

through"

over

the

same

requency

ange,

.e.,

y:GK(io) l t+cx( iu) l

- 'q- l ' r .

(9)

Worse still,

large

loop

gains

can

make

the

control

activity

(variable u in

Fig.

l)

quite

unacceptable.

This

follows

from

u:

K I I+

GKI- 'Q-n -d )

x G - t ( i o t ) ( r - q -

d ) .

(10)

Here we have assumed

G

to be square

and

invertible

for

convenience.

The

result'ng

equation

shows

that

com-

mands, disturbances,

and sensor

noise are

actually

ampli-

fied at

u

whenever

the

frequency

range

significantly

ex-

ceeds

he bandwidth

of G,

i.e., or <o uch

hat

6[G(i<,r)]<<

we

get

G'

does

not stray

too

far

from G.

For

SISO

systems,

he

appropriate

notion

of

smallness

ps(<o)<l l+sj io)k(j@)l

Var(c,r6,

(4)

where

ps(<o) s a

(large)

positive function

and

ars

specifies

tle

active

frequencY

ange'

This

basic

idea

can

be

readily

extended

to

MIMO

problems

hrough

the

use

of

matrix

norms'

Selecting

he

*pa"ttul

nonns

as our

measure

of

matrix

size,

or

example,

the correspondingeedback equirementsbecome

l . l r 1

; f

( r+

G(ja)K(

jo))- '

)

smal l

or conversely

ps(<.r)

o-l t+G(io)x(iot) l

(5)

for the necessary

ange

of

frequencies.

he

symbols

6

and

q

in these

expressions

re

defined

as

follows:

-+i

P

I

L - - J

comrnandsandd is turbancescanbemade. .smal l ' 'by

r"li"g

the

sensitivity

operator,

or

inverse

return

dif-

ir"o"ioperator,

(I+GK)-t,

"small,"

and

(3) shows

hat

loop

sensitivity

s

improved

under

these

same

conditions,

ol

A1

3

max

ll .r l l

l l x l l - l

ol A1

2

min

l l ,4.rl :

l l : l l - l

where

ll'll

is the

usual

Euclidean

norm,

tr[']

denotes

eigenvalues,

and

[']*

denotes

conjugate

transpose'

The

two o's are

called

maximum

and

minimum

singular

values

of ,4

(or

principal

gains

[4]),

respectively,

and can

be

calculated

with available

inear

system

software

8]'

More

discussion

of

singular

values

and

their

properties

can

be

found in

various

exts

[9].

Condition

(5)

on

the

return

difference

I

+

GK

can

be

interpretedas merelya restatementof the common intui'

tion that

large

loop

gains or

"tight"

loops

yield

good

performance.This

follows

from

the inequalities

o- lcK l -

I <q [

I+GK]<o- [G, ( ]

+ I

(8 )

q[c - ' ( ; r ) ]= - , -^ l

>r .

One of

the

major

contributions

of

modern

feedback

he-

ory

is the development

of systematic

procedures or con-

ducting

the above

performance tradeoffs.

We are

refer-

ring, of course, o the LQG theory [11] and to its modern

Wiener-Hopf

frequency

domain

counterpart

12].

Under

reasonable

assumptions

on

plant,

disturbances,

and

per-

formance

criteria,

these

procedures

yield

efficient

design

compromises.

In

fact,

if the

tradeoff

between

command/disturbance

error

reduction

and

sensor

noise

error

reduction

were the

only

constraint

on

feedback

design,

practitioners

would

have

little

to

complain

about

with respect

to

the

relevance

of

modern

theory'

The

problem

is

that

these

performance

rades

are

often

over-

shadowed

by

a

second

imitation

on

high

loop

gains-

namely,

the

requirement

for tolerance

to uncertainties'

Although a controller may be

designed

using

FDLTI

models,

he design

must be

implemented

and

operate

with

a

real

physical

plant. The

properties of

physical systems,

in

particular tle

ways

in

which

they

deviate

rom

finite'

dimensional

linear

models,

put

strict

limitations

on

the

for

the

sensitivity

operator

is well-understood-namely,

o,"

,.qoir.

that

the

complex

scalar

[l

+g(i@)k(itl)l-l

have

small

maCnitude,

r

conversely

hat

l+g(ia)k(i@)

have argemagnituds, or all real frequencies owhere tre

commands,

disturbances

and/or

plant

changes,

AG,

are

significant.

In

fact,

the

performance objectives

of

SISO

feedUact

systems

are

commonly

stipulated

in

terms

of

explicit

inequalities

of

the

form

( l

)

j-r

!

!S1:

,.i-l

:.in

(6)

(7)

Fig.

l. Staadard

fecdback

configuration.

*l,l ,.cf

^^le"q")



III.

UNcrnrervnrs

While

no nominal

design

model

G(s)

can emulate

a

physical

plant

perfectly,

t

is

clear

that

some

models

do

so

with greater

idelity

than

others.

Hence,

no

nominal

model

should

be

considered

complete

without

some

assessment

of its

errors.

We

will

call

these

errors

the

..model

unc€r-

tainties,"

and whatever

mechanism

is used

to

express

then

will

be

called a

"representation

of

uncertainty."

Representations

of uncertainty vary primarily in terms

of the

amount

of

structure

hey

contain.

This

reflects

both

our knowledge

of

the

physical

mechanisms

which

cause

differences

between

model

and plant

and

our

ability

to

represent

hese

mechanisms

n

a way

that facilitates

con_

venient

manipulation.

For

g;ampl€,

a set

membership

statement

for

the

parameters

of

an

otherwise

known

FDLTI

model

is

a

highly

structured

representation

of

uncertainty.

It

typically

arises

from

the,

use

of

linear

incremental

models

at

various

operating

points,

e.g.,

aerodynamic

coefficients

in

flight

control

vary

with

flight

environment

and

aircraft

configurations,

and

equation

coefficients

n

power

plant

control

vary

with

agini,

slag

buildup, coal composition,etc.

In

each

case,

he

amounts

of variation

and

any

known

relationships

betweenparam_

eters

can

be

expressed

by

confining

the

parameters

to

appropriately

defined

subsets

of

parameier

space.

A

specific

example

of

such

a

parameterization

or

the

F_SC

aircraft

is

given

n

[13].

Examples

of

less_structured

epre-

sentations

of uncertainty

are

direct

set

membership

state_

ments

for

the

transfer

function

matrix

of

the

model.

For

instance,

le

statement

IEEE

TRANsAcrroNs

oN AUToMATTC

oNTRoL,

vor.

lc-26.

No.

l. rrsnurnv

lggl

frequency

range

over

which

the

loop

gains

may

be large.

This

statement

confines

G' to

anormalized

neighborhood

In

order to

properly

motivate

these

estrictions,

we

digress

of

G. It is

preferable

over (12)

because

colpensated

in

Section

II

to a brief

description

of

the types

of

system

transfer

functions

have

the

same

uncertainty

representa-

uncertainties

most

frequently

encountered.

he

manner

n

tion

as the raw

model (i.e.,

the

bound (13)

applies

to

GK

which

these

uncertainties

an

be

accounted

or

in

MIMO

as

well

as to

G).

Still

other

alternative

set

membership

design hen forms the basis or the rest of the paper. statementsare the inverse orms of (12) and (13) which

confine (q)-'

to

direct

or

normalized

neighborhoods

about

G-r .

(r2)

The

best

choice

of

uncertainty

representation

or

a

specific

FDLTI

model

depends,

of course,

on the

errors

the

model makes.

n

practice,

t is

generally

possible

o

represent

some

of

these

errors

in

a highly

structured

parameterized

form.

These

are

usually

the low

frequency

error

components.

There

are

always

remaining

higher

frequency

etrors,

however,

which

cannot

be covered

this

way.

These

are

caused

by

such

effects

as

infinite-

dimensional

electromechanical

esonances

[16],

[17],

tim6

delays, diffusion processes,etc. Fortunately, the less-

structured

representations,

12)

or

(13),

are well

suited

to

represent

his latter

class

of

errors.

Consequently, 12)

and

(13)

have

become

widely

used

"generic,'uncertainty

rep-

resentations

or

FDLTI

models.

Motivated

by

these

observations,

we

will

focus

throughout

the rest

of

this

paper

exclusively

on the

effects

of

uncertainties

as represented

by

(13).

For

lack

of

a

better

name,

we will

refer

to tlese

uncertainties

simply

as

"unstructured."

We

will

assume

bat

G,in

(13)

sneins

4

strictly proper

FDLTI

system

and that

G, has

the same

number

of unstable

modes

as

G. The

unstable

modes

of

G'and

G do not need o be identical,however,and hence

Z(s)

may

be an

unstable

operator.

These

restricted

as-

sumptions

on

G' make

exposition

easy.

More

general

perturbations

(e.9.,

time

varying,

infinite

dimensional,

nonlinear)

can

also

be

covered

by

the

bounds

in

(13)

provided

they

are

given

appropriate

"conic

sector,'

inter-

pretations

via Parseval's

heorem.

This

connection

is

de-

veloped

n

[4J, [5]

and will

not

be

pursued

here.

When

used

to represent

the

various

high

frequency

mechanisms

mentioned

above,

1fue

quading

functions

/-(o)

in

(13)

commonly

have

the

properties

llustrated

in

Fig.

2. They

are small

(<<l)

at low

frequencies

And

in-

crease

to

unity

and

above

at

higher

frequencies.

The

growth with frequency inevitably

occurs

because

phase

uncertainties

eventually

exceed

r

180

degrees

nd magni-

tude

deviations

eventually

exceed

le

nominal

transfer

function

magnilud$.

Readers

who

are

skeptical

about

this

reality

are encouraged

to try

a few

experiments

with

physical

devices.

It

should

also

be noted

that the representation

f uncer-

tainty

in

(13)

can

be

used o include perturbation

effects

that

are in

fact not

at

all

uncertain.

A nonlinear

element,

for

example,

may

be

quite

accurately

modele4

but be-

cause

our design

sshniques

cannot

deal with

the

nonlin-

earity

effectively,

t

is treated

as a conic

linearity

[14],

U5].

As another example,we may deliberately choose o ignore

various

known

dlmamic

characteristics

n

order

to

achieve

a

simpler

nominal

design

model.

with

G'

r)

:

G(,r<,r

+

A

G

r)

o [AC( i& , ) ]

1 t " (o)

Vo)0

where

/"(.)

is

a

positive

scalar

function,

confines

the

matrix

G' to

a

neighborhood

of

G with

magnitude

,(co).

The

statement

does

not imply

a

mechanism

or struiture

which gives

ise

to

AG. The

uncertainty

may

be

caused

by

parameter

changes,

as above,

or

by

neglected

dynamics,

or

by

a host

of

other

unspecified

effects.

An

alternative

statement

or

(12)

s

the so-called

multiplicative

form:

G'

jr)

-

|

I +

t(

iot)

c(,r")

ol L(

j0,))

1l-(o)

Vco

0.

with

(

3)



AND

STEIN:

MULTIVARIABLE

FEEDBACK

DESIGN

'7

form of

the

standard

Nyquist

criterion,r

and

in

MIMO

cases,

hey

involve

its multivariable

generalization

18].

Namely,

we

require

that

the encirclement

count

of the

map

det[.I+GK(s)],

evaluated

on

the standard

Nyquist

D-contour,

be equal

to the

(negative)number

of

unstable

open

oop

modes

of GK.

Similarly,

for

requiremetl

2) the

number

of

encircle-

ments of

the

map

det[1+G'K(s)]

must

equal

the

(nega-

tive)

number

of

unstable

modes of

G'K.

Under

our

as-

sumptions

ot

G',

however,

his number

s the

same

as

that

of.GK.

Hence,

equirement

2) is satisfied

f and

only

if the

number

of

encirclements

of det[/+G'K(s)l

remains

unchanged

or

all G'

allowed

by

(13).

This

is assured

ff

det[1+

G'K]

remains

nonzero

as G

is warped

continu-

ously

toward

G',

or

equivalentlY,

ff

o

<

q[

+

|

r+.21s;

c(J)^K(J)

(14)

for all 0(e(1, all r on the D-contour,

and

all

Z(s)

satisfying

(13). Since G'

vanishes

on the

infinite

radius

segment

of

the

D-contour,

and assuming,

or simplicity,

that the

contour

requires

no

indentations

along

the

j<o-

axis,2

14)

reduces

o

the

following

equivalent

conditions:

0

<

q[

1 + G

o)

K(

j

ot)

+ eL

(

@)GUor

,K(

a,

)

]

(

s)

fo r a l l0 (e

(

l ,0 (c . t (o ,

and

a l lL

Another

important

point

is that

the construction

of

l-(<o)

for

multivariable

systems

s

not trivial'

The

bound

'#to-".

a

single

worst

case

uncertainty

magnituds

attli-

cable

to

all

channels.

If substantially

different

levels

of

uncertainty

exist

in

various

channels,

t

may

be

necessary

scale the input- output variables and/or aPply

,frequency-dependent

ransformations

15]

in

such

a

way

thai /- becomes

more

uniformly

tigbt.

These

scale

factors

<+o<s[/+

LGK(I+cK)- ' ]

for all 0

(

r.r

m, and

all

I

(16)

(17)

and

tiansformations

are

here assumed

o

be

part

of

the

;,nominal

model

G(s).

ry.

Fnnosecr

DrsrcN

IN

rr{E Fecr

or

UNsrnucrr-rnno

UNcent

lrvrrrs

Once

we specify

a

design

model, G(s),

and

accept

the

existenceof unstructured

uncertainties

n

the

form

(13)'

the

feedback

design

problem

becomes

one

of

finding

a

. . compensatorK(s) suchthat

i,,,.,

t)-the

nominai

feedback

system,

GKII+GKI-I,

is

i:.stable;

3)

performance

objectives

are

satisfied

Jor

all

possible

,

G'

allowed

y

(13).

...

All

three of

these

requirements

can

be

interpreted

as

!l-

frequency

domain

conditions

on

the

nominal

loop

transfer

i$iilr

matrix,

GK(s),

which

the

designer

must

attempt

to

satisfy.

Stability

Conditions

: '

',

The

frequency domain

conditions

for

requirement

l)

are,

of

course,

well

known.

In SISO

cases,

hey

take

the

e alcxlr+c,()- ' < t̂ (o\

for

all

0

(

o(

co.

The

last

of

these

equations

s the

MIMO

generalization

of the

familiar

SISO

requirement

that

loop

gains

be

small

whenever

the

magnitude

of unstructured

uncertainties

is

large. In

fact,

whenever

.(c,l)>>I,

we

get

the

following

constraint

on

GK:

olcK(i,,,)l

<r

/

t

^(ot)

for

all <^r

uch

hat

/-(<,r)>>1.

(

8)

'q; '

2)

the

perturbed ystem,

G'K[I+G'K7-t,

is

stable

or

,

;'l''

all

possible

'

allowed

y

(13); and

^ - ' l -

'

o - t - - - . ^ - - ^ - ^ ^ ^ L : ^ ^ 4 : . . ^ + i . f i - ; f n , n l l n n c s i h l o

We emphasize

hat

these

are

not conservative

stability

'

conditions. On the contrary, if the

uncertainties

are

truly

unstructured

and

(17)

is

violated,

then

there

exists

a

perturbation ,(s)

within

the

set

allowed

by

(13)

for

which

the system

s

unstable.

Hence,

these

stability

conditions

impose

hard

limits

on

the

permissible oop

gains

of

practi-

cal feedback

systems.

Pe jonnanc

e C

ondi

o

u

Frequency

domain

conditions

for

requirement

3)

have

already

been

described

in

(5)

in Section

II. The

only

lsce aay classicalcotrtrol tcxt.

2If

iadcntations

arc

rcquirc4

(la)

and

(f7)

Fust

hold

in thc

limit

for

aU

,ron tni-i"ac"tcd

patf, as thi

rddius

of

indcntation

is tokcn

to zcro.

LOG

FREOUETICY

Fig.

2.

Typical

behavior

of multiplicativc

perturbations.

i



modification

needed

to

account

for

unstructured

uncer-

tainties

s

to apply (5)

to

G'instead

of

G. i.e..

pr<sL

+( I+L)GKI

e ps(

olt+

tcxlr+GK)-tlo_lt+cxl

*

#b

<qlc.Ku',)l

l ^ ( r )<

o-[

:o_ft+Gx(ja;)- t f

rEEE

TRANsAcrroNs

oN AuroMATrc

coNTRoL,

vor.,qc_26,

No.

l,

FEBRUARv

ggl

COI{OITION

i,

f

.

( le)

fo11[

co uch

hat

I^(<o)

I

and

glcK(j6)l>>t.

This

is

the

MIMO

generalization

oi

another

familiar

SISO

design

rule-namely

that performance

objectives

can

be

met in

the

face

of

unstructured

uncertainties

f

the

nominal

loop gains

are

made

sufficiently

large

to

com_

pensate

or

model

variations.

Note,

however,

hat

finite

solutions

exist

only

in

the

frequency

ange

where

_(co)

l.

stability

and performance

conditlns

derivid

above

illustrate

that

MIMO

feedback

design

problems

do

not

differ fundamentally from their SlSO'counterparts. In

both

cases,

stability

must

be

achieved

nominally

and

assured

or

all

perturbations

by

satisfying

conditions

(17)

and (18).

Performance

may

tlen

be

optimized

by

satisfyl

ing

condition

(19)

as

well

as

possible.

What

distinguishes

MIMO

from

SISO

design

conditions

are

the

functions

used

to

express

ransfer

function

,,size,,,

Singular

values

replace

absolute

values.

The

underlying

concepts

emain

the

same.

We

note

that

the

singular

value

functions

used

n

our

statements

of

design

conditions

play

a

design

role

much

like

s1*1""

Bode

plots.-The

A[i+Cfl

fuo"tion

in (5)

is

the minimunxreturn differencemagnituAeof the closed_

loop

system,

[GK]in

(g)

and

otcKlin

(tg)

are

minimum

and

maximum

oop gains,

and

AtGK(I+'Gkl-rt

i<fa

is

the

maximum

closedJoop

requinry

response.

hese

can

all

be plotted

as

ordinary

frequency

dependent

unctions

in

order

to

display

and

analyzi

the

ieatures

of

a multivari-

able

design.

Such plots

will

here

be

called

o_plots.

One

of

the

o-plots

which

is

particularly

significant

with

regard

to

design

for

uncertainties

is

oUtainea

by

inverting

condition

17),

.e.,

DIT lON

( 1 7 )

Fig.

3.

Thc

dcsign

tradeoff

for

GK.

generally

not

tle

same.

They

can

be

analyzed

with

equal

ease,however,by using the function o_[I+(KG)-rl

in_

stead

of

rII+(GK)-

t1

in

1ZO;.

his

can-

be

readily

veri_

fied

by

evaluating

the

encirclement

count

of

the

map

det

I

+

KG)

under

perturbations

of

the

form

C,

:

G I +

L)

(i.e.,

uncertainties

eflected

o

the

input).

The

mathemati_

cal

steps

are

directly

analogous

o (15)_(lg)

above.

Classical

designers

will

recognize,

of

course,

that

the

difference

between

hese

wo

stability

robustness

measures

is

simply

that

each

uses

a loop

transfer

function

ap-

propriate

for

the

loop

breaking point

at which

robustneis

is

being

tested.

V. TneNsrnn FuNcrroN

LnsterroNs

The

feedback

design

conditions

derived

above

are

pic-

tured

graphically

n

Fig.

3. The

designer

must

find

a loop

transfer

function

matrix,

GK,

for

which

the loop

is

nomii_

nally

stable

and

whose

maximum

and

minimum

singular

values

clear

the higb

and

low

frequency

..disign

boundaries"

given

by

conditions

(17)

and

1fD.

ffre

nigh

frequency

boundary

is

mandatory,

while

the

low

frequency

one

is

desirable

for

good

performance.

Both

aie

in_

GK(r

Gn-'l

fluenced

by

the

uncertainty

bound,

^(r).

(20)

The

o-plots

of

a representative

oop

transfer

matrix

are

also sketched n the figure.As shown, he effectiveband-

width

of

the

loop

cannot

fall

much

beyond

the

frequency

or, or

which

l-(q,):

l.

As

a result,

the

frequency

range

over

which

performance

objectives

an

be

met

is

explicitly

constrained

by

the

uncertainties.

t

is

also

evident

from

the

sketch

that

the

severity

of

this

constraint

depends

on

the

rate

at which

o_lcKl

and

o[GKl

are

attenuated.

The

steeper

hese

unctions

drop

off,

the

wider

the

frequency

range

over

which

condition

(19)

can

be

satisfied.

Unfor-

tunately,

however,

FDLTI

transfer

functions

behave

in

such

a

way

that

steep

attenuation

comes

only

at the

:lp_:lr:

of

small

o[I+GKl

values

and

small

o[.f*

(GK)-

f

l

values

when

o[Gi(]

and.

[GKlxl. This meansthat

while

performance

s

good

at

lower

frequencies

and

stability

robustness

s

good

at higher

requencies,

oth

are

poor

near

crossover.

The

behavior

of

FDLTI

transfer

fo r

a l l0 {ar (oo.

The

function

on

the right-hand

side

of this

expression

s

an

explicit

measure

of

the

degree

of

stability

(or

stability

robustness)

f the

feedback

system.

Stability

is

guaranteed

for

all

perturbations

Z(s)

whose

maximum

sirrgrlu,

values

fall

below

it.

This

can

include

gain

or phase

changes

n

individual

output

channels,

simultaneous

changis

in

several

channels,

nd

various

other

kinds

of

perturbations.

In.effect,

o_[I+(GK)-

11

is

a reliable

multivariable

gen_

eralization

of

SISO

stability

maryin

concepts

(e.g.,

l1191encf

dependent

gain

and

phase

margins).

Unlike

the

SISO case,however, t is important to note that o[1+

(Gr()

-

tl

measures

olerances

br

uncertainties

t the piant

outputs

only.

Tolerances

or

uncertainties

at

the

inpui

are



D6YLE

AND

STEIN:

MULTIVARIABLE FEEDBACK DESIGN

functions,

herefore,

mposes

a second

major

limitation

on

the

achievable

erformance

of feedback

systems'

SISO

Transler

Function

Limitation

For

SISO

cases,

he conJlict between

attenuation

rates

anci

oop

quality

at crossover

s

again

well understood'

We

know

that

any

rational, stable,

proper,

minimrrm

phase

loop

transfer

function satisfies

fixed integral

relations

between

ts

gain

and

phase

components.

Hence,

ts

phase

angle

near

crossover

i.e.,

at

values

of

<.r uch

hat

lgkuor)l

erl)

is

determined

uniquely by

the

gain

plot

in Fig.

3

(for

-a-g=lgeD.

Various expressions

or this angle

were de-

rived

by

Bode

using

contour

integration around

closed

contours

encompassing

he right half

plane

[5,

ch.

13' l4].

One

exPression

s

''

Qs*"

argl

sk(i"")]

tnl

k(i@(v\.)l:tnlsk(io")l

.,

er )

sinh

y

where

v:ln(ot/t's"),

<,r(z):o"expz.

Since

the

sign

of

sinh(z)

is the same

as he sign of z, it follows tlat

f"o"

will

be

large

if the

gain

lg&l

attenuatesslowly and

small if it

attenuates

apidly. In

fact,

qgt"

is

given

explicitly

in terms

of weighted

average attenuation

rate

by

the following

alternate

orm of

(21) (also

rom

[5]):

es*"=+

-ry(r"*tnS)a,.

Q2)

The

behavior

of

qrr"

is sigaificant because

t

defines the

magnitudes

of our

two SISO

design conditions

(17)

and

(19)

at crossover.

pecially,

when

g&l-

l,

we have

l r+sft t : l t+(g/c)-r l : t1""(?)l

Q3)

The quantity

n*Qs*"

is the

plrase

margin

of the

feedback

system.

Assuming

gk

stable,

his margin

must

be

positive

for nominal stability and, according to (23), it must be

reasonably

arge

(=l

rad) for

good

return

difference and

stability

robustness

roperties.

f. riQrr"

is forced to be

very

small

by rapid

gain

attenuation,

he feedback

system

will

amplify disturbances

ll+g&l<<l)

and

exhibit

little

uncertainty

olerance

at and

near orc.

he conflict between

attenuation

rate and

loop

quality

near crossover

s thus

clearly

evident.

It is also

known

that

more

general

nonminimum

phase

andf

or unstable

oop

transfer

functions

do

not

alleviate

this

conflict.

If the

plant

has right

half-plane

zeros,

for

example,

t may be

factored

as

g(s ) : rn

(s )p(s )

(24)

where

n(s) is minimum

phase

andp(s)

is an all-pass

i.e.,

lp(io)l=l

Vo.)

The

(negative)

hase

angle of

p(s)

re-

duces

otal

phase

at

crossover,

.e.,

9

(2s)

:+l:*

Qg*c:Q^* " *Qp"

(Q^* "

and therefore

aggravateshe tradeoff

problem.

In

fact, if

l0r"; it too large,we will be forced to reduce he crossover

frequency.

Thus, RHP

zeros limit loop

gain

(and

thus

performance)

n

a

way similar to the unstructured

uncer-

tainty.

A measure

of severity of

this added

limitation

is

ll-p(j")|,

which can

be used

ust

like l-(a)

to constrain

a

nominal

minimum

phase

design.

If

g(s)

has

right half-plane

poles,

the extra

phase

ead

contributed

by

these

poles

compared

with their

mirror

images

n the

left half-plane

s needed o

provide

encircle-

ments or stability.

Unstable

plants

thus

also do

not offer

any inherent

advantage

over stable

plants

in alleviating

the crossover

onflict.

M ult oariable Generaliz tion

The above

transfer

limitations for SISO systems

have

multivariable

generalizations;

with

some

additional

com-

plications

as

would be expected.

The major complication

is

that singular

values

of

rational transfer matrices,

viewed

as functions

of

the complex

variable

J, are

not analytic

and therefore

cannot be used

for

contour

integration

to

derive relation

such as

(21).

Eigenvalues of

rational

matrices, on

the other

hand, have the necessary

mathe-

matical

properties.

Unfortunately,

they do not

in

general

relate directly

to the

quality

of the feedbackdesign.

More

is said about

this

in

Section

VI.) Thus,

we must combine

the

properties

of eigenvalues

nd singular

values

hrough

the bounding

relations

o_lAl<

^[ r ] l

<a l .q l

(26)

which holds

for any

eigenvalue,

r;, of the

(square) matrix

A.T\e approach

will

be

to derive

gain/phase

relations as

in

(21)

for

the eigenvalues

f

I+GK arid

+(GK)-I

and

to

use tlese

to bound

their

minimt'm singular

values'

Since

good

performance

and stability

robustness

equires

singular

valuesof

both of

these

matrices

o be sufficiently

large near

crossover,

he

multivariable

system's

roperties

can then be no better than ttre properties of their ei-

genvalue

bounds.

Equations

for

the eigenvalues

hemselves

are straight-

forward. There

is

a

one-to-one

colTespondence

etween

eigenvalues

l GK

and eigenvalues

f I

+

GK

such

that

I , l /+Gi ( ] : I + I , [G,<] .

Likewise

or .f+

(GK)

-

t

;

(27)

(28), [ r+1cr;- ' ] :

+

!

r,[Gr(]

Thus,when Ir[GK]l : I for some , and {o:rd", we have

l I , [

+

c/r] l

;r , f

+

1cr;- '

;

:'l''(*)l

@)



Since

this equation

is exactly analogous

o

(23)

for the

scalar

case,

and

since

l,,l

bounds

g, it

follows

that

the

loop

will exhibit

poor properties

whenever

he

phase

angle

(z*0r.") is small.

In oider to derive expressionsor the angle4^," itself,

we

require certain

results

from the

theory

of

algebraic

functions

t201-I261.

he

key concepts

needed

rom

these

references

re

that

the eigenvalues

,,

of

a

rational,

proper

transfer

function

matrix,

viewed as a

function

of

the

complex

variable

s constitute

one mathematical

entity,

tr(s),

called

an

algebraic

unction.

Each eigenvalue

r,

is

a

branch

of

this function

and

is

defined

on

one

sheet of an

extended

Riemann

surface domain.

On

its extended

do-

main

an algebraic

function can

be treated

as an

ordinary

meromorphic

function

whose

poles

and

zeros

are

the

system

poles

and

transmission zeros of

the

transfer

func-

tion

matrix.

It also has additional

critical

points,

called

branch

points,

which

correspond

to

multiple

eigenvalues.

Contour

integration

is

valid

on

the Riemann

surface

do-

main

provided

that contours

are

properly

closed.

In the contour

integral leading

to

(21),

g&(s)

may

therefore

be

replaced

by the algebraic

unction,

l(s)'

with

contour

taken

on

its Riemann domain.

Carrying

out

this

integral

yields

several

partial

sums:

IEEE

RANsAcrtoNs

N

AuroMATIc

oNTRoL,

ol.

rc-26,

xo. l, resnulnv

l98l

of

multivariable

nonminimum

phaseness nd

used

like

/-(<o)

to constrain

a

nominal minimum

phase

design.

VI.

Mwrrvnmesrn

DsslcN

sY

MoorRN

FnreusNcv DouxN Mnrnoos

So

far,

we

have described

he

FDLTI

feedback

design

problem as

a design

tradeoff

involving

performance

ob-

jectives

[condition

(19)],

stability

requirements

n the

face

of unstructured

uncertainties

condition

(17)]'

and

certain

performance

imitations

imposed

by

gan/phase

relations

which

must

be

satisfied

by realizable

oop

transfer

func-

tions.

This

tradeoff

is essentially

he

same

or

SISO

and

MIMO

problems. Design

methods

to carry

it out,

of

course,

are

not.

For

scalar

design

problems,

a

large

body

of

well-

developed tools exists (e.g.,

"classical

control")

which

permits

designers

o conshuct

good

transfer

functions

for

Fig. 3

with

relatively

little difficulty.

Various

attempts

have been

made

to extend

hese

methods

o

multivariable

design

problems. Probably

the

most

successful

of

these

are the

inverse

Nyquist

array

(INA)

[3]

and

the

character-

istic

loci

(CL) methodologies

4].

Both

are based

on

the

idea of

reducing

the multivariable

design

problem to a

sequence

f

scalar

problems.This is done

by

constructing

a set

of

scalar

transfer

functions

which

may be

manipu-

lated more

or

less

ndependently

with

classical

echniques.

In the

INA

methodolory,

the scalar

functions

are

the

diagonal

elements

f a loop

transfer

unction

matrix

which

has been

pre- and post-compensated o be diagonally

dominant.

In the CI-

methodology,

he

functions

are the

eigenvalues

f

the loop

transfer

matrix.

Basedon

tlre design

perspective

eveloped

n the

previ-

ous

sections,

hese

multiple singleJoop

methods

urn

out

to be reliable

design

ools only

for special

ypes

of

plants.

Their restrictions

are

associated

with

the fact

that the

selected

set

of scalar

design

unctions

are

not necessarily

related to

the

system's

ctual

eedback

properties'That

is,

the feedback

system

may

be designed

so that

the

scalar

functions

have

good

feedback

properties

f

interpreted

as

SISO

systems,

ut

the resulting

multivariable

system

may

still havepoor feedbackproperties'This possibility s easy

to demonstrate

or the

CL

method

and,

by

implication,

for the

INA

method

with

perfect

diagonalization'

For

these

cases,

we attempt

to achieve

stability

robustness

y

satisfying

(31)

for all

i and

0(ro(o

and

similarly,

we attempt

to

achieve

performance

objectives

by making

rs(ol)

=ry

<

r,[

+

cK]l:

ll

+I,(clK)l

(33)

l - l ^ ( a )

f o r a l l i a n d 0 ( o ( c o .

As discussed

n Section

V, however,

he eigenvalues

n

the right-hand

sides

of

these expressions

re

only

upper

)

+^,,

:

[*

>

i , ' - - @ i

Iml

,(1"(r)l-ul

,U'")l

r,

sinh

y

(30)

where each

sum

s

over

all branchesof

tr(s)

whosesheets

are connected

by

right half-plane

branch

points.

Thus the

eigenvalues

tr,}

are

restricted n a

way

similar

to

scalar

transfer

functions but

in

summation

form.

The surnma-

tion,

however, does

not alter the fundamental

tradeoff

between

attenuation

ate and

loop

quality

at crossover.

n

fact, if

we

deliberately

chpose o

maximize he bound

(29)

by

making o" and

f^,"

identical

for all i, then

(30) mposes

the

same

estrictionson

multivariable

oops as

(21)

impo-

ses

on SISO

loops.

Hence, multivariable

systems

do not

escape

he fundamental

ransfer

functibn limitations.

As

in

the

scalar case,

expression

30)

is

again

valid for

minimum

phase

systemsonly.

That is,

GK

can

have no

transmission

zeros3 n

the right half-plane.

If this

is not

true, the

tradeoffs

governed

by

(29)

and

(30)

arc aga-

vated because

every

right half-plane transmission

zero

adds

the same

phase

ag as

in

(25)

to one of

the

partial

sums

n

(30).

The matrix GK

may also be factored,

as

in

(24),

to

get

GiK(s):M(s)P(s)

where M(s)

has no

right

half-plane zeros and

P(s) is an

all-pass

matrix

fr1-s;f1s;:L

Analogous

o the

scalar

case,

o-(/-P(s))

can

be

taken as a

measureof the degree

3For

our

purposcs, raasnission

zcros-[41]-q:9--values

i

such that

dcttG(J)K(r-)1j0.

Degcncratc

ystcms

with

dct[GrK]:0

for all 'r are

not

of

ilticit

ieiiusc

thcy canaot

mcct

condition

(19)

in

Fig' 3.

t ^ ( , )< ; r , [ r+1cr ) - ' ] l :11

/ I , (c rK) l

(32 )



l l

DOYLE

AND

STEIN:

MULTIVARIABLE

FEEDBACK DESIGN

los

(t )

Fig.

4. o-plots

for /+G-r.

[G

from

(3a).]

bounds

for

the

true stability

robustnessand

performance

conditions

20)

and

(19).

Hence, o-lI+(GK)-t1

and/or

o_ll

+ GKI

may actually

be

quite

small

even

when

(32)

and

(33) are

satisfied.

An

ExarnPle

These

potential

inadequacies

n the INA

and

CL

meth-

ods

are

readily

illustrated

with a simple

example

selected

specifically

to highlight

the limitations.

Consider

G(s)=t,*fu

|

-ff,*'

;3:.r]

(34)

This

system

may

be diagonalized

exactly

by

introducing

constant

compensation.

et

,:l 'u] '-': l l

-t]

Fig.

5.

stabirityrcgionsror

{r.i?

;]}

"

[Grron(34).]

ity robustness

s

given in Fig. 5.

This

figure

shows

stability

regions

in

"gain

space"

for tle

compensator

K(s):

diag(l

+kbl+/<r).

The

figure

reveals

an unstable

egion

in

close

proximity

to the

nominal

design

point.

The

INA

and

CL

metlods

are

not

reliable

design

ools

because

hey

fail to

alert

the

designer

o

its

presence.

This

example

and

the

discussion

which

precedes t

should

not

be

misunderstood

s a universal

ndictment

of

the

INA

and

CL

methods.

Rather,

it

represents

caution

regarding

leir

use.

There

are

various types

of

systems

or

which the

methods

prove

effective

and

reliable.

Condi-

tions which tlese systemssatisfy can be deduced fiom

(32)

and

(33)-namely

they

must

have tight

singular

value/eigenvalue

bounds.

This includes

naturally

diago-

nal

systems,

of course,

and

also

the class

of

"normal"

systems

28].

The

limitations

which

arise

when the

bounds

are not

tight

have

also

been

recognized

n

[4].

We note

in

passing

hat the

problem

of

reliability

is not

unique to

the

INA and

CL methods.

Various examples

can be constructured

to show

that

other

design ap-

proaches

such

as the

"single-loop-at-a-time"

methods

cornmon

in engineering

practice

and

tridiagonalization

approaches

uffer

similarly.

VII.

Murrrvenresrr

DssrcN

VH LQG

A second

major

approach

to

multivariable

feedback

design

s

the

modern

LQG

procedure

lU'

[12]'

We have

already

introduced

this

method

in

connection

with the

tradeoff

between

command/disturbance

error

reduction

and

sensor

noise

error

reduction.

The

method

requires

that

we select

stochastic

models

for

sensor

noise,

com-

mands

and

disturbances

and

define

a

weighted

mean

square

error

criterion

as

the

standard

of

goodness or the

disign.

The

rest

is automatic.

We

get

an

FDLTI

com-

p"ooto. K(s) which stabilizes he

nominal

model

G(s)

(under

mild

assumptions)

and

optimizes

the

criterion

of

goodness.All

too

often,

of

course'

the

resulting

loop

(35)

i +

o l

G:uGU- t : l

" * '

n l .

t go l

I

o

* t )

If the

diagonal

elements

of

this G

xe

interpreted

as

independent

SISO

systems,

as

in the

INA

approach,

we

could

readiiy

conclude

that

no

further

compensation

s

necessary

o achieve

desirable

feedback

properties. For

6lampl€, unity feedbackyields stability marginsat cross-

over

of

+

oo dB

in

gain

and

greater

than

90

degrees

n

phase.

Thus,

an

INA

design

could

reasonably

stop

at this

point

with compensator

(s): UU

-t:/'

Since

he

diag-

onal

elements

f G

are

also

he

eigenvalues

f G,

we

could

also

be reasonably

satisfied

with this design

rom

the CL

point

of

view.

Singular

value

analysis,

however,

leads

to

an

entirely

different

conclusion.

The

o-plots

for

(/+G-r)

are

shown

in

Fig.

4. These

clearly

display

a

serious

ack of

robustness

with

respect

to

unstructured

uncertainties.

The

smallest

value

of o

is approximately

0.1

neat

u:2

rad/s.

This

means hat multiplicative uncertainties as small as l^(2):

;

0.1

(avl}Vo

gain

changes,

=6 deg

phase

changes)

could

produce

instability.

An

interpretation

of

this

lack

of stabil-



IEEE

TRANsAcrloNs

oN

AUToMATIc

oNTRoL,

vor' lc-26'

No

l'

rrgnuenv

l98l

t2

transfer

functions,

GK

or

KG,

are

entirely

unacceptable

,"i"r,

"*"-ined

against

the

design

constraints

of

Fig'

4'

w,u,. t trenforcedtoi teratethedesigrr_adjustweigbts

in

the

performance

criterion,

change

the

stochastic

dis-

irrrbuoce

and

noise

models,

add

dynamics'

etc'

There

are

so

many

parameters

o

manipulate

that

frustration

sets

n

qrlrny

"ia

the

schism

between

practitioners

and

theoreti-

"i*t L""o.es easier o understand'

Fortunately,

such

design

terations

o!

I-$C

controilers

have

becomi

easier

to

iarry

out

in

the

last

few

years

because

he

frequency

domain

properties-

f

these

con-

;Ji;*

are

bettei

undlrstood'

Some

of

the

key

new

results

a1s

s11--arized

below

and

their

significance

with

respect

to

Fig.

3

are

discussed.

or

our

purposes'

LQG

controllers

oi"oiAio"ty

FDLTI

compensatott

itLl

special

nternal

structure.

This

structure

s shown

in

Fig'

6 and

is

well

known.

It

consists

of

a

Kalman-Bucy

filter

(KBF)

de-

signed

for

a

state

space

realization

of

the

nominal

model

c?ri,

i""fuai"g

uU

upp"ttded

dynamics

for

disturbance

ptI*.t"t, comiands,-integral action'

etc'

The

model

is

* :Ax i

But t ;

x€R ' ,

ue

R^

(37)

y :Cx* t l ;

YeR '

As

noted

eadier,

he

first

three

functions

measure

perfor-

mance

and

stability

robustness

with

respect

o

uncertain-

ties

at

the

plant

outputs

Qoop-breaking

oint

(i)

in

Fig'

6)'

and

the

sicond

three

measure

performance

and

robust-

nesswithrespecttouncertaint iesattheplantinputQoop.

[t.uti"g point (tt) in

Fig.

6)'

Both

points

are

generally

significant

n

design'

-T;;

other

loop-breaking

points,

(i)' and

(ii)"

are

also

shown

n

the

figure.

These

are

nternal

to

the

compensator

una

tfr"t"fote

have

ittle

direct

significance'

However'

they

have

desirable

oop

transfer

properties

which

can

be

re-

lated

to

the

properties

of

points

(i) and

(ii)'

The

proPer-

ties

and

connections

are

these'

Fact

l:

The

loop

transfer

function

obtained

by

break-

ing the

LQG

loop

at

point

(i)'

is the

KBF

loop

transfer

function

CQK..

Fact

2:

thJ

loop

transfer

function

obtained

by

break-

i"g

in"

LQG

loop

at

point

(t)

is

G{('.It

9an

be

made

to

up"ptoo"ft

-CLXrpointwise

in s by designing he LQR in

."aorduo"a

witi

a

"sensitivity

recovery"

procedure

due

to

Kwakenaak

[29].

Fact

3:

The

loop

transfer

function

obtained

by

break-

ing

the

LQG

loop

at

point

(ii)' is

the

LQR

loop

transfer

function

K"QB.

Fact

4:iLe

loop

transfer

function

obtained

by

break-

i"g

tn"

LQG

loop

at

point

(tt) is

t(G'.It

can

be

made

to

up?io*n

k"oa

poi"t*ise

in

s

by

designing

the

KBF

in

"f"ord*""

oritt

u..robustness

ecovery"

procedure

due

to

Doyle

and

Stein

30].

Facts

I

and

3

can

be

readily

verified

by

explicit

evalua-

tion of the transfer functions involved' Facts 2 and 4 take

moreelaborationandaretakenupinalatersection.They'

also

require

more

assumptions'

Specifically'

G(s)

must

be

-i"rt

tt"

phase

with

m)r

lor

Fact2'

mlr

fot

Fact

4'

and

hence,

G(s)

must

be

square

or

both'

Also'

the

n'unes

;r"*i

i"i

y

redvery"

and

"robustness

recovery"

are

oveily

restrictive.

,Full-state

loop

transfer

recovery"

s

perhaps

a

tetter

name

for

both

procedures,

with

the

distinction

that

one

applies

o

points

(t),

(t)'

and

the

other

to

points

(ii)'

(rr)'.

tn"

significance

of

these

four

facts

is

that

we can

.

a"G

i6c

toop

transfer

funcrions

on

a

full-state

feed-

Uaci Uasisand tlen approximate

them

adequately

with

a

recovery

procedure.

For

point

(i), the

full

state

design

must

b;

ione

with

the

KBF

design

equations

(i'e"

its

Riccati

equation)

and

recovery

with

the

LQR

equations'

while

for

point

(ii),

full-state

design

must

be

done

with

tl"

t-qn'equations

and

recovery

with

the

KBF'

The

mathematics

f

these

wo

options

are,

n fact'

dual.

}Ience,

we

will

describe

only

one

option

[for

point

(ii)]

in furt]er

detail.

Results

or

tie

other

are

stated

and

used

later

in

our

examPle'

Full-State

I'ooP

Transfer

Design

The

intermediate

full-state

design

step

is

worthwhile

because

LQR

and

KBF

loops

have

good classical

prop€r-

and

satisfies

with

G ( " ) : C O ( s ) B

d:CO(s) {

@@(s) :

s I^ -A) - l

(38)

(3e)

The symbols { and 4

denote

the

usual

white

noise

pro*r."..

The

filter's

gains

are

denoted

bV

Kt

a1f

ilt :?t"

estimates

by

i.

The

state

esti:nates

are

multiplied

by

full-state

linear-quadratic

regulator

(LQR)

gains'

d'

to

ptoA""" the

contiol

commands

which

drive

the

plant

and

'are

also

fed

back

internally

to

the

KBF'

The

usual

condi-

tions

for

well-posedness

f

the

LQG

problem

are

as-

suned.

In

terms

of

previous

discussions,

he

functions

of

inter-

"ri

io

nig.

6

are

the

loop

transfer,

return

difference'

and

stability

robustness

ulctions

GK,

I ,+GK,

I ,+(GK)- t ,

and

also

their

counterParts

K G ,

I ^ + K G ,

I ^ + ( K G ) - r '

* L i = o ; + g u + ( ! .

Fig.

6.

LQG

fecdback

ooP'



I J

":.

DOYLE

AND

STEIN:

MULTIVARIABLE

FEEDBACK DESIGN

ties

which

have

been

rediscovered

over

the

last

few

years

l31-t331.

The

basic

esult or the

LQR case

s

that

LQR

ioop

transfer

matrices

r(s)

3

r<"o(s)B

(40)

satisfy

he

following

return difference

dentity

[32];

I

t

^

+ T(

a)]

*Rl

r- + r(lco)

:R+

|

naQot)Bl'l

nolir\nl

v0(co(

m

(41)

where

R:Rr

>0

is

the standard

control

weighting

matrix,

a31d

TH:Q>0

is

the corresponding

tate

weighting

matrix.

Without

loss of

generality,

H can

be

of size

(mxn)

[34].

Using

the

definitions

(6)

and

(7)'

(41)

with

R:pI

implies

hat

I-

o,l ^+ r(i"\):\A,L I + - ( HaB)* aB

)

This implies

[35]

that

o_l

^

+

r

-

|

(io;)l

> t z

Hence,

LQR

loops are

guaranteed

o

remain stable

or all

unstructured

uncertainties

reflected to

the input)

which

satisfy

^(a)10.5.

Without

further

knowledge

f

the types

of

uncertainties

present

in the

plant,

this

bound

is the

greatest

obustness

uaranteewhich can

be ascribed

o the

regulator.s

While

it is

reassuring

o

have

a

guaranteeat all,

the

I^<0.5

bound

is clearly

nadequate

or the

requirements

of

condition

20)

with

realistic

^(a)'s.In

order

to

satisfy

condition

(20)

n

LQR designs,

herefore,

t becomes

ec-

essary

o

directly

manipulate

he

high-frequency

behavior

of

Z(s).

This

behavior

can be

derived

from

known

asymptotic

propertiesof the

regulator

as

he scalar

p

tends

to zero

l2gl,l47l,

[48], [34],

[36].

The

result

needed

here

s

tfiat under minimum phase assumptionson H@8, tJl,e

LQR

gains

K"

behave

asymptotically

as

[29]

t/

p

K"--WH

(46)

where

W

is an

orthonormal

matrix.

The LQR

loop

trans-

fer

function,

T(s),

evaluated

at'

high

frequencies,

s:jc/l

p

with c

constant,

s

then

given

by6

r{,i,

|

p

)

:

I

p

x

"(ict

Vi A)-'

B

-+l{HB/jc.

(47)

Since

crossovers

ccur

at o,lrl:1,

this

means that

the

maximum

(asymptotic)crossover

requencyof

the

loop is

, "-* :d[

HB)/{

p

(48)

As shown

in Fig

3, this

frequency

cannot

fall much

beyond co;,

where

unstructured

uncertainty

magnitudes

approach

unity.

Hence,

our

choice

of fl and

p

to achieve

the

performance

objectives

ia

(43)

are

constrained

by

the

stability

robustness

equirement

via

(a8).

Note

also

from

(47)

that

the asymptotic

oop transfer

function

in the

vicinity of crossover

s

proportional

to

|

/

ot

(- I slope on logJog plots). This is a relatively slow

attenuatioo

rate

which,

in view of

Section

V, is the

price

the

regulator

pays

for its

excellent

return difference

prop-

erties.

If l^t(")

attenuates

aster

than

this

rate, further

reduction

of o"

may

be

required.

t

is also

true, of

course'

that no

physical system

can

actually

maintain

a

l/a

characteristic

ndefinitely

[6].

This

is not

a concern

here

since

I(s)

is a

nominal

(design) unction

only and

will

5Tbc

r-<0.5

bound

turos

out to bc

tigbt

for

pure

Sait

changcs,

,c.,

0.5<+6

dg,

which is identical

to

rcgulato/s

cclcbratcd

guarantocd

sain

marsin

t331.

The

bound

is conscrvative

f thc

unccrtainties

are known

to

bc

pirc'pnisc

changcs,

.c., 0.5<+*30

dcg which

fu t63

than

tbc known

-rtiO

dcg guarantcc 33].trnir-ftciri"

Utoitttig

process s appropr-late

or

thc so-callcd..F-oi:

casc

136l

with

full nnk

HB.

Morc

gcneral vcnilons

ot

()))

wrur ra$(

tHBi<;n

are

dcrivcd

n

['141.

V0

(

<^r

co.

(45)

:Vt

*:r , [1nor;*HaBl

(42)

This

expression

applies

to

all

singular

values

o, of

I(s)

and,

hence,

specifically

o

o

and o-. t

governs

he

perfor-

manceand

stability

robustness

roperties

of

LQR

loops.

PerJormance

ropert

es

(

Condition

I 9)

Whenever

g.[f

]>>

, the

following approximation

of

(42)

shows explicitly

how

the

parameters

p

and

.FI

nlluence

(s ) :

o,lr1

"1loo,l

o1

u1

l

fi

.

(43)

We

can

thus

choose

p

and

H

explicitly

to

satisfy

condition

(19)

and also

to

"balance"

the

multivariable

loop

such

that

o[7] and

o[Z]

are

reasonably lose

ogether.4

his

second

objective

is consistent

with our assumption

in

Section II that the transfer unction G(s) hasbeenscaled

and/or

transformed

such

at /-(co) applies

more

or less

uniformly

in all directions.

This is also

the

ustification

for

considering

control

weighting

matrices n

the form

R:pI

only.

Nonidentity

R's

are subsumed

n

G

as GRt/z.

Robustnessroperties

Conditions

17)

and

(20)l

It also follows

rom

(42)

that

the LQR

return

difference

always

exceeds

nity,

i.e.,

s l l ^+Z( jco) ]>

I V0( to(o .

(44)

alt

mav also bc

ncccssarv

to

appcnd

additional

dynamics.

Ia ordcr

!o

achic,vc

z]aro st€ady

statc

cirors,i6r

cxample,

olHABl

must

tetrd

!o €

8s

ar+O.

This may

rcquirc

additional

integratioas

ia

thc

plant"

r+

|"?tuo(io)Bl



1 4

later be approximated

by

one of

the

full-state oop trans-

fer recovery

procedures.

Full-State Loop

Transfer

Recooery

As

described

adier, he

full-state

oop

transfer unction

designedabove

for

point

(ii)'

can

be recovered

at

point

(ii)

by a modified

KBF

design

procedure.

The required

assumptions

are that r)

m and

that

C@8

is

minimum

phase.

The

procedure

hen

consists

of two steps.

l) Append

additional

dummy

columns to .B

and zero

row

to K" to make

COB

and r("O.8

square

(rxr).

CQB

must remain

minimum

phase.

2) Design the KBF

with

modified

noise intensity

matrices.

E(€€r):l

uo+

qran']a1r-

;

E(\ ' f) :Nod(t-r)

where

M6, No

are

1fos

n6minal

noise

intensity

matrices

obtained from

stochastic

models

of

the

plant

and

4

is

a

scalar

parameter.

Under

tlese

conditions,

t is known

that

the

filter

gains

K, have

the following

asymptotic behavior

as

q-+m

[30]:

Ixr.-nwul'r,

(4e)

Here lV'is

another

orthonormal

matrix,

as

in

(46).

When

this K, is

used in the

loop

transfer

expression for

point

(r'i),

we

get pointwise

oop

transfer

recovery

as

q-+co,

i.e.,

rEEE RANsAcrroNs

oN AUToMATTc

oNTRoL,

vor.

ec-26, No.

l, FEBRUARv

9g l

as

the target LQR

dynamics

satisfy Fig.

3's constraints

(i.e.,

as long

as

we

do not

attempt

inversion

n frequency

ranges where

uncertainties

do not

permit

it). Closer in-

spectionof

(50)-(56)

further

shows

hat there s no

depen-

dence

on LQR

or KBF

optimality

of

the

gains

K" or Kr.

The procedurerequires only that Kp be stabitizing and

have

the asymptotic

characteristic

6r.

Thus,

more

gen-

eral state eedback

aws

can

be recovered e.g.,

pole place-

ment), and more

general

ilters

can be used or

the

process

(e.g.,

bservers).7

An Exanple

The

behavior

of LQG

design

terations

with full-state

loop

transfer

recovery

is illustrated

by the following

ab-

stracted

ongitudinal

control

design

example

or

a CH4Z

tandem

rotor

helicopter.

Our

objective

is

to control

two

measured

outputs-vertical

velocity

and

pitch

attitudc-

by manipulating collective

and

dilferential

collective rotor

thrust

commands.

A

nsmins,l

model

for

the

dynamics

relating

thesevariables

at 40

knot

airspeed

s

[a5]

-o.u

0.005

2.4

-32

-0.14

0.4

-

1.3

-30

0

0.018

-

1.6

t .2

0 0 1 0

] '

r

d l

u-:l

Io . r+

-0 .12

I

.13:3!

i'tuJ,

, : [3

3

?,] '

K(s)G(s):K"[O-t

+BK"+KrC7-'K]C@B

(50)

:c[6-or,

Q,

+

coxr)-'.6]

KrceB

(51)

(s2)

(53)

(54)

(55)

(56)

In this

series

of expressions,

D

was

used

to represent

the

matrix (sI,-A+BK)-r,_(49)

was

used

o

get

from (52)

to

(53),

and the identity

QB:@B(I+K"O.8)-t

was

used

to

get

from

there

to

(5a).

The

final

step

shows

explicitly

that the

asymptotic

compensator

K(s)

[the

bracketed

erm

in

(55)l

inverts

the

nominal

plant

(from

the left)

and

substitutes

the desired

LQR

dynamics.

The need

for

minimum

phase

s

thus

clear, and it is also evident that

ttre

entire recovery

procedure

is

only

appropriate

as

long

:x"oxr(t,+cdxr)-tcon

--+r"o,a(co

)-tcon

=KpB(1 ,+KpB) - '

.

lcon14+ "or)

I

]

-'

ceB

:

{r"o,a1cos)-t}coa

:K"98.

Major

unstructured uncertainties

associated with

this

model

are due to neglected

rotor

d5mamicsand unmod-

eled rate limit nonlinearities.

These

are discussed

t

greater

length

in

t46].

For

our

present purposes,

t, suffices

to note

that

they are uniform in

both control channels

and that

I^(ot)) I for

all

co) l0 rad/s.

Hence,

he controller

band-

width

should be constrained

as

in

Fig. 3 to

ro"-o

(

10.

Since

our

objective

s to

control

two

measured

outputs

at

point

(i),

the

design iterations

ut'lize

the

duals

of

(40)-(56). They begin with a full state KBF designwhose

noise

ntensity

matrices,

(€€r):11161r-r;

and, (qf

)

:p/d(r-r),

are selected

o

meet

performance

objectives

at

low frequencies,

.e.,

qlrlxqlcarl/{p

>ps, (s7)

while

satisfying

stability robustness

constraints

at high

frequencies;

o"-*

:o-[

CI ]

/

|

p

(

l0 r/s.

(58)

?Sti[

morc

gcnclally,

thc modificd

KBF

proccdure

will

actually

r+

covsr full-statc fccdback loop transfcr functions at any point, ul, ii thc

systcE

for which

CO.8l is

m-ininum

phasc

t30l.



DOYLE

AND

STEIN:

MULTIVARIABLE

FEEDBACK

DESIGN

FULL

STATE

KBF

l 5

classical

SISO

approaches

o

this

issue

can

be

reliably

generalized

o

MIMO systems,

nd

has

defined

he

extent

to

which

MIMO

systems

are subject

to the

same

uncer-

tainty

constraints

nd

transfer

unction

gain/phase

imita-

tions

as SISO

ones.

Two categories

f

design

procedures,

were then

examined

n the

context

of

these

esults.

There are numerous

other

topics

and

many

other

pro-

posed

design

procedures

which

were

not

addressed'

f

course.

Modal

control,

42]

eigenvalue-vector

ssignments'

[43]

and

the

entire

field of

geometric

methods

19]

are

prime

examples.

These

deal

with internal

structural

prop-

erties

of

systems

which, though

important

theoretically,

cease

o

have

central

mportance

n the

face

of

the

input-

output

nature

of

unstructured

ncertainties.

Hence,

they

were

omitted.

We also

did

not treat

certain

performance

objectives

n

MIMO

systems

which are

distinct

from

SISO

systems.

These

nclude

perfect noninteraction

and

integ-

rity.

Noninteraction

s again

a structural

property

which

loses

meaning

n the

face of

unstructured

uncertainties'

It

is achievedas well as possibleby condition (19)'l Integr-

ity, on

the

other

hand, cannot

be dismissed

as

lightly'

It

"orr""*.

the

ability

of

MIMO

systems

o

maintain

stabil-

ity

in

the

face

of

actuator

and/ot

sensor

ailures'

The

singular

value

concepts

described

here

are

indeed

useful

for

integrity

analysis.

For

example,

a design

has

ntegrity

with respect

o

actuator

failures

whenever

q [1+( , (G)- ' ]>

I

vc , r .

(59)

This

follows

because

ailures

satisfy

^(l'Moreover

it

can

be

shown

37]

that

full-state

control

laws

designed

ia

Lyapunov equations,as opposed o Riccati equations,as

in Section

VII, satisfy

(59). It

is also

worth

noting

that

integrity

properties

claimed

for

design

methods

such

as

INe

ana

Clsuffer

from

the

reliability

problem

discussed

in Section

VI

and,

hence'

may

not

be

valid

in

the

system's

natural

(nondiagonal)

coordinate

system'

The

major

limitations

on

what

has

been

said

in the

paper are

associated

with

the

representation

chosen

n

Section

III

for

unstructured

uncertainty'

A single

magni-

tude

bound

on

matrix

perturbations

s

a

worst

case

epre-

sentation

which

is often

much

too

conservative

i'e', it

may

adrnit

perturbations

which

are

structurally

known

ooito occur).The useof weightednorlns

in

(8)

and

(9)

or

selective

transformations

applied

to

G

(as in

[39])

can

alleviate

this

conservatism

somewhat,

but

seldom

com-

pletely. For

this

reason'

he

problem

of

representing

more

structured

uncertainties

n

simple

ways

analogous

o

(13)

is

receiving

enewed

esearch

attention

[38]'

A

second

major

drawback

is

our

imptcit

assumption

that all

loops

(ail

directions)

of

the

MIMO

system

should

have

equai

band-width

(g close

to

d

in

Fig'

3)'

This

"rro-ptioo

is consistent

*ittt

u

uniform

uncertainty

bound

but

will

no

longer

be

appropriate

as

we

learn

to

represent

more

complex

uncertainty

structures'

Research

long

hese

lines

is

also

proceeding.

0 .

l . O

1 0 .

1 0 0 .

F N E Q U I T C Y

/ 5

Fig.

7.

Full-statc

loop

traasfer

recov€ry.

,

properties

or

condition

19),with

low

gainsbeyond

or:

l0

,t

or condition

(20).e

t

then

remains

to

recover

this

func-

tion

by

means

of

the

full-state

recovery

procedure

for

point

(i). This

calls

or

LQR

design

wth

2:9o+qzcCr

and R=Ro.

Letting

Qo:0,

Ro:/,

the

resulting

LQG

transfer

unctions

or

several

alues

of

q

ate

also

shown

n

Fig.

7. They

clearly

display

the

pointwise

convergence

'

function

will be

considered

o have

the

desired

high

gain

,

properties

of the

procedure.

For the

choice

=8,

(58)

constrains

p

to be

greater han

or equal

to

unity.s

The

resulting

KBF

loop

transfer

for

p:lls shown

n

Fig.7.

For

purposes

f

illustration,

his

VIII.

CoNcrusroN

This

paper

has

attempted

o

present

a

practical

design

petspeciilne

n

MIMO

linear

time

invariant

feedback

con-

trol

problems. t

has

focused

on

the

fundamental

ssue-

feedback

n

the

face

of

uncertainties.

t

has

shown

how

8If

Cf

(or

IfB)

is singular,

(58)

or

(48)

are

still

valid

in lhe

oonzcro

directions.

eibi

tunction

should

not

bc

considercd

final, or

coursc'

Bcttcr

bal-

i

t"""

Ltro"

i

""4"

""a

grcatcr

gain

at

low.frequcncies

via appcndcd

itrtcgraton would bc?esirablc in a seriousdcsip.



l 6

rErE TRANsAcrloNs

oN AuroMATIc

coNTRoL,

vor. lc-26,

No. l.

rrsnuenY

l98l

l35l

[36]

tu

I2l

t31

t41

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the inversc

Nyquist atay,"

Proc. IEEE,

vol. llO

no. ll,

Nov.

1969.

A.

G. J.

MacFarlane and N.

Karcanias,

"Poles

and

zeros of

linear

multivariablc

systeurs:

A

survey

of algebraic,

gcometric,

and

com-

olex

variable heory."

Int.

J.

Contr.,pp.33-74,

July

1976.

i. D.

simon and

-S.

K. Mitter,

"A-

theory of

modal

control,"

Inform,Cont.,

vol.

13,

pp.

316-353,

1966.

B.

C. Moore,

-On

thi

-flexibility

offcred

by state

feedback

in

multivariable systemsbeyond closed loop eigenvalueassignment,"

IEEE

Trans, utottut. Contr.,

vol. AC-21,

pp'

685-692,

1976.

G. St€itr,

"Asymptotic

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functions

of LQ

regulators

and

filten,"

ia

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G.

L.

Hartnann, G.

Stein, and S.

G.

Pratt,

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adaptive

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Landev

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Nov.

1979.

G. $tein

and J. C.

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Design

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n Proc. 1978

Allerton

Conl.,Urbana,

IL, 197E.

Shak;4

"The

asymptotic bchavior

of

-the

root loci

of multivariablc

optimal

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IEEE

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425-430'

June

1978.

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. t

I

I

Iet

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trzl

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Johtr

C.

Doyle

reccived

thc S.B.

and

S.M'

de-

grc€s

in electrical

enginccring

from

the Mas-

sachusetts

nstitutc

of Technologt

n lW.

Hc

now

divides

his time

betwcen

consulting

work for

Honeywell

Systems

and

Research

Center

in Minneapolis,

MN

aad

graduate

stud-

ies

in mathcmatics

at

thc University

of Cali-

fornia

at

Berkclcy.

His

research

nterest

in the

arca

of

automatic

control

is in

developing

theory

that

is relcvant

to

the

practical

problcms

faccd

by engineers.

Gunter

Steln

(S'66-M'69)

received

he B.S.E.E

degrec

rom

General

Motors

Institute,

Flint,

MI'

and the

M.S.E.E.

and

Ph.D.degrces

rom Purdue

University,

West Lafayette,

IN, in

1965 and

1969,

espectivelY'

Sincc

1969

hc has been

with

thc Honcywell

Systems

and

RescarchCetrtcr,

Minn€apolis,

MN.

As

of January

97,be

alsohas

hcld

an Adjunct

Profcssor

appointmcnt

with the

Departmcnt

of

Electrical

Engineering

and Computcr

Scienccs,

Massachus€tts

Institute of

Technolory'

Cam-

bridge.

His

research

nterests

are

to

makc

modern

control

theoretical

methods

work.

Dr. Stein

is a membcr

of

AIAA

Eta

Kappa

Nr.r,and

Sigma

Xi'

Hc has

scrved as

Chairman

of

thc

Tcchnical

Committcc

on

Applications

(1974-

1975), EEE

Tnrxsecnoxs oN AItroMATtc CoNrRor" and as Chairman

of

the l4th

Symposium

on

Adaptivc

Proccssas.

l34l

multivariable feedback design

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