Optimal Controllers for Finite Wordlength Implementation K. Liu, R ...

/ /

b

Optimal

Controllers

for Finite Wordlength

o

Implementation

K. Liu, R. Skelton Purdue

University

West Lafayette,

IN 47907

ABSTRACT When

a controller

is implemented

the numerical

errors

system.

exists realizations

effects have

There

of the computation

from computational a systematic

which

way

take account

precision

to the standard the separation

Since, in general,

of reducing

these round-off

These results LQG controller principle

computer,

can drastically

of a given controller

errors.

of the digital

A/D and D/A converters. reduce

in a diNtal

effects. errors

provide

affect

transfer

Optimum

function

yielding

controller

in the controller extension

when infinite precision

9J

the performance

there is no upper

a natural

does not hold.

with A/D and D/A

bound,

conversion,

of the control arbitrarily

large

it is important

designs

are

to

developed

implementation

and in the

to the LQG theory

since they

computation

is used.

But for finite

I.

LQG controllers will

be perfect

(this

However,

real control

round-off

errors

errors

are normally

should

on the control

subjects

designed

is the infinite systems

INTRODUCTION

wordlength

are subject

not be ignored

system

under the assumption assumption

to the effects in the design

and the optimum

that computer

for

state

variable

of finite wordiength of the controller.

controller

design

implemention computation).

computation.

These

The influence

considering

of these

their effects

are the

of this paper.

We consider of digital

the problems

computers.

that arise with fixed-point

This paper

[1989].

Mulhs

revealed

the fact that the influence

the realization suggest

and

Roberts

was motivated

chosen

a special

by the

and Hwang

[1977]

of round-off

errors

for the filter implementation.

coordinate

This is in stark irrelevant

[1976]

(and hence

transformation

contrast

work

and the finite word length

of Kladiman

in the field on digital

ignored)

round-off

approaches

the state space

Williamson

processing

filter performance

To minimize

domain

and

of signal

T prior to filter (or controller)

to frequency

is completely

arithmetic

first

depends

errors

on

these papers

synthesis.

to control,

realization

which

regard

of the controller

as

transfer

function. The idea applied

of applying

to Kalman [1989].

degradation

in the performance

by Sripad

for LQG controIler by a coordinate controllers transformation optimal

from round-off

Kalman [1981]

design.

T.

to indicate

In these papers to reduce a standard

because

Williamson

without

the effects LQG

to insure

round-off

Riccati

equations

of round-off followed

on the control

errors.

LQG gain is not the optimal

92

percentage error.

of the ideal This approach et. al [1983]

are solved,

followed

We shall call

hence,

and

that the resulting

by an "optimal"

gains,

has been

Kladiman

filters, and later by Moroney,

controller

depends

synthesis [1985],

error is less than a certain

the standard

the standard

to controller

of the computer

filter or LQG controller

This transformation

above in quotes,

prior

problems,

in the design of Kalman

transformation

LQGT

design

One may select the wordlength

of the standard

was adapted

transformation

filter and LQG controller

Williamson

behavior

a coordinate

these

coordinate

we put the word

gain for the round-off

error

problem.

account

The

optimum

the round-off

merely

performing

errors

state estimation

Kalman

filter.

results

with a finite

word

transformation

and Williamson

conditions

algorithm

to reduce

the possibility

of overflow.

shall use is the variance off errors

are additive.

quantization

in the presence

the controller

effects. arbitrarily controller described.

work,

solution.

rather

than

after it is designed.

The

to a modified

in the presence

of round-off

worked

with upper

but their

This paper

for the solution

It is shown large

bounds

algorithm

provides

of this problem.

of the paper

is as follows.

of round-off

errors

does

the

not

necessary

We shall call this

that the portion

of the choice

In Section

conditions

are derived of the optimal in Section

bound

controller.

5. Some conclusions

coordinate

problem.

An

LQG

appear in Section

6.

93

roundwhereas

of LQG

the needed

results

of finite

of the

achieving in terms

controller

of the coordinates

by these errors

from

will range

the lower of chosing The

is then presented

and the LQGFw

of

wordlength

realization

simultaneously.

algorithm

The standard

we

1985].

variation

is discussed

The method

on state quantization,

on upperbounds

the

scaled

We assume

2, the problem

The coordinate

problem

1977].

3 summarizes

with

are properly

available.

The importance

results

the realization

for the optimization LQGv-w

In Section

of coordinates).

and

[-Hwang

of the LQG cost contributed

4, the optimization matrices

criteria

errors [Williamson

3. Section

and our new

lower

controller

by the literature

is formulated.

in Section

1989],

scaling

12-norm scaling

leads to multiplicative

to an achievable

(variation

There are many

This tends to be supported

and Williamson

parameter

compared

earlier

into

This leads

This paper

the states of the LQGFw

procedure,

will be discussed

controller

designs

oriented

of coefficients

The organization

[Kadiman

over

design

takes

LQGFw.

With a fixed point implementation,

design

[1989].

directly

implementation,

[1985].

controller

for an optimal

design

which

length

by Williamson

LQG

improvement

controller

T on the LQG controller

was solved

of optimum

showed

and a controller

controller

associated

by Kadiman

the necessary

conditions

the

problem

The problem

error was studied

provide

is to design

a coordinate

optimal

and numerical

solution

from of the

bound both

is the

necessary for the

controller

are

II. Round-Off

In this section, present.

we formulate

The formulation

and the ideas

be controlled.

the LQG controller

procedure

of Williamson

study of round-off

Error and LQG Controller

error,

follows

Design

design problem

the original

ideas

[1985],

Kadiman

and Williamson

the discrete

controller

is designed

We then introduce

a model

Problem

when round-off

of Mullis

[1976],

[1989].

effects

Hwang

model

are

[1977]

Let us assume,

from a discrete

for finite wordlength

errors

for the

of the plant

into the discrete

to

design

problem. Considering

the following

discrete-time

model of a time-invariant

xp (k + 1) = Apxp(k)

measurement white

state

np-vector,

nz-vector,

Gaussian

noises

The controller

(1)

= C_xp (k)

yp (k)

Xp is the

+ Dpwp(k)

= Mp Xp(k) + Vp(k)

zp(k)

where

+ Bpu(k)

plant:

u, yp and Zp are

vp and Wp are assumed with covariance

matrices

the

control

to be mutually

nu-vector, independent,

output

ny-vector,

zero mean,

discrete

Vp and Wp, respectively.

that one might desire to implement

is described

by following

equations:

(2)

xc(k + 1) = Acxc(k) + Bcz_(k) u(k) Ccxc(k) + Dczp(k)

where

Xc is the controller

described

in the plant model.

measurement quantization is described

state

variable process,

no-vector,

In a finite

z.p will computation

u and zp are the control

wordlength

be quantized

digital computer,

at each

(1) and (2) cannot

by

94

time

and measurement the controller

of computation.

be accomplished.

Instead

vectors

state x c and

Considering

the

the computation

(3a)

zp(k) = Mpxp(k) + vp(k) xp(k + l) = Apxp(k) + BpQ[u(k)] yp(k) = Cpxp(k)

xc(k + 1) = AcQ[xc(k)] u(k) = CcQ[xc(k)]

where

Q[-] stands

for the quantization

error, we can model the quantization

Q[u(k)]

where

assumption

respect

to coefficient numerical

controller

structures

= u(k) + eu(k)

errors.

from D/A conversion,

There

to coefficient

be modeled

covariance

matrix Ex,

errors

are many

ex (k) is the error

All such

resulting

We do not claim in this paper,

One can only evaluate in filter theory,

evidence

points

to state quantization

from

that this since one

designs

with

and we shall add

to a conclusion

tend to also be good

that with

quantization.

_ is the worctlength eu(k)

directly.

(4b) (4c)

assumption

such evaluations

with respect

1977] that, under sufficient

as a zero

mean,

white

Ex = qI,

error

this common

in this paper.

that are good

computer

from A/D conversion.

but we invoke

evaluation

can

where

(4a)

A/D

to coefficient

of the round-off

D/A

= zp(k) + ez(k)

justified,

property

by:

Q[zp(k)]

It was shown [Sripad ex(k)

an additive

control

with respect

own

respect

process

Assuming

= xc(k) + ex (k)

is always

optimize

our

process.

error resulting

(3b)

+ Dc Q[zp(k)]

and ez(k) is the error resulting

cannot

+ Bc Q[zp(k)]

Q[xc(k)]

eu is the round-off

quantization

+ Dpwp(k)

of the control

and the A/D conversion

error

excitation

noise

independent

q _ @2-2 2-2_

computer. ez(k)

conditions, wp(k)

and

vp(k),

,

Similarly,

error with

(5a)

we assume

to be zero mean,

95-

of

the round-off

mutually

the D/A conversion independent

white

noise and also independent

where

13uand 13z are the

obtain

a closed-loop

of wp(k),

vp(k) and ex(k) with covariance

model

Eu and Ez,

Eu = %1,

qu 6__.12_2-21_

(5b)

Ez=qzI,

qz '_ 1 = -_- 2 -213_

(5c)

wordlengths

system

matrices

of D/A

including

and A/D

Substitute

converters.

finite wordlength

(4) into (3) to

effects,

zp(k) = Mpxp(k) + vp(k) xp(k + 1) = Apxp(k) + Bpu(k) + Dpwp(k) yp(k)= Cpxp(k)

(6a)

+ Bpeu(k)

xc(k+ I)= Acxc(k) + Bczp(k)+ Acex(k) + Bcez(k) u(k)= Ccxc(k)+ Dczp(k) + Ccex(k)+ Dcez(k)

(6b)

We seek thecontroller tominimize thefollowingcostfunction

J = lira

E {yp(k)Qpyp(k)

+ u*(k)Ru(k)}

(7)

k ---_ **

where

u and yp are again control After

combining

(6a)

and output vectors,

and

(6b),

and

using

and Qp and R are the weighting the

following

notation

for the

matrices:

[x=(k)J ; y(k)

D=

the closed-loop

;G=

system

= Lu(k) J

Bc

is compactly

Ac

;I°=

described

;II=

by

96

;I2=

;

matrices. vectors

and

x(k+ 1)= [A + BGM]x(k) + Dwp(k)+ BOIl vp(k)+ BGI2ex(k) + BOIl ez(k)+ BI1eu(k) y(k)= [C + IoGM]x(k) + IoGI1vp(k)+ IoGI2ex(k)+ IoGI1ez(k) (9) andthecostfunction(7) maybewritten J= lim E

{y* (k)Qy(k)

}.

(10)

k --._*_

Now,

substitute

(9)

into

(10),

since

eu(k),

ex(k),

ez(k),

wp(k),

and

are

vp(k)

mutually

independent,

J = tr{X[C

+ IoGM]*Q[C

+ loOM]}

+ tr{Ex(IoGI2)*QIoGI2)}

where X is the state covariance

satisfying:

X = [A + BGM]X[A

+ DWpD"

+ BGM]"

+ tr{Ez(IoGI1

(lla)

)" Q(IoGI1 )}

+ (BGI1)Vp(BGI1)*

+ (BGI2)Ex(BGI2)"

We can decomposite

+ tr{Vp(IoGI1)'Q(IoGI1)}

+ (BGI1)Ez('BGI1)*

+ BI1F_,u(BI1)*

(llb)

Jill e,qn. (1 la) into two terms:

J = Jw_ + Je

(12a)

where

Jw-¢ _ tr{ Xl [C + IoGM]* Q[C + loOM] } + tr{ (Vp + Ez)(IoOI1 )* Q(IoOI1 ) } X 1 = [A + BGM]X_ [A + BGM]*

+ DWpD*

+ (BGI1)(Vp

+ Ez)(BGI1)*

(12b)

+ BI1Eu(BI

1)*

(12c)

and

Je _=tr{Xe[C

+ IoOM]*Q[C

Xe = [A + BGM]Xe[A

where

X = X 1 + Xe.

J_

%(k),

ez(k), wp(k) and vp(k).

+IoGM]

+ BGM]*

is the portion

} + tr{Ex (IoGI2)'Q(IoGI2)

}

+ ('BGI2)ExfBGI2)*

of the performance

Je is the portion contributed

97

index

(12d) (12e)

contributed

solely by round-off

by disturbances error ex(k).

To prevent

the overflow

the state variables.

in controller

X1 (2, 2) is the (2.2)

stands for the ith diagonal variables

have

disturbance

variance

subblock

element equal

and measurement

Therefore,

computation,

We use the 12-norm scaling procedure

[X1(2,2)]ii

where

state variable

matrix

i=l,

noise.

""

Equation

(the

the closed-loop

as:

controller

system

subblock),

and [']ii

that the controller

is excited

only

state

by outside

constraint.

is

min J = rain (Jwv + Je) , G G

subject

scale

(13)

(13) requires

We call (13) the scaling

problem

is written

,nc

of X 1 matrix

of the matrix.

to s when

the optimization

=s

which

we must properly

to (12-13).

9B

(ld)

HI.

Contribution

In this section, portion

of the LQG

function large),

of Round-Off

we discuss cost

is coordinate

The lower 1989].

bound

result

G is some given matrix

Given

lower

bound,

any n x n matrix

by eqn. (12d)

errors.

above,

is discussed

which

This portion

(that

can be achieved et_ al. 1983]

Index

is the

of the

cost

is, it can

be arbitm_rily

in an optimal

coordinate.

and [Kadiman in this section,

and Williamson where

we assume

G later).

three key lemmas, 1976, Hwang

M, there

which

coordinate

(we shall optimize

and Roberts

from

by [Moroney

of this optimal

and defined

by round-off

It is unbounded

was obtained

We will first present 1. [Mullis

contributed

dependent.

The consn'uction

Lemma

the Je term in (12a)

function

but it has an achievable

Error to the LQG Performance

which form the basis for the results

of this section.

1977].

exist a (non-unique)

unitary

matrix

U such that (UMU*)jj

=

S

for all j, if and only if tr(M) = sn []

Lemma

2. [well known]

For any two positive

definite

matrices

P and Q, let Li ['] denote

the i th eigenvalue

of matrix

['].

Then,

a)

)_i[QP] >0

for alli

b)

The Li[QP] are invariant

under

the transformation

P = TPT* and 0 = T-*Q T-1 where

T is

nonsingular. []

Lemma

3.

Let a scalar J be defined

by

99

J Atr{rr'pl where

the np x np nonsingular

matrix T is constrained

(TlT-*)ii

and P is a positive

definite

(15a)

matrix.

= s

by

for all i

Then over the set of all nonsingular

(15b)

matrices

T constrained

by (15b),

a)

J is not bounded

b)

J is bounded

from

from above.

below

(J > _ by l r'_J_="--:--[tr('qP)] 2 Snp

(16a)

where

P= _- _-

and "_

is symmetric.

c) J_in (16a) is achievable

by the matrix

T."

T = T A_Wti-ltVt

where

(16b)

Ut, Vt are unitary,

I-It diagonal,

(17a)

satisfying

UtI_It2U_

[VtFlt2Vt]ii

= Snp_-

= s for all i .

(17b)

(17c)

[]

Statementsb) and c) are minor modificationsof the resultsobtainedby [Mullis and Roberts1976]and[Hwang,1977].The proof of a) appearsin AppendixA. An algorithmfor solving(17b),(17c)is givenin AppendixB.

The contributionof finite wordiengtherrorin the costfunctionis describedby equations (12d)and(12e). ThisJetermcanalsobewrittenas: Je= tr{Ke(BGI2)Ex(BGI2)*} + tr{Ex(IoGI2)*Q(IoGI2)} Ke = [A + BGM]*Ke[A

+ BGM] + [C + IoGM]*Q[C

(18a)

+ IoGM]

(18b)

.

Since Ex = qI, we then have:

(19)

Je = q tr{ (BGI2)* Ke (BGI2) + (IoGI2)* Q(IoGI2 ) } •

We can easily

check

that the (2, 2)th subblock

matrix

of Ke (the controller

subblock

Ke(2,

2))

satisfies:

I_(2,

Substituting

2) = (BGI2)*Ke(BGI;)+

(20) into (19) reduces

(IoGI2)*Q(IoGI2)

the minimization

of Je reduces

to (18b), (13) and (12c).

2)]

.

to the problem:

minJe,

subject

Je=qtr{Ke(

From the singular

2,2)1

_h

*

Zx Ux Ke (2, 2)UxZx

Zx, Zk are diagonal

(21)

value decompositions

X 1(2, 2) = UxZxUx

then Ux, Uk are unitary,

(20)

(19) to

J_ = qtr[I_(2,

Hence,

.

aA

and

I01

= U_Z, kUk

(22a) (22b)

Zk. _=diag {... _[Ke(2,

Suppose

we begin our study with the closed-loop

after this coordinate

X l(2,2)=(Uxx --

Ke(2,

If we take equations,

one more

as suggested

*

½

* -1-

tit

1

*

xUk)

2) = (UxX_Uk)

controller

(after we substitute

(22c)

transformation

T as:

, _ , UxZx Uk

transformation

--

2)] ... } .

coordinate

T=

Then,

2)X1(2,

coordinate

*

by Kadiman

Xl(2,2)(UxX

I_(2,

(23)

*

½

xUk

-*

and Williamson

)

-*

[ 1989]:

=I

(24)

* ½ Uk) * = Y--'k• 2)(UxZx

transformation

Tc, the index

(25)

Je and its constraint

(24) and (25) into (13) and (21)), become

Je = qtr[TeT_Zk] [T_lT_*]ii=s,

Since,

from

calculations equation

i=l,

Lemma (18b)

"-"

2, _

,nc.

and (12c)

1. The round-off

constrained unbounded

and concentrate

error

by the scaling from above

(26b)

in (22c) is coordinate

(26), we have following

Theorem

bounded

(26a)

on Tc in (26).

term Je in the LQG eqn. (12c),

the

Then,

performance

by applying

index

(13), is controller

realization

from below by the following

Lemma

coordinate

j- - _ q

by the following

T

(12d)

coordinate varies

and

(12e), and

dependent.

arbitrarily.

Ux, Uk, Ut, Vt are

unitary

It is It

is

tr Z t

(27)

sn c

controller

½

= UxZ x

coordinate

transformation:

*

(28a)

UkUtIitVt

--'C

where

3 on

lower bound:

--e

The lower bound is achieved

we may i_mnore the Ke and X 1

theorem.

constraint when

independent,

matrices,

Zx,

102

I-It are

diagonal

matrices,

subject

to the

constraints.

X 1 (2,

2_UxKe(2

(28b)

2) = Ux2xUx

(28c)

' 2)UxZx " _,_= U_Y--kUk

snc 2k U_FI72U_

-

2

[VtI"lt

Vt

]ii

=

(28d)

tr2k

i = 1, • • • , nc

s,

(28e)

[]

To find the optimal obtain

Ut, Flu Vt. The equations

P is the diagonal needed

coordinate

matrix

transformation

(28d),

T --C

in (28a),

(28e) are, however,

Zk. An algorithm

is given

special

in Appendix

we must

solve

cases

of (17b),

B to compute

(28d), (17c),

(28e)

where

the Ut, I-It, Vt

for (28a). The

conclusion

transformation

of this section

given by (28a).

is that

the problem

min Je is solved Tc

to

by the

coordinate

IV.

LQG Controller

As discussed can

in Section

be decomposed

measurement

into

noise,

not influenced

two

by the coordinate An

requirement

of

optimization

problem

optimal the

without

round-off

terms.

One

coordinate

term

4. Suppose

errors.

design

influence

errors.

the second

prevent

comparing

of disturbance

Although

term is critically

is given

to

Errors

the LQG performance

the

by round-off

variables

now for controller

round-off

contains

transformation

state

of Round-Off

error is present,

of the controller,

controller

Let us first present Lemma

17, when

in the Presence

the other term is conn'ibuted

coordinate.

problem

Design

by

(28a).

overflow,

to the original

In this section, we will discuss

we

and

the first term is dependent

With

the

have

optimal

the controller

index

on the scaling

a different

control

design

design.

a useful result.

Jkx _=_ _ki[Kii,

i)X(j, j)] where K(i, i) and X(j, j) are the (i, i)th subblock

l=1

of K and (j, j)th subblock

of X respectively.

VkJkx

Define

A _ j --"_'_" kx ,

then:

a) VkJ_(p,q)

= 0

when p _ i or q _ i [E -1 (i, j)]]th-row[E(i,

(29a)

VkJkx(P'q)

b)

where

"_[K(i,

q)=0

VxJkx(P,

1 l_ K(i, i)[E -1 (i, j)] _th-row[E(i, j)] q) = _- _ _/_.t[-K(i, i)X(j, j)]

,

of matrix

when p = i and q = i

when p¢jorqcj

q) and VxJkx(p,

matrix

j)

i)X(j, j)]

VxJ_(p,

_TkJkx(P

eigenvector

= 2t_ -

j)]_--colX(j,

(29b)

(29c)

q) are the (p, q)th subblock

l*th---¢o]

of

when

VkJkx

p = j and q = j

(29d)

and VxJkx, E(i, j) is the

K(i, i)X(j, j)

104

[]

The proof of the lemma The account,

LQG

controller

are described

However,

the

coordinates optimal

is given in Appendix

scaling

design

coordinate

we can therefore

problem,

by the equations constraint

of the controUer,

A. when

(12-14).

(13)

can

so the problem

transformation

finite This

be

wordlength is denoted

always

since Jw_ is constant

G,T,

G

in terms of the variation

properly Finding

taking

into

controller.

choosing

the

G and finding

On the strength

of Section

Assume

J _ min Je is given "_

problem

(Jwv + Je)]

of Tc, we have

G

by (27), from Theorem

the equivalent

LQGzw

design

becomes

subject

,

(30a)

to (12c) and (18b) where

Jw_ = trX1 (C + IoGM)*Q(C =

1. Hence,

Tc

G

J

(30)

T,

min[Jw_ + J]

---e

3,

T,

min J = min [Jw,, + rnin Je] G,T,

its

as

rain J = rnin (Jwv + Je) = rain[rain G,T,

by

breaks up into two parts:

problem

are

as the LQGFw

satisfied

Tc to satisfy (12), (13) and (14).

write the optimization

effects

q

(tr

_k)

+ IoGM)

+ tr(Vp + Ez)(IoGI1)*Qfl0GI1)

(30b) (30c)

2

sn¢

where

Zk is defined

algorithm

in Appendix

The following Theorem Necessary

theorem

by (22c),

and the

transformation

B, and may be computed

Tc which

yields

only after the optimal

states the necessary, conditions

J

is given

G is obtained

of the optimization

problem

by

the

from (30).

(30).

2: conditions

for G to be the solution

of the optimal

loft

controller

design problem

(30) are."

[A + BGM]X1

[A + BGM]*

[A + BGM]* I_[A

+ DWpD*

+ BGM]

+ (BGI1)(Vp

+ [C + IoGM]*Q[C

+ IoGM]

[A + BGM]" K2[A + BGM] + [C + IoGM]*Q[C [A + BGM]K3

[A + BGM]*

(I_QIo + B*K2B)G(MX1

where

- X1 = 0 (31a) (31b)

- Ke = 0

(31c) (31d)

- K3 + Vk = 0

+ KeAK3)M*

Vx has 4 subblocks

+ BI1Eu(BI1)"

+ IoGM] - K2 + Vx = 0

M* + I1 (Vp + Ez)I_)

+ B* (K2AX1

+ Ez)(BGI1)*

+ (I_QIo + B*KeB)GMK3M*

+ (31e)

= 0

as

Vx(i, j)=0

i_e2

or j,2

Vx (2, 2) = q--fl-tr 2k

2, 2)[E-11 o rZl ol }

Snc

and V k also has 4 subblocks

as

Vk(i, j)=0

i.2

q

Vk(2, 2)=

or j*2

I-E-1 ] _ow I-E]_colXl (2, 2) tr £k _:g-.

sn---7-

where

E is the matrix

of eigenvectors

1 J

l'=1

of the matrix Ke(2,2)

X1 (2,2). []

The proof of theorem

2 is given in Appendix

Remark

1: The only terms

denoted

by Vx, Vk. Hence

eqs. (31) reduce reduces Riccati

setting

to the standard

to the Kalman equation.

in (31) which

are affected

by q are the two terms

in (31c)

and (31d)

13= *_ gives q = 0, Vk = 0, Vx = 0, K3 = 0, K2 = K_. Hence,

LQG design

filter Riccati

A.

equation,

by setting [3= _.

In this case, the 11 block of (31a)

and the 22 block ] 0

of (31c) reduces

to the control

Remark

2: We shall denote

that the LQGFw

controller

Now, we have following

The LQGFw

the controller requires

LQGFw

an additional controller

(31) as the LQ_w

step; the computation

controUer

to indicate

of T from Appendix me

B.

design algorithm:

Algorithm

Step 1:

Solve G from equations

Step 2:

Compute

(31a)-(31e).

T c = UxZx UkUtI-ltVt

using the G obtained

Step 3:

satisfying

This gives the LQG_

by solving

controller.

Ux,_x,gk,Ut,IIt,V

t

from

(28b)-(28e),

in Step 1.

G =

is the optimal

LQGvw

controller

for implementation.

[]

Remark:

A natural

algorithm

LQGFw

controller

(i)

Solve (31a)-(31e)

(ii)

On the next

to suggest in Step 1 is as follows.

Suppose

one desires

to design

a

for 10 bit arithmetic.

for

iteration

_i =

set

_:,o(hence,

_i

=

the standard

32 (or whatever

LQG controller).

gives

a reasonably

small

number

for

Vx, Vk.

(iii)

Iterate

by indexing

an "answer" guaranteed This is a "natural"

_i. Change

in 32-10

= 22 iterations

to be sufficient homotopy

_i by no more than one bit on each iteration. (but this manner

to yield the optimal

method,

since [3 is a natural

/07

of choosing

This gives

step sizes

answer). choice

for a homotopy

parameter.

in not

V.

We consider

an Euler

Bernoulli

modal

frequencies

and 2 outputs.

The

discrete

is represented

These

model matrices

cost function

are given is given

Computation

beam modeled appear

by the

of the control

A/D conversion

errors

and

measurement

controller

and the LQGFw

we

1. In discrete

C for a uniform

sample

computer

ignore

controller

R = 0.01I

is assumed system

controller

design,

in equation

time At = 0.018

sec.

the (1).

The LQG

these

errors

are computed

.

to be 4 bits.

simply

Frequency

modify

in the

Since

the effects

the effects

of system

example.

Damping

Factor

1

3.4987e+00

9.9994e-03

Mode

2

1.3995e+01

2.1301e-02

Mode

3

3.1488e+01

4.5600e-02

Mode 4

5.5979e+01

8.0400e-02

Mode 5

8.7468e+01

1.2530e-01

1. Frequencies Euler-Bernoulli

Both

for the system.

Mode

TABLE

with 2 inputs

(7) with

on the control

noise,

modes

{Ap,Bp,Cp,Dp,Mp,Wp,Vp}

Qp = 0.991

The wordlength

by its first 5 bending

in TABLE

matrices

in Appendix

by equation

Examples

and Damping

Factors

Beam Example

of the

the

of D/A and disturbance

standard

LQG

The standardLQG controllerof coursewasdesignedwithoutconsiderationof round-off errors(_ = _) andis labeledcontroller"LQG" in the TABLES. Controllersdenoted"LQGTi" i = 1, • ••, 4 arethesameastheLQG, butfor a coordinatetransformation on thecontrollerafter G is computed.The matrices{Ac,Bc,Cc,Dc in Appendix

C. In different

Je in the closed-loop T 1 (controller coordinate wore[length

LQGT1) design of the

computational Coordinates

errors

synthesis

the cost

control

Je is about LQG).

by

Je in two commonly 1988]

Variable

and Phase

Coordinates

[Williamson

in TABLE

unbounded

for some coordinates. or discussed

further.

smaller

improvement

about

5 bits

used coordinates,

Variable

The

2 serves

is

than

the cost

equivalent

increasing

only

to demonstrate

of

Hessenberg 2. The

with other findings

the cost

"1"4was rather

the

effect

in TABLE

of the controller that

The

Observable

is consistent

coordinate

in the original

to

are also given

high costs

of coordinate

In the optimal

Normalized

Coordinates,

are shown contribution

1 (5 = _-- log2500).

extreme

The choice

controller

the finite wordlength

LQG controller.

are bad for computation

1990].

(LQGT4)

with the LQGT1

2 shows

500 times

This

computer

coordinate

be described

Ti, TABLE

cost, using the standard

(controller

[Skelton

fact that Phase filter

system

coordinates

] associated

in

in a particular Je can

arbitrary

become

and will not

Controller ControllerCoordinates

CostJ¢

LQGT1

Optimal

LQGT2

NormalizedObs.Hess. 2.692x 102

LQG

9.793

PlantCoordinates

4.862× 103

LQGT3

PhaseVariable

9.486x 103

LQGT4

Coordinate"X"

1.472x 108

TABLE2. StandardLQGControllerin DifferentCoordinates The LQGFwcontrollerwasdesignedby the LQGFwalgorithmgiven in Section4. The controllermatrices{Ac,Bc,Cc,Dc} of this controlleralsoappearin AppendixC. TABLE 3 showsthe computedcostsof the standardLQG controller,the transformedLQG controller (LQGT1),and the LQGFwcontroller(The "LQGFwwith coefficienterror" will be discussed later). The costsfor three different applicable

disturbances

wordlength

for J,

Jy,

groups

and Ju include

plant

disturbance

for Je,

error e. The applicable

are only wp and vp (no finite wordlength decompositions; Jy + Ju.

and

Jwvy

Jy

effects).

are computed

disturbance Jey,

Hence,

Jeu

J = Jwv + Je.

Jwvu

(the contribution is the control

noise

case.

these sums apply

of vp and wp in J), Ju = Jwvu

The

v, and finite

is only e, and for Jw,¢,

term of J_¢ and

jlO

in each

w, sensor

is the output term of J (the total cost), Ju is the control

is the output term of J_

Je = Jey + Jcu,

of excitations

Jwvy,

Jwvu

to the various

cost

term in J, hence where + Jeu.

Jy =

Jwvy

J = + Jey

As we can

Disturbances

LQG

LQGT1

LQGFw

LQGFw

Controller

Controller

Controller

with coeff,

4.8827e+03

3.0589e+01

2.1207e+01

2.4695e+01

Jy

2.8053e+03

2.3458e+01

2.0798e+01

2.4232e+01

Ju

2.0774e+03

7.1303e+00

4.0941e-01

4.631e-01

Je

4.8621e+03

9.9302e+00

2.0067e-01

1.4071e-01

Jey

2.7850e+03

3.1790e+00

1.3841e-01

1.0275e-01

J_u

2.0771e+03

6.7512e+00

6.2267e-02

3.7961e-02

J,,,,,

2.0659e+01

2.0659e+01

2.1006e+01

2.4554e+01

J,,,,,y

2.0279e+01

2.0279e+01

2.0659e+01

2.0279e+01

Jwvu

3.7912e-01

3.7912e-01

3.4715e-01

4.2514e-01

Costs

Applied

All v, w,

errors

and e

e only

v and w only

TABLE

3. Evaluation

of LQG Controllers

Coordinate

see in the TABLE (LQGT1), 30.589). cost

the Je portion

computer arithmetic

when

compared

In the latter by about

the

standard

to the case,

controller LQGT1

LQG

using

by a reduction

design

this is equivalent

11 bits.

I/)

algorithm, 24,110

to increasing

is in its optimal

we reduce times the

the Je portion

compared

wordlength

this improvement

RMS = _

vs.

coordinate compared

to

of the

to the LQG of the

will give the same performance

Furthermore effort

Controller

controller

and

LQGFw

in control

Optimal

33% of the total cost (9.9302

controller

7 bits, That is, controller

that LQG gives

is accompanied

and of the LQGFw

of the cost is still about

By using the new LQGFw

50 times,

controller.

3, even

in Plant Coordinates,

in output RMS = _.

control

using 4 bit performance To

conclude

this point,

we see that if both

performance

is an order

of magnitude

performance

can

the

mean

difference

controllers

use 4 bits,

('_"6.798

vs. _).

between

feasibility

the difference This

and

in RMS

output

kind of improvement

infeasibility

of some

in

control

missions. With the new controller, as opposed

to 33% for LQG.

the closed-loop suppose

the round-off

system

the LQG_v

are the conditions

Now let us discuss

was only excited

controller

controller

LQGFw

controller

compromise

is a little higher

TABLE

3, that the quantifies

vp and disturbance

than that of standard

and the control

performance

be the total cost if

allows

robustness

that are optimized

These

standard

(under

These

in the evaluation. The output

indicate

that the

LQG controller.

to computational

by the theory

bits.

LQG controller.

term a little lower.

than the designed

wp. That is,

using infinite

since there are no disturbances

is a little more conservative

in nominal

noise

would

for 4 bits, but evaluated

LQG design,

term of the cost is also a little higher

the cost Jw,,, which

by measurement

was designed

of the standard

Jw_ of the LQGFw

portion J_ of the cost is only 0.85% of the total cost

errors.

the given

This

Note

conditions

in are

shaded. In the design by a gradient as the

the

method.

initial

convergence the

process

output

iterative

design

played

input

for

starting

the

algorithm,

role in reducing

as shown

in Fig. 2. problem.

but note from Figs.

(31a) to (31e) were solved

in its optimal iterative

plotting

Ju performances,

a crucial

in the optimization computations,

the equations

LQG controller

for the LQGFw

Jy and

of magnitude)

formulated

The standard

controller

transformation orders

of the LQG_-w controller,

versus

coordinate

process.

Figs.

iteration. errors

This was expected The LQG_-w controller

The

1-3

was used

illustrate

optimal

(reducing

because

the

cost J_, coordinate

the error by 3-4

the transformation

was obtained

1-3 that after 120 iterations

]12

(LQGT1)

the total cost J, the wordlength

the round-off

with little loss.

iteratively

after

was

about

one might have

300

stopped

Coefficient

Errors

In the introduction argued

that

performs errors

even

we promised

though

coefficients.

TABLEs

loop pole locations as marked

arithmatic,

poles

5

transformation

the

before

compared

transformation coefficients

are improved

Finally, quantization

consider

to 4 bits).

all modes is multiplied prior

discussion.

demonstrating

The improved

plant and controller

LQG

closed

only

4 bits.

The

coefficient

controller

4 shows

precision).

in

the closed

The using

system

is

only 4 bit

the unit circle.

by

its

The poles is still

the LQG_v system errors,

errors"

performance,

that all controllers

are implemented

state computation.

with coefficient

coefficient

Hence

the

computer.

loop system

Fig. 6 shows

it

in the controller

Figure

infinite

This is the LQGT1 controller.

optimal (o's)

are in improved

unstable.

The

coordinate

controller

when

controller

is stable, even

confirming

though

for

the controller

The performance

in TABLE

coordinate

3 is about

this has

degradation

in

15% (compared

to

3).

errors

in

both

the

These results

are summarized

by parameter

p. Hence

range

destabilize.

coefficients

in the

in controller

in TABLE

(note

We

quantization

in the state degrades

(using

using a 4 bit control

to controller

"LQGFw

precision

by the o's in Fig. 4, are outside

using

only for errors

we

the controller

but not enough.

robustness

performance

LQG regulator

for state

errors.

this we introduced

of infinite

errors

errors can easily

4, but the

implemented

To show

instead

of coefficient

only

of e is just a disturbance

improvement

to Fig.

J, listed in the column nominal

When

synthesis.

helped

been optimized

the standard

is unstable

shows

as well.

Quantization

Coefficient

as indicated

LQG controller

Fig.

example

using

of the effects

is optimized

4 bit precision

since the effect

by the x's.

some

locations

controller

The key issue here is this.

1 and 2 are stable).

evaluation

quantization

by using

but does not destabilize,

standard

LQGFw

well with coefficient

in the controller

stable

the

some

for

robustness

stability over

plant

and

in TABLE

p=l corresponds

using standard

the LQGFw

controller 4, where

1J3

the modal

to the nominal controller

LQG controllers

coefficients.

coefficients

(due

to

damping

in

plant in all of the

is .729

in the presence

< p < 1.23, of errors

in

Damping

Error Factor

p

LQO

LQGT1

LQGFw

Controller

Controller

Controller

1.5242e+00

unstable

unstable

unstable

1.3717e+00

unstable

unstable

unstable

1.2346e+00

unstable

unstable

STABLE

1.1111 e+00

unstable

unstable

STABLE

1.0000e+00

unstable

9.0000e-01

unstable

unstable

STABLE

8.1000e-01

unstable

unstable

STABLE

7.2900e-01

unstable

unstable

STABLE

6.5610e-01

unstable

unstable

unstable

5.9049e-01

unstable

unstable

unstable

TABLE

4. Robustness

(Fig 4)

Controllers

(4-Bit Wordlength

unstable

with respect Controllers)

(Fig 5)

STABLE

to modal damping

(Fig 6)

VI. Conclusion This papersolvesthe problemof designingan LQG controller to be optimal in the presenceof finite word.lengtheffects(modeledas white noisesourceswhosevariancesare a functionof computerwordlength).Thisnewcontroller,denotedLQG_¢, hastwo computational steps.First the gains are optimized,and then a specialcoordinatetransformationmust be applied to the controller. This transformationdependson the controller gains, so the transformation cannotbeperformeda priori.

(Hence,

LQGvw

to the standard

controller

word.length It was

exists

both

a closed

algorithm achieved

theoretically

influences

an optimal

obtained

requires

reduces

Acknowledgement:

synthesis, and

the effects

form solution significant

by example

in which

for the LQGvw computation.

We gratefully

importance acknowledge

of

that

the

errors

problem,

theorem.)

controller extension

choice

The new an infinite

of the LQG theory. coordinates

system

computations.

design

when

of controller

on the control

to do these

Hence,

of extra computational

The

LQG

so this is a natural

of computational

set of coordinates

at the expense

Williamson.

algorithm

is used for the controller

shown

significantly

design

there is no separation

and that there

Since

we have

of the LQG_-w controller

the improvement

of the new

not

by this

controller

is

effort in design.

this many

this work by NASA grant NAG 1-857, Technical

problem

was

pointed

helpful discussions Monitor

out

to

us by

Darrell

with him, and the support

E.S. Armstrong.

of

Appendix

1.

a)

Proof of Lemma

Using

A

3

the singular

value decomposition

of T = UtFltVt,

then the constraint

equation

(15b)

becomes

(Wtl-lt2V;)ii

from Lemma

1, above equation

= s

is equivalent

for

(32)

all i

to

tr(Ylt 2) = Snp .

Now, let us study the cost '},of (15a).

Using

tr(AA') >

(33)

the inequality

[tr(AB')]2 tr(BB*)

we have a lower

bound on 3'

_= {u, nt ut P} = =(fit, u; _/b-)(ntu; _-)" } 2

*

m

k tr {(Ut [_._]_1 )(Ut [.__]_1 ), }

Now, to prove that 7 is unbounded

tr{p_l }

(34)

from above, we prove that for any large scalar m > 0, we

have y('I') -> m for some '_. Let us choose

a 'r having the following

Irlt = diag (lrli)

such that -

fil =fi2= "'" = nr -2= and

/16

1

1=it:

mtrCP -1 ) ,

I'In- 1 =

lrlr_ = _m

tr(F-I)

%/2mstr(P -1 ) - 1

where

m is so chosen

that 1 m>

2str(P -1 )

Then

2mtr(P -1) - 1 + 1 mtr(P -1 ) mtr(P -1 )

Hence

the chosen

_" satisfies

the constraint

-- Snp.

(33). Now, we have:

nP

_

(End) 2 (tr{M t1) 2

T>

we then conclude

tr{p-1}

the proof

(Mrb)2

-m tr{P -1)

a). The proof of b) and c) follows

of part

bound and the matrix T are found

i=l

---> tr{P -1}

by using following

next.

(35)

holds above when Q*Q = )v2I.

Let us assume are unitary

matrices.

T = UtlitVt, Assume

P

=

UpIIpU;,

where

lit and lip are diagonal,

for the R and Q matrices

Q*Q

(36)

* --IA*- --2* " _IA = U t UpIIp UpUtll t u I Upllp

-*UpU

t

(37)

,

then

(Q*Q)-I

we have:

- *-

--

= tot Upll_"

Ut, Vt, Up

in (35),

R = U;Upli U;U

Hence,

lower

inequality:

(trR) 2 < tr(QRQ* )tr(Q-* RQ -1 )

the equality

The

14-

*-

---2

UpUtllt

//7

"*

" ---1/4"

Ut Opllp

*"

UpU t

.

tr(QRQ*)

= tr(RQ*Q)

* --- tr[UpXlpUpUtl']

tr(Q-*RQ

UpI-[p

* UpUt)(W

2 * tU t] =

tr[PTY*]

= tr[(Ut

*

½

-t) = tr[R(Q* Q)-a] = tr[(Ut *

= tr[-UpUtrlt

From equation

2

*

ty(R)

equalities

¼

* UpUt)

]

* -IA * t Up_"_p UpUtl-]

-2 * -_,4 * t U t UpIIp UpUt)

]

* l

Ut Up] = tr[rlt z]

(33), and the above equation

= tr(g

2 * t U t Uplip

= T

a,6 * UpUt)(U

Upl"lp

tr(Q-*RQ

Now,

* ¼ * t UpIXp UpWtIl

* ½ * t UpIlp UpUt)

back into inequality

we have the following:

-1 ) = tr[Ht 2] = Snp

= tr(I'[_)

(35).

1A * Up)

-- tr(Upllp

We then have:

-- trNlV-.

[tr('_-)]

Substitute

the

2 < Snp3', that is

[tr(',,_)]_

y>

above

(38)

Snp D*

Now, suppose

the matrix

T

= utritV

t

yields the equality

in (38).

Since

the equality

in (35)

holds when Q*Q = )_2I, then we have: *

2

-1A *---U t UpX'ip UpUtl-I

*

__ t U t Uplip

1A * UpU

t

=_.2I

that is

--2-.tUt

Ut/-I

=..

_2-Upllp- .,-* Up

_

_tg72U?--- _.-2Upl'_p ., Up •

(39)

Hence

_= = u, u,,n_v;U, _.2

Substitute

_2

this Fl

into equation

(32) to obtain

*

rio

,_n

_,

(v,ut up-_u;,u#_)_

Then tr [-_)=

Snp, hence X2 = l'_tr(rI_)=snp

i/8

= s.

l"-_tr('_)'Snp

N°w'

substitute

the ab°ve)'2

into (39), to obtain

-- ---2--* snpUpI]p UtFit Ut =

Hence

(38) yields

(40), must satisfy

the lower (17b).

1//2

Up

SnpN/P-

(40)

bound in (16a), and the matrix

(17c) can be easily deduced

achieving

from (15b).

this bound,

This concludes

shown

by

the proof. []

2.

Proof of Lemma

a)

Proof of (29a):

4

Since Jkx does not depend

on K(p,q)

for p ;e i or q ¢ i, we have:

O VkJrx(P'

Proof

of (29b):

We need following

q) -

_K(p, q) Jk,x = 0

equality

(e.g. Page 444 of Skelton

[1988])

to prove

the

we have

by

equation:

_.i[A] = [E -1 ]ith_rowA[E]ith--eol

where

_-i is the ith eigenvalue

taking

A = K(i, i)X(j, j)

Xi[K(i,

of A, and E the eigenvector

i)X(j,j)]

= tr{K(i,

Hence

from the differentiation

rule

= [E-1 ]lth-rowK(i,

i)X(j, j)[E]lth--col

]i9

of A.

i)X(j, j)[-E]tth--col

[E-1]lth-row

OtrAB _ AT we get

OB

matrix

}

Now,

_X/

-: T

OK(i, i)

Then,

T

- FE

•

j)

we have: 8 8Jk.x

1 n OK(i, i) _a[K(i,

0K(i, i) - _t_

-

5/kt[K(i,

i)X(j, j)]

rib--colX(j,• j) [E-1T] it)v-row[E] T

=I_ -

The proof of part b) follows

i)X(j, j)]

_/_a[K(i, i)X(j,j)]

in a similar manner []

3. Proof

of Theorem

minimization

2:

Apply

Lagrangian

Multipliers

K2, K3,

then

(30a)-(30c)

leads

of

= tr{Q([C

+ IoGM]X 1[C + IoGM]* + (IoGI1)(V p + Ez)OoGI:

+ tr{ K 2([A + BGM]X1 [A + BGM]* - X:)} + tr{K3([A

+ DWpD*

+ (BGI:)(Vp

+ BGM]*Ke[A

[C + IoGM]-

I_)}

/20

1)" + (BI1)Eu (BI1)*

+ BGM] + [C + IoGM]* Q + _ snc

Then

+ Ez)(BGI

)* )] }

(trZk) 2

to

g

- [A + BGM]X1

[A + BGM]*

+ DWpD"

+ (BGI1)(Vp

+ Ez)(BGI1)"

+ BI1Eu(BI1

)* - X1 = 0

3K2 - [A + BGM]* Ke[A + BGM] + [C + IoGM]*Q[C

+ IoGM]

- Ke = 0

0K3 - [C + IoGM]*Q[C

+ IoGM] + [A + BGM]'K2[A

+ BGM] - K2 + Vx_ = 0

3X1 - [A + BGM]K3

[A + BGM] ° - K3 + Vk I = 0

_Ke

Applying

Lemma

theorem.

This verifies

_G

4 on the above (31a)-(31d).

- 2I;QCX1M"

two equations,

we can obtain

Vx, and Vkl as stated

in the

Now

+ 2I;QIoGMX

1M" + 2I;QIoGI2(Vp

+ 2B'K2BGMX1M*

+ 2B*K2BGI

+ 2B*K1BGMK3M*

+ 2I_QCK3M*

+ Ez)I_ + 2B'K2AX1

1(Vp + Ez)I_ + 2B'K1AK3 + 2I_QIoGMK;M*

M*

M°

= 0 ,

but since I_QC = O, then,

_G

- 2[I_QIoG(MX + (B'K1B

1M ° + 11('Vp + Ez)I_)

+ IoQIo)GMK3M*

= 2[(I_QI 0 + B'K2B)G(MX1 + (B'K1B

This verifies

+ I;QIo)GMK3M*]

+ B* (K2AX1

+ B*K2BG(MX1 M° + Ii(Vp

+ K2AK 3)M*

M" + I1 (Vp + Ez)I _)]

+ Ez)I_) + B*(K2AX

1 + K1AK3)M*

= 0 .

(31e). []

12I

Appendix

We now present and (17c) Let _

an algorithm

for one set of solutions

in (17b) be written

(originally

B

developed

by Hwang

of Ut, I-It, Vt (The solutions

in terms of its singular

Up unitary,

Algorithm

for solving

(17b)

for Ut, I-It, Vt are not unique).

value decomposition

"_" = UpY-1:,U _

where

[1977])

(4])

Zp diagonal.

(Solving

Ut, I-It, Vt in (17b) and (17c))

I. Take:

(42a)

U t = Up

!

tr(_,p)

n, =

(42b) Snp

Vt = Vn-lVn.-2

where

Vi, i = 1, • • • , n - 1 is computed li.

Compute

" " " Vi " " ' V2V1

as follows:

V 1" Let

Z 1 _FIt 2=diag(.-.

Assume

(42c)

_11 and ch_ are two numbers

clj'")

such that one of them is bigger

than s. Then take V1 as:

122

(43a)

than s, the other is smaller

column fl

0

...

0 gl

0

...

O"

0

1

...

0

0

0

...

0

1

0

0

fl

0

...

0

g 1 =

0 -ga row

0

...

(43b)

--_

0

......

1

............

1

where

fl

:to,0-11 L'G_ll

I

1 - Cll

gl =

'

(_1_ -- (Yll

Compute

1

(43d)

Vi" Let

2i = Vi-I

where

(43c)

Zi

_]R

satisfies

(i-1)x(i-1)

and Zi_ can be written

"_-_-1V_-I

the property

[2il]j

=

j = S, Ei2

_-'i3

cIR (i-1)x(n-i+l)

is a nonzero

matrix,

as

_-43 =

Assume

(Jii and cYia are numbers

than s. Then take V i as

such that one of them is bigger

than s and the other

is smaller

i column "1

o_ column

0

0 0

0

i row

...

fi 0 0

0

...

0 gi

1

0 0

0

tz row -4

0

0

0

Wi =

...

...

(44b)

1 0

gi 0

...

0

0 fi

...

0

0 0

0

Computefi andgi as:

fi =

gi =

I

(_ia

--

(44C)

1 1 la

_ia -