Necessary and Sufficient Conditions for Optimal Fixed-Order Dynamic Compensation of Linear Discrete-Time Systems1

Copyright © IFAC Desi gn Methods of Control Systems, Zurich, Switzerland, 199 I

LQG/LTR-DESIGN

NECESSARY AND SUFFICIENT CONDITIONS FOR OPTIMAL FIXED-ORDER DYNAMIC COMPENSATION OF LINEAR DISCRETE-TIME SYSTEMS! W. L. De Koning and H. De Waard Faculty of Mathematics and Injormatics, De/fr University of Technology, p .a . Box 356, 2600 AI De/ft. The Netherlands

Abstract. The optima l fixed-order dynamic compensa ti on problem is considered in the case of linear discrete-time systems with quadratic criteria. lecessary and sufficient conditions for the existence of a unique optimal stabilizing compensator of given order are derived. A test, explicit in the system matrices, is provided to detcnlline if a given system sat isfies the necessary and sufficient condi tions.

1. Introduction. In thi s pape r the opt imal fixed -orde r dynamic compensation

op timal reduced- o rder dynamic compe n sa tion problem,

problem is considered in the case of linear discrete-time systems.

cond iti o ns for th e existence of an optima l, reduced-order

The fixed (i.e. reduced) -order dynamic compensation problem

dynamic compensator of prescribed order are formulated. In the

has received co nsiderable attention late ly, especially in the field

full-order case these are the usua l stabilizabili ty and detectibility

of design of dynamic contro ll ers for high-order systems, such as

conditions of LQG theory. Next, s ufficiency condition s are

flexible spacec raft. Constraints imposed by available computing

derived, i.e. additional restriction s o n solutions of the first-orde r

power require th e res ultin g cOlllrollers for the se systems to be of

necessary conditions that characterize local minima and th at

reduced order. Several approaches to reduced-order controller

single out the global minimum. In th e full-order case thi s global

design h ave been deve loped in the literature (EI1I1S (1984),

minimum is distingu ished by the unique nonnegative definite

Glover (1984), Yousuff, and Skelton (1984)). They a ll proceed

solutions to the LQG Riccati equations. Finally, a test, explicit in

by first obtainin g the full-ord er LQG compensator design for a

the sys tem matrices, is given, that dete rmines if a given system

high-order state-space model and th en reducing the dimension of

satisfies the necessary and s ufficient conditions or not. The

thi s LQG compensator. A notable exceptio n is the op timal

results are illustrated w it h an examp le.

project io n approach developed by Hyland and Bernste in (1983, 1984). This method differs from the previous ones in tha t it

2.

The

Optimal

Fixed-Order

directly characterizes the quadratically opt imal dynamic

Compensation Problem

compe nsa tor of a given fixed-order, without the necess ity to

Consider the system:

Dynamic

obtain a full-order LQG controller fi rst. The sol uti on of this

xi+t = <1>xi + rUi + Vi

(2 .l a)

opt ima l reduc ed-order dynamic compensat ion problem is

Yi=CXi+wi,

(2. lb)

J< n

i =O, I,.

is the state, ui E .D<

characterized by four matrix equations (two modified RicJtti

where Xi E

equations and two modified Lyapu nov equation s) coupled by a

the observation, Vi E .D<

n

m

is the control, Yi E .D< 1 is

the system noise, wi E .D< 1 the

projection whose rank is prec isely equa l to the order of the

observatio n noi se and <1> , r, C are real matrices of appropriate

compe nsator and which determines the optimal co mpen sator

dime nsion s. The processes {vd, {wd are uncorrelated zero-

gain s. This coupling illu strate s the s uboptimality of standard

m ean white noise seq uences with covariance V

seque ntial LQG design

I controller reduc tion schemes, th at are

~

0 and W > 0

re spectively. The ini tial condition Xo is a stochastic variable with

X;; a nd

based on the separa tion principle.

mean

In their papers H ylJ nd and Bernstein only disc uss the issue of

{wd . It is assumed that m S; n a nd I S; n. System (2. 1) is

covariance Po and is unco rre lated with {vd and

station ary condition s, i.e . first-order necessary conditions.

denoted by (<1> , r , C).

H owever, apart from necessary co ndi tion s, the solution of the optimal redu ced-order dynamic compensat ion probl em also

'This work was sponsored by the Dutch Technology Foundatio n (STW) under grant OWl 77 .1 397.

includes the issues of sufficien t conditions and existence of a solution . This p aper addres ses th ese two iss ues. After a review of the

519

As controller we choose th e fol low ing dy nami c

Proof. Part a is clea r by choosi ng F=K =L=O. The n (
compensator:

is stabl e, Part b follows from the structure of (
Xi+l

= FXj

Uj::::

-Lxi

+ KYi

i = 0, 1. .

(2. 1a )

that ( (
(22b)

KC th en

0]

whe re ~i ERn, is th e compensator sta te, and F, K, L are real matrices of appropriate dimensions. Th e ini ti al con d ition ~

~

Xu is

-I

= [
0

rL]


max(!, m. nu)' where

which proves this part. Fort part d one mere ly has to realize th at

nu is the dimen sion of the unstable subspace of
a compensator of dimension n2ma y be conce ived as a spec ial

(2 .2) is denoted by (F, K,L ). The optimal fi xed-ord er dynamic

case of a compensator of dimension n l .

determini stic. It is assumed that n

'\.

compensation probl em is to find tlie o ptimal stabi li zing compensator (F* , K*, L *) of given dimension ne that mini mi zes

Le t Pi E R

the cri terion

E(x; x;T) then from 0.5)

G QO

· I (F • K , L) :::: 1,1nl N N~

T E{~I L ix Qx. + uT RU 1} ~

\

i

!

1

I

=0

+ n , )(n + n,) denote the c losed- loop cova ri ance

(n

P' i+ 1 =
I;

0.6)

If the closed-loop system is stable, the steady-state covariance P'

(2.3)

-

= lim P'i exists as the uniqu
The closed-loop system may be described by:

,....

P' =
:i+t ] [ xi+t

•

[
(1.4 )

(27)

Furthermore, criterion (1.3) is finit e and independent of initial conditions and can be expressed as

Introducing

cr_ (F,K,L) = tr (Q'P') x, = [;]. v, =

[ KV~.J.


=

I

(j)

,KC

-fL] , Y' = [ F

where Q'

o ] ~ KWK'

Q'

gives for (2.4) +

i = 0, I , .

E

IR (n ,

n, )(n , ' \ )

(2.R)

is given by

=[0 Q LT~J

( 2.5 )

For this re3 son th e admissible se t of compen sa tors C. dm is

where (v'd is a zero -mean whi te noise sequ e nce wi th

re stricted to tho se co mpe nsators for which the closed-loop

x'i + l :::: 'X'j

v'i'

covariance V', and independ ent of the initial condition xlJ'.

sys tem is stable. Funhemlore it is rest ri cted to the set of minimal

System (2.5) is denoted by (
compensators C:lm because the criterion value is independent of the inrema! reali zation of (F,K,L):

Xi' converges as i

~

00

to a value which does not depend on

xo' , I... C a(mIm = {rF,K,L)

Definition I. (
•

A number of properties concerning compe nsatability are stated in

Theorem I,

(
~

(
b) (
=

(
~

( n2 ~

(n -compensatability 2

~

now be refo rmul ated as to find th e op timal compensator (F*, K*,L* )

E

C: m which minimizes (1.8) an d su bject to (2.7)

for given value of ne

~

n, and to find the minimum crit erion

value cr* _ = 0_(F* ,K* ,L *).

the following

a)

C a dIII I (F,K) reachable, (F,L) observable 1 } J

Th e opt im al fixed -orde r dynamic co mpensat ion problem can

th ere exists a compensator (F, K, L) of dim ensio n ne' such th at (
E

nl-compensatability) .

520

For conveni etKe intrexluce the following notation' T WI' =W + CPC

3. Necessa ry Cond itions Fo r Opt imal Fixe d-Order Dynam ic Compensation.

Let p denote spectral radius. It is well known that (<1:>') stab le

~

Rs

p ( ') < I. Bec au se the eigenvalues of (<1:>')

(38)

=R + rTsr

Lp

=PCT W p

L,

(39) ·1

(pc'ly r

(3. 10)

continuously depend on (F,K,L), the set C::tm is open. Therefore, the matrix minimum principle (Athans, 1968) ma y be

p

=(rTS =$ - pcT WptC = - KoC

applied to fi nd nece ssary conditions for the solution of the

<1:>,

=<1:> - rR s·lrTS = - rLo

(313)

' 1.

=In - 1:

(3.14)

op timal fixed-order dynamic compensation problem. For this

(3. 11) (3. 12)

purpose define the Hamiltonian H by: H(F, K, L, P', S') =

utQ'P' + ~'p'
Let;l) de note the set of contragrediently diag o nalizing tran sformati ons defined for nonnegat ive -definite matrices T and U as ;l) (T,U ) = \'P E lR n-n: 'I' . I T 'p ·T and 'I'TU'I' are

(3. 1)

where S'

E

Sn+n, is a Lagrange multiplier. The n the first -o rder

diagona l ), then th e necessary condition s l'harac terizing

necessary conditions are: aH = iLt/'P ' ' TS ' ~ = D aF aF ' .

n. 2a )

aH = ...Q...tr ( 'P' ,T S' + V'S' " = 0 aK aK ' '

CUb)

1'p'c!J"S' + Q'p, L 0 aH = iL aL aL tr " )

C3.2e)

M ap'

= 'T S' , + Q' - S· = 0

M

= 'p' ;r + V' - P' = 0 as ' where S' ;:: 0 and P' ;:: D.

admissible extremals of th e optima l fix ed·ord er dynamic compensation probkm are given by the following theorem.

T heore m 2 ( Be rn stein , Dav is, and I-Iyland, 1986). Suppose (F,K,L)

(3.3a)

C:::1m so lVeS the reduced -orde r dynamic compensator problem. Then there exist non nonn cg:lti vc· definite

CUb)

matrice s, P, S, j'l and ~ suc h th at F. K and L are given by:

Panition (n + n) o (n + nJ S' and P' as

E

F

=H [ - KoC - rLulG T

(3 15)

K

=I-I Ko

(3.16)

L

=- LOGT

(3. 17)

and such tha t p, S,

Pan d ~ sa ti sfy

P

=<1:> pT _$ PCTWp·1 (PCT)+ V+'l. f'>'l.T

(3 18)

S

=T S - (rr S
(3. 19)

j'l

=<1:>,1: j'l1:T ,T + Lp

(3.20)

S = SI - SI2 S 2 St2'

~

=<1:>/ 1:T ~

(3.21)

"

where

according to the partitioning of <1:> ' and define th e n o n nonnegative-defi nite matrices ·1 T

·1 T

:, = SI2 S2 St 2'

t

p + L,

(3. 22 )

Then the followin g lemma is used for stating the mai n i= l

res ult. Le m ma 1 (Richter and Coll ins, 1989). Suppose j'l

for some'!' E

' 1.

Sn

E

;l) (P,~) such that ('IJ·t p~ 'I' )(i.i ) '" O. i = I , ... ,

ne and some projec ti ve fa ctorization G, H of ,.

and § E Sn are symmetric and non negative definite and rank

!> ~ = ne ' Then the followin g statements hold:

It can be shown that the rank of j'l~ is ne and , = p~ (j'>~)#

•

i)

j'l~ is diagonalizable and has nonnegative eige nvalues

ii)

Let (X)# denote the group gene ralized in ve rse of sq uare

matrix

matrix X. Then the non matrix

matrix so tha t (3.18) and <3.19) reduce to th e standard observer

Notice that in the full-order case the optimal projection

(34)

rank 1: = ne

and its factors G and H can be chose n to be the iden tity

and reg ulator Ricatti eq uations. Equations (3. 20) and <3.21) then

is idempotent , i.e. , is an oblique projection and

iii)

t

express the provi so that the compensato r be minim al. Th e (3.5)

coupling of the equa ti ons due to the projection illu strates the

There exists G, H E lR n, -n, and nons in gu lar M E lR n, -n

nonoptimality of standard seq uen tial cont roll er redu ction or

such that

model reduction sche mes, because in the redu ced-orde r case

j'l~ =GTMH

(3.6)

th ere is no longe r separation be tween es timati on and con trol

HGT = In'

(37)

operations .

• 52 1

above mentioned method we define the nonlinear tran sfomlation

4. Sufficient Conditions For Fixed-Order

(De Waard, 19(0) El X: SlIxSnx S nxSn ~ SnxSnxSnxSn by Cl

Optimal Dynamic Compensation.

Necessary co ndit ions for the existence of an optimal fixed-

EluX=

order stabilizing compensator are stated above. In th is section

(<1>X I<1> T_<1>X leT Wx 1· 1 (<1> XICT)T +Y+a 2"C.LX3"C.LT.

we wi ll formulate suffi cient condit ions whi ch are necessary in

<1>T X2 <1> - «(I' Xl <1»T RX2 ·1 (I' X 2 <1> + Q + a2t.LTX4

general. In order to state the main resul t in thi s direction it will

<1>X2 lat + (I-a)InlX3Iar + ( l-a)Inrr <1>X/ + LXI'

appe ar convenient to introd uce th e notion of detectability of a

<1>XIT [at + (l-a) l nITX 4 Iar + (I-a) lnl <1>XI + L X2) ,(4.2)

triple instead of a pair of matrices.

Definition 2. (<1>,

r , C)

"C.L'

where a

is called detectable if (<1>, C) and

•

E

[O, ll . Call (XI. X 2. X 3. X.j) non nega tive definite if

XI. X2. . ~ 0. Denote th e parame terized equation y u = El UyCl by H (YCl , a) = 0, whe re y Cl denotes the nonnegat ive definite soluti o n of X = El Cl X wit h parameter a. Now for a = 1 we

Furthermore we need the followin g lemma.

have th e original reduced -orde r (n,

~

n) case, and for a =

°the

easy full -o rd er (ne = n) case. Th e functi o n H is ca ll ed a

Lemma 2. Either R >0, W>O, (<1>, Y 1/2, Q1 /2) detectable,

homotopy and we may follow the solut ion pat h y Cl if a goes

or Q >0, V >0 => (<1>', (y ' )1 /2, (Q')I!2) detectable.

from

°to I . We are now in a position to state ou r main resu lt.

Proof. This is a spec ial case of th e proof in De Kon ing

( 1989), lemma I , for the while parameter case. The order of th e

•

compensator plays no role in the proof.

Theorem 3. Assume « I>,

r,

C) to be ne-compen sa table

and assume that eit her R>O. W>O, (<1> , VIn , Q1 /2) detectable, or Q >0, V >0. Then Y

= Iim El i (0,

0, 0, 0) exists, Y is the

1.... ~

Now define the nonlinear t ransfoml~Hion

unique nonnegativc definite solution of the eq uation X = El X,

El X: SnxSnxSnxS n~ SnxSnxSnxSn by

(F* , K*, L*) is giwn by (3.15)-(3.17) whe re (P, S, j\~ ) = (Yl, Y2. Y 3. Y4 ), and

Elx = (<1>X I<1>T - <1>X ICT WXI·I (<1> X ICT)T+VH.LX31:.LT. <1>T X2 <1> - «(I' X2 <1»T RX2·1 (I' X2 <1> + Q + 1:.LTX4 1:.L' <1>x2 1: X3 1:T <1>X2T + L XI ' <1> x IT 1:TX4 1: <1>XI + LX2) ,

(4. 1)

where X = (XI. X 2. X 3. X.j), X I. X2. X 3. X4 E Sn Note that (P, S, P,§) = El (P, S, P,§) is equiva lent to (3. 18) - (3.21). Now

(4.3)

co nsider (Xli. X 2i . X 3i . X 4 ) = El i (0,0, 0, 0), i = 0,1,,,,, If ne

Proof. See De Kon in g (1989), theorem 3, for a proof in

= n, then Xli and X3i for i = 0.1,,,. are Ih e iteration s of the

the more ge neral whi te parameter case of th e fac t that the necessary optimality condi tion s (3. 18)-(3 .21) hav e a

well-known un coupled observer and regulator Riccati equation s

non negative definite sol ution, and th at all nonnegative definite

with initial value 0. It is well known th at (Xli) and (X 3d are

so lutions of (3.1 8)-(3.2 1) correspond

monotonic in the sense th at X Ii :,; X Ij and X3i :,; X 3j if i < j.

compensators

This property may be used to prove convergence of (X Ii ) and

to stabi li zing

III

(E

C3dm ) . The order of the compen sator plays no

role sofar. Remain s to prove that (3 .1 8)-(3.2 1), or equivalen tl y

(X 3i ). which gives us an easy way to calculate a solution of the algebraic Riccati eq uations. However in the case that ne < n (Xli

X = El X, has on ly one non nega tive definite solution Y and that Y = lim El i (0,0,0, 0). To that end we consider the homotopy

) and (X 3d are no t monotonic due to the cou pling betwee n the corresponding equations. Fortunatel y it is still possi ble to prove

H(YCl, a) as defined after (4 .2) and follow the solution path yCl

co nverge nce usin g the method of homotopic continuations. This

as a goes from

method is in short: First solve an easy 'si milar' problem, th en

the number of solu tion s along the path remai n co nstant are

continuously deform th is problem into th e original problem and

fulfill ed. For the preci se conditions we refer to Mariton and

follow the path of solutions as the easy problem is deformed into

Benrand (1985). t"o\\' (<1>,

1.... ~

°to 1.

It appears th at the conditions under which

r , C).is

ne-compensa tab le , thus

r , C). is n-compen satable.

the original one. Topological degree th eory th en sta tes under

from th eorem Id (<1>,

what conditions the number of solutions along the path remains

0, \Y > 0, (<1> , V 112 , Q 112) de tecIable , or Q >0. V >0. It is well

constant. For more information on thi s we refer to Rich ter (1983) and Mariton and Benrand (1985). In order to use th e

522

Al so eit her R >

known that under these conditions yO is unique. so yl is also

Example.

unique. Moreover it is well kn own thal yu = lim (; 0 i (0, 0, 0,

Consider the system

I--+~

0). Then, using similar arguments as above, also yl= lim (;Ii I --+~

0.1841

0.2543

0.2004

0.0226

0.2493

0.3222

0.3297

02S~] 0.0441

0.3259

0.3989

0.0037

0.2827

0.3139

0.3261

0.0166

0.1840

0.4466

0.1997

From theorem 4 in the next sect ion it will become clear that the

0.4487

0.0237

0.0321

0.4062

0.3366

conditions of theorem 3 are not only sufficient but also

rT = [0.9103

0.7622

0.2625

0.0475

0.73611 ,

C = (0.3282

0.6326

0.7564

0.9910

0.36531 .

(0,0,0,0). Finally, expression (4.3) may be found by inserting

•

the partitioning of P' and S' in 0.8).

<1>=

necessary in general. If ne = n. then Iheorem 3 gives the well-

['10"

known solution of the standard LQ optimal control problem with infinite horizon and long- te nn average criterion. Finally it can be

V = diag(0.6316 . 0.8847. 0.2727. 0.4364, 0.7665) .

shown that for stability of (<1:>'). or for the compensator to be

W = [0.4777) ,

stabilizing, the second condition in theore m 3 may be weakened

and the criterion

to: R > 0 , (, Q 1/2) detectabl e. or Q >0. or \V > 0 , (T, V 1/2)

Q = diag(0.2470 , 0.9 826, 0.7227 , 0.7534,06515) ,

detectable, o r V > O.

R = [0.0727) . As can be seen n=5, m=l= 1. The spectral radius of is

5. Compensatability test and example.

1.1446, so (
First we may state th e following result concerning nc-

compensatability may easily

compensatability and convergence of (; i

(0, 0, 0, 0) as i ->

is not stable. Choose ne= 1. Now n c be verified

using the

compensatability test. Furthermore Q > 0, V > 0, thu s «I> , vln, Q1I2) is detectable. From theorem 3 we may calcu late Ihe

00.

optimal stabilizing compensator (F, K, L) specified by Theorem 4. Assume that either R > 0, W > 0 , (,

compensatable ~ (; i (0, 0, 0, 0) converges as i ->

F = [-0.5861), K = [0.6401), L = [0.7102J.

r , C).n c -

V1 /2, QI12) de tectabl e, or Q >0. V >0. Then ( ,

The criterion is given by

cr: = 3.5336 .

00.

The spectral radius of <1:>' is 0.5969 , so (<1:>') is stable. Proof. By theore m 3, (, (; i

(0, 0, 0 , 0) converges as i ->

r, 00.

C).nc-compensatable =>

6. Conclusions.

The assumptions in this

theorem are not needed here.

In

this

paper the

optimal

fixed-order dynamic

compensation problem has been considered in the case of li near

Now suppose Y = lim (; i (0, 0, 0, 0) exists. Because I ->~

(; i+1

(0, 0 , 0, 0) = (; (;

i

discrete-time systems. The notion of nc- compensatability has

(0 , 0, 0, 0 ) one ha s, taking the

limits, Y = (; Y. Also Y ~ 0 by definition. Hence X = (; X

been introduced. It has been shown that s uitable conditions of

has a nonnegalive definite Solu lio n. Also from lemma 2 we

nc-compensatability and detectability are sufficient, and

have (, V'I12, Q'1 /2) detectable. Thus, using the same

necessary in general, for the existence of an optimal stabilizing

r,

compensator of given fixed-order ne' A test, explicit in the

arguments as in theorem 3, (<1:>') is stable and therefore (, C).nc-compen sa table.

•

system matrices, has been provided, that determines whether a given system is nc-compensatable or not. Finally, the resu lt s have been illustrated with an example.

From theorem 4 we have the following sufficient an d necessary test, explicit in Ih e syslem malrices, for systems to be compensalable with a reduced-order compensator.

Compensatability test. Choo se Q=V=I an d R=W=O. Then (,

r,

C).is tvcompensa labl e if (; i (0, 0 , 0, 0)

converges as i ->

00 ,

otherwi se it is not.

•

Checking the detectability of «I>, V 112, Q1 /2) means checking the detectability of (, Q 1/2) and (

r. V 112), which can be

done in the usua l way.The optimal stabil izing compensator, if it exists, may be calculated from theorem 3, given a system, a criterion, and the order of the compen sator.

523

References.

Athans, M., "The matrix minimum principle", Information and Control, vo!. 11 , 1968, pp. 593-606. Bernstein, D.S., Davis, L.D., and Hyland , D.e. , "The Optimal Projection Equations for Reduced Order Discrete-Time Modeling, Estimation and Control", J. Guidance, vo!. 9, no . 3, 1986,pp.288-293. De Koning, W.L. , "Compensatability and Optimal Compensation of Systems with White Parameters",

TWI

Report nr. 89-61, Delft University of Technology, Delft, The Netherlands, 1989. Submitted for publication. De Waard, H., "Reduced-Order Control of Diffusion /LPCVD Reactors Using Optimal Projections", T\VI Report nr. 9066, Delft University of Technology, Delft, The Netherlands, 1990. Submitted for publication. Enns, D., "Model Reduction for Control System Design", Ph.D. dissertation, Stanford Universi ty, Palo Alto, CA, 1984. Glover, H., "All Optimal Hankel norm Approximations of Linear Multivariable Systems and their L-error Bounds", Int. J. Control, Vo!. 39, 1984. Hyland, D.e. and Bernstein, D.S., "Explicit Optimality Conditions for Fixed-Order Dynamic Compensation ", Proc. 22nd IEEE Conf. Dec. COIHr., San Antonio, TX, 1983. Hyland, D.e. and Bernstein, D.S., "The Optimal Projection Equations for Fixed-Order Dynamic Compensation", IEEE Trans. Autom. Contr., Vol 29, 1984. Mariton, M., and Benrand, P., "A Homotopy Algorithm for Solving Coupled Riccati Equations", Opt. Control Applic . and Methods, vo!. 6, 1985. Richter, S., "Continuation methods: Theory and App lications", IEEE Trans. Autom. Control, vol AC-28, 1983. Richter, S. and Collins, E.G. Jr. , "A Homotopy Algorithm for Reduced-Order Compensator Design Using the Optimal Projection Equations", Proc. 28th Conf. Dec. Contr., Tampa, Florida, December 1989, pp. 506-511. Yousuff, A. and Skelton, R.E., "A Note on Balanced Controller Reduction ", IEEE Trans. Autom. Contr., Vo!. 29 , 1984.

524