.,,\\\

INT.J. CONTROL,1988, VOL.47, NO. 2, 529-536

Optimal output feedback for non-zero set point regulation:the discrete-time case

WASSIMM. HADDADt and DENNIS S. BERNSTEIN:!:

Optimal discrete-time static output feedback is considered for a non-zero setpoint problem with non-zero mean disturbances. The optimal control law consistsof a closed-loop component for feeding back the measurements and a constantopen-loop component which accounts for the non-zero set point and non-zerodisturbance mean. An additional feature is the presence of state-, control- andmeasurement-dependent white noise. It is shown that in the absence of multipli-cative disturbances, the closed-loop controller can be designed independently of theopen-loop control.

Notation and definitionsR, Rr xs, Rr, E real numbers, r x s real matrices, Rr xI, expectation

In' ()T n Xn identity, transpose(8) Kronecker product

tr Z trace of square matrix Zasymptotically

stable matrixn, m, [, p

x

'\

\i

i

\

\\ u,y

A, Ai; B, Bi; C, CiL,K

b, y, exk

vi(k)WI(k), w2(k)

VI' V2V12

R1, R2R12

A, AiABB.I

matrix with eigenvalues in the open unit diskpositive integersn-dimensional vectorm-, [-dimensional vectorsn x n matrices, n x m matrices, [ x n matrices, i = 1, ..., pr x n matrix, m x [ matrixr-, n-, m-dimensional vectorsdiscrete-time index 1,2,...unit variance white noise, i = 1, ..., Pn-dimensional, [-dimensional white noise processesn x n covariance of WI' [ x [ covariance of w2; VI ~ 0, V2~ 0n x [ cross-covariance of WI' W2r x rand m x m state and control weightings; RI ~ 0, R2 ~ 0r x m cross weighting; RI - R12R2'1RT2~ 0A + BKC, Ai + BiKC + BKCj, i = 1, ..., PI -.4n

Bex+ yBiex

p

w=wl+BKw2+ I BiKw2i= I

Received 9 March 1987.t Department of Mechanical Engineering, Florida Institute of Technolog,', Melbourne, FL

32901, U.S.A.t Harris Corporation, Government Aerospace Systems Division, MS 22/4848, Melbourne,

FL 32902, U.S.A.

,\\

530 W. M. Haddad and D. S. Bernstein

p

V VI + V12KT BT + BKVl2 + BKV2KT BT + L BiKV2KTB;i= 1

Ii LTRIL+ LTR12KC + CTKT RT2L+ CTKT R2KCp

+ L C;KTR2KCii= 1

For arbitrary mE Rn and Q, P E Rnxn define:P

R2s~R2 +BTpB+ L B;PBi'i= 1

P

V2S~ V2 + CQCT+ L Ci(Q + mmT)C;i=1

p

Ps~BTPA+RT2+ L B;PAi'i= 1

p

Qs ~ AQCT + V12 + L Ai(Q + mmT)C;i= 1

P

PSI ~ RT2 + L B;PAi,i= 1

P

QSl ~ V12 + L Ai(Q + mmT)C;i= 1

\

\\

\

1. IntroductionThe quadratic performance criterion

N

J = L xT(k)R1x(k) + uT(k)R2u(k)k=O

(1)

\

i

I\.

\

\!\

expresses the desire to minimize deviations of the state x(k) of the system

x(k + 1) = Ax(k) + Bu(k) + w(k) (2)

from the regulation point x = O. As is well known (Kwakernaak and Sivan, 1972,pp. 504-509), the non-zero set point criterion

N

Jx= L [x(k)-X]TR1[x(k)-x]+uT(k)R2u(k)k=O

(3)

presents no additional difficulty so long as x(k) and u(k) are replaced by x(k) - x andu(k) - ii, where ii satisfies

x = Ax + Bii (4)

Closer inspection, however, reveals that this approach is suboptimal. Specifically,the offset ii in the control may correspond to an unacceptably high level of controleffort when iiTR2ii is large. Hence (3) overlooks design tradeoffs concerning thecontrol effort required for maintaining the non-zero regulation point X. Moreover,such an approach is impossible when ii satisfying (4) does not exist.

A significant advance in extending the full-state-feedback LQR formulation tosteady-state periodic tracking problems (and hence to the special case of non-zeroset point regulation) was given by Artstein and Leizarowitz (1985). Bernstein andHaddad (1987 b) generalize the results of Artstein and Leizarowitz (1985) for the non-zero set point regulation problem to include noisy and non-noisy measurements,weighted and unweighted controls, correlated plant/measurement noise, cross weight-ing, non-zero mean disturbances, and state-, control- and measurement-dependentmultiplicative white noise. They consider the steady-state performance criterion

J = lim E[(Lx(t) _15)TRl (Lx(t) -15) + 2(Lx(t) _15)TR12U(t) + uT(t)R2u(t)] (5)

\"'

\

/\\

Non-zero set point regulation 531

where b is the non-zero regulation point. For full-state feedback with R12= 0 andL= identity, Artstein and Leizarowitz (1985)show that for a constant offsetcontrol law

u(t) = Kx(t) + IX (6)

K and IXare given bv

K= -RiIBTp

IX= -RiIBT(A-!.P)-TR1b

(7)

(8)

where P satisfies the Riccati equation

with

\\

Two features ofthe control law (6)-(8) are noteworthy. First, (6) consists of both aclosed-loop feedback component Kx(t) and an open-loop component IXdependingupon the regulation point b. And, second (and more important), is the observationthat the closed-loop control component is independent of the open-loop component.From a practical point of view this feature is quite useful since it implies that thefeedback gain K can be determined without regard to the set point. Hence a change inthe desired set point b during on-line operation does not necessitate re-solving theRiccati equation in real time; only IXrequires updating. For a new value of b, IXcanreadily be recomputed on-line via the matrix multiplication operation (8). In thepresence of multiplicative disturbances, however, the independence of the closed-loopcomponent from the open-loop component is lost.

The purpose of the present paper is to provide a self-contained derivation of theoptimality conditions for the non-zero set point problem in the discrete-time case. Toobtain a realistic problem setting, we consider the case in which the full state is notavailable, but rather only noise-corrupted measurements of linear combinations ofstates. For greater design flexibility, we also allow the possibility for correlated plantand measurement noise. In addition, we consider the dual design feature, namely,cross weighting in the performance criterion. The presence of a non-zero constantplant disturbance in conjunction with zero-mean white plant disturbances, i.e. a non-zero mean disturbance, is also considered. Our results show that the presence of anon-zero constant disturbance component leads to an additional offset in the open-loop component of the control. Finally, in addition to the above generalizations weallow for the presence of multiplicative disturbances in the plant. The control law thusgeneralizes previous results involving state-, control- and measurement-dependentnoise (Bernstein and Haddad 1987). As shown in Bernstein and Greeley (1986) andHaddad (1987), the multiplicative white noise model can be used for robustness withrespect to plant parameter variations.

\\

II,I

\

\\,

\

2. Non-zeroset point regulation2.1. Non-zero set point problem

Given the nth-order controlled system

x(k + 1)= (A + itl Vi(k)Ai) x(k) + (B + it! Vi(k)Bi) u(k) + WI(k) + Y (9)

.

;.\"'\

/\\

532 W. M. Haddad and D. S. Bernstein

with measurements

(10)

where k = 1,2, ..., determineK and rxsuch that the static output feedbackcontroller

u(k) = Ky(k) + rx (11)

minimizes the steady-state performance criterion

J(K, rx)~ lim E[(Lx(k) - b)TRl (Lx (k) - b) + 2(Lx(k) - b)TR12u(k)+ uT(k)R2u(k)]k-oo

(12)

Using the notation of § 1 the closed-loop system (9)-(11) can be written as

x(k + 1)= (A+ itl Vi(k)A)x(k)+ B + itl vj(k)Bj+ w(k) (13)

\\\

To analyse (13) define the second-moment and covariance matrices

Q(k) ~ E[x(k)xT(k)], Q(k) ~ Q(k) - m(k)mT(k)

where m(k) ~ E[x(k)]. It follows from (13) that Q(k), Q(k) and m(k) satisfy

Q(k + 1) = AQ(k)AT + Am(k)BT + BmT(k)AT + BBTp

+ L [AjQ(k)At+ BimT(k)Al + Ajm(k)BiT + BiBn + jii= 1

Q(k + 1) = AQ(k)AT + f [AjQ(k)Al + Aim(k)mT(k)A(i= 1

(14)

\

\

\\,

+ BjmT(k)Al + Aim(k)Bn + ji

m(k + 1) = Am(k) + B

(15)

(16)

To consider the steady state, we restrict our consideration to the set of closed-loopsecond-moment stabilizing gains

Ss ~ {K: A<8> A + itl Ai <8> Ai is asymptotically stable}

It followsfrom fundamental properties of Lyapunov equations that if KESs, thenA is also asymptotically stable. Hence, for KESs, Q~ lim Q(k), Q ~ lim Q(k) and

k-+C() k-i>aJ

m ~ lim m(k) exist and satisfyk-oo

Note that since A is asymptotically stable, the inverse in (19) exists. For K E S~,it now

,/ \

, \\\

\\\

N on-zero set point regulation 533

follows that J(K, a) is given by

J(K, a) = tr [(Q + mmT)R] + tr [KT R2KV2] + ~TR1~ - 2mTLTR1U

+ 2mTLTR12a - 2~TR12KCm - 2~TR12a + 2mTCTKTR2a + aTR2a (20)

Associated with Q is its dual P ~ 0 which is the unique solution of

P=,.PPA+ f AlPAj+R (21)j= 1

To obtain closed-form expressions for the feedback gain K, we further restrictconsideration to the set

S:!! {K E Ss:R2s > 0, V2s> 0 and 'lis is invertible}where

'¥s~ BTA -TLTR1LA -1 B + BTA -T LTR12(Im+ KCA -1 B)

+ (Im+ KCA -1 B)TRI2LA -1 B + Um+ KCA -1 B)TR2Um+ KCA -1 B)

l\i\,

p - - -+ " [BTA-TA!PA-A-IB+B!PA.A-IB+BTA-TA!PB.L... I I I I I I

j= 1

Furthermore, we assume that

I\.I

\\,.

(22)

i.e. for each i E {I, ..., p}, Bj and Cj are not both non-zero. Essentially, (22) expressesthe condition that the control-dependent and measurement-dependent disturbancesare independent. There are no constraints, however, on correlation with the state-dependent noise. For the statement of the main theorem define

As~ BTA -TLT(RIL+ R12KC)A -1 +(Im + KCA -IB)T(RI2L + R2KC)A-1

p -+ L ([AjA-1B+B;]TPAjA-1 +BTA-TCJKTR2KCjA-l)

j= 1

Theorem 2.1

Suppose K and a solve the non-zero set point problem with K E Ss+. Then thereexist n x n Q, P ~ 0 such that

K = - Rzs1(BT PAQcT + Pst QCT + BT PQsl) V2;1 (23)

a = - '¥s-1[AsY+ fM] (24)and such that Q and P satisfy

T P T T T - T -TQ=AQA +V1+ L [(Aj+BjKC)Q(Aj+BjKC) +BjKV2K Bj + Ajmm Aj

j= 1+ iimTA! + A.miiT + ii.ii! ]I I I I

+ (Qs+ BKV2s)v2;I(Qs+ BKV2s)T- Qs V2sQ~ (25)P

P = AT PA + R1 + L [(Aj + BKCj)T P(Aj + BKCj) + CJKTR2KCjj=1

\\\

534 W. M. Haddad and D. S. Bernstein

ProofThe derivation of the necessary conditions is a straightforward application of the

Lagrange multiplier technique. To optimize (20) over Ss+ subject to the constraints(18) and (19), form the lagrangian

'" [ (- -T P - -T - T-T - T-TL(K, a, Q, P, m) = tr )'oJ(K, a) + AQA + j~l [AjQAj + Ajmm Aj + Bjm Aj

+ AjmBl+ BjBTJ+ V -Q )p+).T(Am+B-m)]

where the Lagrange multipliers ).0;;:;:0, ).E R" and PER" x" are not all zero. SettingaLjaQ= 0 and using the second-moment stability assumption it follows that ).0= 1without loss of generality. Thus the stationarity conditions are given by

\

\\\

aL -T - f -T - --=A PA+ L, A- PA.+R-P=OaQ j=l I I

aL - -T l:p - -T - T-T - T-T - -T --T- = AQA + [A-QA. + A.mm A. + B.m A. + A.mB. + B.B. ]ap j=l I I I I I I I I I I

(27)

+ V-Q=O (28)

I\.

\

\

aL -a).T= Am + B - m = 0 (29)

P

+ 2: [BjPAjQC + BTPAj(Q + mmT)CTJ + BTPVl2 = 0j= I

(30)

aL f T T TIT T

aa = j~l [Bj PAjm + Bj PBjKC + Bj PBja] +"2B ).+ R12Lm

- RI2<5+ R2KCm+ R2a=0 (31)

aL - f -T -T - 1 T T T

am = Rm + j~l [Aj PBj+ Aj PAjm]-"2A ).- L RI<5+L R12a

- cT KT RI2<5+ CTKT R2a = 0 (32)

-T (- P -T - - -T T T). = 2A Rm + j~l [Aj PBj + AjPAj m] - L RI <5+ L R12a

- CT KT RI2<5 + CTKT R2a) (33)

Using the definitions for QSI and PSi along with (33), we obtain (23) and (24).Substituting the expressions for the optimal gains into (27) and (28) yields (25) and(26). 0

Remark 1

Because of the presence of <5in (25) via m in both QSI and V2sand in (25) via B(in m) and Bj, the closed-loop component of the control law (23) cannot be

\\.\

/\\

N on-zero set point regulation 535

determined independently of the open-loop component. As shown in the followingsection, independence is recovered when the multiplicative noise terms are absent.

Remark 2

To specialize Theorem 2.1 to the standard regulation problem, set ~ = 0 and'}' = 0yielding Theorem 2.1 of Bernstein and Haddad (1987 a).

3. Specializationsof Theorem2.1A seriesof specializationsof Theorem 2.1 is now given.We begin by deleting all

multiplicativewhite noise terms, i.e.

Ai>Bj, Cj = 0, i = 1, ..., P

In this case the stabilizing set Ss can be characterized by

S = {K :A is asymptotically stable}

and, furthermore, Ss+becomes

S+ ~ {K E S :R2a> 0, V2a> 0 and '¥a is invertible}

(34)

\ where

'¥ ~BTA-TLTR LA-1B +BTA-TLTR (1 + KCA-IB )a 1 J 2 m

+ (lm + KCA - J B)TRI2LA -1 B + (lm + KCA -I B)TR2(lm + KCA -1 B)

For the statement.of Corollary 3.1 define

Aa ~ BT A -T LT(R1L + RI2KC)A -I + (lm + KCA-I B)T(RI2L + R2KC)A - J

\,\\

I

\hII

\

Corollary 3.1

Assume (34) is satisfied and suppose K and ocsolve the non-zero set point problemwith K E S -t. Then there exist n x n Q, P ~ 0 such that

K = -R1al(BT PAQCT + RI2QcT + BTPVI2)V2~1

oc = - '¥';-I [Aa')'+ Q~]

and such that Q and P satisfy

Q=AQAT + VI +(Qa+BKV2a)V2~I(Qa+BKV2a)T -QaV2aQ~

(35)

(36)

P = ATPA + R1 + (Pa + R2aKC)T Rlal(Pa + R2aKC) - P~R2aPa

Finally, setting

(37)

(38)

'}'= 0, R12 = 0, V12 = 0, r = n, L = 1n (39)

we obtain the discrete-time version of Artstein and Leizarowitz (1985) for the case ofoutput feedback. Define

Si ~ {K E S :R2a > 0, V2a> 0 and '¥ I > O}where

Corollary 3.2

Assume (34) and (39) are satisfied and suppose K and ocsolve the non-zero set

,.\\\

/\\

536 Non-zero set point regulationpoint problem with K E5: . Then there exist n x n Q,P;?:0 such that

K = -Rzal BT PAQCT V2~I (40)

a= -'I'11BTA-TRI<> (41)

and such that Q and P satisfy

Q=AQAT + VI + (AQCT +BKV2a)V2~I(AQCT +BKV2a)T

- AQCTV2aCQAT

P = ATPA + Rl + (BTPA + R2aKC)T Rza1(BT PA + R2aKC)

- ATPBR2aBTPA

(42)

(43)

\...

\

4. Directions for further researchThe extension to fixed-order dynamic compensation for non-zero set point

regulation appears possible using the approach of Hyland and Bernstein (1984)and Haddad (1987). A generalization of Theorem 2.1 to design periodic trackingcontrollers (either static or dynamic) via the parameter optimization approach isbeing developed.

\'\

I\.

\

\\I

ACKNOWLEDGMENT

Dr Bernstein was supported in part by the Air Office of Scientific Research undercontract F49620-86-C-0002.

REFERENCES

ARTSTEIN,Z., and LEIZAROWITZ,A., 1985, I.E.E.E. Trans. autom. Control, 30, 1123.BERNSTEIN,D. S., and GREELEY,S. W., 1986, I.E.E.E. Trans. autom. Control, 31,362.BERNSTEIN,D. S., and HADDAD,W. M., 1987a, Int. J. Control, 46,65; 1987b, I.E.E.E. Trans.

autom. Control, 32, 641.HADDAD,W. M., 1987,Robust optimal projection control-system synthesis. Ph.D. dissertation.

Department of Mechanical Engineering, Florida Institute of Technology, Melbourne,FL.

HYLAND,D. c., and BERNSTEIN,D. S., 1984,I.E.E.E. Trans.autom.Control,29, 1034.KWAKERNAAK,K., and SIVAN,R., 1972, Linear Optimal Control Systems(New York:Wiley-

Interscience).