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 set
point problem with non-zero mean disturbances. The optimal control law consists
of a closed-loop component for feeding back the measurements and a constant
open-loop component which accounts for the non-zero set point and non-zero
disturbance mean. An additional feature is the presence of state-, control- and
measurement-dependent white noise. It is shown that in the absence of multiplicative disturbances, the closed-loop controller can be designed independently of the
open-loop control.
.
,,
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Notation and definitions
R, Rr xs, Rr, E real numbers, r x s real matrices, Rr x I, expectation
In' ()T n X n identity, transpose
(8) Kronecker product
tr Z trace of square matrix Z
asymptotically
stable matrix matrix with eigenvalues in the open unit disk
n, m, [, p positive integers
x n-dimensional vector
u,y m-, [-dimensional vectors
A, Ai; B, Bi; C, Ci n x n matrices, n x m matrices, [ x n matrices, i = 1, ..., p
r x n matrix, m x [ matrix
L,K
b, y, ex r-, n-, m-dimensional vectors
k discrete-time index 1,2,...
vi(k) unit variance white noise, i = 1, ..., P
WI(k), w2(k) n-dimensional, [-dimensional white noise processes
VI' V2 n x n covariance of WI' [ x [ covariance of w2; VI ~ 0, V2~ 0
V12 n x [ cross-covariance of WI' W2
R1, R2 r x rand m x m state and control weightings; RI ~ 0, R2 ~ 0
r x m cross weighting; RI - R12R2'1RT2~ 0
R12
A, Ai A + BKC, Ai + BiKC + BKCj, i = 1, ..., P
A In -.4
B Bex+ y
B.I Biex
w=wl+BKw2+
p
I BiKw2
i= 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.
W. M. Haddad and D. S. Bernstein
530
p
VI
Ii
LTRIL+ LTR12KC + CTKT RT2L+ CTKT R2KC
For arbitrary
+ V12KT
BT
+ BKVl2 + BKV2KT
BT +
p
+
L C;KTR2KCi
i= 1
mE Rn and Q, P E Rnxn define:
P
P
R2s~R2 +BTpB+
,\
L B;PBi'
i= 1
V2S~ V2 +
CQCT+
L B;PAi'
Ps~BTPA+RT2+
i= 1
p
Qs ~ AQCT
+
+
V12
+
L Ai(Q + mmT)C;
i= 1
P
P
PSI ~ RT2
L Ci(Q + mmT)C;
i=1
p
\
L BiKV2KTB;
i= 1
V
L B;PAi,
QSl ~ V12
i= 1
+
L Ai(Q + mmT)C;
i= 1
\
\
\
1. Introduction
The quadratic performance criterion
\
\
N
i
J
I
\.
\
\!
\
= k=O
L
xT(k)R1x(k) + uT(k)R2u(k)
(1)
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)
(3)
k=O
presents no additional difficulty so long as x(k) and u(k) are replaced by x(k) - x and
u(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 control
effort when iiTR2 ii is large. Hence (3) overlooks design tradeoffs concerning the
control 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 to
steady-state periodic tracking problems (and hence to the special case of non-zero
set point regulation) was given by Artstein and Leizarowitz (1985). Bernstein and
Haddad (1987 b) generalize the results of Artstein and Leizarowitz (1985) for the nonzero set point regulation problem to include noisy and non-noisy measurements,
weighted and unweighted controls, correlated plant/measurement noise, cross weighting, non-zero mean disturbances, and state-, control- and measurement-dependent
multiplicative 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)
531
Non-zero set point regulation
where b is the non-zero regulation point. For full-state feedback with R12 = 0 and
L= 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
(7)
(8)
IX= -RiIBT(A-!.P)-TR1b
where P satisfies the Riccati equation
\
"'
\
/\
with
\
\
\
\
\
I
I,
I
\
\
\,
\
Two features ofthe control law (6)-(8) are noteworthy. First, (6) consists of both a
closed-loop feedback component Kx(t) and an open-loop component IXdepending
upon the regulation point b. And, second (and more important), is the observation
that 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 the
feedback gain K can be determined without regard to the set point. Hence a change in
the desired set point b during on-line operation does not necessitate re-solving the
Riccati equation in real time; only IXrequires updating. For a new value of b, IXcan
readily be recomputed on-line via the matrix multiplication operation (8). In the
presence of multiplicative disturbances, however, the independence of the closed-loop
component from the open-loop component is lost.
The purpose of the present paper is to provide a self-contained derivation of the
optimality conditions for the non-zero set point problem in the discrete-time case. To
obtain a realistic problem setting, we consider the case in which the full state is not
available, but rather only noise-corrupted measurements of linear combinations of
states. For greater design flexibility, we also allow the possibility for correlated plant
and measurement noise. In addition, we consider the dual design feature, namely,
cross weighting in the performance criterion. The presence of a non-zero constant
plant disturbance in conjunction with zero-mean white plant disturbances, i.e. a nonzero mean disturbance, is also considered. Our results show that the presence of a
non-zero constant disturbance component leads to an additional offset in the openloop component of the control. Finally, in addition to the above generalizations we
allow for the presence of multiplicative disturbances in the plant. The control law thus
generalizes previous results involving state-, control- and measurement-dependent
noise (Bernstein and Haddad 1987). As shown in Bernstein and Greeley (1986) and
Haddad (1987), the multiplicative white noise model can be used for robustness with
respect to plant parameter variations.
2. Non-zeroset point regulation
2.1. 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)
W. M. Haddad and D. S. Bernstein
532
with measurements
(10)
where k = 1,2, ..., determine K and rxsuch that the static output feedbackcontroller
u(k)
(11)
= Ky(k) + rx
minimizes the steady-state performance criterion
-
J(K, rx)~ lim E[(Lx(k)
;
k-oo
.
b)TRl (Lx (k)
- b) + 2(Lx(k) - b)TR12u(k)+ uT(k)R2u(k)]
\
"'
(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 + BBT
p
\
+
L [AjQ(k)At
+ BimT(k)Al + Ajm(k)BiT + BiBn + ji
i= 1
Q(k + 1) = AQ(k)AT
\
\,
+
f [AjQ(k)Al
i= 1
(14)
+ Aim(k)mT(k)A(
+ BjmT(k)Al + Aim(k)Bn + ji
(15)
m(k + 1) = Am(k) + B
(16)
To consider the steady state, we restrict our consideration to the set of closed-loop
second-moment stabilizing gains
Ss ~ {K:
A A + itl
<8>
Ai
<8>
Ai is asymptotically stable}
It followsfrom fundamental properties of Lyapunov equations that if KESs, then
A 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 satisfy
k-oo
Note that since A is asymptotically stable, the inverse in (19) exists. For K E S~,it now
533
N on-zero set point regulation
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 restrict
consideration to the set
\
\
S:!!
,/
,
{K E Ss:R2s > 0, V2s> 0 and 'lis is invertible}
where
\
\
\
'¥s~ BTA -T LTR1LA -1 B + BTA -T LTR 12(Im+ KCA -1 B)
\
+ (Im+ KCA -1 B)TRI2LA -1 B + Um+ KCA -1 B)TR2Um+ KCA -1 B)
\
- "L... [BTA-TA!PA-A-IB+B!PA.A-IB+BTA-TA!PB.
p
+
l
I
j= 1
\i
\
I
I
I
I
I
,
Furthermore, we assume that
(22)
I
\.
I
\
\,.
i.e. for each i E {I, ..., p}, Bj and Cj are not both non-zero. Essentially, (22) expresses
the condition that the control-dependent and measurement-dependent disturbances
are independent. There are no constraints, however, on correlation with the statedependent noise. For the statement of the main theorem define
As~ BTA -T LT(RIL+ R12KC)A -1 +(Im + KCA -IB)T(RI2L + R2KC)A-1
p
+ j=L1 ([AjA-1B+B;]TPAjA-1
+BTA-TCJKTR2KCjA-l)
Theorem 2.1
Suppose K and a solve the non-zero set point problem with K E Ss+. Then there
exist 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
Q=AQA
T
+V1+
P
L [(Aj+BjKC)Q(Aj+BjKC)
j= 1
T
T
+BjKV2K
T
-
Bj + Ajmm
Aj
+ iimTA!I + A.miiT
+ ii.ii!
I
I
I ]
+ (Qs + BKV2s)v2;I(Qs+ BKV2s)T-
(25)
Qs V2sQ~
P
P = AT PA + R1 +
T -T
L [(Aj + BKCj)T P(Aj + BKCj)
j= 1
+ CJKTR2KCj
W. M. Haddad and D. S. Bernstein
534
Proof
The 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-T
L(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. Setting
aLjaQ= 0 and using the second-moment stability assumption it follows that ).0 = 1
without loss of generality. Thus the stationarity conditions are given by
\
\
\
aL -TPA+
- fL, A--TPA.+R-P=O
- -
-=A
aQ
- -T+
-aL = AQA
\
ap
\\
j=l
(27)
I
- -T+ A.mm
- T-T
- T-T - -T --T
A. + B.m A. + A.mB. + B.B. ]
l: [A-QA.
p
j=l
I
I
I
I
I
I
I
I
I
I
(28)
+ V-Q=O
\
I
-
aL
(29)
a).T = Am + B - m = 0
I
\.
\
P
\
+ 2: [BjPAjQC + BTPAj(Q + mmT)CTJ + BTPVl2 = 0
j= I
aL
aa
f
= j~l
T
T
[Bj PAjm
+ Bj
TIT
T
+ Bj PBja] +"2B ).+ R12Lm
PBjKC
- RI2<5
+ R2KCm+ R2a= 0
aL
am
- f -T
-T -
-T
).
(31)
1
= Rm + j~l [Aj PBj+ Aj PAjm]-"2A
- cT KT RI2<5+ CTKT R2a
= 2A
(-
P
-T -
(30)
T
T
T
).- L RI<5+L R12a
=0
- -T
(32)
T
T
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
535
N on-zero set point regulation
determined independently of the open-loop component. As shown in the following
section, 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'}' = 0
yielding Theorem 2.1 of Bernstein and Haddad (1987 a).
3. Specializationsof Theorem2.1
A 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
(34)
S = {K : A is asymptotically stable}
and, furthermore, Ss+becomes
\
S+ ~ {K E S :R2a > 0, V2a> 0 and '¥a is invertible}
where
\
\
'¥ a ~BTA-TLTR
,
1
LA-1B + BTA-TLTR J 2 (1m + KCA-IB )
\
\
+ (lm + KCA - J B)TRI2LA -1 B + (lm + KCA -I
For the statement.of
I
Corollary
Aa ~ BT A -T LT(R1L
\h
I
I
\
Corollary
B)TR2(lm
+ KCA -1 B)
3.1 define
+ RI2KC)A
-I
+ (lm + KCA-I B)T(RI2L + R2KC)A - J
3.1
Assume (34) is satisfied and suppose K and ocsolve the non-zero set point problem
with 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~]
(35)
(36)
and such that Q and P satisfy
Q=AQAT + VI +(Qa+BKV2a)V2~I(Qa+BKV2a)T -QaV2aQ~
(37)
P = ATPA + R1 + (Pa + R2aKC)T Rlal(Pa + R2aKC) - P~R2aPa
(38)
Finally, setting
'}'
= 0, R 12 = 0,
V12
= 0, r = n, L = 1n
(39)
we obtain the discrete-time version of Artstein and Leizarowitz (1985) for the case of
output feedback. Define
S i ~ {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
Non-zero set point regulation
536
point problem with K E 5: . 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
-
(42)
AQCTV2aCQAT
P = ATPA + Rl + (BTPA + R2aKC)T Rza1(BT PA + R2aKC)
,.
(43)
- ATPBR2aBTPA
\
\
\
4.
/\
Directions for further research
The 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 tracking
controllers (either static or dynamic) via the parameter optimization approach is
being developed.
\
\
...
\
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'\
ACKNOWLEDGMENT
Dr Bernstein was supported in part by the Air Office of Scientific Research under
contract F49620-86-C-0002.
I
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REFERENCES
\
\
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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., 1987 a, 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:WileyInterscience).