..
March 1978
Report ESL-P-811
INVARIANTS OF OPTIMAL MINIMAL-ORDER OBSERVER-BASED COMPENSATORS*
a
P. Blanvillain
Laboratoire d'Automatique et d'Analyse des
Systemes (LAAS)
BP 4036
7 Ave. du Colonel-Roche
31055 Toulouse Cedex, France
T. L. Johnson
Massachusetts Institute of Technology
Electronic Systems Laboratory, Rm. 35-210
77 Massachusetts Avenue
Cambridge, Massachusetts 02139
Abstract
A
Ar --12
The relations characterizing the optimal
minimal-order observer-based compensators for a
linear time-invariant multivariable system in a21
certain canonical form, with a random initial
state (or equivalently, known initial state with
white driving noise) have been reported by
Miller [l] and independently by others [2], [3].
In this note, we establish in general that the
plant transfer function uniquely determines the
optimal compensator transfer function, and that
this characterizes precisely the degrees of
freedom in the compensator design. Any two realizations of the optimal compensator dynamics
yield the same performance.
A
=
Rl
xl
x
B =
2
-
B
1
L
_L
i 1 -12
2
L
I
(3)
The input u is constrained to be generated by a
minimal-order state observer (4] with state z c
, which may generally be expressed in the form
R
z
..
Fz + Gy
-
Hz
z(O)
=
0
(4)
(5)
+
so as to optimize, the standard performance index. *
1.
Introduction
J(F,G,H,M)
The problem of optimally controlling a timeinvariant linear multivariable system
x
=
-
=
Ax + Bu
;
_ N0,(o)
Cx
nith
, control
s
Ru C
Ruldt}
(6)
0
where the expectation is over the distribution
induced by x , and
=O
> 0, R = R' > 0 (in the
>'
-sense of positive definiteness).
(1)
(2)
5
E(Jfx'up
rIand output-
Y E R (m, r < n) and random initial state x(O) =
o on the semi-infinite interval t e [1O,), by
means of a minimal-order observer-based compensator has been solved by Miller ll]in the case
where C has the particular canonical form C =
[I 01. Let A, B, I be parameters of a system
wi;-h-this canonical form, partitioned in accordance with C, as
Let T7
-A,B, ,Z,Q,R denote the parameters
of the optimization problem characterized by (1)(6).
Then under the additional assumptions that
*
{A,1C,}
is minimal and E
> O the optimal compensator of [11-[31 exists, is unique, and may be
characterized as follows: Let
_ _
-R BIP
17
where P is the unique** positive definite solution
of the algebraic Riccati equation
l-.(8)
0 = PA + A'P + Q - PBR-'P
and let
This research was performed at the M.I.T. Electronic Systems Laboratory and the LAAS, with support provided by the National Aeronautics and
Space Administration (Grant NGL-22-009-124) and
by the Centre National de la Recherche Scientifique of France. The latter part of this paper was
prepared while the second author was visiting at
the Department of Computing and Control, Imperial
College.
Note that the parameters of
dependent, but are subject to explicit constraints,
since z must be an observer of +x
**
We assume
- -l/2
A,Q
I completely observable.
..
..
1`
11(9)
2'(WA'2
-+
12
-WA2+ ---.
12-- l. 2)_ 11
L
2
n
(n m )
positive
unique
is
the
where W E R
- isthe
positvedefnitedefinite
solution of the reduced filtering equation
!o
-(A
-
-22
:
Z1A )W + W(A'
- -22
-12 -11-12 -
r'i
~- moi'
A12
-
v-
1-12-
-22
compensators with the same performance (after the
re-transformation (d)) and that such optimal com- pensators are in fact always related by a similarity transformation (in R n-m) ). The conclu-
)the
As tl
-12 -11-12
,l
-12-11-12
(10)
Then the optimal gains are given by
ZE22
-~
12
F
*
I
+ (B1)-~2
- -2LA
{
)i +
2
42-2of
.-
-12)-21
-11
+ (-2
...
l)-(
-L)
(13)
_
_H
-_
t(14)
+ KL
-.And the optimal cost is (3]:
M
H
.
* tr-11a-
lL +;22 -
+ tr-(
z12
-;12-)
(n m)
where
:equation
A well-set compensator problem, P, consists
a sextuple* of matrices {A,B,C,_Z,,R} (dimensioned as above) such that [A,B,C] is minimal,_,
'
> 0, CEC > __, and R > 0
Q, R are symmetric and
in the sense of positive definite matrices. Evidently, this corresponds to the problem (1), (2),
(6) subject to the constraint that u is generated
by an observer (4), (5) of the unmeasured states.
Two well-set problems P , P having equal dimensions (n,m,r) are said to be equivalent if there
exists a nonsingular matrix S E R
,
_+_2_problem
such that
-
-SBC1 1 '
2-=2=
{SA S
___
-1-1
}
where subscript (1) denotes
, R1
,S.1S
,(S')
Q1
P1, Two minimal realizations i{A,B.,C.}i=
2
R(nm)is the solution of the Lyapunov
_22_1_
Equivalence Relations for Compensation
Problems
II.
22
---
that for a given performance index,
Kronecker invariants of the plant uniquely
determine the Kronecker invariants of the optimal
compensator, and that is all: the engineer is
free to construct whatever realization of the
optimal compensator suits him best, e.g. from the
standpoint of reliability, component cost, etc.
Further implications are discussed in Section IV.
(11)
(12)
..--.2.
..
vc /?
sion, then, is
2
22 -12)
(--22 -12)
The first term of (15), which will be denoted ~J*,
°e l.
er o 15
Th frs dity
is the optimal performance assuming full-state
feetdba'k through the IKalman control gains, K, and
the second term [2] is the additional_cost incurred by the observer, denoted CJ-'*
1, 2 of compatible dimensions are equivalent (in
the usual sense) if they are related by a similarwell-set compensation
transform as
above. AA well-set compensation
as above.
ity transform
problem V (m < n) is said to be in state-output
i.e., the first
0
canonical form if C =
states are measured exactly and independently. We
proceed to develop some simple consequences of
In order to apply these results to an arbitrary (non-canonical) problem P = {A,B,C,E, ,R},on
it is necessary tosition
(a) transform the plant to canonical form
(b) transform the performance index in a
'compatible manner
(c) solve for the gains of the transformed
problem using (7)-(14)
(d) retransform plant and compensator to the
original coordinates in an appropriate manner.
One might infer from the prior works that this
procedure is either trivial or uninteresting. We
Partial
shall demonstrate that it is neither.
results pertaining to (a)-(d) are reported inependently in the theses of Blanvillain [3] and Rothschild (see (4]), and perhaps in other documents
not known to the authors.
these definitions.
2.1
2
Our procedure will be to expose in Section II
the appropriate transformations (not uniquel) for
accomplishing (a) and (b). We show that two equivalent realizations of the plant may be reduced to
the same transformed problem (c). Because the
transformation of a problem P to a problem P is
unique. In Section III we show that although the
compensator gains in step (c) depend on this
transformation, that any two transformations yield
~~~~~~ ~~~~~-~--
For any well-set compensation problem, P
there exists an equivalent problem, P, in stateoutput canonical form Cm < n).
Proof
(Yuksel and Bongiorno 15])
S
T
xn is any matrix which renders S
where T E R
nonsingular (i.e., rows of T linearly independent
= I , C = CS
Since SS
of rows of C).
-
(I 01 as desired.
Define the other parameters of
*We have assumed zero-mean initial state for conciseness of exposition. See (3) for the case of
nonzero mean initial state.
---- I~~~~~~~~~~~~~~~~~~~~
P iising the definition of equivalence given above.
Proposition 2.2
Assume that P1 and P are equivalent well-set
Then the outm - n.
compensation problems wit
put feedback problems consisting of (1), (2)
and (4)-(6) with F, G and H set of zero have the
same optimal gains, same closed-loop dynamics, and
numerically equal optimum
optimum performances.
performances
equal
Proof
The optimal gainy for _qis problem are wellknown to be M. - -R -B!P.C.
, i = 1, 2, where P
denotes the solutioi of()with overbars replaced
Let P 1 and P.2 be related by S
by subscripts (i).
as in the definition. Then it is a matter of al1
gebra to verify that P = (St)-lp s- (see [3, p.
17]), and by equivalence we see i-mediately that
roots
closed-loop
the closed-B
2,
and
roots
)I,
Rop
R!2,
and that
that the
M1l =
2 '-2-2
=Z-- .
AE2-
CSs
desir~ed.
ElSt]
,l as
triPi 7I = J 1
0
~~~~desired,~
P1 is (in an appropriate sense)
choice of T
I
.numerically
Equivalence of Observer-Based Compensators
The main result of this section is
'owing:
lwn
Theorem 3.1
The optimal compensators for these prob-
p2
of
of m
m << n.
n.
Then
placed by subscripts 1, 2 respectively).
these optimal compensators are equivalent realizations of (4)-C5), and lead to the same closedloop performance, i.e. there exists a nonsingular
2
} -
Iu
2
UGHu,1.
full-state feedback; in a sense, the following
sections generalize this proposition for the case
The proof of this theorem requires some relinimary
results,
which (see
are stated
in for
the proofs).
followAppendix
of lemmas
ing sequence
Proposition 2.3
Lemma 3.1
Suppose the equivalent well-set compensation
problems P1 and P are related by the transformation S C R°u
and suppose furthermore that P
a transformation S which makes P2 equivalent to
Palso.
! . ieif 2 f~or
Silte
inVerify,
,Simply take R2 = S S
nd so on.
= C -1
-ss-1
Thus, a (commutative) equivalence relation
exists between P1 9 P2 and
F.
T
is nonsingular
(19)
'
M+N
Furthermore,
if the
irl, ~~~~~~~~~~then
'1 t=[ IE
],he
inverse of S is partitioned as
C, 1 have the unique decompositions
C ICC)
-
N'(NN')v
(20)
(21
N(NN
Furthermore, if
T
(hence C-
[£21
Lemma 3.2
R
< n, be given.
has a unique decomposition.
chosen as in Proposition 2.1, there
-l
S
--2 · --1--
such that S
Rm)
with N of full rank, n-i.
-- 2 -1-
=
implies that A
S A Sstance, that A
--i
-- 1
-CS d
S C,
S A
A l S -A SS
-- 2--1 --1 1- -AA-2~2-2--
is
y
is
form through transformation S 1; then there exists
implies t = c
M
of full ra
Let C
in state output canonical
equivalent to problem
-,
exists T 2
the fol--
Let P be a well-set compensation problem
Cn-m)xn
1
2
define (as in
(m <n) and lee T , T £ R
Proposition 2.1) equivalent state-output problems
2
2
2
H2
nm)such that (F ,G ,4
v E Rn~ :~U
This proposition provides justification for
our notion of equivalence in the special case of
= S-
independent of the
lems are defined by C7)-(14) (with overbars re-
1
)IA4 -B R - BP ) are the same (identical closedj=--I
n *
bc l 4a ,-;ekciOS ,
] =
1'op rransfe~--unctions), and J*,
r-2
- --2
l0oa
,2 'rtriP2
tr[ (SC)'P1
problems P2 which may then be reduced to any equiv2
What remains, is merely to show
problem V of P1
that the solution of any problem P derived from
-
-Ip
1S(
-1-1
=1_
=11 )S-
X
of.PI' we may construct an equivalent compensation
atS
C(n-m) xnsch
such thatC
--2
1
T'
Z2
plays the proper role in the proof above. From
these propositions, it is readily seen that for a
given well-set compensation problem, P. there
above) of
are uncountably many ways (choices of
reducing it to an equivalent problem, P (depending
on T i) in state-output canonical form; for any
1IemnaLet
other minimal realization of the plant parameters
0 and ~
I
m , in particular).
1
2
Let T ,T
M < n, be given.
Let C C X
be as in Lemma 3.1 with unique decompositions of
form (19). For any such matrices, there exists a
2
1UN
R(n -m ) such that2
unique nonsglar U
i
Lemma 3.3
C, T be as in Lemma 3.1, and let ] denote
the 'equivalent state-output problem resulting from
3
the transformation S.
Then the dependence of the
parameters in P upon T - VC + N is given by
CAC
CAiTh
CB
-
F
TAC
c2-
TAT
(see (25)).
.22-1
2 2
[+(V -L )C)[IA+BKN
(from (201
(21))
Then
-
N-(NN
.
TB
rioir=[_
-cc, czT1OU(L1
__
~
C L "T
' tIOx
ET'_'QC
t
- DEC,
UC£c
_
QJ
~Also, M
-F12
(using (26))
as desired
Ml, since
8
_-- -2
(22)
R
_R-
(V -L )C IA+BKJN '(NN§ '-1)
independent of V1 , V2
:
em2-
A
I[N
2
+N2 , (N2N 2 ,)1
e"
KC (CC')
'
1-N2 (N2 N2 )-4V
°
K[C (CC')I+N, (N1Nlw)I (Lv
L2 I
with the consequence that in (7)-(10),
K
P
8
em
t-o2
1C)D
1CZ)
as desired. The equality of the costs
H2
H1U
is established in a similar manner; for instance,
N(-'(C-WC')- lC-_)N
1
tr[C
+
0O
10
tr [P
-
1NA'C' (CZC)-CAN'(NN)INW(NN')
'
1
-PC CCC'
1
+ C
PT
tr T' PC CZT1 ' + TpT
1
PTT
(24)
Evidently, we intend to make use of Lemma
3.2, but this requires us to show that the optimal
VI, V2
as
compensator gains are independent of
V,
well as having the desired dependence on N1 or N2
Let 2 W"-2
Let
, W denote the solutions of (24) correspondand N . Then is is readily verified
that W = UT U.'
Applying (22) and the decomposition (19)-(21) to (9), some algebra yields
1
IW _(NN__.
=
I
, twice
along with a trace identity gives
Proof of Theorem 3.1
L
C'J
T'ZT '1]
and using the fact that C1C + TT
ing to N
m
Similar algebra demonstrates that G2 = UG 1 and
J;
C)AN'(NN')-ii +
W(1'
S~NA(I-CI(CZCI)
)
)
(23)
where I denotes the Kalman gains for the original
plant (solution of (17), (18) sans overbars), and
W in (10) is the solution (independent of V) of
N(I-EC'(CrC')'
1 ]
)-NiA' + N3_.]C'
C')
-1
l + V1
(25)
Then applying Lemma 3.2, one finds from a similar
calculation that
J1
=
tr[p(ilTl).] + tr[p(tlT)El
jjp
tr
1
2*
, obviously)
independent of T.* The second
term AJ
is
shown to be independent of N1 using (15), (16)
and (26).
Corollary 3.1
If P1 and P2 are equivalent compensation
problems, then they have equivalent optimal compensators, i.e. any optimal compensator realization of one problem is also optimal for the other.
Proof
z2.
gl
i)
1
+1 2
(26)
2
1 -l
We illustrate the proof that F e UF U .
First
Suppose that P
and P2 are related by S, and
that application of Si to Pi yields a state-output
problem Fi which results in optimal compensator
1
F
-
e
A 2LA 2
2+
; 12
-!22 --
( 2-1
l
parameters
1
-1
1
11
1-1
TAT1 - LlCAT 1 + (TIB-LcB)KT
(from (22),
(23))
ri,GHiMi,
-i-i-u
*Or tr(P
i
1,
Sl) - -
2.
These two
)
rr
giving a shorter proof.
4
compensators must be related by a similarity transformation, since Proposition 2.3 the state-output
problems Pi are then equivalent, and the compen-
159 (Jams 1973).
2.
sator of one merely corresponds to a different
choice of T for the other, and Theorem 3.1 applies.
-
IV.
48
--
Discussion and Conclusions
The degree of non-uniqueness of the solution
of the optimal minimal-order observer-based compensator has been precisely displayed, and is in
fact intuitively pleasing, because the engineer
retains the freedom to choose a realization of the
optimal compensator. Possibly this property is
so pleasing that it has been presumed to hold by
several authors, although in fact it represents a
rather unusual special feature for a constrained
optimization problem.
We could in fact have based our proofs directly on the statement of the constrained optimization problem 13, p. 80-83] rather than on the
(unique) solution of the necessary conditions for
the canonical problem. This approach allows us to
extract a problem formulation with a unique solution prior to the application of direct methods
of computation and hence remove potential convergence problems for the problem stated in the
'original coordinates. In fact, an attempt to express the necessary conditions in the original
coordinates leads to an underdetermined Riccati
2
equation [3, p. 901 of order n rather than
2
(n-m) as in (10):
Oearly
+ C - ~~'
Af~ -+-lA' AX + RA' + Z - (VA'+Z)C'(CEC') C(WA'+)' =- 0
(27)
2
n
where I C R . The fact that equation (10), or
more explicitly, equation (24), does have a
unique solution (under the stated conditions) may
be useful in the study of degenerate Riccati equations; (24) appears to be a type of projection of
(27).
While the observer-based compensator is known
to satisfy the necessary conditions derived by
Newmann [2] and by Levine, Johnson and Athans [61
for the case dim z = n-m, the "separation principle" for lower-order compensators has only been
established by Jameson and Rothschild [4] and
others under the assumption that the compensator
is an observer of the optimal gains. The direct
approach proposed above may be extended to examine
uniqueness of observer-based compensators of any
order. Our results suggest that a major feature
of the observer-based compensator formulation is
to eliminate unwanted degrees of freedom (viz.
equivalent realizations of the plant and of the
compensators) from the design procedure in order
to obtain uniqueness.
Newmann, M.M., "Specific Optimal Control of
the Linear Regulator Using a Dynamical Controller based on the Minimal-Order Luenberger
Observer", Int. J. Control, Vol. 12, pp. 33(J;an
1970).
3.
Blanviliain, P.J., "Optimal Compensator Structure for Linear Time-Invariant Plant with Inaccessible States", MIT Electronic Systems
4.
Jameson, A. and Rothschild, D., "Asymptotically Optimal Controllers", Int. . Control,
Vol. 13, pp. 1041-1050 (June 1971).
5,
Yuksel, Y.O. and Bongiorno Jr., J., "Observers for Linear Multivariable Systems with
Applications", IEEE Trans. Auto. Control,
Vol. AC-16, No. 6, pp. 603-613 (Dec. 1971).
6.
Levine, W.S., Johnson, T.L., and Athans, M.,
"OptimalLimited State Variable Feedback Controllers for Linear Systems", IEEE Trans.
Auto. Control, Vol. AC-16, pp. 785-793 (Dec.
1971).
Appendix
Proof of Lemma 3.1
If S is nonsingular its rows must be linearly iindependent; thus each row of T must be linindependent of all rows of C and all other
rows of T. From linear algebra, the span of the
rows of C forms an m-dimensional subspace of Rn;
(19) is essentially a statement of the projection
theorem, stating that each row of T may be
uniquely decomposed into a projection on the span
of the rows of C (rows of VC) and a vector orWe have explicitly
thongonal to it (row of N).
-1
V
-
TC' (CC,)
N
-
T(I-C'(CC')
-
(Al)
1
(A2
C)
-
is both a left and right inverse of S,
Since S
C, T. e and § must satisfy
(A3)
I
C
TT
=
(A4)
I
-n-m
-
CT
-m,n-m
T
0
-nC + rrT
(A
'
(A6)
.(A7)
I
References
~i, C admit unique decompositions
1. Miller, R.A., "Specific Optimal Control of the
Linear Regulator Using a Minimal Order Observer", Int. J. Control, Vol. 18, pp. 139-
admit unique decompositions
CA
N
a
Ci_
=
O
(A8)
TCN
L
P
CID + R
I
-
C4
0
Using (A9) in (A3) gives i
(AS) gives
(A4), *N
= I
.
imply ! + RV
O
0.
(A9TAT
.-
(CC'))
(A6) gives NW + V
(19) and (A9) in
(AG) in
(A9)
a0E _~~_
.
Using
0.
Using
From (19) and (A8) in
Verify that these three rela-
'C)cT'[T(I-C'(CC')-lC)T-'1
_ [_-_]C_'(cc')
-1
(Al0)
(All)
relating C and T to C and T.
Proof of Lemma 3.2
.
i
Let N C R
, i = 1, 2 correspond to the
decomposition (19) of Ti . The matrices N are not
completely arbitrary, in fact, but must both be of
full rank and satisfy the (n-m)m constraints
Nic IC 0. Thus there are exactly (n-m)2 degrees
of freedom in the choice of Ni so that the conjecturc N =1 UN for some nonsingular U
R(nm)
makes sense. In fact using (A2) one finds that U
is the unique solution of the (n-m) 2 independent
linear relations
(T2-uT )
-C'(CC')
C)
=
0
(A12)
Proof of Lemma 3.3
Equations (22) are a straightforward consequence of the definition of equivalent compensation problems and the partitions of S, S-1 introduced above. Now also
-
'
=
-
__
R 1-B'P,[
1
R B'S'(S ')
=3p1::-
1]
=
K[_
_]
PS
1
(A13)
as desired, where the definition of equivalence
and the expression for P (see proof of Pr 2 position
2.2) have been used. To prove (24), let W be as
in (10) and apply the relations (22) with the decompositions (19)-(21). The first step gives
(TAT-TC'(CEC')
(VC+N)AN'(im')
TZC_ CZC_)
.
1
CA?
.
-VCAN'(NN')-
NEC(CEC'I)
CAN' (NN')
-
Then these results and (A7)
.
tions among W, , N and V are satisfied uniquely
:(in view of '(Al), (A2)) by (20) and (21). We have
in fact rather complicated expressions
(I- CCc(CC
-
CAT)W +
1
W(T' A'T-TT'-AC' (CLC') - CT') - ii'A'C' (CEC')-CATW
+ TET' - TEC'(C.C')
1
CT'
O0
(A14)
We illustrate the procedure for simplifying the
first term of this expression:
since 1l' - 0.
The sum of these two terms is independent of V and identical with the coefficient
term of (24).
Remaining details are
of the first
left to the persistent reader.