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6.1991-2809

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Jean-Franqois MAGNI and Philippe MOUYON
C.E.R.T.-D.E.R.A.
B.P. 4025
F31055 TOULOUSE CEDEX, FRANCE
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Abstract.
In this paper we give, in an eigenstructure assignment setting, a geometric approach to the synthesis of
observers. This formulation is similar to the classical geometric one which is based on the knowledge of a (C,A)invariant subspace. The technique proposed allows the
construction of subspaces which consider various decoupling objectives as for instance Loop Transfer Recovery.
The approach proposed is independent of the kind of observer considered (one dimensional, minimal order, order
larger than the number of sta,tes ...). Some illustrative
examples are given.
We shall consider a linear time invariant control system :
i=Ax+Bv ; y=Cx+Dv
(1)
A , B , C and D are linear mappings A : X --+ X I B :
U -+ X , C : X --+ y , D : U + y , dim S = n , dim y =
p , dim U = m.
For simplifying the notations, it will be assumed
that the dynamics of the observations are real. Complex eigenvalues can be dealt with in a similar way, see
Magni [4].
lernentary and general observers.
1
We ~$11 elementary observers, the one-dimensional dyThe theory of observers introduced by Luenberger [2] namic systems described by the following lemma.
received much attention twenty years ago. The contribution of Wonham [9], where observers are presented in a Lemma 2.1 - The system defined b y :
geometric setting, clarifies this concept. Wonham shows
i = T i - t*y (u*B t*D)v
that t o a given (C,A)-invariant subspace corresponds an
+
observer. Here we popose the step by step construction
of these subspaces, following ideas proposed in Moore
[5] relative t o ( A ,B)-invariant subspaces. The advantage of this stepwise construction, using eigenstructure
assignment, is that we do not have t o consider whether
or not the subspaces we are building are complementary
observability subspaces. This is due t o the fact that the
assigned dynamics are arbitrarily chosen, so, even if it
cannot be modified, there is no problem.
In this paper, we first describe the structure of a
one-dimensional observer, by using eigenstructure assignment concepts. Then observers of any known kind
can be viewed as simple structures mixing together measurements and estimated variables. Our formulation
ma,kes conspicuous the ways to obtain various decoupling properties. The Loop Tra.nsfer Recovery Problem
illustrates such decoupling. The formulation proposed
seems t o be quite efficient for dealing with fault diagnosis. Some illustrative examples are given.
+
where u , t , R satisfjl :
+
u*A t*C = T U
(3)
is an observer of the variable z = u*x. Th.e dynamics of
the observation is given b y T (see Figure 1).
Proof: Clearly from (1) and (2) we have :
i - U* i = ~i-t* (CZ+Dv)+ (U* B+t * D)v-
U* Ax- U* B v
hence from (3)
i - u * i = T ( i - u*x)
This Lemma states that a linear combination of states
u*x can be estimated by a one-dimensional observer provided that the vector u satifies (3) for some vector t and
some complex number
as well as :
Copyright 0 1 9 9 1 by the American Institute of Aeronautics and Astronotics, Inc. All rights reserved.
1748
R.
[
[u* t*]
This equation can be written
]
=0
(4)
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Figure 1: Elementary observer of z = u*x.
Figure 2: Observer of the vector 7 = Lx.
It is dual to the one commonly used for feedback design
by eigenstructure assignment. The main difference is
that for feedback design the assignment of only one vector is generally useless as there are always n eigenvalues
to deal with. Here, there is no need to deal simultaneously with n eigenvalues because (4) gives rise to a device
which is interesting in itself as it is an observer.
The matrix of all vectors u' satisfying (4) will be
denoted T ( n ) . The computation of the matrix T(x) is
done by using the fact that I m T ( r ) T = ((A-.lrI)Ker C)'
or equivalently ImT(x)T = (AT - aI)-lIrnCT.
The use of one elementary observer enables to estimate all linear combinations of z and of the entries of y.
This means that for any row vector Q1 and scalar Q2,
the variable 71 defined by :
Equation (6) relates our approach to the geometrical
one cited in the introduction ([9]) by defining a (C,A)invariant subspace which is KerU.
71 = [Qi
Q2]
[ ]
= LX
s
The observers defined here differ somewhat from usual
observers, so the fundamental property, which says that
the transfer matrix between the input and the system
state vector is equal to the one between the same input
and the estimated state vector must be amended. It
becomes :
Lemma 3.1 In Figure 2 the transfer matrix between v
(5) and ij is equal to the transfer matrix between v and the
vector Lx.
with L = &IC+ Q2u*, can be estimated.
If elementary observers are used in parallel, say a Proof: From Figure :
number equal to q , we can represent the overall observer
as in Figure 1 but replacing u*,1' and x by matricial no- G(s) = Ql(s)(C(sI - A)-lB D - D)v(s)+
tations U , T and II where these matrices denote respecQ ~ ( s (SI
) - II)-l(U - TC(sI - A)-l)Bv(s) (8)
tively [ul,. .. , u q ] * ,[ t l , . . . , t4]*and Diag(n1, ... , r q } ,
where the vectors ui,ti and the scalars rj satisfy equa- Substracting SU to both sides of equation (6), after pretions similar to (2). These q equations can be written and post-multiplication we obtain :
globally as :
( S I - I I ) - l ( U - TC(SI- A)-') = U ( S I - A)-l (9)
U A TC = IIU
(6)
+
+
Here z becomes a q-dimensional vector. Therefore the SO
variables which can be estimated are given as in (5) but
i j ( .) = Q (s ) (C(d- A )- B )v (S ) + Q (~S)u (SI-A )- 1B V (
Q2 becomes a vector.
Now the problem of estimating a vector 71 = Lx is
Therefore from (9)
a problem of finding matrices Q1 and Qz (which might
be constant or transfer matrices) which, for some integer
G ( s ) = (Qi(s)C Q2(s)U)(sI - A)-lBv(s)
number q large enough, satisfy :
which becomes using (7)
S)
+
[ ]
L = [Ql(s) Q2(~)1
(7)
ij(s)
= L(sI - A)-lBv(s)
So the most general observer structure we obtain is depicted in Figure 2. (Note t,liat for a slight generalisation,
Another well known result relative t>oobservers is the
L might be considered as a transfer matrix.)
separation t<heorem:
1749
+
Lemma 3.2 If a feedback v = 1C7j is used, the closed satisfying (4) and u*E t * F = 0 that is :
loop eigenvalues are those of the observer (ie { T I , . . . , rq)
A- TI E
U {poles of Q1(s) and Qz(s))) plus th,ose of A + B K L .
*.[
t*]
C
F
O,
I
[
(12)
Proof: From Lemma 3.1, the transfer function between Similarly, an initial condition to can be decoupled by
v and 7j is equal to the one between v and 9 = L x . So the
observer eigenvalues are made non controllable, therefore
they are in the overall closed loop spectrum. Moreover
this property states that from a transfer matrix point of
It is also very easy to "decouple" the estimation error
view, it is equivalent to closing the loop through q or
from
some kinds of parameter variations. For instance
through 6, which proves the fact that the eigenvalues of
assume
that the variations of matrices A , B , C and D
A BICL also belong to the closed loop spectrum.
can be modelized as AA, A B , A C and A D . In this case
we have :
Comment 3.1 It appears that when a feedback v =
4- u*X = n(2-U* x)- u*AAx-t * ACx - u*ABv-t * A DV
Kij is to be used, the degrees of freedom must be optimized in order to minimize the condition number of the So an observer robust with respect to the above paramematrix L. This is easily understandable as a good con- ter variations corresponds to vectors u and t satisfying :
dition number of this matrix means that the entries of 11
are as independant as possible (in the limit, if two rows
of L are close one another, two measurements almost
merge into a single one). The optimization of such a
These decoupling properties are clearly very useful
condition number is very simple, especially in the Luen- for synthesizing observer based fault diagnosis devices
berger observer cases 56 , where it is performed exactly in order to prevent false alarms caused by model unceras is optimized the insensit,ivity of a control law in the
tainties or exogenous disturbances. If there are too many
state feedback case.
constraints of the forms (lo), (12), (13) or (14), equation
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+
(3) must be solved exactly, whilst other conditions must
be approximately satisfied by mean square approaches.
eeo
#
From Figure 1, decoupling of elementary observers from
Csome linear combinations of inputs or outputs is absolut,ely obvious. For insta.nce, if we do not want v1 the
first entry ov v to act on i ,from (2) it suffices to choose
the vectors u* and t* in such a way that u*B t * D = The observation of a single linear function 9 = L x ( L is
[0, +, . . . , *]. Let B1 and D1 denote the first columns of a row vector) by mea.n of a (UO- 1)-order observer, where
uo denotes the observability index, proceeds easily from
B and D, so the equa.t.ion to solve is :
the fact that
+
dini(TmCT+ImT(nl)T+. . .+ImT(n,,-l)T)
= n (15)
which means that any vector LT can be covered by a
Now let us consider decoupling from disturbances subspace spaned by uo - 1 vectors ui E ImT(ni)T plus
and initial conditions. Assume that the system (1) is a vector in ImCT. This Equation (15) is a special case
subject to a disturbance u
of the following more general result :
X=Ax+Bv+Ev
;
y=Cx+$v+Fu
q
(11)
li
The computation of the estimation error of an elementary observer as in the proof of Lemma 2.1 yields :
z - u* x. =
then
- U*EU- t * F u
So to decouple 2h.e estimation error of z = u * x from the
disturbance u it suffices to choose the vectors u and t
(16)
where 11 2 . . . 2 1, are the observability indices (see [3],
in this formula ImT(nl)T can be repla.ced by I m c ) .
-4
( -zU * Z )
> q}
IiSq
i=l
,
f - u*X = ~
+qCard{li s.t. li
0
0
1 -2
fi;
2 -1
A = - - 2 0 -4
0 -4
0
0 -:];B=[-!
- 1 -2
0 -1
0
- 2 0 0 1 2
-4 - 1
1750
i
[
C=
1
1 0 1 0 0
0 1 0 0 1
finding the zeros of the triple ( A , G , C )where G is any
maximal rank matrix the columns of which span
Its observability indices are { 3 , 2 }, so vo = 3 and
a two-dimensional observer is sufficient t o estimate any
linear function of the states. For instance to estimate
r) = La: with L = [l,1,1,1,1]and with dynamics equal
to {-1,-2}. First we must compute T(-1) and T(-2).
We obtain :
T(-1)
1 0 2.5 -7.5 -1
0
0 1 2.5 -6.5
=:
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T(-2) =
3 0 1.5 - 4.0 1
0 1 2.5 -6.5 0
then we have to solve :
1
1
Ker
1.
[
0
Often in practice, measurements are noisy and estimations must be generated using slow dynamics inducing
bias (in the limit to observe the integral of some measurement, n = 0). Complementary filters might be used
in such a case.
xample 5.4 Let r) be a linear combination of measurements noisy at high frequency, r) = Q11y and ij be
a biased estimate of r), say ij = Q21y
Q22.2. A better
estimate can be obtained by considering complementary
filters = &r)
Which means that the row
vectors Q1 and Qz in (5) are :
+
e
+ &e.
It follows
= [0.962 , 0.7851, (2 = [-0.633 0.1, (3 =
S
Qa(s) = -Qzz
[0.865 ,0.072], and with the notations of $2, ur = (2T(-l)]
l+s
u; = (2T(-2) and Q1 = (1 ; Qz = [l 11. The observer which satisfies (7) because, by hypothesis we have that
is obtained simply by replxing U , TI and II in Figure 2
L = QiiC =: QziC QzzU. 0
by their values : U* = [ul;uz] ; II = Diag{-1, -2}. The
matrix T is obtained by solving two equations similar to
(3).
+
It is often more interestsingto use an approximation
of the observation of r) = Lx.
wg call
stan,dard
observersl the minimum order
state observers ( i e of order equal to n-p) and the n-order
state observers with Kalman filter structure. These obExample 5.2 Consider the matrix L = [2 , 2 , 2 -1
servers correspond to special cases depicted in Figure 2
11. For this example, we can take for u the orthogo- with L = I and a. special choice of the matrices Q1 and
nal projection of LT onto ImT(n)T for some value of R.
Q2 in the second case (in the first case there is only one
With the above numerical example, if R = -10, we obpossibility for these matrices). In the first case we have
tain u* = [0.911 , 2.23 0.622 , - 0.017, -0.7GlI which is
clearly :
not very close to L but a simple optimization of x leads
to much better results, if n = -4.8, u* = [1.94 , 2.31 ,
Procedure 6.1 - Let { T I ] . . . , T , - ~ } be the desired set
1.70 -0.79 , 1.071 which is closer to L. This approach
of observation dynamics. Choose n - p vectors ui satisworks quite well when there are enough measurements,
fying the following conditions
ie when the sizes of the matrices ((T(R()”
are relatively
large. 0
u f ( A - n i I ) + t f C = 0 ; r a n . k [ C T I u ~ l . . . , u , ~=p ]n
]
]
In the above example, we tried to have Q1 = 0. The Then, with U , T and II standing respectively for [u1 . . . ,
consideration of the case when Q1 is non-zero gives ad- u,-~]*, [tl, . . , , t r a p F ] *and Diag(a1,. . .
find
ditional degrees of freedom for finding good a.pproximations. In that way, we have to consider orthogonal pro[Qi, Qzl=
jections of LT onto ImT(n)T + ImCT.
[ f I-’
~~~~~l~ 5.3 considering tile dimensions in ~~~~~l~
5.1, a good choice of n allows to find exactly LT in the
required subspace by considering the values of n such
that
det
[
~
)=O ]
(this choice is possible because t8heabove matrix is square).
More generally] solving this equation is equivalent to
Then the system. illustrated in Figure 2 (with L = I ) , is
Order
observer.
a
In the n-order case we have :
Procedure 6.2 - Let { T I , . . . n,} be be the desired set
of observation dynamics. Choose n vectors uj satisfying
the following conditions
1751
uf(A - niI)
+ tyC = 0 ;
rank[ul,. . . , u,-F] = n
with U , T and II standing respectively for [ U I , . . . , un]+, We have to solve :
Itl, . . . , tn]* and Diag(n1,. . . ,rn},set &1 = 0 and firtd
C
Qz = U - l . Then the system illustrated in Figure 2 (with
[(I (2 &I
T(;l)
L = I ) , is a n-order standard state observer.
[ ]
It is staighforward to manipulate this system in order
With
t o transform it into the standard Kalman filter form. For
this purpose it suffices to introduce the observer output
injection G = U - l T .
21;
= +7‘(-1)
= 0 and
[VI
vz
7731
and us = -vzT(-2)
:
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so
In this section we propose more general examples of partial state observers than in $5. The problem considered The matrices &I and Qz follows by identification,
here, in all its generality, is a “cover problem” which has
not been efficiently solved up to now. So no answer to
this general problem will be given, but some examples of
great practica.1 interest, for which the solution is simple,
are easily exhibited. For example, observers for systems The numerical values are : [I = E0.537, 0.666], (2 =
[0.041, -0.0921, (3 = [-0.409, 0.3001, 771 = [0.546, 0.59G],
with non negligible dynamics of actuators and sensors,
772 = [-0.010, -0.0531, 773 = [-0.363, 0,4581. 0
are usualy synthesised by estimating all the state variables. This is generally not very efficient.
Partial state observers can be also used to decouple the “estimation” of disturbances. Consider a system
Example 7.1 Consider a system with nonmeasured aci = A z +El v1+ E2v2 where v1 and v2 are scalar disturtuator states :
bances. To estimate v1 independently of ~2 we can proceed as follows. Firstly design an observer the estimation
error of which is decoupled from v2, secondly generate
an e&imated variable i j which is identically equal to zero
Our problem is to synthezise an observer for which L = in the absence of perturbation ( L = 0) and thirdly set
[ I ,01. So instead of equations of the form of (4),we have the static gain to one.
to solve :
Example 7.3 Consider a system with 4 states, 2 measurements subject to 2 disturbances v1 and vz. In this
case to generate the required estimated variable i j we
must find Q1 and Q2 such that &IC Q2U = 0. This
or
means that generally, at least three vectors ui satisfying (3) and such that UT E2 = 0 must be selected. The
expected decoupling property is easily cheked by considwhich has a solution if p ( B ) < p(C) or if TT is a zero ering the fact that the transfer matrix between vz and
of the triple ( A ,B , C). It is shown in [4] that, in fact 2-Ux is identically equal to zero (see $4). Therefore, the
to complete the measurements as required, if the above transfer matrix between vz and Q z ( i - U z ) = Q&+&ly
triple has some zeros, they must be considered as obser- is also equal to zero. 0
vation dynamics ; moreover the condition p(CB) = p ( B )
must be satisfied. 0
A-T~IB
+
Assuming t,hat a state feedback gain K is at our
disposal, to implement it when some states are not measured, a standard state observer is usually used. An interesting problem consists of synthesizing a partial state
observer which estimates only the linear combinations of
states required for this purpose, ie we look for L = IC.
osis.
The formulation of observers adopted here is well suited
to unify some fault diagnosis devices based on analytic
redundancy. See for instance [8], [6], [7] and [lo]. To
deal with analytical redundancy, we must be able to distinguish between the effect of external disturbances and
Example 7.2 Consider the numerical values of Exam- of faults. Equations (lo), (12), (13) and (14), tell us
what kind of decoupling of residuals might be generated
ple 5.1. The given state feedback is :
in order t o prevent false alarms. We are now going to
discuss additional advantages associat,ed with the pre- 0.7 1.7 1.3 0.7 - 1.0
sentation of observers given here. Residuals are viewed
0.7 0.4 0.4 0.0
1.0
[
1
1752
1 1 2 0
here as the components of the vector 6. This means that
the matrice L in (7) is equal to zero. The synthesis of
fault diagnosis devices is therefore a problem of solving
for each considered residual an equation similar to (7)
T~ check the Occurence of a fault, it suffices
-;-;; :]
c=[
C
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[El < 2 1 [
with judicious choice of Qli, Q2i and Vi. Let us consider
some illustrative examples. Note that all examples given
below can be considered simultaneously in a single observer. Indeed, each example gives an additional row Qli
and Q2i to matrices Q1 and Q2 (or no additional row if
the estimation of an other residual already uses the same
elementary observer) and one or more additional rows Ui
for U (similar for T ) .
to so]ve :
]
qR) = 0
which has a solution for all values of R . The corresponding observer is given by Figure 2 where Q1 = ( 1 , Qz = 1,
U = & T ( T )and T is obtained by solving one equation
of the form (3). From the numerical point of view, let
R = -1, so :
T(-1)=
[
1 0 0 0 1 0
0 0 1
;]
2
8*1 *lgebraic and kinematic
The corresponding vectors El and E2 are El = [I , 0 , -11
First we give two examples of “degenerate” observers and (1 = [-1 , 1 , 13 or (1 = [0 , 1 , -11 and E1 = [3 , 3 ,
using only measurements (it. Q2 = 0). Consider the 31 (and all their linear combinations). 0
longitudinal motion of an aircraft. We have with usual
More generally, for a solution to the above problem
notations, tr
y = 0. Assuming that these variables
are measured, the fact that this equation is sa.tisfied can with arbitrary choice of R it suffices to have 2p > n, or
be checked by considering a residual given by the row if a fault is to be checked on a specific sensor, p onevector Qli such that Qli?/ = CY y - 8 and Q2i = 0.
dimensionsal observers (as above) must be synthesised
Still considering the longitudinal aircraft motion and considering matrices Ci, i = 1,.. . , p . In this case we
the equation
= q , assuming that both variables are need to have 2 ( p - 1) > n to choose freely the observameasured, to check whether or not this equation is re- tion dynamics (see (16) for a more precise relation to be
ally satisfied, it is convenient to check it within some shisfied).
bandwith in which the derivation is meaningful. For instance we must have &q.=
&e.
Therefore Figure 8.3 Other observers
2 still allows to represent this residual by finding the dy1
namical matrix Qli(s) so that Qli(s)y = 1+75
( q - so) The problem of robustness of fault diagnosis devices is
and Q2i(s) = 0.
often due to the fact that n-dimensional observers are
used. These observers estimate in some cases, irrelevant
states. In the limit the degenerate observers presented
8.2 One dimensional observers
above (no observation at all) are the most reliable. ExAnalytical redundancy can be used in various ways. If amples of partial state observers are given $7. Example
the dynamics used for this purpose must be minimized, 7.3 can be viewed as a fault detection device, where the
it is often quite easy t o use one dimensional observers disturbances are replaced by the effects of faults.
for each actuator or sensor. For instance there is often
In some cases, contrary to what is assumed above, in
a large number of measurements, so the problem of Ex- some cases a n order observer might be necessary to proample 5.3 often has a.n exa.ct solution even considering vide analytical redundancy. For this purpose our presenp - 1 measurements instea.d of p . In this case p one- tation of observers allows t o consider orders even larger
dimensional observers can be considered for estimating than n if it is necessary to have more analytical redunresiduals of the form Lix = 0, with QliCi 92621; = 0 dancy. In fact the order n is a simple special case.
where Ci stands for the matrix C without its ith row,
and UT stands for some vector in intersection of the the
row spans of T(ni)and of Ci.
+
+
+
erY
Example 8.1 Let consider the following triple ( A ,B , C) :
0 0 0 0
A=
’’
0 1 0
O o0 ] ; B = [ ; ; ] ;
0 0 1 0
Consider a feedback u = K L x where K is the gain matrix and LE the linear combination of states used. It
is assumed that this control law presents some robustness properties at the plant input. Assuming that Lx is
1753
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not completely measurable, the Loop Transfer Recovery
Problem consists of finding a dynamic feedback (using
only measurements) which recovers the same robustness
properties at the plant input. This objective is satisfied if the return difference at the plant input is equal in
both cases. From Lemma 3.1, the required property is
satisfied if and only if the transfer matrix from v to K f j
through U and B (see Figure 2) is equal to zero. This
transfer matrix is the following :
Comment 9.1 For an n-order standard state observer
we have Q1 = 0 and L = I, therefore Equation (20)
reduces to K B = 0. If Ii' satisfies this equation, solving
the Algebraic Loop Transfer Recovery problem reduces
to finding the decoupling gain G such that disturbances
entering through B are decoupled from z = I<z. This
problem has generally no solution.
Comment 9.2 The above problem is not independent
of the gain Ii'. An interesting problem consists of conI<Qz(sI- II)-'UB = 0
(18)
sidering a "recovery" problem which works for all Ii'. In
For solving this problem, let us introduce the ob- this case the equation to solve is
server output injection G. It is a matrix satisfying
( L - Q1C)B = 0
(22)
UG=T
which has a solution in Q1 if and only if
so (6) can be written :
+
U ( A GC) = IIU
Ker CB
c Ker L B
(23)
or
(If the linear combinations of states used to synthesize
( S I - II)-'U = U ( s 1 - A - GC)-l
the gain are also linear combinationsof measurements,
From this equation toghether with (7), solving (18) re- (23) says that there is no need to "recover" !) This
duces to finding Q l such that :
equation shows the matrices L corresponding to solvable
recovery problems.
K ( L - QiC)(sI - A- GC)-'B = 0
(19)
This is a disturba.nce decoupling problem, with Q1 as an Example 9.1 The problem treated in Example 7.1 can
additional free parameter.
be viewed as a loop transfer recovery problem where a
Let 8 be a matrix the columns of which span the special matrix L is chosen in such a way that (22) holds.
least (C,A)-invariant subspaces containing B , our prob- It is straightforward to check, assuming that p(CB) =
lem has a solution if and only if there exists a matrix Ql P ( B ) .that
which satisfies
\
I
I
K ( L - Q1C)B = 0
(20)
Assuming that such a matrix Q1 is found, we have to and that the considered choice of L satisfies Equation
compute G solving the disturbance decoupling problem (23). 0
and to compute a matrix Q 2 satisfying (7) ie Q z U =
If a standard minimum order state observer is used
L - &IC. More precisely, the solution to this problem
(whith
solution still independent of I<), the additional
consists of finding vectors ui,t i satisfying
condition L = I must be considered. So the recovery
problem has a solution if and only if K e r C B = 0. In
[us" ts"]
=o
i = 0 which leads to Ker C n
other words, Ker C n Im ?
Im
B
=
0.
Conversely,
if
this
last equation is satisfied, it
in such a way that qi1
Ker ur = Im B. Note that the set
B
is
(C,
A)-invariant
so is equal to Im 8.
means
that
Im
of the zeros of the triple ( A ,B , C) must be a subset of
Therefore
the
necessary
and
sufficient
condition to be
the considered values of n-i. Let U1 denote this matrix.
satisfied
is
Then the matrix U1 is completed by a matrix U2 the rows
KerCnImB =0
of which are additional vectors ui satisfying (3) (instead
[
:]
of (21)), in order to obtain a square invertible matrix U . (this problem was first solved in Apkarian et a1 [l]). In
This means that we consider a nth-order observer, but the considered case, the matrices Q1 and QZO = ( I it will be seen that the order will reduce naturaly.
QIC)V1 are simply given by
Finally, it remains to find Q 2 . Let U-' be denoted
by [VlVz]. Q2 = ( L - Q I C ) [ V ~ V ZClearly,
].
we must
have K ( L - QlC)V2 = 0 therefore Q 2 has a special form
[ & Z O O ] because
because QlC
QzoU1 = QlC
( I - Q l C ) ( l - VzU2)
ICQz = [ K ( L- Q1C)Vl 01
= I.
This means that a part of the estimated variables is not
used. The observer order reduces to n - p(h).
+
1754
+
[7] R.J. Patton and L. Chen. Robust fault detection of
jet engine sensor systems by using eigenstructure assignment. In Proc. AIAA Guidance, Navigation and
Control Conference, New Orleans, August 1991.
Example 9.2 In this example we illustrate the use of a
minimum order observer to solve the Loop Transfer Recovery Problem, simply by decoupling the inputs. Let
consider the triple (A, B , C) defined in Example 8.1 where
the matrix C has only two rows.
c = [-3
[SI R.J. Patton and S . M . Kangethe. Robustness fault
diagnosis using eigenstructure assignment of observers. Chapter $, in Patton, Frank and Clark:
Fault Diagnosis in Dynamic Systems, Theory and
Application, Prentace Hall, pages 99-154, 1989.
-1 0 40 1
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 26, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1991-2809
This minimum phase system has two zeros which are 1 and -2. Therefore to find the second-order observer
which solves exactly the loop transfer recovery problem,
it suffices to find u i , i = 1,2, which satisfy :
(24)
which gives simultaneously the vectors ui and ti. The
following numerical values are obtained : u 1 = [00-2-41,
t i = [3 13, 212 = [0 0 - 2 - 21 t2 = [3 11. Therefore
matrices U , T , II , Q1 and Q2 of Figure 2 follow easily.
W.M. Wonham. Dynamic observers-geometric theory. IEEE Transactions on Automatic Control, AC15:258-259, 1970.
[lo]J. Wunnenberg and P.M. Frank. Model-based residual generation for dynamic systems with unknown
inputs. In Proc. of the 12th IMACS World Congress
on Scientific Computation, Paris, 1989.
0
Comment 9.3 The problem statement corresponding
to (20) or to (22) do not take into account the stability of the observer. To take this into account, it suffices
to consider B as being a ma,trix, the columns of which
span the least (C,A)-invariant subspaces with no unstable fixed poles, containing B.
eferences
P. Apkarian, C. Champetier, and J.F. Magni. Design of a helicopter output feedback control law
using modal and structured-robustness techniques.
Int. J. Control, 50(4):1195-1215, 1989.
D.G. Luenberger. Observers for multivariable systems. IEEE Transactions on Automatic Control,
AC-11(2):190-197, 1966.
131 J.F. Magni. A numerical approach to pole assignment by output feedback. American Control Coiif.,
San Diego, pages 1666-1671, May 1990.
[41 J .F. Magni. Observer synthesis by eigenstructure
assignment - an application to “Loop Transfer Recovery”. IFAC conference, Tallin, 2:155-168, August 1990.
[51 B.C. Moore. On the flexibility offered by state feedback in multivariable system beyond closed loop
eigenvalue assignment. IEEE Transactions on Automatic Control, AC-21559-692, 1976.
R.J. Patton and L. Chen. A parity space approach
to robust fault diagnosis using eigenstructure assignment. In Proc. 1st European Control Conference, Grenoble, July 1991.
1755
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