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State estimation for linear systems with observations partially corrupted by noise
Systems & Control Letters 18 (1992) 139-145 North-Holland
139
State estimation for linear systems with observations partially corrupted by noise * R. G e s s i n g Institute of Automatw Control, Sdestan Technical Unwerstty, ul.Pstrowskiego 16, 44-101 Ghwtce, Poland Received 25 June 1991 Revised 18 September 1991
Abstract This paper exploits the fact that any row vector of the observabihty matrix applied for transforming the state converts the latter to the new state component in the form of some derivative of the output component. Using the same but appropriately chosen vectors for transforming the system with the observation not fully corrupted by white noise we can accurately determine some state components. These vectors create the basis for the l-dimensional subspace of transformaUon vectors to the new accurately determinable state components. Using this basis the state transformation is constructed which in one step converts the singular linear filtering problem to a nonsingular one with state dimension decreased by l.
Keywords State estimahon; Kalman filter; observers; singular problems; continuous-time systems.
I. Introduction It is known that in the conventional filter theory of Kalman [8] the assumption appears that the covariance matrix of the errors of output measurements is nonsingular. In this case the Kalman filter equations can be easily written using known formulas. When this matrix is singular we obtain the singular filtering problem which needs a special treatment. The latter problem has been noticed in the early sixties and appears when some output components are measured accurately, or when measurements errors take the form of colored noise; see e.g. Bryson and Johansen [3], Kwakernaak and Sivan [11]. In [3] is described an intuitive method of solving the problem with nonstationary equations by means of direct, successive differentiation of the accurate output with the appropriate use of the state equation. Leondes and Yonezawa [9] describe some geometric interpretations for the Bryson-Johansen filter in the relatively simple case when the first derivatives of the noise-free outputs contain white noise with full rank of a covariance matrix. For this case they construct a transformation converting the problem to a nonsingular one. For the general case of stationary system equations, transformations of state and output are constructed in [11] which usually must be applied successively several times in order to obtain a nonsingular problem. The method is dual to that applied for solving the singular LQ control problem [4]. A similar idea of successive repeated transformations is utilized by Haas [6] where a method is described of constructing a steady-state reduced order state estimator for the singular case. Another approach for constructing a steady-state estimator by means of the transfer function method is given by Bekir [1] where the perfect measurement situation is shown to be a special case of more general noisy measurements, and the results are the same as in Shaked and Soroka [14]. A steady-state estimation problem is also considered in Schumacher [15]. The singular estimation problems are also treated as the limit of nonsingular ones when the level of measurement noises tends to zero [12,13]. The approach is dual to that applied for the so called cheap control problems. A limiting approach is also used in Halevi and Palmor [7] where a steady-state * Research supported by the departmental program No RP.I.02, coordinated by the Institute of Automatic Control of the Warsaw Technical University 0167-6911/92/$05.00 © 1992 - Elsevler Science Pubhshers B.V. All rights reserved
R Gessmg / State esttrnatum ior linear ~vstems
14(1
estimator for the so called row-reduced systems is considered and some elements of the state transformation of the present paper are appeared. Another group of papers concern the singular discrete-t~me state estimation problems which need a modified treatment and are not discussed here. The contribuhon of this paper is in finding the explicit expressions for state transformation and corresponding system representation which make it possible to obtain a transformed nonsingular filtering problem in one step after applying the transformation.
2. The singular filtering problem The considered system is described by the following equations: .f = A x + u,
(1)
y = C x +e,
(2)
where x and y are n- and m-dimensional vectors of state and output, respectively; A, C are appropriate constant matrices and C has full rank; u, e are Gaussian white noises independent of the Gaussian initial state x ( 0 ) = x0; Eu = O, Ee = O,
E[U(t)te(t) [u'(l-)
e'(r)] =Q•(t-T),
Q=
Q,e'
Qe ]
Exo =~o, E ( x o - X o ) ( X o - ~ 0 ) ' = Qx, where Q,, Q,e, Qe, Qx are given matrices of appropriate dimensions, S(t) is the Dirac function, u' denotes the transposed vector, and we have rank Qe = m - q , O
3. A linear transformation of state Let Og and O h be the subspaces of the m- and n-dimensional row vectors g' and h', respectively, such that O~-- {g': g'Qeg=O} = {g': P r o b ( g ' e = 0 ) = 1},
(3)
O h = {h': h'Q,h = 0} = {h': P r o b ( h ' u = 0) = 1},
(4)
where Prob(g'e = 0 ) = 1 denotes that the equation g'e = 0 is fulfilled with probability 1. The second equalities in (3) and (4) result directly from the definitions of the covariance matrices Qe and Qu. In the present paper, if we say that some component can be determined accurately one should understand 'accurately with probability 1". Let ~ = Ge be a linear transformation with the nonsingular m x m matrix G, such that GQeG' = Qe, where in the last q rows and last q columns of Qe, zeros appear. Let g,', i = 1, 2 , . . . , m, be the i-th row of the matrix G. The formula ~ = Gy defines the output transformation. If we denote r
t . . . . .
then the components of w can be determined accurately.
. . . . .
p+q:m,
R Gessmg / State esttmanon for hnear systems
141
One can notice that the set Og ( O h) contains all the possible transformation vectors with convert the output vector y to some new accurately, determinable output components (the right-hand side of equation (1) to some noise-free scalar expressions). Let us take into account the q following sequences of row vectors: g,'C, g / C A , g, ' C A 2 . . . . , g , , C A m,- 1,
i=p+l,p+2
..... p+q=m,
(5)
where the index m, is determined from the following two relations: g/CAJEOh
(6)
forJ =0, 1,...,m,--2
and (7)
g / C A m,-1 ¢~ O h
or m, ~
,. gmi C, . g ' .C A . . . ,. g~nCA . nm-1
(8)
and denote them appropriately by h k + l ' h,~+2 . . ,h~+~+l,. . . . . . .
(9)
h' - n , , + l , h'n - n m + 2 , . . . , h ' n
where k = n - 1. In the case of nonunique choice of the vectors (9) (when the numbers n, depend on the order of starting with the index i) we should additionally minimize the number max{n,: i = p + 1, p + 2 , . . . , m}. In the sequel, it is shown that n, determines the degree of a derivative used in the filter equations. Let h'l, h i . . . . , h~, be some row vectors which together with the vectors (9) create the nonsingular n × n matrix H = [hi, h 2. . . . . hn]'. The formula £ = H x determines the state transformation. Differentiating successively the transformed output equation ~, = g ' C x and taking into account (1) we obtain y;~J)=g'CAJx+g,'CAJ-lu
t j =g, CA x,
j<~n,-
1 i=p+
1, p + 2 , . . . , m .
(10)
The second equality in (10) results from the relation (6) and the definition (4) (since j - 1 ~
4. The converted equations Let us denote £ = [ z ' , s']', z ' = [ £ 1, X 2 , . . . , . ~ k ] , S ' = [ . ~ k + l , "~k+2 . . . . ,Xn]- The components of the vector s correspond to the transformation vectors (9). Thus, similarly as the vectors (9) also these components may be divided into q groups such that the i-th group contains the n, components of the vector s (i = p + 1, p + 2 . . . . . m).
R Gessmg / State esttmanon for hnear systems
142
Applying to (1), (2) the transformations of state and output determined by the matrices G and H, respectively, we obtain ~=A~7+~,
A=HAH -t,
~=Hu,
(11)
;=flY+Y,
C = G C H -~,
~=Ge.
(12)
Theorem 1. For the transformed equattons (11), (12) we have x
x
x
x
x
X
X
X
-x
x
x
0
X=
0
x
0
x
x
1
0
0
x
x
0 X
0 x
0
0
0
0
0 1 X
..
0
x
0
x
x
0
k x
0
0
0
0
x
x
x
-o---f
k
.....
0 F/p+ 1
/'lp+ 1
0
-o-
(13)
0
F/m
nm
0 X
..
0
0
0
0
0
0
1
X
X
X
X
x
x
x
H m
np+l
X
E=
X
X
X
X
X
X
. . . . . . . . . . . . . . . . . . . . . .
_0_. . . . . 0
0 x
X
o_
l
o
o
0
0
0
0
×
x
x
x
x
x
0___0 . . . . .
_0_
0
0
1
x
~=
x
0
P
(14)
0
k
~p+l
nm
where in the places of x some nonzero elements can appear. The rows of , 4 (components of Yt) corresponding to the transformanon vectors (9) are divided mto q groups, each of which contains appropriately n, rows (components), i = p + 1, p + 2 . . . . . m. For these groups of components of ~ for whzch n, < m,, zeros appear also in the places of ×. The components of the transformed state which create the vector s are determined accurately by the transformed output ~, and their successive n , - 1 derivatives ( i = p + 1, p + 2 . . . . . m). The proof results from an appropriate utilization of equation (10). Remark. The matrices ~zT and C in the last l and q rows, respectively, are the same as in the Luenberger-Brunovsk# observable canonical form [10,2].
5. Construction of the transformed nonsingular filtering problem From (11), (13) we have the new state equation of decreased dimension 2 =AkkZ +Akts + u=
(15)
where Akk and Akt are the k × k and k x I dimensional matrices composed of appropriate elements of the matrix A appearing in the first k rows; u z is the k-dimensional vector composed of the first k components of ~. The state equations containing g in (11), (13) are neglected, since s can be treated as a
R Gessmg / State est,matton for hnearsystems
143
given vector (following the T h e o r e m s 1 and 2, s can be determined accurately). A part of the neglected equations will be used as additional output equations since they contain some additional information concerning the unknown state z. In these equations the vector ~ has nonzero components. Let ij, j = 1, 2 . . . . , r, 0 ~< r ~
Us
=
Ar k z-b Arl
+
(16)
Us
where Cpk and Cpz are composed of the first p rows of the matrix (~; e L, is composed of the first p components of ~; A~k and Arz are composed of these rows with × which appear among the last l rows of A and which correspond to the nonzero components of ~; u s is composed of the r nonzero components appearing among the last l components of ~. Denoting
YC
Iv]'
Cl
tVsJ
[ Cpk
C2=
LArk '
(17)
e~= LArt '
Us ]
we obtain the new output equation (16) in the form
Yc = ClZ + C2s +ec
(18)
with ( p + r)-dimensional output y~. Assumption I. The covariance matrix of the noise e~ is positive definite. Assumption 1 can be checked without calculation of the converted equations. Indeed, let us denote u* = ( q f f A n ' - l ) u for i = i j . Then u s = [u~', u~' . . . . . u*]'. Similarly, if Gp = [g~, g2 . . . . . gr]', then e L,= Gpe and we obtain e c - [el, u ' ] ' without calculation of (11), (12). For the transformed filtering problem (15), (18) we have Eu z = O, Ee c = O, z(O)= z o results from 2 o = Hx o, Ez o = z0, and the covariance matrices
Eu:u':=O~6,
Eece'=ae6 ,
Euze'=Q~6,
E(zo-~o)(Zo-2o)=~)z,
(19)
can be easily determined by choosing and gathering together the appropriate elements of the matrices H Q ~ H ' , HQueG', GQeG' , H Q x H ' . Let Y~(t) = [y~(~-), s(r), 0 ~< ~- ~< t]. The following T h e o r e m 2 is an immediate consequence of Assumption 1. Theorem 2. The problem o f determining the state estimate 2 ( t ) = E { z ( t ) l Y c ( t ) } for the transformed equations (15), (18) with covanance matrices (19) is nonsmgular. We also have 2(t) = E{z(t)l Y(t)}. Let kj, j = 1, 2 . . . . , q - r, be all the indices of the components of the vector w appearing among p + 1, p + 2 . . . . , m for which nk, < mkj.
Conclusion 1. The equations resulting from successive dtfferenttation of the output Ykj, J = 1, 2 . . . . . q -- r, are not taken into account in the new output equations (18). The neglection of these equations causes no loss of information about the state z. One should stress that Conclusion 1 is valid in the case when by means of successive differentiations we obtain the resulting equation both without and with some noise. Thus, using the proposed transfor-
144
R Ges~mg / State esttmauon for hnear ~ystems
mation even the case of infinite order of singularity can be solved [6], which is not possible in other methods.
6. The case when Assumption 1 is not fulfilled
When Assumption 1 is not fulfilled a part (or all) the vectors h'l, h i . . . . . h~ should be chosen not freely, but similarly as the vectors (9). We continue the choice of the vectors h~ starting with the chosen vectors (9). Next, we repeat the derivations (3)-(9) for the system (1) with the new output equation (18) replacing e and Qe in (3) by e, and Qe, respectively. In this way, we choose, say i new vectors which have a form similar to (9) and are linearly independent of l previously chosen vectors (9). The new vectors are adjoined to the previously chosen ones, creating together 1 + i vectors of the form (9). If need be, we repeat the derivations (3)-(9) with appropriately modified output equation (18), completing the chosen vectors by new ones. To sum up, the considered problem can be solved in accordance with the following scheme: 1. Determine the set O~ (3) and related transformation matrix G. 2. Determine the set Oh (4), create the sequences (5), choose the vectors (9) and adjoin them to the previously chosen ones. 3. Check, if the number of chosen vectors is equal to n. If yes, the problem is solved. All the state components can be determined accurately by appropriate output derivatives. 4. Determine the indices ,j, j = 1, 2 . . . . . r, for the lastly chosen vectors (9). 5. Check, if Assumption 1 is fulfilled. If yes, construct the state transformation ~ = Hx. If not, repeat the points 1-4 with Qe replaced by ~)e 6. Derive the transformed equations (11), (12), construct the nonsingular filtering problem (15), (18), (19) using the indices ij, ) = 1, 2 . . . . . r, and apply the usual Kalman filter equations. It ~s seen that the proposed state transformation is created, recurslvely, but is applied only one time. The transformed equation (11) has the matrix ,zT in which some of k first rows take a form similar to those appearing in the last l rows of (13). One can notice that our transformation is a generalization of that appearing in [7], and based on Lemma 3 of [7] (whose unsatisfactory proof can be completed). But in the mentioned paper only a steady-state observer is derived (by using the limiting approach), and in reality, only row-reduced systems are considered.
7. Final conclusions
It is known that the transformation created from the appropriately chosen, hnearly independent row vectors of the observability matrix converts the state equations to the so called Luenberger-Brunovsk~ observable canomcal form. Additionally, in the present paper the fact is exploited that this transformation converts the state to some kind of phase-space form in which the transformed state components are equal to the output components and their appropriate, successive derivatives. The consequence of this is the possibility of instantaneous determination of the observable states on the basis of the instantaneous values of the outputs and their appropriate derivatives. In the proposed state transformation the appropriately modified row vectors of the observability matrix are also used, but only those linearly independent row vectors are taken into account which convert the state to some accurately determinable state components. It is shown that a part of the transformed state equations has the Luenberger-Brunovsk~ observable canonical form. It is also shown that the filtering problem constructed on the basis of the canonical form is nonsingular. It is important that the nonsingular problem is obtained in one step after applying the transformation.
R Gessmg / State esttmatton for hnear systems
145
The proposed transformation replaced in one step several successive transformations of [11]. The transformation can also be applied in the case of infinite order of singularity. In [5] it has been shown that a similar transformation can be applied for discrete-time systems, though in this case a significantly different approach is needed.
Acknowledgment I would like to express my thanks to anonymous reviewers for valuable remarks.
References [1] E. Beklr, A umfled solution to the singular and nonslngular linear mlmmum-varmance estimation problem, IEEE Trans Automat Control 33 (6) (1988) 590-591 [2] F. Brunovsk~, A classification of linear controllable systems, Kybernetlka 6 (3) (1970) 173-187 [3] A.E Bryson and D E Johansen, Linear filtering for time-varying systems using measurements containing colored noise, IEEE Trans. Automat Control 10 (1) (1965) 4-10 [4] D.J Clements and B D O. Anderson, S,ngular Opttmal Control The Lmear-Quadratw Problem (Sprmger-Verlag, Berhn-New York, 1978). [5] R. Gesslng, A transformation for solving a discrete-time singular state estimation problem, in. G Conte, A M Perdon, B. Wyman, Eds, New Trends m Systems Theory (Blrkhauser, Boston, MA, 1990) [6] V B. Haas, Reduced order state estimation for linear systems with exact measurements, Automattca 20 (2) (1984) 225-229. [7] Y. Halevl and Z.J Palmor, Extended limiting forms of optimum observes and LQG regulators, Internat Z Control 43 (1) (1986) 193-212. [8] R E. Kalman, A new approach to hnear filtering and prediction problems, J Bastc Eng 82 (1960) 34-45 [9] C T Leondes and K. Yonezawa, On the geometric structure of Bryson-Johansen filter and stochastic observer, IEEE Trans Automat Control 25(5) (1980) 992-994 [10] D.G. Luenberger, Canonical forms for linear multivanable systems, IEEE Trans Automat Control 12 (1967) 290-293 [11] K. Kwakernaak and R Swan, Linear Control Systems (Wlley-lnterscience, New York, 1972). [12] U Shaked, Optimal nearly singular estimation of continuous linear stationary uniform rank systems, Internat J Control 38 (2) (1983) 275-315 [13] U Shaked, Nearly singular filtering for uniform and non-umform rank linear continuous system, Internat J Control 38 (4) (1984) 811-830. [14] U Shaked and E. Soroka, A simple solution to the singular hnear minimum variance estimation problem, IEEE Trans Automat Control 32 (1) (1987) 81-83 [15] J M. Schumacher, A geometric approach to the singular faltering problem, IEEE Trans Automat. Control 30 (11) (1985) 1075-1082
139
State estimation for linear systems with observations partially corrupted by noise * R. G e s s i n g Institute of Automatw Control, Sdestan Technical Unwerstty, ul.Pstrowskiego 16, 44-101 Ghwtce, Poland Received 25 June 1991 Revised 18 September 1991
Abstract This paper exploits the fact that any row vector of the observabihty matrix applied for transforming the state converts the latter to the new state component in the form of some derivative of the output component. Using the same but appropriately chosen vectors for transforming the system with the observation not fully corrupted by white noise we can accurately determine some state components. These vectors create the basis for the l-dimensional subspace of transformaUon vectors to the new accurately determinable state components. Using this basis the state transformation is constructed which in one step converts the singular linear filtering problem to a nonsingular one with state dimension decreased by l.
Keywords State estimahon; Kalman filter; observers; singular problems; continuous-time systems.
I. Introduction It is known that in the conventional filter theory of Kalman [8] the assumption appears that the covariance matrix of the errors of output measurements is nonsingular. In this case the Kalman filter equations can be easily written using known formulas. When this matrix is singular we obtain the singular filtering problem which needs a special treatment. The latter problem has been noticed in the early sixties and appears when some output components are measured accurately, or when measurements errors take the form of colored noise; see e.g. Bryson and Johansen [3], Kwakernaak and Sivan [11]. In [3] is described an intuitive method of solving the problem with nonstationary equations by means of direct, successive differentiation of the accurate output with the appropriate use of the state equation. Leondes and Yonezawa [9] describe some geometric interpretations for the Bryson-Johansen filter in the relatively simple case when the first derivatives of the noise-free outputs contain white noise with full rank of a covariance matrix. For this case they construct a transformation converting the problem to a nonsingular one. For the general case of stationary system equations, transformations of state and output are constructed in [11] which usually must be applied successively several times in order to obtain a nonsingular problem. The method is dual to that applied for solving the singular LQ control problem [4]. A similar idea of successive repeated transformations is utilized by Haas [6] where a method is described of constructing a steady-state reduced order state estimator for the singular case. Another approach for constructing a steady-state estimator by means of the transfer function method is given by Bekir [1] where the perfect measurement situation is shown to be a special case of more general noisy measurements, and the results are the same as in Shaked and Soroka [14]. A steady-state estimation problem is also considered in Schumacher [15]. The singular estimation problems are also treated as the limit of nonsingular ones when the level of measurement noises tends to zero [12,13]. The approach is dual to that applied for the so called cheap control problems. A limiting approach is also used in Halevi and Palmor [7] where a steady-state * Research supported by the departmental program No RP.I.02, coordinated by the Institute of Automatic Control of the Warsaw Technical University 0167-6911/92/$05.00 © 1992 - Elsevler Science Pubhshers B.V. All rights reserved
R Gessmg / State esttrnatum ior linear ~vstems
14(1
estimator for the so called row-reduced systems is considered and some elements of the state transformation of the present paper are appeared. Another group of papers concern the singular discrete-t~me state estimation problems which need a modified treatment and are not discussed here. The contribuhon of this paper is in finding the explicit expressions for state transformation and corresponding system representation which make it possible to obtain a transformed nonsingular filtering problem in one step after applying the transformation.
2. The singular filtering problem The considered system is described by the following equations: .f = A x + u,
(1)
y = C x +e,
(2)
where x and y are n- and m-dimensional vectors of state and output, respectively; A, C are appropriate constant matrices and C has full rank; u, e are Gaussian white noises independent of the Gaussian initial state x ( 0 ) = x0; Eu = O, Ee = O,
E[U(t)te(t) [u'(l-)
e'(r)] =Q•(t-T),
Q=
Q,e'
Qe ]
Exo =~o, E ( x o - X o ) ( X o - ~ 0 ) ' = Qx, where Q,, Q,e, Qe, Qx are given matrices of appropriate dimensions, S(t) is the Dirac function, u' denotes the transposed vector, and we have rank Qe = m - q , O
3. A linear transformation of state Let Og and O h be the subspaces of the m- and n-dimensional row vectors g' and h', respectively, such that O~-- {g': g'Qeg=O} = {g': P r o b ( g ' e = 0 ) = 1},
(3)
O h = {h': h'Q,h = 0} = {h': P r o b ( h ' u = 0) = 1},
(4)
where Prob(g'e = 0 ) = 1 denotes that the equation g'e = 0 is fulfilled with probability 1. The second equalities in (3) and (4) result directly from the definitions of the covariance matrices Qe and Qu. In the present paper, if we say that some component can be determined accurately one should understand 'accurately with probability 1". Let ~ = Ge be a linear transformation with the nonsingular m x m matrix G, such that GQeG' = Qe, where in the last q rows and last q columns of Qe, zeros appear. Let g,', i = 1, 2 , . . . , m, be the i-th row of the matrix G. The formula ~ = Gy defines the output transformation. If we denote r
t . . . . .
then the components of w can be determined accurately.
. . . . .
p+q:m,
R Gessmg / State esttmanon for hnear systems
141
One can notice that the set Og ( O h) contains all the possible transformation vectors with convert the output vector y to some new accurately, determinable output components (the right-hand side of equation (1) to some noise-free scalar expressions). Let us take into account the q following sequences of row vectors: g,'C, g / C A , g, ' C A 2 . . . . , g , , C A m,- 1,
i=p+l,p+2
..... p+q=m,
(5)
where the index m, is determined from the following two relations: g/CAJEOh
(6)
forJ =0, 1,...,m,--2
and (7)
g / C A m,-1 ¢~ O h
or m, ~
,. gmi C, . g ' .C A . . . ,. g~nCA . nm-1
(8)
and denote them appropriately by h k + l ' h,~+2 . . ,h~+~+l,. . . . . . .
(9)
h' - n , , + l , h'n - n m + 2 , . . . , h ' n
where k = n - 1. In the case of nonunique choice of the vectors (9) (when the numbers n, depend on the order of starting with the index i) we should additionally minimize the number max{n,: i = p + 1, p + 2 , . . . , m}. In the sequel, it is shown that n, determines the degree of a derivative used in the filter equations. Let h'l, h i . . . . , h~, be some row vectors which together with the vectors (9) create the nonsingular n × n matrix H = [hi, h 2. . . . . hn]'. The formula £ = H x determines the state transformation. Differentiating successively the transformed output equation ~, = g ' C x and taking into account (1) we obtain y;~J)=g'CAJx+g,'CAJ-lu
t j =g, CA x,
j<~n,-
1 i=p+
1, p + 2 , . . . , m .
(10)
The second equality in (10) results from the relation (6) and the definition (4) (since j - 1 ~
4. The converted equations Let us denote £ = [ z ' , s']', z ' = [ £ 1, X 2 , . . . , . ~ k ] , S ' = [ . ~ k + l , "~k+2 . . . . ,Xn]- The components of the vector s correspond to the transformation vectors (9). Thus, similarly as the vectors (9) also these components may be divided into q groups such that the i-th group contains the n, components of the vector s (i = p + 1, p + 2 . . . . . m).
R Gessmg / State esttmanon for hnear systems
142
Applying to (1), (2) the transformations of state and output determined by the matrices G and H, respectively, we obtain ~=A~7+~,
A=HAH -t,
~=Hu,
(11)
;=flY+Y,
C = G C H -~,
~=Ge.
(12)
Theorem 1. For the transformed equattons (11), (12) we have x
x
x
x
x
X
X
X
-x
x
x
0
X=
0
x
0
x
x
1
0
0
x
x
0 X
0 x
0
0
0
0
0 1 X
..
0
x
0
x
x
0
k x
0
0
0
0
x
x
x
-o---f
k
.....
0 F/p+ 1
/'lp+ 1
0
-o-
(13)
0
F/m
nm
0 X
..
0
0
0
0
0
0
1
X
X
X
X
x
x
x
H m
np+l
X
E=
X
X
X
X
X
X
. . . . . . . . . . . . . . . . . . . . . .
_0_. . . . . 0
0 x
X
o_
l
o
o
0
0
0
0
×
x
x
x
x
x
0___0 . . . . .
_0_
0
0
1
x
~=
x
0
P
(14)
0
k
~p+l
nm
where in the places of x some nonzero elements can appear. The rows of , 4 (components of Yt) corresponding to the transformanon vectors (9) are divided mto q groups, each of which contains appropriately n, rows (components), i = p + 1, p + 2 . . . . . m. For these groups of components of ~ for whzch n, < m,, zeros appear also in the places of ×. The components of the transformed state which create the vector s are determined accurately by the transformed output ~, and their successive n , - 1 derivatives ( i = p + 1, p + 2 . . . . . m). The proof results from an appropriate utilization of equation (10). Remark. The matrices ~zT and C in the last l and q rows, respectively, are the same as in the Luenberger-Brunovsk# observable canonical form [10,2].
5. Construction of the transformed nonsingular filtering problem From (11), (13) we have the new state equation of decreased dimension 2 =AkkZ +Akts + u=
(15)
where Akk and Akt are the k × k and k x I dimensional matrices composed of appropriate elements of the matrix A appearing in the first k rows; u z is the k-dimensional vector composed of the first k components of ~. The state equations containing g in (11), (13) are neglected, since s can be treated as a
R Gessmg / State est,matton for hnearsystems
143
given vector (following the T h e o r e m s 1 and 2, s can be determined accurately). A part of the neglected equations will be used as additional output equations since they contain some additional information concerning the unknown state z. In these equations the vector ~ has nonzero components. Let ij, j = 1, 2 . . . . , r, 0 ~< r ~
Us
=
Ar k z-b Arl
+
(16)
Us
where Cpk and Cpz are composed of the first p rows of the matrix (~; e L, is composed of the first p components of ~; A~k and Arz are composed of these rows with × which appear among the last l rows of A and which correspond to the nonzero components of ~; u s is composed of the r nonzero components appearing among the last l components of ~. Denoting
YC
Iv]'
Cl
tVsJ
[ Cpk
C2=
LArk '
(17)
e~= LArt '
Us ]
we obtain the new output equation (16) in the form
Yc = ClZ + C2s +ec
(18)
with ( p + r)-dimensional output y~. Assumption I. The covariance matrix of the noise e~ is positive definite. Assumption 1 can be checked without calculation of the converted equations. Indeed, let us denote u* = ( q f f A n ' - l ) u for i = i j . Then u s = [u~', u~' . . . . . u*]'. Similarly, if Gp = [g~, g2 . . . . . gr]', then e L,= Gpe and we obtain e c - [el, u ' ] ' without calculation of (11), (12). For the transformed filtering problem (15), (18) we have Eu z = O, Ee c = O, z(O)= z o results from 2 o = Hx o, Ez o = z0, and the covariance matrices
Eu:u':=O~6,
Eece'=ae6 ,
Euze'=Q~6,
E(zo-~o)(Zo-2o)=~)z,
(19)
can be easily determined by choosing and gathering together the appropriate elements of the matrices H Q ~ H ' , HQueG', GQeG' , H Q x H ' . Let Y~(t) = [y~(~-), s(r), 0 ~< ~- ~< t]. The following T h e o r e m 2 is an immediate consequence of Assumption 1. Theorem 2. The problem o f determining the state estimate 2 ( t ) = E { z ( t ) l Y c ( t ) } for the transformed equations (15), (18) with covanance matrices (19) is nonsmgular. We also have 2(t) = E{z(t)l Y(t)}. Let kj, j = 1, 2 . . . . , q - r, be all the indices of the components of the vector w appearing among p + 1, p + 2 . . . . , m for which nk, < mkj.
Conclusion 1. The equations resulting from successive dtfferenttation of the output Ykj, J = 1, 2 . . . . . q -- r, are not taken into account in the new output equations (18). The neglection of these equations causes no loss of information about the state z. One should stress that Conclusion 1 is valid in the case when by means of successive differentiations we obtain the resulting equation both without and with some noise. Thus, using the proposed transfor-
144
R Ges~mg / State esttmauon for hnear ~ystems
mation even the case of infinite order of singularity can be solved [6], which is not possible in other methods.
6. The case when Assumption 1 is not fulfilled
When Assumption 1 is not fulfilled a part (or all) the vectors h'l, h i . . . . . h~ should be chosen not freely, but similarly as the vectors (9). We continue the choice of the vectors h~ starting with the chosen vectors (9). Next, we repeat the derivations (3)-(9) for the system (1) with the new output equation (18) replacing e and Qe in (3) by e, and Qe, respectively. In this way, we choose, say i new vectors which have a form similar to (9) and are linearly independent of l previously chosen vectors (9). The new vectors are adjoined to the previously chosen ones, creating together 1 + i vectors of the form (9). If need be, we repeat the derivations (3)-(9) with appropriately modified output equation (18), completing the chosen vectors by new ones. To sum up, the considered problem can be solved in accordance with the following scheme: 1. Determine the set O~ (3) and related transformation matrix G. 2. Determine the set Oh (4), create the sequences (5), choose the vectors (9) and adjoin them to the previously chosen ones. 3. Check, if the number of chosen vectors is equal to n. If yes, the problem is solved. All the state components can be determined accurately by appropriate output derivatives. 4. Determine the indices ,j, j = 1, 2 . . . . . r, for the lastly chosen vectors (9). 5. Check, if Assumption 1 is fulfilled. If yes, construct the state transformation ~ = Hx. If not, repeat the points 1-4 with Qe replaced by ~)e 6. Derive the transformed equations (11), (12), construct the nonsingular filtering problem (15), (18), (19) using the indices ij, ) = 1, 2 . . . . . r, and apply the usual Kalman filter equations. It ~s seen that the proposed state transformation is created, recurslvely, but is applied only one time. The transformed equation (11) has the matrix ,zT in which some of k first rows take a form similar to those appearing in the last l rows of (13). One can notice that our transformation is a generalization of that appearing in [7], and based on Lemma 3 of [7] (whose unsatisfactory proof can be completed). But in the mentioned paper only a steady-state observer is derived (by using the limiting approach), and in reality, only row-reduced systems are considered.
7. Final conclusions
It is known that the transformation created from the appropriately chosen, hnearly independent row vectors of the observability matrix converts the state equations to the so called Luenberger-Brunovsk~ observable canomcal form. Additionally, in the present paper the fact is exploited that this transformation converts the state to some kind of phase-space form in which the transformed state components are equal to the output components and their appropriate, successive derivatives. The consequence of this is the possibility of instantaneous determination of the observable states on the basis of the instantaneous values of the outputs and their appropriate derivatives. In the proposed state transformation the appropriately modified row vectors of the observability matrix are also used, but only those linearly independent row vectors are taken into account which convert the state to some accurately determinable state components. It is shown that a part of the transformed state equations has the Luenberger-Brunovsk~ observable canonical form. It is also shown that the filtering problem constructed on the basis of the canonical form is nonsingular. It is important that the nonsingular problem is obtained in one step after applying the transformation.
R Gessmg / State esttmatton for hnear systems
145
The proposed transformation replaced in one step several successive transformations of [11]. The transformation can also be applied in the case of infinite order of singularity. In [5] it has been shown that a similar transformation can be applied for discrete-time systems, though in this case a significantly different approach is needed.
Acknowledgment I would like to express my thanks to anonymous reviewers for valuable remarks.
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