Observers design for linear time-varying systems M. Boutayeb, M. Darouach CRAN CNRS UPRES-A 7039, IUT de Longwy 186,, rue de Lorraine, 54400 Cosnes et Romain, FRANCE Tel : (0033) 3 82 25 91 10 - Fax : (0033) 3 03 82 25 91 17 e-mail:
[email protected]
authors use the reconstructibility condition, which also Abstract : In this note we give some results on the convergence of the Kalman Filter (K.F.) when used as an observer for linear time-varying systems. Based on the block-input / block-output state model, we prove that the state observer given in [7] is equivalent to the K.F. algorithm. One of the main features, however, is that no assumption on the invertibility of the state matrix, namely A in the paper, is needed and the computational k
needs the invertibility of the state matrix, to guarantee global convergence of the Kalman filter. A second approach to build observers for linear timevarying systems consists in using a block-input / blockoutput model from the state space represenation [7] (see also [1] and [6] for control design). The main result of this technique is that only the observability condition is needed. The strong invertibility of the state matrix
requirements are reduced. Furthermore, the obtained result can be extended, by duality, to resolve the state feedback control problem.
assumption is not required. However to prove global convergence of the proposed observer, we have to compute inversion of an pq.pq matrix build upon square of the observability matrix.
1. Introduction In the last decade, many research activities were focused on the theory of linear time-varying systems. One of the main reasons is that most of the works on non-linear systems studied first linearized models. This may be important to establish local properties and to set up analysis and synthesis [2]-[3]-[5]-[8]-[9]-[10]-[11]. In
To avoid such computational requirements, in particular, for large scale-systems or when q is large, we prove by induction that the state observer given in [7] may be written as the K.F. algorithm when used as an observer for linear time-varying systems. On the other hand, no assumption on the invertibility of the state matrix is needed.
this note we deal with state observers design for linear discrete-time time-varying systems. In 1988, Baras et al. [2] have shown that under some strong observability and
2. Problem formulation Consider the linear time-varying system :
controlability assumptions the Kalman filter ensures
x
global convergence when used as an observer for linear
Song et al. 11] have shown that, under invertibility of the
= A x +B u k
k
(1)
k k
y k = C kx k
time varying systems. Similar conditions were already established in the work of Deyst et al. [5]. More recently,
k +1
(2)
where u ∈ Rm , x ∈ Rn and y ∈ Rp represent the input, k
k
k
state matrix and the observability conditions, the Kalman
the state and output vectors at time instant k respectively. A , B and C are known matrices of appropriate
filter is a global and uniform asymptotic observer. In [8],
dimensions.
k
k
k
Following the notations in [7], (1)-(2) can be written in terms of the q-block input / block output state model :
T
Ψq+ k = Ω q+ k Ω q+ k + σI
with
(11)
converges to zero.
x q+ k = Φ q +k x k + Γq + k U k
(3)
I is the pq.pq identity matrix and σ is a small positive real
Yq+ k = Ω q+ k x k + E q +k U k
(4)
number.
where Φ q+ k ∈ R , n.n
Γ
∈R
n.qm
q +k
and Ω q +k ∈ R
qp.n
We notice that the general form of (10)-(11) may be
denote the q-step transition matrix, the q-step reachability
obtained by duality to the optimal state feedback control
matrix and q-step observability matrix respectively,
solution given in [6], the gain matrix for the state
defined by :
estimation is therefore:
Φ q + k = A q+ k-1A q+ k-2 ...Ak
(5)
[
Γq + k = B q+ k-1 A q+ k-1 B q+ k-2 ... A q+ k-1 A q+ k-2 ...A k+1B k
Ω q +k
xˆ q+ k = Φ q+ k xˆ k + Γq+ k U q +k
]
+ L q+ k (Yq +k − Ω q+ k xˆ k - E q+ k U q+ k ) (12)
(6)
Ck C A k +1 k and = . C A ...A q+ k −1 q+ k −2 k
T
-1
L q +k = Φ q+ k Pk Ωq+ k Ψq +k
(13)
T
Ψq+ k = Ω q+ k Pk Ω q+ k + R k
with
(14)
P is a n.n positive semidefinite matrix and R is a pq.pq k
E q +k =
k
positive definite matrix. (10)-(11) is obtained for P = I
0 0 0 0 0 0 : : 0 C q +k −1B q+ k −2
...
0
...
0
... C k+ 2 B k+1 :
:
...
x
C k+1 Bk C k+ 2 A k +1B k : x 0
(7)
k
n
and R = σI . k
pq
In this paper we will show that the Kalman filter (the correct term is in fact the Kalman predictor) is equivalent to the state observer (12)-(14). Therefore we obtain a one-step recursive observer without inversion of the
U
q+k
and Y
q+k
are the q - block input / block output
pq.pq matrix Ψq+ k . On the other hand the strong assumption on invertibility of the state matrix A ,
vectors : U Y
q+k q+k
[ = [y =
u Tq+ k-1 T k
u Tq +k -2
...
uTk
y Tk +1 ... y Tq +k -1
]
k
]
T
frequently used in the literature, is not required.
T
(8)
The following theorem summarizes then the main result of this paper :
Now, let us summarize the main result developed in [7] through the following theorem :
k
Theorem 1 If we assume that A and C are bounded and system (1)k
Theorem 2 If we assume that A and C are bounded, and system (1)k
(2) is q-step observable then the Kalman filter algorithm : xˆ q+ k = A q +k -1xˆ q+ k-1 + Bq+ k-1u q+ k-1
k
(2) is q-step observable, that is Ω q +k has a left inverse,
+ Kq+ k-1 (y q +k -1 − C q+ k-1 xˆ q +k -1 )
then the error dynamics when the state observer is :
(15)
xˆ q+ k = Φ q+ k xˆ k + Γq+ k U q +k
K q +k −1 =
+ L q+ k (Yq +k − Ω q+ k xˆ k - E q+ k U q+ k )
L q +k =
(16)
and (9)
T -1 Φ q+ k Ωq+ k Ψq+ k
−1 Aq +k -1Pq +k -1C Tq+ k-1 Σq +k -1
(10)
Pq +k = A q+ k-1 Pq+ k-1 (A Tq+ k-1 - C Tq+ k-1 K Tq+ k-1 ) (17)
with R
q+k-1
Σ q+ k −1 = C q +k-1Pq +k -1CqT+k -1 + Rq+ k-1
(18)
On the other hand, we have : A k +1K k − K k+1 C k+1 K k = A k+1A k Pk × T −1 −K Tk C kT+1 Σ −k 1+1 C k+1 K k T T T C k Σ k − C k A k C k +1 −1 Σ k +1 C k+1 K k
is a positive definite matrix.
[
is equivalent to the optimal state observer (12)-(14).
]
(26) Proof
or
The proof that we detail here, is by induction. For simplicity and without loss of generality, we set u = 0. k
First, suppose that q = 1, we have :
and
A k +1K k − K k+1 C k+1 K k = A k+1A k Pk × Σ −1 + K Tk C Tk +1Σ −1 k +1C k +1K k C Tk A Tk C Tk +1 k −1 −Σ k+1 C k+1 K k
[
]
Ω q +k = Ω1+ k = C k
the gain matrix in (23) is then written as :
Ψq+ k = Ψ1+k = Σ k
[A k +1K k − K k+1 Ck+1 K k
Φ q+ k = Φ 1+ k = A k
=
Σ−1 k
the 1-step observability condition means that :
rank ( C k ) = n
[
T A k+1 A k Pk C k
+
K k +1 ] T
T
]
(27)
A k C k +1 ×
−1 K Tk C Tk +1Σ k +1C k+1K k −1 −Σ k+1 C k+1 K k
T −K Tk C k+1 Σ −1 k+1 −1 Σ k+1
(28)
it is clear that (12)-(14) is equivalent to (15)-(18). with
For q = 2 we have :
T + R k+1 − Σ k +1 = C k+1 A k Pk A Tk C k+1
xˆ k +2 = A k+1xˆ k+1 + K k+1 (y k +1 − C k+1xˆ k+1 )
−1
C k+1 A k Pk C Tk Σ k C k Pk A Tk C Tk +1
(19) −1
K k +1 = A k+1 Pk+1C Tk+1 Σ k+1
(20)
Now if we set : a = Σ k ( = Ψ1+ k ) 11
and P k+ 2 = (A k +1 - K k+1C k+1 )Pk+1 A Tk+1
T + R k+1 a = C k+1 A k Pk ATk C k+1
(21)
22
T a = C k Pk A Tk C Tk+1 =Ω 1+ k Pk Φ T1+ k C k+1 12
(19) may be written in the extended form as :
and by the use of the inversion lemma of a partitioned
xˆ k +2 = A k+1 (A k ˆxk + K k (y k − C k xˆ k )) + Kk+1 (y k+1 − C k+1 A k xˆ k − C k+1 K k (y k − C k ˆxk )) (22) or xˆ k +2 = A k+1A k xˆ k + [A k +1K k − K k+1 C k+1 K k
We notice that K
k+1
y k − C k xˆ k K k +1 ] ˆ y k+1 − C k+1 A k x k
−1
or
[
T
K k +1 = A k+1 A k Pk C k
as : −1 T T Σ−1 k + K k C k +1Σ k +1C k+1K k −1 −Σ k+1 C k+1 K k −1 T −1 (a 11 − a12 a 22 a12 ) −1 T −1 T −1 −(a 22 − a12 a11 a 12 ) a 12a 11
(23)
or equivalently :
(24)
−1 T T Σ−1 k + K k C k +1Σ k +1C k+1K k −1 −Σ k+1 C k+1 K k
is in the form :
K k +1 = A k+1 A k Pk (A Tk − C Tk K Tk )C Tk+1 Σ k +1
matrix, the right hand side matrix of (28) may be written
−K Tk C Tk +1 Σ−k 1+1 T T A k C k +1 1 Σ −k +1
]
(25)
a11 = T a 12
[
a12 a 22
T −K Tk C k+1 Σ −1 k+1 = −1 Σ k+1 −1 −1 T −1 −a11 a 12 (a 22 − a12 a11 a 12 ) −1 T −1 (a 22 − a12 a 11a 12 )
T −K Tk C k+1 Σ −1 k+1 −1 Σ k+1
−1
]
= Ω 2 +k Pk ΩT2+ k + diag(R k , R k +1 ) −1 = Ψ2+ k
−1
The final form of (23) is then : xˆ k +2 = Φ 2+ k xˆ k + L 2+ k (Y2+k − Ω2+ k xˆ k )
(29)
The extended form of (37), and by the use of (31)-(32), leads to :
with
[
L 2+ k = Φ 2+ k Pk Ω T2+ k Ω2+ k Pk Ω T2+ k + diag(R k , R k+1 )
]
−1
K q +k =
−LTq +k C Tq+ k Σ −1 q+ k Σ−1 q +k
Φ q+ k+1 Pk Ω Tq +k +1
(30) on the other hand we have :
Now assume that :
(
xˆ q+ k = Φ q+ k xˆ k + Lq + k Yq+ k − Ωq +k xˆ k
)
(31)
with L q +k = Φ q+ k Pk ΩTq+ k ×
[Ω
(38)
T q +k Pk Ω q +k
]
+ diag(R k ,..., R q+ k −1 )
−1
−1 A q +k Lq + k − K q +k C q+ k L q+ k = A q+ k Φ q +k Pk Ω Tq + k Ψq +k −LTq + k C Tq+ k Σ−1 q +k T C q+ k L q +k Φ q+ k+1 Pk Ω q +k +1 Σ −1 q+ k
(39)
(32) or
From (15) and (31) at time instant q+k+1, we have :
(
(
xˆ q+ k +1 = Aq +k Φ q+ k xˆ k + Lq +k Yq + k − Ωq +k xˆ k
(
(
(
))+
K q +k yq + k − C q +k Φ q + k xˆ k + Lq + k Yq + k − Ω q +k xˆ k
)))
(33) or xˆ q+ k +1 = Φ q+ k+1 xˆ k +
[A
q +k L q+ k
− K q +k C q+ k L q +k
The gain matrix in (34) may be then written as :
[A
q+ k Lq + k
− K q+ k C q+ k L q+ k
K q +k
]
= Φ q+ k+1 Pk Ω Tq+ k+1 ×
]
K q+ k ×
Yq+ k − Ω q+ k xˆ k y q +k − C q+ k Φ q+ k xˆ k
A q +k Lq + k − K q +k C q + k L q + k = Φ q + k +1Pk Ω qT+ k +1 × Ψq−+1 k + LTq + k C Tq +k Σq−1+ k C q + k Lq + k (40) −Σq−1+ k C q + k Lq + k
(34)
Ψ−1 + LT C T Σ −1 C L q+ k q +k q + k q+ k q + k q+ k −1 −Σ q + k C q+ k Lq + k
−LTq+ k C Tq +k Σ −1 q+ k Σ−1 q +k (41)
where Yq+ k − Ω q+ k xˆ k y = Yq + k +1 − Ωq + k +1 ˆxk q +k − C q+ k Φ q+ k xˆ k
Now if we set : a = Ψq+ k 11
(35)
a = C q + k Φ q+ k Pk ΦTq +k CTk+1 + R q+ k 22 a = Ωq + k Pk Φ Tq+ k C Tk +1 12
Hereafter we will show that the gain matrix in (34) is in
and by the use of the inversion lemma of a partitioned
the form :
matrix as above, we obtain :
[A
q+ k Lq + k
]
− K q+ k C q+ k L q+ k
K q +k =
[
Φ q+ k+1P k Ω Tq+ k+1 Ω q+ k+1 Pk ΩTq+ k+1 + diag(R k , ...,R q +k )
]
−1
= Φ q+ k+1P k Ω q+ k+1 Ψq−+k +1 T
Ψ−1 + LT C T Σ −1 C L q+ k q +k q + k q+ k q + k q+ k −Σq−1+ k C q+ k Lq + k
[Ω
1
First of all, by the use of the recursive form (17), K
q+k
is
written as : −1
K q +k = A q+ k Pq+ k C Tq +k Σq + k
(36)
T q +k +1Pk Ω q+ k+1
−LTq+ k C Tq +k Σ −1 q+ k = Σ−1 q +k
]
+ diag(R k ,..., R q+ k )
(42)
and finally the gain matrix in (34) is equivalent to (13) :
[A
q+ k Lq + k
− K q+ k C q+ k L q+ k
]
K q +k =
Φ q+ k+1P k Ω Tq+ k+1 Ψq−1+k +1
(43)
or
(
)
K q +k = A q+ k A q+ k-1 ...A k Pk A Tk - C Tk K Tk ... ×
(A Tq+ k-1 - CTq+ k-1K Tq+ k-1)CTq+ k Σ−1q + k
(37)
3. Conclusion
In this paper we have shown that the state observer given in [7] is equivalent to the Kalman filter when used as an
[8]
P. E. Moraal and J. W. Grizzle, "Observer design
observer for linear time-varying systems. One of the
for
main features of this technique is that the strong
measurements," IEEE Transactions on Automatic
invertibility condition on the state matrix, as usually used
Control, Vol. 40, No. 3, pp. 395-404, 1995.
in the literature, and the computation of the q-step matrix −1 Ψq+ k
[9]
are not required.
nonlinear
systems
with
discrete-time
P. E. Moraal and J. W. Grizzle, "Asymptotic observers for detectable and poorly observable systems," Proc. IEEE Conference on Decision and Control, New Orleans, USA, pp. 108-114, 1995.
References [1]
P. Albertos, "Block multirate Input-Output model for
sampled-Data
control
systems,"
[10]
least squares algorithm to the observer design for
IEEE
linear time-varying systems,” IEEE Transactions
Transactions on Automatic Control, Vol. 35,
on Automatic Control, pp.1742-1745, 1999.
pp.1085-1088, 1990. [2]
J. S. Baras, A. Bensoussan and M. R. James, "Dynamic observers as asymptotic limits of recursive filters : special cases", SIAM Journal on Applied Mathematics, Vol. 48, No. 5, pp. 1147-
M. Boutayeb and D. Aubry "A strong tracking Extended
Kalman
systems,"
IEEE
observer
for
non-linear
on
Automatic
Transactions
Control, Vol.44, No. 8, pp. 1550-1556, 1999. [4]
G. Ciccarella, M. Dalla Mora and A. Germani, "A robust observer for discrete time nonlinear systems," Systems & Control Letters, Vol. 24, pp. 291-300, 1995.
[5]
J. J. Deyst and C.F. Price, "Conditions for asymptotic stability of the discrete minimumvariance linear estimator," IEEE Transactions on Automatic Control, pp. 702-705, 1968.
[6]
E. W. Kamen, "Study of linear time-varying discrete-time systems in terms of time-compressed models," Proc. IEEE Conference on Decision and Control, Tucson, USA, pp. 3070-3075, 1992.
[7]
[11]
Y. Song and J. W. Grizzle, "The extended Kalman Filter as a local asymptotic observer for discretetime nonlinear systems," Journal of Mathematical systems, Estimation and Control, Vol. 5, No. 1, pp. 59-78, 1995.
1158, 1988. [3]
M. S. Chen and Yen J. Y., “Application of the
E. W. Kamen, "Block form observers for linear time-varying discrete-time systems," Proc. IEEE Conference on Decision and Control, San Antonio, USA, pp. 355-356, 1993.
