⎤
⎨
⎪
w
w
⎪
k − E {wk }
⎢ j − E wj ⎥ ⎬
W
⎢ kδkj Skδkj 0 ⎥
E
⎣
⎦ ⎢
⎥
=
⎪ v
⎣ STδ
⎦ ,
(4)
⎪
k − E {vk }
⎣ v
⎦
k kj Vk δkj 0
⎩
j − E
vj
⎪
x
⎪
0 − x 0
⎭
x
0
0
X
0 − x 0
0
where Wk, Vk and X 0 denotes the noises and initial state covariance matrices, Sk is the cross
covariance and δkj is the Kronecker delta function.
Although the exact values of the means and of the covariances are unknown, it is assumed
that they are within a known set. The notation at (5) will be used to represent the covariances
sets.
Wk ∈ Wk, Vk ∈ Vk, Sk ∈ Sk.
(5)
In the next sub section, it will be presented how to characterize a system with uncertain
covariance as a system with known covariance, but with uncertain parameters.
3.1 The noises means and covariances spaces
In this sub section, we will analyze some features of the noises uncertainties. The approach
shown above considered correlated wk and vk with unknown mean, covariance and cross
covariance, but within a known set. As will be shown later on, these properties can be
achieved when we define the following noises structures:
wk := BΔ w, kwk + BΔ v, kvk,
(6)
vk := DΔ w, kwk + DΔ v, kvk.
(7)
Also here we assume that the initial conditions {x 0 } and the noises {wk} , {vk} are
uncorrelated with the statistical properties
⎧⎡
⎤⎫
⎡
⎤
⎨ wk ⎬
wk
E
⎣
⎦ = ⎣
⎦
⎩ vk ⎭
vk
,
(8)
x 0
x 0
⎧
⎫
⎪⎡
⎤ ⎡
⎤
⎡
⎤
⎨
T
w
⎪
k − wk
wj − wj
⎬
Wkδkj Skδkj 0
E
⎣
⎦ ⎣
⎦
= ⎣
⎦
⎪ v
v
STδ
,
(9)
⎩
k − vk
j − vj
⎪
k kj Vk δkj 0
x
⎭
0 − x 0
x 0 − x 0
0
0
X 0
Kalman Filtering for Discrete Time Uncertain Systems
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5
97
where Wk, Vk and X 0 denotes the noises and initial state covariance matrices and Sk stands for
the cross covariance matrix of the noises.
Therefore using the properties (8) and (9) and the noises definitions (6) and (7), we can note
that the noises wk and vk have uncertain mean given by
E {wk} = BΔ w, kwk + BΔ v, kvk,
(10)
E {vk} = DΔ w, kwk + DΔ v, kvk.
(11)
Their covariances are also uncertain and given by
⎧
⎫
⎪
⎡
⎤
⎨
T ⎪
⎬
w
E
k − E {wk }
⎣ wj − E wj ⎦
= Wkδkj Skδkj
⎪
.
(12)
⎩ vk − E {vk}
v
⎪
STδ
j − E
vj
⎭
k kj Vk δkj
Using the descriptions (6) and (7) for the noises, we obtain
W
T
k δkj Sk δkj
= BΔ w, k BΔ v, k
Wkδkj Skδkj
BΔ w, k BΔ v, k
.
(13)
STδ
D
STδ
D
k kj Vk δkj
Δ w, k DΔ v, k
k kj Vk δkj
Δ w, k DΔ v, k
The notation at (13) is able to represent noises with the desired properties of uncertain
covariance and cross covariance. However we can consider some simplifications and achieve
the same properties. There are two possible ways to simplify equation (13):
1. Set
BΔ w, k BΔ v, k = BΔ w, k 0
.
(14)
DΔ w, k DΔ v, k
0
DΔ v, k
In this case, the covariance matrices can be represented as
Wkδkj Skδkj = BΔ w, kWkBTΔ B
w, k
Δ w, kSk DT
Δ v, k
δ
STδ
D
BT
D
kj.
(15)
k kj Vk δkj
Δ v, kSTk Δ w, k
Δ v, kVk DT
Δ v, k
2. The other approach is to consider
Wkδkj Skδkj = Wkδkj 0 .
(16)
STδ
0
V
k kj Vk δkj
k δkj
In this case, the covariance matrices are given by
W
+
+
k δkj Sk δkj
=
BΔ w, kWkBTΔ
B
B
B
w, k
Δ v, kVk BT
Δ v, k
Δ w, kWk DT
Δ w, k
Δ v, kVk DT
Δ v, k
δ
STδ
D
+ D
D
+ D
kj.
k kj Vk δkj
Δ w, kWk BT
Δ w, k
Δ v, kVk BT
Δ v, k
Δ w, kWk DT
Δ w, k
Δ v, kVk DT
Δ v, k
(17)
So far we did not make any assumption about the structure of noises uncertainties at (6)
and (7). As we did for the dynamic and the output matrices, it will be assumed additive
uncertainties for the structure of the noises such as
BΔ w, k := Bw, k + Δ Bw, k, BΔ v, k := Bv, k + Δ Bv, k, (18)
DΔ w, k := Dw, k + Δ Dw, k, DΔ v, k := Dv, k + Δ Dv, k, (19)
698
Discrete Time Systems
Discrete Time Systems
where Bw, k, Bv, k, Dw, k and Dv, k denote the nominal matrices. Their uncertainties are represented by Δ Bw, k, Δ Bv, k, Δ Dw, k and Δ Dv, k respectively. Using the structures (18)-(19) for the uncertainties, then we are able to obtain the following representation
wk = Bw, k + Δ Bw, k wk + Bv, k + Δ Bv, k vk,
(20)
vk = Dw, k + Δ Dw, k wk + Dv, k + Δ Dv, k vk.
(21)
In this case, we can note that the mean of the noises depend on the uncertain parameters of
the model. The same applies to the covariance matrix.
4. Linear robust estimation
4.1 Describing the model
Consider the following class of uncertain systems presented at (1)-(2):
xk+1 = ( Ak + Δ Ak) xk + wk,
(22)
yk = ( Ck + Δ Ck) xk + vk,
(23)
where xk ∈ R nx is the state vector, yk ∈ R ny is the output vector and wk ∈ R nx and vk ∈
R ny are noise signals. It is assumed that the noise signals wk and vk are correlated and their
time-variant mean, covariance and cross-covariance are uncertain but within known bounded
sets. We assume that these known sets are described as presented previously at (20)-(21) with
the same statistical properties as (8)-(9).
Using the noise modeling (20) and (21), the system (22)-(23) can be written as
xk+1 = ( Ak + Δ Ak) xk + Bw, k + Δ Bw, k wk + Bv, k + Δ Bv, k vk, (24)
yk = ( Ck + Δ Ck) xk + Dw, k + Δ Dw, k wk + Dv, k + Δ Dv, k vk.
(25)
The dimensions are shown at Table (1).
Matrix or vector Set
xk
R nx
yk
R ny
wk
R nw
vk
R nv
Ak
R nx×nx
Bw, k
R nx×nw
Bv, k
R nx×nv
Ck
R ny×nx
Dw, k
R ny×nw
Dv, k
R ny×nv
Table 1. Matrices and vectors dimensions.
The model (24)-(25) with direct feedthrough is equivalent to one with only one noise vector at
the state and output equations and that wk and vk could have cross-covariance Anderson &
Moore (1979). However, we have preferred to use the redundant noise representation (20)-(21)
with wk and vk uncorrelated in order to get a more accurate upper bound for the predictor
covariance error. The nominal matrices Ak, Bw, k, Bv, k, Ck, Dw, k and Dv, k are known and the matrices Δ Ak, Δ Bw, k, Δ Bv, k, Δ Ck, Δ Dw, k and Δ Dv, k represent the associated uncertainties.
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7
99
The only assumptions we made on the uncertainties is that they are additive and are within
a known set. In order to proceed the analysis it is necessary more information about the
uncertainties. Usually the uncertainties are assumed norm bounded or within a polytope. The
second approach requires more complex analysis, although the norm bounded set is within
the set represented by a polytope.
In this chapter, it will be considered norm bounded uncertainties. For the general case, each
uncertainty of the system can be represented as
Δ Ak := HA, kFA, kGA, k,
(26)
Δ Bw, k := HBw, kFBw, kGBw, k,
(27)
Δ Bv, k := HBv, kFBv, kGBv, k,
(28)
Δ Ck := HC, kFC, kGC, k,
(29)
Δ Dw, k := HDw, kFDw, kGDw, k,
(30)
Δ Dv, k := HDv, kFDv, kGDv, k.
(31)
where HA, k, HBw, k, HBv, k, HC, k, HDw, k, HDv, k, Gx, k, Gw, k and Gv, k are known. The matrices FA, k, FBw, k, FBv, k, FC, k, FDw, k and FDv, k are unknown, time varying and norm-bounded, i.e. , FT
A, k FA, k ≤ I, FT
Bw, k FBw, k ≤ I, FT
Bv, k FBv, k ≤ I, FT
C, k FC, k ≤ I, FT
Dw, k FDw, k ≤ I, FT
Dv, k FDv, k ≤ I.
(32)
These uncertainties can also be represented at a matrix format as
Δ Ak Δ Bw, k Δ Bv, k
Δ Ck Δ Dw, k Δ Dv, k
= HA, kFA, kGA, k HBw, kFBw, kGBw, k HBv, kFBv, kGBv, k
HC, kFC, kGC, k HDw, kFDw, kGDw, k HDv, kFDv, kGDv, k
= HA, k HBw, k HBv, k
0
0
0
0
0
0
HC, k HDw, k HDv, k⎡
⎤
GA, k
0
0
⎢
⎢ 0
G
⎥
⎢
Bw, k
0
⎥
⎥
×
0
0
G
diag F
⎢
Bv, k ⎥
A, k, FBw, k, FBv, k, FC, k, FDw, k, FDv, k
⎢
.
(33)
⎢ G
⎥
⎣ C, k
0
0
⎥
0
G
⎦
Dw, k
0
0
0
GDv, k
However, there is another way to represent distinct uncertainties for each matrix by the
appropriate choice of the matrices H as follows
Δ Ak
Δ
:=
HA, k F
C
x, k Gx, k
(34)
k
HC, k
Δ Bw, k
Δ
:=
HBw, k F
D
w, k Gw, k
(35)
w, k
HDw, k
Δ Bv, k
Δ
:=
HBv, k F
D
v, k Gv, k,
(36)
v, k
HDv, k
8100
Discrete Time Systems
Discrete Time Systems
where the matrices Fx, k, Fw, k and Fv, k of dimensions rx, k × sx, k<