1092
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 4, APRIL 1997
Game Theory Approach to Discrete
Filter Design
and Kalman filters in terms of model uncertainty and gives better
estimates.
Xuemin Shen and Li Deng
H1
Abstract—In this correspondence, a finite-horizon discrete
filter
design with a linear quadratic (LQ) game approach is presented. The exogenous inputs composed of the “hostile” noise signals and system initial
condition are assumed to be finite energy signals with unknown statistics.
The design criterion is to minimize the worst possible amplification of
the estimation error signals in terms of the exogenous inputs, which is
different from the classical minimum variance estimation error criterion
for the modified Wiener or Kalman filter design. The approach can show
how far the estimation error can be reduced under an existence condition
on the solution to a corresponding Riccati equation. A numerical example
filter with that of the
is given to compare the performance of the
conventional Kalman filter.
H1
I. INTRODUCTION
The celebrated Wiener and/or Kalman estimators have been widely
used in noise signal processesing. This type of estimation assumes
that signal generating processes have known dynamics and that
the noise sources have known statistical properties. However, these
assumptions may limit the application of the estimators since in many
situations, only approximate signal models are available and/or the
statistics of the noise sources are not fully known or are unavailable. In addition, both Wiener and Kalman estimators may not be
robust against parameter uncertainty of the signal models. Recent
developments in optimal filtering have focused on the H estimation
methods [1]–[10]. The optimal H estimator is designed to guarantee
that the operator relating the noise signals to the resulting estimation
errors should possess an H norm less than a prescribed positive
value. In the H estimation, the noise sources can be arbitrary signals
with only a requirement of bounded noise. Since the H estimation
problem involves the minimization of the worst possible amplification
of the error signal, it can be viewed as a dynamic, two-person, zero
sum game. In the game, the H filter (the designer) is a player
prepared for the worst strategy that the other player (the nature) can
provide, i.e., the goal of the filter is to provide an uniformly small
estimation error for any processes and measurement noises and any
initial states. In this correspondence, we define a difference game
in which the state estimator and the disturbance signals (processes
noise, initial condition and measurement noise) have the conflicting
objectives of, respectively, minimizing and maximizing the estimation
error. The minimizer picks the optimal filtered estimate, and the
maximizer picks the worst-case disturbance and initial condition. We
give a detailed derivation to solve the game that directly produces
the solution for the discrete H filtering problem. A similar design
approach has been proposed in [1] and [2] for the continuous case.
We then give a numerical example to compare the H filter with the
Kalman filter. The comparison includes the magnitudes of the transfer
functions from processes and measurement noises to estimation
errors, which are the estimations of the true signals. It is shown
that the H filter is more robust compared with those of Wiener
1
1
1
1 FILTER DESIGN
II. DISCRETE H
Consider the following discrete-time system
xk+1 = Ak xk + Bk wk
yk = Ck xk + vk
k = 0; 1; 1 1 1 ; N 0 1;
zk
1
1
Manuscript received July 18, 1995; revised December 9, 1996. This work
was supported by the Natural Science and Engineering Research Council
(NSERC) of Canada. The associate editor coordinating the review of this
paper and approving it for publication was Dr. Gregory H. Allen.
The authors are with the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, Ont., Canada N2L 3G1.
Publisher Item Identifier S 1053-587X(97)02592-0.
x0
(1)
=
Lk xk :
(2)
1
The H filter is required to provide an uniformly small estimation
error ek = zk 0 z^k for any wk ; vk 2 l2 and x0 2 Rn : The measure
of performance is then given by
N
01
jjz 0 z^ jj
k
J
=
1
1
=
where
xk 2 Rn state vector,
wk 2 Rm process noise vector,
yk 2 Rp measurement vector,
vk 2 Rp measurement noise vector.
Ak ; Bk ; and Ck are matrices of the appropriate dimensions. Assume
that (Ak ; Bk ) is controllable and (Ck ; Ak ) is detectable. Define the
measurement history as Yk = (yk ; 0 k N 0 1): The estimate of
^k at time k is computed based on the measurement history
the state x
up to N 0 1: We are not necessarily interested in the estimation of
xk but in the estimation of a linear combination of xk
1
1
x0
k=0
jjx 0 x^ jj
0
N
2
0
01
Q
fjjw jj
+
p
2
k
2
k
k=0
W
+
jjv jj
2
k
V
g
(3)
where ((x0 0 x
^0 ); wk ; vk ) 6= 0; x
^0 is an a priori estimate of
x0 ; Qk 0; p001 > 0; Wk > 0 and Vk > 0 are the weighting matrices,
and jjsk jj2R = sTk Rk sk : The optimal estimate zk among all possible
z^k (i.e., the worse-case performance measure) should satisfy
sup J
< 1=
(4)
where “sup” stands for supremum, and > 0 is a prescribed level of
noise attenuation. The matrices Qk 0; Wk > 0; Vk > 0 and p0 > 0
are left to the choice of the designer and depend on performance
requirements. Discrete H1 filtering can be interpreted as a minimax
problem where the estimator strategy z^k play against the exogenous
inputs wk ; vk and the initial state x0 : The performance criterion
becomes
min
z
^
max
(v ;w ;x )
=
J
0 21 jjx 0 x^ jj
0
1
N
01
+
p
1
2
jjz 0 z^ jj 0 1 (jjw jj
k
k=0
2
0
k
2
Q
k
2
W
+
jjv jj
k
2
V
)
(5)
where “min” stands for minimization and “max” maximization.
Note that unlike the traditional minimum variance filtering approach
(Wiener and/or Kalman filtering), the H1 filtering deals with deterministic disturbances, and no a priori knowledge of the noise statistics
is required. Since the observation yk is given, vk can be uniquely
1053–587X/97$10.00  1997 IEEE
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 4, APRIL 1997
1093
determined by (1) once the optimal values of wk and x0 are found.
^k , we can rewrite the performance criterion (5) as
Letting z^k = Lk x
where xk and Pk are undetermined variables. x3k and 3k represent
the optimal value of xk and k , respectively, for any fixed admissible
functions of xk and yk : The optimal values for wk and x0 are
min
x
^
J
max
(y ;w ;x )
=
0 21 jjx 0 x^ jj
2
0 p
0
0 1 (jjw jj
2
k
+
W
1
+
2
N
01
jjx 0 x^ jj
[
k
k
wk3
2
Q
k=0
jjy 0 C
k
k
xk jj2V
)]
(6)
where Qk = LTk Qk Lk : The following theorem presents a complete
solution to the H1 filtering problem for the system (1) with the
performance criterion (6).
Theorem: Let > 0 be a prescribed level of noise attenuation.
Then, there exists an H1 filter for zk if and only if there exists a
stabilizing symmetric solution Pk > 0 to the following discrete-time
Riccati equation:
Pk + CkT Vk01 Ck Pk )01
1 A + Bk Wk BkT
P0 = p0 :
Pk+1
= Ak Pk (I
0 Q
xk+1 + Pk+1 3k+1
k = 0; 1; 1 1 1 ; N
x^k+1
=
Ak x^k + Kk (yk 0 Ck x^k ); x^0
3k
=
Ak Pk (I 0 Qk Pk + C
T
k
V 01 C
=x
^0 :
k
k
Pk
01
) C
T
k
V 01 :
k
=
1
2
+
jjx 0 x^ jj 0 1 (jjw jj
k
2
k
T
k+1
2
k
Q
[Ak xk + Bk wk
+ [Ak xk + Bk wk
=x
^0 + p0 0 ;
+
W
jjy 0 C
k
k
xk jj2V
T
k+1 ]
(11)
=0
T
k
(12)
xk+1
+
0
Bk Wk BkT
ATk
0
Qk x^k + CkT Vk01 yk
k = 0; 1; 1 1 1 ; N
xk
k+1
x0
=x
^0 + p0 0 ;
N
1 [Q
=
xk + Pk 3k
k
T
k
k
k
k
k
1
k
Pk )01
(21)
k+1
0 (Q 0 C V 0 C )P ]0
0 x^ ) + C V 0 (y 0 C x )]
xk
T
k
k
T
k
k
1
k
k
1
k
k
k
1
k
k
=x
^0
Pk+1
(22)
= Ak Pk (I
0 Q
k
Pk + CkT Vk01 Ck )Pk )01 AkT
+ Bk Wk B
T
k
P0
= p0 :
(23)
Equation (23) is the well-known Riccati difference equation. It has
been prooven that if the solution Pk to the Riccati equation (23)
exists 8k 2 [0; N 0 1]. Then, Pk > 08k 2 [0; N 0 1]:
Now, substituting the optimal strategies (18) into the performance
(6), we obtain
min max J =
x
^
y
0 21 jj3 jj
0
2
+
p
N
1
01
T
k
k
k
k
k
jjx
[
2
k
3 0 x^
+ Pk k
k
jj
2
Q
k=0
0 1 (jjW B 3 jj
0 C P 3 jj )]:
k+1
2
+
W
jjy 0 C
k
k
xk
2
(24)
V
In the sequel, we will perform the min-max optimization of J with
respect to x
^k and yk , respectively. Adding to (24) the identically
zero term
[
0
2
p
N
2
P
N
01
+
jj3 jj
(
k+1
0 jj3 jj
2
P
2
k
P
)]
k=0
=0
(15)
(25)
results in the following min-max problem
= 0:
(16)
Since the two-point boundary value problem is linear, the solution
is assumed to be of the form
x3k
T
k
jj3 jj 0 jj3 jj
2
with boundary conditions
x0
k
k
and
1
;
01
k
= Ak xk + Ak Pk [(I
k+1
(13)
T
0
1
= A k+1 + Qk (xk 0 x
^k ) + Ck Vk (yk 0 Ck xk ):
(14)
Ak
Qk 0 CkT Vk01 Ck
k
1
k
For (21) to hold true for arbitrary 3k , both sides are set identically
to zero, resulting in
T
k
These first-order necessary conditions result in a two-point boundary value problem
T
k
k
k
T
k
=
k
k
)
k+1 ]
k+1
:
(20)
1 [Q (x 0 x^ ) + C V 0 (y 0 C x )]
0C
= [0P
+ A P (I 0 Q P + C V
1 A + B W B ]3 :
= Wk B
xk+1
k
3
T
(10)
0x
0x
N
xk 0 x^k ) + CkT Vk01 (yk 0 Ck xk )
1 [Q
k(
Taking the first variation, the necessary conditions for a maximum
are
x0
wk
k
k
+ Ak k+1 ]:
(9)
Proof: By using a set of Lagrange multiplier to adjoin the constraint (1) to the performance criterion (6), the resulting Hamiltonian
is
M
Pk + CkT Vk01 Ck Pk )01
0 Q
k(
(7)
where
Kk
= (I
xk+1 0 Ak xk 0 Ak Pk (I 0 Qk Pk + CkT Vk01 Ck Pk )01
(8)
Kk is the gain of the H1 filter and is given by
(19)
From (19) and (20), we have
01
Lk x^k ;
(18)
Ak xk + Ak Pk 3k + Bk Wk BkT 3k+1
=
k+1
=
3
=x
^0 + p0 0 :
and
k
z^k
x30
Substituting (17) into (15) results in
T
k
The H1 filter is given by
Wk BkT 3k+1 ;
=
(17)
min max J
x
^
y
=
1
2
N
01
jjx 0 x^ jj 0 1 jjy 0 C x jj
[
k=0
k
k
2
Q
k
k
subject to the dynamic constraints (22) and (23).
k
2
V
]
(26)
1094
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 4, APRIL 1997
Fig. 1.
Signal generating mechanism.
Letting
Fig. 2.
rk
=
xk 0 x^k ;
(26) becomes
min max J =
r
q
1
N01
2
k=0
qk
=
yk 0 Ck xk
jjrk jj2Q 0 1 jjqk jj2V
Kalman filter estimate.
(27)
:
(28)
The two independent players rk and qk in (28) affect the variables
xk , but xk does not appear in the performance index, and therefore,
the optimal strategies of rk and qk are
rk3
i.e.
xk
qk3
= 0;
3
=x
^k ;
yk3
=0
=
(29)
Ck xk :
(30)
The value of the game is the value of the cost function (6).
3 3 3
3
^k ; yk ; wk ; and x0 in (18) and (30)
When the optimal strategies x
are substituted into the (6)
J (^
x3k ; yk3 ; wk3 ; x30 ) = 0
(31)
giving a zero value game.
3 3 3
3
^k ; yk ; wk ; and x0 have been assumed to
Thus far, the strategies of x
be optimal, based on the satisfaction of the necessary conditions for
optimality. If the strategies can also satisfy a saddle-point inequality,
they represent optimal strategies. A saddle point strategy can be
obtained by solving two optimization problems:
3
x^ y w x
max max max min J = J3 :
y w x x^
min max max max J = J
(32)
(33)
When J 3 = J3 , the solutions to (32) and (33) produce saddle point
strategies. It can be easily shown that if Pk exists 8k 2 [0; N 0 1], the
3 3 3
3
^k ; yk ; wk ; and x0 satisfy a saddle point inequality
optimal strategies x
J (^
x3k ; yk ; wk ; x0 ) J (^
x3k ; yk3 ; wk3 ; x30 ) J (^
xk ; yk3 ; wk3 ; x30 ):
(34)
Note that the notation J1 J2 means that J1 0 J2 is a positive
semi-definite matrix.
The right inequality can be checked by adding the identically zero
term
3 ^0 jj2 0 jjx3 0 x^N jj2
[jjx 0 0 x
N
P
P
2
N01
3
2
+
(jjxk+1 0 x
^k+1 jj P
0 jjx3k 0 x^k jj2P )] (35)
k=0
to J (^
xk ; yk3 ; wk3 ; x30 ), and the left inequality can be checked by adding
1
the identically zero term
1
2
jjx0 0 x^30 jj2P 0 jjxN 0 x^3N jj2P
[
N01
+
to
3 2
(jjxk+1 0 x
^k+1 jjP
k=0
3
J (^
xk ; yk ; wk ; x0 ):
0 jjxk 0 x^3k jj2P
H1 filter estimate.
Fig. 3.
The optimal strategy of the measurement noise can be obtained by
vk3
=
yk3 0 Ck x^3k
(36)
= 0:
(37)
With (22) and (30), the optimal H1 filter is given by
z^k3
where
x^3k+1
Kk
=
3
Lk x^3k ;
= Ak x
^k + Kk (yk
= Ak Pk (I
k = 0; 1; 1 1 1 ; N
0 Ck x^3k );
x0
01
=x
^0
0 Qk Pk + CkT Vk01Ck Pk )01 CkT Vk01
(38)
(39)
(40)
and Pk is given by (23).
It is important to note that the optimal H1 filter depends on the
weighting on the estimation error in the performance criterion, i.e.,
the designer choses the weighting matrices based on the performance
requitements, whereas both Wiener and Kalman filters are dependent
on the variance of the noises.
For the time-invariant case (N ! 1), the optimal steady-state
H1 filter is given by
z^k3
where
x^3k+1
K
=
3
Lx^3k ;
k = 0; 1; 1 1 1 ; 1
= Ax
^k + K (yk
= AP (I
0 C x^3k ); x0 = x^0
0 QP + C T V 01 CP )01C T V 01
(41)
(42)
(43)
and the Riccati equation becomes
P
=
AP (I 0 QP
+C
T V 01 CP )01 AT + BW B T :
(44)
The solution of the Riccati equation (44) can be obtained by the
following [9]. Let
H
=
A0T
BW B T A0T
A0T [C T V 01 C 0 Q]
:
A + BW B T A0T [C T V 01 C 0 Q]
(45)
Assume that matrix H has no eigenvalues on the unit circle and
that the eigenvector S corresponds to the outer circle (unstable)
eigenvalues of the matrix H: Spanning S as S = [S1T S2T ]T , the
solution of Riccati equation P is given as
P
)]
Ck xk 0 Ck x^3k
=
=
S2 S101 :
(46)
Details of the last result can be found in [11]. Note that in the
limiting case, where the parameter ! 0, the H1 filter given by
(41)–(44) reduces to a steady-state Kalman filter.
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 45, NO. 4, APRIL 1997
1095
over all possible disturbances of finite energy; therefore, they are
overconservative, resulting in a better robust behavior to disturbance
variations. All the simulation results are obtained by using MATLAB
[12].
IV. CONCLUSIONS
A difference game has been formulated and solved for the discrete
H filter design. The existence of a solution to the difference Riccati
equation, over the time interval, is a necessary and sufficient condition
for the existence of the optimal discrete H filter. Since the design
criterion is based on the worst-case disturbance, the H filter is
less sensitive to uncertainty in the exogenous signals statistics and
dynamical model.
1
Fig. 4. Estimation error power spectra—without disturbance.
1
1
REFERENCES
1
Fig. 5. Estimation error power spectra—with disturbance.
1
III. NUMERICAL EXAMPLE
A signal generating system (Fig. 1) is the damped harmonic
oscillator with velocity measurements described by
y
wn
0
x_ =
0wn 02wn
= (0
x+
0
wn
where the state x = (x1 x2 )T with x1 as the position and x2
as the velocity. The natural frequency wn = 1:1, and the damping
coefficient = 0:15: w is a driving signal, and v is the measurement
noise. It is assumed that w and v are uncorrelated, stationary, zeromean, white noise processes of unit intensity. Assuming a zero-order
hold on the input, this system is converted to the following discrete
system with sample time T = 1:
xk+1
yk
=
0
= (0
0:7594
0:2801
k
xk +
0:4921
0:7594
wk
k
1)x + v
where it is desired to estimate
zk
= (1
k:
0)x
(47)
Using (45), (46), and (43), the following Kalman filter gain ( = 0)
and H filter gain ( = 1:24) are obtained.
1
GK
=
0:4236
0:0873
;
GH
=
0:1792
1:1321
:
(48)
The performance of the two filters is compared by simulating their
estimate of zk and the magnitudes of the transfer function Tzw from
noises [wkT vkT ]T to the estimation error ek : The results are depicted
in Figs. 2 to 5. From Figs. 2 and 3, it is observed that the H filter
gives the estimate zk relatively better than that of the corresponding
Kalman filter, even though the statistics of the noises wk and vk
are known. Figs. 4 and 5 give the magnitudes of the estimation error
power spectra using both H filter and Kalman filter when the system
parameters (wn ; ) vary from (0.9, 0.08) to (1.1, 0.3). It is shown that
the error spectra of the H filter have lower peaks, which means
that the error spectra of the H filter are less sensitive to exact
knowledge of the parameters of the system. This can be explained
as follows: H filters guarantee the smallest estimation error energy
1
1
1
1
1
1
1
w
1)x + v
0:5079
0:7594
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1