Linear H-Infinity Tracking Control in Discrete-Time Stochastic
Systems with Uncertain Parameters
SEIICHI NAKAMORI
Professor Emeritus, Faculty of Education,
Kagoshima University,
1-20-6 Korimoto, Kagoshima 890-0065,
JAPAN
Abstract: In linear discrete-time stochastic systems with uncertain parameters, this study proposes an H-
infinity tracking control strategy based on an H-infinity tracking controller and a robust recursive least-squares
Wiener filter. A linear H-infinity tracking control algorithm for quantity , whose components are the
control and exogenous inputs, was proposed for discrete-time deterministic systems without input and
observation noise. Based on the separation principle between control and estimation, this study presents
equations for in linear discrete-time stochastic systems with uncertain parameters as a counterpart to the
equations in deterministic systems. The H-infinity tracking control algorithm in linear discrete-time stochastic
systems with uncertain parameters is derived in the same manner as the H-infinity tracking control algorithm in
linear discrete-time deterministic systems. The filtering estimate of the degraded system state is used
to calculate the estimate of . The robust RLS Wiener filter calculates the filtering estimate of
the system state for degraded stochastic systems with uncertain parameters. With knowledge of the
estimate 1 of 1, the degraded observed value , and the filtering estimate 1 of the
degraded state 1, is updated from 1.
Key-Words: - H-infinity tracking control, control input, exogenous input, robust recursive least-squares Wiener
filter, discrete-time stochastic systems with uncertain parameters.
Received: May 14, 2022. Revised: February 17, 2023. Accepted: March 16, 2023. Published: May 4, 2023.
1 Introduction
Linear quadratic Gaussian (LQG) control has been
studied in e.g., [1], [2], [3], [4], [5], [6]. In addition,
LQG tracking control problems have also been
investigated, [7], [8], [9], [10], [11]. A real-time
transcale LQG tracking control technique for
discrete-time stochastic systems was presented in
[11], and was based on wavelet packet
decomposition (WPD). The system in this scenario
excluded unknown parameters. An output feedback
controller was developed for discrete-time
stochastic systems with uncertainties and missing
measurements, [12]. The parameter uncertainties
were norm-bounded. The probability that the
missing data will occur presupposes that it
is known. Using linear matrix inequalities (LMIs)
solves this problem. In [13], a robust controller
based on a disturbance observer was proposed for
linear continuous-time uncertain systems with a
time delay. The LMI solution determines observer
parameters. It deals with state feedback control. The
H-infinity controller was designed in [14], for a
state-space model with uncertain parameters in
linear continuous-time stochastic systems. A robust
controller has been developed for linear discrete-
time uncertain systems, [15]. In [15], a state
feedback controller is designed using the LMI
technique, and a low-order disturbance observer is
presented. The LMI approach is presented for state
feedback quadratic stabilization in linear
continuous-time uncertain systems in subsection 5.
3 of [16]. For linear discrete-time uncertain
stochastic systems, a combined H2/Passivity
controller was developed, [17]. Some sufficient
conditions are converted into LMIs using the
Lyapunov theory. A repeated-tracking controller for
stochastic time-varying delay systems was designed
in [18].
An H-infinity tracking control technique was
proposed in [19], for deterministic systems without
input and observation disturbances. Robust
recursive least-squares (RLS) Wiener filter and
fixed-point smoother were proposed in [20], [21],
for linear discrete-time stochastic systems with
uncertain parameters in the system and observation
matrices. Signal estimation was the purpose of the
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estimators in [20]. The robust RLS Wiener
estimators in [21], estimate the nondegraded
nominal system state rather than the degraded
system state when utilizing degraded observations.
In this case, the estimators employ the system and
observation matrices from the original system. In
linear discrete-time systems with norm-bounded
uncertainties in the system and input matrices, a
robust filter estimates the degraded state, [22]. A
robust Kalman filter, [23], was designed for the
linear discrete-time state-space model with
multiplicative noise and norm-constrained time-
varying uncertainties both in the system and
observation matrices. In [24], a robust Kalman filter
was proposed for linear discrete-time uncertain
systems with norm-bounded uncertainties in the
system and observation matrices. Recently, an H-
infinity tracking control method using the robust
RLS Wiener filter in [21], was developed to track
the nondegraded nominal signal to the desired value
in linear discrete-time uncertain systems with
uncertainties in the system and observation
matrices, [25]. As seen from [22], [23], [24], these
robust filters estimate the degraded state rather than
the nondegraded nominal system state. From this
fact, this paper aims to newly design an H-infinity
tracking controller for the degraded signal to track
the desired value for linear discrete-time stochastic
systems with uncertain parameters in Theorem 1.
For this purpose, Theorem 2 proposes a new robust
RLS Wiener filter to estimate the degraded state. It
is assumed herein that uncertainties exist in the
system and observation matrices. No norm-bounded
uncertainties are assumed for the uncertain matrices.
The uncertain system and observation matrices are
estimated by (18) and (19), respectively. Using the
uncertain system and observation matrix estimates,
the robust RLS Wiener filter of Theorem 2
recursively calculates the filtering estimate of the
degraded system state. Based on the separation
principle of control and estimation, it is shown in
Section 2 that satisfies (10)-(12) for linear
discrete-time stochastic systems with uncertainties,
corresponding to the deterministic systems in [19].
Here, consists of vector components, control
input, and exogenous input. The filtering estimate
of the degraded system state is used to
calculate the estimate of . The robust RLS
Wiener filter in Theorem 2 computes the filtering
estimate of the degraded system state for
the degraded state-space model with uncertainties.
Information on the estimate 1 of 1,
the degraded observed value , and the filtering
estimate 1 of the degraded state 1 is
used to update from 1. The estimate
of in Theorem 1 uses the filtering
estimate
of the degraded state by the
robust RLS Wiener filter in Theorem 2.
In Section 4, the first numerical simulation
example compares the tracking control accuracy of
the H-infinity tracking controller of Theorem 1 and
the robust RLS Wiener filter of Theorem 2 with that
of the H-infinity tracking controller of Theorem 1
and the RLS Wiener filter, [26], or the robust
Kalman filter, [24]. Compared to the combinations
of the H-infinity tracking controller of Theorem 1
with either the RLS Wiener filter or the robust
Kalman filter, the combination of the H-infinity
tracking controller of Theorem 1 with the robust
RLS Wiener filter of Theorem 2 provides superior
tracking control accuracy. The second simulation
example demonstrates F16 aircraft tracking control
in terms of tracking accuracy.
2 H-Infinity Linear Tracking Control
Problem
Let (1) represent the discrete-time state-space model
in linear stochastic systems.
,,
1
Γ,
,
,
0,,
,
0,00.
(1)
Here, ∈ is the state vector, ∈ is
the input vector, and ∈ is the signal vector.
∈ and ∈, , are
the control and exogenous input vectors,
respectively. The input noise ∈ and the
observation noise ∈ are mutually
uncorrelated with zero mean white Gaussian noise.
Γ is the input matrix, and is the
observation matrix. The auto-covariance functions
for the input noise and the observation noise
are given in (1). This paper considers the state
and observation equations with uncertain parameters
in (2).
,
,
ΔC,
1
Γ,
ΔA,0,
E
v
ks0,E∆Aks0,
ΔC0,00
(2)
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In (2), the degraded system matrix and the
degraded observation matrix are introduced
instead of the system matrix and the observation
matrix in (1), respectively. Here, the matrix
elements of ΔA and ΔC consist of unknown
variables. The initial system state 0 is a random
vector that is uncorrelated with both system and
measurement noise processes. Let represent
the performance output, [27]. The expected value of
‖‖
is given by (3).
‖
‖
(3)
Here, is the desired value, and and
are symmetric positive-definite matrices. As in [19],
the H-infinity optimal tracking control problem is to
find the control input and the exogenous
input when is at its minimum value in the
disturbance attenuation condition (4). 0 is
referred to as the constant-disturbance attenuation
level.
(4)
The H-infinity tracking control problem for a finite
horizon equivalently transforms into a two-person
zero-sum linear quadratic dynamic game, [28], [29].
Namely, given , we investigate the minimax
problem, which minimizes the value function
J,, for the control input and
maximize J,, for the exogenous input
.
J,,
(5)
We assume that the vector contains the
components of the control input and the
exogenous input . Introducing
0
0
transforms (5) into (6).
,,
(6)
In the value function (6) the discount factor is 1.
is expressed as
Φ
,0
11Φ
,1
Γ,
11,0,
0,0,
Φ
,
A
k1A
k2⋯A
s,0,
, .
(7)
Here, Φ
, represents the state-transition matrix
for the system matrix , and 1 represents the
discrete-time unit step sequence. Substituting (7)
into (6), we have
,,
∑
Φ
,0
11Φ
,1
ΓΦ
,0
11Φ
,1
Γ.
(8)
According to the calculus of variations, [19], the
necessary condition for to minimize the value
function (8) for and maximize (8) for ,
is given by (9).
∑∑1
1
1
1Φ
,1
Φ
,1
11Φ
,1
Φ
,0
(9)
If we introduce
,
Φ
,1
Φ
,1,
0,
Φ
,1Φ
,1
,
0
,
(10)
and
1
Φ
,1
(11)
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Φ
,0
,
the optimal satisfies
,
1.
(12)
The sufficient condition for the value function
J,, to be minimal for and maximal
for is given by ,
0, [19].
Note that (10)-(12) of the H-infinity tracking
control problem for the uncertain systems (2) can be
obtained similarly to the equations in [19], for the
H-infinity tracking control problem in linear
deterministic systems. Theorem 1 presents the H-
infinity tracking control algorithm derived from
(10)-(12). In Section 2 of [11], the filtering estimate
used in the LQG tracking control algorithm was
calculated using the Kalman filter. Thus, the
separation principle of control and estimation is
valid for the LQG tracking control problem. In [11],
the filtering estimate is calculated using (11) and
(12), which include the term related to the control
input. In addition, the filter gain was calculated in
the Kalman filter, [11]. The separation principle of
control and estimation is valid for the degraded
uncertain systems (2). Theorem 1 proposes the H-
infinity tracking-control algorithm. The estimate
of uses the filtering estimate of the
degraded state . The robust RLS Wiener filter
in Theorem 2 computes the filtering estimate
of the degraded state using the degraded
observed value . Theorems 1 and 2 utilize the
time-invariant estimates and for the unknown
time-varying system and observation matrices
and , respectively.
3 H-Infinity Tracking Control
Algorithm and Robust RLS Wiener
Filter in Stochastic Systems with
Uncertainties
Fig.1 illustrates the structure of the H-infinity
tracking controller and the robust RLS Wiener filter.
Theorem 1 presents the H-infinity tracking control
algorithm for estimating the control input and
the exogenous input . Using the filtering
estimate instead of the estimates of the
control input and the exogenous input
are denoted by and , respectively. The
robust RLS Wiener filter in Theorem 2 calculates
the filtering estimate of the state by using
the degraded observed value .
Fig. 1: Structure of H-infinity tracking controller of
Theorem 1 and robust recursive least-squares
Wiener filter of Theorem 2.
Theorem 1 Let denote the desired value and
be expressed as 0
0
.
Let have the components of the control input
and the exogenous input as
, (13)
then the estimate of is calculated using
(14)-(19). In (14), is the estimate of the
control input and is the estimate of the
exogenous input .
(14)
1
1
1
1
1
1
(15)
Here, and represent the time-invariant
estimates of the unknown time-variant system
matrix and with uncertain matrix
elements, respectively.
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1
1
,10
(16)
1
1
1
1
,10
(17)
The estimate of the uncertain system matrix
satisfies
1
0,
11
,
0
.
(18)
The estimate of the uncertain observation matrix
is given by
or
.
(19)
In (15), we utilized the filtering estimate for
the state . is computed by the robust RLS
Wiener filtering algorithm of Theorem 2 with the
degraded observed value . and are
computed using (16) and (17) from time L1
in the reverse direction of time until steady-state
values and are reached, respectively. The
estimate of is calculated by (15) using
and . In (15), 1 and 1 are replaced
with their stationary values and , respectively.
Now, let's introduce the robust RLS Wiener filter
in Theorem 2. Suppose that the sequence of the
degraded signal is fitted to an AR model of the
order .
1
2
⋯
,
(20)
is expressed using the state vector as
follows.
,
00⋯00
,
⋮
1
⋮
2
1
(21)
Therefore, the state equation for the state vector
is given by
1
Γ
,
,
0
0⋯0
00
⋯0
⋮⋮⋮⋱⋮
000⋯
⋯
,
Γ
00⋯0
,
.
(22)
The auto-covariance function
, of the state
vector is expressed in the semi-degenerate
functional form of
,ΨΞ
,0,
ΞΨ
,0,
Ψ
,Ξ
,. (23)
The wide-sense stationarity of the auto-covariance
function
,
for the degraded
signal shows that the auto-variance function
, of satisfies (24).
,
1
⋮
1
1 ⋯
1
0
1 ⋯
1
1
0 ⋯
2
⋮⋮⋱⋮
2
3 ⋯
1
1
2 ⋯
0
(24)
Using
, 0, the Yule-Walker equation
for the AR parameters
, 1, is given by
,
⋮
1
2
⋮
1
,
,
0 1 ⋯ 1
1 0 ⋯ 2
⋮⋮⋱⋮
2
3 ⋯ 1
1
2 ⋯ 0
.
(25)
Let
,
represent the cross-
covariance function of with .
,
has the expression
,,0,
Φ
,0,
Φ
,0
,, (26)
with the state-transition matrix Φ
, of the
unknown system matrix for in (2).
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Based on the above preliminaries, Theorem 2
presents the robust RLS Wiener filtering algorithm
for the filtering estimate of with the
degraded observed value in (2). Note that the
filter equation for contains the term 1
on the right-hand side of (28). The benefits of the
technique are that the uncertain observation and
system matrices, and , for the degraded signal
used in the H infinity RLS Wiener filtering
algorithm of Theorem 2 are given by (21) and (22)
respectively, based on the autoregressive model in
equation (20).
Theorem 2 Suppose that, in discrete-time stochastic
systems with the uncertain system matrix and
the uncertain observation matrix , the linear
state-space model of the state is given by (2).
Suppose the degraded signal fits the AR model
of order . Let the variance
, of state
with regards to the degraded signal be
expressed as (24). Let be the variance of the white
Gaussian observation noise . Then (27)-(33)
constitute the robust RLS Wiener filtering algorithm
for the filtering estimate of .
Filtering estimate of the degraded signal :
(27)
Filtering estimate of :
11
ΘA
1,
00 (28)
Filter gain for in (28): Θ
Θ
,
1A
C
,A
1A
C
,
,
or
,
(29)
Filtering estimate of :
A
1
A
1,
00 (30)
Filter gain for in (30):
,
1A
C
,A
1A
C
(31)
Auto-variance function of :
A
1A
,A
1A
,
00 (32)
Cross-variance function of with :
1
Θ
,A
1A
,
00 (33)
Here, the estimates and of the uncertain
matrices and are given by (18) and (19),
respectively. In (28), the term 1 is inserted.
1 is gained from (15), where is
calculated by the tracking control algorithm of
Theorem 1. The equation for uses the filtering
estimate , which is updated from 1 in
(28).
Proof Theorem 2 is derived by modifying the robust
RLS Wiener filter in [21], for estimating to the
estimation of .
,,
,
Fig. 2: Flowchart created by combining the H-
infinity tracking controller of Theorem 1 with the
robust RLS Wiener filter of Theorem 2.
STOP
START
11
11
1
1
10, 10,499
(499 is an example.)
200(Stationaryvalue:)
200(Stationaryvalue:)
00(Initialvalueofat0)
is computed by the robust RLS Wiener filter of Theorem 2.
1
1
111
1
,
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Fig. 2 shows a flowchart combining the H-
infinity tracking controller of Theorem 1 and the
robust RLS Wiener filter of Theorem 2.
Section 4 presents numerical simulation examples
for the tracking control characteristics of the H-
infinity tracking controller using the estimate
of by the robust RLS Wiener filter of Theorem
2 in comparison with the RLS Wiener filter, [26],
and the robust Kalman filter, [24].
4 Numerical Simulation Examples
EXAMPLE 1
Consider the observation and multi-input state
equations given by
,,
0.95 0.4,
,
1AΓ,
,A0.05 0.95
0.98 0.2,
G0.952 0
0.2 1,Γ0.952
0.2 ,
,
0.5.
(34)
The linear time-invariant (LTI) system in (34) is
observable and controllable. In (34), is the
control input and is the exogenous input.
Consider that the degraded observed value and
the degraded signal are generated by the
observation and state equations in (35).
,
,
ΔC,
1
Γ,
,
ΔA,
ΔC0.3∗ 0,
ΔA0.1∗ 0
00.2∗
(35)
Here, “ “ is a MATLAB or Octave function
representing random numbers uniformly distributed
in the interval 0,1. In [24], the uncertain matrices
ΔA and ΔC are subject to the norm-bounded
uncertainty conditions. The robust RLS Wiener
filtering algorithm of Theorem 2 does not directly
use knowledge of the uncertain matrices ΔA and
ΔC. is estimated according to the
relationship 10, where 1
1, 0. The
expectation is approximated in terms of 351
data. That is, 1≅
∑1
,
0≅
∑.
In addition,
, in (29) is approximated as ,≅
∑
. The estimate of is
given by .
Here, is approximated by
∑
. The robust RLS Wiener
filter in Theorem 2 computes the filtering estimate
in (28) to obtain the estimate of in
(15). In this instance, the AR model of order =10
in (20) is applied to a sequence of uncertain signal
. Fig. 3 illustrates the degraded signal
and its filtering estimate
vs. for the white Gaussian observation noise
0,0.3, provided that the desired value
10, 10, 0.0001 and 1. From Fig. 3,
it can be seen that the sequence of filtering estimates
is closer to the desired value of 10 than the
degraded signal . Fig. 4 illustrates the estimate
of the control input vs. for the white
Gaussian observation noise 0,0.3, provided
that 10, 10, 0.0001 and 1.
Fig. 5 illustrates the estimate of the
exogenous input vs. for the white Gaussian
observation noise 0,0.3, provided that
10, 10, 0.0001 and 1. From Fig. 4
and 5, it follows that fluctuations in the sequence of
the estimate of the exogenous input are
much smaller than those in the sequence of the
estimate of the control input . Table 1
shows the mean square values (MSVs) of the
tracking errors ,
and
, , 11200, by the
H-infinity tracking controller of Theorem 1 and the
robust RLS Wiener filter of Theorem 2 for 10
and 0.01, provided that 10,
0.0001 and 1. In this case, the observation
noise is subject to 0,0.1, 0,0.3,
0,0.5, 0,1 and 0,5. The MSV of the
tracking errors is less than that of the
tracking errors for each observation
noise. This indicates that the filtering estimate
accurately tracks the desired value in comparison
with . As the variance of the white Gaussian
observation noise increases, the MSVs of the
tracking errors and
become small gradually. For 10 and 0.01,
the MSVs of the tracking errors are
almost the same for each observation noise.
Similarly, for 10 and 0.01, the MSVs of
the tracking errors
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Fig. 3: Degraded signal
and its
filtering estimate vs. for white
Gaussian observation noise 0,0.3, provided
that the desired value 10, 10,
0.0001 and 1.
Fig. 4: Estimate of control input vs.
for white Gaussian observation noise 0,0.3,
provided that 10, 10, 0.0001 and
1.
are almost the same for each observation noise.
Table 2 shows the MSVs of the tracking errors
,
and ,
, 11200, by the H-infinity
tracking controller of Theorem 1 and the RLS
Wiener filter, [26], for 10 and 0.01,
provided that 10, 0.0001 and
1. From Tables 1 and 2, the tracking controller of
Theorem 1 combined with the robust RLS Wiener
filter of Theorem 2 is superior in tracking control
accuracy to the tracking controller of Theorem 1
combined with the RLS Wiener filter, [26], for each
observation noise. Table 3 shows the MSVs of the
tracking errors and by the
H-infinity tracking controller of Theorem 1 and the
robust Kalman filter, [24], for the observation noise
0,0.1, 0,0.3, 0,0.5, 0,1 and
Fig. 5: Estimate of exogenous input vs.
for white Gaussian observation noise 0,0.3,
provided that 10, 10, 0.0001 and
1.
Table 1. Mean-square values of tracking errors
,
and ,
, 11200, by H-infinity
tracking control algorithm of Theorem 1 plus robust
RLS Wiener filter of Theorem 2 for 10 and
0.01, provided that 10, 0.0001 and
1.
White
Gaussian
observation
noise
10 0.01
MSV of
tracking
errors
MSV of
tracking
errors
MSV of
tracking
errors
MSV of
tracking
errors
0,0.1 0.2469 0.1376 0.2473 0.1382
0,0.3 0.2126 0.1168 0.2108 0.1162
0,0.5 0.1765 0.0968 0.1755 0.0961
0,1 0.1042 0.0572 0.1049 0.0573
0,5 0.0100 0.0019 0.0099 0.0019
0,5. In the computation of the robust Kalman
filter, [24], the program uses the values 100,
10
01
,
11
and 10
01
. From
Table 3, the combination of the H-infinity tracking
controller of Theorem 1 and the robust Kalman filter,
[24], does not track the desired value at all or
diverges.
EXAMPLE 2
Consider linear discrete-time systems for the
observation and multi-input state equations of the
F16 aircraft, [30]. The angle of attack , the rate
of pitch , and the elevator angle of deflection
constitute the state vector
, ,
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Table 2. Mean-square values of tracking errors
,
and ,
, 11200, by H-infinity
tracking control algorithm of Theorem 1 plus RLS
Wiener filter, [26], for 10 and 0.01,
provided that 10, 0.0001 and
1.
White
Gaussian
observation
noise
10 0.01
MSV of
tracking
errors
MSV of
tracking
errors
MSV of
tracking
errors
MSV of
tracking
errors
0,0.1 2.0878e+
003 282.4114 49.9650 51.4861
0,0.3 671.1428 90.6103 49.9649 51.3064
0,0.5
321.3791
42.9632
49.9647
51.2206
0,1 58.7219 8.1504 49.9649 50.9183
0,5 4.1633 0.8318 49.9650 50.4742
Table 3. Mean-square values of tracking errors
,
and ,
, 11200, by H-infinity
tracking control algorithm of Theorem 1 plus robust
Kalman filter, [24], for 10 and 0.01,
provided that 10, 0.0001 and
1.
White
Gaussian
observation
noise
10 0.01
MSV of
tracking
errors
MSV of
tracking
errors
MSV of
tracking
errors
MSV of
tracking
errors
0,0.1 4.8511e+
017
3.7728e+
018
Divergence
Divergence
0,0.3 49.3066 9.5687
Divergence
Divergence
0,0.5
46.2886
9.3179
Divergence
Divergence
0,1 46.1339 9.4984
Divergence
Divergence
0,5 46.0462 9.3938 8.4403e+
235
7.0283e+
238
, . The system output, ,
corresponds to the angle of attack . In (36),
is the control input and is the
exogenous input. The desired value of is set to
0.1. The LTI system in (36) is
observable and controllable.
,,
100
,
,
1AΓ,
,
A0.906488 0.0816012 0.0005
0.0741349 0.90121 0.0007083
0 0 0.132655 ,
G0.00150808 0.00951892
0.0096 0.00038373
0.867345 0 ,
Γ0
0
1,,
0.3
(36)
The order of the AR model in (20) is 10. We
assume that the degraded observed value and
the degraded signal are generated by the
following observation and state equations.
,
,
1
Γ,
,
ΔA,
ΔC,
ΔA000
0 0.05∗ 0
000
,
ΔC0.03∗ 0 0,
(37)
The robust RLS Wiener filter in Theorem 2
computes the filtering estimate in (28) to
obtain the estimate of in (15). A total of
351 datasets are used to calculate the expected
values 1, , and
for and , respectively. Fig. 6
illustrates the degraded signal
and
its filtering estimate vs. for the
white Gaussian observation noise 0,0.3,
provided that the desired value 0.1,
10, 0.0001 and 1. From Fig. 6,
the sequence of the filtering estimates is closer
to the desired value of 0.1 than the degraded
signal . Fig. 7 illustrates the estimate of
the control input vs. for the white Gaussian
observation noise 0,0.3, provided that
0.1, 10, 0.0001 and 1. Fig.
8 illustrates the estimate of the exogenous
input vs. for the white Gaussian
observation noise 0,0.3, provided that
0.1, 10,
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Fig. 6: Degraded signal
and its
filtering estimate vs. for white
Gaussian observation noise 0,0.3, provided
that the desired value 0.1, 10,
0.0001 and 1.
Fig. 7: Estimate of control input vs.
for white Gaussian observation noise 0,0.3,
provided that 0.1, 10,
0.0001 and 1.
0.0001 and 1. From Fig. 7 and Fig. 8,
it can be seen that the amplitude of the exogenous
input sequence is very small compared with
that of the control input sequence. Table 4
shows the MSVs of the tracking errors ,
and , ,
11200, by the H-infinity tracking controller
of Theorem 1 and the robust RLS Wiener filter of
Theorem 2 for 10 and 0.05, provided that
0.1, 0.0001 and 1. Here,
the observation noise is subject to 0,0.1,
0,0.3, 0,0.5, 0,1 and 0,5. The
MSV of the tracking errors is smaller
than that of the tracking errors for each
observation noise. This indicates that the filtering
estimate tracks the desired value more
Fig. 8: Estimate of exogenous input vs.
for white Gaussian observation noise 0,0.3,
provided that 0.1, 10,
0.0001 and 1.
Table 4. Mean-square values of tracking errors
,
and ,
, 11200, by H-infinity
tracking control algorithm of Theorem 1 plus robust
RLS Wiener filter of Theorem 2 for 10 and
0.05, provided that 0.1,
0.0001 and 1.
White
Gaussian
observation
noise
10 0.05
MSV of
tracking
errors
MSV of
tracking
errors
MSV of
tracking
errors
MSV of
tracking
errors
0,0.1 0.1161 0.0095 0.1028 0.0097
0,0.3 0.1318 0.0166 0.1246 0.0164
0,0.5 0.1426 0.0175 0.1524 0.0182
0,1 0.1684 0.0186 0.1383 0.0181
0,5 0.3270 0.0159 0.2600 0.0148
accurately than . For both 10 and 0.05,
the MSV of the tracking errors for each
observed noise is almost identical. As the variance
of the white Gaussian observation noise increases,
the MSV of the tracking errors tends to
increase gradually in both 10 and 0.05 for
0,0.1, 0,0.3 and 0,0.5. Concerning
Fig 6, from Table 4, for the white Gaussian
observation noise
0,0.3, the MSV of the tracking
errors is 0.0166. The MSV of the
tracking errors is 0.1318. This indicates
that the proposed H-infinity tracking control
algorithm improves tracking accuracy by using the
robust RLS Wiener filter in Theorem 2.
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5 Conclusion
This paper developed the H-infinity tracking
control technique combined with the robust
RLS Wiener filter for linear discrete-time
stochastic systems with uncertainties. For linear
discrete-time stochastic systems (2) with
uncertainties, based on the separation principle of
control and estimation, satisfies (12) along
with (10) and (11). The filtering estimate of
is updated from 1 by (28) with the
information of the estimate 1 of 1,
the degraded observed value and the filtering
estimate 1 of the degraded state 1.
The estimate of in (15) uses the filtering
estimate by the robust RLS Wiener filter.
Numerical simulation examples have
demonstrated the characteristics of tracking control
using the H-infinity tracking controller of Theorem
1 and the robust RLS Wiener filter of Theorem 2 in
linear discrete-time stochastic systems with
uncertainties. Tables 1 and 2 show that the tracking
controller of Theorem 1 with the robust RLS Wiener
filter of Theorem 2 is superior in tracking control
accuracy to the tracking controller of Theorem 1
with the RLS Wiener filter for the white Gaussian
observation noise 0,0.1, 0,0.3,
0,0.5,0,1 and 0,5. In addition, from
Table 3, the MSVs of the tracking errors
by the H-infinity tracking controller of
Theorem 1 with the robust Kalman filter show that
the tracking technique either fails to track the
desired value at all or diverges for the observation
noise. In the example for the F16 aircraft, Table 4
shows that the MSV of the tracking errors
is less than that of the tracking errors
for each observation noise. This indicates that
the filtering estimate tracks the desired value
more accurately than .
In particular, as the uncertainties in the system
and observation matrices increase, the accuracy of
the estimates for and is numerically required. In
EXAMPLE 1, in the calculation of ,
is approximated by
∑
instead of
∑
.
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Contribution of Individual Authors to the
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Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received.
Conflict of Interest
The author has no conflict of interest to declare.
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