H-Infinity Tracking Controller for Linear Discrete-Time Stochastic
Systems with Uncertainties
SEIICHI NAKAMORI
Professor Emeritus, Faculty of Education,
Kagoshima University,
1-20-6 Korimoto, Kagoshima 890-0065,
JAPAN
Abstract: - For linear discrete-time stochastic systems with uncertainties, this paper proposes a tracking control
method based on the H-infinity tracking controller and the robust recursive least-squares (RLS) Wiener filter.
In linear discrete-time deterministic systems without input and observation noises, the equations for the
quantity () with the components of the control and exogenous inputs are as previously described. In Section
2, we show that () satisfies the same equations for linear discrete-time stochastic systems with white input
and observation noises as for deterministic systems, based on the separation principle of control and estimation.
The results show that the H-infinity tracking control algorithm for linear discrete-time stochastic systems is the
same as that for linear discrete-time deterministic systems. The filtered estimate () of the system state ()
is used to compute the estimate () of (). The robust RLS Wiener filter of Theorem 2 computes the filtered
estimate () of the system state () for degraded stochastic systems with uncertainties in the system and
observation matrices. () is updated from (1) with the degraded observed value (), the filtered
estimate (1) of the degraded state (1), and the estimate (1) of (1).
Key-Words: - H-infinity tracking controller, control input, exogenous input, robust recursive least-squares
Wiener filter, discrete-time stochastic systems, uncertain parameters, separation principle.
Received: March 21, 2022. Revised: October 19, 2022. Accepted: November 16, 2022. Published: December 31, 2022.
1 Introduction
Linear quadratic Gaussian (LQG) control issues
have been thoroughly studied, for example, in [1–6].
The discrete-time LQG control problem is described
in [2] for stochastic systems with input and
observation noises. The optimal control law for the
stochastic systems is the same as that for the
discrete-time deterministic systems without input
and observation disturbances. For state feedback,
the control law uses the estimate of the state
computed by the Kalman filter. In [6], the discrete-
time LQG control that minimizes Massey’s directed
information from the plant observation output to the
control input is studied to achieve the required
control performance. The tracking control algorithm
based on LQG is described, for instance, in [7-11].
A real-time trans-scale LQG tracking control
algorithm for discrete-time stochastic systems is
described in [11] and is based on wavelet packet
decomposition (WPD). The stochastic systems in
this case do not take into account the uncertain
parameters. The covariance matrices of the input
and observation noises are given. In [12], a
controller with output feedback is studied for
discrete-time stochastic systems with uncertainty
and missing observations. Parameter uncertainty is
bounded by the norm. The probability of the
occurrence of missing data assumes that it is known.
The problem is solved by linear matrix inequalities
(LMIs). Based on the disturbance observer, studies
in [13] propose a robust controller for linear
continuous-time uncertain systems with a time
delay. The observer parameters are determined by
the solution of LMIs. State feedback control is
treated in [13]. In [14], the H-infinity controller is
designed for the state-space model with uncertain
parameters in linear continuous-time stochastic
systems. For linear discrete-time uncertain systems
with nonlinear and unbounded uncertainties, the
robust controller is developed in [15]. The reduced-
order disturbance observer is shown, and the state-
feedback controller is designed based on the LMI
method. Subsection 5.3 of [16] describes the LMI
approach for state feedback quadratic stabilization
in linear continuous-time uncertain systems. The
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feedback gain for the control input is computed by
solving LMIs. [17] proposes a mixed 2/Passivity
controller for linear discrete-time uncertain
stochastic systems. Some sufficient conditions
derived by Lyapunov theory are converted into
LMIs. In [18], the iterative tracking controller is
designed under the influence of an unknown
disturbance with constrained frequency and
parameter variations. State tracking control in
uncertain stochastic time-varying delay systems is
developed. [19] proposes the uncertainty and
disturbance estimator for robust tracking control of
the reference model in linear continuous-time
uncertain stochastic systems.
In linear discrete-time degraded stochastic
systems with uncertain parameters, this paper
proposes an H-infinity tracking control method
based on the H-infinity tracking controller [20] and
the robust recursive least-squares (RLS) Wiener
filter [21], [22]. The robust RLS Wiener fixed-point
smoother is also presented in [21] and [22]. The
robust RLS Wiener estimators in [21] are designed
for signal estimation. The robust RLS Wiener
estimators [22] estimate the state of the system
using the degraded observations generated by the
state and observation equations with uncertainties.
Usually, the robust filter estimates the state of the
system with uncertainties using the degraded
observations [23]. The robust Kalman filter [24] for
uncertain systems assumes multiplicative noise and
norm-bounded time-varying uncertainty. For linear
discrete-time deterministic systems without input
and observation noises, () with control and
exogenous input components satisfies (12), (10),
and (11) in [20]. Based on the separation principle
of control and estimation, it is demonstrated in
Section 2 that () satisfies the same equations for
linear discrete-time stochastic systems with white
input and observation noises as for deterministic
systems. As a result, the tracking control algorithm
for linear discrete-time stochastic systems with
white input and observation noises is the same as for
linear discrete-time deterministic systems. The
filtered estimate () of the system state () is
used to get the estimate () of (). From the
state and observation equations (12) with uncertain
parameters, the robust RLS Wiener filter of
Theorem 2 computes the filtered estimate (),
which is used as the filtered estimate of the system
state () for the state equation (1). The robust RLS
Wiener filter updates () from (1) with the
degraded observed value (), the filtering estimate
(1) of the degraded state (1), and the
estimate (1) of (1) . Then, the
computation of the estimate () of () in
Theorem 1 uses the filtered estimate () of the
state () by the robust RLS Wiener filter.
In Section 4, a numerical simulation example
compares the tracking control accuracy between the
H-infinity tracking controller of Theorem 1 plus the
robust RLS Wiener filter of Theorem 2 and the H-
infinity tracking controller of Theorem 1 plus the
RLS Wiener filter [25]. For the white Gaussian
observation noises (0, 0.12), (0, 0.32),
(0, 0.52), and (0,1), the H-infinity tracking
controller of Theorem 1 plus the robust RLS Wiener
filter of Theorem 2 provides better tracking control
accuracy than Theorem 1's H-infinity tracking
controller plus the RLS Wiener filter [25].
2 H-Infinity Tracking Control
Problem in Linear Discrete-Time
Stochastic Systems
Let the nominal state-space model in linear discrete-
time stochastic systems be given by (1).
() = () + (), () = (),
(+ 1)=()+()+(),
=[12],()=1()
2(),
(0)=,[()()]=(),
[()
()]=(),
[()()]= 0, [(0)()]= 0,
(1)
Here, () is the state vector; () is
the input vector; and () is the signal vector.
The control and exogenous input vectors are,
respectively, 1()1, and 2()2,
1+2=. Both the input noise () and
the observation noise () are zero-mean
white noises that are mutually uncorrelated.
() denotes the Kronecker delta function.
represents the × observation matrix. stands for
the × input matrix for (), while stands for
the × input matrix for (). The auto-
covariance functions of the input and observation
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noises are given in (1). Let the expectation of
()2
2 be given by (2), where () represents the
performance output [26].
[()2
2] = [(()())()
×()()+1
()()1()]
(2)
Here, () is the desired value and the symmetric
matrices () and () are positive definite. The
H-infinity optimal tracking control problem is to
find the control input 1() and exogenous input
2() in the disturbance attenuation condition (3)
for the minimum value of [20]. > 0 is referred
to as the constant disturbance attenuation level.
=0
[(()())()()()]
+
=0
[1
()()1()]
2
=0
[2
()2()]
(3)
An equivalent transformation of the H-infinity
tracking control problem for a finite horizon is a
two-player, zero-sum linear quadratic dynamic
game [27], [28]. That is, given 2, we investigate
the minimax problem of minimizing the value
function (,,) about 1() and maximizing
(,,) about 2().
=0
(4)
Here, the worst-case disturbance 2() is the
exogenous input, and 1() is the control input. (4)
is expressed as (5) by introducing () =
󰇩() 0
022×2󰇪.
=0
(5)
In the value function (5), the discount factor is 1.
() is represented as (6).
()=(, 0)+
1
=0
(,+ 1)
× (()+())
=(, 0)+1
=0
(1)(,+ 1)
× (()+())
1() = 󰇥1,0 ,
0, < 0,
(,)=
, 0 <,
,=.
(6)
Here, (,) represents the state-transition matrix,
and 1() denotes the discrete-time unit step
sequence. Substituting (6) into (5), we get (7).
(,
,
) =
=0
[(()(, 0)
1
=0
(1)(,+ 1)(()
+())()(()
(, 0)1
=0
(1)(,+ 1)
× (()+())) + ()()()]
(7)
Let () be the vector with the components of
optimal control and exogenous inputs. By the
calculus of variations [20], the necessary condition
for () to minimize the value function (7) about
1() and maximize (7) about 2() is satisfied by
(8).
()()
+1(1)1(1)
=0
=0
×(,+ 1)()(,+ 1)()
=1
=0
(1)(,+ 1)
×()(()(, 0))
(8)
By introducing
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(,)
=
=+1 (,+ 1)()(,+ 1),
0
=+1
(,+ 1)()(,+ 1),
0,
(9)
and
(+ 1)=
=+1 (,+ 1)()
× ((, 0)()),
(10)
the optimal () satisfies
()() +
=0
(,)()
=(+ 1).
(11)
Similar to [20], the sufficient condition for the value
function (,) to be minimal for () and
maximal for () is ()() + (,)>
0. In [20], the integral equation is obtained instead
of (11) for linear continuous-time systems. Recently
the analysis of integral equations has been studied in
[29], [30].
We should note that the results obtained in (9)–
(11) for the H-infinity tracking control problem
regarding the state-space model (1) are the same as
the equations in (20) for the H-infinity tracking
control problem in linear deterministic systems.
Therefore, the H-infinity tracking control algorithm
in [20] is equivalent to the H-infinity tracking
control algorithm for the state-space model (1).
Theorem 1 presents the H-infinity tracking control
algorithm obtained from (9)–(11). The Kalman filter
generates the filtered estimate for the discrete-time
LQG tracking control algorithm in Section 2 of [10].
The LQG tracking control problem is solvable based
on the separation principle of control and
estimation. In [10], (11) and (12) compute the
filtered estimate while taking the term of the control
input into account. The filter gain in the Kalman
filter is calculated by (13)–(15) in [10].
Suppose the degraded system of the nominal
system (1) is given by (12).
()=()+(),() =
()(),
() = +(),
[()()]=(),
(+ 1) =
()() + () + (),
() = +(),
[()()]=(),
[()()], = 0,
[()()]= 0, [()()]= 0,
[()()]= 0, [()()]= 0,
[(0)
()]= 0, [(0)
()]= 0
(12)
In (12), the system matrix and the observation
matrix in (1) are replaced with the degraded
system matrix
() and the degraded observation
matrix
(), respectively. It is assumed that ()
and () are uncorrelated with the input noise
() and the observation noise (). The initial
system state (0) is a random vector uncorrelated to
both system and measurement noise processes.
Under these assumptions, the separation principle of
control and estimation can be applied to solve the
H-infinity tracking control problem. In other words,
the H-infinity tracking control algorithm in Theorem
1 for the nominal state-space model (1) utilizes the
robust RLS Wiener filtered estimate in Theorem 2
for the degraded systems with uncertain parameters.
3 H-Infinity Tracking Controller and
Robust RLS Wiener Filter
Fig.1 illustrates the structure of the H-infinity
tracking controller of Theorem 1 and the robust RLS
Wiener filter of Theorem 2.
Fig. 1: Structure of the H-infinity tracking controller
of Theorem 1 and robust RLS Wiener filter of
Theorem 2.
Theorem 1 presents the H-infinity tracking control
algorithm for the estimates of the control input,
1(), and the exogenous input, 2(), with the
filtered estimate () for (). The estimates of
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1() and 2() are represented by 1() and
2(), respectively. The robust RLS Wiener filter
of Theorem 2 calculates the filtered estimate ()
of the state () with the degraded observed value
().
Theorem 1 Assume that () is expressed as
() = 󰇩() 0
022×2󰇪 and let () be the
desired value. Assume that () has the
components of the control input 1() and the
exogenous input 2() as (1).
() = 1()
2()
(13)
The estimate () of () is then calculated by
(14)–(17) regarding the nominal state-space model
(1). In (14), 1() is the estimate of the control
input 1(), and 2() is the estimate of the
exogenous input 2().
() = 1()
2()
(14)
() = 1()()1[(+ 1)
×(1()(+ 1))1+()]
()()+1()()1
×(+ 1)(1()(+ 1))1
×1()(+ 1)+(+ 1)

()()+
1
()
(
)
1
×()()
(15)
()=(+ 1)
× (1()(+ 1))1
(),(+ 1)= 0
(16)
()=(+ 1)
× (1()(+ 1))1
×1()(+ 1)
+
(+ 1)+
()(),(+ 1)= 0
(17)
For the state (), we utilize the filtered estimate
() in (15). The robust RLS Wiener filtering
algorithm of Theorem 2 calculates () using the
degraded observed value () in (12), the filtering
estimate (1) of the degraded state (1),
and the estimate (1) of (1). and are
calculated in the time-reversed direction from
time =+ 1 until the steady-state values,
and , respectively, are reached. The estimate
() of () is calculated by (15) using and .
(+ 1) and (+ 1) in (15) are replaced with
their stationary values, and , respectively. From
the above considerations, the H-infinity tracking
control algorithm of Theorem 1 and the robust RLS
Wiener filtering algorithm of Theorem 2 adhere to
the separation principle of control and estimation.
Let's now quickly review the robust RLS Wiener
filter [21], [22]. Assume that an AR model of order
is used to fit the degraded signal sequence of
().
() = 1(1) 2(2)

() + (),
[()()] = ()
(18)
The state vector () can be used to represent ()
as follows:
() = 󰆾(),
() =
1
()
2()
1()
()
=
()
(+ 1)
(+2)
(+1)
,
󰆾=[×0 0 0 0]
(19)
In light of this, the state equation for the state vector
() is given by
(+ 1) = 󰆾() + (),
[()()]=
(),
󰆾=
0×00
0 0 ×0
0 0 0 ×
12 1
,
=
0
0
0
×
,() = (+).
(20)
The auto-covariance function
(,) of the state
vector ()has the semi-degenerate functional form
of
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(,) = ()(),0 ,
()(),0 ,,
() =
,() =

(,).
(21)
Based on the wide sense stationarity of the auto-
covariance function
(,) = [()()] for the
degraded signal (), (22) provides the auto-
variance function
(,) of the state vector ().
(,) =
()
(+ 1)
(+1)
×[()(+ 1) (+1)]
=
(0) (1) (+ 1)
(1) (0) (+ 2)
(2) (3) (1)
(1) (2) (0)
(22)
Using (), 0, the Yule-Walker equation
for the AR parameters , 1, satisfies
(,)
1
2
1
=
(1)
(2)
(1)
()
,
(,) =
(0) (1) (1)
(1) (0) (2)
(2) (3) (1)
(1) (2) (0)
.
(23)
The cross-covariance function of the state vector
() with () is represented by (,) =
[()()]. (,) has the form of
(,) = ()(),0 ,
() = A,() = A(,)
(24)
with the system matrix for the state vector () in
(1).
Theorem 2 presents the robust RLS Wiener
filtering algorithm based on [22]. The filtered
estimate () is updated from (1) with the
degraded observed value () in (12), the filtering
estimate (1) of the degraded state (1),
and the estimate (1) of (1) . In
comparison with the robust RLS Wiener filter in
[22], the term (1) is inserted on the right-
hand side of (26) for the filtered estimate () of
().
Theorem 2 Suppose (1) provides the state-space
model for the state () in linear discrete-time
stochastic systems. Assume that the sequence of the
degraded signal () is fitted to the AR model of
order . Assume that (22) represents the variance
(,) of the state () concerning the degraded
signal (). Let (24) represent the cross-variance
(,) of the state vector (), for the signal
() in (1), with the state () in (19). Let denote
the variance of the white observation noise ().
Thus, for the filtered estimate () of the state
(), (25)–(31) constitute the robust RLS Wiener
filtering algorithm.
Filtered estimate of the signal (): ()
() = ()
(25)
Filtered estimate of the state (): ()
() = (1) + (1)
+()(()󰆾󰆾(1)),
(0) = 0
(26)
Filter gain for () in (26): ()
() = [
(,)(1)󰆾󰆾]
× {+󰆾[
(,)󰆾0(1)󰆾]󰆾}1
(,) = (,)󰆾
(27)
Filtered estimate of (): ()
() = 󰆾(1)
+()(()󰆾(1)),
(0) = 0
(28)
Filter gain for () in (28): ()
() = [
(,)󰆾󰆾0(1)󰆾󰆾]
× {+󰆾[
(,)󰆾
0
(1)󰆾]󰆾}1
(,) = (,)󰆾
(29)
Auto-variance function of (): 0() =
[()()]
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0() = 󰆾0(1)󰆾
+()󰆾[
(,)󰆾
0
(1)󰆾,
0(0) = 0
(30)
Cross-variance function of () with ():
() = [()()]
() = (1)󰆾
+()󰆾[
(,)󰆾0(1)󰆾,
(0) = 0
(31)
The tracking control algorithm of Theorem 1
calculates () by (15) using the filtered estimate
() calculated by (26).
The flowchart in Fig. 2 is obtained by combining
the robust H-infinity tracking controller of Theorem
1 with the RLS Wiener filter of Theorem 2.
Fig. 2: Flowchart created using the H-infinity
tracking controller from Theorem 1 and the robust
RLS Wiener filter from Theorem 2.
Section 4 presents a numerical simulation
example of the tracking control characteristics of the
H-infinity tracking controller using the estimate
() of () by the robust RLS Wiener filter of
Theorem 2 or the RLS Wiener filter [25].
4 A Numerical Simulation Example
Consider the observation and state equations given
by
()=()+(),() = (),
= [0.95 0.4],
(+ 1) = () + () + (),
() = 1()
2(),
() = 1()
2(),=󰇣0.05 0.95
0.98 0.2 󰇤,
=󰇣0.952 0
0.2 1󰇤,=󰇣0.952
0.2 󰇤,
[()()]=(),
[()()] = 0. 5
2
().
(32)
In (32), 1() is the control input, and 2() is the
exogenous input. Consider the following
observation and state equations, assuming they
produce degraded observations and a degraded
signal.
()=()+(),() =
()(),
() = +(),
() = [0.3 () 0],
(+ 1) =
()+() + (),
()=1()
2(),(0) = 󰇣2.3
2.5󰇤,
() = +(), ,
() = 0.1 () 0
0 0.2 ()
[()()]= 0, [()()]= 0,
[()
()]= 0, [()
()]= 0.
(33)
Here, "()" refers to a MATLAB or GNU Octave
function that generates uniformly distributed
random numbers in the range (0,1) . In (33),
conditions such as norm-bounded uncertainty [18]
are not imposed on the uncertain matrices ()
and (). The robust RLS Wiener filtering
algorithm of Theorem 2 does not use the
information on the uncertain matrices () and
() at all. The robust RLS Wiener filter of
Theorem 2 computes the filtered estimate () in
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(26) to get the estimate () of () in (15). Given
the desired value ()=10 , =10, =
0.0001, and ()= 1, Fig. 3 illustrates the signal
()=() and its filtered estimate () by the
H-infinity tracking controller of Theorem 1 and the
robust RLS Wiener filter of Theorem 2 vs. for the
Fig. 3: ()=() and its filtered estimate ()
vs. for white Gaussian observation noise
(0, 0.32), given the desired value ()=10 ,
=10, = 0.0001, and ()= 1.
white Gaussian observation noise (0, 0.32). From
Fig. 3, the sequence of the filtered estimates () is
closer to the desired value of 10 than the signal
(). Fig. 4 illustrates the estimate 1() of the
control input 1() vs. for the white Gaussian
observation noise (0, 0.32), given ()=10 ,
=10, = 0.0001 , and ()= 1. Fig. 5
illustrates the estimate 2() of the exogenous input
2() vs. for the white Gaussian observation
noise (0, 0.32), given ()=10, =10, =
0.0001 , and ()= 1. In Figs 4 and 5, the H-
infinity tracking controller of Theorem 1 and the
robust RLS Wiener filter are used. Figs. 4 and 5
show that the 2() sequence's amplitude is
considerably smaller than that of the 1()
sequence.
Table 1 shows the mean-square values (MSVs)
of the tracking errors ()(), ()=(),
and ()(), ()=(), 11200, by
the H-infinity tracking controller of Theorem 1 and
the robust RLS Wiener filter of Theorem 2 for
=10 and = 0.01 , given ()=10 , =
0.0001, and ()= 1. Here, the observation noises
are (0, 0.12), (0, 0.32), (0, 0.52), (0,1) and
(0, 52). The MSV of the tracking errors ()
() is fairly smaller than the MSV of the tracking
errors () () for each observation noise. This
indicates that the filtered estimate () tracks the
desired value with high accuracy. For =10 and
= 0.01, the MSVs of the tracking errors ()
() are almost the same for each observation noise.
Fig. 4: Estimate 1() of control input 1() vs.
for white Gaussian observation noise (0, 0.32),
given the desired value ()=10, =10, =
0.0001, and ()= 1.
Fig. 5: Estimate 2() of exogenous input 2() vs.
for white Gaussian observation noise (0, 0.32),
given the desired value ()=10, =10, =
0.0001, and ()= 1.
Similarly, for =10 and = 0.01, the MSVs of
the tracking errors ()() are almost the same
for each observation noise. The MSVs of ()
() and ()() are minimums for the white
Gaussian observation noise (0, 52), respectively.
Table 2 shows the MSVs of the tracking errors
()(), ()=()and ()(),
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()=(), 11200 , by the H-infinity
tracking controller of Theorem 1 and the RLS
Wiener filter [25] for =10 and = 0.01, given
()=10 , = 0.0001 , and ()= 1. Table 2
shows that the tracking errors ()() and
()() diverge for both =10 and = 0.01
in the white Gaussian observation noises (0, 0.12)
and (0, 0.32). The MSVs of the tracking errors
()() and ()() are extremely large in
both
Table 1. Mean-square values of tracking errors
()(), ()=()and ()(),
()=(), 11200, by H-infinity
tracking control algorithm plus robust RLS Wiener
filter for =10 and = 0.01, given ()=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, 0.12)
0.6507
0.0690
0.6485
0.0677
(0, 0.32)
0.6583
0.0779
0.6605
0.0773
(0, 0.52)
0.6114
0.0810
0.6215
0.0816
(0,1)
0.6510
0.0879
0.5806
0.0814
(0, 52)
0.2914
0.0460
0.2894
0.0465
Table 2. Mean-square values of tracking errors
()(), ()=()and ()(),
()=(), 11200, by H-infinity
tracking control algorithm plus RLS Wiener filter
[25] for =10 and = 0.01, given ()=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, 0.12)
Diver-
gence
Diver-
gence
Diver-
gence
Diver-
gence
(0, 0.32)
Diver-
gence
Diver-
gence
Diver-
gence
Diver-
gence
(0, 0.52)
2.2315e+198
4.5986e+198
9.5268e+198
1.9712e+199
(0,1)
7.9888 0.3956
7.7168
0.3835
(0, 52)
0.4185
0.0411
0.4200
0.0411
=10 and = 0.01 for the white Gaussian
observation noise (0, 0.52). The MSV of the
tracking errors ()() by the tracking
controller of Theorem 1 and the robust RLS Wiener
filter of Theorem 2 is smaller than the MSVs by the
tracking controller of Theorem 1 and the RLS
Wiener filter [25] for the white Gaussian
observation noises (0,1) and (0, 52) when
=10 and = 0.01 . In the observation noise
(0,1), for =10 and = 0.01, the MSV of the
tracking errors ()() in Table 1 is smaller
than the MSV of the tracking errors ()() in
Table 2. In the case of white Gaussian observation
noise (0, 52), for =10 and = 0.01, the MSV
of the tracking errors ()() in Table 1 is
almost the same as the MSV of the tracking errors
()() in Table 2. For the observation noises
(0, 0.12), (0, 0.32), (0, 0.52), and (0,1), the
H-infinity tracking controller of Theorem 1 and the
robust RLS Wiener filter of Theorem 2 are superior
in tracking control accuracy to the H-infinity
tracking controller of Theorem 1 and the RLS
Wiener filter [25].
5 Conclusion
This study proposed a tracking control technique
based on the H-infinity tracking controller of
Theorem 1 and the robust RLS Wiener filter of
Theorem 2. In previous research, it has been shown
that () with control and exogenous input
components satisfies (9), (10), and (11) in linear
discrete-time deterministic systems. For the
stochastic systems (1), based on the separation
principle of control and estimation, () also
satisfies (9), (10), and (11). As a result, the tracking
control algorithm of Theorem 1 is applied to the
linear state-space model (1). For the degraded
stochastic system (12), (26) in the robust RLS
Wiener filter of Theorem 2 updates the filtered
estimate () of () from (1) with the
degraded observed value (), the filtered estimate
(1) of the degraded state (1), and the
estimate (1) of (1). Estimating () of
() in (15) uses the filtered estimate () of the
state () by the robust RLS Wiener filter.
WSEAS TRANSACTIONS on CIRCUITS and SYSTEMS
DOI: 10.37394/23201.2022.21.26
Seiichi Nakamori
E-ISSN: 2224-266X
246
Volume 21, 2022
The numerical simulation example compares the
tracking control accuracy of the proposed method
with that of the technique based on the H-infinity
tracking controller of Theorem 1 and the RLS
Wiener filter. As a result, the tracking controller of
Theorem 1 and the robust RLS Wiener filter of
Theorem 2 provide higher tracking control accuracy
for the white Gaussian observation noises
(0, 0.12), (0, 0.32), (0, 0.52), and (0,1). For
=10 and = 0.01 , the MSV of the tracking
errors ()() by the tracking controller and
the robust RLS Wiener filter is almost the same as
the MSV by the H-infinity tracking controller of
Theorem 1 and the RLS Wiener filter in the
observation noise (0, 52), respectively.
A future task is to design an H-infinity tracking
controller with a robust RLS Wiener filter that
estimates the degraded state in linear discrete-time
uncertain systems.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
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 for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
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