IEEE Systems, Man and Cybernetics Magazine - October 2023 - 30

delay compensation strategy could be studied based on
the discrete linear output feedback control system. Consider
the following system:
xk AA Kx kB Bk uk
()=-^h
() ^^ ^^hh
++ +
()
yk Cx kdk
o
uk fy k
dh , h @ is the bounded stochastic output
k ! 6
oo o
12
() ()
= ^h
+=1 TT()hh (9)
(10)
(11)
where (( ))fy k is the feedback strategy to be determined
and
delay. Time-varying system uncertainty is considered as
T A and
the form of
(( ), ())( )( ,)
() () ,
TT T
TT #$
= c
T
6
,,
AK BK Ek HH
kk Ik 0
AB
(12)
where EHA and HB are time-invariant matrices or values
with proper dimensions to describe the amplitude of the
uncertainty and c is a positive scalar to determine the
size of the uncertainty (mainly used to describe the controller's
ability to resist uncertainty). The controllability
and observability of the output feedback system (A, B, C)
are also presupposed.
The previously introduced model describes a linear output
feedback control system with bounded output delay
and bounded model uncertainty. This study aims to design
() ()
uk fy k
= ^h to actively compensate for the negative
effect brought by the delay and model uncertainty while
stabilizing the system.
Delay Compensation Strategy and
Corresponding Controller
In this section, the control strategy to compensate the output
delay is introduced under known and unknown output
delay cases, and the entire system's stability is studied.
TB , which are assumed to be bounded and in
The algorithm to design the stabilized controller and
observer gain is also introduced in this section.
Prediction With Known Delay: Synchronized
Predictor-Observer Method
By utilizing time stamps, it is a plausible assumption that the
output delay can be measured through an output feedback
process. After a certain delay length dk
o
k
o
is attached to each
frame of the output data, the prediction could be conducted
precisely with finite memory of past control input, shown as
d -1
yk Ay kC ABuk didd i
i
r () ()
=+ -+
-k
o
k
o
k
o
=0
where
y(k) is the delayed output. From the previously mentioned
prediction function, it is quite clear that the system
states could be perfectly predicted if the system was linear
time invariant and the system matrix was fully attainable.
However, as introduced in (12), the real system contains
uncertainty, where the prediction error is generated.
Observer-based control [30] could be introduced here to
compensate the prediction error and could perfectly couple
with the predictor and form the celebrated " predictor-
observer " structured controller, shown as
() () () ^ht
tt
xk 1 Ax kBuk ALAy kCxk
+= ++ -
-
ddk
o
k
o
r () () . (14)
It could be figured from (14) that different from the conventional
Luenberger observer, the predictor uses the
delayed observed state to generate a prediction of system
states at the current time step and feed it to the observer,
and that is the reason why we call it the " synchronized " predictor-observer
method hereafter. The process mentioned in
the preceding could also be indicated as Figure 4. Additionally,
as introduced in (12), the real system contains uncertainty,
where the prediction error is generated. Therefore, the
dynamics of the system with the predictor-observer need
Controller
Controller Received
x(k)
x(k + 1) = Ax(k) + Bu(k) + Ado
d
y(k) = Ado
o
k -1
k y(k) + CAdo
i = 0
∑
y(k) = Cx(k - do
x(k - do
Sensor
Output Sent
k)
k)
Delayed (Unsynchronized) Output
x(k)
do
k
k -i-1Bu(k - do
Prediction of " Synchronized " Output
k + i)
x(k) is an
Estimation of x(k)
k LA-do
Time Axle
k (y-(k) - Cx(k))
| ^h (13)
1
Figure 4. The prediction logic of the synchronized predictor-observer method.
30 IEEE SYSTEMS, MAN, & CYBERNETICS MAGAZINE October 2023
IEEE Systems, Man and Cybernetics Magazine - October 2023

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