Exact Command Tracking Control Computations without Integration
TAIN-SOU TSAY
Department of Department of Aeronautical Engineering
National Formosa University
64, Wen-Hua Road, Huwei, Yunlin, TAIWAN
Abstract: - A digital controller for exact command tracking control without integration is derived from a
periodic series. The ratios of adjacent values will be converged to unities after the output has tracked the
reference input command. Integration in control loop usually introduces phase lag to slow command response
and degrade performance.
Key-Words: - Exact command Tracking, digital control, Zero Steady-State Error
Received: July 9, 2021. Revised: November 6, 2021. Accepted: November 20, 2021. Published: January 3, 2022.
1 Introduction
For discrete-time unit feedback control systems, the
control sequences
)( S
jTG
are usually functions of
the difference between the sampled reference input
and output of the plant [1-3].
S
T
is the sampling
interval. They are linear control sequences. In this
literature, ratios of
))1(( S
TkG
to
)( S
kTG
of the
control sequences will be formulated as a nonlinear
function of the reference input command and the
output of the plant. The value of
)( S
kTG
is the
control effort of the plant at time interval between
S
Tk )1(
and
S
kT
. Thus, the considered system is
closed with
)( S
jTG
.The output of the plant will
track the reference input exactly after
))1(( S
TkG
/
)( S
kTG
converged to be unities. It
implies that
)( S
kTG
will be converged to a steady-
state value for a constant reference input applied.
The stability of the closed-loop system is guaranteed
by selecting the proper function of
ratios
))1(( S
TkG
/
)( S
kTG
. It will be proven that the
considered system with the proposed
)( S
kTG
is a
stable negative feedback control system.
2. Propose Method
A series with time period
S
T
[1-3] can be written as
in the form of
,....1,,..,3,2,1),( nnjjTGS
, (1)
where
)( S
kTG
represents a constant value between
time interval between
S
Tj )1(
and j
S
T
. For
simplicity, the representation of
)( S
kTG
will be
replaced by
)( jG
in following evaluations. The
ratios
)(/)1( jGjG
of the series are defined as
,.....1,,...,3,2,1),(/)1()( nnjjGjGjF
, (2)
Eq.(2) gives the value of
)1( nG
approaches to be
a constant value when the value of
)(nF
approaches
to be unity. Now, the problem for closing the
considered system with exact command tracking is
to find the formula of
)( jF
which is the function of
reference input command and output of the plant.
)1( nG
will be used as the input of the considered
system. Considering a possible series to close the
considered system, it is
)()(/)()1(
0
nGnYnRanG i
S
m
ii
; (3)
where
)(nR
is the reference input command and
)(nYS
is the non-zero sampled output of the plant Y
at the sampling interval
S
nT
. Assume that the
reference input command has been tracked by the
control effort
)( jG
, Eq.(3) becomes
)()1(
0
nGanG m
ii
; (4)
For steady-state condition,
)1( nG
approaches to
be a constant value. It gives
1
0
m
ii
a
.
Rearranging the Eq. (3) and taking the derivative of
it with respect to
)(/)( nRnYS
, we have
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i
S
m
iinRnYanF
)(/)()(
0
(5)
and
))(/)((/)( nRnYnF S
1
0
)(/)(
i
S
m
iinRnYia
(6)
The sufficient but not necessary condition for Eq.(6)
less than zero is
0
i
a
for
1)(/)( nRnYS
and
Eq.(5) is rewritten as in the form of
i
S
m
iinRnYanF
)(/)()(
0
(7)
0
i
a
will be used in following evaluations.
Negative value of Eq.(6) represents the closed-loop
system with Eq.(3) activated as a negative feedback
system around the equilibrium condition;
i.e.,
)()( nRnYS
. These statements will be
illustrated by the first order polynomial:
)()(/)()1()1( nGnYnRnG S
; (8)
where
satisfies constrains stated above and
becomes a adjustable parameter. The
ratios
)(nF
becomes
))(/)(/()1()( nRnYnF S
(9)
Taking the derivative of Eq. (9) with respect
to
)(/)( nRnYS
, we have
2
))(/)(/()1())(/)((/)( nRnYnRnYnF SS
(10)
For negative value of Eq.(10), the value of
must
be less than one. The suitability of the proposed
nonlinear digital controller is based upon this
characteristic. Fig.1 shows ratios
)(/)( nRnYS
versus
)(/)1()( nGnGnF
represented by Eq.(8) for
=0.9, 0.7, 0.5 and 0.3; respectively. Fig. 1 shows
that the value of
)(nF
is less than one for
)(nYS
greater than
)(nR
, then the value of
)1( nG
will be
decreased; and the value of
)(nF
is greater than one
for
)(nYS
less than
)(nR
, the value of
)1( nG
will
be increased. This implies that the controlled system
connected with Eq.(8) will be regulated to the
equilibrium point (
)(/)( nRnYS
=1) and gives a
negative feedback control system for deviation from
equilibrium point. One can adjust
to get desired
regulating characteristic. Certainly, other tracking
functions can be formulated and proposed also for
the considered system, if its derivative with respect
to
)(/)( nRnYS
is negative.
Fig.2 shows the connected system configuration in
which Eq.(8) and output of the nonlinear controller
are modified for negative control swing. The C(z) is
the digital compensation for better performance.
Eq.(8) is rewritten as
)(]))(/())(1[()1( nGYnYYRnG OSo
;
(11)
where
o
Y
is the desired negative control swing,
)(nYS
is the sampled value with hold of the plant
output at sampling interval
S
nT
, and U is the
sampled value with hold of the controller output.
The values of
)(nG
and
)(nF
will be all positive for
the summation of
)(nYS
and
o
Y
(or R and
o
Y
) is
greater than zero with a specified value of
o
Y
. All
positive values will give better continuities,
regulating characteristics of the series. Note that
singularity of Eq.(8) for
)(nYS
=0 is avoided by use
of Eq.(11). Eq.(11) implies
]))(/())(1[()(
OSo YjYYRjF
, j=1,2,3,.,
n,n+1,.... (12)
and inputs of the plant
)1()1( nGnu
are replaced
by
)0(/)1()1( PYnGnu o
; (13)
for the negative swing control with positive values
of
,
)( jG
and
)( jF
.
Fig.1. G(n+1)/G(n) Versus
)(/)( nRnYS
for
=0.9,0.7,0.5, and 0.3.
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Fig.2. The Nonlinear Digital Controller.
3. Numerical Examples
Example 1: The design example [4] is
)1.01(
100
)( ss
sPo
(14)
After it has been closed with feed-forward gain 0.03,
the DC gain of the closed- loop subsystem P(s) is
unity. The sampling period
S
T
=1/40 second is
selected. Time responses of the overall system with
the nonlinear digital controller for
=0.95 is shown
in Fig.3. The amplitude of reference
inputs
)( jR
between 0 and 2 seconds are equal to 1;
between 2 and 6 seconds are equal to -0.2, between
6 and 9 seconds are equal to 0.6, and between 9 and
12 seconds are equal to 1.2, in which gives
reference command
)( jR
(solid-line), plant output
)( jYS
(dot-line), control effort
)( jG
(dash-line), and
ratios
)( jF
(dash-dot-line) of
)( jG
. Fig.3 shows
that
)( jG
and
)( jF
are all positive while the value
of output
)( jYS
tracking the negative value of the
reference input
)( jR
exactly. Fig.3 shows also that
ratios
)( jF
are converged to be unities quickly; i.e.,
the controlled output tracks the reference input
quickly and exactly.
Fig.3. Time Responses of the Design Example
for
=0.95.
Example 2: Consider the very high order plant:
20
3)1(
1
)(
s
sP
(15)
Parameters of the nonlinear controller are
5.0
and
.25msTS
Fig.4 shows time response of the
controlled system, in which gives reference input
)(nR
(dash-line), output Y(solid-line), Time series
)(nG
(dot-line), and ratios
)(nF
(dash-dot-line) of
)(nG
. It gives good performance and zero steady-
state errors. The phase-lead filter C(z) is in the form
of
1
12
102.0
18.0
)(
z
z
T
sS
s
s
zC
(16)
Fig.4 shows the considered plant is a large time-lag
system. The high order system model is usually
used to describe the industry process for replacing
pure time-delay(e.g.
sTd
e
). Such that conventional
analysis and design techniques can be applied[5,6].
Fig.4 shows the proposed method can be applied to
a large time-delayed system.
Fig.4. Time responses of Example 2 with C(z)
for β=0.5;
.25msTS
Final results and four other methods are presented
for comparison and show the merit of the proposed
method. They are Ziegler-Nichols method[7-9] for
finding PI and PID compensators, Zhuang et al. [10]
for finding PI compensator and Majhi[11,12] for
finding PI compensator. Parameters of four found
compensators are given below:
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(1)ZN(PI) :
0305.0 585.0 ip KandK
.
(2)ZN(PID):
9135.4 05088.0 ,77256.0 dip KandKK
.
(3)Majhi’s(PI) :
.0443.0 5097.0 ip KandK
(4)Zhuang’s(PI):
058.0 473.0 ip KandK
.
Time responses are shown in Fig.5. Table 1 gives
integration of absolute error(IAE) and integration of
square error(ISE) of them. From Table 1 and Fig.5,
one can see that the proposed method gives better
performance than those of other methods.
Fig.5. Time Responses of Example 2 with
Different Control Methods.
Table 1. IAE and ISE Errors of Example 2 with
Different Control Methods.
Methods
Proposed
ZN(PI)
ZN(PID)
Majji
Zhuang
IAE
16.010
21.227
16.216
20.190
21.814
ISE
18.337
32.708
22.970
26.829
32.912
Example 3: Consider a gas turbine engine with
plant transfer function matrix[13-14].
22
22
3
124000
14925882525880
2.85
68.86428.12268 95150
7.11320941805947
9.14
33.15152533
)(
1
)(
s
s
s
ss
s
s
s
s
sP
(17)
where
432 22.1133.13577.35022525)( sssss
.
It is a
22
multivariable plant. The steady-state
gain of open loop
)(
3sP
is in the form of
3485.100085893.4
2265.71500316.1
)0(
5
P
(18)
A pre-compensating matrix
)0(
1
3
P
is first applied to
decouple the plant in low-frequency band. Then,
two digital filters are used in the diagonal to filter
outputs of two time series for speeding up transient
responses. They are in the form of
1z
1z
T
2
s
1
s
1s0.15
1s0.75
(z)C
(19)
and
1z
1z
T
2
s
2
s
1s0.25
1s0.60
(z)C
(20)
where
msTs25
is the sampling period. Fig.6
shows time responses of this controlled system for
=0.5. It shows that the proposed control scheme
can be applied to the multivariable feedback control
system also.
Fig.6. Time Responses of Example 5 for
=0.5 and
msTS25
.
4. Conclusions
A new nonlinear digital controller has been
proposed for analyses and designs of sampled-data
feedback control systems. It gave exact command
tracking without integration; i.e., zero steady-state
error. The convergence of ratios was illustrated by
one servo system example and two complicated
examples. From simulation results, it can be seen
that the nonlinear digital controller provided another
possible control scheme for exact command tracking
without integration.
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International , CC BY 4.0)
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