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Disturbance observer design for continuous systems with delay.

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ASIA-PACIFIC JOURNAL OF CHEMICAL ENGINEERING
Asia-Pac. J. Chem. Eng. 2007; 2: 517–525
Published online 12 October 2007 in Wiley InterScience
(www.interscience.wiley.com) DOI:10.1002/apj.096
Research Article
Disturbance observer design for continuous systems
with delay
R. W. Jones1 * and M. T. Tham2
1
2
Mads Clausen Institute for Product Innovation, Syddansk University, Sonderborg, Denmark
School of Chemical Engineering and Advanced Materials, Newcastle University, Newcastle of upon Tyne, England, UK
Received 8 January 2007; Revised 16 July 2007; Accepted 13 August 2007
ABSTRACT: Disturbance observers (DOs), which are popularly used for improving the disturbance rejection capability
of mechatronic servo control systems, offer several attractive features that could prove beneficial for process control
systems. The tuning is simple and intuitive and they allow independent tuning of disturbance rejection characteristics,
which is particularly helpful in situations in which gains need to be tuned on-line. This paper is concerned with
examining DO design for continuous-time systems with delay. Two methods for incorporating time delay into traditional
DO design are considered, to assess their relative performance, stability, and noise rejection characteristics. The
performance of one of these time-delay-based DO approaches is then compared with proportional, integral and derivative
(PID) control, designed using a maximum sensitivity approach, on a representative process model. The paper concludes
by briefly examining the incorporation of a DO into a two-degrees-of-freedom (2DOF) control structure.  2007 Curtin
University of Technology and John Wiley & Sons, Ltd.
KEYWORDS: disturbance observers; disturbance rejection; time-delay systems; continuous-time systems
INTRODUCTION
Disturbance observers (DOs), are an extremely popular way of improving the disturbance rejection performance of mechatronic systems.[1,2] The approach has
similarities to the Internal Model Control (IMC) strategy of Morari and Zafiriou[3] in that both approaches
use the inverse of the plant model to reject disturbances as well as to force the input–output characteristics to approximate specified dynamics. DOs have
been applied primarily to motion control systems, usually providing one-degree-of-freedom control within a
two-degrees-of-freedom (2DOF) control structure. With
a DO designed to provide efficient disturbance rejection,
the other controller is designed to provide a desirable
servo response.
DOs do offer several attractive features that could
be beneficial for improving the disturbance rejection
capability of process control systems. They allow independent tuning of disturbance rejection characteristics,
which is particularly helpful in situations in which gains
need to be tuned on-line. Compared to integral action,
DOs also allow more flexibility via the selection of the
order, relative degree, and bandwidth of any low-pass
*Correspondence to: R. W. Jones, Mads Clausen Institute for Product Innovation, Syddansk University, Sonderborg, Denmark.
E-mail: rjo@mci.sdu.dk
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
filtering of the disturbance estimate that takes place.
Although the technique of appending disturbance states
to a traditional state estimator is well known,[4] the
DO structure does allow simple and intuitive tuning of
the DO loop gains independently of the state feedback
gains. In addition, it has been shown that DOs are also
an effective means of suppressing the effects of nonlinearities in a class of nonlinear systems.[5] This property
might also prove extremely useful in the control of process systems.
The inclusion of continuous time delay into DO
design for mechatronic systems was first considered
in the mid-1990s but doubts over accuracy, increased
calculation time requirements, and parameter sensitivity
when a time-delay approximation is used led to a
recommendation that the time-delay term be ignored.[6]
However, the resulting model mismatch necessitates
careful design of the DO and the control bandwidth.
The design and implementation of discrete DOs as part
of a digital 2DOF control scheme do not have the same
implementation problems, as the delay is chosen as a
multiple of the sample time (see for example, Ref. [7])
The use of DOs for process control has been previously considered,[8] though only for the Smith predictortype control of integrator plus time-delay systems. The
control scheme used for this Smith predictor-type control was originally developed in Ref. [9], where the
time-delay term was integrated within the DO design
518
R. W. JONES AND M. T. THAM
procedure by mimicking the implementation used for
discrete-time systems.[7]
This paper introduces the use of DOs for the process
control of stable time-delay processes. The disturbance
rejection case only is considered, except for some
comments relating to set-point response at the end of
the paper. A representative first-order plus time-delay
(FOPTD) process is used throughout to illustrate the
different characteristics of DO design and performance.
As the main emphasis is on examining DO design for
continuous-time systems with time delay, two possible
approaches for the implementation of time-delay within
the DO are assessed. The first implementation is the
approach used in Refs [8 and 9], where the time-delay
term is separated from the nominal plant dynamics,
and used to delay the control signal before it is
used in the DO. The second implementation considers
incorporating an approximation of the time-delay term
directly into the inverse of the nominal model, which
is used in the DO, to filter the plant output signal.
This second implementation is the approach commonly
used throughout model-based process control, see for
example Ref. [3], and should be addressed in the
context of DO design. These different implementations
do have implications for the relative order required for
the low-pass filter, Q(s), which is the major design
parameter for the DO. The limitations placed on DO
design and performance by robust stability- and noiseattenuation considerations are then addressed. The paper
concludes by briefly examining the issues concerning
the incorporation of a DO into a conventional PID
control system.
Asia-Pacific Journal of Chemical Engineering
d
c
+
u
+
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Gp(s)
y
−
Q(s)Gq (s)
−
+
Q(s)Gn−1 (s)
Figure 1. A continuous-time disturbance observer.
Analysis of the DO loop is straightforward and yields
the transfer functions from the signal c, and disturbance
to output, respectively, as:
Gcy (s) =
Gp (s)Gn (s)
y(s)
=
c(s)
Gn (s) + Q(s)(Gp (s) − Gn (s)Gθ (s))
(1)
Gp (s)Gn (s)(1 − Q(s)Gθ (s))
y(s)
Gdy (s) =
=
d(s)
Gn (s) + Q(s)(Gp (s) − Gn (s)Gθ (s))
(2)
Equations (1) and (2) show why Q(s) is chosen
as a unit gain, low-pass filter. In the absence of
unmodeled dynamics, i.e. when Gp (s) = Gn (s)Gθ (s),
as Q(s) → 1 at low frequencies, Gcy (s) → 1 in Eqn (1)
and Gdy (s) → 0 in Eqn (2). Umeno and Hori[2] suggest
the following form for the Q(s) filter:
1+
DISTURBANCE OBSERVER DESIGN
Figure 1 shows the structure of the disturbance observer
for a time-delay process. This implementation is that
previously used in Refs [8 and 9]. The controlled output
is y, while the disturbance input is d. The signal c
is usually provided by an outer loop controller – in
process control systems this would normally be a PI
or PID controller. In this work we concentrate on
the design and disturbance rejection characteristics of
the DO.
The DO makes use of Q(s), Gn (s), and Gθ (s) in
an inner loop around the controlled plant, Gp (s), to
reject disturbances. Gn (s)Gθ (s) is the nominal model
of Gp (s) : Gn (s) contains the gain, lead, and lag components, while Gθ (s) is a pure time-delay term. The
design initially involves choosing the structure of the
Q(s) filter such that Q(s)Gn−1 (s) is proper, followed
by choosing the coefficients of Q(s). The influence of
measurement noise on the system is initially ignored
here, partly for clarification purposes. Noise attenuation
properties are discussed in the section on sensitivity
functions and robustness.
+
Q(s) =
1+
N
−m
k =1
N
ak (sτq )k
(3)
ak (sτq )
k
k =1
where, for the implementation shown in Fig. 1, m represents the relative degree of Gn (s). The filter time constant is τq . A variety of low-pass filter design methods
can be considered for determining the coefficients,ak ,
with Binomial and Butterworth designs being the most
popular.
DESIGN FOR CONTINUOUS-TIME SYSTEMS
WITH DELAY
Consider a continuous-time representation of a linear,
time-invariant plant with time delay, θ , and numerator
and denominator polynomials B (s) and A(s).
Gp (s) =
B (s) −θs
e
A(s)
(4)
Asia-Pac. J. Chem. Eng. 2007; 2: 517–525
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
DISTURBANCE OBSERVER DESIGN FOR CONTINUOUS SYSTEMS WITH DELAY
Two different implementations, which differ in the
way the time-delay term is incorporated into the DO
structure, will be examined. These are:
1. Implementation 1: The time-delay term is separated
from the nominal plant dynamics, and used to delay
the control signal, u, before it is used in the DO;
2. Implementation 2: This incorporates a low-order
approximation of the time-delay term into the nominal model.
These different implementations do have implications
for the minimum relative degree required for the
Q(s) filter. Motion control researchers[2] have found
that the lower the relative degree of Q(s), the better
the disturbance rejection performance. In the second
implementation, an invertible approximation to the
time-delay term has to be used. To minimize the
increase in the order of the nominal model and also
the relative degree required for the Q(s) filter, a (0,1)
Pade approximant of the following form is used:
e −θs ≈
1
(θ s + 1)
(5)
Implementation 1
This is the implementation represented in Fig. 1. The
time delay is separated from the other characteristics
of the model for use in the DO. For a FOPTD process
model, the delay-free plant model component, Gn (s),
and the separated time-delay term, Gθ (s), are
Gp (s) =
kp
e −θs ,
(1 + τ s)
Gn (s) =
Implementation 2
The commonly used approach, in model-based process
control, of incorporating a low-order approximation of
the time-delay term, see Eqn (5), into the FOPTD process model is now considered for the second implementation. The nominal model can be written as:
Gn (s)Gθ (s) G̃n (s) =
kp
(1 + sτ )(1 + θ s)
(9)
The DO is implemented as shown in Fig. 3.
Since Q(s)G̃n−1 (s) has to be proper and G̃n (s) is of
second order, the Q(s) filter should have a minimum
relative degree of 2. With a first-order numerator, the
following Q(s) filter design, with Binomial coefficients,
will be used:
Q13 (s) =
3(sτq ) + 1
(sτq ) + 3(sτq )2 + 3(sτq ) + 1
3
(10)
Figure 4 shows the responses of the DO with Q13 (s)
using time constants: τq = 0.5τ , τ , and 1.5τ , when a
kp
,
(1 + τ s)
Gθ (s) = e −θs
A FOPTD process model with a normalized timedelay value of 0.25 is representative of many process
systems. For the Q(s) filter design, the time constant
τq will be chosen as a function of the time constant of
the process.
Figure 2 shows the response of three different DOs,
corresponding to choices of filter time constants of
τq = 0.5τ , τ , and 1.5τ , when a unit-step change in input
disturbance was introduced at t = 5.0. As expected, the
smaller the value of the filter time constant, the better
the disturbance rejection performance.
(6)
For a first-order nominal model, the relative degree
of Q(s) should be at least 1. Although a Q(s) filter
with a zero-order numerator could be designed, in this
paper only Q(s) filters with a numerator order of at
least 1 will initially be considered. In Ref. [10], simple
first-order Q(s) filters are used to investigate the use
of DOs for reducing interaction in decentralized control
systems. From Eqn (3), and using Binomial coefficients,
the Q(s) filter has the following form:
Q12 (s) =
2(sτq ) + 1
(sτq )2 + 2(sτq ) + 1
(7)
where the subscript ‘12’ denotes a first-order numerator
and second-order denominator for the filter.
For the FOPTD process model, the following parameters will be used:
kp = 1.0,
τ = 2.0,
θ = 0.5
(8)
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
DO performance for a range of τq : Implementation 1. (Solid: τq = 1.0; Dashed: τq = 2.0; Dotted:
τq = 3.0). This figure is available in colour online at
www.apjChemEng.com.
Figure 2.
Asia-Pac. J. Chem. Eng. 2007; 2: 517–525
DOI: 10.1002/apj
519
520
R. W. JONES AND M. T. THAM
Asia-Pacific Journal of Chemical Engineering
d
u
c
+
+
+
Gp(s)
y
−
S (s) = 1 − e −θs Q(s),
Q(s)
Figure 3.
tion 2.
It is straightforward to define the sensitivity, S (s),
and complementary sensitivity, T (s), functions for the
DO-controlled system:
−
+
~
Q(s)Gn−1(s)
Disturbance observer: time-delay implementa-
Figure 4. DO performance for a range of τq : Implementation
2 with Q13 (s) (solid: τq = 1.0; dashed: τq = 2.0; dotted:
τq = 3.0). This figure is available in colour online at
www.apjChemEng.com.
T (s) = e −θs Q(s)
(12)
This direct link between the choice of Q(s) and
the shaping of the sensitivity functions is another
advantageous characteristic of the DO approach. The
time delay term in S (s) places a limitation on the
attainable disturbance rejection performance. The larger
the time delay, the more severe the limitation.
Figure 5 shows the sensitivity, S (s), and complementary sensitivity, T (s), functions for both implementations of the DO investigated in the previous section,
with τq = 1. It can clearly be seen that Smax (s) for
the Q12 (s) filter (solid line) is at a higher frequency
than Smax (s) for the Q13 (s) filter (dashed line). The
wider bandwidth of this filter translates into the superior
disturbance rejection performance demonstrated earlier. The complementary sensitivity function provides
insight into the noise attenuation at higher frequencies.
It can be seen that the high-frequency roll-off of T (s)
for the Q13 (s) filter (dashed line) begins at a lower frequency and is much steeper than T (s) for the Q12 (s)
filter (full line), resulting in better noise attenuation.
The competing requirements of the sensitivity functions
brings out the classical trade-off in feedback control;
namely, good tracking and disturbance rejection (S (s)
small and T (s) large) must be balanced by minimizing
the effect of measurement noise (S (s) large and T (s)
small).
Figures 6 and 7 compare the performance of both
implementations of the DO, τq = 1, in the presence of
zero mean measurement noise. The outputs in Fig. 6
unit step change in input disturbance was introduced at
t = 5.0. Although the response trends were similar and
zero-offset disturbance rejection was achieved, on comparing Figs 2 and 4 it can be seen that Implementation
2 does not provide as tight a control as Implementation
1; peak values and undershoot are generally larger and
settling times longer. Though the τq can be reduced
to improve the comparative performance of the DO,
this approach will always provide slightly inferior performance, in the noise-free case, owing to the higher
relative order of the Q(s) filter. This supports the finding from the use of DOs on mechatronic systems.
SENSITIVITY FUNCTIONS AND ROBUSTNESS
From Fig. 1, and assuming no model uncertainty, the
open-loop transfer function for the DO system can be
found to be:
Gol (s) =
−θs
e Q(s)
1 − e −θs Q(s)
(11)
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 5. Sensitivity and complementary sensitivity func-
tions, S(s) and T(s), for Q12 (s) (solid) and Q13 (s) (dashed)
with τq = 1. This figure is available in colour online at
www.apjChemEng.com.
Asia-Pac. J. Chem. Eng. 2007; 2: 517–525
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
DISTURBANCE OBSERVER DESIGN FOR CONTINUOUS SYSTEMS WITH DELAY
multiplicative perturbation,
Gp (s) = Gn (1 + (s))
(13)
where (s) is due to the unmodelled dynamics. Robust
stability of the DO loop is assured if the following
condition[3] is satisfied:
max |T (s)(s)| ≤ max |T (s)(s)| < 1.
s=j ω
s=j ω
(14)
With Q(s) = T (s), the following limitation on the
design of Q(s) can be defined:
|Q(s)| <
Figure 6. DO performance for τq = 1; Q12 (s) (dashed) and
Q13 (s) (solid). This figure is available in colour online at
www.apjChemEng.com.
1
∀s = j ω.
|(s)|
(15)
If the time-delay term is ignored, it is easily found
from Eqn (13) that:[6]
(s) = e −sθ − 1
(16)
The magnitude plot of 1/ provides the limit for the
robust stability condition, Eqn (15). This is shown as
the upper solid line in Fig. 8 for θ = 0.5. The magnitude
plots for the Q12 (s) and Q13 (s) filters for τq = 1 (solid
line and dashed line, respectively) are also shown. It can
be seen that these designs do not cross the magnitude
plot of 1/ and hence both satisfy the robust stability
condition. For the Q12 (s) filter τq can be reduced to 0.8
before the stability condition is reached (dotted line),
while for the Q13 (s) filter τq can be reduced to 0.58
(dashpot line).
Figure 7. Control values for DOs, τq = 1; Q12 (s) (dashed)
and Q13 (s) (solid). This figure is available in colour online at
www.apjChemEng.com.
indicate that there is little difference between the
responses, though the Q13 (s) filter-based DO (solid
line) appears to be slightly better than the Q12 (s) filterbased DO (dotted line). Comparing the control values
in Fig. 7, the variance of the Q13 (s) filter-based DO
control signal (solid line) is noticeably lower than that
of the Q12 (s) filter-based DO control signal (dotted
line). This is to be expected from the characteristics of
the complementary sensitivity functions demonstrated
in Fig. 5.
The complementary sensitivity function also plays
a key role in determining the robust stability of the
system. If the unmodeled dynamics are treated as a
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Complementary sensitivity functions and the
robust stability limit. Q12 (s) for τq = 1 (solid), Q13 (s) for
τq = 1 (dashed), Q12 (s) for τq = 0.8 (dotted), Q13 (s) for
τq = 0.58 (dash-pot). This figure is available in colour online
at www.apjChemEng.com.
Figure 8.
Asia-Pac. J. Chem. Eng. 2007; 2: 517–525
DOI: 10.1002/apj
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R. W. JONES AND M. T. THAM
Asia-Pacific Journal of Chemical Engineering
A MORE REALISTIC PROCESS EXAMPLE
Table 2. IAE of disturbance rejection responses.
Consider the following high-order damped process
model which has been used as a representative process
system in numerous publications, see Refs [11,12]:
FOPTD-based
DO
Gp (s) =
1
2.989
SOPTD-based
DO
PID with
Ms = 1.4
PID with
Ms = 2.0
2.664
4.020
2.712
(17)
(1 + s)5
Since the implementation of DOs for time-delay systems is the focus of this paper, both FOPTD and secondorder plus time-delay (SOPTD) models of Eqn (17) will
be used for the DO design. In a practical situation, the
high-order model dynamics would be approximated as
an FOPTD or SOPTD model. Implementation 1 is used
for the DO design throughout this section. These models
are shown below and were identified using a relay-based
identification technique.[11]
Ĝp1 (s) =
e −2.93s
,
(1 + 2.73s)
Ĝp2 (s) =
e −1.73s
(1 + 1.89s)2
(18)
The use of the FOPTD for DO design requires a
Q12 (s) filter, while the use of the SOPTD for DO design
requires a Q13 (s) filter. In both of these, the filter time
constant was chosen as τq = 1.365 which is half the
value of the time constant in the FOPTD approximate
model. As before, the coefficients of the respective Q(s)
filters were Binomial coefficients.
To place the performance of the two DOs in some
context, they will be compared with the performance of
a PID controller tuned using the maximum sensitivity,
Ms , the design approach of Panagopoulos et al .[12] This
very powerful design approach uses the high-order
process model Eqn (17) in conjunction with nonconvex
optimization to design the PID parameters. The primary
design goal of the approach is to provide superior
disturbance rejection; hence it will provide an excellent
test for the capabilities of DOs. The structure of the PID
controller used is:
u(s) = kc (r(s) − y(s)) +
kd y(s)
ki (r(s) − y(s))
−s
s
1 + τf s
(19)
where kc is the proportional gain, ki the integral gain,
kd the derivative gain, and Tf is the time constant of the
first order filter on the derivative term. Table 1 shows
two sets of PID controller parameters derived using the
maximum sensitivity design approach on the higherorder model, Eqn (17).[12]
Table 1. PID controller parameters.
Ms
kc
ki
kd
Tf
1.4
2.0
0.7840
1.4700
0.2925
0.6309
0.9722
1.8375
0.0972
0.1838
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 9. Outputs: PID and DO control of a high-order
process. (solid: DO using FOPTD model; dotted: DO using
SOPTD model; dash-pot: PID with Ms = 1.4; dashed: PID
with Ms = 2.0). This figure is available in colour online at
www.apjChemEng.com.
A unit step disturbance was introduced at t = 5.0, and
the responses of the DOs and PID controllers are shown
in Fig. 9. The DO based on the SOPTD model with the
Q13 (s) filter (dotted line) provided the best disturbance
rejection. This is confirmed by the Integral of Absolute
Error (IAE) between controlled output and setpoint of
the responses as listed in Table 2.
In this example, the use of the more accurate SOPTD
model overcame the disadvantage of using a Q(s) filter
with a higher relative degree. Its performance was very
similar to that obtained using the PID controller tuned to
give a maximum sensitivity Ms = 2.0, and significantly
better than the controller tuned to achieve Ms = 1.4.
This is notable, as both PID controllers were designed
using the exact model of the controlled process. Indeed,
the performance of the DO designed using the approximated FOPTD model was only slightly worse than the
PID controller tuned to a specification of Ms = 2.0.
Figure 10 shows the corresponding calculated control
values for each of the output responses.
TWO-DEGREES-OF-FREEDOM CONTROL:
PID AND DO
The use of DOs has, so far, only been considered
for disturbance rejection purposes. In the previous
Asia-Pac. J. Chem. Eng. 2007; 2: 517–525
DOI: 10.1002/apj
Asia-Pacific Journal of Chemical Engineering
DISTURBANCE OBSERVER DESIGN FOR CONTINUOUS SYSTEMS WITH DELAY
for this 2DOF controller structure can be written as:
y(s) =
+
Figure 10. Control values: PID and DO control of a highorder process. (solid: DO using FOPTD model; dotted: DO
using SOPTD model; dash-pot: PID with Ms = 1.4; dashed:
PID with Ms = 2.0). This figure is available in colour online
at www.apjChemEng.com.
section the disturbance rejection performances of both
DOs demonstrate resilience to mismatch in temporal
characteristics between the process and the model. It can
also be seen from Eqn (2) that zero-offset regulation is
independent of the accuracy of the process model and
can be achieved as long as the Q(s) filter has unit gain.
For set-point tracking, however, offset-free performance
can be accomplished only if the model and process gains
are identical. However, any offset that can be predicted
from Eqn (3) can be removed by a scheme similar
to that shown in Fig. 11, i.e. the DO is implemented
as part of a feedback control system that contains an
integral term. The integral term in the controller ensures
an offset-free set-point response behavior. This control
scheme is likely to be the most common way in which
the DO would be used in the process industries.
Although the selection of Q(s) is straightforward
with a clear physical interpretation when implemented
as shown in Fig. 11, the robust stability of the system
now depends on the design of Gpid (s) as well as
Q(s). In the absence of unmodeled dynamics, i.e. when
Gp (s) = Gn (s)Gθ (s), the closed-loop transfer function
r
+
−
Gpid (s)
c
u
+
+
+
Gpid (s)Gn (s)Gθ (s)
r(s)
1 + Gpid (s)Gn (s)
Gn (s)Gθ (s)(1 − Q(s)Gθ (s))
d(s)
1 + Gpid (s)Gn (s)
(20)
The design of the disturbance rejection and set-point
responses in Eqn (20) are obviously not independent.
The control of the FOPTD plant model Eqn (8) will
now be addressed using this control structure. The
implemented DO uses a Q12 (s) filter, Eqn (7), with
τq = 1.0. The disturbance rejection performance using
only the DO is shown in Fig. 2. The feedback controller,
Gpid (s), is in this case a PI controller and designed
specifically to provide a good set-point response, kc =
2.0, ki = 1.0. The full line in Fig. 12 shows the corresponding set-point and disturbance rejection response
of this system with a step input disturbance of −1
entering the system at t = 9. The addition of the feedback PI controller has modified the disturbance rejection
response demonstrated earlier in Fig. 2. The disturbance
rejection response is now slightly more aggressive with
an appreciable undershoot. The performance of this
interacting 2DOF control structure in the presence of
plant/nominal model mismatch is also examined. With
a plant gain of 0.8 and nominal model gain = 1, the
response in Fig. 12 (dotted line) shows an increase
in percentage overshoot for the set-point response and
a longer settling time. With a plant gain of 1.2 and
nominal model gain = 1, faster oscillatory modes are
induced in the set-point response with a similar overshoot as in the ideal case (dashed line). In both cases of
model uncertainty, the addition of a PI controller obviously compensates for any gain mismatch between the
d
Gp(s)
y
−
Q(s)Gq(s)
−
+
Q(s)Gn−1(s)
Figure 11. DO in conjunction with a feedback controller.
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
Figure 12. Interacting 2DOF control of the FOPTD process.
(solid: no uncertainty; dotted: process gain = 0.8; dashed:
process gain = 1.2). This figure is available in colour online
at www.apjChemEng.com.
Asia-Pac. J. Chem. Eng. 2007; 2: 517–525
DOI: 10.1002/apj
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R. W. JONES AND M. T. THAM
Asia-Pacific Journal of Chemical Engineering
plant model and the nominal model to provide offsetfree control. For a SOPTD plant model, the controller
should be chosen as a PID controller to provide offsetfree control in the presence of plant/nominal model
mismatch.
A simple, straightforward approach to decoupling the
set-point response design and the disturbance rejection
design is now implemented, see Ref. [8]. Instead of
using plant output feedback in conjunction with the
controller, the controller output is fed back, filtered
by the nominal model transfer function, and then
compared with the set-point signal. The output of the
comparator, (r(s) − u(s)Gn (s)), becomes the controller
input. For this 2DOF control structure, the closedloop transfer function, in the absence of unmodeled
dynamics, becomes:
C (s)Gn (s)Gθ (s)
r(s)
1 + C (s)Gn (s)
+ Gn (s)Gθ (s)(1 − Q(s)Gθ (s))d(s)
y(s) =
and disturbance rejection response of this system with a
step input disturbance of −1 entering the system at t =
9. The disturbance rejection response can be seen to be
exactly the same as that obtained earlier in Fig. 2. Also,
the decoupled structure allows a more aggressive setpoint response to be designed using C (s). The performance of this structure in the presence of plant/nominal
model mismatch is also examined. With a plant gain
of 0.8 and nominal model gain = 1, the response in
Fig. 13 (dotted line) shows a decrease in percentage
overshoot for the set-point response and a longer settling time. With a plant gain of 1.2 and nominal model
gain = 1, a greater percentage overshoot is accompanied by a shorter settling time (dashed line). Comparing
Figs 12 and 13, the response of the decoupled structure
to plant/nominal model mismatch is preferable.
CONCLUSIONS
(21)
where C (s) represents the forward path controller. With
this decoupled 2DOF control structure the disturbance
rejection and set-point responses can be designed independently. The DO provides the disturbance rejection,
while C (s) is used to shape the set-point response.
For type 0 plants, to ensure that there is no steadystate error in the face of plant/nominal model mismatch,
C (s) needs to include an integral term.
The control of the FOPTD plant model Eqn (8) is
again addressed using the decoupled 2DOF control
structure. The implemented DO uses a Q12 (s) filter,
Eqn (7), with τq = 1.0. The forward path controller is
a PI controller with parameters, kc = 4.0, ki = 4.0. The
full line in Fig. 13 shows the corresponding set-point
Two DO implementations for continuous-time systems
with delay have been investigated. The approach in
which the time-delay term is separated from the nominal plant dynamics and then used to delay the control signal before it is used in the DO provides the
best disturbance rejection performance in a noise–free
environment, though the approach in which the timedelay term is absorbed into the nominal model has
superior noise attenuation characteristics. In a practical situation in which robustness and noise attenuation are important considerations, the more conservative approach, that is the approach where the
time-delay term is absorbed into the nominal model,
would perhaps be preferred. The choice of the Q(s)
filter time constant as half the value of the process
time constant, when the process was modeled as a
FOPTD, was found to provide good disturbance rejection responses. The results were comparable to those
obtained using PID controllers tuned for disturbance
rejection using a powerful maximum sensitivity-based
design approach. It was also shown that the DOs were
quite robust to process–model mismatch in temporal
parameters.
Future work will investigate the use of error-based, as
opposed to output-based, DOs for the control of process
systems.[13] Current work involves investigating the use
of a DO scheme to provide stiction compensation in
flow valves.
REFERENCES
Figure 13. Decoupled 2DOF control of the FOPTD process.
(solid: no uncertainty; dotted: process gain = 0.8; dashed:
process gain = 1.2). This figure is available in colour online
at www.apjChemEng.com.
 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
[1] K. Ohishi, M. Nakao, K. Ohnishi, K. Miyachi, IEEE Trans.
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Asia-Pac. J. Chem. Eng. 2007; 2: 517–525
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Asia-Pacific Journal of Chemical Engineering
DISTURBANCE OBSERVER DESIGN FOR CONTINUOUS SYSTEMS WITH DELAY
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[5] S.M. Sharuz, C. Cloet, M. Tomizuka, Suppression of effects of
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 2007 Curtin University of Technology and John Wiley & Sons, Ltd.
[10] R.W. Jones, M.T. Tham, Proceedings of SICE-ICASE International Conference, Republic of South Korea, Busan, 2006;
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[11] W.K. Ho, C.C. Hang, L.S. Cao, Automatica, 1995; 31(3),
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[12] H. Panagopoulos, K.J. Astrom, T. Hagglund, IEE Proc.Control Theory, 2002; 149(1), 32–40.
[13] K. Yang, Y. Choi, W.K. Chung, I.H. Suh, S.R. Oh, Robust
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DOI: 10.1002/apj
525
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