Tuning a complex multi-loop PID based control system requires considerable experience. In today's power industry the number of available qualified tuners is dwindling and there is a great need for better tuning tools to maintain and improve the performance of complex multivariable processes. Multi-loop PID tuning is the procedure for the online tuning of a cluster of PID controllers operating in a closed loop with a multivariable process. This paper presents the first application of the simultaneous tuning technique to the multi-input–multi-output (MIMO) PID based nonlinear controller in the power plant control context, with the closed-loop system consisting of a MIMO nonlinear boiler/turbine model and a nonlinear cluster of six PID-type controllers. Although simplified, the dynamics and cross-coupling of the process and the PID cluster are similar to those used in a real power plant. The particular technique selected, iterative feedback tuning (IFT), utilizes the linearized version of the PID cluster for signal conditioning, but the data collection and tuning is carried out on the full nonlinear closed-loop system. Based on the figure of merit for the control system performance, the IFT is shown to deliver performance favorably comparable to that attained through the empirical tuning carried out by an experienced control engineer.

1. Simultaneous gains tuning in boiler/turbine PID-based controller clusters
using iterative feedback tuning methodology
Shu Zhang a
, Cyrus W. Taft b
, Joseph Bentsman a,n
, Aaron Hussey c
, Bryan Petrus a
a
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 1206W Green Street, Urbana, IL 61801, USA
b
Taft Engineering, Inc., Harriman, TN, USA
c
EPRI, 1300 West WT Harris Boulevard, Charlotte, NC 28262, USA
a r t i c l e i n f o
Article history:
Received 3 September 2011
Received in revised form
16 April 2012
Accepted 16 April 2012
Available online 26 May 2012
Keywords:
Iterative feedback tuning
PID control
Multi-input–multi-output systems
Boiler/turbine control
a b s t r a c t
Tuning a complex multi-loop PID based control system requires considerable experience. In today’s
power industry the number of available qualified tuners is dwindling and there is a great need for
better tuning tools to maintain and improve the performance of complex multivariable processes.
Multi-loop PID tuning is the procedure for the online tuning of a cluster of PID controllers operating in a
closed loop with a multivariable process. This paper presents the first application of the simultaneous
tuning technique to the multi-input–multi-output (MIMO) PID based nonlinear controller in the power
plant control context, with the closed-loop system consisting of a MIMO nonlinear boiler/turbine model
and a nonlinear cluster of six PID-type controllers. Although simplified, the dynamics and cross-
coupling of the process and the PID cluster are similar to those used in a real power plant. The
particular technique selected, iterative feedback tuning (IFT), utilizes the linearized version of the PID
cluster for signal conditioning, but the data collection and tuning is carried out on the full nonlinear
closed-loop system. Based on the figure of merit for the control system performance, the IFT is shown to
deliver performance favorably comparable to that attained through the empirical tuning carried out by
an experienced control engineer.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction and background
1.1. Sequential and simultaneous tuning of individual PID
controllers in a multi-input–multi-output (MIMO) PID cluster
Power plant control systems have utilized proportional-inte-
gral-derivative (PID) controllers as their primary control algo-
rithm for over 50 years. Researchers have been developing and
testing advanced power plant control laws during the past 20
years and although many of the results are positive, acceptance of
the new technology has been limited. The PID control law is still
the dominant algorithm in the industry. A major difference
between the PID algorithm and most of the advanced algorithms
is that a PID controller is a single input, single output (SISO)
controller and therefore can only control a single control loop at a
time. The more advanced algorithms are inherently MIMO and
can control many loops in an integrated fashion. When using
single loop PID controllers to regulate a process with interacting
variables such as a boiler/turbine system, the process must be
segregated into the SISO loops. The basic arrangement of control
loops is largely done by pairing inputs and outputs that are most
closely correlated. This technique is adequate for many control
loops where the magnitude of the interactions is relatively small.
For larger interactions, ad hoc feedforward signals are sometimes
used to reduce the coupling effects. In addition to this logical
pairing of inputs and outputs, the structure of the control system
also utilizes many cascade control loops. Tuning a complex PID
based control system requires considerable expertise and experi-
ence. While single loop PID tuning is well understood and
supported by several commercial tuning software packages, it is
not clear how to simultaneously optimize several interacting PID
control loops. Thus, there is a need in the industry for better
tuning tools to provide a high performance control of complex
multivariable processes.
The most common method of tuning boiler/turbine control
systems today is the empirical method, also called the trial and
error method. It applies to both SISO and MIMO systems. The
general approach is to start with all the loops in the manual mode
or significantly detuned and begin tuning controllers at the
bottom of the hierarchy, working upwards. After each controller
is tuned it is placed in automatic mode. Where multiple con-
trollers exist at the same level, the controller with the least
interactions is generally tuned first. There are some exceptions
to this rule, but this is the general approach. When the last
Contents lists available at SciVerse ScienceDirect
journal homepage: www.elsevier.com/locate/isatrans
ISA Transactions
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.04.003
n
Corresponding author. Tel.: þ1 217 244 1076; fax: þ1 217 244 6534.
E-mail address: jbentsma@illinois.edu (J. Bentsman).
ISA Transactions 51 (2012) 609–621

2. controller is tuned and placed into automatic mode, the entire
system is expected to be very stable and responsive. Usually it is
necessary to go back and fine tune some of the controllers once
the whole system is in automatic mode.
When tuning a series of controllers sequentially, it is impor-
tant to provide some additional stability margin on the first loops
tuned, because as other loops are later tuned and placed in
automatic mode, the whole system can become unstable or very
oscillatory. This is where the experience of the tuner is most
beneficial. It requires judgment to provide the right amount of
stability margin without sacrificing good performance. For multi-
loop systems the method is carried out in a prescribed sequence,
tuning one controller at a time, and therefore belongs to the class
of sequential tuning methods.
Simultaneous tuning of individual controllers in the MIMO PID
controller clusters in the multi-loop MIMO configurations can
potentially capitalize on the interactions among process variables
and loops to attain better overall performance. Unlike the
sequential empirical loop tuning or sequential loop closure [1],
simultaneous tuning can incorporate, in addition to classical SISO
time domain performance objectives, the true MIMO control
objectives, such as robust performance specified in terms of the
so-called sensitivity and complementary sensitivity functions in
linear case, or gap metric measures in nonlinear case. MIMO PID
cluster tuning is better developed in the linear case; however,
even then it is still highly non-trivial. For example, the fully linear
problem of simultaneous tuning of PID clusters of arbitrary
structure in a MIMO system to attain closed-loop stability belongs
to a class of the so-called computationally NP-hard (nondetermi-
nistic polynomial time) problems characterized by the explosive
computational demand and is in general nonconvex, i.e. not
amenable to simple gradient-type solutions for attainment of
global optimal solution. Nevertheless, a significant progress has
been made in MIMO PID cluster tuning that can parlay specific
application knowledge into successful tuning procedures [1–3].
Simultaneous tuning of individual PID controllers in a MIMO
PID-based closed-loop subsystem can be accomplished in the
closed-loop or the open-loop manner. Closed loop is often pre-
ferable; however, it has limitations and might present difficulty in
re-identifying subsystem dynamics for controller tuning, or more
generally, redesign. As indicated in [1], the simultaneous MIMO
PID tuning techniques can be broadly classified into the model-
free and the model-based methods, with the latter being non-
parametric, partially parametric, and parametric. The model-free
technique is represented by the iterative feedback tuning (IFT)
[4–10] and the controller parameter cycling [11]. The nonpara-
metric, partially parametric, and parametric model-based meth-
ods are represented, respectively, by the relay autotuning
technique [12], the subspace identification technique [13–15],
and the optimal restricted structure design and related methods
[16–28].
Iterative feedback tuning is a model-free direct multi-loop PID
tuning method whose performance is usually superior to sequen-
tial tuning methods. Ref. [4] extends IFT to the case where both
the plant and the controller can be nonlinear. It shows that an
estimate of the gradient can be constructed using only the signal
based information. In [7], an overview of IFT is presented for both
SISO and MIMO linear systems. Stability and robustness aspects
are covered. A survey of existing extensions, applications, and
related methods is also provided. In [8], IFT addresses tuning of
the PID parameters in the applications where the objective is to
achieve a fast response to the set point changes. The performance
of these IFT-tuned PID controllers is compared with the perfor-
mance achieved by four classical PID tuning schemes widely used
in industry. Simulations show that IFT always achieves a perfor-
mance that is at least as good as that of the classical PID tuning
schemes, and often dramatically better: faster settling time and
less overshoot. Nakamoto [9] provides an application of IFT that
adjusts parameters of multiple PID controllers in a MIMO power
plant process. In the IFT applications for the SISO systems,
depending on the controller structure, only two or three closed-
loop experiments are required. However for the MIMO systems
the number of experiments increases proportionally to the
dimension of the controller, making the tuning process time-
consuming. In [10] several methods are proposed to reduce the
experimental time by approximating the gradient of the cost
function.
1.2. Sequential tuning—a simple empirical tuning example
To help clarify the general empirical tuning procedure described
in the previous section, the tuning process will be demonstrated
on a simple boiler control system structure shown in Fig. 1. This
control system is typical of many drum boiler control system
architectures; although a real control system has many additional
details that have not been included in this example. This example
assumes that no prior loop tuning has been done, so that the
system is completely detuned. All tuning should be done with the
unit at steady load with no operational disturbances, such as
sootblowing, in progress. It is preferable for the unit load to be at
or near full load, but if that is not possible, the load should be set
as high as possible.
The control system at hand is seen to have a definite hierarchy
and to include several cascade control loops. On any drum boiler,
the first loop to be tuned should be the feedwater one followed by
the drum level control loop. This is not due to its position in the
hierarchy, but rather because the level is non-self-regulating and
therefore troublesome to operate in manual mode. For this
reason, it is important to get this loop in automatic mode as soon
as possible. The empirical requirements for the controller perfor-
mance are specified in Table 1.
Prior to tuning the feedwater flow controller, the drum level
controller in set to manual or greatly detuned. The feedwater flow
rate dynamics has a fast rise time, usually around 5 s, so it is
relatively quick and easy to tune empirically. Tuning starts with
the integral gain and the derivative gain set to zero and the
proportional gain set well lower than its expected final value. A
step change to the feedwater flow rate setpoint is introduced and
the response is observed. The change in the latter setpoint can be
produced by changing the drum level setpoint. The dynamics of
the drum level controller is much slower than that of the feed-
water system, so it will not interfere with the feedwater loop
tuning. The proportional gain is then increased until a small
amount of overshoot is seen in the response and then it is reduced
to eliminate overshoot. The integral gain is then increased until
the flow rate error returns to zero quickly, but with little or no
overshoot. It may be necessary to reduce the proportional gain
slightly as the integral gain is increased to maintain a quick return
to setpoint with no overshoot. Derivative action is rarely, if ever,
used on the feedwater flow rate control due to noise in the
process measurement and no real need to decrease the response
time of the loop beyond what proportional and integral actions
alone can provide.
Once the feedwater flow rate control is tuned, the drum level
controller should be tuned. The method is similar to the feed-
water flow rate tuning, except that the final loop tuning must be
done by introducing a setpoint change into the loop, usually
through the feedwater flow rate. Also, because the process has an
integrating type response, very little integral action is needed on
the controller. Derivative action may be used on drum level
control, but often is not, and when used, its tuning should be
done prior to that of the integral action. As derivative gain is
S. Zhang et al. / ISA Transactions 51 (2012) 609–621610

3. increased, attention should be paid to the presence of noise at the
controller output. Derivative action tends to amplify any noise
that exists in the process measurement and the noise ultimately
limits the amount of the derivative gain that can be used. As
derivative gain is increased, it may be possible to increase the
proportional gain without introducing any overshoot due to the
damping effect of the derivative gain.
Drum level response may exhibit a nonminimum phase shrink
and swell effect in response to drum pressure and steam flow
changes. This effect causes the level to move in the opposite
direction initially before moving in the expected direction. If this
type of response is observed, less control gain will be possible and
the level response will be worse.
After tuning the drum level, the steam flow rate feedforward
should be scaled so that at full load the steam flow rate and
feedwater flow rate readings agree in the feedwater controller.
It should be checked again at mid-load, and if the two readings
differ by more than 1%, consideration should be given to replacing
the linear scaling with a function generator to allow a better
match between feedwater and steam flow rates over the load
range.
The typical control structure shown in Fig. 1 does not include a
furnace pressure control loop. If it did, that loop should be tuned
PID
PIDPIDPID
Air Flow
Demand
Fuel Flow
DemandFeedwater
Flow Demand
Superheat
Spray Valve
Demand
Heat
Distribution
Demand
(Burner
Tilts)
PID
Turbine
Valve
Demand
X
PID PID
ATTTTT MW PTLT
PID PID
Unit Load
Reheat
Steam
Temperature
Superheat
Steam
Temperature
Drum Level
Excess
Oxygen
Throttle
Pressure
FT
Steam Flow
Typical Control System
Structure for Drum Boiler
3
5 6
41
9 8 2 10
Typical Tuning
Order
K
A
f(x) f(x)
KK
f(x)
K
A
K
A
K
A
K
A
K
K
FT
Air Flow
K
FT
Feeder
Speed
K
FT
0.7 to 1.3
(+/- 30%)
These K’s
convert EUs
to %
FW Flow
PID
7
K
TT
Desuper-
heater
Outlet
Fig. 1. Boiler/turbine control structure in SAMA format.
Table 1
Control performance requirements.
Controlled variable Primary objective Tuning test(s)
Unit load Setpoint following Closed loop load ramp at 5%/min
Throttle pressure Disturbance rejection Closed loop load ramp at 5%/min
Drum level Disturbance rejection Closed loop load ramp at 5%/min
Excess oxygen Disturbance rejection Closed loop load ramp at 5%/min
S. Zhang et al. / ISA Transactions 51 (2012) 609–621 611

4. shortly after the drum level loop, because it is also difficult to
control manually.
The burner air flow rate control loop is tuned next. The air flow
represents total air flow including primary and secondary air
flows that go into the furnace. This task is generally straightfor-
ward, with a time constant of around 20–30 s, depending on the
control means. Fan inlet vanes, fan blade pitch, and fan speed are
all used to control air flow rate. Again, the goal should be to tune
for a small amount of overshoot, and then eliminate it by slightly
reducing the gain.
The fuel flow rate control is tuned next. The loop is often that
of a feeder speed (or flow rate) control, with the bulk of the
control done within the feeder control system. If a heat release
type of fuel measurement is used, the tuning is more difficult and
that system is beyond the scope of this study.
At this point all the low level flow control loops are tuned and
it is time to begin tuning the upper level controllers. The first
upper level controller to be tuned should be the throttle pressure
controller. Again, the tuning procedure is similar to that for the
feedwater flow rate controller described above. Initial tuning can
be done by making setpoint changes at constant load. Because the
pulverizer response is very slow, it is generally necessary to use
derivative action in the throttle pressure controller. During the
initial tuning little or no derivative action should be used. This
should wait until the second round of tuning. The objective at this
point is still stability without using excessive over- or under-
firing. Once the loop is tuned for setpoint response, small load
changes or fuel disturbances should be introduced to check the
response under those conditions.
Excess oxygen loop should be tuned next. The initial tuning
can be done for tracking in response to setpoint changes, but the
final tuning must be carried out for disturbance rejection through
actually generating air or fuel flow rate disturbances. Noise is
usually a factor in excess oxygen measurement, so it may be
difficult to use much derivative action in this loop.
At this point, it would probably be best to do an initial tuning
on the superheat temperature control and the burner tilt control.
The superheat temperature control is another cascade loop with
the inner loop controlled variable being the temperature imme-
diately downstream of the spray injection point, referred to as the
desuperheater outlet temperature. This measurement has a quick
response to changes in spray flow, typically about a 25–30 s time
constant. As a result, it is easy to tune and requires no derivative
action. The outer loop, with the final steam temperature as the
controlled variable, is tuned next. This loop is often the most
difficult to tune due to its slow response time and frequent
disturbances. Initial tuning can be done under setpoint changes,
but final tuning should be done under temperature disturbances.
At this point, it is not necessary to tune the feedforward signals
for the superheat control.
The burner tilt control can be difficult to tune because the
burner tilt response is often inconsistent throughout its operating
range. The response times are also slow. At this point it is not
necessary to tune the feedforward signals for the burner tilt
control.
The final loop to be tuned on the first pass through the control
system is the megawatt control loop. The latter loop may be
tuned for a very quick or a more sluggish response depending on
the operating goals of the unit. If tuned for fast response, the
throttle pressure control will not be as precise as it would if a
more sluggish response were the goal. For this example, the goal
for megawatt control will be a fast tight response. This can be
done by making fairly quick, but small, load ramps in the MW
setpoint. These small ramps approximate step changes. A
response with no overshoot should be the goal for this initial
tuning.
This concludes the initial tuning of the system. At this time all
loops should be in automatic mode and stable under steady load
conditions. Slow load ramps (o0.5%/min) should be possible
without major oscillations in key variables.
The final tuning is primarily focused on improving the unit’s
load ramp response. The most common control objective for the
complete unit is to ramp load at a given rate, while maintaining
key parameters within certain limits. The key loops for maximiz-
ing unit load rates are the pressure control and the steam
temperature control ones. These most often limit a unit response
time, calling for more attention during the second phase of the
tuning. The other loops are certainly important as well; however,
being inherently faster, they do not constitute the limiting factor.
In the final phase of the tuning, the unit load, or MW (mega-
watt), setpoint is ramped up and down, while the response of
individual loops is monitored. In addition to the tuning of the PID
controllers for each loop, the calibration of feedforward signals
for the pressure, air flow rate, and steam temperature loops is
important. If a PID controller is not responding well during a load
ramp, its tuning should be re-examined, with the objective shifting
somewhat towards responsiveness over stability. On drum boilers
it is important to overfire and underfire during load changes to
compensate for changes in energy storage in the boiler and to
reduce the response time of the pulverizers. Dynamic ‘‘kicker’’
functions are added to the firing rate demand signals to accom-
plish this. The setting of these kickers is always a balance between
improving the response and limiting the overfiring to maintain
good combustion and furnace characteristics. With tight tuning
on the MW control, the pressure control needs good feedforward
action to prevent oscillations in its response.
The initial load ramps should be done at ramp rates around
1.0%/min. As the unit is fine-tuned and the response improves,
faster rates can be used. The load change during the ramps should
be large enough to ensure that the system has time to arrest the
initial drop or increase in throttle pressure before the ramp ends.
The feedforward signals for pressure and air flow are generally
just static load index type functions. To calibrate these signals, the
output of the feedforward function generator should agree with
the control loop demand over the load range. Steady state data
must be collected at several load points and used to set the
feedforward signals. If the latter are set correctly, the pressure
controller and the excess oxygen controller outputs under steady
state conditions will be very close to 0.0 throughout the load
range. The steam temperature feedforward signals are usually
more complex than those shown in this example and often
include more than one signal. Tuning these signals involves
determining how each signal affects the steam temperature and
devising a function to counteract that effect. Open loop step tests
or normal load changes can be used to quantify the effects of each
signal. While empirical methods can often find suitable feedfor-
ward functions, this is one area where analytical tools can be used
to good advantage.
1.3. Simultaneous tuning—basic setting for MIMO iterative feedback
tuning
Iterative feedback tuning [4–10] is a model-free direct tuning
method using the closed-loop experimental data. The method is
based on numerical optimization, with the use of an unbiased
gradient estimate in each iteration step. The unbiased gradient
estimates provide convergence of the tuning sequence to a
stationary point of the control criterion provided the closed loop
signals remain bounded throughout the iterations.
The block diagram in Fig. 2 illustrates the feedback loop for a
discrete time linear time-invariant (LTI) system as a suitable
candidate for IFT.
S. Zhang et al. / ISA Transactions 51 (2012) 609–621612

5. The system is described by
yt
wt
!
¼ G
rt
vt
ut
0
B
@
1
C
A ð1:1Þ
where t represents the discrete time instants, G is the (general-
ized) plant consisting of the true plant and possibly some
frequency weighting filters and a reference model and is repre-
sented by a transfer function matrix, rt represents external user
controlled signals such as set-points, vt represents unmeasurable
signals such as (process) disturbances and noise, wt represents
the measured output errors (deviations from the setpoints) that
will be utilized in control laws, and ut represents the control
signals that control the plant. Furthermore, yt represents the
variables of interest to be included in the controller tuning
criterion (e.g. selected measured outputs from wt and control
signals selected from ut if some of them need to be penalized).
Relating these variables to Fig. 1, wt includes the seven incre-
mental outputs (respective deviations from the setpoints): mega-
watt output, throttle pressure, steam flow rate, excess oxygen, air
flow rate, drum level, and feedwater flow rate, whereas four of
them, megawatt output, throttle pressure, excess oxygen, and
drum level, form vector yt in the tuning criterion. None of the
control variables are penalized in the present work, and hence
none are included into yt. The external signal z is used for gradient
estimation.
The controller for this system is given by
ut ¼ cðk,wtÞ ð1:2Þ
where c(k,wt) is a linear time domain operator mapping from wt
to ut parameterized by some parameter vector k. A differentiable
control tuning criterion is chosen as
JðkÞ ¼
1
2N
E
XN
t ¼ 1
½ytðkÞŠT
PtytðkÞ
" #
ð1:3Þ
where E[ Á ] denotes expectation with respect to the disturbance v
and Pt is the weighting matrix. The objective is to minimize this
criterion by the Newton–Raphson type iterative algorithm to
find PID coefficients that best make system track reference
trajectories:
k
iþ 1
¼ k
i
ÀgiRÀ1
i
dJ
dk
ðk
i
Þ, i ¼ 0,1,2. . ., ð1:4Þ
dJ
dk
ðk
i
Þ ¼ À
1
N
E
XN
t ¼ 1
@yt
@k
ðk
i
Þ
!T
Ptytðk
i
Þ
" #
ð1:5Þ
where i’s are the iteration numbers and Ri is some appropriate
positive-definite matrix, typically a Gauss–Newton approxima-
tion of the Hessian of J(k), while gi, the step size, is a positive real
scalar which is adjustable to ensure convergence.
Since the exact calculation of expectation of [(@yt/@k)(ki
)]T
Ptyt(ki
) is impossible for complex systems, it is replaced by finding
the unbiased estimates of the product [(@yt/@k)(ki
)]T
Ptyt(ki
) that
are obtained by performing experiments on the closed-loop
system by using a suitable signal z located as shown in Fig. 2.
Further details of the algorithm are given in Section 2.
In [6], it is shown how the unbiased estimates at each IFT
iteration step (1.5) can be obtained for multivariable LTI systems.
Particular attention is given to the issue of keeping the experi-
ment time to a minimum and several efficient algorithms are
presented. It is shown that, for tuning of an arbitrary LTI MIMO
controller with nw inputs and nu outputs, 1þnu  nw experi-
ments are sufficient in each iteration step of the algorithm. For
the disturbance rejection, an alternative algorithm is proposed
which requires nuþnw experiments. As an illustration, the
method is applied to a simulation model of a gas turbine engine.
1.4. Advantages/disadvantages
The iterative feedback tuning (IFT) method [7] is a model-free
approach, a rather attractive feature that potentially eliminates or
reduces the modeling effort. It uses the closed loop data collection
which is superior to the open loop one. It can handle different cost
functions that can be designed to address the robustness issues.
The method has been extended to be applicable to some common
classes of nonlinear plants. It can be used to tune a known control
structure with no need for making changes to it. The fundamental
drawback of IFT is that it is based on the gradient search, i.e. is an
essentially convex method, whereas the MIMO PID tuning is
known to be a nonconvex problem in general. Thus, for a MIMO
PID cluster tuning, IFT can get stuck in a local minimum that
could be located arbitrarily far from the optimal tuning point. In
this regard, IFT should be supplied with a good starting parameter
value point, possibly obtainable through experience or other
techniques, such as parametric HN restricted structure PID rede-
sign [22]. The operational drawback of IFT is that the number of
experiments required for gradient computation could be large
and quite comparable to the effort needed to perform system
identification. The production of the unbiased Hessian matrix – the
square matrix of second-order partial derivatives of a function –
appears to involve quite a heavy computational load and requires
additional experiments and in some cases two additional identifica-
tion steps along with the one required for gradient estimation.
2. MIMO iterative feedback tuning—techniques and
application
2.1. Iterative feedback tuning of MIMO PID clusters—tuning
procedure
Consider an unknown true closed loop LTI MIMO continuous
time plant/controller system described as follows:
ysðkÞ ¼ GusðkÞ,
usðkÞ ¼ CðkÞðrsÀysðkÞÞ ð2:1Þ
where GG(s) is the p  m plant transfer function matrix whose
elements are Laplace (transform) transfer functions, C(k)C(s,k) is
the m  p controller transfer function matrix whose elements are
Laplace transfer functions parameterized by some parameter
vector kARf
, m and p are the number of inputs and outputs,
respectively. Denote the time domain representations of G and
r
v
y
u
z
w
C
G
Fig. 2. Closed-loop configuration for the iterative feedback tuning.
S. Zhang et al. / ISA Transactions 51 (2012) 609–621 613

6. C(k) as g and c(k), respectively, where both g and c(k) are the
corresponding time domain operators. rs, us, and ys denote the
Laplace transforms of the corresponding continuous time signals
rtARp
, utARm
, and ytARp
representing, respectively, the external
user controlled signals such as the set-points or more general
reference signals, the control signals, and the plant output signals.
The argument k in these signals indicates their dependence on the
changes in the parameter vector k in the controller. The corre-
sponding block diagram is shown in Fig. 3. The external signal
zsARm
, Laplace transform of ztARm
, in Fig. 3 will be used to obtain
the gradient estimate in each iteration step.
We note that for the applicability of the IFT procedure there is
no restriction on the number of plant or controller inputs with
respect to the number of outputs as long as G can be associated
with controllable and observable (or, more generally, stabilizable
and detectable) state space realization [30] and the controller is
stabilizing, i.e. closed loop remains asymptotically stable. In the
application considered, the p  m plant is ‘‘non-square’’ – it has
four inputs (m¼4) and seven outputs (p¼7), while controller,
respectively, has seven inputs and four outputs, having control
signal u as a 4-vector, as indicated above. However, only four of
the seven output components are subjected to tracking setpoints
according to the system tracking performance objectives, while
the remaining three components are used by the controller to
generate control signals making themselves internal signals,
posing only the regulation problem in terms of their stabiliz-
ability. Therefore the plant is essentially a 4 Â 4 system. The
criterion chosen for the present work is the quadratic one:
JðkÞ ¼
1
2N
E
XN
t ¼ 1
½~ytðkÞŠT
Pt ~ytðkÞ
#
ð2:2Þ
where ~ytðkÞ ¼ rtÀytðkÞ is the difference between the setpoints rt
and the achieved output yt. Since yt is the continuous time
function, the criterion J(k) is seen to be based on the samples of
yt at t¼1, 2, y, N s, i.e. the sampling interval of 1 s. This sampling
interval is commensurable with the response of the plant model
used further in the MIMO PID cluster tuning. Pt is the weighting
matrix. The optimal controller parameter kn
is defined by
k
n
¼ arg min
k
JðkÞ: ð2:3Þ
Since the IFT typically performs a local gradient search in the
vicinity of an operating point of a closed loop with the controller
and the plant being both nonlinear, as is the case in the present
work, the problem of minimizing J(k) in the actual IFT application
is in general not convex. To obtain a stationary point of J(k), we
would like to find a solution to the equation:
0 ¼
dJ
dk
ðkÞ ¼ À
1
N
E
XN
t ¼ 1
@yt
@k
ðkÞ
!T
Pt ~ytðkÞ
#
: ð2:4Þ
If the gradient dJ/dk could be computed, then the solution of
the above equation would be obtained by the following iterative
algorithm:
k
iþ 1
¼ k
i
ÀgiRÀ1
i
dJ
dk
ðk
i
Þ, i ¼ 0,1,2,. . . ð2:5Þ
where Ri is some appropriate positive-definite matrix, typically a
Gauss–Newton approximation of the Hessian of J(k), while gi, the
step size, is a positive real scalar which is adjustable to ensure
convergence and whose choice is case-dependent. Large gi usually
gives fast convergence rate but also could lead to instability.
Small gi has slow convergence rate but will ensure stability. In
order to solve this problem, one thus needs to generate the
unbiased estimate of the product
@yt
@k
ðkÞ
!T
Pt ~ytðkÞ: ð2:6Þ
Denoting by y0
(k) the gradient of y(k) w.r.t. an arbitrary
element of parameter vector k, Fig. 4 illustrates how to obtain the
unbiased estimate of y0
ðkÞT
Pt ~ytðkÞ from the closed-loop experiments.
The case when both controller and plant are linear will be
henceforth referred to as the nominal condition. The standard
procedure of IFT is given in this case by the following steps where
the subscript t denotes the time domain signals and the subscript
s denote their corresponding Laplace transform signals.
(0) Initialization: choose some initial set of controller parameters
k0
– these could be arbitrary numbers or the values provided
by an experienced control engineer, both could yield good
results – and specify setpoints rt as some fixed values.
(1) At each iteration i, perform a closed-loop experiment of time
length N using specified rt as the input and with the controller
parameters set to the values ki
for which the derivative
dJ/dk(ki
) is to be computed. Collect measurements ~yt, t ¼
1,. . .,N s. during this experiment, further referred to as the
normal experiment.
(2) Use the output ~yt collected in step (1) as input and calculate
the corresponding output ~yout,t. Here ~yout,s ¼ C0
ðk
i
Þ~ys, where
C0
(ki
)¼@C(ki
)/@k and C(k) is the Laplace transfer function
matrix of the linear controller and C0
(ki
) is the partial
derivative matrix of C(k) with respect to the controller
parameters k at ki
.
(3) Set the setpoints rt to zero and perform another closed loop
experiment referred to as gradient experiment while adding
the signal ~yout,t obtained in step (2) at the position ~yout,t as
shown in Fig. 4. Collect measurements yn
t , t ¼ 1,. . .,N s. It has
been shown in [6] that ½yn
t ŠT
Pt ~ytðk
i
Þ is an unbiased estimate of
½ð@yt=@kÞðk
i
ÞŠT
Pt ~ytðk
i
Þ. However, if we do not introduce noise
into our MATLAB/SIMULINK model for simplicity, they are
exactly equal to the ½ð@yt=@kÞðk
i
ÞŠT
Pt ~ytðk
i
Þ, given that the system
and the controller are both LTI [10]. If controller parameter
vector k has dimension nk, that is k
i
¼ ½k
i
1,k
i
2,. . .,k
i
nk
ŠT
, then for
each k
i
l, l ¼ 1,2,. . .,nk, we need to repeat (2) and (3) once: For
each k
i
l, l ¼ 1,2,. . .,nk, we need to compute C0
ðk
i
lÞ ¼ @ Cðk
i
lÞ=@kl
and then compute ½ð@yt=@k
i
lÞðk
i
lÞŠT
. So, for each iteration, we
need to perform nk gradient experiments.
C k G
r
z
u y
Fig. 3. Laplace transform representation of the feedback system.
G
0r y
( )C k
( )C k
y yout
Fig. 4. Set-up for the exact gradient experiment.
S. Zhang et al. / ISA Transactions 51 (2012) 609–621614

7. (4) Calculate dJ=dkðk
i
Þ as
dJ
dk
ðk
i
Þ ¼ À
1
N
XN
t ¼ 1
½yn
t ŠT
Pt ~yt
#
,
and then follow the iterative algorithm below to calculate the
next optimal controller parameters for i¼iþ1:
k
iþ 1
¼ k
i
ÀgiRÀ1
i
dJ
dk
ðk
i
Þ:
Then repeat (1)–(4) with ki þ 1
, for i¼0,1,2,y until certain
termination criterion regarding J(k) is satisfied. The flow chart of
the IFT algorithm for nominal condition is shown in Fig. 5.
The case when both controller and plant are nonlinear will be
henceforth referred to as the actual condition. Let the nonlinear
controller representation be denoted as f(k,yt,rt), where f(k,yt,rt) is the
time-domain operator mapping output and setpoints vectors into
control signal vector, parameterized by some parameter vector
kARnk
. For simplicity, we omit yt, rt and use f(k) to represent f(k,yt,rt)
for illustration. The procedure of IFT for actual condition is similar to
the nominal condition above, while involving the linearization of the
nonlinear controller around operation condition. The procedure of
IFT for the actual nonlinear system is as follows:
(0) Initialization: choose some initial set of controller parameters
k0
and specify setpoints rt as some fixed values, same as
nominal condition.
(1) Normal experiment: At each iteration i, perform a closed-loop
experiment using specified rt as the input and collect corre-
sponding output response ~yt. The closed-loop system used
here consists of the nonlinear plant and the nonlinear con-
troller f(k) with k set to the value ki
.
(2) Controller linearization: Linearize the nonlinear controller f(k)
around operating condition to obtain the linearized controller
fL(k) and its Laplace transform transfer function matrix
FL(k)FL(s,k). Similarly to the nominal condition above, use
the output ~yt measured in step (1) as input and calculate the
corresponding output according to ~yout,t. Here ~yout,s ¼ F0
Lðk
i
Þ~ys
and F0
Lðk
i
Þ ¼ @ FLðk
i
Þ=@k, where F0
Lðk
i
Þ is the partial derivative
transfer function matrix of FL(k) with respect to k at ki
.
(3) Gradient experiments: Follow the same procedure as the step
(3) of the nominal condition except that the ~yout,t in step (2) is
generated by the partial derivative of the linearized controller
F0
Lðk
i
Þ and all the nk closed loop experiments in this step are
performed in terms of the original nonlinear controller f(k)
and the nonlinear plant.
(4) Controller parameter update: Use the identical update routine
in step (4) of nominal condition to calculate the next optimal
controller parameter set kiþ 1
.
Then repeat (1)–(4) again with kiþ 1
, for i¼0,1,2,y until the
same termination criteria regarding J(k) are satisfied.
The IFT procedure for actual condition follows the same flow
chart in Fig. 5 as well.
Under the actual condition the plant is nonlinear and infinite-
dimensional due to existence of time delays. However, using the
corresponding procedure above for the actual condition, this
poses no problem for the applicability of the IFT technique. This
is due to the nonlinearity around the operating point being
smooth (differentiable) and relatively weak, making the plant
well linearizable. Also, the delay can be accurately finite-dimen-
sionalized, with the resulting linear plant approximation being
controllable and observable. This results in the closed-loop
operation with both the linearization and the original nonlinear
infinite-dimensional plant being asymptotically stable and almost
identical in terms of time domain responses to practically mean-
ingful changes in setpoints.
2.2. Application of IFT to a specific PID cluster
The IFT procedure described in the previous section has been
applied to a typical, but simplified, PID based boiler/turbine
control system cluster shown in Fig. 6. A real control system
has many additional details which have not been included in this
structure. The cluster is seen to consist of four control loops,
namely, the ones for power output, throttle pressure, drum level,
and excess oxygen. The drum level control and excess oxygen
control ones are cascade loops with feedwater flow rate and air
flow rate as the inner controlled variables, respectively. A non-
linear ratio controller is included for the excess oxygen control. A
total of six PID-type controllers are used, with five of them being
tuned as PI controllers and one as a PID controller. This provides a
total of 13 tuning parameters to be adjusted.
The model is an incremental one describing the dynamics of all
the deviation variables with respect to the nominal operating
condition and designed to represent a 250 MW plant dynamics
around 80% of the operating point. The model dynamics were
selected based on [29] and the authors’ experience with similar
plants. Additional details about the process can be found in [29].
The nominal operating point values are specified as follows:
Megawatt output ¼ 200 MW,
Throttle pressure ¼ 12:5 Â 106
Pa,Fig. 5. Flowchart of the standard IFT procedure.
S. Zhang et al. / ISA Transactions 51 (2012) 609–621 615

8. Steam flow rate ¼ 80%,
Excess oxygen ¼ 3%,
Air flow rate ¼ 80%,
Drum level ¼ 0 m,
Feedwater flow rate ¼ 80%,
and all inputs are 80%:
The process outputs in this model are: Dy1—MW, unit load
(megawatts), Dy2—TP, throttle pressure (Pa), Dy3—SF, steam flow
rate (%), Dy4—O2, excess oxygen (%), Dy5—AF, air flow rate (%),
Dy6—DL, drum level (m), Dy7—FW, feedwater flow rate (%).
The controller inputs to the process are: Du1—TV, turbine
valve demand (%), Du2—FR, firing rate demand (%), Du3—FD, FD
fan damper demand (%), Du4—FWV, feedwater valve position
demand (%), and k—controller parameter vector.
This control system model structure provides a simple but
non-trivial testbed for the multi-loop tuning method. We define
the nonlinear controller as containing a lookup table, bias and a
multiplication operator as shown in Fig. 6. Thus, the controller in
Fig. 6 is given by the six-PID cluster that includes one lookup table
and one multiplication operator and two biases, making the
cluster nonlinear. In order to apply IFT, a linearized prefilter
F0
L kð Þ defined in step (2) of the IFT procedure for the actual
condition needs to be constructed that would preprocess the data
for the application of the IFT to the actual closed loop system. In
the present work this prefilter is constructed on the basis of the
linearized PID cluster model fL(k). For this purpose, we linearized
the nonlinear PID cluster f(k) around the system operating point
to obtain fL(k). We indexed the 6 different PIDs from top to
bottom as 1–6. Therefore PID1 is the MW controller, PID2 is the
throttle pressure controller, PID3 is the air flow rate controller,
PID4 is the excess oxygen controller, PID5 is the drum level
controller, and PID6 is the feedwater flow rate controller. Then
the Laplace transfer function matrix of the linearized controller
cluster FL(k) is shown in Fig. 7.
In Fig. 7, we have the following definitions:
us ¼
Du1
Du2
Du3
Du4
2
6
6
6
6
4
3
7
7
7
7
5
, es ¼ DrsÀDys ¼
Dr1ÀDy1
Dr2ÀDy2
Dr3ÀDy3
Dr4ÀDy4
Dr5ÀDy5
Dr6ÀDy6
Dr7ÀDy7
2
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
5
, ð2:7Þ
FLðkÞ ¼
PID1 0 0 0 0 0 0
0 PID2 À1 0 0 0 0
0 PID2 Â PID3 ÀPID3 PID4 Â PID3 PID3 0 0
0 0 ÀPID6 0 0 PID5 Â PID6 PID6
2
6
6
6
6
4
3
7
7
7
7
5
:
ð2:8Þ
where FL(k) is the Laplace transfer function matrix of the linear-
ized controller fL(k) and us and es are the Laplace transforms of
time domain control signals ut and error signals et, respectively.
We denote the three gains in each PID as ki j, i¼1, y, 6, j¼1,2,3,
where i represents the PID number and j represents the corre-
sponding proportional, integral and differential gains. In Laplace
notation each PID has continuous system representation as follows:
PIDi ¼ ki1 þ
ki2
s
þki3s:
The initial controller parameters were chosen empirically and
were deliberately suboptimal. They are given as
k11 ¼ 1, k12 ¼ 0:1, k21 ¼ 0:05, k22 ¼ 0:00001, k23 ¼ 2,
k31 ¼ 4, k32 ¼ 0:1,k41 ¼ 4, k42 ¼ 0:4, k51 ¼ 1, k52 ¼ 0:0001,
k61 ¼ 20, k62 ¼ 0:6 ð2:9Þ
Fig. 6. SIMULINK representation of the nonlinear PID cluster to be tuned.
C k
e r y u
Fig. 7. Laplace transform representation of the linearized controller.
S. Zhang et al. / ISA Transactions 51 (2012) 609–621616

9. Therefore, there are five PIs and only one PID, i.e. PID2, in the
cluster. Henceforth, we drop the D for simplicity unless confusion
occurs. The controller tuning criterion has been selected as
J ¼
1
2 Â 2001
X2000
t ¼ 0
eT
1P1e1 þeT
2P2e2 þeT
4P4e4 þeT
6P6e6
 Ã
ð2:10Þ
where the error signals are ei ¼ri Àyi, i¼1,2,4,6, and weighting
matrices are defined as
P1 ¼ 10I, P2 ¼ I, P4 ¼ 10I, P6 ¼ 100I ð2:11Þ
where I is the identity matrix. The optimal controller parameter
vector is then obtained by
k
nþ 1
ij ¼ k
n
ijÀgn
ijðRn
ijÞÀ1 dJ
dkij
ðk
n
ijÞ, i ¼ 1,. . .,6, j ¼ 1,2,3 ð2:12Þ
where n¼0, 1, y is the iteration number and k
0
ij is given in (2.9).
Then, the estimated gradient of J(k) for each particular kij is
obtained by
dJ
dkij
ðk
n
ijÞ ¼ À
1
2001
X2000
t ¼ 0
@y1
@kij
ðk
n
ijÞ
h iT
P1e1ðk
n
ijÞþ @y2
@kij
ðk
n
ijÞ
h iT
P2e2ðk
n
ijÞ
þ @y4
@kij
ðk
n
ijÞ
h iT
P4e4ðk
n
ijÞþ @y6
@kij
ðk
n
ijÞ
h iT
P6e6ðk
n
ijÞ
0
B
B
@
1
C
C
A:
ð2:13Þ
The Rn
ij is an approximation of the Hessian of J(k), which is
chosen as in [9]:
Rn
ij ¼
1
2001
X2000
t ¼ 0
@y1
@kij
ðk
n
ijÞ
T
P1
@y1
@k
ðk
n
ijÞþ @y2
@kij
ðk
n
ijÞ
T
P2
@y2
@kij
ðk
n
ijÞ
þ @y4
@kij
ðk
n
ijÞ
T
P4
@y4
@kij
ðk
n
ijÞþ @y6
@k
ðk
n
ijÞ
T
P6
@y6
@kij
ðk
n
ijÞ
2
6
6
4
3
7
7
5:
ð2:14Þ
Nine iterations were performed until J(k) stops to decrease.
Constant step size was used for each controller parameter, i.e. gn
ij
are the same for all iterations n¼0,1,y. The step size for each
controller parameter is defined as
gn
11 ¼ 1, gn
12 ¼ 0:01, gn
21 ¼ 1, gn
22 ¼ 1, gn
23 ¼ 1, gn
31 ¼ 1,
gn
32 ¼ 0:1, gn
41 ¼ 1, gn
42 ¼ 1, gn
51 ¼ 1, gn
51 ¼ 0:001, gn
61 ¼ 1,
gn
62 ¼ 0:1: for n ¼ 0,1,2,. . .
In each iteration step, as many gradient experiments are
needed to be carried out as there are parameters to tune, which
is 13 in our case, namely one normal experiment and 13 gradient
experiments. The normal experiment setup is shown in Fig. 8.
In the normal experiment, the setpoints are set as follows:
r1 ¼ 10 MW, r2 ¼ 6895 Pa, r4 ¼ 3%, r6 ¼ 0:0254 m,
r3 ¼ r5 ¼ r7 ¼ 0:
Here, the external signal z is not used. Then, the error signals
et(k) are obtained from this closed-loop experiment.
The setup for gradient experiments is shown in Fig. 9.
In each gradient experiment, the error signals e generated
from the normal experiment are filtered through a linear filter
whose Laplace transfer function matrix is F0
LðkijÞ which is the
derivative of the linearized controller transfer function matrix
FL(k) in Eq. (2.8) with respect to a particular controller parameter
kij where kij ¼ k
n
ij. These filtered signals, working as the external
signal z, are then used as input to the original closed-loop system,
while setting the setpoint r to zero, that is r1 ¼ r2 ¼r3 ¼r4 ¼r5 ¼
r6 ¼r7 ¼0. According to IFT, the corresponding output signal y that
is obtained in each gradient experiment is the estimate of the
ð@yt=@kijÞðk
n
ijÞ. After ð@yt=@kijÞðk
n
ijÞ is obtained, the gradient estimate
dJðk
n
ijÞ=dkij is calculated according to Eq. (2.13).
In Figs. 8 and 9 the closed loop represents the dynamics of the
actual controller and plant, whereas the linearized filter in Fig. 9 is
part of the IFT tuning software. The simulation results of the
above procedure are presented in Section 4.
3. Simulation testbed: process and controller models
Although the process model used in this work is a simplified
one based on transfer functions, it provides a realistic platform for
the comparison between the sequential heuristic and the simul-
taneous optimization-based multi-loop tuning methods. The
process model is shown in Fig. 10 and is seen to compute the
generator power output, the turbine throttle pressure, the drum
level, the excess oxygen, and the fuel, air, and feedwater flow
rates. The inputs to the process model are the turbine valve
position, firing rate demand, forced draft fan damper demand, and
feedwater valve position demand. The model is an incremental
model and is designed to represent a 250 MW plant.
The model is nonlinear as shown in Fig. 10. Deadtimes are
included in the model, i.e. blocks ‘‘TV to MW3’’, ‘‘FR to PT2’’ and ‘‘FR
to FF2’’ to represent the time delays inherent in the processes, such
as coal pulverizer dynamics. There are cross couplings in the model
between several inputs and outputs. The turbine valve position
affects both the power output and the throttle pressure as does the
firing rate demand. The latter also affects the excess oxygen. The
power output (steam flow rate) also affects the drum level.
The control system model structure used in the closed-loop
simulation is given in Fig. 6. To carry out controller tuning, step
change inputs are provided on all setpoints, and the MW load
setpoint also includes a ramp input.
Although the IFT algorithm is developed for LTI systems, it has
been found [7] that IFT can still generate the true gradient up to
the first order if the nonlinear system can be well approximated
by its linearized model around its operating point corresponding
to a specific reference input. Such linearization of the nonlinear
model around its operating point is carried out as follows: In our
case, there are four main nonlinear components: a nonlinear gain,
delays, biases, and calculation of the air-flow/fuel-flow ratio. The
nonlinear gain, described by a lookup table in the model, can be
approximated as linear gain of 0.5 around operating point. Delays
are approximated by the fourth-order Pade approximation, i.e.
eÀ2s
%
s4
À10s3
þ45s2
À105sþ105
s4 þ10s3 þ45s2 þ105sþ105
,
eÀ30s
%
s4
À0:6667s3
þ0:2s2
À0:03111sþ0:002074
s4 þ0:6667s3 þ0:2s2 þ0:03111sþ0:002074
:
Fig. 8. Setup for the normal experiment.
0r
e
y
Nonlinear
PID Cluster
Nonlinear
Plant
Linearized Filter
- generated from normal
experiment in Fig. 8
Setpoint
Fig. 9. Setup for the gradient experiments.
S. Zhang et al. / ISA Transactions 51 (2012) 609–621 617

10. Biases are dealt with by treating all values as deviations about
a nominal value. Finally, the air/fuel flow ratio is approximated as
a summation for small deviations. The air flow and fuel flow units
are both percentage values of full-load nominal value. Since our
model is an incremental one, these units are both deviations from
operating point which is set at 80% load. Therefore, the model
does this ratio calculation as
A=F ¼
AF þ80
FF þ80
À1
where AF and FF are the air flow and fuel flow rate deviations,
respectively. However, when those deviations are small
A=F % 1
80ðAFÀFFÞ:
Thus the linearized system is obtained in the form of a 7 Â 4
transfer function matrix given by
MW
TP
SF
O2
AF
DL
FW
2
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
5
¼
H11ðsÞ H12ðsÞ 0 0
H21ðsÞ H22ðsÞ 0 0
H31ðsÞ H32ðsÞ 0 0
0 H42ðsÞ H43ðsÞ 0
0 0 H53ðsÞ 0
H61ðsÞ H62ðsÞ 0 H64 sð Þ
0 0 0 H74 sð Þ
2
6
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
7
5
TV
FR
FD
FWV
2
6
6
6
4
3
7
7
7
5
ð3:1Þ
where
Fig. 10. Schematic diagram of the simplified process model in Simulink.
H11ðsÞ ¼
900s5
À9000s4
þ4:05 Â 104
s3
À9:45 Â 104
s2
þ9:45 Â 104
s
2760s6
þ2:784 Â 104
s5
þ1:266 Â 105
s4
þ3:007 Â 105
s3
þ3:153 Â 105
s2
þ2:551 Â 104
sþ105
,
H12ðsÞ ¼
0:0001454s4
À9:691 Â 10À5
s3
þ2:907 Â 10À5
s2
À4:523 Â 10À6
sþ3:015 Â 10À7
s6 þ0:6747s5 þ0:2054s4 þ0:03273s3 þ0:00233s2 þ1:762 Â 10À5
sþ6:844 Â 10À8
,
H21ðsÞ ¼
À3:677s2
À0:0352sÀ0:001043
5s3 þ1:042s2 þ0:008645sþ4:9 Â 10À5
,
H22ðsÞ ¼
0:000603s4
À0:000402s3
þ0:0001206s2
À1:876 Â 10À5
sþ1:251 Â 10À6
s6 þ0:6747s5 þ0:2054s4 þ0:03273s3 þ0:00233s2 þ1:762 Â 10À5
sþ6:844 Â 10À8
,
H31ðsÞ ¼
180s5
À1800s4
þ8100s3
À1:89 Â 104
s2
þ1:89 Â 104
s
2760s6
þ2:784 Â 104
s5
þ1:266 Â 105
s4
þ3:007 Â 105
s3
þ3:153 Â 105
s2
þ2:551 Â 104
sþ105
,
H32ðsÞ ¼
2:907 Â 10À5
s4
À1:938 Â 10À5
s3
þ5:815 Â 10À6
s2
À9:045 Â 10À7
sþ6:03 Â 10À8
s6 þ0:6747s5 þ0:2054s4 þ0:03273s3 þ0:00233s2 þ1:762 Â 10À5
sþ6:844 Â 10À8
,
H42ðsÞ ¼
À0:008151s4
þ0:005434s3
À0:00163s2
þ0:0002536sÀ1:691 Â 10À5
25s8
þ226:5s7
þ226:7s6
þ105:6s5
þ28:51s4
þ4:664s3
þ0:4349s2
þ0:01827sþ6:762 Â 10À5
,
H43ðsÞ ¼
0:25
3600s4 þ2040s3 þ409s2 þ34sþ1
,
S. Zhang et al. / ISA Transactions 51 (2012) 609–621618

11. The staircase algorithm [30] is then used to determine the
controllability and observability of the linearized system after
transforming the transfer function into a state-space representation
characterized by 52 state variables. From staircase algorithm, it has
been shown that there are also 52 controllable states and 52
observable states. Therefore, by the definitions of the controllability
and observability [31], the linearized system is both controllable and
observable, although it is essentially a 4 Â 4 system since only four
out of seven outputs need to track the setpoint changes, whereas the
rest three can be viewed as internal variables only.
In most practical conditions, the variables of the nonlinear
process in Fig. 10 will stay in the vicinity of its operating point
where the linearized model above is a good approximation of its
nonlinear counterpart. It is, therefore, likely that IFT would per-
form adequately, showing a degree of robustness to nonlinearities.
This is indeed demonstrated by the simulation results in Section 4.
4. Tuning results and comparisons
The results of the IFT method and their comparison with the
results of the empirical tuning procedure are presented next.
Comparisons are provided for both the control system figure of
merit and the time response plots of the controlled variables.
The IFT procedure was applied to the closed loop system
consisting of the nonlinear plant of Fig. 10 and the nonlinear
PID cluster of Fig. 6, with a linear prefilter outside of the loop, as
shown in Fig. 8 (normal experiment) and 9 (gradient experiment)
in Section 2.2. Table 2 shows the design criteria and controller
parameter values during each iteration step.
The closed-loop system was tuned according to the control
performance requirements given in Table 1. The simulation
results of the closed-loop system response subject to 5%/min
ramp unit load setpoint change from 80% to 90% along with the
initial, the IFT tuned, and the manually tuned PID controllers
are shown in Fig. 11. The ‘‘initial’’, ‘‘Manual’’ and ‘‘IFT’’ in this
figure mean that the simulation results are obtained with the
PID controller parameters set to the original, manually tuned
and the IFT tuned values, respectively. The manually tuned PID
parameters were provided by some experienced field engineer.
The simulation results summarized in Table 2 show that the
minimum for the criterion is achieved after eight iterations, with
the reduction of the value of J from 87.2583 to 7.9814. In the ninth
iteration, the criterion value starts increasing. Therefore, after nine
H53ðsÞ ¼
1
144s2 þ24sþ1
,
H53ðsÞ ¼
1
144s2 þ24sþ1
,
H61ðsÞ ¼
31:5s6
À317:7s5
þ1445s4
À3429s3
þ3591s2
À283:5s
4:347 Â 106
s10
þ4:505 Â 107
s9
þ2:116 Â 108
s8
þ5:297 Â 108
s7
þ6:32 Â 108
s6
þ1:886 Â 108
s5
þ2:323 Â 107
s4
þ1:305 Â 106
s3
þ2:94 Â 104
s2
þ105s
,
H62ðsÞ ¼
5:088 Â 10À6
s5
À3:828 Â 10À6
s4
þ1:308 Â 10À6
s3
À2:455 Â 10À7
s2
þ2:412 Â 10À8
sÀ9:045 Â 10À10
1575s10
þ1498s9
þ653:9s8
þ166:9s7
þ26:18s6
þ2:458s5
þ0:1267s4
þ0:003011s3
þ2:015 Â 10À5
s2
þ6:844 Â 10À8
s
,
H64ðsÞ ¼
À0:0875sþ0:0075
945s5 þ828s4 þ246s3 þ28s2 þs
,
H74ðsÞ ¼
0:5
9s2 þ6sþ1
:
Table 2
Iteration data.
Iteration Criteria, J k11 k12 k21 k22 k23 k31
0 87.2583 1 0.1 0.05 0.00001 2 4
1 47.1814 1.1401 0.1014 0.0541 0.00001 5.7345 3.8590
2 29.9074 1.1372 0.1016 0.0719 0.00001 8.6881 3.5960
3 21.2190 1.0811 0.1014 0.1025 0.00001 10.0482 3.2901
4 15.8913 1.0141 0.1011 0.1415 0.00001 10.7007 2.9827
5 12.1564 0.9504 0.1007 0.1902 0.00001 10.9887 2.6593
6 9.5995 0.8904 0.1001 0.2511 0.00001 11.0151 2.3146
7 8.2462 0.8349 0.0995 0.3485 0.00001 10.8606 1.9491
8 7.9814 0.7853 0.0987 0.3642 0.00001 10.6147 1.5766
9 8.1092 0.7444 0.0981 0.3777 0.00001 10.3517 1.2295
Iteration k32 k41 k42 k51 k52 k61 k62
0 0.1 4 0.4 1 0.0001 20 0.6
1 0.0988 3.8563 0.3666 0.9904 9.9744eÀ5 19.8587 0.5851
2 0.0919 3.5887 0.2465 0.9362 9.9753eÀ5 19.5989 0.5594
3 0.0858 3.2754 0.1873 0.8690 9.9795eÀ5 19.2954 0.5306
4 0.0821 2.9609 0.1794 0.8118 9.9838eÀ5 18.9878 0.5027
5 0.0793 2.6295 0.1826 0.7699 9.9870eÀ5 18.6603 0.4745
6 0.0772 2.2740 0.1903 0.7389 9.9891eÀ5 18.3050 0.4456
7 0.0754 1.8897 0.2003 0.7130 9.9905eÀ5 17.9140 0.4156
8 0.0739 1.4767 0.2126 0.6889 9.9917eÀ5 17.4811 0.3845
9 0.0730 1.0449 0.2265 0.6688 9.9928eÀ5 17.0111 0.3536
S. Zhang et al. / ISA Transactions 51 (2012) 609–621 619

12. steps iterations are stopped and the controller parameters in the
eighth iteration are adopted as the optimal ones and substituted
directly into the original nonlinear PID cluster to test its performance.
If we use the manually tuned PID parameters, the J value is 55.5546
compared to 7.9814 by IFT as shown in Table 3. It can also be seen
from the plots in Fig. 11 that the system dynamic performance
corresponding to ramp unit load setpoint change has improved
significantly with the IFT optimized PID controller parameters,
especially the throttle pressure that has much less oscillation and
settles down much faster compared to manual tunings. While the
excess oxygen time response with IFT tuned parameters, although
satisfactory, is not superior to those with manually tuned gains, we
can improve its performance by putting a greater corresponding
weight matrix in the control criterion Eq. (2.10).
0 1000 2000
0
5
10
15
Time (s)
r1(MegawattOutput/MW)
Setpoint
0 1000 2000
-2
-1
0
1
2
x 105
Time (s)
r2(ThrottlePressure/Pa)
Setpoint
0 1000 2000
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time (s)
r4(ExcessOxygen/%)
Setpoint
0 1000 2000
-0.01
0
0.01
Time (s)
r6(DrumLevel/m)
Setpoint
0 1000 2000
0
5
10
15
Time (s)
y1(MegawattOutput/MW)
Initial
0 1000 2000
-2
-1
0
1
2
x 105
Time(s)
y2(ThrottlePressure/Pa)
Initial
0 1000 2000
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time (s)
y4(ExcessOxygen/%)
Initial
0 1000 2000
-0.01
0
0.01
Time (s)
y6(DrumLevel/m)
Initial
0 1000 2000
0
5
10
15
Time (s)
y1(MegawattOutput/MW)
Manual
0 1000 2000
-2
-1
0
1
x 105
Time(s)
y2(ThrottlePressure/Pa)
Manual
0 1000 2000
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time(s)
y4(ExcessOxygen/%)
Manual
0 1000 2000
-0.01
0
0.01
Time (s)
y6(DrumLevel/m)
Manual
0 1000 2000
0
5
10
15
Time (s)
y1(MegawattOutput/MW)
IFT
0 1000 2000
-2
-1
0
1
2
x 105
Time(s)
y2(ThrottlePressure/Pa)
IFT
0 1000 2000
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Time (s)
y4(ExcessOxygen/%)
IFT
0 1000 2000
-0.01
0
0.01
Time (s)
y6(DrumLevel/m)
IFT
Fig. 11. Comparison of closed-loop responses under 5%/min ramp unit load setpoint change with the original, manually tuned, and IFT tuned PID controllers. The units are:
megawatts for y1 and r1, Pa for y2 and r2, % for y4 and r4, and meters for y6 and r6.
Table 3
Comparison of initial, manually tuned, and IFT tuned PID gains in the nonlinear cluster used in the closed-loop simulation and their
corresponding objective function values.
k11 k12 k21 k22 k23 k31 k32
Initial 1 0.1 0.05 0.00001 2 4 0.1
Manual 0.7 0.06 0.035 0.00004 6 4 0.1
IFT 0.7853 0.0987 0.3642 0.00001 10.6147 1.5766 0.0739
k41 k42 k51 k52 k61 k62 J
Initial 4 0.4 1 0.0001 20 0.6 81.7085
Manual 4 0.55 1.3 0.005 10 1 52.5163
IFT 1.4767 0.2126 0.6889 9.9905eÀ5 17.4811 0.3845 5.2768
S. Zhang et al. / ISA Transactions 51 (2012) 609–621620

13. 5. Conclusions
The general area of automatic multi-loop PID tuning shows an
unexpectedly strong promise. The basic IFT configuration, one of
the simplest techniques in this area, has demonstrated excellent
tuning capability for a nonlinear MIMO PID cluster of the inter-
mediate complexity controlling a simplified nonlinear model
representing power plant dynamics at a single operating point.
The technique shows good flexibility in shaping the time domain
responses of system components according to specifications
through the appropriately chosen static and/or dynamic weight-
ing of the individual terms in the performance index.
While a comparison was done against an empirically tuned
system, a multi-loop tuning method may provide substantial
benefits even if its performance is the same as or only slightly
better than an empirically tuned system. This is because it may be
faster to use the analytical tuning method than the empirical one
which may reduce the time a unit is required to operate off of
automatic generation control. In addition, once the IFT method is
set up for a given unit, it may be relatively easy to repeat the
tuning process as needed, such as after a major outage. This
would encourage and enable plant personnel to maintain the
tuning on their units in a near optimum condition most of
the time.
Acknowledgement
This research was supported by EPRI contract EP-P25830/
C12487 and NSF grants ECS-0501407 and DMI-0900138.
References
[1] Johnson MA, Moradi MH, editors. PID control: new identification and design
methods. London: Springer-Verlag; 2005.
[2] Visioli A. Practical PID control. London: Springer-Verlag; 2006.
[3] Yu C-C. Autotuning of PID controllers: a relay feedback approach. 2nd ed.
London: Springer-Verlag; 2006.
[4] De Bruyne F, Anderson BDO, Gevers M, Linard N. Iterative controller
optimization for nonlinear systems. In: Proceedings of the 36th IEEE
conference on decision and control, vol. 4, San Diego, CA, 1997. p. 3749–54.
[5] Hjalmarsson H, Birkeland T. Iterative feedback tuning of linear time-invariant
MIMO systems. In: Proceedings of the 37th IEEE conference on decision and
control, vol. 4, Tampa, FL, 1998. p. 3893–8.
[6] Hjalmarsson H. Efficient tuning of linear multivariable controllers using
iterative feedback tuning. International Journal of Adaptive Control and
Signal Processing 1999;13:553–72.
[7] Hjalmarsson H. Iterative feedback tuning—an overview. International Journal
of Adaptive Control and Signal Processing 2002;16:373–95.
[8] Lequin O, Gevers M, Mossberg M, Bosmans E, Triest L. Iterative feedback
tuning of PID parameters: comparison with classical tuning rules. Control
Engineering Practice 2003;11:1023–33.
[9] Nakamoto M. Parameter tuning of multiple PID controllers by using iterative
feedback tuning. In: Proceedings of the SICE annual conference in Fukui, 4–6
August, Fukui University, Japan, 2003, p. 183–6.
[10] Jansson H, Hjalmarsson H. Gradient approximations in iterative feedback
tuning for multivariable processes. International Journal of Adaptive Control
and Signal Processing 2004;18:665–81.
[11] Crowe J, Johnson MA, Grimble MJ. PID parameter cycling to tune industrial
controllers—a new model-free approach. In: Proceedings of the 13th IFAC
symposium on system identification, Rotterdam, The Netherlands, 27–29
August 2003.
[12] Wang Q-G, Zou B, Lee T-H, Bi Q. Auto-tuning of multivariable PID controllers
from decentralized relay feedback. Automatica 1997;33:319–30.
[13] Wang J, Qin SJ. Closed-loop subspace identification using the parity space.
Automatica 2006;42:315–20.
[14] Sanchez A, Katebi MR, Johnson MA. Subspace identification based PID
controller tuning. In: Proceedings of the IFAC symposium on system
identification, Rotterdam, The Netherlands, 27–29 August 2003.
[15] Sanchez A, Katebi MR, Johnson MA. A tuning algorithm for multivariable
restricted structure control systems using subspace identification. Interna-
tional Journal of Adaptive Control and Signal Processing 2004;18:745–70.
[16] G ¨undes AN. Simultaneous stabilization of MIMO systems with integral action
controllers. Automatica 2008;44:1156–60.
[17] Kim T-H, Maruta I, Sugie T. Robust PID controller tuning based on the
constrained particle swarm optimization. Automatica 2008;44:1104–10.
[18] Bianchi FD, Mantz RJ, Christiansen CF. Multivariable PID control with set-
point weighting via BMI optimization. Automatica 2008;44:472–8.
[19] Mattei M. Robust multivariable PID control for linear parameter varying
systems. Automatica 2001;37:1997–2003.
[20] Zheng F, Wang Q-G, Lee TH. On the design of multivariable PID controllers via
LMI approach. Automatica 2002;38:517–26.
[21] Saeki M. Fixed structure PID controller design for standard HN control
problem. Automatica 2006;42:93–100.
[22] Hwang C, Hsiao C-Y. Solution of a non-convex optimization arising in PI/PID
control design. Automatica 2002;38:1895–904.
[23] Zhang W, Xi Y, Yang G, Xu X. Design of PID controllers for desired time-
domain or frequency-domain response. ISA Transactions 2002;41:511–20.
[24] Barenthin M, Hjalmarsson H. Identification and control: joint input design
and HN state feedback with ellipsoidal parametric uncertainty via LMIs.
Automatica 2008;44:543–51.
[25] Gevers M, Miˇskovic L, Bonvin D, Karimi A. Identification of multi-input systems:
variance analysis and input design issues. Automatica 2006;42:559–72.
[26] Hildebrand R, Solari G. Identification for control: optimal input intended to
identify a minimum variance controller. Automatica 2007;43:758–67.
[27] Lecchini A, Lanzon A, Anderson BDO. A model reference approach to safe
controller changes in iterative identification and control. Automatica
2006;42:193–203.
[28] Dehghani A, Lanzon A, Anderson BDO. HN design to generalize internal
model control. Automatica 2006;42:1959–68.
[29] Ollat X, Smoak R. Simultaneous control of throttle pressure and megawatts.
Instrumentation, Controls and Automation in the Power Industry 1991:34.
[30] Rosenbrock MM. State-space and multivariable theory. John Wiley; 1970.
[31] Dullerud Geir E, Paganini Fernando. A course in robust control theory: a
convex approach. Springer; 2000.
S. Zhang et al. / ISA Transactions 51 (2012) 609–621 621