This document proposes a cascade model predictive control scheme using generalized predictive control (GPC) for both the inner and outer loops to control boiler drum level. Cascade GPC can effectively reject measured and unmeasured disturbances to maintain drum level at a constant load. It can also handle non-minimum phase characteristics and system constraints in both loops. Simulation results show cascade GPC provides better performance than well-tuned cascade PID controllers. The method was also implemented on a 75-MW boiler plant with improved results over conventional control schemes.

1. Cascade generalized predictive control strategy
for boiler drum level
Min Xu,a
Shaoyuan Li,a,
* Wenjian Caib
a
Institute of Automation, Shanghai Jiao Tong University, Shanghai 200030
b
School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798
͑Received 13 July 2003; accepted 6 December 2004͒
Abstract
This paper proposes a cascade model predictive control scheme for boiler drum level control. By employing
generalized predictive control structures for both inner and outer loops, measured and unmeasured disturbances can be
effectively rejected, and drum level at constant load is maintained. In addition, nonminimum phase characteristic and
system constraints in both loops can be handled effectively by generalized predictive control algorithms. Simulation
results are provided to show that cascade generalized predictive control results in better performance than that of well
tuned cascade proportional integral differential controllers. The algorithm has also been implemented to control a
75-MW boiler plant, and the results show an improvement over conventional control schemes. © 2005 ISA—The
Instrumentation, Systems, and Automation Society.
Keywords: Boiler water level; Cascade control; Generalized predictive control; PID control
1. Introduction
In boiler plants, there are several process vari-
ables, such as main steam temperature, drum level,
and stack gas oxygen content ͓1͔. Among them,
the drum level is an important variable to be con-
trolled such that it is within the pre-specified
safety limits. In practice, this is a difficult control
task due to the existence of constraints on the
amount of feedwater flow dictated by the pump
rating, parameter uncertainties, and modeling in-
accuracies ͓2,3͔. Cascade control schemes are of-
ten adopted to overcome these difficulties ͓4–6͔:
• Disturbances arising in the inner loop are
corrected by the same loop controller before
they influence system output variable.
• The speed and accuracy of system response
is much improved if the inner loop exhibits a
faster dynamic response than that of outer
loop.
• The inner feedback control loop can also re-
duce the effect of parameter variations in the
inner loop.
Conventionally, the cascade proportional inte-
gral differential ͑PID͒ approach is usually imple-
mented for drum level control. However, this con-
trol scheme may be unsatisfactory when dealing
with processes with large time delay and con-
straints.
Today, model predictive control ͑MPC͒ has re-
ceived wide acceptance in the process industries
because of its time domain, optimization-based
formulation, and natural ability to handle con-
straints ͓7͔. The advantage of MPC is that it can
solve simultaneously control and optimization
problems. As an optimizer, it maximizes a produc-
*Corresponding author. E-mail address:
syli@sjtu.edu.cn
ISA
TRANSACTIONS®
ISA Transactions 44 ͑2005͒ 399–411
0019-0578/2005/$ - see front matter © 2005 ISA—The Instrumentation, Systems, and Automation Society.

2. tion index. As a controller, it can be tuned to
achieve stability, robustness, and performance of
plant mismatch, failures, and disturbances ͓8͔.
In this paper, a cascade control structure, in
which both the inner loop and the outer one con-
sist of generalized predictive control ͑GPC͒, for
boiler drum level control is presented. GPC has
several of the desired attributes of MPC. The ob-
jectives for two loops are to maintain drum level
at constant load and to reject measured and un-
measured disturbances of the flow system.
In addition, GPC in both loops can handle non-
minimum phase characteristic and system con-
straints. Through predicting system output via a
receding horizon method over several sampling in-
tervals, an optimal control at every time can be
obtained. Simulation results are provided to show
that cascade GPC gives better performance than
that of well tuned cascade PID. The algorithm has
also been implemented to control a 75-MW boiler
plant; the results showed an improvement over
conventional control schemes.
2. Process description and control scheme
A 75-MW circulating fluidized bed combustion
boiler ͑CFBB͒ mainly consisting of a feedwater
system, a steam temperature system, and a com-
bustion system, is shown in Fig. 1. The separator
feedwater from the deaerator, driven by pumps,
circulates naturally and passes through the mixer
and the water wall to absorb heat from the furnace.
Thereafter, steam generated in the water wall is
separated in a boiler drum from where it flows
through primary and secondary superheaters to a
high-pressure ͑HP͒ turbine. It then re-enters the
boiler to be heated again to increase the energy
and flows through the intermediate and low-
pressure ͑LP͒ turbine to a condenser. Hence the
water goes through the feedwater valve, econo-
mizer, mixer, recycling pump, water wall, and ar-
rives at the steam separator. The key process vari-
ables are the superheated outlet temperature, the
main steam pressure, and the main steam flow,
which are 540 °C, 16.9 Mpa, and 50 t/h, respec-
tively.
The combustion products path is also shown.
Since the water wall absorbs radiant heat in the
furnace, the hot gases leaving the furnace transfer
heat by convection and radiation to the secondary
superheater, reheater, primary superheater, and
economizer in succession.
One control problem is to overcome the shrink
and swell effects, which are more prominent at
start up and the low power range of operation. The
drum level, measured in the downcomer, tempo-
rarily reacts in a reverse manner in response to
water change. This phenomenon is due to the two-
phase mixture of steam and water present in the
boiler drum.
The other problem is the nonlinear relation be-
tween valve position and feedwater flow, which
can be described as a subsection-parabolic func-
tion, as the valve position steps from 0% to 100%.
Because of the large capacity of this type of boiler,
the drum level needs a large time to adjust through
Fig. 1. The schematic diagram of boiler system.
400 Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411

3. valve position variations. Hence the drum level
loop can be considered as a system with a large
time constant and time delay.
Furthermore, some disturbances, such as main
steam flow, feedwater flow, and pressure varia-
tions, etc., contribute to model uncertainty and
nonlinearity.
To sum up, the drum level process is character-
ized by the following aspects:
• the main steam flow acts as the main distur-
bance with respect to the dominant time con-
stant of the process, and affects control re-
sults;
• dynamic response is rather slow and varies
with the operation point;
• there exist shrink and swell effects;
• plant has large time delay and constraints;
• time varying I/O transport delay that de-
pends on manipulated variable ͑feedwater
flow͒.
The control objective is to eliminate drum level
windage quickly and maintain drum level to a pre-
specified set point, however, these process charac-
teristics call for a carefully designed and practical
control strategy, preserving enough robustness to
cope with uncertainty in the plant.
Because GPC control is an on-line optimization
approach to satisfy multiple, changing perfor-
mance criteria, under the existing boiler hardware
structure, once using cascade GPC, both problems
are split apart and render the controller tasks sim-
pler.
As far as the drum level is concerned, the cas-
cade GPC scheme is shown in Fig. 2. The inner
loop is the feedwater flow-valve position system
and the output y2 is the feedwater flow F, while
the outer loop is the drum level water flow system
and the output y1 is drum level H. A specified
drum level H* equals 0 mm that refers to a middle
value of whole drum level, and the pre-specified
set point Hr never surpasses a safety limit drum
level Hmaxϭ50 mm and HminϭϪ50 mm.
3. Cascade generalized predictive control
3.1. Basic generalized predictive control
In recent years, GPC has received much atten-
tion, which emerges as a powerful practical con-
trol technique especially in the process industry
͓9͔. Often, the time-varying boiler drum level dy-
namics can be described as a controlled auto-
regressive and integrated moving average ͑CA-
RIMA͒ model:
A͑qϪ1
͒y͑t͒ϭB͑qϪ1
͒u͑tϪd͒ϩC͑qϪ1
͒␰͑t͒/⌬,
where y(t) and u(tϪd) are the output and input,
respectively, the input is delayed by an assumed
time d, ⌬ is a differencing operator 1ϪqϪ1
. In
most cases, C(qϪ1
) equals 1 ͓10͔. ␰(t) is a uncor-
related random noise sequence with zero mean,
and A and B are polynomials with a backward
shift operator qϪ1
, i.e.,
ͭ A͑qϪ1
͒ϭ1ϩa1qϪ1
ϩ¯ϩanqϪn
B͑qϪ1
͒ϭb0ϩb1qϪ1
ϩ¯ϩbmqϪm.
Assume that predictive horizon is required for a
range of future times ͓tϩN1 ,tϩN2͔, where N1
and N2 is called a minimum and maximum costing
horizon, respectively. Then, define the following
vectors:
Yˆ ϭ͓y͑tϩN1͒,...,y͑tϩN2͔͒T
,
Uϭ͓⌬u͑t͒,...,⌬u͑tϩNuϪ1͔͒T
,
Pϭ͓p͑tϩN1͒,...,p͑tϩN2͔͒T
,
Wϭ͓␻͑tϩN1͒,...,␻͑tϩN2͔͒T
.
Fig. 2. Block diagram of cascade GPC for boiler drum level system.
401Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411

4. Therefore system predictive output can be rep-
resented by the following equation:
Yˆ ϭGUϩP,
where
Gϭ
ͫ
gN1
gN1Ϫ1 ¯ 0 0
gN1ϩ1 gN1
gN1Ϫ1 ¯ 0
Ӈ Ӈ Ӈ Ӈ Ӈ
gN2Ϫ1 gN2Ϫ2 ¯ ¯ gN2ϪNu
gN2
gN2Ϫ1 gN2Ϫ2 ¯ gN2ϪNuϩ1
ͬ.
The elements gi of a matrix G, being points on
the plant’s step response, can be computed recur-
sively from the CARIMA model assuming a zero
noise and a constant unit control input. Moreover,
a free response P(tϩj) can be simply calculated
for all j by iterating plant model, and a future
control equals the previous control variable u(t
Ϫ1).
Considering a multistage cost function
JGPCϭ ͚jϭN1
N2
͓y͑tϩj͒Ϫ␻͑tϩj͔͒2
ϩ ͚jϭ1
Nu
␭⌬u2
͑tϩjϪ1͒,
where ␻(tϩj) is a future reference trajectory,
which is a pre-specified set point yr(t), and ␭ is a
weighting upon future control increments,
ͭ ␻͑t͒ϭy͑t͒ jϭ1,...,N
␻͑tϩj͒ϭ␣␻͑tϩjϪ1͒ϩ͑1Ϫ␣͒yr͑t͒
. ͑1͒
For simplicity, let NϭN2ϪN1 , ␣ is a soften
factor ␣෈͓0,1͔.
Hence the cost function can be written
JGPCϭ͑GUϩPϪW͒T
͑GUϩPϪW͒ϩ␭UT
U.
The solution minimizing JGPC gives an optimal
suggested control increment sequence Uopt :
Uoptϭ͑GT
Gϩ␭IN͒Ϫ1
GT
ϩ͑WϪP͒.
The first output of Uopt is ⌬u(t), and the actual
control to be applied is u(t)ϭu(tϪ1)ϩ⌬u(t).
3.2. Cascade generalized predictive control
Two loops are included in cascade GPC system.
Therefore the cost function of each loop is de-
fined:
JG1͑N11 ,N21 ,Nu1͒
ϭ ͚jϭN11
N21
͓yˆ1͑tϩj͒Ϫ␻1͑tϩj͔͒2
ϩ ͚jϭ1
Nu1
␭1͑ j͓͒⌬u1͑tϩjϪ1͔͒2
, ͑2͒
JG2͑N12 ,N22 ,Nu2͒
ϭ ͚jϭN12
N22
͓yˆ2͑tϩj͒Ϫ␻2͑tϩj͔͒2
ϩ ͚jϭ1
Nu2
␭2͑ j͓͒⌬u2͑tϩjϪ1͔͒2
. ͑3͒
Assume that ⌬u1(tϩj)ϭ0 for jуNu1 and
⌬u2(tϩj)ϭ0 for jуNu2 . In order to get an op-
timal control sequence ͕u1͖ between N11 and N21 ,
we should minimize the cost function JG1 . More-
over, from N12 to N22 , the optimal control se-
quence ͕u2͖ is to be achieved through minimizing
the criterion JG2 . It is true that the time N21 is
larger than N22 .
Even as the standard generalized predictive con-
trol, Eqs. ͑2͒ and ͑3͒ can be written with matrix
form:
ͭYˆ 1ϭG1U1ϩP1
Yˆ 2ϭG2U2ϩP2
, ͑4͒
where
Yˆ 1ϭ͓y1͑tϩN11͒,...,y1͑tϩN11ϩN21Ϫ1͔͒T
,
Yˆ 2ϭ͓y2͑tϩN12͒,...,y2͑tϩN12ϩN22Ϫ1͔͒T
,
U1ϭ͓⌬u1͑t͒,...,⌬u1͑tϩNu1Ϫ1͔͒T
,
U2ϭ͓⌬u2͑t͒,...,⌬u2͑tϩNu2Ϫ1͔͒T
,
P1ϭ͓p1͑tϩN11͒,...,p1͑tϩN21͔͒T
,
P2ϭ͓p2͑tϩN12͒,...,p2͑tϩN22͔͒T
.
As seen in Fig. 2, we know that the outer loop
control variable U2 equals the inner loop set point
402 Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411

5. ␻2 , and the outer loop one Y1 should track set
point ␻1 . G1 and G2 are the step response matrix
of the external and internal system respectively.
The future reference trajectory ␻1 and ␻2 ,
which is similar to Eq. ͑1͒, is
ͭ␻1ϭ͓␻1͑tϩN11͒,...,␻1͑tϩN21͔͒T
␻2ϭ͓␻2͑tϩN12͒,...,␻2͑tϩN22͔͒T.
Substituting Eq. ͑4͒ into Eqs. ͑2͒ and ͑3͒,
JG1ϭ͑G1U1ϩP1Ϫ␻1͒T
͑G1U1ϩP1Ϫ␻1͒
ϩ␭1U1
T
U1 ,
JG2ϭ͑G2U2ϩP2Ϫ␻2͒T
͑G2U2ϩP2Ϫ␻2͒
ϩ␭2U2
T
U2 .
Finally, through computing the equations
‫ץ‬JG1 /‫ץ‬U1ϭ0 and ‫ץ‬JG2 /‫ץ‬U2ϭ0, the generalized
system optimal control variables are to be ob-
tained:
ͭUopt1ϭ͑G1
T
G1ϩ␭1IN11
͒Ϫ1
G1
T
͑␻1ϪP1͒
Uopt2ϭ͑G2
T
G2ϩ␭2IN22
͒Ϫ1
G2
T
͑␻2ϪP2͒
. ͑5͒
The proposed cascade GPC algorithm is given
as follows:
Step 1: Set a sample time of two loops T1 and T2 .
Step 2: Set a maximum, minimum predictive ho-
rizon and control horizon for two loops.
Step 3: Estimate CARIMA model to yield G1 ,G2
and P1 ,P2 .
Step 4: Compute matrix G1 ,G2 and (G1
T
G1
ϩ␭1IN11
)Ϫ1
,(G2
T
G2ϩ␭2IN22
)Ϫ1
.
Step 5: Determine control variable Uopt1 ,Uopt2
based on Eq. ͑5͒.
Step 6: Set kϭkϩ1, go back step 3.
As a matter of fact, there is no particular rule
that enables an optimal choice of N11 ,N21 ,
N12 ,N22 ; Nu1 ,Nu2 ; and ␭1 ,␭2 . Moreover, it is
possible to note following three points:
• It is better to choose N11 ,N12 , so that at
least one element of the first row of G1 ,G2
is nonzero. N11 ,N12 should be greater than
the maximum expected time delay of the
process.
• Very often Nui(iϭ1,2) is chosen so that
NuiӶN2i (iϭ1,2) and we stressed the fact
Nuiϭ1(iϭ1,2) is very interesting.
Fig. 3. Simulation module diagram.
Fig. 4. The output and control variable of inner and outer
loop ͑step reference͒.
403Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411

6. • ␭1 and ␭2 are often hard to determine. If
matrix G1
T
G1 and G2
T
G2 are itself invertible,
even ␭1ϭ␭2ϭ0 can give a solution. But in
most cases, it seems better to choose ␭1 ,␭2
very small but nonzero, so that the matrix
(G1
T
G1ϩ␭1IN11
)Ϫ1
and (G2
T
G2ϩ␭2IN22
)Ϫ1
become invertible.
4. Simulation example
In many processes, the plant can often be re-
garded as a cascade system, in which the transfer
function in the inner loop has no or a negligible
time delay, while the outer loop one has a large
time delay. To show the effectiveness of the pro-
posed method, we use a reduction drum level
model for inner and outer loop transfer functions,
given as follows ͓11͔:
Gp2ϭϪ
0.064
1ϩ80s
and Gp1ϭ
1.23467
͑1ϩ68s͒2
eϪ20s
.
The SIMULINK module in the mathematic soft-
ware MATLAB is shown in Fig. 3. Parameters are
settled for configuring the cascade GPC algorithm:
• The time constant of the inner loop is Ts
ϭ80 s, while that of the outer loop is almost
Tsϭ136 s.
• In order to show interaction effects, multiple
rates sample time should be considered. The
rule of thumb is that outer loop sample time
T1 is five times to ten times than that of
inner loop T2 . Hence let T1ϭ8 s and T2
ϭ1 s.
• Weighting on control variable is ␭1ϭ0.7
and ␭2ϭ0.5, respectively, and a soften fac-
tor for both loops is ␣ϭ0.6. If choosing a
higher value of ␭1 ,␭2 coefficient, the con-
trol system becomes more robust. Therefore
a tradeoff value between control action
weights and soften factors can be designed
to obtain a satisfactory performance.
• In the inner loop, N11ϭ1 and N21ϭ6, while
in the outer loop, parameters are N12ϭ3 and
N22ϭ10.
• Suppose that noise variance is 0.01.
Figs. 4–10 show different responses of cascade
GPC based on different types of set points.
First, we let the set point equal the step se-
quence ͑shown in Fig. 4͒. In order to examine the
Fig. 5. The output and control action of inner loop and
outer loop ͑square wave reference͒.
Fig. 6. The output of inner loop and outer loop with ramp
increase reference.
404 Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411

7. track performance along with operating condition
change, a square wave set point is introduced
͑shown in Fig. 5͒. As seen in Figs. 6 and 7, the
set-point performance is extremely good for not
only positive, but also negative ramps. The results
of Figs. 4–7 show that cascade GPC can stabilize
system output around desired trajectories with mi-
nor oscillation.
Second, a type of white noise, commonly en-
countered in real time systems, is introduced in the
inner loop. Figs. 8 and 9 show simulation results
with three kinds of set-point trajectory. Not only
does the outer controller tackle model uncertainty
problems, but the inner one rejects disturbance.
Finally, cascade PI and cascade GPC are com-
pared at the same operating conditions. Optimal PI
controller parameters using a novel auto tuning
method ͓12͔, which identified model parameters of
cascade loop through a simple relay feedback test,
are obtained as kpiϭ700, kiiϭ1, kpoϭ30, kio
ϭ12.
As seen in Fig. 10, the cascades GPC scheme
exhibits a satisfactory performance that achieves a
fast and nonoscillatory convergence of system out-
put. However, cascade PID has more than 20%
overshoots as set point steps from 0 to 1, which
may cause the actuator to switch frequently. It is
concluded that cascade GPC makes full use of ad-
vance knowledge of future requirements to
achieve improved performance over the well tuned
cascade PI controller.
5. Real-time application
The schematic of a 75-MW boiler is show in
Figs. 11 and 12. Here, serial real-time operation
results are recorded and used as examples to dem-
onstrate the effectiveness of the proposed control
scheme.
The open-loop traditional identification ap-
proach of drum level is adopted, which estimates
parameters on-line by a recursive identification al-
gorithm based on input-output real time data ͑see
the Appendix͒. The practical system block dia-
gram is described as follows:
Gp2ϭ
0.25
1ϩ227s
Gp1ϭ
0.053
͑1ϩ277s͒͑1ϩ104s͒
eϪ14s
.
As shown in Fig. 13, the objective of the inner
Fig. 7. The output of inner loop and outer loop with ramp
decrease reference.
Fig. 8. The disturbance of inner loop and output of outer
loop ͑step reference͒.
405Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411

8. loop is to maintain the inflow of water to the drum
equal to the steam flow leaving the drum by ad-
justing the valve with saturation constraints. The
problem can be formulated as
ͭmin J2ϭ͑Yˆ 2ϪP2͒T
͑Yˆ 2ϪP2͒
s.t.u2 minрA2⌬u2рu2 max
.
While in the outer loop, the system has output
constraints on level control which can be ex-
pressed as follows:
ͭmin J1ϭ͑Yˆ 1ϪP1͒T
͑Yˆ 1ϪP1͒
s.t.y1 minрG1⌬u1рy1 max
.
There are a number of disturbances that can give
rise to varying offsets in the process. For example,
water pipe and water pump may cause the feedwa-
ter bypass valve to switch on and off frequently
and often unnecessarily, load changes in steam tur-
bine, and the quality of pulverized coal, etc. In this
Fig. 9. The output of outer loop based on square wave and
ramp reference.
Fig. 10. Comparison with cascade PID algorithm ͑step and
square reference͒.
Fig. 11. The application plant.
Fig. 12. The schematic of plant operation.
406 Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411

9. application, we consider the main disturbance, i.e.,
main steam flow, which often rounds a center
value 50 tons per hour and vary between 45 and
55 tons per hour. Fig. 14 and 15 show the perfor-
mance of cascade PID and cascade GPC with all
kinds of disturbances.
First, consider drum level setpoint following
problem. Fig. 14͑a͒ shows variance of disturbance.
A set of optimal PID parameters are designed that
is similar to the simulation method. It is shown
that the output under cascade GPC is faster, and
drum level variations are much smaller ͓see Figs.
14͑c͒ and ͑d͔͒. This is because the inner loop can
effectively reject disturbances, and the system out-
put can reach a predefined set-point trajectory with
a finite horizon. As seen in Fig. 14͑b͒, as the inner
loop input, ͑i.e., manipulated variable͒, it is fit for
a practical controller requirement. At the same
time, we can see that a small variation of valve
position is achieved ͓shown in Fig. 14͑d͔͒.
Second, we test set-point regulating perfor-
mance. The main steam flow, control variable, and
a closed-loop response are seen in Figs. 15͑a͒–͑c͒,
respectively. The performance of the cascade PID
scheme is unsatisfactory, as large oscillations are
observed after the main steam flow fluctuated.
Clearly, better performance of the proposed con-
troller is achieved, while the fluctuation in the
level is within Ϯ7% and steady state is reached
with less than 6 min ͓see Fig. 15͑d͔͒.
6. Conclusion
A cascade generalized predictive controller for
boiler drum level was presented in this paper. The
inner loop used an adaptive model based predic-
tive controller, exploiting information conveyed
by accessible disturbances, while the outer loop
used a GPC controller to restrain the error from
nonlinear identification of the generalized system.
Based on drum level models, simulation results
showed that cascade GPC performed better than
the well tuned cascade PID controller. Experiment
demonstrated that a satisfactory system output and
smooth feasible control actions can be achieved.
The novel control scheme, which successfully re-
placed the well tuned cascade PID control algo-
rithm usually adopted in many boiler plants, has
been realized in a 75-MW boiler unit in China for
half a year and the performance of the system is
very good.
The cascade GPC strategy can be easily imple-
mented in other boiler-turbine units of power
plants without much modification. The research
work on the extension of the technology for other
power plant control systems is currently under in-
vestigation and the results will be reported later.
Acknowledgments
This work was supported by the National Natu-
ral Science Foundation of China under Grant No.
60474051 and the Key Technology and Develop-
ment Program of Shanghai Science and Technol-
ogy Department under Grant No. 04DZ11008, and
partly by the Specialized Research Fund for the
Doctoral Program of Higher Education of China
͑Grant No. 20020248028͒. The authors are grate-
ful to anonymous reviewers for valuable recom-
mendations.
Appendix: RLS identification algorithm
Consider the control system is characterized by
G͑s͒ϭ
K
1ϩTs
eϪ␶s
. ͑A1͒
In discrete time, this model can then be described
Fig. 13. Cascade GPC block diagram with constraints.
407Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411

10. Fig. 14. ͑a͒–͑e͒ water level with following under varying main steam flow disturbance.
408 Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411

11. Fig. 15. ͑a͒–͑e͒ water level with regulating under varying main steam flow disturbance.
409Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411

12. y͑kh͒ϭay͑khϪh͒ϩb1u͑khϪh͒
ϩb2u͑khϪ2h͒, ͑A2͒
where h is the sampling period, and
ͭ aϭeϪh/T
b1ϭK͑1ϪeϪ(hϪ␶)/T
͒
b2ϭKeϪh/T
͑e␶/T
Ϫ1͒
. ͑A3͒
For arbitrary time delay ␶, the model becomes
y͑kh͒ϭay͑khϪh͒ϩb1u͓͑kϪn͒h͔
ϩb2u͓͑kϪnϪ1͒h͔, ͑A4͒
where nϭmod(␶/h).
This form can be extended to higher order:
y͑kh͒ϩa1y͑khϪh͒ϩ¯ϩany͑khϪnh͒
ϭb1u͑khϪh͒ϩ¯ϩbnu͑khϪnh͒. ͑A5͒
This equation can be written compactly as
A͑q͒y͑kh͒ϭB͑q͒u͑kh͒, ͑A6͒
where
ͭ A͑q͒ϭqn
ϩa1qnϪ1
ϩ¯ϩan
B͑q͒ϭb1qnϪ1
ϩb2qnϪ2
ϩ¯ϩbn
. ͑A7͒
In the identification experiment, the input/output
pair is normally obtained in each sampling, it is
then convenient to compute the parameter esti-
mates recursively, and all the parameters are
grouped in the vector
␪ϭ͑a1 ,a2 ,...,an ,b1 ,...,bn͒T
,
and introduce the regression vector defined by
␸kϪ1ϭͩϪy͑khϪh͒,...,Ϫy͑khϪnh͒,
u͑khϪh͒,...,u͑khϪnh͒ ͪT
.
The estimate can be calculated recursively by
Ά
ekϭy͑kh͒Ϫ␸kϪ1
T
␪kϪ1
PkϭPkϪ1Ϫ
PkϪ1␸kϪ1␸kϪ1
T
PkϪ1
1ϩ␸kϪ1
T
PkϪ1␸kϪ1
␪kϭ␪kϪ1ϩPk␸kϪ1ek
. ͑A8͒
The RLS algorithm can be extend to the high-
order system whose model is given as follows:
y͑kTs͒ϩa1y͓͑kϪ1͒Ts͔ϩ¯ϩany͓͑kϪn͒Ts͔
ϭb1u͓͑kϪ1͒Ts͔ϩ¯ϩbnu͓͑kϪn͒Ts͔.
͑A9͒
The regression vector and parameters vector is
written
ͭ ␪ϭ͓a1 ,a2 ,...an ,b1 ,b2 ,...bn͔T
␸kϪ1ϭ͕Ϫy͓͑kϪ1͒Ts͔,...,Ϫy͓͑kϪn͒Ts͔
u͓͑kϪ1͒Ts͔,...,u͓͑kϪn͒Ts͔
ͮ
T
.
͑A10͒
Therefore the identify parameters can be ob-
tained by Eq. ͑A6͒ through substituting Eq. ͑A10͒
into Eq. ͑A7͒.
References
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steam generation process. Proceedings of the 12th
World Congress of IFAC, 1993, pp. 395–398.
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model for long term power system dynamic simula-
tion. IEEE Trans. Power Appar. Syst. 14͑1͒, 209–217
͑1999͒.
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͓7͔ Richalet, J., Industrial application of model based pre-
dictive control. Automatica 29͑5͒, 1251–1274 ͑1993͒.
͓8͔ Maciejowski, J. M., Predictive Control with Con-
straints. Prentice-Hall, Englewood Cliffs, NJ, 2001.
͓9͔ Clarke, D. W., Mohtadi, C., and Tuffs, P. S., General-
ized predictive control, Part 1: The basic algorithm,
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410 Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411

13. Min Xu was born in 1977.
She received her M.S. degrees
from Hebei University of
Technology in 2002 and now
she is a doctor candidate in
Shanghai Jiao Tong University.
Her research interests are in
the areas of chemical process
control, predictive control, and
fuzzy systems.
Shaoyuan Li was born in
1965. He received his B.S. and
M.S. degrees from Hebei Uni-
versity of Technology in 1987
and 1992, respectively, and he
received his Ph.D. degree from
the Department of Computer
and System Science of Nankai
University in 1997. Now he is
a professor of the Institute of
Automation, Shanghai Jiao
Tong University. His research
interests include fuzzy sys-
tems, nonlinear system control.
Wenjian Cai was born in
1957. He received his B.S. and
M.S. degrees from Harbin In-
stitute of Technology in 1980
and 1983, respectively, and he
received his Ph.D. degree in
Systems Engineering, Oakland
University, CA, USA in 1992.
Now he is an associate profes-
sor of the School of Electrical
& Electrical Engineering, Nan-
yang Technological University,
Singapore. His research inter-
est includes advanced process
control, fuzzy logic control, and robust control and estimation tech-
niques.
411Min Xu, Shaoyuan Li, Wenjian Cai / ISA Transactions 44 (2005) 399–411