1. ISA
TRANSACTIONS®
ISA Transactions 44 ͑2005͒ 187–198
A simplified predictive control algorithm for disturbance rejection
Futao Zhao, Yash P. Gupta*
Department of Chemical Engineering, Dalhousie University, Halifax, Canada, NS B3J 1Z1
͑Received 5 March 2004; accepted 2 August 2004͒
Abstract
Model predictive control ͑MPC͒ offers several advantages for control of chemical processes. However, the standard
MPC may do a poor job in suppressing the effects of certain disturbances. This shortcoming is mainly due to the
assumption that disturbances remain constant over the prediction horizon. In this paper, a simple disturbance predictor
͑SDP͒ is developed to provide predictions of the unmodeled deterministic disturbances for a simplified MPC algorithm.
The prediction is developed by curve fitting of the past information. A tuning parameter is employed to handle a variety
of disturbance dynamics and a procedure is presented to find an optimum value of the tuning parameter online. A
comparison is made with the commonly used disturbance prediction on three example problems. The results show that
an improved regulatory performance and zero offset can be achieved under both regular and ramp output disturbances
by using the proposed disturbance predictor. © 2004 ISA—The Instrumentation, Systems, and Automation Society.
Keywords: Model predictive control; Disturbance predictor; Disturbance rejection
1. Introduction results in poor disturbance rejection, regardless of
parameter tuning ͓3͔. Several researchers have ad-
Model predictive control ͑MPC͒ has been very dressed this problem. Wellons and Edgar ͓4͔ pro-
successful in the process industries in dealing with posed a generalized analytical predictor by using a
control problems, such as, interactions, time de- first-order or second-order transfer function to es-
lays, and constraints, which are commonly en- timate the effect of disturbance. The disturbance
countered in the chemical and petroleum pro- transfer function need to be known. Because dis-
cesses. For these processes, the objective of most turbances of chemical processes are often diverse
controllers is to regulate the effects of determinis- and time varying, it is a limitation in practice to
tic disturbances on the controlled variables. The obtain the transfer functions of disturbances
standard MPC may do a poor job in suppressing a priori. Shen and Lee ͓5͔ have proposed an adap-
the effects of these disturbances ͓1,2͔ because it tive inferential control to identify an autoregres-
usually assumes that all future unmodeled signals sive model for the disturbances in real time. How-
remain the same as the current prediction error. ever, an effective and reliable on-line parameter
This formulation implicitly assumes the effects of estimation algorithm is needed for this method.
external disturbances to be constant throughout the Ricker ͓1͔ has proposed a MPC with state estima-
prediction horizon. If the unmeasured disturbances tion to improve its regulatory performance. Al-
do not have fast dynamics, this assumption often though some guidelines are provided for the de-
sign of the estimator gain, it is difficult to obtain
*Corresponding author. E-mail address: an appropriate estimator gain for different distur-
yash.gupta@dal.ca bances. Lundstrom et al. ͓3͔ have proposed an
0019-0578/2005/$ - see front matter © 2005 ISA—The Instrumentation, Systems, and Automation Society.

2. 188 F. Zhao, Y. P. Gupta / ISA Transactions 44 (2005) 187–198
observer-based MPC with the consideration of a In this paper, a simple disturbance predictor
disturbance model, but sometimes it may be unre- ͑SDP͒ is proposed for a simplified MPC ͑SMPC͒
alistic to model the disturbances of chemical pro- algorithm described in the next section. The pro-
cesses as filtered white noise. Muske and posed disturbance predictor exploits the advantage
Badgwell ͓6͔ and Pannocchia and Rawlings ͓7͔ offered by the SMPC algorithm since the predic-
have proposed that improved regulatory perfor- tion for the disturbance needs to be made for a
mance can be obtained by the use of state or input single point on the prediction horizon. Initially, the
disturbance models and they have derived condi- effect of the disturbance on the process output is
tions guaranteeing zero steady-state offset. For assumed to be the step response of a first-order
simplicity, Chien et al. ͓8͔ have proposed to pre- system. Then the applicability of the proposed pre-
dict the effects of external disturbances through dictor is extended to other disturbances by em-
linear extrapolation of the slope of the unmodeled ploying a tuning parameter and using the available
signals. A tuning parameter is proposed to handle information on the unmodeled signals. The tuning
different disturbances. However, the adverse effect parameter can be obtained online using an optimi-
of measurement noise on the prediction is not con- zation scheme. The effect of the measurement
sidered. A method for eliminating the steady-state noise on the disturbance prediction is considered.
offset caused in the control of integrating pro- The regulatory performance of the proposed pre-
cesses due to sustained disturbances has been pro- dictor is presented by considering three different
posed by Gupta ͓9͔. transfer functions for the disturbance, namely,

3. F. Zhao, Y. P. Gupta / ISA Transactions 44 (2005) 187–198 189
first-order, second-order, and one containing inte-
gration. A comparison is also made with the com-
monly used disturbance prediction.
2. SMPC algorithm
The standard MPC is composed of a predictor
Fig. 1. Estimation of the effects of unmeasured disturbance.
and an optimizer. The predictor provides the pre-
dictions for the process output through a process
model. Based on these predictions, the optimizer ment noise and deterministic disturbances. Mea-
generates a sequence of control moves to satisfy a surement noise often has high frequency, its detri-
specified objective function. For a SISO system, mental effects can be effectively reduced by low-
the objective function to be minimized may be pass filters. Deterministic disturbances ͑measured
formulated as and unmeasured͒ usually have low frequency and
Np
can cause controlled variables to seriously deviate
J MPC͑ k ͒ ϭ ͚ ͓ R ͑ kϩi ͒ Ϫy ͑ kϩi ͔͒ 2
ˆ from their set points. The measured disturbances
iϭ1
may be handled effectively through feed-forward
Nm control. But chemical processes often experience
ϩ␭ ͚ ⌬u ͑ kϩ jϪ1 ͒ 2 , ͑1͒ the deterministic disturbances which are not mea-
jϭ1
sured or not measurable. The handling of unmea-
where y ( kϩi ) and ⌬u ( kϩ jϪ1 ) are related
ˆ sured disturbances presents a challenge for imple-
through the process model and there are con- menting model-based control schemes because the
straints on process variables. The solution of the future effects of disturbances over the prediction
earlier optimization problem may be obtained horizon are needed. The disturbance transfer func-
through linear programming ͑LP͒. However, the tion, G d , and its input d ͑Fig. 1͒ are often un-
computational effort for solving the LP problem is known. In the standard MPC algorithms, the dif-
a strong function of the prediction horizon N p and ference between the current process output and the
the control horizon N m . Since this optimization current model output, D ( k ) , is calculated and as-
problem needs to be solved at every control in- sumed to be constant over the prediction horizon.
stant, a SMPC algorithm ͓10,11͔ has been pro- The original SMPC algorithm ͓10,11͔ also makes
posed in the literature. It reduces the computa- this assumption and calculates the effect of distur-
tional effort significantly because in this algorithm bances at P steps ahead from the following equa-
only one control move into the future needs to be tion:
calculated and the error is minimized usually at ˆ
D ͑ kϩ P ͒ ϭD ͑ k ͒ ϭy ͑ k ͒ Ϫy m ͑ k ͒ . ͑3͒
one point P steps ahead. P is a tuning parameter in
this algorithm. As P is increased, the control In the following section, a SDP is proposed to pro-
moves become smaller and the robustness of the vide a more reasonable prediction for D ( kϩ P ) .
control system increases. As P is decreased, the
reverse happens. For this algorithm, the objective 3.1. A simple disturbance predictor
function in Eq. ͑1͒ simplifies to
Consider Fig. 1 and assume that the process
J SMPC͑ k ͒ ϭ ͓ R ͑ kϩ P ͒ Ϫy ͑ kϩ P ͔͒ 2 .
ˆ ͑2͒
model is perfect, i.e., G m ϭG p , and G d is a first-
The robust stability of the SMPC algorithm has order transfer function of the form: G d
been analyzed and has been found to be essentially ϭK d / ( ␶ d sϩ1 ) . We make these assumptions to
equivalent to the DMC algorithm ͓11͔. The viabil- derive an expression for a disturbance predictor.
ity of the SMPC algorithm has been demonstrated These assumptions will be relaxed later. If a step
on an industrial distillation column ͓12͔. disturbance of magnitude A enters the block G d at
time instant k 0 , its effect on the process output is
3. Proposed disturbance predictor
given by
Chemical processes are usually operated under D ͑ k 0 ϩi ͒ ϭAK d ͑ 1Ϫe ϪiT/ ␶ d ͒ , iϭ0,1,2,... .
disturbances, which can be classified as measure- ͑4͒

4. 190 F. Zhao, Y. P. Gupta / ISA Transactions 44 (2005) 187–198
From Eq. ͑4͒, the effect of the disturbance on the lim ␣ ϭ1. ͑10b͒
process output at different times can be expressed ␶ d →ϱ
as
When ␣ϭ1, Eq. ͑8͒ provides a linear change in
D ͑ kϪ1 ͒ ϭAK d • ͓ 1Ϫe Ϫ ͑ kϪk 0 Ϫ1 ͒ T/ ␶ d ͔ , ͑5a͒ D ( kϩ P ) .
To relax the assumptions made at the beginning
D ͑ k ͒ ϭAK d • ͓ 1Ϫe Ϫ ͑ kϪk 0 ͒ T/ ␶ d ͔ , ͑5b͒ of this subsection, the parameter ␣ in Eq. ͑8͒ can
be considered a tuning parameter. This parameter
D ͑ kϩ P ͒ ϭAK d ͓ 1Ϫe Ϫ ͑ kϩ PϪk 0 ͒ T/ ␶ d ͔ can be set between 0 and 1.0 for different distur-
bances and for model mismatch. With a value for
ϭg•e ϪT/ ␶ d • ͑ 1Ϫe Ϫ PT/ ␶ d ͒ ϩD ͑ k ͒ , ␣, Eq. ͑8͒ can be used for disturbance prediction if
͑6͒ there is no noise on the process output signals.
Since measurement noise is unavoidable, its effect
where on D ( k ) and D ( kϪ1 ) will make the prediction of
D ( kϩ P ) difficult. To reduce the adverse effect of
gϭAK d e Ϫ ͑ kϪk 0 Ϫ1 ͒ T/ ␶ d . measurement noise on the prediction, the term g in
Because A, K d , ␶ d , and k 0 are unknown, the term Eq. ͑6͒ can be estimated based on a number of
D ( kϩ P ) cannot be calculated directly. However, samples L ( Lу2 ) instead of only two samples:
A, K d , and k 0 can be eliminated by using the dif- D ( k ) and D ( kϪ1 ) , as follows.
ference between D ( k ) and D ( kϪ1 ) . These terms Define: Q ( kϪ j ) ϭD ( kϪ j ) ϪD ( kϪLϩ1 ) ; 0
are known at the current control instant and the р jрLϪ1. Then,
difference between them from Eqs. ͑5a͒ and ͑5b͒ Q ͑ k ͒ ϪQ ͑ kϪ1 ͒ ϭg• ͑ 1Ϫe ϪT/ ␶ d ͒ . ͑11͒
can be expressed as
Assume the time series ͕ Q ( kϪ j ) ,0р jрLϪ1 ͖
D ͑ k ͒ ϪD ͑ kϪ1 ͒ ϭg• ͑ 1Ϫe ϪT/ ␶ d ͒ . ͑7͒ can be fitted by a straight line
By substituting the value of the term g from Eq. Q ͑ kϪ j ͒ ϭ ␦ • ͑ LϪ1Ϫ j ͒ ; 0р jрLϪ1.
͑7͒ into Eq. ͑6͒, D ( kϩ P ) and ␣ can be expressed ͑12͒
as
The coefficient ␦, representing the slope of the
D ͑ kϩ P ͒ ϭ ␣ • P• ͓ D ͑ k ͒ ϪD ͑ kϪ1 ͔͒ ϩD ͑ k ͒ , line, can be estimated by minimizing
͑8͒ LϪ1
J ͑ ␦ ͒ ϭ ͚ ͓ Q ͑ kϪ j ͒ Ϫ ␦ ͑ LϪ1Ϫ j ͔͒ 2 . ͑13͒
ˆ
1Ϫe Ϫ PT/ ␶ d jϭ0
␣ϭ T/ ␶ d
. ͑9͒
P͑ e Ϫ1 ͒
Because Q ( k ) ϪQ ( kϪ1 ) ϭ ␦ , the term g in Eq.
ˆ
If one happens to know ␶ d then the parameter ␣ ͑11͒ can be expressed as
needed in Eq. ͑8͒ can be calculated from Eq. ͑9͒,
otherwise it needs to be set. There are two special ␦
ˆ
gϭ . ͑14͒
cases for parameter ␣: step output disturbance and 1Ϫe ϪT/ ␶ d
ramp output disturbance. If the disturbance causes
a step change in output, then from Eq. ͑9͒: By substituting the value of g from Eq. ͑14͒ into
Eq. ͑6͒, the prediction of D ( kϩ P ) can be ex-
lim ␣ ϭ0. ͑10a͒ pressed as
␶ d →0
D ͑ kϩ P ͒ ϭ ␣ • P• ␦ ϩD ͑ k ͒ .
ˆ ˆ ͑15͒
When ␣ϭ0, Eq. ͑8͒ degenerates to: D ( kϩ P )
ϭD ( k ) . This is the value used in the original The tuning parameter ␣ has the same expression
SMPC and in this case, the assumption made in as shown in Eq. ͑9͒. The physical interpretation
the SMPC is valid because the effect of the distur- for Eq. ͑15͒ is that the slope of the process/model
bance is indeed a constant over the prediction ho- mismatch signals is estimated based on D ( k ) and
rizon. However, if the disturbance affects the out- its historical data. Then the future effect of the
put in a ramp fashion, then external disturbance at next P steps ahead is pre-

5. F. Zhao, Y. P. Gupta / ISA Transactions 44 (2005) 187–198 191
dicted through linear extrapolation. The tuning pa- match, D ( kϩ P ) can be well predicted with an
rameter ␣ is used to counteract the prediction error appropriate ␣ value. However, if the contributions
caused by the linear approximation. If the time to D ( k ) were primarily due to fundamental errors
constant of the disturbance transfer function can in the process model structure or measurement
be estimated, ␣ can be determined from Eq. ͑9͒. If noise ͓2͔, it will be beneficial to set ␣ϭ0 because
this time constant is unavailable, it is proposed the uncertainty in the value of D ( kϩ P ) is high.
that it be set equal to the dominant time constant To achieve an improved regulatory performance,
of the process model. As an alternative, ␣ can be the tuning parameter ␣ can be searched within the
tuned through trial and error. It can be expected range ͑0,1͒ by trying different values of ␣ over
that a large L will result in a smooth but sluggish successive time periods between one ‘‘steady-
prediction of D ( kϩ P ) . A trade off is needed in state’’ to another. The objective function to be
the selection of an appropriate value of L. minimized over the time periods was chosen as
Equation ͑15͒ is the mathematical model for the
Ͳ
M M
proposed disturbance predictor. Considering its 1
simplicity, it is referred to as a SDP, which can be J ͑ ␣ ͒ ϭ ͚ i• ͉ RϪy ͑ i ͒ ͉ ͚ ͉ D ͑ i ͒ ϪD 0͉ ,
iϭ1 M iϭ1
incorporated into the SMPC algorithm directly. ͑16͒
The parameters ␣ and L need to be determined in
using the SDP. A simple way to obtain ␣ and L is where y ( i ) is the controlled process output, and M
to assume that the disturbance model is first-order is the number of control intervals in the time pe-
and estimate ␶ d from the observed values of riod over which a value of ␣ is used. Each time
D ( k ) . Then ␣ and L can be directly obtained from period represents one iteration and D 0 is the
Eqs. ͑9͒ and ͑20͒, respectively. A better value of ␣ model mismatch signal at the beginning of each
may be obtained by using the optimization algo- iteration. The numerator term in Eq. ͑16͒ repre-
rithm proposed in following section if new distur- sents the integral of time-weighted absolute error
bances do not enter the process too frequently. ͑ITAE͒. The denominator term in Eq. ͑16͒ is intro-
duced to allow for different magnitudes of the dis-
turbance that may be encountered. After an opti-
3.2. Determination of ␣ and L online mum value of ␣ is found, one may check if the
number of sample data, L, is appropriate. This de-
The tuning parameter ␣ depends on P and ␶ d as termination may be done as follows.
shown in Eq. ͑9͒. For a certain disturbance, the At each control interval, the disturbance predic-
tuning parameter ␣ decreases/increases as the pre- ˆ
tion, D ( kϩ P ) , is obtained. As time goes on, the
diction length P increases/decreases. However, if a ˆ
prediction forms a time series ͕ D ( kϩ PϪi ) , i
disturbance on the process output produces a step
у0 ͖ , which is estimated based on the noisy signal
change or a ramp change, ␣ should be set equal to
D ( k ) and its historical data. Therefore, the time
0 or 1, respectively, and should not be affected by ˆ
the value of the tuning parameter P. The control series ͕ D ( kϩ PϪi ) , iу0 ͖ will fluctuate if the
interval T and parameter P are set when one sample data for estimating ␦ is not long enough.
implements the SMPC algorithm. So ␣ only de- This fluctuation can be regarded as noise. A prac-
pends on the dynamics of external disturbances. tical method for investigating the noise level of
ˆ
time series, ͕ D ( kϩ PϪi ) , iу0 ͖ , is to estimate
The value of the tuning parameter ␣ decreases/
increases as the dynamics of a disturbance be- its covariance ␴. Considering the time series to be
comes fast/sluggish. Since a fixed ␣ may not pro- non-stationary, one can estimate ␴ as follows ͓13͔:
vide satisfactory disturbance prediction for a time- ˜ ˆ
varying ␶ d , it is better to update ␣ periodically. n ͑ k ͒ ϭn ͑ kϪ1 ͒ ϩ ␨ • ͓ ͉ D ͑ kϩ PϪ2 ͒ ϪD ͑ kϩ P
The SDP is developed based on the available Ϫ2 ͒ ͉ Ϫn ͑ kϪ1 ͔͒ , ͑17a͒
signal D ( k ) and its history data. The contributions
to signal D ( k ) include deterministic disturbances, ␴ ϭ1.78n ͑ k ͒ , ͑17b͒
model mismatch and measurement noise. If the
contributions to D ( k ) were primarily due to low- where n ( k ) is the average noise level of time se-
frequency deterministic disturbances and model ˆ
ries ͕ D ( kϩ PϪi ) , iу0 ͖ , ␨ is an exponential
parameter mismatch, such as the process gain mis- smoothing constant, which is usually chosen be-

6. 192 F. Zhao, Y. P. Gupta / ISA Transactions 44 (2005) 187–198
˜ ˆ
tween 0 and 0.3. D ( kϩ PϪ2 ) , smoothed D ( k Step 3: Does one of the value of objective func-
ϩ PϪ2 ) , is determined from ͓13͔ tions found in step 2 lower than the value found
previously, i.e., using ␣ ( iϪ1 ) ? If yes, go to step
1 5; if not, go to step 4.
˜ ˆ ˆ
D ͑ kϩ PϪ2 ͒ ϭ • ͓ Ϫ3D ͑ kϩ PϪ4 ͒ ϩ12D ͑ k Step 4: Check if, ͉⌬␣͉Ͻ⑀? If yes, an optimized ␣
35
has been found. If not, set ⌬␣ϭ⌬␣/␳ and go to
ˆ
ϩ PϪ3 ͒ ϩ17D ͑ kϩ PϪ2 ͒ step 2.
Step 5: Set ␣ ( i ) ϭ2 ␣ ( i ) Ϫ ␣ ( iϪ1 ) and set L
ˆ ˆ
ϩ12D ͑ kϩ PϪ1 ͒ Ϫ3D ͑ kϩ P ͔͒ . according to Eq. ͑20͒. Then go to step 6.
Step 6: Check if, J ͓ ␣ ( i ) ͔ ϽJ ͓ ␣ ( iϪ1 ) ͔ ? If yes,
͑18͒ set ␣ ( iϪ1 ) ϭ ␣ ( i ) and go to step 5; if not, then go
to step 4.
It is desirable to have a small covariance ␴, which
can be achieved by increasing the data length for
estimating ␦. However, a large L will result in un-
timely predictions of the disturbances. The data 4. Analysis of steady-state offset
length, L, can be determined by satisfying the fol-
lowing requirement: Many model-based control schemes result in
steady-state offset under ramp output disturbances,
␴ / ␴ 0р ␤ , ͑19͒ which often occurs in case of an integrating pro-
cess. In this section, we analyze the offset of the
where ␴ 0 is the covariance of measurement noise, SMPC algorithm, which uses the proposed SDP
␤ ͑␤у1͒ is a threshold. Note from Eq. ͑15͒, the under the following assumptions.
ratio ␴ / ␴ 0 depends on ␣ and P. It is obvious that ͑1͒ The plant-model mismatch is not larger
␴ / ␴ 0 ϭ1 if ␣ϭ0. Simulations show that if one enough to make the system unstable.
selects Lϭ P and ␤ϭ2, Eq. ͑19͒ is always satisfied ͑2͒ Under a deterministic disturbance, the con-
irrespective of the value of ␣. However, if ␣ is trol system reaches a steady state ( y ss ,u ss) ,
small, Eq. ͑19͒ is still satisfied if one selects L where there are no active constraints.
Ͻ P. This is because with a small value of ␣, the ͑3͒ Set point R is a constant.
fluctuation of ␦ will contribute little to increase
ˆ
covariance ␴. So the selection of L depends on ␣. The SMPC algorithm minimizes the predicted
A guideline for selecting L is as follows: error at P steps ahead, that is
J SMPC͑ k ͒ ϭe 2 ͑ kϩ P ͒ . ͑21͒
Set, LϭCeil͓ ␣ ͑ PϪ2 ͒ ϩ2 ͔ , ͑20͒
When the manipulated variable is unconstrained,
the predicted error e ( kϩ P ) will be driven to zero.
where Ceil( x ) is a function that rounds x to the
In other words,
next higher integer. Therefore, with a certain ␣
value, a corresponding L can be determined. Con- RϪy ͑ kϩ P ͒ ϭe ͑ kϩ P ͒ ϭ0.
ˆ ͑22͒
sidering its simplicity and good convergence prop-
erty, the Hooke-Jeeves pattern search method ͓14͔ With the SDP, the predicted output of the SMPC
was used to search for the tuning parameter ␣. The algorithm at P steps ahead is
proposed procedure for determining ␣ and L is as y ͑ kϩ P ͒ ϭy m ͑ kϩ P ͒ ϩD ͑ k ͒ ϩ ␣ • P• ͓ D ͑ k ͒
ˆ
follows.
Step 1: Initialize L ͓ LϭCeil( P/2ϩ1 ) ͔ , ␣, step ϪD ͑ kϪ1 ͔͒ . ͑23͒
size ⌬␣, step size reduction factor ␳ ͑␳Ͼ1͒, termi-
nation tolerance ⑀ on ␣ ͑⑀Ͼ0͒, exponential Now, D ( k ) ϭy ( k ) Ϫy m ( k ) and at steady-state,
smoothing constant ␨ and iteration number of op- y ( k ) ϭy ( kϪ1 ) ϭy ss . By substituting this expres-
timization ( iϭ0 ) . Then find the value of objective sion for D ( k ) , Eq. ͑23͒ can be written as
function using ␣ ( iϭ0 ) . y ͑ kϩ P ͒ ϭy m ͑ kϩ P ͒ ϩy ssϪy m ͑ k ͒ ϩ ␣ • P• ͓ D ͑ k ͒
ˆ
Step 2: Set iϭiϩ1. Find the two values of ob-
jective function using ␣ ( i ) ϭ ␣ ( iϪ1 ) Ϯ⌬ ␣ . ϪD ͑ kϪ1 ͔͒ . ͑24͒

7. F. Zhao, Y. P. Gupta / ISA Transactions 44 (2005) 187–198 193
By substituting y ( kϩ P ) from Eq. ͑24͒, Eq. ͑22͒
ˆ earlier two SMPC algorithms. In this study, the
can be written as performance of a MPC algorithm with objective
function given in Eq. ͑1͒ was also checked. In the
Rϭy ssϩy m ͑ kϩ P ͒ Ϫy m ͑ k ͒ ϩ ␣ • P• ͓ y m ͑ kϪ1 ͒ MPC algorithm, the effects of disturbance were
Ϫy m ͑ k ͔͒ . ͑25͒ assumed to be constant throughout the prediction
horizon as is commonly done. To provide a similar
For a self-regulating process, when the control tuning in the MPC algorithm, the move suppres-
system reaches steady state after regulating a regu- sion ␭ was chosen such that the response of the
lar disturbance, it is obvious that MPC algorithm to a step change in set-point
matched the corresponding response of the SMPC
y m ͑ kϩ P ͒ ϭy m ͑ k ͒ ϭy m ͑ kϪ1 ͒ ϭK m u ss .
algorithm. To avoid confusion among the various
͑26͒
response curves, only the performance indices
Based on Eq. ͑26͒, Eq. ͑25͒ reduces to y ssϭR. In ͑ITAE values͒ for the standard MPC algorithm are
other words, there will be no offset. reported. In finding the optical values of ␣ and L,
For an integrating process, under steady state, the magnitudes of the disturbance and periods be-
the following expressions can be written: tween steady states were allowed to vary within
specified limits. The magnitude A of the distur-
y m ͑ kϩ P ͒ Ϫy m ͑ k ͒ ϭ PK m Tu ss , ͑27a͒ bance was selected randomly to be in the range of
y m ͑ k ͒ Ϫy m ͑ kϪ1 ͒ ϭK m Tu ss . ͑27b͒ Ϯ͑1,2͒ for the SISO examples and to be in the
range of Ϯ͑0.1,0.2͒ for the MIMO example. The
If the disturbance is a regular output disturbance, duration of each new disturbance was selected
then u ssϭ0. Therefore, Eq. ͑25͒ reduces to y ss randomly to consist of 100–200 control intervals.
ϭR, irrespective of the value of ␣. If the distur- The value of M in Eq. ͑16͒ was taken as 100. In
bance is a ramp output disturbance, then ␣ϭ1. general, M is selected to cover the transient por-
With this value of ␣, Eq. ͑25͒ again reduces to: tion for each disturbance. The effectiveness of the
y ssϭR. values obtained for ␣ and L was tested through the
It can be seen from the earlier analysis that the following three examples. In the simulations, the
proposed SMPC algorithm achieves zero steady disturbance and noise were started at kϭ5. The
state offset no matter whether the disturbance is time period for each of these tests consisted of 200
self-regulating or nonself-regulating. It may be control intervals.
noted that a self-regulating process cannot reach a
steady state if it is subjected to a ramp output dis- 5.1. Example 1—Regular SISO process
turbance because the system would be uncontrol-
lable. This example considers the following process
and disturbance models:
5. Control examples Ke Ϫ ␪ s
G p͑ s ͒ ϭ , ͑28͒
͑ ␶ 1 sϩ1 ͒͑ ␶ 2 sϩ1 ͒
The proposed SDP was incorporated into the
SMPC algorithm and the effectiveness of the
SMPC algorithm thus improved was investigated 4.0
G d͑ s ͒ ϭ . ͑29͒
on three example problems. In order to provide an ͑ 10sϩ1 ͒ 2
indication of the performance improvement in a
practical situation, we have considered time de- The values of the process parameters are: K
lays, interactions between variables ͑MIMO case͒, ϭ2.0, ␶ 1 ϭ15, ␶ 2 ϭ10, and ␪ϭ5. The control in-
model mismatch, measurement noise, varying terval Tϭ1. The number of intervals for the open-
magnitudes of disturbances, and different distur- loop response to settle is taken as Nϭ75. The pa-
bance models. The regulatory performance of the rameter Pϭ13. The control parameters for the
proposed SMPC is compared with that of the MPC algorithm are N p ϭ75, N m ϭ2, and
original SMPC where the effect of disturbances at ␭ϭ0.125. A unit step change in disturbance d ( A
P steps ahead is calculated from Eq. ͑3͒. The same ϭ1 ) is considered. The following three cases are
process model and parameter P were used in the considered.

8. 194 F. Zhao, Y. P. Gupta / ISA Transactions 44 (2005) 187–198
Fig. 2. Suppression of a regular disturbance without noise, Fig. 3. Suppression of a regular disturbance with measure-
example 1. ment noise, example 1.
Case 1. In this case we assume that G d is un-
posed SDP provides better disturbance suppres-
known and the process output is noise free. To
sion than the commonly used disturbance predic-
choose ␣, we assume ␶ d ϭ ␶ 1 ϭ15. Then from Eq.
tion.
͑9͒, ␣ϭ0.64 and we set Lϭ2. The results obtained
by the two SMPC algorithms are shown in Fig. 2.
5.2. Example 2—Integrating SISO process
The subscripts p and o on the variables in this and
all the following figures refer to the proposed and
This example considers two disturbance models
original SMPC algorithms, respectively. The ITAE
and the following process model:
values for the proposed SMPC, original SMPC
and the MPC algorithms were 465.9, 1738.2, and Ke Ϫ ␪ s
1784.1, respectively. G p͑ s ͒ ϭ . ͑30͒
s ͑ ␶ sϩ1 ͒
Case 2. In this case we consider noise on the
process output. The noise is assumed to be nor- The values of the process parameters are Kϭ0.5,
mally distributed with zero mean and covariance ␶ϭ20, and ␪ϭ5. The control interval Tϭ1. The
␴ 0 ϭ0.03. Starting from an initial value of 0.2, the measurement noise is normally distributed with
parameter ␣ was found to be 0.49 by using 18 zero mean and covariance ␴ 0 ϭ0.03. The param-
iterations of the proposed optimization procedure, eter Pϭ16. The control parameters for the MPC
and the corresponding L was 8. The results ob- algorithm are N p ϭ75, N m ϭ3, and ␭ϭ0.08. A unit
tained by the two SMPC algorithms are shown in step change in disturbance d ( Aϭ1 ) is consid-
Fig. 3. For the proposed SMPC, ITAEϭ1146.2, ered. The following three cases are investigated.
␴ / ␴ 0 Ϸ1.67 and for the original SMPC, ITAE Case 1. This case considers a second-order dis-
ϭ2143.9. For the MPC, ITAEϭ2186.9. turbance as described in Eq. ͑29͒. Starting from an
Case 3. In this case we investigate the robust- initial value of 0.2, the parameter ␣ was found to
ness of the proposed SMPC for disturbance rejec- be 0.42 by using 17 iterations of the proposed op-
tion under process uncertainties. The process pa- timization procedure, and the corresponding L was
rameters ͑K, ␶ 1 , and ␪͒ were changed by Ϯ20%, 10. The results obtained by the two SMPC algo-
one at a time while the other two parameters were rithms are shown in Fig. 4. For the proposed
maintained at their nominal values. The other con- SMPC, ITAEϭ1631.7, ␴ / ␴ 0 Ϸ1.47 and for the
ditions were the same as in case 2. The ITAE val- original SMPC, ITAEϭ2679.9. For the MPC,
ues for the proposed SMPC, the original SMPC ITAEϭ2786.9.
and the MPC algorithms are presented in Table 1. Case 2. This case considers a disturbance which
The results for this example show that the pro- contains an integration as follows:

9. F. Zhao, Y. P. Gupta / ISA Transactions 44 (2005) 187–198 195
Table 1
ITAE values for example 1—Regular SISO process.
New values of process parameters
Algorithm Kϭ2.4 Kϭ1.6 ␶ 1 ϭ18 ␶ 1 ϭ12 ␪ϭ6 ␪ϭ4
Proposed SMPC 985.1 1438.2 1415.2 1129.4 1231.5 1165.1
Original SMPC 1762.8 2780.0 2203.0 2234.5 2097.5 2191.5
Standard MPC 1796.9 2836.6 2231.2 2278.6 2141.5 2232.7
0.1 Table 2. The results for this example show that the
G d͑ s ͒ ϭ . ͑31͒ proposed SDP provides better disturbance sup-
s ͑ 10sϩ1 ͒
pression than the commonly used disturbance pre-
Starting from an initial value of 0.2, the parameter diction.
␣ was found to be 1.0 by using 14 iterations of the
proposed optimization procedure, and the corre- 5.3. Example 3—MIMO process
sponding L was 16. Figure 5 shows the results of
the two SMPC algorithms. As expected for this This example considers a two-product distilla-
situation, the original SMPC results in an offset. tion column separating a binary feed. Based on
The proposed SMPC brings the controlled vari- energy balance, the column has the following dy-
namics ͓15͔:
ͫ ͬ
able back to its set point. For the proposed SMPC,
ITAEϭ1413.2, ␴ / ␴ 0 Ϸ1.81 and for the original
SMPC, ITAEϭ33587.0. For the MPC, ITAE 0.0747e Ϫ3s Ϫ0.0667e Ϫ2s
ϭ33430.0.
Case 3. In this case we investigate the robust- ͫ ͬ
y1 12sϩ1 15sϩ1
y 2 ϭ 0.1173e Ϫ3.3s Ϫ0.1253e Ϫ2s ͫ ͬ
u1
u2
ness of the proposed SMPC for disturbance rejec- 11.75sϩ1 10.2sϩ1
ͫ ͬ
tion under process uncertainties. Again, the pro-
cess parameters ͑K, ␶ 1 , and ␪͒ were changed by 0.70e Ϫ5s
Ϯ20%, one at a time while the other two param- 14.4sϩ1
eters were maintained at their nominal values. The ϩ 1.3e Ϫ3s •A. ͑32͒
other conditions were the same as in case 2. The
ITAE values for the proposed SMPC, the original 12sϩ1
SMPC, and the MPC algorithms are presented in
Fig. 4. Suppression of a regular disturbance with measure- Fig. 5. Suppression of an integrating disturbance with mea-
ment noise, example 2. surement noise, example 2.

10. 196 F. Zhao, Y. P. Gupta / ISA Transactions 44 (2005) 187–198
Table 2
ITAE values for example 2—Integrating SISO process.
New values of process parameters
Algorithm Kϭ0.6 Kϭ0.4 ␶ϭ24 ␶ϭ16 ␪ϭ6 ␪ϭ4
Proposed SMPC 1268.0 2030.8 1794.6 1529.6 1465.0 1460.2
Original SMPC 28110.0 41648.0 34002.0 33175.0 33655.0 33517.0
Standard MPC 27996.0 41415.0 33859.0 33005.0 33502.0 33357.0
In this example, the manipulated variables are re- bottom product compositions are considered to be
flux flow rate ( u 1 ) and reboiler flow rate ( u 2 ) , and equally important, their ITAE values are weighted
the controlled variables are distillate composition equally and added together. The following two
( y 1 ) and bottoms composition ( y 2 ) . The major cases are investigated.
disturbance is a change in feed composition ͑A͒. Case 1. Starting from an initial value of 0.5, the
The control interval Tϭ1 min. To express the time parameters ␣ 1 and ␣ 2 were found to be 0.550 and
delays in Eq. ͑32͒ as multiples of the control in- 0.587, respectively, by using 59 iterations of the
terval, the time delay 3.3 is changed to 3. All the proposed optimization procedure, and the corre-
initial states are zero. The constraint on the control sponding number of samples were L 1 ϭL 2 ϭ8.
move is ͉ ⌬u max͉р0.2. The control parameters for Considering the interaction between the two con-
the SMPC algorithm for both controlled variables trolled variables, one half of the optimized values
are considered equal: Pϭ12. The control param- of ␣ 1 and ␣ 2 were used, that is, ␣ 1 ϭ0.275, ␣ 2
eters for the MPC algorithm are N p ϭ60, N m ϭ3, ϭ0.294. Figs. 6 and 7 show the control results
and ␭ϭ0.036. The disturbance is a step change, obtained by the two SMPC control algorithms. For
Aϭ0.1, to the light key in the feed. The measure- the proposed SMPC, ITAEϭ55.2 with ␴ / ␴ 0
ment noise is normally distributed with zero mean Ϸ1.15 and for the original SMPC, ITAEϭ76.7.
and covariance ␴ 0 ϭ0.0007. Because the top and For the MPC, ITAEϭ74.0.
Fig. 6. Regulatory response of the controlled variables of the proposed and original SMPC for a MIMO process, example 3.

11. F. Zhao, Y. P. Gupta / ISA Transactions 44 (2005) 187–198 197
Fig. 7. Regulatory response of the manipulated variables of the proposed and original SMPC for a MIMO process, example
3.
Case 2. To investigate the robustness of the pro- Subcase 6: the time constants of main-diagonal
posed SMPC, the following six subcases are con- elements increase by 20% and the time delays of
sidered: first column elements decrease by 33% ͑one con-
Subcase 1: all of the process gains increase by trol interval͒.
40%. The control parameters are kept the same as in
Subcase 2: all of the process gains decrease by case 1. The ITAE values of the proposed SMPC,
40%. the original SMPC and the MPC algorithms are
Subcase 3: the gains of main-diagonal elements presented in Table 3. Again, the results for this
increase by 20% and the time constants of off- example show that the proposed SDP provides
diagonal elements decrease by 20%. better disturbance suppression than the commonly
Subcase 4: the time constants of main-diagonal used disturbance prediction.
elements increase by 20% and the gains of off-
diagonal elements decrease by 20%. 6. Conclusions
Subcase 5: all of the process gains increase by
20% and the time delays of second column ele- A disturbance predictor is developed for the
ments increase by 50% ͑one control interval͒. SMPC algorithm to improve its ability to suppress
Table 3
ITAE values for example 3—MIMO process.
Subcase
Algorithm 1 2 3 4 5 6
Proposed SMPC 48.0 84.0 55.3 60.0 48.3 58.2
Original SMPC 59.0 130.0 73.9 85.2 63.6 76.4
Standard MPC 56.0 126.7 75.5 93.2 61.8 94.3

12. 198 F. Zhao, Y. P. Gupta / ISA Transactions 44 (2005) 187–198
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