CHEMICAL ENGINEERING TRANSACTIONS 

VOL. 61, 2017 

A publication of 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Petar S Varbanov, Rongxin Su, Hon Loong Lam, Xia Liu, Jiří J Klemeš
Copyright © 2017, AIDIC Servizi S.r.l. 
ISBN 978-88-95608-51-8; ISSN 2283-9216

Switch Approach of Control Zone and Optimization Zone for 
Economic Model Predictive Control 

Xin Wan, Bing-Jie Liu, Zhi-Shan Liang, Xiong-Lin Luo* 

Department of Automation, China University of Petroleum Beijing, 102249, China 
luoxl@cup.edu.cn 

The coordination of control and optimization is proposed for economic model predictive control in the paper. 
The characteristic is explained briefly with the field theory. The solution of economic model predictive control is 
gradient descent method. However, economic constraint condition makes the response too long. In order to 
improve the performance and save respond time, control and optimization should be considered separately in 
controller. The method is built for coordinating control and optimization, which separate the whole zone into 
control zone and optimization zone and switch in different zone. The respond time is reduced in control zone, 
and economic benefit is improved in optimization zone. Rapid response and economic optimization are 
obtained. The test and verification are made by the case study of Shell Standard Control Problem. 

1. Introduction

The algorithm of predictive control is usually expressed by a quadratic programming with a lot of constraint 
condition that presented by Kothare et al. (1996) and then Abou-Jeyab et al., (2001).. The solution of model 
predictive control (MPC) is a   typical quadratic programming problem (Kuntze et al., 1986). MPC is applied 
widely to the process with strong interactions and large dead times. In detail, the objective of MPC controllers 
is achieving the following of reference trajectory. Because long response time will influences economic 
benefits, a new control strategy had been proposed, which is called economic model predictive control (EMPC) 
(Ellis et al., 2014). The same constraint conditions are set by EMPC and MPC (Ricker, 1985), but change the 
cost function (Muller et al., 2014). Economic benefits are considered for cost function of EMPC. The economic 
benefits are determined by product output and energy consumption. The objective of EMPC is minimizing cost 
function, which leads to instability of process, so terminal constraint is adopted. However, the coordination of 
terminal constraint and economic indicator is too hard. General method is tentative searching by offline 
simulation, which brings some difficult problems on industrial application (Qin et al., 2003). 
In this paper, the method is built for coordinating control and optimization, which separate the whole zone into 
control zone and optimization zone and switch in different zone. The feasibility verification of the method is 
organized as follows: In section 2, the principle of EMPC is discussed. Further, a detailed description of the 
method is given in section 3. In section 4, the method is applied to a case of Shell Standard Control Problem. 

2. Economic model predictive control

2.1 Typical EMPC  

In general, the chemical process is described as difference equations. 

))(,)(()1( kukxfkx =+  (1) 

))(()( kxgky =  (2) 

For EMPC, P is prediction horizon, use predictive state sequence x(k) = {x(k|k), x(k+1|k), …, x(k+P-1|k)} and 
control sequence u(k) = {u(k|k), u(k+1|k), …, u(k+P-1|k)} to define the cost function L(•). 

=
−+

=

1P
))|(,)|((),,(

k

ki
kiukixlkL ux  (3) 

                               
 
 

 

 
   

                                                  
DOI: 10.3303/CET1761028

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Wan X., Liu B.-J., Liang Z.-S., Luo X.-L., 2017, Switch approach of control zone and optimization zone for economic 
model predictive control, Chemical Engineering Transactions, 61, 181-186  DOI:10.3303/CET1761028  

181



)|()ΔQ()|(Δ)|()R()|())|(,)|(( kiuikiukixikixkiukixl ⊥⊥ += (4) 

And the l(•) in Eq(3) often define as quadric form like Eq(4), the R and Q are economic cost coefficient. 
In order to get optimal control sequence, the implicit expression is presented as follow. 

)},,({min  arg kL uxu =  (5) 

))|( ),|(()|1( s.t. kikukikxfkikx ++=++  (6) 

X∈+ )|(     kikx (7) 

U∈+ )|(     kiku (8) 

)()|(     kxkkx =  (9) 

Ω∈+ )|1-P(     kkx  (10) 

2-P ..., 1, 0,     =i
 (11) 

The optimal control sequence u*(k) is obtained by solving the Eqs(5) - (11). Then the process is manipulated 
by the first element in the sequence. The cost function Eq(3) consists of energy consumption coefficient and 
economic benefit coefficient. In fact, the Eq(5) not includes the information about set point, so the process 
faces with the risk of instability. Eq(10) is terminal constraint that can drive a state close to the set point by 
step. Eq(6) is a state update equation for iterating. The x and u must satisfy relevant constraint, shown as 
Eqs(7) - (8). The initial state is the value of state observer at current time like Eq(9). 

2.2 Disadvantage of EMPC 

EMPC can ensure the economic benefits all the time, but it has some problems about control performance. 
The expression of the cost function is quadratic form like Eq(4). Because the effect of cost function is stronger 
than terminal constraint, the optimal control sequence u*(k) is determined by economic constraint. However, 
economic optimization action should start at an appropriate neighbourhood of set point. Global economic 
optimization will sacrifice the control performance when process states are not nearby the set point. In other 
word, economic optimization should not be considered in controller until the process state moving into an 
appropriate zone.  
The control objective in some petro-chemical and process industries is not a point but a neighbourhood of set 
point. The economic optimization should be considered once the process state moving into this zone in the 
controller. Besides, the state should be ensured moving into this zone quickly if current state has been out of 
the zone. This problem is shown in Figure 1. 

Figure 1: The example for disadvantage of EMPC. 

In Figure 1, the whole zone has two central points. One of them is economic centrality, and for explaining 
clearly, the economic equipotential lines are represented as dotted line. The quadratic programming problem 

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of EMPC is finding a system trajectory that can get minimal economic potential. Similarly, just consider the 
control, the equipotential lines is represented as solid line. In practice, the control objective is a neighborhood 
of set point. The neighborhood, called optimization zone, is defined by Γ(δ) in Eq(12). And the remainder is 
concave called control zone. From theory of convex optimization, the optimization point is on the boundary of 
convex. 

m} ..., 2, 1, ], ,-[|{)( refref =+∈= ixxxx ii δδδΓ
 (12) 

Generally, when changing the set point, the state should move to the control zone quickly. If process state 
moves into optimization zone, economic optimization action is applied to improve economic benefit. The state 
trajectory of system(1) has two stages, firstly it moves to the optimization zone from control zone, and then it 
starts optimization action to arrive optimization point. The system(2) uses EMPC all the time, but the state 
trajectory only has one stage: optimization. The system(2) can also arrive optimization point, but the 
optimizing trajectory is unreasonable because system is not arrive control objective 
untilthe system reaches a steady state. In addition, the system takes long time to along with the unsatisfactory 
trajectory. 

Table 1: Time summary 

System Control time Optimization time Total time 
system(1): proposed tC tO tC+tO 
system(2): EMPC tE tE tE 

The each part time is shown in Table 1 and Figure 1. The system(1) have two stages, tC is control time means 
the time spent in control zone. And tO, named as optimization time, means the time spent in optimization zone 
until process arrives optimization point. The EMPC controller is used in the system(2), and that it only has one 
stage. Control and optimization are considered at the same time in the EMPC strategy, so control time, 
optimization time, and total time is just the same as tE. 

3. Switch approach of EMPC

3.1 Switch method 

In order to improve the performance of controller and make optimizing trajectory more reasonable, this paper 
proposes a switch method to discuss this problem. In the control zone, MPC controller with small prediction 
horizon is selected. In the optimization zone, which is related to parameter δ (δ≥0), the state will drive by 
another MPC controller with large prediction horizon and an optimization layer (OPT). The cost function of 
MPC is Eq(13). 

 Δ+=
−+

=

1P 2

Q

2

Rsp })|(-)|( {),,(
k

ki
kiuxkixkJ ux  (13) 

0)|(  then  ))()|((  if =+∈+ kikukikx δΓ (14) 

0)|(  then  ))()|((  if =+∉+ kikukikx δΓ (15) 

Switch strategy includes two MPC controllers. The controller MPC(1) consist of Eq(13), Eqs(6)-(9), Eq(14), 
and Eq(11). Constraint condition Eq(14) can make sure MPC(1) compute the control sequence when system 
in control zone. Another controller MPC(2) consist by Eq(13), Eq(6)-(9), Eq(15), and Eq(11). Constraint 
condition Eq(15) can make sure MPC(2) compute the control sequence when system in optimization zone. 
The set point of MPC(1) is the central of convex Γ(δ), and small prediction horizon is used for enhancing the 
speed of respond. Optimization point is calculated by OPT, which is the input of MPC(2). Controller MPC(2) 
use a large prediction horizon because optimization is a long-term process. There are many methods to solve 
OPT, and optimization point yopt is calculated by set point ysp and expectation control law uopt (Luo et al., 2014). 

3.2 Directions for switch approach of EMPC 

The system structure diagram is shown in Figure 2. 
In Figure 2, these two controllers are active, and a high selector is used for switching control sequence 
automatically. The boundary of optimization zone is determined by the parameter δ.  

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In Table 2, it shows that the two controllers play different roles. MPC(1) is designed in the control zone, while 
MPC(2) is designed in the optimization zone. 

Figure 2: The system structure diagram on switch approach of EMPC. 

Table 2: Optimal control sequence in different zone 

Controller Control zone Optimization zone 
MPC(1) u (for set point) 0 
MPC(2) 0 u (for optimization point) 

4. Case study

In order to test and verify the feasibility of the method, Shell model is simulated and discussed. The model is 
shown in Figure 3 and Table 3. 

Figure 3: The Shell standard control problem. 

Table 3: The description of variables in shell model 

Manipulated variables Description Model output Description 
u1 Top draw y1 Top end point 
u2 Side draw y2 Side end point 
u3 Bottoms reflux duty y3 Top temperature 
Unmeasured disturbances Description y4 Upper reflux temperature 
d1 Inter. Reflux duty y5 Side draw temperature 
d2 Upper reflux duty y6 Inter. reflux temperature 

y7 Bottoms reflux temperature 

The main purpose is control y1 and y2 to set point, and constraint of y7 is satisfied at the same time. The 
system has strong interactions and large dead times, so the MPC control strategy is better than PID control. 
The essential parameters are given as follow: δ=0.05, y1=0.3, y2=0.3, R=1, Q=1. In addition, prediction horizon 
of MPC(1) is 60, MPC(2) is 180, and EMPC is 120.  

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The result of simulation is as follow: 

Figure 4: Simulation result: state trajectory (system(1) use switch strategy, system(2) use EMPC). 

Table 3: Time summary 

System Control time Optimization time Total time 
system(1): proposed 43 min 125 min 168 min 
system(2): EMPC 218 min 218 min 218 min 

Switch method is applied to system(1). MPC(1) save the running time in control zone, but the aggressiveness 
of MPC(1) leads to the overshoot in optimization zone, shown in Figure 4. The hysteretic aggressiveness is 
eliminated by the large prediction horizon of MPC(2). At the same time, this aggressiveness also assists the 
improvement of rapidity, shown in Table 4. System(2) uses EMPC, although system has strong robustness, it 
takes long time to arrive same cost value, shown in Figure 5.  
For qualitative, the switch method shorten the transition time and reduce waste production caused by 
changing set point. For quantitative, transition time is decreased from 100 minutes to 43 minutes in SHELL 
model. In other word, the method saves more than half a waste production in transition process. 

Figure 5: Simulation result: the value of cost function (system(1) use switch strategy, system(2) use EMPC). 

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5. Conclusions

In this paper, a switch method of EMPC is proposed to improve economic benefits. Typical EMPC consider 
the control and optimization simultaneously, and the respond time is increased by the interaction between the 
two tasks. This paper gives a brief overview of EMPC search principle, and the reason why EMPC have long 
respond time is found. The way to overcome this problem is to separate the whole zone into control zone and 
optimization zone, and different control method is applied in different zones. The result shows that the switch 
method has significant effect on enhancing economic benefit and saving response time. 

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