Feasibility analysis and online adjustment of constraints in model predictive control integrated with soft sensor

    Feasibility analysis and online adjustment of constraints in model predictive control integrated with soft sensor Pengfei Cao, Xionglin Luo, Xiaohong Song PII: DOI: Reference:

S1004-9541(16)30678-4 doi: 10.1016/j.cjche.2016.08.018 CJCHE 655

To appear in: Received date: Revised date: Accepted date:

15 July 2016 18 August 2016 18 August 2016

Please cite this article as: Pengfei Cao, Xionglin Luo, Xiaohong Song, Feasibility analysis and online adjustment of constraints in model predictive control integrated with soft sensor, (2016), doi: 10.1016/j.cjche.2016.08.018

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Feasibility analysis and online adjustment of constraints in model predictive control integrated with soft sensor

IP

T

Pengfei Caoa,, Xionglin Luob, Xiaohong Songc a College of Electrical Engineering and Automation, Shandong University of Science and Technology b Research Institute of Automation, China University of Petroleum, Beijing c State Grid Shandong Electric Power Company Zibo Power Supply Company, Zibo, China

1. Introduction

TE

D

MA

NU

SC R

Abstract Feasibility analysis of soft constraints for input and output variables is critical for model predictive control(MPC). When encountering the infeasible situation, some way should be found to adjust the constraints to guarantee the optimal control law exists. For MPC integrated with soft sensor, considering the soft constraints for critical variables additionally makes it more complicated and difficult for feasibility analysis and constraint adjustment. Therefore, the main contributions are that a linear programming approach is proposed for feasibility analysis, and the corresponding constraint adjustment method and procedure are given as well. The feasibility analysis gives considerations to the manipulated, secondary and critical variables, and the increment of manipulated variables as well. The feasibility analysis and the constraint adjustment are conducted in the entire control process and guarantee the existence of optimal control. In final, a simulation case confirms the contributions in this paper. Keywords: Soft sensor; Model predictive control; Variable constraints; Feasibility analysis

AC

CE P

In chemical processes, some critical variables indicate the production quality directly and play an indispensable role in process control. The lack of real-time measurement technique leads to scare measurements for these variables. As a powerful alternative, soft sensor technique has been proposed and developed rapidly in the last two decades [1-3]. On the basis of the soft sensor model for easily measured variables(defined as secondary variables, such as pressure, temperature, et al.) and the critical variables, it is possible to predict the critical variables in real-time, and in the meanwhile achieve direct control for production quality[4]. Model predictive control(MPC) is an advanced control approach[5-7]. The most significant advantage of MPC is the ability to solve optimal control problem for multi-variable systems with constraints for input and output variables[8,9]. However, little deep research on the combination of soft sensor technique and MPC has been reported so far, and more studies focus on the applications of them[10-12]. Generally, the input and output variables are restricted with hard and soft constraints in actual control. The hard constraints are related with physical condition; the soft constraints reflect control requirements and could be adjusted online. MPC is actually solving quadratic programming with soft constrain conditions. If the soft constraints are feasible, optimal control could be obtained; otherwise, they need to be adjusted until feasible condition is reached. Thus, analyzing the 

Corresponding author. Email address: [email protected](Pengfei Cao). Postal address: College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, 266590, China.

ACCEPTED MANUSCRIPT

TE

D

MA

NU

SC R

IP

T

feasibility of the soft constraints is the preliminary and critical step for MPC. Xi [13] developed the CMMO(constrained multi-objective multi-degree of freedom optimization) method and transformed the soft constraint adjustment into a linear programming problem. However, only the steady-state process was considered, and there was lack of feasibility analysis during the control process. Zhang [14,15] constructed two convex polyhedras first, and analyzed the soft constraint feasibility through judging whether the polyhedras intersected or not. The reasonable constraint adjustment was transformed into a series of linear or nonlinear programming. But the intersectant width was hard to determine in actual process. Although the constraints for the increment of manipulated variables are involved in the optimization process, most approaches do not take these constraints into consideration for feasibility analysis. In MPC integrated with soft sensor, the soft constraints for manipulated, secondary and critical variables should be considered. And they make the feasibility analysis more complicated. Luo [16] considered the related constraints for manipulated variables, and three kinds of constraints appeared as well. Luo verified the approaches in [14,15] were not applicable in this case, and transformed the feasibility analysis into judging whether the convex polyhedron was empty or not. However, high calculation burden is brought and no guidance for the constraint adjustment is proposed. In this paper, combining with linear static soft sensor model and the MPC model, the system model is derived first. Through a simple adjustment for the model, three kinds of soft constraints are converted into two. Based on the adjusted model, a linear programming method is proposed for analyzing the feasibility of these soft constraints, and then the adjustment approach for the soft constraints is given. In final, a simulation case confirms the main contributions of this paper.

2. The effect from three kinds of variable constraints

CE P

The basic structure of MPC integrated with soft sensor is exhibited in Fig.1:

u

ysp

AC

MPC controller

x MPC model

yˆ

y process

Soft sensor model

Measurement at low sampling rate

Figure 1 The structure of MPC integrated with soft sensor Here, y, x and u represent the critical, secondary and manipulated variables respectively. The dashed box indicates the MPC. The MPC model is built for u and x, and soft sensor model is built for x and y; the system model consists of these two models. In regular MPC, only the constraints for manipulated and secondary variables are considered. Integrating soft sensor model brings the critical variable constraint. Consider the following constraints:

ACCEPTED MANUSCRIPT  ymin  y  ymax   xmin  x  xmax u  u  u max  min

(1)

Based on the polyhedral pole theory[14-16], we have three convex polyhedrons as:

IP

P(X)  {x , xmin  x  xmax }

T

P(U)  {u , umin  u  umax }

P(Y)  { y , ymin  y  ymax }

PY(U)

PX(U)

PX(U) PY(U)

P(U)

TE

(a) nonintersect

PX(U)

CE P

(b) PY(U)∩P(U)≠∅

D

P(U)

PY(U)

PX(U)

MA

PY(U)

NU

SC R

Considering the simple case of linear system model, these polyhedrons could be transformed into the corresponding collections within the same space based on system model[14-16], such as transforming P(Y) and P(X) into PY(U) and PX(U) within the manipulated variable space. For two collections P and Q, P∩Q≠∅ indicates P intersects with Q. Then, there exist six situations for PY(U), PX(U) and P(U), as shown in Fig.2.

P(U)

(c) PX(U)∩P(U)≠∅

PY(U) PX(U)

PY(U) PX(U)

P(U)

P(U)

(e) PY(U)∩PX(U)≠∅, (f) PX(U)∩PX(U)∩P(U)≠∅ PX(U)∩P(U)≠∅ Figure 2 Two-dimension space sketch diagram for the intersection of PY(U), PX(U) and P(U) Only satisfying the situation in Fig.2(f) could guarantee the optimal control exist. For regular MPC, only two situations exist: intersection or not. Comparing the number of situation, it is suspected that the feasibility analysis with critical variable constraint tends to be more complicated. As a matter of fact, the analysis difficulty is embodied in two main aspects: some effective methods do not function well with one more kind of variable constraint[16]; nonlinear soft sensor is usually used, which makes it hard for the variable constraint transformation and optimal control solving as well. In [16], Luo considered the related constraints of input variables, and also three kinds of variable constraints were involved in the feasibility analysis. He transformed the feasibility problem into whether the convex polyhedron from the constraints was empty or not. However, this approach could only judge whether the constraints were feasible or not, and no more suggestions on the constraint adjustment was proposed. And the analysis is based on the steady-state process with high calculation burden. Thus, it is necessary to propose applicable methods for feasibility analysis with three kinds of constraints and give corresponding

AC

(d) PY(U)∩PX(U)≠∅

P(U)

ACCEPTED MANUSCRIPT adjustment approach.

3. MPC integrated with soft sensor In this paper, multi-input and single-output(MISO) soft sensor system is studied. For easy discussion, a simple soft sensor model, i.e. linear static model, is adopted: p

T

yˆ (k | k )   ci xˆi (k | k )  Cxˆ (k | k )

(2)

IP

i 1

c p   R1 p , xˆ (k | k )  R p1

SC R

C  c1

in which yˆ(k | k )  R1 represents critical variable estimation, and could be obtained through Eq.(2) between two neighboring slow sampling periods; {xˆi (k | k )} are the secondary variable

NU

estimations; {ci} are the model parameters. We could build the discrete state space representation for the MPC model as: (3) xˆ (k  1| k )  Axˆ (k | k )  Bu(k )

MA

xˆ (k  j | k ) represents the estimation at k+j moment with xˆ (k | k )  x(k ) ; u(k )  Rm1 represents

the manipulated variable vector, and the case of m≤p is discussed; A∈ Rp×p and B∈ Rp×m are the state matrixes. Combining Eqs.(2) and (3), we have the system model as:  xˆ (k  1| k )  Axˆ (k | k )  Bu(k )  yˆ(k | k )  Cxˆ (k | k ) 

D

(4)

AC

CE P

TE

It should be noticed that the MPC model and soft sensor model are built separately. The reason lies in the dual-rate sampling characteristics of the soft sensor system. Generally, the secondary variables are sampled at faster rate than critical variables. For convenience in applications, the MPC model could be built with fast samples of u and x; the soft sensor model could be built and also updated with the samples of x and y at slow sampling period. That is Eq.(2) and Eq.(3) are built in different sampling frequencies. Actually, there is lots of research on these two kinds of models (Eqs.(2) and (3)) for control and prediction[5,7,12,17-19]. Thus, it is reasonable to build the system model in the way of Eq.(4). In order to reduce the complexity of feasibility analysis with three kinds of constraints, we could transform Eq.(4) into xˆ (k  1| k )  Axˆ (k | k )  Bu(k )    xˆ (k | k )   I    yˆ (k | k )   yˆ(k | k )   C  xˆ (k | k )  C xˆ (k | k )     

(5)

in which C'∈ R(p+1)×(p+1), yˆ (k | k )  R( p 1)1 , I∈ Rp×p. Only the constraints for u and y’ are left. Consider the performance function P

min J (k )   yr (k  i )  yˆ (k  i | k ) i 1

2 Q (i )

M 1

  u(k  i ) i 0

2 R (i )

(6)

yr (k  i) represents reference trajectory; P and M are prediction and control domain; {Q(i)≧0}

and {R(i)≧0} represent estimation error and control weights. Assume P≧M, and △u(k+i)=0 when i≧M. Based on Eq.(5), the predictions are derived for xˆ (k  i | k ) (i=1,2,...,P) as: xˆ (k  1 k )  Axˆ (k k )  Bu(k )

ACCEPTED MANUSCRIPT xˆ (k  2 k )  A2 xˆ (k k )  ABu(k )  Bu(k  1)

┇

 ABu(k  M  1)  Bu(k  M  1)

IP

xˆ (k  M  1 k )  AM 1 xˆ (k k )  AM Bu(k )  AM 1 Bu(k  1) 

 Bu(k  M  1)

T

xˆ (k  M k )  AM xˆ (k k )  AM 1 Bu(k )  AM 2 Bu(k  1) 

SC R

┇ xˆ (k  P k )  AP xˆ (k k )  AP 1 Bu(k )  AP  2 Bu(k  1) 

 AP  M Bu(k  M  1)  AP  M 1 Bu(k  M  1) 

and for yˆ (k  i | k ) as:

 Bu(k  M  1)

NU

yˆ (k  1 k )  C Axˆ (k k )  C Bu(k )

MA

yˆ (k  2 k )  C A2 xˆ (k k )  C ABu(k )  C Bu(k  1)

┇

yˆ (k  M k )  C AM xˆ (k k )  C AM 1 Bu(k )  C AM 2 Bu(k  1) 

 C Bu(k  M  1)

D

yˆ (k  M  1 k )  C AM 1 xˆ (k k )  C AM Bu(k )  C AM 1 Bu(k  1) 

TE

 C ABu(k  M  1)  C Bu(k  M  1)

┇

CE P

yˆ (k  P k )  C AP xˆ (k k )  C AP 1 Bu(k )  C AP  2 Bu(k  1)   C AP  M Bu(k  M  1)  C AP  M 1 Bu(k  M  1)  C Bu(k  M  1)

Define

AC

 C A   C A2    ˆ( y k  1| k )      yˆ(k  2 | k )   M  ˆ   Y (k )  , S x   C A  , Y0 (k )  Sx xˆ (k | k ) ,   C AM 1       yˆ(k  P | k )     C A P   

0  C B  C AB C B  u( k )     u(k  1)   , S  C AM 1 B C AM  2 B U (k )   u     C AM B C AM 1 B    u ( k  M  1)     C AP 1 B C AP  2 B 

and we have

0 0 C B C ( AB  B ) C ( AP  M B  AP  M 1 B 

          B ) 

ACCEPTED MANUSCRIPT

T

 yˆ (k  1| k )   yˆ (k  2 | k )    S xˆ (k | k )  S U (k )  Y (k )  S U (k ) Y (k )   x u 0 u      yˆ (k  P | k )  Then, the critical variable vector could be written as Yˆ (k )  D  Sx xˆ (k | k )  SuU (k )   DY0 (k )  DSuU (k )

1

0 0

0

1

0

IP

0

0

Define

0

0    R P P ( p 1) 0  1

0

0 0  ,  0 I 

SC R

0 0 D   0

(7)

NU

U (k )  U (k )  U (k  1)

MA

0  u(k  1)  0    u ( k ) ,E  U (k )   1       u ( k  M  1)   0

0 I

I

0

0    E 21 0  I 

0

E22 

CE P

Then

TE

D

 I I  0 I E2      0

I

Define

AC

0   u(k )     u( k )  u(k  1)     0     U (k ) E1U (k )   u(k  1)     , E2U (k )      U (k )       u(k  M  1)   u(k  M  1) 

0  yr (k  1)  Q(1)  y (k  2)   0 Q(2) , Q  Yr (k )   r        0  yr ( k  P )  Then, we have the performance function as

0

0  0  R(0)   0 R(1) , R  0    Q( P)   0

    0  R( M  1)  0

0

(8)

ACCEPTED MANUSCRIPT J (k ) T

 Yr (k )  Yˆ (k )  Q Yr (k )  Yˆ (k )   U T (k ) RU (k )  Yr (k )  DY0 (k )  DSu U (k )  Q Yr (k )  DY0 ( k )  DSu U ( k )   U T ( k ) RU ( k ) T

 Yr (k )  DY0 (k )  DSu U (k )  Q Yr (k )  DY0 ( k )  DSu U ( k ) 

T

T

T

 u(k  1)    E22     R  E21  U (k )   

 Yr (k )  DY0 (k )  Q Yr (k )  DY0 (k ) 

 u(k  1)   E22     U (k )  

IP

   E21 

(9)

SC R

T

 2 Yr (k )  DY0 ( k )  QDSu U ( k )  U T ( k )  DSu  Q  DSu  U ( k ) T

T

T T T  uT (k  1) E21 RE21u(k  1)  2uT ( k  1) E21 RE22U ( k )  U T ( k ) E22 RE22U ( k )

Assume the soft constraints for U and Y' as

NU

T T  const  U T (k )  E 22 RE 22   DSu  Q  DSu   U (k )   T T T   2 u (k  1) E 21 RE22  Yr (k )  DY0 (k )  QDSu  U (k )  

Here,

T

T T  , U min  umin umax

T

T

T T  , Ymin   ymin umax

CE P

T U max  umax

T

T T  , U max   umax umin

TE

T U min   umin

D

MA

 U min  U (k )  U max   U   U ( k )  E21u(k  1)  E22U (k )  U max  min  Ymin  D 1Yˆ (k )  Y0 (k )  SuU (k )  Ymax 

T  , Ymax   ymax T ymin

(10)

T

T  , umin T

T  , ymax

Then, the soft constraints could be rewritten as

AC

 D1   d1   D  U (k )  d   2  2  D3   d 3 

(11)

S  E  I  D1    , D2   22  , D3   u  ,  I    E22    Su   Y  Y (k )  U   U max  E21u(k  1)  d1   max  , d 2   , d3   max 0    U min   U min  E21u(k  1)   Ymin  Y0 (k ) 

In final, the constrained optimization could be transformed into a quadratic programming: 1 min J (k )  U T (k )ΦU (k )  ΘTU (k ) U (k ) 2

(12)

 D1   d1  s.t.  D2  U (k )  d 2   D3   d 3 

in which Φ  E22T RE22   DSu  Q  DSu  , Θ  uT (k  1) E21T RE22  Yr (k )  DY0 (k )  QDSu  . T

T

T





Once the optimal solution for U(k) is obtained, only u(k) is utilized for the current control. At the next sampling moment, the optimization is conducted again.

ACCEPTED MANUSCRIPT 4. Feasibility analysis and adjustment method for soft constraints

T

When there is no feasible solution for the above quadratic programming, it is unable to achieve control target. The only way to solve this problem is to adjust the soft constraints. In this part, a linear programming method is proposed first, and then the adjustment approach is given. We could rewrite the constraints as

(13)

SC R

IP

 U min  U min,1  U (k )  U max  U max,1   U min  U min,2  U (k )  U max  U max,2 Y  Y  Y (k )  S U (k )  Y  Y min 0 u max max  min

NU

in which △Umin,1, △Umax,1, △Umin,2, △Umax,2, △Ymin and △Ymax represent the soft constraint variation. Let   Ymin  Y0 (k ) , Ymax   Ymax  Y0 (k ) Ymin According to Eq.(8), we have

(14)

D

MA

  0   0   0   0    E1U (k )         U max   U max,1   U min   U min,1   U min  U min,2  E2U (k )  U max  U max,2   0  0   0    0  0 0     E1U (k )           Ymax   Ymin    Ymin  0 Su  Ymax 

TE

Let

CE P

 0   0   0   0  , X2   X1  E1U (k )     E1U (k )    ,   U min   U min,1  U max   U max,1 

0 X3   0

0  0  0  0   0  E1U (k )     , X4        0  Su  Ymin   Ymin  Ymax

0  0  E1U (k )    , Su   Ymax 

Define

AC

X 5  E2U (k )  Umin  Umin,2 , X 6  Umax  Umax,2  E2U (k )

 0   0   0   0          U min , U min,1 , U max , U max,1 ,    U min  U max   U max,1   U min,1  0 Su   0

0  0   0       , Ymin , Ymin      Su  Ymin  Ymin   0   0       Ymax , Ymax     Ymax   Ymax

Then, we obtain  .1  U max  .1  U max   U min   X1  X 2  U min  S  X  X  S  U   Y   Y   S U   u 1 4 u min,1 max max u min         S X  X  S  U   Y  S U  Ymin 3 u max,1 min u max  u 2  X 5  X 6  U max,2  U min,2  U max  U min

Designate

(15)

ACCEPTED MANUSCRIPT

X 4T

X 5T

X 6T

T T Δ  U min,1 U max,1 U T min,2

I S R u 0  0

I 0 Su 0

U T max,2

0 0 0 0 -I 0 I 0 0 -Su I 0 0 0 0 0 0 I I 0

We can write the standard form as

0 0 0 -I

0 0 0 -I

T

T  , Y max

0 0 -I 0

0

0 -I  0  0

(16)

NU

 RZ  b   Z 0

-I 0 -Su

T Y min

T

X 3T

IP

X 2T

SC R

Z   X1T

  U min    U max  Y   S U   T max u min  , ΔT  , b    SuU max   Ymin      U max  U min 

MA

Eq.(16) gives feasible solution collection for the constraints in Eq.(13). Especially when △ =0, the optimal control could be obtained for Eq.(12) with current constraints. The judgment for the feasibility could be summarized as the following theorem. Theorem 1 If the linear programming c6T 

T

(17)

TE

D

min W  c T Δ, c  c T  1   LP  s.t. RZ  b  Z 0  

AC

CE P

has the optimal solution W=0, then the current constraints are feasible [13]. In Eq.(17), c is the weight coefficient vector, and reflects the inclination for the constraint adjustment. If △ i is more preferred to be adjusted than △ j, and ci would be set smaller than cj [13-16]. We have the following adjustment procedure: (1) Set {ci}=[c1T,…,c6T]T according to user’s requirement. (2) Solve the linear programming in Eq.(17) and obtain △ . (3) If W=0, the current soft constraints are feasible and the optimal control exists for Eq.(12), stop; else, adjust c according to user’s requirement and go to step (4). (4) Obtain the [△Umin,1, △Umax,1, △Umin,2, △Umax,2, △Ymin, △Ymax] from △ in step 2. Adjust the soft constraints Umin=Umin-△Umin,1, Umax=Umax+△Umax,1, △Umin=△Umin-△Umin,2, △Umax=△Umax+△Umax,2, Ymin=Ymin-△Ymin, Ymax=Ymax-△Ymax. If some soft constraints reach the hard ones, the constraints hold with them. Modify b in Eq.(17) with new soft constraints. Go to step (2). The process of feasibility analysis and constraint adjustment is shown in Fig.3. Actually, the procedure is a gradually adjusting process. Before the feasible soft constraints are obtained, the constraints will be recalculated (in step 4) with the new △ (in step 2) in one cycle. It seems like expanding the convex polyhedron as seen in [16] to find feasible constraints. The adjustment order will follow the user's requirement with reset c (in step 3). And the small value of ci reflects greater percentage and higher priority for the corresponding variable[13-16]. Once all the constraints reach the hard ones and no feasible constraints exist, more manipulated variables should be involved in this control system. As seen from the Theorem and the procedure, the proposed algorithm embraces several

ACCEPTED MANUSCRIPT

T

advantages: first, the algorithm effectively solves the problem of feasibility analysis with the constraints of manipulated, secondary and critical variables; secondly, the feasibility analysis and constraint adjustment are conducted in the entire control process, which guarantees the optimal control law always exists; thirdly, the increment of manipulated variables are considered in the algorithm, which keeps the control process flat.

SC R

Solve the linear programming in Eq.(17), and obtain △

IP

Set c=[c1T,…,c6T]T according to user’s requirement

Yes Judge whether W=0

MA

NU

No Adjust the constriants: Umin= Umin-△Umin,1, Umax=Umax+△Umax,1, △Umin=△Umin-△Umin,2, △Umax=△Umax+△Umax,2, Ymin= Ymin-△Ymin, Ymax= Ymax- △Ymax.

Current soft constraints are feasible

TE

D

If some soft constraints reach the hard ones, the constraints hold with the values. Modify b in Eq.(17) with new soft constraints.

Figure 3 The diagram for feasibility analysis and constraint adjustment

CE P

5. Case study

AC

Consider a system with one manipulated variable, two secondary variables and one critical variable, and we have the discrete model as:  x (k  1)  Ax (k )  Bu(k )  y (k )  Cx (k )   0.03824 0.009782 2 A , B    , C  0.2 1.072  0  0.01563  1 

The hard constraint for u is set as [0.25, 0.7], and the soft one is set as [0.32, 0.65]. x and y are restricted with [4.6, 22.8] for x1, [75.8, 80.1] for x2 and [67.1, 78] for y, respectively, and the set point of y is set as yr=70. The system model is converted into x (k  1)  Ax (k )  Bu(k ) 0   1    1  ,C   0  x (k )   I      y ( k )   y ( k )   C  x ( k )  C x ( k ) 0.2 1.072      

The proposed feasibility analysis and adjustment methods are conducted based on the above model. For comparison, the CMMO approach is also used for this system [14]. The curves of all the variables are shown in Figs.4 and 5.

SC R

IP

T

ACCEPTED MANUSCRIPT

(b) x2 and its constraints

D

MA

NU

(a) x1 and its constraints

AC

CE P

TE

(c) y and its constraints (b) u and its constraints Figure 4 manipulated, secondary and critical variables and their constraints based on the CMMO method

(a) x1 and its constraints

(c) y and its constraints

(b) x2 and its constraints

(b) u and its constraints

ACCEPTED MANUSCRIPT

MA

NU

SC R

IP

T

Figure 5 manipulated, secondary and critical variables and their constraints based on the proposed method From the CMMO method, the current soft constraints are judged infeasible. The constraints for secondary and critical variables are inclined to be adjusted. Thus, the corresponding weights are set 1 for y’ and 100(some big value, 100 is chosen in the simulation) for u. The new constraints are calculated as [4.6, 24] for x1, [74.8, 80.1] for x2 and [66, 78] for y, which is shown in Fig.4(a-c). However, the manipulated variable exceeds the high soft constraint as shown in Fig.4(d), which is not allowed for this system. The main reason for this is that the CMMO method is realized based on the steady-state model of the system, and the constraints will not be adjusted in the control process. With the proposed method in this paper, the constraints are adjusted as [4.6, 27.8] for x1, [74.1, 80.1] for x2 and [64.6, 78] for y, which is shown in Fig.5(a-c). For this system, only one group of new constraints is obtained. Actually, the feasibility analysis is conducted at each sampling period, and the new constraints are always feasible for the following control optimization. With the adjustment result, the manipulated variable keeps within the constraint, which indicates the effectiveness of the proposed method.

6. Conclusions

AC

CE P

TE

D

The soft sensor technique and MPC are combined for achieving critical variable prediction and advanced control. The main research aims at the feasibility analysis with the constraints for the manipulated, secondary and critical variables. The feasibility of optimal control is transformed into a linear programming problem. The analysis process is conducted at each sampling period, and the adjusted constraint is obtained in real-time. We have done preliminary research, but some issues are worth being studied in further work. A linear soft sensor model is considered in this paper; however, the system has nonlinear characteristics generally. Thus, much more attention should be paid on the combination of MPC and nonlinear soft sensor model, and two issues arise: the realization of nonlinear MPC; the handling of constraints and feasibility analysis with nonlinear soft sensor model.

Acknowledgements This work was supported by the Scientific Research Foundation of Shandong University of Science and Technology for Recruited Talents(2016RCJJ046) and the National Basic Research Program of China (2012CB720500).

References [1] K. Fujiwara, M. Kano, S. Hasebe. Development of correlation-based pattern recognition algorithm and adaptive soft-sensor design[J]. Control Engineering Practice, 2012, 20: 371-378. [2] S. Q. Xu, X. G. Liu. Melt index prediction by fuzzy functions with dynamic fuzzy neural networks[J]. Neurocomputing, 2014, 142(22): 191-198. [3] J. Yu. A Bayesian inference based two-stage support vector regression framework for soft sensor development in batch bioprocesses[J]. Computers and Chemical Engineering, 2012, 41(11): 134-144. [4] Y. G. Li, W. H. Gui, C. H. Yang, Y. F. Xie. Soft sensor and expert control for blending and digestion process in alumina metallurgical industry[J]. Journal of Process Control, 2013, 23: 1012-1021.

ACCEPTED MANUSCRIPT

AC

CE P

TE

D

MA

NU

SC R

IP

T

[5] M. Vishal, H. Juergen. Model predictive control of reactive distillation for benzene hydrogenation[J]. Control Engineering Practice , 2016, 52: 103-113. [6] L. Chang, X. G. Liu, M. A. Henson. Nonlinear model predictive control of fed-batch fermentations using dynamic flux balance models[J]. Journal of Process Control, 2016, 42: 137-149. [7] D. H. Xavier, H. Laurent, R. Jorg, C. Bertrand. Model predictive control for discrete event systems with partial synchronization[J]. Automatica, 2016, 70: 9-13. [8] J. H. Lee. Model predictive control: review of the three decades of development[J]. International Journal of Control, Automation and Systems, 2011, 9(3): 415-424. [9] P. Matej, Z. Eva, R. Rush, C. Sergej, S. Michael. Bridging the gap between the linear and nonlinear predictive control: Adaptations for efficient building climate control[J]. Control Engineering Practice, 2016, 53: 124-138. [10] S. H. Yang, X. Z. Wang, C. Mcgreavy, Q. H. Chen. Soft sensor based predictive control of industrial fluid catalytic cracking processes[J]. Trans IChemE, 1998, 76: 499-508. [11] J. Kortela, S. L. J. Jounela. Model predictive control utilizing fuel and moisture soft-sensors for the BioPower 5 combined heat and power(CHP) plant[J]. 2014, 131: 189-200. [12] J. Kortela, S. L. J. Jounela. Model predictive control utilizing fuel and moisture soft-sensors for the BioPower 5 combined heat and power (CHP) plant[J]. Applied Energy, 2014, 131: 189-200. [13] Y. G. Xi, H. Y. Gu. Feasibility analysis and soft constraints adjustment of CMMO[J]. ACTA AUTOMATICA SINICA, 1998, 24(6): 727-732. [14] X. L. Zhang, S. B. Wang, X. L. Luo. Feasibility analysis and constraints adjustment of constrained optimal control in chemical processes[J]. CIESC Journal, 2011, 62(9): 2546-2554. [15] X. L. Zhang, X. L. Luo, S. B. Wang. Feasibility analysis and on-line adjustment of constraints in process predictive control[J]. CIESC Journal, 2012, 63(5): 1459-1467. [16] X. L. Luo, X. L. Zhou, S. B. Wang. Analysis of constrained optimal control with related constraints of input variables[J]. ACTA AUTOMATICA SINICA, 2013, 39(5): 679-689. [17] W. D. Ni, S. D. Brown, R. L. Man. A localized adaptive soft sensor for dynamic system modeling[J]. Chemical Engineering Science, 2014, 111(8): 350-363. [18] W. M. Shao, X. M. Tian. Adaptive soft sensor for quality prediction of chemical processes based on selective ensemble of local partial least squares models[J]. Chemical Engineering Research and Design, 2015, 95: 113-132. [19] C. Shang, X. L. Huang, J. A. K. Suykens, D. X. Huang. Enhancing dynamic soft sensors based on DPLS: A temporal smoothness regularization approach[J].Journal of Process Control, 2015, 28: 17-26.