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

Pengfei Cao *,Xionglin Luo ,Xiaohong Song

1 College of Electrical Engineering and Automation,Shandong University of Science and Technology,Qingdao 266590,China

2 Research Institute of Automation,China University of Petroleum,Beijing 102249,China

3 State Grid Shandong Electric Power Company Zibo Power Supply Company,Zibo 255000,China

1.Introduction

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(de fined as secondary variables,such as pressure and temperature)and the critical variables,it is possible to predict the critical variables in real-time,and in the meanwhile achieve directcontrolfor 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 forinput and outputvariables[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[10,11].

Generally,the inputand outputvariables are restricted with hard and soft constraints in actual control.The hard constraints are related with physical condition;the soft constraints re flect 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 feasibility of the softconstraints is the preliminary and criticalstep for MPC.Xi[12]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[13,14]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 incrementofmanipulated 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[15]considered the related constraints for manipulated variables,and three kinds of constraints appeared as well.Luo verified that the approaches in[13,14]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 adjustmentapproach forthe softconstraints is given.In final,a simulation case con firms the main contributions of this paper.

2.The Effect From Three Kinds of Variable Constraints

The basic structure ofMPC integrated with softsensor is exhibited in Fig.1:

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:

Based on the polyhedral pole theory[13–15],we have three convex polyhedrons as:

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[13–15],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 that P intersects with Q.Then,there exist six situations for PY(U),PX(U)and P(U),as shown in Fig.2.

Only satisfying the situation in Fig.2(f)could guarantee that the optimal control exists.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 offact,the analysis difficulty is embodied in two main aspects:some effective methods do not function well with one more kind of variable constraint[15];nonlinear soft sensor is usually used,which makes it hard for the variable constraint transformation and optimal control solving as well.In[15],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 whetherthe constraintswere feasible ornot,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 adjustment approach.

Fig.2.Two-dimension space sketch diagram for the intersection of PY(U),PX(U)and P(U).

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:

in which^y(k|k)∈R represents critical variable estimation,and could be obtained through Eq.(2)between two neighboring slow sampling periods;{^xi(k|k)}are the secondary variable estimations;{ci}are the model parameters.We could build the discrete state space representation for the MPC model as:

^x(k+j|k)represents the estimation atk+jmomentwith^x(k|k)=x(k);u(k)∈Rm×1represents the manipulated variable vector,and the case ofm≤pis discussed;A∈Rp×pand B∈Rp×mare the state matrixes.Combining Eqs.(2)and(3),we have the system model as:

It should be noticed that the MPC model and soft sensor model are builtseparately.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 andyat slow sampling period.That is Eqs.(2)and(3)are built in different sampling frequencies.Actually,there are lots of research on these two kinds of models(Eqs.(2)and(3))for control and prediction[5,7,11,16–18].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

in which C′∈R(p+1)×(p+1),^y′(k|k)∈ R(p+1)×1,and I∈Rp×p.Only the constraints for u and y′are left.Consider the performance function

yr(k+i)represents reference trajectory;PandMare prediction and control domain;{Q(i)≧0}and{R(i)≧0}represent estimation error and control weights.AssumeP≧M,and Δu(k+i)=0 wheni≧M.Based on Eq.(5),the predictions are derived for^x(k+i|k)(i=1,2,…,P)as:

4.Feasibility Analysis and Adjustment Method for Soft Constraints

When there is no feasible solution for the above quadratic programming,it is unable to achieve control target.The only way to solve this problemis to adjustthe softconstraints.In this part,a linearprogramming method is proposed first,and then the adjustment approach is given.

We could rewrite the constraints as

in which ΔUmin,1,ΔUmax,1,ΔUmin,2,ΔUmax,2,ΔYminand ΔYmaxrepresent the soft constraint variation.

Eq.(16)gives feasible solution collection for the constraints in Eq.(13).Especially when Δ=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

hastheoptimalsolutionW=0,thenthecurrentconstraintsarefeasible[12].

In Eq.(17),c is the weightcoefficientvector,and re flects the inclination for the constraint adjustment.If Δiis more preferred to be adjusted than Δj,and ciwould be set smaller than cj[12–15].

We have the following adjustment procedure:

(1)Set{ci}=[c1T,…,c6T]Taccording to user's requirement.

(2)Solve the linear programming in Eq.(17)and obtain Δ.

(3)IfW=0,the currentsoft constraints are feasible and the optimal control exists for Eq.(12),stops;else,adjust c according to user's requirement and go to step(4).

The process of feasibility analysis and constraint adjustment is shown in Fig.3.

Fig.3.The diagram for feasibility analysis and constraint adjustment.

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 Δ(in step 2)in one cycle.It seems like expanding the convex polyhedron as seen in[15]to find feasible constraints.The adjustment order will follow the user's requirement with reset c(in step 3).And the small value of cire flects greater percentage and higher priority for the corresponding variable[12–15].

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 Theorem 1 and the procedure,the proposed algorithm embraces several advantages: firstly,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 that the optimal control law always exists;thirdly,the increment of manipulated variables is considered in the algorithm,which keeps the control process flat.

5.Case Study

Consider a system with one manipulated variable,two secondary variables and one critical variable,and we have the discrete model as:

The hard constraintfor u is setas[0.25,0.7],and the softone is set as[0.32,0.65].x andyare restricted with[4.6,22.8]forx1,[75.8,80.1]forx2and[67.1,78]fory,respectively,and the set point ofyis set asyr=70.The system model is converted into

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[13].The curves of all the variables are shown in Figs.4 and 5.

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]forx1,[74.8,80.1]forx2and[66,78]fory,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]forx1,[74.1,80.1]forx2and[64.6,78]fory,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.

Fig.4.Manipulated,secondary and critical variables and their constraints based on the CMMO method.

Fig.5.Manipulated,secondary and critical variables and their constraints based on the proposed method.

6.Conclusions

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.

Acknowledgments

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).

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