Multi-objective nonlinear model predictive control through switching cost functions and its applications to chemical processes☆

Defeng He,Shiming Yu,Li Yu

College of Information Engineering,Zhejiang University of Technology,Hangzhou 310023,China

Keywords:Nonlinear system Model predictive control Multi-objective control Switched control Continuous stirred tank reactor

ABSTRACT This paper proposes a switching multi-objective model predictive control(MOMPC)algorithm for constrained nonlinear continuous-time process systems.Differentcostfunctions to be minimized in MPC are switched to satisfy different performance criteria imposed at different sampling times.In order to ensure recursive feasibility of the switching MOMPC and stability of the resulted closed-loop system,the dual-mode control method is used to design the switching MOMPC controller.In this method,a local control law with some free-parameters is constructed using the control Lyapunov function technique to enlarge the terminal state set of MOMPC.The correction term is computed ifthe states are out ofthe terminalset and the free-parameters of the local control law are computed ifthe states are in the terminal set.The recursive feasibility ofthe MOMPC and stability of the resulted closed-loop system are established in the presence of constraints and arbitrary switches between cost functions.Finally,implementation of the switching MOMPC controller is demonstrated with a chemical process example for the continuous stirred tank reactor.

1.Introduction

Nonlinearmodelpredictive control(NMPC)has been receiving great attention since the 1990s in both industries and academia[1-4],especially in chemical processes[5-9].In general,NMPC uses a model of plants to predict their future evolutions and a control input is determined by online optimization to minimize a certain performance criterion subject to the state and/or the control constraints.The input obtained is applied until the next sampling time and the overall procedure is repeated.Clearly,the cost functions that represent the performance criteria of controllers play an important role in the design of NMPC controllers.

In many MPC applications to chemical processes[5-9],not only one but a number of different performance criteria should be taken into account for the controlled system design.At the same time,chemical processes are usually characterized by nonlinear behavior and strong coupling of various process variables.These features lead to a multiobjective NMPC problem.Different performance criteria to be considered may be associated with different regions in the system state space or with different time instants or stages in processes.Take the grade transition processes of polyole fin plants[8,9]as an example.The prices of produced polymer resins and consumed energy change according to certain schedules dependent on market conditions.Consequently,we should use several different cost criteria to design different controllers for the grade transition process in order to meet the economic optimization for the polyole fin plant.It is also necessary to quickly react to disturbances or faults whenever they occur[10].

To tackle the problem,several new multi-objective NMPC algorithms have been proposed recently.For instance,the MPC control sequence is computed to minimize the max of a finite number of objectives[11].The linear MPC law is derived by minimizing a convex combination of different cost functions and stability is guaranteed by the convex combination,which is close to that desired[12].The lexicographic programming and logic method are used to prioritize multiple cost functions and then design multi-objective NMPC controllers[13-15].The MPC controller is derived by minimizing the distance of the cost function to that of the steady-state utopia point and the nominal asymptotic stability of the NMPC is ensured by the terminal equality constraint and the assumption of strong duality[16].Similarly,we have proposed a utopia-tracking multi-objective NMPC scheme in the dualmode framework using the terminal region constraint and control Lyapunov functions[17].

Different cost functions associated with a certain state region individually and a state-dependent switch between the cost functions have been employed to design the multi-objective NMPC controller[18].Stability ofthe proposed NMPC is ensured by imposing a constraint to the optimization problem such that if a switch occurs at a certain sampling time,the optimal value of the current activated cost function must be smaller than one of the cost functions that are active at the last time.Moreover,general time-dependent switches between different cost functions are exploited to compute multi-objective NMPC law and the average dwell-time method is used to ensure asymptotic stability of the proposed NMPC[19].This means that in average,switches between different cost functions do not occur very often,i.e.,the switches are restricted to guarantee the stability of the controller.

In this paper,we consider a class of multi-objective NMPC problems,where different cost functions to be minimized may be switched and then activated at any sampling time.This implies that at each sampling time,one of the cost functions is selected and the cost functions can be switched arbitrarily.This operation results in a closed-loop switched system since we switch different NMPC controllers designed by minimizing different cost functions.In order to guarantee recursive feasibility of the switching NMPC and stability of the closed-loop system,we use the dual-mode control method[20,21].In this method,a local controllaw with some free-parameters is designed by the controlLyapunov functions(CLFs)concept[22]to enlarge the terminal state set of MPC.Thus the correction termis computed when the states are outofthe terminal setand the free-parameters of the localcontrollaw are computed when the states are in the terminal set.We establish the recursive feasibility of the MPC and stability of the closed-loop system in the face ofconstraints and arbitrarily switched costfunctions.This is a nice result since one can incorporate arbitrary change in the desired cost functions,which allows us to take into account a larger class of multi-objective control problems.The contribution of the paper is then a step forward in the design ofmulti-objective NMPC controllers thatcan take into consideration switches between different cost functions,going beyond the existing modern tools from switched system theory.Finally,the implementation of the proposed multi-objective NMPC controller is demonstrated using a chemical process example of the continuous stirred tank reactor(CSTR).

2.Problem Setup and Preliminaries

We consider a continuous-time nonlinear system described by

for some compact set Z?X×U.Set Z is assumed to contain the equilibrium point as its interior.

where number 0＜Tp＜∞is a prediction horizon,variables x(s|tk)andare the values of states and controls at time s predicted at time tk,respectively,and x(tk)is the state at current time tk,with x(tk|tk)=x(tk).In standard MPC,the goal is to asymptotically stabilize the origin of system(1),while minimizing a single performance function Jj(x).The finite horizon optimal control problem of the standard MPC can be formulated as

where u(Tp,tk)is the predictive control pro file over the prediction horizon window[tk,tk+Tp]at sampling time tk.

Remark 1.In this work,the states ofsystem(1)are assumed to be sampled at each sampling time of the time sequence{tk}with tk=t0+kδ where the discrete-time index k=0,1,… and δ ＞ 0 is the sampling period.Consequently,the control law obtained by minimizing the cost function(3)is applied to the continuous-time system(1)in a fashion of sampling-and-hold with the sampling period δ.For simplicity,let t0=0.

Denote the optimal state and control trajectories obtained by solving optimization problem(4)asandrespectively,where subscript j indicates the optimal state and control trajectories obtained by minimizing the j th cost function Jj(x).In this MPC setup,the MPC law is de fined in the usual receding horizon fashion:only the first part of the computed optimal trajectoryup to the next sampling time tk+1=tk+δ is implemented to system(1),i.e.,

Then the optimization problem(4)is resolved at the next time tk+1.Similar to standard MPC,the asymptotic stability of the closed-loop systems(1)and(5)cannot be guaranteed by the optimality of finite horizon cost functions(3).In addition,the recursive feasibility of Eq.(4)may not be ensured if the cost functions(3)are switched.

In this work,we use the dual-mode control approach and control Lyapunov function technique to achieve the asymptotic stability of the origin of the closed-loop system,while achieving the recursive feasibility of problem(4)for arbitrary switches between cost functions(3).In the following,some well-known notions and results are recalled in order to present our main results.

De finition 1.[23]

De finition 2.[22]

Consider system(1)and a positive de finite function V(x).If V(x)satis fies

Lemma 1.[24]

Let V(x)be a CLF of system(1).For given numbers D1＞0 and D2＞0,there is a controller u(x):=h(x,μ)

Remark 2.The invariant set S in Lemma 1 is clearly dependent on free-parameters μ.One can use the method presented in[24]to extend the size of S as large as possible.By this way,we derive a largest invariant set and denote it as Smax.Then in Smax,the controller(7)and corresponding closed-loop states satisfy constraint(2)and the closed-loop state trajectories are asymptotically convergent to the origin.

3.Switching Multi-objective NMPC

In this section,we present a switching multi-objective NMPC algorithm for the constrained systems(1)and(2)and establish the recursive feasibility and asymptotic stability of the closed-loop NMPC system.Now we consider a set of η ∈ Λ:={1,2,…,N}different cost functions in Eq.(3),out of which one is selected to be minimized at each sampling time.This implies that the switches between different cost functions occur at some sampling time tk.Assume thatthe associated switching signaldenoted by σ:[0,∞)→ Λ is a piecewise constant right continuous function that speci fies the index of the active cost function at time t.

Let Nσ(0,t)be the numberofswitches occurring in time interval(0,t].Denote the switching time in the interval(0,t]by τ1,τ2,…,τNσ(0,t)(by convention let τ0=0)and the index of the cost function active in the interval[τi,τi+1)by ηi.Note that in our problem,the switching time τiatwhich we switch between differentcostfunctions may be arbitrary and can coincide with some sampling time tkwhere the optimal control problem of NMPC is resolved,i.e.,τi=tkfor all i∈ {0,1,…}and some k∈{0,1,…}.

Assume that function V(x)is a known CLF of system(1)and controller(7)with some known numbers D1and D2is constructed in an invariantset Smax.Let ST=Smax.According to the dual-mode idea,we de fine a dual-mode controller

where the constant vector μf∈ [0,D1]× (0,D2]is a set of feasible parameters associated with ST,vectorμ(t)is a setoffree parameters and c(t)is a correction term,both of which are determined online.For the state x(tk)∈STat sampling time tk,we de fine the following optimal control problem to compute vector μ(t):

where Jη(x)is the cost function activated at time tk.The proposed switching multi-objective NMPC algorithm can now be summarized as follows.

Algorithm 1.Switching multi-objective NMPC

(1)Of fline compute STand μfin Eq.(8)for given numbers D1＞ 0 and D2＞0;set i=k=0 and t0=τ0=0.

(2)At time instant tk,measure state x(tk).

(3)Select the cost function to be minimized in Eq.(3).Ifthis cost function is different from the one used at time tk-1;let i:=i+1 and τi:=tk.

(4)Online solve the optimal control problem of NMPC with the cost

function Jηi(x(tk)):

If x(tk)∈ST,online solve Eq.(9)and denote the solution as;go to Step(6).

Else,online solve Eq.(10):

If the optimization problem is feasible,denote the solution asand go to Step(5).

Endif.

In Algorithm 1,we have a switching multi-objective NMPC controller in the form of Eq.(8),i.e.,

which leads to the switched closed-loop system

It should be pointed out that from the method for determining STpresented in[24],the optimal control problem(9)is always feasible.This means that there always exists a solution to satisfy the constraints in Eq.(9).Then the dual-mode NMPC controller applied by Algorithm 1 is said to be feasible for an initial state x if the controller and its corresponding state trajectory generated by the model x=f(x)+g(x)u with initial condition x(t0)=x together satisfy the constraints in the optimal control problem(10).We de fine the feasible set Sf(Tp)as this set of(x,c)pairs

The setofadmissible states Xf(Tp)is then de fined as the projection of Sf(Tp)onto X:

Set Xf(Tp)contains all possible state values that can be steered into the terminal region STin most horizon time Tpwith a feasible control lawThis set also satis fies that

Theorem 1.The switching multi-objective NMPC controller(11)applied by Algorithm 1 is iteratively feasible in admissible states Xf(Tp).

Proof.Consider two cases:

(1)When x(tk)∈ST.The MPC controller(11)is computed by solution ofproblem(9).From the method ofdetermining STgiven by[24],there exists at least a set of parametersthat the control actionwhere x(t;is the evolution of system(1)with input h(x(tk),μf)starting from x(tk)at time t∈ [tk,tk+1].Note that the feasible parameter vector μfis independent of the cost function to be minimized.Then by induction we have that h(x(t),μf) ∈ U and x(t;x(tk),h(x(t),μf))∈ STfor any x(tk)∈ STand t∈ [tk,tk+Tp].Therefore,the constraints in the optimization problem(9)are satis fied ateach sampling time tk,thatis,the optimization problem(9)is feasible at each tk.

(2)When x(tk)∈Xf(Tp)/ST.The NMPC controller(11)is computed by solution of problem(10).From Algorithm 1 and the de finition of admissible state set Xf(Tp),there is at least a correction pro file c(t)for t∈[tk,tk+Tp]such thatthe constraints in Eq.(10)are satis fied at each sampling time tk,that is,problem(10)is feasible at each tk.Also,the feasible correction pro file c(t)for t∈[tk,tk+Tp]is independent of the cost function to be minimized.

Integrating Cases(1)and(2),the conclusion is then obtained.

Theorem 2.Assume that the optimization problem(10)is initially feasible for a given predictive horizon Tp≥δ.Then the switched closed-loop system(12)is asymptotically stable within the domain of attraction Xf(Tp).

Proof.From the assumption ofthe initialfeasibility and Theorem 1,itis known that the overall multi-objective dual-mode MPC controller(11)derived by Algorithm 1 is always feasible in the set of admissible states Xf(Tp).

Remark 3.Itis wellaccepted thatthe computationalburden forsolving optimization problems of NMPC grows exponentially with the number of decision variables due to their con-convexities[3,4].Hence,in order to decrease the computational burden for solving the optimization problem(9),the predictive pro file of parameter vector,μ(t|tk),t∈ [tk,tk+Tp]is frozen at constant vector μ(tk).In this way,the number of decision variables of Eq.(9)is reduced to two regardless of the prediction horizon and the number of control variables.Nevertheless,the above obtained results on fixed μ also hold for μ(t|tk)with t∈[tk,tk+Tp]atthe price ofcomputationaldemand.Note thatalthough the optimalsolutionsare computed separately by solving openloop optimization problems(9)and(10),the multi-objective NMPC controller(11)applied by Algorithm 1 is a closed-loop controllaw.These are bene ficial to lightening the computational load of NMPC and meanwhile,to dealing with uncertainties.In addition,it should be pointed out that the computed parametersare the functions in the current state

Remark 4.From the proof of Theorem 2,it is known that the switched closed-loop system(12)converges to the equilibrium point,i.e.,the steady-state operating point of the process system(1)by allowing switches between different NMPC controllers computed according to one of the multiple performance criteria of interest.It is well known that a switched system consisting of stable subsystems might be unstable[25,26],even for linear exponentially stable subsystems.In particular,a switch between different cost functions leads to a change of the stability constraints of NMPC since these constraints are usually related to the cost functions[1,2,4].Hence,this may cause a loss of feasibility and/or stability of NMPC[18,19].In this work,the terminal state set for guarantee feasibility and stability of NMPC is determined using the model of system(1)and hence,is independent of the cost functions.As a result,the stability of the proposed NMPC is ensured in the case of(arbitrary)switches between different cost functions.

4.Applications to a Chemical Process

Considera CSTR with an irreversible, first-orderexothermic reactionThe process kinetics can be described by the mathematical model

where CAdenotes the concentration of species A,TRis the temperature in the reactor,Q is the rate of heat supplied or removed from the reactor,V is the volume of the reactor,and k0,E,ΔH,cpand ρ are the preexponential constant,activation energy,enthalpy,heat capacity and fluid density in the reactor,respectively.The values of the model parameters are given as follows[27]:V=0.1 m3,R=8.314 kJ·kmol-1·K-1,CA0s=1.0 kmol·m-3,TA0=310 K,ΔH=-4.78 × 104kJ·kmol-1,k0=72 × 109min-1,E=8.314 × 104kJ·kmol-1,cp=0.239 kJ·g-1·K-1,ρ =1000 kg·m-3,and F=0.1 m3·min-1.

In this case study,concentration CAand temperature TRare selected as the controlled variables of the CSTR,and the rate of heat transfer Q and the change in inlet concentration of A,ΔCA0=CA0-CA0s,are selected as the manipulated variables.The constraints on the controlled and manipulated variables are given by 0.41 kmol·m-3≤ CA≤ 0.73 kmol·m-3,392.3 K ≤ TR≤ 398.3 K,|Q|≤ 16.7 J·min-1,and|ΔCA0|≤ 1.0 kmol·m-3.Under the nominal operating condition of Qs=0 kJ·min-1,the CSTR system has an unstable steady-state operating point(CAs,TRs)=(0.57 kmol·m-3,395.4395 K).The control objective of this case is to stabilize the reactor at this unstable operating point in the presence of constraints and to operate the reactor in an economic manner by accounting for changing economic factors of energy and materials.

Let state vector x=[(CA-CAs)(TR-TRs)]Tand control vector u=[Q ΔCA0]T.For this chemical process example,the economic cost criterion of interest is assumed to be

where the weights qi＞0 and ri＞0 for i=1,2 and are viewed as the prices of reactant materials and energy.Hence,the stage cost function in Eq.(16)penalizes the deviations of operating concentration and temperature,and penalizes the energy usage and reactantmaterialconsumption.Due to changes of market demand and/or supply,the prices of reactant resources will vary.In general,we may not know exactly the future prices(and hence,the cost weights)in a practical setting.For this particular study,we take into account two different cost functions

where Tpis the prediction horizon and the weighted matrices

In this example,cost functions(17)and(18)are used to denote the accumulated economic criteria of the CSTR system under differentmarketconditions.Consequently,we can take advantage ofthe switches between Eqs.(17)and(18)to meet the changes of the market demand and/or supply of the reactant resources.

In order to compute terminal state set STand construct controller h(x,μ)in the formofEq.(7),we considerthe linearized modelofsystem(15)at the steady-state operating point

For simplicity,we use a quadratic function V(x)=xTPx to be a local CLF of system(15)with the positive de finite matrix

where P is chosen to satisfy the Riccati matrix inequality of the linearized system(19).Then we obtain the controller h(x,μ)in the form of Eq.(7)with

Let D1=2 and D2=10.Using the method in[24],we have an estimate of the invariant set of the CSTR system,i.e.,ST={x∈R2|xTPx≤0.016},which is shown as the solid ellipse in Fig.1.It should be pointed outthatthe estimate set,i.e.,the terminalstate set ST,is computed using the original nonlinear system(15).

Fig.1.The terminal state set(solid ellipse),equilibrium point(○)and state-phase trajectories with the proposed NMPC(solid lines)and conventional NMPC(dashed lines).

In simulation,we exploit Euler's first-order approximation to calculate the derivatives of the continuous-time CSTR system,with a sampling period δ =0.1 min.Let the prediction horizon Tp=10δ and the simulation time window be 20δ.The switching signal σ(t)is shown in Fig.2,which is assumed to be migrated arbitrarily forevery 2-time sampling periods due to the uncertain variations ofthe prices ofreactantresources.We assume that the cost function(17)is activated if σ(t)=0 and the cost function(18)is activated if σ(t)=1.

Fig.2.The switching signal.

In order to compare the proposed NMPC scheme with the conventional NMPC scheme,we pick two initial points as the initial condition of the CSTR system,i.e.,I(CA,TR)=(0.68 kmol·m-3,394.0 K)and II(CA,TR)=(0.49 kmol·m-3,398.0 K).Fig.1 depicts the state-phase trajectories of the closed-loop systems with the proposed NMPC(solid lines)and the conventional NMPC(dashed lines)schemes.The state and control input pro files of the closed-loop systems with the two NMPC schemes are shown in Figs.3 and 4 for initial points(I)and(II),respectively.For both initial conditions,the CSTR system with the proposed NMPC scheme asymptotically converges to the operating point(CAs,TRs)with changing economic factors of reactant resources represented by the switching signal in Fig.2.However,the conventional NMPC scheme does not necessarily stabilize the CSTR system to the operating point(CAs,TRs)to meet the changing economic factors of reactant resources.For instance,in Fig.3 the closed-loop states with the conventional NMPC steer to the operating state CAsbut the control inputs do not convergence to accommodate the switching signal.Nevertheless,the closed-loop systems with the proposed and conventional NMPC schemes satisfy the constraints on the state and control for all times.

Fig.5 shows the optimalparametersμofcontroller h(x,μ)computed by solving the optimalcontrolproblem(9),with respectto initial points(I)and(II).The closed-loop state starting from(I)enters the terminal state set STat the 2nd sampling time and then the optimization problem(9)is solved online;similarly,the closed-loop state starting from(II)enters the terminal state set STat the 3rd sampling time and then the optimization problem(9)is solved online.

Fig.6 shows the CPU time for online solving the optimal control problems(9)and/or(10)with the proposed NMPC(solid lines)and conventional NMPC(dashed lines)schemes,where the left sub figure corresponds to initial point(I)and the right sub figure corresponds to initial point(II).For both NMPC schemes the computational time is maximal and almost identical at the 1st sampling time.The average computational time for one online optimization with the proposed NMPC is 169 ms and the minimal computational time is 30 ms,while they are 867 and 640 ms,respectively,with the conventional NMPC.Moreover,the average computational time taken to online solve the optimization problem(9),which is activated from the 3rd sampling time(see Fig.5),is 60 ms.These values of computational time are evaluated by the function ‘fmincon’in MATLAB 7.1 on the laptop of Windows 8.0 and an Intel(R)Core?i5-4200 CPU with 1.6-2.3 GHz and 4 GB RAM.We observe that there is a one order ofmagnitude difference in average computational time,compared the proposed NMPC scheme with the conventional NMPC scheme.The reason is that the proposed NMPC has only two decision variables when the states are in the terminal region STwhile the conventional NMPC always has 20 decision variables even for the states entering ST.

Fig.3.The state and control input pro files of the CSTR system with the proposed NMPC(solid lines)and conventional NMPC(dashed lines)schemes for the initial point(I).

Fig.4.The state and control input pro files of the CSTR system with the proposed NMPC(solid lines)and conventional NMPC(dashed lines)schemes for the initial point(II).

Fig.5.The optimal parameters of h(x,μ)with respect to initial points(I)(upper plots)and(II)(lower plots).

5.Conclusions

In this work,we consider the multi-objective controlproblemof stabilization of nonlinear systems subject to state and control constraints.We propose a dual-mode multi-objective NMPC design that guarantees stabilization and constraint satisfaction for arbitrarily switching different cost functions re flecting different performance criteria.A CLF-based analyticalterminalcontrollawwith free-parameters is constructed to enlarge the terminal state set and then increase the size of the stability region of the predictive controller.By a comparison between the proposed NMPC with the conventional NMPC,we demonstrate the application and performance of the proposed NMPC controller design through a chemical process example of CSTR.