Lexicographic MPC with multiple economic criteria for constrained nonlinear systems

Accepted Manuscript

Lexicographic MPC with multiple economic criteria for constrained nonlinear systems Defeng He , Shiming Yu , Linlin Ou PII: DOI: Reference:

S0016-0032(17)30630-0 10.1016/j.jfranklin.2017.11.040 FI 3250

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

27 September 2016 14 November 2017 27 November 2017

Please cite this article as: Defeng He , Shiming Yu , Linlin Ou , Lexicographic MPC with multiple economic criteria for constrained nonlinear systems, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.11.040

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ACCEPTED MANUSCRIPT

Lexicographic MPC with multiple economic criteria for constrained nonlinear systems Defeng He, Shiming Yu, Linlin Ou College of Information Engineering, Zhejiang University of Technology, Hangzhou, 310023 P. R. China E-mail: [email protected], [email protected], [email protected]

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Abstract: Many control problems in process systems feature multi-objective optimization problems that involve several and often conflicting objective functions, such as economic profit and environmental concerns. In this paper, we consider a class of multi-objective model predictive control (MO-MPC) problems where nonlinear systems are subject to state and control constraints and multiple economic criteria are conflicting. Using the lexicographic optimization, we propose a prioritized MO-MPC scheme with guaranteed stability for economic optimization. At each sampling time, the MPC action is computed by solving a set of sequentially ordered single objective optimized control problems. Some sufficient conditions are established to ensure recursive feasibility and asymptotic stability of the MO-MPC in the context of economic criteria optimization. Two examples of multi-objective control of a coupled-tank system and a free-radical polymerization process are exploited to illustrate the effectiveness of the proposed MPC scheme and to evaluate the performance by some comparison experiments. Key Words: nonlinear systems, model predictive control, multiobjective control, economic criteria, asymptotic stability

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Solution of multi-objective optimization problems is not to find a unique locally optimal solution but to compute a set of non-inferior, non-dominated Pareto optimal solutions that satisfy the priorities of the multiple criteria as well as possible [5]. Hence, many efforts have been made to develop multi-objective MPC (MO-MPC) by taking use of various optimization tools. For example, [9] solved an MO-MPC problem for nonlinear systems by minimizing the maximum of a finite number of conflicting objectives. In [10], an MO-MPC scheme was formulated for linear systems, where an additional decision criterion was used to identify specific elements of the Pareto optimal set. The concept of utopia points has been firstly employed to design MO-MPC for nonlinear systems in [8] and the corresponding theoretical results was established in [11]. Moreover, [12] extended the utopia MO-MPC in [8, 11] to cyclic process systems and in [13] we presented an alternative utopia MO-MPC with less computational burden for continuous time nonlinear systems. In principle, utopia MO-MPC online minimizes the distance of its cost vector to a vector of individually optimized steady-state objectives, i.e., the utopia point. Its main advantage is that the computed MPC law deals with tradeoffs among conflicting objectives automatically. Nevertheless, the utopia points are effected by the shape and scaling of the Pareto set and can not explicitly describe the preferences of a decision maker. In general, the various control objectives of systems do not have equal priorities [7]. Using lexicographic goal programming, Meadowcroft et al. in [14] presented a priority-driven framework to solve multi-objective control problems with different priorities. Within this framework, a modular multivariable controller was designed by solving a series of single static-objective problems in such manner that the highest priority objective is solved first and the next highest priority objective is then solved with an additional constraint that used to guarantee optimality of the higher priority objective. The priority-driven framework in [14] has been extended to solve MO-MPC problems with various priorities for linear systems [15-16] and nonlinear process

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Introduction

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Model predictive control (MPC) has been a well accepted feedback control technique in process systems, especially in chemical processes [1-2]. In general, MPC actions are computed by minimizing a certain cost function and the obtained optimal inputs are then applied until the next sampling instant, where the procedure is repeated at subsequent times. One of the main advantages of MPC is the ability to tackle constraints and multiple-input-multipleoutput (MIMO) systems and to naturally incorporate general (multiple) optimization criteria into the design of feedback controllers. Moreover, nonlinear MPC has the added advantage of being able to capture nonlinearities of complex plants and thus provides higher accuracy across a wide range of process systems [3]. In many process control applications, various control objectives to be optimized frequently emerge, such as tracking, robustness, economic profits and environmental concerns [4-6], which are generally conflicting. Since the optimal values of conflicting objectives cannot be achieved simultaneously, one has to compute a tradeoff solution that best suits, a given sense, all the objective functions. Hence, in MPC the common approach used to tackle multiobjective optimization has been to weight the individual objectives such that a single cost function is constructed by summing those objective terms. However, the selection of a desired set of weights is not a trivial task since in general it is done by trial and error or via ad hoc approaches in order that more important objectives are given higher priority [6-7]. Although a weighted single cost function has resulted in a useful framework for theoretical analysis of MPC as well as successful industrial applications, it has been shown to be not a systematic manner to address the design tradeoffs among conflicting objectives [7-8]. * This work was supported by the National Natural Science Foundation of China under Grant 61374111 and 61773345 and Zhejiang Provincial Natural Science Foundation of China under Grant LR17F030004.

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ACCEPTED MANUSCRIPT systems [17-20], which yields the so-called lexicographic MO-MPC. Recently we in [21] established the theoretical results of recursive feasibility and stability for lexicographic MO-MPC of constrained nonlinear systems, where the highest priority objective is a traditional positive definite cost function with respect to some equilibrium points. It has been shown in [12] that for some cases multiple economic criteria optimized over a receding horizon may lead to non-steady process behaviors. This phenomenon is often viewed as a typical feature of (single objective) economic MPC (EMPC) [22-24]. In order to establish stability of EMPC, several single objective EMPC schemes have been proposed in much literature. For example, [25] transformed the economic cost to the rotated cost function and established the monotonic decreasing property of the optimal value function of the rotated cost by the strong duality assumption and terminal equality constraint. The terminal constraint was then replaced by some inequality constraints and the strong duality assumption was relaxed as dissipativity conditions with elaborately chosen supply and storage functions [22,26-27]. The authors in [28-29] presented an unconstrained EMPC under controllability and dissipativity conditions and a sufficiently long prediction horizon. In [30] a Lyapunov-based two-mode EMPC strategy was proposed such that the closed-loop system is ultimately bounded in a small region. This two-mode EMPC was then improved as a dual-layer stabilizing EMPC by controllability assumptions [31]. In [32] we proposed a contractive EMPC by imposing a contraction property of closed-loop behaviors based on the terminal region constraints and penalty functions. Compared with single objective EMPC, there are few available results on stability of multi-objective economic MPC (MO-EMPC). Although utopia MO-MPC has ability to tackle multiple economic criteria with no prioritization, its stability properties can be established under dissipativity conditions and terminal constraints [22]. In the context of MO-EMPC with prioritized economic criteria, however, computation of terminal constraints and choosing supply and storage functions satisfying the dissipativity condition are almost prohibitive because the multiple economic cost functions have to be optimized sequentially according to their prioritization at each sampling instant. Although multi-objective optimization has been a well known field, in this paper we concentrate our attention on the Mo-EMPC problem of nonlinear systems subject to the constraints on inputs and states. The results in [21] are extended to the MO-EMPC problem where the most important criterion is an economic objective and a new lexicographic MO-EMPC strategy is presented to solve the control problem subject to economic objective prioritization. By imposing some contractive constraints [32] to the lexicographic MO-EMPC problem, new conditions are obtained to guarantee recursive feasibility and stability for the lexicographic MO-EMPC. Note that to the best of our knowledge, the existing lexicographic MO-MPC schemes have no ability to address the feasibility and stability properties of multi-objective economic optimized control systems. Two examples of multi-objective control of a coupled-tank system and a free-radical polymerization process are used to demonstrate the effectiveness of the

proposed MO-EMPC scheme by comparing performances with the weighted MO-EMPC controllers. Notation: Let R0 and I≥0 denote the sets of non-negative real and non-negative integer numbers, respectively. I≥a is the set {iI≥0: i≥a } and Ia:b is the set {iI≥0: a≤i≤b} for some aI≥0 and bI≥0. A function (s): RnR0 is positive definite with respect to s=a if it is continuous, (s)>0 for all sa, and (a)=0 if and only if s=a. A continuous function α: R0R0 is a class-К function if it is strictly increasing and α(0)=0, α(s)>0 for all s>0; it is a class-К∞ function if it is a class-К function and α(s)∞ as s∞. Label „T‟ in superscript denotes the transposition of a vector.

Problem description and preliminaries

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Consider a nonlinear process system described by a discrete-time model (1) x   f ( x, u)

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where xRn is the system state, uRm is the control vector, x+ is the successor state, and f: RnRmRn is a locally Lipschitz function in its arguments. The solution of this system for a given sequence of control inputs u and initial state x is denoted as x(t)=(t; x, u) for tI0 with x=(0; x, u). The state of the system and the control action applied at time kI0 are denoted as x(k) and u(k), respectively. The system is subject to constraints on the state and control (2) x(k )  X , u(k ) U , k  I 0 where XRn and URm are compact sets. It is assumed that the state is available for state feedback controllers. In this work we consider multiple economic criteria which are measured by a set of different stage cost functions Lj: XUR for jI1:l and lI2. For the sake of simplicity, we assume that these functions are conflicting between each other; otherwise, we can weight the concordant functions as a single function. Since the conflict of these economic criteria, there generally exists no solution that optimizes all the criteria at the same time. One potential approach used to tackle this case is to apply lexicographic optimization according to the order of priorities of the set of economic criteria. Without loss of generality, we consider the permutation of the economic criteria, {L1, …, Ll}, where L1 is the most important criterion and Ll is the least important criterion to a decision maker. The multi-objective controller design problem is then to develop a control law that simultaneously minimizes all the economic criteria according to the order of their priorities while stabilizing the system (1) at some desired equilibrium points in the presence of constraint (2). Here we exploit the model predictive control technique to solve this problem. In what follows, we recall the lexicographic optimization. The details of the lexicographic optimization can be found in e.g. [4-5]. Consider the multi-objective optimization problem (3) min [ 1 ( ), 2 ( ),, l ( )]  

where Rp is the admissible set of decision variables  and i are the scalar valued functions with respect to . For simplicity let=[1, 2, …, l]. In (3), the objective functions are arranged according to their priorities from the most important 1 to the least important l. We assume that

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ACCEPTED MANUSCRIPT each i attains its minima inside . Note that the optimization of (3) is defined in sense of the dominance notion [5], i.e., an objective function vector (*) is nondominated if and only if there does not exist another vector () such that ()(*) with at least one i()<i(*). Definition 1 [5]: The solution * is a lexicographic minimizer of (3) if and only if there exist no other solution  and an i*I2:l such that i*()<i*(*) and i()=i() for all iI1:i*1. The corresponding value of vector (*) is the lexicographic minima. The standard method for finding a lexicographic solution is to solve a set of sequentially ordered single objective constrained optimization problems. Namely, (4a)

 i*  min{ i ( ) |  j ( )   *j , j  I1:i1}

(4b)

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where set U N  U   U . Moreover, we define two   N

auxiliary continuous functions La: XUR0 and Ea: X R0, where La(x,u) is positive definite with respect to (xs,us) and Ea(x) is positive definite with respect to xs. Hence, there exist some class-K functions i, i=1,…,5 such that 1(||x||)La(x,u)2(||x||)+3(||u||) and 4(||x||)Ea(x)5(||x||) for any (x,u)XU. Namely, functions La(x,u) and Ea(x) are bounded for any (x,u)XU. Note that there are several economic criteria to be optimized simultaneously in the MO-EMPC problem considered here. As a result, the multiple optimal value functions corresponding to economic costs are not exploited as the candidate of Lyapunov functions of the MO-EMPC closed-loop system due to the different priorities of the economic criteria. To this end, in this paper we make use of La and Ea to define a function as N 1 (12) J a ( x, u)  Ea ( x( N ))   La ( x(t ), u(t ))

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 1*  min  1 ( ),

for an initial state xX, a control sequence of N elements, u={u(0), u(1), …, u(N1)}, is feasible if (t; x, u)X and u(t)U for tI0:N-1, and (N; x, u)XT. For a given xX, we define an admissible set of feasible sequence u as   x(t  1)  f ( x(t ), u (t )),   N C ( x)  u  U x(t )  X , u (t )  U , t  I 0:N 1 , (11)   x(0)  x, x( N )  X T  

for all iI2:l. Then we derive a lexicographic solution

 *  arg min{ l ( ) | j ( )   *j , j  I1:l 1}. 

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Note that in order to improve numerical computation of the lexicographic optimization problem (4), the priority constraints j()=j* in (4) or (5) are often relaxed as (6)  j ( )   *j   j , j  I1:l 1

where εj0 are small tolerances determined by decisionmakers. For the sake of simplicity, hereafter these inequality constraints are written as j()≤j*.

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Lexicographic MO-EMPC controller

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Consider the constrained system (1)-(2) and define the set of admissible steady-state points as (7) Z s  {( x, u)  X  U | x  f ( x, u)} .

It is assumed that set Zs is not empty and contains at least a steady-state point as its interior. The set of feasible equilibrium states, Xs, is then defined as the projection of Zs onto X (8) X s  {x  X | u U such that ( x, u)  Z s } .

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which will be showed to be positive definite with respect to (xs,us) under some conditions and then be used to design the candidate of Lyapunov functions of the MO-EMPC closedloop system. Now consider the set of economic objectives J={J1,…,Jl}, where each economic cost function Ji is evaluated over the prediction horizon NI1

J i ( x, u)  t 0 Li ( x(t ), u(t )), i  I1:l

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where x(t)=(t; x, u), tI0:N and x(0)=x. Then the proposed MO-EMPC control law is determined by solving the following lexicographic multiobjective finite horizon optimal control problems:

In order to derive the optimally economic operating point of system (1), we compute the lexicographic optimal steadystate point (xs, us) by solving the lexicographic steady-state optimization problem of the l prioritized economic costs L={L1,…,Ll}, i.e. (9a) L*1  min{L1 ( x, u) | ( x, u)  Z s },

J1* ( x)  min J1 ( x, u) u  C( x), J a ( x, u)   ( x,  ) (14a)

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 u  C ( x), J a ( x, u)   ( x,  ),   (14b) J i* ( x)  min  J i ( x, u)  * u J ( x , u )  J ( x ),  j  I  j j 1:i 1   

( x ,u )

 L j ( x, u )  L*j , j  I1:i 1 , L*i  min  Li ( x, u )  (9b) ( x ,u ) ( x, u )  Z s   for all iI2:l and

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for all iI1:l, where function is given by (15)  ( x,  )  J a ( x, uˆ )  [Va (~ x )  J a ( x, uˆ )] * ~ ~ with some number 0α<1 and x  f ( x , u (0; x )) . Here the control action u* (0; ~ x ) is the first element of sequence * ~ * ~ u ( x )  {u (0; x ),, u* ( N  1; ~ x )} that is the lexicographic optimal solution to (14) with the initial condition x(0)  ~ x,

 L j ( x, u )  L*j , j  I1:l 1 , ( xs , u s )  arg min  Ll ( x, u ) . (10) ( x ,u ) ( x, u )  Z s   Note that there exists at least one steady-state point (xs, us) satisfying (10) since Zs is a non-empty set. We assume that the solution (xs, us) is unique; otherwise, we just use a unique selection map to select one of the multiple minima. Consider the lexicographic optimal steady-state operating point (xs, us) that satisfies (10). Let XTX be a compact set, containing the operating state xs as its interior. It is said that

Va is the value function of (12) on u* ( ~ x ) , i.e. * ~ ~ ~ V ( x )  J ( x , u ( x ))

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and sequence uˆ  {u * (1; ~ x ),, u * ( N  1; ~ x ), loc ( x( N ))}

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ACCEPTED MANUSCRIPT where x( N )   ( N ; ~ x , u* (~ x )) and μloc is a local control law of system (1). In general, μloc is assumed to satisfy the following assumption: A1: Consider the system (1)-(2) and the terminal region XT. There exists a local control law u=μloc(x) such that μloc(x)U and (18) Ea ( f ( x, loc( x)))  Ea ( x)  La ( x, loc( x)) for any xXT. Note that the assumption A1 implies that the region XT is an invariant set of the system (1) in the closed-loop with the local control law μloc if XT is chosen as a sublevel set of Ea. In the literature, many approaches have been proposed to design the law μloc as well as the region XT and the function Ea satisfying A1; see [3,33], etc. At the current time kI0, we derive the lexicographic optimal solution to (14) with the state x(k) at k, i.e.,

 u  C ( x), J a ( x, u)   ( x,  ), u * ( x)  arg min  J l ( x, u)  (19) u J j ( x, u)  J *j ( x), j  I1:l 1   with x=x(k). As usual in standard MPC, the MO-EMPC controller is defined by the receding horizon application of the optimal solution u*(x), i.e., (20) u(k )  mpc ( x(k )) : u * (0; x(k )), k  I 0

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kI≥0 if feasibility of the whole problem at time k implies its feasibility at time k+1. Remark 1: In single objective MPC, feasibility of the optimization problem at the current time is established by constructing a feasible solution using the optimal solution to the optimization problem at the last time [3]. However, feasibility of the lexicographic MO-EMPC at the current time cannot be guaranteed immediately by the optimal solution to the lexicographic optimization problem at the last time. The reason is that the original, prioritized multiobjective problem is transformed into an l-hierarchy of single objective problem. In other words, in the lexicographic MO-EMPC there exists a concept of feasibility in sense of hierarchy. Considering Algorithm 1 with the prediction horizon N, we define the admissible set ZN as this set of (x,u)-pairs   x(t )  X , u (t )  U ,   (23) Z N  ( x, u) x( N )  X T , t  I 0:N 1 ,   J a ( x, u)   ( x,  )   where x(t)=(t; x, u) with x(0)=x. Then the feasible initial state set of the optimization problem (14), XN, is given as the projection of ZN onto X X N  x  X | u U N such that ( x, u)  Z N . (24) Note that in Algorithm 1 the constraint of Ja(x,u)(x,) is inactive at initial time k=0 since η0:=+∞. Hence, the set XN defined as (24) is identified as the set of admissible initial states of the optimization problem (14). It is remarked that the standard Lyapunov‟s arguments used to prove asymptotic stability of standard MPC cannot be directly used in the context of MO-EMPC in Algorithm 1 because each value function Ji* in the MO-EMPC is not necessarily decreasing along the closed-loop trajectories, which is similar to EMPC with a single criterion. In [21], we established asymptotic stability of the MO-MPC by taking into account that the criterion with the highest priority is rigidly positive definite with respect to (xs,us). In this work, we will establish asymptotic stability for the MO-EMPC in Algorithm 1 where all criteria may not be positive definite with respect to the optimal operating point (xs, us). Lemma 1: Under the assumption A1, the lexicographic optimization problem (14) has hierarchical feasibility for iI1:l1 at time kI≥0. Proof: The proof is the same to that of Theorem 1 in [21] and then is omitted. ■ Lemma 2: Suppose that the assumption A1 holds and the first-layer subproblem (14a) is feasible at time kI≥0. Then the lexicographic optimization problem (14) has horizontal feasibility with respect to the invariant region XN. Proof: Consider the assumption that the first-layer subproblem (14a) is feasible at time kI≥0. Setting j=1 and by induction, it is known from Lemma 1 that the whole problem (14) is feasible at the same time k. Let ~ x ) be the x  x(k )  X N be the state at time k and u* (~

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with corresponding the closed-loop system x(k  1)  f ( x(k ), mpc ( x(k ))), k  I 0 .

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The procedure for implementing the MO-EMPC controller (20) is summarized as Algorithm 1. Algorithm 1: (Lexigraphic MO-EMPC algorithm) 1) Rank the prioritization of economic criteria {L1,…,Ll} and offline compute the optimal operating point (xs, us) by solving the optimization problem (10). 2) Pick the functions La and Ea, and offline compute the control law μloc and the region XT satisfying A1. 3) Set factor 0α<1 and let k=0, x=x(k) and η(x,)+∞. 4) Online solve the optimization problem (14) with η(x,) using the following sub-procedure: 4.1) Solve the first-layer subproblem (14a) and obtain one of optimal sequences, u1*; 4.2) Solve the ith-layer of problem (14) for all iI2:l and obtain one of optimal sequences, ui*; 4.3) Determine the lexicographic optimal sequence of the whole problem in (14) as (22) u* ( x(k ))  u*l ( x(k )) .

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5) Implement the first element of u* to the system (1). 6) Let k=k+1 and measure the state x=x(k). 7) Evaluate η(x,) using (15) and (17) with ~ x  x(k  1) and go back to Step 4. ■ In what follows, we analyze recursive feasibility of the MO-EMPC in Algorithm 1 and asymptotic stability of the closed-loop system (21). Definition 2 [21]: At time k, the lexicographic optimization problem (14) has the hierarchical feasibility property for iI1:l1 if feasibility of the ith-layer optimization subproblem implies feasibility of the (i+1)th-layer optimization subproblem. Definition 3 [21]: The lexicographic optimization problem (14) has the horizontal feasibility property for time

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optimal solution to (14) at k. Applying Algorithm 1, the state at time k+1 is measured by x(k  1)  f (~ x , u* (0; ~ x )) . Let x=x(k+1). In order to find a feasible solution to the problem (14a) at time k+1, we consider the control sequence û given by (17). Clearly, by the assumption A1 we have ûC(x).

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ACCEPTED MANUSCRIPT Substituting (25) to (29), we have Va ( x)  Va ( ~ x )  (1   ) La ( ~ x , u * (0; ~ x )) .

Substituting û to (12) and from (16) and the assumption A1, it is derived that (25) Va ( ~ x )  J a ( x, uˆ )  La ( ~ x , u* (0; ~ x )) . Since 0α<1 and La(x,u)0 for (x,u)XU, it is known that J a ( x, uˆ )  J a ( x, uˆ )   [Va ( ~ x )  J a ( x, uˆ )] (26)

Due to the number 0α<1, the inequality (30) shows that Va(x) is a strictly monotone decreasing function along the trajectories of the closed-loop system (21). In order to obtain the bounds of Va(x), we consider the function Ja(x,u) defined in (12). Clearly, Va(x)Ea(x) for all xX and hence there exists a class-K functions 1 such that 1(||x||)Ea(x)Va(x) for xXNX. Moreover, Va(x) is continuous since the assumption A2 and La(x,u) and Ea(x) are continuous on their arguments. Then exploiting the procedure of the proof of Theorem 1 in [32], we can establish an up bound of Va(x) such that Va(x)2(||x||) for xXN with a class-K functions 2. Thus according to the Lyapunov‟s arguments, the steady state xs is an asymptotically stable equilibrium point of the closed-loop system (21) with XN. Since XN is an invariant set of (21), it is a region of attraction of the closed-loop system. ■ Remark 3: From the proof of Theorem 1, it is known that stability of the lexicographic MO-EMPC is guaranteed by the monotone decreasing property of Va, which is a value function of Ja on the lexicographic optimal solution to multiple economic criteria. Note that the value function Ji*(x) of the ith-layer subproblem for any iI1:l is not requested to have monotonicity and hence is not necessarily a Lyapunov function of the closed-loop system. Consequently, stability of the lexicographic MO-EMPC is decoupled with the economic criteria. This will benefit such cases that some objective functions with lower priorities have to be given up due to missing data or actuator faults, whilst maintaining the stability of the closed-loop system [18]. Moreover, compared with the dissipativity condition that plays an important role in the stability analysis for single objective EMPC, the presented contractive constraint can be used to optimized-based multi-objective control of plants in general cases because the dissipativity condition satisfying for linear systems with convex constraints and a single strictly convex economic criterion cost might not true for other cases [22, 25]. Nevertheless, imposing a nonlinear contractive constraint to the optimization problem increases the computational burden of MO-EMPC as well as conservativeness of the obtained controller. Remark 4: Compared with the MO-MPC schemes using a weighted cost function, the proposed MO-EMPC scheme has more computational load of solving the optimization problem. The main reason is that in the lexicographic optimization (14), the number of constraints is increased after solving each single objective optimization subproblem. This leads to the computational load of the whole problem to be more than the sum of loads of the individual objective problem. From the viewpoint of application, one method to reduce the computational load is that one attempts to decrease the number of layers of the whole problem (14) by such methods as uniting the objectives with identical priorities. Another approach of interest is to compute a suboptimal solution to each layer subproblem [21]. Because optimal solutions to each layer subproblem generally cannot be guaranteed or are highly expensive computationally for nonlinear and non-convex programming.

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:  ( x,  ) This suggests that the sequence û is a feasible solution to the first-layer subproblem (14a) at time k+1. By induction and applying Lemma 1 again, it is derived that the lexicographic optimization problem (14) is feasible at k+1, i.e., the whole problem (14) admits the horizontal feasibility with respect to the region XN. Therefore, we have x(k+1)XN for any x(k)XN, which implies that the feasible initial state set XN is an invariant region of system (21). ■ Remark 2: Note that at time k+1, the control sequence (17) is a feasible solution to the first-layer subproblem (14a) but is not necessarily the feasible solution to the subsequent optimization subproblem (14b) due to imposing the priority constraints to the subproblem (14b). It is well known that the optimal path from x(0)=xs to x(N)=xs is often different from x(t)xs for all tI0:N in the context of standard EMPC [22-23]. Meanwhile, in general the sequence u*(x) is not optimal for Ja(x,u) but for the economic criteria Ji(x,u), which may lead to the case that Va(x)>Ja(x,û). As a result, the function Va(x) is not necessarily positive definite with respect to xs unless some additional conditions are imposed. A2: The optimal solution u*(x) is continuous on the terminal set XT and satisfies that u*(t;xs)=us for tI0:N1. From the assumption A2, there exists a class-K function αu such that ||u*(t; x)-us||αu(||x-xs||) for tI0:N-1 and xXT. This assumption bounds the control cost of steering the state xXT to xs. Note that this assumption confines attention to the terminal set XT in which the control actions is not too large. The assumption A2 is satisfied, for example, if XT is returned to the terminal equality. Moreover, as a candidate solution to (14), the sequence u(x) with u(t;x)=μloc(x(t)) for tI0:N1 satisfies the assumption A2. By assumptions A1 and A2, we have the following stability result. Theorem 1: Suppose that assumptions A1 and A2 are satisfied and the first-layer subproblem (14a) is initially feasible in XN. For a given number 0α<1, the steady state xs is an asymptotically stable equilibrium point of the closed-loop system (21) with the region of attraction XN. Proof: Due to initial feasibility of the first-layer subproblem (14a) with the region XN, it is obtained from Lemmas 1 and 2 that the problem (14) is feasible at each time kI≥0 with the invariant region XN of (21). Let ~ x ) be the state and the optimal x  x(k )  X N and u* (~

solution to (14) at time k, respectively, and x=x(k+1)XN and u*(x) be the state and the optimal solutions to (14) at time k+1, respectively, where x  f (~ x , u* (0; ~ x )) . Then from the constraints in (14), we obtain that (27) Va ( x)  J a ( x, uˆ )  [Va (~ x )  J a ( x, uˆ )] which is equal to Va ( x)  (1   ) J a ( x, uˆ )  Va (~ x) . It is derived from (28) that Va ( x)  Va (~ x )  (1   )[ J a ( x, uˆ )  Va (~ x )].

(28) (29)

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are considered to aim at the aforementioned control goal. The first one evaluates the performance of the proposed lexicographic MO-EMPC in Algorithm 1 (here denoted by LEMPC). The second one shows the performance of weighted MO-EMPC with different sets of weights (here denoted by WEMPC). Due to the computational burden of the multi-objective optimization, the predictive control variables in both experiments are parameterized by u=Kx, where the gain K will be computed online by solving multi-objective optimizations. Moreover, let the predictive horizon N=10 and consider an initial state (70%, 70%) for both experiments. In LEMPC, we select the function La as (36) La ( x, u)  ( x  xs ) T Wx ( x  xs )  Wu (u  us ) 2

Examples

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In this section, two nonlinear process examples are presented to demonstrate the implementation and evaluate the effectiveness of the proposed MO-EMPC scheme. The first example is chosen from liquid control of a coupled-tank system, which is a basic problem in the process industries [34]. The second example is an isothermal free-radical polymerization process of methyl methacrylate with four states [35]. For the sake of comparison, we also show simulations with weighted MO-EMPC controllers and the proposed lexigraphic MO-EMPC controller in the first example. For both examples, function fmincon in MatLab V7.1 environment was used to solve the optimization problems on the laptop of Windows 8 and an Inter ® Core ™ i5-4200 CPU with 1.6 GHz & 4 GB RAM.

with matrix Wx=I2 and number Wu=1. Clearly, La is a positive definite function with respect to (xs,us). In order to compute the function Ea, local control law loc and terminal region XT, we consider the system (A, B) being the linearized model of the nonlinear process (31) at (xs,us). Using the LQ optimal control approach, we have a positive definite matrix solution 4.2983 2.7371 to the Riccati equation P   2.7371 3.6347 

Coupled-tank systems

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Consider a coupled-tank system where two tanks are joined together. The process dynamics is described by a nonlinear discrete-time model [17, 34]  x (k  1)  x (k )  b  u (k )  a q (k ) 1 1 1 1 12   (31)  x2 (k  1)  x2 (k )  a1q12  a2 x2 (k )   q12 (k )  sgn( x1 (k )  x2 (k ))  x1 (k )  x2 (k ) where the state variables x1 (%) and x2 (%) are the liquid levels in tanks 1# and 2#, respectively, and control variable u (%) is the pump flow rate fed into tank 1#. The label sgn denotes the signum function, i.e., sgn(s)=1 if s0 and sgn(s)=1 if s<0. The parameters of the model are selected from [17] as a1=0.2232, a2=0.1191 and b1=0.1573. Note that all the variables in (31) are normalized. Thus, the constraints on the states and control are given as (32) 0  x1 (k )  1, 0  x2 (k )  1, 0  u (k )  1 for all time kI0. For liquid control of the coupled-tank system, we consider three criteria of interest (33) L1 ( x, u )  ( x1  H1,sp ) 2

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ATPAPPBBTP+I=0 and local gain Kloc=[0.5512 0.4192]. Then we construct the triplet as Ea(x)=(xxs)TP(xxs), loc(x)=Kloc(xxs)+us and XT={xR2: Ea(x)0.11}, which, together with (36), satisfy the assumptions A1 and A2. In WEMPC, we sum the criteria in (33)-(35) as (37) Lw ( x, u)  w1L1 ( x, u)  w2 L2 ( x, u)  w3 L3 ( x, u)

L2 ( x, u )  ( x2  H 2,sp ) 2

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(35) L3 ( x, u)  u 2 where H1,sp and H2,sp are the set-points of liquid levels in tanks 1# and 2#, respectively. Note that these criteria are not positive definite function with respect to the set-points since L1(x,u)=0 only if x1=H1,sp and regardless of x2; and L2(x,u)=0 only if x2=H2,sp and regardless of x1. Hence, they can be viewed as some “economic” cost functions. In this study case, let H1,sp=75% and H2,sp=60%. It is tested from (31) that the two set-points are not attainable at the same time, namely, they are conflicting. Here the control goal of the coupled-tank system is to keep the level of tank 1# near its set-point as well as possible only if the level of tank 2# can be kept at its set-point and then to minimize the pump flow rate fed into tank 1#. According to the control goal, we arrange the order of priorities of the criteria in (33)-(35) as L2>L1>L3. By solving the lexicographic steady-state optimization problem, we obtain an optimal steady-state point (xs,us)=(77.08%, 60%, 58.65%). In order to compare the performance obtained by applying MO-MPC controllers, two simulation experiments

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with some weights w1, w2, w3>0, which will be used as the stage cost of the WEMPC algorithm. Fig. 1 shows the simulation results for the level profiles of tanks 1# and 2#, obtained by applying LEMPC and WEMPC, respectively. Here the solid profiles are obtained by applying LEMPC with =0.2, the dashed lines by WEMPC with (w1, w2, w3)=(10, 20, 1), the dotted lines by WEMPC with (w1, w2, w3)=(10, 100, 1) and the dash-dotted lines by WEMPC with (w1, w2, w3)=(10, 500, 1). The corresponding regulating errors profiles of the levels of tanks 1# and 2# are also shown in Fig. 2.

ACCEPTED MANUSCRIPT On the other hand, it can be seen from Figs. 1-2 that by different weights the WEMPC leads to different evaluations of the level of both tanks and control inputs. For instance, by an arbitrary set of weights (w1, w2, w3)=(10, 20, 1), there exist some explicit regulating errors of the level of both tanks using the WEMPC (see the dashed lines in Figs. 1-2). In order to reduce the regulating errors of the level of tank 2#, from the heuristic guidance one can increase the weight penalizing the objective L2 to e.g. (w1, w2, w3)=(10, 500, 1). This, however, degrades the objective L3, which is listed as Table 1. Note that in this study case the criterion L2 is the most important objective and need to be attained in prior.

Fig. 1 Level profiles of tanks 1# and 2#, and control lines.

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Fig. 2 Level errors profiles of tanks 1# and 2#.

E1=x1-75

E2=x2-60

Av{L3}

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0

0.34

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-0.53

0.33

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Tab. 1 Steady-state errors and time average values of L3.

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Due to contradiction of the level set-points of tanks 1# and 2#, and the higher priority objective L2 than both objectives L1 and L3, it is observed from Figs. 1-2 that the LEMPC controller regulates the level of tank 2# enter to the setpoint 60% after 20 sec while keeping the level of tank 1# nearest its set-point 75%. Moreover, Fig. 3 pictures the evaluations of value function Va and the three predictive cost functions in LEMPC. From Fig. 3, one can see that the value function Va is an monotone decreasing function along the trajectories of the tank system in the closed-loop with LEMPC and hence, it is acted as a Lyapunov function of the closed-loop system. However, the value functions associated with the three criteria in (33)-(35) may not monotone decreasing functions (e.g. J2p and J3p), which is one of inherent characteristics of economic MPC. As a result, they are not generally selected as the Lyapunov function of the closed-loop system in the context of economic optimization.

Table 1 lists the control performances in terms of steadystate errors of the level regulation of both tanks and the time average values of L3 accumulated over the total simulation window. From the table, one can see that although larger weights should be employed for objectives with higher priority, the set of appropriate weights of the multiple objectives has to be selected by trial and errors through numerical simulations. This will complicate the implementation of MO-EMPC controllers. It is also emphasized that this case study does not show that the weighted method always achieve poor results but show that the different weights used will lead to different control results. Consequently, the results obtained by applying the WEMPC are dependent on one designer‟s experience. Table 2 reports the comparison of the maximal, minimal and average computational CPU time for one online optimization between LEMPC and WEMPC. It can be observed that the computational times for one online optimization of LEMPC are almost three times than those of WEMPC. This difference is mainly resulted from the fact that in LEMPC three single-objective optimization problems are solved online in the order of their priorities whilst in WEMPC only one single-objective optimization problem is solved online. It is remarked that in Table 2 the computational times merely reflect state of the art of optimization solvers in MatLab tool. The research in optimization technique is still active and a greatly speed-ups can be expected in the future. Tab. 2 Computational times of one online optimization.

Fig. 3 The evaluations of value function Va and predictive cost functions in LEMPC.

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Tmax (ms)

Tmin (ms)

Tave (ms)

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930

47

74

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401

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guarantee the quality of polymers, then to maximize the conversion rate and finally to maximize the polymerization profits as well as satisfying the constraints in (39). In order to achieve the control goal, we permute the order of priories of the criteria in (40)-(42) as L1>L2>L3. By solving the lexicographic steady-state optimization problem (10), we derived a lexicographic optimal steady-state point (xs,us)=(5.5018, 0.1358, 0.0020, 49.8756, 0.0171). Note that the optimization problems in this example are nonlinear and non-convex programming problems due to the strong nonlinearities of the polymerization model (38) and economic criterion (42), which leads to some local solutions. Furthermore, let the control horizon Nc be less than the predictive horizon Np in order to reduce the heavy computational burden resulted from solving the nonlinear multi-objective constrained optimization problems. We select the sampling period =0.01 h and use Euler‟s first-order approximation for all derivatives in (38). We construct the function La in Algorithm 1 as (43) La ( x, u)  ( x  xs ) T Wx ( x  xs )  Wu (u  us ) 2

Free-radical polymerization processes

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Consider an isothermal free-radical polymerization of methyl methacrylate using azo-bis-isobutyronitrile as initiator and toluene as solvent [11, 35], where the number average molecular weight (NAMW) of the polymer is controlled by manipulating the inlet initiator flow rate. With some assumptions, the dynamics of the polymerization process can be formulated as the following model [11]: F  dC m (t )  dt  (k p  k fm )Cm (t ) P0 (t )  V (Cm,in  Cm (t ))   dCi (t )  k C (t )  Fi (t ) C  F C (t ) i i i ,in i (38)  dt V V   dD0 (t )  (0.5k  k ) P (t ) 2  k C (t ) P (t )  F D (t ) tc td 0 fm m 0 0  dt V  dD (t ) F  1  M m (k p  k fm )Cm (t ) P0 (t )  D1 (t ) V  dt where Cm(t) is the monomer concentration at time t, Ci(t) is the initiator concentration at t, D0(t) is the zeroth moment at t, D1(t) is the first moment at t and Fi(t) is the initiator flow rate at t. Let Cm, Ci, D0 and D1 be the state variables x1, x2, x3 and x4, respectively, and Fi be the control variable u. The term P0(t) takes into account the total concentration of live polymer chains, i.e., P (t )  2i ki Ci (t ) . The model 0 k td  k tc parameters of the system are taken from [11]. In this example, we consider the state and control constraints 2.7234 kmol/m 3  C m (t )  8.2602 kmol/m 3 ,

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with the diagonal matrix Wx=diag{1, 10, 10, 0.001} and number Wu=100. Clearly, La is a positive definite function with respect to (xs,us). In order to compute the function Ea, local control law loc and terminal region XT, we exploit the discrete-time system being the linearized model of the nonlinear process (38) at (xs,us). Using the command dlqr in MatLab, we have a positive definite matrix solution 0.0207   5.0240  0.7541 0.0015  0.7541 44.4205 0.1361 0.1855  P  0.0015 0.1361 52.6312  0.0005    0.1855  0.0005 0.0045   0.0207 and local gain Kloc=[0.0031 0.2709 0.0008 0.0010]. Then we construct the triplet as Ea(x)=(xxs)TP(xxs), loc(x)=Kloc(xxs)+us and XT={xR4: Ea(x)0.2752}.

0.0665 kmol/m 3  Ci (t )  0.1994 kmol/m 3 ,

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0.9876  10 3 kg/m 3  D0 (t )  0.0030 kg/m 3 , 24.6909 kg/m 3  D1 (t )  74.0727 kg/m 3 ,

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0.0084 m 3 /h  Fi (t )  0.0252 m 3 /h. For this example, we consider three economic objectives of interest. The most important objective is to strictly guarantee the quality of polymers, which is formulated as the criterion (40) L1 ( x, u) | M w  M w,sp | where Mw=x4/x3 is the polymer molecular weight and Mw,sp is its set-point. The second important objective aims at maximizing the conversion function σ(x,u)=(Cm,inCm)/Cm,in, i.e., minimizing the stage cost function (41) L2 ( x, u)  (Cm,in  x1 ) / Cm,in .

Fig. 4 Closed-loop state trajectories with =0.2 from A (solid lines) and B (dashed lines).

The finally objective is to maximize the profit function, i.e., minimize the stage cost function L3 ( x, u)  (a0  a1 0.6  a2 M w0.65  a3u 0.5 ) (42)

In order to demonstrate the implementation of the lexicographic MO-EMPC in Algorithm 1 in the presence of the constraints, we select Mw,sp=2.5104, Np=15, Nc=4 and the simulation time window as 100. We consider two different initial states A(4.2847, 0.1728, 0.0028, 59.2582) and B(6.4534, 0.0764, 0.0010, 34.5673), and pick the

with numbers a0=2.5103, a1=3.5103, a2=8.8210-4 and a3=3103. The optimal values of these criteria are 0, 0.099 and 2899, respectively, and they are conflicting [13]. The control goal of this polymerization process is to firstly

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number =0.2. Figs. 4 and 5 show the closed-loop state trajectories and control profiles obtained by the proposed lexicographic MO-EMPC, where the solid profiles denote the results on initial state A and the dashed profiles initial state B. It can be seen that starting from both A and B, the closed-loop polymerization process is asymptotically stabilized to the lexicographic optimal steady-state point (xs,us) while fulfillment of the state and control constraints at each sampling instant. Correspondingly, it is tested from the Fig. 4 that the criterion of the quality of polymers, which is the most important objective, is attainable after the transient of 70 steps and meanwhile the other criteria automatically close to their optimal values as well as possible.

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Fig. 6 Closed-loop state trajectories with =0.7 from A (solid lines) and B (dashed lines).

Fig. 5 Control input profiles with =0.2 from A (solid lines) and B (dashed lines).

Now consider the effects of number  to the control performance of the MO-EMPC in Algorithm 1. With the same simulation setups to the case of =0.2, Figs. 6 and 7 illustrate the closed-loop state trajectories and control profiles obtained by Algorithm 1 with =0.7, respectively, starting from initial state A (solid lines) and B (dashed lines). As the similar to the case of =0.2, the closed-loop polymerization process is asymptotically stabilized to the steady-state point (xs,us) while satisfying the state and control constraints at each sampling time. However, it is tested from the Fig. 6 that the criterion of the quality of polymers is attainable only after the transient of 61 steps. Moreover, Table 3 tabulates the time average performance of the three economic criteria (40)-(42) in the closed-loop system and Table 4 gives the average computational times of one online optimization. From Table 3, one can see that the time average performances of L2 and L3, starting from initial state A, outperform their steady-state optimal values. The reason of this improvement is that the presented MO-EMPC can employ system dynamics by leaving the Pareto front during process transients and raise process performance. Furthermore, it is observed that for both initial states A and B increasing the value of  improves the performance of the first and third objectives and does not degrade the second objective. In this sense, the obtained average performance is monotonous with respect to the factor . But it is observed from Table 4 that the average computational times of one online optimization in Algorithm 1 have no monotone property with respect to the factor . All these results illustrate the effectiveness of the lexicographic MO-EMPC of constrained nonlinear process systems proposed here.

Fig. 7 Control input profiles with =0.7 from A (solid lines) and B (dashed lines).

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Tab. 3 Time average performance of economic criteria.

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2991.639

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962.901

0.071

2791.406

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0.071

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Tab. 4 Average computational times of one online optimization.

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477

466

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488

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Conclusions

This paper presented a lexicographic MPC formulation for economic optimization of constrained nonlinear process systems subject to various prioritized economic objectives. The MO-EMPC controller was designed by sequentially solving a set of single economic cost functions in the order of their priorities. The corresponding recursive feasibility and stability properties were then established by imposing some conditions. In the new formulation, there is no weight to be arbitrarily selected to reflect the priorities of objectives and no computation of the Pareto front at each MPC step. The multi-objective control results on a coupled-tank system and

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[18]

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a free-radical polymerization process illustrated the effectiveness of the proposed MO-EMPC scheme. The computational burden of the proposed MO-EMPC will be pursued in our future work.

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