Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment

ISA Transactions xxx (xxxx) xxx

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Research article

Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment Jianbang Liu a,b,c , Xin Zhang a,b , Tao Zou d ,

∗

a

State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang 110016, China Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang 110016, China c University of Chinese Academy of Sciences, Beijing 100049, China d School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou 510006, China b

article

info

Article history: Received 13 June 2019 Received in revised form 30 January 2020 Accepted 3 February 2020 Available online xxxx Keywords: Control Two-layer model predictive control Feasibility judgment and soft constraint adjustment Priority ascent strategy Incompleteness

a b s t r a c t The priority ascent strategy of feasibility judgment and soft constraint adjustment has been playing a significant role in two-layer model predictive control for decades of years. Nevertheless, our latest researches indicate the present priority ascent strategy is incomplete in some extent, which could have been repaired through some technical adjustment. Therefore, an augmented method is proposed to fundamentally settle the problem in the paper. Firstly, a special case is presented to verify the incompleteness of the present strategy. Secondly, some existing methods which can partly solve the problem are carried out and discussed. Then, an augmented priority ascent strategy which considers all hard constraints in advance is proposed to solve the incompleteness of the present strategy. Furthermore, a brief proof is given to demonstrate the completeness of the proposed approach. Finally, some simulations are discussed to indicate its feasibility and effectiveness. © 2020 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Model predictive control (MPC) has been extensively applied in various industrial fields because it can effectively deal with multivariate problems with interactions and constraints [1–5]. And industrial MPC can be simply divided into five generations based on their different abilities, as shown in Fig. 1 [6,7]. Among them, a two-layer structure has been widely used in the fourth and fifth generation industrial MPC, which is named as two-layer model predictive control [8,9]. Morshedi et al. and Brosilow et al. introduced a steady-state optimization (SSO) to traditional MPC, which is the prototype of two-layer MPC [10,11]. Yousfi and Tournier and Muske gave a basic presentation of steady-state optimization and target tracking [12,13]. Ying et al. and Nikandrov and Swartz analyzed the steady-state performance, dynamic performance, economic performance, stability and sensitivity of two-layer MPC [14–16]. Zou et al. systematically presented the theoretical architecture of two-layer MPC and gave the detailed theory analysis and algorithm realization [8]. Ding generalized two-layer MPC from input–output models to state–space models, and analyzed the characteristics of integral and pseudo integral, steady-state error, nonlinear transformation and degree of freedom transformation [9]. Sun et al. formulated the conditional minimum ∗ Corresponding author. E-mail address: [email protected] (T. Zou).

variance control and realized the performance monitoring of twolayer MPC [17]. Zou et al. discussed the feasibility and soft constraints of steady-state optimization and gave the detailed procedures of the weighted and priority feasibility judgment and soft constraint adjustment (FJSCA) [18]. Zheng et al. proposed an offline optimization and online table-look-up approach, which effectively decreased the computational complexity of two-layer MPC [19]. Alvarez and Odloak reduced the traditional two-layer structure to a single dynamic layer, which enhanced the stability of relevant algorithms [20]. Pang et al. proposed a point model based steady-state optimization approach for integrating processes, which consummated the architecture of two-layer MPC [21,22]. Pepe and Zanoli introduced a suitable additional module to define the status of each variable, which improved the reliability of two-layer MPC [23]. Zanoli and Pepe analyzed the formulations of two layer MPC and discussed its cooperation and consistency [24]. Wang et al. introduced dynamic trajectory calculation into the two-layer structure based on the present steadystate calculation method, which could bring better economic or tracking performance [25]. Cai et al. presented a two-layer architecture with disturbance suppression using HammersteinWiener models, which could achieve unbiased control with input and output disturbance [26]. Pan et al. discussed the inconsistency between steady-state optimization and dynamic control (DC), and carried out a novel approach to ensure the feasibility of them [27]. Zou considered the dynamic stability property of two-layer MPC and gave an improved offset-free strategy for

https://doi.org/10.1016/j.isatra.2020.02.008 0019-0578/© 2020 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.

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Nomenclature Variables y u d dˆ

ε ∆y ∆u ∆d

Controlled variable Manipulated variable Measurable disturbance variable Unmodeled disturbance variable Slack variable Increment of controlled variable Increment of manipulated variable Increment of disturbance variable

Model parameters Gu Gd Au Ad

Transfer function of u Transfer function of d M-step dynamic prediction matrix of u to y One-step dynamic prediction matrix of d to y

Complex domain and time domain

•(s) •(k) •(k + i|k)

Complex domain variable (y(s), u(s), d(s), . . .) Time domain variable (y(k), u(k), d(k), . . .) Time domain predictive value for time k + i at k

Controller parameters N, P, M cu , cy W Qss , Rss Q, R, S, T

Model length, Optimization horizon, Control horizon Economic self-optimization cost factor Soft constraint adjustment weighting matrix Steady-state optimization weighting matrix Dynamic control weighting matrix

Symbols

•ss •s •h r

• •∗ •¯ •¯

Steady-state (yss , uss , dss , . . .) Soft constraint (y¯ ss,s , y , . . .) ss,s

Hard constraint (y¯ ss,h , y (y1ss,s

Priority rank r ¯

,

ss,h y2 ss,s

, . . .)

, . . .)

Optimal value (J ∗ , ε∗ , . . .) ¯ ss , y¯ , ∆u¯ , . . .) Upper bound (u Lower bound (uss , y, ∆u, . . .)

y˜ PM uM

∆uM wy , wu yT , yT

2. Two-layer model predictive control Two-layer model predictive control containing steady-state optimization and dynamic control has been playing a significant role in industrial MPC [8,9], which is shown in Fig. 2. Consider a continuous multi-input multi-output system shown in Eq. (1): y(s) = Gu (s)u(s) + Gd (s)d(s)

Open-loop prediction of controlled variable Closed-loop prediction of controlled variable M-step value of u M-step value of ∆u Given steady-state trajectories of y and u Given steady-state targets of y and u

(1)

where u(s), d(s) and y(s) represent the dynamic manipulated variable (MV), disturbance variable (DV) and controlled variable (CV), respectively; Gu (s) and Gd (s) are the corresponding transfer functions. The steady-state incremental models used in steady-state optimization can be obtained through stabilizing Eq. (1) and taking the difference between both sides, which is shown in Eq. (2) [8,9]:

∆yss (k) = Gu,ss ∆uss (k) + Gd,ss ∆dss (k)

Others y˜ P0

the present algorithm [28]. Xu et al. applied two-layer nonlinear MPC into a cryogenic air separation system and brought great economic profits [29]. Zou et al. designed a coordinated controller based on two-layer MPC and successfully deployed it into an air separation unit [30]. Above researches reveal that two-layer MPC attracts great interests of many scholars and engineers in the past decades of years, and has been developed into a complete architecture. However, our recent researches indicate that some special situations cannot be correctly solved by the present priority ascent FJSCA strategy, which explains the present policy is incomplete to some extent. The present priority ascent strategy works well in most situations. However, theoretical analysis demonstrates there exist some special situations cannot be perfectly solved by it, which could have been avoided through some technical adjustment. Therefore, a special case is firstly given to indicate the incompleteness of the present priority ascent strategy, and the corresponding figures intuitively present the procedure of the present strategy and show the incompleteness problem clearly. Then an augmented priority ascent strategy is presented to deal with the above problem, which considers all hard constraints in advance and treats the incompleteness problem from the source. Finally, a simple proof is given to demonstrate the completeness of the proposed method, which is used to avoid the same problem to happen again. This paper is organized as follows: Section 2 gives an introduction of two-layer model predictive control, Section 3 introduces the present priority ascent strategy in detail and describes the incompleteness of it with a special case, Section 4 discusses some present solutions to the incompleteness problem, presents an augmented approach for it and gives a completeness proof of the proposed method, Section 5 shows some simulations to verify the effectiveness of the presented strategy, Section 6 discusses some conclusions and future directions.

(2)

where ∆uss (k) and ∆yss (k) denote the steady-state incremental manipulated and controlled variables at instant k, respectively; ∆dss (k) is the steady-state incremental disturbance variable determined by ∆dss (k) = d(k) − d(k − 1), in which d(k) and d(k − 1) are dynamic measured values; Gu,ss and Gd,ss are the corresponding steady-state gain matrixes, respectively; the subscript ss means steady-state. Based on Eq. (2), the new optimal steady-state (uss (k), yss (k)) at instant k can be calculated by: uss (k) = u(k − 1) + ∆uss (k) yss (k) = yss (k − 1) + ∆yss (k)

(3)

where u(k − 1) represents the dynamic manipulated variable at instant k − 1.

Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.

J. Liu, X. Zhang and T. Zou / ISA Transactions xxx (xxxx) xxx

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Fig. 1. The five generations of industrial MPC algorithms.

Fig. 2. The structure of two-layer MPC.

And at instant k, the corrected prediction of last steady-state yss (k − 1) is shown as follow:

ˆ yss (k − 1) = yss (k − 1|k) = Gu,ss u(k − 1) + Gd,ss d(k − 1) + d(k) (4) where Gu,ss and Gd,ss are used to realize the sufficiently long horizon prediction ground on the last manipulated value u(k − 1) ˆ and measured disturbance value d(k − 1); d(k) approximates the estimate of unmeasurable disturbances and unmodeled dynamics [16,18,31,32]. 2.1. Steady-state optimization Steady-state optimization is used to find the optimal steadystate target based on a steady-state model, which mainly contains feasibility judgment and soft constraint adjustment, economic self-optimization and target tracking. The former is applied to search the feasible region of an optimization problem and the latter is devoted to solving the optimal steady-state within the above feasible region. 2.1.1. Feasibility judgment and soft constraint adjustment Feasibility judgment means judging whether a steady-state optimization problem is feasible, in other words, whether there exists a feasible region for an optimization problem. If there exists a feasible region under the limit of all soft constraints, the optimization stage can be directly carried out to solve the optimal steady-state. Otherwise, those soft constraints are required to be relaxed based on a weighted or priority method, to obtain a feasible region which meets the optimization requirements. There are two kinds of popular FJSCA methods called the weighted method and the priority method, which have been detailed described in relevant literatures [8,9,18]. The basic idea of FJSCA is shown in Fig. 3.

As illustrated in Fig. 3(a), consider a two-input three-output system, there is no feasible region under the limitation of the upper and lower boundaries of MV1 , MV2 , CV1 , CV2 and CV3 , which means the optimization problem is infeasible. Therefore, those constraints need to be softened under the guidance of some relaxation rules, to obtain a feasible region. Usually the constraints on u are defined as hard constraints, which on y include hard and soft constraints. The detailed description of hard and soft constraints will be given in Section 3. The optimization problem may have a feasible region or a sole feasible solution (the red point) after feasibility judging and soft constraint adjusting, which is shown as Fig. 3(b). 2.1.2. Economic self-optimization and target tracking Based on different objectives and situations, the optimization stage can be classed into two modes: economic self-optimization and target tracking, which is shown in Fig. 4. Economic selfoptimization means autonomously solving the optimal steadystate in the feasible region based on an economic objective, while target tracking means to find the nearest steady-state away from the external target (uT , yT ) in the feasible region. Combined with Eq. (3), the steady-state constraints of uss (k), yss (k) and ∆uss (k) used in economic self-optimization and target tracking are shown in Eq. (5).

¯ ss uss ≤ u(k − 1) + ∆uss (k) ≤ u y∗ ≤ yss (k − 1) + ∆yss (k) ≤ y¯ ∗ss ss ∆uss ≤ ∆uss (k) ≤ ∆u¯ ss

(5)

¯ ss , ∆uss and ∆u¯ ss represent the steady-state lower where uss , u and upper boundaries of manipulated variables and their increments, respectively; y∗ and y¯ ∗ss denote the confirmed optimal ss steady-state lower and upper boundaries of controlled variables. The formulation of economic self-optimization and target tracking is shown in Eq. (6): min J1 (k) = cTu ∆uss (k) + cTy ∆yss (k)

∆uss (k)

or min J2 (k) = ∥yss (k) − yT ∥2Qss + ∥uss (k) − uT ∥2Rss ∆uss (k)

(6)

s.t. Eqs. (2) and (5)

where J1 (k) and J2 (k) denote the economic and tracking objective functions, respectively; cu and cy are the cost coefficients of ∆uss (k) and ∆yss (k), respectively; Qss and Rss represent the corresponding weighting matrixes.

Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.

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Fig. 3. The basic procedure of FJSCA. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

lower limit of the same controlled variable. However, the lower (or upper) hard constraint and soft constraint of the same controlled variable usually are given the same priority for the requirement of FJSCA. And international convention: the lower the rank, the higher the priority [6]. The core idea of the priority ascent strategy is that the higher priority (lower rank) constraints should be satisfied at first and the lower priority (higher rank) constraints can be considered later. Hard constraints denote those constraints from the physical characteristics of actuators and safety considerations of processes, which cannot be violated at any time. The constraints on manipulated variables usually are defined as hard constraints, such as the opening of a valve is between 0%–100%, which is impossible to be violated. Those constraints on controlled variables being relevant to safety, stability and quality can also be defined as hard constraints, which must be satisfied in the whole control process. Soft constraints are associated with process requirements and economic benefits, which are expected but not required to be satisfied. In some situations, soft constraints can be violated to ensure the safety, stability and quality of the control process. Those constraints on controlled variables which are relevant to economic profits usually are defined as soft constraints to improve the system economic performance. In order to distinctly describe the incompleteness of the present priority ascent strategy, the detailed procedure of it is presented firstly, then a special case is proposed to verify its incompleteness. Unless otherwise specified, all constraints mentioned in the subsequent sections of the article are steady-state constraints.

Fig. 4. Economic self-optimization and target tracking.

2.2. Dynamic control The formulation of dynamic control is shown as Eq. (7): 2 min J(k) = wy (k) − y˜ PM (k)Q + ∥ε(k)∥2S





∆uM (k)

+ ∥wu (k) − uM (k)∥2T + ∥∆uM (k)∥2R s.t. y˜ PM (k) = y˜ P0 (k) + Au ∆uM (k) + Ad ∆d(k) ¯ u ≤ uM (k) ≤ u y − ε(k) ≤ y˜ PM (k) ≤ y¯ + ε(k) ∆u ≤ ∆uM (k) ≤ ∆u¯

3.1. The present priority ascent strategy of FJSCA (7)

where ∆uM (k) is the decision variable, which indicates the Mstep optimal increment of manipulated variable; the remaining symbol definitions can be referred from the nomenclature in this section and some literatures [8,30].

The first step to solve the feasibility problem is initializing the priorities and weighting coefficients of all controlled variable constraints. Subsequently, FJSCA can be taken into account rank by rank with the order from small to large. The detailed procedure of it is shown as below. (1) Rank = 1 The soft constraints with Rank = 1 will be considered firstly, and the formed FJSCA problem is shown as below: min

∆uss ,ε11 ,ε12

3. Problem description

or

Priorities are defined on controlled variables, exactly on the constraints of controlled variables. Different constraints can be configured with different priorities, even for the upper limit and

∆uss ,ε11 ,ε12

min

J 1 = W11 ε11 + W12 ε12 ,

W1j =

[

Wj11

Wj21

···

Wjn1 1

]

J 1 = (ε11 )T W11 ε11 + (ε12 )T W12 ε12 ,

W1j = diag

(

Wj11

Wj21

···

Wjn1 1

)

Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.

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where the superscript 1 and 2 stand for the priority ranks; y1∗ ≤ ss y1ss (k − 1) + ∆y1ss (k) ≤ y¯ 1ss∗ denote the fixed hard constraints of controlled variables which are derived from the relaxed soft constraints of last priority, i.e.

s.t.

∆yss (k) = Gu,ss ∆uss (k) + Gd,ss ∆dss (k)

¯ ss uss ≤ u(k − 1) + ∆uss (k) ≤ u ∆uss ≤ ∆uss (k) ≤ ∆u¯ ss

(8)

− ε12 ≤ y1ss (k − 1) + ∆y1ss (k) ≤ y¯ 1ss,s + ε11 ss,s

y1

− y1ss,h ss,s

where y1 , y¯ 1ss,s , y1 and y¯ 1ss,h denote the lower and upper boundss,s ss,h aries of soft constraints and hard constraints of controlled variables, respectively; ε11 and ε12 denote the upper limit and lower limit slack variables of controlled variables, respectively; uss and ¯ ss are the lower and upper hard limits of uss (k), respectively; u W11 and W12 stand for the weighting matrixes of slack variables; n1 denotes the number of soft constraints in the present priority rank; the subscript ss means steady-state values, s stands for soft constraints, h denotes hard constraints; the superscript 1 stands for the priority rank. Some conclusions can be derived from the optimization results: 1 J = 0: which means there exists a feasible region under the ⃝ limit of all constraints without relaxing the soft constraints; 2 J ̸ = 0: which means no feasible region is found under the ⃝ limit of all constraints without relaxation, but a feasible region can be obtained through relaxing the boundaries of the present soft constraints; J = 0 or J ̸ = 0 means the feasible region under the present priority has been found and the FJSCA of next priority can be taken into consideration. 3 No solution: which illustrates achieving a feasible region of ⃝ feasible solution under the present priority is almost impossible, even if the soft constraints have been exactly relaxed to their corresponding hard constraints. Therefore, the FJSCA of next priority cannot be carried out, and the economic self-optimization or target tracking cannot be executed for there is no feasible region to find the optimal steady-state. In this situation, MPC will be cut out from the controller and traditional proportional–integral– derivative (PID) or other alternative algorithms will be used to carry on the control of the process. (2) Rank = 2 If the FJSCA problem shown in Eq. (8) is feasible (J = 0 or J ̸ = 0), then the next priority rank can be taken into consideration. At first, the relaxed soft constraints of Eq. (8) should be fixed as hard constraints, then the FJSCA of Rank = 2 can be formulated as:

or min

∆uss ,ε21 ,ε22

J 2 = W21 ε21 + W22 ε22 , J = (ε 2

W2j = diag

(

W2j =

ε + (ε

Wj12

Wj22

···

Wjn2 2

]

ε ,

2 T 2 2 1 ) W1 1

2 T 2 2 2 ) W2 2

Wj12

···

Wj22

[

Wjn2 2

)

s.t.

∆yss (k) = Gu,ss ∆uss (k) + Gd,ss ∆dss (k)

¯ ss uss ≤ u(k − 1) + ∆uss (k) ≤ u ∆uss ≤ ∆uss (k) ≤ ∆u¯ ss y1∗ ≤ y1ss (k − 1) + ∆y1ss (k) ≤ y¯ 1ss∗ ss y2 ss,s

− ε22 ≤ y2ss (k − 1) + ∆y2ss (k) ≤ y¯ 2ss,s + ε21

0 ≤ ε22 ≤ y2

− y2ss,h

0 ≤ ε21 ≤ ¯

− y¯ 2ss,s

ss,s y2ss,h

− ε12∗

¯

+ ε11∗

ss,s y1ss,s

=¯

ε11∗ and ε12∗ denote the confirmed optimal slack variables obtained

0 ≤ ε11 ≤ y¯ 1ss,h − y¯ 1ss,s

∆uss ,ε21 ,ε22

y1∗ = y1

ss y1ss∗

0 ≤ ε12 ≤ y1

min

5

(9)

from Eq. (8). The situations of Rank = 3, 4, . . . , N are the same as Rank = 2, so the optimization processes of them will not be given here. As indicated above that “priority ascent” means continuously relaxing and fixing the soft constraints rank-by-rank. The priority ascent strategy can effectively ensure the minimum relaxation of the most important soft constraints, which is quite in line with the actual engineering requirements. 3.2. The incompleteness of the present priority ascent strategy In this section, a special case will be given to verify the incompleteness of the present priority ascent strategy. And for the convenience of analysis, the constraints on the increments of manipulated variables will not be introduced. Given a two-input two-output system shown as Fig. 5, where the x-axis stands for manipulated variable u1 , the y-axis denotes u2 ; the dark solid lines represent hard constraint boundaries on u1 and u2 ; the blue solid and dashed lines denote hard and soft constraint boundaries of y1 respectively, and the priority rank of y1 is 1; the red solid and dashed lines denote hard and soft constraint boundaries of y2 respectively, and the priority rank of y2 is 2. As shown in Fig. 5, there exists a yellow triangle feasible region under the limitation of all hard constraints. Therefore, there should exist feasible solutions in all FJSCA problems with any priority rank. Suppose the initialization has been completed, then the optimization process of the present priority ascent strategy will be given. (1) Rank = 1 Carry on the FJSCA optimization shown in Eq. (8), the hard and soft constraints on y1 whose priority rank equals 1 and all constraints on u should be introduced into the optimization process, so the feasible region is shown in Fig. 6. As shown in Fig. 6 that there exists a feasible region (shadow area) under the limitation of soft constraints of y1 and hard constraints of u, so relaxing the soft constraints of y1 is unnecessary, according to the core idea of the priority ascent strategy. Then the optimal solution of Eq. (8) is ε11∗ = 0, ε21∗ = 0, J 1∗ = 0. (2) Rank = 2 It is known that ε11∗ = 0, ε21∗ = 0, J 1∗ = 0, so the relaxed soft constraints of y1 need to be fixed as new hard constraints in present priority. Then the hard and soft constraints of y2 with Rank = 2 are introduced and the new feasible region analysis is shown in Fig. 7. As shown in Fig. 7, there is no feasible region under the limitation of soft constraints of y2 and (fixed) hard constraints of y1 & u, so the soft constraints of y2 have to be relaxed to achieve a feasible region or feasible solution. However, there is still no feasible region even if the soft constraints of y2 have been exactly relaxed to its hard constraints, which is shown as step 1 in Fig. 7. In this situation, MPC has to be cut out from the ⃝ controller and conventional PID or other alternative algorithms will be introduced to carry on the control of the process.

Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.

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Fig. 5. The feasible region analysis of a special case. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 6. The FJSCA problem of Rank = 1 (present algorithm).

Nevertheless, there should exist a feasible region (yellow triangle region) in this optimization problem, according to Fig. 6. Therefore, the present priority ascent strategy is incomplete because it cannot correctly handle the above optimization problem. The present priority ascent strategy is suitable for most feasibility optimization problems, but there still exist some special situations which it cannot deal with. Remark. In this article, the reason for introducing a two-input two-output system to analyze the incompleteness of the present priority ascent strategy is its feasible region can be given visually in the two-dimensional space, which is of great help to

understand the executing process of the present algorithm. However, the incompleteness of the present priority ascent strategy is very common in multi-input multi-output systems, including nonlinear systems. 4. Solutions to the incompleteness of the present priority ascent strategy 4.1. Conventional solutions There are three kind existing methods which can partly solve the present incompleteness characteristic in practical engineering.

Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.

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7

Fig. 7. The FJSCA problem of Rank = 2 (present algorithm).

4.1.1. Cut out MPC, and introduce PID If there is still no feasible region after many times of optimization, MPC will be sacrificed and cut out from the controller, then traditional PID or other alternative algorithms will be introduced to carry on the adjustment of the process. Usually MPC will be introduced again to get better performance when the system is stably controlled by PID. This method is simple and convenient to implement at the expense of sacrificing MPC optimization ability. 4.1.2. The weighed FJSCA method The weighted method is another way to solve the feasibility problem, which has also been widely used in a variety of control engineering [18,32,33]. In the weighted method, all soft constraints with different weighting factors corresponding to different priorities are relaxed simultaneously in an optimization problem, which makes it do not have the drawback of the previous priority ascent strategy. The weighted method does not indicate so clearly possible ranked relaxation limits, but it works quite satisfactory in most cases. The simulations results in Section 5.2.2 demonstrates the proposed augmented priority method could avoid the infeasibility problem and give the same results as the weighted method. 4.1.3. Configure critical variable and uncritical variable (Critical method) In order to ensure the feasibility in actual engineering applications, the controlled variables will be defined as critical variables and uncritical variables on the basis of their different importance. For the above incompleteness problem: if the controlled variable y2 is set as an uncritical variable, it can be softened again even if its soft constraints has been exactly relaxed to its hard constraint boundaries. The executing process of the above method 2 in Fig. 8, which can form a new feasible is shown as step ⃝ region finally. The yellow point denotes the achieved sole feasible solution. However, if the controlled variable y2 is set as a critical variable, its constraints are unable to be softened again after being exactly relaxed to its hard constraints, which means the above optimization problem will still be infeasible. In conclusion, the first and second methods are two alternative ways to avoid the incompleteness problem, not targeted solutions

for it. The third method cannot fundamentally solve the incompleteness of the present priority ascent strategy because it is not a good choice to relax hard constraints at any time. Therefore, these three methods could just be used as simple engineering strategies to enhance the feasibility and practicability of relevant algorithms.

4.2. An augmented solution to the incompleteness In this section, an augmented priority ascent strategy which in advance takes into account all hard constraints of controlled variables with higher priority ranks is proposed, to fundamentally solve the incompleteness of the present priority ascent strategy. The introduction of all hard constraints of controlled variables ensures the feasibility of all subsequent priorities after realizing the FJSCA of any priority, which avoids the appearance of the above infeasible problem and guarantees the completeness of the augmented priority ascent strategy. The procedure of the presented strategy is given below:

(1) Rank = 1 The FJSCA problem of Rank = 1 can be formulated as below: min

∆uss ,ε11 ,ε12

min

J 1 = W11 ε11 + W12 ε12

∆uss ,ε11 ,ε12

or

J 1 = (ε11 )T W11 ε11 + (ε12 )T W12 ε12

s.t. ∆yss (k) = Gu,ss ∆uss (k) + Gd,ss ∆dss (k)

¯ ss uss ≤ u(k − 1) + ∆uss (k) ≤ u ∆uss ≤ ∆uss (k) ≤ ∆u¯ ss yr ≤ yrss (k − 1) + ∆yrss (k) ≤ y¯ rss,h , r = 2, 3, . . . , N ss,h y1

ss,s

(10)

− ε12 ≤ y1ss (k − 1) + ∆y1ss (k) ≤ y¯ 1ss,s + ε11

0 ≤ ε12 ≤ y1

− y1ss,h

0 ≤ ε11 ≤ ¯

− y¯ 1ss,s

ss,s y1ss,h

Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.

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Fig. 8. The conventional solution to the incompleteness of the present algorithm.

Compared with Eq. (8), all hard constraints of controlled variables whose priority ranks are greater than 1 have been introduced into Eq. (10). Take the special case in Section 3.2, the feasible region of Eq. (10) is shown in Fig. 9. As shown in Fig. 9, there is no feasible region under the limitation of soft constraints of y1 and hard constraints of y2 & u, but a feasible region can be easily obtained through relaxing the 1 soft constraints of y1 . The relaxation process is shown as step ⃝ in Fig. 9, where the yellow point stands for the feasible solution obtained by soft constraint adjustment. The optimal decision variables ε11∗ = 0, ε21∗ ̸ = 0, and the optimal solution J 1∗ ̸ = 0. (2) Rank = 2 Before starting rank 2, the relaxed soft constraints of rank 1 need to be fixed as hard constraints. Then the FJSCA problem of Rank = 2 is shown as Eq. (11). min

∆uss ,ε21 ,ε22

min

J 2 = W21 ε21 + W22 ε22

∆uss ,ε21 ,ε22

or

Remark. No feasible region under original hard constraints indicates the constraint configuration of processes is unreasonable. In this situation, process and control engineers will be required to adjust relevant constraints to make the optimization problem feasible. This problem is not within the scope of the FJSCA problem in the paper.

J 2 = (ε21 )T W21 ε21 + (ε22 )T W22 ε22

s.t.

∆yss (k) = Gu,ss ∆uss (k) + Gd,ss ∆dss (k) ¯ ss uss ≤ u(k − 1) + ∆uss (k) ≤ u ∆uss ≤ ∆uss (k) ≤ ∆u¯ ss

(11)

y1∗ ≤ y1ss (k − 1) + ∆y1ss (k) ≤ y¯ 1ss∗ ss yr ss,h y2 ss,s

≤ yrss (k − 1) + ∆yrss (k) ≤ y¯ rss,h , r = 3, 4, . . . , N − ε22 ≤ y2ss (k − 1) + ∆y2ss (k) ≤ y¯ 2ss,s + ε21

0 ≤ ε22 ≤ y2

− y2ss,h

0 ≤ ε21 ≤ ¯

− y¯ 2ss,s

ss,s y2ss,h

constraints of y2 need to be relaxed to obtain a feasible region. And there exists a feasible solution when the soft constraints are exactly softened to its hard constraint boundaries, which is 1 in Fig. 10. The yellow point denotes the obtained shown as step ⃝ feasible solution locating at the corner of the yellow triangle region shown in Fig. 5, which indicates the relaxing process is correct and feasible. The situations of Rank = 3, 4, . . . , N are the same as Rank = 2, so the optimization processes of them will not be presented here. As indicated above that the augmented priority ascent strategy proposed in the paper can fundamentally solve the incompleteness of the original algorithm, which enhances the feasibility and practicality of two-layer MPC. And the presented augmented method also fits the core idea of the priority ascent strategy to firstly meet the higher priority (lower rank) constraints and minimally relax the most important constraints.

where yr ≤ yrss (k − 1) + ∆yrss (k) ≤ y¯ rss,h , r = 3, 4, . . . , N denote ss,h all hard constraints of controlled variables whose priority ranks are greater than 2. The new feasible region of the aforementioned special case is shown as Fig. 10. As shown in Fig. 10, there is no feasible region under the limitation of soft constraints of y2 and (fixed) hard constraints of y1 & u after introducing the constraints of y2 . Therefore, the soft

4.3. Proof of the completeness of the proposed method Completeness means the priority ascent strategy should have the ability to obtain a feasible region as long as there exists a feasible region under the limitation of all hard constraints of controlled variables and manipulated variables. The present priority ascent strategy has been used for more than ten years and there is no problem is found until a special case is given to verify its incompleteness in the paper. Therefore, to avoid the same situation, a brief proof is carried out to prove the completeness of the proposed augmented priority ascent strategy. Proof. (Proof by contradiction) Suppose the proposed algorithm is incomplete, then the following two conditions need to be satisfied.

Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.

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9

Fig. 9. The FJSCA problem of Rank = 1 (augmented algorithm).

Fig. 10. The FJSCA problem of Rank = 2 (augmented algorithm).

(1) In the stage of FJSCA, the original optimization problem should be feasible when all soft constraints are exactly relaxed to their hard constraint boundaries. Therefore, there should exist a feasible region in Eq. (12)

Or the optimization problem shown in Eq. (13) should be feasible

min

∆yss (k) = Gu,ss ∆uss (k) + Gd,ss ∆dss (k) ¯ ss uss ≤ u(k − 1) + ∆uss (k) ≤ u ∆uss ≤ ∆uss (k) ≤ ∆u¯ ss y ≤ yss (k − 1) + ∆yss (k) ≤ y¯ ss,h ss,h

∆uss ,ε11 ,ε12 ,...,εN ,εN 1 2

J =

N ∑

Wl1 εl1 + Wl2 εl2

l=1

or (12)

min

∆uss ,ε11 ,ε12 ,...,εN ,εN 1 2

J =

N ∑

T

T

(εl1 ) Wl1 εl1 + (εl2 ) Wl2 εl2

l=1

Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.

10

J. Liu, X. Zhang and T. Zou / ISA Transactions xxx (xxxx) xxx

there exists a feasible region under the limitation of all original hard constraints.

s.t. ∆yss (k) = Gu,ss ∆uss (k) + Gd,ss ∆dss (k)

5. Simulation

¯ ss uss ≤ u(k − 1) + ∆uss (k) ≤ u ∆uss ≤ ∆uss (k) ≤ ∆u¯ ss yr − εr2 ≤ yrss (k − 1) + ∆yrss (k) ≤ y¯ rss,s + εr1 , r = 1, 2, . . . , N ss,s 0≤ε ≤ r 2

yr ss,s yrss,h

0≤ε ≤¯ r 1

−

yr ss,h yrss,s

−¯

, r = 1, 2, . . . , N , r = 1, 2, . . . , N

12.8e−s 16.7s+1 6.6e−7s 10.9s+1

(2) There exists l, (2 ≤ l ≤ N) satisfies that the FJSCA of Rank = l is infeasible, but Rank = l − 1 is feasible. Therefore, the optimization problem shown in Eq. (14) should be feasible. min

J l−1 = Wl1−1 εl1−1 + Wl2−1 εl2−1

min

J l−1 = (εl1−1 )T Wl1−1 εl1−1 + (εl2−1 )T Wl2−1 εl2−1

−18.9e−3s 21s+1 −19.4e−3s 14.4s+1

∆yss (k) = Gu,ss ∆uss (k) + Gd,ss ∆dss (k) ¯ ss uss ≤ u(k − 1) + ∆uss (k) ≤ u ∆uss ≤ ∆uss (k) ≤ ∆u¯ ss yr ∗ ≤ yrss (k − 1) + ∆yrss (k) ≤ y¯ rss∗ , r = 1, 2, . . . , l − 2 ss

≤

yl−1 ss,s

−ε

yrss (k l−1 2

0≤ε

l−1 2

− 1) +

≤

≤

ylss−1 (k

yl−1 ss,s ylss−,1h

0 ≤ εl1−1 ≤ ¯

∆yrss (k)

−

− 1) +

yrss,h

≤¯

(14)

u1,ss = −1, u¯ 1,ss = 1, u2,ss = −1, u¯ 2,ss = 1 = −31.7, y1 = −18.9, y¯ 11,ss,s = −12.8, y¯ 11,ss,h = 0 y1 1,ss,h

y2

2,ss,h

1,ss,s

= 0, y22,ss,s = 6.6, y¯ 22,ss,s = 12.8, y¯ 22,ss,h = 19.4

Dynamic constraints:

ylss−,1s

≤¯

+ε

min

yl−1 ss,h

∆uss ,ε11 ,ε21

∆uss ,εl1 ,εl2

min

∆uss ,εl1 ,εl2

[

or

J l = (εl1 )T Wl1 εl1 + (εl2 )T Wl2 εl2

∆y1,ss (k) ∆y2,ss (k)

]

[ =

12.8 6.6

−18.9 −19.4

][

∆u1,ss (k) ∆u2,ss (k)

]

−1 ≤ u1 (k − 1) + ∆u1,ss (k) ≤ 1 −1 ≤ u2 (k − 1) + ∆u2,ss (k) ≤ 1 −18.9 − ε21 ≤ y11,ss (k − 1) + ∆y11,ss (k) ≤ −12.8 + ε11

(17)

0 ≤ ε11 ≤ 0 − (−12.8)

s.t.

0 ≤ ε21 ≤ (−18.9) − (−31.7)

∆yss (k) = Gu,ss ∆uss (k) + Gd,ss ∆dss (k) ¯ ss uss ≤ u(k − 1) + ∆uss (k) ≤ u ∆uss ≤ ∆uss (k) ≤ ∆u¯ ss yr ∗ ≤ yrss (k − 1) + ∆yrss (k) ≤ y¯ rss∗ , r = 1, 2, . . . , l − 1 ss ss,h yl ss,s

J 1 = ε11 + ε21

s.t.

− y¯ lss−,1s

J l = Wl1 εl1 + Wl2 εl2

Suppose the initial state u0 = 0 and y0 = 0, the priority rank of y1 equals 1 and the priority rank of y2 equals 2, all weighting factors of slack variables are 1, and adopt the linear programming objective functions.

(1) Rank = 1 The FJSCA problem of Rank = 1 can be described as:

l−1 1

And the optimization problem shown in Eq. (15) should be infeasible. min

2

5.1.1. The present priority ascent strategy

, r = l, l + 1, . . . , N

∆ylss−1 (k)

(16)

5.1. Steady-state optimization verification (One step)

s.t.

yr ss,h

u(s)

Steady-state constraints:

1

∆uss ,εl1−1 ,εl2−1

]

u1 = −1, u¯ 1 = 1, u2 = −1, u¯ 2 = 1 y = −31.7, y¯ 1 = 0, y = 0, y¯ 2 = 19.4

or

yr

[ y(s) =

(13)

∆uss ,εl1−1 ,εl2−1

Wood berry model is considered to verify the effectiveness of the augmented strategy, which is shown as follows:

The optimal solutions are

ε =0, ε21∗ =0, J 1∗ = 0 1∗ 1

(2) Rank = 2 The FJSCA problem of Rank = 2 is shown as below:

≤ yrss (k − 1) + ∆yrss (k) ≤ y¯ rss,h , r = l + 1, l + 2, . . . , N

− εl2 ≤ ylss (k − 1) + ∆ylss (k) ≤ y¯ lss,s + εl1 0 ≤ εl2 ≤ yl − yl ss,s ss,h

min

∆uss ,ε12 ,ε22

J 2 = ε12 + ε22

s.t.

0 ≤ εl1 ≤ y¯ lss,h − y¯ lss,s

[ (15)

The constraint conditions of Eqs. (14) and (15) are the same if the soft constraints of controlled variables (y¯ lss,s , yl ) shown ss,s in Eq. (15) are exactly relaxed to their hard constraint boundaries. Therefore, the optimization problem shown in Eq. (15) should also be feasible if Eq. (14) is feasible. So, the original hypothesis does not hold. QED. Forward recursion can also prove the completeness of the proposed augmented strategy: the FJSCA of rank 2 will be feasible if the rank 1 is feasible; the rank 3 will be feasible if the rank 2 is feasible; · · ·; finally the rank N will also be feasible. In conclusion, the proposed priority ascent strategy will be feasible as long as

∆y1,ss (k) ∆y2,ss (k)

]

[ =

12.8 6.6

−18.9 −19.4

][

∆u1,ss (k) ∆u2,ss (k)

−1 ≤ u1 (k − 1) + ∆u1,ss (k) ≤ 1 −1 ≤ u2 (k − 1) + ∆u2,ss (k) ≤ 1 −18.9 ≤ y11,ss (k − 1) + ∆y11,ss (k) ≤ −12.8

]

(18)

6.6 − ε22 ≤ y22,ss (k − 1) + ∆y22,ss (k) ≤ 12.8 + ε12 0 ≤ ε12 ≤ 19.4 − 12.8 0 ≤ ε22 ≤ 6.6 − 0 The optimization problem shown in Eq. (18) has no solution, which means the FJSCA of rank 2 is infeasible. Therefore, the MPC controller has to be cut out from the control system.

Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.

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5.2. Complete control comparison

5.1.2. The proposed augmented priority ascent strategy (1) Rank = 1 The optimization problem of FJSCA can be described as: min

∆uss ,ε11 ,ε21

The original priority ascent strategy could not give a feasible steady-state target, which means the dynamic control is impossible to realize. Therefore, the conventional solutions showed in Sections 4.1.2 and 4.1.3 are presented to compare with the proposed priority ascent strategy. Adopt the economic self-optimization algorithm shown in Eq. (6) and the dynamic control algorithm shown in Eq. (7), the remaining parameter configurations are shown as follows:

J 1 = ε11 + ε21

s.t.

[

∆y1,ss (k) ∆y2,ss (k)

]

12.8 6.6

[ =

−18.9 −19.4

][

∆u1,ss (k) ∆u2,ss (k)

]

−1 ≤ u1 (k − 1) + ∆u1,ss (k) ≤ 1 −1 ≤ u2 (k − 1) + ∆u2,ss (k) ≤ 1 −18.9 − ε21 ≤ y11,ss (k − 1) + ∆y11,ss (k) ≤ −12.8 + ε11

(19)

0 ≤ y22,ss (k − 1) + ∆y22,ss (k) ≤ 19.4 0 ≤ ε11 ≤ 0 − (−12.8) 0 ≤ ε21 ≤ (−18.9) − (−31.7) The optimal solutions are

ε11∗ = 6237 , ε21∗ =0, J 1∗ = 970

6237 970

(2) Rank = 2 The optimization problem of FJSCA is shown as below: min

∆uss ,ε12 ,ε22

J 2 = ε12 + ε22

s.t.

[

∆y1,ss (k) ∆y2,ss (k)

]

12.8 6.6

[ =

−18.9 −19.4

][

∆u1,ss (k) ∆u2,ss (k)

]

−1 ≤ u1 (k − 1) + ∆u1,ss (k) ≤ 1 −1 ≤ u2 (k − 1) + ∆u2,ss (k) ≤ 1

(20)

−18.9 ≤ y11,ss (k − 1) + ∆y11,ss (k) ≤ −12.8 +

6237 970 2 1

6.6 − ε22 ≤ y22,ss (k − 1) + ∆y22,ss (k) ≤ 12.8 + ε 0 ≤ ε ≤ 19.4 − 12.8 0 ≤ ε ≤ 6.6 − 0

ε =0, ε =6.6, J

2∗

∆y1,ss (k) ∆y2,ss (k)

]

[ =

2,ss,s

= 6.6

12.8 6.6

−18.9 −19.4

][

∆u1,ss (k) ∆u2,ss (k)

−1 ≤ u1 (k − 1) + ∆u1,ss (k) ≤ 1 −1 ≤ u2 (k − 1) + ∆u2,ss (k) ≤ 1 6.6 − 6.6 ≤

− 1) +

∆y22,ss (k)

]

(21)

−18.9 ≤ y11,ss (k − 1) + ∆y11,ss (k) ≤ −12.8 + y22,ss (k

1,ss,s

Wy¯ 2,ss,s = Wy

(3) optimal feasible region The final constraints on the feasible region which will be used in the optimization stage are shown as below:

[

5.2.1. Comparison between the critical method and the augmented method Fig. 11 shows the steady-state and dynamic trajectories of manipulated and controlled variables, in which the weighting matrix of the slack variable in Eq. (7) is set as S = 0.1 × I. Large punishment on slack variables will make the dynamic trajectories be away from the given steady-state trajectories in the critical method due to the further hard constraint relaxation, which is shown as Fig. 12. In Figs. 11 and 12, “yss,cri ”, “uss,cri ”, “yss,new ” and “uss,new ” stand for the steady-state optimization trajectories of the critical method and the new augmented method, respectively; “ycri ”, “ucri ”, “ynew ” and “unew ” denote the corresponding dynamic trajectories. The simulation results show that the further hard constraint relaxation permit the uncritical variable y2 violate its hard constraints to ensure the feasibility. However, the proposed method can find the actually reachable steady-state while ensuring the feasibility, which is more reasonable and safer. The comparison shows the proposed approach can handle the incompleteness of the original priority ascent method effectively.

Wy¯ 1,ss,s = Wy

The optimal solutions are 2∗ 2

N = 120 [ , P = ]30, M =[ 10 ] cTu = 1 1 , cTy = 0 0 Q = I, R = I, T = I

5.2.2. Comparison between the weighted method and the augmented method The weighting factors used in the weighted FJSCA are set as:

2 1 2 2

2∗ 1

11

6237 970

≤ 12.8

These constraints on the feasible region will be transferred into the optimization stage to calculate the optimal steady-state, including economic self-optimization and target tracking. As a matter of fact, there only exists one feasible solution in the above optimization problem, which is the inevitable result of minimally relaxing the most important constraints of controlled variables. Nevertheless, sometimes there will exist a feasible region with many different feasible solutions after feasibility judging and soft constraint adjusting.

= 10 000 =1

which shows the different importance of controlled variables. The weighting matrix of the slack variable is set as S = 100×I to avoid violation of bounds in dynamic control as much as possible. In Fig. 13, “yss,wei ” and “uss,wei ” are the steady-state trajectories of the weighted method, while “ywei ” and “uwei ” denoting the dynamic trajectories. The simulation results illustrate the proposed method can provide the same reliable steady-state as the weighted method. Therefore, the proposed method repairs the incompleteness problem existed in the previous priority ascent strategy, which makes the priority ascent strategy more reliable and reasonable. 6. Conclusion In this paper, an augmented priority ascent strategy is proposed to deal with the feasibility problem in steady-state optimization, which overcomes the incompleteness of the present strategy. The proposed special case intuitively shows the incompleteness of the present priority ascent strategy, and drawbacks of the existing available methods are presented and discussed. The proposed method fundamentally solves the incompleteness problem, which improves the reliability and practicability of twolayer MPC. The simple proof given in the paper verifies the completeness of the proposed method, which ensures the proposed

Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.

12

J. Liu, X. Zhang and T. Zou / ISA Transactions xxx (xxxx) xxx

Fig. 11. The comparison between the critical method and the augmented method.

Fig. 12. The comparison between the critical method and the augmented method (Large S).

Fig. 13. The comparison between the weighted method and the augmented method.

method works well in all situations. The comparison between the critical method, the weighted method and the proposed augmented method illustrates the feasibility and effectiveness of the proposed approach. The future research will still go on the analysis of the priority and weighted FJSCA, such as the influence of disturbance and noise on them, the zone steady-state targets, aiming at improving the reliability and optimality of relevant algorithms.

Acknowledgments

Declaration of competing interest

References

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

This work was supported by the National Key Research and Development Program of China [grant number 2017YFA0700303]; the National Natural Science Foundation of China [grant number 61773366]; the Natural Science Foundation of Liaoning Province [grant number 2019-KF-03-07]; and the national foundation for studying abroad from the China Scholarship Council.

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Please cite this article as: J. Liu, X. Zhang and T. Zou, Analysis of the incompleteness of the priority ascent strategy of feasibility judgment and soft constraint adjustment. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.02.008.