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REFERENCE TRAJECTORY DESIGN USING STATE CONTROLLABILITY FOR BATCH PROCESSES César Augusto Gómez Pérez, Lina María Gómez Echavarría, and Hernan Alvarez Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/ie504809x • Publication Date (Web): 17 Mar 2015 Downloaded from http://pubs.acs.org on March 22, 2015

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Industrial & Engineering Chemistry Research

REFERENCE TRAJECTORY DESIGN USING STATE CONTROLLABILITY FOR BATCH PROCESSES a

a

a

César. A. Gómez-Pérez *, Lina M. Gómez Echavarría , Hernán D. Alvarez Zapata . [a] Research Group on Dynamic Processes-KALMAN, Facultad de Minas, Universidad Nacional de Colombia, Medellín-COLOMBIA. e-mail address: [email protected] (C. A. Gómez-Pérez), [email protected] (L. M. Gómez Echavarría), [email protected] (H. D. Alvarez Zapata). * Corresponding Author. César Augusto Gómez Pérez; Tel: (57)(4) 4255092; e-mail address: [email protected]. postal address: Carrera 80 #65-223, office: M3-100, Medellín, Colombia; zip code: 050041. Abstract: A proposal for the design of a reference trajectory for batch processes was developed, using set theory state controllability concepts. This proposal introduces the Available Control Action Set, which is an indicator of the state controllability and contains the largest amount of possible control actions that reach the desired final state. The volume of Available Control Action Sets can be maximized to improve the safety of batch processes while maintaining consistently high product quality. Thus a closed loop system has sufficient energy to reject disturbances, ensuring end-of-batch product quality. Case studies are presented to demonstrate the applicability of the approach to batch and semi-batch processes and to illustrate how the proposed method for the design of a reference trajectory for batch processes increases safe and robust operation. Keywords: Batch processes; Set theory; Controllability; Reference trajectory design; Optimization; Dynamic optimization. 1. INTRODUCTION A common practice for batch process operation is the use of a prescribed time sequence of process variables that is considered the best option for driving the process. This sequence is

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called the process operation trajectory. The aim of any process operation trajectory is to produce the best product quality at batch finishing. Due to its main characteristic, which is a set of connected points that form a time sequence, the process operation trajectory is implemented in a process control system as a sequence of set points of the process state 1

variables that are being controlled . Therefore, any process operation trajectory can first be designed in accordance with the available control system. After obtaining some operation data, the process operation trajectory can be optimized for some desirable process 2

objective . Unfortunately, this procedure for obtaining and improving a process operation trajectory is not a current industrial procedure. Generally, reference trajectories are recipes that are empirically constructed in a laboratory or a plant after many tests, and they are used to operate the process during the batch time and to control the process variables. For example, the profile presented in Russell et al. (1998) is a trajectory that was obtained using process measurement data for nylon 3

polymerization in a batch reactor . Azimzadeh et al. (2001) characterized the reference profile using dissolved oxygen measurements to estimate the state of gluconic acid 4

production ; the dissolved oxygen exhibited several process stages that needed to be fulfilled to obtain the final product. Shamekh et al. (2015) use an established temperature profile to 1

evaluate PID control in a exothermic batch process . The trajectory is obtained from the linearization of the different stages to design a control system. While heuristics have been fundamental for batch process operation trajectory design, the improvement in computer processing power and growing market demands have increased the need for solving 2

optimization problems to identify control actions that produce an optimal trajectory . In the literature, the topic of reference trajectory design is not explicitly mentioned; only Junghui Chen and Rong Guey Sheui (2002 and 2003) present an explicit reference 5,6

trajectories design using an adaptive method , to determinate an optimal process operation trajectory for a batch process. Also, Aamir et al. (2012) design an optimal temperature


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7

trajectory using nonlinear optimization in a crystallization batch process , the optimal profile produces a desired targeted crystal size distribution. However, the trajectory design is implicitly incorporated into the batch processes optimization, in which a dynamic optimization problem is formulated to obtain optimal profiles that achieve a particular process performance index such as productivity and batch duration time

8–10

. Dynamic optimization has been used

as the principal tool to obtain the optimal operation of batch reactors in several publications

2,10–25

. Some of these publications use the profile as a reference for designed

control systems, while others want to determine the optimal profile for the operation of a batch process. Unfortunately, the trajectories that are identified by optimization are difficult to follow, even when the process is operating under closed-loop control, because of disturbances or uncertainty of initial conditions and models used for the dynamic optimization

16,23,26

.

Additionally, the irreversible character of batch processes greatly affects the dynamic 27,28

behavior, including the state controllability

. This difficult is hard to handle in batch process

operation, because as it is said in Sontang (1998), irreversible states are no controllable. Trajectories that are calculated under these conditions in real batch processes are difficult to 29

control . According to Luyben (1996) both the steady state economics and the dynamic controllability 30

should be considered at all stages in the design of continuous processes . A similar situation should be considered in batch processes; i.e., in the reference trajectory design of batch processes, both the minimal cost (minimal batch time) and the dynamic controllability should be considered at all stages. However, the literature on batch process optimization does not take into account the fact that the obtained trajectory affects the state controllability of the process. This paper proposes a design method for a reference trajectory that includes a control analysis for batch processes to obtain trajectories, considering process state controllability



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information. This proposal implies that the control actions can properly control the process up to the desired final state, which is related to the final product quality, even with uncertainties in the initial conditions and in the presence of disturbances. As an example, in Section 2, a brief state-of-the-art trajectory design problem is presented. Section 3 introduces a control analysis that uses set theory to present the necessary tools for Section 4, in which the proposal for the design of the reference trajectory, including set theory controllability concepts, is presented. In Section 5, two examples that compare classical trajectory design to the trajectory design that uses state controllability concepts within a set theory framework are presented. Finally, Section 6 shows the conclusions that are reached by this study. 2. OVERVIEW OF THE BATCH PROCESS REFERENCE TRAJECTORY DESIGN PROBLEM. In a batch process, the objective is to obtain a product with a defined specification that is set by conducting a sequence of steps; i.e., the goal is to obtain a set of states within defined final conditions from an initial condition within a given process time. The state sequence goes from the initial states to the final condition along a trajectory. There are many trajectories that connect an initial condition to a final condition. Currently, in the batch process control and optimization area, one of the studied problems is finding an 8,9

optimal trajectory using a given criteria that is expressed as an objective function . This approach is defined as the dynamic optimization of batch process, which is a trajectory design that uses optimization problems. 2.1. Trajectory design as an optimization problem. The aim of this design is to use optimization to determine a productivity maximizing trajectory using a control actions’ vector uk [0, t f ] as the decision variable. Bonvin et al., (2006) 2

presented a general mathematical formulation for this type of problem , as shown by equations (1), (2) and (3):


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min Jk = ϕ ( xk ( tf ) ) + ∫ L ( xk ( t ) ,uk ( t ) , t ) dt tf

uk [0,tf ], ρ

0

(1)

Which is subject to x& k ( t ) = F ( x k ( t ) , u k ( t ) , ρ ) , x k ( 0 ) = x k ,0 ( ρ )

(2)

S ( x k ( t ) , u k ( t ) , ρ ) ≤ 0, T ( x k ( t f ) , ρ ) ≤ 0 (3)

ρ is an additional variable vector in which the initial conditions and the total run time t f could be considered, S represents the run time constraints, and T represents the batch end constraints. This general representation consider the run to run batch operation as a discrete system, Subscript k represents the actual run. The problem can be summarized as follows: find the sequence of inputs uk [0, t f ] that produce the optimal trajectory xkopt [0, t f ] , as determined by the objective function and constraints. Objective function design. The cost function J k in batch processes is aimed to minimizing total run time ( t f ) or maximizing productivity, as indicated in (4) and (5). min J k = min t f (4)

uk [0,tf ], ρ

uk [0,tf ], ρ

min J k = min [ −Prod]= max [Prod] (5)

uk [ 0,tf ], ρ

u k [0,tf ], ρ

u k [0,tf ], ρ

For objective function (4), the idea is to minimize the total run time, which would enable more batches to be produced in a fixed time period and, therefore, more production. For objective function (5), the idea is to find the sequence of control actions that produce the highest productivity for a certain compound or useful product within a fixed batch time. Some authors use both functions; Shah and Madhavan (2001) used an objective function that was the sum 14

of the total run time and the final product with weighting factors , and Martinez et al. (1998) 12

used a relation between the conversion and the total run time . Soroush and Kravaris (1993a) presented several performance indexes that could be used as 8

an objective function , depending on the type of batch process. Some examples include the



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polydispersity index in batch polymerization, the product yield in batch fermentation and the selectivity in complex reactions. Ming et al. (2000) used the nucleated and seed crystal 13

relation , the particle mean size and the coefficient variance to obtain a quality product from batch crystallization. The objective cost functions that are found in the literature are based on several performance process indexes. The use of this kind of cost functions obtains solutions that get near optimization problem constraints. Since it seeks to obtain the maximum yield, together with minimum time, the batch process always needs the extreme feasible actions or the total energy available to success the desired objective. As a result, the designed trajectory is very dangerous to follow, because any disturbance can affect the process and violates the constraint. The use of controllers should help to follow the trajectory, but, with little room for maneuverability, the control will have some problems to fulfill the regulation task

27,28

. This

paper proposes to work with a different paradigm, and use state controllability concepts into the objective cost function to obtain a robust reference trajectory with control purpose. Run Time Constraints. These constraints are determined by the process dynamics and other practical considerations, such as the actuator capacity and limits on variables that are related to the maximum capacity and to required process security margins. The run time constraints define the space in which operation is feasible and safe and are thus needed to obtain an answer that is close to reality. It is important to mention that Bonvin et al. (2006) treated the system dynamics as a 2

constraint for the optimization problem . In fact, the system dynamics is a natural constraint that indicates that the solution cannot operate in regions in which the system does not exist. There are two ways of knowing the dynamic information of a process: by using a model or by implicitly using specific information of the process. In the first case, the optimization problem can be formulated using such model, and in the second case, the optimization formulation 2

uses open-loop process information .


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Generally, optimal trajectory batch process profiles strongly depend on the constraints. From a review of the solutions that were obtained in several papers

14–18,21–25

, it can be concluded

that the solution touch a constraint or several constraints during operation. In terms of state controllability, it could be dangerous to be near a constraint, as in the case of exothermic polymerization, in which a temperature constraint exists to prevent the polymer gel effect; if the process is disturbed, it could reach dangerous temperatures and produce a gel effect 9

problem . Also, constraints can be added to the optimization problem, to make the process to become safe. But generally this kind of constraint is conservative. So, they generally don´t allow to use all process potential. As a result batch process operates with low performance. Also, if disturbances exist there is no guarantee that the process will reach the desire final 27

condition . Batch End Constraints. For the final security conditions and quality requirements, the definition of the batch end constraint is based on the desired product, which must fulfill the 2

characteristics that define the product quality and the operation performance . The batch end constraint defines the ultimate goal of the batch process; sometimes, the characteristic of 26

batch process makes optimization unfeasible. This problem is studied in Welz et al. (2008) , in which an iterative dynamic optimization method is developed and takes into account the uncertainty, and in Almeida and Secchi (2012) who proposed a constraint relaxation method 23

to make the problem feasible . 2.2. Optimization and control in batch processes. Batch process optimization has been closely associated with batch process control. Several authors have worked on dynamic optimization for batch process control with nonlinear predictive controllers

12–15,19,22,26

, in which optimization plays an important role. Also, Zhang et

al. (2012) demonstrate how reachability analysis helps the design of control systems in 31

crystallization batch process . However, the limited number of studies that have been dedicated to the analysis of batch process state controllability indicates that only process



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27

productivity has been considered and that state controllability was neglected . This fact produces systems that are prone to disturbances which can affect the product quality and productivity. Despite this vulnerability, the development of set theory tools makes batch 27

process control plausible for an analysis of state controllability . This work uses such tools to propose a new reference trajectory design to enhance batch process control. 3. SET THEORY IN BATCH PROCESS DYNAMIC ANALYSIS. Modern control theory has started to use set theory for many applications

32–36

, insomuch as

set theory has the advantage of naturally including constraints in the states and the control 27

actions. Particularly for batch process analysis, Controllable Trajectory Sets were proposed . These sets define the controllable space that represents the region over which the states of the dynamic system can evolve. To understand the concept of Controllable Trajectory Sets, certain set definitions are needed. 3.1. Basic Definitions. The definition of the Controllable Trajectory Set is based on two set types: reachable sets and controllable sets. Consider a nonlinear dynamic system, such as x& ( t ) = f ( x ( t ) , u ( t ) )

(6)

u ( t ) ∈ U is the control action vector, U ⊂ Rm is a compact and simply connected set that represents the set of system admissible control actions, x ( t ) ∈ X is the state vector, X ⊂ Rn is a closed and simply connected set of physical states that the system can take, and f (,) is defined in the X × U space. Using this formula, the reachable and controllable set can be 32

defined . Definition: Reachable Set in i-steps. Given a set P , the Reachable Set R

i

(P )

from P in i

steps is the state space subset in which the system evolves from P in i sample times given the admissible control action sequence [u1, u2 , u3 ,..., ui ] of size i .


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R

i

( P ) = {x ( i ) ∈ X∃x ( 0 ) ∈ P ∧ [u1, u2 , u3 ,..., ui ] ∈ U : x ( i ) = f ( x ( 0 ) , u1, u2 , u3 ,..., u i )}

(7)

Observation: The reachable set is the set of all state space vectors that can be reached by system evolution for a given admissible control action. Definition: Controllable Set in i-steps. Given a set S , the Controllable Set C i ( S ) towards S in

i steps is the state space subset in which the system evolves towards S in i sample times given the admissible control action sequence [u1, u2 , u3 ,..., ui ] of size i.

{

}

C i ( S ) = x ( 0 ) ∈ X∃x ( i ) ∈ S ∧ [u1, u 2 , u 3 ,..., u i ] ∈ U : x ( i ) = f ( x ( 0 ) , u1, u 2 , u3 ,..., u i )

(8)

Observation: The controllable set is the set of all state space vectors that for a given admissible control action, can reach set S within a determined time. 3.2. The Controllable Trajectory Set as an analysis tool. Controllable Trajectory Sets can be viewed as a way to make a state controllability analysis. In Gómez et al. (2010), the authors discussed how this tool can provide ideas on batch 27

process state controllability from a set theory viewpoint . Besides, In Gómez et al. (2015), the authors show the algorithm to calculate the sets and discuss its application on irreversible dynamic systems. Because batch processes have irreversible states, it is not relevant to 29

perform a controllability analysis, because, as it is said in Sontag (1998) , the irreversible property makes it impossible to reach a previous state value and classical concept of controllability is not assured. Thus, from the viewpoint of classical control theory, the batch 29

process systems with irreversible states are not controllable . However, it is possible to find a state space region in which an admissible control action sequence exists that reaches the final desired condition from the initial process condition, and this space state region is defined by the Controllable Trajectory Set. The definition by Gómez et al., 2010 is summarized 27

below :



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Definition: Controllable Trajectory Set from Q0 to QN in N steps: This set is the sequence of

Qi sets of states in R n that are reachable in i steps ( i = 1,2,3,..., N − 1) from Q0 and that are controllable in ( N − i ) steps towards QN , An admissible control action sequence exists such that, for every initial condition x ( k ) ∈Q0 , the convergence of the system to the given final condition x ( k + N ) ∈QN in N steps through the state sequence x ( k + i ) ∈Qi is guaranteed (see Figure 1). Observation. The Controllable Trajectory Set is a series of sets that determines the states in which an admissible control action exists such that the states can go from their initial condition to the desired final conditions in a batch process. If the process achieves a state outside of the Controllable Trajectory Set in a given instant, the states will not converge to the desired final condition. Therefore, the ideal situation is that the Controllable Trajectory Set occupies the biggest portion of space state such that more state variables can converge to the final conditions.

X1

N N-1

t

N-2 0

1

2

3

4

N-3

X2 Figure 1. Controllable Trajectory Set.


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3.3. Calculation of the Controllable Trajectory Set. To calculate the Controllable Trajectory Set both the Reachable Set and the Controllable Set for nonlinear systems need to be determined. Pitifully the algorithms used to compute the 35

sets suffer from the curse of dimensionality ; i.e., the computational complexity increases exponentially with the dimension of the system. The difficulty of calculating these sets for nonlinear systems that are subject to constraints has been an impediment in the development and exploitation of this theory for such systems. The same problem has occurred in other control areas in which the complexity of the calculation is also considerable. Blondel and Tsitsiklis (2000) provided a tutorial on computational complexity and discussed from the point of view of computational complexity many open problems in systems and 37

control theory , e.g., stable and unstable polynomials in polynomial families, the stability of matrix families and the null-controllability of hybrid systems. Many problems in theory control that have a considerable computational complexity have been solved using randomized algorithms. These algorithms are simple and fast, e.g., in the analysis and design of robust control systems there are many bibliographies that have used randomized algorithms

38,39

. Other problems that have been tackled by randomized algorithms

are the Controllability/Reachability Analysis of Hybrid Systems

40,41

and the continuous time

42

reachability computation . Based on the aforementioned, a randomized algorithm is used to obtain an estimate of the Reachable Set and the Controllable Set for nonlinear systems that are subject to constraints within a polynomial calculation time. The algorithms is presented in Gómez et al. (2015) and enable the authors to obtain a sub-approximation of the Reachable Set and the Controllable 28

Set with both accuracy and risk of failure . This algorithm has been successfully used in 27

other research . Here the algorithm presented in Gómez et al. (2015) is summarized below: Algorithm 1. Calculation of the Reachable Set



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Given: An initial condition set Ω , an error and a fail risk ( ε and δ ), a state constraint set X and a control variable constraint set U . 1 2 log . δ 2ε 2 Uniform( U) .

−

Find the sample size by the Chernoff inequality N ≥

−

Get N samples xs

−

Uniform(Ω) and us Obtain the prediction xs (t + ∆t ) in accordance with (6).

Return with a probability (1 − δ ) an estimation of the set R i (Ω) such that,

{

PN x ∈ X

N

}

(

)

: P (R ( Ω ) ) − Pˆ (R ( Ω ) ; x ) > ε ≤ 2 exp −2Nε 2 (9)

Observation. For the calculation of the controllable set is enough to solve the system in reverse time, this is xs (t − ∆t ) For calculation of the Controllable Trajectory Set following algorithm is proposed. Algorithm 2. Calculation of the Controllable Trajectory Set Given: An initial condition Q0 , a final condition QN and N . −

Find N reachable sets in i steps ( i = 1,2,..., N ) from Q0 , (R

−

Find N controllable sets in i steps ( i = 1,2,..., N ) to QN , (C 1 (Q N ) ,C 2 (Q N ) ,..., C N (Q N ) ) .

−

Calculate each of the Controllable Trajectory Sets, using the intercepts between the

i reachable set R

i

(Q0 )

and the

( N − i ) controllable

1

(Q 0 ) ,R 2 (Q 0 ) ,...,R N (Q0 ) ) .

set C N − i (QN ) , for every

( i = 1,2,..., N ) . which is shown as follows. (Q0 ,QN ) = Q 0

Q(

0,N )

Q(

1,N −1)

Q(

2,N − 2 )

(Q 0 ,Q N ) = R 1 (Q 0 ) ∩ C N −1 (Q N ) = Q1 (Q 0 ,Q N ) = R 2 (Q 0 ) ∩ C N − 2 (QN ) = Q 2

T Q(

N −1,1)

(Q 0 ,Q N ) = R

N −1

(Q 0 ) ∩ C 1 (Q N ) = Q N −1


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Q(

N ,0 )

(Q 0 ,Q N ) = Q N

Thus, the Controllable Trajectory Set for N steps from Q0 , to QN is the sequence

CTS = (Q0 ,Q1,Q2 ,...,QN −1,QN ) (see Figure 1). 4. REFERENCE TRAJECTORY DESIGN BASED ON STATE CONTROLLABILITY CONCEPTS. An optimization based reference trajectory design, as indicated in Section 2, can lead to trajectories with potential difficulties in batch process control. Attempting to minimize the batch time or to maximize the productivity produces trajectories that move along constraint frontiers; this result makes these trajectories non-controllable for certain events, such as disturbances and uncertainties, which lead to failures in achieving the process objectives within the fixed batch time. This paper suggests finding a safe trajectory in which the closed-loop system has sufficient energy to reject disturbances, thus maintaining an adequate performance. To obtain this effect, a new set is introduced to characterize the batch process state controllability; this new set is called the Available Control Action Set. 4.1. Available control actions. The existence of Controllable Trajectory Sets defines two types of control actions: the actions that can keep the states within the sets and the actions that take the system outside of the sets. Using this idea, the following definitions can be made. Definition: Available control actions: Given a Controllable Trajectory Set at the next sample time, Qi +1 = R

i +1

(Q0 ) ∩ C N −( i +1) (QN ) , and a trajectory point in actual time, x ( t i ) , for

( i = 0,1,2,..., N − 1) . An available control action, u ( ti ) ∈ U , is an action that can drive the point x ( t i ) to a x ( ti + ∆t ) ∈Qi point, as observed in Figure 2.



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u (2) x* (i-1)

Page 14 of 31

* x (i) x (i)

u (1)

* Qi

Figure 2. Available control action: u(1) is an available control action, while u(2) is not. Definition: Available control action set in i ( UA ) . This set is the set of all the control actions

u ∈ U that are Available Control Actions for the state x ( t i − ∆t ) and the Controllable Trajectory Set, Q i . See Figure 3. The available control action is different from the admissible control action ( u∈ U ) , which is the control action that can be achieved in the real process and is specified by the saturation of the actuator. The available control action appears because of the Controllable Trajectory Set. Figure 3 show what occurs for a certain state when all of the admissible control action set is used: the dynamic system reaches the next set, and some of the states stay in the Controllable Trajectory Set, which is the shaded zone in the reachable set. The control actions that correspond to the shaded zone define the Available Control Actions Set ( UA ) that take place in the input space ( U ).


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x2

1

u2

( x (i-1)

(

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


u 2 máx.

x*(i-1)

A

u 2 mín. i

State space

u 1 mín.

x1

u 1 máx. u 1 Input space

Figure 3. Available Control Action Set. 4.2. Optimization problem definition A new cost function is proposed using the Available Control Action Sets, this function tries to select the trajectory whose Available Control Action Sets contain the largest amount of possible control actions, i.e., the largest hypervolume. The general optimization problem can be formulated as follows: N

max ∑ VUAi u

(10)

i =1

Which is subject to x& k ( t ) = F ( x k ( t ) , u k ( t ) , ρ ) , x k ( 0 ) = x k ,0 ( ρ )

(11)

S ( x k ( t ) , u k ( t ) , ρ ) ≤ 0, T ( x k ( t f ) , ρ ) ≤ 0

(12)

VUAi is the hypervolume of the Available Control Action Set of a determined i step, and N is the number of sample steps that it takes to go from Q0 to QN , which is defined by the sample time. Note that this function is opposite to the (4) and (5) cost functions, because this optimization does not lead to an improvement in the efficiency and productivity of the batch processes. This optimization wants to determine a safe trajectory to obtain robust control of



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batch processes, which is the opposite of finding a trajectory that maximize the productivity, 27

as is show in Gomez et al (2010) . Hence, the numerical methods for compute Controllable Trajectory Sets have strong computational cost. Because it is very difficult to solve an optimization problem in which the batch end time is the variable, this time must be defined to compute the Controllable Trajectory Sets and then used as information in the optimization problem. To make a decision, a prudent batch duration time must be chosen using information regarding the Controllable Trajectory Sets and the minimal possible time, as will be shown in the illustrative example section. A possibly interesting future work is the development of a method that takes into account both the state controllability and the productivity in the optimization problem. 5. ILLUSTRATIVE EXAMPLE In this section, two batch process cases are developed using two benchmark models from the literature: a two-state batch reactor from Lee et al. (2000)

43

and a three-state semi-batch

17

reactor from Srinivasan et al. (2003) . These cases are used because the two-state batch reactor is a very simple and intuitive problem that shows how the batch time is important for the state controllability issue that is defined here and how the new proposal method can assure a sufficient control space. The three-state semi-batch reactor is a more realistic case with different constraints and an optimization that has been reported in the literature

17

so can

be used to compare the proposed and the reported batch process optimization. An optimal trajectory is desired for these two processes, and the results from traditional trajectory optimization will be compared with the trajectory that is produced from the optimization problem that is proposed in this study. 5.1. Batch reactor. In a process, a certain amount of product B will be produced from reactive A via the secondorder exothermic reaction A → B . This reaction is performed in a batch reactor that is equipped with a jacket. The objective of the process is to obtain a high conversion of A ;


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therefore, an A concentration of 0.1 mol / L is fixed as the final control objective. A dynamic model of the system is obtained from the mass and energy balances. This model and its 43

parameters are presented in Lee et al. (2000) . E  −  a dC A RT  2 = −k0 e  CA dt

(13)

UA j ( −∆H )V − Ea RT  2 dT =− T − Tj ) + k0e CA ( dt MCP MCP

(14)

CA represents the A concentration in the batch reactor, T is the temperature in the batch reactor, k0 is the frequency factor of the chemical reaction kinetics, Ea is the activation factor, R is the ideal gas constant, T j is the jacket heat exchanger temperature, U is the global heat transfer coefficient, A j is the jacket exchanger area, M is the total reactive mass in the batch reactor, CP is the specific heat capacity, V is the reactive mass volume and

( −∆H )

is the reaction enthalpy. This nonlinear system has two states [ x1, x 2 ] = [C A ,T ] , one T

T

manipulated variable ( u = T j ) and one output variable ( y = CA ) . The dynamics is discretized by splitting the batch time into sample intervals. The objective is to obtain a trajectory that guides the output ( CA ) to obtain a final concentration of 0.1 mol / L .

a) Classical reference trajectory. An offline optimization

10

17

problem is formulated as detailed in Srinivanan et al. (2003) . This

problem consists of determining the control actions that produce the final condition in the least amount of time. A mathematical representation of the problem is presented in equations (15) – (18): min tf u ,tf

(15)

Which is subject to

umin ≤ u ( t ) ≤ umax

(16)



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x& = f ( x, u )

Page 18 of 31

(17)

x1 ( t f ) = C A ( t f ) = 0.1mol

L

(18)

The first constraint represents the real limits that are imposed on the manipulated variable; in this case, 20º C ≤ T j ( t ) ≤ 96º C .The second constraint is associated with the process dynamics, and the third one indicates the final desired concentration. This optimization problem is discretized and solved using the MATLAB® optimization toolbox and the fmincon function. The optimization variables are the discrete control action vector and the batch time. The minimal batch time is 3.29 minutes. Another constraint was added to the discrete problem

∆u ≤ ∆umax ( ∆Tj ≤ 5º C ), in order to force the input variable will not make extreme changes between the samples times. It is important to note that this process is an example that was taken from the batch process literature to serves as a mathematical example because, in practice, the batch times are much longer, frequently running for several hours. Using 10 sample steps, a Controllable Trajectory Set analysis is presented in Figure 4a.The optimal trajectory for T and CA is also presented in Figure 4a, and Figure 5 shows the control action sequence that produces the optimal trajectory (continuous line). In this case, a maximum temperature produces a minimum batch time, which is quite obvious. With sufficient available energy, the reaction would be fast enough, but it should be noted that optimal control action sequence is saturated at its maximum. From Figure 4a, it can be established that the Controllable Trajectory Sets that are associated with this batch time (3.3 min) have notably small areas and that the trajectory evolves close to one of the edges of the set. This result means that there are few safe states for the process and, in turn, implies less freedom for the system to move around the trajectory. Therefore, if a disturbance affects the system, the control actions will not be able to reject this disturbance; the states will exit the controllable sets, making the final control


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objective unattainable. This result can also be observed in Figure 5 and Figure 6a, in which the control action is always saturated, and the Available Control Action Sets are notably poor for every step. If the system is disturbed, the controller will not be able to correct the deviation.

b) Reference trajectory based on state controllability concepts. In this case, a batch time must be defined. Using the previously determined time, a new batch time is selected heuristically. Next, an iterative process is used until the Controllable Trajectory Sets have a considerable hypervolume. The optimization problem is reformulated to maximize the volume of the Available Control Actions: N

max ∑ VUAi u

(19)

i =1

When this equation is subject to equations (16), (17) and (18), the constraints on the manipulated variable T j , the system dynamics and the control objective are maintained. In this case, because there is only one control action, the volume of the sets is measured as the length between the maximum and minimum values.

VUA = ( uA max − uA min )

(20)

This procedure produces the discrete control action vector (see Figure 5, dotted line) and the state trajectory that is presented in Figure 4b. In this example, a batch time of 4 minutes was found. This time is not a better choice for productivity but is prudent, because the Controllable Trajectory Sets hypervolume was considerably increased as observed in Figure 4b.



320

320

Q10

310 305

Q0

300

Q10

315

Temperature [ºK]

315

Temperature [ºK]

310 305 300

Q0

295 1

295 1 3 0.5 Concentration [mol/lt]

2 Time [min]

1 0

4 3

0.5 Concentration [mol/lt]

2 1 0

0

Time [min]

0

(a)

(b)

Figure 4. Controllable Trajectory Sets for: (a) batch time (3.3 min), (b) batch time (4 min).

Tjmax

370 360

Temperature [ºK]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 31

350 340 330 320 310 300 290 0

Tjmin 1

2 Time [min]

3

4

Figure 5. Optimal control actions for: batch time (3.3 min)[], batch time (4 min)[--].


Page 21 of 31

Tjmax 370

370

360

360

350

350

Temperature [ºK]

Temperature [ºK]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


340 330 320 310

340 330 320 310

300

300 Tjmin

290 0

Tjmax

0.5

1

1.5 2 Time [min]

2.5

3

290 0

Tjmin 1

2 Time [min]

(a)

3

4

(b)

Figure 6. Available control action sets for: (a) batch time (3.3 min), (b) batch time (4 min). In this case, the Controllable Trajectory Sets have a larger area than those in the previous case (see Figure 4), thereby providing more freedom to move inside the state space in the process. Figure 4b shows how the trajectory evolves inside the Controllable Trajectory Sets while trying to avoid the set boundaries in a safe process operation. In Figure 6b, the behavior of the Available Control Action Sets can be observed, and the sets are presented for each time sample step. Because the batch process has only one control action, the sets that are presented in Figure 6 are one dimensional. These sets maintain a good margin and decay as the batch end approaches because the Controllable Trajectory Sets decrease in area to converge towards the final condition set. Figure 6a shows the low dimension of the Available Control Action Sets that the classical optimal trajectory produced. It can be noted that the control actions for the trajectory design that took into account the Available Control Action Sets are not saturated, as is the case in the classical trajectory design (Figure 5). This lack of saturation gives a controller a better margin for decision making regarding the events during a batch.

5.2. Semibatch reactor. In a stirred tank reactor, the elemental second order reaction A + B → C occurs. This process is isothermal, and there is a B feed. The process objective is to produce 0.6 moles of



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Page 22 of 31

product C using the A + B → C reaction. To increase the production of C , a high A concentration should be present in the reactor, while B is added continuously during the 17

batch time. This model and its parameters are presented in Srinivasan et al. (2003) . dV =q dt

(21)

dC A q = − kC ACB − C A (22) dt V dCB q = − kC ACB + (CBin − CB ) dt V

(23)

V is the volume of the reactive mass, q is the B feed flow, k is the chemical reaction kinetic coefficient, CA is the A concentration, CB is the B concentration and CBin is the B feed concentration. This nonlinear system has three states, [ x1, x 2 , x 3 ] = [V ,C A ,CB ] and one T

T

manipulated variable ( u = q ) . In this case, the output variable is the moles of product C

( y = NC )

which can be calculated using the states V and CA and equation (24). The

dynamics is discretized by splitting the batch time into 20 sample steps.

NC = CA0V0 + CC 0V0 − CAV

(24)

a) Classical reference trajectory. 17

This model was analyzed in Srinivasan et al. (2003) , who analytically found an optimal trajectory that minimized the batch end time, and their result was a time of 19.8 h . The trajectory that the authors showed is continuous, while the analysis here requires system discretization. To obtain a better analysis possible, the optimization problem was reformulated using similar conditions. The solution of the reformulated optimization problem was obtained numerically using the necessary discretization. min tf u ,tf

(25)

Which is subject to


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umin ≤ u ( t ) ≤ umax x& = f ( x, u )

(26)

(27)

x1 ( tf ) = V ( tf ) ≤ 1 lt

(28)

NC ( t f ) = CC ( t f )V ( tf ) = 0.6 mol

(29)

Constraint (26) describes the actual limits on the manipulated variables; in this case, the limits are 0 lt / h ≤ q ( t ) ≤ 0.03 lt / h . Constraint (27) is associated with the dynamic process system, and constraint (28) limits the working volume to a value that is below the reactor over capacity. Finally, constraint (29) indicates the moles of C that are desired at the end of the batch time. It is important to mention that the original problem has a temperature 17

constraint that is not taken into account in the previous problem . This constraint obliges that reactant B cannot be added to the system from the initial state and must be added during batch operation. This constraint was changed by (26), which assures that the reactant B must be added during the process. The new constraint is necessary because the temperature is not presented as a state, thus the Controllable Trajectory Sets cannot be computed using a temperature constraint, but, when constraint (26) is used, this problem is similar to the original problem. Another constraint was added to the discrete problem ∆u ≤ ∆umax ( ∆q ≤ 0.005 lt / h ), in order to force the input variable will not make extreme changes between the samples times. The optimization variables are the discrete control action vector and the batch time. The minimal batch time is 19.7 hours with 20 sample steps. Figure 8 shows the evolution of the optimal trajectory in the Controllable Trajectory Sets, and Figure 9 shows the optimal control action. It must be noted that the manipulated variable is saturated during most of the process operation.



It can be observed from Figures 7 and 8 that the states have a notably small space to move in at the beginning and end. The resulting Controllable Trajectory Sets at the beginning and end under these conditions are small; therefore, any disturbance could disrupt the final product quality or cause unsafe process operation (such as liquid reactor pouring).

b) Reference trajectory based on state controllability concepts. A batch time must be defined. In this case, the previously calculated minimal time was used as the basis for the heuristic calculation of a better value that produced a larger volume for the sets. The calculated time was 22 hours. As shown in Figure 7b and Figure 8 the sets are larger for this batch process time. Again, the objective is to maximize the volume of the Available Control Action Sets. This optimization problem is formulated in equation (19) which is subject to equations (26), (27), (28) and (29). The volume of the Available Control Action Set is evaluated using equation (20). The results of this optimization are shown in Figures 9 and 10, in which the optimal control actions and the Available Control Action Sets are shown. Because the batch process has only one control action, one dimension sets are presented for each time sample step in Figure 10.

0.7

0.7

Q0

0.65

B Concentration [mol/lt]

B Concentration [mol/lt]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 31

0.6 0.55 0.5 0.45 0.4

1 0.9 0.8

1

0.7

0.4

Q 20 2

1.1 1.5 A Concentration [mol/lt]

0.5

0.3

Q 20

2

Q0

0.6

1.1 1.5 A Concentration [mol/lt]

Volume [lt]

0.8

1

0.7

1 0.9 Volume [lt]

0.6

0.6

(a)

(b)

Figure 7. Controllable Trajectory Sets for: (a) batch time (19.72 hours) (b) batch time (22 hours).


Page 25 of 31

Figure 8. Controllable Trajectory Sets with a optimal trajectory for: (a) Vol vs. [A] (19.72 h) (b) [A] vs. [B] (19.72 h) (c) Vol vs. [B] (19.72 h) (d) Vol vs. [A] (22 h) (e) [A] vs. [B] (22 h) (f) Vol vs. [B] (22 h). 0.03

qmax

0.025

0.02

Flow [lt/h]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.015

0.01

0.005 qmin 0 0

5

10 Time [h]

15

20

Figure 9. Optimal control actions for: batch time (19.72 h) [], batch time (22 h) [--].



qmax

qmax 0.03

0.03

0.025

0.025

0.02

0.02

Flow [lt/h]

Flow [lt/h]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 31

0.015

0.01

0.01

0.005

0.005

0

0 0

0.015

qmin

qmin 5

10 Time [h]

15

0

5

10 Time [h]

15

(a)

20

(b)

Figure 10. Available control action set for: (a) batch time (19.72 h), (b) batch time (22 h). In this case, the optimal trajectory evolves within the Controllable Trajectory Sets (Figure 8); this result implies that there is less risk of not attaining the desired product quality if the process is disturbed. As depicted in Figure 10, the Available Control Action Sets are large until t = 16 min , after which the size is much smaller. This finding indicates that the last part of the process of the trajectory has the highest probability of leaving the sets; however, a strong disturbance could be required to take the trajectory outside of the sets because the control actions still have a sufficient margin to reject a disturbance. By comparing the optimal control actions (Figure 9), it can be observed that the control actions in the classical optimization problem tend to be saturated, while those that were calculated using the current proposal are not saturated. Figure 10 shows a significant difference in the Available Control Actions Sets regarding what was previously yielded. The classic trajectory only provides a small number of possibilities, while the trajectory that is based on state controllability maintains a good margin of control actions that only decays at the end of the process. This result means that, if a controller must make a decision in the case of a disturbance using the classical approach it will be very difficult to find a control


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action that would lead to the desired final state. If the trajectory is calculated using controllability concepts, the controller will have a better margin of action and can thereby exert better control. 6. CONCLUSIONS A current topic in batch process design is determination of the optimal trajectory for process variables that enables the production of adequate product quality with an increased productivity. This work presents a design method for a reference trajectory that takes into account the state controllability of the batch process from a set theory viewpoint to improve the process performance once a controller is applied thus ensuring end-of-batch product quality. For this purpose, the Available Control Action Set is introduced, which is an indicator of state controllability and contains the largest amount of possible control actions that reach the final desired state conditions. Dynamic optimization of batch process seeks the productivity maximization or costs minimization, without taking into account the state controllability. The design of the trajectory using the Available Control Action Set improves the robustness and safe operation of the batch process, then, more opportunities to reach the desired product quality is obtained. When a minimal batch time trajectory is obtained by batch process dynamic optimization, the system does not have sufficient control actions to reject disturbances. Therefore, the industrial concept of determining the minimal batch time must be revised. Moreover, further work should take into account a better compromise between productivity and state controllability, by, for example, using both into the optimization problem considering weight factors. ACKNOWLEDGEMENTS. The authors are grateful to the National University of Colombia Scholarship Program (Programa de Becas de la Universidad Nacional de Colombia) for financial support. which The support helped to successfully complete the research.



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REFERENCES

(1)

Shamekh, A.; Hussein, T.; Altowati, A. Design of Standard PID Controller for Exothermic Batch Process Simulation. In 2013 European Modelling Symposium; IEEE, 2013; pp. 343–348.

(2)

Bonvin, D.; Srinivasan, B.; Hunkeler, D. Control and Optimization of Batch Processes. IEEE Control Syst. Mag. 2006, 26, 34.

(3)

Russell, S. A.; Robertson, D. G.; Lee, J. H.; Ogunnaike, B. A. Control of Product Quality for Batch Nylon 6,6 Autoclaves. Chem. Eng. Sci. 1998, 53, 3685.

(4)

Azimzadeh, F.; Gala, O.; Romagnoli, J. A. On-Line Optimal Trajectory Control for a Fermentation Process Using Multi-Linear Models. Comput. Chem. Eng. 2001, 25, 15.

(5)

Chen, J.; Sheui, R.-G. Using Taguchi’s Method and Orthogonal Function Approximation to Design Optimal Manipulated Trajectory in Batch Processes. Ind. Eng. Chem. Res. 2002, 41, 2226.

(6)

Chen, J.; Sheui, R.-G. Optimal Batch Trajectory Design Based on an Intelligent DataDriven Method. Ind. Eng. Chem. Res. 2003, 42, 1363.

(7)

Aamir, E.; Rielly, C. D.; Nagy, Z. K. Experimental Evaluation of the Targeted Direct Design of Temperature Trajectories for Growth-Dominated Crystallization Processes Using an Analytical Crystal Size Distribution Estimator. Ind. Eng. Chem. Res. 2012, 51, 16677.

(8)

Soroush, M.; Kravaris, C. Optimal Design and Operation of Batch Reactors. 1. Theoretical Framework. Ind. Eng. Chem. Res. 1993, 32, 866.

(9)

Soroush, M.; Kravaris, C. Optimal Design and Operation of Batch Reactors. 2. A Case Study. Ind. Eng. Chem. Res. 1993, 32, 882.

(10)

Bonvin, D. Optimal Operation of Batch Reactors - a Personal View. J. Process Control 1998, 8, 355.

(11)

Ruppen, D.; Benthack, C.; Bonvin, D. Optimization of Batch Reactor Operation under Parametric Uncertainty - Computational Aspects. J. Process Control 1995, 5, 235.

(12)

Martinez, E. C.; Pulley, R. A.; Wilson, J. A. Learning to Control the Performance of Batch Processes. Chem. Eng. Res. Des. 1998, 76, 711.

(13)

Ge, M.; Wang, Q.-G.; Chiu, M.-S.; Lee, T.-H.; Hang, C.-C.; Teo, K.-H. An Effective Technique for Batch Process Optimization with Application to Crystallization. Chem. Eng. Res. Des. 2000, 78, 99.


Page 29 of 31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(14)

Shah, S. S.; Madhavan, K. P. Design of Controllable Batch Processes. In ComputerAided Chemical Engineering, 9: European Symposium on Computer Aided Process Engineering - 11; Gani, R.; Jørgensen, S. B., Eds.; Elsevier Science B.V.: Amsterdam, 2001; pp. 743–748.

(15)

Shah, S. S.; Madhavan, K. P. Design of Controllable Batch Processes in the Presence of Uncertainty. Ann. Oper. Res. 2004, 132, 223.

(16)

Bonvin, D.; Srinivasan, B. Optimal Operation of Batch Processes via the Tracking of Active Constraints. ISA Trans. 2003, 42, 123.

(17)

Srinivasan, B.; Palanki, S.; Bonvin, D. Dynamic Optimization of Batch Processes: I. Characterization of the Nominal Solution. Comput. Chem. Eng. 2003, 27, 1.

(18)

Srinivasan, B.; Bonvin, D.; Visser, E.; Palanki, S. Dynamic Optimization of Batch Processes: II. Role of Measurements in Handling Uncertainty. Comput. Chem. Eng. 2003, 27, 27.

(19)

Arpornwichanop, A.; Kittisupakorn, P.; Mujtaba, I. M. On-Line Dynamic Optimization and Control Strategy for Improving the Performance of Batch Reactors. Chem. Eng. Process. Process Intensif. 2005, 44, 101.

(20)

Guay, M.; Peters, N. Real-Time Dynamic Optimization of Nonlinear Systems: A Flatness-Based Approach. Comput. Chem. Eng. 2006, 30, 709.

(21)

Peters, N.; Guay, M.; DeHaan, D. Real-Time Dynamic Optimization of Batch Systems. J. Process Control 2007, 17, 261.

(22)

Mesbah, A.; Landlust, J.; Huesman, A. E. M.; Kramer, H. J. M.; Jansens, P. J.; Van den Hof, P. M. J. A Model-Based Control Framework for Industrial Batch Crystallization Processes. Chem. Eng. Res. Des. 2010, 88, 1223.

(23)

Almeida, E.; Secchi, A. R. Solving Dynamic Optimization Infeasibility Problems. Comput. Chem. Eng. 2012, 36, 227.

(24)

Gamez-Garci, V.; Flores-Mejia, H. F.; Ramirez-Muñz, J.; Puebla, H. Dynamic Optimization and Robust Control of Batch Crystallization. Procedia Eng. 2012, 42, 471.

(25)

Bolaños-Reynoso, E.; Sánchez-Sánchez, K. B.; Urrea-García, G. R.; RicardezSandoval, L. Dynamic Modeling and Optimization of Batch Crystallization of Sugar Cane under Uncertainty. Ind. Eng. Chem. Res. 2014, 53, 13180.

(26)

Welz, C.; Srinivasan, B.; Bonvin, D. Measurement-Based Optimization of Batch Processes: Meeting Terminal Constraints on-Line via Trajectory Following. J. Process Control 2008, 18, 375.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 31

(27)

Gómez, C. A.; Gómez, L. M.; Alvarez, H. D. An Approach to Stability and Controllability Analysis in Batch Processes Using Set Theory Methods. In ANDESCON, 2010 IEEE; IEEE: Bogotá, 2010; pp. 1–6.

(28)

Gómez, L. M.; Botero, H.; Alvarez, H. Análisis de La Controlabilidad de Estado de Sistemas Irreversibles Mediante Teoría de Conjuntos. Rev. Iberoam. Automática e Informática Ind. 2015, Acepted, 1.

(29)

Sontag, E. D. Mathematical Control Theory: Deterministic Finite Dimensional Systems; Springer, Ed.; Second Edi.; New York, 1998.

(30)

Luyben, W. Process Modeling, Simulation and Control for Chemical Engineers; Beamesderfer, L.; Morriss, J. M., Eds.; Second edi.; McGraw-Hill Inc.: New York, 1996.

(31)

Zhang, K.; Nadri, M.; Xu, C.-Z. Reachability-Based Feedback Control of Crystal Size Distribution in Batch Crystallization Processes. J. Process Control 2012, 22, 1856.

(32)

Blanchini, F.; Miani, S. Set-Theoretic Methods in Control; Başar, T., Ed.; Birkhäuser Boston: Boston-Basel-Berlin, 2007.

(33)

Kerrigan, E. C.; Maciejowski, J. M. Invariant Sets for Constrained Nonlinear DiscreteTime Systems with Application to Feasibility in Model Predictive Control. In Proceedings of the 39th IEEE Conference on Decision and Control; IEEE: Sydney, 2000; Vol. 5, pp. 4951–4956.

(34)

Limón Marruedo, D.; Alamo, T.; Camacho, E. F. Input-to-State Stable MPC for Constrained Discrete-Time Nonlinear Systems with Bounded Additive Uncertainties. In Proceedings of the 41st IEEE Conference on Decision and Control; IEEE: Las Vegas, 2002; Vol. 4, pp. 4619–4624.

(35)

Bravo, J. M.; Limon Marruedo, D.; Alamo, T.; Camacho, E. F. On the Computation of Invariant Sets for Constrained Nonlinear Systems: An Interval Arithmetic Approach. Automatica 2005, 41, 1583.

(36)

Bravo, J. M.; Alamo, T.; Camacho, E. F. Robust MPC of Constrained Discrete-Time Nonlinear Systems Based on Approximated Reachable Sets. Automatica 2006, 42, 1745.

(37)

Blondel, V. D.; Tsitsiklis, J. N. A Survey of Computational Complexity Results in Systems and Control. Automatica 2000, 36, 1249.

(38)

Calafiore, G.; Dabbene, F.; Tempo, R. Randomized Algorithms in Robust Control. In Proceedings of the 42nd IEEE Conference on Decision and Control; IEEE: Maui, 2003; pp. 1908–1913.

(39)

Vidyasagar, M. Statistical Learning Theory and Randomized Algorithms for Control. Control Syst. IEEE 1998, 18, 69.


Page 31 of 31

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(40)

Azuma, S.; Imura, J. A Probabilistic Approach to Controllability/reachability Analysis of Hybrid Systems. In Proceedings of the 43rd IEEE Conference on Decision and Control; IEEE: Atlantis, 2004; pp. 485–490.

(41)

Azuma, S.; Imura, J. Probabilistic Controllability Analysis of Sampled-Data/discreteTime Piecewise Affine Systems. In Proceeding of the 2004 American Control Conference; IEEE: Boston, 2004; pp. 2528–2533.

(42)

Niarchos, K. N.; Lygeros, J. A Neural Approximation to Continuous Time Reachability Computations. In Proceedings of the 45th IEEE Conference on Decision and Control; IEEE: San Diego, 2006; pp. 6313–6318.

(43)

Lee, J. H.; Lee, K. S.; Kim, W. C. Model-Based Iterative Learning Control with a Quadratic Criterion for Time-Varying Linear Systems. Automatica 2000, 36, 641.


Reference Trajectory Design Using State ... - ACS Publications

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