Lexicographic optimization based MPC: Simulation and experimental study

Accepted Manuscript Title: Lexicographic Optimization based MPC: Simulation and Experimental Study Author: Markana Anilkumar Nitin Padhiyar Kannan Moudgalya PII: DOI: Reference:

S0098-1354(16)30021-7 http://dx.doi.org/doi:10.1016/j.compchemeng.2016.02.002 CACE 5366

To appear in:

Computers and Chemical Engineering

Received date: Revised date: Accepted date:

16-11-2015 31-1-2016 2-2-2016

Please cite this article as: Markana Anilkumar, Nitin Padhiyar, Kannan Moudgalya, Lexicographic Optimization based MPC: Simulation and Experimental Study, (2016), http://dx.doi.org/10.1016/j.compchemeng.2016.02.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Highlights

Highlights:

ip t

 Multi-variable prioritized control study is carried out using Model Predictive Control (MPC) formulations (Conventional MPC, Lexicographic MPC and Modified Lexicographic MPC)

us

cr

 Implements explicit prioritization in multiple control objectives using Modified Lexicographic MPC, where tuning of weights in the objective function is not required.

an

 Demonstrates the superiority of the proposed method over conventional MPC and Lexicographic MPC for set-point tracking and disturbance rejection performance.

M

 The effectiveness of MLMPC algorithm is demonstrated on a simulated PMMA reactor for controlling the number average molecular weight and the reactor temperature.

Ac ce p

te

d

 Experimental validation on Single Board Heater System (SBHS) for controlling the temperature of a thin metal plate.

Page 1 of 21

Lexicographic Optimization based MPC: Simulation and Experimental Study Markana Anilkumar Systems and Control Engineering Department, Indian Institute of Technology Bombay, Mumbai, India- 400076

ip t

Nitin Padhiyar∗

Department of Chemical Engineering, Indian Institute of Technology Gandhinagar, Gujarat, India- 382424

cr

Kannan Moudgalya

us

Department of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai, India- 400076

Abstract

ed

M

an

Multi-variable prioritized control study is carried out using Model Predictive Control (MPC) algorithms. The conventional MPC algorithm implements multi-variable control through one augmented objective function and requires weights adjustment for required performance. In order to implement explicit prioritization in multiple control objectives, we have used lexicographic MPC. To achieve better tracking performance, we have used a new MPC algorithm, by modifying the lexicographic constraint, referred to as MLMPC, where tuning of weights is not required. The effectiveness of MLMPC algorithm is demonstrated on a PMMA reactor for controlling the number average molecular weight and the reactor temperature. We have also verified the benefits of proposed algorithm on an experimental Single Board Heater System (SBHS) for controlling temperature of a thin metal plate. These simulation and experimental studies demonstrate the superiority of the proposed method over conventional MPC and lexicographic MPC. Finally, we have presented generalized mathematical solutions to the optimization problem in MLMPC.

Ac ce

pt

Keywords: Lexicographic optimization, model predictive control, multi-objective optimization, single board heater system, PMMA reactor

1. Introduction

Model Predictive Control (MPC) is a multi-variable control methodology to have a significant and widespread impact on industrial process control. It generates optimal control moves by explicit use of the process model for the future predictions of the output variables to be controlled. MPC has gained significant attention in industry and academics for the past couple of decades. Its ability to handle multi input multi output (MIMO) systems, process constraints and to naturally include multiple objective criteria in to design of controllers makes it most effective technique. It was first developed by Shell Oil in early 1970s with an application to multi-variable control of large distillation columns. Later, Cutler and Ramaker [1] presented the unconstrained MPC framework in the form of Dynamic Matrix Control. Since then, there have been numerous studies of the MPC algorithm and its applications. A widespread review on MPC design, development and its applications were presented by Christofides et al. [2], Lee [3] and Qin and Badgwell [4]. ∗ Corresponding

Author, Tel: +91-79-23972583, Fax: +91-79-2397 2622 Email addresses: [email protected] (Markana Anilkumar), [email protected] (Nitin Padhiyar ), [email protected] (Kannan Moudgalya) Preprint submitted to Elsevier

January 31, 2016

Page 2 of 21

Ac ce

pt

ed

M

an

us

cr

ip t

Many control problems of industrial applications involve multiple control objectives to be achieved [5],[6]. These objectives are usually conflicting and they need to satisfy simultaneously. Such kind of control problems can be handled by multi objective optimization techniques. When there are two or more control objectives, conventionally they are augmented as one by assigning appropriate weights to the individual control objectives. It is very difficult, for especially large problems, with significant nonlinear and complex dynamics to translate different control requirements into their weights in a single objective. Even if appropriate weights meeting the control requirements are found, one has to make sure that the system does not become ill conditioned. Further, combining different control objectives does not reflect true performance requirements [7]. Thus, a Multi-objective Optimization (MOO) is preferred over augmented single objective optimization formulations to reflect various control objectives with their priorities. Multi-objective optimization and its role in control have been discussed by Chinchuluun and Pardalos [8], Marler [9], Miettinnen [10], Gambier and Badreddin [11]. Solution of the MOO problem results into a number of solution points called the Pareto-optimal set, also known as non-dominated solutions set. However, in real time situation, it is computationally intensive and time consuming to compute the Pareto front. The resolution of a MOO problem does not end when the Pareto-optimal set is found. In real problems, a single solution must be selected. Ideally, this solution must belong to the non-dominated solutions set and must take into account the preferences of a Decision Maker (DM). During the last decade, several methods are proposed by researchers to solve multi objective control problems in MPC framework. In Bemporad and Munoz [12], optimal control sequence corresponds to one of the Pareto optimal solutions and it is selected based on a time-varying, state-dependent decision criterion. Vito and Scattolini [13] solves multi-objective control problem for discrete time, constrained, non-linear systems by resorting to the receding horizon approach. Tlacuahuac et al. [5] proposed nonlinear MPC design for multi-objective control by computing the utopia point and control design was achieved by minimizing the distance of a set of cost functions to its steady-state utopia point. Zavala and Tlacuahuac [14] proposed a utopia-tracking strategy to handle multiple conflicting objectives in model predictive control and also established conditions for asymptotic stability. A new utopia-tracking multi-objective NMPC design for continuous-time, input-affine nonlinear systems with multiple conflicting objectives was proposed by Defeng et al. [15]. Meadowcroft et al. [16] proposed a priority driven frame of controller, known as Modular Multi-variable Controller (MMC). It is based on the solution of a MOO problem using the strategy of lexicographic goal programming, where objectives have different priorities. This solution strategy can handle static optimization, which is not efficient to solve dynamic process control problems. Other attempts to solve multi objective MPC problem were proposed by Zheng et. al. [17],Ocampo-Martinez et al. [18] and Kerrigan and Maciejowski [19], using lexicographic optimization method. This approach solves the MOO problem by solving a series of single objective optimization problems. Here, the optimization problem with the highest priority is solved first. Optimization problem with the next highest priority is solved next with an additional constraint, which helps maintain optimality of the higher priority objective function. This is continued until all the objectives are exhausted. Padhiyar and Bhartiya [20] addressed the problem of spatial profile control of a process output variable for a distributed parameter system under MPC framework. They proposed the use of Lexicographic Optimization based MPC (LOMPC) to explicitly prioritize different parts of the spatial profile. They presented simulation study of multi variable control using LOMPC. In the lexicographic optimization approach, multiple optimization problems are solved in series in the order of high to low priority of the objectives. The conventional lexicographic constraints on the high priority objective function values can be too stringent while solving low priority optimization problems in MPC. As a result, the solution of low priority optimization problems does not add any value in MPC. To overcome this problem, the conventional lexicographic constraints can be modified for its suitability in MPC framework. We study three MPC strategies for prioritized multi-objective control, namely conventional MPC, lexicographic optimization based MPC, and modified lexicographic MPC in this work. The comparative study has been carried out using two applications. The first application is a continuous polymerization reactor for controlling number average molecular weight and reactor temperature. While this is a simulation case study, the second application is an experimental study for the control of a metal chip temperature using a heating element. The structure of the paper is as follows: The next section depicts the problem statement. Section 3 2

Page 3 of 21

illustrates lexicographic MPC and MLMPC algorithms. Section 4 describes the application to a continuous polymerization reactor. Sections 5 and 6 is devoted to modeling and control of experimental setup (single board heating system). Finally, section 7 is devoted to draw conclusions. 2. Process model description and problem formulation

x˙ =

f (x, u, d)

y

g(x, d)

(1) (2)

cr

=

ip t

The process model considered in this work is given by the algebraic differential equations with the following form:

x,u

such that

M

x˙ = f (x, u)

an

us

where, x is a vector of states, u is the vector of manipulated inputs, y is a vector of outputs and d is the vector of unmeasured disturbance variables. The control problem can be stated as: For a given process with multiple and conflicting control objectives, a closed loop control action is seek to meet the priority of control objectives satisfying system constraints in presence of disturbances. Therefore, in general, multi-objective optimization control problem can be formulated as follows: min J(x, u) = (J1 (x, u), J2 (x, u), ..., JN (x, u)) (3)

h(x) ≤ 0 x

low

low

(5)

high

(6)

high

(7)

≤u≤u

ed

u

≤x≤x

(4)

3. Control methodology

pt

where J is a vector of N objective functions. h is a set of process constraints. The superscripts low and high stand for lower and upper bounds.

Ac ce

MPC is an efficient control algorithm in the process industry for many years. MPC methods are based on the idea that at every sampling time, a set of future manipulated variables are selected in order to minimize an appropriate control objective. The objective function includes deviations of predicted outputs from their reference values over a future prediction horizon, separation moves and the cost of input energy over a control horizon. Only the first control move is implemented and the optimization problem is solved again at the next time step. The future outputs are predicted based on an available linear or non-linear dynamic model of the process. Constraints on manipulated and controlled variables, as well as on control moves can easily be incorporated in the MPC configuration. Also, one may not have measurements of all states of the given process. In such case, reconstruction of the state profile can be carried out using state estimation technique, such as Kalman filtering. 3.1. Conventional MPC In this work, conventional MPC formulation is Extended Kalman Filter (EKF) based nonlinear model predictive control [21], [22]. Here, control moves are calculated by minimizing the following objective function at each discrete time k. min J = ∆u

p X

2

We (yk+j|k − rk+j|k ) +

j=1

q−1 X

Wu (∆uk+j|k )2

(8)

j=0

3

Page 4 of 21

such that, following constraints are satisfied. ∆uk+q = ... = ∆uk+p−1 = 0 ulow k+l ≤ uk+l ≤

uhigh k+l ,

(9)

0≤l ≤q−1

≤ ∆uk+l ≤ ∆uhigh k+l , high low yk+l|k ≤ yk+l|k ≤ yk+l|k , ∆ulow k+l

0≤l ≤q−1

ip t

0≤l≤p

us

cr

where, y and r represent the vector of predicted outputs and corresponding set-point over the prediction horizon p, respectively. ∆u represent control moves over control horizon q, which are optimized at every sampling instant. We is a symmetric positive definite penalty matrix on the error between set-points and outputs. Wu is a symmetric positive definite weighing matrix on the rate of change of inputs in which ∆uk+j|k = uk+j|k − uk+j−1|k . Please note that conventionally, in MPC, the optimal control objective is of set-point error minimization, although economic objective functions have also been demonstrated in the literature [23],[24]. 3.2. Lexicographic optimization

Ac ce

pt

ed

M

an

The optimization problem associated with the MPC is often of multi-objective in nature. A common approach to solve MOO problem as a single objective optimization problem is by scalarization. In this approach, different control objectives are added to form a single objective function with corresponding weights to manifest their priority. The relative selection of these weights on control objectives is tedious and requires proper tuning procedures. Also, tuning will no longer be optimal for different set-points and in presence of disturbances. A quite different approach is followed by the lexicographic ordering procedure, where control objectives are ordered to formulate a hierarchy. The first control objective in the hierarchy is the most important one, while the last objective function has the least priority. After ordering, the most important objective function is minimized subject to the original constraints. If this problem has a unique solution, it is the solution of the whole MOO problem. Otherwise, the second most important objective function is minimized. Now, in addition to original constraints, a new lexicographic constraint is added. This new constraint is added to guarantee that the most important objective function is maintained at its optimal value. If this problem has a unique solution, it is the solution of the original problem. Otherwise, the process of solving optimization problem with additional lexicographic constraints is continued till all the control objectives are exhausted. Thus, using the lexicographic ordering approach, different Pareto optimal solutions can be obtained by modifying the hierarchy of the objective functions. Also, tuning and relative selection of weights on control objectives will not be required in this strategy. For a multi-variable set-point tracking control problem, the mathematical formulation is described in the next section. 3.3. Lexicographic MPC

Consider N number of control objectives ( J1 , J2 , . . . , JN ), in the order of priority, where, J1 and JN are the most and least important control objectives respectively.

Ji

=

p X

i Wei (yk+j|k

−

i rk+j|k )2

j=1

i

=

+

q−1 X

Wu (∆uk+j|k )2

(10)

j=0

1, 2, . . . , N

For the lexicographic MPC, following N number of optimization problems are solved in series at every iteration. Please note that the first control move of the optimal manipulated variables obtained in the N th

4

Page 5 of 21

Opt 1:

min

control objective J1

s.t

constraints (9)

min

control objective J2

s.t

constraints (9)

∆u

Opt 2:

∆u

ip t

optimization problem is introduce to the plant.

Opt 3:

min

control objective J3

s.t

constraints (9)

∆u

Ji ≤ Ji∗ for i = 1, 2 . . min

control objective JN

s.t

constraints (9)

∆u

an

Opt N:

us

.

cr

J1 ≤ J1∗

Ji ≤ Ji∗ for i = 1, 2, ..., N − 1

Ac ce

pt

ed

M

where, Ji∗ is the optimal value of ith objective function. Control objectives that typically arise in design of control systems, using MPC, are quadratic norm of deviations of controlled variables from their set-points and quadratic norm of deviations of inputs. If the objective function is strictly convex, then the lexicographic solution ∆u∗ is unique [19]. This implies that, if the ith objective function of lexicographic MPC is strictly convex, then there is no point in solving (i + 1) to N th optimization problems. Thus, the (i + 1)th prioritized objective function value gets fixed after solving ith optimization problem and it can not be further improved in (i + 1)th optimization problem. This puts a limit to the designer that one can assign strictly convex objective function only at the N th position (least priority) in the hierarchy. Note that for positive definite matrices, We and Wu , the Hessian matrix of J1 is also positive definite. Thus, Opt 1 becomes a convex optimization problem and hence the 2nd onwards optimization problems will not change the lexicographic optimization problem solution. Thus, in this view for the set-point tracking problem, lexicographic constraints are not useful. To address this problem, using lexicographic optimization approach in MPC for set-point tracking, lexicographic constraints are modified, which is discussed in the next subsection. 3.4. Modified Lexicographic MPC (MLMPC) To make a provision of improvement in the (i + 1)th optimization problem, a less stringent, but MPC relevant constraint can replace the lexicographic constraint (Ji ≤ Ji∗ ) as follows: ∗ yk+p = yk+p

(11)

This modified constraint ensures that the controlled variable achieves its value that was obtained from the ith optimization problem at steady state, but not necessarily during transient condition. Thus, MLMPC

5

Page 6 of 21

Opt 1:

min

control objective J1

s.t

constraints (9)

min

control objective J2

s.t

constraints (9)

∆u

Opt 2:

∆u

ip t

problem can be summarized as follows,

Opt 3:

min

control objective J3

s.t

constraints (9)

∆u

yik+p = yi∗k+p for i = 1, 2

us

. . . min

control objective JN

s.t

constraints (9)

∆u

an

Opt N:

cr

y1k+p = y1∗k+p

M

yik+p = yi∗k+p for i = 1, 2, . . . , i − 1

Fi

T

2(Su i Wei Su i + ΛT Wu Λ) ˆ k )T Wei Su i + (Λ0 ∆uk−1 )T Wu Λ] = −2[(ri − Sx i x

=

pt

Hi

ed

Solution to the MOO problem discussed above involves N optimization problems to be solved in series at every MPC iteration. Control objectives Ji , (i = 1 to N ) (10) can be represented in the form of standard QP problem as follows, 1 (12) Ji = ∆uT Hi ∆u + ∆uT Fi 2 where, (13) (14)

Ac ce

Here, Sx , Su , Λ and Λ0 are system matrices and are summarized in [22]. ri is the reference trajectory for the ith controlled output variable. Note that since the lexicographic constraint in the lexicographic MPC (section 3.3) is quadratic in nature, resulting optimization problem is Quadratically Constrained Quadratic Programming (QCQP). On the other hand the modified lexicographic constraint (11) is linear, and hence the corresponding optimization problem is Quadratic Programming (QP) in MLMPC. This modified lexicographic equality constraint can be rearranged in linear form for the second optimization problem onwards as follows, Aeq i ∆u

= beq i

for i = 2 to N

(15)

Here, size of the Aeq i and beq i matrices will increase from i = 2 to N to accommodate additional lexicographic linear constraints from subsequent optimization problems. Unlike the stringent lexicographic constraints on the objective function discussed in section 3.3, the modified constraint is less stringent. Therefore it provides a different solution than that obtained from the previous optimization problem. To see if ∆u∗i+1 can be different from ∆u∗i , the mathematical solution of the ith QP can be obtained in terms of Langrage multipliers. For all active constraints represented in terms of Aeq i and beq i , it is straightforward

6

Page 7 of 21

to show the mathematical solution of the ith QP as follows, ∆u∗i λ∗i

= =

T

i T ∗ H−1 i (−Fi − Aeq λi ) i

iT −1

(Aeq Aeq )

i

(16)

Aeq (−Fi −

Hi ∆u∗i )

(17)

an

us

cr

ip t

where, λ∗i is the optimal value of Lagrange multiplier for the ith optimization problem. Since, contributing rows of Aeq i and beq i corresponding to linear lexicographic constraints for (i + 1)th optimization problem are different from that for the ith optimization problem, the solution ∆u∗i+1 is also different from the ∆u∗i . Thus, modified lexicographic constraints provides additional freedom while solving the optimization problem, yet accounting for the priority of higher order optimization problems. Equality constraint in general is more stringent compared to inequality constraint for finding a feasible solution. However, the setpoint error minimization problem (Opt 1) in Lexicographic MPC (section 3.3) and MLMPC (section 3.4) were found to be convex problems. As a result, the lexicographic constraint, J1 ≤ J1∗ in Opt 2 (section 3.3) is found to be too stringent to leave any freedom for solving Opt 2. On the other hand, every modified lexicographic constraint, yik+p = yi∗k+p consumes only one degree of freedom (DOF) contributed from the total number of decision variables in terms of total number of manipulated variables and the controller horizon. Thus, the modified lexicographic constraint in MLMPC is found to be less stringent compared to the lexicographic constraint in Lexicographic MPC. 4. Application to a continuous polymerization reactor

Ac ce

pt

ed

M

In this case study, we consider the CSTR shown in Figure 1, where, free-redical polymerization of methyl methacrylate (MMA) takes place, with azo-bis-isobutyronitrile (AIBN) as initiator and toluene as solvent [28],[29].

Figure 1: Continuous polymerization reactor (PMMA): schematic diagram.

The dynamic behavior of this process is described by mass and energy balance equations and have been

7

Page 8 of 21

us

cr

ip t

reproduced below,      dCm −Ep −Ef m F (Cmin − Cm ) = − Zp exp + Zf m exp Cm P0 (CI , T ) + dt RT RT V   dCI −EI FI CIin − F CI = −ZI exp CI + dt RT V   dT −Ep (−∆HP ) UA F (Tin − T ) = Zp exp Cm P0 (CI , T ) − (T − Tj ) + dt RT ρcp ρcp V V        −ETc −ETd F D0 dD0 −Ef m 2 = 0.5ZTc exp + ZTd exp [P0 (CI , T )] + Zf m exp Cm P0 (CI , T ) − dt RT RT RT V      dD1 −Ep −Ef m F D1 = Mm Zp exp + Zf m exp Cm P0 (CI , T ) − dt RT RT V Fcw UA dTj = (Tw0 − Tj ) + (T − Tj ) dt V0 ρ w cw V 0 D1 (18) M Wn = D0

an

where,

 12
I 2f ∗ CI ZI exp −E RT     =  −ETd −E + ZTc exp RTTc ZTd exp RT 

(19)

M

P0 (CI , T )

Ac ce

pt

ed

where, Cm and CI represent molar concentrations of monomer and initiator, respectively. M Wn is the average molecular weight of the polymer product. D0 and D1 represent the zeroth and first moments of the molecular weight distribution of dead chains, respectively. T and Tj represent reactor and jacket temperatures, respectively. The system parameters, kinetic parameters and steady-state operating conditions are obtained from Daoutidis [28]. Here, controlled variables are number average molecular weight M Wn of the polymer product and reactor temperature T , and two manipulated variables are initiator flow rate FI and coolant flow rate Fcw . Out of the six states, we assume that T , Tj and M Wn are measured and available at the same sampling rate of 2 mins. We estimate the other states using EKF for all control formulations discussed in this case study. Therefore, we consider a nonlinear measurement model [30] as, yk =

h

D1 D0

T

Tj

iT

+ vk

where, yk represent observation and vk represent measurement noise at time instant k. We assume, initial  T state vector, x ˆ0|0 = 5.5 0.133 334.95 0.00198 49.35 297.097 , process and measurement noise with covariance matrix Q = 10 ∗ I6x6 and R = I3x3 , respectively. Initial error covariance matrix is assumed to be P0|0 = 100 ∗ I6x6 . State Estimation trajectories for all the states is shown in Figure 2.

8

Page 9 of 21

5.9

Estimated true

Ci, kmol/m3

5.8

Cm, kmol/m3

0.2

Estimated true

5.7 5.6

0.15

0.1

5.5 1

2

2.5

3

0.05 0

1.5 Time (hour)

50

D1

D0

1

55

2.5

2

45

40

1.5 Estimated 1 0

0.5

true 0.5

1

1.5 Time (hour)

2

2.5

3

35 0

2

cr

3

x 10

1.5 Time (hour)

0.5

1

2.5

3

ip t

0.5 −3

us

5.4 0

1.5 Time (hour)

Estimated true

2

2.5

3

an

Figure 2: State estimation using EKF in PMMA reactor in presence of measurement noise and initial conditions mismatch.

4.1. Conventional MPC

min J

p X (yk+j|k − rk+j|k )T We (yk+j|k − rk+j|k )

=

(20)

j=1

ed

∆u

M

The performance of the proposed control methodology is tested in terms of set-point tracking performance using simulation parameters as listed in Table 1. The objective function for conventional MPC is as follows:

+

q−1 X

∆uTk+j|k Wu ∆uk+j|k

pt

j=0

Ac ce

where, y is a vector of two controlled variables, namely the number average molecular weight of PMMA, M Wn (y1 ) and the reactor temperature, T (y2 ). r is the vector of future reference trajectories for both the controlled variables. Figure 3 shows simulation results for set-point tracking of average molecular weight and reactor temperature, using conventional MPC, with constraints listed in Table 2. As shown in the figure set-point changes are made in average molecular weight and the reactor temperature. The corresponding manipulated variables profiles (FI and Fcw ) are also shown in the Figure 4. All the three set-point values of average molecular weight are achieved. Set-point changes in reactor temperature are not attained from time 2 hours onwards. Note that the polymer process dynamics (18) indicate that positive step change in Fcw or FI results in increasing M Wn . Similarly, positive step change in FI and negative step change in Fcw result in increasing reactor temperature. Thus, for positive step change in the set-point of M Wn at 2 hours, decrease in FI and increase in Fcw are justified. On the other hand, these movements in manipulated variables resulted in increasing set-point error in reactor temperature. This fact can be attributed to the weighing parameters tuning in (20). Though, it cannot be seen, the reactor temperature approaches to the set-point very slowly after the M Wn set-point error is close to zero. In fact, it took approximately 600 hours of the real time for the temperature to reach to its set-point of 332 K. This was confirmed by a separate simulation run (results not shown). Note that this slow tracking of temperature set-point is not practically useful. This problem can be solved by re-tuning weighing parameters in (20). The corresponding simulation results, with tuned weighing parameters, are shown in Figure 3. After increasing weighing parameter for temperature 10 times and decreasing 1000 times for M Wn in (20), all set-points of M Wn and the two temperature set-points from 0 to 6 hours are achieved with large settling time of around 9

Page 10 of 21

2 hours. The above mentioned issue of re-tuning the weighing parameters can partially be resolved with LOMPC approach for the PMMA application discussed in the next subsection.

Value 30 1 [0.016883, 3.26360] Iq×q 2

Table 2: Constraints in MPC for PMMC reactor

Value 0.001 ≤ u1 ≤ 0.018 0.35 ≤ u2 ≤ 3.5 −0.1 ≤ ∆u1 ≤ 0.1 −1 ≤ ∆u2 ≤ 1 24800 ≤ y1 ≤ 35200 275 ≤ y2 ≤ 340

an

us

Constraints on Volumetric flow rates of the initiator FI , m3 .h−1 Volumetric flow rates of the cooling water Fcw , m3 .h−1 Volumetric flow rates of the initiator moves, m3 .h−1 Volumetric flow rates of the cooling water moves, m3 .h−1 Number average molecular weight Reactor temperature, K

cr

Parameters Prediction horizon, p Control horizon, q Initial inputs, u0 Weighting on inputs, Wu Sampling time, Ts (minutes)

ip t

Table 1: Summery of controller parameters for PMMC reactor

M

4.2. Lexicographic MPC

For lexicographic MPC, we consider two control objectives as J1 and J2 , where minimizing J1 is of higher priority than that of J2 . min J ∆u

=

(y1 k+j|k − r1k+j|k )T We (y1 k+j|k − r1k+j|k )

ed

Opt 1:

p X

(21)

j=1

+

q−1 X

∆uTk+j|k Wu ∆uk+j|k

min J

Ac ce

Opt 2:

pt

j=0

∆u

=

p X

(y2 k+j|k − r2k+j|k )T We (y2 k+j|k − r2k+j|k )

(22)

j=1

+

q−1 X

∆uTk+j|k Wu ∆uk+j|k

j=0

where, y1 and y2 is the number average molecular weight, M Wn and the reactor temperature, T , respectively. Figure 3 refers to the closed loop controller performance for set-point tracking using lexicographic MPC. As shown in the figure, we made two set-point changes in M Wn , one from 25000 to 30000 at 2 hours and other from 30000 to 35000 at 6 hours. We also made set-point change in the reactor temperature from 335 K to 336 K at 4 hours. As can be seen, while all three set-points of M Wn are achieved, the reactor temperature did not achieve set point values during 2 to 8 hours in spite of the fact that manipulated variables are not at their limits. Also, the controller output from Opt 2 is found to be same as that from Opt 1 at every sampling instant and therefore there is no improvement in the solution because of solving opt 2. This can be attributed to lexicographic constraints (10) discussed in section 3.3. Thus, both the manipulated variables adjusted their values such that opt 1 priority is met, ensuring in no compromise even during transient conditions for 10

Page 11 of 21

M Wn . This resulted in poorer set point tracking response in reactor temperature not only during transient conditions, but also at steady state. This issue is taken care by MLMPC as discussed in the next subsection. 4.3. Modified Lexicographic MPC (MLMPC)

an

us

cr

ip t

Figure 3 refers to the closed loop controller performance for set-point tracking using MLMPC. As shown in the figure, set-point changes made in both controlled outputs are achieved without attempting for tuning of weighing parameters, Wu and We . A minor transient deviation in M Wn from their set-points is observed in this case compared to the tuned conventional MPC. This deviation is observed because of the lexicographic constraint (11) applied at the end of prediction horizon, i.e at steady state. Thus, compromising in the transient response of the M Wn for set-point tracking, the set-points for both the output variables are attained. Moreover, the modified lexicographic constraint on y1 at only prediction horizon resulted in marginal poorer transient setpoint tracking performance of y1 as can be seen in the figure. As discussed earlier, Lexicographic MPC and MLMPC requires additional optimization problem to be solved at every sampling instant. Therefore it is also important to know the extra computational efforts required in terms of the time required to solve an extra optimization problem. Note that the optimization problem is formulated as a QCQP in Lexicographic MPC and as a QP in MPC and MLMPC. Computational time required to complete one MPC iteration was observed to be 0.1630 second while it was 0.2808 and 0.2108 seconds for Lexicographic MPC and MLMPC, respectively. Thus, the total computational time was found to be well within the sampling time of 120 seconds for each of the MPC strategies. 4

x 10

3.4

MW

n

3.2

Conventional MPC(tuned) Conventional MPC LOMPC Modified LOMPC setpoint

M

3.6

3

2.6 2.4 0

1

2

337 336

334 333

331 0

4

5

6

7

8

4 Time (hour)

5

6

7

8

Conventional MPC(tuned) Conventional MPC LOMPC Modified LOMPC setpoint

Ac ce

332

3

pt

T,K

335

ed

2.8

1

2

3

Figure 3: Closed loop controller performance for set-point tracking using conventional, lexicographic MPC and MLMPC. We = diag(0.001, 10) for conventional MPC. Tuning of weighting parameters is not required in MLMPC.

11

Page 12 of 21

0.02

3

FI, m /h

0.015

0.005 4

0

Conventional MPC(tuned) Conventional MPC LOMPC Modified LOMPC 1

2

3

4

5

3

4 Time (hour)

5

6

0 0

Conventional MPC(tuned) Conventional MPC LOMPC Modified LOMPC 1

8

cr

2 1

7

2

us

cw

3

F , m /h

3

ip t

0.01

6

7

8

an

Figure 4: Input profiles corresponding to Figure 3

4.4. Disturbance Rejection Performance

pt

ed

M

The performance of all the three MPC control formulations were tested in terms of step changes at the two disturbance variables. The process was initially brought to steady state. At time t = 2 hours a step change at the inlet monomer concentration Cmin was applied, from 6 to 5 kmol m3 and at time t = 5 hours a step change at the inlet temperature Tin was applied from 350 to 347 K. Figure 5 and 6 illustrate the profiles of the two controlled outputs and two manipulated inputs. Disturbance rejection performance is evaluated by calculating ITAE and ITSE performance criteria for all MPC formulations, listed in Table 3. It can be seen that for a temperature variable, both ITAE and ITSE indices are the lowest by MLMPC compared to other MPC formulations. For number average molecular weight, both performance indices, using MLMPC, are also lower compared to the conventional tuned MPC, but are marginally higher when compared to LOMPC. Thus, LOMPC has graceful results for variable M Wn but equally worst results for T .On the other hand, MLMPC maintains better average disturbance rejection performance. Table 3: Disturbance Rejection Performance

Ac ce

Performance Criteria Setpoint Variable Conventional MPC (tuned) LOMPC MLMPC

ITAE(%) M Wn T 1.3352 0.2675 0.7485 4.2144 1.0439 0.0870

ITSE(%) M Wn T 1.3686 0.1054 0.6175 3.5612 1.1810 0.0160

12

Page 13 of 21

4

2.56

x 10

Conventional MPC(tuned) LOMPC Modified LOMPC setpoint

2.52

MW

n

2.54

2.5

2.46 0 337

1

2

3

Conventional MPC(tuned) LOMPC Modified LOMPC setpoint 1

2

3

4

ip t

2.48

5

6

336

cr

334 333 332 331 0

4 Time (hour)

us

T,K

335

7

5

6

7

an

Figure 5: Closed loop controller performance for disturbance rejection. At time t = 2 hours a step change at the inlet monomer and at time t = 5 hours a step change at the inlet temperature Tin was concentration Cmin was applied, from 6 to 5 kmol m3 applied from 350 to 347 K.

−3

x 10

M

20

F , m3/h

15

0 −5 0 4

Conventional MPC(tuned) LOMPC Modified LOMPC 1

3 3

Fcw, m /h

2

ed

5

Ac ce

2 1

0 0

3

4

5

6

7

4

5

6

7

pt

I

10

Conventional MPC(tuned) LOMPC Modified LOMPC 1

2

3 Time (hour)

Figure 6: Input profiles corresponding to Figure 5

.

5. Application to temperature control of Single Board Heater System (SBHS) The SBHS setup, shown in Figure 7, is a well known experimental setup for academicians and researchers of control community [26]. SBHS, designed and developed at IIT Bombay, consists of a heater, fan, temperature sensor for a metal plate, micro-controller for data communication, digital display, instrumentation amplifier and an associated circuitry. Temperature of a thin metal plate can be manipulated by passing current through a heating coil and changing the speed of fan available near the thin metal plate. Amount of power delivered to both, heater and fan can be controlled by passing a command through serial port communication to the system. Detailed description of the hardware of this process is available in Arora et al. [26]. 13

Page 14 of 21

ip t cr us

an

Figure 7: Single board heater system.

y(s) =

M

Open loop SBHS model is identified using experimental step response data. Here, manipulated variables are heater current and fan speed, while temperature being the measured and controlled variable. Note that in the earlier study [27], fan speed is treated as a disturbance variable while it is used as a manipulated variable in this work. Second order transfer function model with respect to both inputs is considered. K2 K1 u1 (s) + u2 (s) (τ1 s + 1)(τ2 s + 1) (τ3 s + 1)(τ4 s + 1)

(23)

ed

Least square fit of step response data is applied to identify the model parameters. These are listed in Table 4. The transfer function model is then converted in to minimal discrete state space model with sampling Table 4: SBHS Model Parameters

Ac ce

time, Ts = 1 sec, as follows:

τ1 63.088

τ2 0.8790

pt

K1 1.2582



xk+1 yk

K2 -0.1237

τ3 37.7112

τ4 0.069

= Φxk + Γuk = Cd xk + Dd uk   0 0.2958   0.022 0  , Γ=  −0.05174  0 0.9756 0

(24)

 0.3112 −0.08535 0 0  0.07395  0.9937 0 0  where, Φ=  0 0 −0.00179 0.01693  0 0
0.03385
0.00797 Cd = 0 0.363 0 −0.3781 , Dd = 0 Above discrete state space model of SBHS process is used in all MPC formulations discussed in Section 3. 6. Experimental study of SBHS In this section, we carry out the closed loop experimental study of conventional, lexicographic MPC and MLMPC strategies discussed in section 3. We compare the set-point tracking performance of these control strategies when applied to the SBHS setup. Controller parameters used in this study are listed in 14

Page 15 of 21

Table 5. ‘quadprog’ and ‘fmincon’ in MATLAB are used for solving QP and QCQP problems in this work. Further, we ensured that the necessary and sufficient conditions for optimality, KKT conditions are met for all optimization problems in this study.

Value 400 1 Ip×p Iq×q 30 ∗ Iq×q 1

Value 0 ≤ u1 ≤ 40 50 ≤ u2 ≤ 250 −2 ≤ ∆u1 ≤ 2 −10 ≤ ∆u2 ≤ 10 12 ≤ y ≤ 70

M

an

Constraints on Heater input (PWM units) Fan speed input (PWM units) Heater input moves (PWM units) Fan speed input moves (PWM units) Output temperature ( ◦ C)

us

Table 6: Constraints in MPC for SBHS

cr

Parameters Prediction horizon, p Control horizon, q Weighing on error, We Weighing on input moves, Wu Weighing on inputs, Wi Sampling time, Ts (sec)

ip t

Table 5: Summery of controller parameters for SBHS

6.1. Conventional MPC

For the conventional MPC, we consider the following augmented objective function to be minimized. Here,∆u is the change in the controller move and u is the input cost to be minimized.

u

=

ed

min J

p X (yk+j|k − rk+j|k )T We (yk+j|k − rk+j|k ) +

(25)

j=1

∆uTk+j|k Wu ∆uk+j|k +

pt

q−1 X j=0

Ac ce

q−1 X

uTk+j|k Wi uk+j|k

j=0

Figure 8 shows experimental results of SBHS system, when controlled using the conventional MPC. Controller parameters and constraints are listed in Table 5 and Table 6, respectively. We have carried out experimental study for controlling temperature with five set-point changes. In fact, none of five set-points are attained, even though manipulated variables were not at their limit values. This is because of the third term in the objective function, namely that of the input cost. The weight of this term dominates in the controller objective function. Thus, the priority of the set-point tracking is not reflected in weight matrices in (8). To verify this hypothesis, we carried out another experimental run with new weighing parameters by relatively increasing the weight of set-point tracking term. With this new weighing parameters, all five set-points of temperature are achieved by appropriate control moves. Though, with this small problem, it is not very difficult to tune controller weights, but it is not trivial when there are more objectives. It will be cumbersome to find relative weighing parameters of the MOO problem when the number of objective function increases. Also, these parameters are sensitive to operating conditions and disturbances. Further, it is not guaranteed that tuned weighing parameters are optimal. To account for prioritized control, we present lexicographic MPC and MLMPC in next sections, wherein tuning of weighing factors is not required.

15

Page 16 of 21

400

600

800

1000

1200

1400

without tuning after tuning

30 20 10 0 0

200

400

600

800

200

400

600

800

1000

100 50 0 0

1200

an

Fan speed PWM units

200

cr

Heater input PWM units

0 0

setpoint without tuning after tuning 1600 1800 2000

ip t

20

us

Temperature °C

40

1000 1200 Time (sec)

1400

1600 1800 2000 without tuning after tuning

1400

1600

1800

2000

M

Figure 8: Experimental study: Closed loop controller performance for set-point tracking using conventional MPC.

6.2. Lexicographic MPC

ed

For the lexicographic MPC strategy, we consider two objective functions J1 and J2 , where, J1 has higher priority compared to J2 . Please note that J1 includes one term, that is set-point error minimization. Further, the second prioritized objective function, J2 is same as J, described in section 6.1. Thus, the following two optimization problems are solved in series: p X

pt

Opt 1:

min J1 = u

(yk+j|k − rk+j|k )T We (yk+j|k − rk+j|k )

(26)

j=1

Ac ce

such that constraints in (9) are satisfied.

Opt 2:

min J2 u

=

p X

(yk+j|k − rk+j|k )T We (yk+j|k − rk+j|k )

(27)

j=1 q−1 X

∆uTk+j|k Wu ∆uk+j|k +

j=0 q−1 X

uTk+j|k Wi uk+j|k

j=0

such that constraints in (9) and additional J1 ≤ J1∗ are satisfied. Here, J1∗ is the optimal value of J1 after solving the first optimization problem. Please note that Opt 1 is a QP problem, while Opt 2 is a QCQP problem because of lexicographic constraints, J1 ≤ J1∗ . Experimental results with lexicographic MPC are shown in Figure 9 with same controller parameters and constraint values listed in Table 5 and Table 6, respectively. Note that we solve a series of two optimization 16

Page 17 of 21

cr

ip t

problems instead of one at every iteration as discussed in section 3.3. As can be observed from Figure 9 that all temperature set-points are attained. Further, it is also observed that optimal values of manipulated variables and J1 obtained after solving Opt 1 do not change after solving Opt 2 at every sampling instant(results not shown). This can be because of the stringent constraint, J1 ≤ J1∗ , in Opt 2. Thus, after solving the first optimization problem, there is no freedom left in manipulated variables while solving the second optimization problem. Moreover, for a negative set-point change from 40 ◦ C to 30 ◦ C, the heater current is expected to decrease while fan speed is expected to increase. Instead, as can be observed , both fan speed and heater current increases. Thus, the effect of the third term, which is available in J2 is not reflected even though we solve the second optimization problem because of the constraint J1 ≤ J1∗ . To facilitate extra freedom of manipulated variables, the lexicographic constraint (J1 ≤ J1∗ ) is replaced ∗ by yk+p = yk+p in MLMPC. The corresponding results of MLMPC are presented in next section. 6.3. Modified Lexicographic MPC (MLMPC)

Ac ce

pt

ed

M

an

us

Experimental results with the MLMPC are presented in Figure 9. As can be observed that the set∗ point tracking is achieved as expected with the modified lexicographic constraint, yk+p = yk+p . With this modification, even after all set-points are achieved, input efforts are also minimized and hence, at the lowest possible fan speed and heater current values, all temperature set-points are attained. Also, it is observed from Figure 10 that in the MLMPC, optimization index for Opt 2 problem is less compared to that of in lexicographic MPC. Though, there is a marginal compromise in terms of settling time before the setpoint is achieved compared to lexicographic MPC. Thus, MLMPC achieves the set-point (highest prioritized objective function) at the cost of larger settling time. Since the current case study refers to the real time implementations of control strategies, it is also important to compute the computational time required for solving an additional optimization problem in Lexicographic and MLMPC strategies. The computational time for one MPC iteration was found to be 0.0160 second while it was 0.0470 and 0.0310 seconds for Lexicographic MPC and MLMPC, respectively. Considering the sampling time of 1 second, computational time was found to be quite affordable with each of the three MPC strategies.

17

Page 18 of 21

0 0 40

200

600

800

1000

1200

800

1000

1200

1400

20

200

400

600

Lexicographic MPC MLMPC

200

0 0

200

400

600

800

an

100

1000 1200 Time (sec)

1600

cr

Lexicographic MPC MLMPC

0 0

Fan speed PWM units

400

ip t

setpoint Lexicographic MPC MLMPC

20

1800

2000

1400

1600

1800

2000

1400

1600

1800

2000

us

Temperature °C Heater input PWM units

40

ed

M

Figure 9: Experimental study: Closed loop controller performance for set-point tracking using lexicographic MPC and MLMPC.

4

x 10

J1

5

LOMPC Modified LOMPC

pt

10

Ac ce

0 −5 0 200 6 x 10 2

J2

1.5

400

600

800

1000

1200

1400

1600

1800

2000

1000 1200 Time (sec)

1400

1600

1800

2000

LOMPC Modified LOMPC

1

0.5 0 0

200

400

600

800

Figure 10: Optimization index values from Opt 1 and Opt 2 problem for lexicographic MPC and MLMPC.

18

Page 19 of 21

7. Conclusions

Ac ce

pt

ed

M

an

us

cr

ip t

Process control with prioritization in objectives has been attempted in this work using lexicographic optimization based MPC and its modified version. Tracking performance of this control strategy is compared with the conventional MPC by considering two engineering applications. The first application is of simulation study for the control of the number average molecular weight of Poly methyl methacrylate (PMMA) and the reactor temperature in a continuous reactor. The second application is of temperature control in the Single Board Heating System (SBHS) using fan speed and heater current. We compare the set-point tracking and disturbance rejection performance of conventional MPC, lexicographic MPC and MLMPC for various number average molecular weight and reactor temperature set-points by simulation in a PMMA reactor. It has been observed that the conventional MPC results in offset, while tracking all set-points of reactor temperature. It requires proper tuning of weights in order to achieve better tracking performance. Lexicographic MPC results in to similar kind of tracking performance to that of conventional MPC. On the other hand, MLMPC gives better tracking performance without requiring tuning procedure and is able to track all set-points with zero offset in the reactor temperature as well as the number average molecular weight. In the second case study, step response based second order process model of the single board heating process has been developed and the same has been validated by open loop experimental data. We compare the set point tracking performance for various temperature set-points by an experimental study in SBHS setup. We observe similar kind of tracking performance of these controllers for controlling the temperature in SBHS setup. Though, the set-point on temperature can be achieved with different controller weighing parameters’ values, it requires tuning of these controller parameters. Lexicographic MPC is able to achieve the highest priority control objective of tracking desired temperature, but requires higher controller efforts when negative step changes in set-points are to be achieved. On the other hand, MLMPC gives better tracking performance and is able to track all temperature set-points with zero offset and at the same time with minimum possible fan speed and heater current values. Though, conventional MPC with weighing parameter tuning can provide the control with priority similar to MLMPC, the tuning will no longer be optimal for different set-points and disturbance. Also, for multiple objectives, appropriate and relative selection of weights would be much cumbersome. Thus, MLMPC provides better control performance when we have multi control criteria with different priority. But, considering QP formulations of these N optimization problems can provide the solution with high computational efficiency. The extra computational time of 0.015 second for solving an additional QP in SBHS process is quite affordable with the sampling time of 1 second. Similarly, for PMMA system, the additional computational time for solving a QP problem was found to be 0.048 second, which is significantly less than the sampling time of 2 Min. References

[1] Cutler, C. R. and Ramaker, B. L. (1980). Dynamic Matrix Control- a computer control algorithm, Proceedings of the joint Automatic Control Conference, San Francisco, CA, Paper WP5-B, 1980. [2] Christofides, P. D., Scattolini, R., Munoz de la Pena, D. and Liu, J. (2013). Distributed model predictive control: A tutorial review and future research directions, Computers and Chemical Engineering, volume 51, 21–41. [3] Lee, J. (2011). Model predictive control: Review of the three decades of development, International Journal of Control, Automation and Systems, volume 9, 415–424. [4] Qin, S. J. and Badgwell, T. A. (2003). A survey of industrial model predictive control, technology. Control Engineering Practice, volume 11, 733–764. [5] Tlacuahuac, A. F., Morales, P. and Toledo, M. R. (2012). Multi-objective Nonlinear Model Predictive Control of a class of chemical reactors, Ind. Eng.Chem. Res., volume 51, 5891–5899. [6] Wojsznis, W., Mehta, A., Wojsznis, P., Thiele, D. and Blevins, T. (2007). Multi-objective optimization for model predictive control, ISA Trans., volume 46, 351–361. [7] Prett, D. M and Garcia, C. E. (1988). Fundamental Process Control. Boston: Butterworths. [8] Chinchuluun, A. and Pardalos, P. M. (2007). A Survey of recent developments in multiobjective optimization, Ann Oper Res, volume 154, 29–50. [9] Marler, R. T. and Arora, J. S. (2004). Survey of multi-objective optimization methods for engineering, Struct Multidisc Optim, volume 26, 369–395.

19

Page 20 of 21

Ac ce

pt

ed

M

an

us

cr

ip t

[10] Miettinnen, K. (1999). Nonlinear Multiobjective Optimization. Kluwer Academic: Boston, MA. [11] Gambier, A. and Badreddin, E. (2007) Multi-objective optimal control: An overview, in 16th IEEE International Conference on Control Applications- Part of IEEE Multi-conference on Systems and Control. Singapore, October 2007 [12] Bemporad, A. and Munoz de la Pena, D. (2009). Multiobjective model predictive control, Automatica, volume 45, 2823–2830. [13] Vito, D. D. and Scattolini, R. (2007). A receding horizon approach to the multiobjective control problem, In Proceedings of the 46th IEEE Conference on Decision and Control, New Orleans, LA, USA, 6029–6034. [14] Zavala, V. M. and Tlacuahuac, A. F.; (2012). Stability of multiobjective predictive control: A utopia-tracking approach, Automatica, volume 48, 2627–2632. [15] Defeng, H., Wang, L. and Li, Y. (2015). Multi-objective Nonlinear Model Predictive Control of process systems: A dual-mode tracking control approach, Journal of Process Control, volume 25, 142–151. [16] Meadowcroft, T. A., Stephanopoulos, G. and Brosilow, C. (1992). The Modular Multivariable Controller: I : Steady state properties, AIChE Journal, volume 38, 1254–1278. [17] Zheng, T., Gang, W., Liu, G. H. and Ling, Q.(2010). Multi-Objective Nonlinear Model Predictive Control: Lexicographic Method, Model Predictive Control, Tao Zheng (Ed.), ISBN: 978-953-307-102-2, [18] Ocampo-Martinez, O., Ingimundarson, A., Puig, V. and Quevedo, J. (2008). Objective prioritization using lexicographic minimizers for MPC of sewer networks, IEEE Transactions on Control System Technology, volume 16, 113–121. [19] Kerrigan, E. C. and Maciejowski, J. M. (2002). Designing model predictive controllers with prioritised constraints and objectives, Proceedings of IEEE international symposium on computer aided control system design, Glasgow, Scotland,U.K., 33–38. [20] Padhiyar, N. and Bhartiya, S. (2009). Profile control in distributed parameter systems using lexicographic optimization based MPC, Journal of Process Control, volume 19, 100–109. [21] Padhiyar, N., Gupta, A., Gautam, A., Bhartiya, S., Doyle,F, J., Das, S. and Gaikwad, S.(2006). Nonlinear inferential multi-rate control of Kappa number at multiple locations in a continuous pulp digester, Journal of Process Control , volume 16, 1037–1053. [22] Lee, J, H. and Ricker, N, L. (1994). Extended Kalman Filter based nonlinear model predictive control, Ind. Eng. Chem. Res, volume 33 (6), 1530–1541. [23] Angeli, D. (2014). Economic Model Predictive Control, Encyclopedia of Systems and Control, Springer London, 1–9. [24] Tran, T., Ling, K-V. and Maciejowski, J. M. (2014) Economic Model Predictive Control - A Review, in 31st International Symposium on Automation and Robotics in Construction and Mining (ISARC), 2014. [25] Jun, J., Chiu, M. and Tien, C. (2000). Multiple objective based model predictive control of pulse jet fabric filters, Trans IChemE, volume 78, Part A, 581–588. [26] Arora, I., Moudgalya, K. and Malewar, S. (2010) A low cost, open source, single-board heater system, in International Conference on E-Learning in Industrial Electronics. AZ, USA: IEEE, November 2010. [27] Moudgalya, K. (2009). Identification of transfer function of a single board heater system through step response experiments, IIT Bombay. [28] Daoutidis, P., Soroush, M. and Kravaris, C. (1990). Feedforward/Feedback control of multivariable nonlinear processes, AIChE Journal, volume 36, 1471–1484. [29] Padhiyar, N., Bhartiya, S. and Gudi, R. (2006). Optimal grade transition in polymerization reactors: a comparative case study, Ind. Eng. Chem. Res, volume 45, 3583–3592. [30] Kottakki, K. K., Bhartiya, S. and Bhushan, M. (2014). State estimation of nonlinear dynamical systems using nonlinear update based Unscented Gaussian Sum Filter. Journal of Process Control, volume 24, 1425–1443.

20

Page 21 of 21