Robust tuning for classical MPC through the Multi-Scenarios Approach

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Process Systems Engineering

Robust tuning for classical MPC through the Multi-Scenarios Approach José Eduardo Weber dos Santos, Jorge Otávio Trierweiler, and Marcelo Farenzena Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.8b05485 • Publication Date (Web): 20 Jan 2019 Downloaded from http://pubs.acs.org on January 20, 2019

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Industrial & Engineering Chemistry Research

Robust tuning for classical MPC through the Multi-Scenarios Approach José Eduardo W. Santos*, Jorge Otávio Trierweiler, and Marcelo Farenzena Group of Intensification, Modeling, Simulation, Control, and Optimization of Processes (GIMSCOP). Chemical Engineering Department, Federal University of Rio Grande do Sul (UFRGS), R. Eng. Luiz Englert, s/n. Campus Central, Porto Alegre, Rio Grande do Sul 90040-040, Brazil

ABSTRACT: In the literature, the available techniques for MPC tuning usually consider a specific operating point (OP), while in real plants, controllers should be robust in a wide operating region facing different plant behaviors that arise due to disturbances, saturations, and nonlinearities. In this work, a method for MPC tuning proposed in our previous work is extended for a robust tuning for classical (square) MPCs. This technique applies to any predictive control algorithm, and it considers multi-scenarios based on the closed-loop attainable performance of the system. The sequential procedure is applied, where initially the attainable performance for each scenario (herein, different operating points (OPs) are used) is determined, and an estimate of the closed-loop potential is computed. In the end, the optimum scaling for the model and the MPC tuning parameters are calculated, solving an optimization problem that uses the attainable trajectories for each scenario as a reference. The selection of the controller’s process model is also determined according to constraints of the attainable performance determination. This robust and constrained strategy is applied to the

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controller design of two nonlinear systems: (i) the Quadruple-Spherical Tank (QST) system and (ii) a CSTR reactor with separation column and recycle. The proposed approach was successful to tune the MPC capable of working with all proposed OPs, including those with critical operational conditions for the process controllability, such as non-minimum phase dynamics and model-plant mismatches.

1.

INTRODUCTION

Model Predictive Control (MPC) is one of the most used advanced control strategies that shows success in industrial applications mainly due to its capacity to work with input saturation in a multivariable system with more controlled variables (CVs) than manipulated variables (MVs)1,2. Besides that, the formulation of the control problem, the explicit use of the model to predict the outputs and the ability to handle input and output constraints in the optimization problem make the implementation easier and provide a reliable measure of economical performance3. As part of a multilevel hierarchy of control functions4, the MPC is responsible for leading the plant to the best operational condition, minimizing the dynamic error along the path. The tuning of this type of controller is responsible for ensuring that the controlled variables will achieve the reference values without exceeding manipulated efforts. However, due to a large number of parameters available in the predictive control algorithms, that differ from one to another, and the influence of each parameter in the MPC cost function, basic methodologies for tuning cannot succeed to provide a good tuning. Therefore, there is a gap between the theoretical and actual MPC application5. Garriga & Soroush6 divide the methodologies of tuning for MPCs into two groups: (i) explicit formulations for the tuning parameters based on approximation/simulation of the process model or (ii) bounds (maximal or minimal values) on where the tuning parameters lie based on the process dynamics. Among the formulations, i.e., analytical expressions, for


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tuning for the MPC parameters Hinde Jr. & Cooper7, Banerjee & Shah8 and Shridhar & Cooper9 compute these parameters based on an approximation of the process model by a first order plus dead time (FOPDT) function. These and other methodologies10,11 provide good guidelines for tuning, but they focus on single and accurate plant models, which are implemented in a specific MPC algorithm, that restricts its industrial application even further. Besides, as the robust tuning strategies are not straightforward they are easily replaced by trial and error heuristics. As a way to understand the behavior of MPC algorithms, which differs from GPC and DMC controllers, tuning strategies based on optimization problems were developed. These multi-objective techniques, as developed by Gous and de Vaal12, Van der Lee et al.13, Suzuki et al.14, and Nery Júnior et al.15, solved through nonlinear algorithms, are used for tuning of countless particular parameters combining properties of the closed-loop system (overshoot rates, settling and rise-times, steady-state errors, etc.) evaluating desired performance indexes. These and other methodologies16,17 propose tuning based on nonlinear optimization problems aiming to obtain the controller parameters. These available methods, when applied to industrial controllers, can present a large computational demand and nonconvex problems leading to a poor closed-loop performance since the closed-loop performance potential is unknown. A more reliable tuning approach can be achieved if the real potential of the closed-loop system is available considering a set of representative dynamic models of the different OPs, including models with performance limitations like those imposed by RHP transmission zeros. Based on these assumptions, a method for MPC tuning for zone region tracking and non-square systems proposed by Santos et al.18 is adapted in this work for robust tuning of control problems with the same number of MVs and CVs. This methodology is based on the attainable performance of the system, considering a set of operating points (scenarios) and is


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divided into three parts: (1) determination of the best attainable performance for the closedloop process for the set of scenarios, (2) scaling of the plant through the global minimized condition number of the linearized scenarios, and (3) controller weights tuning. The determination of the best process model used in the controller is also settled and justified. The paper is structured as follows: in Section 2 are presented the basic concepts necessary for understanding the proposed methodology. In Section 3 the QST process is described and the tuning procedure previously stated is employed. In Section 4 the CSTR with separation column and recycle system together with the representative scenarios is used for robust controller design. In Section 5, the impact of the three-step tuning and the selection of the controller model are evaluated, and at the end, in Section 6, the conclusions are drawn.

2.

MPC TUNING STRATEGY BASED ON MULTI-SCENARIOS FOR A ROBUST OPERATION

The methodology18 proposed for MPC tuning is based on representative scenarios of the non-square process model operating in ranges. The scenarios originally proposed were defined as a square association of all combinations of inputs and outputs of the system. In this work, in order to provide a robust tuning approach, i.e., a tuning approach capable of handling the influence of OPs, disturbances and model-plant mismatches, the scenarios are considered as different OPs of the process. Moreover, the methodology proposed here will be employed in classical MPCs with the same number of inputs and outputs (square systems) and set point tracking. Initially, the MPC tuning procedure consists of obtaining more than one OP (𝑘) for the process, where 𝑘 = 1…𝑛𝑐 is the number of scenarios in which the controller will be robustly tuned. These scenarios should be selected based on plant dynamics, giving preference to those conditions that cause performance limitations that must be handled by proper tuning.


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Thus, the first step is to estimate the best achievable performance for the closed-loop system based on each OP (𝑘). It could be obtained through Blaschke factorization19 (Eq. 1), that is based in an unique complementary sensitivity function 𝑇𝑑(𝑠) (Eq. 2). † 𝑇𝑘(𝑠) = 𝐵𝑂,𝑧(𝑠)𝐵𝑂,𝑧 (0)𝑇𝑑(𝑠)

𝑇𝑑(𝑠) ≜

(1)

1 ― 𝜀∞ 2

( ) 𝑠 𝜔𝑛

⟺𝑇𝑑(𝑟𝑡,𝑀𝑝[%],𝜀∞) 2𝜁𝑠 + +1 𝜔𝑛

(2)

† where 𝐵𝑂,𝑧(𝑠) is the zero output Blaschke factorization, 𝐵𝑂,𝑧 (0) is the pseudoinverse of 𝐵𝑂,𝑧

(𝑠), 𝑟𝑡 is the rise time, 𝑀𝑝 is the maximal overshoot and 𝜀∞ is the steady-state error. This unique 𝑇𝑑(𝑠), relates the output with the reference signal and, is estimated through an optimization problem that minimizes the rise times of each channel of this transfer matrix considering it as a second order function with 5% of overshoot ratio and no allowed offset constrained by robustness metrics related to each scenario (𝑘): relative Robust Performance Number (𝑟𝑅𝑃𝑁), Maximal Sensitivity Function (𝑀𝑠) and Maximal amplitude in control actions (𝑀𝑘) (Eq. 3). 𝑛

min 𝜙(Ω) = min Ω

Ω

∑(𝑟𝑡 ) 𝑖

2

𝑖=1

s.t. 𝑟𝑅𝑃𝑁(𝑘) ≤ 1.0

(3)

𝑀𝑠(𝑘) ≤ 2.2 𝑀𝑠(𝑘) ≥ 1.2 𝑀𝑘(𝑘) ≤ 10 𝑇𝑑(𝑠) = 𝑑𝑖𝑎𝑔[𝑇𝑑,𝑖(𝑠)]

(4)

where Ω = [𝑟𝑡1,𝑟𝑡2,…,𝑟𝑡𝑛], being 𝑛 the number of outputs.


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The attainable performance function, 𝑇𝑘 (𝑠), which computes the limitations of each OP, will be used as achievable reference trajectories in the final tuning of the parameters. The scaling procedure is an important step of the tuning problem, because it provides the matrices that are capable of handling the different magnitudes present in the input and output variables through the well-conditioned system. In this manner, the system scaling matrices are calculated through the global minimized condition number (Eq.5), and they will be used to scale the controller’s process model and the plant (Eq. 6). 𝛾∎ ≝ min ‖𝛾𝑘(𝐿𝑠𝐺𝑘(𝑗𝜔𝑀𝐴𝑋,𝑘)𝑅𝑠)‖∞

(5)

𝐺𝑠 = 𝐿𝑠𝐺(𝑠)𝑅𝑠

(6)

𝑅𝑠,𝐿𝑠

where 𝐺𝑘(𝑠) are the unscaled process model, 𝐿𝑠 and 𝑅𝑠 are the diagonal scaling matrices for the outputs and inputs, respectively – used for the controller design, 𝛾𝑘(𝑀) is the condition number of the matrix 𝑀, and 𝑘 = 1…𝑛𝑐 is the number of scenarios. The value of the 𝛾𝑘(𝑀) represents the ratio between the maximal and minimal singular value of the model system 𝑀. This calculation allows the determination of the best scaling matrices considering all scenarios. The last step is the tuning of the parameters which will penalize the error of the outputs and the move suppression of the control actions. These parameters are obtained considering an optimization problem that minimizes the difference between the attainable trajectory estimated for each scenario (𝑇𝑘(𝑠)) and the simulated outputs (Eq. 7). 𝑛𝑐

min 𝜓(Θ) = min Θ

Θ

∑ ‖𝑇 (𝑠)Δ𝑦 𝑘

𝑘=1

𝑠𝑒𝑡

Δ𝑦

2

‖

― (𝑦 ― 𝑦𝑏𝑖𝑎𝑠)

(7)

where Θ are the tuning parameters for a given MPC implementation. By having 𝑘 scenarios that will be handled in the robust tuning approach, the question that arises is: Which model will be used in the controller design? The constraints in the calculation of the attainable performance function (Eq. 3) will be active in the model that


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presents most of the performance limitations for the process. This model will be used in the controller design because of these features and its ability to handle the worst operational case, and consequently the others. Figure 1 summarizes the proposed tuning procedure.

Figure 1. Multi-Scenarios tuning procedure.

In Figure 1 the 𝑂𝑃(𝑘) corresponds to each model 𝐺𝑘.that will be used in the constraints for desirable transfer function determination. The model used for the controller design will be the one which makes at least one of these constraints active. This tuning technique can be applied to any MPC algorithm because the number of tuning parameters can be adjusted according to the number of particular parameters provided by different controller cost functions. Considering more than one scenario that the problem occurs or that are critical for the operation, together with robust operation metrics as maximal


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sensitivity function (𝑀𝑠), relative Robust Performance Number (𝑟𝑅𝑃𝑁)20 and, control actions amplitude (𝑀𝑘), the controller will robustly handle, through optimal tuning, the influence of these critical issues in the optimal operation of the system. Besides that, the scaling procedure will consider all the input and output variables with the same size, avoiding conservative values for the controller weights. For more details see Santos et al.18 . The choice of the controller’s process model based on that which makes at least one of the constraints active will be responsible for the stability of the process, since it represents the worst operational condition among the scenarios. Besides, the determination of the controller parameters will guarantee that the process is running close to the best attainable trajectories that the OPs can provide. 3.

CASE STUDY – THE QUADRUPLE-SPHERICAL TANK SYSTEM

As a case study, a nonlinear QST system is proposed, whose schematics is illustrated in Fig. 2.


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(1-x2).F2 (1-x1).F1

h3

h4

x1.F1

x2.F2

h1

h2

F1

F2

Figure 2. Quadruple-spherical tanks system scheme.

This system, adapted from Escobar and Trierweiler21, has a crossfeed for the top tanks and direct feed for the lower tanks. The feed is divided into two three-way valves that send to the lower tanks a portion 𝑥1 and 𝑥2 and the complementary (1 ― 𝑥1) and (1 ― 𝑥2) to the top tanks. The nonlinear model is:

𝑑ℎ1 𝑑𝑡

=

𝑥1𝐹1 + 𝑅3 ℎ3 ― 𝑅1 ℎ1 𝜋ℎ1(𝐷1 ― ℎ1) (8)

𝑑ℎ2 𝑑𝑡

=

𝑥2𝐹2 + 𝑅4

ℎ4 ― 𝑅2 ℎ2

𝜋ℎ2(𝐷2 ― ℎ2)


9


𝑑ℎ3 𝑑𝑡

𝑑ℎ4 𝑑𝑡

=

=

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(1 ― 𝑥2)𝐹2 ― 𝑅3 ℎ3 𝜋ℎ3(𝐷3 ― ℎ3) (1 ― 𝑥1)𝐹1 ― 𝑅4 ℎ4 𝜋ℎ4(𝐷4 ― ℎ4)

where ℎ1,ℎ2,ℎ3, and ℎ4 are the levels, 𝐹1 and 𝐹2 are the feed flow rates, 𝑅1,𝑅2,𝑅3, and 𝑅4 are the discharge coefficients and 𝐷1,𝐷2,𝐷3, and 𝐷4 are the diameters of the tanks. Three operational representative scenarios were selected, where the parameters (diameter and discharge coefficient) are considered the same. Stationary conditions for all the operation scenarios are shown in Table 1.

Table 1. Stationary conditions for the operating points (scenarios) of the QST system. Parameters

Scenario 1

Scenario 2

Scenario 3

𝑥1

0.35

0.40

0.30

𝑥2

0.25

0.20

0.30


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𝐹1 [cm3/s]

13.0

13.0

13.0

𝐹2 [cm3/s]

13.0

13.0

13.0

ℎ1 [cm]

14.54

17.23

11.96

ℎ2 [cm]

9.71

7.50

11.96

ℎ3 [cm]

6.76

7.62

5.84

ℎ4 [cm]

5.06

4.22

5.84

This model shows special and important dynamic behavior. Besides the nonlinearities caused by the spherical shape, this MIMO system when linearized has a nonminimum phase behavior for the specific subcase 2 × 2, i.e., the levels ℎ1 and ℎ2 controlled by the feed flow rates 𝐹1 and 𝐹2, with 𝑥1 + 𝑥2 < 1. This arrangement exhibits a right half plane (RHP)-zero that limits the attainable performance of the system. The step responses of the considered OPs are illustrated in Fig. 3. Step Response From: F1

2

From: F2

To: h1

1.5

1 G1 G2 G3

Amplitude

0.5

0 1.5

1 To: h2

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


0.5

0

0

2000

4000

6000

80000

2000

4000

6000

8000

Time (seconds)


11


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Figure 3. Step response of the considered scenarios (𝐺1,𝐺2 and 𝐺3) of the QST system.

Table 2 presents the values of the RHP-zeros and the output (𝑦𝑧) and input (𝑢𝑧) directions, the value of the diagonal of the Relative Gain Array (λ1,1) and the determinant of the matrix gain. It is possible to see that all scenarios considered for the robust tuning have RHP transmission zeros and there is no inversion in the gain signal, according to the determinant of the matrix gain. Table 2. Characteristics of the OPs of the QST system. RHP-zero

𝑦𝑧

𝑢𝑧

𝜆1,1

det (𝐺)

𝐺1(𝑠)

0.0030

[ ―0.6411 0.7675 ]

0.6353 [ ―0.7723 ]

―0.2195

―1.3476

𝐺2(𝑠)

0.0033

[ ―0.5581 0.8298 ]

[ ―0.5550 0.8318 ]

―0.2060

―1.2490

𝐺3(𝑠)

0.0029

[ ―0.7071 0.7071 ]

[ ―0.7071 0.7071 ]

―0.2283

―1.3416

3.1 SET-POINT TRACKING FOR CLASSICAL MPC IN QST SYSTEM The simulation of the MPC and the nonlinear process was accomplished by using MATLAB R2012b, using Simulink, and the optimization problems related to the tuning were solved with the Nelder-Mead algorithm. The 2 × 2 subcase was considered in the QST system, where the levels ℎ1 and ℎ2 are controlled variables and the feed flow rates 𝐹1 and 𝐹2 are manipulated variables. The tuning of the sampling time (𝑡𝑠), prediction horizon (𝑃) and control horizon (𝑀) were accomplished as proposed by Trierweiler and Farina22, i.e., based in the characteristics of the open-loop behavior of the system (rise time and settling time). Initially, the desired closed-loop transfer matrix (𝑇𝑑) for the system was estimated. It consists of a diagonal matrix composed by a second order transfer function. The estimated rise-times are 𝑟𝑡 = [633.63;242.5] in seconds (keeping the overshoot rates constant at 5% for each output). This desired closed-loop transfer matrix will be used to compute the


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attainable performance functions (𝑇𝑘) related to each scenario. The scaling matrices, based on global minimized condition number evaluated in the frequency of RPN are 𝐿𝑠 = 𝑑𝑖𝑎𝑔[0.9759;1.2047] and 𝑅𝑠 = 𝑑𝑖𝑎𝑔[0.8081;1.0]. For MPC tuning evaluation, the nonlinear plant starts its operation in scenario 1 and the system is changed so that the plant passes through all operating points. The linear plant model used by MPC controller is corresponding to the scaled scenario 1, which is responsible for the constraint of the optimization problem being active, i.e., 𝐺𝑠,1 = 𝐿𝑠𝐺1(𝑠)𝑅𝑠. The hardconstraints used for the simulation are based on the physical limits of the tanks and actuators being

ℎ𝑚𝑎𝑥 = 25 cm,

ℎ𝑚𝑖𝑛 = 0.0 cm,

𝐹𝑚𝑎𝑥 = 30 cm3/s,𝐹𝑚𝑖𝑛 = 0.0 cm3/s

and

𝛥𝐹𝑚𝑎𝑥 = 𝛥𝐹𝑚𝑖𝑛 = 3 cm3/s. Sequences of steps in the output RHP-zero directions, which represent the worst operational condition were implemented. The changes in the operating point over the simulation in the nonlinear process are considered as a disturbance in the problem, because the three-way valves are not used as manipulated variables. The tuned weights estimated by the method are 𝑄 = 𝑑𝑖𝑎𝑔[1.8611;1.6582] and 𝑊 = 𝑑𝑖𝑎𝑔

[0.0263 × 10 ―3;0.6387 × 10 ―3] and will be used to penalize the deviation between the outputs and the reference trajectory and excessive movement of the manipulated variables. In a general way it is given by Eq.9.

[∑

𝑀

𝑃

min 𝐽 = min Δ𝑢

Δ𝑢

‖𝑦(𝑦 + 𝑗│𝑡) ― 𝑟(𝑡 +

𝑗=1

𝑗)‖2𝑄

+

∑‖Δ𝑢(𝑡 + 𝑗 ― 1)‖

]

2 𝑊

𝑗=1

(9)

where Δ𝑢 are the controlled actions sent to the plant, 𝑃 is the predictive horizon and 𝑀 are the control horizon. Figure 4 shows the set-point tracking for the nonlinear plant tuned by the multi-scenarios approach.


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Nonlinear model 4 OP 1

OP 2

OP 3

 (h1) [cm]

2 0

set-point CVs Scenario 1 Scenario 2 Scenario 3

-2 -4 0

0.5

1

1.5

2

Time [sec]

2.5 4

x 10

4 2  (h2) [cm]

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

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0 -2 OP 1 -4

0

OP 2 0.5

OP 3 1

1.5 Time [sec]

2

2.5 4

x 10

(A)


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Nonlinear model

3  (F1) [cm /s]

15

OP 1

OP 2

OP 3

10 5 0 -5 -10 0

0.5

1

1.5

2

Time [sec]

20

OP 1

2.5 4

x 10

OP 2

15 3  (F2) [cm /s]

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


OP 3

10 5 0 -5 MVs

-10 0

0.5

1

1.5

2

Time [sec]

2.5 4

x 10

(B) Figure 4. Controlled (A) and Manipulated (B) variables for set-point tracking (Nonlinear Plant) tuned by multi-scenarios approach for the QST system.

Figure 4 presents the changes in the OPs for the nonlinear system and the trajectories estimated for each scenario, separately. It is possible to observe through simulation that the outputs of the nonlinear process followed the sequence of set-point and OPs changes in a robust manner, presenting its behavior quite similar to the estimated achievable performance of each scenario. Besides that, the system does not show excessive control actions to achieve the path, due to the correct scaling procedure and tuning parameter estimation.

4. CASE STUDY – CSTR WITH SEPARATION COLUMN AND RECYCLE The system is composed of a CSTR reactor with separation column and recycle as shown in Fig. 5.


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CA,RO, CB,RO, FRO CA,R, CB,R, FR

CA,in, Fin

LIC

LIC

Fc,in,Tc,in

AIC

TIC

CA, CB, CC, CD, F LIC

LIC

CA,P, CB,P, CC, CD, FB

Figure 5. CSTR with separation column and recycle

This process, as proposed by Schultz et al.,23 has four components (𝐴,𝐵,𝐶,𝐷) and inside the CSTR reactor occurs a reaction that follows the kinetics of Van de Vusse: 𝑘1 𝑘2

𝐴 𝐵 𝐶 𝑘3

2𝐴 𝐷 Component A is fed into the reactor, and the unreacted component 𝐴 is separated in the distillation column and recycled back to the reactor. The recycle is composed only of the components 𝐴 and 𝐵 (the heavy components – 𝐶 and 𝐷 – are totally removed from the bottom stream of the column). The process model is given by: 𝑉𝑅

𝑑𝐶𝐴 𝑑𝑡

= 𝐹𝑖𝑛𝐶𝐴,𝑖𝑛 + 𝐹𝑅𝑂𝐶𝐴,𝑅𝑂 ― 𝐹𝐶𝐴 ― 𝑉𝑅(𝑘1𝐶𝐴 + 2𝑘3𝐶2𝐴)

𝑉𝑅

𝑑𝐶𝐵 𝑑𝑡

= 𝐹𝑅𝑂𝐶𝐵,𝑅𝑂 ― 𝐹𝐶𝐵 ― 𝑉𝑅(𝑘2𝐶𝐵 ― 𝑘1𝐶𝐴) (10) 𝑉𝑅

𝑑𝐶𝐶

𝑉𝑅

𝑑𝑡 𝑑𝐶𝐷 𝑑𝑡

= ―𝐹𝐶𝐶 + 𝑉𝑅 𝑘2𝐶𝐵 = ―𝐹𝐶𝐷 + 𝑉𝑅𝑘3𝐶2𝐴


16

Page 17 of 33 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


(

0.1

(

0.1

)

𝑑𝐶𝐴,𝑅𝑂

(1 ― 𝑦𝑎)0.7

𝑑𝐶𝐵,𝑅𝑂

)

(1 ― 𝑦𝑎)0.7 (0.1)

𝑑𝑡

𝑑𝑡

𝑑𝐹𝑅𝑂 𝑑𝑡

= 𝐶𝐴,𝑅 ― 𝐶𝐴,𝑅𝑂

= 𝐶𝐵,𝑅 ― 𝐶𝐵,𝑅𝑂

= 𝐹𝑅 ― 𝐹𝑅𝑂

where 𝐶𝐴, 𝐶𝐵,𝐶𝐶, and 𝐶𝐷 are the concentrations of the components 𝐴,𝐵,𝐶, and 𝐷, 𝐶𝐴,𝑅𝑂, 𝐶𝐵,𝑅𝑂 are the concentrations in the recycle stream of the components 𝐴 and 𝐵, respectively, and 𝐶𝐴,𝑖𝑛 is the feed concentration of 𝐴. 𝐹𝑖𝑛, 𝐹, and 𝐹𝑅𝑂 are the flow rates in the feed, from the reactor and from the recycle, respectively. 𝑉𝑅 is the reactor volume, 𝑦𝐴 is the molar fraction of the component 𝐴 in the reflux stream and 𝑘1,𝑘2, 𝑘3, 𝐶𝐴,𝑅,𝐶𝐵,𝑅 and 𝐹𝑅 are given by Eq. 11.

(

9758.3 𝑇[°𝐶] + 273.15

(

9758.3 𝑇[°𝐶] + 273.15

(

8560 𝑇[°𝐶] + 273.15

𝑘1 = 1.2870 × 1012exp ―

𝑘2 = 1.2870 × 1012exp ―

𝑘3 = 4.5215 × 109exp ―

) )

)

(11)

𝐶𝐴,𝑅 = 𝑦𝐴(𝐶𝐴 + 𝐶𝐵 + 𝐶𝐶 + 𝐶𝐷)𝐶𝐴 𝐶𝐵,𝑅 = ―

𝐹𝑅 =

𝑦𝐴(𝐶𝐴 + 𝐶𝐵 + 𝐶𝐶 + 𝐶𝐷)𝐶𝐴(𝑦𝐴 ― 1) 𝑦𝐴 𝑦𝐴(𝐶𝐴 + 𝐶𝐵 + 𝐶𝐶 + 𝐶𝐷)𝐶𝐴𝐹 𝑦𝐴(𝐶𝐴 + 𝐶𝐵 + 𝐶𝐶 + 𝐶𝐷)𝐶𝐴

Three operational scenarios for the development of the tuning procedure were selected. The stationary conditions for the selected OPs are presented in Table 3. Table 3: Stationary conditions for the scenarios of the CSTR with separation column and recycle system.


17


Page 18 of 33

Parameters

Scenario 1

Scenario 2

Scenario 3

𝑉𝑅 [m3]

0.611

0.83

1.5

𝐹 [kmol/h]

30

25

20

𝑇 [°C]

135

135

135

𝑦𝐴

0.977

0.977

0.977

𝐶𝐴,𝑖𝑛 [kmol/m3]

5.1

5.1

5.1

𝐶𝐴 [kmol/m3]

1.2786

1.0334

0.6535

𝐶𝐵 [kmol/m3]

0.6796

0.6688

0.5257

𝐶𝐶 [kmol/m3]

0.7369

1.1820

2.0989

𝐶𝐷 [kmol/m3]

0.1173

0.1249

0.1128

𝐶𝐴,𝑅𝑂 [kmol/m3]

1.8742

1.7429

1.4713

𝐶𝐵,𝑅𝑂 [kmol/m3]

0.0443

0.0412

0.0348

𝐹𝑅𝑂 [kmol/h]

20.4655

14.8226

8.8831

This MIMO system was linearized, and the subcase 2 × 2 was analyzed. The concentrations of the components 𝐴 and 𝐵 (𝐶𝐴 and 𝐶𝐵, respectively) were considered as controlled variables, and the molar fraction of 𝐴 (𝑦𝐴) and temperature (𝑇) as manipulated variables. It was considered that the volume of the reactor is perfectly controlled. This model is a realistic representation of an industrial process that covers important problems (parallel reaction, distillation column, and recycle stream) that can be handled by a robustly tuned MPC. The step response of the system OPs are illustrated in Fig. 6.


18

Page 19 of 33

Step Response

From:  y A 0.7

From:  T

0

0.6

To:  CA

H1 H2 H3

-0.01

0.5 0.4

-0.02 0.3 0.2

-0.03

0.1 0

0

5

10

15

0.2

-0.04

0

5

10

15

0

5

10

15

0.01 0.005

0.1

0

To:  CB

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 -0.005 -0.1

-0.01

-0.2 -0.3

-0.015

0

5

10

15

-0.02

Time (seconds)

Figure 6. Step response of the three operating scenarios (𝐻1,𝐻2 and 𝐻3) of the CSTR with separation column and recycle system.

In Table 4 are presented the characteristics of the OPs. It could be seen that all scenarios present RPH-zeros responsible for limiting the achievable performance of the system and there is no inversion in the multivariable signal gain. Table 4. Characteristics of the CSTR with separation column and recycle system OPs. RHP-zero

𝑦𝑍

𝑢𝑧

𝜆1,1

det (𝐻)

𝐻1(𝑠)

0.460

[0.1435 0.9896]

[0.0463 0.9989]

―0.3946

―0.0057

𝐻2(𝑠)

0.946

[ ―0.1495 0.9888 ]

[ ―0.0794 ―0.9968]

0.4540

―0.0050

𝐻3(𝑠)

2.596

[ ―0.4673 0.8841 ]

[ ―0.1915 ―0.9815]

1.2833

―0.0021


19


4.1.

Page 20 of 33

SET-POINT TRACKING FOR CLASSICAL MPC IN THE CSTR CASE WITH SEPARATION COLUMN AND RECYCLE SYSTEM

The implementation and simulation of the process and controller were the same as employed in the QST system. The tuning procedure described in Fig. 1 was employed, and the rise-times were calculated of the desired transfer function matrix, constrained by achievable robustness metrics. The values are 𝑟𝑡 = [9.997;8.853] in seconds (keeping the overshoot rates at 5%). These second order functions are factorized according to the OPs and they represent the attainable performance transfer functions. The scaling procedure, responsible for normalization of the system resulted in 𝐿𝑠 = 𝑑𝑖𝑎𝑔[0.7916;1.9359] and 𝑅𝑠 = 𝑑𝑖𝑎𝑔[0.0696;1.0]. The nonlinear plant was used for the simulation and simulation started at scenario 1. The parameters were changed during the operation, along with the scenarios, to evaluate the robustness of the method. In the controller, implementation was used the scaled model system 𝐻𝑠 = 𝐿𝑠𝐻1(𝑠)𝑅𝑆, that corresponds to the scenario 1, which become the constraints active. The hard constraints for the outputs are 𝐶𝐴,𝑀Á𝑋 = 𝐶𝐵,𝑀Á𝑋 = 6 kmol/m3, 𝐶𝐴,𝑀𝐼𝑁 = 𝐶𝐵,𝑀𝐼𝑁 = 0 kmol/m3, 0 ≤ 𝑦𝐴 ≤ 1, and 95 °C ≤ 𝑇 ≤ 135 °C. The tuned weights are 𝑄 = 𝑑𝑖𝑎𝑔 [2.3321;0.5736] and 𝑊 = 𝑑𝑖𝑎𝑔[0.3757;0.0012]. Figure 7 shows the set-point tracking of the simulated system tuned by the multi-scenarios approach.


20

Page 21 of 33

Nonlinear model 0.3

3  (Ca) [kmol/m ]

OP 2

OP 1

0.2

OP 3

0.1 0

set-point CVs Scenario 1 Scenario 2 Scenario 3

-0.1 -0.2 -0.3 -0.4

0

50

100

150

200

250

Time [sec]

0.3 OP 2

OP 1

0.25

OP 3

3  (Cb) [kmol/m ]

0.2 0.15 0.1 0.05 0 -0.05

0

50

100

150

200

250

Time [sec]

(A) Nonlinear model 0 OP 2

OP 1

OP 3

-0.1

 yA

-0.2 -0.3 -0.4 MVs -0.5

0

50

100

150

200

250

Time [sec]

10 OP 2

OP 1

OP 3

0

 T [°C]

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


-10 -20 -30 -40

0

50

100

150

200

250

Time [sec]

(B) Figure 7. Controlled (A) and Manipulated (B) variables for set-point tracking (Nonlinear Plant) for the CSTR process.


21


Page 22 of 33

Figure 7 shows that the outputs follow the trajectory stated by the set-points without the need of excessive control moves, with the dynamic error similar to the estimated scenarios. Besides that, the changes OPs do not present instability zones or saturation in the manipulated variables. 5.

IMPACT OF CLOSED-LOOP PERFORMACE SELECTION

The importance of the attainable trajectory and the scaling procedure, estimated through the multi-scenarios approach, is evaluated in this section. Instead of following the steps proposed in this paper, the difference between the set-point and the outputs for the weights tuning calculation without the scaling procedure were employed. The simulation was accomplished for the same scenarios used in the previous analysis for the nonlinear systems. The results are presented in Figure 8 for QST system, and in Figure 9 for CSTR with separation column and recycle.


22

Page 23 of 33

Nonlinear model

 (h1) [cm]

18

OP 1

OP 2

OP 3

16 14 set-point Multi-scenarios tuning set-point tuning

12 10

0

0.5

1

1.5

2

Time [sec]

2.5 4

x 10

14

 (h2) [cm]

12 10 8 OP 1 6

OP 2

0

0.5

OP 3 1

1.5

2

Time [sec]

2.5 4

x 10

(A) Nonlinear model 30 OP 1

OP 2

OP 3

3  (F1) [cm /s]

20 10 0 -10

0

0.5

1

1.5

2

Time [sec]

2.5 4

x 10

40 OP 1

OP 2

OP 3

30 3  (F2) [cm /s]

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


20 10 Multi-scenarios tuning set-point tuning

0 -10

0

0.5

1

1.5

2

Time [sec]

2.5 4

x 10

(B) Figure 8. (A) Controlled and (B) Manipulated variables for the comparison between the multiscenarios tuning and set-point tuning in QST system.


23


Nonlinear model 0.4 OP 1

OP 3

OP 2

3  (Ca) [kmol/m ]

0.2 0 -0.2

set-point Multi-scenarios tuning set-point tuning

-0.4 -0.6

0

50

100

150

200

250

200

250

200

250

Time [sec]

0.3 OP 1

0.25

OP 2

OP 3

3  (Cb) [kmol/m ]

0.2 0.15 0.1 0.05 0 -0.05

0

50

100

150 Time [sec]

(A) Nonlinear model 0.2 OP 1

0.1

OP 3

OP 2

 yA

0 -0.1 -0.2 -0.3 -0.4 -0.5

0

50

100

150 Time [sec]

10 OP 1

OP 2

OP 3

0

 T [°C]

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 33

Multi-scenarios tuning set-point tuning

-10 -20 -30 -40

0

50

100

150

200

250

Time [sec]

(B) Figure 9. (A) Controlled and (B) Manipulated variables for the comparison between the multiscenarios tuning and set-point tuning in CSTR with separation column and recycle system.


24

Page 25 of 33 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


As seen in Figure 8, the controlled outputs follow the reference trajectory similarly as the multi-scenarios approach until the end of the operation point 2. In the operation point 3 (scenario 3), the change in set-point causes excessive oscillations in the process. The manipulated variables calculated for the MPC, tuned through the set-point tuning (without scaling procedure) presented a similar behavior in the steady-state, for operation points 1 and 2, but in the transient state shows an oscillation pattern mainly in operation point 3. In the same way, the set-point tuning was evaluated in the CSTR with separation column and recycle. As observed in Figure 9, a faster response in the set-point tracking due to the excess of movement and oscillations in the manipulated variables was obtained. This tuning was not capable of reducing the variability of the plant, making difficult the control and optimization of the plant. As a way to compare the proceedings, the Integral of the Square Errors (ISE) in the controlled variables and the Variance index in the manipulated variables (VAR) were calculated. Table 5 shows the results. Table 5. Integral of the Square Errors and Variance Index. ISE

VAR

Multi-scenarios tuning

1112.1

67.05

Set-point tuning

1151.5

67.96

CSTR with separation

Multi-scenarios tuning

0.721

109.555

column and recycle

Set-point tuning

1.502

74.0907

QST system

5.1. IMPACT ON SELECTION OF THE CONTROLLER’S PROCESS MODEL In this section the importance of the selection of the controller’s process model was evaluated. The choice of the model is based on that which makes at least one of the constraints active, i.e., the worst operational scenario considered in the tuning.


25


Page 26 of 33

In the previous sections the model related to OP 1 was employed for both case studies. Here, in order to evaluate the choice of the controller’s model, OP 2 in the controller design was considered. The tuning procedure was employed in each case study following the steps as proposed in the Section 2. The results are presented in Figure 10 and 11.


26

Page 27 of 33

Nonlinear model 20

 (h1) [cm]

18 16 14 12 10

0

0.5

1

1.5

2

Time [sec]

2.5 4

x 10

14 set-point Controller Model - G1 Controller Model - G2

 (h2) [cm]

12

10

8

6

0

0.5

1

1.5

2

Time [sec]

2.5 4

x 10

(A) Nonlinear model 30 25

3  (F1) [cm /s]

20 15 10 5 0 -5

0

0.5

1

1.5

2

Time [sec]

2.5 4

x 10

40 30 3  (F2) [cm /s]

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


20 10 Controller Model - G1 Controller Model - G2

0 -10

0

0.5

1

1.5 Time [sec]

2

2.5 4

x 10

(B) Figure 10. (A) Controlled and (B) Manipulated variables for the comparison between the tuning using the controller’s process model G1 and G2 for the QST system.


27


Nonlinear model 0.6 0.4

 (Ca)

0.2 0 -0.2 -0.4 -0.6 -0.8

0

50

100

150

200

250

Time [sec]

0.6 set-point Controller Model - H1 Controller Model - H2

0.5

 (Cb)

0.4 0.3 0.2 0.1 0 -0.1

0

50

100

150

200

250

150

200

250

Time [sec]

(A) Nonlinear model 0

 (ya)

-0.2

-0.4

-0.6

-0.8

0

50

100 Time [sec]

0 Controller Model - H1 Controller Model - H2 -10

 (T)

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 28 of 33

-20

-30

-40

0

50

100

150

200

250

Time [sec]

(B) Figure 11. (A) Controlled and (B) Manipulated variables for the comparison between the tuning using the controller’s process model H1 and H2 for the CSTR reaction with separation column and recycle system.


28

Page 29 of 33 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


As seen in Figure 10 the system follows the trajectory settled by the set-point in a similar way as proposed by the multi-scenarios tuning, but presents oscillations when changed to OP 3. In Figure 11, the controller’s tuning parameters provide a faster response when compared to the multi-scenarios tuning. However this behavior causes excessive oscillations in the system with regard to OP 1. 6.

CONCLUSIONS

A tuning strategy for MPC, originally used for nonsquare systems controlled by range, is extended in this work for a robust tuning of square MPC. The procedure is based on the determination of an attainable performance function for each scenario that the controller should be robustly tuned. The best scaling matrices are estimated, and in the final step, the MPC weights are tuned, through an optimization problem, leading the process to the best operational condition handling in the best way the disturbances and changes in OPs. Besides that the selection of the controller’s process model was proposed and justified. The case studies QST system and CSTR with separation column and recycle were used to illustrate the proposed approach. Both systems exhibit a RHP transmission zero, that limits the closed loop performance and are critical for the operation. The examples confirm that the method provides a reasonable closed-loop performance for all scenarios, allied with a robust behavior. When an abrupt trajectory is used as reference for tuning, the behavior becomes worse due to the oscillations. Moreover, it also shows the importance of the scaling of variables and the selection of the correct model for the controller design.

AUTHOR INFORMATION Corresponding Author *Tel.: +55 (51) 3308-4167. E-mail: [email protected]


29


Page 30 of 33

Notes The authors declare no competing financial interest. REFERENCES (1)

Godoy, J. L.; Ferramosca, A.; González, A. H. Economic Performance Assessment and Monitoring in LP-DMC Type Controller Applications. Journal of Process Control 2017, 57, 26.

(2)

D’Jorge, A.; Ferramosca, A.; González, A. H. A Robust Gradient-Based MPC for Integrating Real Time Optimizer (RTO) with Control. Journal of Process Control 2017, 54, 65.

(3)

Ferramosca, A.; Limon, D.; González, A. H.; Odloak, D.; Camacho, E. F. MPC for Tracking Target Sets. In Proceedings of the IEEE Conference on Decision and Control; 2009; pp 8020–8025.

(4)

Qin, S. J.; Badgwell, T. A. A Survey of Industrial Model Predictive Control Technology. Control Engineering Practice 2003, 11 (7), 733.

(5)

Holkar, K. S.; Waghmare, L. M. An Overview of Model Predictive Control. International Journal of Control and Automation 2010, 3 (4), 47.

(6)

Garriga, J. L.; Soroush, M. Model Predictive Control Tuning Methods : A Review. Ind. Eng. Chem. Res. 2010, 49 (8), 3505.

(7)

Hinde Jr., R. F.; Cooper, D. J. A Pattern-Based Approach to Excitation Diagnostics for Adaptive Process Control. Chemical Engineering Science 1994, 49 (9), 1403.

(8)

Banerjee, P.; Shah, S. L. Tuning Guidelines for Robust Generalized Predictive Control. Proc. 31st IEEE Conf. on Decision and Control 1992, 4, 3233.

(9)

Shridhar, R.; Cooper, D. J. A Tuning Strategy for Unconstrained Multivariable Model Predictive Control. Ind. Eng. Chem. Res. 1998, 37 (10), 4003.

(10)

McIntosh, A. R.; Shah, S. L.; Fisher, D. G. Performance Tuning of Adaptive


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Generalized Predictive Control. Preprints 11th IFAC World Congress 1990, 4, 174. (11)

Rani, K. Y.; Unbehauen, H. Study of Predictive Controller Tuning Methods. Automatica 1997, 33 (12), 2243.

(12)

Gous, G. Z.; de Vaal, P. L. Using MV Overshoot as a Tuning Metric in Choosing DMC Move Suppression Values. ISA transactions 2012, 51 (5), 657.

(13)

Lee, J. H. Van Der; Svrcek, W. Y.; Young, B. R. A Tuning Algorithm for Model Predictive Controllers Based on Genetic Algorithms and Fuzzy Decision Making. 2008, 47, 53.

(14)

Suzuki, R.; Kawai, F.; Nakazawa, C.; Matsui, T.; Aiyoshi, E. Parameter Optimization of Model Predictive Control by PSO. Electrical Engineering in Japan (English translation of Denki Gakkai Ronbunshi) 2012, 178 (1), 40.

(15)

Júnior, G. A. N.; Martins, M. A. F.; Kalid, R. A PSO-Based Optimal Tuning Strategy for Constrained Multivariable Predictive Controllers with Model Uncertainty. ISA Transactions 2014, 53 (2), 560.

(16)

Davtyan, A.; Hoffmann, S.; Scheuring, R. Optimization of Model Predictive Control by Means of Sequential Parameter Optimization. IEEE SSCI 2011 - Symposium Series on Computational Intelligence - CICA 2011 - 2011 IEEE Symposium on Computational Intelligence in Control and Automation. 2011, pp 11–16.

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Exadaktylos, V.; Taylor, C. J. Multi-Objective Performance Optimisation for Model Predictive Control by Goal Attainment. International Journal of Control 2010, 83 (7), 1374.

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Santos, J. E. W.; Trierweiler, J. O.; Farenzena, M. Model Predictive Control Tuning Strategy for Non-Square Systems and Range Controlled Variables Based on MultiScenarios Approach. Industrial & Engineering Chemistry Research 2017, 56 (40), 11496.


31


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Trierweiler, J. O. A Systematic Approach to Control Structure Design, Universität Dortmund, 1997.

(20)

Trierweiler, J. O. Application of the RPN Methodology for Quantification of the Operability of the Quadruple-Tank Process. Brazilian Journal of Chemical Engineering 2002, 19 (2), 195.

(21)

Escobar, M.; Trierweiler, J. O. Multivariable PID Controller Design for Chemical Processes by Frequency Response Approximation. Chemical Engineering Science 2013, 88 (0), 1.

(22)

Trierweiler, J. O.; Farina, L. A. RPN Tuning Strategy for Model Predictive Control. Journal of Process Control 2003, 13, 591.

(23)

Schultz, E. S.; Trierweiler, J. O.; Farenzena, M. The Importance of Nominal Operating Point Selection in Self-Optimizing Control. Industrial & Engineering Chemistry Research 2016, 55 (27), 7381.


32

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


MPC controller

set-points

Step Response

2.5

OP 1 OP 2

2

cvs Amplitude

Page 33 of 33

MPC tuning

1.5

OP 3

1

0.5

0

0

5

10

15

Time (seconds)

Suitable model CV1 CV2

MV1

20

OP 2

25

mvs

MV2

Figure 12. For Table of Contents only.


33

Robust tuning for classical MPC through the Multi-Scenarios Approach

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