Model Predictive Control Tuning Strategy for Non-Square Systems

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MPC Tuning Strategy for Non-Square Systems and Range Controlled Variables Based on 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.7b01531 • Publication Date (Web): 16 Sep 2017 Downloaded from http://pubs.acs.org on September 18, 2017

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Industrial & Engineering Chemistry Research

MPC Tuning Strategy for Non-Square Systems and Range Controlled Variables Based on MultiScenarios 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: A novel strategy for MPC tuning considering non-square systems, with more outputs than inputs, and set points or ranges for controlled variables is introduced. This methodology is applicable for any predictive control algorithm, since it has adjustable parameters according to the implementation, and is divided in three parts that consists of, initially, the determination of the best attainable closed-loop performance function for the multiscenarios, generated from the complete model, including performance and robustness metrics as additional constraints. Based on the attainable closed-loop function, the second step is the calculation of the optimal scaling for the process and, finally, the controller weights are tuned leading the process to the best operating condition. This constrained strategy was applied in the

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controller design of the Shell heavy oil fractionator benchmark showing good results for zone region tracking.

1. INTRODUCTION Model predictive control (MPC) is a control technique, implemented in continuous or discretetime domain1, which uses an open-loop process model to describe the behavior and the evolution of the plant through a pre-defined horizon and, based on this model, it calculates control actions that minimize an objective function2. This strategy has a broad application and success in the process industry mainly due to the formulation of the control problem and the explicit use of the model to predict the evolution of the plant, considering the constraints in the optimization problem3, 4. It allows handling plants with complex behaviors and multivariable nature better than standard proportional–integral–derivative (PID) controllers and, since it takes into account the physical limits of the actuators and process variables, it allows operating close to the process constraints. However, due to the number of tuning parameters and the different MPC implementations, an efficient tuning procedure is still an open problem. Qin & Badgwell5 and Holkar & Waghmare6 highlight the main differences among the MPC implementations related to: (i) the model type used to describe the process (i.e., linear, nonlinear, empirical or phenomenological models); (ii) the use of soft-constraints instead of set-point tracking; (iii) keeping the controlled variables in zones or regions of control; and (iv) the consideration of non-square systems with negative degrees of freedom, i.e., more controlled than manipulated variables. Non-square systems with more outputs than inputs are quite common in chemical processes and others engineering branches, also occurring in systems that are originally square when they saturate one or more actuators7. For a system that operates in this manner, the soft-constraint


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operation is a way to keep the controlled variables in the specifications, which becomes the control problem achievable and ensures flexibility for the controller8. The role of the MPC, as part of a lower level in a control function hierarchy5, is to lead the plant to the best operating condition, respecting the constraints and minimizing the error along the trajectory given by a reference signal or a zone region, where the outputs should lie most of the time. According to Campos et al.9, the tuning of this type of controller has crucial importance in operation, because poor tuning may not contribute to reducing the variability of the controlled variables, increasing the oscillations and interfering in the optimization of the plant. Trierweiler & Farina10 report that each tuning parameter has a particular importance in the control problem. The correct choice of the sampling time and the prediction and control horizons result in not exceeding computational effort (the horizons are responsible for the size of the matrices that will calculate the control actions) and capturing the plant evolution in a representative way. Furthermore, the weighting matrices will be responsible for giving priorities among the considered variables and scaling the process model, i.e., making the units of measurement comparable, ensuring robustness and performance in the operation. The available methodologies for MPC tuning are ranked according to the determination of the parameters11: (i) explicit formulations or bounds based on the process dynamics and (ii) optimization problems, minimizing desirable indices. Trierweiler & Farina10, Hinde Jr. & Cooper12, Shridhar & Cooper13 and Rowe & Maciejowsky14 are examples of (i) providing good guidelines for theoretical and practical perspectives for MPC tuning parameters through the open-loop model, however, they are dedicated only to particular algorithms, specific plant models, squared systems, and set point tracking, restricting their industrial applications.


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Gous & Vaal15, Davtyan et al.16, Exadaktylos & Taylor17, and Júnior et al.18 (ii) proposed a way to consider a large variety of MPC algorithms through non-linear optimization problems for unconstrained and constrained systems. These techniques are based on the minimization of desirable performance indices and they were presented as an alternative for covering different MPC implementations. However, they fail at considering only square systems in set point tracking. Among the several works that consider MPC tuning, none consider non-square systems, variables controlled by the range and the use of different ways to implement the model of the process. In addition, achieving the attainable performance robustly rejecting disturbances and considering limitations imposed by constraints, non-minimum phase factors and other are not taken into account to give priority between the input and output variables. Based on that, the method for MPC tuning introduced in this study allows the tuning for non-square MPCs (with more controlled than manipulated variables) considering both set points and zone region tracking for the controlled variables, capturing the achievable performance for the process. The paper is structured as follows: in Section 2 is presented a background containing the basic concepts necessary for understanding the proposed methodology. In Section 3, the methodology employed for MPC tuning based on multi-scenarios is described. In Section 4, the tuning strategy is applied to a Shell heavy oil fractionator (SHOF) problem, with cases proposed originally and, finally, the conclusions are stated. 2. BACKGROUND 2.1. Model Predictive Control MPC usually considers a discrete open-loop model of the process. Based on this model, control actions that minimize the error between the outputs and the reference trajectory are


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calculated, penalizing the excess of movement of the manipulated variables until a given interval, called control horizon2. In some cases, the difference between the manipulated variables and a target value, usually provided from an optimization layer, is also penalized. In practice, constraint violations might be unavoidable and, in the controlled variables, it is employed a nonnegative slack variable , which quantifies the worst-case constraint violation, making the constraints soften, allowing a feasible solution. The moves are calculated at each sample time, sending to the process only the first calculated control action, characterizing the “receding horizon”. A general form to represent the usual optimization problem solved by an MPC is:

= ‖ + | − + ,

,

‖

subject to Δ

()*

()*

#

+ ‖ + − 1‖" + $% &

≤ ≤

(1a) (,-

≤ − − 1 ≤ Δ

()* − ≤ ≤ (,- +

(,-

(1b)

where + | is the predicted output steps into the future, + is the reference

trajectory, Δ = 1 − / 0 = − − 1, is the slack variable at control interval 1

and ‖2‖" is the weighted Euclidean norm of 2 ∈ ℜ* defined as ‖2‖" = 2 5 62 with 6 ∈ ℜ*×* positive definite. In (1b), the constraints are presented for the optimization problem being

hard-constraints for the manipulated variables and maximal rate of change of manipulated variables and a mixture of soft and hard-constraints for the controlled variables.

The tuning parameters are: prediction horizon 8, control horizon 9, sampling time : ,

controlled variable weight ;, move suppression weight 6, and soft-constraint violation

penalty weight $% .


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The prediction horizon 8 limits the time for the outputs, following the reference trajectory,

and the control horizon 9 specifies the degrees of freedom in selecting the control actions

(after 9 samples of time, the increment in manipulated variables are assumed as zero). Sampling

time : must be compatible with the plant dynamics, being careful in not increase the

computational effort, because the horizons 8 and 9 are dependent of this choice and will affect

the dimension of the optimization problem solved in each sample time. The weighting matrices

; and 6 are usually diagonal matrices, whose elements are tuned to achieve the desired closedloop performance, prioritizing the most important outputs and scaling the system. When the exact values of the controlled outputs are not important, they could remain within specified

boundaries, or ‘zones’, by setting as null the values of the matrix ;. The soft-constraint violation

penalty weight $% is used to guarantee that the outputs will remain in an acceptable zone control at the steady state. 2.2. RPN methodology

The Robust Performance function ?G>?H IJ ∗ D=?H + L∗ DMNHO P8Q ≝ , ? = T2N % U V, ?W

(2a) (2b)

where J ∗ D=?H is the minimized condition number of =?, BCEF − >G> is the maximal singular value of the transfer function matrix EF − >G> and >X is the attainable closed-loop 0

function for the model =X and the controller YX, i.e., >X = DF + =XYXH =XYX.


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The condition number is minimized, for all possible scaling, aiming to avoid an ill-conditioned matrix and preventing that the variable units confuse the controller20.

The attainable performance function >X is computed using Blaschke factorization, i.e.

] 0>_ X >X = Z[,\ XZ[,\

(3)

] 0 denotes the pseudo-inverse of where Z[,\ X is the zero output Blaschke factorization, Z[,\

] 0 = F, and >_ X is the desired closed-loop performance Z[,\ , in such way that Z[,\ 0Z[,\

function. This factorization computes the attainable performance function considering nonminimum-phase factors caused by presence of zeros in right half plane (RHP). These constraints are factorized according to the output directions of positive zeros. It is also possible to factorize the system according to the dead time and the RHP-poles19.

The P8Q, developed by Trierweiler21, is obtained as a relative measure of the deviation

between the area under the curve of the function < `5 and the area under the curve of the function _,) X = gh :h ijg:i, where the parameters k and l can be calculated from specifications in the

time domain, such as overshoot ratio and rise time . To find a good compromise between regulatory and servo performance, it is recommended19,

22

to set the overshoot ratio in 5%,

whereas the smaller the rise time the faster is the closed-loop controller response. To determine the smallest achievable value for the rise time, respecting the control action limitation and considering maximal sensitivity robustness criterion, it was formulated the following optimization problem: *

no = ) m

m

)

(5a)

subject to

P8Q1 ≤ 1.0 ACS Paragon Plus Environment

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9X1 ≤ 2.2

9X1 ≥ 1.2

911 ≤ 10

(5b)

where o = E , , … , ) G, being = 1 … the number of outputs of the complete transfer

matrix, 1 = 1 … t is the number of possible square subsystems originated from the full model,

P8Q1 is the relative robust performance number of the scenario 1 , 9X1 corresponds to the maximal sensitivity value and 911 to the maximal value of the function uX that relates to

= uX:vw for each scenario. Considering for each scenario 1 a value for P8Q1 ≤ 1.0, it is granted that the area under the robust performance function are two times the area under the reference robust performance function and, considering 911 ≤ 10, it is ensured that no abrupt control actions are sent to the plant. The values set for the maximal sensitivity D9X1H ensure a

compromise between robustness and performance (according to Aström & Hägglung23).

The minimization of the index nΩ, presented in Eq. 5a, aims to obtain the faster desired

closed-loop performance function, respecting the attainable constraints existing in each scenario. Here, the quadratic objective function was used as a convenience, avoiding that the optimization problem lead to negative values. If the desired performance is not attainable in one or more

scenarios for a given channel, the corresponding rise time is increased until P8Q1 ≤ 1 is

fulfilled. It is important to highlight that the constraints, for each scenario, are the most important part of the optimization problem (Eq. 5) mainly because they will lead the system to an achievable trajectory. The rise times will be minimized, but the constraints will be responsible for the commitment between performance and robustness in the process operation. If a slower controller response is desired, a lower bound should be provided for the rise time determination.

When the P8Q number is computed for each scenario, it also provides the value of the

frequency ?#RS, where this value is obtained. According to Trierweiler & Farina10, this ACS Paragon Plus Environment

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frequency has an important role in the optimal scaling of the system, allowing a more robust and efficient tuning. The scaling matrices for the system are then obtained by minimizing the condition number for the entire model, which is defined by: J ∎ ≝ minJ D: =D?#ÁS, HP: H U} ,~}

(6)

where P: and : are diagonal scaling matrices for inputs and outputs, respectively, =D?#RS, H

is the original model of the process evaluated in the frequency ?#RS, where occurs the P8Q of

the attainable trajectory for each scenario 1, J9 is the condition number of the matrix 9 and

‖2‖ is the infinity norm of 2.

The matrices P: and : are determined and are used to scale the plant as =: = : =XP: . The

scaled model is then used for simulation and controller design. Figure 2 illustrates the scaled control loop. The scaling matrices are computed solving the minimized condition number

problem, evaluated in the frequency that occurs the maximal P8Q in each scenario. This procedure brings into the control problem a well-conditioned model with robust and performance factors.

Figure 2. The scaled control loop.


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The final step consists of an optimization problem where the tuning controller parameters are used in order to approximate MPC performance to the attainable performance calculated in the first step. The decision variables depends on the controller implementation and, for MPC defined

by (1), they correspond to the weighting matrix 6, ; and $% , which will penalize the move suppression, the error between the outputs and set-point signal, and the soft-constraint violation, respectively. The optimization problem can be described by:

*

*

− ),: = > X:vw − + ̂

(7)

where, for the MPC given by (1), the decision variables are = E$% 6 ;G, > X is the

attainable closed-loop transfer function for the system, :vw is the set-point variation, is the

MPC simulated outputs, and ̂ is the maximal violation of the soft-constraint given by the

controlled variables using the scaled system and controller based on the model =: . When is

considered only tracking zone region, the portion of the Eq. 7 that penalizes the error between the outputs and the achievable trajectory is neglected. It is important to highlight that the objective function of the proposed optimization problem is convex. Nevertheless, due to the constraints (5b), the optimization problem (5) can become nonconvex, but this situation has never occurred in all analyzed case studies. The steps for the tuning of the non-square MPC are summarized in Table 1.


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Table 1. MPC tuning procedure. Specification of the squared scenarios for the process model:

= X, 1 = 1 … t

Determination of the desirable closed-loop transfer function D>_ XH, through the optimization problem: *

no = ) m

s.t.

m

)

P8Q1 ≤ 1.0 9X1 ≤ 2.2

9X1 ≥ 1.2 911 ≤ 10

Converting the desirable D>_ XH into attainable D> XH closed-loop transfer matrix: ] 0>_ X > X = Z[,\ XZ[,\

Calculation of the scaling matrices:

J ∎ ≝ J D: =D?#ÁS, HP: H U} ,~}

Tuning of the weighting matrices for the actual MPC implementation:

*

*

= > X:vw − − ),: + ̂

4. CASE STUDY As a case study, it was considered the SHOF process proposed by Prett & Morari24, as shown in Fig. 3. This problem is composed of a distillation column, and the main feature of this process is the high interaction among the variables as well as large time delays.


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PC

y3

TI 3

LC

FIC 1

d2 y4

u1

AI 1

y1

TI 4

d1 TI 5

TI 6

y5

y6

LC

u3 u2

FIC 2

FIC 3

LC TI 7

AI 2

y7 FEED

y2

Figure 3. Schematic representation of SHOF. The control system is composed originally by seven measured outputs and five inputs, three of them are manipulated variables (control inputs) and two are disturbances. The objective is to

control the Top End Point Composition , Side End Point Composition , Top Temperature , Upper Reflux Temperature , Side Draw Temperature , Intermediate

Reflux Temperature and Bottoms Reflux Temperature through the manipulation of the

Top Draw

,

Side Draw

and Bottoms Reflux Duty

,

rejecting the disturbances

Intermediate Reflux Duty and Upper Reflux Duty . The process model = is given by:


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0: :i .¡i.¡ 0¢: :i .i.¡ 0: ¡:i .¡i. 0: = = :i .i. 0: ¢:i .i.¡ 0¢: :i .¢i. 0: :i

.i.

.i.¡h

:i .i.h :i .i.h :i .i.h

0¢:

0: 0: 0:

¡:i .¢i.h :i

:i .¡i.¢¡£

0:

:i .¢i.¡h

:i .i.h

0: ¦ 0: ¥ :i ¥ .i.£ 0: ¥ :i ¢.i.£ 0: ¥ ¥ :i .i.£ 0: ¥ :i ¥ .i.£ 0: ¥ ¡:i ¥ .i.£ ¤ ¡:i

.¢¢i.¡£

0:

0:

(8)

where §) are the uncertainties in the gain model and the time constants are given in minutes. The focus of the control performance specifications is on rejecting disturbances, satisfying the

constraints. The controlled variables and should be within ± 0.005 and , , , ,

should be in ± 0.5 in the steady-state. All manipulated variables have ±0.5 as hard-constraints.

The maximum move size limitation in the manipulated is 0.20 per sampling time, using the

sampling time ª = 4 .

The disturbance transfer function ¬ is given by:

0: :i .i. 0: :i .i.¢ :i .i. ¬ = :i .i. :i .¡i.¢ ¡:i .i.¢ :i

.i.

24

.i.® :i .¢i.®

0: ¦

0: ¥ :i ¥ .i.¢® :i .¡i.®

¡:i .i.® :i .i.® :i .i.¢® :i

¥ ¥ ¥ ¥ ¥ ¥ ¥ ¤

(9)

Prett & Morari propose solving, through simulation, to satisfy the control objectives without

violating the constraints for the following plants within the uncertainty set, considering five case studies, as seen below. The motivation behind this case study is based on the fact that it presents


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an originally non-square system that are used for several papers18, 25, 26 in an adapted way (3 × 3

or 2 × 2 ), because of its complexity.

Case 1: § = § = § = § = § = 0, = = 0.5

Case 2: § = § = § = −1, § = § = 1, = = 0.5

Case 3: § = § = § = § = 1, § = −1, = = −0.5

Case 4: § = § = § = § = § = 1, = 0.5, = −0.5

Case 5: § = −1, § = 1, § = § = § = 0, = = −0.5

In the simulations were assumed that all inputs and outputs are initially at zero and, in the time

= 100 , a step in the entries of the disturbance matrix, as indicated for each case, was imposed to the system. The implementation of the model predictive controller and the process were made at

MATLAB® R2012b and Simulink®. This MPC application has the same tuning parameters as proposed in this study, and this software version allows to implement the hard-constraints and to make them soften, through an inclusion of a slack variable. However, it is not possible to set different weights for the soft-constraint violation, being then the scaling procedure an important step capable of differentiating the weights for each allowed violation. The MPC simulation time is approximately 118 X tX for each case. The optimization problems were solved through

Nelder-Mead algorithm (fminsearch function) using about 400 iterations and the time for each

iteration was approximately 5 X tX (Intel®CoreTM i7 – 4770S CPU @ 3.10GHz and 12 GB

RAM).

The prediction horizon 8 and control horizon 9 are tuned as proposed by Trierweiler &

Farina10, being 8 based on the settling time of the open-loop process model and 9 = 8/4. For

SHOF tuning, in tracking zone regions, were considered 98 square scenarios decomposed by the


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original model (63 scenarios 2 × 2 and 35 scenarios 3 × 3). The 2 × 2 scenario occurs when a manipulated variable are saturated and two controlled variables are active, i.e., violating the soft-

constraint. The frequencies that occur the P8Q in each scenario were used for the scaling

procedure. The values obtained for the scaling matrices are P: = T´E1.5151; 0.7575; 1.0G and : = T´E0.0012; 2.0854; 0.0013; 1.3287; 5.51 × 100 ; 0.0002; 2.6241G. These matrices are

responsible for the normalization between the inputs and outputs, considering the same size for each variable. The system was configured as Fig. 2, and the MPC weights calculated. The simulation results are shown in Figs. 4, 5, 6, 7, and 8.

2

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(b) Figure 4. (a) Controlled and (b) manipulated variables for multi-scenarios tuning in case 1.

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(b) Figure 8. (a) Controlled and (b) manipulated variables for multi-scenarios tuning in case 5. As seen through Figs. 4 to 8, the proposed methodology kept the controlled variables in the tracking zone region at the steady-state achieving the desirable performance of the process. Therefore, the scaling matrices were responsible for bringing all the inputs and outputs to the same scale, not exceeding the movement of the manipulated variables for reaching the best operational condition. In the case 5, the influence of the model-plant-mismatch and the disturbance make the control actions for the manipulated variable

more demanded, because of the violation of the soft-

constraint, however, a stable operation is obtained, and the control actions do not saturate at the steady-state. As a way to compare the tuning procedure proposed in this article, considering the fact that there is no other study available for tuning considering non-square systems, it was proposed a


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“single step tuning”. This procedure consists of the determination of the weights through a minimization of the soft-constraint violation, without considering the other steps (determination of attainable performance and scaling procedure.). The results for the Case 1 are presented in Fig. 9.

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(b) Figure 9. (a) Controlled and (b) manipulated variables for “single step tuning” in case 1. It could be seen that the three steps proposed in the multi-scenarios approach methodology were important for the stable operation of the system, since that the “single step tuning” is not capable of keeping the controlled variables inside the allowed zone region. Moreover, the manipulated variables presented excess of movement, which is not desirable for the operation. The same behavior was obtained for the other four cases. 5. CONCLUSIONS The article proposed a novel tuning strategy for MPC based on operating scenarios for nonsquare systems operating in zones of control. These scenarios were obtained from the complete process model through the association of two or more channels of the entire system. The attainable performance function aims to obtain the faster closed-loop function constrained by the relative robust performance number P8Q, the value of the maximal sensitivity 9: and the


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maximal value of the function u 9 . These constraints provide a barrier for non-robust controllers and allow the calculation of the achievable function with limited control actions. The system was then scaled, in the critical frequency, and the weights of the tuning matrices were computed. The SHOF benchmark problem was used as case study, which is a non-square system composed by seven controlled, three manipulated and two disturbance variables. The benchmark control problem considers to reject the disturbances, keeping the controlled variables in zones of control in five cases, as suggested by Prett & Morari24 with uncertainty sets and disturbance pattern. All input and output variables are constrained as well as the variation of the manipulated variables. The methodology proposed for MPC tuning presented an excellent performance for the nonsquared system, since a stable and robust operation was achieved without exaggerating or saturating the use of manipulated variables. The violations of soft-constraints are allowed and occur because of the thin controlled zone region. Here again, the proposed tuning methodology was capable of handling the disturbances and the influence of model-plant mismatch ensuring a stable operation. The importance of the three steps procedure was proved, since an unstable operation was achieved considering a “single step tuning”. As far the authors know, the present article was the first one where the original and complete control objective of the Shell benchmark problem was satisfied entirely. AUTHOR INFORMATION Corresponding Author *Tel.: +55 (51) 3308-4167. E-mail: [email protected] Notes


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

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(2)

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(3)

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Maciejowski, J. M. Predictive Control with Constraints; Prentice Hall: Harlow, 2002.

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Shead, L. R. E.; Anastassakis, C. G.; Rossiter, J. A. Steady-State Operability of MultiVariable Non-Square Systems: Application to Model Predictive Control (MPC) of the Shell Heavy Oil Fractionator (SHOF). In Mediterranean Conference on Control & Automation; 2007.

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Campos, M. C. M. M. de; Gomes, M. V. de C.; Perez, J. M. G. T. Controle Avançado E


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Rowe, C.; Maciejowski, J. Tuning MPC Using H∞ Loop Shaping. In Proceedings of the American Control Conference; 2000; Vol. 2, pp 1332–1336.

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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.

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Aström, K. J.; Hägglund, T. PID Controllers: Theory, Design, and Tuning, 2nd ed.; Instrument Society of America: Durham, 1995.

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Prett, D. M.; Morari, M. The Shell Process Control Workshop; Butterworth Publishers: Houston, 1987.

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Roshany-Yamchi, S.; Escaño, J. M.; Canty, N.; Cornelio, A. A. Game Theory-Based Distributed Predictive Control: Application to Shell Oil Fractionator. In 26th Irish Signals and Systems Conference (ISSC); IEEE, 2015.

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Yamashita, A. S.; Zanin, A. C.; Odloak, D. Tuning of Model Predictive Control with Multi-Objective Optimization. Brazilian Journal of Chemical Engineering 2016, 33 (2), 333.


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Model Predictive Control Tuning Strategy for Non-Square Systems

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