Model-based robust H∞ control of a granulation process using Smith predictor with Reference updating

Journal of Process Control 77 (2019) 38–47

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Journal of Process Control journal homepage: www.elsevier.com/locate/jprocont

Model-based robust H∞ control of a granulation process using Smith predictor with Reference updating T. Shaqarin a,∗ , A.E. Al-Rawajfeh b , M.G. Hajaya c , N. Alshabatat a,d , B.R. Noack e,f,g a

Mechanical Engineering Department, Tafila Technical University, 66110 Tafila, Jordan Chemical Engineering Department, Tafila Technical University, 66110 Tafila, Jordan Civil Engineering Department, Tafila Technical University, 66110 Tafila, Jordan d Mechanical Engineering Department, King Abdulaziz University, Rabigh, Saudi Arabia e Laboratoire d’Informatique pour la Mécanique et les Sciences de l’Ingénieur, LIMSI-CNRS, Rue John von Neumann, Campus Universitaire d’Orsay, Bât 508, F-91403 Orsay, France f Institut für Strömungsmechanik und Technische Akustik (ISTA), Technische Universität Berlin, Straße des 17. Juni 134, D-10623 Berlin, Germany g Institute for Turbulence-Noise-Vibration Interactions and Control, Harbin Institute of Technology Graduate School Shenzhen, 58800 Shenzhen, PR China b c

a r t i c l e

i n f o

Article history: Received 10 May 2018 Received in revised form 4 December 2018 Accepted 12 March 2019 Keywords: Granulation Robust control Smith predictor Mixed sensitivity Reference updating

a b s t r a c t Model-based feedback control is developed for a continuous granulation process addressing the challenge of time delay and physics-based input-output constraints. The process plant is a multi-input multi-output (MIMO) linear model with time delay. A robust H∞ controller is designed using the mixed sensitivity loop shaping design. A framework has been laid down to insure the robustness of the Smith predictor by incorporating the model mismatch as an additive uncertainty in the predictor’s structure. The control performance and robustness is assessed by simulations for regulation and reference tracking problems. We show significant performance gains by employing a Smith predictor and the technique of reference updating: The control is coping significantly better with time delay, physical constraints and model mismatch. The proposed control approach is more efficient as compared to other widely used methods such as model predictive control (MPC); obtaining a stable behaviour of the response and control effort while forcing them to remain within the desired bounds. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Continuous granulation processes are considered to be among the least understood industrial processes. In fact personal experiences and skills of expert operators remain vital in many aspects of the design and operation of the process [1,2]. Granulation is the process of purposely assembling fine particles into larger, multiparticle clusters called granules. It is used in several processes in the chemical and pharmaceutical industries for the purpose of enhancing handling and flow characteristics of the material. Many challenges exist in the field of process design, optimization, and control, and as a result research in the aforementioned process components has grown extremely over the last 50 years [2]. In the work presented here an advanced control method will be established for the control of a continuous granulation system. The granulation system model used here was originally presented by Pottman et al. [3]. Fig. 1 shows a schematic diagram for the studied

∗ Corresponding author. E-mail address: [email protected] (T. Shaqarin). https://doi.org/10.1016/j.jprocont.2019.03.003 0959-1524/© 2019 Elsevier Ltd. All rights reserved.

continuous granulation process. The model represents an industrial wet granulation system where dry feed is sprayed with liquid binding agents, and granules grow in size during processing. Produced granules are then dried and classified based on their granule size. Undersized and oversized granules are recycled to be reprocessed. Product quality indicators in the industrial granulation process are mainly related to two fundamental quantities: bulk density and particle size distribution (establishing upper and lower limits for the granule size). Process control for the system in hand is primarily achieved by adjusting the flow rates through each of the aforementioned spray nozzles, and the specific amount introduced at each location, affecting the mechanism and the rate of particle growth. Typically, the flow rates through each nozzle are easily available for manipulation, while the nozzle locations are fixed by design [3]. The nozzles flow rates are inherently limited to upper and lower values, related to the operational physical constraints of the process itself [4] (i.e. each nozzle has a maximum flow rate capacity therefore has an upper limit, and a minimum flow rate were the spray rate will have no effect on granule size therefore a lower limit). Efficient process control plays an essential role in realizing the preferred attributes of granules. However, it requires complex feed-

T. Shaqarin et al. / Journal of Process Control 77 (2019) 38–47

39

Fig. 1. Granulation process [3].

back control approaches and in-line or offline monitoring of the process. One important challenge in granulation process control is finding an accurate model that is capable of simulating the entire process. This is attributed to the fact that from a systems-theoretic point of view, particulate processes represent a form of infinitedimensional processes. [5]. Thus, in general, a process control needs to be achieved through unified use of characterization methods. These methods dynamically track changes made to the granule characteristics, and modify the process parameters accordingly [6]. Furthermore, the requirements of input-output constraints in order to achieve a smother change in the controlled variables introduces longer operational time [2]. Additionally, complexities and difficulties encountered in inline/online measurement of product parameters in continuous granulation make the control of the processes difficult, and small changes in one variable will affects a number of process outputs [7]. Different control schemes have been discussed in the literature for the granulation process. Pottmann et al. [3] suggested using the Model Predictive Control (MPC) for a drum granulation process, where experimental data was fitted to linear discrete-time model for the process. Their control objectives are tracking the bulk density to a reference value and keeping the particle diameters within limits. Gatzke and Doyle [8] controlled the same model of [3] using MPC with relaxed output constraints and prioritized control objectives. Sanders et al. [9] presented a linear MPC valid for a nonlinear granulation process. The linearized state-space model was derived from a nonlinear discretized population balance model. They showed that MPC provides more stability than generic PID controller. Burggraeve et al. [10] designed a feed-forward control strategy for a top-spray uid bed granulation process. They also presented a partial least squares (PLS) model to predict the end product density. Closed-loop feedback control for continuous wet granulation in SILICO was tested by Singh et al. [11]. They coupled controller parameter tuning strategy with an optimization strategy. The tuning strategy includes an integral of time absolute error method. The disturbance rejection capabilities were investigated. Finally, model free control approaches such as fuzzy logic and artificial neural network were implemented to model and control granulation process [12]. Despite the different approaches used in controlling granulation processes, there is still some opportunities for improvement. Process complexities dictated the use of a MPC approach, but the resulting fluctuations in the controlled variables can have an expensive outcome on the process being controlled [13]; increasing power requirements and reducing the efficiency. Control of the continuous granulation process suffers from the issue of time delay and requirement of input-output constraints. The multi-input multi-output (MIMO) linear model of the granulation

process in hand involves a time delay of the same order of magnitude of the time constants of the model itself. The significance of this time delay cannot be ignored, because it will more likely cause an unstable response of the closed-loop controller. An attractive approach to overcome the time-delay problem is using the Smith predictor (SP)[14]. The Smith predictor was initially introduced to enhance the closed loop performance of single-input single-output (SISO) systems. For the nominal case, the following attributes are introduced by the prediction strategy: (a) the ability of dead-time compensation; removal of the time delay from the characteristic equation, (b) the ability of output prediction for changes in the setpoint; future outputs are estimated by the predictor, and (c) the ability of ideal dynamic compensation; the system is divided into a product of two parts: a delay-free part and a pure time-delay part [15]. The application of the SP was originally extended to be used in multiple-input multiple-output (MIMO) problems in [16]. However, in [17] it was shown that this extension [16] is not unique. Any of the aforementioned SISO attributes can be attained individually or collectively for a specific MIMO dead-time compensation structure. The property of dead-time compensation, also referred to as delay free characteristic equation property, can also be found in several works that include multiple time-delays [18–20]. In this work, the complex control problem of a wet granulation process is addressed utilizing the efficient approach of cancelling the time delay effect in the feedback loop by utilizing a SP, and designing a robust H∞ controller using the well documented mixed sensitivity approach [13]. In addition, the reference updating method is used to cope with the input/output constraints dictated by the nature of the process. 2. Problem formulation A granulation process aims to enlarge the particle size of granular materials to a desired value. As originally discussed by Pottmann et al. [3], for the purpose of a control system design, an approximate linear model can be considered appropriate and thus used to connect the manipulated inputs to the desired outputs. In their original work, the authors used process data to obtain a linear discretetime model and further deduced based on individual step response data that a simple first-order plus time-delay transfer function will effectively represent the behavior of each of the responses. Additionally, the same authors found that changes in the process outputs take place jointly, and by assuming that the times required to obtain measurements of the three outputs are also equal, there is a single time delay associated with each input. Finally, the authors were able to determine individual step response gains and time constants from an open-loop step response data obtained from a

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simulation of the granulation process. As seen in Fig. 1, the process has three inputs (u1 , u2 and u3 ) that can be used for controlling the process. These inputs are the spray nozzles flow rates of the liquid mixture containing binder which are introduced into the granules. The outputs of the process are the bulk density of the product slurry (y1 ), and the product particle size distribution represented by the sizes of granules at the 5th percentile (y2 ) and 90th percentile (y3 ) in it. The aforementioned model variables are defined in terms of the deviation of actual process variables from their respected steady state values. This model was also used by Gatzke and Doyle [8]. As mentioned above, bulk density and particle size distribution are considered among the fundamental product quality indicators in the granulation process, and they are chosen as outputs accordingly. The aforementioned percentiles with their associated sizes represent the lower and upper limits in the desired product particle size distribution, in which at most 5% of produced granules are undersized and at most 10% of produced granules are oversized, resulting in a product yield of at least 85% (i.e. at least 85% of produced granules will have sizes within the required spec). Particular values for the size of granules in both 5th and 90th percentiles are dictated by the size requirements for the final product, and it is related to the specific application where the granulation process is applied. The model is represented by:

⎡

⎤

The control of Granulation process requires the regulation of the bulk density to a certain desired value while keeping the product 90th percentile below 1650 and the product 5th percentile above 350. Meanwhile, all the nozzles flow rates must be maintained between 100 and 340. The control of this granulation process suffers from the time delay problem and requirement of input-output constraints. These constraints are, as mentioned above, related to the maximum and minimum flow rates of the binders (input), and the upper and lower limits on the particle sizes with their respected percentiles dictating a specific particle size distribution (outputs). The multi-input multi-output (MIMO) linear model of the granulation process in hand involves significant time delay, which is of the same order of magnitude of the time constants of the model. Ignoring such time delay in the controller design process will decrease the phase margins of the controller, deteriorating the closed-loop response, and shifts the response toward instability. In this framework we will use Smith predictor (SP) to cope with this time delay. Furthermore, the mixed sensitivity approach is used in the H∞ controller design. That will be implemented on the nominal model without time delay. Finally, a reference updating technique is employed to cope with the constraints problem.

G2,3 (s) ⎦

⎥

3.1. Mixed sensitivity controller design

G3,3 (s)

The mixed-sensitivity problem [21] can be formulated in the general form as depicted in Fig. 2. The reference r is a single exogenous input w, z is a vector of the regulated outputs, e is the error signal and u is the controller output.

G1,1 (s)

G1,2 (s)

G1,3 (s)

G(s) = ⎣ G2,1 (s)

G2,2 (s)

G3,1 (s)

G3,2 (s)

⎢

 Gij (s) =

⎡

ki,j i,j s + 1

0.35

0.30

0.70

1.20

⎤

1.10 1.30 ⎦ ,





(1)



0.58

j = 3 3 3



= gi,j (s)e−j s

0.20

ki,j = ⎣ 0.25



e−j s

⎡

2 2 2

i,j = ⎣ 3 4

z

⎤

e

3 3⎦,

0.22

0.64

(2)

ki,j = ⎣ 0.25 1.10



0.33 0.77

j = 2 3 4



0.38

⎤

1.30 ⎦ , 1.32

⎡

2.2 2.2 2.2

i,j = ⎣ 3

4.4

3

 w

=P

u



 ⇒

z

e

=

P11

P12

P21

P22

 w

(4)

u

u = Ke

4 4

where k is the steady state gain,  time constant,  time delay, ˆ i number of outputs and j number of inputs. Let us define G as the system without time delay. The model steady state values, inputs, outputs, and constraints were assigned dimensionless values as presented in the original model [3] The dimensionless presentation of the aforementioned parameters was maintained in the work of Gatzke and Doyle [8]. In this work the above mentioned parameters were assigned dimensionless values in order to allow for a better comparison of the results in the suggested controller approach to the results of previously published ones. The steady state values for y1 , y2 and y3 are 40, 400, and 1620, and for u1 , u2 and u3 are 175, 175, and 245, respectively. This nominal model is used for the controller and Smith predictor design. To test the controller robustness, the nominal model parameters are perturbed by ∼10% of its nominal values. Moreover, to check the efficiency of the predictor time delays are perturbed by ∼33% of its nominal values. The mismatched model parameters are as follows:

⎡

3. Controller design

3

⎤ ⎦,

4.4 4.4 (3)

(5)

Here, P(s) is the generalized plant and K(s) the controller. Specific transfer functions are denoted by their subscript: For example, Pzw is the transfer function from w to z. The plant reads

⎡ P=

P11

P12

P21

P22

=

Pzw

Pzu

Pew

Peu

W1

⎢ ⎢ 0 =⎢ ⎢ 0 ⎣ I

ˆ −W1 G

⎤

⎥ ⎥ ⎥ ˆ ⎥ W3 G ⎦ W2

(6)

ˆ −G

were W1 , W2 and W3 are frequency-dependent weights. Then, the closed-loop transfer matrix Tzw is defined by Tzw = P11 + P12 K(I − P22 K)−1 P21

(7)

ˆ This matrix can be rewritten with the loop gain L = GK

⎡ ⎢

W1 So

⎤ ⎥

Tzw = ⎣ W2 KS o ⎦

(8)

W3 To where So = (I + L)−1 is the output sensitivity transfer function, To = I − So is the output complementary sensitivity transfer function and Ro = KSo is the output control sensitivity function. The mixed sensitivity technique computes a controller K that minimizes the H∞ norm of the closed-loop transfer function Tzw of the weighted mixed sensitivity problem, such that: Tzw ∞ ≤ 

(9)

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41

Fig. 2. Mixed sensitivity problem.

Fig. 3. Block diagram of the closed-loop system.

3.2. Smith predictor Feedback loops tend to become unstable when the time delay is large as compared to the time constant of the system (see Fig. 3b). In this case, the controller attempts to react for past events and the control authority is reduced for many reasons. Even worse, the controller may overreact on past events, thus amplifying deviations from the desired output. The Smith predictor [14] is the most common technique for time delay compensation. The structure of this predictor is depicted in Fig. 3a. The controller output u is fed through a model of the process and through the same model without dead time This way, the

controller acts on a simulated process which behaves as if there was no dead time in the process—of course in the ideal situation of a perfect model. Consider the model in Eq. (1), for which we seek the following decomposition









ˆ ˆ −s ˆ G(s) = gi,j (s)e−j s = gi,j (s) e−s = G(s)e

(10)

ˆ is the rational delay-free part. Such simple decomposition where G is very useful for our model, since the nominal time-delays  j all have been assigned the same value as can be seen from Eq. (2).

Fig. 4. Block diagram of the uncertain closed-loop system.

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T. Shaqarin et al. / Journal of Process Control 77 (2019) 38–47

and the family of the plant associated with this uncertainty type is as: A :

ˆ Gp (s) = G(s) + WA (s)A (s);

A ∞ ≤ 1

(14)

where A is an arbitrary stable transfer function and WA is a stable proper rational weighting transfer function. Since the class of uncertainty employed corresponds to the additive uncertainty, the associated robust stability according to [21] is given by the following condition: WA KS o ∞ ≤ 1

Fig. 5. Smith predictor with Reference updating.

Based on Fig. 3b, the closed-loop transfer function is written as follows: H(s) =

Y (s) K(s)G(s) = ˆ R(s) −s ˆ ˆ 1 + K(s)G(s) + K(s)[G(s) − G(s)e ]

(11)

From Eq. (11), it can be seen that stability of the closed-loop system depends on the time delay and model mismatch. In the nominal case, when there is no model mismatch and the actual ˆ −s ˆ G(s) = G(s)e ˆ resulting in a stable time delays  j are all equal to , closed-loop system. The closed-loop transfer function in Eq. (11) can be rewritten as H(s) =

Y (s) K(s)G(s) = R(s) ˆ 1 + K(s)G(s)

(12)

The closed-loop transfer function in Eq. (12) is limited to the nominal case in ideal conditions, where there is no model and/or time-delay mismatch. To overcome such limitations, it is necessary to compensate for the model parameters or time-delay uncertainties. Instead of only designing the controller K(s) based on the ˆ rational part of the nominal model (G(s)), the Smith predictor is reorganized as shown in Fig. 4b. In such formulation of the SP, the model mismatch can be easily modelled as additive uncertainty as shown in Fig. 4b. Therefore, the generalized system can be represented as: A :

ˆ Gp (s) = G(s) + EA (s);

ˆ EA (s) = G(s) − G(s)e

ˆ −s

(13)

(15)

It is obvious from Section 3.1 that the term WA KSo is equivalent to the term W2 KSo in the mixed sensitivity problem where the control sensitivity function is involved. Consequently, minimizing the infinity norm in the mixed sensitivity problem implies minimizing the infinity norm of WA KSo . Hence, making the system robust against additive uncertainty. 3.3. Selection of the frequency-dependent weights The selection of the weights improves the controller design significantly. In general, the sensitivity function S is made small at low frequency. This yields a good disturbance rejection and small tracking error. In the high-frequency region, the complementary sensitivity function T is made small too. This gives rise to a good noise rejection and large stability margin. More details can be found in [21,22]. According to [22], The sensitivity and complementary sensitivity weightings are designed as





W1 = diag WS11 , WS22 , WS33 ;

WSii =

s MSi

+ ωSi

s + ASi ωSi

and W3 = WT I3×3 ;

WT =

s+

ωT MT

AT s + ωT

where ωS and ωT are the closed-loop bandwidth based on S and T, respectively. MS , MT ,AS and AT define the upper bounds in high and low frequency ranges for the sensitivity functions S and T.

Fig. 6. Robustness control design and analysis of additive uncertainty: (6a) WA11 as an upper bound of the singular values of EA1 by varying the time delays ␪j by ± 30% and the time constants ␶ij by ±10%; (6b) Checking robust stability with additive uncertainty. Solid lines, from top to bottom, correspond to y3 , y2 and y1 .

T. Shaqarin et al. / Journal of Process Control 77 (2019) 38–47

43

Fig. 7. Closed-loop response of the granulation system with/without Smith predictor. Dashed-dotted lines represent: The upper and lower constrains for u1 , u2 , and u3 , and the minimum size for the 5th percentile and maximum size for 90th percentile.

As was discussed in Section 3.2, W2 = WA . The proposed design of WA consists of a square diagonal matrix of transfer function:



WA = diag WA11 , WA22 , WA33



where the transfer function WAii must be stable, minimum phase and with module greater than the maximum singular value of the uncertainty previously discussed for each frequency, that is, |WA (jω)| ≥ ¯ (EA (jω))

∀ω

The condition can be relaxed such as: |WAii (jω)| ≥ ¯ (EAi (jω))

∀ω

where EAi is the ith output of EA . The inverse of WAii is the upper bound of KSoii as a result of the robust stability condition in Eq. (15). 3.4. Reference updating The granulation process is an excellent candidate for reference updating technique, since only one state should be regulated (y1 ) and the other two states (y2 , y3 ,) can be varied without exceeding

certain limits. The basic idea of reference updating (see Fig. 5) is that the control effort can be limited between upper and lower bounds via changing the reference simultaneously. Intuitively, one of the unregulated references can be a function f(u) or one of its derivatives in order to obtain faster dynamics, such that: ˙ runreg. = Kr f (u, u)

(16)

where runreg. represents the unregulated references (r2 , r3 ), and Kr is a proportional gain that is designed from the steady state inputoutput relationship.

⎡

⎤ ⎡ u1 ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ y2 ⎦ = ⎣ 0.25 1.10 1.30 ⎦ ⎣ u2 ⎦ y1

y3

⎤

ss

⎡

0.20

0.58

0.35

0.30

0.70

1.20

u3

(17)

ss

where ss stands for steady state, Kr is chosen in such a way that when the control effort u is approaching towards one of the bounds (decreasing or increasing), the unregulated reference is changed accordingly in a way that drives the control effort in the opposite direction, to maintain the bounds of the control effort.

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T. Shaqarin et al. / Journal of Process Control 77 (2019) 38–47

Fig. 8. Closed-loop response of the granulation system with Smith predictor and reference updating. Dashed-dotted lines represent: The upper and lower constrains for u1 , u2 , and u3 , and the minimum size for the 5th percentile and maximum size for 90th percentile.

By left-multiplying both sides of Eq. (17) with the inverse of the gain matrix, and assuming zero steady state error (r = yss ):

⎡

u1

⎤

⎡

5.1282

⎢ ⎥ ⎣ u2 ⎦ = ⎣ 1.1257 u3

ss

−1.9387

−5.6410 1.6886 0.4253

⎤ ⎡ r1 ⎤ ⎢ ⎥ −2.1576 ⎦ ⎣ r2 ⎦ 4.6154

0.9381

(18)

r3

rlower < runreg. < rupper ulower < u < uupper

As an example, if we are interested in minimizing u1 for positive r1 , one possible solution is the following: r2 should be positive and r3 should be negative as it appears from Eq. (18). Eq. (18) and (16), will yield the following objective function: u = f (Kr , runreg. )

The optimization is as follows: Design Kr such that the objective function (19) is minimized, and subject to the following constraints:

(19)

The lower and upper limits refer to the limits discussed in Sec. 2. The aforementioned optimization routine is terminated when the following criteria is satisfied: o (runreg. − runreg. ) < ı

T. Shaqarin et al. / Journal of Process Control 77 (2019) 38–47

45

Fig. 9. Closed-loop response of the nominal/mismatched granulation system with Smith predictor and reference updating. Dashed-dotted lines represent: The upper and lower constrains for u1 , u2 , and u3 , and the minimum size for the 5th percentile and maximum size for 90th percentile.

where ı represent the maximum deviation in the final product specification.

4. Results In order to assess the performance of the suggested control approach, typical simulations of the granulation process are presented here. The H∞ controller is designed using the MATLAB function ”mixsyn” that solves the mixed sensitivity problem discussed in 3.1, where the minimum  was found 1.6813. The controller k was designed such that H∞ norm of the closed loop transfer function is minimized through the minimization of gamma using the bisection technique. The sensitivity and complementary

sensitivity weightings were selected with the appropriate bandwidths as follows:

WS11 = WT =

s + 0.6 , 2s + 0.0006

WS22 = WS33 =

s + 0.4 , 2s + 0.0004

s+1 0.001s + 2

The additive uncertainty weights were designed as the upper bound of the maximum singular values of the additive uncertainty EA . The Uncertainty was generated via varying the time delays  j by ± 30% and the time constants  ij by ± 10% for the actual model

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T. Shaqarin et al. / Journal of Process Control 77 (2019) 38–47

(G(s)). For example, the design of WA11 is shown in Fig. 6a. Then, the uncertainty weights were selected as follows: WA11 =

0.0024638(s + 3746)(s + 0.0007296) (s + 0.2541)(s + 37.44)

WA22 =

0.0061594(s + 3746)(s + 0.0007296) (s + 0.2541)(s + 37.44)

WA33 =

0.012319(s + 3746)(s + 0.0007296) (s + 0.2541)(s + 37.44)

The robust stability is illustrated in Fig. 6b. The Figure shows the singular values plot for WA KSo . It can be seen from the figure that the stability robustness is generally satisfied for the condition (||WA KSo ||∞ < ). However, a slight violation for the condition in Eq. (15) existed for the outputs y3 and y2 within the range of frequencies (0.1 < ω < 38) and (0.15 < ω < 0.5) rad/s, respectively. This can be attributed to the fact that  is slightly higher than 1. The designed controller is acceptable for the problem in hand, since the regulation of the output y1 is the major concern. Initially, the closed-loop simulation of the granulation process with and without Smith predictor is depicted in Fig. 7. The Figure shows the time histories of the bulk density (y1 ) of the product slurry, the product particle size of particles in the 5th percentile (y2 ), the product particle size of particles in the 90th percentile (y3 ), and the flow rate of the three nozzles (u1 , u2 , and u3 ); i.e. the control effort, to a step reference for the bulk density. As it seen in the Figure, steady state values were reached and maintained within the controller limits with and without the implementation of SP. However, with implementation of SP did in fact eliminate any oscillations from the responses of y1 , y2 , and y3 . In addition, steady state values were reached sooner for the aforementioned variables compared to the response without implementation of SP. Despite improvements in the performance, the H∞ controller with and without implementation of SP significantly violate the quality constraints. As seen in Fig. 7, It is clear that u1 exceeds the upper bound resulting in a violation of the required constraints. In order to overcome any violation in constraints from the controller response, as seen above, the reference updating method is employed as a good choice to limit the controller action u1 via the unregulated references (r2 and r3 ), as presented in Sec. 3.4. In order to simplify the optimization routine for the 2 unregulated references one of them can be selected following Eq. (18), while the other is optimized. Thus the following is assumed: r2 = 0.6r1 and r3 = Kr u1 , and Kr was designed to be Kr =−0.1, so that u1 have a negative action on r3 and consequently u1 is decreased. Fig. 8 depicts the closed loop response of the granulation process for the H∞ controller with SP and reference updating. As is it shown, the implementation of the latter was able to limit the control action regarding u1 and limiting it to its bounds, with the steady state values reaching the desired values. In addition, the control effort for u1 , u2 , and u3 was efficient, achieving the desired values without any oscillations. Fig. 8 also compares the result from the current technique with the closed-loop results for the same granulation system using prioritized objective MPC control approach [8]. It is clear that the results from the current method are more efficient in the following two main aspects: stability of the controlled variables (y1 , y2 , and y3 ) and following the input/output constraints. The ability for reference tracking is also tested for this control technique as shown in Fig. 9. All responses (y1 , y2 , and y3 ) were able to track the imposed references efficiently using the proposed controller. Meanwhile, all the control efforts (u1 , u2 , and u3 ) remained within bounds. The Figure also compares the closed-loop response based on the nominal model (Eq. (2)) with closed-loop response based on a mismatch model (Eq. (3)) using the same controller.

In order to test the controller robustness and the ability of the predictor to cope with the mismatched time delays, a comparison was carried out to test the performance of the controller against nominal and mismatched models. The simulation results are shown in Fig. 9. It can be seen that the controller was robust against 10% of model parameter variations and also the predictor successful dealing with 33% of time delay variation. The results in Fig. 9 agrees with results from Fig. 6b, since the deteriorations due to modelmismatch was relatively higher in the case of the output y3 than the outputs y2 and y1 . It must be added that the closed loop response for y3 in the mismatched case settled faster than the nominal case, because the reference of the mismatched case is deferent than the nominal case as seen in control effort u1 (r3 = Kr u1 ).

5. Conclusions An efficient, successful closed-loop control for a continuous wet granulation process was achieved. The control approach involved the design of H∞ controller, Smith predictor and reference updating to cope with model mismatch, time delay problem and input/output physically imposed constraints, respectively. Closed-loop simulations showed that the use of SP eliminated the oscillations from the response and the control effort. In addition, the proposed reference updating method was proven to keep the response and control effort within the desired bounds. The simulations also showed that the proposed controller was robust against model and time delay mismatch. The suggested control approach was found superior regarding its efficiency compared to other widely used control techniques such as model predictive control.

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