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H∞ Loop Shaping Control for Continuous Fluidized Bed Spray Granulation with Internal Product Classification Stefan Palis*,† and Achim Kienle†,‡ †

Otto-von-Guericke-Universität, Universitätsplatz 2, D-39106 Magdeburg, Germany Max-Planck-Institut für Dynamik Komplexer Technischer Systeme Sandtorstrasse 1, D-39106 Magdeburg, Germany

‡

ABSTRACT: This paper is concerned with stabilizing open loop unstable fluidized bed spray granulation with internal product classification by means of feedback control. Using the results of a detailed model analysis, that is, numerical bifurcation, controllability and observability analysis and a study of the gap metric, an appropriate combination of manipulated and control variables is proposed. Using a reduced transfer function model a robust controller is designed by means of H∞ loop shaping methods. The controller is validated by simulation of the nonlinear plant model using an appropriate start-up strategy.

1. INTRODUCTION Fluidized bed spray granulation is a particulate process, where a bed of particles is fluidized, while simultaneously injecting a solid matter solution. As process air temperatures are typically high, the fluid evaporates and the remaining solid material either forms new nuclei or contributes to growth of already existing particles. For a continuous production an ongoing nuclei production is required. In Vreman et al.4 the authors proposed a model for a continuous fluidized bed spray granulator with internal product classification, where nuclei production is due to spray drying. For certain ranges of the operating parameters regions of instability, resulting in nonlinear oscillations of the particle size distribution, have been identified. These oscillations give normally undesired time behavior of product quality. Similar patterns of behavior have been observed for other particulate processes as continuous fluidized bed spray granulation with external product classification and material recycles2,3 and crystallization processes.13 So far the main emphasis has been on crystallization processes. Here several approaches for stabilizing control design have been proposed, using linear (e.g., ref 10) and nonlinear (e.g., ref 11) lumped models or linear (e.g., ref 12) and nonlinear infinite dimensional models (as in ref 9). For fluidized bed spray granulation with external product classification a linear and nonlinear control design were proposed recently in refs 7 and 8. In this contribution a control design for fluidized bed spray granulation with internal product classification is investigated. Owing to the different mechanism of oscillation, different handles and control structures have to be used. Therefore, an appropriate control structure selection for a fluidized bed spray granulation with internal product classification is a major goal to be achieved in this paper. In a first step, linear robust control theory based on a lumped plant model is used. For this purpose, the population balance model is discretized, resulting in a set of nonlinear ordinary differential equations. For this lumped model set points are calculated by means of continuation methods. Linearizing the lumped system around the calculated set points, an observability and controllability analysis is performed. Taking into account the results from an analysis of the gap metrics an appropriate controlled variable is identified. For a certain set point a model reduction is performed, resulting in a low order © 2012 American Chemical Society

linear transfer function. Using H∞ loop shaping theory a controller is derived for this reference point guaranteeing robust stability. In combination with an appropriate start-up strategy the proposed controller is tested by means of simulation studies. Table 1 gives the values for the plant parameters used in this study. Table 2 defines the notations used in the calculations. Table 1. Plant Parameters 5 × 106 mm2 440 mm 0.5 1.67 × 105 mm3/s 0.028 0.3 mm 0.7 mm 1.92 × 10−4 1/s

A hnoz ϵ V̇ e b∞ L0 L1 K

2. CONTINUOUS FLUIDIZED BED SPRAY GRANULATION WITH INTERNAL CLASSIFICATION The process scheme of the continuous fluidized bed spray granulation with internal classification is depicted in Figure 1. The granulator consists of a granulation chamber, where the particle population is fluidized through an air stream and coated by injecting a suspension with a volume flow V̇ e. The associated particle growth has been described in ref 1. To account for internal formation of nuclei the growth rate G has been modified in Vreman et al.4 There, it is assumed that only a certain part of the injected suspension ((1 − b)V̇ e)) contributes to the particle growth, while the rest (bV̇ e) results in new nuclei. 2(1 − b)Vė 2(1 − b)Vė = G= ∞ 2 πμ2 π ∫ L n dL

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0

Received: Revised: Accepted: Published: 408

October 11, 2011 November 19, 2012 November 21, 2012 November 21, 2012 dx.doi.org/10.1021/ie202327x | Ind. Eng. Chem. Res. 2013, 52, 408−420

Industrial & Engineering Chemistry Research

Article

is reached, when the bed reaches the height of the nozzle. For bed heights beyond the nozzle the nucleation parameter b is assumed to remain constant at b∞. In the limit of vanishing bed height it is assumed that 100% of the injected suspension forms new nuclei giving a nucleation parameter b = 1. As can be seen in Figure 2 the nucleation parameter b is interpolated linearly between the two limiting situations h = 0 and h = hnoz resulting in the following relationship

Table 2. Notation μi V̇ e G B ṅ* n L A W1(s) W2(s) G(s) ρ δ(L) σ(L) δg A,B,C,D M,N product s r Δ

ith moment, i.e., ∫ 0∞Lin dL suspension injection rate growth rate birth rate particle flux number density distribution diameter overall surface precompensator postcompensator transfer function spectral radius Dirac delta function Heaviside step function gap metric system matrices of state space model denominator and numerator of transfer function Subscripts product particles steady state reduced model uncertain model set

⎛ h − h⎞ b = b∞ + max⎜0, (1 − b∞) noz ⎟ hnoz ⎠ ⎝

(4)

Figure 2. Dependence of the nucleation parameter b on bed height h according to Vreman et al.4.

In the continuous configuration of the fluidized bed spray granulation with internal classification, particles are continuously removed through an air sifter with counter current flow, which separates smaller from larger particles. The large particles pass the air sifter while the small particles are reblown into the granulation chamber. The sifting diameter L1 gives the minimal particle diameter for particles, which can pass through the sifter. The classifying product removal therefore reads n prod = Kσ(L − L1)n ̇

where K is the drain constant. To describe the process, a population balance model for the particle size distribution has been proposed recently in ref 4 consisting of the following particle fluxes: (i) ṅprod particle flux due to product removal, and (ii) B particle flux due to nuclei formation, and particle growth associated with the size independent growth rate G according to eq 1.

Figure 1. Process scheme.

Nucleation results from spray droplets, which completely dry before hitting existing particles in the bed. Here, it is assumed that nuclei are formed with a characteristic diameter L0. bV̇ B = 1 e 3 δ(L − L0) πL 0 (2) 6

∂n ∂n = −G − n prod +B ̇ ∂t ∂L

V (1 − ϵ)A

(6)

Starting with an initial particle size distribution as depicted in Figure 13, which is the steady state particle size distribution for V̇ e = 1.67 × 105 mm3/s, this model shows interesting dynamical behavior. For sufficiently high suspension injection rates and an associated bed height higher than the nozzle height, transition processes decay and the particle size distribution reaches a stable steady state as shown in Figure 3. When the suspension injection rate is decreased below a critical value the steady state becomes unstable giving rise to nonlinear oscillations as depicted in Figure 4. Here, the associated mechanism is as follows: (1) For a bed height smaller than the nozzle height an increased nuclei production takes place due to spray drying. (2) This results in a large number of small particles and a reduced growth rate. (3) After a certain time the bed height reaches the nozzle height,

The nucleation parameter b, which determines how much of the injected suspension results in new particles, is assumed to depend only on the bed height h, which can be obtained from h=

(5)

(3)

where ϵ is the bed porosity. In the following the bed porosity ϵ is assumed to be constant. With increasing bed height h the free distance for the spray droplets decreases resulting in a decreasing nuclei formation. The minimum of the nucleation parameter b∞ 409

dx.doi.org/10.1021/ie202327x | Ind. Eng. Chem. Res. 2013, 52, 408−420


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Figure 3. Open loop simulation in the stable region V̇ e = 0.96V̇ e,0.

version of the population balance eq 6 with 150 equidistant spatial grid points. At first the systems steady states are calculated depending on the suspension injection rate V̇ e by continuation methods. Therefore, an initial steady state particle size distribution ns,0 is generated by time integration for a nominal suspension injection rate V̇ e,0 = 1.67 × 105 mm3/s, for which a stable steady state is found. Then, using this initial steady state solution ns,0 as a prediction for a new steady state solution np,1 for a different suspension injection rate V̇ e,1, a steady state solution ns,1 is derived by a corrector step involving numerical minimization of the L1-norm of the residuals of dni/dt. This is repeated for a successively decreasing V̇ e from V̇ e,0 to 0.8 × V̇ e,0. Along this branch the local stability of the computed steady states is determined by solving the eigenvalue problem for the linearized system. At a certain point V̇ e,BP two conjugated complex eigenvalues occur in the right-half plane. Beyond this point the steady states solutions are unstable. Further investigations of the time behavior in this region show that a stable limit cycle occurs. The described behavior is depicted in Figure 5, where thick continuous lines represent stable stationary solutions, dashed lines unstable stationary solutions and dots the

resulting in a small and constant production of nuclei and a higher growth rate. (4) When the peak of the particle size distribution reaches the particle radius, where the particles pass the air sifter with counter current flow, L1, the associated particles are removed from the granulator. This is connected with a decrease of the bed height below the nozzle height and hence the process repeats. In contrast, a high suspension rate results in a permanent high nuclei production, a higher growth rate, and therefore a bed height being bigger than the nozzle height. Hence, after a transition time the steady state particle size distribution is reached and no oscillations occur.

3. BIFURCATION ANALYSIS It is well-known, that qualitative properties as stability of equilibria, existence of limit cycles etc., can change for a nonlinear dynamic system under parameter variations.5,3 In particular this phenomenon has been observed for the current process configuration.4 Therefore, before designing an appropriate control structure a detailed bifurcation analysis is useful. Here, we are using the finite volume method to derive a semidiscretized 410



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Figure 4. Open loop simulation in the unstable region V̇ e = 0.9V̇ e,0.

rank(R ) = rank([ B AB A2 B ... AN − 1B ]) = N

maximal and minimal amplitudes of the observed oscillations, that is, limit cycles. As can be seen from the bifurcation diagram in Figure 5 the moments μ0, μ1, and μ2 are not monotone functions of V̇ e, which would result in uniqueness problems when choosing V̇ e as a control input and one of these moments as the controlled variable. This phenomenon, that is, the abrupt change in the slope of the bifurcation curves and the loss of monotonicity, is connected to the nozzle height hnoz and occurs when the bed reaches the height of the nozzle as can be seen in Figure 5. 3.1. Controllability and Observability Analysis. Before deriving a control law, which is capable of stabilizing the steady state solution in the unstable region of V̇ e, the appropriate control in- and outputs have to be chosen. As a candidate for a control input one could choose the suspension injection rate V̇ e, which has a direct influence on the nuclei production. A criterion to check, whether this choice is appropriate for this configuration, is to check for controllability of the state n by the input u = V̇ e. Here, we use the family of linear systems derived along the steady state continuation path. Then, for each state space model the controllability matrix R has to have rank N, where N is the dimension of the A matrix.

(7)

As the numerical rank evaluation is sensitive, the staircase algorithm has been used to transform the system into its controllability staircase form17 ⎡ Ac A12 ⎤⎡ xc̅ ⎤ ⎡ Bc ⎤ ⎥⎢ ⎥ + ⎢ ⎥ u ẋ = ⎢ ⎣ 0 A uc ⎦⎣ xuc̅ ⎦ ⎣ 0 ⎦

(8)

⎡ xc̅ ⎤ y = [Cc Cuc ]⎢ ⎥ + Du ⎣ x uc ̅ ⎦

(9)

Here, the rank of Auc is equal to the number of uncontrollable states. Checking that all states are controllable for the family of linear systems derived along the steady state continuation path yields, that the state n is linear controllable by V̇ e. As the intention is to design a controller, which uses only moment measurements, one has to check for observability of the state n with respect to the candidate measurement μ0, ..., μ3. This is done again for the family of linear models using the dual criterion for observability, that is, the observability matrix B has to have rank N, where N is the dimension of the A matrix. 411



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Figure 5. One parameter bifurcation diagrams depending on V̇ e.

⎛⎡ C ⎤⎞ ⎜⎢ ⎥⎟ ⎜⎢CA ⎥⎟ rank(R ) = rank⎜⎢CA2 ⎥⎟ = N ⎜⎢ ⎥⎟ ⎜⎢⋮ ⎥⎟ ⎜⎢ N − 1 ⎥⎟ ⎝⎣CA ⎦⎠

that the state n is linearly observable using an arbitrary value of the moment measurement μ0, ..., μ3.

4. CONTROL DESIGN The detailed one-parameter bifurcation analysis gives that the process time behavior strongly depends on the chosen suspension injection rate V̇ e. For sufficiently high values of V̇ e the particle size distribution reaches a stable steady state. Decreasing the suspension injection rate V̇ e to a critical value and below leads to a loss of stability for the steady state solution and a stable limit cycle occurs. The stable limit cycle is associated with self-sustained oscillations, which are undesirable in process operation. Therefore, in the following a control structure will be designed, which is able to stabilize the process. As we showed by controllability analysis the suspension injection rate V̇ e can be used as a control input. This is somehow in accordance with our earlier results on control of fluidized bed spray granulation with external classification and mill cycle, where we choose the mill grade as the principal parameter for nuclei production as control input.7,8 In both cases, the nucleation process can be directly controlled by the corresponding handle.

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Again, the staircase algorithm has been used to transform the system into its observability staircase form,17 due to sensitivity of the numerical rank evaluation. ⎡ Ao 0 ⎤⎡ xõ ⎤ ⎡ Bo ⎤ ⎥⎢ ⎥ + ⎢ ⎥ u x∼̇ = ⎢ ̃ ⎦ ⎣ Buo ⎦ ⎣ A 21 A uo ⎦⎣ x uo

(11)

⎡ xõ ⎤ y = [Co 0 ]⎢ ⎥ + Du ̃ ⎦ ⎣ x uo

(12)

Here, the rank of Auo is equal to the number of unobservable states. A check that all states are observerable for the family of linear systems derived along the steady state continuation yields 412



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Figure 6. Gap metric G1(s) (left) G2(s) (right).

Observability analysis gives that the particle size distribution is observable through any moment μ0, ..., μ3. So, from this point of view we could use any of them as the controlled variable. To decide which measurement is most appropriate for a control design a similarity analysis using the concept of gap metrics is done. 4.1. Analysis of the Gap Metrics. From a control perspective, it would be desirable if the family of linear systems derived along the steady state continuation could be embedded

into a set of perturbed plants. Where the set of perturbed plants can be described by a nominal system Gnom and a set of bounded, stable uncertainties. As this set should reflect the change of stability, the most appropriate set of uncertainties is the set of normalized coprime factor uncertainties. Here, the plant is represented by its normalized left coprime factorization with additive uncertainties ΔM(s), ΔN(s) in each factor. GΔ(s) = (M(s) + ΔM (s))−1(N (s) + ΔN (s)) 413

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Figure 7. Control scheme.

As, in contrast to multiplicative or additive model uncertainties, coprime factor uncertainties do not give a unique realization for ΔM(s) and ΔN(s). Hence, for a given plant to be embedded, the choice of ΔM(s) and ΔN(s) is an additional degree of freedom. Therefore, one could derive a ΔM(s) and ΔN(s), which gives a minimal H∞-norm for [ΔN ΔM]. δg⃗ (Gs , G):=

inf

{||[ΔN ΔM ]||∞ :

[ΔN ΔM ] ∈ /∞

G = (M(s) + ΔM (s))−1(N (s) + ΔN (s))}

(14)

As δ⃗g is not symmetric in its arguments, the gap metric δg is introduced.16 δg(Gs , G) = max{δg⃗ (Gs , G), δg⃗ (G , Gs)}

(15)

Using this gap metric two systems G1(s) and G2(s) are close if the associated value of the gap metric δ⃗g(G1(s), G2(s)) is close to zero, implying that both can be embedded in a family of linear models using a nominal model and a small, with respect to H∞norm, coprime factor uncertainty. The maximum gap metric is 1. In the following, four families of linear models are generated, each of them using a different measurement y = μ0, ..., μ3. For

Figure 8. Hankel singular values of G(s).

The normalized left coprime factor uncertainty is assumed to be stable with ∥[ΔN ΔM]∥∞ < ϵ.

Figure 9. Full transfer function G(s) (solid black) and reduced model Gr(s) (dotted gray). 414



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Figure 10. Open loop shape (dotted gray) and desired loop shape (solid black).

Figure 11. Bode plot of H∞ loop shaping controller.

result in additional uniqueness problems. Therefore the choice of μ3 as the controlled variable is the most appropriate, as μ3 is monotone with respect to V̇ e and the family of the associated linear systems can be embedded into a set of plants consisting of a nominal model and a coprime factor uncertainty. For better control performance, an additional feed forward component has been added. Here, the steady state value of the suspension injection rate V̇ e associated to the desired third moment of the particle size distribution μ3,d is calculated using

each family two nominal models associated to two different suspension injection rates V̇ e,1 = 16059 mm3/s (stable region) and V̇ e,2 = 14709 mm3/s (unstable region) are used as a nominal model. Then the gap metric with respect to each nominal model G1,· and G2,· is calculated. As can be seen in Figure 6 choosing μ0, μ1, or μ2 as a measurement would result in configurations where only parts of the family of linear systems could be embedded. In addition, it has been shown earlier that μ0, μ1, and μ2 are not monotone with respect to the potential input V̇ e, which would 415



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Figure 12. Desired (solid black) and achieved loop shape (dotted gray).

Using the following abbreviations H = −(BDT + ZCT)R−1

R = I + DDT

(19)

the normalized left coprime factorization can be obtained by Laplace Transformation of the following state space model

Using this factorization each coprime factor is reduced by standard balanced truncation.6 Gr(s) = M r−1(s) Nr(s)

Figure 13. Initial condition.

As can be seen from the diagram of the Hankel singular values σi in Figure 8 a truncation up to order 5 is reasonable. The additive approximation error |G − Gr| is small over the whole frequency range of interest as shown in the Bode diagram in Figure 9. The reduced transfer function model Gr(s) is

the stationary solution. The overall control scheme is depicted in Figure 7. 4.2. Model Reduction. For a direct control design, the linear system used so far is too complex and would result in a control law of unnecessary high order. To reduce the order of the design model a reduction procedure based on the normalized coprime factorization is used.6,14 Therefore the nominal plant with V̇ e as control input and Δμ3 as the controlled variable is transformed into its normalized left coprime factor realization. G(s) = M −1(s) N (s)

Gr(s) = [2(s + 9 × 10−5)(s + 1.87 × 10−6) (s 2 + 0.0003s + 1.5 × 10−7)] /[(s + 1.91 × 10−6)(s 2 − 0.00013s + 1.2 × 10−7)(s 2 + 0.0007s + 2.9 × 10−7)]

(16)

(22)

Using the state space realization of the transfer function G(s) the normalized left coprime factorization can be calculated by solving an algebraic Riccati equation in Z

4.3. H∞ Control Design. To stabilize the particle size distribution in the whole range of V̇ e, that is, the family of linear models derived by linearization along the steady state continuation, in the following a control law is derived using H∞ loop shaping methodology.14,15 Here the plant is represented by its normalized left coprime factorization with additive uncertainties ΔM(s), ΔN(s) in each factor.

(A − BS −1DTC)Z + Z(A − BS −1DTCT)T − ZCTR−1CZ + BS −1BT = 0

(17)

where S = I + DTD

(21)

GΔ(s) = (M(s) + ΔM (s))−1 (N (s) + ΔN (s))

(18) 416

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Figure 14. Start up with and without control.

G(s) = M(s)−1 N (s)

The normalized left coprime factor uncertainty is assumed to be stable with ∥[ΔN ΔM]∥∞ < ϵ. It is well-known that a controller K robustly stabilizes the perturbed feedback system if and only if it stabilizes the nominal system

(24)

and 417



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Figure 15. Disturbance response.

⎡K ⎤ −1 −1 ⎢⎣ ⎥⎦(I + GK ) M I

≤ ∞

1 ϵ

as it is not restricted to perturbations which preserve the number of right half-plane poles of the plant. This fact is crucial for the control of continuous fluidized bed spray granulation as stability behavior changes depending on the specific operating conditions, that is, suspension injection rate V̇ e.

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A coprime factor uncertainty representation is in general superior over others, for example, additive or multiplicative uncertainties, 418



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For a given plant the maximal achievable stability margin ϵmax is given by ϵmax = (1 + ρ(XZ))−1/2

mm3/s); as soon as the stable steady state is reached the control loop is closed and the set point is shifted to the desired open loop unstable operating region (here V̇ e = 13360 mm3/s). The second approach has been implemented for the system without additional feedback control resulting in increasing oscillations, which would end in the associated limit cycle and for the system with the proposed H∞ controller. As can be seen in Figure 14 oscillations occurring during the shifting are damped in closed loop operation. The particle size distribution and all its moments μ0, μ1, μ2, μ3 are stabilized with reasonable control effort. To investigate the closed loop disturbance behavior the rate of product removal K is increased by 10%. As can be seen in Figure 15 the proposed control scheme can reject this disturbance.

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where X and Z are the solutions of two algebraic Riccati equations (A − BS −1DTC)Z + Z(A − BS −1DTCT)T − ZCTR−1CZ + BS −1BT = 0

(27)

(A − BS −1DTC)T X + X(A − BS −1DTCT) − XBT S −1BX + CR−1CT = 0

(28)

with R = I + DDTS = I + DTD

5. CONCLUSION AND FUTURE WORK A controller for continuous fluidized bed spray granulation with classifying product removal has been developed. For the control design the main steps are model discretization, bifurcation analysis, and linearization of the discretized model, design of the control structure, model reduction, control design and the development of a start-up strategy. The proposed control strategy was validated by means of simulation of the nonlinear plant model. A good control performance was observed with relatively small deviations from the reference trajectory during start-up, which demonstrates good reference tracking for reasonably fast changes of the reference value. Applying other tradional linear control methods, for example, conventional PI control, would require an a posteriori robustness analysis. The proposed H∞-loopshaping controller guarantees stability over the full range of operating conditions in a small neighborhood of the steady state owing to linearization. This motivates the investigation of more advanced nonlinear control concepts. First steps in this direction have been presented in ref 7 for a fluidized bed spray granulation with external product classification.

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Moreover given an ϵ < ϵmax a controller K achieving a stability margin of ϵ can be calculated by

where F = −S −1(DTC + BT X )

L = (1 − ϵ−2)I + XZ (31)

As the controller K∞ gives only robust stability, usually the preand postcompensator W1(s) and W2(s) are used in order to shape the open loop singular values of the plant before calculating a robustly stabilizing controller K∞. The overall controller K using the pre- and postcompensator W1(s) and W2(s) therefore reads K (s) = W1(s) K∞(s) W2(s)

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The application of H∞ loop shaping design procedure for a shaped plant Gs(s), that is, calculating a controller K(s) for Gs(s) = W2(s) G(s) W1(s) with given ϵ, results in a controller stabilizing all plants G(s) with δg(Gs,G) < ϵ, where δg is the gap metric.16 In the single input−single output case the postcompensator W2(s) can be set to one designing only the precompensator W1(s). For the precompensator the design requirement of zero steady state error requires integral action. The desired loop shape is therefore realized by choosing the following pre- and postcompensator W1(s) = 8 × 10−6

W2(s) = 1

(500s + 1)2 s(200s + 1)

■

AUTHOR INFORMATION

Corresponding Author

*Fax: +49-391-67-11186. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

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REFERENCES

(1) Mörl, L.; Mittelstrass, M.; Sachse, J. Zum Kugelwachstum bei der Wirbelschichttrocknung von Lösungen oder Suspensionen. Chem. Technol. 1977, 29, 540−542. (2) Heinrich, S.; Peglow, M.; Ihlow, M.; Henneberg, M.; Mörl, L. Analysis of the start-up process in continuous fluidized bed spray granulation by population balance modeling. Chem. Eng. Sci. 2002, 57, 4369−4390. (3) Radichkov, R.; Müller, T.; Kienle, A.; Heinrich, S.; Peglow, M.; Mörl, L. A numerical bifurcation analysis of continuous fluidized bed spray granulation with external product classification. Chem. Eng. Process 2006, 45, 826−837. (4) Vreman, A. W.; van Lare, C. E.; Hounslow, M. J. A basic population balance model for fluid bed spray granulation. Chem. Eng. Sci. 2009, 64, 4389−4398. (5) Pathath, P. K.; Kienle, A. A numerical bifurcation analysis of nonlinear oscillations in crystallization processes. Chem. Eng. Sci. 2002, 57, 4391−4399. (6) Obinata, G.; Anderson, B. Model Reduction for Control System Design; Springer: London, 2001.

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For the shaped plant Gs(s) = W2(s) G(s) W1(s) a robustly stabilizing controller K∞ of order 7 is derived with ϵ = 0.5. The open loop Bode diagram of the original and shaped plant can be seen in Figure 10. The Bode diagram of the calculated H∞ loop shaping controller is depicted in Figure 11. As can be seen in Figure 12 the difference between achieved and desired loop shape is small. 4.4. Start-up Strategy and Results. As the controller is designed for a specific set point, the remaining task is how to bring the process sufficiently close to the set point. One possibility might be to start the process with an initial particle size distribution near to the desired set point, which is obviously not favorable for a practical implementation. An alternative is to start the open loop in the region of stability (for example V̇ e = 16700 419



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(7) Palis, S.; Kienle, A. Stabilization of continuous fluidized bed spray granulationA Lyapunov approach. In NOLCOS, 8th IFAC symposium on nonlinear control systems, Bologna, 2010. (8) Palis, S.; Kienle, A. Stabilization of continuous fluidized bed spray granulation with external product classification. Chem. Eng. Sci. 2012, 70, 200−209. (9) Palis, S.; Kienle, A. Diskrepanzbasierte Regelung der kontinuierlichen Flüssigkristallisation. Automatisierungstechnik 2012, 60, 145−154. (10) Eek, R., Control and dynamic modeling of industrial suspension crystallizers. Ph.D. Thesis, TU Delft, 1995. (11) Christofides, P.; El-Farra, N.; Li, M.; Mhaskar, P. Model-based control of particulate processes. Chem. Eng. Sci. 2008, 63, 1156−1172. (12) Vollmer, U.; Raisch, J. Population balance modeling and H∞controller design for a crystallization process. Chem. Eng. Sci. 2002, 57, 4401−4414. (13) Randolph, A. D.; Larson, M. A. Theory of Particulate Processes; Academic Press, Inc.: New York, 1988. (14) McFarlane, D. C.; Glover, K. Robust Controller Design Using Normalized Coprime Factor Plant Descriptions; Lecture Notes in Control and Information Sciences, Vol. 138; Springer: Berlin Heidelberg, 1989. (15) McFarlane, D. C.; Glover, K. A Loop Shaping Design Procedure using H∞ Synthesis. IEEE Trans. Autom. Control 1992, 37, 759−769. (16) Vinnicombe, G. Measuring Robustness of Feedback Systems. Ph.D. Thesis, Department of Engineering, University of Cambridge, U.K., 1993. (17) Zhou, K.; Glover, K. Robust and Optimal Control; Prentice Hall: Upper Saddle River, NJ, 1996.

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