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Latent Variable Model Predictive Control for Trajectory Tracking in Batch Processes: Internal Model Control Interpretation and Design Methodology Honglu Yu* and Jesus Flores-Cerrillo Praxair Inc., 175 East Park Drive, Tonawanda, New York 14150, United States ABSTRACT: In this paper, a theoretical analysis of the latent variable model predictive control (LV-MPC) algorithm, originally proposed by Flores-Cerrillo and MacGregor,1 is presented. The properties of the algorithm in terms of stability, robustness, and control performance are analyzed. An off-line design methodology, using Internal Model Control framework (IMC), to determine the adequate range of the LV-MPC parameters is proposed. The excellent control performance and robustness of the off-line design methodology is illustrated for the temperature tracking of an emulsion polymerization process and an exothermic chemical reaction system.

1. INTRODUCTION It is well-known that batch and semibatch processes are used in many industries because of their flexibility to manage many different grades and types of products. In these processes, one of the requirements to achieve consistent final quality specifications and adequate operation is to ensure the setpoints, determined by master controllers or optimizers, are closely followed. Proportional-integral-derivative (PID) controllers are by far the most common approach used in industry. However, batch processes usually exhibit large time constants and time varying dynamics, and sometimes it is necessary to track complex set-point trajectories. Under these situations the standard PID controllers might not achieve adequate control performance. Enhancements to conventional PID controllers have proven to lessen some of these deficiencies. Examples include PID-feed forward controllers (Clarke-Pringle and MacGregor2), adaptive PID controllers (Lontra3), and selftuning PID controllers (Altintena et al4). Several advanced control approaches based on nonlinear theoretical models of the batch process have been proposed to improve PID performance. Kravaris and co-workers5,6 used globally linearizing control (GLC). Wang et al.,7 Cott and Macchietto,8 and Aziz et al.9 used generic model control (GMC) to track reactor temperature set-points. Nonlinear model predictive control (NMPC) methods have also been presented to tackle this problem. Garcia 10 implemented a NMPC strategy for the temperature control of synthetic rubber production in a semibatch process. Gattu and Zafiriou11 extended the work of Garcia by incorporating Kalman filter estimation. Multi-input multioutput (MIMO) NMPC was addressed by Peterson et al.12 for the control of temperature and average molecular weight in the solution polymerization of methyl methacrylate (MMA). Ö zkan et al.13 implemented NMPC to track optimal reactor temperatures for the solution polymerization of styrene. Lee et al.14 proposed a model predictive control technique (MPC) for trajectory tracking aided with iterative learning. The methodology was illustrated for the temperature control of a laboratory batch reactor. © 2013 American Chemical Society

Statistical controllers for continuous processes based on principal component analysis (PCA) have also been proposed (Chen and McAvoy,15 McAvoy,16 Lu and Skelton,17 Shah et al.18). These controllers express the control objective in the score space of a PCA model. However, their objective is mainly to regulate the controlled variables around a fixed operating setpoint, and the use of PCA is mainly to reduce the dimension of the control variable space. Flores-Cerrillo and MacGregor1 proposed LV-MPC for batch process trajectory tracking based on dynamic PCA and a missing data algorithm. LV-MPC method utilizes easily obtainable historical closed-loop data for model building and excellent control results were obtained in three simulation examples. Several publications followed the original LV-MPC: Shamekh and Lennox19 investigated the performance of LVMPC in the reactor temperature control of an exothermic batch process. The article considers both constrained and unconstrained LV-MPC methods. The authors found that very good control performance can be obtained with this method but it is dependent on the data set and magnitude of PRBS signals used for model building. Golshan et al.20 proposed a multiphase latent variable-based MPC approach. They studied different data arrangements and evaluated their effects on the performance of the control in an exothermic batch process system. In a follow-up paper, Golshan and MacGregor21 studied the impact of different latent variable modeling approaches on MPC for trajectory tracking in a nylon polymerization simulation. Wan et al.22 explored the application of multiway PLS models for batch trajectory tracking and compared it with multiway PCA model-based methods in the simulation of a fed-batch Special Issue: John MacGregor Festschrift Received: Revised: Accepted: Published: 12437

January 13, 2013 April 7, 2013 April 10, 2013 April 10, 2013 dx.doi.org/10.1021/ie400060j | Ind. Eng. Chem. Res. 2013, 52, 12437−12450

Industrial & Engineering Chemistry Research

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fermentation process. Pla23 adapted the LV-MPC method to continuous processes. Despite the good control results obtained by Flores-Cerrillo and MacGregor,1 their research was mainly observatory and no fundamental analysis was performed. Furthermore, even though an approach based on “reconstruction error” was proposed for parameter selection, this approach did not offer insight on how the choice of parameters may affect the performance and stability of the controller. Furthermore, in some situations, the controller designed using this method may lead to unacceptable control performance, as observed in Shamekh and Lennox.19 To address the issues arising from improper parameter value selection, in this research, we propose an of f-line design strategy to select the adequate values of the parameters combinations in the LV-MPC models. The proposed design strategy, leads to good and robust control performance with guaranteed stability. To facilitate the analysis of the LV-MPC algorithm properties, we rewrite the LV-MPC control law into the two-degree-offreedom internal model control (IMC) structure. In this study we focus on single-input single-output (SISO) systems. The control design for multi-input multioutput (MIMO) systems will be addressed in a later publication. The outline of the paper is as follows: in section 2 the LVMPC algorithm is briefly reviewed; in section 3, the IMC interpretation of the LV-MPC algorithm is discussed; in section 4 the LV-MPC design metrics is presented; and in section 5 the design framework is summarized. Two examples are presented in section 6 to illustrate the method: an emulsion polymerization process and a chemical reaction system. Conclusions are given in section 7.

Flores-Cerrillo and MacGregor1 used a PCA missing data estimation algorithm as base of their control algorithm. In this missing data algorithm, each row of the X matrix is separated into two parts, xT1 and xT2 . xT1 contains past manipulated variables as well as both past and future controlled variables, while xT2 contains all future manipulated variables. x1T = [ui − Hp ··· ui − 1

p

yi + 1 ··· yi + H ] f

x 2T = [ui ··· ui + Hf ]

The objective of the missing data algorithm is to estimate xT2 by using xT1 and loading matrix P. When no hard constraints in the control system are considered, xT2 is calculated by solving the following optimization problem:

where the tT̂ is the estimated vector of scores and Q is the weighting matrix. The loading matrix P is decomposed into two corresponding parts for the xT1 and xT2 vectors respectively, P1 and P2. ⎡ P1 ⎤ P=⎢ ⎥ ⎣ P2 ⎦

When Q = I the solution to the eq 1 is x̂ 2T = x1TP1(P1TP1)−1P2T

(2)

To calculate controller output at time i, the future controlled variables (yi+1 ... yi+Hf) in xT1 are replaced with their desired setpoint trajectories (ri+1 ... ri+Hf).

2. LV-MPC ALGORITHM 2.1. Algorithm formulation. The LV-MPC algorithm was proposed by Flores-Cerrillo and MacGregor.1 It is based on empirical models that are obtained using dynamic principal component analysis (PCA). The database used for building the dynamic PCA model, denoted as X0, consists of K matrices, Zk (k = 1, 2, ..., K),

x̃1T = [ui − Hp···ui − 1

yi − H ···yi

x̂ 2T = x̃1TP1(P1TP1)−1P2T

p

ri + 1···ri + Hf ] (3)

As in standard MPC algorithms, though all future Hf steps of manipulated variables are calculated, only ui is implemented. As pointed out in Flores-Cerrillo and MacGregor,1 in practice it is possible that the matrix PT1 P1 is ill-conditioned. This situation can be easily handled by implementing SVD based pseudoinverse,

⎡ Z1 ⎤ ⎢ ⎥ ⎢⋮⎥ X 0 = ⎢⎢ Zk ⎥⎥ ⎢⋮⎥ ⎢ ⎥ ⎣ ZK ⎦

(P1TP1)+ = VM ΣM −1VM T

Where VM is the singular matrix and ΣM contains the M singular values. 2.2. Model Parameters. The calculation of LV-MPC controller, eq 3, is solely dependent on four parameters: past horizon Hp, future horizon Hf, the number of PCA components N, and the number of pseudoinverse singular value components M. Flores-Cerrillo and MacGregor1 proposed an approach based on “reconstruction error” to select the value of the LV-MPC parameters. However, this approach did not offer insight on how the choice of parameters may affect the stability and performance of the controller. Furthermore, in some situations, the controller designed using this method may lead to unacceptable (even unstable) control performance, as illustrated in the example of section 6.1. In this research, we analyze the LV-MPC controller to understand the relationship between model parameters and

where K is the number of batches used for model building. The method uses closed-loop data from a few batches, in which small pseudorandom binary sequences have been added on the output of the PI controller to ensure the identifiability. For single-input single-output (SISO) systems, each row of X0, xT0 , contains information of the manipulated variable u and controlled variable y at time i, including Hp steps of past data, current data, and Hf steps of future data. x 0T = [ui − Hp···ui···ui + Hf

yi − H ··· yi

yi − H ···yi ···yi + H ] p

f

PCA is then applied on X, which is the mean centered and unit variance scaled matrix of X0. (For readers not familiar with PCA, please refer to Jolliffe24 and Wold et. al25 for additional background). X̂ = TPT 12438

dx.doi.org/10.1021/ie400060j | Ind. Eng. Chem. Res. 2013, 52, 12437−12450


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Figure 1. Two degree-of-freedom IMC framework.

closed-loop control performance. In the following sections, we will show that by utilizing the two-degree-of-freedom internal model control (IMC) structure, we are able to derive several metrics that provide a quantitative estimation of closed-loop performance for a given combination of parameters. We then propose a framework to select LV-MPC designs that can achieve both good performance and robustness.

We can convert eq 6 into a two-degree-of-freedom IMC structure (Figure 1). The details of the derivation can be found in Appendix A. IMC controller design is a well developed area and a more in-depth explanation can be found in Morari and Zafiriou.26 In Figure 1, Gp and Gd are the transfer functions of the process and disturbance respectively, and Gm is the process model. There are two controllers in this structure: Cf is designed to reject disturbances and handling model mismatch while the set-point controller Cs is designed to shape the response to set-point changes. Also note that this control system focuses on the rate of change of the set-point rather than the set-point itself. From Figure 1, we have

3. IMC INTERPRETATION OF LV-MPC Based on eq 3, control action ui is a linear combination of the past manipulated variables, past controlled variables, and future set-point reference trajectories. We can simply write the LVMPC control law as i−1

i

∑

ui =

∑

αj + Hp− i + 1uj +

j = i − Hp

βj + H − i + 1yj p

j = i − Hp

i + Hf

∑

+

βj + H − i + 1rj p

j=i+1

(4)

where [α1 α2...αHp β1 β2 ... βHp+Hf+1] is the first column of P1(PT1 P1)+PT2 . Equation 4 can also be expressed in transfer function form T

βH

u(z) =

p+ 1

+ βH z−1 + ··· + β1z−Hp p

1 − α Hpz−1 − ··· − α1z−Hp +

βH

p+ Hf + 1

+ βH

p+ Hf

(1 − α Hpz

−1

z

−1

βH + H + 1 + ··· + βH + 2z −(Hf − 1) S(z ) p f p Cs(z) = = −1 A (z ) (1 − α Hpz − ··· − α1z −Hp)z −Hf

(7)

βH + 1 + ··· + β1z −Hp F (z ) p Cf (z) = =− A (z ) 1 − α Hpz −1 − ··· − α1z −Hp

(8)

Gm(z) =

1 − α Hpz −1 − ··· − α1z −Hp A (z ) − 1 z =− z −1 F (z ) βH + 1 + ··· + β1z −Hp p

(9)

y(z)

The closed loop response is

+ ··· + βH

− ··· − α1z

−H p

p+ 2

)z

z

−(Hf − 1)

−H f

r(z)

yΔ (z) =

Cs(z)Gp(z) 1 + [Gp(z) − Gm(z)]Cf (z)

(5)

+

Denote A(z) = 1 − α Hpz −1 − ··· − α1z −Hp p

βH + H + 1 + ··· + βH + 2z −(Hf − 1) p

f

p

z

−H f

Then eq 5 becomes u(z) = −

S(z ) F (z ) r (z ) y(z) + A (z ) A (z )

[1 − Cf (z)Gm(z)]Gd(z) d (z ) 1 + [Gp(z) − Gm(z)]Cf (z)

(10)

Decomposing the single LV-MPC equation into multiple IMC components allows us to utilize the theory and insight of the well-researched IMC area. We will show below that by using the IMC structure, we are able to derive several metrics that provide an estimation of closed-loop performance for a given combination of parameters. We then propose certain criteria to screen all the possible LV-MPC designs off-line to obtain controllers with proper parameter values that can achieve both good control performance and robustness. Usually the design of IMC controller starts with obtaining the best possible process model Gm (Morari and Zafiriou26). This could be time-consuming and may require a great deal of

F(z) = −(βH + 1 + ··· + β1z −Hp)

S(z ) =

rΔ(z)

(6) 12439



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the missing data reconstruction error ε for ui. We can express ui by rewriting eq 2 in the form of eq 4,

effort, which is particularly true for batch processes since they could be nonlinear and complex. Most of the effort is then on the design of controllers Cs and Cf to achieve a reasonable trade-off between controller performance and robustness. Generally this is a two-step approach. In the first step the controller Cs is selected for good set-point response, while Cf is selected for good disturbance rejection using the invertible part of the model. In the second step, the controllers Cs and Cf are augmented by a low-pass filter to provide milder action of the manipulated variables movements to improve robustness. It is worthy of mention that we are not directly using IMC structure to design the LV-MPC controller; rather we are only using the IMC structure to interpret the LV-MPC approach and to give a general guideline on the adequate range of the parameters values when designing the LV-MPC controller.

i−1

z→1 Hp+ Hf + 1

=

∑

f

+

u(z) = −

p

j=1

f

yj + 1 − yj = rj + 1 − rj ⇒ ej + 1 = ej ,

(18)

j = i + 1, ···i + Hf

⇒ei + 1 = ··· = ei + Hf = e

Equation 18 becomes (βH + 2 + ··· + βH + H + 1)e = −εi p

(13)

e=−

p

βH + 2 p

(14)

⇒σe =

Substitute 8 and 9 into 13:

f

εi ε =− i KS + ··· + βH + H + 1 p

f

σε |KS|

On the basis of the derivation above, σε/|KS| is an indication of set-point tracking error. This implies that the closed-loop control performance of LV-MPC controller depends on the magnitude of |KS| and the missing data reconstruction error σε. Therefore as long as the covariance information of the dynamic PCA matrix is consistent with the training data set, good control performance can be achieved regardless of the shape of the reference set-point. This is why we can see from FloresCerrillo and MacGregor,1 and the examples in this paper, that very good set-point tracking is obtained even when the setpoint trajectory patterns are very different from the set-point trajectory used in the training data set, from which the model was obtained. Of course, the smaller σε/|KS| ratio means tighter control. However, very small σε/|KS| ratio is often at the expense of very aggressive manipulated variable movements. This may make the

lim(Gm(z)Cf (z)) = lim z −1 = 1 z→1

which means LV-MPC always satisfies eq 13. Substitute 7 and 8 into 14, z→1

p

As we mentioned earlier, LV-MPC is tracking the change of reference trajectory. Assuming that from time point i + 1 to i + Hf, yΔ follows perfectly rΔ, then

and

lim

(16)

(17)

βH + 2ei + 1 + ··· + βH + H + 1ei + Hf = −εi

Following the basic property of two-degree-of-freedom IMC controller (Morari and Zafiriou26), the control error vanishes asymptotically for all asymptotically constant inputs (i.e., rΔ and d), if

z→1

F (z ) S(z ) 1 y(z) + y(z) + ε( z ) A (z ) A (z ) A (z )

where e(z) = y(z) − r(z) is the controller set-point tracking error. We can write eq 17 in time series form:

(12)

C (z ) lim s =1 z → 1 Cf (z)

(15)

⇒S(z)e(z) = −ε(z)

−H f

p

lim(1 − Gm(z)Cf (z)) = 0

p

⎧ F (z ) S( z ) 1 y(z) + y(z) + ε( z ⎪ u(z) = − A (z ) A (z ) A (z ) ⎪ ⎨ ⎪ F (z ) S( z ) u(z) = − y(z) + r (z ) ⎪ A (z ) A (z ) ⎩

Hp + 1

z→1

βj + H − i + 1yj + εi

We can obtain the relationship between controller variable y and reference set-point r by combining eq 6 and 16,

(11)

z→1

p

where εi is missing data reconstruction error when using uj(j = i − Hp, ..., i − 1) and yj(j = i − Hp, ..., i + Hf) to estimate ui. We can write eq 15 in transfer function form:

KF = lim F(z) = − lim(βH + 1 + ··· + β1z −Hp) = − ∑ βj z→1

∑ j=i+1

βj

j = Hp+ 2

βj + H − i + 1yj

j = i − Hp

i + Hf

p

z

z→1

∑

αj + Hp− i + 1uj +

j = i − Hp

βH + H + 1 + ··· + βH + 2z −(Hf − 1) p

∑

ui =

4. METRICS FOR LV-MPC DESIGN 4.1. Performance Related Metrics. We propose two metrics to evaluate the closed-loop control performance of LVMPC. 4.1.1. Metric 1: KS/KF ratio. The KS/KF ratio is for evaluating the zero-offset property of the controller. Here, KS and KF are the steady state gain of S(z) and F(z), respectively. KS = lim S(z) = lim

i

Cs(z) K = S =1 Cf (z) KF

Therefore, when KS/KF =1, the control error vanishes asymptotically for all asymptotically constant inputs rΔ and d, which means the control achieves zero offset in tracking ramp input r and step input d. In practice, we propose an acceptable condition for this metric as |KS/KF − 1| < 0.05. 4.1.2. Metric 2: σε/|KS| ratio. σε/|KS| ratio is for evaluating the LV-MPC closed loop control performance. KS, as in metric 1, is the steady state gain of S(z). σε is the standard deviation of 12440



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the manipulated variables and then choose the designs with adequate manipulated variable movements. 4.3.1. Metric 5: |KS/KA| ratio. .|KS/KA| ratio is to measure the magnitude of manipulated variable movement responding to a reference set-point change. KS/KA is the steady state gain of CS, the controller for trajectory tracking. Consider uS as the output of CS,

control system vulnerable to measurement errors and model mismatch. In this work, since y is mean-centered and unit-variance scaled, we simply choose σε/|KS| < 0.05 to be the criteria for acceptable control performance, which is equivalent to 1.25% tracking error relative to the range of y (measured by ±2 standard deviation). The derivation of this metric also shows that the reconstruction error alone (Flores-Cerrillo and MacGregor1) cannot guarantee good control performance. 4.2. Stability Related Metrics. We propose two metrics for assessing the stability of the controller. 4.2.1. Metric 3: |ρA,max| ratio. |ρA,max| is the largest absolute root of A(z). This metric is for ensuring the stability of Cf. On the basis of the IMC theory (Morari and Zafiriou26), when we have a perfect model (Gp = Gm) and Gp is stable, the IMC structure guarantees closed-loop stability of the system if Cf is stable. It is well-known that the condition to ensure Cf to be stable is that all the roots of A(z) must lie within the unit circle, which means |ρA,max| < 1. 4.2.2. Metric 4: KA/KF ratio. KA/KF ratio is another metric related to stability of the controller. KA/KF is the steady state gain of Gm, where KF is defined in eq 12 and KA is the steady state gain of A(z),

uS(z) = CS(z)rΔ(z)

Normally in set-point change there is no problem with noise amplification; however, a large gain of the CS controller is not recommended because of the likelihood of manipulated variable saturation. Assume reference input is a constant ramp, r∇ = δ and when uS is at steady state,

uS =

In order that |uS| does not exceed bound |uS,max|, we have |uS,max | KS KS δmax < |uS,max | ⇒ < KA KA |δmax|

where |δmax| is the max rate of change of the reference trajectory. |δmax| can be determined based on the training set and/or the anticipation of other possible input trajectories. 4.3.2. Metric 6: σ∇uf,a/σa ratio. σ∇uf,a/σa is a measure of the noise amplification factor of Cf. In order the control system be robust, we would like Cf is such that the noise amplification effect is small. Denote a(z) as the noise of controlled variable y. The effect of Cf on the noise a(z) can be calculated as

KA = lim A(z) z→1

= lim(1 − α Hpz −1 − ··· − α1z −Hp) z→1

Hp

=1−

∑ αj j=1

Metric 3 assumes that we can obtain a perfect model. Of course, in practice the model is essentially never perfect and in order to reduce the effects of model mismatch, classic IMC design (Morari and Zafiriou26) suggests including a filter in Cf. This filter is designed to have a decreasing magnitude as frequency increases, which can help stabilize the closed-loop system. On the basis of IMC control theory, it is possible to design a feedback controller which guarantees a zero tracking error for step changes and closed-loop stable if and only if the process steady-state gain has the same sign as the model steadystate gain (Morari and Zafiriou26). When the process and model gain have a different sign, the result is a positive feedback loop and the system is then unstable. Therefore we have the following condition for stability:

u f, a(z) = Cf (z)a(z)

L

σ∇ uf , a/σa =

where L is the length of the noise signal a(z). The upper limit on σ∇uf,a/σa can be obtained from, σ∇ uf , a ,max σ∇ uf , a/σa ≤ σa ,0 where σ∇uf,a,max is the maximum change in the movement of manipulated variable caused by noise and σa,0 is the measurement error level which can be estimated based on process knowledge or historical data. Metrics 5 and 6 have a similar purpose as the move suppression factor, R, frequently used in conventional optimal MPC controllers as well as the filters incorporated in the classic IMC design. 4.4. Summary of Metrics. In summary, six metrics were proposed to help select the four parameters values of the LVMPC controller: Hp, Hf, N and M. All six metrics can be easily calculated of f line given any of the combination of the parameters. These metrics were derived to describe different aspects of the controller performance, including trajectory tracking performance, stability, and robustness. Once these

Generally the sign of Gp is easy to obtain, either by process knowledge or simply by using the sign of Kc of the existing PI controller. Therefore, it is needed when K C > 0

KA /KF < 0,

when K C < 0

∑ (uf,a ,i − uf,a ,i− 1)2 /L i=1

z→1

KA /KF > 0,

(19)

Assuming a(z) is a normal distributed signal with zero mean and σa = 1, we can directly calculate uf,a using eq 19. σ∇uf,a/σa is then calculated as

lim Gp(z) × lim Gm(z) > 0

z→1

KS δ KA

4.3. Robustness Related Metrics. Classic IMC controller design makes emphasis on the robustness of the controller. It uses filters on both Cs and Cf to reduce the movement of manipulated variables. In LV-MPC, we do not use the same filtering technique to handle uncertainty; instead we propose the following two metrics to determine the aggressiveness of 12441



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Table 1. LV-MPC Design Metrics metrics performance stability robustness

1. 2. 3. 4. 5. 6.

KS/KF ratio σε/|KS| ratio |ρA,max| KA/KF ratio |KS/KA| ratio σ∇uf,a/σa ratio

function

criterion

zero steady state offset closed-loop set-point tracking performance controller stability controller stability indicator of manipulated variable movement from Cs noise amplification factor from Cf

|KS/KF − 1| < 0.05 σε/|KS| < 0.05 |ρA,max| < 1 sign(KA/KF) = sign(KC) |KS/KA| < |uS,max|/|δmax| σ∇uf,a/σa ≤ (σ∇uf,a ,max)/σa,0

(3) For each combination of Hp and Hf, rearrange the training trajectories to form dynamic matrix X0. (4) Normalize X0 and apply PCA on X. A filter may be applied on controlled variable y to further reduce the influence of noise in modeling if necessary. The number of PCA components N is usually determined by cross-validation procedures, and a principal component is considered significant if certain criteria are met. However, since we only need an initial range of PCA component numbers, it is no need to be precise in this step. A sufficient method is to select the number of PCA components such that R2 falls within a certain range. On the basis of our observations, often a high value of R2 (99% to 99.99% is used in the examples) is needed due to the nature of the lagged dynamic matrix. However, this does not necessary indicate that a perfect or near perfect model reconstruction is needed. The general guideline for choosing R2 is that the value should be high enough to explain most of the system variation while avoiding overfitting. (5) Apply singular value decomposition on PT1 P1 and select a range of singular values M to compute pseudoinverse of PT1 P1. In the simulation examples shown in section 6, the number of singular values M is chosen between M0 and N, where M0 is the smallest singular value larger than 0.001. The threshold value here is the one used in the examples. In general, if this number is too big, then the pseudomatrix inversion may be too inaccurate because too few components are included in the model, and if the number is too small then the pseudomatrix inversion may have too much impact by the ill-conditioned components. (6) Calculate the six metrics for each combination of Hp, Hf, N and M (Table 1). (7) Screen all the designs and select the ones within the criterion of all six metrics (Table 1). (8) Create the σε/KS ratio vs σ∇uf,a/σa ratio scatter plot for the selected designs and pick the ones on the outer bound of the curve. (9) For each selected combination of Hp, Hf, N and M, the control action is obtained using eq 3 or 4.

metrics are computed, it is easy to screen all the potential designs and pick the ones with desired performance. All metrics calculations and screening process are done offline and are very fast since most of the metrics calculations only involve linear operations. After screening out the designs that do not satisfy the metrics selection criterion, there are often a handful of possible designs left. For the final choice, it often comes down to the trade-off between Metric 2, which is an indicator of control performance in the controlled variable and Metric 6, which is an indication of robustness (movement of the manipulated variable). For SISO systems, we can draw a chart with σε/|KS| ratio vs σ∇uf,a/σa ratio and select the designs falling on the outerbound of the curve based on the desired control performance, as can be seen in the examples of section 6. Table 1 provides a summary of the proposed metrics for assessing the control performance, stability, and robustness of the LV-MPC controller. 4.5. Additional Remarks. It is important to mention that the advantage of the LV-MPC combined with the proposed design approach versus IMC and other controller methods is that in the LV-MPC design approach, there is no need to sequentially identify, design, and tune the different controller components such as model and filters. Instead, the design and the tuning is performed in a single step and determined by four parameters. The only decision is the adequate selection of the parameters ranges, which is in any case a necessary step when building the LV-MPC controller. Once the initial parameters ranges are decided, we can easily select the proper combination of the parameters by calculating the six proposed metrics. Another interesting point is that from our analysis, LV-MPC is actually designed to track the change of set-points. Therefore, as we can see from the simulation examples, it has superior performance particularly for tracking transient set-points. However, it may have a small offset in steady state value when the new set-point trajectory is very different from the training trajectory. Furthermore, in the situation where the system has high nonlinearity, the LV-MPC method can be easily extended to utilize multiple linear models (Flores-Cerrillo and MacGregor,1 Aumi and Parskar,27 and Aumi et al.28).

6. SIMULATION EXAMPLES In this section, we present two examples to illustrate the proposed design methodology. Both examples are the same as the ones presented in Flores-Cerrillo and MacGregor.1 The first example is to control the reactor temperature trajectory by adjusting the inlet jacket temperature in a nonlinear styrene emulsion polymerization process. In this example, we will show step-by-step the parameter selection design and control results. The second example is a nonlinear chemical reaction system where the control objective is to track the reactor temperature by adjusting the inlet jacket temperature. In this example, we will demonstrate the consistency of the design methodology for

5. DESIGN FRAMEWORK SUMMARY The step- by-step procedure of designing LV-MPC is given as follows: (1) Collect training data set. LV-MPC usually requires only a few batches as training set for model identification. The batches are under PI control and have low level of PRBS excitation. Further discussion on the data set requirements is given in the examples section. (2) Select ranges for past horizon Hp and future horizon Hf. 12442



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Figure 2. Data set used for model identification: (---) inlet jacket/reactor temperature control obtained for the three batches in which PRBS signals were added; (−) set-point trajectory.

Figure 3. Example of unacceptable controller performance when the minimum reconstruction error approach is used to select controller parameters: (···) controlled or manipulated variable; () controlled variable set-point.

20 different data sets. We will also illustrate the capability of the LV-MPC to handle change of process dynamics. 6.1. Example 1: Emulsion Polymerization of Styrene. A nonlinear model for the styrene emulsion polymerization, originally developed by Lynch and Kiparissides29 for tubular reactors with full recycle was adapted for use in batch and semibatch processes. For a complete description of the model and model parameters the reader is referred to the original publication. The control objective is to perform reactor temperature trajectory tracking by adjusting the inlet jacket temperature. Online reactor temperature measurements, considered to be available every 30 s, are corrupted by a normally distributed random error with standard deviation of 0.15 K (Kelvin degrees). Simulation time for one batch is 300 min. Control action is taken every 30 s. 6.1.1. Training Trajectory. The training data set consists of three batch runs under PI closed-loop control. As suggested in Flores-Cerrillo and MacGregor,1 small PRBS signals were added on the manipulated variable. The PI parameters are KC = 30 and TI = 100. The mean absolute error (MAE = ∑i L= 1|yi − ri|/L) of the CV for the data set used in the model building is only 30% higher than when no PRBS signal is added. The data used for the model building is shown in Figure 2. In this Figure, the dotted line (···) indicates the inlet jacket/reactor temperatures for the three batch runs obtained when a small PRBS has been added to the MV (inlet jacket temperature) throughout

the run, while the solid line (−) indicates the set-point trajectory. 6.1.2. Control Results When Using Minimum Reconstruction Error Criteria. Flores-Cerrillo and MacGregor1 proposed an approach using minimum missing data reconstruction error. However, though a controller with good control performance may be obtained using this approach, it also can lead to a controller with unacceptable control performance or even an unstable controller. One example of this situation is shown in Figure 3, where the controller parameters were obtained using the missing data reconstruction error approach. As can be seen in Figure 3, the manipulate variable is unstable and the setpoint tracking performance is quite poor. 6.1.3. LV-MPC Design and Control Results. Following the steps outlined in section 5, we begin by choosing a range of initial values for the future control horizon Hf, the past data horizon Hp, the number of PCA components N, and the number of singular values M in the pseudo-inversion calculation. Without prior knowledge of the process dynamics, initial Hp and Hf are set to be from 1 to 20 for both parameters, which result in 400 Hp and Hf pairs. For each pair of Hp and Hf values, PCA is performed on the corresponding dynamic matrix. The number of PCA components N, as mentioned in section 5, is chosen so that R2 is between 99% and 99.99%. For the pseudo-inversion calculation, the number of singular values M is chosen between M0 and N, where M0 is the smallest singular value that is greater than 0.001. 12443



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Table 2. Example of Calculated Metrics and Selection of Controller Parameters Range

Figure 4. Metric 2σε/|KS| ratio vs Metric 6 (σ∇uf,a)/σa ratio for selected controllers. Each circle (○) represents a feasible LV-MPC controller design. Each circled group (A, B, C, and D) contains three designs with similar metrics values within the group.

design parameters are highlighted, for example the combinations Hp = 20, Hf = 20, N = 36, and M = 24 and 25 satisfy all the desired design metrics criterion. In this example, it takes less than 3 min to compute all the metrics, including the PCA modeling, for all 43 885 combinations of parameters on a HP laptop (2.5G DuoCPU and 2.9Gb of Ram); 6778 of these 43 885 combinations satisfy criteria of stability, that is, metric 3 and metric 4. However, only 4457 controllers satisfy both performance and stability requirement. After screening with all the desired design conditions, 443 controllers remain. From the 443 selected controllers, we can then plot Metric 2 σε/|KS| ratio vs Metric 6 σ∇uf,a/σa ratio, as shown in Figure 4. Each circle (○) in Figure 4 represents a selected control design. Twelve points, circled in four groups (group A, B, C and D; bigger circles in the plot), are chosen to illustrate the trade-off between controlled variable performance and manipulate variable aggressiveness. The ones located at the outerbound of the curve are the best candidates. The three points in group A, grouped with the smallest σε/|KS| values and the largest σ∇uf,a/σa values, are expected to have the best closed-loop controlled variable performance but the most aggressive manipulated variable behavior. On the other hand, the points

The calculation of the six metrics is straightforward. From Table 1, we need to decide the desired range for metrics 5 and 6. For Metric 5 |KS/KA|, the desired range is determined by |KS/ KA| < |uS,max|/|δmax|. |δmax| is estimated from the maximum rate of set-point change (in the scaled space). Here |δmax| = max|ri − ri−1| = 0.032. |uS,max| is the maximum movement of manipulated variable (in the scaled space), which is set as |uS,max| = 0.65. Therefore the range for |KS/KA| is less than 0.65/0.032 ≈ 20. In Metric 6, σ∇uf,a/σa, the desired range is determined by (σ∇uf,a)/σa ≤ (σ∇uf,a,max)/(σa,0). Here we use σ∇uf,a,max = 0.2, which is maximum movement of manipulated variable (scaled) for responding to the measurement noise. σa,0 is the estimated standard deviation of measurement noise (scaled) which can be obtained from process knowledge or data. In this example, we know that the noise level is 0.1K, dividing it by the standard deviation of y, σy = 17.54, we have σa,0 = 0.1/17.54 = 0.0057. Therefore the upper bound for σ∇uf,a/σa is 0.2/0.0057 = 35.1. Table 2 shows an example of the calculated metrics when Hp = 20, Hf = 20, N = 36 and various values of M. The second row of the table lists the desired value range for each metric. The shaded cells highlight the values satisfying the criterion of the metrics. If all the criterion of the metrics are satisfied then the 12444



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Table 3. Metrics Values for the 12 Selected Designs (A, B, C, and D Sets) A-1 A-2 A-3 B-1 B-2 B-3 C-1 C-2 C-3 D-1 D-2 D-3

Hp

Hf

N

M

KS/KF

σε/|KS|

|ρA,max|

KA/KF

|KS/KA|

σ∇uf,a/σa

13 14 16 17 20 20 18 20 20 11 11 18

7 20 7 10 9 9 13 12 20 15 17 12

22 36 25 27 30 31 31 32 36 19 22 31

16 19 19 20 23 23 21 23 24 14 16 23

1.004 1.004 1.004 1.006 1.005 1.005 1.010 1.007 1.010 1.005 1.005 1.006

0.014 0.013 0.013 0.027 0.026 0.028 0.050 0.048 0.049 0.047 0.047 0.046

0.966 0.987 0.985 0.991 0.996 0.969 0.974 0.984 0.989 0.991 0.997 0.975

0.067 0.053 0.068 0.121 0.116 0.123 0.213 0.196 0.122 1.385 1.717 1.193

14.897 18.913 14.782 8.245 8.593 8.162 4.698 5.113 8.211 14.837 17.098 18.005

30.904 32.253 31.534 10.186 9.492 9.644 6.347 5.338 5.978 31.169 30.679 33.094

in group C have the largest σε/|KS| value and the smallest σ∇uf,a/ σa, so we would expect these designs have smaller manipulated variable movement but not as good closed-loop controlled variable performance as the points in group A. The designs in group B represent a trade-off between both control objectives: tight manipulated variable control and reasonable manipulated variable movement. Group D is expected to have the worst control performance as well as large manipulated movements. Table 3 lists all the metrics values for these 12 designs. Notice that group A also has large |KS/KA| value, which is in indication that these designs are more aggressive than groups B and C. The closed loop simulation is then conducted on these 12 chosen designs to verify the expected control performance for 2 different set-point trajectories (training data sets originated from set-point 1). The control results are shown in Table 4. In

the performance of the PI controller (from which the model building data set was generated) is also listed in Table 4. Figures 5 and 6 show the excellent control performance that is achieved with the LV-MPC for two different set-points. It can be seen that for a design in group A, the control tracking performance for the controlled variable is “almost perfect” at the expense of relatively large manipulated variable variability. In designs belonging to group C, the controlled variable has small tracking deviations from the set-point but the manipulated variable is less aggressive when compared to a group A. In Figures 5 and 6, it can be also seen that a design belonging to group B has very good control tracking performance without large manipulated variable variability and therefore designs falling into this group are recommended. For comparison, the tracking performance of a design from group D is also presented (Figure 5d). This design, as expected, has relatively large manipulated variable movements with a relatively large controlled variable tracking error. 6.2. Example 2: Chemical Reaction System. The chemical reaction system is a nonlinear model based on the work of Aziz et al.9 and Cott and Macchietto.8 For a complete description of the model the reader is referred to the original publications. Model parameters are the same as the ones reported in Aziz et al.9 except for the time constant of the inlet jacket temperature, which was increased from 3 to 5 min in this work. The control objective is to track the reactor temperature setpoint by adjusting the inlet jacket temperature. Online reactor temperatures, considered to be available every 30 s, are corrupted by a normally distributed random error with standard deviation of 0.2 C. Each batch run is 150 min. Control action is taken every 30 s. 6.2.1. Training Trajectory. In this example, we would like to test the consistency of the proposed design methodology to different data sets. Twenty data sets were built under the same PI controller (KC = 15 and TI = 10) with similar level of PRBS excitation. Each data set contains three batches, sampled at 0.5 min, in which a small PRBS signal has been added to the inlet jacket temperature (manipulated variable). The mean absolute error (MAE) for the CV is only about 20% higher with the added PRBS when the MAE is under pure PI control. Figure 7 shows the training data for all 60 batches. 6.2.2. LV-MPC Design and Control Results. In this example, we use exactly the same procedure presented in the previous example to determine the initial sets of parameters. Like in example 1, we use the default values for the desired range of Metrics 1−4 (Table 1).

Table 4. Control Performance for the 12 Selected Designs of Figure 4 set-point trajectory 1 A-1 A-2 A-3 B-1 B-2 B-3 C-1 C-2 C-3 D-1 D-2 D-3 PI

set-point trajectory 2

Hp

Hf

N

M

MAEu

MAEy

MAEu

MAEy

13 14 16 17 20 20 18 20 20 11 11 18

7 20 7 10 9 9 13 12 20 15 17 12

22 36 25 27 30 31 31 32 36 19 22 31

16 19 19 20 23 23 21 23 24 14 16 23

6.457 8.405 7.421 3.541 2.984 2.558 2.433 2.033 3.646 7.823 6.131 6.793 4.132

0.173 0.210 0.191 0.273 0.299 0.319 0.372 0.422 0.483 0.394 0.422 0.462 1.069

5.707 7.407 6.300 3.027 2.607 2.246 1.877 1.486 2.203 7.316 6.002 6.341 3.634

0.117 0.141 0.135 0.170 0.178 0.189 0.230 0.230 0.317 0.278 0.292 0.307 0.508

this table, it can be seen that the designs within the same group have similar performance in terms of the MAE for u and y, and that, as predicted in the parameter design step, group A has tighter control at expenses of larger and more frequent manipulated variable movements. Group C is the opposite to group A, that is, larger tracking error (MAEy) but smaller MAEu. Group B, as expected, is a trade-off between both groups and perhaps the most recommendable controller design. These results are true for both of the set-points in Table 4 and for many other different set-point patterns tested. For comparison, 12445



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Figure 5. Control results for set-point trajectory 1: (···) controlled or manipulated variable for a design; () controlled variable set-point. 12446



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Figure 6. Control results for design B-1 in group B for set-point trajectory 2: (···) controlled or manipulated variable for a design; () controlled variable set-point.

Figure 7. Data set for model identification in example 2: (···) inlet jacket/reactor temperature control obtained for the 60 batches with added PRBS; () controlled variable set-point.

For Metric 5 |KS/KA|, the desired range is determined by |KS/ KA| < |uS,max|/|δmax|. Like in example 1, we use |uS,max| = 0.65. |δmax| is estimated from: |δmax| = max|ri − ri−1| = 0.065. Therefore the upper bound for |KS/KA| in this example is 0.65/ 0.065 = 10. For Metric 6 σ∇uf,a/σa, the desired range is determined by (σ∇uf,a)/σa ≤ (σ∇uf,a,max)/(σa,0). Here we use σ∇uf,a,max = 0.1. We know the noise level of the reactor temperature is 0.2 C, dividing it by the standard deviation of y, σy = 18.3, we have σa,0 = 0.2/18.3 = 0.01093. Therefore the upper bound for σ∇uf,a/σa is 0.1/0.01093 ≈ 10. Using the 20 data sets and the metrics of Table 1 (metrics range as described above), we generated designs for each one of the data sets. Table 5 shows the total number of initial designs and the number of designs that fall within the criterion range of the metrics for 5 of the 20 data sets. Figure 8 shows the σε/|KS| ratio vs σ∇uf,a/σa ratio plot for the designs within the desired range for 3 of the 20 data sets (data set 1, 2, and 3). The rest of the data sets have similar results. To compare the controller performance obtained from different data sets, we choose a reference point on σε/|KS| ratio versus σ∇uf,a/σa ratio plot. The reference point is indicated by a star (★). For each data set, we pick up the design which is the closest to the reference point where σε/|KS| = 0.025 and σ∇uf,a/ σa = 3. The distance is measured as shown in the following; 0.005 is the weight to account for the different scales of the two axes.

Table 5. Total Number of Designs and Number of Designs Satisfying Criterion of All Six Metrics for 5 of the 20 Different Data Sets data set

total number of designs

number of designs satisfying criterion of all six metrics

1 2 3 4 5

31121 30811 31410 31141 30720

565 587 490 613 502

dj = |(σε/|KS|)j − 0.025| + |(σ∇ uf , a/σa)j − 3| × 0.005

For each one of the 20 data sets, we select one design that is the closest to the fixed reference point (★). Since these designs are all close to the same reference point, we expect that similar control performance should be achieved regardless of the values of the LV-MPC parameters. Table 6 shows the design parameters, the value of the metrics, and the closed loop control performance for the five designs (the ones closest to the reference point) belonging to the five data sets of Table 5. As expected, the control performance is very similar regardless the value of the parameters, and the data set from where the model was originated. This corroborates that the design methodology is robust and independent of the data sets used for model building. As an illustration, Figure 9, shows the control results for three data sets (data set 1, 2, and 3). The control 12447



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Figure 8. Feasible designs for three data sets of Table 5 (data sets 1, 2 and 3). Reference point is indicated by a star (★), from which the closest design to this reference point is selected.

Table 6. Metrics and Control Performance for the Five Designs (the Ones Closest to the Reference Point) Belonging to the Five Data Sets of Table 5 design parameters

metrics

control results

data set

Hp

Hf

N

M

KS/KF

σε/|KS|

|ρA,max|

KA/KF

|KS/KA|

σ∇uf,a/σa

MAEu

MAEy

1 2 3 4 5

15 18 20 19 19

17 18 16 20 12

33 38 38 38 31

20 23 25 25 23

1.018 1.012 1.005 1.011 1.013

0.025 0.027 0.025 0.025 0.025

0.954 0.932 0.947 0.963 0.977

0.318 0.381 0.400 0.395 0.427

3.141 2.623 2.500 2.535 2.345

2.901 3.049 3.006 2.965 2.808

2.417 2.128 2.239 2.554 2.803

0.223 0.227 0.196 0.206 0.212

Figure 9. Control results for three different data sets (data sets 1, 2 and 3) using designs closest to the reference points (★) shown in Figure 8.

ance would be affected when the heat transfer coefficient, U, is decreased. Table 7 shows the control results when a multiplying factor in U changes from 0.7 to 0.9 (the LV-MPC controller parameters are obtained from data set 1 of Table 6). Figure 10 shows the control results for 0.7U. As we can see from this

performance and the manipulated variable movements, as expected, are very similar. 6.2.3. Control Results for Change in Process Dynamics. To further assess the robustness of the controller, we modify the process dynamics. Here we studied how the control perform12448



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The design methodology and the performance of the resulting LV-MPC designs are illustrated using two nonlinear batch systems: A styrene emulsion polymerization process and a nonlinear chemical reaction system. The consistency of the design methods for many different data sets and different setpoint trajectories is also illustrated. By combining the proposed control design procedure with the simplicity of the LV-MPC algorithm, it gives a practical (and powerful) solution with substantial improvement over the existing PI controllers. The current methodology was demonstrated for a SISO system, and its extension to MIMO processes will be presented in a later publication.

Table 7. Effect that Process Dynamic Changes Has on Control Performancea heat transfer coefficient

MAEu

MAEy

0.7U 0.8U 0.9U 1U (nominal)

2.831 2.702 2.586 2.417

0.492 0.338 0.248 0.223

a

The process dynamic changes are represented by a multiplying factor in the heat transfer coefficient, U (controller obtained from data set 1of Table 6).

■

APPENDIX A In this appendix, we show a step-by-step derivation for converting eq 6 to 2-degree-of-freedom IMC structure (i.e., eq 7−9). Multiply both sides of eq 6 by (1 − z−1), F (z ) (1 − z −1)y(z) A (z ) S( z ) (1 − z −1)r(z) + A (z )

(1 − z −1)u(z) = −

(20)

From Figure 1 (two-degree-of-freedom IMC structure) u(z) is computed as u(z) = Cs(z)(1 − z −1)r(z) − Cf (z)[(1 − z −1)y(z) − Gm(z)u(z)] ⇒[1 − Cf (z)Gm(z)]u(z) = Cs(z)(1 − z −1)r(z) − Cf (z)(1 − z −1)y(z)

(21)

By comparing 20 and 21, we can obtain eq 7−9.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: 716-879-2577. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

Figure 10. Control results when the process dynamics changes. Heat transfer coefficient is set to 0.7U (controller is obtained from data set 1of Table 6).

■

REFERENCES

(1) Flores-Cerrillo, J.; MacGregor, J. F. Latent variable MPC for trajectory tracking in batch processes. J. Process Control 2005, 15 (6), 651−663. (2) Clarke-Pringle, T.; MacGregor, J. F. Nonlinear adaptive temperature control of multi-product, semi-batch polymerization reactors. Comput. Chem. Eng. 1997, 21 (12), 1395−1409. (3) Lontra, H. C. Modelagem e Simulaçaõ de um Reator Batelada de Polimerizaçaõ de Estireno em Suspensão, Anais do Congresso Brasileiro de Polı ́meros São Paulo, Brazil, 1991; pp 619−623. (4) Altıntena, A.; Ketevanlioğlua, F.; Erdoğana, S.; Hapoğlub, H.; Alpbaz, M. Self-tuning PID control of jacketed batch polystyrene reactor using genetic algorithm. Chem. Eng. J. 2008, 138, 490−497. (5) Kravaris, C.; Wright, R. A.; Carrier, J. F. Nonlinear controllers for trajectory tracking in batch processes. Comput. Chem. Eng. 1989, 13, 73−82. (6) Kravaris, C.; Saroush, M. Synthesis of multivariate nonlinear controllers by input/output linearization. AIChE J. 1990, 36, 249−264. (7) Wang, Z. L.; PLA, F.; Corriou, J. P. Nonlinear adaptive control of batch styrene polymerization. Chem. Eng. Sci. 1995, 50 (13), 2081− 2091.

Figure, the control performance degraded but is still acceptable when there is a 30% reduction in the heat transfer coefficient. For situations when there are drastic changes in the process dynamics and the control performance is no longer acceptable, we can use the data obtained from the most recent batches and then rebuild the LV-MPC model. This model will then contain the most recent process behavior and disturbances effect.

7. CONCLUSIONS In this work, a design methodology to select the design parameters for the latent variable model predictive controller (LV-MPC) is proposed. The off-line design methodology uses six metrics to determine the adequate parameter range that results in LV-MPC controllers with good control and robust performance. IMC principles are used to facilitate the analysis of the LV-MPC control law and the properties of the LV-MPC algorithm. 12449



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(8) Cott, B. J.; Macchietto, S. Temperature control of exothermic batch reactors using generic model control. Ind. Eng. Chem. Res. 1989, 28 (8), 1177−1184. (9) Aziz, N.; Hussain, M. A.; Mujtaba, I. M. Performance of different types of controllers in tracking optimal temperature profiles in batch reactors. Comput. Chem. Eng. 2000, 24, 1069−1075. (10) García, C. E. Quadratic Dynamic Matrix Control of Nonlinear Processes. An Application to a Batch Reaction Process. AIChE Annual Meeting, San Franciso, CA, 1984. (11) Gattu, G.; Zafiriou, E. Nonlinear dynamic matrix control with state estimation. Ind. Eng. Chem. Res. 1992, 31, 1096−1104. (12) Peterson, T.; Hernandez, E.; Arkun, Y.; Schork, J. F. A nonlinear DMC algorithm and its application to a semi-batch polymerization reactor. Chem. Eng. Sci. 1992, 47 (4), 737−753. (13) Ö zkan, G.; Ö zen, S.; Erdoğan, S.; Hapoğlu, H.; Alpbaz, M. Nonlinear control of polymerization reactor. Comput. Chem. Eng. 2001, 25, 757−763. (14) Lee, K. S.; Chin, I.; Lee, H. J.; Lee, J. H. Model predictive control technique combined with iterative learning for batch processes. AIChE J. 1999, 45 (10), 2175−2187. (15) Chen, G.; McAvoy, T. J. Process control utilizing data based multivariate statistical models. Can. J. Chem. Eng. 1996, 74, 1010− 1024. (16) McAvoy, T. J. Model predictive statistical process control of chemical plants. Proceedings of the 15th IFAC World Congress, Barcelona, Spain, 2002. (17) Lu, J.; Skelton, R. E. Covariance Control using Closed Loop Modeling for Structures. Proceedings of the ASCE Structure Congress, Portland, Oregon, April 13−16, 1997. (18) Shah S.; Randy, M.; Takada, H.; Morinaga, K.; Satou, T. Modeling and Control of a Tubular Reactor: A PCA Based Approach. Proceedings of the 5th IFAC Symposium on Dynamics and Control of Process Systems, Corfu, Greece, June 1998. (19) Shamekh, A.; Lennox, B. Latent Variable MPC and its Consistency in Temperature Control of Batch Processes. Proceedings of the 9th International Symposium on Dynamics and Control of Process Systems, Leuven, Belgium, July 5−7, 2010. (20) Golshan, M.; MacGregor, J. F.; Bruwer, M.-J.; Mhaskar, P. Latent Variable MPC for Trajectory Tracking in Batch Processes: Role of the model structure. American Control Conference, St. Louis, MO, USA, June, 2009. (21) Golshan, M.; MacGregor, J. F. Latent Variable Modeling of Batch Processes for Trajectory Tracking Control. Proceedings of the 9th International Symposium on Dynamics and Control of Process Systems, Leuven, Belgium, July 5−7, 2010. (22) Wan, J.; Marjanovic, O.; Lennox, B. Trajectory tracking in batch processes using latent variable models. Preprints of the 18th IFAC World Congress, Milano, Italy, August 28−September 2, 2011. (23) Pla, D. L. MPC: Relevant Identification, and Control in the Latent Variable Space. Ph.D. Thesis, Universidad Politecnica de Valencia, 2012. (24) Jolliffe, I. T. Principal Component Analysis, 2nd ed.; Springer: New York, 2002. (25) Wold, S.; Esbensen, K.; Geladi, P. Principal Component Analysis. Chemom. Intell. Lab. Syst. 1987, 2 (1−3), 37−52. (26) Morari, M.; Zafiriou, E. Robust Process Control; Prentice Hall: Englewood Cliffs, NJ, 1989. (27) Aumi, S.; Mhaskar, P. Integrating Data-based modeling and nonlinear control tools for batch process control. AIChE J. 2012, 58, 1374−1386. (28) Aumi, S.; Corbett, B.; Mhaskar, P. Model predictive quality control of batch processes. Proceedings of the American Control Conference, 5646−5651, Montreal, Canada, 2012. (29) Lynch, D.; Kiparissides, C. Numerical simulation of a tubular polymerization reactor. J. Appl Polym. Sci. 1981, 26, 1283−1293.

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