# Nonlinear quadratic dynamic matrix control of a fluidized catalytic cracking unit

A.T. Boum, D Bingong, Jean Pierre Corriou

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A.T. Boum, D Bingong, Jean Pierre Corriou. Nonlinear quadratic dynamic matrix control of a fluidized catalytic cracking unit. International Journal of Control Systems and Robotics, 2017, 2, pp.141-148. [hal-02383495](https://hal.science/hal-02383495)

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Distributed under a Creative Commons Attribution 4.0 International License
Nonlinear quadratic dynamic matrix control of a fluidized
catalytic cracking unit

A.T. BOUM*, D. BINGONG**, J.P. CORRIOU***

* University of Douala,
ENSET, B.P 1872 Douala,
CAMEROON
boumat2002@yahoo.fr

** University of Douala,
EDSFA, UFD MIAPF,
CAMEROON
davidbingong@yahoo.fr

*** University of Lorraine
LRGP, UMR 7274, Nancy,
FRANCE
jean-pierre.corriou@univ-lorraine.fr

*Abstract:* This paper present the application of a nonlinear model predictive control strategy to an
fluidized catalytic cracking (FCC) unit. The FCC is a complex nonlinear process that has been the
subject of many models and control studies. The present dynamic model of the FCC process, inspired
from Blacken model, uses three lump kinetics to describe the cracking reactions in the riser considered
as a plug-flow system. The riser behaviour is described by gasoil, gasoline, coke and energy balances.
The separator is considered as a CSTR. The regenerator mainly constituted by a dense and a dilute zone
is also considered as a CSTR to represent the catalyst and its dynamic behaviour is described by coke,
oxygen and energy balances. This model is sufficiently complex to capture the major dynamic effects
that occur in an FCC and to control the key variables that are the riser outlet temperature and the
regenerator dense bed temperature. The manipulated inputs are the air inlet flow rate in the regenerator
and the regenerated catalyst flow rate. Hard constraints are imposed with respect to the manipulated
variables. In spite of the important nonlinearity of the FCC, Nonlinear quadratic dynamic matrix control
is able to maintain a smooth multivariable control of the plant, while taking into account the constraints.
The comparison study with linear quadratic dynamic matrix control shows a better following of the set
point.

*Key-Word:* Nonlinear, Predictive control, fluid catalyst

**1. Introduction**

Fluid Catalytic Cracking (FCC) is one of the
most important processes in a refinery, in
particular due to its economic importance [1].
An FCC is used to crack heavy atmospheric
residues and vacuum distillate into lighter
molecules that yield more valuable products
such as gasoline, kerosene, light gas oil ...The
FCC unit is a complex process due to the
composition of the feed and the dynamic mass
and heat interactions between its main
components, namely riser, separator and
regenerator, with recycling effects. Thus, while
respecting the environmental regulation and

operating constraints, the gains that can come
from an improvement in the optimal control of
this process make it a challenge to science and
engineering in the field of automation and call
for an advanced control tool. Nonlinear Model
Predictive Control (NMPC) is presently used
due to its capacity to easily handle
multivariable processes while also taking into
account hard and soft constraints with respect to
the manipulated variables, their moves and the
controlled variables and also the better
following of set point.

FCC control has been performed by means of
different approaches [10, 12, 13, 15]. In
practice, FCC units are frequently regulated
using PID controllers based on the knowledge
and experience of operators in the refinery.
In this work, Nonlinear Model Predictive
Control with the algorithm named nonlinear
quadratic dynamic matrix control (NQDMC) is
used to control the FCC using two manipulated
variables and two controlled variables. Section
3 describes the FCC model. Section 4 deals
with NQDMC principles. Section 5 deals with
the control of the FCC unit and presents the
simulation results. Section 6 deals with the
comparison study between QDMC and
NQDMC. Finally, the conclusions of the paper
are presented in Section 7.

## 2. Description of a modern FCC process

A modern FCC unit mainly consists of three
units [3]. The cracking reactions of the
hydrocarbon feed take place in the riser while
the catalyst is reactivated in the regenerator by
combustion of the coke deposited on the
catalyst in the riser reactor.

The feed is preheated at a temperature in the
range 450-600K. Thereafter, this feed is
injected in the bottom part of the riser with a
small quantity of vapor. The feed is vaporized
at the contact of the hot catalyst. The
hydrocarbon vapors undergo an endothermic
reaction while rising to the top of the riser. The
residence time of the catalyst and the
hydrocarbon vapors in the riser is a few
seconds. The temperature at the top of the riser
is between 750 and 820K. The disengagement
part of the reactor is used to separate the
catalyst from the vapors, then the vapors enter
the main fractionator. The spent catalyst is
separated from the vapors by cyclones and
flows in the extraction part where the remaining
hydrocarbons on its surface are removed by
stripping steam. The catalyst flows through a
transport line to the regenerator.

In the regenerator, the catalyst is reactivated by
burning the deposited coke using air entering at
the bottom of the regenerator. This partial or
total exothermic combustion reaction
reactivates the catalyst and maintains the bed
temperature between 950-980K for future
gasoil cracking. The regenerated catalyst flows
continuously in the riser through another circuit
and the heat transported by the catalyst is used
to compensate the endothermic reactions in the
riser.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic diagram of FCC unit.</figcaption></figure>

## 3. Mathematical model of the FCC

The FCC model that is used in this work is
inspired and adapted from [2, 8, 9], this model
derives in a great part from the model proposed
by [11].

### 3.1 Riser model

The riser is considered as a plug flow reactor.
The residence time of catalyst and feed in the
riser is supposed to be a few seconds.
Consequently, the riser is only described by
spatial equations and considered as an algebraic
system. The kinetic model makes use of a three
lump scheme [16] to describe the cracking in
the riser [5].

The feed temperature $T_{ris}(z = 0)$ at the entry
of the riser results from the energy balance. $z$
is a dimensionless height $\in [0,1]$.

#### Mass balance of gas oil

$$ \frac{dy_{go}}{dz} = -k_1 y_{go}^2 C_{owr} \phi t_{c1} \quad (1) $$

where $k_1$ is the kinetic constant for gasoil
consumption, $COWR$ is the catalyst to oil ratio,
$y_f$ is the mass fraction of gasoil in the riser, $t_{c1}$
is the residence time of the catalyst in the riser
at top. $\phi$ is the deactivation factor of the
catalyst due to coke deposition

#### Mass balance of gasoline

$$ \frac{dy_g}{dz} = (\alpha_2 k_1 y_{go}^2 - k_3 y_g) C_{owr} \phi t_{c1} \quad (2) $$

#### Energy Balance

$$ \frac{dT_{ris}}{dz} = \frac{\Delta H_{crack} F_{feed}}{(F_{regcat} C_{pcat} + F_{feed} C_{po} + \lambda F_{feed} C_{psteam})} \frac{dy_{go}}{dz} \quad (3) $$
Where $\Delta H_{crack}$ is the heat of reaction, $F_{feed}$ and $F_{regcat}$ are the flow rates of gasoil and catalyst respectively. The kinetic constants follow Arrhenius law. The catalyst deactivation by coke deposition is given as

$$ \phi = (1 - m C_{cokereg}) \exp ( - \alpha t_{c1} z C_{owr}) \quad (4) $$

The produced coke concentration is empirically given by

$$ C_{cokeprod} = k_c \sqrt{\frac{t_{c1}}{C_{rc}^N} \exp (\frac{-E_{acf}}{RT_{ris,1}})} \quad (5) $$

where $T_{ris,1}$ is the temperature at the riser outlet. The amount of coke concentration leaving the riser is

$$ C_{cokeris,1} = C_{cokereg} + C_{cokeprod} \quad (6) $$

## 3.2 Separator model

The residence time of catalyst in the separator is frequently of the order of one minute. This separator can be modelled as a perfectly mixed tank.

### Mass balance of coke on catalyst

$$ \frac{dC_{cokesep}}{dt} = \frac{F_{regcat}(C_{cokeris,1} - C_{cokesep})}{m_{catsep}} \quad (7) $$

### Energy balance

$$ \frac{dT_{sep}}{dt} = \frac{C_{pcat} F_{regcat} (T_{ris,1} - T_{sep})}{m_{catsep} C_{pcat}} \quad (8) $$

## 3.3 Regenerator model

The regenerator model is inspired from [7]. The regenerator is a fluidized bed where air bubbles cross the dense bed formed by the catalyst. This bed is considered as a CSTR where the residence time of catalyst is frequently between ten to twenty minutes. The air bubbles through

the bed could be modelled as a plug flow, but for ease of simulation, they are modelled as a CSTR. The temperature and amount of coke are considered uniform throughout the dense bed as well as the oxygen concentration. An important feature of the FCC is that the reactions in the riser are mainly endothermic whereas those in the regenerator are exothermic, thus the heat released in the regenerator is used by the riser by means of the transported catalyst. As the FCC process involves catalyst recycling, its behavior is difficult to simulate correctly in steady state and transient state.

### Mass Balance of coke on the catalyst

$$ \frac{dC_{cokereg}}{dt} = \frac{(F_{spentcat} C_{coksep} - F_{regcat} C_{cokereg}) - \mathcal{R}_{cb}}{m_{catreg}} \quad (9) $$

### Energy balance in the regenerator

$$ \frac{dT_{reg}}{dt} = \frac{1}{(m_{catreg} C_{pcat})} [(T_{sep} F_{spentcat} C_{pca} + T_{air} F_{masregair} C_{pair} - T_{reg} (F_{regcat} C_{pcat} + F_{masregair} C_{pair}) - \Delta H_{cb} \frac{\mathcal{R}_{cb}}{M_{wcoke}})] \quad (10) $$

The kinetics of coke combustion is given by

$$ \begin{aligned} \mathcal{R}_{cb} &= k_{cb} \exp ( - \frac{E_{acb}}{RT_{reg}} ) x_{O2} C_{cokereg} m_{catreg} \end{aligned} \quad (11) $$

### Mass balance of oxygen in the dense bed

$$ \frac{dx_{O2}}{dt} = \frac{1}{m_{airreg}} \left[ \begin{aligned} & F_{masregair} \\ & /M_{wair} (x_{O2,in} - x_{O2,reg}) \\ & - ((1 + \sigma) n_{CH} + 2 \\ & \quad + 4\sigma)/(4(1 + \sigma)) r_{cb} \\ & \quad /M_{wcoke}) \end{aligned} \right] \quad (12) $$
<table><caption>Table 1. FCC data</caption><thead><tr><th>Symbol</th><th>Meaning</th><th>value</th></tr></thead><tbody><tr><td>Fregcat</td><td>mass flowrate of catalyst (kg.s<sup>-1</sup>)</td><td>294</td></tr><tr><td>Ffeed</td><td>mass flow rate of feed (kg.s<sup>-1</sup>)</td><td>40.63</td></tr><tr><td>Fmasregair</td><td>mass flow rate of air to regenerator (kg.s<sup>-1</sup>)</td><td>25.378</td></tr><tr><td>Tair</td><td>tempeura of air to regenerator (K)</td><td>360</td></tr><tr><td>Tfeed</td><td>feed temperature (K)</td><td>434.63</td></tr><tr><td>Tboil</td><td>boiling temperature of the feed (K)</td><td>700</td></tr><tr><td>Hvap</td><td>heat of feed vaporization (J.kg<sup>-1</sup>)</td><td>1.5610<sup>5</sup></td></tr><tr><td>Cp<sub>o</sub></td><td>heat capacity of oil (J.kg<sup>-1</sup>.K<sup>-1</sup>)</td><td>2671</td></tr><tr><td>Cp<sub>steam</sub></td><td>heat capacity of steam (J.kg<sup>-1</sup>.K<sup>-1</sup>)</td><td>1900</td></tr><tr><td>Eaf</td><td>activation energy for cracking of gas oil (feed)<br>(J.mol<sup>-1</sup>)</td><td>101.5 10<sup>3</sup></td></tr><tr><td>Eag</td><td>activation energy for cracking of gasoline<br>(J.mol<sup>-1</sup>)</td><td></td></tr><tr><td>t<sub>c</sub></td><td>residence time in the riser(s)</td><td>112.6 10<sup>3</sup></td></tr><tr><td>E<sub>act</sub></td><td>activation energy for coke formation (J.mol<sup>-1</sup>)</td><td>9.6</td></tr><tr><td>C<sub>pair</sub></td><td>heat capacity of air (J.kg<sup>-1</sup>.K<sup>-1</sup>)</td><td>2089.5</td></tr><tr><td>Hcrack</td><td>heat of cracking (J.kg<sup>-1</sup>)</td><td>1074</td></tr><tr><td>a2</td><td>fraction of gas oil that cracks to gasoline</td><td>506.2 10<sup>3</sup></td></tr><tr><td>mcatsep</td><td>holdup of catalyst in separator (kg)</td><td>0.75</td></tr><tr><td>mcatreg</td><td>holdup of catalyst in regenerator (kg)</td><td>17500</td></tr><tr><td>mairreg</td><td>holdup of air in the regenerator (mol)</td><td>175738</td></tr><tr><td>n<sub>H</sub></td><td>number of moles of hydrogen per mole of carbon in the coke</td><td>20000</td></tr><tr><td>Mw<sub>coke</sub></td><td>molar weight of coke (kg.mol<sup>-1</sup>)</td><td>2</td></tr></tbody></table>

The parameters of the present FCC model are given in Table 1. A complete description can be found in [9]. The model is simulated in Fortran90.

## 4. Nonlinear quadratic matrix control

The principle used for nonlinear quadratic dynamic matrix control is shown in (Figure 2). The principle of operation of this nonlinear model predictive strategy is based on a successive linearization of a nonlinear model and the use of a linear model predictive control algorithm namely the quadratic dynamic matrix control during each samplingperiode to generated control inputs.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Nonlinear mpc algorithm</figcaption></figure>

For explanation purposes, first DMC is presented in a SISO framework [6]. A quadratic criterion taking into account the difference between the estimated output and the reference on the prediction horizon $H_p$ is given by

$$J = \sum_{i=1}^{H_p} (\hat{y}(k + i|k) - y^{ref}(k + i))^2 \quad (13)$$

This criterion is minimized with respect to the variation of $\Delta u(k)$ of the input considered over a control horizon $H_c$.

The prediction of the ouput based on past and future inputs is

$$\hat{y}(k + l|k) = y_{ss} + \sum_{i=l+1}^{H_m-1} h_i \Delta u(k + l - i) + h_M (u(k + l - M) - u_{ss}) \\ + \frac{\text{past inputs effect}}{\sum_{i=1}^{l} h_i \Delta u(k+l-i)} + \frac{\text{future inputs effect} + \hat{d}(k+l|k)}{\text{predicted disturbances}} \quad (14)$$

where $h_M$ is the model horizon that must be larger than or equal to the prediction horizon. The output prediction based on past inputs is defined as

$$y^*(k + l|k) = y_{ss} + \sum_{i=l+1}^{M-1} h_i \Delta u(k + l - i) + h_M(u(k + l - M) - u_{ss}) \quad (15)$$

The vector of output predictions $\hat{y}(k + l|k)$ is related to the vector of ouput predictions $y^*(k + l|k)$ based on past inputs, to the vector of inputs $\Delta u(k)$ and to the vector of predicted disturbances as

$$\begin{bmatrix} \hat{y}(k+1|k) \\ \vdots \\ \hat{y}(k+H_p|k) \end{bmatrix} = \begin{bmatrix} y^*(k+1|k) \\ \vdots \\ y^*(k+H_p|k) \end{bmatrix} + A \begin{bmatrix} \Delta u(k) \\ \vdots \\ \Delta u(k+H_c-1) \end{bmatrix} + \begin{bmatrix} \hat{d}(k+1|k) \\ \vdots \\ \hat{d}(k+H_p|k) \end{bmatrix} \quad (16)$$

where A is the dynamic matrix made of step response coefficients $h_i$ of the plant outputs to the manipulated inputs.
$$A = \begin{bmatrix} h_1 & 0 & \cdots & 0 \\ h_2 & h_1 & & \vdots \\ \vdots & \vdots & \ddots & \vdots \\ h_M & h_{H_c-1} & \cdots & h_1 \\ \vdots & \vdots & \ddots & \vdots \\ h_M & h_{M-1} & \cdots & h_{M-H_c+1} \\ \vdots & \vdots & \ddots & \vdots \\ h_M & h_M & \cdots & h_M \\ \vdots & \vdots & \ddots & \vdots \\ h_M & h_M & \cdots & h_M \end{bmatrix} \quad (17)$$

For a multivariable system of dimension $n_u \times n_y$ the dynamic matrix is simply composed of submatrices. As

$$A = \begin{bmatrix} A_{11} & \cdots & A_{1n_u} \\ \vdots & \ddots & \vdots \\ A_{n_y 1} & \cdots & A_{n_y n_u} \end{bmatrix} \quad (18)$$

According to past equations, the vector of future input moves is given as

$$\Delta u(k) = [\Delta u_1(k)^T \quad \dots \quad \Delta u_{n_u}(k)^T]^T \quad (19)$$

which is the least-squares solution of the following linear system

$$\begin{bmatrix} y^{\text{ref}}(k+1) - y^*(k+1|k) - \hat{d}(k|k) = e(k+1) \\ \vdots (20) \\ y^{\text{ref}}(k+H_p) - y^*(k+H_p|k) - \hat{d}(k|k) = e(k+H_p) \end{bmatrix} = e(k+1) = A\Delta u(k) \quad (21)$$

In the absence of constraints, the least-squares solution is

$$\Delta u(k) = (A^T A)^{-1} A^T e(k+1) \quad (22)$$

In order to take into account the constraints, QDMC is used instead of DMC. Furthermore, a modification of the quadratic criterion as the sum of a performance term and an energy term is introduced in QDMC. Hard constraints affecting the manipulated variables and their moves are taken into account

$$\begin{align} u_{\min} &\le u \le u_{\max} \tag{23} \\ \Delta u_{\min} &\le \Delta u \le \Delta u_{\max} \notag \end{align}$$

These constraints can be gathered as a system of linear inequalities incorporating the dynamic information concerning the projection of constraints

$$B \Delta u(k) \le c(k+1) \quad (24)$$

In the presence of constraints, the QDMC problem can be formulated as quadratic programming such as

$$\min_{\Delta u(k)} \left[ \frac{1}{2} \Delta u(k)^T H \Delta u(k) - g(k) + (1)^T \Delta u(k) \right] \quad (25)$$

subject to constraints. H is the Hessian matrix which is equal to

$$H = A^T \Gamma^T \Gamma A + \Lambda^T \Lambda \quad (26)$$

where A is the dynamic matrix, Γ is a diagonal matrix of weights for the outputs, Λ is a diagonal matrix of weights for the inputs, g is the gradient vector that is equal to

$$g(k+1) = A^T \Gamma^T \Gamma e(k+1) \quad (27)$$

This quadratic programming problem can be solved efficiently by available subroutines [14]. The present MPC code has been developed in Fortran90 and is able to take into account any number of inputs and outputs, any type of constraint, with respect to the inputs, their moves or the outputs [6]. The version used in this article is based on step responses. The strategy to implement nonlinear quadratic matrix control is to carry out at each sampling time the identification of the process and use this algorithm to come out with a control law that is applied to the process.

# 5. Control of the FCC Process

In order to identify the FCC process, step responses are used. These coefficients of the step responses are generated in each sampling period and are used to build a dynamic matrix during each sampling period. As the process is multivariable and as imposed constraints on the manipulated variables are considered, the nonlinear quadratic dynamic matrix control algorithm is used. The controlled outputs are the riser temperature and the regenerator temperature. The manipulated inputs are the air flow rate to the regenerator and the regenerated catalyst flow rate

## 5.1 Identification

Step inputs are successively applied to the FCC model [3, 4] and the outputs are sampled with a sampling period 250s in order to obtain the step response coefficients according to Figure 3. The model horizon is equal to 60. The identified step responses are represented in

Figures 4, 5. The indicated normalized time is the number of sampling periods. The manipulated inputs are respectively the
regenerated catalyst flow rate $u_1$ and the air flow rate to the regenerator dense bed $u_2$. The controlled outputs are the temperature at the top of the riser $y_1$ and the regenerator temperature $y_2$. The step responses to $u_1$ show an algebraic effect followed by an inverse response. The algebraic effect is due to the immediate influence of the catalyst flow rate variation as the dynamic influence is neglected in the riser. The inverse response is more complex, the temperature first decreases due to the endothermic reactions in the riser, then it increases due to the exothermic reactions in the regenerator, but with a larger time constant in this latter. The influence of $u_2$ is simpler and the responses are close to first-order transfer function responses. For nonlinear quadratic dynamic matrix control, the identification process will be carried at each ampling period.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Schematic diagram of Identification Process</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 4. Coefficients of the step responses between $u_1$ and $y_1$ (left) and between $u_1$ and $y_2$ (right)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 5. Coefficients of the step responses between $u_2$ and $y_1$ (left) and between $u_2$ and $y_2$ (right)</figcaption></figure>

## 5.2 Selection of the manipulated controlled variable pairings

The relative gain array is a valuable tool for the pairing of manipulated and controlled variable in a multivariable process. The selection of the manipulated and controlled variable are described in [4].

## 5.3 Simulation results and discussions

The process is operated in partial combustion mode, opposite to complete combustion, i.e. the catalyst is not completely regenerated in the regenerator and some coke remains on the catalyst before entering the riser. The

algorithms used to control the FCC unit are NQDMC and QDMC. The sampling period is 250s, the prediction horizon 60 (sampling periods) and the control horizon 3. Constraints are imposed on the manipulated variables ($u_1 \in [270, 320]$, $u_2 \in [24, 50]$). The two algorithms are executed and the results are compared. The simulation results (Figures 6, 7, 8, 9) show that, despite changes in the set points, the outputs follow the set points with small deviations, lower than 1 or 2K at the most, while the manipulated variables remain within the constraints.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Flow rate of regenerated catalyst $u_1$ (kg.s⁻¹)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 7. Flow rate of air to the regenerator $u_2$ (kg.s⁻¹)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. Temperature at the top of the riser $y_1$ (K)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 9. Temperature in the regenerator $y_2$ (K)</figcaption></figure>

Decoupled set points were imposed to put into evidence the coupling effects between inputs and outputs. Thus, the NQDMC controller used with the successive identification through step response coefficients is able to maintain the complex process outputs close to their respective variable sets points with very acceptable deviations.

The weights used in the criterion ($\Gamma = 10$, $\Lambda = 1$) were introduced to give more importance to the performance part than to the energy part of the criterion. The consequence is that the tracking is very correct, but that at the same time some rather steep variations of the inputs are imposed, such as clearly shown around set point changes.
## 6. Comparison between QDMC and NQDMC

The results of NLQDMC algorithm and QDMC algorithm are presented in 10, 11, 12, 13, 14, 15, 16, 17. These results shows a great improvement in the control of the FCC unit by the nonlinear quadratic dynamic matrix control algorithm compare to quadratic dynamic matrix control which is a linear mpc algorithm.

<figure><img src="image_11.png" /><figcaption>Fig. 10. Temperature at the top of the riser with nlqdmc $y_1$ (K)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 11. Temperature at the top the riser with qdmc $y_1$ (K)</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 12. Temperature at the top of the riser with nlqdmc $y_2$ (K)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 13. Temperature at the top of the riser with qdmc $y_2$ (K)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 14. Flow rate of regenerated catalyst with nlqdmc $u_1$ (kg.s$^{-1}$)</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 15. Flow rate of regenerated catalyst with qdmc $u_1$ (kg.s$^{-1}$)</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 16. Flow rate of air with nlqdmc $u_2$ (kg.s$^{-1}$)</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>Fig. 17. Flow rate of air with qdmc $u_2$ (kg.s$^{-1}$)</figcaption></figure>

<table><caption>Table 2. Quantitative comparative data</caption><thead><tr><td>Parameters</td><td>NLQDMC</td><td>QDMC</td></tr></thead><tbody><tr><td>Average time of simulation for one period of sampling (s)</td><td>5.21</td><td>0.07</td></tr><tr><td>Maximum going beyond on output 1 (K)</td><td>0.4</td><td>2.7</td></tr><tr><td>Maximum going beyond on output 2 (K)</td><td>1.5</td><td>2.2</td></tr></tbody></table>

## 7. Conclusion

In this work, a multivariable nonlinear quadratic dynamic matrix control algorithm based on successive step responses was implemented to control the FCC process with the regenerated catalyst flow rate and the flow rate of air to the regenerator as manipulated variables. The simulation results show a very good tracking of the temperatures at the top of riser and in the regenerator, despite strong interactions between the riser and the regenerator. The yield of products such as gasoline and, more generally, the overall yield of an FCC unit can be improved while taking into account the different constraints of the process. A comparative study was carried out with the quadratic dynamic matrix control algorithm and the results show an improved control of the FCC process with the use of the NLQDMC algorithm rather than the QDMC algorithm.

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