**4. Design and implementation of advanced control strategies**

#### **4.1. Internal model control**

The internal model control has emerged as an alternative to traditional feedback control algorithm feedback output as the simulation methods, mathematical modelling and model validation techniques developed [10]. This method provides a direct link between the process model and the controller structure. The IMC control structure is presented in **Figure 14** where *p* represents the transfer function describing the interconnections between process inputs *u* and outputs *y*, *pd* represents the transfer function that describes the effects of disturbance on the output, *pm* is the mathematical model of the process and *q* is the transfer function of the IMC controller.

The model of the process is assumed to be equal to the process transfer function matrix presented before in Eq. (9) inferring the need of a Smith predictor structure:

$$
\begin{vmatrix} T\_{out} \\ \mathcal{C}\_{out} \end{vmatrix} = \begin{vmatrix} H\_{f11m}(\mathbf{s}) \cdot e^{-\sigma s} & H\_{f12m}(\mathbf{s}) \cdot e^{-\sigma s} \\ H\_{f21m}(\mathbf{s}) \cdot e^{-\sigma s} & H\_{f22m}(\mathbf{s}) \cdot e^{-\sigma s} \end{vmatrix} \begin{vmatrix} Q\_{\text{oct}} \\ T\_{\text{ln}} \end{vmatrix}; \tag{12}
$$

In order to ensure the process decoupling it is necessary to determine the pseudo‐inverse matrix [11] of the steady state gain matrix:

$$H\_m(0) = \begin{vmatrix} H\_{f11m0} & H\_{f12m0} \\ H\_{f21m0} & H\_{f22m0} \end{vmatrix} = \begin{vmatrix} -2.1 & 1 \\ -0.0049 & 0.0111 \end{vmatrix} \tag{13}$$

Dynamic Mathematical Modelling and Advanced Control Strategies for Complex Hydrogenation Process http://dx.doi.org/10.5772/65336 341

$$H\_m^\# = H\_m\{0\}^H \cdot (H\_m\{0\} \cdot H\_m\{0\}^H)^{-1} = \begin{vmatrix} -0.6044 & 54.945\\ -0.2692 & 115.384 \end{vmatrix} \tag{14}$$

The decoupled process is obtained as: <sup>=</sup> C # <sup>=</sup> 11 12 21 22 . Due to the static

decoupling, steady‐state matrix *HD*(*s* = 0) will be equal to the unit matrix, all the elements which are not on the first diagonal being equal to zero. Having the decoupled process the next step is to approximate the elements on the first diagonal of the matrix HD(s) with simple transfer functions using identification methods:

$$H\_{f11d}(\mathbf{s}) = \frac{{}^{1\text{e}}\text{e}^{-1650\text{s}}}{{}^{1055\text{ s}}\text{e}^{2} + 55.05\text{ s} + 1}; H\_{f22d}(\mathbf{s}) = \frac{{}^{1\text{e}}\text{e}^{-1650\text{s}}}{{}^{2204.1\text{ s}}\text{e}^{2} + 69.64\text{ s} + 1} \tag{15}$$

The last step consists of the IMC controller design using: <sup>=</sup> 11 \*() ∙() where f(s) is the IMC filter:

$$f(\mathbf{s}) = \frac{1}{(\lambda\_i s + 1)^n} \tag{16}$$

where *λi* is the time constant of the filter associated to each output and *n* is chosen such as the final IMC controller is proper.

The obtained IMC controllers are:

**Figure 13.** Output temperature evolution for reference step variation—nominal case vs. uncertain case (multivariable

A decrease in the control system performance can be observed, inferring reduced robustness,

The internal model control has emerged as an alternative to traditional feedback control algorithm feedback output as the simulation methods, mathematical modelling and model validation techniques developed [10]. This method provides a direct link between the process model and the controller structure. The IMC control structure is presented in **Figure 14** where *p* represents the transfer function describing the interconnections between process inputs *u* and outputs *y*, *pd* represents the transfer function that describes the effects of disturbance on the output, *pm* is the mathematical model of the process and *q* is the transfer function of the

The model of the process is assumed to be equal to the process transfer function matrix

In order to ensure the process decoupling it is necessary to determine the pseudo‐inverse

(12)

(13)

presented before in Eq. (9) inferring the need of a Smith predictor structure:

an aspect that may be improved by using advanced control strategies.

340 New Advances in Hydrogenation Processes - Fundamentals and Applications

**4. Design and implementation of advanced control strategies**

PID control).

**4.1. Internal model control**

matrix [11] of the steady state gain matrix:

IMC controller.

$$H\_{RIMC1}(\mathbf{s}) = \frac{{}^{1059 \cdot \mathbf{s}^2 + 95.85 \cdot \mathbf{s} + 1}}{{}^{400 \cdot \mathbf{s}^2 + 40 \cdot \mathbf{s} + 1}}; \ H\_{RIMC2}(\mathbf{s}) = \frac{2204.1 \cdot \mathbf{s}^2 + 69.64 \cdot \mathbf{s} + 1}{^{100 \cdot \mathbf{s}^2 + 20 \cdot \mathbf{s} + 1}}\tag{17}$$

**Figure 14.** IMC control structure.

The IMC control system evaluation and testing are presented in comparison to the conven‐ tional multi‐variable PID control strategy in order to conclude the results. To this end, the first test scenario consists in the set point tacking analysis. The simulation results are presented in **Figure 15**.

**Figure 15.** Output temperature evolution for reference step variation: PID vs. IMC.

The second simulation scenario is focused on the disturbance rejection analysis considering a 0.25 [Kmol/m3 ] disturbance in the 2 ethyl‐hexanal flow input flow (**Figure 16**).

**Figure 16.** Output temperature evolution for a disturbance in the 2 ethyl‐hexanal input flow: PID vs. IMC.

The third simulation scenario evaluates the IMC control system capability to counteract the effect of the catalyst deactivation (**Figure 17**).

**Figure 17.** Catalyst deactivation: IMC control.

The IMC control system evaluation and testing are presented in comparison to the conven‐ tional multi‐variable PID control strategy in order to conclude the results. To this end, the first test scenario consists in the set point tacking analysis. The simulation results are presented in

The second simulation scenario is focused on the disturbance rejection analysis considering a

] disturbance in the 2 ethyl‐hexanal flow input flow (**Figure 16**).

**Figure 16.** Output temperature evolution for a disturbance in the 2 ethyl‐hexanal input flow: PID vs. IMC.

effect of the catalyst deactivation (**Figure 17**).

The third simulation scenario evaluates the IMC control system capability to counteract the

**Figure 15.** Output temperature evolution for reference step variation: PID vs. IMC.

342 New Advances in Hydrogenation Processes - Fundamentals and Applications

**Figure 15**.

0.25 [Kmol/m3

As in the previous case the last test scenario consists in the robustness evaluation of the IMC control system by considering the same variation of the process parameters (**Figure 18**).

**Figure 18.** Output temperature evolution for reference step variation—nominal case vs. uncertain case (IMC control).

By analysing the comparative results between the MIMO SP-PID and SP-IMC control strategies it can be concluded that the SP-IMC control strategy outperforms the classical control. Another advantage is that it is easy to design and implement. However, even if the performances are clearly the robustness to process parameter variations can be improved.
