**4.2. Robust control**

Robust control can be as an to control the uncertain systems (uncertainties). This approach accepts the idea of incomplete knowledge of the process, which has an uncertain dynamic and is by disturbances known. If, however, for these uncertainties can be established a mathematical norm, by using the robust control theory a robust, unique, able to meet certain performance (hard or relaxed), controller

can designed, respecting the uncertainty domain. Regardless of the method used to determine the mathematical model it is necessary to impose simplifying assumptions so that the obtained model is suitable for controller design. The differences between the real plant and the mathe‐ matical model represent modelling uncertainties or errors. Precisely from this view point the choice of robust control algorithms for the hydrogenation process 2‐ethyl hexanal is justified. The robust controller design is a laborious task itself, but as the computational tools evolved the only difficult part left is the process parameter variation range determination. The same process operational model presented in Eq. (9) is used for multi‐variable robust controller design based on *H*infinity approach to ensure robust stability for both the nominal model and the entire class of systems that exist in a particular area of uncertainty around the nominal model. A Smith predictor MIMO structure is also necessary due to the large time delays presented by the hydrogenation process like in the previous sections.

$$
\begin{vmatrix} T\_{out} \\ C\_{out} \end{vmatrix} = G\_{\mathbb{S}} \begin{vmatrix} Q\_{oct} \\ T\_{tn} \end{vmatrix} + G\_{\mathbb{z}} \begin{vmatrix} Q\_{real} \\ P \end{vmatrix} \tag{18}
$$

The first step was to determine the process state space representation:

$$\begin{array}{l} \dot{X} = A \cdot X + B \cdot U \\ Y = C \cdot X + D \cdot U \end{array}; X = \begin{bmatrix} T\_{out} \\ C\_{out} \end{bmatrix}; U = \begin{bmatrix} Q\_{oct} \\ T\_{in} \end{bmatrix} \tag{19}$$

$$A = \begin{bmatrix} a11 & a12 & 0 & 0 & 0 & 0 & 0 & 0 \\ a21 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & a33 & a34 & 0 & 0 & 0 & 0 \\ 0 & 0 & a43 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & a55 & a55 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & a65 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & a77 & a78 \\ 0 & 0 & 0 & 0 & 0 & 0 & a87 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & a16 & 0 & 0 \\ 0 & 0 & 0 & 0 & 24 & 0 & 0 & 0 & a128 \\ 0 & 0 & 0 & 0 & 24 & 0 & 0 & 0 & a28 \\ \end{bmatrix}; B = \begin{bmatrix} b11 & 0 \\ 0 & 0 \\ b31 & 0 \\ 0 & 0 \\ b51 & 0 \\ 0 & 0 \\ 0 & b72 \end{bmatrix} \tag{20}$$

The nominal values of the process parameters are:

$$\overline{a11} = -0.046, \overline{a12} = -0.0225, \overline{a33} = -0.0522, \overline{a34} = -0.0279, \overline{a55} = -0.0434,$$

$$\overline{a56} = -0.0207, \overline{a77} = -0.0361, \overline{a78} = -0.0176, \overline{a87} = 0.0313, \overline{b31} = 0.0156,\tag{21}$$

$$\overline{b72} = 0.0156, \overline{c12} = -0.1890, \overline{c16} = 0.1657, \overline{c24} = -0.0087, \overline{c28} = 0.0124$$

It is a well‐known fact that, in real control systems, uncertainties are unavoidable and can negatively affect the stability and the performance of the whole control system. Usually, the uncertainties can be classified in two main categories: disturbance signals (input/output disturbance, sensor/actuator noise) and dynamic perturbations (difference between the actual dynamics of the process and the mathematical model) [12]. The dynamic perturbations are usually caused by inaccurate characteristic description, torn and worn effects and shifting operating points. They are also called 'parametric uncertainties' and are represented by certain process parameter variation over a certain value range. In a control system the dynamic uncertainties can be represented in multiple ways. For this particular case the output multi‐ plicative representation is considered showing the relative errors (between the actual system Gp(s) and the nominal model Go(s)) not only the absolute errors: Gp (s) = [I + ∆(s)] Go(s). No matter what type of uncertainty representation is chosen, the actual, perturbed system can be represented like a standard upper linear fractional transform, where the uncertainties are lumped in a single block ∆, a diagonal matrix corresponding to parameter variations (**Figure 19**).

**Figure 19.** Generalized structure of the closed loop system.

can designed, respecting the uncertainty domain. Regardless of the method used to determine the mathematical model it is necessary to impose simplifying assumptions so that the obtained model is suitable for controller design. The differences between the real plant and the mathe‐ matical model represent modelling uncertainties or errors. Precisely from this view point the choice of robust control algorithms for the hydrogenation process 2‐ethyl hexanal is justified. The robust controller design is a laborious task itself, but as the computational tools evolved the only difficult part left is the process parameter variation range determination. The same process operational model presented in Eq. (9) is used for multi‐variable robust controller design based on *H*infinity approach to ensure robust stability for both the nominal model and the entire class of systems that exist in a particular area of uncertainty around the nominal model. A Smith predictor MIMO structure is also necessary due to the large time delays presented by

It is a well‐known fact that, in real control systems, uncertainties are unavoidable and can negatively affect the stability and the performance of the whole control system. Usually, the uncertainties can be classified in two main categories: disturbance signals (input/output disturbance, sensor/actuator noise) and dynamic perturbations (difference between the actual dynamics of the process and the mathematical model) [12]. The dynamic perturbations are usually caused by inaccurate characteristic description, torn and worn effects and shifting operating points. They are also called 'parametric uncertainties' and are represented by certain process parameter variation over a certain value range. In a control system the dynamic

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the hydrogenation process like in the previous sections.

344 New Advances in Hydrogenation Processes - Fundamentals and Applications

The nominal values of the process parameters are:

The first step was to determine the process state space representation:

The interconnection matrix for the considered multiplicative perturbation is:

$$\mathcal{M} = \begin{bmatrix} o & G\_0 \\ I & G\_0 \end{bmatrix} \tag{22}$$

The uncertainty description is determined in an unconventional manner [13]. By using the experimental identification methods several second order models were determined using experimental data from different points of operation. In this way one can determine the interval for nominal model parameters variations.p11, p12, p33, p34, p55, p56, p77, p78, p87, pb31, pb72, pc12, pc16, pc24 and pc28 represent the computed possible, relative perturbation of the nominal process parameters. Each parameter *aij*, *bij* and *cij* (*i*, *j* = 1.8) may be represented as a linear fractional transformation (LFT) considering multiplicative uncertainties.

The next step is to determine the process mathematical model that takes into account also the model parameter uncertainties, Gmds having the following form [12]:

$$G\_{mds} = \begin{bmatrix} \underline{A} & \underline{\mid} & \underline{B1} & \underline{B2} \\ C1 & \mid & D11 & D12 \\ C2 & \mid & D21 & D22 \end{bmatrix} \tag{23}$$

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The block diagram of the closed loop system in Smith predictor structure including the robust multi‐variable controller and the uncertainties bloc is presented in **Figure 20**.

**Figure 20.** The MIMO‐SP closed loop control structure using robust controller.

As it can be observed Gmds is nominal model of the process, *d*1, *d*2 represent the disturbances, Δ is the matrix uncertainties, Wu1, Wu2 and WP1, WP2 are weighting functions which reflect the relative significance of performance requirements. For good mitigation of disturbances and to ensure a certain response time and overshoot Wu1, Wu2 and WP1, WP2 have the following form:

The next step is to determine the process mathematical model that takes into account also the

The block diagram of the closed loop system in Smith predictor structure including the robust

As it can be observed Gmds is nominal model of the process, *d*1, *d*2 represent the disturbances, Δ is the matrix uncertainties, Wu1, Wu2 and WP1, WP2 are weighting functions which reflect the relative significance of performance requirements. For good mitigation of disturbances and

multi‐variable controller and the uncertainties bloc is presented in **Figure 20**.

**Figure 20.** The MIMO‐SP closed loop control structure using robust controller.

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model parameter uncertainties, Gmds having the following form [12]:

346 New Advances in Hydrogenation Processes - Fundamentals and Applications

$$\mathcal{W}p1(\text{s}) = 0.9 \frac{\text{s}^{\text{3}} + \text{2s} + 5}{\text{s}^{\text{2}} + 1.1 \text{s} + 0.02}; \mathcal{W}p2(\text{s}) = 0.1 \frac{\text{s} + 3}{\text{s} + 0.1}; \mathcal{W}u1 = 10^{-5}; \mathcal{W}u2 = 10^{-5} \tag{25}$$

It should be noted that choosing the suitable weighting functions is a crucial step in the synthesis of robust controller and usually requires several attempts. By applying the presented method, using MATLAB/SIMULINK environment – *hinfsyn* function, the following robust H8 controller was determined:

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Like for the previous control strategies the first simulation test scenario consists of closed loop simulation evaluation under nominal parameter values. The same simulation scenario was performed for the whole class of systems in the uncertainty domain for a reference step variation for the output temperature. The simulation results show good performances of the developed control structure even considering the process parameter uncertainty domain (**Figure 21**).

**Figure 21.** Output temperature evolution—robust controller: nominal vs. uncertain considering a reference step varia‐ tion.

The second simulation scenario will evaluate the capability to reject the disturbance effect and also to test at the same time the robustness of the system. To this end, a step variation of the 2

ethyl-hexanal input is considered along with the process parameter variations. Good performances are reached even in the case of the uncertain plants (**Figure 22**).

**Figure 22.** Output temperature evolution—robust controller: nominal vs. uncertain considering a reference step variation of the 2 ethyl-hexanal input

Another performance that needs to be evaluated is the ability to counter act the catalyst deactivation (**Figure 23**).

**Figure 23.** Catalyst deactivation: robust controller.

By analysing all the results obtained in the previous it can be concluded that even considering the catalyst deactivation steady-state errors of 0.006% and 0.18% are achieved for the output temperature and 2 ethyl-hexanol concentration, which are clearly within acceptable limits making the robust control strategy the most suitable for 2 ethyl-hexanal hydrogenation process control.
