¹⁸O Isotope Separation Process Control

Abstract

The paper presents an original solution for the nonlinear Gaussian control of the ¹⁸O isotope concentration at the output of a separation column. The design of the nonlinear controller is based on the proposed separation column mathematical model which is an accurate one. In order to highlight the advantages of using a nonlinear controller, the comparison with using a linear Proportional Integral Derivative controller is made. Both the architecture and the tuning procedure of the nonlinear controller represent original elements introduced in the paper. It is proved, through simulation, that the nonlinear controller improves significantly the control system performances, especially the settling time, which is a dominant parameter in analyzing the separation processes work. Also, a solution to determine the instantaneous value of the separation column length constant is proposed, solution which opens the possibility to implement new control strategies.

1 Introduction

The separation column, together with the mandatory refluxing system’s equipment used for the ¹⁸O isotope separation process is depicted in Fig. 1 [1]. In this case study, the ¹⁸O isotope separation is obtained by means of isotopic exchange in the system made up of NO, NO₂-H₂O and HNO₃ [2]. In Fig. 1, the authors have noted FSC as the Final Separation Column (the term “Final” relates to the fact that this specific column operates as the end column of a separation cascade which consists of two separation columns). The aim of this paper is to design a control strategy for the FSC when it operates independently from the separation cascade. The results of this research activity will be further used in order to find an appropriate method to control the entire separation cascade.

Fig. 1.

The separation column for the ¹⁸O isotope and the equipment of the refluxing system.

In the following, we describe the constructive details of the FSC, shown in Fig. 1. The height is h = 10 m (the cascade is composed of 5 sectors (S1 – S5), each having equal heights of h/5 = 2 m). The corresponding sectors contain steel packing [3], hence the hatching. The column diameter (d) equals 14 mm. The process of the ¹⁸O isotope separation is based on the circulation, in counter-current, of the nitric oxides (NO, NO₂) and the nitric acid (HNO₃) – solution, introduced in the lower part, respectively in the upper part, of the column.

In the above figure, the input flow of nitric oxides is represented as F_i and the output flow of nitric acid is represented by F_W, as the isotopic waste of the separation process. F_o represents the output flow of the nitric oxides from FSC which is circulated to the arc-cracking reactor (ACR – containing steel packing [3]). The result gathered at the ACR’s output is the flow F_N of both nitrogen (N₂) and nitric oxides (NO, NO₂), which has an increased concentration of NO₂. Inside the absorber A, the process of nitric oxides absorption in water takes place due to the ceramic packing, generating the nitric acid solution that supplies the upper part of FSC (thus, F_H2O represents the supplied water flow and F_A represents the flow of nitric acid solution). The amount of nitrogen and nitric oxide from the absorber’s output, the flow F_NN (NO, N₂), is sent to the catalytic reactor CR (also containing steel packing [3]). Here, the water supplied to the absorber is the result of nitric oxide (NO) reacting to the hydrogen (H₂), as the CR is supplied with the hydrogen flow F_H. The excess of nitrogen and hydrogen is released in the form of F_NH.

Eventually, from the top of FSC, the product is extracted in the form of nitric acid with an increase concentration of ¹⁸O isotope (HN¹⁸O₃). The concentration of the ¹⁸O isotope at the output of the separation column is measured using a mass spectrometer. The pipe designated for the product flow (F_P) is represented graphically by a dashed line since, this paper’s research only deals with the total reflux working regime (thus, F_P = 0).

2 Modeling of the Isotope Separation Process

Throughout the separation exchange, the main output signal (in this case the ¹⁸O isotope concentration) depends on two independent variables, thus the ¹⁸O isotope separation process is considered to be a distributed parameter process [4, 5]. The independent variables are represented by time (t) and the position in FSC height (s), highlighted by the (0s) axis (see Fig. 1). The origin (0) of the (0s) axis is given by the centre of the Column Base Section (CBS). The independent variable (s) starts from the column base and increases its value as it reaches the top, having a maximum value at s = h, corresponding to the Column Top Section (CTS). The main input signal for this process is represented by the input flow of nitric oxides F_i(t) and, as it was previously mentioned, the main output signal is represented by the ¹⁸O isotope concentration, notated as y(t, s). The mathematical modeling is made in the hypothesis that FSC operates in total reflux regime (F_P = 0). This regime is used in the first part of the plant functioning, aiming at increasing the ¹⁸O isotope concentration until the imposed value, at the column top, is reached. The specified working regime has been selected by the authors since it is the most relevant for the FSC dynamics.

The mathematical model which describes with high accuracy the functioning of FSC was previously proposed in [6], and is briefly presented in the following equations, starting with the final form of the output signal as:

$$ \text{y(t,}\,\rm{s) = y}_{\text{0}} \rm{ + y}_{\text{f}} \rm{(t)} \cdot \frac{{\text{F}_{{\rm{s1}}} \text{(s)} - \rm{y}_{\text{0}} }}{{\rm{F}_{{\text{s1}}} \rm{(s = s}_{\text{f}} \rm{ = h)} - \text{y}_{\rm{0}} }}\text{ = y}_{\rm{0}} \text{ + y}_{\rm{f}} \text{(t)} \cdot \frac{{\rm{e}^{{\frac{\text{s}}{\rm{S}}}} - \text{1}}}{{\rm{e}^{{\frac{{\text{s}_{\rm{f}} }}{\text{S}}}} - \rm{ 1}}} $$

(1)

where y₀ = 0.204% represents the natural abundance of the ¹⁸O isotope. The function that models the separation process dynamics in relation to time (t) is represented by y_f(t), as the solution of the ordinary differential equation:

$$ \frac{{\text{dy}_{\rm{f}} \text{(t)}}}{{\rm{dt}}} = - \frac{\text{1}}{{\rm{T(F}_{\text{i}} \rm{)}}} \cdot \text{y}_{\rm{f}} \text{(t) + }\frac{\rm{1}}{{\text{T(F}_{\rm{i}} \text{)}}} \cdot \rm{u}_{\text{f}} $$

(2)

where T(F_i) represents the time constant of the isotope exchange process (that depends on the input flow F_i(t)); u_f(t) represents the increase in steady state regime of the ¹⁸O concentration over the y₀ value. The previously presented differential equation is solved considering the remark that firstly the value of the time constant is singularized for a certain (F_i). The linear variation of the time constant in relation to the input signal (T(F_i)) is given by:

$$ \text{T(F}_{\rm{i}} \text{) = T}_{\rm{2}} \text{ + K}_{\rm{T}} \cdot \text{(F}_{\rm{i}} - \text{F}_{\rm{2}} \text{)} $$

(3)

where F₂ = 185 l/h (implying the time constant T₂ = 84.703 h, experimentally determined) and $\text{K}_{\rm{T}} = - \text{3}\rm{.9262}\frac{{\text{h}^{\rm{2}} }}{\text{l}}$ represents the gradient of the ramp T(F_i). The form of the u_f(t) signal is presented below:

$$ \text{u}_{\rm{f}} \text{ = y(t}_{\rm{f}} \text{,s = s}_{\rm{f}} \text{ = h)} - \rm{y}_{\text{0}} \rm{ = y}_{\text{0}} \cdot \rm{(S(F}_{\text{i}} \rm{)} - \text{1) = y}_{\rm{0}} \cdot {(\alpha }^{{\text{n}_{{\rm{tp}}} \text{(F}_{\rm{i}} \text{)}}} - \rm{1)} $$

(4)

where t = t_f is the final value for time (given by the reach of the steady state regime) and s = s_f is the final value for the independent variable (s) (corresponding to the top of the column). Also, the term $\text{S(F}_{\rm{i}} \text{)} = \alpha^{{\rm{n}_{{\text{tp}}} \rm{(F}_{\text{i}} \rm{)}}}$ represents the separation, a function that depends on the input signal. In the separation equation, α = 1.018 represents the ¹⁸O isotope elementary separation factor, for the isotopic exchange procedure in the system NO, NO₂-H₂O, HNO₃. The theoretical plates number is given by the equation:

$$ \text{n}_{{\rm{tp}}} \text{(F}_{\rm{i}} \text{) = }\frac{\rm{h}}{{\text{HETP(F}_{\rm{i}} \text{)}}} $$

(5)

where HETP (F_i) is the Height of Equivalent Theoretical Plate. Both n_tp and HETP functions depend on the input signal (F_i). The HETP function has a linear variation over small intervals, but it has a nonlinear variation over the entire domain of F_i values, and it is described by the system of equations given below:

$$ \left\{ {\begin{array}{*{20}l} {\rm{HETP(F}_{\text{i}} \rm{) = HETP}_{\text{0}} \rm{ + K}_{{\text{H1}}} \cdot \rm{(F}_{\text{i}} - \rm{ F}_{\text{0}} \rm{),}{\kern 1pt} {\kern 1pt} \text{if}{\kern 1pt} {\kern 1pt} \text{F}_{\rm{i}} \, \le \,\text{140}\,\rm{l/h}} \hfill \\ {\text{HETP(F}_{\text{i}} \rm{) = HETP}_{\text{0}} \rm{ + K}_{{\text{H2}}} \cdot \rm{(F}_{\text{i}} - \text{F}_{\rm{0}} \text{),}{\kern 1pt} {\kern 1pt} \rm{if}{\kern 1pt} {\kern 1pt} \text{F}_{\rm{i}} \,\text{ > }\,\rm{140}\,\text{l/h}} \hfill \\ \end{array} } \right. $$

(6)

where HETP₀ = 8.6 cm, obtained for the input flow F₀ = 140 l/h; $\text{K}_{{\rm{H1}}} = - \text{0}\rm{.08}\,\;\frac{{\text{cm}\, \cdot \,\rm{h}}}{\text{l}}$ for $\text{F}_{\rm{i}} \le \,\text{140}\;\,\rm{l/h}$, and $\text{K}_{{\rm{H2}}} \text{ = 0}\rm{.0333}\;\frac{{\text{cm} \cdot \rm{h}}}{\text{l}}$ for $\rm{F}_{\text{i}} \rm{ > }\,\text{140}\;\,\rm{l/h}$ represent the gradients of the two experimentally determined ramps described by (6). Based on the above, the function HETP (F_i) is plotted in Fig. 2 [1].

Fig. 2.

The HETP dependency in relation to F_i.

Simplifying the use of the two equations from (6) is achieved by training a neural network to learn the behaviour of the HETP(F_i) function. The neural network in use is a feed-forward fully connected one [7, 8], having one input signal (F_i), one hidden layer (containing 20 nonlinear neurons, with hyperbolic tangent activation function) and one output signal (HETP(F_i)) (containing a linear neuron). Following the training process, the neural network’s output (the HETP(F_i) function) results as:

$$ \text{HETP(F}_{\rm{i}} \text{) = WL} \cdot \rm{tanh(W} \cdot \text{F}_{\rm{i}} \text{ + B) + b}_{\rm{o}} $$

(7)

where the training solutions are: WL - the row vector which contains the input weights, W - the column vector which contains the layer weights, B - the column vector which contains the bias values of the neurons from the hidden layer and b_o - the output neuron bias value. The authors used the notation “tanh” to represent the hyperbolic tangent function.

The function $\text{F}_{\rm{s}} \text{(s) = }\frac{{\rm{e}^{{\frac{\text{s}}{\rm{S}}}} - \text{1}}}{{\rm{e}^{{\frac{{\text{s}_{\rm{f}} }}{\text{S}}}} - \rm{ 1}}}$ given in (1), highlights the proportion of the ¹⁸O isotope concentration at a certain position from the FSC height, in relation to the isotope concentration at the top of the column, meaning that F_s(s) models the concentration evolution on the column height. The F_S(s) function is derived from the $\text{F}_{{\rm{s1}}} \text{(s) = y}_{\rm{0}} \cdot \text{e}^{{\frac{\rm{s}}{\text{S}}}}$ function (the approximation of the ¹⁸O isotope concentration distribution on the FSC height, for a certain (F_i) (implicitly for a certain HETP(F_i))). In F_s1(s), (S) is the constant “length” of the column, depending on the input signal F_i. The function S(F_i) is given by the system:

$$ \left\{ {\begin{array}{*{20}l} {\rm{S(F}_{\text{i}} \rm{) = S}_{\text{1}} \rm{ + K}_{{\text{S1}}} \cdot \rm{(F}_{\text{i}} - \rm{F}_{\text{1}} \rm{),}{\kern 1pt} {\kern 1pt} \text{if}{\kern 1pt} {\kern 1pt} \rm{F}_{\text{i}} \le \,\rm{140}\,\text{l/h}} \hfill \\ {\rm{S(F}_{\text{i}} \rm{) = S}_{\text{2}} \rm{ + K}_{{\text{S2}}} \cdot \rm{(F}_{\text{i}} - \rm{F}_{\text{2}} \rm{),}{\kern 1pt} {\kern 1pt} \text{if}{\kern 1pt} {\kern 1pt} \text{F}_{\text{i}} \text{ > }\,\text{140}\,\text{l/h}} \hfill \\ \end{array} } \right. $$

(8)

where S₁ = 751.124 cm is obtained for F₁ = 80 l/h and S₂ = 565.5 cm is obtained for

F₂ = 185 l/h. The gradients of the two ramps from (8) are experimentally determined as $\text{K}_{{\rm{S1}}} = -\text{ 4}\rm{.4804}\;\frac{{\text{cm} \cdot \rm{h}}}{\text{l}}$ for $\rm{F}_{\text{i}} \le \,\rm{140}\,\;\text{l/h}$ and $\rm{K}_{{\text{S2}}} \rm{ = 1}\text{.8489}\;\frac{{\rm{cm} \cdot \text{h}}}{\rm{l}}$ for $\text{F}_{\rm{i}} \text{ > }\,\rm{140}\;\,\text{l/h}$. From (8), it can be remarked that S(F_i) has a linear evolution over intervals, but a nonlinear variation over the entire domain of F_i values. The same neural structure as in the case of HETP(F_i) function is used to learn the S(F_i) function, but with 50 neurons in the hidden layer. From the training process it results the neural network output (the S(F_i) function as:

$$ \text{S(F}_{\rm{i}} \text{) = WL} \cdot \rm{tanh(W} \cdot \text{F}_{\rm{i}} \text{ + B) + b}_{\rm{o}} $$

(9)

where WL, W, B and b_o notations have the same meaning as above, noting that they are the solutions of the S(F_i) function’s neural network training.

3 The Proposed Control Strategy

The control structure proposed for the ¹⁸O isotope concentration is a complex one, and is depicted in Fig. 3. The SDPP block from Fig. 3 corresponds to the Separation Distributed Parameter Process which has an input signal that consists of F_i(t) (flow of nitric oxides) and, also, the independent variable (s) that it depends on. The output signal from the SDPP block represents the ¹⁸O isotope concentration y(t, s). The output signal is not affected by the disturbance effect just yet.

Fig. 3.

The proposed control structure [1].

The main disturbance signal (y_d(t)) is added, and it influences directly the value of SDPP output’s signal. This implies that the resulted value of the ¹⁸O isotope concentration (meaning the final output signal) is given by y₁(t, s) = y(t, s) + y_d(t). The DD block represents the disturbance delay element. The role of this block is to model the disturbance propagation into the process and its input y_d0, is the steady state value of the disturbance. The main disturbance in the control system is represented by the product extraction flow (F_P), since it can be seen as a disturbance signal due to the fact that extracting (HN¹⁸O₃) from the separation column, with an increased concentration of ¹⁸O, leads to a decrease in the ¹⁸O isotope concentration in the column. The CS block represents the transducer, the concentration sensor (a mass spectrometer) used to measure the concentration of ¹⁸O isotope, and it generates the feedback signal noted as r₁(t). The automation equipment [9, 10] from the proposed control system uses the range 4–20 mA for current signals, corresponding to the unified current’s range An example of such a signal is represented by r₁(t). The actuating signal (F_i(t)) is generated by the actuator (the nitric oxides pump P). The MSFP Block represents the reference Model of the System Fixed Part (consisting of mathematical models of elements such as: the pump P, the SDPP, the sensor CS, elements that have a serial connection). MSFP is run on a process computer parallel to the real process, having its initial behavior not affected by the exogenous disturbances (y_d(t) = 0) and also, not affected by parametric disturbances. MSFP also integrates the SDPP, by implementing the mathematical model presented in Sect. 2. Also, the mathematical models of P and CS are integrated in MSFP by implementing for each a neural network. The two neural networks have nonlinear autoregressive structure with exogenous inputs (NARX), each containing 9 linear neurons in the single hidden layer and one linear neuron in the output layer, respectively they have, each, two unit delays (one on the input signal and one on the output signal, due to the fact that P and CS are first order systems).

The main concentration controller NCC (Nonlinear Concentration Controller) processes the main error signal e₁(t) = w(t) – r₁(t) (where w(t) is the concentration setpoint signal), and generates the main control signal noted as c₁(t). The secondary error signal e₂(t) = r₁(t) – r₂(t), where r₂(t) is the feedback signal generated at the output of MSFP. This feedback signal represents the measure of the effects of all disturbances, exogenous and parametric, which occur in the system. Thus, e₂(t) becomes a measure of the deviation of the output’s signal value in relation to its reference value. For proper comparison between the real plant behaviour (referring to the fixed part) and its reference model behaviour, the same input signal c_f(t) must be applied to both blocks. There are two types of possible parametric disturbances, such as small variations in relation to time of the separation column structure parameters, or small variations of the independent variable (s). The latter may occur due to a shift of the CS position and of the product extraction point, in relation with the column height, thus the reference model is simulated for s = s_f. The disturbances compensator block (DC), of Proportional – Derivative type (PD), computes the error signal e₂(t) and generates the necessary compensation control signal c₂(t). The total control signal c₃(t) is a composed control signal, such as c₃(t) = c₁(t) – c₂(t). The final control signal c_f(t) results as c_f(t) = c₀ – c₃(t), where c₀ represents the value proportional to the value of the reference flow, in unified current. The reference flow is referring to the initial flow at which the separation column starts to operate. From (3) it results that the lowest value of the separation column’s time constant is obtained for the flow F₂. Consequently, in order to generate an acceptable value of the concentration control system settling time, the reference flow is fixed at the value of F₂. This means that other values could be chosen for the reference flow (such as F₁), however this would generate slower control systems, since it increases the settling time. The element IF (Initial Flow) generates the value F₂, scaled to the c₀ value using the proportionality constant K1. In case of the real separation plant, the following elements from Fig. 3: NCC, DC, MSFP, LIVIE, PM, IF, K1, NA and K2 have to be implemented on numerical equipment.

Since the proposed control structure is a complex one, to not further complicate it in a non-understandable manner, the analog to digital and the digital to analog converters are not figured. The role of such blocks is to connect the numerical equipment to the separation plant to be controlled.

4 Controllers Tuning and Simulations Results

The computer simulations presented in this paper are carried out via MATLAB/Simulink. The NCC tuning is made neglecting the disturbances action (meaning y_d0 = 0 and s = s_f) and, also, without considering the DC control action (c₂(t) = 0). Moreover, in the primary tuning procedure, we singularize the NCC controller as being a linear Proportional Integral Derivative one (PID CC – PID Concentration Controller). Due to the fact that the isotope separation process, besides being a distributed parameter process, is a strong nonlinear one, the relay method is used for the PID CC tuning. This method is based on generating oscillatory variation on the output signal y(t, s), and for that to occur the PID CC controller is replaced with a two position relay. The relay characteristics are: the output commutation between the value b_min = 0 mA and the value b_max = 6.4 mA (without considering the initial value of 4 mA for the unified current signals), respectively the hysteresis set to ±1 mA.

The simulation is carried out for a step type setpoint signal (proportional with the imposed value of 1.5% for the output, in this case the ¹⁸O isotope concentration). The resulted control system response (the variation in relation to time of the y(t, s) signal) is presented in Fig. 4 [1].

Fig. 4.

The control system response in the case of the relay tuning method application.

In Fig. 4, the fact that oscillation with constant amplitude are obtained, after 1000 h from the simulation start, is highlighted. From Fig. 4, the oscillation period T_OSC = 552 h and the oscillation amplitude A_OSC = A/2 = 0.0907% result. Using the values obtained for T_OSC and A_OSC, the parameters of the PID CC controller can be computed quite easily using the Ziegler – Nichols equations. The PID controller transfer function is:

$$ \text{H}_{{\rm{CC}}} \text{(s) = K}_{{\rm{CC}}} \cdot \text{(1 + }\frac{\rm{1}}{{\text{T}_{{\rm{ICC}}} \cdot \text{s}}}\rm{ + }\frac{{\text{T}_{{\rm{DCC}}} \cdot \text{s}}}{{\rm{T}_{{\text{fCC}}} \cdot \rm{s + 1}}}\text{)} $$

(10)

where K_CC is the controller proportionality constant, T_ICC is the controller integral time constant, T_DCC is the controller derivative time constant and T_fCC is the time constant of the first order filter used to make the PID controller feasible. The controller parameters values that result from Ziegler – Nichols equations, are the benchmarks for fine tuning the controller, in order to improve the control performances. The fine tuning consists in modifying the initial form of the obtained controller parameters and testing the control performances variation, until the best set of performances is obtained. At the tuning procedure end, the following parameters are obtained: K_CC = 0.358, T_ICC = 197.143 h, T_DCC = 33.12 h and T_fCC = 0.625 h. If the setpoint concentration is set to 1.5%, the comparative graph between the separation process open loop response and the control system responses for the cases of using the best PI controller that could be obtained, respectively using the proposed PID CC controller, is presented in Fig. 5 [1]. In Fig. 5, the notation CSt signifies the minimum concentration value of the steady state band near the steady state value of the output signal (the steady state band is fixed at ±2%, more exactly CSt = 1.47%).

Fig. 5.

The comparative graph between the open loop separation process response and the control system responses using two different controllers.

From Fig. 5, it results that in all three simulation cases, the steady state error e_st = 0%, due to the fact that, in steady state regime the three curves reach the imposed value for ¹⁸O concentration (y_st = 1.5%). Also, in the case of using both controllers, a negligible overshoot occurs, having the value smaller than 1% (the maximum imposed limit). The PID CC controller generates an overshoot of σ₁ = 0.94%, insignificantly higher than σ₂ = 0.9% (generated by the PI controller). Obviously, in the case of simulating the separation process open loop response, the overshoot is σ₃ = 0%. Due to the fact that, in all three cases, the overshoot is included in the steady state band, the settling time (which is a critical performance) is determined as the first moment when the curves plotted in Fig. 5 reach the CSt value. From Fig. 5, the fact that the PID CC controller generates the smallest settling time t_s1, results. In order to distinguish the three values of the settling times, a detailed view of Fig. 5, around the CSt value is depicted in Fig. 6 [1].

Fig. 6.

The settling times associated to the three curves from Fig. 5.

In Fig. 6, the fact that the PID CC controller generates the best settling time (t_s1 = 719.58 h) is highlighted. It can be remarked that t_s1 < t_s2 < t_s3 (t_s2 = 740.21 h and t_s3 = 819.92 h). Consequently, the PID CC controller’s usage improves significantly the separation plant settling time, leading to important economic advantages. Also, using the PID CC controller, even better settling times were obtained in simulations, but only if the actuating signal exits from the allowed variations limits. Practically, t_s1 represents the best settling time that could be obtained if the actuating signal (F_i(t)) remains enclosed between the saturation limits of the unified current’s range. The open loop separation process simulation is made by simulating the mathematical model presented in Sect. 2, for the input flow F₅ = 150.255 l/h, value which generates the steady state value of 1.5% for the output ¹⁸O isotope concentration (according to (3) and (6), the best settling time for the open loop process, if we impose a certain value of the output signal, is obtained if F_i(t) > 140 l/h).

The main disadvantage in the case of using the separation process in open loop is the fact that the effect of the disturbances cannot be rejected, aspect highlighted in Fig. 7. The simulations from Fig. 7 are carried out in the same manner and conditions as the simulations from Fig. 5, but the exogenous disturbance with the value y_d0 = −0.1% is added in the system at the moment t₁ = 2500 h. The y_d0 value is propagated in the system through the DD element having the transfer function $\rm{H}_{{\text{DD}}} \rm{(s) = }\frac{\text{1}}{{\rm{T}_{{\text{DD}}} \cdot \rm{s + 1}}}$, where the time constant equals T_DD = 50 h. Also, the negative value of y_d0 is physically interpreted as the output ¹⁸O isotope concentration decrease, due to the exogenous disturbance effect. From Fig. 7, it results that in the case of the PID CC controller usage, the disturbance effect is rejected, thus in steady state regime the system response reaches again the imposed value (y_st = 1.5%). The open loop response of the separation process, after the disturbance occurrence in the system, settles at a different steady state value of y_st1 = 1.4% (with y_d0 smaller than y_st, resulting that the disturbance effect is not rejected), fact which highlights again the necessity of using a controller for the concentration control. An important problem which results from Fig. 7 [1] is the slow reaction of the control system in disturbance rejecting regime (the duration of the transitory regime caused by the disturbance effect is over 800 h).

Fig. 7.

The exogenous disturbance effect.

As it is presented in Figs. 5 and 6, the linear PID concentration controller represents the best solution of controlling the separation process that could be obtained using linear controllers, obtain through classical methods. But from these figures, the necessity to improve even more the control system performances occurs, too. Hence, even if the obtained settling time value is consistently improved using the PID controller, it remains higher than 700 h. In this context, the implementation of a control strategy which can generate a settling time for the ¹⁸O isotope concentration, in starting regime, lower than 700 h, would improve significantly the separation plant efficiency.

A solution to obtain higher control performances is to use the nonlinear control and implicitly to use a NCC (Nonlinear Concentration Controller). The main idea of implementing the NCC is based on the simulation presented in Fig. 8. In Fig. 8, the comparative graph between the open loop response of the separation process, if the input signal has the value F₂ = 185 l/h and the closed loop response of the control system, if the determined PID controller is used, is presented. From Fig. 8, the much faster increase of the ¹⁸O isotope concentration until the value of 1% results, for the case of the open loop response (the proportion between the two increasing times, being: t_i1 = 138.35 h << t_i2 = 213.1 h). This phenomenon is explained by Eq. (3), the separation process time constant having a smaller value as higher the input nitric oxides flow is. Consequently, the lowest value of the process time constant and implicitly, the fastest response, is obtained for the highest value of the input flow (F₂ = 185 l/h). The problem in using this value for the input flow is the fact that its corresponding HETP value is HETP(F₂) = 10.1 cm, which is higher than the column optimal one: HETP₀ = 8.6 cm (aspect shown in Fig. 2). Due to this aspect, the steady state value of the blue line from Fig. 8 is only y_st2 = 1.1933%, value lower than the imposed one (1.5%). Being an open loop response, the purpose of this simulation is not to obtain a null steady state error, but to prove that the increase of the ¹⁸O isotope concentration can be made, using the same separation column, much faster than in the case of using the previously determined PID controller.

Fig. 8.

The input nitric oxides flow effect on the ¹⁸O isotope concentration increase.

Another important aspect, is presented through the simulations from Fig. 9.

Fig. 9.

The HETP(F_i) effect on the ¹⁸O isotope concentration increase.

Practically, Fig. 9 resulted by adding in Fig. 8 the separation process open loop response for the input flow F₀ = 140 l/h. As it can be remarked from Fig. 2, for the input flow F₀, the column optimum from the HETP(F_i) function perspective, is obtained. This aspect explains the fact that, for the red curve from Fig. 9, the steady state value y_st3 = 1.62% is obtained, practically the highest concentration value for the ¹⁸O isotope that can be obtained using this separation column (obviously using the considered separation column as a separate entity and not working in a separation cascade). Even if the usage of F₀ has the previously mentioned advantage, from Fig. 9 it can be remarked that, until the concentration value of 1%, the open loop response in the case of applying the F₂ input signal generates a much lower increasing time (based on the explanations associated to Fig. 8, the red and the green curves from Fig. 9 being almost superposed until the ¹⁸O isotope concentration of 1.1%). But, an important advantage generated by the F₀ input signal application is obtained in the domain of high concentrations. From Fig. 9, it results that, considering as reference point the concentration value 1.4%, the obtained increasing time t_i3 = 476.1 h, in the case of using the F₀ input signal and the open loop simulation, is smaller than the obtained increasing time t_i4 = 538.6 h in the case of using the PID controller and implicitly the closed loop simulation. This phenomenon occurs due to the higher steady state value of the response obtained for the open loop separation process, in the case of applying the F₀ input signal. This remark proves the possibility to obtain a better increasing time of the ¹⁸O isotope concentration in the domain of high concentrations, if we apply at the separation process input, nitric oxides flows having the values in the F₀ neighborhood.

Based on the above two ideas, the following architecture for the nonlinear concentration controller is proposed in Fig. 10.

Fig. 10.

The proposed architecture for NCC (Nonlinear Concentration Controller).

As it is highlighted in Fig. 10, the NCC has a complex architecture containing the following elements: GPE – Gaussian Processing Element, CE – Comparison Element, PID CC – linear PID controller from the NCC structure, TPRL – two position relay, SRE – Signal Routing Element. The input signal in NCC (e₁(t)) is applied at the input of both GPE and PID CC elements, respectively the output signal c₁(t) is generated at the output of SRE element.

The GPE is used both to control the TPRL work and to generate a variable input signal for the logic implemented by CE. A viable solution for these two objectives is the usage of the Gaussian functions, due to their properties of generating a maximum value for the optimum input variable value, respectively minimum values if the input variable value differs significantly by the optimal one. The proposed form of the Gaussian function used in the case of NCC and implemented by GPE, is:

$$ \text{g(t) = e}^{{ - \frac{{\rm{(e}_{\text{1}} \rm{(t)} - \text{e}_{\rm{0}} \text{)}^{\rm{2}} }}{{\text{DS}}}}} $$

(11)

where g(t) is the Gaussian function (signal) generated by GPE, e₀ is the imposed reference error for which the Gaussian function has the maximum value (value 1) and DS is a constant through which we control the “width” of the Gaussian curve (DS/2 represents the variance of the Gaussian curve). As it is previously mentioned, the main purpose of the g(t) function is to control, through its value, the commutation of TPRL. As consequence of the Fig. 8 analysis, until the y₁(t, s) output signal (from the proposed control system) reaches the value 1%, it is recommendable that at the process input, the F₂ input signal to be applied (in order to obtain the fastest response in the first part of the separation column starting regime). In this context, it is necessary to study the error signal (e₁(t)) dynamics in the first 200 h, if the hypothesis when the F₂ input signal is applied. The corresponding simulation is presented in Fig. 11.

Fig. 11.

The dynamics of the e₁(t) signal, if the F₂ input signal is used as control signal.

Studying the error dynamics presented in Fig. 11, the conclusion that the value of the reference error e₀ has to be imposed at a relative high limit. In order to highlight the procedure for imposing the value of e₀, the parallel to the time axis, which passes through the value e₁(t) = 10 mA is figured with dashed line.

Also, the intersection points I1 and I2 of the parallel to the time axis with the e₁(t) signal are highlighted. Analyzing the positions of the I1 and I2 intersection points, we have concluded the following aspects: the I1 point is very closed by the ordinate; the point I2 is placed at “medium” distance in relation to the ordinate; between the I1 and I2 points, a distance, on the time axis, of approximately 50 h can be measured. Considering these conclusions, the value of e₀ = 10 mA is appropriate to be imposed for this application. If DC is initialized with the value 1, the g(t) signal, computed for the error variation form presented in Fig. 11, is presented with red line in Fig. 12.

Fig. 12.

The g(t) signal variation for the initial and for the determined values of DC constant.

From Fig. 12, the fact that the g(t) signal for DC = 1 (figured with red line), reaches the maximum value (equal to 1 mA) in two points, corresponding to the positions in relation to time of the I1 and I2 intersection points from Fig. 11. Under this form, the Gaussian curve for DC = 1 cannot be used in the case of the NCC controller presented in Fig. 10 (implicitly neither in the case of the control structure presented in Fig. 3). The two main problems of the Gaussian curve figured with red line, are: the free space between the two peaks, which does not allow the possibility to activate the relay in the corresponding domain of values, respectively the fact that, practically after 100 h, its value tends at 0 mA (not allowing the relay control until the moment t_i1 (on the domain (0−t_i1) h), in the context that until t_i1, the ideal concentration increase can be made according to Fig. 8). The solution to obtain a usable form of the Gaussian signal, for this application, is to increase its variance by increasing the value of the DC constant. Increasing the value of the DC constant, the “width” of the Gaussian curve on the initial time domain (associated to the approximately first 500 h), is increased, too. The value of the DC constant is determined iteratively, following the procedure: the initial value of DC is set to 1; at each iteration, the value of DC constant is increased with 1; at each iteration, the Gaussian curve is simulated and its properties are analyzed, until they satisfy the imposed quality indicators (the imposed quality indicators for the Gaussian curve, are: the minimum value of the curve, on the time domain from the two peaks, to have a higher value than 0.7 mA, respectively, for higher values of time, the curve to have insignificant values only after 500 h); the algorithm is stopped at the first iteration when the two quality indicators are satisfied, the corresponding minimum value of DC for which this objective is accomplished, being considered the searching solution. After applying the described procedure, the value DC = 20 is determined. If the new value, DC = 20 is considered, the g(t) signal, computed for the error variation form presented in Fig. 11, is presented with blue line in Fig. 12. From Fig. 12, it can be remarked the fact that the blue Gaussian curve satisfies the imposed quality indicators. Also, from Fig. 12, it results that the value of the g(t) signal (for DC = 20), corresponding to t_i1, is g₀ = 0.4 mA (highlighted as the intersection point between the two dashed lines from Fig. 12), value which is a significant one. Considering these aspects, the proposed solution for the Gaussian function which will be following used in the NCC tuning, is:

$$ \rm{g(t) = e}^{{-\frac{{\text{(e}_{\rm{1}} \text{(t)} - \rm{10)}^{\text{2}} }}{{\rm{20}}}}} $$

(12)

The Two Position Relay (TPRL) work is based on the g(t) signal value, g(t) being its input signal. Also TPRL is a relay with hysteresis, being defined by the following equation:

$$ \text{c}_{{\rm{RL}}} (\text{t}) = \left\{ \begin{aligned} 0\;\rm{mA},\;\text{if}\;\rm{g}(\text{t}) \ge \,0.75\;\rm{mA} \hfill \\ 6\;\text{mA},\;\rm{if}\;\text{g}(\rm{t})\, \le \;0.4\;\text{mA} \hfill \\ \end{aligned} \right. $$

(13)

As it can be remarked from Fig. 3, if we consider c₂(t) = 0 mA, than c_f(t) = c₀ − c₁(t). The direct implication of this equation is represented by the fact that, when c₁(t) has minimum value (0 mA) than c_f(t) = c₀ has maximum value, which implies the maximum nitric oxides flow (F₂) applied at the separation process input (practically, the separation column is started applying at its input, the maximum allowed value of the nitric oxides flow). Technically, the constrain that c₁(t) = 0 mA can be obtained if, at the NCC output, the c_RL(t) control signal is generated (c₁(t) = c_RL(t)), and c_RL(t) = 0 mA. Following this procedure, the ideal concentration increase can be made until the value 1%, according to Fig. 8. After the concentration increase over the value of 1%, according to the conclusions obtained by Fig. 9 analysis, the input nitric oxides flow should be decreased at a value from the neighbourhood of the F₀ optimum flow. In this context, considering the nitric oxides pump (P) proportionality constant value and, also, considering c₂(t) = 0 mA, the value c₁(t) = c_RL(t) = 6 mA implies a constant input flow F₃ = 145.625 l/h. It results a sensible higher value than the value of F₀ in order to satisfy the following compromise: after the concentration value of 1%, we have to reduce the flow value near F₀, but, it is, also, recommended to maintain a higher value than F₀ in order to preserve a process time constant as much small as it possible (according to (3)) and implicitly a dynamics as much faster as it possible for the separation process. The TPRL hysteresis is imposed through the g(t) Gaussian function values from (13), 0.4 mA and 0.75 mA. The g₀ = 0.4 mA value determining is made as it was above presented in the case of the DC constant computation algorithm. The g₁ = 0.75 mA value is determined, in order to obtain a fast commutation of the relay output to the value of 0 mA (after 3 h, analysing the blue curve variation from Fig. 12, on the first 10 h of its evolution (obviously applying the zoom options)). Practically, in this context, in starting regime on the first 3 h of work, at the process input the F₃ input signal is applied, and after that, until t_i1, the F₂ input signal is applied. The first period of 3 h is insignificant in relation to the column dynamics (dynamics of hundreds of hours, as it is proved through (3)) and we can approximate the fact that its start is made applying F₂ input signal. Also, the first 3 h of the plant work, in which we physically avoid the application of the maximum input flow, are important for the column protection against faults.

Based on the Gaussian curve variation figured with blue line in Fig. 12, the TPRL output signal (c_RL(t)) is presented in Fig. 13.

Fig. 13.

The c_RL(t) simulation, for the Gaussian curve presented with blue line in Fig. 12.

In Fig. 13, the commutation points of TPRL are highlighted through the intersection points between the blue Gaussian curve and the parallels with the time axis which pass through the values of g₀, respectively of g₁, according to (13). The TPRL output signal is figured with purple continuous line. The usage of TPRL represents a very efficient method in order to obtain a much lower settling time of the output signal y₁(t, s) from the control structure presented in Fig. 3. The main problem of TPRL is the fact that it usage cannot generate null steady state error (e_st = 0%). Considering this remark, the usage of the PID controller in the control structure becomes a necessary one in order to impose the y₁(t, s) signal proper stabilization. With the purpose not to decrease significantly the advantage introduced by the TPRL usage through the consistent decrease of the settling time value, the PID controller action will be introduced only to increase the concentration of the y₁(t, s) signal in the final domain [(y_{st_sp} − 0.1%); y_{st_sp}], where y_{st_sp} represents the steady state value of the output signal (y₁(t, s)) imposed through the setpoint signal.

The Comparison Element from Fig. 10 is used to compare the value of its input signal g(t) with a certain imposed value and to generate, based on the comparison, the logical signal l(t) which control the SRE element commutation. The CE element work is modeled by the equation:

$$ \rm{l(t) = }\left\{ \begin{aligned} \text{0,}\;\rm{if}\;\text{g(t)}\, \ge \,\rm{g}_{\text{2}} \hfill \\ \rm{1,}\;\text{if}\;\rm{g(t)}\,\text{ < }\;\rm{g}_{\text{2}} \hfill \\ \end{aligned} \right. $$

(14)

where “0” and “1” are logical variables. The g₂ value represents the value of g(t) given by (12), if the error signal value corresponds to a difference equal to 0.1%, between the imposed value of the ¹⁸O isotope concentration through the setpoint signal and the instantaneous value of the ¹⁸O isotope concentration measured by CS. Considering the mathematical model of CS, more exactly its proportionality constant, the error value corresponding to the mentioned concentration difference (0.1%) is e₁₀ = 1.1282 mA. Applying (12), for e₁(t) = e₁₀, the value g₂ = 0.0195 mA results.

The Signal Routing Element (SRE) works as a “switch”. In relation to its input signal value (l(t)), it commutes between the TPRL output signal and the PID controller output signal, allowing the passing to the NCC output only of one between these two signals. The SRE element work is modeled by the equation:

$$ \rm{c}_{\text{1}} \rm{(t) = }\left\{ {\begin{array}{*{20}l} {\text{c}_{{\rm{RL}}} \text{(t),}\;\rm{if}\;\text{l(t) = 0}} \hfill \\ {\rm{c}_{{\text{PID}}} \rm{(t),}\;\text{if}\;\rm{l(t) = 1}} \hfill \\ \end{array} } \right. $$

(15)

where c_PID(t) is the output signal of the PID CC controller.

The linear PID controller from the structure of NCC has the same form as in (10), and its parameters are, in primary form, considered the same with the parameters determined above using the relay method. In order to improve the performances of the control system, the K_CC proportionality constant has to be adjusted as value (this aspect is due to the fact that the PID control action is introduced only in the domain of high concentrations). The new form of the PID controller proportionality constant is given by: K_CC1 = q ∙ K_CC, where q represents a constant, more exactly a weighting coefficient. The value of q weighting coefficient is determined using the following iterative procedure: the q constant is initialized to the value 1; the increase of the q constant, from an iteration to the next one, is considered ∆q = 0.05; the simulation of the control structure from Fig. 3, containing the NCC controller, is repeated (for c₂(t) = 0 mA) until, from an iteration to the next one, the settling time of the y₁(t, s) signal starts to increase; the q value determined at the previous iteration is considered as the searching solution. After applying this procedure, the value q = 1.3 is determined. In this moment, the structure of NCC is completely defined.

If the setpoint concentration is set to 1.5% (the same as in the case of Fig. 5 case), the comparative graph between the control system responses for the cases of using the initial linear PID CC controller, respectively of using the proposed NCC, is presented in Fig. 14. In Fig. 14, the fact that in both cases, the steady state error e_st = 0%, due to the fact that in steady state regime, the red, respectively the blue curves have the value 1.5% (the imposed setpoint concentration). Also, from Fig. 14, it results that the system response is faster in the case of using NCC than in the case of using the linear initial PID controller (implicitly, the settling time is shorter in the case of using the NCC controller than in the case of using the initial linear PID controller). In Fig. 14, the open loop response of the separation process is presented (with purple line), also, for the case when the input signal is F₂ = 185 l/h (step type signal). This response is figured in order to prove the fact that, until a concentration higher than 1%, the system response in the case of using the NCC controller is almost superposed over it. This aspect proves the accomplishment of the NCC efficiency in obtaining the best possible response in the range of concentration values enclosed between 0.204%–1−1.1%.

Fig. 14.

The comparative graph between the control system responses, in the case of using the initial linear PID controller, respectively in the case of using the NCC controller.

In order to prove the advantages of using the NCC controller, the comparative graph from Fig. 15 is considered. Practically, in Fig. 15, the same responses of the system in the cases of using the two controllers are presented as in Fig. 14, but only in the domain of values which highlight the entering of them in the steady state regime. The linear initial PID controller, obviously, generates the same settling time as in Fig. 6 case, more exactly t_s1 = 719.58 h and the NCC controller generates the settling time t_s4 = 662.4 h. It results that t_s4 < t_s1 with 46.82 h which represents a consistent settling time improve introduced through the NCC usage and implicitly a big obtained technological advantage. Another aspect which is highlighted in Fig. 15 is the difference between the overshoots of the two responses. Consequently, the overshoot generated by NCC controller σ₄ = 0.28% is lower than the overshoot generated by the linear initial PID controller σ₁ = 0.94%, fact which represents another advantage of using the NCC controller. The settling time is imposed by the same CSt = 1.47% concentration value as in the case of Fig. 6. Also, the overshoots, they being included in the steady state band of ±2% near y_st = 1.5%, do not influence the two settling times values.

Fig. 15.

The comparative graph between the control system responses, in the case of using the linear initial PID controller, respectively in the case of using the NCC controller.

In Fig. 16, the actuating signal (the nitric oxides input flow) variation, in relation to time, is presented, generated as effect of the NCC control action (practically the actuating signal from Fig. 16 is associated with the red response from Fig. 15). In Fig. 16, it can be remarked that the separation column work is started using the F₃ = 145.625 l/h nitric oxides flow, and after the moment t_c1 = 3 h, the actuating signal, due to the TPRL action, commutates to the value F₂ = 185 l/h.

Fig. 16.

The actuating signal generated as effect of the NCC control action.

At the moment t_c2 = t_i1 = 138.35 h, the TPRL commutates again (after the ¹⁸O isotope concentration increase over the value 1%), generating as effect the value F₃ = 145.625 l/h for actuating signal. After the moment t_c3 = 440.62 h, when the control system response reaches at the value 1.4% (y_st – 0.1%), the linear modified PID controller (having K_CC1 proportionality constant) is activated, respectively TPRL becomes inactive. The actuating signal, after the moment t_c3, is generated as effect of the modified linear PID controller action.

In the context of the slow reaction of the control system in disturbance rejecting regime (proved in Fig. 7), the necessity of using the DC element occurs. For the structure of the Disturbance Compensator (DC) a PD transfer function is proposed, it having the form:

$$ \text{H}_{{\rm{DC}}} \text{(s) = K}_{{\rm{DC}}} \cdot \frac{{\text{T}_{{\rm{DDC}}} \cdot \text{s + 1}}}{{\rm{T}_{{\text{fDC}}} \cdot \rm{s + 1}}} $$

(16)

The tuning procedure of DC follows the stages: firstly the proportionality constant of DC is initialized at the value K_DC = 1, its derivative time constant is initialized at the value T_DDC = 10 h, respectively the time constant of the first order filter used to obtain the feasible form of DC is fixed at the value T_fDC = 1 h; secondly, T_DDC is iteratively determined by increasing its value from an iteration to the next one with ∆T_DDC = 1 h until the system response overshoot in disturbance effect rejecting regime occurs (the overshoot values are determined repeating the simulation at each iteration); finally, K_DC is iteratively determined by increasing its value from an iteration to the next one with ∆K_DC = 0.01 until the system response overshoot in disturbance effect rejecting regime equals the system response overshoot in starting regime (σ₁). After the application of the presented algorithm, the structure parameters of DC result: K_DC = 1.1, T_DDC = 50 h. The comparative graph between the control system response in the case of using and the case of not using the DC element, is presented in Fig. 17.

Fig. 17.

The effect of using DC.

The simulations from Fig. 17 are made in the same conditions as the simulations from Fig. 7. From Fig. 17, it results that the control system generates much better performances in disturbance rejecting regime in the case of using DC than in the case without DC. Relative to the moment t₁ (when the disturbance signal occurs in the system), in the case of using DC, the ¹⁸O isotope concentration reaches again the steady state regime after 219.4 h (corresponding to the moment t_s5). This transitory regime is much shorter than in the context of not using DC (in this case the ¹⁸O isotope concentration reaches again the steady state regime after 842.7 h (corresponding to the moment t_s6)). Consequently, the usage of DC implies a consistent improvement of the system performances in rejecting the disturbances effects, fact which represents a major technological and economical advantage.

In Fig. 18, the control system response is presented in the case of the (s) independent variable variation. The variation occurs at the moment t₂ = 2500 h from the simulation start, when (s) independent variable value is changed from the initial value s = s_f (s_f = 1000 cm) to the value s = 970 cm. This variation is due to the concentration sensor position change in relation to the column height. In Fig. 18, the fact that the two control elements (NCC and DC) reject the effect of the (s) independent variable value “jump”, is highlighted. Only after 237.8 h (corresponding to the moment t_s7) from the moment t₂, the ¹⁸O concentration equals the CSt value, the imposed steady state regime (near the value y_st = 1.5%) being reached again.

Fig. 18.

The system response, in the case of the (s) independent variable variation.

This aspect proves the fact that the proposed control structure can efficiently manage the effect of the parametric disturbances, too.

The instantaneous value of (s) independent variable is determined using LIVIE (Length Independent Variable Identification Element), more exactly the sensor position in the column. This procedure is necessary in order to determine if the separation plant can be physically used when the sensor position in the column is changed. Another advantage of this procedure is the possibility to equivalate the disturbances effects with the change of the CS position in the column. The work of LIVIE is based on obtaining the F_S(s) function from the ratio between the feedback signals r₁₀(t) and r₂₀(t), resulting the signal a(t) ($\text{a(t) = }\frac{{\rm{r}_{{\text{10}}} \rm{(t)}}}{{\text{r}_{{\rm{20}}} \text{(t)}}}$) (Fig. 3). The feedback signals r₁₀(t) and r₂₀(t) result as differences between the primary feedback signals r₁(t), respectively r₂(t) and r₀, which is the value in unified signal corresponding to the ¹⁸O natural abundance (y₀). The value of (y₀) is generated by NA (Natural Abundance Element), after that it being scaled with the proportionality constant K2, resulting (r₀). Obviously, at the plant work starting, r₂₀(t) is initialized at a value different than 0, in order to make possible the computation of a(t). The instantaneous value of the independent variable (s), notated with (s₁), can be obtained from the equation of F_S(s), being given by:

$$ \rm{s}_{\text{1}} \rm{ = S(F}_{\text{i}} \rm{(t))} \cdot \text{ln[a(t)} \cdot \rm{e}^{{\frac{{\text{s}_{\rm{f}} }}{{\text{S(F}_{\rm{i}} \text{(t))}}}}} \rm{ + (1} - \text{a)]} $$

(17)

The numerical implementation of (17) is made by LIVIE. The two input signals in LIVIE are a(t) and F_ia(t) which is the approximated value of the actuating signal F_i(t). The signal F_ia(t) is generated by the simulation of PM (Pump Model) element, which is implemented using the same procedure as in the case of MSFP and having as input signal the c_f(t) signal. The natural logarithm and the exponential function from (17) are numerically implemented using feed-forward fully connected neural networks, too. The Decision Element (DE) generates the “DECISION” in relation to the identified value of (s₁). The “DECISION” can be the generation of an alarm if the (s₁) variable tends to reach the limit value or the separation plant stop if (s₁) variable equals the imposed limit. The variation in relation to time of the LIVIE output signal (the approximation (s₁) of the (s) instantaneous value), corresponding to the response from Fig. 18, is presented in Fig. 19.

Fig. 19.

The variation in relation to time of the (s₁) signal.

It can be remarked the fact that the approximation of the (s) independent variable is an exact one both before and after the variation moment (t₂).

In the case when the concentration sensor remains in the position given by s = 970 cm, the ¹⁸O isotope concentration evolution, in relation to time, for the value s = s_f, is presented in Fig. 20. From Fig. 20, it results that the ¹⁸O isotope concentration increases at the column top (s = s_f). This aspect is explained through the increasing evolution of the output signal in relation to (s) independent variable (according to (1)). The change of the concentration sensor position has an important application, more exactly when, immediately after the extraction of the necessary ¹⁸O isotope quantity at the imposed concentration (in the example 1.5%), the isotope extraction at a higher concentration is necessary (for example 1.6%). In this case, firstly the extraction is made from the point s = 970 cm, where the control is made and secondly, when the ¹⁸O isotope concentration has to be increased, the extraction point is changed to s = s_f = 1000 cm (the sensor position can be, also, changed to the column top). The main advantage consists in the avoidance of a new transitory regime and implicitly in a much faster extraction of the ¹⁸O isotope at an increased concentration. The procedure of identifying the (s) independent variable value can be, also, used in order to determine if the separation plant can be physically used. The constrain regarding the possibility of usage the separation plant consists in limiting the maximum deviation of the (s) independent variable with more than 40 cm in relation to (s_f) value. In this context, the equivalent value of (s) has to be approximated (containing the superposition of the physical value of (s) and also the equivalent value of (s) (which represents a measure of the exogenous disturbances equated as a position variation in the column height)).

Fig. 20.

The output signal evolution, in relation to time, for s = sf.

The computation of the instantaneous value of the (s) independent variable has a very high importance due to the fact that it is interpreted by the DE element in order to assure the separation plant working in safety regime. Hence, the DE element from Fig. 3 is set to take the DECISION of emitting an Alarm at the detection of the value (s_f – s = 37 cm) and it is set to take the DECISION of stopping the separation plant at the detection of the limit value (s_f – s = 40 cm). Obviously, the difference (s_f – s) is always positive due to the fact that the sensor position can be changed only in the column volume (and only near its top).

5 Conclusions

In order to obtain the imposed performances for the concentration control structure, both a linear PID and a nonlinear (NCC) controller (based on the usage of Gaussian functions) are proposed. The performances obtained using NCC controller are consistently better than the performances generated by the linear PID controller (especially the settling time which is a dominant parameter in the case of separation processes), fact proved through the simulations from Sect. 4. In order to obtain high control performances in disturbance rejecting regime, a compensation loop, based on the process reference model, is designed.

Another original element proposed in the paper is the design of an identification system of the exact concentration sensor position in the neighbourhood of the column top. Also, a procedure to equivalate the disturbances effects, with a change of (s) independent variable value, is proposed. These mathematical mechanisms can be used in order to optimize the product extraction from the separation plant, when the future orders (for ¹⁸O isotope) can be predicted.

A future possibility to improve the control system performances can be the implementation of a more complex structure for the NCC controller (the obtained performances using the NCC are much better than in the case of using the linear initial PID controller, but they are not optimal). Another future possibility to improve the control system performances is given by the possibility to replace the main NPC controller with other types of advanced controllers [11, 12]. Also, the proposed model can be extended in order to obtain the mathematical model of a separation cascade, which contains two separation columns in its structure. Obviously, for the case of a separation cascade, the proposed control strategy has to be adapted.