The multilinear model control design approach is based on the approximation of the nonlinear model of the system by a set of linear models. The paper presents the method of creation of a bank of linear models of the two-pass shell and tube heat exchanger. The nonlinear model is assumed to have a Hammerstein structure. The set of linear models is formed by decomposition of the nonlinear steady-state characteristic by using the modified Included Angle Dividing method. Two modifications of this method are proposed. The first one refers to the addition to the algorithm for decomposition, which reduces the number of linear segments. The second one refers to determination of the threshold value. The dependence between decomposition of the nonlinear characteristic and the linear dynamics of the closed-loop system is established. The decoupling process is more formal and it can be easily implemented by using software tools. Due to its simplicity, the method is particularly suitable in complex systems, such as heat exchanger networks.
Most physical systems are inherently nonlinear. Nonlinearity is especially pronounced in systems with wide ranges of desired behaviours and variable set points. There are a lot of such systems in the process industry [
Linear control theory provides many confirmed methods and tools for controller design with desired performances and robustness. Unfortunately, that theory is limited to strictly linear systems or certain classes of nonlinear systems with small deviations around the nominal operating point. In real systems with a wide operating range and multiple operating points, where nonlinearity cannot be ignored, different control strategies are necessary. In order to use the vast potential of linear control theory, at the same time taking into account nonlinearities, various modifications of classical control design are proposed.
The multilinear model (or multimodal) control approach which has been shown to be suitable for strongly nonlinear systems with multiple operating points, tracking control, and wide operating ranges [
There are two important questions which should be answered in the multilinear model control design approach: how to decouple a nonlinear system into a bank of locally linear subsystems and how to design a global controller according to the desired performances of a nonlinear system.
This paper gives the answer to the first question. The literature mainly uses the gap-metric based method where the minimal linear model set is determined for the given threshold value in order to span the expected operating range of a nonlinear system [
Nowadays, a lot of dynamic models defined to study transient responses of HE [
The model of HE development by Chen et al. [
Functional view of two-pass shell and tube heat exchanger [
It is assumed that the HE has a Hammerstein structure (Figure
Hammerstein structure of heat exchanger model.
Iterative identification with the pseudorandom binary sequence (PRBS) input in the closed loop with the switching time of 1 min is used for determination of the model [
The collected data were the basis for obtaining the model which, after translation into the continuous domain (sampling time 12 s, zero-order hold on the input), has the following form:
Model (
Figure
Estimated static nonlinearity and steady-state operation points of heat exchanger [
In the next stage of multilinear model control design approach, the goal is to approximate the nonlinear plant (
The change of gain of the heat exchanger as a function of the control signal
The change of gain of the heat exchanger.
The main question which should be answered is the following: how many and which linear models are required to span the expected operating region of a nonlinear system? An approach based on the IAD method [
(1) Prescribe (2) Distribute along the steady-state IO curve evenly (3) Compute (4) for (5) for (6) if (7) (8) else (9) (10) end (11) end (12) end (13) compression subregion of a small range (14) In the middle of
After the previous procedure of 14 steps, a set of linearized segments and the corresponding operating points,
In order to obtain a set of linear models, it is necessary to identify the local linear model or linearize the nonlinear first-principle model for each segment, around its operating point.
The method is simple and easily realized. It is only necessary to know the nonlinearity of the steady-state characteristic.
The paper proposes two modifications of the IAD method. The first modification refers to the algorithm shown in Algorithm
(1) Prescribe (2) Distribute along the steady-state IO curve evenly (3) Compute (4) for (5) for (6) if (7) (8) else (9) (10) end (11) end (12) for (13) if (14) (15) else (16) (17) end (18) end (19) end (20) compression subregion of a small range (21) In the middle of
The header of the new loop has the form for
The second modification refers to two weaknesses of the IAD method: Selection of the threshold value requires certain experience and a priori knowledge. Therefore, it is difficult to make a procedure (e.g., software tool) for a systemized approach. Decomposition of the plant model is performed only on the basis of the nonlinear steady-state characteristic; that is, the dynamics of the linear part are not taken into account.
The starting point in solving these problems is the desired dynamics of closed-loop systems in order to make the basis for defining the criterion for segmentation of the nonlinear steady-state characteristic. In other words, the threshold value
In [
Step response of the HE for different values of linearized gains.
On the other hand, step responses (
Step response of the closed-loop system with unit gain.
Let us observe the multilinear model based control system shown in Figure
Multilinear model based control system.
At the second level, the local controllers are switched by means of the global controller (GC) using hard-switching or soft-switching methods [
It should be noted that
The linear models
Since
The parameters
If the value
Inequality (
If the current gain is now defined by using the slope angle of the steady-state characteristic (
If
Slope angle of the HE steady-state characteristic.
It can be seen from the figure that
Figure
Threshold value for different values of
In selecting values for the parameter Parameters of a linearized model depend on the operating conditions. Hammerstein model is just an approximation of a real system. The parameter Transfer from one linear model to another (hard-switching or soft-switching) can result in oscillations in the system. Linear dynamics of the system depend not only on the dynamics of the plant but also on the parameters of the local controller (Figure Value of the parameter
The
Nyquist plot with the stability margin.
Figure
Number of segments depending on
The modified IAD method was used. It can be seen that in this method the number of linear models practically depends to a small extent on the number of steady-state points. That dependence is more pronounced in lower values of the parameter
Figure
Linearization of the heat exchanger static nonlinearity.
Two segments
Five segments
The corresponding set points are shown in Tables
Operating points for 2 segments.
| −0.09 | 0.21 |
| −2.72 | −4.61 |
| −23.53 | −7.84 |
Operating points for 5 segments.
| −0.03 | 0.11 | 0.17 | 0.21 | 0.22 |
| 0.93 | −3.15 | −4.25 | −4.60 | −4.68 |
| −35.58 | −21.23 | −12.82 | −8.16 | −6.21 |
The paper presents the method for creating a set of linear models of heat exchanger described by the Hammerstein model type. Decomposition of the model is based on linearization of the nonlinear steady-state characteristic which is assumed to be a priori known. The method is intuitive and can be easily implemented by using software tools. It is shown that the number of linear segments practically depends only on the a priori given threshold value but that it does not depend on the number of steady-state points. The threshold value can be determined based on the allowed changes of the open-loop gain, which establishes the correlation between decomposition of the nonlinear model and the linear dynamics. The proposed method can easily be extended to other SISO Hammerstein-like systems with a memoryless nonlinearity.
The authors declare that there are no competing interests regarding the publication of this paper.
This research has been supported by the Serbian Ministry of Education, Science and Technological Development through Project TR 33026. Also, this research has been supported by the European Commission through IPA Adriatic and Project ADRIA-HUB.