This article provides a comparative analysis of two common control configurations used to control the side-stream distillation used to separate benzene, toluene and xylene as suggested by Doukas and Lyben. Their under-actuated model has been considered as the model of distillation column and the internal model controller is designed considering a Singular Value Decomposition (SVD) and Virtual Inputs (VI) techniques. An internal controller design based on VI is proposed in this article for this kind of underactuated systems. This design is used to control in parallel the distillation process and its model in real time. The proposed controller design is simple and systematic in relation with the desired closed loop specifications of the internal model control structure. Furthermore, the controller obtained ensure robustness to process variations. The SVD technique can realize the decoupling of under-actuated processes and wipe out unrealizable factors by introducing compensation terms, affecting the dynamic of the system. The aim of this article is to make a comparison between our proposed VI controller and the SVD approach. The results we obtained confirmed the potentials of the proposed controller based on VI considering the set point tracking and its robustness.

The contribution of this article is study of two different control strategies for a class of chemical process industries. The model of the distillation column used in this plant, which is characterized by an underactuated structure. According to the diversity and complexity of these systems, it is important to emphasize that none of the technique proposed and developed for fully actuated systems can be applied directly to any underactuated system. Therefore, it is meaningful to develop control methods for this class of systems, and more precisely to develop the most optimized controller design.
The control objective for systems characterized by the fact that there are more degrees of freedom than actuators, is to obtain a desirable behavior of several output variables by simultaneously manipulating several inputs channels. Under-actuated systems are less sensitive to modelling errors, so it has to be controlled in its original form to obtain robust stability and performance

The process to be controlled is assumed linear and discrete-time governed by the
following equation

Where

(i) If the model contains time delays, its inverse involves predictive terms, which make the controller unrealizable.

(ii) If the zeros of the transfer matrix of the model outside the complex unit circle yield an unstable perfect controller.

(iii) System equipped with a perfect controller is extremely sensitive to modelling errors and time delays.

(iv) The direct model inversion is also impossible in the case of underactuated systems.

In fact, the model must provide an accurate description of the process dynamics and characteristics. Therefore, the model expression must be very close to that of the plant. For underactuated systems, the number of control inputs is less than the number of outputs and therefore we will have a rectangular matrix that is invertible. This represents the major problem encountered. In this article, to resolve this problem, it is proposed to develop some methods of inversion in the case of under-actuated systems.

The development of the IMC structure has progressed in recent years in order to design an optimal feedback controller. In this section we present the general IMC structure and we describe its basic principles and properties. Due to the fact that the traditional IMC methods cannot solve the control problem of non-square systems, we introduce two design technique to realize the internal controller but, when using a matrix to describe a non-square system, the issue of inversion often emerges.

THE GENERAL IMC DESCRIPTION

The general IMC structure of multivariable systems adopted in this article is shown
in Figure 1, where

We begin by reviewing the properties of the IMC structure. This structure is
equivalent to a conventional feedback loop with controller. From Figure 1 the inputs
vector u(z) and the system outputs vector y(z) are expressed by

Property 2. Perfect Control; Assume that the controller

Property 3. Zero Offset; Assume that the steady state gain

The application of the internal model structure to under-actuated systems is considered like our main target. In this section, we will describe the design phase of the internal controller based on SVD and the implementation at the level of the structure IMC. This approach of design can realize the decoupling of under-actuated processes and eliminate the unrealizable factors by inserting compensated terms. Meanwhile, a non-diagonal filter is designed based on SVD matrix theory. We discuss with more details this approach in the following.

The IMC structure of an under-actuated systems based on SVD is shown in Figure 2,
where CSVD(z) is the IMC controller.

In the nominal case,

According to the equation (17), the robustness of the

Step 1: Use the SDV in the inverse of the steady-state gain matrix of

Step 2: Let

Step 3: The improved internal model controller is as follows

The robustness of the system can be greatly enhanced by adding a
filter

Focusing now on the inversion method of the under-actuated systems based on Virtual
Inputs. To successfully apply our approach, firstly we need to modify the IMC basic
structure mentioned in Figure 1, so that it becomes applicable to underactuated
systems with more outputs than control inputs. Secondly, we design an approximate
inverse of the model plant which is inspired by the studies of

The inserting columns mentioned previously can be chosen as first-order transfer
functions, which verify the stability criterion, and in order to simplify the study
and avoid inversion problems

- Monovariable system where

- Overactuated system where

- Underactuated system where

To analyze the comparative study on the above control technique, we considered a
chemical process industry. In order to test the control effect of discrete
underactuated internal model controller based on SVD and VI, a side-stream
distillation control problem suggested by Doukas and Lyben will be used

Figure 5 shows the side-stream distillation scheme that serves to separate benzene, toluene and xylene. The problem posed in this case was to control the concentrations of four impurities in three product streams with only three manipulated variables: reboiler duty, reflux ratio and side stream flow rate.

In this section, we evaluate the controller performance using the above-mentioned approach.

SIMULATION RESULTS USING THE SVD CONTROLLER

Starting with the internal controller based on Singular Value Decomposition, and
using the whole approach seen in section 4. Using equation (23),

Using equation (24),

SIMULATION RESULTS USING THE VIRTUAL INPUT CONTROLLER

Dealing now with our proposed approach, the Virtual Input methods applied on the same
system studied previously. Considering the same underactuated system with three
control and four outputs. The system transfer matrix G(z) is given by (33). Let us
consider the case of perfect modeling

The gain matrix

COMPARISON ANALYSIS

After using both Singular Value Decomposition and Virtual Inputs approaches, it seems to be quite interesting making a comparison between them showing their effectiveness in terms of stability, robustness, precision and tracking signal. From the data presented in table 1, it is relevant to note that the desired specifications of the closed-loop responses are not met with the SDV controller and this is expected because of the problem of the interaction which are too strange as illustrated by the decoupling of the controller and the simulation results obtained with the SVD controller in Figure 6. SVD method has a problem if we do not make the right choice of the parameter of the filter, in this case, the approximation cannot be made with effectiveness and the system can be diverging. Added to that this approach towards the under-actuated process is to square the system by make the decoupling of the model affecting the characteristic of the system. Unfortunately, this operation can decrease the performance of the system and making it so poor by neglecting some information’s.

The internal model controller using a Virtual Inputs as inversion approach of the
model

Parameters to control | Outputs of System | ||||

Y1 | Y2 | Y3 | Y4 | ||

IMC controller based | Rise Time (s) | 93 | 160,6 | -6 | 240 |

on SVD | Steady state error (%) | 12,99 | 1,96 | 66,24 | 21,36 |

Settling time (s) | 560 | 320 | 247 | 400 | |

IMC controller based on VI | Rise Time (s) | 53,95 | 23,92 | 30,02 | 30,14 |

Steady state error (%) | 0 | 0 | 0 | 0 | |

Settling time (s) | 96,02 | 42,05 | 42,05 | 47,97 |

In order to verify the robustness of the proposed internal model controller, we added a white noise with a variation of 0,2 acting on the outputs of the system in the interval of time [100s, 150s, 200, 250s]. The influence of this noise is observed in Figure 8.

To control an under-actuated industrial process demands not only to know all the characteristic of his model, but also to achieve the desired performance when there exist load disturbances. Both Virtual Input and SVD approaches are applied to distillation column process, to prove the controller achievability for this industrial process and that it is the most efficient approach in terms of rapid response, set-point tracking and disturbance rejection. Through a comparative analysis, the Virtual Inputs approach avoids the complex calculation, such as calculate the inverse of the matrix, the controller structure is simple, has a few tuning parameters, and is easy to be accepted by operators. The simulation results show that our proposed method (Virtual Inputs) ensures suitably the set-point tracking and the disturbance rejection for the external disturbance as illustrated in distillation control problem suggested by Doukas and Lyben and gives better results compared to SVD approach. The modified internal model control scheme gives more degree of freedom in the controller design technique in order to improve the performance of the controlled system. As future works, it will be interesting to handle with unstable multivariable non-square systems and non-linear systems as proposed in order to design an internal controller with more flexibility and to apply the proposed Virtual Input controller to real processes.