Prelucrarea Hartiei
CONTENTS
Chapter 1 INTRODUCTION
1.1 Purpose of the paper
1.2 Objectives
1.3 State of the-art
1.4 The Wood and Berry distillation column
1.5 Summary
Chapter 2 CONTROL STRATEGIES FOR MULTIVARIABLE SYSTEMS
2.1 Decentralized control
2.1.1 The analysis of the controller interaction
2.1.2 Pairing of controlled and manipulated variables
2.2 Decoupling Control
2.2.1 Static Decoupling
2.2.2 Dynamic Decoupling
Chapter 3 IMPLEMENTATION OF MIMO CONTROL STRATEGIES FOR A WOOD AND BERRY DISTILLATION PROCESS
3.1 Decentralized control
3.1.1 Phase margin method
3.1.2 Guillemin Truxal method
3.2 Static decoupling
3.2.1 Phase margin method
3.3 Dynamic Decoupling
3.3.1 Phase margin method
Chapter 4 COMPARATIVE RESULTS AND ROBUSTNESS ISSUES
4.1 Comparative results
4.2 Robustness of the designed controllers
Chapter 5 CONCLUSIONS
REFERENCES
ACRONYMS
LIST OF FIGURES
Figure 1.3 Schematic Diagram of Distillation Column………………………………………………………….4
Figure 2.1.1 Decentralized control structure……………………………………………………………………..10
Figure 2.2.1 General Structure of Decoupling Control System…………………………………………….12
Figure 2.2.2 Designed Decoupler into the Decoupling Control System…………………………………13
Figure 2.2.3 Decoupling Control Structure……………………………………………………………………….13
Figure 3.1.1 Simulink representation for Decentralized control structure. Phase Margin……….21
Figure 3.1.2 First output of the process…………………………………………………………………………….21
Figure 3.1.3 Second output of the process………………………………………………………………………….22
Figure 3.1.2.1 Simulink representation for Decentralized control structure. Guillemin Truxal……………………………………………………………………………………………………………………………24
Figure 3.1.2.2 First output of the process………………………………………………………………………….24
Figure 3.1.2.3 Second output of the process……………………………………………………………………….25
Figure 3.2 Decoupling control structure……………………………………………………………………………26
Figure 3.2.1 Transfer function identification for static decoupling, first output………………………27
Figure 3.2.2 Transfer function identification for static decoupling, second output………………….28
Figure 3.2.1.1 Simulink Static Decoupling control structure………………………………………………..30
Figure 3.2.1.2 First output of the process………………………………………………………………………….30
Figure 3.2.1.3 Second output of the process……………………………………………………………………….31
Figure 3.3.1 Dynamic Decoupling Controller Structure………………………………………………………32
Figure 3.3.2 Transfer function identification for dynamic decoupling, first output………………….33
Figure 3.3.3 Transfer function identification for dynamic decoupling, second output……………..34
Figure 3.3.1.1 Simulink Dynamic Decoupling control structure…………………………………………..36
Figure 3.3.1.2 First output of the process………………………………………………………………………….36
Figure 3.3.1.3 Second output of the process……………………………………………………………………….37
Figure 4.1.1 First outputs of the process in comparison………………………………………………………39
Figure 4.1.2 Second outputs of the process in comparison…………………………………………………..41
Figure 4.2.1 First outputs of the process with robustness tests. Decentralized Control, Phase margin…………………………………………………………………………………………………………………………..43
Figure 4.2.2 Second outputs of the process with robustness tests. Decentralized Control, Phase margin…………………………………………………………………………………………………………………………..44
Figure 4.2.3 First outputs of the process with robustness tests. Decentralized Control, Guillemin Truxal……………………………………………………………………………………………………………………………45
Figure 4.2.4 Second outputs of the process with robustness tests. Decentralized Control, Guillemin Truxal…………………………………………………………………………………………………………….46
Figure 4.2.5 First outputs of the process with robustness tests. Static Decoupling Control, Phase Margin…………………………………………………………………………………………………………………………..47
Figure 4.2.6 Second outputs of the process with robustness tests. Static Decoupling Control, Phase Margin…………………………………………………………………………………………………………………48
Figure 4.2.7 First outputs of the process with robustness tests. Dynamic Decoupling Control, Phase Margin…………………………………………………………………………………………………………………49
Figure 4.2.8 Second outputs of the process with robustness tests. Dynamic Decoupling Control, Phase Margin…………………………………………………………………………………………………………………50
Chapter 1 INTRODUCTION
1.1 Purpose of the paper
The purpose of the paper is the study of the distillation columns along with the control methods for Wood and Berry. We will also compute and implement in Matlab our obtained control structures for MIMO processes: decentralized control and decoupling control. For decentralized control we proposed two control methods, them being Phase Margin, Guillemin Truxal and Dahlin. For decoupling control we went on the part where we used the static and dynamic decoupling, and for each of them being proposed two control methods, Phase Margin, Guillemin Truxal, and Dahlin.
The obtained control structures will be tested and compared to each other in order to find out, which one is the best to use in a distillation process. Not only that, but we will apply robustness tests too in order to check if the designed control strategies can meet the performances criteria despite the modeling uncertainties.
1.2 Objectives
The main objectives of the paper are the following:
The study of the simplest multivariable control methods
Analysis of the interactions in Wood and Berry distillation columns
Design of the decentralized control strategy using Phase Margin criteria and Guillemin Truxal
Design of the static decoupling control strategy using Phase Margin criteria and Guillemin Truxal
Design of the dynamic decoupling control strategy using Phase Margin criteria and Guillemin Truxal
Comparison of the designed methods for reference tracking and interactions
Robustness analysis of the designed methods
1.3 State of the-art
For the control of the Wood and Berry distillation column several approaches have been proposed over the years. Some of these approaches are summarized as
follows.
Starting with “Design and Development of Model Predictive Controller for Binary Distillation Column”, in which the goal would be the understanding of the Model Predictive Controller for constrained and unconstrained input/output, on SISO system as well on MIMO system. MPC uses an explicit dynamic plant model to predict the effect of future reactions of the manipulated variables on the output, by minimizing the cost function. The schematic Diagram of the Distillation Column on which they worked is the following one:
Figure 1.3 Schematic Diagram of Distillation Column
The Wood and Berry distillation column was taken as the primary system model. The system is controlled with multi loop PID and MPC controller. The performances indicators such as settling time, overshoot, and the errors of the MPC controller were compared with the multi loop PI controller. The results were that MPC is far better than the conventional controller, due to the smooth reference tracking it provides, the reduced peak overshoot, and better closed loop performances. [7]
The next paperwork would be “Effect of Tuning Parameters of a Model Predictive Binary Distillation Column”, which circles around the effects of tuning parameters with the help of MPC. First they implemented an ideal MPC by taking the unit step function. After that they implemented a general MPC for the binary distillation column, meaning they’ve taken the unit step input without the disturbances, and in the last part they managed to find a way to remove the ringing effect in manipulated variables for MPC. One way that they’ve found in order to remove the ringing, was to change the design parameters. The result of the paper if we can say so, was that they found out that at a certain tuning parameter, the performance of the control system are better than in any classical control system. [8]
In “Decoupling Smith Predictor Design for Multivariable Systems with Multiple Time Delays”, the main idea is centered on the new design approach for the multivariable Smith Predictor controller, which will be used for decoupling and stabilizing multivariable processes with multiple time delays. With the frequency domain approach to the decoupler design and the model reduction of the resultant decoupled process, the multivariable Smith Predictor was decomposed to a single-loop Smith Predictor design. [6]
“MIMO Smith Predictor: Global and Structured Robust Performance Analysis”, presents an extension of the classical Smith Predictor to MIMO systems with multiple uncertain delays. They presented different conditions under which a model could be separated into diagonal left and right pure delays, and some rational matrix. All these conditions were satisfied by a multiple pool open flow canal system. They found out that there is not right way or exact way to perform a computation for both structured robust analysis, and performance conditions. They computed two upper bounds, based on the delays uncertainty separated in two ways, globally and individually. They computed a controller for the two-pool canal system, and they evaluated the obtained performances and robustness. They ended up needing more research, in computing the exact robust stability and the performance margins. [5]
The last approach on Wood and Berry distillation columns is “Design of PID Controllers for Delayed MIMO Plants using Moments Based Approach”. The paper deals with the design and the optimization of a robust PID controller for MIMO plants. The technique is based on the choice of a reference model, gathering the desired performances of a closed loop, as all the singularities which the system could have such as unstable zeros or time delays. The ideal controller which should achieve all the performances and have a complex structure is offered by IMC. The IMC is reduced through a nonlinear programming algorithm in order to ensure low frequencies performances, from which it leads to a PID controller that represents a simple structure for implementation providing a good tracking and robustness properties. [1]
1.4 The Wood and Berry distillation column
The distillation column control is lately seen as a dead filed of research. But due to that, in the last ten or so years, a lot of papers appeared with reh”. The paper deals with the design and the optimization of a robust PID controller for MIMO plants. The technique is based on the choice of a reference model, gathering the desired performances of a closed loop, as all the singularities which the system could have such as unstable zeros or time delays. The ideal controller which should achieve all the performances and have a complex structure is offered by IMC. The IMC is reduced through a nonlinear programming algorithm in order to ensure low frequencies performances, from which it leads to a PID controller that represents a simple structure for implementation providing a good tracking and robustness properties. [1]
1.4 The Wood and Berry distillation column
The distillation column control is lately seen as a dead filed of research. But due to that, in the last ten or so years, a lot of papers appeared with regards to this matter. The results that appeared in this researches are related to multiple steady states and instability in simple or complex columns, the fundamental problems in identifying models from open-loop responses, simple formulas in order to estimate the dominant time constants. This issues apply to all sort of columns. [10]
Chemical plans such as distillation columns are by nature multivariable systems, which implies that they frequently have time delays and the complexity of controlling such process it’s high.
The distillation is the separation method in the petroleum and chemical industries for purification of final products. They are used to transfer heat energy. [7]
In many industrial plants, the basic extension of classical PID controller design, implementation and tuning is the decentralized approach, where structural concepts are used to decouple the interactions between variables. The use of standard equipment and the ease of hand-tuning or understanding by non-specialist technicians are the main advantages of this approach. Nevertheless, the control effort is decomposed into two stages: first to decouple the different subsystems and then to control them. The extra effort rewards consist in similar subsequent design, implementation and tuning. [5]
The majority of complex engineering systems contain actuators that can influence the systems static or dynamic behavior. “A multivariable system (multiple inputs, multiple outputs) it is a system which has a number of process variables that are controlled, and a number of variables that can be manipulated”. The control objective for MIMO system is to obtain a desirable behavior for the output variables by controlling in the same time the input variables.
The simplest method to control a multivariable system it’s the decentralized control or also called multi-loop control strategy, which implies decomposition of the MIMO system into SISO control loops. With a limited number of sensors and actuators in a system, a centralized controller carrying out matrix operations will be able to obtain better performance from the processes with strong couplings or conditioning problems. [5]
For processes that are strongly coupled, a decoupling control is preferred instead of a decentralized control. Decoupling is a procedure that aims to pair the input and the output signals in such way that the multivariable interactions are at their minimum, and sets the premises for an improved design of the decentralized control. The algorithm to decouple a MIMO system consists of transforming the process matrix in diagonal one, which is realized by using an additional controller called a decoupler, with the role of compensating process interactions. After we obtained the pseudo-plant, which consists of the initial MIMO process and the decoupler, SISO techniques can be applied in order to design the controllers.
Wood and Berry (1973) reported a transfer function as being for methanol-water separation in a distillation column. The controlled variables on the process are expressed on weight percentage of methanol. The manipulated inputs are expressed in lb/min and they are the reflux and the reboiler steam flow rates, and the inputs of the process refer to distillate and bottom concentrations of a volatile compound. The transfer function used in the process is:
[9]
1.5 Summary
Two control strategies for multivariable processes are proposed that are based on decentralized and static/dynamic decoupling approach. The controllers for the control strategies were computed with Phase Margin method and Guillemin Truxal method. Despite of being a complex system to control, we managed to obtain stable structures in all cases with good performances.
First control strategy was decentralized approach, where after obtaining the controllers we implemented them into the decentralized control structure in Matlab Simulink in order to obtain the results for the method. In both cases the process was stable and the obtained performances were satisfying.
Even if decoupling control structures are used more as theoretical points of view, for both static and dynamic controls we obtained stables structures, but because of having strong couplings and high interactions the obtained performances were not that good. After obtaining the controllers in both cases static and dynamic, yet again we implemented them into Matlab Simulink in order to be able to compare the results that we got.
As a comparison among the resulted structures after computing the controllers for all of them, we managed to see that decentralized control strategy works better than the decoupling strategy, because using the decouplers into a control structure, it complicates the structure, even if it is made in order to remove the interactions from it.
We moved on applying some robustness tests upon the obtained control structures, in order to see if we obtain similar results in terms of closed loop performance and interactions. We did obtain such things with the robustness test in which we lowered the initial parameters from the nominal transfer matrix with 20%. The results were much better than in the nominal case, in some cases being with almost 50% better.
The Wood and Berry distillation columns can be controlled with multivariable structures, preferably decentralized control structures due to the better performances obtained within our tests.
Chapter 2 CONTROL STRATEGIES FOR MULTIVARIABLE SYSTEMS
The aim of a control system which can be multivariable or not is to force a given set of processes variables to function in the prescribed way by trying to fulfill the best performance as expected. The main goal always in such systems is to keep the processes running at nominal conditions.
If we select a set number of p-variables that we want to control then at least a number of m-variables should be manipulated (m ≥ p) and both these types of variables should be independent, so they won’t produce the similar effects on the variables. If in a process we have more manipulated then controlled variables, it results in more options to control such process, and it should obtain better performances.
When we are working with multivariable systems, we need to take in consideration some concepts such as grouping and pairing, conditioning. [5]
Grouping and pairing. Usually, one or more inputs may be “attached” to each controlled variable or group of variables, and the choice of groups influence the interaction and dominance. Interaction means that the manipulated variables that are attached to one controlled variables, may influence the other variables. Dominance, if the effect of the corresponding attached manipulated variable is greater than the others, the coupling presents dominance.
Conditioning it refers to the different “gains” a multivariable system can present according to the combination of inputs, if the “gains” are very different, the process can be called as being ill conditioned, and it results in a difficult process to control.
The two control strategies used for multivariable systems are the decentralized control and decoupling control. [5]
The schematic representation of the process is presented into Figure 1.3.
2.1 Decentralized control
This is the simplest method in controlling MIMO systems, which means the decomposition of MIMO systems into simpler SISO control loops. It has the main advantage of easy implementation and tuning if in the system has a sufficient number of actuators and sensors. The decentralized control method is highly reliable and it is typically used in the industry because of its simplicity. The steps to design a decentralized control strategy include the selection of the right manipulated and controlled variables, the pairing of such variables, the selection of the types of feedback controllers and the actual tuning of these controllers.
When we want to decompose the multivariable systems into several SISO control loops some interactions occur due to the fact that the subsystems are not independent. Those types of interactions are interference also called feed-forward interaction and coupling. The control loop interactions are due to the presence of a new feedback loop and they have two effects on the output, a direct one and an indirect one. The problems that arise from the control loop interactions are the following: closed loop may become destabilized and the tuning of the controller is more difficult.
2.1.1 The analysis of the controller interaction
To analyze the interactions that occur in a multivariable system, we will consider two situations: an open loop and a closed loop case. The aim of this analysis is to show that the hidden feedback loop, presented by the thick black line in the figure below, modifies the gains of the process transfer function for which the controller are computed. We set the second controller in manual which means that the controller is equal to 0. Then, there is no interaction from the second loop, and in the Figure 2.1.1 the output y1 is the result of the influence of u1 upon G11:
Figure 2.1.1 Decentralized control structure
From here we design the controller , bases solely on .
Second controller will be in automatic now which means that is non-zero. Using some basic mathematic, we obtain the relation between the output and the input .
Compared to the previously obtained relation, in open loop, we can now easily observe that the transfer function that links to is no longer . In closed loop, the relation between and depends also on all the other process transfer functions, , and , and also on the second controller . This obviously shows that designing the controller solely based on will result in poorer performance once we close the loop.
Some strategies in order to reduce the loop interactions are to detune one or more control loops, to choose different manipulated or controlled variables, to consider a Decoupling Controller or to consider a multivariable controller. So our aim it’s to tune multi-loop controllers which will be done by following the next steps: we will tune single loops with all other loops in manual, then we will close all the loops and if the performance of the process is close to what we want to obtain we will do nothing else to the tuning, if the performance are not right, we will detune the less critical loops by trying to reduce the controller gains, and increase in the same time the integral times, or if we see that one loop from the system it’s more important than the rest then we will tune the important loop first, and we will leave the first loop in automatic and we will tune the other loops keeping the gain low to try and avoid side effects on important loops.
2.1.2 Pairing of controlled and manipulated variables
In order to pair the controlled and manipulated variables we need to set proper inputs/outputs pairs which can be formed by calculating the Relative Gain Array of the system to reduce the impact of the interaction.
The RGA it’s an interaction measure, being one of the most common used, developed by Bristol in 1966. The Relative Gain Array methodology indicates which pairings should not be made due to the possible stability and performance problems. [3]
The formula from which we will determine the RGA for the set system is:
After we determined the RGA of the system we will move on into choosing the right pairings which will be done following the few rules of pairing recommendations, all depending on λ:
λ=1, is an ideal case when we have no interactions between the loops, the pairing should be along the diagonal ( -,-);
λ=0, we have an ideal situation as above just that the pairing now it is on the off-diagonal(( -,-);
0<λ<1, in this situation the gain increases when the loops are closed, which means that there is an interaction, being one of the worst interaction when λ=0.5;
λ>1, now the gain decreases when the loops are closed and the interactions gets worse the larger λ is;
λ<0, even the sign changes when the loops are closed in this case, and this is highly undesirable. The more negative λ gets, the interaction gets worse.
After pairing the inputs and the outputs of the process we move on calculating the regulators.
2.2 Decoupling Control
The general term of decoupling can be divided into two sub-categories such as dynamic/transient and static decoupling. The dynamic or transient decoupling is more general and it makes sure that under any conditions, the manipulated variables influence independently the targeted outputs. On the other hand the static decoupling is more concerned with the problem of steady-state, meaning that decoupling is guaranteed only for a special set of inputs.
The term decoupling usually means diagonal decoupling, meaning that each input/output channels are independent. When different input/outputs are grouped together, we obtain variants of decoupling regimes such as block decoupling, when some subsystems are independent, or triangular decoupling when we have some outputs that influence other outputs but not vice versa. [4]
Decoupling may be done using several techniques like restructuring the pairing of variables, minimizing interactions by making sure we detune the conflicted control loops, opening loops, and putting them in manual control, or using linear combinations of manipulated controlled variables. If the systems cannot be decoupled then, there are other methods such as neural networks or model predictive control which will be used to control the system, that not being the case in this paperwork.
We have two main ways to determine if a system can be decoupled, first one with mathematical models and the second one are based on intuitive educated guessing method. The mathematical method in simplifying MIMO control schemes include the RGA method which we will use further on. [11]
The general structure of a decoupling system looks like the following one:
Figure 2.2.1 General Structure of Decoupling Control System
The principle design of a decoupler it’s to succeed in cancelling the effects of the cross blocks, such that there is no interaction between – and – signals. This is shown in Figure 2.2.2.
Figure 2.2.2 Designed Decoupler into the Decoupling Control System
Further on resulting into the scheme of the controlled process with a decoupler:
Figure 2.2.3 Decoupling Control Structure
2.2.1 Static Decoupling
Mainly designed to eliminate the Steady State interactions. The main idea of steady state decoupling is that the transfer matrix of the process which is:
and plus the decoupling element results into the diagonal selection of the interactions but in steady state.
The main steps for computing the static decoupling are the following:
First we must compute the transfer matrix into its steady state meaning s=0, and the obtained steady state matrix is:
The second step is about computing the inverse of the steady state matrix, because we will need to multiply it with the steady state matrix and in the ideal case we will obtain the unity matrix. The equations for computing the inverse of the matrix and the results are:
inv( multiplied with G(0) we obtain:
the unity matrix
Because in reality, we deal with a dynamic system, the multiplication of the process transfer function matrix and the inverse of the steady state gain matrix, results in a decoupled matrix that is different from the unity matrix.
2.2.2 Dynamic Decoupling
The difference between dynamic and static decoupling is that at dynamic decoupling, the main diagonal of the decoupled matrix is composed of dominant elements of the transfer matrix of the process, plus the decoupling element but in transient regime.
The main steps for computing the dynamic decoupling are the following:
First we have the equations for the processes output:
Second step being the introduction of the decouplers into the structure which results into obtaining inputs, and , and a new equation:
Now multiplying them into transient regime we obtain a set of equations from which we will obtain the decouplers, the equations being:
As a last step we choose and of being one, for simplicity, and we obtain the equation for and :
Chapter 3 IMPLEMENTATION OF MIMO CONTROL STRATEGIES FOR A WOOD AND BERRY DISTILLATION PROCESS
In this chapter we will present the actual results for each method calculated and the simulations for them too. For each main method in part meaning Decentralized, Decoupling (static/dynamic) the implementation was done with Guillemin Truxal and with the Phase Margin method.
Guillemin Truxal is used for fast systems, and the usual performance set used on this
method is:.
The fundamental hypothesis that this method is using, is based on considering the closed system equivalent to a second order system meaning that the transfer function for such process will be:
= =
Knowing the fixed part which is usual written as we can
determine easily the regulator:
. [2]
Obviously will be chosen accordingly with the imposed performance set.
A problem that could appear would be approximating the control structure calculated, meaning that it could be – complicated or hard to be achieved – but we could use a simpler version that could lead into fulfilling the performance set. The simpler relationship for calculating the regulator will be:
If , then we have a P regulator, meaning it’s much simpler to fulfill the approximated performance set. The problem in this case would be a verification to see if the approximation can be accepted. [2]
In the processes where we have more than two poles, we could use in some conditions different approximations for the transfer functions in order to bring them to the usual second form transfer function. The simplest method would be, to simplify the transfer function, by identifying the dominant poles.
Some equations used in order to obtain the closed loop transfer function that meets all the imposed performance criteria, that will be used in Guillemin Truxal and Phase Margin methods:
[2]
These equations will be used in order to check if the system fulfills the imposed performance set.
Phase margin method is a complex method, for the fact that it is used for processes with dead time, and the characteristic transfer function for such processes is where ,that represents the rational part of the transfer function, and is the value of the dead time. [2]
Depending on the value of the ratio between the dominant time constant of the process and the dead time, there will be more situations such as:
If the value of the dead time can be neglected, and the controller design will be done in the classic way, using for the fixed part: .
If 0.2<<1, the dead time cannot be neglected anymore, which means that in the controller design we will take it in consideration.
If >1, the dead time is predominant, meaning that for the controller design will be used special structures such as Smith predictors.
Phase margin method cannot specify the type of the regulator. That has to be picked accordingly to some imposed performances in the closed loop, such as for the steady state position error to be zero, a PI regulator has to be used, for a small settling time, a PD regulator is recommended, while for both, a PID regulator is a better choice. [2]
We have separate algorithms for obtaining each type of regulator, and I will present the steps required to obtain the PI and PD ones.
Starting with PI, the algorithm means to determine the parameters of the PI controller,, with the following transfer function :
.
The next step would be representing the transfer function of the process with the Bode diagram with phase and modulus. The phase margin being give , the systems phase on the direct loop will be :
180+15+= -165+
The phase of the PI regulator being considered as: .
From Bode diagram of (phase diagram) we read the frequency on which the phase is equal to . For the frequency we read from the Bode diagram the modulus which is in decibels. We impose the frequency to be from which it results that and (the modules are in decibels). [2]
Now we obtain that . We impose that the modulus of =k, resulting in and we can get that k= and to determine the integration constant we use the relationship .
Moving on with the algorithm for PD regulator, the starting transfer function from which we have to find the PD parameters is:
, where [2]
We represent the transfer function of the process with Bode diagram, and the phase of the direct loop, will be given by: that if we impose the value of and also . In the case when we know only the value of we consider having = from where we get .
The phase of the regulator PD is considered to be .From Bode diagram we read the frequency at which the phase is equal with . For the frequency we read from the Bode diagram the module of , after which we consider that the frequency and we will obtain and (the modules are in decibels).
To calculate the gain of the controller factor k, we impose the module of resulting in =20 and further =20.The value of k will be obtained from the next relation k=.
To calculate the time constant we use from which we obtain . [2]
3.1 Decentralized control
First of all we started with the steady-state gain of the process matrix that being:
From which we moved on to calculate the Relative Gain Array (RGA) of the process with the formula:
and we obtained
The selection of the pairs can be achieved now after obtaining the RGA and that will be:
=>> resulting into the first process transfer function that will be used to tune the first controller
=>> resulting into the second transfer function that will be used to tune the second controller.
After setting the pairs we can continue with calculating the regulators of the process.
3.1.1 Phase margin method
The type of the regulator is PI and we imposed a =60
After calculating the Bode diagram for first transfer function we obtained the cutting frequency as , after we obtained || at the cutting frequency 0.404 to be 1.8772.
Finding the as being 0.5327 and =9.9010 resulted into finding the first regulator:
.
After calculating the Bode diagram for second transfer function we obtained the cutting frequency as after we obtained || at the cutting frequency 1.15 to be 1.1655.
Finding the as being 0.8580 and =3.4783 resulted into finding the second regulator:
.
After obtaining both controllers for our process we implemented them into Simulink into the following structure which represents the decentralized control structure where we have presented the controllers calculated along with the four transfer functions and their dead time, the coupling is done along the main diagonal which means that first controller is coupled with and acts on the second output too, and the second controller is coupled with and acts on the first output of the system:
Figure 3.1.1 Simulink representation for Decentralized control structure. Phase Margin
The results are displayed in the following figures for each of the process outputs and they are as it follows. The simulated scenario is the following:
At the time t=0, the reference r1for the first output is modified from zero to one, while the reference r2 for the second output is kept at zero.
At time t=150, the reference r2 for the second output is modified from zero to one, while the reference r1 for the first output is kept to one
Graphic for first output:
Figure 3.1.2 First output of the process
In Figure 3.1.2 we have the signal for the first output. The reference was set to one and the controller takes the corresponding output to that reference value. The maximum amplitude for the second part beyond 150sec, which is due to the interactions, is 1.58 and the settling time is 80sec, meaning the system stabilizes at around that time. For the first part of the graphic between zero and 150sec, which corresponds to the reference tracking, we have a settling time of 80sec and an overshoot of 13%. On the second part the signal should have stayed on the value one. But due to the interactions the system’s output oscillates.
Graphic for second output:
Figure 3.1.3 Second output of the process
The second graphic is related to the second output. We can see that the first part until 150sec we have transitory regime due to the interactions that occur in the system, meaning that the process starts at zero and it comes back to zero in the end. The reference value was set to zero and, the output should have stayed permanent to zero. Due to the couplings into the multivariable system, because of the change in the first output reference signal, the second output also reacts, but in time it is brought back to zero. In this case we have a settling time of 80sec and the maximum amplitude of the signal is 0.2. On the second part of the graphic from 150sec beyond we have an overshoot of 13% and a settling time of 80sec.
3.1.2 Guillemin Truxal method
We imposed an overshoot of 15% and settling time of 60 sec, thus a lower settling time then the one obtained with the phase margin criterion.
The first step into calculating our first controller is to obtain the damping factor which in our case is . After obtaining the damping factor we calculate the natural frequency and we obtain =0.1105. After finding both coefficients and we calculate the transfer function for the closed loop system and we obtain the following form:
We know that the transfer function without dead time is = and now with all the information that we have we can calculate the first controller and we obtain it as being:
Moving on with the second controller same as on the previous one we calculate the damping factor and the natural frequency and we obtain: and =0.1290 and so we calculate the transfer function for the closed loop:
We know that the transfer function without dead time is and with the computed parameters we can move on and calculate the second controller:
Now that both controllers are computed we can implement them into the control structure for our system where we have the calculated controllers, the matrix of transfer functions that are imposed along with the dead time for each one. The first controller it is coupled with and acts on the second output too, and same for the second controller is coupled with and acts on first output of the system.
Figure 3.1.2.1 Simulink representation for Decentralized control structure. Guillemin Truxal
The results of the control structure are displayed on the following graphics, where we can see the settling time and the overshoot of the process. The same scenario was used as indicated above, with the phase margin criterion.
Graphic for first output:
Figure 3.1.2.2 First output of the process
In the Figure 3.1.2.2 we have the first output of the process. We have reference tracking meaning that the controller takes the output to the set reference value which is one. We have maximum amplitude of 1.538 and the settling time is 145 sec for the second part, which is beyond 150 sec. On the first part from zero to 150 sec we have settling time of 145 sec and overshoot of 3%.
Graphic for second output:
Figure 3.1.2.3 Second output of the process
In this graphic the signal is related to the second output, and what happens on the first part until 150 sec is due to the interactions. The reference value was set to zero until 150 sec, but the output should have stayed permanent to zero, but due to the couplings into the multivariable systems, it makes the output to react, but in time it is brought back to zero. We have a settling time of 145 sec and maximum amplitude of 0.36. On the second part we have an overshoot of 5.1% and a settling time of 145 sec.
As a remark we can see that with Guillemin Truxal method we obtained processes with higher settling time and this is due to neglecting the dead time on the transfer functions even if in one case the value of the ratio between the dominant time constant and the dead time of the transfer function was close to 0.2. About the overshoot is almost the same in both cases, but Guillemin Truxal seems to reduce it a bit more than the Phase Margin method. Obviously, this comes with the drawback of increasing the settling time even twice as much.
3.2 Static decoupling
Starting with the process transfer matrix:
which we will bring it to steady state form in order to be able to calculate the decoupling elements. The steady state matrix is:
=
In order to understand the calculation of the decoupling elements we will illustrate the general case of the decoupling control structure:
Figure 3.2 Decoupling control structure
From where we have a few equations in that will help us to deduce the decoupling elements formulas:
and
From the above equations resulting:
The product of the two transfer matrices would result into the unity matrix . After calculating the product of the two transfer matrices we obtain that:
From here we consider and equal to one.
The first element we will calculate is resulting in and so we will calculate from which .
Further on we need to obtain the process transfer function from which we will calculate the controllers, and to do so we have the following equations:
The equation above relates the output to the new input
The next step was to identify and after that approximate in order to obtain a single transfer function so that we can calculate much more easily the controller transfer function for the first output.
The plot of the identification is:
Figure 3.2.1 Transfer function identification for static decoupling, first output
Where ; We calculated 63.4% from the stationary value which is 6.4 and the result was 4.0576, from which accordingly to the graphic the time constant T is equal to 12.6 and minus 1 the dead time, gives that T=11.6.
And the obtained transfer function after the approximation:
Moving on with the next equation, that relates the second output to the new input :
From the sum of transfer functions, we had to approximate and identify a single transfer function so we can obtain the second controller, the following graphic being the identification:
Figure 3.2.2 Transfer function identification for static decoupling, second output
Where ; We calculated 63.4% from the stationary value which is 9.633 and the result was 6.1073, from which accordingly to the graphic the time constant T is equal to 16.3 and minus 3 the dead time, gives that T=13.3.
And the obtained transfer function after the identification and approximation is:
With both transfer functions calculated we can proceed and obtain the controllers for the decoupling structure.
3.2.1 Phase margin method
We will impose for both controllers but we will use for first one and for the second controller in order to obtain them.
After calculating the Bode diagram for first transfer function we obtained the cutting frequency as , after we obtained || at the cutting frequency 0.448 to be 1.2078.
Finding the as being 0.8279 and =8.9286 resulted into finding the first regulator:
.
After calculating the Bode diagram for second transfer function we obtained the cutting frequency as after we obtained || at the cutting frequency 0.207 to be 3.2734.
Finding the as being 0.3055 and =19.3237 resulted into finding the second regulator:
.
In both cases the was equal to -105.
After obtaining both controllers for our process we implemented them into Simulink into the following structure which represents the static decoupling control structure where we have presented the controllers calculated along with the four transfer functions, the decoupling elements and their dead time, the coupling is done along the main diagonal which means that first controller is coupled with and acts on the second output too, and the second controller is coupled with and acts on the first output of the system. We did not show and on the figure because both of them in this case were considered to be 1.
Figure 3.2.1.1 Simulink Static Decoupling control structure
The results of the static decoupling control structure are displayed on the following graphics, using the same scenario, as indicated previously:
Graphic for first output:
Figure 3.2.1.2 First output of the process
In Figure 3.2.1.2 we have the signal for the first output and the first controller. We can observe that in this case the oscillations are really high, but the system manages to stabilize in the end on the value one, meaning we have perfect reference tracking. The controller manages to take the output of the system to the set reference value being one. In this graphic we have the signal for the first output. We have maximum amplitude of 1.25 and the settling time is 130 sec for the second part after 150 sec, and we have a settling time of 130 sec, and overshoot of 58.2% for the first part of the graphic between zero and 150 sec.
Graphic for second output:
Figure 3.2.1.3 Second output of the process
Here we have the signal for the second output and the second controller, which is in transitory regime on the first part until 150sec due to the interactions in the system. The output starts on zero but it comes back to zero. The output should have stayed permanently to zero due to the set value of the reference to zero, the couplings in the multivariable systems make the output to react, but it time is brought back to zero. We have a settling time of 145sec, and maximum amplitude of -2.395. On the second part from 150sec we have a settling time of 145sec, and overshoot of 51%.
We did try two more methods in order to try and control the decoupling structure but none of them worked, first was Guillemin Truxal method and it did not work due to fact that the ratio of the dead time and time constant was 0.22 and the result was unstable control structure. The same thing happened to Dahlin algorithm, the structure was unstable.
I tried to impose a too but the oscillations were even bigger than in the current graphics.
3.3 Dynamic Decoupling
Starting with the process transfer matrix:
In dynamic decoupling we do not work in steady state anymore meaning that the decoupling elements will be now transfer functions. In order to understand the calculation of the decoupling elements we will illustrate the general form of the decoupling control structure:
Figure 3.3.1 Dynamic Decoupling Controller Structure
From where we have a few equations in that will help us to deduce the decoupling elements formulas:
and
from here resulting:
The product of the two transfer matrices would result into the unity matrix . After calculating the product of the two transfer matrices we obtain that:
From here we consider and equal to one.
The first element we will calculate is resulting in ,
and so we will calculate from which .
Further on we need to obtain the process transfer function from which we will calculate the controllers, and to do so we have the following equations, that relate the output to and the output to , respectively:
The next step was to identify and after that approximate in order to obtain a single transfer function so that we can calculate a lot easier the controllers.
The plot of the identification is for the first relation, between and is given in the figure below:
Figure 3.3.2 Transfer function identification for dynamic decoupling, first output
Where ; We calculated 63.4% from the stationary value which is 6.377 and the result was 4.043, from which accordingly to the graphic the time constant T is equal to 15.6 and minus 1 the dead time, gives that T=14.6.
And the obtained transfer function after the approximation:
Moving one with the next equation:
From the sum of transfer functions, we had to approximate and identify a single transfer function so we can obtain the second controller, the following graphic being the identification:
Figure 3.3.3 Transfer function identification for dynamic decoupling, second output
Where ; We calculated 63.4% from the stationary value which is -9.684 and the result was -6.1397, from which accordingly to the graphic the time constant T is equal to 12.3 and minus 3 the dead time, gives that T=9.3.
And the obtained transfer function after the approximation:
With both transfer functions calculated we can proceed and obtain the controllers for the dynamic decoupling structure.
3.3.1 Phase margin method
We will impose for both controllers but we will use for first one and for the second controller in order to obtain them.
After calculating the Bode diagram for first transfer function we obtained the cutting frequency as , then obtain || at the cutting frequency 0.418 equal to 0.272.
Finding the as being 0.9855 and = 9.3897 resulted into finding the first regulator:
.
After calculating the Bode diagram for second transfer function we obtained the cutting frequency as , then obtain || at the cutting frequency 0.49 to be 1.7427.
Finding the as being 0.2455 and = 17.1764 resulted into finding the second regulator:
.
In both cases the was equal to -105.
After obtaining both controllers for our process we implemented them into Simulink, in the following structure which represents the dynamic decoupling control structure where we have presented the controllers calculated along with the four transfer functions, the decoupling elements and their dead time. The coupling is done along the main diagonal which means that first controller is coupled with and acts on the second output too, and the second controller is coupled with and acts on the first output of the system. We did not show and on the figure because both of them in this case were considered to be 1.
Figure 3.3.1.1 Simulink Dynamic Decoupling control structure
The results of the dynamic decoupling control structure are displayed on the following graphics:
Graphic for first output:
Figure 3.3.1.2 First output of the process
As we can see in the above graphic the signal is for the first output, and due to the strong coupling and interactions the output is very oscillating. The system has perfect reference tracking because the controller takes the output to the set reference value which is one. The maximum amplitude in this case is 1.12 and we have a settling time of 150 sec. On the first part from zero to 150 sec we have a settling time of 150 sec and an overshoot of 33.1%, corresponding to the reference tracking closed loop performance.
Graphic for second output:
Figure 3.3.1.3 Second output of the process
In this graphic we have the signal for the second output. The signal is very oscillating due to the interactions and the strong coupling of the process. On the first part of the graphic between zero and 150sec we have a transitory regime due to the interactions, meaning that the output starts on zero but it comes back to zero. The output signal should have stayed permanently to zero due to the set value of the reference to zero. Due the strong couplings in the multivariable systems, they are making the output to react, but in time is brought back to zero. We have a settling time of 150sec, and maximum amplitude of -2.59. On the second part from 150sec we have a settling time of 150sec, and an overshoot of 13.5%.
As a remark at the end of this chapter, we can easily see that decoupling control structures are not performing so well as decentralized control structures. It is known that decoupling controllers are not so desirable in practice, they are mostly used from a theoretical point of view. It also yields insights into the limitations that the multivariable systems are imposing on the achievable performance.
Some disadvantages of decoupling would be that it is really complex, and it requires accurate process models, the problem of adequately tuning the SISO controller, it is really sensitive to modeling errors, especially when the coefficients of the RGA (Relative Gain Array) are too large.
Chapter 4 COMPARATIVE RESULTS AND ROBUSTNESS ISSUES
In this chapter I will mainly speak about which method from the presented ones gives the best results, therefore would be the best to use into the multivariable control structure. By best I mean which one has the lowest settling time, the lowest overshoot and the lowest interactions.
4.1 Comparative results
First I will speak about the first controller for each control structure, decentralized or decoupling, calculated with either Phase Margin or Guillemin Truxal criteria. The control loop interactions occur due to the presence of a new feedback loop in all cases.
Figure 4.1.1 First outputs of the process in comparison
Starting with the red signal we have the decentralized control structure and the first output signal influenced by the first controller of the process, the controller being calculated with phase margin method. We can see that on the first part until 150sec it has an overshoot of 13% and a settling time of 80sec and for the second part we have maximum amplitude of 1.584 while the settling time remains the same. What happens on the second part of the graphic is due to the interactions in the process.
Moving on with the comparison we have the green signal which is for decentralized control structure and the first output signal influenced by the first controller of the process, the controller being calculated this time with the Guillemin Truxal method, meaning that we had to ignore the dead time of the transfer functions. That resulted into having a settling time of 145sec and an overshoot of 9% on the first part until 150sec. On the second part beyond 150sec we have maximum amplitude of 1.538 and the settling time remains the same.
Until now on the decentralized control we can see that Guillemin Truxal method works better due to the lower overshoot in both parts of the graphic, and phase margin is better in obtaining a lower settling time. In terms of couplings, the controllers computed using the phase margin method ensure a reduced settling time in rejecting the interactions, as compared to the Guillemin-Truxal situation.
Onto decoupling control, first we start with static decoupling and the phase margin method. The signal belonging to this method is the black one, which on the first part until 150sec has an overshoot of 58.2% and the process stabilizes in 130sec, and for the second part it has maximum amplitude of 1.25 and the settling time remains the same.
The blue signal is for the first output of the dynamic decoupling structure, controller being calculated with phase margin method. We can see that the signal on the first part until 150sec has an overshoot of 33.1% and settling time of 150sec meaning that the system barely stabilizes in the imposed set time, and on the second part we have maximum amplitude of 1.12 and the settling time is 150sec.
As a conclusion for decoupling control in these cases, we can say that the static decoupling offers higher overshoot but lower settling time while the dynamic one is better for the lower overshoot that it can provide. In terms of interactions, the output oscillates quite heavily around the reference signal, but overall the interactions are rejected.
Overall decentralized control structures are more often used in practice, due to the good performances that they can offer, because the system is much simpler to implement and to control then the decoupling structure. In our case the best control structure and the best method used to calculate the controller would be decentralized control structure with method Guillemin Truxal due to the lower interactions in the system, lower overshoot better settling time than in other cases.
In Figure 4.1.2 the second output is presented. The first part up to 150 seconds, corresponds to a reference equal to zero, while the second part, after 150 seconds, corresponds to a reference signal equal to one.
Figure 4.1.2 Second outputs of the process in comparison
Up to t=150 seconds, the reference value was set to zero and the output should have stayed permanently to zero. But due to the strong coupling that occur in our process, they are making our output to react, but in time it is brought back to zero.
Starting with the red signal we have the decentralized control structure and the first output signal influenced by the second controller of the process, the controller being calculated with phase margin method. We can see that on the first part until 150sec it has maximum amplitude of 0.2 and a settling time of 80sec and for the second part we have an overshoot of 13% while the settling time remains the same.
Moving on with the comparison we have the green signal which is for decentralized control structure and the second output signal influenced by the second controller of the process, the controller being calculated this time with the Guillemin Truxal method, meaning that we had to ignore the dead time of the transfer functions. That resulted into having maximum amplitude of 0.3 and a settling time of 145sec on the first part until 150sec. On the second part beyond 150sec we have an overshoot of 12% and the settling time remains the same.
Until now on the decentralized control we can see that Guillemin Truxal method works better due to the lower overshoot in the second part of the graphic, and phase margin is better in obtaining a lower settling time. The interactions are stronger in the Guillemin Truxal case obtaining higher maximum amplitude then in the phase margin case with 0.1.
Onto decoupling control, first we start with static decoupling and the phase margin method. The signal belonging to this method is the black one, which on the first part until 150sec has maximum amplitude of 1.45, and the process stabilizes in 145sec, and for the second part it has an overshoot of 51% and the settling time remains the same.
The blue signal is for the first output of the dynamic decoupling structure, controller being calculated with phase margin method. We can see that the signal on the first part until 150sec has maximum amplitude of 0.78, and settling time of 150sec meaning that the system barely stabilizes in the imposed set time, and on the second part we have an overshoot of 13.5% and the settling time is 150sec.
As a conclusion for decoupling control in these cases, we can say that the static decoupling offers higher maximum amplitude but lower settling time while the dynamic one is better for the lower maximum amplitude that it can provide, meaning that the interactions are lower in the dynamic decoupling control structures. In the second part of the graphic the dynamic decoupling offers a better overshoot then the static decoupling and a slower settling time.
The best control structure is the decentralized one, with the method of calculating the controller being phase margin due to the lower settling time it offers, lower interactions than in almost all the cases presented and lower overshoot on the second part of the graphic then the rest cases with exception being the Guillemin Truxal method, which provides better overshoot but fails on the interaction part and the settling time one.
4.2 Robustness of the designed controllers
We will start with the description of the robustness control, which means obtaining performance and stability in the presence of modeling errors. Some early methods like Bode or state-space were lacking robustness, meaning that they had to do some research in order to improve them. That was the start of Robust control if we can say so, which started in around 1980s and 1990s and is still active even today. The robust control policy is known to be static.
A definition of robustness control could be stated as: “Design a controller such that some level of performance of that controlled system is guaranteed irrespective of changes in the plant dynamics within a predefined class”.
A main purpose of the robustness design if we can say so, is to retain assurance of a systems performance in spite of model changes. A system can be robust when it can accept changes in performance due to the model changes and inaccuracies. One key issue for design and control structures is the good “communication” between robustness and the performance of the system.
We can say that a controller which is designed to work on a set of parameters, if it manages to work well under another set of assumptions the controller is robust.
I did try robustness control on each method I’ve tested, meaning I changed the transfer matrix coefficients with +20% and -20% and analyzed the results. The new transfer matrix after the robustness adjustments are:
The matrix with +20% modeling errors in the gains of the process:
The matrix with -20% modeling errors in the gains of the process:
Starting from here each method was tested accordingly to the new set of data. It is important to mention that only the state matrix was modified in the robustness tests the controllers remained the same.
First robustness that I’ve tested was on the decentralized control structure, method phase margin and the results for the +20% and -20% tests for the first output and the second one:
Figure 4.2.1 First outputs of the process with robustness tests. Decentralized Control, Phase margin
As we can see we have 3 signals: the red one for the nominal controller, the green one for the robustness test with 20% variation of the parameters and the black signal with-20% variation. In the case which we increased the robustness we can observe an increase in overshoot in both parts of the graphic and a small increase of interactions meaning the system it’s a bit more oscillating then the nominal signal.
On the other hand when we adjusted the parameters with -20% we can for sure say that the process should work better due to the lower overshoot that we obtained and not only that, we can see a settling time reduction too, and smaller interactions meaning that the system is not oscillating that much with the new adjusted parameters.
We can stay that with robustness control the system would work better in the case when we reduce the parameters values by-20%.
All in all the system is stable after the robustness tests, which means that the designed multivariable controller would stabilize the process, despite +20% and -20% modeling uncertainties.
Moving on with the second controller of the decentralized control structure, calculated with phase margin method and with applied robustness tests the result is the following:
Figure 4.2.2 Second outputs of the process with robustness tests. Decentralized Control, Phase margin
The first part of the graphic until 150sec remains still the same in both cases of robustness tests the +20% and -20%, meaning that the designed controller achieves similar interaction attenuation in both nominal, as well as modeling errors.
We can see in this case too that the +20% robustness test brings along higher overshoot, higher maximum amplitude and higher settling time compared to the nominal output. The system is stable but the performances are not better than in the initial nominal.
But again for the -20% robustness test, the closed loop performance is improved, in all ways, meaning we have better maximum amplitude then in the nominal case, better settling time and a better overshoot on the second part of the graphic. Which means the performances are improved overall and, the control structure is close to the ideal case then as it was before in the nominal case.
In both cases for the first and the second output of the system, the performances have been improved when considering the decreasing of the parameters with 20% resulting into more stable structure and lower interactions.
Now that we saw the improvements on the decentralized control structure with the method of controller calculation being phase margin, we move on still on the decentralized structure but we changed the method with which we calculated the regulator this time is the Guillemin Truxal method. The results are:
Figure 4.2.3 First outputs of the process with robustness tests. Decentralized Control, Guillemin Truxal
In the above graphic we have displayed the first output of the structure, the nominal signal is the red one, the first robustness test with +20% in parameters value is the green signal and the second robustness test with -20% in parameters value is the black signal.
We can say that yet again the robustness test in which we lowered the given parameters with 20% is better, because we can see an improvement in settling time, in overshoot that being in both parts of the graphic.
If we take a look at the robustness test, where we increased the parameters value with 20% the system is still stable but the performances are not the desired ones, being even worse than the initial nominal process. We obtained higher overshoot, an increased settling time and the interactions are also increased due to the lower performances that we obtained.
Here is the second output of the decentralized control structure where the controller was calculated with Guillemin Truxal method. The following graphic shows the results of the robustness tests:
Figure 4.2.4 Second outputs of the process with robustness tests. Decentralized Control, Guillemin Truxal
The signals elaboration are, the red one being the initial signal for the second output, the green one that represents the robustness test with +20% in the parameters value, and the last one the black signal, the one for the -20% robustness test.
Now starting with the green signal, we can see that the performances are not as good and in the initial control structure, meaning that they remain stable but the maximum amplitude increases along with the settling time, and the overshoot on the second part of the graphic.
The black signal which represents the second output of the control structure on which we applied the robustness test with lower parameters values by 20%, seems to be better in this case too. Because as we can see on the graphic the system does not improve by much but we have smaller settling time, maximum amplitude and on the second part of the graphic we have a smaller overshoot. The first part of the signal is still in transient regime due to the interactions but we are pleased with obtaining better performances even if they are not that visible.
Once again the robustness test in which we lowered the initial parameters values with 20% seems to be the best one to use, due to the better performances obtained and the lower interactions in the control structure. In both cases of first and second output on the Guillemin Truxal method, the better performances were obtained with lowering the initial values from the transfer matrix.
Overall, with both decentralized control structures, the performance specifications (overshoot/settling time) are not severely affected, neither the interactions, when considering the +20%/-20% variation in the process parameters, which indicates that the proposed controllers are robust to modeling errors.
Moving on with the decoupling control structures, the method used to calculate the controllers being, Phase Margin method on both static and dynamic decoupling. We applied robustness tests on all methods, in order to check if the performances are changing, meaning if we manage to obtain similar performances as in the initial case. In the following graphics we will present the results obtained after the robustness tests.
We will start with static decoupling, method used to calculate the controllers for the structure being Phase margin with. The following graphic is for the first output of the static decoupling structure.
Figure 4.2.5 First outputs of the process with robustness tests. Static Decoupling Control, Phase Margin
As we can see in this case we did not use the robustness test in which the parameters got increased with 20%, because the structure resulted into an unstable one, therefore we just used the test in which the initial transfer matrix had lower values of the parameters with 20%.
The results were really good, meaning that the structure reacted very well to the new parameters, obtaining much better settling time, and overshoot on both parts of the graphic. To be more specific the overshoot obtained on the first part is 15% compared to 58.2%, a settling time reduced from 130sec to 110sec and really reduced interactions into the structure due to the new parameters. On the second part from maximum amplitude of 1.25 we managed to reduce it to new maximum amplitude of 1.05, the settling time as on the previous part from 130sec to 110sec.
Even with the new signal that we obtained, the signal still has reference tracking, meaning that our reference value was set to one, and the controller manages to take the output of the system to that set reference value.
Compared to decentralized control the results are still not that great, because as we can see on this graphic the interactions into the static decoupling are much higher than in the decentralize structure. Also, as compared to the decentralized control strategy, in this case, a +20% modeling error will cause the system to destabilize. The proposed controller in this case is nor robust to modeling uncertainties.
Now that the first output of the static decoupling was presented, we will move on with the second graphic, which is specific for the second output of the calculated static decoupling structure. The robustness tests used are as in the previous case, meaning that we used only the test in which we decreased the values of the parameters, because in the other test the controlled structure resulted into an unstable one.
Figure 4.2.6 Second outputs of the process with robustness tests. Static Decoupling Control, Phase Margin
Yet again the new signal obtained with the robustness test when we decreased the values of the initial parameters seems to work better than the nominal ones. The red signal represents the signal for the nominal controller/output when the black signal is for the signal obtained with the robustness test.
The results are obvious. We managed to obtain a new structure in which we managed to reduce the overshoot, the maximum amplitude of the signal, the settling time and the interactions in the structure which could be one of the most important parts.
On the first part of the graphic we still have a signal in transient regime due to the existing interactions and strong couplings, but we managed to reduce them and not only them. We succeeded to obtain better maximum amplitude, reduced from -2.395 to -2.16, and a better settling time reduced from 145sec to 138sec, the improvement not being so big but it is there.
Onto the second part we have reduced the overshoot from 51% which was the initial one without the robustness test to a new overshoot of 33%, the settling time staying around 138sec as in the first part of the graphic.
As a small conclusion for the static decoupling control structure on which we applied robustness tests, them being into reducing the initial values of the parameters, we could say that the tests were successful because we managed to maintain the system into a stable point and not only that, we managed to obtain better performances.
The next step after finishing with the tests on the static decoupling control structure, was to perform some tests of robustness on the dynamic decoupling control structures and see how they perform with the new conditions. I must state that yet again we could not obtain any results for the robustness test in which we increased the values of the parameters with 20% due to the unstable process that we obtain.
Now we will check the results for the applied robustness test for the dynamic decoupling control structure, the method of calculating the controller being the Phase margin method where the imposed was of 60.The first graphic is for the first output of the system.
Figure 4.2.7 First outputs of the process with robustness tests. Dynamic Decoupling Control, Phase Margin
First thing that we can observe on the graphic is that the black signal which is the signal obtained after we applied the robustness test is much better than the red signal, which represents the nominal process output with the values for the parameters unchanged. We say that the black signal is less oscillating than the red one due the better performances obtained after we applied the robustness test.
We decreased the initial transfer matrix parameters with 20% and the results are visible. On the first part of the graphic from zero until 150sec we have a better overshoot reduced from 33% to 19% which might not seem that much, but the less the better because the system will act more stable. The settling time is also reduced from 150sec to 114sec.
On the second part of the graphic from 150sec and further we have as expected reduced maximum amplitude from 1.12 to 1.01, and the settling time as in the previous case from 150sec to 114sec.
The structure has a visible reduction too in interactions, meaning that the system is not oscillating as in the initial case, considering a perfect modeling of the process. We still have reference tracking as in all cases for the first output, meaning that the controller takes the output to the set reference value.
The last signal on which we applied the robustness test was the signal for the second output of the dynamic decoupling control structure, on which we were also impressed with the obtained results. The method with which we calculated the controller for this output was Phase margin method with an imposed of 60. The following graphic shows the displayed results of the test.
Figure 4.2.8 Second outputs of the process with robustness tests. Dynamic Decoupling Control, Phase Margin
As in previous cases the red signal is for the nominal controller without robustness tests and the black one is for the output signal but with applied robustness tests. Yet again we did not use the test in which we should have increased the values of the initial parameters with 20% due to unstable structure that we obtained after testing it. But we managed to obtain a stable structure and not only that, but with improved performances. It was obtained with the robustness test in which we lowered the values of the initial parameters from the transfer matrix with 20%.
At start, both signals on the first part of the graphic are in transient regime due to the strong interactions and coupling existing in the control structure, but we manage to stabilize the system into the set value of one in the end. In the first part we have better maximum amplitude obtained from -2.59 to -2.32, a better settling time being reduced from 150sec to 140sec and to not forget reduced interactions that being on both parts of the graphic.
On the second part from 150sec and further we managed to get better overshoot from 13.5% reduced to 1% and the settling time remains the same as on the first part of the graphic and more exactly, reduced from 150sec to 140sec.
The conclusion for applying robustness tests on the dynamic decoupling control structure is a positive one due to the increased performances we managed to obtain after applying certain tests over the structure we had. Nevertheless, as in the case of the static decoupling, this also resulted in poor performance in terms of +20% modeling errors. This indicates that the proposed control strategy is not robust to +20% variations of the process parameters.
Chapter 5 CONCLUSIONS
This paper presented two solutions for controlling a distillation column. Being a multivariable system we had to use decentralized control which is one of the easiest methods to control a multivariable system, and it implies the decomposition of a MIMO system into a SISO one. The other solutions would be decoupling control, which is mostly used when we have structures with strong couplings, decoupling being more used in this case than decentralized control. Decoupling aims to pair the input and the output signals in such way that the multivariable interactions are at their minimum, and sets the premises for a possible improved design of decentralized control.
First step was to start with decentralized control and study the control structure obtained. The RGA that we obtained resulted into a very strong coupled system with the parameters in the RGA that should have been close to one, being in fact around two. That means the higher the value of the RGA coefficients, the stronger the couplings and the harder the process to control.
But we did manage to obtain some good results with decentralized control, after we computed the controllers with Phase margin and Guillemin Truxal criteria. The obtained controller with Phase margin method resulted into a better control structure then the one obtained with Guillemin Truxal due to the better set of performances that were obtained in closed loop. We noticed that the settling time was reduced more in the Phase margin chase, the overshoots were fairly the same in both cases, but a bit better in Guillemin Truxal criteria. The maximum amplitude due to the interaction was again almost the same but, better in Phase Margin method.
Guillemin Truxal did not work so well due to neglecting the dead time of the process. Even if the ratio between the dominant time constant and the dead time was higher than 0.2, for some loops, we chose to still ignore it and try the control structure, which was stable, but provided slightly poorer performances compared to the Phase Margin criteria.
Next step was to move onto static decoupling, where we had to identify a transfer function for both outputs of the system, due to the complex sum of transfer functions obtained when we obtained the decoupler and the decouple process transfer function matrix. That resulted into a transfer function of order one with dead time for both outputs, from which we computed the controllers. Static decoupling was hard to control, the only method that managed to work for us being Phase Margin, the other ones like Dahlin and Guillemin-Truxal resulting into an unstable control structure.
The same happened to dynamic decoupling, where only Phase Margin worked again. As in the previous case we had to do some identification in order to determine the transfer function which will further on be used to compute the controller for both outputs. The obtained transfer functions were first order ones, with dead time. From here we computed the controller for our control structure, and we obtained a stable process with high oscillations due to the strong couplings that occur in decoupling control.
As a conclusion for decoupling control in these cases, we can say that the static decoupling offers higher overshoot but lower settling time while the dynamic one is better for the lower overshoot that it can provide.
After obtaining all the control structures for decentralized and decoupling control, we moved on with doing some comparison between them in order to see which one on them would be better to be used in a distillation process.
We managed to see that overall decentralized control structures are more often used in practice, due to the better performances that they can offer, because the system is much simpler to be implemented than the decoupling structure. In our case, the best control structure and the best method used to calculate the controller would be decentralized control structure with the Guillemin Truxal method, due to the lower interactions in the system, lower overshoot better settling time than in other cases.
Next we moved to robustness issues, where we tested if the controllers would work better/similar if the process was poorly modeled. We simulated this by increasing or decreasing the initial gains of the process transfer function matrix, with either +20% or -20%. The results were really good, meaning that for the robustness test in which the parameters were lowered with 20% in all cases, the decentralized and decoupling control structures we computed, managed to maintain the nominal performance criteria, or even better performances then in the nominal cases. In decentralized control we managed to test the robustness test with an increase of 20% in the initial parameters but the system was not as stable as in the nominal case or the other test. The second test, meaning the one with the increase of the parameters, did not work in decoupling structures, resulting into an unstable control process.
As a general conclusion, for Wood and Berry distillation column, among the proposed control strategies, the best one proved to be decentralized control structures, both under nominal, as well as under modeling errors.
As a future scope for this paper, we have in plan to test some other methods in order to control such multivariable process, meaning the Wood and Berry distillation columns. But this will require further research and testing.
REFERENCES
[1] Abdelmadjid Bentayeb, Nezha Maamri, Jean- Claude Tigeassou, “Design of PID Controllers for Delayed MIMO Plants Using Moments Based Approach”, Journal of ELECTRICAL ENGINEERING, 2006;
[2] Cristina I. Pop, Eva H. Dulf, Clement Festila, “Ingineria Reglarii Automate 1”, UTPRESS, 2012
[3] Jeffrey Carrey, Ben van Kuiken, Curt Longcore, Angela Yeung, “MIMO Control Using RGA”, 2006. Available online at:
https://controls.engin.umich.edu/wiki/index.php/RGA
[4] M. Fikar, “Control Systems, Robotics and Automation- Vol VIII – Decoupling Control”;
[5] P.Albertos, A. Sala, “Multivariable Control Systems: An Engineering Approach”, Springer, 2003;
[6] Qing-Guo Wang, Biao Zou, Yu Zhang, “Decoupling Smith Predictor Design for Multivariable Systems with Multiple Time Delays”, Institution of Chemical Engineers Trans IChemE, May 2000;
[7] R.Sivakumar, Shennes Mathew, “Design and Development of Model Pretictive Controller for Binary Distillation Column”, International Journal of Science and Research, 2013;
[8] Rakesh Kumar Mishra, Rohit Khalkho, Brajesh Kumar, Tarun Kumar Dan, “Effect of Tuning Parameters of a Model Predictive Binary Distillation Column”;
[9] S.Lakshminarayanan, Sirish L. Shah, K. Nandakumar, “Modelling and Control of Multivariable Processes: The Dynamic Projection to Latent Structures Approch”, Department of Chemical Engineering University of Alberta Edmonton, Canada, 1997;
[10] Sigurd Skogestad, “Dynamics and control of distillation columns a critical survey”, University of Trondheim, 1992;
[11] Wen Chung Lim, James Bennett, Jamila Grant, Ajay Bhasin, “Determining if a System can be decoupled”, 2006. Available online at:
https://controls.engin.umich.edu/wiki/index.php/Decouple
ACRONYMS
MIMO – Multivariable/ Multiple Inputs Multiple Outputs
SISO – Single Input Single Output
MPC – Model Predictive Controller
IMC – Internal Model Control
PID – Proportional-Integral-Derivative Controller
PI – Proportional-Integral Controller
RGA – Relative Gain Array
PD – Proportional-Derivative Controller
REFERENCES
[1] Abdelmadjid Bentayeb, Nezha Maamri, Jean- Claude Tigeassou, “Design of PID Controllers for Delayed MIMO Plants Using Moments Based Approach”, Journal of ELECTRICAL ENGINEERING, 2006;
[2] Cristina I. Pop, Eva H. Dulf, Clement Festila, “Ingineria Reglarii Automate 1”, UTPRESS, 2012
[3] Jeffrey Carrey, Ben van Kuiken, Curt Longcore, Angela Yeung, “MIMO Control Using RGA”, 2006. Available online at:
https://controls.engin.umich.edu/wiki/index.php/RGA
[4] M. Fikar, “Control Systems, Robotics and Automation- Vol VIII – Decoupling Control”;
[5] P.Albertos, A. Sala, “Multivariable Control Systems: An Engineering Approach”, Springer, 2003;
[6] Qing-Guo Wang, Biao Zou, Yu Zhang, “Decoupling Smith Predictor Design for Multivariable Systems with Multiple Time Delays”, Institution of Chemical Engineers Trans IChemE, May 2000;
[7] R.Sivakumar, Shennes Mathew, “Design and Development of Model Pretictive Controller for Binary Distillation Column”, International Journal of Science and Research, 2013;
[8] Rakesh Kumar Mishra, Rohit Khalkho, Brajesh Kumar, Tarun Kumar Dan, “Effect of Tuning Parameters of a Model Predictive Binary Distillation Column”;
[9] S.Lakshminarayanan, Sirish L. Shah, K. Nandakumar, “Modelling and Control of Multivariable Processes: The Dynamic Projection to Latent Structures Approch”, Department of Chemical Engineering University of Alberta Edmonton, Canada, 1997;
[10] Sigurd Skogestad, “Dynamics and control of distillation columns a critical survey”, University of Trondheim, 1992;
[11] Wen Chung Lim, James Bennett, Jamila Grant, Ajay Bhasin, “Determining if a System can be decoupled”, 2006. Available online at:
https://controls.engin.umich.edu/wiki/index.php/Decouple
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