Control of a Fluidised Catalytic Unit (FCCU)
Process Control Coursework 2018
Group 120
Table of Contents
Introduction Page 3
Part A Page 6
Part B Page 11
Part C Page d
Part D Page e
Part E Page f
Part F Page g
Part G Page h
University of Manchester School of Chemical Engineering and Analytical Science CHEN30091/CHEN30191 Process Control
Control of a Fluidized Catalytic Cracking Unit (FCCU)
1. Introduction to FCCUs
Most refineries include a fluidized catalytic cracking unit. Heavy hydrocarbons are decomposed to lighter ones, of higher value. This “upgrade” in composition is performed by thermal cracking. Gasoline is largely produced from FCC units. Figure 1 below illustrates a simplified schematic of a typical FCCU, including the control loops.
Figure 1: Schematic diagram of a typical FCCU
The riser tube brings in contact the recirculating catalyst with the feed oil, which then vaporizes and splits to lighter components as it flows up the riser, forming the desired gasoline fraction. Coke is a by-product of this process, which deposits on the catalyst reducing its activity. The spent catalyst is separated from the rest of the mixture in the 'reactor' unit shown in Figure 1, which is practically a separator of staged cyclones and is only called reactor for historical reasons. In a part of the unit not shown in the Figure, steam is used for the stripping of volatile components from the catalyst. The latter is then fed to the regenerator, where air is used to burn off the coke. Typically, partial combustion is employed, although some units do perform complete combustion. The regenerated catalyst is recirculated back to the FCCU by mixing it with the inlet feed oil.
2. The control problem
FCCUs are considered to be complex units and give rise to VERY challenging control problems. The selection of manipulated and control variables as well their pairng is crucial.
In this case, the important measured variables are chosen to be the reactor temperature/riser outlet temperature (T1), the regenerator gas (cyclone) temperature (Tcy) and the regenerator bed temperature (Trg). The manipulated variables are the catalyst recirculation rate (Fs) and the regenerator air rate (Fa).
In Figure 1, you can see the pairing of the variables: the reactor temperature/riser outlet temperature (T1) can be controlled by manipulating the catalyst recirculation rate (Fs), whereas the regenerator
gas temperature (Tcy) can be regulated using the regenerator air rate (Fa). The regenerator bed temperature (Trg) is not part of any control loop, but it is a quantity that needs to be monitored.
3. The assignment
You will focus on controlling the cyclone Temperature (Tcy) by manipulating the regenerator air rate (Fa). You are given a Simulink file, containing a quite detailed model of an FCCU. You can download the Simulink model from blackboard. Even numbered groups should download file FCCUa.mdl. Odd numbered groups should download file FCCUb.mdl. The region of operation of the unit is the vicinity around the steady-state provided. You need to design a robust working controller for the input output pair that you have chosen. To help you in this process a number of tasks are provided below
Investigate the dynamics of your system and extract an approximate first order transfer function model with delay. Clearly discuss and justify the procedures you follow as well as all your findings. [2 marks]
Demonstrate the robustness of the approximate model you obtained in (b) by illustrating that it works for a range of different (not only step) inputs. Clearly illustrate, discuss and justify all your findings. [2 marks]
Design stable controllers based on your approximate system using different tuning methodologies
(you will have to consult the literature for part of this task):
(i) Ziegler-Nichols (PI & PID)
(ii) Cohen-Coon (PI & PID)
(iii) IMC, where the delay is approximated by
a. 1st order Taylor expansion
b. Padé
Test your controllers on the real system (i.e. the detailed model). Check if they produce a stable output and justify your findings appropriately by simulating the closed-loop system you have designed using appropriate step changes to the set point. Clearly show all your work. [7 marks]
Explain clearly which parts of your block use deviation variables and which real variables by using appropriate diagrams. [3 marks]
For the closed-loop simulation of the real system with the controllers you designed in part (c), plot the inputs as well as Trg as a function of time. Discuss the range of values of the inputs and of Trg
that are physical/realistic. Clearly justify your answers. [3 marks]
Provide a meaningful comparison of the controllers you have designed, by discussing their features
and choose your best two controllers clearly justifying your choices. [2 marks]
G. For the two best controllers you have chosen you need to find:
i. What is the range of set points for which your controller can be trusted? (i.e. all inputs and outputs remain realistic and the controller performs as it should)? [3 marks]
ii. For the nominal set point, what is the range of input disturbances that your controller can reject, while all inputs and outputs remain realistic? Clearly justify all your answers. [3 marks]
Remarks:
Note that the final time of all simulations has to be chosen so that it allows the system to reach steady- state.
References
Lee, W., and V.W. Weekman, "Advanced Control Practice in the Chemical Process Industry: A View from Industry", AIChE J., 22, 27 (1976).
Grosdidier, P., A. Mason, A. Aitolahti, P. Heinonen, and V. Vanhamaki, "FCC Unit Reactor-Regenerator Control", Computers Chem. Eng., 17, 165 (1993).
PART A
Investigate the dynamics of your system and extract an approximate first order transfer function model with delay. Clearly discuss and justify the procedures you follow as well as all your findings. [2 marks]
A schematic diagram of model that was simulated is shown below.
Figure 2 – A simulation of the FCCU on SIMULINK.
After running this simulation, the output variable Tcy can be modelled by the graph shown in Figure 3.
Figure 3 – Graph showing Tcy as a function of time.
For the process dynamics to be evaluated, it is essential to induce a step change for the regenerator air rate (Fa). The step time was kept at 1 and the initial value of 25.35 kg s-1 was increased to 26.35 kg s-1. After running the simulation, the graph of Tcy as a function of time was now modelled as show in figure 4.
Figure 4 – Graph showing Tcy as a function of time with a unit step change for F¬a.
From assessing the shape of the graph shown in figure 4, it can be seen that the system can now be modelled as a first order transfer function with a time delay. The values that are needed in equation can be obtained from the graph.
A first order transfer function with delay can be approximated by the following;
G(s)=K_p/(τ_p s+1) e^(-t_d s) (1)
Where:
KP = Steady state gain
τp = Process time constant
td = Time delay constant
The steady state gain can be determined from equation 2.
K_p=(∆(output) at steady state)/(∆(input)at steady state)= (∆T_cy)/(∆F_a ) = (999-988)/(26.35-25.35) (2)
Kp = 11.0
The process response time (τp) can be found by subtracting the time delay from the time taken to reach 63.2% of the new steady state. The 63.2% comes from analytical data on many first order reactions and became a part of the definition of first order transfer functions.
τ_p=t_(63.2%)-t_d (3)
So in order to be able to determine t63.2%, T63.2% has to be calculated.
T_( 63.2%)=988+0.632(999-988)=995K
Figure 5 – Finding t63.2% by finding the time taken to get to 995K (T63.2%).
From figure 5, it can be seen that it takes approximately 28.9 seconds to reach T63.2%. Therefore, t63.2% = 28.9 s. Now the time delay needs to be calculated.
Another snapshot is shown in Figure 7 which Is the same as figure 5 with scaled x axis. This has been done to determine the time delay. The time delay is explained further via a diagram of a graph in figure 6.
Figure 6 – Graph highlighting the time delay and higher order models being modelled as first order with delay.
So now with the graphical representations of td. From figure 7 it can be seen that td is approximately 11 seconds. Subbing this into equation 3;
τ_p=t_(63.2%)-t_d = 28.9-11 = 17.9s
Figure 7 – graph showing the time delay td.
Now all the parameters are known they can be substituted into G(s) which is equation 1.
G(s)=K_p/(τ_p s+1) e^(-t_d s)=G(s)=11.0/(17.9s+1 ) e^(-11s)
Before simulating this on Simulink, since the dynamic model is dealing with deals with deviation variables in comparison to the real model which is dealing in real variables. Equation shows how to change between the variables.
T_cy^'=T_cy-T_(cy,s) (4)
Where T’a is the deviation variable
Where Ta is the real variable
Where Ta,s is the variables at steady state.
So, by adding the steady state temperature to the deviation variable, we can get the real variable. The steady state temperature is shown in figure 1 and was approximately 998K. A schematic of this process done on Simulink is shown in figure 8.
Figure 8 – Schematic diagram of the first order transfer function with delay, modelled on Simulink.
Plotting both of these models both verifies that approximating this model as a first order with a delay is a good approximation. MAYBE ADD HERE?
Figure 9 – Simulation of the real and dynamic models, showing that this model is a good approximation.
PART B
Demonstrate the robustness of the approximate model you obtained in (b) by illustrating that it works for a range of different (not only step) inputs. Clearly illustrate, discuss and justify all your findings. [2 marks]
To be able to determine the robustness of the model that was obtained, the input to the system was varied. The different inputs that were decided to show the systems robustness besides the step function shown in figure 9 was;
Sine wave function
Ramp function
Uniform random number function
In the graphs below and in figure 9 the yellow line represents the real system whereas the blue line represents the approximated first order system.
Sine Wave Function
The sine wave function uses a time-based model varying the input continuously over a given time. Figure 10 shows the model was simulated in Simulink and Figure 11 shows how the approximate model follows the real model.
Figure 10 – Sine wave as an input and how it was modelled on Simulink
The properties of the sine wave were just the default settings when the sine wave input function was selected.
Figure 11 – How Tcy varies over time with a sine input change of Fa
As it can be seen in figure 11 the approximate model follows the real model well, however the real model fluctuates with a higher amplitude than the approximate mode. This is due to disturbances.
Ramp Function
A ramp function generates a signal which increases steadily with time. Figure 12 shows how the ramp input function was modelled on Simulink and Figure 13 shows how the approximate model compares to the real model.
Figure 12 – Ramp function as an input and how it was modelled on Simulink
The properties of the ramp function were just the default settings when the ramp input function was selected.
Figure 13 – How Tcy varies over time with a ramp function input change of Fas.
As it can be seen in the graph, the approximate model is reasonable for a certain period of time before diverging away from the real model further as time increases. Figure 14 magnifies this image so the time that the time that the model is suitable can be seen.
Figure 14 – Figure 13 amplified to during what time period is the approximate model close to the real model.
From Figure 14 it can be seen that up until 37 seconds this is a good model and then it starts to diverge as it does in figure 13.
Uniform Radom Number Function
The Uniform Random Number block generates uniformly distributed random numbers over an interval that is specified. Figure 15 shows how the ramp input function was modelled on Simulink and Figure 16 shows how the approximate model compares to the real model.
Figure 15 – Uniform Random Number function as an input and how it was modelled on Simulink
For the Uniform Random Number, all the properties besides the sample time were the default properties. The sample time was changed from 0.1 to -1.
Figure 16 – How Tcy varies over time with a Uniform Radom Number Function
input change of Fas.
From Figure 16 it can be seen that approximate model very closely follows the real model. The main difference is that the amplitude of the real model is larger, this is due to real model taking disturbances into account.
PART C
Design stable controllers based on your approximate system using different tuning methodologies. Test your controllers on the real system (i.e. the detailed model). Check if they produce a stable output and justify your findings appropriately by simulating the closed-loop system you have designed using appropriate step changes to the set point. Clearly show all your work. [7 marks]
Ziegler-Nichols (PI & PID)
PID controllers are mainly used due to their simplicity and how effective they are in the majority of cases. It can also be tuned easily without needing a great depth of knowledge in controls. The 4 major characteristics of the closed loop response is;
Rise time – Time taken for output to rise beyond 90% of desired level for the first time.
Overshoot – How much the peak is higher than steady state.
Settling time – Time taken for system to converge to steady state.
Steady state error – Difference between the setpoint and steady state output.
Figure 17 shows how each parameter affects the system dynamics. The typical steps for designing PID controllers is determining what characteristics of the system need to be improved. Then use Kp to decrease rise time. Secondly, use KD to reduce the overshoot and settling time. Finally use KI to eliminate steady state error [2].
Figure 17 – How PID parameters effect system dynamics for closed loop [2]
The Ziegler-Nichols method is a trial and error method to tune controllers and is one of the most common ways to tune a closed loop PID controllers. The controller is designed by setting the integral gain (I) and derivative gain (D) to zero and then increasing the proportional gain (P). In table 1 below, the equations that can determine the tuning parameters for PID controllers are shown.
Type of controller Controller gain, KC Integral time, τI Derivative time, τD
P τ_P/(k_p 〖 t〗_d ) – –
PI (0.9τ_P)/(k_p t_d ) 3.3〖 t〗_d –
PID (1.2τ_P)/(k_p 〖 t〗_d ) 2.0〖 t〗_d 0.5〖 t〗_d
Table 1 – Zieglar-Nichols tuning parameters [3]
A PI controller combines proportional and integral control and a PID combines all 3.
PI=K_c+τ_P/(τ_I s) (5)
PID=K_c+τ_P/(τ_I S)+ K_c τ_D s (6) PI Controller
To calculate the tuning parameters for a PI controller〖 K〗_C and τ_I need to be calculated. The approximate model transfer from part A was determined to be:
G(s)=K_p/(τ_p s+1) e^(-t_d s)=G(s)=11.0/(17.9s+1 ) e^(-11s)
Therefore K_p = 11.0, τ_p= 17.9 s and t_d = 11s
Now K_c and τ_i can be calculated.
K_c=(0.9τ_p)/(K_p t_d )=(0.9×17.9)/(11×11)=0.133
τ_i=3.3t_d=3.3×11=36.3 s
Henceforth, by substituting these parameters into (5) the PI controller is given below:
PI controller = 0.133+0.133/36.3s
PID Controller
Now K_c , τ_i and τ_D are calculated for a PID controller.
K_c=(1.2τ_p)/(K_p t_d )=(1.2×17.9)/(11×11)=0.178
τ_i=2t_d=2×11=22 s
τ_D=0.5t_d=0.5×11=5.5 s
Therefore, by substituting the parameters into (6) the PID controller is given below:
PID controller = 0.178+0.178/22s+0.572s
In table 2 there is a summary of the values obtained for the PI and PID controllers.
Type of controller Controller gain, KC Integral time, τI Derivative time, τD
P – – –
PI 0.133 36.3 –
PID 0.178 22 5.5
Table 2 – Summary of calculated tuning parameters via Ziegler-Nichols method.
These controllers were simulated on Simulink as shown in Figure 18. This was to see if both the controllers were stable.
Figure 18 – Simulation of PI and PID on approximate system
To work out the parameters for the controllers in Simulink, the following was done (the example is for the PID controller but the same method was applied to the PI controller.
P=K_c=0.178
I=K_c/τ_I =0.178/22=0.00809
D=K_c 〖×τ〗_D=0.178× 5.5=0.979
For the PID controller there was a filter coefficient that was needed. The default value was N = 100. As the equation was:
N/(N+1 )=100/101=0.99
The filter value is essentially 1.
On Figure 19, the blue line is the setpoint, yellow line is the PI controller and the PID controller was the red line.
Figure 19 – Graph of the PI and PID controllers’ effects on the approximate system, using the Ziegler-Nichols method
From Figure 19 it can be seen that the PI controller reaches steady state within 300 seconds and therefore is stable. However, the PID controller oscillates above and below the setpoint throughout the 1000s, this indicates it is not stable in this case.
This was then simulated his onto the real system with initially zero step change in Tcy as shown on figure.
Figure 20 – Simulation of Tcy on the real system to compare PI and PID controller against setpoint for the Ziegler-Nichols method
Figure 21 – PI vs PID for Ziegler-Nichols. Tcy constant step input of 988K.
On Figure 21, the blue line represents the PID controller, the yellow line represents the PI controller and the setpoint is orange. The step input of Tcy was at a constant 988K.
From the graph it can be seen that the PID controller is very unstable as it is oscillating below and above the setpoint. However, The PI controller seems to have very good control and hardly oscillates away from the setpoint. This indicates that the best controller to use on the real system with a constant step input is the PI controller. On figure 22, the step input is increased from 988K to 999K.
Figure 22 – PI vs PID for Ziegler-Nichols. Step change from 988K to 999k
On Figure 22, the blue line represents the PID controller, the yellow line represents the PI controller and the setpoint is orange. The step input changed from 988K to 999K.
This figure has similar results to figure 21. However, this time the PI controller takes around 130 s to reach steady state. Once again, the PID controller fails to reach steady state and oscillated above and below the setpoint with a large amplitude. Therefore, the PI controller using the Ziegler-Nichols tuning method is the stable and best controller to use.
Cohen-Coon (PI & PID)
The Cohen-Coon method tuning rules are more suited to a wider variety of processes than the Ziegler-Nichols. This is due to Z-N rules only working well on processes where the dead time is half the time of the time constant [4].
The tuning parameter equations that can determine the tuning parameters for PID controllers for the Cohen-Coon method are shown below in table 3.
Type of controller Controller gain, KC Integral time, τI Derivative time, τD
P τ_P/〖K_P t〗_d (1+t_d/(3τ_P )) – –
PI τ_P/(K_P t_d ) (0.9+t_d/(12τ_P )) t_d (30+(3t_d)/τ_P )/(9+(20t_d)/τ_P ) –
PID τ_P/(K_P t_d ) (4/3+t_d/(4τ_P )) t_d (32+(6t_d)/τ_P )/(13+(8t_d)/τ_P ) t_d 4/(11+(2t_d)/τ_P )
Table 3 – Cohen-Coon tuning parameters
PI controller
K_p = 11.0, τ_p= 17.9 s and t_d = 11s
K_c= τ_P/〖K_P t〗_d (1+t_d/(3τ_P ))=17.9/(11× 11) (1+11/(3×17.9))=0.178
τ_I=t_d (30+(3t_d)/τ_P )/(9+(20t_d)/τ_P )=11((30+(3×11)/17.9)/(9+(20×11)/17.9))=16.45
Henceforth, by substituting these parameters into (5) the PI controller is given below:
PI controller = 0.178+0.178/16.45s
PID controller
K_c=τ_P/(K_P t_d ) (4/3+t_d/(4τ_P ))=17.9/(11×11) (4/3+11/(4×17.9))=0.220
τ_I=t_d (32+(6t_d)/τ_P )/(13+(8t_d)/τ_P )=11((32+(6 × 11)/17.9)/(13+(8×11)/17.9))=21.91
〖τ_D=t〗_d 4/(11+(2t_d)/τ_P )=11(4/(11+(2×11)/17.9))=3.60
In table 4 there is a summary of the values obtained for the PI and PID controllers.
Type of controller Controller gain, KC Integral time, τI Derivative time, τD
P – – –
PI 0.178 16.45 –
PID 0.220 21.91 3.60
Table 4 – Summary of calculated tuning parameters Cohen-Coons method.
Therefore, by substituting the parameters into (6) the PID controller is given below:
PID controller = 0.220+0.220/21.91s+3.60s
Figure 23 – Simulation of PI and PID on approximate system
These controllers were simulated on Simulink as shown in Figure 23. This was to see if both the controllers were stable.
Figure 24 – Graph of the PI and PID controllers’ effects on the approximate system, using the Cohen-Coon method
On Figure 24 the blue line represents the setpoint, the yellow line is the PI controller and the PID controller was the red line.
Similar to figure 19, the PI controller reaches steady state again despite taking a few more oscillations than using the Ziegler-Nichols method. Once again, the PID controller fails to reach steady state in the 1000s that lapsed. This suggests that the PI controller is stable for this process whereas the PID controller is not.
The same simulation was used as Figure 20 but now using the Cohen-Coon method for the tuning parameters for the PI and PID controllers. This is to test the controllers on the real system with a constant step change and step change from 988K to 999K.
Figure 25 – PI vs PID for Cohen-Coon. Tcy constant step input of 988K.
In Figure 25 and Figure 26, the blue line represents the PID controller, the yellow line represents the PI controller and the setpoint is orange. The step input of Tcy was at a constant 988K.
Once again the PID controller is very unstable and oscillated about the setpoint whereas the PI controller hardly moves from the setpoint.
Figure 26 – PI vs PID for Cohen-Coon. Step input change from 988K to 999k
In Figure 26, the step input change is from 988K to 999K. This model behaves very similar to the Ziegler-Nichols process in Figure 22. The PI controller has a couple more oscillations but reaches steady state in the same time. From the graph it again be deduced that the PID controller is unstable and the PI controller is the stable controller and the best to use in this case.
(iii) IMC, where the delay is approximated by
1st order Taylor expansion
IMC is a model-based procedure, where a process model is embedded in the controller. For this method, a controller is designed to force the output of a stable process to respond in a desired manner to a set point change (servo) and counter the effects of disturbances that enter directly into the process output (regulatory). This model assumes the process is perfect [1].
In order to design a realisable controller for first order processes the polynomial in the denominator of the controller must have an order that is at least equal to that of the numerator. If the numerator is = denominator it is described as semi-proper. If the denominator > numerator it is known as proper. It is possible to make an improper controller a proper controller by adding a filter. Furthermore, for the system to be stable the pole of the denominator has to be negative. If it isn’t, there are two methods to make it stable, the good part analysis and the all pass analysis.
Good Part Analysis – Can completely ignore the positive pole and add a filter if required.
All Part Analysis – Turns the positive pole into a negative pole.
This can be done because it is your controller.
Good Part Analysis
G_p (s)=G ̃_p (s)=K_p/(τ_p s+1) e^(-t_d s) (7)
Using Taylors expansion:
e^(-t_d s)=(1-t_d s) (8)
Henceforth, substituting equation 8 into equation 7 gives:
G_p (s)=G ̃_p (s)=(K_p e^(-t_d s))/(τ_p s+1)=(K_p (1-t_d s))/(τ_p s+1)
G_q (s)=1/(G ̃_p (s) ) (9)
Therefore, Inversing G ̃_p (s) to get G_q (s) gives:
1/(G ̃_p (s) )=(τ_p s+1)/(K_p (1-t_d s))
Removing the bad pole and adding a filter where a filter is:
F(s)=1/(λs+1) (10)
G_q (s)=(τ_p s+1)/(K_p (λs+1))
The equation for the controller is given by:
G_c=G_q/(1-G_q G ̃_p ) (11)
Such that Gc is:
G_c=((τ_p s+1)/(K_p (λs+1) ))/(1-((K_p (1-t_d s))/(τ_p s+1))((τ_p s+1)/(K_p (λs+1) )) )
Next the equation needs to be Simplified. the τ_p s+1 on the denominators cancel out and then from algebraic manipulation you get the next few steps.
G_c=((τ_p s+1)/(K_p (λs+1) ))/(((K_p (λs+1)-K_P (1-t_d s))/(K_p (λs+1) )) )=(τ_p s+1)/(K_p λs+K_P t_d s)=(τ_p s+1)/(K_p s(λ+t_d ) )
The IMC method has deduced that the controller that is needed is a PI Controller.
PI=(τ_p s)/(K_p s(λ+t_d ) )+1/(K_p s(λ+t_d ) )
All-Part Analysis
For this method the positive pole can be assumed to be negative.
G_q (s)=(τ_p s+1)/(K_p (1+t_d s))
Subbing into equation 11:
G_c= ((τ_p s+1)/(K_p (1+t_d s) ))/(1-((τ_p s+1)/(K_p (1+t_d s)))((K_p (1-t_d s))/(τ_p s+1)) )
Simplifying this equation:
G_c= ((τ_p s+1)/(K_p (1+t_d s) ))/(1-((K_p (1-t_d s))/(K_p (1+t_d s) )) )=(τ_p s+1)/(2K_p t_d s)
This is again a PI controller in the form
PI=(τ_p s)/(2K_p t_d s)+1/(2K_p t_d s)
PLOT GRAPHS AND EXPLAIN
WHAT TO USE AS λ????
b. Padé
For the Padé model:
e^(-t_d s)≈(-0.5t_d s+1)/(0.5t_d s+1) (12)
Subbing (12) into (9) gives
G_p (s)=G ̃_p (s)=(K_p e^(-t_d s))/(τ_p s+1)=(K_p (-0.5t_d s+1))/((τ_p s+1)(0.5t_d s+1))
And
G_q (s)=1/(G ̃_p (s) )=((τ_p s+1)(0.5t_d s+1))/(K_p (-0.5t_d s+1))
Removing the positive pole and adding in two filters to make it into a semi proper function gives:
G_q (s)=((τ_p s+1)(0.5t_d s+1))/(K_p (λs+1)(λs+1))
Subbing this into (11) for Gc gives:
G_c=G_q/(1-G_q G ̃_p )=(((τ_p s+1)(0.5t_d s+1))/(K_p (λs+1)(λs+1)))/(1-(((τ_p s+1)(0.5t_d s+1))/(K_p (λs+1)(λs+1)))((K_p (-0.5t_d s+1))/((τ_p s+1)(0.5t_d s+1))) )
By doing algebraic manipulation Gc becomes:
G_c=((〖(τ〗_p s+1)(0.5t_d s+1))/(K_p (λs+1)(λs+1) ))/(1-((-0.5t_d s+1))/(λs+1)(λs+1) )=((〖(τ〗_p s+1)(0.5t_d s+1))/(K_p (λs+1)(λs+1) ))/(((λs+1)(λs+1)+0.5t_d s-1)/(λs+1)(λs+1) )
G_c=(〖(τ〗_p s+1)(0.5t_d s+1))/(K_p [(λs+1)(λs+1)+0.5t_d s-1] )= (〖(τ〗_p s+1)(0.5t_d s+1))/(K_p s[λ^2 s+2λ+0.5t_d ] )
For the denominator the filter is ignored, therefore only K_p s(2λ+0.5t_d ) is used for the denominator and the numerator becomes 0.5τ_p t_d s^2+(τ_p+0.5t_d )s+1. So the final equation becomes:
G_c=(0.5τ_p t_d)/(K_p (2λ+0.5t_d ) ) s+((τ_p+0.5t_d ))/(K_p (2λ+0.5t_d ) )+1/(K_p (2λ+0.5t_d ) ) 1/s
Using the Padé the IMC has derived that a PID controller with a first order filter is needed.
PLOT GRAPHS AND EXPLAIN
WHAT TO USE AS λ????
REFERENCES
[1] Theodoropoulos, K. (2018). Process Control. [online] Available at: https://online.manchester.ac.uk/bbcswebdav/pid-6539422-dt-content-rid-27166454_1/courses/I3021-CHEN-30091-1181-1SE-026818/lecture03%285%29.pdf [Accessed 22 Nov. 2018].
[2] Faculty.mercer.edu. (2018). [online] Available at: http://faculty.mercer.edu/jenkins_he/documents/TuningforPIDControllers.pdf [Accessed 26 Nov. 2018].
[3] IN BOOK. ALAN LAWS BOOK
[4] Blog.opticontrols.com. (2018). Cohen-Coon Tuning Rules | Control Notes. [online] Available at: http://blog.opticontrols.com/archives/383 [Accessed 27 Nov. 2018].