MControl of a Fluidized Catalytic Cracking Unit DECEMBER 4, 2019 1 TABLE OF CONTENTS Task a ............................................................................................................................. 2 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 Task b ............................................................................................................................. 7 Demonstrate the robustness of the approximate model you obtained in (a) illustrating that it works for a range of different (not only step) inputs. Clearly illustrate, discuss and justify all your findings. ..........................................................................................................................7 Sine Function...........................................................................................................................8 Ramp Function........................................................................................................................8 Pulse Generator Function........................................................................................................9 Uniform Random Number Function .......................................................................................9 Task c ............................................................................................................................10 Design stable controllers based on your approximate system using different tuning methodologies (you will have to consult the literature for part of this task). ..........................10 Ziegler-Nichols Method (PI & PID).......................................................................................10 Cohen-Coon Method (PI & PID)...............................................................................................14 Internal Model Control With Delay Approximated By 1st Order Taylor Expansion .....................18 Internal Model Control With Delay Approximated By pade......................................................23 Summary................................................................................................................................28 Task d.............................................................................................................................29 Explain clearly which units of your block use deviation variables as inputs and outputs and which real variables, by using appropriate diagrams................................................................29 Task e.............................................................................................................................31 Choose your two best controllers. Explain why they are the best. For the closed-loop simulation of the real system with these best controllers, plot the inputs as well as Trg as a function of time. Do you think the values of the inputs and of Trg are physical/realistic? If they are not you need to retune your controllers so that they produce realistic outputs. Clearly justify your work and answers...........................................................................................................................31 Task F.............................................................................................................................33 For the best two controllers you have designed you need to find:............................................33 I. What is the range of set-points that your controller can be trusted, i.e. all inputs and outputs remain realistic and the set-point is SATISFIED?..........................................................33 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. .........38 Nomenclature.................................................................................................................46 Bibliography...................................................................................................................47 0 0 2 TASK 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. The two manipulated variables, which are present within the Fluid Catalytic Cracking Unit (FCCU), that provide an input to the system are Fs (the catalyst recycle rate) and Fa (rate of regenerator air). There are also three measured variables: the regenerator’s gas cyclone temperature (Tcy), the regenerator bed temperature (Trg) and the riser outlet temperature (T1). In order to develop the dynamic model for a system, the following procedure is followed: 1. Identify input and output variables. 2. Select an input variable and introduce a step change. 3. Select and output variable and trace its response. 4. Develop a first order model, with delay, to approximate the response. Firstly, it was necessary to induce a step change in the regenerator air rate, in order to produce a 1st order transfer function from the FCCU system. Prior to a step change, the output Tcy at steady state is showing in Figure 1a; however, after a step change of 1 is introduced at time zero (from 25.35 kg s-1 to 26.35 kg s-1), the resulting output is highlighted in Figure 1b. Figure 1a – Tcy at steady state. 0 0 3 Figure 1b - Tcy with a step change of 1 K. Firstly, it is apparent that the system reaches steady state after the step change at approximately 120 s, attaining a maximum value of 999.0 K. As well as this, there is a significant degree of fluctuation within Figure 1b, due to the noise within the system; and therefore, the first order approximated transfer function will have a degree of error associated to it. In order to approximate the first order transfer function, the below derivation must be transferred into Laplace time. Equation 1 is the first order differential, modelling the output in the time domain, with y representing the output and f(t) representing the input. Kp and 𝜏 representing the steady state gain and the process time constant respectively (both being characteristic of the system at hand). 𝜏 𝑑𝑦 𝑑𝑡 + 𝑦 = 𝐾𝑝𝑓(𝑡) (1) Equation 2 shows the conversion of the variables to deviation variables, a necessary step prior to conversion to Laplace time. y’ represents the process output and output its deviation from steady state. The same principle is equally applied to input, f(t). 𝜏 𝑑𝑦′ 𝑑𝑡 + 𝑦 ′ = 𝐾𝑝𝑓′(𝑡) (2) Between Equation 2 and 3, there has been a transfer to Laplace time. 𝜏𝑠𝑦̃(𝑠) + 𝑦̃(𝑠) = 𝐾𝑝𝑓′(𝑠) (3) 0 0 4 Equation 4 is a re-arrangement of Equation 3, in terms of output divided by input – the definition of the transfer function. 𝐺(𝑠) = 𝑦̃(𝑠) 𝑓̃(𝑠) = 𝐾𝑝 𝜏𝑠 + 1 (4) The transfer function can be better approximated accounting for the time delay, which is defined in Equation 5. 𝐺(𝑠) = 𝐾𝑝 𝜏𝑠 + 1 𝑒 −𝑡𝑑𝑠 (5) There are two methods available in which the delay can be calculated, one graphically, and the other is through calculating back after determining the gradient. However, before this occurs, the value for Kp can be found. Figure 2 highlights the change in output; necessary to determine the Kp, with the definition outlined by Equation 6. Figure 2 - Tcy with a step change of 1 K with measured output change. 𝐾𝑝 = ∆ (𝑂𝑢𝑡𝑝𝑢𝑡)𝑆𝑡𝑒𝑎𝑑𝑦 𝑆𝑡𝑎𝑡𝑒 ∆ (𝐼𝑛𝑝𝑢𝑡) 𝑆𝑡𝑒𝑎𝑑𝑦 𝑆𝑡𝑎𝑡𝑒 (6) Having changed the output for a step change of 1, Kp can be calculated, shown in Equation 7. Using the Simulink software, the output value was determined through measurement at times 0 (at steady state, prior to the step) and 650 s (at steady state after the step) to calculate the change in output. 𝐾𝑝 = 999.0 − 988.2 26.35 − 25.35 = 10.80 (7) ΔOutput (Temperature / K) 0 0 5 The graphical method is depicted in Figure 3, with points chosen to calculate the initial gradient of the approximated first order curve. This would allow extrapolation back to the yaxis origin to determine the time delay. The values chosen are illustrated within the figure. Figure 3 – Method to obtain the time delay of the system. Therefore, the gradient is calculated using the points (10.185, 988.70) and (11.829, 989.70). This is shown in Equation 8. However, after shifting the curve by 988.2 (the temperature at time 0), the points are (10.185, 0.500) and (11.829, 1.500). 𝑚 = 1.500 − 0.500 11.829 − 10.185 = 0.608 (8) At the point (10.185, 0.500), and using the previously calculated gradient, the y-intercept can be determined as 𝑐 = −5.692. Equation 9 and 10 are then used to determine the time delay, when 𝑦 = 0. 𝑦 = 0.608𝑥 − 5.692 (9) 𝑥 = 𝑡𝑑 = 5.692 0.608 = 9.363 𝑠 (10) When comparing this to the result determined graphically, in which a value of 9.615 s was determined, there is a minimal difference of 0.252 s; however, carrying forward, the previous value listed in Equation 10 will be used. After this, the value of the process time constant 𝜏𝑝 must be determined, through finding the time that is required for the system to reach 63.2 % of the new steady state, minus the time 0 0 6 delay td. The corresponding temperature of the system at T63.2% was found to be 995.026 K and is shown in Equation 11. 𝑇63.2% = 998.2 𝐾 + 0.632(999.0 − 988.2)𝐾 = 995.026 K (11) Using this value, the time at which the temperature of 995.026 K is achieved can be determined graphically. The value is found to be 27.600 s; however, there is a degree of error in reading this value, due to the noise present within the system. Following this the process time constant 𝜏𝑝 is calculated as previously described and shown in Equation 12 and 13. 𝜏𝑝 = 𝑡63.2% − 𝑡𝑑 (12) 𝜏𝑝 = 27.600𝑠 − 9.363𝑠 = 18.237𝑠 (13) The time delay is due to a lag effect within the system, in which the gas cyclone temperature does not increase, with increasing regenerator air rate, thereby proving the existence of dead time. Using the calculated values, the approximated transfer function can then be found through implementation of Equation 5. Therefore, the final approximated first order transfer function is defined in Equation 14. 𝐺(𝑠) = 10.8 18.237𝑠 + 1 𝑒 −9.363𝑠 (14) Figure 4 shows this approximated transfer function in comparison to the actual system dynamics curve. Figure 4 highlights that the calculated transfer function is a representative approximation of the actual system dynamics. Figure 4 – Approximated transfer function and real system with a step change of 1 K at t=0. 0 0 7 TASK B DEMONSTRATE THE ROBUSTNESS OF THE APPROXIMATE MODEL YOU OBTAINED IN (A) ILLUSTRATING THAT IT WORKS FOR A RANGE OF DIFFERENT (NOT ONLY STEP) INPUTS. CLEARLY ILLUSTRATE, DISCUSS AND JUSTIFY ALL YOUR FINDINGS. The definition of the robustness of the approximated model is its ability to cope with disturbances or varying system parameters (Smith, 1995). To test the robustness of the approximated model obtained in part (A) various inputs were connected to the step, these include; the sine function, pulse generator function, uniform random number function and the ramp function. The robustness of the approximated model will be verified through the addition of each of these inputs whilst monitoring the output, evaluating the similarity of the approximated model against that of the actual system dynamics. Figure 5 depicts the addition of these inputs, with each added separately as in individual investigation. Figure 5 – Simulink model used to induce different input functions. 0 0 8 SINE FUNCTION A sinusoidal wave is produced when implementing the sine function to the system. The amplitude for the test was set at a value of 1, as well as the frequency at 1 rad s-1. The bias, phase and sample time were kept at a value of 0. This ensures that the block is operating in a continuous mode. Studying Figure 6, it shows that the approximate model is a good approximation to the system dynamics. Figure 6 – Real and approximated model showing change to Tcy with time inputting sine function. RAMP FUNCTION Secondly, a test was conducted to see the robustness of the approximated model using the ramp input. A ramp of 0.5 was set, with a start time of 0 and an initial output of 0 also. The results from this test can be seen in Figure 7. The model can be seen as a good approximation within the range of 0 to 55 s; however, beyond this, there is a significant degree of deviation between the approximated model and the system dynamics, with the approximated model showing a linear increasing rate of overshoot. Figure 7 – Real and approximated model showing change to Tcy with time inputting ramp function. 0 0 9 PULSE GENERATOR FUNCTION A test was also conducted using the pulse generator function using an amplitude of 0.5, with a set period of 10 s; a pulse width of 5%; and a time delay of 0 s. Figure 8 highlights the results of this, with the output of the model generating continuous waves. Despite the fluctuation around the model, it can be considered to be a good approximation. Figure 8 – Real and approximated model showing change to Tcy with time inputting pulse generator function. UNIFORM RANDOM NUMBER FUNCTION The uniform random number input provides a randomly uniformly distributed output signal, with the output being repeatable for a given seed. The conditions set were a minimum and maximum of -1 and 1 respectively, with a sample time of 0.5 s and a seed of 0. The model is seen to be robust, as the system dynamics do not deviate from the approximated model. This is highlighted in Figure 9. Figure 9 – Real and approximated model showing change to Tcy with time inputting random number function. 0 0 10 TASK C DESIGN STABLE CONTROLLERS BASED ON YOUR APPROXIMATE SYSTEM USING DIFFERENT TUNING METHODOLOGIES (YOU WILL HAVE TO CONSULT THE LITERATURE FOR PART OF THIS TASK). ZIEGLER-NICHOLS METHOD (PI & PID) The Ziegler-Nichols tuning method is one of the oldest and most used methods for PID parameter tuning, with methods being available for closed and open loop control (Amit and Garg, 2015). Table 1 defines the Ziegler-Nichols tuning parameters (Noris, 2006). Table 1 – Ziegler-Nichols tuning parameters. 𝑲𝒄 𝝉𝒊 𝝉𝑫 P 𝜏𝑝 𝐾𝑝𝑡𝑑 = 0.1803 PI 0.9𝜏𝑝 𝐾𝑝𝑡𝑑 = 0.1623 3.3𝑡𝑑 = 30.8979 PID 1.2𝜏𝑝 𝐾𝑝𝑡𝑑 = 0.2164 2𝑡𝑑 = 18.7260 0.5𝑡𝑑 = 4.6815 However, these values must be manipulated prior to inputting into Simulink. This is described by the following Equations 15-17, with the results displayed in Table 2. 𝑃 = 𝐾𝑐 (15) 𝐼 = 𝐾𝑐 𝜏𝑖 (16) 𝐷 = 𝐾𝑐𝜏𝐷 (17) Table 2 – Ziegler-Nichols Simulink coefficients. 𝑲𝒄 𝝉𝒊 𝝉𝑫 P 0.1803 PI 0.1623 0.0053 PID 0.2164 0.0115 1.0046 0 0 11 Using the Ziegler-Nichols PI and PID results and inputting this to Simulink for a step change in the output Tcy from 988.1 K to 999.0 K for the real system, is highlighted in Figure 10. A filter value of 1 is set for the PID controller. This is due to large values of N, this will ensure that there is an ideal derivate term; however, an ideal derivative has a high gain, meaning noise will generate large fluctuations in the control signal. To ensure this does not occur, N is set to a low value chosen in this work to be 1. Figure 10 – Ziegler-Nichols PI and PID controllers for a step change from 988.1 K to 999.0 K for the approximate system. Figure 10 highlights the effect of both the PI (blue) and PID (red) controllers, as well as referencing the set-point (yellow). Both controllers are seen to have overshot the set-point of 999.0 K, prior oscillating around the value. Each oscillation, as time progresses, diminishes in amplitude, until reaching steady state. It must be noted that there is still an element of white noise affecting the system; therefore, reducing the ability of the controller to maintain an output of 999.0 K after the step change. For the PID controller, the system initially overshoots to an output value of 1007.0 K at approximately 27 s; however, it responds quicker to the step change in comparison to the PI controller, which only experiences an overshoot to the value of 1001.0 K at approximately 30 s. The time as to which both the PID and PI controller reached steady state were approximately 170 s and 140 s respectively. In order to check the stability of the controllers, both were subjected to step changes of 1K, 2K, 5K, 10K and 20K. The results can be seen in Figures 11-12. With both controllers, as the step change is increased, the overshoot of the set point also increases. Comparing the PI and PID controllers it can be seen that the PID controller overshoots to a greater degree and also experience a longer period of oscillations about the set point in comparison to the PI controllers, with the PI controllers reaching steady state in a shorter amount of time. In addition to this, further investigation was required to be carried out on the PID controller with regards to the filter coefficient, N, that can be varied. It was prudent to carry out multiple 0 0 12 simulations across a range of filter coefficient values to see if this effect corresponds with the previously mentioned hypothesis regarding the differential aspect of the controller. Values of N=1, 10 and 100 were chosen to be investigated. Figures 12-14 display the results of said simulations. Figure 11 – Ziegler-Nichols PI controller with step changes of +1 K, +2 K, +5 K, +10 K and +20 K. Figure 12 – Ziegler-Nichols PID controller with step changes of +1 K, +2 K, +5 K, +10 K, + 20K and N=1. 0 0 13 Figure 13 – Ziegler-Nichols PID controller with step changes of +1 K, +2 K, +5 K, +10 K, + 20K and N=10. Figure 14 – Ziegler-Nichols PID controller with step changes of +1 K, +2 K, +5 K, +10 K, + 20K and N=100. As the filter coefficient is increased, the following changes are identified: • The amplitude of the oscillations dramatically increases about the set point. • The number of oscillations increases with a simultaneous decrease in wavelength. The changes displayed and identified highlight the effect that changing the filter value, N, has on the system, with lower N values resulting in greater stability of the system due to the decreased number and amplitude of oscillations; therefore, proving the aforementioned predictions. 0 0 14 COHEN-COON METHOD (PI & PID) Similar to Ziegler-Nichols, Cohen-Coon developed a tuning method which is considered more complex but there is a difference in the fact that the Cohen-Coon method provides a controller with a faster rise time (Amit and Garg, 2015). The values obtained during part A, are those described in Table 3. Table 3 – Approximate first order transfer function parameters found in part A. 𝝉𝒑 𝒕𝒅 𝑲𝒑 18.237 9.363 10.8 These values were used with the equations found in Table 4 (Sen et al., 2014), as taken from literature, with the resultant values displayed below, again Equations 15-17 are required prior to inputting the data into Simulink. Table 4 – Cohen-Coon tuning parameters. Controller 𝑲𝒄 𝝉𝑰 𝝉𝑫 P 1 𝐾𝑝 𝜏𝑝 𝑡𝑑 (1 + 𝑡𝑑 3𝜏𝑝 ) - - PI 1 𝐾𝑝 𝜏𝑝 𝑡𝑑 (0.9 + 𝑡𝑑 12𝜏𝑝 ) 𝑡𝑑 30 + 3 ( 𝑡𝑑 𝜏𝑝 ) 9 + 20 ( 𝑡𝑑 𝜏𝑝 ) - PID 1 𝐾𝑝 𝜏𝑝 𝑡𝑑 ( 4 3 + 𝑡𝑑 4𝜏𝑝 ) 𝑡𝑑 32 + 6 ( 𝑡𝑑 𝜏𝑝 ) 13 + 8 ( 𝑡𝑑 𝜏𝑝 ) 𝑡𝑑 4 11 + 2 ( 𝑡𝑑 𝜏𝑝 ) Tuning for the PI controller, Proportional: 𝐾𝑐 = 1 𝐾𝑝 𝜏𝑝 𝑡𝑑 (0.9 + 𝑡𝑑 12𝜏𝑝 ) = 1 10.8 18.237 9.363 (0.9 + 9.363 12(18.237) ) = 0.1700 (18) 𝑃 = 𝐾𝑐 = 0.1700 (19) Integral: 𝜏𝐼 = 𝑡𝑑 30 + 3 ( 𝑡𝑑 𝜏𝑝 ) 9 + 20 ( 𝑡𝑑 𝜏𝑝 ) = 9.363 30 + 3 ( 9.363 18.237) 9 + 20 ( 9.363 18.237) = 15.3264 (20) 0 0 15 𝐼 = 𝐾𝑐 𝜏𝐼 = 0.1700 15.3264 = 0.0111 (21) Tuning for the PID controller Proportional: 𝐾𝑐 = 1 𝐾𝑝 𝜏𝑝 𝑡𝑑 ( 4 3 + 𝑡𝑑 4𝜏𝑝 ) = 1 10.8 18.237 9.363 ( 4 3 + 9.363 4(18.237) ) = 0.2636 (22) 𝑃 = 𝐾𝑐 = 0.2636 (23) Integral: 𝜏𝐼 = 𝑡𝑑 32 + 6 ( 𝑡𝑑 𝜏𝑝 ) 13 + 8 ( 𝑡𝑑 𝜏𝑝 ) = 9.363 32 + 6 ( 9.363 18.237) 13 + 8 ( 9.363 18.237) = 19.2000 (24) 𝐼 = 𝐾𝑐 𝜏𝐼 = 0.2636 19.2000 = 0.0137 (25) Derivative: 𝜏𝐷 = 𝑡𝑑 4 11 + 2 ( 𝑡𝑑 𝜏𝑝 ) = 9.363 4 11 + 2 ( 9.363 18.237) = 3.1140 (26) 𝐷 = 𝐾𝑐𝜏𝐷 = 0.2636(3.1140) = 0.8209 (27) Therefore, the controller settings for the PI and PID controller tuning as displayed in Table 5. Table 5 – Cohen-Coon Simulink coefficients. P I D PI Controller 0.1700 0.0111 - PID Controller 0.2636 0.0137 0.8209 These values were then inputted into Simulink in order to carry out analysis and comparisons of the controller’s effect on the real system. The flowsheet of the Cohen-Coon model can be seen in Figure 15. 0 0 16 Figure 15 – Cohen-Coon Simulink flowsheet. Figure 16 highlights the comparison between the PI and PID controllers, after using the CohenCoon model. A step change from 988.1 K to 999.0 K was tested, similar to what was previously conducted for the Ziegler-Nicholas model. The figure highlights the greatest overshoot is for the PID controller, achieving a value of 1009.0 K, with the PI controller only achieving a peak overshoot of 1005.0 K. In addition to this, the PI controller converges at the steady state setpoint at a time of approximately 151 s; however, the PID controller achieves steady state at approximately 200 s. Due to the reduced amplitude and oscillation around the set point, as well as achieving steady state faster, the PI controller is presumed to be the optimal controller for the Cohen-Coon model in comparison to the PID controller. This is a similar result to that achieved after modelling from the Ziegler-Nicholas model; however, this will be discussed in further detail later. Figure 16 – Cohen-Coon PI and PID controllers with a step change from 988.1 K to 999.0 K. 0 0 17 Secondly a test was conducted to see the response of the PI and PID controllers with varying step changes of 1 K, 2 K, 5 K, 10 K and 20 K. This is depicted in Figures 17-18. It can be seen that for both controllers, there is a subsequent increase in the overshoot with increasing step change, with the largest overshoot for both controllers being exhibited in the 20 K step change. As well as this, a longer time is required in order for the system operating with a PID controller to achieve the steady state output for all step changes. In conjunction with this, a test was conducted to understand the effect of the filter coefficient on the response of the system output, for the PID controller. N was set as 1, 10 and 100 (similar to the test conducted for the Ziegler-Nicholas method), with the noise within the system being accounted for by the film coefficient. The results for this are highlighted by Figures 18-20. Similarly, to the previous test for increasing the filter coefficient, it can be seen that the oscillations decrease in wavelength and increase in amplitude; and therefore, this reduces the stability of the system when using a PID controller, reaffirming the trend seen for ZieglerNichols and in theory. Figure 17 – Cohen-Coon PI controller with step changes of +1 K, +2 K, +5 K, +10 K and +20 K. Figure 18 - Cohen-Coon PID controller with step changes of +1 K, +2 K, +5 K, +10 K, +20 K and N=1. 0 0 18 Figure 19 - Cohen-Coon PID controller with step changes of +1 K, +2 K, +5 K, +10 K, +20 K and N=10. Figure 20 - Cohen-Coon PID controller with step changes of +1 K, +2 K, +5 K, +10 K, +20 K and N=100. INTERNAL MODEL CONTROL WITH DELAY APPROXIMATED BY 1 S T ORDER TAYLOR EXPANSION Internal Model Control (IMC) is an amalgamation of a controller and of a simulation of a process, the internal model. The internal model allows the computation of the difference between the process output and of the internal model (Zhang, 2010). Using an internal model, based on the process model, it is theoretically possible to attain perfect control of the system (Mokhatab and Poe, 2012). The delay for the process within the transfer function has previously been defined as an exponential, highlighted in Equation 28. 𝐺𝑝 (𝑠) = 𝐺̃𝑝 (𝑠) = 𝐾𝑝 𝜏𝑝𝑠 + 1 𝑒 −𝑡𝑑𝑠 (28) 0 0 19 However, using 1st order Taylor expansion, the delay can be approximated as described in Equation xx. This is necessary in order to linearise the expression, allowing for the formation of an IMC. 𝑒 −𝑡𝑑𝑠 ≈ 1 − 𝑡𝑑𝑠 (29) Therefore, the transfer function can be represented as: 𝐺𝑝 (𝑠) = 𝐺̃𝑝 (𝑠) = 𝐾𝑝 𝜏𝑝𝑠 + 1 (1 − 𝑡𝑑𝑠) (30) In order to develop the IMC with perfect representation, the inverse of the process transfer function is developed, as highlighted by Equation 31. 𝐺𝑞 (𝑠) = 𝜏𝑝𝑠 + 1 𝐾𝑝(1 − 𝑡𝑑𝑠) (31) The denominator now contains one pole, which can be considered unstable. This is due to the controller transfer function containing a positive, real pole; and therefore, this part must be removed. However, when removing this, the transfer function is no longer physically realisable, as the order of the denominator is less than that of the numerator. In order for the controller to be semi-proper, or proper, the denominator must be of equal or greater order than that of the numerator. In order to solve this problem, a filter is added, ensuring the model is now considered semi-proper. This is shown in Equation 32, with 𝜆 representing the filter tuning parameter. 𝐺𝑞 (𝑠) = 𝜏𝑝𝑠 + 1 𝐾𝑝 (𝜆𝑠 + 1) (32) Finally, the expression for 𝐺𝑐 (𝑠) can be represented by Equation 33, leading to substitution and re-arrangement to achieve the controllers transfer function. 𝐺𝑐 (𝑠) = 𝐺𝑞 1 − 𝐺𝑞𝐺̃𝑝 (33) 𝐺𝑐 (𝑠) = (𝜏𝑝𝑠 + 1) 𝐾𝑝 (𝜆𝑠 + 1) 1 − 𝐾𝑝(1 − 𝑡𝑑𝑠) (𝜏𝑝𝑠 + 1) (𝜏𝑝𝑠 + 1) 𝐾𝑝 (𝜆𝑠 + 1) = (𝜏𝑝𝑠 + 1) 𝐾𝑝 (𝜆𝑠 + 1) 1 − (1 − 𝑡𝑑𝑠) (𝜆𝑠 + 1) (34) 0 0 20 Equation 34 shows the substitution of both 𝐺𝑞 (𝑠) and 𝐺̃𝑝(𝑠) into Equation 33. The denominator of Equation 34 then needs to be homogenised to simplify the transfer function further. 𝐺𝑐 (𝑠) = (𝜏𝑝𝑠 + 1) 𝐾𝑝 (𝜆𝑠 + 1) (𝜆𝑠 + 1) (𝜆𝑠 + 1) − (1 − 𝑡𝑑𝑠) (𝜆𝑠 + 1) = (𝜏𝑝𝑠 + 1) 𝐾𝑝(𝜆𝑠 + 𝑡𝑑𝑠 + 1 − 1) (35) Therefore 𝐺𝑐 (𝑠) can finally be represented as: 𝐺𝑐 (𝑠) = 𝜏𝑝𝑠 𝐾𝑝𝑠(𝜆 + 𝑡𝑑) + 1 𝐾𝑝𝑠(𝜆 + 𝑡𝑑) (36) The transfer function for the controller is shown in two distinct parts, with the first representing the proportional aspect of the controller, and the second representing the integral. Using this, 𝐾𝑐 can be determined, as shown in Equation 37. 𝐾𝑐 = 𝜏𝑝 𝐾𝑝(𝜆 + 𝑡𝑑) (37) Subsequently, 𝜏𝑖 can be calculated through re-arranging Equation 38, as depicted in Equation 39. 𝐾𝑐 𝜏𝑖𝑠 = 𝜏𝑝 𝐾𝑝(𝜆 + 𝑡𝑑)𝑠 (38) 𝜏𝑖 = 𝐾𝑐𝐾𝑝(𝜆 + 𝑡𝑑) (39) With the value of 𝜏𝑖 , 𝜆 and 𝐾𝑐 all being unknown, the substitution of Equation 37 is needed into Equation 38 in order to remove 𝜆. This is represented by Equation 40. 𝜏𝑖 = 𝜏𝑝𝐾𝑝(𝜆 + 𝑡𝑑) 𝐾𝑝(𝜆 + 𝑡𝑑) = 𝜏𝑝 (40) With 𝐾𝑐 equal to the proportional aspect, and Equation 41 equal to the integral aspect, neither can be calculated without first determining the optimum value for 𝜆. 𝐼 = 𝐾𝑐 𝜏𝑖 (41) 0 0 21 Table 6 – IMC (delay approximated by Taylor) tuning parameters and Simulink coefficients. Table 6 shows the three values for 𝜆 that were considered (5, 10, 15) and the proportional and integral controller parameters were calculated and set within Simulink, to achieve a setpoint of 999.0 K. These values were selected because, for an ideal controller, the value of 𝜆 must be less than the process time for the controller to respond quicker. The results obtained for each of these 𝜆 values is depicted in Figure 21. For a 𝜆 value of 5, the system overshoots the desired set-point, whilst the response for a 𝜆 of 15 can be considered overdamped. Despite still having a small overshoot, the 𝜆 of 10 can be considered the optimum and will be considered for the remainder of this section. Figure 21 – IMC (delay approximated by Taylor) with filter coefficient values, N, of 5, 10 and 15. After determining the value of 𝜆, consideration into the effectiveness of using the IMC model can be conducted. Firstly, the flowsheet described in Figure 22 was set prior to analysis. λ Kc τi P I 5 0.118 18.273 0.118 0.006 10 0.087 18.273 0.087 0.005 15 0.069 18.273 0.069 0.004 0 0 22 Figure 22 – IMC (delay approximated by Taylor) flowsheet. Repeating the control loop highlighted in Figure 23, the response to a variety of step changes could be observed for the derived PI controller. The step changes that were tested include 1 K, 2 K, 5 K, 10 K and 20 K. As the step change is increased, the overshoot of the systems output also increases. Figure 23 – IMC PI controller (Taylor) with step changes of +1 K, +2 K, +5 K, +10 K, +20 K and N=100. 0 0 23 INTERNAL MODEL CONTROL WITH DELAY APPROXIMATED BY PADE The perfect controller is mathematically attainable through design based upon the inverse of the process; however, perfect control is unattainable in practicable applications due to constraints that will be outlined below. The process transfer function is described by Equation 42. 𝐺𝑝 (𝑠) = 𝐺̃𝑝 (𝑠) = 𝐾𝑝 𝜏𝑝𝑠 + 1 𝑒 −𝑡𝑑𝑠 (42) Where the delay of the process is represented by the exponential term. In order to simplify this exponential factor, the Pade approximation may be used, shown by Equation 43. 𝑒 −𝑡𝑑𝑠 ≈ −0.5𝑡𝑑𝑠 + 1 0.5𝑡𝑑𝑠 + 1 (43) Substituting this value into Equation 44, the process transfer function becomes: 𝐺𝑝 (𝑠) = 𝐺̃𝑝 (𝑠) = 𝐾𝑝 𝜏𝑝𝑠 + 1 −0.5𝑡𝑑𝑠 + 1 0.5𝑡𝑑𝑠 + 1 (44) As described a perfect controller could be produced in theory through use of the inverse of the process. 𝐺𝑞 (𝑠) = 1 𝐺̃𝑝 (𝑠) = (𝜏𝑝𝑠 + 1)(0.5𝑡𝑑𝑠 + 1) 𝐾𝑝(−0.5𝑡𝑑𝑠 + 1) (45) The first constraint is highlighted above, with a positive pole being present in the denominator. The presence of just one positive pole leads to a destabilised system and, therefore, action must be taken. As the designer of the controller, it is within our power to remove this positive pole from the equation entirely. This is provided that the remaining equation has a denominator polynomial in the controller, of which the order is higher than or equal to that of the numerator. The method of enabling this to be the case is through the addition of one or multiple filters depending on the process, as shown in Equation, into the controller. The filter tuning parameter is given as 𝜆 and is described in Equation 46. 𝑓(𝑠) = 1 𝜆𝑠 + 1 (46) Analysing Equation 45, it is seen that the numerator is second order and the denominator, following the removal of the positive pole, is zero order. Therefore, to make the denominator 0 0 24 at least of equal order, two filters must be added. The resultant controller is shown in Equation 47. 𝐺𝑞 (𝑠) = (𝜏𝑝𝑠 + 1)(0.5𝑡𝑑𝑠 + 1) 𝐾𝑝 (𝜆𝑠 + 1)(𝜆𝑠 + 1) (47) Finally, the equation for the controller transfer function is given in Equation 48. 𝐺𝑐 (𝑠) = 𝐺𝑞 1 − 𝐺̃𝑝𝐺𝑞 (48) Substituting the previously derived equations for 𝐺𝑝 (𝑠) and 𝐺𝑞 (𝑠) into Equation 48 yields Equation 49. 𝐺𝑐 (𝑠) = (𝜏𝑝𝑠 + 1)(0.5𝑡𝑑𝑠 + 1) 𝐾𝑝 (𝜆𝑠 + 1)(𝜆𝑠 + 1) 1 − 𝐾𝑝 𝜏𝑝𝑠 + 1 −0.5𝑡𝑑𝑠 + 1 0.5𝑡𝑑𝑠 + 1 . (𝜏𝑝𝑠 + 1)(0.5𝑡𝑑𝑠 + 1) 𝐾𝑝 (𝜆𝑠 + 1)(𝜆𝑠 + 1) (49) Homogenising the denominator produces Equation 50. 𝐺𝑐 (𝑠) = (𝜏𝑝𝑠 + 1)(0.5𝑡𝑑𝑠 + 1) 𝐾𝑝 (𝜆𝑠 + 1)(𝜆𝑠 + 1) (𝜆𝑠 + 1)(𝜆𝑠 + 1) − (0.5𝑡𝑑𝑠 + 1) (𝜆𝑠 + 1)(𝜆𝑠 + 1) = (𝜏𝑝𝑠 + 1)(0.5𝑡𝑑𝑠 + 1) 𝐾𝑝(𝜆 2𝑠 2 + 2𝜆𝑠 + 1 + 0.5𝑡𝑑𝑠 − 1) (50) Simplifying Equation 50 and expanding the bracket leads to Equation 51. 𝐺𝑐 (𝑠) = 0.5𝑡𝑑𝜏𝑝𝑠 2 + (𝜏𝑝 + 0.5𝑡𝑑)𝑠 + 1 𝐾𝑝 (𝜆 2𝑠 + 2𝜆 + 0.5𝑡𝑑 )𝑠 (51) Finally, the controller transfer function can be represented by Equation 52. 𝐺𝑐 (𝑠) = 0.5𝑡𝑑𝜏𝑝𝑠 2 + (𝜏𝑝 + 0.5𝑡𝑑)𝑠 + 1 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 )𝑠 ( 𝜆 2 2𝜆 + 0.5𝑡𝑑 𝑠 + 1) (52) 0 0 25 From Equation 52 the expressions 𝐾𝑐 , 𝜏𝐼 and 𝜏𝐷 can be obtained. This is done through the removal of the filter term, shown in Equation 53, separation of the fraction. 𝐹𝑖𝑙𝑡𝑒𝑟 𝑇𝑒𝑟𝑚 = ( 𝜆 2 2𝜆 + 0.5𝑡𝑑 𝑠 + 1) (53) Which leads to the values of 𝐾𝑐 , 𝜏𝐼 and 𝜏𝐷 as shown in Equations 54 and 55. 𝐾𝑐 = (𝜏𝑝 + 0.5𝑡𝑑) 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) (54) 𝐾𝑐 𝜏𝐼𝑠 = 1 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 )𝑠 (55) Therefore: 𝜏𝐼 = 𝐾𝑐𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) = (𝜏𝑝 + 0.5𝑡𝑑) 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) = 𝜏𝑝 + 0.5𝑡𝑑 (56) 𝜏𝐼 = 𝐾𝑐𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) = (𝜏𝑝 + 0.5𝑡𝑑) 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) = 𝜏𝑝 + 0.5𝑡𝑑 (57) 𝐾𝑐𝜏𝐷𝑠 = 0.5𝑡𝑑𝜏𝑝𝑠 2 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 )𝑠 (58) Therefore. 𝜏𝐷 = 0.5𝑡𝑑𝜏𝑝 𝐾𝑝𝐾𝑐 (2𝜆 + 0.5𝑡𝑑 ) (59) 𝜏𝐷 = 0.5𝑡𝑑𝜏𝑝 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) . 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) (𝜏𝑝 + 0.5𝑡𝑑) = 0.5𝑡𝑑𝜏𝑝 (𝜏𝑝 + 0.5𝑡𝑑) (60) The P, I, D aspects of the controller are calculated through the use of Equations 61-63. 𝑃 = 𝐾𝑐 = (𝜏𝑝 + 0.5𝑡𝑑) 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) (61) 0 0 26 𝐼 = 𝐾𝑐 𝜏𝐼 = 1 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) (62) 𝐷 = 𝐾𝑐𝜏𝐷 = 0.5𝑡𝑑𝜏𝑝 𝐾𝑝 (2𝜆 + 0.5𝑡𝑑 ) (63) It can be seen that each of these functions depend upon the filter tuning parameter, 𝜆, which is currently unknown. Therefore, several values of 𝜆 were inputted into Equation 53 and Equations 61-63 and across various simulations, with the results analysed to determine the correct value of 𝜆. Again, these values of 𝜆 were chosen to be less than the process time for the reasons aforementioned. Table 7 shows the values inputted into Simulink when varying the 𝜆 value, with Figure 24 denoting the system output response for each controller with a varying 𝜆 value and a constant filter coefficient of 1. For a 𝜆 of 5, the response is seen to be underdamped, with a quicker initial response but with a degree of overshoot. There is also a degree of oscillation, prior to reaching steady state, a distinct feature of an underdamped system. Therefore, the optimal 𝜆 value is seen to be a value of 10, reaching steady state faster than that for a value of 15, as well as attaining no overshoot. Consequently a 𝜆 value of 10 was used for the remainder of the analysis of the IMC (Pade) controller. Table 7 – IMC (delay approximated by Pade) Simulink and filter coefficients for a range of filter tuning parameter values. λ P I D Filter 𝝉𝒑 5 0.1445 0.0063 0.5384 1.7028 10 0.086 0.0038 0.3203 4.0516 15 0.0612 0.0027 0.2279 6.4876 Figure 24 - IMC (delay approximated by Pade) with 𝜆 values of 5, 10 and 15. 0 0 27 Similar to the studies conducted previously, the system response for a system including an IMC controller with delay approximated by Pade, was tested for different step changes, ranging from 1 K, 2 K, 5 K, 10 K and 20 K. This is depicted in Figures 25a, Figures 25b and Figures 25c, showing the change in response for varying filter values: 1, 10 and 100. Firstly, it can be seen that as the step change is increased, there is little difference in the response of the system prior to achieving steady state after the step change, with the time taken to achieve steady state remaining approximately constant, as well as seeing little to no oscillation in the system response for the IMC controller. Secondly, as the filter coefficient, N is increased from 1 to 100, there is also little change in response, with only the larger step changes seeing some degree of oscillation prior to achieving steady state; however, there is no overshoot of the set-point for this IMC controller. Figure 25a - IMC PI controller (Pade) with step changes of +1 K, +2 K, +5 K, +10 K, +20 K and N=1 . Figure 25b - IMC PI controller (Pade) with step changes of +1 K, +2 K, +5 K, +10 K, +20 K and N=10 . 0 0 28 Figure 25c - IMC PI controller (Pade) with step changes of +1 K, +2 K, +5 K, +10 K, +20 K and N=100. SUMMARY In conclusion, Figure 26 shows the system response for each of the four methods used for a step change from 998.1 K to 999.0 K at t=0, with the best controller chosen for both the Cohen-Coon and Ziegler-Nichols methods (PI Controllers). For the PID controller described for the IMC Pade method, a filter coefficient of 1 was used, despite the result varying minimally when the filter coefficient was changed between 1 and 100. It can be seen from Figure 26, that both the Ziegler- Nichols and Cohen-Coon methods have the largest overshoots, with both appearing to possess the largest degree of oscillation around the set-point as well. Choosing between these two controllers, it is apparent that the Ziegler-Nichols PI controller is the best, due to it possessing an overshoot approximately 4 K lesser than the Cohen-Coon PI controller, as well as achieving steady state after less time. However, neither of these controllers compare to the two IMC controllers. Despite the IMC Taylor PI controller showing some degree of overshoot, it is significantly less than that for the Ziegler-Nichols and Cohen-Coon PI controllers. When comparing the two IMC methods, it can be seen that the Pade controller is the optimal, as despite both achieving steady state at approximately the same time, the Pade controller possesses no overshoot. 0 0 29 Figure 26 – Response in Tcy for each of the controllers with a step change from 998.1 K and 999.0 K when N=1. TASK D EXPLAIN CLEARLY WHICH UNITS OF YOUR BLOCK USE DEVIATION VARIABLES AS INPUTS AND OUTPUTS AND WHICH REAL VARIABLES, BY USING APPROPRIATE DIAGRAMS. Within a process, values fluctuate about a set-point value across a period of time. In order to ensure that the set-point is maintained, it must be guaranteed that the fluctuations about the setpoint are compensated and accounted for. It requires less computation to calculate the change required to maintain the set-point rather than tracking the total value of the variable. Therefore, during control of systems, it is typical to convert variables to deviation variables (also referred to as perturbation variables), observing the difference between the current actual value and that of the steady state set point value. Prior to this conversion the variables are defined as real process variables. Equation 64 depicts the previously stated point and defines the deviation variable. 𝑦 ′ (𝑡) = 𝑦(𝑡) − 𝑦𝑠 (0) (64) Where 𝑦′(𝑡) represents the deviation variable, 𝑦(𝑡) represents the variables actual value at time t and 𝑦𝑠 (0) represents the steady state value at time 0. A transformation of variables into deviation variables is a necessity when converting from time domain to the Laplace domain. This greatly simplifies process control, with a deviation value of 0 desired as this shows that there is no deviation about the set-point. 0 0 30 Figure 27 – PID controller for the approximate model; pink represents deviation variables and blue represents real variables. Figure 27 depicts the approximated system highlighting the deviation variables (magenta) and the real variables (blue). In Figure 28, Fa and Fs are the real measurable inputs with Tcy, T1 and Trg the measurable outputs also considered to be real. The Laplace transfer function is used to link the input Fa to the output Tcy with deviation variables. Figure 28 – PID controller for real system; pink represents deviation variables and blue represents real variables. For the simulation of the FCCU, deviation forms of the regenerator air rate and regenerator gas cyclone temperature are formed as shown in Equations 65 and 66 respectively. 𝐹𝑎 ′ = 𝐹𝑎 − 𝐹𝑎𝑠 (65) 𝑇𝑐𝑦 ′ = 𝑇𝑐𝑦 − 𝑇𝑐𝑦𝑠 (66) The difference between the real and desired Tcy is calculated by the comparator, the result of which is called the error, also a deviation variable, and this is sent to the controller. Dependent upon the value received, the controller can make a decision of whether to manipulate the value of Fa. The constant initial value of Fa, set as 25.35 kg s-1, which is received by the comparator is a real value; however, the input from the controller is a deviation variable. The two values are summed together to produce a new Fa value, a real variable, which is then used to produce the desired output, also all real. 0 0 31 TASK E CHOOSE YOUR TWO BEST CONTROLLERS. EXPLAIN WHY THEY ARE THE BEST. FOR THE CLOSED-LOOP SIMULATION OF THE REAL SYSTEM WITH THESE BEST CONTROLLERS, PLOT THE INPUTS AS WELL AS TRG AS A FUNCTION OF TIME. DO YOU THINK THE VALUES OF THE INPUTS AND OF TRG ARE PHYSICAL/REALISTIC? IF THEY ARE NOT YOU NEED TO RETUNE YOUR CONTROLLERS SO THAT THEY PRODUCE REALISTIC OUTPUTS. CLEARLY JUSTIFY YOUR WORK AND ANSWERS. As aforementioned, the two best controllers designed during Task C, for a closed loop system, were the IMC Controllers with the delay approximated using the Taylor expansion and Pade methods. This was due to the Ziegler-Nicholas and Cohen-Coon PI and PID controllers both showing a large overshoot in their initial responses, as well as achieving steady state slower than the IMC controllers. In addition to this, when using a PID controller, the response was greatly affected by the filter coefficient – as the coefficient was increased, the wavelength and amplitude of the oscillations increased around the set-point, reducing system stability. This was not the case with the IMC controllers, with both showing little variation with increasing filter coefficient. For analysis of the IMC Controller with delay approximated by Taylor expansion, the step change chosen for this simulation was 20 K, occurring at a time of 10 seconds, setting an initial temperature of Tcy as 988.1 K with a final output of 1008.1 K, with both the IMC controllers with delays approximated by Taylor expansion and Pade modelled for their inputs; step change of Tcy, Fa and Fs, and outputs; Tcy and Trg. Further to this the error for each of the controllers was also modelled. The inputs and outputs to the system are shown in Figures 29 and 30, on different graphs, such that the y-axis is correctly scaled to read the results. Figure 29 – Inputs to the system using IMC (Taylor) with a step change from 998.1 K to 1008.1 K. 0 0 32 The step change in the output Tcy is from 998.1 to 1008.1, a 20 K increase. The output for Tcy in Figure 30 shows this, with a slight overshoot of approximately 1 K before achieving steady state after approximately 81 s. As the temperature of Tcy tends toward steady state after the step change, there is a decrease in the error, labelled in Figure 29. This is due to a slight increase in Fa (the hot air flowrate) ensuring that the temperature the regenerator can increase. Subsequently, this causes an increase in Trg, highlighted in Figure 30 – a larger increase occurring in Trg in comparison to Tcy. However, the increasing temperatures within the regenerator also cause an increase in the temperature in the reactor, as T1 increases from a value of 776.7 K prior to the step change to 796.8 K after the step change; however, there is no effect to the input Fs, as it maintains a value of 294 kg s-1 Figure 30 – Outputs to the system using IMC (Pade) with a step change from 998.1 K to 1008.1 K. Similarly, the analysis was conducted for the IMC controller with delay approximated by the Pade method. The same step change of 20 K was chosen, with the change occurring at 10 s to test the effect of changing Tcy on the other inputs and outputs. The results are shown in Figure 31 for inputs and Figure 32 for outputs. The same trend as for the Taylor expansion PI controller is seen in this scenario, with all three outputs seeing an increase in temperature, after the 20 K step change in Tcy. The input Fa is increased to facilitate this increase in Tcy; however, the PID controller ensures there is an overshoot of hot air flowrate into the system, as depicted in Figure 31. This did not occur in the prior case, as instead there was an overshoot in the output Tcy. In addition to this, the time taken to reach steady state for Tcy after the step change is decreased for the PID controller to approximately 71 s – a quicker response using the Pade PID controller than the PI Taylor controller. Finally, similar to the previous case, there is an increase in the temperature T1, with no increase or decrease in the Fs input. 0 0 33 Figure 31 - Inputs to the system using IMC (Pade) with a step change from 998.1 K to 1008.1 K. Figure 32 - Outputs to the system using IMC (Pade) with a step change from 998.1 K to 1008.1 K. These results can be considered realistic for the two IMC controllers. In theory, in a FCCU the regenerator gas temperature is greater than the bed temperature. This is due to a temperature variation existing in the vertical section of the regenerator vessel. The highest and lowest values of these temperatures can be found at the flue gas line and the regenerator bed respectively. Therefore, Tcy is greater than or equal to Trg. TASK F FOR THE BEST TWO CONTROLLERS YOU HAVE DESIGNED YOU NEED TO FIND: I. WHAT IS THE RANGE OF SET-POINTS THAT YOUR CONTROLLER CAN BE TRUSTED, I.E. ALL INPUTS AND OUTPUTS REMAIN REALISTIC AND THE SET -POINT IS SATISFIED? 0 0 34 The limiting set points of the controller are determined by the physical limitations of the system, in terms of equipment capacity, and flowrates not being able to attain negative values. For example, the IMC controllers are limited by the inability for the hot air flowrate to go below zero; and therefore, there will be a minimum achievable temperature that this controller can realise. In order to attain the maximum value for the IMC Controller with delay approximated by Taylor expansion, a step change from 988.1 K to 1500 K was induced at a time of 10 s (this value could be increased if the Tcy temperature was able to attain this set-point and achieve steady state). The result for this is described in Figure 33 and Figure 34 showing the inputs and outputs respectively. Figure 33 – Inputs to system using IMC (Taylor) with a step change from 988.1 K to 1500.0 K at a time of t=10s. As the temperature Tcy increases, to meet the new set-point, the input Fa increases, meaning a larger flow of hot air enters the regenerator to increase both Tcy and Trg. Despite the flowrate of hot air increasing beyond 1000 kg s-1, the temperature Tcy is maximised at a value of 1206 K. Additionally, the temperature Trg also increases, surpassing Tcy and maximising at a value of 1363 K after approximately 300 s. 0 0 35 Figure 34 - Outputs to system using IMC (Taylor) with a step change from 988.1 K to 1500.0 K at a time of t=10s. Equally, the minimum temperature for Tcy for this controller can be determined, through decreasing the set-point until Fa equates to 0 kg s-1. Any value beyond this will be negative flow and this is not physically possible, as aforementioned. The results of this study are shown in Figures 35 and 36 for both the inputs and outputs respectively. The set-point at which the flowrate of hot air equates to zero; and consequently, the minimum attainable temperature of Tcy is found to be 803.2 K. This is represented in Figure 35 for inputs and Figure 36 for outputs, with Figure 37 showing clearly that the value for input Fa does not reduce below 0. Figure 35 - Inputs to system using IMC (Taylor) with a step change from 988.1 K to 803.2 K at a time of t=10s. 0 0 36 Figure 36 - Outputs to system using IMC (Taylor) with a step change from 988.1 K to 803.2 K at a time of t=10s. Figure 37 – Magnified input data showing Fa does not fall below zero. This analysis is conducted in the same manner for the IMC controller with delay approximated using the Pade method; and therefore, the minimum and maximum temperatures achieved were 778.0 and 1206.0 K. This is similarly shown in Figures 38-41. 0 0 37 Fi gurFigure 38 - Inputs to system using IMC (Pade) with a step change from 988.1 K to 1500.0 K at a time of t=10s. Fi gurFigure 39 - Inputs to system using IMC (Pade) with a step change from 988.1 K to 1500.0 K at a time of t=10s. Figure 40 - Inputs to system using IMC (Pade) with a step change from 988.1 K to 778.0 K at a time of t=10s. 0 0 38 Figure 41 - Outputs to system using IMC (Pade) with a step change from 988.1 K to 778.0 K at a time of t=10s. 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. An initial test was set up to understand the effect of disturbances on the cyclone temperature, Tcy when set at a value of 988.1 K with no step change. The input used to create the disturbance was the ‘band limited white noise’ set for values of 1, 5, 10, 25, 50 and 100 for both the PI (Taylor) and PID (Pade) controllers. It must be noted the PID controller was set with a filter coefficient value of 1. The results for this are shown in Figures 42-47. Figure 42 – PI and PID controller output responses for Tcy and set point 988.1 K for a noise power of 1. 0 0 39 Figure 43 – PI and PID controller output responses for Tcy and set point 988.1 K for a noise power of 5. Figure 44 – PI and PID controller output responses for Tcy and set point 988.1 K for a noise power of 10. Figure 45 – PI and PID controller output responses for Tcy and set point 988.1 K for a noise power of 25. 0 0 40 Figure 46 – PI and PID controller output responses for Tcy and set point 988.1 K for a noise power of 50. Figure 47 – PI and PID controller output responses for Tcy and set point 988.1 K for a noise power of 50. In summary, the results for increasing the noise power and the resulting effect on Tcy can be summarised by Table 8. Table 8 – Oscillations and deviation for Tcy from set point when varying the noise power. Noise Power Oscillation Due to Noise Around SetPoint 988.1 K % Deviation From SetPoint 1 ±0.6 K 0.06% 5 ±0.8 K 0.08% 10 ±1.0 K 0.10% 25 ±1.4 K 0.14% 50 ±1.6 K 0.16% 100 ±2.5 K 0.25% 0 0 41 As the noise power increases, so did the amplitude of the oscillations around the set-point: however, despite seeing this increase in % deviation from the set-point, a noise power of 100 only has a deviation of 0.25 % - a minimal value. In addition to this, the effect of increasing noise power can be examined for the output Fs. For a constant Fs at a set-point of 294 kg s-1, the results can be seen in Figures 48-53. It can be seen that as the noise power increases from a factor of 1 to a factor of 100, the oscillations increase from ± 11 kg s-1 to ± 121 kg s-1. Even considering these fluctuations, the values produced are still physically possible, with Fs all being a positive value. However; in reality operations exhibiting such fluctuations would not be feasible. A pump would not be able to sustain operation with such rapid deviations; a typical pump would alter the flow passing through it through the use of a VSD, requiring a large heat sink for excess energy, and as such the rapid deviations could lead to overheating and subsequent maintenance issues (Brumbach and Clade, 2017). Figure 48 – Changing response of input Fs, for a set-point of 294 kg s-1, with noise factor of 1. Figure 49 – Changing response of input Fs, for a set-point of 294 kg s-1, with noise factor of 5. 0 0 42 Figure 50 – Changing response of input Fs, for a set-point of 294 kg s-1, with noise factor of 10. Figure 51 – Changing response of input Fs, for a set-point of 294 kg s-1, with noise factor of 25. Figure 52 – Changing response of input Fs, for a set-point of 294 kg s-1, with noise factor of 50. 0 0 43 Figure 53 – Changing response of input Fs, for a set-point of 294 kg s-1, with noise factor of 100. Table 9 – Oscillations and deviation for Fs from set point when varying the noise power. Noise Power Oscillation Due to Noise Around SetPoin 294 kg s-1 % Deviation From SetPoint 1 ± 11 3.74% 5 ± 27 9.18% 10 ± 39 13.27% 25 ± 63 21.43% 50 ± 87 29.59% 100 ± 121 41.16% The noise power was increased to the point at which the result is no longer physically feasible as the flowrate drops below zero. This is highlighted in Figure 54 for a noise power of 657. 0 0 44 Figure 54 - Changing response of input Fs, for a set-point of 294 kg s-1, with noise factor of 657. Similar to the work previously done to Fs, a noise block was introduced to understand the effect of the disturbance of Fa, at a set-point value of 988.1 K. The flowsheet for this is described in Figure 55. Figure 55 – Flowsheet used for varying noise input of Fa. 0 0 45 When noise is introduced to the input Fa, the noise cannot be increased above 5, due to the input decreasing below a flowrate of 0. This result is shown in Figure 56, as well as showing the response for a noise power of 1 in Figure 57. Figure 56 - Changing response of input Fa, for a set-point of 25.35 kg s-1, with noise factor of 5. Figure 57 - Changing response of input Fa, for a set-point of 25.35 kg s-1, with noise factor of 1. The resulting effect on the outputs Tcy and Trg for a noise factor of 5, at a set-point value for Fa of 25.35 kg s-1. This was conducted for both the PI and PID IMC controllers and is described in Figure 58. 0 0 46 Figure 57 - Changing response output Tcy and Trg for PI and PID IMC controllers with a noise factor of 5. NOMENCLATURE Symbol Definition Units 𝑭𝒔 Catalyst Recycle Rate 𝑘𝑔 𝑠 −1 𝐹𝑎 Regenerator Air Rate 𝑘𝑔 𝑠 −1 𝑇𝑐𝑦 Gas Cyclone Temperature K 𝑇𝑟𝑔 Regenerator Bed Temperature K 𝑇1 Riser Outlet Temperature K 𝜏 Process Time Constant s 𝐾𝑝 Steady State Gain Dimensionless 𝑡𝑑 Time delay S 𝐾𝑐 Controller Gain Dimensionless 𝐺𝑝 Process Transfer Function Dimensionless 𝐺𝑐 Controller Transfer Function Dimensionless 𝜆 Filter Parameter Dimensionless N Filter Coefficient Dimensionless 0 0 47 BIBLIOGRAPHY Amit, K. and Garg, K. . (2015) ‘Comparison of Ziegler-Nichols , Cohen-Coon and Fuzzy Logic Controllers for Heat Exchanger Model’, International Journal of Science, Engineering and Technology Research (IJSETR), 4(6), pp. 1917–1920. Brumbach, M. and Clade, J. (2017) Electronic Variable Speed Drives. 4th Editio. Boston: Cengage Learning. Mokhatab, S. and Poe, W. A. (2012) ‘Process Control Fundamentals’, Handbook of Natural Gas Transmission and Processing, pp. 473–509. doi: 10.1016/b978-0-12-386914-2.00014-5. Noris, F. B. M. (2006) Comparison Between Ziegler-Nichols and Cohen-Coon Method for Controller Tunings. University College of Engineering & Technology Malaysia. Sen, Rajat et al. (2014) ‘Comparison Between Three Tuning Methods of PID Control for High Precision Positioning Stage’, Mapan - Journal of Metrology Society of India, 30(1), pp. 65–70. doi: 10.1007/s12647-014- 0123-z. Smith, R. S. (1995) ‘Model validation for robust control: an experimental process control application’, Automatica, 31(11), pp. 1637–1647. doi: 10.1016/0005-1098(95)00093-C. Zhang, P. (2010) Industrial control system simulation routines. First Edit, Advanced Industrial Control Technology. First Edit. Peng Zhang. doi: 10.1016/b978-1-4377-7807-6.10019-1.