User:Smzoha

''''Shams closed-loop tuning method''''

Abstract—

The objective of this study is to develop a new online controller tuning method in closed-loop mode. The proposed closed-loop tuning method overcomes the shortcoming of the well-known Ziegler-Nichols (1942) continuous cycling method and it can be an alternative for the same. This is a simple method to obtain the PI/PID setting which gives the acceptable performance and robustness for a broad range of the processes. The method requires closed-loop step setpoint experiment using a proportional only controller with gain Kc0. Based on simulations for a range of first-order with time delay processes, simple correlations have been derived to give PI/PID controller settings. The controller gain (Kc/Kc0) is only a function of the overshoot observed in the setpoint experiment. The controller integral and derivative time (τI and τD) is mainly a function of the time to reach the first peak (tp). The simulation has been conducted for a broad class of stable and integrating processes, and the results are compared with recently published paper of Shamsuzzoha and Skogestad (2010). The proposed tuning method gives consistently better performance and robustness for broad class of processes.

Introduction

The proportional, integral and derivative (PID) controller is widely used in the process industries due to its simplicity, robustness and wide ranges of applicability in the regulatory control layer. A recent survey of Desborough and Miller [1] reported that more than 97% of the regulatory controllers utilize the PI/PID algorithm. Although the PI/PID controller has only few adjustable parameters, they are difficult to be tuned properly in real processes. One reason is that tedious plant tests are required to obtain improved controller setting. Due to this reason, finding a simple PI/PID tuning approach with a significant performance improvement has been an important research issue for process engineers. Therefore, the objective of this brief paper is to develop a method that should be simple thumbnail r with enhanced performance in closed-loop mode. There are varieties of controller tuning approach and among those two are widely used for the controller tuning. It can be based on either open-loop or closed-loop plant tests. Most tuning approaches are based on open-loop plant information; typically the plant’s gain (k), time constant (τ) and time delay (θ). The direct synthesis (Seborg et al. [2];) and IMC based PID (Shamsuzzoha and Lee [3, 4]) tuning method are very popular among them. However, these approaches require that one first obtains an open-loop model of the process and then tuning of the control-loop. There are two problems here. First, an open-loop experiment, for example a step-test, is normally needed to get the required process data. This may be time consuming and may upset the process and even lead to process runaway. Second, approximations are involved in obtaining the process parameters (e.g., k, τ and θ) from the data. The alternative of the open-loop approach is a two-step tuning procedure based on closed-loop setpoint experiment with a P-controller. It was originally proposed by Yuwana and Seborg [5]. They identified a first-order with delay model by matching the closed-loop setpoint response with a standard oscillating second-order step response. In next step for the controller setting they used the Ziegler-Nichols [6] tuning rules, which may give aggressive setting but one can use other tuning rule. It is important to note that often it is difficult to carryout open-loop test, and there are always possibility that control variable may drift away and operator needs to intervene in order to prevent products qualities from off-specification. In case of closed-loop test, one can easily keep control on the process during experiment and reduces the effect of disturbance to process operation. The other alternative approach of the above two-steps procedure is to use closed-loop experiments. One very popular approach is the classical method of Ziegler-Nichols [6] which requires very little information about the process; namely, the ultimate controller gain (Ku) and the period of oscillations (Pu) which are obtained from a single experiment. However, there are several disadvantages. First, the system needs to be brought to its limit of instability and a number of trials may be needed to bring the system to this point. Not that if we try to save time by making large adjustments in the search for the Ku, it becomes much more likely that we will actually go unstable, at least for a brief period. Second disadvantage is that the Ziegler-Nichols [6] tunings do not work well on all processes. It is well known that the recommended settings are quite aggressive for lag-dominant (integrating) processes (Tyreus and Luyben [7]) and quite slow for delay-dominant process (Skogestad [8]). To get better robustness for the lag-dominant (integrating) processes, Tyreus and Luyben [7] proposed to use less aggressive settings (Kc=0.313Ku and τI=2.2Pu), but this makes the response even slower for delay-dominant processes (Skogestad [8]). A third disadvantage of the Ziegler-Nichols method is that it can only be used on processes for which the phase lag exceeds -180 degrees at high frequencies. For example, it does not work on a simple second-order process. Recently Shamsuzzoha and Skogestad [9, 10] have developed a new online controller tuning method in closed-loop mode. This closed-loop tuning method overcomes the shortcoming of the well-known Ziegler-Nichols continuous cycling method and gives consistently better performance and robustness for broad class of processes. The PI/PID controller design method has been discussed extensively in the literature and it shows that most of the tuning method is based on the two steps procedure. First step is to find the process parameters (e.g., k, τ and θ) by using an open-loop or closed-loop test. Second step is to use suitable tuning method to obtain the PI/PID controller setting. Therefore, the goal of the present study is to find simple and direct controller tuning method in closed-loop for the broad class of the processes. No detail prior information of the plant (process parameters k, θ and τ) is required to obtain the robust controller setting from the closed-loop setpoint experiment.

2. Objectives•

       'Method should be in closed-loop mode.' 

• The PI/PID tuning rule should be simple, analytically derived and applicable to different types of process with a wide range of process parameters in a unified framework. • Remove the shortcoming of the Ziegler-Nichols continuous cycling method. • It should be applicable to the wide range of the overshoot (approximately 10-60%) with the initial controller gain Kc0.

3. Shams Closed-Loop Tuning Method

The proposed procedure is as follows: 1. Switch the controller to P-only mode (for example, increase the integral time τI to its maximum value or set the integral gain KI to zero). In an industrial system, with bumpless transfer, the switch should not upset the process. 2. Make a setpoint change that gives an overshoot between 0.10 (10%) and 0.60 (60%); about 0.30 (30%) is a good value. Record the controller gain Kc0 used in the experiment. Most likely, unless the original controller was quite tightly tuned, one will need to increase the controller gain to get a sufficiently large overshoot. Note that small overshoots (less than 0.10) are not considered because it is difficult in practice to obtain from experimental data accurate values of the overshoot and peak time if the overshoot is too small. Also, large overshoots (larger than about 0.6) give a long settling time and require more excessive input changes. For these reasons we recommend using an “intermediate” overshoot of about 0.3 (30%) for the closed-loop setpoint experiment. 3. From the closed-loop setpoint response experiment, obtain the following values (see Figure 1): • Controller gain, Kc0 • Overshoot = (Δyp - Δy∞) /Δy∞ • Time from setpoint change to reach peak output (overshoot), tp • Relative steady state output change, b = Δy∞/Δys.

The output variable changes are given as: Setpoint change  : = ys – y0 Peak output change (at time tp)  : = yp – y0 Steady-state output change after setpoint step test  : = y∞ - y0 To find Δy∞ one needs to wait for the response to settle, which may take some time if the overshoot is relatively large (typically larger than 0.4). In such cases, one may stop the experiment when the setpoint response reaches its first minimum and record the corresponding output, Δyu. Δy∞ = 0.45(Δyp + Δyu) (1)

4. Selection of Proportional Controller Gain (Kc0)

It is mentioned earlier that the proposed method is valid for the overshoot between 0.1 to 0.6. However, an overshoot of around 0.3 is recommended for a better response. Sometimes achieving the P-controller gain (Kc0) via trial and error that gives the overshoot around 0.3 can be time consuming. Therefore, one can use direct equation for the gain of the next closed-loop test

(2)

Note: It is not so important to achieve the precise fractional overshoot of 0.3, so few trial is sufficient to get the desire overshoot around 0.3 from above Eq.(2).

Figure 1. Closed-loop step setpoint response with P-only control 5. Summary of the Shams Closed-Loop Tuning Method A simple approach has been developed for PI/PID controller tuning by the closed-loop setpoint step experiment using a P-controller with gain Kc0. The PI/PID-controller settings are obtained directly from following three data from the setpoint experiment (see Figure 1): • Controller gain, Kc0 • Overshoot = (Δyp - Δy∞) /Δy∞ • Time from setpoint change to reach peak output (overshoot), tp • Relative steady state output change, b = Δy∞/Δys.

If one does not want to wait for the system to reach steady state and speed-up the closed-loop experiment, it is recommended to use the estimate Δy∞ = 0.45(Δyp + Δyu). In conclusion, the final tuning formula for the proposed “Shams closed-loop tuning method” is summarized as:

where, A=[1.55(overshoot)2 -2.159 (overshoot)+1.35] F is a detuning parameter. F=1 gives the “fast and robust” PI/PID settings corresponding to τc=θ. To detune the response and get more robustness one can selects F>1, but in special cases one may select F<1 to speed up the closed-loop response. An overshoot of around 0.3 is recommended for the better response in the Shams method. The initial controller gain (Kc01) which gives overshoot around 0.3 in the closed-loop test can be obtained from equation below:

The Shams method works well for a wide variety of the processes typical for process control applications, including the standard first-order plus delay processes as well as integrating, high-order, inverse response, unstable and oscillating process. 6. Simulation Study To show the effectiveness of the Shams method three different cases of the simulation are shown below, which covers wide range of the processes. The simulations illustrated in figures are for the overshoot around 0.3 is compared with the setpoint overshoot method. Examples: E5: E8: E24: Figure 2-4 present a comparison of the proposed method by introducing a unit step change in the set-point and an unit step change of load disturbance at plant input. It is clear from these three figures that the proposed method constantly gives better closed-loop response for several type of processes. The Shams method has been compared with setpint overshoot method and results show that it has significant performance improvements in all the cases for the disturbance rejection while maintaining setpoint performance.

Figure 2. Responses of high-order process , Setpoint change at t=0; load disturbance of magnitude 1 at t=10.

Figure 3. Responses of third-order integrating process , Setpoint change at t=0; load disturbance of magnitude 1 at t=100.

Figure 4. Responses of integrating process with time delay , Setpoint change at t=0; load disturbance of magnitude 1 at t=50. 7. Application to the Distillation column The case study demonstrates the application of the Shams tuning method in the distillation column temperature control loop. The dynamic model of the distillation column in Aspen-Hysys® is selected from Luyben [11] to show the simplicity and effectiveness of the proposed method. The depropanizer column considered in this case study produces a distillate product that is 98 mole% propane. At 110˚F the vapor pressure of propane is slightly higher than 200psia. Therefore, an operating pressure of 200 psia is kept in the condenser. The boiler pressure is estimated by assuming a pressure drop over each tray of 5 inches of liquid in this high-pressure column. The liquid density of this hydrocarbon system is about 30lb/ft3. The column has 30 trays and is fed on tray 15, and the pressure in the reboiler is 202.6 psia. The column is feed 100 lb-mol/hr of a mixture of propane (30 mol%), isobutene (40 mol%) and n-butane (30 mol%) at 90˚F. The specified purity of distillate is 98 mol% propane. The specified impurity of propane in the bottoms is 1.0 mol%. The design reflux ratio is 3.22 and the design reboiler heat input is 1.02×106 Btu/hr. Luyben [11] suggested Reflux-Vapor Boilup (RV) control structure of the depropanizer and is shown in Figure 5. The suggested tuning parameters of the different loops are kept unchanged except temperature loop. The flow controller has Kc=0.5, τI=0.3 minutes, and two level controllers Kc=2.0. The pressure controller is tuned using normal slow setting with Kc=1.0 and the integral time is τI=20.0 minutes. For the temperature loop, Luyben [11] applied relay-feedback test and found ultimate gain (Ku=32) and the ultimate period (Pu=7.3 minutes). Finally he obtained the PI setting using the TL [7] method as Kc=10.0 and τI=16.0 minutes. In the proposed method, overshoot around 0.30 gives satisfactory performance and robustness. Start the test in closed-loop using a P-controller with gain Kc0. The magnitude of the gain Kc0 should be selected such that it gives overshoot around 0.30 for a setpoint change of magnitude Δysp. From the setpoint experiment, read off the maximum response, yp, the steady state response y∞, and the time to reach the first peak (tp). It is assume that the process output has value y0 before the setpoint change occur. Step test in temperature loop is shown in Figure 6. Process output before the setpoint change (y0) = 125.7˚F, and manipulated variable (OP) = 50.60%, A step test is conducted for setpoint change (Δys)= ys – y0=130.7-125.7=5.0, with the P-controller of Kc0= 8. Note: It is important to eliminated the impact of the integral action in the step test and for that substitute τI =1000 (sufficiently large value). Based on the closed-loop setpoint response to a step changes of amplitude ∆ys =5oF as shown in Figure 6, the overshoot and other parameters are calculated as

The relative steady-state change of the process output is

It shows that process is almost integrating and the value of peak time tp=107.83-100.0=7.83 minutes. The PID parameter settings can be calculated as A=1.55(OS)2 – 2.159(OS)+1.35= 1.55(0.334)2 -2.159(0.334)+ 1.35=0.801

For the integral time, τI

τD=0.14*tp=0.14*7.83=1.10 minutes

The effectiveness of the proposed method has been checked for the setpoint change in the temperature loop and closed-response is shown in Figure 7. The response is significantly fast and smooth without any oscillation. The proposed closed-loop method has been also tested for the disturbance rejection. The results for two disturbances in feed flowrate are shown in Figure 5. At 15 minutes the feed is increase from 100 to 120lb-mol/hr and at 120 minutes a large change in the feed flowrate is made, and is finally dropped to 80 lb-mol/hr. Figure 8 clearly shows the advantage of the proposed method for the disturbance rejection. It gives smooth and fast disturbance rejection with sufficiently less control effort.

Figure 5. Depropanizer column flowsheet with controllers installed, pressure controller is not shown in main flowsheet, and it is installed in sub-flowsheet.

Figure 6. The closed-loop responses with a P-controller (controller gain Kc0 = 8.0) of a depropanizer temperature loop.

Figure 7. The closed-loop setpoint responses of the depropanizer temperature loop with a PID-controller, setpoint change of magnitude +5˚F at t=100 minutes; reverse setpoint change of magnitude -5˚F at t=150 minutes.

Figure 8. Closed-loop response for step changes in feed flow rate as a disturbance at t=100 minutes from 100 to 120 lb-mol/hr, at 200 minutes from 120 to 80 lb-mol/hr. 8. Conclusions Shams closed-loop method works well for a wide variety of the processes typical for process control, including the standard first-order plus delay processes as well as integrating, high-order, inverse response, unstable and oscillating process. References

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