Lecture 5: Controller Tuning PDF

Lecture 5: Controller Tuning Control Systems – ENGI 24495 Outline of Presentation Tuning methods: − Trial-and-error − Ziegler-Nichols continuous cycling − Ziegler-Nichols reaction curve − Autotuning [email protected] Lecture 5 – Controller Tuning...

Lecture 5: Controller Tuning Control Systems – ENGI 24495 Outline of Presentation Tuning methods: − Trial-and-error − Ziegler-Nichols continuous cycling − Ziegler-Nichols reaction curve − Autotuning [email protected] Lecture 5 – Controller Tuning Page 1 of 21 PID Controller Controller SP error 𝐾" CV + 𝐾! + + 𝑇# 𝑠 Actuator Process - 𝑠 FB Sensor PID controllers are everywhere! Due to its simplicity and optimal performance in many applications. They can be tuned by operators without extensive background in controls. Tuning is the procedure of adjusting the controller parameters i.e., K ! , 𝐾" and 𝑇#. Before tuning, control system is “sluggish” (reacts slowly to disturbances), but after tuning, the system becomes “responsive” (reacts quickly to disturbances) [email protected] Lecture 5 – Controller Tuning Page 2 of 21 Unit-Step Response of a Closed-Loop System 1. Rise Time: the time it takes for the process variable to rise to 90% of the setpoint. 2. Overshoot: how much the peak value is higher than the steady state, normalized against the steady state. 3. Settling Time: the time it takes for the system to converge to its steady state. 4. Steady-state Error: the difference between the steady-state output and desired output. [email protected] Lecture 5 – Controller Tuning Page 3 of 21 How do the PID parameters affect system dynamics? 𝐾" 𝐶𝑂 = 𝐺$"# 𝐸 𝑠 = 𝐾$ + + 𝑇# 𝑠 𝐸(𝑠) NB: 𝐾" = 1/𝑇" 𝑠 The effects of increasing each of the controller parameters 𝐾$ , 𝐾" and 𝑇# can be summarized as, Response Rise Time Overshoot Settling Time S-S Error 𝐾! Decrease Increase NT Decrease 𝐾" Decrease Increase Increase Eliminate 𝑇# NT Decrease Decrease NT NT: No definite trend. Minor change. NB: Use 𝐾$ to decrease the rise time. Use 𝑇# to reduce the overshoot and settling time. Use 𝐾" to eliminate the steady-state error. [email protected] Lecture 5 – Controller Tuning Page 4 of 21 How do we tune the PID controller? Typical preliminary steps to be done before tuning are: 1. Study the diagram of the control loop to be familiar with its function and components. 2. Obtain the proper clearance for tuning activities. Make sure the changes you are going to make doesn’t affect the product. 3. Verify that each component in the loop is operating correctly. Make sure that enough energy is supplied to the actuator, calibrate the sensor, etc. Now, can we find a good set of initial PID parameters easily and quickly? Ziegler and Nichols conducted numerous experiments and proposed rules for determining values of 𝐾$ , 𝐾" 𝑎𝑛𝑑 𝑇# based on the step response of the system. [email protected] Lecture 5 – Controller Tuning Page 5 of 21 Ziegler-Nichols Continuous-Cycling Method Performs a closed-loop response test (or controller in automatic), and – Begin with a low gain, 𝐾$ , and with integral and derivative disabled. Potential of this method is that the loop goes unstable, and damage may result. SP error CV + 𝐾! Actuator Process - FB Sensor [email protected] Lecture 5 – Controller Tuning Page 6 of 21 Ziegler-Nichols Continuous-Cycling Method Procedure: A measured approach is to start with a low gain 𝐾$ , and with integral and derivative disabled. Gradually increase 𝐾$ until a steady-state oscillation occurs, note this gain as ultimate gain, 𝐺%. Measure the period of one cycle and note that 𝑃% (ultimate period). Sustained oscillation with period 𝑃$. (𝑃$ is measured between two successive peaks) 𝑃$ [email protected] Lecture 5 – Controller Tuning Page 7 of 21 Ziegler-Nichols Continuous-Cycling Method Now, use the gain estimator chart to calculate PID parameters. Table (1): Ziegler-Nichols continuous-cycling chart. Reset Time 𝑇& Type of Proportional Proportional Derivative (Minutes per controller gain 𝐾$ Band PB Time 𝑇' Repeat) P 0.5 𝐺% 2 𝑃𝐵% ∞ 0 1 PI 0.45 𝐺% 2.2 𝑃𝐵% 𝑃 0 1.2 % 1 PID 0.6 𝐺% 1.7 𝑃𝐵% 0.5 𝑃% 𝑃 8 % Proportional Band 𝑃𝐵 = 1/𝐾$. Reset Rate (Repeats per Minutes) 𝑇( = 1/𝑇&. [email protected] Lecture 5 – Controller Tuning Page 8 of 21 Example This example shows a slow-acting temperature process. The process- identification steps determine that the ultimate proportional gain (Gu) is 2 when the process begins to oscillate, as shown in the Figure. The ultimate period (Pu) of the waveform is 10 minutes. Using the continuous-cycling method with a PID controller that has a proportional gain, reset time, and derivative time adjustments, determine the proper settings for each mode. From the system response: G) = 2, P) = 10 min Use Table (1) for PID mode: K ! = 0.6 x G) = 0.6 x 2 = 1.2 T* = 0.5 P) = (0.5)(10 min) = 5 minutes per repeat P) 10 min T+ = = = 1.25 minutes 8 8 [email protected] Lecture 5 – Controller Tuning Page 9 of 21 Another Method: Ziegler-Nichols Reaction Curve Tangent line at inflection point S-shaped reaction curve Time Applies to 1st order system whose unit-step response resembled S-shaped curve. Start with open-loop system, or controller in manual mode. The S-shaped reaction curve can be characterized by two constants, dead time D and reaction rate (slope of the curve) 𝑅 = 𝐵/𝐴, which is determined by drawing a tangent line at the inflection point of the curve and finding the intersections of the tangent line with the time axis and the steady-state level line [email protected] Lecture 5 – Controller Tuning Page 10 of 21 Ziegler-Nichols Reaction-Curve Method Now, use the gain estimator chart to calculate PID parameters. Table (2): Ziegler-Nichols reaction-curve chart. Reset Time 𝑇& Type of Proportional Proportional Derivative (Minutes per controller gain 𝐾$ Band PB Time 𝑇' Repeat) 1 P 100𝑅, 𝐷 ∞ 0 𝑅, 𝐷 0.9 PI 110𝑅, 𝐷 3.33𝐷 0 𝑅, 𝐷 1.2 PID 83𝑅, 𝐷 2𝐷 0.5𝐷 𝑅, 𝐷 𝑅, : Unit reaction rate. Reaction rate per actuator step input, 𝑅, = 𝑅/𝑋. [email protected] Lecture 5 – Controller Tuning Page 11 of 21 Example For the reaction curve shown in the Figure, determine the proper settings for a PID controller with proportional band, reset time, and derivative time adjustments. [email protected] Lecture 5 – Controller Tuning Page 12 of 21 Solution Process reaction rate, 𝑅, which is the slope of the reaction curve. B 27% R= = = 27 %/min A 1 min. Unit reaction rate, 𝑅,. R 27 %/min R, = = = 2.7 /min X 10 % Effective Delay (Dead Time): D = 1.5 minutes Use Table 2 for PID mode: Proportional Band: PB = 83R, D = 83 x 2.7 x 1.5 = 336.15 Reset Time: T* = 2 x D = 2 x 1.5 = 3 minutes per repeat Derivative Time: T+ = 0.5 x D = 0.5 x 1.5 = 0.75 minutes [email protected] Lecture 5 – Controller Tuning Page 13 of 21 Ziegler-Nichols Tuning Method Although the reaction-curve method uses the results of a single test, the test is performed with open-loop (or controller in manual mode). So, if a major disturbance occurs during the test, no corrective actions will be taken. Consequently, the test results may be misleading. The parameters will typically give you a response with an overshoot on the order of 25% with a good settling time. You may then start fine-tuning the controller using the basic rules that relate each parameter to the response characteristics, as noted earlier. [email protected] Lecture 5 – Controller Tuning Page 14 of 21 Other Tuning Methods Autotuning: Usually included with industrial process controllers as part of the set-up utilities. – Some controllers have additional autotune routines. Trial-and-Error: No mathematical formulae. – Begin with a low gain 𝐾$. Increase 𝐾$ until a steady-state oscillation occurs, note this gain as 𝐾-( (critical gain). , – Set the gain at 𝐾$ = 𝐾-(. Add integral to eliminate steady-state. error. Add derivative to reduce overshoot and settling time. – Although this method gives acceptable results in many situations, it is time consuming and relies on the operator’s experience. [email protected] Lecture 5 – Controller Tuning Page 15 of 21 Verify the Controller Parameters With PID in automatic mode, perform the bump test i.e, apply 5-10% step input, and monitor the controlled variable. Desired response is known as 1/4 decay ratio, which implies that the size of a successive peak is one-fourth the size of the previous peak. % ' One quarter decay ratio. (𝐵 = ; 𝐶 = ) & & [email protected] Lecture 5 – Controller Tuning Page 16 of 21 Fine Tuning the Controller Parameters You start fine tuning if the ¼ decay ratio is not accomplished. Proportional Gain, 𝐾$ : – If the step response of the system dampens out quickly, the proportional gain is too low and should be increased. – If the step response oscillates too much, the proportional gain is too high and should be decreased. Derivative Time, 𝑇' : – If the PID is overshooting much, 𝑇' is too low and should be increased. – If the response is too slow, 𝑇' is two high should be reduced. [email protected] Lecture 5 – Controller Tuning Page 17 of 21 Derivative Time Effects: Example The figure shows the step response for two different controllers (PI and PID). With derivative, the step response curve has peaks with less amplitude and short period. [email protected] Lecture 5 – Controller Tuning Page 18 of 21 Questions 1. The term ultimate gain (or ultimate proportional band) refers to the controller adjustment that _________. a. causes the process to continuously cycle b. is the proportional setting when the controller is tuned 2. If a process reaction curve produced when the controller is tuned does not display a proper 1⁄4 decay ratio because it dampens out too quickly, the proportional gain is set too _____. a. low b. high 3. The process-identification information for the Ziegler-Nichols reaction-curve method is observed on a chart recorder with the controller in the ______ mode, and the 1⁄4 decay ratio is observed when the controller is in the ________ mode. a. manual b. automatic [email protected] Lecture 5 – Controller Tuning Page 19 of 21 Questions 4. Determine the proper settings for a two-mode controller using the Ziegler- Nichols continuous-cycling method and Table (1). Given: Ultimate Proportional Band = 3, Ultimate Period = 2 minutes Proportional Setting ______________ Integral Setting (Reset Rate) _____________ Table 1: Ziegler-Nichols continuous-cycling gain chart [email protected] Lecture 5 – Controller Tuning Page 20 of 21 Summary Two things to take away from Ziegler-Nichols tuning: 1. Relations between 𝐾$ , 𝐾" and 𝑇' and important response characteristics, of which these three are most useful: § Use 𝐾$ to decrease the rise time. § Use 𝐾" to eliminate the steady-state error. § Use 𝑇' to reduce the overshoot and settling time. 2. The Ziegler-Nichols tuning rules are good for initial estimate of parameters. [email protected] Lecture 5 – Controller Tuning Page 21 of 21