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**بسم الله الرحمن الرحيم Advanced Control Lecture four**

Tuning of PID Controllers 1- Off-line methods (Ziegler-Nichols, Tyreus-Luyben, Ciancone-Marlin, IMC) 2- On-line methods (Ziegler-Nichols, Relay feedback) Ref: C. C. Yu, Autotuning of PID controllers, 2nd ed., springer, Chapters 2, 3 & 4 Lecturer: M. A. Fanaei Ferdowsi University of Mashhad

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**Off-Line Tuning of PID Controllers**

About 250 tuning rules are exist for PI and PID Controllers What is the suitable tuning rule? It really depends on your process (Type, Order, Parameters, Nonlinearity, Uncertainty, etc) Ziegler-Nichols(1942): Recommended for 0.1< D/t <0.5 ( )

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**Off-Line Tuning of PID Controllers**

Tyreus-Luyben(1992): Recommended for time-constant dominant processes ( D/t <0. 1 ) Ciancone-Marlin(1992): Recommended for dead-time dominant processes ( D/t > 2.0 )

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**Off-Line Tuning of PID Controllers**

PID tuning based on IMC (Rivera et al., 1986)

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**Recommended Tuning Formulas**

The following formulas are recommended by Luyben and Yu for tuning of PI controllers:

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**On-Line Tuning of PID Controllers**

Ziegler-Nichols Test (1942) Set the controller gain Kc at a low value, perhaps 0.2. Put the controller in the automatic mode. Make a small change in the set point or load variable and observe the response. If the gain is low, then the response will be sluggish. Increase the gain by a factor of two and make another set point or load change. Repeat step 4 until the loop becomes oscillatory and continuous cycling is observed. The gain at which this occurs is the ultimate gain Ku , and the period of oscillation is the ultimate period Pu.

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**On-Line Tuning of PID Controllers**

Relay Feedback Test (Astrom & Hagglund, 1984) Luyben popularized relay feedback method and called this method “ATV” (autotune variation).

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**On-Line Tuning of PID Controllers**

Relay Feedback Test Bring the system to steady state. Make a small (e.g. 5%) increase in the manipulated input. The magnitude of change depends on the process sensitivities and allowable deviations in the controlled output. Typical values are between 3 and 10%. As soon as the output crosses the SP, the manipulated input is switched to the opposite position (e.g. –5% change from the original value). Repeat step 3 until sustained oscillation is observed . Read off ultimate period Pu from the cycling and compute Ku from the following Equation: Ku = 4h/(πa) , ωu = 2π/Pu

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**On-Line Tuning of PID Controllers**

Advantages of Relay Feedback Test It identifies process information around the important frequency, the ultimate frequency (where the phase angle is -π). It is a closed-loop test; therefore, the process will not drift away from the nominal operating point. The amplitude of oscillation is under control (by adjusting h ). The time required for a relay feedback test is roughly equal to two to four times the ultimate period. If the normalized dead time D /t is less than 0.28, the ultimate period is smaller than the process time constant. Therefore the relay feedback test is more time efficient than the step test. Since the dead time can not be too large, the temperature and composition loops in process industries seem to fall into this category.

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**On-Line Tuning of PID Controllers**

Advantages of Relay Feedback Test

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**On-Line Tuning of PID Controllers**

Process model identification from a Relay Feedback Test In theory, the steady state gain can be obtained from plant data (Kp = Δy/Δu). For highly nonlinear process, the change in input (Δu) must be small as 10-3 to 10-6 % of the full range. Therefore for highly nonlinear processes, trying to obtain reliable steady state gains from plant data is usually impractical. Luyben showed that the simple relay feedback test provides an effective way to determine the linear model for such processes. If necessary, the dead time (D) in the transfer function can be easily read off from the initial part of the relay feedback test.

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**On-Line Tuning of PID Controllers**

Process model identification from a Relay Feedback Test Model 1 (FOPDT) :

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**On-Line Tuning of PID Controllers**

Process model identification from a Relay Feedback Test Model 2 (integrator plus dead time) : Model 3 (pure dead time):

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**On-Line Tuning of PID Controllers**

Relay feedback responses of FOPDT processes Assume an integrator plus dead time (Time constant dominant processes) Assume a FOPDT (Most slow processes) Assume a pure dead time (Dead time dominant processes)

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