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TMA Assignment

TMA Assignment

Question 5 (a)

Controller Principles

The Control Units are in general build on the control principles

proportional controller

integral controller

derivative controller

Proportional Controller (P-Controller)

One of the most used controllers is the Proportional Controller (P-Controller) who produce an output action that is proportional to the deviation between the set point and the measured process value (Astrom & Hagglund, 2002, pp. 364- 380).

OP = -kP Er (1)

where

OP = output proportional controller

kP = proportional gain or action factor of the controller

Er = error or deviation between the set point value and the measured value

The gain or action factor - kP

influence on the output with a magnitude of kP

determines how fast the system responds. If the value is too large the system will be in danger to oscillate and/or become unstable. If the value is too small the system error or deviation from set point will be very large.

can be regarded linear only for very small variations.

The gain kP can be expressed as

kP = 100 / P (1b)

where

P = proportional band

The proportional band P, express the value necessary for 100% controller output. If P = 0, the gain or action factor kP would be infinity - the control action would be ON/OFF.

Note! A proportional controller will have the effect of reducing the rise time and will reduce, but never eliminate, the steady-state error (Qin & Badgwell, 1997, pp. 232- 256).

Integral Controller (I-Controller)

With integral action, the controller output is proportional to the amount of time the error is present. Integral action eliminates offset.

OI = - kI S(Er dt) (2)

where

OI = output integrating controller

kI = integrating gain or action factor of the controller

dt = time sample

controller's integral action time

With integral action, the controller output is proportional to the amount of time the error is present. Integral action eliminates offset that remains when proportional control is used.

controller output = (1/Ti)*int(error)

where the parameter Ti is called the integral time. Integral action is also know as reset and the parameter Ti as reset time. Integral action gives the controller a large gain at low frequencies that results in eliminating offset. Integrals give information concerning the past. That is why integrals are always late. Integrals provide stability but have a tendency to get stuck in the past. In most controllers the proportional and integral action are combined. The output of the combined proportional and integral action (in s-domain) is then (Astrom & Hagglund, 2001, pp. 1163-1175):

Question 5 (b)

Question 6 (a)

Question 6 (b)

Question 6 (c)

Question 7 (a)

Ziegler and Nichols have developed PID tuning methods back in the early fourties based on open loop tests (less known than for example the Cohen-Coon formulas) and also based on a closed loop test, which is maybe their most widely known achievement.

The open loop method allows to calculate PID parameters from the process parameters. The procedure:

Step 1: Make an open loop plant test (e.g. a step test)

Step 2: Determine the process parameters: Process gain, deadtime, time constant (see below: draw a tangent through theinflection point and measure L and T as ...

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