Mumbai University > Mechanical Engineering > Sem 5 > Mechanical measurements and control

Marks: 5M

Year: Dec2014 ,Dec2015 ,May 2015 ,May 2016

PID Controller

The composite controller including the combination of the propositional, integral and derivation control mode is called PID control mode and the controller is called three mode controllers. It is very much complete to design but very powerful in action. Mathematically such a control mode can be expressed as, P(+)=$k_p$ e(+) +$k_p$ $k_i$ $∫_0^t e(t) dt$ + $k_p$ $k_d$ $\frac{de(+)}{dt}$ + P(0) Where P(0)=Initial value of the output This mode has advantages of all the modes. The integral mode eliminates the offset error of the proportional mode and the response is also very fast due to derivative mode. Thus it can be used for any process condition.

With the PID control action; there is no offset, no oscillations with least settling time. So there is improvement in both transient as well as steady state response. Figure (A) shows the response of PID control for a particular error signal, assuming direct action.

The figure (b) shows the response of various control modes to unit step load change. The proportional and PD control produces the offset error. It requires significant time to attain the steady state. The PI control eliminates the offset but at the expense of higher maximum overshoot, a long period of oscillations and more settling time. The PD control produces the steady state very quickly with least oscillations and smallest maximum overshoot but offset is significant with PID control, there is no offset & system achieves the steady state with less settling time. Thus PID is the ultimate process composite controller.

PI controller

This is a composite control mode obtained by combining the proportional mode and the integral mode. The mathematical expression for such a composite control is, P(+)=$k_p$ e(+) + $k_p$ $k_i$ $∫_0^t e(t) dt$ + P(0) Where P(0)=Initial value of the output at +=0 The important advantage of this control is that one to one correspondence of proportional mode is available with the offset gets eliminated due to integral mode, the integral part of such a composite control provides a reset of the zero error output after a load change occurs. Consider the load change occurring at t=$t_1$ and due to which error values as shown in the fig(a). The controller output changes suddenly by amount $v_p$ due to the proportional action. After that the controller output changes linearly with respect to time at a rate $\frac{k_p}{T_i}$. The reset rate is defined as the reciprocal of $T_i$.

ON-OFF Controller

This is one of that most common and simplest mode of controller. It has to control two positions of control elements, either on or off. Hence this mode is also called ON-OFF controller mode. It is the cheapest controller and often used if its limitations are well within the tolerance. This controller mode has two possible output states namely 0% or 100%. Mathematically this can be expressed as,

The P is the controller output and $e_p$ is error based on the percent of span. This if the error rises above a certain critical values the output changes from 0% to 100%. If the error decreases below certain critical value, the output falls from 100% to )%. The best example is a room heater. If the temperature drops below a setpoint, the heater is turned ON and if the temperature increases above a setpoint, the heater is turned OFF. In all the practical implementations of the ON-OFF controller there is an overlap as the errir increases through zero or decreases through zero. Such an overlap creates a span of error in which there is no chane in the controller output. This span is called neutral zone, dead zone or dead band.

This is shown in the figure. It cab be seen that till the erroe changes by ∆$e_p$ there is no change in the controller output. Similarly while decreasing also the error must decrease beyond ∆$e_p$ below 0 to change the controller output. Hence during the range of 2∆$e_p$, there is no change in the controller output. This zone is also called the differential gap. In such a controller, the control variable always oscillates with a frequency which increases with decreases width of the dead band. Hence dead band is purposely designed to prevent the oscillations in ON-OFF controllers.

In a two position controller, there exits undershoot and overshoot in the output. Figure shows the output with the position controller which shows undershoot and overshoot. Such undershoot and overshoot are inherent in two position mode.

The applications of such two position control mode are room air conditions, ON-OFF of a heater, liquid level control in large volume tank etc. this controllers are preferred for a large scale systems with relatively slow process rates.