Proportional Integral Derivative Controller Course Work Example

Published: 2021-06-25 22:30:04
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In a control system, the interest is maintaining the system in a steady state irrespective of the input into the system. In achieving this aim, the system is designed to constantly check for deviations from this goal and apply the corrective measures to keep the system steady when there are deviations from the steady state. The device that ensures this desired output is a controller. The overall aim of using a controller in a system is thus to maintain the stability of such a system in the face of deviations (errors) from a desired output.

Controllers are used in a feedback system because of the need to constantly check the system output for variations from the goal state. This output is fed back into the input for comparison and appropriate action.

The PID controller is a controller that implements three different controllers that can each be used independently in a control system. The Proportional, Integral and Derivative controllers are combined in one single controller – PID controller. This is achieved by tuning the value of the error with the proportionality constants and derivatives of the errors.

The figure above shows the block diagram of a PID controller with all the individual output of the P-, I- and D- controllers added together as the control signal of the PID.
The input to the plant (the system to be controlled) which is the output of the PID controller in time domain is given by ut=Kpe+Kie dt + Kddedt
where, u(t) is the output of the PID controller, Kp is the proportional gain, Ki is the integral gain, Kd is the derivative gain and e is the error.

The transfer function of the output is Kds2+ Kps+ Kis

Before going into the details of explaining how a PID controller works, let us consider the effect of each of the individual Proportional, Integral and Derivative controllers on a closed loop system as presented in the table below.

When an error is discovered by the controller, the difference between the desired output value and the actual output value, the Proportional component produces an output which is the product of the error value and the proportionality constant Kp; the Integral component produces an output which is the product of the integral gain Ki and the integral of the error while the Derivative component produces an output that is the derivative gain Kd multiplied by the derivative of the error. All the three outputs are combined together as the control signal to the plant. After sending the control signal, the new output of the plant is obtained, fed back to the controller to determine the error and the new value of the control signal is obtained. The process of obtaining the output and determining the error and the control signal continues on-and-on.

In the proportional controller, the output depends on the value of the proportionality constant Kp. If the constant is too high, the system will give a large change in output for a small error change and this can make the system to become unstable. If on the other hand, the proportionality constant is too small, a small output response will be produced by the system in response to a large system error. Take note that the proportional controller works on a non-zero error condition. This is why it is always operated with a steady state error in the system.

The figure above shows the action of a P- controller. From the table shown above, the P controller causes an increase in overshoot and a little change to settling time, these can be remedied with the use of a derivative controller.
Figure 2: The action of a Derivative controller

The derivative controller reduces the overshoot and gives a better settling time as compared with a Proportional controller. This it achieves by using the slope of the error over time dedt to better predict the system behavior.
The integral controller eliminates the steady state error. Since the I controller also increases overshoot and reduces the rise time, the proportional constant Kp is reduced accordingly.

Shown in the figure below is the graph showing the effect of the integral controller in eliminating steady state error. Remember that the steady state error is needed to operate a proportional controller. This bias is taken care of by the Integral controller.
The use of a PID controller thus will enable a component to compensate for the inadequacies of another to produce a controller that combined all the advantages of each of the component.

The PID controller thus helps achieve a control system with no steady state error, no overshoot and fast rise time.

PID controllers are used in systems where the output and the input to the plant can be measured, and the desired output value is also known.
PID controllers find various uses in industry. They are implemented as mechanical devices or analog electronic components. As mechanical devices, they are made up of levers, springs and masses energized by compressed air. As electronic devices, they are implemented with solid state amplifiers, capacitors and resistors.
They are used in chemical reactors using temperature PID controllers, biological reactors or bioreactors used by pharmaceuticals to produce proteins that are too complex for chemical processes to produce. They are also used in the control of distillation columns and crystallizers. Early use of the PID controller was in the control of ships.

REFERENCES

McMillan G, (2008). Effect of sample delay on standard PID tuning and loop performance. Advanced Application Note 5. http://www.modelingandcontrol.com/repository/AdvancedApplicationNote005.pdf
McMillan, G.,(2011). Industrial Applications of PID Control. In Vilanova,R. and Visioli,A. (Eds.), PID Control in the third Millennium: Lessons Learned and New Approaches,ch 14. Springer.
PID Controllers Explained (2011).Control Notes. Retrieved on October 19, 2013 from http://blog.opticontrols.com/archives/344

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