Proportional Plus Integral Controllers with Equation
Till now we have seen the definition of a Proportional Integral controller. Now time to understand the mathematical formula behind it. We can represent a PI controller in mathematical expression from the definition. And from that a block diagram is made for better understanding the working of a PI controller. Below is the elaboration of equation and diagram of a PI controller.
Mathematical Expression of PI Controller
From equation number (i) and (ii) we can write by adding both of them,
Co(t) = Kp . e(t) + Ki . ∫ e(t) dt ———-(1)
This equation indicates that PI controller works like a simplified PI controller without a derivative action.
Taking laplace in both side of the equation,
L{Co(t)} = Kp.Le(t) + Ki.L{∫ e(t) dt }
The Laplace transform of the integral term is expressed using the property L{ ∫f(x)dx} = (1/s).L{f(x)}
L{Co(t)} = Kp.L{ e(t) } + Ki.(1/s).L{ e(t) }
In the Laplace domain, C(s) represents the transformed output, [ L{Co(t)} ], E(s) represents the transformed error signal, [ L{ e(t) } ]. So we can reform the equation below,
C(s) = Kp. E(s) + Ki. (1/s). E(s)
C(s) = Kp. { 1+(ki/s.kp)}. E(s)
We can say, Kp/Ki = Ti, or Ki/Kp = 1/Ti,
C(s) = Kp. {1 + (1/s.Ti)}. E(s)
C(s)/E(s) = Kp. {1 + (1/s.Ti)}
where, Kp is proportional gain, and Ti is the integral time constant.
WE can express C(s) as M(s). Now the equation will be,
M(s)/E(s) = Kp. {1 + (1/s.Ti)} ———(2)
Proportional Integral Controller – Control System
The proportional controller commonly known as PI controller is an essential part of the Industrial Automation and Control system. It is a closed-loop feedback control mechanism that aims to adjust the process variable by manipulating the variable based on the error between the setpoint and the process variable. It strikes a balance between quick response to deviations and long-term error elimination. Tuning the controller allows adjustment to meet the desired value.
Table of Content
- Definition
- Equation and Diagram
- Block Diagram
- Effects
- Tuning
- Applications
- Advantages and Disadvantages
- Comparison with Other Types of Controllers