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.

What is Proportional Integral Controller?

PI controller or Proportional controller is a combination of Proportional controller action and Integral controller action which is designed to regulate a process variable based on its setpoint and manipulated variable. Also, it can be identified as a combination of proportional and integral controllers.

What are Proportional and Integral Controllers?

Proportional Controller: Proportional Controller in electronics engineering continuously adjusts the output based on the current error signal. The p term or proportional term calculate the current error from the setpoint and the current manipulated variable output. It generates a control signal to regulate the error.

Co(t) ∝ e(t)

Co(t) = Kp . e(t) -----------(i)

Where,

Co(t) = controller output at time T,

e(t) = error at time t(SP-PV),

Kp = tuning constrains for proportional action.

Integral Controller : In Integral Controller, the I term or Integral term actuates the past error over time and generates a control action to eliminate the accumulated steady-state error. It ensures that even small errors are eventually corrected. It eliminates offset but can lead to sluggish responses and overshooting if too aggressive.

This is represented as :

It can be represented mathematically as :

Co(t) ∝ ∫ e(t) dt

Co(t) = Ki . ∫ e(t) dt -----------(ii)

Where,

ki = tuning constrains for integral action,

others are same as equation number (i).

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)

Block Diagram of PI Controller

Explanation of the Block Diagram

A Proportional controller operates in step by step manner to regulate a control system. It starts by calculating the error, which is the difference between the desired setpoint and current process variable. The proportional(P) part of the controller multiplies this error with proportional gain(Kp) and generates an immediate action. which is directly proportional to the error.

The Integral(I) part computes the cumulative sum of past errors by integrating gain(Ki). The two components P and I are then added together to determine the control output, which is applied to the system. This output adjusts the ultimate output of the system by minimizing the error over time and maintain the setpoint.

Effects of Proportional Integral Controller with Equation

A proportional-integral controller is a feedback control system component with an equation Co(t) = Kp . e(t) + Ki . ∫ e(t) dt , combining proportional band integral action. The proportional component responds to the current error, while the integral component accumulates past errors to eliminate steady-state errors. Properly tuning the Kp and Ki gains is crucial to achieve stable control with fast response and minimal error in various engineering applications. The PI controller diagram illustrates its functioning with error input, proportional, integral actions and the output controlling a process.

The system equation with the PI controller is

M(s)/E(s) = Kp. {1 + (1/s.Ti)} from equation no(2). so,

M(s)= Kp. {1 + (1/s.Ti)} . E(s)

Here M(s) is equal of the system , and E(s) is the error signal.

The error signal E(s) is difference between the reference input (R(s)) and the system output (Y(s)):

E(s) - R(s) - Y(S)

Now, let's substitute the expression for E(s) into the equation for M(s)

M(s) = Kp{1 + (1/s.Ti)} (R(s) - Y(s))

The output (Y(s)) of the system can be expressed as the product of the plant transfer function (G(s)) and the controller output C(s) :

Y(s) = G(s).C(s)

Now, let's substitute this expression for Y(s) into the equation

M(s) = Kp{1 + (1.s.Ti)}(R(s) - G(s).C(s))

C(s)=\frac{K_p(1+\frac{1}{ST_i})}{1+k_p(1+\frac{1}{ST_i})G(s))}R(s)

Tuning of PI Controller

Tuning of a PI controller means adjusting the Kp and Ki term to achieve the optimized controller output by minimizing error and avoid overshooting and oscillation. Their are some rules that follows for tuning a PI controller.

• Tuning controller involves adjusting Kp and Ki for optimal system control.
• Start with initial guesses for Kp and Ki.
• Set Ki value to '0'
• Gradually increase Kp to achieve a fast response without excessive overshoot.
• Introduce Ki and incrementally raise it to eliminate steady-state error.
• Continuously fine-tune Kp and Ki, considering their impact on overshoot, setting time and steady state error.
• Test the controller under various condition.
• Tuning is an iterative process, and optimal parameters may vary for different systems, requiring practice and experience to achieve stable and effective control.

Applications of Proportional Integral Controller

• PI controllers are often used in temperature control systems, such as ovens, furnaces, and HVAC systems.
• PI controllers are used in motor control applications to control speed and position.
• In robotics, PI controllers are used to control the movements and positions of robot arms and manipulators.
• PI controllers are an essential part of the flight control systems in aircraft.
• PI controllers can be used in audio systems for applications like noise cancellation.
• PI controllers are used in medical devices, such as infusion pumps and respirators, to regulate parameters like drug delivery rates or patient ventilation.

Advantage and Disadvantage of PI Controller

There are some list of Advantage and Disadvantage of PI controller given below :

Advantages

• It responses fast as compared to other controllers like PD controller or PID controller.
• PI controller enhance system stability.
• They eliminate steady state error.
• PI controllers are versatile to use in various system.

Disadvantages

• PI controller may not suitable for highly dynamic or complex system.
• In some cases it can produce overshoot before settling.
• Proper tuning can be challenging for a PI controller.
• PI controller might not be the best choice for rapidly changing setpoints.

The Proportional-Integral Controller(PI) is a key component in control system. It offers stability by eliminating steady-state errors, improves settling times, and is robust in various applications. It is simplicity, with only two tunable parameters, makes it a practical choice for regulating process efficiently.

Difference between with other types of controller and PI controller

The PI controller is one of several types of feedback controllers used in control systems. Let's briefly differentiate it from other common controllers