P, PI, & PID Controllers | Control Systems - Electrical Engineering (EE) PDF Download

Automatic Controller

An automatic controller is a device which compares the actual value of the plant output with the desired value (setpoint) and produces a control signal to reduce the deviation. The difference between the reference input and the measured output is called the error or deviation. The method by which the controller produces the control signal is called the control action.

Elements of an Industrial Automatic Controller

The main elements commonly present in an industrial automatic controller are explained below.

• Measuring element: Converts the process variable (for example temperature, pressure, level, speed) into a form suitable for comparison (electrical signal, displacement, etc.).
• Error detector (comparator): Compares the measured signal with the reference input (setpoint) and produces the actuating error signal e(t).
• Controller (control element): Processes the error signal by a chosen control action to produce the manipulated variable M(t).
• Final control element: (for example a control valve, motor, heater) which changes the process input to steer the output towards the setpoint.

The actuating error is

e = r - b

where r is the reference (setpoint) and b is the feedback (measured output in feedback form).

Classification of Industrial Controllers (by control action)

• ON-OFF controller
• Proportional controller (P)
• Integral controller (I)
• Proportional + Integral controller (PI)
• Proportional + Derivative controller (PD)
• Proportional + Integral + Derivative controller (PID)

Proportional Control Action

In a proportional controller the controller output is directly proportional to the instantaneous value of the actuating error signal.

The algebraic relation in time domain is

M(t) = Kp E(t)

Applying the Laplace transform (assuming zero initial conditions) gives

M(s) = Kp E(s)

where Kp is the proportional gain (also called proportional sensitivity). The proportional action produces a control signal that is immediate and proportional to the magnitude of the error.

Effects of proportional control:

• Reduces rise time.
• Increases system responsiveness but may increase overshoot.
• Does not, in general, eliminate steady-state error for many types of plants (a residual offset or steady-state error usually remains).
• Excessive Kp can reduce stability or cause sustained oscillations.

Integral Control Action

In an integral controller the controller output changes at a rate proportional to the actuating error; the controller integrates the error over time.

Mathematically, in the time domain

M(t) = Ki ∫ e(t) dt + M(0)

where Ki is the integral constant and M(0) is the controller output at t = 0.

Taking Laplace transforms (zero initial condition for the integral action) gives

M(s) = (Ki / s) E(s)

Therefore the transfer function of a pure integral controller is

M(s)/E(s) = Ki / s

Alternative notation often used in control practice relates integral action to an integral time constant Ti, with

Ki = Kp / Ti

Effects of integral control:

• Eliminates steady-state error (offset) for many types of systems by accumulating error until it is driven to zero.
• Improves accuracy at steady state.
• Can increase overshoot and settling time, and may reduce damping-too much integral action can cause oscillations or instability.

Derivative Control Action

In a derivative controller the controller output is proportional to the rate of change (time derivative) of the actuating error. Derivative action provides a predictive (anticipatory) element based on how quickly the error is changing.

In the time domain

M(t) = Kd (d/dt) e(t)

Taking Laplace transforms gives

M(s) = Kd s E(s)

Thus, the transfer function of a pure derivative controller is

M(s)/E(s) = Kd s

Effects of derivative control:

• Improves transient response by damping the rate of change, reducing overshoot and improving stability margins.
• Acts as a predictor and therefore helps to counter rapid error changes.
• Highly sensitive to high-frequency noise in the measured signal; often derivative action is implemented with filtering to reduce noise amplification.

Combined Controllers: PI, PD and PID

Combining the basic actions gives controllers that exploit advantages of multiple actions.

• PI controller: combines proportional and integral actions. Transfer function:

M(s)/E(s) = Kp + Ki / s

• PD controller: combines proportional and derivative actions. Transfer function:

M(s)/E(s) = Kp + Kd s

• PID controller: combines proportional, integral and derivative actions. Transfer function:

M(s)/E(s) = Kp + Ki / s + Kd s

Alternative unified notation is

M(s)/E(s) = Kc (1 + 1/(Ti s) + Td s)

where Kc is the controller gain, Ti the integral time constant and Td the derivative time constant.

Typical effects of combined controllers:

• PI: Eliminates steady-state error (due to integral action) while keeping simpler implementation than full PID. Often used where derivative action is not required or measurement noise is high.
• PD: Improves transient response (derivative) and provides some reduction of error (proportional), but does not eliminate steady-state error unless the plant type provides sufficient inherent integration.
• PID: Offers the most general control: proportional for immediate response, integral to remove steady-state error, derivative to improve transient behaviour. Widely used in industrial processes when tuning and measurement noise permit.

Tuning and Practical Considerations

• Tuning: Controller parameters (Kp, Ki, Kd or Kc, Ti, Td) are adjusted so that the closed-loop system meets desired performance: acceptable rise time, overshoot, settling time and steady-state error. Common tuning methods include trial-and-error, Ziegler-Nichols rules, relay feedback, and model-based design methods.
• Noise sensitivity: Derivative action amplifies measurement noise; practical implementations include filtering or using a PD term with a small lead filter.
• Anti-windup: Integral action can cause integrator windup when actuators saturate. Anti-windup schemes are used to prevent excessive integral buildup.
• Implementation: Controllers can be implemented as pneumatic, hydraulic, electronic analogue circuits, or in digital form using microcontrollers/DSP/PLCs; the mathematical actions remain the same.

Applications and Examples

• Temperature control in furnaces and ovens: PI or PID controllers maintain setpoint and eliminate steady-state error despite load disturbances.
• Speed control of DC/AC motors: PID controllers are common to provide fast response and suppression of disturbances.
• Level control in tanks and flow control in process industries: PI controllers are widely used where derivative action is not helpful due to measurement noise.
• Chemical process control and robotics: PID controllers are foundational, often as starting point before using more advanced controllers.

Summary

Proportional, integral and derivative actions form the basic building blocks of industrial controllers. Proportional action produces an immediate response proportional to error, integral action removes steady-state error by accumulating error over time, and derivative action anticipates error changes to improve transient performance. PI, PD and PID combinations are selected according to process requirements, noise level and desired performance. Practical implementations require proper tuning, anti-windup measures and filtering where necessary to ensure robust and stable closed-loop control.