Process Control System

In a process control system, the objective is to maintain a process variable (such as temperature, pressure, or level) at a desired value called the setpoint. The controller continuously compares the setpoint with the actual process variable and generates an output based on the error, which is:

$$Error=Setpoint-Process\:Variable$$

The controller action may consist of Proportional (P), Integral (I), and Derivative (D) modes.

(a) Proportional Control

Proportional control is the simplest form of feedback control. The controller output is directly proportional to the present value of the error. This means that the magnitude of corrective action depends on how large the error is at that instant.

The mathematical expression is:

$$Controller\:Output=K_{P}\times e\left(t\right)$$

Where:

• ( K_p ) = Proportional gain
• ( e(t) ) = Instantaneous error

If the error increases, the controller output increases proportionally. A higher value of ( K_p ) makes the system respond more strongly and quickly to deviations.

However, proportional control alone usually results in a steady-state error (offset). This means that even after the system stabilizes, a small error remains because the controller requires some error to produce an output. Increasing ( K_p ) reduces this offset but too high a gain can cause oscillations or instability.

(b) Integral Control

Integral control is introduced to eliminate the steady-state error produced by proportional control. It works by accumulating (integrating) the error over time and adjusting the controller output accordingly.

The mathematical expression is:

$$Controller\:Output=K_{i}\int e\left(t\right),\:dt$$

Where:

• ( K_i ) = Integral gain

As long as an error exists, even if it is small, the integral action continues to increase or decrease the output. This ensures that the process variable eventually reaches the exact setpoint, thereby eliminating steady-state error.

However, if the integral gain is too high, the accumulated error may become excessive, leading to integral windup. This can cause overshoot and sustained oscillations before the system stabilizes.

(c) Derivative Control

Derivative control acts on the rate of change of the error, rather than the error itself. It predicts the future trend of the error by measuring how fast the error is increasing or decreasing.

The mathematical expression is:

$$Controller\:Output=k_{d}\frac{de\left(t\right)}{dt}$$

Where:

• ( K_d ) = Derivative gain

Because it responds to the slope of the error curve, derivative control provides a corrective action before the error becomes large. For this reason, it is often called anticipatory control.

Derivative control does not eliminate steady-state error, but it improves the dynamic performance of the system.

(d) Necessity of Derivative Control

Derivative control is necessary in systems where stability, fast response, and reduced oscillations are important.

Its key contributions are:

1. Reducing Overshoot

• As the process variable approaches the setpoint rapidly, derivative action reduces the controller output. This braking effect prevents the system from exceeding (overshooting) the desired value.

2. Damping Oscillations

• Derivative control provides a damping effect, reducing oscillatory behavior. This allows higher proportional gains to be used without causing instability.

3. Improving Response in Systems with Lag

• In processes with significant inertia or time delay, such as temperature control systems, derivative action reacts to rapid changes and improves recovery from disturbances.
• In practical applications, the three modes are combined as a PID controller, which balances responsiveness (P), accuracy (I), and stability (D) to achieve optimal control performance.