Digital control implementation: how is integral (I) action realized on a digital controller?

Introduction / Context:
Integral action removes steady-state error by accumulating past error. In digital control (PLC, DCS, microcontrollers), I action must be implemented numerically because signals are sampled in time.

Given Data / Assumptions:

• Sampling period Ts is fixed and sufficiently small.
• Error e[k] = setpoint − process_variable at sample k.
• Controller state stores the accumulated sum.

Concept / Approach:
Discrete integral is a running sum (or sum with scaling), e.g., I[k] = I[k−1] + Ki * Ts * e[k] The integral state I[k] contributes to the control output. Practical implementations include anti-windup (clamping, back-calculation) to prevent integrator growth during saturation.

Step-by-Step Solution:
Represent continuous integral as a discrete sum over samples.Accumulate the error term each cycle to form the integral state.Add the term to controller output with proper scaling and limits.

Verification / Alternative check:
Controller vendor function blocks (PI/PID) describe I action exactly as a sum/accumulator with tuning parameter Ti or Ki and options for anti-windup.

Why Other Options Are Wrong:
(a) Digital controllers do implement integral routinely. (b) Dividing error without accumulation is not integral action. (d) Cannot be true since (a) and (b) are incorrect.

Common Pitfalls:
Ignoring anti-windup; poor scaling of Ki with Ts; changing Ts without retuning.

Final Answer:
The controller sums the error over discrete sampling intervals and accumulates it