Control Engineering Basics
A digital PI controller with a sampling period Ts = 1 ms calculates the integral with the rectangular approximation Σe·Ts. With an integral gain Ki = 10 s⁻¹, which of the following is the increase in the integral term when an error e = 0.2 lasts for 50 samples?
- A0.1
- B2
- C100
Show answer and explanation
Answer: A. 0.1
The increase in the integral term is Ki × Σe·Ts = 10 × 0.2 × 50 × 0.001 = 0.1. Fifty samples correspond to 0.05 s, so thinking of it as "integral gain × error × time" also gives 10 × 0.2 × 0.05 = 0.1. Programs often combine Ki·Ts into a single coefficient in advance so that each update takes a single multiply-accumulate. In that case the coefficient must be recalculated when the sampling period is changed.
- A. Correct. It grows at a rate of 2 per second for 0.05 s.
- B. This is Ki × e, the rate of increase per second. It must be multiplied by the duration, 0.05 s.
- C. This forgets to multiply by the sampling period Ts.