Which expression corresponds to the ISE performance criterion?

• A. Integral of the Squared Error
• B. Integral of the Absolute Error
• C. Integral of the Error over Time
• D. Integral of the Squared Error Rate

Answer: (A)

In control evaluation, the ISE criterion sums up how far the output is from the setpoint by integrating the square of the error over time. Mathematically, it’s J = ∫ e(t)^2 dt, where e(t) is the difference between desired and actual output. Squaring the error makes large deviations especially costly, which drives the controller to correct big mistakes more aggressively while still rewarding small residual errors as time passes. This differentiable, quadratic form is what makes ISE a common choice for tuning and performance assessment.

The other options correspond to different metrics: the integral of the absolute error uses |e(t)| and is called IAE; the integral of the error over time without squaring would just measure the net area and could cancel positive and negative errors; the integral of the squared error rate would penalize rapid changes in error rather than the magnitude of the error itself.