Which of the following defines 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)

The key idea is that ISE measures how much the error deviates from zero over the entire response by summing the square of the error as time progresses. Squaring the error emphasizes larger deviations, so the performance index grows quickly if big mistakes occur and pushes control designs to minimize those large excursions while also promoting smooth behavior. This also makes the measure nonnegative and straightforward to optimize, since the objective responds to the magnitude of error rather than its direction.

In contrast, using the integral of the absolute error treats deviations in magnitude similarly but without the extra penalty for especially large errors, and the integral of the error over time could cancel positive and negative errors, masking actual performance. The integral of the squared error rate would focus on how quickly the error changes rather than how large the error is, which is a different quantity altogether.