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ITAE

As explained in Optimization Algorithms: AI techniques for design, planning, and control problems

ITAE is an integral performance metric that weights absolute control error by elapsed time.

ITAE (integral time absolute error) measures control performance by multiplying absolute error by elapsed time before integrating. It therefore penalizes errors that persist, especially when they occur later in a response. For a reference signal r(t) and output y(t), e(t) = r(t) − y(t), and ITAE = ∫t|e(t)| dt. section 8.7

Use ITAE when settling behavior matters, and distinguish it from related criteria:

  • IAE integrates absolute error without weighting duration. section 8.7
  • ITSE also weights time but squares error, emphasizing larger deviations. section 8.7
  • ITAE can favor a slower rise with more oscillatory behavior while penalizing late error strongly. section 8.7

PID tuning

For a heater’s step response, calculate the error between desired and actual temperature, integrate its time-weighted absolute value, and minimize the result while searching for PID parameters. A Python implementation evaluates ITAE numerically with NumPy’s trapezoidal integration and can use a genetic algorithm to tune Kp, Ki, and Kd. section 8.7

How Optimization Algorithms: AI techniques for design, planning, and control problems covers it

How deep it goes
Defines the formula, compares related criteria, and demonstrates numerical evaluation, but does not derive the metric formally.
Examples
Uses a heater step response and a Python genetic-algorithm objective for evaluating PID parameters.
What it assumes
Assumes control error, reference signals, actual outputs, and basic PID terminology while defining the metric from zero.
What you can do after
Apply ITAE as a numerical objective when tuning PID parameters for reduced persistent and late response error.
How it connects
Connects ITAE with PID tuning, control error, step response, IAE, ITSE, ISE, overshoot, and settling time.
Cover of Optimization Algorithms: AI techniques for design, planning, and control problems
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