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ISE

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

ISE is an integral performance metric equal to the accumulated squared control error over time.

ISE (integral squared error) measures control performance by numerically integrating the squared error across a system’s response over time. Lower ISE generally indicates less accumulated error during PID tuning. section 8.7

Compare ISE with other response criteria when choosing a tuning objective:

  • IAE integrates absolute error, while ITAE and ITSE additionally weight error by time. section 8.7
  • ISE fits optimization workflows that simulate a step response, calculate error, and assign each PID parameter set an objective value. section 8.7

PID step-response evaluation

For a heater or other closed-loop system, compute the error between the desired step and simulated output, then evaluate ISE with numerical integration before comparing candidate Kp, Ki, and Kd values. section 8.7

def objective_function(t, error, Kp,Ki,Kd, criterion):
    
    if criterion == 1:
        ITAE = np.trapz(t, t*error)                                    #1
        objfnc= ITAE
    elif criterion == 2:
        ITSE = np.trapz(t, t*error**2)                                 #2
        objfnc= ITSE
    elif criterion == 3:
        IAE =  np.trapz(t, error)                                      #3
        objfnc= IAE
    elif criterion == 4:
        ISE = np.trapz(t, error**2)                                    #4
        objfnc= ISE
    elif criterion == 5:
        T, _, _ =transfer_function(Kp,Ki,Kd)
        info = control.step_info(T)
        beta = 1
        Mp = info['Overshoot']
        tr = info['RiseTime']
        ts = info['SettlingTime']
        ess = abs(1-info['SteadyStateValue'])
        W = ((1-math.exp(-beta))*(Mp+ess))+((math.exp(-beta))*(ts-tr)) #5
        objfnc=W;      
        
    return objfnc

The implementation lists ISE alongside ITAE, ITSE, and IAE as selectable objective functions. section 8.7

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

How deep it goes
Defines ISE and implements numerical integration, but does not derive integration methods or analyze the metric’s deeper control-theoretic trade-offs.
Examples
Uses heater temperature control, simulated step responses, and NumPy-based objective-function code.
What it assumes
Assumes familiarity with PID parameters, tracking error, and simulated closed-loop responses.
What you can do after
Evaluate PID parameter sets with a numerical squared-error objective during controller optimization.
How it connects
Connects ISE with PID control, step response, genetic algorithms, IAE, ITAE, and ITSE.
Cover of Optimization Algorithms: AI techniques for design, planning, and control problems
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