# How to describe the ISE perfome index in discrete time?

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Emmanuel Bastar Torres il 23 Mag 2022
Risposto: Sachin Lodhi il 28 Dic 2023
Hi everyone,
I am working with the Integrate square error (ISE) performe index, but the formula that I found was this:
But I want to translate it in a discrete time because I am working with a MPC. Does anyone knows how it is matematically expressed?
Thanks in advance and have a good day.
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### Risposte (1)

Sachin Lodhi il 28 Dic 2023
Hello Emmanuel,
The Integral of Squared Error (ISE) performance index is commonly used in control theory to evaluate the quality of a control system. For a continuous-time system, the ISE is defined as:
where ( e(t) ) is the error signal, which is the difference between the desired output (reference) and the actual output of the system at time ( t ).
In a discrete-time setting, such as when working with a Model Predictive Control (MPC) system, the continuous integral is replaced by a summation over all time steps, since the system is evaluated at discrete intervals. The discrete-time ISE can be expressed as:
or, if the system is evaluated over a finite number of samples ( N ):
where:
( e[k] ) is the error at the kth time step,
( Ts ) is the sampling period (the time interval between successive samples).
For realizing the above equation, we can use a Discrete-Time Integrator block or a Sum block with enabled state to accumulate the squared error over time.
Hope this helps.
Best Regards,
Sachin
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