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How to describe the ISE perfome index in discrete time?

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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.

Answers (1)

Sachin Lodhi
Sachin Lodhi on 28 Dec 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.
Please refer to the following link for more information on Discrete-Time Integrator block - https://www.mathworks.com/help/simulink/slref/discretetimeintegrator.html
Hope this helps.
Best Regards,
Sachin

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