How to describe the ISE perfome index in discrete time?

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.

Answer 1

[Sachin Lodhi]

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