Dear All,

This could be a solution for this delay system, however, it should be remembered that Smith's predictor is tested by the lack of robustness with respect to the uncertainty on the delay, while the robustness with respect to the parameters is less problematic. You will have to deal with this problem by choosing a particular configuration of the controller.

Best regards

Dears Zeashan H. Khan

Thanks for your comments. Although at the first glance, it seems they are in direct correspondence, whereas, I have another viewpoint. I think error (e) and delay (d), are of different conflicting dimensions, time and space, which are both independent absolute variables of our objective reality. Hence, logically we cannot make a direct correspondence between them. It is like two cars, which are moving with the velocities v1=1 m/s, and v2=5 m/s. Their distance (error) is continually increasing, however, there is no delay in their response !

Hence, we can only logically say that, an irregular quadrilateral could be constructed at any instance of time, as depicted in the image attachment to this comment. What we can do to decrease error and delay together, is to only decrease the area of this irregular quadrilateral, which is an infinitesimal representation of the problem geometry.

Warning: although a spring is mathematically modelled by the relation F=-k*x, and if k=1, we have F=-x, and we can compare x and F together, while they are of different dimensions, whereas for delay and error such a situation does not at all exist, due to some subtle considerations between the command and output signals when we look at the illustrations. At any instance of time, we can imagine irregular quadrilaterals, with respect to a non-binding specific time interval we select. Meanwhile error and delay variation over time has a specific dynamics with regard to any infinitesimal evolution of command and output signals.

Delay time may (or not) be of first order. The magnitude of the error that results from identifying the observed/measured variable (e.g. concentration; affected by first order delay) with the corresponding actual (real) concentration, can be estimated accordingly eq. 2.58 of the following reference (p. 25):

Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...

Dear Dr. Carlos Araújo Queiroz

Thank you for your contribution to this discussion.