Electronic – Control theory: is there any actual application for D in ABCD matrix

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Recall that a LTI state space of a dynamical system is given as:

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Is there any actual purpose for D in the output equation?

If D was not zero, it would mean that the output is "directly" dependent upon the input. Does anyone know any system that behave or designed in such a way?

Best Answer

A lag-lead compensator in state-space form will have non-zero \$D\$.

In \$H_2\$ or \$H_{\infty}\$ control design if the performance variables include the input (e.g. keep it 'small') then again \$D\$ will be non-zero.

They can also appear when a continuous-time system is approximated as a discrete-time system (e.g. convert \$\frac{1}{s+p}\$ using Tustin's transform).