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Recall that a LTI state space of a dynamical system is given as:
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?
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).
That is a feedforward term. As long as the plant model is understood & accurately captured within a controller, a feedforward term can provide a performance boost that can mitigate aspects of a PID controller.
Take for instance a controller to regulate the speed of a car. If correctly modeled, a given amount of accelerator-pedal would be needed for a given speed to overcome drag. Rather than waiting on a closed-loop PID to increase the accelerator force to overcome the present drag + acceleration, the present drag component can be used as a feedforward term into the overall control topology.
"If D was not zero, it would mean that the output is dependent upon the input."
Even if D is zero, the output is dependent on the input (and the initial condition of your integrator(s) ). The input affects the state dynamics, and the output is observed from those same dynamics.
For a state space system where the D matrix (or vector or scalar) as you have drawn just means that the equivalent transfer function(s) contain zeros. The presence of D means that there is a component of the output that changes instantaneously when the input changes.
