The expression is referred to as the closed-loop transfer function of the system.

An example of a closed-loop transfer function is shown below:

The closed-loop transfer function is measured at the output.

The closed-loop transfer function may also be obtained by algebraic or block diagram manipulation.

The characteristic equation is nothing more than setting the denominator of the closed-loop transfer function to zero (0).

When this suppression vanishes, the parasympathetic system may overreact, as typical for any Closed-loop transfer function.

The solution is to apply feedback such that all poles of the closed-loop transfer function are at the origin of the z-plane.

The output signal waveform can be calculated from the closed-loop transfer function and the input signal waveform.

Once the closed-loop transfer function is obtained for the system, the closed-loop poles are obtained by solving the characteristic equation.

As a first step, suppose we only consider (the plant without a delay) and design a controller with a closed-loop transfer function that we consider satisfactory.

Closed-loop poles are the positions of the poles (or eigenvalues) of a closed-loop transfer function in the s-plane.

Having the PID controller written in Laplace form and having the transfer function of the controlled system makes it easy to determine the closed-loop transfer function of the system.

The closed-loop transfer function is obtained by dividing the open-loop transfer function by the sum of one (1) and the product of all transfer function blocks throughout the feedback loop.

A closed-loop transfer function in control theory is a mathematical expression (algorithm) describing the net result of the effects of a closed (feedback) loop on the input signal to the circuits enclosed by the loop.

The additional pole at the origin is desirable because when considering the closed-loop transfer function of the PLL, this pole at the origin integrates the error signal and causes the system to track the input with one more order.

If the feedback loops in the system are opened (that is prevented from operating) one speaks of the open-loop transfer function, while if the feedback loops are operating normally one speaks of the closed-loop transfer function.