The open-loop transfer function is equal to the product of all transfer function blocks in the forward path in the block diagram.
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.
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.
Therefore, a crucial design parameter is the location of the eigenvalues, or closed-loop poles.
Each point on the locus satisfies the angle condition and magnitude condition and corresponds to a different value of K. For negative feedback systems, the closed-loop poles move along the root-locus from the open-loop poles to the open-loop zeroes as the gain is increased.
For this reason, the root-locus is often used for design of proportional control, i.e. those for which
For this system, the open-loop transfer function is the product of the blocks in the forward path,
The product of the blocks around the entire closed loop is
In general, the solution will be n complex numbers where n is the order of the characteristic polynomial.
The preceding is valid for single-input-single-output systems (SISO).
are matrices whose elements are made of transfer functions.