 Closedloop pole

Closedloop poles are the positions of the poles (or eigenvalues) of a closedloop transfer function in the splane. The openloop transfer function is equal to the product of all transfer function blocks in the forward path in the block diagram. The closedloop transfer function is obtained by dividing the openloop transfer function by the sum of one (1) and the product of all transfer function blocks throughout the feedback loop. The closedloop transfer function may also be obtained by algebraic or block diagram manipulation. Once the closedloop transfer function is obtained for the system, the closedloop poles are obtained by solving the characteristic equation. The characteristic equation is nothing more than setting the denominator of the closedloop transfer function to zero (0).
In control theory there are two main methods of analyzing feedback systems: the transfer function (or frequency domain) method and the state space method. When the transfer function method is used, attention is focused on the locations in the splane where the transfer function becomes infinite (the poles) or zero (the zeroes). Two different transfer functions are of interest to the designer. If the feedback loops in the system are opened (that is prevented from operating) one speaks of the openloop transfer function, while if the feedback loops are operating normally one speaks of the closedloop transfer function. For more on the relationship between the two see rootlocus.
Closedloop poles in control theory
The response of a system to any input can be derived from its impulse response and step response. The eigenvalues of the system determine completely the natural response (unforced response). In control theory, the response to any input is a combination of a transient response and steadystate response. Therefore, a crucial design parameter is the location of the eigenvalues, or closedloop poles.
In rootlocus design, the gain, K, is usually parameterized. 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 closedloop poles move along the rootlocus from the openloop poles to the openloop zeroes as the gain is increased. For this reason, the rootlocus is often used for design of proportional control, i.e. those for which .
Finding closedloop poles
Consider a simple feedback system with controller , plant and transfer function in the feedback path. Note that a unity feedback system has and the block is omitted. For this system, the openloop transfer function is the product of the blocks in the forward path, . The product of the blocks around the entire closed loop is . Therefore, the closedloop transfer function is
.
The closedloop poles, or eigenvalues, are obtained by solving the characteristic equation . 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). An extension is possible for multiple input multiple output systems, that is for systems where and are matrices whose elements are made of transfer functions. In this case the poles are the solution of equation:
Categories:
Wikimedia Foundation. 2010.