Closed-loop pole

Closed-loop pole

Closed-loop poles are the positions of the poles (or eigenvalues) of a closed-loop transfer function in the s-plane. 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 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. The closed-loop transfer function may also be obtained by algebraic or block diagram manipulation. Once the closed-loop transfer function is obtained for the system, the closed-loop poles are obtained by solving the characteristic equation. The characteristic equation is nothing more than setting the denominator of the closed-loop 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 s-plane 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 open-loop transfer function, while if the feedback loops are operating normally one speaks of the closed-loop transfer function. For more on the relationship between the two see root-locus.

Closed-loop 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 steady-state response. Therefore, a crucial design parameter is the location of the eigenvalues, or closed-loop poles.

In root-locus 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 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 \textbf{G}_c = K.

Finding closed-loop poles

Consider a simple feedback system with controller \textbf{G}_c = K, plant \textbf{G}(s) and transfer function \textbf{H}(s) in the feedback path. Note that a unity feedback system has \textbf{H}(s)=1 and the block is omitted. For this system, the open-loop transfer function is the product of the blocks in the forward path, \textbf{G}_c\textbf{G} = K\textbf{G}. The product of the blocks around the entire closed loop is \textbf{G}_c\textbf{G}\textbf{H} = K\textbf{G}\textbf{H}. Therefore, the closed-loop transfer function is

\textbf{T}(s)=\frac{K\textbf{G}}{1+K\textbf{G}\textbf{H}}.

The closed-loop poles, or eigenvalues, are obtained by solving the characteristic equation {1+K\textbf{G}\textbf{H}}=0. 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 \textbf{G}(s) and \textbf{K}(s) are matrices whose elements are made of transfer functions. In this case the poles are the solution of equation:

det(\textbf{I}+\textbf{G}(s)\textbf{K}(s))=0


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • Loop antenna — A shortwave loop antenna A loop antenna is a radio antenna consisting of a loop (or loops) of wire, tubing, or other electrical conductor with its ends connected to a balanced transmission line. Within this physical description there are two very …   Wikipedia

  • Magnetic pole strength — (symbol: p) is a physical quantity used to measure the strength of the pole of a bar magnet (or a hypothetical magnetic monopole). If there is an infinitely long wire where the electric current is I, then the magnetic pole strength is defined as… …   Wikipedia

  • Root locus — In control theory, the root locus is the locus of the poles and zeros of a transfer function as the system gain K is varied on some interval. The root locus is a useful tool for analyzing single input single output (SISO) linear dynamic systems.… …   Wikipedia

  • Full state feedback — (FSF), or pole placement, is a method employed in feedback control system theory to place the closed loop poles of a plant in pre determined locations in the s plane. Placing poles is desirable because the location of the poles corresponds… …   Wikipedia

  • Operational amplifier applications — This article illustrates some typical applications of operational amplifiers. A simplified schematic notation is used, and the reader is reminded that many details such as device selection and power supply connections are not shown. Contents 1… …   Wikipedia

  • Angle condition — In mathematics, the angle condition is a constraint that is satisfied by the locus of points in the s plane on which closed loop poles of a system reside. In combination with the magnitude condition, these two mathematical expressions fully… …   Wikipedia

  • Magnetic field — This article is about a scientific description of the magnetic influence of an electric current or magnetic material. For the physics of magnetic materials, see magnetism. For information about objects that create magnetic fields, see magnet. For …   Wikipedia

  • Control theory — For control theory in psychology and sociology, see control theory (sociology) and Perceptual Control Theory. The concept of the feedback loop to control the dynamic behavior of the system: this is negative feedback, because the sensed value is… …   Wikipedia

  • Operational amplifier — A Signetics μa741 operational amplifier, one of the most successful op amps. An operational amplifier ( op amp ) is a DC coupled high gain electronic voltage amplifier with a differential input and, usually, a single ended output.[1] An op amp… …   Wikipedia

  • Nyquist stability criterion — The Nyquist plot for . When designing a feedback control system, it is generally necessary to determine whether the closed loop system will be stable. An example of a destabilizing feedback control system would be a car steering system that… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”