• Open Loop Transfer Function
• Closed Loop Transfer Function Examples
• Closed Loop Transfer Function Block Diagram

I'm trying to convert it to a closed loop system with a gain of 0.987 and damping ratio of 0.7071 and find the closed loop transfer function. I know open loop transfer functions are given by KG(s)H(s) and closed loop transfer functions are given by KG(s)/1+KG(s)H(s) but I'm not sure how to relate this to the equation I have here.

1. Meet the Instructors. So when we talk about an open-loop transfer function, what we mean is the transfer function of the original power stage before introducing a feedback loop. On the other hand, the closed-loop transfer functions, are what we get with the feedback loop present. So, they're the transfer functions like these.
2. Frequency Domain Representation of PID Controller. In Frequency Domain (after taking Laplace Transform of both sides),the control input can be represented as. Thus,PID controller adds pole at the origin and two zeroes to the Open loop transfer function. The Closed loop Transfer Function of the system can be written as.

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.

Overview[edit]

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

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

The summing node and the G(s) and H(s) blocks can all be combined into one block, which would have the following transfer function:

${\displaystyle {\displaystyle \frac{Y(s)}{X(s)}}={\displaystyle \frac{G(s)}{1+G(s)H(s)}}}$

Open Loop Transfer Function

Derivation[edit]

We define an intermediate signal Z shown as follows:

Using this figure we write:

${\displaystyle Y(s)=G(s)Z(s)}$

${\displaystyle Z(s)=X(s)-H(s)Y(s)}$

${\displaystyle Y(s)=G(s)(X(s)-H(s)Y(s))=G(s)X(s)-G(s)H(s)Y(s)}$

${\displaystyle Y(s)+G(s)H(s)Y(s)=G(s)X(s)}$

${\displaystyle Y(s)(1+G(s)H(s))=G(s)X(s)}$

${\displaystyle \Rightarrow {\displaystyle \frac{Y(s)}{X(s)}}={\displaystyle \frac{G(s)}{1+G(s)H(s)}}}$

See also[edit]

References[edit]

• This article incorporates public domain material from the General Services Administration document 'Federal Standard 1037C'.

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 (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[edit]

The response of a linear and time invariant 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 gainK 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 ${\displaystyle {\mathbf{\text{G}}}_c=K}$.

Finding closed-loop poles[edit]

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

Closed Loop Transfer Function Examples

${\displaystyle \mathbf{\text{T}}(s)=\frac{K\mathbf{\text{G}}}{1+K\mathbf{\text{G}}\mathbf{\text{H}}}.}$

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

${\displaystyle det(\mathbf{\text{I}}+\mathbf{\text{G}}(s)\mathbf{\text{K}}(s))=0.}$

Closed Loop Transfer Function Block Diagram