Time response Analysis in Control system

The evaluation of the system’s performance in relation to time is the main goal of the time response analysis. The following time-response graph illustrates:

It comprises of two parts, transient part and the steady state part.

When a control system receives an input, it takes some time for the output to stabilize. This stage of the response is referred to as the transient response, and the transient portion of the graph is depicted above. The graph’s steady portion is known to occur when the transient portion ends and the steady start occurs.

The common input signal is shown below:

A test signal r(t) is applied as the input to the system that results in the response c(t). The input signal of a system can take many forms.

Note: The input and output are functions that vary with time.

Here, we will discuss the transient response, steady state response, and the standard signals of the control system.

Transient Response

It is a component of the time response that, when the time is very long, reaches 0 (zero). The poles lying on the left half of the s-plane in the graph analysis with poles and zeroes produce the transient response. The part of the response where output continuously rises or falls can also be described in this way. The temporary component of the response is another name for the transient response.

Or

The difference in the system’s response from the equilibrium state is known as the transient response.

For example,

The switching time of a bi-polar transistor

The characteristics BJT or Bi-polar Junction transistor depicts the transient nature.

Steady-state response

The steady-state response is the one that follows the transient response. The poles on the fictitious axis provide the steady-state response in the graph analysis with poles and zeroes. Another way to put it is that it is the portion of the response where the output is constant. Additionally, the output can change periodically with equal amplitude and frequency. The steady-state component of the response is another name for the steady-state response. Since it is a result of the input signal, the forced response of the system is another name for it.

The transient and steady-state terms of the given equation are illustrated by a few examples.

Examples

Example 1: 5 + 2e^-t

Solution:

Here, the transient part of the equation is 2e^-t because as t approaches to infinity, the term becomes 0. Hence, 2e^-t is the transient term. In the case of the first term 5, it will remain same when t approaches infinity. Hence, 5 is the steady-state term of the equation.

Example 2: 10 + 5e^t

Solution:

Here, the first term, 10, is the steady-state term of the equation because it will remain the same when t approaches infinity. In the case of the second term, 5e^t, the result is infinity when t approaches infinity. Hence, it is not a transient term. It is because something to the power infinity is always infinity.

So, there is only a steady-state term in the equation.

Standard signals

1. Step Input signal
2. Ramp input signal
3. Sinusoidal input signal
4. Impulse input signal

Step Input signal

The step input displays constant time values for the positive value. For the negative value of the time signal, it has zero value. The transition takes the form of a step size with a constant value, and the initial value of the signal is. The signal is referred to as a step input signal if its constant value is 1, which is denoted as follows:

The value of the signal is:

0 for t=0 and

1 for t>0

The graph is a function of one variable named t.

Ramp input signal

The ramp input signal’s graph has a ramped shape. It shows the linear increase that starts at a particular location. The ramp signal’s value exhibits the continuous change with respect to time. For negative values, the signal’s value is 0. It indicates that the output is displayed for positive inputs.

The ramp function is represented as:

The value of the signal is:

At for t>0 and

0 for t<0

If the value of A is 1 when t>0. The signal is known as unit ramp signal.

Sinusoidal input signal

A sinusoidal input is one whose oscillations can be modeled by a sine-wave equation. Sinusoidal is the response of the linear process to a sinusoidal. The signal is sent out by:

The sinusoidal signal in the control system is represented as:

The sine wave starts from zero, covers positive value, reach zero, covers negative values, and again reaches zero, as shown above.

Impulse Input signal

The impulse signal is a particular kind of short-duration, high-amplitude signal. It implies that when the time reaches zero, the magnitude is approaching infinity. As a result, we can state that the signal’s value is infinite at time zero. If not, its value is zero.

Its integration from -infinity to infinity is 1, as shown above.

It is a purely physical non-existent signal that is based on the idea of an area. It is not predicated on the idea of amplitude. The representation of the impulse input signal is:(Time response Analysis in Control system)