- Time Domain Analysis Definition: Time domain analysis of control system involves studying system behavior over time using linear differential equations.
- Step Function: A sudden change in input voltage, starting from zero and jumping to a constant value.
- Ramp Function: A function that increases linearly over time, starting from zero.
- Impulse Function: A brief, sudden input to the system, often used to study system response.
- First Order Control Systems: Systems where the highest power of s in the transfer function’s denominator is one, determining the system’s time response.
In a control system, energy storing elements like inductors and capacitors are often present. These elements affect the system’s energy state, causing it to take time to transition from one state to another. This transition period is called the transient time, and the changes in voltages and currents during this time are known as the transient response.
A transient response often involves oscillations, which can be sustained or decaying. The nature of the response depends on the system’s parameters. Any control system can be described with a linear differential equation, whose solution gives the system’s response. This approach is known as time domain analysis of the control system.
Step Function
Let us take an independent voltage source or a battery which is connected across a voltmeter via a switch, s. It is clear from the figure below, whenever the switch s is open, the voltage appears between the voltmeter terminals is zero. If the voltage between the voltmeter terminals is represented as v (t), the situation can be mathematically represented as
Now let us consider at t = 0, the switch is closed and instantly the battery voltage V volt appears across the voltmeter and that situation can be represented as,
Combining the above two equations we get
In the above equations if we put 1 in place of V, we will get a unit step function which can be defined as
Let’s examine the Laplace transform of the unit step function. To find the Laplace transform of any function, multiply it by e-st and integrate from 0 to infinity.
Fig 6.2.1
If input is R(s), then
Ramp Function
A ramp function is represented by a straight line starting from the origin and inclining upwards. It begins at zero and increases or decreases linearly over time.
Here in this above equation, k is the slope of the line.
Fig 6.2.2
Now let us examine the Laplace transform of ramp function. As we told earlier Laplace transform of any function can be obtained by multiplying this function by e-st and integrating multiplied from 0 to infinity.
Parabolic Function
Here, the value of function is zero when time t<0 and is quadratic when time t > 0. A parabolic function can be defined as,
Now let us examine the Laplace transform of parabolic function. As we told earlier Laplace transform of any function can be obtained by multiplying this function by e-st and integrating multiplied from 0 to infinity.
Fig 6.2.3
Impulse Function
Impulse signal is produced when input is suddenly applied to the system for infinitesimal duration of time. The waveform of such signal is represented as impulse function. If the magnitude of such function is unity, then the function is called unit impulse function. The first time derivative of step function is impulse function. Hence Laplace transform of unit impulse function is nothing but Laplace transform of first-time derivative of unit step function.
Fig 6.2.4
Time Response of First Order Control Systems
If the highest power of s in the denominator of a transfer function is one, it represents a first order control system.
Time Response for Step Function
Now a unit step input is given to the system, then let us analyze the expression of the output:
Fig 6.3.2It is seen from the error equation that if the time approaching to infinity, the output signal reaches exponentially to the steady-state value of one unit. As the output is approaching towards input exponentially, the steady-state error is zero when time approaches to infinity.
Let us put t = T in the output equation and then we get,
This T is defined as the time constant of the response and the time constant of a response signal is that time for which the signal reaches to its 63.2 % of its final value. Now if we put t = 4T in the above output response equation, then we get,
When the actual value of the response reaches to the 98% of the desired value, then the signal is said to be reached to its steady-state condition. This required time for reaching the signal to 98 % of its desired value is known as setting time and naturally setting time is four times of the time constant of the response. The condition of response before setting time is known as transient condition and condition of the response after setting time is known as steady-state condition. From this explanation, it is clear that if the time constant of the system is smaller, the response of the system reaches its steady-state condition faster.
Time Response for Ramp Function
In this case, during the steady-state condition, the output signal lags behind the input signal by a time equal to the time constant of the system. If the time constant of the system is smaller, the positional error of the response becomes lesser.
Time Response for Impulse Function
In the above explanation of time response of the control system, we have seen that the step function is the first derivative of ramp function and the impulse function is the first derivative of a step function. It is also found that the time response of step function is the first derivative of time response of ramp function and time response of impulse function is the first derivative of time response of step function.
