Control System - First order system

In summary, the time constant for the first order system is 15 seconds and the response to a changing linear temperature input can be determined by calculating a and setting K to eliminate steady state error.
  • #1
kloong
36
0
Q:
1) A thermometer requires 1min to indicate 98% of the response to be a step input. Assuming the thermometer to be a first order system, find the time constant.

2)If the thermometer is placed in a bath, the temperature of which is changing linearly at a rate of 10degrees/min, how much error does the thermometer show?

for the first part, i think that it takes 4 time constant tor for the system to reach 98%. So I am thinking that if i divide the 1min by 4 (i.e, 1min/4), id get the time constant, T. is this correct?

and for the second part, I am not sure how should i tackle it. can you give me an outline of what i should do?

thanks.
 
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  • #2
You got the first part right.The time constant is 15s.
For the second part, your system is of the form [tex]G(s)=\frac{k}{s+a}[/tex].
You can calculate a from the time constant. K can be determined requiring that the termometer has no steady state error.
Now, what is the response of your system to an input [tex]\frac{A}{s^2}[/tex]?
A = 10/60 degrees/s.
 
  • #3


1) Yes, your approach is correct. The time constant, T, can be calculated by dividing the response time, t, by 4. This is because in a first order system, it takes approximately 4 time constants for the system to reach 98% of the response to a step input. So, T = t/4 = 1min/4 = 0.25min.

2) To calculate the error, we need to first determine the change in temperature of the bath over the response time of the thermometer. This can be found by multiplying the rate of change (10 degrees/min) by the response time (1 min). So, the change in temperature is 10 degrees.

Next, we need to determine the change in temperature that the thermometer showed. This can be found by multiplying the change in temperature of the bath (10 degrees) by the thermometer's response to a step input (98%). So, the thermometer would show a change of 9.8 degrees.

Therefore, the error is the difference between the actual change in temperature (10 degrees) and the change shown by the thermometer (9.8 degrees), which is 0.2 degrees.
 

What is a first order control system?

A first order control system is a type of control system that has a single input and a single output, and its behavior can be described by a first order differential equation. It is often used to model simple systems with a linear response, such as a spring-mass system or an RC circuit.

What are the components of a first order control system?

The three main components of a first order control system are the input signal, the system, and the output signal. The input signal is the command or reference signal that is given to the system, the system is the physical or mathematical representation of the process being controlled, and the output signal is the result of the system's response to the input signal.

How is the time constant of a first order system calculated?

The time constant of a first order system can be calculated by taking the reciprocal of the system's gain. In other words, it is the time it takes for the output to reach 63.2% of its final value when the input is a step function. It can also be calculated by dividing the time constant of the system's transfer function by the system's gain.

What is the steady-state error in a first order control system?

The steady-state error in a first order control system is the difference between the desired output and the actual output of the system when the input is a constant or step function. It is a measure of the system's accuracy and can be minimized by adjusting the system's gain or using a different type of control system.

What are some real-world applications of first order control systems?

First order control systems are used in a variety of real-world applications, such as temperature control systems, speed control systems in vehicles, and process control systems in manufacturing. They are also commonly used in electronic devices, such as thermostats, to maintain a desired output based on a given input.

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