Time-Variant and Invariant Control System

Control systems play an important role in engineering, they help in regulating and controlling a process or a system to obtain controlled output. There are different types of control systems such as Linear and non-linear systems, Causal and Non-causal systems. Time variant and Time invariant control systems are one of them. In this article, we'll learn about the time-variant and invariant systems.

Time-Variant System

A system is said to be a time-variant system if its input /output characteristics change with time. If the control system is time-dependent then it is known as a time-variant control system.
In a time variant system time shift in the output signal differs from the time shift in the input signal.
Examples: Medical Ventilators, Agricultural Robots, Air Conditioner

Advantage of Time-Variant System

• These systems can easily adapt the changes in parameters or external disturbances.
• When the reference signal changes over time, the time-variant control system can provide better tracking.
• Time-variant control system can optimize the input to reduce energy consumption.
• These systems are used in biomedical applications such as to monitor patient condition.
• These systems are used in aerospace because the operation conditions change dynamically.

Disadvantage of Time-Variant System

• Time-variant control systems are often more complex to design and implement.
• Adaptive nature of these systems needs more computational resources, make them less suitable for real time applications.
• If these systems are not properly designed and tuned then they are less stable.
• Time-variant control systems are expansive.

Applications of Time-Variant System

• Time-variant control system is used in spacecraft and aircraft for guidance and control.
• Time-variant control system is used in Unmanned Arial Vehicles(UAVs).
• Time-variant control system is used in various industries like chemical processing and manufacturing.
• These systems are used in smart gird and in energy management.
• Used in agriculture.

Time-Invariant System

A system is said to be time-variant if it's input/output characteristics does not change with time. if a control system is time independent then it is known as time-variant control system.
In time-invariant control system time shift in output signal is same as the time shift in the input signal.
Examples:- Thermostats, Traffic lights, Elevator Control, Water Tank Level Control

Characteristics of Time Invariant System

• Time-invariant system follow the principle of superposition.
• The output of the system depends only on the past and present input
• The Time-invariant system is unaffected from the shift in input signal it means whenever the input signal is applied system produces the same output.

Advantage of Time-Invariant System

• Time-invariant control systems are simple to design and implement.
• Time-invariant control systems are more stable due to fixed control parameters.
• These type of system required less specialized knowledge to operate.

Disadvantage of Time-Invariant System

• Time-invariant control systems are less adaptive. It is not suited for the applications where the system dynamics change over time.
• Time-invariant systems are relatively stable system but over a long time changes can occurs due to the environmental variations.
• Time-invariant systems have limited robustness, when they faced external disturbances.
• Time-invariant control system is used in traffic signals.

Application of Time-Invariant System

• Time-invariant system is used in Signal Processing and Filtering to remove unwanted frequencies from signal.
• Time-invariant system is used in control system and feedback loops.
• Time-invariant system is used in communication system.

Determination of Time-Variant and Time-Invariant Control System

Consider the following figure-

In this figure, x(t) is input signal and y(t-T₀) and y'(t) both are output signal,

if, y(t-T₀) is equal to the y'(t) then the system is time invariant control system.
y(t-T₀) = y'(t)
otherwise,
if y(t-T₀) ≠ y'(t) , then the system is time-variant control system.

Mathematical Equation of a System

If the mathematical notation of a control system is given then follow the following equation to determine time-variant and time-invariant control system.
Let the given mathematical equation of a control system is -

y(t)=kx(2t)

1. first find the value of y(t,t₀), to find y(t,t₀) replace the value of x(t) by value of x(t-t₀).
2. second find the value of y(t-t₀) by replacing the value of 't' by t-t₀.
3. compare the value of y(t,t₀) with the value of y(t-t₀)
if, y(t,t₀) = y(t-t₀), then the given system is time invariant.
if, y(t,t₀) ≠ y(t-t₀), then the given system is time-variant.

y(t,t₀) is the calculation of response of system with input x(t-t₀).
y(t-t₀) is the calculation of response of system with time delay t₀ means at time (t-t₀).

Solved Examples of Time-Variant and Invariant Control System

Some pf the examples of this system are mentioned below:

Determine the following system is time-variant or time-invariant.

• y(t)=x(t)+x(t-3)
  solution:- y(t,t₀)=x(t-t₀) + x(t-t₀-3).
  y(t-t₀)=x(t-t₀) + (t-t₀-3).
  y(t,t₀)=y(t-t₀),
  so the system is time-invariant.
  
• y(t) = x²(t) +3t.
  solution:-y(t,t₀) = x²(t-t₀)+3t.
  y(t-t₀) = x²(t-t₀)+3(t-t₀)
  y(t,t₀) ≠ y(t-t₀),
  so the system is time-variant.
  
• y(t) = tx(t) - x(t+3)
  solution:- y(t,t₀) = tx(t-t₀) - x(t-t₀+3).
  y(t-t₀) = (t-t₀)x(t-t₀) - x(t-t₀+3)
  y(t,t₀) = y(t-t₀)
  so the system is time-variant.

Difference Between Time-Variant and Time-Invariant System

Conclusion

In this article, we learned the basic concept of two type of control system one is Time-variant and Time-invariant control system,
Advantages, disadvantage and applications of these type of control system. Also we see the main differences between them. These two system are also a part of control system which helps in controlling a system to give a certain output.