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-

Block diagram of Time-Variant and Time-Invariant Control System

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

Time-Variant Control System	Time-Invariant Control System
Complex to design and implement.	Simple to design and implement.
Behavior change dynamically over time	Behavior remains relatively stable.
Less stable .	More stable.
Real time adjustment is needed.	Static control parameters.
Response is less predictable.	Response is more predictable.

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.

Frequently Asked Questions

1. What do you understand by control system?

Control system is a system in which output of the system is controlled by the varying it’s input.

2. Define time-variant and time-invariant control system.

If a control system is time dependent then it is known as Time-variant control system, and if control system is time independent then system is known as Time-invariant control system.

3. Describe types of control system.

There are two types of control system :-

1. Open loop control system: In open loop control system input is independent from output, means there is no feedback system.
2. Close loop control system : In close loop control system input is dependent on output, means there is feedback system. Output is given as input to the control system.