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.
Table of Content
- Time-Variant System
- Time-Invariant System
- Determination of Time-Variant and Time-Invariant Control System
Mathematical Equation of a System- Solved Examples of Time-Variant and Invariant Control System
- Difference Between Time-Variant and Time-Invariant System
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
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.
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