Control systems can be broadly classified into two categories based on their behavior with respect to time: time-invariant systems and time-variant systems. These terms refer to how the system's parameters or characteristics change in relation to time.
Time-Invariant Systems:
A time-invariant system is one in which the system's parameters or characteristics do not change with time. In other words, the system's behavior remains constant regardless of when it is observed. This property simplifies the analysis and design of control systems because the underlying dynamics do not change over time.
Mathematically, a time-invariant system can be described by the following property:
If an input signal
(
)
x(t) produces an output
(
)
y(t), then for any time delay
T, the input signal
(
−
)
x(t−T) will produce the output
(
−
)
y(t−T).
Examples of time-invariant systems include electrical circuits with fixed components, mechanical systems with constant parameters, and systems with linear time-invariant (LTI) characteristics.
Time-Variant Systems:
A time-variant system is one in which the system's parameters or characteristics change with time. This means that the behavior of the system varies as time progresses. Time-variant systems can be more complex to analyze and design because their dynamics change over time, introducing additional challenges in control.
Mathematically, a time-variant system can be described as follows:
If an input signal
(
)
x(t) produces an output
(
)
y(t), there may not be a direct relationship between the input signal
(
−
)
x(t−T) and the output
(
−
)
y(t−T) due to changing system parameters.
Examples of time-variant systems include systems with components that degrade or change over time, systems affected by external factors like temperature or humidity variations, and nonlinear systems with changing characteristics.
In control system analysis and design, it's often desirable to work with time-invariant systems because they offer more predictable and manageable behavior. Many classical control theories, like those based on Laplace transforms, assume time-invariant system properties. However, modern control techniques and adaptive control strategies have been developed to address time-variant systems, allowing for control in situations where parameters may change dynamically.
In summary, the distinction between time-invariant and time-variant systems lies in whether the system's parameters remain constant or change with time. Time-invariant systems have constant behavior over time, while time-variant systems exhibit changing behavior as time progresses.