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This article mainly introduces the difference between time-varying system and time-invariant system, the article is very detailed, has a certain reference value, interested friends must read it!

Difference: one or more parameter values in a time-varying system change with time, so that the whole characteristic changes with time, while the property of the time-invariant system does not change with time. The behavior of the system response only depends on the behavior of the input signal and the characteristics of the system, but has nothing to do with the time when the input signal is applied.

According to whether the system contains components whose parameters vary with time, automatic control systems can be divided into two categories: time-varying systems and time-invariant systems.

Time-invariant system is also called time-invariant system, which is characterized by that the nature of the system (the essential properties of the object studied, such as mass, moment of inertia, etc.) does not change with time. Specifically, the response behavior of the system only depends on the behavior of the input signal and the characteristics of the system, but has nothing to do with the time when the input signal is applied, that is, if the input u (t) produces an output y (t), it is applied to the system after the input delay τ, and the output produced by u (t-τ) is y (t-τ).

Time-invariant system is also called time-invariant system

That is, the nature of the system does not change with time. Specifically, the response behavior of the system only depends on the behavior of the input signal and the characteristics of the system, and has nothing to do with the time when the input signal is applied. That is to say, when I input u in T1, the output is y, then I input u in T2, and the output value is still y.

Time-varying system

A system in which one or more of the parameter values change over time, so that the whole characteristic changes over time.

A rocket is a typical example of a time-varying system whose mass decreases with time due to fuel consumption in flight; another common example is a manipulator whose moment of inertia around the corresponding axis is a complex function with time as an independent variable.

A time-invariant system is a system whose output does not change directly with time.

If the input signal

Generate the output y (t), then for the input with arbitrary time delay

You will get the output with the same time delay.

This property can be satisfied if the transfer function of the system is not a function of time. This feature can also be described in schematic terms.

If a system is time-invariant, then the system block diagram and the block diagram of any delay time are interchangeable.

To show how to determine that a system is a time-invariant system, let's look at two systems:

System A:

System B:

Because system An is explicitly dependent on t in addition to x (t) and y (t), it is a time-varying system, while system B is not explicitly dependent on time t, so it is time-invariant.

Mathematical analysis:

Suppose that the input of a system is u (t) and the corresponding output is y (t).

When the input is delayed by τ, that is, when the input is u (t-τ), if the output delays τ accordingly, that is, the output y (t-τ), then the system is a time-invariant system.

That is, when the input signal u (t) advanced row time shift τ is u (t-τ), then the value H [u (t-τ)] is obtained by system transformation H [▪].

To put it bluntly, a system runs for a period of time T from the initial time, and the input and output of this period of time has a corresponding trajectory.

If the state of the T moment of the system is re-run for a period of time as the initial moment, the input changes from the initial time to the same as before, and see if the output is the same as before.

(for example, in the case of a rocket, the input refers to the push energy, and the output refers to the acceleration.

The first time the rocket is launched under normal conditions, when running T time, the input energy, acceleration and time can draw a three-dimensional curve Q1.

The second rocket is launched in the same state as the T moment of the first time, when running T time (the input energy varies with time the same as the first time). At this time, the input energy, acceleration output, and time change draw another three-dimensional curve Q2Q Q1 and Q2 these two curves do not coincide in the output acceleration (it will certainly change, because the mass of the T moment becomes smaller).

It is equal to the input signal u (t) by system transformation H [▪] to get y (t), and then to time shift to get the same value y (t-τ), that is, H [u (t-τ)] = y (t-τ).

Such as:

1. Judge whether the system y (t) = coss [u (t)], t > 0 is a time-invariant system:

1)。 The input signal u (t) is first time shifted to u (t-τ), and then the value obtained by system transformation is cos [u (t-τ)], t > 0.

2)。 The input signal u (t) is first transformed into coss [u (t)], and then the value obtained by time shift is cos [u (t-τ)], t > 0.

The two are equal, so the system is time-invariant.

two。 Determine whether the system y (t) = u (t) ▪ cost is a time-invariant system:

1)。 The input signal u (t) is first time shifted to u (t-τ), and then the value obtained by system transformation is u (t-τ) ▪ cost,t > 0.

2)。 The input signal u (t) is first transformed into u (t) ▪ cost, and then the value obtained by time shift is u (t-τ) ▪ cos (t-τ), t > 0

U (t-τ) ▪ cost ≠ u (t-τ) ▪ cos (t-τ), so the system is time-varying.

In the end, we will see whether the final image of the two paths (that is, the image in the lower right corner) coincides.

Note: the above analysis method can also use the method I said to test whether it is a steady system, that is, suppose that T1 input u, output is y, see T2 time input u, output value is still y

These are all the contents of this article entitled "what's the difference between time-varying systems and time-invariant systems". Thank you for reading! Hope to share the content to help you, more related knowledge, welcome to follow the industry information channel!

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