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STEM 隨筆︰古典力學︰轉子【五】《電路學》四【電容】IV‧Laplace‧D‧後

時至棒擊球飛之際︰

 

且已明白『時移算子』意義知其確指︰

Time-invariant system

A time-invariant (TIV) system has a time-dependent system function that is not a direct function of time. Such systems are regarded as a class of systems in the field of system analysis. The time-dependent system function is a function of the time-dependent input function. If this function depends only indirectly on the time-domain (via the input function, for example), then that is a system that would be considered time-invariant. Conversely, any direct dependence on the time-domain of the system function could be considered as a “time-varying system”.

Mathematically speaking, “time-invariance” of a system is the following property:

Given a system with a time-dependent output function \displaystyle y(t) ,and a time-dependent input function \displaystyle x(t) ; the system will be considered time-invariant if a time-delay on the input \displaystyle x(t+\delta ) directly equates to a time-delay of the output \displaystyle y(t+\delta ) function. For example, if time \displaystyle t is “elapsed time”, then “time-invariance” implies that the relationship between the input function \displaystyle x(t) and the output function \displaystyle y(t) is constant with respect to time \displaystyle t :
\displaystyle y(t)=f(x(t),t)=f(x(t))

In the language of signal processing, this property can be satisfied if the transfer function of the system is not a direct function of time except as expressed by the input and output.

In the context of a system schematic, this property can also be stated as follows:

If a system is time-invariant then the system block commutes with an arbitrary delay.

If a time-invariant system is also linear, it is the subject of linear time-invariant theory (linear time-invariant) with direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. Nonlinear time-invariant systems lack a comprehensive, governing theory. Discrete time-invariant systems are known as shift-invariant systems. Systems which lack the time-invariant property are studied as time-variant systems.

Abstract example

We can denote the shift operator by \displaystyle \mathbb {T} _{r} where \displaystyle r is the amount by which a vector’s index set should be shifted. For example, the “advance-by-1” system

\displaystyle x(t+1)=\,\!\delta (t+1)*x(t)

can be represented in this abstract notation by

\displaystyle {\tilde {x}}_{1}=\mathbb {T} _{1}\,{\tilde {x}}

where \displaystyle {\tilde {x}} is a function given by

\displaystyle {\tilde {x}}=x(t)\,\forall \,t\in \mathbb {R}

with the system yielding the shifted output

\displaystyle {\tilde {x}}_{1}=x(t+1)\,\forall \,t\in \mathbb {R}

So \displaystyle \mathbb {T} _{1} is an operator that advances the input vector by 1.

Suppose we represent a system by an operator \displaystyle \mathbb {H} . This system is time-invariant if it commutes with the shift operator, i.e.,

\displaystyle \mathbb {T} _{r}\,\mathbb {H} =\mathbb {H} \,\mathbb {T} _{r}\,\,\forall \,r

If our system equation is given by

\displaystyle {\tilde {y}}=\mathbb {H} \,{\tilde {x}}

then it is time-invariant if we can apply the system operator \displaystyle \mathbb {H} on \displaystyle {\tilde {x}} followed by the shift operator \displaystyle \mathbb {T} _{r} , or we can apply the shift operator \displaystyle \mathbb {T} _{r} followed by the system operator \displaystyle \mathbb {H} , with the two computations yielding equivalent results.

Applying the system operator first gives

\displaystyle \mathbb {T} _{r}\,\mathbb {H} \,{\tilde {x}}=\mathbb {T} _{r}\,{\tilde {y}}={\tilde {y}}_{r}

Applying the shift operator first gives

\displaystyle \mathbb {H} \,\mathbb {T} _{r}\,{\tilde {x}}=\mathbb {H} \,{\tilde {x}}_{r}

If the system is time-invariant, then

\displaystyle \mathbb {H} \,{\tilde {x}}_{r}={\tilde {y}}_{r}

 

故曉拉普拉斯變換實論述『當下眼前』也◎

Laplace Transform

Properties

Time Delay

The time delay property is not much harder to prove, but there are some subtleties involved in understanding how to apply it.  We’ll start with the statement of the property, followed by the proof, and then followed by some examples.  The time shift property states

We again prove by going back to the original definition of the Laplace Transform

Because

we can change the lower limit of the integral from 0– to a– and drop the step function (because it is always equal to one)

We can make a change of variable

The last integral is just the definition of the Laplace Transform, so we have the time delay property

To properly apply the time delay property it is important that both the function and the step that multiplies it are both shifted by the same amount.  As an example, consider the function f(t)=t·γ(t).  If we delay by 2 seconds it we get (t-2)·γ(t-2), not (t-2)·γ(t) or t·γ(t-2).  All four of these function are shown below.

 

The correct one is exactly like the original function but shifted.

Important: To apply the time delay property you must multiply a delayed version of your function by a delayed step.  If the original function is  g(t)·γ(t), then the shifted function is g(t-td)·γ(t-td) where td is the time delay.

© Copyright 2005 to 2015 Erik Cheever    This page may be freely used for educational purposes.