拉普拉斯变换

拉普拉斯变换（Laplace Transform）可帮助简化计算。

定义

信号（函数）\(f\)的拉普拉斯变换为函数\(F = \mathcal{L}(f)\)并且其定义为：

\[
F(s) = \int_{0^-}^{\infty}f(t)e^{-st}\mathrm{d}t
\]

其中\(s\)为复数并被称为频率参数（frequency variable）。

对于一些信号使用上述方程会很复杂。所以通常使用拉普拉斯变换的性质和常见的拉普拉斯变换对来进行求解。

常见的拉普拉斯变换对

\(f(t)\) \(F(s)\)
\(\delta (t)\) \(1\)
\(U(t)\) \(\frac{1}{s}\)
\(t\) \(\frac{1}{s^2}\)
\(t^k\) \(\frac{k!}{s^{k+1}}\)
\(e^{-at}\) \(\frac{1}{s+a}\)
\(te^{-at}\) \(\frac{1}{(s+a)^2}\)
\(\frac{1}{(k-1)!}t^{k-1}e^{-at}\) \(\frac{1}{(s+a)^k}\)
\(\sin (bt)\) \(\frac{b}{s^2+b^2}\)
\(\cos (bt)\) \(\frac{s}{s^2+b^2}\)
\(e^{-at}\sin (bt)\) \(\frac{b}{(s+a)^2+b^2}\)
\(e^{-at}\cos (bt)\) \(\frac{s+a}{(s+a)^2+b^2}\)

拉普拉斯变换的性质

  • Linearity: if \(f\) and \(g\) are any signals and \(\alpha\) and \(\beta\) are any scalar, we have

    \[
    \mathcal{L}\{\alpha f(t) + \beta g(t)\} = \alpha F(s) + \beta G(s)
    \]

  • Derivative: if a signal \(f\) is continous at \(t = 0\), then

    \[
    \mathcal{L}(f^{\prime}(t)) = s F(s) - f(0^-)
    \]

    • higher-order derivatives\[
      \mathcal{L}(f^n(t)) = s^n F(s) - \sum_{k = 0}^{n - 1}s^{n-1-k}f^{k}(0^-)
      \]

  • Integral:

    if

    \[
    g(t) = \int_0^t f(\tau)\mathrm{d}\tau
    \]

    then

    \[
    G(s) = \frac{1}{s} F(s)
    \]

  • Derivative of Transform (Multiplication by time):

    \[
    \begin{align}
    F^{\prime}(s) &= \mathcal{L}\{-tf(t)\}
    \newline
    \frac{\mathrm{d}^n}{\mathrm{d}s^n}F(s) &= (-1)^n\mathcal{L}\{t^nf(t)\}
    \end{align}
    \]

  • Time delay (shift in time-domain):

    if a function \(f(t)\) is delayed by \(t_0\) unit of time

    \[
    g(t) =
    \begin{cases}
    0,\ &0 \leq t < t_0
    \newline
    f(t - t_0),\ &t \geq t_0
    \end{cases}
    \]

    its Laplace transform is then given by

    \[
    G(s) = e^{-st_0}F(s)
    \]

  • Shift in frequency (shift in s-domain):

    \[
    \mathcal{L}(e^{-at}f(t)) = F(s + a)
    \]

  • Final value theorem:

    \[
    \lim_{s \rightarrow 0} sF(s) = \lim_{t \rightarrow \infty} f(t)
    \]

    Allows us to compute the constant steady-state value of a time function given its Laplace transform without having to solve the differential equation or perform inverse laplace transform.

    Note: the final value theorem is only applicable to stable system.

  • Convolution:

    The convolution of two signals \(f(t)\) and \(g(t)\) (denoted by \(h = f * g\)) is given by

    \[
    \begin{align}
    h(t) &= \int_0^t f(\tau)g(t - \tau)\mathrm{d}\tau
    \newline
    h(t) &= \int_0^t f(t - \tau)g(\tau)\mathrm{d}\tau
    \end{align}
    \]

    It can be shown that the Laplace transform is given by

    \[
    H(s) = F(s)G(s)
    \]

    Laplace transform turns convolution into multiplication.

汇总成表…

Time Function Laplace Transform Comments
\(\alpha f(t) + \beta g(t)\) \(\alpha F(s) + \beta G(s)\) Superposition
\(f(t - t_0)\) \(e^{-st_0}F(s)\) Time delay
\(f(at)\) \(\frac{1}{|a|}F(\frac{s}{a})\) Time scaling
\(e^{-at}f(t)\) \(F(s + a)\) Shift in frequency
\(f^{n}(t)\) \(s^nF(s) - \sum_{k = 0}^{n-1}s^{n-1-k}f^k(0^-)\) Differentiation
\(\int_0^t f(\tau)\mathrm{d}\tau\) \(\frac{1}{s} F(s)\) Integration
\(f(t) * g(t)\) \(F(s)G(s)\) Convolution
\(tf(t)\) \(-\frac{\mathrm{d}}{\mathrm{d}s}F(s)\) Multiplication by time
\(f(0^+)\) \(\lim_{s\rightarrow \infty} sF(s)\) Initial value theorem
\(\lim_{t\rightarrow \infty} f(t)\) \(\lim_{s\rightarrow 0} sF(s)\) Final value theorem

拉普拉斯逆变换

The easiest way to find \(f(t)\) from \(F(s)\), if \(F(s)\) is rational, is to expand \(F(s)\) as a sum of simpler terms that can be found in the tables.

The basic tool to perform this operation is called partial-fraction expansion.

传递函数

定义

The transfer function, \(G(s)\), of a system is the transfer gain from input, \(U(s)\), to output, \(Y (s)\).

It is the ratio of the Laplace transform of the output to the Laplace transform of the input:

\[
G(s) = \frac{Y(s)}{U(s)}
\]

这里有个重要的假设是零初始条件。

如果\(u(t)\)是一个 unit impulse，那么\(U(s) = 1\)，所以可得\(Y(s) = G(s)\)。说明传递函数\(G(s)\)是 unit impulse response 也就是\(h(t)\)的拉普拉斯变换。

Non-zero initial conditions

Most real-world problems do not satisfy the zero-initial conditions assumption which is key to derivation of the transfer function.

It is possible to relax the zero initial condition assumption. Output signal may be derived from transfer functions as long as system is initially at rest, i.e., \(\left(y(0)\neq 0;\ \frac{\mathrm{d}^ny}{\mathrm{d}t^n} = 0\right)\)

For systems with non-zero initial conditions, we introduce the following dummy output and input variables:

\[
\begin{align}
\tilde{y}(t) &= y(t) - y(0)
\newline
\tilde{u}(t) &= u(t) - u(0)
\end{align}
\]

其中\(y(0)\)和\(u(0)\)分别是输出和输入信号的初始状态。现在\(\tilde{y}(0)\)和\(\tilde{u}(0)\)就都为零了。