Steady-state Performance

DC gain 被定义为 zero frequency 处的 gain。

对于上面这个反馈系统：

  • Open-loop t.f.: \(L(s) = G(s)K(s)\),

    the open-loop DC gain is \(G(0)K(0)\).

  • Closed-loop t.f.: \(G_{\mbox{cl}}(s) = \frac{G(s)K(s)}{1 + G(s)K(s)}\),

    the closed-loop DC gain is \(G_{\mbox{cl}}(0) = \frac{G(0)K(0)}{1 + G(0)K(0)}\).

Consider steady-state errors in stable systems for general polynomial inputs

\[
r(t) = rt^{n - 1}
\]

它的拉普拉斯变换为 \(R(s) = \frac{Cr}{s^n}\) where \(C = (n - 1)!\)。

Steady-state error is given by

\[
e_{\mbox{ss}} = \lim_{s \rightarrow 0} \frac{Cr}{s^{n - 1} + s^{n - 1}G(s)K(s)}
\]

If \(G(s)K(s)\) contains \(n\) or more integrators, then the closed loop system will track an input given by \(r(t) = rt^{n−1}\) without any steady state error.

用表格表示：

\(r(t) = rt^{n - 1}\) Constant input
\[\begin{align}r(t) &= r\newline R(s) &= \frac{r}{s}\newline n &= 1\end{align}\]		 Ramp input
\[\begin{align}r(t) &= rt\newline R(s) &= \frac{r}{s^2}\newline n&= 2\end{align}\]		 Parabolic input
\[\begin{align}r(t) &= rt^2\newline R(s) &= \frac{2r}{s^3}\newline n &= 3\end{align}\]
0 integrator in the loop \(\frac{r}{1 + p}\) \(\infty\) \(\infty\)
1 integrator in the loop \(0\) \(\frac{r}{p}\) \(\infty\)
2 integrator in the loop \(0\) \(0\) \(\frac{2r}{p}\)
3 integrator in the loop \(0\) \(0\) \(0\)

其中 \(p = P(0)\) 而 \(G(s)K(s) = \frac{1}{s^m}P(s)\)。

用 system type 来分类的话：

Number of integrators in the loop (System Type) Constant input
\[\begin{align}r(t) &= r\newline R(s) &= \frac{r}{s}\end{align}\]		 Ramp input
\[\begin{align}r(t) &= rt\newline R(s) &= \frac{r}{s^2}\end{align}\]		 Parabolic input
\[\begin{align}r(t) &= rt^2\newline R(s) &= \frac{2r}{s^3}\end{align}\]
Type 0 \(\frac{r}{1 + K_p}\) \(\infty\) \(\infty\)
Type 1 \(0\) \(\frac{r}{K_v}\) \(\infty\)
Type 2 \(0\) \(0\) \(\frac{2r}{K_a}\)
Type k ≥ 3 \(0\) \(0\) \(0\)

其中

  • Position Error Constant: \(K_p = \lim_{s \rightarrow 0} G(s)K(s)\)
  • Velocity Error Constant: \(K_v = \lim_{s \rightarrow 0} sG(s)K(s)\)
  • Acceleration Error Constant: \(K_a = \lim_{s \rightarrow 0} s^2G(s)K(s)\)

Transient Performance

Time constant

Consider the first-order system

\[
G(s) = \frac{K}{s\tau + 1}
\]

The larger the \(\tau\), the closer the pole to the origin → the slower the response.

Damping ratio

Consider the standard second-order system

\[
G(s) = \frac{K \omega_n^2}{s^2 + 2\zeta \omega_n s + \omega_n^2}
\]

  • \(\zeta < 1\), underdamped system, complex poles: as \(\zeta\) ↑, the response is more sluggish and less oscillations.
  • \(\zeta = 1\), critically-damped system, repeated real poles with no oscillations.
  • \(\zeta > 1\), over-damped system, distinct real poles.

Natural frequency

Consider the standard second-order system

\[
G(s) = \frac{K \omega_n^2}{s^2 + 2\zeta \omega_n s + \omega_n^2}
\]

The smaller the \(\omega_n\), the more sluggish is the response.