Nyquist图分析结论与根轨迹矛盾的错误排查请求

Hey there, let's break down where things might be going wrong with your Nyquist analysis vs. root locus observation. First, let's restate your setup clearly to align on the basics:

• Open-loop transfer function:
  $G(s) = K \frac{s+3}{s(s+1)}$
• Your Nyquist path analysis led you to conclude there's 1 counter-clockwise (CCW) encirclement of the $-1$ point, so $N=1$.
• You noted there's 1 open-loop pole enclosed by the Nyquist path (the pole at $s=0$), so $P=1$.
• Using the Nyquist stability criterion $N = P - Z$, you calculated $Z=0$—but your root locus sketch shows closed-loop poles in the right-half plane (RHP), which contradicts this.

Let's walk through the most likely mistakes that caused this mismatch:

1. You incorrectly counted open-loop poles for $P$ #

The Nyquist stability criterion defines $P$ as the number of open-loop poles strictly inside the open right-half plane (RHP)—poles on the imaginary axis (like your $s=0$ pole) are not included in $P$. For your system, both open-loop poles are at $s=0$ (imaginary axis) and $s=-1$ (left-half plane), so $P=0$, not $1$.

2. You misidentified encirclement direction or missed the detour around the imaginary-axis pole #

When dealing with poles on the imaginary axis, we have to add a small semicircular detour around them (usually into the RHP) to avoid passing through the pole. For your $s=0$ pole:

• As $s$ travels around this detour ($s=\epsilon e^{j\theta}$, $\epsilon\to0$, $\theta$ from $-\pi/2$ to $\pi/2$), $G(s)$ traces an infinite semicircle in the upper half-plane, connecting $+j\infty$ to $-j\infty$ via $+\infty$.

Combined with the main Nyquist plot (along the imaginary axis from $-j\infty$ to $j\infty$):

• For large $K$ (where the plot crosses the real axis left of $-1$), the combined path makes one clockwise (CW) encirclement of $-1$, not CCW. Since CW encirclements count as negative, $N=-1$, not $+1$.

3. You may have ignored the impact of $K$'s value #

Encirclements only happen when $K$ is large enough that the Nyquist plot crosses the real axis left of $-1$. The crossover point occurs at $\omega=\sqrt{3}$, and the magnitude there is $|G(j\sqrt{3})| = K \frac{\sqrt{2}}{2}$. So:

• If $K > \sqrt{2}$: the plot crosses left of $-1$, leading to encirclements.
• If $K < \sqrt{2}$: the plot crosses right of $-1$, no encirclements.

Correcting the calculation #

Using the corrected values:

• $P=0$ (no open-loop poles in the open RHP)
• For large $K$, $N=-1$ (1 CW encirclement)
• Plug into $N = P - Z$: $-1 = 0 - Z \implies Z=1$

This matches your root locus observation—one closed-loop pole in the RHP for large $K$. For small $K$ ($K < \sqrt{2}$), $N=0$, so $Z=0$, which also aligns with root locus (all closed-loop poles in the left-half plane).

备注：内容来源于stack exchange，提问作者eet