Question 4 of 8: Polar Plot and Nyquist Stability with an Unstable Open-Loop Pole

Question 4: Polar Plot and Nyquist Stability with an Unstable Open-Loop Pole (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. $G_{open}(s)=K(s+1)/[s(s-1)]$ — unity negative feedback, ONE open-loop pole in the right-half plane ($s=+1$) in addition to the pole at the origin.

Find. 1) The real/imaginary crossover coordinates of the normalized polar plot and the sketch. 2) The range of $K_p>0$ for closed-loop stability via Nyquist.

Approach. Split $G_{open}(j\omega)/K$ into real and imaginary parts as explicit functions of $\omega$ and solve $\text{Im}=0$ for the real-axis crossover. Because there is an open-loop RHP pole ($P=1$), simple gain/phase-margin shortcuts do not apply; use the full Nyquist criterion $Z=N+P$ (closed-loop RHP poles = CW encirclements of $-1$ plus open-loop RHP poles), then cross-check the resulting $K$-range directly against Routh–Hurwitz on the closed-loop characteristic equation.

1. Part 1) — real/imaginary parts and the crossover. $$\frac{G_{open}(j\omega)}{K}=\frac{1+j\omega}{j\omega(j\omega-1)} =\frac{-2}{\omega^2+1}+j\,\frac{1-\omega^2}{\omega(\omega^2+1)}.$$ As $\omega\to0^+$: $\text{Re}\to-2$, $\text{Im}\to+\infty$ (a vertical asymptote at $\text{Re}=-2$). As $\omega\to\infty$: both parts $\to0^-$ (the locus spirals into the origin). Setting $\text{Im}=0$: $1-\omega^2=0\Rightarrow\omega=1$ rad/s, where $\text{Re}=-2/(1+1)=-1$. So the normalized plot crosses the negative real axis at exactly $\boxed{(-1,0)\text{ at }\omega=1\ \text{rad/s}}$ — the ACTUAL (non-normalized) locus of $G_{open}(j\omega)$ therefore crosses the real axis at $\boxed{-K}$ at that same frequency. The sketch above shows the branch for $\omega>0$ (solid, arrow showing $\omega$ increasing from the $-2+j\infty$ asymptote down through $(-1,0)$ into the origin) and its complex-conjugate mirror for $\omega\lt0$ (dashed).
2. Part 2) — Nyquist contour and stability range. The standard $\Gamma$ contour runs up the $j\omega$-axis, is indented by a small clockwise-avoiding semicircle bulging INTO the right-half-plane around the pole at $s=0$ (so that pole is excluded from the enclosed region, matching $P=$ count of RHP poles only), and needs no large arc at infinity since $G_{open}(s)\to0$ as $|s|\to\infty$ (relative degree $\ge1$). There is $P=1$ open-loop pole in the RHP ($s=+1$); for a stable closed loop we need $Z=0$ closed-loop RHP poles, i.e. $N=Z-P=-1$: exactly ONE counter-clockwise encirclement of the $-1$ point. From Part 1, the actual locus crosses the negative real axis at $-K$ (at $\omega=\pm1$) and is otherwise confined to $\text{Re}\lt0$; the $-1$ point is encircled (counter-clockwise, satisfying $N=-1$) precisely when the crossing $-K$ lies FARTHER left than $-1$, i.e. $K>1$. Cross-check via Routh–Hurwitz on the closed-loop characteristic equation $1+G_{open}(s)=0\Rightarrow s(s-1)+K(s+1)=0\Rightarrow s^2+(K-1)s+K=0$: for a 2nd-order polynomial, stability needs every coefficient positive, i.e. $K-1>0$ AND $K>0$, giving the identical result: $$\boxed{K>1\ \text{for closed-loop stability}}$$ (at exactly $K=1$ the real-axis crossing lands ON $-1$, matching the marginal case $K-1=0$ in the Routh test).