The loop-gain function L(s) of a control system with unity feedback is given to be $L\left(s\right)=\frac{k}{\left(s+1\right)\left(s+2\right)\left(s+3\right)}$, where k > 0. If the gain cross-over frequency of the loop-gain function is less than its phase cross-over frequency, the closed-loop system is

Understanding Control System Stability via Frequency Response

This question explores the stability of a closed-loop control system. We are given the loop-gain function $L(s)=\frac{k}{\left(s+1\right)\left(s+2\right)\left(s+3\right)}$ for a system with unity feedback. A crucial condition is provided: the gain cross-over frequency (${\omega }_{gc}$) is less than the phase cross-over frequency (${\omega }_{pc}$), i.e., ${\omega }_{gc}<{\omega }_{pc}$. We need to determine the stability of the closed-loop system under this specific frequency relationship.

**Analyzing the Loop-Gain Function $L(s)$ **

The provided loop-gain function is:

$$L(s)=\frac{k}{(s+1)(s+2)(s+3)}$$

For frequency response analysis, we substitute $s$ with $j\omega$:

$$L(j\omega )=\frac{k}{(j\omega +1)(j\omega +2)(j\omega +3)}$$

The phase and magnitude of $L(j\omega )$ determine the system's frequency response characteristics.

- Phase Calculation: The phase angle is given by: $$Arg(L(j\omega ))=Arg(k)-Arg(j\omega +1)-Arg(j\omega +2)-Arg(j\omega +3)$$ Since $k>0$, $Arg(k)=0^{\circ }$. The phase becomes: $$Arg(L(j\omega ))=-\arctan \left(\frac{\omega }{1}\right)-\arctan \left(\frac{\omega }{2}\right)-\arctan \left(\frac{\omega }{3}\right)$$ This phase starts at $0^{\circ }$ when $\omega =0$ and decreases monotonically towards $-270^{\circ }$ as $\omega$ approaches infinity.
- Magnitude Calculation: The magnitude is: $$|L(j\omega )|=\frac{k}{|j\omega +1||j\omega +2||j\omega +3|}=\frac{k}{\sqrt{{\omega }^2+1^2}\sqrt{{\omega }^2+2^2}\sqrt{{\omega }^2+3^2}}$$ The magnitude starts at $|L(j0)|=k/6$ and decreases monotonically towards 0 as $\omega$ approaches infinity.

Understanding Crossover Frequencies

Frequency domain analysis uses specific frequencies to assess stability:

- **Phase Cross-over Frequency (${\omega }_{pc}$):** This is the frequency at which the phase shift of $L(j\omega )$ is exactly $-180^{\circ }$. $$Arg(L(j{\omega }_{pc}))=-180^{\circ }$$
- **Gain Cross-over Frequency (${\omega }_{gc}$):** This is the frequency at which the magnitude of $L(j\omega )$ is unity (or 0 dB). $$|L(j{\omega }_{gc})|=1$$

For the given $L(s)$, since it's a product of terms like $(s+a)$ with $a>0$, both the magnitude and phase change monotonically with frequency. This implies that ${\omega }_{pc}$ and ${\omega }_{gc}$ are unique positive frequencies (assuming $k$ is such that the magnitude actually reaches 1 for $\omega \ge 0$, which requires $k\ge 6$).

**Stability Implications of ${\omega }_{gc}<{\omega }_{pc}$ **

The relative values of ${\omega }_{gc}$ and ${\omega }_{pc}$ are directly related to the stability margins of the system:

- Phase Margin (PM): Defined as $PM=180^{\circ }+Arg(L(j{\omega }_{gc}))$. A positive PM indicates stability.
- Gain Margin (GM): Defined as $GM=\frac{1}{|L(j{\omega }_{pc})|}$. A positive GM (magnitude ratio > 1) indicates stability.

Let's analyze the given condition ${\omega }_{gc}<{\omega }_{pc}$:

1. Phase Margin Analysis: Since the phase $Arg(L(j\omega ))$ decreases as frequency $\omega$ increases, and we know ${\omega }_{gc}<{\omega }_{pc}$, it follows that the phase at ${\omega }_{gc}$ must be greater (less negative) than the phase at ${\omega }_{pc}$. $$Arg(L(j{\omega }_{gc}))>Arg(L(j{\omega }_{pc}))$$ Substituting the definition of ${\omega }_{pc}$: $$Arg(L(j{\omega }_{gc}))>-180^{\circ }$$ Therefore, the Phase Margin is positive: $$PM=180^{\circ }+Arg(L(j{\omega }_{gc}))>180^{\circ }+(-180^{\circ })=0^{\circ }$$ So, $PM>0$.
2. Gain Margin Analysis: Similarly, the magnitude $|L(j\omega )|$ decreases as frequency $\omega$ increases. Since ${\omega }_{gc}<{\omega }_{pc}$, the magnitude at ${\omega }_{pc}$ must be less than the magnitude at ${\omega }_{gc}$. $$|L(j{\omega }_{pc})|<|L(j{\omega }_{gc})|$$ Substituting the definition of ${\omega }_{gc}$: $$|L(j{\omega }_{pc})|<1$$ Therefore, the Gain Margin is positive: $$GM=\frac{1}{|L(j{\omega }_{pc})|}>\frac{1}{1}=1$$ So, $GM>1$ (a positive gain margin).

The condition ${\omega }_{gc}<{\omega }_{pc}$ guarantees that both the Phase Margin and the Gain Margin are positive for this system.

Final Conclusion on Closed-Loop Stability

In control system theory, a system is generally considered stable if both its Phase Margin and Gain Margin are positive. As demonstrated above, the condition that the gain cross-over frequency is less than the phase cross-over frequency (${\omega }_{gc}<{\omega }_{pc}$) ensures positive margins for the given loop-gain function.

Thus, the closed-loop system is stable.