For a long time, I’ve been puzzled by this question: why can we analyze closed-loop stability using the open-loop transfer function? Why not directly use the closed-loop transfer function? And how do Bode plots, root locus, and Nyquist plots of the open-loop system correspond to the closed-loop system?
Here are my humble thoughts—please feel free to point out any mistakes.
Imagine you are pushing a swing.
The closed-loop system represents the whole process: you observe the swing’s position and velocity (feedback), then decide when and how hard to push (control), aiming for smooth and high oscillations (system objectives, e.g., stable operation).
The open-loop transfer function L(s)=G(s)H(s) (or including a controller, C(s)G(s)H(s)) can be understood as the effect of one full “loop” of your push. It describes how the signal evolves after being applied, passing through the plant G(s), and the feedback H(s), and returning as the observed output.
So why does analyzing this “single loop effect” tell us if the whole push process (closed-loop) will remain stable?
The key lies in the closed-loop characteristic equation:
1+L(s)=0
This equation determines the poles of the closed-loop system. Stability is dictated by whether all poles are in the left half-plane (LHP). If any pole enters the right half-plane (RHP), the system becomes unstable.
Rewriting the characteristic equation:
L(s)=-1
This −1 point is crucial:
Magnitude: 1
Phase: -180° (π radians)
Understanding this connection explains why open-loop analysis works. Let’s go through the common tools:
1. Nyquist Plot and Criterion
The Nyquist plot maps
in the complex plane, showing the signal’s “final state” after one loop at different frequencies.
Core idea: check whether the Nyquist curve encircles the critical −1 point.
Why −1? Because L(s)=-1 indicates the system is on the verge of instability. If the loop gain equals 1 with a 180° phase shift, the feedback effectively becomes positive, potentially leading to sustained oscillations. The closer the trajectory is to −1, the more sensitive the system is to instability.
2. Bode Plot: Gain and Phase Margins
The Bode plot is another way to visualize the open-loop frequency response. It shows magnitude (dB) and phase (degrees) versus frequency.
Phase Margin (PM): At the gain crossover frequency (where |L(jω)| = 1), how far is the phase from -180°? Larger PM → more stable.
Gain Margin (GM): At the phase crossover frequency (where ∠L(jω) = -180°), how far is the magnitude from 1? Larger GM → more stable.
Bode plots let us quantify “how far” the system is from the critical -1 point, making stability assessment intuitive.
3. Root Locus
The root locus shows how closed-loop poles move in the complex plane as a parameter (usually the open-loop gain K) varies from 0 to infinity.
Start points: open-loop poles
End points: open-loop zeros
The root locus directly maps how the closed-loop pole locations depend on open-loop dynamics. Crossing into the RHP signals instability. Hence, the entire trajectory is determined by the open-loop transfer function L(s).
Why Open-Loop Analysis Works
Mathematical connection: Closed-loop stability depends on poles, which are roots of 1+L(s)=01 + L(s) = 01+L(s)=0.
Practical convenience: L(s) is usually simpler to analyze than the closed-loop transfer function T(s).
Design intuition: Controller design (C(s)) modifies L(s) to achieve desired closed-loop performance. Open-loop tools directly show how these changes affect stability margins.
Summary:
Although our ultimate interest is in closed-loop behavior, analyzing the open-loop transfer function gives a clear, intuitive way to assess stability. By understanding the relation of L(s) to the critical -1 point, we can predict and ensure closed-loop stability effectively.
