Pole location vs. Tran behavior

https://web.mit.edu/2.14/www/Handouts/PoleZero.pdf

The location of the poles in the s-plane therefore define the n components in the homogeneous response as described below:

1. A real pole pi = −σ in the left-half of the s-plane defines an exponentially decaying component , Ce−σt, in the homogeneous response. The rate of the decay is determined by the pole location; poles far from the origin in the left-half plane correspond to components that decay rapidly, while poles near the origin correspond to slowly decaying components.

2. A pole at the origin pi = 0 defines a component that is constant in amplitude and defined by the initial conditions.

3. A real pole in the right-half plane corresponds to an exponentially increasing component Ceσt in the homogeneous response; thus defining the system to be unstable.

4. A complex conjugate pole pair σ ± jω in the left-half of the s-plane combine to generate a response component that is a decaying sinusoid of the form Ae−σt sin (ωt + φ) where A and φ are determined by the initial conditions. The rate of decay is specified by σ; the frequency of oscillation is determined by ω.

5. An imaginary pole pair, that is a pole pair lying on the imaginary axis, ±jω generates an oscillatory component with a constant amplitude determined by the initial conditions.

6. A complex pole pair in the right half plane generates an exponentially increasing component. These results are summarized in Fig. 2.

引申：

If the open loop T(s) contains no RHP poles, the close loop 1/(1+T) and T/(1+T) contain no RHP poles when PM >0.

If T contains RH poles, the PM test cannot be used. (the original OL system is unstable)