Understanding Poles, Transients, and Key Concepts
Given Output:
c(t) = 1 + 0.25e-50t - 1.25e-10t
1. Poles (s₁, s₂)
The values s₁ = -50 and s₂ = -10 are:
- Roots of the characteristic equation
- Poles of the transfer function
- Eigenvalues of the system
Poles determine the transient behavior and stability.
2. Transient Response
The transient response comes from the exponential terms:
0.25e-50t - 1.25e-10t
| Pole | Time Response Term | Meaning |
|---|---|---|
| s₁ = -50 | e-50t | Fast decaying transient |
| s₂ = -10 | e-10t | Slow decaying transient |
3. Steady-State vs Transient
Steady-State Part: 1
This is the final value as t → ∞.
Transient Part: 0.25e-50t - 1.25e-10t
These decay to zero over time.
4. Key Control System Concepts
- System Stability: All poles must have negative real parts for a stable system. Here, s₁ and s₂ are negative → system is stable.
- Damping Ratio (ζ): Determines overshoot and oscillation. ζ > 1 → overdamped (non-oscillatory).
- Natural Frequency (ωₙ): Determines the speed of response.
- Time Constant (Ï„): For each pole, Ï„ = -1/s. Fast decaying transient (Ï„₁ = 1/50 = 0.02 s), slow decaying (Ï„₂ = 1/10 = 0.1 s).
-
Overdamped / Underdamped / Critically Damped:
- ζ > 1 → Overdamped (no oscillation)
- ζ = 1 → Critically damped (fastest non-oscillatory)
- 0 < ζ < 1 → Underdamped (oscillatory)
- Final Value Theorem: Steady-state = limt→∞ c(t) = 1 (matches the constant term in output).
5. Summary
- s₁, s₂ → system poles / characteristic roots
- e-50t, e-10t → transient response components
- 1 → steady-state response
- System is stable because all poles are negative
- System is overdamped because ζ > 1