MAE 4182 · November 11 · Quick recap
Recall · time domain
Poles: left → decay · right → grow · simple ±iω → sustained oscillation
Transient response
Delay · rise · overshoot · settling
Steady-state error
eₛₛ = lim(t → ∞) [r(t) − y(t)]
MAE 4182 · November 11 · Quick recap
Recall · frequency domain
Sinusoidal response
Same frequency · gain |G(iω)| · phase ∠G(iω)
Closed loop · T = Y/R
Mᵣ = max |T| · |T(i BW)| = |T(0)|/√2
ωₙ ↑ → faster · BW ↑ (fixed ζ)
ζ ↑ → Mₚ ↓ · Mᵣ ↓ toward 1 (fixed ωₙ)
MAE 4182 · November 11 · Quick recap
Recall · stability margins
F = KG · G = 1/[s(s + 1)²] · K = 0.2
K = 0.2
Gain margin
GM = 1/|F(iωₚ)|
Phase −180° → magnitude.
Phase margin
PM = 180° + ∠F(iωg)
Magnitude 1 → phase.
Boundary: |F| = 1, phase −180° · same ω.
Next · Use these ideas to design a controller.
MAE 4182 · November 11 · Control System Design
From analysis to control design
Modeling
System → equations
Analysis
Predict the response
Design NOW
Choose the controller
Stable
Fast
Accurate
Today P → PD → temperature control
Later PID · Bode · state feedback · estimation
MAE 4182 · November 11 · Control System Design
Can a flying robot catch a pole?
Watch the catch.
- Goal?Upright.
- Measure?Motion.
- Input?Motors.
Choose the next command.
MAE 4182 · November 11 · Control System Design
Ball Balancing System
r · positionv = ṙ · velocityθ · tilt
Real system
r = 0, v = 0
Center · stop
1D: left ↔ right
You control the tilt.20-second warm-up
θ 0°r +12.0 cmv 0.0 cm/s
Hold ← / →. Release: level.
±2.5 cm · ≤3 cm/s · 2 s
θ → motion → r
Another example · a marble maze
Camera → two tilts → finish.
CyberRunner · learned control.
MAE 4182 · November 11 · Control System Design
Simplified 1D Dynamics
MAE 4182 · November 11 · Control System Design
Linearization · near the center
MAE 4182 · November 11 · Control System Design
Transfer function · angle in, position out
MAE 4182 · November 11 · Control System Design
Open-loop stability
MAE 4182 · November 11 · Control System Design
Control System
MAE 4182 · November 11 · Control System Design
Proportional Control
MAE 4182 · November 11 · Control System Design
Proportional–Derivative (PD) Control
MAE 4182 · November 11 · Control System Design