Real-world "Stiff system" Example for ODE page

[tordnat]

Stiff System: Voltage-Controlled DC Motor Arm

State & Dynamics

State vector: $x = [\theta, \dot{\theta}, i]^T$

Subsystem	Equation	Time Constant
Electrical	$\dot{i} = (v - Ri - K_b\dot{\theta})/L$	$\tau_e = L/R \sim 10^{-4}$ s
Mechanical	$\ddot{\theta} = (K_t i - b\dot{\theta})/J$	$\tau_m = J/b \sim 10^{-1}$ s
Control	$v = -K_p(\theta - \theta_d) - K_d\dot{\theta}$	—

Student implementation note: Implicit methods allow ~1000× larger $\Delta t$, but each step requires solving a nonlinear system (Newton iteration with Jacobian). A naive implementation may only see ~10–50× wall-clock speedup. Production solvers (Radau, ESDIRK) achieve greater gains through adaptive stepping, efficient linear algebra, and Jacobian reuse.

Demo Outline

1. Sweep $K_p \in [10, 10^4]$
2. Compare step counts: explicit RK4 vs student-implemented backward Euler vs Radau
3. Plot Jacobian eigenvalues vs gain
4. Measure wall-clock time to show implicit overhead vs stability benefit

Notes

• Equations follow standard DC motor model (Kirchhoff + Newton-Euler)
• Voltage-mode control assumed (not torque-mode)
• $K_d\dot{\theta}$ term couples into electrical dynamics via $v$, contributing additional stiffness
• Production solvers: SciPy's Radau (implicit RK) and MATLAB's ode15s (BDF, a multistep method) both handle stiffness; this course focuses on RK formulations