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Connor Johnstone
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roadmap/features/04-pid-controller.md
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roadmap/features/04-pid-controller.md
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# Feature: PID Controller
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## Overview
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The PID (Proportional-Integral-Derivative) step size controller is an advanced adaptive time-stepping controller that provides better stability and efficiency than the basic PI controller, especially for difficult or oscillatory problems.
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**Key Characteristics:**
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- Three-term control: proportional, integral, and derivative components
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- More stable than PI for challenging problems
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- Standard in production ODE solvers
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- Can prevent oscillatory step size behavior
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## Why This Feature Matters
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- **Robustness**: Handles difficult problems that cause PI controller to oscillate
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- **Industry standard**: Used in MATLAB, Sundials, and other production solvers
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- **Minimal overhead**: Small computational cost for significant stability improvement
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- **Smooth stepping**: Reduces erratic step size changes
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## Dependencies
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- None (extends current controller infrastructure)
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## Implementation Approach
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### Mathematical Formulation
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The PID controller determines the next step size based on error estimates from the current and previous steps:
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```
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h_{n+1} = h_n * (ε_n)^(-β₁) * (ε_{n-1})^(-β₂) * (ε_{n-2})^(-β₃)
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```
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Where:
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- ε_i = error estimate at step i (normalized by tolerance)
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- β₁ = proportional coefficient (typically ~0.3 to 0.5)
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- β₂ = integral coefficient (typically ~0.04 to 0.1)
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- β₃ = derivative coefficient (typically ~0.01 to 0.05)
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Standard formula (Hairer & Wanner):
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```
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h_{n+1} = h_n * safety * (ε_n)^(-β₁/(k+1)) * (ε_{n-1})^(-β₂/(k+1)) * (h_n/h_{n-1})^(-β₃/(k+1))
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```
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Where k is the order of the method.
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### Advantages Over PI
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- **PI controller**: Uses only current and previous error (2 terms)
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- **PID controller**: Also uses rate of change of error (3 terms)
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- **Result**: Anticipates trends, prevents overshoot
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### Implementation Design
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```rust
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pub struct PIDController {
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// Coefficients
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pub beta1: f64, // Proportional
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pub beta2: f64, // Integral
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pub beta3: f64, // Derivative
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// Constraints
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pub factor_min: f64, // qmax inverse
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pub factor_max: f64, // qmin inverse
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pub h_max: f64,
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pub safety_factor: f64,
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// State (error history)
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pub err_old: f64, // ε_{n-1}
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pub err_older: f64, // ε_{n-2}
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pub h_old: f64, // h_{n-1}
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// Next step guess
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pub next_step_guess: TryStep,
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}
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```
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## Implementation Tasks
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### Core Controller
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- [ ] Define `PIDController` struct
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- [ ] Add beta1, beta2, beta3 coefficients
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- [ ] Add constraint fields (factor_min, factor_max, h_max, safety)
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- [ ] Add state fields (err_old, err_older, h_old)
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- [ ] Add next_step_guess field
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- [ ] Implement `Controller<D>` trait
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- [ ] `determine_step()` method
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- [ ] Handle first step (no history)
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- [ ] Handle second step (partial history)
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- [ ] Full PID formula for subsequent steps
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- [ ] Apply safety factor and limits
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- [ ] Update error history
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- [ ] Return TryStep::Accepted or NotYetAccepted
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- [ ] Constructor methods
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- [ ] `new()` with all parameters
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- [ ] `default()` with standard coefficients
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- [ ] `for_order()` - scale coefficients by method order
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- [ ] Helper methods
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- [ ] `reset()` - clear history (for algorithm switching)
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- [ ] Update state after accepted/rejected steps
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### Standard Coefficient Sets
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Different coefficient sets for different problem classes:
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- [ ] **Default (H312)**:
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- β₁ = 1/4, β₂ = 1/4, β₃ = 0
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- Actually a PI controller with specific tuning
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- Good general-purpose choice
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- [ ] **H211**:
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- β₁ = 1/6, β₂ = 1/6, β₃ = 0
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- More conservative
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- [ ] **Full PID (Gustafsson)**:
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- β₁ = 0.49/(k+1)
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- β₂ = 0.34/(k+1)
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- β₃ = 0.10/(k+1)
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- True PID behavior
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### Integration
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- [ ] Export PIDController in prelude
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- [ ] Update Problem to accept any Controller trait
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- [ ] Examples using PID controller
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### Testing
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- [ ] **Comparison test: Smooth problem**
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- [ ] Run exponential decay with PI and PID
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- [ ] Both should perform similarly
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- [ ] Verify PID doesn't hurt performance
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- [ ] **Oscillatory problem test**
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- [ ] Problem that causes PI to oscillate step sizes
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- [ ] Example: y'' + ω²y = 0 with varying ω
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- [ ] PID should have smoother step size evolution
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- [ ] Plot step size vs time for both
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- [ ] **Step rejection handling**
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- [ ] Verify history updated correctly after rejection
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- [ ] Doesn't blow up or get stuck
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- [ ] **Reset test**
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- [ ] Algorithm switching scenario
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- [ ] Verify reset() clears history appropriately
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- [ ] **Coefficient tuning test**
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- [ ] Try different β values
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- [ ] Verify stability bounds
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- [ ] Document which work best for which problems
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### Benchmarking
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- [ ] Add PID option to existing benchmarks
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- [ ] Compare step count and function evaluations vs PI
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- [ ] Measure overhead (should be negligible)
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### Documentation
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- [ ] Docstring explaining PID control
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- [ ] When to prefer PID over PI
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- [ ] Coefficient selection guidance
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- [ ] Example comparing PI and PID behavior
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## Testing Requirements
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### Oscillatory Test Problem
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Problem designed to expose step size oscillation:
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```rust
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// Prothero-Robinson equation
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// y' = λ(y - φ(t)) + φ'(t)
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// where φ(t) = sin(ωt), λ << 0 (stiff), ω moderate
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//
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// This problem can cause step size oscillation with PI
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```
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Expected: PID should maintain more stable step sizes.
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### Step Size Stability Metric
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Track standard deviation of log(h_i/h_{i-1}) over the integration:
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- PI controller: may have σ > 0.5
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- PID controller: should have σ < 0.3
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## References
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1. **PID Controllers for ODE**:
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- Gustafsson, K., Lundh, M., and Söderlind, G. (1988)
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- "A PI stepsize control for the numerical solution of ordinary differential equations"
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- BIT Numerical Mathematics, 28, 270-287
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2. **Implementation Details**:
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- Hairer, E., Nørsett, S.P., and Wanner, G. (1993)
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- "Solving Ordinary Differential Equations I", Section II.4
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- PID controller discussion
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3. **Coefficient Selection**:
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- Söderlind, G. (2002)
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- "Automatic Control and Adaptive Time-Stepping"
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- Numerical Algorithms, 31, 281-310
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4. **Julia Implementation**:
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- `OrdinaryDiffEq.jl/lib/OrdinaryDiffEqCore/src/integrators/controllers.jl`
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- Look for `PIDController`
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## Complexity Estimate
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**Effort**: Small (3-5 hours)
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- Straightforward extension of PI controller
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- Main work is getting coefficients right
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- Testing requires careful problem selection
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**Risk**: Low
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- Well-understood algorithm
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- Minimal code changes
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- Easy to validate
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## Success Criteria
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- [ ] Implements full PID formula correctly
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- [ ] Handles first/second step bootstrap
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- [ ] Shows improved stability on oscillatory test problem
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- [ ] Performance similar to PI on smooth problems
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- [ ] Error history management correct after rejections
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- [ ] Documentation complete with usage examples
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- [ ] Coefficient sets match literature values
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## Future Enhancements
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- [ ] Automatic coefficient selection based on problem characteristics
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- [ ] More sophisticated controllers (H0211b, predictive)
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- [ ] Limiter functions to prevent extreme changes
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- [ ] Per-algorithm default coefficients
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