EVOLV/third_party/docs/pid-control-theory.md

# PID Control for Process Applications

> **Used by**: `mechanical-process-engineer` agent, `node-red-runtime` agent, `generalFunctions/src/pid/`
> **Validation**: Verified against Astrom & Hagglund (ISA, 2006) and MATLAB/Simulink documentation

## Continuous PID Controller

### Standard (ISA/Ideal) Form

```
u(t) = K_p · [e(t) + (1/T_i) · ∫e(τ)dτ + T_d · de(t)/dt]
```

Where:
- u(t) = controller output
- e(t) = error = setpoint - process variable (SP - PV)
- K_p = proportional gain
- T_i = integral time (seconds)
- T_d = derivative time (seconds)

### Parallel Form

```
u(t) = K_p · e(t) + K_i · ∫e(τ)dτ + K_d · de(t)/dt
```

Where K_i = K_p/T_i and K_d = K_p · T_d

## Discrete PID Implementation

### Positional Form (absolute output)

```
u[k] = K_p · e[k] + K_i · Δt · Σe[j] + K_d · (e[k] - e[k-1]) / Δt
```

- Computes the absolute output value each cycle
- Requires tracking of the integral sum
- Subject to integral windup
- Used when the controller output directly sets an actuator position

### Velocity Form (incremental output)

```
Δu[k] = K_p · (e[k] - e[k-1]) + K_i · Δt · e[k] + K_d · (e[k] - 2·e[k-1] + e[k-2]) / Δt
u[k] = u[k-1] + Δu[k]
```

- Computes the **change** in output each cycle
- Inherently bumpless on mode transfers (auto→manual→auto)
- Less prone to windup (but still needs anti-windup for output limits)
- Preferred for incremental actuators (VFDs, variable valves)

## Anti-Windup Strategies

Integral windup occurs when the controller output saturates (hits actuator limits) but the integral term continues to accumulate, causing large overshoot when the error changes sign.

### 1. Clamping (Conditional Integration)

Stop integrating when the output is saturated **and** the error has the same sign as the integral term:

```
if (u_raw > u_max || u_raw < u_min) && sign(e) == sign(integral):
    freeze integral (do not accumulate)
else:
    integral += e * Δt
```

- Simple to implement
- Effective for most process control applications
- The approach used in the EVOLV generalFunctions PID implementation

### 2. Back-Calculation

When the output saturates, feed back the difference between the saturated and unsaturated output to "unwind" the integrator:

```
integral += (K_i · e + K_b · (u_saturated - u_raw)) · Δt
```

Where K_b = 1/T_t (tracking time constant, typically T_t = √(T_i · T_d) or T_t = T_d).

- More sophisticated than clamping
- Better performance for systems with large disturbances
- Recommended by Astrom & Hagglund for demanding applications

### 3. Integrator Reset

Reset the integrator to a value that would produce the saturated output:

```
if u_raw > u_max:
    integral = (u_max - K_p · e) / K_i
```

- Simple but can be aggressive
- May cause discontinuities

## Derivative Filtering

The derivative term amplifies high-frequency noise. Always filter it:

### First-Order Low-Pass Filter on D-Term

```
D_filtered[k] = α · D_filtered[k-1] + (1-α) · D_raw[k]
```

Where α = T_f / (T_f + Δt) and T_f = T_d / N (N typically 5-20, default 10).

### Derivative on PV (not Error)

To avoid derivative kick on setpoint changes:
```
D = -K_d · (PV[k] - PV[k-1]) / Δt   (instead of using error)
```

This is the standard approach for process control.

## Cascade PID

Two nested loops where the outer (master) loop's output is the setpoint for the inner (slave) loop.

```
[SP_outer] → [PID_outer] → [SP_inner] → [PID_inner] → [Actuator] → [Process]
                 ↑                            ↑
            [PV_outer] ←──── [Process] ←── [PV_inner]
```

### Design Rules
- Inner loop must be **5-10x faster** than outer loop
- Tune inner loop first (with outer loop in manual)
- Then tune outer loop
- Anti-windup on outer loop essential (its output is bounded by inner loop's SP limits)

### EVOLV Application
- Outer: Level controller → outputs flow setpoint
- Inner: Flow controller → outputs pump speed
- pumpingStation node coordinates this cascade

## Tuning Methods

### Ziegler-Nichols (Ultimate Gain Method)
1. Set I and D to zero, increase K_p until sustained oscillation
2. Record ultimate gain K_u and ultimate period T_u
3. Apply: K_p = 0.6·K_u, T_i = T_u/2, T_d = T_u/8

### Cohen-Coon (Process Reaction Curve)
1. Apply step change, record process reaction curve
2. Identify: gain K, dead time L, time constant τ
3. Apply formulas based on K, L, τ

### Lambda Tuning (IMC-based)
1. Identify process as FOPDT: K, L, τ
2. Choose closed-loop time constant λ (typically λ = max(3·L, τ))
3. K_p = τ/(K·λ), T_i = τ

- **Preferred for process control** — gives non-oscillatory response
- Directly specifies desired closed-loop speed
- Robust to model uncertainty when λ is chosen conservatively

## Authoritative References

1. Astrom, K.J. & Hagglund, T. (2006). "Advanced PID Control." ISA — The Instrumentation, Systems, and Automation Society.
2. Astrom, K.J. & Hagglund, T. (1995). "PID Controllers: Theory, Design, and Tuning." 2nd ed., ISA.
3. Seborg, D.E. et al. (2011). "Process Dynamics and Control." 3rd ed., Wiley. (Chapter on PID control)
4. MATLAB/Simulink documentation — "Anti-Windup Control Using PID Controller Block"
5. Smith, C.A. & Corripio, A.B. (2005). "Principles and Practices of Automatic Process Control." 3rd ed., Wiley.