---
id: wiki-2026-0508-pid-controllers-in-ai
title: PID Controllers in AI
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Proportional-Integral-Derivative Control, PID Loop]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [control-theory, robotics, RL, tuning]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: simple_pid/control
---

# PID Controllers in AI

## 매 한 줄
> **"매 error = setpoint - measurement 에 P/I/D 항을 적용해 actuator 를 제어"**. 1922 Minorsky ship steering 에서 시작, 매 산업 control 의 80%+ 사용. 매 2026 의 AI hybrid 사용: drone attitude (PX4), robot joint, LLM token-budget control, RLHF KL-coefficient tuning, training schedules.

## 매 핵심

### 매 PID 공식
- **u(t) = Kp·e(t) + Ki·∫e(τ)dτ + Kd·de/dt**.
- **P (Proportional)**: 매 현재 error 에 비례 — 매 fast response, 매 steady-state error 남김.
- **I (Integral)**: 매 누적 error — 매 steady-state error 제거, 매 windup 위험.
- **D (Derivative)**: 매 변화율 — 매 overshoot 감소, 매 noise 증폭.

### 매 tuning methods
- **Ziegler-Nichols**: 매 Ku, Tu (oscillation) 측정 후 공식 적용.
- **Cohen-Coon**: 매 process reaction curve 기반.
- **AutoTuning**: 매 relay feedback (Åström-Hägglund).
- **Bayesian optimization**: 매 simulation rollout + GP.
- **RL-based**: 매 PPO/SAC 으로 Kp, Ki, Kd 학습.

### 매 응용
1. Drone attitude (PX4, ArduPilot).
2. Robot joint position control (ROS 2).
3. Cruise control, HVAC, 3D printer extruder.
4. LLM serving — token-rate controller (vLLM 의 batch sizing).
5. RLHF KL-coefficient adaptive tuning.
6. Training: gradient norm clipping with adaptive coefficient.

## 💻 패턴

### Plain PID (discrete)
```python
class PID:
    def __init__(self, kp, ki, kd, dt, output_limits=(-1, 1)):
        self.kp, self.ki, self.kd, self.dt = kp, ki, kd, dt
        self.lo, self.hi = output_limits
        self.integral = 0.0
        self.prev_error = 0.0

    def __call__(self, setpoint, measurement):
        error = setpoint - measurement
        self.integral += error * self.dt
        derivative = (error - self.prev_error) / self.dt
        output = self.kp*error + self.ki*self.integral + self.kd*derivative
        output = max(self.lo, min(self.hi, output))
        self.prev_error = error
        return output
```

### Anti-windup (clamp + back-calculation)
```python
class PIDAntiWindup(PID):
    def __init__(self, *args, kt=0.5, **kw):
        super().__init__(*args, **kw)
        self.kt = kt  # back-calc gain

    def __call__(self, setpoint, measurement):
        error = setpoint - measurement
        derivative = (error - self.prev_error) / self.dt
        u_unsat = self.kp*error + self.ki*self.integral + self.kd*derivative
        u_sat = max(self.lo, min(self.hi, u_unsat))
        # back-calculation: discount integral when saturated
        self.integral += self.dt * (error + self.kt * (u_sat - u_unsat))
        self.prev_error = error
        return u_sat
```

### Derivative on measurement (D-kick 회피)
```python
class PIDDerivOnMeas(PID):
    def __init__(self, *args, **kw):
        super().__init__(*args, **kw)
        self.prev_meas = None

    def __call__(self, setpoint, measurement):
        error = setpoint - measurement
        if self.prev_meas is None:
            d = 0
        else:
            d = -(measurement - self.prev_meas) / self.dt  # 매 setpoint step → spike 없음
        self.integral += error * self.dt
        out = self.kp*error + self.ki*self.integral + self.kd*d
        self.prev_meas = measurement
        return max(self.lo, min(self.hi, out))
```

### Ziegler-Nichols tuning
```python
def ziegler_nichols(Ku, Tu, kind="classic"):
    if kind == "classic":
        return dict(kp=0.6*Ku, ki=1.2*Ku/Tu, kd=0.075*Ku*Tu)
    elif kind == "no-overshoot":
        return dict(kp=0.2*Ku, ki=0.4*Ku/Tu, kd=Ku*Tu/15)
    elif kind == "PI-only":
        return dict(kp=0.45*Ku, ki=0.54*Ku/Tu, kd=0)
```

### RLHF adaptive KL (PPO-style β)
```python
def adaptive_kl_pid(kl_observed, kl_target=0.02, state=None):
    """β ↑ when KL too large; β ↓ when too small. PI controller on log(β)."""
    if state is None:
        state = {"integral": 0.0, "log_beta": 0.0}
    error = kl_observed - kl_target
    state["integral"] += error
    state["log_beta"] += 0.1 * error + 0.01 * state["integral"]
    return float(np.exp(state["log_beta"])), state
```

### LLM token-rate controller (server batch)
```python
def batch_size_pid(target_tps, current_tps, pid_state):
    """Maintain target tokens/sec by adjusting batch size."""
    delta = target_tps - current_tps
    pid_state["integral"] += delta
    adj = 0.05*delta + 0.001*pid_state["integral"]
    return max(1, int(pid_state["batch"] + adj))
```

## 매 결정 기준
| 상황 | Controller |
|---|---|
| no steady-state error needed | P only |
| eliminate steady-state, slow process | PI |
| fast + minimal overshoot | PID with D-on-measurement |
| highly nonlinear / complex | MPC or RL (not PID) |
| discrete-event / queue | PI on rate |
| training schedule (KL, lr) | adaptive PI |

**기본값**: 매 80% 의 경우 PI 면 충분. 매 D 매 noise 환경에서 신중히.

## 🔗 Graph
- 부모: [[Control-Theory]] · [[Feedback-Control-Systems]]
- 응용: [[Robotics]] · [[RLHF]]
- Adjacent: [[Model-Predictive-Control (MPC)]] · [[Kalman-Filter-and-State-Tracking]] · [[Reinforcement-Learning]]

## 🤖 LLM 활용
**언제**: 매 inference server batch sizing, 매 RLHF KL coefficient, 매 lr schedule under loss target.
**언제 X**: 매 highly nonlinear delays — MPC. 매 strong constraints — RL/MPC.

## ❌ 안티패턴
- **No anti-windup with saturation**: 매 integral 폭주 → overshoot.
- **D term on noisy raw signal**: 매 noise 증폭 → 매 LPF (low-pass filter) 필수.
- **Derivative on setpoint (D-kick)**: 매 setpoint step → spike. 매 D-on-measurement 사용.
- **One-size-fits-all gains across operating regions**: 매 nonlinear 시스템 — gain scheduling 필요.
- **Tuning by 임의 가이드**: 매 system 마다 다름 — Ziegler-Nichols 또는 simulation 사용.

## 🧪 검증 / 중복
- Verified (Åström & Hägglund, *PID Controllers*; Franklin et al. *Feedback Control of Dynamic Systems*; ArduPilot/PX4 firmware).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — PID FULL with anti-windup, D-on-meas, RLHF/serving applications |