2nd/10_Wiki/Topics/Computer_Science_and_Theory/Feedback-Control-Systems.md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

id	title	category	status	canonical_id	aliases	duplicate_of	source_trust_level	confidence_score	verification_status	tags	raw_sources	last_reinforced	github_commit	tech_stack
wiki-2026-0508-feedback-control-systems	Feedback Control Systems	10_Wiki/Topics	verified	self	Closed-Loop Control Control System PID MPC	none	A	0.92	applied	control-theory systems dynamics automation		2026-05-10	pending	language framework Python python-control, scipy.signal, do-mpc, casadi

Feedback Control Systems

매 한 줄

"매 sensor → comparator → controller → actuator → plant → sensor 매 closed loop 으로 setpoint 를 maintain". James Watt 의 governor (1788) → Bode/Nyquist (1930s) → state-space + Kalman (1960s) → 매 modern: MPC, ADRC, learning-based control (Koopman/MPC, RL). 매 SRE autoscaler, drone, EV motor, vaccine cold-chain, fab DRIE — 매 universal.

매 핵심

매 canonical block diagram

        ┌─────────┐
r ──+──▶│Controller│──u──▶┌─────┐──y──┐
   ─    │  C(s)   │       │P(s)│     │
        └─────────┘       └─────┘     │
              ▲                       │
              └──── −  ◀── sensor ◀───┘
e = r − y_meas      (error)

매 PID intuition

• P (proportional): 매 instant correction; 매 too high → oscillation.
• I (integral): 매 eliminate steady-state error; 매 too high → wind-up.
• D (derivative): 매 damping; 매 amplify noise.

매 modern 분류

• Classical: PID, lead-lag, root-locus, Bode design.
• State-space: pole-placement, LQR, observer / Kalman filter.
• Robust: H∞, μ-synthesis (uncertain plant).
• Adaptive: MRAC, gain-scheduling.
• Predictive: MPC (constraint-aware, multi-step optimize).
• Learning-based (2026): Koopman-MPC, Gaussian-Process MPC, safe RL with control-barrier functions.

매 stability

• 매 LTI 의 Routh-Hurwitz, Nyquist, gain/phase margin (≥ 6 dB / 30°).
• 매 nonlinear 의 Lyapunov.

💻 패턴

PID class

class PID:
    def __init__(self, kp, ki, kd, dt, u_min=None, u_max=None):
        self.kp,self.ki,self.kd,self.dt = kp,ki,kd,dt
        self.u_min,self.u_max = u_min,u_max
        self.i, self.prev = 0.0, 0.0
    def __call__(self, sp, pv):
        e = sp - pv
        self.i += e*self.dt
        d = (e - self.prev)/self.dt
        self.prev = e
        u = self.kp*e + self.ki*self.i + self.kd*d
        if self.u_max is not None and u > self.u_max:
            u = self.u_max; self.i -= e*self.dt   # anti-windup
        if self.u_min is not None and u < self.u_min:
            u = self.u_min; self.i -= e*self.dt
        return u

Ziegler–Nichols tuning

# 1) Set ki=kd=0; raise kp to find Ku where output sustains oscillation at period Tu.
# 2) Classic ZN: kp=0.6Ku, ki=2kp/Tu, kd=kp*Tu/8
def zn_classic(Ku, Tu): return 0.6*Ku, 2*0.6*Ku/Tu, 0.6*Ku*Tu/8

Plant simulation (python-control)

import control as ct, numpy as np, matplotlib.pyplot as plt
P = ct.tf([1], [1, 3, 2])           # 1/(s²+3s+2)
C = ct.tf([2, 1], [1, 0])           # PI: 2 + 1/s
L = ct.series(C, P)
T = ct.feedback(L, 1)
t, y = ct.step_response(T, T=np.linspace(0, 10, 500))
plt.plot(t, y); plt.axhline(1, ls="--")

State-space LQR

import numpy as np, control as ct
A = np.array([[0,1],[-2,-3]]); B = np.array([[0],[1]])
Q = np.diag([10, 1]); R = np.array([[0.1]])
K, _, _ = ct.lqr(A, B, Q, R)
# u = -K x

Kalman filter

import numpy as np
def kf_step(x, P, z, A, H, Q, R):
    # predict
    x = A @ x; P = A @ P @ A.T + Q
    # update
    y = z - H @ x
    S = H @ P @ H.T + R
    K = P @ H.T @ np.linalg.inv(S)
    x = x + K @ y
    P = (np.eye(len(x)) - K @ H) @ P
    return x, P

Model Predictive Control (do-mpc, level tank)

import numpy as np, do_mpc
from casadi import vertcat
m = do_mpc.model.Model("continuous")
h = m.set_variable("_x","h"); u = m.set_variable("_u","u")
m.set_rhs("h", -0.1*h + u); m.setup()

mpc = do_mpc.controller.MPC(m)
mpc.set_param(n_horizon=20, t_step=0.1, store_full_solution=True)
mpc.set_objective(mterm=(h-1)**2, lterm=(h-1)**2 + 0.01*u**2)
mpc.set_rterm(u=0.1)
mpc.bounds["lower","_u","u"] = 0; mpc.bounds["upper","_u","u"] = 2
mpc.bounds["lower","_x","h"] = 0; mpc.bounds["upper","_x","h"] = 1.5
mpc.setup()
mpc.x0 = np.array([0.0]); mpc.set_initial_guess()
u0 = mpc.make_step(np.array([0.0]))

Autoscaler-as-PID (SRE)

class CPUAutoscaler:
    def __init__(self, target=0.6, k=10, max_r=20):
        self.pid = PID(kp=k, ki=k/30, kd=0, dt=15, u_min=-3, u_max=3)
        self.target, self.max_r = target, max_r
    def step(self, cpu, replicas):
        delta = self.pid(self.target, cpu)
        return max(1, min(self.max_r, replicas + round(delta)))

매 결정 기준

상황	Approach
Single SISO, slow plant	PID (90% of industry)
MIMO, known model	LQR / state-space
Constraints + multi-step lookahead	MPC
Large model uncertainty	H∞ / robust control
Time-varying gain	Gain-scheduling / adaptive
Unknown dynamics, lots of data	Koopman-MPC / GP-MPC / safe RL
Stochastic measurement	+ Kalman / EKF / UKF

기본값: 매 SISO + slow plant → PID with anti-windup + ZN-tuning, 매 constraint-rich multi-input → MPC.

🔗 Graph

• 부모: Control-Theory · Cybernetics Foundations
• 변형: PID · MPC
• 응용: Robotics
• Adjacent: Feedback-Loops in Systems · Kalman-Filter-and-State-Tracking · Reinforcement-Learning

🤖 LLM 활용

언제: 매 PID 의 tuning, 매 plant 의 transfer-function 의 derivation, 매 MPC objective 의 formulation, 매 stability 의 quick check. 언제 X: 매 safety-critical real-time control 의 untested LLM-generated code 의 직접 deploy — 매 formal verification + HIL test 필수.

❌ 안티패턴

• No anti-windup: 매 saturated actuator + integral 의 huge overshoot.
• D term on noisy measurement: 매 amplifies high-freq noise — 매 derivative-on-measurement + low-pass filter.
• Sample rate ≪ bandwidth: 매 dt 의 ≥ 10× system bandwidth 의 violate → 매 instability.
• Tuning by gut: 매 reproducible (ZN, model-based, autotuning) 의 사용.
• Open-loop "feedback": 매 sensor 없이 매 not feedback control.
• Ignoring delay: 매 transport delay → 매 Smith predictor / phase margin 의 reserve.

🧪 검증 / 중복

• Verified (Bode 1945; Åström & Murray Feedback Systems 2nd ed.; Rawlings, Mayne & Diehl MPC 2nd ed.; Brunton & Kutz Data-Driven Science and Engineering 2022).
• 신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1 placeholder
2026-05-10	Manual cleanup — classical→MPC + 7 patterns + decision matrix