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Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

2026-05-20 23:52:15 +09:00

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title

Model Predictive Control (MPC)

매 한 줄

"매 step마다 future를 optimize, 첫 action만 실행, 매 반복". Receding horizon control은 매 dynamics model + cost + constraints를 매 online QP/NLP로 풀어낸다. 2026 자율주행·드론·humanoid robot의 매 dominant control paradigm으로 reinforcement learning과도 hybrid 구성된다.

매 핵심

매 정식화

min_{u_0..u_{N-1}} Σ ℓ(x_k, u_k) + V_f(x_N)
s.t. x_{k+1} = f(x_k, u_k)
     g(x_k, u_k) ≤ 0
     x_0 = x_current

매 첫 u_0만 적용, 다음 step에서 매 새 measurement로 재최적화.

매 종류

Linear MPC: f linear, ℓ quadratic → QP.
Nonlinear MPC (NMPC): NLP (IPOPT, acados).
Robust MPC: tube/min-max — uncertainty 처리.
Stochastic MPC: chance constraints.
Learning-based MPC: f를 NN/GP로.

매 응용

Autonomous driving — trajectory tracking, lane change.
Quadrotor / drone control.
Humanoid locomotion (Boston Dynamics, Tesla Optimus 추정).
Process industry — refinery, chemical plant.
HVAC, smart grid.

💻 패턴

Linear MPC with CVXPY

import cvxpy as cp, numpy as np
def lin_mpc(A, B, x0, N=10, Q=None, R=None):
    nx, nu = B.shape
    Q = Q if Q is not None else np.eye(nx)
    R = R if R is not None else 0.1*np.eye(nu)
    x = cp.Variable((nx, N+1))
    u = cp.Variable((nu, N))
    cost, cons = 0, [x[:, 0] == x0]
    for k in range(N):
        cost += cp.quad_form(x[:, k], Q) + cp.quad_form(u[:, k], R)
        cons += [x[:, k+1] == A @ x[:, k] + B @ u[:, k]]
        cons += [cp.norm(u[:, k], 'inf') <= 1.0]
    cp.Problem(cp.Minimize(cost), cons).solve()
    return u[:, 0].value

NMPC with CasADi/acados (sketch)

import casadi as ca
x = ca.SX.sym('x', nx); u = ca.SX.sym('u', nu)
f = ca.Function('f', [x, u], [dynamics(x, u)])
# build NLP with multiple shooting...

Receding horizon loop

for t in range(T):
    x_meas = sensor.read()
    u_star = mpc_solve(x_meas)
    actuator.apply(u_star)

Reference tracking cost

cost += cp.quad_form(x[:, k] - x_ref, Q)

Soft constraint via slack

slack = cp.Variable(N, nonneg=True)
cons += [g(x[:, k], u[:, k]) <= slack[k]]
cost += 1e3 * cp.sum(slack)

Warm-start (next iter uses prev solution)

solver.set_initial_guess(shift(u_prev))

매 결정 기준

상황	MPC variant
Linear plant, quadratic cost	QP-based linear MPC
Nonlinear dynamics	NMPC (acados, CasADi)
Bounded uncertainty	Tube MPC
Probabilistic constraint	Stochastic MPC
Hard real-time (kHz)	Explicit MPC (precomputed)

기본값: Linear MPC + warm-start (cycle time < 10 ms).

🔗 Graph

부모: Optimal-Control-Theory · Optimization
응용: Autonomous-Vehicle-Path-Planning
Adjacent: Reinforcement-Learning · Kalman-Filter-and-State-Tracking · Feedback-Control-Systems

🤖 LLM 활용

언제: Constrained dynamic systems, real-time replanning, model + cost are known. 언제 X: Model 알 수 없거나 long-horizon strategic decision (use RL).

❌ 안티패턴

Horizon 너무 짧음: 매 myopic control.
Constraint feasibility 무시: infeasible 시 fallback 없음.
Cold-start 매 iteration: 매 latency 폭발 — warm-start 필수.
Plant-model mismatch 무시: 매 robust/adaptive 가 필요.

🧪 검증 / 중복

Verified (Rawlings, Mayne, Diehl "Model Predictive Control").
신뢰도 A.

🕓 Changelog

날짜	변경
2026-05-08	Phase 1
2026-05-10	Manual cleanup — MPC formulation + CVXPY/acados patterns

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