Pemodelan Matematis
Sistem Infrastruktur Kritis

Tanpa model yang akurat, merancang kontroller sama seperti merakit mesin tanpa gambar teknik — bisa berhasil, tapi berisiko fatal untuk infrastruktur kritis.

Order Sistem

Contoh: Motor DC dalam Sistem Pompa

Diagram Blok Sistem Kontrol Sederhana

Contoh Umum Fungsi Transfer

Kebanyakan sistem fisik nyata bersifat nonlinier. Contoh: debit air melalui keran ∝ √h (akar kuadrat tinggi air, bukan linier). Tapi untuk kontrol, kita perlu model linier. Solusinya: linearisasi di sekitar titik kerja.

1. Tentukan Operating Point (x₀, u₀)
   Titik di mana sistem biasanya beroperasi (steady-state). Contoh: suhu reaktor 150°C, tekanan 5 bar.
2. Hitung Turunan Parsial (Jacobian)
   Hitung ∂f/∂x dan ∂f/∂u di titik operasi untuk mendapatkan matriks A dan B.
3. Buat Model Deviasi
   Ganti variabel asli dengan deviasi: Δx = x − x₀, Δu = u − u₀
4. Validasi
   Pastikan model linier akurat di sekitar operating point dengan simulasi.

Kasus 1: Sistem HVAC Gedung Kritis (Data Center)

Kasus 2: Sistem Distribusi Tenaga Listrik

Kita akan gunakan Python Control Library untuk mensimulasikan respons sistem HVAC data center.

# Install terlebih dahulu (jalankan di terminal):
# pip install control matplotlib numpy scipy

import control as ct
import numpy as np
import matplotlib.pyplot as plt

# ==========================================
# MODEL HVAC DATA CENTER
# ==========================================

# Parameter sistem termal
C  = 5000    # Thermal capacitance (J/°C)
UA = 200     # Heat loss coefficient (W/°C)

# Fungsi Transfer: G(s) = 1/(Cs + UA)
# = (1/UA) / ((C/UA)s + 1) = K/(τs+1)
K   = 1/UA         # Steady-state gain
tau = C/UA          # Time constant (detik)
print(f"Gain K     = {K:.4f} °C/W")
print(f"Time const = {tau:.1f} detik ({tau/60:.1f} menit)")

# Buat transfer function
num = [K]
den = [tau, 1]
G = ct.tf(num, den)
print(f"\nFungsi Transfer:\n{G}")

# --- RESPONS STEP (AC menyala tiba-tiba) ---
t   = np.linspace(0, 600, 1000)   # 0 sampai 600 detik
t, y = ct.step_response(G, T=t)

# Konversi: step = 1000W AC → perubahan suhu
Q_AC   = 1000        # 1 kW cooling
delta_T = y * Q_AC    # perubahan suhu (°C)

# --- PLOT 1: Step Response ---
fig, axes = plt.subplots(1, 2, figsize=(12, 5))

axes[0].plot(t/60, delta_T, 'orange', lw=2)
axes[0].axhline(K*Q_AC, color='red', ls='--', label=f'Steady-state: {K*Q_AC:.1f}°C')
axes[0].axvline(tau/60, color='gray', ls=':', label=f'τ = {tau/60:.1f} menit')
axes[0].set_xlabel('Waktu (menit)'); axes[0].set_ylabel('ΔSuhu (°C)')
axes[0].set_title('Respons Step: HVAC 1 kW')
axes[0].legend(); axes[0].grid(alpha=0.3)

# --- SIMULASI LENGKAP: Server Panas + AC ON/OFF ---
dt   = 1       # 1 detik per langkah
T_sim = 3600   # 1 jam simulasi
time = np.arange(0, T_sim, dt)
T_room = np.zeros(len(time))
T_room[0] = 25.0   # suhu awal 25°C
T_target = 20.0
T_ambient = 30.0

for k in range(1, len(time)):
    # Panas server (fluktuasi realistis)
    Q_server = 800 + 200*np.sin(2*np.pi*time[k]/1800)  # 600-1000W
    
    # Controller on/off (bang-bang)
    Q_AC = 1200 if T_room[k-1] > T_target else 0
    
    # Update suhu
    dT = (Q_AC - Q_server - UA*(T_room[k-1] - T_ambient)) / C * dt
    T_room[k] = T_room[k-1] + dT

axes[1].plot(time/60, T_room, 'tomato', lw=1.5, label='Suhu ruangan')
axes[1].axhline(T_target, color='lime', ls='--', lw=2, label=f'Target {T_target}°C')
axes[1].fill_between(time/60, T_target-0.5, T_target+0.5, alpha=0.2, color='lime', label='±0.5°C toleransi')
axes[1].set_xlabel('Waktu (menit)'); axes[1].set_ylabel('Suhu (°C)')
axes[1].set_title('Simulasi 1 Jam: HVAC Data Center')
axes[1].legend(); axes[1].grid(alpha=0.3)

plt.tight_layout()
plt.savefig('hvac_simulation.png', dpi=150)
plt.show()
print("\n✅ Simulasi selesai! Grafik disimpan sebagai 'hvac_simulation.png'")

import control as ct
import numpy as np

# ==========================================
# STATE-SPACE: Motor DC untuk Pompa Air
# ==========================================
R  = 1.0   # Resistansi armature (Ω)
L  = 0.5   # Induktansi (H)
Kb = 0.01  # Back-EMF constant
Kt = 0.01  # Torque constant
J  = 0.01  # Momen inersia (kg·m²)
B  = 0.1   # Koefisien gesek (N·m·s)

# Matriks state-space
A = np.array([[-R/L,  -Kb/L],
              [ Kt/J, -B/J ]])
B_mat = np.array([[1/L], [0]])
C_mat = np.array([[0, 1]])   # output = kecepatan angular ω
D_mat = np.array([[0]])

sys = ct.ss(A, B_mat, C_mat, D_mat)
print("State-Space Motor DC:")
print(f"A = \n{A}")
print(f"\nEigenvalues (poles): {np.linalg.eigvals(A)}")

# Konversi ke fungsi transfer
tf = ct.ss2tf(sys)
print(f"\nFungsi Transfer (V → ω):\n{tf}")

# Respons step (tegangan 12V)
t, y = ct.step_response(sys)
import matplotlib.pyplot as plt
plt.figure(figsize=(8,4))
plt.plot(t, y*12, 'orange', lw=2)
plt.xlabel('Waktu (s)'); plt.ylabel('Kecepatan ω (rad/s)')
plt.title('Respons Motor DC: Tegangan 12V → Kecepatan Putar')
plt.grid(alpha=0.3); plt.tight_layout(); plt.show()