real-time process control

# Real-time Process Control

## Introduction & Motivation

Real-time process control through ML enables dynamic parameter adjustment and optimization during manufacturing operations.

Motivation: Achieve optimal control in real-time.

Applications: Adaptive control, feedback regulation, dynamic optimization, quality assurance.

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## Core Concepts & Theory

### Feedback Control

Closed-loop regulation systems.

### Process Dynamics

Temporal response characteristics.

### Control Strategies

PID, MPC, and advanced methods.

### Optimization

Real-time parameter tuning.

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## Mathematical Formulation

PID Control:
$$u(t) = K_p e(t) + K_i \int e(τ) dτ + K_d \frac{de}{dt}$$

Model Predictive Control:
$$u_t^* = \arg\min_u \sum_{i=1}^{H} ||y_{t+i} - y_{ref}||^2$$

Stability Criterion:
$$|G(jω)H(jω)| < 1$$

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## Advanced Theory & Extensions

### Adaptive Control

Parameter adjustment algorithms.

### Nonlinear Control

Handling process nonlinearity.

### Robust Control

Uncertainty handling.

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## Computational Considerations

Latency: O(1) processing.

Control: O(D²) network.

Throughput: High-frequency updates.

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## Summary & Key Takeaways

ML enables real-time process control.

Principles: 1. Feedback, 2. Dynamics, 3. Stability, 4. Optimization, 5. Robustness.

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## Appendix: Practical Labs

### Lab 1: PID Control Simulation

import numpy as np

class PIDController:
 def __init__(self, Kp=1, Ki=0.1, Kd=0.1):
 self.Kp = Kp
 self.Ki = Ki
 self.Kd = Kd
 self.integral = 0
 self.prev_error = 0
 
 def control(self, error, dt):
 self.integral += error * dt
 derivative = (error - self.prev_error) / (dt + 1e-6)
 u = self.Kp * error + self.Ki * self.integral + self.Kd * derivative
 self.prev_error = error
 return u

controller = PIDController()
for _ in range(10):
 error = np.random.randn()
 u = controller.control(error, 0.01)
 assert isinstance(u, (float, np.ndarray)), "Control failed"
print(f"✓ PID control working")

### Lab 2: Process Response

import numpy as np

def simulate_process(setpoint, control_input, process_gain=1.0):
 """Simulate first-order process"""
 tau = 1.0
 dt = 0.01
 y = 0.0
 dy = (process_gain * control_input - y) / tau
 y += dy * dt
 return y

setpoint = 100
for i in range(100):
 u = 50
 y = simulate_process(setpoint, u)
 assert -500 < y < 500, "Process simulation failed"
print(f"✓ Process simulation working")

### Lab 3: Control Loop

import numpy as np

def control_loop(setpoint, measurements, gains):
 """Execute control loop"""
 Kp, Ki, Kd = gains
 integral = 0
 prev_error = 0
 outputs = []
 
 for y in measurements:
 error = setpoint - y
 integral += error
 derivative = error - prev_error
 u = Kp * error + Ki * integral + Kd * derivative
 outputs.append(u)
 prev_error = error
 
 return np.array(outputs)

setpoint = 100
measurements = np.random.normal(100, 10, 50)
outputs = control_loop(setpoint, measurements, (1, 0.1, 0.1))
assert len(outputs) == 50, "Control loop failed"
print(f"✓ Control loop executed")

### Lab 4: Stability Analysis

import numpy as np

def check_stability(gains, process_gain=1.0):
 """Check control stability"""
 Kp, Ki, Kd = gains
 gain_margin = 1 / (Kp * process_gain + 1e-6)
 phase_margin = np.arctan(Kd / (Kp + 1e-6))
 stable = gain_margin > 0.5 and phase_margin > 0.2
 return stable

stable = check_stability((1, 0.1, 0.1))
assert isinstance(stable, (bool, np.bool_)), "Stability check failed"
print(f"✓ Stability: {stable}")

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