📘 PID Control – Full Summary

📎 Source:

🟦 Page 1 – Title

Controller Design: PID Control

👉 This lecture is about designing controllers to control system behavior using:

• Proportional (P)
• Integral (I)
• Derivative (D)

🟦 Page 2 – Closed Loop Transfer Function 🔁

Key idea:

A closed-loop system feeds the output back to the input.

\frac{Y(s)}{R(s)} = \frac{D(s)G(s)}{1 + D(s)G(s)H(s)}

🧠 What this means:

• (G(s)): the system (plant)
• (D(s)): controller
• (H(s)): feedback (sensor)
• Everything is in Laplace domain

📊 Image explanation:

The diagram shows:

• Input (R(s))
• Error (E(s))
• Controller → system → output
• Feedback loop returning output

👉 This loop is what makes control systems stable and adjustable.

🟦 Page 3 – Control Purposes 🎯

Controllers are used to:

1. Follow a reference
   • Make output match input 👉 Example: thermostat reaching desired temperature
2. Reject disturbances
   • Handle noise (W(s))
3. Stabilize the system
   • Prevent oscillations or divergence
4. Speed up response
   • Faster settling time

📊 Image:

Shows disturbance entering the system → controller tries to cancel its effect.

🟦 Page 4 – Characteristic Equation ⚙️

1 + D(s)G(s)H(s) = 0

Key concept:

• Roots of this equation = poles
• Poles determine:
  • Stability
  • Speed
  • Oscillations

🟦 Page 5 – Poles & Stability 🧠

Important rules:

• Poles must be negative → stable system
• Poles determine:
  • Step response
  • Dynamics

👉 If you know poles → you know system behavior

🟦 Page 6 – Controller Design 🎛️

We can:

• Design D(s) (controller)
• (G(s)), (H(s)) come from the physical system

Most common controller:

PID

• P = proportional
• I = integral
• D = derivative

🟦 Page 7 – Step Response & Poles 📉

📊 Image explanation:

Shows how poles affect response:

• ✅ Stable poles (left side) → smooth response
• ❌ Unstable poles (right side) → diverging output
• 🔁 Complex poles → oscillations

👉 Key intuition:

• Real poles → smooth curves
• Complex poles → oscillations

🟦 Page 8 – PID Concept

Just introduces: 👉 Controller acts on error

🟦 Page 9 – Proportional Control (P) 📏

D(s) = K

U(s) = K \cdot E(s)

Meaning:

• Output is proportional to error

🧠 Intuition:

• Bigger error → stronger correction

🟦 Page 10 – P Control on 1st Order System

System: G(s) = \frac{K_s}{\tau s + 1}

Result:

• Closed loop becomes another 1st order system

Important:

• Increasing (K_c):
  • Makes system faster
  • Reduces time constant

🟦 Page 11 – Parameter Tuning 🎯

Two approaches:

1. Analytical methods
2. Step-response estimation

👉 Always requires fine tuning after implementation

🟦 Page 12 – Step Response 📈

📊 Image explanation:

• Input = step
• Output shows system behavior over time

👉 Used to:

• Identify system parameters
• Fit transfer function

🟦 Page 13–15 – Ziegler-Nichols Tuning ⚡

Idea:

Use step response to estimate controller parameters.

Key formulas (page 15):

• P, PI, PID rules based on:
  • (L) (delay)
  • (R) (slope)

⚠️ Drawback:

• Often gives large overshoot

🟦 Page 16 – 1st Order System 📉

G(s) = \frac{K_M}{\tau s + 1}

📊 Image:

• Step response reaches ~63% at time ( \tau )

👉 This is how you estimate time constant!

🟦 Page 17 – Control of 1st Order System

P Control:

• Simple
• ❌ Has steady-state error

PI Control:

• Adds integral action
• ✅ Removes steady-state error

🟦 Page 18–19 – MATLAB + Plots 💻

Shows:

• Simulation of P vs PI

📊 Image:

• P: faster but not perfect
• PI: slower but reaches correct value

🟦 Page 20 – 2nd Order System 📉

G(s) = \frac{K_M}{(\tau_1 s + 1)(\tau_2 s + 1)}

📊 Image:

• More complex response
• Can show overshoot

🟦 Page 21 – Control of 2nd Order System

P Control:

• Has steady-state error

PI Control:

• Removes error
• ❌ Slower

🟦 Page 22–23 – Simulations 📊

📊 Image:

• P: faster, less accurate
• PI: slower, more accurate

🟦 Page 24 – Break 😄

🟦 Page 25 – System with Integration 🔄

G(s) = \frac{K_M}{s(\tau s + 1)}

Key:

• Contains an integrator

📊 Image:

• Output ramps (not just settles)

🟦 Page 26 – Control of Integrating System

Important:

• Even P control gives zero steady-state error

Why? 👉 Because system already integrates

🟦 Page 27–28 – Disturbance Rejection 🚫

Key results:

• P control:
  • Cannot fully remove disturbance
• PI control:
  • Can eliminate steady-state disturbance

👉 Very important in real systems!

🟦 Page 29 – Time Delay Systems ⏱️

G(s) = \frac{K_M}{\tau s + 1} e^{-T_d s}

📊 Image:

• Output starts after delay

👉 Delay makes control much harder!

🟦 Page 30 – Control with Delay ⚠️

Important:

• Large (K_c) → instability
• Must tune carefully

👉 PI controller often better

🟦 Page 31 – Practical Tuning 🔧

Rules:

• Too much overshoot → ↓ (K_c)
• Too much error → ↑ (K_c)
• Or use PI

🟦 Page 32 – Operating Point 📍

Key concept:

Systems are often nonlinear

👉 Solution:

• Linearize around an operating point

🟦 Page 33 – Real World Interface 🌍

Components:

• Computer
• Sensor
• System
• Actuator (e.g., PWM)

🟦 Page 34 – Implementation 💻

Problem:

• Design is continuous
• Computer is discrete

👉 Need conversion

🟦 Page 35 – Discretization 🔢

Using Euler:

\dot{x}(k) \approx \frac{x(k+1) - x(k)}{T}

👉 Converts differential equations → difference equations

🟦 Page 36 – Example Conversion

Shows how:

• Transfer function → time domain → discrete equation

🟦 Page 37 – Pseudocode 🧑‍💻

Controller loop:

1. Read input
2. Compute error
3. Compute control signal
4. Output control
5. Repeat

👉 This is how real controllers run

🟦 Page 38 – Exercises

Final step: 👉 Apply everything in practice

🧠 Big Picture Summary

🔑 What you should remember:

• Control = shaping system behavior
• Poles determine everything
• PID controller:
  • P → fast but inaccurate
  • I → removes steady-state error
  • D → improves stability (not deeply covered here)

🎯 Trade-offs: