Control Systems Lecture Notes

Introduction to Control Systems

• Control Systems involve a mapping between input (U(t)) and output (Y(t)).

Problem Statements in Control System Analysis

1. Synthesis Problem

   • Objective: Given U(t) and Y(t), find the mapping S.
   • Example: Regulating temperature in a room with an air conditioner.
     • Input: Amount of cold air.
     • Output: Room temperature.
   • Purpose: To derive a mathematical model for the system by varying input and measuring output.

2. Prediction Problem

   • Objective: Given the mapping S and a specific U(t), find Y(t).
   • Example: Predicting how temperature changes with different AC units without installation.
   • Purpose: Helps in making informed decisions on system design.

3. Control Problem

   • Objective: Given the mapping S and a desired Y(t), find the appropriate U(t).
   • Example: Regulating air temperature to a setpoint, e.g., 25°C.
   • Purpose: Determine the required input to achieve the desired output.

Examples of Control Problems

• Room Temperature Control
• Motor Speed Control
• Biological Control Systems:
  • Human body temperature regulation.
  • Blood pressure and blood sugar level regulation.
  • Heart functioning as a pump.

Classification of Control Systems

1. Open-Loop Control

   • Characteristics:
     • No feedback mechanism.
     • Cannot compensate for disturbances (e.g., voltage fluctuations).
     • Lower cost and complexity.
   • Example: Ceiling fan control (fixed speed, factory calibrated).

2. Closed-Loop Control

   • Characteristics:

     • Incorporates feedback mechanisms.
     • More robust to disturbances and uncertainties.
     • Higher cost and complexity.

   • Example: Measuring RPM of a fan and adjusting input based on desired performance.

   • Feedback Mechanism: Corrects output by comparing the desired output to the actual output (error calculation).

     • Negative Feedback: Involves subtracting the feedback from the desired input.

Components of Closed-Loop Control Systems

• Controller: Calculates necessary input based on deviations from desired output.
• Sensors: Provide feedback measurement (e.g., speed sensor).
• Actuators: Convert control signals into actual physical actions (e.g., motor movements).
• Disturbances: External factors affecting system performance.

Key Concepts in Control Systems

• Feedback Path: Path where output is measured and fed back into the control system.
  • Can involve additional dynamics, such as sensor response or noise filtering.

Course Overview

• Focus on Closed-Loop Feedback Control of LTI (Linear Time-Invariant) Causal Dynamic Systems.
• Mathematical models will typically consist of Linear Ordinary Differential Equations (ODEs) with constant coefficients.
• Models are Spatially Homogeneous (temporal variations only; spatial variations lumped into a single value).
• Types of Models studied will be:
  • Continuous time.
  • Deterministic.

Prerequisites for the Course

• Basic knowledge of the following mathematical concepts:
  • Complex Variables
  • Ordinary Differential Equations
  • Laplace Transforms

Plan of Action

• Brief recap on necessary mathematical foundations before delving into control analysis.

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Summary

The lecture covers the foundational concepts of control systems, highlighting the synthesis, prediction, and control problems, and contrasting open-loop and closed-loop systems. The course will focus on closed-loop feedback control of linear dynamic systems characterized by deterministic mathematical models.