ELEN 30123 Feedback Control Systems

Course Overview

• ELEN 30123 focuses on Feedback Control Systems at the Polytechnic University of the Philippines, targeting fundamental control theories applied in engineering.
• The grading system encompasses attendance, assignments, quizzes, major exams, and research, contributing to a balanced evaluation.

Course Structure

• Introduction to electrical systems, control systems, and fundamental mathematical methods such as Differential Equations and Laplace Transforms.
• Control System Analysis includes mathematical modeling, time and frequency domain analysis, and system behavior assessment.
• Design and Implementation covers control applications and system design for desired outputs.

Weekly Breakdown

• Week 1: Introduction to course objectives, classroom rules, and instructor profiles.
• Week 2: Basics of control systems, historical context, and role of engineers.
• Week 3: Introduction to modeling in frequency and time domains.
• Week 4: Control system behavior and analysis.
• Week 5: Time response analysis and subsystem integration.
• Week 6: Steady-state error analysis and error specification.
• Week 7: Design techniques for frequency response in control systems.

Instructor Profile

• Benson G. Pulga serves as the instructor with experience in industrial motor controls and electrical equipment maintenance.
• Connections can be made through social media or email for academic inquiries.

Control System Basics

• Control systems are designed to manage the output of processes based on inputs, with different terminologies for input (stimulus) and output (response).
• Applications span multiple domains: biological systems, space exploration, transportation, manufacturing plants, and automation technologies.

Control System Advantages

• Power amplification for enhanced output.
• Remote control capabilities for managing hazardous situations.
• Simplified input formats for user convenience.
• Compensation for disturbances, ensuring accurate operations despite external factors.

System Characteristics

• Transient Response: The system's immediate reaction to input changes.
• Steady State Response: How close the system's output aligns with the desired result over time.
• Stability: System characteristic defined by minimal to no natural responses to inputs.

Classification of Control Systems

• Systems can be classified based on signal types (continuous vs discrete), input/output numbers (SISO vs MIMO), and feedback mechanisms (Open Loop vs Closed Loop).
• Open-Loop Systems: Achieve control without feedback; less accurate and easier to design.
• Closed Loop Systems: Utilize feedback for corrective action; typically more precise but complicated.

Open vs Closed Loop Control Systems

• Open Loop:
  • No feedback mechanism.
  • Economical but inaccurate.
  • Simpler design.
• Closed Loop:
  • Feedback path present.
  • More expensive but accurate due to continuous monitoring and adjustments.
  • Complex to design and implement.

Key Concepts in Control System Analysis

• Assessing transient and steady-state responses is essential for system design and optimization.
• Analyzing current system responses helps in adjusting designs to achieve desired outcomes.

Preparation and Assignments

• Review advanced mathematics, including Differential Equations and Laplace Transforms.
• Familiarize with Circuit Analysis topics to support control system concepts.
• Short quizzes will precede lectures to reinforce learning.### Steady-State Errors and System Analysis
• Steady-state error can be reduced through analysis and corrective actions.
• Achieving stability involves ensuring the natural response approaches zero or exhibits controlled oscillations.
• Total response of a system consists of natural response plus forced response.

Natural vs. Forced Response

• Natural response reflects how a system dissipates or acquires energy, determined by system properties only.
• Forced response relates to external disturbances or errors, influenced by inputs.
• Transient response is primarily governed by natural responses, while steady-state response focuses on the smaller natural responses over time.

Control System Principles

• The design process begins with schematic creation and development of a mathematical model using block diagrams.
• Mathematical modeling often utilizes differential equations transformed by Laplace Transform.
• Block diagrams simplify complex systems into manageable components, depicting interconnections.

Analyzing System Specifications

• Identifying physical systems and their specifications is vital for effective design.
• Evaluate time responses, including transient and steady-state behaviors, to analyze errors and stability.

Types of Feedback in Control Systems

• Positive feedback enhances performance by adding reference input to feedback output.
• Negative feedback also combines reference input and feedback output but functions to stabilize the system.

Transfer Functions

• A transfer function relates the Laplace transform of output to that of the input under zero initial conditions.
• Form is given as T = Output = C(s) and Input = R(s) with adjustments for feedback.

Laplace Transform

• The Laplace Transform converts time-domain functions into the frequency domain, enabling easier manipulation of differential equations.
• It simplifies convolution into multiplication, benefiting various applications in science and engineering.

Partial Fraction Expansion

• This technique breaks down complex functions into simpler components that are easier to transform.
• Conditions include ensuring N(s) is of lesser degree than D(s) for direct expansion.

Application in Electric Networks

• Transfer functions model electric circuits by relating voltage and current under steady-state conditions.
• Components such as resistors, inductors, and capacitors are characterized by specific relationships between voltage and charge.

System Response Problems

• Finding the transfer function involves manipulating differential equations and applying Laplace transforms.
• The system response can be determined through analysis of the differential equation's components and initial conditions involved.

Key Formulae for Derivatives

• Laplace derivatives relationship: For first-order, L{f′(t)} = sL{f(t)} - f(0), and further for second and third-orders by increasing powers of s.
• The integral of a function can be converted through the Laplace Transform to facilitate resolution of complex equations.

Summary Quiz Preparation

• Know alternative terms for input (reference input) and output (controlled variable).
• Understand advantages of control systems, e.g., improving accuracy and stability.
• Root Locus Technique credited to a specific pioneer in control theory.
• Familiarize with distinctions between open-loop and closed-loop systems regarding feedback and accuracy.### Circuit Components
• Capacitor voltage relationship: ( V(s) = C s I(s) )
• Resistor voltage relationship: ( V(s) = R I(s) )
• Inductor voltage relationship: ( V(s) = L s I(s) )

Impedance

• Impedance formula: ( Z(s) = \frac{V(s)}{I(s)} = C s + R + L s )
• For parallel circuits, apply: ( \frac{1}{Z(s)} = \frac{1}{C s} + \frac{1}{R} + \frac{1}{L s} )
• Series circuit involves summation of individual impedances.

Transfer Function Basics

• Conduct analysis by transforming circuit variables to Laplace domain: ( v(t) \to V(s), i(t) \to I(s), vc(t) \to Vc(s) )
• Replace component values using their impedance values.

Steps to Find Transfer Function

• Example query: Find transfer function relating capacitor voltage ( Vc(s) ) to the input voltage ( V(s) ).
• Capacitor voltage derived as: ( Vc(s) = I(s) C s )
• Impedance addends result in higher-order linear equation for circuit behavior: ( V(s) = Vc(s)(s^2LC + sRC + 1) )

Analyzing DC Circuits

• Identify current and voltage signs:
  • Current flows from "+" to "-" (lift), indicating a "+" sign.
  • Current flows from "-" to "+" (drop), indicating a "-" sign.
• Define the current loop direction, and maintain consistency across loops.

Problem Solving Using Loop Rule

• Apply Ohm’s Law for calculations in circuits:
  • ( I = \frac{V}{R} ) and voltage drop calculated as ( V_D = I \times R )
• In any circuit: ( \text{Sum of } I_{\text{in}} = \text{Sum of } I_{\text{out}} )
• Voltage rule states: ( \text{Sum of Voltages in the circuit} = 0 )

Sample Circuit Problem Steps

• Example: To find current ( I1 ):
  • Compute ( I1 = \frac{12 \text{ volts}}{60 \text{ ohms}} )
  • Results in ( I1 = 0.2 \text{ amps} )
  • Voltage drop calculated as: ( V_D = 0.2 \text{ amps} \times 60 \text{ ohms} = 12 \text{ V} )

Additional Circuit Problem Focus

• Tasks may involve identifying multiple currents and voltage drops.
• Consider node voltages across given nodes for comprehensive circuit analysis.