Linear Control Systems
(EEN-412)
Week: 01→Lecture: 01, 02, 03
Engr. Dr. Ghulam Fareed Laghari
Robotics and Intelligent Systems (RIS)
Department of Electrical Engineering
Bahria University (BU)
H-11 Campus, ISLAMABAD
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linear control system, Summaries of Linear Control Systems
summarize notes from control system engineering by norman nise 7 edition chap 2
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Linear Control Systems
(EEN-412)
Week: 01 → Lecture: 01, 02, 03
Engr. Dr. Ghulam Fareed Laghari Robotics and Intelligent Systems (RIS) Department of Electrical Engineering Bahria University (BU) H-11 Campus, ISLAMABAD
COURSE CONTENTS
[continued…]
COURSE CONTENTS
[continued…]
COURSE CONTENTS → Course Objective
➢This course provides students with a strong foundation in linear control systems ▪ Focusing on their role in robotics and intelligent systems ➢Students will: ▪ Explore the modeling of dynamic systems using transfer functions and state-space methods ▪ Study system response in time and frequency domains ▪ Apply classical stability analysis techniques. ➢Emphasis is placed on: ▪ Designing and implementing feedback controllers to:
- Enhance stability , accuracy , and performance in: o Robotic Manipulators , Mobile Robots , and Autonomous Systems [continued…]
COURSE CONTENTS → Resources
➢ TEXTBOOK:
▪ Control System Engineering by Norman Nise , 7
th
th
Edition
➢ REFERENCE BOOK:
▪ Modern Control Engineering by Katsuhiko Ogata , 5
th
Edition
▪ Control Systems by Anand Kumar , 2
nd
Edition
CONTENT
▪ Linear Systems ▪ Linear and Nonlinear Systems ▪ Control Systems → Definition ▪ Control Systems → Components / Constituents ▪ Control Systems → EXAMPLE: Elevator System ▪ Control Systems → Performance Measures of a Control System ▪ Control Systems → Advantages of Control Systems ▪ Control Systems → Applications of Control System ▪ Control Systems → System Configurations → Open-Loop Control Systems ▪ Control Systems → System Configurations → Closed-Loop/Feedback Control Systems ▪ Control Systems → Real-World Examples → Automated vehicles ▪ Control Systems → Real-World Examples → Rotating disk speed control
Linear Systems
What is a Linear System?
Linear Systems → Explanation ➢A linear system satisfies the properties of both superposition and homogeneity. ➢These properties define how inputs relate to outputs in a predictable, proportional way. 1 ) SUPERPOSITION: ▪ The response of a system to multiple inputs is the sum of the responses to each input individually. ▪ Mathematical form: If 𝒙𝟏 → 𝒚𝟏 and 𝒙𝟐 → 𝒚𝟐 then: 𝒙𝟏 + 𝒙𝟐 → 𝒚𝟏 + 𝒚𝟐 ▪ Advantage: Helps us analyze complex inputs as the sum of simpler ones. 2 ) HOMOGENEITY (SCALING): ▪ Scaling ( multiplying ) the input by a factor scales ( multiplies ) the output by the same factor. ▪ Mathematical form: If 𝒙𝟏 → 𝒚𝟏 then for any constant 𝜶 : 𝜶𝒙𝟏 → 𝜶𝒚𝟏 ▪ Advantage: Ensures the system responds proportionally to changes in input.
Linear and Nonlinear Systems
What is the difference between
Linear and Nonlinear Systems?
Linear and Nonlinear Systems → Difference ➢ NONLINEAR SYSTEM: Any system that does not satisfy superposition or homogeneity. ❑ Nonlinear systems have other types of properties , depending on their nature: ▪ Stability properties
- Lyapunov stability
- Asymptotic stability ▪ Time-domain properties
- Bounded Input–Bounded Output (BIBO) stability: Output remains bounded for bounded inputs. ▪ Phase-space behavior
- analyzed via phase portraits , limit cycles , and attractors. ▪ Nonlinear phenomena unique to these systems:
- Bifurcations: Small changes in parameters cause sudden qualitative changes in behavior.
- Chaos: Sensitive dependence on initial conditions.
- Hysteresis: Output depends on history, not just current input.
Control Systems → A Simple Explanation
❑A control system is a system that automatically manages or regulates how
something works.
❑It takes an input , processes it, and produces a desired output.
▪ Input-output relationship describes the cause-and-effect behavior of a
process that transforms an input signal into a desired output.
❑The goal is to make things work correctly , stably , and efficiently.
What is a Control System? FIGURE 1: Process to be controlled by the system
Control Systems → Definition
➢A control system is an interconnection of subsystems ( components ) and
processes ( plants ), forming a system configuration assembled for the purpose of
obtaining a desired output with desired performance , given a specified input.
➢ In simple words:
▪ A control system is a collection of parts (subsystems) that work together to
produce a desired output for a given input.
▪ The goal of a control system is to achieve the output accurately , effectively ,
and with good performance.
▪ Briefly, a control system makes sure that what we want happens the right way.
Control Systems → EXAMPLE: Elevator System ▪ You are on the 1st floor and press the 4th-floor button. ▪ Pressing the button is the input (what you want). ▪ The elevator moving upward is the system. ▪ Reaching the 4th floor stably and accurately is the output. ➢The elevator is designed to:
- Move at a comfortable speed
- Stop exactly at floor level [continued…]
Control Systems → EXAMPLE: Elevator System ➢ Desired Output as a Step Input ▪ When you press the button, the request is sudden and fixed.
- This type of input is called a step input , as shown in FIGURE 3(a). ▪ The elevator’s movement over time is called the response. ▪ The performance of the elevator can be seen from the elevator response curve in FIGURE 3(a). FIGURE 3(a): Elevator Response