Introduction to Control Systems - Part 1

Introduction to Control Systems

Overview of the Course

• The course focuses on classical control theory, aimed at first-level undergraduate students.

• It will cover the design of controllers for dynamic systems encountered in practice.

Definition of a System

• A system is defined as an entity that receives an input u(t) and produces an output y(t) .

• Dynamic systems are characterized by variables (inputs and outputs) that change over time, with time being the independent variable.

Understanding Dynamic Systems

• A system or process can also be referred to as a "plant," especially in fields like chemical engineering.

• Various terms may describe systems, but they all refer to entities that transform inputs into outputs.

Classification of Dynamic Systems

Types of Systems

• The course will focus on linear time-invariant causal dynamic systems, primarily single input single output (SISO).

Mathematical Representation

• A dynamic system can be visualized mathematically as a mapping S: u(t) rightarrow y(t) .

Input/Output Classifications

• Classification based on inputs and outputs includes:

• SISO: Single Input Single Output.

• MIMO: Multiple Input Multiple Output.

Characteristics of SISO and MIMO

• In SISO systems, both input u(t) and output y(t) are scalar-valued functions of time.

• In MIMO systems, inputs and outputs are vector-valued functions mapping to multiple dimensions.

Linear vs Non-linear Systems

Understanding Linearity

• The second classification examines whether a system is linear or non-linear.

Understanding Linear Systems

Definition of a Linear System

• A system is defined as linear if it satisfies two conditions: additivity and homogeneity. If the input u_1 results in an output that is a scalar multiple of the corresponding output y_1 , then the system is linear.

Alternative Definitions

• The definition can also be expressed using arbitrary linear combinations of inputs, such as c_1 u_1(t) + c_2 u_2(t) . If this leads to the same linear combination of outputs y_1 and y_2 , then the system remains classified as linear. This principle is known as superposition.

Practical Implications

• In real-world applications, most systems are non-linear; thus, linearity often serves as an approximation for practical problems. It’s important to recognize this limitation when analyzing systems in various contexts.

Time Invariance vs Time Variance

Concept of Time Invariance

• A time-invariant system produces the same output for a given input regardless of when that input is applied. For example, if an input signal is provided at different times but yields identical outputs (delayed by a constant time interval), it indicates time invariance.

Example: Ceiling Fan

• An example illustrating time invariance involves a ceiling fan: if switched on today with a specific electrical input resulting in 100 RPM, switching it on tomorrow with the same input should ideally yield similar results unless wear affects performance over longer periods.

Considerations Over Time

• Over extended periods, factors like wear and tear may alter system responses; hence, while analyzing systems over short durations (e.g., minutes), one can assume they behave consistently (time invariant). However, for long-term operations (e.g., months), variations must be considered leading to potential classification as time-varying systems.

Course Focus

Mathematical Definition of Time Invariant Systems

Understanding Time Invariance

• The mathematical definition states that if y(t) = S(u(t)) , then time invariance implies y(t - tau) = S(u(t - tau)) = y(t - T) for all T in mathbbR .

• This means shifting the input by a time T results in the same output shifted by the same amount, illustrating the concept of time invariance.

Limitations and Considerations

• It is acknowledged that no real system maintains consistent output throughout its lifetime, indicating that time variance may be significant.

• A critical question arises regarding the relevant timescale over which a system's characteristics change when deciding between approximating as time invariant or varying.

Causal vs Non-Causal Systems

Defining Causality

• A causal system's output at any moment depends only on past and current inputs, making it non-anticipative.

• Conversely, a non-causal system anticipates future inputs, which is not considered in this course focused on causal systems.

Course Overview: Types of Systems

Key System Classifications

• The course will focus on dynamic systems categorized as:

• Causal Systems

• Time Invariant Systems (LTI)

• Linear Systems

• Single Input Single Output (SISO)

Importance of LTI Systems

• LTI systems are deemed useful for practical problems and serve as reasonable approximations in many case studies discussed later in the course.

Understanding Control in Dynamic Systems

Concept of Control

• Control refers to making a system behave according to design specifications. This involves regulating how a system operates based on desired outputs.

Example: DC Motor Control

• An example provided is controlling a DC motor where voltage input V(t) affects shaft speed output omega(t) .

Designing Controllers

• The primary question in control design is determining what voltage input achieves a desired angular speed for the motor.

Goals of the Course

Learning Objectives

Mathematical Modeling of Dynamic Systems

Developing a Mathematical Representation

• The initial focus is on developing a mathematical representation for the mapping S , which connects input and output in dynamic systems.

• Various methods exist for obtaining these mathematical models, with an emphasis on physics-based models in this course.

• A semi-empirical approach may be employed, combining physics-derived elements with experimental data to enhance model accuracy.

• The process involves three main steps: creating a mathematical representation for S , analyzing system response, and designing the controller.