Introduction to Control Systems - Part 2

Understanding System Models and Their Applications

Introduction to System Models

• A system is defined as a mapping between input u(t) and output y(t) .

• The discussion revolves around understanding how to obtain these models and their practical applications.

Problem of Synthesis

• The first problem, termed synthesis, involves finding the mapping S that relates input u(t) to output y(t) .

• An example provided is an air-conditioned room where the temperature (output) is regulated by varying amounts of cold air (input).

• By measuring the output for different inputs, one can derive a mathematical model for the system. This process helps in predicting how temperature will vary with changes in cold air supply.

Problem of Analysis or Prediction

• Once the mapping S is established, the second problem allows us to predict output y(t) given a specific input u(t) .

• This predictive capability enables informed decisions about which air conditioner to purchase based on its capacity without needing physical testing.

Control Problem

• The third problem involves determining what input u(t) is necessary to achieve a desired output y(t) , known as the control problem.

• For instance, if one wants to set room temperature at 25 degrees Celsius, this requires calculating how much cold air must be supplied. This represents an inverse approach compared to synthesis and analysis problems.

Examples of Control Systems

• Various examples illustrate control systems:

• Room temperature regulation through air conditioning.

• Motor speed control.

• Human body functions such as maintaining body temperature within a narrow range despite external conditions. These include blood pressure regulation and heartbeat maintenance, showcasing natural biological control mechanisms.

Understanding Control Systems: Open Loop vs Closed Loop

Introduction to Control Problems

• The discussion begins with the marvel of maintaining desired values accurately over time, leading into various case studies on formulating practical problems as control issues.

Classification of Control Systems

• Control systems are broadly classified into two categories: open loop control and closed loop control. This classification helps in understanding how different systems operate under varying conditions.

Open Loop Control

• An example is given using a ceiling fan, which operates as an open loop controller where speed settings are adjusted without feedback mechanisms.

• In open loop systems, disturbances such as voltage fluctuations can affect performance, leading to variations in output (fan RPM).

• Ceiling fans typically do not require precise control; hence they function adequately within the limitations of open loop control despite its lack of robustness against disturbances.

Characteristics of Open Loop Control

• Key characteristics include no feedback mechanism and lower cost and complexity compared to closed loop systems. However, this results in poor tolerance for disturbances.

Transitioning to Closed Loop Control

• For applications requiring precise RPM maintenance (e.g., industrial uses), a closed loop system is necessary. This involves measuring actual RPM and adjusting input based on error calculations.

Feedback Mechanism

• Closed loop control incorporates feedback by measuring variables that need regulation, allowing for corrective actions based on discrepancies between desired and actual outputs.

Advantages and Disadvantages of Closed Loop Control

• While closed loop systems are more robust against uncertainties and disturbances, they come with increased costs and complexity compared to their open-loop counterparts.

Example of a Closed Loop System

• A DC motor is introduced as an example where voltage input leads to desired RPM output. The process involves comparing actual output with the desired reference input.

Error Calculation Process

• The difference between measured output and desired value constitutes the error, which is processed through a controller that adjusts the input accordingly.

Negative Feedback Concept

• The layout illustrates negative feedback in closed-loop systems where the feedback signal is subtracted from the reference input at a summing junction.

Conclusion on Feedback Path

Understanding Feedback Control Systems

Sensor Dynamics and Feedback Mapping

• The discussion begins with the importance of sensor dynamics in feedback systems, highlighting that sensors measuring speed may have unique dynamic characteristics.

• When a mapping is introduced in the feedback path, it can either be unity (1) or non-unity. Non-unity feedback affects system analysis significantly.

Actuators and Control Signals

• Controllers calculate control signals which are executed by actuators; for instance, an electric motor drive system translates voltage signals into motion.

• The controller's output may specify a force (e.g., 10 Newtons), but there is often a response time before this force is realized through the actuator. Understanding actuator dynamics is crucial in design processes.

Disturbances and System Complexity

• Disturbances such as sudden loads on motors must be modeled to understand their impact on system performance and how to mitigate them effectively.

• The complexity of feedback systems can vary, and while basic loops will be studied initially, more complex scenarios will be explored later in case studies.

Course Overview: SISO LTI Causal Dynamic Systems

• The course focuses on closed-loop feedback control of Single Input Single Output (SISO) Linear Time-Invariant (LTI) causal dynamic systems, characterized by linear ordinary differential equations (ODEs) with constant coefficients. This forms the foundation of the study material.

Modeling Assumptions: Spatial Homogeneity

• Models used will assume spatial homogeneity, meaning variations are only considered over time rather than space; for example, temperature measurements in a room are averaged into a single function of time rather than multiple spatial points.

• This approach simplifies modeling by treating variables as lumped parameters and leads to continuous-time deterministic models that focus solely on temporal changes without stochastic effects.

Characteristics of Dynamic Models

Introduction to Deterministic Models in Control Systems

Overview of Course Content

• The course focuses on closed-loop feedback control of Single Input Single Output (SISO) Linear Time-Invariant (LTI) causal dynamic systems, utilizing deterministic mathematical models.

• Emphasis is placed on treating all variables as deterministic rather than random, which shapes the nature of the models used throughout the course.

Mathematical Background Recap

• A brief recap of essential mathematical concepts will be provided, necessary for understanding the analysis involved in this course. This includes topics such as complex variables, ordinary differential equations, and Laplace transforms.

• It is assumed that students have completed relevant mathematical courses prior to enrolling in this course, ensuring a foundational understanding of these tools.