Dynamical Systems: At the core, a dynamical system is any system that evolves over time according to a set of fixed rules. This encompasses a broad range of systems, from mechanical systems like vehicles to electrical systems like circuits and even natural systems like the weather. Understanding dynamical systems is fundamental because control theory is all about influencing these systems' behavior over time.
Control Systems: A control system is a subset of dynamical systems with mechanisms (controllers) designed to manipulate the system's inputs to achieve a desired effect on its output. Control systems are everywhere, from automatic temperature controls in air conditioners to cruise control in vehicles.
State Spaces: This concept provides a framework for analyzing dynamical systems by defining a "state" comprising all the necessary information to describe the system at any time. The state space is the set of all possible states. Understanding state spaces is crucial for designing controllers that can navigate a system from one state to another desired state efficiently.
Theoretical Frameworks
Control Theory: This is a theoretical framework for understanding how to manipulate the inputs of a dynamical system to obtain the desired output. It encompasses a wide range of techniques and methodologies, from simple feedback loops to complex, predictive algorithms.
Linear Systems and Linear Control: These terms refer to a special class of systems and control methods where the rules governing the system's dynamics (and the control applied) are linear. This means the system's responses are directly proportional to the inputs. Linear systems are simpler to analyze and control, making them a fundamental starting point for learning control theory.
Specific Techniques and Tools
PID Controller: One of the most common and simplest forms of feedback controllers, PID (Proportional, Integral, Derivative) controllers, are widely used due to their simplicity and effectiveness in a broad range of situations. They adjust the control input based on the current error, the integral of the error, and the derivative of the error.
Kalman Filters: This is an algorithm that uses a series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone. Kalman filters are essential in control theory for dealing with uncertainties and are widely used in applications like navigation and tracking systems.
Optimal Control: This area of control theory is concerned with finding a control law for a given system such that a certain optimality criterion is achieved. It involves complex mathematical calculations to determine the best course of action from a set of possible control actions.
