Notes in Control 1

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Published	10/15/2024	A critical concept in dynamical systems is the {{c1::Laplace Transform}}, which is used to analyze {{c2::linear time-invariant systems}}.
Published	10/15/2024	In control systems, {{c1::PID control}} is a common feedback mechanism that stands for {{c2::Proportional-Integral-Derivative}} control.
Published	10/15/2024	The {{c1::state space model}} represents a system using a set of input, output, and state variables related by first-order differential equations.
Published	10/15/2024	The {{c1::root locus}} method is used to determine the stability of a control system by examining the poles of the transfer function.
Published	10/15/2024	{{c1::Bode Plots}} are used in control system design to analyze the frequency response and assess gain and phase margins.
Published	10/15/2024	The concept of {{c1::reachability}} in control theory refers to the ability to move the system from any initial state to any desired final state withi…
Published	10/15/2024	{{c1::Observability}} is the ability to infer the internal state of a system based solely on its output.
Published	10/15/2024	The {{c1::Final Value Theorem}} provides a way to predict the steady-state value of a system’s output as time approaches infinity.
Published	10/15/2024	{{c1::Stability}} in control systems refers to the system's ability to return to its equilibrium state after a disturbance.
Published	10/15/2024	The process of {{c1::modeling}} involves describing the relationship between input and output, which can be either analytical (math/physics-based) or …
Published	10/15/2024	The {{c1::Good Regulator Theorem}} states that any regulator that is maximally both successful and simple must be isomorphic with the system being reg…
Published	10/15/2024	{{c1::State-space representation}} is a control-oriented representation of a differential equation that includes sensor dynamics and provides a more c…
Published	10/15/2024	The {{c1::state}} of a system is the set of variables that collectively describe the current condition of the system completely, allowing for predicti…
Published	10/15/2024	A good {{c1::controller}} should be able to effectively manage the system, ensuring it operates as desired under various conditions.
Published	10/15/2024	In {{c1::mathematical modeling}}, dynamic systems are often represented in state-space to simplify analysis and control design.
Published	10/15/2024	The {{c1::modelling problem}} in control systems is essential for developing strategies to predict and control system behavior.
Published	10/15/2024	A {{c1::Proportional Controller}} is a simple feedback law that drives a dynamical system to your desired state.
Published	10/15/2024	A {{c1::Linear Time Invariant (LTI) system}} is a system whose response to a given input signal remains the same over time, regardless of when the inp…
Published	10/15/2024	{{c1::Linear Dynamical Systems}} are a special class of dynamical systems that satisfy the superposition principle and are easier to work with.
Published	10/15/2024	The {{c1::Taylor Series Expansion}} of a function is an infinite sum of terms expressed in terms of the function's derivatives at a single point, used…
Published	10/15/2024	{{c1::Linearization}} is the process of approximating a nonlinear system locally by a linear system, which can then be analyzed and controlled using l…
Published	10/15/2024	In a {{c1::time-invariant system}}, the system's response to a given input does not change over time.
Published	10/15/2024	The {{c1::impulse response}} of a Linear Time Invariant (LTI) system contains all the information about the actual system and is crucial for understan…
Published	10/15/2024	In {{c1::state-space representation}}, a system is represented in terms of a set of first-order differential equations that describe its behavior over…
Published	10/15/2024	The principle of {{c1::superposition}} states that in a linear system, the response to a sum of inputs is the sum of the responses to each individual …
Published	10/15/2024	A {{c1::nonlinear system}} can often be approximated as a linear system locally through the process of {{c2::linearization}}.
Published	10/15/2024	In the {{c1::time domain}}, signals are represented as functions of time, while in the {{c2::frequency domain}}, signals are represented in terms of t…
Published	10/15/2024	The {{c1::Fourier Series}} is a technique for approximating periodic functions by decomposing them into a sum of sine and cosine functions.
Published	10/15/2024	The {{c1::Fourier Transform}} converts a function from the time domain to the frequency domain, allowing analysis of the frequency components of the s…
Published	10/15/2024	An {{c1::integral transform}} maps a function from its original function space into another function space via integration, often making it easier to …
Published	10/15/2024	The {{c1::Laplace Transform}} converts a function from the time domain to the complex frequency domain, simplifying the analysis of linear systems.
Published	10/15/2024	The {{c1::Initial Value Theorem}} and the {{c2::Final Value Theorem}} are properties of the Laplace transform that allow determination of the initial …
Published	10/15/2024	The {{c1::transfer function}} is a mathematical representation of the relationship between the input and output of a system in the frequency domain.
Published	10/15/2024	The {{c1::Laplace transform}} is particularly useful for converting linear Ordinary Differential Equations (ODEs) into algebraic equations.
Published	10/15/2024	The {{c1::Fourier Transform}} provides a way to analyze the frequency components of a signal, while the {{c2::Laplace Transform}} extends this analysi…
Published	10/15/2024	The {{c1::convolution operation}} in signal processing represents the relationship between the input signal and the system's impulse response.
Published	10/15/2024	{{c1::Lyapunov Stability}} refers to a system where, for any small perturbation \( \delta \), the system's state remains close to the {{c2::equilibriu…
Published	10/15/2024	A system is said to be {{c1::asymptotically stable}} if it is Lyapunov stable and, over time, the system's state converges to the equilibrium point.
Published	10/15/2024	{{c1::Bounded-Input, Bounded-Output (BIBO) stability}} means that for any bounded input to the system, the output remains bounded.
Published	10/15/2024	In the {{c1::s-plane visualization}}, poles on the left side indicate stability, while poles on the right side indicate instability.
Published	10/15/2024	{{c1::Poles}} determine the homogeneous response (zero input response) of a system, and their location in the s-plane affects system stability.
Published	10/15/2024	{{c1::Zeros}} are the frequencies at which the output of the system is zero, regardless of the input, and can influence the system's frequency respons…
Published	10/15/2024	The {{c1::Laplace Transform}} is used to convert a function from the time domain to the s-domain, facilitating the analysis of linear systems.
Published	10/15/2024	{{c1::Routh’s Stability Criterion}} is a method used to determine if all roots of a polynomial equation have negative real parts, indicating stability…
Published	10/15/2024	In a {{c1::homogeneous differential equation}}, the output is dependent only on the initial conditions, not on any external input.
Published	10/15/2024	A system is {{c1::unstable}} if not stable, which means at least one pole lies on the right side of the s-plane.
Published	10/15/2024	{{c1::Poles and Zeros}} are used to characterize the dynamic behavior of a system, with poles affecting the system's stability and zeros affecting its…
Published	10/15/2024	A {{c1::Bode Plot}} is a graphical representation of a system's frequency response, showing the magnitude and phase across a range of frequencies.
Published	10/15/2024	{{c1::Resonance}} occurs when a system amplifies an input signal at a particular frequency, leading to large oscillations.
Published	10/15/2024	The {{c1::Gain Margin}} and {{c2::Phase Margin}} are metrics used in Bode plots to measure how close a system is to instability.
Published	10/15/2024	In control systems, {{c1::stability margins}} provide a measure of how far the system is from instability, with larger margins indicating better stabi…
Published	10/15/2024	{{c1::Controller parameters}}, {{c2::physical parameters,}} and {{c3::control frequency}} affect the gain and phase plots, influencing the system's st…
Published	10/15/2024	{{c1::Zeros}} in a transfer function correspond to frequencies where the system's output is zero, and they can be used to shape the frequency response…
Published	10/15/2024	The {{c1::Phase Margin}} indicates the additional phase lag required to bring the system to the verge of instability.
Published	10/15/2024	{{c1::Poles}} determine the {{c3::system's natural response}}, while {{c2::zeros}} can modify this response by {{c4::canceling out certain frequency c…
Published	10/15/2024	A {{c1::step input}} is commonly used in control system analysis to test the system's transient and steady-state response.
Published	10/15/2024	{{c1::Kinematic controllers}} transform the mid-level control objective (e.g., controlling the tip position) into a low-level control problem (e.g., c…
Published	10/15/2024	In {{c1::Forward Kinematics}}, given the joint angles, we can determine the position and orientation of the robot's end-effector.
Published	10/15/2024	{{c1::Inverse Kinematics}} involves calculating the joint angles required to achieve a desired end-effector position.
Published	10/15/2024	{{c1::Iterative Learning Control (ILC)}} improves control performance by learning from previous iterations, making it more stable and efficient than r…
Published	10/15/2024	{{c1::Impedance Control}} regulates the dynamic interaction between a robot and its environment by controlling force and motion.
Published	10/15/2024	{{c1::Optimal Control}} seeks to minimize a cost function, such as energy consumption or time, while achieving the desired system behavior.
Published	10/15/2024	{{c1::Trajectory Optimization}} is an open-loop predictive control method that {{c2::optimizes the path of a system}} over time but is not {{c3::robus…
Published	10/15/2024	{{c1::Model Predictive Control (MPC)}} is a closed-loop control strategy that uses a model of the system to predict future behavior and optimize contr…
Published	10/15/2024	The {{c1::Jacobian Matrix}} is used in inverse kinematics to relate the velocities of the joints to the velocities of the end-effector.
Published	10/15/2024	{{c1::Newton's Method}} is a root-finding algorithm often used to solve the inverse kinematics problem iteratively.
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