Notes in Week 1 - Intro to MPC

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Published	10/15/2024	In the receding horizon strategy of MPC, {{c1::feedback}} is introduced to handle unexpected events.
Published	10/15/2024	In MPC design, constraints are {{c1::explicitly included}} in the optimization process.
Published	10/15/2024	MPC provides a systematic approach for handling {{c1::constraints}} and achieving high-performance control.
Published	10/15/2024	A key challenge in MPC is that the optimization problem must be solved in {{c1::real-time}} within the system's sampling interval.
Published	10/15/2024	One of the risks in MPC is that {{c1::closed-loop stability}} is not guaranteed automatically.
Published	10/15/2024	{{c1::Feasibility}} in MPC refers to the ability to satisfy all constraints at each time step.
Published	10/15/2024	The MPC optimization problem is typically solved at each {{c1::sampling interval}}.
Published	10/15/2024	{{c1::State constraints}} are part of the MPC optimization problem, limiting the system's state values.
Published	10/15/2024	In the prediction model of MPC, the state at future steps is denoted as {{c1::xt+k}}.
Published	10/15/2024	The stage cost function in MPC, q(xt+k, ut+k), penalises {{c1::deviations}} from the desired state.
Published	10/15/2024	The {{c1::control actions}} in MPC are computed based on solving a constrained optimization problem.
Published	10/15/2024	The receding horizon strategy in MPC means only the {{c1::first control action}} is implemented, and the process repeats.
Published	10/15/2024	Advanced Process Control (APC) is a term often used interchangeably with {{c1::MPC}}.
Published	10/15/2024	The mathematical formulation of MPC involves the optimization of {{c1::decision variables}} over a prediction horizon.
Published	10/15/2024	In classical control, the main focus is on {{c1::disturbance rejection}} and {{c2::noise insensitivity}}.
Published	10/15/2024	The {{c1::plant model}} in MPC represents the system dynamics, linking the input to the state evolution.
Published	10/15/2024	In MPC, {{c1::physical }} constraints, like actuator limits, are incorporated directly into the optimization.
Published	10/15/2024	{{c1::Safety constraints}} in MPC refer to limits such as temperature or pressure that must not be exceeded.
Published	10/15/2024	In real-time applications, the MPC optimization problem must be solved within the {{c1::sampling interval}}.
Published	10/15/2024	A major advantage of MPC is its ability to operate systems near their {{c1::optimal limits}}.
Published	10/15/2024	One challenge of MPC is ensuring {{c1::robustness}} against uncertainties and disturbances in the system.
Published	10/15/2024	If an MPC problem becomes {{c1::infeasible}}, the system may fail to find a valid control action.
Published	10/15/2024	The {{c1::receding horizon strategy}} ensures that MPC reacts to real-time changes in the environment.
Published	10/15/2024	{{c1::Explicit MPC}} refers to a version of MPC where the optimization problem is pre-solved, and solutions are stored.
Published	10/15/2024	MPC stands for {{c1::Model Predictive Control}} and is widely used in modern control systems.
Published	10/15/2024	In MPC, the {{c1::prediction model}} is used to estimate future states based on current inputs and conditions.
Published	10/15/2024	The {{c1::stage cost function}} in MPC is used to penalize errors and control efforts at each step of the prediction horizon.
Published	10/15/2024	MPC solves an {{c1::optimization problem}} at each time step to compute the best possible control action.
Published	10/15/2024	The {{c1::receding horizon}} refers to MPC's strategy of repeatedly solving the optimization problem at every time step.
Published	10/15/2024	The prediction horizon in MPC is denoted by {{c1::N}} and represents the number of future steps considered in the optimization.
Published	10/15/2024	The control actions in MPC are influenced by {{c1::state constraints}} and {{c2::input constraints}}.
Published	10/15/2024	The {{c1::closed-loop stability}} of MPC refers to the ability of the system to converge to a desired state over time.
Published	10/15/2024	{{c1::Robustness}} in MPC is the system's ability to maintain performance in the presence of disturbances or model inaccuracies.
Published	10/15/2024	In practical applications, the MPC problem must be solved in {{c1:: real-time}} and within the sampling interval of the system.
Published	10/15/2024	One key advantage of MPC is its ability to handle {{c1::multivariable systems}} with interacting constraints.
Published	10/15/2024	In MPC, {{c1::constraints}} can include limits on actuator movement, system states, and performance metrics.
Published	10/15/2024	In an MPC problem, the {{c1::input constraints}} ensure that the control signals do not exceed the system's physical limits.
Published	10/15/2024	{{c1::Explicit MPC}} refers to a form of MPC where the optimization problem is pre-solved and the control laws are stored.
Published	10/15/2024	The {{c1::plant model}} is a mathematical representation of the physical system being controlled by MPC.
Published	10/15/2024	One major challenge of MPC is ensuring that the optimization problem remains {{c1::feasible}} at all time steps.
Published	10/15/2024	MPC differs from classical controllers like PID because it optimises control actions over a future time horizon rather than responding to {{c1::curren…
Published	10/15/2024	MPC’s ability to {{c1::handle constraints}} explicitly in the design process is one of its main advantages over classical control.
Published	10/15/2024	{{c1::Multivariable systems}} involve multiple inputs and outputs that interact with each other.
Published	10/15/2024	The {{c1::sampling interval}} in MPC refers to the time between each optimization and control update.
Published	10/15/2024	MPC allows for dynamic adjustments based on {{c1::feedback}} from the system’s current state.
Published	10/15/2024	{{c1::Disturbance rejection}} is a key goal of both MPC and classical control, but MPC handles it by predicting future disturbances.
Published	10/15/2024	In MPC, {{c1::input variables}} represent the control actions that can be applied to influence the system.
Published	10/15/2024	The prediction horizon in MPC determines how far {{c1::into the future}} the system predicts its behavior.
Published	10/15/2024	The {{c1::plant output}} in MPC is the measurable result of the control actions, such as speed, temperature, or pressure.
Published	10/15/2024	{{c1::Closed-loop control}} in MPC refers to using feedback from the system to continually update the control actions.
Published	10/15/2024	{{c1::Control constraints}} in MPC ensure that the control actions do not exceed allowable limits, such as maximum force or energy usage.
Published	10/15/2024	MPC handles {{c1::nonlinearities}} by using models that can approximate or directly represent the system’s nonlinear behavior.
Published	10/15/2024	In an {{c1::open-loop system}}, control actions are applied without using feedback from the system.
Published	10/15/2024	MPC computes an {{c1::optimal sequence}} of inputs to minimize the cost function while satisfying constraints.
Published	10/15/2024	The {{c1::objective function}} in MPC defines what the controller is trying to achieve, such as minimizing energy consumption or tracking a reference …
Published	10/15/2024	The {{c1::control law}} in MPC is the mathematical rule that determines the control actions based on the current state and predictions.
Published	10/15/2024	MPC is particularly useful for systems where {{c1::constraints}} play a major role, such as in robotics or automotive control.
Published	10/15/2024	{{c1::Tuning parameters}} in MPC, such as the weight of the cost function, affect how aggressively the system responds to changes.
Published	10/15/2024	MPC can handle {{c1::multiple time scales}} by considering both short-term and long-term effects in its optimization.
Published	10/15/2024	The {{c1::cost function}} in MPC often includes terms for both tracking performance and control effort.
Published	10/15/2024	{{c1::State estimation}} is often necessary in MPC when not all system variables are directly measurable.
Published	10/15/2024	The {{c1::plant model}} used in MPC can be linear or nonlinear, depending on the system dynamics.
Published	10/15/2024	One of the key strengths of MPC is its ability to anticipate and respond to {{c1::future events}} based on predictions.
Published	10/15/2024	MPC can incorporate {{c1::soft constraints}} that allow for temporary violations if necessary, unlike hard constraints that cannot be breached.
Published	10/15/2024	{{c1::Recursive feasibility}} ensures that once the MPC problem is feasible, it remains feasible at future time steps.
Published	10/15/2024	{{c1::Robust MPC}} accounts for uncertainties in the system model and disturbances, ensuring that control actions are reliable even under unpredictabl…
Published	10/15/2024	The rotation matrix for a 2D rotation by an angle \( \theta \) is \[ R(\theta) = \begin{pmatrix} {{c1::\cos \theta}} & -{{c1::\sin \theta}} \\ \si…
Published	10/15/2024	In 3D, a rotation about the z-axis is represented by the matrix \[ {{c1::R_z(\theta) = \begin{pmatrix} \cos \theta & -\sin \theta & 0 \\ \sin …
Published	10/15/2024	The {{c1::determinant}} of a rotation matrix must always equal 1.
Published	10/15/2024	The inverse of a rotation matrix is its {{c1::transpose}}, i.e., \( R^{-1} = R^T \).
Published	10/15/2024	The rotation matrix for a 3D rotation about the {{c1::x}}-axis is \[ {{c1::R_x(\theta)}} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos \theta &a…
Published	10/15/2024	The rotation matrix for a 3D rotation about the y-axis is \[ {{c1::R_y(\theta)}} = \begin{pmatrix} \cos \theta & 0 & \sin \theta \\ 0 & 1 …
Published	10/15/2024	The matrix product of two rotation matrices represents the {{c1::composition of rotations}}.
Published	10/15/2024	In rigid body motion, both {{c1::rotation}} and {{c2::translation}} are necessary to describe the motion of an object in 3D space.
Published	10/15/2024	The form of a homogeneous transformation matrix is \[ H = \begin{pmatrix} R & d \\ 0 & 1 \end{pmatrix} \], where {{c1::R}} is the rotation mat…
Published	10/15/2024	The {{c1::special orthogonal group}} SO(3) refers to the set of all 3x3 rotation matrices.
Published	10/15/2024	The orthogonality condition for a rotation matrix is that the transpose of the matrix equals its inverse: {{c1::R^T = R^{-1} }}.
Published	10/15/2024	In the axis-angle representation, the rotation matrix is given by Rodrigues' formula: \[ R = I + \sin \theta K + (1 - \cos \theta) K^2 \], where …
Published	10/15/2024	The determinant of a rotation matrix is always {{c1::1}}, ensuring it is a proper rotation.
Published	10/15/2024	A rotation matrix in 2D transforms vectors by rotating them about the {{c1::origin}} of the coordinate frame.
Published	10/15/2024	The {{c1::rotation group SO(3)}} consists of all matrices that represent proper rotations in 3D space, preserving orientation.
Published	10/15/2024	The Euler angle representation involves rotations around the principal axes, typically performed in the {{c1::z-y-x}} order.
Published	10/15/2024	Rotations in 2D can be represented using a single {{c1::angle}} parameter, \( \theta \).
Published	10/15/2024	In 3D, every rotation can be described by an axis and an {{c1::angle}} of rotation.
Published	10/15/2024	The special orthogonal group {{c1::SO(2)}} represents all 2D rotation matrices.
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