Notes in Week 3 - MPC and State Feedback

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Published	11/26/2024	In model predictive control (MPC), the goal is to minimize the cost function over a prediction horizon \( N \) by minimizing {{c1::state and input cos…
Published	11/26/2024	In the MPC formulation, the input vector is denoted by {{c1::\( u(t) \)}}, representing the control input at time \( t \).
Published	11/26/2024	The cost function for MPC is often defined as \( q(x, u) = {{c1::x^T Qx}} + u^T Ru \), where \( Q \) and \( R \) are weighting matrices.
Published	11/26/2024	The objective of MPC is to ensure the system's state \( x(t) \to 0 \) as \( {{c1::t \to \infty}} \).
Published	11/26/2024	In state feedback control, the control input is a linear function of the state, given by \( {{c1::u(t) = -Kx(t)}} \).
Published	11/26/2024	The system dynamics in MPC are modeled by the equation \( x(t+1) = Ax(t) + Bu(t) \), where \( A \) and \( B \) represent the {{c1::system matrices}}.
Published	11/26/2024	The horizon in MPC refers to the number of future steps \( {{c1::N}} \) over which the control sequence is optimized.
Published	11/26/2024	The feedback gain \( K \) is computed to minimize the {{c1::quadratic cost}} function in MPC.
Published	11/26/2024	The system state at time \( t \), denoted by \( x(t) \), is used as the input to compute the {{c1::optimal control input}} in MPC.
Published	11/26/2024	The cost function is minimized in MPC by solving an optimization problem at each time step to determine \( {{c1::U_t^*}} \), the optimal control seque…
Published	11/26/2024	The cost function for MPC is designed to penalize both the {{c1::deviation from the desired state}} and the control effort.
Published	11/26/2024	In MPC, the control input sequence is denoted by \( U_t \), and only the first control input \( {{c1::u(t)}} \) is applied at each time step.
Published	11/26/2024	The stage cost function in MPC is often expressed as \( q(x, u) = {{c1::x^T Qx}} + u^T Ru \), where \( Q \) and \( R \) are weight matrices for state …
Published	11/26/2024	In state feedback, the control law is determined by the feedback gain matrix \( K \), such that \( {{c1::u(t) = -Kx(t)}} \).
Published	11/26/2024	The prediction horizon in MPC represents the number of future time steps \( {{c1::N}} \) for which the control input is optimized.
Published	11/26/2024	The feedback gain \( K \) is used to stabilize the system by minimizing the {{c1::quadratic cost function}} in MPC.
Published	11/26/2024	The MPC optimization problem is solved at every time step to find the control input that minimizes the {{c1::cost function}}.
Published	11/26/2024	In state feedback control, the control input is a function of the system state, expressed as \( {{c1::u(t) = -Kx(t)}} \).
Published	11/26/2024	The feedback gain \( K \) is computed to minimize a cost function that penalises {{c1::state deviation}} and control effort.
Published	11/26/2024	The tracking problem in MPC aims to minimize the {{c1::error}} between the system output and a desired reference signal.
Published	11/26/2024	In tracking MPC, the objective is to follow a reference trajectory while minimizing both the tracking error and the {{c1::control effort}}.
Published	11/26/2024	The control law in state feedback is designed to drive the system state \( x(t) \) toward the {{c1::desired reference}}.
Published	11/26/2024	State feedback control requires full knowledge of the system's state, represented by the vector \( {{c1::x(t)}} \).
Published	11/26/2024	The tracking problem in MPC is addressed by minimizing the error between the output \( y(t) \) and the reference \( {{c1::r(t)}} \).
Published	11/26/2024	The cost function in tracking MPC penalizes both the {{c1::tracking error}} and the control input \( u(t) \).
Published	11/26/2024	The feedback gain matrix \( K \) is used to compute the {{c1::control input}} that minimizes the tracking error.
Published	11/26/2024	In MPC, the control law is implemented by solving an optimisation problem to compute the optimal sequence of {{c1::control inputs}}.
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