Notes in Week 5 - Constrained MPC

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Published	11/26/2024	In constrained MPC, the objective function is minimized over a {{c1::finite horizon}}, predicting future states.
Published	11/26/2024	Constrained Model Predictive Control (MPC) is formulated as an {{c1::optimization problem}} subject to constraints.
Published	11/26/2024	The core principle of MPC involves updating control actions at each step based on {{c1::current state measurements}}.
Published	11/26/2024	MPC optimizes control signals by solving a quadratic programming (QP) problem using a {{c1::cost function}}.
Published	11/26/2024	Constraints in MPC include both {{c1::state}} and {{c2::input}} constraints.
Published	11/26/2024	State constraints limit the operational range of plant states, expressed as {{c1::xmin ≤ x(t) ≤ xmax}}.
Published	11/26/2024	Input constraints restrict control inputs, commonly within bounds like {{c1::umin ≤ u(t) ≤ umax}}.
Published	11/26/2024	For systems with a single-input, rate of change constraints on \( \Delta u(t) \) prevent sudden spikes, formulated as {{c1::∆umin ≤ ∆u(t) ≤ ∆umax}}.
Published	11/26/2024	In soft constraints, slack variables are added to constraints, allowing for slight violations at a {{c1::penalty cost}}.
Published	11/26/2024	Physical constraints ensure that the system adheres to {{c1::hard limits}}, such as maximum voltage or actuator range.
Published	11/26/2024	Safety in MPC means all constraints must hold throughout system operation, or \( (x(t), u(t)) \in {{c1::X \times U}} \).
Published	11/26/2024	Recursive feasibility ensures that if the problem is feasible at \( t = 0 \), it remains feasible for {{c1::all time steps}}.
Published	11/26/2024	Initial feasibility is checked by ensuring the initial state \( x(0) \) satisfies the {{c1::constraints}}.
Published	11/26/2024	Lyapunov stability in MPC is achieved by constructing a function \( V(x) \) that {{c1::decreases over time}}.
Published	11/26/2024	Recursive feasibility implies safety, ensuring that all states and control inputs satisfy {{c1::constraints over time}}.
Published	11/26/2024	The MPC problem is solved using {{c1::quadratic programming (QP)}}, a structured optimization approach.
Published	11/26/2024	In QP formulation, the cost function combines state and control variables as a quadratic function of {{c1::control inputs}}.
Published	11/26/2024	The QP for MPC minimizes \( J_t = \sum_{k=0}^{N-1} (x_{t+k}^T Q x_{t+k} + u_{t+k}^T R u_{t+k}) \), balancing {{c1::state}} and {{c2::control costs}}.
Published	11/26/2024	Constraints in the QP are expressed as \( MU_t \leq \gamma(t) \), where \( M \) and \( \gamma \) represent {{c1::state and input bounds}}.
Published	11/26/2024	Using a terminal constraint, \( x_{t+N} = 0 \), improves {{c1::stability}} by ensuring the end state remains within limits.
Published	11/26/2024	Stability in MPC relies on a Lyapunov function, \( V(x) \), that decreases at each time step, converging to {{c1::zero as time approaches infinity}}.
Published	11/26/2024	The Lyapunov function in MPC is defined as the cumulative cost function \( V(x(t)) = J_t^*(x(t)) \), ensuring {{c1::stability}}.
Published	11/26/2024	Recursive feasibility supports stability by ensuring that solutions at \( t \) remain feasible for {{c1::future time steps}}.
Published	11/26/2024	In Lyapunov stability, if \( V(x(t+1)) - V(x(t)) \leq -x(t)^T Q x(t) \), then the state {{c1::converges to zero}}.
Published	11/26/2024	Terminal constraints help maintain stability by forcing the system state toward the {{c1::origin over time}}.
Published	11/26/2024	The prediction horizon in MPC represents the future steps, over which predictions for {{c1::states}} and {{c2::controls}} are made.
Published	11/26/2024	Increasing the prediction horizon \( N \) improves performance but also increases {{c1::computational complexity}}.
Published	11/26/2024	In MPC, only the first element of the optimal control sequence is applied at each time step, ensuring {{c1::real-time control}}.
Published	11/26/2024	At each update, MPC uses the latest measurement \( x(t) \) to re-solve the optimization problem for {{c1::feedback control}}.
Published	11/26/2024	Feedback and prediction in MPC involve updating parameters based on {{c1::real-time data}}.
Published	11/26/2024	Recursive feasibility in MPC implies that if feasible at \( t \), feasibility holds for {{c1::t+1}} and beyond.
Published	11/26/2024	Ensuring recursive feasibility is essential for safety, as it maintains constraints throughout {{c1::system operation}}.
Published	11/26/2024	Closed-loop MPC involves updating the control sequence at every time step, forming a {{c1::feedback loop}}.
Published	11/26/2024	Adding terminal equality constraints ensures recursive feasibility by forcing future state predictions to {{c1::zero}}.
Published	11/26/2024	The control sequence in MPC is updated in a closed-loop fashion, adapting to {{c1::measured state changes}}.
Published	11/26/2024	The cost function balances tracking performance and control effort, with terms like {{c1::Q for state penalties}} and {{c2::R for input penalties}}.
Published	11/26/2024	The total cost function in MPC is minimized over the prediction horizon to optimize the {{c1::control actions}}.
Published	11/26/2024	Minimizing the QP cost in MPC, \( J_t = X_t^T Q X_t + U_t^T R U_t \), ensures desired {{c1::state behavior}} with minimal control effort.
Published	11/26/2024	Higher values of \( Q \) in the cost function penalize deviations from desired states, ensuring {{c1::tight tracking}}.
Published	11/26/2024	Adjusting weights in \( Q \) and \( R \) helps tailor the MPC response, balancing {{c1::state accuracy}} and {{c2::control smoothness}}.
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