Notes in Week 4 - Stability and Closed-loop Behaviour

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New Card	10/27/2024	Closed-loop stability in control systems can often be assessed by checking if all eigenvalues of the system matrix fall within {{c1::(−1, 1)}}.
New Card	10/27/2024	The closed-loop system matrix, incorporating the MPC controller, is given by {{c1::A − B K_{MPC} }}.
New Card	10/27/2024	The general equation for closed-loop state evolution is \( x(t) = (A - B K_{MPC})^t x(0) \), where \( x(0) \) is the {{c1::initial state}}.
New Card	10/27/2024	For closed-loop systems, if \( Q \) is an invertible matrix, we can transform states using \( \chi = Q^{-1}x \) to analyze convergence based on {{c1::…
New Card	10/27/2024	In a system with matrix \( A \), the eigenvalue decomposition is \( A = Q \Lambda Q^{-1} \), where \( \Lambda \) contains the {{c1::eigenvalues}}.
New Card	10/27/2024	If all eigenvalues are within {{c1::(−1, 1)}}, the state vector \( x(t) \) will converge to zero as \( t \) approaches infinity.
New Card	10/27/2024	Lyapunov stability for a discrete-time linear system is achieved if there exists a {{c1::positive definite}} matrix \( P > 0 \) such that \( A^T P …
New Card	10/27/2024	The Lyapunov function \( V(t) = x(t)^T P x(t) \) represents a generalized {{c1::energy}} measure in stability analysis.
New Card	10/27/2024	For Lyapunov stability, \( V(t) = x(t)^T P x(t) \) decreases over time if the system is {{c1::globally asymptotically stable}}.
New Card	10/27/2024	In Lyapunov stability, the scalar function \( V(x) \) is used to show that the system’s energy decreases to {{c1::zero}} as \( x(t) \) converges.
New Card	10/27/2024	The objective in MPC is to minimize a cost function of the form \( \sum_{k=0}^{N-1} (x_t^T Q x_t + u_t^T R u_t) \), balancing {{c1::state error and co…
New Card	10/27/2024	An MPC with prediction horizon \( N = \infty \) approaches the solution of a {{c1::Linear-Quadratic Regulator (LQR)}}.
New Card	10/27/2024	The MPC controller output \( u(t) = -K_{MPC} x(t) \) aims to bring the system state toward {{c1::a desired reference}}.
New Card	10/27/2024	Increasing the MPC prediction horizon \( N \) generally improves stability but increases {{c1::computational cost}}.
New Card	10/27/2024	In control theory, the term {{c1::terminal cost}} is often added to finite-horizon MPC to mimic the behavior of an infinite horizon OCP.
New Card	10/27/2024	In infinite horizon problems, the Bellman equation ensures that the value function at each state is minimized to achieve {{c1::optimal control}}.
New Card	10/27/2024	The recursive nature of the Bellman equation helps in solving {{c1::optimal control}} problems by breaking them into smaller stages.
New Card	10/27/2024	For an MPC, if the prediction horizon \( N \) is too short, stability may not be guaranteed without adding a {{c1::terminal cost}}.
New Card	10/27/2024	An optimal MPC solution minimizes the sum of state and control costs over the prediction horizon, ensuring the system operates {{c1::efficiently and s…
New Card	10/27/2024	In MPC, the control law \( u(t) \) results from solving an optimization problem that considers both {{c1::current and future states}}.
New Card	10/27/2024	The solution to {{c1::an LQR}} problem is defined by \( u(t) = -Kx(t) \), where \( K = (R + B^T P B)^{-1} B^T P A \).
New Card	10/27/2024	The matrix \( P \) in LQR is obtained by solving the {{c1::Discrete Algebraic Riccati Equation (DARE)}}.
New Card	10/27/2024	The stability of the closed-loop system in LQR depends on the {{c1::eigenvalues}} of \( A - B(R + B^T P B)^{-1}B^T P A \).
New Card	10/27/2024	An LQR controller provides optimal performance in terms of balancing {{c1::state error and control energy}}.
New Card	10/27/2024	In LQR, the cost function \( J = \sum (x_t^T Q x_t + u_t^T R u_t) \) measures both {{c1::control effort}} and {{c2::state deviation}}.
New Card	10/27/2024	The optimal feedback gain \( K \) in LQR is designed to minimize the {{c1::quadratic cost function}} over time.
New Card	10/27/2024	A higher \( Q \) matrix in LQR emphasizes state minimization, while a higher \( R \) matrix emphasizes {{c1::control effort minimization}}.
New Card	10/27/2024	For LQR, the discrete-time Riccati equation, known as DARE, is essential in computing the matrix {{c1::P}}.
New Card	10/27/2024	In LQR, the optimal control law \( u(t) \) is linear in state, expressed as \( u(t) = -Kx(t) \), and provides {{c1::feedback stability}}.
New Card	10/27/2024	The feedback gain \( K \) in LQR changes the system dynamics to ensure {{c1::desired stability and performance}}.
New Card	10/27/2024	In stability analysis, the term {{c1::Lyapunov function}} represents a form of energy that decreases over time if the system is stable.
New Card	10/27/2024	A positive definite matrix \( P \) ensures that the Lyapunov function \( V(x) = x^T P x \) is {{c1::non-negative}}.
New Card	10/27/2024	In Lyapunov stability, if \( V(x) \) can be shown to decrease over time, the system is deemed {{c1::stable}}.
New Card	10/27/2024	Lyapunov’s stability criterion involves finding an energy-like function that decreases as the system evolves, indicating {{c1::asymptotic stability}}.
New Card	10/27/2024	If the derivative \( \dot{V}(x) < 0 \) for all non-zero \( x \), the Lyapunov function \( V(x) \) implies {{c1::global stability}}.
New Card	10/27/2024	In discrete-time systems, the Lyapunov function \( V(x) = x^T P x \) must satisfy \( V(x(t+1)) < V(x(t)) \) to ensure {{c1::stability}}.
New Card	10/27/2024	For global asymptotic stability, the eigenvalues of the matrix \( A \) in \( x(t+1) = A x(t) \) should lie strictly within {{c1::the unit circle}}.
New Card	10/27/2024	If a positive definite \( P \) matrix exists, then \( x(t) = 0 \) is a globally asymptotically stable equilibrium for the {{c1::system}}.
New Card	10/27/2024	In energy-based control, if the energy function converges to zero, then all states of the system will also converge to {{c1::zero}}.
New Card	10/27/2024	The Lyapunov function is a core method in control theory for proving {{c1::system stability and convergence}}.
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